Seepage Hydraulics

Seepage Hydraulics

DEVELOPMENTS IN WATER SCIENCE, 10 advisory editor VEN TE C H O W Professor of Civil and Hydrosystems Engineering Hydrosystems Laboratory University of Illinois Urbana, IL.. U. S. A. OTHER TITLES IN THIS SERIES I G. BUGLIARELLO A N D F. GUNTER COMPUTER SYSTEMS A N D WATER RESOURCES 2 H. L. GOLTERMAN PHYSIOLOGICAL LIMNOLOGY

3 Y. Y. HAIMES, W. A. H A L L and H. T. FREEDMAN MULTIOBJECTIVE OPTIMIZATION IN WATER RESOURCES SYSTEMS: THE SURROGATE WORTH TRADE-OFF METHOD

4 J. J. FRIED GROUNDWATER POL LUTlON 5 N. RAJARATNAM TURBULENT JETS

6 D. STEPHENSON PIPELINE DESIGN FOR WATER ENGINEERS

7 V. HALEK A N D J. SVEC GROUNDWATER HYDRAU LlCS 8 J. BALEK HYDROLOGY A N D WATER RESOURCES IN TROPICAL AFRICA 9 T. A. McMAHON A N D R. G. MElN RESERVOIR CAPACITY A N D YIELD 10 G. KOVACS SEEPAGE HYDRAULICS

11 W. H. GRAF A N D C. H. MORTIMER (EDITORS) HYDRODYNAMICS OF LAKES

12 W. BACK A N D D. A. STEPHENSON (EDITORS) CONTEMPORARY HYDROGEOLOGY

Seepage Hydraulics Gyorgy K o v k s Corresponding Member of the Hungarian Academy of Sciences Director-General of the Research Centre of Water Resources Development (VITUKI) Budapest, Hungary

Elsevier Scientific Pubtishing Company Amsterdam

Oxford

New York i981

This book is the revised English version of the "A sziv6rgds hidraulikdja" published by Akademiai Kiad6. Budapest Translated by Katalin Kovdcs The distribution of this book i s being handled by the following publishers and Canada Elsevier North-Holland, Inc. 52 Vanderbilt Avenue New York. New York 10017, U.S.A.

for the U.S.A.

for the East European countries, Democratic People's Republic of Korea, People's Republic of China. People's Republic of Mongolia, Republic of Cuba and Socialist Republic of Vietnam

Akademiai Kiad6, The Publishing House of the Hungarian Academy of Sciences, Budapest for a11 remaining areas

Elsevier Scientific Publishing Company 335 Jan van Galenstraat P. 0. Box 211, lo00 AE Amsterdam, The Netherlands Library of Congress Cataloging in Publication Data Kovdcs. Gyorgy, 1925Seepage hydraulics. (Developments in water science ; 10) Translation of A szivdrgds hidraulikdja. Bibliography: p. Includes index. 1. Seepage. 2. Groundwater flow. 1. Title. II. Series. TCI 76.K6813 627'.17 ISBN 0 -444- 99755 5 ISBN 0-444-41669-2 (Series)

-

0 Akadimiai

80-20236

Kiad6, Budapest 1981

Joint edition published by Elsevier Scientific Publishing Company, Amsterdam, The Netherlands and A k a d h i a i Kiad6, The Publishing House of the Hungarian Academy of Sciences, Budapest, Hungary Printed in Hungary

Preface

Seepage hydraulics, a relatively small branch of hydro- and aeromechanics, deals in general with the properties, the behaviour and the motion of liquid and gmeous media. Its methodology differs basically, however, from that applied to investigating other transport processes of fluids or gases. The main differences result from the special structure of the flow-domain in which the seepage develops. Most of this flow-domain is filled with solid particles; the movement of water or other fluid is only possible through randomly interconnected channels. These channels are of a size which varies considerably and are composed either of the pores surrounded by the grain3 or of fissures and fractures. From this definition of seepage i t can be seen that seepage hydraulics covers bordering fields of the earth sciences, physics and technical sciences. The role of the earth sciences is to determine the properties of the solid matrix. The principles of physics are utilized to describe the transport and storage processes effected by means of the versatile system of water-conveying interstices and to derive the basic relationships between the kinematic parameters by using applied mathematics. The solution of practical problems by simultaneously satisfying the kinematic equations and the boundary conditions developing along the perimeter of the flow-domain (which condition may represent either natural effects or man-induced actions) belongs to the technical sciences. There are numerous handbooks already published concerning the problems of flow through porous media. Some of them give excellent summaries of applied mathematics related to seepage. Others give similarly good physical and mathematical treatments of flow theory. I n another group, the geological approach ie emphasized by dealing mostly with the characterization of the solid matrix and with the description of its behaviour. The new feature of this book is that i t provides an integrated system covering the entire field of seepage hydraulics and i t presents solutions for practising engineers without neglecting the scientific background. It is hoped that the different approaches (theoretical and practical; mathematical and physical) and the various scientific aspects (hydraulics, hydrology, geology, soil science) a m presented in a balanced form in relation to:

6

the characterization of porous media, the dynamic analysis of the acting processes, and the kinematic description of the transport of fluids. It is also hoped that the book will provide guidance towards the complete solution of the problems, ensuring thereby not only a firm theoretical basis for the investigations but also their practical application. Large numbers of experimental and field data are summarized in the tables; these should be of value to those who wish to use the book for further research. This manualwaa written primarily forhydraulic engineers; apart from them it can be used also by others who need to solve various problems related to the movement o i subsurface water in geology, mining, civil and geotechnical engeneering, agricultural sciences (land reclamation, irrigation and drain&ge) etc. The book was first published in Hungarian in 1972. For the English version the original version has been completely revised and up-dated to include the latest developments, which have been extremely rapid in the first half of the 1970s. Important new parts concerning seepage through fractured rocks and unsaturated media have been added. -

Dr. Gyiirgy Kovdcs

Contents

Part 1 Fundamentals for the investigation of seepage

Chapkr 1 . 1 General characterization of subsurface water and its movement 1.1.1 Classificationof the various types of subsurface water 1.1.2 Characterization of flow through various aquifers 1.1.3 T h e definition of seepage References to Chapter 1.1

....

........ ........... ................................ ........................................

Chapter 1.2 Physical and mineralogical parameters of loose clastic sediments influencing permeability 1.2.1 The size and shape of grains forming a layer 1.2.2 Investigation of grain-size distribution 1.2.3 Mineralogical composition of sediments and its influence on the determination of the effective diameter 1.2.4 Relationship between porosity and the geometrical parameters of grains 1.2.5 Physical model representing the irregular network of pores and channels between grains References to Chapter 1.2

...................................... ................. .................... ..................

............................................. ................................

.......................................

.................

Chapter 1.3 Dynamics of soil moisture above the water table 1.3.1 General characterization of soil moisture and processes acting in the soil-moisture zone 1.3.2 Parameters characterizingthe various moisture content8 of soil samples 1.3.3 Interpretation of field capacity, gravitational porosity and soil-moisture retention curve 1.3.4 Description and determination of the soil-moisture retention curve References to Chapter 1.3

................................ .............................................. ............................

................................................ ....................................... Chapter 1.4 Investigation of the balance of the ground-water space ............ 1.4.1 General hydrological characterization of water exchange between soil moisture and ground water ....................

13 13 20 28

37 38 39 43

60 76

87 94 98 99 104 109 123 146 147 148

8

Contents

............................. .................... ............................... .......................................

1.4.2 Storage capacity of aquifers 1.4.3 Analyeis of horizontal ground-water flow for the determination of the vertical water exchange 1.4.4 Interpretation and determination of the characteristic curve of the ground-water balance References to Chapter 1.4

153 178 188 199

Part 2 Dynamic interpretation and determination of hydraulic conductivity in homogeneous loose clastic sediments

.................................. ................. ....................................... ................... ........................................

Chapter 2.1 Dynamic analysis of seepage 202 203 2.1.1 Forces influencing the flow between grains 2.1.2 Dimensionless numbers characterizing the various validity zones of seepage 226 2.1.3 Numerical limits of the zones of seepage 230 References to Chapter 2.1 238

..................... ........... ...........................

Chapter 2.2 Hydraulic conductivity of saturated layers 239 2.2.1 Determination of D ~ C Y ’hydraulic B conductivity 240 2.2.2 Investigation of the turbulent and transition zones of seepage 251 2.2.3 Investigation of microseepage 269 References to Chapter 2.2 281

.......................................

........................... 283 284 ............ 293

Chapter 2.3 Seepage through unsaturated layers 2.3.1 Movement equations characterizing unsaturated flow ........ 2.3.2 Physical soil paremeters used in diffusion theory 2.3.3 Theoretical analysis of hydraulic conductivity in unsaturated porousmedia References to Chapter 2.3

.......................................... .......................................

301 314

Part 3 Permeability of natural layers and processes influencing its change in time

Chapter 3.1 Characterization of special behaviour of hydraulic conductivity in loose claatic sediments 3.1.1 Laboratory and field methods to determine permeability .... 3.1.2 Evaluation of pumping tests with more observation wells 3.1.3 Characterization. determination and practical consideration of anisotropy References to Chapter 3.1

.......................................

.... ............................................

........................................

318 319 326 338 348

........ 349 ...................... 351 ............................. 362 ................. 369 ........................................ 378

Chapter 3.2 The motion of grains in cohesionless loose claatic sediments 3.2.1 The motion of f h e grains (suffusion) 3.2.2 T h e liquidization of the layer 3.2.3 Design of protective filters and well screens References to Chapter 3.2

9

Contents Chapter 3.3 Investigation of clogging .................................... 3.3.1 The change of concentration of the percolating water depending on time and place ................................... 3.3.2 Application of the capillary tube model of the porous medium to characterize the clogging process ...................... References to Chapter 3.3

380

381 387

........................................ 395

Chapter 3.4 Hydraulic conductivity and intrinsic permeability of fissured and fractured rocks ........................................ 3.4.1 Characterization of various non.carbonate, water.bearing, fissuredandfracturedrocks .............................. 3.4.2 Hydraulic properties of carbonate rocks ................... 3.4.3 Models for the characterization of flow through the openings of solid rocks ............................................ 3.4.4 A conceptual model for the determination of hydraulic conductivity of fissured rocks .............................. References to Chapter 3.4

........................................

396 397 417 437 454 470

Part 4 Kinematic characterization of seepage

Chapter 4.1 Kinematic relationships characterizing laminar seepage ........... 475 4.1.1 Interpretation of velocity-potential and potential water 476 movement ............................................ 4.1.2 Interrelation between stream-function and potentia1.function Theflownet ........................................... 482 4.1.3 Geometrical and kinematic classification of seepage .......... 493 References to Chapter 4.1 ....................................... 504

.

Chapter 4.2 Boundary and initial conditions of potential flow through porous media ............................................... 505 4.2.1 Characterization of external boundary conditions ........... 506 4.2.2 T h e investigation of the layered seepage field ............... 519 4.2.3 Application of hodograph image and other special transformations for the characterization of boundary conditions ........ 529 4.2.4 Consideration of initial conditions ........................ 539 References to Chapter 4.2 ........................................ 541 Chapter 4.3 Kinematic characterization of non-laminar seepage .............. 4.3.1 Differential equations equivalent t o Laplece’s equations for the various zones of seepage ............................. 4.3.2 Consideration of the continuous change of the flow condition within the seepage field ................................. References to Chapter 4.3

........................................

542 543 549 555

10

Contents Part 5 Solution of movement equations describing seepage

............ ....................

Chapter 6.1 Characterization of two-dimensionalpotential seepage 660 6.1.1 Complex potential Conjugate velocity 663 6.1.2 Solution of seepage problems by applying mapping .......... 667 6.1.3 Basic mapping functions applied most frequently ........... 676 6.1.4 Application of Schwartz-Christoffel’s mapping ............. 686 References to Chapter 6.1 ........................................ 695

.

Chapter 5.2 Combined application of various mapping functions .............. 6.2.1 Application of a series of mapping functions within the flow plane of a two-dimensional seepage 6.2.2 Combination of mapping functions applied on two different planes of the flow space ................................. 6.2.3 Combined application of hodogreph and conformal mappings 6.2.4 Characterization of unconfined field by the application of Zhukovsky’s function Referencesto Chapter 6.2 ........................................

.......................

696 697 610

. . 616 ................................... 626

Chapter 6.3 Horizontal unconfined steady seepage (Dupuit’s equations and the limits of their application) 6.3.1 Derivation of Dupuit’s equations ......................... 6.3.2 The influence of the capillary water conveyance 6.3.3 Characterization of horizontal unconfined seepage influenced by accretion 6.3.4 Local resistance occurring in the vicinity of the entry and exit faces References to Chapter 6.3

626

627 ............................... 628 ............. 648 .......................................... 661 ................................................. 670

........................................

686

.................... 687 ........................ 688 ............... 696 ............ 704 ........................................ 712 Chapter 6.6 Model laws for sand box models ............................... 713 6.6.1 General derivation of model laws for hydraulic models ....... 714 6.6.2 Geometrically distorted sand box models ................... 719 722 References to Chapter 6.6 ....................................... Subject index ................................................... 723

Chapter 6.4 Investigation of horizontal unsteady seepage 6.4.1 Derivation of Boussinesq’s equations and the problems in connection with their linearization 6.4.2 Application of the differentialequation of unsteady flow to the characterization of seepage in an infinite field 6.4.3 Unsteady seepage in a horizontally limited field References to Chapter 6.4

Part 1

Fundamentals for the investigation of seepage

The greatest amount of water on the globe is stored in the basins of oceans and seas. This water however, is salty, and therefore, unsuitable for direct congumption. Fresh water comprises only 2.8% of the total amount of water (i.e. about 37 million cubic kilometres) and is stored on and below the surface of the earth. It occurs in several different forms: ice and snow in polar areas and mountainous regions, ground water in the pores of aquifers, water in the beds of lakes and rivers, etc. Water stored below the surface of the earth in the form of soil moisture and ground water makes u p more than 20 % of the earth’s total fresh water resources. It is distributed almost equally over the continents, and its exploitation is a fundamental basis of water management in every country. Although only the smaller part of the total amount is available for direct use, (because the exploitation of water stored in clay, or at great depth is not yet economical) even this amount would be enough to meet the increased world demand for a very long period. It is not permissible however, to use up large quantities of stored water, because it is one of the basic elements of the biosphere, and some very serious changes would be caused by its absence (e.g. heat balance would be disturbed by the decrease in evaporation, or vegetation would be destroyed by lowering the water table over large areas). The tltored water - the so-called stutic water resource - haa t o be considered therefore, as a safety reservoir, from which water may be exploited on a limited scale, when its recharge is expected. For this reason, the amount of water replenished per annum on the surface of the continents in the form of precipitation is more important from the aspect of water reaources development, than the stored quantity. The detailed investigation of these dynamic resources involves the determination of parameters characterizing the subsurface branch of the hydrological cycle. This topic belongs to hydrology, especially ground-water hydrology. Another baaic requirement for this investigation is the study of the movement of water within the cycle, namely hydrodynamics. The investigation of the subterranean part of the hydrological cycle - the dynamics of flow through porous media - is also a fundamental part of the scientific background of water resources development. Apart from the analysis of this general flow system, many hydrodynamical problems have to be solved when designing the structures (wells, shafts, or

12

1 Fundamentals for the investigation of seepage

drains) through which the exploitation of ground water becomes possible. I n other cases, hydraulic structures (reservoirs, dams, canals) raise the level of surface waters above thenatural ground-water table, and thus, create seepage. Sometimes the position of the water table, and the moisture content of the upper soil have t o be controlled, either t o ensure higher agricultural production (land drainage, irrigation), or to create better environmental conditions (marsh reclamation). Similarly, the permanent or temporary modification of the position of the phreatic ground-water surface and/or the pressure condition of ground water might be the aim of special operations (dewatering of construction pits, protection of mines against water intrusion). The solution of all the problems related either to the design or the operation of the hydraulic structures listed, requires the investigation of water movement in various flow domains, the boundaries of which are determined by the geological structure of the layers. The boundary conditions are influenced by hydrological and hydrogeological phenomena, and the flow conditions are governed by the soil- or rock-mechanical properties of the beds in question. This short summary of the problems already indicates the multidisciplinary character of seepage hydraulics. It is basically part of hydromechanics, starting with the theoretical discussion of the kinematics and dynamics of water movement, and also including the practical application of the theoretical results. The fundamental topics of hydromechanics (e.g. interpretation and determination of mass, density, specific weight and viscosity of fluids, or that of hydrostatic pressure and its distribution) may be regarded as wellknown, and need not be repeated here. The reader is referred to existing manuals dealing with these basic concepts of hydrodynamics (e.g. NBmeth, 1963; Bear, 1972). At the same time, because of the multidisciplinary character of seepage hydraulics, i t is necessary to deal with some aspects of the interrelated sciences as well (e.g. hydrology, hydrogeology, mineralogy, soil phyeics, soil sciences, etc.), whichcan be regarded as fundamental to our discussion. Naturally, the same basic philosophy as mentioned in connection with hydromechanics is followed here, that is, that the principles of these sciences are not explained - only some special points having close connection with seepage hydraulics are considered. Those aspects are emphasized, which may influence the discussion of flow through porous media. Sometimes the method of discussion of topics taken from other scientific fields and summarized in Part 1 does not follow that generally used in the original literature. Slight modifications are made, or ideas further developed, to facilitate the easy application of the results in seepage hydraulics. It is hoped that these changes do not alter the basic concepts of the related sciences, and that contradictions are not caused by them.

1.1 General characterization of subsurface water

13

Chapter 1.1 General characterization of subsurface water and i t s movement The first problem which hau to be solved before starting the detailed discussion of the various scientific topics is the establishment of a common terminology, so that misunderstandings can be avoided. The clarification of the definitions of terms used generally is necessary not only because the meanings of some expressions differ in different languages, but also because differences can be found in the literature of the various sciences written in the same language. To create a suitable framework for the comparison of the various expressions, and to clarify the defhitions used in this book, a short summary of basic hydrogeological concepts, and of expressions concerning the flow of water through porous media, will be given here as an introduction. This chapter includes, therefore, the analysis of the various effects influencing the character of water movement below the surface, and finally, the defhition of seepage itself. When establishing the proposed terminology, attempts were made to follow the definitions accepted in recently published international manuals and glossaries (UNESCO, 1972;FAO, 1972;FAO-UNESCO, 1973;IAHS, 1974; UNESCO-WMO, 1974). Changes - or more precisely supplementation were made only in a few cases, where the special character of some aspects of seepage hydraulics required them.

1.1.1 Classification of the various types of subsurface water

A considerable amount of the water falling on the continents in the form of precipitation infiltrates through the surface. This infiltrating water becomes part of the subsurface water, which is stored for either a short or a long period in the various interstices of the crust (pores, hollows, joints, fissures and fractures), and moves along the channels composed of these pores and openings. When investigating the hydrological processes occurring under the surface of the earth, the most important aspects to be considered are the part of the subsurface water plays in the entire hydrological cycle, and the characterization of both flow and storage in the layers as influenced by their structure, and physical characteristics. The subsurface waters can be classified in many different ways, taking their various properties into consideration as the basis of the classification (e.g. temperature, chemical composition, origin, character of movement, etc.). As explained earlier, the two most important aspects from the hydrological point of view are the character of the water-bearing layers and their connection with the meteorological and hydrological processes on and above the surface. According to the claasification based on these two parameters (Table 1.1-l),three different levels of subsurface water can be distinguished (soil moisture, shallow ground water and deep ground water). Each level can

14

1 Fundamentals for the investigation of seepage Table 1.1-1. Hydrological classification of the various types of subsurface water B

L o w elastic sediment

I

Position

of aquifer

-(limestone, (dolomite) Karstio

gravel, send, silt, clay

Zone of aeration

I

Nan Laretic

moisture content of rocks

i Zone of eaturation

M i d rocka

cohive

2

3

-

shallow ground water

shallow karstic water

-~ deep ground water

deep kerstic water

Iw

~ ~ ~ ~ Of t e n t

in shallow position in deep position

be further divided into two main groups, considering those characteristics of the water-bearing layers, which influence the flow of water, and the process of its storage to the greatest extent (loose clastic sediments, or fissured and fractured solid rocks). Soil moisture is the water content of the pores in the soil above the water table, retained in various forms against gravity. The most important forces influencing the development of soil moisture are adhesion and capillarity. Adhesion creates a thin fllm of water covering the wall of the pores (adhesive water) while capillarity completely fills some of the pores with water (or all the pores, depending on the height above the water table of the point considered) (capillary water). In fissured and fractured solid rocks, the same type of water is called the moisture content of rocks. For the complete clarification of this definition, it is necessary to analyze the term water table as well. According t o the generally accepted meaning, it is the surface of unconfined ground water bodies dividing the ground water and the soil moisture zones. Another characteristic feature of the water table is that the pressure of the water included in the pores is equal here to that of the atmosphere. It is necessary t o note that the water table is not a real surface, because there is no rapid change in the saturation of pores at this level. Both the ground-water zone and the lower part of the capillary zone are completely saturated. The decrease in water content related to porosity starts gradually only in the upper reaches of the closed capillary zone, and it is continued in the open capillary zone. Thus the level of the water table is indicated only by the change in the sign of the water pressure - it is positive in the ground-water zone and negative (suction, tension) above the water table. It should be noted here that water pressure is always expressed in the form of excess pressure (p),i.e. the difference between the total ( p l ) and the atmospheric ( p o )value:

P = Pt

- Po.

(1.1-1)

This parameter becomes negative where and when the total pressure is lower than that caused by the atmosphere a t the time and place of the

1.1 General characterization of subsurface water

15

investigation. The change in the sign of the excess pressure is naturally a gradual quantitative modification, and no qualitative difference occurs a t the boundary. Hence, the most important common characteristic of any type of soil moisture (and of the moisture content of rocks) is the negative pressure governing the behaviour and the movement of water in this zone. (A more detailed explanation of the various subzones of the soil moisture zone, and the description of the forces acting there, will be given in Chapter 1.3.) Having established a definition of “water table”, i t is relatively eaay to define the term most commonly used in this book. Qround water is that part of the subsurface water which is stored (or moves) in the interstices of the crust below the water table. Hence its total pressure is higher than the atmospheric one. This pressure may be hydrostatic, or influenced by the movement of water (hydraulic pressure), but its excess value is always positive. In the ground-water zone, all the pores and other types of interstices are completely filled with water (or with other fluids, and in exceptional cases, with gases under pressure). Considering the connection of ground water with the hydrological and meteorological processes on and above the surface, two groups may be distinguished: shallow and deep ground water. Shallow ground water is stored near the surface, below the water table and above the first, largely extended, continuous, impervious formation. It is directly influenced by meteorological and hydrological events (recharged by precipitation, drained by evaporation and transpiration, having interactions with surface waters). Deep ground water can be found in aquifers, lying below continuous, impervious beds, which hinder the direct contact between surface and ground water over a very large area, as well aa the influence of atmospheric prosesses on ground water. It hm no direct recharge from precipitation or surface waters, and is drained only through the shallow ground water. It is very difficult to make clear distinctions between the two types of ground water. Shallow ground water may be partly covered by impervious strata, but if the covering layers are not continuous or large enough, the effects of the meteorological and hydrological events can be clearly observed in the regime of the ground water. On the other hand, i t is also possible, that the aquifer covered by impervious formations over a large area may occur near the surface far from the place of investigation. Thus, the same groundwater body may be regarded aa deep within the investigated area, and shallow elsewhere. It is not possible therefore, to draw a sharp line between the two types. There are several transition forms, and the distinction can be baaed only on the investigation of the dominating factors influencing the ground-water regime. The means of this investigation are the observed ground-water data. Considering the rough sketch of the vertical section of a sedimentary basin given in Fig. 1.1-1, deep ground water may be divided into several subgroups: e.g. closed ground water, when the aquifer is completely surrounded by impervious layers; artesian water, which hm a recharge area where the water table is above the artesian area, and thus, the water level in wells tapping the artesian layer rises above the surface. The adjective

16

1 Fundamentals for the investigation of seepage

Fig. 1.1-1. Rough vertical section of a sedimentary basin for representing the Various types of aquifers

confined (or unconfined) also indicates whether the water-bearing layer is covered by impervious formations or not. This term does not however, express the extension of the cover. Hence, it is suillcient for characterizing the behaviour of water movement in the aquifer, but not for that of its hydrological contact. It can well be used therefore, in seepage hydrtGulics without providing su5cient information for hydrogeologists. It can be seen from the foregoing that such expressions aa confined, closed or artesian aquifer have special meanings, and they do not completely characterize the hydrological behaviour of the water-bearing layers. It is advisable, therefore, to use the adjectives of shallow and deep for distinguishing ground water directly influenced by meteorological and hydrological phenomenon from that recharged or drained only through another part of the ground-water space. Several expressions have already been used to characterize the layers containing water, or hindering its movement: e.g. aquifer or water-bearing layer, impervious formation etc. Four terms are generally used t o describe the behaviour of strata with respect to the flow of water. These are as follows:

1.1 General characterization of subsurface water

17

(a) Aquifers - permeable geological formations having interconnected interstices, which permit an appreciable quantity of water t o move through them under ordinary field conditions. Ground-water reservoir, waterbearing or permeable layer are commonly used synonyms. impermeable strata, which may contain a great quan(b) Aquicludes tity of water (in some cmes more than aquifers, e.g. clay). Some aquicludes do not transmit water at all, and others only a very small quantity. (c) Aquifuges - also impermeable formations but neither contain nor transmit water. These rocks have no interconnected pores or fissures and cannot therefore, either absorb water or allow it to pass through. Aquicludes and aquifuges are the two opposite types of impermeable geological formations. (d) Aquitard - a transition form between aquifers and aquicludes. This type of layer contains interconnected pores, but the water-conveying capacity of the channels made up of pores is relatively small compared to that in aquifers. However, if the direction of flow is perpendicular to the large bordering surface of such a layer, the amount of water conveyed by seepage is not negligible, because of the great extent of the area. These strata are often called semi-pervious or leaky formations. The hydrogeological terms listedin the previous paragraphs are not entirely sufficient for clwsifying the layer hydrologically, or from the point of view of seepage hydraulics. They indicate only the existence, or the lack of interconnected pores, and thus, the possibility of water movement, but do not describe the structure of the network composed of the interstices, which is one of the most important factors influencing the character of flow. The latter information is basically required t o classify the layers from the aspects of hydrology and hydraulics. Considering the origin of the various rocks, the geological classification distinguishes the following types:

-

Sediments - of mechanical origin: loose clastic sediments or cemented sediments; - of chemical origin; - of biological origin; Igneous rocks - effusive ( h e grained) volcanic rocks; - intermediate (porphyriiic) rocks; - intrusive (coarse grained) plutonic rocks; Metamorphic rocks. Because the water-conveying network in some of these formations (and hence, the character of the movement in or through the layers) are very similar, several geological groups may be combined when deciding the h a 1 form of the hydrological classification. The most important difference is between the network of pores in loose clastic sediments, and that of the fissures and fractures of solid rocks. Hence, these two main groups were selected as the basis of the hydrological classification. Within both groups, two further subgroups were distinguished: the non-cohesive loose clastic sediments are pervious formations, and the cohesive ones may be regarded 2

18

1 Fundament& for the investigation of seepage

as impervious, or semi-pervious layers, although there is a continuous transition between the two subgroups without any sharp borderline. In the group of solid rocks, it is reasonable to deal with the karstic formations separately. In these, the water-conveying openings are generally enlarged by chemical solution and mechanical erosion cawing high permeability.The other remaining subgroup is composed of non-karstic solid rocks. It can easily be seen that the structure of the basic classification summarized in Table 1.1-1 is also suitable to determine the relationship between

Fig.

absorbed fossil ,juvenile water water water _.Grouping c vsriouS types of mbsurface water according to their O L ~ J in he system of h~dmlogicalc l d o a t i q n

the hydrological c l d c a t i o n of subsurface water, and some other types of grouping baaed on different parameters of the water, or those of the layers containing it. The origin of water can be used as an example to show this (Fig. 1.1-2). It is well known that generally three types of subsurface water can be distinguished amording to their origin: internal water (juvenile water) is derived from the interior of the earth as anew resource; external water originates from atmospheric or surface water and may be trapped in rocks when the constituent material was deposited (fossil or connate water), or i t may be absorbed into the interstices of the layers some time after deposition, even quite recently (absorbed water) (UNESCO,1972). The amount of juvenile water is negligible from the point of view of water resources development. Coming from very great depths, its occurence in deep rather than shallow ground water is more probable. Because the upward movement of juvenile water generdy follows faults, larger amounts may be expected in solid rocks, where it sometimes rises into shallow positions. I n most cases, fossil water preserves its original characteristics (e.g. chemical composition) derived from the water-bearing layers in which it formed or into which it idltrated during their development. Changes may occur aa a result of water exchange between aquifers, for instanoe by migration and diffusion. For this reason, a higher ratio of fossil water may be expected

1.1 General characterization of subsurface water

19

in layers where the movement of water is considerably hindered. Thus, the highest amount is probable in cohesive, loose, clastic sediments. Adsorbed water indicates the recent influence of meteorological and hydrological processes. It is quite evident that soil moisture is entirely, and shallow ground water almost completely composed of this type of water. Similar to the previous example, the interest of the various water oriented sciences in the multidisciplinary field of subsurface water can also be clearly represented in the proposed system of classification (Fig. 1.1-3).

nydrohydrology hydropedology mechanics geologi Fig. 1.1-3. The grouping of various types of subsurface water according to the sciences involved in the system of hydrological claasifkation

Soil moisture and part of shallow ground water ensuring the recharge of the former are the most important sources of water supply to plants, and they take part in the development of the various types of soils. Thus, the investigation of this part of subsurface water is an important field of soil science: i.e. hydropedology. The most important taak of hydrology is to explore the interrelationships between the various forms of water on the globe and to analyse its continUOUE movement along the hydrological cycle. When investigating subsurface water from the hydrological point of view, its contact with atmospheric and surface water aa well as the flow of water along the subsurface branch of the cycle have to be studied. The field of interest of hydrology is concentrated, therefore, mostly on soil moisture and shallow ground water, these having direct contact with hydrological and meteorological processes on and above the surface. This interest also extends partly t o deep ground water, which may join the hydrological cycle in the form of water exchange between deep and more shallow layers. The pores and other interstices of strata developed in geological ages are m e d with water. The presence, pressure, chemical composition and movement of water also influence the behaviour of the layers. Thus, the water itself is also part of the crust, and a topic of geology, a science which investigates the structure and composition of the crust. That part of geology which investigates the extension of water-bearing layers, their parameters in con2*

20

1 Fundamentals for the investigation of seepage

nection with the storage and movement of water, and the characteristics of ground-water is called hydrogeology. All three sciences mentioned in the previous paragraphs require the investigation of the movement of various types of subsurface water (infiltration; flow of water and air through unsaturated pores; confined or unconfined seepage of both shallow and deep ground water, etc.). The analysis of movement is part of physics, and, within this large field, that of mechanics. When dealing with the flow of fluids it is called hydromechanics. This branch of science - together with hydraulics which is a form of hydromechanics simplified and applied in engineering practice - supply important supplementary information for soil science and geology, as well as hydrology, and covers the entire domain of subsurface water.

1.1.2 Characterization of flow through various aquifers

As explained in the previous section, hydromechanics and hydraulics include the investigation of movement of all types of subsurface water. When further distictions have to be made according to the different forms of flow, it is necessary to analyse the parameters characterizing the movement of the previously selected groups of subsurface water. It has also been mentioned that the character of the flow is basically determined by the structure of the water-conveying network composed of the interconnected interstices of the layers. Apart from its structure, the instantaneous conditions of the network at the time of the movement (e.g. the rate of saturation, or the prevailing pressure condition) also provide important aspects for the classification of flow. The greater part of subsurface water (at least that water available for practical utilization, and economical exploitation) is stored in loose clastic sediments. The flow of ground water is, therefore, closely related t o the physical character of these aquifers. The pores between grains form randomly interconnected, complicated pipe- or channel-systems in such a layer. The resistance of these continuous but very tortuous channels has t o be overcome when water flows through the layer. The network of the waterconducting channels is generally equally distributed in these sediments, the contact between the pores being ensured in each direction with equal probability. Hence, in most cases these aquifers can be regarded as the homogeneous flow space of a porous medium limited by the neighbouring impervious formations. In connection with the previous statement, it can easily be seen from the description of the character of the channel network composed of the pores between grains, that the loose clmtic sediments satisfy the requirements set up in the definition of porous media by various authors (Bear et al., 1968; Bear, 1972), mostly in the form of relative terms. The solid matrix surrounds the pores, the significant majority of which are interconnected. The pores are filled with liquid and/or gases, and the pore size is relatively small compared to the whole aquifer. The specific surface (the internal surface dividing the different phases of the heterogeneous medium

1 . 1 General characterization of subsurface water

21

related to its total volume) is relatively high, and the phases of which the medium is composed, are present in each representative elementary volume. [The detailed interpretation of the representative elementary volume is based on the continuum approach of porous media (Bear, 1972). This is defined as the limit of the size of an elementary cube or sphere around an arbitrarily chosen point of the domain, above which the numerical parameters describing the structure of the multiphase medium (e.g. ratio of pores to the total volume, or specific surface) do not change considerably as a function of the size of the investigated body, but only have random fluctuation. Below this limit, the same parameters determined for a volume of the same shape may change in magnitude depending on the size in question (see Fig. 1.1-8).] The constancy of characteristic values determined for bodies larger than the representative elementary volume makes the elementary volume values characteristic of the whole aquifer. This reasoning, which is generally valid for loose clastic sediments, is the basis of the practice, which regards the aquifers as unified flow spaces. It has also been mentioned that in most cases, the pores are equally distributed in these formations, and the development of channels composed of the interconnected pores may be observed in all directions with equal probability. This is the basis of the theory which states that loose clastic sediments usually form homogeneous flow domains. Solid rocks having equally distributed pores, or a dense network of fine fissures, form unified flow domains similar to that of loose clastic sediments (sandstone, fractured dolomite). This is because the structure of the channels built up by these pores and fissures never or rarely differs from the network composed of the interconnected pores between individual grains. However, some differences may be caused by the different size of the pores (e.g. in sandstones the cementation decreases the size of the pores), or in fissured rocks the closed and open fissures may be oriented according to the main directions of geomechanical forces. Inhomogeneity may also result from porous solid rocks (sandstone, basaltic lava flows) because the permeability of the fractured zones is generally higher, than that of the whole mass of the formation. The development of fractures causes not only inhomogeneity of the flow domain, but also changes some of the characteristics of the formation. If the characteristics of the solid matrix are the function of the direction of flow, the domain is called anisotropic. Orientation of the interstices is more dominant in beds where water movement develops along the contact planes of thin layers. In this way, the flow in laminated rocks (e.g. slate, marl) may become two-dimensional, even if the movement extends to the total volume of the layer. Sometimes, the flow in the space of the aquifer may be further simplified and investigated as a one-dimensional movement in solid rocks, where flow can develop only along large openings or dissolved channels (e.g. karstic limestone), when the channels are not interconnected. Naturally these types of movement may be present simultaneously as well. For instance in limestone in addition to one-dimensional flow, or in sandstone in addition to the three-dimensional movement, there may develop two-dimensional flow along fractured zones. The latter may even become dominant. As another example, loose clastic

22

1 Fundamentals for the investigation of seepage

sediments can be mentioned, which may have anisotropic properties caused by varying conditions during the process of their deposition. In the previous paragraphs the structure of the flow domain, or that of the water conveying network was described only qualitatively. Even when describing porous media, relative terms were used and not numerical parameters. Naturally, quantitative values are also applied t o characterize the various aquifers, e.g. specific surface which has already been mentioned, or porosity, which is the most commonly used parameter. Porosity (or volumetric porosity) is defined a,s the ratio of the volume of void space to the bulk volume of the porous medium. When determining this parameter, one of the basic requirements is, that the investigated sample should be larger than the representative elementary volume (Bear, 1972). Another important aspect is t o distinguish the pores acting as part of the interconnected water-conveying network from those not taking part in water transport. The pores may be inactive, either because they are completely separated from the interconnected channels, or the same pores may form dead ends of the network, or the opening may be so small that the fluids contained in them are rendered essentially static by the close proximity of the force fields at the surface of the solid matrix (De Wiest et al., 1969). Taking into account the difference between the numerical values of the pore volume used as the basis of the calculation of porosity, different terms may be distinguished such as: (a) Total porosity, when the volume of all pores are related to that of the sample; (b) E8ective porosity considering only the pores available for the transmission of fluid in the porous medium; (c) Isolccted porosity due to separated and inactive interstices. The detailed analysis of porosity in loose clastic sediments will be given in Chapter 1.2, and some special aspects of determining porosity in solid rocks (e.g. interpretation of primary and secondary porosity; statistical evaluation of porosity in karstic formations) will be discussed in Part 3. It is also necessary to note that there are other interpretations of porosity which differ from that given previously as the ratio of pore volume to total volume, and which is called volumetric porosity. The other possible msthods of determining porosity are the calculation of linear and areal values (Fig. 1.1-4).

Analysing the porosity of a sample along a line, the lengths of the pores are related to the total length of the sample. The calculated ratio is the parameter called linear porosity: k

n , =-1-1

Ax

(1.1-2) *

Similarly, the sum of the arew of the pores in a given section has to be related t o the total area of ths cross section t o obtain areal porosity:

1.1 General characterization of subsurface water

23

(1.1-3)

Special relationships exist between the various parameters of porosity, on the basis of which they can be substituted by each other. The average areal porosity can be calculated aa the mean value of the same parameters

------Fig. 1.1-4. Interpretation of linear, areal, and volumetric porosities

determined for several parallel sections. If the distances between the sections are not equal t o each other, the areal porosities measured in various sections have to be adjusted according to the length of the sample belonging to the section in question. Assuming that the element distance is small ( d z ) and the change in areal porosity in the direction of the z axis is given by a continuous function m(z) the calculation of the average value can be expressed in the form of an integral:

where S is the total area of the cross section (S = Ax Ay in the figure) and the s h ( 2 ) fundion expresses the sum of the area of pores in the sections depending on its position [thus, sh(z)= Sm(z)].It is evident, therefore, that the length of the sample multiplied by the area of the cross section is equal to the volume of the sample ( V ) ,in prismatic or cylindrical samples. The integration of the s&)function dong the length of the sample similarly gives the total pore volume of the sample (V,,).Because the ratio of these

24

1 Fundamentals for the investigation of seepage

two volumes is equal to the volumetric porosity, i t can be stated, that the average areal porosity is equal to the volumetric parameter in the case of randomly distributed pores [the latter hypothesis originates from the condition where the m(z) function is continuous]. The equality of the average linear porosity and areal porosity can similarly be proved. Hence, it can be

Y

Fig. 1. 1-5. Ih e a r , areal and volumetric porosities of a structure having interstices composed of equidistant spaces between cubes

assumed, that the three different interpretations of porosity calculated for samples larger than the representative elementary volume, pro\-ide the same numerical value. A condition of the previous derivation was the random distribution of pores. In the case of a regular structure of fissures [when the m(z) function is not continuous], 8ome complications are caused by the difference between total and effective porosity. The orthogonal system of equidistant spaces between cubes can be used as an example (Fig. 1.1-5) for investigating the problems arising. Two different values of linear porosity can be calculated parallel to one of the main axes of the structure. Depending on the position of the line in question, it may cross the cubes or run inside the spaces: (1.1-5)

Similarly, two parameters characterize areal porosity as well:

25

1.1 General characterization of subsurface water

while total volumetric porosity is determined by the following equation:

(a + b)3- a3 3ab2+ 3a2 b + b3 (a b)3 (a b)3.

n=

+

+

(1.1-7)

Comparing the parameters, which are not equal to unity, an apparent contradiction occurs because these values are not equal, although equality would be expected for randomly distributed pores. At the same time, it is evident that the areal porosity expressed by Eq. (1.1-6) assumes a flow, perpendicular to one of the main planes of the structure lying along two axes of the coordinate system. In this case, however, a considerable portion of the pores (those spaces perpendicular to the direction of flow) do not transmit water, because there is no pressure difference within them. After decreasing the pore volume by the volume of the inactive spaces, effective porosity can be calculated [(a b ) 2 - u'](u b ) 2ab + b2 neff= = n;. (1.1-8) ( a bI3 (a"b)2

+

~

+

+

It can be proved in this way, that the equality of areal and volumetric porosity is valid also in the case of a regular network of fissures, but effective porosity has to be determined for such systems, considering only interstices taking part in water transport. After describing the structure of water conveying networks, the acting forces have also to be analysed for the complete characterization of flow. According to Newton, movement can only develop as a result of a driving force or forces, and, if there is movement in the system, forces are also acting as a reaction to the accelerating forces to stop, or slow down the motion. Steady movement can only develop, if the entire system of forces (both those accelerating and retarding the motion) is in a balanced condition. The most important accelerating force in the case of ground-water flow is gravity. Apart from this force, the pressure of the upper layers may be taken into account. These cause compression and force water to flow out of the pores due to their reduced volume. Sometimes there may be a further accelerating force as well: the pressure of vapour and gases enclosed in the layers. These are generally only significant at greater depth. Their effects become important, therefore, in mining of hydrocarbons, and also in the case of water exploitation from deep aquifers, and should be taken into consideration, when determining hydrodynamic relationships related to deep wells. There is a special case due to gravity caused by the Werent specific weights of water stored in various layers. The specific weights of water can differ from each other, because of differences in the amount of dissolved salts and/or temperature. These effects may be important, when investigating water exchange between deep and shallow aquifers and when studying sea-water intrusion. Soil moisture adheres to the wall of the solid matrix and is kept in an elevated pobition above the water table against gravity by adhebive forces.

26

1 Fundamentals for the investigation of seepage

The dynamics of this type of water differs from the water movement in the saturated zone, because the adhesive forces created by the interaction of the solid particles and water molecules are also accelerating forces apart from gravity. On the basis of the hydrodynamic aspects described above, the movement of subsurface waters can be divided into the following main groups:

( A ) Flow through saturated porous media, wherein the network of the pores is equally distributed and connected at random This and the following groups can be subdivided according to the main accelerating forces: (a) Gravity which is the solely dominant accelerating force; (b) The pressure of overlying layers, the difference i n the specific weights of water, or the pressure of vapour and gases also have important roles apart from gravity. Further clrtssification within this group considers the three possible retarding forces, and their combined effects related to one another. These forces are: inertia, friction and the adhesive forces between solid grains and water molecules. This type of subdivision can of course be applied in both previously mentioned subgroups baaed on the weight of the accelerating forces. Thus, the following secondary subgroups can be distinguished:

( i ) All other retarding forces can be neglected except inertia, the flow being turbulent ; ( i i ) There are two retarding forces to be taken into consideration, inertia and friction. The condition of flow can be regarded m a transition from the turbulent to the laminar zone; ( i i i ) The solely dominant retarding force is friction; the flow is laminar (the greater part of ground-water movement pertains t o this group, which is the so-called Darcy’s flow condition); ( i v )Adhesive forces createdby tension, and acting between the solid walls of pores and water molecules, also have an important role in the system as retarding forces apart from friction. This type of flow is called micro-seepage. Ground-water movement in loose clmtic sediments can mainly be characterized by the types of flow through the saturated porous medium previously described. There are, however, some solid rocks aa well (sandstone, fractured dolomite, basaltic lava, etc.), the water movement in which can be considered in this main group. The importance of the latter is, however, s m d e r , than that in sedimentary baains filled with loose claetic sediments, which are generally the major possibilities for the large-scale use of ground-water resources. For this remon, the determination of the hydrodynamic parameters of flow through porous media is baaed on the investigation of samples built up from grains and saturated pores.

( B )Flow through saturated, fracturated and fissured r o c b In solid rocks, water movement can develop mostly along the contact planes of layers, fractured zones and fbsures, or through dissolved openings

1.1 General characterization of subsurface water

27

and channels. The flow is usually turbulent, or belongs to the transition zone, because the size of the water conducting channels is generally larger, than that of the pores between the grains. The interstices are not equally distributed through the whole volume of the layer, but are concentrated in fractured zones. Their interconnections do not occur at random, but are directed in most cases by the structural planes of the strata. A classification based on the accelerating forces, is unreasonable for this group. There are only very rare cases, when the second most important accelerating force, the pressure of overlying layers, may have any role in addition t o gravity. Adhesion can be neglected in almost every cme, because the ratio of the internal specific surface, or the surface of the solid matrix related to the volume of water contained in the pores, is considerably smaller than that in loose cla.stic sediments. There are many cams, when inertia has a dominant role in the relatively largeopenings. Hence, Darcy’s equation can hardly be applied, or only in a modified form. I n other cases, where the interstices are narrow, the supposition of laminar flow is acceptable, but the non-evenly, or non-randomly distributed pores make the application of the hypothesis of united flow domain more difficult. There is another aspect, according to which some distinction between the various types can be made. This is the character of the water conducting channels or structures: (a) One-dimensional flow in openings, channels, and conduits connected in only a few places, the investigation of which is similar to the hydraulics of water movement in networks of pipes. This type of movement is characteristic of karstic formations, most important of all in limestones. (b) Two-dimensional flow along the contact planes of layers, structural planes, and in fractured zones. This type of movement can develop both in karstic and non-karstic formations. The most important types of rocks in the latter group are mark and slates. (c) Flow through the interstices of solid rocks where no definite relationship exists between the material of the layer and the structure, or character of the water conducting network.

( C ) Flow through unsaturated porous layers

M fractured rocks Among the accelerating forces creating and maintaining this type of water movement, two have a dominant role: gravity and the tension diflerence between two points on the surface of water films created by adhesive forces. The size of the effective area of the flow cross section is essentially influenced by the amount of air entrapped in the sample. Some subgroups within this type of flow can be distinguished according to, whether the air bubbles in the flow space are in contact with the atmosphere, or not. In the former case, it is advisable to distinguish between porous sediments and fractured rocks. As a result, the following classification can be given:

(a) In the case of flow through unsaturated porous sediments above the water table, the air included in the layer is in direct contact with the atmosphere, its pressure is determined by the atmospheric pressure and it may also be

28

1 Fundamentals for the investigation of seepage

influenced by the air-permeability of the formation (Morel-Seytoux, 1973; Morel-Seytoux and Khanji, 1974).The surface of the solid grains is relatively large, compared to the volume of the moving water, and the influence of adhesive forces as a retarding effect is, therefore, considerable. The decrease in these two parameters (e.g. in coarse-grained gravel) forms a transition to the second subgroup. (b) In fractured zones or conduits of solid rocks above the water table, contact with the atmosphere is similar t o that mentioned in the previous case, but the flow is less affected by adhesion. Infiltration is relatively rapid, storage capacity is small, and the evaporation of ground water through this unsaturated zone can be neglected. As a special form of this type of water movement, the open channel flow in karstic conduits can he mentioned, the hydraulic description of which is similar t o that of water courses on the surface. (c) Unsaturated layers can develop at great depths as well, where the gaseous phase originates from vapours and dissolved gases in the water, when the temperature is raised considerably. The pressure of this phase is determined by the local conditions prevailing at depth.

1.1.3 The definition of seepage

On the basis of the clausification of subsurface water and that of the hydrodynamical aspects explained in connection with the analysis of groundwater flow, those types of subsurface water whose detailed investigation is the objective of seepage hydraulics can easily be selected. It wm clearly demonstrated that the scope of hydrodynamics may include the investigation of the movement of any kind of ground water or soil moisture. Sometimes the whole section of hydraulics dealing with subsurface flow is called seepage hydraulics. This use of the term seepage is, however, not quite correct, because some examples of this water movement (e.g. turbulent flow in large karstic conduits either under pressure or with a free water surface) are far from the idea generally connected with the word: seepage. The best way to understand the clear concepts of seepage is to go back into the past and analyse the results of the original experiments by Darcy, who established the scientific study of seepage hydraulics. He investigated the resistance to flow of sand filters, and on the basis of the empirical evaluation of the measurements, in his famous book (Darcy, 1856) he evolved a relationship between the hydraulic parameters of flow through a sandy layer, thought to be generally valid for the characterization of water movement in porous media. The sketch of equipment used for repeating Darcy’s baaic experiments is shown in Fig. 1.1-6. A tube of constant cross-sectional area A (cylindrical or prizmatic) is flled with the porous medium. The sample is bordered at both ends by planes, perpendicular to the axis of the tube, and its length (the distance between the two borders) is L. Flow through the sample is created by a constant difference ( A H ) between the pressure heads of headand tail-water, maintained by recharging the upper space of the system,

29

1.1 General characterization of subsurface water

reference level Fig. 1.1-6. Sketch of equipment used for repeating Darcy's experiments

and draining the lower one, ensuring a constant water level at both places. The sample is in a horizontal position in the figure, although the original experiments were executed with vertically standing samples, and the literature also describes equipment generally using vertical samples. This difference, however, does not cause any discrepancy, because the potentials at any point of 110th ends of the sample are calculated as the sum of potential- and pressure-energy ( z y p ) , or, when expressed as the height of an equivalent water column, that of the height of the point above an arbitrarily chosen reference level, and the height of the water level above the point (z h ) . The reistance against the flow before and behind the sample is negligible because the cross section of the tube is relatively very large. The pressure is proportional to the difference in level, and hence independent of the po,4tions of the points of meaurement, depending on the vertical difference between the head- and tail-water levels only. The constant pressure head difference creates steady flow, the discharge (&) of which can be determined as the amount of water flowing through the system. This is measured in unit time at the spill way of the tail water. Darcy found, that the specific discharge (the ratio of the measured discharge to the total area of a section of the sample perpendicular to the direction of flow which is equal to the cross section of the tube containing the sample) is linearly proportional to the difference between the pressure heads and inversely proportional to the length of the sample:

+

+

(1.1-9)

The dimeiision of the specific discharge is equal to that of velocity [LT-l], and, therefore, Darcy named this parameter as seepage velocity. This is,

30

1 Fundamentals for the investigation of seepage

however, an apparent value, because a considerable part of the cross section of the tube is occupied by the solid matrix, and the area which is free for conveying water, is only a portion of the total area ( Z a ) .The ratio of free and total surfaces is equal t o areal porosity ( n l ) , which, considering Eq. ( 1 . 1 4 ) , can be substituted by effective porosity ( n ) .The actual mean velocity of water in the pores can be calculated as the quotient of the discharge and the sum of the free areas of the pores. This parameter is called eflective velocity and is proportional to Darcy’s seepage velocity, being l / n , the factor of proportionality: & & 1 V,ff = -= --- - v ; because Za = n A . (1.1-10) Za n A n

(To avoid misinterpretation of seepage velocity and effective mean velocity, the former is often called flux or specific flux in the literature, and is indicated by q instead of v.) Both terms in the quotient on the right handsideof Eq. (1.1-9) (difference between pressure heads and the length of the sample) have a dimension of [L], and hence their ratio I is dimensionless. This parameter, the hydraulic gradient expresses in an equivalent water column the energy loss necessary t o overcome the resistance against the flow along a unit length of the porous medium. In the case of steady flow, it is constant in time, and independent of the position of the investigated section, if the flow domain is homogeneous, and the area of the cross section does not change. Since the publication of Darcy’s book, Eq. (1.1-9) became the basic law of seepage hydraulics, often referred to in the literature as Darcy’s law. It states that seepage velocity is linearly proportional to the hydraulic gradient, and the factor of proportionality ( K D , hydraulic conductivity) is independent of either velocity or gradient, being a material constant which summarizes all the parameters of both solid matrix and moving,fluid, influencing the resistance against seepage. Hydraulic conductivity has a dimentjion of velocity [LT-l], and according to the dehition, i t is equal to seepage velocity created by a hydraulic gradient of unity in a given porous medium and a given fluid. Research workers, having further developed Darcy’s basic ideas, determined the dependence of conductivity on the parameters of the transported fluid. They found that hydraulic conductivity is proportional to Che ratio of specific weight ( y) and dynamic viscosity ( p ) of the fluid, or t o acceleration due to gravity (9) divided by the kinematic viscosity ( v ) of the fluid, which is equivalent to this quotient. The remaining part of the parameter i.e. E , the factor of proportionality in the mentioned relationship, depends only on the properties of the solid matrix of the porous medium, and is called intrinsic permeability, matrix permeability, rock permeability or sometimes only permeability. The dimension of k is consequently [L2]:

K D -- k - =Y k - . 9 P

(1.1-11)

V

Since the establishment of Darcy’s law, many research workers have tried to prove its accuracy and validity on the basis of their own measurements

1.1 General characterization of subsurface water

31

and theoretical studies (Hazen, 1893, 1895; Carman, 1956; Koieny, 1953; Zamarh, 1928; Zauberei, 1932; Zunker, 1930). Others - on the basis of experimental data - have denied the accuracy and applicability of this equation, and proposed other formulae instead of Darcy’s law (Forchheimer, 1886,1924; Lindquist, 1933; Lovaas, 1954). On the baais of the dynamic analysis of the movement of fluids through porous medium, recent investigations have proved that there is a large zone of flow conditions (i.e. laminar seepage), where Darcy’s equation is acceptable aa a good approximation As a further result of these investigations, (Kovhcs, 1969s; 1969b; 1969~). it can be stated that the formal application of Eq.(1.1-9) can be extended to non-laminar flow conditions aa well, if Darcy’s hydraulic conductivity is multiplied by a function depending on either seepage velocity or hydraulic gradient, and the non-linear proportionality between the two variables can be followed in this way: v =KI; where

R = R D @ I ( v ) = k -@9I

( v );

V

or

R

= KD@2(I)= k -9a 2 ( I ) .

(1.l-12)

V

Consequently, the generalized hydraulic conductivity is composed of three factors: (a) Intrinsic permeability, including all the parameters of the solid matrix influencing the development of seepage; (b) Properties of the flowing fluid, expressed a,a the ratio of acceleration due to gravity and kinematic viscosity of the fluid in question; (c) Function depending on either seepage velocity or hydraulic gradient for characterizing the mtml flow condition. A further expansion of the concept of hydraulic conductivity is its application t o the description of flow through unsaturated porous media (Averjanov, 1949a, 1949b; Bear et al., 1968; Bear, 1972; Irmay, 1954; Kovhcs, 1971a, 1971b). It is a generally accepted assumption that the specific flux through an unsaturated sample is smaller but proportional to that flowing through the same porous matrix in a saturated condition, assuming that hydraulic gradients are equal in both systems. It follows from this relationship that the apparent hydraulic conductivity of an unsaturated sample, generally called unsaturated conductivity can be determined by multiplying Darcy’s hydraulic conductivity, or more precisely, the generalized hydraulic conductivity, by a factor depending on the rate of saturation of the sample. Unsaturated conductivity includes not only the resistance of the solid matrix, but also the effect of the decrease in area available for the transport of fluid, when part of the pores is occupied by air. In the equation

32

1 Fundamentals for the investigation of seepage

below the rate of saturation s is the quotient of volumetric water content W and effective porosity n:

K,

9

= K f ( s ) = K D @ z ( I ) f ( s ) = k -@ 2 ( 1 )

f(8).

(1.1.13)

V

It is necessary to note here that in the caae of unsaturated flow, apart from friction, the molecular forces become dominant as retarding forces. I n this zone, the modification of Darcy’s parameter can be better expressed aa the function of hydraulic gradient, than that of seepage velocity. For this reason, the second form of the generalized hydraulic conductivity waa used in Eq. (1.1-13). In connection with the validity of Darcy’s law in most cases only the interpretation of hydraulic conductivity and the applicability of the linear relationship is queried in the literature. As summarized in the previous paragraphs and aa will be explained in detail in Part 2 of the book, the solution of these problems can eaaily be found in the form of the dynamic interpretation of the generalized hydraulic conductivity, and the relationships between the various parameters can be determined. In Part 2, the practical application of certain factors will be discussed, together with their numerical determination from known physical parameters of soil. There is, however, a further hypothesis in the derivation of Eq. (1.1-9) which has to be considered when applying either Darcy’s relationship, or any other type of seepage laws. This supposition is included in Eq. (1.1-9) in the form of Darcy’s seepage velocity, although this fact is not explicitly explained either in Darcy’s publication or by other research workers having followed his experiments. The exact wording of the condition is aa follows: any type of seepage law can be applied only if the water-conveying openings in the cross sections, perpendicular to the main direction of flow, are randomly distributed, and small, compared to the entire area of the crosa section. If this condition is fulflled, the effective mean velocity in the pores can be substituted by seepage velocity (specific flux) calculated for any subarea of the cross section. Here a further condition arises: the subarea used as a unit should be larger in magnitude, than the representative elementary unit. In thitj caae, the ratio of seepage and effective velocities can be regarded aa equal to areal porosity, and the porous medium, bordered by pervious or impervious boundaries having geometrically fixed positions, can be dealt with aa a continuous flow domain. The basic physical concept of this special approach t o the investigation of seepage summarized in the previous paragraph is explained in various ways in recently published manuals dealing generally with hydromechanics (NBmeth, 1963), or especially with the hydrodynamics of flow through porous media (De Wiest, 1969). One of the latest and most comprehensive explanations is that given by Bear and discussed as a continuum approach in his book (Bear, 1972). For this reason, this main line of thought is pursued in the following paragraphs. To achieve the objective, of the characterization of the porous aquifer as a continuous flow field, the double application of statistical averaging has

1.1 General characterization of subsurface water

33

to be performed. The flowing water has to be described first of all as a continuum, instead of a mass of numerous, separately moving molecules. The molecular structure of water is substituted by water particles of a conveniently chosen size, ensuring that the material behaviour of the maas and the average parameters of movement can be regarded as homogeneous within the particles. Using these particles as parts of the water continuum, their dynamic and kinematic analysis provides the investigators with all the necessary information and means to solve the movement equations, and to determine in this way the parameters of flow if the boundary conditions of the flow domain are known. I n a porous matrix however, the actual flow field is composed of the very complicated and ramifying channels formed by the interstices. It is almost impossible to apply to seepage hydraulics the methods of solution, generally accepted in hydrodynamics and easily applicable to relatively simple forms of flow domains, because the consideration of the boundary conditions in the microscopic network of channels results in a large group of complex movement equations which cannot be dealt with mathematically. A second statistical averaging enables investigators to neglect the microscopic flow pattern of the actual water movement, and describe the entire domain of the porous medium as a unified and continuous flow space, the behaviour of which is influenced by both the solid matrix and the transported fluid. On this macroscopic scale the simplified parameters of flow can be easily determined by applying the basic movement equations for the new continuum, substituting the statistically averaged macroscopic properties of the domain, and considering the boundary conditions of this complex system. Thus any investigation of seepage requires the substitution of the molecular structure of the water with the Continuum of water particles, and the microswpic flow pattern followed by these particles with the macroscopic continuum of the porous medium. The application of the continuum approach is based on the theory of random variables, where the expected value of a quantity scattered randomly around a given value can be characterized by the average of the observed data, if the number of observations is large enough for statistical evaluation. As an initial example, the investigated random variable may be the velocity vector of the water molecules. It is well known that they have random movement (Brownian movement), even if the mass of water is in a static state. The resultant average of the velocity vectors scatters around zero in this case, and the variance can be lowered below a given limit, assuming, that the number of molecules included in the investigation is large enough. Ahove this limit the variance is practically constant, but below it the scattering gradually increases with the decreasing number of molecules, and achieves its highest value, equal to the fluctuation of the velocity of one molecule, if a single molecule is considered. Because the velocity of the Brownian movement depends on temperature, it is evident, that the smallest possible number of molecules, or the smdlest volume of water including this number of molecules - which is called the representative elementary unit - will change with temperature, and also with any other variables influencing the Brownian movement. Therefore, the size of the water par-

34

1 Fundamentals for the investigation of seepage

ticles has t o be determined, so, that they are hrger than the largest representative elementary unit occurring under the conditions of the investigation. The expected value of the resultant of Brownian velocities determined for such particles is zero, Therefore, the velocity vector of a particle characterizes the actual flow in the continuum. There is an upper limit to the particle size as well. The purpose of the concept of continuum approach is to describe some behaviour of the flow domain: e.g. the movement in the present caBe. The parameter to be determined may continuously change from point to point and i t may be represented by a series of discrete data calculated for each particle. Thus, the approximation of the continuum would be very rough, when applied to a particle of large size. The average value of a parameter determined for the particles still haa a random variance, compared to the expected value. It is necessary, therefore, to evduate statistically the number of particles for a reliable calculation of the expected value. Hence, a set of some tens of particles are regarded a~ a unit. These units can then characterize the behaviour of the flow space, a condition which requires the particle size to be smaller in magnitude, than the investigated domain. The transformation of a set of molecules into homogeneous particles ia the basis of the interpretation of most of the physical parameters of water e.g. the mass or density can be described by the number of molecules within a representative elementary volume. The latter is also a random variable, which gives an approximate value only, when a particle larger than a given limit is analyzed. The ratio of the jointed dipole molecules and the freely moving ones within a particle also may be statistically evaluated as a random variable influenced by Brownian movement. This property is viscosity. As already explained, the use of a regular system of discrete continuous functions for describing the properties and behaviour of the domain, instead of individual molecules randomly filling in the space, makes possible the application of hydrodynamic movement equations. These equations can be solved however, only in the case of relatively simple boundary conditions, and it is hardly regarded a~ a likely method to determine the parameters of flow through the microscopic channels of porous media, even if the capacity of today’s computers could be considerably enlarged. The microscopic view should be changed to the macroscopic characterization of the flow domain, following the same general concepts a~ previously used. It is necessary to note that the parameters calculated in this way are only statistical averages for a unit area or volume (larger than the representative elementary

Fig. 1 .l-7. Actual and average flow direction of watm particles t,hroughporous medium

35

1.1 General characterization of subsurface water

unit), and do not represent the actual extreme values: e.g. a continously changing velocity characterizes the flow domain although the effective velocity in a channel changes rapidly and periodically, following the random variation of the cross section. I n Fig. 1.1-7, the possible paths of water particles are shown through a porous matrix. Two particles following each other, may choose different paths at a bifurcation. Thus this process itself is a random variable, and the effective velocity at a given point may fluctuate with time. It can be stated, therefore, that the movement is time variable, even if the boundary conditions are time invariant, and theoretically, an unsteady flow is created by steady conditions, the average parameters of which (mean direction of flow, average flow rate through a unit area, etc.) are constant, and the movement in the flow domain can be regarded zw a quasi steady flow. The interpretation of porosity is a good example of the application of continuum approach for the determination of the parameters of a macroscopic flow domain. Similar to the process explained in connection with the averaging of molecular velocities, the porosity may also be determined for gradually increasing volumes. The extremes occur when using infinitesimal volumes. They are one or zero depending on whether the centre of the investigated volume is in a pore or within the solid matrix, and are independent of the actual character of the porous medium. When increwing the volume, the scattering decreases and achieves a constant minimum at the representative elementary volume. The calculated porosity fluctuates randomly around the expected value when the volume is above this limit. Naturally, the size of the representative elementary unit is strongly influenced by the structure of the solid matrix of the porous medium in question.

i

R

. . . -, ,..-. - .. .....,. ... .. : ,.... ... . -.,.'><;->.L< . ,/: ..;. ' .

t.;:

.

,.TTfVT S T ? r n f f T '

'

. : 4 .

f 7 ffff

,

I .

3;

,

,

of a homogenpors layer

0

maximum hoinoge~eausvolume Fig. 1.1-8. Interpretation of representative elementary unit using the example of determination of porosity representative elementary volume

3*

36

1 Fundamentals for the investigation of seepage

Porosity represents a good example of this statement as well. In a loose clastic sediment, the limit above which porosity fluctuates randomly, may be expected to be low, especially in sand and gravel. If the formation is layered, a slow change in the average porosity is probable after crossing an upper limit, indicating that above a given size, the investigated volume becomes inhomogeneous (it includes also a layer having properties different to the basic one), and the average parameter is more and more influenced by the second layer. Figure 1.1-8 shows the interpretationof porosity asa function of the volume size, based on the figures in Bear (1972). In fissured and fractured rocks, the distribution of the interstices is more uneven than that in a loose clastic sediment. The representative elementary length is generally longer in such formations and greatly varies with the structure of the rock type. To substantiate this statement, the linear porosity of a dolomite covered with dense fine joints is compared to the same parameter in karstic limestone. Four lines of various lengths in the dolomite were investigated, and the two extremes are represented in the figure. 1.6

(a) dolomite

a.9

( 2 samples of 50 rm) 0.8 0.05

n.7 ~

0.6

& B

$

a5

o

L

s

5

IO

{5

za

30

25

length af the separafely enalysed part of the fine rcm3

RJ

0.4

(b) limesfone

a3

( I sample of 25 m)

$& 2%

frequency disfribufian of the parasity of samples

0.2

8s -* & p 3 sz

a.i

a

-._._._

b.

I

I

1

1

I

I

I

2

4

6

8

ID

12

44

16

length of the separatply analysed part of the line [mi Fig. 1.1-9. Comparison of porosity distribution of limestone and dolomite

1.1 General characterization of subsurface water

37

Although the joints are relatively well distributed, the range of porosity determined for various stretches of 10 cm is still about 30-707; of the average parameters, while it is reduced below 10-20% if the investigated length is 25 cm. In karstic limestone, where chemically dissolved openings as large aa 50 cm may be found, the length of the acceptable representative elementary unit is 50-100 times longer than in dolomite (Fig. 1.1-9). On the basis of continuum approach and considering the interpretation of the general seepage law determining the relationship between seepage velocity and hydraulic gradient, the definition of seepage can be clarified. T h e movement of any type of subsurface water i s called seepage, if seepage velocity, as a characteristic parameter of the flow space, can be calculated and applied to any point of the domain, as a function of the local hydraulic gradient. In loose clastic sediments and solid rocks having similar structures (e.g. sandstone), the flow may be called seepage almost without any restriction according to the extension of the flow domain. In the case of fissured and fractured rocks, the minimum size of the flow space above which the aquifer can be substituted by a continuous domain may be determined and thus, the movement can be investigated by applying the seepage law. However, if the size of the investigated layer is smaller than the limite, the water transport through the separate channels has to be calculated. The applicability of both the term seepage and the relationships determined for the characterization of seepage depends on the size of the flow domain, if the latter is composed of fissured and fractured rocks as will be discussed in detail in Part 3.

References to Chapter 1.1 AVERJANOV, S. V. (1949a): Relationship of Permeability of Soil with Air Content (in Russian). D.A.N., No. 2. AVERJANOV, S. V. (1949b): Approximative Evaluation of the Role of Seepage in the Capillary Fringe (in Russian). D.A.N., No. 3. BEAR,J. (1972): Dynamics of Fluids in Porous Media. Elsevier, New York, London, Amsterdam. S. (1968): Physical Principles of Water PercolaBEAR, J., ZASLAVSKY,D. and IRMAY tion and Seepage. UNESCO, Paris. CARMAN, P. C. (1966): Flow of Gases through Porous Media. Butterworth, London. DAROY,H. (1866): Publio Water Supply of Dijon (in French). Dalmont, Paris. DEWIEST,R. J. M. (1969): Flow through Porous Media. Academic Press, New York, London. F A 0 (1972): Glossary and Multilingual Equivalents of Karst Terms. Paris. PAO-UNESCO (1973): Irrigation, Drainege and Salinity. Hutohison and Co., London, Southampton. FOROEH&IMER, PE.(1886): On the Yield of Wells and Drains (in German). Hannover Z&chrift dea Archdekten und Ingenieure. Vol. 32. FOROHHEIBIER, PH. (1924): Hydraulios (in German). Teubner, Leipzig, Berlin. HAZEN,A. (1893): Some Physioal Properties of Sands and Gravels with Reference Lo Tbeir Use in Filtration. 24th Annual Report of the Massachusetts State Board of Health, Boeton. HAZEN, A. (1896): The Filtration of Publia Water Supplies. New York. IAHS (1974): Hydrologid Investigation of the Unsaturated Zone. 2nd Circular. International Glossary. Budapest.

38

1 Fundamentals for the investigation of seepege

IRMAY, S. (1964): On the Hydraulic Conductivity of Unsaturated Soil. Trunaactwn of ABU, Vol. 36, No. 1. KovAcs, G. (1969a): General Characterization of Different Types of Seepage. 13rd COngreee of IAHR, Kyoto, 1969. KOVACS, a. (1969b): Rdationahip Between Velocity of Seepage and Hydraulic Gradient in the Zone of High Velocity. 13th Congreee of IAHR, Kyoto, 1969. KovAcs, G. (19690): Seepage Law of Microseepage. 13th Cmgreaa of IAHR, Kyoto, 1969. KovAcs, G. (1971a): Seepage through Saturated and Unsaturated Layer. Bulletin of IAHS, No. 2. KovAcs, G. (1971b): Seepage through Unsaturated PorousMedie. 14th Cbngrese of IAHR, Paria, 1971. KO-, (1963): Hydraulics (in German). Springer, Wien. LINDQUIST,E. (1933): On the Elow of Water through Porous Soils.1atCbrrgreee of ICOLD, Stockholm, 1933. LOVASS,L. (1964): New Reeulta of Investigations Related to the Permeability of Soils (in Hungmian). Hidrol6giai Ko&ny, No. 9-10. MORE~SEYTOUX, H. J. (1973): On a Modified Theory of Infiltration (in French). Cahiere ORSTOM, Sene Hydrolo ie, Part 1, No. 2 and Part 2, No. 3. M O ~ E Y T O T J XH. , J. and !CE.AN~I, J. (1974): Derivation of an Equation of Infiltration. Water Reaoumes Reaearch, No. 4. N ~ T HE., (1963): Hydromeahmica (in Hungarian). Tankonyvkiad6, Budapest. UNESCO (1972): Ground-water Studies. Paris. UNESCO-WMO (1974): International Glossary for Hydrology. Geneva. ZAXARIN, J. A. (1928): Calculation of Ground-water Flow (in Ruesian). Trudey I.V.H. Twkent. ZAUBEREI, I. I. (1932): On the Problem and Determination of Hydraulio Conduotivity (in Ruesian). Iaveetia V.N.I.I.G. M n g r a d , No. 3-6. ZUNKER, F. (1930): Bebviour of Soils in Connection with Water (in Germen). Handbook of Soil Science, Springer, Berlin. Vol. VI.

Chapter 1.2 Physical and mineralogical parameters of loose clastic sediments influencing permeability During the process of seepage, the resistance of channels formed! by pores between grains connected almost continuously and distributed at random in the flow space, must be overcome by the forces accelerating and maintaining the movement. The resistance of the network depends mostly on the size and shape of the pores forming the channels. It is also influenced however, by the adhesive forces acting between the solid walls of the channels and water molecules. It is necessary, therefore, t o investigate and determine all the geometrical factors describing the size of the channels, and the other characteristics influencing the conditions of their w d s . The geometrical parameters of the network depend on the size and shape of the grains, the degree of sorting of grain sizes (grain distribution), and the porosity. Apart from the latter, the adhesive forces are affected by the mineralogical and chemical character of the grains. These factors and their influence on the flow of water are discussed in this chapter, which summarizes some physical and mineralogical fundamentals of seepage.

1.2 Physical and mineralogical parameters of clmtic sediments

39

1.2.1 The size and shape of grains forming a layer Grains and their main parameters (surface, area volume, etc.) can be characterized by one or more dimensions of the particle only in the case of regular geometrical forms. In nature, the grains forming the various layers differ considerably from regular forms, especially from that of the sphere, which is generally used in soil physics to substitute the actual shape of grains, because this is the simplest form, all the parameters of which can be determined knowing only one datum, i.e. the diameter of the sphere. For this rertson, the introduction of a shape coefficient is also necessary in addition t o the diameter of the grains to characterize the difference between the sphere and the actual form. There are two different definitions of the diameter of a sphere equivalent to the actual grain. Both are related to the methods used in soil physics for measuring the size of grains. If the grain is larger than 0.1 mm, this measurement is determined using sievee. Thus, the diameter of the equivalent sphere in this range is equal to the size of the holes in the sieve (which is the diameter for a circular hole, or the length of the side of a square), through which the grain can fall. I n the case of finer grains, whose clize is determined by sedimentation (hydrometric method), the equivalent diameter is the size of a sphere settling in the water at the same velocity as the actual grain in question. In the first case, the diameter of the grain is equal to the diameter of the clphere encircling the grain. In the smaller grain-size range such a clear relationship between the grain size and the substituting sphere cannot be determined. A further problem is caused in the latter case by the fact that the colloid grains may sink separately or joined together as a flake, according to the colloidal character of the suspension (morphology of the surface of grains, coagulation, peptization). The settling velocity of an aggregate composed of many grains is equal to that of a single grain having the same diameter rts the flake. Taking into consideration this uncertainty, it is also acceptable t o assume that a grain size measured by sedimentation is approximately equal to the diameter of the encircling sphere, and t o ensure in this way the uniformity of the interpretation of grain size. There are several shape coefficients for characterizing the difference between the actual grain and the equivalent sphere. These are generally, calculated from the three main dimensions of the grain, perpendicular to each other. A further basic requirement is that the shape coefficient should characterize the difference mentioned above from aspects relevant to the investigation in question. Thus there are shape coefficients used in geology to determine the origin of the grains and the distance they have travelled (Hagerman, 1938; MihBltz and UngBr, 1954; Sztideczky-Kardoss, 1933) and in hydraulics concerning sediment transport and sedimentation (Heywood, 1938; Ivicsics, 1957; Stelczer, 1967). To investigate seepage, a shape coefficient has to be chosen, which suits the physical description of the process in question (KovBcs, 1968a). Friction is the most important force among those retarding the velocity of seepage. Apart from this inertia and adhesion may be dominant. TWOof

40

1 Fundamentala for the investigation of seepage

these forces (friction and adhesion) are proportional to the contact surface between the solid and liquid phases of the system. Acceleration due to gravity is proportional to the volume of water. The aftect of solid grains can be accounted by using the porosity. Finally the ratio of retarding and accelerating forces, which physically, characterize the flow, can be approximated by the ratio of the surface area of a grain (A) to its volume (V). The dimension of this ratio is [L-'1. Hence it can be expressed as the quotient of a dimensionless shape coefficient (a) and a characteristic diameter ( d ): (1.2- 1)

As shown by the equation, a is a function of the diameter chosen to characterize the grain. According to the physical determination of the soil, this diameter can be that of the encircling sphere (D). The corresponding value of the shape coefficient ( a D )for some particles having regular geometrical form is as follows: Sphere a D = 6; Cube tcg = 10.4; Octahedron aD= 10.4; Tetrahedron aD= 18. There are some crystals, where the ratio of the two main axes is constant (those determining its cross section) and the third varies considerably. Thus, the grains may have any form from a laminated plate to the shape of an elongated nail (pyramid, tetragonal, or hexagonal prism, circular cylinder, tetragonal or hexagonal pyramid). The shape coefficient of such grains can be expressed as a function of the quotient calculated from the diameter of a circle encircling the main section of the grain (d,) and the height of the grain ( 1 ) (Fig. 1.2-1). The dotted lines show the values of a,,.(the shape coefficient being calculated not from the diameter of the encircling sphere but from dl). The comparison of the graphs reveals that there is practically no difference between the two types of shape coefficient (i.e. aD and adl) in the zone of the disc-shaped grains, but the two values differ considerably in the case of nail-shaped crystals. Another conclusion which can be drawn from the figure is that the shape coefficients of prismatic forms having a square and oblong cross section respectively may also differ only in the caae of elongated grains, and only where the ratio of the sides of the oblong is significant (e.g. 1 : 10). The shape coefficient depends on the mineralogical character of the grains, which in turn may relate to the size of the grains. I n gravels, the dominant material of this grain size being quartz, the partioles are generally stubby, the lengths of the main axes not differing considerably, and the quotient l/d, is nearly 1. Their form can be approximated by a sphere, or a cube, depending on the amount of abrasion whioh has occurred during transport from the place of origin to the sedimenbation location. Thus, their shape coefficient is generally about aD =

1.2 Physical end mineralogical parametere of clastic sediments

ratio of hetghf to tbe dhneief of the circle surrounding the cross section ofa gmn1

C/q

Fig. 1.2-1. Shape coefficient of grains as a function of the ratio of their length to the circle encircling their cross seotion

E

42

1 Fundamentals for the investigation of seepage

Table 1.2-1. Shape coefficients of clay minerals

1 Kaolinite Illite Montmorillonite Na-montmorillonite Nail-shaped grains (e.g. halloysite, attapulgite)

1

I~UI

I

4~

I

W,

aD

0.16-0.40 O.OpO.03 gO.01
0.5-0.15 0.7-0.30

0.10-0.030 0.20-0.060 0.03-0.016 0.003-0.0016

30-70 20-60 70-100 700-1000

2.0-0.6

0.25-0.04

10-30

40-100

1.4-0.40

= 7 - 11. The cme of gravels originating from laminated rocks (sandstone, slate) is an exception; the shape coefficient of such particles may be = 20. high ZiS Quartz is the main component of sand as well. The shape of sand grains is very similar to that of gravels. I n this group, abrasion depends mostly on whether the investigated sediment is a fluvial or wind borne deposit. In the &st cme the edges of the grains are generally sharp and hence, the shape coefficient aD = 9 - 11, while the quartz grains of a wind borne sand can be characterized,bby a coefficient of about aD = 7 - 9. In sands, and especially the finer grained ones, a considerable amount of mica may

' \

I I

\ \

Gravel

'' i\

I R

30

NI

40

0

14

-

a

\

B

\ \

I

\

I!

% 20

12

\

/

1

t t

..................

d

I h

I

d

\

I

quartz .......... andesife (slope deposit)\ k l u v i a l and marine) 'f4. *'

slate (slope deposit)

..........

\ \

'\.- ---*.-..............

...................

\

1.2 Physical and mineralogical parameters of clastic sediments

48

also be found. This is a characteristically laminated mineral, and hence its shape coefficient is much higher than that of quartz (aD = 20 - 50). The clay minerals form a quite separate group from the point of view of shape. Their diameter is smaller than 2 p and they generally form very thin plates (illite, kaolinite, montmorillonite), but some special types can exhibit the form of a nail, which is really a tube rolled from thin plate (halloysite). These minerals have the highest shape coefficients (Albert, 1967; Kuhn, 1963) (Table 1.2-1). According to this summary, the shape coefficient of clay minerals generally varies between 30 and 100. Na-montmorillonite is an exceptional case, where the lattices are separated into very thin independent lamellae and thus, the internal surface of the crystals becomes active. The probable value of the shape coefficient for various grains of different sizes, origin, and mineralogical character can be determined by statistical evaluation of numerous measurements (Fig. 1.2-2). Such studies furnish suitable data for the approximation of the actual shape coefficient, which is applicable in practice without any further detailed investigation.

1.2.2 Investigation of grain-size distribution All parameters discussed in the preceding paragraphs are related only to a single grain. In nature, however, no layer is built up from particles of identical size and shape. It is necessary, therefore, to find methods by which the layer can be characterized a,s a mixture of various grains. For comparing the amount of grains of various sizes, the so-called grainsize distribution curwe is generally used, the dehition of which is well known in soil physics (Fig. 1.2-3) (JAky, 1944; KBzdi, 1972). Presented in some manuals are comparisons showing the different names used to indicate the Y O

/OD

90 80 70 L7U

50 40

30 20 I0

0

Fig. 1.2-3. The ratio of grains having different diameters in the sample characterized by thg grain-size distribution curve

44

1 Fundamentals for the investigation of seepage

various grain sizes in different countries or organizations (Bear, 1972). Many attempts have been made to fkd only one parameter by which the sample can be described, to identify it numerically, and to characterize its most important physical behaviour. Such parameters are the various characteristic diameters (e.g. maximum diameter Dmax;diameter corresponding t o the 50 % value of the distribution curve Ds0;diameter of grains of maximum weight present in the sample i.e. the inflexion of the distribution curve D,; etc.). Hazen’s design diameter, which pertains to the 10 % value of the distribution curve (Dlo),has an important role in seepage hydraulics (Hazen, 1895). Previously, its use wag generally accepted when taking into comideration that both the size of pores and the inner surface of the sample are influenced mostly by the fine grains. Thus, the physical character of seepage can be related t o a value measuring the amount of fine grains in the sample. This parameter is usually supplemented by the coeficient of uniformity which is the ratio of the grain diameter corresponding t o the 60 yovalue of the distribution curve (Dao)to Hazen’s diameter:

u = -QO.

(1.2-2)

DlO

There are some other design diameters, whose authors wanted to combine both Hazen’s diameter and the coefficient of uniformity, and construct one parameter from D,, and Dlw ,An example of these is Fedorenko’s diameter (Melentev, 1960). (1.2-3)

On the basis of recent investigations, theuse of Kozeny’s effective diameter

(Dh)can be recommended as a parameter, which can accurately describe the heterodisperse sample in seepage hydraulics (Koieny, 1953). According to the original definition, this is the diameter of a sphere, whoxe homodisperse sample (a sample of particles having the same diameter) has the same surface-volume ratio as the investigated heterodisperse spherical sample. The determination of the effective diameter using the symbols in Fig. 1.2-4, can be given aa follows. The distance between the maximum and minimum diameter is divided into n equidistant intervals. The average diameter in thei-thinterval is Di, and the weight of grains in this interval compared to the total weight of the sample is represented by ASi. Aswming the grains t o be spheres in both systems, and the surface-volume ratio of the solid phase to be identical for both the model and the investigated sample, the following relationship can be given: Qi

ND; n ND#n/6

-

6

=y,g 2% ; YS

Dh=

1

(1.2.- 4)

1.2 Physical and mineralogical parameters of clastic sediments

45

I00

90 80 70 QI

$

60

.% 50

0

Fig. 1.24. Symbols used for the calculation of effect,ivediameter

where N is the number of spheres in the homodisperse sample, and ys is the specific weight of the solid grains. A similar relationship can be determined in the c5tse of grains with shapes differing from the sphere, using the shape coefficient described in the preceding section. It has to be assumed that the shape coefficient of grains in each intercan be apprnximately characterized by an average value (ai) val. Following the same steps taken to derive Eq. (1.2-4), the quotient of the effective diameter and the average shape coefficient can be calculated: D h

1

(1.2-5)

The determination of the shape coefficient for every fraction requires mineralogical and microscopic investigation. For the practical application of Eq. (1.2-5) however, some informative data can be given on the basis of statistical evaluation of numerous samples. An example is given in Fig. 1.2-5, which indicates the probable shape coefficient of grains according t o their origin (alluvial or aeolian), mineralogical composition (quartz, feldspar, mica, clay mineral), and size, in terms of the forms generally used to represent grain-size distribution. Using a simple mathematical equation to approximate actual grain-size distribution, relationships can be determined between the various charac-

46

1 Fundamentals for the investigation of seepage Table 1.2-2. Relationships between varioua characteristic diameters Coefecient of uniformity (V)

Ratio of characteristic diameters on the baais of linear distribution

1.0

I

1.5

I

2.0

dculated

1.00

1.30

1.69

calculated

1.00

1.07

1.13

1.00

1.33

1.69

Ratio of characteristic diameters on the baais of logarithmic distribution 1.00

1.03

1.09

teristic diameters aa t h e functioncj of the coefficient of uniformity. I n Table 1.2-2,the results of two different approximations are given. It was first assumed that the relationship between the diameter and the integrated weight is linear. The second (so-called logarithmic) approximation which is generally used aasumes that a linear relationship exists between the integrated weight: and the logarithm of the corresponding diameter. The calculated

grain diameter, 0

[mmJ

Fig. 1.2-6 The average shape coefficients represented in the heading of forms used for plotting distribution c w w

47

1.2 Physical and mineralogical parameters of clastic sediments

aa the functions of the coefficient of uniformity 2.5

I

I

3.0

1.79

1.95

1.23

1.33

4.0

1

I

5.0

2.10

1.92

2.10

2.20

1.62

2.19

1.62

7.0

1

1

10.0

15.0

I

ao.0

1

25.0

2.25

2.25

2.25

2.25

2.25

1.92

2.58

3.67

5.42

7.18

8.90

1.77

1.94

2.13

2.28

2.48

2.64

2.73

2.77

2.78

1.17

1.26

1.40

1.58

1.87

2.42

3.12

3.85

4.42

ratios of D,/Dl0 and Ds,,/Dh were compared with meamred values (Fig. 1.2-6). It can be seen that the actual ratios generally lie between the two approximations nearer the logarithmic one. The figure is supplemented by another curve aa well. Paladin has used the diameter corresponding t o the integrated weight of SO%, to calculate the hydraulic conductivity of the sample. He has however, applied a corrective U=- OEO ’

I

2

3 4 5678910

40

W 30 405OSa.B0/0~ rtWionsb@ c u m on tbg b m s of lineur sipproximetion corrected lineer 8pproxirnution lajariihmic Bpproximation Palsdin formulls

3.5

3.0

2-2.5. 6 2.0

1.5 1.0

I

2

3 4 5 s?;Ssta

24 30 M ~ O ~ O ~ O O U

-

/J= Dl90

40

Fig. 1.2-6. Relationships between the ratios of different,characteristic diameters and the coefficient of uniformity

48

1 Fundamentals for the investigation of seepage

factor t o include the heterogeneity of the sample (Juhhsz, 1966; Paladin, 1964). If permeability really depends on the effective diameter, this factor is proportional to the ratio of D3,,/Dh.Reconstructing this relationship, it can be perceived that the curve corresponds to the measured data and is of similar character t o that based on logarithmic approximation. This fact verifies not only the accuracy of the described relationship between various characteristic diameters, but also the statement according to which Koieny's effective diameter is the most suitable parameter to characterize a heterodisperse sample in seepage hydraulics. The problem of aggregation of very fine grains has already been mentioned in connection with the determination of grain size. This process also makes less accurate the section of the distribution curve measured by sedimentation. The flakes settle with a velocity corresponding t o their diameter, ' and not to that of the individual grains. A higher weight is measured in intervals including the diameter of flakes and this amount is missing at the lower section of the curve (Szilvtigyi, 1966, 1967). Figure 1.2-7. represents some examples showing the deformation of the distribution curve. The first sample is a bentonite with no grains above the limit of 0.1 mm. All but one curve were determined by sedimentation, and the last by computing the weight of grains smaller than 2 p, using mineralogical methods (electron microscope, X-ray analysis, differential thermoanalysis). The first sedimentation was executed in distilled water and the others by adding various chemicals to the suspension. The sedimentation in distilled water measures the weight of grains finer than a given diameter (e.g. D = 2 p ) , and not bound to flakes in the natural condition. This is the lowest curve in the example in question. The mineralogical method determines the total weight of fine grains independent of aggregation (fully peptized conditions), and naturally, the corresponding curve gives the highest value at a given diameter. The other curves are between these two, showing the various grades of peptization created by the chemicals used. In the second example, where the sample contains mostly kaolinite among the clay minerals, the effect of the chemicals is somewhat different. Some of these chemicals do not increase the grade of dispersity, but on the contrary promote coagulation. Thus, the curve determined in distilled water is not the lowest in this case. Finally, the third sample was an illite. Its distribution curve was not altered by the chemicals used. From these examples the following question arises: when the sample contains clay minerals which distribution curve should be used to determine the effective diameter for the purposes of investigating the seepage? I n answering this question it must be realized that there are two kinds of bonds between the grains. The i%st is loose, the grains bound in this way can easily be separated either mechanically or by using chemicals. There are, however, irreversible bonds as well. In the latter case, from the point of view of seepage, the aggregated grains behave as one solid grain. The purpose of the investigation is, therefore, to measure the amount of individual and loosely bound grains separately from those irreversibly aggregated. The ratio of mobile or active (individual or loosely aggregated) clay minerals to those irreversibly bound to aggregates (immobile or inactive) is called the I

49

1.2 Physical and mineralogical parameters of clastic sediments

.bentonite (Mid) nOnfn7mifJOUXe 44%

illite keotinh’e 17% others 39 % (quartz, orthoclase)

too 4 -.

c

80

*%

2

60

8

40

s

20

-

0

.s

grain diameter, D. CmmJ kaoljnite ( Bombo/y) monfmoritlonite -

4s

$

100

p‘ 80 .9

3

G QJ

2

60

montmorillonite

-

40

h

!s

20

ki0 grain diameter, D tmnl

Fig. 1.2-7. Deformat,ionof grain-size distribution curves caused by the presence of colloid particles

morphological condition of colloid grains. This value has t o be determined and characterized by the distribution curve. Intense drying, frost, or chemicals change the structure of the crystals considerably, and can alter the morphological condition. For this reason, treatment cauning such alternation (e.g. drying) must be avoided during the preparation of the sample. 4

50

1 Fundamentals for the investigation of seepage

1.2.3 Mineralogical composition of sediments and its influence on the determination of the effective diameter

The mineralogical character of grains has a close correlation with their size. Naturally, it also depends on the type of rocks from which the particles in the sediments have originated. Grains larger in diameter than 2 mm (gravels) are mostly of quartz and feldspars. There are, however, many rock fragments in this fraction composed of several minerals. The amount of calcite may also be high in some places. I n sand (2 mm < D < 0.1 mm) the grains are composed predominantly of one mineral. Among the minerals, quartz is the most common. The amount of feldspars and calcite is less significant. The presence of mica may also be characteristic in this fraction. The next size interval (0.1 mm < D < 20 p ) is called mo or rock flour in soil physics. The mineral composition of this fraction does not differ from that of sand, except perhaps, in the larger amount of mica. The size of silt particles is 20 p < < D < 2 p. The considerable change in the physical behaviour of such samples (i.e. plasticity) is due mostly to clay minerals, which may already be present in this fraction in the h e r grainsizeportion. Finally, the character of clays ( D < 2 p ) is determined mostly by clay minerals and micas, although the other minerds (quartz, feldspar, and carbonate) may remain dominant in quantity. To illustrate the relationship between the size of grains and the mineralogical composition within a heterodisperse sample, Fig. 1.2-8. represents the distribution of various minerals according to the diameters of the particles (Kdzdi, 1972). As shown in the foregoing, the interaction between water and grains is mostly influenced by the surface-volume ratio of the latter. I n a homodisperse sample of spheres, this ratio is inversely proportional t o their diameter. I n the case of spheres of 2 p diameter (clay), this value is 10 000 times greater than that of a sample of particles having a diameter of 20 mm (gravel). The surface related to volume is further increased in the zone of small grains by the fact that the laminated forms are more frequent here than in the zone of gravel and sand, as clearly indicated by the higher probable shape coefficient of the small particles. Finally the mineralogical structure of the minerals having small diameters (mica and clay minerals) VJ

gfaiin dhmeter, 0 r,W Fig. 1.2-8. Mineralogical composition of a semple aa the function of the diameter of grains

51

1.2 Physical and mineralogical parameters of clestic sediments

may also increase the internal surface of the samples. At the same time however, the aggregation of these small particles can cause contrary effects, as mentioned in the previous section. Before investigating the determination of the effective diameter of samples containing colloid particles, a short summary of the structure of clay minerals will be given. Clay minerals originate mainly from the decomposition of feldspars in eruptive rocks. Generally three main groups of these minerals are distinguishable. The main representatives of these groups are kaolinite, illite and

the silicium net

A- A .section scheme o f the aluminium net

E -B section structure

of kuoknite cr&9tuI

structure of tnanttnarillonL+etrystd 60

4 Si

.

2(OH) + 40 4 A/ 2 (OHP40 4 Si

6 0 Fig. 1.2-9. The probable crystal structures of kaolinite and montmorillonite 48

52

1 Fundamentals for the investigation of seepage

montmorillonite. A common feature of clay minerals is the laminated structure of their crystalu. One type of lamella is composed of the plain net of one silicium and four oxygen atoms bound to each other in the form of tetrahedrons, while the other lamella is a net of octahedrons consisting of one aluminium and six oxygen atoms, or hydroxide radicals (Fig. 1.2-9) (Baver, 1948). In the caae of kaolinites [A14(Si401,)(OH),]the peaks of the tetrahedra in the silica net are aligned in the direction of the alumina net (the number of silica nets and aluminanets is the same), and thus, the oxygen atoms here are built into the octahedra with their free charges. The charges of the silica net are therefore, neutralized by the A l 3 + and OH- ions of the other lamella and the whole system of kaolinite haa free charges only at the edges of the crystals, where ions can be absorbed from the water surrounding the particles. H,0] In the crystals of montmorillonite [n(Ca, Mg)O Al,O34SiO, * H,O all lamellae of aluminiumhydroxide are surrounded by two silica nets (the aluminium can be substituted by iron or manganese in the octahedron net). Owing to the probable structure of the crystal, only every second tetrahedron of the silica net is joined to the alumina net, and the peaks of the others are turned outwards. Hence, the crystal grid has free negative charges on the surface of each group of lamella composed of three nets (one alumina and two silica nets). Ions can therefore, be absorbed at the internal surfaces of the crystals, and the type of the binding ions influences the behaviour of the montmorillonite. Water molecules can also be inserted between the crystal grids and absorbed by the free charges, causing the swelling of montmorillonite. The third type of clay mineral is illite (hydromica). Its structure and also its behaviour in connection with water and thus its role in seepage, can be considered aa a transition between kaolinite and montmorillonite. Summarizing the foregoing, it can be stated that clay minerals create electrostatic fields due to the free charges of the crystals. Thus, they can absorb ions from the surrounding solution. The binding of these ions however, is not permanent, and the so-called ion exchange can therefore, alter the chemical character of the minerals aa well. It is not yet clear whether the interaction between water and grains is also modified by the ion exchange or not. According to investigations on the adhesive forces, the tension is inversely proportional to the sixth power of the distance measured from the wall of the grains, and both the character of this relationship and the numerical parameters of the latter are constant, and is not influenced by the absorbed ions. The most acceptable hypothesis is therefore, to asume that the adhesive force binding the water to the grains is independent of the mineralogical and chemical character of the particles (as generally stated in the literature under the so-called Van der Waals force). The interaction between grains in the suspension, however, is influenced by the adhesive and repulsive forces (KovBcs, 1968b). Thus, the relationship between the distance from the wall and the total energy (which is the result of attractive and repulsive forces)will be considerably modified by the change in the repulsive force (Fig. 1.2-10).

-

+

1.2 Physical and mineralogical parameters of clmtic sediments

53

4

2 e 3!

,

-

' .

3 -

P

\

\ \

\

\

n\\

",

r ~ l s e\

f I

resu/tent

ettraction

l/tc

thickness of the diffuse double layer

Fig. 1.2-10. Dist,ributionof total electrostatic energy around a grain

grains (the ratio of independent, loosly aggregated, or irreversibly bouiid grains), and thus, alter the effective diameter, which depends on the active surface of the grains related t o their volume. Since the interaction between water and grains - accepting a constant adhesive force - is influenced only by the active surface of grains, change in this paramefer modifies the hydraulic character of the sample (water content, hydraulic conductivity, etc.). This line of thought returns t o the problem mentioned at the end of the last section: i.e. the correct method of determining the effective diameter (the grain-size distribution curve, or some of its characteristic points: e.g. the ratio of the weight of grains smaller than 2 ,u t o the total weight of the sample), when colloid particles are present in the sample. If the actual morphological character (the ratio of irreversible aggregates t o the total amount of colloid grains) can be taken into consideration using the granulometric curve, separate investigation of the influence of mineralogical and chemical

54

1 Fundamentals for the investigation of seepage

character is not necessary, because the effective diameter includes all the existing effects. It has already been mentioned in connection with Fig. 1.2-7, that the total amount of fine grains can be measured by mineralogical methods, and therefore, the clay content so determined is always greater than the actual amount of active particles, or equal to the latter if no grains are bound together by irreversible aggregation in the sample. (Clay content represented by the symbol Szccmeans the weight of grains smaller than 2 p related to the total weight of the sample). The clay content measured by sedimentation can only be smaller than, or equal to, the amount of active grains smaller than 2 p. I n the suspension prepared for sedimentation, some loose aggregation may occur apart from the irreversible bonds, and hence some active particles will be measured among the larger grains. It is necessary therefore, to find suitable methods for the direct determination of the actual active clay content. Previous attempts have been made to use the plasticity of the sample to characterize its clay content. It is obvious that plastic behaviour is caused by the clay content, and hence, a relationship must exist between the two phenomena. At the same time, the parameters characterizing plasticity (e.g. those used in soil physics: limit of plasticity wpr and liquid limit wL)can easily be measured the procedures being well known and standardized all over the world. Numerous data can be found in the literature, which give the corresponding values of plasticity ( w pand wL) and clay content (#,+) in the same sample. Some of these data are listed in Table 1.2-3. Data obtained by various measuring methods greatly differ from each other, and can be divided into two main groups. The first includes data measured by mineralogical methods (i.e. electron microscope, X-ray, DTA), and the second includes those determined by Sedimentation. The diversity of results is further increased by the use of different chemical treatment of the soil sample before or during sedimentation. There are also numerous articles investigating and describing the direct relationship between plasticity and clay content (Herczog et al., 1966; Seed et al., 1964; Szilviigyi, 1966, 1967), which differ widely, especially in the measurements used in the determination of the clay content. One article (Dumbleton and West, 1966) has to be dealt with in detail because the results they achieved and explained provided the Grst steps for further investigations (KovBcs, 1971). The first result in their paper is represented by four curves, which characterize the w p - Szp,and w L - Szp relationships for montmorillonite and kaolinite respectively. The clay content of pure montmorillonite and kaolinite samples determined by mineralogical methods was found to be nearly 100%. The limit of plasticity and the liquid limit of samples with a lower clay content have been measured using mixtures of these pure clay mineral samples with quartz sand and silt in various predetermined ratios. The second part of the paper presents the measured parameters of numerous natural clays. The clay contents of the latter have been determined by sedimentation. It is very interesting to note that the relation curves con-

1.2 Physical and mineralogical parameters of clastic sediments

55

Table 1.2-3. Corresponding values of plasticity and clay content of various samples (A) Data measured by mineralogical methods K - kaolinite, M - montmorillonite, 0 - others or undetermined 1 - illite, ~~

~

Clay content and the ratio of the various clay minerals

OharacteristiCa

of plasticity [%3

[%I

The type of the sample

Kaolinite HR. Kaolinite Ub. E.

88

-

-

96

6

64

33

84

-

-

96

69

30

Montmorillonite I. Illite-montmorillonite B.

93

86

-

10

187

63

88

60*

-

46

6 6 6

103

39

Illite-montmorilloniteS. Illite F. Halloysite N.

92

20*

20

60

10

70

30

82

-

90

-

10

63

66

76 _ _ ~ ~ _ _ ~ 77 Natural Ce bentonite 76

-

SO**

20

66

32

-

6

18

114 94

34 36

40

7

93

-

-

48 43

30 26

Kaolinite Mixture of the former three samples

26

-

6

91

4

26

23

60

38

2

48

12

67 60

63

68

1

26

16

76

38

19

3

70

8

49 48 67

26 42 60

70 46 23

1 2 3

4 10 14

74 42 46 61 67 82

21 21 29 26 20

60

60

26 28

32 30

26 12

16 30

49 47

27 29

87

44

17

39

80

26

70

-

96

-

6

93

36

-

82

18

27

23

46 61 74 43 63 61 74 78 83

11 22 33

-

66

60

24 28 34 16 10 9 14 21 31

61

-

60 72 66 79 82 84 80 78

22 26 30 24 32 36 33 32 31

100

96

6

76

96

168 116

68 40

60 26

96 96

6 6

82

28 17

Kaolinite Mixture of the former three samples

Ca bentonite Its mixture with quartz sand

36

-

-

11 22 33

Craft, 1967

~

Illite

Natural bentonite Illite

Referenma

~

24 46 71 71 48 23

33 61 44 20 4 9 13

_ _ _ -

_ _ -

60

Szilvhgyi, 1966

21 29 27 30

Dumbletonand West, 1966

56

1 Fundamentals for the investigation of seepage Table 1.2-3. (cont.) Clay content and the ratio of the varions clay miner&

Characteristics of plasticity [%I

[%I

The type of the sample

Kaolinite

96

-

-

96

6

82

42

Its mixture with quartz sand

72

-

-

96

6

64

31

48 24

-

-

-

-

96 96

6 6

44 23

22 14

100 100 100 100 100 100 100 100 100 100 100 100

30 34 36 33 38 40 49 48 66 46 47 69 66

16 20 18 20 20 23 26 29 34 27 28 34 40

Keuper marl

6 62 6 68 66 6

7 77

0

-

-

-

4

-

-

-

-

-

-

-

~

9

0

-

-

-

-

-

-

81 87 87 88 9 4 -

-

-

100

-

-

-

.-

-

-

-

-

DWEPK kaolinite

60

-

96

6

29

26

Clay

20

10

10

16

66

67

28

Mixture

13

10

-

-

90

30

16

Clay

63

--

46

-

66

38

19

Kaolinite

60

-

-

96

6

62

31

CLY

86

86

6

-

10

160

46

Bentonite

41

*

-

-

-

92

18

Na bentonite

63

*

-

-

__

391

31

Ca bentonite Kaolinite

62

*

-

-

-

122

28

-

78

32

ChY

68

-

78

19

Activated Na bentonite Its mixture with quartz sand

86

94

-

6

-

71 66 42 34 28 21 10

94 94 94 94 94 94 94

-

6 6 6 6 6 6 6

-

398 310

34 32

234 179 143 130 94 46

26

72

96

48 36 29

96 96 96

Ca bentonite Its mixture with quartz

56

~

*

-~

*

~

-

~

-~

~

-

-

--

-

23 14 14 14 16

6

-

87

35

6

-

67

6 6

-

46 49

21 19 16

References

Schrnertmann, 1962

Arcrtn, 1966

author's measurements

1.2 Physical and mineralogical parameters of claatic sediments

Table 1.2-3.

(COT&)

Clay content and the ratto of the various clay minerale

Charactaristicaof P l d i d t Y 1%1

The type of the sample

18 7

96 95

-

-

32

-

-

Clay (Oligocen)

36

-

71

29

46

22

Its mixture with quartz sand

29 23 17

-

71 71 71

29 29 29

38 32 26

16 16 13

Silt (Oligocen)

20

-

33

19

17 13 10

--

76 76

25

Its mixture with quartz sand

26 25 25

28 26 21

16 15 14

36 17

43 50

43 50

14

30

-

28

22 20

Silt (Alluvium)

-

75

76

13

-

Clay (Alluvium)

60

60

30

-

44

24

I t s mixture with quartz sand

41 33 25

50 50 50

30 30 30

-

42 31 26

21 17 16

*

References

Illite-montmorillonite

** Halloysite

(B) Data measured by sedimentation

The type of sample

the

Clay

Clay content and the type of the various clay minerals ~.

-

52

0

Method used for peptization

Na hexametaphosphate

Characteristics of plasticity [yo] Referelices

75

30

-

30

0

62

26

Na bentonite

-

87

M

530

120

Clay

-

50 46

0

44

0

80

21 30

20

M

114

34

Natural Ca bentonite

10

Boiled with N%CO3

Illite

2

14

I

N%Si03

Kaolinit e

6

23

K

Boiled with NaZCO3

Lo, 1961

Szilvhgyi, 1966

4843 26

30-25 23

57

58

1 Fundamentals for the investigation of seepage

Table 1.2-3. (omt.)

The type of the sample

Olay oontent snd the type of the various clay mlnerals without with peptiza- peptization tion

-~

Mixture of the former three

1 2

0 0 6 0 4 16

&y

minerals

0 7

Natural bentonite The former sample after cation exchange Illite

13

60

-.

60 42 28

M M M

16

27

The former sample after cation exchange Kaolinite

-

20 16 18

I I I I

3

18

K

42

0

62

0 0 I 0 M 0 M M 0 M 0 0 M M M 0 0 0

Silty clay

silty soil

aharaoteristica of PMiCits I% 1

-

WL

I

References WP

17 14 16 21 24 29 6 18 7 12

-

Method osed for peptiration

39 46

Silty soil

19 38 30 34 20

silty clay-soil

30 34

Silty soil

29 62 19 32 21

Clay

26 26

silty soil

16

NqCO, (NH,),CO, Cacl,

N%CO,

PO*

CaC1,

116 88 64

27 27 22

93

36

70 70 62

36 40 29

37

23

68

36

63

24

69

20

49

22

33 64 34 63 32

26 22 19 22 20

50 64

29 24

46

63 41 60 32

28 24 30 21 22

36 33

14 18

26

24

~

Na hexemetaphosphate

Thompson, 1966

_

_

1.2 Physical and mineralogical parameters of clastic sediments

59

Table 1.2-3. (cont.) (Ray content and the type Of the various

The type of the sample

Silty soil

clay minerals

I I

-

Clay

-

soil

-

B level of soil Silty soil

-

Limely loess

-

Loess Silty soil

-

-

Clay

Soil on slate

Soil (alluvium) Soil on basement ChY

26 29 31 27 18 17 28 14 18 11 26 23 7 11 29 40 36 31 39 27 34 64 69 71 76 72 83 74 67 62 68 41 63 73 81 83 83 86 87

Oharactarlatics of P W t Y C%1

osed tor

Without with oky peptiza- pepth- minw tion tion ale

-

Silty clay soil Silty soil

Method

0 M 0 0

M 0 0 I I 0 0

0 M M 0 M 0 0 M M M M M M M M

M M

M M M K R K K K K K R

Referenoes

pepthation

N8 hemmetephosphete

41 44 46 39 34

24 24 17 18 16

26 37 26 26 26 28 29 28 30 36 64 61 63 63

13.. 18 14 17

41 49 81 94 92 100 98 118 122 116 64 68 62 77 94 80 86 89 78 87

-

16 14 26 26 20 29 27 29 29 19 Dumbleton end 22 West, 1966 47 62 61 67 44 47 39 42 30 36 27 36 38 39** 42**

a** 38**

48**

60

1 Fundamentals for the investigation of seepage Table 1.2-3. (cont.) Clay content and the type of the various

The type of the sample

~

without] with

I

clay

~

I

lCL

I

I

-

30

K

90

36

-

47 52 69 61

62 97 106 62

26 38 40 28

Kmlinite

-

39

K K K K K

50

21

Kaolinite with bauxite

-

67

K

88

39

Kaolinite and Goethite

-

38

K

38

23

.-

54 80 89 41 20 79 18

0

60 106 103 69 titi 104 67

34 70 66 43 46 70 60

23 24 28 36 27

0 0 0 0

30 34 35 33 38

15 20 18 20 20

-

29

0

40

23

-

32 12 24 22 30 36 26

0 0

49 48 56 46 47 59 66

26 29 34 27 28 34 40

103

67

Kmlinite and Goethite

Bauxite Halloysite

-

-

-

Keuper marl

-

-

__ -

-~

Ca bentonite

10

78

Its mixture with quartz sand

10 8

60 43 38 32 11

7 5 3

Illite Its mixture with quartz sand

49

61

40 28 24

44 33 28

K* K* K* K* K* K* 0

0

0 0 0 0

M M M M M M I I I I

NaOH

c

of plesticity [yo]

Method

6o

83 61 55 47 32

36 28 24 21

87

43

76 68 50

32 24 21

Derab et al., 1967

a

61

1.2 Physical and mineralogical parameters of clastic sediments Table 1.2-3. (cont.)

I The type of the sample

Clav content and the type of the vartons clay minerals

i I

without with peptiaa- peptition tion

18 9 0

clay mine-

0 0 0 0

Kaolinite

-

48

K

IUite

-

36

I M

Its mixture with quartz sand

0

Na bentonite

Na bentonite Ca bentonite Clay (Oligocene)

78 11 0 0 3 0

Silt (Oligocene) Silt (alluvium) Clay (alluvium)

**

8

96

80 29 11 7 13 17 12

Method d for pepthtion

Ohteristics of plasticity [%] References

I I

23 10 18 14 11 10 8 4

Kaolinite

1

42 29 33 28 22 19 17 11

22 16 22 17 14 13 11 11

46

36

48 622

24 48

398 87 46 33 30 28 44

34 35

K K

K K K K

M

Na hexametaphosphate

Na$iO,

M I 1 0 0 M

Seed et al., 1964

author’s measurement

22 19 22 20 24

Halloysite The parameter increased due to keeping the sample still.

( C ) Data of brown forest soils from clayey Pliocene marine sediments measured by sedimentation

(The types of clay minerals were not determined) Clay content

12 12 10 16 17 20 10 16 13 37 20

Characteristics

34 36 39 56

89 86 42 60 38 73 44

21 19 22 23 27 26 20 25 20 38 28

Chanrcteristics

14 12 13 10 11 22 24 21 11 18 16

46 48 41 34 39 48 68 61 37 55 48

18 20 19 20 19 24 27 27 25 26 20

11 10 12 12 18 23 17 16 28 13 18

37 34 44 40 44 42 66 73 73 24 37

19 16 26 23 23 24 26 34 33 16 19

1 Fundamentals for the investigation of seepage

Table 1.2-3.(con$.)

16 13 12 12 11 12 13 10

12 13 11 17 12 11 9 14 28 20 19 23 16 14 14 12 20 13 19 22 16 23 16 18 16 19 16

36 60 66 48 46 46 41 60 48 38 46 43 47 47 48 46 70 66 71 61 67 62 46 46 67 40 44 64 46 60 60 62 61 68 46

22 26 23 23 22 21 21 26 22 21 24 24 26 23 24 24 28 29 34 24 33 27 24 26 21 23 20 29 23 23 20 26 26 24 24

6 13 10 8 16 18 16 20 26 19 6 27 18 34 11 20 12 16 18 30 9 13 7 6 6 7 6 11 10 8 10 6 10

9 9

42 43 60 40 66 61 40 67 62 76 46 63 66 71 38 48 43 47 66

66 34 33 63 43 32 30 33 48 43 38 66 32 42 60 62

29 24 24 24 24 21 24 23 23 36 21 26 28 26 24 22 20 18 22 20 18 18 19 18 13 16 20 23 22 22 23 16 20 22 20

10 14 16 7 9 13 7 10 12 14 17 10 14 16 14 8 11 6 7 14 6 6 10 10 9 8 11 8 8 12 10 14 6 27 17 8

48 60 82 70 48 49 46 46 42 48 66 61 43 60 60 34 32 26 27 47 27 26 46 41 41 38 43 42 38 37 33 32 38 67

46 37

21 26 39 32 23 27 26 20 23 20 31 29 23 24 24 22 18 18 20 21 17 16 20 20 19 18 19 20 20 19 18 21 21 23 31 26

1.2 Physical and mineralogical parameters of clastic sediments

63

(D) Data of alluwicrl loess eods meaeured by sedimentation (Peptization with NaPO,; the types of clay minerals were not determined)

24 12 7 4 2 4

6 6 3 8

6 6 6 3 4

6 6 6 4 14

6 6 16 10 1 6 6 6 3 2 33 2 1 2 37 21 40 6

6 4 10 23 13 30 13 26 3 14 16 24 16

32 31 29 23 16 20 36 23 12 37 19 18 31 18 18 18 29 18 14 30 30 23 23 16 14 26 37 23 37 26 42 19 19 38 31 43 47 34 17 16 39 31 40 29 32 39 16 36 24 32 24

67 69 68 43 31 40 62 39 32 73 37 42 67

40 46 36 68 33 32 66 61

44 44 33 28

60 66 48 79 47 76 41 29 62 60 67 86 73 63 38 67 61 64 86 46 67 28 68 46 63 42

26 26 24 23 19 22 28 22 22 30 21 20 26 22 23 21 26 20 20 28 21 20 20 18 19 21 22 23 26 21 23 28 14 21 22 24 19 24 22 19 22 24 27 28 22 28 21 21 22 26 26

16 6 21 12 17 24 10 17 14 6 16 28 11 18 11 8 37 8 30 29 10 10 41 33 30 16 23 19 16 20 26 13 22 10 24 19 10 22 9 26 10 3 6

7 3 20 12 2

3 18 1

33 11 31 17 32 36 17 18 28 28 27 39 12 34 19 26 41 30 30 46 24 31 46 46 26 28 40 20 33 29 39 24 32 28 32 22 17 22 13 39 28 23 19 26 33 34 30 16 21 34 22

60 29 60 38 62 68 31 37 46 48 49 67 36 60 39

42 82 49 42 63 40 49 86 71 63 60 62 68 48 72

60 49 64 49 68 38 41 68 36 71 22 42 40 46 48 67 69 30 40 67 44

26 19 23 20 30 28 24 19 22 24 24 28 24 22 23 20 30 23 21 28 23 26 36 29 20 21 28 26 21 28 18 23 24 23 23 21 21 26 18 26 18 20 27 24 28 24 27 18 19 26 19

6 6 1 6 36 7 4 6 10 9 20 13 11 31 19

44 6 7 6 6 11 16 10 4 0 26

6 6 10 20 6 20 7 26 7 4 3 3 2 17 30

6 6 11

6 11 2

0 2 38 10

20 28 13 29

60 32 23 10 26 14 21 27 27 60 27 41 11 30 26 24 18 23 22 12 27 32 23 26 20 23 17 31 26 37 19 16 9 12 26 27 41 21 26 19 14 11 18 14 20 62 26

40 48 34 68 92 66 41 40 47 30 49 47 68 83 60 88 21 48 47 39 38 47 60 29 61

66 48 44 38 49 43 60 41 66 37 62 28 27 42 44 41 64 64 34 48 26 40 30 38 86 60

19 24 22 26 36 23 20 20 22 22 26 20 24 30 26 36 16 22 30 20 19 24 26 24 26 23 24 23 19 24 24 23 22 24 20 23 23 23 21 24 23 24 23 20 23 19 20 12 20 30 32

64

1 Fundamentals for the investigation of seepage

structed from these data, do not show such marked differences between the types of clay minerals, as those based on the results of artificial samples: e.g. the curve of natural kmlinites more closely resembles the curve of artificial montmorillonite, than the curve representing artificial kmlinite samples (Fig. 1.2-11). The series of measurements quoted in Dumbleton and West’s paper reveals that the relation curve changes according to the methods used to determine

0

0

0.5

Determined wifh

minerological method

-- ------

monfmorillonife

-.-._.-kaoJihfe ......_._.......[puppr marl

sedirnenfsfiun

-----...- ...-...-

(siflgle data)

Fig. 1.2-1 1. Relationship between clay content and the parameters of plasticity (after Dumbleton and West, 1966)

the clay content. In their investigations, the authors used various samples of Keuper marl and the results are summarized in a figure which is reproduced here aa Fig. 1.2-12. The data obtained by mineralogical measurements indicate a close relationship. The scattering of points representing the data measured by sedimentation is greater, and the clay content corresponding to a given plasticity according to these data is considerably smaller than that obtained when the mineralogical data are used. When data measured in different countries and by various investigators are plotted on one figure (the w p - Szprelationship in Fig. 1.2-13. and the w L - S, relationship in Fig. 1.2-14), the separation of the points bears a direct relation to the methods used to determine the clay content, and not to the various types of clay minerals. This statement is verified by Fig. 1.2-13a and 1.2-14a, which represent almost all data listed in Table 1.2-3.

1.2 Physical and mineralogical parameters of clestio sediments

65

Only a few data had to be omitted, their results being very different from that of the majority of the samples. The different behaviour of these samples, however, can always be explained by the presence of special clay minerals (Na-montmorillonite, halloysite, or amorphous colloids). The left-hand side of each of these figures constructed from more than 500 data is covered with points measured by sedimentation, while on the right-hand side, the data determined by mineralogical methods can be found.

s?

*

% g

30 20 10

* 50 Q

g

o

io 20 30 40 50 60 70 80 90

loo

creg confent, SZp I%3 o

The S2/1 determinetion with sedimenfation wifh mineralogical

metbads Fig. 1.2-12. Modification of relationships between clay content and the parameters of plasticity caused by the methods used for determination of clay content (after Dumbleton and West, 1966)

There is quite a well defined border between these two regions in each cme. For characterizing these borders the following curves expressed by mathematical equations can be used:

+ 0.42Sg , w L = 0.175 + 1.26 s v .

wP = 0.125

(1.2-6) (1.2-7)

Only a few points from each group do not lie on their own particular side of the border, and the distance of these points from the given curves is generally smdler than the probable error in the original measurements (differences of -+lo% are indicated in the figures). The existence of a border between the two different groups of data is clearly demonstrated by the figures, and its sigdicance cannot be ignored. The first question however, concerning the evaluation of the figures is, what do the lines characterized by Eqs (1.2-6)and (1.2-7)mean? To answer this 5

66

1 Fundamentals for the investigetion of seepage

I aj Arrangement of data according to methods

used for measurement 0

50 -

dota determined by sedimentation data. determined by mineralogical methods

. . .

25

0

*

50

75

J

ib) Arrangement of data determined by mineralogical methods

?I

25

50

75

SZP Kb'?lCD

Fig. 1.2-13. Relationship between clay content and the limit of plasticit,y

question, it is necessary to recall that the total amount of clay content can be divided into three groups: '

(a) Individual grains (the total surface is active); (b) Grains united by loose bonds (almost the entire surface of each grain is active);

1.2 Physical and mineralogical pmxneters of clastic sediments

67

c) Arrangement of dsfa dztemioed b j seri'lmeffiafiou according to methods usEd for pepfisafion

x

English standard

o

Na CO,

8

wifbout peptisation

9

(c) Grains bound by irreverbible aggregation (the inner surface of these aggregates is not active).

It has already been mentioned that the total amount of fine grain&is measured by mineralogical methods, and therefore, the clay content determined in this way is always greater than or equal to the actual amount of active grains, if there are no grains bound together by irreversible aggregation in the sample. Meanwhile, the clay content measured by sedimentation can only be smaller than or equal to the amount of active grains smaller than 2 p, depending on the coagulation in the suspension. The right-hand sides of the figures include data greater than (or equal to) the active clay content, and the left-hand sides, those which are smaller than (or at the limit equal to) it. Thus, it is evident that the border line indicates the relationship between the really active clay content (including the fist two groups of fine grains, but excluding the grains bound by irreversible aggregation), and the parameter of plasticity in question. Consequently, i t can be stated that Eys (1.2-6) and (1.2-7) give the correct weight of active grains of 2 p diameter which is needed for the determination of the effective diameter. This value (azp) isafunctionof the limit of plasticity and the liquid limit, respectively. 5*

1 Fundament& for the investigation of seepage

68

(a)Arrangemenenf of data according to methods of measurement used 0

0

data determined by sedimenttion data determined by minerafogicaf meihods

Q

( illite monfmori:ionife)

0

mixed or undetermined samples

25

.

50

75

Fig. 1.2-14. Relationship between clay content and liquid limit

1.2 Physical and mineralogical parameters of clestic sediments

69

[c) Arrangeme& of dafa defermined by sedlinenfaff'on according to methods used for pe@WiOn x

W$

J

too

75

50

x

25

English stirndaro

wifhouf peptisation

I0

0

1

I

,

25

50

75

sz/lfil

1

A second question has also to be considered in connection with the present investigation: i.e. do the various types of clay minerals have any influence on the proposed relation curves ? Contrary to previous investigations, the answer is negative and requires further explanation. The points in the figures are not separated according to the clay minerals. The various points representing kaolinite, montmorillonite, illite, or other minerals can be found situated along the border with almost the same probability, which indicates that the border line can stand for all types of clay minerals (except for Na-montmorillonite, halloysite, and amorphous colloids). On the other hand, some separation can be observed on the right-hand side of the figures, among data determined by mineralogical methods (see Figs 1.2-13b and 1.2-14b). I n general the points for kaolinites, or other small grains that are not members of the three main clay-mineral groups (e.g. Keuper marl) are dispersed farthest away from the border line, while points for montmorillonite are located along it.

70

1 Fundamentals for the investigation of seepage

Recalling the energy curves shown in Fig. 1.2-10, some explanation of the differences can also be given. The energy curve of montmorillonite is similar to type A in the figure, while that of kaolinite is similar to type B. In the first case, the repulsive force is higher than the adhesive one over a considerable distance, and their difference increases as the distance decreases. The sign of the resultant changes only in the very close vicinity of the wall. The number of irreversible aggregations is, therefore, very small. The energy curve of type B shows, however, a higher adhesive force at each point, that of A . A considerable number of kaolinites - represented by this type of curve - can therefore easily be aggregated, so that the total clay content is always higher than the active one. The effects of various chemicals used to disintegrate the flakes in the suspension can also be investigated (Figs 1.2-13c and 1.2-14c). It can be seen that the use of sodiumhexametaphosphate (English standard) is the most reliable method. In general, however, the sedimentation involves a very high uncertainty. As a result it can be stated that the relationship between plasticity and the active clay content is not influenced by the type of clay mineral as long as Na-montmorillonite, halloysite, and amorphous colloids are excluded from the investigation. The probable ratio between active and total clay content, however, depends on whether montmorillonite, illite, kaolinite, or other minerals are dominant in the sample. The next step is the necessity t o control the validity of Eqs (1.2-6) and (1.2-7) and to provide a method for determining the presence of interfering clay minerals in the investigated sample. The simplest control is to calculate the active clay content, using different parameters of plasticity, and to compare the values obtained. When the two values of clay content determined from w p and wL are close to each other, the proposed method is applicable, while a difference higher than a given limit (e.g. 10 15%) refers to the presence of clay minerals, which exclude the application of the derived relationships. Some results of this type of comparison are given in Table 1 . 2 4 . Another method of control mentioned is the use of Casagrande’s A line. It is well known that the equation of this line gives a linear relationship between the index of plasticity (i.e. I , = wL - wp) and the liquid limit: I = w L - wp = 0.73(wL - 0.2). (1.2-8) A similar relation curve can be derived from Eqs (1.2-6) and (1.2-7): TV

,

I , = wL - wP = 0.05

+ 0.84 Sir = 2/3(wL - 0.1).

(1.2-9)

To evaluate the difference between Eqs (1.2-8) and (1.2-9) the possible scattering of points wm determined by assuming that there may be an error of &loyoand f 1 5 % respectively in the measurements of both the liquid limit and the limit of plasticity. 84.2% of points representing the sampleo listed in Table 1.2-3 are within the zone of 10% error covering this field almost uniformly. 32 points are located above this zone and 49 below it, from which 26 and 35 respectively are within the zone of f 1 5 y 0 error (Fig. 1.2-15). The figure was supplemented by the results of some previous

71

1.2 Physical and mineralogical parameters of clastic sediments

Material

euthor’ameasurement SzilvAgyi (1966) Craft (1967)

Kwlinite

Dumbleton and West (1966) Dumbleton and West (1966) Craft (1967)

Y

Silt

0.33 0.43 1.03

0.19 0.23 0.39

0.20 0.40 0.88

0.07 0.14

0.26

-

0.79

0.28 0.39 0.74

0.62

0.26

-

0.41

0.42

0.44

0.82 0.69

0.42

0.96 0.88

-

0.64 0.47

0.79

0.30

-

I

Actdm day wntent

0.36

0.66

Schmertmann (1962) Dumbleton and West (1966)

0.38

0.19

0.63

-

0.30

0.29

0.77

0.36

-

0.63

0.60

0.66

author’s measurement Thompson (1966)

0.33 0.64

0.19 0.24

0.20

0.07 0.34

0.26 0.43

0.28 0.41

’

Material

Claycontent determined by

Investigator

Illite

a

almm&z%ics of plasticity

Oharacteristica o! plasticity

IIlVestigator VL

I

-

Clay wntent determined by mtnereaedimentation

WP

:zz81

0.38

-

Active day wlltent

2;

I 2:

..

the preparation of the sample (probably amorphous colloids)

(1966)

0.78

0.86

0.61

0.74

72

100

rpPLI

90

70 .

number of data above the line o f 15 % error 60 50

number of data below the line of 45% error 14 (2.7 %)

40

30. I0f.S~

A

(DUn~~V8fOS)

20

Casagrande's A line A

Dumbleton and Wesf

(0

'

0-

Fig. 1.2-15. Represontation of the investigated samples in 8 coordinate system compoaed of the index of plaeticity and liquid limit (Ca88grande's A line)

1 Fundarnentale for the investigation of seepage

ao

1.2 Physical and mineralogical parameters of clastic sediments

73

investigations as well (Rkthhti, 1971; Herczog et al., 1966; Dumbleton and West, 1966). The comparison shows the reliability of Eqs (1.2-6) and (1.2-7). It is also reasonable in practice t o use Eq. (1.2-9) instead of Caagrande’s equation [i.e. Eq. (1.2-8)]. Not only is the accuracy of Eqs (1.2-6) and (1.2-7) supported by the existence of such a linear relationship, but at the same time the explanation of these equations, which was missing until now serves aa a physical background of Casagrande’s A line. The linear relationship of I , and wL is based on the fact that both values (more precisely w p and wL) depend on the active clay content of the samples, and this common cause creates close contact between the parameters of plasticity. Excluding some special minerals (Na-montmorillonite, halloysite, amorphous colloids), this relationship is not influenced by the type of clay minerals. Knowing the memured w p and wL values of a sample, the latter can be represented by a point in the I , - wL system. If this point is within the zone determined by the possible error in measurements, Eqs (1.2-6) and (1.2-7) can be used to calculate the active clay content of the sample. Considering the results of these investigations, the grain-size distribution curves, the lower stretches of which were either measured by sedimentation or determined by mineralogical analysis, can be corrected so that they represent the actual morphological character of the colloid particla. The method of corredion is as follows. Firstly, the distribution of grain diameters of particles larger than 0.1 mm has to be determined by sieving. The method generally used for memuring the lower section of the distribution curve is the sedimentation of the bample, from which the grains larger than 0.1 mm have already been separated. It is advisable to execute this analysis both with distilled water and by adding sodiumhexametaphosphate t o the suspension (English standard). The difference between the curves determined by these two methods indicates the morphological character of the colloid grains. If the ordinates of the two curves belonging t o the D = 2 p diameter differ considerably, the sample contains large amounts of rtctive colloid particles. It is expected that the active clay content is equal to, or near the value determined by the analysis of the chemically treated suspension. When there is a possibility of detailed investigation of the mineralogical composition of the sample (X-ray, electron microscope, DTA), informa,t’ion can be obtained not only on its total colloid content, but aldo on the ratio of different minerals (montmorillonite, kmlinite, illite, amorphous colloids) in the fraction of the sample, smaller than 2 p. This type of analysis is relatively expensive and therefore, not generally applied in every day routine work. After measuring the two plastic properties of the sample (liquid limit and limit of plasticity) the active clay content (the ratio of the weight of colloid particles having a smaller diameter than 2 p , and not bound irreversibly in aggregates to the total weight of the sample) has to be determined from Eqs (1.2-6) and (1.2-7). If the two values calculated in this way do not differ considerably, or the point representing the measured wL and w p value, lie near or along the corrected Casagrande’s A line in Fig. 1.2-15. the average

74

1 Fundamentals for the investigation of seepage

of the two calculated S2,, values can be accepted as the most probable active clay content. This value indicates the height of the grain-size distribution curve belonging to the diameter of 2 p. Hence, the curve may be corrected to pass through this point. The corrected distribution curve can be the bmis for the determination of the effective diameter, the calculation of which is explained in the previous section [Fig. 1 . 2 4 and Eq. ( 1 . 2 4 ) or (1.2-5)]. It is hoped that this diameter correctly expresses the morphological character of the colloid particles, being calculated from the grain-size distribution curve corrected by considering the active clay content of the sample. According to the hypothesis explained previously, the mineralogical composition of the grains modifies the behaviour of the sample in relation to water only through influencing the development of irreversible or loose aggregates. It is assumed therefore, that the effective diameter calculated in this way is suitable for the characterization of the resistance to flow through the pores of the sample, without any further mineralogical investigations if the colloid particles behave normally. It has already been mentioned that there are some minerals, the presence of which considerably changes the interaction between water and grains in the sample. The most important minerals causing such “abnormal7’ behaviour are Na-montmorillonite, halloysite and the amorphous colloids. The interfering effects of these minerals are indicated by the large difference calculated from Eqs (1.2-6) and (1.2-7), or between the two values of S2,, by the fact that the point representing the measured wL and w p parameters of the sample does not lie along the Cmagrande’s A line. In this case, detailed mineralogical investigation of the sample is advisable. Examples of the correction to the grain-size distribution curve according to the calculated active clay content are shown in Fig. 1.2-16. The distribution curves of two samples are represented in the figure. The samples (Ca bentonite and clay) were analyzed both mineralogically (curve I ) and by sedimentation, the latter with distilled water (curve 2) and after treating the suspension with NaCO, (curve 3). Thus, originally three curves were determined for each sample. The correction was executed on the basis of the calculated active clay content (curve a), and the effective diameters were determined. The ratio of the effective diameters determined from the corrected distribution curve, and that calculated on the basis of the analysis of the chemically treated suspension is 1/300 TV 1/200. (N.B. I n the examples, the probable active clay contents were determined from a third parameter of plasticity, i.e. maximum molecular water retention capacity, which also hm a close relationship with the active clay content.) The examples can also be used to demonstrate the change occurring as the result of any modification of the morphological character of the colloid particles. For this purpose, the influence of intense and repeated drying was investigated. The water films surrounding the grains are decreased by drying, and hence the distance between the grains also decreases, a process which increases the number of irreversible aggregates in the sample. The consequence of drying is therefore, the lowering of plasticity, and the increase in the hydraulic conductivity. The latter parameter can be mea-

1.2 Physical and mineralogical parameters of claatic sediments

75

groin diameter CmmJ Fig. 1.2-16. Change in grain-size distribution curves aa a result of drying

sured for the control of the accuracy of the distribution curves constructed on the basis of the previously explained method. The numerical results of the investigation are summarized in Table 1.2-5, and the probable grain-size distribution curves are represented by dotted lines. The clecreme in the parameter used for the characterization of plasticity, clearly indicates the increase in the number of the irreversibly bound Table 1.2-6. The change in maximum molecular water retention capacity and hydraulic conductivity aa the results of drying

Cs bentonite

37.6

1.3 x 10-o

29.0

9.3 x 10-0

Clay

20.2

4.6 x

16.9

4.1 x 10-7

particles. The new distribution curve shows the increase in the effective diameters, which is the expected result of aggregation. In the case of the first sample, the effective diameter is 2.65 times greater than that of the original sample, while for clay this ratio is 3.16/1. This result can be checked by comparing hydraulic conductivities measured before and after drying. As will be proved further on, hydraulic conductivity is linearly proportional to the square of the effective diameter. Thus, the expected ratio of the

16

1 Fundamentals for the investigation of seepage

increase in the former parameter is 7/1 and l o l l , while the measured quotients are 7.1 and 9.1, respectively. This investigation proves both the reliability of the method proposed to determine the relationship between active clay contents and pmameters characterizing plasticity, and the influence of the change in the number of irreversible aggregates on active clay content.

1.2.4 Relationship between porosity and the geometrical parameters of grains Apart from the size and shape of grains and the grain-sizedistribution, there is a further important variable influencing the size of pores and channels between the grains i.e. porosity. This is the volume of pores (V,) related t o the total volume (V,) of the sample*: (1.2-10)

[the meaning of the symbols used in Eq.(1.2-10) is indicated in Fig. 1.2-17, KBzdi, 19721. In some investigations, the use of void ratio instead of porosity is reasonable. This parameter is the volume of pores related to that of the solid grains:

V V

e =2, and

consequently e =

n

.

~

1-n

(1.2-11)

ti0

3

p = - =VPp ' + e "

V

n=-=-G V,+V

e e+t

Fig. 1.2-1 7. Symbols used in connection with the determination of porosity and void ratio

* The problems occurring in connection with the correct interpretation of porosity have already been mentioned in Section 1.1.2, but the differences between total and effective porosities caused by the presence of pores not interconnected with the network of the other interstices can be neglected when loose clastic sediments are investigated.

1.2 Physical and mineralogical parameters of clastic sediments

77

The scattering of porosity is considerable. The porosity of the loosest structure of a homodisperse sample of spheres (Fig. 1.2-18), when the centres of the spheres form a hexahedron in space (a= go"), and each sphere is in contact with six others, is n = 0.476. I n the case of the most compact sample, the angle with the same configuration being a = 60°, horizontal seri7on

vertical section

Fig. 1.2-18. Position of grains in a homodisperse sample of spheres having various porosities

each sphere is in contact with twelve others, and the porosity is n = 0.259. I n general, when the angle mentioned previously lies between the two limits (a = 90 + 60),the porosity can be calculated from the following equation (Slichter, 1899): n=l-

nl 6(1 - cos a) 1/1

+ 2 cos a

(1.2-12) *

The upper limit of porosity, however, is not absolutely invariable. A stable structure of spheres can be built up where the particles form arches, and 0.7 (Jkky, porosity is higher than the theoretical upper limit: n = 0.6 1944). The variability of porosity is even greater in the case of natural layers. It is influenced by the origin of the layer, the size and shape of the grains, the grain-size distribution, and the pressure affecting the layer a8 well. Fluvial sediments (sand and gravel) are generally well compacted, while sand of lacustrine origin is usually deposited in less compact layers. The porosity of cohesive materials (clay) is influenced by chemicals dissolved in the water. When these chemicals increase coagulation, the settled flakes build up a loose layer, while a more compact layer is formed from suspensions in which the development of flakes is hindered. The relationship between porosity and the size of particles can be explained by the fact that the number of contacts between grains related to the unit weight of the sample increases when the size of particles decreases. Thus, the resistance against the movement of a grain attempting a more stable position, is also increased, the layer of fine grains being generally less compacted. This effect is strenghtened by the shape of the grains as well. Among fine grains there are more laminated particles and, therefore, their surface-

-

78

1 Fundamentals for the investigation of seepage

Homodiapersesamples consisting of following grains

Material

I

The porosity of the sample

I

I

D i ~ e r d cShape ient

inloose incompacted ~condition

Ground rocksalt

1.60

Ground mica

1.60

90*

Lead balls

2.42 2.03 1.40

6.0 6.0

6.0

0.369 0.369 0.370

Round beans

3.43

10*

0.376

Poppy seeds

0.61

8*

0.398

2.24 0.36 0.12 0.046

8* 8* 8* 8*

0.377 0.382 0.386 0.426

Sand

0.218 0.281 0.366 0.448

9.4 8.0 8.3 7.6

0.437-0.428 0.408-0.424 0.412-0.426 0.424-0.41 9

0.372-0.384 0.374-0.373 0.384-0.386 0.377-0.374

Sand after making i t glow

0.218 0.281 0.366 0.448

8.4 7.8 8.0 8.1

0.462-0.436 0.468-0.440 0.439-0.466 0.436-0.436

0.392-0.382 0.389-0.393 0.406-0.419 0.384-0.386

Ground flint

0.068 0.106 0.163 0.264 0.386 0.63 0.81 1.40 2.12

Washed round sand

Flint sand

0.2-0.4

20-26*

20-26*

Referenen

0.620

0.436

0.924

0.873 Westmann and Hugill, 1930

Chardabellaa, 1964

0.698 0.693 0.686 0.670 0.660 0.660 0.660 0.640 0.626

0.4& 0.484 0.479 0.460 0.460 0.460 0.460 0.460 0.460

Andreasen, 1960

0.642

0.462

Donat, 1929

1.2 Physical and mineralogical parameters of clagtic sediments

79

Table 1.2.-6. (cant.)

Sand

0.010 0.003 0.001

Sand

0.096 0.162 0.483 0.611

Sea sand

0.016 0.030 0.040 0.060 0.136 0.190 0.300

Sea sand

0.010 0.030 0.040 0.060 0.070 0.080 0.090

diSCS

Aluminium pins

1958

0.406 0.460 0.376 0.369 0.376 0.376 0.373 0.384 0.396 0.41 8 0.443 0.442

8*

0.467 0.363 0.250 0.177 0.126 0.089 0.061

Aluminium

Fuchtbauer and Reineck,

0.776 0.760 0.615 0.630 0.606 0.604 0.676 0.490 0.426 0.426 0.446 0.430 0.446 0.420 0.410 0.436 0.436

0.600

36.4 18.6 14.3 10.4

39.0 23.1 20.4 23.9

0.608 0.663 0.670 0.686

0.601 0.467 0.403 0.410

10.0

36.6

0.686

0.600

* Estimated values

King, 1899

Hamilton and Menard, 1966

0.676 0.610 0.720 0.612-0.610 0.476 0.467 0.410

0.120 0.130 0.140 0.160 0.160 0.170 0.190 0.200 0.210 0.220 0.230 1.386 0.693

1962 0.386 0.368 0.329 0.362

0.110

Sand

Emery and Rittenberg,

0.730 0.800 0.890

Stakman, 1966a

author's measurements

80

1 Fundamentala for the investigation of seepage

volume ratio is higher, a fact which further increases the number of contacts, and helps in forming arches. Table 1.2-6 shows measured data collected from the literature representing the porosity of homodisperse samples (or samples built up from grains having very small differences in size) as a function of the diameter and the shape of the grains. !. 0

0.3 0.8

e 0.7 9 % 0.6

8 a c cx

0.5 0.4

0.3 0.2

5

6

7

8

910

20

30

40

50

60 70 BD 90 110

shape coefficient, a Fig. 1.2-19. Relationship between porosity and shape coefficient

The graphs in Fig. 1.2-19 constructed from these data show the relationship between porosity and shape coefficient. Only measurements concerning grains larger than 0.2 mm were used. I n this zone the influence of the grain size can be neglected, and thus, the data characterizingtherelation of porosity to shape coefficient can be regarded as a homogeneous set, suitable for the unambiguous representation of the relationship between the two variables. In the semi-logarithmic coordinate system used for plotting the data, the curves - representing both compacted and loose conditions - can be approximated by a parabola of the second order [see Eq. (2.1-14)]. The lower limit of validity, above which this approximation can be applied, is the 0.2 mm grain diameter as mentioned previously. A set of data from the same series can also be chosen representing grains of approximately the same shape. According to graphs constructed from these data (Fig. 1.2-20), the porosity of homodisperse samples does not depend on the diameter of grains, if the latter is greater than 0.2 mm. Below this limit porosity increases with decreasing diameter. It has also been shown, that some measurements (Hamilton and Menard, 1956; Fuchtbauer and Reineck, 1958) contain data of samples differing from each other not

1.2 Physical and mineralogical parameters of clestic sediments

RO~

0102 alu

005

41

nz

81

lo

Fig. 1.2-20. Relationship between porosity and the diameter of grains 0.40

e

0.38

.s2

0.36

0

0.34

0.32 0.30 WLY

,

o

dz

i4

ols

la

ihemt~oofthe two mtked wmples, rn

r'o

mu-vd, vd2

0

02

0.4

0.8

0.8

10

the ratio of the two nixed samples, m Fig. 1.2-21 .. The porosity of a mixture of two homodispanm munples and the modification of porosity es a fmction of the ratio of the weights of the two samples

6

82

1 Fundamentals for the investigation of seepage

only in particle size, but in the shape of grains as well. The probability of this statement is proved not only by the origin of the samples (natural samples from the sea bed were investigated, where the grains may be separated according to both size md shape), but also by comparison with data of grains with a known shape coefficient. For this reason, the rapid chamge in porosity, represented by a dotted line in the figure, cannot be accepted, although some authors have drawn this conclusion from the observations mentioned (Chardabellas, 1964; Engelhardt, 1960). A further well-known relationship between porosity and grain-size distribution exists. Porosity is generally higher in a homodisperse sample than that in a layer composed of grains of various sizes. In the latter case, the very fine grains can fill the pores between the larger particles, thus decreasing the volume of the pores without changing the total volume of the sample, King’s experiments exhibit this effect very well, representing the change in porosity of a mixture of two homodisperse samples as the ratio of the two samples when the mixture was modified (Fig. 1.2-21) (King, 1899). Chardabellas has made very detailed analyses during investigations on the way in which porosity is influenced by grain-size distribution. He has found 4

2

3

4

5

6

7

8

9

1

2

3

4

5

6

7

8

9

coefficient of uniformity, U Fig. 1.2-22. Relationship between porosity and the coefficient of uniformity

1.2 Physical and mineralogical parameters of clastic sediments

83

that the shape of the distribution curve, the location of its maximum, whether the curve has one or more maxima, all affect the porosity in both compacted and loose conditions (Chardabellas 1964). I n practice, however, it is preferable to give an equation which can describe the existing relationship numerically as a function of only one or a few known parameters with acceptable accuracy, rather than use a classification based on the comparison with standard curves, as proposed by Chardabellas. Figure 1.2-22 constructed from data listed in Table 1.2-7, shows that porosity is closely related to the coefficient of uniformity. Although there is some scattering of the points, the relationship can be well approximated by an exponential equation in both cases representing compacted and loose conditions [see Eq. (1.2-15)]. As a summary of the results shown in Figs 1.2-19 and 1.2-22, the calculation of the expected lowest and highest porosity can be given for samples having a greater effective diameter than 0.2 mm as follows: (a) The probable extreme values of porosity for a homodi.yperse sample of spheres can be taken in practice as no mIn = 0.38; and no max = 0.43;

(1.2-13)

(b) The relationship between porosity and shape coefiient can be approximated by a parabola of the second order in the system used in Fig. 1.2-19. Thus, the porosity of a homodisperse sample composed of irregular grains is

[+

n , = no 1

I al'l

Ion! log-

,

(1.2-14)

which gives the porosity of a compacted and a loose sample respectively, according to the substituted value of no; (c) The porosity of a heterodisperse sample can be determined as a function of the coefficient of uniformity and the corresponding n, value: 2 n=-n 3

u-1

(1.2-15)

As the investigation was limited to grains larger than 0.2 mm the influence of the grain diameter could therefore, be neglected. For the analysis of the relationships between porosity and the geometrical parameters of grains, the cases of compacted and loose samples were investigated separately. These two extreme values indicate a relatively large interval, which covers the probable porosity of the natural layers. As already mentioned, the pressure acting on the layer also affects porosity. I n addition porosity is influenced not only by present day pressures, but by all pressures in force during previous geological ages, due to the lack of elasticity of the layers. The relationship between porosity, or more precisely, void ratio, and the present acting pressure can be expressed in terms of the depth of the layer below the surface, the latter being proportional to the pressure. The equation generally used to represent this relationship is based on the hypothesis 6*

Table 1.2-7. Data characterizing the porosity of heterodispeme samples The pomsity of the eample

The investigated @ample

Heterodiepem eemple of send

Mixture of homodisperse semples of send

10.9 9.7 10.4 9.6 9.1 9.0 8.3 10.4 10.3 9.2 9.1 9.6 9.6 9.2 10.0 9.1

0.466 0.466 0.676 0.662 0.378 0.387 0.739 0.162 0.162 0.366 0.366 0.478 0.404 0.404 0.466 0.430 0.460 0.464 0.138 0.802 0.810 0.306 0.261 0.211

9.9 10.4 9.2 8.3 10.6 10.3 10.7

10 10

0.246 0.246 0.320 0.320 0.401

8.3 8.0 7.6 7.6 8.0

10 10 10 10 10

8.7

10 10 10 10 10 10 10 10

2.10 2.00 3.16 3.03 1.81 1.86 6.40 1.66 1.66 2.09 2.09 3.36 2.46 2.46 2.67 2.39 3.83 3.38 1.81 4.38 3.76 2.80 2.00 2.89

0.384 0.377 0.364 0.352 (0.390) (0.391) (0.318) 0.429 0.416 (0.364) (0.367) 0.310 0.399 (0.398) 0.364 (0.349) (0.360) 0.366 0.414 (0.329) (0.334) 0.344 0.346 0.374

1.27 1.27 1.26 1.26 1.26

(0.436) (0.426) (0.429) (0.436) (0.426)

0.396 0.397 0.326 0.368 0.370 0.403 0.363 0.367 0.333 0.342

0.446 0.437 0.444 0.449 0.436

0.343 0.336 0.308 0.308 (0.348) (0.349) (0.266) 0.368 0.366 (0.324) (0.333) 0.290 0.364 (0.362) 0.328 (0.321) (0.304) 0.287 0.378 (0.284) (0.292) 0.308 0.308 0.324 (0.386) (0.398) (0.388) (0.386) (0.381)

chardabellas, 1964

2

1 W

0.361 0.363 0.271

R

E ?

?

0.327 0.336

w.

1

0.366 0.323 0.309 0.286 0.297

0.392 0.406 0.397 0.394 0.387

Chardabellas, 1964

0.401 0.277 0.277 0.349 0.349 0.300 0.295 Mixture of homodisperse samples of sand

0.185 0.288

Heterndisperse samples of sand

0.016 0.028

Glass bells Sandy gravel

7.6 7.6 7.0 7.3 7.0 6.9 7.3

0.173 6.0 0.136 6.0 _____ 10 10 10

10 10 10 10 10 10 10

1.26 1.41 1.41 1.45 1.45 1.59 1.69

(0.429) (0.433) (0.431) (0.421) (0.431) (0.411) (0.429)

0.443 0.447 0.447 0.435 0.447 0.426 0.444

5.06 4.00 -____ 3.0 0.390 1.3 0.430 10 10

1.45 1.59 37.0 48.0 10.4

(0.426) (0.406) 0.307 0.300 0.318

(0.386) (0.328) (0.377) (0.371) (0.368) (0.368) (0.372)

0.394 0.390 0.385 0.381 0.377 0.377 0.382 0.255

c1

ia

5!

King, 1899

0.303

0.442 0.422

E

5

Weweling and Wit, 1966

n4 !3 5'

Sine and Bentz,

1

1966 0.240 0.263 0.285

2.

author's measurements

k 2. 1 W

c

B 3 1 %

86

1 Fundamentals for the investigation of seepage

which assumes linear correlation between the modulus of compressibility and pressure (Terzaghi, 1943): %+0 0 . e = e , - clog---,

(1.2-16)

0,;

where 0, is the actual effective normal stress at a depth of z ; and 0 , is the actual effective normal stress at a starting level, near the surface, where the porosity of the layer is characterized by e , (loose condition). There is some contradiction in this equation, because it gives a negative value at very great depth. To eliminate this problem a modified equation wm proposed (Juhhz, 1966): ( 1.2-17)

although there is no significant difference between the two formdm in the range of small depths, and, therefore, Terzaghi’s equation can be used, when the investigated depth is not greater than 100 m. The curves representing the relationship characterized by Eqs (1.2-16) or (1.2-17) are compared t o measured data in Fig. 1.2-23. Both equations can give only rough estimates, because the material of the layer is not elastic and only partial expansion can develop after decreasing the pressure. 0

0.25

0.50

0.75

ele,

la

Fig. 1.2-23. Relationship between porosity and the depth of the investigated point

1.2 Physical and minerelogicel parameters of clestic sediments

87

Thus, compression is not an unambiguous function of the present load (i.e. the depth of the layer in question below the surface at the time of investigation), since porosity has preserved the influence of previous pressures as well. The difference between actual and theoretically calculated porosity may possibly result from the slow development of consolidation. The total stress caused by the load on the layer can be divided into two parts: one taken by the solid skeleton of grains (effective stress), and the other balanced by water pressure (neutral stress). The latter does not cause any compression, and hence, porosity may be a function of the effective stress only. The rate of the effective and neutral stresses changes in time, ae the surplus water is drained from the system. The process depends on the size of loads, and on the permeability of layers. The actual porosity is therefore, the function of all these factors. Finally, it can be stated that the probable zone of the expected porosity can be determined on the basis of Eqs (1.2-13), (1.2-14) and (1.2-15), knowing the geometrical parameters of the grains in the investigated layer. The probable location of the actual value within this interval can also be estimated using Eqs (1.2-16), or (1.2-17). The reliability of the latter, however, is very low because of the effects of many unknown factors.

1.2.5 Physical model representing the irregular network of pores and channels between grains Several attempts have been made previously to substitue the irregularly connected channels formed by the pores of a loose clastic sediment with a regular physical model, in which the relationship between the geometric and hydraulic variables can be determined theoretically. Naturally, the models provide only the most probable form of the movement equations, and carefully executed experiments are necessary to prove the reliability of the model, to examine its validity zone and the accuracy of the equations achieved in this way, and to determine the numerical constants of the formulae. The best known models are those composed of capillary tubes (Scheidegger, 1953, 1960). The most simple form of these models is a bundle of straight pipes. I n other cams the tortuosity of the capillary tubes is also considered. According to another proposal, straight tubes composed of small sections of different diameters are very suitable to simulate flow through the water conveying channels of the sediments. Such bundles can be composed of tubes having only two different diameters (Wyllie and Spangler, 1952), or many different sections ensuring only a constant area free for water transport in a cross section perpendicular to the main direction of flow (Wyllie and Gardner, 1958). Bear and Bachmat’s model (1967) is a generalized form of the capillary tube models, because it is composed of a spatial network of interconnected random passages of varying lengths, cross section and orientation, and of junctions, where the channels (a minimum of three) meet (Bear, 1972).

88

1 Fundamentals for the investigation of seepage

Another method usually applied for determining the resistivity of the porous medium is to consider the analogy between the drag force acting on a sphere settling in water and the resistance against flow in the interconnected pores (Goldstein, 1938; Ward, 1964). Rumer gives a detailed summary of the results achieved by applying this type of model to characterize the relationship between seepage velocity and hydraulic gradient in the various zones of seepage (De Wiest, 1969). Irmay’s proposal can also be included in this group of models (Bear et al., 1968). In this theory, the resistance of spheres located in steady flow along a line perpendicular t o the flow direction is determined by averaging the solution of Navier-Stokes’ equations, considering also the inertial terms within them. For simulating water movement through fissured and fractured rocks, the various forms of Hele-Show’s model is used (narrow slit between two parallel plates). The slit may have a constant or a varying width. A network of such slits is used many times, assuming either the same width of the slits in all three main directions (see Fig. 1.1-5),or considering the possible anisotropy by applying different gaps in different directions (Louis, 1970;Bocker, 1973). A network of pipes may also be used for the characterization of fractured rocks, especially karstic formations. A large number of geometrical parameters are needed, however, for the description of such a system. It is not euitable therefore, for the general characterization of the resistance of aquifers, but is used for solving special problems in the form of small scale models (Mijatovib, 1970). Although Rumer has stated in the previously quoted 1,ook (De Wiest, 1969),that the capillary tube model is not suitable to characterize non-laminar flow, it can be proved that the difficulties he encountered can be avoided by a conveniently chosen form of pipe (see Chapter 2). On the other hand, from a practical point of view, it is desirable to select the most simple model, from which however, the largest possible amount of information can be gained. For this reason, a bundle of capillary tubes aligned in the main direction of flow will be used as a physical model of the water conveying channels. To ensure the reliable simulation of both non-laminar flow and capillary effect, it is necessary t o use tubes of varying diameters. As will be seen, the simplest of these types of models (that of Wyllie and Spangler using two different diameters) is adequate for this purpose and, therefore, the nurnerical determination of the geometrical parameters of this model will be summarized here. On the basis of the movement equation valid for the investigated condition of flow and determined for a model pipe (e.g. in the case of laminar flow, when the dominating forces are gravity and friction, this equation is the well-known Poiseuille’s formula), the relationship between the hydrodynamic parameters of seepage can be determined, or the previously established empirical formulae can be controlled, and their numerical factors determined (in the case of laminar movement mentioned before as an example, the validity of Darcy’s equation can be determined, and its factor, i.e. hydraulic conductivity can be calculated). Previous investigations have shown that in the caae of homodisperse samples, the diameter of the pipes forming the models is proportional to the

1.2 Physical and mineralogical parameters of clastic sediments

89

diameter of grains. Authors investigating heterodisperse samples proposed various design diameters to characterize the sample (e.g. Hazen Dlo; Seelheim Ds0; etc.). The basis of these proposals was the hypothesis that this diameter is proportional to the pipe diameter of the bundle of capillary tubes, which from a hydraulic point of view, is equivalent to the channels composed of the pores in a heterodisperse sample, even if this assumption was not mentioned by the authors (Hazen, 1895; Forchheimer, 1924). Koieny’s work is a very important corner stone of all investigations related to the determination of the characteristic diameter for a heterodisperse sample, and to that of the hydrodynamically equivalent pipe-diameter as well (Koieny, 1953). He has proved that the surface-volume ratio has to be identical in the original sample and the model systems, because in these cases, the sufficient ratio of the two most important forces (i.e. gravity and friction) is ensured. This is the basis of the determination of his effective diameter, as explained in connection with Eq. ( 1 . 2 4 ) . Since the surfacevolume ratio is influenced by the shape of grains as well, the original formula was supplemented by the use of the shape coefficient. Following Koieny’s line of thought Eq. (1.2-5) was finally proposed to determine the quotient of the effective diameter and the average shape coefficient. The same result was achieved by Carman, who proposed the use of the actual surface-volume ratio instead of a characteristic diameter (Caiman, 1956; Engelhardt, 1960). This value is, however, identical to the quotient of Dh/a proposed here, as can easily be proved taking Eq. (1.2-1) into consideration. The advantage of the latter method is that this parameter can be approximated, using average values determined by statistical evaluation of numerous measurements, and thus, it does not require special measurements or investigation for practical purposes. Assuming that the parameter Dh/a is known [e.g. i t was calculated from Eq. (1.2-5)], two conditions can be found to determine the average diameter of the model pipe (do),and the number of pipes crossing the unit area of the section, perpendicular to the main direction of flow ( N ) .According to the first condition, the surface of the pipe-wall related to its inner volume should be equal to the ratio of the grain surface ( A ) in the sample to the pore volume ( V p ) :

consequently, do=--.

4 n D,

(1.2-19)

a The second condition states that the cross section of flow should be identical in both the original and the model system. It follows from this condition that the crobs section of one model pipe multiplied by the number of l-n

90

1 Fundamentals for the investigation of seepage

pipes crossing a unit area should be equal to areal porosity, and therefore, the number of pipes can be calculated

N = - =4n” -, din

4n

din

.

(1.2-20)

because according to Eq. (1.1-4), areal porosity is equal to effective porosity. It is necessary to note that some difference can be caused here by the various possible interpretations of the effective and total porosity. I n the case of a porous medium composed of equal cubes (see Fig. 1.1-5), areal and effective porosity values are equal only if the gaps perpendicular to the flow direction are regarded as inactive ones, and thus, the effective porosity is about 0.7 times smaller than the total porosity, although there are no dead ends, isolated pores, or very thin joints. It can be assumed that in a sample composed of individual grains, a few channels me directed at right angles to the flow direction, and therefore, the velocity in them is very smdl, or it may even be zero. These channels only play a role in equalizing the pressure between the stretches of the water conveying network, but not in water transport, as from this aspect, they are inactive. Considering the calculated ratio of the inactive pores in the system of cubes mentioned before &e a lower limit, it may be supposed that the porosity value n used in Eqs (1.2-18), (1.2-19) and (1.2-20), respectively is 0.85-0.95timessmallerthantheparameter determined in soil physics. Knowing the diameter and the number of the pipes, the discharge carried by the model system under a given pressure head can be calculated by using Poiseuille’s equation, assuming the flow t o be laminar. This value can be compared to the results of some very carefully executed experiments (Zunker, 1930; Lindquist, 1933; Carman, 1956).The actual hydraulic conductivity is 2.5 times smaller than that recalculated from the resistance of the model pipe system. (The constant proposed by Carman to be used inKoieny’s equation, i.e. 1/5, means similarly a multiple of 0.4 compared to the theoretical value calculated for straight tubes.) The difference can be caused by three possible factors (Fig. 1.2-24): (a) The cross section of the actual channels is not circular; (b) The channels in the network are longer than the length of the sample, and the tubes do not cross the cross section (which is directed at right angles to the main flow direction) perpendicularly; (c) The cross-sectional areas of the channels are not constant, but change continuously. The discharge of laminar flow through pipes having a non-circular cross section (triangle, ellipse, etc. Fig. 1.2-24b) was calculated (Forchheimer, 1924; Engelhardt, 1960), and compared with that in a circular pipe. This showed that the effect of the pipe shape can be expressed by a multiplying factor from 0.8 to 1.2, and thus, the’difference in question cannot be explained by this cause.

1.2 Physical and mineralogical parameters of clestio sediments

91

Fig. 1.2-24. Physical model to substitute the system of irregular channels composed of pores between grains

The two phenomena summarized as the second possible cause are generally included in one term, i.e. tortuosity. As explained by Bear (1972), this effect can be correlated to the ratio of the average length of the channels (1') t o the length of the sample in the flow direction ( I ) , but this relationship is not linear, as supposed by several authors. The linear relationship considers only the difference in length, but the change in the area free for water transport has to be taken into account as well, and this influence may also be related t o the ratio of lengths. Thus, i t can be assumed that the parameter of toris proportional or equal t o the square of the ratio of the length tuosity (T) of the sample to the average length of the channels (Fig. 1.2-24c): (1.2-21)

Many authors have proposed numerical values for considering tortuosity. Thus Irmay's parameter (Bear et al., 1968) i.e. 0.4 as a multiplying factor, eliminated the difference mentioned earlier (hydraulic conductivity of the actual sample is 2.5 times smaller than that of a bundle of straight capillary tubes). The tortuosity can also be investigated theoretically, if the average length of the water conveying channels can be determined. Bear haa pointed out that two different averages can be obtained depending on the aspects used as the basis of the calculation. If only the pore geometry were considered, it could be assumed that the average angle between the flow direc-

1 Fundamentals for the investigation of seepage

92

tion and the axes of the channels was n/4.These hypotheses result in a parameter, T = 0.5. It is necessary, however, to investigate the kinematic differences between the various directions, a requirement which needs the determination of the mean velocities of the channels as well (weighing the channels according to the mean velocities in them). It is evident that channel velocity is higher in tubes parallel to the flow direction than in those directed at different angles, and therefore, the average position of the tubes considering kinematic aspects forms a smaller angle with the flow direction than the geometricaverage. As alreadymentioned, even inactive channels may be expected, and hence, the overall result can be characterized by a considerably higher tortuosity factor than 0.5. Considering the T = 0.5 value with the influence of the non-circular tubes, these two effects could explain the observed difference. Because of the kinematic character of the channels, the numerical results, however, can probably cover only half the expected value and therefore, a third cause has also to be considered. It would be possible to take into account all three effects separately, and finally to summarize the numerical parameters. Effort is made, however, to apply only a very simple model and to base the total explanation of experimental observations, as well as the determination of the numerical factors, on the basis of this model. It is considered, therefore, that the best method of eliminating the difference mentioned previously, is the use of model pipes constructed from short stretches of various diameters (Fig. 1.2-24d). The tubes are parallel to the main flow direction (the ratio of the inactive channels perpendicular to this direction, may be considered by decreasing total porosity by a factor smaller than unity). The tubes are located so that the area free for water transport is constant. This model is suitable not only for the verification of the difference mentioned, characteristic of laminar movement, but the numerical comparison indicates also good agreement between measured and calculated data in the case of non-laminar flow conditions. According to Lindquist’s investigation, the ratio of the largest to the narrowest cross-sectional area of the channels in a homodisperse sample of spheres can be afj high as 10. This value is however, a very extreme one, which is valid only at a few points in the channel. To determine a model pipe having two different stretches (with a diameter of d , and d,, respectively), the average ratio of the area of large to narrow sections in the channel should be investigated. A pipe was chosen as a final model (Fig. 1.2-24d), the volume of which is the same, as that of the pipe with a constant d,, diameter, but constructed from short stretches of d , and d , diameters (d, < d o < d,) in such a way that half the length has the smaller and the other half, the larger diameter. The @scharge of this pipe was calculated and plotted against the ratio of the two cross-sectional a r e a (dg/dI) (Fig. 1.2-25). Similar graphs were determined not only for circular pipes but for other shapes as well. It is shown by the result of this investigation that the discharge becomes equal to that of the pipe with a constant diameter multiplied by 0.4 (i.e. the actual discharge of the original seepage according to the experiments represented in the figure), when the ratio of the two areaa is 113 1/3.5. On the basis of the foregoing, the following equations

-

1.2 Physical and mineralogical parameters of clastic sediments

93

Fig. 1.2-26. The flow rate of a pipelcomposed of two types of short stretches with different diameters, as a function of the ratio of the areas of the two different cross sections

can be given to calculate the characteristic diameters of the pipe: d, = 1.86 d,;

d d, = 2 ; d, = 1.25 d o ; 1.5

(1.2-22)

while the length of each stretch can be approximated, taking into consideration the actual character of the channels, by the following formula

I , = (1

-

1.5) d o .

(1.2-23)

The accuracy of the proposed ratio of the cross-sectional areas can be checked by other measurements as well. The air bubbling pressure method is the usual test to determine the capillary behaviour of samples. For this investigation, the pressure required to push the air bubbles through a thin saturated layer of the sample is measured. Stakman has found that this 20 pressure is constant when the thickness of the layer is greater than 15 mm. Below this limit, the necessary pressure decreases with decreasing thickness. He has assumed, that the constant value of the required pressure is proportional to the narrowest cross section of the capillary pores, while the pressure necessary to push the air bubbles through a layer of 1 mm thickness has the same correlation with the largest pores. According t o his 1 : 3.5, similar numerous memurements, the ratio of the two areas is 1 : 3

-

-

94

1 Fundamentals for the investigation of seepage

t o the results of the hydraulic deduction, and the scattering of the data points is insignificant. On the basis of Stakman’s measurements, the average pipe diameter could be recalculated as well, and his results are in good agreement with the parameter calculated from Eqs (1.2-19) and (1.2-22) (Fig. 1.2-26).

210(50/057550-

300 210 I50 105 75

/6- 50

-3 -%

-k --&9r

---8

D

b equivalent pare diameter

[p 1

Fig. 1.2-26. Relationship between pore size and grain diameter (after Stakman, 1966)

Although this model was developed, and has served the purpose of describing the flow of water through pores, d, and d, were found to be numerically equal to the actual smallest and largest ,pore diameters, determined by Stakman with air bubbling experiments. Thus, the diameters calculated in this way can also be used a,s parameters, characterizingthe expected actual size of pores in a given sample, when studying the movement of fine grains or the capillary rise in the pores. References to Chapter 1.2 ~LLBEET, J. (1967): The Materials of Brick and their Use in Rough Ceramics (in Hungarian). Mffizaki Konyvkiadb, Budapest. A L B E R ~ O N , M. L. (1962): Effect of Shape on the Fall Velocity of Gravel Particles. 5th Hydraulic Conference, Iowa, 1952.

References

95

ANDEREASEN, A. H. M. (1960): Adhesive Forces between Grains within their Pores in Clastic Sediments (in German). Kolloid-Zei$ung. h a m , L. (1966): The Determination of Clay Com onent of Soil by Infrared Spectrophotogrammetry (in French). Sympoeium on fiater i n Unaaturated Zone, Wageningen, 1966. BAVER,L. D. (1948): Soil Physics. John Wiley, London, New York. BEAR,J. (1972): Dynamics of Fluids in Porous Media. Elsevier, New York, London, Amsterdam. BEAR,J. and BACHMAT, Y. (1967): A Generalized Theory on Hydrod namic Dispersion in Porous Media. I A S H Sympoeium on Artifical Recharge and &anageme& o f Aquifere, Haifa, 1967. BEAR, J., ZASLAVSKY,D. and IRMAY, S. (1968): Physical Prhciples of Water Permlation and Seepage. UNESCO, Paris. BIDLO,G., KLEB,B. and TBROK,M. (1967): Analysis of Aggregatee of Concrete of Hydraulic Structures (in Hungarian). ZpitSanyag, No. 11. BOCKJCR, T. (1973): Theoretical Model for Karetic Rocks. Karezt- b Barlangkutatci.8, Budapest. Vol. VII. C a w , P. C. (1966): Flow of Gases through Porous Media. Butterworth, London. P. (1964): Standardization of k-value Determination by FieldTests CEWRDABELLIL~, through Standardizing the Grain-size Distribution Curve of Water-bearing Clastic Sediments (in German). Mitteilurrg InatitUte8 f i r WmeerWirtechoft, No. 20. Chum, J. (1967): The Influence of Soil Mineralogical Composition on Cement Stabilization. Ueotechnique, 2. DARAB, K., RBDLIN~, and R E ~ N Y I N I(1967): ~ Development of Methods for Determination of Clay Minerals (in Hungarian). (Manuscript).MTA Talajtani 6s Agrokemiai IntAzet, Budapest. DE WIESTJ. M. (1969): Flow through Porous Media. Academic Press, New York, London. DOLOOV, S. I. (1948): Studies Related to the Determination of Soil Moisture and its Availability for Plants (in Russian). Izd. A d . Nauk SSSR, MOSCOW,Leningrad. DONAT,J. (1929): Study on the Permeability of Sands (in German). Waeeerkraft ~ n d w a S 8 d & C h a f t , No. 17. DUMBLETON, M. J. and WEST, G. (1966): Some factors Affecting the Relation between the Clay Minerals in Soils and their Plasticity. Clay Minerale, No. 3. ELRICK,D. E. (1966): The Mineralogic Characterization of Soils. Sympoeium on Water i n the Unaaturated Zone, Wageningen, 1966. EMERY,K. 0. and RITTENBERO, S. C. (1962): Early Diagenesis of California Basin Sediments in Relation to Origin of Oil. Bulletin of American Aseociation for PetrolGeology. ENOELHILBDT, W. (1960): Porosity of Sediments (in German). Springer, Berlin, Gottingen, Heidelberg. FEKETE, Z. (1960): Comparative Analysis of Parameters Describing the Water Content of Soils (in Hungarian). HidroMgiai Kodiiny, No. 7-8. FOROHHEIMER, PH.(1924): Hydraulics (in German). Teubner, Leipzig, Berlin. FRASER, H. J. (1935): Experimental Study of the Porosity and Permeability of Clastic Sediments. J o u m l of Ueology. FUCHTBAUER, H. and REINECK, H. (1968): Edoggze geologiae. Helv., (in German). GOLDSTEIN,S. (1938): Modern Developments in Fluid Dynamics. Oxford University Press, Vol. 11. London. HAOERB~AN, T. H. (1938): About the Relation between the Distribution Field of the Relative Width of the Particles and the Genesis of the Sediment. UeologdskaForeningens F o r h a d . , No. 3. HAMILTON, E. L. and MENARD,H. W. (1966): Density and Porosity of Sea Floor Surface Sediments of San Diego, California. B d e t i n of A m e r k n Aeeociation for Petrol -Ueology HAZEN,A. (1896): The Filtration of Public Water Supplies. New York. HERCZOO, H., Kovdcs, G., NAOY,L., PAPR~LVI, F. and C)LL&, G. (1966): Applied Soil Mechanics in Hydraulic Engineering (in Hungarian). M6rnoki TovtibbkBpz6 I n 6 zet, Budapest.

.

96

1 Fundamentals for the investigation of seepage

HEYWOOD, H. (1938): Measurement of the Fineness of Powdered Materials. Proceedinga of Instdute of Mechanical Engineers. IVICSICS, L. (1967): Hydraulic Design of Sedimentary Basins and Determination of Velocity of Sedimentation (in Hungarian). ViziiglJi Kodemknyek, 3. J ~ YJ., (1944): Soil Mechanics (in Hungarian). Egyetemi Nyomda, Budapest. J T J H ~ Z , J. (1966): Available Deep Ground Water (in Hungarian). Doctoral Thesis (Manuscript). Budapest. J u ~ h z , - J(1967): . Hydrogeology (in Hungarian). Tankonyvkiadb, Budapest. BZDI, A. (1972): Soil Mechanics (4th edition in Hungarian). Tankonyvkiad6, Budapest. KINQ,F. H. (1899): A Study on Porosity and Grain Relationship in Experimental Sands. U . S. Geological Survey. KLEB,B. (1968): Geological Investigation of Pliocene Formations Southwards from Mecsek Mountains (in Hungarian). Foldtani K o d h y , No. 2. KOVACS,G. (1966): Hydraulics (in Hungarian). VITUKI, Budapest, Vol. 3. KovAas, G. (19688): Characterization of Shape of Grains in Seepage Hydraulics (in Hungarian). g p i t h - 6s Kozlekedkatudomanyi Kozlemknyek. No. 1-2. KovAcs, G. (196813): Characterization of the Molecular Forces Influencing Seepage with the Help of the p P Curve. Agrokkmia ks Talajtan, (Supplementum). KovAcs, G. (1971): Relationship of Plasticityand Clay content of Soils (in Hungarian). Agrokkmia ks Talajtan, No. 1-2. K O ~ N YJ. , (1953): Hydraulics (in German). Springer, Wien. KREYBIQ, L. (1951): Heat and Water Balance of Soils (in Hungarian). Budapest. K R I V ~P., (1957): Geological Evaluation of Hagerman's Grainshape Method (in Hungarian). Poldtani Kodony, No. 3. KUHN,A. (1963): HBndbook of Colloid Chemistry (in Hungarian, translated from German) MGszaki Konyvkiad6, Budapest. KURON,H. (1932): Zeitschrift fiir PfleJlzen (in German). Berlin. Vol. 24A. LEBEDIEV.A. F. f19631: Soil Moisture and Ground Water (inRussian). I d . Akad. Nauk SSSR; Moscow. LINDOUIST. E. 11933): On the Flow of Water through ., Porous Soil. 1st C w r e a 8 of ICOLD,- Stockholm, 1933. Lo, K. Y. (1961): Secondary Composition of Clays. Proceedings of ASCE, SM. 6. LOUIS,C. (1970): Three-dimensional Flow in Fractured Rocks (in French). Comitk FranprLis de Mkcunique des R o c k , Paris. MADOS,L. (1939): Studies Related to Irrigation and Water Management in the Area of the Irrigation Scheme of Tiszafured (in Hungarian). dntozhi Kodemdnyek. MADOS,L. (1941): Application of Soil Science in Connection with Drainage (in Hungarian). dntozksi Kodemdnyek. MELENTEV. V. A. f19601: Hvdraulic Dams from Gravels and Sands (in Russian). Gostekhizdat; Moscow. MIHALTZ.J. and UNQAR.T. 119541: SeDaration of Alluvial and Wind-borne Sands (in Hungkian). Foldtani Kodony, No. 1-2. MIJATOVI~, B. F. (1970): Hydraulic Mechanism of Karst Aquifers in Deep L e g Coastal Collectors. Bulletin of Institute fw Geological and Geophyeical Reaeurch, Beograd, Series B. No. 7. MITSCHERLICH,E. A. (1932): Soil Science for Agriculture and Forestry (in German). Berlin. PALADIN, J. A. (1964): Determination of Permeability of Grained Non-cohesive Soils (in Russian). Gkhotechnkheskoye Strodektvo, 3. R ~ T ~ TL.I (1971): , Correlation Associated with Liquid and Plastic Limits of soils. Proceeding8 of 4th Conference of SMFE, AkadBmiai Kiad6, Budapest. RODIE,A. A. (1962): Soil Moisture (in Russian). Izd. Acad. Nauk SSSR, MOBCOW. ROSE, D. A. (1966): Water Transport in Soils by Evaporation and W t r a t i o n . Symposium on Water in the Ulzllaturated Zone, Wageningen, 1966. SCHEIDEQQER, A. E. (1963):Theoretical Models of Porous Matter. Producere Monthly, No. 10, 17. SCHEIDEQQER, A. E. (1960): The Physics of Flow Through Porous Media. Univereity of Toronto Press, Toronto. SCHLICHTER, C. S. (1899): Annual Report of the US. Geological Survey. I

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SCHAUCRTMANN (1962): Comparison of One and Two-specium CFS Tests. Proceedings of ASCE, SM. 4. SEED H. B., WOODWABD R. J. and LTJNQEN R. (1964): Clay-Mineralogical Aspecte of the Atterberg's Limits. Proceeding8 of ASCE, SM. 4. SINE,L. and BENTZ,A. (1966): Differences between Experimental and Theoretical Values of Capillary Diffusivity (in French). Symposium on Water i n the Unaaturated Zone, Wageningen, 1966. STAKMAN,W. P. (19668): The Reletion between Partiole Size, Pore Size and Hydraulic Conductivity of Sand Separate. Symposium on Water in t k Umaturated Zone, Wageningen, 1966. STAKMAN,W. P. (196613): Determination of Pore Size by the Air Bubbling Pressure Method. Symposium on Water i n the Unaa4urated Zone, Wagdngen, 1966. STELCZER, K. (1967): Abrasion of Bed-load Material (in Hungarian). VCzQigyi KWemknyek. No. 2. SZADECZKY-KARDO~~, E. (1933): Determination of the Coefficient of Roundness (in German). Centralblatt fiir Minerdogie. SZILVBQYI,Z. (1966): Experiments to Characterize the Rheological Behaviour of Clays (in Hungarian). F&&aniKukz&, No. 4. S Z I L V ~ YI. I , (1967): T h e Stability Problems of Clayey Valley Slopes in North Hungary. 9th Congrese of ICOLD,Iatambul, 1967. TERZAQHI, K. (1943): Theoretical Soil Mechanics. John Wiley, New York. THOMPSON (1966): Lime Reaotivity of Illinois So&. Proceedings of ASCE, SM. 6. VAaHAm, G. (19668): General Verification of Deroy's Law and Determinetionof Capillary Conductivity (in French). Symposium on Water i n the Unaaturated Zone, Wageningen, 1966. VAamuD, G. (1966b): Study on Soil-moisture Distribution after stopping Horizontal Infiltration (in French). Symposium on Water i n the Unaaturated Zone,Wapningen, 1966. V ~ R W Y A IG. , (1942): Experiences on Irrigation at Mkrialiget (in Hungarian). ~ntozksiiyyiKozlemdnyek, No. 2. WARD, J. (1964): Turbulent Flow in Porous Media. Journal of ASCE, HY. 6 . WENTWORTH, CH. (1922): A Scale of Grade and Clam Terms for Clastic Sediments. Journal of Geology. WESSELTNG, J. and WIT, K. (1966): An Infiltration Method for the Determination of the Capillary Conductivity of Undisturbed Soil Cores. Symposium on Water in the Unsaturated Zone, Wageningen, 196% ' WESTMANN, A. E. R. and HIJQIZL,H. R. (1930): The Packing of Particles. Journal of American Qeran-Society. WILLIAXS,N. M. (1966): A Method of Indicating Pebble Shape with one Parameter. Journal of Sedimentary Petrology, No. 4. WIND. G. P. 11966): CaDillftrv Conductivitv D a t a E s t h t e d bv a Simple Method. Symposium on lVa8er'in th> U&durdtd Zone; Wagencngen, 1966. " WYLLIE,M. R. J. and SPANQLER, M. B. (1962): Applioetion of Eleotrical Resistivity Measurements to Problem of Fluid Flow in Porous Media. Bulletin of American Assock~iwnfor Petrol-Chlogy, No. 36. WYLLIE,M. R. J. and GARDNER, G. H. F. (1968): The generalized Kofeny-Carman Equation. World Oil Prod. Sect., 11, No. 4. ZINQQ,TH. (1936): Characterization of Claatio Sediments According to Grain-size Distribution (in German). Schweizcrkche Mineralogkche und Petrographkchc Mdteilungen, No. 36. ZUNKER, F. (1930): Behaviour of Soils Related to Water (in German). Springer, Handbook of Soil Science, Berlin. Vol. VI.

7

98

1 Fundamentals for the investigation of seepage

Chapter 1.3 Dynamics of soil moisture above the water table The investigation of hydrological processes occurring in the soil above the water table concerns a boundary field of surface-water hydrology, groundwater hydrology (hydrogeology) and pedology boil science and agronomy). This series of related sciences has t o be supplemented by seepage hydraulics, which investigates the flow of water through the very complex system of the unsaturated soil. Knowledge of the dynamic characteristics of water movement provides a firm baais for the study of the other sciences, as all the phenomena have to be analyzed as a continuous process, and hence, seepage hydraulics plays an important auxiliary part in any kind of investigation of the top soil. Although the thickness of the soil-moisture zone is relatively negligible when compared to 'the entire thickness of the crust of earth, its hydrological importance is extremely high. The amount of water stored in this zone is easily accessible, and the water-conveying ability of this layer influences the ratio of actual and potential evapotranspiration (the latter is a function of climatic and surface conditions only). Surface run-o# and infiltration depend on the actual condition of the soil-moisture zone (on its storage capacity and permeability influenced by the moisture content of the soil at any instant). Thus, the behaviour of this zone af€ects the water regime of both surface and ground water. Apart from the hydrological role mentioned in the previous paragraph, the investigation of the zone of soil moisture also provides valuable information on the study of seepage hydraulics. Hence, there is the potisibility of recalculating hydraulic parameters from observed moisture content and its change. Thus, knowing the dynamic equilibrium of the vertical distribution of the soil moisture, it is possible to determine the forces of attraction between the grains and water aa well as the negative capillary pressure and, from this value, the probable pore distribution. The process of soil-moisture redistribution after a rapid change in moisture content (e.g. infiltration of precipitation) can be used in the determination of soil permeability. Owing to the close relationship between parameters characterizing the soil-moisture zone, and those influencing flow through the unsaturated medium, the hydrological investigation of this zone will be dealt with in detail in this chapter. From this very large field, those phenomena, which assist in the understanding of the hydraulic processes above the water table will be analyzed first. The reader is referred to a recently published glossary (IASH, 1974) for the interpretation of other basic terms.

99

1.3 Dynamics of soil moisture

1.3.1 General characterization of soil moisture and processes acting in the soil-moisture zone To simplify further explanations, the various processes influencing water movement through and acting i n the unsaturated zone are grouped and claasified. For this grouping the model sketched in Fig. 1.3-1 can be used. This model includes not only the unsaturated zone, but also the plant zone overlying it. I n addition, part of the gravitational ground-water space, which is the lower boundary of the zone of aeration is included. The path of the

.Sa

F

Fig. 1.3-1. Model for representing hydrological processes in the soil-moisture zone 7*

100

1 Fundamentals for the investigation of seepage

water moving in, or through the unsaturated zone can easily be followed in the sketch. Regarding the model as a closed system, the amount of water moving through the boundaries of the separate columns is the input and output of the system. These effects, which can be regarded as the boundary conditions of the model, are indicated by mrows in the figure. The volume under investigation is divided into three main subsystems: i.e. m e occupied by air, soil-moisture zone and ground-water zone. Two further subsystems may also be distinguished, i.e. the soil surface and the active root zone, the latter being part of the soil-moisture zone. (The soil surface has a special role in the system, being the boundary between the atmospheric and soil-moisture zones, although roots do cross it. At the same time, part of the precipitation reaching the surface is returned t o the atmospheric zone by evaporation, and surface run-off represents special input or output amounts. For this reason, the surface has to be regarded aa an independent subsystem. The separation of the root zone as a H t h subsystem is justified, because its structure differs from that of the remaining part of the soil-moisture zone.) Apart from input and output, a system is characterized by the operation within the system itself. The storage within the zones of plants and soil moisture as well as the fluctuation of the water table (which represents the storage in the ground-water zone) are indicated in the figure as internal operations. Some characteristic values governing the operation of the system are also represented (i.e. soil-moisture retention curve, actual moisture distribution, etc.). The column investigated is in contact with the atmosphere at its upper boundary. The input through this surface is precipitation (either liquid or solid), while the output is considered under the term of actual evapotranspiration. Along the vertical faces, the model is bordered by the same zones, which are also included in the volume under investigation: i.e. atmospheric zone, the lower part of which is the vegetation zone above the surface, soilmoisture zone between the surface and the water table, and finally, the gravitational ground-water space below the water table. The lower horizontal boundary of the column is similar t o the lower part of the vertical face, and is in contact with the ground-water zone. By separating the model from its surroundings a discontinuity is created along the boundaries, which has to be replaced by the inputs and outputs acting on and originating from the system. Such replacement factors are as follows: (a) Horizontal vapour movement in the atmospheric zone; (b) Horizontal moisture movement in the soil-moisture zone; (c) Surface run-off (d) Ground-water flow.

Further groups of processes can be distinguished according to the subsystems mentioned previously. Storage of water in the vegetal zone and its evaporation may be included in the investigation of interception. The

1.3 Dynamics of soil moisture

101

storage on and the direct evaporation from the surface is closely related to the former group, although these processes belong to the next subsystem (i.e. to the surface itself). This is characterized by surface run-off as an input (inflow from the surface to the investigated space) and also as the output of this subsystem (outflow on the surface). Water exchange between the atmospheric and soil-moisture zones is the result of infiltration through the surface, as well as evaporation and transpiration from soil moisture. The two depleting processes mentioned (i.e. evaporation and transpiration from the soil moisture) are only part of the actual evapotranspiration, because the latter includes the direct evaporation from the surface of both plants and the earth. The next subsystem is the soil-moisture zone, where the storage process is governed by direct evaporation from soil moisture, transpiration of plants from the root zone, and the water retaining capacity of the unsaturated soil, which can be characterized by the soil-moisture retention capacity compared to the actual moisture content of the layer. As already mentioned, the structure of the top layer (root zone, or cultivated zone) has to be dealt with separately because its structure differs basically from that of the remaining part of the soil-moisture zone. Thus, the soil-moisture retention curve cannot be applied here, viz. the secondary porosity created by the large channels produced by roots and ploughing maintains the atmospheric pressure in deeper levels aa well. The h a 1 group of phenomena creating a link between the subsystems is the water exchange between ground water and soil moisture (accretion). Vertical recharge and drainage of ground water are the components of this group, and their results may be indicated by the change in the amount of stored ground water indicated by the fluctuation of the water table. Surveying these processes, those most closely related to the hydrological investigation of the unsaturated zone are: (a) Infiltration through the surface; (b) Evapotranspiration from soil moisture; (c) Storage in the unsaturated layer; (d) Water exchange between soil moisture and ground water.

After giving a general picture of the whole model and surveying the processes acting in the soil-moisture zone, a more detailed analysis of the various forms of soil moisture is given and further subzones between the cultivated zone and the water table are distinguished. This classification can be based either on consideration of the forces in action, or on the rate of saturation of the pores. Because the dynamics of soil moisture and the saturation of the soil axe closely related to each other, the two aspects result in the same subdivision of the soil-moisture zone. Near the surface the water is kept in an elevated position, against gravity, by adhesion, which develops between the solid matrix and the water molecules at the surfaces of contact between the solid and liquid phases (Fig. 1.3-2). In loose clastic sediments this force creates a film of water around the grains, which may be divided into t w o parts. The inner layer is the adsorbed film, the tension of which is so high that the suction of the roots is not able

102

1 Fundamentals for the investigation of seepage

Fig. 1.3-2. Various forms of soil moisture above the water table

to draw off the water molecules from the grain surface. The adhesive force decreases with increaing distance from the surfaces of contact and, therefore, the upper layer of the film is bound to the grain by a lesser force. At the point of contact of two neighbouring grains - or, more precisely, of their water films - the effects of the adhesive forces are superimposed and a thicker water film is formed. This is the pellicular water or intergrenular water film. I n the interstices the surface tension of the water increases the amount of stored water, but this influence (capillarity) becomes dominant only near the water table. The upper part of the soil-moisture zone is the zone of adhesion, because adhesion is the dominating force acting against gravity as explained in the previous paragraph. Another common condition prevailing in this zone is that soil moisture occurs only in the form of films of water, and in dynamic equilibrium, few pores are completely filled with water. Considering the aspect of saturation this domain is therefore called the unsaturated zone.

1.3 Dynamics of soil moisture

103

Moving further downwards, the pores are under the influence of different forces and from the point of view of saturation their condition may be regarded as a transition form between the saturated and unsaturated state (transition zone). Some of the pores are completely filled with water, and in others water films develop under the influence of adhesion, similar to those in the unsaturated zone. This change in the character of saturation is caused by the fact t h a t in this zone the other molecular force also becomes dominant: i.e. capillarity (the more detailed dynamic interpretation of both adhesion and Capillarity will be given in Chapter 2.1).The effect of capillarity is proportional to the elevation of the point of investigation above the water table. The transition zone (capillary zone) can be divided, therefore, into two parts. Its upper part is known as the open capillary zone, where only the narrow pores are completely filled with capillary water, while in the other pores only the walls are covered by adhesive water. The upper boundary of this zone (which also divides the adhesive from the capillary zone) is the maximum capillary height (see Section 1.3.4).Below this zone and above the water table lies the closed capillary zone, where all the pores (or almost all) are completely filled with water by capillarity. The thickness of this zone is equal to the capillary height belonging to the diameter of the largest pores. The method of determining this minimum capillary height will be explained in Section 1.3.4. The lower boundary of the capillary zone is the water table, below which the pores are completely filled with water (zone of saturation). As already mentioned, the pressure of water is higher here than that of the atmosphere. Capillarity does not act in this zone because its existence is closely related to the boundary of the liquid and gaseous phases. The influence of adhesion hinders the flow only in the close vicinity of the walls of the solid matrix, whilst inside the pores water can easily be mobilized by gravity (or, more precisely, by its gradient). For this reason this part of the model may be called a gravitational ground-water zone as well. Both the dynamic characterization of soil moisture and the zoning of the layer lying between the soil surface and the water table is based on the supposition of a dynamic equilibrium. External influences can however, disturb this balanced condition, causing temporary changes in the soil-moisture conditions. Infiltration of water (precipitation, irrigation) can completely f2.l the pores near the surface and in this cme capillarity also acts in the unsaturated zone (pending capillary zone). As a result of evapotranspiration, the amount of soil moisture may be less than that belonging to the dynamically balanced state. In both cases there exists a vertical gradient of the total energy (Buckingham’s potential), the latter being composed of the gravitational (potential) energy and the difference in tension of the neighbouring points of the surface of the water film (Buckingham, 1907; Richards, 1931; De Wiest, 1969).The gradient creates a flow causing the redistribution of soil moisture which results in the development of the dynamic equilibrium described above (Fig. 1.3-3). Naturally, the balanced state can only be achieved if there is no new external action which may disturb the equilibrium of the internal forces. Buckingham’s potential is always related to the position of the water table. It is natural, therefore, that the change in depth of

104

1 Fundamentals for the investigation of seepage

percent of volume 1- 5 distribution of moisture content with depth at varlous tirn~,mflAFig. 1.3-3. Change in the vertical distribution of soil moisture in time (soil-moisture redistribution)

the water table has to be regarded as an external influence similar to infiltration or evapotranspiration, and also followed by the redistribution of soil moisture.

1.3.2 Parameters characterizing the various moisture contents of soil samples As mentioned before, the pores of loose claatic sediments are filled with water either completely (saturated system) or partially (unsaturated system). It is necessary, therefore, t o define numerical parameters for the quantitative characterization of the amount of stored water and for the rate of saturation of the layer. The amount of water stored in a unit volume of the sample is called the specific water or moisture content. According to its definition, water content is the weight or volume of water related to the weight of the dried sample or its total volume. To standardize the measurement, the sample should be kept at a temperature of 105OC, until no alteration in its weight can be observed. The water content expressed atj the ratio of weights can be calculated from the following equation: W =

Rv

-Go. Y

QO

(1.3-1)

1.3 Dynamics of soil moisture

105

where Q, is the wet weight of the sample in its original condition and Qo is the weight of the same sample after drying. To calculate the water content as a percentage of the volume, the specific weight of the dry sample must be known:

W=

Q, - Qo Yw

.. c, - Qw Yo

-00

Yo =w Yo ;

(1.3-2)

QO

where yw is the specific weight of water, which is unity, and y o = ys (1 - n ) , if the specific weight of the solid grains is indicated by ys. This method of measurement requires a sample from the layer, which disturbs the upper layer and excludes repetition at the same point of both single measurement and the continuous observation of the change in soil moisture. Equipment can be obtained which measures soil moisture, thus eliminating these disadvantages. The various types of equipment are based on electrical resistivity or capacity (e.g. gypsum block); on the direct measurement of the properties of soil moisture using the absorption of radioactive rays by hydrogen atoms or by the maas of grains and water (neutron or gamma probes); and on the me of infra-red photographs. None of them can be regarded aa a fbal solution for the following reasons. Electrical measurements are disturbed by the change in temperature and chemical composition of water. The operation of tensiometers is hindered by various uncertain phenomena. Neutron probes measure hydrogen ions and hence the results are influenced by the organic content of soil, especially in the root zone. Gamma rays measure the total mass, and hence their sensitivity is relatively low, and the correct quantitative evaluation of airborne infra-red photographs has not yet been completely solved. It can be stated, however, that these methods indicate very important developments in the continuous observation of soil moisture and also in the general characterization of large areas. Discussion on the methods of measurement of soil moisture would, however, digress from the original topic-the hydrological investigation of the water retention capacity of the soil. It is necessary, therefore, to continue the discussion of the physical interpretation of water content. The water content determined by the method explained in Eqs (1.3-1) and (1.3-2) represents the amount of water present in the sample at the time of the investigation. This value cannot be regarded, therefore, as a physical parameter of soil. It only gives information on the instantaneous condition of the sample. When determining the water content related to a given condition of the sample, this value becomes a parameter suitable for the classification of soils. Examples of such parameters are found in the various values of moisture content used to characterize the cohesive samples in soil physics such as the liquid limit (w,; or W,) and the limit of plasticity ( w p ;or W p ) ,or the index of plasticity, which is the difference between the two former values expressed BB the ratio of weights ( I p = wL - y p ) . Although these parameters belong to arbitrarily chosen con&tions (the liquid limit based on water content measured in Casagrande’s apparatus, in which a cut sample is made to close again by a series of small physical shocks, and the limit of plwticity, when a string of 3 mm diameter can be

106

1 Fundamentals for the investigation of seepage

rolled from the sample) they were proved to be useful by means of comparing various cohesive soils. The maximum molecular water capacity (w,,,, or Wmol)is also an arbitrary parameter (viz. the water content of a sample the measurement of which is achieved if the diameter of the sample is 50 mm, its thickness is 2 mm and it is loaded by a pressure of 6.55 N/mm2after the development of consolidation). This method of measurement, however, is only an approximation to standardize the parameter. According to the original concept, maximum molecular water capacity is the water content of a soil column remaining after drainage by gravity and, therefore, it characterizes a general physical property of the soil (see also Chapter 1.4, Lebediev, 1936). [It is necessary to note, that the symbols applied here to indicate the two different types of moisture content (i.e. the amount of water stored in the sample and expressed aa the ratio of weights, or that calculated as the volume of the stored water related to the total volume of the sample) are different from those generally used in the literature. It was felt that the similar character of both parameters has to be emphasized by being represented by the same letter. At the same time, the necessary distinction can be ensured by using small and capital letters, respectively. It was decided, therefore, that the symbol of gravimetric moisture wntent should be a small w , while the volumetric moisture wntent should be represented by a capital W.] The pores below the water table are completely saturated ( n = W ) . Above this surface (where 0 < W < n ) , the water content indi,cates the degree of saturation. Two samples having the same water content have differing rate of saturation when the porosity of one of them is high and that of the other is low. For this reason, the weflcient of saturation (or the rate of saturation) is also used to characterize the condition of the sample. This parameter is the quotient of the volumetric moisture content to porosity:

H'

s=-;

n

(1.3-3)

the value of which is unity in a saturated sample (9, = 1) while in a completely dry sample, it is zero. The total range of saturation lies, therefore, between zero and unity (0 < s < 1). The practical lower limit however, differs from the theoretical value, because a completely dry sample does not occur in Nature. If the minimum natural water content (the volumetric moisture content of a so-called air-dry sample) is W,, the possible range in variation of the coefficient of saturation is so < s < 1 , where the lower limit is so = Wo/n. One can e a d y argue that knowledge of the ratio of saturation still gives insufficient information as to the expected behaviour of the soil, as the change in saturation in a sand does not involve much modification, whilst in a clay soil considerable modification may occur. Measuring the water content for a given state of the sample facilitates the complete understanding of the effects of the actual or instantaneous moisture content of the soil. Knowing the specific values of water content of the soil, and comparing

1.3 Dynamics of soil moisture

107

them with the actual amount of soil moisture, the behaviour of the soil can be estimated. When selecting the specific parameters, the most important requirement is that the state they describe should be characteristic and easily repeatable. In thia respect, the parameter of plasticity used in soil physics is a useful value to determine, although it belongs to arbitrarily chosen states of the sample as already explained. At the same time, the index of consistency, which is the ratio of two differences (that between the liquid limit and instantaneous water content related to the difference of w L and w p ) : (1.34)

iu a good example of how to compare the actual water content to the selected limit values. Although the specific soil moisture values generally used as parameters in soil science belong to more natural conditions (describing the real physical character of the sample), than the physical parameters of soil, their determination causes some difficulties, because the conditions described are not known precisely enough. The parameters generally applied in soil science are as follows: (a) Field capacity (wre, or Wre)is the water content retained against gravity in the sample. Thus, the value is approximately equal to the difference between the total porosity and gravitational pores; (b) Wilting point (wwp,or W w p )is the amount of water which cannot be used or removed by the roots of plants; (c) Usable (disposable) water (wd, or W,) is the difference between the two parameters mentioned above.

Field capacity is a very important parameter, but it is influenced by numerous undetermined factors (e.g. temperature, relative humidity of the air, etc., among which the position of the point of investigation related t o the actual water table is perhaps the most important, aa will be demonstrated later on). This can hardly be reproduced, or physically interpreted. The same statement can refer to the wilting point as well, because the water, not freely available for the plants depends on the suction of the roots, which differs from plant to plant, and also changes according to the growth phase. The most stable and repeatable parameters among the specific moisture content values used in soil science are the various types of hygroscopic moisture content (or hygroscopicity), which meaaures the water content retained by the sample against the suction of sulphuric acid of different concentrations in a closed space at a given temperature. Thus, Mitscherlich’s hygroscopicity ( w H y , or W H Y )is determined with sulphuric acid of 10% concentration, while Kuron’s hygroscopicity (w,,, or Why)is measured with sulphuric acid of 50 % concentration (Mitscherlich, 1932; Kuron, 1932). Because of the high stability of these parameters (assuming that a constant temperature, 2OoC for example, is ensured for all the measurements), it is reasonable in future, to accept their general use and to regard the others

108

1 Fundamentals for the investigation of seepage

only aa rough approximations, which can be estimated aa the functions of hygroscopicity using linear relationships. According to Mados’ investigations, the characteristic values of moisture content mentioned previously can be approximated aa the functions of Kuron’s hygroscopic moisture content by the following equations (Mados, 1939; 1941):

+ 0e.12; w w p = 4why + 0.02; uyc= 4why

(1.3-5)

w, = wf: - wwp= 0.10.

According to these relationships, the usable water content is 10% of the weight, independent of the type of soil. Excluding very extreme cases (coarse sand and very cohesive clay) this average value can be accepted a~ a rough approximation. Similar relationships were determined between Mitscherlich’s hygroscopic moisture content and the various water contents used aa parameters in soil physics and soil science (JuhQsz, 1967): wg

= 0.54 WHY;

wWp= 1.44 WHY

+ 0.02; (1.3-6)

wp = 1.7 WHY

+ 0.065.

On the basis of Fekete’s (1950) measurements shown in Table 1.3-1, a further relationship can be established to connect Mitscherlich’s and Kuron’s hygroscopicities and to construct a transition between Mados’ and Juhbz’ equations w follows: why = 0.45 WHY.

(1.3-7)

Apart from his own observations and measurements, Juh&sz bwed the relationships given in Eq. (1.3-6) partly on previously published data in various papers (Dolgov, 1948; Fekete, 1950; Rodie, 1952; Kreybig, 1951; Lebediev, 1963) and also on formerly established relationships such ah( those published in papers by Procerov and Karasev (1939), Kreybig (1951), Vhrallyai (1942) regarding wilting point, and Kreybig (1951) on field capacity. Comparing Mados’ and JuhBsz’ equations their good accordance can be seen. Naturally, these linear relationships cannot substitute direct measurements. They give, however, very useful information on the contact between the parameters characterizing the various states or conditions of soils. Thus,

1.3 Dynamics of soil moisture

109

Table 1.3-1. Some data characterizing the relationship between the moisture content of air-dry samples and Kuron’s hygroscopicity (after Fekete, 1960)

Type of the sample

i

as percentage of weight

w.

Send

0.6

Fine sand

1.2 1.8 2.6

Rock flour (loess)

3.1 3.6

Silt

4.2 4.8 6.4

I

6.0

Heavy clay

6.6 7.8

why

0.6 1.0 1.6 2.0 2.6 3.0 3.6

4.0 4.6 6.0 6.6 6.6

it can be recognized that the limit of plasticity and maximum molecular moisture content are very similar parameters, and the liquid limit and field capacity are also almost identical characteristics.

1.3.3 Interpretation of field capacity, gravitational porosity and soil-moisture retention curve

As already mentioned, the water retention capacity of a soil prism depends on the position of the investigated sample. It is necessary to know, therefore,

whether the sample to be characterized is separated from its surroundings, or if the prism is situated as part of a continuous soil profile. I n the latter case, its elevation above the water table also influences the moisture retention capacity. The reliability of this statement can easily be understood. I n the case of a sample separated from its surroundings, the eflect of gravity i s expressed by the weight of water incorporated in the pores. If the soil prism (the water retention capacity of which is under investigation) is a part of a continuous soil profile, the sample id fitted into a space of gravitational potential, the datum (reference level) of which is the water table, where the surplus pressure is zero (the gravitational potential is zero at this level). Everywhere in this space, the effect of gravity should be expressed with reference to this datum. Thus, above the reference level, gravity m w e s suction proportional to the height of the investigated point above the water table. Physically this process can be explained by imagining that the continuous chain of water films composes a closed system, in which the pressure on a water particle (in this case, negative pressure, i.e. suction, because the particle is

110

1 Fundamentals for the investigation of seepage

above the water table with a zero pressure) is proportional to the weight of a straight water column between the particle and the reference level, and independent of the form of the container (the form of the chain composed of water films). For this reason, the point value of field capacity will be larger near the water table where the suction caused by gravity is smaller, than at a higher point of the profle, and the water retention capacity measured in a separated sample can be regarded as a meaningless parameter when investigating the field condition of a complete soil profile. There is another parameter generally used in the investigation of the Unsaturated zone, which becomes questionable if we accept the interpretation of field capacity given previously, and when determining the latter as a function of the elevation above the water table. This parameter is the free or gravitational porosity (generally indicated by the symbol no ).Its clavaical definition indicates that it is part of the total porosity, where, under the given conditions the water is not bound to the grains by adhesion or under the influence of capillarity, and therefore, moves freely. Consequently, it was previously proposed that gravitational porosity should be calculated as the difference between total porosity and field capacity, the latter expressed as the volumetric ratio. Accepting that field capacity is a function of the podtion of the investigated point, it has to be stated that the gravitational porosity can also be expressed only as a function of the elevation above the water table. In all types of soil profile, a line can be determined which divides the total porosity into two parts: one is field capacity and the other is gravitational porosity. It is evident that field capacity is relatively larger near the water table and its value decreases moving upwards in the profile, while gravitational porosity changes inversely. It is also obvious that the sum of the two parameters at a given elevation is equal to total porosity and is conxtant if porosity does not change in the profile (Fig. 1.3-4). There is a further condition which has also to be considered when determining the position of the line dividing the porosity of the soil profile into two parts. If the part covered by field capacity is filled with water and the remaining part is occupied by air, the soil moisture i n the profile is i n dynamic equilibrium. There is no water movement, except in the case where i t is induced by an external force (e.g. evaporation from or infiltration to the top soil, change in the position of water table, etc.). Dynamic equilibrium can develop only when the acting internal forces are balancing each other. Investigating the adhesive (unsaturated) zone the internal force8 are the tension ( p ) on the surface of water films, and gravity. Their balanced state can be expressed by equalizing the total hydraulic gradient ( I )to zero (both the horizontal and vertical components of the gradient should be equal to zero). Investigating the process in a horizontal plane, this condition requires that the vertical moisture distribution should be identical at each profile in a homogeneous medium, while in the vertical direction, the condition gives a relationship between tension and gravity:

+

I = d ( h Y ) - dy dx dx

dy d W dW dx

1 =---

1 =0 ;

(1.3-8)

111

1.3 Dynamics of soil moisture

1

n

__c

fotal porosity Fig. 1.3-4. Graphical representation of the vertical distribution of field capacity and gravitational porosity

considering that h = H - x and the tension divided by the specific weight of water gives the potential caused by suction (y, = p l y ) . The total (Buckingham's) potential ie achieved by adding the gravitational and suction-potential (Richards, 1931). From Eq. (1.3-8), field capacity (water content belonging to the state of dynamic equilibrium at different elevations in the profile) can be calculated as a function of the height of the investigated point above the water table ( k ) or that of the depth below the surface (2)if the relationship between tension and moisture content ie known. The equation (1.3-9)

W j , = f ( h )= f ( H - 2)

describes the position of the dividing line above the lower boundary of the unsaturated zone. This relationship is valid, where h h,, max. Among the symbols h, max is the capillary rise and W j , indicates the field capacity in the unsaturated zone (in the closed and open capillary zones, this parameter will be indicated by the symbols W{ and W$ respectively, see Fig.

>

1.34).

-

The conditions in the transition zone differ from those in the unsaturated zone, because gravity is here balanced by two forces (adhesion end capillarity). As already mentioned, the closed capillary zone is al.most completely

112

1 Fundamentals for the investigation of seepage

saturated, because capillarity causes the water to rise in all the pores to the top of this zone. The vertical distribution of field capacity in homogeneous media (having constant porosity) can be characterized, therefore, by a constant value equal to porosity

W;,= n = const.,

(1.3-10)

instead of the solution found in Eq. (1.3-9). The assumption of the equality of field capacity and porosity is valid theoretically only if h = 0, but it is is the miniacceptable aa a good approximation if h < h, &, where h, mum capillary rise belonging to the diameter of the largest pores. The dynamics of the movement also differs from that in the zone of adhesion. The balanced condition cannot be described by Eq. (1.3-8). I n a capillary tube having a capillary rise h, (which is inversely proportional to the diameter d of the tube) and filled with water to a level h above the water table, the gradient is proportional to the difference between the gravitational and capillary forces expressed by the height of the water columns: (1.3-1 1)

where p is the pressure created by the weight of the water column having a height h, and p , is the capillary suction (the capillary pressure is regarded here as constant, although its value may change as the meniscus is raised or lowered as will be explained in Section 2.1). The same relationship can be accepted for the characterization of the closed capillary zone, because the influences of adhesion compared to that of capillarity can be neglected. If the gradient expressed in Eq. (1.3-1 1) is zero, the system is in a state of equilibrium, while if it is positive (the effect of gravity surpasses the capillary force), the draining of the tube to the gravitational ground-water space starts at its lower end and the water level in the tube sinks immediately until reaching the balanced conditions. Similarly, in the case of negative gradient, the water rises in the tube are recharged from the gravitational ground water. some of the pores are Within the open capillary zone (h, m,n < h h, under the influence of capillarity (e.g. those pores at an elevation which are smaller than the diameter of a capillary tube having a capillary rise, h ) . In this fraction of pores, the water content is equal t o the total volume of these pores, and the dynamics of flow is characterized by Eq. (1.3-11). In larger pores, water fllms can develop around the grains under the influence of adhesion, and both the vertical distribution of moisture content and the balanced condition can be described in a manner similar to that in the unsaturated zone [Eqs (1.3-8) and (1.3-9)]. The ratio (R)of capillary fllled pores (V,) related to the total volume of pores ( V p ) can be determined as a function of the elevation above the water table ( h ) .The same ratio can be achieved by using porosity instead of the volumetric values of pores, since porosity is the specific value of pore volume related to the total volume of the sample. Thus the ratio R of capillary filled pores can be calculated

< A,),

113

1.3 Dynamics of soil moisture

as the quotient of capillary porosity (n,) and total porosity ( n ) :

where

R and

N

0

R

at the level N

h = h, max ;

1 if h = h, ,,,

.

(1.3-12)

Hence, the vertical distribution of moisture content in the open capillary zone can be determined by combining Eqs (1.3-9),(1.3-10) and (1.3-12)(see Fig. 1.3-23);

W;: = R(h) W;c+ [l

- R(h)]Wj,= n,(h)

+ [l-

-]Ah) n (h) n

if

;

(1.3-13)

hc min

< h < hc max.

The interpretation of capillary porosity used in Eq. (1.3-12)can be given as follows. At an elevation h, the total pore volume (V,,.)can be divided into and those remmming under the sole two parts: capillary filled pores (7,) influence of adhesion (Va).Any partial value divided by the total volume of the sample (V,)gives a parameter similar to porosity. The first is called cupillury porosity (n,) and the second adhesive porosity (nu): v c -n,;

therefore

vt

Va-na ;

vt

n = n,

v, = vc+ v u ;

+ na.

(1.3-14)

As already mentioned, the water content does not yield sufficient information on the behaviour of the soil. It is necessary to compare these data to parameters belonging to special conditions in the sample. This restriction should be enlarged upon now, showing that the point measurements - even compared to such specific characteristics as hygroscopic moisture content, liquid limit, or limit of plasticity - are insufficient t o describe field conditions. The complete vertical distribution of moisture content has always to be determined in a profile and compared to soil-moisture distribution belonging to the dynamic equilibrium. Only the differences between the actual distribution and the matching curve shows where water is deficient or where excess water occurs in the profile. The only remaining problem is the numerical determination of the equations [Eqs (1.3-9),(1.3-10) or (1.3-13) which gives the superposition of the other two]. These describe the position of the line dividing the field capacity and gravitational porosity, which characterizes the state of equilibrium of soil moisture in a profile as proved earlier. It can be called the characteristic vertical distribution of soil moisture belonging to the dynamic balance of water content, or the soil-moisture retention curve. As already 8

114

1 Fundamentals for the investigation of seepage

shown in the unsaturated zone of homogeneous media, this problem can be reduced to the investigation of the relationship between tension and water content [Eqs (1.3-8) and (1.3-91, while in the transition zone, the analysis haa to be supplemented by the determination of the ratio of the capillary influenced pores to the total pore volume [Eqs (1.3-12) and (1.3-13)]. This condition immediately gives rise to the idea that a p F curve could be used pF= log W

U

lU

20

7

30

moisture in percent by volume, W

[%I

Fig. 1.3-6. Characteristic form of p F curves determined from large number of measurements (after Wind)

to determine sought after matching curve, since - according to its definition - a pF curve also represents the relationship in question. The usually accepted method of determining a pF curve is to apply vacuums of various values to the investigated sample, and to meaaure the retained soil-moisture values against the pressure in the pores. After plotting the points of vacuum vs. water content t h e p F curve can easily be constructed (Fig. 1.3-5; Wind, 1966). Different types of equipments have been proposed t o create suction on the sample, the construction of which depends mostly on the range of pressures to be applied. The two curves (i.e. p F and soil moisture retention) are identical if the osmotic forces (chemical-, electro-, or thermo-osmosis) are negligible, because the pF curve expresses the relationship between water content and tension, the latter including the effects of all the forces. Neglecting osmosis, the remaining acting forces are gravity and adhesion supplemented by capillarity in the partially saturated transition zone. Gravity, adhesion and capillarity are also forces, the balanced condition of which (or more precisely the water content belonging $0 this condition) is characterized by the water retention curve as well.

115

1.3 Dynamics of soil moisture

Differences occur only in the determination and the interpretation of the two curves. The p F curve describes the relationship of tension and water w n tent of a sample determined by applying various tensions to a separated soil sample. Considering that the tension expressed in an equivalent water column is the same as suction caused by gravity at an elevation above the water table equal to the water column mentioned, i t is evident that the two curves have to be identical if the profile is homogeneous and composed of the same material (with the same porosity) aa the sample invetitigated (supposing, once again, that osmotic forces are negligible). For this reaaon, the physical interpretation of the p F curves is dealt with in detail further on, which gives - at the same time - the explanation and understanding of the physical character of the soil-moisture retention curve. There are two special problems to be discussed in connection with the proposed application of p F curves: (a) The change in the bulk volume of the investigated sample aa a result of increasing moisture content; (b) The structure of the cultivated layer (root zone) which makes dificult the interpretation of the water retention capacity on the basis of the p F curve. A common feature of the soil-moisture retention curves is that they are composed of three clearly recognizable sections (Kovkcs, 1968; Fig. 1.3-6a). The upper part of the curves is almost vertical (zone of adhesion) which is followed by a nearly horizontal section, where the water content increases rapidly with decreasing tension (open capillary zone). Finally, the curves have to be closed by a vertical line where the moisture content equals porosity ( W = n ; s = l),this value being the upper limit of the amount of titored

I

0.5

l.0

s=W n

Fig. 1.3-6. Characteristic stretches of p P curves (a) General form; (b) Curve representing very fine-grained sample

8*

116

1 Fundamentals for the investigation of secpage

water in the pores (closed capillary zone). There are only a few exceptions, always in the case of very fine material when the first two sections are substituted by a line evenly curved with a decreasing slope as the water content increases (Fig. 1.3-6b). However, the vertical closing section below the minimum capillary rise intersects the horizontal axis a t W = n in this case as well. In contrast to this concept based on the physical upper limit of the water retention capacity (which states that the moisture content cannot be higher than that belonging to total saturation), most of the p F curves published in the literature show gradually increasing water contents, considerably exceeding the original porosity of the sample, as the tension approaches a zero value. The sloping character of the closing part of the p F curve is especially characteristic in cohesive materials (clay, silt), but many similar curves are also published for fine sand. The theoretically correct, vertical, closing part is generally measured only for coarse grained sands. Instead of repeating these curves from other publications (Klute, 1952; Sudnitsin, 1966; Vkallyai, 1974), three p F curves were carefully measured and these are shown in Fig. 1.3-7, together with the original porosity of the samples. It can clearly be seen that the total moisture content belonging to a low tension is higher than the original porosity of the samples. This contradictory result can only be explained by the fact that the bulk volume of the samples, and thus their porosity as well, increased considerably as the moisture content was increased. In Fig. 1.3-8, the observed change in volume of the three samples is also presented to prove the reliability of the previous statement (Zotter, 1975).These measurements indicate that in the '

0

02

0.4

06

0.8 W ( 0

Fig. 1.3-7. p F curves with increasing moisture content in the close capillary zone

117

1.3 Dynamics of soil moisture

case of cohesive samples, the change in porosity could also cause some uncertainties, even in the zone of adhesion. The porosity of the material can only change if the investigation is carried out on a separated sample because, in a soil profile, the weight of the overlying layer hinders the expansion of the medium. It is also evident that p F values belonging to different porosities must not be used for the construction of a unified soil-moisture retention curve. The basis of this statement 07-

9 0.003-

*.P.

116- \*

.s 0.NlZ-

i2 F

115-

9,

s. 9

s8

\

Q

s

8

i

'?

F!$

i

k

how

& 04-92

0.001--

.' \.

,.'

',

0.1-

1 2

\

-.-. ------------_____ .---.

.- ._

4.5

I

I

I

I

1

50

55

80

65

70

PF

clay

.,' -. -.

. * .

*\.

sand

1

u

clay

--. ---------___

loess

'-+

-.

\.

-*-

Q g

0.2 - '\

\.\.

0 *\.

,sQJ

8h

\.

loess .-*.

*\

PL Q F J8 0.3-

s

.'

a 45

I

I

1

I0

. 6 ,.a6

60

I

~75of

1

ZD

Fig. 1.3-8. Change in the volume of the samples with increasing moisture content

is the fact that the relationship between tension and water content includes porosity as an independent variable, because both the size of the pores (capillarity) and the relative internal surface (adhesion) change with the relative pore volume. The physical interpretation and the mathematical approximation of the dependence on porosity of the relationship in question will be discussed in detail in the next Section. It is advisable, therefore, either to ensure a constant pore volume of the sample when measuring p F values, or to correct the observed data on the basis of the measuredchange in pore volume in the open capillary zone and to close the soil-moisture retention curve with a vertical face in the closed capillary zone at values of W = n (Fig. 1.3-9). It is necessary to note here that there is a special structure of soil, which may result an elongated closing part of the retention curve: i.e. the secondary pores enclosed by aggregated particles. In such soils the p F curve developes according to the pattern characteristic for the evenly distributed pores ehowing an almost vertical section, where water content is equal

118

1 Fundamentals for the investigation of seepage

to the primary pores. Lowering the suction the saturation of the secondary pores starts; this process raises the water content until total porosity, and results in the development of a second sloping strech of the curve. Another problem arises when the p F curve is applied to the determination of the soil-moisture retention capacity. This is caused also by the fact that the cultivated zone has a special structure composed of aggregates, while at a greater depth the layer is generally built up from evenly distributed, indichange in Porosity, An

Fig. 1.3-9. Correction of meaaured p P values considering the change in pore volume

vidual grains and pores. The development of primary porosity created inside the aggregates and secondary porosity created by the large channels between the clods is represented in Fig. 1.3-10. It is the general opinion that the physical investigation of the unsaturated zone has to exclude the analysis of the processes occurring in the root zone, because physiological processes are there predominant, whereas below the layer affected by roots the water movement is solely a physical process (van Bavel, 1966). Some investigations have proved that the channels of earth-worms could also modify the water movement through the top soil (Peterson and Dixon, 1971). Apart from these influences of biological origin, the development of the normal soil-moisture retention curve is also disturbed by the special structure previously mentioned, which is created partly by

119

1.3 Dynamics of soil moisture

roots and the aggregation of grains influenced by the organic content of the soil, and partly by ploughing. Thus, the separation of the root zone, or the cultivated zone, from the other part of the unsaturated zone is absolutely necessary from the physical aspect as well, was the case in the construction of the model represented in Fig. 1.3-1. The comparison of grain-, aggregate-, and pore-size distribution in the different soil layers (horizons) can be used to demonstrate and characterize the structural differences resulting from cultivation and the activity of roots. In the case of the example represented in Fig. 1.3-11 a chernozem soil developed on a loess terrace was investigated (VuEic, 1966). The almost aggregate romposea ofgrains end enclosed primary pores

7

secondary pores

u---aggregates Fig. 1.3-10. Primary porosity of aggregates and secondary porosity composed of the channels between the clods identical granular composition of the Iayer was indicated by the grain-size distribut'ion. Thus, their similar physical behaviour could be expected. There was only a slight difference between the lowest layer and the others, showing that the former was somewhat finer. This was probably caused by the irreversible aggregation of fine grains in the upper part of the profile where the physico-chemical properties of the particles had been influenced to a greater extent by drying and wetting or freezing and thawing. The distribution of the aggregate-size was determined only for the A horizon (where the aggregates could be well distinguished), by sieving the samples in both wet and dry conditions. The wet sieving did not show any difference between the ploughed Iayer and that beneath it, while in the dry sieving, the aggregates of the ploughed layer were considerably coarser indicating that the large clods created by ploughing are less stable. The most important information was provided by the pore-size distribution curves. The probable maximum and minimum pore sizes were calculated from the physical characteristics of soil in the samples [Eqs (1.2-19 and (1.2-22)J It was found that the diameter of the primary pores is smaller than 3 p and their total volume is about 30-32% of the bulk volume (pri0.32), which agrees well with the measured mary porosity n, = 0.30 data. The measured total porosity of the ploughed layer is n 0.55, that

-

-

,

120

1 Fundamentals for the investigation of seepage

aggregate diameter CcmJ 4,

F 50P

'i

POre-sile

calcuJafed from grain-size __ ~

x

pore-size disfributiun I

1 Z

2

A Z

f

h$iiii7-3 A;iiio-2 pore diameier [ e m ] Fig. 1.3-11. Grain-, aggregate- m d pore-size distribution of the different horizons of chernozem soil developed on a loess terrace 10-4

2

1.3 Dynamics of soil moisture

-

--

121

of the A and AC horizon is n 0.5, that of the C horizon is n 0.45, while 0.38.Thus, secondary porosity (n,= n - n,) in the basal layer (CQ), n decreases with depth. Ploughing raises this value from n, 0.18-0.19 to n2 70.24, a parameter which is characteristic of the top soil. At the C horizon, a secondary porosity of n, 0.16 was found. I n the basal layer the secondary porosity is only a few percent and itsdevelopment is probably due to the deeply penetrating roots. According to thlt continuum approach, porosity (more precisely its change) can be characterized by selecting a suitable representative elemen, tary volume (see Chapter l . l ) ,and most of the hydraulic processes can be described (even in the case of a porous medium having a special structure) in a macroscopic way by applying this parameter. The relationship between tension and moisture content is, however, an exception because its development is basically influenced by the structure of the pores as well. The process has to be investigated, therefore, on a microscopic level, and the macroscopic approach is not applicable. The channels composing the secondary pores are generally large and capillary menisci cannot develop in them. Hence, atmospheric pressure prevails throughout the aggregates. For this reason all the clods have to be regarded as a separated system in which the relationship between tension and moisture content develops independently of the others, influenced only by the boundary conditions of the clod under investigation. As already mentioned, the boundary condition along the large secondary channels is characterized by atmospheric pressure. Only a small fraction of the surface is in contact with other aggregates where different boundary conditions prevail, governed by the moisture content and tension condition of the neighbouring clods. The aggregation of the particles usually occurs only in very fine grained (cohesive) soils. Thus, the tension belonging to the saturated condition (height of the closed capillary zone) is considerably greater than the size of the clods and, therefore, the complete saturation of the primary pores is expected if the boundary condition is described by p = p , on the surface of the clod (even if there is a small suction caused by other clods in contact with the one under investigation). Only a few pores contain trapped air, which slightly decreases the water content of the aggregates (see Fig. 1.3-10). From this condition the moisture content of the structured top layer ( W s l ) can be calculated assuming that suction on the surface of the clods is smaller than the air entry pressure at that point (which is approximately equal to the height of the closed capillary zone):

-

W,, = VPl--n, ;

vt

SSl=

~

WSl n

n1 n1+ n2

(1.3-15)

If the moisture content is higher than this limiting value, the secondary pores also contain water, a condition which causes the disaggregation of the clods and destroys the structure of the soil. The moisture content of the aggregates is constant until the pressure on their surfaces is smaller than a given limiting value characterizing the porous medium of the clods. The pressure determining the boundary condition can be raised above the given

122

1 Fundamentals for the investigation of seepage

limit either by evaporation (around the clods near the surface), or by the roots encircling the aggregates and even penetrating into them. Because the maximum pressure created by the usual growth of crops is about 16 atm ( p F = 4.2), water will be avdlable for the plants if the water content is higher in the clod than that belonging to p F = 4.2. Thus, the lower limit of moisture content ( W,), below which agricultural production requires artificial water recharge, can also be calculated:

where qaa2) is the saturation of a homogeneous sample from the material of the clods under the influence of pressure described by the p F = 4.2 value. Using Eqs (1.3-15) and (1.3-16) in addition to Fig. 1.3-11 in Table 1.3-2, the probable characteristic values of the porosities of the different horizons in the chernozem soil under investigation are summarized, together with the range of moisture contents within which saturation has to be maintained in arable land. The last line of Table 1.3-2 [which was calculated Table 1.3-2. Porosity and characteristic moisture content of structural chernozem soil (as percentage of volume) (saturation belonging to p F = 4.2: S(4.2)

DiEerent horisoiis of chernmem mil covering loess terrace

I

= 0.40)

I

Porosity

upper limit

secondcontent

A (ploughed) A AC C

CG

I

Required

0.66 0.60 0.60 0.46 0.38

0.31 0.31 0.32 0.29 0.32

0.24 0.19 0.18 0.16 0.06

CB (assuming that the structure can be neglected)

ration

0.31 0.31 0.32 0.29 0.32

0.66 0.62 0.64 0.64 0.84

0.12 0.12 0.13 0.11 0.12

0.22 0.26 0.26 0.26 0.32

0.19 0.19 0.19 0.18 0.20

0.38

1.00

0.16

0.40

0.23

assuming that the basic layer hae no special structure (secondary porosity, large channels), but that the total porosity ( n = 0.38) is evenly distributed in the layer] shows that in this very fine material, agricultural production can hardly achieve good results without secondary porosity because total 1.8 2.0. Hence, plants cannot be provided saturation occurs at the p F with the necessary air.

<

'V

123

1.3 Dynamics of soil moisture

1.3.4 Description and determination of the soil-moisture retention curve

In the unsaturated zone, the water in the form of a thin film covers the solid wall ,formed of grains. The force, retaining this water against gravity, is adhesion. Without investigating the character of this force in detail (which will be given in Chapter 2.1), its existence can be explained by the attraction between the wall of the solid skeleton and water molecules. This force is created by the electrostatic charges of the graim, and the orientation of water molecules having dipole character (Van der Wads force). These types of forces are generally approximated in theoretical physics by relating the tension at the surface of the water film ( p ) to the thickness of the film ( 6 ) in the form of a hyperbola of the sixth order:

.=It) 6

(1.3-17)

It ie quite evident that the thickness of the water film is closely correlated to the water content of the sample. As a first approximation of the dynumic equilibrium of the moisture content of a sample in the zone of adhesion ( W;) can be determined by multiplying the grain surface ( A )by the thickness of the water film and by forming the ratio of the obtained value to the total volume: A6 A a C (1.3-18) w;;=-= -6(1 - n ) = -__ (1 - n ) ; Vt V D,, P1lS because the ratio between the volume of the solid grains ( V ) and the total volume of the sample ( V , ) is equal to (1 - n ) , and the ratio of the grain surface to the grain volume can be expressed as the ratio between effective diameter and shape coefficient [see Eq. (1.2-l)]. A similar result can be obtained by calculating the moisture content adsorbed to the walls of the pipes composing the physical model of the water transporting channels (see Section 1.2.5). In a pipe having a length 1 and a radius ro, the amount of water stored in a film of thickness 6 is

W, = In($ - r2) = h 6 ( d 0 - 6);

(1.3-1 9)

became the internal radius of the water film can be calculated as the difference between the radius of the pipe and the size of the water film ( r = ro- 6 ) and double the value of the pipe radius is equal to the average diameter (2r0 = d o ) which can be calculated from Eq. (1.2-19). The number of pipes crossing the unit area of the sample ( N ) is determined by Eq. (1.2-20). The water content of a sample with a cross section of unity and a length I (the total volume of which is, therefore, V t = I ) is the product of the W ,value and the number of model pipes. This product divided by the total volume of the sample gives the moisture content, being in a balanced condition under a pressure of p . Thus, the second approximation of the balanced moisture content in the zone of adhesion (W$*) aa the func-

1 Fundamentals for the investigation of seepage

124

tion of the physical parameters of soil can be estimated on the bwis of the proposed physical model of samples composed of individual grains: ~

~

NW 6 n = 2 = 4 I-v* d0

~

( d”,)

=--(l-n)

;hpt6

[

1

_ _1_a_ _C _1 _- n 4 Dhp116

n (1.3-20)

From Eq. (1.3-20), the rate of saturation in a balanced state can also be determined for the zone of adhesion, depending on the prevailing tension and the physical parameters of the soil. According to the definition of the coefficient of saturation, this value can be achieved, if the volumetric moisture content is related to porosity:

As already mentioned in the state of dynamic equilibrium the tension on the surface of the water film balances the effect of gravity which is proportional to the elevation of the investigated point above the water table: (1.3-22) p = yy = h y . Substituting this value into Eqs (1.3-20) and (1.3-21), the vertical distribution of soil moisture belonging to the balanced condition ( W, or ajc) can be determined in the unsaturated zone, giving the correct form of Eq. (1.3-9):

(1.3-23)

where the constant Co is composed of the Cconstant of Eq. (1.3-17) and the specific weight of the water. It is necessary t o note that accepting Eq. (1.3-20) as a basic principle, any kind of y = f ( W) function will obviously satisfy the condition given in Eq. (1.3-8). This equation cannot serve, therefore, as a control and the reliability of the proposed relationship between tension and water content has to be checked by measurements. For this reason, measured data which gave the related values of porosity, effective diameter, shape coefficient, water content and tension were collected from the literature (Table 1.3-3). Following the structure of Eq. (1.3-20), some of the measured parameters were combined into one characteristic value ( Wy116/l - n ) and this is listed in the last but one column of the table. The fact that the Wy1/6 product calculated for a sample from several measurements (sometimes the number of data concerning one sample was as high as 9 or 10) shows only a random fluctuation, justifies the approximation used to characterize the relationship between the balanced tension and the thickness of the water film [Eq. (1.3-17)].

125

1.3 Dynamics of soil moisture

Table 1.3-3.

Data characterizing the relationship between moisture content and adhesion

I I

Dh

Porosity

Material

1

Effective diameter DhIcml

I 1 1 shape caW$'* cirt

1-n

Referenoes ~-

Glass beads Glass beads Sand Sand

0.37 0.40 0.38 0.36

7.26 x 4.36 x 2.00 x 10-3 1.20 x 1 0 - 2 3.16~ 2.60~ 6.87~10-~6.87~10-~

Glass beads

0.38

2.00 x 10-3

Sandy, clayey loam

N

0.G

N

6 x 10-5

1.20 x

N

6 6

7.6 x

10 6

0.130

Peck, 1966

16

1.676

Rose, 1966

8

10-2

-

Johnson etal.,

0.062 0.076 0.200 0.110

1963;

PrillUaZ., 1966

Sand Sand

0.467 0.382

6 . 8 0 ~ 1 0 - ~ 6.6 x ~ O - ~ 1.92~ 1.6 x

8.1 8.3

0.380 0.233

Wind, 1966

Coarse sand Sand Sand Fine sand Fine sand Loess Light clay Fine sand Sand

0.363 0.363 0.386 0.404 0.434 0.490 0.430 0.360 0.340

3.96~10-~ 2.30~ 1 . 4 5 ~10-3 6 . 0 0 ~10-4 6.22 x 10-4 9.30 x 10-5 6.70 x 10-6 8.00~10-4 3.10~10-~

10

Wesseling and Wit, 1966

6.0 x 10-3 4.16 x 1.4 x 10-3

0.107 0.167 0.208 0.243 0.276 1.600 1.170 0.381 0.129

Fine sand Sand separates Sand separates Sand separates Sand separatet Sand separates Sand separates Sand separates Sand separates Sand separates Sand separates Sand separates

0.460 0.440 0.440 0.450 0.440 0.410 0.430 0.360 0.360 0.360 0.340 0.340

2.57 x 10-4 2.60 x 1 0 - 4 4 . 6 4 ~10-4 8.18 x 10-4 1 . 1 10-3 ~ ~ 1.64~ 2.32 x 10-3 4.00 x 10- a 5.11 x 10-3 6.67 x 10-3 8.33 x 10-3 1.67 x

2.67~10-3 3.00~10-~ 6.60~10-~ 9.00~10-3 1.27x10-* 1.80~10-~ 2.66~10-~ 3.60 x 4.60 x 6 . 1 0 ~lo-* 7.60 x lo-* 1.60 x lo-'

Stakman, 1966

9 9 9 9 9

0.396 0.679 0.268 0.238 0.179 0.163 0.140 0.109 0.097 0.096 0.064 0.046

Sand

0.372

2.76 x

2.20 x

8

0.148

Kastanek, 1971

Very fine sand Fine sand

0.438 0.438

6.80 x 1.45 x

6.80 x 10-4 1.46 x

10 10

1.207 0.961

Sand

0.420

1.45 x 10-3

1.76 x

12

0.369

Vachaud, 1966 Boreli and Vachaud, 1966 Vachaud et a1 ,

Sand

0.400

2.48 x 10-3

2.48 x

10

0.160

Vachaud and Thony, 1971

Silty loam

0.6000.678 0.440 0.470 0.390

6.30 x

7.67 x

12

Elric, 1966

9.98 x 8 . 6 4 ~10-5 8.21 x 10-5

9.98 x lo-' 8 . 6 4 ~10-4 8.21 x 10-4

10 10

1.761.94 1.143 1.396 1.041

10-4

4.0 X ~ O - ~ 1.8 x lo-* 1.2 x 1 0 - 2

1.0 x 1 0 - 3 8.0 ~ 1 0 - 3 3.1 X ~ O - ~

8 8 8 8

16 16 10 10 10 12 11 11 11 11 11

1974

Sandy loam Silty loam Fine sandy loam

10

126

1 Fundamentals for the investigation of seepage

Table 1.3-3. (cont.)

Sand

0.380

2.75 x

2.20 x

Sand

0.29

1.91 x 10-3

Send

0.36

Send Loess

0.36 0.40

8

0.113

Abmmova et al., 1966

2.10~ 10-2

11

0.148

1.91 x

2.10 x 10-2

11

0.176

Watson and Whialer, 1969 Wetson, 1967

1.79~ 1.22~

2.16~ 1.46~

12 12

0.177 0.217

Zotter, 1976

Continuing the theoretical analysis, the constant of Eq. (1.3-23) can be expressed 88 the function of the physical parameters of soil (Dheffective diameter; a shape coefficient of grains; n porosity). This theoretical relationship agrees well with the empirical result gained from the measurements (Fig. 1.3-12).

Fig. 1.3-12. Relationship between grain diameter and the parameter including moisture oontent, tension and porosity

1.3 Dynamics of soil moisture

127

When the data from Wy1/8/1- n in Table 1.3-3 was previously analyzed as a function of the effective diameter (KOV~CS, 1968), only some discrepancy waa found between the theoretical and the measured data. In theory, the power of D , is 1 but the experiments gave a value of 0.8. This discrepancy was explained by the fact that the shape coefficient was neglected. For this reaaon, some further data with known shape coefficients were collected, and in other cases the parameter waa estimated on the bwis of the physical description of the samples. According to Fig. 1.3-12, the theoretical curve lies along the measured points, although the scattering of the points is considerable. It is acceptable, therefore in practice, to substitute the theoretical relationship with an equation having a more simple form (which is represented in Fig. 1.3-12 by a dotted line): 1/6

wy = 2 . 5 ~ 1 0 - ~ 1-n

. ..

valid if 4~

Dh << 2~

a

(1.3-24)

To indicate the reliability of this relationship, the curves lying 30 % above and 30% below the original curve were also represented in the figure. They prove that by using a numerical factor in Eq. (1.3-24) ( 1 . 7 5 ~ or 3.25 x the probable extreme position of the soil-moisture retention curve can be determined with a probability of 90% in the zone of adhesion (only 4 data have larger scattering than *30% from the total number of 44). The indicated width between the limits of confidence is relatively low if the uncertainties influencing the process are considered. In the empirical formula, the numerical factor and the limits of the validity zone have certain dimensions, and the effective diameter has, therefore, to be substituted in cm to obtain the tension expressed by the height of the equivalent water columns also measured in cm. On the basis of the equations given, the position of the upper part of the soil-moisture retention curve can be determined, knowing the physical soil parameters of the sample or alternatively these parameters can be calculated from the measured upper section of the curve. Considering that two unknown parameters (n and the ratio of D&) have t o be determined, it is sufficient to have two measured points, or if more, there is a possibility of a control as well. The number of measurements necessary, immediately suggests the use of Kuron’s and Mitscherlich’s hygroscopic moisture content to characterize the upper part of the curves. Knowing that the suctions created by sulphuric acid (ensuring a constant temperature of 20°C) in concentrations of 10 and 50% are pF=4.62 and pF=6.16, respectively, and the measurement of the water content belonging to these conditions is accurate and easily repeatable, the curve and the characteristic physical parameters of soil can be measured using simple instruments. For a control, the use of concentrations of 20 or 30% can be proposed, which give points at pF = 5.28 or 5.62, respectively. Chemicals other than sulphuric acid can similarly be used. The relationship between tension and moisture content for the three

128

1 Fundamentals for the investigation of seepage

Table 1.3-4. Interrelated p F values and moisture contents determined as hygroscopic moisture in the zone of adhesion (after Zotter, 1976) p P value

0 0.4 1.0 1.6 2.0 2.3

I-

Measuring method

Box with sand plate

Box with kaolinite plate

2.7

3.4 4.2

High pressure equipment

Measured soil moistum in cluy

80.7 74.1 74.1 71.0 68.4 66.0 63.1 62.6 36.2

I

loess

1

send

46.64

31.8

-

30.9 30.1 28.3 20.1 6.9 4.0 2.9

-

39.67 32.81 28.77 16.3 11.6

Exsiccator Applied chemical

Relative moisture content of air

[%I

materials (the p F curves of which are represented in Fig. 1.3-7) was determined in this way and the results are listed in Table 1 . 3 4 (see also PBczely and Zotter, 1973). There are a few cases where the curve described by the equations previously listed remains valid in the whole zone of 0 < W < n. Examples of these are of very fine grained samples having a monotonously decreasing pF curve (see Fig. 1.3-6b). The intersection of the y = f ( W ) curve with the vertical W = n indicates the limiting value of tension ( y , ) . If the suction caused by gravity (hy,) is smaller than p , = ynyu;( h < y , ) , the soil remains saturated because the tension (even in the centre of the pores) is greater than the suction created by the weight of the water column:

In a layer composed of coarsergrains (which is generally the characteristic condition) complete saturation can alao occur in the cwe of higher suction than the limit given in Eq. (1.3-25). The new upper limit of this saturation

129

1.3 Dynamics of soil moisture

is the height of the water column equal to that of the closed capillary zone. The physical cause of this phenomenon is the existence of another molecular force acting against gravity i.e. capillarity. The character of this force is well known. Attraction between water molecules is balanced in the interior of the medium, because the same forces act on a molecule from each direction. The unbalanced mass-attraction caused by the asymmetrical position of water molecules at the surface, however, creates stress (surface tension), which becomes observable in the form of a curved surface where the water surface is in contact with the solid wall. If the horizontal section of the container of water is small enough, the unbalanced surface stresses around the solid wall can be summarized in the form of suction acting against gravity and raising the water above the water table of zero pressure (i.e. capillary suction). For the numerical characterization of the capillary force (the detailed analysis of which is given in Section 2.1.1) the capillary height is generally used, which is linearly proportional to the surface tension (u) and inversely proportional to the horizontal size of the capillary pore (in the case of a circular cross section, to its diameter). Both the surface tension and the coefficient of proportionality depend on the nature of the materials in cohtact. Thus, in the case of the contact of quartz, water and air as solid, liquid and gaseous media respectively, (and this is the general case in soils), the capillary height in a tube of d diameter can be calculated from the following simple equation [neglecting the dependence of surface tension on temperature and using average parameters acceptable in the most probable range of water temperatures ( 10-20°C)]: 0.30 [cm2] (1.3-26) hc [cm] = -d [cml Substituting the probable small and large pore sizes [d, and d2 on the basis of Eq. (1.2-21)] as well as the mean diameter [do from Eq. (1.2-19)] into Eq. (1.3-26), the maximum, minimum and average expected capillary heights can be calculated: Height of the open capillary zone:

.

h,,,,

0.30 =--0.11--.

1-n

a Dh

dl

Height of the closed capillary zone:

h,

=

-0.30

1-n u 0.06 -_ . Dh

d2

Average capillary height: 0.30

1-n

u

hCo= -= 0 . 0 7 5 ~ -

.

(1.3-27)

n Dh do Stakman’s air bubbling pressure data can be used to check the reliability of Eq. (1.3-27) (Stakman, 1966). The measured data (already represented 9

130

1 Fundamentals for the investigation of seepage

in Fig. 1.2-26), and the probable maximum and minimum capillary heights calculated from the physical parameters of soil me compared in Table 1.3-5. Considering Whisler and Bouwer's (1970) results which showed that air entry pressure is roughly equal to half the minimum 'capillary height, the Calculated parameters seem to agree well with the meaaured values. Table 1.3-6. Comparison of Stalnnan's air bubbling data and the probable minimum and maximum capillary heights calculated from physical parameters of soil Physlcal parameten ot soil

D~[cm]

3 . 0 0 ~10-3 6 . 6 0 ~10-3 9.00 x 10-a 1.27 x 1.8OX 10-2 2.66 x 3.60 x lo-* 4.60~ 5.10 x 7.60 x 1.60 x 10-l

I

a

n

12 11 11 11 11 11 9 9 9 9 9

0.44 0.44 0.46

0.44 0.41 0.43 0.36 0.36 0.36 0.34 0.34

I

[cm water] sampleheight lmm asmm

I

160 76 60 39 27 16 13 9 7 6 4

190 126 88 66 54 39 23 18 16 13 6

I

Minimum

306 129 90 66 67 34 28 21 20 14 7

I

hlaximum

660 337 164 121 97 63 51 38 36 26 13

The dependence of the probable maximum and minimum cupillary rise on porosity wm previously investigated in a more detailed form (Smith et al., 1931; Smith, 1933). The theoretical basis of his analysis was built on the probable structure of spherical particles with different porosities. The experiments were carried out with glass beads and sand separates using various fluids. For this reason, the results were expressed as a product of capillary height, grain radius (Rh)and the specific weight of the fluid ( yr) divided by surface tension (u),the latter parameter being related to porosity. The same parameter can also be expressed from Eq. (1.3-27) as the function of porosity ( n ) and the shape coefficient ( a ) : yf

hc max Rh,., 0.75 a 1-n, . _ Yf hCrnI"Rh _ _ U n a

~

0.40 a

1-n

n

.

(1.3-28)

Smith's measured data and the results of his proposed equation- are compared to the probable maximum and minimum capillary heights calculated from Eq. (1.3-27) in Fig. 1.3-13. The comparison proves that the dependence of the capillary height on porosity and shape coefficient is adequately considered in Eq. (1.3-27). The minimum value isin good accordance numerically with the observed data while the calculated maximum capillary rise 20% than the measured one. This difference is seems to be higher by 10 regarded aa acceptable, considering the uncertainties of the observation of the actual capillary height in a granular sample.

-

131

1.3 Dynamics of soil moisture

curves calculatedusing

+ I

0.33

035

037

039

0.41

043

0.45 .33

035

I

037

I

I

l73g

0.41

Oh3

*

0.45

porosity> n porosity, n Fig. 1.3-13. Comparison of Smith's data and capillary height calculated from Eq. ( 1.3-27)

The size of the pores can be considered as a random value. A given probability can be attached, therefore, to both d , and d,, while the ratio of the various pore-sizes to the average value can be characterized by a probability distribution curve. From the measured values of the cross-sectional area of every pore in various samples a statistical analysis was made proving that the ratio of the individual areas ( f ) to the average value ( f o ) can be well approximated by a distribution function. The frequency distribution and its integral form determining the probability of the number of pores having a given pore-size is as follows:

r

and (1.3-29)

The particular character of the function applied to the description of poresize distribution can be expressed by considering special parameters : (a) The smallest possible area tends to zero, and thus the lowest limit of the distribution curve is zero (xo70); (b) T h e random variable is the ratao of the area of pores related to their mean d u e (x = f / f o ) . The mean of this variable is, therefore, equal to unity ( G l e a n = 1); (c) I t follows from the previous condition that the two further parameters of the functional are equal to one another (k/A= m; m = 1; and k = A). Hence, the final form of the frequency distribution function is

9*

132

1 Fundamentals for the investigation of seepage

where

x = f/f,.

(1.3-30)

I n this special caue, direct relationships exist between the I and k parameters of the distribution function and the m and u parameters of the normal distribution

r

Thus, not only the author’s own measurements (four different samples and from each more different packing, namely, 7, 6, 8 and 4 measurements respectively) were available for further analysis but also the data of a previous publication which summarizes the mean value and the variation in the pore-size distribution measured on seven different samples and approximated with the normal distribution function (Murota and Sato, 1969). The graphical representation of the measurements in the I , vs. n coordinate system (Fig. 1.3-14) and the statistical investigation of the data have shown that the mean value of I , is 1.14 with a variance of o = 0.42 (this variance being 37 %). Considering the observed relationship between porosity and the 1, parameter, the variance can be slightly reduced (a, = 0.34), proving that the use of the equation determined by regression analysis

I,

= 2.19 - 3.75n & 0.34

(1.3-31)

can eliminate only about 20 yoof the uncertainties. The scatter of the points belonging t o the same sample with different porosities indicates that the remaining variance is caused mainly by sorne physical parameter of the soil not considered in the investigation (lines joining the points have similar slopes

4l 2.3 -

15 -

4.g -

c.5-

0-

Fig. 1.3-14. Relationship between porosity end the 1, parameter of the I’distribution function

1.3 Dynamics of soil moisture

133

to each other in the direction of increasing porosity and intersect the vertical axis at different heights). Considering the relatively small discrepancy between I, mean and unity and also the position of the line representing Eq. (1.3-31) in the range of the most probable porosity (0.3 < n < 0.4), the simplification by applying the I , = 1.0 value seems to be acceptable. In this case the r distribution function is reduced to an exponential function:

I(1.3-32)

and

Representing the P ( z ) functions belonging to ‘various I, values (0.5; 1.0; 1.5 and 2.0, Fig. 1.3-15), it can be seen that the probability of pores having a diameter of d , is between 20 % and 30 %, while the same parameter characterizing the pores with a diameter of d, is 80%. Thus, half of all the pores are those having diameter between the small and large diameters calculated from Eq. (1.2-12), the sample hawing about 30% smaller and 20% larger pores than those represented in the hydraulic model. Knowing the probable number of pores of various sizes, the total area of the pores having a smccller area than a given limit can easily be calculated. within a unit section of the sample can be The total number of pores (N) divided into m groups, according to the size of the pores. The basis of all intervak is df and the average area of the i-th interval (in which there are m

N I number of pores) is f i ;

2 N i= N lkl

1

.The specificfrequency distribution

can be characterized by dividing the number of the pores in each interval by the total number of the pores. When approximating the experimental

Fig. 1.3-16. Probable distribution of the number of pores as a funct)ionof the specific surface

134

1 Fundamentals for the investigation of seepage

frequency distribution by continuous function, the parameters of the latter can be calculated as the area above the horizontal axis and below the graph which is equal to the area below the curve (Fig. 1.3-16):

mAf Fig. 1.3-16. Theoretical sketch represent.ing the frequency distribution of the pore size

and

-

[ v(f; df)df = Of where

0

(if d f

-

0);

(1.3-33)

and v ( f ; df)is the continuous approximation of the frequency distribution. If the df interval is small enough, the total area of pores (which is equal t o the areal porosity where the total cross section of the sample is unity) can be expressed as the sum of the fiNi product:

1.3 Dynamics of soil moisture

135

therefore (1.3-34)

and

-

J ’ f v ( f ; A f ) d f = 1N2 A A f = f O A f ; 0

because the areal porosity can also be determined as a product of the total number of pores and their mean value (nA= Nf,). Further investigations can be simplified by introducing dimensionless variables. This goal can be achieved if the variables having an areal dimension are divided by the mean value of the measured pore sizes

The function describing the frequency distribution of the number of pores of different sizes has also to be transformed using the E(z;As)function instead of v(f; df). Introducing the new variables, the integrals in Eqs. (1.3-33) and (1.3-34) are as follows:

-

J E(s;

dx)dz = AS;

(1.3-35)

0

~ z [ ( z ;AX)& = AS.

0

Considering the results of the statistical analysis of the measured data, and accepting the simplified form of the distribution functions [see Eq. (1.3-32)], the mathematical description of the E(z;dz) function satisfying the conditions specified in Eq. (1.3-35) can be given in the following form: t ( z ; Ax) = Ax exp (-x)

.

(1.3-36)

Investigat,ing the rate of capillary saturation of a sample, the network of pores can be substituted b y vertical straight capillary tubes as a fist approximation. The distribution of the number of tubes with different diameters is the same as that determined by Eq. (1.3-30) [or by its simplified form - i.e. Eq. (1.3-32)]. In this case at an elevation of h, above the water table all the tubes having smaller cross-sectional area than f, are completely saturated, while the larger sections are sled with air. The rate of capillary saturation can be determined, therefore, by relating the sum of the area of pores having a smaller area than fc to the total free area in the cross section of the sample:

where

136

1 Fundamentals for the investigation of seepage

Substituting Eq. (1.3-36) and integrating the rate of capillary saturation is expressed as the function of the limiting vdue of the pore size:

+

(1.3-38) = 1 - (fdfo 1) exP ( I - f J f o ) * The ratio between the cross-sectional areaa of two circular tubes is equal to the square of the rate of their diameters. Thus, the coefficient of capillary saturation can be expressed aa the function of a given limiting value of the diameter related to the average pore diameter: sc

because

sc = 1 -

+ 11exp [-(dJdo)21;

[(dJdo)a

(1.3-39)

fdfo= (dc/~o)a. The soil-moisture retention curve describes the relationship between suction (elevation above the water table) and the water content. If it is divided into two parts (i.e. the rate of capillary and adhesive saturation, respectively) and only the effect of capillarity is investigated, the interrelation indicates which part of the pores can be saturated by capillarity at a n elevation of hc. This is the ratio of pores having a capillary height equal to or greater than he [see Eqs (1.3-12) and (1.3-13)]. Considering Eq. (1.3-26), this ratio depends only on the pore diameter and, therefore, the capillary component of the soil-moisture retention curve can also be approximated by a probability distribution function (Rbthgti, 1960). The distribution function describing the relationship between the rate of capillary saturation and capillary height can be characterized mathematically by applying a further transformation in Eq. (1.3-39). Considering that the capillary height is inversely proportional to the pore diameter, the ratio of the limiting diameter to the average value can be substituted by the quotient of the average capillary height and the height of the investigated elevation above the water table: 8,

because

=I1

-

+ 11exp [ - ( ~ c d h J 2 1 ;

( 1.3-40)

-=dc hco do hc

The curves described by Eqs (1.3-38), (1.3-39) and (1.3-40) arerepresented in Fig. 1.3-17. The figure shows that among the three capillary heights calculated from Eq. (1.3-27) one belongs to the s = 0.5 value of the rate the second to s = 0.01 0.05 (h, while the third of saturation (he (hcocalculated from do average pore diameter) gives the position of the capillary component of the soil-moisture retention curve where the rate of saturation is equal to s = 0.25. The basis of the previous derivation is a model composed of straight capillary tubes having different diameters, though the diameter of each tube is constant. The network composed of the pores is, however, a system of channels with a changing cross section and not that of straight pipes with constant diameters. The capillary height is influenced, therefore, not only by the

-

1.3 Dynamics of soil moisture

137

Fig. 1.3-17. Distribution of pore size, pore diameter and capillary height determined by applying a model composed of capillary tubes of constantfdiameters

distribution of the pore sizes in a horizontal section but by the vertical change of the pore diameters as well. For this reaaon some pores are not saturated when the dry sample is made wet from the direction of the water table, while the same pores can retain capillary water when the process starts with the complete saturation of the sample and dynamic equilibrium is achieved by drainage. This phenomenon is the hysteresis of the soil-moisture retention curve (the nearly horizontal section of the curve haa a lower position if i t is determined by wetting and a higher one in the case of drainage). In the literature, the phenomenon of hysteresis is generally explained by two different physical causes (Bear, 1972). The f i s t possible reason may be the rain drop egect, according to which the angle of contact of the meniscus with the wall of the capillary tube (8,) is larger when the water advances and the decreme in the in the tube than when in a static condition (8,) angle can be observed (0, < 0,)in the case of a receding water column (see Fig. 2.1-17). The same difference in the contact angles can be observed between the front and the rear of a rain drop moving along a sloping glass plate (the name of the phenomenon is based on this similarity). The water level may remain a t a higher position when a capillary tube with changing diameter is drained, compared to the rising condition, because the downward movement can be stopped when the meniscus reaches a small cross section (where capillarity can balance the weight of a relatively higher water column), while, similarly, the rise of the meniscus stops at a large diameter. This phenomenon is called the ink-bottle egect, because in old ink bottles this effect was used to keep the level of ink near the mouth of the bottle. From the two possible causes, the rain drop effect cannot be accepted as an explanation of hysteresis. The soil-moisture retention curve represents the vertical moisture distribution in a static condition. On the contrary, the

138

1 Fundamentals for the investigation of seepage

rain drop effect is a process occurring only in the case of flow in capillary tubes. Thus, a phenomenon characterizing the movement of water cannot be the cause of a phenomenon (the two possible positions of the retention curve) which describes the static condition of the moisture distribution.

i Fig. 1.3-18. Distribution of capillary height in the case of a bundle of identical wavy capillary tubes

For this rewon, the change i n diameter of the chccnnels will be used to explain the existence of hysteresis. It is necessary, therefore, to use a model composed of tubes, the longitudinal section of which is represented by two undulating lines having a minimum distance of d,, a maximum one of d2, and a wave length of 2a (Figs 1.3-18 and 1.3-19), instead of the straight capillary tubes mentioned previously. Even if the wave length is constant (all the tubes are identical with each other as in Fig. 1.3-18), the pore-size distribution previously described in a horizontal cross section can eaaily be ensured by placing the sections of tubes of different diameters at the level of the water table. The same result can be achieved with the other model, where the amplitude (the fluctuation of the diameter of the tubes) is the same as in the previous case, but the wave length varies from tube to tube (the parameter of a is also a random variable, described by a distribution function; Fig. 1.3-19). Investigation of the model represented in Fig. 1.3-18, shows that the maximum capillary height (belonging to diameter d,) can never be achieved by wetting the tubes from the direction of the water table, if the difference between the maximum and minimum capillary rise is greater than the wave h, < (h, min 2a), all the tubes length. This is because in the zone of h, and the have a section of diameter d, whose capillary rise is equal to h,

<

+

1.3 D y n d a of soil moisture

139

t

wetting

draining Fig. 1.3-19. Distribution of capillary height in a group of different wavy capillary tubes

rising water, therefore,cannot surpass the level of the cross section of diameter d2. The opposite process occurs in the system when it is saturated and the dynamic equilibrium is achieved by drainage. I n this caae, the capillary level should be i n the zone h, > h, > (h, - 2a) because all these tubes have a section of diameter d,, where the capillary tension can balance a water column of 11, max. In the case of the more realistic model composed of diflerent wavy tubes, the wave length is a random variable (Fig. 1.3-19). In investigating the wetting process of the sample, the water can be raised by capillarity to the level of h, max in some of the tubes [the wave length of which is greater than the difference of (h, max - h, mln) and if the capillary rise is unaffected by a large section of the tube below the maximum level]. In the tubes having shorter wave lengths, the capillary rise is limited to the level (h, mln+2a)tm described in the previous' paragraph. If the wave length is long enough but has a large diameter below the highest capillary rise, the elevation where the weight of the water column is balanced by capillarity depends on the diameter at this level. In this case, the average (mostprobable) capillary height is between the minimum capillary rise and the level characterized by the sum of the former and the most probable wave length (2a). Only the vertical shifting of the nearly horizontal stretch of the soilmoisture retention curve can be explained by the ink-bottle effect. The

1 Fundamentals for the investigation of seepage

140

observations have shown, however, that the y vs. W relationship is not a single-valued function in the zone of adheeion either. One of the remons for the different water contents at a given elevation above the water table is the d e r a b l e time lug between the time points of the change of the position

(b)

10

-

0

I

0

1

0.1

1

I

I

0.2

1

I

0.3

water content, W Fig. 13-20. The development of the soil-moistureretention curve in time

of the water table and the development of the dynamic equilibrium. Figure 1.3-20 .

shows the refjults of an experiment illustrating the observation of the process in time. The data indicate the extremely slow development of equilibrium in the adhesive zone when the column is drained, viz. in this case, the upper part of the sample has very small hydraulic conductivity because of the low

141

1.3 Dynamics of soil moisture

water content, and thus, the excess water percolates downwards with a very small velocity. The wetting is generally a faster process, but a considerable difference can be observed between the initial wetting (when the process starts with a completely dry sample) and rewetting (when the sample waa previously drained and some soil moisture was retained in the pores). In the first caae, the development of the dynamic equilibrium in the adhesive zone has, perhaps, a greater time lag than that of the column drainage. Thus, (b)

variation of moisture content

v s. sucfion bead relatiunship with

time on the basis of fteld measurements at a given depth (Bourm, z = 40 cm)

0

0.l0

0.20

0.30

water confeflf, WI

1 '

0

0.1

water

I

I

0.2

a3

content, W

Fig. 1.3-21. Characterization of hysteresis by laboratory and field meaaurementa (after Vachaud and Thony, 1971; Royer and Vachaud, 1975)

in many cmes the difference between the water contents observed at the 8ame elevation in a wetted and drained column respectively can be explained by the insufficient length of the observations. The horizontal shifting of the vertical stretch of the retention curve is, however, also indicated by data determined directly from simultaneous measurements of tension and soil-moisture values (Fig. 1.3-21). In this case the corresponding data are not influenced by the time-dependent develop- . ment of the dynamic equilibrium. The cause of hysteresis is the water retained in the corners of the pores after draining. The explanation of the phenomenon of hysteresis indicates many uncertainties influencing the form of the soil-moisture retention curve and cause its multi-valued character. It is not expected, therefore, that the history of

142

1 Fundamentals for the investigetion of seepage

Toucbetsilty loam

a

az

0.4

06

I

1

1

1

02 n.4 06 .a3 Fig. 1.3-22. Comparison of the theoretical soil-moisture OR s

{a a

1 , -

1.3 Dynamics of soil moisture

0

0.4

D.6

143

00 s lo 0

3.0h -

3.0.

(h) sand

nco 2.5 n

=

o m

n

= 0.346

2.0 -

1.5 -

1.0 -

0.5 0

P

D-

o

02

04

06

20

retention curve with measured data

s

ID

o

02

ao

05

JB s

/c

144

1 Fundamentals for the investigation of seepage

the wetting and draining processes can be followed theoretically. The random position of the retention curve can be characterized only by comparing the calculated average curve and measured data. For this comparison observations published by three different laboratories supplemented by the author’s own measurements were used (Figs 1.3-22a and 1.3-22b, Johnson et d., 1963; Prill et al., 1965; Figs 1.3-22c and 1.3-22d, Ldiberte et al., 1966; Figs 1.3-220 and 1.3-22fJ Vachaud et aZ., 1975; Vauclin, 1975; Khanji, 1975; Vachaud et al., 1970; Vachaud and Thony, 1971; Figs 1.3-22g and 1.3-22h the author’s own measurements). The measured data prove the reliability of the theoretical relationships [Eqs (1.3-24) and (1.3-40)] and considering the composition of adhesive and capillary retentions (Fig. 1.3-23), the final form of the equation describing the average retention curve is as follows:

(1.3-41) To characterize the probable zone of hysteresis, it can be proposed that separate enveloping curves should be applied along the capillary and adhe-

Fig. 1.3-23. Composition of soil-moistureretention curve from adhesive and capillary retention

145

1.3 Dynamics of soil moisture

sive stretches respectively. The scattering of data in the zone of adhesion has already been discussed in connection with Eq. (1.3-24). Considering uncertainties explained there, the enveloping curves can be given in the following: form: ( 1.3-42)

In the capillary zone the enveloping curves can similarly be given by using a multiplying factor in the distribution fnnction which characterizes the rate of capillary saturation

sc2= 1 -

[i z I : 1.5-

+1

1

[[

exp - 1.5-

,

These curves are also represented in Figs 1.3-22g and 1.3-22h. Their position proves that Eqs ( 1 . 3 4 2 ) and (1.3-43) are suitable to characterize approximately the expected scattering caused by hysteresis. References to Chapter 1.3 BEAR,J. (1972) Dynamics of Fluids in Porous Media. Elsevier New York, London, .Amsterdam G. (1966): Certain Problems of Infiltration in Unsaturated BORELI,M., VACHAUD, Porous Media. Saopstenja Inetittda za VodorpriiVp.eduJaroslav Cerni,N o 38. BncKwaHAM, E. (1907): Studies in the Movement of Soil Moisture. USDA Bureau of So& Bulletin, No. 38. L)E WIEST,J. M. (1969): Flow through Porous Media. Academic Press, New York, London. DoLaov, S. I. (1948): Investigation of the Movement and the Availability of Soilmoisture (in Russian). Izd. A d . Nauk SSSR, Moscow, Leningrad. ELRICK, D. E. (1966): The Microhydrologic Characterization of Soils. I A S H Symposium on Water i n the Unaaturated Zone, Wageningen, 1966. FEKETE, Z. (1950): Comparative Investigation of Parameters Characterizing the Water Balance of Soils (in Hungarian). Hidroh5qiai Kozlony, No. 7-8. IASH (1974): Hydrological Investigation of the Unsaturated Zone (Second circular). Budapest. JOHNSON, A. I., PRILL,R. C. and MORRIS, D. A. (1963): Specific Yield-column Drainage and Centrifuge Moisture Equivalent. Qeologiccrl Survey Water Supply Paper, 1662-A. JwAsz, J. (1967): Hydrogeology (in Hungarian). Tankonyvkiad6, Budapest. KASTANEK,F. (1971): Numerical Simulation Technique for Vertical Drainage from a Soil Column. Journal of Hydrology. p. 213. KHANJI,J. D. (1976): Investigation of the Recharge of Ground Water with Free Water Table by Infiltration (in French). Doctoral Thesis Univereity o f arenoble. 10

146

1 Fundamentals for the investigation of seepage

CUTE, A. (1962): Some Theoretical Aspects of the Flow of Water in Unseturated So&. Proceedings of American Sod SciencS Society, Vol. 16, No. 2. KOVAOS,G. (1968): Cheraoterization of the Moleoular For- Muencing Seepage with the He1 of the p F Curve. Agrokhia b Tdajtan, (Supplementum). KOVAOS, (19718): Seepage through Unsaturated Porous Media. 14th Uongrei?s of IAHR, Pa&, 1971.

Cf

KOVACB, G. (1971b): Seepage through Saturated and Unsaturated Layers. Bdletin of IASH, No. 2. KREYBIU,L. (1961): Heat- and Water-balanoe of Soils (in Hungarian). Budapest. KWON, H. (1932): Characterization of SoilMoisture (in German). Zeitschrijt fiir Pflanzen, Berlin. LALIBERTE,G. E., COREY,A. T. and BROOKE,R. H. (1966): Properties of Unsaturated Porous Media. Hydrology Papers, Colorado State University, November, No. 17. LEBEDIEV,A. F. (1963): Sog Moisture and Ground Water (in Russian). Izd. Akad. Nauk SSSR, Moscow. LUTZ,J. F. (1966): Proceeding of American Soil Science Society, pp. 330-361. MADOS, L. (1939): Studies Related to Irrigation and Water Management in the Irrigation Scheme of T i e z a f i i (in Hungarian). hiidaiigyi Kiizikrnknyek. MADOS, L. (1941): Use of Soil Moisture Related to Surface Drainage (in Hungarian). c)ntGz&i&yi K6demdnyek. MITEIOHERLIOH, E. A. (1932): Soil Scienoe for Agriculture and Forestry (in German). Berlin. MUROTA, A. and SATO,K. (1969): Statietical Determination of Permeability by the Pore-size Distribution in Porous Media. 13th Congress of IAHR, Kyoto, 1969. P E ~A., J. (1966): Diffueivity Determination by a New Outflow Method. IASH Sympeeium on Water in the Uneaturated Zone, Wageningen, 1966. P~OZELY, T . and ZOTTEB, K. (1973): Summary of Inte mtation and Measuring Methods of p F Curve (in Hungarian). (Manuscript). V d , Budapest. PETERSON, A. E. and DIXON,R. M. (1971): Water Movement in Large Soil Pores: Validity and Utility of the Channel SMem Concept. Reaearch Report University of WieconSin, June, No. 76. A. J. and MORRIS,D. A. (1966): Specific Yield-laboratory PRILL, R. C., JOHNSON, ExDerimenta Showinn the Effect of Time on column Drainage. Geoloaical survew. "W&er Supply Paper,"l662-B. PROOEROV. A. V. and KARASEV.N. K. 11939): Land Reclamation and Water Control (in Russian): MeZior a i gidrotekhniki, No: 12. RbTHdm, L. (19603ngineering Aspects of the Capillarity of Soils (in Hungarian). Vcziisyi Kodemknyek. No. 1. RICEARDB, L. A. (1931): Capillary Conduction of Liquids through Porous Media. PhyeicS, Nov. RODIE,A. A. (1962): Soil Moisture (in Russian). Izd. Acad. Nauk SSSR, MOSCOW. ROSE, D. A. (1966): Water Transport in Soils by Evaporation and In6ltration. IASH S y m p o h m on Water in the Unsaturated Zone, Wageningen, 1966. ROYER,J. M. and VAOHILUD, Q. (1976): Field Determination of Hysteresis in Soil Science Characteristics. Proceeding8 of Soil Science Soc&ety of America, No. 2. SMITE, W. 0.(1933): Minimum capillary Rise in an Ideal Uniform soil. Phyeics, May. SMITE,W. O., FOOTE, P. D. and BUBANU, P. F. (1931): Capillary Rise in Sands of Uniform Spherical Grains. PhyeicS, July. STAKBUN,W. P. (1966): Relations Between Partiale Size and Pore Size and Hydraulio Conductivity of Sand Separates. IASH Symposium on Water in the Unsaturated Zone, Wageningen, 1966. SUDNITBIN,L. I. (1966): Soil-moisture Pressure in Some Climatic Zones. IASH Sympeeium on W&r in the Uneaturated Zone, Wageningen, 1966. VAOEAUD, G. (1966): Study on Redistribution after Stoppin Horizontal Infiltration (in French). IASH Sympoeium on Water in the Unsaturated &me, Wageningen, 1966. VACJUUD,G. (1966):Verification of Generalized Darcy's Law and the Determination of CapilLary Conductivity by Analysing Horizontal Infiltration (in French). IASH Sympoeium on Water in the Uneaturated Zone, Wageningen, 1966.

-

1.4 Balance of the ground water

147

VACHAUD, G., CISLER, J., THOM, J. L. and DE BACKER,L. (1970):Utilization of Gamma Emission of Americium-241 for Measuring the Soil-moisture Content of Unsaturated Soils (in French). Isotope Hydrology - 1970. I A E A . VACEAUD, G. and THONY,J. L. (1971): Hysteresis during Infiltration and Redistribution in a Soil Column of Different Initial Water Contents. Water Resources Reeearch. No. 1. VACHAUD, G., GAUDET, J. P. and KURAZ, V. (1974):Air and Water Flow during Ponded Infiltration in a Vertical Bounded Column of Soil. Journal of Hydrology, NO. 1-2. VACEAUD, G., VAUCHIN, M. and HAVERKAMP, R. (1976):Towards a Comprehensive Simulation of Transient Water Table Flow Problems. Seminar on Modeling and Simuldion of Water Reaourcea System, 1975. VAN BAVEL,C.H.M. (1966):Three-phase Domain in Hydrology. IASH Symposium on Water in the Uneaturated Zone, Wageningen, 1966. V-YAI, G. (1942): Experiences in Irrigation a t MAtraliget (in Hungarian). C)ntoz&igyi Kodemdnyek, No. 2. V ~ L L Y A G. I , (1974):Investigation of Water Movement Developing in Unsaturated Soil Layer (in Hungarian). Agrokkmia 6s Talajtan, Vol. 23, No. 3-4. VAUOLIN, M. (1976):Experimental and Numerical Inveatigation of the Drainage of Ground Water with Free Water Table. The Infiuence of the Unsaturated Zone (h French). University of Grenoble (Dootoral Thesis), A ril VTJ~IC, N. (1966): M u e n o e of Soil Structure on h t r a t i o n and p P Values of Chernozem and Chernozem-like Dark Meadow Soil. IASH Symposium on Water in the Uneaturated Zone, Wagewingen, 1966. WATSON,K. K. (1967):Experimental and Numerical Study of Column Drainage. P r o d i n g s of ASCE, ELY.2. WATSON,K.K. and WHIBLER, F. D. (1969):Analysis of Infiltration into Draining Porous Media. Proceedings of ASCE, IR.4. WESSELINU, J. and WIT, K. E. (1966):An Infiltration Method for the Determination of the Capillary Conductivity of Undisturbed Soil Cores. IASH Symposium on Water in the Unaaturated Zone, Wageningen 1966. WHISLIER,F. D. and BOUWER,H. (1970):Comparison of Methods for Calculating Vertical Drainage and Infiltration for Soils. Journal of Hydrology. No. 1. WIND, G. P. (1966):Capillary Conductivity Data Estimated by Simple Method. IASH Sympoeium on Water in the Unaaturated Zone, Wageningen, 1966. ZOTTER, K. (1976):An Instrument for Determining the Upper Stretch of the PP Curve (in Hungarian). (Manuscript). VITUKI, Budapest.

Chapter 1.4 Investigation of the balance of the ground-water space In the previous chapter the hydrological processes occurring in the soilmoisture zone were analyzed. It has been shown that infiltration and evapotranspiration through the surface (both modified by the change of storage in the unsaturated pores) affect also the regime of the gravitational ground water. The resultant of these influences can be summarized aa the water exchange between soil moisture and ground water (accretion). This combined effect can be divided into two vertical components. The first is directed downwards, and it is that part of infiltration which reaches the water table. The second component is the vertical natural drainage of the ground-water space through the water table, which raises water from a deeper posit,ion by capillarity and tension difference to replenish the storage capacity of the soil-moisture zone emptied by evapotranspiration. The interaction between the two zones in question can be investigated as a pure hydrodynamical process, when the instantaneous water transport has 10*

148

1 Fundamentals for the investigation of seepage

to be calculated, convidering the actual flow conditions. For such an analysis, detailed knowledge of the acting forces, as well as that of the resistance of the unsaturated porous medium is necessary and the prevailing boundary conditions have to be taken into account. These aspects will be summarized later on in this book; therefore, the hydrodynamic investigation of flow through an unsaturated zone will only be discussed after the explanation of all the necessary basic concepts. There is, however, a more general way of investigating the water exchange between soil moisture and ground water, when this process i s regarded as a part of the hydrological cycle, and its average numerical parameters are determined from the water balance equations of the relevant water bodies. This hydrological analysis provides important information also for solving numerous hydraulic problems; the average parameters obtained in this way can be used for checking the hydrodynamic calculations, st8 the amount of water transport integrated for a longer period has to be equal to the result determined from the water balance equations. Another important field of application of the hydrological parameters in seepage hydraulics is the investigation of the horizontal flow of the shallow ground water, when accretion influencing the flow field through the water table has to be considered. I n this case the average parameters calculated from the balance equations can be used as independent variables describing this special boundary condition. This close contact between hydrodynamic and hydrological investigations gives ground for including the hydrological analysis of the ground-water balance in a book dealing with seepage hydraulics, as it is done in this chapter.

1.4.1 General hydrological characterization of water exchange between soil moisture and ground water

The basis of the present investigation is the model of the soil-moisture zone represented in Fig. 1.3-1 ; the same symbols are used here as those indicated there (Kovbcs, 1959a, 1959b, 1971). Let us &st consider a limited soil column in the vicinity of the surface, where evapotranspiration can affect the gravitational ground-water space owing to capillarity and the suction of the roots of vegetation. The column includes a part of the ground-water space (below the phreatic ground-water surface) and the soil-moisture zone (both the unsaturated and the transition zones between the water table and the soil surface) (Fig. 1.4-1). The equation of continuity can be determined for this column by considering the recharging and draining processes. The amount of water contained within the investigated column is increased by infiltration through the surface ( I , ) . Evapotranspiration, which includes direct evaporation from the soil moisture (Em)and transpiration by plants (T,)causes losses in the column. Ground water may flow either in or out of the space observed, recharging or draining it (Diand Do). Horizontal moisture flux, either into

1.4 Balance of the ground water

149

Fig. 1.4-1. Sketches representing the components of the water balance equations characterizing the hydrological processes in the soil-moist,urezone

the column (Mi), or directed out of it ( M o ) ,is usually very small compared with the other recharging and draining processes; it is, therefore, neglected in the following discussion. The sum of water transported through the boundaries of the column is equal to the change of the water content stored within a given period (T) in the soil column. The latter includes the change i n amount of both soil moisture ( A W ) and ground water ( A V ) from the beginning to the end of t,he period (0 < t < T ) : T

2 [+ I,(t) + D,(t)- E,(t) - T,(t)

-

Do(t)]At = [-+AM'& AV],T. (1.4-1)

n

The members on the left-hand side of the equation are functions of time, and they are expressed the water amount related to the surface of the column and to the unit of time. Thus their dimension is equal to that of velocity [LT-11. Infiltration through the surface originates from precipitation, and it can be betermined by subtracting from the total precipitation both surface run-off dnd the amount of water evaporated directly from the surface, or retarded ay the vegetation. Apart from climatic parameters (amount and distribution

150

1 Fundamentals for the investigation of seepage

of precipitation, factors affecting evaporation), this value depends on the condition of surface (degree and direction of slopes, covering vegetation), as well as on the type and condition of soil influencing the process of infiltration. One part of the infiltrating amount is stored in the unsaturated zone (Iw) filling this layer up to its water retention capacity (or sometimes, temporarily, above this limit). The other part reaches the gravitational ground-water space ( I , ) , either during the rain, or with a time lag (Fig. 1.4-lb):

Irn(t) = Iw(t)

+ Ig(t).

(1.4-2)

As it has been mentioned, evapotranspiration is composed of direct evaporation ( E m ) and transpiration through plants (T,,,).Although both draining processes reduce the moisture content in the unsaturated zone, the total amount of water subtracted from the soil may be divided into two parts: the first is the prolonged decrease of the moisture content ( E , and T,, respectively), while the second part is replenished from the water table below; thus this amount ( E g T , = ET,) is a negative component of the ground-water balance (Fig. 1.4-lb):

+

Ern(t)

+ Trn(t)= E d t ) + Tw(t) + Eg(t) + Tg(t)= ETw(t) + ETg(t)* (1.4-3)

The amount of the soil moisture evaporated and transpired depends on the potential evapotranspiration (temperature, moisture deficit of air, wind velocity), on the parameters characterizing the evaporating surface (mostly conditions of vegetation, such as depth of the root zone, total surface of leaves, water consumption of the plants) and on the type and condition of the soil. The third group of the influencing factors is the ground-water flow rechargand outflow (Do) ing or draining the gravitational space. Both inflow (Di) depend on the hydraulic gradient developing in the vicinity of the column investigated and on the transmis8ibiZity (the product of thickness and hydraulic conductivity) of the water conveying layer. The two processes may by investigated jointly, by considering their difference:

*AD(t) = Dj(t) - D,(t).

(1.44)

Although the listed components of the balance equation are time-dependent variables, when the period investigated is long enough, their average multiplied by the length of the period can be used instead of their integrated value. _

m

1 "

- J E,(t) dt = Em ; T i 1 T -J T,,,(t) dt = F,,, ;

TO

1.4 Balance of the ground water

151

1 T J AD(t)dt = AD ;

TO

(1.4-5) 1 T J ET,(t) dt = EF,,, ;

TO

1 T J I,(t) dt = 1,;

TO

1 T

- J ETg(t)dt = ETg.

TO

Substituting these averages into Eq. (1 4-1), the simplified balance equation, valid for a long period, can be written in the following form:

[ ( I g- ET,)

+ ( I , - ET,)

f ADIT = [ f d V f AW):.

(1.4-6)

The two members on the right-hand side of the equation express the change of the water amount stored in the column between the beginning and the end of the period investigated, indicated by 0 and T s u f i e d outside the bracket. The first symbol (dV) refers to the change of the stored gravitational ground w t e r , while AW indicates the diflerence i n the soil moisture. These two values are closely interrelated, because if the change i n gravitational storage is calculated as the product of the change of the water level ( A H ) and t,he effective porosity of the layer ( n ) (Fig. 1.4-lc)

f AV = f d H n ,

(1.4-7)

this parameter includes a part of soil-moisture storage aa well, while the stored amount of soil moisture also depends on the position of the water table. According to its definition, the change of storage in the soil-moisture zone is equal to the difference of the amount of moisture above the water table at t w o subsequent points of time. Thus the change may be caused not oiily by the variation of the point values of moisture content, but also by the change of the position of the water table, because the thickness of the .soil-moisture zone changes in this case as well: 0

0

f A W = J J W ( t = 0 ) dA dm - J J W ( t = T)dA dm = W 1- Wz; (1.4-8) mi ( A )

m (4

where A is the area of the horizontal cross section of the investigated column (which is supposed to be unity) and dA is its elementary part. In most areas the water table shows a regular annual fluctuation, and generally still another one, with a longer period. Under natural conditions, Iwwever, a state of equilibrium can be expected, if the period considered is long

152

1 Fundamentals for the investigation of seepage

enough. (Here all effects caused by human activity, excluding direct artificial recharge and pumping, are regarded as “natural”; e.g. change of cultivation, control of surface run-off, etc.). This assumption is evidently correct, because these effects act continuously over a number of decades, even centuries, almost without m y considerable changes, thus providing enough time for the development of a dynamic equilibrium. The average elevation of the undisturbed water table is, therefore, at a level, where all the recharging effects are balanced by the various draining actions. The equlibrium of the natural influences (and those acting indirectly over a long period) can be characterized with a time-invariant average depth of the water table (mav). This is the reason why the hydrographs of such water tables (the graphs representing the change of the position of the water table in time) are generally closed, if the period is long and its limits were properly chosen. (The closing of the hydrograph means, that the initial and end points are at the same elevation). Assuming the soil column t o be i n a state of equilibrium, and supposing that the determination of the investigated period satisfies the conditions mentioned in the previous paragraph, the change of the amount stored i n the gravitational ground-water space is zero (AV = 0 ) , because the position of the water table at the end of the period is the same as it was at the beginning ( A H = 0). If the starting and closing time points are in the same season, the change of the stored soil moisture m y also be neglected (AW = 0). In this special case the water balance can be limited to the gravitational ground water below the average water table (mav)(Fig. 1.4-Id). The equation of ground-water balance [Eq. (1.4-1) or (1.4-6)] remains valid also for this case, but the recharge from infiltration refers to the amount of water which reaches the water table (I,), and not to that crossing the surface. Similarly, the amount drained from the system has to be considered as the water raised from the gravitational ground water (ET,) to replenish evaporated and transpired soil moisture, and not the total amount of the latter. This modified form of the balance equation expresses that the vertical water exchange between soil moisture and ground water is eqml to the total effect of ground-water flow (the difference of outflow and inflow), if the position of the water table is practically the same at the beginning and the end of the period used for the calculation of the averages of the parameters in question: (1.4-9) I , - ETg = Do - Di. There are many cases, however, when a shorter period has to be analyzed (e.g. when studying the seasonal fluctuation of water exchange between ground water and soil moisture) and, therefore, storage cannot be neglected. The investigation can yet be limited to the balance of the gravitational space in this case as well, but the changing volume of the investigated column has to be considered; this can be expressed in the form of the change of the stored amount of ground water : (1.4-10) ( I , - ET,) (Di- Do) f AVO = 0.

+

It is necessary to note here that the AVO value is not equal to A 7 in Eq. (1.4-7), because the latter gives a part of the total change of both ground

1.4 Balance of the ground water

153

water and soil moisture, which can be determined arbitrarily, considering that the calculation of the other part ( A W ) is suitable to supplement the first part and to give the correct amount of the total change. If the investigation is limited to the gravitational space, the physically correct interpretation of the storage capacity of the layer has to be determined. I n the literature the storage coeficient (storistivity, S ) or specific storage (S,) is generally used to characterize this property of confined aquifers, and the specific yield (n,) is the accepted parameter for unconfined layers. In addition to the topic discussed here, the characteristics listed have an important role in many other fields of seepage hydraulics. Their detailed analysis is given, therefore, in the next section, together with the methods proposed for their numerical determination. Supposing that the column, whose balance is investigated now, is separated from an unconfined aquifer, its storage capacity will be characterized by the specific yield (n,) of the layer in this analysis; thus the change of storage as a result of the modification on the position of the water table ( A H ) can be expressed as A V O = AHn,. ( 1.4-1 1) Finally, it can be stated that the balance equations describe the relationships between two [Eq. (1.4-9)] or three [Eq. (1.4-lo)] independent processes occurring in the gravitational ground-water space (i.e. vertical water exchange, horizontal flow and change i n storage). Owing t o the numerous uncertainties generally associated with the determination of hydrological parameters, i t is advisable to measure, or calculate, all the members of the balance equations separately, and to use the formulae only for checking the reliability of the measurements, or for determination of the closing errors (use of the implicit form of balance equations). It is also possible to express one member of the equation as a function of the others (explicit form),and calculate the former from the measured data of the known parameters. It is necessary, however, to consider in this case that all errors in the measured data are burdens to the calculated value. Considering the required implicit use of balance equations, the three processes listed previously will be analyzed separately in the following, starting with the storage in both confined and unconhed layers.

1.4.2 Storage capacity of aquifers The storage capacity of a layer is characterized i n general by the volume of water released from, or taken into storage as a result of a unit change i n pressure, or piezometric head. The volume is related to the horizontal arm of the layer, and thus it has a dimension of length [L]. This amount depends on - The compressibility of solid grains; - The compressibility of water; - The change of porosity of the saturated layer; - The change of the ratio of the saturated and unsaturated thicknesses. The f i s t component is very small in comparison with the others, therefore it is negligible. In confined aquifers (theoretically, only if it is covered with

154

1 Fundamentals for the investigation of seepage

aa absolutely impervious formation) the system will remain always saturated, thus the fourth effect does not influence the process. In the investigation of shallow ground water with a phreatic water table, the change of the thicknesses of the saturated and unsaturated zones has the most pronounced role. The effect of consolidation and water compressibility can therefore be neglected in this case. It is the reason why the characterization of a confined aquifer differs basically from that of unconfined water-bearing layers. In the first case the parameter defined previously is called storage coepient (S), and it d e p e d on the consolidation of the layer and on the cumpressibility of water. This value represents the total storage capacity of the layer having a thickness of b. Sometimes this parameter is related to the unit volume of the aquifer (water released from, or taken into storage in a volumetric unit of the layer). This specific storage (S,) multiplied with the thickness provides the storage coefficient:

S = bS,.

(1.4-12)

The specific yield (n,) characterizing the storage capacity of an unconfined aquifer is generally determined as the volume of water drained from a completely saturated sample related to the bulk volume of the latter. It is supposed that this parameter is equal to the amount released or stored in a column of unit horizontal cross section, if the position of the water table changes by a height of unity. The process occurs at the surface of the gravitational ground-water space, therefore, this parameter cannot be related to the thickness of the aquifer. According to its definition, the storage coefiient is composed of two parts: the change of the volume of water caused by the modification of pressure, and the change of the pore volume, where the water is stored:

s = s w + s,.

(1.4-13)

The compressibility of water is expressed by the change of density with pressure, which is proportional to the density itself (1.4-14) where B ( w e f i i e n t of compressibility) is the proportionality factor; for pure water having a temperature of 2OoC at atmospheric pressure, its numerical value is /3 = 4.6 x lo-* mm2/N.The maw being unchanged, the modification of V, volume of water can be expressed from Eq. (1.4-14):

The volume of water in a saturated layer is equal to the volume of the pores; thus in a unit volume of the aquifer it is equal to the numerical value of porosity. In a column of unit area of a homogeneous layer having a thickness of b, the total change of the water volume can be calculated con-

1.4 Balance of the ground water

155

sidering the total pore volume nb, because all the parameters are constant:

r31=

- Bnb ;

[dV,]:

= - Bnbdp.

(1.4-16)

The storage coefficient meaaures this change if the pressure difference, expressed in equivalent water column, is unity: dp/r = dh = 1 [L]. Combining this condition with Eq. (1.4-16), the first component of the storage coeflcient can be determined Sw = - - y n b .

(1.4-17)

The process of compression, which is well known in soil mechanics, may be used for the determination of the second component of the storage weflcient. For simplifying the mathematical description of the phenomenon, some approximative hypotheses have to be accepted. The first is the supposition of the linear vertical compression of the layer, neglecting the possible horizontal displa,cementof particles, aa well aa the shearing stress caused by unequal comprevsion along two neighbouring vertical sections. Loading the sample gradually, relationships between the load and various physical parameters of the soil (specific linear compression, dry specific

Fig. 1.4-2. Graphs representing the results of experiments investigating compression (KBzdi, 1972)

156

1 Fundamentals for the investigation of seepage

weight, water content, porosity, void ratio) can be observed and represented (Fig. 1.4-2) (KBzdi, 1972). (Itis necessary to note here that the pore volume is generally characterized in this investigation with the void ratio instead of porosity, because the latter is given aa the ratio of pore volume related t o the bulk volume, and thus compression modifies the basic value aa well; void ratio, on the other hand, is the ratio of pore volume and the volume of

I

-\

Fig. 1.4-3. Sketch showing the symbols used in the interpret,ationof the relationship between load and compression

solid matrix, the latter remaining unaffected by compression.) Terzaghi (1926, 1943) established an empirical relationship, according to which the change of void ratio over the change of load is inversely proportional to the value of load ( p ) increased with a surplus load ( p o ) ,determined experimentally (Fig. 1.4-3): Ae - de 1 -- - - = = a =

4

dP

C(P

+ Po)

(1.4-18)

*

The coefficient of consolidation a can be expressed as a function of the modulus of compressibility

(a):

(1.4-19)

dP Ah where M = - , and E = - is the relative change of the vertical length of de h the loaded sample (the interpretation of Ae and Ah, as well as a comparison of the two types of compression curves is shown in Fig. 1.4-4; some informative values of M are given in Table 1.4-1). The integration of Eq. (1.4-18) gives Z’erzughi’s equation of compression [Eq. (1.2-16)].

1.4 Balance of the ground water

157

Table 1.4-1. Informative values of the modulus of compressibility (KBzdi, 1972) M [kp/cm'I

I

Type of the sample

1

A) Coarse grained sPdiments

Sandy gravel Sand Very fine sand (mo)

Condition of the sample

I

IOOSe

300-800 100-300 80-120

I

(B) Cohesive sedimenls

Silty very fine eand silt Light clay Heavy clay Organic silt Organic clay Peat

800-1000 300- 600 120- 200

compacted

1000-2000 600- 800 200- 300

Condition of the sample

soft

1

I

plastic

60-80 30-60 20-60 16-40

100-160 60-100 60- 80 40- 70 6- 60 6- 40 1- 20

hard

160-200 100-160 80-120 70- 120

pressure, p [ k N / m 2 ] 200 300 400

500

pressure, p [ k N / m 2 J 200 300 400

500

400

0

I

medium

1

very hard

200-400 160-300 120-200 120- 300

0.8 t

0

m I

I

I

I

I

I

I

--+-+---Fig. 1.4-4. Comparison of the representation of compression as a function of the void ratio or the change of linear size

158

1 Fundamentals for the investigation of seepage

The next problem to be discussed is the way of application of the listed equations to calculate storistivity. I n a loose clastic aquifer the weight of the overlying layers (the latter related to a unit horizontal area is characterized with the total vertical stress a) is transferred only partly grain by grain (this part of the total stress is the eflective stress, a);the other part is balanced by water pressure (neutral stress, u). Creating depression ( A h ) by draining the layer, water pressure decreases, and therefore the effective stress will increase, while the total stress remains unchanged: UZ=U~-YA~; a = u1

+ al = ug+ ZZ;

(1.4-20)

+ ydh;

-

az = Z1

(sufbes 1 and 2 indicate the condition before and after draining, respectively). Eq. (1.4-20) shows that the decrease of the pressure head of water increases the eflective vertical stress and this change can be regarded as a n external load causing compression : d p = da = y d h .

(1.4-21)

According to the definition, the second component of the storage coeficient is the change of the total pore volume in a column having a unit horizontal area and a height equal to the thickness b of the layer, if the pressure head is modified with unit height. In a unit volume of the layer, the pore volume is equal to porosity. Its change over the change of pressure can be expressed from Eq. (1.4-18): dn - - dnde __-

-

dP

dedp

dn =

(1-n)2

.

C(P+PO)’

(1 - n)2

+

d p;

(1.4-22)

C(P Po) or for unit change of the pressure head:

where the pressure is equal to the effective stress ( p = Zl)in the initial state. Thus the final form of S, is b

where

b

1.4 Balance of the ground water

159

is a special variable, combining all the depth-dependent parameters in one factor. Considering the numerous uncertainties influencing both the process itself and the parameters, as a &st approximation it is a generally accepted supposition that this factor is independent of the depth. The form of the storage coefficientusually published in the literature (Bear et al., 1968; Bear, 1972) is based on this hypothesis: S = y b(a - np). (1.4-24) I n highly compressible layers the effect of water compressibility can be neglected, but the change of both porosity and effective stress along a vertical section has to be considered, This can be done either by integrating Eq. (1.4-23), or in tabulated form (see Table 1.4-2), supposing the constancy of the parameters within short stretches of the vertical section (Kovhcs, 1972b).

Q6

D

I

ZDO

400

EUO

BOO

,

ill00 0 20

50 /OO 200 500 U80 effective pressure, d [kiV/m2J effective pressurer B [kN/mzJ Fig. 1.4-6. Compression curves represented on arithmetic and logarithmic scales

A further problem is caused by the fact that the layer is not completely elastic. Thus the extension following the increase of neutral stress is not equal to the compression caused by the decrease of water pressure of the same size, and the first loading creates greater compression, than reloading (Fig. 1.4-5). Although the definition of the storage coefficient supposes that the amounts of water released from, or stored in the layer as a result of a unit change in the pressure head are equal t o each other, it is quite evident from the figure that this statement is only a very rough approximation. When solving practical problems in seepage hydraulics, it is impossible to analyze the whole history of loading and reloading of the layer, therefore, a.n average parameter is acceptable to characterize the storage capacity of aquifers. Detailed analysis of compression and the direct application of soil mechanical results are also hindered by the time lag occurring between loading and the total development of compression. The curves representing the relationship between changes of both load and void ratio (see Figs 1.4-2 - 1.4-5) were

160

1 Fundamentals for the investigation of seepage

obtained with gradual loading, waiting after all changes of the load until no further compression or extension could be observed. When investigating the time-dependent process called consolidation, many further parameters and conditions have to be considered. A mathematical treatment of the problem needs therefore the application of further simplifying approximations. For showing the complexity of the problem to be solved, the representation of the most simple example of linear vertical compression will be continued here, supposing that the a parameter in Eq. (1.4-23) is constant, and thus the relationship between the change of porosity and that of effective stress can be expressed from Eq. (1.4-22), also considering Eq. (1.4-20): dn _ _- _ _dn = a . d3 du

(1.4-25)

Accepting the validity of Darcy’s law the specific flow rate i n verticul direction crossing a horizontal area of unity can be calculated as follows: (1.4-26)

because the hydraulic gradient can be determined as the change of pressure and potential energy along the z axis (the negative sign indicates, that the flow is directed upwards, while the z axis downwards). The modification of the flow rate within an elementary dz stretch expressed from Eq. (1.4-26) is (1.4-27)

The flow rate can increase only if there is a source in the elementary volume, and in the present case only the decrease of the pore volume can be such a source. Thus the change of porosity i n time should be equal to the change of flow rate along an elementary length. This condition combined with Eq. (1.4-25) provides the following relation: aq -

K 8%

a2

y

a22

- -an - . - -ap;

au

at

at

8% c-=--.* a22

au at ’

where c=-*

K

(1.4-28)

YQ

Thus finally a differential equation is obtained for describing the change of neutral stress and consolidation, which can be solved by integration, if the

boundary conditions are known at both horizontal boundaries of the layers. This condition adds to the compleJrity of the process, indicating that the investigation of consolidation together with that of storage capacity cannot be limited to one layer, because the neighbouring formations may recharge or drain also the aquifer in question.

1.4 Balance of the ground water

161

This short summary of consolidation clearly indicates that the storage coeficient, influenced by the compressibility of the layer, depends mainly on the following factors: (a) Compressibility of water (B); (b) Compressibility of the solid matrix ( a , being the function not only of materid constants, but also of parameters describing the initial state of the layer, which may change with the depth of the point investigated; thus the

time. t

Fig. 1.4-6. Characteristic curves showing the development of consolidation u = const. hypothesis is only a rough estimation);

(c) History of previous loadings of the layer; (d) Time elapsed between the start of consolidation and the time point for which the storage capacity is qletermined; (e) Hydraulic conductivity of the layer ( K , the effect of which can be characterized with the time dependency of the absolute or relative compression, the former being the change of the height of the investigated ssmple, and the latter is the same value related to the total compression after the completion of consolidation. Fig. 1.4-6); (f) Boundary conditions (hydraulic conductivity and pressure conditions of the neighbouring layers) and initial condition (pressure distribution in the layer in question) prevailing along the boundaries of and within the investigated layer [for demonstrating this influence Fig. 1.4-7 shows the dependence of relative compression on time, considering three different boundary and initial conditions; after KBzdi (1972)].

As it has been mentioned, when the storage capacity of a shallow unconfined uquifer is investigated, the effect of compressibilities of both water and the solid matrix can be neglected, compared with the amount of water released from, or stored in the pores following the change of the position of the water 11

162

1 Fundamentals for the investigation of seepage

time farfor 3 O ! r

0.2

0.4

I

I

0.6 I

oe

LO

I

I

T= Ct h2 1.2 14 I J

layer drained synmetrical/y at bffofh sides

x

lagers drained on& along

n.

'

e-

.? 40 h

I

\x

initial candii'ion :

triangular pressure diStribuTiffn.?empressure

zero-pressure at tbe basis Fig. 1.4-7. Relative compressionas a function of time for different boundary and initial conditions (KBzdi, 1972)

table. To justify this statement, the values of the two components were calculated, supposing that a depression of 10 m is created in a layer, 200 m deep, bordered below by an impervious, non-deformable formation (KovAcs, 1972a). Two different sediments were considered (loose fke sand, and very compressible clay) and the physical parameters of soil were selected accordingly. The results are shown in Table 1.4-2. Although it is seen that the volum e of water released from the pores immediately after drainage is less than the total amount stored above the level of depression, the numbers indicate that - even in a thick layer, such as the one investigated - the effect of consolidation is relatively small, especially in the case of pervious formations, which are important in practice.*The statement according to which the storage capacity of unconfined aquifers should be characterized with the specific yield, seems to be acceptable. Considering the definition of specific yield, this parameter can be calculated aa the difference between porosity ( n ) and the volumetric water content retained i n the sample against gravity ( W , ) [the latter used to he a1)proximated with Lebediev's maximum molecular water capacity ( Wmol),or field capacity (WfJ as well]: (1.4-29) n, = n - W,. The difficulties arising in connection with the application of this concept are caused by the incomplete physical interpretation of the water content in question, and by the uncertainties of the respective memuring methods. The most commonly applied method of determining both specific yield and water retention capacity of the sample is column drainage, when a column composed of the material to be investigated is completely saturated,

1.4 Balance of the ground water

163

Table 1.4-2. Comparison of water amounts gained by water releatie from the pores and by consolidation, respectively

(a: natural condt tion

(0)

drained condfflon

surface wafer ladle

I Total available water Water stored above the level of depression

Water gained by consolidat ion

in [ml

in [ml in [yo] of the total available water

I 4.00 3.71

93.2

6.26 4.88

78. I

in [m]

0.29

1.37

in [yo]of the total available water

7.2

in [m]

0.35

21.9 __ 1.60

__

Expected land subsidence

may

and the volume of water drained from the sample is then measured, determining the water content retained by the sample as well. Schoeller (1962) collected numerous data from the literature representing the results of such measurements. This information together with some recently published materials are summarized in Table 1.4-3. Representation of the data in the form of graphs (Fig. 1.4-8) clearly indicates the tendency of the relationship between water retention capacity and grain diameter (the former slightly increases with decreasing diameter t o D , 2 0.02 em, and below this limit the increase is more rapid), although the scattering of the points is very large. The data, especially those measured by Hazen, also indicate the influence of porosity (or uniformity, which is closely related t o poroHity), but the uncertainties hinder the numerical evaluation of this relationship. 11*

Table 1.4-3. Specific yields of various samples determined by column drainage Diameter given by

Effective diameter

(loefflcierltof

Porosity

unirormity

dcalated orestimated

[=I

4 [=I

0.003

Weter retention

specie0

field

-pacity

w,

References

n

U

0.0061

0.44

2.3

0.19

0.26

0.006 0.017 0.036 0.048 0.14 0.6

0.0102 0.0272 0.0876 0.0864 0.252 0.760

0.42 0.42 0.326 0.40 0.416 0.46

2.3 2.0 7.8 2.4 9.4 1.8

0.16 0.11 0.096 0.080 0.076 0.070

0.26 0.31 0.23 0.32 0.34 0.37

0.0204 0.036 0.009 0.139

0.020 0.036 0.070 0.140

0.413 0.406 0.391 0.382

near unity

0.061

0.302 0.362 0.360 0.346

Wollny (1886)

0.0082 0.0118 0.0166 0,0186 0.0473

0.008 0.012 0.016 0.018 0.060 0.008 0,012 0.016 0.018 0.060

0.397 0.406 0.408 0.401 0.389 0.397 0.406 0.408 0.401 0.389

nearunity

King (1898)

0.069

0.331 0.366 0.367 0.366 0.366 0.208 0.268 0.294 0.330 0.320

0.014 0.03 0.07 0.14 0.30

0.404 0.406 0.418 0.404 0.401

near unity

0.061 0.048 0.040 0.028 0.032

0.363 0.367 0.378 0.370 0.369

Atterberg (1911)

Comments

na

Haeen (1892)

The characterktic diameter k gken in the form of D,,

0.0139 0.0306 0.0094

0.139 0.303

0.044

0.041 0.037 0.066 0.060 0.041 0.035 0.034 0.180 0.148 0.114

0.080

The e*perimenb were made with elmoat homodisperee samples

IF

zB

?i

t 5'

The experiments were made with almost homodisperse samples

Directly published values for specific yield and water retention capacity Parameters recalculatedfrom dstapubliehed for the ahracterimtion of timedependency of the process

The experiments were made with almost homodisperse eamples

d

6w-

g @

9

s

H

7i 9%

0.0014 0.0047 0.0067 0.022 0.046 0.072 0.112

0.0016 0.0060 0.0060 0.022 0.046 0.072 0.112

0.422 0.406 0.381 0.372 0.362 0.360 0.366

0.0436

0.36-0.38

near unity

0.282 0.174 0.134 0.062 0.043 0.040 0.038

0.140 0.292 0.347 0.320 0.309 0.310 0.317

Zunker (1930)

The experiments were made with elmoet homodisperse samples, although the smaller porosity of more coarse materials indicates thet their coefficient of uniformity was higher than three

Johnsonetd.

GMeSs beads

~~

Given in the form of grain-size distribution

0.0120

0.39-0.41

0.0016

0.40-0.42

0.0687

0.36

0.016* 0.031** 0.120*** 0.030* 0.043** 0.280** * 0.062** 0.290*** O.W* 0.003** 0.186***

0.0212

0.37

0.0114

0.40

O.OSl* 0.090** 0.142*** 0.076* 0.100** 0.213***

0.36 0.36 0.33 0.36 0.24 0.26 0.36 0.38 0.36 0.37 0.11 0.13 0.36 0.38 0.11 0.13 0.310 0.287 0.164 0.289 0.280 0.228 0.326 0.300 0.187

(1963) Prill d al. (1906)

* at the top of sample ** around the top of the open capillery Zone *** average for the column

20 Del Monte sand

Fresno medium sand ~ ~ t u ralluviei el send (Holbrook, Arizona)

166

1 Fundamentals for the investigation of seepage

a32 4r sF2 aze

0.24

B P J2 5 0.20 %

s

* % s

,

I

de t P r m in P d an average as the moisture mffted of of the column the dry part ofthe rofumi

+ o.--- .-o

A----A

++ .-.-.0

0

.............x

Johnson ef al. King

Hazen

x

...-..-..+

0.16 0.12

2 c2

2

y

am

L

2

0.04

0 effective diameter, --cirFig. 1.4-8. Relationship between water retention capacity and grain diameter

The most probable reason causing the large discrepancy between the meavured data, is the development of the soil-moisture retention curve i n the drained sample. The U.S. Beologicul Survey, in cooperation with the California Department of Water Resources, made a series of experiments (Johnson et al., 1963; Prill et al., 1965), the results of which are also listed in Table 1.4-3, showing the possible large differences between the parameters determined in different ways from the same observations. The soil columns were drained either by gravity or by suction. (In the latter case a semi-permeable film was placed below the column, the water level was maintained at a given height, and it was connected with the experimental column with a tube). I n Fig. 1.4-9 the results are compared, showing the correctness of the previous statement, t h a t field capacity (in this case water retention capacity, being here equal t o field capacity) is the function of the elevation of the investigated point above the water table. The development of the soilmoisture retention curve is also well demonstrated in the figure. The time-dependent character of the specific yield was also investigated in the framework of USGS’ experiments. It was found that even 100 hours were not enough for the attainment of complete dynamic equilibrium, which can be characterized with hydrostatic tension distribution (the tension being everywhere equal t o the product of the height above the water table and the specific weight of water). Figure 1.4-10 shows the change of tension as

167

1.4 Balance of the ground water

p..2,?

g 220

$& 0

5

280

10 /5 20 25

moisture contentin percentayeof r o h e

5

moisture content in percentage of volume

10

i5

20

25

moisture content in percentage of mlume

Fig. 1.4-9. Development of soil-moistureretention curves in drained soil columns

time elapsed ChoursJ I 2 5 /0/53i 7099

20

40

60

80

i00

i2G

lension in cenfimeiersof wafer Fig. 1.4-10. Graphs showing the distribution of tension during drainage of 0.12 mm glass beads (Johnson el al., 1963)

168

1 Fundamentala for the investigation of seepage

a function of time, by way of example, as memured in one of the experimental columns. It can be seen that in a very f i e sand dynamic equilibrium is achieved only in the lower 80 cm after 100 hours, while at the top-of the column (about 150 cm above the water table) the hydraulic gradient remdns almost unity. At the same time, saturation of the sample is very OW at this elevation and, therefore - aa it is well known and will be proved later on in Chapter 2.3 - unsaturated conductivity is also very low. Thus the IOU

eo 60 1532rm long column

+

40

20 0

fo

130 Ma {jo time ChuursJ Fig. 1.4-11. Relationship between speoific yield and drainage time on the basic of USOS data

a

lo

20

30

40

50

70

ED

90

IOO

iiu

120

amount of water transported by this gradient is negligible. This supposition can also be substantiated by the graphs showing the rehtionship between the drained amount of water and the time elapsed from the beginning of drainage (the graphs were constructed also from the data of USGS’ experiments; Fig. 1.4-11). Among the previous investigations, King’s measurements also afforded data representing the dependem of 8pecific yield on time (Fig. 1.4-12). The length of his experiments waa two and a half years, and water wa,s still released from the sample at the end of this very long period. The result is in good agreement with the conclusions mentioned previously, showing that the process is very slow, but the amount drained at the end of the prolonged experiments is almost negligible. Another interesting experience can be gained from King’s data as well: i.e. about 13-18% of the total stored water was lost during the long experiments, probably through evaporation (the sum of the drained water and the soil moisture retained at the end of the investigation in the sample is smaller than the total pore volume). As a consequence of the numerous uncertainties, occurring in connection with the determination of both specific yield and water retention capacities, as noted previously, there have been attempts in the literature to give average figures as rough estimations for the parameters in question. Such

169

1.4 Balance of the ground water

proposah are summarized in Fig. 1.4-13 (Castagny, 1967; Major, 1972b; De Wiest, 1968). Another way to avoid the uncertainties caused by both the development of the soil-moisture retention curve and the slow process of drainage, is the use of a centrifuge for dewatering the sample. The retained moisture content may be numerically different from water retention measured with column drainage, thus i t is called centrifuge moisture equivalent. This parameter is also influenced by many factors. It was necessary therefore to standardize 0.5 9,

B E?

B

R s?

0.4

porosity

0.3

specific yield

s?

0.2

2

0.1

wafer retention capacity ,

chg fine sand sandy grave/ boulder silt sand marel Fig. 1.4-13. Average porosity, water retention capacity and specific yield as a function of grain size

170

1 Fundamentals for the investigation of seepage

its measurement. The American Society for Testing Materials (1958) defked the centrifuge moisture equivalent, it8 follows: “water content retained by a soil, which has been first saturated with water, and then subjected to a force, equal to one thousand times the force of gravity for one hour”. The detailed investigations performed by USGS (Johnson et al., 1963) also included an analysis of the moisture content measured by centrifuging. They have found that this parameter depends on the

farce times gravity

o

300

600 goo m a 1500 force Times gravflg

i000

z

90

Fig. 1.4-14. Effect of the acting force on the centrifuge moisture content (Johnson et al., 1963)

Temperature ; Length of period of centrifuging; Size and structure (porosity) of the sample; Size of the acting force (angular velocity); and, since the iorce changes with the distance from the centre of rotation, the moisture content also changes within the sample. This latter effect can be well demonstrated if the meaaured moisture content is represented as a function of the acting force (Fig. 1.4-14). The relationship can be characterized by considering that the tension created by rotation (hCR)expressed in centimeters of the equivalent water column depends on the angular velocity ( 0 ) [T-l], it8 well aa on the distances of both the bottom of the sample ( r l )and the specified point (r2)from the centre -

171

1.4 Balance of the ground water

of rotaticm measured in centimeters: (1.4-30)

In Table 1.4-4 the centrifuge moisture equivalents (data measured with the standardized method) are listed for each sample investigated, and in the further columns the range of moisture content is given within which the parameter was found, when one of the influencing factors waa modified. The data listed in Table 1.4-4 are also graphically represented in Fig. 1.4-15, similarly to the water retention capacity. The relationship with grain diameter shows the same tendency, although the numerical values are different. The advantage of the use of a centrifuge is that the process is much quicker than column drainage, although the scattering of the measured data cannot be decreased considerably. At the same time, the real physical meaning of specific yield is lost in this way. After demonstrating all the difficulties in connection with the determination of the specific yield, it is necessary to recognize that most of the problems are caused by the insufficient physical interpretation of the term. In an unconfined system, the capillary zone develops always above the water table, hence the dynamic balance i s characterized by the soil-moisture retention curve. The correct way of determining the specific yield i s , therefore, to calculate the diflerence of the amount of water stored above a reference level (chosen arbitrarily, lmt below both the original and the modified water table), if the change in the elevation of the phreatic surface is unity, according to the definition of specific yield. Redistribution of the soil moisture following a change i n the position of the water table needs some time, the length of

g

70

6 ,$D

o

8 &, 50

+ uncontrolled temperafure

%

s

. B %

standard measurement

I scattering caused bg the modification of some of the influencing condltions

40 1

B

9 39 b

1

3.% zn B

Q

3 I0

~

5

&

s o

~

8

Range of ds*r scattering under vdoas hiSki8l

Db

la1

(I

modllied conditions

ZZ?

U n C o n W d

equivalent

Mpsnave

-

Del Monte send 80 meah 30 meah 20 mesh

0.0166 0.0300 0.0687

2.42 1.40

Glese be& 0.47 mm 0.12 m m 0.036 m m

0.0436 0.012 0.0016

1.11 1.30 1.90

69 CAL 191-244

0.0212

4.0

67 CAL 122

0.0026

6.43

4.0

67 CAL 66

0.0045

2.66

4.2

67 CAL 110

0.0030

26 00

6.4

67 CAL 118

0.0022

8.00

6.0

59 CAL 268-261

0.0017

16.66

9.8

13.0

69 CAL 261-262

0.0013

22.60

10.6

13.1

11.0

1.4 0.4 0.6 1.2 1.3 3.4

1.4 0.5

1.3 1.9

N

-

1.2 0.3 0.4

1.6 0.6

1.8 0.6 0.6

1.9

0.9 N 1.3 1.0 1.7 3.1 3.9

2.2

1.6

--

6.0

7.0

13.6

69 CAL 263-264

0.00084

61.11

59 CAL 137-138

0.00071

33.00

13.2

14.4 N 16.1

67 CAL 72

0.00227

66.00

14.1

68 OKL 16

0.00067

26.66

16.6

-

Loess (Bonny Dam, Colo.)

0.00074

6.16

17.3

67 CAL 106

0.00071

30.86

17.G

18.0

-

2.0

13.6

12.4

17.3

---

14.6

8.0 21.1

Loess (Trenton, Nebr.)

0.00076

14.0

4.69

19.0 20.6 18.8

N

20.0

69 CAL 248

0.00070

30.00

20.2

68 OKL 42

0.00091

18.76

20.4

67 CAL 166

0.00092

30.40

21.6

69 G A L 130-131

0.00038

16.00

23.1

67 G A L 139

0.00149

16.66

24.8

26.6

67 CAL 162

0.00061

38.88

28.8

21.1

67 GAL 162

0.00029

66.66

30.0

20.2

;P

67 CAL 173

0.00048

13.33

32.3

27.2

F

68 ARK 146

0.00014

4.16

37.2

67 CAL 112

0.000 17

6.43

40.9

34.6

%

67 CAL 123

0.00063

11.10

43.2

32.8

67CAL

2

0.00020

8.33

46.4

41.4

f 3

67 CAL 206

0.00013

8.33

49.3

46.9

Kaolin (N.F)

0.00029

3.36

49.6

Kaolin (EPK)

0.00007

2.86

61.2

42.0

N

48.0

Fuller's mirth

0.00048

16.66

109.0

76.0

N

80.0

N

23.4 12.4

21.9

N

26.9

F

31.0 29.4

N

N

36.0

33.6

f

5a

174

1 Fundamentals for t,he investigation of seepage

which depends on the hydraulic conductivity of the 1ayer.Thus the comparison of the stored amounts has to be made after the development of the dynamic equilibrium, hence the determination of specific yield can be reduced to the calculation of the difference between the areas of two soilmoisture retention curves (Fig. 1.4-16). Let us indicate two positions of the water table with sufhes 1 and 2, respectively. It is quite indifferent, which is the original condition, because the specific yield must have the same numerical value whether lowering or raising the water table. To f d i 3 the condition required by the definition, the distance between the two water tables is equal to unity. The stored amount of water in a column having a cross section of unit area can be easily calculated above the datum. Executing the calculation for both positions of the water table, and forming the difference of the two determined values, the physically correct specific yield is obtained:

because 21

- z2 = 1.

Comparing Eq. (1.4-31) with Eq. (1.4-29), i t is readily seen that t,he difference of the two expressions to be integrated is equal to the water retention capacity. Although this concept is theoretically correct, difticulties are raised, however, when it ix applied in the practice. The t w o most important diffi-

c Fig. 1.4-16. Calculation of the specific yield from the difference of two soil-moisture retention curves

1.4 Balance of the ground water

175

culties are caused by the hysteresis of the soil-moisture retention curve and by the time lag between the modification of the water table and the final development of the dynamic equilibrium. Further problems are encountered when the unsaturated zone is layered, but details of this will not be discussed now. As it was explained in Chapter 1.3, the relationship between tension and water content is not unambiguous, owing to the phenomenon called hysteresis. Dynamic equilibrium develops at a lower position, if the soil-moisture zone is wetted from the direction of the water table, while the highest curve can be achieved if the zone had been completely saturated previously and drained vertically later. Between these extreme positions the balanced condition can develop a t any elevation. In Fig. 1.4-17 four different cases are represented:

Fig. 1.4-17. Influence of hysteresis on the specific yield

176

1 Fundamentals for the investigation of seepage

(a) A wetting process is followed by further riee of the water table, thus both balanced conditions are characterized by the lower stretch of the hysteresis loop. The curves are pardel, and their distance is unity, equal to the modification of the water table. (b) The original wnditwn w m achieved by drainage, and the lowering of the water table is investigated. Both soil-moisture retention curves follow the upper branch of the hysteresis loop, thus their distance is equal to case (a). (c) After wetting the soil-moisture zone,the water table is lowered once again. The lower limit of hysteresis describes the initial condition, and the upper one characterizes the final state. Thus the difference between the two areas of the retention curve is smaller than that calculated from the average curves. Theoretically the difference can also be negative, but in this case the upper stretch of the hysteresis curve is never achieved; the dynamic equilibrium develops at an intermediate position, hence the lower limit of the calculated difference is zero. (d) The case opposite to (c) is the rise of the water table in a previously drained system. Now the final state is characterized with a low retention curve, and the initial condition with a high one. The result is also a decrease of the difference between the two areas, and its lower limit can be again zero.

Further special cases may also be expected (e.g. wetting of the soil-moisture zone previously, and rise of the water table afterwarrds due to infiltration from the surface) when the distance between the initial and final retention curves is greater than unity, its upper limit being the sum of the change of the water table and the thickness of the hysteresis loop. It is obviously very difficult to investigate each of the above processes separately, when the difference between the actual and average values of the specific yield is to be determined. Only one possible way can be proposed: to accept the specific yield calculated from the average soil-moisture retention curves, and to neglect the influence of hysteresis. Realizing the uncertainty introduced in this way, some further approximations can be applied to derive a formula amenable to mathematical treatment, and suitable for describing the relationship between water retention capacity and the physical parameters of soil. When analysing the soil-moisture curve, it was shown that the relationship between tension and moisture content in the adhesive zone can be well approximated with a hyperbola of sixth order [see Eq. (1.3-24)]. At this very steep stretch, the curves belonging to theinitial and the final state practically cover each other (Fig. 1.4-18). Accepting the use of the average curve (without hysteresis) for the characterization of both conditions (before and after the modification of the water table), the difference of the area8 covered by the retention curves can be calculated a,s the free gravitat i m l porosity at the adhesive zone multiplied with the change of the elevation of the water table. The latter being equal to unity, if the purpose is the determination of the specific yield or water retention capacity, the only remaining problem is to select the numerical value suitable for the characterization of both water content and gravitational porosity at high tension.

1.4 Balance of the ground water

177

. surface

Fig. 1.4-18. Sketch for the interpretation and calculation of the water retention capacity

It can be proposed that the parameter considered in the calculation should be that specific amount of adhesive soil moisture, which belongs to the highest capillary rise. The water retention capacity can thus be calculated by substituting the possible highest capillary rise [see Eq. (1.3-28)] into Eq. (1.3-25):

because

h,,,,

1 - n a . = 0.11 --, n Dh

I%(

or the same parameter expressed with the rate of saturation: so = 3.6 x 10-3

;

[l n)518

-

(1.4-33)

In Fig. 1.4-19 the relationship given by Eq. (1.4-32) is compared with the measured data listed in Tables 1.4-3 and 1 . 4 4 , and represented in Figs 1.4-8 and 1.4-15. The comparison shows that the proposed equation provides an acceptable and reasonable compromisebetween the various meaaurements, giving the lower limit of data measured with column drainage (which is acceptable considering the development of the capillary zone), but running above the centrifuge moisture equivalent. At the same time it affords an easily applicable method for the calculation in practice of the water retention capacity and specific yields as the ficnction of the physical parameters (n,Dh, a) of soil. 12

178

1 Fundamentals for the investigation of seepage

effective diameter, Dh [em/ Fig. 1.4-19. Comparison of the proposed theoretical value of water retention capacity with meesured data

Naturally, the time lag preceding the development of the dynamic equilibrium also disturbs the direct application of the parameter calculated from Eq. (1.4-32). I n a permeable layer, the ratio of the amount of water drained or stored at the beginning of the process is relatively high; only a few percent of specific yield occurs in the later period, when the water has t o be transferred through the upper part of the column, where the unsaturated conductivity is very small owing to the low saturation. In fine grained material the hydraulic conductivity is relatively low, therefore the whole process is considerably prolonged. Figures 1.4-1 1 and 1.4-12 give sufficient information about the magnitude of the expected time lag, although it is also mentioned in the literature that the delay will be longer if the balance is achieved by wetting the unsaturated zone than in the case of drainage (Smith, 1933).

1.4.3 Analysis of horizontal ground-water flow for the determination of the vertical water exchange The second component of Eq.(1.4-10) to be investigated is the horizontal ground-water flow. For a complete study of this movement, the knowledge of some basic concepts of seepage hydraulics is necessary, which will be dis-

1.4 Balance of the ground water

179

cussed only later on in this book. This very simple form of flow can be analysed, however, if some approximative hypotheses are accepted as starting conditions: 1. Seepage velocity is linearly proportional to the hydraulic gradient [Darcy’s law, see Eq. (1.1-9)]; 2. The flow is horizontal, therefore, the velocity is constant along a vertical section, hence the numerical value of velocity can be calculated as a function of the slope of the water table (Dupuit’s hypothesis). Following Kamensky’s (1940, 1943) investigations, let us consider the water balance of an elementary part of the gravitational ground-water space, whose lower boundary is a horizontal impervious layer, and which is bordered in the horizontal plane by a rectangular oblong (Fig. 1.4-20). There are five observation wells for recording the position of the water table, one inside the oblong, and four outside its borders. The wells form a rectangular cross, and the borders are fixed that way that they divide the

12*

180

1 Fundamentals for the investigation of seepage

distances between the internal and external wells into halvee. Let b, indicate the length of the different sides [ b , = b, = 1

L

1

1

. . . b,

(Ax, + Ax4) ;

Y

b, = b, = - ( d X , + dX,) the time dependent areas of the vertical cross 2

sections at the borders are then

where H,(t) indicates the elevation of the water level in the i-th well above the impervious lower boundary at the time t. The average slope of the water table can also be calculated from the data of the observation wells:

I , = AHlX ( t ) ; I , = AH,, ( t ) . 4 A% ’

(1.4-351

where dHi,(t) is the difference of water levels registered at time t in the i-th well and in the centre well, respectively [AH,(t) = H,(t) - H,(t); . . . * * AH444 = H4(t) - Hp(t)l. Accepting the two basic hypotheses mentioned previously, the flow rates crossing the vertical boundaries of the investigated prism can be calculated a5 follows:

-

There is, however, a H t h bordering surfme of the prism, through which a flow can develop, i.e. the waater table. Its area is

A = b, b, ;

(1.4-37)

and the flow rate can be expressed here as the product of the area and the specific vertical water exchange [ ~ ( t ) ] :

&o(t)= Ae(t)*

( 1.4-38)

Investigating a finite time period (At = t, - t,; and t, > t,), the water balance of the prism requires that the product of the elementary time unit (dt) and the sum of flow rates summarized for the entiretimeperiod, At, should be equal to the change of the stored water amount, which is equal to the change of the position of the water table [characterized with the recorded water level (or ground-water depth) in the central well AHot = HO(t3)- Ho(t,)] mul-

181

1.4 Balance of the ground water

tiplied with the horizontal area of the prism and the specific yield of the layer:

%[ &o(t)+ Z & i ( t )1dt 4

tl

= m o t An,

-

(1.4-39)

4-1

[Eq. (1.4-39) is a special form of Boussinesq's differential equation (which will be discussed in detail in Chapter 5.4) expressed with bite differences.] Substituting the time-dependent variables with their averages, which can be regarded aa the values of the function in question belonging to a point of time tz within the investigated period (tl < tz < t3), and applying transmissibility [which parameter is equal to the product of the depth of flow and conductivity

the simplified form of Eq. (1.4-39) can be given as follows:

The geometrical parameters (bi and x l ) and those calculated from water level records ( H , and H o t )can be determined with relatively high accuracy. The methods applicable for the calculation of the specific yield were already discussed in the previous section. Thus transmissibility is the remaining variable the knowledge of which is necessary, if Eq. (1.4-40) is to be used for the direct calculation of the time-dependent water exchange through the water table. If the position of the impervious lower boundary is known, only hydraulic conductivity has to be determined as a basic data. The equation can be also used for the calculation of transmissibility by substituting recorded water level data of a period when the vertical flow was surely negligible, E(t) = 0, (e.g. in winter, when frozen top layer hindered both infiltration and evaporation). If the cross of the observation wells is fixed so that one line of wells (1, 0, 3) is parallel to the dip of the water table, and the other (2, 0 , 4 , being perpendicular to the former) is parallel to its strike, there is no flow in the second direction. The data of the wells with even numbers (2, 4) need not be taken into account, and the investigation can be restricted to the analysis of the water balance within a stripe of unit width:

There are several other proposals to simplify the calculation, e.g. to choose very short At periods and substitute the time-dependent variables with their values belonging to the beginning of the period (Kamensky, 1943); to use

.

182

1 Fundementah for the investigation of seepage

continuous time-dependent functions and divide the solution into three parts, because in this case the mathematical functions approximating the separated parts can be determined in closed form by integration (Szdkely, 1973); to give graphical interpretation of the solution with finite differences (Lebediev, 1963), etc. Other authors have made efforts to generalize the

Fig. 1.4-21. Map of the Koml6si Imre Ground-waterResearch Station

method, making it applicable for an existing irregular network of observation wells, when the geometrical conditions used as the basis of the derivation are not satisfied (Rowe, 1955; Major, 1972b). It is a general opinion that the solution of water balance equations of this type with computers does not raise any special problem and, therefore, the most suitable form of the relationship has to be determined always according to the local conditions (distribution of wells within the network, type and reliability of the recorded data, information concerning the depth of flow, hydraulic conductivity and transmissibility, etc.). This is the reavon why the theoretical basis of this method will not be dealt with further on, but a practical example will be presented here, to illustrate the special features of vertical water exchange between soil moisture and gravitational ground-water zones.

183

1.4 Balance of the ground water

In an experimental area (KomMsi Imre Ground-water Research Station, Hungary), where long ground-water records were available from several wells, detailed investigations were made by applying the method described in the previous paragraphs (Major, 1972b). The layer containing the shallow ground water is a relatively homogeneous h e sand, the transmissibility of which is known from pumping tests. The specific yield of the sand was estimated on the basis of the water retention curves measured in the field with neutron probes. The surface is plain, covered partly with forest surrounding grassy clearings, as indicated in the map (Fig. 1.4-21). Four areas were chosen, for which the water balance of six-day intervals were determined from the records covering a period of 18 years (1954-71). Some characteristic data of the areas and the annual sums for the decade 1961-70 of the two components of vertical water exchange (i.e. I , infiltration to the groundwater and ET, evapotranspiration replenished from the ground water) are listed in Table 1.4-5. Table 1.4-5. Annual values of the components of vertical water exchange, determined for various areas by using the water balance method

I B l c

Denotation of the arem

Extension of the area [m2]

10 000

260 000

13 640

9 200

89

78

68

43

Rate of the forested area

1% 1

1961 1962 1963 1964 1966 1966 1967 1968 1969 1970

188 88 148 66 131 261 163 4 137 191

319 303 287 242 199 264 346 384 366 330

196 112 183 103 191 313 231 30 188 292

260 196 196 146 146 230 286 226 219 266

346 236 319 307 460 632 472 212 291 386

139 27 19 13 16 43 37 32 60 89

277 260 347 362 669 604 611 306 346 386

40 22 18 16 0 30 14 16 36 62

42 1 478 616 669 662 844 618 384 693 690

To characterize the seasonal fluctuation of the components of the vertical water exchange, the monthly averages of both I , and E T , were calculated from the data. As an example, the parameters determined for area A using a period of twelve years (1960-71) are represented in Fig. 1.4-22. As a further example, the monthly data of an arbitrarily chosen year (1963) are listed in Table 1.4-6 for the areas A and B, and the basic data of the same year, calculated for six-day intervals, are also represented in Fig. 1.4-23 for the area A .

184

1 Fundamentals for the investigation of seepage 60 50

40

30

za la

0

Nor Jan. March May July Sept Fig. 1.4-22. Monthly averages of I , and ET, calculated for a 12-year period (1960-71) from the data of area A

There is a special condition characterizing the area investigated: just before the beginning of the recorded period a young forest was planted there, thus the influence of the growing trees on the hydrological cycle can be well observed. Since practically there is no surface run-off from the area because of the very flat and sandy surfwe, and since the change of moisture content Table 1.4-6. Monthly values of the components of vertical water exchange for 1963, determined for areits A and B Month

Monthly preoipiestion

Positive monthly amretion 1,

forarea A

1962

November December

[-I

I

forarea B

-

-

0.76

2.24

72.2 66.2 46.7 36.6 39.9 44.8 30.7 76.3 80.8 26.2

16.10 17.08 76.97 37.09 0.74

21.43 24.66 78.68 49.99 3.47

616.6

147.74

86.6 24.0

Negative monthly accretion ETI C m l

for area -4

13.67 7.43

I-KEZT 2.80 2.46

1963

JmUery Februery March April

W Y June JdY Ausust Ekptember October hnUd

EUm

2.60

182.94

-

0.86

1.26

-

-

-

14.94 66.77 76.03 61.27 33.63 22.19

8.99 67.37 69.19 32.81 20.67 10.12

287.18

195.26

1.4 Balance of the ground water

Fig. 1.4-23. Annual tluctuation 01 I g ana

~

2 CaiCuIaTda ' ~

185

xor six-aaymmrvmu (ivoo;

area A )

over a year-long period is negligible, it can be supposed that the difference of annual precipitation and the yearly sum of infiltration to the water table is equal to the amount of water stored temporarily either by interception, or in the soil-moisture zone, and evaporated from this position. Thus the annual actual evapotranspiration can be calculated as the sum of this difference and the vertical withdrawal from the ground-waterspace. These parameters are listed in Table 1.4-7 together with their averages calculated for periods of six years. The averages clearly indicate the increasing trend of actual evapotranspiration, corresponding to the growth of the forest.

186

1 Fundamentals for the investigation of seepage

Table 1.4-7. Annual values of precipitation and actual evapotranspirationfor a period of 18 years A

m

water level in the central obl3ervation well below the wvface

Year

€ I

I4 Ann.

1964 1966 1966 1967 1968 1969

683 614 664 622 491 626

1960 1961 1962 1963 1964 1966

662 421

1966 1967 1968 1969 1970 1971

844 618 384 693 690 429

778 616 669 662

2 g

146 184 208 178 146

2 2

83

2

4

132 188 88

148 66 131 261 163 3 137 191 69

2

2

2 $:

436 430 346 444 346 442 432 233 390 468 603 621 683 466 381 466 499 370

2

3

3

2

$!

189 160 240 187 2 34 162 219 319 303 287 242 199 264 346 384 366 330 411

2

2

2

'?

$$

626 690 686 631 679 604

I

Av.

392 382 362 369 378 409

4

*

846 720

422 399 431 429 436 403

847 811 766 821 928 781

362 316 376 390 348 37 1

661 662 693

2 8

2

766

2 3

cI)

Some conclusions which can be drawn from the summarized data of the analyzed example are the following: It is desirable to select a homogeneous area for the investigation of the ground-water balance. Homogeneity should be ensured for each element of the processes affecting the regime of the ground water [structure of the water bearing layer (permeability, transmissibility, specific yield); surface conditions (type of soil, covering vegetation, slopes influencing the development of surface run-off); climatic factors (precipitation, temperature, radiation, wind velocity), etc.]. In areas covered by inhomogeneous vegetation, only the average characteristics of the vertical water exchange can be determined instead of the actual influence of the different plants. At the same time, if these average values can be correlated with the ratio of the areal extension of the various types of vegetation, the data can be used to characterize the effect of a special cover (in the example presented the influence of forest can be investigated). The water balance is influenced not only by the type of the covering plants, but also b y their phase of growth. Thus the development of trees gradually

1.4 Balance of the ground water

187

decreases the vertical recharge of the ground water; this is the result of the increase of interception and soil-moisture storage. These two components are readily available for evaporation and transpiration, thus the part of evapotranspiration originating from interception and soil moisture is increased by the growth of trees. At the same time, the amount of water withdrawn from the gravitational ground water to replenish the evaporated and transpired soil moisture is also increased, probably aa a result of the development of the root structure. Hence the increase of the actual evapo15 transpiration is very rapid (in the example it surpaases the value of 10 mm/year). At the beginning of the investigated period the yearly amount of precipitation was equal to the actual evapotranspiration, but (owing to the increase of the latter) at the end of the period the average yearly deficit was greater than 200 mm. I n the investigated area, where the water table practically had no slope at all, and thus there had been no horizontal groundwater flow, the development of the forest lowered the water table and created a depression cone. The deficit i n water balance of the forest is replenished now by the ground-water flow developing along the slopes of the cone, and the surrounding arem recharge the ground water of the forest. A very rapid decrease of ET,.can be observed as the ratio of the forested area decreases within the investigated area. Such very strong effect can only be expected, if there is no ground-water withdrawal at all under the clearings. This supposition is supported by the circumstance described in the previous paragraph: the water table is considerably lowered under the forest, and thus the vertical flow directed upwards can convey only negligible amounts of water from the ground water to the shallow root zone of the clearings. It can be stated, therefore, a~ a general rule, that the depth of the water table (measured from the lower boundary of the active root zone) is one of the most important factors influencing the amount of water withdrawn vertically from the ground water. Infiltration reaching the water table shows an increasing trend with the decrease of the rate of forested areas. The most likely cause of this change is the smaller interception at clearings. At the same time, it is also probable that the moisture content is generally smaller at the end of dry periods under a forest due to greater evapotranspiration by the trees; thus, during a rainy period, a larger soil-moisture storage occurs above the water table in the forest, than outside. It can be concluded that I , is a function of both the infiltration through the surface and the available storage capacity of the soilmoisture zone. A regular seasonal fluctuation of both infiltration and ground-water withdrawal can be observed, which naturally follows the fluctuation of the climatic 1)arameters.Under the conditions of the investigated area, there is practically 110 vertical ground-water recharge during the summer half-year (from May to October). Infiltration to the water table originates mostly from autumn and winter precipitation, and reaches its maximum with a time lag after the melting of the snow. The seasonal fluctuation of vertical drainage has R more regular pattern, than infiltration. The difference between the actual graph in Fig. 1.4-22 and a sine curve used generally to describe the fluc-

-

188

1 Fundamentals for the investigation of seepage

tuation of climatic parameters (which has its maximum in June and its minimum in December) can be.wel1 explained by the fact that before the maximum of potential evapotranspiration the water table has a higher position than its average, therefore, a surplus withdrawal occurs as compared with the sine curve. Similarly, a deficit can be observed at late summer, because of the greater depth of the water table. It has to be emphasized that both the amount and the seasonal pattern of I, and E T , also depend on the type and the phase of growth of the ctovering vegetation. Thus the conclusions (especially quantitative statements) are valid only for the investigated case (a gradually growing forest with a few percent of clearings). For other surface conditions only rough qualitative approximations can be given on the basis of the example analyzed. However, several of the results summarized here may be applicable for the better characterization of hydraulic processes developing in flow fields influenced by accretion.

1.4.4 Interpretation and determination of the charaderistic curve of the ground-water balance

Returning to Eqs (1.4-9) and (1.4-lo), the remaining component to be investigated is the vertical water exchange between the soil moisture and ground-water zones. It wm already shown in the previous section, that its components (infiltration to, and vertical withdrawal from, the ground water) is a function of the depth of the water table below the surface. In addition to this parameter, both I, and E T , depend on some other variables, which can be divided into three main groups: (a) Climatic conditions: amount and annual distribution of precipitation and potential evaporation. The latter includes the effects of temperature, humidity, wind velocity, etc. (b) Characteristics of the surface: angle and direction of slopes, method of cultivation, vegetation covering the surface, etc. The resultant of some of these effects can be measured by surface run-off. (c) Behaviour of the soil within the unsaturated zone: geological character and structure of the upper lying layer, water retention capacity and the instantaneous soil-moisture content of the top soil. Thus the two function, in general form, characterizing the parameters in question are as follows:

Ig=

( m ; climate; surface; soil);

(1.4-42)

E T g = G2 ( m ; climate; surface; soil). The factors describing the influence of climate, surfaee and soil are timeand space-dependent variables. It has been mentioned that instead of the examination of their modification in time the values averaged for a suficiently

189

1.4 Balance of the ground water

long period are used. Similarly, equalization of local effects can be assumed in a large area, thus the individual parameters cun also be averaged over the area. Regarding all variables, except the depth of the water table, 88 constants for a given area, and describing these by their average over the period investigated, Eq. ( 1 . 4 4 2 ) can be simplified: 1, = f Am); ET, = fi(m).

(1.4-43) '

These two relationships - their parameters in any case, but perhaps their structure as well - are functions of the natural conditions (climate, surface, soil), which were regarded as constants for a given site. Some general rules describing the character of the functions can be, however, determined. Infiltration through the surface is the difference of precipitation and the sum of evaporation, run-off and retention on the surface. According to Eq. (1.4-2), i t can be divided into two parts, one stored above the water table, and the other part reaching the ground water. The ratio of these two values depends on the storage capacity of the unsaturated zone influenced by the physical parameters of the soil and the instantaneous water content of the upper layer at the time of infiltration. Although the amount of the stored water changes from case to c a e and, therefore, the amount of water stored in a year also varies every year, an average of the yearly retention of water in the soil-moisture zone can be calculated. Thus graphs can be constructed representing the average yearly storage for various soils as a function of the depth of the water table (Fig. 1.4-24).

For a given soil and in an average year a series of parallel curves can be obtained graphically to represent the relationship between infiltrating recharge storage rapacity of the dry layer rm3/m3J 0

0.0J

alo

at5

azo

0.25S(Z)

Fig. 1.4-24. Characterization of the storage capacity of the soil-moisture zone function of depth

190

1 Fundamentals for the investigation of seepage

and the depth of the water table. The parameter of the individual curves is the infiltration through the surface, which value gives the intersection of the inEltration curve with the horizontal axis (Fig. 1.4-25). To make the mathematical treatment of the relationship more convenient, the curve can be approximated by an exponential function, whose mymptote is parallel to the vertical axis; the position of the asymptote depends

m#z Fig. 1.4-25. Relationship between idltration and the depth of the water table

on the infiltration through the surface. I n a special case (indicated in the figure with a dotted line) the asymptote is the vertical axis itself. The water balance of ground water is very complicated when the water table is near the surface, since very small precipitations may also exert an influence. Another reason why the upper few decimeters of the soil should be disregarded in the characterization of the process of infiltration is the undisturbed structure of the cultivated and the active root zone, as mentioned already in Chapter 1.3. It is advisable, therefore, to fix another point of the curve, than its intersection with the surface. For this purpose the average infiltrating yearly recharge ( I , ) has to be determined at a given level (m,). Thus the final mathematical form of the investigated relationship is ( 1.4-44) I g ( m )= ( I , - a ) exp [ - a ( m - m , ) ] + a ; where the parameters ( I , , m,, a , a ) are functions of the natural conditions. The curve representing the relationship between the amount of ground-water withdrawn and evaporated or transpired, and the depth of the water table is very similar to the curve described previously. This also decreases monotonously with depth. It is evident that the amount is approximately equal to the potential evapotranspiration originating below the land surface, when the water table is near the surface (its average depth 1)eing practically zero).

1.4 Balance of the ground water

191

Actual evapotranspiration (ET,+ ETg)decreases with increasing depth of water table, and i t becomes constant at a given depth, supposed to be 2-3 m. under temperate conditions. In the case of such a water table, actual evapotranspiration is only 50-70% of the potential value, depending on the type of soil and vegetation. The total actual evapotranspiration must be divided into two parts; one supplied from the soil moisture and the other

/

/ /

I I ETw+ E G

I

I

pP

i

41

I

6

I

-X-

0

++

calculated from /ysimeefer dar3 (Najor) Blaney bare steppe oasis Kovda steppe Altovszkij Konopljancev W. Friedrich Juhisz J. (ralrulated)

1

Fig. 1.4-26. Relationship between the amount withdrawn from ground water to replenish evaporated and transpired soil moisture and the depth of t,he water table

from ground water (Fig. 1.4-26). The latter, being of interest from the mpect of the present investigation, is also a function of the depth of the water table, and there exists a level from where no water can rise to the root zone. The relationship between the depth of ground water and evaporated losses can also be approximated with an exponential function of a structure similar to Eq. (1.4-44): ETg(m)= ETo exp [ - p ( m - m,)]. (1.445) The process of evaporation depends on the capillarity of the soil, the tension between its grains and water, and the influence of vegetation. Thus the parameters of Eq. (1.4-45) (ET,, m,, p) are also functions of the natural conditions. It is necessary to note here that in lysimeters with constant water table a different character of the ET,(m) function was observed (Fig. 1.4-27) (Major, 1972a). In this case, however, the influenceof the fluctuating water table and the relationship between this latter process and the seasonal variation of ET, is left out of consideration. Supposing a regular fluctuation

192

1 Fundamentals for the investigation of seepage

0

20

Fig. 1.4-27. The ETg(m) function determined from date measured in lysimeters with constant water table

in both the position of the water table and evapotranspiration, which can be approximated with a sine function, the expected ETg(m)function was recalculated from the lysimeter data, and is also represented in Fig. 1.4-26. In this investigation it was also considered that the lysimeter was not covered by vegetation, thus the depth had to be measured below the root zone, and the thickness of this zone also increased with the depth of the water table. The agreement between this curve and that calculated from Eq. (1.445)is acceptable. It must be mentioned that the basic idea of the present analysis is only the qualitative characterization of the processes. The principles concluded in this way do not alter, if another equation is used for the description of the relationships existing between the variables. The character of the relationship supposed to exist between the average depth of the water table and the anwunt of ground-water withdrawn vertically is also supported by the water quality data. It is evident that in a ground-water basin the concentration of salts dissolved in the ground water is higher if the water table is near the surface, because the horizontal inflow is balanced by vertical drainage, and the salts transported by the water there accumulate. It is also expected that the salt concentration has to be proportional to the amount of water drained vertically. Indeed, the data collected by Hazen (1892) at Tafilalet in Morocco fit a very similar curve (Fig. 1.4-28) as the one shown in Fig. 1.4-26. Hazen’s data also indicate that the thickness of the layer influenced by evaporation is greater in the arid zone of Morocco, than it is in the temperate climatic zone, although it is also known that this depth depends not only on the climatic conditions, but also on the capillary properties of the soil.

1.4 Balance of the ground water

193

concen:rs?ml; grams uer llfer ll?

0

10

( total oissolved sohds) 20 30 40 50

60

Fig. 1.4-28. The amount of total ealt content in the und water aa a function of the depth of the water ta%

According to Eqs (1.4-9) and (1.4-lo), vertical water exchange is characterized by the difference between recharge and water withdrawal. Both variables are given aa a function of the average ground-water depth, thus their difference can also be plotted against the same depth. This is the characteristic curve of the ground-water balance (Fig. 1.4-29). At the level where the curve intersects the vertical axis, recharge is equal to vertical drainage (equilibrium level). The water table can only develop at this level if there is no horizontal flow, or the inflow is q w l to the outflow. Above this level the zone of horizontally recharged ground water can be found. In the case of such water tables vertical drainage is grbater than recharge, and the differenceis balanced by the surplus of horizontal inflow compared with the outflow. Reversed, when the invwtigatd column i s drained by horizontal flow (outflow being greater than inflow), this difference should be balanced by the surplus of recharge from infiltration compared with the amount of ground-water withdrawn vertically; thus the water table can only develop below the equilibrium level. If ground water had no horizontal movement, its table would develop everywhere at the equilibrium level, thus approximately parallel to the 13

194

1 Fundamentals for the investigation of seepage

infiltration [mm] to0 1-

vertical drainage [ m m l

300 400 500 I I I difference of infitration and vertical drainage C mmJ + 5a a -50 -{an -150 -200 -250

50

100.

0

I

200 I

-

Fig. 1.4-29. Characteristic curve representing ground-water balance as a function of the depth of the water table

surface

I

At

Fig. 3.4-30. Development of water table under a sloping surface

195

1.4 Balance of the ground water

surface. The terrain, however, is not a horizontal plain, hence the ground water would have a gradient, if its table were parallel to the slope of the surface. A horizontal ground-water flow would then be started by this gradient in accordance with the permeability of the layers involved, rising the water table in the valleys and lowering it under the hilly areas. Regarding this process in a static way, it would be expected that the water table becomes horizontal in every ground-water catchment. Water tables are actually neither parallel t o the surface nor horizontal, but rather follow the slope of the terrain; thus a dynamic balance develops as shown in Fig. 1.4-30. At higher areas the water table sinks below the equilibrium level, thus fotal salt content 0 0

,

~

"

"

"

/ "

"

'

"

'

2 " "

L %J 3

tbe iota1 salt confenfis negltgible in profiles 0 and @ in profi/e I, @ it is pracfica/lly fbe same as in @ %nd Q

0

- i

140-

: :

i

0@ @

CaCO, content in % of the fofal amount of salf 20 30 40 50

I I I

1' p

---.-.-.

e

65 150

490

........... 230 0300 @ --- 300 @

1 3 (8

450 700

Fig. 1.4-31. Relationship between the salt content of the soil and t,he position of the wmter teble

13*

196

1 Fundamentals for the investigation of seepage

ground water is recharged by surplus infiltration. This amount of water will be transported by horizontal ground-water flow towards lower situated areas, and a dynamic balance will develop a t the levels where the recharge and discharge summarized within the catchment are equalized. The development of this dynamic balance is also indicated by the chemicd composition of the soils. Where the water table lies a t a great depth, the soil is leached, the total amount of salts is very low and none of the well soluble sodium is found in the soil profile, whereas above a high water table surplus evaporation causes salt accumulation, as shown not only by the high salt content, but also by the high Na/Ca ratio (Fig. 1.4-31) (Vhrallyai, 1967, 1970).

A further special point of the characteristic curve of ground-water balance i s its maximum, which divides the range of the possible water table into two zones. Utilizing the ground-water resources continuously a new equilibrium

rn depth b l o w the surf&e K m l Fig. 1.4-32. Classification of shallow ground water according to the iduencing effects

1.4 Balance of the ground water

197

can only develop in such cases when the original water table was above this level, because at greater depths the difference between natural recharge and vertical drainage decreases with depth. Assuming the m, parameter in Eqs (1.444) and (1.4-45) t o be equal to the depth of the equilibrium level, the depth of the maximum point can also be calculated (Fig. 1.4-32): 1 BET, m=m,+In 8-a aI,-a or (when a = 0 and therefore ET, is equal to I, a t the equilibrium level): ml=m,,+-ln-. 1 B-a

B a

(1.4-46)

The numerical data of the characteristic curve of the ground-water balance can be determined by lysimeters, measuring the recharge infiltrating to various levels and also evaporation from the water tables of M e r e n t depths (Altovsky and Konopljancev, 1954). When using lysimeter data, the fluctuation of the natural water table should also be taken into consideration. Another possible method of measuring the characteristic curve is the construction of closed ground-water columns (Kovhcs, 1966a). I n the absence of measured data, the curve can be approximated by the use of the relationship representing the connection between the depth and the fluctuation of ground-water (Kovhcs, 195%). Comparing the data collected from semi-arid moderate climatic zones (see Fig. 1.4-26), the average depth of the equilibrium level (naturally influenced by the local climatic, soil and surface conditions) was estimated to be around 2 m in Central Europe (Hungary). This value is in good agreement with the observations of the salt content of soils showing that salt accumulation can be expected in this area, if the average depth of the water table is less than 2 m from the surface (Vhrallyai, 1967, 1970; Szabolcs et al., 1969a, 1969b). It has been mentioned that this characteristic depth is probably greater in arid zones (see Fig. 1.4-28), whereas in humid regions the same parameter may be considerably smaller. This latter supposition is proved by Fig. 1.4-33, which represents a characteristic curve of groundwater balance determined for a swampy area in Byelorussia (Lavrov et al., 1972). Ground-water balance was characterized by the yearly averages in the foregoing. Sometimes the analysis of periods shorter than a year may become necessary. I n such cases the monthly or seasonal mean values of the parameters (depth of water table, evaporation, infiltration) must be determined. Assuming that the general character of the relationship does not vary with the specified part of the year investigated, only the numerical values of the parameters do, the same equation can be used applying a factor expressing the ratio of the parameters (infiltration and evaporation) belonging to the investigated interval (e.g. a month) and the yearly average (Fig. 1.4-34). The final point in the present discussion is the explanation of the way of application of the characteristic curve in seepage hydraulics.

198

1 Fundamentals for the investigation of seepage

inflltra fion to grotind water

I-

tflfll

vertical drainage of grotind wd:er EG t m m l

Fig. 1.4-33. Characteristic curve of ground-water balance for a swampy area in Byelorussia

difference between infiltration and evaparation (Is- E 51 5

B 8 lE

b P

1

0

sa

1.0

3

20

9

L

-2

P

k

sQ

3.0

%

4.0

Fig. 1.4-34. Seasonalfluctuationof the characteristiccurve of ground-waterbelance

When facing the task of the investigation of a ground-water flow influenced by accretion, a method must be found for the numerical determination of the E ( X ) function describing the surface effects. It is already evident from the foregoing that both infiltrating recharge and vertical drainage caused by evapotranspiration and, consequently, also their resultant are functions of the depth of the water table below the surface. It is also an obvious statement that the change of the water table

1.4 Balance of the ground water

199

produced by the artificial recharging or draining of a system cauyes surplus evaporation and infiltration, respectively, along the influenced flow space, and these surplus surface effects are proportional to the change of groundwater depth. The surplus surface effects can be numerically determined using the characteristic curve of ground-water balance. The latter represents the relationship between accretion and, the depth of ground water. Thus the difference of the values belonging to the original and the modified water tables yields the surplus effects in question. This way of calculation is valid in both cases, if the water table was originally at equilibrium level and, therefore, the surface effect pertaining to the Btarting condition was zero, and if there was already some surface effect present influencing the groundwater space tjefore the changing of its level, and the original ground-water table wazj at tin elevation different from that of the equilibrium. The detailed description of the method proposed for the hydraulic consideration of accretion will be discussed in Section 4.1.3,where the practical hydraulic application of the characteristic curve derived now from hydrological data will be presented. References to Chapter 1.4 ALTOVSICY, M. E. and KONOPLANCEV, A. A. (1954): Methodological Handbook for the 1nvestigat)ion of the Ground-water Regime (in Russian). Gosgeoizdat, Moscow. ASTM (1958): Procedures for Testing Soils. American Society for Testing Materkk, Philadelphia. A. (1911): Plasticity of clays (in German). Overeatt fr. evemka UT ATTERBERO, Kungl. Lantbruksakademkm Handlinger och Tidakon, 60. 1911. BEAR, J. (1972): Dynamics of Fluids in Porous Media. Elsevier, New York, London, Amsterdam. BEAR,J., ZASLAVSKY,D. and IRMAY, S. (1968): Physical Principles of Water Percolation and Seepage. Arid Zone Research, UNESCO, Paris. G. (1967): Practical Investigation of Subsurface Waters (in French). CASTAQNY, Dunod, Paris. DE WIEST, J. &I. (1968): Flow through Porous Media Chapter 2. Porosity and Permeability of Natural Materials by S. N. Davis. Academic Prees, New Ymk-London. HAZEN, A. (1892): Experiments upon the Purification of Sewage and Water at Lawrence Experimental Station. Health Publication of Ma.gaachueetta State Board, NO. 1. 1889-1891. JOHNSON, A. I., PRILL, R. C. and MORRIS, D. A. (1963): Specific Yield-Column Drainage and Centrifuge Moisture Content. Geological Survey, Water-mpply Paper, 1662-A. Washington. KAMENSKY, Q. N. (1940): Differential Equations of the Nonsteady Ground-water Flow and their Application to Calculate Backwater Effects (in Russian). Izvestiia of Akademii Nauk, USIS'R, No. 4. KAMENSKY.0. N. f1943): PrinciDles of Ground-water Dvnamics lin Russian). Gosgeoltekljz;lat, Modow. ' ~ Z D I A. , (1972): Soil Mechanics (4-th edition) (in Hungarian). Tankonyvkiad6 Budapest. KI~ZDI, A. (1974): Handbook of Soil Mechanics Vol. I. Soil Physics. Elsevier-Akad6miai Kiad6, Amsterdam, Budapest. KINQ,F. H. (1898): Principles and Conditionsof theMovements of Ground Water. 19th Annual Report of US. QS. 1897-98. Koviics, G. (1959a): Determination of the Flow Rate of Ground-water flow on the Basis of Analizing Water Balance (in Hungarian). V'iziigyi Kozlemdnyek, No. 3. KovAcs, G. (1959b): Ground-water Household. ICID Annual Bulletin, New-Delhi,

200

1 Fundamentals for the inveetigation of seepage

KOVACS,G. (1971): Salt Accumulation in Ground Water and in Soil. I A H S Sympoeium cm Qrcyund-wder Pouution, Moecow. KOVACS,G. (1972a): Ground-water Reeources and their Contact with Surf..-water Resources. EUE Seminar on Methodology for the Compilation of Balancse of Water Resources and Needa, B u d a p t , 1972. KOVACS,G. (1972b Asseamnent of Ground-water Re~ources.Sympoeium on Water Resources Planning, e x k o City, 1972. LAVROV,A. P., FADEJEVA, M. V., SACEOK, 0. I. VAXHOVSKY, A. P. and VASILJEV, V. F. (1972): Problems of Underground Water Regimea in Byelorueeian P o W c . UNESCO-IAHS Sympoeium on the Hydrology of Mamh-ridden A r e a , Minek, 1972. LEBEDIEV,A. V. (1961): Investigation of theRegimeand Balance of Ground Water on the Basis of Continuous Observations (in Run~im).Paper in the Volume: Methods of I n v e a t i g a t k and 0 - k in Engineering Geology and Hydrogeology, Gosgeoltekhizdat, M m w . LEBEDIEV,A. V. (1963): Method6 to Inveetigate the Balance of Ground Water (in Russian). Gosgmltekhkdat, Moscow, 1963. MAJOR, P. (1972a): Determination of Infiltration and Evaporation Considering the Capillary Condition of the Covering Layer (Manuscript in Hungarian). SciSntifiO Conierenm of V I T U K I . 1972. &OR, P. (1972b):'Investi ation of Balance Parametere of Ground Watere in Plain6 (in Hungarian). Scientific Repor&, No. 2162. (Manuscript). PRILL, R. C., JOHNSON,A. I. and MORBIE,D. A. (1966): Specso Yield - Laboratory Experiment8 Showing the Efleot of Time on Column Drainage. Geological Survey, Water-eupplyPaper, 1662-B. Washington. ROWE,P. R. 1966): Differenae Approximations to Partial Derivative for Uneven Spacings in the etwork. T r a m a d o m of AGU, 36. SCEOEI&ER, H. (1962): Ground Watere (in French). Masson, PEL&. Srdrr~,W. 0.(1933): Minimum Capillary Rise on Ideal Uniform Soil. P h y s b , May. SMITE, W. O., FOOTE, P. D., BUSANU,P. F. (1931): Capillary Raise in Sands of Uniform Spherical grain^. P h y s b , July. SZABOLCS, I., DARAB,K. and V~ALLYAI, G. (1969a): T h e Tisza Irrigation Syetema and the Fertility of the Soils in the Hungarian Lowland (in Hungarian). Agrokdmiu 4% Talajtan, No. 2. SZABOLCS, I., DAEAB,K. and V~ALLYAI, G. (196913): Alkalization Process Due to Irrigation on the Hungarian Plain. A p k d m i a 4.8 Talajtun, Supplementum. S ~ L Y F. , (1973): Determination of Ground-water A m t i o n and Seepa e Parameters on the Bask of Ground-water Obeervations (in Hungarian). Hidro&iai Rozl h y , No. 6 . TERZAUHI, I(. (1926): Soil Physical Basis of Mechanics of Earth Structures (in German). F. Deuticke, Wien. TERZAunx, K. (1943): Theoretied Soil Mechanice. John Wdey, New York, London, 1943. V~ALLYAI,G. (1967): Salt Acoumulation P'rocea in Soils in the Danube Valley (in Hungarian). Agrokdmia ke Talajtan, 1967. No. 3. VARALLYAI, G. (1970): Salt Movement, Salt Balance and their Importance on Irrigation S tmm on Lowland6 (in Hungarian). Agr6rtudomdnyi Kodemknyek. WO-,~. (1886): Investigation of Water Retention Capacity of Soils (in G m a n ) . Formhungen auf dem Gebiete der Agrikdtur Phyeik.. ZWNXER, F. (1930): Behaviour of Soils in COMection with Water (in German). Handbook of Soil Sciences, Springer, Berlin. Vol. VI.

li

d,

b

Part 2

Dynamic interpretation and determination of hydraulic conductivity in homogeneous loose clastic sediments

I n every case, where the purpose is to describe a special type of movement and to discover the interrelationships of physical processes for the mechanically correct (so-cdled deterministic) characterization of the motion, the detailed investigation of the acting forces is necessarily the first step. There are accelerating and retarding forces forming a balanced system in the case of steady movement. Distinguishing the various types of movement so that in each group the same forces have a dominating role, the limits of the validity zone can be determined, outside of which the motion is affected by other forces. The equilibrium of the accelerating and retarding forces gives an equation in each zone, on the basis of which the correct form of the movement equation can be constructed and even some numerical factors of the latter can also be calculated. The other important step in the theoretical investigation is to find a physical (geometrical) model of the analyzed system. When selecting this model, two contradictory aspects act against each other. The boundaries and structure of the model should be as simple aa possible, t o ensure the mathematical solution of the relevant movement equations. On the other hand, it is necessary to include in the model the largest possible number of parameters from the actual system, because each parameter may describe a special feature of the prototype. Following the determination of the acting forces and the geometrical model, the combination of results provides a conceptual model, which is suitable for the theoretical analysis of the process under investigation. The basic relationships between the interrelated variables may be determined in this way, and the variables having negligible effects can also be selected. The investigation has always t o be finalized with carefully executed experimental research, which serves a dual purpose: firstly it is necessary for the verification of the theoretical relationships, together with the indication of their validity zones, and, secondly some numerical constants of the formulae have to be determined from the data measured. The theoretically derived equations include some experimentally determined components, and thus these relationships are semi-empirical formulae. They have, however, many advantages compared with the purely empirical methods. They are based on the actual dynamic character of the motion

202

2 Determination of hydmdk conductivity

and their application, therefore, reflects the real physical contacts between the variables under consideration. The use of these formulae prevents particular mistakes which are very often caused by applying empirical equations t o practical situations. An example of such a mistake is where an equation derived for a given character of movement is applied in another zone without considering the basic differences between the physical processes and using only some corrective factors, which, though valid in the vicinity of the measurements used for their calculation, are not so in the whole zone. This possible mistake can well be illustrated by an example taken from seepage hydraulics, i.e. where Darcy’s equation is used to describe all types of seepage. Although it is clearly proved that the linear relationship between velocity and gradient is valid only in the case of laminar movement (when the dominating forces are gravity and friction), the same function is very often applied to describe other types of movement as well, using only a numerical factor to correct the resulting error. Naturally, this factor differs from cam to case and investigations determining a factor from a limited number of measurements cannot, therefore, generally be accepted. The method of investigation previously explained, and proved to be physically correct, will be appliedin this part of the book for the characterization of seepage through an homogeneous porous nzedium, the solid m t r i x of which is composed of evenly distributed grains of loose clastic sediments. Using the continuum approach the main aspects in the determination of the geometrical model were explained in Section 1.1.3. It was followed by the analysis of the physical parameters of soil in loose clastic sediments, on the bmis of which the effective diameter ( D J , the shape coefficient of grains (a) and the effective porosity ( n ) were selected as the most important characteristics to be incorporated into the model as parameters. Comparing the various possible types of models, it was found that a bundle of straight tubes lying parallel to the direction of flow and composed of small sections of two different diameters, is simple enough t o be amenable to mathematical treatment, and at the same time is suitable for the accurate reproduction of the most important hydraulic processes occurring in homogeneous loose clastic sediments. The h a 1 result is summarized in Section 1.2.5 where all the equations, from which the model-parameters can be calculated aa functions of the characteristic physical soil data are listed. After having determined the geometrical model, the next step necessary to complete the conceptual model, is the selection and characterization of the acting forces. It has to be followed by the experimental verification of the model. This dynamic analysis and experimental research is discussed in the following part of the book.

Chapter 2.1 Dynamic analysis of seepage As already stated, the creation and maintenance of movement requires the action of an accelerating force, or of a system of such forces. The retarding forces act against the accelerating ones, and they generally increase with the

203

2.1 Dynamic analysis of seepage

velocity of the motion. Thus, there is a limit where the two groups of forces reach a state of equilibrium and a steady movement develops. When investigating the dynamics of seepage, the fist task is to analyze the acting forces. It is necessary t o determine their physical properties and to find the most suitable way of incorporating them in the conceptual model. There are many cases where the dynamic laws established in theoretical physics are not suitable for direct application, because the model would be too complicated, making the mathematical solution more difficult. Acceplable approximations have to be determined in these cases, which ensure both the necessary simplicity and the accuracy required in practice. One important method of approximation is to dbthguish between the dominant and the negligible forces. The ratio of the considered and neglected forces gives numerical values which indicate the limits of the validity zones of the various approximations applied by neglecting certain factors. Similar dimensionless quotients can be derived as the ratios of the accelerating and retarding forces. These parameters provide the theoretical form of the bwic movement equations, which will later be verified by experiments. The stages in the dynamic investigation laid down in the previous paragraphs will be discussed in detail in this chapter.

2.1.1 Forces influencing the flow between grains In seepage hydraulics the most important accelerating force is gravity and in most cmes i t is sufficient to consider only this force, as indicated in connection with the definition of seepage. Apart from this force the pressure of the covering layer may be a dominant factor in deeper lying aquifers, if the comprewibility of the latter is significant. I n the vicinity of the water table, capillarity, and in the unsaturated zone, adhesion between water and grains may also have an important accelerating role. The latter should also be included in the group of retarding forces in he-grained sediments, where the ratio of the internal surface related to the volume of the sample is high. Among the retarding forces, three generally become dominant: adhesion, friction and inertia. Hence, there are six forces, detailed knowledge of which is required for the dynamic investigation of seepage. Four of them are well known and generally used in other branches of hydraulics and soil mechanics. These are: gravity, the pressure of the overlying layer, inertia, and friction. The other two forces (i.e. capillarity and adhesion, both created by molecular interaction) have important roles, mainly in the investigation of seepage, and their more detailed discussion is, therefore, justified here. Qravity is a maas-force (body-force) having potential and caused by the attraction of the Earth. Its size (a)is equal to the product of the acceleration due to gravity ( 9 )and the moving mass ( M ) and is directed vertically downwards: (2.14) c)L = Mg = Veg; where V is the volume of the investigated body, and

e is its density.

a04

2 Determination of hydraulic conduotivity

A force has potential when there exists a potential-function - a singlevalued scalar function of the space coordinate system - whose gradient is equal to the force in question (the differential quotient of the potential function in any direction is equal to the component of the force vector in the same direction). I n the case of a mass-force, the potential may be related to the unit of the mass, and thus, the gradient of thepotentialfunction supplies the vector of acceleration. In a gravitational field, choosing one axis of the coordinate system as vertical, the vector of acceleration has no component normal to this axis, and the derivate of the potential-function, therefore, i n a vertical direction is equal to the acceleration due to gravity, and the potential can be calculated from the following equation: and for a unit mma

u = Mzg; u1= zg.

(2.1-2)

Consequently, the potential is proportional to the distance (21 . , measured from an arbitrarily chosen horizontal plane used as a datum of reference.

unloaded layer

drained layer

Fig. 2.1-1. Hydimstatic pressure distribution and its modification caused by surface load

m5

2.1 Dynamic analysis of seepage

When the total weight of the grains in the layer and that of the covering layer, is transferred from grain to grain at the points of contact, the water in the pores is loaded only by the overlying water body, and hence, the vertical distribution of water pressure can be characterized by the hydrostatic pressure-distribution curve (Fig. 2.1-la). I n this case, the sum of potentialand pressure-energy is constant at each point, and water movement can be created, therefore, only by outside action. The volume of pores in the layer decreases when the latter is d e c t e d by a surplus load placed on its surface. If the loaded layer is drained (a given amount of water equivalent to the decrease of pores can be immediately discharged from the layer), the surplus load is transferred through the grains as well, and hence, the pressure distribution in the water is not modified by this outside load (Fig. 2.1-lb). Another extreme case is, where the borders of the aquifer are impervious, the layer is compressible and the total load has to be taken over by the water itself (Fig. 2.1-lc) (KBzdi, 1963, 1974). I n an aquifer within a stratum composed of pervious and impervious layers, the outside load is created by the overlying layers. Assuming the stratum to be completely filled with water (the water table is at the level of the surface), the probable water pressure can be limited by two extreme vdues (i.e. the hydrostatic pressure and the loaded condition), in the form of the following unequality (Fig. 2.1-2):

(t=

plane 0 I,g+b, T~pV=Zg plane I Zlg+hl 7sQ~ = =ZY+~I crs-rv)PV

1

pressure differem between plane 0 and plane !related to the datum level

Fig. 2.1-2. Water pressure in natural layer influenod by the weight of overlying layere

206

2 Determination of hydraulic conductivity

(2.1-3)

where yI, is the specific weight of water and ys is that of the solid grains. When the pressure is higher than the lower limit, there is a potential difference between the upper and lower surfaces of the layer covering the aquifer, which may create movement through the impervious, or semiimpervious covering layer (and drain the water from the aquifer). Thus, the upper limit in Eq. (2.1-3) describes the condition at the point of time when the load WM placed upon the layer, while the lower limit characterizes the total consolidation, the development of which can be theoretically expected after an infinite time. The actual water pressure depends on the time elapsed since the loading, the size of the load and the permeability of the layers (that of both the aquifer in question and the bordering formations). A further uncertainty is caused by the fact that many loadings follow each other before the total development of consolidation caused by the earlier loading. A decrease in the load can also sometimes be observed (erosion, exfoliation), and since the layer is not elastic, this is not always followed by complete expansion. For this reason, the expected value of pressure cannot be calculated, only its possible extreme limits can be estimated (Fig. 2.1-3). When calculating the parameters of seepage, the pressure diflerence between the investigated and the draining space can be taken into consideration as an active force. This value divided by the specific weight of water is expressed a.s the equivalent height of a water-column and thus, i t can be taken into account as a supplementary part of gravity. The presence of this type of force requires special investigation, therefore, only in the care of unsteady movement, when the development of consolidation influences the character of movement as well. Inertia is also a body-force similar to gravity. Consequently, it is the product of mass and acceleration. I n this case, the latter is the acceleration of the motion, which is retarded by this force: (2.1-4)

Internal friction is a retarding force i n a viscous fluid, obstructing the relative displacement of two water particles moving along two neighbouring paths (Fig. 2.1-4). This force is directed against the flow and its size is proportional to the surface ( A ) ,parallel to the flow line and perpendicular to the direction of the velocity gradient. The friction acting on a surface of unit size (specific value) is the shearing stress (z), which is the product of dynamic viscosity (7)and the change in velocity in a direction normal to the area investigated (velocity gradient):

S=Az;

z=q-.

dv dn

(2.1-5)

2.1 Dynamic analysis of seepage

207

pressure bead in wafer column CmJ

Fig. 2.1-3. Values of water pressure measured at various depths

g&yq

- --

2

Fig. 2.1-4 Sketch for the interpretation of viscosity and shearing stress

The ideal (non-viscous) fluid is a theoretical concept. Its molecules are absolutely movable, and their displacement related to each other is not affected by any resistance. The actual water molecules are dipoles, the positive and negative charges being located asymmetrically within the molecule

208

2 Determination of hydraulic conductivity

(Fig. 2.1-5). This structure creates an electrostatic field around the molecules, the find result being the attraction between the molecules and the binding of positive and negative poles (Newtonian fluid). The shearing stress between two layers of molecules acting on a unit area is proportional to the velocity difference of the layers. The factor of proportionality is the dynamic

yofy+ Fig. 2.1-5. Dipole structure of water molecules

viscosity, the technical dimension of which was [kp sec m-2]. I n practice, the physical dimension (poise)was also often used, which is roughly 1/100 of the former: 1 poise = l[dyn sec cm-21 = 0.01019 [kp sec m-2]. I n the S I system the physical and the praotical units are the pascal seconds (Pas) and the Nsm+ (1 Pas = lNsm-'= lop). I n many cams the physical behaviour of the fluid is characterized by viscosity and density. It is reasonable, therefore, to introduce a combined parameter, which is the quotient of dynamic viscosity and density v = q/e;

(2.1-6)

known M kinematic viscosity, and its dimension is [L2T-l]. The orientation of the molecules is not constant. The bonds are disintegrated by Brownian movement and new groups are continually developing. Thus, viscosity is a probability parameter formed by a number of randomly bound molecules, which always represents the equilibrium between attraction and Brownian movement. Thus, viscosity can be regarded aa an example of the application of the continuum approach when an arbitrarily chosen parameter is applied to represent a property of the actual group of molecules and to investigate the continuum of a fluid instead of the real molecular structure. It is evident, therefore, that viscosity decreases with inereusing temperature. The equation generally applied in practice to describe this relationship is as follows: 71 (2.1-7) 1 0.0337 t 0.00022 t2 ' qo

+

+

209

2.1 Dynamic analysis of seepage

where t is the temperature of the fluid at "C, and q,, is the viscosity of the fluid when t = OOC. The latter is the function of the electrostatic field created by the dipole molecules, and hence, this parameter is a material constant of the fluid, which is T o = 1 . 7 7 10-3 ~ [pa 81

in the case of water at a pressure of 1 atm (100 kPa). The theory that viscosity is a result of molecular attraction, is supported by the fact that the relationship between viscosity and absolute temperature ( T ) can be characterized by an equation having the same structure as that between vapour pressure ( p ) and absolute temperature, and thus, viscosity can be directly related to vapour pressure (Madge, 1934):

[:I

In - = A -

BIT;

Inp = A' - B'fT ;

and thus qpB"B

(2.1-8)

c.

1

It is also quite evident that the increasing pressure acting on the fluid decreases the distance between the molecules, and hence, incremes viscosity (although in the case of water some irregularity can be found, as shown in 1.0016 Fig. 2.1-6). Since the compressibility of water is very small (1.0010 times the increase of viscosity as a result of 1 kp/cm2 increase in pressure), this dependence can be neglected in seepage hydraulics. As with the increase of pressure, all kinds of external eflects which increase the probability of the orientation and binding of dipoles, increase the viscosity

-

0.0'

'I 2' 3'

I

4

I

5

pressure lo^

'

6

I

7

'

8

I

'

9 t O

N/mm 2.3

Fig. 2.1-6. Relationship between temperature, pressure and viscosity of water 14

210

2 Determination of hydraulic conductivity

of the fluid. Among these influences, the most important is the electrostatic charge of the solid wall surrounding the fluid. The investigation of this effect leads to the study of the next influencing force, i.e. adhesion. The solid wall has electrostatic charges related to the water, which polarize the water molecules along the wall. The molecules are fully orientated in the first layer. They are bound by their opposite charge t o the wall and their charge, which is the same as that of the wall, is orientated towards the

(-)

(-1 (-1 (->

(-) (-)

(-1 (-)

Fig. 2.1-7. Distribution of polarized water molecules along the walls of grains

interior of the water body. The effect of polarization can also be observed inside this double-layer within the water body. The number of orientated molecules, however, is inversely proportional t o the distance measured from the wall (Fig. 2.1-7) (Erdey-Grliz and Schay, 1954; Khzdi, 1972, 1974). It is not yet absolutely clear, whether the adhesive force is influenced by the mineralogical and chemical character of the grains forming the solid skeleton of the seepage field. On the basis of the explanation given in Section 1.2.3 concerning the interaction between water and grains, it is very likely that this relationship is determined by the Van der Waub force, the effect of which can be characterized by a tension ( p ) inversely proportional t o the sixth power of the distance from the wall (a), independent of the mineralogical and chemical character of the latter [see also Eq. (1.3-15)]:

C ' 6

p=-.

(2.1-9)

There are, however, a few exceptions, when the presence of some minerals having a special structure (Na-montmorillonite, amorphous colloids) modi-

2.1 Dynamic analysis of seepage

211

fies the normal relationship between the tension in question and the water content of the sample. Another observable phenomenon is that the repulsive force influencing the interaction of colloid grains i n the suspension, isagected by the mineralogical and chemical character of small particles (see Chapter 1.2). Since the repulsive force determines the morphological character of the grains (the ratio of the irreversible bonds to the total amount of colloid particles), the mineralogical and chemical character of the grains indirectly influences the size of the active surface of the sample, and hence, the interaction between the water and grains as well. This influence, however, can be neglected in seepage hydraulics, because the actual active surface is always taken into consideration by the application of the effective diameter. There are some publications which state that adhesion is inversely proportional to the third power of the water content, or sometimes to its second power, when the free charges of the grains are bound by monovalent cations (JuhAsz, 1967; Mattson, 1932; Vageler, 1935). The detailed explanation of the soil-moisture retention curve is given in Chapter 1.3, where the meaning of the two separated sections of this curve is also explained. The validity of Eq. (2.1-9) is also proved for the uppermost section of this curve, where the interaction between the grains and water is influenced only (or more precisely, mostly) by adhesion. The lower power mentioned in the papers, is valid in the open capillary zone, where the water content is jointly determined by capillarity and adhesion. I n this zone, however, the relationship between water content and tension can be described more easily by the use of the probability distribution of capillary head, than by formulae expressed in the form of a direct function. The detailed study of such equations, therefore can be neglected here. The tension is perpendicular to t e solid wall, and hence, it has no component in the direction of flow. It is necessary, therefore, to investigate the way in which the motion of water can be influenced by this action. The accelerating froce displaces polarized water molecules in the electrostatic field of the solid grains, and hence,thepaths of the dipoles cross the forcelines of the electric field. It is well known, that this process consumes energy and can be expressed by the action of a force parallel to the flow line (bimilar to friction). The magnitude of this force naturally decreases with increasing distance from the wall (KovAcs, 1957, 1958). A very similar result can be achieved when the eflect of the adhesive force is indirectly taken into account by considering its influence on the flow as a surplus shearing stress. The application of this method means a deviation from the Newtonian fluid, and requires the investigation of a material having a shearing stress in static condition as well. This kind of medium is called Bingham’s body or Bingham’s plastic (Bingham, 1922; Reiner, 1952). In this cme, the sum of friction and the secondary effect of adhesion acting in the direction of flow (S E ) can be expressed by a common equation using a shearing stress (or yield stress z’) having a member independent of velocity, instead of that expressed by Eq. (2.1-5). The existence of the static shearing stress (zo) is proved by the fact that in cohesive layers (and also in capillary tubesof verysmall diameter), there

+

14*

212

2 Determination of hydraulic conductivity

is a limiting value of gradient (I,,, threshold gradient), below which there is no movement at all. Thus, zero velocity can occur with a finite gradient. The measured value of the threshold gradient can, therefore, be used to characterize the static shearing stress (Kar&diand Torok, 1955; Kutilek, 1965). Bondarenko and Nerpin (1966, 1967)andBondarenko (1966)havepublished the results of a series of experiments designed to determine the threshold gradient and the static shearing stress. They measured the corresponding velocity and gradient values of the flow of water and ethyl alcohol, respectively, through glass and quartz capillary tubes of known constant diameter. Figure 2.1-8a shows the results measured using water and Fig. 2.1-8b those using ethyl alcohol. The data represented in a v vs. I coordi. nate system prove that the relationship between the two variables cart be well approximated by a linear equution over almost the entire range of velocity. The straight line representing this relationship does not cross the origin of the coordinate system [as it should if Darcy's equation, Eq. (1.1-9) were

(a) wafer

(b) ethyl alcohol 0.3 a2

0.1

0

0

b

r

0.1

I

I

0.2 0.3

d4

a's

d.6

I a'8

d7

0.1

a2

0.3 0.4

0.5 0.6

0.7

b5

LO 0.5 0

2.5

2.0 1.5 1.0

0.5

0

Fig. 2.1-8. Relationship between velocity and hydraulic gradient in the zone of small velocities (after Bonderenko and Nerpin. 1967)

2.1 Dynamic analysis of seepage

213

valid], but intersects the horizontal axis at a point I , . The gradient of I , is not equal, however, to the threshold value, because the movement of fluid also begins under the influence of smaller gradients inthezoneof very low velocities, where the linear relationship is not valid, as shown by Fig. 2.1-8c, which is the enlarged form of the lower parts of a velocity vs. gradient graph. The figure is also supplemented by a theoretical sketch (Fig. 2.1-8d) showing the interpretation of the threshold gradient ( I , ) and the I , value characterizing the position of the linear relationship, and fkally, that of a third parameter ( I , , ) , applied by Bondarenko and Nerpin for the evaluation of the measurements. The basis of the evaluation is the Buckingham-Reiner’s equation, or more precisely, its simplified form, in which the static shearing stress is assumed to be constant. This constant can be calculated from the velocity and gradient data: v =KI

I,,

-

[ 3 M 4

3

4 I,, + 1 3 L i

[

(Buckingham-Reiner’s equation);

- K I 1 - - Lo ; O-

:rII]

(simplifiedform);

(2.1-10)

and the I , , constant proportional to the threshold gradient: 1

to=IryI,,; 2

where K is Darcy’s hydraulic conductivity, r is the radius of the capillary tube, y is the specific weight of the fluid and I , , is the characteristic gradient, the interpretation of which is given in Fig. 2.1-8d. An important result of this analysis is that the tovalue calculated in this way was found to be independent of the material of both the tube and the fluid and of the diameter of the tube, and it is a single-valued function of the temperature. The intersection of the straight line characterizing the v vs. I relationship and the horizontal axis (Il),is proportional to the I , , value. It can be expected, therefore, that the r y I , product behaves similarly, and the data determined from the graphs of Figs 2.1-8a and 2.1-8b prove this expectation (Table 2.1-1). Bondarenko (1973) has also proved that the non-Newtonian behaviour of the fluid is cawed mostly by the hydrogen bounds between the molecules. The comparison of the virtual hydraulic conductivity values (the ratio of corresponding data of velocity and gradient) measured with various fluids (Fig. 2.1-9) testifies the fact that the rapid decrease of this parameter was observed in the zone of small gradientsonly,if the fluid applied during the experiments had dipole molecules. At the same time, the change in concentration of the dissolved material in the water (KC1in the experiments) did not modify the calculated t oparameter as shown in Table 2.1-1. This gives the r y I , products recalculated from the shearing stress given in the original publication. This result is a negative proof of the previous statement,

214

2 Determination of hydraulic conductivity Table 2.1-1. The characteristic I , gradient measured I,xlO' parameter if temperatun, is r C ]

Pipe Type of fluid

Distilled water y = l.O[pcm-S]

61 103.6 166 198 266

6.4 -

-

-

4,l 2.6 1.3 1.3 1.0

-

-

-

2.80 1.26 1.76

1.30 0.66 0.26

-

-

0.69

0.16

-

-

0.08 -

-

-

-

-

0.40 0.16

0.00

-

average Ethyl alcohol y = 0.8[pcm'*]

67 109.26 166

-

-

6.4

3.66 2.22

3.3

-

-

-

1.1

-

0.68 0.31

0.75

-

0.10

-

-

-

-

-

average

KC1 solution y = 1.0 [ p ~ r n - ~ ] N N N 1 p = 9.807 X

198 -

1.2 1.3 1.3

-

-

-

-

-

-

average

lo-' N

showing that the dissolved minerals do not influence the behaviour of the fluid. It was also found that the heat value of the H-bonds (a),can be related to the shearing stress in the form of an equation having a similar

0 0.2 0.4 a6 0.6 LO 1.2 64 !.6 hydraulic gradient, I Fig. 2.1-9. Comparison of the virtual h draulic conduotivitiea determined by various fluids (after 6ondarenk0, 1973)

2.1 Dynamic analysis of seepage

215

in capillary tubes by Bondarenko and Nerpin (1967) 1 . ~ ~ 1 0 ~pcm-'] 4

15

3.26

1

20

1

a5 -

-

2.09 2.69 2.14 2.67 2.66

-

3.26

2.41

-

-

-

__ -

1

30

I

40

I

60

1

55

1

58

0.66 0.67 0.41

0.20

0.04

-

-

-

0.00

-

1.43 1.30 1.24

0.28

-

-

1.66

-

0.40

-

-

-

-

-

1.38

0.64

0.24

0.04

0.00

-

2.46

1.66

1.01

-

-

-

0.33

2.88

-

-

-

-

1.47

1.12

0.41

0.13

-

-

2.88

1.97

1.39

0.71

0.23

-

-

-

remlouleted from z,

structure to those giving this parameter as a function of other properties caused by the dipole character of the molecules. The relationship expressing the value of G as a function of zo is the following: (2.1-1 1) where T is the absolute temperature, A and C are constants and the two subscripts ( 1 and 2) indicate the corresponding values of T and zo. Since the G value can be regarded as a constant within a limited range of temperatures, Eq. (2.1-11) can also be used to determine the general pattern of the relationship between the r y I , product and temperature. Choosing a fixed T, value (T,= 273OC, i.e. the temperature in centigrade is equal to can be neglected in the zone of praczero), the dependence of r y I , on TITz tical importance and the (T,*- T,)difference can be supplemented by the temperature expressed in centigrade. The neglect of the T,Tzproduct can be partially compensated by the appropriate selection of the C parameter. Thus, a linear relationship can be expected between the variables under investigation ( t OC and r y 11)in a semi-logarithmic coordinate system. In Fig. 2.1-10 the data summarized in Table 2.1-1 were plotted, resulting in a straight line. The constants and the corresponding basic values of the temperature and the r y I , products, were found to be as follows: C =

216

2 Determination of hydraulic conductivity

- - 5 x 10-5; G/AT,T, = 1/50; T , = 273OC; (ry I & = 7 x p cm-2. It is also shown in the figure that by shifting the line horizontally by &5OC

two enveloping curves can be achieved, which border a stripe incorporating tmhewhole range of scatter of the measurements. The probable zone of the

3 --

2--

8verage Wues of tbe measurements determibedw#h o wfer + efhjl alcohol KCI S d U f h R I

I

I

points measured and the I, vs. t relationship within the 10°C < t < 6OoC range of temperature can be described by the following equations: tOC

*

5oc

50

1

--

[log ( r y I,

+

+5x

+ 3.1251 ;

(2.1-12)

I, = -{exp [- 4.6(t°C 156.3)10-2] - 5 x 10-5). V Equation (2.1-12) characterizes the flow of fluids having dipole molecules through a capillary tube with a contltant radius r . The application of this dynamic principle to describe low velocity seepage through fine grained, loose, claatic sediments will be discussed in Chapter 2.3. The interrelation between the threshold gradient (I,)and the I, parameter will also be explained.

2.1 Dynamic analysis of seepage

217

It is necessary to analyze here the various opinions regarding the character of the t ostatic shearing stress t o clarify the influence of the non-Newtonian fluid on seepage. Most of the authors dealing with this problem assume that the static shearing stress is a function of the distance ( 6 ) measured from the interface between the solid bordering material and the non-Newtonian fluid (Kovbcs, 1957,1958; J u h h z , 1958; KarM and V. Nagy, 1960; Childs and Tzimas, 1971). On the contrary, Bondarenko (1973) argues that this parameter has t o be a constant throughout the entire interior of the fluid, because he does not take account of any physical influence which may modify the property of the fluid. It appears to be proved that the special behaviour of the fluid is caused by the fact that the chains composed of dipole molecules are longer and more stable in the electrostatic field of the wall than in the interior of the fluid. It is reasonable, therefore, to assume that the physical parameter chosen to characterize this property should be proportional to the attraction exerted by the solid material on the molecules of the fluid. Therefore, the hypothesis of a static shearing stress decreasing with increming distance from the wall [zo(6)] is acceptable. On the other hand, the relationship between the attraction and zo is not necessarily linear and a more gentle decrease than that of the attraction indicated by Eq. (2.1-9), can also be assumed. A suitable approximation can be determined only by experiments (as will be ghown in Chapter 2.3) and it must not be linkedto,or influenced by, the high power usedinEq. (2.1-9) to describe the attraction vs. distance relationship (Thirriot, 1969). On the basis of the dynamic analysis of the non-Newtonian behaviour of water, the total shearing stress can be expressed as the sum of the static value t o , depending on the distance from the wall ( 6 ) and the Newtonian parameter being proportional to viscosity, and the velocity gradient normal to the direction of flow. Applying this total shearing stress, the combined retarding effect of friction and the molecular forces can be described by one equation:

consequently,

(2.1-13)

S

+E =A

[

to(6)

3

+ 7-

.

It is very likely that the second member on the left-hand side of Eq. (2.1-13), depending on viscosity and velocity gradient, is also a function of the distance measured from the wall. As shown, however, in connection with the investigation of friction, the viscosity is doubled when pressure is increased 1000 times. Combining this condition with the relationship between tension and the distance from the wall, it can be proved that the modification of viscosity is insignificant in a very large zone, and hence, this effect can be neglected in seepage hydraulics. The sixth dominant force influencing the flow between grains is capillarity, which occurs on the surface of contact between water and air. The most

218

2 Determination of hydraulic conductivity

simple explanation of this force can be given by taking into consideration the attraction between water molecules. In the interior of the water body the attractive forces act equally from every direction, and thus, all molecules are in a balanced condition. A molecule on the water surface is attracted only by molecules below the surface. Thus, in an elementarily thin layer along the contacting surfaces of two different media there is free energy, which has to be balanced by pressure differences existing between the two sides of this layer (within the two contacting media). To characterize the amount of free energy its specific value is used, which is equal to the ratio of the surface energy P(A)and the surface ( A ) .Relating this quotient to an infinitely small surface element the interfacial tension is achieved (aik),which is a material constant for any pair of media, i and k, depending only on absolute temperature (T) : ( 2.1-14)

The special value of the interfacial tension developing on the surface of a fluid covered by its own vapour, is called surface tension (a).According to Eq. (2.1-14), its dimension can be calculated as the ratio of energy (work) over surface, which is equal to force over distance. Thus, surface tension can be interpreted m a force acting within the surface along a line having a length of unity [FL-l]. Eotvos law has established a relationship between surface tension and the other molecular parameters of the medium (molecular volume, molecular weight, latent heat, absolute temperature, vapour pressure). The relation between surface tension and temperature can be expressed from this law as follows: (2.1-15) u v213 = c(To- T ) = ct; where v is the molecular volume of the medium and T o is the critical temperature, where the product of av213 becomes zero. Eotvos (1886) proved that the c factor is constant for fluids having R socalled simply composed molecular structure (Table 2.1-2) : (2.1-16)

Some media (especially water, alcohols, fatty acids) exhibit exceptional behaviour. The c factor of ethyl alcohol increases with temperature and only reaches the constant of Eq. (2.1-16) near the critical temperature. The parameter of acetic acid is considerably lower (0.134) than the general constant, but the expected 0.227 value can be achieved if a double molecule (2C2H40,) is considered. The data for water listed in Table 2.1-2 were also calculated assuming the binding of two molecules. As seen in this case, normal behaviour prevails above l O O O C , but below this limit the c factor decreases with decreasing temperature. This relationship also proves the previously mentioned hypothesis in connection with the probable molecular structure of water: i.e. the number of bound water molecules increases with decreasing

2.1 Dynamic analysis of seepage

219

Table 2.1-2. Experimental data characterizing the relationship between surfaoe ternion o f various fluids and temperature (after Eotvos, 1886) Type of fluid

Ethyl ether Ethyl ether Ethyl ether Ethylene bromide Ethylene bromide Chloroform Mercuric methyl Carbon tetrachloride Carbon dioxide Carbon bisulphide Sulfuric acid Ethyl alcohol Ethyl alcohol Ethyl alcohol Ethyl alcohol Ethyl alcohol Ethyl alcohol Water ( 2 S O ) Water Water Water Acetic acid (C,H,O,) Acetic acid Acetic acid Acetic acid (2 C,H,O,)

6- 62 62-120 170-190 20- 99 99-213 20- 60 20- 99 3- 63 3- 13 22- 78 2- 60 21- 78 78-108 108-138 138-168 168-199 199-236 3- 40 40-100 100-160 160-210 21-107 107-160 160-230 21-160

0.228 0.226 0.221 0.227 0.232 0.230 0.228 0.231 0.228 0.237 0.230 0.104 0.136 0.169 0.183 0.202 0.226 0.169 0.180 0.228 0.227 0.132 0.132 0.138 0.211

temperature. As a final result of this discussion the change in the surface tension of water in relation to temperature is given in the form of a graph (Fig. 2.1-11). Let us investigate the points of contact of three immiscible fluids (or gmeous media). I n a section normal to the three surfaces of contact, the balanced condition of the acting interfacial tensions necessitates the closure of the vector triangle formed by the three tensions (Fig. 2.1-12). From this condition, the angles ( aik)of the contact surfaces measured from an arbitrarily chosen axis can be calculated as a function of the numerical values of the interfacial tensions, or - if the axis is equal to the tangent of one of the contacting surfaces - the so-called contact angles (Oik)can be determined (NBmeth, 1963; Bear, 1972):

+

aI2cos aI2 or

023

cos a23

uI2 + a,, cos O,,

+

+ 013 cos a13 = 0; (2.1-17)

613

cos

013

= 0.

This condition indicates at the same time, whether or not the development of the balanced condition is possible. It is evident that if u12 (u23 yU), there is no possibility of fulf3ling Eq. (2.1-17) and, therefore, equilibrtum cannot be achieved at the contact of these media.

>

+

220

2 Determination of hydraulic conductivity

75

5 70

.$ 65

8

S! 60

& 55 $

50 -10

u

10 20 30 40 50 60 70 80 90 i o o m

temperature, t tecl

Fig. 2.1-11. Relationship between the surface tension of water and temperature

Fig. 2.1-12. Balanced condition of surface tensions at the points of contact of three immiscible fluids

Equation (2.1-17) can be simplified if one of the contactingmaterials is solid. The molecular forces do not modify its form, and hence, the interfacial tensions between the solid and the two liquid (or gaseous) media lie along a straight line (Fig. 2.1-13): ~ 2 cos 3

0 + 613 -

= 0;

cos 0 =

- u13

.

(2.1-18)

012

It follows from Eq. (2.1-18) that if a,,

> al3, then

cos 0 is positive and

0 < 5 .Fluid 1 is called the wetting fluid in this caae, while in the opposite 2

case

lz < uI3),cos 0 < 0 and 0 > , and fluid 1 is a non-wetting one.

2

2.1 Dynamic analysis of seepage

221

Fig. 2.1-13. Angle of contact between the surface of two liquid media and a solid wall

Knowing the parameters characterizing the surface tension of a fluid and its contact with the solid wall ( 0 and O ) ,the pressure difference between the two sides of the surface (capillary pressure p,) and the form of the surface near the wall can be determined. The change of free energy when an elementary part of the surface df = = dZ, * dZ, moves in a direction normal to its position by an infinite dC distance, serves as the basis of the derivation of capillary pressure (Fig. 2.1-14) (Levich, 1958). If R, and R, indicate the radii of curvature in the main directions, the modified area of the investigated df element after dC displacement is: (2.1-19)

The total change of free energy can be calculated by integrating individual elements over the whole surface. At the same time, it is evident that equi-

\

Fig. 2.1-14. Sketch for the derivation of capillary pressure

222

2 Determination of hydraulic conductivity

librium can develop only if the free energy is minimal and, therefore, the elemental change is zero:

Equation (2.1-20) h w a solution for any infinite d5 displacement if (2.1-21 )

$

-

-

c of Q gX Y of == C017Sf.C017Sf.- Q gX Y

Fig. 2.1-16. Vertical section of free water surface near a vertical wall

For simple geometrical forms, the position of the free surface of a fluid (capillary surface or meniscus) can be determined by integration. Along this surface, the pressure of the upper medium is the atmospheric value (p, = = p A = const.). Considering this condition, the pressure of fluid in a static condition can easily be calculated in the force field of gravity: p , = const. - e g x .

(2.1-22)

For example in the vicinity of a vertical, solid wall (the radius of the curvature being infinite in one direction, R, =-), the condition of the equilibrium is (Fig. 2.1-15):

d2Y -

Equation (2.1-23) can be solved by double integration. The two boundary conditions necessary t o determine the two integration constants are given dY = tan 0 ; and if y = at the two extremities of the surface if y = 0, = oo,x= 0

I.

[

dx

I n a space surrounded by solid vertical walls, the develop-

ment of the meniscus is influenced from each direction by the presence of

2.1 Dynamic analysis of seepage

223

a wall. If the distance between the opposite walls is small enough, it can be assumed, that the curvature is constant in the section normal to the walls. Thus, in capillary slits or tubes having regular forms in a horizontal plain, the meniscus can be approximated by a cylinder (in a slit), ellipsoid of rotation (in an elliptical tube) or sphere (in a circular tube) (Fig. 2.1-16). In these cases, the curvature and the capillary pressure can be calculated from the geometrical parameters of the horizontal section of the surrounding space and from the contact angle:

Fig. 1.1-16. Approximative forms of menisci in slits and tubes

224

2 Determination of hydraulic conductivity

(a) I n a slit having a width of b (Fig. 2.1-16a):

(b) In an elliptical tube the two main axes of which are a and b (Fig. 2.1-14b): a 2

R 1-

l COY

, R , = - -b

1 2 cos

.

0

o

1 ; p,=2a cos 0 -

a

1 +; b

(c) In a circular tube with a diameter of d (Fig. 2.1-16c): d 1 4acosO R , 1R2= R = ; P,= d 2 cos o

(2.1-25)

(2.1-26)

It can be seen that the sign of capillar pressure is determined by the size n of the contact angle: if 0 < - ; p , 2 opposite case 0

> 0 , and, therefore, p 1 > p 2 ; in the

II >-; p , < 0 , and p , < p,;

2

n and finally, if 0 =-,pc= 2

0,

and p 1 = p,. The consequence of the capillary pressure not being equal to zero is that the position of the meniscus i n the mpillary tube or slit will be diflerent from the level of the contacting water body having a large surface. Choosing this latter level as a datum where the surplus pressure is zero, the average height of the meniscus (mpillary rise) above this level can be calculated from capillary pressure:

h, =

4acos @

P

es

; in a circular tube h, = --

Y

.

(2.1-27)

d

Hence, the meniscus is above the reference level if the contact angle is 76

smaller than - , while the surface is lowered in the capillary tube in the case 2

3

of non-wetting fluid 0 >-

[

.

Investigating the contact between water, air and quartz (as fluid, gaseous and solid media), it was found that the contact angle is near zero, although this value changes, t o agreat extent, with the condition of the solid surface. Substituting cos 0 = 1 end the value of the surface tension belonging t o the most commonly occurring temperature (i.e. 15 20°C) into Eq. (2.1-27) a simple approxjmation for the capillary rise of water in a glass tube of diameter d can, in practice be proposed [see also Eq. (1.3-26)]:

-

h, [cm] =

0.30 [cm2]

d [cml

(2.1-28)

225

2.1 Dynamic analysis of seepage

Some observations show that the contact angle depends not only on the quality of the contacting media and the condition of the solid surfaces, but that it is also influenced by the movement of the meniscus and by the pressure conditions prevailing in the capillary tube. This phenomenon can easily be recognized when a raindrop moves along the slope of a glrtss plate, the drop having a large contact angle at its front and an angle approaching zero at the back (Fig. 2.1-17a), the phenomenon is, therefore, called, the raindrop

(a)

(Q7

(b)

raindrop

wetting

drainage

Fig. 2.1-17. Development of different angles of contact in capillary tubes

effect. The same change in 0 characterizes the meniscus of a water column moving in a capillary tube. When the tube is first saturated and the balanced is condition is approached by lowering the meniscus, its contact angle (0,) smaller than that belonging to the static state (B0),while the opposite inequality is valid in the cme when the meniscus advances in the tube (0, > 0,)(Fig. 2.1-17b). The change in the contact angle enables a drop in a capillary tube to equalize the pressure difference without creating mvement, this phenomenon is, therefore, a special type of threshold gradient in a capillary tube. Let us suppose that a pressure of p , prevails at one end of a horizontal tube and p , at the other ( p , < p z ) . The two sections under different pressure are divided by a drop bounded by menisci at both ends (Fig. 2.1-17c). The pressure within the fluid, p , can be determined from the external pressures and the capillary pressures acting on the menisci: 40

p = p , - pc1=p,- -COB 0, ; d and 4a

p = p , - p , = p , -cos d

0,;

therefore,

p, 16

-p, =

40

-(cos 0,- cos 0,). d

(2.1-29)

226

2 Determination of hydraulic conductivity

Thus, a static balance is achieved if the contact angles can develop so that the difference between their cosine values multiplied by 4a/d is equal to the pressure difference. It follows from the condition where p , < p,, that

0,> 0,.

Considering the same phenomenon in a vertical tube, the conditions can also be determined where a drop, having atmospheric pressure on both sides, does not move downwards under the influence of gravity (Fig. 2.1-17d). The surplus pressure just below the upper meniscus is equal to the capillary pressure belonging to this surface. Moving downwards inside the drop, the pressure increases in proportion to the length measured from the upper surface, [the curvature of the surface is neglected here, because of the very small horizontal area, and the length is measured from the centre of gravity of the meniscus which is approximated as d / 3 above the deepest point of the meniscus (Swartz, 1931)l. The surplus pressures inside the drop can be calculated from two directions (the upper capillary pressure increased by the weight of the water column, or the lower capillary pressure decreased in proportion to the distance from the point), and the two valueb have to be equal. Hence, a relationship can be established between the two capillary pressures: 40

40

pcl = -COB 0,; p,, = - COB 0, ; d d d +pie - l y = p,,; -1y = cos0, - co.0, ; 4a

(2.1-30)

+

where 0, = 0 being the lower limit of the contact angle at the upper meniscus (cos 0, = 1). Movement will not develop if d c o s 8 , ~1 - - 1 y 4a

;

(2.1-3 1)

and

021,mit = arc cos It is necessary to note that this condition indicates only the posbibility of static equilibrium, because the contact angle can develop anywhere in the 8, 021imit aa well, and in this caae the 8, < '8,inequality does range 8 not stop the movement, only decreases its velocity.

>

2.1.2 Dimensionless numbers characterizing the various validity zones of seepage As already mentioned in Chapter 1.1, seepage can be classified according to the accelerating forces and further mb-groups can be determined considering the acting retarding forces. Thus, the following types of movement have to be characterized when seepage through loose claatic sediments is investigated:

2.1 Dynamic analysis of seepage

227

(a) Gravitational seepage through a saturated medium: - Turbulent seepage (retarding force is inertia); - Transition zone (retarding forces are inertia and friction); - Laminar seepage (retarding force is friction); - Microseepage (retarding forces are friction and adhesion); (b) Seepage through a saturated medium influenced by gravity and the pressure of the overlying layers (further sub-groups are the same as those of the previous group) ; (c) Seepage through the unsaturated zone above the water table (accelerating forces are gravity and adhesion, and capillarity has3 to be taken into account as well; retarding forces are friction and adhesion). Considering the fact that the pressure of the overlying layer can be regarded as a supplementary part of gravity (as already mentioned), and because seepage through an unsaturated medium is influenced by numerous forces and, therefore, i t must be investigated separately, the following combinations of dimensionless numbers can be constructed as quotients of the dominating accelerating and retarding forces (KovQcs, 1966; Spronck, 1932; Mosonyi and KovQcs, 1952): Laminar zone

S -

a

Turbulent zone

dv Arl-

dn - v v -

-MK ;

Veg

(2.1-32)

1%

dv

V e-

T

dt - v2 - Fr ;

a Transition zone

dv

T + - S--

(2.1-33)

Is

Veg

dv

ve-+Arldt dn

a

-

(2.1-34)

Veg

(2.1-35)

Microseepage

1, V E + S -- --

es + A r l -

dv

vv dn - I o +-=KO,. 9 Veg la9 When only one retarding force is considered and the others are neglected, similar dimensionless numbers can be constructed as3 the ratio of the forces neglected to the considered retarding force. It is quite evident that these dimensionless numbers can be used, therefore, to characterize numerically the limits of the validity zones of the various types of seepage. The limit between the turbulent and transition zones is evidently characterized by the fact that friction can be neglected as compwed with inertia, ~

16*

228

2 Determination of hydraulic conductivity

while the lower limit of the transition zone (the upper limit of the laminar zone) falls where friction becomes dominant over, inertia. Thus, both limits can be expressed as a ratio of inertia and friction: dv VeT dt lv -Re. * (2.1-36) A q -d v v dn

The lower limit of the laminar zone can similarly be expressed by the given numerical value of the ratio of adhesion to friction:

E - I-o-V e g- I , - =lag -_ Ko S

a---

I,

-

(2.1-37)

dv vv MK Arldn Finally, there is a further possible combination of the forces in question i.e. the ratio of adhesive force to gravity El@ = I , = KO,. This dimensionless number characterizes a static condition below microseepage, where friction is zero. Some of the dimensionless numbers given are well known from general hydraulics as the bases of the various model-laws, such aa the Froudenumber from Eq. (2.1-33), the Reynolds’ number in Eq. (2.1-36) or the Mosonyi-Kovhcs’ number [Eq. (2.1-32)], which were previously derived to determine the model-law valid for the characterization of seepage models, and for calculating the transforming factors between the corresponding measurements of the prototype and the model respectively (Mosonyi and KovQlcs,1952, 1956). The structure of the other dimensionless numbers is the same, or very similar to those given before. The only significant difference occurs in the numerator of KO, and KO,, where the sum of two forces instead of one is evident. For this reason, these numbers cannot be used as model-laws, because dynamic similarity can only be ensured in the cme of two dominating forces, when the identity of the ratio of these two forces can be represented in the prototype and the model. When there are three dominating forces (i.e. two retarding forces in the numerator and the accelerating gravity force in the denominator), the identity of the dimensionless number for the two systems ensures only the same ratio of the retardingandaccelerating effects. The ratio of the two retarding forces may, however, be quite different in the model from that in the prototype. The dimensionless numbers characterizing the various zones of seepage [Eqs (2.1-32) to (2.1-35)] can also be used to determine the general forms of movement equations, valid for each zone. It is necessary to take into account that seepage is not a free movement. Only a part of the potential energy (increased with pressure energy) accelerates the movement. The actuul accelerating force can be characterized numerically by the product of gravity and hydraulic gradient:

G’= IMg

= IVeg.

(2.1-38)

2.1 Dynamic analysis of seepage

229

The substitution of this value instead of gravity, in the equations in question, is in accordance with the theory of similarity as well. The total mechanical similarity includes both the geometrical similarity and the dynamic one. The former is ensured when the ratio of two corresponding lengths in the two systems is constant. Thus, the gradient determined as a quotient of two lengths (pressure expressed in equivalent water columns and the length of seepage) must be the same in the prototype as in the model. Mechanical similarity in Darcy’s zone necessitates the identity of both Mosonyi-Kovkcs’ number and the hydraulic gradient in the two systems. Thus, the general form of the movement equations can be derived as follows: (2.1-39)

if

K = M K ’ la - .9 V

Similarly, a well-known formula can be derived for the turbulent mne (i.e. Chezy’s equation): (2.1-40)

if

A=-.

2

Pr’

In the transition zone the general form is also the same as that proposed empirically to describe this type of movement (Forchheimer, 1924):

if 1 a = --

K 0; lg

; and b =

Y

Ko; l a g

(2.1-41)

Finally, the general form of the movement equation in the zone of microseepage is:

KO$= I , +l v v ; v = - la (K 9oj;II 1129 V

Io)=C1I-c2I0; (2.1-42)

129

c1 = -Koj; V

la 9 1 and c2 = - = K -. V MK’

The accuracy of such application of dimensionless numbers calculated as quotients of retarding and accelerating forces, is also proved by K a r a ’ s measurements (Kartidi, 1963). The points representing his data were plotted

230

2 Determination of hydraulic conductivity

on a coordinate system with the logarithm of Reynolds’ number on the horizontal axis and 2/MK‘ on the vertical one (Fig. 2.1-18). The fact that the points lie along a horizontal line in the laminar zone verifies the constancy of the M K ’ value, and also that of Darcy’s hydraulic conductivity.

Fig. 2.1-

-KOV&CB’

2.1.3 Numerical limits of the zones of seepage As already mentioned, the ratio of retarding forces considered and neglected respectively is a suitable parameter to calculate the limits of the validity zones of each movement equation, 88 this quotient meaaures the magnitude of the two forces in question. It is possible, therefore, to determine such fixed numerical values of these dimensionless numbers that indicate where one retarding force becomes negligible compared to the one under consideration. For this investigation, it is necessary to define the characteristic length and velocity from which the dimensionless number in question should be calculated. The Reynolds’ number is proportional to the ratio of inertia to friction. If this number is smaler than a given limit, inertia is small and friction is high. The validity of Darcy’s law can probably be accepted below this limit. When this parameter is above another limit (which is higher than the previous one), the inertia of the numerator is dominant over friction and the

231

2.1 Dynamic analysis of seepage

latter is negligible. This is the turbulent zone. Between these two limits, there is the zone of the transition condition, where both inertia and friction have to be considered. In seepage hydraulics, according to general practice, the Reynoldd number is calculated by using the eflective diameter as a characteristic length, and substituting Darcy’s seepage velocity. The latter is the discharge of flow divided by the total cross section, including the area of both pores and solid grains [see Eq. (1.1-9)]. Consequently, the Reynolds’ number generally used in seepage hydraulics can be calculated from the following formula: R e , = - Dh .V

(2.1-43)

-

Experimental results, give the upper limit of Darcy’s zone as 2 5 expressed by the parameter calculated from Eq. ( 2 . 1 4 3 ) (Lindquist, 1933; Koieny, 1953; Kar&di andV. Nagy 1960; Kar&di, 1963). According to these 200 as measured papers, the lower limit of the turbulent zone is 100 by Re,. There are, however, some explanations in these papers, indicating that this value gives only the lower limit of the second transition zone, and the real turlbulent zone starts at a higher limit. Some of the physical parameters of the soil (porosity, shape-coefficient) are excluded from the Reynolds’ number calculated from Eq. ( 2 . 1 4 3 ) .It is more correct t o use the Reynolds’ number determined for the model pipe rather than Re,. I n this parameter (Re,),the characteristic length i s the average pipe diameter, and the mean velocity %nthe pores (or in the model pipes) is substituted as the characteristic velocity. Thus, the relationship between the two Reynolds’ numbers can be determined as well:

-

Re because

,-

dn -

veff -

v

Dh

l - n a

v

Re,.

- --,4

l-n

u

(2.1-44) 4

n Dh ; and u l-n

=

-.n V

Using this parameter, the following numerical limits between the various zones can he given, as will be proved in detail in the following chapter:

Re, < 10 laminar (Darcy’s) zone; < Re, < 100 first transition (Lindquist’s) zone; < Re, < 1000 second transition zone; 1000 < Re, turbulent (Froude’s) zone. 10 100

The purpose of all the other proposals concerning the calculation of Reynolds’ number, is to select, in a similar manner, the characteristic length and velocity, so that further information can be obtained by including physical data of the soil other than the effective diameter. The velocity value generally used is either the seepage velocity (v) (Ward, 1964; Harleman et al., 1963; Perez Franco, 1973; Collins, 1961) or the effec-

2 Determinetion of hydraulic conductivity

232

tive mean value in the pores (vetf = v/n) (Maasarani, 1967; Kovhcs, 1969a, 196913; Chauveteau and Thirriot, 1967; Thirriot, 1969; Thirriot and Habib, 1970). Considering the possible difference between volumetric and areal porosity, the n value waa introduced,sometimes with a power lower than unity [as explained by Thirriot (1969) the power can be between 1 and 2/3 and Zampaglione (1969) used a characteristic velocity of the Reynolds’ number of vetf= ~ n - ~ l ~ ] . There is an even larger variety of parameters applied as characteristic lengths for the calculation of the numerical value of the Reynolds’ number. Several attempts have been made to find a length directly proportional to the resistance of the solid matrix. The most natural method is to use the square root of intrinsic permeability for this purpose (Ward, 1964; Harleman et al., 1963; Perez Franco, 1973; Valentin, 1970; Massarani, 1967). Some authors have related this parameter to the square root of porosity

(V,

[E)

(Collins, 1961). Thirriot (1969) haa derived a characteristic length,

assuming that the resistance of the porous medium is equivalent to that of a capillary tube having this length as diameter. According to his result, this length (d) is equal to the product of d*=

K

:v

- and a COmhmt (2.145)

32-.

Substituting the value of intrinsic permeability, according to the modified Koieny-Carman equation (the validity of which will be proved in the next chapter), it is easy to prove that the d o average pore diameter, used in Eq. ( 2 . 1 4 3 ) m a characteristic length, is similarly proportional to

k=-

5 (1 - n)a

d-4--01-n

51-n

(2.146) a

-.

D h 80 -rc

a

The characteristic length used by Zampaglione (1969) is similarly derived from the total resistance of the sample. The most important deviation from the other method, is the consideration of the difference between areal and volumetric values of porosity. The square of the length in question ( A 2 ) is defined as a factor of proportionality between mean pore velocity and hydraulic gradient v l f f = C I ; C = - A9’ ; V

and because

2.1 Dynamic analysis of seepage

233

therefore (2.1-47)

A grouping according to the two applied parameters, gives an excellent summary of the various proposals analysed in the foregoing paragraphs (Table 2.1-3). The summarization clearly shows that the proposed dimensionless numbers differ only in the effect due to porosity (in the applied power of this parameter), because the use of various numerical constants Table 2.1-3. Grouping of the various Reynolds’ numbers proposed for the characterization of seepage

Square-mot of intrinaic permeebility

fi

Square-root of the ratio of intrimia permeability and porosity

Harleman et d . Masserani (1967)

Rew = v/v

(1963) Ward (1964) Perez (1973)

ReM = v/v

fi

Collins (1961)

rn

Equivalent tube diameter d* or do Parameter derived from the total resistance of the sample

Thirriot (1969) K o v h (1969)

Rey = f Z v / v

fk/n8

Rep = )I&/v Zampaglione (1969)

Rez = v / v

fw

A Square root of the Kovh product ofintrinsjo (unpublished permeability end result) porosity

fax

does not change the character of the Reynolds’ number. On the basis of the comparison,the transforming factors between the various Reynolds’ numbers can also be determined together with the ratio of these numbers to the classical seepage Reynolds’ number (Re,) (Table 2.1-4).

234

‘

I

2 Determination of hydraulic conductivity

Relp

Rec

Rex

=

Rez

I

Rer

1

1

1

After comparing the different Reynolds’ numbers, Valentin (1970) proposed that the use of intrinsic permeability should generally be accepted for the calculation of the characteristic length. The only remaining problem is the method of consideration of porosity and the choice of the applied constant. In the first instant, i t seems that this problem is only a question of decision, because knowing the various transforming factors, the numerical characteristics of the limits between the various types of seepage served by the application of Reynolds’ number can easily be transferred from one system to another (e.g. the limits previously fixed in the form of the given Re, values). However, aiming at the establishment of a general seepage law describing the relationship between seepage velocity and hydraulic gradient without any restriction of the validity zone, the power of porosity has to be chosen so that the points representing the measured resistances in a coordinate system based on the applied form of Reynolds’ number, should lie along a curve. Some guidance concerning the constant parameter to be used, can be gained from the constant of the modified Koieny-Carman equation proposed for the calculation of Darcy’s hydraulic conductivity. Considering these objectives, it was found that the multiplication of intrinsic permeability with porosity, is more suitable than the previous proposals, and the application of a constant of 5 is advisable:

Re, = 5-2) V

.

(2.1-48)

Tables 2.1-3 and 2.1-4 are supplemented with this finally proposed parameter.

2.1 Dynamic analysis of seepage

235

the various Reynolds’ numbers Re8

(using the modifled Rep

Rq

n = -Rep

m

1

Kokny-Cannan equation for the calculation of permeability)

Reg

1 1

Rec = - -ReC 6 n

Rep =

,.

16 1

ReK

Rec=--

I

n

1

-Res

61-n

a

4

1

1-n

a

Rep=--

ReS

In the z.o)ie of microseepage, where seepage velocity is smaller than that pertaining to the laminar flow, the water movement i s also influenced by two retarding forces i.e. adhesion and friction. This zone can also be regarded aa a transition one, which develops between laminar movement and the static condition. As in the case of the use of the Reynolds’ number, the limit between the laminar zone and microseepage can be indicated by a given numerical value of the dimensionless number, proportional to the ratio of adhesion and friction [KO,in Eq. (2.1-37)]. Because of the lack of sufficient measured data, this limit can only be determined on the basis of theoretical consideration. Both Eq. (2.1-39) and Fig. 2.1-18 indicate the constancy of the M K ’ value in Darcy’s zone. Substituting the average pipe diameter in Eq.(2.1-39) as the characteristic length and hydraulic conductivity in the form of the modified KoSeny-Carman equation (aa already shown and as will be explained in detail in Chapter 2.2), this constant can be calculated: n MK~=--I (2.1-49) 80 The dimensionless number M K ‘ can also be determined from grain diameter instead of from the diameter of the model pipe: 1

MK:= 5

a2

n3 (1 - n)2

(2.1-50)

As in the various forms of Reynolds’ number, the square root of intrinsic permeahility can also be substituted as a characteristic length. In this ca-ae

236

2 Determination of hydraulic conductivity

the dimensionless parameter is equal to unity, or using the 5 m v a l u e , is inversely proportional to porosity

or 1 MKk=-. 25 n

(2.1-5 1)

Assuming that porosity, n = 0.40, MK; is approximately for a sample of spheres (a = 6), a value which agrees with Karsdi's data (see Fig. 2.1-18). Combining Eqs (2.1-37) and (2.149),the dimensionless number giving the limit in question can be expressed in Darcy's zone as a product of the ratio of threshold and actual gradient, and a constant depending only on Dorositv: (2.1-52) The same parameter if v k o r 5

v& is substituted for characteristic length is I K o,, = 2 ; I

or (2.1-53) This type of relationship must be valid not only within the laminar zone but also at its lower limit. Thus, the numerical value in question can also be determined by a constant value of the ratio between the threshold gradient and the actual one: I , = 1210; (2.1-54)

is the actual gradient a t the limit between the laminar zone and where microseepage, and the constant of 12 is based on the detailed investigations of microseepage given in Chapter 2.3. Comparing this result with Eq. (2.1-12), it can be seen that this limit is about Ih = 0.003 in sand (Dh > 0.01 cm), while in clay (Dh 2 p ) , the limit gradient is higher, about I , = 0.5, although the effect of adhesion is still very small here - smaller than the probable error in measurement of the gradient. The validity zones of the various types of seepage can be represented graphically as the function of either velocity or hydraulic gradient (Fig. 2.1-19). The first graph shows the laminar zone and the zones having a higher gradient. Here the Reynolds' number is linearly proportional to velocity. Its given numerical values indicate the borders between the subsequent zones. This number being proportional to the ratio of inertia and friction, indicates that the ratio of the two retarding forces can also be

-

2.1 Dynamic analysis of seepage

--

237 3

3

2s I

0 zm2

a

~~

retarding forces

0

retardipQforces Fig. 2.1-19. Graphical representation of the validity zones of the various types of

-age

determined by this parameter. By definition, Mosonyi-Kov4cs' number is independent of velocity in the laminar zone. Its value decreases gradually with increwing velocity in the transition and turbulent zones. The I(v) function is represented in two ways. The solid line shows the connection between dynamically correct relationships in each zone, while the dotted line assumes the validity of the linear function within the whole range of movement. The difference between the two curves indicates the error which may arise by the unjustified application of Darcy's law. The second graph summarizes the validity zones of the types of seepage having small velocity (including the laminar zone as well). Here the w ( l ) function is similarly represented as in the first part but with the gradient now chosen as an independent variable (measured on the horizontal axis). The linear relationship gives a higher velocity than the actual one in this range of seepage. From the dimensionless parameters the Mosonyi-Kov&cs' number is represented as a function of hydraulic gradient. It is nearly constant in the laminar zone and gradually decreases below the lower limit of the latter. It becomes zero at the threshold gradient.

238

2 Determination of hydraulic conductivity

References to Chapter 2.1 BEAR,J. (1972): Dynamics of Fluids in Porous Media. Elsevier, New York, London, Amsterdam. E. C. (1922): Fluidity and Plasticity. London. BINOHAM, BONDARENKO, N. F. (1966): Nature of Seepage-anomaly in Fluids (in Russian). Symposium on Seepage and Well Hydraulics, Budapeat, 1966. BONDARENKO, N. F. (1973): Physics of the Movement of Ground-water (in Russian). Gidrometeoizdat, Leningrad. BONDARENKO, N. F. and NERPIN,S. (1966): Influence of Viscous-Plastic Properties of Water on its Equilibrium and Transfer in the Unsaturated Zone. Symposium on Water i n the Unsaturated Zone, Wageningen, 1966. BONDARENKO, N. F. and NERPIN, S. (1967): Sheering Stress of Fluids and its Consideration in the Investigation of Surface Phenomena (in Russian). Publication of the Institute of Physico-Chemistry of the Academy of Sciences, USSR. CHAWETEAU, G. and THIRRIOT,C. (1967): Regime of Flow in Porous Media and the Limit of Darcy’s Law (in French). La Houille Blanche, No. 2. CHILDS, E. C. and TZIMAS,E. (1971): Darcy’s Law at Small Potential Gradients. Soil Science No. 3. COLLINS, R. E. (1961): Flow of Fluids Through Porous Materials. Reinhold, New York. DE WIEST, R. J. M. (1969): Flow through Porous Media. Academic Press, New York, London. EOTVOS,L. (1886): On the Itelationship between Surface Tension and Chemical Properties of Various Fluids (in Hungarian). Matematikai 6 Termbzettwlomanyi grtekezkaek, Vol. N ,p. 34. ERDEY-GR~z, T. and SCHAY,A. (1954): Theoretical Physico-Chemistry (in Hiingarian). Tankonyvkiad6, Budapest. FORCHHEIB~ER, PH. (1901): Watermovement through Soil (in German). ZeitschriJt dea Verbundea der deutechen Ingenieurs. FORCHHEIMER, PH. (1924): Hydraulics (in German). Teubner, Leipzig, Berlin. HARZEMAN, D. R. F., MEHLHORN, P. F. and RUDR, It. R. (1963): Dispersionpermeability Correlation in Porous Media. Proceeding of ASCE, HY. 2. JUJL~SZ, J. (1968): Investigation of Seepage (in Hungarian). Hidroldgiai Kiizlii?ay, No. 1. JWASZ,J. (1966a): Dynamic Characterization of Seepage (in Hungarian). Symposion on seepage and Well Hydraulics Budapeat, 1966. J ~ ~ sJ. z(196613): , Calculation of Static Ground-water Resources Available as a Result of Consolidation (in Hungarian). Symposwn on Seepage and Well Hydraulics, Budapeat, 1966. Joadsz, J. (1967): Hydrogeology (in Hungarian). Tankonyvkiad6, Budapest. -I, G. (1963): Hydraulics of Linear Drainage Systems (in Hungarian). (Thesis; Manuscript), Khartum, Budapest. KARADI,G. and TOROX, L. (1966): Applicability of Seepage Law (in Hungarian). HdroMgiai KOd&y, NO. 6-6. KAR-~DI, G. and V. NAGY,I. (1960): Investig&ion ofivalidity of Darcy’s Law (in Hun mian). Conference on Hydradics, Budapest, 1960. ~ Z D I A. , (1963): Neutral Stress and Pressure of Flow (in Hungarian). v k i i g y i Kodemdnyek, 1. K h D I , A. (1972): Soil Mechanics. 4th edition (in Hungarian). Tankonyvkiad6, Budapest. K ~ z D I ,A. (1974): Handbook of Soil Mechanics (Vol. 1. Soil Physics). ElsevierAkademiai Kiad6, Amsterdam, Budapest. KOV~CS, G. (1957): Theory of Micro-seepage (in Hungarian). HidroMgiai KOdizl&ry, No. 3. KOV~CS, G. (1968): Theoretiad Investigation into Micro-seepage. Acta Technica Acudemiae Scientiarum H u n g a r k e , Tom. XXI. 1-2. Kovkcs, G. (1966): Dynamic Investigation of Seepage by Invariant Numbers. Symposium on Seepage and Well Hydraulics, Budapest, 1966.

2.2 Hydraulic conductivity of saturated layers

239

Kovlics, G. (19694: General Characterization of Different Types of Seepage. I A H R Congress, Kyoto, 1969. KovAcs, G. (1969b): Relationship between Velocity of Seepage and Hydraulic Gradient in the Zone of High Velocity. 13th I A H R Congreae, Kyoto, 1969. KO~ENY, J. (1953): Hydraulics (in German). Springer, Wien. KIJTILEK, M. (1965): Influence of Interface on Filtration of Water in Soil (in French). Science du Sol, (Prague), 1. KUTILEK,M. and SUINOEROVA,J. (1966): Flow of Water in clay Minerals as Influenced by Absorbed Quinolimium and Pyridinium. Soil Science, 6. LANDAU, L. D. and LIPSCHZTS,E. M. (1974): Textbook of Theoretical Physics (in German). Akademie Verlag, Berlin. LEVICH, V. G. (1968): Physico-chemical Hydrodynamics (in Hungarian, translated from Russian). Akademiai Kiad6, Budapest. LINDQUIST,E. (1933): On the Flow of Water through Porous Soil. ICOLD let Congreee, Stockholm, 1933. MADQE, E. W. (1934): The Viscosities of Liquids and their Vapour Pressures. PhY8k8, N O . 2. MASSBANI, G. (1967): Critics of the Validity of Darcy's Law in Heterogeneous Porous Media (in French). CRAS, July. MATTSON, S. (1932): Soil Science. Baltimore. MOSONYI,E. and KovAcs, G. (1952): Model Law by Joint Consideration of Gravity and Friction (in Hungarian). Hidroldgiai Kodony, No. 7-8. MOSONYI, E. and KovAcs, G. (1956): Model Law of Filtration (in French). I A H R Congreee, Dijon, 1956. N~METH E., (1963): Hydromechanics (in Hungarian). Tankonyvkiad6, Budapest. PEREZ FRANCO, D. F. (1973):Non-Darcian Flow of Ground Water towards Wells and Trenches. Particularly within the Turbulent Range of Seepage. Technical University of Budapest (Manuscript). REINER,M. (1962): Theory of Rheology (in French). Dunot, Paris. SPRONCK, R. (1932): Hydrodynamic Similarity and the Investigation of Models (in French). Annalea dea Travaux Publiqua de Be2gique. SWAE~TZ, C. A. (1931): The Variation in the Surface Tension of Gas-saturated Petroleum with Pressure of Saturation. P h y e k , vol. l. THIRRIOT, C. (1969): Hydrodynamics of Flow in Porous Media (in French). 13th IAHR Congress, Kyoto, 1969. THIRRIOT,C. and HABIB,J. (1970): Experimental Study of Flow with Low Reynolds' Number in Clays (in French). Sciences Agrmomique, R e n w . VAGELEB, P. (1936): Cations and Water Content of Mineralogical Soils (in German). Berlin. F. (1970): Non-linear Resistance of Porous Media (in German). MitteilunVALENTIN, gen, Inatitut fiir Hydraulik und ae&eeerkzl.n.de, Miinohen, No. 6. WABD,J. C. (1964): Turbulent Flow in Porous Media. PV'OCeedilage of ASCE, H Y . 6. ZAMPAQLIONE, D. (1969): Turbulent Seepage Flow in Filters 13th IAHR Conpeas, Kyoto, 1969.

Chapter 2.2 Hydraulic conductivity of saturated layers The interpretation of Darcy's seepage velocity [Eq. (1.1-9)] and that of hydraulic conductivity [Eq. (1.1-ll)] h w already been discussed. It hw also been stated that the simple v=KI (2.2-1) relationship can be accepted for the characterization of all types of seepage

240

2 Determination of hydraulic conductivity

through saturated media, if the concept of hydraulic conductivity instead of Darcy’s original parameter (KO), is used in a more generalized form. The Darcy coeflcient is composed of two factors: one characterizing the properties of the solid matrix (intrinsic permeability) and the other describing the behaviour of the transported fluid. These two components have to be supplemented by a third one, depending on the condition of the flow. The third factor should be the function of either the seepage velocity or the hydraulic gradient, because the apparently linear relationship between v and I in Eq. (2.1-1) should be modified in such a way and for each zone of seepage that a suitable mathematical expression may be achieved connecting the two basic variables. Investigating seepage through loose clastic sediments, the intrinsic permeability is influenced by two properties of the solid matrix: (a) The size and form of grains from which the layer is built up; (b) The packing of the grains, which determines the size of the pores between the grains.

Thus,four components have to be investigated and determined for the complete characterization of the hydraulic conductivity of saturated, loose claatic sediments: (a) Geometrical parameters of the grains in the layer (size, shape and distribution); (b) Porosity of the layer; (c) Physical parameters of the propagating fluid (the ratio of specific weight and dynamic viscosity, or that of acceleration due to gravity and kinematic viscosity) ; (d) Flow condition of the seepage investigated (considering the dominant forces among those listed in the previous chapter). In Chapter 1.2, a geometrical model waa proposed to characterize the water transporting channels between the grains of the layer. Combining this simplified structure with the movement equations based on the consideration of the acting forces, a conceptual model can be established. According t o the explanation given in the introduction of this part of the book, the next phase is the verification of this model and the determination of its numerical constants on the basis of experimental data to ensure the practical application of the theoretical results. The h a 1 verification, which is the object of this chapter, will be based partly on data from publications, and partly on the author’s own measurements.

2.2.1 Determination of Darcy’s hydraulic conductivity Accepting the hypothesis whereby there are only two main forces to be taken into account in the zone of laminar movement (i.e. gravity and friction), their equilibrium can be expressed in a mathematical form for a model pipe with a diameter of do. The well-known Poiseuille’s equation can be

2.2 Hydraulic conductivity of saturated layers

241

derived in this way. The equilibrium of a cylinder concentric about the axis of the pipe having a radius r and a length bf I , gives the following equation (Fig. 2.2-1):

I y r Z n l + 2 r n l q -d=v O . dr

(2.2-2)

Fig. 2.2-1. Symbole used for deriving Poiseuille's equation

After solving this differential equation with a boundary condition, where the velocity at the wall of the pipe is zero (w = 0 ; where r = ro),the velocity at a point at a distance of r from the axis can be determined: (2.2-3)

Integrating the product of the velocity and an elementary area (U) along the total surface of the cross sect,ion, the flow-rate through one pipe with a radius of ro can be achieved r.

This value, divided by the total area, is the mean velocity:

v = -Qo- - r zI. Y (2.2-5) A 8rl The number of pipes in the model system crossing the unit area of the sample [see Eq. (1.2-20)] is known, and thus, the total discharge and the virtual seepage velocity can also be calculated:

'

vs -- - =

As

Q o N = - *I ;Y

3211 where A , is the total cross-sectional area of the sample. 16

(2.2-6)

242

2 Determination of hydmulic conductivity

Substituting the effective grain diameter of the sample instead of the pipe diameter [see Eq. (1.2-19)], the following relationship can be determined: vS --_1 y n3 [?)'I. (2.2-7) 2 7 (l-nn), The hydraulic conductivity of the model pipes with constant diameter cdculated from Eq. (2.2-7) is 2.5 times greater than the actual value deter-

4

Fig. 2.2-2. Comparison of flow rate in pipes having constant or changing diameter

mined by reliable experiments as mentioned in Chapter 1.2 in connection with the detailed discussion of the establishment of the model system. It was also explained that this difference can be eliminated if the pipe is constructed from short sections with diameters of d1 and d, so that the volume of the pipe remains the same as that of the pipe with a diameter of do, and the 1 : 3.5 [Eq. (1.2-22)]. The aboveratio of df and df is equal to 1 : 3 mentioned increase in flow-resistance or the decrease in discharge can also be proved mathematically (Fig. 2.2-2):

-

Qi = Qz

m(I1

= QI-2 = a&,;

+ I,) = I , 1;

(2.2-8) Q1+

= p I l d : = BIzda = 2 P I ,

2.2 Hydraulic conductivity of saturated layers

243

d, =1 ; 3.5 and d o 1.5 then a

N

0.4

On the basis of this explanation, the determination of the virtual seepage velocity can be based on the proposed geometrical model and can be calculated from Eq. (2.2-7) multiplying its right-hand side by 0.4. Comparing this result to Darcy’s equation, the theoretical value of hydraulic conductivity can be determined, which agrees with the dynamic analysis and includes all the effects of the influencing factors: (2.2-9)

The variables influencing this parameter can be divided into three groups:

”] [I

(a) The parameter expressing the behaviour of the fluid -, or using the

, the equivalent ratio is - ;

kinematic viscosity

V

(b) The characteristics of the grains in the layer (Dh/a) expressing the size, distribution and shape of the grains; (c) The effect of porosity

ns

(1

).

- n)2

To verify the proposed formula, i t is of value to investigate the ways in which these groups of variables were considered in the previously developed equations. If the relationship between the hydraulic conductivity and one of the groups is similarly described in a formula aa in Eq. (2.2-9) all measurements and observations used to determine the formula, also prove the validity of the relationship. The first investigations aimed a t exploring the relationship between permeability and the behaviour of fluids, have expressed their results in a form which illustrates the dependence of the K D parameter on temperature. Thus, Hagen (1869) observed a modification of 3 yoper degree centigrade between 12.5”Cand 23.4OC, while Havrez measured an increase of six times from 0°C to 100°C. This relationship waa later expressed in the form of mathematical equations (Seelheim, 1880; Hazen, 1895; Forchheimer, 1924). Among these formulae, Hazen’s equation is the best-known:

K D = llSD?,(0.7 16*

+ 0.03t).

(2.2-10)

244

2 Determination of hydraulic conductivity

The observations listed do not contradict Eq. (2.2-9). Temperature is only an indirect parameter to characterize the property of fluids and the correct method of expressing this relationship is by viscosity, which is inversely proportional to hydraulic conductivity. All the observations listed and formulae proposed, supply practically the same numerical ratio between two given values of temperature, as does the use of viscosity for the same temperatures. This statement is supported by the fact that in recent investigations, every research worker used the ratio of specific weight to dynamic viscosity (or the equivalent of acceleration due to gravity to kinematic viscosity) to express the dependence of hydraulic conductivity on the behaviour of the transported fluid (Forchheimer, 1924; Lindquist, 1933; Terzaghi, 1926, 1943; Koieny, 1953; Carman 1956; Chardabellas, 1964). On the basis of the above, the relationship between permeability and the parameters characterizing the properties of the fluid, can be regarded as an accurate and reliable one in the form expressed by Eq. (2.2-9). The variables used to describe the effect of porosity in the same form as they are applied here were first found in Koieny’s equation (Koieny, 1953): n3 e3 (2.2-11) Pk = (1-n)2 lfe Most of the recently published formulae use the same expression (Leibenson, 1947; Zamarin, 1928; Carman, 1956; Zauberei, 1932). There are, however, several proposals which differ from the above. From a theoretical point of view, the most important is Schlichter’s equation. This is the first attempt to combine all variables, independent of viscosity and grain diameter, into one factor, which is therefore named Schlichter’s number (Schlichter, 1899). Recalculating the effect of porosity from his formula, the relationship in question can be expressed by a parameter approximately proportional to

Ps=-.n2

(2.2-12)

1-n

The same formula expressing the influence of porosity can also be achieved theoretically by using the drag force model (Goldstein, 1938; Ward, 1964; De Wiest, 1969). More recent publications generally accept Koieny’s formula theoretically and suggest other equations only from a practical point of view to make the calculation easier. These proposals are mainly based on experimental measurements and can be summarized in the following general form:

P = ea;

(2.2-13)

where P is the parameter expressing the effect of porosity; e is the void ratio; and a is a constant determined from various measurements (a = 2 Terzaghi, 1926, 1943; a = 3 Chardabellas, 1964). JuhBsz (1967) published two different formulae to calculate this parameter. One is a member of the group

2.2 Hydraulic conductivity of saturated layers

245

described by Eq. (2.2-13) with a power of a = 2.8 and the other has the following form:

Pi= n4.

(2.2-14)

The various proposals cannot be directly compared by using their absolute values, because the latter also include some constants, which may differ in each case. Good results can be achieved, however, by investigating the

porosiiy, n Fig. 2.2-3. Comparison of the results of the various methods proposed to calculabe the relationship between porosity and hydraulic conductivity

relative change of parameter P with porosity. In Fig. 2.2-3, the ratio of PIPs8 (i.e. the P parameter at a given porosity divided by that belonging to the most probable porosity n = 0.38) was plotted against porosity. On the basis of the graphs representing the proposals of several authors, it can be stated that the relationship determined by Eq. (2.2-11) and also used in Eq. (2.2-9), occupies a central position among the different formulae. This form has been accepted by most research workers and the other equations proposed do not give an appreciable difference for the probable values of porosity. Thus, the accuracy of Koieny's parameter in Eq. (2.2-9) can also be accepted. The investigation of the various methods published in the literature to calculate the influence of size, distribution and shape of the grains on hydraulic conductivity, is the most complicated study because of the numerous publications and the wide variety of basic ideas. There is, however, one point on which all the authors are in general agreement, i.e. hydraulic conductivity is proportional to the square of a characteristic diameter. This statement is proved by the formulae listed in Table 2.2-1 and by the graphs in Fig. 2.2-4. The latter combines the results given by Romer (see De Wiest, 1969) and Bear (1972). In both publications, intrinsic permeability is related t o the mean diameter of nearly homogeneous samples. The memurements used

Table 2.2-1. Comparison of formulae proposed for the calculation of permeability coefficient Author

I

Form

--

of the formula

KrBber

Temperature considered,,i the formula or m e d in the

[“a

KD=41 Di0

10 20

Seelheim

KD = 37 Di0

12

Hagen

KD KD

10

Hezen

Jhky KarAdi

DZo = 116(0.7

= 36

+ 0.03t)Df0

10 20

KD = 100 D:o

10

KD = 100 D2 (W balls) KD = (90 140) D:, (heterodisperse gravel) KD = 200 eS D:o

10

N

Terzaghi

2 2600 v

Dz (lead balls)

20 10 20

The given or aswmed validity =oneof the c0eta-t of miformity

-

--

1.2

m e multiplying factor valid in the c88e of the Itse of effectivediameter

43

m e recalculated shape wetacient assuming n 0.38 aa porosity

-

7.16

The

Q,

we Of diameter [cml 0.064-0.210

8.16

1.6

40

7.68

0.016-0.068

1.6

39

7.63

0.028

1.6-2.6

46 60

6.92 6.92

2.0

39

7.63

1.0

100

6.10*

2.0

40-62

1.6

46

1.0

10

6.31-7.86

%

7.00 8.00

Y

61

6.00

G

69 32 43 60 63

6.10 8.36 7.16 6.66 6.60 6.90-10.8

r

; e.

Lindquist

KD=

Chardabelles

KD = 266 e3 D2 (lead balls) KD = 140 ea De K D = 186 ea D 2 KD = 216 e3 D2 KD = 230 e3 Dz direct measurements

10 10 10 10 10

Zeuberei

330 N 3 6 h a KD = 0 (1 - n)*

18

47-60

Veroneae

K D = ~-g D 2 2300 v

10

34

7.96

Romer

K, = 6 . 6 4 ~1 0 - 4 s Dz

10

1.0

49

6.24

0.01-

Bear

K,= 6.17 x 10-4 S DZ

10

1.0

46

6.80

0.008- 0.1 J

Nola: n = 0.49

2

N

-

N

0.016-0.026 0.026-0.060 0.06O-O.100 0.123 0.022-0.081 ,

-

7.37-7.62

0.66

8

P a

x u

2.2 Hydraulic conductivity of satuxated layers

o

247

Krumbeh and Monk (lQ43)

particle size, D rcml

particle size, 17 Ccnl

Fig. 2.2-4. Relationship between intrinsic permeability and grain diameter (after Romer me De Wiest, 1969 and Bear 1972)

by Romer are listed in Table 2.2-2 while Bear has based his relationship on data measured by Krumbein and Monk and by the Institute Franpis Du Petrol. The opinions related to the interpretation of the characteristic diameter in the cttse of a heterodisperse sample, are more diverse. Assuming that friction is proportional to the surface of the grains and gravity to the volume of water, which can also be substituted by the volume of grains using porosity values, a result can be achieved according to which Kogeny's effective diameter is the most reliable parameter of a heterodisperse sample. This is because it is based on the identity of the specific surface (the ratio of the surface to the volume of grains) in the prototype and the model system. The same result was previously expressed by Koieny (1953) and Carman (1956). The proposal of Zamarin (1928) for the calculation of another characteristic diameter will first be compared to Kogeny's effective diameter. According to his formula, every fraction of the grain-size distribution curve has to be weighed by different factors when calculating the characteristic diameter. The reciprocal value of the latter is equal to the sum of the products composed of the weight of the grains belonging to a fraction and the total weight of the sample (A S f ) ,the reciprocal value of the mean diameter (D f ) and the previously mentioned factor of the same fraction (ai): (2.2-15)

248

2 Determination of hydraulic conductivity

Table 2.2-2. Experimental reaults between intrinsic permeability and diameter for nearly homogeneous porous media (after Romer) Diameter of mean particle siee D

t-4

u-ty we5cient

poroefty

IntrlnsiC penneability

U

n

k [em']

Investigator

Spherical shape 0.63 0.30 0.20 0.20 0.20 0.14 0.092 0.0646 0.0468 0.039 0.0383 0.0322 0.0273

0.646 0.14 0.092 0.046 0.0116

1.0 1.0 1.0

0.40 0.37 0.38 0.38 0.37

1.0 1.0 1.16 1.08 1.o 1.0 1.13 1.0 1.0 1.0

-

0.36

-

0.33 0.39 0.40 0.39 0.33

1.16 1.08 1.12 1.0

16.1 x 64.6 x 34.6 x 22.0 x 10-0 24.6 x 16.7 x 10-0

Brownell et al. (1960) Sunada (1966)

Harleman et al. (1963) Harleman et al. (1963) Harleman et al. (1963) Harleman et al. (1963) Harleman et al. (1963) Ward (1964) Ward (1964) Harleman et al. (1963) Ward (1964) Ward (1964) Ward (1964)

6.7 x 26.2 x 10-7 18.7 x 10-7 10.3 x 10-7 11.6 x 10-7 9.0~10-7 5.77 x 10-7

2 . 6 6 ~lo-' 9.46 x 10-@ 4.83~ 1.10 x lo-" 8.10~

Banka et al. (1962) Harleman et al. (1963) Harleman el al (1963) Harleman el al. (1963) Raimondi et al. (1969)

The detailed analysis of this equation shows that this parameter is very similar to the ratio of the effective diameter to the average ghape coefficient. = 10 mm, ai= 1; The factor ai,increasing with decreasing diameter (Di and Di= 0.001 mm, ai= 1.8), expresses practically the same effect, which is represented by the increasing shape coefficient in the zone of fine grains. To prove this statement, the shape coefficients were recalculated from Eq. (2.2-15) using the constant in Zamarin's formula, and are listed in Table 2.2-3 as a function of the diameter. The data, compared to those repreTable 2.2-3. Probable shape coefficients belonging to various diameters recalculated from Z~MI&S formula Shape cOe5OiEnt Qmln diameter

[a1 1.0 0.1 0.01

0.001 0.0001

reoelculated from Zemarin's formula

8.0 9.2 10.6 12.1 14.4

average value in Fig. 1.2-5

8-1 1 9-1 2 10-13 12-16 16-18

2.2 Hydraulic conductivity of saturated layers

249

sented in Fig. 1.2-5, verify the statement that Zamarin’s formula gives practically the same result as the theoretically derived equation [Eq. (2.2-9)] thus all measurements which served as the basis of Zamarin’s method, simultaneously, testify the validity of the latter as well. Although Paladin (1964) uses D5, as the characteristic diameter, his investigation also proves the accuracy of the use of the effective diameter. He applies a corrective factor depending on the coefficient of uniformity, to supplement D50and to express the eflect of grain-size distribution. This factor is proportional to the ratio of D50 to Dh, as shown in Chapter 1.2 (see Fig. 1.2-6). This fact can be accepted as proof of the accuracy of using effective diameter. Further comparisons are hindered by the fact that the various authors use different characteristic diameters (mostly D5, or D,,,). The relationship between these values and the effective diameter given in Chapter 1.2 (Fig. 1.2-6 and Table 1.2-2) make it possible, however, to recalculate the probable value of the shape coefficient included in the constant factor of the various formulae proposed. If the parameter remains within a realistic range of values, the investigated empirical equations and all the measurements used for their development prove the validity of Eq. (2.2-9). This recalculation wm easy in those cases where all the other variables were included in the formula in question, or the validity zone of the neglected variables was given. In other cmes these values were taken into consideration on their probable average i.e. porosity at n = 0.38 and temperature at t = 10°C and t = 20”C, the latter including the possible range of temperature. The result of this comparison is shown in Table 2.2-1. Because of the large number of formulae, it is not possible to list all the methods presented in the literature. The equations shown in Table 2.2-1 represent examples over a long period of time and from as many different countries m possible, ensuring the reliability of the presently proposed method by the investigation of a very large number of observations. As shown by the result of the comparison, a shape coefficient of a = 6 can be recalculated from equations, the bmic data of which were measured by using a sample of spheres, while the samples composed of natural sand or gravel generally resulted in a shape coefficient between 7 and 8, which agrees well with the expected theoretical value. The analysis of Beyer’s data also yields very reliable verification of the theory developed. Beyer has collected numerous data and presented them in a table which gives the expected coefficient of permeability as a function of D,, and D6,. The data in this table can be expressed in the same form as the general equation proposed by Hazen KD = CG,;

(2.2-1 6)

where the coefficient of C can be calculated from Beyer’s data atj a function of the coefficient of uniformity (U= D6dDl0),since both the corresponding Db0 and D,, values are given in the table. The best method of comparing Beyer’s data with Eq. (2.2-9) is to represent the recalculated C factor as a function of U (Fig. 2.2-5). The figure was supplemented by further values:

250

2 Determination of hydraulic conductivity

<

i.e. Hazen’s coefficient C = 116 valid in the zone of 1.5 U < 2.5, and the probable extreme values of C belonging to U = 1 (C = 150) and to a very large number of U (C = 60) (Forchheimer, 1924). Finally’Eq. (2.2-9) was transformed into a form similar to Hazen’s equation assuming a shape coefficient of a = 7 and a temperature o f t = 20°C. The ratio of D, to D1, and the probable extreme values of porosity, were chosen a~ a function of the coefficient of uniformity on the basis of Figs 1.2-6 and 1.2-22. The path of the curve representing Eq. (2.2-9) and that of the series of points r e d -

coefficien f of unifarm?y, U Fig. 2.2-5 Comparison of data from Beyer’s table with hydraulic conductivity oelculated by using the proposed equation

culated from Beyer’s table, are very similar and the numerical difference between the corresponding C values is also insignificant. Thus, the numerous measurements used to establish Beyer’s table also prove the reliability of the theoretical relationship. Summarizing the above, it can be stated that Eq. (2.2-9) given to c d culate hydraulic conductivity and based on theoretical investigations is in very good agreement with the previously proposed empirical formulae and experimental measurements as well. This statement is proved by Table 2.2-1. The proposed equation is practically identical to Koieny-Carman’s equation. The similarity with Zamarin’s formula shows that the shapecoefficient was used correctly and the comparison with Paladin’s equation proves the reliability of the use of the effective diameter. The analysis of Beyer’s table involves a very large number of measurements for the verification of the theoretical relationship. It can be stated, therefore, that the proposed equation is an improved form of Koieny-Carman’s formula, which supplies the most reliable hydraulic conductivity in the zone of laminar seepage.

2.2 Hydraulic conductivity of saturated layers

251

2.2.2 Investigation of the turbulent and transition zones of seepage The investigation of laminar seepage began with the dynamic analysis of the movement, and the theoretical relationship established in this way was compared to the results of empirical studies. This comparison has proved the reliability and practical applicability of the method developed by theoretical means. For the investigation of seepage with higher velocity (i.e.turbulent and transition zones) the method of study was reversed: a relationship between velocity and hydraulic gradient was determined first on the basis of experimental data, and the accuracy of this method was then proved theoretically. Lindquist proposed a method of representing experimental data in a coordinate system using the Reynolds’ number for seepage on the horizontal axis [Eq. (2.1-l)] and the product of the coefficient of resitivity [see Eq. (2.1-40)] and the same Reynolds’ number on the vertical axis (Lindquist, 1933). This product can be calculated directly from the data measured experimentally in a simple example of Darcy’s model:

(2.2-17)

Apart from his own measurements, Lindquist presented Zunker’s data as well in this system (Zunker, 1930). Both experiments were executed with lead balls, and they resulted in straight lines parallel to each other in the figure. The curves became horizontal only in the zone of very small Reynolds’ numbers (according to Lindquist the limit is Re, = 4 ) (Fig. 2.2-6). This presentation corresponds well with Nikuradze’s graphs, which represent the resistance of pipes as a function of Reynolds’ number. The only difference is that the latter has the A value on the vertical axis and not one multiplied by Reynolds’ number. Thus, the corresponding sections of the two graphs can be determined [i.e. the horizontal stretch of Lindquist’s graphs is identical to the sloping straight line in the laminar zone of Nikuradze’s system. The sloping line constructed by Lindquist is practically the same as the so-called Blasius’ line characterizing smooth pipes. Finally, the horizontal section ( A = const.) of Nikuradze’s curve corresponds to a straight line through the origin of the Lindquist’s system although his original figure did not show this zone, because his highest measured datum was only Re, = 2001. As previously mentioned the use of the Reynolds’ number calculated from the data of the proposed model system [Re,, see Eq. (2.1-43)] is more convenient than that of the Reynolds’ number for seepage (Re,),because the influence of porosity and grain-shape can also be taken into

252

2 Determination of hydraulic conductivity

consideration. To take into account the effects of these factors, it seemed advisable to execute some further modification of Lindquist's system, by multiplying the value represented on the vertical axis by a factor determined by the porosity and the shape coefficient (Kovhcs, 1969a). Thus a new parameter was determined measured along the vertical axis: n3 1 (1-n)2 a2

v

(1-n)2

(2.2-18)

Several series of data have been collected from the literature and supplemented by my own measurements, giving corresponding values of the Reynolds' number and the coefficient of resistivity. Because of the very large number of data, only the general characteristics of each series are listed in Table 2.2-4, indicating the number of measurements included in the various series.

253

2.2 Hydraulic conductivity of saturated layers

Table 2.2-4. Characteristic data of experimental series used for the investigation of transition and turbulent zones (A) Homodiaperae aamples The shape and shape' coefticient of the p i n

1

The diameter of theencircling sphere D [mm]

1

PorMlitY ~

I

The range of memurements ni,,,& < Rep <

< Rcma.

EgEEnE

Spheres (a= 6)

1.870 0.930 0.791

0.3910 0.3796 0.3690

0.681-44.60 0.0643-7.60 0.096-4.66

18 23 16

Spheres (a= 6)

4.92 4.02 2.99 2.04 1.06

0.381 0.390 0.381 0.391 0.371

6.13-199.0 10.70-163.8 6.66-107.3 2.62-49.6 0.683-10.6

29 29 34 24 16

Spheres (a= 6)

2.60 2.76 3.76 4.26

0.42 0.42 0.42 0.42

6.33-646.0 22.70-622.0 26.70-1266.0 30.60-1666.0

17 16 21 17

Spheres (a= 6)

2.60 2.60

0.413 0.366

11.10-21.40 12.90-60.30

14 20

0.679 0.686 0.643

6 9 9 8

Aluminium pins (a= 36.7)

10.0

0.600

14.1-68.8 8.7-36.3 7.9-33.2 7.2-30.0

Aluminium discs (a= 24.0)

10.4

0.606 0.643 0.669 0.486 0.410 0.314

9.9-170.0 8.9-178.0 9.3-39.7 10.4.46.6 6.3-63.1 6.9-88.2

24 24 8 9 10 13

Aluminium discs (a= 22.4)

14.3

0.483 0.403 0.376 0.290

13.8-1 79.0 12.0-166.0 6.4-64.2 10.1-130.0

26 17 9 22

Aluminium discs (a= 23.6)

18.6

0.663 0.467

39.1-267.0 32.9-198.0

6 14

Aluminium discs (a= 39.0)

36.4

0.608 0.691 0.620 0.601

62.0-337.0 104.0-420.0 21.1-276.0 11.4-96.2

11 10 26 12

A t first, the samples composed of homodisperse spheres were investigated. 14 series were used, involving 300 data measured in four different laboratories. The zone covered by these measurements extends from Re, = 0.064 to Rep = 1655 and the range of investigated porosity was n == 0.37-0.42.

The points representing these data in the corrected coordinate system are situated, ;t8 expected, along a continuous line (Fig. 2.2-7). The curve can be approximated mathematically by a hyperbola, the equation of which is: (2.2-19)

254

1

2 Determination of hydraulic conductivity

aluminium diSCS

36.4 18.6

39.0 23.6

14.3 10.4

22.4 24.8

1

2

aluminium discs

$

::

2 3

aluminium diSCS

36.4 18.6 10.4

39.0 23.6 24.8

33.3 33.3 33.3

36.4 18.6 14.3 10.4

39.0 23.6 22.4 24.8

26

3 4

aluminium discs

g:

19.0-146

16

0.498

16.4-147

9

0.636

11.4-103

9

0.392

8.6-69.2

8

0.646

13.6-122

18

0.460

11.2-102

20

0.648

13.3-120

19

0.443

10.9-108

11

0.664

8.7-78.0

17

0.476

7.2-67.6

16

26

4 6

0.678

aluminium

Piqs alurmnium discs

10.0

35.7

14.3

22.4

6o 60

6

where Y is the parameter given by Eq. (2.2-18) and (2.2-20)

This relationship can be regarded tw the general equation of movement, characterizing seepage when it is influenced by two retarding forces: friction and inertia. Substituting this hyperbola by its tangent, the hydraulic gradient can be expressed m the function of velocity. The form of this equation is similar to the general formula achieved from the dynamic analysie of this zone [see Eq. (2.1-41)]:

cx'lx_,.

Y = Y1+(X-X1) Y = AX where

+ B;

;

zunker

+ n = 0.391 e n= 0.374 x n= 0.369 Lindquiist Q n = 0.381 t[ n = 0.390 n= 0.38/ 0 n = 0.39/ n = 0.371 v: hegy A n = 0.42 A n= 0.42 0 n = 0.42 n = 0.42 n = 0.385 AurhOf 3 veasuremfi-0 n = 0.413 @

2

2

3 45678 !/

2

3 4 56704

D = 1.87 mm

D = 0.93 mm D = 0.79 mm D = 4.92 mm D = 4.02 mm D = 299 n m D = 2.04 nlfl D = 1.05mm n = 250 mm D = 2.75 mm 0 = 3.75 mm D = 3.75 mm D = 250 nm D = 2.50 mm

2

3 451

Fig. 2.2-7. The resiutance of homodisperae samples of spheres represented aa the function of Reynolds’ number

256

2 Determination of hydraulic conductivity

Most of the equations previously proposed for characterizing the relationship between velocity and hydraulic gradient in the transition zone are also linear on the Y-X plane, as in Eq. (2.2-21). Some of the empirical formulae are compared to the general hyperbola in Fig. 2.2-8 and Table 2.2-5 also offers a possibility for such comparison. A and B parameters listed in this table, are recalculated from various equations. Average values of porosity, grain shape and viscosity were used. The result of the comparison is that the previously proposed formulae generally give the secant of the hyperbola. Hence, these equations prove the accuracy of Eqs (2.2-19) and (2.2-21).They could also be used as an approximation of the more complicated formula within those zones where the line represent,ing the equation in question, is near the general hyperbola. Instead of using the validity zones for the different empirical formulae on the basis of Fig. 2.2-8,it is more profitable to determine a unified system of secants to the general curve. The limits of the validity zones of each line can be chosen as round values of the Reynolds' number, so that the maximum difference between the curve and the secants should not be higher than 15%. This system of approximation is represented in Fig. 2.2-9 and the separated zones are as follows: Laminar (Darcy) zone First transition (Lindquist) zone Second transition zone Turbulent (Froude) zone

(Re, < 10); (100 > Re, > 10); (1000 > Re, > 100); (Be, > 1000).

The laminar zone can be characterized dynamically by two dominating forces. Gravity is the accelerating force and among the retarding forces, 3 2 8% a

100 8

".I 4 2 ; 2

lo

e

$

4 3

2

I I

2 3 4567810

2 3 4 5678100

2 3 4 56781UOU 2 3 45676MOUO

X=Rep Fig. 2.2-8 Comparison of equations previously proposed for the characterization of transition and turbulent zones with the general hyperbola

Table 2.2-6. Comparison of formulae proposed for the characterization of turbulent and transition *Ones /

Panunetem considered in the comparison

Range of validity

Lindquiat v =

2sD2 I 4OvD

2600v

+

vs D Re I Veroneee - = - 2g 4 1160

Re,< 6

V'

D R8.7' I

2g

4

-=-

-

4 < Rep< 200

Rep< 4

6 < Re, < 200

0.38

6

-

-

-

0.38

8

-

-

-

4 < Re, < 160

720

-v2 =-D

2g

4 < Re,< 100

I I 4 16.6

Re, < 200

Re,, > 160

+

Forchheimer I = 0 . 0 3 3 ~ 0.79 v'

+

Z

dI

= 0.09 (v

Irmay I = t -

+ 0.8vf)

v 1 (1-n)' g Dz (n - no)3

-

-.

0.1 < D[cm] < 0.3

1 (1 - n)' + so (n -

60 < Rep< 1000

0.38

7

20

0.3

16-300

Vl

Thh equation is identical to the one proposed if nois neglected compared to n and the coefticienta are 88 follows:

Ooetaoiente of the qnation

z

10.0

a 2

10.3

ij.

the relationship is nothear

8B a 8

0.16 0

0.17

0

E; 2 . v

1.68

2.3

sk 3

0.16

6.0

r

21

258

2 Determination of hydraulic conductivity

Fig. 2.2-9. Representation of relationships approximating the general hyperbola in various zones of seepage

inertia can be neglected when compared t o friction. Theoretically, this condition is valid when the ratio of inertia to friction (i.e. the Reynolds’ number) tends to zero. Thus, laminar seepage can be characterized by the horizontal tangent to the general hyperbola at the intersection of the curve with the vertical axis, the equation of which is

Y = Y,

[EL

+ (X- XI)-

= Y , = 9.3.

(2.2-22)

To increase the range of validity of this approximation, the use of a horizontal secant is advisable, instead of the previously mentioned tangent: (2.2-23)

from which ( 2.2-24)

The equation given to calculate Darcy’s coefficient of permeability [Eq. (2.2-Q)], can be derived in this way aa well. This fact proves, simultoneously,

the reliability of this part of the general hyperbola at the same time. Taking into account the maximum acceptable difference mentioned previously (15%), the upper limit of the validity zone can be extended to Re, = 10. When marking the limits of the first and second transition zones, the main consideration was that both the limits and the factors of the equations approximating the continuous solution, should be whole numbers. Thus, the

2.2 Hydraulic conductivity of saturated layers

259

above-mentioned limits were chosen and taking into account the maximum acceptable difference of 15 yo,the following formulae were determined: First transition (Lindquist’s) zone

Y =8 where

+0.2X;

I = a , $ + blv;

Re 1 al=---,a=0.02-J-; 0.4 a 1 n -

KD

and

consequently

+ 0.8 -;

V

V

I = 0.02 Re, K D

v=KDI

+

0.8

KD

1 0.02 Re,

(2.2-25)

K D is Darcy’s hydraulic conductivity in the equation, which can be calculated from Eq. (2.2-9). Second transition zone Y = 20 0.08 X ; I = a2 v2 b2 v; where 0.16 a 1 - n Re 1 a2=---- 0.008 p- - ; g D n3 and

+

+

consequently

I = 0.008 Re,-

V

KD

v = K D I -2

+

+ 2 -; V

KD

1 0.008 Re,

(2.2-26)

The equat.ions are empirical formulae. There is only one theoretical aspect supporting them: i.e. their form is identical to that derived by dynamic analysis [Eq. (2.1- 4 1 ) ] . Their good numerical agreement with the earlier formulae also proves the reliability of Eqs (2.2-25) and (2.2-26). As in laminar seepage, the turbulent condition is also valid theoretically at only one point where friction is zero. Consequently, the Reynolds’ number tends t o infinity. Thus, the equation characterizing the turbulent zone correctly is that of the asymptote of the general hyperbola. The asymptote intersects the origin of the coordinate system and its slope is

Elxmm = 0.093.

17;

(2.2-27)

260

2 Determination of hydraulic conductivity

It is advisable, however, to use a straight line a little different from the asymptote, to extend its validity range as in the laminar zone. The contact of this line with that representing the second transition zone can be ensured in this way at the limit of Re, = 1000. The movement equation derived from this line is as follows: Y = 0.1

x;

0.2 a 1 - n v2 ; n3

I=-----g D

(2.2-28)

It can be seen that the Chezy's equation, valid for a model pipe of diameter d o can be achieved in this way. Apart from the comparison represented in Fig. 2.2-8, this fact also proves the reliability of the method. Further theoretical evidence of the validity of Eq. (2.2-28), can be given by the numerical analysis of the coefficient of resistivity. This value is much higher here (A, = 1.6),than Dupuit's average parameter given for a pipe ( A = 0.03), but this difference can be explained by the character of the channels between the grains. The model pipe, hydraulically equivalent to the network of pores, is composed of short stretches of diameters d , and d,. For this reason, apart from the resistivity of a straight pipe, the change in cross section has to be taken into consideration as well. Applying the formulae generally used in pipe hydraulics, the total resistance can be given by the following equation:

(0.10 j0.06)

I1

;

(2.2-29)

where do, d,, d, and I , are the characteristic sizes of the model pipe [see Eqs (1.2-19), (1.2-22) and (1.2-23)], and v,, v2,A , and A, are the velocities and areas at the narrow and broad stretches. The ratios of velocities, areas, and diameters indicated in Eq. (2.2-29) can be calculated from the geometrical data of the model pipe:

[%Iz

5

= 0.3 ; = 1.5 ; % = 0.8. (2.2-30) A2 dl d2 The constant factors, whose extreme limits are given in the equation, depend on the character of the change in cross section (i.e. sharp or round,

[3)' 5.0= ; veff

Veff

= 0.4 ;

2.2 Hydraulic conductivity of saturated layers

261

sudden or gradual). Finally, the average of the coefficient of resistivity, recalculated from Eq. (2.2-29) is

ll = 1.4

-

1.8;

(2.2-3 1)

which is in good agreement with the factor determined from the general form [Eq. (2.2-28)]. It is worthwhile to note here, that Krischer (1962) has measured the ill value in the transition and turbulent zones, by using a sample composed of spheres, and the range of the parameter was o.5
Seepage through samples composed of homodisperse spheres was investigated in the previous paragraphs. To generalize these results it is necessary to prove that the equations proposed are acceptable approximations in the case where grain shapes differ and in heterodisperse samples as well. It is quite natural that the scattering of points will be much larger than in the case of Fig. 2.2-7. The reason for the larger uncertainty is that the total surface remains active in the sample when the grains are in contact with each other at one point, which can be ensured only in the case of spheres. If the grains are plate-shaped, their shape coefficient is high and the porosity is small, it is very probable that the grains are in contact with each other over a larger surface. Thus, theactivesurface of the sample and the shape coefficient calculated as the ratio of surface and volume, is smaller than the theoretical value. The scattering of the measurements represented in Figs 2.2-10 and 2.2-11 is caused partly by this effect. The alignment of points representing the data measured, follows the same trend as that of the theoretical curve. The position of the enveloping curves parallel to the curve determined from data measured in homodisperse samples of spheres and bordering the stripe covered by the points measured, can be characterized by the average value multiplied by three and divided by two. It was thought that this discrepancy was approximately equal to the uncertainties generally characterizing the determination of hydraulic conductivity, and, therefore, the method was regarded as acceptable. Thirriot (1969) has, however, drawn attention to the fact that the scattering can be considerably decreased by executing some transformation on the axes of the coordinate system. Thus, increasing the accuracy of the method was one reason why the analysis of seepage with high velocity was continued. The purpose of the second phase of the investigation was to establish a general relationship between seepage velocity and hydraulic gradient, without any restriction on the validity zones [from this aspect, the equation sought is similar to Eq. (2.2-19)], which is simple enough for practical application and simultaneously provides good approximation of the values measured. Before commencing the analysis of the proposed general formula, it is worthwhile summarizing the earlier attempts made to achieve this objective. There are two different methods generally applied to approximate the relationship in question: (a) Binomial form: I = uw + bv2; (b) Potential form: I = Cwm;or w = IW".

262

2 Determination of hydraulic conductivity

2 3 4567810 2 3 4 5678tOO 2 3 456781000 Fig. 2.2-10. Resistmce of grains having sheJ,mooeffioient higher than 6, as a function of Reynolde number

263

2.2 Hydraulic conductivity of saturated layers 2

1000

B

4

e*

5

;

Q-pu3 II

L

2

m

B7

+ 2 mixture n = 0.536

4

o 3 mixture n = 0.546

6 5

x 2 mixture

n = 0.392

+ 3 mixture n = 0.460

3 0

’

l0

8

+ +

4 mixture n = 0.548 4 mixture n -0.443 5 mixture n = 0.564 5 mixture n = 0.476 I

I

2

3

4 56789IO

2

3

4 56789/&?

2

l

I

I

l

I

I

1

3 4 56785

ID0

Re, Fig. 2.2-1 1. Resistance of heterodisperse samples aa the function of Reynolds’ number

Although there were previous proposals assuming that the binomial form can be generalized by considering a polynom having members with a higher order of velocity than two (Forchheimer, 1924) it was proved by both the dynamic analysis in the previous Chapter (Eq. 2.1-41) and by the linear approximation of the curve based on experimental data (Eq. 2.2-21), that the binomial form is the correct approximation and its extension with further members cannot be justified theoretically. Recently, scientists dealing with the problem have therefore concentrated their efforts to determine the a and b parameters as the functions of the physical parameters of the soil (Table 2.2-6). The purpose of Valentin’s investigation ( 1970) was similarly, the survey of various proposals to find the basis of a general seepage law. He has compared various equations chosen not only from hydrodynamics but also from other scientific fields, where the problem of the flow of flu& or gaseous media through porous material has to be solved (e.g. filters composed of metal fibres). The comparison was also supplemented by special experiments. According to the final conclusion, the investigation of seepage with high velocity can be based on generally applicable principles, but the establishment of a general seepage law is hindered by two facts: (a) The structure of the channels formed by the pores depends on the character of the solid matrix (e.g. if it is composed of grains or fibres);

264

2 Determination of hydraulic conductivity

Table 2.2-6. Paremeters of binomial and potentiel formulae depending on phflcel cheraoteristics of mil

I = av

Imay (1964)

1 (1 - n ) 1 (n - no)*5

a-- v (1 - n)* 1 g (n- no)* DZ

Scheidegger (1957)

5a v (1-n)* f7

Ward (1964) .

v 1 --

na

sol

8 g

Ahmedendsunada (1969)

no ineffective

porosity; a end B fectom h d h t h g the ratio of losses a w e d by viscous and kinetic energy dissipation respectively; S , specific &face of grains

nS

c 1 --

g kD

Sunaxle (1966)

+ bv'

SUPpoSins

C = 0.66 = const.

K D

v 1 --

for glass spheres of 3 m m

kD

o = 0.146 b = 0.0648

v 1 --

1

g kD

g m G

p---1 1-n

Englund (1963)

g

l a gv bz

C=-1

na D

I = 0,m Author

Hatch (1940)

I

Chdor

I

Syrnbola

a undetermined coefficient so Spf3CifiC Sllrf8W Of Bra& nA areal porosity

Slepicka (1969)

surface tension q dynamic visaxity

(I

(b) The a and b factors themselves and hence their ratio, are the functions of the character of flow, and they cannot be determined as material constants. The first objection is absolutely acceptable. The conceptual model always has to describe the actual character of the pores by using the parameters of the solid matrix in question. Thus, two different models have to be applied for the investigation of a filter composed of grains or fibres and a third one to describe seepage through fissured rocks. Efforts to find a generalized seepage law have to be limited, therefore, to the characterization of similar porous media (in our caae to that of samples built up of grains).

2.2 Hydraulic conductivity of saturated layers

265

The factors of the binomial form cannot be material constants, even in the restricted case of similar solid matrices, is also true. This is also proved by the fact that most of the formulae listed in Table 2.2-6 use factors in the relationship proposed for the calculation of the a and b parameters, expressing the ratio of head losses caused by viscous resistance and kinetic energy dissipation respectively (e.g. a and /?Englund, 1953; Scheidegger, 1957; and Irmay, 1954; C Sunada, 1965; Ahmed and Suna.de, 1969; and Ward, 1964). This result, however, only indicates that the general relationship sought cannot be based on the theoretically well founded, binomial form. There were also attempts to apply the potential form to achieve a general equation describing the relationship between seepage velocity and hydraulic gradient (Hatch, 1940; Slepicka, 1969; Perez Franco, 1973). It is well known that laminar seepage is described with a power of m = 1. The power gradually increases as the effect of inertia becomes stronger, achieving an m = 2 value in the case of turbulent flow. It is quite evident that the most important parameter of the potential form (i.e. the power m ) ,depends on the character of seepage. Thus, this approximation cannot be applied with unified numerical parameters, but a validity zone has always to be attached to a given value of the power. Following Thirriot’s advice and applying coordinate transformation, the scattering of the points measured can be minimized. It was found by using trial and error methodsthat the best results can be achieved if the two parameters represented on the axes of the system are as follows:

because

[Eq. (1.2-19)];

v Re, = 5 -l/nkD;

[Eq. (2.1-47)];

V

1 g

kD

and

V

n3

=-KD; 9 = Re, n 2 .

x

’;

[Eq. (2.2-9)]; [Eq. (1.1-ll)]; (2.2-32)

It can be seen that the Y coordinate is identical to that applied in Fig. 2.2-7, while the new X abscissa can be obtained by multiplying the Re, abscissa of that figure by r g n 4 (see Table 2.1-4). Plotting all the points

266

2 Determination of hydraulic conductivity

measured from Figs 2.2-7, 2.2-10 and 2.2-11 on one graph (Fig. 2.2-12), the scattering of the data determined using samples made up of spheres (and having, therefore, a higher degree of accuracy) is negligible. The points resulting from measurements executed on non-sphere-shaped particles cover a larger area, the enveloping curves of which indicate that the lowest probable value is twice as small and the highest one is twice as high as the average value. Accepting the general form of the approximate mathematical equation aa (2.2-33)

the parameters can be determined from the figure. The most accurate correlation was found in the case of sphere-shaped grains (a= 6) if B = 8.4; A = 2.1. The ratio of the main axes of this hyperbola is B / A = 4. The scattering of these points (represented by solid points in Fig. 2.2-12) is greater in the transition zone between laminar and turbulent flow. The physical basis of this discrepancy is the gradual development of turbulent seepage, which is influenced during the experiments by other conditions not considered as independent variables. It is reasonable, therefore, t o use two different values of the power in Eq. (2.2-32) and to determine in this way two curves which border the possible zone of points. The width of the enclosed belt is negligible in the laminar and turbulent zones, and shows the greatest scattering in the transition zone. Suitable values of the power were found to be m, = 1/2 and m2 = 3/4. Thus, the h a 1 form of the general equation for sphere-shaped particles is: Upper enveloping curve: (8.-)’”-

[$]l/,=

(gyp)

Lower enveloping curve: - 3/4 -

1; (2.2-34)

- 3/4=

.

In the case of grains having a higher shape coefficient than a = 6, less accuracy is expected, because the uniform packing of the sample and hence, the identical structure of pores cannot be ensured as accurately as in the case of spheres. Let us suppose that a discrepancy of 15% is acceptable for the three parameters of Eq. (2.2-32). The extreme values of the parameters in this case are as follows:

A,, = A(1 f0.15) = 2.1(1 Jr 0.15); A,,, = 2.56; Amin= 1.68;

B,, = B(lf0.15) = 8.4 (1+0.15) ;

Bmax= 10.08 ; Bmln= 6.72 ; mle2= m(1 f0.15) = (0.5 f0.75) (1 & 0.15) ; mmax= 0.8625; mmin= 0.425.

(2.2-35)

2.2 Hydraulic conductivity of saturated layers

N

C

II

3 a

x= ReKn2 5

L

Y

.*0

4

2

ci I

3

2

G

a

C

0

C

d

8 2a

do

Fig. 2.2-12. Representation of the relationship between resistance and the Reynolds’ number on the basis of all measured data

267

268

2 Determination of hydraulic conductivity

After selecting those numerical values which give the extreme position of the mathematically described curve, the enveloping curves for all measurements represented in Fig. 2.2-12 can be determined: Upper enveloping curve : (2.2-36) 0.8625

Lower enveloping curve:

0.8625

=l.

There are only a few points outside these two curves, indicating that the scattering of points causes an error not larger than &15% in the parameters of the general equation. The comparison of the graph with the general equations usually proposed (Ward, 1964; Zampaglione, 1969; Ahmed and Sunada, 1969; Perez Franco, 1973), also indicates the accuracy of the method (Fig. 2.2-13). To construct this figure, the parameters applied by other authors were transformed into the coordinate system used here. I n those caaes where the effect of porosity waa not considered in the original publication (or where i t waa used in a different way), the graphs belonging to the borders of the possible range of porosity (or to the actual extreme values measured) were determined. The only remaining problem is to determine an equation describing the average value within the field covered, instead of that of the enveloping

Fig. 2.2-13. Analysis of previously proposed general equations and the representation of the final result

2.2 Hydraulic conductivity of saturated layers

269

curves, and to express its parameters as functions of the physical data of soil. To ensure the agreement with Darcy’s hydraulic conductivity, it is advisable to select a higher B value than that used in Eq. (2.2-34) ( B , = 10). To balance the influence of this change the B / A ratio has to be increased aa well ( B, / A , = 5; A , = 2). Applying the parametera listed, the upper limit of the range of m determined for Eq. (2.2-34) (i.e. m, = 3/4), characterizes the curve running through the middle of the zone covered by the points in Fig. 2.2-12. This curve can be accepted, therefore, aa the general relationship between resistance and the Reynolds’ number (aa also indicated in Fig. 2.2-13). Substituting the Y and X values from Eq. (2.2-33), the general equation expressing hydraulic gradient aa the function of velocity and physical parameters of the soil can also be determined:

It can be seen that the parameters of v and v2 are the same or very similar, respectively, to those listed in Table 2.2-6. The application of factors, independent of the flow condition [and, consequently excluding the undetermined constants (a,8, C) dependent on the ratio of friction and inertia] becomes possible by introducing a power different from unity (m, = 3/4) in all the members of the equation. 2.2.3 Investigation of microseepage The third main group of seepage through a saturated porous medium has a smaller velocity than that pertaining to Darcy’s zone. This type of Beepage, so-called microseepage is influenced by two retarding forces, gravity is balanced by: (a) Internal friction; (b) Adhesion between the grains and water. The difference between microseepage and laminar flow can be characterized by a graph representing the relation between velocity and hydraulic gradient (Fig. 2.2-14). This graph is a straight line which passes through the origin and its slope is proportional to the hydraulic conductivity (tan a oc R if the seepage is characterized by Darcy’s law. As already explained, there exists a threshold gradient ( I , ) ,in the case of microseepage. If the gradient is smaller than this value, there is no movement in the system, i.e. the velocity is zero. Thus the relation-curve intersects the horizontal axis at a point where I = I,. Several studies dealing with the investigation of this type of seepage have recently been presented: Kovhcs, 1957,1958; J u h b z , 1958,1967; Karhdi and V. Nagy, 1900; Kutilek, 1965,1967; Swartzendruber, 1962, 1908; Childs and Tzimas, 1971; Habib, 1971; Bondarenko, 1973. An important

270

2 Determination of hydraulic conductivity

Fig. 2.2-14. Relationship between velocity and hydraulic gradient in the zone of microseepage

theoretical hypothesis generally accepted to explain the physical character of microseepage assumes that water behaves as the Bingham-body where i t is influenced by adhesion. As proved by experiments, the non-Darcian behaviour of seepage in the range of small velocities can be sufficiently justified by assuming the static shearing stress as the basic property of fluids being composed of dipole molecules (see Section 2.1.1). As also discussed in Section 2.1.1, the distribution of the t oparameter in a direction normal to the solid wall, cannot be determined from the measurements. The various hypotheses applied to approximate the actual to(d) function, lead to different theoretical descriptions of microseepage. There are also empirical formulae proposed in the publications, to characterize the relationship between hydraulic gradient and seepage velocity, below the lower limit of the validity zone of Darcy’s law. The variables (v and I) being small in this zone compared to the accuracy of the measurements, the uncertainty of the experimental data causes the large discrepancies between the empirical results. The simplest theoretical assumption is to assume a static shearing stress, being independent of the position of the investigated point, or more precisely, of the distance of the point from the solid wall [ro(d) = const.]. This hypothesis leads to the Buckingham-Reiner’s equation (Reiner, 1949) which

2.2 Hydraulic conductivity of saturated layers

271

gives the mean velocity of the flow through a capillary tube in the following form : (2.2-38)

where K Ois the hydraulic conductivity of the capillary tube, assuming the flow of a Newtonian fluid, and I , , is the characteristic gradient, the interpretation of which is given in Fig. 2.1-8. Since this characteristic gradient is generally small compared to the actual gradient in most practical cases, Bondarenko and Nerpin ( 1 9 6 6 , 1967) have proposed the simplification of this equation by neglecting the term incorporating the I,& ratio with a high power. Thus, they calculate the virtual hydraulic conductivity by using the following; equation (2.2-39)

Combining this relationship with the capillary tube model where the channels are formed by the pores of loose clastic sediments (see Section 1.2.5) the hydraulic parameters of micro-seepage can be determined. As explained, however, in connection with the dynamic analysis of the forces acting on the percolating fluid, the assumption of a static shearing stress decreasing with increasing distance from the solid wall, is more reasonable than the application of a constant parameter, since the non-Newtonian behaviour of the fluid is caused by the orientation of the dipole molecules, which is stronger along the wall than in the interior of the fluid. For this reason, a theoretical velocity vs. gradient relationship will be derived in detail by assuming a parameter, dependent on the position of the point investigated. The fundamental equation of this theory is the Poiseuille’s formula [Eq. ( 2 . 2 - 2 ) ] , in which the retarding force has to be taken into account according to Eq. (2.1-13): Iyr=2

[

$1 .

zo(r)-q-

(2.2-40)

After differentiating both sides of the equation, the gradient representing the accelerating force (i.e. gravity) can be divided into two parts, one balanced by the static shearing stress and the other by viscous resistance caused by friction

I y =(IT+Is)y = 2 where

d r2 (2.2-41)

Using only a general symbol, for the time being to indicate the dependency of the static shearing stress on the distance of the point investigated from

272

2 Determination of hydraulic conductivity

the wall or, more precisely, from the axis of the pipe, the velocity distribution can be determined by double integration (Fig. 2.2-15):

(2.2-42)

because the constant of the integration can be determine4 uy conside1 ng dv the boundary condition where -= 0, and r = 0.The result of the second dr integration is

IY v = -(TI - ra) 47

-Sf(.) rl

1

11

dr ;

r

(2.2-43)

distribff tjun of sheaf/ngstress

Fig. 2.2-16. Interpretation of symbols used to derive the relationship describing miorfmeep~e

2.2 Hydraulic conductivity of saturated layers

273

because, now, the boundary condition to be considered is v = 0 if r = r,. In the equation, r , is the radius of the pipe where velocity is zero, because at this distance from the axis, the gradient becomes equal to the vdue that can be balanced by the static shearing stress (I,=I)and, therefore, there is no movement at all. The amount of the gradient taken up in overcoming the static shearing stress is proportional to the change of the static shearing stress [Eq. (2.2-41)]. At the same time, this parameter can be related to the threshold gradient as well, where the flow through a capillary tube of constant diameter is investigated. In this case, movement can develop only if the actual gradient is greater than the I, value, at least in a small part of the cross section. Accepting the explanation given in Section 2.1.1, where t o(and probably its derivative &s well) is a decreasing function of the distance measured from the solid wall, it can be supposed that in a circular tube, the smallest value of the I, parameter belongs to the centre of the cross eection ( r = 0). As a final conclusion i t can be stated, therefore, that flow through the pipe does not start if the actual gradient is smaller than that required t o overcome the resistance originating from the static shearing stress at the centre of the tube. Thus, this parameter is equal to the threshold value (Fig. 2.2-15b). There is no movement in the tube if (2.2-44) Another hypothesis has to be put forward at this stage in the derivation: is proportional to the I, value charmterizing i.e. the threshold gradient (I,,) the intersection of the horizontal axis and the straight line approximating the relationship between velocity and hydraulic gradient in the range of higher velocities (see Fig. 2.1-8). The accuracy of this supposition will be proved by the derivation given in the following paragraphs and even the numerical value of the factor of proportionality will be determined (testifying that this value is equal to 2): and substituting the probable numerical value

I, = 21,.

(2.245)

It was proved, however, by Bondarenko and Nerpin’s measurements that the I, parameter can be calculated as a B parameter divided by the radius of the tube and the specific weight of the fluid and the B Bondarenko’s constant was found to be a single-valued function of temperature. Thus, the threshold gradient in a tube having a radius of r can also be calculated knowing the physical parameters and the temperature of the flowing fluid:

I , = - ,B .

(2.246) 2ro Y where the R value can be determined from the relationship given in Eq. (2.1-12). 18

274

2 Determination of hydraulic conductivity

The combination of Eqs (2.2-44) and (2.2-48) gives a condition which has to be satisfied by the function used to approximate the unknown relationship between the static shearing stress and the distance measured from the solid wall of the tube: (2.2-47) The most simple mathematical relationship which ful% this condition, is a hyperbola of first order: d t L _B dr 8S

--

.

(2.2-48)

I n a pipe, the influence of the two opposite walls has to be taken into consideration, because the superposition of the two actions can be assumed (Fig. 2.2-16c). At the same time, both S1 and 6, characterizing the distance of the point investigated from the two walls can be expressed as the functions of the distance measured from the axis of the pipe (r) ( 8 , = ro - r and

S, = ro + r). Thus the- dzo(r) differential quotient and the general form of

dr the T o ( r ) function can be determined:

B

tO(r= ) -:arth 4

1

+ C = -In B 8

TO+

To-

IrI+C* Irl

(2.2-49)

( 4 function [which can It can be seen, that the minimum value of the dz 0

dr be achieved by substituting the r = 0 value, satisfies Eq.(2.247)]. On the baais of the last equation the velocity distribution can be determined from the general formula given as Eq. (2.2-43):

(2.2-50) Using the formula generally applied in calculating the flow rate through a capillary pipe rl

Q=

2nrw(r)dr;

(2.2-51)

0

and substituting the point value of the velocity according to Eq. (2.2-50),

2.2 Hydraulic conductivity of saturated layers

275

the mean velocity in a capillary tube influenced by adhesion can be calculated by integrating Eq. (2.2-51):

-In---

IIo

i ’IIll . 1 - 2

(2.2-52)

where the v, symbol is applied to indicate that this relationship characterizes the mean velocity of microseepage in the tube. The characteristic radius rl was substituted by the t,hreshold gradient on the basis of the following relationship:

I

= I , where:

r =r,;

consequently I0 - 6 - 4 . ---,

I

4

-rl=

(2.2-53)

r0

It can be supposed that the change of velocity caused by the influence of adhesion (i.e. the ratio of the velocify of microseepage and that of laminar flow) is independent of the structure of the solid matrix. Thus, the quotient of the mean velocity, aa determined from Eq.(2.2-52) and related to Darcy’s velocity calculated for the same capillary tube, can be assumed to be equal to the ratio of the hydraulic conductivity of a soil sample valid in the zone of microseepage to that of one in Darcy’s zone. This hypothesis is based on the fact that all the geometrical characteristics of the model system,substituting the actual channels between the pores (i.e. pipe diameters, number of pipes), are unchanged and the comparison of two flows with the same hydraulic gradient is also msumed. Thus, the seepage velocity in microseepage can be calculated as a product of Darcy’s velocity and a ratio of the two mean velocities calculated for the same tube but assuming different flow conditions.

This ratio tends to unity as the gradient tends to i n h i t y , and hence the difference in Darcy’s value becomes negligible in the range of high gradients (Fig. 2.2-14). Since the basic value (related to which the difference is calculated) increases with increasing gradient, the absolute differenoe above a given limit is practically constant and the two relation-curves (i.e. the Darcy’s line and that representing microseepage) become parallel to each other (Fig. 2.2-14b). This limit can be chosen at a value of I = 12 I , 18*

276

2 Determination of hydraulic conductivity

which is, therefore, the upper limit of micro-seepage and the lower one of the laminar zone. Above this limit the second member on the right-hand side of Eq. (2.2-54) tends to zero, and even the quadratic member of the first part can be neglected: - - 1 --2 2 " + I - - 1 - 2 0 I. (2.2-55) I VD I

[?IZ

This is because the intersection of the horizontal axis and the straight line characterizing the laminar flow waa indicated by I , . The equation of this line is v, = KD(I - 11),whichoncomparison with Eq. (2.2-55) proves the I , = 21, relationship given in Eq. (2.2-45). Figure 2.2-14 and Eq. (2.2-53) indicate that the influence of adhesion should be taken into consideration in the laminar zone aa well. The correct relation-curve is not Darcy's line, but another straight line p a r d e l t o the former and intersecting the horizontal axis a t the point I = 21,. In practice, the factor given in Eq. (2.2-52) can also be simplified. Thus, the actual relationship in the zone of microseepage can be approximated by a parabola with a vertical axis, the minimum point of which is v = 0; I =I,. Another condition which determines the parameters of the parabola is that the tangent of the latter at the point v=lOKJ; I = 121,, is the line previously mentioned aa v = K ( I - 21,). Taking into consideration the derivation presented in the previous paragraphs, the following equations can be given to describe the relation between velocity and hydraulic gradient for laminar movement and microseepage: I n the zone of laminar &ow (2.2-56)

where

In the zone of microseepage

I

The threshold gradient used in these equations can be calculated from Eq. (2.2-46) combining it with the data of the model pipe and considering Eq. (2.1-12): B 1 - n a

I , = --47 n D

1 l - n a {exp [- 4.6(t°C+156.3) 10-a]-5 4 y n D

- ---

x (2.2-58)

2.2 Hydraulic conductivity of saturated layers

277

The applicability of the proposed method in the practice was proved originally by comparing calculated graphs with measured data (Kov&cs, 1969~). Kutilek’s measurements were used for the comparison, having been at that time the most detailed series published in the literature (Fig. 2.2-16) (Kutilek, 1967).This investigation has shown that the calculated values are in good agreement with the measured data, and the proposed method can follow both the actual character of the relationship and the influence of the water temperature as well.

C? - kaoliflife Fig. 2.2-16. Comparison of 6he theoret,icalrelationship with Kutilek’s measurements

278

2 Determination of hydraulic conductivity

Only a slight difference between the measured and calculated data can be observed in the zone of very small velocities. The measured points seem to contradict the existence of the threshold gradient, especially in the case of kaoliqite samples, as Thirriot (1969) has pointed out. At the same time the measurements executed in capillary tubes clearly indicate that there exists a limit to the gradient below which the velocity is zero, and there is no flow in the tube (see Fig. 2.1-8c). This discrepancy between the results of experiments with glass tubes and actual soil can be explained by the fact that the pores are considered in the derivation as having an average diameter [do; Eq. (1.2-19)] and hence, the ca1culated.threshold gradient is also an average characteristic of the sample. This parameter can be accepted, therefore, if the phenomenon extends to a volume of the sample large enough to be characterized by average parameters, on the basis of the continuum approach. The completely static state can be disturbed, however, if there are within the solid matrix, only a few larger pores through which movement can be initiated by an even smaller gradient than the threshold value. Considering the size of the large pores [d,; Eq. (1.2-22)], some correction can be made, to eliminate most of the differences (see the dotted line in Fig. 2.2-16). The observed discrepancy between the measured and calculated data draws attention to the fact that the commencement of seepage is governed by a property of the sample which cannot be characterized by applying the continuum approach and, therefore, the derived relationship gives only the general character of the phenomenon, and the actual data may differ from the calculated averages. Bondarenko (1973) has also collected some data, measured on soil samples concerning the I , gradient. Among the physical soil parameters, porosity ( n )

KI

and the specific surface - are given in this publication (Table 2.2-7). According to Eq. (1.2-1), the specific surface is equal to the ratio of the Table 2;2-7. Comparison of measured and calculated values of the threshold gradient of soil samples on the basis of Bondarenko's (1973) data

I

Medium sandy clay Medium sandy clay Heavy sandy clay Clay

Light sandy clay Medium sandy clay Fine rock flour Fine rock flour Light sandy clay

a s

Rock flour Light mndy clay

1.66 1.68 1.60 1.40 1.63 1.68 1.66 1.66 1.46 1.67 1.46 168

0.38 0.37 0.40 0.47 0.42 0.37 0.42 0.42 0.46 0.41 0.45 0.41

130 124 116 118 !32 83 76 70 61 131 42 69

0.8 1.9 1.9 4.0 0.6 0.7 1.9 0.8 1.0 3.6 1.1 0.4

1.32 1.32 1.09 0.82 0.79 0.88 0.66 0.80 0.38 1.18 0.32 0.53

average

0.60 1.44 1.76 4.86 0.76 0.79 2.88 1.33 2.60 3.06 3.40 0.76 2.02

2.2 Hydraulic conductivity of saturated layers

279

a shape coefficient and effective diameter - . Assuming a given temperaDh

ture (e.g. 2U°C) the threshold gradient can be calculated from Eq. (2.2-58), as listed in the last but one column of the table. Finally, the last column gives the ratio between the measured I , and the calculated I , values. It can be seen that the spread of this quotient, caused by the uncertainties in meaaurements is enormous, but the average agrees well with the theoretical values (2.02 instead of 2.0). For complete informtition on the methods applicable to characterize the interrelation between seepage velocity and hydraulic gradient in the zone of microseepage the empirical relationships have to be mentioned. These methods approximate the points determined on the w vs. I field, in most cases, from data obtained by measurement, by exponential formulae, the general form of which, according to Habib (1971) is:

v = KD ( I

- I , 11 - exp(-I/I,)]).

(2.2-59)

Examples of this type of equation are:

v = K , { I - A [l - exp(-GI)]};

1;

+ exp(BI)] - I ,

(Swartzendruber, 1962);

I

v = K D -1n

[A

v =K, 9 {I

- 11[1 - exp(-I/I,)]};

; (Kutilek, 1964);

(2.2-60)

(Hansbo, 1960);

V

where the constants are the functions of the threshold gradient.

(l&

(4011 (416

Fig. 2.2-17. Comparison of the results of various methods on the basis of Habib’s meesurements

280

2 Determination of hydraulic conductivity

Habib (1971) compared the data, determined by carefully executed experiments using kmlinite samples, with the calculated graphs, and found that the experimental relationships are more reliable than the theoretical ones, aa both Eq. (2.2-38) and Eq. (2.2-57) give smaller velocities than the measured values. This contradiction originates, however, from the fact that Habib did not distinguish between the I,, I, and I,, values (see Fig. 2.1-8), but substituted I, into the theoretical equations as well. After recalculating the necessary basic data from the measurements published, the same comparison was executed (Fig. 2.2-17). As shown in the figure, there is no conmeasured values

- curves calculated ---

'\?@:Po+,

0

. 0

\.em

from Eq.(2.2-56)

Q .-

0

a.8 0.6 0.4

I

0.45

I

0.50

I

I

0.55

I

I

0.6D porosity,

n

Fig. 2.2-18. Relationship between the I , gradient and porosity on t,he basis of Habib's measurements

siderable difference between the calculated intrinsic permeabilities, except in the vicinity of the threshold gradient, if the correct interpretation of I,, I, and I,,.is used. The smallest permeability measured belongs to a gradient of six times greater than the threshold value. The points determined from the observations lie along each of the calculated curves. On the basis of the detailed data published by Habib it was also posoible to check the accuracy of Eq. (2.2-58). The trend of the I, gradient vs. porosity, determined from the equation, agrees with the yo8ition of the measured points (Fig. 2.2-18). The scattering of the measurements is considerable. Two curves indicated by dotted lines and also calculated from Eq. (2.2-58) by using different multiplying factors were constructed to envelop the zone of scattering. The factors of the enveloping curves related to the constant of the average curve, indicate an uncertainty (or error in measurement) of about +30%. The curves show the influence of only one independent variable (i.e. porosity), while the effects of the others are incorporated in the multiplying factor which is assumed to be constant. Taking into account a temperature of 20 "C maintained during the experiments, the probable effective diameter of the kaolinite can be recalculated. The value determined from the constant of the average curve 1 p), is smaller than that calculated from the published grain-size (i.e.

-

281

References

distribution curve, but considering the probable high degree of coagulation of the kaolinite, the results of the experiments do not contradict the theory, and Eq.(2.2-58) can be accepted as a realistic approximation of the threshold gradient. References to Chapter 2.2 AHMEND, N. and SWNADA, D. I(.(1969): Non-linear Flow in Porous Media. Proceedings of ASCE, H Y . 6. November. S. and OWSTON, R. (1962): Studies on Die eraion in Porous BANKS, R. B., JEXWATE, Media Flow. Technicd Report Northwestern Univerdy andSEAT0 ScLol of Engineers, Bankok. BEAR,J. (1972): Dynamics of Fluids in Porous Media. Elsevier, New York, London, Amsterdam. BONDARENKO, N. F. (1973): Physics of the Movement of Groundwatem (in Russian). Gidrometeoizdat, Leningrad. BONDARENKO, N. F. and NEWIN,S. (1966): Influence of Visclous-Phtic Properties of Water on its Eauilibrium and Transfer. IASH S v m m i u m ma Wder in Unsaturated Zone, Wageningen: 1966. BONDLRENKO. N. F. and NERPIN.S. (1967): Shearine Stress of Fluids and its Consideration & the Investigation of.Surfbe Phenomen; (in Russian). Publiccation of the Inutitute of Physico-Chemistry of the Academy of Sciences, USSR. BROWNELL,L., DOMBROWBKI, H. and DICKEY,C. (1950): Premure Drop through Porous Media. Chemical Engineering Prop., Vol. 46, CARMAN, P. C. (1966): Flow of Gases through Porous Media. Butterworths, London. CE~ARDABELLAS, P. (1964): SpecScation of Field Tests by Standardizing the Grainsize Distribution Curve of Water-bearing Clastic Sediments (in German). Mitteilung dea InatitUtes Fiir Waeserwitechaft, No. 20. CHILDB, E. C. and T z m , E. (1971): Darcy's Law at Small Potential Gradients. Soil Science, No. 3. DARCY,H. (1866): Public Water Supply of Dijon (in French). Paris. DE WIEST,R. J. M. (1969): Flow through Porous Media. Academic Press, New York, London. ENQELJ~ARDT, W. (1960): Porosity of Sediments (in German). Springer, Berlin, Gottingen, Heidelberg. ENOLUND, F. (1963): On the Laminar and Turbulent Flows of Ground-water through Homogeneous Sand. Technical Univerdy of Denmark, Bulletin No. 4. FORCEHEIMER, PH. (1924): Hydraulics (in German). Teubner, Leipzig, Berlin. GOLDBTEIN, S. (1938): Modern Developments in Fluid Dynamics Vol. 11.Oxford University Press, London. HABIB, J. (1971): Flow-rate of Seepage of Water through clay, and Darcy's Laws (in French). Doctorale Thesis. University of Toulouse. HAQEN, G. (1869): Handbook of Hydraulic Structures (in German). Berlin. HANBBO, J. (1960): Consolidation of Clays with Special Reference to Influenceof V e r t i d Sand Drain. Proceedings of Sw&h Qeotechnicol Irretdute, No. 18. HILRLEMAN, D. R. F., ME-ORN, P. F. and R m a , R. R. (1963): Dispersionpermeability Correlation in Porous Media. Proceedings of ASCE, HY. 2. March. HATCH,L. P. (1940): Flow through Granular Media. Journal of Applied Mechanics, ASME, Vol. 62. HAZEN, A. (1896): The Filtration of Public Water Supplies. New York. IRBIAY. S. 11954): On the Hvdraulic Conductivitv of Unsaturated Soils. Transcrctione of AQU, Vol. 36, No. 1: JAKY, J. (1944): Soil Mechanics (in Hungarian). Egyetemi Nyomda, Budapest. J. (1958): An Investigation into Percolation (in Hungarian). HidroMgiai JUHABZ, Kodony, No. i. Jwksz, J. (1969): An Investigation into Percolation. Acta Technica Academiae Scientiarum Hungaricae, Tom. 24. "

a

282

2 Determination of hydraulic conductivity

JUHASZ, J. (1967): Hydrogeology (in Hungarian). Tankonyvkiadd6, Budapest. JWAsz, J. (1976): Hydrogeology (in Hungarian). Akademiai Kiad6, Budapest. KARLDI,G. (1963): Hydraulics of Linear Drainage Systems (in Hungarian). Khartum, Budapest, (Thesis; Manuscript). KARADI, G. and TOROP, L. (1966): Applicability of Seepage-law. (in Hungarian). Hidroldgiai Kozlony, No. 6-6. KARADI, G. and V. NAQY,I. (1960): Investigation of the Validityof Darcy’s Law (in Hungarian). Conference 011 Hydraulics, Budapest, 1960. KovAcs, G. (1967): Theory of Micro-seepage (in Hungarian). Hidroldgiai Kozlony, No. 3. K O V ~ C G. S , (1958): Theoretical Investigation into Micro-seepage. Acta Technica Academiae Scientkrum Hungaricae, Tom. XXI, No. 1-2. Kovbcs, G. (1966): Dynamio Investigation of Seepage by Invariant Numbers. Symposium on Seepage and Well Hydraulic%, Budapest, 1966. Kovlics, G. (19694: General Characterization of Different Types of Seepage. 13th I A H R Congress, Kyoto, 1969. KovAcs, G. (196913): Relationship between Velooity of Seepage and Hydraulic Gradient in the Zone of High Velocity. 13th I A H R Congress, Kyoto, 1969. Kovlics, G. ( 1 9 6 9 ~ ) :Seepage Law for Micro-seepage. 13th I A H R Congress, Kyoto, 1969. KOEENY,J. (1963): Hydraulics (in German). Springer, Wien. KRISCHER, 0. (1962): Development of Material Transport through Earthworks and Porous Media by Diffusion. Molecular Movement aa well as Laminar and Turbulent Flow (in German). Chemie-Ing. Techn., No. 3. KUTILEP,M. (1964) :The Filtration of Water in Soils in the Region of Laminar Flow. 8th Congress of ISSS, 1964. KUTILEK, M. (1966): Influence of Interface on Filtration of Water in Soil (in French). Science du Sol, (Prague), 1. KUTILEP,M. (1967): Temperature and Non-Darcian Flow of Water. International S o i l Water Symposium, Praha, 1967. KUTILEP.M. and SALINQEROVA. J. (1966): Flow of Water in Clay Minerals aa Influenced by Adsorbed Quinolini& k d Pbdinium. Soil Science, Vdi. 101, No. 6. LEIBENSON, L. S. (1947): The Flow of Natural Fluids and Gases in Porous Medium (in Russian). Geotekhizdat, Moscow. E. (1933): On the Flow of Water through Porous Soil. 1st Congress of LINDQUIST, ICOLD, Stockholm, 1933. MOSONYI,E. and KovAcs, G. (1962): Model Law by Joint Consideration of Gravity and Friction (in Hungarian). Hidroldgiai Kozlony, No. 7-8. MOSONYI,E. and KovAcs, G. (1966): Model Law of Filtration (in French). Congresr, of I A H R , Dijon, 1956. PALADIN, I. A. (1964): Determination of Permeability Coefficient of Grained Noncohesive Soils (in Russian). Gidrotechnicheakoe Stroitelstwo, No. 3. PEREZ FRANCO, D. F. (1973): Non-Darcy Flow of Ground-water towards Wells and Trenches. Particularly within the Turbulent Range of Seepage. (Manuscript). Technical University of Budapest. RAIMONDI, P., GARDNER, G. H. F. and PETRICP, E. (1969): Effect of Pore Structure and Molecular Diffusion on the Mixing of Miscible Liquids Flowing in Porous Media. Symposium on Fundamental Concepts of Mkcible Fluid Dieplacement, San Franekco, 1959. REINER,M. (1949): Deformation Strain and Flow. Lewis, London. A. E. (1967): T h e Physics of Flow through Porous Media. University SCIHEIDEQQER, Toronto Press, Toronto. SCHLICHTER, C. S. (1899): Theoretical Investigation of the Motion of Ground-water. Annual Report of the U.S.Geological Survey. SEELHEIM (1880): Zeitschrift fiir analitkche Chemie, ,(in German). Wien. SLEPIOKA, F. (1969): The Linear end Non-linear Regimes of the Filtration Law and their Consequences in Geohydraulic Problems 13th I A H R Congress, Kyoto, 1969. SUNADA, D. (1966): Turbulent Flow through Porous Media, Water Resources Centre Contribution No. 103. Univ. of California.

References

283

SWARTZENDRUBER, D. (1962): Modification of Darcy’s Law for the Flow of Water in Soils. Soil Science, No. 1. SWARTZENDRWER, D. (1968): The Applicability of Darcy’s Law. Proceedings of the American Society of Soil Scientists, 1. TERZAQEU, K. (1926): Soil Physical Bases of Soil Mechanics (in German). F. Deuticke, lien. TERZAQHI, K. (1943): Theoretical Soil Mechanics. John Wiley, New York, London. THIRRIOT,C. (1969): Hydrodynamics of Flow in Porous Media (in French). 13th I A H R Congrem, Kyoto, 1969. VALENTIN, F. (1970): Non-linear Resistance of Porous Media (in German). Mitteilungen, ImtitUt fiir Hydraulik und Gewllsserkunde, Miinchen, No. 6. V. NAGY,I. (1967): Laboratory Measurements to Determine the Coefficient of Permeability of Samples (Oral information) (in Hungarian). Budapest. WARD,J. C. (1964): Turbulent Flow in Porous Media. Proceedings of ASCE, HY. 6. September. ZAMARIN,J. A. (1928): Calculation of Ground-water Flow. (in Russian). Trudey I. V . H . , Taskent. ZAMFAQLIONE, D. (1969): Turbulent Seepage Flow in Filters. 13th I A H R Congrees, Kyoto, 1969. ZAWEREI, I. I. (1932): On the Problem and Determination of the Permeability Coefficient (in Russian). Izvestia VNIIQ, Leningrad, No. 3-6. ZUN~ER, F. (1930): Behaviour of SoiL in Connection with Water (in German). Handbook of Soil Science, Springer, Berlin, Vol. VI.

Chapter 2.3 Seepage through unsaturated layers The most complicated form of flow among the various types of seepage in layers composed of loose clastic sediments is the water movement through an unsaturated porous medium. The flow is affected by the highest number of major forces, since two retarding and three accelerating forces have t o be considered. These are friction and adhesion on the one hand, and gravity (or pressure difference substituted by gravity in the form of the height of equivalent water column), tension difference caused by adhesion and that created by capillarity, on the other. The theoretical description of the movement is made more difficult by the fact that water exchange may occur between the liquid and gmeous phase through the surface of the water film (evaporation or condensation), depending on the vapour pressure in the gaseous phase filling part of the pores, related t o the tension of the water surface. Thus, water movement may develop in the vapour phase as well. In both phases (liquid and gaseous), water can be transported not only by convective flow, mechanically driven by the accelerating forces (naturally, in the vapour phase pressure difference substitutes gravity), but also by di8usion, caused by the uneven distribution of some intensive properties in both the liquid and gas (i.e. temperature, chemical concentration, etc.). Thus, the total mass flux is fhally composed of four members (Klute, 1952): (a) Diffusion in the liquid phase; (b) Piffusion in the gaaeous phase; (c) Convective transport of the fluid; (d) Convective transport of the vapour.

284

2 Determination of hydraulic conductivity

The diffusive component can, of course, influence wate? transport through saturated media as well. Under unsaturated conditions, however, its influence is considerably higher, because in this case, the flow rate of the convective flux is smaller. To simplify the hydraulic investigation of the process, diffusion will be neglected in further discussions, assuming that an isothermal condition of the unsaturated layer is investigated and the differences between other intensive properties of the various points of the system are also negligible. I n connection with vapour flux,it was found that this component can be significant only if the water content of the medium is low enough to provide a continuous gas phase open for vapour movement. In this case, however, the differences in vapour pressure at various pointa are relatively small and, therefore, neither convective nor diffusive vapour flux can play a considerable role, except on the upper few cm of the soil profile, and under non-isothermal conditions. At the same time, the rise in pressure of the gaseous phase can be significant in pores closed by the fluid, and this phenomenon can indirectly influence the propagation of the fluid in the unsaturated medium. Considering the aspects given previously, the objective of the investigations discussed in this chapter, is to analyze the convective flux of water through unsaturated porous media. The chain of water films and saturated pores is, therefore, regarded as a closed system, through the border of which water exchange does not occur, and water movement in the vapour phase can be neglected. Diffusive water transport is also regarded as insignificant. The investigation of this simplified system is further complicated by the unsteady character of seepage. The thickness of the water film is modified by the flow. The tension on the water surface, however, depenh on this changing parameter. Thus, the resultant of the acting forces is also altered by the flow and, therefore, the movement is generally unsteady. There are numerous cases where the determination of some basic relationships needs the application of a further approximation (e.g. the moisture content to be constant at both ends of the investigated sample), to ensure that the assumption of steady movement is acceptable. For the general description of this type of movement, however, the time-dependent character of the flow has to be considered.

2.3.1 Movement equations characterizing unsaturated flow The movement of water through the unsaturated, upper zone of the soil is of very high practical importance, because it determines the ratio between surface run-off, infiltration and evaporation. Thus, i t governs the continental branch of the hydrological cycle. For this reason, the investigation of this phenomenon has a very long tradition. At the same time the versatile character of the influences makes the physical description of the process difficult and, therefore, a complete solution which is well founded theoretically and is easily applicable in the practice, cannot be proposed yet.

2.3 Seepage through unsaturated layers

285

The first investigations were limited to the analysis of infiltration, through the surface, although this process describes only one of the boundary conditions of the whole system. As this component of the system waa relatively easy to observe and meaaure, numerous attempts were made to approximate the flow rate inatrating through the surface and decreasing continuously with time. The proposed models express the flow rate as the function of the physical parameters and the existing condition of the soil. These empirical formulae generally consider only a few limiting conditions when selecting the mathematical form of the approximation (e.g. after a very long time having elapsed from the beginning of infiltration, the flow rate has to be equal to bhe flux transported by the saturated medium under the influence of unit gradient). Horton’s equation (Horton, 1939) can be mentioned aa an example of the empirical formulae, which assumes that the time-dependent inatration through a unit area of the surface, [ f ( t ) ] , has to be divided into two parts. The first decreases exponentially, with increasing time and the second is constant, representing the water conveyance of the saturated soil (Fig. 2.3-1): f ( t ) = (a - f o ) exp (-W

+

(2.3-1)

fo.

In the above equation f is the flux of the saturated medium, whichis approximately equal t o the saturated hydraulic conductivity, if the water table is deep, because in this case the final gradient tends t o unity. The other constants ( a and 13)depend on the physical parameters and the instantaneous conditions of the soil at a particular time. The next development in the theoretical analysis of infiltration was the supposition that there is a well d e h e d horizontal boundary (wetting front) between the upper and lower domains of the vertical column under investigation. Above this front, the medium is completely saturated from the water on the surface, while below it the original characteristic water content is constant and the rate of saturation is not influenced by it. The conditions considered in the mathematical description of this so-called piston fiow, are represented in Fig. 2.3-2. The hydraulic parameters (the flow rate and the propagation of the wetting front) can be determined theoretically by the integration of the basic differential equations, derived from the kinematic

‘W

time

t

Fig. 2.3-1. Characterization of the relationship between infiltration and time using Horton’s equation

286

2 Determination of hydraulic conductivity

concept of the laminar flow through the continuous field (Laplace’s equation). This will be demonstrated in the introduction to Chapter 5.1 [see Eq. (5.1-5)].The same result can be achieved, however, by substituting the gradient into Dmcy’s equation. In this cme the gradient is equal to the total pressure diflerence between the surface and the wetting front, related to the length of the seepage. The numerator (the total pressure difference) is comthe length of posed of three members: the depth of the ponded water (a), , the average capillary sucthe seepage, being a function of time [ z j ( t ) ]and

.\

{xh, capi(1ary% ;

, , , , . I . .

r:gn‘c-i* “.,-” sucf/on&& i,*-y:-+$;:&;: 10

I.-‘,

,.I,

,-..*--A.

I.

Fig. 2.3-2. Parameter chersoterizingthe piston flow

tion at the front (h,).Thus, the flux of the piston flow (the infiltration through a unit area of the surface, q) and the velocity of the propagation of the zvetting front ( v d f )can be cdculated from the following equations (Green and Ampt, 1911):

and

. (2.3-2) n, Considering the continuity of the movement, a further condition can be determined: the total amount of water crossing the surface from the beginning of infiltrationuntil a time point, t , should be equal to the storage of a prism having a base of unity and located between the surface and the wetting front. Thus, the variables have to satisfy the following condition: V,ff

9 =-

where n, is the specific yield of the soil. There are three parameters in Eq. (2.3-2) characterizing the soil through which the infiltration takes place ( K D , ns, hc). Among them the hydraulic conductivity of the saturated medium and the specific yield are well defined

2.3 Seepage through unsaturated layers

287

characteristics, although, in the case of infiltration the observed value may differ from that determined in a completely saturated sample, because the air bubbles trapped above the wetting front may result in K D and ns being smaller than those parameters belonging to the saturated condition. The cupillary suction is even more uncertain than the other two characteristics. The actual wetting front is never a horizontal plane. The water runs ahead through the large pores and propagates more slowly in the small ones, while the capillary suction is larger in the narrow channels. This action of capillarity, varying from point to point, hm to be taken into account, therefore, with an average value. If the suction is expressed by the probable equivalent height of a water column, the average capillary rise lies between the probable maximum and minimum capillary height. In spite of the numerous uncertainties and the rough approximations applied to derive the mathematical model, the latter method has proved to be quite reliable in practice if the physical parameters of the soil are carefully chosen. It is quite evident, that parameters derived from the observation of a processes and evaluated by assuming a given relationship between the variables measured, will provide an acceptable description of the phenomenon, if they are substituted into the same equation from which they were derived. The application of such mathematical models and physical parameters of the soil always requires, however, the execution of local measurements which hinder the generalization of data determined elsewhere. Another limitation arising in connection with the application of the Green-Ampt’s equation, is caused by the fact that its form, given as Eq. (2.3-2), describes the vertical movement of the water through the unsaturated zone only if the flow is directed downwards, and the pores are completely saturated at the surface and the infiltrating water is continuously replenished on the surface. Similar equations can be determined to characterize the upward movement of water from the water table towards the assumed average capillary height, or the lowering of the upper surface of the saturated zone to a height h, above the water table [see Eq. (5.1-4)]. A condition of the application of this relationship is the assumption of a bharp border between the dry and completely saturated domains. Thus, this approximation is not suitable to describe those types of movements which also occur in an unsaturated medium without requiring the complete saturation of any part of the profile (e.g. the effect of evaporation on ground water and the redistribution of soil moisture). A further important development concerning the investigation of water movement in unsaturated porous media was the combination of Buckingham’s potential and Darcy’s law (Richards, 1931). As already explained in Chapter 1.3, the water is not under hydrostatic pressure i n the unsaturated zone, but a suction (negative pressure) prevails on the surface of the water film. This suction which depends on the water content of the solid matrix in the vicinity of the point investigated, is generally expressed by the equiva-

”)

. Thus the total energy available lent water column suction head y = Y (also expressed in units having a dimension of length) to create and maintain seepage is determined by two conditions: the height of the point related to the

[

288

2 Determination of hydraulic conductivity

arbitrarily chosen reference level, and the suction head. The hydraulic gradient vector can be calculated aa the divergence of the total energy (the change of the energy head in the direction of movement):

d(h - Y ) - a@ - Y ) + ds ax

-Y ) (2.3-4) az Substituting this gradient into Darcy’s equation, the flux (the flow rate through a unit area normal to the main direction of flow) can be achieved: I=v(h-y)=

- y)

+

aY

According to the equation of continuity, the local change of flux is equal to the change in the water content with time within a prism of unit volume

aW vq=--. (2.3-6) at Considering that hydraulic conductivity is also a function of water content, [ K (W ) ] ,in the unsaturated zone and using the relationship between suction head and water content, [y(W)], all the terms can be given aa the functions of either volumetric moisture content, (W), or suction, (y). A differential equation can be determined by combining Eqs (2.3-5) and (2.3-6)) which gives a relationship between time, the space coordinates and either W or y. Choosing the positive z axis directed vertically upwards, the various forms of the differential equation are aa follows:

where

8W

A ( y ) = - (specific water capacity; Richards, 1931); aY

(2.3-7)

or

where

(

:;)

D( w,:=K( W ) - - (soil-water diffusivity; Childs and Collis-George, 1950).

The great advantages of both forms of Eq. (2.3-7) are that the seepage i n any direction can be described by them and they do not include any restrictions concerning the rate of saturation of the medium. Thus, they can be applied to the determination of the hydraulic parameters of infiltration, such aa those produced by the upwards movement of water which starts from ground water replenishing the moisture content evaporated from the

2.3 Seepage through unsaturated layers

289

unsaturated zone, as well aa those of the horizontal seepage of soil moisture (or its flow in any direction) created by the tension differences between the various points of the medium. Difficulties are, however, caused by the fact that there is no unique relationship between the suction head and water content, because of the hystereais of the soil-moisture retention curve, aa explained in Chapter 1.3. Assuming that the hydraulic conductivity of the unsaturated solid matrix is an unambiguous function of the water content, the K ( y ) relationship and the specific water capacity are not single-valued functions in Richards’ formula, while in the other form of the differential equation, the diffusivity term has the same uncertainty. Apart from the problems caused by hysteresis, the very complicated structure of the equation also hindered the practical application of the method. Those simplifications, which could ensure the analytical solution of the differential equation, would be very rough physical approximations and, therefore, reliable results can be achieved only by the numerical handling of the formula. For this reaaon, this theory became widely used after large computers were developed. One of the pioneers in preparing the various numerical methods was Philip (1955, 1957, 1966). Supposing that both D and K are single-valued functions of W and considering the boundary 0, and W = W , if z = 0. t 2 0 , he has conditions W = W 1if t = 0 , z solved the differential equation simplified for the one dimensional movement of infiltration

>

8W

-

8

i3W

8K

a+-az(Dd+x

(2.3-8)

by expanding it into a power series of PI2. One of the most important results of this analysis is the statement that - excluding the vicinity of the ) be assumed time point t = 0 - the propagation of the wetting front ( x ~can to be proportional to the square root of the time Zf(t)= qt1’2.

(2.3-9)

In the last twenty years the number of proposed solutions has increased very rapidly. Some publications are listed here aa examples: Bouwer, 1964; Liakopulos, 1966; Boreli and Vachaud 1966a; Vachaud, 1966; Kobayashi, 1966; Rubin, 1966; Whisler and Klute, 1966; Irmay, 1966; Watson, 1967; Whisler and Watson, 1969; Freeze, 1969; Whisler and Bouwer, 1970; Hornberger and Remson, 1970; Kastanek, 1971; Braester, 1973; Mein and Larson, 1973. The differences between the various methods are mostly due to the different boundary conditions considered in the derivations and to the approximations applied to achieve a numerically solvable form of the differential equations. Theoretically, they do not differ considerably from each other. For this reason these methods are not analyzed separately in detail here but reference is given only to Swartzendruber’s excellent summary (De Wiest, 1969), where the application of the methods based on the theory explained (so-called digusion theory) is put forward in connection with different practical problems (one dimensional, horizontal flow; vertical infiltration downwards; vertical steady-state flow from the ground water 19

290

2 Determination of hydraulic conductivity

having a water table of constant position; change of moisture content caused by constant drainage through the surface; influence of the flow having a decreasing flow rate with time; vertical drainage of the soil moisture to the ground water space). There are several theoretical objections in connection with Eq. (2.3-7). In recent publications, the influence of air compressed in the lower lying pores is investigated (Adrian and Franzini, 1966; Morel-Seytoux, 1973; Morel-Seytoux and Khanji, 1974; Vachaud et al., 1974; Brustkern and Morel-Seytoux, 1975). The basis of these analyses is the supposition that a part of the total available energy is consumed by the flow of air moving against the infiltrating water and only the remaining part is used in maintaining the seepage. This action can be observed in experimental columns if the lateral escape of the air is not ensured. The infiltrating water seals the surface and the pressure of the entrapped air gradually increases as the wetting front propagates downwards. Thus, the total potential between the free surface of the ponded water and the front decreases, thus retarding the seepage. A further consequence of the air compression is that after achieving a given pressure the air starts to move upwards escaping through the pores to the surface. Thus the upper part of the profile can only he partially saturated, because a proportion of the pores have to remain unsaturated to ensure the free passage of air. The theory of the investigation of simultaneous air and water flow is still being developed. It would be premature to give a detailed analysis of the Eoblem. Referring to the publications previously listed, only one example is presented here, where the development of saturation in two experimental boil columns is compared. The mantle of the first was perforated to ensure the lateral escape of air, while in the second case, the air could leave the pores only through the surface (Fig. 2.3-3 after Vachaud et al., 1974). Although some discrepancies between the theoretical results and the observed practical values can be explained by considering the influence of the air entrapped and compressed in the pores, there are further uncertainties making some hypotheses applied in the derivation of Eq. (2.3-7), questionable. The opinion, is that the actual physical processes have to be further investigated to find a more precise description than that achieved by the diffusion theory. The more accurate approximation of the dynamic interactions between the solid matrix, the flowing water and the air in the unsaturated zone, will also facilitate the answer to some questions which are still open in connection with the analysis of the simultaneous air and water movement in the pores. One of the most important hypotheses accepted for the derivation of Eq. (2.3-7) is the application of the continuum a p p r m h in the unsaturated medium. This assumption makes it possible to characterize the field by macroscopic average parameters instead of the analysis of the actual processes occurring in the real structure of the pores having micro sizes. As explained in Section 1.1.3, the basis of the continuum approach is the existence of a representative elementury unit, large enough to ensure that averaging the microscopically changing physical properties or phenomena within this unit, the macroscopic parameter has only random variation depending on the position

291

2.3 Seepage through unsaturated layers

moisture content, w / 02 0.3 04

a

t 012 a3

4

I0 20

30 40

5u

air compression

60 Fig. 2.3-3. Comparison of infiltration with and without compressed air (afterVachaud et d.,1974)

of the centre of the unit. At the same time, the representative elementary unit should be small compared to the size of the field, because the continuously changing macroscopic behaviour of the field has to be described aa a function of the parameters determined for the representative elementary units, and considering the random character of the averages calculated, a number of units lending themselves to statistical evaluation are required for this purpose within the field. It is evident, that the field can be arbitrarily extended horizontally, if a naturaI layer is investigated, and hence the size of the field in this direction cannot limit the validity of the relationships derived on the basis of the continuum approach. The vertical measurement of the unsaturated m e may, however, hinder the application of this method in many cases. As already explained, the root zone (or the cultivated zone) hars to be excluded from the investigation of the unsaturated zone, not only because the structure of the pores is very uneven, but i t also has to be considered that the biological processes strongly influence the water movement here and, therefore, the Beepage cannot be investigated as a purely physical phenomenon. The remaining part of the unsaturated medium has to be divided into further zones according to the dynamic diflerences occurring at diflerent elevations above the uater table (adhesion or capillarity is the dominating force acting against gravity or the influence of both has to be taken into account). The differences between the dynamically separated zones are not comidered in the diflusion theory and all Me parameters [ K (W ) ;y( W )and D( W)] 19*

292

2 Determination of hydraulic conductivity

are described by continuous functions in the whole unsaturated zone. Since the approximation of the relationships with single-valued functions already includes some uncertainties (e.g. those caused by the hysteresis of the retention curve), this type of function seems to be generally acceptable, although the separate determination of the relationships between the variables in the different zones is advisable, to find the best continuous approximation. There are cases, however, when the dynamic differences cannot be neglected. If the closed capillary zone reaches the surface, the hydraulic conductivity of the whole layer (where the water has a lower pressure than that of the atmosphere) is equal to that belonging to the saturated condition. In the open capillary zone, some pores are completely saturated, whereas in others only a thin water film develops. Thus, the hydraulic conductivity of the pores under the influence of different condition, have to be averaged. Finally, in the adhesive zone, the movement occurs through the water film or through the development of several capillary diaphragms at the narrow stretches of pores. From these water propagates downwards by gravity when the weight of the overlying water breaks through the diaphragm. The dynamic models of the various possible forms of water movement are summarized in Fig. 2 . 3 4 . If the continuum approach is acceptable, the flow of water through the pores of the solid matrix can be described by applying the seepage law. The nonlinear character of velocity vs. gradient relationship has to be taken into account, by using the generalized form of Darcy’s equation, which can be achieved by substituting a hydraulic conductivity depending on either velocity or gradient [see Eq. (1.1-13)]. It can be seen, therefore, that the applicability of the continuum approach is a pre-condition of the combination of the Buckingham’s potential and Darcy’s law (the combination wm used by Richards to develop the theoretical basis of the diffusion theory). In contrast to Eq. (1.1-13), the unsaturated hydraulic conductivity is regarded in most cases, as a single-valued function of saturation, neglecting its dependency on the local gradient. For this reaon, the data memured indicate considerable scattering not only in the K(y) relationship (which is generally explained by the hysteresis of the soil-moisture retention curve), but even in the cme where unsaturated conductivity as a function of the moisture content [K( W ) ]is investigated. The uncertainty caused by the influence of gradient on conductivity has also therefore, to be considered when the reliability of Eq. (2.3-7) is analyzed. The most important factor usually mentioned to prove the reliability of the diffusion theory is the agreement between the measured and theoretical values. It has to be considered that the physical parameters of soil were generally determined on the same experimental columns for which the observed and calculated data were compared. Thus, the agreement between measured and calculated data testifies to the accuracy of the whole system composing the model and the related parameters. It draws attention to the importance of the determination of the physical parameters of the soil, the detailed discussion of which will be given in the following sections. Another consequence of the uncertainties in the physical description of water movement in unsaturated media, has also to be emphasized. The models derived

2.3 Seepage through unsaturated layers

293

E-i 1 1

capillary tube with gradually changing diametw Fig. 2.3-4. Dynamic models of the possible forms of water movement through the pores of the unsaturated zone

from the diffusion theory do not give a completely accurate image of the actual process and, therefore, they can only be applied with locally determined parameters, and hence thme parameters cannot be generalized as material constants. Research has to be continued in two directions: the determination of the physical soil characteristics applied in existing models and the better understanding of the actual physical process.

2.3.2 Physical soil parameters used in diffusion theory There are three physical soil parameters, which have to be determined for the characterization of water movement in an unsaturated medium, by applying diffusiontheory, these are: soil-moisture retention curve [relationship between the suction head and the moisture content: y( W ) ] ,unsaturated hydraulic conductivity [which can be expressed in relation to either suction head or moisture content; K(y) or K( W ) ]and soil-water diflzlsivity [the product of hydraulic conductivity and the derivative of the retention curve; D( W ) see Eq. (2.3-7)]. The physical interpretation of the soil-moisture retention curve waa discussed in Chapter 1.3, where it waa proved that the volumetric moisture content ( W ) or its value related to the total porosity of the solid skeleton

294

2 Determination of hydraulic conductivity

(s rate of saturation) can be expressed as the function of the suction head ( y ) prevailing i n the sample. This may be quantified depending on the height of the point investigated above the water table ( h ) , assuming that the soilmoisture distribution belongs to the dynamic balance. One of the assumptions applied in the derivation of the equations describing this relationship, is that the osmotic forces may be neglected, because the cultivated zone is excluded from the investigation. Hence, the conditions can be assumed to be isothermal and the concentration of the dissolved salts in the soil moisture is relatively low. For this reason the two parameters can be regarded as equal to one another ( y = h). Since a double system of the forces (adhegion and capillarity) acting against gravity, has to be taken into account, the rate of saturation is defined by two conditions, one created by capillarity (9,) and the other by adhesion (so). Thus, the h a 1 formof the equation describing the average relationship between saturation and suction head is as foll o w ~[see Eqs (1.3-13); (1.3-24); (1.3-39) and (1.3-40)]: s = s,

where

+ s,(l - s,); (2.3-10)

and so=2.5

x

[In the equation, Dh is the effective diameter and a the shape coefficient of the grains, n is the porosity of the sample and h,, is the average capillary height, which can be calculated from the physical soil parameters by using Eq. (1.3-26.)] Equation (2.3-10) can be regarded as an approximation of the relationship investigated not only because the equation itself contains many approximations, but also because the interrelation between suction head and moisture content is rendered uncertain by the hysteresis of the soil-moisture retention curve .The function describing the contact between the two variables is not a single-valued one because it is also influenced by the history of wetting and drying of the profile. Analyzing the water movement in the unsaturated zone mathematically, it is not possible to consider the differences caused by the present state and the previous conditions of the solid matrix and, therefore, only average relationships can be used to characterize the y ( W ) function. The uncertainty of such approximations must not be forgotten, however, and the mathematical methods applied to solve movement equations, have to be selected so that the accuracy of the method and that of the basic data should be in bdance. As indicated by Eq. (2.3-7), the physical characters differing in the cme of water movement through an unsaturated solid matrix, from those of seepage under saturated conditions can be expressed by the special behaviour of two parameters, depending on the moisture content. These are: the hydraulic gradient (of two t e r m , one characterizing the potential energy and the other depending on the suction digerences) and the hydraulic conductiv-

2.3 Seepage through unsaturated layers

295

i t y being also a function of the water content. The determination of the total hydraulic gradient is bmed on the suction head vs. moisture content relationship already discussed. The next physical process to be investigated is, therefore, the development of hydraulic conductivity in an unsaturated porous medium. According to experimental measurements, the resistance of an unsaturated solid matrix is greater than that of the same medium in saturated conditions. A simple explanation of this observation can be given when i t is considered that hydraulic conductivity is equal to the flux related to the gradient maintaining the movement. The actual cross section through which the water movement develops within a unit area is measured by the areal porosity of the sample, if the latter is saturated. Under unsaturated conditions, only part of the pores are filled with water and only these pores convey water. The actual cross section transporting the water decreaes, therefore, as saturation decreases. This simple geometrical rewon testifies, that both the flux and the hydraulic conductivity have to be smaller through the unsaturated porous medium than that in saturated conditions even if the dynamic differences are not conbidered. It was aleo proved by experimental measurements, that the two valuas of hydraulic conductivity [i.e. those belonging to the saturated ( K )and unsaturated ( K , ) conditions] are proportional to one another. The coefficient of proportionality can be given w the function of either saturation (depending 011 the rate of saturation) or that of the suction head (y)prevailing at the

(2.3-11)

Naturally, t,he same proportionality is also valid for the ratio of intrinsic permeability values, because the fluid is the same in the two systems. Since the resistance of the solid matrix is supposed to depend mostly on the rate at which those pores are sled with water, the general opinion is that the K,( W )function is a more realistic approximation than the use of the K , ( y ) relationship. In the second case the y( W) function is used to create contact between hydraulic conductivity and suction head and since this equation is multi-valued (because of the hysteresis of the soil-moisture retention curve) i t is not possible to determine a unique relationship between the two variables in question. It is necessary, however, to mention that the K,( W ) equation is not a single-valued function either, because there are special dynamic processes acting (change of the capillary contact angle, influence of the actual gradient on adhesion) which are not considered in the relationship. The form most generally applied to describe the K , vs. W relationship is:

K,=K

s-so m* -

11-80)

’

(2.3-12)

where s is the rate of saturation and so is the parameter characterizing the

296

2 Determination of hydraulic conductivity

layer when i t contains only strongly adsorbed water. The m power is determined from the experimentally measured data and hence i t is different in the various publications [m = 3.0 (Irmay, 1954)m = 3.5 (Averjanov, 1949s; 1949b; Boreli and Vachaud 1966b) m = 4 (Johnson and Kunkel, 1963). The relationship described by Eq. (2.3-12) is represented in Fig. 2.3-5. Other research workers have published their observations in the form of the actually measured data (Rose, 1966; Vachaud, 1966; Elrick, 1966; Peck, 1966; Elrick and Bowman, 1964). Some of these measurements are summarized in Fig. 2.34%. The observations indicate the uncertainty of the data. As it was already mentioned, one of the processes which may influence the scattering of conductivity is the change in the capillary contact angle. Wladitchensky’s (1966)experiments prove the reliability of this hypothesis. Figure 2.3-6 shows the change in the cosine of the contact angle resulting from various kinds of treatment of a given sand. The relationship between this value and

Fig. 2.3-5. Relationship between hydraulic conductivity and water content

1.

Fig. 2.3-8. Incomplete saturation of a wetted soil column

2.

This Page Intentionally Left Blank

2.3 Seepage through unsaturated layers

297

a parameter being proportional to hydraulic conductivity and called the water-uptake index ( K * ) is represented in the figure. The same publication also contains data proving the change in the wetting angle depending on the moisture content of sand samples (Table 2.3-1). The values observed testify

K* f40 7

120 100

80 60 40

20 0

1

0.8 cos 8 Fig. 2.3-6. Relationship between the water-uptake index and the contact angle of the capillary zone. Table 2.3-1. Relationship between the moisture content and wetting angle in two sand samples Fraction size

[a1

0.06-0.026

Moisture content

[%I

0.036 0.049 0.068 0.09 8.27 11.97

0.66 0.69 0.20 0.36 0.44 1.00

0.03

0.62 0.40 0.17 0.36 0.69 1.00

0.08

0.02&0.01

0.12 2.49 6.19 8.32

that in unsaturated media, the rate of saturation influences the dynamic conditions as well, and this change effects the conductivity of the sample. Another investigation (Vachaud et al., 1974) has drown attention to the errors which may occur in connection with the determination of the basic parameter (conductivity under saturated condition) to which all the measured capillary conductivity values are related. It was found that complete

298

2 Determination of hydraulic conductivity

saturation could hardly be achieved in the unsaturated soil column invmtigated, because of the entrapped air bubbles. The highest rate of saturation after rapid wetting of the column was about s, = 0.8 and, therefore, only the lowest part of the K,/K vs. W curve sould be determined (Fig. 2.3-7). The same result is shown in Fig. 2.3-8, which includes two photos. I n the photos

6 4 2

a m ozto

0

a315 a4zW WO

o

0.2 013 i4 moisture confenf, W

0.1

Fig. 2.3-7. Special measurements showing the relationship between hydraulic conductivity end water content

the saturated zones are white while the grey spots indicate the air remaining in the sample, the darkness being proportional to the air content. When the first picture was taken, the water table waa at the bottom of the column and a closed capillary zone of about 8 cm was observed. The water table waa raised at a velocity of 1 m/hour up to 21 cm. It is clearly indicated by the second picture, that the soil remained unsaturated between 10 and 23 cm (Koening’s verbal information, Laboratorium voor Groundmechanica, Te Delft). The expression of hydraulic conductivity as a function of suction head is more uncertain, than the characterization of the K J W ) function. Two examples are shown in Fig. 2.3-9. The data of the first were determined by Wind (1966) on samples including two different peats, one clay and one sand (Fig. 2.3-9a). Wesseling and Wit (1966) have used disturbed and undisturbed samples of granular sediments (Fig. 2.3-913). The second publication gives also a complete and clearly arranged survey of the various formulae proposed for the characterization of the K,(y) function:

K , = ay-b; (Weaseling, 1967); K , = a(yb + c)-l; (Gardner, 1958); R, = K exp(-ay); K , = K exp[-a(y - yo)]; (Rijtema, 1965).

(2.3-13)

Q

9

8

.

2.3 Seepage Seepage through through unsaturated unsaturated layers layers 2.3

s. r

ruimat

299

Fig. 2.3-9. Relationship between hydraulic conductivity and suction head

300

2 Determination of hydraulic conductivity

Among the parameters applied in the diffusion theory, the soil-water diffusiwity is the least reliable. It has no physical interpretation, its definition being only a formal mathematical operation giving the parameter the product of hydraulic conductivity and the negative tangent of the soilmoisture retention curve [see Eq. (2.3-7)]. Even its name “diffusivity” does not refer to the physical process of diffusion. The parameter has only a similar place and character in Eq. (2.3-7) as the real coeficient of diffusivity in . ? l

Q ID

;*

5

2 2

9-10 2

.s 5 32

$$,a L

P

5

32 B

.& moisture content, W

moisture content, W

(c) 0

0.20

0.30 0.40 moisture content, W O./O

(a) Fig. 2.3-10. Relationship between soil water diffusivity and water content

the general transport equations. For this reason, the parameter was named the soil-water diffusivity. The uncertainty of this parameter is also shown by the publications generally giving the data measured in a direct graphical or tabulated form without trying to approximate them with some mathematical formulae. Even the trends of the D vs. W curves fitted to the data points are different according to the results of the various experiments. I n some cases, diffusivity was increasingly monotonous with increasing soil moisture (Fig. 2.3-10a) (Boreli and Vachaud 1966a), while other observations showed inflection in the middle of the range of soil moisture investigated (Fig. 2.3-lob) (Black et al., 1969) or the experiments indicated a local minimum on the curve (Fig. 2.3-1Oc) (Rose, 1966). Considering the fact that both the hydraulic conductivity and the soilmoisture retention curve are multi-valued functions of the water content and even the character of these relationships is different in the adhesive and capillary zones, it is quite obvious that soil water diffusivity being composed of the other two parameters, cannot be determined a,? a unique func-

2.3 Seepage through unsaturated layers

301

tion of the moisture content. Afuller understanding of the roleand behaviour of this special term has to be based, therefore, on the theoretical investigation of the y vs. W and K vs. W relationships. The detailed explanation of the first contact has already been given in Chapter 1.3, while the physical analysis of the interrelation between hydraulic conductivity and water content is the topic of the next section.

2.3.3 Theoretical analysis of hydraulic condictivity in unsaturated porous media The soil-moisture zone is not a homogeneous field as the dynamic character of the main forces in action differ in the upper and the lower part of the zone and hence, both the rate and type of saturation also di8er near the soil surface and i n the vicinity of the water table. As already explained in Chapter 1.3, within a given distance from the phreatic surface (or if a separated sample is investigated, below a given limit of suction prevailing in the sample), the main force acting against gravity is capillarity. Here a part of the pores are completely filled with water (i.e.thosepores, which are smaller than the diameter of the capillary tube corresponding to the height or suction considered as a capillary rise). I n the remaining larger pores (and at a higher position above the water table in all the pores), a water film develops on the walls of grains due to adhesion. Below the probable minimum capillary height, all pores are completely saturated by capillarity (closed capillary zone) and, therefore, the hydraulic conductivity is equal to that of a saturated sample under pressure. As an exception, the rate of saturation may be lower than unity if air bubbles are entrapped in the pores due to the relatively rapid wetting of the sample. I n the theoretical investigation, this influence can be neglected. Moving upwards in the open cupillury zone (between the probable maximum and minimum capillary rise), the ratio of the completely saturated pores to the total areas of pores gradually decreases. Here, water transport develops through the entire section of the smaller pores which are saturated, while in the larger pores water movement can develop only through the water films. The hydraulic conductivity has to be determined, therefore, aa the resultant of the influences of this double system. Finally, above the probable maximum capillary height (zone of adhesion), practically all the pores contain air, amd the water can flow only through the continuous water films surrounding the grains. Thus, hydraulic conductivity is determined here by the water transporting cupaciiy of the films. Considering the various parts of the soil-moisture zone, two different dynamic models have to be developed for the investigation of the hydraulic conductivity of an unsaturated porous medium. The previously derived geometrical model, which substitutes the actual network of channels formed by the pores with straight capillary tubes, is accepted as the basic concept for the determination of further relationships. The difference between the two dynamic models is that the first assumes an annular cross

302

2 Determination of hydraulic conductivity

section attached to the wall and suitable for water transport within each tube, while the second model asumes that some pores are saturated and the others are completely dry. Hence, the water transporting capacity is decreased a a result of the smaller area of the active pores (Fig. 2.3-11). The first model directly characterizes the conditions in the adhesive zone. If the

Y!P/

romposedofmp;llary

.ibes wifh ajerage dlameier

F n r characerizing adhesive

rwQucf/vify

nodel coiiiposed of rap .3ry r:tes wit0 aifferent dtame?e:s CF w- “?I by fhe pore-size aistr.our 01: cu.”/e‘7Cnaracfermngz,?it;’dry cunCoc:,V I 7;

combination of fbe fwo models for characterizingunsafurated CoflductiYiifY

Fig. 2.3-11. Physical models for characterizing unsatureted flow

open capillary zone were under the sole influence of capillarity, the second model would prove suitable to simulate water movement. In the )ores not filled by capillarity, however, adhesion forms water films, and, herefore, this section of pores may convey water a well, aphenomenon which can be described by uxing the first model in the caae of large pores. The hydraulic conductivity of the open cupillary zone can be d e t e r m i d , therefore, by combining the two models in a suitable form. Finally, the characterization of the closed, capillary zone does not need any special investigation, because, theoretically, the pores are completely saturated here, and the hydraulic conductivity i s not injluenced by the fact that the water in the pores is under positive or negative pressure.

d

2.3 Seepage through unsaturated layers

303

Before starting the theoretical analysis of the models, tjome problems have to be clarified in connection with the terminology used further on. The topic investigated here is the difference between the hydraulic conductivity values of a given sample belonging to its saturated and unsaturated conditions, respectively. In the literature, these parameters are generally referred to as“ saturated and unsaturated Conductivity”. It is quite evident that these terms are incorrect from a linguistic aspect, because conductivity cannot be either saturated or unsaturated. To avoid the adjective form, some authors proposed the use of the term capillary conductivity (Buckingham 1907; Richards, 1931). This proposal cannot be accepted in our investigations. As explained in the previous paragraphs two different dynamic models will be used for the characterization of the hydraulic conductivity in the soil-moisture zone: i.e. one considering only adhesion ae a force acting against gravity, and the other regarding only capillarity and gravity as the dominant forces. For this reason, the results of the two models will be culled adhesive and capillary conductivity respectively. To shorten the description of the correct form (i.e. hydraulic conductivity of an unsaturated porous medium or hydraulic conductivity of a medium in an unsaturated condition) the term “unsaturated conductivity” will be accepted although linguistically it is not absolutely correct. The opposite cme, the term ‘Lsaturatedconductivity” will also be used. The terms accepted and used in the following part of this section are also summarized in Fig. 2.3-12, indicating the various parts of surface

Fig. 2.3-12. Characterization of hydraulic conductivity in the different zones above the water table

304

2 Determination of hydraulic conductivity

the soil-moisture zone, the characterization of which can be given by applying the different parameters. As also shown in the figure, unsaturated conductivity is equal to adhesive conductivity in the zone of adhesion. I n the open capillary zone adhesive and capillary conductivity has to be combined to obtain unsaturated conductivity. Finally, in the closed capillary zone and below the water table, the water transporting capacity of the solid matrix ie described by saturated conductivity. The model constructed for the theoretical characterization of adhesive conductivity is a tube in which the water coats the wall in the form of a thin film, and hence, the water conveying cross section has an annular form. When investigating the dynamic conditions in this geometrically fixed system, two different methods were followed (KovBcs, 1971a). At f i s t , it was assumed that adhesion could be neglected as a retarding force. This hypothesis characterizes the flow which corresponds to Darcy’s condition in a saturated state. In Poiseuille’s equation, this condition can be considered by assuming the velocity to be equal to zero at the wall of the pipe (v = 0, where r = ro). I n the case of the second approximation, adhesion is considered in a similar way to microseepage or more precisely to the extended form of laminar flow. The modified boundary condition is v = 0 where r = rl. (The symbols used are shown in Fig. 2.3-13).

’1 4 Fig. 2.3-1 3. Symbols used for deriving unsaturated hydraulic conductivity

2.3 Seepage through unsaturated layers

305

The basic equation in the first cme is (2.3-14)

From Eq. (2.3-14) a formula for velocity distribution can be determined by integration and taking the previously given boundary condition (v = 0 ; r = r o ) into consideration: (2.3-1 5)

The flow discharge in the pipe is: (2.3-16)

because the space filled with air does not take part in water conveyance. When calculating mean velocity, however, this part of the cross section must also be taken into account because the final aim is not the calculation of effective mean velocity, but that of Darcy’a seepage velocity which is related to the total cross-sectional area and, therefore, the mean velocity determined for the whole area of the pipe combined with the number of pipes, gives the parameter sought. Thus, seepage velocity can be determined from the following formula:

The volume of water in the pipe is proportional to the annular area of z(ri-rz), and the total volume of pores to the areaof the cross section. Thus, the coefficient of adhesive saturation can be calculated aa a function of the two geometric data: (2.3-1 8)

Substituting this relationship into Eq. (2.3-17) the following formula, is achieved for the ratio between adhesive conductivity and Darcy’s hydraulic conductivity ( K , ) :

When repeating the derivation and considering the retarding effect of adhesion m well, the basic equation is

20

306

2 Determination of hydraulic conductivity

It is now necessary to apply some further approximations. As shown in connection with microseepage, both local and mean velocity are made up of two members, one of which originates from the dw/dr component of the basic equation and the other from t&. The latter tends to zero when the hydraulic gradient increases, and as proved, the second member could be 121,. Accepting this condition in the caae of the neglected when I present investigation as well, the expression of local velocity differs from Eq. (2.3-15) only in the boundary condition (w = 0; r = rl):

>

w ( r ) = -IY (r!47

rz)--riln-. IY 27

rl

(2.3-2 1)

T

Consequently, when determining the flow rate of a pipe, the limits of integration also have to be modified: rx

Q = 2nJ vrdr;

(2.3-22)

r.

and the mean velocity is:

( 2.3-23)

As in Eq. (2.3-la), the minimum saturation can also be expressed as a function of the characteristic radii:

) :1

so = 1 -

2

.

(2.3-24)

Substituting this relationship and considering also the saturated conductivity influenced by adhesion and determined for the zone of 1 121, (laminar seepage) by combining Eqs (2.2-51) and (2.2-53):

>

(2.3-25)

the ratio of the two values of hydraulic conductivity (i.e. that characterizing the unsaturated condition in the zone of adhesion and that belonging to the relevant dynamical zone of saturated seepage) can be calculated:

KO = KMl

%--so [l-sO)

2

-2

(l-sO)(sO-so)

(1-s,)2

2

1'

- 'a))'],, 1 -so

('-'a)

.

(2.3-26)

(1-so)

The first member of this expression is of the same form as Irmay-Averjanov's formula. Even its power can be raised to three (as in Irmay's equation) by developing the logarithmic members into series. The influence of further members necessitates the use of the higher power (Averjanov m = = 3.5). The numerical comparison of the result of Eq. (2.3-26) with the empirical formulae is very good (Fig. 2.3-5). It can be seen that even a higher power ( m = 4.0), would still be acceptable.

2.3 Seepage through unsaturated layers

307

Summarizing the above, the empirical formulae for calculating the hydraulic conductivity in the unsaturated zone and its relative value related to Darcy's hydraulic conductivity, can be accepted as accurate approximations in the following form :

where values, which most closely correspond to the theoretical and measured data, can be obtained by substituting m = 3.5; a more simple approxima4) is, however, also acceptable. tion ( m = 3 Equations (2.3-26) and (2.3-27) are only valid theoretically in the range of I > 1210, since some approximations had to be applied for the derivation, which are acceptable only in this zone. Because of the relatively small difference and also taking into consideration some other factors, whose influence cannot be considered, the use of these relationships can in practice, be extended to the whole range of velocity. This extension means that the line with a slope of tan a oc K,, representing the velocity vs. gradient relationship is accepted without any lower limit. This line intersects the horizontal axis at a point I = 21,. Thus, the approximation assumes that there is no movement in the system when the gradient is smaller than the double value of the theoretical threshold gradient. On the basis of this hypothesis and taking into consideration Eqs (2.2-51) and (2.3-24) as well, a relationship can be determined, given the rate of saturation of a sample having only strongly adsorbed water as a function of the actual and threshold gradient:

-

I---

>

-

- =I-

(;J2

so; s o = - I0

I

.

(2.3-28)

This equation shows that the minimum saturation is not an unambiguous parameter of the sample, but it depends on the acting hydraulic gradient. The second model applied for the characterization of the hydraulic conductivity of an unsaturated porous medium has to represent the capillary zone, mpecially its open part, because the closed capillary zone is supposed to be completely saturated (except those pores in which trapped air bubbles remain). Thus, here the hydraulic conductivity is equal to that describing the saturated condition. In the zone of adhesion, the possibility of some pores being fully saturated was neglected and the model was based on the aasumption of annular-shaped water conveying cross sections in each straight tube, substituting the actual network of the channels between the grains. Investigating the capillary zone, the opposite hypothesis will be made: i.e. only the water transport through pores filled by mpillarity will be determined, neglecting the possibility of flow which may also develop through the water films bound by adhesion to the walls surrounding large pores. 20*

308

2 Determination of hydraulic conductivity

The series of straight capillary tubes w a accepted as the geometrical model of the water transporting system composed of pores (see Section 1.2.5). It was also proved by the investigation of the pore-size distribution of loose clastic sediments, that the frequency distribution of the number of the pores within a given range of the cross-sectional area (5f)can be approximated with an exponential function [see Eq. (1.3-31)]. At a height of h, above the water table [or if the sample is under a suction of p , expressed in an equivalent water column (y) which is equal to this height], the rate of capillary saturation (8,) can be related to a limiting area of the pores (f,) which divides the total number of pores into two parts [see Eqs (1.3-34) and (1.3-35)]. The smaller pores ( f f,) are completely saturated, the f,) are filled with air, if the presence of the adhesive water larger ones ( f films is neglected. Since the water transporting capacity of the sample cannot be characterized by the hydraulic conductivity of the average tube, as in the caae of a saturated medium, the original geometrical model has to be modified. The model assumes that within a unit area normal to the flow direction, the system is composed of as many straight capillary tubes as the total number of pores, and the distribution of the number of the tubes according to their areu i s equal to the pore-size distribution. The limiting area off,, therefore, also divides the number of the tubes into two groups, the smaller tubes taking part in water transport, and the larger ones being inactive. The flow rate (Pi)through a capillary tube is proportional to its cross The coefficient of proportionality section (fi) and the hydraulic gradient (I). is the hydralic conductivity of the tube ( K i ) ,which is the function of the second power of the diameter, and is linearly interrelated, therefore, to the f r areas of the cross section:

<

>

(2.3-29 )

If the f, limiting area is equal to the upper limit of the k-th Af interval, the total flux through a capillarly saturated sample (q,) can be determined by multiplying qr by the number of tubes in the i-th df interval and summarizing the products calculated for the first k intervals. Introducing the dimensionless specific area of the pore, whichisits size related to the average value (x = f/fo) and using those symbols applied for the derivation of Eq. (1.3-36), the necessary mathematical operation can also be expressed in the form of a definite integral:

or q,=*JPv(f; 5f 0

A f ) d f = c2rs'fxaE(x; dx 0

dx)dZ.

(2.3-30)

309

2.3 Seepage through unsatumted layers

The same integral equation is also valid for saturated media, but in this case the water transporting capacity of each tube has to be taken into account, and therefore, the upper limit of the integral is infinity. Thus, the jlus through a unit a r m of the saturated sample (q,) is aa follows:

0

0

The flux is equal to the seepage velocity. Its value divided by the gradient gives, therefore, the hydraulic conductivity. Consequently the ratio of capillary conductivity (K,) and saturated conductivity ( K , ) , can be calculated as the quotient of Eqs (2.3-30) and (2.3-31). Introducing the simplified form of the C(s;A s ) distribution function, according to Eq. (1.3-31) the ratio is as follows: x,

Js2exp (- s )dz

x

--=3-! K, -

c-

OD

Ks

"

Jzzexp(-s)dz' (2.3-32)

0

x-1-

,-

(1 -s2+s+1 12

Repeating the equation for the rate of capillary saturation (9,) rn a function of the dimensionless specific area of the pore ( s ) ,[see Eg. (1.3-37)]: sc = 1 - (z

+ 1)exp

(-2)

;

(2.3-33)

the relative capillary wnductivity ( x , ) can be related to capillary saturation using a parameter (5) which may have any value between zero (completely dry medium) and positive infinity (total saturation) (0 z 00). The

< <

0

D.?

04

Fig. 2.3-14 Comparison of xc vs.

06 Sa OR Sc l 0 8c and

xu VS.

8,

functions

810

2 Determination of hydraulic conductivity

correspondingvalues of x , 8, and xc are listed in Table 2.3-2. The relationship between the rate of capillary saturation and capillary conductivity is compared to the curve showing the 8, vs. xu relationship in Fig. 2.3-14. Assuming that the same rate of saturation is only created by either adhesion or capillarity, the two values of relative conductivity do not differ consider0.6, while below this limit, the absolute value of x is ably in the zone s

>

Table 2.3-2. Corresponding values of

2,

ac and x, variables Relative

Upper h i t of thesatarated pore-size related to the avexuge a m of pores 2

Chpillary

- folf

0 0.10 0.20 0.30 0.60 0.70 1.o 1.3 1.6 2.0 2.6 3.0 4.0 6.0 7.0 10.0

0 0.0047 0.0176 0.037 0.090 0.166 0.264 0.373 0.476 0.694 0.713 0.801 0.908 0.960 0.993

0 0.00016 0.00116 0.00360 0.0144 0.0341 0.0803 0.143 0.217 0.323 0.466 0.677 0.762 0.876 0.970 0.997 1.0

0.999 1.0

m

so small that the difference (although relatively high) does not surpass the possible error caused by the uncertainty in the determination of hydraulic conductivity. The real problem is not the comparison of the two parameters, but the superposition of adhesive and capillary conductivities by considering the ratio between the rates of adhesive and capillary saturation respectively since the two forces (i.e. adhesion and capillarity) act together in the open capillary zone. To solve this problem in closed mathematical form, both saturation and hydraulic conductivity huve to be expressed as the function of the suction head, although it i s known that these relationships are not singlevalued functions because of the hysteresis of the soil-moistureretention curve. Accepting the rough approximation from the average p vs. W relationship, the rate of saturation depending on the prevailing suction head can be given in the following form [see also Eq. (2.3-lo)]: 8

where a,=

1

-[($I2+

= 8,

+ s u ( l- sc) ;

l]exp[-

;I):(

[see Eq. (1.3-39)]; (2.3-34)

311

2.3 Seepage through unsaturated layers

and

[see Eq. (1.3-24)] ; therefore

To express the unsaturated conductivity aa a function of suction head, it must be understood that the flux through a unit area (4,) is composed of two parts. These are cupillary flux (q,) through the completely filled pores, which is equal to the product of capillary conductivity and gradient; and adhesive flux ( p a ) ) developing through the films of water in the remaining part of the pores [n(1-s,)]. Therefore, it can be calculated aa a product of adhesive conductivity and gradient aa well as taking the areal reduction [i.e.(l-s,)] into account: qu = q c pa = [Kc (1 - sc) Ka1 I ; and (2.3-35) K , == K, (1 - sC)K,.

+

+

+

To simplify the h a 1 result, the influence of the rate of initial saturation (so) on adhesive conductivity is neglected and the empirical form proposed

by Irmny (1954) [see Eq. 2.3-12)] is used instead of the theoretically derived relationship [see Eq. 2.3-26)]. Thus, the final form of the relative unsaturated conductivity can be given 88 the function of the suction head:

1 - 2.75 x

[ [$I2+

]'I2

2

h%2]

13)

.

(2.3-36)

Some conclusions drawn from Eqs (2.3-34) and (2.3-36) are as. follows: (a) Both the rate of saturation and the relative unsaturated conductivity can be expressed &s a function of the relative suction. This dimensionless parameter is the average capillary height ( hco) related to the prevailing suction (y). (b) Neither s nor x is a single-valued function of the h,,,/y quotient, but in both relationships the absolute value of a material constant has to be introduced. According to the theoretical investigation a combination of the average capillary height and porosity gives a parameter suitable for the characterization

rin)

of the material of the porous medium - h%'. (c) Using this material constant aa a parameter the p vs. W or s relationship (soil-moisture retention curve; Pig. 2.3-15a) and the contact between

312

2 Determination of hydraulic conductivity

3.0-

s

1

2.5-

2.0 -

i.5

1.0 .

05.

0-

..

I .

V l nco

Fig. 2.3-15 Theoretically derived relationships between the v/hcoand x and 8 parameters

8:

x and v/hm:

unsaturated conductivity and suction head (Fig. 2.3-1 5b) can be represented in coordinate systems having dimensionless quantities memured along their axes. Using the corresponding values of s and x (Table 2.3-3) even the relationship between these two parameters can be graphically reconstructed depending on the absolute value of the average capillary height and porosity (Fig. 2.3-15c).

Table 2.3-3. Corresponding values of the rste of saturation, relative suction head and relative conductivity '

aw81w

psaq no!pne

I p

X

*/Y 3.8

01

2.0 2.2 2.4 2.6 2.8

I

1.8

OOOl

1.2 1.4 1.6

I

1.o

0.0156 0.0239 0.0392 0.0638 0.0991 0.147 0.206 0.264 0.430 0.689 0.729 0.837 0.910 0.966 0.979 0.991 0.997 0.999

oO0001

u

0.8 0.9

2p a w p m aqi ~ q z p q m e q : , aqamw8d e

0.7

*I

0.3 0.4 0.5 0.6

3.00~ 10-4 1.10x 1 0 - a 2.70 x 5.80 x 10-3 1.16 x 10-' 2.10 x 10-8 4.90 x 10-2 8.00x 10-2 0.176 0.312 0.472 0.628 0.761 0.861 0.926 0.967 0.984

0.994

0.0287 0.0376 0.0632 0.0774 0.113 0.160 0.219 0.286 0.444 0.696 0.734 0.840 0,910 0.965 0.979 0.991 0.997 0,999

3.14 x 1.12x 10-8 2.72~ 8.82x lo-* 1.36~ 2.70 x lo-* 4.90 x lo-* 8.00 x 10-2 0.176 0.312 0.472 0.628 0.761 0.861 0.926 0.967 0.984 0.994

0.0572 4.49 x lo-' 0.0670 1.27 x 10-8 0.0834 2.89 x 1 0 - 3 0.108 6.00 x 0.145 1.57 x lo-= 0.189 2.70x 0.247 4.90 x 10-2 0.312 8.00x 10' 0.176 0.461 0.312 0.612 0.472 0.744 0.846 0.628 0.761 0.916 0.861 0.967 0.926 0.979 0.967 0.991 0.984 0.997

0.505 0.646 0.767 0.860 0.923 0.961 0.982 0.991 0.997

0.999

0.999

0.994

0.119 0.131 0.149 0.173 0.208 0.262 0.306

0.367

1.8Ox 10-8 2.82~ 4.59~ 7.82~lo-* 1.66~lo-' 2.9 1 x lo-* 6.11 x 8.20 x lo-' 0.178 0.313 0.473 0.629 0.762 0.861 0.926 0.967 0.984 0.994

0.261

M 3 X 10-2

1.83~ lo-' 0.889 416x i0-' 0.318 2 . 6 0 ~lo-' 0.346 3.43 x 10-z

0.268

0.388 4.83 x lo-* 0.434 1.00x 10-2 0.486

0.601 0.716 0.818 0.889

0.939 0.970 0.986 0.994 0.997 0.9Q9

0.100 0.193 0.326 0.482

0.634 0.766 0.863

0.927 0.967 0.984 0.994

0.536 0.662 0.691 0.819 0.647 0.677 0.711

0.160 0.173 0.192 0.208

0.742 0.808 0.869 0.919 0.963 0.976 0.988 0.996 0.998 0.099 0.999

0.282 0.360 0.448 0.668 0.689 0.797 0.880 0.936 0.970 0.986 0.994

2 f

0.224 0.240 0.269

F J

5 5ip"3 n,

2

a

314

2 Determination of hydraulic conductivity

References to Chapter 2.3 ADRIAN, D. D. and FRANZINI, J. B. (1966): Im edence to Infiltration by Pressure Build up ahead of the Wetting Front. Journal of 8eophysical Research. AVERJANOV, S. A. (1949a): Relationship of Permeability of Soil with Air Content (in Russian). DAN., No. 2. AVERJANOV,S. A. (194913): Approximative Evaluation of the Role of Seepage in the Capillary Fringe (in Russian). DAN., No. 3. BLACK,T. A., GARDNER, W. R. and THURTESS, G. W. (1969): The Prediction of Evaporation, Drainage and Soil Water Storage for a Bare Soil. Proceedings of American Soil Science Society, No. 6. BORELI,M. and VACHAUD, G. (1966a): Certain Problems of Infiltration in Unsaturated Porous Media. Seepstenja Inatituta za Vodoprivredu Jaroslav Cerni, No. 39. BORELI,M. and VACHAUD, G. (1966b): On the Determination of the Residual Water-content and on the Variation of Relative permeability of the Unsaturated Soil (in French). C. R. Academy of Sciences, Par& Series A, Tom. 263. BOUWER, H. (1964): Unsaturated Flow in Ground-water Hydraulics. Proceedings of ASCE, HY.5 . BRAESTER, C. (1973): Moisture Variation a t the Soil Surface and the Advance of the Wetting Front during Infiltration at Constant Flux. Water Resources Research, No. 3. BRUSTKERN, R. L. and MOREL-SEYTOUX, H. J. (1976): Deaoription of Water and Air Movement during Infiltration. Journal of Hydrology. No. 1. BUOKINQEAM, E. (1907): Studies in the Movement of Soil Moisture. USDA Bureau of Soils Bulletin, No. 38. CHILDS,E. C. and COLLIS-GEORQE, N. (1960): Movement of Moisture in Unsaturated Soils. 4th ISSS Congress, Amsterdam, 1960. DE WIEST,J. M. (1969): Flow through Porous Media. Academic Press, New York, London. ELRICK, D. E. (1966): The MicrohydrologicCharacterization of Soils. I A S H Symposium on Water in the Unaaturated Zone, Wageningen, 1966. ELRICK, D. E. and BOWIUN,D. H. (1964): Note on an Improved Apparatus for Soil-moisture Flow Measurements. Soil Science Society Proceeding, No. 3 . FREEZE, R. A. (1969): The Mechanism of Natural Ground-water Recharge and Discharge. Water Resources Research, No. 1. GARDNER, W. R. (1968): Some Steady State Solution of the Unsaturated Moisture Flow Equation with Application to Evaporation from a Water Table. Soil Science, No. 4. GREEN,W. H. and A m , C. A. (1911): Studies on Soil Physics, 1. Flow of Air and Water through Soils. Journal of Agricultural Sciences, No. 4. HORNBERQER, G. M. and REMSON, J. (1970): A Moving Boundary Model of a Onedimensional Saturated-Unsaturated Transient Porous Flow System. Water Resources Research, No. 3. HORTON, R. E. (1939): Analysis of Runoff-plot Experiment,s with Varying Infiltration Capacity, Tranaactwna of ABU, No. 20. IRMAY, S. (1964): On the Hydraulic Conductivity of Unsaturated Soils. Tramactions of AGU, No. 1. IRMAY, S. (1966): Solution of the Non-linear Diffusion Equation with a Gravity Term in Hydrology. IASH Symposium on Water in the Unaaturated Zone. Wageningen, 1966. JOHNSON 4.I. and KUNEEL, F. (1963): Some Research Related to Ground-water Recharge (Progress re rt from the USGS). Conference on Ground-water Recharge and Ground-water Basin &&zgement, University of California, Berkeley, 1963. KASTANEK, F. (1971): Numerical Simulation Technique for Vertical Drainage from a Soil Column. Journul of Hydrology. %UTE, A. (1962): Some Theoretical Aspects of the Flow of Water in Unsaturated Soils. Proceedings of American Soil Science Society, Vol. 16, No. 2. KOBAYASHI, H. (1966): A Theoretical Analysis and Numerical Solution of Unsaturated Flow in Soil.I A S H Sympo&um on Water in the Unaatura&d Zone,Wageningen, 1966.

References

315

Kovkcs, G. (1971a): Seepage through Unsaturated Porous Media. 14th Congress of I A H R , Paria, 1971. KovAcs, G. (1971b): Seepage through Saturated and Unsaturated Layers. Bulletin of I A H S , 2. . LIAKOPULOS, A. G. (1966): Theoretical Approach to the Solution of the Infiltration Problem. I A S H Bulletin, No. 1. MEIN. R. G. and LARSON.C. L. 119731: Modelinn " InGltration durinn " a Steadv Rain. Water Resources Research, 2. MOREL-SEYTOWX. H. H. 11973): On the Modified Theorv of Infiltration f 1st and 2nd parts) (in French). 'Cahiers'of ORSTOM, No. 1-2. MOREL-SEYTOUX, H. J. and KHANJI, J. (1974): Derivation of an Equation of Infiltration. Water Resources Research, No. 4. PECK,A. J. (1966): Diffusivity Determination by a New Outflow Method. I A S H Symposium on Water in the Unsaturated Zone, Wageningen, 1966. PHILIP, J. R. (1966): Numerical Solution of Equations of the Diffusion Type with Diffusivity Concentration-dependent. Tramaction of Faraday Society, No. 61. PETLIP,J. R . (1967): Numerical Solution of Equations of the Diffusion Type with Diffusivity Concentration-de endent 11. Auatralian Journal of Physics, No. 10. PHILIP, J. R. (1966): A Einearization Technique for the Study of Infiltration. I A S H Symposium on Water in the Unsaturated Zone, Wageningen, 1966. RICEURDS,L. A. (1931): Capillary Conduction of Liquida through Porous Media. Physics. Nov. RIJTEMA,P. E. (1966): An Analysisof Actual Evapotranspiration. (Doctoral Thesis). Wageningen, VLO. 669. ROSE, D. A. (1966): Water Transport in Soils by Evaporation and Infiltration. I A S H Symposium on Water in the Unsaturated Zone, Wageningen, 1966. RUBIN, J. (1966): Numerical Analysis of Ponded Rainfall Mltration. I A S H Symposium on Water in the Unsaturated Zone, Wageningen, 1966. VACHAUD, G. (1966): Study on Redistribution after Inhibiting Horizontal InGltration (in French). I A S H Symposium on Water in the Unsaturated Zone, Wageningen, 1966. VACHAUD, G. (1968): Verification of the Generalization of Darcy's Law and the Determination of Capillary Conductivity by Analysing Horizontal Infiltration (in French). I A S H Symposium on Water in the Unsaturated Zone, Wageningen, 1966. VACHAUD, G., GAUDET,J. P. and KURAZ, V. (1974): Air and Water Flow during Ponded Infiltration in a Vertical Bounded Column of Soil. Journal of Hydrology. No. X

I

1-2.

WATSON.K. K. 11967): ExDerimental and Numerical Studv of Column Drainage. Proceedings of ASCE, 2: WESSELING,J. (1967): Some Aspects of Water Movement in Soils (in Dutch). Verst. Landb. Onderz., 6. WESSELINQ. J. and WIT, K. E. 11966): An InGltrakion Method for the Determination of the Capillary Conductivit of Undisturbed Soil Cores. I A S H Symposium on Water in the Unsaturated Zone, d q e n i n g e n , 1966. WHISLER, F. D. and KLUTE, A. (1966): Anal sis of Infiltration into Stratified Soil Columns. I A S H Symposium on Water in the gmaturated Zone, Wageningen, 1966. WHISLER, F. D. and WATSON, K. K. (1969): Analysis of Infiltration into Draining Porous Media. Proceedings of A S C E , I R . 4. WHISLER, F. D. and BOUWER,H. (1970): Comparison of Methods for Calculatinp Vertical Drainage and Infiltration in Soils. Journal of Hydrology. No. 1. WIND, C. P. (1966): Capillary Conductivity Data Estimated by a Simple Method. I A S H Symposium on Water in the Unsaturated Zone, Wageningen, 1966. WLADITCHENSKY, S. A. (1966): Moisture Content and Hydrophility aa Related to the Water Capillary Rise in Soils. I A S H Symposium on Water in the Umaturated Zone, Wageningen, 1966.

*.

Y

Part 3

Permeability of natural layers and processes influencing i t s change in time

The resistance against flow through the network of channels formed by the pores in loose clastic sediments, was investigated in Part 2 of the book. It was assumed that this resistance is the same at every point within the flow field and i t is also independent of the direction of flow, viz. the field is homogeneous and isotropic. The first assumption is of homogeneity, so that there is a uniform distribution of grains and pores within the layer. In this case the parameters characterizing the size and shape of grains and pores calculated for elementary reference volumes, do not depend on the position of the point as the centre of the reference volume. It follows from this that intrinsic porosity is also constant, because its numerical value is an unambiguous function of the listed physical parameters of the soil. As already proved, however, the resistance depends not only on the behaviour of the solid matrix, but also on the properties of the propagating fluid and the character of flow, and, in some special cases, on the saturation of the pores as well. Thus the prerequisite of homogeneity is that both the viscosity and density of the fluid should be independent of the space coordinates (constancy of temperature, salt concentration, etc. within the considered space) and the flow condition must not change in the seepage field. The latter condition means theoretically, the constancy of Reynolds’ number or that of the rate of actual and threshold hydraulic gradients. It is quite evident, that this very rigid requirement can be only fulfilled if seepage velocity is constant at every point of the field. This is because it was previously anssumed, that the physical parameters of the soil and the fluid properties are independent of the position of the point investigated, and thus the constancy of both Reynolds’ number and the hydraulic gradient is equivalent with constant seepage velocity. Accepting larger validity zones for each type of seepzge, the limits of the zones indicate those ranges of velocity, within which the seepage velocity has to remain in the whole seepage field, for seepage to be regarded as homogeneous. It follows from this definition of homogeneity that the constancy of the generalized hydraulic conductivity [given in the form of Eq. (1.1-12) and for unsaturated conditions by Eq. (1.1-13), the detailed interpretation of which can be found in Chapters 2.2 and 2.31 is the necessary and sufficient

3 Permeability of natural layers

317

requirement of homogeneous seepage. This parameter summarizes all the influencing variables (intrinsic permeability, fluid properties and flow conditions). In the general form of the movement equations, the use of a timedependent hydraulic conductivity, which is a function also of the space coordinates, does not cause any problem. The basic differential equations, however, cannot be solved mathematically, except with the use of a constant K parameter. There are only a few special cams when the change of hydraulic conductivity in space and time can be considered. Examples are the investigation of unsaturated flow (which was discussed in Chapter 2.3), and the clogging of the porous medium (the detailed investigation of which can be found in Chapter 3.3), when the R ( z , y , z , t )function can be approximated with a relatively simple mathematical equation. I n other cases, inhomogeneity h w t o be neglected in practice, and one of the most serious difficulties is the det,ermination of the average hydraulic conductivity. Consequently, the solution of the equations of movement with this constant parameter should provide acceptable results. The supposition of isothermal conditions and negligible change in density of the fluid, is a generally accepted approximation. In most cases the seepage may be regarded as being laminar, and thus inhomogeneity caused by the change iii the flow conditions can also be neglected.However, there are methods to solve the equations of movement in flow field with gradually changing flow conditions (see Chapter 4.3),but the mathematical treatment of these relationships is very complicated in practice. Thus the basic problem remaining to be solved is the characterization of an inhomogeneous solid matrix. Homogeneity of intrinsic permeability can be achieved only with the careful packing of a loose claatic sample. In nature, the distribution of grains and pores in a layer is generally irregular. I n solid rocks even the use of the model composed of capillary tubes is questionable. It is quite natural, therefore, that numerous attempts are made, to find methods for the determination of hydraulic conductivity in laboratories and in the field, assuming that these values can more accurately approximate the average resistivity of the natural layers. Another problem to be investigated in connection with the homogeneity of intrinsic permeability, is its change in time (the modification of the structure of the solid skeleton under the influence of the transported fluid). I n most cases the matrix is regarded aa a non-deformable medium, but there are some special conditions, when changes in the structure also have to be considered. For instance consolidation of aquifers (see Chapter 1.4), or the movement of fine grains in loose claatic sediments. These can destroy the stability of the layer (if the fine grains are washed out) or reduce its permeability (clogging), as will be discussed in Chapter 3.2 and 3.3. reepectively. As mentioned in the first paragraph of this introduction, the dependence of resistivity on the direction of flow is also neglected. If any property of a medium can be expressed aa a scalar function of the space (because this property is independent of the direction of theinvestigation), the medium is called isotropic. In seepage hydraulics, the condition of isotropy is the ran-

318

3 PermeabiliCy of natural layers

dom character of the direction of water transporting channels formed by the pores, fissures and fractures. Considering the development of the interstices of the layers (either the settling of individual grains of loose clrtvtic sediments, or tectonic forces creating fksures and fractures in solid rocks and weathering enlarging them) i t can be expected that most of the aquifers are anisotropic and even anisotropy is inhomogeneous. This latter condition means that the rate of permeability in various directions at a point, depends on the location of the investigated point. Similar to inhomogeneity, the anisotropy of the resistance can also be expressed giving hydraulic conductivity in the form of a special function or intrinsic permeability can similarly be modified. This is because neither the properties of the fluids nor the flow conditions depend on the direction of flow. The general form of the equations of movement, however, is not amenable for mathematical treatment. It is necessary, therefore, to accept simplifying approximations, if practical problems have to be solved and the anisotropy of the seepage field is not negligible. These concepts will also be discussed in this part of the book, together with the methods suitable for the numerical characterization of anisotropy in natural layers. According to this short introductory summarization of the problems, this part of the book dealing with the special behaviour of natural layers includes the following topics: (a)Laboratory and field measurements of hydraulic conductivity and intrinsic permeability; (b) Interpretation, determination and practical consideration of anisotropy; (c)The movement of fine grains in loose clastic sediments rtv a result of aeepage, including the problems of their stability and clogging; and (d)The characterization of the permeability of solid rocks.

Chapter 3.1 Characterization of special behaviour of hydraulic conductivity in loose clastic sediments On the basis of the analogy between seepage and electrical current, an interesting experiment can be executed to demonstrate the influence of the inhomogeneity of the flow space. It is well known, that a two dimensiond seepage field can be modelled by a grid of electrical resistivities. If the model is built up by randomly placed blocks of different resistances (which are also randomly selected within a predetermined range) the resultant conductivity of the field can b9 measured. The experiments executed in this way, prove that knowledge of the local resistivity at numerous points is necessary for the correct characterization of an inhomogeneous system. To eliminate the uncertainties caused by inhomogeneous layers, the use of field tests were generally proposed in the past. Various laboratory and

3.1 Behaviour of in-situ layers

319

field methodv were developed to measure permebility, either on samples or by using field measurements. Even though previous calculations of the coeficient of permeability from physical parameters of the soil was only a rough estimate due to the neglect of some important factors, the reliability of these methods was higher than that of a simple calculation, although both laboratory and field tests are also affected by many unknown and uncertain phenomena. At present, with the reliability of the available formulae considerably improved by various theoretical investigations, t here is practically no difference between the accuracy of various method?, and sometimes the use of formulae supplies even more accurate data than some laboratory or field tests. A further advantage of the numerical methods is their simplicity. In most cases inhomogeneity can be more accurately characterized by numerous data pertaining to different points of the flow space than by a single measurement, even if the latter describes the in-situ condition. The number of the investigated points can be increased, however, when the method used to determine the permeability of the layer is simple, quick, and cheap. From this point of view the best methods are the numerical formulae. As explained in the previous chapter, the influence of the flow conditions can also be expressed by various equations. Measuring the parameter in question the investigation of only one flow condition is assumed in most cases. In most field tests, even the character of the actual condition is not known or, as often occurs, the flow conditions at various parts of the flow space created by the test (e.g. by pumping), are different. It can be stated, therefore, that the use of laboratory and field methods is only reasonable when such parameters of the layer are determined in a way that cannot be calculated from equations (e. g. anisotropy, development of large channels, direction of recharge, storage capacity, etc.). For this reason, the application of several laboratory and field methods is not advisable, although they are generally used in practice. Thus the laboratory test with a disturbed sample and the so-called “quick field methods” should be avoided, because their uncertainty is higher than that of the calculated parameters.

3.1.1 Laboratory and field methods to determine permeability The laboratory measurements and the field tests are based on simultaneously measured data of the gradient and the flow rate in a system, where the flow is created and maintained artificially. Because of the uncertainties mentioned in the introduction, only those methods, the reliability of which is satisfactory and which provide supplementary informations on the properties and behaviour of the field apart from permeability are discmsed. For this reason the detailed analysis of the quick field tests is neglected here. Laboratory t a t s can be performed with two different types of equipment. The simple one operates with constant pressure head. This test is practically

320

3 Permeability of natural layers

the repetition of Darcy’s experiments (Fig. 3.1-1). The length of the sample is I and the pressure head is h. I n a time interval t , the total amount of water percolating through the sample is V . The discharge (& = V / t )divided by the cross section of the sample ( A ) ,yields seepage velocity, (v), which is proportional to the hydraulic gradient (I= h/l).The coefficient of proportionality is independent of the gradient if the movement is laminar (the

Fig. 3.1-1. Permeabimeter with constant pressure head

latter condition must be ensured in any case), and this coefficient is equal to the hydraulic conductivity, which can be calculated from the measured parameters : (3.1-1)

This method cannot be used for measuring the permeability of a cohesive sample, because the amount of water percolating through the sample is very small in this cwe and, therefore, therelative error of the measurements may be considerably high. Equipwnt operating with changing pressure head can be used to measure the permeability of such a sample (Fig. 3.1-2). The difference between the two methods is that in the second case only the lower water level is constant and the pressure is maintained by a water column sinking gradually during the experiment in a calibrated glass tube. Both the pressure head and the flow rate can be calculated by measuring the changing watkr level in the tube. Because of the change of the pressure head the movement is unsteady. The flow rate is equal t o the change of the stored water in the glass tube having a cross section of A,. This condition can be expressed by the following differential equation:

dh

A,--= dt

h K-A.

I

(3.1-2)

321

3.1 Behaviour of in-situ layers

wef cotton n

IF-----

.

.

Fig. 3.1-2. Permeabimeter with changing pressure head

After separating the variables and integrating the equation, a relationship can be established to calculate the hydraulic conductivity:

h

K = l - A0 - ; '1 n I A tz - t , h,

(3.1-3)

where two boundary conditions are taken into account i.e. at time t , the height of the upper water level above the lower constant level is h,, and the same pressure head at the time point of tais h,. Finally, the results of experiments to meaaure the consolidation of a sample are sometimes also used to determine the permeability of very cohesive clays. The development of compression with time also depends on the permeability of the sample, aa clearly expressed in Terzaghi's differential equation:

where

(3.1-4)

M C=K-;

Y

where Ae is the decreaae of the void ratio until time t ; de, is the t o t d expected decrease, h is the thickness of the layer and M is the coefficient of 21

322

3 Permeebility of natural layers

compressibility. To calculate the consolidation, the index of consolidation (C)is meaaured by consolidation apparatus (oedometer), from which hydraulic conductivity can be calculated knowing the coefticient of compressibility and the specific weight of the sample. Laboratory tests have a common feature: a sample 7m to be taken from the layer and built into the apparatus. As already mentioned, disturbed samples are not convenient for this purpose. From bore holes the drilling tools bring up the material, selected according to grain size (from coarsegrained layers) or in kneaded condition with altered structure (from clays). The porosity of thedisturbedsample in the apparatus depends on the preparation of the test and not on the natural condition of the layer. It is only an estimated value, similar to the parameter used in the case of numerical formulm. Consequently the investigation of such a sample, which does not preserve any special character of the layer, is unjwti$ed. The determination of a parameter representing some natural conditions of the layer, which cannot be calculated from physical parameters of the soil, can only be expected using an undisturbed sample, which is built into the apparatus together with the core-tube. To take an undisturbed sample is, however, very difficult, especially from saturated sediments. There are special types of apparatus designed to get an undisturbed core from bore holes or from trenches dug for exploration. The core-tube, however, always causes some deformation along its wall. A further error may be caused by the discontinuity between the core and the wall of the tube. The influence of the amount of water percolating along the wall may be significant, compared to the total discharge in the caae of clay samples, where the total amount percolating through the sample is relatively small. To decrease this possible error the crow section of the core should be as .?urge as possible. I n spite of all these difficulties, the measured permeability may 6upply some additional information on the natural structure of the layer. Thus there are cases, when this method is one of the most reliable to determine some special parameters of the sediment e.g. anisotropy, aa will be disoussed in detail later. One way to avoid the difficulties arising in connection with taking undisturbed samples is to measure the permeability under natural conditions (in-situ). For this purpose various field rnethods were developed. The permeability of the unsaturated layer near the surface can be measured by the simplest method. The original form of this method, called infiltration test, is the use of the so-called Miintz-Laink’s cylinder (Nhmeth, 1942). After cleaning the surface of the soil, a cylinder is sunk into the layer. The internal volume of the cylinder is filled with water and its infiltration rate is recharged, so that a constant water level can be maintained inside. Infiltration velocity ( v ) can be calculated as a quotient of the amount of water recharged during a time unit (&) and the cross section of the cylinder (A). This velocity is equal to the product of the hydraulic gradient and hydraulic conductivity. The former can be determined from the pressure head divided by the length of seepage. The length of seepage is equal to the depth of infiltration (It), which is a function of time, while the total pressure head is composed of three members [i.e. the height of t.he water column

323

3.1 Beheviour of in-situ layers

above the surface (h,),the depth of infiltration ( h ) ,and the capillary suction (h,?)].It is evident that h, and h,, can be neglected, compared to the depth of infiltration, if the latter increases, the gradient tends to unity, and the infiltrating recharge becomes constant w indicated in Fig. 3.1-3. The socalled Green-Ampt's equation describing this process is w follows (Green and Ampt, 1911):

v = - Q= K I = K A

ho

+ h + hco . h

9

and

v + K if I = ho +

h

hco -1;

because h

+

(3.1-5)

00.

The theoretically described infiltration (piston flow) is disturbed by the process of saturation along the wetted front. This phenomenon is only partly considered in Eq. (3.13) by including the cupillary head. Both the disturbing effects and capillary suction are terminated when the wetted front reaches the water table, or more precisely, the closed capillary zone, which is indicated by the break in the curve representing the recharge against time relationship (Grishkan, 1966). Observations during the first period of the process (when the gradient is higher than unity) can also be used to calculate permeability. In this case, the complete form of Eq. (3.1-5) has to be applied, whichrequires knowledge of the depth of infiltration. The instantaneous position of the wetted front can be determined with small hand-drilling equipment. The layer i s not completely saturated during infiltration. Thus the permeability determined by this method would be generally smaller than the correct parameter of the saturated layer. This regular error is, however, partly compensated by the fact that the form of the infiltrating jet is not cylindrical, it extends sideways.

t

around wafer Fig. 3.1-3. Infiltration test to determine the hydraulic conductivity of unsaturated soil 21*

324

3 Permeabbility of natural layers

To decrease the second effect, double cylinders are generally used (e.g. Nesterov’s method). In this case two concentrical cylinders are sunk into the soil, maintaining the same water level separately in both (i.e. in the inner cylinder and between the two cylinders). Only the recharge to the inner cylinder is measured. The purpose of the outer cylinder is only to supply water for the sideways extension, ensuring in this way the approximately cylindrical form and vertical direction of the internal jet. The decrease in the sideways flow related to the total amount of recharge can also be achieved by using large infiltration basins. Their further advantage is that a large soil column is investigated in this way and thus a better average of the conductivity can be memured. The larger size of the apparatus and the greater amount of water to be supplied, however, hinder the application of this method (Makaricheva, 1966; Szab6, 1954). Field tests to measure the permeability of layers below the water table can be divided into three groups: (a) Recharge of the bore hole; (b) Short period of pumping, observing the rate of lowering and - after stopping the pumping - the rise of the water level in the well; (c) Pumping test.

The common basis of the methodv pertaining to the first group is that an excess pressure related to the natural water table is created in the bore hole, and the rate of /low induced by this pressure is measured. Differences between the various methods are caused by the position and size of the surface throught which infiltration takes place, and which may be either at the bottom or at the side wall of the bore hole, or perhaps, in both places. A further difference can be caused by the character of the flow. Sometimes unsteady state movement is observed. I n other cases the development of the balanced condition is sought, and the parameters of steady seepage measured. Khafaghi’s and Lugeon’s methods belong to this type of test. A characteristic test illustrating the second group is Porchet’s method. The water from the well or bore hole is pumped with constant yield for a short period. The level in the well is observed during and after pumping until the well is refilled. From the discharge and the graph of the water level change with time, the expected yield of the well or the hydraulic conductivity of the layer is calculated. A general fault of the methods described (so-called quick pumping tests) is that the boundary conditions of the flow field are not known and, therefore, neither the condition of flow nor the influencing factors can be determined. Lack of this information causes many uncertainties and, therefore, the reliability of these methods is much lower than that of the simple mathematical formulae. In the case of pumping, theuncertaintyis further increased by the development of the free exit fwe viz. the level in the pumped well is always lower than that in the layer. For this reaaon, the use of the quick pumping tests is not advisable. The only reliable method for in-situ investigation of the hydraulic wnductivity or permeability of layers below the water table is the pumping test, using

3.1 Behaviour of in-situ layers

325

more than two observation wells. The pumped well penetrates the aquifer to its full depth if possible. Using this method the pumped yield is constant and the hydraulic conductivity is calculated from the observed depression cone. The depression is determined by observation wells generally after awaiting the balanced condition, when the depression curve attains its constant position. There are, however, methods for calculating the K-value from the modification of the depression, using the parameters of unsteady flow.

Fig. 3.1-4. Symbols used in the derivation of Dupuit's equations for the characterization of axial symmetrical flow

The basic equation for this calculation is Dupuit's formula, according to which the yield of a pumped well can be calculated from the following relationships [the application of Eqs (5.3-8) and (5.3-3) for axial symmetrical flow] : If the flow field is unconfined Q=nK

Hi

H: . R '

-

(3.1-6)

In r0

I n the case of a confined /low field Q = 2nKm H2 - HI R ' In -

(3.1-7)

TO

(The meaning of the symbols used in the equations is given in Fig. 3.1-4).

326

3 Permeability of natural layers

There are two problems hindering the application of these equations: (a)The development of the free exit face at the screen of the well; (b) The fact that the length of the zone affected by the depression (A)is generally unknown. Both difficulties can be eliminated using two observation wells in addition to the pumped one. Dupuit’s equation was derived from the differential equation of movement by double integration. The constants of the integrations were determined by using the water level in the well and that at the end of the depression as boundary conditions. The integration can also be executed in the same way, with other boundary conditions. To avoid errors caused by the uncertain flow conditions around the well and near the end of the depression, the water levels measured at the observation wells have to be used as boundary conditions. Equations determined on the basis of this assumption, or, more precisely, the hydraulic conductivity expressed from these equations, can be given as follows:

If the flow field is unconfined In

X

(3.1-8)

I n the case of a confined flow field 51 In -

(3.1-9)

Summarizing the discussions of the various laboratory and field tests, it can be stated that there are only three feasible methods: (a)Laboratory tests with undisturbed samples; (b) Infiltration tests; (c) Pumping tests with more than two observation wells. All other methods evaluate parameters measured in a seepage field, the boundary and flow conditions of which are unknown. This includes pumping tests executed without any or with only one observation well. Here the observed level of the pumped well and the length of the affected zone have to be taken into account. Therefore, the hydraulic conductivity determined in this way is very uncertain. For this reason tests, apart from the three above, should be avoided.

3.1.2. Evaluation of pumping tests with more observation wells

It can be stated on the basis of the critical evaluation of laboratory and field tests given in the previous paragraph, that the use of formulae is preferable to the many types of field tests and the application of experimen-

3.1 Behaviour of in-situ layers

327

tul methods is reasonable only i n those cases when some special feature of the layer can be determined i n this way. Among the advisable methods there is only one field test which is suitable for investigating the permeability of layers below the water table, i.e. the pumping test with several observation wells. The evaluation of the data observed in this way is very diEcult because they are influenced not only by the natural conditions of the flow bpace but also by the special flow conditions created by the pumping itself. By grouping the collected data carefully and evaluating those of a hydrodynamical aspect, very valuable information can be gained, however, on the properties of the layer and on the conditiom affecting the system. To clarify the theoretical basis on the evaluation of the pumping tebts, the geometrical, flow, and boundary conditions have first to be summarized, which are the fundamental assumptions for the derivation of Dupuit’s equations [Eqs (3.1-6) and (3.1-7)]. These approximations are aa follows: (a)The flow space is homogeneous and isotropic; ( h ) The base of the aquifer is formed by a horizontal impervious layer while its upper surface is either unconfined [Eq. (3.1-6)lor confined, or semipervious [Ey. (3.1-7)]; (c)The flow space is neither recharged (infiltration) nor drained (evaporation) along its surface; (d)The area affected by the cone of depression can be characterized by a radius R,a condition, which is ensured only when the well is situated in the centre of a circular island and the pumped water is recharged without any resistance along the bank of the latter. Hence, an axial symmetrical flow can develop; (e)The seepage remains within the validity zone of Darcy’s equation at every point of the flow space; (f)The water is in a state of equilibrium before pumping and hence its table is horizontal under natural conditions; (g)The seepage is a steudy flow. This assumption also requires that the recharging Hection of the flow space should be within a finite distance from the well and the level of the recharging surface-water haa to be constant. If any one of these hypotheses is not satisfied the movement is unsteady; (h) Finally, the laat approximation, is the so-called Dupuit’s hypothesis, which assumes that each cylindrical surface, concentric around the well, is a potential surface, and the velocity is constant at every point on this surface and proportional to the gradient of the water table or the piezometric burface determined at this distance from the well. In other words Dupuit presumed that the stream lines can be approximated by horizontal lines, and the vertical component of the velocity vector can be neglected.

Analyzing the applicability of these hypotheses in the case of a pumping test, it can be observed how the unsatisfied assumptions affect the observed data. According to Eqs (3.1-8)and (3.1-9),the hydraulic conductivity can be calculated knowing the distances of two observation wells from the pumped well and the depressions at these two points. Having more than two observation wells a parameter can be determined from the data of each pair of

328

3 Permeability of natural layers

wells. If all the assumptions above were satisfied, these parameters would be identical within an acceptable scattering caused by the errors of measurement and calculation. The differences between the various values of hydraulic conductivity calculated from any pair of wells, being higher than that which can be explained by these errors, indicate the inhomogeneity of the layer, or the fact that one or more of the above hypotheses are not satisfied. If the calculated data are not constant but their change shows some regular pattern, this will give some idea of the basis on which the type of the unsatisfied assumption can be estimated. Some important information can also be gained on the natural conditions of the layer (e.g. the main direction of the flow, the type and amount of recharge, the flow condition, etc.). The inhomogeneity of the layer is indicated by the irregular but high scattering of data. A regular arrangement of the variation of permeability may also be caused by inhomogeneity, if the change in the physical behaviour of the layer is regular. The position of the surface of the lower impervious layer is determined by using geological and geophysical methods. Thus its declination from the horizontal plane can be directly checked, or knowing the actual position of the lower boundary, the change caused by the actual thickness of the layer can be studied. Natural effects influencing the seepage along the flow space (accretion) cannot be avoided. There are methods of study which include these influences and consequently, some corrections can be calculated when the vertical recharge or drainage is significant compared to the horizontal flow. The relationships describing this special condition are, however, more complicated. It is advisable, therefore, to choose a time for the execution of the pumping tests when the accretion is relatively low (e.g. in winter the frozen surface hinders both infiltration and evaporation). An important aspect which must be considered is that the flow space should not be recharged artificially. The water pumped during the test should be conveyed through pipes aa far away from the pumping well as possible. The exuct radial flow and the circular recharge can be ensured only in the laboratory. When the natural recharge comes from several directions, the unified distribution of flow around the well can be assumed. I n some other caaes, when the flow has one main direction (e.g. in the case of a well near the bank of a river or a lake), the calculated permeability coefficients differ depending on the direction and where the observation wells are located (Fig. 3.1-5). Evaluating the measured data, it is assumed that the observed slope of the depression cone has developed under the influence of a uniformly distributed flow. I n contraat, the flow i n the m a i m direction is higher than the average and creates, therefore, a steeper slope. Thus the pair of wells in this direction gives a lower K-value than the actual one, while the permeability calculated from the data of other wells is higher than average. Some special types of non-uniformly distributed flow can be studied by evaluating equations, which were derived by considering relevant boundary conditions. This analysis draws attention to the importance of the radius of the influenced zone ( R ) .The use of empirical formulae to determine some characteristic

3.1 Behaviour of in-eitu layers

329

main direct, 0

pumped well

4-

observation well

of ffm

braffcb of Danube

-

Fig. 3.1-5. Modification of hydraulic conductivity calculated from the date of pumping testa having uneven recharge

radius (e.g. Sickhart's equation) is unacceptable, because such arbitrarily chosen parameters do not describe the actual geometry of the flow space. The fictive values calculated, for example, on the basis of an assumed permeability, have no physical meaning at all. When there is a river or lake near the well, the distance of its bank can be considered as R. It is more reliable, however, to choose a method which eliminates the application of this parameter. For this reason, the use of Eqs (3.1-8) and (3.1-9) is preferable to Eqs (3.1-6) and (3.1-7) which require at the same time the application of two observation wells at least. When the investigated aquifer is very thick, drilling of the pumped well down to the lower impervious layer would be costly and the yield of a well perforated along the entire depth of such a layer would be very high compared to the generally applied pumping capacity. I n this case the pumping test m y be executed on a partially penetrating well. It is general practice to evaluate the data collected from such a system by using Dupuit's equations, assuming in most cases that the influence of the layer below the bottom of the well can be neglected. The y variables are measured, therefore, from the

330

3 Permeability of natural layers

T m

.4 .

.

-

-

Flownet bt?lOUJ/flJ to the partial/y penetratiny well

, dehtcrn

Flowflet of the model system

-

covering / a G

bottom of the well as a reference datum (Fig. 3.1-6). It is evident, that smaller flow discharge moves through this part of the layer than the total yield of the well and, therefore, the actual slope of the depression cone is smaller than it would be in the case of an aquifer with an impervious layer at this depth. For this reason, the calculated permeability is always higher than the actual one, independent of the location of the pair of observation wells. There is, however, a regular arrangement of the determined parameters. The ratio of the flow moving above the horizontal plane of the bottom of the pumped well, to the total yield, increases nearer the well, aa shown in Fig. 3.1-6. The regular error caused by this effect is smaller when the observation wells are near the pumped one and the calculated K-values show an increasing trend as a function of the distance from the centre of the system. The case of a pumped well penetrating to 22 m below the surface of a very thick layer of gravel is shown in Fig. 3.1-7 as an example. Using a method (Kovhcs, 1966) suitable to calculate the hydrodynamic parameters of a flow system around a partially penetrating well [see Eq. (5.3-56)] for

331

3.1 Behaviour of in-situ layers

dLstance from the punped we//

rmJ

Fig. 3.1-7. The correction of hydraulic conductivity calculated from the daIt&O f pumping test executed in a partially penetrating well

8

evaluating the observed dam, the hydraulic conductivity was calculated as a function of the distance from the well, assuming various actual depths of the aquifer. According to the dependency explained previously, the K-value incremes with the distance and decreases with the assumed actual depth. A t the same time the modification in horizontal direction becomes less and less when the fictive thickness of the layer is increased. Theoretically, the calculated peymeability must be independent of the location of the observation wells uhed to determine the hydraulic conductivity, if the fictive depth is equal to the actual one. In the example, this condition is attained at a depth of 200 m , which agrees with the real thickness of the layer known from some deep bore holes in the vicinity. The comparison of the hydraulic conductivity calculated from the observation wells assuming this depth (which is practically constant) to that determined on the basis of the physical parameters of the soil, gives K = 0.6-0.7 x 10-3 m sec-l and K = 0.9 x x 10-3 m sec-I, respectively.This proves the reliability of both the method of evaluation and the method proposed for calculating the yield of a partially penetrating well. Neglecting the development of the free exit surface causes error only when the flow space is unconfined, since, in the cwe of seepage under pressure the water level in the well is above the upper surface of the aquifer,

332

3 Permeability of natural layers

and thus free exit surface cannot develop. Excluding the water level measured in the pumped well from the data used in the calculation, this error is practically eliminated because it can be proved hydrodynamically that Dupuit’s equation is an acceptable approximation to determine the yield of a well [Charnyi, 1951; see Eq. (5.3-8)]. There is, however, a small regular error i.e. the actual yield may be slightly higher than the calculated one depending on the capillary water conveyance of the layer. The maximum difference may reach about 10%) when the water column in the well is zero (El, = 0). The error decreases with increasing H2.Thus the hydraulic conductivity determined by pumping tests may be 10% greater than the actual value, in the case of total draw-down. This difference is smaller than the expected error caused by other uncertainties and may be neglected, therefore, in practice. It gives, however, the explanation of an observed irregularity. It has been reported in some papers (Ubell, 1954, 1958) that the hydraulic conductivity calculated from the same pumping well, increases using observations pertaining to higher yields. The example given in Fig. 3.1-8 demonstrates not only the existance of the phenomenon mentioned and the influence of the free exit surface on the results of pumping tests, but the numerical identity of the hydraulic conductivity when allowance is made for the capillary effect is equal to that calculated from physical soil parameters. This suggests that the ttssumed physical model for this process is correct. The next hypothesis is the validity of Darcy’s law. To show the possible error caused by not satisfying this condition, the results of a laboratory experiment are given. A well wm placed in the centre of a circular shaped basin, recharged along its outer edge. The sandy aquifer was covered by horizontal glass plates at both sides and the flow remained under pressure I +K (data calculafed fi -om

-

originai level of ground

the pumping test cuns/de‘p/:Tg the reductm necessary f ’7 consider fbe iflfluence of

I level of draw down

water

tbe free exit surfaces) K (wlcuiefed froin

3 level of dmwdown

4 level of draw down I

I

I

I

T

Fig. 3.1-8. Reduction of the calculated hydraulic coefficient considering the effect of the free exit face

333

3.1 Behaviour of in-situ layere

during the whole experiment. Thus all the other hypotheses applied for the derivation of Dupuit’s equation were correctly satisfied. Fig. 3.1-9 shows a vertical radial section of the basin indicating both the total area of the cross section ( A ) and the Reynolds’ number [Re see Eq. (2.1-42)] as a function of the distance from the axis. Similarly the Jaw-down curve and the hydraulic conductivity calculated from the latter, also depending on the distance, were plotted on the graph. The calculated permeability incremes g r d u d y with increasing distance from the well and becomes constant where Re, decreases below 10, where the validity of Darcy’s law can be assumed. Outside this section, the average of K determined on the basis of the observed

B

5

-

44-4-

8s

k

3n/f4-

8*

p/oi

I parameters

kc1

$ e

.1.10-4

I I I

1

1

I

I

1

1

-

Fig. 3.1-9. Decrease of the calculated hydraulic conductivity caused by non-laminar seepage

3 Permeability of natural layers

334

piezometric line is K = 437x10-4 m sec-l, which agrees with the parameter calculated from the grain-size distribution curve (K = 4 . 5 ~ x 10-4 m sec-l). If one of the wells whose data are used to calculate permeability, were in the zone of non-laminar movement, the determined parameter would be only about half of the realistic value ( 2 . 4 ~ lo-' m sec-l). When there is ground-water flowin nutural conditions, the water table has a natural slope. In this case the difference in level due to the slope of the water table and that due to the draw-down caused by pumping must be measured. This makes the observations more difficult because the value to be measured may be smaller than the error expected. It is a general policy, therefore, to choose a location for the pumping tests where the natural slope of the water table can bo neglected. There are many reported pumping tests indicating a constant hydraulic conductivity around the well and of varying value only at a greater distance. If the permeability were really smaller here than the constant parameter, the change could be explained by t,he development of microseepage, although the effect of adhebion is generally small in coarse-grained sedimentd in which the experiments quoted were executed. In the reported cases, however, a higher permeability is always indicated at a greater distance from the well (Fig. 3.1-10) (Szilhgyi, 1954; Ubell, 1958). This irregularity can be caused by the fact that the emptying of the stored water does not follow dtsfance from the pumped well r m l 3 4 5 878910'

2

3 4 5~778910~ 2

3

0.03 0.07

5 .$

0.09

8

$

0.Il

2 c.

*

0.13

0.7

~

$ 0.6

3

Q

4

5

v,

0.5

e

0.15

3 0.4

b

8

.$ 0.3

B

$0.2

Dl 3

4 5 6789/0

2

3 4 5 6789100

2

3

distance from the pumped well LmJ Fig. 3.1-10. Increase of the calculated hydraulic conductivity far from the pumped well

335

3.1 Behaviour of in-situ layers

the lowering of the water level instantaneously and, therefore, in the areas far from the well the depletition of the pores is delayed. The amount of water drained from these pores becomes part of the horizontally transported flow only after a considerable time lag has elapsed. To avoid the limit of application caused by the assumption of steady flow, which is a fundamental hypothesis of Dupuit’s equat,ion, formulae bayed on th,e hydrodynarnicul analysis of axial symmetrical unsteady flow were developed [see Eq. (5.4-19)]. These can be used to evaluate the unsteady period of movement around the pumped well and to calculate the permeability and storage coefficients from the rate of draw down data. The basis of the method generally used is the Theis’ equation (Theis, 1935): K=-

‘

4n sH

Ei(u);

(3.1-10)

where s is the depression in an obscrvation well at a distance, r froin the axis after a time, t having elapsed from the beginning of pumping. and H is the original depth of the ground water (or the thickness of the aquifer, if the system is confined). The integrl-exponential function [ E i ( u ) ]can be expressed by expanding the term to be integrated into series:

-

I

= - 0.5772 - lnu

+--+ u -2-21 3.3! U2

u3

u4

4.4!

+...;

U

(3.1-1 1)

where the u variable ie the function of the storage coeficient: u=-.

r2S 4KHt

(3.1-12)

The storage coefficient in the case of shallow unconfined ground-water flow can be substituted by specific yield (ns)as discuesed in Chapter 1.4. Knowing its numerical value, the hydraulic conductivity can be calculated by the above method from one pair of the corresponding values of the observed depression and time. Having more pairs of data, both storage capacity and permeability can be determined using the system of equations above. The calculation cannot be directly solved because of the integral-exponential function. Theis applied, therefore, a graphical method. Jacob explained, however, that the parameter u is inversely proportional to time, thus the third and further members on the right-hand side of Eq. (3.1-11), are negligible as compared to the first two members, when timeislong enough (Jacob, 1940). Considering this simplification, the corresponding time data and depression at a given distance from the well, can be represented in a semi-logarithmic coordinatesystem (i.e. log t and s), by a series of points along a straight line. Each line characterizing a given distance of r , intersect,s the horizontal axis at t,,(r),which is the fictive starting lime of the depres-

336

3 Permeability of natural layers

sion at the distance in question (Fig. 3.1-11). On the basis of the slope of these lines both hydraulic conductivity and storage coefficient can be calculated:

K=

‘

4nH(s, - 91)

ln-;t2 t,

(3.1-13) (3.1-14)

Fig. 3.1-11. Representation of the relationship characterized by Jacob’s equation

Comparing the various values of hydraulic conductivity calculated from Dupuit’s, Theis’ and Jacob’s equations, it can be stated that the evaluation of steady and unsteady movement gives practically the same result in the vicinity of the well, where the calculated parameters are constant (in the case of the example shown in Fig. 3.1-10 this distance is about 50 m). A short time after the start of pumping, this section of the draw down curve is lowered parallel to its previous position, because the relatively rapid depression and high suction cause a quick emptying of the gravitational pores. The similarity of results in the two basically different methods can be explained by the fact that their fundamental assump%ions are equally satisfied (or can be regarded as acceptable approximations) in the vicinity of the pumped well. I n the caae of Dupuit’s equation this hypothesis is the steady condition of flow and in TheisJakob’s method it is the parallel change of the draw down curve. Outside this zone the application of hydrodynamic equations describing unsteady seepage is hindered by the slow anddelayeddepletion of the pores. Thus storage capacity is not an unambiguous physical parameter of the soil in this case, but is the function of time. To demonstrate this relationship the result of an experiment is shown in Fig. 3.1-12, where storage capacity (specific yield) of a column of gravel is plotted against the velocity of the lowering of the water table (LBczfalvy, 1966). Because of the variation of storage capacity in time, this parameter determined by the TheisJacob’s methods for a given time point, is also a

3.1 Behaviour of in-aitu leyers

337

function of the distance of measurement from the pumped well. The coefficient near the well can be regarded as a good approximation of the actual value. A t a greater distance, water originating from the delayed depletion of the upper lying pores raises the increase of the horizontal flow rate compared with the theoretical value. This process results in higher storage c u p ity than is the actual case and makes the calculated parameter very uncertain &s also indicated in Fig. 3.1-10.

velocity of lowering of wafer &He icm/spc/ Fig. 3.1-12. Experimental relationship between the velocity of the lowering of the water table and the storage capecity

As a summary of this analysis it can be stated that both steady and unsteady methods are applicable only when data observed near the pumped well (where the draw down curves are practicdly p a r d e l to each other in different points of time) are used. Thus the same pumping test can be evaluated hy both methods. Generally, the use of Dupuit’s equation is preferable because of its simplicity. The application of the TheisJacob’s method is advisable in cases where the final objective is the study of an unsteady process and parameters (includingdso storage capacity or specific yield) for this investigation are t o be determined by pumping tests. The so-called Dupuit’s hypothesis was previously mentioned aa a h a 1 approximation. There are methods (e.g. Jmger, 1949) using other assumptions for eliminating the possible error in the application of vertical potential surfaces. It can be proved, however, that significant differences are caused by Dupuit’s method only in the case of large draw down, and the occurrence of this extreme condition is improbable because of the development of the free exit surface.

22

338

3 Permeability of natural layers

3.1.3 Characterization, determination and practical consideration of anisotropy

As already mentioned in the introductory part of this chapter, permeability often depends on the direction of the investigated /low. Because of this anisotropy of hydraulic conductivity, the vectors of velocity and gradient are generally not parallel to each other and Darcy's law can be given in the following form: v1 = K l l l l KZIIZ K3113; v2

v3

+ ==KIZIl + = + K1311

K2212

+ +

KZ312

f

K3213;

(3.1-15)

K3313;

where vl, v2, v3 and I,, I,, I , are the components of velocity and gradient vectors respectively in an arbitrarily chosen orthogonal, three-dimensional, coordinate system, and permeability is represented by a symmetrical tensor of the second order (because K , , .= K,,) having nine members ( K l l . . . . . . K,, ..Ka).Readers interested in further details concerning the mathematical treatment of flow equations characterizing seepage through a field having such general anisotropic behaviour , are referred to the following publications dealing with this theoretical problem (Samsioe, 1931; Ferrandon, 1948; Irmay, 1951; Biot, 1955; Bear et al. 1968; Bear, 1972). It can be proved that even in this general cme there exist three main directions perpendicular to each other (z, y , z ) , in which the components of the gradient and velocity vectors are proportional and thus Eq. (2.1-15) can be simplified (Ollendorf, 1950):

.

(3.1-16)

A further simplification becomes possible if two of the coefficients of proportionaJity, e.g. K , and K,, are equal and only the third, K , is different. This assumption means that the permeability of the layer is constant in a plane determined by two main directions (z and y ) , while the parameter in a direction perpendicular to this plane ( z ) is different. This condition is generally characteristic in fine laminated sediments, which represent the most important examples in seepage hydraulics. This special type of anieotropic hydraulic conductivity is called transverse anisotropy. Using the symbol KH to indicate the hydraulic conductivity in the r y plane and K v to characterize that in the direction 2 , normal to the z y plane, K , = K, = K,; and K , = K , , Darcy's law for a field having transvers anisotropy can be given in the following form: (3.1-17)

The b a i s of the application of these symbols originates from the fact that in sedimentary basins, the hydraulic conductivity does not depend on the direction of the investigation along the bedding planes of the fine layers or lamellae, which are usually horizontal (or almost horizontal), while in a vertical direction (perpendicular to the bedding plane) this parameter may

3.1 Behaviour of in-situ layers

339

be completely different from that measured in the horizontal direction. Thus K Hstands for horizontal and K v for vertiml hydraulic conductivity. Investigating a two dimensional seepage with a vertical flow plane in a space having transverse anisotropy, it has to be remembered that the corresponding velocity and gradient vectors are parallel only in the horizontal and vertical directions. In other cases, the proportionality is valid only for their components i n the two main directions. Let us construct an ellipse in the

I

Fig. 3.1-1 3. Representation of transverse -anisotropic hydraulic conductivity in a vertical flow plane

vertical flow plane, the two main axes of which are horizontal and vertical respectively (Fig. 3.1-13). The horizontal axis should be proportional to KH (it is the major axis if KH> K , ) and the vertical one to K v . Knowing the direction of the velocity vector the ratio of its components can be determined. These components have to be divided by KH and K v respectively to get the components of the corresponding hydraulic gradient. Assuming that the minor axis proportional to K v but being chosen arbitrarily, is equal to unity, the operation described previously (dividing the components by K H and K v respectively). is equivalent to the horizontal contraction (or enlargement if KH
v = v,

+ ivy = veia ;

where

v = VvE + v: ; and tan a = 2 ; V

VX

I =I,

+ i I y = IeiB ;

where (3.1-18) 22*

340

3 Permeability of natural layers

where

I,=

2 ; and KH

therefore

tanB=--.vy K H

Kv

vx

The tangent values of the a and B angles can also be expressed from the equation of the ellipse and the circle respectively:

(g)2 I+: [

2I:[

+

2

(the ellipse); = A2 2

If)

(the circle); =A2

and tan

/I =

=I I . 2 2'

(3.1-191

[3J2 - 1.

v

Assuming the velocity vector intersects the ellipse at point P,2 = 3 . A f u r v,

x

ther hypothesis was the horizontal contraction of the ellipse, according to which x' =KV x and y' = y . Substituting these relationships the accurctcy I

KH

of the /I angle constructed graphically can be proved:

Y v H K=XRH= V X K ,

x Kv

'- 1 =.tan B.

(3.1-20)

Thus this simple graphical method is sumient to determine the direction of the gradient in a field having transverse anisotropy of the hydraulic conductivity, if the direction of the velocity vector is known. In practice, a further assumption is made to ensure an easy mathematical handling of the problem, i.e. the homogeneity of anisolropy. According to this hypothesis, the ratios of horizontal and vertical permeabilities are the same at every point of the flow space and the absolute values of the parameters are also practically constant. This approximation is generally acceptable in loose clastic sediments, because the system of more and less pervious thin layers has developed almost uniformly within a sedimentary basin affected by identical external processes. Supposing homogeneous transverse anisotropy, this special physical behaviour of the layer can be characterized by a constant parameter i.e. the

3.1 Behaviour of &-situ layers

341

quotient of the horizontal and vertical hydraulic conductivity, called w e B i e n t of anisotropy : #

A=-.KH

(3.1-21 )

KV

The determination of this coefficient ( A ) is a very important feature in the exploration of aquifers and since some irregularity (inhomogeneity) may occur in natural layers, it is necessary to measure this parameter at &s many

Fig. 3.1-14. The location of observation wells in the case of 8 pumping test for the determination of an anisotropy coefficient

points a8 possible. The most wmmon inethod of measurement requires a slight modi@ztion of the pumping test (Fig. 3.1-14). The perforated length of the pumped well is shortened to ensure a quasi-radial flow in the aquifer. The perforations i n the observation wells are located at diflerent depth. From the difference between the depressions observed at the same distance from the pumped well but draining the aquifer at different levels, the coefficientof anisotropy can be estimated. A numerical method for evaluating the data observed in this system wm proposed by Babushkin (1954) :

and

The symbols used in the equations are explained in Fig. 3.1-14. The 1 coefficient can be calculated only by successive approximation and after having determined this value the horizontal permeability can be determined.

342

3 Permeability of natural layers

Another possible method of evaluation of the data measured by this special pumping test is the graphic method, which is easier but yields only a very rough approximation. The equipotential lines characterizing the pressure distribution at a given time point are constructed around the perforated length of the pumped well from the observed depressions. In the cwe of

8msotmpic leyer Fig. 3.1-15. Deformation of equipotential lines around a pumped well caused by anisotropy

an isotropic layer these lines are concentric circles, while in anisotropic layers they are elliptical. The ratio of the axes of the ellipse is proportional to

[+I

r% ;

=

(Fig. 3.1-15).

The horizontal permeability is calculated in this case by the method generally applied to evaluate pumping tests, using the data of observation wells drailling the aquifer at the level of the perforation of the pumped well. The coeficient of anisotropy can be determined by tracers (dye or isotope) as well (&&a and Juh&ez, 1966). A tracer is put into an observation well (after drawing the tube out for a short length t o ensure the flow through the bore hole and awaiting the development of an approximately steady flow) at a distance of r from the centre of the depression (i.e. the centre of the short perforated section of the pumped well). The time of propagation along this distance is then measured (Fig. 3.1-16). The relationship between the time of propagation ( t ) , the yield of the well (Q) and the anisotropy coefficient ( A ) can be given in the following form: t = - 4 r S n [l

where

+(A-

l)sin2a]-,1 .

B

Q

(3.1-23)

B=

v;1.-1 arc tan a-1

3.1 Behaviour of in-situ layew

343

I

Fig. 3.1-16. Symbols used for deriving f i W u h & z ’ s equation to calculate the coefficient of anisotropy

The formula can be simplified by measuring the time of propagation along a flow line having the following special positions:

Along a horizontal atreum-line (point 1) 1 3Qh. -=B 4r3n’

(3.1-24)

or

Along a vertiml stream-line (point 2) I 3Qt2 -=(3.1-25) B 4r3n There is another method applied in France, using tracers to determine the anisotropy coefficient. After having executed the normal pumping test, a sheet-pile is introduced between the pumped well and an observation well near the former. The time of propagation from the observation well to the pumped one divided by the vertical length along the sheet-pile gives the effective velocity of seepage (veit).It has to be multiplied by porosity (n) to get the seepage velocity ( v ) . The ratio of the latter to the hydraulic gradient (I,calculated from the difference in water levels and the seepage length) gives the vertical permeability (Fig. 3.1-17):

where

and

(3.1-26)

+

1, 12 . veff= t

344

3 Permeability of natural layers b

7 impervious layer Fig. 3.1-17. Sketch represent,ingthe flow condition around a aheet-pile applied for the determination of the coefficient of enisotropy

The inaccuracy of the methods wring tracers is increased by t w o possible errors i.e. the diffusion and adsorption of the tracer. Diffusion conveys the tracer in the aquifer even without ground-water flow, and this phenomenon modifies the time of propagation. To decrease its effect small concentrations should be used. In this case, however, the tracers would be absorbed by the grains of the layer along a short stretch of the flow, and hence the process would hinder observations. It is more advisable, therefore, to use corrective factors (determined theoretically or experimentally) to avoid errors caused by diffusion. Finally, a further and relatively reliable method to determine the coeficient of anisotropy, is the execution of l a h a t o r y flow rnm8urernen.k on undisturbed wre-samples in two diaerent directions. One poMible method of measurement generally applied is to cut a cube from the core and meamre the hydraulic conductivity of this sample in different directions parallel to the edges of the cube. This method, however, can be applied only in the case of cohesive sediments. To ensure the undisturbed character of the bample i t is advisable that the sample should be placed together with the core tube into the apparatus. A method has been developed in the Netherlands to execute this measurement directly on the core (Wit, 1962). The core tube including the sample is first placed into the permeabimeter in the usual way and the parameters of flow through the sample parallel to its axis provide the vertical permeability. I n the following step the two ends of the tube are sealed and two holes with a diameter of a few mm are drilled along the opposite walls of the core-tube (Fig. 3.1-18). After creating a pressure head difference between the two holes flow develops in the plane perpendicular to the axis of the core. The horizontal permeability of the layer can be cal-

3.1 Behaviour of in-situ layers

345

ffownef wben deferming veHicaf permeabihfy Fig. 3. 1-18. Measuring the anisotropy coefficient on an undisturbed aample

culated from the meaaured parameters of this flow by considering the flow pattern shown in the figure. The homogeneous and transverse anisotropy is the only type of irregularity occurring in the seepage field, the consideration of which does not need the application of numerical solutions, but the hydraulic parameters of the flow can be determined by using relatively simple analytical methods. Because of the fine layered structure of the sediments, the mathematical models providing solutions in these special cases, are widely applied. Investigating an example of two-dimensional seepage with a vertical flow plane, it can be seen that the flownet has a considerable deformation in these seepage fields, compared to the flow pattern developing in an isotropic medium having the same boundary conditions (Samsioe, 1931; KovBcs, 1966). I n a homogeneous and isotropic layer an orthogonal net composed of curvi-linear squares is formed by the system of stream and potential lines. The flowmet is elongated i n the direction of higher permeability in the case of an anieotropic porous medium, and hence the squares are substituted by oblongs. When anisotropy is homogeneous, the rate of this lengthening is the same at every point of the seepage field. Hence, a regular quadratic flow pattern can he created by transforming the original net. In most cases, when the two main directions are the horizontal and the vertical and the horizontal permeability is higher, the transformation is a simple vertical enlargement of the flou: field. The condition from which the necessary size of the vertical enlargement can be determined, expresses the identity of the flow discharge in the original anisotropic system and in the new isotropic field determined by this transformation. The components of the flow (or those of the flux) have to be identical in the two systems in both vertical and horizontal directions, naturally assuming the same boundary conditions around both fields. To meet this requirement the original seepage field should be enlarged in the vertical direction by multiplying and the hydraulic conductivity of the isotropic

vn

346 new system should be conductivity:

3 Permeability of natural layers

times smaller than the natural horizontal hydraulic (3.1-27)

as will be proved in the following paragraphs. The area of the cross section normal to the flow - in the cam of two-dimensional seepage investigating a space of unit width - is proportional t o the horizontal length when the flow is vertical and similarly in the case of horizontal seepage, the cross section can be characterized by a vertical length. Considering this relationship the flow rate in the two main directions can be calculated for both the original and the transformed systems (Fig. 3.1-19):

I n the m e of vertical seepage In the original system

Ah qv= A , K v -.

(3.1-28)

1,

In the transformed isotropic system

misotropic system

4' K'1

I

' &am

t K ,.

tube aftpr mRpRing periioui &r

impervious leyer Fig. 3.1-19. Interrelation between the parameters of the orighal &tropic equivalent htropio flow fields

and the

347

3.1 Behaviour of in-situ layers

I n the case of horizontal seepage I n the original system Ah

(3.1-30)

q H = A v KH -; IH

In the transformed isotropic system

The identity of Eqs (3.1-28); (3.1-29) and Eqs (3.1-30); (3.1-31) verifies the accuracy of the transformation chosen. Another way of proving the applicability of this model is the comparison of the pressure head diflerences needed to overcome the resistance of two flow tubes - one in the vertical and the other in the horizontal direction - having the same area of cross section (AH = A,= A ) and the same length ( I H = lv= I ) . The ratio of pressure head differences must be inversely proportional to the similar quotient of the hydraulic conductivity in both the original and the transformed systems. This ratio in the anisotropic system is as follows: -; lv

Ahv=therefore

AH

and A h H = - q

-,I H .

(3.1-32)

KH

KV

\

---=A. -

KH

KV

The same ratio calculated for the isotropic model can be given as follows:

and

therefore (3.1-33) Isotropic

Anisotropic

Fig. 3.1-20. Comparison of flownets in isotropic and anisotropic seepage fields

348

3 Permeability of natural layers

As shown by these comparisons both flow discharges and pressure head differences are proportional to each other in the investigated seepage field. Hence, the transformed system can be regarded aa the isotropic model of the original anisotropic flow space. All the hydrodynamic parameters can be determined mathematically in the transformed model and retransformed to the natural system. An example representing the determination of the flownet of an anisotropic seepage field is shown in Fig. 3.1-20.

References to Chapter 3.1 BABUSHKIN, V. D. (1964): Determination of Anisotropic Permeability by Pumping Test (in Russian). Razvedka i ochrana nedr., No. 6. BEAR, J. (1972): Dynamics of Fluids in Porous Media. Elsevier, New York, London, Amsterdam. Y IRMAY, S. (1968): Physical Principles of Water Percolation BEAR,J., Z A ~ L ~ V S Kand and Seepage. UNESCO, Paris. BIOT, M. A. (1966): Theory of Elagtioity andConsolidation for Porous Anisotropic Soil. Journal of Applied Phy&, p. 182. CEARNYI,J. A. (1961): The Proof of the Correctness of Dupuit’s Formula in the Case of Unconfined Seepage (in Russian). Dokl. A W . Nauk SSSR, No. 6. $ L I ~ S , E. and J d s z , J. (1966): Determination of the Anisotropy Coefficient (in Hungarian). Symposium on Seepage and Well Hydradim, Budapeat, 1966. FERRANDON, J. (1948): Mechanics of Permeable Layers (in French). La Horille Blanche, No. 9. GREEN,W. H. and A m , C. A. (1911): Studies on Soil Physics; 1. Flow of Air and Water through Soils. J o u d of Agricultural Science, No. 4. GRIS-, S. A. (1966): Permeability Coefficient as an Important Factor of Groundwater Flow (in Russian). Symposium on Seepage and Well Hydradim, Budapeat, 1966. p. 17. IRMAY, S. (1961): Darcy’s Law for Non-isotropic Soils. ]ASH Qeneral Assembly, Bruseels. JACOB, C. E. (1940): On the Flow of Water in Elastic Artesian Aquifer. Tramaction of AQU, No. 2. JAEGER, CH. (1949): Technical Hydraulics (in German). Birkhauser, Basel. J u d s z , J. (1968): Seepage under Structures in Pervious Layer of Great Thickness (in Hungarian). HidroMgiui KOzBny, No. 8. KEAFAOHI,A. (1961): Permeability of Layers in Natural Conditions (in German). Schweizerkche Zeitechrijt fiir Vermesaunqswmen und Kdurtechnik. KovAcs, G. (1966): Seepage Problems Arising in Connection with Construction of a . Barrage on the Danube (in Hungarian). VIZITERV Tanulmhyok, Buda KOVACS,G. (1966): Yield of Partially Penetrating Wells. Symposium on E&ge and WeU Hydradice, Budapest, 1966. KovAc~,G . (1968): Seepage to Groundwater Created by Hydraulic Structures. A d a T e c h n b Academiae ScientiamLm Hungaricoe. Tom. 60, No. 3-4. Lbozma~vy, S. (1966): Water Exploitation, Water Supply from Bore-holes (in Hungarian). Mffizaki Kiinyvkiad6, Budapest. LOVASS,L. (1962): Two New Methods t o Determine the Permeability of Soils (in Hungerian). H i d d d g i a i KOzBny, NO. 11-12. M~K~RIT~FIEvA, E. A. (1966): Determination of the Porosity of Layers in Field Conditions with Large Basins. (in Russian). Moecow. NBMETH,E. (1942): Water Problems of Modern Agriculture (in Hungarian). MBrnoki Tovtibbk6pz6 IntAzet, Budapest. OLCIENDORF, F. (1960): The World of Vectors (in German) Wien. SAMSIO~~, A. F. (1931): Influence of Tube-wells on Ground-water Flow (in German). Zeitechrift f i r Angewandte Mathematik und Mechanik, No. 11.

3.2 Motion of grains in cohesionless soils

349

S-4 L. (1964): Influence of Sideway Seepage on the Nornatives of Surface Irrigation (in Hungarian). Hidroldgiai KO&y, No. 7-8. S ~ Q Y IG., (1954): Yield of Wells Caloulated with Variable Coeffioient of Perneability (in Hungarian). Viziigyi KOilCmchyek, No. 3. TERZAQHI, K. (1943): Theoretiad Soil Mechanics. John Wile New York, London. THEIS,C. V. (1935): The Relation between the Lowering of tz; Piezometric Surface and t h e Rate and Duration of Discharge of a Well Using Ground-water Storage. Transaction of AQU, Vol. 16, p. 619. on of Methods Serving to Determine the Water Yielding UBELL,K. (1964): Corn Capacity of Aquifers (in ungarian). Vhtigyi K6&mdnyek, No. 2. UBELL,K. (1968): Practical Applioation of Theoretioal,Well Hydraulics (in Hungarian). Viziigyi K6demknyek 3. VISSER,W. C., RIJTEB~A, P. E., WIT, I(. E., STAKMAN, W. P. and STOL,PII. TH. (1962): Conception and Methods in Relation to the Desoription of Soil structure by P h y s i d y Signifioant Parameters. Ine. Vow. Culturtechnik en WuterhuiehouJding, Wugeningen, 1962. WIT, K. E. (1962): An Apparatus for Coring Undisturbed Samples in Deep Boreholes. Soil Science, No. 2. ZOUR, J. (1963):Approximative Determination of the Yield of W e b According to Pomhet's Method. (in Hungarian). H i d ~ ~ l d + iK6&y, NO. 1-2.

f;I"

Chapter 3.2 The motion of grains in cohesionless loose clastic sediments Apart from the hydrostatic pressure prevailing in the pores of a saturated layer, the water moving through the interstices creates an additional dynamic force on the grains of the loose clastic sediments. The vector of this force is parallel to the direction of flow and hence to the vector of the hydraulic gradient. Its magnitude on a unit area being perpendicular to the flow, is calculated as the product of the specific weight of water and the hydraulic gradient. When this force is greater than the forces holding a grain within the soil, the grain starts to move along a path determined by the internal structure of the layer. The process commences with the movement of the finest grains. These very small grains can move in the interior of the layer, where the local velocity surpasses a given limit. Concerning further movement, there are two possibilities. The grain is either &topped after a certain distance, or it is washed out of the layer. In the first case the overall solid volume of the layer is not altered by the movement of the grains. There is only a structural redistribution, which causes a decrease in both porosity and permeability of the soil in the zone of accumulation and an increase of the same parameters in the zone of removal. This structural change causes the flow to concentrate in the zones of increased permeability, this process strengthens the action of the percolating water on the grains. In caaes where the small particles moving along a flow path can leave the layer, the total solid volume of the latter is decreased, accompanied by an increase in porosity and permeebility. In such cases, zones with a permeability higher than the original may develop as a result of flow concentration. If the layer is composed of a skeleton of coarse grains which are capable of carrying the load of the overlying layers at their contact points, and the

350

3 Permeability of natural layers

percolating water conveys the fine ones through the pores of the coarse fraction without causing any change in the total volume of the solid nirttrix, the movement of grains alters only the flow conditions in the layer. When the hydrodynamic force, however, is high enough to remove even coarse grains from the skeleton, the process can influence the stability of the layer as well. Arching may balance the lack of a few coarse grains for a while, but with increasing removal, the subsidence of the layer can be expected. A special example of this process is known EMthe boiling or pipiiig of the soil, where the movement of coarse grains occurs near the surface. The result is the development of a channel with high permeability within the layer, which has an opening at the surface, where the movement of grains similar to a boiling action can be observed. The liquidization of the layer is &o a characteristic failure condition of the grains near the surface. In this c u e the forces caused by the flow exceed the weight of the soil particles over a large area and these begin to float. This process is indicated on a horizontal terrain by heaving and by the complete loss of the load-bearing capacity because the soil has no stability left. On sloping surfaces, the downward flow of grains becomes apparent. Another example of the motion of grains may occur primarily in layered systems. The second layer in the direction of flow may stop all grains washed out of the first causing the development of an impervious boundary layer and stopping the entire process of particle movement. Again the percolating water may be capable of carrying the h e grains through the pores of the second layer, but the coarse grains forming the skeleton of the first are unable to penetrate those pores. The second layer may then be regarded as a protective filter of the first. Finally, the case where the coarse grains are not retarded by the second layer, is called internal liquidization. Because of the manifold effects, the very complicated process explained previously was classified in the literature according to different aspects (e.g. Izbas, 1933; Botchkov, 1936; Tstomina, 1957). Taking all the expected changes in the hydrodynamic parameters, as well as in the stability of the layer, into consideration tho following grouping is proposed: (a) Suflusion (the motion of h e particles) Redistribution of fine grains within the layer (internal suffusion), when the solid volume of the layer is not changed only the local permeability is altered; Scouring of fine grains (external suffusion) when the volume of the solid matrix is reduced, accompanied by an increase in permeability, but the stability of the skeleton composed of the coarse grains is unaffected;

(b) Destruction of the skeleton Subsidence of the layer, when some of the coarse grains are removed from the solid matrix, and thus the load of the overlying layers causes the total volume of the layer to decrease; Piping or boiling effect when the considerrcble movement of grains along a flowline creates a channel of high permeability within the layer and the solid particles are observed to boil at the exit of this channel;

3.2 Motion of grains in cohesionless soils

351

(c)Liquidimtion of the layer On horizontal surfaces the loss of load-bearing capacity due to a hydrodynamic uplift being greater than the weight of the grains, which start floating; Soil flow on slopes is a similar lack of equilibrium along sloping surfaces. Before going into further detail, it is necessary to emphasize that these investigations are limited to the case of cohesionless samples. In cohesive layers the process is fundamentally influenced by the aggregation of soil particles. I n the latter case distinction must be made between primary porosity (inner porosity of the aggregates) (KBzdi, 1969a). The percolating water may remove thin lamellae from the aggregates and these groups of particles may move through the secondary pores. The process is, therefore, much more complicated, influenced by the development of aggregates and secondary porosity, as well as by the adhesive forces between soil particles.

3.2.1 The motion of fine grains (suffusion) There are two basic problems to be investigated concerning the motion of h e grains. First of all, it is necessary to know whether the structure of the layer makes any movement possible at all, or the size of the pores is smaller than the smallest particle. This limitation may be regarded &t) the geometrical condition of nzovement. If the structure of the solid matrix does not exclude the possibility of suffusion, the hydraulic conditions must be studied which determine the critical velocity or hydraulic gradient, above which the fine grains start to move. It seem very easy to determine whether suffusion can occur in the layer or not. This can be expressed in a simple form stating whether the pore diameter ( d ) is smaller, than the smallest grain diameter (Dmln) or not. In the case of no movement: d < Dmln * (3.2-1) Recalling the discussions regarding the development of a physical model with the pores composed of straight cupillary tubes of varying diameter (see Section 1.2.5) which is hydraulically equivalent to the network of interstices of the loose clastic sediments, it can be stated that: not only the average diameter of the capillary tubes [do;Eq. (1.2-19)] but also their probable large and narrow diameters [d, and d,; Eq. (1.2-22)] can be calculated from the known physical soil parameters of the layer (Dheffective diameter of the grains, a their shape coefficient and n , the porosity of the sample). As also mentioned, the comparison of these computed parameters with the results of Stakman’s (1966) air-bubbling meargurements show great similarity. Hence, d , and d, can be regarded as not only the characteristic diameters of the capillary tubes modelling the channels composed of the pores, but the probable sizes of the large and narrow pores aa well. As also proved by the statistical analyses of the pore-size distribution of several samples of loose clastic sediments, the probability of having a snmller pore diameter than d , is

352

3 Permeability of natural layers

about 30%, while the probable rate of the number of pores smaller in diameter than d , related to the total number of pores, is 80% (Chapter 1.3). In Eq. 3.2-1 the variable pore-size has to be taken into account by a single figure only. Substituting the smallest expectedpore-size Id,) ,the inequality signifies that the particles finer than this may move, as long aa they do not reach a narrow pore with diameter d,. Introducing the hrgmt pore (d2) into Eq. (3.2-1) no movement is expected to take place. Since, during their movement the fine grains can form arches in the pores, this second approximation is too strict. In practice, therefore, the use of the smallest diameter (d,) is generally accepted, or where a higher degree of safety is required because of other uncertainties (e.g. inhomogeneity) the substitution of the average pore diameter (do)can be proposed:

or

(3.2-2)

According to this equation the possible occurence of suffusion in a layer can be estimated from an inequality involving the ratio of the smallest p r tide to the eflective particle diameter. I n Patrashev’s formula (1938) even the numerical constant agrees well with Eq. (3.2-2), if the average shape-coef0.30) is introficient (a = 10) and the most probable porosity (n = 0.25 du ced . Lubochkov (1965)has investigated separately the eflect of porosity on the limit characterizing the geometrical condition of suffusion. He found that the ratio of the pore diameters belonging to porosities n = 0.4 and n = = 0.3 is about 1.6, which is practically equal to the result calculated from Eq. (3.2-2). The condition given in Eq. (3.2-2) can also be expressed by claiming that the ratio of two grain diameters belonging to two given percentage values on the grain-size distribution curve, should be smaller than the number depending 012 porosity. The form of this ratio is similar to Hasen’s uniformity coefficient (U = D6dDl0).This fact is presumably the basis of the classifications in which the geometrical condition of suflzcsion is given in terms of the uniformity coeficient (Istomina, 1957):

-

There is no suffusion if Transition condition Suffusion is liable if

u < 10; 10 < u < 20; u220.

(3.2-3)

In laboratory experiments with sandy gravels having a continuous grainsize distribution curve, no grain-movement occurred if the uniformity coefficient was smaller than 8 or 10. In the transition zone very high hydraulic gradients produced suffusion, while in the range of U > 25, the fine grains started to move readily under the influence of relatively small gradients (Cistin, 1965).

3.2 Motion of grains in cohesionless soils

353

The same basic idea is reflected by the proposals for the design of filters, according to which, to ensure the stability of the flter, the uniformity coefficient of the filter material (Uf) should be lower than a given value:

Uf< 6 (Moscow-Volgostroj construction in 1935); U f < 16 20 (Creager et d.,1945); Uf< 20 ( U . S . Army Corps of Engineers, 1955).

-

(3.2-4)

The results of Lubotchkov's investigations (1965) are in slight contradiction to the previous statements. He has found that the possibility of movement depends to a great extent on the shape of the grain-size distribution curve, and hence not all materials having uniformity coefficients higher than 20 are liable to suffusion. He has pointed out at the same time that the condition satisfying Eq. (3.2-1) does not completely exclude the possibility of movement. If the layer is composed of the skeleton of coarse grains and the pores are fUed with fine particles, the latter can move through the voids of the skeleton, although the smallest grain is larger than the pore diameter calculated from the distribution curve of the mixed sample of coarse and h e grains. For this reason, Lubotchkov has proposed the investigation of the shape of the distribution curve and has prepared standard curves as a second limitation of the geometrical condition (Fig. 3.2-1). On the basis of the same investigation, Lubochkov has elaborated an analytic method as well. Its theoretical basis is the hypothesis that the Zuyer

am

0.1

4 relative diameter 47l5X

Fig. 3.2-1. Matching curves of relmtive grain-size distribution cheracterizingthe upper mnd lower boundmry of samples not sueceptible to suff'usion (mfter Lubochkov, 1962) 23

354

3 Permeability of natural layers

is not susceptible to su&&m when the slope of the distribution curve is eqml to, or smaller than a given limit in each diameter-interval. A simplified mathematical form of this condition can be written as follows (Fig. 3.2-2):

AS,IA& < l ; if Dn-1 - Dn 4.0 D, D"+l

AsJAs, < I ; 2.6

Fig. 3.2-2

if%=--

DrI

- 10

(tolerance safety factor 1.0);

Dn - 5 (tolerance safety factor 1.5);

Dn+1

of grains

355

3.2 Motion of grains in cohesionless soils

. ASJ AS1 1.7

Dn< 1; if -Dn

D

1

= 2.5 (tolerance safety factor 2.3).

Dn+l

(3.2-5) Here and in Fig. 3.2-2, the symbols used are as follows: D, is an arbitrary diameter on the distribution curve; Dn-l and Dn+l can be determined from D, by multiplying or dividing it by 10; 5 or 2.5 according to the safety desired; AS1 and AS, are differences between the percentage in weight be0.05 01 az

0.5

D77L7J

Fig. 3.2-3. Application of KBzdi’s method to determine the self-filtering ability of a sample 23.

356

3 Permeability of natural layers

longing to the particular diameters:

AS1 = S,-I - S,; AS, = S , - Sn+l.

(3.2-6)

In his method, K6zdi (1969a) hus investigated the whole distribution i n a similar way. After dividing the sample at an arbitrary point of the distribution curve (D,) into two parts, the coarse skeleton can be ragarded as the filter of the remaining fine particles (Fig. 3.2-3). Applying Terzaghi’s filter law (according to which DidDS, < 4 < D{S/DS, where the symbols f and s refer to the filter and the soil respectively, while the subscript indicates the percentage by weight of particles of a certain diameter, it is possible to determine, whether the h e particles are able to pass through the pores of the skeleton. The 15 percentage by weight of the skeleton (Si;)and also the 15 and 85 percentages of the h e particles (Si5and S’&) can be expressed as a function of prcentage belonging to the selected D, diameter on the original distribution curve (S,): sF5= 0.858, 0.15; Sf5= 0.158,; (3.2-7) si5= 0.858,. Thus, the material forms a self filtering system (the geometrical condition excludes the possibility of movement), if the following inequality is valid at every point of the distribution curve:

+

‘D(OSSS) > D(OsSn+0.15) > 4 D t ~ . 1 5 S n ) *

(3.2-8)

It should be noted here that the lower grain size limit of the filter in Terzaghi’s equation, ensures the highest possible permeability, and the condition of stability i s expressed by the upper limit only. All three members of the inequality can be represented in the form of a graph as a function of the D, diameter chosen arbitrarily [i.e. the

4D(0.s5Sn)= 4@5 VE4*D n ; the D ( o . ~ S , , + o * U= ) DL VS. Dn and the = 4 0 4 , ) vs. D , curves respectively].

4D(0.1sS,)

-

The last curve gives the requiredlower limit of the representative diameter of the skeleton ( D k ) ,but this limit has no practical importance in this investigation as already mentioned. The critical particle size (Dcr)can be indicated where the upper limit (40$ vs. D, curve) intersects the Di; vs. D , curve. The grains being smaller than this size (in the range where D;, vs. D, curve runs above the upper limit) are able to move through the large pores of the skeleton. If the two curves have no point of intersection and the Di; VS. D, curve is everywhere below the upper limit, no suffusion is expected. There is a further limitation to be investigated. Uding the same division of the sample as before, the porosity of the remaining skeleton (n.) can also be determined depending on the original porosity ( n ) : V V - V,(1 - 8,) nu= =n S, = n S,(l - n ) ; (3.2-9)

V

+

V

+

3.2 Motion of grains in cohesionless soils

357

where V is the total volume of the sample, V , is the volume of the solid particles and two simplifying msumptions are made. Firstly, the specific weight of the grains does not vary with grain size, and secondly the skeleton is uniformly distributed in the total volume of the layer after the fine particles are removed. As in Eq. (3.2-7), the diameters of the skeleton belonging to the 10 and 60 percentages by weight can be determined and their ratio yields the uniformity coefficient of the remaining coarse grains:

because and

+ 0.1; Sio = 0.45, + 0.6. &,

(3.2-10)

= 0.9s,

According to a previous investigation, the highest porosity (that'belonging to the loose condition of the sample) is a function of the uniformity coefficient and the shape coefficient, and it is independent of the diameter, as long as the latter is greater than 0.2 mm. Assuming an average shape coefficient of a = 10, the highest possible porosity of the skeleton is:

nhax= 0.3 + 0.15 exp

[

-

~

(3.2-11)

Comparing Eqs (3.2-9) and (3.2-11), it is possible to determine whether the grains of the remaining skeleton contact each other after removing particles smaller than the chosen D, diameter, or the new porosity (nu)is greater than that belonging to the loosest possible condition (n;,,). I n the latter case, part of the weight of the overlying layer is carried by the fine grains and the forces at their contact points also act against suffusion. Considering the geometrical condition of grain movement, the upper limit of unhindered suffusion has to be determined first, calculating the diameter Do below which the expected porosity [Eq. (3.2-9)] is lower than the possible maximum [Eq. (3.2-ll)] (Fig. 3 . 2 4 ) . The possibility of suffusion can be investigated only along the lower part of the distribution curve, where

D,

< Do.

(3.2-12)

Any critical value (calculated on the basis of methods given by Lulmtchkov, KBzdi, or by the method to be subsequently presented) above this limit, does not imply high susceptibility to suffusion, because the forces at the points of contact of the fine grains hinder their movement and, therefore, a relatively high gradient is necessary to initiate movement. It is worthwhile to mention here, however, that in this case a high velocity will cause the subsidence of the layer, displacing the fine grains, relative to each other, and resulting in a loss of stability. 'IB To ensure the uniformity of the mathematical treatment, the self-filtering bility of the layer can also be expressed using the model system of cupillary

3523

3 Permeability of natural layers

0.05 0.1 0.2

a5

I

2

5

lo

0.7

0.6 0.5 0.4

q 6 20 el; fa

u'

.-Y* @

10

0

005 0l L?Z 05 1.0 20 50 M O 20.0 D,,cmm_7 Fig. 3.2-4. Consideration of the maximum possible porosity in the determination of the self-filteringcapacity of a sample

tubes aa in the derivation of Eq. (3.2-2). Dividing the sample at a given diof the skeleton can also be calculated. ameter D,, the effective diameter (0;) Considering now the movement of a group of fine grains it is not necessary to compare the smallest particle to the pore diameter because the development of arching effect may be expected. According to the literature on filter design, the diameter belonging t'o 85% of the moving mms measured by weight appears to be a representative parameter (Cedergren, 1968). For safety, it may be compared to the average diameter of the pipes of the

3.2 Motion of grain8 in cohesionless soih

359

capillary tube model (see Eq. (3.2-2)]. Substituting n = 0.3 and a = 10 values as average parameters, the following relationship is obtained

n D;

4.0 -- < @s

l-n

a

:

where 4.0 n - 0.17 ; a l-n

(3.2-13)

thus

-

Di: < 5.0 D(0SesS").

Accepting the D,/D;, 1.25 ratio on the baciis of the analysis of the distribution curve (see Fig. 1.2-6) it is easy to see that this theory corresponds well with Terzaghi's and KBzdi's methods, not only formally, but also numerically. From the above, it is concluded that the first step in investigations concerning cluffusion is to check whether the geometrical conditions permit movement in the layer or not. Three conditions must be considered in this invest,igation: (a)The smallest grain is capable of movement through the pores of the layer [Eq. (3.2-2)]; (b) A group of fine grains below a given limit (D,) can move through the pores of the remaining skeleton [Lubochkov, Eq. (3.2-6); KBzdi, Eq. (3.2-8); or using the capillary tube model, Eq. (3.2-13)]; (c) Whether or not aftar removing fine grains smaller than the critical diameter determined according to the second condition (Dcr= D,) the etabili t y of the skeleton is e m r d [for comparison see Eqs (3.2-9) and (3.2-11).

Tf the second condition indicates the possibility of movement, but D,,

>

> D , according to the third condition, the subsidence of the layer can be

expected, rather than the development of suffusion. After checking the geometrical possibility of suffusion in the layer, it must be determined, whether or not the expected seepage-velocity is high enough to move the fine grains. The purpose of this investigation is the determination of the hydraulic condition of the motion of fine grains. The emieet way to express this hydraulic condition of movement is to establish the balance between the acting forces. In the case of a grain at reat on a horizontal surface, this Condition can be calculated by considering the weight of the grain (a)less the upholding force (af=a-U)and the drag force created by the flowing water (8):

af = V(y,- y o ) ;

s = Ivy,;

(3.2-14)

where V is the volume of the grain; yr and yo are the specific weights of the grain and the water respectively; and I is the hydraulic conductivity maintaining the movement of the water.

360

3 Permeability of natural layers

In the critical condition the angle formed by the resultant vector of the two forces with a horizontal direction, is equal to the angle of friction (KCzdi, 1969a): - = tan&, = tan@ = ___. I c r ~a (3.2-1 5 ) at Yt- Yo Accepting the validity of this relationship aa the first rough eatimation for characterizing the movement of small grains through the pores of a layer, and assuming the validity of Darcy’s law, and knowing the permeability of the layer, the critical value of both the seepage velocity and the effective velocity can be calculated

s

Yt - Y v ., vU,-ct= K tan @ -

and

K tan @ Yt - Y o . =n Yv

(3.2-16)

It is quite evident, that the velocity calculated in this way is considerably smaller than the actual critical value, because in the layer the grain is compelled to move in a complicated conduit including rising sections and not on a plain surface. The only consequence which can be drawn from Eq. (3.2-16), is that the critical seepage velocity is a function of permeability and friction, assuming the specific weights of both the grain and the water are constant. Another simplified model is the investigation of a flow directed u,pwards i n a vertical pipe (Cistin, 1966).The balanced condition is expressed in this case by the equality of the settling velocity ( w ) and the effective flow velocity. It is well known, however, that the velocity of settling is decreased by the wall of the container, if the diameter of the latter is small. The collision of grains with each other and with the wall has the same effect. Cistin has estimated that the critical effective velocity is smaller than half the settling velocity because both negative effects in the pores are of considerable magnitude: veffer 0 . 5 ~ and ; vsWcr Q.5nw. (3.2-17)

<

Hence, tho critical seepage velocity may depend on porosity and grain diameter, the settling velocity being a function of the latter. A further conclusion of this analysis may be that the critical velocity is a function of the flow directions aa well, because i t is clearly indicatedby the basically different results of the two simplified models, that other forces have to be considered when investigating the horizontal movement of a grain, compared to those when analyzing its floating condition. Other experiments and the analysis of the forces acting on the particles contacting the screen of a well (vertical exit face), have shown that the crib ical seepage velocity is proportional to porosity (n) as well as to the square root of an expression formed from the specific weights of the grains and water

3.2 Motion of graim in cohesionless soils

361

( y Sand ye), and the effective diameter (Dh)(Schmieder et al., 1975). Because t.he grain size can be related to the square root of hydraulic conductivity, the critical velocity can also be given aa the function of K and the other variables are generally included into one constant in which the combined physical parameters of the soil are considered with their average values: (3.2-1 8)

where hydraulic conductivity is substituted in m sec-l. This theoretical investigation provides the basis of the most simple relationships which are generally used in practice and which give the allowable limit (it being smaller than the critical value) of the seepage velocity entering a well. The limit velocity (vIlm) is expres8ed i n these equations as the function of hydraulic conductivity:

vr

vIIm= -= 6 . 6 10+VK ~ ; (K in m sec-l); (Sichardt, 1928);

15

3-

vIIm= 63 1/K ; (K in m day-l); or

(3.2-19) 3

vIlm= 3.2 x

; (K in m sec-l); (Abramov, 1952);: 4

vIlm = 5 x 10-2

VZ;

(K in m sec-l) (Schmieder, 1966).

hydrauhc cDnductiyify, K cm/secJ Fig. 3.2-6. Comparison of the entry velocity into wells and the relationships proposed to calculate its allowable higher limit (after Schmieder, 1966)

362

3 Permeability of natural layers

In Fig. 3.2-5 the three equations are compared with data from wells operated over a longer period of time, without observing the silting of the screen. The points determined by the hydraulic conductivity of the layer and the calculated entry velocity into the well, cover the shaded area. The lines determined by Sichardt’s and Abramov’s equations cross this area indicating that there are numerous w e b operating with higher velocity than the limit given by these equations. For this reaeon, Schmieder has proposed the use of the third equation, which can be achieved by dividing the theoretical Eq. (3.2-18) by a safety factor of 4. The line calculated from this relationship, gives the upper tangent of the shaded area.

3.2.2 The liquidization of the layer The movement of the fine particles through the pores formed by the coarse grains has been discussed. As demonstrated, the first condition of such movement depends on the ratio of the pore and particle size, or considering the interrelation of the probable pore diameters and the physical parameters of the Roil this geometrical condition can also be expressed as the function of the characteristics generally applied in the investigation of seepage (effective grain diameter, grain-size distribution, porosity). If movement is geometrically possible, there is a second criterion for the development of the motion of fine particles: i.e. the actual velocity of water in the pores has to be higher than a critical limit value. If the velocity is considerably higher than the limit mentioned, the hydrodynumical pressure exerted by the flowing water on the surface of the grains composing the solid matrix may surpass the resistance of the coarse particles. The resistance hindering the movement of particles is the result of several forces: i.e. the component of the weight of the grain parallel to the main direction of flow but directed in the opposite direction, the friction between the contact points of the particles which depends also on the weight of the overlying layer, and the weight of other grains to be pushed away from the direction of the moving particle. This action is negligible if the grain in question is on the outside of the seepage field. When the velocity of seepage reaches this second critical value, the development of which deteriorates the stability of the layer, the process of liquidization (or quickening) of the sediment starts by washing several grains from the surface. It is followed by disturbance of the solid skeleton. The subsidence of the layer can be observed and the boiling of the particles starts at several points, if the surface is nearly horizontal. On sloping surfaces the slow, downward creep of the grains is a sign of the development of the critical condition. These phenomena indicate only a transition state in most cases, which is suddenly followed by the complete deterioration of the layer, when i t loses its loading capacity and the whole mass of grains achieve a floating condition. There are no unambiguous criteria to characterize the occurrence of the boiling or piping effect, because i t depends mainly on the local seepage velocity. Hence, apart from the form of the seepage field, which generally de-

3.2 Motion of grains in cohesionless soils

363

termines the local velocity in the sense of continuum approach, the small inhomogeneities within the layer influence also the development of piping, locally creating great velocity high enough to move the grains on the surface. Many laboratory tests to determine some criteria of boiling have been recorded. For example, in the soil mechanics laboratory in Delft, the water pressure necessary to create piping in a dune sand (average diameter 0.18 mm; porosity 0.37; hydraulic conductivity 1 2x m sec-l) waa

-

, 4

I 1 -

J ?

120 cm ~~~~~

Fig. 3.2-6. Section of the seepage field of an experiment used to determine the critical condition of boiling

Fig. 3.2-7. Investigation of stability of a sloping exit face under the influence of hydrodynamic force

measured in the seepage field, the vertical section of which is represented in Fig. 3.2-6. It wa.s found, that initial boiling started when the pres~urehead waa 14 cm. At 20 cm head, the cotinuous movement of the sand waa observed and reaching 25 cm the boiling became severe. These results and those of other similar experiments cannot provide a theoretical basis for the investigation of the boiling process. The discussion will be continued, therefore, with the analysis of the development of liquidization as the final stage of the change of the solid matrix of the porous medium. Considering a soil column of unit area and having a thickness of As perpendicular to the surface, the balance of the acting forces gives the limit beyond which the loss of stability can be expected. Let us investigate, as a general case, a column attached to a sloping surface, which forms an angle of with the horizontal (Fig. 3.2-7). The weight of the column reduced by

364

3 Permeability of natural layers

the uplift force (a')and its components parallel (Gi) and perpendicular (a;) to the slope can be expressed as follows:

G' = (1 - n) A z ( y t - y.) ; therefore and

G; = 0' sin B ;

(3.2-20)

G,t, = G' cos B.

The force created by the percolating water on the same column (hydrodynamic force) acts in the direction of stream t!ines crossing the column. The direction of flow (and that of the stream lines) forms an angle of a with the horizontal. Thus the forces P and its components parallel ( P t )and perpendicular (P,) to the slope are:

P=(l-n)Idzy,; therefore

P,= P cos (B - a) ;

and

(3.2-2 1 )

P, = P s i n ( B - a ) .

When reaching the critical condition of stability, all the forces acting in, the direction of the slope are in equilibrium. The components of the weight and the hydrodynamic force parallel to the slope can be taken from Eqs (3.2-20) and (3.2-21) respectively. Apart from these the friction of the grains along the lower boundary of the column has to be considered, which hinders the development of the movement. The value of the friction can be calculated aa the product of the resultant of the normal components of G' and P' and the coefficient of friction, which can be expressed as the tangent of the angle of friction. Thus the final form of the balanced condition is as follows:

G;

+ P , = t a n @ (GA - P,); B

+I

(B - a) = = t a n @ [ ( y t - yo) COB j3 - I y,sin ( B - a ) ] . (rt - y o ) sin

Y a cos

(3.2-22)

Naturally, the hydraulic gradient can be replaced by the seepage velocity in Eq. (3.2-22), to obtain the same relationship as a function of seepage velocity: cos ( B - a ) = t a n @ [ ( y , - yo) CO8 B v

- - yo sin ( B -a)]. K

(3.2-23)

Equations (3.2-22) and (3.2-23) give the critical value of hydraulic gradient and seepage velocity respectively, above which liquidization is liable to occur in the caae of a sloping surface without any load upon it. The same relationship is found in dmost all publications dealing with this problem of soil mechanics (Bernatzik, 1940; Taylor, 1948).

3.2 Motion of grains in cohesionless soils

365

From this general form, special equations can be derived to characterize simplified cases when the angles of the slope and the main flow direction have special given values. By substituting the relevant parameters e.g. in the caae of a horizontal surface ( B = 0 ) , where the stream lines are vertical (perpendicular to the surface and, therefore, B-a = n/2), Eqs (3.2-22) and (3.2-23) can be reduced to Yt - Y v . [Icrlp 0 = Yo and Yt - Y v . (3.2-24) [vc,lp-o=K-, Ya

-

9

which corresponds well with the equations derived to describe this special condition (Terzaghi, 1943; Domjhn, 1950; Lampl, 1959). Another possible simplification is to assume, that the stream lines are perpendicular to the surface at th,e critiml point. It will be proved when discuseing the kinematics of seepage (Chapter 4.1), that the stream lines are perpendicular to the exit face, where the seepage field is contacted by surface water because this stretch of the exit face is a potential surface. It can also be demonstrated by hodograph mapping (see Section 4.2.3), that there exists a free exit surface (and a capillary exit face) above the upper level of the surface water. Here the direction of the stream lines changes from perpendicular to parallel to the slope, (normal at the elevation of the level of the tail water and tangent at the upper boundary of the seepage field which may be either the phreatic surface or the upper surface of the capillary fringe). The solution can be completed by expressing the required tan @ value from Eq. (3.2-22), and determining the pair of corresponding I and (/?-a) values at which the maximum required angle of friction occurs (Kovhcs, 1970). The extreme value problem can be solved either mathematically or graphically. The second method is shown in Fig. 3.2-8 (KBzdi, 1969b). A vertical ( ~ ~ - 7vector ~ ) starts from an arbitrarily chosen point on the slope (P). Its lower end is the centre of a circle with aradius of I ya. Any position of the radius is also regarded aa a vector. Adding the two vectors, namely the vertical one and that radius of the circle which is parallel to the stream-line intersecting the starting point P, the sum of the two vectors is proportional to the resultant of the acting forces (weight, uplift, and hydrodynamical force). Since the tangent to the angle of friction required to ensu,re stabiljty can be calculated aa the ratio of the forces parallel and perpendicular to the slope, respectively,

and the determination of the coefficient of proportionality is not necessary. The tan @ value in question can be directly calculated aa the cotangent of

366

3 Permeability of natural layers

the angle of 6 between the slope and the resultant constructed in Fig. 3.2-8: tan @ = cotan 6 . (3.2-26) The possible zone along the circle at the end of the resultant is limited to a quarter of the circle indicated by a thicker line in the figure, because the possible magnitude of the angle formed by the slope and the stream line in question, remains within the zone determined by the following inequality: 0

< /I - a

nl2.

(3.2-2 7 )

Fig. 3.2-8. Vector-diagram to determine the Stability of a sloping exit face (after KBzdi 1969b)

The critical direction of flow is indicated by the radius from the point of P,because the smallest 6 gives the largest required t a n @ value. The size of the radius being ly,, the minimum value of the 6 angle is a function of the hydraulic gradient. When the circle is tangent to the slope, there exists a limiting value of the gradient above which seepage causes liquidimtwn on the slope if the stream-lines are perpendicular to the slope. In this cme Fig. 3.2-8 gives infinite tan @ aa necessary for stability. The magnitude of this critical gradient is

, intersection of the circle and i b tangent drawn from point

- - = COB /I Yt - Ya .,

[4r3ta"o

(3.2-28)

Ya

which is the generalized form of Eq. (3.2-24) for sloping surfaces and which can also be obtained by substituting tan @ = 00 and sin ( p - a) = 1 in Eq.(3.2-22). As already mentioned, the gradient increases downwards from the exit point and reaches a loml maximum at the intersection of the exit surface with the level of the surface water, where the gradient theoretically tends to infinity. On the other hand, other investigations have shown that its value remains finite everywhere, because of the development of non-laminar seepage (see Section 5.3.1). It is also necessary to consider that the stability of the particle is not determined by the point value of the velocity or the gradient, but the average value over a distance equal to the size of the grain has to he taken

3.2 Motion of grains in cohesionless soils

367

into account. Hence, the assumption of infinite gradient at a single point is not acceptable in practice. The variation of the gradient can be represented by a series of circles instead of the one circle in Fig. 3.2-8. Describing the continuous variation of the gradient by a step-by-step increase using finite differences, the location of the end-points of the resultant of the vectors can be indicated as shown in Fig. 3.2-9. It is evident from the figure that in the

Fig. 3.2-9. Modified vector-diagram to consider the changing value of exit gradient along the slope

presence of tail water, the maximum of the required tan @ value should be very near the value calculated, assuming that the slope and the stream line intersect each other at right angles. - a) = x/2;and thereAccepting the approximation mentioned above, [(/I fore cos ( B - a) = 0 sin ( B - a) = 11 the critica.1 gradient and seepage velocity can be calculated as a function of the internal friction of the layer : and the angle of slope or

Yo

(3.2-29)

This simplified form of Eqs (3.2-22)and (3.2-23) can be used to characterize the stability of a sloping exit face of the Seepage field if it is partly covered by .surface water. It, is quite evident that substituting t a n @ = w into the last equation, the upper limit of the gradient is determined [Eq. (3.2-28)], above whish a slope with nii angle of B cannot be stable. Similarly, the aub= 0; tan B = 0: cos B == stitution of the condition of a horizontal surface (/I

368

3 Permeability of natural layers

= 1)leads back to Eq. (3.2-24),indicating that the angle of friction of the material does not influence the stability in this case. If the whole sloping exit face is free (i.e. i t is not in contact with surfacewater) and the seepage field is underlain by a horizontal impervious boundary, the largest angle between the exit face and the stream lines can be observed at the toe of the slope. Here, the lowest stream line lies along the contour of the seepage field and, therefore, i t is horizontal (a = 0; j3 - a = j3). The

I

I

Fig. 3.2-10. Velocity distribution along an exit face above an impervious horizontal boundary

practical example of thiq caae is an earth dam on a horizontal impervious foundation (Fig. 3.2-10). The detailed investigation shows that the maximum hydraulic gradient also occurs at the intersection of the exit face and 2.5, the impervious boundary, if the former is vertical anditsvalue is I,,, independent of the geometrical parameters of the field (see Section 5.3.1). In the case of a sloping exit face, the toe point is theoretically a stagnation point with zero gradient and the maximum exit velocity can be observed at a higher level, where the stream-line forms a positive angle with the horizontal [(j3 - a ) < 83. It increases the safety factor, however, if the maximum possible gradient is combined with the smallest angle of the stream lines. Substituting the a = 0 value into Eqs (3.2-22)and (3.2-23),the critical gradient and exit velocity on the slope of an earth d a m having a horizontal foundation and not contacted by tail water, can be calculated:

-

[Icr11-0 =

yt - yo t a n @ - tan j3 ~

yo

or [Vcrla

-

0

=R

1

+ t a n @ tan B ’

yt - yo tan @ ~

yo

.

1

- tan j3

+ t a n @ tan j3

(3.2-30)

3.2 Motion of grains in cohesionless soils

369

Finally, this equation can be further simplified by assuming the investigation of a vertical exit face ( p = n / 2 ; tan p = 00). I n this case the supposition applied previously (i.e. the combination of the highest gradient and the lowest horizontal stream line) is not an approximation, but a correct expression of the actual condition, if the exit face penetrates the lower horizontal impervious boundary. Substituting this condition, the critical values can be calculated from the following equations:

[ crI 8]-

n=---

~-

2

.

Yf - Y a

1

y,,

tan@ ’ (3.2-31)

or

The negative sign in the equations indicates that the stability of a vertical face can only be ensured if the water infiltrates through th section and hence the hydrodynamic force increases the stability, not decreases it. A vertical exit face of a seepage field composed of loose clastic sediments, can be stable only if i t is protected by external forces (e.g. by the screen in the case of wells). The necessity of having a structure to protect the exit face against liquidization and to ensure its stability leads to the investigation of flters which Till be discussed in the next section.

3.2.3 Design of protective filters and well screens The conclusion drawn from the investigation of liquidization, is that there exists a group of parameters, corresponding to the materid of the seepage field, (angle of friction @), and the hydraulic characteristics of seepage (the critical combination of the exit gradient I and the a angle of the stream-line to the horizontal direction at the point where the gradient is chosen) and the slope of the exit face (its angle with the horizontal being p), which determine the stability condition of the layer. These parameters cannot be arbitrarily chosen, and they have to remain below the linlit given in the previous section [Eqs (3.2-22 and (3.2-23)] or in special cases they have to satisfy the conditions determined in the simplified forms of the basic relationship. The material of the seepage field is generally determined by natural conditions. The hydraulic conditions can be modified between given limits by applying special structures (larger foundation, sheet-piles, etc.), but naturally this modification increases the cost of the whole hydraulic structure. The application of a flatter slope has similar kancial implications and liquidization can occur even on horizontal surfaces. There are even cams when neither the slope can be chosen arbitrarily nor the hydraulic conditions modified (e.g. the vertical exit face of a well). A possible method to ensure the stability of the layer is to cover the exit face with coarse material (protective filter), which allows the water to be drained, but retains the fine grains and which has itself the required stabil24

370

3 Permeability of natural layers

ity against the expected exit velocity. According to this short dehition the ater must be deeigned to meet three criteria: (a) The stability of the filter requires the filter itself to be self-filtering, i.ei its fine particles should not move through the pores of the coarse fraction. Hence, this condition determines whether or not a given material can be used in protective filters. (b) The geometrical condition of filters expresses the requirement that the small grains of the protected layer should not be washed through the filter. It gives, therefore, a relationship between the grain size of the layer and the pore size of the filter. (c) The hydraulic condition includes those requirements which ensure the high permeability of the filter and its structural resistance against percolating water. There is no need to discuss the first condition in detail. It is only necessary to aacertain, whether the material used CM a protective filter i s self-filtering or not, a~ discussed in Section 3.2.1. Ae mentioned there, the condition can be expressed where the ratio of two characteristic diameters of the grain-size distribution curve are smaller than a given limit. The analogy between this ratio and the coefficient of uniformity U (at least their similar structure) might be the basis of the f a t that in the literature: this condition is generally expressed by the upper limit of the uniformity coefficient [Eq. (3.2-4)]. It is advisable to use a relatively uniform material (Uf< lo), but this condition is not sufficient to ensure the internal stability of the filter. It ie necessary to exclude the possible movement of the smallest particle [Eq. (3.2-2)], and that of the group of h e grains [Eq. (3.2-13)], as well aa the subsidence of the filtering layer [Eqs (3.2-9) and (3.2-ll)]. Most of the proposals (so called filter-laws) wmbine the geometrical and hydraulic conditions,although those published more than 50 years ago considered the geometrical condition separately. These early publications chose one chartracteristic diameter of the grain-size distribution curves of both the filtering material and the protected layer, and expressed the geometrical condition aa the limit of these two diameters:

g* D

4.42

Of,"' 5 5

(Prim, 1919); (Terzaghi, 1922).

(3.2-32)

%em

Prinz did not define precisely, which characteristic diameter should be used, while Tenaghi applied the mean values. I n his later works, which have already been quoted, Tenaghi (1922,1943)haa developed further principles and proposed the combination of the two conditions: (3.2-33)

371

3.2 Motion of grains in cohesionless soils

[Here in Eqs (3.2-32) and (3.2-33) the symbol f indicates the material of the filter and s that of the protected soil. The same symbols will be used further on.] The left hand side of this double inequality expresses the geometrical condition, signifying that the snmEl grains of the protected layer cannot penetrate into the filter, if the ratio between the diameter of the filtering material at 15% by weight and the diameter at 850/, on the distribution curve of the soil, is smaller than 4. According to the hydraulic condition on the right hand side of the same inequality, the characteristic filter diameter (15 yo)should be four times aa large aa the same parameter of the soil. The permeability, being proportional to the second power of the characteristic diameter, increases considerably (about 16 times) a t t,he boundary of the two layers. Further investigations related to the geometrical condition necessitated some negligible changes in the numerical constant (the use of a value between 4 and 5 instead of 4) (Bertram, 1940; beager et al., 1945). Others proved the reliability of the method experimentally (Cedergren, 1968), or theoretically (Istomina, 1957). Although the theoretical investigations have shown that the ratio I)$/@! ia not an absolute constant, i t depends to a small extent, on the grain-sue distribution of both materids, the geometrical condition, aa expressed in Terzaghi's filter law, can be accepted without any restriction in practice. on the I n this form the geometrical condition gives only one point grain-size distribution curve of the atering material. The stability condition applied in the form of the up r limit of the uniformity coefficient, provides an additional parameter (Up" ). I n practice, it is advisable to supplement Terzughi'e criterion and determine the required limits of the coefficient of uniformity:

(a)

2
(3.2-34)

In this way a range of the materials acceptable for filtering the layer in question can be determined as indicated in Fin. 3.2-11. YO

100

ran e of fl'lterin materia/ according +o Bureau of&/amatm -

IS

40

i)

30 20

QOO/ QO! at lo tomm Fig. 3.2-11. Application of filter-laws to determine the required range of the grainsize distribution C U N ~ of the filtering materiel 24*

372

3 Permeability of natural layers

The same theory is the basis of the proposal made by the US Bureau of Reclamation when they fix the upper and lower limits of two points of the grain-size distribution curve of the filtering material. Accordmg to this condition, the ratios of D{dD~5and D$JDS,.should remain within a given range. The limits can even be modified in t h s very practical proposal according to the degree of safety required. A basic value for open drains €or example is,

DL < 4 0 . 12 < -

(3.2-35)

Dt It is evident from Fig. 3.2-11, where the result of this proposal is also indicated that there is no considerable difference between the appropriate materials determined by using the two different methods. There is one danger, however, if Eq. (3.2-35) is used without k i n g the upper limit of the uniformity coefficient of the filter. The grain-size distribution curve of the layer is shifted and lies almost parallel, to limit the zone of the distribution of the materials applicable aa protective Nters. Even the upper limits of D&D, and DfdD& can be equal to one another because the limits can be modified, and in this caae the two curves are completely parallel. At the same time most of the layers requiring protection are not self-filtering materials, a fact which is indicated by their high uniformity coefficient. Eq. (3.2-35) gives acceptable results only if it ia combined with the condition of the stability of the filter and at least the upper limit of Uf is fixed. It is quite evident that the uncertainties concerning the determination of the filtering material can be reduced, if more information is available on the required structure of the filter (Lubochkov, 1955). It is reasonable therefore, to further supplement Terzaghi’s filter law. By using the capillary tube model, the application of which waa explained in connection with Eq. (3.2-13), the geometrical condition of the filter can be given in the following form : 4.0 --

1-n

(3.2-36) a

Because of its simplicity the original Terzaghi’s equation is most suitable for the design of the filter. This formula can be proposed for checking t h e

material selected, taking into consideration the porosity of the filter, the shape of the grains and also the whole distribution curve of the filtering material, using its effective diameter instead of D&. Finally, the interpretation of the hydraulic condition needs more detailed investigation because these principles have been neglected by many authors. As already mentioned, the upper limit of Terzaghi’s law takes into account only one aspect of the hydraulic process occurring in the mter, when the percolating water carrying some fine particles passes through it. The upper limit of Eq. (3.2-33) ensures the desired permeability of the filter, which can be obtained by using a material having a suitable characteristic diameter

3.2 Motion of grains in cohesionless soils

373

(I)&,). There are, however, more important hydraulic conditions to be coiisidered when designing filters. Some authors proposed basing this condition on the requirement that the very fine grains washed out of the soil should pass the filter without clogging. Lubochkov (1955) has pointed out, that this requirement cannot be accepted aa a design criterion, because the pores of the filter are larger than those in the protected layer. Since the seepage velocity (as well as the effective velocity if the porosity of the filter does not differ from that of the layer), is the same because of continuity, the filter is not liable to clogging. It has to be ensured, however, that the critical velocity for the filter (or for the topmost layer of multi-layered filters) should be lower, with the required safety factors, than the velocity expected i n the exit section (COMECON, 1969). This hydraulic condition excludes the possibility of liquidization of the filter. Inaamuch a8 the first condition, called the stability conditon of the filter, ensures that no suffusion will occur in the filter itself, its combination with this interpretation of the hydraulic condition will guarantee the total stabilit’y of the filtering material. A filter designed according to the upper limit of Terzaghi’s law will have high permeability but in this condition there is no information about the safety of the filter against liquidization. It may happen, therefore, that a filter has sufficient permeability to protect the fine l~articlesof the layer and its small grains are not expected to move, but the exit velocity is still greater than that allowed for the material of the filter and liquidization occurs on the surface of the filter. This special hydraulic requirement emphasizes the necessity for the maximum exit seepage velocity (urnax)to be smaller than a limiting value critical for the filtering material ( v i r ) : vmax

<

(3.2-37)

If this criterion is compared to the upper limit of Terzaghi’s law, it can be stated that Eq. (3.2-37) shows how the hydraulic conductivity has to be increased between the surface of the protected layer and that of the filter. Considering continuity, the exit velocity does not change considerably, while the critical value is proportional to hydraulic conductivity; see Eqs (3.2-24), (3.2-20) and (3.2-30). Equation (3.2-33) determines the required change of the hydraulic conductivity at the contact surface of the layer and the filter (or a t the contact of two layers of the filter if the latter is composed of more lamellae with increasing permeability). Jn this way the combination of ihe two conditions enable the designer to determine whether the filter can be formed of one material only, or if the application of a multi-layered structure is required. This uncertainty in connection with Eq. (3.2-33) could be the reason why many research workers accepting the lower limit of Terzaghi’s law, tried to give some different interpretation for the upper limit. The most important results of these investigations are those which give the allowable exit velocity aa a function of both the protecting and the filtering material (Izbar, 1933; Kozlova, 1934). Taking into consideration the results of investigations related to the liquidization of a layer as previously discussed, the critical exit velocity can

374

3 Permeability of natural layers

be determined by using Eq. (3.2-23). In general caaes, this critical value is the function of the slope of the exit surface and the angle of the streamlines at the critical point. It depends also on the permeability and the angle of friction of the filter. If the geometrical and flow parameters of the seepage field make i t possible, the simplified forms of this equation can be used [for horizontal surfaces Eq. (3.2-24) for cases the stream-lines are normal to the surface Eq. (3.2-29); and at the exit face of a dam on horizontal foundation Eq. (3.2-30)]. Summarizing the above, the hydraulic condition of filter design can be expressed in the following form: f

Vcr -2

B;

(3.2-38)

Urnax

indicating. that the ratio of the critical velocity of the filter to the expected exit velocity (which can be calculated aa a hydraulic parameter of the seepage field investigated) should be greater than a given safety c o e p i e n t ( B ) . Assuming that. the angle of friction of the filter is equal to that of the layer and the surface of the filter is parallel to its plane of contact with the protected material, information on the required hydraulic conductivity of the uppermost layer of the filter can be obtained from Eqs (3.2-23) and (3.2-38) (3.2-39)

Once the permeability is accepted as proportional to the second power of the effective diameter, a relation can be derived between the eflective grain size of both the protected soil and the upper layer of the filter from Eq. (3.2-39): (3.240)

Here, some further very rough approximations were necessarily applied, assuming that the coefficient of proportionality between the permeability and the second power of the effective diameter (which is actually a function of the porosity and the shape coefficient) is the same for the soil and the filter. Although many important variables were neglected when deriving Eqs (3.2-39) and ( 3 . 2 4 0 ) , the latter is suitable for the rapid calcuhtion of the grain size required a t the upper surface of the filter. This value can be subsequently checked using the correct relationships considering the actual values of friction, shape and porosity in the soil aa well as in the filter. If Eq. ( 3 . 2 4 0 )indicates a filter considerably coarser than that determined by the upper limit of Terzaghi’s law, the application of a multi-layered filter becomes necessary. The hydraulic condition, aa established by Terzaghi [the upper limit in Eq. (3.2-33)] should be regarded only aa the limit of the change in grain diameter at the interface of two layers. Thus Eq. (3.2-40) or its more exact form, gives the grain size in the most coarse upper layer of the filter, while Terzaghi’s law tells the designer how many layers should

3.2 Motion of grains in cohesionless soils

375

be applid i n the filter between the protected soil and the surface of the filter. If the layer to be protected has no large surface, but the aquifer is drained along a line, a solid structure (perforated drain tubes, well screens) can be applied instead of protective flters. There are many c w s when filtering material has to be placed around that stable structure of the flter, to bridge the gap between the size of the holes of the tube and the particle size of the very fine-grained layer. The required ratio of a special grain diameter of the material contacting the perforated tube and the size of the draining 8.Army Corps of Engiholes can also be given in the form of flter-laws (U. neers, 1955). In the case of elongated slits, if the width of the slit is b: 0 85

> 1.2 ;

b

In the case of circular holes of diameter d: 0 8 5 -

d

>1.0.

(3.2-41)

Hence, in both cases. the diameter belonging to the 85% ordinate of the grain-size distribution curve of the material contacting the tube, has to be considered as the design value. The same guide-line can be followed if the perforated tube is covered by a screen, or if a solid screening skeleton having regular square-shaped openings is constructed in some other way (Johnson well screen). In this cme DS hm to be related to the length of the side of the square and the ratio should be larger than unity. Investigating a well screen i t may also be considered that the normal forces acting on the particles increase with the weight of the overlying layer, and hence, the friction obstructing the movement of the grains is higher at a greater depth (Kesserfi, 1967,1968, 1970). As already shown, the vertical exit face of a seepage field cannot be stable without the support of external forces [Eq. (3.2-31)]. In the presence of a protecting structure, the stability of a grain sinaller than the opening of the screen has to be studied, instead of that of an elementary prism, as in the derivation of Eq. (3.2-22). The symbols applied now are summarized in Fig. 3.2-12. It can be supposed, that in the internal part of the layer the stresses acting on the sides of an elementary prism are equal in a horizontal direction ( a , = as), while the vertical stress is different (al).I n staticequilibriumthe stresses prevailing on the opposite sides of the prism are equal to each other, because the difference caused by the weight of the material within the prism is negligible when a small elementary volume is investigated. If the same prism is in contact with the well screen at one side, the supporting effect of the screen substitutes the continuity of the solid matrix in this direction. It can be assumed, however, that a new static condition develops after having disturbed the structure of the layer by drilling the bore hole and then the pressure of the screen on the prism is equal to the original stress (a’ = = a&.

376

3 Permeability of natural layers

la,

Fig. 3.2-12. Investigation of the stability of the grains protected by well screens

I n connection with the determination of the acting forces, i t is necessary to mention that in natural conditions the total weight of the overlying layers is balanced by the efiective and the neutral slresses [Eq. (1.4-19)] and their ratio depends on the rate of consolidation. In the present caae, however, a drained layer has to be investigated in the close vicinity of the well, and hence, the assumption of the total development of consolidaiion is acceptable. The effective strem can be calculated, therefore, 88 the difference between the total weight of the overlying layers and the static water pressure: and where

(2

I:

I=tm - - - ;

(3.2-42)

where h, is the depth of the investigated point below the surface and JL, the depth of the water table. 3, is the approximative value of the so-called active earth pressure coefficient. Tf there is no coneiderable difference between h, and ho Eq. (3.2-42) can be simplified 01

= 4%- Y o ) ;

and a, = u3 = Aa1;

(3.2-43)

3.2 Motion of grains in cohesionless soils

where h

-h,

377

h,

Investigating the balance of a particle, the following forces have to be considered: (a) Its weight decreased as a result of upward forces (GI); (1)) The horizontal friction originating from the weight of the grain (S,,); (c) The hydrodynamic force ( P ) ; (d) The friction (8,)caused by the vertical stresses acting 88 normal forces ( N , ) on both the upper and the lower side of the particle; (e) The friction (S,) caused by horizontal stresses perpendicular to the direction of movement and acting similarly on two sides of the grain (AT2), (t,he stress u3 from the third horizontal direction is inactive, the particle being separated by the movement from the whole of the layer and pressure does not act at this side of the particle). The hydrodynamic force is just balanced by the friction prevailing on the various sides of the particle at the beginning of the movement:

p where

= 8,

+ -+ 8 1

S o = G' tan @; and GI=

82;

n

- D3(y f - yo);

6 S, = 2N, tan@ = 2 tan@aD2h(yf- yo);

S, = 2N2 t a n @ = 2 tan@aD2Ah(yf- yo); and (3.2-44)

In the determination of N , and N , normal forces the a multiplying factor expresses the uncertainty in the method of transfering the weight of the overlying layers. If the solid matrix were composed of uniformly packed spheres arranged in a hexagonal system, this factor would be equal to unity. If the weight carried by one sphere is assumed to be proportional to its horizontal n area, a = - . Since some arching effects can also occur, this factor may be 4

even smaller, and because the shape of the grains differs generally from the sphere, the a value can surpass unity. Considering these possibilities the a factor should be included in the calculation of the normal forces:

and (3.2-45)

Substituting all the relationships derived in this way into the balance equation, the critical grudient or the critical exit velocity can be calculated

378

3 Permeability of natural layers

as a limiting value, above which the grains smaJler than the diameter of the openings of the screen will move from the layer into the well:

Icr= =tan@ Y

8a 1 + --h(1 + A ) 6 n D

4

or

vcr=~-tan~ YO

-4 + - -8a h (1l + ~ ) 6

nD

1 1. ;

(3.2-46)

References to Chapter 3.2 ABRAMOV, C. K. (1962): Methods of Seleotion and Caloulation of Filtem of Boreholm (in Ruseirm). Goageoltekhizdat, Mosoow. BERNATZIX, W. (1940): Limit Values of Sand Slopes in Case of Simultaneous Gmundwater Flow (in German). Die Bat&chmik, No. 66. BERNATZIK, W. (1947):Foundation and Physios (in German). Sohweizer Druok- und Verlaghaua, Ziirioh. BER-, Q. E. (1940): An Experimental Inveati ation of Protmtive Filters. Publication of of the Qraduate School of Engineering, Ifarward Universdy, No. 267. BOTOEKOV, N. M. (1936): Mechanioal Suffusion of the Soil(in Russian). ONTI, Moscow. CEDERQREN, H. R. (1968):Seepage, Drainage and Flow Nets. John Wiley, New York, London, Sidney. ~WQAJEV, R. R. (1936): Approximative Calculation of Stability of Bodiea of Earth Darns (in Russian). Izvatia VNIIU, No. 18. CISTIN, J. (1966 : Problems of Deformation of some Mechanio Filters (in Czeoh). V o d o m r 8 k a haeopk, No. 2. CISTIN, J. (1966):A Contribution to the Problem of Inner Suffusion of Non-ooheaive Layers (in Hungarian). Leoture in the Hungarian Hydrologioal Sooiety, Budapest, (Manuscri t). COMEJON (1969):Reeearah Re rt concerningSeepage around Hydraulic etruoturea and Proteotive Actions against &&age. (Manuscript), B u d a p t . J. D. and HINDS,J. (1946): Engineering for Dams. John C ~ U A Q EW. R , P., JUSTIN, Wiley, New York, London. DOMJLN,J. (1960): Meohboa of Sand Boil Formation (in Hungarian). HidroZdgiai K & Z h y ,No. 6-6. ISTOMINA, V. S. (1967): Filtration Stability of Soils (in Russian). Oostroizdat, Mmoow, Leningrad. Ims, K. J. (1960a): Rational Deaign of Filters. Proceeding8 of Inetituthn of C i d Eq$neer8, 1960. IVES,K. J. (1960b):Filtration through a Porous Se tum: a TheoretioalConsideration of Boncher’s Law. Proceeding of InetitUtiOn of ~&?En&e8r8, 1960. IZSAS,S. V. (1933): Deformation of So* Caused by Permlation (in Russian). Izveutia VNIIff, No. 10. KESSERB, as. (1967): New Aspeots and Methods of the Inveatigation of Critical See age Velooity (in Hungarian Hidroldgiai Rod ,No. 10. & S S E R ~ , Zs. (1968): On the veetigation of the quilibrium of Slopes in the Case of Peroolating Water. (in Hungarian). B h y h z a t i Kutat6 Inthzet, B u d a p t , No. 1. KESSER~, Zs. (1970): Equilibrium of Saturated Loose Clastio Sediments in the Surrounding of Well Screens (inHungarian).-VI. Bdnya&v&lmi Konjerencia, Budap a t , 1970. K ~ ~ JA. I ,(1969a): Inoreese of Proteotive Capwity of Flood Control Dikes (in epartment of Geoteohniquea, Budapeat, Teohnioal University, Report

c)

. 1,

3

References

379

~ Z D A. I , (196913): Comments to the Paper “Effects of Ground-water Flow on the Stability of Slo es of Loose Clastic Materials” by Dr. G. Kovhs (in Hungarian). Banyciazati ds &hciezati Lapok, Banyciezat, No. 6. KOVACS, G. (1968a): See age to Ground-water, Created by Hydraulic Struotures. Acta Technica Academiae &ien8iarum H u n g a r k , Vol. 60, No. 3 4 . K O V ~ CG. S , (1968b): Effwts of Ground-water Flow on the Stability of Slopes of Loose Clastic Materials (in Hungarian). Banyciezati ds Kohciezati Lapok, Bhyciezat, No. 10. KOVLCS, G. (196813): Characterization of the Molecular Forces Muencing Seepage with the Help of the p F Curve. Agrokdmia 4.s Talajtan, (Supplementum). KovLcs, G. (1968d): Characterization of the Grain Shape in Studies Concerning Seepage (in Hungarian). gpit4.s- 4.s Kodekedhtudornanyi K6;zle&yek, No. 1-2. Koviics, G. (1969): Relationship between Velocity. of Seepage and Hydraulic Gradient in the Zone of High Velocity. 13th Congreas of IAHR, Kyoto, 1969. KovLcs, G. (1970): Response to the Comments Made by Prof. Dr. A. KBzdi (in Hungarian). Banyciezati ds Kohciezati Lapok, Bhnydazat, No. 6. KOZLOVA, L. J. (1934): Experimental Investigations on the Deformation of Two Adjacent Layers, Caused by Seepage (in Russian). Izvestkz VNIIG, No. 14. LAMPL,H. (1969): Sand Boil Formation and the Liquidization of Soils (in Hungarian). V’iziigyi R o h d n y e k No. 1. LECLERO, E. and BEUSJEAN,P. (1966): The Seepage of the Water from Rivers and the Index of Seepage Capacity (in French). Cdedeau. L I P T ~F., (1966): Laboratory Investigation on Clogging (in Hungarian). Conference on Seepage and Well Hydraulics, Bwhpest, 1966. LWOOHKOV, E. A. (1966): Design of Protective Filters of Hydraulic Structures (in Russian). Candidate Thesis Leningrad. LWOCHKOV, E. A. (1962): The-Self-filtering Behaviour of Non-Cohesive S o b (in Russian). Izvestka VNIIG, No. 71. LWOCHKOV, E. A. (1965): Graphical and Analitical Methods for the Determination of the Properties of Non-cohesive Soils Characterizing Suffusion (in Russian). Izveatia VNIIG, No. 78. PATRASEIEV,A. N. (1936): Confined Ground-water Flow in Layers Composed of Fine Sand and Clay Particles (in Russian). Izvestia VNIIG, No. 16-16. PATRASHEV, A. N. (1938): Confined Ground-water Flow Causing the Movement of Fine Particles (in Russian). Izveatia VNIIG, No. 22. PATRASHEV, A. N. (1967): Methodology of Selection of Grain-size Distribution of Protective Filters (in Russian). Sb. T d o v Lengiprochtranaa. PRINZ,E. (1919): Handbook of Hydrogeology (in German). Springer, Berlin. SCHMIEDER, A. (1966): Problems Concerning the Critical and Allowable Velocity around Wells (in Hungarian). HidroZdgiai Kodony, No. 10. SCHMTEDER, A., KESSER~~, Zs., Jwdsz, J., WILLEMS,T. and MARTOS, F. (1976): Water-risk and Water Management in Mines (in Hungarian). MGszaki Konyvkiad6, Budapest. SICHARDT, W. (1928): Filtering Ability of Tube Well (in German). Springer, Berlin. SICHARDT, W. (1962): Gravel and Sand Filters in Soil and Hydraulic Engineering (in German). Die Bautechnik. STAKMAN, W. P. (1966): Determination of Pore Size by Airbubbling Pressure Method. Symposium on Water in Unaaturated Zone, Wageningen, 1966. TAYLOR, D. W. (1948): Fundamentals of Soil Mechanics. John Wiley, New York. TERWOHI, K. (1922): Soil Failure a t Barrages and its Prevention (in German). Die Waasevkraft. TERZAOHI,K. (1943): Theoretical Soil Mechanios, John Wiley, New York, London. U S . Army C q 8 of Engineers (1941): Investigation of Filter Requirements for Underdrains. Technical Memorandum, No. 183- 1. December. U.S . Army C o r p of Engineers (1966): Drainage and Erosion Control - Subsurface Drainage Facilities for Airfields. Engineering Manual, Military Construction, Washington.

380

3 Permeability of natural layers

Chapter 3.3 Investigation of clogging In the previous Chapter the motion of the fine particles of the solid matrix of a loose, clastic, porous medium waa investigated. As a result of the processes discussed the moving grains are transferred from one place to another within the layer or, are washed out of the layer. Thus, the total volume of the solid phase remains constant or decreases. If the percolating water curries a suspended load, the fine grains are deposited along the path of flow. This process increases the volume of the solid matrix and because the bulk volume of the layer is not altered at the same time, the porosity and intrinsic permeability is decreased. The process of settling of fine particles transported by the percolating water i n the original pores of the porous medium is called clogging (colmatation). Numerous research workers have dealt with the theoretical investigation of clogging, Ornatsky et al. (1955), Fenin (1953), Sehtmann (1961), Izblts (1933), Patraahev (1935) in the Soviet Union, Djankov (1961) in Bulgaria, V. Nagy (1965), Starasolszky (1966, 1967), Kov&cset al. (1973) in Hungary, endeavoured to give a theoretical solution of this phenomenon. A method of calculating the parameters describing the development of clogging, was derived by Ives (1960, 1961, 1963) in England, based on the filter theory. Boreli and Jovasevid (1961) in Yugoslavia, De Keustner (1964) and Frenette (1964) in France, have studied the theory of clogging as well. The theoretical description of the process of clogging is generally based on the analysis of the change in concentration of the transported, suspended load in place and time. The aim of the investigations is to define the quantity of fine materials in the layer settling i n a prism of unit volume. This quantity also depends on the position of the prism (distance measured from the entry face along a flow path) and on the elapsed time from the beginning of the process. It is necessary to note here that clogging is not a pure mechanical process. Particles smaller than the narrowest stretches may be retarded in the pores by the attraction between the grains and by electrostatic and electrochemical forces acting on their surfaces (Konanov, 1959). However, only the kinematics of the process will be discussed here, and not the dynamic analysis of the forces involved in the development of clogging. It is quite evident, that clogging a8ects the hydraulics parameters of the seepage field (porosity and hydraulic conductivity) and in this way i t reacts on the water movement, which induces and maintains the clogging process itself. There has therefore, to be a limiting condition, when the settling of the h e particles in the pores is terminated either because the seepage is stopped by the complete cloggingof thepores, or because the effective velocity in the pores is increased by the increase of the gradient and the decrease of porosity. Hence, the percolating water is able to transfer the suspended material further without depositing it. Considering the interaction between the two processes (seepage and clogging), the determination of the hydraulic behaviour of the field influenced by clogging is one of the final aims of the investigation, which is necessary not only for the prediction of the change of seepage but also for the proper description of the clogging process.

3.3 Investigation of clagging

381

Attempts can be made to utilize the methods characterizing the silting of the pores and those deriving the hydraulic parameters of the field, by Considering the size and structure of the pores. The amount of decrease i n a function of the hydraulic conductivity by clogging can be determined distance from the entry section, and time assuming a permeability changing monotonously with the distance measured from the entry face.

3.3.1 The change of concentration of the percolating water depending on time and place

Some parameters describing the complicated phenomenon of clogging can be expressed in the form of mathematical functions only by applying approximate assumptions to simplify the process. The most important among these hypotheses are: (a) The homogeneity of the layer at the beginning of the investigation; (b) The comtant grain-size distribution of the suspended load transported by the water; and (c) The constant concentration of the percolating suspension expreaaed either or its volume (Vs) related to the by the weight of the suspended load (Gs) volume of water ( Vv): B S

C A = - [FL-3]

V

= const.;

or C A= V S [L3 L-3 = 11 = conut.

v,

(3.3-1)

Since the grain-size distributions of both the solid matrix of the layer and the suspended sediment carried by the percolating water are statistically determined parameters, the effect of clogging and the change of hydraulic conductivity have also to be regarded a.a random events and they can be characterized only statistically. The basic relationships describing the process of clogging can be achieved by analyzing the rnass conservation of the transported fine particles following Ives’ (1960b) derivation. The investigation starts with the determination of the mws balance of the suspended load within an elementary prism of the solid matrix, the edges of which are dz,dy and d z (Fig. 3.3-1). The flow having a seepage velocity v(t) is parallel to the y axis of the coordinate system. Thus the concentration of the percolating suspension expressed in volumetric units, is the function of the y ordinate and the time only. The relationship has to satisfy special boundary and initial conditions:

c = C(Y’ t ) and

C = C,(t) if y = 0; 0 < t < 00 (boundary condition); 0; 0 < y < 00 (initial condition). C = 0 if t

(3.3-2)

382

3 Permeability of natural layers

Fig. 3.3-1. Derivation of the equation of continuity (mass conservation) for h e particles

The difference between the volume of the suspended load entering the prism during a time period of dt, and that leaving its exit section, has to be equal to the volume of the fine grains deposited within the prism during the same dt time period. If n,(y, t) indicates the volume of the fine particles silted i n the poreswithinaprim of unit volume around apoint being at adistance of y from the entry section from the beginning of the process until the time point t, the continuity equation for the suspended load expressing the maas conservation can be written in the following form: v(t)dxdz[C(y,t) - C(y

+ dy t)]dt = -[nc(y, t ) - n,(y, t + dt)]d x dy'dz;

(3.3-3) dC dn, v(t) - -= 0. dy dt It is necessary to draw attention to the difference between C, in Eq. (3.3-1) and C, in Eq.(3.3-2). One of the hypotheses generally applied in the literature, is that the concentration of the suspension contacting the surface of the layer (C,)should be constant. This aasumption is not a basic requirement in the derivation and as will be proved it does not correspond to the actual condition. Even if the concentration of the water entering is constant, one part of the particles may be larger than the maximum pores of the solid mutrix, and these grains will be deposited on the surface of the layer. Only those particles enter the pores whose geometrical condition o f D < dmax (3.34)

+

-

-

is satisfied, where D is the diameter of the suspended material and d,,, is the maximum pore diameter. Naturally, this criterion can be interpreted only statistically, because i t is a r a n d m event, whether a given grain is transported by the flow to an opening which is larger than its size or to a smaller one so that the particle enters the layer, or is stopped on the surface. It follows

383

3.3 Investigation of clagging

from this, that the concentration of the water infiltrating through the surface (C,) will be equal to or smaller than the original concentration (C, CA). Considering the clogging of the pores near the surface, and thus the decrease of the free openings, C, has to be regarded as a time-dependent variable, as indicated by Eq. (3.3-2) [C,(t)]. Here a basic distinction haa to be made between two types of clogging. There are systems, where the entire amount of water transporting the suspended load has to percolate through the layer (e.g. sand filters of water treat-

<

influence of clogging

Fig. 3.3-2. The infiuence of surface deposit on the flow rate of Beepage

ment plants). In other caaes the water currying the fine particles flows above the porous medium, and only some of the particles enter the layer, always depending on the actual hydraulic gradient and the resistivity of the solid matrix near the surface. The example of the second process is the clogging effect of rivers and canals recharging ground water. In the caae of a closed system (sand flter) the surface deposit always creates a membrane of high resistivity hindering both the seepage and the entrance of h e particles into the pores. This process causes the rapid decrease in the efficiency of filters. In river beds, the horizontally flowing water can, however, take away the sediments deposited on the surface and hence the development of the thin, very impervious membrane delays the process of clogging and hinders the infiltration only temporarily. In Fig. 3.3-2 the effect of the surface deposit is demonstrated on the baais of experimental data. In the system investigated in this caae, the water transporting the fine particles waa flowing over a sandy gravel layer1 of 1 m thickness, and a constant pressure difference was maintained between the upper andthe lower surface of the layer. It is almost impossible to consider theoretically, :the time-dependent influence of the membrane composed of the surface deposit. For this reaaon only the deep clogging will be analyzed in the following discuasion, aasuming that the surface deposit is continually taken away. The calculation on the basis of this hypothesis provides the quickest possible development of deep

384

3 Permeability of natural layers

clogging and the smallest hydraulic conductivity within the layer, but not the lowest flow rate of seepage. The second step generally applied in the derivation of the basic equations of clogging, is the study of the distribution of the amount of the deposited material along the path of the flow (in the system represented in Fig. 3.3-1 along the y axis). Since the silting of the pores is regarded as a random event, the n,(t, y ) function is usually approximated by a probability distribution curve, assuming that the form of distribution is independent of time. The n, function is, therefore, divided into two parts expressed as the product of two members, i.e. the first depends on time and the second is the distribution function describing the relationship between the deposited fine materials and the distance measured from the entry section. Most of the research workers use exponential distribution functions to approximate the second relationship and consider also the initial and boundary condition characterizing the process. Thus the general form of the equation describing clogging and giving the amount of fine particles deposited i n the pores as. the function of time and place is as follows:

where n,((t) is the specific amount of the suspended material deposited in a unit volume of the solid matrix in contact with the entry section. This parameter is time variant and expresses the dependence of the process on time. The initial and boundary conditions to be satisfied are as follows: if t = 0 (initial condition) n,(y, 0 ) = 0; nc(O,t ) = n,,(t); if y = 0; and n,(oo, t ) = 0 ;

if y = 00 (boundary conditions).

(3.3-6)

After having differentiated Eq. (3.3-5) according to y , i t can be combined with Eq. (3.3-3) (3.3-7)

It is assumed in this derivation that A is a constant parameter characterizing the clogging process and is independent of time. The complete solution of the problem requires, however, not only the determination of A , but also that of the two interrelated time-dependent functions 1i.e. w(t) and n c o ( t ) ]The . various methods proposed in the literature to describe clogging, generally differ in the assumptions applied at this point to simplify the relationship and to make i t mathematically amenable. Concerning the numerical value of the A parameter, assumed to be a constant characteristic value of the process depending only on t*hematerials of both the solid matrix and the suspension, some estimations can be made on the

385

3.3 Investigation of clagging

basis of various experiments. Boreli and Jovasevid (1961) have published the results of an experiment where the whole suspension percolated through the flter. The parameters of the sand used for this measurement were as follows: D,, = 0.84 mm; U = 1.54; nA = 0.44; K , = 0.7 cm sec-l (the grainsize distribution curves of both the solid matrix and the transported sediment are given in Fig. 3.3-6). A constant velocity of 0.0415 cm sec-l was maintained in the system and the concentration of the suspension was C, = 45 mg/l. The distribution of the fine particles deposited in the pores

0

I0

20

30

40

5R

distance from the entry fare CcnJ Fig. 3.3-3. Distribution of the deposited materiel along the path of flow (after Boreli and Jovasevi6,1961)

was given in the form of agraph (Fig. 3.3-3). Exponentialdistribution curves of different A values applied to the averages of the n ( y , t ) values, are also plotted in the figure. The value characterizing the curvea was found to be A = 0.03 - 0.04 cm-l. Starasolszky’s experiments (1967) were carried out in a system where the water transporting the fine particles flowed horizontally over the porous layer. There were five series of measuremenla using the same materials, the characteristic parameters of which are summarized in Fig. 3.3-4. The A value recalculated from the measured hydraulic conductivity of the clogged materid is A = 0.05 cm-l. For the further investigation of the interrelation between the time-

(-21

dependent variables one may assume, that the velocity of clogging i e

i s proportional to the concentration of the transported suspension (Mackrle and Mackrle, 1961): 25

386

3 Permeability of natural layers

grain diameter, D tmmJ Fig. 3.3-4. Parameters characterizing the materials applied in Starasolszky's experiments (1967)

(3.3-8)

An analytically amenable solution can be a 4 e v e d by this approximation if not only the constancy of the C,, concentration is assumed, but also that of the seepage velocity is ensured [ v ( t )= const.]. This condition can be satisfied, however, only in systems where the dioerence between the pressure heads prevailing at the two sides of the porous material is proportional to the resistunce of the solid matrix at each time point. Therefore, i t is continually increased as the hydraulic conductivity decreases as a result of clogging. Accepting the supposition v = v o = const. during the whole period of the process, Boreli and Jovasevi6 (1961) have proposed characterizing the proportionality between the concentration of the suspension and the rate of clogging by the following equation:

anC=

I

I 31.

A 1-exp - I , ; (3.3-9) at where A is the coeficient of proportionality; I,, is the maximum gradient within the sample, when further development of clogging is terminated, because the high seepage velocity is able to transfer all the fine particles through the pores; and K is the instantaneous hydraulic conductivity at the investigated point, also changing aa the result of clogging K K A = l

n A - n, nA

rn

1'

(3.3-10)

In this latter equation K , and nA are the constant hydraulic conductivity and porosity of the layer before the start of the clogging process.

3.3 Investigation of clagging

387

The assumption of the constant seepage velocity is accept,able only for filters with automatic operation, where the pressure head is continually increased to achieve constant flow rate. A more suitable approximation of the actual boundary conditions of the natural processes in which the development of clogging is investigated, is the supposition of a constant di#erence of the pressure heads between the two sides of the porous material. Although this assumption accurately simulates the real situation along rivers and canals for considerably long periods, i t has also to be considered that the fluctuation of the water level can cause a large discrepancy from the theoretically assumed conditions. The modification of the process becomes especially important, if the flow direction can be temporarily altered in the layer by the lowering of the water level in the river below the elevation of the water table. In this case, the water flowing back from the aquifer into the river can disturb the already settled fine particles, transporting them out of the pores. A method assuming the constancy of the pressure head and the change of the seepage velocity, has been developed by Sehtmann (1961). The basis of his derivation waa similarly the equation of continuity and the random character of the distribution of clogging. From these relationships, the expected reduction of hydraulic conductivity is estimated and t.he fhal result is summarized in an equation giving the time-dependent velocity v ( t ) as a function of the constant velocity ( v A ) characterizing the solid matrix (depending also on the prevailing pressure head) before the beginning of the clogging process

I

v ( t ) = v A -a exp -b - ; t:l where a , b and t o are empirical constants.

(3.3-11)

3.3.2 Application of the capillary tube model of the porous medium to characterize the clogging process The capillary tube model developed for the characterization of porous media in Section 1.2.5 at the same time provides the probable maximum and minimum diameter of the pores between the grains [Eq. (1.2-22)]. It also opens the way to determine the hydraulic conductivity of the seepage field depending on the physical parameters of the soil. This model can be combined with the basic equations of clogging. After having determined the decrease of both the porosity and the characteristic grain diameter a8 a result of ~ettlingof the fine particles into the solid matrix, the hydraulic w n ductivity, changing within the field, can he calculated and the inhomogeneity produced by clogging can be dcqcribed. Considering the maximum, probable pore diameter of the layer before the start of clogging, the maximum particle size entering the pores ( D o )can be determined numerically as the function of the physical parameter of the soil of the original solid matrix initial porosity and DhAinitial effective 26*

388

3 Permeability of natural layers

grain diameter). Assuming that the shape coefficient is taken into account by its probable average value (a= lo), the combination of Eqs (3.3-3) and (1.2-22) gives the following result: (3.3-12)

where dZAis the probable maximum pore diameter of the original layer. If the maximum diameter of the fine particles transported by the water (Dmax) is smaller than the upper limit calculated from Eq. (3.3-12) (Dmax< < Do)all the grains of the suspended sediment can enter the pores and the concentration of the percolating water at the entry section (C,) is equal to that of the suspension (CA).I n the other case, the particles larger than Do are deposited on the surface of the layer. It is supposed, however, that these particles are continually carried away by the horizontally flowing water, because only the investigation of deep clogging was previously mentioned as a basic assumption. Thus the concentration of water entering the solid matrix is smaller than that of the suspension moving above the layer, and it can be roughly approximated by using the following proportionality: (3.3-13)

where S(D,) is the percentage by weight of the particles in the suspension having a smaller diameter than the D o limiting value. By the separation of the transported fine particles at the surface of the layer into two groups (i.e. grains larger and smaller than Do),the grain-size distribution curve of the suspended load taking an active part in the clogging process differs from that determined by sampling the water above the layer. The particle distribution, interesting from the point of view of deep clogging, can be approximated by using the simple proportionality as shown in Fig. 3.3-5. The second part of the figure represents the expected grain-size distribution of the mixed material composed of the original skeleton and the h e particles deposited in the pores. The curves were constructed by assuming various rates of clogging (i.e. different amounts of the h e particles were added to the grains of the basic layer). The distribution curves determined by this very simple method can be accepted as the characteristics of the layer influenced by clogging at a given time point and at a determined place, if one supposes, that the grain-size distribution of the fine particles deposited at various points i n the field and at di8erent times is constant. As already mentioned, this parameter haa a random character. More precise wording of this hypothesis indicates, therefore, that the distribution vazies around a mean value, which is independent of time and place. Boreli’s and Jovasevid’s experiments (1961) proved this assumption to be acceptable (Fig. 3.3-6). Larger discrepancy was observed only at the bottom of the soil column used in the experiments, indicating coarser material than that applied in the suspension. If a selection of grains along the path of flow were observed, h e r material would be expected a t a greater distance from the entry face. Thus, the probable reason for the

3.3 Investigation of clagging

389

grain diamter, D L m J Fig. 3.3-6. Grain-size distribution curves characterizing the suspended load, the original layer and the solid matrix including the settled fine particles supposing various rates of clogging

grain diamefer, D rmm3 Fig. 3.3-6. Comparison of the grain-size distributions of the suspension used in the experiments and the fine particles deposited in various depths (after Boreli and JovaseviO 1961)

390

3 Permeability of natural layers

observed difference is the suffusion of fine grains through the exit face of the column. As explained in connection with the derivation of the general equations describing the clogging process, the most important change in the character of the physical properties of the soil caused by the settling of the fine particles, is the decrease of porosity. Considering Eq. (3.3-5) the modified porosity can be calculated aa the difference between the original value (nA)and the amount of the deposited grains nc(y, t ) , thelatterdepending on time and place :

- nc(y,t ) = nA - nco(t)exp (-

(3.3-14) AY) ; while the same parameter at the entry section changes with time according to the following relationship n ( y , t ) = nA

no ( t ) = nA

- "co

(t)*

(3.3-1 5 )

Figure 3.3-5 shows, however, that the other physical parameter of soil also influencing the hydraulic conductivity of the solid matrix (i.e. eaective diameter), i s a k o changed by clogging. Considering the dehition of the effective diameter [Eq. (1.2-4)] and supposing that the average shape coefficient (a = 1 0 ) is also acceptable for the characterization of the suspended particles, the effective diameter of the changing solid matrix depending on time and place [Dh(y,t ) ] , can be expressed as the function of the amount of the deposited grains [nc(y, t ) ] , initid porosity (nA),and the effective diameters of both the original layer (&A) and the transported material (DhS):

DhA ' DhS Introducing the d, = DhA/DhS symbol, the simplified form of Eq. (3.3-16) and the relationship characterizing the decrease in the effective diameter with can be given aa follows: time at the entry section [DhO(t)],

and

-1-nn,

The latter equation can be used to determine the parameters of the b y e r after the development of the total clogging, when no more fine grains can enter the porous medium. At this time point, the amount of the suspended par-

391

3.3 Investigetion of clagging

ticles silted in the pores at the entry surface, reaches its possible maximum value [nco(t)= ncomax]. The pore diameters near the surface decrease gradually due to the h e grains occupying part of the pores. When the probable maximum pore size at the surface (d20),becomes smaller than the smallest diameter of the suspended particles (Dmin),only clean water can enter the layer and, therefore, the process of deep clogging terminates. This is the parameter and since the maximum condition described by the ncomax probable pore size can be expressed aa a function of the effective diameter and porosity [Eq. (1.2-22)], the following relationship can be given for the characterization of total clogging: d

,

20rnln

because

-1 -

nomin

2 1 - nomin

Dhomin=--

1 n~-ncomax DhA = Dmin ; 2 1 - nn,+ dl 12comax

and

+ + A i nco

1 - %A

DhO min

Applying the

= DhA 1 -

= A2and the

N A=

Dmin

nA

1 - nn,

~

(3.3-18)

ncomax max

symbols, the ncomax value

can be expressed from Eq. (3.3-18): 2 4 1 - -2 nco max = n~

NAA2,

1+2'

A

(3.3-19)

A2

If the task is the complete characterization of the time-dependent process, the change in the concentration of the percolating water has also to be considered. To simplify the relationship expressing the change of concentration with time, a further hypothesis is applied: i.e. the C(t) value is proportional to the diameter of the particle which can enter the pores at a given time point [D(t)]: C(t)=C0 D ( t )- Dmin (3.3-20)

Do - Dmin The time-dependent particle size in the equation can be regarded aa equd to the probable maximum pore diameter near the surface at each time point

392

3 Permeability of natural layers

n ( t ) symbol is introduced to simplify the equation. where the N ( t ) = Al A 1 -%A Among the parameters describing the process of clogging, the change in the water conveying capacity of the field is perhaps the most important from the point of view of seepage. Accepting the validity of Darcy's law and using the equation derived for the determination of hydraulic conductivity [Eq. (2.2-9)], the change of this parameter in time and place within the field influenced by clogging can be followed mathematically;

(3.3-22)

where KA is the initial hydraulic conductivity of the layer. Substituting the possible maximum amount of the fine particles deposited in a distance of y from the entry surfme [Eq. (3.3-19)] into Eq. (3.3-22), the hydraulic cmuluctivity developed after complete clogging (the expected minimum water conveying capacity of the field), can be determined

1 - exp ( - 1 y ) 1+2'

A (3.3-23)

I n a field having changing hydraulic conductivity in the direction of flow and also in time [K(y, t ) ] ,the time-dependent flow-rate (the flux or seepage velocity if a cross-sectional area of unity is considered) can be calculated if the length of the flow (L)and the difference of the pressure head assumed to be time invmiable ( A H ) are known:

-dY K(Y4

-

g. v(t) '

therefore (3.3-24)

393

3.3 Investigation of clagging

The initial water conveying capacity of the same system is characterized dH by the initial seepage velocity ( w A = __ K A ) . Hence, the ratio of the two

L

fluxes can be expressed in the following form:

v(t)=

LfKA

L

WA

-

L 2

0

I’

n (4 1 - “exp(-ly) nA

0

dy.

(3.3-25)

The solution of the expression to be integrated in the denominator of Eq. (3.3-25) is L

N A exp(l L) - N(t) N - N(t) 0

-

NA+1

N,exp(lL) - N(t)

1’ 2[ [

NA

-

+

NA+1

NA - N(t))’- ( N AeXp(LL)- N(t) (3.3-26)

Substituting this value into Eq. (3.3-25),the time-dependent seepage velocthe function of a series of variables characterity v ( t ) can be calculated izing the original layer and the process of clogging D h A , nA and dl, A,, nco(t).Among these variables there is only one which depends on time, i.e. the decrease of porosity at the surface nco(t).For the complete description of clogging this ncous. time relationship has to be determined. The total amount of fine particles deposited up to a time point t within a flow space having a cross-sectional area of unity [ Vf(t)], can be determined by integrating the nc(y,t) function along the flow path from the entry section to infinity;

/.

-

”

1

J

V,(t) = n,(y, t) dy = nco(t) exp(- ly) dy = nco(t)L .

J 0

(3.3-27)

?L

0

The change of this value with time is equal to the product of the concentration of the suspension entering the layer [C(t) in Eq. (3.3-20)] and the seepage velocity [w(t) in Eq. (3.3-25)]: (3.3-28) or L

t 7

394

3 Permeability of natural layers

where time is substituted as a variable in both functions, v ( t ) and C ( t ) ,by n,, using Eqs (3.3-21), (3.3-25) and (3.3-26). The solubion of the integral provides the required relationship between time and the amount of fine particles deposited i n a unit volume of the porous medium at the surface of the layer. The reliability of the method explained above was checked by comparing the calculated parameters with experimentally measured data. It was found that the hydraulic conductivity is the most representative characteristic which shows the combined effect of clogging, and, i t can be easily measured. For this comparison, Starasolszky’s (1967) experiments, quoted already, were used, and the K ( y , t ) functions calculated for three time points by using the parameters of the materials applied in these experiments were constructed and are indicated in Fig. 3.3-7 by solid lines. The &st is the expected minimum hydraulic conductivity (curve 1), determined by using Eq. (3.3-23).Curve 2 characterizes the condition belonging to the time point, when the deposited amount of the fine particles is half the possible maximum

.The third full line, (curve 3), represents the hydraulic coefficl’en f of permeabi/ify too

I

,-I

CCm/s..C J

-4

to -2

to-5

- coefficient of

permeabilify before cloggjng average . K = 4 x {o-cm/sec x--x 20-hours after me beginning of tbe experimenl -_-- tz-bOUrS af ter tbe beginning of the experimeni f

I’ I

Fig. 3.3-7. Comparison of the measured and calculated values of hydraulic conductivity decreased by clogging

395

References

1 4

conductivity if nco(t)= - n,,

max. According to the

data measured, the exper-

iments of relatively long duration (experiments having a duration of 166 and 140 hours, respectively) compare well with curve 1 and 2, approximately halving the distance between them.The final data of the shorter experiments (experiments with a duration of 52, 68 and 57 hours respectively) are grouped around curve 2. The hydraulic conductivity measured during intermediate time points (one experiment after 20 hours from the beginning of clogging and another one after a lapse of 12 hours) waa also plotted in the figure. Below 40 cm the measured data lie close to curve 3, while in the vicinity of the surface the experiments ehow a decrease in the hydraulic conductivity, i t being only slightly faster than the calculated rate. I n general, the close agreement between the measured and the cdculated values of hydraulic conductivity, proves the reliability of the method proposed to characterize the influence of cloggingon the hydraulic behaviour of the seepage field.

References to Chapter 3.3 BLEIFUSY, D. I. (1964): Unwatering Akosombo Cofferdams. Preceding8 of ASCE, SM. 2, March. D. (1961): Clogging of the Porous Media. 9th Congress BORELI,M. and JOVASEVI~, of IAHR Dubrovnik, 1963. CISTIN,J. (1966): Investigation of Clogging Problemsof theReservoir at H.usov (in Czech). (Manuscript) Brno. CISTIN,J. (1965): Some Problems of Mechanioal Deformation Influenced by Seepage in Cohesionless Soils of Earth D m (in Czeoh). VodohoepaEa7etvy ChaSopM, No. 2. DE KEUSTNER, J. L. (1964): Dootoral The& (in French). University of Urenoble. DJANUOV, Z. J. (1961): Colmatation of Soil P o r n with cley Greine Effeoted by Seepage in Canals (in Bulgarian). Publication of Water Manugerned and Water Errgineming Inatitute of the Bulgarian Academy of Sciences, Sofia. FENIN, N. K. (1963): Artiiicial Clogging of Canals Using Hydmmeohanization Construction Methods (in Russian). Gidrotechnica i Mdiwacia, NO. 6 . FRENETTE,M. (1964): Measurements of Clogging in Laboratory (in French). Report Ma . P. A. (1962): Application of Clogging as a Method of Protection against Seepage in the Construction of theGreat Irrigation Canal Network of Dneper (in Russian). Gidrotechnica i Melioracia No. 2. IVEB, K. I. (1960a): Simulation of Filtration on Electronic Digital Computer. Journal of American Water Worka Assoc&Aun, Jul I n s , K. I. (1960b): Rational Design of Filters. haceedings of ImtitUtion of Civil Engineers, Vol. 16. p 189-193. I n s , K. 1.(1960~):Filtration through a Porous Se tum: a Theoretiad Conaderetion v . 6469. of Boucher's Law. Proceedings of the Institution ofCi.il E n g i ~ ~ e ~ a . N oNo. 1-8, K. I. (1961): New Conoepts in Filtration. Water and W* Engin~Sring, July, Aug., Sept. IvEs. K. I. 11963): SimDlified Retionel Analysis of Filter Beheviour. Proceedkg8 - of. the In&&ution'of Civil E&kers. July 1963, No. 6661. IZBAE, S. V. (1933): Seepage Effeoted Deformation of Soils (in Russian). IzveutM: of VNIIG, No. 10. KONANOV, I. V. (1969): Calculation of Clogging as a Method of Protection against Seepage (in Russian). Gidrotechnica i Mdioracia, No. 6 .

ASSA AN,

.

3 Permeability of natural layers

396

0. and SZILV~SSY, Z. (1973):Changes in the materials Kovkcs, G., STARASOTSZKY, of Earth Dams and their Influence on Permeability. V I T U K I Pub~icationsin Foreign Languagea, Budapest, No. 7. LECLERC,E. and BESUJEAU,P. (1966): Infltration of Riverflow and Index of Filtration and Infiltration Filtrability (in French). CEBEDEAU, Mar&. MAC~RLE,V. and MACKRIJZ, S. (1961): Adhesion in Filters Proceeding8 of ASCE, No. Sa. 6 . Sept. 2940. NOVIKOVSKY, V. E.and FEDOSEV, J. G. (1960):Clogging of Canals in Conditions of Kara-Kum (in Russian). Gdrokhnica i Methacia, No. 2. ORNATSKY, N. V., SEROEJEV, E. M. and SEHTMA", T. M. (1966):Investigation of clogging Process of Sand soils (in Russian). h ~ h t d e t v oof Mo8cow Univereity. PATRAEHEV, A. N. (1936):Confined Seepa e of Water Saturated with Fine Grains of Sand and Clay (in Russian). Izveetija VNfIG, No. 16-16. PDULOV, F. J. (1960): Use of Clay Cover, Clogging and Packing in Protection against the Percolation from Irrigation Canals (in Russian). Gdrotechnica i Melwracia, No. 11. SEET.MA", I. M. (1961): Seepage of Water Solutions (in Russian). Dokl. Akad. Nauk SSSR. SWATSKOV, E. A. (1966): Graphical and Analytical Method of Determination of Suffusion Characteristics of Cohesionless Soils (in Russian). Izweatija VNIIG, No. 78. STARASOL~ZKY, 8. (1966): Investigation on the Sealing Effect of Colmatation. Sympoaium on See age and Well Hydradie?, Budapeat, 1966. STARAEOLSZXY,(1967): Decrease of See age due to Clogging (in Hungarian). Beeximold a VITUKI 1965. dvi tevdkeny8&6r6[ Budapest, 1967. V . NAOY,I. (1966):Calculation of Clogging of Canal Soils (in Hungarian) Hidroldgiai Kod&y, No. 11. V. NAOY,I., STAF~ASOL~ZXY, 8.and HAMPAS,F. (1966):Investigation into the Operation of Irrigation Canals in Hungary. Conpea8 of ICID, New Delhi, 1966, Question 20, Report 22.

8.

Chapter 3.4 Hydraulic conductivity and intrinsic permeability of fissured and fractured rocks The structure of the interconnected water transporting channels in the solid phase of the porous medium, basically determines the hydraulic resistivity of the latter. For this reason, the conceptual model established for the characterization of seepage through loose clastic sediments composed of individual grains, cannot be applied for the determination of the hydraulic parameters of flow along the fissures and fractures of solid rocks. There are areas, however, where such a water-bearing formation is the only source of available water, and layers of this type are regarded as the best aquifers (i.e. limestone and dolomite, commonly known aa carbonate rocks or karstic formations), and have an important role in the water supply of large regions. I n some oil fields, the source rocks of hydrocarbons are also hsured and fractured rocks, and production from these fields requires the determination of the hydraulic parameters of seepage through such formations. Thus great practical importance is at,tached to the determination of a conceptual model suitable for the characterization of this type of seepage. The dynamic principle of the construction of the model is the same as that applied for the derivation of the relationship8 between seepage velocity

3.4 Fissured and fractured rocks

397

and hydraulic gradient in loose clastic sediments. The difference is caused by the different structure of the water conveying openings. Another geometrical model has to be found based on parameters describing the size, form, and distribution of the channels within the solid matrix of the rocks. The geometrical model has to be combined afterwards with the dynamic relationships, to achieve the new conceptual model. In this chapter, a general description of the various fissured and fractured rocks will be given first. Also summarized here are data collected from the literature to give aa much general information aa possible on the porosity, permeability and hydraulic conductivity of the various formations, aa well as on the water yielding capacity of wells drilled in these rocks. This section is followed by the comparison of various mathematical models proposed in the literature for the determination of the hydraulic parameters of seepage which develop through openings in solid rocks. Finally, a geometrical model developed recently on a statistical baais and its combination with dynamic principles will be presented. The opinion of the author is that this model is suitable for the characterization of water movement in fractured rocks.

3.4.1 Characterization of various non-carbonate, water- bearing, fissured and fractured rocks

The fissures and fractures occurring in hard, solid rocks are of different form and nature depending on the type of rocks. The main types are igneous and metamorphic rocks , indurated sediments (sandstone, quartzites, shales etc.), and carbonate rocks, the latter group forming a special type of chemical and biological sediments. The form of fractures and fissures remains more or less unchanged after their development in the caae of igneous and metamorphic rocks, while they are enlarged due to solution and erosion in the case of limestones and dolomites. The porosity, the hydraulic conductivity of the fissures and the yield from the wells penetrating the fissured structure, are different and vary widely depending on the type of rock, in which these network of fractures are formed. After listing the definitions of some basic terms, a general description of the usual network of openings is summarized on the baais of Schoeller’s work (Brown et d.,1972), grouped according to the types of rocks. The difference between fractures and fissures has to be noted. A fracture is the result of a break of the rock maases caused by tectonic forces, while a fissure is a small split, or a narrow opening of varying length, with no restriction concerning its origin. Faults are planes of fractures, where obvious signs of differential movement of the rocks maas on either side of the plane can be observed. They may be vertical or oblique. Fault zones are sometimes intersected by a network of other faults forming interconnected drainage systems. In other caaes the movement of rock maases along the faults interposes impervious blocks between the separated parta of an aquifer, causing the

398

3 Permeability of natural layers

development of a barrier obstructing continuous ground-water flow. Water circulation in a fault is not uniform and is frequently very localized, since faults are not open in all places. Joints are fractures in the rock, along which there has heen extremely little or no movement. Fractures and fissures used t o be divided into further groups according t o the processes creating them. The names of the different types immediately indicate their origin. Hence detailed explanation is not given here, only a simple list of the various fissures and fractures: (a) Tectonic joints, distinguishing closed shear joints, and generally open tension joints; (b) Weathering fissures originating frequently from one of the various types of fissures; (c) Relmse fractures or sheet joints. The degree of fracturing depends on the intensity of the tectonic movement creating the fractures. The development of weathering fissures is related t o the climate to which the rocks have been expoged and the length of time of this exposure. The depth t o which fracturing occurs also depends on their origin: i.e. faults may run t o a very great depth, and joints generally transform the solid rocks into water-bearing formations only near the surface (to a depth of 100 mi or less). Both characteristics of fracturing and fissuring (degree and depth) are greatly influenced by the type of rock in which the network of interstices occurs. It is reasonable, therefore, to summarize the more detailed description of the probable structure of openings according t o the main groups of solid rocks. Numerical data characterizing the hydraulic properties of various rocks (porosity, hydraulic conductivity, intrinsic permeability, yield of well, bore holes and springs) are listed in tabulated form (Tables 3.4-1, 3.4-2, 3.4-3 and 3.4-4). Intrusive rocks (Table 3.4-1)

The openings in intrusive igneous rock (granite, syenite, diorite) may have the following nature: (a) Joints parallel t o the surface of the rock mass, marking limits between different sheets; (b) Concentric joints forming conical surfaces, and running towards the I centre of the mass; (c) Radial joints emanating from the centre of the incam. The weathering action at the surface of granite, disintegrates the feldspars t o form a soft, friable mass of greater porosity and permeability under certain climatic conditions, or i t may cause disintegration and form clayey substances of very low permeability. The depth of the weathered zone depends to a large extent on climate and exposure time. I n the upper part of the granite mws, where sheet joints are open and intersect the concentric and radial joints, water can flow freely through the channels and aquifer features are exhibited. These openings become narrower and more scarcely distributed at greater depths. I n a temperate climate, the weathering zone

Table 3.4-1. Parametere characterizing the hydraulic properties of intrusive rocks POmSitY

Type of m k

Location av.

Type of rock

1

[%I

max.

miu.

Hydraulic conductivity [cm sec-'1

-__

Location av.

1

mu.

I

2.0 x 10-'0

__

-

Comments

0.5 x 10-lo

Louis

1 . 8 ~10-3

3 . 7 ~10-4

LOX 10-4

LOX

Morris and Johneon French

10-6

j

Av. depth

2;

84 83 66

40 47 61

98 76 61 16

37 51 48 56

Location av.

Old grenite

@/dl max.

1

Number of

Failore

11 12 11

62 497 130

14 26

12 6 4 8

404 202 136 16

10 36 37 31

min.

Author

19/01

South Afrim

Pretoris, Johannesburg North "mmmd Rustenburg south-w est

Granite

I

Author

min.

Yield of wells Type of rock

Author

Comments

I

Transwad Mafeking Vryburg Van Rhysndrop Connecticut (USA)

48

Uganda

Bred &Urita& Sweden USSR Dnepr river Selenga river

-

-

45

0.1

26

-

11

330

60

27

ElliE

20-60 30-40

Q0m

11

Suseczynski Archambault

22-43

3.4x 10-3 12.6

Du Toit

Meyer and Peterson 10-30 5-60

38

. . .

53

-

Rats and Chernyeshov

3 Permeability of natural layers

400

may go down 30 m or more, and water-bearing joints reach depths of over 100 m. Although factors other than geological conditions influence the yield of wells (diameter, casing, testing method, etc.), data collected from a given region are suitable to characterize the probability of an expected yield for a given geological formation and its dependency on the drained depth. Turk’s (1963) analysis is shown here as an example, which is based on the data of 239 wells in Sierra Nevada (California). Most of these wells are in

0 30

sp

20

I 1median

mean

total well yield in 6PU

Fig. 3.4-1. Frequency distribution of yields of 239 w e b in Sierra Nevada (California) (after Turk, 1983)

granodiorite, but a number of methamorphic rocks are also involved. Figure 3.4-1 shows the frequency distribution of the yields of wells without taking into account the depth of wells. The median (most probable value) is about 34 l/min and the mean 90 l/min. It is worthwhile to note here that the form of the frequency distribution curve suggests that it can be approximated mathematically using logarithmic normal distribution, which is generally applied in the literature to characterize the probability of a given yield, or hydraulic conductivity in fissured and fractured rocks. About 8.4 yo of the wells were unsuccessful and 16.3 % yielded less than 14 l/min. Grouping the data according to the depths of the wells (using arbitrary depth-intervals) the same statistical analysis can be executed for each interval. Representing either the median or mean values determined in this way as the function of depth, the decrease of permeability of the intrusive rocks with increasing depth, can be accurately demonstrated (Fig. 3.4-2). A similar relationship can be achieved between depth and water yielding capacity, if the latter is characterized by a certain recharge rate measured by pressure tests at various depths (Davis and Turk, 1964), aa shown in Fig. 3.4-3. The graph was constructed on the basis of 412 teets executed in granite in California.

401

3.4 E'issured and fractured rocks

to

aoi

0.05 0.1 0.5 10 well yield in GPM per foot

Fig. 3.4-2. Relationship between depth and yield of 239 wells in Sierra Nevada (California) (after Turk, 1963)

I"

DO!

0.05 0.i

water

intake

0.5 i.0

50

GPM per foot

Fig. 3.4-3. Recharging rates in granite related to depth (California) (after Davis and Turk, 1964)

Metamorphic rocks (Table 3.4-2)

Gneiss, well crystallized schists, slatey schists, amphibolites, quartzites and crystalline limestones belong t o the group of metamorphic rocks. Fissures occurring in these rocks are due t o schistosity (parallel layers of flakes having the same petrographic composition), or foliation (parallel layers of flakes of different mineralogical composition), or sheet joints forming a network of fractures normal t o the ones mentioned above. Weathering Gsures and faults are also common. Comparing schistosity, foliation and sheet joints in gneiss and slatey schists t o the flasures in granite, i t can be seen t h a t the openings of the former group are generally smaller. Due t o their very steep dip, however, the infiltration is not negligible, especially if the openings are enlarged by weathering. Hence, the recharge may be considerable, the number of joints being higher than that in granite. Sheet joints resulting from decompression, occurring one below another in metamorphic rocks, are known t o be capable of storing significant volumes 26

rnofrOcL

1

wtim

1

P O d Y av.

0.13.0 0.66.0

Brunawiok shale

Slete Merble

New Yemy (USA)

2.6 x 10-2

I

1%1

mar.

-

I

oommente

Author

mtn.

-

-

-

-

-

i!

estimation etorege capeaity from

L Menger

Author

Sht.0 sohiat (weathered) schist Slate Mica-schist Gneiee Granita-gneise

USA

1.0 x 10-8 2.1 x 10-4 -

kenoe Michigan (USA)

1 . 3 10-6 ~ -

QuertZit43

Sht.0 M i a WhiEt

0.9 x 10-6

4.6 x 10-8 6.0 x

6.0 x 1 0 - 1 O 2.1 x 10-5

8.0 x lo-' 4 . 0 ~lo-' 3 . o ~10-4

2.0 x 10-7 6.0~10-7 6.0 x 10-7

Ukraine

-

1.6 x 10-1

6.4 x 10-6

-

4.2 x 10-3

7.1 x 10-4

Graywaoke

(f-tyw

1.2 x 10-7

-

Fefiuginous echiet Meteeedunent (fractured) Qumrts-mica schist (weathered) Grayweoke

2.1 x 10-10 2.1 x 10-8

Brazil

S h ~ S O m

6.3 x 1 . 3 ~lo-'

4.6 x

-

-

8 samples 17 samplea

M o d and Johnson

Lo& 9 samples 2 samples 34 Lugeon t a t s 72 Lupon teats

Stuart et al. Franc&

Rudenko

11 samples

Lewis d al.

3.1 x 10-6

-

-

3.3 x 10-5

-

-

21 samples

Stewart

9.7 x 10-4 3.0 x lo-' 1.9 x lo-' 1.a x 10-8

-

-

field teeta 6 samples

S t u d et ai.

2.1 x 10-0

-

6 teeta

I

-

9 eemplee

w IP

a

Table 3.4-2. (colzt.) Trammhibility [m* sec-'1

Location

Type of ruck

av.

I

mar.

I

1-

~~

comments

Yield of wells @/min]

Type of rock

Location

av.

I

mar.

I

Author

min.

min.

1 I

Av. drew-

down

~

Number ofwells

1 1 Failure

[%]

Author

Connecticut (USA) Gneiss Quartzite Schist Slate

Magnetite

Sweden Hallan Viistergotland Bohnalan VIirmlrtnd SBdermaland Norrland

Gneiss

USA

Gneiss

46.6 27.4 62.6 0

Ell&

46 63 18 40 12 42

Meier and Peterson

46 64 34 91 46

Davis and de Wiest

77 26 23

Susvxzynski

Slate

Maryland Virginia New England Maryland Virginie New England Brazil Monteiro Petrolina Petrolina

Slate

Rajesthan (India)

37

Taylor et al.

schist

Gneiss

38

3.4 Fissured and fractured rocks

405

of water. The weathered top mantle is often reasonably permeable, allowing greater infiltration of water, which can be recovered by digging wells down to the hard rock. At greater depths, the fissures close more rapidly than in granite, and they are unidirectional and less interconnected with less possibility of yielding sustained amounts of ground water. The water yielding capacity aa a function of depth can be similarly chmacterized as in the caw of intrusive rocks. The analysis of water-injection data from Oroville dam site (California) is shown w an example in Fig. 3.44 (Davis and Turk, 1964), where the frequency distribution curve of

Fig. 3.4-4. Water-injection data from Oroville dam site (California) (after Davis and Turk, 1964). (a) Frequency distribution of data measured between 24 and 30 m; (b) Water-injection rate aa a function of depth / oou

I

I

SOD

-

-

400

-

-

-

-

I . p1

c 2

3

-0

50

(0

0

grmte

0

schist

I

I

\ \

\

\ \ \

406

3 Permeability of natural layers

one depth interval (from 24 to 30 m) and the relationship between the depth and the injection rate, are shown. Although porosity and permeability are supposed to be higher in hard intrusive formations than in metamorphic r o c h , the water yielding capacity of the weathered mantle of the two types of rocks (the two formations being frequently combined as crystalline rocks) does not differ at all. This fact is proved by the comparison of the yield of 1,522 wells in the emtern part of the United States, of which 814 drain granite and the others penetrate l0OR

500

I

I

-

granite

I

.s

-

/

rocks

c

i

/

o metamorphic h

!

I

/

100

Fig. 3.4-6. Percentage of tests having zero water intake in granite and metamorphic rocks (after Davis and Turk, 1964)

60

40

I

I

I

I

I

I

I

I

I

I

I

-

h

r=

9,

2PI

30

-

Q

40

-

0

I

0

20

40

I

6U

'& 80

depth in feet Fig. 3 . k 7 . Porosity and specific yield as the function of depth in metamorphic rock north of Atlanta (Georgia) (after Stewart, 1962)

3.4 Fissured and fractured rocks

407

schist (Fig. 3.4-5) (Turk, 1963). The same similarity can be found if the percentage of water-injection tests having zero water intake in granite and in metamorphic rocks are compared (Fig. 3.4-6) (Davis and Turk, 1964). Normal porosity of gneiss is about 0.1 to 3y0, and that of schists is 0.5 to 5%. Greater porosity (up to 500/,) is exhibited by rocks subjected to a very high degree of weathering. The relationships of depth vs. porosity and specific yield respectively, in weathered metamorphic rocks are shown in Fig. 3.1-7 based on measurements made in Georgia, north of Atlanta (Stewart, 1962). Due to schistosity, metamorphic rocks have highly anisotropic behaviour. Permeability along the fracture planes may be many times greater, than the average. Non-carbonate, indurated sediments (Table 3.4-3)

Sandstones generally have a double system of interstices. The first part is composed of the original pores between the grains, the second one is created by fissures, fractures and stratification joints. Primary porosity is very variable ranging from less than 1 to 35% and depends on the grain-size distribution and the packing of the particles, as well as on the material and degree of cementation. The number of jointe is generally high, and they are more open, if the rock is more indurated. I n most c w s the secondary porosity is lower than the primaq one, but the joints are larger, than the original pores. Thus hydraulic conductivity is greatly influenced by the degree of fissuring and fracturing, while the storage capacity of sandstones depends mostly on primary porosity. It follows from the structure of the interstices explained in the previous paragraph, that wells sunk in sandstones have a high initid yield by draining the network of joints, but the yield characteristic for a long period of operation i s determined by primary porosity and hydraulic conductivity (by the parameters representing the continuous blocks without joints). The same role of secondary porosity can be observed in natural conditions as well, viz. infiltration and water transport through the joints is rapid, which causes high yield of springs after rains, while the second part of the recession curve (the graph showing the decrease of the yield of a spring in time after a rainy period) is flat, indicating the relatively low permeability and high storage capacity of the solid matrix. The influence of the double system of interstices decreases with depth as the degree of fissuring is low below a given depth (which, for example, is 100-200 m on average in Bunter sandstone in the Saar region). As in the case of crystalline rocks the general decreasing yield with depth can be similarly observed although this relationship is not so close in sandstones. The influence of primary porosity and the change of the latter in the layered system may perhaps, be the reaaons for this difference. In Fig. 3.4-8 an example is shown bawd on the yield of wells draining an Oligocene sandstone formation of several hundred metres within a relatively small region in Northern Hungary. Three different parametera were inveeti-

Table 3.4-3. Parameters characterizing the hydraulio properties of non-carbonate indurated sediment8 TypeIvf wck

I

Location

Sandetone (tine)

(d-1 Miocene sandstone Sandstone Devonian Miseisipien ordoviCi8n

crefecsoua Pliocene Eocene Conglomerate (coarse) Ark- ( k e ) (medium)

Tunieia

POdtY

av.

33 37 -

I

I

19/01

msx.

I

mln.

66 samples

10 eemples 2 samples

Anthor

Morris and J o h n Sahoeller

USA

Bradford Bere8 Oil Creek Woodbine Repetto wiloox

1-(

Siltetone Sandsfone

USA

Sandstone

Cromwell uilcrest Prue wilmx USA

Cambrian Pennsylvanien Shale Oligocene and Miocene Silurian

Permian sandstone

I

s00

Brumunddal (Nomay)

14.8 19.0 6.7 26.6 19.1 16.3 17.3 14.4 16.6 10.9 9.7

%via and

0

de Wiea

Rims et d.

E: Mueket

16.6 27.4 11.4 16.6-12.4

k

2

1

11.2 17.4

24 samples 687 samples

21.1 6.2

9 samples 6 eemplee

15.2

9 ssmplea

Manger

Englund and J0rgemn

Type of rock

Location

Sandstone (fine) (medium) Sandstone Miocene Oligocene

Tunisia

Sandstone Devonian Missisipian Ordocivian Cretaceous Pliocene Eocen

USA Brandford Berea Oil Creck Woodbine Repetto wilcox

j

Hydraulicconductivity [cm/sec]

I

av.

1

Author min.

4.2 x 10-7 2.6 x 10-6

2 . 6 ~ 4.1 x 10-3 __

20 samples

1.ox 10-3

Morris and Johnson Schoeller

3.0~10-~

Davis and de Wiest

2 . 6 10-6 ~ 3.5~ 3.9 x 10-6 4.3 x 10-3 3 . 6 10-5 ~ 2.9 x 10-7

Sandstone Carboniferous Devonian

2.9 x lo-'*

Louie

2.1 x 10-12

Conglomerate (coarse)

3.8 x 10-7 4.9 x 10-7

Arkose (fine)

1.6 x 10-7 1.6 x 10-6 5.5 x 10-7 1.1 x 10-6 3.8 x 10-7 5.6 x 10-7 1.6 x 10-7 1.2 x 10-7

(medium) (coerse) Siltstone Sandstone

max.

USA Cromwell ailcrest Prue Wilcox

-

-

1.7 x 10-4 4.1 x lo-' 5.8 x 1 0 - 4 8.0 x 10-6 4.7~ 10-7 3.4 x 10-6 3.6 x 10-4 8.6~ 7 . 6 10-5 ~ 8.8 x 10-6

-

-

horizontal

-

vertical horizontal vertical horizontal vertical horizontal

__

-

~

Rima et al.

5a

vertical

-

-

-

vertical horizontal vertical horizontal vertical horizontal vertical horizontal

Musket

-

-

vertical horizontal

f

Table 3.4-3. (cont.) Author

Tspe of rook

4.0 x lo-" 9.0 x 10-11 Oligocene BBndetone

N6gr&d (Hungary)

Permian sandatone

!l!ypa of rock

Brumunddal (NOWaY)

I

I

Ordovician argillite

Angara basin USSR

Brumunddd (NOWBY)

Permian sandstone

I

I lppe of rock

1

Location

I Sendatone Oligocene eendatone

I I

Sear Region N6W (Hungary)

Scele

7.8 x 10-I

2.9 x lo-'

6.2 x

6.1 x lo-'

1.8 x lo-'

8.7 x 10-3

7.9 x lo-'

1.6 x

1.4 x 10-4

3 . 0 ~lo-'

1.6 x lo-'

10-200 m depth interval, 32 w e b 200-300 m depth interval, 37 wells 300460 m depth interval,23 w e b

Lorberer

W

Englund and

9 samples

Jergeneen

Yield of wells @/mh]

I

-

I

Author

*

45.7

-

-

3.1

-

-

71.3

1260.0

I

-

6-26 m depth interval 2 W 6 m depth interval 40-100 m depth interval, 13 wells

9.0

I

Specleo yielding capacity of wells

BV.

0

GIondoUtn and

3.0 x 10-2

BV.

2!

I

*

12.0 6.8 x 10-1

0.6 3.9 x lo-*

1.0 x 10-1

3.8 X 10-l

6.4 x

8.1 x lo-'

3.0 x 10-I

2.2 x lo-'

Englund and Jergenaen

I

I

I

1.6 1.7 x 10-1

Rats and Chernyeshov

Author

I

10-200 m depth interval, 32 wells 200-300 m depth interval, 37 w e b 300460 m depth intend, 23 w e b

Sailer Lorberer

411

3.4 Fissured and fractured rocks

gated: the yield related to the draw down of the well,

m

the former parameter divided by the length of the hydraulic conductivity calculated from the yield, screened length of the well [m sec-l] (Lorberer, 1975). A similar relationship between depth and yield was also observed in the Brumunddal Sandstone, in Norway (Englund and Jrargensen, 1975). There have also been attempts to charmterize the density of joints and their hydraulic conductivity, statistically. It waa found by Rate and Chernyaahov (1965) in sandstones of the Ordovician flysch in Central Kazakhstan, that the average distance between neighbouring fractures (1) in metres and hydraulic conin ductivity along the fractures perpendicular to the stratification (Kf) m sec-l can be expressed aa functions of the thickness of the layer (m) given also in metres: (3.4-1) log I = 0.41 log m 0.45 ; K f = 0.82 log m - 0.19. K im/sec~

+

IU-~

10”

Fig. 3.4-8. Parameters of wells (rate of yield and draw-down, yield divided by drawdown and length of filter; calculated hydraulic conductivity) netrating into Oligocene sandstone in N 6 W (Northern Hungary) (after grberer; 1976)

412

3 Permeability of natural layers

Shales usually have very low primary porosity and water circulates mainly through fissures, joints and planes of stratification. I n general, the yield of wells in shales is, therefore, poor. A well-developed network of joints can be expected only near the surface due to the high degree of weathering. The reported average porosity ranges from 1yo t o 25% and hydraulic concm sec-l. ductivity was observed t o be ;t8 low as 5 x 10-l' t o 4 x Extrusive volcanic rocks (Table 3.4-4)

The most important water bearing formations are the basalts having low silica and high magnesium and iron content. The mineral composition of rhyolites is opposite in character, while andesites and trachytes form an intermediate type between the other two. The water-bearing capacity of rhyolites, trachytss and andesites is generally poor. Basalts cover extensive areas in India, North America, Brazil and South Africa providing large ground-water resources. Basalt formations can be grouped as follows: (a) Aa lavas with a hardened Rcoriaceous surface crust due to the escape of gases; (b) Pahoehoe lavas with it smooth, undulat,ing ropy surface; (c) Pillow l a v a formed by outflow below water and composed of blocks (pillows) of a few decimetres in diameter with radially arranged vesicles inside them. Lava flows may be superimposed on each other, varying in thickness and number. There may be a layer of ash or fossil soil, or alluvial deposits between the flows, if the dormant phase of the volcano was fairly prolonged. Basaltic rocks are known as good aquifers due t o numerous openinqs within them. I n addition t o the interstitial porosity caused by scoriae and breccia flows, the fissure porosity in basalt may be created by the following elements : (a) Cavities between pahoelioe lava flows; (b) Shrinkage cracks occurring as fissures normal t o the isothermic surfaces brought about by surface cooling and as fissures caused by cooling from beneath resulting in prismatic structures: (c) Gas vesicles; (d) Lava tubes within the flow; (e) Fractures caused by mechanical forces acting on the cooled lava; (f) Tree moulds. Horizontal permeability is generally greater than the vertical permeability, and pahoehoe flows are more permeable than the aa flows. Thelatter are reasonably permeable when they consist of breccias. I n pillow lavas water can circulate between the pillows, if these channels are not sealed by vitreous material and secondary minerals. Because of the high porosity and permeability of basalts, the natural infiltration is relatively high. It is estimated t o be near 25% of yearly precipitation in the island of Oahu in the Hawaian Group, and it can reach

3.4 Fissured and fractured rocks

413

107; even in hot and arid climates (e.g. in Afaras and Issm in Africa). For this reason, the basaltic terrains are known t o contain aquifers as good as those of carbonate (karstic) formations, with good natural recharge and high available ground-water resources, which occur in the form of large springs. Some characteristic data of such springs are also listed in Table 3.4-4. I n a thick basalt layer covering a large area (trap), the enlargement of the fissures and fractures by weathering can be characterized by the decrease of permeability with depth. There is, however, another process acting near the surface against this effect of weathering: i.e. the sealing of the openings by clay. The sealing material may be transported onto the surface of the bare rock from neighbouring areaa. This process waa also observed in the upper part of a dolomite formation covered by younger marine sediments (Transdanubian Mountain Range, Hungary). The sealing is meet common on the surface of extrusive volcanic rocks, where clay is locally formed by the decomposition of feldspars as a result of weathering, and in the upper part of indurated sediments, the solid matrix of which is composed of very fine particles (e.g. argillite). The examples shown in Fig. 3.4-9 t o demonstrate the permeability vs. depth relationship summarize observations from

Fig. 3.4-9. Permeability vs. depth relationship in sandstone-argilliteseries and in trap of the Middle Angara Basin (after Rats and Chernyashov, 1965)

POdtY

[%I

n P o f d

Type of mck

m t i m

I Andezite Weethered basalt end fiesured compacted eeh

Location

M&tra (Hungary)

1

I

Hydraulic conductivity [cm aecr']

I

87.

mar.

I

min.

I

Author -h

TransmWbility [m' f3ec-']

av.

I

max.

I

Author min.

20 wells (36% failures) 8 . 6 10-7 ~

6.2~

Morocco -

3.0

x

Schmieder

1 . 4 10-a ~

1.1x

Mortier et al.

e5

Beeeltic tuffs and alightly fhured compact baaalt

Nuns of the spring

Locstion ~roapofspringa

I

comments

Author

hmddsprine

84

Meinmr

KsLeUM m-pu

Oaluu Inland (Hawai)

-

0.4-1.0

0.7-1.0

E

cn

416

3 Permeability of natural layers

the Middle Angara Basin of the Far Eastern Region, USSR (Rats and Chernyashov, 1965). It represents the logarithm of the ratio between yield and draw down at3 a function of depth in traps and in a formation built up from sandstones and argillites. The effect of sealing can be well observed in the latter group. The extrusive volcanic rocks, other than basalts, (rhyolite, trachyte, andesite) are generally very poor aquifers. Their permeability is also the result of joints dislocating the maas of the rock, and hence, i t decreases with depth. To demonstrate this relationship, permeability data determined by pumping and water injection tests were collected in the andesite layers of the M&tra Mountain (Northern Hungary). There were two obstacles hindering the usual statistical evaluation of the data: i.e. the small number of observations, and the great open length of the bore holes below the water table (which waa necessary to achieve measurable results). In Fig. 3.4-10 the position of the open bore holes is, therefore, represented, as a function of the calculated hydraulic conductivity. The general trend of the hydraulic conductivity vs. depth relationship was determined by averaging the data observed between 0-400 m; 400-800 m and 800-1200 m. The frequency dis-

Fig. 3.4-10. Relationship between hydraulicconductivity and de th in andesite of the Mhtra Mountain (Northern Hungary) (SchmiederPEI data)

3.4 Fissured end fractured rocks

411

tribution determined from all observed data can be accurately approximated by a logarithmic normal distribution function, while after separating the data observed above and below 700 m, both empirical frequency distribution curves become asymmetrical, even using a logarithmic scale on the horizontal axis. It is worthwhile to note, that hydraulic conductivity of this rock is the smallest among those mentioned in the previous examples (e.g. at 500 metres, sandstone investigated in connection with Fig. 3.4-8 haa a m sec-l, and andesite of only 3 x hydraulic conductivity of 2 x m sec-l. The high percentage of impermeable bore holes (30% above 700 m and 35% below this level) also indicates the poor water yielding capacity of these rock-types.

3.4.2 Hydraulic properties of carbonate rocks The sediments composed mostly of different carbonates (generally calcium and magnesium carbonates) of chemical or biological origin are called carbonate rocks. The main representatives of this group are the various limestones and dolomites. The term karstic rock is used many times aa a synonym of carbonate rock, although this application is not absolutely correct, because the adjective “karstic” indicates the development of chemically enlarged openings in the rock maas, a process which may occur in noncarbonate sediments aa well (in evaporites, e.g. gypsum). There are also carbonate rocks in which the development of dissolved openings is not common (e.g. carbonate marla). The carbonate rocks and especially those having karstic behaviour are known to be the best aquifers, apart from the coarse grained loose claatic sediments. On terrains covered by carbonate rocks, the recharging rate of ground water is generally high anditsevaporation is low, because of the wide vertical openings through which precipitation infiltrates rapidly and in which capillarity is negligible. The storage capacity of carbonate rocks depends on their porosity, because the compressibility of the solid matrix is practically negligible, and hence, both important hydraulic parameters (i.e. storage capacity and hydraulic conductivity) are basically influenced by the structure, volume and size of the interconnected pores. For this reason, the chemical and mechanical processes enlarging the openings increase theAe parameters considerably. Similarly, &B in the case of other indurated sediments, primary and secondary, porosity can be distinguished. The former is the volume of the original pores within the rock-mass related to the bulk volume. These interstices are the residual pores of the sediment from which the rock wm formed, and are frequently reduced by calcite developing within the pores during or after the process of lithogenesis. Primary porosity is generally very low in limestones and dolomites. In young coarse grained limestone, however, primary porosity may be aa high aa 20%. If the dolomite minerals have developed from calcite after the general lithification, the primary porosity of the rock is about 12-13%, because the transition from calcite to dolomite results in this reduction of 27

418

3 Permeability of natural layers

volume (Davis, see De W e s t , 1969). Some special carbonate rocks may have even higher primary porosity (chalks u p t o 500/,, tuffs up t o 60%). Secondary porosity is the result of discontinuities of different origins (e.g. fractures due t o tectonic movements, bedding planes, expansion fissures). The joints generally form a dense network throughout the whole mass of the rock. The size of the openings generally decreases with increasing depth, due t o the burden of the overlying layers. The effects of weathering and exfoliation result in high porosity near the surface. Secondary tectonic movements may cause either the closing or the widening of the joints, the result of which is the inhomogeneous and anisotropic permeability of carbonate rocks. Openings are larger and also more frequent in the fractured zones along cross faults, than in the mass of the rock, while the joints are closed, although numerous in compressed zones. Complete or partial sealing of the fissures and fractures may occur near the surface from silt and clay transported by the infiltrating water, and inside the rock-mass a similar result is caused by the development of calcite crystalls on the wall of the openings from the carbonates dissolved in the percolating water. Another effect of water percolating through the masses of carbonate rocks (more important than sealing) is the enlargement of the openings and the development of cavities. Water having free CO, content dissolves the material of carbonate rocks. The process is more rapid in limestone, than i n dolomite. Impurities in the carbonate rocks (e.g. clay in carbonateous marl or quartz in cherty limestone) hinder the development of dissolved openings, while suspended materials carried by the water accelerate the development of large cavities, because the chemical process is supplemented by niechanical erosion. The strongest enlargement of secondary porosity can always be observed in the zone of fluctuation of the water table of the ground water stored in carbonate rocks (karstic water), where the water having continuous contact with the atmosphere, always contains free dissolved CO,, and the water movement is also considerable in the direction of draining. Usually gently sloping passages and interconnected galleries are formed here, the intersections of which are enlarged t o form chambers and domed caves. Above the water table in the zone of aeration, the dissolved openings are generally vertical or steeply sloping. Shafts, chimneys, and sink-holes are created by the infiltrating water. The elevation of the zone of fluctuation related to the mass of the rock, might be changed many times during geological ages due t o the erosion of valleys crossing the mountain blocks, or t o the vertical tectonic movement of the rock mass. This explains the development of multi-storied caves having more than one network of galleries above each other interconnected by vertical shafts. The same effect of the changing of drainage level can be observed, when the porosity vs. depth relationship is investigated in carbonate rocks, especially in limestones. Apart from the generally decreasing trend of porosity characteristic for any other fissured and fractured rocks, local irregularities can be found a t special levels, where the rapid increase of porosity is observed. An example is shown in Fig. 3.4-11. The graph is constructed from data observed in the northern part of the Ural-Mountains,

3.4 Fissured and fractured rocks

419

USSR (Shevyakov and Mankovsky, 1963), and similar conditions were also explored by mining activities in the Transdanubian Central Mountain Range, Hungary (Schniieder et al., 1970). I n most cwes, because the primary porosity of carbonate rocks is generally very low, and they have poorly interconnected pores, the hydraulic behaviour of these aquifers is hardly affected by primary porosity (the exceptions are those types, which may have extremely high porosity, such as chalks 0

05

LO

i.5

porosity [%I 2.0 25

"1\

Fig. 3.4-11. Porosity of limestone as a function of depth (Ural-Mountaim,USSR)

and tuffs). Limestones and dolomites still exhibit the double, even multiple hydraulic character (high initial yield and low but continuous yield after the depletion of large pores), the development of which was explained by the different character of primary and secondary porosity in the case of sandstones. In carbonate rocks having negligible primary imrosity, the different characters of the openings explain the special hydraulic behaviour. The first system of interstices is composed of narrow joints (bedding planes, tectonic fractures) near the surfaces supplemented by expansion joints due to exfoliation. Their hydraulic conductivity i R small, but they represent a larqe part of the storage capacity of the layer, because their volume generally surpltsses half the total volume of the pores. The other part of the openings were widened by later tectonic niovements in the fault zones and further enlarged by the chemical action of the percolating water. The fractures belonging to this group have high hydraulic conductivity ensuring the high initial yield, but the depletion of water stored in their network is generally rapid. The influence of the different sized fractures is generally characterized by the recession curve o f springs, which represents the yield of a spring as a 27*

420

3 Permeability of natural layers

function of time. The numerical interpretation is generally based on a very simple mathematical model, which dewribes the depletion of a cylindrical reservoir having constant cross section whose area is F. The water leaves through a pipe, which is filled with porous material. The length and area of the cross section of the pipe are indicated by 1 and f respectively, and the

(a) sinjle reservoir

(b) double reservoir -

- - -Fz - - - - -- --6

inittat water l e d

6

Fig. 3.4-12. Models for the derivation of the equation of recession curves

hydraulic conductivity of the material filling the pipe by K (Fig. 3.4-12a). At a time point t the amount of the outflowing water is equal to the change of storage: h F dh = Qdt = f K -dt ; 1 (3.4-2) h = h, exp [-a(t - t o ) ] ;

Q = Qo exp [-a@ - t o ) ] ; where the initial flow rate at a time point to

(4,

= h,

3) , depends on the 1

height of the water level (h,) at the same time point and the hydraulic

421

3.4 Fissured and fractured rocks

tfl

transmissibility of the outlet - . The constant a = -f is inversely pro-

I F

portional to the storage capacity of the syAtem (in the model to the horizontal area of the reservoir) and the coefficient of proportionality is the reciprocal value of the parameter characterizing transmissibility. If two reservoirs are drained through the same outlet (Fig. 3.4-12b), their flow rates have to be added to get the instantaneous discharge at a time point t : (3.4-3) & = Q O l exp [-a& - t o ) ] &02 exp [ - a 2 ( t - t o ) ] . The comparison of the mathematical model with observed yields of karstic springs proves the accuracy of such approximation of the recession curves, and even the constants can be calculated from the measured data (Schoeller, 1965; Forkaaiewicz and Paloc, 1965). The example shown in Fig. 3.4-13 waa constructed and the numerical parameters of Eq.(3.4-3) indicated in the figure were calculated on the basis of the data from a karstic spring called Nagytohonya spring in the Bukk Mountain (Northern Hungary, Maucha, 1972). The ratio of the yield belonging to the time point t o and the a constant

+

80 70 60 50

40 30 20

10 9

8

7

6 5

4 3

2

I 0

20

40

60

80

100

t-to dSy Fig. 3.4-13. Recession curve of Nagytohonya spring (Bukk Mountain, Northern Hungary) (after Maucha 1972)

422

3 Permeability of natural layers

of Eq. (3.4-2), is proportional to the storage capacity of the system. This ratio can be calculated for the two components of the recession curve, and supposing that the initial water level was equal in both reservoirs (hol = h,,), the quotient of these parameters is approximately equal to the rate of the porosities originating from narrow and large fractures. Data are published in the literature showing the ratio of the volumes of narrow and large pores calculated in this way or by using similar mathematical models. The data listed below give the porosity calculated from narrow openings (n,) and that from wide fractures (nJ, related to total porosity ( n ) :

Nagytohonya Spring (Hungary) (Maucha, 1972) Fountaine de Vaucluse (France) (Schoeller, 1965) 0.80 Spring of Kef (Tunisia) (Schoeller, 1948) Spring of Loz Herault (Schoeller see Brown et al., 1972)

-

nn -

nW -

n

n

0.71 0.69 0.75 0.20 0.76

-

0.29 0.31 0.25 0.24

There is also a great deal of published data concerning the total secondary porosity of carbonate rocks, althoughit is well known that the determination of this parameter is difficult, and hence the numerical values can be accepted only as rough estimations because of the uncertainties involved in their measurements. The determination is generally based on the investigation of core samples taken from bore holes, on the evaluation of pumping tests, on the comparison of the change of water level and the recharge or discharge of the aquifer during a given time period, or on the investigation of the openings along a free wall of the rock either on the surface or in mines. Some collected data are summarized in Table 3.4-5. It can be seen from the data in the table, that the hydraulically active secondary porosity of carbonate rocks is generally between 1 and 3 % . Schmieder (1970) has proposed that the average porosity of Triassic limestones in the Transdanubian Central Mountain Range (Hungary) might be characterized by n = 1.5 & l.O%, and the dolomite of the same age by n = 3.0 I, 1.0%. There are several investigations, however, showing much larger parameters. In the same area Venkovits has found as high a porosity as 6-23% (one local value was 65%) using core samples. In the southwestern part of Spain, the porosity of Silurian-Devonian limestone ranges up to 10-120//,, and that of Middle Triassic dolomite to 6-7% (Samper and Navarro, 1965). The explanation of these very high parameters can be given on the basis of the investigation by Schoeller and Aigrot. They have found that in the surroundings of the Fontaine de Vaucluse (Southwestern France) the porosity of the limestone in the zone of fluctuation of the water table, can be estimated as 13-14%. Immediately below this zone this parameter is only 0.2-0.8 %. Considering this observation and also those represented in Fig. 3.4-11, i t can be stated that the average porosity of carbonate rocks is only 1-5 yonear the surface, and it decreases with increasing depth. There are, however, levels where the conditions were and still are favourahle for

1

,

Type of rock

~

Porosity

[%I

Comments

Measuring method

Author

Transdanubian Central Mountain Range (Hungary) Dolomite Middle Triassic upper. Tnegsic

Iszkaszentgyorgy

samples

Nyiriid

samples pumping tests samples samples

4.6 6.2 2.0 9.0 2.2-2.6

pumping tests samples

1.32 0.66 1.66 1.4 1.3 2.6

Nagylengye1 Tatabhnya

Limestone UPpe'. Triassic Cretaceous Eocene

Marble SilurianDevonian Dolomite Middle Triassic Limeatone Miocene

Dorog

P&pa Ajka Halimba Riibasornjh South-eastern Spain

Schmieder et al.

1.6

10-12

0.6-0.7

at the surface between 60-80 m

Triassic

Willems Schmieder et al.

fluctuation of the yield

1.2

Ip

2

FL

:

p1

z

-

-

.-

-

Navarro and Samper

Plandrin and Paloc pumping-tests

w

?

areal porosity

Limestone Fontaine de Vaucluse South-eastern France

Hi3riszt Schmieder et al. Uerber

-

-

zone of saturation

13-14

0.2-0.8

at deeper level

Schoeller and Aigrot

424

3 Permeability of natural layers

the development of karstic formation. Here extremely high poroeity can be observed locally. It is not unusual, for the local maxima to reach 10-15%, and even as high a value m 30 yocan be expected in some places. The extremely wide fluotuation of poroaity within ct relatively small area needs the investigation of the correct definition and determination of this parameter in carbonate rocks. It is quite evident that the value depends on the size of the bulk volume of the rock-maas, within which the volume of the pores was measured. One can choose a very small volume, which does not include any pore if i t is located inside ct block and then n = 0, while the same volume can be placed in a large opening (a cave) and in this caae the calculated porosity is one hundred percent (n= 1).Thus, porosity also depends on the size and location of the investigated sample, and the fluctuation of n decreases with the increasing volume considered for the calculation. There is, however, a size (the representative elementary volume; Bachmet, 1965), above which the calculated porosity has only random variation depending on the location of the centre of the investigated volume. Thus the determination of the porosity of carbonate rocks is a good example to demonstrate the practical application of the representative elementary unit, the theory of which waa discussed in Chapter 1.1 and which 4 ,

,

dbwok, dolomite

Fig. 3,4-14. Comparison of curves enveloping the linear porosity values calculated along independent lines having the w

e direction

425

3.4 Fissured and fractured rocks

16

Fig. 3.4-16. Determination of average linear porosity and representative elementary length (dolomite, bbarok, Hungary) (after BalsshAzy and KovAcs J., 1976)

will be discussed on the basis of a field study carried out by Ral&shBzyand Kov&cs (1975). Four different carbonate rocks of Triassic age were investigated: (a) Middle Triassic (Karni) dolomite (6barok);

(b) The same rock type in a fractured zone along a cross fault with well observable karstic forms (chemically enlarged openings) (Sz&r); (c) Upper Triassic (Nori) bedded limestone; the beds having dolomite content to a varying degree (Csolnok); (d) Upper Triassic (Dachstein) karstic limestone (Dorog). At each place, three directions approximately normal to each other were fixed on the wall of the rock (one vertical and two horizontal). The width of and the distance between the openings were measured along these lines, so that linear porosity could be calculated for different lengths in each direction. The scattering of the data determined for independent stretches having equal lengths, decreased gradually as the investigated length was increased.

426

3 Permeability of natural layers

Szlir, dolomi fe vertical direction

Fig. 3.4-16. Determination of average linear porosity and representative elementary length (dolomite, SzBr, Hungary) (after BalBshBzy and KovBcs, J., 1976)

The enveloping ciirves enclosing the points constructed from the calculated data in a coordinate system representing the length on the horizontal and the linear porosity on the vertical axis, tend to a horizontal line indicating the average linear porosity of the rock. A limit can be determined, above which the distance between the enveloping curves and the average value is smaller than 10% of the latter. The length belonging to this scattering ( I ) can be regarded aa the representative elementary length of the rock, belonging to the investigated direction. Ten pairs of enveloping curves determined for ten independent stretches at one investigated place, for one given direction, are presented in Fig. 3.4-14 (Obaroli, dolomite vertical direction). The comparison of these curves proves that they are very Rimilar to each other, and both average linear porosity and representative elementary length have only random variation. Hence, these parameters can be regarded as probability variables characterizing the rock at a given place and in a given direction. The curves enveloping all the calculated data grouped according to the measured directions are shown in Figs 3.4-15; 3.4-16; 3.4-17 and 3.4-18 for the four investigat'ed rocks. If there were more than ten independently determined, average, linear porosity values in a particular direction, the data were evaluated

427

3.4 Fissured and fractured rocks

Table 3.4-6. Porosity and representative elementary length of four different carbonate rocks (after Balhshhzy and Kovhcs, J. 1976) Porosity [%I Type of ruck

I

Direction

Middle Triassic x horizontal dolomite (6barok) 6O-186' y horizontal 96O-276' z vertical Middle Triassic dolomite in strongly faulted zone (SzBr)

Upper Triassic bedded limestone (Csolnok)

Upper Triassic karstic limestone (Dorog)

are11

1.46

4.34

2.61 1.88

3.27 3.86

8.02

7.39

6.63 1.89

9.76 13.16

1.55

1.79

0.80 1.06

2.48 2.29

1.96 non measurable 0.66

-

I

1 [ml

volytrio

0.24 6.69

0.24 0.24

x horizontal

120'-300' y horizontal 100-1900 z vertical

30.0 14.80

6.0

N

x horizontal

6Qo-249O y horizontal 169O-339O z vertical

1.2 3.26

x horizontal

24O-204O y horizontal z

vertical

---

-

2.60 -

1.6 0.8

0.32 0.60 0.32

34.0 30.0 6.0

3.2 2.4 2.4

30.0

8.0

statistically as well. The empirical distribution curves are also shown in the figures in these cases. The distribution could always be well approximated by a normal distribution function. The results of the investigation (average linear porosity and representative elementary length) are listed in Table 3.4-6. Considering the relationships existing between linear, areal and volumetric porosity, as explained in Chapter 1.1, all the parameters can be calculated if the linear porosity values are known in three orthogonal directions (nkx;nLy;nLz).Areal porosities in planes normal to the 2 axis (nAx),the y axis ( n A y )and the z axis (nAz)are: nAx = 1 - ( 1 - n ~ y (1 ) -~

=1-

- nLz) ( 1

=

- nLz)

-(

+ + +

L Z )

- %LX)

~ L Y ~ L Z ;

nLZ

nLX

;

(3.4-4a)

( l - %y) nLx nLy * The volumetric porosity ( n )can be similarly calculated ib

+

= 1 - (1 - nLx)(1 - nLy)(1 - nLz) nLxnLynLz.(3.4-4b)

The areal and volumetric porosity values determined on the basis of the investigation discussed are also listed in Table 3.4-6.

428

3 Permeability of natural layers

a

ant

aaz

a03

am

005

n,

Csolnok, bedded limestone

o

I

ant

aoz

I

I

a03

a04

I

a m n,

I

Fig. 3.4-17. Determination of linear porosity and representative elementary length (bedded limestone, Caolnok, Hungary) (after BalhhBLzy and KOVACEIJ., 1976)

Dorog, karstic liinestone 24 204" borizontW O -

a04

007

L! 006

0.05 004

003

ant 0

a 2.5 50 z5 /on 12.5 150 L [ m i Fig. 3.4-18. Determination of linear porosity and re resentative elementa length (karstic limestone, Dorog, Hungary) (after B a d & z y and Kodcs J.,?hS)

3.4 Fissured and fractured rocks

429

Apart from porosity the other parameter important for the hydraulic characterization of the seepage field is hydraulic conductivity or intrinsic permeability. There are several sets of data available in various publications giving directly the Permeability of carbonate rocks. Some of them are listed in Table 3.4-7. The uncertainty of these data, however, is very high. The execution of a correct pumping test with more than one observation well - as described in Chapter 3.1 - is very expensive and time consuming, because of the high expected yield, the great thickness of the carbonate layers, and the high cost of drilling in hard rocks. The simplified field testa are, therefore, applied in most cases, e.g. observation of the lowering of the water level in the pumped well during a short period of pumping, and recording the rise of the water until recovery to the original level; the water injection test (water take in, Lugeon test) in an open bore hole or through the wall of a separated stretch of the hole; recording the corresponding yield and draw-down data of existing wells operated for any other purposes; etc. There are, however, attempts to calculate the permeability of the rock even from these indirect data. Boreli and Pavlin (1965) have found, for example, that Lugeon's number can be expressed in the form of equivalent hydraulic conductivity 1 Lugeon = 0.07

-

0.15 [m day-l] = 1

-

2x 10-8 [m sec-'1.

Although the transformation of the directly observed data into the form of hydraulic conductivity or matrix porosity is possible, there are many authors, preferring the publication of the original measurements, who know the very uncertain character of the parameters derived from simplified field tests. It is necessary, however, to provide information on the conditions influencing the measurements. Thus the yields of various wells are comparable, only if the draw down, the length of the screen (or that of the open stretch of the bore hole), and the diameter of the well is known. I n many cases the yield (&) related to the draw-down (s)is used to obtain the required parameter ( q being the specific yield of the well). The latter is also divided sometimes by the length of open stretch of the well ( I ) t o obtain q' the specific yield of the rock (or double specific value) (Table 3.4-8): q = &Is;

q' = q/E =

9 . sl

(3.4-5)

It is necessary to note here, that all the problems explained in connection with the determination of permeability, occur not only in the case of carbonate rocks, but in the investigation of other fissured and fractured rocks as well. For this reason, data, other than for permeability were also collected in the previous section concerning the various aquifers composed of hard rocks. The detailed explanation of the problems is given here not only because carbonate rocks are the most important aquifers, but. because the development of dominantly water conducting channels is the most probable occurrence in this type of rocks. In addition the chemical action of the water may have varied effects on the network of joints which in turn has uncertain effects an the permeability.

I Type of rock

Location

~

l

Hydraulic conductivity [cIn/sec-’]

Measuring method

W

’d

Middle Triassic

Nagyegyhhza

water injection

Upper Triassic

Nyirhd

Pumping tests

Limestme Cretaceous

Nagyegyhhza

Eocene

NagyegyMza

water injection water injection

Karstic Busko Blato limestone (Yugoslavia) Gypsum and anhydrite Jezireh (Syria) anhydrite

Lugeon tests P”Pi% tests

Tertiary limestone

pumping tests

Limestone Campanian Eocene

Sout,h Carolina (USA) Beaufort County Jasper County Savannah Tunisia

8.4~10-5

1.7 x 10-5

1.3 x 10-6

6.0 x 10-2

2.5

x 10-1

2.0 x 10-2

1.5 x 10-4

10-5 4.5~

7.6 x 10-8

2.5 x 10-4

1.1 x 10-2

3.0 x 10-6

lo-?

3.0 x 10-1

1.5~ 10-3

1.5 x

Schmieder et al.

1.1 x 10-1 6.10 - 3 4 . 1 0 - 4 1.10-~4.10-3

in fault zones

-

8.8 x lo-’ 1.2 x 10-2

6.5 x 10-2 G.O x 10-4

eD

I.

G %

Pavlin Mortier and Safadi Siple

6.10 - S2.10 - 3

pumping tests

$

Goseelini and Schoeller

b-

2

1

431

3.4 Fissured and fractured rocks Table 3.4-8. Yield of wells draining carbonate rocks in Hungary

-

Typeofruck d doloPnite

Budapest Vhosliget Budapest Zugl6 Biikk-1 HarkBny-3 Harkhny-4 Magyersdk MiSkOlC-1 Miskolc-2 Szentendre Budafok LeBnyfalu VisegrBd Bia KomBrom-1 Tapolca (HGN-39) SBrosptak-2 HBv~z-3 Koml6-17 VBC TorokbBlint PBpa-2

Trkzssic d 1

d 1

open'-

Depsion

ofqoFm 64

6 6 6 20 138 28 40 -

470

37

9.6

960 9000 2600 800 1000 1000 4860 60 60 1040 686 460 2600 600 860 1000 610 2600 860

960 3600 1000 800 960 923 4860 60

100 40 666 400 142 76 404 4

60 660 686 460 2600 410 860 1000 610 1176 860

45 632 14 86 31 173 3333 136 196 286

3260

3260

120

76.0

-.

1 1

._

1 1 1 1

-

27.1

1

1 1 1

Cretaceous 1

-

capacity

470

1 1

1

speciac gridding

12.3

1.6 2.0 6.7 12.2 12.0 14.0 1.0 12.2 1.1 34.0 29.0 1.3 4.9 0.3 4.6 6.0 3.0

1 1 1

Yield a/minl

Author

Korim verbal information

60

Tho determination of hydraulic conductivity is hindered even when data from correctly executed pumping tests are available, or the draw-down of mining activity is observed by more observation wells and the water amount drained through the galleries of the mine is measured (mining can be regarded in this case as a large scale pumping test, and the area of the mine as a fictive well). The difficulties are caused by two factors in these cases: (a) The unknown depth of the layer taking part in the transport of water; (b) The role of large openings in water conveyance near the draining structures (wells, shafts, galleries). The thickness of the carbonate rocks is generally very high, a layer of several hundred metres is not rare. The total thickness of carbonate formations composed of more layers and forming a unified reservoir of karstic water may surpass thousand metres in many cases. At the same time the decrease of porosity as well as permeability is observed in the carbonate rocks as in other hard rock aquifers. The approximation of the relationship of hydraulic conductivity vs. depth as an exponential function is proposed by many author:

K = K Oexp (-/3z);

(3.4-6)

432

3 Permeability of natural layers

where KOis the hydraulic conductivity at the surface, and z is the depth. The ?! , factors was found by Boreli and Pavlin (1965) to be 0.0052 in karstic limestone (Busko Blato, Yugoslavia), while from data presented in Fig. 3.4-1 9 and characterizing Middle Triassic dolomite (Transdanubian Central 0.0030 waa cdculat%d. It is also Mountain Range, Hungary), a value of /l= shown in the figure, that the probable distribution of hydraulic conductiv-

h-5

10

Km/secl Ktm/ser? Fig. 3.4-19. Frequency distribution of hydraulic conductivity in Middle Triassic dolomite (6barok) Hungary) (Schmieder's data)

ity can be approximated by using a logarithmic normal distribution function in the case of carbonate rocks, as is generally proposed in the study of fissured and fractured rocks, although the frequency distribution presented in a semi-logarithmic coordinate system has some asymmetry. Some other measurements show, however, that there are areas where the distribution can be better described by normal rather than logarithmic normal distribution functions (Fig. 3.4-20) (Kirhly, 1973). From Eq. (3.4-6) an average depth can be calculated, the transmissibility of which (assuming a constant K Ohydraulic conductivity) will be equal to that of an infinite layer with hydraulic conductivity decreasing with depth:

-

-[Koexp(-Bz)dz=KO-;

0

b

T =Korn,,=JKdz=

1 rnae = - .

1

B

(3.4-7)

B

It is shown, however, by Fig. 3.4-11, that porosity may have high local values at elevations where the conditions are at present or were in the past

3.4 Fissured and fractured rocks

433

n Fig. 3.4-20. Frequency distributions of hydraulic conductivity showing normal and logarithmic normal character (after KirBly, 1973)

suitable for the development of large dissolved openings. Hydraulic conductivity being closely dependent on porosity, may cause this parameter t o have local changes superimposed on the decreasing with depth trend. Thus the depth of the flow field estimated or calculatedon the basis of any theoretical method is very unreliable, and. therefore, many investigators do not try t o divide the influence of depth and hydraulic conductivity, but publish the value of transmissibility (T= K m ) as calculated from pumping tests or from the depressions developed around mines draining a karstic reservoir (Table 3.4-9). The influence of large openings around the draining structure waa mentioned as the second cause hindering the evaluation of the data gained from pumping tests. Discussing the determination of the representative elementary unit in carbonate rocks, it was mentioned that in the close vicinity of wells or shafts, the water movement cannot be investigated as a threedimensional seepage extending t o the whole space of the fractured rock. The draining structure crosses only a few water conveying elements which may be large channels or fractured zones. The relationship between the discharge and the head loss is determined here, therefore, by the resistance of these openings, and turbulent movement frequently occurs in the channels because the movement of a large amount of water is concentrated into a few openings. Farther from the draining structure, the flow is better distributed, especially if a fractured zone normal t o that is in contact with the water t,ransporting channels penetrated by the well. It can be assumed, therefore, that outside a given distance from the well, the whole network of openings take part in water transport at an almost even rate (Fig. 3.4-21). An example is shown in Fig. 3.4-22, to demonstrate the influence of the water transporting channels on the development of draw down in the vicinity of a pumped well. The drilling crossed a steeply dipping fractured zone. The water intake was 1600 l/min. After some fluctuation, a constant draw down of 73.0 m waa maintained by pumping water at 900 llmin (the total duration of the testing period was two months). From the water level data observed in the observation wells located in a relatively dense network around the pumped well, the draw down cone was constructed, as shown on the map. About 99% of the total head loss occurred within a distance of 28

Table 3.4-9. Tranemissibility of carbonate rocks

Dolornite Middle Triaasio Upper TrisSaic

depression of mine water injection Pumping depreseion of mine depreeaion of mine depreaaion of wells

3.0 x 10-3 3.4 x 10-6 7.0 x 10-8 7 . 8 ~10-3 2.2 x 10-2 7.7 x 10-2

6 . 4 ~10-8 8.0 x 10-4

depression of mine Pumping

8.Ox 10-8 2.3 x 10-4

2.1

depreeeion of mine depreaaion of mine water injection water injection depression of mine

3.0 x lo-; 42 . x lo-' 1.1x 10-6 4.4x 10-6 2.0~10-4

pumping tests

-

-rag

TatebBnp Negyegyhem NEbgpgyhb Ajke

Illinois (USA) Silurian dolomite Cook County D u Page County Kane County Lake county MoHenry County will county

-

1.8 2.6

x 10-3 x 10-7 -

-

-

x 10-2 -

-

-

-

6.3 x 10-8 3 . 6 ~10-8

2.2 6.0

-

9.ox 10-8 1.7 x lo-* 2.4~ 1.6 x 10-0 2 . 4 ~10-0 4.1 x

-

-

1.8~ lo-' 1.4~ 10-4 1.7 x 10-8

3.0 x 10-s 11 .3x 10-4 2.6 x lo-'

-

W

x 10-8 x 10-7

-

-

Schmieder et al.

-

Limestone amend

pumping testa

Intelmediete Kerstified

Gypsum and anhydride

Jezireh (Syria)

pumping tests

SSW

pumping tests N

Jmper County Sevenneh

Burdon Mortier end

8.8 x 10-2

Tertiary limestone South Carolina (USA) Beellfort county

1.0 x 10-8 9.1 x 10-6 1.0 x 10-8

4.3 x 10-0 1.8x lo-' 8.0 x 10-2 4.3 x lo-;

N

Siple

3.4 Fissured and fractured rocks

435

zone of flaw through I! zoneaf seepaje individual channels Fig. 3.4-21. Water conveyance of large openings in the vicinity of a well

between 150 and 400 m from the well depending on the direction. Only the remaining 0.70 m draw down caused the general lowering of the piesometric surface of a reservoir further away, where the flow could be investigated aa seepage. The transmissibility calculated from the outer part of the draw down cone (T,= 7 . 5 l~o h 3 [m2 sec-l]) is practically equal to that calculated from the influence of a nearby mine (TatabSnya) (T,= 8.2 x [m2 sec-l]), while the transmissibility of the drained fractured zone (T, = 5x [m2 sec-l]) is between the average and maximum value of ths parameter determined by water injection tests for the same layer (see Table 3.4-9).

A very similar result waa achieved by KirBly (1973). Investigating the hydraulic conductivity of carbonate rocks in the La Brkvine syncline (Switzerland). He found that the parameter determined by field tests ranges from 5 x to 1 x lo-' m sec-l, while a value higher than lo-* is characteristic of large regions aa calculated from the yields of springs and the slope of the piezometric surface. In his opinion this deviation of the order of two magnitudes or more is caused by the water conveying capacity of large openings spaced at a considerable distance from each other, the exploration of which by drilling is very improbable. The example and the explanation given in connection with the role of individual water transporting elements around the draining structurea draws attention to the fact that water movement cannot be characterized aa seepage within a given distance from a well or a shaft. It is necessary, therefore, that this limit should be determined for each layer aa a further hydraulic parameter supplementing porosity and hydraulic conductivity. A further consequence of this condition is that the total head loss maintaining the flow, has to be divided into two parts: i.e. the local loss near the draining structure calculated aa the resistance of the water conveying chan28*

436

3 Permeability of natural layers

?%

-10-

u

faults and fractured zone lowering of the water fable in cm

wafer levef in hie pumped well im.a. s.22' [

I

, 1 1 1 1 1

1

3.4 Fissured and fractured rocks

437

nels and zones crossed by the structure, and the seepage loss along the outer part of the influenced area where the development of a three-dimensional movement through an almost homogeneous flow space can be supposed.

3.4.3 Models for the characterization of flow through the openings of solid rocks As already discussed, the construction of a conceptual model t o characterize seepage through porous media has t o be composed of several steps i.e.:

- Determinationof a geometrical model describing the flow space in the simplest possible form suitable for the approximation of the effects of the complicated network of water transporting channels; - Tnvestigation of the dynamic character of flow in the geometrical model and the application of physical laws to describe the movement; - Combination of the geometrical model and the movement equations into a conceptual model; Comparison of the theoretical values with data measured under natural or experimental conditions to prove the reliability of the model and t o caldulate its constant factors, if necessary. ~

The geometrical models proposed t o characterize the flow through the openings of fissured and fractured rocks can be divided into two large groups : (a) Network of interconnected pipes with constant or varying diameters; (b) One or more series of slits between parallel plates. The first model is very similar to the network of the mains and distributing pipes of an urban water supply scheme. Generally, the water transporting channels are supposed t o have a circular cross section. The joints of the stretches may be distributed regularly in apace (e.g. forming an orthogonal network) or randomly. The diameter of the pipes may be constant between the joints, or they may be variable. Using a constant diameter between the joints, there are two further possibilities, either all the stretches are modeled from pipes of the same diameter, or their sizes are different. The change of the diameter may be either random or i t can be determined according t o some known behaviour of the solid matrix (e.g. t o simulate the anisotropic character of permeability, larger diameters are used in one direction). Some possible variations are represented in Fig. 3.4-23. The dynamics of flow in a closed conduit is a thoroughly investigated process in hydromechanics. Relatively simple solutions are known for networks composed of circular pipes having constant diameters between the joints. For this reason, this simplified form is applied generally t o simulate the karstic openings through carbonate rocks (OllBs, 1963). There is a problem, however, hindering the application of these types of models: i.e. too many parameters have t o be determined for the complete description of the models and these cannot be derived from easily measurable physical

'

438

3 Permeability of natural layers

general arrangement of pipes andjoints

m

regular networks spacing

variations of akmeters differmt spacing

irflgdar network

,

Fig. 3.4-23. Sketches showing various networks of pipes simulating the interconnect,ed channels of carbonate rocks

rock characteristics of the aquifer. Even using the most simple model (constant pipe diameter at every stretch and regular distribution of the joints), the pipe diameter, the length of stretches, the density of the joints in various directions have to be estimated, and the data depend on the position and the distribution of the large water conveying channels and they cannot be interrelated with such simple physical parameters aa porosity or the statistical distribution of openings. For this reason, the models composed of pipes aro generally applied only to solve a given problem in a welldefined and not too large part of the carbonate rock terrain. I n this caae the hydraulic parameters which develop under the influence of known boundary conditions (e.g. the discharge, the position of the water table or piesometric surface and its change in time) can be measured, and the best fitting network of pipes can be determined by the method of trial and error. Hence, the network of pipes is usually proposed for small scale modelling of a special process occurring in karstic zones of carbonate rocks (Mijatovid, 1970a),and not generally to describe the hydraulic conductivity of the large maw of a fissured and fractured aquifer. A further reason can also be given to support the previous statement. In the entire rock maas there are generally only a few large openings, while it

3.4 Fissured and fractured rocks

439

is divided by thousands and thousands of h e joints, and smaller or larger fractures, especially in fault zones. Thus, the total permeability of the rock is more strongly influenced by the latter type of openings, than by karstic channels. The role of dissolved galleries becomes dominant mostly in the vicinity of draining structures, ;here the resistivity of the aquifei is basically determined by the character of individual water transporting elements one setof slits parallel to the

various networks of slits

of differ& widths

anisotropic model

,A

of differentspacinj

Fig. 3.4-24. Sketches showing various networks of slits simulating the interconnected joints of fractured rocks

(karstic channels or strongly fractured zones). The joints, fractures and faults can, however, be better simulated by narrow slits, than by pipes. It is the general opinion, therefore, that the hydraulic conductivity of a large mass of carbonate rocks (or any other type of fissured and fractured aquifer) can be described by using a geometrical model composed of a network of narrow slits running through the solid mass in different directions and intersecting each other (Lomidze, 1951; Louis, 1970). This model may be composed of either regularly or randomly distributed slits, and the width of the slits may either vary or be constant. The anisotropy of hydraulic conductivity may be simulated either by choosing different spacing of the slits ro by applying different widths in different directions (Fig. 3.4-24).

440

3 Permeability of natural layers

The dynamic characterization of water movement in a narrow slit bordered by two smooth walls parallel to one another, is well known if the flow is laminar (gravity and friction are the dominant forces). Let us investigate the flow in a vertical slit (Fig. 3.4-25a). Because of the small width of the slit, the component of the velocity vector [v(v,, v,, v,)] normal to the wall

Fig. 3.4-25. Interpretation of symbols used for the derivetion of the Hele-Shaw equation

(v,) is zero, while i t follows from the laminar condition of flow that the change of velocity in the x and z directions compared t o that in the y direction is negligible: vy = 0 ;

Considering the condition where gravity is the only accelerating force, the total potential acting on the unit maas of the fluid can be composed of two terms: i.e. potential and pressure energies: rp=

1

3

z+-.

(3.4-9)

3.4 Fissured and fractured rocks

441

Substituting the approximations summarized in Eqs (3.4-8) and (3.4-9) into the Navier-Stokes equation, the simplified form of the latter expressed in the three directions of the coordinate system is aa follows:

(3.4-10)

Integrating these equations, the result is

where the integration constant is determined from t.he condition given at the symmetrical axis of the slit: avx - -avz =o. y=o; -

aY

aY

(3.4-12)

Executing a further integration and taking the velocity as zero a t the wall (if y = +a, then v, = v, = 0 ) ,the velocity distribution normal to the wall can be determined

(3.4-13)

or supposing that the direction of the 2 axis is parallel to the main direction of flow, and using the I = dP, relationship (the hydraulic gradient is equal ds

t o the change of potential in the direction of flow), an equation is obtained describing the velocity distribut,ion in the slit: 9 (y2 - a y . v(y) = - -

(3.4-14)

2v

From Eq. (3.4-14), the amount of water transported during a given time in a section of the slit having unit length normal t o the flow direction, can be determined: +a

q = j v ( y ) d y = - -1 -9- P I ; 12 v -a

(3.4-15)

442

3 Permeability of natural layers

where b = 2a is the width of the slit. The mean velocity of the fluid is equal to the ratio of discharge and area (3.4-16) and, using the analogy of Darcy’s law, the hydraulic conductivity of a slit can also be calculated (3.4-17) Investigating the flow in a horizontal slit (Fig. 3.4-25b) a modification of velocity distribution can be expected because the gravity force acts at right angle to the walls. I n practice, however, the gravity term of potential ( z ) is very small compared to the pressure (ply), and, therefore, its influence can be neglected. Hence, all equations [Eqs (3.4-15); (3.4-16) and (3.4-17)] derived for vertical slits can be applied for all fractures, independent of their direction. It is necessary to note here, that there is a basic difference between the parameter determined by Eq. (3.4-17)and Darcy’s hydraulic conductivity. The latter, multiplied by the hydraulic gradient, gives the seepage velocity (or flux), which is equal to the ratio of the flow rate and the total area normal to the flow direction and thus is smaller than the mean velocity in the pores [see Eqs (1.1-9)and (1.1-lo)]. In contrast, Eq. (3.4-16) immediately gives the mean velocity, and, therefore, the hydraulic conductivity of the slit does not characterize the rock, but only one opening. It is possible, however, to derive Darcy’s parameter also from the relationships given previously, if it is assumed that an s number of slits having b width and parallel to the flow direction cross the unit area (normal to the flow) of the rock (Fig. 3.4-26). (Because 8 is the number of slits counted within a unit length, its dimension is [L-l].) The total flow rate through this area is the product of the number of slits and the water conveying capacity of each opening. This flow rate is equal to the flux (vD seepage velocity), because the area investigated perpendicular to the flow has an extension of unity. Hence, Darcy’s L

t

--!

A = / total erea a = SbI free area Sb . nL= - tinear poros?y

I

Fig. 3.4-26. Interpretation of Darcy’s hydraulic conductivity in fractured rocks

3.4 Fissured and fractured rocks

443

hydraulic conductivity can be calculated as that of the slit multiplied by linear porosity v D = - =QS q = s b 1 -gb z I ; A 12 v (3.4-18)

K D = nLK,; where nL = sb; because sb is the total width of the slits, and it has to be related to the total investigated length to get the linear porosity. In the present caae this length is equal to unity, and, therefore, the product sb immediately gives the linear porosity. There are many conditions, chosen arbitrarely, in deriving Eq. (3.4-18) for the characterization of the conductivity of a fractured medium and they hinder the direct application of this model in the description of the behaviour of natural rocks. . The most important hypotheses are a8 follows: (a) The walls of the openings are parallel to one another; (b) The slits are straight and parallel to both the flow direction and each other; (c) The flow is laminar; Id) The openings have smooth walls and are not intersected by other fractures; (e) the network of openings is composed of joints having equal width. The errors caused by the approximations and the methods suitable to consider the actual natural conditions have been investigated by different authors and the results will be summarized in the following paragraphs. The effect of walls not being parallel to one another waa investigated by Wilson and Witherapoon (1974). They have found that the width (beH)of an opening with parallel walls and hydraulically equivalent to a fracture having a distance [b(Z)] between its walls varying along its length ( L ) ,can be calculated from an integral equation: L

L

(3.4-19)

If the walls of the actual opening are straight forming an angle of 2 8 (Fig. 3.4-27), the solution of the integral is (3.4-20)

which does not differ considerably from the mean value (for an opening having a length L = 20 cm, the smaller width b , = 0.205 cm and the larger b, = 0.415 cm, the equivalent width is beff= 0.286 cm). The comparison of systems, assumed to be equivalent hydraulically, has shown that the network of fractures of varying width can be simulated by openings having parallel walls and using the equivalent width.

444

3 Permeability of natural layers

L

el'

I .

slit of v%ryingwdtb

bydraulicallly equivalent slit between parallel walls Fig. 3.4-27. Symbols used for the derivation of the width of a slit wit.h parallel walls, equivalent to an opening of varying width

The second hypothesis includes two cbnditions i.e. ( a ) The network of openings is composed of one system of straight fractures running parallel t o each other; and (b) They are also parallel to the flow direction, and thus normal t o the investigated cross section, which is perpendicular t o the flow direction.

To consider conditions other than those proposed in the first hypothesis, the linear porosity has t o be determined in two orthogonal directions of the cross section normal to the flow direction. If the network of openings is composed of two orthogonal systems of fractures and linear porosity wa8 determined in the two main directions, the total flux and conductivity of the solid matrix can be determined by simple superposition (Fig. 3.4-28a):

K , = nL.,K,+ n,,K,.

(3.4-21)

If there is only one system of parallel straight fractures forming an angle of

" ;

/? with one direction of linear porosity and - - /? with the other, the two

I

directions being normal t o one another , the flux can be calculated consid-

3.4 Fissured and fractured rocks

L

M5

J

nLi=nL;tnL;"=S, 'bfiSyb; Fig. 3.L-28. Characterization of the position of the slits in a plane normal to the flow direction

446

3 Permeability of natural layers

ering only one of the measured linear porosities (Fig. 3.4-28b): 1 g b: sin2p; Q = Q1= slqll, = I slbl 12 v

or

Q = Q2 = ~ ~ q , 1== , IS,b,- 1 g b2 c0s2/3; 12 v

(3.4-22)

K , = nL1K , sin2 /I = nL2K , cos2p; 8

1 1 because the length of a fracture within a unit area is I , = -or I , = sin /I cosp ’ and the width of the opening normal to the wall is b, = b, sin 9, = b, COB /I. The third possible variation is where two systems of parallel fractures form angles of p‘ and /I” respectively, with the first measured line, and where n:,), the total linear porosity is composed of two members (nL1= nil one belonging to the first system (including s; fractures having a width of b; and a hydraulic conductivity of K i ) , and the other to the second system (8:) b; and K i ) . The parameters along the other line can be similarly deterni2; s; and s;; b; and b;: KI and K:). Using the premined (nL2= 4 2 vious derivation and Fig. 3.4-28c, the total flux can be determined from the data measured along one of the fixed lines:

+

+

12 v

or

Assuming that the normal widths of the slits are equal in the two systems of fractures (bh = b i ) , the following simplifications can be executed

b A = b: = b; sin p’ = b; sin ,!I” = b; cos p’ = b; cos @” ;

nL1= 8; bi

+ sf b; ; nL2=

S;

b;

+ s; b; ;

(3.4-24)

€2 = & I = nL1Kn21 = Q 2 = nLzK,,I; 1

Q =(nL1Kn,+ 2

1

nL2Kn2)I = -Kn (nL1+ nL2) 1. 2

The contradiction between Eqs (3.4-21) and (3.4-24) is only virtual, because the value of linear porosity measured in the caw of openings normal to the investigated line, is smaller than when the slits form an acute angle with it. Hence, the use of a dividing factor of 2 is reasonable. It is necessary to note that under natural conditions, the parameters can be measured only

3.4 Fissured and fractured rocks

447

on the free surface of the rock and both the /?‘and /I” angles are random variables. It can be easily understood that their most probably value is /?’= /I” = nI4, and hence, sin2 /?‘ = sin2 8” = COG8’ = cos2/I” = 0.5. Considering b; and b; as statistically determined probable values of widths measured along the first fixed line, they can be regarded as equal to one another and substituted as the most probable parameter 6,. The hydraulic conductivity of the slit is calculated from this value (El). The same approximation can be followed along the second line resulting in the characteristic Hence, the final value of the flux determined from parameters of h2 and g2. data memured on the surface of the rock is (3.4-25)

It can be seen that the discrepancy between the flux through an orthogonal network of fractures, if the values of linear porosity are measured along the main directions [Eq. (3.4-21)], and the discrepancy if the fractures are distributed randomly and linear porosity is measured dong lines chosen arbitrarily [Eq. (3.4-25)], is still higher (4 : 1). Absolutely regular distribution of openings cannot be expected in nature. On the other hand the most probable declination of n/4 is not Characteristic for all fractures. The two values of flux [those calculated from Eqs (3.4-21) and (3.4-25)] can be regarded as probably the highest and the median parameters, respectively. The reduction explained in the previous paragraph is due only to the declination of the slits from the measuring lines in a plane normal to the flow direction. Further decrease of flux is caused by the fractures possibly forming acute angles with the flow direction and consequently they are not normal to the cross section of flow in each caae, and hence, correction in the third dimension becomes necessary. This effect can be studied as in the tortuosity of the channels composed of the pores of clastic sediments (Bear, 1972). Double correction becomes necessary in this case because the normal width

I

plan? normal

C to the flow direction

Fig. 3.4-29. Characterization of the position of the slits related to the general flow direction

448

3 Permeability of natural layers

is smaller than that measured in the cross section normal to the flow and the length of the fracture (L,) is longer than that of the sample ( L J . The ratio of the corresponding values can be calculated, if the angle a formed by the axis of the opening and the general flow direction is known (Fig. 3.4-29):

b, = b cos a ;

(3.4-26)

1 cos a

L,= L s - .

Considering the pressure head between the t w o faces of the sample to be constant ( A H ) , the gradient along the fracture ( I , = AH/L,) is smaller than that calculated for the sample ( I = AHIL,). The water amount transported through a slit can be calculated from the following expression: q = - - 1b 3 g- 12 v

AH L,

-_-

12 v

H b3 -cos4a ; Ls

(3.4-27)

and thus the factor of reduction is C O S ~a. The direction of fracture and thus the a angle as well can be regarded as a random variable, and on a purely geometrical basis, the most probable parameter is a = n/4.The reduction factor calculated in this way is c0s4 a = 114,which can be regarded as the median value, while the upper limit is unity. Combining the two necessary reductions, the probable flux through a unit area of the rock can be determined from linear porosity and the average width of the openings measured along two orthogonal lines on the surface. The value depends, however, on the direction of the fractures related both to the direction of the measured lines and to the general flow direction. The position of the fractures is a random variable, and, therefore, the possible range of flux has to be determined. It was shown that the equations derived on the basis of the geometrical model give the median and the probable highest value of flux. Considering that the smaller values are more probable (the distribution is not symmetrical), a lower limit calculated as a quarter of the median can be proposed for practical use: 12 v

64

9 < nL1-1 b!i + nL2

[

12v

(3.4-28)

12 v 12 v

I .

The next process, which may cause discrepancy between the actual resistivity of the solid matrix and that calculated from the theoretical model, is the development of non-laminar flow in large openings. As explained'in Chapter 2.1, the dynamic reason for the development of the various types of seepage in the zone of high velocities, is the change in ratio of the influence

3.4 Fissured end fractured rocks

449

between friction and inertia, and, therefore, the limifs of laminar, transition and turbulent seepage can be determined with a given value of Reynolds' number. In the case of fractured rocks, the mean velocity in the openings is generally used as a characteristic velocity, while the characteristic length may be either the average width of the fracture ( b ) , or its hydraulic radius

( R ) , which is equal to half the width

, in the case of a long slit

between two parallel walls. L l = 18.6 cm

Fig. 3.4-30. Sketch of the Hele-Shew model used for measuring the hydraulic conductivity of slits

There are several publications proposing slightly different limits on the basis of experiments executed either in laboratories or in the field (e.g. Lomidze, 1951). It was discussed, however, in connection with the investigation of loose claatic sediments that the establishment of a general relationship between velocity and hydraulic gradient is more suitable for the practical characterization of non-laminar flow, than the determination of validity zones and different equations to describe this relationship within the zones (see Section 2.2.2). For this reason, experiments were carried out to determine a general movement equation for a slit with parallel walls [similar to Eq. (2.2-36) valid for porous medium], instead of the investigation of the limits dividing the various zones of seepage (KovAcs, 1973). The equipment used for the experiment waa the most simple form of Hele-Shaw models: i.e. confined horizontal flow of viscous fluid between two glass plates (Fig. 3.4-30). The length of the flow field waa 20 cm. Two variations in height were applied, 5 and 10 cm. Eight different widths were investigated between 1.1 and 0.1 cm. After having developed the permanent flow at fixed levels of the head and the tail water, the flow rate, the temperature of the fluid and the piezometric head at three points were measured. The next step waa the calculation of the corresponding values of the mean 29

450

3 Permeability of natural layers

velocity and the hydraulic gradient. Assuming a similar relationship between these two variables, to that described by Eq. (2.2-36), i.e. (3.4-29)

the data were graphically represented in a coordinate system having z&$ on the horizontal and

on the vertical axis (Fig. 3.4-31). The graphs

0

o

i

z

3

4

5

6

7

e

i u ~ i/ z 1 3 / 4 / 5 1 6

g

X,"314

Fig. 3.4-31. Relationship between velocity and hydraulic gradient in slits witb parallel W d B

proved the reliability of Eq. (3.4-29). The factors A and B were determined ae the intersections of the lines with the vertical axis and the slopes of the lines respectively. These data were analyzed afterwards ;t9 the functions of the width of the slit (Fig. 3.4-32). The m s ~ l t can s be summarized in the form of a general relationship between mean velocity and hydraulic gradient:

A=

B= 1314

=

(3.4-30)

0.0087

3.4 Fissured end fractured rocks

451

The structure of the equation is the same aa that determined for loose claatio sediments. The laminar term (i.e. the h t member on the right-hand side) is numerically equal aa well, while the second term (the turbulant one) is smaller in the slit (the ratio is about 1 : lo), than in the pores. This difference may be explained by the fact that the development of turbulent flow is slower between smooth glms plates, and the role of inertia is smaller, than in the pores, where the actual velocity changes from section to section. It waa, therefore, felt necessary to investigate the character of flow not only in a single slit, but a180 through a network of fractures. The comparison of the results of such experiments with data determined in a single fracture,

Fig. 3.4-32. Factors of Eq. (3.4-29) aa the functions of the width of slits 29 *

452

3 Permeability of natural layers

can characterize the surplus resistance caused by the vortices developing at the edges of openings intersecting each other. Bocker’s experiments (1972) were carried out with 5 cm cubes, arranged regularly (Fig. 3.4-33). Three variations were investigated, in which the widths of the straight slits between the cubes and forming three orthogonal systems of openings were 1, 2 and 3 mm, respectively. The flow was maintained by a constant pressure head between the upper and lower faces of the model. Its main direction was parallel to the vertical slits and normal to the horizontal ones. Thus the model creates basically the same water

Fig. 3.4-33. Sketch of the model used for measuring the hvdraulic conductivity of three sets of orthogonal slits (Bocker, 1972)

movement as that developing between parallel walls, but simultaneously the yield of two systems of slits intersecting one another at right angles is investigated (in the model, the water conveyance of two times seven slits). There was no flow in the horizontal slits. They served only to equalize the pressures between neighbouring vertical openings and to decrease the smoothness of the walls by intersecting them and forming sharp edges at the intersections. of the developing steady flow was measured. This value The discharge (Q), divided by the horizontal area of the vertical openings gives the actual mean velocity and Darcy’s seepage velocity can be calculated by multiplying 0.7n). Further the latter with areal porosity (in this special case nA measured or calculated parameters were: temperature (to calculate the viscosity of water); pressure head at six points inside the model (to check pressure distribution); and hydraulic gradient (calculated as the ratio of head loss to the length of the model). The interrelated values of mean velocity and hydraulic gradient were evaluated in the form of graphs, as in the case of single slits (Kovbcs. 1973). The results are represented in Fig. 3.4-34, giving the factors A and B in Eq. (3.4-29) as the functions of the width of slits. Although the scattering of the points is larger, than that in Fig. 3.4-32, the trend of the relationships is the same as that determined for a single slit, and thus the structure of the

-

3.4 Fissured and fractured rocks

-8 - 7 -6 -5 -4

-3

-2

6 - 10-3 -8 - .7 -6 -5

-4 -3 .

-2

- 10-4

Fig. 3 . k 3 4 . Factors A and B &B the functions of the width of slits in the case of intersecting networks of openings

general movement equat.ion is unchanged, only the numerical values have to be slightly modified (considering the scattering observed in the graphs, the limits of the possible ranges of the factors are given below):

(3.4-31)

454

3 Permeability of natural layers

Considering that in nature, the walls of tho openings are even less smo0t.h than in the applied model, not the average values but the upper limits of the indicated zones can be used for establishing the general movement equation: 314 0.01 I3/4= 1 252)ert 3’4 (3.4-32) . K,J * On the basis of explanations given in connection with the probable difference between the laminar flow developing in a straight slit, and the actual seepage through the fractures of hard rocks, the water conveying capacity of the latter can be determined, if the width of the openings is a constant value, (or if this parameter is variable, but it can be substitutedwith an average value) and the number of the fractures within a given size of the rock-mass is known. The hydraulic conductivity ( K , ) of one opening of b width is calculated from Eq. (3.4-17). The mean velocity (vef,) vs. hydraulic gradient (I)relationship for one fracture, can bedetermined by substituting this value into Eq. (3.4-32). If the development of non-laminar flow is not expected, even this equation can be simplified by neglecting the second term on its right-hand side and obtaining a linear relationship between the two variables in question. Measuring the values of linear porosity and average width of slits along two orthogonal lines on the surface of the rock perpendicular to the general flow direction, the expected flux (or the hydraulic conductivity) of the solid matrix can be calculated by combining the parameters determined in the two directions according to Eq. (3.4-28). The obstacle most difficult to overcome, when the theoretical models have to be applied in practice, arises from the last hypothesis mentioned on which the model was baaed: i.e. the model is composed of openings having equal width, while in nature a large variation of the fractures can be observed from very narrow joints to large karstic channels. There are some publications aiming at the determination of an average width and number of the uniform slits within a given maas. This investigation is baaed generally on the statistical analysis of the probable distribution of both the widths of openings and the distances between the fractures (e.g. Kirtily, 1973; Snow, 1970; Romm, 1966). These methods included, however, some uncertainties in connection with the determination of the hydraulically equivalent slits. An effort was made, however, to characterize the resistance of fractured blocks by considering the water transport through all the openings of various sizes. The application of this new model will be discussed in the following section.

I

+

lmvq

3.4.4 A conceptual model for the determination of hydraulic conductivity of fissured rocks The method described in this section combines three elements i.e. (a) The geometrical model, substituting the actual openings with straight slits, considering also their position on the basis of linear porosity values measured in at least two orthogonal direction [Eq. (3.4-24)];

3.4 Fissured and fractured rocks

455

(b) Dynamical analysis of flow in slits between two parallel walls, the result of which described the mean velocity vs. hydraulic gradient relationship, independent of the type of flow in a general form [Eq. (3.4-32)], or in the case of laminar movement with a linear formula, neglecting the second

ueffl

I

term of the equation mentioned earlier I = 1.25 - ;

Kl

(c) The statistical analysis of the frequency distribution of both the width and spacing of the fractures, which creates the necessary transition from the theoretical to the practical characterization of the actual openings of natural rocks. The derivation is based on the research carried out by BalBshBzy and Kov&cs (1975). Hence, the same four types of carbonate rocks, (Middle Trimsic dolomite and its karstic form, Upper Triassic bedded limestone and karstic limestone of the same age) are investigated here, the representative elementary lengths of which were given previously. The measurements were executed on the surface of the rock (mostly in mines) in three orthogonal directions. The values mertsured were: (a) The number of the openings (S)within a given length (L); (b) Their width ( b ) ; (c) Their distance from the origin of the fixed line (the latter data were used only for the determination of the representative elementary length). The total range of the measured widths (0 < b < bmax) wm divided into (db,where rndb > b,,,), and the number of openings belonging to each interval was counted. The total number of openings and the numbers belonging to each db interval were divided by the meaaured length to get their specific values (average numbers of openings within a unit length). The specific numbers in the intervals were also related to the specific value of the total number of openings to determine the frequency distribution of the data according to the width of the openings. The symbols used in the statistical analysis of the data are as follows: S total number of fractures observed along a line of length L; Si number of openings belonging to the i-th interval of Ab range along m number of uniform intervals

I

the h axis z S i = S ; b, mean width of openings within the i-th interval, thus the limits of the

interval are determined by b, 5

Ab values; 2

s total specific number of the openings

I 3 8

=- ;

si specific number of openings belonging to the i-th interval

456

3 Permeability of natural layers

uifrequency distribution value of the specific number of openings belonging to the i-th interval m i- 1

a(b; d b ) is a mathematical form to approximate the empirical frequency distribution curve, which depends on the random variable of the width and on the length of the db interval chosen arbitrarily (Fig. 3.4-35).

-Bihjintprra/ m Ab Fig. 3.4-36. Theoretical sketch to represent the frequency distribution of the widths of openings

Apart from the upper and lower limit of the a value mentioned previously, there are two further conditions to be considered in determining the mathematical approximation of the empirical frequency distribution: (a) The area covered by the empirical frequency distribution is equal to the d b interval, and the area enclosed by the horizontal axis and the approximating curve should be equal to this value

A i= a, Ab m

A =

m

2A, =2 I-1

i-1

; m

0,

Ab = A b z a i = Ab ;

(3.4-33)

i-1

A = fa(b, Ab)db = Ab. 0

(b) The linear porosity can be expressed as the sum of the measured width of openings related to the total length, and, if the d b interval is small enough, the sum can be substituted with the product of the frequency distribution values and the mean widths in each interval. The relationship derived

3.4 Fissured and fractured rocks

457

on this way should be valid also for the approximating curve:

2

bk

k-I . nL = L ’

zsibi nL =

i- 1

L

rn

rn

=zsibi=szaibi; i- 1

(3.4-34)

i- 1

rn

nLAb=s z a i b i A b ;

-

I- 1

nLAb = s

a(b,Ab)b db.

0

6b3

0

a3

R6

a9

12

,-,

L5 b

6b

a./a 0.09 a.08

0.07 0.06

R05

a.04 0.03

aaz 001

0

a 0.3 06 09 12 15 b a 0.3 a.6 0.9 LZ 1.5 b 18 Fig. 3 .4-36. Frequepcy distribution of the widths of openings in Middle Triassic dolomite (Obarok,Hungary) (after BalbhBzy and KOV&XJ., 1975)

458

3 Permeability of natural layera

Csolffok,

I. 0

B 0.9

0.8

0.7 0.6

0.5 0.4 1

3.3

1 1

',

\

1

1

0.2 0. I n

U

3

2

0

Gb 0la

6

4

Gb

1i

bb

6b

0

3

4 b

I!

o.,"l

g02

Gb

0.09 0.08 3.07

0.06

0.05 0.04 0.03 0.02 0.u

0

n I 2 3 b 4 0 1 2 3 4 b Fig. 3.4-37. Frequency distributionof the widths of openings in U per Trieseio bedded limestone (Csolnok, Hungary) (after Balbh&zyand Kov&)oeJ., 1976)

459

3.4 Fissured and fractured rocks

After plotting the actual, memured data (see Figs 3.4-36; 3.4-37; 3.4-38 and 3.4-39), and considering the conditions listed, a first approximation by an exponential distribution function appears to be very suitable:

-

u(b, db) = a exp ( - c b ) ;

.s

a a exp ( - c b ) d b = - - [exp (-&)I,, = - = Ab; C

C

0

n Ab

L

=

u

S

s

bexp(-cb)db=-a

0

c=-;

S

a=&-;

nL

S

nL S

u(b,d b ) = d b -exp nI.

Instead of the usual statistical investigation, the forms of the empirical frequency distributions of bu, baa and b3a were compared to theoretical curves determined by the proposed approximation. The application of this Bb

0.07

Dorog, karsfic limestone

006

vertical

OD3 1

0.05

0.04 5

003

)I

Gb

0.32 mi D

‘

00/2

1

2

9

4

b

66

oa/o DOOB ODD6

;

om4 \

1

1

I

2

-

a002 1

3

I

j

1

4 6 0

Fig.3.4-38. Frequency distribution of the widths of openings in Upper Triassic karstic limestone (Dorog, vertical direction, Hungary) (after Bel&hBzy and KovBcs J., 1976)

460

3 Permeability of natural layers no2

6b2

l77L-l

Dorog, karstic limestone

Fig. 3.4-39. Frequency distribution of the widths of openings in Middle Triassic karstic limestone (Dorog, horizontal direction, Hungary) (after BaltiahBzy and Kov&cs J., 1976)

method was chosen, because these products of the higher power of the width, will be used in future calculations and it is also necessary to see the reliability of the listed values. It is evident that each of the product would have only one local maximum, if the values were approximated by Eq. (3.4-35) 1 [b3a],,, if b= - ; [b2a],,,if

31

i

i.e.:

2 b = - ;[ b 3 ~ ] , , ,if, ~b = - .The measured values

C

C

C

show, however, two or sometimes three local maxima, indicating that the approximation is not acceptable. Considering also the conditions mentioned above, it is proposed, therefore, to use a series of exponential functions composed of r members instead of the simple exponential frequency distribution: r

a(b;db) =

r

2 aj = z a j e x p ( - c j b ) . j= 1

j= 1

(3.4-36)

461

3.4 Fissured and fractured rocks

From Eqs (3.4-33) and(3.4-34),considering also the upper limit of a, the following conditions can be determined for the constants ( a j and c j ) in Eq. (3.4-36).

(3.4-37) i a j e x p [-cj$)

I

j- 1

Some physical explanation can be found to support the use of more than one exponential function for describing the frequency distributions of the widths of openings. There are many geological processes taking part in the creation of joints and governing the development of their size (lithification and initial tectonic stresses in the rocks, exfoliation and weathering, later tectonic movements, chemical and mechanical enlargement of the fractures). It may be assumed, therefore, that the total number of openings observed at present, reflects the influences of these actions in a combined form. It is reaaonable to divide the whole system into sets of openings each described by an exponential distribution function, obtaining a series composed of aa many members aa the number of dominating factors in the development of the recent conditions. Equation (3.4-37) does not contain sufficient information to calculate the constants in closed form, if more than one exponential function is used. Their values can be estimated, however, after the construction of the empirical distribution function, by using the trial and error method (Fig. 3.4-36). The accuracy of the method can be increased by applying smaller Ab intervals in the zone of very narrow joints (Fig. 3.4-37). The interrelation between the position of the local maxima of the ba, b2a and b3a curves and the c parameters, assists in determining the numerical values of the latter (Fig. 3.4-38). If the range of the measured widths is large, it is advisable to use a relatively high Ab value for the determination of the smallest c parameters, and repeat the investigation with the residual part of the frequency distribution [i.e. a, - a, exp (-c,b); where a, indicates the a value determined experimentally] by using a smaller interval (Fig. 3.4-38). In this case it is

[:

necessary to consider the a parameter as linearly proportional to Ab -=

=""I.

After the determination of the first estimation of the parameters in Ab this way, the values can be checked and corrected to satisfy the conditions expressed in Eq. (3.4-37) (Table 3.4-10). It is quite evident that the parameters depend on the material of the rock and on the processes governing the development of the openings. It is interesting to note, however, that the grouping of the c values can be observed in the cases investigated. A member of the series with a parameter c1 of

Qraphically estimated parameters

8pecifla n u m b Direction

Type of rock

1

Middle Trieseic dolomite (6-k) Middle Trisseic dolomite in strongly

l

a

l

3

vertical

0.1347 0.2361 0.1929

3.30 2.60 2.60

2.7 x 2.7 x

x

12O0-30O0

0.1076

3.60

1.6~

1 . 0 ~10-3

y

100-1900

0.2661

4.30

4.0~

-

X

y z

6O-186O 96'-276'

L

l

Upper T r i d c bedded limestone (Csolnok) Upper Triassic karstic limestone (Dorog)

-

-

2.6 2.6

-

-

11.0 11.0 11.0

-

-

1.6~

12.0

1.2

0.20

0.01

4.0X10-'

16.0

1.2

-

0.01

-

-

-

-

-

-

vertical

0.2198

4.00

4.6~

-

-

13.6

1.2

-

z y

69'-249' 169O-339O

1.8 x 1.8 x 8 . o ~10-3

-

14.0 14.0 14.0

1.4 1.4 1.4

-

vertical

4.10 4.10 4.10

-

z

0.1146 0.0761 0.0960

x

24O-204O

0.0662

2.66

1.6~ lo-'

1 . 8 ~10-4

11.6

2.6

0.30

0.06

y

non meesurable vertical

-

-

-

-

2.80

-

-

0.0388

11.0

2.6

0.30

-

Linear POroeitY nL

d h k

5

Y Z

SZSr

Ceolnok

Dorog

-

z

z

C%l

I

nr. Ab

I

-

1.7 x

lo-'

0.0326 0.0320 0.0292

3.00 3.08 3.34

9.4 x 10-2 3.1 x 3.0 x

8.02 6.80 1.88

0.2238 0.0663 0.0263

3.00 3.76 3.80

1.6 x 4 . 0 ~lo-' 4 . 6 ~lo-'

1.66 0.80 1.06

0.0406 0.0320 0.0329

3.16 3.76 3.20

2.0x 10-2 1.4~ 8.0 x 10- 3

1.96

0.1060

2.76

1.4 x

0.66

-

0.0610

1.2~ -

1.6 x

-

-

2.82

lo-'

-

1.7 x

lo-'

-

-

-

-

-

-

1.0 x

-

1.6 x 4.0~10-6

-

-

-

-

-

1.2 x

-

1.7~ lo-'

-

1.8 x

lo4

11.0 10.7 11.7

3.6 2.6 2.6

10.6 12.7 12.8

1.2 1.2 1.2

11.0 13.0 10.9

1.2 1.2 1.2

11.8

2.6 __ 2.9

-

-

-

11.8

0.20

-

-

-

0.01 0.01

-

-

0.30

0.06

-

0.38

W

F

- i ! z 5

Paremeters corrected by considering the condition to be antiseed

1.46 2.61 1.88

-

ta

4

feulted zone

( S 4

--

a

-

a

% s

3.4 Fissured and fractured rocks

463

approximately 11, was characteristic for all places. This set of openings might be created by initial tectonic stresses. The second member is also 3.5 due perhaps to generally characteristic with a parameter c2 = 1.5 weathering and exfoliation. Smaller c values were determined only for karstic layers and strongly faulted zones. The number of cases investigated is, however, very low. It would, therefore, be premature to draw further consequences from the data, but the investigation opens a promising path in this direction. The following step is the hydraulic application of the statistical model. The linear porosities measured in three orthogonal directions (nu, nLy,nLr) and probable distribution of the specific number of openings,

-

are the basic data required. Here s,, sy and s, are the probable specific numbers of opening within an interval d b , with a mean value b in the x,y and z directions respectively; sx0, syo and sz0 are the total specific number of openings in the same directions; the symbols u,, ay and IS,stand for the sum of the determined series of exponential function exp ( -cj b) in x direction

1

.

The flow rate through the fractures observed along one line of unit length, can be calculated m the spocific number of openings multiplied by the flow rate of one slit having a width corresponding to the relevant Ab interval: (3.4-39)

Supposing we first investigate only the laminar flow, and substituting the movement equation accordingly, the following relationships can be given to calculate the water transporting capacity of the fractures in question m

0

(3.4-40)

alexp(-cjb)db=

s l g r ---I26 db15 v 1-1

a. 2. cj4

0

The method of summarizing the influences of the fractures observed on a surface in two orthogonal directions, has already been explained, when the construction of the geometrical model was discussed [Eq.(3.4-28)]. Following the same steps, the most probable (median) flow rate in the z direc-

464

3 Permeability of natural layers

tion through a unit area in the zy plane (i.e. flux),and the virtual hydraulic conductivity or intrinsic permeability in the same direction, can be calculated from the following equations:

(3.4-41)

The same parameters in the other two main directions can be cdculated by substituting the parameters ( 8 , a, c) determined in the plane normal to the direction in question. The permeability parameters calculated from the measured data are listed in Table 3.4-1 1. The magnitude of the data seems to be very realistic. Their reliability can also be proved by comparing them to data measured in the same layer. For this reatjon Fig. 3.4-19 is partly repeated here because the field tests represented in the figure were carried out in the same layer aa one of those investigated here (Middle Triassic dolomite, Obarok; Fig. 3.4-40). The hydraulic conductivity vs. depth relationship is shown in the figure. The data calculated by this method indicated a higher value [KCa,= (2.4-3.4)x x m sec-l] than extrapolated from measurements carried out atvarious hydraulic c~?nducl~vt ty at the surface

K rm/secj Fig. 3.4-40. Comparison of measured and calculated values of hydraulic oonduotivity

Table 3.4-1 1. Hydraulic conductivity of carbonate rocks calculated by using the conseptual model Hydraulicoonductivity [m see1]

Miniiumlength Direction

!Cype of rock

Consideringthe non-laminar cheracter Of flow -a

Middle Triassic dolomite (6barok) Middle Triaasic dolomite in strongly faulted zone (Szhr) Upper Trimic bedded limestone (Csolnok) Upper Triassic karstic limestone

2

Y

6O-186O 96O-276'

2

vertical

X

12O0-30O0 100-1900

Y z

vertical

X

69O-249O 169'-339'

107.0 132.2 167.2 2260.6 3660.6 326.1

z

vertical

263.0 400.9 708.6

2

24'-104'

2337.6

Y Y

(Doros) 2

non meaaurable vertical

3.43 x 10-4 2.41 x 10-4 2.86 x 10-4

21.7 292.6 314.2 2.89 x 10-4 4.37 x 10-4 6.07 x 10-4

17.8 241.6 269.3 2.89 x 10-4 4.36 x 1 0 - 4 6.06 x 10-4

3 . 4 1 9 ~lo-' 2.403 x 2.842 x lo-'

3.371 x 2 . 3 7 3 ~lo-* 2.806 x 1 0 - 4

3.098 x 2.238 x 2 . 8 0 0 ~10-4

0.32 4.20 4.62

outside the

validity zone

2.87 x 10-4 4.33 x 1 0 - 4 6.03 x 10-4

2.77 x 1 0 - 4 4.17 x 10-4 4 . 8 4 ~10-4

2.17 x 1 0 - 4 3.26 x 10-4 3.78 x 10-4

6'

F

FL P

F3

g 2

f2 0.628

2230.0

3.428 x lo-' 2.409 x 2.848 x

0.611

0.633

0.097

outaide the validity zone

466

3 Permeability of natural layers

levels (K,,,, = 3 . 5 ~ m sec-l). If the calculation is repeated by using only the fht member of the series of exponential functions (assuming that the second one characterizes a process acting only on the surface i.e. weathering, exfoliation) reasonable agreement is achieved [Kred= (5.0-7.5) x m sec-l]. The reliability of the assumed frequency distribution can also be checked in the same figure. Originally, the empirical frequency distribution of measured data of hydraulic conductivity waadetermined. Because the conductivity of a rock mass is proportional to the third power of the width of openings (R= n,R,oc b3) the b3 vs. b3a relationship plotted on a suitable scale can be compared to the empirical distribution curve. The figure also proves the good agreement between the two results achieved in different ways. Kir&ly’s ideas (1973) concerning the role of very large openings in the development of the relatively high regional hydraulic conductivity of carbonate terrains, are also supported by the following method. He has found - as already mentioned - that it is necessary to aasume a network of relam sec-l spaced tively narrow zones with a conductivity higher than a distance of 2 4 km, to explain the large flow rate and relatively flat slope in a region, where the measured conductivity ranges from 5 x 10-a to 1x lo-’ m sec-l. Hydraulic conductivity of the Middle Triassic dolomite was determined by the statistical method in two places: (1)where the rock is relatively intact (6barok) and (2) in a strongly faulted zone (Szhr). The ratio of the two values is 1 : lo6 which agrees well with Kirhly’s ohservations. When explaining the concept of the continuum approach and representative elementary unit and also when discussing the application of the latter for the characterization of h u r e d and fractured rocks, it was observed, that the flow through the channels can be described by using the seepage equations, if the flow field is many times larger, than the representative elementary volume. The model is also suitable to determine the limit of the size below which the water conveying capacit,y of individual water transporting elements has to be investigated. The basic concept of this analysis is to set up a limit of the flow rate related to the total discharge of the system (e.g. 95’73, considering the very large but rare openings conveying the amount of water above this limit (e.g. 5%) to be negligible. The error caused by neglecting the very large fractures having a width larger than the limit value (bh),can be calculated, and the requirement is that i t should be smaller than or equal to a given value ( E ) :

-

(3.4-42)

where Q is the total flow rate [Eq. (3.4-40)], while the limit value is br

exp (- c j b ) db = 0

(3.443)

467

3.4 Fissured and fractured rocks

_-_-

Ab15 v

i-1

Calculating the possible error, the members of the series of exponential functions having high c parameters can be neglected, taking into account only the last member with parameters a, and c,. Thus, the simplified form of Eq. (3.4-42) is as follows

($bR + -bg c:

1

2

c:

+ bh c, + 1

exp (- c, bh)

<

E

.

(3.4-44)

After a numerical value hasl been given to E (0.05 can be proposed, which neglects only 5 % of the total flow), Eq. (3.4-44) can be solved numerically by expressing the product of b, c,. Using the limit of 5"/0, the result is 7.75

c, bh> 7.75; and at the limit b, = -.

(3.4-45)

Cr

The probable number of the openings having length is

R

width of

bh

within a unit

or neglecting once again the members of the series having higher c parameters and considering only the last one 8,'-

sa, exp (- 7.75) = 4 . 3 ~

sa,.

(3.4-47)

It is necessary to note, that the a parameter and thus, the probable specific number of the openings having a width of b h aa well, is the function of the Ab interval chosen arbitrarely. It is reaaonable, therefore, to deter, to the limiting mine a design value for the interval [ ( A b ) h ]proportional width, and transform all the results calculated by using any Ab interval to this system. The determination of such a design value is an arbitrary step once again, and i t can be proposed that the b, width should be the mean value of the 5-th d b interval. Considering the listed conditions, the probable specific number of openings having the limiting width and also those within a length of L can be calculated 1

( A b ) , = -b, = 0.22 b, ; 4.5

(3.4-48)

30*

468

3 Permeability of natural layers

If a characteristic length is defined in such a way aa to be the shortest size of a field regarded as a continuous seepage field, the requirement is that the probable number of an opening having a width of (1 f O.ll)bh has to be 1 within this length s h = 1; consequently (4.4-49) ~ b 1 0 4 ~b c, 104 L=--=---* ab, a, s a, 7.75 The data calculated in this way are also listed in Table 3.4-1 1 , and they agree well with the observations around pumped wells (see Fig. 3.4-22), and also with Kir4ly’s hypothesis, where the spacing of large water transporting 4 km. formations has to be about 2 The final step still to be solved is the combination of the general movement equation [Eq. (3.4-32)] and the model. Unfortunately, the equation gives the hydraulic gradient as a function of mean velocity, while in the model the velocity haa to be substituted in explicit form. For expressing the velocity from the general equation in an easily amenable form,some terms have to be expanded into series, which require the application of some new restriction of the validity zone. Thus, the equation loses its general character, but it remains suEcient for the combined characterization of the laminar and transition zones. For this derivation Eq. (3.4-32) has to be modified

-

where

and

I

1;

4B

<

A2

413=

20]/5.

(3.4-50)

A2

This is the condition for use in expanding the term in question into series. This limit is sufficiently high to ensure the application of the result in solving almost all practical problems. Following the derivation and expanding the right-hand side once again into series, neglecting the terms with higher powers,

I A= v=-( A 413 1 - -1314

) --4/3

IK,

[

0.03 K, ( 7 ) 3 1 * 1 7 1 4 ] . (3.4-51) 1.25 Consequently the virtual hydraulic conductivity of a slit depending on hydraulic gradient, if the flow is non-laminar is as follows:

Ktrans=

I

=K ,

[=

- 0.03 (I

1-

K3 314

(3.4-52)

469

3.4 Fissured and fractured rocks

The water transporting capacity of one slit having a width of b is (in the second term its power is considered to be 4 instead of 4.125 for simplification): Q

= bv,ff

--

0.8b3 - 1 . 8 3 ~

12 v

y3/4

[-

b4] ; (3.4-53)

and that of a set of fractures within a unit lengt,h can also be calculated:

Q =1 9I 12 v

-[ 5

Ab

0.8

b3aj exp ( -cj b)db -

j- 1 0

Equation (3.4-54) provides the flow rate calculated on the basis of fractures measured along one fixed line. Applying once again the geometrical model [Ey. (3.4-24)], ;t8 in the derivation of flux in the case of laminar flow, the result is the seepage velocity (or flux), the hydraulic conductivity and the int,rinstic permeability in one of the main directions (after rounding up some figures):

The parameters i ~ other i directions can be similarly calculated from the data measured in a plane normal to the direction in question. It is shown by this derivation that the non-laminar character of flow can be considered with a member, depending on hydraulic gradient, to be sub-

470

3 Permeability of natural layers

tracted from the parameters determined for laminar movement. For characterizing the effect of inertia numerically the hydraulic conductivities calculated by assuming various gradients are compared also in Table 3.4-11. References to Chapter 3.4 BAOHMET, Y. (1966): Basic Transport Coefficients as Aquifer Characteristics. I A S H Sympoaium on Hydrology of Fracturerl R o c b , Dubrovnik, 1965. BALbsHdZY, L. and Kovdos, J. (1976): Determination of Hydraulic Conductivity of Triaesio Carbonate Rocks by Statistioal Analysis of Slits (in Hungarian). (Manuscript). Budapest. BEAR, J. (1972): Dynamics of Fluids in Porous Media. Elsevier, New York, London, Amsterdam. BOOKER, T. (1972): Theoretical Model for K m t i c Rocks. Ram& i a B a r l a n g M d , VOl. VII. BORELI, M. and PAVLIN, B. (1966): Approaah to the Problem of the Underground Water, Leakage from the Storages in Karst Regions, Karat Stora es Busko Blato, , Peruca and Krusoica. I A S H Sympoaium on Hydrology of Fractured & . ~ k aDubrovnik, 1965. BRACE,W. F. (1966): Some New Measurements of Linear Compressibility of Rocks. Journal of Geophysical Research, Vol. 70, p. 393 BERETON, N. R. and SKWNEB,A. C. (1974): Ground-water Flow Cbaracterhtics in the Triassic Sandstone in the Tylde Area of Lanoashire, Water Service%,August. A. A., INESON, J. and KOVALEVSKY, V. S. (1972): BROWN, R. H., KONOPLYANTSEV, Ground-water Studies Chapter 9. Analytical and Investigational Techniques for Fissured and Fractured Rocks bv M. Schoeler. UNESCO S W and Rev& in Hydrology, Pa&, 1972. BURDON.B. J. 11966): Hvdroneolow of Some Karstic Areas of Greece. I A S H Sympocrium on Hydrology of h-actUuured"kocb,Dubrovnik, 1965. C a m , S. (1966): The Hydraulic Properties and Yields of Dolomite and Limestone Aquifers. I A H S Symposium on Hydrolosy of Prractecrad R o c b , W m i k , 1965. DAVIS, S. N. and TURK,L. J. (1964): Optimum Depth of Wells in Crystalline Rocks. Ground-water, No. 2. DAVIS, S. N. and DE WXEST,R. J. M. (1966): Hydrogeology. John Wiley, New York. DE WIEST (1969): Flow through Porous Media. Academic Press, New York, London. ENQLUND, J. 0. and JPIILOENSEN, P. (1976): Weathering and Hydrogeology of the Brumunddal Sandstone, Southern Norway. Nordic Hydrology, No. 1. FORKASIEWIOZ, J. and PALOO, H. (1966): Tbe Procees of Recession of "La Toux de la Vis", Preliminary Study (in French). I A S H Symposium on Hydrology of Fractured R o c b , Dubrovnik, 1965. FRANOISS, F. 0. (1970): Study of Ground-water Movement through Fissured Media (in Frenoh). Doctoral Thesis, University of Grenoble. GONDOUM,M. and So-, C. (1968): Streaming Potential and the SP log. T r a m action of A m . Inat. of Minw and Metallurpiat Engineera, Paper 8023. GOSSELINI, M. and SOEOEIJBB, H. (1939): Observations of the Yield of Artesian Wells (in French). I U W General Aaaernbly, Wahington. KLRLLY,L. (1973): Explanatory Notes for the Hydrogeological Map of NeuchLtel Country (in French). Bulletin de la Soc&SNeucWd&edeeSciencee Natwdles,Tom. 96. KOVACS,G. (1973): Determination of the Hydraulic Parametere of Seepage Developing through Homogeneous Earth Dame (in Hungarian). (Manusoript). -, Budapest, No. 6. LEWIS,D. C., KRIZ, G. H. and BWOY,R. H. (1966): Traoer Dilution Sempling Technique to Determine Hydraulio Conductivity of Fraotured Rock. Water Resources Research, No. 2. LOMIDZE, T. M. (1961): Peroolstion in Fissured Rocks (in Russian).Cosgeoltekhizdat, Moscow.

References

471

LORBERER, A. (1975): Investigation of the Hydraulic Parameters of Oligocene Sandstone Having Double Porosity on the Basis of Well Data (in Hungarian). HidroMg i a i Kozliiny, No. 5 . Loms, C. (1967): Study of the Flow of Water in Fissured Rocks and its Influence on the Stability of Mining Structures and Slopes in Rocks (in German). Doctoral Thesis, University of Karlsruhe No. 1. Loms, C. (1969): A Study of Ground-Water Flow in Jointed Rock and its Influence on the Stability of Rock Masses. Rock Mechanics Centre, Imperial College, London. Loms, C. (1970): Three Dimensional Flow in Fissured Rocks (in French). Rock Mechanics Centre, Imperial College, London. MANUER,G. E. (1963): Porosity and Bulk Density of Sedimentaq Rocks. US. Geological Survey Paper, No. 1144-E. MAWCHA,L. (1972): Investigation of Natural Processes Influencing the Yield of Springs (in Hungarian). (Manuscript). VITUKI Budapest, No. 2162. MIJATOVI~,B. F. (1970a): Hydraulic Mechanism of Karst A uifem in Deep-lying Coastal Collectors. Bulletin of the Imtitute for Geological and 8eophysical Research, 1967. Beograd. No. 7. MIJATOVI~, B. F. (1970b): A Method of Studying the Hydrodynamic Regime of Karst Aquifers by Analysis of the Discharge Curve and Level Fluctuation during Recession. Bulletin of the Institute for Geological and Geophysical Reaearch, 1968, Beograd. No. 8. MORTIER,F., QUANQ, N. T. and SADEK,M. (1965): Hydrogeology of Volcanic Rock Formations in Northeast Morocco (in French). I A S H Symposium on Hydrology of Fractured Rocks, Dubrovnik, 1965. MORTIER, F. and SAXADI,CH. (1966): Karstic Phenomena in Gypsum of Jezireh (in French). I A S H Symposium on Hydrology of Fractured Rockt?, Dubrovnik, 1965. MUSKAT, M. (1937): The Flow of Homogeneous Fluids through Porous Media. McGrow-Hill, New York. ( h ~ t J s ,G. (1963): Hydraulic Investigation of Flow Rate in Fractured Rocks by Small Scale Model (in French). I A S H Bulletin, No. 2. RATS,M. V. and CHERNYASHOV, S. N. (1966): Statistical Aspect of the Problem on the Permeability of the Jointed Rocks. I A S H Symposium on Hydrology of Fr&ured Rocks, Dubrovnik, 1965. RIMA,D. R., MEISLER, H. and LONOWILL, S. (1962): Geology and Hydrology of the Stockton Formation in Southeast Pennsylvania. Penmylvania Topographic and Geological Survey, CFrOund-water Repwt, W-14. R o m , E. S. (1966): Flow Phenomene in Fissured Rocks (in Russian). Moscow. SAMPER, A. A. and NAVARRO, A. (1966): Problems of the Storage of Water in South-east Spain (in French). I A S H Symposium on Hydrology of Fractured Rocks, Lhbrovnik, 1965. SCHMIEDER, A., WIUEMS,T., SZILAQYI,G. and KESSER~, ZS.(1970): Investigation of Laws Governing the Movement of Water and Rocks in Aquifers (in Hungarian). B h y b z a t i Kutat6 IntBzet, Budapest, (Closing report, manuscript). No. 13-2/69 K. S C H O E ~H. R (1948): , Hydrogeological Regimo of Eocene Limestone in the Synclinal of Dyr-El-Kef (Tunisie) (in French). Bulletin de Society Geological de France Tom. 18. SCHOELLER, H. (1962): Ground Water (in French). Masson, Paris. SOHOEUER, H. (1966): Hydrodynamics in the Karst (in French). I A S H Symposium on Hydrology of Fractured Rocks, Dubrovnik, 1965. SHEVYAKOV, L. D. and MANXOVSKY,G. I. (1963): Experiences Gained by Dewatering Mining Sites of Productive Minerals Having Complicated Hydrogeological Conditions (in Russian). Gosgeoltekhizdat, Moscow. SIPLE, G. E. (1966): Salt-water Encroachment of Tertiary Limestone along Coastal South Caroline. I A S H Symposium on Hydrology of Fr&ured Rocks, Dubrovnik, 1965. SNOW,D. T. (1970): T h e Frequency and Aperturea of Frmtures in Rock. International Journal of Rock Mechanzcs and Mining Science. 1. STEWART, J. W. (1962): Water-yielding Potential of Weathered Crystalline Rocks. Stanford University Student M. S. Report. STEWART, J. W. (1964): Infiltration and Permeability of Weathered Crystalline Rocks. U.S. Geological Survey, Paper No. 1133-D.

472

3 Permeability of natural layers

STOABT,W. T.,BBOW,E. A. and RHODAMEL, E. C. (1964): Ground-WaterInvestigations of the Marquette Iron-MiningDistrict. Michigan Geological Survey, Technica2 Report, No. 3. TURK,L. J. (1963): The Omurrence of Ground Water in Crystalline Rocks. Stanford University, Student M. S. Report. VECCEIOLI, J. (1966): Direotional Hydraulic Behaviour of a Fractured Shale Aquifer in New Jersey. IASH Spptmkrn on Hgdrology of Fr&red Rock, Dubrovnik, 1965. WILSON,C . R. and W~TEEIWPOON, P. A. (1974): Steady State Flow in Rigid Networks of Fractures. W e Reeourw Reaearch, No. 2.

Part 4

Kinematic characterization of seepage

The practical problems of seepage to be solved in connection with the design and operation of hydraulic structures (dams, canals, wells, etc.) require knowledge of the main hydraulic parameters of water movement developing through porous media aa a result of pressure differences prevailing between two points or sections. The most important hydraulic parameters are the amount of water transported, as well aa the pressure and velocity values developing at various points inside the system. Theoretically the complete solution of various problems needs the calculation of certain hydraulic parameters at each point of the flow field. In practice, however, the investigation of some special points, or sections is quite sufiicient, for example the total flow rate through the field; the uplift pressure along the upper boundary of the flow field, or the position of the phreatic water table; the exit velocity, where water leaves the seepage field; and naturally the change of these parameters with time. The seepage f i l d or seepage space (flow field) is separated by geometrically well defined boundaries, and filled with porous material. Some of the boundaries are fixed, such aa the contact of the seepage field with impervious layersor structures, and the surface through which water can either enter or leave the investigated field. The position of others can change in time (e.g. phreatic surface). The first requirement for the determination of the hydraulic parameters js the complete characterization of the geometry of the seepage field (geornetrim1 conditions), which includes knowledge of geometric data sufficient to describe the position of the fixed boundaries, and methods for calculating the space coordinates of the phreatic surface (the latter being itself a type of hydraulic parameter to be determined). The second large group is composed of data describing the flow conditions within the seepage field. To solve practical problems the porosity [ n ( z ,y , z ) ] and hydraulic conductivity (or intrinsic permeability) [ K ( z ,y, z,) or k(z,y , z ) ] have to be known aa the function of the space coordinates. Conductivity and permeability may depend also on the direction of flow (anisotropy). In this case the parameter haa to be given in the form of a second-rank symmetrical tensor composed of nine members, as was explained in Chapter 3.1. Further flow conditions influence the water movement only in special case l e g the

474

4 Kinematic chmrtcterization of seepage

distribution of specific weight or density of the fluid, if i t is inhomogeneous; the change of porosity and permeability with time when the compressibility of the solid matrix within the flow space is not negligible; etc.). Considering the practical purposes of this book (which is also expressed by its title using the term hydraulics and not hydromechanics) , the listed special properties are not discussed further on, because the relationships derived in this way are not amenable to mathematical treatment, and, therefore, they are generally neglected in practice. Readers interested in studying the effects of either inhomogeneous fluids or compressible solid skeletons are referred to recently published books dealing with the hydromechanics of seepage (De Wiest, 1969; Bear, 1972). It is necessary to stress again that the two most important parameters characterizing the flow conditions (i.e. porosity and permeability) can be described as the functions of the space co-ordinates within the seepage field if the microscopic properties of the porous medium are considered in a combined form (accepting the continuum approach to determine the macroscopic parameters). As discussed in Chapter 1.1, the macroscopic concept is used not only to express the flow conditions with continuous functions of space instead of their randomly changing value, but on the basis of the continuum approach the kinematic parameters of the water movement can be described with mathematical formulae. The actual randomly varying path is approximated by a smooth curve and the effective velocity which changes rapidly with the cross section of pores, may be described as the average within the elementary volume (see Fig. 1.1-7). Various procedures and methods of the kinematic investigation are summarized in this part of the book. Apart from the already mentioned geometrical and flow conditions of the seepage field, the dynamic law summarized in Part 2 governing the relationship between seepage velocity and hydraulic gradient is the basis of the kinematic analysis. Knowing the two interconnected variables at every point of the flow field, all the hydraulic parameters can be easily determined. Seepage velocity is identical with the flux (flowrate through unit area), thus the total water transport can be calculated by integrating seepage velocity multiplied by an elementary length along a cross section. According to the definitions, the local value of the average actual velocity (as determined by the continuum approach) is the ratio between seepage velocity and porosity. Finally the hydraulic gradient is equal to the local change of pressure, thus its value multiplied by an elementary length and integrated along a line from a point where the pressure is known to the point in question, provides the pressure value at any arbitrarily chosen point. The existence of these relationships justifies the statement given previously that knowledge of seepage velocity and hydraulic gradient is a sufficient condition for the solution of any practical problems. The dynamic investigation, however, provides us with means suitable only to calculate one of the two interconnected variables (i.e. seepage velocity and hydraulic gradient) as a function of the other. The results of an analysis of the acting forces would assist the solution of the problems only if either the velocity or the gradient were previously given for each point of

4.1 Laminar seepage

475

the field and for every instant of the investigated period. This requirement is met only along the boundaries of the seepage field, where either the prevailiiig pressure or the developing seepage velocity (at least one of their components). is determined by the behaviour of the boundary (for example there is no component of velocity perpendicular to impervious boundary). By the nature of the given problem the pressure and its change in time along the entry and exit faces have always to be known to solve]the problem. The predetermined seepage velocity, gradient or pressure along the perimeter of the seepage field i s called a boundary condition. In the case of the investigation of a time-variant procees, either the velocity or the gradient at a point in time chosen as the beginning of the process must be known at each point within the field (apart from the boundary conditions, which have to be given in this case as functions of time). The declaration of the starting conditions of one of the two interconnected variables within the whole field is also a prerequisit for the determination of the hydraulic parameters of a time-variant seepage, and they are summarized in the form of initial conditions. While analysis of the forces acting (dynamics) ensures only the establishment of the connection between velocity and gradient, the investigation of the movement of water particles through the continuous flow field (kinematics) provides us with results which enable us to transfer the boundary conditions to an internal point of the field, and to follow the change of the hydraulic parameters starting from the initial conditions. The method of this investigation is summarized in Part 4 of the book.

Chapter 4.1 Kinematic relationships characterizing laminar seepage As already explained in the introduction, some basic approximations are accepted for the kinematic analysis of seepage. The most important among them is the application of a continuum approach which leads to the investigation of the movement of water particles through a continuous field, instead of fluctuating water molecules through randomly distributed and interconnected pores. The homogeneity of the fluid and the incompressible property of the solid matrix were also mentioned as basic hypotheses to simplify the investigation. I n connection with the solid skeleton it has to be supposed also, that the grains composing the porous medium are immobile, or at least their movement does not start in the range of velocities encountered in the investigation. Otherwise, the dynamic and kinematic balance is disturbed and the gradual or rapid deterioration of the solid skeleton may be expected (see Sections 3.2.1 and 3.2.2). The hypotheses listed in the previous paragraph are usually acceptable in practice. There are only few cases (i.e. water movement in strongly compressible sediments; development of boiling sand, or liquidization of the layer, etc.) when the consideration of special effects is necessary. For the

476

4 Kinematic characterization of seepage

determination of general kinematic relationships these approximations will be applied, therefore, further on. In this Chapter one more supposition will be accepted: i.e. the seepage is laminar, a n d thus the linear relationship between seepage velocity and hydraulic gradient is valid in each case (Darcy’s law). Jt is the general opinion, that this approximation - as with the previously listed ones - does not cause any major errors in the solution of practical problems. There are circumstances, however, where this supposition results in conclusions different from the kinematic conditions actually prevailing in the seepage field, although the number of such cases is not high, and in most cases the hypothesis of laminar movement R i acceptable. Some of the investigations included here will be repeated in Chapter 4.3 using a general lion linear relationship between velocity and gradient to demonstrate the possible solution of problems encountered in cases when the Beepage does not follow Darcy’s law.

4.1.1 Interpretation of velocity-potential and potential water movement

The general movement equations used for the determination of the hydraulic parameters of seepage do not differ from those generally applied in hydromechanics to describe other types of water movement if the paths of the water particles through the seepage field are characterized by average directions. The velocity is also an average value determined for the elementary representative volume (the local ratio of seepage velocity and porosity). Naturally the special properties of the seepage field and those of the movement discussed previously have to be considered in the equations. The equation of continuity is one of the basic formulae which expresses the conservation of mass in the form of the continuity of water movement. Applying this equation to a seepage field it must be borne in mind that a considerable part of the space is occupied by the solid matrix, and only the remaining part is filled with water (under saturated conditions). Thus the volume of water within the system is equal to the product of the volume of the seepage space and porosity. There are even cases (if the influence of attractive forces is not negligible), when the amount of moveable water is reduced, and, therefore, specific yield (n,) has to be used instead of porosity ( n ) in the product previously mentioned. The other basic equation of hydromechanics is based on the principle of energy conservation. There are many formulae generally used i n hydromechanics, which express this principle either in the form of a balance between the accelerating and retarding forces acting, or they may state the equality of the work performed by the accelerating forces along a given part of the path and the energy consumed by the resistance within the same stretch. Any of these formulae can be used for the characterization of seepage, if those forces are considered which are dominant in the various types of seepage on the basis of the dynamic investigation of the processes (see Part. 2). Another application of the principle of energy conservalion for

477

4.1 Laminar seepage

describing seepage is the expression of the energy consumed for the maintenance of movement on the basis of the previously discussed dynamic analysis, in the present case (when the laminar character of seepage is accepted as a basic hypothesis) by using Darcy’s law. The general form of the equation of continuity is

which can be simplified if the fluid is homogeneous and the system is incompressible :

aw, -+ av, -+

av, - divv= 0.

--

(4.1-2)

ax ay a2 This equation is valid only if there is no independent source or sink in the sj-stem (a point where a given amount of mass is added to or taken out of the system either continuously or temporarily). This supposition is generally acceptable except at those boundaries of the field, where the seepage space is recharged or drained. For these contours Eqs (4.1-1) and (4.1-2) have to be altered 80 that the right-hand sides of the equations are equal to the relevant amount of recharge or drainage. Excluding the drained or recharged singular points and contours from the investigated field, and analyzing a one dimensional flow insbad of the three dimensional one, the equation of continuity can be further simplified. If A indicates the cross section of the field perpendicular to the direction of flow, and the total flow rate is calculated as the product of this area and the mean value of seepage velocity (which is supposed to be constant within the cross section Q = Av),the new form is as follows: a v - -dQlA 1 (8 Q-and

a

as

A

as

- 1 (842 as A as

as

at

=o; (4.1-3)

8Q a A -=as at Physically this relationship states, (in the cwe of one dimensional flow, if the change of seepage velocity within a cross section is negligible, the mean value of velocity can be used) that the change of flow rate along the seepage field is equal to the change of the area of the cross section in time. It is quite evident, that the free surface of the wetted section has to be raised or lowered, if the amount of water entering a given stretch is not equal to that leaving it. The flow rate increases or decreases along the stretch, and - according to the principle of mass conservation - the difference has to be stored within, or released from storage in the investigated stretch. The change of storage is interrelated with the change of the position of the free surface in an incompressible system. The product of the elementary time interval ( A t ) and the difference between the entrant and exit flow rate

478

4 Kinematic characterization of seepage

(A&) [which has a dimension of volume (L3)] has to be equal to the change of the stored volume [the product of the change of cross-sectional area (AA)and the length of the stretch (As)].This explanation gives the differential form of Eq. (4.1-3),and at the same time indicates, how the special character of the seepage field has to be considered in the equation of continuity. In the case of seepage only a part of the space is active from the point of view of storage, which is not occupied by the solid skeleton. Considering also the amount of water bound strongly to the grains by adhesion the storage capacity of the layer can be characterized by specific yield (n,; see Section 1.4.2).The product of the change of the cross-sectional area and the elementary length of the stretch (AA As) has to be multiplied by specific yield t o get the change of the stored volume as a result of the lowering or raising of the phreatic water surface within the porous medium (the water table, where the surplus pressure is zero). Thus the equation of continuity in the m e of incompressible fluid and solid matrix, and one dimensional unconfined seepage is 88 follows:

-

aQ - n,-8 A , and !?! = n,-a Y respectively -as at ax at

.

(4.1-4)

The second simplified form can be applied if the parameters of seepage are constant horizontally at right angles to the direction of flow, and the investigation can be limited, therefore, to the analysis of the movement in a strip having unit width. I n this case the area of cross section is equal to the depth of the seepage ( y ) ,and the flow rate conveyed within this strip is the so-called specific flow rate ( q ) .It is also a special condition that the paths of the water particles are curves fitted to the vertical, parallel flow planes (in the case of one dimensional seepage they are straight lines parallel to each other). It is suficient, therefore, to investigate the movement in a twodimensional coordinate system instead of space coordinates, and if the x axis is parallel to the flow direction, the s length along the path can be substituted by the x coordinate value. Among the movement equations describing energy conservation, the Navier-Stokes equation is the formula used most frequently. Its general form is as follows:

+

(4.1-5) e p - gradp qv2veft+ 3 g r a d d i v v e f f . 3 at If the flow is continuous, i.e. there is no sink or source inside the field, and the fluid is incompressible [div v = 0; Eq. (4.1-2)],the laet member of Eq. (4.1-6)is equal to zero. The equation can be simplified by dividing i t by density. Considering the special character of seepage, the actual velocity has to be substituted by the ratio of the flux vector (seepage velocity) and porosity. Thus the following new form is achieved:

e -avew-

A 2 = P - grad- P + V Va v . n at

e

n

(4.1-6)

479

4.1 Laminar seepage

The dynamic conditions for the validity of Darcy’s law which was accepted as a basic hypothesis is, that only two dominant forces are taken into account i.e. gravity as an accelerating force and internal friction as a retarding one. It follows from this condition, that there is only one body force to be considered, i.e. gravity, which is a potential force, and thus its value can be expressed as the gradient of the gravitational potential [P = grad U = = grad (-29); see Eqs (2.1-1) and (2.1-2)]. The h a 1 result of this interpretation is the possibility of including the efiects of gravity and pressure i n one combined term (this fact also justifies the previously given statement that pressure can be considered as a supplementary part of gravity): 1 av --= ng a t

-grad

I:

z+-

V

+-V2v. n9

(4.1-7)

Equation (4.1-6) or (4.1-7) can be applied to describe seepage by comparing i t with Euler’s differential equation, which has a very similar form, but which waa derived for the characterization of the movement of any ideal fluid, i.e. without internal friction in contraat with the viscous Newtonian fluid. It is evident from this comparison, that the last member of Eq. (4.1-6) Y

(i.e. -V2 v) expresses the viscous resistance of the fluid caused by friction. Fric12

tion being the only retarding force, this resistance related to unit maas is equal to the total energy loss along a unit length of the path: (4.1-8)

where gh, is the potential loss (energy loss) expressed aa the product of head loss and the acceleration due to gravity. In the caae of seepage the potential loss can be expressed also aa a function of seepage velocity and hydraulic conductivity [aa is indicated in Eq. (4.1-8)], because according to Darcy’s law the head loss along a unit length is equal to the ratio of seepage velocity and hydraulic conductivity. Thus the final form of the Navier-Stokes’ equation for laminar seepage of homogeneous incompressible fluid through an incompressible and non deformable porous medium is aa follows: (4.1-9)

Equation (4.1-9) can be further simplified, if the movement is time invariant (steady flow) : v=-Kgrad

(

z+-

I;

=-Kgradh,

or expressed in the Descartian (orthogonal) coordinate system

(4.1-10)

480

4 Kinematic characterization of seepage

It is necessary to refer here once again to the practical purpose of the book. This is the reason, why the more general forms of the two basic equations are not discussed in this Chapter. Both equations (i.e. continuity and Navier-Stokes) can be determined for an inhomogeneous compressible fluid and solid matrix, and the inhomogeneity or anisotropy of the latter can also be considered. These generalized forms, however, cannot be handled mathematically in closed form to achieve analytical solutions. There are only a few cases when the equations can be solved numerically using large computers (e.g. if inhomogeneous and anisotropic permeability is given for the whole seepage field with tabulated parameters as the function of space coordinates). In engineering practice the fluid is always supposed to be incompressible and homogeneous. A few exceptions could be mentioned, for example the movement of the boundary surface of two different, but homogeneous fluids (e.g. seawater intrusion); or the compressibility of air is considered when the combined flow of two immiscible fluids (water and air) is the topic of the investigation (e.g. inatration through the unsaturated zone). T h e inhomogeneity of the layer cannot be explored in such detail, that the change of the properties could be characterized within the entire flow field in a sufficiently dense grid. If the flow field is inhomogeneous, i t is sometimes necessary to divide i t into more layers, so, that in one unit the variation of the soils physical parameters remains between previously determined limits. To solve the practical problems, whether the seepage field is composed of one or more layers, efforts should be made to have a large number of parameters measured at various points of each layer, and to ensure in this way the possibility of statistical characterization of both the properties of the layer and the hydraulic parameters of the developing seepage (some examples of the application of statistical methods were given in Part 3 of the book). Anisotropy also is considered only very rarely in practice, although i t may influence seepage considerably. This property can be also inhomogeneous (the change of permeability in various directions from point to point within the seepage field). Its characterization would require, therefore, the similarly close exploration of the layer as was mentioned in connection with inhomogeneitp. In general i t is sufficient in practice to estimate from statis tically evaluable number of measurements the probable rate of the two principal hydraulic conductivities, supposing that the layer has transverse anisotropy. In this case, the anisotropic flow field can be transformed into an isotropic by applying one non conformal mapping (linear enlargement in one direction; see Chapter 3.1). The hydraulic parameters of seepage can be determined in the new field on the basis of the relationship derived for isotropic media. The compressibil2y of the layer generally has to be considered in reservoir engineering, when the most important problems are the determination of the available oil or water in a closed system, the estimation of the change of pressure energy with time, and - on the basis of information concerning the parameters mentioned before - the preparation of the plan of operation for the exploitation of the stored fluid. The tasks listed heresre verypractical ones indeed, but the solution of the basic equations differs considerably from

481

4.1 Laminar seepage

that of other engineering problems. Thus, concerning the movement equations in the case of a compressible porous medium, reference is made only to Section 1.4.2, where the storage capacity of a confined aquifer waa discussed, and where the basic relationships describing this special type of seepage can be found. I n connection with the restrictions listed in the previous paragraphs except the compressibility of the layer (inhomogeneity and anisotropy of the solid matrix; inhomogeneity and compressibility of the fluid) the opinion is, that the determination of the physical parameters (and especially their changes both locally and in time) of the porous medium, and the fluid is a more dificult and serious practical problem, than the derivation and application of the generalized basic equations. Considering the uncertainties of the parameters, the approximation of the process of seepage with possibly simple mathematical formulae seems to be sufficient. However, readers interested in studying the more complicated methods can consult some recently published more theoretical books referred to already (De Wiest, 1969; Bear, 1972). Referring to Eq. (4.1-11) there is a scalar quantity determined as a single-valued function of the space coordinates, the diflerential quotient of which in each direction gives the component of the flux vector (seepage velocity) i n the direction in question at every point of the flow space. According to our defiition, this behaviour is the characteristic property of potential movement, and the scalar quantity t t f i e d to t>hepoints of the seepage space by continuous function is called velocity-potential:

V(X,

I + 71 .

y, Z) = K h = K z

-

(4.1-12)

Since in the case of continuous flow of incompressible fluid the divergency of seepage velocity [the sum of its three differential quotients determined in the directions of three coordinate axes perpendicular to each other (Eq. 4.1-2)] is equal to zero, i t can be stated that the sum of the sewnd differential quotients of the velocity-potential according to the axes of an orthogonal coordinate system has to be similarly equal to zero (Laplace’s equation):

There are numerous characteristic properties of the potential movement of fluids known from general hydromechanical analyses, the consideration of which makes the kinematic investigation of the flow easier and more simple (Milne-Thomson, 1955; Kellog, 1929; Nhmeth, 1963; Courant and Hilbert, 1937; Bear et al., 1968). The potential flow is always irrotational (rot v = 0). The apace distribution of the potentid function can be characterized by equipotential or potential surfaces, along which the change of the potential is equal to zero (Arp = 0). 31

4 Kinematic characterization of seepage

482

The velocity vector is perpendicular to the potential surfaces (v ds = 0; where ds is an elementary vector tangent to the surface). The potential-function is a continuous, and derivable function of the space coordinates at every point except, at the so-called singular points. The potential-function is homogeneous and linear concerning cp, thus if cpl and vz respectively are arbitrarily chosen solutions satisfying Eq. (4.1-13);and c1 and cp are also arbitrary constants y = c!cpl czcp2 also fulfls the differential equation. It is also necessary to note in connection with the characteristic properties of the velocity-potential that - according to the definition given in Eq. (4.1-12) - its dimension is [LT-l], and its numerical value can be calculated aa the product of the head (the difference in elevation between head and tail water) maintaining the seepage and hydraulic conductivity. I n many cases the cp value divided by hydraulic conductivity is used instead of velocity-potential (the dimension of which is [L]). The derivate of the latter in a given direction gives the component of the hydraulic gradient vector ( I ) since the quotient of seepage velocity and hydraulic conductivity is equal to hydraulic gradient (Darcy’s law):

+

‘-=Iz. a@ (4.1-14) Y’ ay 82 The time invariance of flow has been supposed up to this point to be a basic hypothesis (steady seepage), because turning from Eq. (4.1-9) to (4.1-10) the member expressing the change of velocity with time was neglected (it was supposed to be equal to zero). If an unsteady flow (timedependent movement) is also irrotational the physical interpretation of its velocity potential can also be correctly determined. In this cme, however, the velocity potential ie not a single-valued function of the space coordinates, but its value depends also on time: (4.1-15) rpf = f ( s y, , 2, t ) . @=-;

cp

K

grad@=-I;--

a@ -- 1 -a@ =-I ax

x’

I n the caae of unsteady seepage the characteristic properties of velocity potential listed before have to be slightly modified. Thus the constancy of the potential along a potential surface is valid only for one instant. The whole scalar space determined by the potential-function changes from moment to moment, following the change of the time variant boundary conditions along the borders of the seepage space, and, therefore, the position of the equipotential surfaces can be characterized with a continuous movement.

4.1.2 Interrelation between stream-function and potential-function. The flownet

When investigating laminar flow, i t is necessary to distinguish and define three different terms to describe the movement of the water particles within a given period, i.e. stream line (or pow line), path (or path line of a water particle) and streak line. All three are well-known terms generally applied in hydromechanics. For this reason, only short definitions of them are

4.1 Laminar seepage

’.\,

3

“.point 5! stream line throyhpbinf3 rCt- at the time point t3

the path of a wafer padicle betweenpoints 0 and 5

J ,lh

streak line

Fig. 4.1-1. Interpretation of stream line, path, and streak line

31 *

483

484

4 Kinematic characterization of seepage

given here. The reader is referred to handbooks of hydromechanics where more details may be found (Nhmeth, 1963; Bear et al., 1968; Bear, 1972). Flow line (stream line) is the course of a number of neighbouring water particles. Thus, the velocity vectors are tangents to this line at every point (Fig. 4.1-la). I n the case of steady flow the position of the flow lines is a permanent characteristic of the flow space, while if themovement is unsteady i t changes in time. Path is the course taken by a n arbitrarily chosen water particle, thus each particle has its own path line (Fig. 4.1-lb). If the flow is time-independent (steady flow), path and flow line are identical curves. saddie pomf (V-U) (internal corner)

vortex p m f

nodal point (source) nodalDolnt(siuk) ( v = +-) [ v = --1

I Y= 0 )

comer point ( Y = -) (external corner)

Fig. 4.1-2. Various types of singular points

Each curve of the system of streak lines runs through those points of the flow space, where the water particles which have crossed an arbitrarily chosen point previously, are at the moment of the investigation. The streak line therefore, indicates one point of each path fitted to the given point (Fig. 4.1-lc). It is evident, therefore, that in the case of steady flow (when the position of path lines does not change) the streak line is identical with the path and thus with the flow line as well. Since velocity potential determines one and only one velocity vector at each point, i t follows from the definition of the flow line (tangent to velocity vector), that each point of the flow space can be crossed only by one flow tine, and the flow lines, therefore, do not intersect each other. The singular points of the flow space are the exceptions, where the gradient of the potential function is undetermined (its derivate is multi-valued). According to Eq. (4.1-11) the velocity has no finite value at these points, i t has to be either zero (stagnation point) or i n h i t e (point of cavity). The main types of singular points are as follows (Fig. 4.1-2): saddle point or intPrna2 corner ( v = 0 ) ; source ( v = 00); sink (w = -00); vortex point (v = 0 ) ; corner point (external corner) ( v = 00) (K&rm&nand Biot, 1940).

+

485

4.1 Laminar seepage

For analytic characterization of flow lines the relationship expressed by their deihition can be considered according to which the velocity vector i s tangent to the investigated curve at each point, consequently: dx

dY dz . (4.1-16) vy(x,y, 2) vz(x,y, 2) ’ or if the flow lines of an unsteady seepage have to be determined at a timepoint t : ~~

~

VJX,

y, 4

dx V,(G

y,

-

2, t )

dY vyuy(z,y, z , t )

-

dz v,(x, y,

(4.1-17) 2, t )

The flow lines intersecting a closed curve chosen arbitrarily inside the flow space surround an irregular cylindrical body, since the flow lines must not cross each other. This body is called stream or flow tube (Fig. 4.1-3). In the case of time-independent seepage, the flow tube can be analyzed as a separate closed system, without causing any modification in the kinematics of the system, because velocity has no component normal to the flow lines. Hence, water particles cannot cross the border of the flow tube, there is no interaction between the parts of the flow space inside and outside the tube. For further simplification of the kinematic analysis of seepage, the investigations will be limited to the study of two-dimensional flour. In this cme all vectors of the vectorial quantities characterizing the movement (deplacement, velocity, gradient, acceleration) are parallel to a given plane ($ow plane) at every point of the seepage field, and their magnitudes are

Fig. 4.1-3. Flow tube surrounded by stream lines

486

4 Kinematic characterization of seepage

constant along a straight line perpendicular to the flow plane. A further characteristic feature of this type of movement is that the special curves describing the flow kinematically (i.e. stream line, path and streak line) are also two-dimensional ones fitted to the parallel flow planes, and two such interrelated curves (crossing the intersections of the flow planes and the same perpendicular straight line) are identical. At the same time the flow planes can be regarded a8 the borders of flow tubes, because all the

Fig. 4.1-4. Characterization of two-dimensional seepage

stream lines included into a flow plane necessarily intersect another arbitrarily chosen curve of the plane. Thus, their system forms a boundary of the flow tube. Hence, the whole flow space can be divided into many independent subsystems along the flow planes, and the kinematic investigation of such a stripe with unit width between two flow planes is quite sufficient for the complete characterization of the seepage space (Fig. 4.1-4). The mathematical discussion can also be simplified in the case of twodimensional seepage by fixing two axes ( x and y) of the coordinate system within a flow plane, while the third axis ( z )indicates the direction perpendicular to the plane. In this third direction both velocity and acceleration have to be equal to zero according to the definition of the two-dimensional flow: (4.1-18)

487

4.1 Laminar seepage

Thus, the kinematic relationships defined previously take more simple fornis; i.e. the condition of the irrotational flow is:

av, (4.1-19) ax ay and that of continuity can be expressed with the following equation:

-+-?!=o avx ax

av ay

(4.1-20)

The eyuipotential surfaces are straight cylinders, the generatrices of which are normal to the flow planes. Their position is unambiguously fixed if the curves formed at the intersection of the equipotential surfaces and the 2, y p1:tne of the coordinate system (the basic flow plane) is known. These curves are called potential lines (or equipotential lines). The equations describing the potential lines can be easily determined by simplifying Eqs (4.1-1 I ) , (4.1-12) and(4.1-13), respectively considering the two-dimensional character of the flow:

It follows from Eq. (4.1-21), that the two-dimensional potential flow is irrotational. The total change of the potential (the total differential of velocity potential) along an elementary vector in the flow plane being composed of dx and dy components is determined with the following relationship :

This differential equation always has a solution, if the flow is continuous [Eq. (4.1-20)]. Knowing that the potential is rpo(x0,yo) at a, given point P o ( z o yo) , the p, value at an arbitrary point P(x, y) can be calculated (Fig. 4.1-5): m

~ o ( ~YO) 0 ,

- ~ ( xY), = J vx dx XO

Y vy dy;

+J

~ ( xy), = Cia

(4.1-23)

Yo

Any given constant value of C , variable indicates a potential line on the flow plane, where the velocity potential is constant and equal to the numerical value of It follows from the constancy of the potential that the change of the potential is zero along such lines, thus

c,.

d V = - ( ~ , d x f ~ ~ d y ) = Ov ;* d n = O ;

(4.1-24)

where cln is an elementary vector on the flow plane tangent to the potential line. The relationship proves, that the potential lines have always to be normal to the velocity vector, and, therefore, they have to intersect the

488

4 Kinematic characterization of seepage

..,C@

X

Fig. 4.1-6. Interpretation of potential line and potential-functionin the case of two dimensional seepage

stream lines perpendicularly (because the velocity vector is always tangential to the stream line). Equation (4.1-16) fixing the position of the stream lines in the space can also be used in two dimensional form. The simplified formula gives the equations of the stream lines in the basic flow plane, these stream lines being intersections of the x, y coordinate plane and the stream surfaces perpendicular to the flow plane: dx - dY or

VX(X9

Y)

Vy(X9

Y)

'

(4.1-25)

vxdy - vYdx= 0. Supposing that the components of the velocity vector are continuous and derivable functions of the coordinates, the differential equat,ion haa a solution, and its general form can be determined by the Y(S9

Y) = (72

(4.1-26)

stream-function (or flow-function). The conetant C, originates from the integration of the differential equation, and its given numerical value determines one stream line on the flow plane (Fig. 4.1-6). The tangent of the stream line is the velocity vector, thus the angle a closed by the tangent and the horizontal coordinate axis at any point P(x, y) can be calculated

tana=m,=

[21 2 - =-.

(4.1-27)

Let B be the angle of the curve perpendicular to the stream line at the same point. Its tangent can a180 be expressed from the previous equation (4.1-28) VY

489

4.1 Laminar seepage

X

Fig. 4.1-6. Interpretation of stream-function

which relationship gives the equation of the potential lines [v,& + vYdy= 0; Eq. (4.1-24)] proving once again that the series of stream lines and potential lines form orthogonal trajectories on the flow plane (they intersect each other always at right angles). This system of the two types of curves is called a $ownet (Fig. 4.1-7). The value of the stream-function does not change along the stream lines, consequently (4.1-29) dv = -(v,dy - v,dz) = 0 ,

+

if both ends of the elementary vector (the components of which are dz and dy) are fitted to the same stream line. In an opposite case a finite dy value belongs to the dz, dy displacement. Let us investigate the flow rate through a flow tube bordered by two flow planes a distance of unity from each other, and by two stream surfaces, the intersections of which with the coordinate plane are two stream lines a distance A n , apart. The parameter to be determined is the integrated value of the product of the velocity vector and the elementary surface perpendicular to it. The integration has to be executed along a potential surface (which indicates the normal section of the flow tube) between the two bordering s h a m surfaces. In the case of two-dimensional movement the operation is simplified to an integration along a curve, because the width of the flow tube is equal to unity: q

=s

(F)

dq =

s vdf = J vdn

(F)

(An)

B

(vxdy - vydz).

=

(4.1-30)

A

The integral can be solved if (4.1-31 )

This condition is the same as that of the continuity [Eq. (4.1-20)]. It can be stated, therefore, that the amount of water conveyed through a

490

4 Kinemat,ic characterization of seepage

flow tube of unit width can be calculated if the seepage i s continuous. T h e flow rate i s equal to the dioerence of the values of stream-functions belonging to points A and B respectively: B

A

A

because

(4.1-32)

-a Y_ - -vx; aY

and

aw

-- Vy.

ax

On the basis of the foregoing it is an important property of stream-function, that - in the case of continuous flow - the components of the velocity vector can be determined at every point of the field by differentiating the stream-function. The specific flow rate (the amount of water transported through a unit width of the flow space of a two-dimensional seepage during a time period of unity), is equal to the difference of the numerical values of the stream-functions belonging to the two stream lines bordering the seepage field. It follows from the relationship between the components of the velocity vector and the stream-function, that the Dotentid- and stream-functions are also interrelated (Cauchy-Riemann's differential equations):

(4.1-33)

After deriving the first equation according to x and the second according to y , their sum gives the Laplace's equation, while their difference is identical with the condition of irrotational flow. It can be stated, therefore, that the movement i s continuous and irrotational, if the potential- and stream-functions describing the seepage in question satisfy Cauchy-Riemnn's condition. The inverse interpretation of this statement is also valid. If a steady seepage is laminar, (Darcy's law is valid) continuous, (the singular points are excluded from the seepage field) and irrotational, it can always be characterized by potential- and stream-functions,and these functions fulfil theCauchy-Riemann's equations. Considering the definition of the stream-functions and Eq. (4.1-33), the condition of the irrotational flow [Eq. (4.1-19)] can be expressed in a form, very similar to that of the Laplace's equation but depending on y value

4.1 Laminar seepage

491

It can be seen, that both p7 and y are harmonically interrelated to each other, viz. knowing one of these functions the other can be determined: x

Y

YO

(4.1-35) V

xa

Y.

Equation (4.1-33) provides us with important means to construct a flow net. The x and y directions being the axes of an arbitrarily chosen orthogonal coordinate system, dx: and dy values can be substituted by the elementary lengths of two vectors being normal (An) and tangent (As)to the stream line at a given point. When the two vectors are at right angles, s is the direction of the stream line and n is that of the potential line. Thus, the following equation is achieved:

4-3 As An

(4.1-36)

Applying this condition, and using numerically the same differences between the represented stream lines and potential lines (dp, = dy), the flownet can be easily constructed, because i t is composed of squares with curved sides. The neighbouring sides (but not the opposite ones) have approximately the same size, they and aJBo the diagonal of the s q u a w intersect each other at right anglee. Considering the conditions listed previously, and knowing the borders of the seepage field the flow net can be approximated graphically, and its correctness can be checked either by constructing the internal tangent circles or by checking the angles of the diagonals (see Fig. 4.1-7) (Forchheimer, 1924; Leliavsky, 1955). =l p*- (0, = 93-472= .... * A q ; n=IO W , - & = @ - W = ~ ~ - W ~ = . . . . = A Wj 177~6

impervious boundary Fig. 4.1-7. Construction of flownet

492

4 Kinematic characterizetion of seepage

The flownet (either constructed graphically or determined by any other means e.g. using a continuous electrical analogue model) can be used for the calculation of the hydraulic parameters of seepage aa well. Let us suppose, that the total potential of the system (9 = K A H ) was divided into n equal

I :I

parts Arp = - and the potential lines belonging to the numerical values of the subsequent potential steps were determined (PO

0;

1

~0

+

~2

= TI+

-

-

rpn-1+

A q ; * . . vi+1= pi $- AT; . Arp = .n&).

pn = p~

This condition fixes not only the position of the potential lines, but that of the stream lines a8 well, because the equality of Arp and Ap is a prerequisite. Thus after having constructed the flow net the m number of the elementary flow tube (conveying a flow rate of dy) can be counted from the graph. The pressure head (which is constant along apotentialline) can be calculated, knowing the total pressure difference between the entry and exit faces of the seepage field ( A H ) aa well 88 the pressure along the exit face ( H , ) and counting the number of the potential intervals (i) from the exit face to the investigated Doint (4.1-37)

The local seepage velocity at a point is the product of the hydraulic gradient and hydraulic conductivity. The gradient can be interpreted, however, as the change of velocity potential along a unit length of a stream line, consequently AH 1 vI .-- - - Arp _ _- K -Ah, =-K-(4.1-38) Asi ASi n Asi Finally the flow rate through a stream tube is equal to the difference of the stream-functions belonging to the t,wo bordering stream lines. According to the conditions listed previously, this value is the same as the difference of the potential functions determined by two neighbouring potential lines: (4.1-39)

The flow rate of one stream tube multiplied by the number of the tubes gives evidently the total discharge of a stripe of the seepage space having a width of unity (specific flow rate):

1 p I = K A H -m. n

(4.1-40)

There is one further point, which has to be mentioned in connectmion with the determination of flow nets. Considering Eq. (4.1-33), it can eilsily be understood, that the role of potential- and stream-functim are interchangeable.

4.1 Laminar seepage

493

This property of the net is very usefully applied in the construction of the net, when such methoda are used, which are suitable only for the determination of one type of curve (e.g. in a continuous electric analogue model in which only the position of the pot,ential lines can be measured). In this case the inverse model of the field has also to be prepared (viz. the impervious boundaries have to be changed with the exit and entry faces), and its potential lines are identical with the stream lines of the original system.

4.1.3 Geometrical and kinematic classification of seepage

The forces creating and maintaining the seepage together with those characterizing the resistance of the field and retarding the movement have already been analyzed in the preceding chapters. On the basis of the dominating forces the dynamic classification of seepage was also discussed. Some 13stinctions were made according to the character of the porous media, of which the seepage field is composed (loose claatic sediments, or fissured and fractured solid rocks). I n the fkst group investigated in greater detail, the two main dynamic types were the seepage through saturated layers, and the water movement developing in an unsaturated porous matrix. Further subgroups could be indicated, when distinctions were made according to the dominating retarding forces as well [turbulent flow, two transition zones, laminar water movement and microseepage). These latter subgroups are especially important in the case of saturated seepage, while in the other group these aspects can be neglected as the basis of classification. Within such dynamically distinguished group further classification can be made using the geometrical position of the various kinematic configurations (stream line, path and streak line) aa the basis of grouping. The external effects (their position in space and their change in time) acting along the boundaries of the seepage field and governing the kinematic behawiour of the movement can also be used as a basis of classification. This kinematic (and geometrical) classification is the topic of this section (Kov&cs, 1975). The most simple form of geometric classification is based on the position of flow lines or paths within the flow space. From this aspect the forms below can be distinguished (Fig. 4.1-8): (a) One dimensional seepage: all flow lines are parallel to each other, or as an approximation they are supposed to be parallel. (b) Two-dimensional seepage: the flow lines are plane curves fitted to parallel flow planes and the flow nets in two flow planes are identical. (c)Three-dimensional seepage : the flow lines are generally space curves and according to their special positions further subgroups can be created.

( i )Axial symmetrical seepage: the flow lines are plane curves, the flow planes are not parallel, but they intersect each other at a common axis. ( i i )Seepage along straight cylindrical surfaces: the flow lines are space curves fitted to cylindrical surfaces having straight - mostly vertical generatrices.

494

4 Kinematic characterkstion of seepage

su

tsilwsfer h the drein

Fig. 4.1-8. Position of flow surfaces of two-dimensional seepage, axial symmetrical flow and seepage along straight cylindrical surfaces

(iii) Qeneral three-dimensional seepage :along flow lines having no specially defined position in the flow space. The detailed definition of the two-dimensional seepage has previously been given. The one dimensional flow is a simplified form of the former, when not only the flow planes are parallel, but also the stream lines within the plane. It is necessary to emphasize that this most simple seepage is the only one form of flow, when the Laplace’s equation can be integrated in closed form without applying approximative hypotheses. This is the reaaon

4.1 Laminar seepage

495

why efforts are always made, when the objective is the analytical solution of the problem that the more complicated flow patterns should be transformed with exact mathematical means into one dimensional seepage, or the conditions are sought which approximate the actual movement with the simplest geometrical form of water movement. Conformal mapping can be mentioned as an example of the former caae. This method can be used t o transform the flownet of the two-dimensional flow t o one dimensional seepage. Applying the same method in the horizontal plane, the flow surfaces of both the axial symmetrical flow and the seepage along cylindrical surfaces can be transformed into parallel planes. This fact justifies the distinction of the special three-dimensional flow types, and this is the rewon, why the general flow patterns are also sometimes approximated with one of the more simple systems of flow. The position of the boundaries of the flow space is also a geometrical characteristic of the seepage developing within a given field. It affects, however, the flow together with the boundary conditions acting along the borders, which influence the kinematic behaviour of the movement aa well (determining, whether the flow is time-invariant or not). The further classification of seepage can be based, therefore, on the boundary conditions, which include both geometrical and kinematic aspects. If at the entry and exit faces of the flow space the potentials do not change in time (and the period elapsed since the start of movement is long enough for the development of dynamic equilibrium between the accelerating and retarding forces in the entire flow space) the kinematic parameters of seepage (discharge, velocity, acceleration) are constant in time. This type of movement is called steady flow. Its opposite is the unsteady flow which is created by time-variant boundary conditions along the two bordering potential surfaces (or at one of them). There are e m s , when the seepage is unsteady, although the boundary conditions are constant in time. This situation is characteristic of the period after the development of the actual influencing boundary conditions, and before that of the steady state in the whole interior of the flow system. The kinematic parameters of unsteady flow change mtinwuusly in time in the entire /low space or only in a part of it. The steady seepage can be described with the time invariant flownet, while in the case of unsteady flow the paths to be determined, became the position of both potential surfaces and flowlines change in time, the potential- and /low-functions describe only the instantaneous condition [see Eqs (4.1-15) and (4.1-17)]. Until now only the influences of boundary conditions along the entry and the exit faces were investigated. The upper surface of the flow space is another important boundary from the kinematic point of view. If the water conveying aquifer is covered with an impervious layer and the pressure at the upper surface of the former is higher than atmospheric, the flow system is confined, while the upper boundary of the unconfined flow space i s the water table, where pressure is eqzml to the atmospheric value. The latter type of boundary is only a fictive surface, because there is no sharp border between the saturated and unsaturated zones. The unconfined ground-water zone is always covered with a capillary fringe, the lower part (close capillary zone) of which is completely, or almost completely saturated - although the pres-

496

4 Kinematic characterization of seepage

sure in the water is negative - and the rapid decrease of saturation starts only in the open capillary zone. In the case of steady flow the character of the upper boundary does not influence the kinematic behaviour of the seepage. The determination of mathematical models for describing steady confined seepage is easier than that for an unconfined system. The geometrical parameters (the position of the upper boundary and the area of the vertical cross section) are a priori known values in the first case, while the position of the free water table is also a hydraulic variable, where a given boundary condition has to be satisfied, the potential should be proportional to the elevation of the surface above a horizontal reference level. There is a transition form between confined and unconfined flow spaces i.e. a water conveying layer covered with semi-pervious material (semi-confined system). Investigating the horizontal seepage in such a flow space, the hydraulic conductivity of the covering layer (and thus the water conveyance here) can be neglected, but the vertical movement - which follows the change of the piezometric head -has to be considered. Thus the mathematical models for steady confined flow can be applied to investigate semi-conh e d systems as well. Special relationships are necessary only in the case of studying unsteady flow in such a flow space. If the flow is unsteady, the type of the covering layer (confined, semiconfined or unconfined) causes basic differences in the interpretation of the dynamic equations, i t even determines what type offlow (steady or unsteady) can develop (KovQcs, 1966c, 1968). In a completely closed flow space, both the bed and the cover of which are impervious, the actual gradient cannot create seepage through these layers and the developing pressure does not cause any deformation. The change of the potential at any end of the system will travel with pressure-wave velocity through the flow space. Although there is information about experiments to determine the velocity of the propagation of pressure-waves through POrous media, i t is not yet clear, how the effectsof the solid matrix (and those of their special physical properties) modify this velocity. It seems certain, however, that the propagation does not differ essentially from that in a closed water body, and the velocity, therefore, is so high, that a very short time after the change of the potential difference between the entry and exit surfaces is enough to develop new balanced conditions in the entire system. The unsteady flow can be approximated, therefore, with the series of steady seepage, regarding the discharge as the function of time only and supposing that i t is constant at every section along the whole length of flow at each time point. A further characteristic of the system is that the depth of the flow (area of the cross section) is time invariant, being a value given a priori by the geometric parameter determined by the position of the bordering impervious layers (Fig. 4.1-9a). To represent an example for the comparison of kinematic equations pertaining to the various types of seepage fields, the most simple flow field is investigated, when the thickness of the pervious layer is constant [nz(s) = = m = const.]. As was already mentioned this is the simplest one dimensional flow pattern when the Laplace’s equation can be directly integratad

497

4.1 Laminar seepage

heud

imoervous lower boundaru

Fig. 4.1-9. Vertical section of confined, unconfined and semi-conhed seepage fields

in the caBe of steady seepage:

AH q = m v = Km-= L

K m HI - -H, L

. 7

because

'v =o

-

(the potential lines are vertical);

Y' and if x = 0 ; p = KH, and x = L; y = KH,. 32

(4.1-41 )

498

4 Kinematic characterization of seepage

The system is c o n h e d and, therefore, the unsteady seepage can be approximated with the series of steady conditions. Thus for unsteady movement Eq: (4.1-41) is only slightly modified q = K m -a, t ) . (4.1-42)

L

both H , and H , being time dependent, their difference (the total head) has to be expressed also aa a function of time. It is necessary to note here that Eq. (4.1-41) can be achieved in a more simple way, than the integration of the Laplace’s equation. It follows from the fact that the flownet is composed of straight and vertical potential lines and of equidistant horizontal stream lines, that the velocity is constant within a vertical section and can, be calculated as the product of hydraulic conductivity and the slope of the p i e z m t r i c line at the investigated vertical section. This relationship called Dupuit’s hypothesis is absolutely exact in this very simple form of seepage field (straight flow tube with parallel borders), when the velocity is constant at every point of the field, because the area of the cross section ( A = 1 m) does not vary dong the field. It can be applied also as an approximation in the case of a confined aquifer with slightly changing thickness of the pervious layer, or even for the characterization of horizontal seepage through an uncodned field. In the case of the very simple field investigated presently, Dupuit’s hypothesis results in exactly the same relationship as Eqs (4.1-41) and (4.1-42). Its application will be demonstrated in connection with the analysis of unconfined seepage. In unconfined flow-space the change of potential created by the varying boundary conditions, is transferred from section to section in the form of change of the elevation of the water table. The rise of the water table requires the increase of the stored ground water, while lowering the water surface, the stored amount decreases. The discharge is, therefore, a function of both time and distance (the latter measured from the entry or exit face). The travel time of the action is much longer than that in c o n h e d space, because wave propagation goes together with m s transport. The surplus storage has to be conveyed from the entry face to the section, where the water table is raised or the water drained by the lowering of the free water surface is carried away along the flow space until the exit face is reached. Thus, the propagation velocity is limited by the velocity of water transport. The position of the water table in this system is the function of both time and distance and, therefore, the depth of flow at a given section (the area or the cross section) varies with the time (Fig. 4.1-9b). Applying Dupuit’s hypothesis the specific discharge can be expressed as the product of the flow depth ( y ) varying with the distance from the starting section (and also in time, if the flow is unsteady), the hydraulic conductivity

-

( K ) ,and the slope of the water table

(3 :

(4.1-43)

4.1 Laminar seepage

499

I n steady state the discharge is constant and the differential equation can be solved, considering that yo = H , if x = 0 and yo = H , if 5 =I;. q=K

H:

- Hg

2L

(4.1-44)

If the flow is unsteady Eq. (4.143) has t o be combined with the equation of continuity simplified for one dimensional seepage [Eq. (4.1-4)]. The result is Boussinesq's differential equation (Boussinesq, 1904): (4.1-45) the solution of which will be discussed in Section 5.4.1. The semi-wnfined flow space is a transition form between the two basic cases (Fig. 41.1-9c). The horizontal flow is practicallylimited to the pervious layer, the hydraulic conductivity of the covering material being of smaller magnitude than that of the main body of the flow space. Thus, the geometry (depth and area) of the croas section is time-invariant and similar t o the case of c o n h e d flow. The change of the pressure at a section is followed, however, by the modification of the stored amount of water. This process needs only a very short vertical flow in the semi-pervious material and thus the boundary condition (perhaps with some delay) develops similarly, as in the case of a free surface. Considering these conditions the basic kinematic equations describing the seepage through the field analyzed as an example (one dimensional flow in a horizontal layer having constant thickness) results in more simple relntionships than Eqs (4.143) and (4.1-45). On the basis of Dupuit's hypothesis a linear differential equation is achieved (similar to that derived for a confined system): (4.1-46) which can be integrated directly if the flow is steady ( q = const.). When combining Eq. (4.146) with the equation of continuity, the result is as follows: (4.1-47) The solution of this relationship is much easier than that, of Eq. (4.1-45).

It will be also demonstrated in Chapter 5.4 that the various methods applied for the linearization of Eq. (4.1-45) lead back to this simplified form by supposing, that the change of the flow depth (yo)along the field is negligible (KovAcs, 1962a, 1966b, Verigin, 1949, 1952). According t o the definition of steady flow i t develops only after a period of time has elapsed following the establishment of a constant boundary condition. This period is necessary for the travel of the actions created by the change of the potential conditions. The first consequence of this fact is (as 32 *

500

4 Kinematic characterization of seepage

was already mentioned) that in an absolutely confined system (even in a semi-infinite flow field) all the movements can be approximated using mathematical models derived supposing steady conditions, because the propagation velocity is very high. The other consequence is that steady conditions can never be achieved theoretically in a semi-infinite unconfined or semi-confined system. Considering the relatively low propagation velocity, the steady state models can be applied only, if the position of both the entry and exit faces are known and the distance between them is not so high that i t might be regarded aa infinite. If only one of the faces is known, the propagation of the influenced zone (draw down or backwater surface), can only be investigated as a function of time using unsteady state mathematical models. Equations giving the length of this zone depending only on physical soil parameters (e.g. Sickhart formula) have no practical use. There is only one special caae when steady movement can develop in an unconfined semi-infinite flow field, i.e. if the field is vertically recharged or

x I

. * , - . . ..... .* - > . * . :. . .,. . _.'.; .; ., ,I. .'. . *. .: ....,. . ... ..I :;,..\ .: . .' ' : . . . . :.'I' ;;;.._ - . .,., ' ,

*

.

,I

.

,

*

Fig. 4.1-10. Vertical sections of vertically recharged and drained seepage fields

4.1 Laminar seepage

501

drained (leaking aquifer). Investigating a horizontally drained system, the water amount withdrawn from the layer is balanced by positive accretion (vertical surplus recharge) caused by the lowering of the water table (or piezometric level), while in the caae of artificial recharge, water crossing the entry face may be drained vertically within the zone of influence (negative accretion) (Fig. 4.1-10). To explain the baaic relationships of this type of investigation i t is necessary to suppose that there is only one external influence which horizontally drains or recharges the flow field and acts through the investigated entry or exit face. Another supposition is that before the development of the actual boundary condition here, static equilibrium prevailed in the system (the level of the surface water and the horizontal water table were at the same elevation). After the instantaneous rise (or lowering) of the level of the surface water, it is kept at a constant elevation and an unsteady flow commences. The border of the influenced zone propagates slowly from the face to the semi-infhite field. In the first period the water amount recharged through the entry face (or drained through the exit one) is balanced mostly with the change of the stored ground-water amount and the vertical effects have little influence. The importance of this increaaes aa the zone of influence extends and the change of the elevation of the water table increases, because both processes increase the total sum of the vertical effects. The propagation of the influenced zone reaches a limit, when the integrated value of accretion within the border of the influenced zone becomes equal to the amount of water crossing the entry or the exit section. At this time a dynamic balance develops and the steady state flow is characterized with a time invariant flow rate, which depends only on the distance from the starting section. This is the only form of movement, when the limited extension of the influenced zone can be theoretically interpreted in a semi-infinite flow space. It is necessary to note that the change of the depth of water table changes the boundary conditions in both vertical directions. Upwards water exchange will develop between the saturated and unsaturated zones, while at the lower boundary flow may commence from, or to the underlying aquifer because of the change of pressure. According to the hypothesis of static equilibrium, it haa to be supposed that the resultant of the vertical flow is zero before the start of the horizontal flow. At and outside the border of the influenced zone, it remains similarly zero, while inside this limit the lowering of the water table causes vertical recharge and its rise is followed by vertical drainage. Accepting, that accretion is proportional to the change of depth of the water table, these e8ects can be expressed as a function of either the distance from the starting section (exit or entry face), or that of the verti.ml clmnge i n elevation of the water table (Fig. 4.1-11). Note here that surface effects are generally the dominant ones because they always act in unconhed and semi-confined system. The water exchange of the lower layer depends on the permeability of the underlying layers. The dependence of accretion on the position of the water table excludes the application of the most simple mathematical model applied generally in the literature, which suppose constant accretion. Considering the fact, that both vertical recharge and drainage is the result of t,he change of

502

4 Kinematic Characterization of seepage

Fig. 4.1-11. The change of the vertical recharge and drainage depending on the position of the water table

depth of water table, it would be logical to express the changing value of these parameters as a function of the depth of the phreatic surface [or more ) f(y yo)]. precisely its change related to the static water table: ~ ( y= It was found, however, that the relationship derived in this way is very difficult to handle mathematically, even if the E( y ) function is approximated with very simple (linear or exponential) formulae (Ldczfalvy, 1958; Kovhcs, 1962b). In practice it can be proposed, therefore, that the accretion should be described as a function of the horizontal distance measured from the starting section (the entry or exit face) (Juhhsz, 1953). Considering, however, the dependency of accretion on the change of the depth of water table, its distribution along the seepage space [the E ( X ) function] has to be approximated always by a smooth curve, and becomes zero at the border of the influenced zone. The distance of this border from the starting section depends, therefore, also on the probable relationship between accretion and the depth of the water table. On the basis of the explanations given in the previous paragraphs, the hydraulic (mathematical)models illustrating the characterization of seepage within a geometrically well defined (or approximated) flow field with known boundary and flow conditions, can be easily classified. The first distinction has to be made according to whether the position of both the exit and the entry faces are known (together with the boundary conditions along them) or whether the flow space may be regarded as a semi-infinite stripe with only one of bhe faces is fixed. The second case is characteristic, if the water regime of ground water is investigated near surface water (in a river plain) while the models of the first type are used for calculating the hydraulic parameters of seepage between two surface-water bodies (development of the water table between two canals; drains or wells along rivers; etc.). Within both main groups models based o n the supposition of the steady state and those describing unsteady flow can be included. A further distinction has to made because differences are caused in the models (and in the charac-

-

503

4.1 Laminar seepage

Table 4.1-1. Classification of hydraulic models describing the interaction between surfme and ground water ~~~~~

I StedY state in semi infinite flow space

Steady state between two surface water bodies

Unsteady state

Neglecting accretion

1

Considering accretion

Dupuit's equation for confined, Extended Dupuit's equation for semi-confined and unconfined u n c o h e d and semi-confined aquifers aquifers - with constant accretion - with aocretion being the function ofthe vertical change of water table; - with accretion being the function of the distance from the starting section Consideration of the curved Development of the water table character of the flow net in between two parallelanahif the vicinity of the entry or - both can& have the same exit face. character (rechargingor drainSpecial structures along the ing the ground water) bank of surface water - canah having individually - fully penetrating draina the same charmter but be- partially penetrating drains tween the two canah this char- trenches acter is maintained only in - singlewell one of them, while the other - seriesofwells effect is changed -- canals with different character Boussinesq's equation and its Extension of Boussinesq's simplified form for semi-conequation for considering surfined aquifers. Application of face effects (Verigin's equathe simplified model for chartions). acterizing the effect of - rapidly - gradually - periodically changing surface-water level. Relationship between semi-confined and unconfined aquifers. The superposition of the unsteady effects created at two different sections

ter of movement as well) by considering or neglecting the vertical effects (accretion). Finally the upper boundary (confined, unconfined, or semiconfined) of the flow field has to be considered together with the geometry of the flow planes (two dimensional, axial symmetrical or other types of three dimensional seepage). Considering all the aspects listed in the previous paragraph, one of the possible groupings of the various hydraulic models is presented in Table 4.1-1.

504

4 Kinematic characterigation of seepage

References to Chapter 4.1 BEAR, J. (1972): Dynamic of Fluids in Porous Media. Elsevier, New York, London, Amsterdam. BEAR, J., ZASLAVSKY, D. and IRMAY, S. (1968): Physical Principles of Water Percolation and Seepage. UNESCO, Paris. BIOT, M. A. (1941): General Theory of T h e e Dimensional Consolidation for a Porous Anisotropic Soil. Journal of Applied Physica, Vol. 12, p. 166. BIOT, M. A. (1966): General Solution of the Equation of Elasticity and Consolidation for a Porous Material. Journal of Applied Mechunics. Trans. ASME, Vol. 78, p. 91. BOU~SINESQ, J. (1904): Theoretical Research on the Flow Rate of Groundwater Percolating in the Soil and on the Yield of Springs (in French). C. R . Acad. Sci., Paris. COURANT,R. and HILBERT, D. (1937): Methods of MathematicalPhysics (in German). Springer, Berlin. DE WIEST, R. J. M. (1969): Flow Through Porous Media. Academic Press, New York. FLORIN. V. A. (1948): Theory of Consolidation of Earth Masses (in Russian). Gostroizdat, MOSCOW. ' FORCHHEIMER, PH. (1924): Hydraulics (in German). Teubner, Leipzig, Berlin. IRMAY, S. (1963): Saturated Steady Flow in Non-homogeneous Media and its ADDlication t o Earth Embankments, Wells, Drains. 33rd Conference of S M F E , Z&ich, 1953. IRMAY,S . (1968): On the Theoretical Derivation of the Darcy and Forchheimer Formulae. Tramactiom o f AGU, No. 4. JACOB, C. E. (1960): Flow of Groundwater (Chapter 6. of Engineering Hydraulics edited by H. Rouse). John Wiley, New York. J d s z , J. (1963): Groundwater Data on the Hungarian Lowland with Special Aspects on the Backwater Effects of Barrages (in Hungarian). V'iziigyi Kozlemdnyek. No. 2. KARMAN, TH. and BIOT, M. A. (1940): Mathematical Method in Engineering. McGraw-Hill, New York. KELLOO, 0. D. (1929): Foundation of Potential Theory. Dover Publ. New York. KovLcs, G. ( 19624: Dimensioning Flood-Control Levees for Underseepage. Acta Technica Academiae Scientiamm Hungaricae, Tom. 41. No. 1-2. KOVACS,G. (1962b): Design of Canals Draining Groundwater (in Hungarian). EpLtds is Kozlekeddstudomcinyi Kozlemdnyek, No. 2. KovAcs, G. (1963a): Practical Method for Hydraulic Dimensioning of Canals Draining Groundwater (in Hungarian). &pitdsi 6s Kozlekeddstudomcinyi Kozlemdnyek, NO. 1-2. KovAcs, G . (1963b): Hydraulic Characterization of Steady Seepage from Jrrigation Canals (in Hungarian). Hidroldgiai Kozlony, No. 1. KovAcs, G. (1963~):The Development of Water Table in the Vicinity of Canals with Nearly Constant Water Level. 5th Congress of ICID, Tokyo, 1963. KovAcs, G. (1966): Seepage Developing as a Result of the Backwater Effect of a Barrage on the Danube (in Hungarian). VIZITERV Studies, Budapest. KovAcs, G. (1966a): Hydraulics, (in Hungarian). VITUKI, Budapest, Vol. 111. KcvAcs, G. (1966b): Physical Interpretation of Linearization of Differential Equations Characterizing Unsteady Seepage. Symposium on Seepage and Well Hydraulics, Budapest, 1966. KovAcs, G . (1966~):Seepage Problems in Connection with Hydraulic Structures (in Hungarian). Symposium on Seepage and Well Hydraulics, Budapest, 1966. KovAcs, G. (1968): Seepage to Groundwater Created by Hydraulic Structures. Actu Technicu Academiae Scientiumm Hungaricae, Tom. 60, No. 3-4. KovAcs, G. (1976): Interaction between Rivers and Ground Water. Symposium o f I A H R / S I A on Ground Water, Rapper.mil, 1975. LI~CZFALVY, S. (1968): Determination of the Yield of Water Collecting Galleries (in Hungarian). Hidroldgiai Kozlony, No. 1. LELIAVSICY, S. (1965): Irrigation and Hydraulic Design. Chapman & Hall, London. VOl. 1.

4.2 Boundary and initial conditions

505

MID-THOMSON, L. M. (1966): Theoretical Hydrodynamics. MacMillan, London. N~METH, E. (1963): Hydromechanics (in Hungarian). Tankonyvkiadb, Budapest. A. F. (1931): Influenceof TubeWells on Ground-water Flow (in German). SAMSIOE, Zeitschrift fii. angewandte Mathematik und Mechanik No. 11. TERZAUHI,K. (1926): Soil Mechanics and the Basis of Soilphysics (in German). Deuticke, Leipzig, Wien. K. (1943): Theoretical Soilmechanics. John Wiley, New York, London. TERZAUHI, VERIUIN,N. N. (1949): Unsteady Ground-water Flow in the Vicinity of Reservoirs (in Russian). Dokl. Akad. Sci. USSR. No. 6. VERIUIN,N. N. (1962): Ground-water Movement in the Vicinity of Reservoirs (in Russian). Gidrotechnichakoye Stroyteletvo, No. 4.

Chapter 4.2 Boundary and initial conditions of potential flow through porous media The kinematic relationships summarized in the previous chapter facilitate the determination of the hydraulic parameters (i.e. velocity, pressure, flow rate) of potential water movement through porous media a t any point of the flow field, and the solution of the practical problems of seepage. The mathematical models describing the contact of the different variables can be solved analytically in some cases. I n the investigation of more complicated systems, numerical computation or graphical approsimation can be applied. I n other cases physical models (small scale or analogue models) are used. I t has already been mentioned that knowledge of the actions influencing the development of flow along the borders of the seepage field (boundary conditions) is a prerequisite for solving the kinematic equation. I n the case of steady seepage the boundary conditions are characterized with time-invariant values depending only on the space coordinates, while, if the flow is unsteady, these conditions change with time as well. Thus, they have to be expressed as a, function of the time elapsed after a time point chosen arbitrarily at the start of the process. When characterizing time-dependent flow, further information is necessary for the complete solution of the problem. The kinematic equations provide us only with the changes of hydraulic parameters a t every point. The actual parameters at a given time point can be determined only if the starting conditions are known everywhere inside the investigated field (initinl conditions) and the changes calculated from the relevant formulae can he added t o the initial values. If the flow field is composed of more than one layer, some special conditions have to be satisfied along the internal surfaces dividing the physically different parts of the field. These internal boundary condifions are not known a priori characteristics of the developing seepage. In cont,rast t o the external boundary conditions, they do not describe or represent any action creating and maintaining the flow, but they develop according t o the influence of

4 Kinematic characterization of seepage

506

these external effects. They have to be considered, however, to ensure the continuity between the different parts of the seepage field. The detailed analyaia of both external and internal boundary conditions, as well as initial conditions is the topic of this chapter.

4.2.1 Characterization of external boundary conditions The example in Fig. 4.2-1 represents the conditions along the borders of the seepage apace which can be divided into four main groups. The following types of boundary surfaces (in the case of two-dimensional seepage the contours of the section of flow space on the flow plane) can be distinguished S, Rear et al. 1968; Bear, 1972): (Nemeth, 1963; K O V ~ C1966; la) Impervious boundary; (b) Surface where the seepage field i s i n contact with a free water body (with surface waters) ; (c) Water table (phreatic or piezometric surface); (d) Free exit Surface, where the seepage field is contacted by air. The term impervious boundary includes three different surfaces: i.e. the lower boundary of the field, where the latter is generally closed by an impervious bed (CJ; the upper boundary of confined aquifers, where the flow field is covered by an impervious layer ((7;); the contour of any hydraulic structure, which penetrates into the seepage field ((7:). A common geometrical property of these boundaries is that their position is fixed in space, does not change in time, and i t is generally known or at least well approximated.

draulic .h4ructure

irrigated stripe

rem

Fig. 4.2-1. Sketch for interpretation of the various types of boundary conditions

4.2 Boundary and initial conditions

507

There is no water movement through the impervious boundaries, the seepage space is neither recharged nor drained here. Thus, the velocity vector has no component perpendicular to this boundary; i t is always tangential to the latter. In the case of two-dimensional flow the impervious contours have to be, therefore, stream lines. Their analytical characterization can be given in the form of the following mathematical relationships (Fig. 4.2-2):

imper v

i m boundary

Fig. 4.2-2. Boundary condition along impervious boundarim

For steady seepage aY = aP, y = const.; -0 ; as

an

(4.2-1)

The surfaces contacted by free water bodies could be either entry faces

(C2,where the seepage field is recharged from the surface water) or exit where the field is drained). It is necessary to note here that only faces ((26, a part of the exit faces belongs to this type of boundary (that covered with water). The field can be drained also along borders, where the porous medium iu contacted by air and not water. To make a distinction this latter type of border is called a free ezit face (CJ. The boundary conditions characterizing the free exit will be analyzed later on. The position of exit and entry faces is fixed and time-invariant similar to the impervious boundaries, and the geometrical parameters describing i t are known. These parameters (or their suitable approximation) are basic data to characterize both the system and the seepage developing through it. For the kinematic characterization of this boundary condition, the velocity potential has to be determined along the entry and exit faces. It can be found that the sum of elevation and pressure head is constant along these faces (if the slope of the free water body can be neglected, which supposition is generally acceptable, since the flow rate through the porous medium is relatively small compared to that which could cause considerable head losses in surface waters). The numerical value of the total head (elevation plus pressure head) depends on the selection of the reference level,

508

4 Kinematic characterigation of seepage

but the latter does not influence the constancy of this parameter along the investigated face. It has already been proved that velocity potential is proportional to the total head [Eq. (4.1-12)]. It can be stated, therefore, that this important kinematic parameter is constant along those contours of the seepage field, which are contacted by free water bodies, and its numerical value depends only on the elevation of the level of the surface water above the arbitrarily chosen datum :

Along the entry face

+C=KH,+C; Along the exit face

(4.2-2)

”y”i + C = K H 2 + C .

v2=K h 2 + -

The velocity potential being constant along the entry and exit faces, these planes (or curved surfaces) are equipotential surfaces (in the case of twodimensional flow the contours are potential lines). On the basis of the fundamental relationships between the potential- and stream-function, i t can be stated, that the change of the potential-functions is equal t o zero along these contours, while that of the stream-functions is zero in the perpendicular direction (Fig. 4.2-3):

8P aw -

=o.

(4.2-3)

8s an It follows from Eq. (4.2-2), that the potential difierence between the two bordering potential surfaces is independent of the position of the reference level and proportional to the height of the head water above the level of the tail water ( A H ) : Ag, = - p2 = Ap = K A H . (4.2-4) I n the case of unsteady flow, the relationships given by the previous equations are only slightly modified. The potential along the entry and exit faces is independent of the position of the investigated point, but is depends on time. These contours are potential lines always belonging t o the insfan-

inpervious boundary Fig. 4.2-3. Boundary condit,ion along entry and exit, faces

4.2 Boundary and initial conditions

509

taneous flownet. The total potential difference maintaining the seepage is also a time-dependent variable. Its numerical value changes in time either because the level of the head water is changed, or with the change of the level of the tail water, or finally the position of both levels can be timevariant: K d H ( t ) = &(t) = K[H,(t) - H , ] ; or = K [ H l - H,(t)]; (4.2-5) or

4 4 t ) = K[H,(t) - HAt)]' There are many possible conditions prevailing along the upper border of the seepage field. It can be an impervious boundary (Ci;Ci),entry face (C,) and both types of exit faces, covered with water (Ci), or free (C4),but the most general and from the kinematic point of view very special boundary is the phreatic surface or water table (C3). This is the upper boundary of the unconfined seepage field, where the prevailing pressure is constant$ and equal to the atmospheric value. The position of the water table is not previously fixed geometrically, but i t develops according to the character of the movement. I n the case of unsteady movement the depth of the field and the form of the piezometric surface changes even in time. This is the reason, why the boundary conditions are different along the water table, if steady or unsteady seepage is investigated. When solving practical problems one of the most important hydraulic parameters to be determined is the position of the water table, the geometrical data of which lave to be calculated from the kinematic equations. I n the case of steady flow there are two special processes, which may have to be considered along the water table for the correct characterization of the boundary conditions: i.e. the development of the capillary fringe together with the water conveyance through it and the water exchange between the soil moisture zone and gravitational seepage field through the water table (accretion). The characterization of the boundary conditions along a phreatic surface of steady seepaye is the most simple, if the influence of both capillarity and accretion can be neglected (C, stretch of the water table). I n this case the surface has to be a stream line, because the velocity vector is always tangential to the water table (it does not cross the phreatic surface, and, therefore, has no component perpendicular to the latter) (Fig. 4.2-4): y=const.;

av

aY ---=O.

8.3

an

(4.2-6)

Another condition to be satisfied determines the velocity potential at every point of the water table. As has already been proved, the potential is proportional to the total head above an arbitrarily chosen datum [Eq. (4.1-12)]. A t the water table the excess pressure is zero [the total pressure (ptot)above atmospheric one (p,) is called excess pressure ( p ) and this is always implied further on, when the term pressure is used without any

510

4 Kinematic characterization of seepage

water table

Fig. 4.2-4. Boundary condition ir: +he caae of steady seepage along the water table without accretion and capillary influence

adjective ( p = ptot - p,,)]. Thus velocity potential has to be proportional to the height of a certain point on the water table above the reference level, and the potential difference between two points on the phreatic surface is independent of the position of the reference datum, but it is proportional to the difference in elevation of the points: (4.2-7)

Equation (4.2-7) multiplied by the derivative of the potential-function along

El

the water table - provides a relationship between the components of the velocity vectors:

consequently

V,2

+ V$ =

-

Kv,;

because (~)'=V'=V$+V$;

and K -8P - - =dY- K v , . as ds

(4.3-8)

Considering this relationship, and also the validity of Darcy's law various formulae can be proposed to determine the relationships between the orthogonal components of either the velocity or the gradient vectors at any points of the phreatic surface of the seepage field. (Note: the gradient vector gives at the same time the slope of the water table in the case of two dimensional Bow) :

I;+I;=-I,;

I:+

I [ y

:I2

+-

:

=-

(4.2-9)

511

4.2 Boundary and initial conditions

If the influence of the capillary zone cannot be neglected, the phreatic surface with a pressure of p = 0 does not remain a stream line, because there is water exchange through this surface between the gravitational field to the capillary zone within the Grst stretch of the flow field and in the opposite direction near the tail water. The contour of the field from the intersection of the head water level and the entry face to the exit point (i.e. the intersection of the water table with the exit face) is composed of two different types of boundaries. The stretches between the top of the capillary fringe capillary surface (P = - h

C h )

+

& I

--y

&/-T

4-

&

-,

-

.'..//TI

\

>.

I_ -i----;--L-

capillary exposed

1

'

exit face

\.A\

/'

tad water exiii ibcp Fig. 4.2-5. Boundary condition in the case of steady seepage along the borders of the capillary fringe

and the head water and the exit point respectively are called capillary exposed faces (Cg),and the capillary surface is situated between them (Ca). These two types are different even from a geometrical point of view. The position of the capillary exposed faces is fixed, known and time-invariant (they are the continuation of the entry and exit faces, respectively). The capillary surface has the same role and undetermined character as the phreatic surface (Fig. 4.2-5). The kinematic characterization of the capillary surface is identical with that of the water table without capillary influence [Eqs (4.2-6) and (4.2-7)]. It is a stream line and the pressure is constant along this contour although it is not zero, but a negative value, equal to the capillary suction ( p c = = yh,). Thus the boundary conditions are determined here partly by Eq. (4.2-6), and the second condition to be satisfied is

(4.2-10)

4 Kinematic characterization of seepage

512

The final form of Eq. (4.2-10) being the same as that of Eq. (4.2-7), it can be stated that Eq. (4.2-9) is suitable for the determination of the relationships between the orthogonal component8 of either the velocity or the gradient vector along the capillary surface, as in the analysis for the water table (unaffected by capillarity). The capillary exposed faces are stream lines. Equation (4.2-6) is valid, therefore, for their characterization as well. The pressure, however, is not constant along them, i t varies between zero (at the intersection with the level of the head water and at the exit point) and p , = - yh, (the maximum suction is achieved where the capillary exposed faces join the capillary surface). An important parameter characterizing the capillary zone is its height ( h , ) . It is shown in Chapter 1.3 that this value cannot be described with a definite numerical quantity, because saturation changes gradually in the open cupillary zone. When the water amount carried in the Capillary fringe has to be analyzed i t is necessary to take into account the rapid decrease of hydraulic conductivity with decreasing saturation (see Chapter 2.3). In the literature (Youngs, 1966; Childs, 1959; Luthin, 1966) integration is generally proposed of the product of unsaturated hydraulic conductivity (as a function of height above the water table) and the elementary vertical lenght up to the maximum capillary rise. The result divided by the saturated conductivity gives the estimated value of h,, because the water conveyance through a saturated capillary zone of height hc is approximately equal to that of the actual capillary fringe:

p".sat (4

h-x

1

h, = Ksat

(4.2-11)

dz.

0

Considering also the fact that generally the closed capillary zone is not completely saturated, i t can be supposed as a very rough practical estimation that the equivalent capillary height should be equal to or only slightly higher than the minimum capillary rise: h,

-

1 (0.25 f 0.30) - = (0.25 d0

11-n a + 0.30) -4

n

Dh

;

(4.2-12)

where d o is the average pore diameter, D, is the effective grain diameter, a is the shape coefficient of the grains and n is porosity [see Eq. (1.2-19)]. On the basis of the foregoing, and considering also the resistance against flow along the capillary exposed faces, a qualitative statement can be made in connection with the position of the cupillarysurface, according to which the vertical distance between the head water level and the top of the capillary zone (hcl)is smaller than the value determined from Eq. (4.2-11) or Eq. (4.2.-12), while at the exit face (hcz)the opposite inequality is valid

-

(4.2-1 3) hcl < hc = pcly < hcz However, a capillary surface parallel to the water table is an acceptable approxiniation (except at the entry face, where the capillary surface

4.2 Boundary and initial conditions

513

should be horizontal, see Section 4.2.3.). This is because of the great uncertainties embracing the determination of the equivalent capillary height itself, and also because of the relatively small absolute values of h,, h, and h,, and especially that of their differences. Water exchange between soil moisture and, ground water through the water table also modifies the boundary conditions along the phreatic surface considerably. Such section can be found in Fig. 4.2.-1 between point A and B indicated as stretch C;. The numerical value of accretion is generally given related to the unit of horizontal surface, thus in the caae of two-dimensional seepage i t can be ] characterized as a function of the horizontal coordinate [ E = ~ ( z ) (Fig. 4.2-6). The inequality E > 0 means i d t r a t i o n to the ground water while E < 0 indicates its vertical drainage. Accretion related to the unit area of the phreutic surface ( E , ) can be calculated from the parameter given for a horizontal surface ( E ) and the /3 angle between the water table and the horizontal plane: En = E cos p, (4.2-1 4) although the difference between E, and E is generally neglected, because usually the slope of the water table is very small. The amount of accretion is equal to the flow rate through a unit area (e.g. m3sec-1 over m2), thus its

+&

-&

Y

-u-

X

Fig. 4.2-6. Development of stream lines in seepage field influenced 33

accretion

514

4 Kinematic characterization of seepage

dimension is equal to that of velocity [LT-11. When the purpose is to determine its influence on the boundary conditions, the flow rate crossing the water table (either recharging or draining the seepage space) has to be considered first. The flow rate or flux is equal to the seepage velocity perpendicular to the phreatic surface, thus it can also be expressed depending on the component of gradient vector in the same direction: 8,

=V, =K

as, I , = - -. an

(4.2-1 5)

The angle between the internal normal of the water table and the vertical axis is B (equal to the declination of the water table), while the angle closed by the internal normal and the horizontal direction is n/2 - B. The I , component of the gradient can be expressed, therefore, as the function of the I , and I , projections of the gradient: I , = I , sin p I , cos B . (4.2-16) Combining the last two equations, and considering that accretion is positive in the case of infiltration, and thus its direction is opposite to that of the y axis, the following equation may be written:

+

sin B I ,

+ COB?!,

I + ;I I,

- =0.

(4.2-17)

The left-hand side of this equation is a scalar product of two vectors: the first is the unit vector perpendicular to the phreatic surface, the coinponent of the other are I , (in z direction) and I ,

+ - (in y &

K

direction). The

scalar product being zero indicates that the two vectors are perpendicular

to one another, the second vector has to be tangential, therefore, to the phreatic surface. It follows from this relationship that the scalar product of the pressure gradient vector and this new vectorial parameter are also zero on the surfaces with constant pressure, where the pressure gradient is normal to the surface: (4.2-18) I , -8P + I -=o.

ax

1, + -A:

The surface where Eq. (4.2-18) is valid can be either the water table ( p = 0 ) if capillarity can be neglected, or the capillary surface, because the pressure is also constant along the latter ( p , = - yh,). From this relationship the boundary conditions can be expressed in the form of a function between the components of the hydraulic gradient or the velocity vector on a surface having constant preaure and influenced by accretion, similar to Eq. (4.2-9):

or

(4.2-1 9)

4.2 Boundary and initial conditions

515

The boundary condition along such a border can also be expressed by using the stream-function. It is evident that the change of the flow rate between point A and B is equal to the sum of recharge or drainage along this stretch. T h e digerence of the stream-functions belonging to the two points in question has to be equal, therefore, to the product of the accretion function and the elementary horizontal length integrated from A to B (Polubarinova-Kochina, 1952, 1962): B Y B -YA

=

(4.2-20) A

Until now, when investigating the boundary conditions along the free surface of the seepage field, i t was always supposed that the seepage is in steady state, and that the kinematic parameters, and thus the position of the investigated surface as well, are time-invariant. In the case of unsteady flow the conditions are basically different, because the surface i s raised or lowered in time. When the surface propagates, dry pores of the layer become completely or partly saturated, though when the surface is withdrawn the water stored in the pores above the new position of the water table is drained into the seepage field. The amount of water stored during the first process or released from storage in the second case, is calculated generally as the product of the change of the water table and specific yield, although the latter parameter is neither exact, nor completely characteristic. This is because of the prolongated process of drainage and thus the time lag occurring between the lowering of the water table and the complete release of the stored water (see Section 1.4.2).It is for this reason that proposals can be found in the literature to use the instantaneotks dewatering weflcient ( m ) instead of specific yield, which is the amount of water released from the pores just after the lowering of the water table, and which is, therefore, always smaller than the specific yield (Bear et al., 1968). The most important difference in the boundary conditions along a timevariant free surface and those for steady seepage [see Eqs (4.2-6) and (4-2-7)], is the fact that the surface is not a stream line, thus the constancy of the stream-function is not valid. The instantaneous stream lines intersect this border at angles different from n/2 the border is not therefore, a potential line either. Two examples are shown in Fig. 4.2-7 to represent the development of the flow net in the case of a falling water table. The first is the generally quoted case of an earth dam neglecting the effect of capillarity above the water table. The second example is constructed on the basis of experimental data (Vauclin e f al., 1975) and clearly indicates the influence of the soil-moisture zone. Investigating the boundary conditions along the water table and assuming the negligible effect of capillarity and accretion, it can be stated that the potential is proportional to the elevation of the investigated point fitted to the water table, because the pressure is zero at this surface. When the total differential value of the potential has to be determined, i t is necessary to consider that the potential changes also with time, aa the phreatic surface 33.

516

4 Kinematic characterization of seepege

wafer level at the sfarflng time point of the process

wiibouf capillary effect

impervious lager with capillary effect

/no

0

0

-1 1

~

'

~

'

1

'

1

cm 300

2011 '

1

'

'

1

1

capillery urpce

0

1

'

1

1

1

~

1

~

~

'

'

1

~

'

~

water table

200

400

cm 300

velocity vector Instantaneous flow net in an intermediate time point of the draining process of earth dams cm

Fig. 4.2-7.

'

T=I.O h

-I

moves upwards or downwards. Eq. (4.2-9) has to be supplemented with a term, expressing the change of the potential with time:

or

(4.2-2 1)

v:+v;4,+Kvy=ns-.av at

4.2 Boundary and initial conditions

517

Equation (4.2-21) i s valid also for the characterization of the cupillary surface but capillary suction has to be considered as a component of the total potential at the water surface. Similarly the influence of accretion can be considered also in the characterization of boundary conditions along the phreatic surface of unsteady seepage: + E+E n as, I ; + I; + I , K -. = 3- ; K K K 2 at

vi

+ v$ + v,(K + + K E= n, aP, . E)

(4.2-22)

at

impervious layer

1 exit &-e Fig. 4.2-8. Development of seepage and boundary conditions in the surroundings of the free exit faoe

Finally the forth main type of surface bordering the seepage field is the free exit surface (CJ, where the percolating water leaves the porous medium and enters into a space of free air characterized by atmospheric pressure (ptot= p o ; p = 0 ) . The position of the free exit face is geometrically fixed, time-invariant and known a priori, as that of the exit face covered with water. There is one basic difference between the two types of exit faces caused by the different materials contacting the seepage field i.e. the pressure along the free exit face is constant ( p = 0). Consequently this contour is not a potential line, but the velocity potential changes similarly to the water table (the potential is proportional to the elevation of the point in question above the reference level) (Fig. 4.2-8):

v = KYk.

(4.2-23)

The free exit surface is not a stream line either because water leaves the field through this surface, thus the stream lines cross it. Another boundary condition can be derived, however, from the fact, that the pressure is constant along the contour, thus the pressure gradient has to be perpendicular to it at every point. From this condition, and considering also the relationship between velocity potential and pressure gradient, an equation can be determined, which gives the boundary condition in the form of interrelated horizontal and vertical components of the hydraulic gradient vector (Bear et al., 1968): (4.2-24)

518

4 Kinemah characterization of seepage

and because

(4.2-25)

consequently

I,

--

-

sin

-Iy-l. cosp

’

or

(4.2-26)

I, =

-

(1

+ I , cotan p) ;

where fi is the angle between the tangent of the investigated point at the free exit face, and the positive direction of the horizontal x axis. There is one further important aspect, which has t o be mentioned concerning the influence of the free exit surface on the development of seepage within the flow field. Along stream lines crossing this stretch of the contour, the total potential i s smaller than that between the head and tail water, and its value changes depending on the height of the intersection of these stream lines and the free exit face (KovBcs, 1965). As already shown, the total potential of the system of stream tubes reaching the exit face below the level of the tail water is constant and equal t o the product of hydraulic conductivity and the elevation of the head water above the tail water [AT = K d H ; see Eq. (4.2-4)]. In the case of stream tubes having higher position, the total potential has t o be calculated aij the difference of the constant entry potential and the changing value of the exit potential, e.g. choosing the level of the tail water as datum, ;t8 i t is shown in Fig. 4.2-8:

I n connection with the kinematic classification of seepage as previously mentioned, the seepage field covered by a semi-permeable layer can be regarded as a transition form between the c o n h e d and unconfined systems. The same behaviour can be observed in the characterization of the boundary conditions as well. I n the case of steady seepage the upper contour of the field (the lower boundary of the covering layer) i s a stream line, the boundary conditions being described as those along a n impervious boundary. If the seepage is unsteady, this border does not remain a stream line, because the water originating from the change of storage crosses this contour. Its positioii does not change, however, in time. I t s role is similar, therefore, to the free surface with accretion, but the latter changes in time. Another difference is that the pressure is not constant either in time or along the border between the seepage and the covering semi-pervious layer.

4.2 Boundary and initial conditions

519

4.2.2 The investigation of the layered seepage field As already mentioned, in the determination of the hydraulic parameters of seepage the knowledge of the boundary conditions around the contour of the field, the initial condition for the whole field in the case of unsteady flow, and the flow conditions of the field (hydraulic conductivity or intrinsic permeability and porosity) is necessary. If the flow conditions diger considerably within the field, so that they can be regarded as constant in a region, while in other regions the parameters are different (but similarly constant within the separated regions), the seepage field has to be divided into so many parts (layers) as is necessary to have homogeneous (or almost homogeneous) regions. When investigating the seepage through such layered fields, the special conditions along the internal boundaries have to be considered a.s well. The layered structure is called regular, if the bordering surfaces between the different layers are parallel planes, and thus the thicknesses of the layers are constant. This type of structure is quite common in nature, especially in the water transporting systems of the large sedimentary basins, where the aquifers may be well approximated with such regular structures. I n regularly layered flow fields two basic cases of flow can be emphasized: seepage parallel to the bordering surfaces of the layers, or flow perpendicular to these surfuces. If the flow is parallel to the layers, the total resistivity of the formation against seepage can be calculated on the basis of an analogy between this process and electrical resistances connected in parallel (Fig. 4.2-9) :

(4.2--28)

where K , is the hydraulic conductivity of the i-th layer, mi is its thickness and K , , is the resultant hydraulic conductivity parallel to the dividing planes of the formation composed of n layers. In the c a e of seepage perpendicular to the surfaces bordering the layers, the combined hydraulic conductivity can be determined similarly, consider ing now its analogy with the electrical resistivity of a system connected in series:

i.

li

K1= -q1 . 9

(4.2-29)

2-

Kf

where li indicates the thickness of the i-th layer, which is measured in this case in the direction of flow (Kamensky et al., 1935; Kamensky, 1943; Shea and Whitsett, 1958). It is evident from Eqs (4.2-28) and (4.2-29), that even a thin layer having very high hydraulic conductivity determines basically the total

520

4 Kinematic characterization of seepage

t

L

L

t

i

1

TT

--7

CL

inpervlbus boundary

..

x

.. . . . . . _ .- . .., .

..... .

impervious boundary impervious bmdrlry Fig. 4.2-9. Characterization of resistivity parallel or perpendicular to the borders of the layers in layered seepage field

permeability of the formation, if the seepage is parallel to the layers. Similarly the total formation can be regarded as impervious in the caae of normal flow, if there is one (even a relatively thin) very impervious layer in the formation (Bear et al., 1968). When solving practical problems numerically, the seepage field may be characterized by its original geometrical data. In this caae the flow condition is described with one of the resultant hydraulic conductivities calculated from Eqs (4.2-28) and (4.2-29) respectively, depending on the direction of flow. Another possible way, which may he applied, is to choose an arbitrary , hence, determine the hydraulically value of hydraulic conductivity ( K O )and equivalent length of the jield (lflctive)belonging to the chosen permeability in the case of perpendicular seepage, or the hydraulimlly equivalent thickness of if the flow is parallel to the layers (see Fig. 4.2-9): the formation (mflctlve)

Seepage perpendicular to the layers:

(4.2-30)

(4.2-31)

4.2 Boundmy and initial conditions

521

When the investigated seepage is normal to the surfaces dividing the of layers, the ratio of the thickness (li) and the hydraulic conductivity (Ki) the layers can be applied to simplify the numerical characterization (Galli, 1959).This parameter has a dimension equal to that of time [TI,and it is called seepage resistance. The total resistance of a, formation expressed with these parameters can be written in the following form:

li

n

etot = 2 ei = 2 - . 1

1

Ki

(4.2-32)

If the direction of seepage closes an angle with the surfaces bordering the layers, different from 0 and nI2, the formation can be characterized as a layer having transverse anisotropy. In large sedimentary basins the development of this type of anisotropy is generally characteristic, because the pervious formations are composed of layers of only slightly differing permeability, with nearly horizontal bedding planes. Thus hydraulic conductivity in each horizontal direction is the parameter representing the flow parallel to the layers (KN= KII)while the vertical permeability is described by the parameter calculated for perpendicular flow ( K , = K l ) . The coefficient of anisotropy can be calculated aa the ratio of parallel and perpendicular hydraulic conductivities (or intrinsic permeabilities) considering that the thickness of the layers is identical in Eqs (4.2-28) and (4.2-29), although they are there indicated by different symbols (mi = l j ) :

(4.2-33)

There are, however, some restrictions to be considered, when the layered formation is characterized as an anisotropic field: (a)The variation of permeability of the layers should not be high, because extreme permeability can basically influence the behaviour of the formation as already mentioned. In practice it is a generally accepted limit that the ratio of permeability of any single layer should not be higher or lower than the average value multiplied or divided by a constant from 3 to 5. (b)The layers should be relatively thin, because in opposite cases the approximation with an anisotropic field is inaccurate, and, therefore, the investigation of the actual layered system is preferable. I n solving practical problems layers having a thickness more than one metre are generally considered separately, but the limit depends also on the total size of the seepage field, and also on the relative permeability of the neighbouring layers. (c) When the formation is composed of groups of layers, the field has to be divided into more than one anisotropic units (an example is shown in Fig. 4.2-10, where a double layered anisotropic field is considered). The requirement concerning the ratio of permeability should be satisfied in each unit but the difference is larger between them.

522

4 Kinematic characterization of seepage

3' B

;s"

8%

$ Lsg

Fig. 4.2-10. Characterization of layered flow fiold with anisotropy

Because of the limitations mentioned in connection with the application of anisotropy as the simplest characterization of layered seepage fields, there are many cases, when the consideration of the layered structure and thus that of the internal boundary conditions along the surfaces dividing two layers becomes necessary. These internal boundary conditions are very similar to the law of light refraction i.e. if a stream line enters a porous medium having smaller permeability than the preceding layer, and it crosses the bordering surface at an angle different from 4 2 the line breaks so, that its angle t o the normal is greater than that in the more permeable layer. Going in the opposite direction (from the less permeable medium into a more permeable one) the ratio of the angles with the normal is also opposite, the entrant angle is larger than that of the emergent one. The basis of the mathematical characterization of the internal boundary condition is fhe continuity of both pressure and flow at the intersection of the stream line with the bordering surface. T h e pressure and the potential energy (elevation above the reference level) have to be the same on both sides of a point on the surface. It follows from this condition that the ratio of the velocity potentials in the two contacting layers at the same point on the surface is equal to the ratio of the permeabilities or hydraulic conductivities (Fig. 4.2-1 1 ) :

K" z" = z'; p' = p"; consequently rp" = -rp' . K'

(4.2-34)

If the two velocity potentials are proportional to one another along the dividing surface, their differential quotients in this direction have t o follow

4.2 Boundary ark1 initial conditions

523

Fig. 4.2-11. Break of st,ream lines at the border between two layers

this pro1)orti oiiality : 1

W f-

1 alp'

(4.2-35)

K" as K' as On the basis of the definitions of velocity potential and hydraulic gradient. i t is quite evident that Eq. (4.2-35) determines a condition for the hydraulic gradients in the two contacting layers a t the intersection of the stream h i e with the bordering surface; viz. their components parallel t o the surface have to be equal: I," = I;. (4.2-36) Tlie other basic condition is the continuity of flow,which requires that the compoitenls of secpugc velocities pcrpendiculur to the surface should also be epcal : v; = VA. (4.2-37) Accepting the vdidity of Darcy's law the ratio of the normal components of the hydrnulic gradients can also be determined from Eq. (4.2-37), i.e.

K"I; = KfIL.

(4.2-38)

I n mi isotropic, homogeneous medium the gradient vector is tangential t o the path (in the case of steady flow t o the stream line as well). Therefore, the angles closed by the path (or stream line) and the normal of the bordering ?' can be calculated in both layers from the ratio of the surface (i.e. ,iand coml)onents of the gradient vectors:

r)

I' = tan

I:,

p'

I," I;

; -- - tan

r.

(4.2-39)

Combining Eqs (4.2-36), (4.2-38) and (4.2-39) the h a 1 result can be achieved: the rutio of the tangents of the two angles closed by the path and the n o r m 1 of the dividing surface i s equal to the ratio of the hydraulic conductivities of ihe contacting layers: tan K" -=(4.2-40) tan p' K'

524

4 Kinematic characterization of seepage

It follows from Eq. (4.2-40) that if the resistance of the second layer is very high compared to the previous one, the path (or stream line) does not enter i t (more precisely it hardly enters) and, therefore, the second laver can be regarded as impervious. This interpretation gives a further e s l h i a tion why the approximation of the layered field with an anisotropic medium is not acceptable if there is a very impervious or very pervious member i n the series of layers. Here the ratio of hydraulic conductivities of two neighbouring layers is too high. However, there is no complete refraction a t the borders of the layers (as it can be observed in optics), but according to the correct characterization, the stream lines hardly enter the layer having low hydraulic conductivity. It is the reason why a sedimentary layer is never absolutely impervious. This is only a relative term, the acceptance of which depends on the character of flow, the size and form of the seepage field, and also on the proportion of the total flow rate, which can be regarded as negligible. To prove the last statement, let us consider as an example the seepage field below the horizontal impervious foundation of a dam (Fig. 4.2-12). The difference between the elevations of the head and tail water maintains a steady seepage under the dam. The first supposition should be that the depth of the pervious layer is infinite. The flow net can be mathematically determined in this very simple case, and the vertical distribution of seepage velocity along the symmetrical axis of the flow field can be calculated. If the field is not homogeneous, but composed of two layers, the original flow net is distorted. The modification of the vertical distribution of seepage permeable layer of infinite depfh

practica/& impervious lower layer

2b

w8ter transporf through the lower layer is noi'n~l@ible 2b

Fig. 4.2-12. Interpretation of the depth of seepage field, in layered Hystem

525

4.2 Boundary and initial conditions

velocity depends evidently on the depth of the second layer, and on the ratio of hydraulic conductivities. The lower lying layer conveys a portion of the total flow in any case (even if i t has low conductivity and its surface is in a deep position). It is necessary, therefore, to decide first, what ratio of the two flow rate is regarded as negligible. After determining the total flow rate (e. g. by integrating the graph of vertical velocity distribution in both layers) and comparing the water conveyance of the lower layer to it the relative imperviousness of the lower layer may be judged. When this limit is set e.g. at 10% of the total flow rate, the lower layer is called impervious, when the upper layer transports more than 90% of the total flow rate. It follows that a layer can be regarded as impervious in a deep position, while the same layer conveys more than the limiting rate if its surface is in a higher position. If the criterion of imperviousness is determined by the percentage of total flow rate, even the lower part of a homogeneous layer can be neglected from the point of view of water t,ransport, as i t would be impervious. It can also be seen in Fig. 4.2-12 that the critical depth of a given layer (where its water transport can be neglected) cannot be determined with absolute values of depth given in metres, because i t depends not only on the ratio of hydraulic conductivities, but also on the size and form of the seepage field. On the basis of the similarity of flownets the maximum depth of a given stream line is influenced by the geometrical data of the system aa well (in the example by the length of the foundation i.e. 2b). This fact draws attention to insufficient investigations where an attempt is made to determine the active depth of a pervious layer, below which neither natural effects nor human activity on the surface can initiate seepage. Such approximation can be accepted only, if they consider the whole structure (form and

0.0

0.50

m, m

LOO

0.0

0.5

m, m

-

l.0

Fig. 4.2-13. Ratio of total water conveyance of a double layered seepage field and the fictive flow rate of a similar homogeneous field

526

4 Kinematic characterization of seepage

size) of the seepage field and its contact with the actions creating and maintaining the flow. For numerical characterization of the relationships explained only qualitatively in the previous paragraphs some of the results of an experiment are shown in Figs 4.2-13 and 4.2-14. The object of the experiment was to determine the influence of the layered flow field under different foundations composed of horizontal and vertical elements (concrete blocks and sheet piles) (ojfaludy, 1974). I n Fig. 4.2-13 the measured points and the constructed curves are shown, which characterize a double layered system of two dimensional seepage when sheet piles are not applied, and the foundation is a horizontal plane placed on the surface of the layer. The measured total flow rates (9) are related to a fictive value calculated by supposing only one layer below the foundation, the thickness of which is equal t.0 the total thickness of the two different layers ( m = m1+m2) and its hydraulic conductivity is equal to the parameter of the more pervious original layer [qol if the upper layer is more permeable ( K , > K,) and qo2in the opposite case ( K , < K , ) ] . These fictive values are calculated from Eq. (5.3-67) hy substituting m aa the thickness and K1 or K2 as hydraulic conductivities. The first graph of Fig. 4.2-13 represents the case K ,

>K ,

and the second one shows the q/qo2 value as the function of

,-' ] I : [

[

q/qoL=

m

m

f

if K ,

< K,.

The curves approximating the measured points can be espressed in the form of mathematical equations as well:

---

l0-

--g 05-

00

0.5

LO

2 m

Fig. 4.2-14. Ratio of water conveyance of the upper layer and the total flow rate in double layered system 88 a function of the rate of the hydraulic conductivities

4.2 Boundary and initial conditions

527

> K,;

if K ,

q=-

K*AH x

where

1.5-

and 901=

if K ,

KIAH -

(4.241)

7c


[

q = - K*AH ar sh 1.5%) ;

x where

K* = K ,

and 902

=K,AH ar sh (1.5

x

TI.

The flow rate through the upper layer (ql) related to the total discharge K , > K , (Fig. 4.2-14):

(q) can also be approximated mathematically and, if

(4.242)

Supposing that a is the negligible fraction of flow rate (e.g. if the limit, mentioned previously is lo%, then a = O . l ) , the condition of the imperviousness of the lower lying layer can be expressed also in mathematical form from Eq. (4.2-42):

52 m

[ :[ 1.67 -sh

(1 - a)ar sh

528

4 Kinematic characterization of seepage

Applying one or more sheet piles the situation becomes more complicated e.g. if the thickness of the upper lying pervious layer is smaller than the depth of the sheet piles, all stream lines, therefore, have to penetrate into the second layer, and the water transport is determined by the water conveying capacity of the less pervious material. Although Eqs (4.2-41), (4.2-42) and (4.2-43) are valid only in the case of a very simple form of the seepage field (as shown in the figures), they are suitable to visualize the problems arising in connection with the use of the term impermeable. They can be applied as rough estimations in many practical cases to characterize the expected development of flow through a layered system. Another special type of layered seepage fields is the case of a vertical (or almost vertical) position of the contacting surfaces, if a nearly horizontal unconfined flow develops through the system. The special aspects to be considered in this cme can be well demonstrated in the section of a multilayered earth dam (Fig. 4.2-15), which is one of the most frequently occurring practical examples of this type of seepage field. Where the water enters a less pervious layer from a pervious one, the less pervious layer causes a backpressure effect in the preceding part of the seepage field.’ At the contacting surfaces, where the change of hydraulic conductivities is the opposite (the first layer is less and the second more permeable) a free exit surface has to develop in a similar way to the contact of the seepage field with the tail water (Kov&cs 1968). It is necessary to emphasize here that a stretch of the seepage field must never be investigated separately, always the whole field has to be considered from the entrant potential surface to the final exit face. The latter is also a potential surface below the level of the tail water. Also resistances to flow within the entire section have to be determined. It is for this reason, in the example given in the figure that the dynamic equilibrium can be described within the body of the dam, including not only the free exit face where the water enters into the atmosphere, but also the internal free exit face (or the internal free exit faces, if there is more than one impervious layere, from which the water flows into more permeable layers). The equations characterizing this dynamic equilibrium have to be solved simultaneously, satisfying the given boundary conditions.

Fig. 4.2-16. Development of phreatic surface in a vertically layered earth dam

529

4.2 Boundary rand initial Conditions

4.2.3 Application of hodograph image and other special transformations for the characterization of boundary conditions Among the various kinematic formations, the path (or the stream line in the case of steady movement) can be used for the characterization of seepage velocity. It gives information, however, only on the direction of the velocity vector (the latter being tangential to the path). For characterizing the size of the vector the hodograph curve i s frequently applied on the basis of Kirchhoffs proposal. This curve runs through the end points of velocity vectors represented in a coordinate system independent of the field of the actual movement. All the vectors start from the origin of this new coordinate system (hodograph space), and the curve follows the order of the points along the real path. When investigating a two-dimensional movement, the hodograph (which is generally a three-dimensional curve) becomes also two-dimensional, and it can be represented on a so-called, hodograph plane (Fig. 4.2-16). In seepage hydraulics the use of hodographs assists us in representing the phreatic surface, the position of which is not known a priori, but considering the boundary conditions along this surface its hodograph image can be easily constructed. It is for this reason that the application of this method is very wide in this subject. It is necessary to note also that in homogeneous and isotropic seepage fields the velocity vector is proportional to the vector of the hydraulic gradient and the factor of proportionality is the hydraulic conductivity (accepting the validity of Darcfs law). The velocity hodograph can be supplemented, therefore, with gradient hodograph, which differs from the former only in the unit size measured on the axes of the coordinate system. The advantage of the application of the gradient hodograph is that in this case dimensionless quantities are represented on the axes (the components of the gradient vectors). Tf a hodograph is constructed for each point of the seepage space, the end points of the velocity or gradient vectors will completely fill a limited space,

t"

hodograph plane

"X

Fig. 4.2-16. Interpretation of hodograph curve 34

530

4 Kinematic characterization of seepage

bordered by the end points of those vectors which belong to the contour surface of the actual seepage space. Limiting the discussion to the investigation of two dimensional flow, the end points of the vectors, belonging to the contour line of the seepage field, determine a closed curve on the hodograph plane, which is called hodograph contour. The construction of the limited hodograph field surrounded by the hodograph contour is a special type of mapping of the actual seepage field, because contact can be established between the relevant points of the two fields. If this contact can also be determined analytically and expressed in the form of mathematical formulae, the equations can be used for the calculation of the hydraulic parameters, as will be described in Chapter 5.1 and 5.2. In many cmes there are possibilities only for the graphical construction of the hodograph contour, and not for the analytical description of the relationship between the flow plane and the hodograph plane. This method, however, is also very useful in solving practical problems, because valuable information can be gained in this way on the velocity conditions along surfaces, the positions of which are not known on the actual seepage plane (as was already mentioned it gives the hodograph image of the phreatic surface, and even the velocities at special points of this surface can be determined). For this reason, the literature of seepage hydraulics deals generally in great detail with the construction of hodograph contours and with their practical application (Hamel, 1934; Vedernykov, 1934; Polubarinova-Kochina, 1952; 1962; Aravin and Numerov. 1953; Harr, 1962; Bear et al., 1968; Bear, 1972). The reader interested in special problems of the construction of hodographs is referred to the listed publications. Here only the main rules of the method will be summarized and some examples will be given to demonstrate its application. Along a potential line (or if three dimensional flow is investigated, along an equipotential surface) the gradient is always perpendicular to this line. The hodograph images of the points of a potential line are fitted. therefore. to straight lines,starting from the origin of the hodograph plane and perpendicular to the potential line at the point in question. The sections of entry a n d exit faces contacted by surface-water bodies are always potential lines. If these contours are straight lines, the relevant hodograph points form a line crossing the origin and normal to the contour in question (Fig. 4.2-17; lines a' and a"). If the entry or exit face is not a plane (in the case of two dimensional seepage its contour is not a straight line) the position and the form of the hodograph contour cannot be determined, because only the direction of the velocity (or gradient) vector is known at every point, but not its length. In the case of an earth dam, shown as an example, it is evident that the toes of both the upstream and downstream slopes are singular points with zero velocity (saddle points). Their hodograph image8 are fitted, therefore, to the origin of the hodograph plane. Along a stream line of a flow net the direction of the velocity vector is similarly known, the vector being everywhere tangential to the stream line. On the basis of this condition the hodograph images of impervious contours can be constructed, these borders being stream lines. In general cases these hodograph contours are also undetermined curves, the position and form of

4.2 Boundary and initial conditions

531

if the influence of the caoillary fringe has to be mnsiderd Fig. 4.2-17. Construction of a hodograph field characterizing the seepage field of an earth dam

which is not known. But if the bordering impervious surface of the seepage space is a plane (the contour of the flow field is a straight line) its hodograph image is also a straight line crossing the origin and parallel to the contour in question. In the example the lower impervious boundary of the earth dam is such a contour (Fig. 4.2-17: line b). Because this border of the seepage field is horizontal, and crosses the toes of the slopes (where the velocity is equal to zero) the relevant hodograph contour has to be horizontal as well, and i t has to start and end at the origin of the hodograph plane. The ho34*

532

4 Kinematic characterization of seepage

dograph line moves from the origin along the horizontal axis of the hodograph plane ( I , or v, axis). Velocity or gradient reaches its highest value at point 8 (the position of which is not known either in the seepage field or in the hodograph plane), from where the hodograph contour turns back to the origin following the same path. If the role of the capillary zone has to be considered, as is indicated in the figure, the hodograph images of the capillary exposed contours (faces) have also to be determined. They arealso stream lines, the geometrical position of which is fixed and known. In the exemple they are straight lines, the relevant hodograph contours are, therefore, also straight lines crossing the origin of the hodograph plane (Fig. 4.2-17 lines c' and c"). Both ends of the hodograph lines can also be determined, but we shall return to this problem after characterization of the hodograph image of the phreatio surface. There is one further section of the of seepage field contour, the position of which is geonietrically determined and independent of the kinematical character of the movement; i.e. the free exit face. The boundary condition along this contour is described by Eq. (4.2-24). Considering this equation, it can easily be proved, that the end points of the gradient vectors, starting from the origin of the hodograph plane have to be fitted on lines crossing the vertical axis at a point Iy = -1 and being perpendicular t'o the free exit contour at the relevant points. I n the general caae the position of this hodograph contour is also undetermined, but if the free exit surface is a plane, the lines transforming the various points of this stretch onto the hodograph plane become identical. Thus, only one straight line (crossing the point I , = -1; I , = 0 and perpendicular to the freeexit face) characterizes the position of the hodograph contour (Fig. 4.2-17; line d ) . Knowing the velocity or gradient values at the two bordering points of the free exit surface the validity zone of the hodograph line can also be indicated. There is only one further part of the contour of the seepage field which remains to be investigated, i.e. the phreatic surface. It is necessary once again to make distinctions according to the conditions along this surface. There are three aspects which have to be taken into account: whether the capillary eflect has to be considered or can be neglected, whether the flow i s steady or unsteady and whether accretion influences the seepage field through this surface or not. The most simple case is the steady seepage without capillary influence and accretion. The boundary conditions are given in this caae by Eq. (4.2-9). Considering the form of the equation giving contact between the components of the hydraulic gradient vector, i t can easily be seen, that the image of the phreatic surface o n the gradient hodograph plane is a circle with a radius of R = 112, the centre of which is at a point fitted to the vertical axis C(0:-112) (Fig. 4.2-17; line e). The position of the circle indicates that the gradient along the phreatic surface can never have a vertical component directed upwards, if the seepage is steady, capillarity is negligible and there is no accretion. As already proved, Eq. (4.2-9) characterizes the upper surface of the capillary zone and not the phreatic surface if such a zone develops above the water table. This is because in this case the surface with constant pressure

533

4.2 Boundary and initial conditions

( p = 0 ) is not a stream line, while the capillary surface satisfies the two conditions given by Eqs (4.2-6) and (4.2-10): i.e. i t is the stream line with constant pressure ( p = - yh,). It can be stated therefore, that if the capillary effect is not negligible the hodograph circle [R = 1/2: C(O:-l/2)] describes the hydraulic gradient along the capillary surface (Fig. 4.2-17; line e l ) , while the image of the water table is a curve with undetermined form and position within the hodograph field. I n the caae of unsteady seepage without capillary eflect and accretion the hodograph image of the phreatic surface i s similaTly a circle, the centre of which is also at the point C(0:-1/2), but its radius changes from point t o point and from time t o time (Morel-Seytoux, 1961). The numerical value of the radius can be determined from Eq. (4.2-21) (4.2-44)

It follows from Eq. (4.2-44), that the radius is greater than 112, if-89, >0, at

and the upper stretch of the circle is above the horizontal axis of the hodograph plane, indicating that t,here is a stretch where the movement is directed upwards along the water table. I n the opposite case, if-aQ1< 0 , at the circle is smaller than that representing steady conditions and, therefore, the water particles composing the phreatic surface can move only downwards (Fig. 4.2-18). Posifive accretion (recbaue)

neg8itve accrefion(draimje)

unsteady seepale

+f t

Fig. 4.2-18. Modification of the hodograph circle characterizing the water table unde, the influence of accretion and the unsteady state of seepage

534

4 Kinematic characterization of seepage

Changing both the radius of the hodograph circle and the position of its centre, the hodograph curve of a water table influenced by accretion can also be determined. I n this cam both parameters depend on the ratio of accretion and hydraulic conductivity. On the basis of Eq. (4.2-19) their numerical characterization can be given by the following formulae:

consequently the hodograph circle intersects the vertical axis of the coordinate system a t a point I , = -1: I , = 0. If the seepage field is vertically recharged (positive accretion E > 0), the radius of circle is smaller and its centre is a t a deeper position, compared t o the uninfluenced condition. I n the case of vertical drainage (negative accretion E < 0), the circle having a larger radius and higher centre rises above the horizontal axis indicating the possibility of vertical upward movement of the water particles situated along the water table (Fig. 4.2-18). Knowing the position of the various stretches of the hodograph contour, the images of the corner points of the seepage field can also be determined. It has already been shown that the two toe-points of the slopes are transformed t o the origin of the hodograph plane (singular points with zero velocity) and point 8 (maximum velocity along the horizontal impervious boundary) is fitted to the horizontal axis, but its position along this line is unlmown. The transformation of these points is not influenced by the conditions prevailing along the water table, their images may be fixed, therefore, a priori. The further form of the hodograph contour depends, however, o n the character of the phreatic surface. Two different conditions of steady seepage will be analyzed further on aa examples, one without capillary influence and the other considering the effect of the capillary zone, but both without accretion. The hodograph has the most simple form, if the steady seepage i s not influenced either by capillarity or accretion (Fig. 4.2-17c). Starting from point 1, the image of which is the origin, the hodograph contour goes along line a’. Point 2 is the intersection of the entry face and the phreatic surface, its image has t o be fitted, therefore, t o bath line a‘ and line e (hodograph circle). Thus, the gradient (or velocity) is completely determined at this point. The hodograph contour from here follows line e upwards having smaller and smaller gradient, uiitil reaching the inflection of the water table (point 9), the position of which is not known either in the seepage field or in the hodograph l’lane. It turns back from this point and goes once again along line e because this contour of the seepage field is still a phreatic surface. The next corner point is that, where the water table meets the free exit surface (point 5). Its image haa t o be, therefore, the intersection of line e and line d. The gradient along the water table is equal t o the slope of the latter, the position of point 5 determines, therefore, not only the size of the gradient here but also the apgle of the slope. It follows, however, from Thales’ law that the line is perpendicular to line d . Knowing that the latter is perpendicular to the

4.2 Boundary and initial conditions

535

exit face, i t can be stated, that the free exit face is tangential to the water table at their intersection. After point 5 the hodograph contour has to follow line d , but it is also clear, that below the level of the tail water i t is represented by line a", so that point 7 should be fitted to the origin once again. Lines d and a" being parallel to one another, the hodograph contour can be continuous and closed only if the image of point 6 is at an infinite distance in the direction of lines a" and d. After point 7 the hodograph contour follows the horizontal axis, reaches the point of maximum velocity along the impervious boundary (point a), the position of which is undetermined, turns back to the origin andcloses the hodograph field, when reaching point 1 once again. When the capillary zone has to be considered as a part of the seepage field (Fig. 4.2-17d) the hodograph contour starts similarly from the origin (point 1 ) and moves along line a'. Point 2 is, however, a singular point in this caae with i n h i t e velocity (external corner point). Its hodograph image is, therefore, at an i n h i t e distance, characterized by a section of circle with infinite radius, extending from the direction of line a' to that of line c' (because point 2 has to be fitted to both lines). The hodograph contour comes from infinity along line G' and reaches the origin, because point 3 is a saddle or internal corner point (a singular point with zero velocity). Point 3 being the intersection of the capillary exposed entry face and the capillary surface, the hodograph contour has to be continued along the hodograph circle which is the image of the capillary surface (line e'). This condition states that the capillary surface always joins horizontally to the capillary exposed face. Because point 4 is the meeting point of the capillary surface and the capillary exposed exit face, its image is the intersection of lines e' and c". But the same point is also the intersection of lines c" and d on the hodograph plane, thus the images of points 4 and 5 cover one another. It follows from this condition that the capillary exposed exit face is also a tangent to the capillary surface, as is the free exit face to the water table in the previous example. The stretch of the border of the seepage field is a stream line, where the pressure changee from -p, to zero, thus the potential at a point here is 9 = K(Y - PI; where

(4.246)

Pc > P > 0. This contour being a straight line in the example, and if the linear change of pressure p can also be assumed, the derivate of the potential along this line is constant, and, therefore, both velocity and hydraulic gradient are also constant. Thus, the hodograph image of this stretch is reduced to one point (point 4 identical with point 5 ) . The continuation of the hodograph contour is the same as it was in the previous caae. It goes to infinity along line d (point 6) turns back to the origin along line a" (point 7) and after following the image of the horizontal impervious boundary through point 8, it is closed at the origin (point 1).

536

4 Kinematic characterization of seepage

There are some other special functions, which can be applied in a similar way to the hodograph. Mapping the original physical seepage field into a new plane according to these functions, the position of the images of some undetermined contours of the flow field can easily be constructed in the transformed system and also some hydraulic parameters along the contours can be calculated. The well-known, and widely applied mapping method among these functions, i s the Zhukovsky’s function (Zhukovsky, 1923). In the case of steady, two-dimensional seepage the product of the pressure and a constant value can be expressed as a linear function of velocity potential [see Eq. (4.1-12):]

(;1

because p = K - + y . (4.2-47) Y The 8,function is harmonic in x and y, because both g, and y are harmonics. The conjugate of 0,can also be determined:

01 --K-=g,-Ky; P

0 , = y + Kx.

(4.2-48)

The complex number composed of the two variables is called Zhukovsky’s function, which can be expressed also as a function of the position of the investigated point on the actual phjsical plane (described by the complex form of the position vector: z=z+iy) and the potential- and stream-functions belonging to this point, (these may be included in a summarized form into one function called complex potential, t,he detailed definition of which will be discussed in Section 5.1.1 w = g, iy):

+ 0 = 0,+ i0,= w + iKz.

(4.2-49)

Considering the Zhukovsky’s function as a complex number and representing i t in a coordinate system the horizontal axis of which is 0,and the vertical a, the boundaries of the flow field can be determined on the new plane, if both the position and the boundary conditions at each point of the contour are known (the position determines the z value, while the boundary condition gives g, and y from which w can be composed). The images of special boundaries can be constructed on the Zhukovsky’s plane, even if some of the basic data are undetermined. Thus, the water table (independent from the influence of the capillary fringe) and the free exit face are mapped to the vertical axis of the new plane, because the pressure is zero along those lines, and the velocity potential is proportional, therefore, to the elevation above the reference level: 0 , = 0 ; i f p = O ; andg,=Ky.

(4.2-50)

Along a stream line the 0, value is a unique function of the position of the point in question, because the stream-function is constant here:

0,= C, + Kx ; if y = C1= const.

(4.2-51)

Equation (4.2-51) is valid for the water table if the influence of capillarity is negligible, and accretion does not affect the flow. The same relationslip

4.2 Boundary and initial conditions

537

describes the image of the impervious boundaries of the field, (although the numerical value of the constant is different if the lower rather than the upper contour is investigated). Thus the Zhukovsky’s function €or the phreatic surface of a steady two-dimensional seepage without capillary effect und accretion is: (4.2-52) 0 = i(C, K x ) .

+

As in equation (4.2-51), the O1value can be expressed a8 a unique funct,ion of the position of the investigated point along potential lines (e.g. entry and exit faces), where the potential-function is constant:

0,= C, - K y ; if

v = C, = const.

(4.2-53)

T h e image of the capillary surface is parallel to that of the water table, because the pressure is constant along this contour as well, although it is not zero [see Eq. ( 4 . 2 - l o ) ] , and thus Zhukovsky’s function can also be used to describe this contour, the capillary surface being always a stream line: 0 1

= -Kh,;

if and

(4.2-54)

To characterize the capillary exposed exit face it may be supposed once again that the pressure changes linearly between - p c and zero [see the explanation given in connection with Eq. (4.2-46)]. Accepting this approximation the image of the contour in question on Zhukovsky’s plane can be described with the following equation:

because and

q=K

[i

y1--

3 ; +-y

, O J .

9, = C , + K z ; [see Eq. ( 4 . 2 - 5 1 ) ] . where

and

(4.2-55)

538

4 Kinematic characterkation of seepage

Special hydraulic interpretation can also be given to Zhukovsky’s function if Eq. (4.2-49) is differentiated according to the z complex variable: d@ - d(w +iKz)dw --dz dz dz

+ i K = - V* + i

~ ;

(4.2-56)

[because the derivate of w complex potential according to z gives the negative conjugate vector of velocity (-v*) as will be proved in Section 5.1.11. Thus, knowing the interrelation between Zhukovsky’s function and velocity along a contour (e.g. by constructing the images of the contour for Zhukovsky’s and the hodograph planes) the position of the contour on the real physical field can also be determined. z=

J -v* d@+ iK-.

(4.2-57 )

Any other functions differing from Eq. (4.2-49) only by multiplying it with a constant factor can be similarly applied to characterize special boundary conditions, and they are known generally as Zhukovsky’s functions (Polubnrinova-Kochina, 1952, 1962). Thus, the complex pressure where

-+ ip’ ; Yo,;

P =p p’=

(4.2-58)

K

is also a type of Zhukovsky’s functions. Similarly the following generally applied forms @=i(w iKz) = Kz - iw ;

+

or

. U’ G=z-~-;

(4.2-59)

K

can also be applied instead of Eq. (4.2-49) (Aravin and Numerov, 1953, 1965).

Examples showing the application of Zhukovsky’s function can be found in the books of Polubarinova-Kochina (1952, 1962); Aravin and Numerov (1953, 1965); Harr (1962) andBear (1972). The role of Hamel’s mapping function (4.2-60)

is the same aa that of the previous methods (i.e. to characterize the geometrically uncompletely defined contours of the flow field) (Hamel, 1934). The special properties of this function are described and examples of its application are given by Muskat (1937) and Bear (1972). Readers interested in studying further details in connection with this method are referred t,o these publications.

4.2 Boundary and initial conditions

4.2.4 Consideration of initial conditions The definition of the initial or starting conditions has already been given in the introduction of this chapter. It waa explained that the kinematic equations can describe only the change of the hydraulic parameters in the m e of unsteady movement. Therefore, the calculated data have to be related always to an initial situation, when the parameters (velocity, pressure, flow rate) are known for the entire seepage field, because the time-variable values of these quantities can only be obtained by adding the changes to the initial values. The most simple solution is to choose the static equilibrium as an initial cmditwn, supposing that at the beginning of the investigated period the total potentials were equal along the entry and the exit faces of the seepage field. Thus, the initial condition is characterized by zero velocity, zero potential difference and zero flow rate at every point of the field. The pressure may be determined from the position of the head and tail water which are supposed to be a t the same elevation at the start of the process. If the seepage is created and maintained by continuously changing potential diflerence, a time point can be determined in most caaes, which can be regarded aa the start of the movement, when the static equilibrium can be applied as for the initial condition. The method can be used even in those cases, when the period in which the seepage is actually investigated, does not include this starting time. Knowing the time variant boundary condition, they can be extrapolated either backwards or forwards in time outside the investigated period to determine the time point, when the static state can be accepted, as the most simple initial condition. There are cases, when the mvement has an initial transition period before developing that state of seepage, in which the changes of boundary conditions can be characterized (or at least approximated) with monotonous mathematical functions. An example of this type of seepage is the unsteady movement around a well, the detailed analysis of which is given in connection with Fig. 3.1-11. If a mathematically easily amenable relationship is determined for that period of the process when i t can be considered to be regular, a starting point can be calculated by extrapolation, which really does not fit the curve describing the actual variation of the parameters in time. This fictive initial time point can beaccepted, however, if the characterization of the transition period can be excluded from the investigation, because starting at the fictive time point from the supposed initial condition (e. g. static state), and following the estimated change of boundary conditions, the hydraulic parameters can be well approximated at every moment after the elapse of the critical transition period. It happens many times that such approximate hypotheses have to be applied, which give infinite value for one or more parameters at the beginning of the movement, e.g. if an unsteady seepage maintained by a given and constant draw down of the contacting surface-water body has to be investigated. Choosing the static state aa the initial condition, i t has to be supposed that the draw down was created instantaneously at the beginning of the in-

540

4 Kinematic characterization of seepage

vestigated period. It then incrertsed from zero to a definite value during an infhitesimally small period, which hypothesis gives infinite velocity and flow rate at the first instant. Although this discrepancy of the method could cause considerable error, if the purpose were the characterization of the process from the beginning of the movement, good approximation of the later stage of seepage can be achieved in this way, if the calculation of the parameters in the first short period can be neglected. Investigating seepage with periodically fluctuating time-variant parameters, i t is very probable that after starting from static equilibrium the first few periods are not yet regular, the parameters show a continuously changing trend apart from the fluctuating variation. It is necessary to have some time elapsed after the beginning of the process to achieve a state of movement, when the characteristics have the same numerical values at the corresponding time points of two subsequent periods. The calculation has to be continued, therefore, until a considerable difference occurs between the periods, and the values describing the hydraulic parameters in the last period can be accepted as the solution of the given problem. The static state has been analyzed up to now as an initial condition and various approximate hypotheses have been mentioned to facilitate the application of this simplest characterization. If the actual movement cannot be approximated, however, in this way other initial conditions can be chosen. A general state of the unsteady movement at a moment chosen arbitrarily is not suitable for this purpose, because both the determination and the mathematical description of this condition are hindered by numerous difiiculties. The other way is, therefore, to accept the steady state seepage as an initial condition, supposing that before the boundary conditions started to change in time, they had been constant, and the movement had developed as the result of those time-invariant conditions. Naturally, the situation is very similar to the application of the static state, a8 initial condition. The steady seepage (which is fully described for the entire seepage field by knowing velocity, pressure or potential, and flow rate at every point) may be characteristic at one time point of the continuous series of changing stages, or i t may be a hypothetic stage at a fictive time point providing a good approximation of the actual parameters in the investigated period. The time-variant period of movement can start from the steady state with continuous. smooth transition, or i t may be necessary to assume an instantaneous change. The latter case results in infinite parameters at the rapid change of the process, the period surrounding this time point should be excluded, therefore, from the investigation. Finally periodically fluctuating movement can be superimposed over a steady seepage as well. I n this crtse the calculation of more periods is necessary to check whether the difference occurring between two subsequent periods is negligible or not, and to determine the parameters of the stabilized waves, a8 is done when the investigation starts from static equilibrium. As a final consequence, i t can be stated that in practice either the static state, or the steady movement is applied as an initial condition, because the complete characterization of the seepage field cannot be expected in other cases. Attempts have to be made always to determine how the actual time-

References

541

variant movement could have originated from one of the possible initial condtions. Generally the methods and approximations listed in the previous paragraphs are applicable, but universal rules and laws cannot be stated. References to Chapter 4.2 ARAVIN,V. I. and NUMEROV,S. N. (1953): Theory of Movement of Fluids and Gaseous Material through Non-deformable Porous Medium (in Russian). Gostekhizdat, Moscow. S. N. (1966): Theory of Motion of Liquids and Gases in ARAMN,V. I. and NUMEROV, Undeformable Porous Media. I P S T , Jerusalem. BEAR,J. (1972): Dynamics of Fluids in Porous Media. Elsevier, New York, London, Amsterdam. BEAR, J. and DAOAN,G. (1962): The Use of Hodograph Method for Groundwater Investigation. Dept. of Civil, Eng. TECHNION, Haifa. S. (1968): Physical Principles of Water BEAR, J., ZASLAVSKY, D. and IRMAY, Percolation and Seepage. UNESCO, Paris. CHILDS,E. C. (1959): A Treatment of the Capillary Fringe in the Theory of Drainage. Journal of S o i l Science. GALLI,L. (1959): Approximating Method for the Calculation of Seepage under Hydraulic Structures through Layered Formations (in Hungarian). Viziigyi KozZemdnyek, No. 3. HAMEL, G. (1934): On Ground-water Flow (in German). Zeitschrift jiir Angewandte Maternatik und Mechanik, No. 3. HARR,M. E. (1962): Groundwater and Seepage. McGraw-Hill, New York. KAMENSKY, G. N. (1943): Principles of Ground-water Dynamics, (in Russian). Gosgeoltekhizdat , Moscow. G. N., KORCHEBOKOV, N. and RAZIN,K.J. (1935):Flow of Ground Water KAMENSKY, in Heterogeneous Strata (in Russian). Gosizdat, Moscow. KovAcs, G. (1962): Consideration of the Covering Layer Situated before the Dam when Calculatin: the Parameters of Seepage under the Dam (in Hungarian). Viziigyi Kozlemdnyek. No. 3. Kovbcs, G. (19ti5): Influence of Development of Free Exit Face on Flow Rate Percolating through an Earth Dam with Vertical Faces (in Hungarian). HidroZdgiai Koz16ny, 9. KovAcs, G. (1966): Hydraulics. (in Hungarian). VITUKI, Budapest, Vol. 111. KovAcs, G. (1968): Seepage to Ground Water Created by Hydraulic Structures. Actu Techriaco dradernirce Scientinrum Hungaricae, Tom. 60. No. 3-4. LUTHIN,J. N. (1966): Some Observations on Flow in the Capillary Fringe. IASH Symposium wk JVrzter in Unsaturated Zone, Wageningen. MOREL-SEYTOUX, H. J. (1961): Effect of Boundary Shape on Channel Seepage. Stanford Unia. Dep. C. E . Technical Report, No. 7 . MUSKAT, M. (1937): The Flow of Homogeneous Fluids through Porous Media. McGraw-Hill, New York. NEMETH,E. ( 1 963): Hydromechanics (in Hungarian). Tankonyvkiad6, Budapest. POLUBARINOVA-KOOF~INA, P. Ya. (1952): Theory of Ground-water Movement (in Russian). Gostekhizdat, Moacow. POLUBARINOVA-KOCHINA, P. Ya. (1962):Theory of Ground-water Movement. Princeton University Press, Princeton. PRANDT, L. and TIENTJENS,0. G. (1934): Fundamentals of Hydro- and Aerodynamics. McGraw-Hill, New York. H. E. (1958): Predicting Seepage under Dams on SHEA, P. H. and WHITSETT, Multilayered Foundations. Proceeding8 of ASCE, Vol. 84. p. 1. TERZAQHI, K. (1943): Theoretical Soil Mechanics. John Wiley, New York. ~JFALUDY, L. (1974): Investigation of the Length of the Floors of Weirs and the Depth of Sheet Piles from the Aspects of Seepage (in Hungarian). V I T U K I Report, No. 2639. Buclaprst.

542

4 Kinematic characterization of seepage

VAUCLIN, M., VACHAUD, G. and KHANJI,J. (1975): Two Dimensional Numerical Analysis of Transient Water Transfer in Saturated-unsaturated Soils. Modelling and Simulation of Water Reeourcee Systeme. North-Holland PublishingCompany, New York. VEDERNYIKOV, V. V. (1934): Infiltration from Canals (in Russian). Gostroizdat, Moscow. Yomas, E. G. (1966): Horizontal Seepage through Unconfined Aquifers Taking into Account Flow in the Capillary Fringe. IASH Sympoeium on Water i n Umaturated Zone, Wageningen. ZHWOVSKY, N. E. (1923): Seepage through Dams (in Russian). Collected Work9 Vol. 7. Gostekhizdat, Moscow.

Chapter 4.3 Kinematic characterization of non-laminar seepage Substituting the relationships describing the resistance of porous medium to laminar flow into the two basic differential equations of hj-dromechanics expressing the conservation of both mass and energy (i.e. equation of continuity and Navier-Stokes’ equation), Laplace’s differential equation is achieved as the final mathematical form of the kinematic charwcterizution of seepage. Investigating this relationship in detail, it was proved that any flow satisfying the Laplace’s equation has velocity potential and fulfils also the Cauchy-Riemann conditions, thus i t is irrotational. A further result of the analysis is the justification of the fact, that the Laplace’s equation is valid for the characterization of all laminar seepage having a linear relationship between seepage velocity and hydraulic gradient within the whole flow field except at the singular points. The further kinematic terms (stream-, and potential-functions, flownet, etc.) derived on the basis of tlie Laplace’s equation can, therefore, be directly applied when investigating this type of seepage. In most practical cases the assumption that the linear relationship between velocity and gradient is valid, is acceptable. Thus, the solution of the Laplace’s equation satisfying at the same time the boundary and initial conditions describing the influences along the border of the seepage field (as well as the starting state inside the latter if the flow is unsteady) provides the hydraulic parameters of seepage (flow rate, pressure and velocity) with adequate accuracy. This is the reason, why handbooks dealing with the flow through porous media discuss generally the analytical, numerical or analogue solution of these mathematical systems, as is done in the nest part of this book. There are, however, few cases when the application of the hypothesis previously mentioned leads to larger error, than the generally acceptable one. However, its application may neglect some of the important characteristics of seepage. Perhaps the most common example of this situation is the investigation of the unsaturated flow, which was analyzed in Chapter 2.3. It may happen sometimes, however, that the use of Laplace’s eqmtion hccs to be excluded even f r m the study of saturated seepage as well. It is necessary, therefore, to find relationships suitable to substitute Laplitce’s equa-

543

4.3 Non-laminar seepage

tion, when other than laminar flow in a saturated field is analyzed, and to determine the differences between the corresponding parameters calculated on the basis of different hypotheses. The objective of this chapter is to summarize the methods applicable to the characterization of non-laminar seepage through saturated porous media.

4.3.1 Differential equations equivalent to Laplace’sequations for the various zones of seepage

When the seepage belonging to the validity zone of Darcy’s law was investigated, Laplace’s equation was derived by simplifying the Navier-Stokes’ equation. The dynamic principles characterizing laminar seepage [i.e. there is only one accelerating body force (gravity), and one retarding force (internal friction), thus the resistance can be expressed by Darcy’s equation]. The general kinematic relationships substituting Laplace’s equation could be derived similarly, if the forces found to be dominant in the validity zones of the various types of seepage were taken into account in the Navier-Stokes’ equation. The necessary dynamic analysis was, however, already executed in Chapter 2.1, and the results were summarized in the form of relationships between seepage velocity and hydraulic gradient, valid in the different zones of seepage. The combination of these movement equations with the equation of continuity gives, therefore, the same result, and ensures at the same time an easier method of investigation. The movement equations in question are analyzed and summarized in Chapters 2.1 and. 2.2. Expressing the differential change of head (i.e. hydraulic gradient) as the function of seepagevelocity, the following formulae can he repeated here: dh For turbulent flow --- bv2 ; ds In the transition zones

-

On the basis of Darcy’s law

-

For laminar flow

dh - - av ; ds dh__ -av+c; ds

(considering threshold gradient) For microseepage _ -dh = c1 00.9 ds

(4.3-1)

+ cp.

These equations have to be combined with the equation of continuity valid for steady seepage [using the two-dimensional form here to ensure more simple representation, see Eq. (4.1-20)] to get the relationships equiva-

544

4 Kinematic characterization of seepage

lent with Laplace’s equation and to characterize the different types of seepage. Before performing this step, i t is necessary, however, to analyze, whether some elements of the flownet determined by assuming the potential character of flow remain applicable in the case of non-laminar seepage, or not. For the interpretation of the stream-functions and the derivation of mathematical formulae describing the position of stream lines, there was only one hypothesis applied: i.e. the velocity vector i s tangent to the stream lines at every point of the field, except at the singular points. This supposition is fixed by Eqs (4.1-16) and (4.1-25) in mathematical form and this interpretation of the stream lines is acceptable for any type of seepage. Only one restriction has to be mentioned, viz, in the case of turbulent flow when at a given point the direction of the velocity vector varies in time even in the case of steady seepage, the instantaneous position of the actual fluctuating vector will not be considered, but its average value, which determines the main direction of flow, is regarded as the tangent of the stream line. The next principle used for the derivation of the stream-function was the statement, according to which Eq. (4.1-25) has a solution, if the velocity i s continuous and derivable at every point of the seepage field, the solution results in the stream-function [Eq. (4.1-26)]. Excluding the singular points from the investigated field, as, when the potential seepage waa analyzed, the continuous and derivable character of seepage can also be assumed in all the zones of seepage. Once again the restriction mentioned before is taken account of, i.e. that in the case of turbulent flow the fluctuating velocity is characterized by its mean value. The result of this explanation is that the concept of stream-function can be generalized. On the basis of the relationship between its total and partial differential quotients the components of the velocity vector can be expressed depending on the stream-function: (4.3-2)

The fulfilment of this equation ensures at the same time, that the equation of continuity is also satisfied. In constrast to the stream-functions, the concept of potential-functions cannot be applied, if the flow i s not laminar, because the basis of their derivation was the existence of seepage potential, the prerequisit of which is the linear relationship between velocity and gradient. The equipotential lines, however, can be substituted by curves interconnecting the points of the seepage field, where the hydraulic head (the sum of elevation above a reference level and pressure head) has the same numerical value. The development of hydraulic head within the field is determined independently of the type of seepage and it is governed by the same principles as the development of potentials in the case of laminar flow. Along the entry and exit faces (except the free stretch of the exit face) the value of hydraulic head is constant and determined by the boundary conditions. The integration of the change of head along a stream line results in the total available head (the difference between the levels of the head and tail water). Movement

4.3 Non-laminar seepage

h

AH

a9

0.8

0.7 860.5 0.4 0.3

DIZ

- - -- - -- - -

545

0.1

calculated from L@?lace’squation calculafed porn O h ’ s equafion observed curve I

I

Fig. 4.3-1. Sketch of the flow field, investigated by Oka (1969)

being possible between two points only, if the hydraulic heads prevailing there are different, the velocity vector cannot have components parallel to the curves interconnecting the points having the same head. It follows from this condition that the two system of curves (i.e. stream lines and lines of equal heads) make an orthogonal network (the two types of curves intersect each other everywhere at right angles). This system, however, is not a net of orthogonal squares because the equality of the elementary changes of the potential- and the stream-function is ensured only in the caae of laminar movement (see. Fig. 4.3-1). Substituting the concept of velocity potential with hydraulic head in this way, relationships have to be looked for between this parameter and the components of the velocity vector instead of Eq. (4.1-12), which gives the 35

546

4 Kinematic characterization of seepage

same relationship between potential and velocity. It has already been proved, that in a homogeneous field the stream lines are normal to the lines of equal heads. It follows from this condition, that the gradient vector of the head is parallel to the velocity vector at every point of the field. Considering this fact the relationships in question can be determined from Eq. (4.3-1):

For turbulent flow

ah --=

8X In the transition zones

i3h blvl v,; - - -- b 8Y 1v

--vx+b

(4.3-3)

1%; (4.3-4)

On the basis of Darcy’s - ah ah = av, ; - - = avy ; law ax aY For laminar flow C (considering threshold - av, -v, ; gradient) I V I

(4.3-5)

+

C

=avy+-vy;

(4.3-6)

I V I

For microseepage

+v, ; Iv I C

0.1%

(4.3-7)

Recalling the results of the dynamic analysis of the various types of seepage, i t can be stated that Eq. (4.3-5) describing the potential seepage and derived on the basis of Darcy’s law does not provide an accurate relatiow ship for any zone, because theoretically the existence of the threshold gradieiLt ought to be considered in the laminar zone as well, and therefore. Eq. (4.3-6) would be the exact solution. It was, however, also proved that the difference between velocity values calculated either from Darcy’s equation or by using the correct relationship [Eq. (2.2-56)] is negligible over almost the entire range where practical problems occur. The potential theory and Laplace’s equation derived from the linear relationship is acceptable in practice for the characterization of laminar seepage. Equation (4.3-7) relates f o microseepage, the validity zone of which is determined with the ratio of the actual and threshold gradients ( I o I < 121,,). The error caused by the use of Darcy’s law in this zone is not too high, except in the close vicinity of the threshold gradient. It has to be considered also that the flux transported through the mass of porous media composed of very h e grains (e.g. clay, the threshold gradient of which may be considerable) is relatively small. At the same time these materials have

<

547

4.3 Non-laminar seepage

special structures (cracks and other dislocations), the water conveyance of which is generally higher than the mass flux. In most cases the consolidation cannot be neglected either in fine grained sediment,s, which fact excludes the application of the simplified form of the equation of continuity. It follows from previous explanations that the application of Laplace’s equation is acceptable in practice aa an adequate approximation, if the velocity is equal to or smaller than that belonging to the upper limit of the laminar zone. Only the water transport through cohesive fine grained material requires special and very careful1 attention, because in this case the seepage through the special structures and the influence of consolidation have to be studied as well. The existence of a threshold gradient has also to be considered, which may result in the decrease of the seepage field, because in the vicinity of the points where the gradient would be smaller than the threshold value no water movement is expected. This is the reason, why the general kinematic equation (applicable instead of Laplace’s equation) will be determined only for those types of seepage, which have higher velocity, than that in the laminar zone. The number of variations of the analyzed type can be further decreased, because only the relationships describing the movement in the transition zones have to be investigated, viz. from Eq. (4.3-4) all the other conditions can be achieved (substituting a = 0, Eq. (4.3-3) is gained while the substitution of b = 0 results in Darcy’s equation). Let us differentiate both lines of Eq. (4.3-4), the first should be differentiated with respect to y , and the second line with respect to x . After changing the sign of one of the equations their combination results in (4.3-8)

The hydraulic head (h) is excluded from the equation in this way. As the next step the stream-function should be substituted instead of the components of the velocity vector using Eq. (4.3-2). It has to be considered as well that the absolute valueof the velocity vector can also be expressed with the streamfunction on the basis of Eq. (4.3-2): (4.3-9)

The final form of the simplified equation is as follows:

+ 35*

aaw (!q2+ 8x2 a x

I-

(4.3-10)

4 Kinematic characterization of seepage

548

Following similar steps another relationship can be achieved, if the first line of Eq. (4.3-4) is differentiated with respect to x , andthesecondequation with respect to y . I n addition the sum of the two new equations is formed without changing their signs:

Considering Eq. (4.3-2), i t is evident that the expression in brackets and multiplied by a in the first member on the right-hand side of the equation is equal to zero. Executing the same substitutions aa in the caae of Eq. (4.3-8). the result gives a relationship between hydraulic head and streamfunction: aY aY, 8% -+-j-$=b P h 8% 8x2

””[

a x ay

[2y-2I]:[

-

zay [G

VI2)’+1 3

-

$1

.

(4.3-12)

For the calculation of the hydraulic parameters the system wmposed of two diferential equations [i.e. Eqs (4.3-10) and (4.3-12)] has to be solved, ensuring at the same time, that the result satisfies the boundary and initial conditions as well. The determination of the boundary conditions is exactly the same as discussed in the previous chapter. The only modification is that the hydraulic head has to be used everywhere instead of velocity potential as a boundary condition. Thus, the head is constant and a given value along the entry and exit faces. Along other boundaries either the constancy of the stream-function is a condition (which can be expressed also by stating that the change of the head is zero perpendicular to the border), or the numerical value of the head can be fixed (phreatic surface). Substituting the b = 0 value (which condition leads back to the application of Darcy’s law) into Eqs (4.3-10) and (4.3-12), the differential equations are considerably simplified:

-8% + - = o ;8% ax2

ay2

(4.3-13)

and

a2h a2h -+-=o.

ax2

ay2

It is evident that the first equation is the condition of the irrotational flow, [see Eqs (4.1-19) and (4.1-33)], while the second is identical with Laplace’s equation, since after multiplying i t with the hydraulic conductivity (which is supposed to be constant within the homogeneous field in the caae of the validity of Darcfs law), and substituting the equality, Eq. (4.1-13) is reconstructed. Some simplifications can also be made if the general system of the kinematic differeritial equations is used for the characterization of the turbulent

549

4.3 Non-laminar seepage

seepage. In this case a = 0 value has to be substituted in Eq. (4.3-4) to get Eq. (4.3-3). This substitution can be considered also in Eqs(4.3-10) and (4.3-12). The new system of differential equations, valid if the flow is turbulent within the whole seepage field, is as follows:

1.3-14)

ax2

' ay2

-

.

(4.3-15)

[Note: Eq. (4.3-15) is equal to Eq. (4.3-12), because the latter has no member depending on a . ] It is quite natural, that the diflerential equations describing nowlaminar Seepage are more complicated, than the Laplace's equations. Their solution is not expected, therefore, in closed form. Even in the ca8e of arelatively simple flow field the application of numerical methods (e.g. the use of finite differences) and thus the use of large computers is required. It is for this reimon that only a few publications can be found in the literature referring to the results of such investigations (e.g. Oka, 1969; McCorquodale andNg, 1969). These examples always prove the possibility of large errors caused by the application of the linear approximation. For example in the case of seepage around a sheet pile, the head loss necessary to transport the same flow rate was greater by 50 percent if the seepage waa in the transition zone than that calculated by using Darcy's and Laplace's equations.

4.3.2 Consideration of the continuous change of the flow condition within the seepage field The examples mentioned previously raise at the same time a further problem to be solved for the correct characterization of the kinematics of non-laminar seepage. Let us analyze the flow conditions in the flow field investigated by Oka (1969, Fig. 4.3-1). The purpose of the study is the determination of the hydraulic parameters of the seepage developing under a horizontal impervious foundation, which is supplemented with a vertical sheet pile. The two pressure head values necessary between the head and tail water to convey

4 Kinematic characterization of seepage

550

a given flow rate through flow field assuming a laminar and transition state of seepage respectively, were compared. In the latter case the required head was 1.5 times greater, than in the laminar state, if the relationsip between hydraulic gradient and seepage velocity waa described by the equation I = = 0.23~ 0 . 1 2 at ~ ~each point of the flow field. Although the hypothesis used aa the basis of the comparison h e . the transition state is characterized with a homogeneous flow condition in the whole seepage field) is acceptable for the purpose of the investigation in question (when the task is the determination of the average resistivity of the field), theoretically this approximation is not absolutely correct. Both types of singular points (stagnation points with zero velocity and points of cavitation with infinite velocity) can be found within the field, aa indicated in the figure. It has already been explained in Chapter 2.2, that the constant factors in the binomial formula describing the velocity vs. gradient relationship depend on velocity. T t would be necessary, therefore, to take into account the change of velocity within the whole possible range from zero to infinit, and to consider, that the seepage field has to be divided into zones of laminar, transition and turbulent flow. Alternatively the resistance could be described with a relationship giving the hydraulic conductivity as a continuous function of local velocity. The limits between the varioua types of seepage were given by the determination of numerical values of the Reynolds' number. Supposing both the solid matrix and the transported fluid to be homogeneous, neither the effective grain diameter nor the viscosity depend on the location of the investigated point within the seepage field, thus the flow condition is a singlevalued function of the local velocity. This latter parameter can be calculated from Laplace's equation as a first approximation, and the validity zones of the various movement equations can be determined by indicating those parts of the whole field, where velocity remains between given limits. Following this method, and applying the corresponding forms of the movement equations in the different zones, the second approximation of velocity can be achieved. The result of this second step is, that the limits between the separated parts of the field have to be corrected according to the new value of the local velocity. For example crossing the border between the laminar and the first transition zone, the second approximation of velocity will be smaller than the first one, because the resistance is greater in the transition state than in that of laminar flow. Thus, thelaminar zone becames larger i n the second step than it was i n the first approximution. The same process is repeated a t the border of the first and second transition zones, and where the turbulent zone is reached. Applying the method of successive iteration in this way, the h a 1 form of the validity zones of the various movement equations can be determined. A special difficulty is caused by the fact that the boundary conditions are known only around the contour of the field, but not along the internal border, where the different validity zones contact each other. The movement equation valid for a given part of the field can be solved only, if the boundary conditions are known along both external and internal borders, thus the successive iteration haa to include the determination of the internal bound-

+

4.3 Non-laminar eeeprtge

551

ary conditions as well, because the latter depends also on the velocity developing at different points of the field. Seeing all the difticulties which arise in connection with the above method a question arises immediately whether i t is worth-while in practice to consider the development of the different flow conditions within the seepage field, or not. In most cases the seepage is laminar in the largest part of the field, or if the solid matrix is coarse grained, and the hydraulic gradient is high, the transition state is characteristic almost of the whole field. Different conditions (very high or very low velocity) prevail only around the special singular points. This discrepancy from the average flow type extending only to relatively small parts of the field does not affect the average hydraulic parameters (e.g. the total head loss required for the conveyance of a given flow rate), as was mentioned in connection with the example in Fig. 4.3-1. There are many cases, therefore, when the inhomogeneity of the seepage conditions is really negligible, but in other cases the determination of special parameters in the vicinity of singular points is the purpose of the investigation (e.g. maximum exit velocity, which develops at cavitational points along the exit face), when the assumption of a uniform flow condition might cause considerable error. The practical importance of the correot consideration of non-laminar zones will be presented in an example. Let UB investigate a relatively simple seepage field: i.e. flow through a very deep (practically infinite) pervious layer under the horizontal foundation of a dam (Fig. 4.3-2). The seepage is maintained by the difference between the levels of the head and tail water. To design a dam the determination of the vertical exit velocity directed upwar& is a very important parameter. Where this value surpasses a given limit depending on the physical soil properties of the layer, the hydrodynamic pressure may disturb the stability of the layer (the development of the liquidization of the grains, boiling of quicksand). If the development of such unstable conditions is expected protective measures (sheet piles, protective filters) have to be applied. The basic data for the determination of the required sizes of these structures are the absolute value of exit velocity (or gradient) and its distribution along the exit face. If the distribution of exit velocity is determined by applying Laplace’s equation, the result indicates at the fist instant: the assumption of laminar movement does not provide an acceptable solution. The edge of the horizontal foundation is a cavitational point, thus the exit velocity is infinite and independent of the size of the applied sheet pile (except the case when the sheet pile is placed a t the lower edge of the fundation). The necessary grain size of the protective filter depends also on the maximum value of the exit velocity, but no filter can be designed ensuring sufficient protection against infinite velocity. At the same time, many hydraulic structures of similar type were constructed in the past, and their stability indicates, that the theoretically derived infinite velocity does not exist in nature. The explanation of the discrepancy between theory and practice can be based on the development of non-laminar seepage in the vicinity of the cavitational points. Approaching a cavitational point the velocity increases gradually, thus the deviation from the laminar state becomes larger and

552

4 Kinematic characterization of seepage

i\

v

distribution curve supposing / / f i n i t e pervious layer

impervious layer U

i

'0

-

I

I

I

1

2

4

6

B

I

1 0 ;

-

calcu/Wed theoretical velocity distribution

0.4

02

0.I X b

In

1 '/

7theoretical curve

theoretical curve

ii

velocity distribution determined by sand box model

- , xh Fig. 4.3-2. Velocity distribution dong the horizontal exit face behind horizontal imperviousfoundation

4.3 Non-laminar seepage

553

larger, which causes the continuous and rapid increase of the resistance of the solid matrix against seepage (because the relative importance of the second member of the binomial formula depending on the square of velocity becomes higher). The total pressure difference available for the maintenance of seepage along a flow line is constant. If there is a stretch where the resistance is considerably higher than that at other points of the field, the flow rate through a stream tube surrounding the flow line in question and determined on the basis of potential theory (laminar flow) will be relatively low. Cross-flow between the theoretical stream tubes will even develop, equalizing the velocity distribution in the vicinity of the cavitational points. The whole seepage field can be regarded, therefore, as a self regulating system, which restricts the development of extremely high velocity. It is very probable, therefore, that the field can be characterized with an upper limit of velocity, which has to be determined as the design value for the calculation of the main parameters of protective flters. It was already proved that the approximation achieved by applying Laplace’s equation does not provide an acceptable solution in this special case. Models (either sand box or electric analogue) are suitable only for the determination of the averages of exit velocity within arbitrarily divided but finite stretches of the exit face, as indicated in Fig. 4.3-2 (KovAcs, 1968). The difficulties arising when attempts are made to achieve the solution by dividing the seepage field into parts having homogeneous flow conditions, have already been mentioned. Easily applicable methods may be expected only if the system of differential equations substituting Laplace’s equation can be derived from an equation, so that it gives a continuous relationship between hydraulic conductivity and seepage velocity without the limitation of the validity zones. The derivation follows the same steps, through which Eqs (4.3-10) and (4.3-12) were achieved from Eq. (4.3-4), but now the general relationship between hydraulic gradient and seepage velocity [see Eq. (2.2-36)] is the basis instead of the binomial formula: (4.3-16)

or

The gradient vector of the pressure head is parallel to the velocity vector in a homogeneous field, thus its components in the direction of the axes of the coordinate system can also be expressed from Eq. (4.3-16): (4.3-17)

and (4.3-18)

554

4 Kinematic characterization of seepage

After differentiating Eq. (4.3-17) according to y and Eq. (4.3-18) according to x,their difference gives the first member of the system of differential equations, while the second member is achieved as the sum of the same equations if the first is differentiated by x and the second by y:

a [v,(A +B I v 13/4)u3]- a [Dy ( A aY 8%

+ B ) IV

13/4)4/3]=0 ; (4.3-19)

Substituting the relationships between the, differential quotients of the stream-function and the components of seepage velocity in the directions of the two axes, and expressing the absolute value of the velocity vector depending also on the stream-function [Eqs (4.3-9) and (4.3-2)], the final form of the system of differential equations is achieved:

and

The two differential equations compose a non-linear system of the second order having two variables. Since the sign and the numerical value of the multiplying factors may change from point to point, the character of the differential equations can be elliptic, parabolic or hyperbolic. The boundary conditions indicate a boundary value problem of mixed Dirichlet-Neumann’s type. The general form of the equations can be simplified by functional mapping (excluding for example mixed partial differential quotients), but their structure remains unchanged. Analytical solutions, therefore, can be found only after neglecting some terms. Hence, the assumption of B -+ 0 leading back to the Laplace’s equation wau shown. This is the reaaon why the numerical handling of the equations is proposed to determine the hydraulic parameters. The method applied for such solution would be either the use of finite difference or the expansion of some terms into series. The most suitable method has always to be chosen according to the character of the actual problem, because the most appropriate method of solution is strongly influenced by the boundary conditions which have to be satisfied.

References

555

References to Chapter 4.3 KovAcs, C. (1968): See ge to Groundwater Created by Hydraulic Structures. Actu Technica Acaderniue f?%m8tiarurn Hungaricae, Tom. 60. 3-4. Kovtics, G. (1971): Charaoterizationof Non-laminar Seepage (in German). Symposium on Irbvestigation of Seepage by Models, ‘Varna, 1971. MCJCORQUODALE, J. -4.and NQ. H. S. (1969): Non Darcian Flow Solved by Finite Element Analysis. 13th I A H R Congreas, Kyoto, 1969. O u , T. (1969): A Study on the Seepage Around a Sheet Pile by Applying Forchheimer’s Law. 13th I A H R Congress, Kyoto, 1969.

Part 5

Solution of movement equations describing seepage

As already mentioned the complete description of seepage through a geometrically defined field of a porous medium requires the determination of the hydraulic parameters (velocity, flow rate, pressure) at every point of the field, and at each time point, considering all the external influences acting around the border of the field, and the physical properties of both the porous medium (porosity, intrinsic permeability) and the percolating fluid (viscosity, density) influencing the flow. To solve practical problems this rery complex and laborious task can be simplified, and the investigation can be limited to the determination of special parameters at several points or alonbr b' riven lines (e.g. pressure distribution along the contour of a hydraulic structure; the maximum exit velocity; or the total flow rate through the field). The determination of the hydraulic parameters (either to find their 1-alues at every point in the field, or to accept the analysis of the flow conditions at specific locations aa sufficient information) can be achieved by solving the relevant movement equation, which describes the seepage through the porous medium and considers the actual conditions of both the movement and the field [these conditions include the character of flow, type of seepage (such aa turbulent, transition, laminar or microseepage depending on the dominant forces acting), inhomogeneity, anisotropy, compressibility etc.]. Although the simplifying hypotheses of incompressibility and homogeneity of both the fluid and the solid matrix are accepted generally for practical purposes, and anisotropy is also neglected in most caaes (at most transverse anisotropy is considered by applying special transformation and the problem is led back in this way to the solving of seepage through the isotropic field), the general form of the movement equation [Eqs (4.3-16) and (4.3-20)] is hardly amenable mathematically. A further commonly accepted approximation in the practical investigation of seepage is, therefore, the supposition of laminar pow, and this hypothesis will also be applied in the following chapters of the book together with homogeneity, isotropy and incmnpressibility mentioned previously. Even in this very simplified caae, when the seepage is described by Laplace's equation, the solution of an elliptic (in the caae of steady seepage), or a parabolic (if the flow is unsteady) differential equation of second order haa to be determined, which simultaneously satisfies the boundary and

5 Movement equations describing seepage

557

initial conditions. There are only very rare cases, when this equation can be solved in closed form by integration. It was stated earlier that absolutely correct and accurate mathematical handling can be applied only for one dimensional flow, having straight stream lines and constant cross-sectional area (piston flow). It is for this reason that all the analytic solutions aim to achieve this basic form either by mathematically transforming the seepage field, or by applying physically reasonable approximations. These analytic methods will be discussed in detail in the first four chapters of this part of the book. There are, however, seepage fields having complicated geometry, when the analytic methods are not applicable, because they lead to very complicated mathematical relationships, even to unsolvable forms, or the approximations would cause basic divergence from the actual physical character of the process in question. In these cases the application of hydraulic or analogue models, or the numerical solution of the differential equation using digital computers are in practice commonly accepted methods. A hydraulic (or sand box) model is a seepage space geometrically similar to the original system, having proportionally decreased size (small scale model), and filled with porous material. After applying the same (or sufficiently distorted) boundary conditions as those acting along the borders of the prototype, similar physical processes develop in the model to those, which ought to be investigated on the original scale. The hydraulic parameters of seepage through the sand box model can be measured directly, and using model laws to describe the proportionality between the parameters of the two systems, the variables sought for the solution of the problem can be recalculated. Accepting the explanation, according to which the purpose of the models is either the substitution of the differential equations or the determination of their factors, and the exploration of the relationships between the variables, not only a physically similar process can be used to perform the necessary measurements, but the solution of the differential equation can be reconstructed from the measured data of any other phenomenon if its development is characterized by a physical law identical to that of seepage. These phenomena are called analogue physical processes. According to this definition, every type of transport of extensive quantities is analogue with seepage, if the potential difference creating and maintaining the movement is linearly proportional to the transported quantities (e.g. viscous fluid in HeleShow model; electric current in continuous or discrete electric analogue model, etc.). Applying the relevant boundary and initial conditions in such analogue models, and measuring the parameters of the process induced in this way, the data assist us with the complete, or partial solution of the differential equation of seepage. It was also shown, when analyzing the kinematic characteristics of the seepage field, that the stream lines are always perpendicular to the equipotential surfaces, in the case of two dimensional flow the stream and potential lines compose an orthogonal network. Knowing the geometry of the flownet, the hydraulic parameters of seepage can be culculcated. It follows from this fact that the scope of the analogue models can be considerably enlarged.

558

5 Movement equations describing seepage

Not only can the measured parameters of transport processes be used for the determination of the variables of the differential equation of seepage but all the phenomena producing an orthogonal network of trajectories in a geometrically defined field with given boundary conditions can be used, if the position of the two systems of curves normal to one another can easily be determined, made visible or measured (e.g. displacement in a membrane model, stresses in an elastic optical model, etc.). On the basis of this interpretation the various mapping methods, which will be discussed in detail in the following chapters and which also produce

Y

contour @pproxim&d by

Yi+I Yi Yi-l

Fig. 5.0-1. Regular grid covering the seepage field used to determine the hydraulic paramet,era by applying finite differences

the orthogonal network of interrelated curves, can also be regarded as an analogue model. It is absolutely correct, therefore, to call these methods rnathematicul models of seepage, although this term is used generally in a broader sense, including all equations proposed for the calculation of the hydraulic parameters of seepage. The most general solution of the problem can be achieved by applying numerical methods, the rapid development of which was greatly supported by the use of digital computers. The large memory capacity of the computers makes i t possible, to solve Laplace’s equation, even in an inhomogeneous field, by satisfying the boundary conditions instantaneously. The usually applied numerical method is the approximation of integration in the form of summing up finite differences. The field is covered with a network of two sets of lines parallel to the two coordinate axes of the plane chosen arbitrarily (gtjnerally an orthogonal system is used). A t the intersection of the grid the parameters characterizing the flow field (e.g. hydraulic conductivity or transmissibility, storage capacity) have to be known (Fig. 5.0-1). The contour of the field is approximated by a stepped poligon, the stretches of which are parallel either to the 5 or to the y axis. A t the points located along the perimeter the boundary conditions should also be determined. As the h a 1 result the ~ ( sy,,t ) potentials are calculated (more precisely approximated) at the discrete (xi,y i ) points of the grid and for the similarly discrete tk time points within a given [O; t ] interval.

559

5 Movenient equations describing seepage

a(9 = a2q differential Investigating a one dimensional unsteady flow the -

at

ax2

equation has to be solved. Supposing that the [O; t] interval is divided into equal At parts and the x direction is similarly separated into equal Ax stretches, three different approximations can be applied to determine the velocity potential prevailing at the end of the ( n 1)-th time interval, from the value belonging to the end of the n-th interval calculated previously (at the start of the procedure the known value is the initial condition):

+

Explicit formula (or forward in time scheme)

Implicit formula (or backward in time scheme)

Du Fort-Frankel (1953) (scheme with alternating direction) pj(tn+i/J-

+

Vj(tn+l)

.

V j ( t n ) - V j + l ( t n + l / Z ) - 2~j(tn+1/2) ~ j - ~ ( t n + ~ h Z )

At12

9

AX2

+

-Vj(tn+I/A = Vj+l(tn+l/z) - 2 ~ j ( t n + l / z ) At12

Ax2

+

(5.0-3)

.

~j-l(tn+~/A 9

(the subscripts j - 1, j and j 1 indicate the position of the neighbouring points at a distance of Ax from one another). The formulae determined to solve a one dimensional problem can be easily supplemented to characterize a two-dimensional flow. It is advisable, to use a regular grid (i.e. Ax = Ay = A ) in this case. There is only one unknown value [rpj(tn l)] in the explicit formula, its solution is, therefore, relatively easy. At the same time stability (convergence of the series) can be achieved only if dx is smaller than a given value depending on the At time interval, and thus the number of the required grid points is very high. The implicit formula has to be solved simultaneously for all the points. This procedure is very time consuming and requires large computer capacity, although the solution is stable. The alternating scheme unifies the advantages of the other two methods. Apart from the method of finite differences the method of finite elements and the relaxation method can be similarly used for solving Laplace's equation. Readers interested in these other possible solutions are referred to Bear's publication (1972). There are some recently published books in English literature (Bear et al., 1968; Bear, 1972) giving very complete and excellent description of models (both sand box and analogue) including both their theoretical background and aspects of their practical application. We felt, therefore, that to repeat

+

6 Movement equations describing seepage

560

this material here is unnecessary. It is for this reason that there is only a short chapter dealing with the explanation of the model laws of sand box models, as a supplement to the books quoted in Chapter 5.5, and because this topic was somewhat neglected previously. Similarly the use of numerical methods to solve Laplace’s differential equation is not diecussed here in detail, but the reader is referred to a book dealing separately with this special topic (Kovkcs, 1978).

Chapter 5.1 Characterization of two-dimensional potential seepage The basis of any analytic determination of the hydraulic parameters of laminar seepage is the solution of Laplace’s second order differential equation in the cam of a flow space having constant cross section and a straight axis (piston flow). It is necessary, therefore, to recall here the simplest relationship gained as the solution in the case of steady flow [when the differential equation is elliptic, see Eq. (4.1-41); and symbols applied in the following equations are indicated in Fig. 5.1-11: u2p=

8%

-0 ;

p(x) = a

+ bx ;

where

b

=

--

v

Q

AH L

= -- = - K - ;

A

and a

2

KH 1’

(5.1-1)

This relationship can also be applied as an approximation to characterize the stream tubes with curved axes if the size of the field normal to the flow direction is relatively small compared to the length of the field and thus, the differences between the lengths of stream lines are negligible. If the area of cross section is not constant but changes gradually, and the difference between the extreme values is small, Eq. (5.1-1) can also be applied, by substituting the average area and seepage velocity instead of the actual changing parameters (Bear et al., 1968). In the case of unsteady flow the parabolic form of the differential equation can be similarly solved. The hypotheses already listed and accepted for the description of piston flow have to be supplemented by amuming a wetted front (potential surface dividing the saturated and dry parts of the porous medium and moving ahead or being drawn back depending on the character of the movement) normal to the axis of flow along which the change of potential is negligible (Fig. 5.1-lb). I n this case the potential at a given time point is a linear function of the location of the point investigated (linear function of x distance measured from the starting section). The position of the wetted front is time-variant and its relationship with the timedependent flow rate can be calculated: p(x, t) = 01 p x ; xf = f ( t ) ; &(t) = -A- av = An,- axr (5.1-2)

+

ax

at

5.1 Two-dimensional potential seepage

561

The a and ,b constants can be determined from the boundary and initial conditions:

t = 0 ; Zf = f ( t ) = zfo; (initial condition); z = 0 ; c p ( ~ ,t ) =

v1 = K

(boundary condition a t the entry face); z = Zf = f ( t ) ; cp(z, t ) = cpf = R z r ; 36

562

6 Movement equatiom describing seepage

(boundary condition at the wetting front if capillary suction is negligible); (5.1-3) x = xf = f ( t ) ; V ( X , t ) = K ( z - h,) ; (boundary condition of the wetting front if capillarity is considered with an average h, suction head). In the case of seepage in a vertical tube filled with porous material, when the hypothesis of constancy of potential along the wetted front is satisfied (because both the pressure and the elevation above the reference level are constant at every point of the front) the solution of Eq. (5.1-2) is as follows: The tube is contacted with free surface water at its bottom (Fig. 5.1-lc): ~

I ( t )= H +

f

=

; &(t) = K A I ( t ) ;

hc--f(t) Xf ( t )

if t

--f

00

; zf(t.) -,H

+ h, ; I ( t )

(5.1-4) +

0 ; and &(t) -,0 .

Equation (5.1-4) characterizes both the upward and the downward movement of the wetted front. The direction of the movement depends on the initial condition. If zf t ) belonging to a zero time point is smaller than the h,] a wetting process is initiated, if sum of H and h, [z t = 0 ) < H xf(t = 0) > H h, t i e drainage of the soil column starts. The tube is contacted with free surface water at its top (Fig. 5.1-ld):

+

if t

--f

00

;H

t

+ h, + x f ( t )-+

+

z f ( t ); I ( t ) + 1 ; and &(t) + R A .

In this case only the wetting process (infiltration) can develop, because both gravity and capillarity act in the same direction. Uncertainties are caused even in the characterization of this very simple form of unsteady seepage by the fact that the saturated zone and the dry part still undisturbed by seepage can never be divided by a plane with a well determined position. There are channels between the pores, in which water moves faster and, therefore, the wetting front is always a partly saturated zone of finite length with hydraulic conductivity smaller than that belonging to the saturated condition, and changing from point to point in both directions. The capillary suction is also a random variable within the cross section, which can be only roughly approximated with its average value. For a seepage field different from the most simple form investigated in the previous paragraphs, there always exists a single valued continuous solution of the differential equation, if the boundary and initial conditions are known. The only problem is, how to find this solution. The graphical solution (Leliavsky, 1955) has already been discussed in Chapter 4.1 [see Fig. 4.1-5 and Eqs (4.1-37), (4.1-38) and (4.140)]. In the introduction to this part of the book models and numerical methods were mentioned as auxiliary means, and the analytical methods will be discussed here in detail.

6.1 Two-dimensional potentional seepage

563

The possible analytical solution can be divided into two parts. The application of diflerent mupping methods to simplify the flownet until the orthogonal system of straight stream and potential lines is achieved, is the first one (Muskat, 1937; Milne-Thomson, 1955; Polubarinova-Kochina, 1952, 1962; Bear et al., 1968; Bear, 1972). The second part of the solution is the approximation of the actual seepage with a one dimensional flow (Dupuit, 1863; Boussinesq, 1904; Forchheimer, 1924). I n this case the validity limits of the approximations and the probable errors of the methods have to be determined. The theoretical bases of the first group of analytical methods will be explained in Chapter 5.1 while its practical application and some other methods will be dealt with in the subsequent chapters.

5.1.1 Complex potential. Conjugate velocity The mapping methods are based on the mathematical analysis of complex numbers and functions of complex variables. Here the basic concepts of complex and conjugate complex numbers, the symbols and terms applied generally, as well as the use of algebraic operations with complex numbers are assumed t o be known. Consequently only a short summary of the application of functions of complex variables is given, as an introduction to the conformal mapping. If z is a complex variable (which can have any z = x + iy value on the z complex plane), the result of special algebraic operations executed with z and some constants is a new complex variable which is the algebraic function of z : 2u = f ( z ) ;

w=cp+iy;

z=x+iy;

(5.1-6)

Each value of w complex number indicates similarly a point on a complex plane (now on plane w having a real axis and an imaginary axis y). Thus the function w = f ( z ) creates a contact between the points of the two planes, which can be either single- or multi-valued, depending on the algebraic operations applied within the function. After having determined the contacts between the real and imaginary parts of the variables (by dividing either their real and imaginary parts or their absolute values and the angles of the complex vectors), the relationship between the coordinates of the points on the two interrelated planes can be characterized. To assist this operation, the real and imaginary parts as well as the absolute values and the angles of the complex vectors determined by some algebraic funct,ions of complex variables are listed in Table 5.1-1. A curve on the z plane can be regarded as a line of points. Each point has its corresponding pair on the w plane. Thus, the i m a g e of a two-dimensional curve y = y ( s ) can be determined on the w plane, and also its equation tp = y(v) can be calculated. It can be stated similarly, if the y = y(x) curve 36 *

564

5 Movement equations describing seepage

Table 5.1-1. Real and imaginary parts as well as the absolute values and the angles of vectors determined by some functions of complex variables Function w =f(z t i Y )

w z $. iY)

+ iy)

sin (z COB

(zI iy)

tan

(3:

1

Real part

Imaginary part

1 y*) 2 sin z ch y - ln(z*

+

*arc tan

X

+cos z sh y

cos z ch y

$.sinsshy

sh(z i iy)

sin 22 cos 22 ch 22 sh z cos y

sh 2y 22 ch 22 k c h z siny

ch(z I iy)

c h x cosy

+shz s i n y

th(z

t iy)

iy)

Absolute value

Rew

+

sh 22 ch 2z COB 2y

+

COB

+

+ sh*y ~ccos*z+ sh*y )/sin2z

+ sinzy )/sh?z + cos*y )/&z

1

angle arg w

*arc tan (cotan x t h y ) *am tan (tan z t h y )

+arc tan (cth a tan u ) +arc tan (th z ten y) " I

sin 2y ch2z cos2y

+

is a closed one surrounding a single contacted continuous field of the x plane, the image of this field can also be constructed on the w plane. When determining the borders of the interrelated fields it i s necessary to divide and select those parts of the planes, inside which the mapping function in question creates a single-valued reationship between the corresponding points. Thus, the singular points (where the mapping function is undetermined) have to be excluded from the field, or if the applied function is multi-valued (e.g. square root), one solution has to be indicated as a main value and the image determined by this value has to be regarded as the transformed form of the field investigated originally. Limit value, continuity and differential quotient have the same formal interpretation for functions of complex variables, as given in the case of functions with real variables. If the w = f ( z ) function is single valued, continuous and differentiable at each internal point of a continuous field, this function is analytic (or regular) in this field. The necessary and sufficient condition for a function to be analytic is, that it has to satisfy CauchyRiemann's condition:

w =p

+ i y = f(x) = f(x + iy) ;

p = f d x , y ) ; y = f2(% Y ) ;

(5.1-7)

aw . -&L&.%L _ ax a y ' ay aY This idea leads to the hydrodynamic interpretation of functions of complex variables. It was proved earlier [see Eq. (4.1-33)] that velocity potential and stream-function are interrelated functions of the x and y coordinates of the seepage field, and their relationship is characterized by the fact, that their differential quotients satisfy in all cases the Cauchy-Riemann's condition repeated here as Eq. 5.1-7.

565

6.1 Two-dimensional potential seepage

It can also easily be proved that the fulfilment of Cauchy-Riemann’s condition ensures a t the same time the conformal character of mapping executed by an analytic function (the mapping is proportionate concerning both distance and angles; Nbmeth, 1963). The total differentials of z and w variables are as follows: (5.1-8) dz = dx idy ;

+

d w = dg,

89 + idy = -dx + ax

Using this relationship the differential quotient of w according t o z can be written in the following form:

dz

dx

dx

+ idy (5.1-9)

+ idy

A function is differentiable within a field if its differelitid quotient dw ix single-valued a t every point of the investigated field. The - ratio, dz however, would be infinitely multi-valued depending on the choice of the pairs of dx and dy, except in only one case, when the numerator choosing any pair of dx and d?j - can be divided by (dx + idy). This condition is satisfied if

9 + i - aw = aw - m a p , 2-

;

(5.1-10)

ax

ax ay ay i .e. if Cauchy-Riemann’s condition is fulfilled. It follows from Eqs (5.1-9) and (5.1-10) that if the w function can be dw differentiated with respect t o z within the investigated field, the -quotient

dz

is constant in the vicinity of a point and independent of the investigated direction (dx and dy elementary lengths can be chosen arbitrarily).This fact proves that the ratio of two interrelated elementary lines (dz and dw) is also independent of the direction of the investigated lines, and thus the mapping executed by an analytic function ensures the proportionality between the original field and the image. At the same time the proportionality of the elementary lengths is a SU& cient condition of the mapping to be proportionate also concerning the angles, viz. an arbitrarily chosen curvilinear triangle and its image have t o be similar t o one another, if the lengths of the sides are elementarily small, because dii) the - ratio can be regarded as constant, independent of the position of dz

566

6 Movement equations describing seepage

the point within the surrounding area. The similarity of triangles ensures the identity of the corresponding angles at the same time. Further hydrodynamic consequences can be drawn from the fact that the mapping executed by analytic function is conformal. Let us construct an orthogonal network of squares on the w plane composed of straight lines characterized by v1,rp2, . . . p,,, constant values having a difference of d = q i - 'pi- ; and another set of straight lines being normal to the former, having the same distance of d and described by yl,y2, . . . yk constant ( A = - yi - yi-l relationship is also valid). Applying the inverse function of w = f ( z ) (which is also analytic, if the original function is regular), an orthogonal network of two sets of curves is reproduced on the z plane, the network being composed of curvilinear squares. A part of the z plane bordered by the curves corresponding to straight lines on the uf plane determined by the constant values of vA,v B and yA, pB respectively, can be regarded as the flow net covering a given seepage field. This is because a constant q value belongs to each of its potential lines, and each stream line is characterized with constant y value. At the same time these two functions (p, and y) satisfy Cauchy-Riemann's condition, and, therefore, the kinematic conditions explained earlier are also necessarily fulfilled. For this reaaon, using the analogy of the name of p,(x, y) potential-function, the y ( x , y) streamfunction is frequently called conjugate potential, and the name of the uy = f ( z ) function is complex potential. It follows from the hydrodynamic interpretation of potential- and streamfunctions that the di#erential quotient of complex potential according to the z variable can be used to determine the vector of seepage velocity. If only dz the inverse function of the complex potential [ z = f(w)] is known, the dw differential quotient gives the same result, providing us with the reciprocal value of the conjugate of the velocity vector. Combining Eqs (5.1-9), (5.1-10) and (4.1-33) the result is as follows:

dw

---

dz

ap, ay +i-=--

ax

ay

ax

. & = - (v, - i v y ) .

8-

ay

(5.1-11)

ay

The velocity vector can also be regarded as a complex number

v = v,

+ ivy = 1 v I eie.

(5.1-1 2)

I v I e-'",

(5.1-1 3)

Its conjugate vector is

v* = v,

- ivY --

the negative,value of which is identical with the differential quotient of the complex potential. Differentiating the negative inverse function of the complex potential according to the latter, the reciprocal value of the conjugate velocity is achieved as previously mentioned:

567

6.1 Two-dimensionel potential seepage

Thus, the angle of this vector is identical with that of the velocity vector itself, while its absolute value is equal to the reciprocal value of the absolute value of the velocity vector. Consequently, it can be proved that by differentiating the complex potential, or its inverse function, the conjugate velocity vector or its reciprocal value can be determined at each point of the seepage field. The conjugate velocity vector provides us with the absolute value and the angle of the velocity vector at the same time. It can be stated, therefore, that the velocity can be calculated at each point of the field, if the analytic mapping function of complex variables is known, which transforms the flow net into an orthogonal network of squares composed of straight lines on the plane of complex potential. The problem can be similarly solved, if the inverse of this mapping function is given.

5.1.2. Solution of seepage problems by applying mapping The main rules of mapping by using analytic functions of complex variables were summarized in the previous section. It was also shown that a hydrodynamic interpretation can be given for the mapping function which is called complex potential. Considering the relationship between the complex potential and the hydraulic parameters, the seepage velocity can be culculated from the mapping function at euch point of the field of a two dimensional seepage. The problem can be regarded tt8 completely solved, if the other parameters (flow rate, pressure and its distribution) can also be determined at any point or at least along special lines and at given points. When discussing the kinematic characteristics of seepage, it was previously proved that hydraulic gradient and velocity are closely interrelated, thus the knowledge of velocity simultaneously determines the gradient as well, and from these two parameters all the others can be calculated. The most direct application of conformal mapping is, therefore, the determination of the complex potential (or its inverse function), and after differentiating it, the vector of seepage velocity can be calculated for every point of the field by using Eq. (5.1-11) [or Eq. (5.1-14)]. By integrating the productof seepage velocity and an elementary length normal to the former along a potential line the discharge transported through a porous medium of unit width by the two dimensional seepage is achieved (Fig. 5.1-2):

pv dn = q.

(5.1-15)

A

Similarly the integration of the product of the velocity and an elementary length can be used to determine the potential OT pressure mlue at a given point P.I n this case, however, the elementary length is parallel to the velocity vector and the integration is executed along a stream line:

(5.1-16)

568

6 Movement equations describing seepage

impervious boundwy Fig. 6.1-2. Velocity distribution along a potential line and a stream line, respectively within a two-dimensional seepage field

This method is frequently used if the only purpose is the determination of seepage velocity at a few points, because knowing the mapping function, this parameter can be directly calculated. It would be, however, very laborious to determine velocity and gradient for many points of the field- to facilitate the calculation of pressure and flow rate in this way. To solve these types of problems some other application of the mapping method is preferable. The use of the mapping function to construct the flow net by determining the relevant lines on the x plane belonging to constant values of q ~ a n d y respectively, also requires very long calculation, although the hydraulic parameters can be easily calculated after the flow net is determined [see Eqs (4.1-37), (4.1-38) and (4.140)]. Cases are very rare, therefore, when this method is followed to determine the hydraulic parameters. A t the same time the essential part of this method is the graphical representation of curves given in analytic form. Consequently, this method dots not need any further detailed investigation. There are problems - especially when an unconfined system ie investigated and thus the position of the upper boundary of the field is undetermined - when the hodograph mapping discussed in the previous part of'the book can be used to determine the complex potential (Hamel, 1936; Vedernikov, 1934). Considering the boundary conditions, the hodogralh contour can be constructed even in the case of unconhed flow. Knowing the hodograph image representing the velocity vector, the contour of the field on the plane of conjugate velocity (v*)or on that of its reciprocal value also be determined. If an analytic function 1 v* = fl(w) ; or - = f,(w) V*

(5.1-17)

5.1 Two-dimensional potential seepage

569

can be found, which contacts the corresponding points and contours of the new plane and the w plane of complex potential, the problem of mapping [the determination of w = f ( z ) function] is solved. It is known from the foregoing that dw 1 dz v*=-= fl(z) ; and - = - - = f2( w )9 (5.1-1 8 ) dz V* dw respectively. From one of these relationships, after separating the variables, the w = f ( z ) function (or its inverse) can be calculated by integration. The solution is achieved also, if a n intermediate auxiliary complex plane ([ = E ill) can be found instead of the direct relationship between the complex potential and the hodograph of the conjugate vector of the seepage velocity. I n connection with this intermediate plane, there is a requirement that, after mapping by analytic functions both the complex potential and

+

I

the v* hodograph or the hodograph of - onto this new [plane, the images V* ll of the corresponding points either transformed from the w plane or from the hodograph should be identical. After determining the mapping functions establishing the required con-

I

tact from the w to the [ plane and from the v* or - to the [ plane respecv7 tively, the basis of the further analytical investigation is once again the relationship between the conjugate velocity vector and the differential quotient of the complex potential:

or using the inverse function of the complex potential as the basis of t,he investigation

(5.1-20)

After separating the variables and integrating the equation in question the z complex variable can be determined as a function of the 5 complex number describing the position of the points on the intermediate complex plane. The images of the contours bordering the seepage field are known on the plane. Thus, the previously undetermined position of the phreutic surface can be recalculated using z = z ( [ ) function by transforming its image to the

570

6 Movement equations describing seepage

actual seepage field. The knowledge of the relationship between w and complex numbers was also a prerequisite for the application of this method [see Eqs (5.1-19) and (5.1-20)]. If the two equations [i.e. z(l) and w ( Q ] can be combined the direct mapping function between the actual seepage field and the plane of complex potential can also be determined. Here only the main concept of the application of hodograph mapping was summarized. Examples to show the practical use of this method will be given in Chapter 5.2.The application of Zhukovsky’s function will also be demonstrated there. The theoretical basis of this latter method is very similar to that of hodograph mapping. Similarly an analytic function (or a, series of such functions) has to be found, which transforms the contour of the field determined on the Zhukovsky’s plane onto the plane of complex potential (where the contours are also determined by the boundary conditions). This short summary has demonstrated, however, the complexity of the problem and the large amount of calculation, which has to be executed, if the application of one of these methods is intended to be used to determine the hydraulic parameters at every point of the field. This method leads back practically to the same type of work, which ought to be performed in the case of a geometrically fully determined seepage field, if all data are to be calculated on the basis of the complex potential. As will be shown, in the latter case an easier way can be followed by transforming the flow net into a plane, where the stream lines become parallel to each other and, therefore, Eq. (5.1-1)can be applied - at least as a good approximation - to calculate the necessary parameters. When investigating unconfined flow the hodograph mapping or Zhukovsky’s transformation can be wed only to determine the position of the water table, and afterwards the fixed contours can 1963,1964). be mapped similarly, as those of a confined field (KOV~CS, The easiest application of mapping for the determination of the hydraulic parameters of steady two-dimensional seepage is the reduction of the problem to the investigation of one dimensional seepage characterized by parallel straight stream lines. It was also mentioned in the introductory part of this chapter that this purpose can be achieved in many cases by using exact mathematical methods if the original /low net can be transformed into the system of orthogonal straight lines. Applying this method, the mapping is regarded as only a pure geometrical process. Its result is a rectangular flow field on the new plane which is characterized in this cam by u real and v imaginary axes [w(u,v ) ] to emphasize that the hydrodynamic interpretation of the complex potential (and that of potential and y stream-functions) i s not considered. In such a system the Laplace’s equation can be directly solved, and the result of integration is the set of characteristics looked for. Naturally in this case the parameters are given as the functions of the geometrical data of the field determined by mapping. The mapping functions express, however, the relationships between the geometrical parameters of the actual seepage field and those of the transformed system. These functions can be used, therefore, to introduce the original data into the hydraulic formulae, and h a l l y the interesting kinematic parameters can be related to the sizes of the actual field, which means the practical solution of the problem.

6.1 Two-dimensional potential seepage

571

To demonstrate the application of the method, a very simple seepage field is chosen as an example, the flow net of which can be easily transformed into the required form, using only one mapping function (Weaver, 1932). The purpose of the investigation should be the calculation of the discharge below the horizontal foundation of a dam having a width of 2b, and the determination of the pressure distribution along the contour of the foundation, which is placed on the horizontal surface of a permeable layer. Further hypotheses are: (a) Both the entry and exit faces are horizontal planes; (b) The field is homogeneous, isotropic and bordered by an impervious bed; (c) The vertical section of the lower contour of the pervious layer is an ellipse, the half axes of which are interrelated to the half width of the foundation by the following equations (Fig. 5.1-3) :

N2 - M2 = b2; N =nb;

(5.1-21)

M=bvn2-l.

It is also supposed that the difference between the levels of head and tail water ( A H total pressure head) as well as the hydraulic conductivity of the

head water

Fig. 6.1-3. Mapping of elliptic seepage field

572

5 Movement equations describing seepage

permeable layer ( K ) are known, thus the total potential digerenee inducing and maintaining the seepage can be calculated:

dv = pi-

~ 1= 2

K ( H 1 - H,) = K d H .

(5.1-22)

This very simple seepage field can be mapped onto a w plane, (where the contour surrounds a rectangular field) by the following analytic function : w =u

+ iv=

arc sin-

. x+iy

z

b

= arc sin-.

(5.1-23)

b

To simplify the mathematical contact, the analysis of the inverse function is preferable : z =x

+ i y = b sin w = b sin (u + iv).

(5.1-24)

The first step of calculation is toprove, that the mapping function or its inverse satisfies Cauchy-Riemnn’s condition. It is necessary, therefore, to determine thc relationships between the coordinates of z and w by selecting the real and imaginary parts:

x = b sin u ch v; y =b

COY

(5.1-25)

u sh 2).

The derivatives of x and y with respect t o u and v respectively, have to he determined to check the correctness of Eq. (4.1-33):

8X = b cos

ZL

8ZL

ch v = - = b cos u ch 8V

V;

(5.1-26) 8X -b

sin

ZL

sh

= - @= b sin u sh

v;

aU

aV

:~nd thus Cauchy-Riemann’s condition is satisfied. Consequently, the map-

z 7E ping function is analytic and single-valued within the field - - < u : +2 2 z rxcept the singular points w - - . 0 j ; w2[+;. 011.

[ ,[

The next step of the investigation is the determination of thc coordinates of the corner points of the field on the w plane by using the relationships given in Eq. (5.1-25):

P,(O, 0); P,(-b, 0);

W,(O, 0);

573

5.1 Two-dimensional potential seepage

~ ~ (arshVn2 0 ,

- 1);

or considering that sh2y = n2 - 1 ; n2 = sh2y

+ 1 = ch2y;

W5 (0 ,arch n).

(5.1-27)

The seepage field is mapped in this way onto an orthogonal quadrangle bordered by parallel straight lines. The images of the entry and exit faces are parallel t o the imaginary axis, and their distance (the length of the field) can be calculated either from the positions of the W , and W , points or from the W, and W 4positions:

AU = u,

-

u1 = u

-

u, = z.

(5.1-28)

The stream lines indicating the upper and lower impervious boundaries are also straight lines parallel t o the real axis and, consequently, they are normal to the Imtential lines determined previously. T h e width of the field between these stream lines is:

LIV

=

v3 - vl = v5 - v o = v4 - v2 = arch n .

(5.1-29)

0 1 1 the w plane the internal stream and potential lines are straight. Identical lines are parallel while the two different groups of lines are perpendicular to each other. The investigation is reduced, therefore, in this way t o the analysis of the most simple flow pattern i.e. the one dimensional seepage tlirough a rectangular flow field. Applying the boundary and flow conditions according t o the data of the original system, the problem can be solved by using Eq. (5.1-1). The necessary data are as follows (Fig. 5.1-4):

(a) Hydraulic conductivity K (original flow condition); (b) Total pressure head AH (original boundary condition); (c) Area of the cross section in the transformed system (considering that a unit width of the two-dimensional seepage is investigated)

A = Av = arch n ;

(5.1-30)

(d) Total length of the transformed field AU = X

;

(5.1-3 1)

574

6 Movement equations describing seepage

head warer

pressure bead

I

impervious boundary

'

potentiat surnce Fig. 5.1-4. Flow model of on0 dimensional straight stream tube of unit width

(e) Distance of a point fitted to the upper impervious boundary of the field in the transformed system from the entry section 7d

u =2

+ arc sin-.5b

(4.1-32)

The parameters t o be determined can be directly calculated as the functions of the geometrical data of the new simplified system:

Specific discharge (flow rate through a section of unit width of the Dermeable laver): AH AH 4: AK= AvK; (5.1-33) Au Au Pressure distribution along the foundation: -P= H,+ h ;

Y

(5.1-34)

Equations (5.1-33) and (5.1-34) are valid for any stream tube having a cross section of A = 1. Av and a lengthof Au if the flow condition is characterized with homogeneous, isotropic hydraulic conductivity of K , and the boundary conditions are given in the form of a constant pressure head of A H . These relationships can be used, therefore, in every case after transforming the actual flow field of two-dimensional seepage into the new system

575

6.1 Two-dimensional potential seepage

characterized by straight, one dimensional flow. The contact between the geometrical parameters of the field achieved by mapping (i.e. A u ; Av; 2 4 ) and those of the original system will d$er from case to case. I n this example the width of the foundation is 2b; the position of the impervious bed is charact-erized by the M and N parameters; and the x parameter describes the position of the investigated point along the foundation. This contact, however, is always determined by the mapping function (or by the series of such functions, if the transformation is executed in more than one step), and Au, Av and u variables can be substituted by considering the relationships determined from the mapping functions [in the present caae these relationships are listed in Eqs (5.1-30); (5.1-31) and (5.1-32)]. Thus, the final solution of the investigated example can be given in the following form: (a) Specific discharge ar ch n . n

p = K A H ~,

(5.1-35)

(b) Pressure distribution along the foundation

=AH

2

n

5.1.3 Basic mapping functions applied most frequently

It can be stated, from the results of the previous section that the easiest and most generally applicable method of determination of the hydraulic parameters of steady two-dimensional seepage is to transform the original contour of the field into a rectangle by using mapping purely geometrically without considering its hydrodynamic role. T h e hydraulic problem i s simplified in this way to a geometrical one: to find an analytic function of complex variables (or a series of such functions) which execute the expected transformation of the original contour. It is quite evident that the actual borders of the flow field (bedding planes, river beds, etc.), which are generally irregular in nature, have to be approximated at first with straight lines or other mathematically amenable curves. For the application of this method i t is necessary t o know the basic mapping functions and the results achieved by each of them (the two formations on the original and new planes interrelated by a given analytic function). Various combinations of these basic functions enable us to map even a very complicated flow net into a rectangular field (Muskat, 1937; Polubarinova-Kochina, 1952, 1962; Nkmeth, 1947, 1963; Gruber and Blah6, 1973; Harr, 1962; Rear et al., 1968; Bear, 1972).

Table 6.1-28

rnappgng functions and

relationships between the coordinates of the interrelated plane

the orfhogoml tt?ijectories interconnected by mapping functions

unambiguously interrelated fields

fie two typesoffucfionsareinversefinc(ionq they contacf berefore, the sane ortnogona/tmjecforfes, but the role of the planes is changed . .

&-+-&!!q-+ ..

.!

Y

c;

G

II

II

u

ISS.

t

6.1 Two-dimensional potential seepage

v

N

Y

578

5 Movement equations describing seepage

Some of the simple mapping functions are listed in Table 5.1-2, where the interconnected orthogonal trajectories on z and w planes are also shown. If the mapping function is multi-valued, the transformed image of a plane covers many times the other plane which describes a multi-fold Riemann’s surface. I n this caae - as already mentioned - one of the results of the multi-valued function has to be selected at3 a main value, and thus the fields of the planes can be indicated within which the mapping function ensures a single-valued relationship between the corresponding points. The interconnected fields are also shown in Table 5.1-2. The explanation of the result of mapping by the various functions is summarized in the following paragraphs. B) depends on the character of The effect of linear function (w = Az the constants A and B respectively. They may be real, imaginary or complex numbers. In special cases A may be unity or B equal to zero, when they do not influence the mapping, because the position of the field does not change if the position vector is multiplied by unity or if zero is added to it. The effect of the B additive member is the shifting of the points a definite distance parallel to a given direction, corresponding to the vector described by the B complex number. If the B is a real number the shifting occurs in the direction of the real axis and similarly, the points move along the imaginary axis if an imaginary B is added to the x complex variable. Multiplication with a real A value causes the linear extension or shrinking of the plane [the result is extension if 1 A I > 1, while I A I < 1 indicates shrinking (negative extension)]. If A multiplying factor is an imaginary number its effect is a similar extension and the plane is also turned anticlockwise by an angle of n/2.The factor may be generally a complex number ( A = a + ib). In this cam its effect is a combination of extension and turning. The size of the components can be calculated from the following equations: The angle of turning b B = arc tan ;

+

a

The rate of extension m = vu2 + b2

I 3

.

(5.1-37)

The fractional function w = - can be characterized by two reflection steps. The points of the original plane are reflected at first to the unit circle (its centre is the origin and its radius is unity), and the result is reflected once again to the real axis. The image of the origin is a circle in the infinite distance, while the infinite points are crowded in the origin of the image plane. There are two points the position of which do not change (i.e. +1 and -1 on the real axis). If a point is described with a complex number of Reie on the original plane, its image is determined by another complex number (re‘*), the angle of which is the negative value of the former, while

5.1 Two-dimensional potential seepage

579

its absolute value is equal to the reciprocal value of that of the original vector: z = Rei@; and w = reie ; where (5.1-38) .B=-@andr=-.

1

R

The z = 0 point is a singular point. The mapping method creates contact between the u = const. lines (parallel to the imaginary axis of the w plane) and the circles tangent to the z axis in the originof the z plane (their centre being fitted to the y axis). Straight lines on the w plane parallel to the real axis and determined by v = const. values are also interrelated to circles crossing the originofthe z plane, but now they are tangent to the y axis and their centres line-up along the z axis. The flownet composed of these two sets of curves is called dipole or dublett in the literature (Fig. 5.1-5). az b The linear-fractional function w = - can be regarded as a combi-

+

I

cz+d

)

nation of three other mapping functions: i.e. t = cz + d (linear function): a bc-ad 1 s (linear function), and, theres = - (fractional function): w = t c C fore, the effect of this mapping can also be characterized as the superposition of the result of the three components. The point characterized by the cz - d = 0 value is a singular point. The simple fractional function being

+

Fig. 5.1-5. Flow nets of simple and turned dipolee 37

5 Movement equetions describing seepage

580

one of the components, the flownet transformed into an orthogonal system of straight lines on the w plane by this mapping is also a dipole. Its centre is shifted and its main axes are turned according to the constants of the linear mapping. The power function (w= zn) is generally multivalued. A sector of the z 2n plane having a central angle a = - (the field closed by the real axis and n

2nl

a straight line crossing the origin and having a slope a = - is mapped n onto the total w plane cut along the positive real axis. That part of the field of the z plane, which is closed by the real axis and the line having an angle a n = - = - , is interrelated, therefore, to the positive imaginary half of the

2 n w plane. To calculate the corresponding complex number in the w system, the absolute value of the original vector ( z ) has to be raised to the n-th power, while its angle should be increased n times. Special cases of this mapping function are achieved by substituting determined even numbers instead of n. For example the function of w = z2 maps the flow net in a right-angledcorner onto the upper half of the w plane. Using any value of n the function is irregular at the point z = 0 (singular point). *-

The root function (w= v z = z'/") is also a special case of the powerfunction (with a fractional number as a power). Its effect is, therefore, identical with that explained in the previous paragraph. From the point of view of the investigation of seepage the square-root ( n = 2 ) is important, which maps the positive imaginary half of the z plane into a quadrate of the w plane bordered either by the + u and + v or by the +u and -v axes. (The square root being double-valued the total z plane is mapped either onto the upper half or onto the lower half of the w plane, and, therefore, i t is always necessary to indicate, whether the positive or the negative result of the square root is regarded as the main value in the calculation.) This mapping transforms the seepage field having the positive real axis (+x) as an impervious boundary and drained along the negative real axis (-x) into the field of one dimensional flow. This mapping being a type of power-function, the z = 0 point is a singular point. The logarithmic function (w= f l n z ) transforms the total z plane onto a horizontal stripe of the w plane extending in both directions into the infinite distance and having a height of 2n. Depending on whether the multifolded z plane is cut along either the positive or the negative real axis. The primary field on the w plane lies between the real axis and the straight line parallel to that characterized by the u = +2n value, or i t is bordered with the lines u = -n and u = +n. The further possible stripes interconnected with the other folds of the z plane join one of the stripes mentioned previously (depending on the place of the contact of the further folds on 2: plane) shifted by a distance of 2kn along the imaginary axis (where k is an arbitrarily chosen positive or negative integer), according to the z = = r exp [i(@ 2kn)l = r exp (i0relationship. )

+

581

5.1 Two-dimensional potential seepage

By such mapping the = const. lines of the image are interconnected with the radii of the z plane running into the i n h i t e from the origin, if the sign of the function is positive (source), while the negative function is represented by radii coming from the infinite and submerging at the origin as flow lines into a sink. The mapping function is called, therefore, the transformation of a source or a sink, which has a singular point at the origin. The exponential function (w = eZ)is the inverse of the logarithmic function discussed previously. Its effect can, therefore, easily be understood, if the roles of the prototype and of the image are exchanged. The trigonometric functions i.e. the sine function (w = sin z ) and its inverse (w = arc sin z ) were already analyzed in the example given in the previous section (see Fig. 5.1-3). As waa shown there, the arcus sine function maps the half plane into a stripe parallel t o the imaginary axis, having a width of two units bordered by the real axis at one side and extending into infinity in the other direction. The cosine mapping (w = cos z ) can be derived from the former. According to the relationship

COB

L 1

z = sin - f z

,this transn

formation is a combination of the sine function and a shifting of - along 2

the real axis. The practical application of further trigonometric functions (tangent or cotangent mapping), as well aa that of hyperbolic functions is generally of less importance, and they can be constructed aa the superposisin z ez - e-z tion of other functions e.g. uj = tan z = -; or w = s h z = cos z 2 Their detailed explanation is, therefore, neglected here. The linear-fractional function can be regarded aa the overall result of three mapping exercises executed after each other, aa waa proved previously. By the superposition of the eoects of other simple mapping functions in a similar way very complicated seepage fields can be transformed into straight flow tubes of constant area by the different series of functions. The number of possible variations is increaaed by the fact t h a t the potential and stream lines are conjugated trajectories, thus their role can be exchanged. This operation can also be expressed mathematically as mapping characterized by multiplication with the imaginary unit. An example of this transformation is ahown in Fig. 5.1-6, where two sets of flow Lines are compared i.e. that encircling a foundation of 2b width, and the other characterizing a seepage recharged in infinity and discharged along a linear drain also having a width of 2b. The other principle providing baais for the superposition of various mapping methods is the additive character of the complex potential. If there are two analytic functions of complex variables and the relationships are determined between their components

i

w1 = f l ( z ); and w, = f 2 ( z ) ; where u1 = ul(x. y) ; and v1 = q ( z , y) ;

1.

582

5 Movement equations describing seepage

zi plane

Fig. 6.1-6. Mapping of elliptic and hyperbolic stream lines

and similarly

(5.1-39) = u ~ ( xY) , ; and ~2 = v ~ ( xy) , ; the same relationships can also be calculated for the new function formed 88 the sum of the previous two functions: ~2

where and

29.

= w1

+

w2

u = Ul(X,

v = Vl(G

= fl(4

+ f&) ;

+ u2(x, Y) ; Y) + Y) -

(5.140)

Y)

V2(G

After h d i n g the derivatives of the individual members of the equations, it can be proved that the sum of the two functions satisfies Cauchy-Riemann’s condition provided the original functions also fulfilled the condition: -+-=8% 8% 8% I 8% ; ax ax ay ay and au, au, av1 av2 . ----+-= aY aY ax ax ’ if (5.1-41) 8% - av1 -8% - av1 . ax ay ’ a y ax ’ and au, av2 . au, 8% -=---ax ay ay ax

. 9

583

5.1 Two-dimensional potential seepage

It can be stated, therefore, that the sum of analytic functions (even if they are multiplied by different constants) is also always regular. It is quite evident, that innumerable variations of mapping functions can be determined in this way. Only a few of them are listed in Table 5.1-2 and those explained in the following paragraphs have the greatest practical importance. Superposition of a source and a sink having identical discharge (strength) + a = ln(z a ) - ln(z - a ) . The flow can be formulated by to = In z2-a net of the system can be graphically constructed locating the source at + a point and the sink at -a point of the real axis (Fig. 5.1-7). At these two points the mapping function is irregular (singular points). The stream lines are circles crossing both the source and the sink, their centres are fitted to the imaginary axis. The potential lines are also circles, the centres of which are on the real axis outside f a points going gradually to infinity ;t8 the size of the radii of the circles increases. (The imaginary axis is also a potential line being a circle with infinite radius.) The upper half of the z plane is interrelated to an infinite strip on the w plane parallel to the real axis and having a width of n (between the lines u = 0 and u = n). Since any potential or stream line can be regarded as an impervious boundary, the image of the field recharged along a potential line and drained at the sink (or along a potential line surrounding the close

I

w plane

1

+

irnaje of the fird

quarier of the z plane

impervious bounda,-q

Fig. 5.1-7. Flow net characterizing the superposition of a source and a sink having aame strength

584

5 Movement equations describing seepage

vicinity of the sink) can be determined also by this mapping function. It is quite evident, that one need to consider only that stretch of the infkib strip which lies between the images of the two bordering potential lines. The practical example of the application of this mapping function is the horizontal flow net of a single well along a river, when the bank of the river is substituted by a straight line (the imaginary axis of the original plane) and the wall of the well by a small circle around the sink. Naturally, a recharging well in the vicinity of the river bank can be similarly characterized, considering the other half of the z plane (that between the imaginary axis and the source). Infinite series of sources having identical discharge and located at equal

distances along the imaginary axis. The mapping function transforming this very complex flow net into one dimensional flow can ah0 be determined aa the sum of simple analytic functions. The number of the sources being infinite, the mapping series is composed also of an infinite number of functions, the limiting value of which can, however, be determined thus: w=

+ [lnz + ln(z + ih) + ln(z - ih) + ln(z + 2ih) +

+ ln(z - 2ih) + . . .] = +In s h yZhZ .

(5.142)

There are two groups of singular points. The sources ( z = 0; f i h ; h 2 i h ; . . .) are cavitational points with infinite velocity, while the points dividing the h distances of the sources into two halve z = &i - ; -f 3i - . . . are stagna2 ( 2 tion points with zero velocity. The field of the z plane bordered by the lines h h y = - y = - - and x = 0,being open in the fourth direction extending 2’ 2 into infinity and having a width of h , is transformed on the w plane into a strip, infinite in both directions, parallel to the real axis and bordered n n by u = -and u = - - lines (its width is, therefore, equal to n). This 2 2 mapping function is applied in practice, when the task is the characterization of the horizontal flownet of a series of wells located along a river. Naturally, in this case the series of sinks is investigated instead of sources, but i t does not change the character of the mapping function, the more detailed analysis of which will be given in Chapter 5.2 where the practical application of the composed mapping functions will be discussed. It has also to be noted here, that infinite series can be composed of the superimposed flownets of a source and sink, or those of sources and sinks located alternately along an axis. The number of variations can be further increased by multiplying the functions which are the members of these series by different constants. This means in practice that the yields of the sources and the discharges of the sinks are different.

h l

+

+

5.1 Two-dimension'al potential seepage

585

+

The sink in a uniform flow (w= -az b In z ) is also a flownet having an infinite number of variations depending on the ratio of the two constants a and b. The position of the line dividing the part of the field recharging the sink from that, where the flow is continued in the main direction, is determined also by this ratio (Fig. 5.1-8). The stream-function is zero along this

Fig. 6.1-8. Sink in uniform flow

water divide, this fact providing us with the basis for the calculation of the position of this line, and also of its intersection with the x axis (this latter point and the sink itself are singular points of the field): a

x = y cotan - y ; b

(5.143)

b xo=-. a The half width of the recharging part of the field (the distance of the asympt,oteof the water divide line from the real axis) can also be calculated from this relationship by substituting an infinite value instead of 2:

h, = n-

b a

= n x, .

(5.1-44)

586

5 Movement equations describing seepage

5.1.4 Application of Schwartz-Christoffel’s

mapping

When two-dimensional seepage is investigated, the section of the seepage space and the flow plane is a two-dimensional field, the contour of which is composed of straight lines, or the stretches of the contour can be well approximated by such lines. For this reason, Schwartz-Christoffel’s mapping is frequently used in seepage hydraulics. This method transforms a polygon composed of arbitrarily chosen straight lines into one line without any corner point (Schwartz, 1869; Christoffel, 1867). Schwartz-Christoffel’s transformation gives the inverse function of the mapping (i.e. the z complex number is expressed as a function of the w vector representing the position of a point on the image plane) in the form of an integral equation (Fig. 5.1-9): z =5 z =AS-

+ i y = P(w)= P(u + iv) ; dw

(w- u p n (w- U Z ) +

. . . (1c - u n ) a +

(5.145)

+B.

As shown in the figure, this mapping transforms the polygon given on the z plane into the real axis of the w plane. The denominator of the expres-

tY

w plme

Fig. 6.1-9. Representation of Schwartz-Chhtoffel’s mapping

587

5.1 Two-dimensional potential seepage

sioii to be integrated has as many factors as the number of corner points of the polygon. The basic value of the factors is the difference of the independent variable w and the real coordinate of the image of the relevant corner point ( u l ,u 2 .. . u,). The power of this basic value can be calculated m the angle of the polygon at the corner in question related to n. The angle is inemured between the elongated line of the previous side and the following stretch of the polygon, if the direction followed along the contour is determined by the increasing order of the coordinates of the corners on the w plane (ul< u2 < . . . < un).The sign of the angle is negative if the line of the previous side covers the following one, when the former is turned clock-wise. In the other case a is positive. This interpretation gives the possible limits of the angles and also that of the powers of the factors: (5.146) The A and B constants in Eq. (5.1-45) can be any complex number. Their value is determined by choosing the positions of three uicoordinates arbitrarily considering only the possible easy execution of the integration. Fixing the posit,ion of fewer corner points than three, the problem is undetermined, while i t becomes too determined if the coordinates of more than three points are known. As a first example of Schwartz-Christoffel mapping a polygon composed of three sides will be transformed into the real axis of the w plane. The polygon is the contour of a stripe parallel to the y axis, having a width of 2b symmetrically located on both sides of the y axis (bordered by the lines x = + h and x = --b respectively) closed below by the real axis x and extending into infinity in the other direction (Fig. 5.1-10a). It haa two corner and P J , where the angles are equal to one another points (PI The position of the two corner points on the 10 plane can be arbitrarily chosen : WA-1; 0 ) ; P,(--b; 0 ) ;

Pz(+b; 0 ) ;

W,(+l; 0). Because of the symmetry, it is evident, that the two origins are corresponding points: Po(0; 0 ) ; Wo(0; 0). The coordinates of the two corner points are consequently u1= -1 and u2= + l . Thus, the equation to be integrated is as follows: z=AJ

(w

+

A dw 1)1/2 (w - 1)1/2 + B =

dw

7J v1 -

___ f w2

A B = Tart sin w 8

-+ B.

(5.1-47)

The B constant is determined by the condition given for the origins ( z = 0 and w = 0 are interrelated points), and it is equal to zero ( B = 0). The other

588

5 Movement equations describing seepage

la)

z plane

4+y

Fig. 5.1-10. Application of Schwartz-Chrktoffel’smapping

F

pair of interconnected points is w = 1 and z = b. From this condition the multiplying factor can be calculated: A n - A 2b b = - A a r c s.i n l = - - , -=-. (5.148) i i 2 i n Thus, the final form of the mapping function is 2b . z = - arc sin w . (5.1-49) 7c

This mapping is the inverse function of that applied previously [see Eq. (5.1-23) and Fig. 5.1-31. Comparing the results of the two transformations, i t can easily be seen, that the only difference is the exchange of roles of the z and w planes respectively. The second example is the straightening of the polygon forming a step on the z plane (Fig. 5.1-lob). The coordinates of the corner points on the z and w planes, the angles at the corners and the relevant powers of the factors can be listed as follows: Symbols of points

1

2 0 0

Coordinates on the plane Coordinates on the w plane

X

v

0 t a -1 0

Angles

a

_ _II

f,

Powers

a/n

-112

+1/2

z

Y U

2

+1 0

n

589

6.1 Two-dimensional potential seepage

Consideringthese values the mapping function can be given in the following form:

A i

= -(arc

sin w -

V m )+ B .

(5.1-50)

The constants have to be calculated from the coordinates of the interrelated points: a.

A=-; x

and B = - - .

a 2i

(5.1-5 1)

Combining Eqs (5.1-50) and (5.1-51) the final form of the mapping function is achieved: 2 a

1 n

- = -(arc

1 sin w - v 1 - w2) - -. 2

(5.1-52)

More complicated contours can also be mapped by Schwartz-Christoffel’s method, as shown in the third example, where a horizontal foundation having a width of 2b supplemented with a sheet pile located symmetrically and having a depth of 1 is investigated. It is assumed that the foundation is laid on the horizontal surface of a permeable layer with i n h i t e depth (Fig. 5.1-1Oc). The original contour has five characteristic points, among them only three are corners, because the direction of the contour does not change at points 1 and 5, and, therefore, a, = a5= 0. Thus, it is not necessary to fix the position of these points on the w plane, but its coordinates will be calculated from the mapping function. The a priori determined data required for the derivation of the mapping function are as follows: Symbols of points

Coordinates on the z plane Coordinates on the w plane Angles Powers

1 2

Y

-b 0

2 0

0

21

+1

2)

0

3

4

0

0 0

-1 0 0

a

0

+n/2

-n

ah

0

+1/2

-1

6

+b 0

-1

0 fR/2

+112

0 0.

The integral equation and the solution of the mapping function can be written in the following form:

z=Af

w dw (w - 1)1/2(w+ 1)’/2

-+ B = i A v 1 - w2 + B . (5.1-53)

590

5 Movement equations describing seepage

The next step is once again the determination of the constants in the equation. Now the interconnected coordinates of points 3 and 4 can be used for this purpose. The result is

A=-1;

and B=O.

(5.1-54)

Thus the mapping function transforming the given contour into a straight line is determined by the following equation: 2 = - ill/l-

(5.1-55)

w2;

from which the coordinates of points 1 and 5 on the w plane can be calculated:

Wl(u1; w1); W5(u5; us);

u.1

= 71@1'+1;

VI+]+

w,=o; (5.1-56)

2

u5 =

--

1 ; w5 = 0.

If the z plane is a seepage field, the entry face of which is the x < --b stretch of the horizontal real axis, and the water percolates around the foundation and the sheet pile until it reaches the exit face, which extends along the x > b stretch of the real axis, the transformation analyzed above does not result i n one dimensional flow, only simplifies the original flow net. On the new plane the seepage equivalent to the original system can be characterized by horizontal entry and exit faces located along the u real axis (the entry face is the u

r[!)'

>+

extends from the point of u = -

+ 1 stretch of the axis, while the exit face

[+I' +

1 to negative infinity). Between

the faces, a horizontal impervious boundary with a width of 2

VIY+

can be found and the seepage develops above this boundary. The image on the w plane is evidently double-valued, because both the negative and the positive value of w provides the same result. By considering negative w as the main value the seepage field from the z plane can be mapped onto the lower half of the w plane. Thus, i t can be stated, that a seepage under a foundation composed of a horizontal block and a vertical sheet pile is simplified to the investigation of a horizontal foundation without a sheet pile (in both systems an infinite depth of the permeable layer is assumed). As already shown, however, the stream lines are confocal ellipses having their foci at point W ,and W,in the simplified system which can be mapped as straight lines by applying the sine transformation [see Eq. (5.1-24)]. Combining the two steps, considering the recently achieved form only as an intermediate plane (using in Eq. (5.1-55) the 5 variable instead of w), and mapping the newly formed flownet into straight by using Eq.

5.1 Two-dimensional potentional seepage

591

(5.1-24) (where the 5 plane is used instead of the original z plane) the complete solution of the transformation is aa follows (Fig. 5.1-11): z=-iil~1-~2=-iil

5 = Bsin w ; where (5.1-57)

After the determination of the geometrical parameters of the seepage field on the w plane and substituting them into Eqs (5.1-33) and (5.1-34), the discharge and the pressure distribution along the foundation can be calculated. Some difficulty will arise when applying this relationship for solving practical problems, because the infinite depth of the permeable layer was presupposed in the example. The cross-sectional area of the field is also infinite, which evidently also results in infinite discharge.

5

I

Fig. 5.1-1 1. Mapping of seepage field with infinite depth below a foundation composed of horizont,al block and vertical sheet pile into the system of one-dimensional flow

592

6 Movement equations describing seepage

When investigating the seepage under a foundation i t is necessary, therefore, to find a transformation which also considers the position of the lower impervious boundary of the field. This requirement makes the mapping function more complicated. The integral equation can be derived similarly on the baais of Schwartz-Christoffel’s theory but there are many caaes when the integral cannot be solved in closed form, or it leads to functions, which cannot be handled in analytic form (elliptic functions, integral-sine, integrallogarithm, exponential-integral, etc.). In these cMes the numerical solutions can be used either by having expanded the expression to be integrated into series, or by using the pretabulated values of the special functions. The investigation of the seepage under a foundation i n a pervious layer of finite depth, provides a good example to demonstrate the problems occurring in connection with the application of Schwartz-Christoffel’s mapping. For the characterization of such a seepage field Pavlovsky’s mapping methods are generally applied (Pavlovsky, 1922, 1956; KovAcs, 1960; Nkmeth, 1947, 1963; Leliavsky, 1955). The simplest form will be discussed here that is when the foundation contacts the surface of the permeable layer along a horizontal plane, and the mapping function will be derived for this cwe (Fig. 5.1-12). The seepage field is an infinite strip bordered b y two parallel lines. The upper is the real axis, which is divided into three parts: i.e. entry face (z -b), upper impervious boundary (--b < 2 <+b), and exit face (z > + b ) . The lower boundary is the surface of the impervious bed, the depth of which

<

an

impervious boundarg

Fig. 5.1-12. Mapping of a seepage field of finite depth below horizontal foundation

593

5.1 Two-dimensional potential seepage

is m. The t,wo parallel lines form a closed polygon, the corner points of which are in negative and positive infinity and the sides close at both ends with angles of a = IC.Let us map this field onto a 5 intermediate complex plane, the real axis of which is E and the imaginary one 7.The coordinates of the corresponding points and the other parameters required for mapping are listed aa follows: Symbols of points Coordinates on z plane Coordinates on 5 plane Angles Powers

X

Y

5 rl

a a/n

1

2

3

0 0

-b

+b 0

0 0

+1 0

0

0 0

0

0

-1

0

4 --m

6 f - m

Oand -m Oand --m respectively respectively +l/P -1/P 0 0 f n +1

0 0

f n +1

7

6

0

0 --m

--m

+w

--m

0 0 0

0 0 0

Thus, the integral to be solved for obtaining the mapping function is

(5.1-58) Considering that z = 0 and 5 = 0 are interrelated points, the B = 0 condition can be calculated. To determine the multiplying constant Al, i t has 1 1 -and - -points on the [ plane to be taken into account that both the c1 P have t w o corresponding values on the z plane: i.e. P4( -00; 0); P4( -00; - m ) and P5( cu;0);P5( 00; - m ) respectively. Passing along the original contour the change occurring at point 5 is Az = -im. The corresponding change on the C plane can be characterized by the modification of the direction of the vector 1 - p 5 = r exp (iv) with an angle of n. It can be stated, therefore, that

+

+

+

- - m* = 2P

and A 1 -_ _ 2_ w_ n

The interrelated values of point z = b and its image considered, providing the following relationships:

(5.1-59)

5 = 1 can also be bn

n

1-p

1-p

2m

(5.1-60) 2m A , = -th n 38

bn

-.

2m

594

6 Movement equations describing seepage

Hence, the h a 1 form of the first mapping function:

To transform the flow and potential lines into the system of straight lines on the w plane a further tranafornuctwn ?umto beapplied. Now once again a field bordered by a polygon composed of straight lines (on the w plane) has to be mapped onto a half plane, the contour of which is the real axis of the 5 plane. Thus, the solution can be achieved by the second application of Schwartz-Christoffel’s theory. I n this caae the basic data of the mapping function are ~EIfollows: Symbole of points Coordinetes on the C plene

1

c

0

rl

0

Cmrdin8tm on the w plene

u

+K

v

0

Anglee

a

0

3 -1

2 +1

0 +2K

0

4

+-P1 0

6

1 -0

P

61 +on

0

+2K

0

+K’

+K’

+K +K’

II II II +T + p +T

+f

0

0

0 0

Powers

7 --oo

0

+K,

+K 0 0

Considering the parameters listed above, the relationship between the two complex planes (i.e. w and 5 planes) can be given in the form of the following integral equation:

=pWC;

w +B.

(5.1-62)

Thus, the final result is the first type of Legendre’s elliptical integral, the modulus and the supplementary modulus of which can be expressed by the following equations:

(5.1-63)

and

m 2

References

595

According to the definition of the elliptic integrals, their total value (belonging to C = 1) and the supplementary total value can also be calculated: K = F(l ; k,) and K' = F(l ; ki) . (5.1-64) Using these values, the coordinates of the corner pointe of the field on the w plane, 88 well 88 the A and B constants can be determined: 1

and (5.1-65)

consequently

p A = l ; and B = K .

As shown in Fig. 5.1-12, the length of the straight flow tube (the mhievement of which was the objective of the transformation) is 2K, while the area of a cross section of the stream tube having a unit width perpendicular to the flow plane, is equal to the supplementary total value of the elliptical integral (K'). After substituting these parameters into Eq. (5.1-33) the specific dischccrge through the seepage field can be calculated: K' q=KAH-.

2K

(5.1-66)

The pressure distribution along the foundation can be similarly determined using Eq. (5.1-34) as well aa the geometrical parameter describing the position of the investigated point on the w plane (5.1-67)

References to Chapter 5.1

BEAR, J. (1972): Dynamios of Fluids in Porous Media. Elsevier, New York, London; Amsterdam. B-, J., Z A ~ L A V ~ KD. Y , IRMAY, S. (1968): Physical Principles of Water Percolation and Seepage. UNESCO P&. BOUSSINESQ, J. (1904): Theoretical Research on the Flow Rate of the Ground Water Percolating in Soil and on the Yield of Sources (in Frenob). J o u d Mathematiquee Puree e.t Appliqudee, Vol. 10. . 1. CIEIBTO~EL,E. B. (1867): 8 n the Problem of Steady Temprature and the Representation by Surfaae Data (in Italian). Anndee di Matematiccr. No. 2. DUPWIT,J. (1863): Theoretical and Pmtical Studies on Water Movement in Open Canals and through Pervious Soil (in French). b o d , Paris. 38*

596

5 Movement equations describing seepage

FOROIIHEIMER, PH. (1924):Hydraulics (in German). Teubner, Leipzig Berlin. GRIJBER,J. and BLAH^, M. (1973): Mechanics of Liquid Media (in Hungarian). Tankonyvkiad6, (8th edition), Budapest. HAMEL, G. (1936): On Ground-water Flow (in German). Zeitschrift fiir Angewandte Mathematik und Mechanik, No. 3. HaRR, M. E. (1962):Groundwater and Seepage. MoGraw-Hill, New York. Kovbcs, G. (1960):Calculation of Disoharge Percolating under D a m s (in Hungarian). Viziigyi Rddemdnyek, No. 2. Kovbcs, G. (1963):Free Seepage from Irrigation Canals. F‘III. Convegno di Idraulica, Piaa. Kovbcs, G. (1964): Hydrauiia Characterization of Free Seepage from Irrigation Canals (in Hungarian). v$zizgYiKodemdnyek, No. 2. Kovbcs, G. (1978):Mathematical Modelling of Ground-water Flow. V I T U K I Proceedings. No. 8. LELIAVSKY, S. (1966):Irrigation and Hydraulic Design. Vol. 1. Chapman & Hall, London. MILNE-THOMSON, L. M. (1966): Theoretical Hydrodynamics. MacMillan, London. MUSXAT, M. (1937): The Flow of Homogeneous Fluids Through Porous Media. McGraw-Hill, New York. NEMETH, E. (1947):Flow Problems in Engineering Practice (in Hungarian). MBrnoktovBbbk6pzd IntBzet, Budapest. N~METH, E. (1 963): Hydromechanics (in Hungarian). Tankonyvkiad6, Budapest. PAVLOVSKY, N. N. (1922): Theory of Ground-water Movement around Hydraulic Structures (in Russian) (lithographic). Leningrad. PAVLOVSKY, N. N. (1966): Collected Works (in Russian). Gostekbizdat, MOSCOW, Leningrad. POLWARINOVA-KOCH,P. YA. (1962): Theory of Ground-water Movement (in Russian). Gostekhizdat, Moscow. POLIJBARINOVA-KOCHINA, P. YA.(1962);Theory of Ground-waterMovement. Princeton University Press, Princeton. SCHWARTZ,H. A. (1869): On Some Mapping Problems (in German). Berlin. V. V. (1934): Infiltration from Canals (in Russian). Gostroitizdat, VEDERNIHOV, Moscow. WEAWER, W. (1932):Uplift Pressure on Dams. Journal of Mathemdim and Physics, No. 2.

Chapter 5.2 Combined application of various mapping functions The final purpose of mapping is to produce by mathematical means a flow pattern of one dimensional seepage from the original flownet. As explained in the previous chapter, this objective could rarely be achieved by applying only one mapping function, and the combination of variozcs functions becomes necessary in many cases, and even different mapping methods (conformal, hodograph, or Zhukovaky’s mapping) had to be used in combination. The most simple form of the combined application of mapping functions occurs, when a two-dimensional seepage is investigated, and the contour of the seepage field in the flow plane cannot be mapped into the required rectangular form by one analytic function, but a series of such relationships has to be used. In this case all steps are performed in the same plane and, therefore, the mapping can be summarized in one group of transforming functions. The stream lines of a special three-dimensional seepage having straight cylindrical flow surfaces can also be transformed info the flownet of a one

5.2 Combined application of various mapping functions

597

dimensional flow by mapping. I n this case the b t transformation has to be executed in a plane perpendicular to the flow surfaces. A mapping function (or a series of such functions) should be found which creates parallel straight lines from the curves characterizing the intersections of the mapping plane and the flow surfaces these being perpendicular to the former. The seepage is reduced in this way to a two-dimensional flow, which can be simplified afterwards by applying further mappings in the flow plane produced by the first transformation. This double mapping can be used as an approximation even in the caae of general three-dimensional seepage. When investigating unconfined seepage the unknown form and position of the phreatic surface hinder the application of conformal mapping, even in the case of very simple two-dimensional, flow. As already mentioned, the hodograph image or Zhukovsky’s mapping can be used in the vertical section of the seepage space to determine some parameters characterizing the position of the water table. After knowing the complete contour of the flow field, the investigation can be continued by the application of conformal mapping to achieve the one dimensional flownet aimed at originally. Sections of this chapter will show practical examples of the combined application of the various mapping functions and methods.

5.2.1 Application of a series of mapping functions within the flow plane of a two-dimensional seepage Drains along the bonk of surface water

The two-dimensional seepage to a drain running parallel to the bank of a surface water will be investigated as an example of mapping executed within the flow plane by applying a series of analytic functions. Apart from the hypothesis of the validity of Darcy’s law (laminar nwvement) further simplifying approximation are also applied. These suppositions are as follows: (a) The water conveying layer is confined (this approximation can also be accepted in the case of unconhed flow, if the slope of the water table is relatively small, and, therefore, a constant vertical depth of the groundwater flow can be assumed); (b) There is no free exit surface among the various stretches of the contour of the field; this is absolutely satisfied condition in the case of confined seepage (the level of the tail water has to be above the upper contour of the field) while i t is, in some cases, an acceptable approximation of unconfined flow aa well (partially penetrating drains when the free exit face is relatively small); (c) The entry face (the contact surface between surface and ground water) can be approximated with a horizontal or vertical plane.

Considering the last supposition, and dividing the draining structures into three groups (i.e. fully penetrating drains, partially penetrtting drains and draining trenches), the hydraulic characterization of six different seepage

598

5 Movement equations describing seepage

horizonfsl

vertical entry

me

entryfire

I

II

Fig. 6.2-1. The main type of seepage field developing between surface water and drains

fields haa to be solved aa indicated in Fig. 5.2-1. The various combinations of mapping function6 providing the hydraulic parameters necessary to design the drains are explained separately in the listed types of flow fields. Horizontal entry face; fully penetrating drain (Fig. 6.2-2)

The flow field is identical to the first half of the one characterizing the seepage under a horizontal impervious foundation of a dam (see Fig. 5.1-12). Thus, the transformation proposed by Pavlovsky (1922), baaed on SchwartzChristoffel’s theory and resulting in the use of Legendre’s first typa of elliptical integral, can be applied to solve this problem aa well. The derivation of this mapping method is summarized in Eqs (5.1-58) to (5.1-65), and. therefore, it is not repeated here. There is only a slight difference in the final results: the flow field investigated here being half of that indicated in Fig. 5.1-12. Therefore instead of Eqs ( 5 . 1 4 6 ) and (5.1-67), the following equations have to be usedfor the calculation of specific flow rate (9) and surplus pressure above the water level in the drain:

599

6.2 Combined application of various mapping functions

impervious boufld31-y

I

Fig. 6.2-2. Transformation of flow field between horizontal entry face and fully penetrating drain

(5.2-1)

and

To avoid the difficulties arising from the use of the tabulated values of elliptical integrals, more simple formulae can in practice be proposed to approximate Eq. (5.2-1) (Kovtica, 1960s). q=2-

KAH II

m

ar sh 1 . 5 - ; L

and if

m -
1 2 (5.2-2)

or

h=dH

[

1--

[:+arcsin-

L

if

m

1

L

2

->-.

600

5 Movement equations describing seepage

(It is necessary to note here that the theoretical basis of Eq. (5.2-2) is similar to mapping executed by Pavlovsky’s transformation, and the more simple form is achieved by a mathematical approximation of the elliptical integrals.) Horizontal entry face; partially penetrating drain (Fig. 6.2-3)

There is a further hypothesis applied in the derivation of this model, i.e. the approximation of the contour of the drain. It is fixed only with points 1 and 5, while the position of the contour line is determined with u = cost

A +.u

z plane

Fig. 5.2-3. Transformation of flow field between horizontal entry face and partially penetrating drain

5.2 Combined application of various mapping functions

601

value on the w plane, supposing that its retransformed image on the x plane is suitable to substitute the actual contour. The subsequent steps of the mapping are w follows (Kovhcs, 1961b):

+

(a) Mapping of the z plane to the t = r is plane by applying Schwartzpolygon Christoffel’s theory, which transforms the 1-2-3-4-5-6-7-8-9-10 into the real axis of the t plane, and the flow field into its positive imaginary half t = c o s -- r t h 2 4 n + tan2--, d’ II

Id’

.

m 2

m 2

m 2

(5.2-3)

where d’ = d - bl2 indicates the position of point 3. (b) The position of the points fitted to the real axis of the z plane and especially those indicating the edge of the entry face and the sides of the drain (points 10, 1 and 5 respectively) can be determined from the following equations: r = c o s ~ ~ 5 ) l / t h P ( ~ % ) + t a n ~ ( ~ ~ ) ; (5.2-4)

ro = [WlX-; :

and e = [r(41x-~ (c) Shifting the t plane horizontally to have the origin of the new t’ = = r’ is‘ plane at the fictive centre of the drawn down a

+

(5.2-5)

t’=t+d

where A(the size of the shifting) is determined by the condition that, on the final image (w plane) the real coordinates of points 1 and 5 should be equal to one another, and its value after applying some approximations is

Am?$. (d) Application of the mapping by fractional function of the t ‘ plane to get 5 = 6 i q plane

+

1 [=--;

and t=-

t

1

r+A’

.

(5.2-6)

(the use of the exponential form of the complex number and the expression of its imaginary member is not necessary, because the characterization of the relationship between the corresponding points of the real axes is sufficient to solve the practical problems). (e) The width of the entry face is (5.2-7)

and the next phase of mapping is to shift the origin to the middle of this width: 5’ = 5 6;

+

602

6 Movement equations describing seepege

therefore

5’ = 5 and

+ 8;

q‘ = q ; where

8=

1-A

1

Tv-.

( e - d ) ( l- A )

1 - ri

(5.2-8)

(f) The final mapping transforms the contour of the flow field of th0 C’ plane into an orthogonal rectangle on the w plane having flow lines parallel to the real axis

.

w = z arc sin-,C’. Q

therefore

5 ’ = usin(-w)chu;

and 7’

(5.2-9)

= Q cos ( -v) sh U .

By this last mapping method the required rectangular form of the seepage field is achieved, and its geometrical parameters substituted into Eqs (5.1-33) and (5.1-34) provide the necessary hydraulic data. These are the specific flow rate and the pressure distribution along the upper boundary of the field, or applying the method to approximate the caae of an unconfined system, the expected position of the water table aa the height of the water column equivalent to the pressure and measured above the water level of the drain: (5.2-10)

and

The relationship between the geometrical parameters of the original field and those on the w plane, can be determined from mapping functions. The total length of the mapped field is (5.2-11)

and the position of a point along the upper boundary is characterized by the following equations: 1 utn=arch or

arch[^[^+^]]; 1

Q

r+ro

1 4 ;

(5.2-1 2)

5.2 Combined epplicetion of verious mapping functions

603

depending on whether the pressure is calculated a t an internal point (between the surface-water and the drain), or a t a point outside the drain (external point). The exit velocity is constant in the transformed one dimensional system

v w = - - Q - K - .AH n UO

(5.2-13)

In the original system this parameter changes along the contour of the drain and it is proportional to the ratio of the stretches corresponding to each other on the z and the w planes respectively

Aw AH dw v(2)=vw-=K------. Az uo dz

(5.2-14)

Horizontal entry face; draining trench (Fig. 5 . 2 4 )

In the case of a very shallow drain (e.g. a draining trench) the previous 1960b, 1961a). The specific flow rate, the model can be simplified (KOV~CS, surplus pressure head and the exit velocity, can be calculated from the same equations aa in the previous cam [Eqs (5.2-10) and (5.2-14)], but in the first mapping function the d = 0 condition can be considered, and thus the

:[

t = th

z)

(5.2-15)

relationship can be used instead of Eqs (5.2-3) and (5.2-5). The final result is a simplified form of the equations proposed to calculate the geometrical parameters of the field of one dimensional seepage:

[:

u O = a r c h -cth--

;]

;

and uIn=arch

-

cth- x n - d)];

[:(

m 2

or u,,=arch where

[:[-

:; I]

cth--+d

;

(5.2-1 6)

604

6 Movement equations deecribing seepage

! Fig. 6.2-4. Transformation of flow field between horizontal entry face and draining trench Vertical entry face; fully penetrating drain (Fig. 5.2-5)

In this case the application of conformal mapping is unnecessary, because the original flow net is composed of orthogonal straight lines, the seepage being one dimensional. Thus, the hydraulic parametera can be calculated directly from the integrated form of Laplace's equation:

(5.2-17)

605

5.2 Combined application of various mapping functions

impervious boundacy Fig. 5.2-5. Flow field between vertical entry face and fully penetrating drain

and L

m

Vertical entry face; partially penetrating drain (Fig. 6.2-6)

For the characterization of this flow field, the mapping method derived for the calculation of the yield of partially penetrating wells may be used, although there is a basic difference between the two ewes. The well is in

most cases, recharged symmetrically from each direction and, therefore, i t is a generally accepted hypothesis, that the flow lines do not cross the axis of the well. In the case of the asymmetrically recharged field investigated, this approximation does not cause any appreciable error if the penetration of the drain is larger than half the thickness of the aquifer d

3

> - . In the

opposite case the model describing the seepage into a draining trench can be used. Accepting this approximation, the task is the determination of the thickness of u virtual layer ( m > M > d ) in which a fully penetrating well yields the same flow rate as the investigated field (KovAcs, 1966). The first step in mapping is the application of the sine transformation to obtain the image of the flow field on the auxiliary plane 2 C = sin n;

m

therefore 5 Y 5 = sin-nnh-n;

m

m

(5.2-18)

and

It' is necessary afterwards to shift the origin to the centre of the image of the exit face

C' = 5 - 6;

606

6 Movement equations describing seepage

z plane

w plane

VA

+

'

.

L

Fig. 6.2-6. Transformation of flow field betweeavertical entry face and partially penetrating drain

therefore

E'

and

= 5 - 6;

(5.2-19)

q' = 7;

where

6=-

7

2

dl

l+cosn-.

m

The field can easily be retransformed into a s e m i - i n f i b strip between two impervious boundaries which is fully penetrated by the vertical exit face. During this mapping i t can be ensured, that the length of the new field should be equal to the original distance between the vertical exit face and the drain (L):

L arc Bin . C' ., w =A

1-6

therefore

E' = 51.11 A u Av -ch - ; 1-6

L

L

6.2 Combined applicetion of various mapping functions

607

and

Av ; '' - cos-A u sh -L

1-6

(5.2-20)

L

where

1-6

As shown by the figure, the entry face is not absolutely vertical after mapping, but the approximation is acceptable, if the-

L

ratio is greater

m

L a than unity; this limit can even be decreased to - > 0.5, if ->0.7. Among m

m

the hydraulic parameters the specific flow rate can be calculated as the function of the geometrical data of either the new or the original field

M L

q= KAH-;

where

M = -nL. ,

(5.2-2 1)

A

:1

while within the validity zone mentioned previously - >-

the pressure 1 ) distribution along the upper boundary may be approximated 'with a linear relationship X

h= AH-.

L

(5.2-22)

The exit velocity is always constant in the transformed system, and its original local value can be calculated by considering the ratio of the corresponing stretches of the exit face in the two systems [see Eq. (5.2-14)].

Vertical entry face; draining trench (Fig. 6.2-7)

Similar to the investigation of flow fields with horizontal entry face8 the actual confour line of the trench is approximated in this case also with the retransformed image of a straight line on the w plane characterized by u = u,,= const. value (Kovhcs, 1975). At first the flow field (which is a semiinfinite strip) should be mapped onto the positive imaginary half of an auxiliary plane ( t plane): .

e

t = sin -x ;

m

608

5 Movement equations describing seepage

4

3

i4

5

Fig. 6.2-7. Transformationof flow field between vertical entry face and draining trench

therefore X Y r = s i n --rich-n; m m

(5.2-2 3)

and X Y s = c0s-nnh-n. m m

The seepage field in this plane is identical to that on the t plane, when the flow between a horizontal entry face and a draining trench is mapped (see Fig. 5.2-4). After shifting the origin into the centre of the exit section t' = t

- 6,;

5.2 Combined application of various mapping functions

609

therefore r’ = r - 6,; and s’ = 5;

where 1

(5.2-24)

the steps of mapping are the same as those described previously [Eqs (5.2-6), (5.2-8) and (5.2-9)]. Thus the relationship between the corresponding geometrical paramehm of the original field and its image on the w plane can be determined: 61 ch -

m

and (5.2-25)

or

On the basis of the combined transformation functions [Eq. (5.2-25)] the hydraulic parameters can be directly calculated from Eqs (5.2-10) and (5.2-14).

The influence of both the position of the closing faces and the main geometrical parameters on the hydraulic characteristics can be well demonstrated by cumparing the results of the various models. Thus, Fig. 5.2-8 shows the flow rate as the function of the ratio of the two main geometrical parameters (Llm) determined by considering different positions of the closing faces. Supposing that the difference between the results of the models is negligible, if it is smaller than 5 % of the flow rate, the following conclusions can be drawn from the figure: (a) In the caae of partially penetrating drains the hydraulic parameters can be calculated by the more simple models characterizing the fully penetrating drains, if the ratio of penetration dlm > 0.8; 39

610

5 Movement equations describing seepage

-.......... ...-...

--- -+-

R50J 08 10

3.0 4.05080 80 IOU L/m Fig. 5.2-8. Comparison of the results of models established for t.he de1,ermination of hydraulic parameters of seepage between surface-water and drain 20

(b) I n the range of Llm > 1 the difference between the results of models for horizontal and vertical entry faces is generally smaller than loo/,, thus the actual form can always be substituted with one of these approximations (which is nearer the actual form) without causing a larger error than 5%; (c) In the zone of Llm > 4 5 the digerewe of /low rates determined by horizontal or vertical entry face is smaller than 5%, the more simple and accurate models can be applied; (d) If Llm > 8 10, the largest dioerence between the results of the various models becomes less than 5 % and, therefore, the flow rate above this limit can be calculated from the simplest Dupuit's equation [Eq. (5.2-li')]. The differences caused by the form and position of the closing faces are more considerable if the pressure head distribution is investigated instead of the flow rate. In this case a suitable model has to be selected more carefully, especially if the piezometric head has to be determined at special points (along the entry face, in the vicinity of the draining structure, on the protected side of the ground-water space, etc.).

-

N

5.2.2 Combination of mapping functions applied on two different planes of the flow space Yield of riparian wells

The flownet of three-dimensional seepage cannot be transformed into one dimensional flow by the simple application of mapping functions. There are, however, special cmes, when the stream lines are fitted to cylindrical surfaces (see Fig. 4.1-8). A section investigated perpendicular to these flow surfaces,

611

5.2 Combined application of various mapping functions

reveals a two-dimensional orthogonal net composed of the curves representing the intersections of the plane of section with both the flow surfaces and the equipotential surfaces. The conformal mapping has at first to be applied on this plane, thus obtaining a system of parallel flow planes. As a second step the contour of the seepage space on the new flow plane has to be mapped onto the plane of complex potentials to achieve the required one dimensional seepage equivalent to the original system. It is necessary to note, that after executing the first mapping in the plane perpendicular to the flow surfaces, the geometrical parameters of the new system have dimensionless values, and they have to be combined with the flow nets on the flow surfaces. These nets however, are distorted because the lengths in one direction parallel to the intersection with the plane of mapping, were transformed and are characterized by dimensionless quantities, while in the direction normal to the former the original measurements remained unchanged having a dimension of length. Before starting the second mapping this contradiction has to be eliminated. It is necessary to determine, therefore, the most characteristic length in the plane of the first mapping, and to multiply the result of the first transformation so that the image of this length should be equal to the original size. The multiplying factor also has a dimension of length, thus the geometrical parameters of the mapped system will be homogeneous having the same dimension in the plane of the transformation aa in the normal direction. The best example to demonstrate the application of two conformal mappings executed along two surfaces normal to one another, is the determination of the yield of riparian wells. If the wells fully penetrate the aquifer and the entry face is vertical, the stream lines are really fitted to vertical cylindrical surfaces, and thus the method of double conformal mapping gives the accurate solution. In the case of a non-vertical entry face the difference is negligible, while larger discrepancy is caused if the well is a partially penetrating one, because the stream lines coming from the direction of the river bed may cross the axis of the well in the lower part of the aquifer and turning back they approach the screen from behind. Thus the model can be applied as an approximation for any position of the entry face, and it is even acceptable for the calculation of the yield of partially penetrating riparian wells, if the ratio of penetration is high enough ( d / m > 0.5 0.6). The transformation of the flow field between the bank and a single well, aa well as the application of the method for the characterization of one member of a series of wells running parallel to the bank, will be demonstrated here as practical examples.

-

Single well near the bank of a river (Fig. 5.2-9)

There is more than one method of solving this problem (e.g. the method of images), but here the use of double conformal mapping will be shown. The horizontal sections of the flow surfaces are identical to the stream lines of the well-known flow pattern characterizing the superposition of a source and a sink having equal flow rates. With conformal mapping (see Fig. 5.1-7) 39*

612

5 Movement equations describing seepage

Fig. 5.2-9. Transformation of the cylindrical flow surfaces of a single riparian well into parallel flow planes, applying mapping in a horizontal plane

this flow field is transformed into a semi-infinite stripe of 2n width (the image of the centre of the well is at infinity). The length of the transformed active field is determined by the distance between two parallel straight lines, which are the images of the bank and the horizontal section of the well screen. The multiplying factor, which has to be applied after the mapping in the horizontal plane, can be determined where the distance between the images of the bank and the well screen (the length of the flow field) is equal to the distance of the centre of the well from the bank intheoriginal system. Thus, the width of the new seepage space (the length of drain equivalent to that of the well) can be calculated from the following equation:

B-L-

2n 2L’

(5.2-26)

In r0

Considering the explained relationships, the yield of the fully penetrating well can be calculated as the flow rate to the drain of B width and being of L distance from the bank. In the case of unconfined flow this value is determined by Eq. (4.1-44). After substituting the geometrical parameter on the basis of the transformations, the yield of a riparian well is:

H2, - H L n K H : - H ; . Q=Bq=KB

2L

2L

2111-

7

(5.2-27

5.2 Combined application of various mapping functions

613

where H , is the height of the water level in the river above the lower impervious boundary of the aquifer, and H , is the elevation of the water level in the well above the same datum. If the seepage field is confined having a thickness of m Eq. (5.2-17) can be used for the determination of the yield:

& = Bq=KBm

- H Z = Km 2 4 H ,

L

- H,) 2 L

(5.2-28)

In r0

In the case of partially penetrating wells (when the method can be applied as an approximation) the flow net of the vertical flow plane obtained by the first mapping has to be transformed to get a one dimensional flow. This second m p p i n g depends on the position of the entry face and o n the rate of penetration. The steps of the necessary transformation have already been discussed in the previous section for each profile of practical importance. Thus, the task is the combination of the mapping executed in the horizontal section with one of those models. O n e well from a series of riparian wells (Fig. 5.2-10)

Supposing that both the spacing ( B )of the wells and the distance between the wells and the bank ( L )are constant, the horizontal section of the vertical flow surfaces shows a pattern identical with the flow lines of the infinite series of sources or sinks. The mapping function derived for the character-

z plane

Fig. 5.2-10. Transformation of the cylindrical flow surfaces of a member of a series of riparian wells into parallel flow planes applying mapping in a horizontal plane

614

5 Movement equations describing seepage

ization of this flow net [seeEq. ( 5 . 1 - 4 2 ) ]can be used, therefore, for the determination of the geometrical parameters of the flow field belonging to a drain hydraulically equivalent to the seepage space occupied by the flow from the river bed to the well in question: tu = [In z

+ ln(z + iB)+ ln(z - iB)+ ln(z + i 2 B ) + ln(z

-

i2B)

+ . . .] =

ZZ

= lnsh-;

B

therefore

zc = -In2

and

’[

2

X

ch 2 n - - cos

B

tan

(5.2-29)

j.;)

v = arc tan

Applying this mapping method, the geometrical parameters of the new system having parallel flow planes can be determined. The basis of the determination of the necessary multiplying factor is that the width of the hydraulically equivalent drain should be equal to the spacing of the wells (KovQcs,1 9 6 2 ) : B, = B ;

(5.2-30)

n

2

where the last relationship gives the distance between the bank and the image of the corner point (the intersection of the axis of the wells and the line normal to the bank and half the distance between the wells) on the transformed plane. Knowing the geometrical parameters of the flow space after the first mapping method executed in the horizontal plane, a system of parallel flow planes is achieved. The vertical dimensions of the seepage field in this flow plane and the position of the entry and exit faces are determined by the original data, and the horizontal by mapping. Considering the form of the new seepage field, the model suitable to characterize the flow pattern in question can be selected from those analyzed in the previous section. After combining i t with the mapping explained above the hydraulic parameters characterizing the seepage from the river to one member of a series of wells along its bank can be calculated.

5.2 Combined application of various mapping functions

615

5.2.3 Combined application of hodograph and conformal mappings Infiltration from canals

The Dupuit's hypothesis can be used for the approximate determination of the hydraulic parameters of unconfined seepage only i n the m e of horizontal or nearly horizontal flow, when the supposed constancy of the velocity along a vertical section is acceptable. Investigating the other types of unconfined seepage and even in the case of nearly horizontal flow if more accurate analysis is required, the position of the seepage line (the intersection of the water table and the flow plane) has to be determined first. This analysis can be followed by conformal mapping, aiming at the determination of the field of the one dimensional seepage equivalent with the original system. As already mentioned, one of the possible methods of determining the seepage line is hodograph mapping. In this section an example will be discussed to demonstrate the combination of this special mapping and the conformal transformation. The practical example chosen for this purpose is the infiltration from irrigation canals. Before going into details of the application of hodograph mapping, it is necessary to investigate the general character of this type of seepage varying with time. On the basis of this analysis the validity zones of the various methods applicable to the description of the different flow conditions can be clearly indicated (KovAcs, 1963b). The process of infiltration from irrigation canals starts with the filling up of the canals. The first stage is the saturation of layers around the canal, when the infiltrating water has no contact with the continuous ground water, but the wetted front moves slowly downwards until i t reaches the water table, or more correctly, the upper surface of the closed capillary zone. During this propagation the water entering from the canal is used for the saturation of the wetted zone, where the pores were previously empty. After reaching the water table only a part of the infiltrating water is stored within the extending saturated zone, the other recharges the ground-water space. A unified seepage field develops in this way and the horizontal water conveyance of the groundwater space aflects the process of infiltration. On the basis of the explanation given in the previous paragraph, the process of infiltration can be separated into two main phases, i.e. the free seepage, and that influenced by the ground water. Both main types can be further divided by distinguishing unsteady and steady conditions. The process starts with the saturation of the layers around the canal which is an zcnsteady free seepage (Fig. 5.2-lla). Theoretically, the steady state of the free infiltration can develop only if the wetted front can propagate until reaching an infinite depth without the influence of the water table (Fig. 5.2-11b). In practice there is a point in time when the front just contacts the lower saturated zone and the process changes from the free stage into the influenced one. This t'ransition phase can be regarded as the quasi-steady state of the free infiltration (Fig. 5.2-11c). If there is a relatively very permeable layer below the water table, which is able to ensure the horizontal transport of the

616

6 Movement equetions describing seepage

(a) free unsteady infiltration

7 (c)quasi-steady free infiltmfion

h e depth ofthe pervious lager! is infinite

(d)influenced

unste8dy infl'ltration

(influenced steady infiltratiou)

Fig. 6.2-11. Verioua flow conditione characterizing infiltration from irrigation canal

5.2 Combined application of various mapping functions

617

infiltrating water without considerable rise in the gradient and the elevation of the water table below the canal, the quasi steady state of the free movement can become stable. The amount of infiltrating water is small because of the relatively low hydraulic conductivity of the layer between the canal and the water table. In other cases the water conveying capacity of the ground-water space in a horizontal direction hinders the further development of the infiltration. The seepage line indicating the continuous upper surface (contacting the points having pressure equal to atmospheric) of the combined flow field of the infiltration, extends gradually sideways. Part of the infiltrating water is stored in the pores becoming saturated as a result of the extension of the ground-water mound, and the remainder is drained by the horizontal flow (Fig. 5.2-11d). Thus this phase is also unsteady, but the movement is of the influenced type. The sideways propagation of the slope stops when the drainage becomes equal to the gradually decreasing amount of infiltration and a dynamic balance develops within the seepage field (steady influenced infiltration, Fig. 5.2-lle). It is necessary to note here, that drainage may be caused not only by the nearby canals having a lower water level, but also by the negative accretion resulting from the rise of the water table. The analysis of the various phase of infiltration clearly indicates that in the case of influenced seepage the largest part of the seepage field is characterized by nearly horizontal flow and, therefore Dupuit’s approximation is acceptable, though it may have to be supplemented by the determination of the local resistance caused by the strong curvature of the stream lines in the vicinity of the canal. Thus, the free infiltration (its unsteady and steady state) is the form of seepage which will be used as an example to demonstrate the combined application of hodograph and conformal mapping. The computation of the unsteady free infiltration has been dealt with by few authors. Perhaps the most frequently quoted relationship is given by Averjanov (1950). He has expressed the time-dependent flow rate of infiltration as a product of the water loss belonging to the free steady state ( t = 00) and a factor greater than unity and determined the latter partly on the basis of some theoretical consideration and partly from experimental measurements. In contrast to the previous type of movement steady free infiltration has been investigated by many research workers and a great number of theoretically well established results are known (Koieny, 1931; Vedernikov, 1934; Pavlovsky, 1936a, 1936b; Riesenkampf, 1940). The listed investigations are equally based on the application of both hodograph and conformal mapping. The differences between them are caused, in general, by the diflerent profiles of the canal considered aa boundary condition. They can differ further, according to whether the capillary eflect is taken into account or not. Among the various methods, which all characterize the steady free infiltration, Verigin’s derivation (1949) is presented here. The advantage of this method is not only the relatively simple method of studying the capillary water transport and the rapid convergence of the series into which some expressions are expanded, but also the possibility of further developing the method

618

5 Movement equations describing seepage

for the description of the unsteady state of free infiltration on the basis of the results achieved in the analysis of the steady state (Kov&cs,1963a). The basis of Verigin’s derivation is a diagram specifying the bondary conditions around the seepage field of the free steady infiltration (Fig. 5.2-12). These conditions along the various stretches of the boundary are as follows:

1

1-

potential line

1-

Fig. 5.2-12. Seepage field characterizing free steady infiltration from irrigation canal

h

(a) Stream line bldl: stream-function y = 9 4 2 ; pressure head p l y = - -h, = const.; (b) Stream line b,d,: stream-function y = -qd2; pressure head p l y = - - h, = const.; (c) Special boundary blcl: stream-function ly = 942; pressure head varies from p l y = 0 at point c1 to p l y = -h, (at point bl); (d) Special boundary b,c,: stream-function ly = -942; pressure head is similar to that of the previous stretch p l y = 0 at point c, and p l y = -h, at point b,; n (e) Potential line c1c3c2: the potential is constant or choosing the water h

h

level as datum it is equal to zero cp = K (The symbols are defined as follows: h, capillary height, go specific discharge infltrating from one metre length of the canal; velocity potential.) The seepage field of the two dimensional flow is represented on the complex potential plane (w = rp i y ) aa well aa on the hodograph plane of the

+

reciprocal vector of the conjugate velocity stripe parallel to the real axis (Fig. 5.2-13). between these two planes, an auxiliary plane (( = E

.

) by an infinite

the relationship

+ i q ) has to be intro-

619

5.2 Combined application of various mapping functions

Fig. 5.2-13. Conformal and hodograph mapping to determine the hydraulic parameters of free steady infiltration

duced. Both stripes can be easily mapped onto the positive imaginary half of the 5 plane by applying Schwartz-Christoffel's transformation:

In 6

2

(5.2-3 1 )

dz dz dw 1 Considering the relationships between - , - - and -[see d5 dw ' dC V* (5.1-20)] the following equation can be given dz dC

-

h, -1n--, 1 1+C. nln6 C 1-C

Eq.

(5.2-32)

620

6 Movement equa.tions describing seepage

the solution of which (after expanding the expression t o be integrated into series) yields z=

[ c + ~ c ~1 + - ~1 ~ + - : ' + . .(5.2-33) .

1 -2 2h

nln 6

52

72

Substituting the corresponding parameters characterizing the width of the water in the canal ( z = S/2 and 5 = 6 ) a formula is achieved t o calculate the unknown 6 parameter as a function of the ratio of the width and the capillary height: SJ2=?

1 1 6+-63+-65f-67+ nln 6 9 25

1

...

49

(5.2-34)

The other important geometrical parameter is the fictive width of the water level ( S othe distance of the two extreme stream lines at the elevation of the water level between points b, and b,), which characterizes the influence of the capillary water transport. Its size can also be determined from Eq. (5.2-33), by substituting the 5 values belonging to point b, or b,: s o / 2 = - z

n2h ln6

[

1 1 1+-+-+-+ 9 25

1

...

49

(5.2-35)

Using Eqs (5.2-34) and (5.2-35) the 6 parameter and the So/S ratio can be presented either in tabulated form (Table 5.2-1) or by graphs (Fig. 5.2-14). Table 5.2-1. Determination of effective water table width. Relationship between actual water table width and capillary suction pertaining to saturat,ion Ratio of mtual water Ratio of effective and actual water table width saturation

d

0.00 0.01 0.02 0.04 0.06 0.08 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.92 0.94 0.96 0.98 1.00

f%lS

0 0.0028 0.0065 0.0158 0.0272 0.0404 0.0564 0.159 0.318 0.565 0.948 1.56 2.66 5.25 12.3 16.6 22.5 34.4 71.5

00

123.4 61.7 30.8 20.6 16.4 12.3 6.1 4.1 3.0 2.4 1.96 1.66 1.40 1.20 1.16 1.12 1.09 1.06 1.00

6.2 Combined application of various mapping functions

621

After determining all the auxiliary variables, substituting the [ values belonging to points b, and b, into Eq. (5.2-31) and calculating the difference between t,he w(b,) and the w(b,) vectors, the specific discharge can be calculated. By using Eq. (5.2-35) this hydraulic parameter can also be expressed as the function of the virtual width ( S o ) :

nKh,

qo=-- 2KS0. In 6

(5.2-36)

There is an important conclusion which can be drawn from this formula. At infinite depth, where the gradient is equal to unity, the area of the cross section having unit size perpendicularly to the flow plane has to be equal to 25,. Namely, t,heflow field is twice aa large at infinite depth, aa at the elevation of the water level in the canal. One point (i.e. b,) and the vertical asymptote of the bordering stream line is fixed in this way. Considering these two fixed conditions the border of the flow field can be approximated by curves, which can be easily mapped aa parallel straight lines to get a one dimensional flow pattern (Fig. 5.2-15). The first stage of mapping changes the distance between the two aaymptotes from 25, to n:

therefore (5.2-37)

622

5 Movement equations describing seepage

Fig. 5.2-15. Conformal mapping to determine the hydraulicparameters of free unsteady filtration

and

This dimensionless plane is further mapped t o transform the t w o asymptotes onto the imaginary axis of the t plane. This result can be achieved by applying the logarithmic functions z’=lnt=lne+it9; therefore z‘= In e; and (5.2-38) y’ = 6 ; where

--

e = 1 / 1 2 + s2;

8

and 6 = arc tan - . r

The basic hypothesis of the derivation is that the contour of the seepage field can be approximated by a curve, the image of which on the t plane is a straight line parallel to the vertical axis and running through the images of the points b, and b,. The ordinates of these two points in the polar system

6.2 Combined application of various mapping functions

623

. After shifting the origin to the intersection of the real axis and the vertical image of the contour,

t‘ = t therefore

Ic

- cos - ;

4

r’ = r - 0.7071;

(5.2-39)

and s’=s;

a second supposition is made. The actual contour of the canal is approximated by a curve, the image of which i s a semicircle, passing through the images of points c1 and c, and having its centre fixed to the origin. It should be noted here, that the original seepage field assumed by Verigin is enlarged to simplify further investigation.Two curvilinear triangles (with 4-11-14 and 5-12-15 as corner points) are attached to the field. This enlargement is negligibly small and does not cause appreciable differences in the results. Verigin has already pointed out the uncertainties occurring in the boundary conditions along the lines 4-11 and 5-12. It is evident that the water rises in the soil above the water level due to capillarity. It can be assumed, therefore, that the actual conditions are better characterized by the enlarged field than by the original one. Applying one more mapping function the contours of the flow field become parallel straight lines: w = In t ’ ; therefore u = lne’; and v = 6’; where e’ = = I / ( r - 0.7071)2 s2 ; (5.240) and

v

v

9’

+

S

6‘ = arc tan - = arc tan r’ r - 0.7071 Some further approximations have to be accepted using the mapped field to determine the hydraulic parameters of the unsteady free infiltration. It has to be supposed, that the form of the wetted zonedoesnot differ considerably from that of the potential line intersecting the y axis at the same point aa the wetted front at the moment of the investigation. A further assumption is that the change in the length from the canal to the wetted front as well as the decline i n the actual flow direction from the vertical are negligible. On the basis of these two hypotheses the piston flow in the mapped system can be investigated instead of the actual unsteady free seepage. Comparing the forms of experimentally determined wetted fronts to the calculated potential lines and considering the uncertainties occurring in the vicinity of the front (random character of capillary suction, uneven velocity i n the pores having different

6 Movement equations describing seepage

624

sizes, etc.) the suppositions can be accepted. Thus, the hydraulic parameters can be calculated aa the functions of the y depth reached by the wetted front at a time point t :

v = K Y+hC.

Seepage velocity

9

uy-

Velocity of propagation

K Yfhc. vefl = n uY-uA q=nK

Specific flow rate

uA

Y uy

(5.2-41)

9

+ hc - uA

The geometrical parameters of the mapped field depending on measurements of the original system are determined by the transformation functions: uA= In eX = In

1.5- 1.4142cos

u B = In & = In 0.7071 = -0.3466; uy = h e ;

[

KO

= I n exp - - -0.7071

(5.242)

1.

The relationship between the position of the wetted front and the time elapsed since the start of the process can be determined if the velocity of propagation is taken into consideration. Thus, all the hydraulic parameters can be expressed as time-dependent variables:

therefore

where 2 yo = h',-ln x

+ 0.70711;

[exp (uA)

(5.2-43)

the last parameter being the virtual depth of the canal. The second expression to be integrated can be solved directly, while the first one leads to an integral exponential function. Therefore, tabulated values, numerical solutions or the expansion into series have to be applied.

5.2 Combined application of various mapping functions

625

5.2.4 Characterization of unconfined field by the application of Zhukovsky’s function Development of the water table beside a draining trench

As already explained in Section 4.2.3 the position of the seepage line of an unconfined field can be determined not only by the hodograph mapping but also by applying other special mapping functions. The application of Zhukovsky’s mapping function in the following example demonstrates the determination of the hydraulic parameters of seepage around a trench draining the ground water. Recalling Eq. (4.249) it can be seen that the ground-water table drawn down by the trench, the free exit face and the bottom of the trench (it is supposed that the water depth is equal to zero) is mapped onto the vertical axis of the Zhukovsky’s plane, because the pressure along these contours is equal to zero at each point (Fig. 5.2-16). The hodograph plane of velocity, or that of its conjugate value, can also be easily constructed (v = v, + ivy or v* = v, - ivy plane) to determine the vertical ordinates of points 3 and 5 on 0 plane, while the parameters of points 2, 4 and 6 are known from Eq. (4.2-48)

Fig. 5.2-16. Application of Zhukovsky’s mapping to determine the hydraulic parameters around draining trenches 40

626

6 Movement equations describing seepage

(5.2-44)

P =2

+ -2b -

Applying the 1

t= V*

- iK

(5.2-45)

relationship the mapped image of the hodograph field becomes a triangle being slit at its lower peak. This form can be mapped onto the lower half iq) by applying the Schwartz-Christoffel’s of an auxiliary plane ( 5 = 5 formula:

+

After solving the integral numerically for a given a angle of the slope, the ilparameter can be calculated. Considering the identity of the 0 and the 5 planes and the corresponding ordinates of points 2 and 6 on the two planes, the e2ordinates of point 3 and 5 can be also determined: @=im[;

where (5.2-47)

and, therefore,

Combining the relationships derived here and the equations establishing contacts between the z vector of the original system, the conjugate velocity and the Zhukovsky’s potential [seeEqs (4.2-56) and (4.2-57)], the hydraulic parameters can be calculated, as analyzed by Polubarinova-Kochina (1952, 1962) for special values of the angle of the slope. References to Chapter 5.2 AVEBJANOV, S. V. (1960): Seepage Losees from Irrigation Canals (in Russian.) Wrotechnica i M d b a c i a , No. 9-10. KOVLCS,G. (1960a): Calculation of the Flow Rate of Seepage under Dam (in Hungarian). Viziigyi EozZem&nnyek, No. 2. KOVAW,G. (1960b): Design of Draining Trenches along SurfaceWaters. Part I (in Hungarian). Hidroldgiai Kodony, No. 6.

5.3 Horizontal unconfined steady seepage

627

Kovilcs, G. (1961a): Design of Draining Trenches alongsurface Waters. Part I1 and I11 (in Hungarian). Hidroldgiai Kozlony, No. 1. KovAcs, G. (1961b): Yield of Riparian Drains (in Hungarian). Hidroldgiai Kozlony, No. 4. KOVACS,G. (1962): Yield and Influence on Pressure Head of the Series of Riparian Wells (in Hungarian). Hidroldgiai Kozlony, No. 2. Kovlics, G. (1963a): Free Seepage from Irrigation Canals. V I I I . Concegno di Zdraulica. Pisa. KovAcs, G. (196313): Characterization of the Steady Influenced State of Seepage from Recharging Irrigation Canals (in Hungarian). Hidroldgiai Kozlony, No. 1. K O V ~ C G. S , (1966): Yield of Partially Penetrated Wells (in English). Symposium on Seepage and Well Hydraulics, Budapest 1966. Kovilcs, G. (1976): Interaction between Rivers and Ground-water (in English). I A H R Symposium o n Groundwater, Rapperswil, 1975. KOZENY,J. (1931): Infiltration from Rivera and Canals (in German). Wusserkruft und Wasserwirtschajt, No. 3. PAVLOVSKY, N. N. (1922): Theory of Ground-water Movement around Hydraulic Structures (in Russian). Leningrad. PAVLOVSKY, N. N. (1936a): Basis of the Solution of See age Problems Concerning Free Infiltration from Canals (in Russian). Izwestia V N I I 6 No. 19. PAVLOVSKY, N. N. (193613): Free Infiltration from a Circular Shaped Canal until I n h i t e Distance (in Russian). Izvatiu VNIIQ, No. 19. POLUBARINOVA-KOC, P. YA. (1962): Theory of Ground-water Movement (in Russian). Gostekhizdat, Moscow. P. YA. (1962): Tbeory of Ground-water Movement. POLWARINOVA-KOCHINA, Princeton University Press, Princeton. RIESENKAMPF.B. K. (1940): Ground-water Hvdraulics (in Russian). Proceedinm, - . University of Sbatov, Vol. XV. No. 6. VEDERNIKOV. V. V. (1934): Infiltration from Canals (in German). Wmserkraft und Waseerwirtschajt, NO. ii-12: VERIQIN,N. N. (1949): Infiltration of Water from the Canals of Irrigation Systems (in Russian). Dokl. Akad. Nauk SSSR, No. 4.

Chapter 5.3 Horizontal unconfined steady seepage (Dupuit’s equations and the limits of their application) In the previous chapters the application of those mapping methods, which can be used to transform the flownets of two and three dimensional seepage space into a new system, where the stream lines are straight and parallel to each other waa discussed. These methods are necessary to determine the hydraulic parameters of seepage, because the one dimensional confined flow is the only form of seepage, for which Laplace’s equation can be directly integrated in closed form, as shown in Chapter 4.2 [see Eq. (4.1-41)]. It ww also explained there and repeated once again in the introduction of Part 5 that there are several cmes in practice, when the actual seepage field (or at least a part of it) is approximated by the simple flow pattern of horizontal confined flow. Also the hydraulic parameters calculated in this way are accepted as good approximations of the real values, if the actual conditions are within the validity zones of the hypotheses applied for the derivation of the simplified relationships. 40*

628

5 Movement equations describing seepage

The use of Dupuit’s equations to describe the unconfined steady seepage was mentioned ;t9 an example of this approximation [see Eq. (4.1-44)]. Since the publication of this method (Dupuit, 1863) based on the laminar character of seepage (Darcy’s law) and on the supposition of the constancy of seepage velocity along a vertical section of the seepage space being normal to the flow direction (Dupuit’s hypothesis), this equation is indeed the most frequently applied relationship in seepage hydraulics. Its field of application wm enlarged by deriving further relationships suitable to characterize axial symmetrical flow (Thiem, 1870) and to consider the accretion of the seepage field (Hantush, 1964). It was proved recently that the hypotheses seeming at first to be very rough approximations, can be justified theoreti1973). cally as well (Charnyi, 1951; KOVBCS, Further extension of the examples investigated by using Dupuit’s equations, is ensured by dividing the seepage field into separate stretches. There are many practical problems, when the unconfined horizontal seepage is characteristic in the largest part of the field but special conditions in the vicinity of the entry and exit faces (e.g. strong curvature of the stream lines) exclude the application of Dupuit’s hypothesis for the whole system. Separating these disturbed stretches and calculating the local resistances byconsidering the actual conditions, the seepage through the remaining part of the field can be described by Dupuit’s equations. The combination of the results determined for the various stretches provides us with the required hydraulic parameters by considering the superposition of the head losses and the continuity of the flow rate. This important practical application justifies the detailed discussion of Dupuit’s method. The derivation of the basic equations, the validity zones of the applied hypotheses, the special boundary conditions to be considered and the determination of local resistance (in some cases of divided seepage fields), will be summarized in this chapter.

5.3.1 Derivation of Dupuit’s equations The basic hypotheses used for the derivation of Dupuit’s formulae are as follows: (a) The validity of Darcy’s law in the homogeneous flow field; (b) The fixed geometry of the contour lines of the two dimensional seepage field with known parameters, [horizontal lower boundary; vertical entry and exit faces, the distance between them being L; constant elevation of both the head water ( H , ) and the tail water (H,) above the impervious lower boundary; and in the case of a confined system the horizontal impervious boundary above the flow field having constant thickness ( m ) ] (Fig. 5.3-l), (c) The steady state of flow as a result of the constant boundary conditicns (Hl and H2); (d) The constant velocity along a vertical section which can be expressed in mathematical form as the product of hydraulic conductivity and the deri-

5.3 Horizontal unconfined steady seepage

629

Fig. 5.3-1. Symbols used for the derivation of Dupuit’s equations in the case of twodimensional seepage

vative of the seepage line [yo(x)]with respect to the flow direction: V(X)

= - K-,dY0

.

(5.3-1)

dx

where the seepage line (yo)indicates either the depth of the flow field (unconfined seepage) or the elevation of the piezometric line above the lower impervious boundary. Combining Eq. (5.3-1) with the equation of continuity, a differential equation for the determination of the percolating flow rate is obtained: For confined and semi-confined systems: q(x) = mw(x) = const. ; q = - K mdYO ; ax For unconfined systems q(x) = yo(x)w(x) = const.; q = - Kyo-. dY0

ax

(5.3-2)

The differential equation can easily be integrated in the case of a confined (or semi-confined)aquifer, because the boundary conditions are well determined (if x = 0; yo = H , and if x = L; yo = H 2 ) .The resulting integration is

9=

and

mK L

-w,

- H,) ;

(5.3-3)

630

5 Movement equations describing Reepage

Originally, the differential equation describing the unconfined flow waa similarly solved, using the same boundary conditions. Since the existence of the free exit face and its influence on the flow developing in an unconfined field became known, the accuracy of the boundary conditions had to be revised. It can easily be proved by hodograph mapping that the intersection of the exit face and the tail water level is theoretically a singular point. The same conclusion can be drawn from the simple physical study of the boundary conditions. Below the tail water the stream lines are normal to the exit face, the latter being a potential surface. The section of the phreatic surface with the flow plane is also a stream line having a slope determined by the boundary condition which characterizes the stream lines of constant pressure in general. If this seepage line joined the lower water level, this point would be the intersection of two stream lines and stream lines intersect each other only at singular points with infinite velocity. Therefore, the exit point (intersection of the seepage line and the exit face) cannot reach the level of the tail water, because the hydraulic gradient on the surface always haa a finite value (the slope of the water table cannot be infinite), while the intersection of the seepage line and the tail water level would require an infinite value. In a negative way this analysis proves the necessary development of the free exit face between the exit point and the tail water (Muskat, 1937). The influence of the special boundary conditions along the free exit face on the whole seepage field can be well demonstrated by comparing the potential differences along the stream lines, supposing at first the lack of the free exit face and its development in further cams (Fig. 5.3-2). The Dupuit’s hypothesis supposes not only the constant velocity along a vertical section, but involves at the same time the constant potential diflerence along euch stream line, which corresponds to the first version of the lower boundary conditions (Fig. 5.3-2a). This approximation is not acceptable because of the development of the free exit face and, therefore, the consideration of x = L,yo = H z boundary condition does not give the correct solution of the basic differential equation. If there is no tail water ( H z = 0), the potential distribution along the exit face ( y k = Kyk) is linear and the potential difference can be represented with a trapezium shaped distribution (Fig. 5.3-2b). The condition is considerably modified by the existence of the tailwater, because below its level, the potential is constant (fpk = KHz if yk < H2), while above the tail water the potential remains unchanged (fpk = K y , if yk > H z ) . The graph representing the distribution of the p t e n tial d i f l e r e m is divided, therefore, into two parts: i.e. along the exit surface contacted by the surface water the diflerence is w m t a n t , while along the free exit face the relationship between the potential diflerence and the height of the investigated point is linear (Fig. 5.3-2c). A method to calculate the specific flow rate by studying the lower boundary conditions was derived by Charnyi (1951). The symbols used in the derivation are shown in Fig. 5.3-1. The flow rate is determined by integrating the product of the horizontal seepage velocity

(

2)

vx = -- and the

6.3 Horizontal unconfined steady seepage

potential difference along flowlines related fo the entry

pofential didribufion ~Iongthe entry surf8ce

seepage field

631

poiential distri- poteniial dfference butiffn along the along flowlines exit suM8ce reiaied to ibe exit points of the latter

(cl With fail wafer and free exit surface Fig. 6.3-2. Distribution of the potential difference measured along the stream lines and represented along the entry and exit faces

thickness of an elementary horizontal layer (dy) between the impervious lower boundary and the water table: (5.34) y=o

Y-0

The velocity potential along the surface [cp = Kyo(x)]is known.Thus, the previous integral can be transformed by using the relationship between the total and the partial derivatives of defhite integrals:

0

thus

0

(5.3-5)

632

6 Movement equations describing seepage

Integrating Eq. (5.3-5) with respect to x the constant can be determined by considering the boundary condition along the entry face (5= 0; y o = H,; rpB = K H , = const.). The result is aa follows: (5.3-6) 0

The h a 1 form of the equation derived to calculate the specific flow rate can be achieved by substituting the lower boundary condition (x = L; y o = H 3 ) , where the integral of the potential haa to be divided into two parts expressing the difference of the potential distribution below and above the tail water level (see Fig. 5.3-2c):

r 7 . r 0

~ ( ydy) = KHZ dy 0

K (Hi - Hi) . +H,Ky dy = 2

(5.3-7)

Substituting this relationship into Eq. (5.3-6), the surprising result is that the theoretically correct relationship to calculate the flow rate is identical to Dupuit’s original equation determined by the uncorrect consideration of the lower boundary condition :

K q =-(H, 2L

- H,).

(5.3-8)

This virtual contradiction can be explained by the fact that the two approximations applied in Dupuit’s derivation (the supposition of the constancy of the potential difference along each stream line and the substitution of the depth of the tail water instead of the height of the exit point) compensate one another. It is quite evident that the position of the seepage line cannot be determined by using this incorrect substitution of the lower boundary condition, because the water table has to intersect the exit face at the height of the exit point, and not at the level of the tail water. Considering only the geometrical conditions (i.e. the water table has to be fitted to the x = 0, y o = H , and x = L, yo = H 3 points) and neglecting the hydraulic interpretation of the boundary conditions, the differential equation given for unconfhed systems [Eq. (5.3-2)] can be solved by expressing y o as the function of the horizontal distance from the entry face yo =

vm).

(5.3-9)

Dupuit’s parabola is, however, only a very rough estimation of the actual position of the water table. Analyzing the unconfined seepage field between vertical entry and exit faces and the hodograph fields of both the gradient and the reciprocal value of its conjugate, as well as the image on the complex

5.3 Horizontal unconfined steady seepage

633

potential plane (Fig. 5.3-3), i t can be seen that the parabola does not satisfy the boundary conditions at the entry and the exit faces (it is not normal to the entry face and i t does not join tangentially the exit face at the exit point). There is, however, an internal part of the field (indicated by hatching) exluding the vicinity of the closing faces, where the proportionality 1

between the fields on the complex potential and the - planes may be

I*

assumed aa an approximation: w-KHj

Q

K

K .

V*

v;x

- ---

’ (5.3-10)

and considering Eq. (5.1-14) 1 H j l w - - - -- -dz _--dw v* P v;x K9

After substituting the corresponding values belonging to the border of the stretches supposed to be proportional, the solution of this differential equation gives a relationship similar to Dupuit’s curve. This result indicates that the p = ky, boundary condition is approximately satisfied along the internal &retch of the parabola.

( w plane)

( ~ p l m ?) Fig. 6.3-3. The unconfined seepage field between vertical entry and exit, faces and its images on the hodograph and complex potential planes

634

6 Movement equations describing seepage

Using Hamal’s mapping function an exact mathematical relationship can be derived to determine the position of the water table (Muskat, 1937).This method is, however, very complicated and, therefore, the use of Dupuit’s equation by substituting H , as the lower boundary condition [Eq. (5.3-9)] is generally accepted in practice. It is necessary, however, to remember the uncertainties of this approximation, and to apply more exact methods for the determination of the seepage line, if its position is critical from the point of view of the investigation. The same uncertainties justify the use of other simple geometrical approximations [e.g. the substitution of the seepage line by an ellipse passing through the x = 0, yo = H , and 2 = L, yo = H , points and having a vertical minor axis ( b = H 1 - H,) fitted to the x = 0 coordinates axis, also a horizontal major axis ( a = L ) at the elevation of the exit point, whose curve satisfies the required conditions at the two closing faces]. The acceptability of the very rough approximation of the seepage line can be justified by considering the relatively small differences, which may occur between two smooth curves passing through two fixed points (i.e. x = 0, yo = H , and x = L, yo = H,). A further factor supporting this hypothesis is that the position of the water table has practical importance only in the vicinity of the exit face (here i t influences the stability of the solid matrix) and, therefore, the height of the exit point is the parameter, which has to be determined very carefully. There are two theoretically based methods published in the literature for the calculation of the exit height. Polubarinova-Kochina (1952, 1962) gives mathematically correct derivation, in which the acceptance of Dupuit’s flow rate is a basic hypothesis, while the Hungarian method (Kov&cs,1973) uses some approximations, but provides a general relationship between the flow rate and the exit height using the fixed geometrical parameters of the field as independent variables. The comparison of the two methods proves their satisfactory accuracy and also gives important information about the character of seepage. head water ,,; .................................. ................................... . . . . ...... . . . ..... ..: . .:.... . . . .;.:.:. .: ......: T I . .. . :..:..,.; ....:. .

\ . .‘.:.I:

exit joint 7 -~ -

Fig. 6.3-4. Mapping of the flow field by wing Polubarinova-Koohma’s method

635

6.3 Horizontal unconfined steady seepage

Poluburinova-Kochina’s derivation is based on the analytic theory of linear differential equations. This highly sophisticated mathematical method is used t o map the curvilinear polygon surrounding the seepage field into the upper half of an auxiliary plane (Fig. 5.3-4). It has, therefore, many similarities to Schwartz-Christoffel’s mapping, which solves the same transformation in the case of polygons with straight sides. The angles formed by the stretches of the contours of the seepage field a t the corner points are an essential feature. It is necessary, therefore, to analyze the hydraulic and geometrical conditions determining the internal angles of the field, as in the case of conformal mapping. Because the relationships are more complicated in the case of a curvilinear polygon than those achieved by mapping tt field bordered by straight lines, hypergeometrical functions have to be applied instead of the elliptical integrals characteristic to the solution of Schwartz-Christoffel’s mapping. The general form of the hypergeometrical functions is a8 follows:

P( a, b, c, 2) = 1

+ ab + a(a-1.2+- 1)c(cb(b++1) 1) --2 C

22

+

. ..

(5.3-11)

By using various auxiliary functions, going through a very complicated chain of mathematical relationships and applying the results of hodograph mapping, the geometrical parameters and the specific flow rate through the various sections of the seepage field can be expressed by a series of integral equations: b 1

H2

H,

I

=A b

- H2 = A

i -

CLdC

5 CV(5 - 1) ( 5 - a ) ( 5 - b ) C- 1 (1 -

1-C

=AJ,;

d5

5) Y(a - 5 ) ( b - 5 )

-=A J4 ;

(5.3-1 2)

636

6 Movement equations describing seepage

To solve thifi system, the consideration of a further hydraulic condition is necessary. As dready mentioned, the validity of Dupuit's flow rate [Eq. (5.3-8)] is assunzed as a supplementary hypothsis. The relationship can also be expressed by the integral equations listed earlier: 2 JIJ, = Jt - 4 . (5.3-13) corresponding symbols in PafubariflovaKochina's orgina/ publicafton and in fnis texf

:

10

3 Hf

015

0

-

4

Fig. 5.3-5. Graphical representation of Polubarinova-Kochina's resultascharacterizing the height of t,he exit point

5.3 Horizontal unconfined steady seepage

637

From Eqs (5.3-12) and (5.3-13) the general solution can be achieved by using numerical methods after expanding the relationships to be integrated into series. The final results are given in the form of graphs (Fig. 5.3-5). To ensure the unified discussion of the various problems the symbols and the quantities here differ from those applied by Polubarinova-Kochina in the original derivation. Figure 5.3-5 represents, therefore, the height of the exit point as the function of the geometrical parameters of the seepage field in both systems, and also lists the corresponding symbols. The original description emphasized that, if the seepage field is not influenced by tail water ( H , = 0 ) , the reliability of the relationships is higher than that of the other curves characterizing the hydraulic parameters of seepage under the influence of tail water of various depths. This is because in the latter case the calculation requires the use of successive iteration. Apart from this difference there are special caaes, when substituting the geometrical parameters of the flow field simplifies the equations: e.g. in the case of a very large relative length (if L } H , tends to infinity) both the height of the free exit face ( H , - H,) and the specific flow rate percolating through this section [(J(~,-~,)],can be expressed as functions of the same constant. Thus, a relationship can be established between the two variables: and (5.3-14)

consequently q(H,-H,)

= 1.3469 K(H3 - H,) ;

where 1.8319 = 2G and G is Catalan's constant. If the entire height of the exit face is in contact with the air (there is no tail water, H , = 0 ) Eq. (5.3-14) gives a direct relationship between the total specific flow rate and the height of the exit point. Thus, the latter can also be expressed as a function of known geometrical parameters:

consequently

[2l0

(5.3-1 5 )

H = 0.371 -2; L

where the subscript 0 indicates, the parameter H , = 0. As already mentioned the baaic hypothesis of this derivation is the aasumption that Dupuit's flow rate according to Eq. (5.3-8) is valid. A further approximation applied in determining Eqs (5.3-14) and (5.3-15) is the supposition that the seepage field is of infinite length. Polubarinova-Kochina has found that the hyperbolic relationship between the relative height of the exit point ( H 3 / H 1 )and the relative length of the field ( L / H , ) is valid, if the latter is greater than 1.5. Comparing the results with those of the other theoretical method and to measured data the validity zone can even be

638

5 Movement equations describing seepage

extended and Eq. (5.3-15) can be accepted as a good approximtion in the zone LIH, 2 1. The mathematical basis of the other method is conformal m p p i n g of the seepage field assuming the position of the water table to be approximated by Dupuit’s parabola [Eq. (5.3-9)]. After a suitable shifting of the origin of the coordinate system represented in Fig. 5.3-1, the new vertical axis intersects the focus of Dupuit’s parabola. The equation of the seepage line in the new system is (Fig. 5.3-6): yo=A Bz; where

+

B=

- H2

a= 2p;

(5.3-1 6)

Fig. 5.3-6. Conformal mapping of the flow field of unconfined horizontal seepage

639

5.3 Horizontal unconfined steady seepage

and p=--.

H:

-

Hi

2L Using an auxiliary pkane c(5, q), whose contact with the origiiial system is expressed by a well known analytic mapping function,

consequently (5.3-17)

and and the image of the water table becomes a straight line parallel to the re a axis. Its distance from the latter is:

;i!V! - H 2 = va.

qo = const. =

(5.3-18) 4L A further approximation in this method is the hypothesis, that the series of curve8 intersecting the parabola at its maximum point ( P o )and covering the whole field between the water table and the impervious lower boundary, can be regarded as a system of stream lines independently of the position of the entry and exit faces. These lines are not perpendicular to the vertical closing faces, although the whole entry face and the exit face below the tailwater level are potential surfaces, which have, theoretically to he intersected by the stream lines at right angles. The images of the curves, supposed to be the approximative stream lines compose a series of almost parallel curves from infinity on the 5( 5 , q) plane. They are bent upwards near the imaginary axis and intersect it at the point Zo(O, qo). This system is identical to a strip of the flownet characterizing the infinite series of sources (or sinks) located along the imaginary axis at points Zol(O, qo);Z,(O, -q,,); Z,,(O, 3q0); Zo,(O, - 3 q 0 ) ; . . at a distance of 2n, from each other. This svstem of stream lines can be transformed into parallel straight lines by appliing the following mapping functions [see also Eq.(5.2-29)]:

.

."

w =u

+ i v = lnsh-([x

consequently and

- iqo);

2rlO

u=-ln-[ 2

2

-E

ch

) +cos [

x- 7 ~ O q ~ . ) ] ;

(5.3-19) x

tan -(rl - 7 0 ) v = arctan

270

9

~

JC

t h -5 2rlO when the image of point P o is transferred to infinity.

640

5 Movement equations describing seepage

By this form of mapping the original seepage field is transformed into a series of elementary lamellae having a thickness of Av. The length of these lamellae depends on their distance from the horizontal axis of the w(u, v) plane : Au(v) = u ~ ( v) uK(v). (5.3-20) Knowing the potential diflerence along each lamellae (which depends on the position of the corresponding stream line in the original field, im explained in Fig. 5.3-2) it can also be expressed as the function of the imaginary ordinate of the I I ~plane and

Avl = K ( H , - H,) = const. ; if 0 5 Y k dv2 = K ( H l - Y k ) = & ( V ) ; if H ,

H2 ;

(5.3-2 1)

5 Y k 2 H3.

The water conveyance through a lamella can be calculated from the listed parameters in both zones and summarizing these quantities, the total specific flow rate is achieved:

and dq, = K

- Y k ( v ) d v ; if H , 5 yk

H, ;

(5.3-22)

fwv) consequently

where the limits vl, v, and v3 can be determined from the mapping functions, substituting yk = 0, yk = H , and Y k = H , values respectively: 7c

vl=

- -;

2

v2

= f ( H 2 ); and

w3 = 0.

( 5.3-23)

If the seepage field is not influenced by tail water ( H , = 0 ) Eq. (5.3-22) can be simplified: 0

(5.3-24)

When solving the integrals in Eqs (5.3-22) and (5.3-24), i t has t o be considered that both Y k and Au have t o be substituted as the functions of the v imaginary variable. The result gives the specific flow rate depending on four

641

6.3 Horizontal unconfined steady seepage

geometrical parameters or, using dimensionless quantities, i t can be expressed as function of three independent ratios of the four parameters: or

(5.3-25)

Among the four types of geometrical data (i.e. length, depth of the head and tail water, and height of the exit point) only three are known a priori ( H l , H2,L). Eg.(5.3.-25) does not give, therefore, a single-valued solution but a series of interrelated q and H , values. Thus rating curves can be constructed

El

showing the relationship between the relative exit height - and the flow rate. It is advisable that this latter value should be related to Dupuit’s flow rate calculated from Eq. (5.3-8), to get a dimensionless quantity independent of hydraulic conductivity. If a seepage field which is not influenced by tail water is investigated and the relative length of the field is used as a parameter, one set of rating curves can be determined (Fig. 5.3-7).

qL7 Fig. 6.3-7. Relationship between the relative height of the exit point and the dimensionless charaoteristicsof flow rate if the field is not intluenced by tail water 41

642

6 Movement equations describing seepage

In general, there are two variables to be considered (LIH, and H,/H,) apart from the interrelated values of H $ H , and q / q D . The rating curves belonging to different given values of LIH, can be represented using H,IH, aa the parameter (Fig. 5.3-8). The result of this method can be compared with Polubarinova-Kochina’s derivation only if the same hydraulic hypothesis is used i.e. the water conveyance is equal to Dupuit’s flow rate. Applying this supposition the exit height belonging to various LIH, and H J H , parameters can be determined from the graphs in Figs 5.3-7 and 5.3-8 aa the intersections of the curves and the vertical line belonging to the q / q D = 1 abscissa. These data are plotted in Fig. 5.3-5 and show good agreement between the two methods, and thus the reliability of both derivations. A further important conclusion can be drawn. Although Churnyi’s theoretical analysis has proved that the correct specific flow rate can be calculated from Eq.(5.3-8), the rating curves show that the field could transport higher amounts of water. The possible increase in flow rate is relatively small.

‘:r %? 0.4

0.2

0

fl!

h? 0.8

0.6 0.4 0.2

0 Fig. 6.3-8. Relationship between the relative height of the exit point and the dimensionleea characteristics of flow rate if the field is influenced by tail water

643

6.3 Horizontal unconfined steady seepage

The difference between the absolute maximum belonging to the infinite relative length (resulting H d H , = 0.5 relative exit height) and Dupuit’s value is only 12.5%, but the least increase in the discharge causes considerable rise of the exit point, the rating curve being very flat. Considering the principle of “lex minimi”(which is a basic law of theoretical physics) i t is expected that the movement oould reach a steady condition only in a system having a given energy content (represented by the difference between the levels of head and tail water), if the maximum possible flow rate were transported. Combining this principle with the rating curves, the probable flow rate and the exit height can be determined &B the functions of LIH, and H,IH, by determining the position of the maximum points of the rating curves.

Figure 5.3-9 shows the points representing both the maximum possible flow rate and the exit height belonging to this value, aa the functions of the relative length of the field, supposing that the field is not influenced by tail water ( H , = 0). The equations presented in the figure, mathematically describe the curves through the points in the range LIH, > 1. A third curve H also constructed in the figure shows the interrelation between -2and L / H , Hl on the baaia of Polubarinova-Kochina’s investigation [Eq. (5.3-15)] compared to that of the points determined at the intersection of the rating curves and the q/q, = 1 ordinate in Fig. 5.3-7 (a dotted line passes through these points). The first conclusion drawn from the figure is the accurate correlation between the results of the two methods, as shown in Fig. 5.3-5.

0

2.5

50

z5 % 8

Fig. 5.3-9. Representation of the maximum possible flow rate and the exit height belonging to this value and to Dupuit’s flow rate, if the seepage is not influenced by tail water 41*

644

5 Movement equations describing seepage

It can also be seen, that the ratio of qma&D does not differ considerably from unity, while the position of the exit point belonging to the maximum water transport is much higher than that determined by taking Dupuit’s flow rate into account. There is an interesting question to be answered in connection with this analysis: i.e. why the flow rate determined by Charnyi’s theoretically correct derivation is smaller than that calculated on the basis of the principle of lex minimi”. The approximations applied to construct the rating curves (especially the use of Dupuit’s parabola) can cause some discrepancy but it may be neglected since the results achieved in this way are similar to those derived from the hypergeometrical functions. There is another possible reason: a part of the available energy i s consumed by resistances not considered in the investigation and, therefore, the water transport haa to be lower than the maximum water conveying capacity of the field. Some interesting aspects in connection with this problem can be made clearer by analyzing the development of the exit velocity. The hydraulic model described in Eq. (5.3-22) can be used to calculate the exit velocity by dividing the elementary flow rate by the thickness of the corresponding lamella measured on the original seepage field: 66

and

(5.3-26)

where v k is the exit velocity (the subscript is used to make a distinction between velocity and the imaginary ordinate of the w plane). The symbol du(yk) is used to indicate that the du value has now to be expressed aa the function of the height of the investigated point along the exit face of the dv original seepage field and- is the differential quotient of the imaginary du part of z calculated on the &is of the mapping functions. The velocity distribution calculated by using Eq. (5.3-26) can be represented in the form of graphs. An example is shown in Fig. 5.3-10. In this case the relative length of the field is LIH, = 2 , each graph representing the exit velocity belonging to a given value of H31H, using H d H , as a parameter. Hence, i t is evident that the highest velocity develops when seepage is not influenced by tail water. The distribution above the surface of the tail bater does not change, as in the non-influenced caae, while below this level the velocity is decreased. The numerical values do not indicate the theoretically expected increase i n velocity in the vicinity of the singular point. The velocity distribution can also be characterized by the maximum, mean, and minimum velocity values. The mean can be calculated as the ratio of the flow rate and the exit height:

645

5.3 Horizontal unconfined steady seepage

Thus, the dimensionless parameter of the mean exit velocity

[

]

6: L H, ‘ H be expressed depending on the - , -, -and3- ratios. The curves in Fig. q

qD

Hl

Hl

Hl

5.3-11 represent this relationship if H , = 0 and are calculated from the corresponding values determined by the rating curves in Fig. 5.3-7. Two special

cams are emphasized in the figure, namely, those belonging t o Dupuit’s flow rate and the mean exit velocities belonging t o the pomible maximum

=O K

Fig. 6.3-10. Velocity distribution along the exit face calculated from Eq. (6.3-26)

646

6 Movement equations describing seepago QQQQQb

eKl.+jw-<

\

Q

%.

Q

A

c;

cs

H3

1.0

-

0.9 -

0.8 0.7 -

mean vWoclfj in tbe case of tbe conveyame of Uupuit ‘s discbarge

C.6 0.5 -

0.4 0.3 -

0.2 0.t -

0 Y

I

I

1

I

I

0.5

/. 0

1.5

20

2.5

‘k mean K Fig. 6.3-11. Mean exit velocity as the function of relative exit height and field length mUming H I = 0

water transport. It is interesting to note that in the former case (q = q D ) the mean velocity is practically constant if LIH, > 1. This result can also be derived mathematically from Eq. (5.3-15):

Another possible summarization of the calculated velocity data, is the construction of graphs representing the minimum (developing at the exit point) and the maximum (at the level of the lower impervious boundary if H2 = 0) velocity values as the functions of the relative height of the exit

13

point - and using LIH,

&B

a parameter (Fig. 5.3-12).

The curves characterizing the minimum exit velocity have a vertical enveloping line, which indicates that in the probable zone of the exit point, this parameter is constant, independent of any geometrical parameter

-

v k mln (5.3-29) 0.55. K On the curves constructed to show the maximum exit velocity, the points belonging to Polubarinova-Kochina’s exit height (Eq. 5.3-15) can be

determined. These points indicate a practically constant maximum exit velocity. If the same exit height is characterized by the intersections of the rating curve8 with the q / q D = 1 ordinate in Fig. 5.3-7 and the maximum

6.3 Horizontal unconfined steady seepage

647

Fig. 6.3-12. Maximum and minimum exit velocity as the functions of relative exit height and field length assuming H z = 0

exit velocity is determined as the function of this height, the numerical values of the maximum velocity do not differ considerably from the previous data, the points being scattered along the vertical ordinate determined by Polubarinova-Kochina’s method. Thus, this parameter can also be characterized by a constant, independent of the geometry of the field: v k max

--2.5.

K

(5.3-30)

The interesting result of this analysis is that in the case of seepage not influenced by tail water all parameters (mean, maximum and minimum values) of the ezit velocity are constant. They do not depend on the relative length of the field, or on the absolute depth of the head water, if i t is assumed that Dupuit’s flow rate is the correct water transporting capacity of the system, and the exit height is determined by Eq. (5.3-15). The contradiction between this result and the potential theory, can be explained by the development of,?tonlaminarflow.In the vicinity of points which ought to be singular points according to the potential theory, the velocity would be high, approaching an infinite value and in any case, higher than indicating the upper limit of the validity of Darcy’s law. The laminar character of flow is a basic principle of potential theory and, therefore, i t can be supposed that in this zone of the field, seepage is not a potential movement and the resistance is higher than that calculated from the equations generally applied. The seepage field can be regarded as a selfregulating system, in which the increase in velocity raises the resistance, and this process hinders the further rise in velocity. There exists, therefore, an upper limit of velocity, and the height of the exit point develops at the elevation to which this limit value belongs, as a maximum exit velocity. This explains why part of the energy

648

5 Movement equations describing seepage

has to be consumed to overcome the resistance occurring above that calculated a8 the retarding effect against potential seepage. It is reasonable, therefore, that the flow rate is smaller than the maximum possible water transporting capacity of the field. Thus, Dupuit’s discharge is not only a flow rate, the validity of which can be proved by potential theory, but the exit point hawing this value also has a special role, viz. the maximum velocity developing i n this m e is equal to the upper limit, the existence of which was explained previously.

5.3.2 The influence of the capillary water conveyance The investigation of the hodograph of an unconiined.seepage field, clearly indicates that the development of the capillary fringe above the gravitational field considerably changes the boundary d i t i o n s of the latter (see Fig. 4.2-17), and thereby influences seepage i n the whole field. Eq. (5.3-8) gives only the flow rate percolating through the gravitational part of the system. The water transported in the capillary zone has to be added to this value to get the total flow rate. The two amounts (i.e. the gravitational and the capillary discharge) are not idependent of one another, because, at the entry and exit faces, the total flow rate crosses the sections of the gravitational field. Water cannot enter into the porous medium above the level of the head water, or exit into free air from the capillary zone having lower pressure than atmospheric. Thus the capillary fringe is recharged and drained in the form of water exchange between the two zones of the seepage field (Fig. 5.3-13). As an approximation, it can be supposed that the regular error caused by considering the capillary wafer tramport independently, is negligible (KOV&CS, 1973). The capillary fringe can be regarded, therefore, as a closed, nearly horizontal tube, and its water conveyance can be calculated as the product of the cross-sectional area and seepage velocity. This is equal to the capillary height since a unit width of the flow space of the two-dimensional seepage is investigated (A=h,) multiplied by the seepage velocity. It can also be supposed that the total potential is proportional to the difference in eleva-

Fig. 5.3-13. Water transport through the capillary zone

649

5.3 Horizontal unconfined steady seepage

tion of the lowest points of the capillary exposed faces Thus, the capillary flow rate is

['p

= K(H, -H,)]. (5.3-31)

Accepting the supposition of independent capillary and gravitational water conveyance, the total amount transported through the field is equal to the sum of flow rates calculated from Eqs 5.3-8 and 5.3-31:

H2 = 0

or, if

(5.3-32)

Hyl+--' I:]-

((&=K--

1--

2L

The possible range of the factor by which the parameter is multiplied can easily be limited. Its highest value occurs if there is no tail water and in this case 1--< H3 1;

(5.3-33)

HI while in general the following inequality has to be considered: HI > H 2 , and, therefore, if H2 H 3 H I ---f

1--

>H3 >

--t

a 3

HI

+ 0.5.

(5.3-34)

Because the water cannot leave the capillary zone through the capillary exposed exit face, it has to percolate downwarrds into the gravitational field. Thus, the exit velocity would be increased by the capillary water transport if the exit height remained unchanged. On the basis of the explanation given in the previous section in connection with the existence of an upper limit of the exit velocity [Eqs (5.3-28), (5.3-29) and (5.3-30)] it is expected that the exit point would have to be raised by the capillary water conveyance to keep the exit velocity below this limit. Experiments executed to measure the height of the exit point (Muskat, 1937; Polubarinova-Kochina, 1952, 1962; Maione and Franzetti, 1969; Ching-Ton Kuo et al.,1969; Kov&cs,1973) have proved that the exit points are always at higher levels than those calculated from Eq. (5.3-15) in the case of the absence of tail water (but below

650

5 Movement equations describing seepage

the maximum possible height) and the difference becomes greater aa the rate of the capillary water transport increases (Fig. 5.3-14). The measured values are suitable to check the reliability of the theories of both the determination of the rating curves showing the flow rate vs. exit height relationship (Figs 5.3-7 and 5.3-8) and the calculation of the capillary flow rate [Eq. (5.3-31)].

Fig. 5.3-14. Comparison of the measured and calculated values of the exit height

All the characteristics of 60 measurements executed without and 43 with tail water influence were carefully determined ( H I ,H,, H,, L, h, and 9).

The frequency distributions for both the 919, and the

'

-C ' values

were cal-

qD

culated separately for both sets (Fig. 5.3-15). The mean of the original data,

I:([

uninfluenced by tail water shows some difference from unity

I

= 1.041

[r 2 I

o mean = while after correction a normal distribution around unity

ot i a n

1

= 1.004 indicates only the influence of errors in measure-

ment. If the exit face is partly covered by tail water the original data have no larger discrepancy from unity

3

= 1.007 than the corrected

651

5.3 Horizontal unconfined steady seepage

number of dafa

distributfuff of fbe measured dafa .

-

--

... ...

-----

without fail water with &il wafer complete set of dafa

distribution of the corrected dafa .............. witbuuf faif water wifb tail water complete set of dafa

-----

600

b 02

106

108

1/0

q and q - %. values Fig. 5.3-15. Frequency distribution of qD

qD

l , [i

(I - QC

= 0.993]. The capillary reduction, however, mean values decreases the variance in the measured data and thus the more accurate correlation justifies the correction. The accordance of the mean calculated

- Qc by combining the two corrected sets with unity, is excellent Q=o.gg~]. qD Imean

[(

Checking the theory by measurements can be continued by comparing the actual exit height with the calculated quantities determined a s the ordinates of the relezwat rating curve belonging to the measured qfqDvalues. This control

652

5 Movement equations describing seepage

proves the reliability of the rating curves [ q / q D = f(H$H1; H,IH,; LIH,)] determined by using the mapping method. The AH$Aq=

H3

- (H3)D

Q ~- Qn ratios calculated from the measured data were practically identical with those indicated by Fig. 5.3-7 and 5.3-8 [the symbol ( H J Dmeans the exit height belonging to Dupuit’s flow rate]. The conclusions drawn from this analysis are as follows:

(a) Dupuit’s flow rate is acceptable aa an accurate value characterizing the water transport through the gravitational seepage field ; (b) The total discharge can be calculated aa the sum of the gravitational and capillary water conveyance, and the latter may be determined assuming the complete independence of the capillary zone of the gravitational field [Eq. (5.3-32)]; (c) The influence of capillarity is relatively smaller in the presence of tail water, because the rise in the exit point decreases the gradient in the capillary zone; (d) The actual height of the exit point has to be calculated by considering the total flow rate and using the rating curves in Fig. 5.3-7 and 5.3-8. The relationships given previously ensure the complete characterization of horizontal unconfined seepage through a seepage field with vertical faces, even in the case of considerable capillary influence. I n practice however, Eq. (5.3-32) cannot be applied directly because the H 3 variable is not a n a priori known basic parameter, but i t haa to be determined as a function of the flow rate. Successive approximation would be necessary for the simultaneous calculation of the two unknown values ( H 3 and 9). To avoide this laborious process, a further approximation haa to be applied, which results in a method convenient for the computation of both the capillary water transport and the position of the exit point modified by this surplus flow rate, if the seepage is not influenced by tail water. Substituting the relative height of the exit point from Eq. (5.3-15) into Eq. (5.3-32) (supposing that the influence of capillarity on the position of the exit point is negligible) the first approximative value of the capillary flow rate can be calculated:

(Ej0=2( --‘ 1 - - H3 -

L

H,)

This equation indicates a linear relationship between the capillary discharge and the relative length of the field. The actual height of the exit point is greater than that determined from Eq. (5.3-15)and the capillary water transport is, therefore, smaller than this calculated value. It can be presumed, however, that the linear relationship remains valid in the following general form :

L

L

(5.3-36)

653

5.3 Horizontal unconfined steady seepage

In the case of the first approximation, both the tangent to this line and its intersection with the vertical axis of the coordinate system are known values calculated theoretically ( m = 1 and b = -0.371). I n general, these parameters have to be determined from the measured data, after representPC

-

L

ing the experimental results in a system having - -and -as ordinates. qD

2hc

The graphical representation of the measurements proves that the linear relationship is an acceptable approximation for each series (Fig. 5.3-16). The scattering of the b additive constant is relatively small and irregular, while the tangent to the lines shows a definite contact with the parameter of 2nc -,

L

as indicated in the figure. The rn and b parameters were calculated by applying regression analysis (Table 5.3-1). The coeEcient of correlation ( r > 0.999) indicates that the linear relationship is almost a functional contact. The intersections of the straight lines with the vertical and the horizontal axes ( b and blm respectively) show a scatter from -0.221 to -0.376 and

Fig. 6.3-16. Relationship between the5!! qD

L L and - parameters Hl

654

6 Movement equations describing seepage

Table 6.3-1. Results of the regremion analysis concerning the relationship between capillary height and the position of the exit point Symbol of the

serieaof measurements

1

m

0.873 0.834 0.818 0.910 0.898 0.838 0.960 0.916

1

b

-0.286 -0.239 -0.298 -0.221 -0.207 -0.369 -0.376 -0.320

1

0.9992 0.9995 0.9995 0.9997 0.9999 0.9982 0.9999 0.9999

0.010 0.012 0.016 0.023 0.017 0.026 0.035 0.021

0.328 0.275 0.364 0.243 0.231 0.428 0.392 0.360

0.0196 0.0218 0.0330 0.0100 0.0111 0.0240 0.0047 0.0062

from $0.231 to $0.428. Although this variance is relatively high, its influence does not appreciably affect the position of the line, because the

L > 1.0 and the HI

application of the whole method is limited to the range-

horizontal discrepancy caused by the variance of b is negligible. This fact as well as the irregular scattering independent of the capillary height and probably caused by errors of measurement, support the hypothesis that the intersection of the line and the horizontal axis can be considered as a constant equal to the theoretical value blm = 0.371. For the calculation of the m parameter, the following empirical relation(5.3-37)

Substituting these values into Eq. (5.3-36), a relationship can be determined for the calculation of the capillary flow rate. The equation contains only a priori known independent variables [i.e. geometrical ( L ,H I ) and physical ( K ,hc) parameters] and therefore, it can be directly applied to design :

consequently

(5.3-38) 0.05

L Combining Eqs (5.3-32) and (5.3-38), a direct relationship can be determined between the relative length of the field and the height of the exit point, considering the capillary influence as well:

6.3 Horizontal unconfined steady seepage

\

Fig. 6.3-17. Comparison of measured and calculated exit heights if the field is not influenced by tail water, and the capillary water transport is not negligible

655

656

6 Movement equations describing seepage

Fig. 5.3-18 . Measured exit heights aa the functions of the relative length of the field in the case of seepage influenced by tail water

This equation is the modified form of Polubarinova-Kochina's hyperbola, supplemented with the egect of capillary water transport. Its reliability can be proved by comparing the measured and calculated values, which exhibit conformity in the range LIH, > 1.0 (Fig. 5.3-17). Analyzing the influence of capillary water transport on the exit height in seepage fields influenced by tail water, the measured data can be com-

5.3 Horizontal unconfined steady seep8ge

c

i

4

k

8

in

P

14

657

L I H ~10

pared with the curves calculated from Eq.(5.3-39)(Fig. 5.3-18). As already mentioned, the egect of capillarity is not so prominent if a part of the exit face is covered by water, and, therefore, the establishment of a separate method for the characterization of this condition is unnecessary. Instead of further investigation, the division of the total height of the exit point into t w o parts is proposed: (a) The height calculated from Eq.(5.3-39) neglecting the effect of the tail water; (b) The surplus height caused by the presence of the tail water. The second part (5.340)

Hl 42

Hl

6 Movement equatione dmoribing eeepage

Fig. 6.3-19. Empirical approximation of A -Ha values HI

659

6.3 Horizontal unconfined steady seepage

H l and 2 H by using an empirical forcan be approximated aa a function of -

L L mula determined from measured data (Fig. 5.3-19). It is quite evident that H$Hl has to tend to unity, when H l / L -tends to H,IL. To unify the data i t H

is reasonable, therefore, to relate the d3-

H,

value to the difference between

values belonging to the Hl/L = H,/L variable,

Hl

and express this ratio as the function of the difference of

1% $1 -

. After

representing the calculated points in the form of graphs, it was found that - as paramthe points queue along curves characterized by different values eters, while if the independent variable is divided by

r2,

the points

indicate only one curve with a negligible scattering. The jinal form of the empirical formula determined in this way is aa follows:

y =a

+ exp[(2.7 + b ) x ] ;

where

Y=

2-[2Io

., x =

-[(2IolH,-

L

and ’

H l- -H ,

L

L .

H.

L

a = f0.03; b = 50.4.

(5.342)

This relationship can be controlled theoretically by comparing the curves calculated from Eq. (5.3-42) with Polubarinova-Kochina’s graphs (see Fig. 5.3-5), or with data determined by using the rating curves in Fig. 5.3-8. The capillary influence has to be neglected in this comparison, because it is not considered by Polubarinova-Kochina. Hence, this comparison serves only to show the reliability of the determination of the second separated part of the exit height. Figure 5.3-20 shows the final result of the comparison, indicating that the empirical formula agrees not only with the measured data but with the theoretical methods as well. It can also be seen in the figure that the mean 42*

660

5 Movement equations describing seepage

of the strip covered by the measured points does not describe the relationship best fitted to the theoretical results, but a curve running a little bit higher is more suitable for general characterization. The final form of the equation proposed to calculate the exit height influenced by both tail water and capillarity is, therefore, as follows: 2.5 L

L

wHb ihe pmumed "...... 1 . Lcalculated . ewation

0

+

L

values ra/cufated fbeorefim,@ using Pufubarihow -KOfcbha's neibod, and by conformal mapping

Fiy. 6.3-20. Comparison of empirical and theoretical values of the exit height,assuming the seepage field is influenced by tail water

661

5.3 Horizontal unconfined steady seepage

5.3.3 Characterization of horizontal unconfined seepage infI uenced by accretion

As already explained in connection with the kinematic classification of seepage, a dynamic equilibrium can develop in a semi-infinite unconfined field only, if the water crossing the starting section of the field (recharging the field through the entry face, or drained by the surface water at the exit face) is balanced by the accretion of the ground water along the flow space (Section 4.1.3). This type of seepage field (generally termed a leaking aquifer) is the characteristic case, when the width of the influenced zone has a physical explanation: i.e. there is a limit to which the influence of the surface-water (either recharge or drainage) may extend to achieve a balanced state between the horizontal flow and the sum of the vertical accretion. The equations describing the relationships between the hydraulic parameters of horizontal unconfined seepage have to be further developed by supplementing them with the influence of accretion. Dupuit’s modified and extended formulae are suitable to calculate the time invariant flow rate as the function of the distance meamred from the starting section [q(z)] and to determine the width of the influenced zone (L).The flow equation baaed on Dupuit’s hypothesis remains unchanged [Eq. (5.3-l)]. I n this case, the equation of continuity states that the change of flow rate along an elementary horizontal length (dz)is equal to the accretion [ ~ ( z )prevailing ] within the same dx distance [see Eq. (4.2-20) and Figs 4.1-10 and 4.2-6)]:

dgo = E ( 2 ) ; dz

consequently X

Q(4 = Qo - J I 4 4 1 dx = A 4 4 = y o ( 4 4 4 ;

(5.3-44)

0

because in the cam of two-dimensional unconfined field the area of the cross section ( A ) is equal to the y o depth of the ground-water flow. The equation can also be applied to characterize serni-wnfined seepage. In this special case the area has to be substituted by the thickness of the pervious layer, which may be approximated with a constant value ( m ) ,or its change along the field m ( z ) , may also be considered: X

0 4 = 90 - J 1 “(4I dz = m ( 4 4 4 .

(5.3-45)

0

Knowing the numerical value of accretion as a function of the distance measured from the starting point [ ~ ( x ) ]the , hydraulic parameters can be determined. When solving Eqs (5.3-44) and (5.3-45) the boundary conditions to be considered are as follows: (a) In the case of recharging the ground water from the surface water: at the section x = 0 the depth yo = H I ;

the accretion ~ ( zis) negative;

662

5 Movement equations describing seepage

at the end of the influenced zone ( x = L ) the depth is equal to the original ground water depth yo = H,; (b) Investigating the drainage of ground water by surface water: yo = H,, at the point x = 0, assuming that the influence of the free exit face is negligible; E ( Z ) is positive; and the original depth prevailing at the end of the influenced zone is yo = H I if x = L. The width of the influenced zone ( L ) can be determined from the equations knowing that the total interaction between the surface water and the seepage field is balanced by accretion within this distance, and, therefore, the horizontal flow rate is equal to zero. It is necessary to analyze the practical determination of the numerical value of accretion. As already discussed in Section 4.1.3,the accretion is generally proportional to the vertical change in the position of the water table (or the piezometric surface). The earlier proposals, giving accretion aa a constant value depending for example only on the yearly precipitation and the physical soil parameters of the covering layer, as in Kriiger’s table (NBmeth 1942),cannot be accepted. It would be logical to determine the vertical @ e& ? as the function of the depth of water table below the surface, or m r e precisely as that of the change in this depth [ ~ ( d y ) ] Using . this concept the numerical value of accretion can be determined by the application of the charactmistic curve of the ground-water balance (see Fig. 4.1-ll), the interpretation and construction of which is discussed in Section 1.4.4.The basic differential equation can easily be established in this case as well:

d2y dx2

yo-+

dY0

1 K

- +--(dy)=O;

(dx)

(5.3-46)

but solving of this relationship raises considerable problems even in caaes, ) when the most simple functions are used to approximate the ~ ( d yrelationship. LBczfalvy (1958)solved Eq.(5.3-46) by using linear relationship, and Kovhcs (1962)by applying an exponential function, but their results cannot be proposed for practical use, because they are very complicated, and their use would not be in keeping with the low accuracy of the numerical factors in the equations. Because of the difficulties emerging in connection with the integration of Eq. (5.3-46),it is generally accepted that the accretion be approximated by a function describing its horizontal distribution. This function should be a steadily decreasing one with its maximum at the starting point ( E ~ ) and , it should have zero value at the limit of the influenced zone (if z = L; E = 0). In the first publication, in which the vertical effects were described in this way (JuhQsz, 1953),a linear relationship and/or a parabola of second order were used. A broader generalization can be achieved, however, by applying a parabola of the n-th order (Kovhcs, 196313):

I z)”

e=&E01--.

(5.3-47)

6.3 Horizontal unconfined steady seepage

663

Substituting this relationship into Eq. (5.3-44), the latter can be integrated in closed form. The solution is shown in Fig. 5.3-21 in the investigation of a horizontally drained seepage field. The parameters to be determined are aa follows: - L the width of the influenced zone; - go the drained specific discharge; - q(s) the steady flow rate changing along the seepage field; - y o ( z )the curve describing the position of the water table or the piezo-

metric surface. The solution is given separately for semi-conhed and unconfined fields. In the m e of a semi-confined system:

::Y

L=

-(n+1)(n+2)(H,-H2);

(5.3-48)

unconfined system Fig. 6.3-21. The development of the water table (or the piezometric surface) of a horizontslly drained ground water, considering the iduenoe of accretion

664

5 Movement equations describing seepage

For an unconfined system (assuming that the development of the free exit face can be neglected, thus H , = H a ) :

The n parameter of these equations has to be determined considering the local conditions. The same relationships can also be applied for the charac-

unconfined system Fig. 5.3-22. The development of the water table (or the piezometric surface) of a horizontally recharged ground water considering the influence of accretion

6.3 Horizontal unconfined steady seepage

665

terization of a horizontally recharged flow field. It waa found, however, that in this case a more rapidly decreasing horizontal distribution better approximates the actual conditions. A proposal was made, therefore, to use another type of E ( Z ) function instead of Eq. (5.3-47) (Fig. 5.3-22) (KovBcs, 1963a): E(Z)=-E0

[

1-

where

li-ia] 1-

1--

=-EO[l-l/l-Z"]; (5.3-50)

Using this approximation the integral can be solved only after expanding Eq. (5.3-50) into series. Applying the following symbols:

y(n;z ) = [I

-

v l - z"] ;

1 a(n;z ) = 1B(n;z ) dz ;

B(n;z ) = y(n;z ) dz ;

t ( n ) = y(n;z = 1 ) = 1 ;

o(n)= B(n;z

= 1) ;

(5.3-51)

e(n)= a(n;z = 1) ;

the results of the integration can be summarized in the form of the following equations: For a semi-confined system:

(5.3-52)

For a n unconfined system:

(5.3-53)

6 Movement equations describing seepage

666

As a numerical example, the parameters summarized in Eq. (5.3-51) are given in Table 5.3-2 where n = 12 in Eq. (5.3-50) (which was found to be reliable to characterize the vertical surface effect, i.e. the increase of evapotranspiration aa a consequence of the rising water table). It can be seen from the table of values, that at a distance longer than one third of the length of the influenced zone from the entry point both the change in the position of the water table and the accretion caused by this change, can be practically neglected, if the indicated horizontal distribution of the vertical drainage is used. Table 6.3-2. Numerical values of the parameters listed in Eq. (6.3-51) using n = 12 power to the Eq. (6.3-10) e (12) = 3 . 0 2 ~ (I (12) = 4.76 x lo-' T (12) = 1.00 z -

a(e; 12)

L

e (12)

0.00 0.01 0.02 0.06 0.10 0.20 0.30 1.00

1.00

0.99 0.98 0.96 0.90 0.80 0.70 0.00

1.ooo 0.868 0.733 0.464 0.213 0.041 0.007 0.000

B 0;12) 0

Y(G 1%

(12)

1.000 0.839 0.714 0.462 0.214 0.046 0.008 0.000

1.000 0.668 0.632 0.322 0.163 0.036 0.007 0.000

When investigating the development of the water table and ground-water flow between two canals, the ratio of the distance between the canals to the depth of the flow field is generally greater than 10, mentioned previously as the limit, above which the non horizontal character of seepage around the recharging and draining structure can generally be neglected, especially if the influence of the curved stream lines on the flow rate is investigated (see Fig. 5.2-8). The same condition (the relatively large length of the field) decreases the horizontally transported amount of water and increases the total accretion along the field if these two values are compared to one another. For this reaaon, accretion has to be considered in most cases, in the investigation of the development of ground-water flow between two canals. If seepage is influenced by accretion, the interaction of the two canals within the enclosed seepage field cannot be characterized by the superposition of the separate influences of the canals, because the accretion is itself the function of the new position of the water table. The original differential equation [Eq. (5.3-44)] has to be solved considering simultaneously the boundary conditions at both ends of the flow field as well aa the horizontal distribution of accretion along the field (Kovhs, 1963c, 1964a). If the interaction of a recharging and draining canal is investigated, the new water table crosses the horizontal surface of the ground water, which develops under static condition (Fig. 5.3-23). The dynamic equilibrium can be

6.3 Horizontal unconfined steady seepage

r

667

lb

Fig. 5.3-23. Interaction of recharging and draining canals if the seepage field is influenced by accretion

characterized by considering that the inflowing recharge is equal to the sum of the negative accretion accumulated from the starting section until the cross section, where the new water table crosses the original horizontal ground-water surface, and the horizontal flow rate through this vertical section. The same relationship can be determined for the other part of the field, namely, that the amount discharged is equal to the sum of the integrated positive accretion and the horizontal flow rate at the section, which divides the two parts of the field having been influenced by opposite accretions: and

(5.3-54)

where the definition of a(n)is similar to that in Eq. (5.3-51). Its numerical value depends on the approximation of the horizontal distribution of accretion and may be different in the two parts of Eq. (5.3-54). Taking this condition into account and accepting a given e(%) function the differential equation can be solved and the position of the influenced water table determined. Between two canals having the 5ame character, (two recharging canals or two drains) two different conditions can be distinguished. I n the first case the role of the canals does not change in the combined system (Fig. 5.3-24). In the other case the action of one of the two canals is modified under the influence of the other canal (Fig. 5.3-25). If i t wasoriginally a drain, it will recharge the ground-water space between the two canals and, inversely, the original recharging canal will drain it. This case occurs when the water level in one of the canals is relatively close to the original water table. The character of the canals remains unchanged, as indicated in Fig. 5.3-24, if the accretion between the canals can balance the action of the two canals.

668

5 Movement equations describing seepmge

Fig. 5.3-24. The development of the water table between two canals having the same character, Considering in addition the influence of accretion

This condition is fulfilled if the water table, calculated by considering the single effect of one of the canals, intersects the vertical section of the other canal, between the water level of the latter and the original horizontal water table. Calculating the balance of the flow rates, it is necessary to take into account the fact that accretion is not zero at that point of the seepage field, where there is no horizontal flow, because the position of the water table may here be changed Q1= + d n ) A&]; and (5.3-55)

+

Y2

=

&I&,+)

+ 4.

When the previous condition is not fulfilled (when the water level in one of the canuls is between the original water table and the water surface calculated by assuming that the seepage space is influenced only b y the other canal) the role of the canal (the level of which is a greater distance from the original water table) remains unchanged and its effect is balanced partly by accretion and partly by the other canal. Following the sketch given in Fig. 5.3-25, the balance of the field in this case can be expressed by the following equations: r/1= l e 1 4 4

and

+ q2;

(5.3-56 )

where the interpretation of the p(z, n ) function is similar to that in Eq. (5.3-51). Naturally the action of the second canal (the water level of which lies near the original ground-water surface) changes only in the direction of the common flow field, while outside i t maintains its original character.

669

6.3 Horhontal unconfined steady seepage

wafer iable, which would devehpunder fhe influence ofwnal f. only

----

h

I

-

A

I

e

I-

-

_

1

,

.

,

_

,

__

L

~

,

l

.I

I

__

-

.

Fig. 5.3--25. The development of the water table between two similar canals if the role of one canal is modified within the common field

Equations (5.3-54), (5.3-55) and (5.3-56) do not give the final explicit form of the mathematical models; they are [together with Eq. (5.3-44)] only the starting points for the derivation of the models. Many different formulae can be determined in this way according to the horizontal distribution of accretion considered in the investigation. The correct approximation depends, however, on the local conditions of the leaking aquifers. It is advisable, therefore, t o develop the models by always taking into account the most probable local influences.

670

6 Movement equations describing seepage

5.3.4 Local resistance occurring in the vicinity of the entry and exit faces An important hypothesis for the derivation of Dupuit's equations is the assumption that the lower boundary o€ the seepage field is horizontal and the vertical entry and exit faces cross the pervious layer at full depth. If the latter condition is not satisfied (the closing faces are not vertical or they do not penetrate until the lower impervious boundary) the seepage cannot be approximated as a horizontal flow because high resistance may occur within the curved zone of the flownet, consuming a considerable part of the total available energy. This decreases the amount of water transported through the field and influences the position of the developing water table. Investigating the flow between two surface-water bodies in Section 5.2.1, it was found, that the position and the depth of the closing faces do not influence the water transport of the field, if the ratio of length and depth of the latter is greater than 8 to 10 ( W m > 8-10 see Fig. 5.2-8). A similar result can be achieved by comparing the yield of fully and partially penetrating wells. The equation for the transformation of the vertical flow plane of a partially penetrating well was derived in Section 5.2.1 [see Eqs from (5.3-18) to (5.3-21), and Fig. 5.2-61 (Kov&ca,1966). Using those relationships, the yield of a well, penetrating to a depth d below the upper boundary into the aquifer having a thickness m and recharged horizontally around the well at a distance R , can be calculated and compared to other values determined by using more simple approximations. Such approximations may be the substitution of the actual aquifer by a pervious layer having an impervious boundary at the bottom of the well, or to neglect the partial penetration of the well and to calculate the yield by supposing that the well is screened along the whole depth of the aquifer (Fig. 5.3-26). The yields to be compared can be calculated from the following equations: (a)By using the mapping functions, the result of which can be considered in the form of a modified thickness of the aquifer (Eq. 5.2-21):

R

&=2nKAH

Rn

13

1

R

=KAHR--

ch - - 8 lnarch ro 1-8

nm

m 2n

R

R

("n3

ch - - - 6

ln-

arch

where

1--6

(5.3-57)

(b) Assuming fully penetrating wells:

m m 2n =K A H R - - . R R R In In -

(&)DL = 2n K A H --

(5.3-58)

5.3 Horizontal unconfined steady seepage

671

Fig. 6.3-26. Comparison of the yields of partially penetrating wells determined by various approximations

(c) Finally, if the partially penetrating character is considered in the form of an aquifer closed by an horizontal lower boundary at the level of the bottom of the well d d 2n (&)02= 272

KdH -= K A H R - - . R R In TO

R

(5.3-59)

In TO

The figure clearly indicates that the second approximation (the limitation of the aquifer at the lower level of the well) is not acceptable,if the rate of pen-

d etration - < 0.8 while in the other cme its application is unnecessary, m because the first approximation provides acceptable results. Thus the use of the (Q)mvalue can be a priori excluded.

672

5 Movement equations describing seepage

Another interesting result is, that in the case of axial symmetrical flow, the

R depth of the field has to be compared to In- and not to the length of the field. r0 If the same comparison were executed in connection with partially penetrating drains recharged from both directions, the result would be the same aa that achieved in Section 5.2.1 [i.e. if Llm > 8 10, the curvature of the

-

impervious layer Fig. 6.3-27. Flow nets in the vicinity of recharging canals

flownet is negligible if the flow rate is calculated]. I n the caae of axial symmetrical flow

R

In --r 0 > 8 - 1 0 m 2n

(5.3-60)

is the condition indicating the limit, above which the local resistance around the well is negligible compared to the total resistivity of the field. This result shows that the yield is more influenced by the rate of penetration in

R

the case of axial symmetrical flow if the - ratio is smaller than 550. r0 If the ratio of the length and depth of the seepage field does not allow the simple, direct application of Dupuit’s equations, or the task is the more accurate determination of the water table (or the piezometric surface), which requires more detailed investigations, a frequently applied method is to divide the field into stretches. Then within the main part, the flow may be apprixomated as horizontal seepage, while the local resistances characterizing the remaining stretches can be determined separately using other suitable formulae. One good example to demonstrate the application of this method is the investigation of the seepage i n the vicinity of recharging canals (Kovhcs, 1964b). The flow nets determined in sand box models (Fig. 5.3-27) clearly shows that the supposition of vertical potential lines is acceptable in almost the whole field except close t o the canal. To determine the hydraulic parameters of the seepage the position of the dividing section has first t o be fixed (Fig. 5.3-28). The first stretch having a length of L, between the axis of the canal and the dividing section has to be characterized by local resistances, while the second stretch ( L J extends t o

673

5.3 Horizontal unconfined steady seepage

second sfrpcb of seepage Fig. 5.3-28. Interpretation of the separation of the various stretches of the flow field

the draining structure, which is now supposed to be a fully penetrating canal with vertical faces, to simplify the lower boundary condition. The aeepage in the secondstretch can bedescribed by Dupuit’s equation (using either the most simple form or that which considers accretion, depending on the upper horizontal boundary condition of the field). The dividing section, substituting the entry face of the second stretch, is called, therefore, the starting section. As already emphasized, in every cme the total field has to be investigated from the actual entry face to the exit face, where the water leaves the porous medium. This requirement determines two conditions, which have to be considered in the investigation. These are: the sum of the head losses along the

the various stretches

[5Aht) has to be equal to the total available pressure head i- 1

( A H ) ,and the continuity has to be expressed in the form of equal rates i n each stretch (ql = q2 = . . . = q., - . . . = qn), if there is neither recharge nor drainage along the field. In the field investigated here which have only two stretches, these conditions are expressed by the following equations: (5.3-61)

Considering that the width of the wetted zone below a recharging canal is a well determined parameter, if the seepage is in a free steady state, i t can be proposed, that the starting section should be identical with the vertical asymptote of the wetted field. Thus the length of the first stretch should be equal to the virtual width of the canal [see Eq. (5.2-35)]: (5.3-62) 43

674

6 Movement equations describing seepage

In this way the position of the starting section also depends on the capillarity of the material composing the field. It is evident, however, from S Fig. 5.2-14, that the influence of capillarity is negligible, if - > 25. In this hC case the length of the first stretch may be independent of the physical soil parameters of the field S

L , = S ; if - > 2 5 . hC

(5.3-63)

It will be demonstrated in connection with the analysis of the local resistance in the vicinity of the canal, that Ah, decreases at first if the length of the first stretch (L,)related to its depth (H) increases. The Ah, = f LI

l.Ul ,

4 =4 value, and relationship has a local minimum somewhere around the H 3 L, parameter indicating, that a increases afterwards with the increasing H

considerable length is also included in the first stretch, where the flow could be characterized by horizontal flow. It has to be noted that below large beds if 35 > 4H and, therefore, - >- the flow net becomes horizontal within H L1

4 31

the part of the field covered by the bed. Therefore, the starting section can be located at the edge of the water surface (or that of the virtual width if capillarity is not negligible). In this case the water table joins the level of the surface water and thus the head loss within the first stretch is negligible:

L; = S d 2 ; and Ah, = 0 if 35,

> 4H;

S and -hC

< 25 ;

or

(5.3-64)

Li = 512; and Ah, = 0; if

35

S > 4H; and >2 5 .

hC The four possible variations in the determination of the starting section are represented in Fig. 5.3-29. It is quite evident that the total pressure head is consumed along the stretch characterized by horizontal flow if 3 5 > 4H (or 35, > 4 H ) , and, therefore, Dupuit’s equation can be directly applied using the vertical section at the edge of the surface water as the starting section, without studying the local resistances. In the other cme, the remaining problem is to determine the head loss between the perimeter of the canal and the starting section. Acceptable approximation is obtained in the investigation of the field lying along the contour, which is symmetrical about the axis of the canal and bounded by the impervious lower boundary, the starting section and the

6.3 Horizontal unconfined steady seepage

(5)

(b)

starting section

675

starting section

/

L’-

=I? V $1 -4 & 313 V

WlCu

B

K =s

imprviaus ager k~ -4

impervioos ayer

$. -.J

~ ~ 1 - s ~

Fig. 5.3--29. Possible variations in the determination of the starting section

water table. By comparing the stream lines of this field to those developing in the surrounding of a rectangular internal corner of a large continuous field the similarity is obvious (Fig. 5.3-30). The mapping function transforming this latter flow net into two orthogonal sets of stra.ight lines can be used, therefore, to calculate the local resistance near the recharging canal:

w =u

+ iv =

22

= (x

+ iyy;

therefore u = 5 2 - y2;

(5.3-65)

and 2,

contour of the canal

impervious layer

= 2xy.

t-

border ofthe rectangular mrner

(b) Fig. 6.3--30. Comparison of stream lines in the vicinity of a recharging canal and in the surrounding of a rectangular internal corner of a continuous field 43*

676

5 Movement equations describing seepage

Applying this mapping method and approximating the images of some uncertain contours with straight lines, a field is achieved, the cross-sectional area ( A )of which is a linear function u of the real ordinate of the w plane (Fig. 5.3-31). Accepting this relatively very rough approximation the specific flow rate can be expressed depending on the change of pressure head along contour o f

Fig. 6.3-31.

the u axis, and after integrating the elementary pressure head according to the u variable between two vertical sections. Thus, relationship is achieved between the local head 108s and the flow rate: dh du

q= KA(u)-;

Ah,

=

(5.3-66)

7

dh -du= du

du. -[H'-&)']

-[H'-er]

Substituting the linear relationship previously mentioned between A(u) and u the result of the integration is: 3

H'+7LI

Ah1 =

s

1 Q 1 du =--[ln(b-au')]o K(b - au') K a

Q

0

where

.'=.+H2-($)

2;

and 2 -Ah1 -1

Ll A(u') = L1H - -

H

3

Ha+-Lf

u' = b - au';

;

677

6.3 Horizontal unconfined steady seepage

consequently 2--1 Ah1

[yqq

H LH H In2 1---, i ) = K H L , 4 H

Ah,

[

3+

.

(5.3-6 7)

The numerical determination of the three dimensionless quantities

r+

;

among which only Ll is an a priori known geometrical parameter,

H

H ’ KH

has to b e executed by using successive iteration together with the characterization of the second stretch to ensure that Ah, and q satisfy the conditions given in Eq. (5.3-61). To demonstrate that the results are acceptable in practice, although very rough approximations are applied, the calculated local resistance values are compared in Fig. 5.3-32 to those of the water tables observed in the three sections of a recharging canal. The conclusions drawn from the figure are aa follows:

I

0 -

---__-.

\\

/I.

1 3

@ @

.

.-.

‘Cy

I

s

Go

I

/-;4 /

f!ucfuaflon of the water !eve! in the investigated strefch

R:

intersectionof the starting section and tbe theoretical water hble calculated by considering the O l C31 losses

8 8 ,observation well

SI

@ QR

static water fable before fiM?Jg the canal Fig. 5.3-32. Comparison of calculated local resistances of natural water tables observed in the vicinity of a recharging canal

678

5 Movement equations describing seepage

(a)There are many cases, when the local resistances cannot be neglected, as they considerably influence both the water transport and the position of the water table; (b) Within the stretch characterized by almost horizontal seepage the fiuctuation is very similar to that of the natural water table, due to the seasonal fluctuation of accretion (infiltration and evapotranspiration), but in the vicinity of the canal, where local resistances occur, these eflects can be neglected because the length of this stretch is very short, and thus the influence of accretion is not considerable; (c) Although the accuracy of the method described here is not high, it can be accepted in practice to give rough information on the development of flow conditions near the recharging canals.

Another example frequently used to show the application of the separation of the various parts of the seepage field to simplify the hydraulic characterization of the flow, is the seepage under horizontal impervious foundations (Fig. 5.3-33). The mapping method suitable for mathematically correct determination of the hydraulic parameters of this seepage field was derived by Pavlovsky (1922) and discussed in Section 5.1.4. The results, on the basis of which the specific flow rate and the pressure distribution along the horizontal foundation can be calculated were summarized in Eqs (5.1-66)and

2.5

5.0

z5

a m

MD

Fig. 5.3-33. Comparison of the various approximations proposed for the calculation of the specific flow rate under horizontal foundation

5.3 Horizontal unconfined steady seepage

679

(5.1-67). In Section 5.2.1 an example of the practical application of this method was also shown, because the flow field represented in Fig. 5.2-2 is half of that analyzed below, and considering the symmetry of the field the equations applied for the characterization of confined seepage between the horizontal entry and vertical exit face can also be applied directly here. Only the geometrical parameters have to be considered precisely. Thus, the original relationship for the calculation of the specific flow rate composed of the total and supplementary total elliptical integrals can be substituted in practice, by a more simple formula (Kovbcs, 1960) [see also Eq. (5.2-2)]: q=-

KAH 32

m

arsh 1.5 -. b

(5.3-68)

Before the derivation of Pavlovsky’s method it was generally accepted that the actual water transporting layer could be substituted by an elliptical seepage field, the depth of which is equal to that of the pervious layer and its length determined so that the edges of the foundation should be the foci of the ellipse. The equation for the calculation of the water conveying capacity of this field waa also previously derived as well [see Eqs (5.1-30) and (5.1-33)]. Using the geometrical parameters of the actual seepage field the specific flow rate is in this case approximated by

KAH m qL-4 = -arsh b

32

.

(5.3-69)

Finally, the most simple approximation is the supposition that the flow is horizontal below the foundation between two vertical faces: m qo = K A H - .

2b

(5.3-70)

The comparison represented in Fig. 5.3-33 shows that the use of Eq. (5.3-69) is unreasonable because i t causes a regular error generally higher than 10 yo and its structure is no simpler than that of Eq. (5.3-68). As in the previous results, the difference in the flow rate between the mathematically correct calculation and the simple use of Dupuit’s approximation, is not higher than 5 yoif the ratio of the length anddepth of the field is higher than 8-10

if-> 1 . 8-10

In this special caae even the pressure distribution can be approximated by simple relationships. In Fig. 5.3-34 the results of the various methods used for the determination of the pressure distribution below the horizontal foundation are compared. Dupuit’s approximation results in a linear relationship, while the distribution can be expressed by a trigonometric function, if an elliptical (or infinite) seepage field is assumed [see Eqs (5.1-32)

680

5 Movement equations describing seepage

Fig. 5.3-34. Comparison of the various approximations of the pressure distribution below horizontal foundation

and (5.1-34)]. These two curves envelop the set of distribution curves bem

longing to various -ratios and are Calculated from the theoretically correct b equation [Eq. (5.1-67)]. The discrepancy, however, between the accurate and approximative values, is relatively small. It can be proposed, therefore, m 1 that the linear relationship is applicable if - < - ;while Eq. (5.1-34)gives b 2

;1 3 .

a better approximation in the opposite case - >-

These aspects were

also considered when the confined seepage from surface water having a horizontal bed to a fully penetrating drain was analyzed [see Eq. (5.2-2)]. The summary of the theoretical and practical investigation of seepage under a horizontal foundation proves that the practical methods bawd on the theoretically correct conformal mapping provide the designers with relationships suitable for the accurate determination of the hydraulic parameters. Nevertheless, i t is worthwhile to analyze the previously generally applied methods, to attempt to approximate the correct parameters by dividing the field into separate stretches. Using these assumptions is still the easiest way to calculate the /low rate and prassure distribution i n special layered fild8. As mentioned in the previous discussion (Fig. 5.3-33), the difference between the correct flow rate and that calculated by the simple use of Dupuit’s 2b equation is larger than 5% if - < 8-10. To eliminate this error i t was m proposed earlier that the field should be divided into three stretches, assuming a vertical stretch to have the same cross-sectional area as the horizontal section before and after the horizontal foundation. Its length being equal to four tenths (Dachler, 1936)or half (Kamensky, 1943)of the thickness of

6.3 Horizontal unconfined steady seepage

o

r

2

3

4

5

1

681

_ rn

Fig. 6.3-35. Comparison of the flow rates calculated by using Dechler’sand Kemensky‘s approximation to the theoretically correct value

the pervious layer (Fig. 5.3-35). Considering these hypotheses the flow rate can be calculated from the following equations q1= K A H

m

2b

+ 0.8m (Dachler) ; (5.3-71 )

or q2 = K A H

rn (Kamensky) . 2b + m

The comparison of the results of these equations with the theoretically correct flow rate (calculated from Eq.5.3-68) proves the accuracy of these methods (especially that of Dachler’s approximation). Although in the case of a homogeneous field there are similarly simple and theoretically better baaed models, the basic idea of this approximation can be usefully applied, if the field i s covered by a semi-pervious layer. In the case of layered fields, where they are composed of a series of semipervious lamellm covering the aquifer (which can also be built up from several layers differing slightly in permeability) the whole field has first to be substituted with a double layered system, in which a covering layer of lower permeability lies above the water transporting formation I n Fig. 5.3-36 the first sketch shows the original system and the second one represents the

682

5 Movement equations describing seepage

(b) ~

X

Q

$1 4-

eomin.o............. . .lauer . . . .I:.I .I I. ..................... ................ ............ ...........:.. .. :::........................ ..... .

.

parameters characterizing the members of tbe covering forma fion :t,I K,, i t2I h2 .,,.. ; tn, hn; parameters characterking the members ofthe pervious formation :n/I KO, j mz I Ko2 I ..... j mn ,KO,,

Fig. 5.3-36. Transformation of the layered water transporting system to an homogeneous field

result of the transformation. In the next step the field has to be homogenized by enlarging the thickness of the covering layer by multiplying the original size by the ratio of the average permeabilities of the two layers (Fig. 5.3-36c):

t , = t - K" . (5.3-72) Kt Since the covering layer, having relatively high resistance, is substituted by a long vertical stretch before and behind the horizontal foundation, the vertical flow within the water transporting layer can be neglected, compared to the resistivity of the upper lying layer. Thus, the equation proposed for the calculation of the specific flow mte is aa follows (Kamensky, 1943; Galli 1959): (a) If the covering layer is different before ( t l ;K , ) and behind (t2; K,) the horizontal foundation:

6.3 Horizontal unconfined steady seepage

683

AHmK,

(b) Having thesamecovering layer ( t ;K,) on both sides of the foundation:

AHmK,

!7=

2b

+2

vk

(5.3-73)

K,p

The same aspects applied in subsequent steps can be followed if the water transporting field below the foundation is composed of more pervious and semi-pervious layers (Shea and Whisett, 1958). In this case horizontal flow is always assumed in the pervious layers and vertical flow is assumed in the semipervious ones. There is one rough approximation in Dachler’s method: i.e. the supposition, that the active width of the vertical flow is equal to the thickness of the horizontal water transporting layer. This hypothesis may cause considerable error in the case of a layered field. To eliminate this source of error Juh&sz (1968) has developed a method combining the basic aspects of Dachler’s derivation (separation of vertical and horizontal stretches of flow) with the use of finite elements (Fig. 5.3-37). Let us first consider the water conveyance through a horizontal stripe of thickness dz, at a depth z below the lower boundary of the covering layer. The vertical stretches joining the stripe have a thickness of a dz,where the a

separated sfreches of the seepage field

:

(1) vertical tbrough tbe covering layer before the dam ( 2 ) vertical in tbe pervious layer before the dam

(3) borizontal in the pervious layer ( 4 ) vertical in tbe pervious layer bebind the &m (5) vertical tbmugb tbe cwering layer oebind the dam

Fig. 5.3-37. Application of Dachler’s method for the cheracterization of finite stripes of the seepage field

684

5 Movement equations describing seepage

factor is provisionally unknown. This water transporting element of the field can be divided into five stretches: (a) Vertical jlow through the covering layer before the foundation area A = a,dz; length 1 = t,; hydraulic conductivity K,; (b) Vertical flow in the water transporting layer before the foundation

A = a&; 1 = z; K O ; (c) Horizontal flow

A = dz; 1 = I,

+ a,z + a,z;

KO;

(d) Vertical flow in the water transporting layer behind the foundation

A = a,&; 1 = z; KO; (e) Vertical flow through the covering layer behind the foundation

A = a&; 1 = t,; K,. Considering continuity and the requirement that the sum of the head losses along a stream line should be equal to the total available pressure head [see Eq. (5.3-61)] a relationship can be determined between the dq and dz elementary variables:

where

B = lo+ K O

[- tl

+

a1K1

1-

t2 ; a2K2

and 1

C =a1

1 + -+

a,

a2

+ a,.

(5.3-74)

By integrating this relationship between the z = 0 and z = m limits, the specific jlow rate can be determined as the function of geometrical and physical soil parameters, but i t also depends on the undetermined a, and a, values, the effects of which are included in the B and C factors: rn

--JB+Gz=Cln 9 dz AHK, 0

1

B+Cm B

(5.3-75)

685

6.3 Horizontal unconfined steady seepage

If the thickness and the material of the covering layer on the two sides of the foundation do not differ, the a1 value is equal to a, and the two factors can be simplified if

t , = t, = t ; K , = K , = K , ; B , = 1, 2-,t K O . aK+

a, = a2= a;

+

(5.3-76)

and

The a value has to be determined by trial and error method considering the principle of “lex minimi”, by which the highest possible water transport has to develop as the result of a given energy content. Thus the -pstrameter AHK,

‘ [

has to be calculated as a function of a, and the highest value aa well as the a parameter belonging to -

can be accepted aa the final result. If the

cA:K.,m.d

similarity of the covering layers in front of and behind the foundation is an acceptable assumption [see Eq. (5.3-76)] this trial and error method only re-

‘

quires the calculation of one -vs. a curve to determine its local maxi-

AHK, mum. If different t , and t2,K , and K2,as well as a, and

a2 values

‘

have to be

considered, u, and a2 combination which gives the highest -value, has

AHK,

to be used. Even the anisotropy of the water transporting layer can be taken into account in this way. If the horizontal hydraulic conductivity is indicated by the K O rr

symbol ( K H = KO), and ;z = A-isH

K”

the coefficient of anisotropy, the hydraulic

conductivity of the vertical sketches within the water transporting layer haa to be considered by using

K K v = 2 parameter. Thus, the C factor in I

Eqs (5.3-74) and (5.3-75) has to be modified: (5.3-77)

or the simplified form, if the same layer covers both sides of the aquifer (5.3-78)

Suitable a values have to be determined by a trial and error method in this case as well, as previously explained.

686

5 Movement equat,ions describing seepage

References to Chapter 5.3 CHARNYI, J. A. (1961): The Proof of the Correctness of Du uit's Formula in the Case of Unconfined Seepage (in Russian). Dokl. Akad. Nauk. &SR No. 6. CHENQ and Y A N O - SSHA ~ (1969): An Experimental CHINO-TON KuO, M O W - S O ~ O Study on the Boundary Conditions for the Flow t h o u g h Porous Media. 13th Congress of I A H R , Kyoto, 1969. DACHLER,R. (1936): Ground-water Flow (in German). Springer, Wien. DUPUIT, J. (1863): Theoretical and Practical Studies on Water Movement in Open Channels and through Permeable Layers. (2nd edition) (in French). Dunod, Paris. G m , L. (1969): Approximative Method to CalculatetheParameters of See age through a Layered Field below Hydraulic Structures (in Hungarian). Viziigyi &zlem h y e k , No. 3. HANTUSH, M. S. (1964): Hydraulics of wells. (From Advances in Hydrosciences edited by V. T. Chow). Academic Press, New York. J ~ s z J. , (1963): Data about the Ground Water in Plains Especially Regarding Aspects of the Backwater Effects Caused by Barrages (in Hungarian). V'iziigyi Kozlemdnyek, No. 2. JWASZ, J. (1968): Investigation of Seepage Developing under Dikes in the Case of a Thick Water Transporting Layer (in Hungarian). HidroMgiai Kozlony. No. 8. KAMENSKY, G. N. (1943): Bases of the Dynamics of Ground Water (in Russian). Gosgeoltekhizdat, Moscow. KOVLCS,G. (1960): Calculation of the Flow Rate of Seepage under Dams (in Hungarian). Viziigyi Kozlemdnyek, 2 . KOVLCS,G. (1962): Design of Drains Discharging the Ground Water (in Hungarian). $pit&- ks Kozlekedkstudomanyi Kozlemknyek, No. 2. KOVACS,G. (1963a): Characterization of the Steady Influenced State of Seepage from Recharging Irrigation Canals (in Hungarian). Hidroldgiai Kozlony, No. 1. KOVACS,G. (196313): Practical Method for Hydraulic Design of Drains Discharging the Ground Water (in Hungarian). &pitks- 6s Kozlekeddstudomanyi Kozlemknyek. NO. 1-2. KovAcs, G. (1963~): The Development of the Water Table in the Vicinity of Canals with Nearly Constant Water Level. 5th Congress of ICID, Tokio, 1963, Vol. 17. KOVACS,G. (1964a): The Development of Ground Water under the Influence of More Canals (in Hungarian). HidroMgiai Kodony, No. 4. KOVLCS,G. (1964b): Local Seepage Resistances in the Vicinity of Canals Recharging and Draining the Ground Water (in Hungarian). Hidroldgiai Kozlony, 10. KOVACS,G. (1966): Yield of Partially Penetrating Wells. Symposium on Seepage and Well Hydraulics, Budapest. KOVACS,G. (1973): Characterization of Steady Seepage through Homogeneous Earth Dams With Vertical Faces. V I T U K I Publications in Foreign Languages. Budapest, 6. LkOma-, S. (1968): Determination of the Yield of Drains (in Hungarian). HCdroldgiai Ko.z.?ony, 1. -ONE, U.and FRANZETTI, S. (1969):Unconfhed Flow Downstream of an Homogeneous Earth Dam with Impervious Sheetpiles. 13th Congress of I A H R , Kyoto, 1969. MUSEAT,M. (1937): The Flow of Homogeneous Fluids through Porous Media. McGraw-Hill, New York. Nh?E!t%, E. (1942): Water Problems of Modern Agriculture (in Hungarian). Budapest. PAVLOVSKY, N. N. (1922): Theory of Ground-water Movement around Hydraulic Stmatures (in Russian). Leningrad. POLUB~LI~INOVA-KOCHINA, P. YA. (1962): Theory of Ground-water Movement (in Russian). Gostekhizdat, Mosaow. POLWARINOVA-KOCHINA,P. YA. (1962): Theory of Ground-water Movement. Princeton University Press, Princeton. S ~ AP., H. and WEISETT, H. E. (1968): Predicting Seepage under Dams on Multilayered Foundations. Proceedings of ASCE, Vol. 84. ! ~ I E M , A. (1870): The Yield of Artesian Boreholes, Shaft-wells and Filter-galleries (in German). Journal fur Gasbeleuchtung und Wasservermrgu~g,Vol. 14.

6.4 Horizontal unsteady seepage

687

Chapter 5.4 Investigation of horizontal unsteady seepage In the foregoing chapters, when the conditions of development of steady seepage and those of the application of Dupuit’s equations for the description of this state of flow were discussed, the known position of both the entry and exit faces was mentioned rn a basic requirement. Analyzing the kinematic classification of seepage, i t was emphasized that in a field having i n h i t e length, steady seepage can develop only in a confined water transporting layer, where the actions caused by the changes of the boundary conditions propagate with very high velocity. In such systems the time-variant movement can be approximated with the series of parameters calculated by supposing a steady state. If the field is unconfined or it is covered by semi-pervious materials, and thus the change of pressure modifies the stored amount of water the velocity of propagation is restricted by the velocity of water transport. Steady seepage can develop in these systems only if the horizontal flow is balanced by the accretion of the ground water. In other cases the actions created at the starting section of the field (i.e. the entry or the exit face, the position of which is known) propagate slowly until infinity, and this state of seepage can only be characterized by methods describing the unsteady flow. Analyzing the methods generally applied in engineering practice to solve seepage problems the errors most frequently occurring and the largest differences between the calculated and actual values, are caused by the fact that the equations derived by assuming steady state conditions were also applied to determine the hydraulic parameters of unsteady seepage. The main ambition of engineers always was and is to find simple, practical solutions. For this reason, Dupuit’s equations are generally applied to characterize even unsteady flow and the obstacles caused by the unknown position of one of the closing faces are overcome by using arbitrarily chosen limits of the influenced zone without any physical meaning or explanation. Thus, most of the practical problems are solved by approximating the movement as a steady flow, although the natural character of seepage in most cases is unsteady. It is necessary to develop a system of mathemtically easily applicable methods having a firm theoretical basis so that the equations composing this system ensure at the same time the uniform treatment of the largest part of the various types of unsteady seepage frequently arising in practice. Since any method considering at least the unsteady character of flow, may bridge the gap existing between theory and practice, the hypotheses applied to simplify the solution of the differential equations can be based on relatively rough approximations, because the errors caused in this way are probably smaller than those made by using steady state equations. Considering the above, the objective of this chapter is that after the derivation of the differential equations describing the horizontal (or nearly horizontal) unsteady seepage, their solution should be given in a unified form. The application of this method should be demonstrated by practical examples as well.

688

6 Movement equations describing seepage

5.4.1 Derivation of Boussinesq‘s equations and the problems in connection with their linearization Let us first investigate the continuity of flow within an elementary prism having a nearly horizontal impervious lower boundary (Bear et al., 1968). In a coordinate system having x,y and z orthogonal axes the position of the lower boundary is described by the functions [ = [ (x,y). The height and form of the water table can be similarly characterized by Z = Z (x,y) function. The depth of the flow field (h) can be calculated a t each vertical line in this way (Fig. 5.4-1):

h=Z-c.

(5.4-1)

The change of the depth between two points is

az (dh), = U -d[ = -& ax or where

(dh), = - I , d s

az dy - -& ac +aY

ax

ac

- -dy ;

8Y

+ ix& + i y d y ;

- IYdy

(5.4-2)

Fig. 6.4-1. Interpretation of the symbols used to derive Bouseinesq’s equations

6.4 Horizontal unsteady seepage

689

indicate the slopes of the two surfaces in the x and y direction respectively. The flow being unsteady, the depth changes not only according to location, but i n time as well. This change is equal to the modification of the position of the water table in time, because the height of the lower impervious boundary is a fixed geometrical parameter. Thus, the partial differential value of the depth according t o time is

(5.4-3) The determination of continuity requires the analysis of the water balance in the prism. Considering that the water occupies only the pores of the column, the total amount of water stored within the investigated space can be calculated for the time points o f t and t At:

+

V,(t) = n ( t ) Ax d y h ( t ) ; and

V,(t

+ At) = n(t + At) Ax A y h(t + At);

(5.4-4)

where n(t ) is the time-dependent value of porosity. Neglecting the compressibility of water, the difference of the two volumes has to be equal to the difference of the water flowing in or transported out of the column during the investigated At time-period. Further simplification can also be made by supposing a non-deformable solid matrix and substituting, therefore, the n, specific yield as a time invariant parameter characterizing porosity. The final form of the balance equation is

where F is the accretion having a dimension [LT-l], positive if the ground water is recharged vertically, and negative in the opposite case. A further assumption is the validity of Darcy’s law and the vertical position of the potential surfaces (Dupuit’s hypothesis). Thus relationships can be determined between the components of seepage velocity and the slopes of the water table

v , = K I , = - K K - ;az ax

and

vY= K IY 44

az

- - K--.

-

aY

(5.4-6)

690

6 Movement equatiom describing seepage

By substituting these values into Eq. (5.4-5) the general form of Boussinesq’s equation can be achieved:

(5.4-7)

The final hypothesis applied to simplify the basic differential equation is the aasumption, that the lower impervious boundary i s horizontal, its plane can, therefore, be chosen as a reference level, and thus the height of the water table is equal to the depth of the field (2 = h). The new form of Eq. (5.4-7) is a2h2 a2h2 -+72 ---(5.4-8) ;)* ax2 ay -

(x:

If the investigation is limited to the characterization of one flow plane (twodimensional investigation, but the flow being horizontal the movement is really one dimensional) the form of Boussinesq’s equation already quoted in Chapter 4.1 is attained: 1a2h2 n, a h E - __- - -

2 8x2

K at

K

(5.4-9)

There are only two slight differences between this equation and Eq. ( 4 . 1 4 5 ) . Here the influence of accretion is also considered, which was pre-

viously neglected. The other is a difference in the symbols applied. The depth of the field was previously indicated by y, while here h is used, because one of the axes of the three-dimensional coordinate system is indicated by y. After restricting the analysis to a flow plane, the y symbol will again be applied as flow depth. A further frequently used version of the differential equation of unsteady seepage is the form of Eq. (5.4-8) expressed in polar coordinates, which is more suitable for the characterization of axial symmetrical flow: (5.4-10)

Before discussing the approximations necessary for the analytical solution of these differential equations it is worthwhile summarizing the hypotheses used so far and in this way survey the reliability of the basic equations: (a) Both the water and the solid skeleton of the flow space are incompressible. (b)The change in the position of the water table is immediately followed by the change in storage and, therefore, the storage mpacity is time-invariant constant, equal to the specific yield ( n J : (c)The impervious lower boundary is horizontal, and (d)Both Darcy’s law and Dupuit’s hypothesis are valid.

691

5.4 Horizontal unsteady seepage

In spite of the detailed approximations there are only a few very special forms of seepage when Boussinesq’s equation can be solved directly by analytical methods. Such special solution may be achieved by the separation of the x and t variables by expressing the h ( x , t ) function as a product of two functions one depending only on x and the other on t. This method was applied by Boussinesq (1904),to determine the parameters of two-dimensional flow, when the field is drained by an infinite series of sinks located along the lower horizontal impervious boundary. It was also his proposal to combine the x and t variables into one a parameter, within which the interrelated role of x and t was determined a priori. This method was used by Polubarinova-Kochina (1952, 1962) in the form of h ( z , t ) = f(4;

where (5.4-1 1)

to determine the specific flow rate crossing the vertical starting section (where the surface water and the seepage field contact one another), if the level of the surface water was instantaneously raised or lowered from an elevation of HI to H , (Fig. 5.4-2). The result of this derivation gives the flow rate as a function of the geometrical and physical soil parameters depending also on time: n q= i H , H , K A (5.4-12) 2t‘

li

Following a similar derivation the propagation of a ground-water wave can also be determined, if the original level of the surface water was below

.or@inal

impervfous boundary

A

Fig. 5.4-2. Interpretation of the symbols used in the examples demonstrating the direct solution of Boussinesq’s equations 44*

692

5 Movement equations describing seepage

the lower impervious boundary of the field and, therefore, the wave pene$rates into dry soil after suddenly raising the level of the surface water. The length of propagation in this cwe is (Fig. 5.4-2c):

s o ( t )= 1.62

V T -.

(5.4-13)

A special type of propagation of the ground-water wave is also described by Irmay’s method (1961). The basis of this derivation is similarly the separation of the x and t variables. The h ( x , y ) function is not composed, in this case as the product of the two new functions but as their sum. The final relationship is suitable for the characterization of the action of surface water, the level of which rises gradually and is described by a linear function of time (Fig. 5.4-2d). It can be stated, however, that both the direct methods listed here and those described in the various publications (Polubarinova-Kochina, 1952; 1962; Aravin and Numerov, 1953, 1965; Bear et al., 1968; and Bear, 1972) apply special hypotheses t o find the possible solution of the relevant differential equation and, therefore, they always satisfy those boundary conditions which correspond to the assumed approximations. Thus, their use in practice is limited to the cases characterized by the special conditions studied. To obtain more generally applicable solutions, various approximations are proposed to linearize the basic differential equations. Several linearization techniques are described in the literature (apart from those publications listed in the previous paragraph Jacob and Lohman, 1952; Charnyi, 1951; Bochever and Verigin, 1961; and Kar&di, 1963). The method most frequently used, assumes that the change in the depth of the ground-water flow along the field and in time is negligible compared t o the average depth of flow ( h 9 H I - H,; where h is the average depth, H , is the natural depth and H , is the highest point or smallest depth along the field and within the investigated time period; Fig. 5.4-3). Accepting this approximation the area of cross section is supposed to be constant and thus, this value multiplied by the slope of the water table and hydraulic conductivity gives the specific discharge according to Dupuit’s hypothesis. This product differentiated according t o the horizontal distance, has to be equal t o the change of depth in time multiplied by the specific yield. Thus, the

impervious boundary Fig. 5.4-3. Interpretation of symbols used for the linearization of Ronssinesy’s equation

693

5.4 Horizontal unsteady seepage

simplified form of the Boussinesq’s equation valid for one dimensional flow after neglecting the effect of accretion is (5.4-14)

There are also proposals as t o how the average depth of the field should be calculated. Verigin (1949; 1952a) for example gives twice the weight t o oiezometrich e

pf(-FJu layer

h

,,\,

I

‘X

/

I

imperiwus bourdqf

Fig. 5.4-4. Derivation of Roussinesq’s equation for semi-confinedfield

the origiilal depth ( H , ) and consider the minimum depth ( H , ) as having only a weighting factor equal to unity: (5.4-1 5)

Another possible method of linearization could be to raise the power of h in the case where i t is differentiated by the time and substitute here the average depth of flow: (5.4-16)

Finally, special functional mapping can also be used to simplify the differential equations. The application of such solutions depends, however, on the boundary conditions t o be satisfied. Summarizing the above, i t can be stated that the general solution of Boussinesq’s equation requires its linearization, the most frequently applied basis of which is the supposition of a constant, time-invariant cross-sectional area along the field. This hypothesis is equivalent to the application of the models determined for semi-confined aquifers [see Eqs (4.1-16) and (4.1-17)J For this system, Boussinesq’s equation can be directly derived, resulting in a linear differential equation. Using the symbols represented in Fig. 5.4-4, where the height of the piezometric line above the lower impervious boundary is indicated by y to distinguish between the semi-confined and unconfined systems, the combination of the equation of continuity and Dupuit’s hypotheses provides the new form of the differential equation. I n the most

694

6 Movement equations describing seepage

simple caae, if there is no accretion along the field or its influence isnegligible the equation is as follows:

and

therefore

m K -a2Y - n s-- .

a22

aY at

(5.4-17)

Considering that the basis of linearization is the supposition of a constant average depth which is equivalent to the thickness of the layer in the new model ( h = m) and the role of the water table is taken over by the piezometric surface ( h = y ) , it is evident that Eqs (5.4-16) and (5.4-17) are identical. Thus the use of this made1 provides the same results and the same accuracy, as the various methods of linearization (Kov&cs,1966). The application of this model will be demonstrated by using a practical problem as an example, the solution of which is given by the equation most frequently applied among those derived when assuming unsteady flow: i.e. TheisJacob’s equation for the characterization of unsteady seepage around a pumped well (Jacob and Lohman, 1952) (Fig. 5.4-5). The suppositions applied in the derivation are aa follows: (a)the horizontal water transporting layer extends to infinity is each direction and the boundary condition does not change there; (b) the storage capacity characterized by the specific yield (n,) is time-invariant and does not depend on the position of the investigated point either.

i impervious boundary Fig. 6.4-6. Axial symmetrical unsteady flow around a pumped well

695

6.4 Horizontal unsteady seepage

Thus, the equation of continuity for an annular area of the aquifer around the well and Dupuit’s hypothesis can be combined in the following form: - &(r

and

+4

1 At

= n, 2r iz

[Y@) - Y(t

+41;

& ( r ) = 2 r n m K - aY ; ar

thus

a& = ns2rn-;aY at

ar

and (5.4-18) which is equal to Eq. (5.4-10). if it is linearized by aasuming the constant depth of the field, and the influence of accretion is neglected. Considering that the piezometric level was at H, above the impervious lower boundary before pumping and supposing that the pumped yield is the solution of the differential equation results in the vertical constant (Q0), coordinate of the draw-down surface above the reference level as the function of r and t independent variables:

-

t ) = Hi -

4n mK U

where u=-

r2 n,

(5.4-19)

4mKt

is a new independent variable combining r and t . The solutions contains the exponential integral function, which can be expanded into series:

-

j ~du=-o.5772--ln

u+u+

u2 2.2I

u3 u4 +- -+ . .. . 3.31 4.41

-

(5.4-20)

0

To solve Eq. (5.4-19) Theis (1935) proposed the application of a graphical method, while Jacob has proved that i t can be considerably simplified, if u < 0.01, because in this case the square and the higher powers of u can be neglected. Consequently, there is a time point to each value of r above which the draw down (the difference between the original horizontal piezometric level and its value changing at a certain time and location) can be expressed in a relatively simple form: Q0 1.5(mKt/ns)1/2. 8 ( r , t) = H , - y ( r , t) = ’ 2nmK r if n 1 t>r2---S--. ( 5.4-2 1) mK 0.04.

5 Movement equations describing seepage

696

The numerical application of this method is demonstrated in Chapter 3.1 where the determination of hydraulic conductivity by the evaluation of the unsteady state pumping tests is discussed (Fig. 3.1-11).

5.4.2 Application of the differential equation of unsteady flow to the characterization of seepage in an infinite field

For the general application of the simplified model, based on the supposition of a semi-confined water transporting layer (which directly provides a linear form of the basic differential equation) in an infinite seepage field Eq. (5.4-17) has to be integrated between two time points ( t l and t z ) along the entire length of a semi-infinite field (0 < 5 < 00).The draw-down [s(z, t ) = = H l - y ( z , t ) ] is substituted instead of the depth of the ground-water flow, which is naturally negative, if the level of the surface water is raised at the starting section. The resulting integration is expressed by the following relationship: tl

n , [ s ( z ;t l ) - s(s; tz)]dz. I,

(5.4-22)

0

It is worth-while to note the physical interpretation of Eq. (5.4-22). It states that the product of the fiow rate through the starting section (the area multiplied by hydraulic conductivity and hydraulic gradient) and the elementary time unit summarized between two time points, is equal to the change in the stored volume. The latter can be calculated as the difference between the two water tables corresponding to the time points f , and f , multiplied by storage capacity. The total change in the volume of the seepage space can be characterized by the area enveloped by the two seepage lines in the c u e of two dimensional flow, when a unit width of the space normal to the flow plane is investigated. The solution of Eq. (5.4-22) is complicated by the fact, that unsteady seepage may be created by different influences and, therefore, the boundary conditions (the change in the level of the surface water at the starting section) have always to be considered i n the solutions of practical problems. To achieve a generally applicable solution (or at least one which satisfies the boundary conditions occurring most frequently in practice), a combination of the approximations mentioned in connection with the direct solution of the original differential equation is proposed. That is the separation of the variables, and the use of an a priori determined relationship between time and distance (KovAcs, 1966):

45,t ) = f l ( t ) q l ( Z , t ) ;

(5.4-23)

where f l ( t ) is t ~ known e boundary condition [ f l ( t ) = s ( 0 ; t ) ] , influencing

697

5.4 Horizontal unsteady seepage

the investigated unsteady flow. Thus, for a semi-infinite flow field the q~ function has t o satisfy the following conditions: ifz=O; if

2

v ( x , t ) = l ; (O
= 0 0 ; qI(2,t)= 0; ( 0 < t

if t = 0 ; p(5, t ) = 0;

( 2 ,t )

( 0 0 ) ;

< (0< 5 <

00);

(5.4-24)

-2).

Apart from the suppositions previously mentioned a further approximation is applied here. Instead of investigating the propagation of the action caused by the change in the boundary condition, i t is assumed, that the modification of the water level at the starting section is immediately followed by a change in the position of the water table in the entire infinite flow field. The static water table remains unchanged only at infinity. The most simple mathematical formula of ( 2 , t ) satisfying this hypothesis and the conditions in Eq. (5.4-24) is:

~ ( zt ),= exp - f;t)l*

[

(5.4-25)

111 this relationship some restriction has to be made in connection with the f2(t) function, as according t o the third condition in Eq. (5.4-24) q~ (5,t ) = = 0, if t = 0 , and, therefore, f2(t)has t o be equal t o zero a t the start [ f z ( t )= =0, if t = 01. Substituting Eqs (5.4-25) and (5.4-23) into Eq. (5.4-22) a relationship can be achieved between the known boundary condition [ f l ( t ) ]and the other time-dependent function [f 2 ( t ) ] :

and ~

rnK .~ f l ( t ) - d ‘ f 1 ( t ) f 2 ( t ) 1 = f l ( t )f ; ( t ) + f;(t) f 2 ( t ) . n s fz(t) dt

(5.4-26)

This is the general form of the unified solution, which can be applied if the f 2 ( t ) function can be expressed from the f l ( t ) boundary conditions given by the character of the practical problems investigated. I n the following part of this section the methods suitable for the determination of this function will be demonstrated in the special cases occurring most frequently in practice. The hydmulic parameters (known as boundary conditions or to be determined) are as follows: so(t) the cltange in the water level at the starting section; .s (5,t ) the draw-down (or the rise of the pressure head) related t o the natural horizontal level of the piezometric surface (water table a t diflerent disiance from the starting section depending on the time elapsed after the initial time point of the investigation;

698

6 Movement equations describing seepage

qo(t) specific flow rate through the starting section as a function of time; q ( x , t ) specific flow rate depending also on time, through the various vertical sections of the field, with the distance measured from the starting sections. Rapid change in the level of the surface water (Fig. 6.4-6)

The boundary condition is characterized by constant draw-down (or recharging head) : so(t)= f l ( t ) = 0 ; if t = 0; and (5.4-27) sO(t) = f ( t ) = ,s = const.; if t 0.

>

I

I

I

I

Fig. 5.4-6. Hydraulic parameters of flow created by rapid change in the level of the surface water

Substituting this condition into Eq. (5.4-26) the hydraulic parameters to be determined can be calculated. The fz(t)function has to satisfy the (5.4-28)

condition. One of the suitable and simple formulae which can be proposed as the solution of the problem is

(5.4-29)

Consequently the hydraulic parameters to be determined can be expressed in the following forms:

699

5.4 Horizontal unsteady seepage X

q ( x , t ) = - mK

Unsteady flow created by constant discharge or recharge (Fig. 6.4-7)

In this case the boundary condition is given in the form of an a priori determined value of the flow rate crossing the starting section qo(t) = 0; if t = 0 ;

qo(t) = qm = const.; if t

I

I

1

(5.4-3 1)

> 0. I

1

I

Fig. 5.4-7. Hydraulic parameters of unsteady seepage created by constant discharge or recharge

Considering Eqs (5.4-23) and (5.4-25) this flow rate can be expressed as the ratio of f l ( t ) and fz(t)functions introduced in this method: (5.4-32)

Selecting one pair of the possible corresponding f l ( t ) and fz(t) functions, the following solution can be proposed

fz(t) = v

e

t ;

n S

fl(t)= and

SVF

(5.4-33)

700

5 Movement equations describing seepage

Thus, the final result in the form of the hydraulic parameters to be determined is aa follows:

and

Gradual change in the level of the surface water (Fig. 5.4-8)

The boundary condition cannot be changed by a finite value within an infinitely small time interval1 in practice. Dracos (1963) haa found that the exit point cannot follow the rapid lowering of the surface water, even if the rate of change is finite. According t o his investigation the highest possible velocity of the lowering of the exit point is

K

-sin2 B ;

(5.4-35)

n where B is the slope of the surface dividing the seepage space from the free water body. It is advisable, therefore, t o approximate the boundary condition with a smoothly changing function t o avoid the occurrence of an infinite flow rate a t the time point t = 0. With a suitably determined function i t can also be proved, that the rate of lowering should be less, than the limit value determined by Eq. (5.4-35). One of the possible approximations may be the use of the following function: t (5.4-36) sn(t)= f l ( t ) = S , t h 2 - ; tn max

*

I

.~ -_L-

e:X=O)

I

I

I

t

t

.-L

3 -

l

l

s,x=71t

pig. 5.4-8. Hydraulic parameters of flow created by the gradual change in the water level

5.4 Horizontal unsteady seepage

701

where t o is the time point after the beginning of the operation, when the required s, draw-down has to be practically achieved. The substitution of this relationship into Eq. (5.4-26) provides the other time-variant function:

(5.4-37) to Considering the f l ( t ) and f 2 ( t ) functions determined in this way the following formulae give the final solution:

Prom Eg. (5.4-38) the expected maximum flow rate can also be calculated, which is an important parameter, for example in the operation design of (5.4-39)

-

The development of this flow rate is expected to occur around the time 0.5 to. point t If the limiting velocity of lowering has to be considered e.g. in the form of the Dracos' equation, the corresponding s, and t o values have t o be chosen by taking into account the fact that the derivative of the 42, t ) function, according t o time a t time point t = O and at section x = 0, should not be higher than a given value 1

-s ,

2

-

K . <sin /3.

(5.4-40)

702

6 Movement equations describing seepage

This condition determines the t o value depending on physical ( K , n) and geometrical (/?) parameters, when the a priori given s, draw-down can be achieved. The combination of constant flow rate and constant draw down (Fig. 5.4-9)

In practice the lowering of the water table generally starts with a pumping of the constant flow rate, and after achieving the required depth, the water level is maintained in the well with decreasing pumping. The parameters. belonging to this method of operation can be determined by combinc L

Fig. 5.4-9 Combination of constant discharge and constant drawn-down

ing the two boundary conditions previously analyzed and described in Eqs (5.4-27) and (5.4-31). In this derivation it has to be considered that after a t o period of pumping at constant flow rate, a water table develops in the seepage field, which is identical with that created by the maintenance of a constant lowered level at the starting section after a period of t o / 2 having elapsed, measured from the fictive starting point of the rapid change of the water level. Thus, at this time point the two processes can be combined to one another without discontinuity, and, therefore, both the boundarv conditions and the results have to be determined separately for the periods t < t o and t > t o respectively: for t

< to m

Kn,

for t > t o so(t) = a m ; -~

qo(t)= smv> f 2t - t o (5.4-41)

(5.4-41)

703

6.4 Horizontal unsteady seepage The influence of surface water with fluctuating level (Fig. 5.4-10)

The unsteady ground-water flow created by the periodically changing level of surface water (flood waves on a river) can be characterized by this type of mathematical model as well, if the hydrograph of the river (its level determining the boundary condition and changing in time) can be well approximated with a periodic mathematical function (e.g. with some combination of trigonometric functions), (KovBcs,1962). Such time variant boundary condition may be (5.442)

where 2h is the amplitude of the wave and 2t0 is the wave length. Let us suppose that the change of the water table at each time point can also be well approximated with an exponential function. The function which expresses the draw down depending on the 5 and t variables, has to be divided into two parts aa before but one part should now be a single valued function of 5,while the other remains time- and space-dependent. Applying the boundary condition given in Eq. ( 5 . 4 4 2 ) , and accepting the approxima-

Fig. 5.4-10. Hydraulic parameters of flow created by periodically fluctuating boundary condition

704

6 Movement equations describing seepage

tion mentioned before, the position of the water table depending on time and distance can be calculated:

h s(x,t)=-exp(-Az) where

vz

(5.4-43)

There are two important parameters characterizing this type of unsteady ground-water flow, firstly the curves enveloping the possible extreme positions of the water table as the function of the distance measured from the starting and secondly the time lag between the culminations of the section [smax(x)]; wave at the starting section and at a distance of x from i t ( A t ) : z (s ,),

= -f he-AX;

(5.4-44)

and

t At = 3!A x . n

5.4.3 Unsteady seepage in a horizontally limited field In the previous section a unified and simplified method was discussed for the determination of the unsteady seepage developing in a semi-infinite field considering different boundary conditions at the starting section, the position of which is geometrically determined and known. Apart from the hypotheses generally applied in seepage hydraulics two further basic suppositions were accepted: i.e. the field is covered by a semi-permeable layer and the vertical section of the draw down surface can be approximated with an exponential curve. The first hypothesis is the physical interpretation of the approximation usually applied for the linearization of Boussinesq's equation, and thus i t can be used for the characterization of limited seepage fields ~ t 8 well. By using an exponential function to describe the piezometric line (or the water table) the actual propagation of actions originating from the change in the boundary condition is neglected and it is supposed that the influence of the change can be observed in the first moment along the entire field, (naturally at a very rapidly decreasing rate as i t corresponds to the exponential relationship). This second supposition is not acceptable, therefore, when a horizontally limited seepage field is investigated and another approximation has to be sought, which satisfies the special boundary wnditions occurring at the external vertical border of the field. At the end of the seepage field opposite to the starting section two extreme boundary conditions can develop (Fig. 5.4-11). The field may be bordered either by an impervious layer or contacted by surface-water where there is no resistance against the water exchange between the layer and the surface

5.4 Horizontal unsteady seepage

705

seepagefield' having free ror~tac?'w7b fbe suVace waters a f both sides (a)consiaanf mter level at one side (s,,,~=0) (b) different influems at tbe sides (smff s , ~ (c) s,mmetrical& infiuenced fieid (sml =s;nz) seepagefield bavng impervious border &one side (d)

~I

impervious boundary

(Cl ..

fb) \-I

impervious boundary

mpervious bounddry

Fig.6.4-1 1. Possible boundary conditions along the external vertical border of a limited seepage field

water (entry or exit face). Between these two basic conditions a continuous series of transition forms may developif water exchange is partially restricted by semi-pervious material covering the contacting surface. Investigating only the absolutely closed field and that undergoing water exchange without the development of any resistance at the contacting surface, further cases have to be distinguished within the second group. The level of the surface-water recharging or draining the field at the external face may be constant (steady boundary condition), or it m y change in time (unsteady boundary condition). The change of the potential along the contacting surface may be the same aa that at the starting section or may differ from it. Although there are many possible variations, the hydraulic investigation of one type is sufficient, (the field drained or recharged by the external contacting surface water having constant level) because the other conditions can be derived from that as indicated in Fig. 5.4-11. If the water level changes not only at the starting section, but the external boundary condition is also 46

706

5 Movement equations describing seepage

unsteady, the influences of the two actions have to be determined separately and the results can be superimposed. Section 1 is regarded at h t as a starting section with given boundary condition, assuming constant water level at Section 2. I n the second step the actual change of the water level at Section 2 is the action.creating the unsteady flow in the field, which has constant water level at Section 1. If the changes of the two boundary conditions are identical, the draw-down curve is symmetrical. It is evident that the flow pattern in a field with an impervious vertical boundary is equal to half of the flownet developing a symmetrically influenced layer having double length compared to the original field.

impervious boundary Fig. 6.4-12. Development of unetedy eeepage in horizontally limited seepage

field

On the basis of the explanation given in the previous paragraph the task is the determination of a method suitable for the approximation of the hydraulic parameters of the unsteady i b w in a finite, two-dimensional seepage @ld having a length of L,bordered below by a horizontal impervious bed, covered by a horizontal semi-pervious layer and having in this way constant thickness of m (Fig. 5.4-12). The boundary condition at the starting section is given as an a priori known parameter of the flow [ H , ( t ) ] ,while i t is supposed that the field is recharged or drained without resistance at the other and by a surface water having constant level (HZ = const.). The basic equation [i.e. Eq. (5.4-17)] remains unchanged, while its integrated form [Eq. (5.4-22)] has to be slightly modified: 11

n,[s(s;t l ) - s(z; t z ) ] d s . tl

(5.445)

0

In this case the draw-down curve is approximated by the product of two functions the h t depending only on time and being equal fo the given boundary condition, and the second combining the influence of both variables s and t (Kovhcs, 1975). Equation (5.4-23) can be used, therefore, without any change from the previous investigation. The initial and boundary conditions determining the form of the possible approximation have to be revised and modified, according to the actual data of the field investi-

6.4 Horizontal unsteady seepage

707

gated. Thus instead of Eq. (5.4-24) the following conditions have to be satisfied : if x = 0; rp(x, t ) = 1; ( 0 < t < 00); if x = L; rp(z,t)= 0 ; (0 < t if t = 0; p ( z , t ) = 0;

< (0 < z < L ) . 00);

(5.4-46)

There is a further special condition, which has to be considered in a finite field. In the cases discussed in the previous section the flow remains unsteady for an infinite time, and, therefore, the validity zone of the equations derived for the characterization of that type of flow is 0 < t < 00. Thus, the V(Z,t ) = = 1 if t = 00 requirement, has to be taken into account. The validity zone in time is, however, limited if the seepage field is not infinite, because the steudy state of flow m y develop at a finite time point between the closing sections, if the levelis constant at one side of the field, while at the other end it is drained or recharged to a constant level or by a constant flow rate. Indicating the time point of the development of the steady state by t,, the validity of the equations describing the unsteady seepage is 0
(5.447)

One of the possible approximations satisfying the conditions given in Eq. (5.446)may be (5.448)

It follows from the third line of the conditions that fi(z)has to be equal to

1’ L ”)

zero if t = 0, because the -value is always smaller than unity, and i t becomes zero independently of x only if its power is equal to infinity. Considering that the position of the piezometric level changes linearly with the distance in the case of steady flow through a semi-confined field having constant m thickness, the upper limit of the validity zone is determined by the condition

t = t,; if f2(t)= 0.5. 45*

(5.449)

708

6 Movement equations describing seepage

Combining Eqs (5.4-23) ( 5 . 4 4 5 ) and (5.4-48) the relationship between f l ( t ) and f2(t) can be determined for a horizontally limited field: (5.4-50)

This equation has to be solved by considering the special boundary conditions for the derivation of the practical relationships. Rapid change of the surface-water level at the starting section (Fig. 6.4-13)

The characteristic boundary condition at the starting section is s,(t) = f l ( t ) = 0; if

t = 0;

and s,(t) = f l ( t ) = , s = const.;

if

t > 0.

(5.4-5 1)

It follows from this condition, that f;(t) = 0, except at the time point t = 0. Substituting these values into Eq. (5.4-50), a relationship can be determined in implicit form: (5.4-52)

11 - f2(t)I exp [f2(t)l = exP 0

0.1

0.2

0.3

0.4

0.5

firt)

Fig. 6.4-13. The change in the piezometric level in time in limited flow field caused by rapid draw-down

6.4 Horizontal unsetady seepage

The corresponding values of thef2(t)and

709

mK t parameters are represented in Ln,

Fig. 5.4-14. From the graph, the upper time limit of the validity zone of the unsteady state can also be determined

mK

if f2(t) = 0.5; -t, = 0.193; L2n, consequently L2ns t, = 0.193 -.

mK

(5.4-53)

After determining f2(t),all the hydraulic parameters (the position of the piezometric surface, the flow rate depending on time and distance, as well as the flow rate through the starting section), can be calculated:

go@)= 8 ,

"[

- -- 11.

L

f2W

I

Substituting the f 2 ( t ) = 0.5 value, which belongs to the upper limit of the validity zone ( t J , the parameters become equal to those describing the steady

mk

Fig. 6 . k 1 4 . Relationship between the f&) and -t parameters in the cam of n, unsteady seepage through finite field caused by the rapid change in the surface-water level

710

5 Movement equations describing seepage

seepage created by a pressure difference s, = HI - H,:

and

because (5.4-55) and

[-&

- 23 =O; if fi(t) = 0.5.

Unsteady flow created by constant discharge or recharge (Fig. 6.4-15)

I n this case the boundary condition at the starting section is given in the following form: qo(t) = 0; if t = 0; and

mK L

= qm = const.; if t

POU) = -ffl(t)

----

0.0905

-........... 0.179

0.290

10

0

smax8

...--... 0.414

y

- 0.5

> 0.

(5.4-56)

R.009 0.039 0.I18 0.282 0.5

Fig. 6.4-16. The change in the piezometric level in time in limited flow field caused constant discharge

5.4 Horizontal unsteady seepage

711

From this condition the following relationships between the two timedependent functions [ f l ( t ) and f 2 ( t ) ] can be derived:

qrnL fA4 = -

mK 1

f2(t)

.

-f2(t) '

(5.4-57)

Substituting these values into Eq. (5.4-50), expressions are achieved for the calculation of the f l ( t ) and f 2 ( t ) functions depending on the time elapsed since the start of the process. The stages of this derivation are as follows: (5.4-58)

consequently

mK 1 -t = z + -+c; L2n, 2 where z = 1 - f2(t) and the C constant of integration can be determined from the condition, where, at the time point t = 0 the power in Eq. ( 5 . 4 4 8 ) has to be i n h i t e to achieve the p (2,t ) = 0 condition independent of z. Thus the final forms of the two time-dependent functions are:

(5.4-59)

where smaXis the maximum possible draw down defined by Eq. (5.4-63). Further hydraulic parameters can be calculated by substituting f l ( t ) and f2(t) into the corresponding basic relationship:

(5.4-60)

712

6 Movement equations describing seepage

As in the previous caae the t, upper limit of the validity zone of t,he unsteady seepage can be determined from the condition, where the f 2 ( t ) function has to be equal to 0.5 at this time pont:

2L2n,

mK t, - -t

2~2n,

2~2n,

2 -

(5.4-61)

2

1 L2ns

t,=TmK. When the unsteady movement reaches this upper limit, the hydraulic parameters become equal to these characterizing the steady state q&)

= qm = const. = so, TaK.,

L s,(5,

t ) =,s

L-x. ~

L

,

because (5.4-62)

It also follows from the relationships derived here that there is a maximum poasible lowering of the water level belonging to a given discharge, because the development of the steady state ensures the continuous recharge at the other end of the field. This maximum draw down is ,,,s

qmL . = so, = TaK

(5.4-63)

The discharge to be pumped has in practice to be determined, therefore by considering the required lowering of the water table.

References to Chapter 5.4 ARAVIN, V. J. and NUMEROV, S. N. (1963):Theory of Motion of Liquids and Gases in Undeformable Porous Media (in Russian). Moscow. S. N. (1966):Theory of Motion of Liquids and Gaaes in AVARIN,V. J. and NUMEROV, Undeformable Porous Media. Jerusalem. BEAR,J. (1972):Dynamics of Fluids in Porous Media. Elsevier New York, London, Ameterdam. BEAR,J., ZAELAVSKY, D. and IRMAY, S. (1968): Physical Principles of water Percolation and Seepage. UNESCO, Arid Zone Rmearch, Park. BOOHEVER, F. M. and VERIQIN,N. N. (1961): Methodological Guide to Calculate Ground-water Resources for Water Supply (in Russian). Moscow. BOUSSINESQ, J. (1904):Theoretical Research on the Flow Rate of the Ground Water percolating in Soil, and on the Yield of Sources (in French). Journal Mathbmatique pure et Appliqde, Vol. 10.

5.5 Model laws for sand box models

713

CHARNYI,J. A. (1951): Methods of Linearization of Non-linearDifferential Equations of Heat-transport Type (in Russian). Id. A d . Nauk USSR, 6. DRACOS, T . (1963): Two-Dimensional Unsteady Unconfined Ground-water Flow (in German). VAWE-Mitteilung, No. 67. IRMAY, S. (1961): Unsteady Flow Through Porous Materials. 9th Cmgress of IAHR, Dubrovnik, 1961. JACOB, G . E. and LO-, S. W. (1962 : Unsteady Flow to a Well of Constant Draw-Down in an Extensive Aquifer. T A U, 33. -1, G. (1963): Hydr8dics of Linear Draining Structures. (Doctoral Thesis, manuscript in Hungarian). Kbartum, Budapest. KOV~CS, G. (1962): Dimensioning Flood-control Levees for Underseepage. A d o Technica Acaokmiae Scientkmm Hungaricae. KOVACS, G. (1966):Physical Int retation of Linearization of Differential Equations characterizing Unsteady Seepage~ympos~um m Seepage and WeU Hydraulice, Budapest, 1966. KOV~CS, G. (1967): Practical Characterization of Unsteady Seepage in the Vicinity of Drains (in Hungarian). Hidroldgiai KOd6my. Kovdos, G. (1976): Interaction between Rivers and Ground Water (in English) IAHR Symposium, Rapper&, 1975. POLWARINOVA-KOCI, P. YA. (1962): Prop ation of the Ground-water Wave in the Case of Infiltration from Canals (in Russian)%oU. Akad. Nauk SSSR. 6. THEIS,C. V. (1936): T h e Relation between the Lowering of the Piezometric Surfam and the Rate and Duration of Discharge of a Well UsingGround Water Storage. TABU. VERIQIN,N. N. (1949): Unsteady Ground-water Flow in the Vicinity of Reservoirs (in Russian). Dokl. Akad. Nauk USSR, 6. VERIQIN,N. N. (1962a): Ground-water Movement along Reservoirs (in Russian). Bidrotechnicheskcye Stroitdstvo. 4. VERIQIN, N. N. (1962b): The Regime of Ground Water under the Influence of Raising and Lowering the Water Levels in Reservoirs (in Russian). Gidrotechnicheskcye Stroiteletvo. 11.

B

Chapter 5.5 Model laws for sand box models

As already mentioned in the introduction of Part 5 models are used in many cases to measure the hydraulic characteristics of the investigated seepage, or at least to determine some parameters of the relationships describing the flow, instead of the direct solution of the differential equations. The models may be a small scale form of the actual physical process (hydraulic, OT sand box models). Sometimes the transport of quantities other than water or even other phenomena producing the network of orthogonal trajectories within a given field are used for this purpose (analogue models). Reference was made to some recent English publications (Bear et al. 1968; Bear, 1972)) where the detailed description of the application of both hydraulic and analogue models can be found. It is well known that constructing a small scale model of the seepage field to be investigated, flling i t with porous material and water, and h a l l y applying similar boundary conditions along the entry and exit faces of the model to those prevailing in the prototype some of the hydraulic parameters of seepage through the model can be measured, or visualized. The stream lines are generally determined by dyeing the water at given points, while

714

5 Movement equations describing seepage

the pressure inside the sand box is measured and the potential lines are constructed from the observations (Fig. 5.5-1). Originally, this method waa only used to determine the flownets of complicated seepage fields, and the hydraulic parameters were calculated from the geometry of the net [e.g. by using Eqs (4.1-37), (4.1-38) and 4.1-40)].

Fig. 5.5-1. Observation of pressure in sand box models

The discharge percolating through the model can also be measured. This value aa well aa the observed pressures, can be recalculated for the prototype, if the model laws valid in the case of hydraulic seepage models, are known. There were, however, very few publications on this topic. It waa felt necessary, therefore, that a short summary of the model law should be given in this chapter, while the reader interested in other problems of modelling is referred to the books quoted already.

5.5.1 General derivation of model laws for hydraulic models The hydraulic model is a scale (generally decreased) copy of the flow space (and its boundaries), where the hydromechanical process to be investigated actually develops. There are special caaes, when the proportionality is not required for all the sizes of the prototype, and some measurements are deliberately distorted and modelled in a predetermined rate unlike those in the other models (distorted models). Filling the model with fluid and applying the same boundary conditions aa those prevailing in the prototype a hydrodynamics1 process develops in the small scale system, the hydraulic parameters of which can be directly measured. The fluid and the other materials transported in the model (e.g. bed load in the caae of open channel models) may be the same aa those moving in the original system, or different materials may also be used. In the latter cwe the differences between the physical properties of the transported material have to be considered when evaluating the data measured in the model.

715

6.6 Model laws for sand box models

The hydraulic parameters of the original system (which are not meaaurable, because of the large size of the prototype or because i t is only a designed condition and not existing in reality) can be calculated from the data of the model, if the similarity between the two hydrodynamic processes is ensured. This similarity is expressed by the model laws, which give a relationship between the interrelated data of the two systems. The first requirement of the total hydromechanical similarity is the constant rate of the corresponding geometrical parameters of the prototype and the model (geometrical similarity). If this condition is satisfied, any size of the original system (1) divided by the corresponding size in the model (1’) gives a constant value called the transforming factor of length ( A ):

1 - = A = const. I’

(5.5-1)

The next condition is the kinematic similarity, which ensures the constancy of the rate of the interrelated kinematic parameters. This condition requires that the form of paths of two moving particles (one in the prototype and one in the model) should be similar, and the time ( t ) needed for a particle to move along an 1 stretch of its path in the prototype related to the model 1 time ( t ’ )necessary for a particle in the model to paas an I’ = -stretch of the 3,

corresponding path should be conatant, independently of the place and the time point of the investigation: t t’

- = z = const. ;

(5.5-2)

where t is the transforming time factor. Finally, the hydromechanical similarity is regarded as t o t d , if the ratio of all the external and internal forces acting at the interrelated points of the prototype (P)and the model (F’) at the corresponding time points is also constant (dynamic similarity) :

F = z = const.

F‘

(5.5-3)

and this ?G constant is called the forces transforming factor. From the three baaic transforming factors the constant ratio of the interrelated values of any other original and model parameters can be calculated, as indicated in the first column of Table. 5.5-1. The role of model laws is to create contact between the basic factors, making the calculation of any transformation possible if one of the ratios (generally the A transforming factor of the length) is chosen arbitrarily and determined a priori. Only those relationships, which provide the determinaton of such contact between the basic factors can be called model law. The ratio of the forces acting on the moving particles in the original system and in the model respectively can be determined separately for each

716

6 Movement equations describing seepage

Table 6.6-1. Transforming fwtors of some physical quantities moat frequently used in hydr8ulics on the basis of vdous model laws

Length 111' Area AIA' Volume VlV' Time tlt' Velocity v/v' Acceleration a/a'

&chsrBe QIQ' Force F/F' Work W/W' Power PIP'

type of force. Thus the numum of e q u a h n s describing the rate of forces, is equal to the number of forces having a dominating role in the system. These equations can be used to derive the relationships between the baaic transforming factors. As already mentioned, the three most important forces acting within the seepage field are gravity, internal friction and inertia. Even in the cam of other hydrodynamic processes these forces are generally regarded aa dominating wtions. Three forces provide three equations expressing the ratio of the corresponding forces within the two interrelated systems. The ratio of gravitational forces: (5.5-1)

assuming that acceleration due to gravity is the same in both systems, and expressing the rate of densities of the two fluids by a =.:e The ratio of forces caused by internal friction:

e

where = q/q' is the ratio of the dynamic viscosities of the two fluids (i.e. those percolating in the original system and those applied in the model, respectively). The ratio of inertia (5.5-6)

5.5 Model laws for sand box models

717

The equations give the following conditions for the relationships between the three main transforming factors, A, z and n if the mentioned three forces are dominant only in the system:

f G ( A , z, n,a ) = aA3 - n = 0 ; A2

fs(1, z , n , p ) = p - z f T ( A , z,

n = 0;;

(5.5-7)

14

n,a ) = a - - n = 0. 22

The three equations contain five variables, among which two, ( a and /l) depend on the physical properties of the fluids moving in the two systems. Their values are, therefore, determined by choosing the fluids. The other three can be calculated from the three equations. None of them, therefore, can be chosen arbitrarily, if the parameters of the fluids are given as a priori determined quantities. The most common case is where the two fluids are identical. The movement of water is investigated in the natural system, and water has also to be used as the model fluid, where the use of another medium would be too costly because of the great quantity required. Neglecting the di#erences in density and viscosity which may be caused by the temperature difference, the conditions of Eq. (5.5-7) can be simplified: fp&,

z, n) = A 3 - n = 0; A2

ps(2, z, n) =- - n = 0; z

&A,

(5.5-8)

14

z , n ) =-- n = 0. z

The solution of this system of equations gives the simplified result

A=z=n=l;

(5.5-9)

which is the mathematical expression of the statement, that in the case where the same fluid is used in both the prototype and the model and having three or more dominant forces, the effect of which cannot be neglected, the total hydromechanical similarity can only be achieved if the sizes of the two systems are equal to each other and naturally in this case the corresponding time periods, and the interrelated forces are also equal. Three or more dominant forces can be considered, therefore, in a small scale model, only if the fluid applied in the model is different from that of the prototype (Hank6, 1965). The application of small scale models using the same fluid is limited to the cases when the influences of only two forces are dominant in the system, and thus the effects of the others can be neglected or their actions can be considered in a special way. The similarity ensured by taking only two main

718

6 Movement equations describing seepege

forces into account is called partial mecbnicul similarity, and the wellknown model laws always give the relationships between A, z and n for these cases. Only two main forces being dominant, two conditions of Eq. (5.5-8) have to be considered simultaneously. One of the basic transforming factors can be chosen arbitrarily (generally that of the lengths, A) and the other two can be calculated as its functions [e.g. z(A) and 7441. Knowing these relationships the transforming factors of the other characteristic quantities can also be expressed, depending on the A value (see further columns of Table 5.5-1). The model laws describing the relationships between the basic transforming factors are generally given in the form of dimensionless numbers, and the requirement is that the dimensionless number valid in the case of the process in question, should have the same numerical value in the prototype and the model, if i t is calculated from corresponding quantities. The dimensionless numbers can be determined as the ratio of the two forces regarded clcs dominant in the development of the process investigated, as already shown in Chapter 2.1. In the caae of water movement in an open channel, the accelerating force is gravity and the most important retarding force is inertia. From the ratio of these two forces the Froude-number can be derived. Thus, partial mechanical similarity is ensured in this caae, if this value is the same calculated from the data of the prototype and the model respectively: ( 5.5-10)

Relationships determined from this equation between the basic transforming factors, fulfill both the h t and second members of Eq. (5.5-8) simultaneously (i.e. those describing the ratio of gravitational forces and the ratio of the forces caused by inertia respectively): z=

v>; and

7c

= 13.

(5.5-11)

I n the caae of a closed system of pipes the effect of gravity can be neglected, compared to the pressure differences along the pipes, and this latter accelarating force can be modelled arbitrarily by choosing suitable pressure values applied at the open sections of the pipes. However, in this case the geometrical similarity of the piezometric lines is not satisfied, and from this point of view the model is a distorted one. Because the main accelerating force can be freely modelled, two retarding forces, in this special case, can be considered aa dominating actions when calculating the transforming factors: i.e. friction and inertia. Their ratio results in the Reynolds' number, the equality of which haa to be achieved in the two systems to ensure the partial mechanical similarity: vl vtlt - - -= Re = const. (5.5-12) v

v

To express the transforming factors z and 7c as the function of A , the same relationship can be derived from Eq. (5.5-12) a~ from the combination of

6.5 Model laws for sand box models

719

the second and third conditions of Eq. (5.5-8), (i.e. conditions determined aa the ratios of forces created by friction and of those caused by inertia in the two similar systems): t = A2; and n: = 1. (5.5-13) Finally a third variation can also be formed from two of the conditions listed in Eq. (5.5-8), i.e. the consideration of gravity and friction. On the basis of the dynamic analysis of seepage given in Chapter 2.1 it is well known that the dominant role of these two forces is characteristic in the case of the investigation of laminar seepage (the theoretical derivation of Darcy's law is also based on this hypothesis). The dimensionless number achieved by dividing friction by gravity is the Mosonyi-Kov&cs number (Mosonyi and Kov&cs, 1952; 1956): (5.5-14)

Investigating this type of movement, the relationships between the basic transforming factors can be expressed by the following equations :

t i = - ;1 A

'

and

3 ~ ~ 1 3 .

(5.5-15)

Considering the analysis of the model laws it can be stated that the partial mechanical similarity i s ensured in two geometrically similar seepage spaces if the Mosonyi-Kovcics numbers calcwlated from corresponding data of the two systems are equal to each other. Here all sizes of the systems not only the length of the contours but the sizes of grains and pores as well, are proportional to each other, 1 being constant for any size. The transforming factors of the various parameters calculated from Eq. (5.5-15) are listed in the fourth column of Table 5.5-1.

5.5.2 Geometrically distorted sand box models

As explained in the previous section only partial mechanical similarity can be achieved between the prototype and the hydraulic model. The small scale process may differ considerably from the original one, if the effects of the forces not taken into account are not negligible. Thus the characteristics calculated from the data observed in the model may also differ from the actual parameters. It is also evident that better result can be achieved, if the size of the model is nearer that of the prototype (1tends to l ) , because in this case the effects of the neglected forces become more and more similar in the two systems. This phenomenon is called scale eflect. Moving in the other direction by increasing the ratio of the corresponding geometrical parameters, the differences caused by the change of the neglected forces become greater and greater. There exists a certain limit, and if the model i s decreased below i t , the data measured in the small scale process cannot be used

720

6 Movement equations describing seepage

for the determination of the original parameters, because of the uncertainties arising from the various effects of the neglected forces. There are cases, when the proportional decrease in the size of the process investigated causes a sudden change in the basic physical properties of the phenomenon. The scale belonging to this state indicates an absolute limit of the model size, because above this limit not only are the possible errors increased by uncertainties but the character of the movement is also completely changed. These limits generally referred in classical hydromechanics as Eisner’s limits, are as follows: (a)Development of cavitation; (b) Development of capillary waves instead of gravitational ones; ( c )Limit between turbulent and laminar flow; and (d)Limit between tranquil and shooting flow.

These limits are generally supplemented in special cases by considering the effects of both surface roughness and the development of bed load movement. As is shown, however, by the character of the limits, these were determined for hydraulic models of open channel flow. It is necessary, therefore, to add, a further limit to the previous one, if seepage is investigated by using small scale sand box models. To ensure geometrical similarity each size of the prototype has to be decreased in the same proportion determined by the length transforming factor A. As already mentioned, this proportionality should also be ensured between the grain sizes (or the pore sizes) of the two systems. It is quite evident that the decrease in the grain size of the solid matrix applied in the model is physically impossible below a certain limit, the high ratio of colloid particles can cause the modification of the physical character of the seepage (development of microseepage instead of laminar flow; large stagnant parts of the field because of high threshold gradient). If the available facilities (place, water supplying system, etc.) require a smaller model than that prescribed by one of the limits, the general procedure is t o distort one or more geometrical parameters i.e. by building the model on a small scale, proportionally reducing all but one (or sometimes more) sizes of the system. In sand box models the geometry of the boundaries of the seepage space is always decreased proportionally, but the grain size is distorted applying generally coarser particles than those which would be needed on the basis of geometrical similarity. It is the usual practice, to fill the seepage space of the sand box with the same material (having the same size and distribution of grains and even the same porosity) as the prototype. T o apply the original porous medium in the model is a special cme of the distorted model, when the transforming factor of the grain size is equal to unity ( A D = 1). As discussed in Chapter 2.1 the characteristic velocity in Mosonyi-KovAcs number, can be either Darcy’s seepage velocity (v) or the effective mean velocity in the pores (vet,). There are similarly more than one choices when selecting the characteristic length. It can be either Koieny’s effectivediameter (Dh),or the average diameter of pipes hydraulically equivalent to the pores ( d o ) ,or the square root of intrinsic permeability (k).In the case of distorted

5.5 Model laws for send box models

721

models all geometrical parameters concerning the boundaries (and also boundary conditions) of the seepage space, have t o be proportionally decreased and, therefore, (5.5-16)

stating that the total pressure head has t o be determined, as does any other size of the seepage space. It follows from this condition, that the average hydraulic gradient has to be the same in both systems: (5.5-17)

At the same time the internal flow conditions of the seepage field depend on. the decrease in the grain size of that of the equivalent pipe diameter

hydraulic conductivity and intrinsic permeability being linearly proportional to the square of the grain size, or pipe diameter. Combining Eqs (5.5-17) and (5.5-18), the velocity transforming factor can be determined for distorted sand box models: V=

1

K I ; vefr=-KI; n

and V’

1

= K ’ I ’ ; v,ff = 7K’I’;

n

therefore -- %ff = A :. 0’

(5.5-19)

v:tr

The ratio of Mosonyi-KovBcs numbers calculated for the two systems can also be determined. The results will be the same, independent of the characteristic velocity and length, supposing that the latter has always t o be a length describing the i;iternal flow conditions (i.e. effective grain diameter, equivalent pipe diameter, or the square root of intrinsic permeability): (5.5-20)

It cari be stated, therefore, that the equality of Mosonyi-Kovdcs numbers calculated from the parameters of the prototype and the model, respectively, provides the model law for each type of sand box models. It ensures the partial mechanical similarity in the case of a geometrically similar model, while i t is thc condition of the required hydraulic similarity if the model is a distorted one. 46

722

6 Movement equations describing seepage

From Eq. (5.5-19) the basic transforming factors can also be determined

therefore

A and A2

n =- (Eq. 5.5-8); therefore n = A A&.

(5.5-21)

2

Eq. (5.5-21) can be further simplifiedin thespecial case when the original material is applied in the model and, therefore, AD = 1:

_v -- 1 ; % = A ;

n=A;

vt

(5.5-22)

while the other transforming factors are listed in the last column of Table 5.5-1.

References to Chapter 5.5 BEAR,J. (1972): Dynamics of Fluids in Porous Media. Elsevier New York, London, Amsterdam. BEAR,J., ZASLAVSKY,D. and IRMAY, S. (1968): Physicd Principles of Water Percolation and Seepage. UNESCO, Arid Zone Reaearch, Peris. EISNER,R. (1926): Hydraulic Laboratories in Europe (in German). Leipzig. -6, Z. (1966): Fulfilment of Similarity when Investigating Seepage Process in Small Scale Models (in Hungarian). M T A MzZszaki T u d o d n y o k Oeztrilycirrak K o d e mdnyei, Tom. 36. 1-4. HORVLTH,J. (1961): Model Law for Considering Friction and Capillarity Simultaneously as Dominant Forces (in Hungarian). Viziigyi Kozlemdnyek, 4. HORV~TH, I. (1962): Similarity of Seepage Process Considerin the Simultaneous Effects of Ca illarity, Friction and Gravlty (in Hungarian). Hic$roMgiai Kodony,3. MOSONYI, and KOV~OS, a. (1962): Model Laws for Considering Gravity and Friction Simultaneously (in Hungarian). HidroMgiai K o a n y , 7-8. MOSONYI, E. and Kovbos, G. (1966): Model Law of Seepage (in French). IASH Symposium, Dijon, 1956. SPRONCH, R. (1932): Hydrodynamic Similarity and Investigation of Models (in French). An& dea Tmvaux Publiquerr de Belgique, 1932. WEBER,M. (1919): Basis of Mechanics of Similarity and its Evaluation in the Investigations of Models (in German). Jahrbuch der Schiffbautechmbchen Qeaellschaft, 1919.

8.

Subject index

absorbed water 18 accelerating forces 202 acceleration 206 - due to gravity 30, 203 accretion 180-181,184-186, 609, 613, 661 active clay content 67-70 active root zone 99 active surface 63, 211 adhesion 101, 123, 211, 304 adhesion V.S. moisture content relationship 125 adhesive flux 3 11 adhesive force 26, 211, 269 adhesive (hydraulic) conductivity 303, 307 adhesive porosity 113 adhesive saturation 124, 294, 307 adhesive water content 124 adhesive zone 102 aggregates 39, 48, 66-67 aggregate-size distribution 120 aggregation of fine grains 48, 121, 351 airbubbling pressure 94, 129-130 air compression 290-291 amorphous colloids 70 analogue models 667; 713 analytic complex function 664 andesite 416 angle of friction 364 anisotropy 338-348, 621 aquicludes 17 aquifers 17 8guifUgeS 17 aquitards 17 areal porosity 22-23, 427 artesian water 16 artesian well 16 atmospheric zone 99 attractive force 63 average shape coefficient 46 axial symmetrical seepage (flow, movement) 328, 694

basalt 412 basic mapping functions 676 bedding planes 41 8 Bingham’s plastic 21 1 binomial form of seepage law 261 body force 203 boiling effect 350, 362 Bondarenko’s constant 273 boundary condition 496, 605 - external 6 0 6 6 1 8 internal 6 2 2 6 2 8 Boussinesq’s equation 688 break of stream lines 623 Brownian movement 33 Buckingham’s potencial 287 Buckingham-Reiner’s equation 2 13, 270 bulk density 166

-

calcite 60 Capillarity 103, 217-224 capillary (hydraulic) conductivity 303,309 capillary contact angle 137, 219, 226 capillary exposed faces 611 capillary flux 311 capillary fringe 609, 648 capillary height 129, 224, 294, 611, 648 capillary meniscus 222 capillary porosity 113 capillary pressure 221 capillary rise 130 capillary saturation 136-136, 294, 309 capillary slit 224 capillary suction 286, 323 Capillary SUl‘faCe611 capillary tube 224 capillary tube diameter 89 capillary tube model 87-94, 302, 387 capillary water content 135 capillary water transport 612, 648-660 capillary zone 99, 103, 612 carbonate rocks 417-437

724

Subject index

Casagrande’s - Aline 70, 72 - apparatus 105 Cauchy-Riemann’s condition 490, 565 cave 418 cavitational point 484 cemented sediments 17 centrifuge moisture equivalent 169-173 change in grain-size distribution 75 characteristic curve of ground-water balance 188-199 characteristic grain diameters 44 characteristic length 231 characteristic velocity 231 chernozem soil 122 Chezy’s equation 229 classification - of hydraulic models 503 - of subsurface waters 13-18 clay content 53-69, 74 clay minerals 4 2 4 3 , 50-52 clogging 317, 380-395 closed capillary zone 99, 103, 111, 301 closed ground water 15 coagulation 39, 48 coefficient - of anisotropy 341 - of compressibility 154, 321 - of consolidation 156 of resistivity 251 - of saturation 106 - of uniformity 44, 47, 352 coefficient of uniformity V.B. porosity relationship 82 colloid particles 4 8 4 9 , 53-65, 73 colmatation 380 column drainage 162-168 combined application of mapping functions 596 complex number 563 complex potential 536, 563-567 complex variable 564 compressibility - of solid matrix 155-159, 480 - of water 154-155 compression 86, 155 - of air 290 - V.S. loading relationship 156 uoncentration - of dissolved salts 193 - of suspension 381 concentric joints 398 conceptual model for characterizing fissured rocks 454 conductivity of one slit 442 confined flow 495 confined seepage field (domain, space) 325, 497 confined water bearing layer 16 conformal mapping 565

-

conjugate directions 581 conjugate velocity 566, 568 connate water 18 consolidation 160, 205, 376 constant draw down 698, 702, 708 contact angle 137, 219, 226 continuity (equation) 476-478, 661 continuum approach 32-36, 424, 474 corner point 484 covering layer 682 critical gradient 360, 365, 378 critical velocity 360, 365, 373, 378 crystalline rocks 398, 401 Darcy ’s - hydraulic conductivity 30, 240-250 - law 28-31, 179 - veloci9 29-30, 239 - zone of seepage 231, 256 deep ground water 15 depression cone 187, 436 depth of seepage field in layered systenis 524 development of water table a t draining trenches 625 differential-thermo-analysis (DTA) 54, 73 diffuse duble layer 210 diffusion theory 283, 289 dimensionsless numbers 226-237 dipole (flow net) 579 dipole molecule 207 direct solution of Boussinesq’s equation 691 distance from the solid wall 210, 217, 223 distorted model 714, 721 disturbed samples 322 dolomite 36, 42, 417 dominant forces 203, 227 double infiltrometer 324 double layer 210 draining trench 603, 607, 625 drains along the bank of surface water 59 7 dry sieving 120 Dupuit ’8 - condition (hypothesis) 179, 628, 661 - equations 628-648 - parabola 632 dynamic analysis of seepage 202 dynamic balance (equilibrium) 98, 103, 110 dynamic similarity 715 dynamic viscosity 30 dynamic water resources 11 dynamics of soil moisture 98 effective dianieLer (Kozeny’s) 4 4 4 5 , 231 effective porosity 22

Subject index effective stress 158, 204, 376 effective velocity (mean velocity) 30 effect of free exit face 332, 517, 631 Eisner’s limits 720 elasticity of layers 206 electrostatic charge 52, 123, 210 electrostatic field 52-53 entry face (section) 508 equilibrium level 193 equipotential line 487 equipotential surface 481 evapotranspiration 98, 101, 150, 183188 exit - face, free 510, 630 - face, covered by water 508 - height 634, 649, 655, 656, 660 - point 629 - velocity 551, 644-647 expansion 159 expansion fissures 418 exponential function 581 exponential integral function 695 external boundary condition 506-518 external corner 484 extrusive volcanic rocks 17, 412417 fault zone 419 fictive starting time 335 field - capacity 107, 109-114 tests 322 filter law 370 finite difference method 558-559 first transition zone 231, 256, 258 fissured and fractured rocks 17, 397417 fissures 396 flow - conditions 31, 473 - function 488, 544 - line 482, 488, 544 - net 489, 568 - plane 485 - through saturated porous media 26 - through saturated solid rocks 26 - through unsaturated porous media 27, 283-284 - through unsaturated solid rocks 28 - tube 485 flux veator 478 foliation 401 Forchheimer’s equation 229 fossil water 18 fractional function 576, 578 fractures 396 free electrostatic charges 52 free exit face 510, 517 free porosity 109-114 free seepage (flow, movement) 616

-

725

friction 26, 206, 716 Froude’s law 718 - number 227 - zone 231, 256 fully penetrating drains 598, 604 generalized form of seepage law 266, 553 geometrical application of mapping 570 geometrical classification of seepage 493 geometrical condition, of suffusion 351, 370 geometrical similarity 229, 715 gradient 29-30, 338 gradient balanced by static shearing stress 273 gradual change in water level 700 grain - size (diameter) 39, 81 - size distribution 4 3 4 9 , 353, 371 - shape 3 9 4 3 grain size V.B. porosity relationship 81 granite 397 graphical approximation of flow nets 49 1 gravel 40, 50 gravimetric water (moisture) content 104, 106 gravitational flow 649 - ground water 111 - porosity 109-114 - seepage field 648 - storage 151 gravity 26, 203, 716 Green-Ampt’s equation 287, 562 ground-water 15-18 - balance 148-153, 193-194 - flow 26, 100, 150, 178-181 - resources 11 - surface 14 - table 14 - zone 99 halloysite 42, 66, 70 Hamel’s mapping function 538 Hazen’s design diameter 44, 89 H-bonds 214 head water 629 height of exit point 652 Hele-Show’s model 88, 440-442 heterodisperse sample 44, 83-84, 263 hodograph image 529-535 - contour 530 - field 531 - mapping 531, 368 - plane 529 homodisperse sample 45, 77, 83, 262 homogeneity 327 horizontal foundation 678

726

Subject index

horizontal moisture flux 148 Horton’s equation 283 hydraulic condition, of suffusion 369, 370 hydraulic conductivity (Darcy) 30 hydraulic conductivity (generalized) 3 1 hydraulic consuctivity, of solid rocks 437,

-

470 V.S.

tension relationship 288, 298,

intrusive rocks 17, 398-401 irrotationd movement 481 isolated porosity 22 jet, infiltrating 323 joints 398 juvenile water 18

312

- V.S.

water content relationship 288,

296, 312

hydraulic gradient 29-30, 229, 294 hydraulic head 644 hydraulic model 667, 713 hydraulic parameters of seepage 666 hydraulic similarity 721 hydrodynamic force 364, 377 hydrological classification of subsurface waters 14 hydrometric method 39, 64, 66, 68 hydromica 62 hydrostatic pressure 204, 287 hygroscopic moisture content 107-108,

Kamensky‘s equation 179-181 kaolinite 42, 6162, 64, 64, 277 karstic rocks 417 karstic water 14 Keuper marl 64 kinematic classification of seepage 493, 496

kinematic similarity 7 16 kinematic Viscosity 30 Kohny-Carman’s equation 243 Kohny’s effective diameter 44-46, 89 Kuron’s hygroscopicity 107-108, 127

127-128

hygroscopicity 107-108, 127-128 hypergeometrical function 636 hysteresis ,137, 141-146, 174-176, 294 ideal fluid 207 igneous rocks 17, 42 illite 42, 62 impervious boundary 607 index of consistency 107 of Dlaticitv 70. 106 indura&d clasdc &iments 17, 407-417 inertia 26, 206, 716 infiltration 98, 101,- 149-160, 183-188,

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3R.? - -- basin 324 - curve189-190 - from canals 616, 672 - test 322 infiltrometer 322 infinite series of sources:677, 684 influenced seepage 616 infrared photos 106 inhomogeneous anisotro y 480 inhomogeneous layer 32,! 480 initial condition 639-641 ink-bottle effect 137 integral-exponential function 336 interaction of rechargingland draining can& 667 interfacial tension 218 intermediate (porphyric) rocks 17 internal boundary condition 6 2 2 6 2 8 internal corner 484 intrinsic permeability 30-31, 232 429,

laboratory tests 319 laminar seepa e (flow, movement) 26 laminar zone 827, 231, 266, 643 landsubsidence 163, 360, 362 Laplace’s equation 481, 660 lateral air flow 291 layered see age field (domaine, plane) 6 19-62f leaching 196 leaking aquifer 600, 661 Legendre’s elliptical integral 694 limestone 42, 417 limit of plasticity 64, 106 Lindquist’s zone 231, 266 linear extension of seepage field 346 linearization of Boussinesq’s equation 696

liquid limit 64, 106 liquidization of layers 360, 362-369 load v.8. compreseion relationship 166 locd resistance 670 logarithmic function 677, 680 loose clastic sediments 17, 20 Lugeon’s number 324, 429 lysimeter 191 macroscopic characterization of flow domains 34 mapping 667 functions 664 Mariotte’s bottle 123 mathematical model 668 maximum capillary height 103, 129, 177,

-

301

maximum flow rate 643

Subject index maximum molecular water capacity 74, 106 mechanical similarity 716 meniscus 222-223 metamorphic rocks 42, 401-407 mica 42, 46, 60 micro seepage 26, 227, 269-281, 643 microscopic flow pattern 33 mineral composition of samples 60-76 mineralogical analysis 48, 64 mineralogical character of grains 60, 211 mineralogical composition of sediments 60 minimum capillary height 103, 129, 301 Mitcherlich’s hygroscopicity 107-108, 127 modulus of compressibility 86, 167 moisture content 104-109 - of rocks 14 v.8. adhesion relationship 126 V.B. hydraulic conductivity relationship 288, 296, 312 V.S. tension relationship 116, 312 moisture flux 99, 148 molecular forces 217, 227 montmorillonite 42. 61-62, 64. 64. 277 morphological character of colldd particles 73-74 Mosonvi-KovBics - rn‘bdel law 719 number 227, 236-236 motion of fine grains 349 multi-layered filter 373 multi-storied cave 418 Muntz-LainB’s infiltrometer 322

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-

Na-montmorillonite 42, 277 Navier-Stokes’ equation 478, 643 Nesterov’s infiltrometer 324 neutral stress 158, 204, 376 Newtonian fluid 206 nodal point 484 non-laminar seepage 642-664 non-Newtonian fluid 213 non-wetting fluid 220 numerical method 668 observation well 327, 341 one-dimensional seepage (flow, movement) 27, 477, 493 open capillary zone 99, 103, 112, 301 orientation of dipoles 209 parallel plates model 88, 440442 partial mechanical similarity 718 partially penetrating drains 600, 606 partially penetrating wells 329, 671 path 482 Pavlovsky’s analogy 667 mapping 692

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727

pelJicular water 102 pending capillary zone 103 pendular water 102 periodic fluctuation 640, 703 permeabimeter with changing head 320 with constant head 319 pF curve 114-117 pF value 114 physical soil parameters 293 piezometric level 497 piezometric Line 498, 629 piping effect 360 piston flow 286, 323, 660 point value of field capmity 110 Poiseuille’s equation 240-241 polarization of molecules 210 pore diameter 94 pore-size distribution 120, 131-136, 308 porosity 22-26, 169, 366 of carbonate rocks 422-428 of loose cleetic sediments 76-87 - of solid rocks 399, 402, 408, 414 V.S. coefficient of uniformity relationship 82 V.S. g r b - s i z e relationship 81 V.S. shape coefficient relationship 80 potential difference 631 potential force 204 potential form of seepage law 261 potential function 482, 487, 644 potential line 487 potential seepage 481 potential surface 482 potential velocity 481 power function 676, 680 pressure 204 distribution 204, 674 head 168, 671 of overlying layers 203, 206 wave 496 primary porosity 119, 407, 417 propagation velocity 286, 498 of wetting front 289 protective filter 369-376 pumped well 326 pumping test 324, 326-337, 436

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-

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quartz 42, 60 quesi-steady free seepage 616 quasi-steady seepage (flow, movement) 36 quick sand 362 radial joints 398 raindrop effect 137, 226 rapid change in water level 689, 708 rate of consolidation 162

728

Subject index

rate of narrow and large openings 422 rate of saturation 106, 294 - v.8. hydraulic conductivity relationship 288, 296, 312 - V.S. tension relationship 115, 312 recession curve 421 recharge of bore holes 324, 401, 405, 429 redistribution of fine grains 349 Reimann’s surface 578 relative compression 155 release fractures 398 representative elementary - length 426, 466 - unit 34-35, 424 - volume 35, 424 repulsive force 53 resistivity coefficient, in pipes 251 retarding forces 202 retention curve 113, 123 Reynolds’ - model law 718 - nuniher 228, 231-235, 333 Richards’ equation 288 riparian well 611 root - function 576, 580 - zone 99 saddle point 484 safety coefficient 374 salt - accumulation 196 - content 196 sand 42, 50 sand-box model 557, 713 sandstone 42, 407 scale effect 719 schistosity 401, 407 Schlichter’s number 77 Schwartz-Christoffel’s mapping 586-595 sealing of fissures 41 8 seasonal fluctuation 183, 185, 198 secondary porosity 119, 407, 417 second transition zone 231, 256, 268 sedimentation 39, 54, 66, 68 seepage 28-37 - law ( D t ~ ~ y ’28-31 s) - plane485 - parallel to layers 520 - perpendicular to layers 520 - resistance 521 - space (domain, field) 473 - under impervious foundation 588, 678 velocity (Darcy’s) 29-30, 32, 231, 239 - zones 231 self filtering character of layers 356 semi-confined flow 496, 661

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semi-confined seepage field (domain, space) 497, 693 series of riparian wells 613 shallow groundwater 15 shape coefficient 39-43, 80 - V.S. porosity relationship 80 shearing stress 206 sheet - joints 398 pile 545, 589 sieving 39 silt 43, 50 simplified solution of Boussinesq’s equution 696 sine function 577, 581 single well near the bank of a river 611 singular points 477, 484 sink - flow net 484, 581 - holes 418 - in uniform flow 585 slope of the terrain 195. soil - horizons 121 - moisture 14, 99-104 - - redistribution 98, 104, 287 - - retention curve 109-114, 123145, 166, 167 - - zone 99-104 - structure 418-422 - water diffusivity 288, 293, 300 solid matrix 20, 382 solid rocks 17, 21 source 484, 581 specific discharge 29, 574 specific flow rate 489 specific flux 32 specific soil moisture values 107 specific surface 40 specific water capacity 288 specific water (moisture) content 104-109 specific weight 30 specific yield 154, 286, 335 specific yielding capacity of wells 41 1, 429, 431 stability - of filters 370 - of layers 317, 350, 359 - of slopes 363 Stagnation points 484 starting section 501, 661, 675, 697 static shsaring stress 211, 217, 271 static water resources 11 statistical distribution - of fissures 455 - of pore-size 131 steady flow 29, 327, 479, 499 steady free infiltration 616 steady influenccd infiltration 616

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Subject index steady seepage (flow, movement) 29, 327, 479, 499 storage - capacity 153-178 coefficient 154-159 stream - function 488, 544 - line 482, 488, 544 - tube 485 suction (tension) head 287, 294 suffusion 350, 351-362 superposition - of a source and a sink 577, 583 - of 6 uniform flow and 6 sink 585 - of infinite series of sources 584 - of mapping functions 681 surface - deposit 383 - runoff 98 - tension 218, 220 - volume ratio 40, 50, 89 suspended load 380 swelling 116-1 17

729

turbulent zone 227, 256, 543 two-dimensional seepage (flow, movement) 27, 339, 485, 493

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tail water 630 tectonic: joints 398 tension (suction) 110, 123 - V.S. water content relationship 116, 312 - V.S. hydraulic conductivity relationshii, 288, 294, 312 Terzaghi’s - filter law 370 - equation of compression 86, 156 Theis’s equation 336, 694 three-dim~nsioniL1(flow, movement) 493 threshold gradic:rt 212, 216, 236, 273, 546 tortuosity 91-92 total rlogging 390 total draw down 333 total elliptic integral 594 total energy 287 total mechanicnl similarity 71F total porosity 22 total salt content 193 total stress (vertical) 158 transient flow 335 transition zone of seepage 26, 227, 251259, 543 transniiseibility 181, 432, 434 transporting capacity of water films 301, 304 transverse anisotropy 338, 621 trap 413 trigonometric functions 581 turbulent flow in fractures 468 turbulent hydraulic conductivity 260 turbulent seepage (flow, movement) 26, 260, 468

unconfined flow 478, 495, 661-669 unconfined seepage field (domain space) 325, 497 unconfined water bearing layer 16 undisturbed Sam les 322, 344 uniformity coefKcient 44, 47, 82, 352, 371 unsaturated (hydraulic) conductivity 31, 288, 293, 295, 301-313 unsaturated seepage (flow, movement) 27-28, 227 unsaturated zone 99, 102 unsteady seepage (flow, movement) 499, 516, 539, 560, 687-712 unsteady effect - of constant discharge 699, 710 - of constant recharge 699, 710 unsteady seepage in horizontally limited field 704 validity of Dupuit’s hypothesis 630 validity zones of seepage 226, 230 vapour flux 99, 284 van der Weals’ force 62, 123, 210 velocity - Darcy’s or virtual 29-30 - effective 30 gradient 206 - hodograph 629 potential 476-481 vertical distribution of soil moisture 98 vertical exit face 369 vertically drained seepage field 601, 666 vertically recharged seepage field 600, 662 virtual width of canals 620, 673 viscosity - dynamic 30, 208 - kinematic 30, 208 viscous fluid 206 volcanic rocks 412-417 volumetric porosity 22-23, 427 volumetric water (moisture) content 105, 106, 288, 293 vortex point 484

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water - balance 148 - balance equation 149, 179 - content 104-109 - - v.8. hydraulic conductivity relationship 288, 296, 312 - - V.S. tension reIationship 115, 312 - exchange 101, 152, 178-188

730 water film 101-102 intake 401, 406, 429

Subject index

- mining 196 - molecules 208 - pressure 204-206 - resources 11 - retention capacity 162, 169, 177 - table 14, 99, 149, 189, 194, 609 - - between two canals 668

wave prop ation 496 weetherhg?bures 398, 406 well screen 3 7 6 3 7 8 wet sieving 120 wetting fluid 220 front 286

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vetting - process 138-140, 143 wilting point 107

X-ray investigation 64, 73 yield of riparian w e b 610 Zhukovsky’s mapping 636, 570, 626 zone - of adhesion 99, 102, 123, 301 - of cultivation 99 of plants 99 - of saturation 99, 102

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