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ix xiii xv
Chapter 1 Introduction
1
1.1 Importance and distinction of sediments
1
1.2 Outline of the development of cohesive sediment behavior 1.3 Objectives of the book and outline of its contents
Chapter 2 The Mineralogy and the Physicochemical Properties of Cohesive Sediments
3 6
11
2.1 General properties of cohesive sediment suspensions and of cohesive sediment deposits 2.2 The bonding mechanisms
11 14
2.2.1 Interatomic or Primary Bonds
14
2.2.2 Secondary Bonds
22
2.3 The nature and mineralogy of clay particles
24
2.3.1 Introductory Remarks
24
2.3.2 The Basic Clay Minerals
25
2.4 Origin and occurrence of clay minerals and formation of clay deposits
Chapter 3 Forces between Clay Particles and the Process of Flocculation
43
47
3.1 Introductory remarks
47
3.2 The electric charge and the double layer
47
3.2.1 Isomorphous Substitution
48
3.2.2 Preferential Adsorption
48
3.3 The theoretical formulation of the double layer
49
3.3.1 The General Case
49
3.3.2 Surfaces of Constant Potential
55
3.3.3 Surfaces of Constant Charge Density
55
3.3.4 Illustrative Applications
56
Cohesive Sediments in Open Channels Copyright© 2009
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GJ
Contents
3.4 Interaction of two flat double layers
59
3.4.1 Force and Energy Interaction
59
3.4.2 Illustrative Examples
62
3.4.3 Potential Energy of Interaction between Two Flat Double Layers 3.4.4 Illustrative Examples
64 65
3.4.5 Potential Energy of Interaction of Two Flat Double Layers Due to van der Waals Forces
68
3.4.6 Total Potential Energy for Two Particles and the Process of Flocculation
70
3.5 Some important properties of fine particles and aggregates
75
3.5.1 The Counterion Exchange
75
3.5.2 Limitations of the Gouy-Chapman Theory and the Stern Layer
75
3.5.3 The Water Phase
78
3.5.4 Sensitivity and Thixotropy
80
3.6 Internal structure and fabric of floes, aggregates, and cohesive sediment deposits
81
3.6.1 Particle Arrangements within Floes
81
3.6.2 The Microstructure of Deposited Cohesive Sediment Beds
87
Chapter 4 The Hydrodynamic Transport Processes of Cohesive Sediments and Governing Equations
89
4.1 The fundamental transport equations for cohesive sediments
89
4.1.1 The Development of the General Transport Equations 4.1.2 Discussion of the Developed Equations
89 94
4.2 The process and dynamics of flocculation
95
4.2.1 Collisions Due to Brownian Motion
95
4.2.2 Collisions Due to Velocity Gradients
100
4.2.3 Collisions Due to Differential Settling
103
4.2.4 Concluding Remarks 4.3 Review of fundamental properties of turbulent flows
106 106
4.3.1 Significant Stresses and Parameters
106
4.3.2 Collision Rates in Turbulent Flows
116
Contents
4.4 The properties of the aggregates and the aggregate growth equation
118
4.4.1 The Properties of Aggregates and Their Relation to the Controlling Flow Variables
118
4.4.2 Quasi Steady-State Aggregate Distribution and Maximum Aggregate Size
128
4.4.3 Some Additional Research Work on Flocculation and Aggregate Properties 4.4.4 Discussion and Concluding Remarks
Chapter 5 Rheological Properties of Cohesive Sediment Suspensions 5.1 Importance of the subject
140 152
155 155
5.2 Basic properties of sediment suspensions and methods of evaluations 5.3 Concluding remarks
Chapter 6 Erosion of Cohesive Soils
156 169
173
6.1 Introductory remarks
173
6.2 Erosion of consolidated cohesive soils
174
6.2.1 Early Empirical Information
174
6.2.2 More Recent Field and Laboratory Studies
178
6.3 Erosion of soft cohesive sediment deposits
183
6.4 Summary and concluding remarks
200
Chapter 7 Deposition and Resuspension of Cohesive Soils 7.1 Deposition of cohesive sediments 7.1.1 Early Experiments and Preliminary Conclusions
203 203 203
7.1.2 Detailed Studies on Deposition. Part A: The Degree of Deposition
208
7.1.3 Detailed Studies on Deposition. Part B: The Rates of Deposition
224
7.1.4 Variation of Depositional Parameters as the Sediment Sorts during Deposition in Open Conduits
234
7.2 Hydrodynamic interaction of suspended aggregates with the deposited bed
244
Contents
7.3 Resuspension of deposited cohesive sediments
252
7.3.1 Introductory Remarks
252
7.3.2 Fundamental Considerations
253
7.3.3 Experimental Results 7.4 Summary and closing comments
Chapter 8 Engineering Applications of Cohesive Sediment Dynamics
254 271
275
8.1 Areas of application
275
8.2 Design of stable channels
276
8.2.1 Design for Safety Against Scouring 8.2.2 Design for Safety Against Deposition 8.3 Shoaling in estuaries 8.3.1 Fine Sediment Transport Processes in Estuaries 8.4 Illustrative case histories
276 278 278 279 293
8.4.1 The Savannah Estuary
293
8.4.2 The Delaware River Estuary
301
8.4.3 The River Thames Estuary
303
8.4.4 The Maracaibo Estuary
306
8.4.5 The San Francisco Bay Estuary
315
8.4.6 Closing Remarks
317
8.4.7 Applications to Estuarine Modeling
318
8.5 Control of environmental pollution
319
References
323
List of Symbois
333
Greek Symbols
341
About the Author
345
Author Index
347
Subject Index
351
Preface
Cohesive sediments, which consist predominantly of silt and clay with size ranging from a few micrometers to a fraction of a micrometer, enter frequently in several areas within the civil engineering domain as, for example, in soil mechanics and foundations. In the field of water resources, the beds and banks of natural and artificial channels often consist of cohesive sediments subject to erosion. Such channels may also carry such sediments in suspension, and that sediment may eventually deposit in some areas of the canal or in reservoirs. In sanitary and environmental engineering, water purification and sewage treatment involve handling of cohesive sediments. In all the preceding cases and in some others, the physicochemical and colloidal properties of cohesive sediments are of primary importance in their response to external loads, as in the case of soil mechanics and foundations. The same is true in the control of erosion, deposition and transport in open channels, in the maintenance of reservoirs, in water and sewage treatment, and in the control of the environmental quality of natural water systems, such as lakes, rivers, and estuaries. The primary difference between coarse and cohesive sediments lies in the capacity of the latter to form, under the influence of interparticle attractive forces, agglomerations with size, density, and strength much different than those of the original particles. Moreover, these properties are not even constant, but they are functions of the acting forces during their formation and can even vary with time and quality of pore or ambient water. Therefore, any simplified description and modeling of such sediments based only on some gross quantities without taking into consideration the effect of their intricate physicochemical, colloidal, and mineralogical properties may lead to erroneous results. The need to incorporate these properties into engineering design first became obvious in foundations and various soil mechanics problems. This need motivated extensive research, both fundamental and applied, since the early part of the 20th century, which led to significant interdisciplinary advances for a better and more reliable design of foundations and earth structures and to an estimate of their bearing capacity and settlement under external loading. There is indeed a large volume of publications on this subject. Application of the same properties in hydraulic design and/or solution of sedimentation problems started much later, predominantly motivated by the design of open channels safe against scouring and deposition and for the control of
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Preface
shoaling in estuaries. Therefore, it is a relatively new field. Related fundamental research originated about 50 to 60 years ago. A large volume of knowledge has already been obtained while active research is still going on in a number of institutions in the world. The current existing knowledge, however, is sufficient for the formulation of a rigorous hydrodynamic framework for the overall hydraulic behavior of these kinds of sediments. This framework can be and should be used as the basic guideline for a rational approach to hydraulic problems involving cohesive sediments, for the planning of the necessary laboratory experiments and field measurements, and also for further research on these subjects. The objective of this book is to present in a coherent form the entire spectrum of the behavior of cohesive sediments in a flow field, and more specifically in open channels, starting from the process of flocculation and proceeding to the processes of erosion, deposition, resuspension, and transport, always in relation to their physicochemical properties. The main subject is preceded by a brief treatment of the fundamentals of clay mineralogy and clay colloid chemistry for the sake of readers with inadequate background in these fields and as a starting point for those wishing to expend their knowledge with further studies. The subject matter was selected and arranged in a way to contribute to three objectives: first, as an introduction to undergraduate and graduate students of hydraulic, coastal, and environmental engineering; second, as a guideline to practicing engineers; and third, as a starting point and/or an aid for further research. It is hoped that the book will meet all these three goals. Several people from various engineering and scientific areas have so far contributed to the present state of knowledge of cohesive sediment behavior. The subject matter of the book is based on a selection from work related in some way to its primary objective, which is the presentation of a rigorous framework with direct applications. The first major contribution started by Professor R. B. Krone at the University of California in Berkeley in the 1950s with his work on the effect of flow-induced shear stresses on the density and strength of flocs and floc aggregates and his laboratory and field studies on estuarial sediment transport processes. The early work of the author followed in 1960, also at the University of California in Berkeley, focused primarily on the processes of erosion of dense and deposited cohesive estuarine sediments. Work on the deposition phase continued at MIT by Professor J. F. Kennedy, the author and their graduate research assistant from 1963 to 1966. Their fundamental work was significantly enhanced by simultaneous field research on estuarine shoaling in the Bay of Maracaibo in Venezuela by the same people. A special research apparatus was developed and used in that research phase. The latter was furthermore developed and improved at the University of Florida in Gainesville, in 1968 and 1969, and was used from 1968 to 1983 for studies of deposition and resuspension by Mehta, the author, and a number of graduate assistants. Many researchers, mostly sanitary and coastal specialists, made important contributions to the hydrodynamics of floc formation and on the aggregate properties. The work of
Preface
xi
Watanabe, Hozumi, Tambo, and Kusuda and his colleagues in Japan is particularly noteworthy from the practical aspect, and their results are incorporated in this book as most directly related to its main theme and objectives. Additional important work of several other researchers is also mentioned and commented even with only indirect relation to the subject matter of the book.
Acknowledgments
The work by the author and his colleagues on the hydraulic behavior of cohesive sediments has been supported by the following agencies: Ford Foundation supported the author in his doctoral studies and his research at the University of California through a special predoctoral scholarship. The latter part of his research was also partly supported by the Corps of Engineers of the U.S. Army in 1961–1962. Ford Foundation also supported part of his research at the Massachusetts Institute of Technology (MIT) from 1963 to 1965 under a postdoctoral fellowship. The same fundamental research and field investigations in the Gulf of Maracaibo in Venezuela were supported by the U.S. Agency for International Development (AID) from 1963 to 1966. The work at the University of Florida in Gainesville was first supported by the Environmental Protection Agency (EPA) from 1968 to 1970 and from then on by the National Science Foundation until about 1980. Substantial simultaneous support was also provided during that period by the Waterways Experiment Station of the Corps of Engineers in Vicksburg, Mississippi. Finally, the College of Engineering of the University of Florida provided funds for the building of a special room for the housing of the experimental apparatus used for all the fundamental research experiments on the deposition and resuspension of cohesive sediments. All this support, thanks to which the field of cohesive sediment hydrodynamics was brought to its present stage of development, is gratefully acknowledged.
Cohesive Sediments in Open Channels Copyright © 2009
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Dedications
The author wishes to respectfully dedicate this book to the memory of the following outstanding professors with whom he had the privilege to be associated and who had a profound influence in his overall academic work. First, to the memory of Professor H. A. Einstein, founder and one of the most important contributors to Sediment Transport Mechanics, and the PhD thesis supervisor of the author at the University of California at Berkeley. Second, to the memory of Professor Arthur T. Ippen, director of the Hydraulics Laboratory at MIT during the author’s work there for his encouragement and support of his research and for introducing him to the Estuarine Hydrodynamics and shoaling in estuaries. Third but not least, to the memory of Knox Millsaps, Chairman of the Department of Engineering Science at the University of Florida from 1974 to 1986 and in which the author has been a faculty member from 1974 to now, for inspirational leadership and his commitment to academic principles and values.
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Chapter 1
Introduction
1.1 Importance and distinction of sediments The importance of sediments in hydraulic engineering and, in general, in the technical development of water resources is well known. In rivers the total amount of sediment discharge is the most obvious and direct concern. Sediments also affect the roughness and the frictional resistance of natural waterways, thus raising the question of stage-discharge-sediment transport relationships. The stability of beds and banks against scouring and deposition is another important subject, particularly for manmade canals and waterways. The useful life of reservoirs depends on the sediment load of the contributing natural streams. The extent and frequency of maintenance of navigable waterways in estuaries are determined by the rates of deposition of sediments, particularly fine, by the discharging river or rivers into that specific estuary. Another serious problem in estuaries, bays, and lakes is sediment-induced pollution either by increasing the water turbidity or by depositing highly contaminated sediment on ecologically sensitive zones. There are indeed cases where the environmental damage is so severe that the restoration of the original quality becomes either impossible or extremely difficult and expensive. For all these reasons, sedimentation has been the subject of intensive fundamental, applied, and field research since the 19th century and many theories, empirical formulas, and semitheoretical equations have been developed for the prediction of sediment transport rates and the control of channel stability. Sediments have been distinguished into two broad classes: coarse or cohesionless and fine or cohesive. The division has been arbitrarily placed on the grain size distribution. The first class refers to sediments ranging from fine sand to coarse gravel, whereas the second contains silt and clay. It has been observed that in river flows coarse sediments in transport are represented in appreciable quantities in the bed and that their rates of transport are functions of the flow conditions. In contrast, fine sediments are encountered in the bed in only very small quantities in proportion to their total load, and their transport rates appear to be unrelated to the flow parameters and to depend only on their supply rates [25, 26]. The boundary between the Cohesive Sediments in Open Channels Copyright © 2009
Cohesive Sediments in Open Channels
two sediment classes has been established at 50 m. It should be noticed at this point that even the finest sediments eventually deposit under sufficiently low bed shear stresses and, therefore, at some stage their transport and deposition rates have to depend on the flow parameters. The major difference between coarse and fine sediments lies not so much in their size but primarily in the mutual interaction of grains in a water environment. Coarse grains in suspension behave independently from each other, with the exception of mechanical interaction in highly dense suspensions, while as a bed material only forces of interlocking and friction enter into the picture. The settling unit in the sediment transported in suspension and the eroded unit from the coarse bed is the individual sediment unit; therefore, the sediment can be introduced in the relationships describing either the bed stability or sediment transport rates by a representative grain size distribution or by an equivalent distribution of settling velocities. Cohesive sediment grains, which range in size from 50 m to a small fraction of 1 m, are subjected to a set of attractive and repulsive forces of an electrochemical and atomic nature acting on their surfaces and within their mass. These forces are the result of the mineralogical properties of the sediment and of the adsorption of ions on the particle surfaces. The fine sediment grains have in general a flat plate or a needle shape and a high specific area, which is a high surface to volume ratio, so that the total magnitude of the surface forces becomes dominant in comparison to the submerged weight of the particle. Dispersed particles have such a low settling velocity that the finer portion of them can stay in suspension almost indefinitely, whereas even a slight degree of agitation is sufficient to keep the coarser part of them in suspension. When, under certain conditions, the attractive forces exceed the repulsive ones, colliding particles stick together, forming agglomerations known as flocs with size and settling velocities much higher than those of the individual particles. Rapid deposition may then take place. This phenomenon is known as flocculation. In a flocculated cohesive sediment suspension, the settling unit is the floc rather than the individual particle. The same physicochemical forces are responsible for certain properties of consolidated cohesive sediment deposits, such as cohesive strength and plasticity. Flocs join together to form floc aggregates of various orders of magnitude. In a quiescent water environment, the Brownian motion of the water particles provides the only mechanism for interparticle and interaggregate collision. In flowing waters the shear rates and the turbulent velocity fluctuations affect the collision frequency to a much greater extent than the Brownian motion, so that a much higher rate of aggregate formation is expected. At the same time, however, the same forces induce disrupting stresses within the aggregates, thus limiting their maximum size and controlling their basic properties. A quasi steady-state aggregate size distribution is reached, which is a function of the flow parameters themselves. Settling units develop similar bonds with the cohesive bed that have to be broken for the units to be resuspended. These surface forces constitute the enormous difference between coarse and fine cohesive sediments. The flow conditions, which control the degree and rates
Chapter | 1 Introduction
of deposition, also determine the size distribution and the important properties of the aggregates. The deposited cohesive bed is composed of flocs and/or higher order aggregates whose properties have been molded by the flow-induced stresses. Therefore, their erosional resistance and the rates of resuspension and erosion as well as their gross mechanical properties are expected to also be functions of the flow conditions. The same physicochemical forces may attract other suspended matter in the water, such as organic and inorganic toxic substances, so that a highly contaminated bed is formed after deposition. Upon resuspension the polluted sediment may contaminate the entire water environ with detrimental consequences to the aquatic life. To make the situation more complicated, the surface interparticle forces are not even constant, but they may change drastically with small changes in the water quality, temperature, and time. At first glance this interdependence of flow, water quality, aggregate properties, deposition, and erosion gives the impression that a quantitative description of the hydraulic behavior of cohesive sediments constitutes an insurmountable problem. Fortunately, extensive fundamental and applied research, particularly since 1950, has led to a much better understanding of the dynamics of cohesive sediment behavior in a turbulent flow field. Quantitative equations have been developed for the initiation, degree and rates of deposition, erosion, and resuspension in terms of readily determinable flow variables and parameters representing the overall effect of the interparticle physicochemical forces. These relationships and an understanding of the dynamics of floc formation and of the processes of erosion, deposition, resuspension, and transport of cohesive sediments supplemented with some simple laboratory tests and field measurements may lead through mathematical and physical models to reasonable answers to problems involving cohesive sediments. The terms “fines” and “cohesive sediments” have been used in literature meaning essentially the same thing. Both terms will also be used alternatively in this book.
1.2 Outline of the development of cohesive sediment behavior Because of the outlined importance of sedimentation, the hydraulic behavior of sediments has been a subject of concern and investigation since the inception of the field of hydraulic engineering. In fact studies on sedimentation followed closely the developments in that field and in fluid mechanics in general. A good summary of the historic development of sediment transport mechanics was given Graf [37]. Like many engineering disciplines, hydraulics in general and sedimentation in particular started from pure empiricism and gradually proceeded together with advances in fluid mechanics to more fundamental and universal relationships. This is particularly true for cohesive sediments, whose behavior is much more complicated than that of cohesionless soils. The development of cohesive sediment hydraulics is summarized and discussed in Chapter 6, which deals with the mechanics of erosion of these sediments.
Cohesive Sediments in Open Channels
An earlier analysis and discussion was presented by Paaswell and the author [92, 93, 116, 117]. Only a brief outline of representative examples of the evolution of the hydraulics of cohesive sediments will be given here. The first and earliest phase of the subject consisted in establishing guidelines for the design of stable canals through empirical formulas and/or tables for limiting velocities as the only criterion. The soil properties were described by a mere classification or, at most, by some measure of their density. For instance, in the table of critical velocities recommended by Schoklitsch in 1914, the soil was described only by its type and “the degree of compaction” [125, Vol. I, p. 232]. Similar critical velocities were given by Etcheverry in 1916, also based on a very general soil classification [30]. The average velocity continued to be used as the stability criterion into the early part of the 20th century in spite of the fact that, as early as in 1816, Du Buat introduced for the first time as a criterion of sediment transport the concept of “shear resistance,” that is, essentially the force per unit area of the bed of the stream [19]. This was, in fact, the greatest contribution of Du Buat to the field of sediment transport. A similar concept of tractive force or bed shear stress was introduced in 1879 by Du Boys [18]. Still, however, in 1926 the Special Committee on Irrigation Hydraulics presented estimates of “experienced irrigation engineers” for critical design velocities reported by Fortier and Scobey again on the basis of soil classification [33]. It was only in 1955 that Lane reported data by Russian engineers giving both critical velocities and critical shear stresses for channels with cohesive boundaries and of various densities [70]. Another school of approach within that first phase is known as the “regime theories.” These theories aimed at the development of empirical formulas for velocities for the design of channels in specific areas on the basis of extensive and numerous field data on canals which exhibited various degrees of stability. No laws of mechanics were introduced in the derivation nor were any specific soil data reported. Kennedy’s work on irrigation canals in India is representative of this school of thought. It was presented in 1895, and the channel depth was the only parameter representative of the channel geometry [55]). In 1959 Leliavsky reported later investigations introducing, in addition to the depth of flow, some other variables describing the channel geometry and the boundary resistance [73]. Kennedy’s formulas for stable canal design do not specify whether they imply safety against scouring or shoaling. An analysis in Chapter 6, though, indicated that these formulas and rules really apply for safety against deposition or siltation. Like almost all empirical laws and formulas, these early representative results and criteria may be valid for soil types and soil properties and for general environmental conditions very similar to those they were based on. Otherwise, the results of their application may be erroneous. This is particularly true in the case of cohesive sediments in which even apparently minor changes in one or more aspects may drastically affect their erosional and depositional characteristics. The realization that the flow-induced shear stresses on the bed rather than the average flow velocity is the controlling flow variable for channel stability and the recognition of the related importance of soil properties in addition to its
Chapter | 1 Introduction
composition and density led to the second phase of cohesive sediment research in both the field and in the laboratory. In all studies the objective was to relate the critical bed shear stresses, also referred to as critical tractive forces, to some soil mechanics parameters representative of the soil structure and the gross shear strength. Field investigations conducted by the U.S. Bureau of Reclamation presented in 1953 revealed little correlation between critical tractive force for erosion and mean grain size. The same studies indicated a strong effect of some external factors, such as desiccation, on the critical tractive force [148]. At about the same time, the field studies by Sundborg on the Klarälven river suggested that the critical velocity decreases with decreasing particle size down to the silt range of 50 m, but it increases as the sediment becomes finer [130]. This is the limit below which the interparticle physicochemical forces start causing flocculation. These field investigations may supply valuable data on erosive and depositional trends, but seldom lead to a basic understanding of the particular process, much less to formulations of equations of general validity. The laboratory research of this second phase aimed at the derivation of experimental relationships linking a critical tractive force or boundary shear stress to some readily determinable soil parameters representative of the macroscopic shear strength and other mechanical properties of the soils. The mechanism of erosion and the details of bed structure were not considered. The work by Dunn in 1959 [20], of Smerdon and Beasley in the same year [128], of Moore and Mash in 1962 [90], of Espey in 1963 [29], of Flaxman in 1965 [32], of Berghager and Ladd in 1964 [5], and of Grissinger in 1966 [40] are representative examples of this kind of research effort. In 1966 the Task Committee on Cohesive Sediments of the ASCE published an annotated bibliography on the subject containing the results and conclusions of several other laboratory studies [136]. Most of these studies utilized small, improvised experimental setups, such as cylinders within which an interior cylindrical sample of the soil was subjected to shear by rotation, or jets impinging on a soil sample or relatively small samples of cohesive sediments placed over a section of an open flume. Some other researchers, like Smerdon and Beasley [128] and Abdel-Rahman in 1962 [1] did use a cohesive bed over the entire length of an open flume. The latter work, though, overlaps the third phase. The experimental research of the second phase constitutes an important and indispensable part of the overall research effort in cohesive sediment dynamics. Its direct contribution, however, has been limited by two facts. First, because of their shape and small size, some of the experimental devices do not generate a flow field similar to that in open channels. As a result, the boundary stresses on the sample and the flow structure near the wall may deviate substantially from that in real conduits. In addition, the difference in the shape of the equipment with the associated disparity of boundary conditions makes any comparison between the results of various investigators even more difficult. Nevertheless, these early laboratory experiments on cohesive sediments led to some important conclusions. It was
Cohesive Sediments in Open Channels
made clear that neither the shear strength nor the Atterberg limits could be used as a unique parameter for cohesive soil erodibility. Indeed, samples with comparable strength determined by any standard test used in soil mechanics, and/or comparable Atterberg limits, were found to display erosive resistance differing by orders of magnitude. The third and more detailed phase started from about 1950 and was motivated by the need for a rational control of shoaling in navigable waterways. The erosion and deposition processes were studied in conjunction with flow structure, the dynamics of flocculation, and the interaction between settling suspended flocs and the bed. For the first time the erosion and deposition rates were introduced in addition to the critical limits. Equations have been derived for the initiation, degree, and rates of erosion and deposition of generalized validity, which, used with the appropriate evaluation of certain parameters through field measurements and/or laboratory tests, may lead to reliable estimates for erosion, deposition, and resuspension. This phase started with the pioneering laboratory and field work of Krone on the relationship of floc properties to the flow-induced stresses as well as on the shoaling processes in the San Francisco Bay [61–64]. Studies, primarily on the erosion of dense and deposited cohesive sediment beds, were initiated by the author in 1960 [102, 105]. This work was followed by fundamental studies on erosion, deposition, and resuspension by the author, Kennedy, Mehta, their associates and graduate students, as well as by others along similar lines of approach for several years and are still being continued [16, 31, 81–86, 94, 95, 99–101, 103, 104, 107–109, 112–115, 118, 154, 155]. Parallel research work was developed about the effect of flow-induced stresses on the properties of flocs and higher order aggregates. Many of these studies were conducted by sanitary engineers in their effort to improve the efficiency of water purification and sewage treatment. Their results, though, are equally applicable to any cohesive sediment suspension. A fundamental framework for cohesive sediment dynamics has been thus developed and formulated [101]. This framework can be used as a guideline for rational approaches to problems involving cohesive sediments. It can also serve as a basis for future research in the field of cohesive sediment dynamics. The three outlined phases of cohesive sediment research do overlap, and in fact some of them interact. However, they define three distinct philosophies of approach and the various steps such a complicated subject has gone through to its present state of the art.
1.3 Objectives of the book and outline of its contents The objective of this book is to present in a unified framework current fundamental and applied knowledge on the hydraulic behavior of cohesive sediments in a turbulent flow field and specifically in open channels. It is based on extensive theoretical and laboratory research and on field investigations over the second half of the 20th century. A number of scientists and engineers from various
Chapter | 1 Introduction
specialties have contributed to the present state of knowledge on this subject. The main emphasis is focused on the processes of erosion, transport, deposition, and resuspension of cohesive sediments, and it addresses the following important and frequently encountered problems in water resources projects: 1. Erosion of natural and manmade canals with cohesive beds and/or banks and ways to stabilize them. 2. Control of deposition of fine sediments in suspension in canals and prevention of shoaling. 3. Fine sediment transport processes in tidal estuaries and bays and the relation of these processes to the salinity and the overall regimen of the estuary with the ultimate objective of control of shoaling in navigable waterways. 4. Control of sediment deposition in reservoirs. 5. Design of sedimentation basins in water purification and sewage treatment plants for optimum performance. 6. Prediction and control of turbidity in river and estuarine waters that may have undesirable effects on the marine life. 7. Proper planning of dredging and filling operations in estuaries and bays to avoid sediment pollution by spreading contaminated fine sediments in ecologically sensitive areas. For a rational approach and a successful design of related operations and structures, the behavior of cohesive sediments in a flow field and the basic physicochemical properties of the sediments, which control and determine that behavior, have to be understood. Following are some typical, major questions involved in the problems and the kind of engineering operations listed previously: 1. Under which conditions and water quality do suspended fines flocculate? 2. What are the relevant properties of the flocs and of higher order aggregates, such as density, strength, and settling velocities, and how do they relate to the erosional and depositional processes and criteria? 3. Which hydraulic parameters determine the critical flow conditions for scouring and siltation in open channels? 4. Which sediment properties determine the erodibility and/or the zones of potential shoaling and what kind(s) of test(s) would be representative of these properties? 5. Is the erosive resistance of a cohesive bed related to the gross properties of the soil, such as the macroscopic shear strength and Atterberg limits, and how? One may recall from the brief outline of the second phase of development in the previous section that any correlation to such soil parameters may lead to erroneous results. 6. Are the sediment properties implied in question 4 constant, or do they change with time and/or environmental conditions and how? The fundamental and applied research in the past 40 to 50 years was planned and conducted to provide guidelines, analytical principles, and suggestions for
Cohesive Sediments in Open Channels
laboratory tests and field measurements as a basis for rational answers to these problems and questions. The model of the hydrodynamic behavior of cohesive sediments in a flow field to be presented in this book is based on the results of extensive fundamental and applied research as well as field investigations from 1950 to about the present time by Krone, the author, Mehta, their numerous collaborators and associates, and several others. This book is not meant to be a compilation of all the work related to cohesive sediment behavior. Work even remotely related to the main theme of the book has been analyzed, commented, and integrated in the overall picture. However, work either inconclusive or unrelated or far removed from the primary objectives outlined above is not included. This does not mean that the omitted work is unimportant by any means. It was simply felt in the present phase of the art and in addressing the listed objectives and questions that the inclusion of a large volume of data without an obvious connection and bearing to these objectives and questions would distract rather than benefit the reader. The book is written by a hydraulic and coastal engineer and is addressed to hydraulic and coastal engineers but also to some extent to environmental engineers. A basic knowledge of fluid mechanics and open channel flow at an undergraduate level as well as an introductory course in soil mechanics are considered sufficient for the study of the subject matter. No knowledge in clay mineralogy and colloidal chemistry is assumed from the part of the readers. Since, however, an understanding of the basic properties of the clay minerals and of the clay suspensions is necessary to fully appreciate the hydraulic behavior of cohesive sediments, Chapter 2 and Chapter 3 have been devoted to these subjects. As the title suggests, the subject matter is focused on the hydrodynamic interaction of cohesive sediments with the flow in open channels, and it includes the processes of flocculation.: This book is designed to serve three main objectives: first, to provide hydraulic engineers and scientists with a rigorous basis for rational decisions regarding the appropriate tests and their application to hydraulic problems involving fine cohesive sediments; second, to be used as a textbook for upper division undergraduate and graduate students in civil and/or environmental engineering either for a special course or as a supplement to a course on the general subject of sediment transport; and third, to assist present and future researchers in this field by providing to them a rigorous basis as a guidance to identify other related areas of research and decide about the best methods of approach. There are several special subjects and problems related to cohesive sediments not addressed in this book. Sediment transport by waves, for instance, albeit an important subject to coastal engineering and currently under investigation, is not included in the present edition. Likewise, biological factors affecting flocculation and certain mechanical properties of sediments are excluded. This is a very specialized field requiring extensive additional fundamental and applied research because its present state of knowledge is very inadequate for practical purposes. There have been many field observations regarding in situ behavior of flocs and aggregates of various orders in lakes, estuaries, bays, and rivers, which,
Chapter | 1 Introduction
if properly analyzed, could add substantially to our knowledge on cohesive sediment dynamics. For the same reason these investigations have been left out of the present edition of the book. The subject matter of the book is divided into eight chapters. Chapter 1, the introduction, outlines the importance of the subject, the philosophy of the book, and a brief summary of the historic development of the field. Chapter 2 covers the basic physicochemical and mineralogical properties of clays. Hydraulic engineers need to have a good understanding of these properties and of the nature and the origin of the interparticle physicochemical forces. The chapter starts with an outline and discussion of the bonding mechanism and of the various types of bonds. Some frequently used elementary concepts and definitions regarding the structure of matter are briefly summarized. The mineralogy of clays follows with a section on the most important clay minerals. The chapter closes with the origin and occurrence of clays and clay deposits. Chapter 3 covers the forces between clay particles and the process of flocculation; therefore, it provides the very important background for the understanding of the hydraulic behavior of clays. Particular emphasis is given to the concept of the double layer, which is of primary importance in the process of flocculation, and to the effect of water chemistry on it. All these subjects are treated through mechanistic pictures, when appropriate, and simplified albeit rigorous models representing the bonding mechanisms. The chapter continues with some of the most important properties of fine particles and aggregates and concludes with the internal structure and fabric of flocs and the microstructure of deposited cohesive beds as they are related to erosion and deposition. The material in both Chapters 2 and 3 has been selected from the cited references so as to provide the nonexpert readers with the minimum necessary background to (a) understand the processes and dynamics of flocculation and the hydrodynamic behavior of cohesive sediments; (b) communicate with geologists and clay mineralogists, as the case may be; and (c) expend their knowledge in these fields through additional studies starting from the suggested references. Readers with sufficient background in the areas of clay mineralogy and clay colloid properties may decide to skip these two chapters. Chapter 4 presents and explains the transport processes of sediments by flowing water and related equations. The process and the dynamics of flocculation are treated next with particular emphasis on the relationship of the aggregate properties to the pertinent parameters describing the flow-induced stresses responsible for the molding of fine particles into aggregates of various orders. This chapter also includes an extensive section on the fundamental aspects of turbulent flows that are of direct importance to flocculation. Chapter 5 deals with the special subject of the rheological properties of cohesive sediment suspensions and compares these properties with the erosion process of cohesive sediment beds. Chapter 6 is the first of the three most important chapters of the book covering the essence of the subject matter. For the reasons explained there, the presentation
10
Cohesive Sediments in Open Channels
of the erosion process was treated separately from the deposition and resuspension in spite of the overlap and the fact that both processes constitute two phases of one and the same process. The chapter starts with a critical review of the early empirical information, briefly outlined in Section 1.2, and it proceeds with presentation of the results of the first fundamental research effort on erosion. The material is predominantly based on the early work by Krone [62] and of the author [102, 105]. For the first time it revealed the detailed process of cohesive sediment erosion in relation to the microstructure of the bed and led to the first analytical model and equations describing the hydrodynamic interaction between the surface layer of the bed and the near bed flow structure. Chapter 7 is concentrated on the subject of deposition and resuspension of beds deposited from suspension from flowing waters. The subject matter is primarily based on the work of Mehta, the author, Kennedy, and several of their graduate students and collaborators. The depositional behavior and resuspension processes have been related to the microstructure of the bed with a clear distinction in the erosional behavior of deposited and artificially placed beds of uniform consistency. Analytical equations have been developed for the degree and rates of deposition and resuspension in terms of readily determinable flow variables and sediment parameters representing their cohesive properties. The special experimental equipment for the study of deposition and resuspension of cohesive sediments, developed first by Kennedy, the author, and collaborators at MIT [31, 109, 114, 118] and later on improved at the University of Florida, is outlined together with its operation [31, 81–86, 109, 114, 118]. Similar setups have been developed since then in some other countries for the same purpose. This chapter also contains a section on the hydrodynamic interaction of suspended and bed sediment in general and explains certain differences in the depositional and erosional behavior between coarse and cohesive sediments. Chapter 8 is devoted to engineering applications of the hydraulics of cohesive sediments. Suggestions are presented first, and guidelines are given for the design of stable channels safe against both scouring and siltation. The emphasis, however, is concentrated on the shoaling control of estuarial waterways, which motivated the initiation of more detailed research on erosion and deposition of cohesive soils, as mentioned earlier. Although, as has already been pointed out, mathematical modeling is not part of the objectives of this book, some hints as to the use of the developed analytical expressions to mathematical and particularly physical models are presented and discussed. The chapter closes with a description of five illustrative case histories of partially mixed estuaries with severe shoaling problems and with their associated field investigations and remedial works. There are still many aspects of cohesive sediment behavior of both academic and practical interest that remain inadequately understood and in need of future research. It is hoped that this book will be of help to any such future research effort.
Chapter 2
The Mineralogy and the Physicochemical Properties of Cohesive Sediments
2.1 G eneral properties of cohesive sediment suspensions and of cohesive sediment deposits As pointed out in Chapter 1, in hydraulic engineering the term cohesive sediment implies a mixture of silt and clay with settling diameter less than 50 m and as small as a fraction of 1 m with various degrees of organic matter. The same term applies to such mixtures containing a substantial percentage of sand provided that they still display cohesive properties. Two of the most common properties of clay masses subjected to various degrees of consolidation are plasticity and cohesion. The first is the property of a clay mass to undergo substantial plastic deformation under stress and within a certain range of water content without breaking. The Atterberg limits are used as a measure of this property. Cohesion is the ability of a clay sample to withstand a finite shear stress within its mass without confinement. The concept of cohesive strength of soils can be demonstrated by the following example and in Figure 2.1. If a sample of cohesive soil is submitted to shear stresses under various confining normal stresses to the point of yield or failure, the plot of the ultimate shear stress at failure or at yield, Sh, versus the normal stresses, ph, will fall approximately on a straight line as shown in Figure 2.1. In this diagram the abscissa indicates the normal stresses, ph; and the ordinate, the shear stresses, . This line, if extrapolated, intersects the axis at a point, ch, in general different from the origin. The shear strength, Sh, of the sample can be described by Coulomb’s equation: Cohesive Sediments in Open Channels Copyright © 2009
Sh ch ph tan φh
(2.1) 11
12
Cohesive Sediments in Open Channels
τ �h
ch 0
ph
Figure 2.1 Shear strength envelop for cohesive soils.
The intercept ch is commonly known as the cohesion and φh is defined as the angle of internal friction. It should be noted that the parameters ch and φh depend on the type of the shear test, the drainage conditions, the rate of application of the shear forces, and the degree of saturation. For instance, for undrained conditions, φh becomes approximately zero and ch attains its highest value. For a completely drained triaxial shear test with a slow application of the shearing load, the angle of internal friction attains its highest value, being more representative of the mechanical resistance due to friction and interlocking among the grains, while the value of ch becomes minimum. The latter has been used as a measure of that part of the soil strength, which is due to the interparticle physicochemical bonds. However, the actual strength due to these forces is expected to be somewhat lower due to the expansion of the soil under very low confining pressures. It is doubtful whether any of the classical soil mechanics tests for shear strength give the true measures of these forces. Nevertheless, ch has been used as a convenient parameter and a reasonable measure. The plastic and cohesive properties of fine sediments are due to that part of the soil mass that is fine enough and of specific area sufficiently large for the surface physicochemical forces to become dominant. The size of these particles, also known as colloids, varies from a few micrometers to a small fraction of 1 m, and they normally have the shape of little flat plates or needles or laths, depending on their mineralogical composition. The behavior of colloidal suspensions is well described by van Olphen in the first chapter of Ref. [150]. The most important aspects of this behavior have been briefly summarized and explained by the author elsewhere [93, 99, 117] and are quoted here. In dry form, clays look like a fine powder of various colors, depending on their mineralogy and impurities. When mixed with water, this powder seems to dissolve like a common salt. However, this is not a real solution but actually a dispersion of very small clay particles. Only some of the larger suspended particles
Chapter | 2 The Mineralogy and the Physicochemical Properties
13
can be observed through an ordinary microscope, whereas an ultramicroscopic arrangement is needed for the observation of the finer particles. If light is transmitted to the sample under the microscope in a way that the light beam hits the particles without entering into the objective lens, the particles scatter the light in all directions. Part of this light enters into the objective lens so that the particles appear to the observer as light specs on a dark background. The light specs display a vivid random motion in all directions, known as Brownian motion. This motion is caused by the thermal activity of the water molecules. This phenomenon takes place in the following way. Consider an infinitesimal water particle of mass m close to the surface of a flat solid clay particle. The latter, however small it may be, is by orders of magnitude larger than the group of water molecules composing the particle. Let unB be the average particle velocity normal to the clay surface due to the thermal energy. The collision will impart on the clay particle a force, F, of a magnitude given by the impulse-momentum equation:
F (m)unB
(2.2)
A suspended particle will receive a large number of such random kicks from all directions at any time. Some of the kicks will cancel each other, allowing only a number of unbalanced impulsive forces on the clay particle, which will determine the imparted motion. It is reasonable to assume that, because of the randomness of the impacts, the net unbalanced force on the particle, F, will have little dependence on the size of the particle. The resulting acceleration as a result of that net impact will be
aB
F M
(2.3)
where M is the mass of the solid clay particle. It follows from Equation 2.3 that the Brownian acceleration, aB, decreases rapidly and in inverse proportion to the third power of the particle diameter. In a quiescent water environment, particles with settling velocity smaller than the average velocity imparted by the Brownian motion appear to stay indefinitely in suspension. This is normally the case of clay particles with Stokes diameter equal to or smaller than 1 or 2 m. But even a very slight degree of agitation would be sufficient to keep the coarsest range of fine sediments in suspension. A homogeneous dispersion of clay fine particles is commonly referred to as a clay solution. van Olphen defines this dispersion as colloidal solution or sol when no measurable deposition takes place within a long period of time [150]. Otherwise, he defines it as clay suspension. Regardless of the definition, the Brownian motion will cause particles to collide. In any microscopic arrangement, it will be observed that, under certain conditions, colliding particles will tend to move away from each other, while under some other conditions they will
14
Cohesive Sediments in Open Channels
stick together, forming larger agglomerates. The first is mostly but not always the case of clays dispersed in distilled water, and the sol is defined as stable, peptized, or deflocculated. If, in such a sol, a small amount of an electrolyte, such as ordinary salt, is introduced to the water, the picture changes drastically with colliding particles sticking to each other forming continuously growing agglomerations. These agglomerates eventually grow large enough for rapid deposition to take place. This phenomenon has been defined as flocculation, and the sol is termed as unstable. The described behavior of suspended fines suggests the existence of repulsive interparticle forces in a stable sol and of attractive forces in an unstable one. In reality, however, both sets of interparticle forces coexist in both types of sols. The electrolyte simply changes the relative magnitude of these forces in a way that, depending on the sol, the net effect can be either attraction or repulsion. In an unstable sol in still water, the process of agglomeration will continue at rates increasing with increasing suspended sediment concentration, thus generating larger and larger agglomerates. Eventually a continuous aggregate network will be formed near the bottom of the container, which, if left undisturbed, will continue slowly consolidating with the water escaping through its pores. That network possesses some shear strength. In moving waters, the flow will induce disruptive shear stresses within the flocs and aggregates, thus preventing their volume increase beyond a certain limit. These stresses, as well as the interparticle physicochemical forces, control not only the size distribution of the flocs and aggregates, but also their density, strength, and settling velocity. The interrelationship between the acting and resisting forces and the preceding properties is examined in Chapter 3. Since a cohesive bed is composed of deposited flocs and aggregates, it follows that the same acting and resisting forces responsible for their formation and properties will also determine not only the depositional behavior but also the density and the resistance to erosion of the bed. For this reason the nature and generation of both forces has to be first understood in order to develop rational criteria for the prediction of the degree and rates of erosion and deposition of cohesive sediments.
2.2 The bonding mechanisms The bonding forces between atoms and/or material particles can be distinguished into two general categories: (a) interatomic or primary bonds and (b) secondary bonds. The first are by far the strongest, and they act between all atoms and molecules of any matter; the second are the ones acting between material particles and are much weaker than the former. The attractive and repulsive forces among colloidal particles belong to the second category.
2.2.1 Interatomic or Primary Bonds Interatomic or primary bonds are the bonds that hold the atoms and molecules of any matter together. Their nature lies in the atomic structure of the matter itself.
Chapter | 2 The Mineralogy and the Physicochemical Properties
15
This section summarizes some of the fundamental concepts regarding the origin and the nature of these bonds. According to the Bohr model, developed in 1913, an atom consists of three basic components: electrons, protons, and neutrons. The electron is a particle possessing a negative electric charge equal to 16 1020 coulombs or 4.81010 esu (electrostatic units). The proton is a particle with a positive electric charge equal in magnitude and opposite in sign to the total charge of its electrons. The neutron is a particle without any electric charge. The proton and the neutron have essentially the same mass, which is taken as the unit mass. The mass of an electron is about 1/1648 the mass of one neutron or a proton and, for practical purposes, can be neglected. Neutrons and protons are packed tightly together at the center of the atom to form the nucleus of the latter. The mass of the nucleus is about 99.95 percent of the total mass of the atom. It is surrounded by electrons moving around it in spherical trajectories known as shells with the nucleus as their center, as outlined in Figure 2.2 ([88], Fig. 2.2). The diameter of the nucleus is about 104 times the diameter of the atom. That is, 1012 of the total volume of an atom is occupied by the nucleus. An introduction to the fundamental concepts of the atomic and molecular structure can be found in any elementary textbook of chemistry and physics. The brief summary of these concepts and definitions herewith presented are based on the book titled Chemistry Made Simple by Hess [50]. Two examples of atomic structure are shown in Figure 2.3: the first represents a hydrogen atom, which has the simplest structure; and the second, the much more complex carbon atom. The situation is similar to that of the planets revolving around the sun or of the satellites revolving around the earth. In both cases a steady circular or elliptical trajectory is being maintained, defined by the balanced
Nucleus: contains protons, neutrons, and other particles 99.95% of mass centered in nucleus Electron shells Diameter of atom about 1 Å
Protons and neutrons have same mass Diameter of nucleus about 10�4 Å
Electronic charge � �16.0 � 10�20 coulomb �4.8 � 10�20 esu No. protons � No. electrons � Atomic number No. protons � No. neutrons � Atomic weight Figure 2.2 Simplified representation of the structure of an atom [88, Fig. 2.2].
16
Cohesive Sediments in Open Channels
e– e– e– e–
e–
e–
e–
(a)
Hydrogen At. no. 1 At. wt. 1
(b)
Carbon At. no. 6 At. wt. 12
Figure 2.3 Examples of atomic structure [50].
action of gravitational and centripetal forces. The sum of the potential and the kinetic energies remains constant; therefore, any change of the distance of a satellite from the earth involving an increase or a decrease of its potential energy is accompanied by a corresponding decrease or increase of its kinetic energy. The electronic energy is not continuous but, according to the quantum theory, an electron can have only certain levels of energy. Transition to a different energy level is discontinuous and can be achieved by either absorption or emission of radiant energy. Only two electrons can be at the same energy level spinning in opposite directions. The combined effects of energy quantization and the limitation of the number of electrons at each energy level are responsible for the different bonding mechanisms that develop when the energy level of the electrons of the interacting atoms composing the various molecules is lowered. Atoms are electrically neutral. That means that the number of protons in the nucleus must be equal to the number of the revolving electrons. This number has been defined as the atomic number, and it is one of the main characteristics of each atom. The mass of the atom is determined by the total number of protons and neutrons in the nucleus and is defined as the atomic weight of the atom. The distribution of the electrons around the nucleus generates the necessary bonds between the atoms to form molecules. The electrons revolve in definite distances about the nucleus and in specific patterns. The latter is surrounded by surfaces of electrons, defined as shells, each one of which is capable of containing a definite number of electrons. The shells are designated by order numbers 1, 2, 3, etc. The maximum number N of electrons in a shell of order n is given by the relationship:
N 2 n2
(2.4)
According to this relationship, the first shell can have a maximum of 2 electrons; the second, a maximum of 8; the third, a maximum of 18; and so on.
Chapter | 2 The Mineralogy and the Physicochemical Properties
17
The table of distribution of elements according to the order of their shells can be found in any book or handbook of chemistry and will not be reproduced here. Only a few observations will be made instead. Hydrogen has only one shell. In the first 18 elements, the higher order shell is formed as soon as the one of lower order obtains the maximum possible number of electrons. From then on a higher order shell is formed as soon as the shell of the immediate lower order obtains 8 electrons. In this way in the higher numbered elements, there can be two or even three unfilled shells of electrons, but there can never be more than 8 electrons in their outmost shell. Some elements, known as inert elements, have their shells filled so that, as the name suggests, they cannot form compounds with other substances. The elements with only one shell unfilled are classified as simple elements, whereas elements with two or three unfilled shells are referred to as transition elements or rare earth elements. Only simple elements are of interest for our purposes. In the formation of compounds, normally electrons of the outermost shell are involved although occasionally electrons from the second shell may be affected in some of the higher number elements. In this formation, a rearrangement of the electronic structure takes place so that the structure can obtain an electronic configuration similar to that of a nearby inert element. The property of the elements to form compounds is called valance, and the manner in which they combine is known as the valance mechanism. The number of electrons involved in the process is referred to as the valance number. There are three types of interatomic bonds referred to as primary bonds: (a) electrovalent or ionic bonds, (b) covalent bonds, and (c) metallic bonds. The last is of little importance to the formation of flocs and aggregates and will not be discussed here.
2.1.1.1 Electrovalence or Ionic Bonds Ionic bonds are developed by the electrostatic attraction of elements with opposite electric charges generated in the following way. Consider as an example a sodium and a chlorine atom. In their interaction to form a compound, these two elements will undergo a rearrangement of their electronic configuration similar to that of a nearby inert element. Sodium has to give up the single electron of its outermost shell, thus forming a positively charged sodium cation or simply a cation with electronic structure similar to that of its nearest inert element neon. Chlorine, on the other hand, being closest to the inert element argon, has to add an electron to its outer shell to obtain the electronic structure of the latter, thus forming the negatively charged chlorine ion or anion. In this way two ions of opposite charge can be formed by the transfer of one electron from the sodium to the chlorine atom, resulting in the formation of the electrostatically neutral sodium chloride, NaCl, known as common salt. The valance of each element is indicated as a superscript on the symbol of the element with a sign for anions and a sign for cations. Thus, the symbol of the sodium ion is Na and that for the chlorine ion is Cl. In the same
18
Cohesive Sediments in Open Channels
way symbols for the ions of the bivalent cations calcium, Ca; magnesium, Mg; and aluminum, Al, are indicated by Ca, Mg, and Al or by Ca2, Mg2, and Al3, respectively. For the negatively charged anions, such as sulfur, nitrogen, and oxygen, the symbols are S, N, and O, or S2, N2, and O2, respectively. Compounds formed by ionic bonds can be viewed as ionic agglomerates. Such agglomerates consist essentially of oppositely charged ions packed and held together by forces of electrical attraction with each cation attracting all neighboring anions. It follows that ionic bonds are nondirectional so that compounds formed by such bonds do not display any preferred direction nor any characteristic geometric pattern. For this reason they are defined as amorphous. For example, a cation of sodium chloride may attract as many chlorine anions as will fit around it. Bonding involves energy changes, because it takes energy to remove an electron from one atom and force it into another. Associated with this energy is the concept of the activity of the element. The term indicates the degree of ease with which this transfer takes place. Since this energy change increases with the number of transferred electrons, atoms with one electron in their outer shell are expected to be the most active in forming compounds, followed in activity by atoms with two electrons and so on. By the same reasoning, elements lacking one electron to fill their outer shell, such as chlorine and fluorine, are more active than others lacking two or more electrons in that shell.
2.2.1.2 Covalence and Covalent Bonds Covalence and covalent bonds develop when the shell of one atom penetrates into the shell of another atom in such a way that the electrons of the interpenetrated shells would be affected by the nuclei of both atoms. This is equivalent to a sharing of one or more bonding electrons among a number of atoms in order to complete the outer shell of each of the combining atoms. Depending on the number of shared atoms, two of the same kinds of ions may combine into more than one way to give different substances. Figures 2.4a and 2.4b give such an example of combination for carbon and hydrogen ions to form the two different compounds methane and acetylene. The methane (CH4) consists of one carbon atom with four electrons, indicated by dots, in its outer shell and four hydrogen atoms, each one with one electron in its outer shell, indicated by the symbol x. If the hydrogen shells interpenetrate the outer carbon shell, each electron of the hydrogen atom will be shared also with the carbon atom, and vice versa, each electron of the carbon atom will be shared with the hydrogen atom. In this way each hydrogen atom completes its outer shell with two electrons, thus reaching a stable configuration, and the carbon atom does the same with eight electrons. In acetylene (C2H2) three pairs of electrons are shared between two atoms of carbon forming a triple bond, while a single pair of electrons is shared between a carbon atom and a hydrogen atom. In this way each hydrogen shell has two electrons, and the second shell of each carbon atom has eight electrons.
19
Chapter | 2 The Mineralogy and the Physicochemical Properties
H
H
C
H H
C
C
H
H (a)
Methane
(b)
Acetylene
Figure 2.4 Examples of covalent bonds [50].
In contrast to compounds formed by ionic bonds, molecules formed by covalence contain a definite number of atoms and possess specific properties. In summary, electrovalence leads to ionic agglomerates through a complete transfer of electrons and formation of ions, whereas covalence produces molecules by sharing pairs of electrons. Ionic conglomerates do not display any preferred direction in their internal structure, but covalent bonds are directional with atoms and molecules combining according to specific geometric patterns. This difference is of particular importance to the clay minerals. Ionic bonding causes separation between the centers of positive and negative charges in the compound, thus forming a dipole. The latter is a system of a positive charge, -ne, where n is an integer and e is the unit charge of one electron, separated by a distance d. In an electric field the dipole will orient itself accordingly. The product
Mo dne,
(2.5)
known as the dipole moment, is a measure of its strength. In electrovalent compounds the valence number of an ion is numerically equal to the charge of the ion. In covalent compounds or molecules, the valence number of an atom is numerically equal to the number of electrons shared by the interacting atoms. However and contrary to the ionic compounds, the valence number may vary from molecule to molecule, depending on the number of shared electrons. This is illustrated in the case of the carbon atom in the two examples of Figure 2.4. In methane, the carbon shares all its four electrons in its outer shell with hydrogen; therefore, it has a valence of four. In acetylene, however, it shares only two of its four electrons with two hydrogen atoms and, therefore, it has a valance of two. This property of covalent bonds is responsible for the complexity of organic chemistry. Primary electrovalent and covalent bonds are much stronger in comparison to the secondary bonds to be discussed in Section 2.2.2. The energies of primary bonds per mole of bonded atoms range from 60 103 J to more than 400 103 J, an equivalent range between 15 and 100 kcal ([88], Section 2.3). Considering that every mole of a substance contains 6.03 1023 molecules
20
Cohesive Sediments in Open Channels
(Avogadro’s number), the energy per molecule ranges from 1019 J to a maximum of 66 1019 J. These energies may appear small in absolute terms, but they are very large if the mass of the molecule is taken into account. It follows that high energy is needed to break the primary bonds in comparison to the secondary bonds. The first are significant in understanding the structure and properties of clay minerals, but the latter are the ones of importance to flocculation and to cohesive sediment behavior.
2.2.1.3 Some Basic Concepts and Definitions of Chemistry This section will close with the definition of certain basic terms, laws, and concepts of chemistry, which are frequently encountered in soil technology and, therefore, in cohesive sediment behavior. The following are some of the most important of these terms and concepts, which can be found in any introductory book on chemistry: 1. The law of definite proportions. Whether combined by ionic or covalent bonds, the number of electrons involved in the process determines the relative mass of each element in the compound product. That means that a given compound contains the same elements combined in the same proportions by mass. This is the well-known law of definite proportions. As an example, consider the combination of oxygen and hydrogen to produce water. Oxygen has an atomic weight of 16 and 6 electrons in its outermost shell, and hydrogen has an atomic weight of about 1 and 1 electron in its only shell. Since an atom of oxygen needs 2 electrons to fill its outermost shell, it has to combine with two atoms of hydrogen to produce one molecule of water. With the oxygen atom having 16 times the mass of one hydrogen atom, it follows that 8 mass units of oxygen have to combine with one mass unit of hydrogen to produce water. The proportion of atoms in the compound is indicated by subscripts in its constituent element with the subscript 1 omitted. Thus, the chemical formula of water is OH2. From the chemical formula and the atomic weight of the constituent elements, the molecular weight of the compound can be defined and evaluated. The latter is the sum of all protons and neutrons in the elements of the compound. For example, the molecular weight of water is equal to 18; that is 2(atomic weight of hydrogen) one atomic weight of oxygen 2(1) 16 18. 2. The mole. A quantity of a compound equal in weight to its molecular weight is called a mole indicated by M. A mole is furthermore characterized by the units of weight or mass used. For example, it may be indicated as a gram mole, a kilogram mole, and a pound mole. Thus, the gram mole of water is 18.016 grams, and its pound mole is 18.016 lbs. The number of moles that a substance may contain can be found by dividing the actual weight of the substance by its molecular weight. 3. The equivalent weight. This property is defined as that weight or mass of an element which combines with or displaces from any substance a weight or mass of hydrogen equal to the atomic weight of the latter. The equivalent
Chapter | 2 The Mineralogy and the Physicochemical Properties
21
weight is very important in determining the proportions of the various elements that form a specific compound. Since a mole of an element combines with a number of moles of hydrogen equal to their valence number, it follows that the equivalent weight of an element is equal to its atomic weight divided by its valence number. Therefore, one equivalent weight of any element reacts with one equivalent weight of any other element to produce one equivalent weight of the resulting compound. The equivalent weight of a compound is defined as the molecular weight of the substance divided by the net positive valence, that is, by the product of the valence of the positive element of the compound or molecule times its subscript in the chemical formula. This last product represents the number of replaceable hydrogens. In summary:
Equivalent Weight of Element
Equivalent Weight of Mole
Atomic Weight Valence Number
Compound Molecular Weight Net Posiitive Valence
Net Positive Valence (Valence Number ) (Chem. Subscript )
The equivalent weight of oxygen is, for example, 16/2 8, since one mole of oxygen combines with two moles of hydrogen to produce one mole of water. Therefore, the equivalent weight of an element can also be defined as the mass or weight of the element which combines with or displaces eight parts by weight or mass (in this case 8 unit weights) of oxygen. Many elements can manifest more than one valence number, depending on the compound under consideration. Such elements can have more than one equivalent weight, which is shown by the element in a particular reaction. Iron, for example, depending on the conditions of the reaction, may form two oxides: FeO and Fe2O3. In both oxides the oxygen atom has a valence of 2; therefore, the valence of Fe in FeO is 2 but in Fe2O3 is 3. 4. Solutions and concentrations. Substances very often appear in a dissolved state in liquid media. The dissolved substance is defined as the solute and the liquid medium as the solvent. An enormous number of chemical reactions take place in solution. In the domain of cohesive sediment dynamics, the only solvent to be considered is water. The amount of solute per unit volume or per unit mass of solvent is defined as the concentration of the substance. There are various ways to express the latter quantitatively, the following being the most commonly used:
(a) Weight per unit volume. The most frequent units for such a concentration are one gram of substance per liter of solution and one gram of substance per cubic meter of solution. The latter is normally referred to as parts per million symbolized by ppm. Obviously, one gram per liter is equal to 1,000 ppm.
22
Cohesive Sediments in Open Channels
(b) Molarity. This term indicates the number of moles of a solute per liter of solution, and its symbol is M. (c) Normality. This is the number of equivalents of solute per liter of solution, and its symbol is N. Molarity and normality are related by the following equation:
N M(Net Positive Valence) (d) Molality. The number of moles per 1000 grams of solvent is defined as molality, and its symbol is ML. For very low concentrations in water, molality is numerically very close to normality; in general, however, molality and molarity are two distinct terms and they may differ substantially—first, because 1000 grams of a liquid is not necessarily a liter and, second, molality indicates concentration with respect to the weight of solvent, whereas molarity defines concentration with respect to the total volume of solution. (e) Percent of composition. This term may indicate either percentage by weight or percentage by volume according to the relations: Percent by weight Wt. of Dissolved Solute/Wt. of Solvent
Percent by volume Volume of Solute/Volume of Solvent
In solids dissolved in liquids, the first definition is almost exclusively used, while the second is common to solutions of gases in gases or liquids in liquids. Solutes may be distinguished into electrolytes and nonelectrolytes. When dissolved in water, electrolytes produce a solution that conducts electric current, whereas nonelectrolytes do not. Acids, bases and salts are electrolytes. Solutes may affect the properties of the solvent. For example, they may lower the vapor pressure and the freezing point and raise the boiling point. Discussion of these effects is beyond the scope of this limited review; interested readers are referred to any introductory book of chemistry. As already mentioned, ionic and covalent bonds, also known as primary bonds, are the strongest of all bonds and the ones primarily responsible for the formation of compounds and the internal structure of clay minerals. There is a third primary bond, the metallic bond. This is a nondirectional bond generated in metals by loosely held valence electrons that hold the positive metal ions together while at the same time are free to travel through the solid material ([98], Chapter 2). Metallic bonds are irrelevant in most soils and they are mentioned here only for the sake of completion.
2.2.2 Secondary Bonds In addition to the primary bonds, there are other weaker bonds which may affect the final arrangement of atoms in solids and cause attraction between very small particles as well as between solid particles and liquids. These bonds are termed
Chapter | 2 The Mineralogy and the Physicochemical Properties
23
secondary bonds. The following are the most important secondary bonds for cohesive sediments and will be briefly discussed.
2.2.2.1 Hydrogen Bond A hydrogen bond may develop when a hydrogen ion forms the positive end of a dipole being attracted to the negative charge of a molecule at the surface of a particle. These bonds require strongly electronegative atoms, such as oxygen, to produce strong dipoles. The electron transferred from the hydrogen to the oxygen spends most of its time between the two atoms, thus leaving the hydrogen proton to act as the positive end of the dipole and the negative oxygen atom as its negative end. A hydrogen bond is, therefore, a permanent directional dipole and can readily be formed between clay minerals with oxygens on their surfaces; they constitute the mechanism for water adsorption on the surfaces of clay minerals and play an important role in determining the fundamental properties of cohesive sediments. The small size of the hydrogen atom makes it particularly suitable to fit between the spaces of surface atoms of clay particles. For this reason, hydrogen bonds are stronger than any other secondary bond. 2.2.2.2 van der Waals Bonds In contrast to the permanent dipole of hydrogen bonds, the van der Waals bonds amount to a fluctuating dipole generated by the mutual influence of the motion of electrons of atoms. At any time there may be more electrons on one side of the atomic nucleus than on the other, thus creating instantaneous dipoles with oppositely charged ends, which attract each other. The van der Walls forces are always attractive and exist between all units of matter. They are the primary mechanism of flocculation but are one order of magnitude smaller than the hydrogen bonds. There are three reasons for their importance. First, as already mentioned, they exist between any two material units, whereas the development of the hydrogen and cation bonds, to be discussed later, requires special conditions. Second, they are for all practical purposes independent of the water quality and of other physicochemical factors. And third, although individual van der Waals forces are relatively weak, they are nondirectional and additive between atoms, and the overall attractive force and its potential decrease more rapidly with distance than ionic and primary valence bonds. Specifically, for two atoms the van der Waals attractive force is inversely proportional to the seventh power of the interparticle distance, and its potential is inversely proportional to the sixth power of that distance. The net force and potential between two spherical particles are inversely proportional to the third and second power of the distance, respectively. The equation describing the variation of the potential energy of the van der Waals forces is presented in Chapter 3. 2.2.2.3 Cation Bonds Such bonds amount to cations attracting negatively charged particles, as in Figure 2.5. Cation bonds, for example, occur between montmorillonite units
24
Cohesive Sediments in Open Channels
�
�
�
�
�
�
�
�
Figure 2.5 Schematic picture of a cation bond [99].
(Section 2.3) with sodium or potassium as the bonding cations. Cation bonds are weaker than hydrogen bonds and can easily be broken by water adsorption and swelling.
2.2.2.4 Chemical Cementation The type of bonding known as chemical cementation actually amounts to links between clay particles generated by chemical compounds. Such compounds, for example, are used for the treatment of soils as subgrade materials ([88], Chapter 4, Chapter 9, and Chapter 11). Iron oxides, which carry a net positive charge, are strong cementing agents with an iron oxide being attracted by two negatively charged clay particles.
2.3 The nature and mineralogy of clay particles 2.3.1 Introductory Remarks So far we have been using loosely the term fines and clays without any qualification other than their size. For an understanding of their hydrodynamic behavior, these sediments have to be distinguished into certain groups, each one of which is characterized by special mineralogical and physicochemical properties. These properties control the interparticle forces and, therefore, the engineering behavior of aggregates and of cohesive sediment deposits. One may recall that the various atoms within a solid particle are held by strong primary valence bonds, whereas the much weaker secondary valence bonds, specifically the van der Waals, hydrogen, and cation bonds, are responsible for the interparticle attractive forces whether in suspension or in a deposited sediment bed. That means that any particle agglomerate, when subjected to stresses, will deform and eventually break through failure of the interparticle joints rather than by breaking of the particles. Fine sediments have been defined as mixtures of silt and clay possibly with some fine sand and organic matter exhibiting colloidal properties. These colloidal properties are due almost exclusively to the clay fraction of the total sediment. On the basis of their size and according to the most widely accepted classification, silts range in size from 2 to 74 m, whereas the size of clays is below 2 m. However, a mechanical classification by itself does not reveal the basic properties of flocs and of cohesive sediment deposits. The colloidal properties of the clay fraction of the sediment do not depend only on the nominal grain size range, but also on a variety of other factors, such as mineralogy and surface activity.
Chapter | 2 The Mineralogy and the Physicochemical Properties
25
In fact, particles larger than 2 m may display colloidal properties, while particles smaller than 2 m may be virtually inert as far as physicochemical forces are concerned. It is more realistic to distinguish fine sediments into clay and nonclay with silts belonging to the second group. Clay particles have the shape of small flat plates, needles, and tubes with high specific area and are subjected to physicochemical forces that are very large in comparison to their weight. These forces, which are the subject of the following chapter, are determined by the mineralogical characteristics of the clay particles, their crystalline structure, and the dissolved ions in the water.
2.3.2 The Basic Clay Minerals Modern electron microscopy, X-ray diffraction, and differential thermal analysis have revealed that clays are composed essentially of one or more members of a small group of clay minerals [88, 99, 150]. These minerals have a predominantly crystalline arrangement as a result of the directional covalent bonds. That means that the composing atoms are arranged in such an orderly way as to form a definite three-dimensional geometric network defined as the lattice of the crystal. The points of a lattice in which atoms or atomic groups are located are termed lattice points. Only 14 different arrangements of lattice points have been identified and are known as Bravais space lattices. If a clay crystal is continuously subdivided, a minimum size will eventually be reached that has all the characteristics and atomic arrangement of the crystal. That minimum size element is defined as a unit cell. A clay particle, therefore, is composed of unit cells. The 14 unit cells, according to Bravais classification, are shown in Figure 2.6 with dots indicating location of atoms ([88], Chapter 2, Figure 2.4). Each of these arrangements can be described by its three sides a, b, and c and the three angles , , and between them. Crystals have been classified in 32 distinct classes based on the arrangement of atoms and the orientation of their faces. Some of these crystal classes bear close similarities and relationships to each other so that they can be grouped into six crystal systems illustrated in Figure 2.7. The indicated crystallographic axes aa, bb, and cc are parallel to the edges of intersection of the prominent crystal faces, and the relationship between the sides a, b, and c and the angles , , and are indicated for each crystal system. It should be noted that five out of the six crystal systems have three crystallographic axes, a, b, and c, whereas the hexagonal system has four. The following is a brief description of each crystal system with some examples ([88], Chapter 2). 1. Isometric or cubic system. The three axes of this system are perpendicular to each other with equal lengths. Galena, halite, magnetite, and pyrite belong to this system. 2. Hexagonal system. This system has a hexagonal base characterized by three axes of equal length intersecting at 60° and a fourth c axis normal to the hexagonal base. Quartz, brucite, calcite, and beryl are examples of this group.
26
Cohesive Sediments in Open Channels
� � 90�� a�b�c
c
c a a a a p
a
a p
b
c
P a
� � � 90� a�b�c
b P
c
c
a
a I
b C
Tetragonal
a
c
a
� � � 90� a�b�c
a
a b C
F a
a
c a R
Monoclinic a
b F
Rhombohedral
a
� � a�b�c
c a a
c
a
c
I Isometric: a � b � c: � � � 90�
a
a P
a
� � 90� a�c
a
b b I
Hexagonal Cubic
� 120�
Orthorhombic
P Triclinic
a� � � 90� a�b�c
Figure 2.6 Lattice classification according to Bravais [88, Fig. 2.4].
3. Tetragonal system. This system has three mutually perpendicular axes: two horizontal of equal length and one vertical of different length. Zircon belongs to this group. 4. Orthorhombic system. In this system the three mutually perpendicular axes have different lengths. Examples of minerals belonging to this system include sulfur, anhydrite, barite, diaspore, and topaz. 5. Monoclinic system. Two of the three axes of this system are unequal and inclined to each other at an oblique angle, while the third axis is perpendicular
27
Chapter | 2 The Mineralogy and the Physicochemical Properties
c
c
α
β
a
a3
60°
60°
b
b
60°
a2
γ a
a2 a1
abc αβγ90°
a3 a1a2a3c
c
c
Isometric (cubic)
Hexagonal
c
c
α
β
a
b
α
β b
b γ
a1
abc αβγ90°
abc αβγ90°
c
c
Tetragonal
Orthorhombic
c
c
α β
a
b
α
β b
a
b
b
γ a
a1
b
γ a
a1
abc αγ90° β90°
γ cab αβγ90°
a
c
c
Monoclinic
Triclinic
Figure 2.7 The crystalline systems [88, Fig. 2.5].
to the other two. Orthoclase, feldspar, gypsum, muscovite, biotite, gibbsite, and chlorite are among the most common minerals belonging to this group. 6. Triclinic system. All three axes of this system are unequal intersecting each other at oblique angles. Typical examples of this group are plagioclase, feldspar, kaolinite, albite, microcline, and turquoise.
28
Cohesive Sediments in Open Channels
An interesting question is how and why the various crystal structures develop. The answer lies in the principle that atoms tend to take the most stable arrangement in a structure and this arrangement is the one that minimizes the energy per unit volume. The geometry of the lattice determines also the cleavage planes, that is, the planes along which a particular crystal breaks, as well as the external form of a piece of mineral. The cleavage plane lies between planes having the most densely packed atoms because in this case the center-to-center distance between atoms on opposite sides of the plane is greater than across any other plane through the crystal. Therefore, the shear strength along the cleavage plane is smaller than in any other direction. Finally, the orderly atomic arrangement within the crystals allows diffraction of light, X-rays, and electron beams. These properties have been used to identify the minerals. For a more detailed description of crystals and discussion of their properties and their identification, the reader is referred to ([88], Chapter 2). Clay minerals appear as small particles ranging in dimensions from a fraction of 1 m to a few micrometers. Each particle is built by a number of crystal unit cells held together by secondary valence bonds. Depending on the nature and strength of the bonds, clay particles will disintegrate in water to various degrees. Those with the more resistant bonds will form sediment with grains composed of several unit cells in the form of a book or packet, while those with relatively weak bonds and, in particular, with swelling cations between them may disintegrate to almost single unit cells. The clay mineral kaolinite belongs to the first case, while the sodium montmorillonite mineral belongs to the second. Chemically, clay minerals are silicates of aluminum and/or magnesium and iron. From the structural standpoint there are two fundamental building blocks. Each one of these blocks forms two dimensional arrays: the silica tetrahedral unit and the aluminum, magnesium, or iron octahedral unit [88, 99, 150]. The sequence of superposition of these blocks, the cations in the lattice of the mineral, and the adsorbed cations between the unit cells determine the nature of the various clay minerals. The following are the two fundamental building blocks: 1. The silica tetrahedral unit. This unit consists of four oxygens having the configuration of a tetrahedron enclosing a silicon atom (Figure 2.8a). The tetrahedral units are combined in a sheet structure, known as a tetrahedral sheet or a silica sheet, so that the oxygens of their bases are in a common plane with each oxygen shared by two tetrahedral units while all the tips point in the same direction (Figure 2.8b). In this way three of the oxygen atoms of each tetrahedron are shared with three neighboring tetrahedra. Figure 2.9 shows a plan view of such an array of tetrahedra in which the hexagonal symmetry is clearly discerned. The unit cell area is indicated by the dashed line rectangle. This unit has 4 silicon atoms and 10 oxygen atoms with an electric charge of 4, and a composition of (Si4O10)4. Electrical neutrality can be obtained by replacement of four oxygens by hydroxyls or by a union with a positively charged sheet of different composition. In the array,
Chapter | 2 The Mineralogy and the Physicochemical Properties
(a)
Single unit and
(b)
� Oxygens
� Silicons
Figure 2.8 Silica tetrahedral units [88, 99].
(b) (a)
Legend Oxygen Slicon (c)
Combination
(d)
Figure 2.9 Top view of a tetrahedral sheet [150, Ch. 6].
29
30
Cohesive Sediments in Open Channels
OH OH
OH
OH
OH
OH
OH
OH OH
OH OH
OH OH (a)
OH Single unit
OH OH (b)
OH and OH � Hydroxyls
OH
OH OH
Combination � Aluminum, Iron and Magnesium
Figure 2.10 Octahedral units [88, Ch. 3; 99].
the oxygen-to-oxygen distance is 2.55 Å, the space available for the silicon atom is 0.55 Å, and the thickness of the sheet is 4.63 Å ([150], Chapter 6). The hexagonal oxygen arrangement in the silica sheet leaves holes within each hexagonal ring of six oxygens. These holes play a dominant role in the adsorption of ions and in the stability of clay minerals in water. 2. Aluminum, iron or magnesium octahedral. This second building block consists of six hydroxyls and/or oxygens having the configuration of an octahedron enclosing an aluminum, iron, or magnesium atom, as shown in Figure 2.10a The octahedral units combine together into a sheet structure, known as an octahedral sheet or an alumina or a magnesia sheet, with each hydroxyl shared by two units as in Figure 2.10b. The sheet may be viewed as two layers of densely packed hydroxyls with the cation between them in octahedral coordination. The neutral three-layer structure is the basic model known as pyrophyllite. If the cation is trivalent aluminum, only two of the three cation spaces are filled. Such a structure is called dioctahedral with composition Al2(OH)6 and the corresponding mineral is known as gibbside. If the cation is divalent, such as magnesium, normally all three cation spaces are filled with the cation, and the structure is defined as trioctahedral with composition Mg3(OH)6. The mineral corresponding to this trioctahedral structure is defined as brucite ([88], Chapter 3). Figure 2.11 shows a plan view of an octahedral sheet together with a horizontal projection of an octahedral unit. It is observed that the oxygen atoms and hydroxyl groups form a hexagonal close packing with “holes” considerably smaller than those in the tetrahedral sheet. Cations other than aluminum, Al3, and magnesium, Mg2, can also be present in the octahedral like Fe2, Fe3, Mn2, Ti4, Ni2, and Cr3. The oxygen-tooxygen distance in the octahedral sheet is 2.60 Å, the space available for the cation is 0.61 Å, and the thickness of the octahedral sheet is 5.05 Å. The various clay minerals are formed by superposition of these two fundamental building blocks. The structure of each clay mineral is determined by the cation in the octahedral layer, the sequence and number of building blocks in the unit cell
Chapter | 2 The Mineralogy and the Physicochemical Properties
31
(b)
(a)
Legend Oxygen
Hydroxyl
(c)
Al or Mg
(d)
Figure 2.11 Top view of an octahedral sheet [150, Ch. 6].
of the mineral, and the cations between the unit cells. A unit cell of a clay mineral is the smallest part of the structure, which repeats itself in both directions of the sheet and normal to it. Tetrahedral and octahedral sheets combine by sharing oxygen atoms in a common plane. Depending on whether two layers (one tetrahedral and one octahedral sheet) or three layers (two tetrahedral and one octahedral sheet) are involved in the unit cell, the minerals are distinguished into two-layer minerals or 1:1 layer minerals and three-layer minerals or 2:1 layer minerals, respectively. Unit layers are stacked parallel to each other (Figure 2.12). There are many factors involved in the development of the clay minerals. The most important of these factors are the cations in the lattice of the building blocks, the unit layer of adsorbed cations and water between the layers, and the development of the crystal structure. The latter may range from very poor to almost perfect. There is, therefore, an enormous variety of clay minerals, so that clay mineralogy amounts to a distinct field in itself. This discussion is limited to the description of a simplified structural model to be used for an explanation of the fundamental characteristics and engineering behavior of the most commonly encountered groups of clay minerals. For an in-depth study, Refs. [88, 150] are recommended.
32
Cohesive Sediments in Open Channels
Charge 6 O 12 4 Si 16
Tetraheral sheet octahedral sheet
4O 2 OH
10
4 Al
12
6 OH 6
(a)
Surface of the unit cell: 5.15 8.9 Å2 Formula of unit cell: [Al2 (OH)4 (Si2 O5)]2 Unit cell weight: 516
O OH Si Al
28 28 C–spacing 7.2 Å
Distance between atom centers Tetraheral sheet
Charge
0.60 Å
6O 4 Si
1.60 Å
16
4 O 10 2 OH
Octahedral sheet
2.20 Å
Tetraheral sheet
1.60 Å
4 Al
O OH Si Al
12
2 OH 10 4O
0.60 Å
(b)
12
Surface of the unit cell: 5.15 8.9 Å2 Formula of unit cell: [Al2 (OH)4 (Si2 O5)2]2 Unit cell weight: 720 Hydroxyl water: 5%
4 Si 6O
12
16
44 44 C–spacing 9.2 Å
Figure 2.12 Outline of building blocks: (a) two-layer mineral; (b) three-layer mineral [150, Ch. 6].
Clay minerals can be grouped according to their crystal structure and stacking sequence of the various layers. Those belonging to the same group have similar engineering properties and are expected to display similar sedimentological behavior. In relation to the lattice cations, there is an important process known as isomorphous substitution. This term implies replacement within the lattice of the unit layer of one cation by another of the same or different valence without altering the crystal structure of the mineral. In this way, some of the tetravalent silica, Si4, in the tetrahedral sheet may be replaced by trivalent aluminum, Al3. It is more common, however, to have a substitution in the octahedral sheet of trivalent aluminum by divalent magnesium, Mg2, and of magnesium by ferrous iron. The cation distribution in the octahedral and tetrahedral layers may develop either
33
Chapter | 2 The Mineralogy and the Physicochemical Properties
OH
OH
OH
OH
OH
Oxygens
7Å
OH Hydroxyls
Aluminiums OH
OH OH
OH
Silicons
OH
Figure 2.13 Kaolinite structural unit [99, 100].
during the initial stage of formation of the clay mineral or later on. An isomorphous substitution of a higher valence cation by one of lower valence leads to the development of a negative electric charge on the clay particle. This charge plays a dominant role in the process of flocculation. As already discussed, the unit layer of a clay mineral consists of a sequence of tetrahedral and octahedral sheets with two such sheets sharing oxygens and hydroxyls on the same plane with both covalent and ionic bonding of the primary valence type. Unit layers are held together by much weaker secondary valence bonds, some of which may vary widely from relatively strong to very weak; in addition, they can be drastically affected by physicochemical changes in the water environment. van der Waals forces and hydrogen bonds are relatively stable; cation bonds, however, can be significantly affected by water adsorption and swelling, which may push the unit layers apart. Potassium ions provide the most stable cation bonds because they fit well into the holes of the bases of the silicon tetrahedron sheet. In contrast, sodium between octahedral units does not fit in any base holes, hydrates, and swells in the presence of water, thus resulting in extreme cases in an almost complete disintegration of the clay mineral to its unit cells. There are two basic structural groups of tetrahedral and octahedral sheets: the kaolinite and the montmorillonite. 1. The kaolinite structural or two-layer group. The unit of this group consists of an aluminum (or iron or magnesium) octahedral layer with a parallel superimposed silica tetrahedral layer so that the tips of the silica sheet and one of the layers of the octahedral unit form a common plane as in Figure 2.13. The unit is about 7 Å thick and may extend indefinitely in the other two directions. Kaolinite, dictite, nacrite, and halloysite are members of this group. That mineral
34
Cohesive Sediments in Open Channels
may thus be viewed as a succession of oxygens, silicons, hydroxyls plus oxygens, aluminums (or irons or magnesiums), and hydroxyls. Particles are formed by a regular stacking of such 7 Å units in a book-like form. 2. The montmorillonite structural unit. This structural unit is made up of sheetlike layers with each of them composed of two silica tetrahedral layers and one octahedral layer between them like a sandwich. The oxygen atoms of the tetrahedral tips combine with the hydroxyls of the octahedral layer bonded by primary valence bonds into a continuous plane. The order of the elements in the unit layer of this group is as follows: oxygens, silicons, oxygens plus hydroxyls, aluminums (or irons or magnesiums), oxygens plus hydroxyls, silicons, and oxygens (Figure 2.14). The thickness of the sheet is approximately
OH
9.5 A
OH OH
Exchangeable cations and water
Oxygens OH Hydroxyls
Aluminum, Iron, and Magnesium
Silicon, Occasionally Aluminum
Figure 2.14 Montmorillonite structural unit [88, Section 3.7].
35
Chapter | 2 The Mineralogy and the Physicochemical Properties
9.5 Å with unlimited dimensions in the other two directions. The sheets are stacked one above the other like the leaves of a book in an arrangement similar to that of the kaolinite group. The various principal mineral groups to be described and discussed here consist of different combinations of these two fundamental structural groups stacked one on the other and held together by secondary valence bonds. The kinds of bonds, the nature of the bonding cations, and the cations within the structural group determine the type of clay mineral, its physicochemical characteristics, and its engineering properties, specifically its hydrodynamic behavior in a flow field. The following are the most important clay minerals. 1. The kaolinite-serpentine group or the 1-1 minerals ([88], Chapter 3, Section 3.6). The minerals of this group are composed of the structural kaolinite unit, which consists of alternating layers of silica tetrahedral and octahedral sheets with the tips of the tetrahedra coinciding with one of the planes of atoms of the octahedral sheet and pointing toward the center of the latter. Two thirds of the atoms in that plane are oxygens shared by both the tetrahedral and the octahedral layers. The remaining atoms in that plane are hydroxyls (OH) with each one of them located below the hole of the hexagonal network formed by the bases of the silica tetrahedra, as shown in Figure 2.9. The mineral is called kaolinite if the octahedral layer is dioctahedral gibbsite, and it is called serpentine if the octahedral layer is trioctahedral brucite. The structural formula of the kaolinite is (OH)8Si4Al4O10 and is outlined schematically in Figure 2.15. As shown, the unit cell has zero charge and its exact dimension is 7.2 Å. Dioctahedral minerals of the kaolinite group are by far the most common; in contrast, trioctahedral 1:1 minerals are rather rare, and they are normally encountered mixed with kaolinite and illite.
B
G
B
G
7.2 Å
7.2 Å B
G
(a)
(b)
Figure 2.15 Atomic arrangement in a unit cell of (a) kaolinite, (b) (serpentine) [88, Fig. 3.9].
36
Cohesive Sediments in Open Channels
Successive kaolinite layers are held together by strong hydrogen and van der Waals bonds. Therefore, very little water can enter between the individual unit layers, and very few cations can be adsorbed between them. Because of this bonding, kaolinite clays do not swell, have a stable structure and, although they cleave fairly easily along the plane surface of the 7.2 Å unit cell, do not disintegrate and split in water like other clay minerals. They form the largest subgroup of clay minerals in nature in the form of six-sided flat plates with lateral dimensions ranging from 0.1 to 4 m and with a thickness ranging from 0.05 to 2 m. The specific area of a kaolinite particle ranges from 10 m2 to 20 m2 per gram of dry clay. This is the lowest specific area in clay minerals; therefore, kaolinites are characterized by the lowest surface activity, which is the lowest intensity of interfacial physicochemical forces in comparison to their weight. A good measure of surface activity is the cation exchange capacity. As it will be shown in Chapter 3, the surfaces of the clay particles carry a negative charge either from adsorption of anions or from isomorphous substitution. To preserve electrical neutrality, cations are attracted on the surfaces and the edges of clay particles. Some of these cations can be replaced by others of a different type in the solution. The quantity of such exchangeable cations per unit mass of clay is defined as the cation exchange capacity, it is symbolized by CEC, and it is commonly expressed in milliequivalents (meq) per 100 g of clay. Kaolinite has the lowest CEC ranging from 3 to 15 meq/100 g. Although isomorphous substitution in kaolinites is believed to be rather rare and uncertain, it takes the replacement of one trivalent silicon by a bivalent aluminum in every 400 units to account for an exchange capacity of the above order ([150], Chapter 6). Moreover, broken bonds around the edges of the clay particles may give rise to unsatisfied charges that are also balanced by adsorbed cations. It is noteworthy that, because separation does not take place between unit layers of kaolinite, the balancing cations can be adsorbed only on the faces and edges of particles but never in the inbetween layers. A variation of the kaolinite subgroup is the mineral halloysite [88, 150]. It is structurally similar to kaolinite, except that its unit layers are separated from each other by a single layer of water molecules, when hydrated. Its composition is (OH)8Si4Al4O10.4(H2O), and the spacing of the unit layer is 10.1 Å, which is considerably higher than the 7.2 Å spacing of the kaolinite. The difference of 2.9 Å is the approximate thickness of one water molecule. Because of this layer of water molecules, the interlayer bonds are reduced, resulting in a curvature of the mineral and the formation of tubes with the hydroxyls of the octahedra on the inside and the basis of the tetrahedra on the outside. The outside diameter of the tubes ranges from 0.05 to 0.20 m, and the average wall thickness is 0.02 m. Dehydration of halloysites at low temperature removes the interlayer water and reduces the mineral to its kaolinitic form; however, this may cause splitting and unrolling of the tubes. The separation of the unit layers by water molecules makes possible the adsorption of cations there, thus increasing the cation exchange capacity of the mineral to a range between 40 and 50 meq/100 g. This is quite higher than that
37
Chapter | 2 The Mineralogy and the Physicochemical Properties
of kaolinite. The specific area of halloysites ranges from 35 to 70 m2g1. This is about 3.5 times the specific area of kaolinites. Therefore, from the standpoint of surface activity, halloysites are more active than kaolinites. 2. The smectite or montmorillonoid structural subgroup. This subgroup includes expending three-layer clays of the montmorillonite structural group shown in Figure 2.14. The theoretical composition in the absence of isomorphous substitution is (OH)4Si6Al4O20.n(interlayer)H2O [88, p. 30]. The hydroxyls of the octahedral sheet fall directly above and below the hexagonal holes formed by the bases of the silicon tetrahedral units. This neutral three-layer structure represents the mineral pyrophyllite, and it has a net spacing of 9.2 Å. When the octahedral layer is dioctahedral gibbsite with aluminum as its cation, the subgroup is defined as montmorillonite. When the octahedral layer is trioctahedral brucite with magnesium as its cation, the mineral is defined as saponite. The structures of these two subgroups are shown schematically in Figure 2.16. In either case the unit layer has a thickness of about 9.6 Å with indefinite dimensions in the other two directions. The unit layers are stacked one above the other like the leaves of a book in a way similar to the kaolinite. This higher spacing is due to the adsorption of cations and water molecules between the unit layers. There is, however, an important bonding difference between the kaolinite-serpentine group and the smectite group. Unlike kaolinites with strong interlayer hydrogen bonds, both sides of a smectite unit layer end in oxygens, so that the only possible interparticle bonds are the weaker van der Waals and cation bonds. As a result, cations and water can enter between the unit cells. The thickness of the water layer can vary depending on the nature of the adsorbed cations and on the hydration energy
G
(a)
B
G
9.6 Å → ∞
B
G
n-H2O � cations in interlayer regions
B
9.6 Å → ∞
(b)
Figure 2.16 Schematic diagram of smectite structure: (a) montmorillonite, (b) saponite [88, Fig. 3.15].
38
Cohesive Sediments in Open Channels
involved. If that energy is able to overcome the energy of the van der Waals attraction, the layers will split apart, leading in extreme cases to complete disintegration of the clay to its individual unit layers of a thickness of 9.6 Å. In smectites, there is an isomorphous substitution of silicon and aluminum by other cations. Thus, aluminum in the octahedral layer can be replaced by magnesium, iron, zinc, nickel, lithium, or other cations ([88], Chapter 6). The resulting structure is almost either exactly dioctahedral (montmorillonite subgroup) or trioctahedral (saponite subgroup). Figure 2.17 shows the charge distribution in a unit cell for the structural type of montmorillonite, where the indicated electric charges are in terms of the charge of an electron ([88], Chapter 2). These substitutions cause a positive charge deficiency and an excess of negative charge in the range of 0.5 to 1.2 with an average of 0.66 electronic unit charges per unit cell. Montmorillonite clays usually occur in thin flakes with a minimum thickness ranging from that of the unit layer of 10 Å to about 1/100 of their width and a maximum length of 1 to 2 m. The specific area of the smectites is even more illustrative of their activity. Their primary surface, excluding the interlayer zones, ranges from 50 to 100 m2g1, while the specific area including the interlayer zones, which are exposed to water penetration and to cation adsorption, can be as high as 840 m2g1 ([88], p. 31). Therefore, smectite minerals are electrochemically the most active. A special type of montmorillonite with commercial applications is the bentonite. It is the main clay mineral in the bentonite rock, which originates from the volcanic ash. It is the most active clay, it can form highly colloidal glue type suspensions, and it has the highest degree of swelling with a liquid limit of 500 and higher. Because of these properties, it has several practical applications, specifically as a stabilizing material in drilling in loose soils. That is why it is commonly 6O
�12
4 Si
�16
6O �10 2 (OH) 4 AI
�12
4O �10 2 (OH) 4 Si
�16
6O
�12
Net charge �44�44�0 Figure 2.17 Charges in a montmorillonite unit [88, Fig. 3.17].
9.6 to 21 Å
Chapter | 2 The Mineralogy and the Physicochemical Properties
39
referred to as driller’s mud. It should be noted that the name bentonite refers to the rock and not to the mineral itself. Montmorillonites constitute the most common mineral of the smectite group. The electric charge deficiency results from replacement in the octahedral sheet of every sixth trivalent aluminum atom (Al3+) by a divalent magnesium atom (Mg2+) with no substitution in the tetrahedral layer. The excess charge is balanced by exchangeable cations adsorbed not only on the surfaces of the clay particles, as in kaolinites, but also between unit cell layers. The chemical formula for the montmorillonite is [150, Appendix II]
Si8 (Al333 Mg067 )O20 (OH)4 ↓ Na 066
with the downward arrow indicating the deficiency and the balancing cation. In the case of sodium in the saponite subgroup, there is isomorphous substitution of silicon (Si4) by aluminum (Al3) in the tetrahedral sheet and iron (Fe3) for magnesium (Mg2) in the octahedral sheet. Its formula per unit cell is ([88], Chapter 3, Table 3.2):
(OH)4 (Si734 Al066 )Mg6 O20 ↓ Na 066
These formulas are indicative of the fundamental crystal structure of the clay mineral. A number of variations may exist within the same crystal structure ([88], Chapter 3, Table 3.2). 3. The mica group or nonexpending three-layer clays. Minerals of this group have the basic pyrophylite structure and consist of stacks of the three-layer units of Figure 2.16 held together by van der Waals forces. Since these forces are weak relatively to the primary valence bonds, cleavage parallel to the faces of the unit layers is easier. As a result, these minerals occur in the form of flakes. However, in the absence of interlayer swelling, these flakes are rather thick and large in comparison to clay particles of the kaolinite and smectite groups, and they do not disintegrate to a size typical for the clay minerals. For this reason they are classified as mica-type minerals. There are two main subgroups of the mica group: (a) the moscovite and illite and (b) the vermiculite. Their structures are schematically outlined in Figure 2.18. In moscovite the octahedral layer is gibbsite ([88], Section 3.8). About onefourth of the silicon positions in the pyrophylite unit are filled with aluminum, thus resulting in a negative charge balanced by potassium between the unit layers. Potassium atoms have a radius of 1.33 Å, which is about equal to the 1.32 Å radius of the holes in the bases of the tetrahedral sheets; therefore, the potassium
40
Cohesive Sediments in Open Channels
G K
K
Ca
K
K
K
K
G K
K
B
Mg
Exchangeable Ca
10 Å K Fixed
Mg
B
10 to 14 Å (14 Å as shown)
Ca
Mg
G B (a)
(b)
Figure 2.18 Structure of mica type minerals [88, Fig. 3.19]: (a) Moscovite and illite, (b) vermiculite
atoms can be buried into the structure of the pyrophylite. The unit cell formula then becomes (OH)4K2(Si6Al2)Al4O20. Moscovite is the dioctahedral end member of the micas with only Al3 in the octahedral layer. There are other types of micas, such as phlogopite or brown mica, which is the trioctahedral end of the mica subgroup with the octahedral layer filled entirely by magnesium. In between these two extreme clay minerals, there is the black mica or biotite, which is trioctahedral with the positions in the octahedral layer filled mostly with magnesium or iron. Since micas are relatively large flakes, they do not display colloidal behavior; however, they may contribute to the compressibility of the soil. On the other hand, although of a structure basically similar to that of micas, illites differ from the latter in certain details, which cause a drastically different formation of particles. Specifically, fewer of the silicon (Si4) positions are filled with aluminum (Al3) in the tetrahedral layer of illites and, consequently, there are fewer potassium atoms between the unit layers. There is also some randomness in the stacking of the unit layers. Some illites may contain magnesium and iron in the octahedral sheet. The charge deficiency in illites ranges from 1.3 to 1.5 electronic units per unit cell located primarily in the tetrahedral sheet ([88], Section 3.8). This deficiency is balanced by nonexchangeable potassium atoms between the unit layers, as in micas. However, because of the lesser charge, the number of the potassiums between the unit layers is lower in illites than in micas so that the bond between the former is weaker than that in the latter. Illite clays may expand and break into small flaky particles with a long axis ranging from 0.1 m to a few micrometers and a plate thickness as small as 30 Å. The only potassium ions that can be exchanged by other cations are the ones adsorbed on the external surfaces of the illite particles. In contrast, in smectites, such an exchange can take place also between the unit layers. The specific area of illites ranges from about 65 to 100 m2g1, and their cation exchange capacity (CEC) varies from 10 to 40 meq/100 g.
41
Chapter | 2 The Mineralogy and the Physicochemical Properties
Table 2.1 Clay Mineral
Lateral Dimensions (microns)
Thickness (microns)
Specific Area (m2/gm)
CEC (meq/ 100 gm)
Kaolinite
0.1–4
0.05–2
10–20
3–15
Illite
0.1–few
0.03
65–100
10–40
Montmorillonite
1–2
0.01– 1/100width
840
80–150
In vermiculites the octahedral layer is biotite with magnesium and calcium between the layers as charge compensating cations. In addition and very important, there is a double layer of water between the unit layers giving a basal spacing of 14 Å. The charge deficiency in vermiculites varies between 1 and 1.4 electronic units per unit cell, which is slightly less than that of the illites. In vermiculites the cations are exchangeable even between the unit layers due to spacing and the water layer; therefore, their cation exchange capacity (CEC) is higher than that in the illites, ranging from 100 to 150 meq/100 g. The specific area of the particles ranges from 40 to 80 m2g1, but the secondary or interlayer specific surface may be as high as 870 m2g1, that is, about equal to that of montmorillonites. Illite is the most common mineral found in soils of interest to sedimentation and to soil engineering. It is sometimes referred to as hydrous mica. In summary, the three most important clay minerals in the order of diminishing size and increasing physicochemical activity, as measured by their CEC, are kaolinites, illites and montmorillonites with the properties shown in Table 2.1. 4. Other clay minerals. In addition to the described primary clay minerals, there are some others less frequently encountered. Only the following most important of these minerals will be outlined: (a) Chlorites. The structure of the chlorites is similar to that of the vermiculites, and it is outlined in Figure 2.19. The mineral consists of alternate layers of mica and brucite with a basal spacing of 14 Å. An octahedral sheet replaces the water layer that exists between the mica layers. There is substitution of Mg2 by Al3, Fe2, and Fe3 as well as Al3 for Mg2 in the brucite layer. Chlorites are normally encountered mixed with other clay minerals in the soils in the form of plate-like microscopic grains of undetermined size and with a cation exchange capacity (CEC) between 40 and 100 meq/100 g ([88], Section 3.9). (b) Chain structure of clay minerals. These minerals include attapulgite and imogolite. They are formed of double chains of silica tetrahedra, and they have a thread-like or lath-like morphology with a diameter between 50 and 100 Å and a maximum length up to 5 m.
42
Cohesive Sediments in Open Channels
B–G B B–G
14 Å
B B–G
Figure 2.19 Structure of chlorite mineral [88, Fig. 3.23].
5. Amorphous (noncrystalline) materials. This group contains all materials encountered in soils without any definite crystalline structure. They are distinguished into (a) allophanes and (b) oxides. The first have no definite shape or composition, and their physical properties may vary widely. The most common oxides, contained to some degree in almost all soils, are hydroxides of aluminum, silicon, and iron. Oxides, as will be seen in Chapter 6, may affect the properties of the flocs and of the soils; however, at present, little is known about these effects. 6. Summary and general remarks. In this section the fundamental mineralogical properties of the basic clay minerals have been presented in a concise way and discussed. These properties are related to the interparticle physicochemical forces and to the mechanical properties of flocs and of cohesive sediment deposits; therefore, an understanding of them is essential in the study of the flocculation process and of the engineering behavior of cohesive soils. Specifically, clay mineralogy plays a dominant role in determining size, shape, and surface physicochemical properties of clay particles as well as their interaction with water. In engineering practice, mineralogical analysis is seldom performed except as a general description of the nature of the sediments. Instead, the mineralogical effects on the sediment and the soil properties can be represented by some parameters that can be readily evaluated by a few simple tests. The background material presented here is needed in order to appreciate the significance and reliability of these parameters, to properly control and interpret the related tests and, equally important, to communicate with the specialists in clay mineralogy and soil science whenever they may be involved in special cases.
Chapter | 2 The Mineralogy and the Physicochemical Properties
43
2.4 Origin and occurrence of clay minerals and formation of clay deposits Clays originate from the chemical weathering of rocks of various mineralogical compositions. The chief agent for weathering is the atmosphere, which uses energy derived from the heat of the sun. The gases of the atmosphere penetrate into the openings of the rocks and come into contact with a large internal surface of the latter. Oxygen and moisture are the most important ingredients of the atmosphere responsible for weathering. The process is furthermore enhanced by temperature variations ([88], Chapter 4, Section 4.2). Rock weathering can be distinguished into mechanical and chemical. The former amounts to a breaking of the rocks through mechanical forces generated mainly by temperature variations, frost action, stresses induced by drying and shrinking, abrasion, and by growth of plant roots in existing fractures. The product of mechanical weathering is sand and gravel. Chemical weathering, on the other hand, can eventually produce clays by chemically dissolving and removing various parent substances or through chemical reactions, bond breaking, and freeing cations. Practically all chemical weathering processes require the presence of water. The products of chemical weathering can readily be eroded by water and wind. In particular, clays, after having been subjected to flocculation, are transported by the flowing water and eventually are deposited in areas of favorable hydraulic conditions, such as flats, lakes, and estuaries. Such clay deposits can attain great depths. Throughout their geological history, clay deposits have been subjected to overburden pressures and high temperatures due to sediment accumulation, action of glaciers, and geotectonic processes. As a result, various degrees of metamorphism and changes of mechanical properties of the original clay deposits have taken place. Other clays, such as bentonites, have been formed essentially by alteration of in situ volcanic ash. Whatever the origin of the clay deposits, the geotectonic changes and the lifting of lands have brought above the water level older clay strata previously deposited under water and subsequently subjected to high overburden pressures. Erosion by rain, overland flow, waves, and rivers have exposed enormous masses of cohesive soils to the surface. The stability of these masses against further erosion may be very important in many cases. Such, for example, is the case of soil erosion by overland flow and of the stability of natural and artificial channels against scouring and/or siltation. If erosion continues, the eroded fine material will be transported and eventually deposited in other places, and the cycle will continue. In summary, there is a continuous process of decomposition, erosion, transport, deposition, and resuspension, known as the geologic cycle, outlined in Figure 2.20. A simplified version of the rock cycle is outlined in Figure 2.21 ([88], Chapter 4).
44
Cohesive Sediments in Open Channels
Denudation
Sediment formation
Deposition
Crustal movements Figure 2.20 Geologic cycle [88, Ch. 4].
Weathering environment Erosion, transportation, and deposition
Uplift and erosion
Metamorphicigneous environment
Sedimentary environment
Deep burial or igneous intrusion
Burial and heating Diagenetichydrothermal environment
Figure 2.21 Rock cycle [88, Ch. 4].
The geologic cycle consists of the following phases: (a) Denudation, which indicates the process of wearing down the land masses. (b) Weathering, which includes all the mechanical and chemical processes that break down in situ rock masses. (c) Transport, of eroded soil from its original position to the areas of deposition.
Chapter | 2 The Mineralogy and the Physicochemical Properties
45
(d) Deposition, which is the accumulation of sediment from its site to places where favorable conditions exist for deposition. (e) Sediment formation, which includes all the processes by which accumulated sediments are consolidated, densified, altered in composition, and eventually transformed into metamorphic rock. (f) Crystal movement, involving two processes: (i) the epirogenic movement that amounts to a gradual rising of unloaded areas and slow subsidence of basins where deposition takes place; (ii) the abrupt movements associated with faulting and earthquakes defined as tectonic movement. It should be noted, however, that more than one process may act simultaneously. For example, erosion can take place during periods of crystal movement; likewise, regional subsidence may occur during periods of filling of the basin with sediment. Hydraulic and soil engineers are almost exclusively interested in recent sediment deposits subjected to various degrees of consolidation and to a depth affected by externally imposed stresses. The engineering behavior of these surficial cohesive soils, and specifically their resistance to erosion, depends not only on the physicochemical and mineralogical characteristics of the clay itself, but also on the previous stress history of the deposits. Related to the latter is the fabric of the clay mass, which is the relative orientation and spacing of the clay particles. The soil fabric is determined first by the fabric of the depositing aggregates and, second, by the stresses and the degree of disturbance of the soil after deposition. It is reminded that the properties of the flocs and their aggregates depend also strongly on the flow-induced shear stresses during deposition. This aspect of sedimentation together with the types of flocculation and the soil fabric will be presented after the discussion in the following chapter of the forces between clay particles and of the process of floc formation.
Chapter 3
Forces between Clay Particles and the Process of Flocculation
3.1 Introductory remarks The basic properties of colloidal suspensions in water, the process of flocculation, the factors affecting flocculation, and some important and frequently encountered definitions were outlined and discussed in Section 2.1. The section showed that suspended particles may either repel or attract each other depending on the chemical properties of the solvent. This change from repulsion to attraction indicates the existence of both attractive and repulsive interparticle forces acting simultaneously on each colloidal particle. The interparticle forces are essentially secondary bond forces of an electrochemical nature and were outlined in Section 2.2. The origin of the repulsive forces lies in the net electric charge of the sediment particles. Although a sol is electrically neutral, suspended particles may have a negative or positive net electric charge. This charge can readily be demonstrated by applying an electric field to the sol, in which case negatively charged particles move toward the positive electrode and vice versa. This transport is known as electrophoresis. Almost without exception, clay mineral particles carry a negative electric charge on their faces and sometimes a positive charge on their edges. Positive charges occur in ferric hydroxide sols and other metal hydroxide sols. The latter, however, are not of direct importance to cohesive sediment transport mechanics, unless they interfere with the clay minerals themselves.
3.2 The electric charge and the double layer The electric charges on the surfaces and edges of clay mineral particles can be generated in two ways: (1) by isomorphous substitution, outlined in Section 2.3, and (2) by preferential adsorption of certain specific ions on the surfaces and the edges of clay particles. Cohesive Sediments in Open Channels Copyright © 2009
47
48
Cohesive Sediments in Open Channels
3.2.1 Isomorphous Substitution This origin of electric charges lies in the imperfections in the crystal lattice of the mineral. A single plane of atoms exists in both the tetrahedral and the octahedral sheets, which compose the clay mineral. The sheets are bonded together by strong primary bonds and the various structural units by secondary valance bonds. The latter are of several types and may be sufficiently weak to allow some chemical changes in these bonds. Specifically, a layer of one cation can be replaced by another of lower valence without altering the structure of the mineral, resulting in a deficiency of positive charge and in excess of negative charge. This substitution was discussed in Section 2.3. It was defined as isomorphous substitution, and it is a permanent feature of the clay mineral. Thus, in the tetrahedral sheet tetravalent, silicon, Si4, can be replaced by trivalent aluminum, Al3, and in the octahedral sheet, aluminum can be replaced by divalent magnesium, Mg2. The negative electric charge created by the isomorphous substitution is compensated by the adsorption on the surface layer of cations that are too large to be accommodated within the lattice. Isomorphous substitution is very common in montmorillonite but seldom in kaolinite. In the same section, an important property of the clay particles associated with the isomorphous substitution was introduced as the cation exchange capacity, symbolized as CEC. When clays are submerged in cation containing water, the compensating cations on the particle surfaces may be readily exchanged by other cations dissolved in the suspension known as exchangeable cations. This capacity can be determined chemically, and it is measured in milliequivalents per 100 g of clay mineral (meq/100 g). CEC is characteristic of the activity of the clays and their potential for flocculation. The following are some typical values of CEC: kaolinite, 3–8 meq/100 g; illite, 40 meq/100 g; montmorillonite, 80 meq/100 g. Montmorillonite is the most active and kaolinite the least active of these clays. The replacement depends on the valance, the relative abundance of ions of different types, and the ion size. Normally, small cations tend to replace larger ones. The following order is representative of the replacement ability of the cations [88, Chapter 7]:
Na , Li , K , Rb , Cs , Mg2 , Ca 2 , Ba 2 , Cu2 , Al3 , Fe3 , Th 4
The rate of exchange depends on the type of clay and on some other factors. It increases with decreasing physicochemical activity and vice versa. In kaolinite minerals the exchange is almost instantaneous; in illite clays a few hours are needed for complete exchange, whereas for montmorillonites it takes the longest time because the major part of the exchange capacity is located in the interlayer regions.
3.2.2 Preferential Adsorption This charge is created by the adsorption of specific ions on the particle surfaces. The valences of the lattice atoms that are exposed to the surface are not as completely compensated as those in the interior atoms of the lattice. These uncompensated
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
49
valances, referred to as broken bonds, are the ones responsible for the adsorption of the specific ions defined as peptizing ions, because they contribute to the creation of a deflocculated sol. The peptizing ions constitute an outer coating of the particle, and adsorption can take place by three processes: (1) by chemical bonds (chemisorption), (2) hydrogen bonds, and (3) by van der Waals attraction. The last two mechanisms are known as physical adsorption and are particularly responsible for the adsorption of organic ions. The adsorbed ions can be either negative or positive. Normally, negative stabilizing ions are adsorbed on the surfaces of the clay particles and positive ones on their edges. This last type of electric charge generation is important for kaolinites, where little isomorphous substitution takes place. There are several theories for ionic adsorption well discussed by Mitchell [88, Chapter 7]. In systems containing both monovalent and bivalent cations, the ratio of divalent to monovalent cations is much higher in the adsorbed layer than in the solution. A suggested practical guide is the Gapon equation:
1 Na Na k k (SAR)(meq / liter ) 2 2 (3.1) 2 1 [(Ca 2 Mg 2 ) / 2] 2 Ca Mg s e
The quantity on the right side in parentheses is defined as the sodium adsorption ratio (SAR) and can be determined by chemical analysis. The subscript s refers to the exchange complex of the clay and e to the equilibrium solution. k is the selectivity constant and has a value of 0.017 for most soils. Thus, if the composition of the pore fluid is known, the relative amounts of monovalent to divalent ions in the adsorbed cation complex can be estimated [88, p. 129]. Regardless of the origin of the electric charges, any clay particle in an ion containing water will attract ions of opposite charges. These ions are defined as counterions or gegenions. At the same time the ions tend to diffuse away from the surface because of their thermal activity. The diffusion takes place from a zone of high concentration to a zone of lower concentration in a way analogous to the diffusion of air molecules in the atmosphere. A clay particle can then be visualized as a thin rectangular plate surrounded on either side by a diffused layer of counterions with a distribution determined by the balance between the electrostatic attraction and their thermal activity. This layer is defined as the double layer and plays a dominant role in flocculation and in aggregate properties. The system of the particle and the double layer, defined as clay micelle, is electrically neutral and is outlined in Figure 3.1.
3.3 The theoretical formulation of the double layer 3.3.1 The General Case The mathematical description of the double layer is based on the distribution of the electrical potential and electric forces around the surfaces of the clay particles, and it is well presented by van Olphen [150, Appendix III]. The theoretical model was
50
Cohesive Sediments in Open Channels
Diffused Double layer Figure 3.1 The clay micelle [99].
first developed by Gouy in 1910 and Chapman in 1913. It is known as the GouyChapman theory, and the diffused ionic layer is also referred to as the Gouy layer. The theory has been developed for both planar (one-dimensional) and spherical surfaces. Only the one-dimensional case will be presented here based on the assumptions of (a) zero volume for ions (point charges), (b) uniform distribution of electric charge on the surface of the particle, and (c) infinitely large particle surface in comparison to its thickness (one-dimensional assumption). The derived equations have been shown to hold strictly for smectite particles and for monovalent solutions at low electrolyte concentrations, specifically less than 100 M/m3. Nevertheless, the model is adequate for the understanding of the flocculation process in general and of the behavior and properties of cohesive soils. A concise summary of the GouyChapman theory and some examples are presented in this section. A more complete treatment can be found in Ref. [88, Chapter 7] and in Ref. [150, Appendix III]. The counterions in Figure 3.1 are subjected to two opposing tendencies: electrostatic attraction by the negatively charged particle surface, which tends to bring them close to it, and diffusion due to thermal activity, which tends to drive them away from the particle surface. The result of these two opposing tendencies is a counterion distribution that decreases with distance from the surface reaching asymptotically the mean water concentration. The opposite is true for the negatively charged ions. However, at any distance within the double layer, both positive and negative ions are present. The ionic distribution within the double layer is given by the following equations developed by Boltzmann:
e * n n exp kT
(3.2a)
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
e * n n exp kT
51
(3.2b)
where n and n are the local concentrations of the positive and negative ions, ∗ ∗ respectively; n and n are the corresponding concentrations within the equilibrium liquid and sufficiently far from the particle; is the electric potential with respect to the water medium at a distance x from the particle surface; T is the absolute temperature in degrees Kelvin (K); and k is the Boltzmann constant. On the SI system this constant has a value equal to 1.38 1023 J°K1. and are the valances of the positive and negative ions, respectively, and e is the elementary ionic charge equal to 16 1020 coulomb (C) or 4.8 1010 esu. It is observed that when is negative, n n and vice versa. The concentrations are expressed as numbers of ions per cm3. The local electric charge, e, is then given by
e en en
(3.3)
The relationship between the electric density and the electric potential was developed by Poisson on the assumption of uniform charge distribution and of a flat plate shape. Its general form for ions of several valances was presented and discussed by Mitchell [88, Chapter 7]. For the simplest case of monovalent ions and for n∗ n∗ n∗ Poisson’s equation obtains the simplified form d 2
dx
2
e e ed eo Dr
(3.4a)
in which d 2 /dx 2 is the rate of change of the strength of the electric field, d/dx , and ed is the dielectric constant of the medium given by
ed eo Dr
(3.5)
In Equation 3.5, Dr is the relative dielectric constant of the medium with respect to that of the vacuum and eo is the dielectric constant for vacuum equal to 8.8542 1012 C2J1m1 with C indicating coulomb units. In the esu system, eo has the value of 1/4. For water, Dr has the value of 80. The dielectric constant is defined in terms of the electric force, Fe, between two particles with electric charges, Q and Q9, and at a distance d by the relationship
Fe
QQ9 ed d 2
(3.6)
If the units in Equation 3.3 are in the esu-CGS system, Equation 3.4a can be written as 4 e Dr dx
d 2
2
(3.4b)
52
Cohesive Sediments in Open Channels
Introduction of Equation 3.2a, Equation 3.2b, and Equation 3.3 into Equation 3.4a leads to the following equation for the ionic distribution within the double layer: 2 n∗ e e e sinh kT 2 dx d
d 2
(3.7a)
for the SI-Coulomb system of units and 8 n∗ e e D sinh kT 2 dx r
d 2
(3.7b)
for the esu-CGS system of units. The preceding two equations can be transformed to a more convenient form by introducing the following dimensionless variables:
y
eo e ,z kT kT
and x
(3.8)
o is the potential energy at the surface of the particle and 2 n∗ e2 2 2 m ed kT
(3.9a)
8 n∗ e2 2 in cm2 Dr kT
(3.9b)
2 for the Coulomb-SI system, and 2
for the esu-CGS system. Introduction of Equation 3.8, Equation 3.9a, and Equation 3.9b to the appropriate Equation 3.7a or Equation 3.7b leads to the following dimensionless equation, which is independent of units: d2y
(d )2
sinh y
(3.10)
A first integration taking into consideration that for , y 0, and dy /d 0 – that is, the potential energy tends asymptotically to zero – yields
y dy 2sinh (2cosh y 2)½ 2 d
(3.11)
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
53
The minus sign in Equation 3.11 was selected because y diminishes with increasing . Integration of Equation 3.11 subject to the boundary conditions for 0, o, or y z leads to the final equation: e
2
( 1 (e
) 1) e
e 2 1 e 2 1 e z
y
e
z
2
z
z
2
(3.12)
in which e is the basis for the Neperian logarithms. A more simplified equation can be derived for small surface potentials, that is, for o ,, 25 mV, z ,, 1, and y ,, 1, which for practical purposes is a case of interest. Under this condition, the hyperbolic sine in Equation 3.7a can be approximated by y, in which case the equation reduces to d 2 dx 2
2
(3.13)
whose solution is o exp(x )
(3.14)
Equation 3.14 means that for small surface values the electric potential is approximately an exponentially decaying function of the distance from the surface of the particle, as represented in Figure 3.2. A good measure of the double layer thickness is the center of mass of the electric charge, defined as the characteristic length of the double layer: xc
1
(3.15)
Of importance is the total charge on each side of the double layer, , per unit area of the particle surface. This charge can be evaluated as follows, considering that at infinity d/dx 0 :
∫
∞ 0
e dx
D d Dr ∞ d 2 dx r ∫ 2 0 4 4 dx x0 dx
(3.16a)
for csu-CGS units and
d ed dx x0
(3.16b)
54
Cohesive Sediments in Open Channels
Imaginary plane of infinite charge and potential
Surface of clay plate
y�
υeΦ kT
ξ0 �� x0 ξ�ξ0 �� (x�x0) ξ�� x ξ0 Figure 3.2 Variation of electric potential with distance from the particle surface [88, Fig. 7.4].
for C-SI units. It can be thus concluded that the initial slope of the potential distribution determines the charge on the surface of the particle. To express the derivative d/dx x 0 , we have to go back to Equation 3.11; substitute y and in terms of , , e, k, and T, as given by Equation 3.8; and use the chain rule of differentiation (d / d ) (d/dx )(dx /d ) (1/) d/dx . Next, in the resulting equation, we introduce from Equation 3.9a for the case of Equation 3.16a and from Equation 3.9b for the case of Equation 3.16b. Finally, the expressions thus derived for d/dx x 0 are introduced to Equation 3.16a or Equation 3.16b, depending on the units, to obtain
for the esu-CGS system, and
D n∗ ½ r ( eo ) 2 kT
(3.17a)
2n∗ e ½ d ( eo ) kT
(3.17b)
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
55
for the SI-Coulomb system of units. For the general case of large values of electric potential, can be readily derived from Equation 3.11 and the proper substitutions to yield
n∗ D kT ½ ∗ ½ r ( 2 cosh z 2 )½ 2 n Dr kT sinh z 2 2
(3.18a)
for the esu-CGS system and
(
2 n∗ ed kT
½
)
½
z
½ ( 2 cosh z 2) ( 8 n∗ kT ed ) sinh (3.18b) 2
for the Coulomb-SI system. These equations reduce to Equation 3.17a and Equation 3.17b for sinh z z. In both equations, the coefficients of o are equal to the square root of 1/, that is, the characteristic length, which can also be interpreted as the capacity of the double layer. There are the following two cases of electric charges on the particle surfaces [150, Chapter 3]:
3.3.2 Surfaces of Constant Potential Such surfaces have a constant concentration of potential determining ions due to adsorbed cations. According to Equation 3.17 or Equation 3.18, the charge, , increases in proportion to the square root of the electrolyte concentration, n*, while, according to Equations 3.16a and Equation 3.16b, is also proportional to the gradient of the potential density, d/dx at x 0. Therefore, as indicated in Figure 3.3a, the initial slope of the potential curve increases with increaseing surface charge, o. Also the distance at which the becomes zero, that is, when the electrolyte concentration becomes about equal to n*, decreases with decreasing n*.
3.3.3 Surfaces of Constant Charge Density The density of the electric charge in this case is determined by the imperfections in the interior of the crystal lattice of the clay particle and is independent of the electrolyte concentration. According to Equation 3.16a and Equation 3.16b, the gradient d/dx remains constant since is a constant quantity. The variation of the electric potential with distance from the surface of the particle is schematically outlined in Figure 3.3b. In both cases, according to Equation 3.9, the double layer thickness, as measured by 1/, decreases with increasing n*. In the first case, starts from a constant value, o, but the slope d/dx increases with n*. In the second case, the potential energy, , decreases with n*, but the near-surface potential gradient remains constant so that the curves for high and low n* remain qualitatively parallel.
56
Cohesive Sediments in Open Channels
�0 � ��0 Electrical potential, �
�
High concentration (n*�)
0
0
�0 Electrical potential, �
�
Low concentration (n*)
(a)
(b)
� ∝ �1 � � ∝ �2 �0 � ��0
Distance from surface
�
��0
�∝�∝� �0 � ��0
� Low concentration (n*) High concentration (n*�)
0
0
Distance from surface
Figure 3.3 (a) Potential energy distribution in the double layer for constant surface potential; (b) potential energy distribution in the double layer for constant surface charge [150, Ch. 3].
3.3.4 Illustrative Applications The use of the developed equations for the estimate of the effect of electrolyte concentration on the double layer thickness are demonstrated by the following examples for both sets of units and for normal ambient conditions. The following constant quantities enter into the equations: Na Avogadro’s number 6.02 1023 ions/mole k Boltzman’s constant 1.38 1023 J°K1 or 1.38 1016 erg°K1 kT 4 1014 erg or 4 1021 J for room temperature e electron charge 1.602 1019 coulomb (C) or e 4.8 1010 esu dielectric constant ed eoDr 8.8542 1012 C2J1m1Dr or 1 ed Dr esu2erg1cm1 4 Dr is the relative dielectric constant, which for water has the value of 80.
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
57
The salt content in the water is normally given in terms of normality, N, that is, number of equivalents per liter. To convert it to ions per unit volume, n*, we have to multiply by Avogadro’s number 6.02 1023 moles per liter according to the relation: n* N 6.02 1023 (ions per mole) The cation exchange capacity (CEC) given in meqL1 should be divided by the specific area of the clay mineral to obtain the surface charge density, , which can be converted to coulombs through multiplication by Faraday number Fa 96.5. It follows from Equation 3.9 and Equation 3.15 that the center of mass of the double layer thickness is inversely proportional to the square root of normality, N, and inversely proportional to the valence, . Therefore, by increasing the first by an order of magnitude of 100, we reduce the thickness of the double layer by a factor of 10. Such an increase is very easy to achieve in relatively fresh river and lake waters with very low original salinity by adding small amounts of salt. The following numerical examples using both the esu-CGS and Coulomb-SI units will further demonstrate the use of the double layer equations. For a constant charge density, , of 15 coulombs cm2 on a flat surface and a normality N 0.001 of NaCl in the solvent water, determine the representative thickness of the double layer, 1/. Solution: The surface density in esu cm2 is
esu / cm2
(
)
mmC / cm 2 106 4 0.3 10 10 3.33 10
where on the right side is in coulombs cm2 and the denominator is the factor converting Coulomb units to esu units. For the present example, 15 106 0.3 1010 4.5 104 esu/cm2 and n* (concentration) (Avogadro’s number). The solute concentration, Cs, is expressed in number of equivalents per liter of solution. Therefore, it has to be multiplied by 103 to give the mole concentration per cm3. The ion content per cm3 then becomes n* N 103 6.02 1023 6.02N 1020 with N in expressed in terms of normality. For N 0.001, n* 6.02 1017. (8 )6.02 N 1020 ½ 4.8 1010 From Equation 3.9b, we obtain 80 4 1014 33 106 N . For 1 and N 0.001, we have 1.04 106 cm1 and 1/ 100 Å.
58
Cohesive Sediments in Open Channels
Next, from Equation 3.18a, it follows that z sinh ∗ 2 2 n Dr kT
1
2
20 14 2 6.02 N 10 80 4 10
0.0082 107 z sinh 2 N
1
2
0.0286 10−3
N
1
2
or
,
z where 0.3 104(15) 4.5 104, sinh 40.7, z 8.80. From the 2 definition of z in Equation 3.8, we obtain o
zkT z 300 103 25 220 mV , e
where mV is in mV for a valence, , equal to 1. For a solute concentration Cs 0.1 N. n* 6.02 1019 ions cm3 and 33 106 N 33 106 0.1 10.4 106 0.1 1 or xc 107 cm 10 Å That is a tenfold decrease of the double layer length. From Equation 3.18, we observe the sinh(z/2) is inversely proportional to the square root of n*. Therefore, a hundredfold increase of n* will decrease sinh(z/2) by a factor of 10. So for Cs 0.1 N, sinh(z/2) 4.07, z 4.22, and o 105 mV. 1 Likewise, for Cs 105 N, = xc = 1000 Å, sinh(z/2) 406.9, z 13.4, and o 335 mV. In conclusion, for a particle surface of constant electric charge, an increase of salinity by a factor of 100 decreases the thickness of the double layer by a factor of 10 and the surface potential by a factor of about 2. Both effects are very important for flocculation. The same problem will be now solved using the Coulomb-SI units. Here, n* has to be expressed in terms of ions m3; that is, n* N 6.02 1023 with N again expressed in equivalents per liter. Therefore, the right side of the last equation for n* has to be multiplied by 103 to become equivalents per m3; that is, n* 0.001 103 6.02 1023 or n* 6.02 1023. Also kT 4 1021 J and from Equation 2.29a for Coulomb-SI units, we obtain
59
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
1
2 6.02 1023 (16 1020 )2 12 2 1.043 108 m1 , 8.8542 1012 80 4 1021 1 xc 0.96 108 m 100 Å From Equation 3.18b the surface potential can be obtained as follows: ½ z 1 sinh 271 2 8 6.02 1023 4 1021 8.8542 1012 80 15mmC /cm 2 15 106 z 10 4 C /m 2 0.15C /m 2
and
z sinh 40.65, z 8.80 2 and from Equation 3.8 the following value of o is obtained as follows: o
8.80 4 1021 1 16 1020
0.22 V 220 mV
For surfaces of constant potential energy, the procedure is the same except that is evaluated from Equation 3.18a or Equation 3.18b for a known z given from Equation 3.9b.
3.4 Interaction of two flat double layers 3.4.1 Force and Energy Interaction The interaction of force and energy is based on the assumption of the validity of the Gouy-Chapman theory and of a double layer thickness much smaller than the particle size. The electric potential distribution between two such layers together with xc 1/ is indicated schematically in Figure 3.4, where 2d is the distance between the clay particles and d is the midway potential. The theory was first developed by Verwey and Overbeek [151] and is well presented in an abbreviated form by van Olphen in Appendix III of Ref. [150]. Using the dimensional variables defined by Equation 3.8, we indicate by ϕd the dimensionless potential at midpoint as
ϕd
ed kT
(3.19)
60
Cohesive Sediments in Open Channels
0
0
(x)
d
(x)
0
x d
2d
0
x
1/�
Figure 3.4 Schematic distribution of electric potential between two clay particles [151, Fig. 8].
The governing differential equation is still 3.10, but with the boundary conditions set at the midpoint between the plates, that is, at x xd d, dy/d 0, d, and y ϕd, because at that point the potential attains its minimum value. Equation 3.10 can also be written as 2 1 d dy d (cosh y ) 2 d d d
(3.20)
which, upon integration, introduction of the stated boundary conditions, and the fact that the gradients of y and normally have a negative value, takes the following form: 1 dy (2 cosh y 2 cosh ϕd ) 2 d
(3.21)
A second integration yields d
∫ d −d 0
ϕd
∫0
1
(2 cosh y 2 cosh ϕd )
2
dy
(3.22)
Equation 3.22 gives the midway dimensionless potential, ϕd, for any specific value of surface potential, as represented by z, and a given electrolyte concentration, which determines the value of in Equation 3.9a and Equation 3.9b. For most practical cases, knowledge of only ϕd is sufficient. The integral in Equation 3.22 is elliptic of the first order and cannot be evaluated in closed form, but only numerically. Table 3.1 and Figure 3.5, developed by Vervey and Overbeek, can be used for an estimate of d for given values of z and ϕd and, vice versa, an estimate of ϕd from z and xd. In Table 3.1 the first line gives d from the simplified Equation 3.24 that follows, and the second line is based on the numerical integration of
61
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
Table 3.1 κd as a function of z and ϕd according to Equation 3.23 and 3.24 [151, Table X, p.71] z 6, y 0.905 ϕd
0.1
0.25
xd (eq. 3.24) 4.28
0.50
3.365 2.67
z 8, y 0.964 1.0
2.0
0.1
0.25
0.50
1.0
2.0
1.98
1.29
4.35
3.43
2.73
2.04
1.35
xd (eq. 3.22) 4.280 3.354 2.635 1.876 1.061
d �
4.343 3.417 2.698 1.939 1.124
υe d kT
10 9
z � 10
8
z�8
7 6 5 z�6
4
z�4
3
z�2
2
z�1 z � 0.5
1 0
0
0.5
1.0
2.0
3.0
4.0
5.0 6.0 d
Figure 3.5 Graphical solution of Equation 3.22 [151, p. 72].
Equation 3.22 [151, Table IX] Table 3.3 [151, Table XI] gives d for a wider range of z and ϕd. In either case, interpolation will be needed, and the approximation is satisfactory for most practical cases [151, pp. 67–72]. For large values of electrolyte concentrations and, therefore, large values of d, Equation 3.11 for y ϕd can be reduced to the much simpler form [151, Appendix III]: z
d
ϕd 4 γ e
where γ
e 2 1 z
e 2 1
(3.23)
62
Cohesive Sediments in Open Channels
For this case, the two interacting double layers can be assumed as practically unperturbed and, therefore, linearly superimposed. Accordingly, the dimensionless midway potential is about double of that for a single particle at the same distance d, that is:
ϕd 2 yd 8γ ed
(3.24)
3.4.2 Illustrative Examples Case 1: Constant surface potential. As an application of Table 3.1 and Figure 3.5, we consider two particles with surface charge of the previous example in Section 3.3.4, except that instead of a constant charge, the potential is taken as constant and equal to 200 mV or 0.200 volts. We want to find the distance for which the potential d reaches the value of 0.05 volts for three electrolyte concentrations: 103, 105, and 101 N. In the first case, it was found that 1.043 108m1 and the center of mass of the double layer, xc, is 1/ 100 Å. For normal ambient conditions, e / kT ≈ 40 ; therefore, z 40 (0.200) 8.00 and for d 0.05 volts ϕd 40 0.05 2.00. From Table 3.1, we obtain d 1.124 and d 1.124 100 Å 112 Å. For the second concentration, 1/ 1000 Å and d 1124 Å, while in the last case of the highest concentration, d is about 12 Å. Very close but more approximate values can be verified from Figure 3.5. So far, we have discussed the case of constant surface potential. The same procedure could be followed when the surface charge rather than the surface potential is constant, as in Section 3.3.4. The latter case is the most common for clays whose surface charge is determined by isomorphous substitution. The approach could be the same as in the previous example but in reverse, like in case 2, which follows. Case 2: Constant surface charge. The data are the same as in the previous case but with a constant surface charge of 15 C cm2, as in the example in Section 3.3.4. For an electrolyte concentration of 103 N, it was found that o was about 0.220 V, z 8.8, 1.043 108m1, 1/ 100 Å, and e / kT 40 . The distance, d, at which the potential energy becomes d 0.05 V can be found, as in the case of constant surface potential. ϕd is equal to 2. From Table 3.1, at z 9 from extrapolation as an approximation and in the column of ϕd 2, we obtain d 1.14 and d 114 Å. For electrolyte concentration of 101 N, was found to be equal to 10.4 106, 1/ equal to about 10 Å, z 4.22, and o equal to 105 mV. ϕd is still the same at 2. Then from Table 3.3 or from Equation 3.22 for z approximated by 4, we obtain d 0.884 and d 8.8 Å. Finally, for electrolyte concentration equal to 105 N, 1/ 1000 Å, z 13.4, and o 335 mV, while ϕd still has the value of 2. From Table 3.3 and Equation 3.22, we then find d 1.15 and a distance d 1150 Å.
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
63
Similar approximate results can be found from Figure 3.5, as in the case of constant potential. One can easily observe from both Table 3.3 of Section 3.4.3 and Figure 3.5 that the effect of z on d becomes negligible for z higher than 9. Moreover, xc for a single particle is the same for both constant surface charge and constant surface potential because it depends only on the electrolyte concentration. A final important observation is the comparison of d with xc, that is, the center of mass and the representative double layer length. For the 103 N case, d somewhat exceeds xc, the center of mass of the double layer; for the 101 N case, d is somewhat shorter than xc; while for the last case of 105 N, d is longer than xc. For the latter case of constant surface charge, van Olphen developed a different approach [150, p. 260]. Using Equation 3.11, he computed dy/d for several values of the charge, , from Equation 3.16a and Equation 3.8, that is d
∫ e dx 0
D n∗ kT r 2
1
2
Dr d D dy kT r 4 dx x0 4 d 0 e
(2 cosh z 2 cosh ϕd )
1
2
(3.25)
Therefore, for a constant surface charge and, consequently, constant electrolyte concentration, n*, (n*)½(2 cosh z 2 cosh ϕd)½ is constant and independent of d. The problem of evaluating ϕd for different distances, d, and for a given electrolyte concentration, amounts to finding a set of values for z and ϕd that satisfy the following equation:
dy d
0
(2 cosh z 2 cosh ϕd )
1
2
2 D r n∗ kT
1
2
constant (3.26)
For the values of Dr and kT, given for normal conditions and used in the illustrative example in Section 3.3.4, we obtain the derivativze (dy/dξ)ξ50 5 4peυs/DrkTk, which, upon substitution of the appropriate values, becomes (dy/dξ)ξ50 5 1885(s/k). In Equation 3.26, is in esu cm2, in cm1, Dr in (esu)2 (erg)1 cm1, and kT in erg. van Olphen developed tables for dy/d of 840, 560, 280, 187, 94, 84, 56, 28, 18.7, 9.4, 8.4, 5.6, 2.8, 1.87, and 0.94 [150, Appendix III]. The table for the first three values of (dy/d) is reproduced here as Table 3.2. The procedure is as follows. From the known charge, , in esu cm2 and the computed value on n* from the electrolyte concentration, we evaluate (dy/d) at the surface of the particle. We then enter the table at the two values of (dy/dξ)ξ50 containing the actual value. For each value and for the closest value of z, going horizontally, we determine both ϕd and d, from which the
64
Cohesive Sediments in Open Channels
Table 3.2 Surface potential, z, and midway potential, ϕd, for three surface charges, , at various electrolyte concentrations, , and plate distance, 2d [150, p. 262] (dy/d)0 9.4 d
ϕd
(dy/d)0 8.4 z
(dy/d)0 5.6
d
ϕd
z
d
ϕd
z
8.7760
0.001 4.50
15.659
1026
4.28
15.545
1026
3.51
6.4734
0.01
4.50
13.356
25
10
4.28
13.242
25
10
3.51
3.2424
0.25
4.51
11.054
1024
4.28
10.940
1024
3.51
2.5240
0.5
4.51
8.7513
10
4.28
8.637
23
10
3.51
2.0868
0.75
4.51
6.4487
0.01
4.28
6.335
0.01
3.51
1.7650
1.0
4.52
4.1436
0.1
4.28
4.030
0.1
3.51
1.5080
1.25
4.52
2.4993
0.5
4.29
2.386
0.5
3.51
1.2939
1.5
4.53
1.7406
1.0
4.30
1.6292
1.0
3.54
0.9540
2.0
4.56
1.2700
1.5
4.32
1.1621
1.5
3.58
0.6979
2.5
4.61
0.9309
2.0
4.36
0.8289
2.0
3.66
0.5029
3.0
4.69
0.6760
2.5
4.41
0.5824
2.5
3.78
0.3551
3.5
4.80
0.4827
3.0
4.51
0.4003
3.0
3.94
0.2449
4.0
4.96
0.2684
3.5
4.17
0.1646
4.5
5.18
0.1756
4.0
4.45
0.1079
5.0
5.47
0.1124
4.5
4.80
23
estimate of d follows. Interpolation will be required for the best approximation of the last two parameters.
3.4.3 P otential Energy of Interaction between Two Flat Double Layers By definition, the potential energy of interaction is the work necessary to bring two charged particles from infinity to a distance 2d. A quantitative expression for this energy is important in determining the net force between the two interacting double layers because the net force of interaction is the negative derivative of the potential energy. Before proceeding with the attractive potential due to the van der Waals forces, we will elaborate a little on the repulsive potential, VR, at the middle of the distance 2d between the two layers. Associated with the latter is the free energy, indicated by VF, which, according to Verwey and Overbeek, “represents the amount of work associated with some isothermal and reversible process of building up the double layer” [150, Chapter 3]:
VR 2(VF d VF ∞ )
(3.27)
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
65
where VF is the free energy per cm2 of the single (noninteracting) double layer and VFd is the free energy per cm2 of the double layer when the distance is 2d. An analysis for the free energy was developed by Verwey and Overbeek and is well presented in Chapter 5 of Ref. [151] with a table and some graphs for an estimate of VR for constant surface electric potential. This table is reproduced here as Table 3.3. For a given electrolyte concentration and known and z, the table gives the function f ( z, ϕd ) ( 2 /)VR in 107 dynes. All the variables in the table are expressed in esu-CGS units. Therefore, the value of VR obtained is in erg cm2. For given concentration, can be evaluated from Equation 3.9a. z is determined by the constant surface charge. Then entering Table 3.3 at the appropriate z and proceeding horizontally, we determine both ϕd and d for any particular value of the dimensionless midway potential ϕd. Repeating the procedure for sufficient number of values of ϕd, we can obtain a graph of potential versus distance 2d or half distance d. Remember that the same table can be used instead of Table 3.1 for a more accurate estimate of the distance between two flat particles at which the electric potential has a prescribed value, as in the examples of Section 3.4.2. One should observe that for any given value of f(z,ϕd), VR is inversely proportional to the square of the ion valence . Therefore, for a divalent ion at the same distance d, VR will have one-quarter of the value corresponding to monovalent ions with the other variables remaining the same.
3.4.4 Illustrative Examples Example 1. Let us consider two flat plates under the data and the conditions of the illustrative example in Section 3.3.4, which is for a concentration of 103 N and a surface charge of 15 C cm2 or 4.5 104 esu cm2. The following values were found for standard conditions and for a value of Dr of 80, which is quite close to 78.55 of Table 3.3: kT 4 1014 erg 1.04 106 and 1/ 106 cm or 100 Å z 8.8 then for a ϕd equal to 2, for example, and a z about equal to 9, we obtain f(z, ϕd) 1.26 107 dynes and d 1.139. The following values of VR and d are then computed: V R 1.26 107 106 0.126 ergs cm2 d 1.139 106 114 Å.
Table 3.3 Repulsive potential f (z, ϕd)VR of double layers around two parallel flat plates at a distance 2d for different values of z and ϕd 25 °C temperature and a value of Dr equal to 78.55 [151, p. 82] ϕd z z 10 f(ϕd·z) 268.3 κd z 9
0.0000
f(ϕd·z) 161.5 κd
0.0000
z 8
f(ϕd·z) 96.52
z 7
f(ϕd·z) 57.13
κd κd z 6
0.0000
0.0000
f(ϕd·z) 33.27 κd
0.0000
z 5
f(ϕd·z) 18.83
z 4
f(ϕd·z) 10.13
z 3
f(ϕd·z) 4.962
z 2
f(ϕd·z) 1.993
κd κd κd κd z 1
0.0000
0.0000
0.0000
0.0000
f(ϕd·z) 0.4575 κd
0.0000
8
7
6
5
4
3
2
1
0.5
0.25
0.1
127.1
75.4
44.1
25.4
14.1
7.36
3.42
1.26
0.26
0.06
0.015
0.0023
0.0204
0.0437
0.0813
0.143
0.244
0.412
0.690
1.148
1.962
2.721
3.440
4.366
95.6
76.3
44.3
25.4
14.1
7.36
3.42
1.26
0.26
0.06
0.015
0.0023
0.0337
z 0.1
z 0.3
z 0.6
228.2
192.6
160.0
0.00434
0.00836
0.0134
135.2
115.2
9
0.0073
0.0138
0.0221
0.0721
0.134
0.236
0.403
0.679
1.139
1.953
2.712
3.431
4.357
80.56
68.56
56.60
44.8
25.4
14.1
7.36
3.42
1.26
0.26
0.06
0.015
0.0023
0.0555
0.0121
0.0227
0.0364
0.119
0.221
0.388
0.665
1.124
1.939
2.698
3.417
4.343
47.46
40.18
32.89
25.8
14.17
7.36
3.42
1.26
0.26
0.06
0.015
0.0023
0.0915
0.0199
0.0375
0.0600
0.196
0.364
0.641
1.101
1.915
2.674
3.393
4.318
27.47
23.04
18.66
14.38
7.39
3.42
1.26
0.26
0.06
0.015
0.0023
0.1509
0.0327
0.0618
0.0990
0.323
0.601
1.061
1.876
2.635
3.354
4.280
15.32
12.69
10.07
7.52
3.43
1.26
0.26
0.06
0.015
0.0023
0.2488 0.533
0.995
1.811
2.570
3.290
4.215
1.26
0.26
0.06
0.015
0.0023
0.0541
0.1018
0.1632
8.07
6.51
4.97
3.50
0.4105 0.884
0.0891
0.1680
0.2692
3.793
2.913
2.061
1.291 0.681
1.702
2.462
3.181
0.4107
0.26
0.06
0.015
0.0023
0.1471
0.2774
0.4455
1.518
2.280
2.998
3.924
1.413
0.966
0.584
0.265
0.06
0.015
0.0023
1.178
0.2435
0.4643
0.751
1.915
2.680
3.608
0.280
0.135
0.0348
0.063
0.015
0.0023
0.4353
0.855
1.537
1.283
2.035
2.971
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
67
For the same data but for a divalent cation, the value of becomes, according to Equation 3.9a, twice that for monovalent ions, that is, about 2 106; and the value of 1/ or xc becomes equal to 50 Å or 0.5 108 cm. z remains the same and equal to about 9. Then for ϕd 2, f(z, ϕd) and d have the same values as for monovalent cations so that VR 1,26 107 2 106 22 0.063 102 erg cm2 at a distance of 50 Å, which is one-half of the previous value. This change is indicative of the rapid rate of decrease of the repulsive potential with increasing ionic valence. As already mentioned, the outlined table and procedures are valid only for constant surface electric potential, whereas in the most common clay minerals, in which the charge is generated by isomorphous substitution, the surface charge is constant rather than the surface potential. For a direct computation of the interaction force between two double layers, van Olphen outlined an analysis by Langmuir [71] based on the concept of the osmotic pressure [150, Appendix III]. The assumption was that the repulsive force is given by the osmotic pressure at the midpoint of the distance between the two plates, which is determined by the excess of the midway ionic concentration. Using Equation 3.2a, Equation 3.2b, Equation 3.3, and the dimensionless electric potential, ϕd, from Equation 3.19 for equal valence and concentration of cations and anions, the total excess concentration of cations at the midpoint is given by
ϕdd
n* (eϕd 1) n∗ (e
1) 2 n∗ (2 cosh ϕd 1)
(3.28)
The repulsive force per unit area is
pre 2 n∗ kT (cosh ϕd 1)
(3.29)
In contrast to the first method based on the free potential energy, Equation 3.29 is valid for both constant charge and constant potential surfaces. Thus, the pressure pre can be evaluated from the solute ion concentration and the midpoint potential for a single surface. For small interactions and high ion concentrations, the analysis can be further simplified by introducing Equation 3.23 and Equation 3.24 for ϕd into Equation 3.29 and integrating to find the following expression for the repulsive potential energy per unit surface:
64 n∗ kT γ 2 exp(2d ) V R
(3.30)
Example 2. As an illustration of the application of Equation 3.30, we will compute the interaction energy of two double layers at a distance of 100 Å for an electrolyte concentration 103 N of NaCl and a surface charge density 0.50 104 esu cm2 assuming week interaction.
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Cohesive Sediments in Open Channels
In the example in Section 3.3.4 for the same electrolyte concentration, was 106, 1/ xc was 100 Å or 106 cm, with all the other variables remaining the same. Then, according to Equation 3.26, we have dy d
0
0.50 10 4 1885 1885 9.42 106
Next, we enter Table 3.2, [150, Appendix III] in the column for 9.4 (a good approximation) to d of 0.954, which is the closest to the actual 1, and moving horizontally, we find the values ϕd 2 and z 4.56. Equation 3.23 then gives
γ
e e
z z
2 2
1 1
e2.28 1 e2.28 1
0.81 and γ 2 0.663
e2z e2 0.1353 VR
64 103 103 6.02 1023 0.4 1013 106
(0.663 0.1353)
0.138 erg/cm 2 and pre 2 103 103 6.02 1023 0.4 1013 (3.762 1) 13.3 10 4 dynes/cm 2 0.127 atm
3.4.5 P otential Energy of Interaction of Two Flat Double Layers Due to van der Waals Forces As explained earlier, the van der Waals forces are the dominant attractive forces with a long range of action. For very small distances, particularly when two particles are about to come into contact, other factors, such as residual chemical valence fields, may also have a measurable influence. However, as far as flocculation is concerned and for the reasons to be analyzed shortly, the long-range effect is the one of primary importance. Moreover, these forces are universal acting between all atoms, molecules, and ions and are generated by the mutual influence of the electronic motion on two interacting atoms. The fluctuating charge in one atom causes, as a first approximation, a fluctuating dipole in the second atom. The latter induces a dipole in the first, and the net result of the interaction of these dipoles is attraction. The attractive potential between two dipoles is proportional to r3, where r is the distance between the molecules. The attractive force, therefore, is proportional to r4. Then, according to London’s
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
69
theory, the van der Waals forces and attractive potential between two atoms will be proportional to r7 and r6, respectively. The attractive potential energy for two atoms is given by the equation VA p
r6
(3.31)
in which the constant depends on the properties of the atoms or molecules under consideration [151, Chapter 6]. The attractive potential between two particles can then be evaluated by integrating that potential between all pairs of atoms [151, Chapter 6 and Chapter 11]. The integration involves some constants that are very difficult to evaluate precisely. For particles of the order of clay particles, and more specifically between two unit layers of three-layer clay minerals, the following approximation was accepted as satisfactory by van Olphen [150, Appendix III]:
Aa 1 1 2 VA 48 d 2 (d )2 2 d 2
(3.32)
is the plate thickness and Aa is a constant whose value depends on the number of atoms contained in 1 cm3 of the substance building up the particles, and 2d is the distance between the two particles. For practical purposes Aa has been approximately accepted as 1012. Equation 3.32 can be reduced to the following simpler approximations for special cases [151, Chapter 6]:
VA VA
2 Aa 32 d 4
for d >>
Aa 1 7 2 2 for d < 48 d
VA
Aa 48 d 2
for d <<
(3.33a)
(3.33b)
(3.33c)
According to Equation 3.33a, the attractive potential decays in proportion to d4. This is practically true when the particle thickness is much smaller than the interparticle distance, but it is approximate otherwise. The same equation suggests that the potential reaches infinity as the distance tends to zero. However, beyond a minimum distance, other forces enter into the picture, such as Born repulsion, to be discussed later, which prevent complete contact and interpenetration
70
Cohesive Sediments in Open Channels
of particles. Moreover, and as will be explained in the following section, the interaction of the van der Waals attractive potential with the ionic repulsion is far more important to flocculation than VA itself. An example of variation of that potential with the half distance d, for a plate thickness 6.60 Å and for Aa equal to 1012, is shown in Figure 3.6 in Coulomb-SI units [150, Appendix IV].
3.4.6 Total Potential Energy for Two Particles and the Process of Flocculation The interaction of the two potential energies will determine the nature of the net interparticle forces and, therefore, the stability of the suspension. According to Section 3.4.3, the repulsive potential energy, VR, starts from a finite value at the surface of the particle and decreases approximately exponentially with distance. In contrast and according to Equation 3.33a, the attractive potential energy, VA, 0.10
VA ergs/cm2
0.01
0.001
0.0001
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 Half-distance, Å units
Figure 3.6 Variation of van der Waals attractive potential with interparticle distance 2d [150, App. IV].
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
71
decreases almost with the fourth power of distance and reaches very high values near the particle surface. One also is reminded that for two atoms or molecules the van der Waals attractive force is inversely proportional to the seventh power of the distance and the attractive potential energy proportional to the sixth power of the latter. For two spherical particles, the attractive force and the attractive potential are inversely proportional to the third and the second power of the distance, respectively [150, Chapter 3]. The dominance of the attractive or the repulsive potential and, therefore, the dominance of the corresponding forces depends on whether the net potential is attractive or repulsive at any interparticle distance 2d. Moreover, since VA can theoretically reach infinitely high values at the surface, it will always predominate to the VR sufficiently close to the surface. Therefore, sufficiently near the surface, the net interparticle forces will be attractive. Approaching particles will coagulate if the concentration-controlled double layer is short enough for the attractive potential to become dominant because it is independent of the electrolyte concentration. Figure 3.7 shows a simplified picture of the interaction of the two potentials for high and low electrolyte concentrations and of high and low ionic valence. At sufficiently low concentrations, the repulsive potential extends well beyond the range of the attractive one, whereas for high concentrations the opposite is true. For the first case, the net potential is negative at relatively large distances, reaching an energy hump at a distance x, beyond which it becomes positive. It should be noted that VR is pointing to the positive and VA to the negative direction of the ordinate so that the corresponding force being the negative derivative of the potential will be negative for the repulsive potential and positive, that is attractive, for the positive potential. So, for a low electrolyte concentration, two approaching particles will experience repulsion as soon as their double layers begin to interact. To approach a distance smaller than that corresponding to the energy hump, a force has to be applied to provide the necessary energy. Therefore, in the absence of such a force, the particles will separate. In contrast, for the case of high electrolyte concentration and small double layer length, the net potential is everywhere attractive, the two particles will be subjected to an attractive force as soon as they come sufficiently close to each other, and flocculation will take place. It was shown in Section 3.3 that the thickness of the double layer varies in inverse proportion to the square root of the electrolyte concentration and in inverse proportion to the ionic valence. The mode of double layer interaction and the properties of the van der Waals potential led many engineers to the assumption that, under sufficiently low electrolyte concentration conditions, fine suspended sediment will remain dispersed and stable with negligible flocculation. As a result it was taken for granted by hydraulic engineers that salinity in estuaries is the main cause for flocculation and shoaling and that fine river sediment reaches the salinity intrusion zone in a dispersed state. This, however, is not necessarily true. In fact, certain sediments, such as kaolinite, may even flocculate more readily in distilled water. Flocculation and deposition of cohesive sediments still can take place at low or
72
Cohesive Sediments in Open Channels
VR Double layer repulsion High salt concentration Low salt concentration
Net potential Vt Low concentration X Particle separation 2d
Van der waals attraction
Net potential Vt High concentration
VA
Figure 3.7 Interaction of energies for high and low electrolyte concentrations [99].
even zero electrolyte concentration. There are two ways that flocculation occurs under such conditions: (a) attachment of negatively charged surfaces to positive charged edges, a case common in kaolinite minerals; and (b) by the kinetic energy of the particles due to Brownian motion and the turbulent velocities. Indeed, considering one of two particles as stationary and the second approaching the first with a velocity v, the total of the potential and the kinetic energy remains constant. If the latter is sufficiently high, it may overcome the potential energy hump and push the particle to the zone of attractive potential energy. It may be concluded, therefore, that there is no absolute stability and the difference between stable and unstable clay suspensions lies in the relative rates of coagulation. A commonly used criterion is the coagulation time, which is the time required for the number of dispersed particles to be reduced by 50%.
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Chapter | 3 Forces between Clay Particles and the Process of Flocculation
z 10 z 9 z 8 z 7
Potential energy
0.5
Unit of abscissa Unit of ordinate
z 6 z 5 z 4 0
10
Value of Aa
20
7
10
8
cm · 10 7
2
erg 10 1/cm2
10
30
cm · 10
erg/cm
12
2 10
40
2 10
6
11
50
d half of distance between the plates in Å 0.5
Figure 3.8 Distribution of total potential energy per cm2 for o 25.6 mV and 1–1 electrolyte [151, p.111].
The particle size also has an important effect on the rates of coagulation. It has indeed been observed that for the same environmental conditions the smaller the range of the particle size, the faster is the coagulation rate. This behavior is due to two effects: (a) the smaller lower total particle repulsion for the smaller particles and (b) their higher kinetic energy per unit volume that will push them through the energy hump. The reason for this increase is that the total net force on the particle surface by the water molecules induced by either the Brownian motion or the turbulent velocity fluctuations is proportional to the particle surface, which is proportional to the square of the particle size; whereas the imparted acceleration is proportional to the particle mass and, therefore, to the third power of the particle size. As a result the smaller particles will acquire higher velocities. van Olphen cites a lower particle limit for flocculation of 10 Å, which is much smaller than the size range of the normally encountered fine cohesive sediments. Figure 3.8 shows a more detailed set of total potential curves for a surface potential o 25.6 mV, Aa 2 1012, univalent electrolyte and the indicated values of z [151, p.111]. According to Equation 3.17a and Equation 3.17b, o is inversely proportional to the square root of the electrolyte concentration n*. Therefore, z will decrease with increasing n* and vice versa. It is observed that for z 4, that is, for high electrolyte concentration, both the total potential and its gradient are negative and, therefore, the interparticle forces are everywhere attractive. The first hump appears for z 5 with a maximum at about d 1 for 107 cm1: from then on the height of the hump increases with increasing values of z, which is for decreasing electrolyte concentration. Moreover, it is observed that for z approximately equal to 5.6 the maximum point of the hump just touches the
74
Cohesive Sediments in Open Channels
abscissa at d 1. For higher values of z the maximum rises above the abscissa, and the corresponding values of d become smaller than 1. Another interesting point of the curves is the energy minimum, which represents the point at which the repulsive and attractive forces are about equal. This point shifts to greater distances as z increases and the electrolyte concentration decreases. Of particular practical importance is the flocculation concentration, which is the minimum electrolyte concentration of a specific type necessary for flocculation to take place. This concentration can be estimated by considering the curves of each potential energy versus the interparticle half distance, d, and making use of the simplified Equations 3.30 for VR and Equation 3.33c for VA. As stated earlier, Equation 3.30, which is of an exponential form, is valid for large values of d, but it has been shown to be applicable with reasonable degree of approximation down to d 1. The flocculation concentration can be conceived as the concentration for which, at the point of the hump, the two potential energies and the two forces become equal, that is, VR VA
and
dVR d (VA ) dd dd
where d is the half distance. On the basis of these assumptions and considerations, Verwey and Overbeek derived the following approximate equation for the flocculation concentration, nf, as ions cm3 in terms of the valence [151, pp. 117–118]. nf
107 Dr3 k 5T 5 γ 4 Aa2 (e)6
(3.34)
In spite of the approximations involved and the uncertainty about the value of the coefficient Aa, this equation reveals the degrees of influence of the controlling variables. Most important, nf varies in inverse proportion to the sixth power of the ionic valence. This means that we must expect the concentrations of 1–1, 2–2, and 3–3 electrolytes needed to flocculate a clay suspension or sol 1 6 1 6 : or 100 : 1.6 : 0.13. These ratios are consistent with 1 : to be in a ratio 2 3 the Schultze-Hardy rule, which states that the value of the flocculation number is predominantly determined by the valance rather than the type of the ions with a charge opposite to that of the particle surface. In fact, van Olphen reported the following orders of magnitude for flocculation values: for monovalent ions, 15 to 50 mmoleL1; for divalent ions, 0.5 to 2 mmoleL1; and for trivalent ions, 0.01 to 0.1 mmoleL1. These ranges are in close agreement with the ratios obtained from Equation 3.34. In practice, however, where we deal with mixtures of clay minerals, the flocculation concentration has to be determined by a series of tests
75
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
of sediment samples under various electrolyte concentrations according to the method described by van Olphen [150, Chapter 2].
3.5 Some important properties of fine particles and aggregates 3.5.1 The Counterion Exchange We pointed out earlier in the discussion of the Gouy-Chapman theory that the ions in the double layer that compensate for the electric charges on the surfaces of clay particles can easily be exchanged by other cations when available in the solution [150, Appendix III]. The relationship between the concentrations of two species of monovalent cations within the double layer and in the equilibrium solution (sufficiently far from the particle surface) is given by K
(C A )i (C B )e (C A )e (C B )i
(3.35)
in which CA and CB are the concentrations of the two species and where the subscripts i and e refer, respectively, to their equilibrium concentrations in the double layer and the original solution. Theoretically, for cations of the same valence, K should be equal to unity; however, because of the deviation of some of the assumptions involved in the Gouy-Chapman theory and according to some corrections introduced by Bolt, the value of K may range between 1 and 4 [150, Appendix III]. One major reason for the deviation of K from unity is the interaction between the counterions, the clay particles, and some specific ionic effects related to the ion size. For two ions of different valence, the theory predicts that the ion with the higher valence predominantly accumulates within the double layer. It is thus possible to replace cations of lower valence with others of higher valence and reduce the extent of the double layer. Because of this property, the counterions are also known as exchangeable ions. As mentioned previously, the number of cations required to compensate the charges in the double layer in milliequivalents per 100 g of dry clay is defined as the cation exchange capacity, or CEC. This is a measure of the degree of isomorphous substitution in the crystal lattice.
3.5.2 Limitations of the Gouy-Chapman Theory and the Stern Layer The Gouy-Chapman theory, outlined here, is based on several assumptions, some of which do not accurately represent the real situation. This theory provides a general conceptual model for the gross interaction of clay particles but fails to explain certain aspects of that interaction. Therefore, it should be viewed only as a basis for an understanding of the fundamentals of flocculation and
76
Cohesive Sediments in Open Channels
of the observed hydrodynamic behavior of cohesive sediments with only limited quantitative applications. The following are the most important of these limitations: (a) The clay particles are not infinite thin plates, as assumed, but have very irregular shapes. An analogous theory has been developed for spherical particles; however, because a spherical particle is as unrealistic as the previously assumed, the analysis for the latter case will not be presented. Interested readers are referred to Ref. [150, Chapter 3]. (b) The size of the ions and the possibility of their interaction with the charges on the surface of the particles have been neglected. A finite ion size, though, would limit the distance between them and the particle. Stern modified the Gouy-Chapman model by introducing a layer of thickness, s, known as the Stern Layer, in which there is no charge and which separates the counterion charge from the charge of the particle surface [150, Chapter 5]. Stern’s model is shown schematically in Figure 3.9. According to van Olphen, “the total counterion charge is divided between the charge at an atomic distance (1) and the charge in the diffused remaining double layer (2).” The electric potential at the surface of the particle is still o; however, at the end of the Stern layer, this energy is δ o and is known as Stern Potential. The adsorption forces between the counterions and the surface of the particle are taken into account by introducing the specific adsorption energy of the counterions, which must be added to the electrostatic
1
2
s 0
Electric potential
Stern layer (molecular condenser)
� Diffuse or Gouy layer Position of Stern ions Distance from surface Figure 3.9 Stern’s model for electric double layer [150, Fig.14].
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
77
energy. In Stern’s model the addition of electrolyte not only compresses the double layer but also shifts the counterions from the diffused layer to the Stern layer, thus further decreasing the Stern potential δ. The computed distribution of the ionic charges and potential are more realistic than the ones based on the Gouy-Chapman model. These distributions are presented in summary in Ref. [150, Appendix III] and in more detail in Ref. [151, Chapter 7]. (c) A repulsion of a very short range is attributed to two possible causes: (i) the resistance in extruding lattice points as two particles approach each other, known as Born repulsion, and (ii) a short-range repulsion due to specific adsorption forces between the crystal lattice and the water (or any other liquid) medium. Because of this repulsion, one or two monomolecular layers of water are tightly held by the particle surface. For two particles to approach each other at a distance smaller than the thickness of the adsorbed water layer, work has to be done, which manifests itself as the short-range repulsive potential shown in Figure 3.10 for low electrolyte concentration. The work required for such a desorption becomes appreciable for particle distances of the order of 10 Å. It should be added at this point that the nature of the adsorbed water and its effect on interparticle forces is a controversial subject to be discussed in the following section. The overall effect of all the repulsions of short range is a sharp dip in the potential curves of Figure 3.10 where Vt is the total potential VR VA. This means that for the separation of two coagulated particles, forces considerably
VR
Vt Particle separation VA
Vt min Figure 3.10 Net interaction energy distance for low electrolyte concentration [150, Fig. 13a].
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Cohesive Sediments in Open Channels
higher than those which caused flocculation are required; therefore, once flocculation has taken place, flocs are quite stable. It will be shown in Chapter 6 and Chapter 7 that the flow-induced forces could hardly break the original first-order flocs, but they can easily break the higher order agglomerates. There is no sharp line of demarcation between stable and unstable cohesive sediment suspensions. The distinction is only a matter of degree. In a completely quiescent liquid medium, eventually almost all suspended sediment will flocculate and will deposit given sufficient time. In flowing waters the flow-induced forces, in addition to enhancing flocculation, affect the internal structure and properties of the flocs. Contrary to earlier widespread assumptions held by hydraulic engineers, suspended cohesive sediments can readily be flocculated in the river water long before reaching the brackish estuarine region. Certain sediments, such as kaolinite, can even flocculate more readily in distilled rather than salt water due primarily to the attraction between their negatively charged surfaces and positively charged edges.
3.5.3 The Water Phase Water constitutes an integral part of the sediment and soil structure. A saturated soil with a void ratio greater than 1.00, which is the rule rather than the exception, contains a greater volume of water than solids. There is an interaction between solid particles and water molecules because none of them are chemically inert. Therefore, these interactions will affect the physicochemical properties of cohesive sediments and their behavior as flocs as well as a deposited soil mass under any degree of consolidation. The solid-water interaction is due to both the molecular structure of the water and the described physicochemical properties of the clay particles. The atomic nuclei in a water molecule (H2O) are arranged in a V shape H-O-H with the oxygen at the summit and the hydrogens at the two ends. The angle of the lines joining the two hydrogens with the oxygen atom is about 105 degrees. The water molecule thus forms a dipole in the form of a tetrahedron with two positive corners occupied by the hydrogen protons and two negative corners located symmetrically above and below the plane connecting the nuclei of the oxygen with those of the two hydrogen atoms. This polar water structure explains the behavior of water and ice discussed in Ref. [88, Chapter 6]. The question here is if and to what extent the water phase can affect the engineering properties of flocs and of fine sediment deposits. There is a strong adsorption of water molecules on the surfaces of the clay particles as a result of the properties mentioned, known as hydration. A similar hydration may also take place with dissolved ions, provided that the hydration energy is less than the energy between molecules in normal water. More specifically, because the surfaces of the clay particles end in either oxygens or hydroxyls, hydrogen bonds are expected to be one of the main mechanisms of
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
79
adsorption. The following are the possible mechanisms for clay-water interaction [88, Chapter 6]: 1. Hydrogen bonding, as already explained; 2. Cation hydration, with hydrated cations attracted to the surfaces of clay particles; 3. Attraction by van der Waals forces; 4. Attraction by osmosis, which is important for water content above the threshold needed for the formation of the first few molecular layers. The practical implications of the water phase are not completely understood, and considerable controversy exists even to date. The following are the main categories of the water phase in a clay-water system: 1. Absorbed water held very strongly by the clay particles; 2. Water with a high ion concentration held mostly by the cation positive charge, which is defined as double water layer; 3. Free pore water subject to the laws of hydrodynamics. Of particular interest to cohesive sediment behavior is the nature and properties of the adsorbed water. At the present time, this is subject to much speculation. Terzaghi and Freolich speculated that each clay particle is coated with a film of tightly held water in a semisolid state with very high viscosity, which provides an additional glue kind of a link between the particles [140]. Rosenquist adapted a similar idea and presented evidence that clay particles are surrounded by a thin layer of crystalline water of very high viscosity, which was assumed to yield plastically when two minerals are pressed together [123]. This yield, according to Rosenquist, causes the permanent deformation of cohesive soils after the load is removed. He also postulated that the same layer is responsible for the time deformation of clays known as creep. These concepts of a semisolid or crystalline adsorbed water have been challenged by other investigators. Data presented by Martin on sodium montmorillonite suggest a water structure to a distance of a few molecular layers from the particle surface, which differs from that of pore water, but not having the structure of ice [78]. Low presented a view similar to that of Rosenquist and claimed that in the vicinity of the clay particle the water has a quasi-crystalline structure with the following properties [76, 123]: 1. It has a yield value that leads to a threshold hydraulic gradient. For hydraulic gradients below that threshold, there is negligible flow velocity, whereas for higher gradients the flow velocity increases rapidly with the hydraulic gradient. 2. After flow begins, there should be a range of hydraulic gradients over which non-Newtonian flow would take place. 3. Near the surface of a clay particle, the viscosity should be higher than that of free water. Data have been obtained that corroborate this argument. In addition, Martin claimed that reasons other than a quasi-crystalline water structure can lead to the preceding flow conditions, and that the first few water
80
Cohesive Sediments in Open Channels
molecule layers can be simply closely packed liquid. He claimed that the density of the adsorbed water increases rapidly with decreasing distance from the particle surface, particularly for distances less than 10 Å, and may exceed 1.2 g cm3. This claim is supported by more recent findings, according to which the viscosity and the diffusion properties of adsorbed water in clays are essentially the same as those for the pore water [88, Chapter 6, Section 6.7]. That means that Fick’s law for diffusion and Darcy’s linear force-flow rate relationship are applicable for clay masses of high plasticity. In conclusion, it seems unlikely that the adsorbed water acts as a solid substance with a gluing action. There may be some hydrogen bonding through the water molecules providing an additional link between clay particles; however, such a link appears to be rather weak in comparison to the other bonding mechanisms discussed. Therefore, for practical purposes, the contribution of adsorbed water to the interparticle bonding should be disregarded, at least until further evidence to the contrary. For a more thorough discussion of the subject of adsorbed water, the reader is referred to Ref. [88, Chapter 6].
3.5.4 Sensitivity and Thixotropy It is well known that all clay masses have higher shear strength when they are in their undisturbed state than after they are in some way disturbed or remolded. The ratio of the original undisturbed clay strength to that after remolding is defined as sensitivity. The sensitivity varies widely. Slightly sensitive and medium sensitive clays have sensitivity ranging between 1 and 4; the range increases to between 4 and 8 for very sensitive clays. The most sensitive clays are known as quick with a sensitivity ranging from 16 to over 64 [88, Chapter 11]. The last clays, in fact, lose practically all their strength and behave like a thick viscous fluid upon remolding. This phenomenon is known as liquefaction. Such quick clays are encountered in previously glaciated areas of Scandinavia and North America. The degree of sensitivity depends primarily on the geotectonic processes that took place after the deposition of cohesive sediments and not so much on the physicochemical properties of the clay minerals. Mitchell [88] listed the following six causes of sensitivity: (1) metastable fabric; (2) cementation; (3) weathering; (4) thixotropic hardening; (5) leaching and ionic changes in the pore water; and (6) addition of dispersive agents. The first amounts to a change of clay fabric due to the overburden that in turn changes the effective stresses. These changes cause a change of the interparticle distances and orientations and, therefore, in the strength of the soil mass. Cementation, as discussed in Section 2.2.2, is caused by deposit of carbonates, iron oxides, alumina, and organic matter around the interparticle joints, thus increasing the overall strength. Part of these cemented joints break when the soil is remolded. The weathering process changes the types and relative proportions of ions in solution. These changes may affect the flocculation-deflocculation tendency after remolding. Thixotropic hardening amounts to unbalanced internal forces within the clay mass so that attraction after consolidation and/or compaction exceeds repulsion.
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
81
A slow rearrangement of clay particles then takes place, resulting in gradual increase of strength. Any disturbance, which would destroy this new balance, would inevitably result in a reduction of strength. The fifth cause of sensitivity is perhaps the most important because it is primarily the one for the development of quick clay deposits. Removal of salt and other flocculants from cohesive sediment beds deposited under saline conditions through diffusion may lead to a pore water quality favoring deflocculation, while the interparticle forces at the joints remain the same as long as the soil remains undisturbed. However, destruction of these joints due to any disturbance from the in situ position may lead to a dispersed system with double layers too large for flocculation. The system then becomes partially or wholly dispersed, retaining relatively little or none of its initial strength. Finally, the sixth and last cause amounts to something similar to the fifth. The sensitivity of consolidated clays is not particularly important to sedimentation engineering other than its effects on the erodibility of exposed clay masses. It may be important, though, in recent flocculated cohesive sediment deposits. Such deposits are composed of aggregate networks, have a very high void ratio, high water content, and very low strength, resembling a gel. Although they may appear as having practically no bearing capacity in a structural sense, these deposits may have substantial resistance to erosion by flowing water. The yield strength of gels can also be reduced by agitation to various degrees. Most clays and gels gradually regain all or at least a substantial part of their lost strength when they are left in a quiescent state after remolding. This regaining of strength is known as thixotropy. Both sensitivity and thixotropy are two phases of the same process, which can be explained by considering the mechanism of the physicochemical particle interaction. More specifically, thixotropy is defined as an isothermal, reversible, time-dependent process occurring under conditions of constant composition and volume, whereby a cohesive sediment mass or a colloidal suspension in the state of a gel develops higher interparticle bonds and, therefore, higher strength while at rest, and loses part or all of its interparticle bonding forces and its strength when remolded [88, p. 201]. Thixotropy is caused by a rearrangement of the fine particles due to their internal energy and unbalanced internal stresses resulting from the removal of external forces that caused the remolding [67]. Thixotropic effects will take place if the initial cohesive sediment structure is dispersed artificially to an extent greater than that dictated by the interparticle forces. Since this rearrangement involves water flow through very small pores, the thixotropic process also is expected to be very slow.
3.6 I nternal structure and fabric of flocs, aggregates, and cohesive sediment deposits 3.6.1 Particle Arrangements within Flocs The internal arrangement of individual particles within the flocs, their distances, and the orientations with respect to each other constitute the fabric and the structure of the flocs. These properties are determined by the nature of the
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Cohesive Sediments in Open Channels
interparticle forces and the flow-induced forces as well. They play a dominant role in the depositional and erosional characteristics of cohesive sediments. Figure 3.11 shows the following three types of aggregate and floc structure, according to Lambe [67, 68]: 1. Salt water flocculation, with a relative random particle-to-particle attachment that is face-to-face, edge-to-edge, and edge-to-face. In this case the van der Waals forces provide the main bonding mechanism (Figure 3.11a). 2. Nonsalt flocculation or flocculation containing low cation concentration. The dominant interparticle bonding is electrostatic attraction between negatively charged faces and positively charged edges. This structure resembles a house of cards and is referred to as card-house flocculation; it is particularly common in kaolinite clays because they have been shown to carry positive electric charges on their edges (Figure 3.11b). 3. Dispersed structure. Remolding, which is essentially the result of repeated shear deformation throughout the clay mass, tends to destroy the flocculated fabric and bring the particles into a more or less parallel orientation known as dispersed fabric (Figure 3.11c). This last fabric may also result from compression of deflocculated clay deposits. The microstructure of flocs and floc aggregates has been the subject of extensive studies and of several speculations. Terzaghi in 1925 [140] and Casagrande in 1932 [10], the fathers of soil mechanics, were the first to present a mechanistic model for the fabric of clay flocs. According to their model, numerous single fine particles are held together by the interparticle attractive forces to form units or cells in the form of a honeycombed structure. Several studies and theories have followed since that time, a discussion of which is beyond the scope of this book. A good and concise summary of the development of the various models and theories was presented by Quinn [121]; nevertheless, the Terzaghi-Casagrande model remained the most widely accepted one until recently, albeit with some modifications, with the single particle considered as the basic unit of the flock fabric. In fresh water deposits, the dominant fabric was believed to be more open and of the card-house type, with a prevailing edge-to-face attachment, whereas
(a)
Salt flocculation
(b) Non-salt flocculation
(c)
Dispersion
Figure 3.11 Sediment fabric according to Lambe [67, 68]. (a) Salt flocculation; (b) Non salt flocculation; (c) Dispersion.
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
83
in salt water, or in the presence of an electrolyte, the structure was assumed to be more random with edge-to-edge, face-to-face, and edge-to-face attachments. The studies that followed led to the rejection of the single particle model. Instead, the basic unit of the floc fabric was considered to be the clay packet, which is a domain of several individual clay particles mostly in a stepwise fashion in the form of a book. This arrangement is known as turbostratic structure or multiple aggregate particle fabric. Figure 3.12 shows schematically a clay packet floc structure proposed first by Moon [89], with the single particle replaced by the packet. In flocculated suspensions, the packets are joined randomly face-to-face, edge-to-face, and edge-to-edge, in a way similar to that in the single particle model. In a nonflocculated suspension, the dominant arrangement is edge-to-face with a more open structure in comparison to that of flocculated systems. The arrangement indicated as dispersed represents a deposited nonflocculated system having been subjected to external pressure. The packets owe their origin and formation to the processes of rock weathering and of the subsequent transport. For
Flocculated
Dispersed
Fresh clay
(a)
(b)
Consolidated clay
(c)
(d)
Figure 3.12 Consolidated floc structure based on packet of particles [121, p. 23].
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Cohesive Sediments in Open Channels
example, kaolinite forms originally as books or plates. Some of these books are subsequently dispersed during transport by the flow-induced shear stresses, but they may reform as multiple particle aggregates in a face-to-face stair-step structure or possibly in other types of associations [121]. These reformed agglomerations together with the original books constitute the basic building blocks of the flocs and of the deposited cohesive beds. The multiple particle aggregate fabric in quiescent water and its relation to the concentration of suspended clay and salt was investigated by Quinn in 1980 for kaolinite particles between 1 and 2 m using a scanning electron microscope [121]. The only stresses exerted on the flocs were those due to settling, which are very small. Regardless of clay concentration or electrolyte content, no single particle behavior was observed. Instead, even in the most dilute suspensions, the dominant unit was the multiple particle book in the form of steps with only a small number of single particles. The concentration of clay and electrolytes had some effect on the floc structure, but not as much as was anticipated. Indeed, flocs formed in fresh water were found to become larger and structurally more complex with increasing clay concentration in suspension. A tendency was detected for fresh water flocs to form elongated networks made up of platelets and packets in edge-to-face and face-to-face stair-step configurations. The differences between flocs formed in fresh and salt water were not as drastic as expected. Indeed, apart from a somewhat more open structure in fresh water flocs, both face-to-face and edge-to-face flocculation occurred in either case. The only conclusion is that clays in fresh water display a tendency to form open book-house type randomly structured flocs with edge-to-face and edge-to-edge packet junctions, whereas in salt water there is a greater tendency for more subparallel face-to-face packet junctions. Figure 3.13a and Figure 3.13b represent examples of salt and fresh water flocs formed at the lowest suspended sediment concentration of 100 ppm. The clay packets are obvious in both figures; however, a single particle can be observed in the fresh water floc, which tends to produce a more open structure similar to the traditional card-house one. The proportion of packets with more than five individual clay platelets per packet increased with increasing suspended sediment concentration, as observed in Figure 3.13c and Figure 3.13d. As pointed out earlier, flocs may join together to form higher order aggregates. The order and number of the latter depends on the suspended sediment concentration, the time of flocculation, but most particularly on the flow-induced stresses, because it is much easier to break an aggregate at its interfloc joints than to break a first-order floc. Figure 3.14a and Figure 3.14c display examples with the interfloc joints clearly indicated for fresh and salt water, respectively. Figure 3.14b shows another example of fresh water flocculation at high sediment concentration. The aggregates display only a small difference from the one corresponding to Figure 3.14c but a higher degree of randomness in particle orientation, causing a more open and voluminous microstructure. The fabric of aggregates may affect considerably the resistance to erosion and other mechanical properties of deposited and/or artificially formed cohesive
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
(a)
Salt water floc at 100 ppm clay concentration.
(b)
Fresh water floc at 100 ppm clay concentration.
(c)
Salt water floc at 400 ppm clay concentration.
(d)
Fresh water flocc at 400 ppm clay concentration.
85
Figure 3.13 Flocs in salt and fresh water at a high and a low concentration [121].
soils. In the case of compacted soils, for example, compaction on the dry side of the optimum water content tends to produce a flocculated fabric, whereas compaction on the wet side results in a more dispersed fabric. Finally, two kinds of aggregate formation of engineering importance deserve attention: taktoid formation and gelation. Tactoids are spindle-shaped agglomerations of clay particles that occasionally separate as bottom sediment [71, 150]. The origin of these formations lies in the first minimum of the total energy curves of Figure 3.7 and Figure 3.8. This minimum indicates the point at which the two potentials and the two forces are equal. Beyond that point, work is required to bring the particles closer to each other. Therefore, in the absence of external agents, such as turbulence-induced kinetic energy, to provide the required work, there will be some kind of loose, near-parallel particle agglomeration of relatively high density. According to van Olphen, the occurrence of equilibrium distance between clay particles is of the order of 100 Å [150]. It seems that tactoids are more likely to form in quiescent or very low velocity waters. Cohesive sediment beds formed by tactoids are expected to possess very low resistance to erosion. Gelation amounts to a quasi-homogeneous-looking system known as gel that possesses a small but measurable degree of both rigidity and elasticity. Gels are
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Cohesive Sediments in Open Channels
(a)
Fresh water aggregate at 500 ppm clay concentration.
(c)
(b)
Fresh water aggregate at 500 ppm clay concentration.
Salt water aggregate at 500 ppm clay concentration.
Figure 3.14 Floc aggregates in fresh and salt water for high and low concentrations [121].
formed in sufficiently high concentrations of colloidal particles and amount to a continuous network extending all the way to the boundaries of the container. Microscopically, the gel structure is highly nonuniform, ranging from elementary flocs to higher order aggregates. The porosity and other properties of gels depend on the type of flocculation. For example, when the dominant flocculation is edge-to-face, the gel is expected to have the most open structure, whereas a random flocculation corresponds to the densest structure. Gels have some peculiar properties. Up to a certain degree of imposed shear stresses, they act as rigid bodies displaying a yield strength and some elastic deformation. Beyond a limit, however, they break down and behave from then on as non-Newtonian fluids. The viscosity and yield strength of gels vary with time and the degree of agitation. The former, in particular, is a measure of the average interparticle forces. These forces can be significantly reduced upon agitation, but they normally rebuild gradually after the system is left to rest. The degree of strength reduction due to agitation is known as sensitivity, while the regaining of the strength is defined as thixotropy. Both aspects were discussed in the previous section and will be discussed again in Chapter 5 on the rheological properties of cohesive sediment suspensions.
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Floc aggregate
Failure plane Σ Figure 3.15 Flocculated bed at its loosest state A [102, 105].
3.6.2 The Microstructure of Deposited Cohesive Sediment Beds Suspended flocs and floc aggregates eventually deposit. As will be explained later, the properties of the deposited sediment depend to a great extent on the flow parameters. As pointed out earlier, in a quiescent environment, flocs in suspension join together to form aggregates of higher orders. For simplicity, we will consider only one characteristic aggregate: the floc aggregate. As the flocs and floc aggregates settle, they eventually join together into a continuous network with the properties of a gel defined as phase A [102, 105]. It has a highly nonuniform honeycombed structure, the highest void ratio, and the minimum possible density (Figure 3.15). Its basic unit is the floc with a density much higher than the average density of the bed. When these layers are gradually compressed by either the overburden pressure or by any external load, the bonds between the aggregates will be first broken, and the aggregates will move to a new, more stable equilibrium position, indicated as phase B (Figure 3.16). The aggregate density remains about the same; however, the overall density of the bed increases considerably. A further pressure increase will break the bonds between the flocs, which will come to a new denser equilibrium position, indicated as phase C (Figure 3.17), in which the flocs still maintain their original density. Finally, upon a further increase of the pressure, the inter-floc voids will gradually disappear, the flocs will deform to fill the interfloc voids, and the entire clay mass will reach a quasi-uniform fabric and density of the order of the original flocs, indicated as phase D (Figure 3.18). From then on, any further consolidation will result in a decrease of the average interparticle distance and an increase of the average density and of macroscopic strength.
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Figure 3.16 State B of a cohesive bed [102, 105].
Figure 3.17 State C of a cohesive bed [102, 105].
Surface particle
Failure plane Figure 3.18 State D of a cohesive bed [102, 105].
The structure of deposited cohesive sediment beds will be examined in more detail in Chapter 4, Chapter 6, and Chapter 7 in relation to the microstructure of suspended aggregates and to erosion, deposition, and resuspension of deposited cohesive sediment beds.
Chapter 4
The Hydrodynamic Transport Processes of Cohesive Sediments and Governing Equations
4.1 T he fundamental transport equations for cohesive sediments 4.1.1 The Development of the General Transport Equations A theoretically rigorous study of the hydrodynamic behavior of cohesive sediments in a turbulent flow field has to start with the formulation of the fundamental transport equation for these kinds of sediments. The process is controlled by two groups of variables: (a) the relevant sediment properties and (b) the set of large- and small-scale flow parameters which fully define and control flocculation, erosion, deposition, and resuspension. As already indicated, the properties of coarse sediments can be adequately represented by either an average grain size or by an appropriate grain size distribution. In contrast, the size and other properties of cohesive sediment aggregates depend on the stresses induced by the turbulent velocities. The latter generate the mechanism for interaggregate collision while at the same time limiting the maximum aggregate size. In addition, aggregates are subjected to a continuous process of growth and disintegration, which has to be properly introduced in the transport equations. The first general cohesive sediment transport equation was developed by McLaughlin [80] and was based on the following continuity principle: The net rate of inflow of settling units with a diameter di through a control surface enclosing a control volume plus the rate of generation of such units within the control volume is equal to the rate of concentration increase of the same settling units within the control volume.
Cohesive Sediments in Open Channels Copyright © 2009
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It is more convenient to describe the elements in suspension by their settling velocities, ws, rather than by their diameters, because the former are the relevant parameters in sedimentation. In the foregoing analysis, ws is subdivided into n classes with an n number of equal increments differing by w. Thus, ws1 refers to the class of settling velocities between 0 and w, ws2 refers to velocities between w and 2w, and, in general, wsi refers to values of settling velocities between (i–1)w and iw. Particles with settling velocities wi are called i particles, and their concentration is indicated as Ci. Denoting the total concentration by C and the average settling velocity by wsi, the continuity principle leads to the following relation: n
C ( x,y,z,t )
∑ Ci ( x,y,z,t )
(4.1)
i1
and n
ws (x ,y ,z ,t )C (x ,y ,z ,t )
∑ wsi (x ,y ,z ,t )C i (x ,y ,z ,t )
(4.2)
i 1
It is noted that, in general, settling velocities and concentrations are functions of space and time and that the left side of Equation 4.2 represents the total rate of settling of suspended sediment. For the derivation of the general fundamental transport equation for a threedimensional flow field, we consider an infinitesimal control volume dx.dy.dz in a Cartesian coordinate system x, y, z, where x and y are the two horizontal directions and z is the direction of gravity. Then the transport and accumulation of sediment through the control surface and within the control volume, respectively, are evaluated. The transport of i particles takes place through the following two processes: (a) by convection due to the temporal mean flow velocity, V uiˆ vjˆ wkˆ, where u, v, and w are the velocity components in the x, y, and z directions, respectively; and iˆ, ˆj , and kˆ are the three unit vectors in the same corresponding directions; the total convective sediment transport is equal to VC ; and (b) by turbulent diffusion, which follows Fick’s law and can be expressed as Dij (∂Ci / ∂x j ), where j (1, 2, 3) and where Dij is the turbulent diffusion coefficient of i particles in the j direction. Within the control volume there is a net rate of positive or negative concentration change by coagulation and destruction, expressed as (∂Ci / ∂t )dx.dy.dz. There can also be a possible source generating i particles per unit time and unit volume at the rate of (x, y, z, t); that is, within the control volume, there may be a rate of i particle production equal to .dx.dy.dz. Applying the continuity principle to all i particles, adding the resulting equations for all i values, and introducing Equation 4.1 and Equation 4.2 into the sum
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and considering the diffusion coefficients independent of particle size, we obtain the following equation:
∂Ci ∂C ∂(wsC ) ∂C ∂C ∂C ∂ n u v w ∑ Dtx ∂t ∂z ∂x ∂y ∂z ∂x i1 ∂x n ∂Ci ∂Ci ∂ n ∂ n ∑ Dtz ∑ i 0 ∑ Dty ∂z i1 ∂y i1 ∂y ∂z i1
(4.3)
where Dtx, Dty and Dtz are the turbulent diffusion coefficients in the x, y and z, directions, respectively. If the i particles are created solely by flocculation of smaller particles and/or by destruction of larger agglomerates, that is, if 0, and if the turbulent diffusion coefficients, are assumed to be independent of the agglomerate size, Equation 4.3 reduces to the following simpler form:
∂C ∂C ∂C ∂C ∂(ws C ) u v w ∂t ∂x ∂y ∂z ∂z ∂ ∂C ∂ ∂C ∂ ∂C Dtz Dtx Dty 0 ∂x ∂ x ∂y ∂y ∂z ∂z
(4.4)
where the effect of flocculation and/or hindered settling is included implicitly in the term wsC. The two-dimensional near uniform flow case is of particular interest since it approximates the conditions in a wide variety of open channel flows. If x is considered to be the direction of flow, then v w 0 and, moreover, u is a function of z only; that is, u f(z). Equation 4.4 then simplifies into the following one:
∂C ∂C ∂C ∂(wsC ) ∂ ∂C ∂ u Dtx Dtz 0 ∂z ∂t ∂x ∂z ∂x ∂x ∂z
(4.5)
If it is further assumed that only the mean sediment concentration changes spatially while the average settling velocity of the aggregates remains constant, Equation 4.5 takes the following form:
∂C ∂C ∂C ∂ 2C ∂Dtx ∂C ∂ 2C ∂Dtz ∂C 0 u ws Dtx 2 Dtz 2 ∂t ∂x ∂z ∂x ∂x ∂z ∂z ∂x ∂z
or
∂C ∂C ∂C ∂ 2 C ∂D ∂ 2C ∂Dtx ∂C (4.6) u Dtx 2 tz ws Dtz 2 ∂z ∂z ∂t ∂x ∂x ∂x ∂x ∂z
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For a near uniform flow, the diffusion coefficients can be assumed independent of the coordinate x and dependent only on z. Furthermore, as will be explained later in Chapter 6 and Chapter 7, the effect of suspended sediment on the diffusion coefficients is negligible for the range of concentrations encountered in rivers and estuaries. Only when in excess of 20,000 ppm do sediment concentrations appear to affect the diffusive and dispersive properties of the fluid system. The gradient of Dtx with respect to x is, therefore, practically zero, so that Equation 4.6 reduces to its final simplest form:
∂C ∂C ∂C ∂ 2C ∂D ∂ 2C u Dtz 2 tz ws Dtx 2 ∂z ∂t ∂x ∂z ∂z ∂x
(4.7)
Equation 4.7 is similar to that developed by Dobbins [17] for the case of twodimensional steady flow without the time derivative of C. For the solution of Equation 4.7, appropriate initial and boundary conditions are needed. The obvious initial condition is At t 0, C constant and equal to a prescribed value Co. For the free surface the no sediment flux condition can be expressed by
Dtz
∂C wsC at z yo ∂z
(4.8)
where yo is the depth of flow. The most complicated and crucial boundary condition is that at the bed. Its general form is [100, 103]
E (1 pr )wsC Dtz
∂C at z 0 ∂z
(4.9)
where E is the rate of erosion and/or resuspension, pr is the proportion of all near-bed settling sediment that reaches the bed coming into contact with the latter and sticking to it, and z is the distance from the bed. Equation 4.9 states that the amount of bed sediment eroded per unit bed area and unit time plus the amount of deposited sediment that cannot reach the bed must be entrained back to the main flow by turbulence. This equation leads to the following special cases: (a) Net erosion without simultaneous deposition: E 0 and pr 1; (b) Neither erosion nor deposition: E 0, pr 0. This is a case of wash load transport (Section 7.2) and Refs. [100, 103]; (c) Net deposition without erosion: E 0, 0 pr 1. There was considerable speculation in earlier studies as to the actual nearbed sedimentation processes. For instance, Dobbins hypothesized that, under
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93
steady conditions, the rate of particle and/or floc pickup is equal to the rate of deposition, an assumption which can be stated by Equation 4.9 with pr 0 [17]. Krone claimed that during deposition there is an exchange between bed flocs and suspended flocs [62]. This hypothesis appeared to originate from the basic idea behind Einstein’s bed load function for coarse sediment [25] and from some limited and fragmentary experimental evidence. The more recent and extensive experimental investigations, to be discussed in detail in Chapter 6 and Chapter 7, revealed that this is not the case and that for cohesive sediments, apart possibly from a very short transient time period, no simultaneous erosion and deposition occurs. Equation 4.3 describes the most general three-dimensional unsteady case with sediment input from outside. Equation 4.4 is a simpler form of the previous one. Both equations are significant and have to be solved in cases in which the sediment distribution is of importance. Such is the case, for example, in environmental studies, where turbidity plays a significant role. In many cases, however, it is sufficient to know the average cross-sectional distribution of the concentration, the location of shoaling, and the rates of deposition and/or resuspension. In estuarial shoaling control, for instance, primary attention is focused on the shoaling sites and rates without much concern about the vertical concentration distribution. In such cases, moreover, we only need the average over the depth flow velocities or the total discharge over either the depth of flow per unit width for the two-dimensional flows or over the entire cross-section for the one-dimensional flows. The first apply to relatively wide estuaries and bays and the second to relatively narrow channels with width-to-depth ratios of the order of unity. The sediment transport equations in either case are developed by an over-the-depth integration of Equation 4.4 and an introduction of the overall dispersion coefficients in place of the turbulent diffusion coefficients. The analysis for both cases has been presented elsewhere [100, 103] and in abbreviated form in Chapter 11 of Ref. [110]. The two-dimensional sediment transport equation is
∂C ∂C ∂C ∂ ∂C ∂ ∂C u v Dy Dx ∂t ∂x ∂y ∂x ∂x ∂y ∂y Fe ( x, y, C , t ) Fd ( x, y, C , t )
(4.10)
where u and v are the average over-the-depth velocities in the direction of flow, x, and in the normal to x horizontal direction, y, respectively, given by u v
qx yo qy yo
(4.11)
(4.12)
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where C is the average over-the-depth suspended sediment concentration; qx and qy are the flow rates per unit width in the x and y directions, respectively; and Fe and Fd are the sediment source and sink functions representing the resuspension and deposition rates per unit bed area. These last two functions depend on the location, as defined by the coordinates x and y, on the suspended sediment concentration, C, and on the time. Dx and Dy are the dispersion coefficients in the x and y directions, respectively. For the one-dimensional case, the sediment transport equation reads
∂C ∂C ∂C 1 ∂ V Fe ( x, C , t ) Fd ( x, C , t ) ADx ∂t ∂x A ∂x ∂x
(4.13)
where A is the cross-sectional area of the channel, being in general a function of the flow direction x; V is the instantaneous average over the cross-section velocity, i.e., V Q/A, where Q is the discharge; Dx is, as in the previous case, the dispersion coefficient in the direction of flow, x, while the sediment source and sink functions, Fe and Fd, representing the corresponding rates per unit length of the channel, depend now only on x, C, and t.
4.1.2 Discussion of the Developed Equations Both Equation 4.7 and Equation 4.13 are integral forms of the most general fundamental Equation 4.4, and they both apply to near one-dimensional flows in prismatic channels. Equation 4.7 follows from an integration of 4.4 over the channel width. Equation 4.13 is derived from the integration of the same equation over the channel depth. In Equation 4.7 the sediment properties enter in the settling velocity, ws, and in the bed conditions in the form of the rates of deposition and/or resuspension. In Equation 4.13, which essentially describes the mass balance within and through a control volume h(x)b, where b is the channel width and h is the channel depth, the settling velocity of the aggregates does not appear explicitly, but it is included implicitly in the sediment source and sink functions, Fe and Fd, respectively. The observation is true for the two-dimensional depth-integrated Equation 4.10. The source and sink functions represent the net effect of suspended and bed flocs and/or higher order aggregates on the sedimentation process. The first represents the rate of aggregate erosion and the second the rate of aggregate deposition on the bed. The initial and boundary conditions for both equations are not anymore to be found at the water surface and at the bed, but instead they have to be prescribed as the distribution of the average over the cross-sectional area concentration C and by a known sediment input at a specific location of the system. However, because the sediment concentration is the dependent variable in these equations, that input has to be expressed as Co; i.e., the reference concentration at the specific location, xo, assuming that the sediment which enters the channel per unit time, is immediately spread uniformly over the entire section. The developed sediment transport equations are essentially diffusion and dispersion equations and can be solved either analytically for simplified cases
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or, more generally, numerically. The subject of numerical models is beyond the objective of this book. Examples of such models are those developed by Arathurai [2], Arathurai and Krone [3], and Hayter [49]. These fundamental equations have been presented for the sole purpose of demonstrating the importance and potential application of the research results presented and discussed in Chapter 6 through Chapter 8. For an abbreviated presentation, the reader is referred to Refs. [100, 110, 111]. The difference of these equations in comparison to similar transport equations for dissolved substances lies in the aggregate settling velocity, ws, for the three-dimensional model and in the sediment source and sink functions for the two depth-integrated equations. The main volume of recent and current fundamental research on cohesive sediments has been directed toward a reliable formulation of these variables. The difference between cohesive sediments and other transportable substances lies in the dependence of the aggregate properties on the flow parameters themselves and in the fact these properties are in a dynamic process of continuous change. These changes take place as the sediment first settles from the main flow to the bottom, is later resuspended, and is subsequently transported and deposited again along the water system. The dynamics of floc formation in a turbulent flow field is the subject of the following section.
4.2 The process and dynamics of flocculation As pointed out in Chapter 1, particles and/or aggregates are brought into contact by (a) Brownian motion, (b) velocity gradients, and (c) differential settling. The formulation of a theory for rapid coagulation has been the objective of several investigators and constitutes a major field of fluid mechanics by itself. A good treatment of this field can be found in Chapter 5 of Physicochemical Hydrodynamics by Levich [74]. The mechanism of each one of these processes will now be presented, and their relative importance will be discussed in an abbreviated form along the line of presentation by the cited reference. The process of coagulation has been treated as a diffusion problem.
4.2.1 Collisions Due to Brownian Motion As explained in Section 2.1, this motion of the suspended units is caused by the unbalanced impulses of the water molecules on the surfaces of the solid particles according to Equation 2.2 and Equation 2.3. These impulses are the result of the thermal motion of the fluid molecules and depend, therefore, on the temperature. The theory of Brownian motion was developed by Smoluckowski and was well summarized by Levich in Chapter 5 of Ref. [74]. Being of a random nature, this motion can be described by a diffusion coefficient, Dbr, known as the Brownian diffusion coefficient, not to be confused with the molecular diffusion coefficient, D. Dbr is isotropic and a property of the fluid. In the analysis, the concentration of suspended particles is assumed to be low enough so that collisions occur only
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in pairs; that is, no more than two particles or units collide at the same time. If no is the number of suspended particles per unit volume, then in a Cartesian system the transport rate is given by Equation 4.4, by replacing C with the number n of particles per unit volume and by setting all the diffusion coefficients equal to Dbr. The resulting equation is ∂n Dbr ∇2 n ∂t
(4.14)
In spherical coordinates, which are more appropriate for our purpose, Equation 4.14 takes the following form: ∂n 1 ∂ 2 ∂n Dbr 2 r ∂t r ∂r ∂r
(4.15)
Equation 4.15 will now be derived directly using a spherical control volume. The simplest case presented here considers a dispersion of spherical particles of equal diameter, dp, with a volume concentration no. Each particle is visualized as surrounded by a sphere of radius, Rpc, defined as radius of coagulation, such that any other particle entering the sphere will be attracted to the first particle and will be attached to it. Obviously, over a spherical surface of radius Rpc, the particle concentration is zero. In the special case of noninteracting particles, the radius of coagulation Rpc will be equal to the particle diameter dp; that is, Rpc dp 2Rp where Rp is the particle radius. We take next an infinitesimal control volume enclosed between the surfaces of radius r and r dr. The volume of a sphere of radius r is V (4/3)r 3; therefore, the volume of the control volume will be equal to (dV/dr)dr 4r 2dr. The continuity principle states that the net rate of particle inflow into the control volume (CV) through the control surface (CS) must be equal to the net rate of particle accumulation inside the CV. According to the diffusion law, the transfer rate of transfer of particles through a control surface, S, is SDbr ∂n ; because the ∂t 2 surface of a sphere is 4r , the conservation law for the particles can be written as
(
)
∂n ∂n ∂ ∂n 4r 2 dr dr Dbr 4r 2 ∂t ∂r ∂r ∂r ∂n ∂ ∂n 4 Dbr r 2 4 Dbr r 2 ∂r ∂r ∂r 4 r 2
(4.15a)
from which Equation 4.15 follows. The following boundary conditions have to be satisfied for the solution of Equation 4.15: For r Rpc and t 0 n no For r Rpc and t 0 n 0 For r n no
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The second condition states that no particle can stay at the boundary of the sphere of influence. The solution to Equation 4.15 subject to these boundary conditions is
R pc 2 R pc n n0 1 r r
∫0
r R pc 2 Dbrt
exp( 2 )d
(4.16)
where is a dummy variable. It follows from Equation 4.16 that the flux, fp, of particles across the surface of the sphere is
∂n D n f p Dbr br o ∂r R pc r R pc
1 R pc Dbr t
(4.17)
Considering that the elemental surface of the CS is dS 8rdr, the total number of collisions, Mp, with the reference particle per unit time, which is equal to the total rate of influx through the CS, is M p 8 ∫
R ij 0
4 no R pc
8 no Dbr R pc 1 R pc Dbr t
f p dS
2 1 + R pc R pc Dbrt 2 (4.18)
2 / Dbr ), the last For a time t sufficiently large, and specifically for t (R pc term in the bracket of Equation 4.18 becomes insignificant in comparison to unity, in which case Equation 4.18 obtains this simpler form:
M p ≈ 4 Dbr R pc no
(4.19)
This is the number of collisions of particles with only one stationary particle. Therefore, to find the number of contacts per unit volume, we must multiply the right side of Equation 4.20 by no, the total number of particles per unit volume. In addition, because the reference particle is actually subjected to the Brownian motion as well, the total number of contacts per unit volume of dispersion per second, Mpt, will be twice the value given by Equation 4.19; i.e.,
M pt 8 Dbr R pc no2
(4.20)
One can easily extend the same analysis to the general case of particles with various diameters and various radii of coagulation. Referring to Figure 4.1b, the
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Rj r � dr
r
dr
Ri Rij � Ri � Rj (a)
(b)
Figure 4.1 Spherical particle and sphere of influence around it and infinitesimal control volume.
i order particle is the stationary one and the j order particle the colliding. The two particle diameters are dpi and dpj, and the radius of coagulation is Rij. The number of collisions of j particles with the i particles is
Rij Mij 4 n j Dbr Rij 1 Dbr t
(4.21)
For the case of t Rij2 / Dbr , Equation 4.21 reduces to
Mij ≈ 4 Dbr Rij n j
(4.22)
and the total number of collisions per unit volume and per second between i and j order particles becomes
Mij 4 Dbr Rij ni n j
(4.23)
in which ni and nj are the concentrations of i and j order particles, respectively. Equation 4.23 expresses the total frequency of collision of i and j order units assuming that the i units are stationary and that only the j units are subjected to Brownian motion. However, the first units are also subjected to the same motion. Therefore, and according to the same argument regarding uniform particles, the frequency of collisions and the reflecting diffusion coefficient should be double. Equation 4.19 then becomes
Mij 8 Dbr Rij ni n j
(4.23a)
We return next to Equation 4.20 for the case of equal size particles. Because in rapid flocculation every collision results in a conglomeration, the frequency of
Chapter | 4 The Hydrodynamic Transport Processes
99
collision, Mpt, will be essentially the rate of reduction of primary particles from the suspension, Mo dn/dt, leading to the equation
dn 8 Dbr R pc n 2 dt
(4.24)
where Rp Rij. Integration under the boundary condition n no at t 0 yields no
n
1
t tc
(4.25)
where tc is the coagulation time given by tc
1 8 Dbr R pc no
(4.26)
According to Levich, the approximate treatment of the coagulation process due to the Brownian motion presented here leads to the same results as the more rigorous solution by A. N. Kolmogorov and M. A. Leontovich outlined by Levich [74, Chapter 5]. Furthermore, if we consider that, for all practical purposes in sedimentation engineering, the frequency of Brownian collisions is negligible in comparison to that caused by velocity gradients and flow-induced shear stresses, the approximate analysis presented here is more than adequate. The Brownian diffusion coefficient, Dbr, has to be estimated for the solution of the derived equations. According to Levich, Einstein’s formula is entirely satisfactory for order of magnitude estimates: Dbr
kT 3R ij
(4.27)
where T is the absolute temperature in degrees Kelvin, is the fluid viscosity, and k is the Boltzmann constant equal to 1.38 1023 J°K1. One should note that the actual particle diameter is used rather than the radius of coagulation, Rp. For colloidal particles in water and for normal room temperature range, Dbr has an order of magnitude [74, Chapter 5]: Dbr ≈
1013 cm 2 for d p in cm s dp
(4.28a)
and for aerosol particles in air [74, Chapter 5]: Dbr ≈
1011 cm 2 for d p in cm s dp
(4.28b)
Cohesive Sediments in Open Channels
100
These values correspond to the following Prandtl numbers for water ( 102 cm2s1) and for air ( 0.01 cm2s1):
For water PR
102 ≈ 13 d p 1011 d p Dbr 10
and for air PR ≈
0.1d p 1011
1010 d p .
For a clay particle on the order of 1 m, i.e., 104 cm, the Prandtl number is on the order of 106. Therefore, even for the smallest particles in suspension, the Brownian diffusion coefficient is smaller than the molecular diffusion coefficient by six orders of magnitude. The model presented involves certain idealized assumptions. First, the inertia effect of the mass of the particles has been neglected. The specific gravity of the primary sediment particles normally ranges between 2.00 and 2.5, whereas the density of the flocs and of the higher order aggregates is much lower, being slightly above that of the water. This density difference may cause a deviation of the aggregate trajectories from those of the water molecules. In practice, however, we seldom have to deal with original clay particles except, perhaps, at the first stage of flocculation. Moreover, as we already pointed out and will show in the next section, in moving waters the effect of the velocity gradients on the collision frequency and on flocculation is much more important than the Brownian motion. Therefore, the omission of the inertia effect is not expected to be of practical importance. The solid particle inertia, though, may have a significant effect in aerosols, where the suspended particle density is much higher than the gas density.
4.2.2 Collisions Due to Velocity Gradients Particles and aggregates in a flow field are brought into contact if they are moving with different velocities. To understand this mechanism of collision and to develop the basic equations, we consider first a laminar flow, as in Figure 4.2. In this figure, we have a velocity profile (Figure 4.2a) and two spherical units (Figure 4.2.b; particles and/or aggregates), i and j, with radii Ri and Rj. The corresponding water velocities are vi u and vj u u, respectively, at the center of the unit. Figure 4.2.c shows the same units at the moment of contact. Considering that the distance y between the centers of the units is very small, we may assume a linear variation of the velocity within y and equal to u = Gy
∂u ∂u y where G ∂y ∂y
G is the velocity gradient or the shear rate.
(4.29)
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101
y u � G∆y ∆y
Rj
Rj
u
θ G � du dy
0 (a)
∆y Ri
Ri
u Flow profile
(b) Approaching aggregates
(c) Aggregates in contact
Figure 4.2 Spherical aggregate collision in laminar flow.
When the two particles or, in general, two units come into contact, the distance between their centers is Rij Ri Rj. A sphere with a radius Rij is known as the sphere of influence for the units i and j, because any j unit whose center crosses the surface of that sphere will be attached to the i unit. At contact y (Ri Rj)sin. Therefore, before contact y (Ri Rj)sin. To derive the number of collisions of j order and the i order units, we consider the zone of the spherical surface of radius Rij contained between the angles and (Figure 4.3). The radius of that zone is (Ri Rj)cos and its thickness d(y) (Ri Rj)cosd. The relative velocity of the j unit with respect to the i unit at the moment of contact is rx G( Rij )sin
(4.30)
The j units enter the sphere of influence through a surface equal 2Rij(cos)d(y) 2Rij(cos)Rijcos(); therefore, when we substitute with the differential d, the total rate of collision of j units with i units will be
N ij 4n j ∫
/2 0
G( Ri R j )sin ( Ri R j ) cos ( Ri R j ) cos d (4.31)
which gives
N ij
4 n j G( Ri R j )3 3
(4.32)
and the total rate of collision between all i and j units in the entire flow field will be
N ijt
4 ni n j G (R i R j )3 3
(4.33)
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Rij cos �
dy ∆q u
dy � (Ri � Rj) cos � d �
q
Rij sin � du ∆u � R sin � � GRij sin � dy ij
Rij
Figure 4.3 Infinitesimal sector of a sphere of influence.
where ni and nj are the number of i and j units, respectively, per unit volume of the fluid. For the simple case of uniform size agglomerates, i.e., when Ri Rj R average agglomerate radius, Equation 4.34 simplifies into the form
N grad
32 2 3 4 3 no GR p no2GR pc 3 3
(4.34)
where R is the average radius of the unit and Rpc 2Rp. The ratio of the collision rate due to velocity gradients to that due to the Brownian motion (Equation 4.20) is
N grad N br
32 no2GR p3 48 Dbr R p no2
=
2GR p2 3 Dbr
(4.35)
For the estimate of Dbr given by Equation 4.28 for solid particles in water, the ratio becomes
2GR p3 (1013 ) 3
(4.36)
For a representative clay diameter dp equal to 1 m, i.e., 104 cm, becomes approximately equal to the representative shear rate 10G. Therefore, even at low values of G higher than 101 s1, 1 and the gradient effect predominates. For the same particle size for Brownian motion to predominate, G should be less than 0.1s1, which is a very low velocity gradient. For particle sizes on the order of 106 cm, the gradient effect will predominate for G 105. Taking into consideration that even in nonsalt water systems there is almost always a certain degree of flocculation and assuming the smallest floc size to be on the order of 10 m, i.e., 103 cm, the ratio should be higher than about 103G, or that G should be larger than 105 for the predominance of velocity gradients. The latter limit is very small and indicative of almost stagnant waters. Because in natural water
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103
Ratio of number of collisons by velocity gradients and brownian motion 1000
Particle diameter d (µm)
1,000,000 100
100,000 10,000 1000
10
100 10 1
1.0
0.1 0.01 0.001
0.1 0.01
0.1
1.0
10
100
1,000
Velocity gradient G (s�1) Figure 4.4 Relative importance of velocity gradients and Brownian motion on the frequency of collisions [149].
systems the agglomerate sizes range well above 10 m, one can conclude that, apart perhaps from an initial stage, in moving waters carrying cohesive sediment in suspension, the gradient effect is by far the dominant and the Brownian effect can be neglected for all practical purposes. Figure 4.4 shows the relationship of the ratio as a function of the particle size and the velocity gradient G [149, p. 356]. The solid line separates the region of dominance of the velocity gradients (above) from that of the Brownian motion (below). Thus, for a range of G between 1 and 10 s1, which is a representative range for estuaries [149], the Brownian motion becomes more important than the velocity gradients only for particles smaller than 2 to 6 microns. For 10 m diameter flocs, the ratio ranges between 100 and 1000; whereas for 100 m aggregates, the same ration varies between 100,000 and 1,000,000.
4.2.3 Collisions Due to Differential Settling This collision process takes place by faster moving units entering the radius of influence of slower moving ones. We consider again the two aggregates i and j of Figure 4.2 and their sphere of influence with its infinitesimal sector, shown in Figure 4.3, contained between the angles and . Let wsij be the relative
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104
settling velocity of the aggregates. Then, according to the analysis in Section 4.2.2, the area of the differential sector is
d (A) 2 Rij (cos ) Rij sin d
(4.37)
Indicating the collision frequency between the i and j order units by Hij, the rate of passage of j units through d(A) will be Hij (A)(wsij)nj and the total rate through the entire sphere of influence will be
Hij 2 Rij2 wsij n j ∫
/2 o
cos sin d Rij2 (wsij )n j
(4.38)
and the total collision frequency between all i and j order units becomes
H tij Rij2 (wsij )ni n j Rij2 (wsj wsi )ni n j
(4.39)
The relative importance between the effects of the velocity gradient and the differential settling is given by the ratio of Equation 4.33 and Equation 4.39, i.e., 1
4 Rij G 3 wsij
(4.40)
Therefore, if RijG wsij, the shear rate will be the dominant mechanism of flocculation and vice versa. The size of flocs and aggregates varies normally from a few micrometers to at most a few millimeters. However, and as shown in Section 4.4 about the aggregate properties, the aggregate density decreases rapidly with increasing order of aggregation and size so that larger agglomerates are associated with smaller settling velocities. For example, for a 100 m floc with a settling velocity of 10 mms1, the Reynolds number is on the order of unity, which is well within the Stokes range. So the relative settling velocity of flocs is expected to be on the order of 1 mms1. In flowing waters, particularly in the zone of high gradients, which is the critical zone of molding the aggregate properties, the velocity difference is on the order of several centimeters per second. Although this point will be further discussed in the appropriate chapter, we can safely state at this point that in flowing waters the shear rate effect is the dominant one in flocculation. Differential settling may become the controlling parameter, though, at times of very low velocities, such as the slack times in tidal estuaries. At such times, differential settling may lead to the formation of relatively large higher order aggregates, resulting in rapid deposition. For Reynolds numbers less than 10, the settling velocity can be approximated by the Stokes equation:
ws
1 s 2 gd p 18 w
(4.41)
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105
which, introduced in Equation 4.39 with the change Rtij (1/2)(dpi dpj), leads to this special form: H tij
g (d i d j )2 ( j d 2j i d i2 )ni n j 72w
(4.42)
The three frequency collision equations (Equation 4.23, Equation 4.33 and Equation 4.42) can be condensed into the equation Fij K ij (d i , d j )ni n j
(4.43)
where Kij is defined as the collision frequency function being a function of the particle and/or aggregate diameters di and dj, and where Fij can have the values of Mtij, Ntij, and Htij. Table 4.1 gives the collision frequency function for the three collision mechanisms [149]. In the presented simplified model, the colliding units have been assumed spherical, the volume of the aggregate formed by two smaller units was considered to be equal to the volume of the two constituent units, and the suspended sediment concentration was supposed to be low enough so that no more than two particles and/or aggregates collide at a time. Finally and most important, each collision results in a permanent bonding of the two colliding units. In reality, however, the shape of the aggregates is quite different than spherical. Only in completely destabilized cohesive sediment suspensions corresponding to the case of rapid flocculation is the assumption of permanent bonding after each collision satisfied. The remaining two assumptions can be considered valid, with the exception perhaps of the rare cases of rivers and estuaries with very heavy load
Table 4.1 Collision frequency function Collision Mechanism
Collision Frequency Function
Equation Number
Brownian motion
Kbrij
Velocity gradients
K sij
4 (Ri R j )3G 3
(4.33)
Differential settling
Kdij
g (di d j )2 ( j d 2j i di2 ) 72w
(4.42)
8kT 3
From Eqs. 4.23 and 4.27
Cohesive Sediments in Open Channels
106
in suspension. One must remember that a substantial degree of flocculation can take place even in so-called stable suspensions. Therefore, in general, not all collisions will result in the formation of larger aggregates, and only a fraction of the collisions will thus be effective in achieving further coagulation. To account for the collision effectiveness in achieving coagulation, a stability factor should be introduced in Equation 4.21, Equation 4.33 and Equation 4.42 or, more generally, in Equation 4.43, which thus obtains the form
Fij K ij (d i , d j )ni n j
(4.44)
where can take on values from 0 to 1. The first value corresponds to the unrealistic case of a completely stable suspension and the second to a completely destabilized one.
4.2.4 Concluding Remarks Section 4.2 presented and compared the three processes of floc and higher order aggregate formation. The conclusion was that in natural water systems, even with low velocities, the velocity gradients or shear rates constitute the dominant mechanism of flocculation, with the exception perhaps of water slack periods and of rivers and estuaries with very heavy fine sediment load. The developed equation for velocity gradient collision was based on the assumption of uniform laminar flow. This section did not consider the stresses acting in the developing aggregates by both the velocity gradients and the stresses which are generated by settling that limit the size of the aggregates. The ultimate volume of a particular aggregate will be determined by the physicochemical properties of the constituent particles and by the local shear rates. In natural flow systems, however, the flow not only varies spatially and with time but is also highly turbulent. The question is, therefore, what can be taken as a representative velocity gradient and how such a gradient can be related, if possible, to some relevant and readily determinable gross flow variables? The process is further complicated by the deposition and resuspension of aggregates. The flow-induced stresses on the flocs and aggregates and the ultimate size of the latter are the subject of the next section.
4.3 Review of fundamental properties of turbulent flows 4.3.1 Significant Stresses and Parameters As pointed out earlier, during the process of flocculation the ultimate size of the aggregates will be determined by the flow-induced shear stresses on these aggregates. Thus, the shear rates perform the dual function in promoting aggregate growth and in limiting the aggregate size. However, in natural water systems,
Chapter | 4 The Hydrodynamic Transport Processes
107
and more specifically in open channels and estuaries, the flow is not only turbulent, but also highly nonuniform in the time-average sense, with the time-average velocities varying from a maximum very near the top to zero at the bottom. The following two questions arise: (1) Which parameter or parameters are representative of the turbulence-generated shear rates that control the size and the relevant physical properties of the aggregates? (2) Because the time-average velocity is nonuniform, which zone of the flow field is the most crucial in the formation of the aggregate properties? Many researchers have investigated these questions experimentally and analytically, and a number of quantitative equations have been obtained. This section discusses the development of the equations in the light of the fundamental properties of turbulent flows. Although the reader is expected to have some familiarity with the dynamics of turbulent flows, this section will briefly review some of their fundamental properties related to cohesive sediment dynamics. For a more detailed study, interested readers are referred to Hinze [51] and Tennekes and Lumley [139]. A good summary is also given in Chapter 1 and Chapter 3 by Levich [74]. It is well known that at a certain critical value of the Reynolds number, a steady laminar flow changes into a highly irregular motion, which at first glance appears to be totally chaotic. The motion of any small mass of fluid as well as other flow properties appears to be random and unpredictable so that any description of turbulent flow should be approached from a statistical viewpoint. This means that any property averaged over a sufficiently long time should give the same value. Thus, in turbulent flows, we speak of time-averaged rather than instantaneous values of the various parameters. This review is limited to the case of fully developed turbulence. According to Section 4.2.2, the first two questions are: (1) How can we determine a representative value of the shear rate G? and (2) How can we describe the turbulence-induced shear stresses that will limit aggregate growth? Both controlling parameters have to be expressed in terms of some readily determinable gross flow variable or variables if they are to be of practical use. It will be shown that the rate of local energy dissipation per unit mass, , can be related to both G and the controlling shear stresses. So, before proceeding any further, we will have a look into the ways energy is distributed and lost through friction in fluid flows. The internal energy in fluids has been studied by some of the pioneers of fluid mechanics, such as Lamb [66] and G. I. Taylor [138]. However, Bakhmeteff and Allan were the first to present a practical description and a mechanistic model of the energy process [4]. In any conduit with a discharge Q, energy is lost in overcoming frictional resistance at a rate Eo per unit length and time given by
Eo SeQ
dp Q dx
(4.45)
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108
where is the unit weight of the fluid and Se is the slope of the energy grade line, which is equal to the total loss of head p / z o v (V 2 / 2 g ), per unit length and time and per unit weight of flowing fluid and v is the velocity head correction coefficient. For the simple case of an open channel with steady uniform flow, Se is equal to the bottom or water surface slope, which is equal to the elevation change of either the bottom or the water surface divided by the length of the channel segment considered, l. That is, because the velocity does not change, potential energy is withdrawn from the flow and spent to overcome the frictional resistance of the channel and eventually dissipated into heat. However, the withdrawal of energy and its final dissipation do not coincide spatially because, as is well known, the energy dissipation into heat takes place predominantly near the solid boundaries. Therefore, a process must exist through which energy is borrowed from the flow and transferred to the boundary to overcome the resistance and to be dissipated into heat. Indicating by (Eb)0 the rate at which energy is being borrowed from the entire flow, the following relationship exists:
E0 ( Eb ) 0 ( E s ) 0
(4.46)
where (Es)0 is the total rate of energy loss due to friction per unit length and unit time. Equation 4.46, however, is valid for the entire flow field. Local values of borrowed and spent energy per unit volume and unit time, indicated by b and s , respectively, are not in general equal and not in balance with each other. In fact, the distributions of the local borrowing and spending rates follow quite different and, in a sense, opposite trends. Specifically, the borrowing rate starts with a maximum value at the surface and decreases toward zero at the bottom, while the spending rate starts from zero at the surface and reaches its highest value near the bottom. This trend is clearly displayed in Figure 4.5, taken from Ref. [4]. Following the analysis of Bakhmeteff and Allan [4], the local rate at which energy is withdrawn (borrowed) per unit volume from the flow is
b u
d dy
(4.47)
and the local rate of energy spending per unit volume is
s
du dy
(4.48)
which for laminar flow becomes
du 2 b dy
(4.49)
Chapter | 4 The Hydrodynamic Transport Processes
109
According to the preceding two equations, energy spending is concentrated primarily near the walls of any conduit, where the velocity gradients attain their highest values. Next, according to Equation 4.47, the borrowing of energy is highest near the surface of the channel, where the velocity u has its highest value, while the y y d/dy remains constant. The cumulative values Eb o and Es 0 of the borrowed and spent energies, respectively, to a depth y’ below the free surface in an open channel are given by Equation 4.50 and Equation 4.51 and are shown in Figure 4.5:
y 0
y
∫0
y
y
∫0
du
(4.51)
um 1.5 U a
y 0
B
τ
0
a1
o
u
1.5 E0 E0
01
y
y τ f(y )
y 0
Axis or surface of two dimensional streaming 0
y
U
m
s1
C (a)
τ0
T1
u f(y )
m1
A 0
1 u
Values of
2
1
S1
2
E Values of E0
(b)
U
m
Es
0
1
O
b1 M
Et
3
B1
t1
Eb Wall
(4.50)
0
Es Unit length
y
b dy Se ∫ udy Se q 0
y M 0.577 y 0 y 0 �1/3
Eb
0
y
E0
0.4
b2 t2 y’0
s2
a2 Values of y /y0
y M 0.577 y 0
0.2
y M 0.6
Es
Et
Eb
0.8
1.0
A2 0
S2 B 2 0.2
0.4
0.6
0.8
1.0
y
(c)
E 0 Values of E0
Figure 4.5 Schematic variation of energies, velocity, and shear stress in an open channel [4].
3
Cohesive Sediments in Open Channels
110
where q is the discharge per unit width through that part of the cross-section from the surface to a distance y. Of course, for the total depth of flow, the two enery gies become equal. In Figure 4.5, there is a point M at a depth yM above which the rate of borrowed energy, b, exceeds that of the rate, , s at which energy is spent; whereas below it, and particularly near the wall, s exceeds by far b. It follows that there must be some mechanism of transmittance of energy from the upper region to the lower one represented by a transmittance function, t . The local energy balance then becomes b s t
(4.52)
Obviously, the corresponding cumulative quantities satisfy the following equation:
Et
y 0
Es
y 0
Eb
y 0
(4.53)
Of the three local energies, b and s are given already by Equation 4.47 and Equation 4.48, respectively. t was shown to be
t
d d ( u ) ( u) dy dy
(4.54)
These three forms of energy satisfy the obvious equation: u
d du d ( u ) dy dy dy
(4.55)
Equation 4.47, Equation 4.48 and Equation 4.55 for the three local energies are valid for both laminar and turbulent flows because no restrictions were imposed on the agency that generates the shear stresses and the velocity distribution. The local energies can readily be derived for laminar flow in open channels and circular pipes. Bakhmeteff and Allan [4] derived such equations for the latter case. For a turbulent flow in a uniform open two-dimensional channel, the local velocities and shear stresses enter into the preceding equations by their timeaveraged local values, u and , respectively. The latter variable is known as the Reynolds stress, being equal to
u.
(4.56)
where u and v are the instantaneous turbulent velocity components in the x and y directions, respectively. Equation 4.55 then takes the time-average form
s b t u
d d u du ( ) dy dy dy
(4.57)
Chapter | 4 The Hydrodynamic Transport Processes
111
One of the characteristic aspects of turbulence is the high concentration of energy spending near the wall. Thus, while for laminar flow in an open channel, the M point (Figure 4.5) is located at a distance of yM 0.577 y0 , where yo is the depth of flow, below the surface and in circular pipes at a distance rM 0.293r0 from the wall, in turbulent flows in circular pipes the corresponding distance is 0.053r0. The energy spent within the zone between the wall and rM amounts to about 83% of the total energy E0, while the borrowed energy within the same region is about only 8% of E0. The process by which the energy transmittance takes place is the next important question. In laminar flows, the transmittance of momentum and energy takes place by viscous action. In contrast, turbulent motion is characterized by a cascade of eddies of various magnitudes ranging from the largest size on the order of the size of the conduit to the smallest one determined by the viscosity of the fluid. These eddies are associated with velocities of extremely varied magnitudes, which are superposed on the time-averaged local velocity, u. The distance over which the eddy velocities change significantly is known as the eddy size or the scale of eddy motion. Thus, scale and velocity are the two defining properties of the turbulent eddies. Of the entire spectrum of eddies, the largest have a size on the order of the conduit size. Therefore, their corresponding eddy velocity will be of the order
u ≈ U
(4.58)
where U is the change of the average velocity over the distance . In a circular tube, for example, the largest scale is on the order of the radius of the tube. Likewise, the eddy velocities are on the order of the difference of the average velocities within ; that is, they are on the order of the maximum velocity, Vmax, at the center of the tube. The Reynolds number for these largest eddies, U./, is of the same order as that for the total flow. The latter is much larger than unity; therefore, the large eddies are not affected significantly by viscosity, and their motion does not involve significant energy dissipation. On the other end of the eddy spectrum, we have eddies of a size small enough for the viscous forces to have a noticeable effect on the fluid motion. If we designate the scale of these smallest eddies by 0 and the corresponding velocity by v0, then their Reynolds number should be on the order of unity if the inertia and viscous forces are to be of the same order of magnitude. That is, Re0 (v 00 / ) 1 . Therefore, the motion of these eddies is accompanied by energy dissipation. Between the largest eddies of a scale and the smallest of scale 0, there are many eddies of scales smaller than and of smaller velocities v. The larger eddies draw energy from the total flow, and the smaller in-between eddies transfer gradually and continually that energy to the smallest eddies, where the energy is ultimately dissipated into heat by friction. Thus, the motion of the large eddies constitutes the energy-borrowing mechanism. The borrowed energy
Cohesive Sediments in Open Channels
112
is subsequently transferred by the intermediate eddies, acting as the tranfer agency, to the smallest eddies, where the largest portion of the borrowed energy is dissipated into heat. The large-scale eddies are transferring energy and momentum from zones of the flow field, where these quantities have high values, to near the wall regions of low energy and momentum, where energy is dissipated into heat through viscous shear stresses. The wide spectrum of eddies of various scales and velocities is generated by the nonlinear terms of the equations of motion. Energy is being transferred from larger to smaller scale eddies continuously with relatively small energy dissipation until they reach their lowest limit. In the absence of viscosity, the eddy scale could become infinitesimal. The viscous stresses, however, as they appear in the equations of motion, limit any further reduction of eddy scales to a minimum size. The small-scale eddies are related to the fluid viscosity and to the energy dissipation. They are expected, therefore, to be independent of the relatively slow large-scale turbulent eddies, and their governing parameters should be functions of the rate of the local energy dissipation and of the kinematic viscosity. On this basis Kolmogorov established his universal equilibrium theory of the small-scale structure and, using dimensional analysis, derived the three microscales of turbulence: one for the length (0), another for the time (), and a third for the velocity (). A discussion of Kolmogorov’s theory can be found in Chapter 1 of Ref. [51] and Chapter 15 of Ref. [139]. The three microscales of turbulence are as follows: 3 1 / 4 0
(4.59)
(4.60)
1 / 2
()1 / 4
(4.61)
The Reynolds number corresponding to the three microscales is / 1 , as expected. is the rate of energy dissipation per unit mass with dimensions (L2/T3), while the rate of energy dissipation per unit volume was indicated by with dimensions (F/L2T). is given by [139]:
u3
(4.62)
The following relationships exist between the microscales and the macroscales:
u 3 / 4 0 Re3 / 4
(4.63)
Chapter | 4 The Hydrodynamic Transport Processes
113
(4.64)
u 1 / 2 u Re1 / 2 t
(4.65)
u 1 / 4 Re1 / 4 u
Considering that for turbulent flows the Reynolds number is much higher than unity, one can conclude that the order of magnitude of the microscales is also much smaller than that of the macroscales. Moreover, according to Equation 4.59 and Equation 4.60, the length and time microscales diminish with increasing rates of energy dissipation, whereas the velocity scale increases with . Considering that the shear stresses within the small eddies are on the order of /0, it follows that the smaller the scale of the smaller eddies, the higher will be the corresponding shear rates. This aspect is of particular importance in the formation of flocs and floc aggregates in a turbulent flow. In the large-scale eddies, the velocity u in Equation 4.58 is on the order of the velocity change over a distance equal to the eddy scale ; that is, u u
∂u ∂y
(4.66)
and the corresponding rate of energy dissipation is [139]
(u)3
(4.67)
Eddy scales 0 belong to the so-called inertia subrange, while those with scales 0 belong to the viscous subrange. In practically all flow fields, either natural or manmade, the flow parameters vary significantly with location from zero to maximum values; as a result, the scales and other properties of the small eddies, which control flocculation and the general behavior of sediments, will also vary widely. It follows that the time-averaged velocity distribution in any real conduit is of primary importance in determining the previously mentioned crucial parameters for flocculation. In pipes and open channels, the velocity distribution is given by the well-established logarithmic law:
u y 1 u y u ln ∗ 5.5 2.5 ln ∗ 5.5 u∗
(4.68a)
In Equation 4.69a, is Karman’s constant, approximately equal to 0.4, and u* is the so-called friction velocity defined as u∗
where is the fluid density.
(4.68b)
Cohesive Sediments in Open Channels
114
According to Equation 4.49, it is expected that by far most of the borrowed energy will be dissipated in the immediate vicinity of the bed, where the velocity gradients attain their highest values. A bed formed by deposition of fine sediments, which is normally the case in estuaries and other conduits with cohesive boundaries and in which fine sediment transport and deposition take place, can be reasonably assumed as smooth. In every smooth boundary, there is a zone of thickness, t, within which the dominant stresses and the velocity distribution are controlled by viscosity rather than by turbulent velocity fluctuations. That zone is known as the viscous sublayer, and its thickness is given by the following equation:
t
4 u∗
(4.69)
The zone above the viscous sublayer, for which Equation 4.68 is valid, is known as the inertia sublayer. One should note that the preceding value of t should be considered as an estimate. The dimensional relation is the one of primary importance. Within the viscous sublayer, the velocity distribution is linear, given by the following equation:
yu u ∗ u∗
(4.70)
The linear Equation 4.70 indicates that the velocity is proportional to the distance from the wall. However, turbulence is not absent. Turbulent eddies do penetrate inside the viscous sublayer, transferring energy from the outer or inertial sublayer, reaching the solid boundary instantaneously. These imposed fluctuations, though, are rapidly suppressed by the viscous stresses within the viscous sublayer so that the time-averaged velocity within the latter follows the law given by Equation 4.70. This aspect will be discussed again in greater detail together with the process of flocculation. A treatment of the viscous sublayer is given in Refs. [28, 51]. The turbulence energy production rate in the inertia sublayer is given by [139]
i u ′v ′
u3 ∂u ∗ y ∂y
(4.71)
where the time-averaged velocity gradient was taken from Equation 4.68. That means that the local turbulent energy is inversely proportional to the distance from the wall. Within the viscous sublayer, the local rate of energy dissipation per unit mass, v, is given by Equation 4.48, which, combined with Equation 4.70, becomes
du 2 u4 v ∗ dy
(4.72)
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115
The ratio v/i is on the order of the frictional Reynolds number, Re* u*y/v, which for y t becomes much larger than unity. In a very wide open channel, the average value of energy dissipation is given by av gVSe
(4.73)
where V is the average over the cross-section velocity and Se is the energy grade line. The friction velocity for the same channel is given by u∗ =
gy0 Se
(4.74)
where yo is the depth of flow. In terms of Manning’s coefficient, n, av obtains the following form: av
y10 / 6 u∗3 u∗3 . = n g y0 n gy05 / 6
(4.75)
It follows that
v u n gy05 / 6 ∗ av
(4.76)
This last ratio is on the order of the frictional Reynolds number for the entire depth and, therefore, it is much larger than unity. Consider now a channel with yo 2.00 meters, V 0.50 m/s, and n 0.012. The corresponding value of Se is 1.420 105, the bed shear stress is b gyoSe 0.28 Nm2, and the friction velocity is u* 0.0167 ms1. Equation 4.75 and Equation 4.76 then give av 0.0000701 m2s3, v 0.076 m2/s3, and v / av 1,120 . The corresponding microscales of turbulence, o, are 350 m and 60 m, respectively. This example gives an idea of the difference on the order of energy dissipation and of microscales of turbulence in an open channel. Both of these variables will be shown to be of primary importance to the process of flocculation and to the properties of the aggregates formed in a turbulent flow field. Section 4.2.2 showed that the velocity gradients, G, constitute the dominant mechanism of interaggregate collision in the vast majority of real flow cases. Equation 4.30 through Equation 4.34 were valid for a uniform shear field. Smoluchowski extended the validity of these equations to any flow, which can be described locally as a laminar flow with a velocity gradient, G, by postulating that the aggregate diameter, da, is small enough for the velocity gradient to be considered as linear over a distance da [74, Chapter 4]. According to the previous example, the order of magnitude of the microscale of turbulence varies between 50 and 1000 m; experimental studies revealed the cohesive sediment aggregates range in size from 5 to 2000 m. Since the rate of energy dissipation
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116
in both the inertia and the viscous subrange depends on the velocity scale of the eddies, it is logical to use the rate of energy dissipation as a measure of the governing velocity gradients or shear rates. We can define the shear rate, G, as follows: G=
=
(4.77)
This definition was originally introduced by Camp and Stein in 1943 and presented by Camp in 1955 [9]. Since then, it has been adopted as a design parameter for flocculation chambers in water and sewage purification systems with and interpreted as the total energy dissipation divided by the mass or the volume of the flocculation chamber, respectively. The same parameter, G, was used by Kusuda et al. in 1981 [65] as a measure of flow-induced shear stresses. The preceding definition of G presupposes a uniform turbulent flow field. Flocculation chambers, however, operate with rotating pads, thus generating a highly nonuniform flow pattern. The same is true for many laboratory experiments using rotating paddles as a mixing mechanism, such as in the Kusuda et al. experiments and in many others to be discussed later. To account for this nonuniformity of the flow, Levich [74] introduced an effective energy dissipation rate, 0 and 0 o , of about 10% to 20% of the mean value as representative of the eddies far from the blades. This, however, should be considered as just an empirical approximation.
4.3.2 Collision Rates in Turbulent Flows As stated earlier, in almost all practical cases, the shear rates are the primary and dominant agents of flocculation while at the same time limit the maximum aggregate size by the shear stresses imposed on the aggregate surfaces as well as within their mass. The same stresses mold the internal structure of the aggregates, thus determining not only their sizes but also their density and strength. The first of these last two properties affects the settling rates, and the second will be shown to control the degree and rates of erosion, deposition, and resuspension. We limit the discussion of flocculation here to that due only to velocity gradients. In Section 4.2.2, Equation 4.34 and Equation 4.32, for the frequency of collisions of units with two different diameters and equal diameters, respectively, have been derived in terms of the velocity gradient or shear rate G. Introducing Equation 4.77 as a measure of G, these equations obtain the form
N ijt
4 o ( Ri R j )3 ni n j 3
(4.78)
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117
and N turb
4 o 3 2 da no 3
(4.79)
Equation 4.79 was further modified by Levich on the basis of the concept of locally isotropic turbulence and on the assumption of viscous subrange diffusion control [74]. For particles of uniform diameter, da, the Levich equation is N turb 12
o 3 2 da no
(4.80)
where n indicates the number of aggregates of diameter da per unit volume and is an experimental coefficient. One must remember that the effective rate of energy dissipation, o, was taken equal to 10% to 20% of the actual average in a paddle flocculator. Levich’s Equation 4.80 was based on the following assumptions: (a) that the particle radius is much smaller than the microscale of turbulence; and (b) that there is complete containment of colloidal particles within the turbulent eddies and that the mechanism of contact occurs on a scale smaller than the size of the eddies. From the previously cited example, one can state that in open channels and estuaries the microscale is expected to be on the order of a few hundred micrometers in the outer region and a few tens of micrometers in the viscous sublayer. Because the shear rates within the latter sublayer determine the critical size and other properties of the aggregates that control erosion, deposition, and resuspension, one can conclude that the microscale within the viscous sublayer is of primary importance. Experimental studies revealed aggregate sizes ranging from 5 to 2000 m; therefore, aggregates of sizes larger than the microscale of turbulence can be present in the viscous sublayer. Because of these arguments and in view of the uncertainty in determining the effective , Equation 4.80 should be regarded only as a functional relationship with the constant 12 taken as a numerical factor. Nevertheless, the foregoing analysis is along the lines used by Tambo and Watanabe [135], and Equation 4.80 will be used for the collision frequency with 1/ 15 , as determined experimentally by Fuchs [34]. If, in an enclosed flow system, we start with suspended primary particles of even constant size, the process of flocculation will soon result in an aggregate population of a wide range of sizes. This brings into question the problem of the floc growth process and related equation. First, Equation 4.80 has to be modified to account for collisions between i-fold and j-fold aggregates with diameters di and dj, respectively. An i-fold aggregate is defined as one composed of i primary particles. Then, according to Equation 4.32 and under the assumption of constant aggregate density, Equation 4.80 takes the form N tij
12 o
di d j 3 1/ 3 1/ 3 3 2 ni n j A (i j ) ni n j
(4.81)
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where A
3 2 15
o 3 d1 (cm 3s1 )
(4.82)
and d1 is the diameter of the primary particles. The developed equations describe the rate of collision of two particles with the same size or with two different diameters. Once the aggregation process starts, the size distribution of aggregates changes continuously, because collisions lead to the generation of larger aggregates. This growth eventually stops by the flowinduced shear stresses on the aggregates. The density and the shear strength of the latter diminish rapidly with an increasing order of aggregation. In addition, not every interaggregate collision necessarily leads to a larger aggregate. It is expected, therefore, that in a turbulent flow field a quasi steady-state aggregate size distribution will eventually be reached. The picture becomes even more complicated in a nonuniform flow with time averaged and turbulence parameters varying widely. It becomes apparent from the preceding considerations that, for the formulation of any equation describing the aggregate growth and its ultimate size distribution, reliable equations are needed governing the size-density and size-strength relationships as well as a measure of the flow-induced forces on the aggregates. This is the objective of the following section.
4.4 T he properties of the aggregates and the aggregate growth equation 4.4.1 T he Properties of Aggregates and Their Relation to the Controlling Flow Variables The ultimate properties of the aggregates, and specifically the density, size distribution, and settling velocities, will be controlled by the flow-induced shear stresses. These stresses develop as a result of different instantaneous velocities at two extreme points of an aggregate, as shown in Figure 4.6.
�1�∆�
ur2 � u2�u0
da
u0
ur1 � u1�u0
∆r �1
Figure 4.6 Stress generation in an aggregate by differential velocities.
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119
The preceding stresses are resisted by the internal strength of the aggregate, which is a function of its density and the interparticle and/or interaggregate forces, which in turn depend on the mineralogy and the properties of the claywater system outlined in Chapter 2. The first experimental studies on the strength of flocs and aggregates were conducted by Krone [62]. A laminar flow field was generated in a space between two concentric cylinders with the gap much smaller than their average diameter. The outer cylinder was fixed, and the inner one was rotating at various speeds. A near uniform shear rate, G ∂u / ∂r , was thus developed with a constant shear stress G ∂u / ∂r . The following equation was derived:
ras
16r as 3
(4.83)
in which as is the aggregate strength, defined as the shear at the floc surface at which failure occurs; ras is the radius of the largest aggregate; and r is the depth of the surface roughness [62, Appendix A]. For an assumed value r about equal to 2 m, that is, the order of an individual clay particle, as was found to be equal to 2.7 dynes/cm2 or 0.27 Nm2. In a turbulent flow the controlling breakup force is expected to be generated by the instantaneous velocities of the turbulent eddies acting at two extreme points of the aggregate one diameter apart; therefore, they should be expressed in terms of the local rate of energy dissipation, . According to the analysis of Tambo and Hozumi [133], the difference of the acting force, f, per unit area of aggregate induced by the difference of the turbulent fluctuating velocities over a distance equal to the aggregate diameter can be expressed as
f
v1 v2 2
2
(4.84)
where v1 and v2 are the turbulent fluctuating velocities of the water at two points separated by a distance equal to the aggregate diameter da, and is the water density. Next, according to Equation 4.67 for the inertia subrange, da , and to Equation 4.72 for the viscous subrange, the difference of the instantaneous turbulent velocities over a distance y da can be represented by the following equations:
d 1 / 3 vda (o da )1 / 3 o a for da o (inertiia subrange) (4.85)
o (viscous subrange) (4.86) vda o da o da for da
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where the microscale of turbulence, o, is given by Equation 4.59 and where and are numerical constants, which, according to Fuchs [34] and reported by Tambo and Hozumi, are equal to 1 and 1 / 15 , respectively. Introduction of the preceding values for into Equation 4.84 leads to the following results for the force, F, over the aggregate area, which is proportional to da2 , and gives 2/3 8/3 1/ 3 2/3 8/3 nge) (4.87) F o da ( ) da for da o (inertia subran
F o da4 o
4 da for da o (visccous subrange)
(4.88)
where o ο . An aggregate will break if the breakup force is at least equal to the binding force. The latter is proportional to the net sectional area, An, of the aggregate at the plane of rupture and to the average strength, , representative of the interparticle cohesive forces. That is, B An
(4.89)
with B in dynes, An in cm2, and in dynes/cm2. It should be pointed out that this differs from Krone’s aggregate strength, as, in Equation 4.84 because the latter is an average figure over the gross area of the aggregate, whereas represents the real cohesive strength among the constituent particles. Indicating the porosity of the aggregates by e, that is, the ratio of the volume of its voids divided by the total aggregate volume, then the net volume of solid particles within the aggregate will be Vn da3 (1 e)
(4.90)
and, because An (Vn)2/3, the total area of solid contacts within the plane of rupture will be
An da2 (1 e)2 / 3
(4.91)
B da2 (1 e)2 / 3
(4.92)
and
The evaluation of the porosity requires knowledge of the aggregate density. This aggregate property is complicated by the fact that it changes drastically with aggregate size. More specifically, the higher the order of the aggregate, the smaller is its density and, consequently, also the smaller is its gross strength, as defined in Equation 4.83. Indeed, as shown in Figure 4.7, there is a large and
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121
Particle aggregate, pa
Particle aggregate aggregate, paa
Particle aggregate aggregate aggregate, paaa Figure 4.7 Two-dimensional representation of aggregates of various orders [61, Fig. 37].
discontinuous increase in the percentage of voids as we proceed from the original floc to floc aggregates of the first order, and then to the second order and so on. For the same reason, the larger the aggregate diameter, the fewer the points of contact per unit area over a potential plane of failure and the lower the aggregate strength. Krone conducted experimental studies of the rheological characteristics of cohesive sediment suspensions presented in Chapter 5 using both rotating cylinder and capillary viscometer [61]. These studies gave values of densities and strengths for a number of cohesive sediments from five estuaries of the United States and for four orders of aggregation. In his sequence of orders, zero order represents the original floc composed entirely of single particles; the number one aggregate is the earlier defined floc aggregate, and it is composed of flocs and possibly some individual particles; the order two aggregate is composed predominantly of first-order aggregates and so on. The densities and strengths of the indicated sediments and for a water density 1.025 gcm3 are presented in Table 4.2. It is observed that the densities decrease rapidly with increasing order of aggregation approaching the water density. The effect of the order becomes more striking for the submerged densities, particularly because the latter controls the settling velocities. A similar rapid decrease of the shear strength with increasing
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Table 4.2 Properties of suspended cohesive sediments [61] Sediment Sample
Order of Aggregation
Density (gr/cm3)
Submerged Density (gm/cm3)
Shear Strength (dyn/cm2)
Wilmington District
0 1 2 3
1.250 1.132 1.093 1.074
0.225 0.107 0.068 0.049
21 9.4 2.6 1.2
Brunswick Harbor
0 1 2 3
1.164 1.090 1.067 1.056
0.139 0.065 0.042 0.031
34 4.1 1.2 0.62
San Francisco Bay
0 1 2 3 4 5 6
1.269 1.179 1.137 1.113 1.098 1.087 1.079
0.244 0.154 0.112 0.088 0.073 0.062 0.054
22 3.9 1.4 1.4 0.82 0.36 0.20
White River (salt)
0 1 2 3
1.212 1.109 1.079 1.065
0.187 0.084 0.054 0.040
49 6.8 4.7 1.9
order of aggregation is observed, particularly between the zeroth and the firstorder aggregates. The conversion factor to Nm2 is Nm2 Pa 10 dynescm2, where dynecm2 0.1Pa. One must remember that the shear strength is the gross one as defined by Equation 4.83. Tambo and Watanabe investigated experimentally the aggregate density of clay aluminum sediment [134]. Their experimental setup consisted of a flocculation chamber with rotating paddles, at the bottom of which was a long tube with a floc trap door, allowing the passage of only a few aggregates at a time. The time of settling and the size of each unit were observed and recorded, and with a suitable settling equation, the submerged density was estimated. For an average observed sphericity of 0.8, the settling velocity becomes
ws
g (a )da2 34
(4.93)
In this way a direct density estimate was achieved. The effect of the coagulant dosage on the aggregate density for the neutral PH range was also investigated. This dosage was represented by the ALT ratio defined as the ratio of aluminum ion concentration divided by the suspended particle concentration. The kaolinite
Chapter | 4 The Hydrodynamic Transport Processes
ALT 1:60 rb
Floc effective density, rb, 10 3g cm 3
100 80
ALT 1:12 rb
60
123
0.0012 da1.05 0.00030 da1.40
40 30 20
ALT 1:60 ALT 1:12
10 8 6 4 3 2
1
2
3
4 5
8 10
20
30
Floc diameter, da 10 2 cm
Figure 4.8 Example of buoyant density-aggregate diameter relations [134, Fig. 4].
particles were suspended in distilled water with 50 mL1 NaHCO3 as alkalinity. Aluminum sulfate (Alum) or polymerized aluminum chloride (PAC) was used as a coagulant with a dosage of 0.5–100 gL1 as aluminum. Figure 4.8 shows the test results for ALT ratios 1:60 and 1:12 in a log-log plot with the buoyant or effective density of the aggregate as an ordinate. The data fall approximately on a straight line, suggesting a power law. Similar plots were obtained for several ALT ratios. From this plot, as well from all others, the following relationship, defined as the aggregate density function, between the aggregate diameter and the buoyant or effective density, b , is derived: b a
K
da
(4.94)
The density is expressed in gcm3, the diameter da in cm, and and Kr are constants depending on the ALT ratio. One should note that for the aggregate diameter in cm, also must have dimensions in gcm2. Figure 4.9 shows the relationships of and K with the ALT ratio. The results appear to be quite consistent.
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124
ALT ratio
10�1
ALT ratio
10�1
10�2
10�2 0
2
4
6
8 10 12 14
α � 10
�4
0.8
1.0
1.2
1.4
K�
Figure 4.9 Relationships between the ALT ratio and constants in the aggregate density function 4.94 [134, Fig. 8].
This aggregate density function is continuous in contrast to Krone’s model, which gives densities as a step function on the order of aggregation. Therefore, Equation 4.94 does not represent the real process of development of aggregate properties. Nevertheless, it gives a good fundamental theory for the rapid decrease of density with increasing aggregate size. From Figure 4.8, one can see that for an ALT ratio of 1:12, the effective density decreases from about 0.065 gcm3 for a 0.02 cm or 200 m aggregate to about 0.0025 gcm3 for a 0.2 cm or 2000 m aggregate. For an ALT ratio of 1:60, the densities for the same diameters are 0.070 gcm3 and 0.0065 gcm3, respectively. This clearly shows the effect of coagulant concentration on the aggregate density and, therefore, on their strength; this effect becomes higher as the aggregate diameter increases. There is a difference between Tambo and Watanabe’s experiments and Krone’s investigations. In the first case, each experiment started with dispersed sediment in the same state, whereas in the second, flocs were already formed and higher order aggregates developed as the experiment progressed. The latter approach was also followed in the studies by Kusuda et al. [65], Burban et al. [8], and Tsai et al. [147] to be discussed in Section 4.4.2. This difference in experimental approach was due to the objective of each study. Tambo and Watanabe were interested in settling rates in water and sewage treatment plants, whereas Krone had in mind flocculation and sedimentation processes in estuaries, where cohesive sediments are continuously subjected to variable stresses. Nevertheless, the rapid reduction of density with aggregate size in Tambo and Watanabe’s experiments does suggest some ordering of aggregation, although there is no clear distinction among orders. Returning now to the subject of aggregate strength and maximum aggregate size, the critical condition will be reached when the breakup force, expressed by Equation 4.87 and Equation 4.88, will be equal to the binding force, B, as represented by
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125
Equation 4.92. The latter involves the porosity e, which in turn is a function of the aggregate density; therefore, it can be expressed in terms of the aggregate density function, i.e., Equation 4.94. Indicating the aggregate volume by Va; the volume of solids within the aggregate by Vs; the interaggregate volume of water by Vw; and the densities of the aggregate, water, and solids by a, , and s, respectively, the equations of volume and mass equality are
Va Vs Vw
(4.95)
aVa sVs Vw
(4.96)
The last two equations together with Equation 4.94 give 1 e
a K s da (s )
(4.97)
Equation 4.92 for the binding force B takes the form 2 (1− K / 3)
B 1da
where 1
2 / 3 ( s )2 / 3
(4.98)
The parameters , 1, s, and K are constants for given sediment properties and water quality. Equating the last form of B with F, as given by Equation 4.87 and Equation 4.88, leads to the following equations, which include some suitable coefficients: d max
(4.99)
d max
3 / 2(1K ) (3 /(1K )) 3 / 2(1K ) (1 /(1K )) 3 3 1 o o o 3
3 / 2 ( 3K )
3 / 2(3K ) 4 o
3 / 2 (3K ) 3 / 2(3K ) 4 o o′
(4.100)
In Equation 4.99 and Equation 4.100, the coefficients of viscosity and are, for all practical purposes, constant. The parameter , which is included in the coefficients 3 and 4, and K, in the aggregate density function, depend on the type of clay and on the water quality, specifically on the type and concentration of the coagulant. Therefore, for the same type of sediment and the same water quality, all these variables can be considered constant and represented by a
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126
single factor. In this case Equation 4.99 and Equation 4.100 can be written in the simpler form (1 /(1K ))
d max 5o
3 / 2 (1K )
d max 6o
5 o 6 o
(1 /(1K ))
for o
3 / 2 (1K )
for o
(4.101) (4.102)
where
3 / 2(1K ) 3 / 2(1K ) 5 3 , 5 3 1/3
3 / 2(1K ) 3 / 2(1K ) 6 4 , 6 4
(4.103)
(4.104)
The preceding four equations give the maximum aggregate size for the two extreme cases of o and o. Nothing is said about the intermediate range for which, at present time at least, one has to make an interpolated guess. However, the situation is simpler than it appears if we look at some of Tambo and Watanabe’s experiments. These experiments were conducted in paddle flocculators where rotating paddles generated a quasi-uniform turbulent field. According to the same authors, the value of K in the aggregate density relation (Equation 4.94) ranges between 1.0 and 1.5. The effective rate of energy dissipation is proportional to the third power of the rate of rotation, Nr, of the blades of the flocculator. Therefore, Equation 4.101 and Equation 4.103 can be written in the simpler approximate form
( 0.50 → 0.4 ) dmax 5 ≈ N r(1.5→1.4 ) for 0 o
(4.105)
( 0.38 → 0.33) dmax 6 ≈ N r(1.1→1.0 ) for 0 o
(4.106)
Thus, the exponent of o ranges from 1.0 to 1.5 for both cases; therefore, for any intermediate range of , some average value of K would seem reasonable. Additional research by Tambo and Hozumi on contact flocculation provided more details on the agglomeration process [132]. Contact flocculation is defined as the process by which larger and well-grown aggregates, designated as grown flocs, adsorb minute individual fine sediment particles or original zero order flocs designated as microflocs. This process is sometimes used in sanitary engineering in solid contact clarifiers and in conventional flocculators. The grown
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127
flocs are circulated at high concentration in a turbulent flow field with incoming microflocs. Collisions between these two groups of aggregates result in adsorption of the second by the first and a gradual clarification of the water. In the modeling of contact flocculation, Equation 4.81 and Equation 4.82 have been modified for two aggregate diameters, da for the grown flocs and df for the microflocs, with corresponding concentrations na and nf, so that the collision frequency equation becomes N taf
12 15
o
3 d a d f n n a f 2 2
(4.107)
Next, considering that df da and on the basis of the assumptions stated in [132], the preceding equation can be reduced to the form N taf
dn f dt
3 2 15
P
o 9 na n f da3 P o Va n f 15
(4.108)
in which P is the mean collision agglomeration factor, that is, the fraction of collisions that result in an adsorption, and Va is the grown floc volume assumed as spherical with diameter da, i.e., Va (/6)nad3a. Tambo and Hozumi’s experiments indicated an average value of P of about equal to 1/45, in which case Equation 4.108 takes the form dna K c1na dt
(4.109)
where
K c1
9 15
P
o Va
(4.110)
The actual total rate of energy dissipation, o, was evaluated from torque measurements of the rotating blades, according to the equation
o
To g c VT
(4.111)
in which To is the torque; VT is the water volume of the flocculator tank, both in CGS units; gc is the value of the gravitational acceleration, that is 981; and is the angular velocity of blade rotation equal to 2Nr, where Nr is the rate of rotation in rps. The value of o was estimated from experiments on the basis of Equation 4.109 and Equation 4.110 and the total energy dissipation was determined from Equation 4.111. From these measurements the relationship between
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128
and the effective o (0.1–0.2), reported earlier and used in the subsequent studies, was established.
4.4.2 Quasi Steady-State Aggregate Distribution and Maximum Aggregate Size Returning now to Section 4.3.2, we proceed to introduce the aggregate density function (Equation 4.94) into Equation 4.81. Assuming all aggregates to have a spherical shape and considering that by definition an i order aggregate consists of i primary fine particles, then Equation 4.95 becomes
3 di d13i Vw 6 6
(4.112)
where di is the diameter of an i-fold aggregate and Vw is the water volume in the aggregate in cm3. Introducing next the buoyant density of the aggregate, bi i , into Equation 4.96, the following equation is obtained:
3 di (bi ) d13i(b1 ) Vw 6 6
(4.113)
After we multiply Equation 4.112 by , the combination of Equation 4.112 and Equation 4.113 yields di3 bi id13 b1
(4.114)
Introducing the aggregate density function (Equation 4.94) for the two submerged densities in Equation 4.114, we obtain 1 /(3 K )
di i
d1
(4.115)
which directly relates the particular aggregate diameter to the number of primary particles it contains. We return next to Equation 4.81 for the collision frequency between i and j order particles, and we substitute the diameters di and dj using Equation 4.115. The resulting equation is
(
1/ (3 K )
N tij A i
j
) nn
1/ (3 K ) 3
i j
(4.116)
with A given by Equation 4.82. Equation 4.107 and Equation 4.108 are special cases of Equation 4.81 with the addition of the mean collision agglomeration factor P. The rate of generation of aggregates of a particular order, R, can be derived from Equation 4.116 under the following considerations. The number of R order
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129
aggregates is increased by every effective collision of i order particles with j order ones provided that i j R, and it is decreased by every effective collision of R order aggregates. The resulting equation is 3
dnR 1 R 1 A ∑ i1 /(3K ) (R i )1 /(3K ) ni nR i dt 2 i 1 ∞
(
A∑ i i 1
3
1 /(3K )
R
1 /(3K )
) ni nR
(4.117)
where n is the number of R order aggregates per unit volume. It is obvious that Σini is equal to no, where no is the original total number of particles per unit volume equal to the concentration of the primary particles. Tambo and Watanabe introduced the collision agglomeration coefficient 8R, into Equation 4.117, which is essentially the same as the coefficient P in Equation 4.108 and which was assumed to be a function of the aggregate size [135]. They also assumed that the aggregates can grow to a maximum order S without any breakup. The modified equation is dnR 1 R 1 A ∑ R dt 2 i 1
i1 /(3K ) (R 1)1 /(3K ) 3 n n i R i
S R
A nR
∑ R i ( R
1 /(3K )
i
) nn
1 /(3K ) 3
i 1
i R
(4.118)
By introducing next the dimensionless variables ni
ni , no
S
∑ ini 1
(4.119)
i1
m Ano t 1.22
o 3 d1 no t
(4.120)
Equation 4.118 can be transformed into its dimensionless equivalent: dn R 1 R −1 A ∑ R i1 /(3 K) ( R 1)1 /(1 K ) dm 2 i =1
(
SR
(
n R ∑ Ri i1 /(3 K) R1 /(3 K) i1
3
)
3
)
ni
ni n Ri
(4.121)
A numerical iterative integration of Equation 4.121 under the assumption R 1 led to a strong disagreement, with experimental results reproduced in Figure 4.10.
Cohesive Sediments in Open Channels
130
There are two main reasons for this disagreement. First, according to Equation 4.99 and Equation 4.100, the flow-induced shear stresses impose an upper limit for the aggregate diameter; second, the interparticle physicochemical forces vary over a wide range. Therefore, coagulation in an originally dispersed suspension of fines will generate an aggregate population of variable strength and maximum size. Figure 4.10 is quite informative on the process of flocculation. For a maximum aggregate order of S 5000, aggregates of an order R 270 and lower start with a relatively high concentration, and they decline rapidly to zero, being readily adsorbed by the larger aggregates. On the other hand, larger-sized aggregates start from zero or at very low concentrations and increase rapidly with time to their ultimate value. The approximate volume percentage of each class of aggregates amounts to up to 100%, as expected. In this case, the aggregates were divided into four groups on the basis of their size range and of the corresponding R values. The lowest group suggests that the lower limit of the interparticle forces is sufficiently high so that no aggregate of an order smaller than R 1000 can survive. Equation 4.121, therefore, has to be modified to incorporate the maximum aggregate order, S, and the related shear stresses. Moreover, in a turbulent flow, the collision agglomeration factor is expected to vary; however, this variation can affect only the rate of flocculation, but not the ultimate distribution of aggregate order and size. For given physicochemical properties of the sediment-water system, the latter depend on the strength of the aggregates as defined by Equation 4.89 and Equation 4.92. Therefore, each curve in Figure 4.10 corresponds to a particular value of of the system. For the proper modification of Equation 4.121 Tambo and Watanabe introduced first the following empirical equation for R: R R o 1 S 1
Floc volume percentage
40 30
0.1 mm (R � 270)
S � 5000 K� � 1.25
(4.122)
0.5–0.7 mm (R � 4000, 5000)
0.15 mm (R � 550)
0.4–0.45 mm (R � 3000) 0.3–0.35 mm (R � 2000)
20
0.2–0.25 mm (R � 1000) 10
0
5 Dimensionless flocculation time, (m)
Figure 4.10 Experimental aggregate growth pattern [135, Fig. 2].
10
Chapter | 4 The Hydrodynamic Transport Processes
131
where is a numerical constant and o is the initial agglomeration factor with approximate numerical values 6 and 1/3, respectively. The exponent was assumed to be a universal constant, whereas o may change with the coagulation conditions. Thus, for R 0, R o and for R S 1; that is, the probability of an S order aggregate to grow into an order S 1 is zero. With the introduction of the preceding relationship, Equation 4.121 changes into the following form: 6 3 dn R R 1 /(3K ) 1 R 1 1 /(3K ) n n ∑ o 1 (R − i ) i i R i dm S 1 2 i 1 3
6 R i 1 /(3K ) 1 /(3K ) 1 R ∑ o S 1 i n i i 1
S R
n R
(4.123)
An example of the change of the collision agglomeration factor with pH and with coagulant concentration is presented in Figure 4.11. If we want to introduce in Equation 4.123 the controlling rate of energy dissipation , S has to be related to the maximum aggregate size, dmax, given by Equation 4.99 and Equation 4.100, in which the aggregate strength is included in the coefficients 5 and 6. S shows the number of primary particles in a maximum-sized aggregate formed under a particular agitation intensity; then, according to Equation 4.115 for di dmax, S
1 /(3K ) Sm
where S m
d max 3 d
(4.124)
1
Collision agglomeration factor, �
One is reminded that in Ref. [135], all magnitudes are in CGS units; that is, dmax is in cm, 0 is in cm2s3, and the kinematic viscosity is in cm2s1. Equation 4.123 can thus be numerically integrated if the parameters S, K, and o can be estimated. The dimensionless parameter m in Equation 4.120 is proportional to 0.02 A : pH 5.75
A
B : pH 5.5
0.015
C : pH 5.25 D : pH 5.0
0.01
B
0.005
C D 0
2.5
5
7.5
10
12.5
Coagulant dosage (mol
15
L�1)
Figure 4.11 Variation of collision agglomeration factor [135, Fig. 4].
17.5 (10�4)
Cohesive Sediments in Open Channels
132
GCot, where G is the representative shear rate, du/dy; Co is the total sediment concentration; and t is the time. This product is referred to as the GCoT value, and it should be used instead of Camp’s parameter GT [9] for the attainment of aggregate growth operation in a flocculator. As indicated earlier, the coefficients K and in the aggregate density function (Equation 4.94) depend on both the sediment properties and the water chemistry, as represented by the ALT ratio and the pH (Figure 4.9). Figure 4.12 shows the experimental results by Tambo and Watanabe for kaolinite clay aluminum aggregates [135]. One can observe that, for the same effective rate of energy dissipation, the maximum aggregate diameter increases with both ALT and pH [135]. Numerical integration of Equation 4.123 by Tambo and Watanabe is shown in Figure 4.13 for the indicated values of K and S. The set of curves thus obtained, albeit based on the assumption of constant interparticle strength, , nevertheless reveal certain fundamental characteristics of the flocculation process that will be presented and discussed in Chapter 7. Each curve represents the distribution as a volume percentage of aggregates of order R versus the aggregation number R* with the dimensional time o m as a parameter, with m given by Equation 4.120. The following general observations can be readily made. In the early stages of flocculation, which is for small values of ο m, flocculation proceeds relatively rapidly but also with a relatively narrow range of aggregate diameters or R values. With the increase of the flocculation time, the rate of aggregate growth becomes slower and slower, while the spread of the aggregate diameter becomes increasingly larger. An ultimate distribution is rapidly reached, beyond which no noticeable change is observed. Finally, for sufficiently large time, the distribution curves
Maximum floc diameter dmax (mm)
The rate of agitation blades rotation Nr, rev min�1 10 25 30 40 15 20 2.0
1.0 0.8 0.6
50
60
pH 8.0 pH 7.5 pH 7.1 �0 pH 6.5
0.4 0.2 0.1 0.01
ALT � 10�1 4 � 10�2 2 � 10�3 0.05 0.1
0.5
1
5
Effective rate of energy dissipation ε�0 (erg cm�3, s�1) Figure 4.12 Maximum aggregate diameter versus effective energy dissipation [135, Fig. 6].
10
.1 �0
80
2
�0 m
Cumulative floc volume, %
90 70 60
�
50
m 0
�
0.
R50 � 7.8
40
m �0
�
0.3 m �0
�
0.4
m �0
7.6
�
Ultimate equilibrium floc size distribution
0.5 .6
34
.4
7.1
40
23.1 .8 2 30 1.0 .5 3 0.6 8 . � �1 0 � 2.0 m � m � � 0 � 0m m �0 m �0 �0
16.4
30 20
42.3 R50-ult � 42.3
S � 100 Kp � 1.2
10 0
2
0
4
6
8
10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 Agglomeration number, R 0.25
0.5
0.75 1.0 Normalized agglomeration number, R x
Chapter | 4 The Hydrodynamic Transport Processes
100
46 48 50 52 54 56 58 60 62 64 66
1.25
1.5
Figure 4.13 Cumulative aggregate size distributions [135, Fig. 7].
133
Cohesive Sediments in Open Channels
134
50% cumulative floc number, R50
50 42.3 40
R50-ult
30
20
10
0
0.2 0.4 0.6 0.8 1.0
1.4
1.8 2.0
2.4 2.6
Dimensionless flocculation time, a0 m Figure 4.14 Variation of the 50% cumulative aggregate [135, Fig. 8].
seem to approach the typical S-shape of the normal distribution. In the same figure, there is a second abscissa, R* R/R50ult, where R50ult is the R value of the ultimate curve corresponding to 50% volume accumulation, equal to 42.3% or 0.423 in the present case. This last parameter can be used as a measure of the degree of flocculation. Its variation with the dimensionless time οm is shown in Figure 4.14, from which it is again concluded that most of the aggregation takes place at the early stage of the flocculation process, dropping rapidly with time afterward. An even more revealing relationship is achieved by plotting the percentage of the ultimate cumulative aggregate volume for various combinations of S and K against the normalized floc volume, R*3/(3K). The result is shown in Figure 4.15 with all points falling on the same curve, which again resembles the S normal probability curve. It appears, therefore, that Tambo and Watanabe’s studies point to the existence of a self-preserving aggregate size distribution at the final stage of the aggregate growth process. One should note that Swift and Friedlander found a similar self-preserving tendency for flocculation of hydrosols under Brownian motion [131]. Moreover, Tambo and Watanabe claimed that a single normalized curve can be obtained not only for the ultimate aggregate size distri* bution, but for each aggregate size distribution with the same R50 , as shown in Figure 4.16. The dimensionless flocculation time, m, required for a cohesive sediment suspension to reach an aggregate size distribution identified by the param* eter R50 is given by
m ≈ S m
(4.125)
Chapter | 4 The Hydrodynamic Transport Processes
135
100
Cumulative floc volume, %
90 80 70
Ultimate floc size distribution
60 50
S�50 S�50 S�50 S�200 S�100 S�75
40 30 20 10 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.5 3 Normalized floc volume R*3–K� � (R/R
K��1.0 K��1.2 K��1.4 K��1.2 K��1.2 K��1.2
2.0 3 3 )3–K� � (R/0.42S)3–K�
50-ult
Figure 4.15 Normalized ultimate aggregate size variation [135, Fig. 9].
100
Cumulative floc volume, %
90 80 70 60 50 40 30
R*50�0.4
0.5
0.6 0.7
0.8
0.9 R*50�1
S�100 S�50 S�200 S�50 S�50
K��1.2 K��1.2 K��1.2 K��1.4 K��1.0
20 10 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
1.5 3 Normalized floc volume R*3–K�
2.0
2.3
Figure 4.16 Normalized general aggregate size variation [135, Fig. 10].
where the exponent was found to vary between 0.15 and 0.20 in Tambo and Watanabe’s experiments [135]. Hunt also concluded from theoretical and experimental studies that aggregate size distributions were self-similar. However, his experiments were limited to flocs of a diameter between 0.6 and 1.2 m, and his analysis was based on the assumption of a constant floc and aggregate density [52]. His results were obtained in a laminar flow field between two co-centric cylinders and, therefore, cannot be compared very well to those by Tambo and Watanabe. The dependence of the aggregate size distribution on the flow-induced shear stresses is further confirmed by the experiments of Tsai et al. [147] and Burban et al. [8]. These experiments made use of a Couette-type viscometer to apply a uniform laminar shear stress to a sediment suspension. Figure 4.17 shows the time history of the median aggregate diameter, and Figure 4.18 shows the average
Cohesive Sediments in Open Channels
136
120
1 dyne/cm2
1 dyne/cm2
Median floc diameter (µm)
100 2 dynes/cm2
80 2 dynes/cm2 60
4 dynes/cm2
40
4 dynes/cm2
20
0
Stress change 0
100
200
Time (min) Figure 4.17 Time variation of aggregate size for 100 ml/L concentration [147, Fig. 4].
1 dyne/cm2
% by volume
30
2 dynes/cm2
20
4 dynes/cm2 10
0
1
10 Floc diameter (µm)
100
Figure 4.18 Distribution of aggregate diameter for 100 ml/L concentration [147, Fig. 5].
Chapter | 4 The Hydrodynamic Transport Processes
137
steady-state distribution by volume of aggregate sizes for a sediment concentration of 100 ml L1 and for the three indicated shear stresses. The following conclusions can be reached from Figure 4.17 and Figure 4.18: 1. After a rapid initial increase, the aggregate size reaches a quasi steady-state distribution, albeit with significant fluctuations. 2. The steady-state average diameter decreases with increasing shear stress. Moreover, it is independent of the manner in which this steady-state size is reached, that is, whether under one and only one shear stress or after having been subjected to different stresses before the final one has been applied. 3. The distribution of aggregate diameter seems to fit the normal distribution (Gaussian) curve. This distribution shifts to the right for decreasing shear stresses, thus suggesting that the lower the shear stress, the higher is the peak of the aggregate size with smaller proportion of smaller-sized aggregates and higher proportion of larger-sized aggregates. 4. Similar distributions were found for sediments of other concentrations tested up to 800 mgL1. However, as shown in Figure 4.19 and Figure 4.20, the steady-state median aggregate diameter decreases with increasing sediment concentration for the same shear stress, while at the same time the size distribution curve shifts to the left. This is indicative of a larger proportion of smaller-sized aggregates and of a smaller proportion of aggregates with larger diameter.
Median aggregate diameter (µm)
The first three conclusions are consistent with the models of flocculation and aggregate properties presented, particularly with the relationships between aggregate size and shear stress. At first glance, no explanation seems obvious for the reduction of the steady-state diameter with increasing sediment concentrations.
120 100 50 ppm 80
100 ppm
60 40
400 ppm 800 ppm
20 0
0
50
100
150
Time (min) Figure 4.19 Distribution of aggregate diameter for a shear stress of 2 dynes/cm2 [147, Fig. 8].
Cohesive Sediments in Open Channels
138
30 50 ppm
20 % By volume
100 ppm
400 ppm 10 800 ppm
0
1
10
100
Floc diameter (µm) Figure 4.20 Variation of aggregate size for a shear stress of 2 dynes/cm2 [147, Fig. 7].
One must remember that in all models previously mentioned, only collision between two aggregates was considered. To explain this observation, Burban et al. introduced in their analysis a disaggregation process due to three-aggregate collision [8]. No such collisions were considered in the models presented, and no similar effect of the sediment concentration was reported in the earlier presented studies. The gap of the viscometer in which the last-discussed experiments were conducted was only 2 mm wide, and such a small width may have something to do with the aggregate size distribution, although this is not obvious at this time and needs further investigation. Kusuda et al. conducted experimental studies on the density and settling velocity of flocs and aggregates of kaolinite and montmorillonite clay with various dosages of coagulants [65]. The turbulent coagulation took place in a paddle flocculator similar in principle to the one used by Tambo and Watanabe and by Tambo and Hozumi, whereas for the study of coagulation due to differential settling, a specially designed column was used. Their devices permitted direct measurement of the weight of the aggregates. The G value given for turbulent flow in Equation 4.77 was used as a measure for shear rates. The work by Kusuda et al. reconfirmed the main previous conclusion that the aggregate size decreases with increasing shear rates [65]. It also reconfirmed that the aggregate density decreases with increasing aggregate diameter. In addition, they found that aggregates composed of smaller flocs formed previously under significantly higher stresses are less dense than aggregates formed under
Chapter | 4 The Hydrodynamic Transport Processes
139
10
1
Floc dry weight (µg)
5
G value s�1 50 50 100 50 200
2.5
1
0.5 0.1
0.5 Floc diameter (mm)
1
Figure 4.21 Aggregate-size relationships [65, Fig. 5].
the same final stress from the original clay particles. Figure 4.21 shows the relationship between the aggregate dry weight and the aggregate diameter for these two modes of flocculation. In all tests, the final G value was 50 s1. The points for the experiments under constant G clearly fall above those corresponding to sediments previously flocculated under a higher shear stress. Both sets of points fall on straight lines on the log-log plot, with a slope of 1:2.5 indicating decreasing density with increasing aggregate diameter. Their experiments also seem to suggest that the settling velocities are independent of the mixing intensity. Figure 4.22 displays the test results for the three G values indicated together with the equations for the average line and the two extreme envelopes containing the data points. The plots suggest that the effect of the aggregate size on the settling velocity is offset by the decrease of the aggregate density. Images indicated that aggregates formed under a single stress are denser and consequently possess higher strength than aggregates previously submitted to higher stresses. The authors concluded that the aggregate density can be estimated from its settling velocity and diameter by using the following drag coefficient corresponding to a quasi-spherical particle with 1.1 correction factor: CD
24 3 0.34 Re Re
(4.126)
Cohesive Sediments in Open Channels
140
10 G value s 1 50 100 200
Settling velocity (mm/sec)
5
CD 24/Re CD 24/Re 3/�Re 0.341
CD 45/Re 1
0.5 0.1
0.5
1
Floc diameter (mm) Figure 4.22 Settling velocity of kaolinite aggregates [65, Fig. 3].
These studies lead to the conclusion that in any flow field the primary building block of the aggregates is the minimum size floc or aggregate that has been formed under the highest shear stress the sediment was subjected to. More specifically, in estuaries with river-borne sediment, this building block has to be formed in the river because it is there that shear rates are expected to attain their highest values. This subject will be further discussed in Chapter 7 on the deposition and resuspension of cohesive sediments.
4.4.3 Some Additional Research Work on Flocculation and Aggregate Properties The references cited and discussed in the preceding section were selected primarily because they were judged as best describing the fundamental process of flocculation and the relationship between the pertinent flow parameters and aggregate properties. In addition, they were thought to be more relevant to the estuarine sedimentation processes, although the work by Tambo and Watanabe [134, 135] and Tambo and Hozumi [132, 133] was motivated by the needs of sanitary engineering. There have been several additional investigations on both the flocculation process and on the aggregate formation and their properties. Most of these investigations were directed toward the needs of water purification and sewage treatment. In essence, they are based on the same fundamental concepts as those previously
Chapter | 4 The Hydrodynamic Transport Processes
141
discussed, except that they have been modified for special cases and for specific aspects, such as effects of polymers and other additives on the enhancement of flocculation and on the increase of the aggregate strength. Most of them reconfirmed the previously presented functional conclusions and provided additional information of specific aspects. Since these additional references do not contribute significantly to the understanding of the flocculation process and to the cohesive sediment transport mechanics, they will not be discussed extensively; only a brief summary of some of them will be given for the information of the readers. In 1950, Manley and Mason studied theoretically and experimentally the motion of glass spheres in suspensions subjected to shear flow generated in a Couette apparatus with the cylinders rotating in opposite directions [77]. Their model amounted essentially to Equation 4.31 through Equation 4.35, with the assumption of inelastic collision between spherical particles. Their attention was focused on doublets and their rotation under the action of flow. Substantial contributions to flocculation of solids have also been made by researchers of aerosol behavior in spite of the fact that the relative density of aerosols with respect to air is by orders of magnitude higher than that of fine sediment particles, flocs, and aggregates to water. Swift and Friedlander were among the first to observe the coagulation of hydrosols by Brownian motion and shear flow [131]. Their theory is along the same lines as Smoluchowski’s theory of coagulation as modified and presented by Levich, which was outlined in Section 4.3.1 and Section 4.3.2. However, by introducing a similarity hypothesis, they simplified the equations for the rate of change of aggregate concentration to a solvable form for both Brownian motion and shear coagulation. They claimed that the course of a coagulation process can be completely described by the variation of the aggregate size distribution function, which they defined in terms of the aggregate volume. A continuous distribution function, n(cv,t), was introduced corresponding to ni, defined such that n(cv,t)dv represents the aggregate concentration in the volume range from cv to cv dcv. Then the total number of aggregates per unit volume (per cm3 in this case), n, is given by
n∞
∞
∫0
n(cv )dv
(4.127)
and the number of particles per cubic centimeter larger than a given volume v is defined by the following integral:
nv
∞
∫c
n(cv ) dcv
(4.128)
v
The total volume fraction of aggregates is given by
φv
∞
∫0
cv n(cv )dcv
and it is constant as long as there is no material lost or gained.
(4.129)
Cohesive Sediments in Open Channels
142
The introduction of the preceding parameters to make the equations for the collision frequency dimensionless led to the so-called self-preservation of the aggregate size distribution; that means that the aggregate volume distribution curves made dimensionless by the factors φv n∞ and n approximately coincide for any time during the flocculation process. The derived self-preservation equation has the form n(cv , t )
2 n∞ ψ1 ( ) φv
(4.130)
where
cv n∞ φv
(4.131)
Substitution of Equation 4.131 into Equation 4.127 gives ∞
∫0
ψ1 ( )d 1
(4.132)
Figure 4.23 shows an example of self-preservation by the authors with oilin-water emulsion. The data seem to fall close enough to each other for the assumption of self-preservation to be valid although there is some slight distinction of the points for various times. One is reminded that Tambo and Watanabe
1.0
t min 0 15 30 45 60 75 105 150
0.8
nν/n∞
0.6
0.4
0.2
0 0.01
0.1
1.0 η
Figure 4.23 Self-preservation volume distributions for oil-in-water emulsions [131].
10.0
Chapter | 4 The Hydrodynamic Transport Processes
143
found a similar self-preservation of the aggregate volume distribution curves for clay particles (see Figure 4.15 and Figure 4.16). Hunt [52] extended the results on emulsions obtained by Friedlander and by Swift and Friedlander [131] to kaolinite and illite clays subjected to coagulation by Brownian motion, laminar shear stresses, and settling. Only the results of the first two cases will be outlined here. Hunt assumed that the aggregate size distribution is in local equilibrium maintained by a flux of primary particles through the size distribution. First, he changed the equation for the rate of change of the concentration of aggregates of volume Va into a dimensionless form by using the parameters K
kT for Brownian motion
(4.133)
where T is the absolute temperature in degrees Kelvin. A K value of 4.28 1012 cm3s1 was used. The following size distributions were then obtained by dimensional analysis: For Brownian motion:
F 1 / 2 n(cv ) Ab V cv1 / 2 K
(4.134)
F 1 / 2 n(cv ) Ash V cv2 G
(4.135)
For shear:
where G is the shear rate for shear coagulation as defined by Equation 4.77, in Section 4.3.1 and FV is the volume flux of the particles defined by
FV
dVa dt
(4.136)
where Va is the total volume of aggregates, and its negative derivative indicates the rates of their removal from the suspension by settling. Ab and Ash are dimensionless constants. The sediment was tested in a laminar shear field generated by two concentric rotating cylinders with water at ocean salinity between them and positioned vertically to allow larger aggregates to settle. The available instrumentation did not allow larger aggregates to be sized, and the sedimentation prediction was not tested. Only particles smaller than 2 m were tested. Aggregate size distributions were measured with a Coulter Counter type system, and the aggregates were counted and sized by going through a small aperture. Because the system caused a breakup of larger aggregates, aggregate size distributions were measured only for diameters from 0.6 to 1.2 m.
Cohesive Sediments in Open Channels
144
Hunt chose to present his data in the form of a volume distribution dVa/d(logda), which is the suspended aggregate volume in a logarithmic interval of aggregate diameter. The equation describing such a distribution is dVag d ( log da )
2.3 2 6 da n(cv ) 12
(4.137)
where 2.3 is the ln(10). The total aggregate volume, Vag, is related to the volume distribution, n(cv), by the following equation: Vag
∞
∫0
cv v n(cv )dcv
(4.138)
Introducing n(cv) from Equation 4.134 and Equation 4.135 for Brownian motion and shear coagulation, respectively, into Equation 4.137, we obtain the following expression: For Brownian motion:
F 1 / 2 dVa 5.0 Ab v da3 / 2 K d ( log da )
(4.139)
F 1 / 2 dVa 6.9 Ash v G d ( log da )
(4.140)
For shear:
The preceding two equations were subsequently made dimensionless by introducing the reference variables (K/G)1/2 for the diameter and (FV/G)1/2 for the total volume, Va. The dimensionless quantities da and Va corresponding to da and Va are given by
dVa 5.0 Ab da3 / 2 for Brownian motion d ( log da ) dVa
d ( log da )
6.9A sh constant for shear
(4.141)
(4.142)
The estimated dimensionless coagulation constants were Ab 0.046 and Ash 0.026. This normalization suggests that all data pertaining to Brownian motion and shear coagulation collapse on a single line. Figure 4.24 shows the results for kaolinite for the indicated times and for G 1 s1, and the ultimate data for the indicated shear rates are plotted in Figure 4.25. In Figure 4.24, all the curves seem to have a similar shape, but there is a distinct curve for each
Chapter | 4 The Hydrodynamic Transport Processes
Normalized volume distribution
1
3/ 2
T � 25min 40 60 90 130
op
e
10�1
Sl
dVa /d (log da)
1
10�1
145
10�2 �1 10
10�2 10
1 da
Figure 4.24 Normalized volume distributions for kaolinite At G 1s1 [52, Fig. 4].
dVa /d (log da)
0.4
Normalized volume distribution
0.4
0.1
0.1
G � 1 ( ), 2 ( ), 4 ( ), 8 ( ), 16 ( ), and 32 S�1 (�). 0.03 0.2
1
4
0.03
da Figure 4.25 Normalized ultimate volume distributions for kaolinite in sea water [52, Fig. 5].
Cohesive Sediments in Open Channels
146
time. The left branch of each curve is very nearly straight with a slope of about 3/2 consistent with Equation 4.141 corresponding to Brownian motion. The right side, though, corresponding to shear coagulation, is by no means horizontal, as it should be according to Equation 4.142 and is in contradiction to Figure 4.25 [52]. In Figure 4.24 a straight line at a slope of 3/2 can be fitted at the left-hand group of the points predominantly controlled by Brownian motion, and a horizontal one for the right-hand group of points controlled predominantly by shear rates. The scatter of data points seems to be rather random and on the order of 50% of the average value. So, for the ultimate steady-state volume distribution, the assumption of similarity for various shear rates seems to be plausible. In 1963 Vold [152] presented a computer model for the simulation of aggregate formation in a colloidal suspension. Model aggregates were generated through successive addition of primary particles, taken as spheres of unit radius, to the growing aggregate in a completely random way. The aggregate thus formed was characterized by a roughly isometric core with a gradually decreasing density from the center toward the edges. Additional particles not included in the core form projected tentacles, giving the aggregate a rough surface and an average extent of about five particle diameters larger than the core diameter. This is in qualitative agreement with the experimental results presented and discussed in the previous section. In 1967 Hannah et al. [41] studied experimentally the strength of flocs formed in a Couette-type flocculator. The flocs were counted by a particle counter after passing through a 70 m diameter aperture, thus forcing the flocs through a highvelocity gradient zone. The objective of their study was to find ways to produce flocs of the maximum possible strength for the most efficient water treatment. No quantitative relationships linking the floc strength to flow variables were derived, but the following important qualitative conclusions were reached: 1. The highest rate of flocculation does not necessarily produce flocs most resistant to shear. 2. Alum flocs alone are relatively weak, and they can be strengthened by incorporating solids into them. 3. The coagulants in carefully planned limited dosages have a pronounced effect on the formation of large high-strength flocs. Thomas [141] studied the disruption of aggregates in turbulent flows on the basis of the rheological behavior of concentrated cohesive sediment suspensions as described by the Bingham plastic model. He considered the Bingham plastic strength, y, to be representative of the aggregate shear strength, and he related empirically the shear strength to the dynamic viscosity of the suspension, s, and of the water , using the following equations: y 210
φs3 da2
(4.143)
Chapter | 4 The Hydrodynamic Transport Processes
s 14 exp φs 2.5 da
147
(4.144)
In the preceding two equations, φs is the dimensionless volume fraction of solids in the suspension, da is in micrometers, and the coefficients of viscosity are in pounds per sq. ft. If w is the ratio of immobilized water volume within the individual aggregates to the solid volume of the latter, then the volume fraction of solids within the aggregate is
Vas
1 1 w
(4.145)
and the volume fraction of aggregates in a suspension is
φas (1 w )φs
(4.146)
From hindered settling measurements, the aggregate diameters were empirically correlated to the original particle size or to the original minimum floc diameter by
da (1 w )2 dp
(4.147)
To be able to apply concepts developed for breakup studies in emulsions to solid particle aggregates, Thomas introduced a pseudo surface tension, a, for the aggregates related to the Bingham yield stress of the suspension using
a
y da
(4.148)
One can easily show that w can be related to the aggregate porosity, pae, i.e., is the ratio of the volume of water (or of pores) within the aggregate to the aggregate volume, through the relationship
1 1 pae 1 w
(4.149)
in which case Equation 4.147 takes the form
1 pae
d p 1 / 2 da
(4.150)
Equation 4.150 confirms once more the previously established fact that the aggregate density decreases rapidly with increasing size. For example, for da equal to 16 times the particle diameter, dp, pae is about equal to 0.75; for da 100dp, pae 0.90,
148
Cohesive Sediments in Open Channels
and for da 1000dp, pe 0.97. Thomas’s further development was based on the assumption that either Equation 4.147 or Equation 4.148 could be applied if the volume fraction w or the porosity pae is equated to the volume fraction of the suspension, that is, when φs 1 pae. This is a very restrictive assumption, because with the rare exception of extremely heavy fine sediment load in rivers and/or estuaries, the overall volume fraction is much smaller than that within the aggregates. On the basis of these equations Thomas gave the empirical relation
3/2 1/2 y ≈ d dp a
(4.151)
indicating that the aggregate strength, as represented by the Bingham yield stress of the suspension, decreases rapidly with increasing aggregate diameter and with increasing order of aggregation. Using the preceding rheological properties, Thomas derived the following relationships for the aggregate size for homogeneous isotropic turbulence and for the inertia and viscous subrange:
da ∝ y9 / 25/2 f 1 (φas ) for da
(4.152)
1/2 da ∝ 1/2 f 2 (φas ) for da y
(4.153)
where is the rate of energy dissipation per unit mass of fluid. In the preceding equations, f1 and f2 are functions of the volume fraction of the suspension 1 but in terms of aggregates, because φas φs (1 pe ) . Substitution of the yield strength from Equation 4.151 leads to the following equations:
da ∝ 0.32 f 3 (φas ) for da o
(4.154)
da ∝ 0.29 f 4 (φas ) for o da
(4.155)
where is the macroscale and o is the microscale of turbulence. These equations are similar to Equation 4.101 through Equation 4.104 derived by Tambo and Watanabe [135] and presented in Section 4.4.1. Specifically, in Equation 4.101 and Equation 4.103, which correspond to Equation 4.154 for the inertia subrange, the exponent of the energy dissipation ranges between 0.4 and 0.5, whereas in Equation 4.154 the same exponent is 0.32. The same exponents in Equation 4.106 for the viscous subrange vary between 0.38 and 0.33 in comparison to 0.29 in Equation 4.155. Parker et al. [97] modified Thomas’s work to account for surface erosion of particles from aggregates rather than aggregate disruption by shear. According to a summary by Hunt [52], their analysis, based on several assumptions, led to equations relating the aggregate size to the turbulent kinetic energy dissipation per unit mass, , raised to the powers of 1, 1/2, and 1/4, depending on the subrange the aggregate was falling into and on whether the latter was composed
Chapter | 4 The Hydrodynamic Transport Processes
149
of particles or filaments. Considering aggregate strength, the appropriate coefficients involved aggregate yield terms and aggregate porosity; however, these properties were assumed independent of aggregate size. This assumption contradicts the experimental conclusions previously discussed. Matsuo and Unno conducted a good review of some previous work on aggregate disaggregation [79]. They attempted next to improve the model of Parker et al. by first pointing to the inefficiency of their equations for the forces acting on nearly buoyant aggregates. They subsequently assumed “forces acting on suspended aggregates to be approximately equal to those on comparable fluid particles from the surrounding fluid.” From a numerical comparison of their method to that of Parker et al., they concluded that “the effect of difference in density was minor compared to that of the difference in deformability as a cause of break-up of flocs whose density was less than 1.1 gcm3.” They developed formulas for the forces caused by differences in deformability for both homogeneous and isotropic turbulence and for nonisotropic wall turbulence. All these formulas give a yield stress involving the rate of energy dissipation and an aggregate Reynolds number. They also conducted experiments in cylindrical pipes. To compare the strength of different aggregates, they assumed a “pseudo-surface tension caused by a thin shell-like structure near the floc surface.” On the basis of their aggregate strength analysis, they claimed “that optimum alum dosage for the strongest floc is smaller than the one which produces the minimum turbidity.” Matsuo and Unno’s work and conclusions were questioned by Clark [11] and Parker [96] and, in particular, the validity of their stress-strain relationships, their interpretation of the various terms in the Navier-Stokes equations, and their assumption that “forces acting on a suspended floc unit are to be approximately equal to those on a comparable fluid particle from the surrounding liquid.” Tomi and Bagster [144] studied the aggregate size in stirred vessels both theoretically and experimentally. According to a summary by Hunt [52], their work was based on the assumption of constant aggregate strength although they recognized that strength decreases as the aggregate size increases. Their research led to the conclusion that the aggregate diameter is related to the rate of energy dissipation per unit mass, , through the following equations:
da 1 for da o for the inertia subrange
(4.156)
while for the viscous subrange of isotropic turbulence, they claimed that aggregate sizes seemed to be independent of the kinetic energy dissipation rate, in spite of their recognition that such a situation was not observed in practice. However, because most flocs observed in their work were on the order of Kolmogorov’s microscale, they combined viscous and inertial effects of turbulence to arrive at the suggested formula:
da 1/2 for da ∝ o
(4.157)
150
Cohesive Sediments in Open Channels
A review of experimental data conducted by others indicated a power dependence of aggregate size on with the exponent ranging from 0.83 to 0.13 with an average of 0.50. Smith and Kitchener investigated the properties of aggregates by photographing the suspensions reported by Smith [129] while they were subjected to specific laminar shear stresses generated in a Couette-type apparatus. They used three different procedures for measuring the strength of adhesion between particles and flat surfaces. The force measurements extended over four orders of magnitude. The tested materials were crushed glass and Ballotini beads of a size greater than 2 m mixed with varying amounts of MgSO4. They found the following relationship between aggregate diameter and shear rate:
da 2 G0.2
(4.158)
where 2 is a parameter found to increase qualitatively with increasing adhesion force. Motivated by the needs of water treatment procedures, Leentvaar and Reblun [72] studied the strength of ferric oxide flocs and aggregates formed in batch coagulation tests in rectangular tanks with the intention of obtaining a clearer understanding of the aggregate breakup mechanism. They also studied the dependence of this strength on an inorganic coagulant, ferric chloride, combined with organic polymer. Their experimental results were found to fit the expression
da max Ca G2
(4.159)
where the exponent was found to vary between 0.3 and 0.5 and where the coefficient Ca depends on the coagulant dosage. It was found that logCa ranged between 1.87 and 2.54. They also observed that aggregates would not reform following disruption by higher turbulence-induced shear stresses. Glasgow and Hsu [35] conducted experimental studies on the disintegration of individual inorganic flocs and aggregates subjected to disruption by water jets. Their primary objective was to observe the aggregates photographically and to deduce the magnitude of the force responsible for the breakup. They also tried to study the strength of kaolin-polyacrylamide aggregates, but they also investigated kaolin-Fe3 flocs. According to their interpretation of the photographic records, “the break-up mechanism entails massive splitting” and the “primary particle erosion did not contribute significantly to the size reduction.” For large aggregates formed at an ionic strength of 0.03M and a PH of 9.5, their data yielded the following relationship for the binding force Fb in dynes and the aggregate diameter in millimeters:
Fb 13.56 da2.4
(4.160)
A similar relationship was derived for other cases. Considering that the average shear stress on the aggregate is proportional to the square of its diameter, it
Chapter | 4 The Hydrodynamic Transport Processes
151
follows that this stress will be proportional to da0.4 ; i.e., it increases with aggregate size. This conclusion contradicts the previously discussed experimental results, and in particular those by Krone [61, 64], Tambo and Watanabe [134], and Kusuda et al. [65] as well as the most probable model of aggregate structure based on these very detailed and systematic studies. However, their data suggested that the strength per unit mass rather than per unit area decreases with increasing aggregate size, and they supported the multiple-level model for aggregation. Glasgow et al. [36] studied the size reduction of biological aggregates in a turbulent field generated in a stirred flocculator. They claimed that microbial flocs are related to the mean rate of turbulent energy dissipation by a relationship similar to Equation 4.157 with the exponent of ranging between 0.03 to 0.09. This range suggests a low sensitivity of microbial aggregates to turbulent energy dissipation. A recent development in colloid science is the elastic model summarized by Hunter [53] and reported by Hunt [52]. This model relates the rheological properties of coagulated colloidal dispersions to fluid parameters and sediment properties. According to the abbreviated report by Hunt, the model has been verified “with an experimental system requiring initially high shear rates of 2000 to 3000 s1 to form dense primary flocs.” For a laminar shear rate, the data suggested a relationship similar to Equation 4.158 suggested by Smith and Kitchener but with an exponent of the shear rate G equal to 0.42 instead of 0.2. Delichatsios and Probstein developed a model based on simple binary collision mean-free path concepts for the prediction of coagulation rates in isotropic turbulent flows of destabilized nondispersed particles less than and larger than the Kolmogorov microscale of turbulence [12, 13]. They also conducted experiments in pipe flows with particles smaller than the microscale of turbulence and reported an excellent agreement with their theory. No specific relationships were derived linking aggregate size to the rate of energy dissipation and, therefore, their theory appears to be difficult to apply to practical sedimentation problems. Klimpel and Hogg [57] investigated experimentally the effect of polymer concentration, agitation intensity, excess mixing time, solid concentration, and primary particle size on the density, porosity, and structure of flocs and aggregates. The sediment used was pure quartz properly ground and classified to give a fairly narrow particle size distribution in the range of 2 to 10 m. Porosities and densities were evaluated on the basis of measured sizes and settling velocities by means of photographic techniques. The flocculent used was a nonionic polyacrylamide of very high molecular weight, and a standard tank mixing unit was used for flocculation. Their experimental observations led to the following empirical expression: log (1 pae )
log(1 pae1 ) ac
d 1 a d
(4.161)
152
Cohesive Sediments in Open Channels
where pae is the porosity of the aggregate, as in Thomas’s Equation 4.149 and Equation 4.150 pae1 is the limiting porosity of large aggregates; dac is a characteristic floc or aggregate size; and is a constant. However, these last two parameters varied within a certain range; specifically, dac varied from 31 to 277 m and from 0.96 to 1.56, depending on the conditions of each test. In addition to this empirical formula, the following important conclusions were reached: 1. A multistage aggregate development was verified in agreement with Krone’s model [64]. Specifically, they proposed the following three aggregate growth regions: one of low size, low porosity, and high density approaching asymptotically unity corresponding to solid fraction volume; a second on the other extreme of relatively large size and high porosity, tending asymptotically to a minimum solid volume fraction and to a maximum porosity pae1; and a third for the intervening region. 2. They confirmed the earlier mentioned conclusion that aggregate density decreases with increasing size. 3. The basic unit in the aggregates is not the primary solid particle but a floc of high density whose size and density depends on the test conditions. 4. The effect of polymer dosage was found to be insignificant. 5. The aggregate density increases and the size decreases with increasing agitation intensity, a conclusion also reached by previous investigators. 6. The porosity was found to decrease with increasing size of primary solid particles. Differences in these particles were most significant in the changes between density-size relationships. For any given aggregate size, the density increases with increasing initial particle size. Variation in the floc structure could not be explained entirely by differences in the number of primary particles contained in the aggregate. 7. Aggregates grow through random addition of small flocs of relatively high density rather than initial solid particles. This conclusion is in agreement with the experimental results by Kusuda et al. [65].
4.4.4 Discussion and Concluding Remarks In Section 4.2, the processes of flocculation were presented with emphasis on flocculation due to velocity gradients or shear rates because they provide the dominant mechanism of aggregate formation in estuaries and other natural and manmade open channels. We showed that in turbulent flows, the shear rates can be expressed as functions of the energy dissipation per unit mass or per unit volume of fluid. Moreover, the same factors responsible for interparticle and interaggregate collisions mold the size, density, strength, and related properties of the flocs and aggregates. The various theories and experimental results relate these properties to the rate of energy dissipation. However, there are significant variations in the associated coefficients and the associated exponents. For the associated coefficients, these differences depend on the type and concentration of the coagulants, and for the associated exponents they depend on the assumptions regarding the details of the
Chapter | 4 The Hydrodynamic Transport Processes
153
flocculation process and, in general, the differences depend on the types of experimental equipment and procedures. For example, the coagulation factor, that is, the probability of two units sticking together upon contact, and the estimated value of effective energy dissipation, , are two sources of considerable controversy and uncertainty. One is reminded that in the commonly used paddle flocculators, as well as in other mixing systems, varies tremendously within the flow field. Therefore, the presented theories and formulas should be taken in a qualitative rather than in a strict quantitative sense and as guide for the hydrodynamic process of flocculation. In spite of these differences, the cited theories and experimental results lead to some very fundamental and important conclusions regarding the formation and the properties of flocs and aggregates that can be used as a basis for sediment-related predictions. The following are some of the most important of these conclusions: 1. There is ample evidence for a discontinuous structure of multiple-stage aggregates rather than a continuous aggregate growth. The smallest unit of zero order, known as floc, is the densest and is composed of individual sediment particles. Flocs combine into first-order aggregates which, in turn, combine into second-order ones and so on. 2. The density and strength of the aggregates decreases rapidly with increasing size and order of aggregation. 3. In turbulent flows the controlling shear rate can be represented by Equation 4.77. 4. The basic unit of an aggregate is the floc whose properties have been determined by the highest shear rate the sediment has been subjected to. Considering estuaries, in particular, it seems logical to assume that the basic building unit has been formed in the near bed zone of the river, where the shear rates are expected to attain their maximum value. 5. The upper limit of aggregate size is controlled by the forces induced by the velocity gradients rather than by collisions between aggregates. 6. Aggregate breakup takes place predominantly through shear within the mass, i.e., is through breaking of the joints of the composing smaller order aggregates and/or flocs. 7. It follows from the preceding conclusion that the Bingham yield stress, which has been used in some studies as a criterion for aggregate strength, is not a representative measure of the latter. Indeed, excluding the rather rare cases of very high fine sediment load, suspended aggregates behave as independent units subjected individually to the disrupting shear rates. In contrast, the Bingham yield stress is generated in suspensions with sufficiently high concentrations of cohesive sediment for the development of a continuous network of flocs and aggregates. This aspect will be discussed in more detail in Chapter 5 on the rheological properties of cohesive sediments.
Chapter 5
Rheological Properties of Cohesive Sediment Suspensions 5.1 Importance of the subject It has been well known that suspended sediment causes an increase of the coefficient of viscosity of a mixture. In addition, cohesive sediments at concentrations above a certain limit impart shear strength, B, to the suspension, known as yield shear strength or Bingham strength. For shear stresses below B the suspensions behave as a plastic solid, whereas for shear stresses higher than B they behave like fluids with constant or variable viscosity. Fluids exhibiting such strength are known as Bingham plastics and belong to the general category of non-Newtonian fluids. They were first studied in detail by Bingham, who described the shear strength-velocity gradient relation by Equation 5.1 [6, 61]:
B B
du dy
(5.1)
where is the applied shear stress, B is the yield strength of the suspension, B is its dynamic coefficient of viscosity, and du/dy is the velocity gradient or shear rate in a direction normal to the direction of the flow. Bingham conducted an extensive study on the viscous properties of different fluids, mixtures, and suspensions using various types of Oswald capillary viscometers. Cohesive sediment suspensions at concentrations high enough to develop yield strength, usually above 20,000 ppm or 20 g/L, are rather rare in estuaries. They have been observed, however, in undercurrents in lakes and reservoirs as a result of sediment-generated density currents and in coastlines caused by mudflows. Such high concentrations with their associated yield strengths may have a significant effect on the flow pattern. For example, a mud layer may develop around the interface within which the flow-induced shear stresses are smaller than B so that there is no relative motion within that layer. The latter then moves as a solid plug, thus preventing any mixing between the muddy undercurrent and the clean water layer above [24]. In estuaries, and in other open channels, the Cohesive Sediments in Open Channels Copyright © 2009
155
156
Cohesive Sediments in Open Channels
sediment concentration is normally low enough for the suspension to behave as a Newtonian fluid. The yield strength is controlled by the same interparticle forces that are responsible for flocculation and for the hydraulic behavior of cohesive sediments. However, the aggregate interaction in a viscometer is quite different from the process of surface erosion of a deposited bed. The development of yield strength requires a continuous honeycombed structure of aggregates extending all the way to the walls. Low-concentration suspensions contain isolated flocs and higher order aggregates. In contrast, as will be shown in Chapter 6, surface erosion takes place by removal of individual flocs and/or aggregates from the bed. Yield strength depends on concentration, whereas some erosion and deposition parameters do not. Therefore, the yield strength could not be used as a criterion of erodibility or deposition. Nevertheless, the dynamics of Bingham fluids and their associated yield strength reveal some important aspects of cohesive sediment behavior. For example, the viscosity and the yield strength for various concentrations have led to estimates of the order of aggregation, the density, and the strength of the aggregates. Table 4.2 in Section 4.4.1, for four different sediments, was based on analyses of rheological data [61]. For this reason this chapter on the dynamics of Bingham fluids is introduced.
5.2 Basic properties of sediment suspensions and methods of evaluations In 1911 A. Einstein derived the following theoretical equation for the viscosity of dilute rigid and wetted spherical particles in an incompressible liquid [23]:
s 1 2.5CV
(5.2)
where CV is the fraction of the total volume of the suspension occupied by the spheres, s is the viscosity of the suspension, and is the viscosity of the liquid medium. The factor 2.5 applies to spheres. The equation is valid for CV 1. Dispersed deflocculated cohesive sediment suspensions behave like Newtonian fluids with a coefficient of viscosity increasing with sediment concentration. H. A. Einstein was among the first to study the rheological properties of such suspensions as part of a study of highly concentrated underflows using a modified version of an Oswald-type capillary viscometer [24]. Figure 5.1 shows the experimental results for dispersed South Carolina clay at the indicated concentrations. The abscissa, p, is the driving pressure in cm of water, while the product of the viscosity coefficient, B, and the average velocity, vm, has been used as a measure of the shear rate. Figure 5.2 shows an example of similar experimental results for a constant concentration of about 120,000 ppm and for the indicated flocculent content. It is observed that even the minimum addition of flocculent causes a significant
Chapter | 5 Rheological Properties of Cohesive Sediment Suspensions
Distilled water 10°C Distilled water 16°C Distilled water 24°C 12gr.solids in 100cc mixture 23°C 6 gr.solids in 100cc mixture 23°–31°C 3 gr.solids in 100cc mixture 25°C
1.1 1.0
157
λ
0.9
B.Vm�g/s2
0.8 λ
0.7 0.6 0.5
λ
0.4 0.3
0.2 0.1
λ 13
0.0 20 10
0
10
20
30
40
50
60
70
80
∆p-Centimeters of water Figure 5.1 Stress-shear rate results for dispersed cohesive sediment suspensions [24, Fig. 2].
increase of the yield strength as measured by the maximum water head at zero velocity. The yield strength increases with increasing flocculent content but at decreasing rates, and it appears as if there is a certain limit beyond which any more flocculent addition does not have any effect on the yield strength. In all cases the suspensions behave like Newtonian fluids for pressure heads above those corresponding to the yield strength given by Equation 5.1. Figure 5.3 shows a schematic shear stress distribution in a uniform density underflow in a deep reservoir or lake. Within a distance hs above and below the zero stress point, the shear stress is less than the yield strength of the suspension. Therefore, there is no relative motion within that zone of thickness 2hs, which moves as a solid plug, thus preventing any mixing between the two layers [24]. The viscosities and the yield strengths of sediment suspensions and, in general, Bingham plastic fluids, have been studied by two types of experimental setups: the capillary tube and the rotating-cylinder viscometer. The first is the oldest system. The fluid is forced through a capillary tube; and from the applied pressure head and the flow rate, the shear stress, the viscosity coefficient, and
1.1
Sample No. 8 with 11.97 g of solids in 100°C mixture Dispersed 1 Drop NaCl solution added 2 Drops NaCl solution added 3 Drops NaCl solution added 12.5°C 3 Drops NaCl solution added 22.5°C 3 Drops NaCl solution added 32.5°C
1.0 0.9
te
r
0.7
sti
lle
d
wa
0.6 0.5
Di
�B.Vm�g/s2
0.8
0.4 0.3 0.2 0.1
�13
0.0 �20 �10
0
10
20
30
40
50
60
70
80
∆p-Centimeters of water Figure 5.2 Stress-shear rate results for flocculated cohesive sediment suspensions [24, Fig. 3].
Lake-surface �,� Interface �i �y
hs
� � ��, �B
hs
�
�y
A
�
B
yo
S � tan �
h
y �b
Figure 5.3 Shear stress in a uniform underflow [24, Fig. 4].
Unde
rflow
Lake-bed
Chapter | 5 Rheological Properties of Cohesive Sediment Suspensions
159
the yield strength are deduced. Figure 5.4 shows the main part of the Oswald viscometer described in Ref. [6]. The flow rate is evaluated by measuring the time necessary for water volume Vw, contained between points c and d, to pass through d. Figure 5.5 shows a more detailed setup used by Einstein in his studies on clay suspension [24]. The flow through the capillary tube is laminar. For fluids and/or suspensions possessing yield shear strength, there is a core at the center of the tube moving as
c Vw d
Figure 5.4 The Oswald viscometer [6].
Filling-tube Valves D
E
Stirring-tube F Marks
A C Airchamber
Thermometer B
Constant temperature water-bath Ostwald viscosimeter
Watermanometer Figure 5.5 Capillary viscometer system [24, Fig. 1].
Cohesive Sediments in Open Channels
160
a plug. Therefore, the shear rate can be represented by the average flow velocity, va, or the volume flow rate Vw /t. The following equation developed by Babbitt and Caldwell in terms of shear stresses was used by Einstein [24]:
4 D 1 y 4 Vw cc c w y 3 3 3 t w
(5.3)
The term , defined as the coefficient of rigidity, is the slope of the shear stressvelocity lines in Figure 5.1 and Figure 5.2, Dc is the diameter of the capillary tube, w is the wall stress, and cc is a proportionality constant. The original Bingham equation for the same viscometer in terms of pressure is [61] Vf
t
py4 πrc4 4 p py 3 8 3 3 p
(5.4)
where p is the applied pressure, py is the pressure necessary to overcome the yield strength of the suspension, rc and are, respectively, the radius and length of the capillary tube, and Vf is the volume of the fluid passing in time t. In both Equation 5.3 and Equation 5.4, the last term in the parentheses on the right side is negligible because p is much higher than py, with the exception of pressures very close to the yield pressure. Therefore, this last term can be neglected. Krone used the same equation for the analysis of his experiments on estuarine cohesive sediments, but in terms of the hydraulic head instead of the pressure and without the last term. The modified equation has the form [61]
2 h s vv 8v B 4 h s h 4 B o 3 2 g πrc g t
(5.5)
where h is the total head, vv2 / 2 g is the velocity head, hB is the head required to overcome the Bingham yield strength, vv is the flow velocity and ho is the average head in the viscometer. The ratio of the sediment density, s, to the fluid density, , was used to convert all heads into heads of water. The term in parentheses on the right side of Equation 5.5 contains the geometrical characteristics of the system and remains constant after it is set and determined. Figure 5.6 shows some of Krone’s experimental results using the capillary viscometer and San Francisco Bay mud as sediment with the mass concentration, C, in gcm3 as the abscissa and the ratio of the coefficients of viscosities B / as the ordinate for the indicated three salinities [61]. The semilogarithmic plots are straight lines with slope ki. It is observed that the coefficient of viscosity for the suspension, B, increases with salinity, but that
Chapter | 5 Rheological Properties of Cohesive Sediment Suspensions
161
3.0 Salinity, g/L 33.8 20.8 �0.012
Relative differential viscosity, µB/µ�
2.5
2.0
1
Ki
6.3
�
5
5.9 7 .2
1.5
4
1.0 0.9 0.8
0
0.02
0.04
0.06
0.08
0.10
Sediment concentration, C, g/cm3 Figure 5.6 Relative viscosity of San Francisco Bay mud suspensions from capillary viscometer [61, Fig. 7].
above a certain salinity limit the effect of the latter diminishes rapidly. The suspension viscosities fit the power equation, which differs from Einstein’s equation (5.2).
B exp(ki C )
(5.6)
Krone developed the following model to reconcile these last two equations. Equation 5.2 is ideally valid for infinitesimal volume concentrations. The total volume concentration of sediment, Cv, was divided into n fractions, each with concentration Cv Cv /, nwhere n is an integer tending to infinity. For each Cv Equation 5.2 was assumed to be valid. Then for the first increment we have
s1 (1 2.5Cv )
(5.7)
The viscosity for the second increment is
s 2 s1 (1 2.5Cv ) (1 2.5Cv )2
(5.8)
It follows that for the final increment the suspension viscosity will be
sn (1 + 2.5Cv )n
(5.9)
Cohesive Sediments in Open Channels
162
For infinitesimal values of Cv n becomes infinite, in which case the last equation reduces to
B exp(2.5Cv )
(5.10)
According to Equation 5.7, it follows that the mass concentration C is related to Cv by 2.5Cv ki C
(5.11)
Next, Krone estimated the aggregate density from the value of C for which Cv 1, that is, when the entire suspension volume is occupied by aggregates of density a. The total suspension volume, Vs, then becomes equal to the aggregate volume Va. If Vp is the total volume of solid particles of density p, then the mass continuity equation gives
Va a Vp p (Va Vp )
(5.12)
The left side of Equation 5.12 is the total mass of the aggregates, the first term of the right side is the mass of solids, and the last term is the mass of the fluid whose net volume is Va Vp. Equation 5.12 can be written as
a
Vp p Va p
( p )
(5.13)
Next, by definition, the ratio Vp p / Va is the mass concentration of suspended sediment, C, so that the sediment mass conservation equation obtains its final form: a
C ( p ) p
(5.14)
C 2.5 / ki for Cv 1 and with ki determined from plots like the one in Figure 5.6. For most sediments p can be taken equal to 2.65 g/m and the saline water density equal to 1.025 g/cm3. On the basis of the preceding considerations, Krone obtained the data in Table 5.1 for seven estuarine muds, where Cvp and Cvf are the volume concentrations of the particles and flocs, respectively, and f is the density of the flocs. In the third column of Table 5.1, Cvp Cv f 2.5 / ki and the numbers represent sediment concentration within the aggregates. Because of the small diameter of the capillary tube, one should note at this point that only aggregates of zero order, that is flocs, could be formed. In the fourth column Cvp Cvf C / pCvf and this represents the fractions of flocs that are composed of mineral particles. Finally, the floc densities were estimated from Equation 5.14. It is observed that the primary or floc densities for all
Chapter | 5 Rheological Properties of Cohesive Sediment Suspensions
163
Bingham shear strength, τB, dynes/cm2
Table 5.1 Floc densities in the capillary viscometer [61, Table IV] Sample
ki cm3/gr
C/Cvf gr/cm3
Cvp/Cvf
f gr/cm3
Wilmington District
6.83
0.366
0.138
1.250
Brunswick Harbor
11.1
0.225
0.085
1.164
Gulfport Channel
8.56
0.292
0.110
1.205
San Francisco Bay
6.31
0.396
0.150
1.269
Delaware River
5.49
0.455
0.172
1.305
Potomac River
3.73
0.670
0.253
1.437
White River
8.24
0.304
0.115
1.212
Salinity, g/L 12
33.8 20.8 20.8 �0.116
10 8 6 4 2 0
0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 Sediment concentration, C g/cm3
Figure 5.7 Bingham yield shear strength of San Francisco Bay mud from capillary viscometer measurements [61, Fig. 14].
seven sediments are quite close to each other with an average of 1.263 and a standard deviation of 0.089. Figure 5.7 shows Krone’s results for San Francisco Bay mud obtained from viscometer measurements of Bingham shear strength versus sediment concentration. There seems to be a linear increase of the yield strength with sediment concentration for most points as well as an increase with salinity up to 20 gL1 or 20,000 ppm. However, the data on the salinity effect seem to be rather inconclusive. The latter is more clearly displayed in Figure 5.8 for four different cohesive soils and for the indicated range of sediment concentration [61].
Cohesive Sediments in Open Channels
164
Bingham shear strength, τB, dynes/cm2
For all soils, the critical salinity limit above which there seems to be negligible effect is in the neighborhood of 1 gL1 or 1000 ppm. This is an additional proof that salinities above that value have negligible effect on the mechanical properties of suspended aggregates and deposited beds in estuaries. However, salinity may have a very significant effect on the density currents, which, in turn, determine the shoaling sites and the rates and extent of shoaling. Referring now to Figure 4.7 and Table 4.2 in Section 4.4.1, it can be concluded that only flocs or zero-order aggregates could be formed where capillary viscometer experiments reveal shearing rates of 100s1 or more. The shear strengths of these flocs, summarized in Table 5.2, are of a magnitude similar to
12
Shear strength at 20.8 g/L 33.8 g/L
10 8 Legend 6
Wilmington Dist. Brunswick H. Gulfport CH. San Francisco Bay Sediment concentration 0.065–0.070 g/cm3
4 2 0
0
1
2
3
4
5
6
7
8 20.8
Salinity, g/L Figure 5.8 Changes of Bingham strength with salinity [61, Fig. 18].
Table 5.2 Estimated shear strengths of aggregates in a capillary viscometer [61, Table V] Sample
Shear Strength dynes/cm2
Wilmington District
21.0
Brunswick Harbor
33.7
Gulfport Channel
45.9
San Francisco Bay
21.8
Delaware River
11.0
Potomac River
19.0
White River
48.6
33.8
Chapter | 5 Rheological Properties of Cohesive Sediment Suspensions
165
the strengths of recently deposited and naturally consolidated cohesive estuarine sediments. According to Krone, it appears unlikely that aggregates could develop a closer pack. Krone’ test results led to a number of speculations, which will not be discussed here as they are not directly related to the primary objectives of this book. However, he derived an estimate of the shear strength of the flocs by extrapolating the Bingham strength plots to concentration values equal to those in Table 5.2 and using his model described by Equation 5.7 through Equation 5.13. These strengths are substantially higher than the shear-stress-causing erosion of deposited mud composed from the sediments of Table 5.2. This is to be expected, because, as will be demonstrated in Chapter 6 and Chapter 7, deposited beds have a loose honeycombed structure, so that erosion by flowing water takes place by breaking of aggregates at their joints and not by shearing the primary flocs themselves [102, 104, 105]. Krone also showed a strong correlation between the shear strength of the flocs and their cation exchange capacity (CEC), shown in Figure 5.9. With the exception of the Potomac River, all other data seem to fall on a straight line indicating a rapid increase of floc shear strength with CEC [61]. As was pointed out earlier, the capillary viscometer is good in determining the properties of primary aggregates or flocs in a simple way; however, because of its small diameter and the subsequent high shear rates, it does not allow the formation of higher order aggregates. For the study of higher order aggregates, therefore, it was necessary to utilize a rotating cylinder type viscometer with a flow field of relatively large thickness to allow formation of aggregates of higher order. The cylinder viscometer was
Bingham shear strength, τB, dynes/cm2
70 60 50
WR GC
40 BH 30 20
DR
10 0
SF PD
PR
0
10
20 30 40 50 60 Cation exchange capacity, meq/100g
70
80
Figure 5.9 Relation between floc shear strength and cation exchange capacity for sediments in Table 4.2 [61, Fig. 19].
Cohesive Sediments in Open Channels
166
Micrometer dial
Torsion wire Mirror
Light lever null indicator Inner cylinder positioning tube
Magnetic damper
Bearings
Suspension Inner cylinder Rotating cylinder
Stepped pulley
Stopper Air Pulley & turntable Bearings
Variac & rectifiers
Drive motor
Cam & switch for revolution counter
Figure 5.10 Rotational cylinder viscometer [61, Fig. 20].
designed by Krone based on the viscometer described by Mallock. It is outlined in Figure 5.10 and described in detail in Ref. [61]. The system consists of two co-centric cylinders, the outer one of which is supported by the rotating turntable, while the inner one remains stationary hanging by a torsion wire from which the torque is measured. The radii of the cylinders were 3.08 cm and 1.88 cm, thus allowing an intercylinder annular thickness of 1.20 cm. The turntable speed ranged from 8 to 180 rpm. The flow was found to be laminar for speeds of rotations less than 100 rpm, while flow instability was observed for speeds between 100 and 110 rpm.The following relationship between shear stresses and shear rates for the rotating cylinder viscometer was developed by H. A. Einstein [24]. In cylindrical coordinates, this equation reads
Chapter | 5 Rheological Properties of Cohesive Sediment Suspensions
,
22
, ,
Increasing rpm Decreasing rpm Change of symbol indicates mixing
20 Dial reading, D
167
18 16 H � 24.3 cm T � 22.7°C
14
Sediment concentration 0.112 g/cm3
12 11
0
20
40
60
80
100
120
140
160
180
Outer cylinder rotation, rpm Figure 5.11 Example of rotating cylinder viscometer data for Wilmington District sediment [61, Fig. 23].
du u B B r r dr r
(5.15)
where ur is the tangential velocity and r is the radius. The torque Tr, which remains constant throughout the flow field, is given by Tr 2π Hr 2 (5.16) in which H is the height of the fluid in the annular space. Integration of Equation 5.15 leads to the following final equation relating torque, angular velocity, (), Bingham shear stress, (B), the radii, (r1 and r2), of the inner and outer cylinders, and H:
4π H ln r2 4π H r1 Tr B 1 1 1 1 B r2 r 2 r22 r22 1 1
(5.17)
In Equation 5.17, the quantities in the brackets represent the geometrical characteristics of the system, and, therefore, they are constant. The angular velocity, , of the outer cylinder and the torque are proportional to the rotation per minute and to the wire torsion, respectively, so that both can be readily determined through calibration. Therefore, from a plot of dial reading, D, versus , the values of B as the slope of the curve and of B, as the intercept on the ordinate axis, can be obtained. Figure 5.11 shows an example of such a plot. It is observed that several slopes are possible for each sediment suspension. Certain branches correspond to increasing and some others to decreasing rates of strain. The suspended
Cohesive Sediments in Open Channels
168
sediment was mixed at intervals between branches, and more specifically, at around 43 and 180 rpm. The experimental details are presented in reference [61]. Each line of Figure 5.11 corresponds to a particular order of aggregation. For each order, the corresponding value of ki can be estimated from the plot of the relative differential viscosity versus the sediment concentration, as in Figure 5.6. Figure 5.12 shows a similar plot for Wilmington district sediment, but from rotating cylinder viscometer measurements. Each ki value corresponds to a particular order of aggregation. Using ki from Figure 5.12 the corresponding volume concentration for each order of aggregation can be determined from Equation 5.11. Thus, for a given value of the mass concentration, C, the higher ki, the higher will be the volume concentration, Cv. Since a higher degree of aggregation is associated with a higher porosity and proportion of voids, for the same sediment concentration by weight, Cv is expected to also increase; therefore, each higher value of ki corresponds to a higher degree of aggregation. 1.0 0.9 0.8 0.7
14 .4
22.
0.4
8
1.1
0.5 ki � 3
Relative differential viscosity µB/µ�
0.6
0.3
0.2
0.1
0
0.02
24
26
28
0.10
Sediment concentration, C,
0.12
0.14
g/cm3
Figure 5.12 Relative viscosities of Wilmington District sediment determined by rotating cylinder [61, Fig. 25].
Chapter | 5 Rheological Properties of Cohesive Sediment Suspensions
169
The viscosity of the suspension, B, and the Bingham shear stress, B, can be evaluated from a plot of the measured torque and the angular velocity of the outer cylinder, . The measured torque is given from the torque versus dial reading plot, while the angular velocity, , in rad/s is related to the number of revolutions per minute, N, by the relation:
2π N 60
(5.18)
Finally, the aggregate density for each order of aggregation can be evaluated from Equation 5.14 for a value of sediment concentration, C, corresponding to Cv 1, in exactly the same way as done for the capillary viscometer experiments. From the analysis of these last data, Krone derived the aggregate densities of Table 4.2 (Section 4.4.1). Figure 5.13 shows a plot of Bingham yield shear stresses versus suspended sediment concentration for a White River sediment sample. The lines represent the envelope of the data points. The slope of these lines, as well as similar lines for the other sediments, is of the order of 5/2 leading to the general equation
B K yC 5 / 2
(5.19)
The coefficient Ky was considered to be a function of the energy required for the dispersion of 1 g of sediment. From the data of Figure 5.13, and from similar data from other sediments, Krone estimated the aggregate shear strength for each order of aggregation by extrapolating the appropriate straight line to the concentration, C, corresponding to Cv 1. The latter value of C can be found from Equation 5.11. The aggregate shear strengths, thus estimated, have been classified in Table 4.2. It is noted that in Table 4.2 the densities and strengths of zero-order aggregates or flocs are the ones determined by the capillary viscometer and are also shown in Table 5.1 and Table 5.2.
5.3 Concluding remarks In this chapter, the viscosities and Bingham yield strengths of cohesive sediment suspensions were related to the density and strength of aggregates of various orders. For studies of similar nature, the rotating cylinder viscometer is the appropriate tool for the development of higher order aggregates. The capillary viscometer can only generate zero-order aggregates, i.e., flocs. These studies led to the following conclusions: 1. Salinity higher than 1 g/L or 100 ppm seems to have negligible effect on both the viscosity and the yield strength of the suspensions. 2. Both viscosity and yield strength increase with increasing sediment concentration. The Bingham strength, in particular, was found to be proportional to the 5/2 power of the sediment concentration.
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170
4.0 3.0
Bingham shear strength, �B, dynes/cm2
2.0
1.0 0.9 0.8 0.7 0.6
5
0.5 2
0.4 0.3
0.06 0.07 0.08 0.09 0.10
0.05
0.04
0.03
0.01
0.1
0.02
0.2
Sediment concentration, C, g/cm3 Figure 5.13 Bingham shear strengths of White River sample in salt water from rotating cylinder tests [61, Fig. 35].
3. There is also a strong increase of the Bingham strength with CEC. This is to be expected, since higher CEC values are associated with higher physicochemical bonds, which are responsible for the yield strength and the differential viscosity of the suspensions. 4. Krone’s model for the evaluation of the densities and strengths of the various-order aggregates, particularly the extrapolation to densities for which Cv 1, is based on certain reasonably realistic speculations. Therefore, any values obtained from a similar model should be considered as approximate. Nevertheless, the model provides a powerful tool for estimating the order of magnitude of the aggregate properties and for comparing the various sediments. 5. The shear strength of the flocs is much higher than the erosive shear stresses on a deposited bed. This observation led initially to the conclusion, confirmed
Chapter | 5 Rheological Properties of Cohesive Sediment Suspensions
171
by subsequent detailed studies, that erosion takes place by the rupture of interaggregate bonds. Moreover, according to the same studies on erosion and deposition of cohesive sediments, the shear stresses for the initiation, degree, and rates of erosion are independent of the mass density and the mass strength of the bed, whereas the Bingham shear strength increases rapidly with sediment concentration. This means that neither the differential viscosity nor the Bingham strength can be used as a criterion for erodibility of deposited cohesive sediment beds. They can be helpful, however, for comparative studies of aggregate properties and for an estimate of aggregate densities, settling velocities, and strengths to be used in mathematical models for cohesive sediment transport studies.
Chapter 6
Erosion of Cohesive Soils
6.1 Introductory remarks In Section 4.1.1, the general three-, two-, and one-dimensional equations for cohesive sediment transport processes were developed. That section showed that the process involves both erosion (including resuspension) and deposition. We also concluded that there could be three special cases: (a) erosion without simultaneous deposition, (b) deposition without simultaneous erosion and/or resuspension, and (c) neither erosion nor deposition. Early laboratory investigations on the resuspension of deposited San Francisco Bay mud in a flume revealed that the rate of increase of suspended sediment, coming from resuspension, decreases as the resuspension continued, approaching asymptotically a constant value [62]. This eventual steady-state situation was interpreted as the stage at which the rates of deposition become equal to the rates of resuspension of deposited sediment. This interpretation was based on the concept of coarse sediment transport as described by the Einstein’s bed load function [25]. The subsequent fundamental research proved that this was not the case. It was shown in fact that, apart perhaps from an initial stage, erosion and deposition of cohesive sediments do not take place simultaneously; instead, the entire process of cohesive sediment transport in tidal estuaries takes place by successive steps of deposition and resuspension following the tidal cycle. There are, however, cases where we are concerned only with either deposition or erosion exclusively. These are the cases of stability of natural and artificial channels with cohesive boundaries subjected to flow-induced shear stresses and channels carrying water with a heavy sediment load. The long-term stability of such channels is of primary concern to the hydraulic designer. But there are two distinct cases: safety against scouring and protection against shoaling. The first applies to channels with boundaries possessing substantial cohesion but carrying sediment-free water at velocities high enough to cause scouring; the second is the case of canals with relatively stable boundaries under the expected flow-induced shear stresses but with water carrying a heavy sediment load. Cohesive Sediments in Open Channels Copyright © 2009
173
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For the stated reasons and in spite of the fact that erosion and deposition constitute two phases of the same process and they are controlled by the same sets of hydrodynamic and physicochemical forces, they will be discussed separately. In fact, each phase has been investigated individually. Moreover, there is substantial empirical and semi-empirical information, particularly for erosion, based on laboratory and field data. Some of this early information can still be quite useful, if properly interpreted; however, the limitations of each formula and table need to be well understood before any design application to avoid major errors.
6.2 Erosion of consolidated cohesive soils 6.2.1 Early Empirical Information Some of the most well-known early work on channel stability and early formulas based on limited small-scale laboratory experiments were discussed in Refs. [116, 117], and some representative examples were briefly summarized in Section 1.2. This early information was presented either in tabular form or by totally empirical formulas. The first attempts to establish safe design criteria for canals for various purposes go back to the early part of the 19th century. These criteria were given in terms of critical velocities and/or critical shear stresses for scouring and shoaling while the canals were described by geometry and some gross soil properties. The first values of critical velocities for erosion control, based on field observations, were given by Du Buat in 1816 for various soil types [19]. These soils were vaguely described by a general consistency. For example, the only cohesive soil was classified as “dark clay fit for pottery” with specific gravity 2.64, i.e., is highly dense, for which he recommended a critical velocity for scouring of 0.35 ms1 and for deposition 0.27 ms1. In 1916 Etcheverry gave a set of critical velocities ranging from 2.75 to 3 ft.s1 for average loams and from 4 to 5 ft.s1 for stiff clays. At about the same time, a special committee of the American Society of Civil Engineers (ASCE) published estimates by experienced engineers of that time for critical design velocities reported by Fortier and Scobey [33]. These velocities ranged from 1.5 to 3.0 ft.s1 (or 0.5 to 1.0 ms1) for fine sandy loam and from 2.5 to 5.0 ft.s1 (or 0.75 to 1.50 ms1) for stiff clays. Similar values were given later by the Irrigation Hydraulics Committee of the ASCE on the basis of field data for canals with clear water and water carrying cohesive and noncohesive silts [33]. The highest recommended permissible critical velocities were for cohesive clays in suspension and the lowest for suspended noncohesive silts. The recommended critical velocities for channels carrying clear water were in between the previous two. In all cases the soil properties were represented only by a classification. In 1936 Russian engineers published critical velocities and critical bed shear stresses [70]; however, in addition to soil classification, they also included the soil density of the cohesive bed as the main soil parameter. These values were
175
Chapter | 6 Erosion of Cohesive Soils
comparable but somewhat higher than similar values obtained by the American engineers. For example, for stiff clays the Russians suggested limiting velocities between 4.0 and 5.5 ft.s1 (1.2 to 1.7 ms1). For a similar soil classification the recommended American values ranged between 3 and 5 ft.s1 (0.9 to 1.5 ms1). One should note that, although, with the exception of Russian engineers, the critical average velocities were used as the only or main parameter for channel stability as late as the middle of the 20th century, the significance of the boundary shear stress was recognized much earlier. In fact, it was Du Boys who introduced the concept of tractive force in 1879 [18]. Also, as early as 1816 Du Buat introduced the concept of the shear resistance for the first time, which means essentially the force per unit area of the canal boundary [19]. The previously mentioned data can be found in the quoted references and in a number of classical books on sedimentation and hydraulic design [125]. Table 6.1 is presented here with the critical velocities in meters per second and the shear stresses in Pascal (Pa). Section 6.3 shows that classification and density alone mean very little as far as erodibility and deposition are concerned. It became clear from the extensive summary of the physicochemical properties of cohesive sediments in Chapter 4 that the interparticle and interaggregate forces are the ones controlling both erosion and deposition. Moreover, the mechanism of failure of a soil mass by shear applied through a large plane is drastically different than the mode of failure by surface erosion. Therefore, the recommended safe values, as well as the empirical formulas based on field data, which will be discussed later, can be used only under conditions similar to those the recommended values have been based on. Otherwise, serious errors may occur. Empirical formulas for safe canal design were also developed as early as the 19th century. They are often referred to as regime theories, and they are essentially based on field data from irrigation and other water conveyance canals that displayed various degrees of stability over the years. Several such theories and formulas can be found in a number of textbooks, papers, and reports [30, 37, 73]. However, their discussion is beyond the scope of this book. A good example of
Table 6.1 Critical velocities and shear stresses for open channels (after the Russian engineers, [70]) Void Ratio
2.01.5
Clay Type
Vc
c
Vc
c
Vc
c
Vc
c
Sandy clays (50% clay)
0.45
1.92
0.90
7.50
1.30
15.7
1.80
30.2
Heavy clays
0.40
1.50
0.85
6.75
1.25
14.6
1.70
27.0
1.20.6
0.60.3
0.30.2
Clays
0.35
3.30
0.80
5.94
1.20
13.50
1.65
25.40
Lean clays
0.32
0.96
0.70
4.6
1.05
10.25
1.35
17.00
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Cohesive Sediments in Open Channels
such an approach is Kennedy’s theory and formula developed by the end of the 19th century and based on a large volume of data from several irrigation districts in India [55]. His original formula is
Vc k yok
(6.1)
in English units, where Vc is the critical velocity for stability, yo is the depth of flow, k is a dimensional empirical coefficient, and k is an exponent. The coefficient and the exponent depend on the channel geometry, soil composition, and on the general environmental conditions. Therefore, they are expected to change from place to place. Kennedy initially recommended k 0.84 and k 0.64; later he suggested values ranging from 0.67 to 0.23 for k and 0.53 to 0.64 for k. More recent studies reported by Leliavsky included the effect of other variables describing the channel geometry and boundary resistance [73]. Kennedy’s regime formula, as well as other similar formulas, may be useful for conditions sufficiently close to the ones they have been based on. They should by no means be generalized and applied indiscriminately to any other soil and environment. Moreover, Equation 6.1 does not indicate whether the term stable channel means safety against scouring or avoidance of objectionable shoaling. Apart from the direct usefulness of the regime formulas and the quantitative design criteria, these early irrigation engineers provided the following important qualitative information from their field data, which much later was proven to be consistent with fundamental research results: 1. Critical velocities against deposition are, as a rule, lower than critical velocities against scouring of beds composed of sediment of similar consistency to that of the transported sediment in suspension. This difference was detected in 1883 by Suchier and reported and confirmed in 1926 by Fortier and Scobey [33] as well as by Hjulstrom in 1835 [117]. Fortier and Scobey also recognized that Kennedy’s equation 6.1 was intended to be a criterion for safety against siltation and that its use to provide safety against scouring would lead to an over-design. 2. The gap between the lower limit of velocities for deposition and the upper limit for erosion increases with decreasing sediment size. 3. Cohesive materials with the same classification and very similar grain size distribution may display widely different resistance to erosion. Since that early time, it was thus recognized that other factors, in addition to the mechanical composition, also control erosion of cohesive soils. 4. The resistance to erosion of artificial channels may increase considerably with time. 5. Although the average velocity has been used as a criterion for canal design, it had been known that shallower canals have higher sediment transporting capacities.
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Chapter | 6 Erosion of Cohesive Soils
The last observation suggested at that time that erosion and deposition are controlled by the flow-induced shear stresses rather than by the average velocities. Moreover, the same recommended safe velocities obtained for a particular depth can be extended to a different depth so that the shear stresses corresponding to the second depth will be similar to those corresponding to first one [116, 117]. As an example, we consider two wide channels with a hydraulic radius equal to their depth of flow, yo1 and yo2, respectively, but with equal bed shear stresses b1b2. Application of Manning’s equation gives
y 1/6 V2 o 2 yo1 V1
(6.2)
It is obvious that for yo2 yo1 then V2 V1. Therefore, for the same bed shear, the canal with the smaller depth should have a lower velocity to maintain the same shear stress than the canal with the larger depth, and vice versa. Consider next the same canals with equal velocities. The relation between the two boundary shear stresses will then be
yo1 1/3 b 2 yo 2 b1 V
(6.3)
where the subscript V indicates constant velocity. For a constant discharge per unit length, q, the relationship of the boundary stresses are given by
y 7/3 b 2 o1 y b1 o2 q
(6.4)
where, again, the subscript q means constant discharge. So, for either constant velocity or constant discharge, the channel with the lower depth will be subjected to higher boundary stresses. Moreover, it can be shown that the coefficient k in Kennedy’s Equation 6.1 can be related to the critical boundary shear stress, bc, by k
1.486 bc n w
(6.5)
where w is the unit weight of the water and n is Manning’s friction coefficient with English units. As an example, consider n 0.024, which is a representative value for irrigation canals in silty clay and similar soils; then for Kennedy’s recommended range of k between 0.84 and 0.23, Equation 6.5 gives a corresponding range of critical shear stress values between 0.0009 and 0.0107 psf. According to recent research results on the deposition and resuspension of cohesive sediments, to be presented and discussed in Chapter 7, this is the order of magnitude
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Cohesive Sediments in Open Channels
of the range of the boundary shear stress at which a substantial portion of suspended sediment starts depositing. Considerably higher stresses are needed for erosion of even recently deposited clays. We can conclude, therefore, that indeed Kennedy’s Equation 6.1 should be used as a criterion for safety against siltation only and not for safety against erosion.
6.2.2 More Recent Field and Laboratory Studies It became evident from these early empirical studies that a mere soil classification and/or density are insufficient criteria for resistance to erosion and that other parameters representative of the soil consistency and internal structure have to be taken into account. This second phase, so to speak, attempted a correlation between critical stresses for erosion with some soil parameters. It was natural to start first with the Atterberg limits and the macroscopic strength as such correlation parameters. Field studies by the U.S. Bureau of Reclamation in 1953 showed that there is little correlation between critical tractive force and the mean grain size of the sediment [148]. The same studies revealed an increase of the resistance to erosion with an increase of the plasticity index and a higher erosive resistance of ephemeral channels subjected to periodic drying compared to continuously wet ones. An erosive resistance from 20 to 30 Pa was recommended for the first category of channels, while for channels with vegetative cover, the allowable erosive resistance was about 60 Pa. A similar range of allowable critical boundary stresses for moderate strength clays was recommended by Flaxman in 1963 [32]. One should note that these last two sets of values are quite close to the recommendations of the Russian engineers. In his study of fluvial processes, Ake Sündborg demonstrated that the critical velocity decreases with decreasing particle size down to a limit of 50 m, which is the boundary between fine sand and silt [130]. For particle sizes below that limit, the critical velocity starts increasing with decreasing particle size. This observation makes sense because cohesive forces appear only for fines below the silt limit, whereas above that limit the resistance to erosion is provided primarily by the submerged weight of the individual particles. This observation led Sündborg to the conclusion that the critical velocity for erosion is a function of the macroscopic strength of the soil. These field studies, like the earlier ones as well as some others, constitute a valuable first step in establishing erosion criteria, but they are essentially of an empirical nature with very little relation to the highly complex nature of cohesive sediments. Some of their conclusions are based on intuitive expectations rather than on systematic laboratory experiments and, therefore, their applicability is limited. In addition, field investigations, where the controlling variables cannot be changed at will, show whether a given soil fails or resists an estimated boundary shear stress, but they do not reveal whether a stable channel can still resist higher stresses or the stress under which an eroded channel began failing. The linking of erosive resistance to the macroscopic cohesive strength appeared a
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179
reasonable assumption because both strengths depend on the interparticle forces. However, neither the difference between the mechanism of mass failure of cohesive soils and by surface erosion nor the mineralogy and other physicochemical properties of the soil were considered. The following phase of laboratory research attempted to go a step further. The efforts were directed toward the establishment of working relationships between some parameters representative of the flow-induced shear stresses and some gross clay mechanical properties, such as the mass shear strength as determined by any conventional shear strength test, the Atterberg limits, and the percentage of silt and clay. The following are some of the most representative efforts of this category of research, which were concentrated in the late 1950s and the early 1960s. Dunn was the first to conduct small-scale laboratory tests on the erodibility of remolded clay samples [20]. His apparatus consisted of a water jet impinging on the surface of a small sample, forming part of the bottom of a small cylindrical container. The criterion of failure by erosion was defined as the flow at “which the water becomes cloudy and no subsequent clearing occurs.” The critical shear stress was evaluated from the total measured force exerted on the sample. The failure criterion corresponded not to the initiation of scouring but to a stage of mass erosion. The critical shear stress, thus defined, was then plotted versus the vane shear strength of the soil, Sv. The plot of a limited number of points gave a straight line whose slope was correlated to the plasticity index, PI, as the only sample identification parameter, which ranged from 6 to 16, and to some statistical parameter representative of the mechanical composition of the cohesive material. No density measurements were made. The PI range suggests that the test material was rather inert and that the specimen was subjected to swelling. The critical shear stress seems to increase with the vane shear strength. For strengths below 100 psf (4800 Pa), the critical shear stress, c, ranged from 0.03 to 0.5 psf (1.44 to 24 Pa). It appears, therefore, that the critical stress for erosion is three to four orders of magnitude smaller than the macroscopic strength of the soil, as determined by any standard test. This difference raised the first doubts as to whether the critical scouring stresses can be correlated to the macroscopic shear strength of the soil. Smerdon and Beasley conducted erosion experiments in an open flume with loosely placed cohesive beds [128]. They defined the critical shear stress as the stress for which “the bed material was in general motion.” This stress was subsequently correlated to the PI of the bed material according to the equation
c 0.0034 (PI )0.84
(6.6)
with c in psf. The strength of the bed material was not measured. For a range of PI between 10 and 20, the critical shear stress in their experiments ranged between 0.02 and 0.047 psf. In contrast, for comparable PI ranges, Dunn obtained critical strength values between 0.35 and 0.50 psf for low-strength clays. In general, for
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Cohesive Sediments in Open Channels
comparable PI values and bed strengths, the last two investigations gave values for the critical boundary stresses differing by a factor of 10 to 15. A little later, Moore and Mash [90] and Espey [29] also attempted to develop small-scale tests for estimating the erosive resistance of cohesive soils. The apparatus of the first two investigators utilized a submerged jet impinging on a clay sample, in a way similar to that used by Dunn. The second used a system of rotating coaxial cylinders with the clay sample as the inner cylinder. At the stage of severe failure, characterized by removal of large pieces of the soil sample, the measured critical stresses ranged from 0.18 to 2 psf or 8.6 to 64 Pa. No correlation with sample strength or any parameter describing physicochemical properties was attempted in either of these two studies. In 1964, Rectoric [122] conducted experiments by an approach similar to that of Espey. He presented a linear correlation of vane shear strength, Sv, with critical shear stress, c. For a strength range between 20 and 200 psf (960 to 9600 Pa), he reported a corresponding range of critical boundary stresses between 0.3 and 2 psf (14 to 96 Pa). This range is very close to the one obtained by Espy. However, both ranges are well above the recommended design values for low- to mediumstrength cohesive soils. Rectoric also attempted to correlate c with the plasticity index, clay content, and calcium/sodium ratio without conclusive results. In 1962 Abdel-Rahman presented results of research on the erosion of artificially compacted cohesive beds with a PI equal to 23 in an open flume [1]. Unlike the previous four investigators, Abdel-Rahman did not define a critical limit of the shear stress. Instead, he tried to correlate the mean depth of erosion and the roughness of the bed surface to the flow-induced shear stress, the soil strength, and the properties of the water. In his experiments, the erosion eventually ceased. It was at that stage where the depth of erosion was defined and measured. It appears from the description that the bed material was not fully saturated and that expansion took place during testing. Slacking or flaking was also observed. The latter amounts to a gradual disintegration of the soil to small pieces or flakes due to a loss of strength when a particular soil is submerged in water. Air entrapped in the pores of the soil is known to be a cause of this phenomenon. The bed shear stress ranged between 0.015 and 0.09 psf or 0.72 Pa to 4.3 Pa. For about the same PI, Smerdon and Beasley’s [128] Equation 6.6 gives c 0.047 psf or 2.27 Pa, which is substantially lower than the stresses of AbdelRahman at maximum depth of scouring. In general, the depth of erosion and the bed roughness at that depth increased with increasing shear stress and decreasing bed strength, as intuitively expected. In 1964, Berghager and Ladd presented limited results of experiments of erosion in a small flume of Boston blue clay with a PI of 12.5% and mean size of 3.5 μm [5]. Saturated samples consolidated to 2,000 psf (96,000 Pa) displayed no visible signs of scouring even under shear stresses of 0.7 psf (34 Pa), whereas unsaturated samples subjected to the same consolidation pressure failed by slaking even under the lowest applied stresses. For the same PI, Equation 6.6 gives a critical stress of 0.03 psf or about 14 Pa. No correlation was attempted between the flow and soil parameters.
Chapter | 6 Erosion of Cohesive Soils
181
In 1966, Mirtskhulava presented results of experimental studies conducted in Russia on the erosional stability of cohesive soils [87]. He investigated the controlling soil properties and tried to establish permissible velocities for safety against erosion instead of permissible bed shear stresses. Unlike work reported earlier at that time, Mirtskulava’s work used photographic techniques to register erosion and a model of interaction of bed units with flow-imposed drag and lift forces. He claimed that the dominant criterion for the erosional stability of cohesive soils is “cohesion at full moisture capacity of the soil and the size of aggregates carried away.” Neither the bed shear stress nor the erosion rates were considered. The comparison of the test results, discussed earlier, suggests that there is no correlation between the macroscopic shear strength of the soil and its resistance to erosion. Mirtskulava’s equations contain a number of parameters and constants related to some soil properties, such as the angle of internal friction. The permissible velocities range from 0.60 m/s to 1.00 m/s. The water depth in the experimental channel varied from 0.29 m to 0.95 m, and the cohesion ranged from 2.06 104 Pa to 0.59 104 Pa without a very good correlation to the permissible velocities. The cohesive strength range is on the order of macroscopic shear strength for clay masses of low to moderate density and, therefore, according to the preliminary conclusions from the earlier experimental studies, is not representative of erosive resistance. In addition, the “size fragments washed away” assumed spherical were more or less uniform in size with a diameter between 3.5 and 9.5 mm, indicative of mass erosion rather than surface erosion. The summarized research work can be viewed as the second phase of the effort, after the first purely empirical approach, to derive erosion criteria for cohesive soils. Almost all of them correctly used the bed shear stress rather than the average velocity as a measure of the eroding agent, and they tried to relate its critical stress to the macroscopic strength, clay content, and plasticity index of the soil. Their results vary widely. For samples with similar characteristic parameters, the reported corresponding critical stresses differ by orders of magnitude. The criterion of failure was arbitrary and subjective and in most cases corresponded to a stage of rapid scouring, a case supposed to be avoided by all means. The boundary conditions were dissimilar, so the results of the various tests cannot very well be compared. However, this early laboratory research under controlled conditions constitutes a significant step of progress in the cohesive sediment dynamics. It led, moreover, to some very important albeit qualitative conclusions regarding the significant soil properties, which control erosion and opened the way for the recent more systematic and more detailed research on the subject. The following are the most important of these conclusions: 1. The critical shear stresses are smaller by two to three orders of magnitude than the macroscopic shear strength of the bed material. 2. The critical shear stresses may also vary widely for soils with very similar strengths, evaluated by any macroscopic shear strength apparatus, such as the vane, and with comparable plasticity indices and clay content. 3. It can be concluded, therefore, that the macroscopic shear strength of the soil cannot be used as a unique criterion of soil resistance to erosion.
182
Cohesive Sediments in Open Channels
4. The same conclusion applies to the plasticity index and to the mechanical composition of the soil. These parameters are secondary indices not necessarily representing the detailed internal structure of the cohesive bed or the interparticle physicochemical forces, which are responsible for flocculation and which are expected to provide the main resistance to erosion. These indices should be viewed, at most, only as some type of reflection of the primary control indices, that is, the internal structure and the internal force system of the cohesive soil. 5. It became clear that the mechanism of erosion by flowing water is drastically different than the failure by shear stresses along a plane through the soil mass. 6. Erosion may eventually stop. This is indicative of an increase of the erosive resistance of the bed with depth and, possibly, with time. 7. The way the bed was prepared, and specifically the water content during compaction, may affect drastically the resistance to erosion. Under certain conditions, a submerged bed may lose its strength entirely and disintegrate through flaking even without any applied shear stress. 8. None of the previous research work addressed the initiation of erosion and the erosion rates after a critical shear stress has been exceeded. Grissinger took into consideration the preceding factors and tried to observe the influences of relevant and definable soil properties on erosion [40]. He specifically studied the effect of bulk density, water temperature, type, and orientation of clay minerals, and percent of clay on erodibility for constant flow conditions. The soil erodibility was determined in terms of the erosion rates rather than an arbitrarily defined critical shear stress. The latter corresponds to the defined shear rates and was found to be equal to 0.400 psf or 19 Pa, which is near the order in Smerdon and Beasley’s experiments [128]. The following are Grissinger’s main findings [40]: 1. The orientation of the clay particles has a pronounced effect on erodibility. Specifically, the erosion rates decrease with increasing degree of orientation, and vice versa. 2. The erodibility decreases with increasing clay content for all soil mixtures with the exception of calcium montmorillonite additive. 3. The resistance to erosion was found to increase slightly with increasing density of the soil although the results were considered as inconclusive. Bulk density is essentially the average density of a soil sample, whereas erosion is a surface phenomenon depending on the density of the surface layers and even more so on the density and structure of the surface aggregates and on the forces binding them to the bed. The latter do depend on the compaction techniques, the water content during compaction, and the degree of expansion under saturation. The range of surface densities may vary significantly from the average soil density. 4. The antecedent water content, defined as the water content of the sample after compaction and prior to testing under flowing water, has a significant
Chapter | 6 Erosion of Cohesive Soils
183
effect on erosion rates. It was found that for mixtures of illite and montmorillonite and for loessial soils, the erosion rates decrease with increasing antecedent water content down to a minimum rate, but beyond that point they increase with further increasing antecedent water content. The ratio of the maximum to the minimum erosion rates ranged from 6 to 20. In contrast, the erodibility of kaolinite mixtures was found to be rather insensitive to the antecedent water content. 5. The aging of a cohesive soil sample, which is the time after compaction, also has an effect on the erosion rates. In fact, the ratio of erosion rates before and after aging was found to be as high as 10. This aging effect diminishes rapidly with time, particularly four hours after compaction. 6. The erosion rates increase with increasing water temperature. For instance, the rates at 35°C were about twice the rates for the same sample at 20°C. In spite of the fact that the outlined work did not result in establishing specific criteria for erosion and in spite of the contradictions among the various test results, it did succeed in elucidating certain fundamental aspects and in indicating the reasons for the observed discrepancies. These conclusions were taken into consideration in the planning and conduct of the more fundamental research on erosion and deposition of cohesive soil that followed. A cohesive sediment system is highly complex and its properties as far as erosion and deposition are concerned have to take into consideration the interparticle physicochemical forces presented in Chapter 3. These forces, which are much higher than the submerged weight of either the individual particles or of the aggregates, provide the main resistance to erosion and constitute the primary agent for deposition. When the bonds among flocs and/or aggregates fail, erosion begins, and when settling flocs reaching the bed can form resisting bonds with the latter, then deposition takes place. Thus, the critical bed shear stress is essentially the summation of a finite number of minute forces per unit surface of the bed exerted on aggregates of various orders. The soil parameters used so far for soil characterization with respect to its resistance to erosion, and in particular density, shear strength, plasticity index, and type and content of clay minerals, do not adequately represent these governing forces. Nor do they provide any information about the internal soil structure, i.e., fabric, like the degree of particle orientation, which was shown to be important.
6.3 Erosion of soft cohesive sediment deposits The more recent fundamental research work on the erosion and deposition of cohesive sediments was motivated by the need for shoaling control in navigable waterways in tidal estuaries. Such a shoaling takes place through a number of deposition resuspension cycles until the sediment is deposited to a place where the flow-induced shear stresses are below the critical limit for the initiation of resuspension [100, 103]. The first laboratory experiments were conducted by Krone in an open flume at the University of California at Berkley [62]. The sediment bed was formed by
184
Cohesive Sediments in Open Channels
deposition of suspended San Francisco Bay mud at low velocity 12 days prior to the beginning of testing. This mud is composed of about 60% in the clay range, and the remaining material is silt with traces of fine sand and some organic matter. Its liquid limit is about 99% and its plasticity index about 55%. Figure 6.1 shows the time variation of suspended sediment concentration under the indicated velocity of 1.14 ft.s1 or 0.35 ms1. The plot follows closely a power law: C (t ) Co t 0.316
(6.7)
Co is some initial or reference value of C. It follows that the time rate of concentration increase, which represents the erosion rate, is given by the following equation: dC (t ) 0.316Co t0.684 dt
(6.8)
Equation 6.8 suggests that the concentration will eventually reach a constant value with no more net resuspension. At that time, this steady-state situation was interpreted as the stage at which the rate of aggregate resuspension becomes equal to the rate of deposition of suspended aggregates. The basis of this assumption was similar to the one in Einstein’s bed load function for coarse sediment transport with the aggregate replacing the cohesionless sediment unit. More detailed experiments on erosion indicated that there is no such interchange between bed and suspended flocs and aggregates. The observed gradual decrease
Suspended solids, g/L and optical density
100
10
Mare island strait sediment velocity, 1.14 ft. s 1
C(t) Cot 0.316 1
0.1 0.01
Optical density Suspended solids
0.1
1.0
10
100
Time after velocity change, hr
Figure 6.1 Variation of suspended sediment concentration due to erosion of deposited bed [62, Fig. 30].
1000
Chapter | 6 Erosion of Cohesive Soils
185
of the rates of suspended sediment concentration was shown to be due to the diminishing erosion rates of the deposited material and the increased resistance to erosion of the bed with the depth of scouring. The first phase of experimental research by the author aimed at providing answers to some of the questions raised by the previously discussed work and to develop a model for sediment-bed interaction in a turbulent flow field. The erodibility of a cohesive soil should be represented not only by a critical stress for the initiation of erosion, but also by the rates of erosion under a range of bed stresses. The effect of the macroscopic shear strength of a soil on its resistance to erosion was also studied. In addition, there are questions as to the effect of suspended sediment concentration on erosion rates, the changes of resistance to erosion with the depth of scouring and the possibility of changes of that resistance with time and by dissolved chemicals in the water [102, 105]. A cohesive bed of uniform consistency was required to investigate the first two questions. The bed material used was mud from the San Francisco Bay of a consistency similar to that used in Krone’s earlier experiments already mentioned. Its classification falls between CH and OH according to the United Soil Classification System. It is highly cohesive, and its primary clay mineral is montmorillonite with some illite. The pure clay portion of settling diameter less than 2 m was 60% by weight, and 40% was silt with sizes between 2 m and 50 m with very small amounts of fine sand and organic matter. Its liquid limit was 99%, its plastic limit was 44%, and its plasticity index was 55%, just as in the material used by Krone [62, 63]. It contained 33 mg of iron and 0.93 mg of magnesium per gram of dry weight. Its natural water content was 110%, and its natural dry density 40 lb ft3 or 640 kgm3. The material was tested in an open rectangular flume 60 ft (18.3 m) long, 1 ft (30.5 cm) wide, and 1.5 ft (45.8 cm) deep with recirculating water at ocean salinity. Two types of beds were tested: one at natural water content and a second composed of the same material but deposited directly from suspension at a velocity low enough for the entire suspended sediment to deposit. The thixotropic strength increase after one month was about 100%. A special strength-testing device had to be made to obtain the variation of the strength of the bed with its depth. It consisted basically of a 6 in. (15.2 cm) diameter screen made of 1/16 in. thick brass wires with two reinforcing brass bars of the same thickness. Several 1/32 in. wires were soldered at intervals of approximately 0.2 in., which provided the main resistance to penetration. The system was driven into the soil by adding small weights on a plate fastened to its top, and at every depth the weight that caused a penetration of 1 mm was taken as indicative of the strength of the bed at that depth. The description of the experimental device and procedures can be found in Ref. [105] and in more detail in Ref. [102]. With the outlined apparatus, the strength could be estimated at any desired depth because the thickness of the local failure zone is about the same as the diameter of the penetrating wire. Very low strengths, like those of a highly flocculated deposited bed, could also be measured. Figure 6.2 shows the strength variation of both the uniform dense bed and of the deposited bed.
186
Cohesive Sediments in Open Channels
The strength of the first bed, indicated as Series I, is very nearly uniform and independent of the depth. The deposited bed, indicated as Series III, displays a drastic increase of strength with depth, which cannot be explained by the consolidation effect. The flow rates in the flume water were gradually increased, and the suspended sediment concentration for each velocity was determined at regular time intervals. From these measurements, the erosion rates in grams of dry weight of material per liter of suspension were determined, as indicated in Ref. [102]. The results of tests 2 to 6 are shown in Figure 6.3. The velocity of the first erosion test was 0.32 ft.s1 (0.098 ms1) and caused no scouring. The first signs of erosion were observed for a velocity of 0.80 ft. s1 or 0.244 ms1 corresponding to a bed stress of 1.10 dynes cm2 or 0.11 Pa.
Depth below bed surface, inches
0 0.2 0.4
Flocculated bed series lll
0.6
Dense bed series ll
0.8
Dense bed series l
1.0 1.2 1.4 1.6 10
100
1000
10000
Load on penetration screen, grams
Figure 6.2 Strength variation of cohesive beds [105, Fig. 1].
Suspended sediment concentration, grams/liter
2.5 Run 6
Run 5 Run 4
2.0 1.5 1.0
Run
0.5
3
Run 2
0
0
20
40
60
80
100
120
140
160
Time after start of run, hours Figure 6.3 Suspended sediment variation in the water of the flume [105, Fig. 3].
180
200
187
Chapter | 6 Erosion of Cohesive Soils
The results of all 11 tests of Series I are summarized in Table 6.2. In all tests, the points fell on straight lines, suggesting an independence of erosion rates from suspended sediment concentration and from the depth of scouring. This independence is furthermore confirmed by the second and sixth tests with the same velocities, for which the erosion rates were the same in spite of the difference in suspended sediment concentration. In runs 4, 5, and 6, the velocities were gradually decreased. Apart from a slight deposition at the beginning of these last three tests, the erosion rates remained constant for each flow, suggesting no further deposition during the erosion process. In all six runs, no visible clay pieces and no surface movement were observed. The erosion took place by removal of flocs and/or aggregates, which went immediately into suspension. After the first six erosion tests, the velocity was reduced to 0.20 ms1. This velocity is a little lower than the minimum erosion velocity of 0.244 ms1. Deposition of the suspended material took place at diminishing rates asymptotically reaching a constant value of about 2.7 gL1 defined at that time as equilibrium concentration. One should note that, although under this last velocity no erosion could take place with clear water, nevertheless, the corresponding shear stresses prevented the previously eroded material from depositing. This was the first direct indication that the minimum boundary stress for the beginning of erosion is higher than the stress under which all the previously eroded material settles. This last inference was further confirmed by allowing the remaining suspended sediment in run 7 to settle at zero velocity and then rescour it also under the same velocity (run 8). After a rapid initial increase, the suspended sediment concentration converged to a constant value of 1.2 gL1.
Table 6.2 Summary of erosion rates of uniform dense bed of Series I [105, Table 1] Concentration Increase (grams/ liter/hr)
Erosion Rates (grams/ft2/hr)
1.10
0.00128
0.050
4.92
0.00885
0.338
3.34
0.00442
0.170
0.290
2.15
0.00255
0.096
0.244
1.10
0.00130
0.050
9
0.451
5.38
0.0118–0.00650
0.720
10
0.580
9.34
0.01520
1.670
11
0.714
13.30
0.0256–0.0152
2.821.67
Run No.
Average Velocity (m/sec)
2
0.244
3
0.409
4
0.345
5 6
Bed Stress (dynes/cm2)
188
Cohesive Sediments in Open Channels
Runs number 9, 10, and 11 followed at the velocities and stresses indicated in Table 6.2. However, after complete deposition, the erosion rates at velocities comparable to those of runs before deposition were substantially lower. In addition, in run number 11, at the highest velocity, the erosion rates started diminishing after 50 hours of continuous run. It was observed that erosion was concentrated within the one-third central portion of the bed, as shown in Figure 6.4. This phenomenon was attributed to wall-generated secondary currents, as explained in Refs. [102, 105]. The diminishing erosion rates of run 11 were attributed to the fact the eroded zone started reaching the solid bottom of the flume. After Series I of erosion, the first deposition phase was conducted. This phase consisted of eight experiments under various velocities, and its results will be presented and discussed in the following chapter on the deposition of cohesive soils. After complete deposition, the bed surface was examined and recorded prior to Series II on erosion. The surface of the eroded bed to the right and left of the central zone of deep scouring was wavy and considerably rougher and harder than the original bed material having the appearance of a thin crust, with sandstone texture. However, the bed resistance to erosion did not measurably increase. Below this surface crust, the material looked like the original. Part of the crust was bluish-gray with a coarser sandpaper texture, and part was reddish-brown, fine textured, and brittle. These differences in texture, hardness, and color are indicative of mechanical and physicochemical changes in the bed material during the process of erosion and deposition. The clay, silt, and sand composition and the content of iron and magnesium in the surface crust, the underlining material after the first erosion phase, and in the original Bay mud bed are summarized in Table 6.3. The strength changes of the bed during Series I were investigated in Series II. Prior to testing, the surface crust was removed, the bed was remolded and
Thickness of clay bed in feet
0.2 Facing downstream 32 feet from entrance
0.1 0
0.5
0.2
0.4
0.3
0.2
0.1
0
0.1
0.2
After run no. 18 After run no. 21
0.3
0.4
0.5
Bottom of flume
Facing downstream 37 feet from entrance 0.1 0 0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
Distance from centerline in feet Figure 6.4 Sections of eroded bed, Series I [105, Fig. 8].
0.3
0.4
0.5
189
Chapter | 6 Erosion of Cohesive Soils
leveled, and it was left undisturbed for 15 days with salt water. No measurable expansion or contraction was observed during that period of time. The strength of the upper portion of the bed was again measured and, as shown in Figure 6.2, it was found to be 40% to 25% lower than the strength of the original bed. The water content was increased from 110% to 120%. Four erosion experiments were conducted, with results shown in Figure 6.5 and summarized in Table 6.4. In Table 6.4, the erosion rates of two runs from Series I at velocities very close to those of runs 22 an 23 of Series II are indicated together with the corresponding erosion rate ratio. In spite of the lower macroscopic shear strength and the higher water content of the bed, the erosion rates of Series II were substantially lower than the corresponding ones of Series I. This is another strong indication that, even for the same soil, the macroscopic strength of a cohesive bed cannot be correlated with the erosion resistance of the latter. Series II showed how drastically the erosive resistance of a cohesive bed can be changed with time by various factors. In the present case, two such factors affected the bed: deposition of sand, which formed a protective layer, and iron Table 6.3 Mechanical composition and chemical characteristics of cohesive bed [105, Table 3] Sample
% Clay % Silt (dp 2 ) (2 dp 50 )
% Sand (dp 50 )
Iron content Magnesium mg/gr of dry content mg/gr of dry soil soil
Bluish-gray rough texture crust
53
39
8
36
1.2
Reddish-brown 59.5 fine texture crust
40.5
0
120
0.71
Original Bay mud
37
0
33
0.93
63
Table 6.4 Erosion rates of uniform dense bed of Series II [105, Table 5] Run No. Average Bed Stress Erosion Rates Erosion Rates EII/EI for similar velocities to Velocity (dynes.cm2) (g ft2 h1) of Series I Series I (ms1) (g ft2 h1) 22
0.403
4.78
0.174
0.338
0.515
23
0.705
13.00
0.674
2.82
0.240
24
0.854
18.80
2.160
–
–
25
0.982
28.30
6.25
–
–
190
Cohesive Sediments in Open Channels
8
Suspended sediment concentration, grams/liter
7
6 n
Ru
25
5
4 Run
3
24
2 Run 23
1 Run 22 0
0
10
20
30 40 50 60 Time after start of run, hours
70
80
Figure 6.5 Variation of suspended sediment concentration due to erosion [105, Fig. 11].
oxides dissolved from the flume by the saline water. The effect of the latter was obvious by the reddish color of part of the bed. As explained in Chapter 2, iron oxides can be cementing agents between particles and aggregates of various orders. Moreover, these results suggest that the resistance to erosion of a channel with cohesive boundaries can be significantly enhanced by depositing sand on its surface and/or letting cementing chemical agents, such as iron oxides, in stagnant water in the channel for some time. The necessary type, quantity, and time for these stabilizing constituents can be determined only by laboratory experiments for each particular case. The objective of Series III that followed was to test the erosional behavior of a cohesive bed composed of the same sediment as that of the previous two series, but deposited directly from suspension. The bed material was first completely suspended at the highest possible flow rates, and it was then allowed to deposit on the bottom of the flume at velocities low enough for complete deposition but still not allowing deposition in the return pipe. The new bed thus formed had the highly flocculated honeycombed structure at its loosest state displayed in Figure 3.15 in Section 3.6.2. It can be represented by a network of second-order aggregates, which, in turn, are composed of first-order aggregates and zero-order
191
Chapter | 6 Erosion of Cohesive Soils
Suspended sediment concentration, grams/liter
1.2
Run 26
1.0 0.8
Run 30
0.6 Run 29
0.4 0.2 0
0
10
20
30
(a)
40
50
60
70
80
90
6.0
3.5
ale
sc 8(
A)
31 Run
n2
Ru
3.0
(scale
B)
5.0 4.5
2.5 Run 27
2.0
cale A)
(s
4.0
1.5
cale
2 (s un 3
R
1.0
3.5
A)
3.0 2.5
0.5 0
5.5
(Scale B)
Suspended sediment concentration, grams/liter (Scale A)
4.0
(b)
100
Time after start of run, hours
0
10
20
30
40
50
60
70
80
90
2.0 100
Time after start of run, hours
Figure 6.6a and Figure 6.6b Variation of suspended sediment concentration of Series III due to erosion [105, Fig. 16].
flocs in a way similar to Krone’s model of Figure 4.7 in Section 4.4.1 [61]. The strength of its upper 1.5 cm layer, tested by the same penetrating screen device, was found to be on the order of 1/100 the strength of the first dense bed, as already mentioned. The erosion results are shown in Figure 6.6a and Figure 6.6b. Scouring was first observed at velocities of 0.60 and 0.75 ft.s1 (0.18 and 0.23 ms1). These
192
Cohesive Sediments in Open Channels
values are approximately the same as the minimum scouring velocity for the first dense bed, estimated between 0.65 and 0.80 ft.s1 (0.20 and 0.24 ms1) in spite of the 100-fold higher macroscopic strength of the first. Seven erosion experiments were conducted with velocities of 0.87, 1.07, 1.27, and 1.46 ft. s1 (0.27, 0.3, 0.39 and 0.45 ms1) corresponding to bed shear stresses of 1.53, 2.56, 4.35, and 5.74 dynes/cm2. The last three flows were applied to two runs for comparison purposes. The erosion rates of the deposited bed were not constant except for the last one under the highest bed shear stress. These rates, calculated from the time rate of increase of suspended sediment concentration, either displayed abrupt changes in slope, or they gradually diminished, as in Figures 6.6a and Figure 6.6b. In order to examine the possibility of simultaneous resuspension and deposition as the cause of the decrease of the rates of suspended sediment concentration increase, two experiments, 31 and 32 under the same bed stress of 5.74 dynes/cm2, were conducted, and their results are shown in Figure 6.6b. In run 31 the erosion rates were initially high, probably due to mass scouring. They suddenly diminished when the suspended sediment concentration exceeded 5.1 gL1 and remained constant to the end of that run. After 100 h of run and when the suspended sediment concentration reached the value of about 5.6 gL1, the muddy water was gradually replaced by clean salt water at rates equal to the withdrawal rates until a low concentration of about 0.7 gL1 was reached. The rates of concentration increase that followed in run 32 under the same stress were the same as in the previous run, corresponding to an erosion rate of 0.252 g/ft2h1. This direct observation leads again to the conclusion that simultaneous erosion and deposition does not take place in cohesive sediments. The erosion rates of the cohesive bed were diminishing with time but always remained lower than the erosion rates for the first dense bed. The ratio of the final erosion rates of the deposited bed to those of the first bed at approximately equal velocities ranged from 0.37 to 0.016. It appears, therefore, that the resistance to erosion of a deposited bed at its lowest density increases with the depth. This increase can only be attributed to an increase of the interparticle and/or interaggregate physicochemical forces, which control erosion. A cohesive sediment suspension is composed of flocs and aggregates of various orders formed in a turbulent flow field. The process of formation and the properties of the aggregates were discussed in Section 4.4. The surface forces connecting the clay particles among themselves and, possibly, to physicochemically inert particles, discussed in Section 3.4, are expected to vary. If we consider an initially dispersed system in a turbulent flow, flocs and aggregates will be formed with sizes, densities, and strengths varying statistically. As indicated in Section 4.4, higher aggregate strengths are associated with higher densities and higher settling velocities. Thus, aggregates with the stronger bonds are expected to settle first and those with weaker bonds to settle last. A stratified bed is thus formed with interparticle and interaggregate forces and, therefore, resistance to erosion increasing with depth. The deposited bed stratification and its properties will be discussed in detail in the next chapter on deposition.
193
Chapter | 6 Erosion of Cohesive Soils
The experimental studies on the deposited bed revealed two modes of erosion: (a) surface erosion that takes place only through the removal of surface fine particles, flocs, and aggregates; (b) failure by mass scouring of relatively large pieces of soil when the flow-induced stresses within a finite plane of the bed material exceed the macroscopic strength of the latter. The second type of erosion actually represents a case of failure, which, of course, is to be avoided. Moreover, its occurrence is unpredictable because it may depend on various other factors. The foregoing discussion is limited to surface erosion. A rigorous model for cohesive sediment erosion has to take into consideration the microstructure of the bed material and the details of the turbulent flow near the bed surface. The first was outlined in Section 3.6.1 and Section 3.6.2 and in Figure 3.13 through Figure 3.18. In Section 4.3, some fundamental properties of turbulent flows related to the flocculation process were discussed. That section showed that the rate of energy dissipation within the immediate neighborhood of the bed exceeds the average rate of energy dissipation by three orders of magnitude. It is in that zone where the erosion process takes place and where the shear stresses attain their highest values. A cohesive sediment bed can reasonably be assumed as hydraulically smooth. The time average shear stress at the bed is given by the equation
b Rb Se w
(6.9)
in which w is the unit weight of the water, Rb is the hydraulic radius pertaining to the bed, and Se is the slope of the energy grade line. The bed shear stress depends on the thickness of the near-bed viscous sublayer, within which shear stresses are transmitted predominantly by viscosity. In 1958 Einstein and Huon Li developed a theory of a periodically forming and disintegrating viscous sublayer. The theory explains the transition from a predominantly laminar near-bed flow to the turbulent one and several associated phenomena, such as the transportation of dye and sediment from the bed to the main turbulent flow [28]. At a certain time, turbulence reaches all the way to the bed, at which time the bed shear stresses attain their highest value. A very high shear flow results, which slows down layer after layer, thus building up a sublayer within which the flow is controlled primarily by viscous action. They called this variable thickness layer the viscous sublayer to distinguish it from the older concept of a constant thickness laminar sublayer. The viscous sublayer breaks down as soon as its thickness becomes large enough to make it unstable. At this time, turbulence reaches the bed, the instantaneous shear stresses become maximum, and the whole process of building and disintegration starts all over again. Einstein and Huon Li developed the following equation for the instantaneous bed shear stress:
b µ
U vb πν t
(6.10)
in which Uvb is the instantaneous turbulent velocity near the sublayer and t is the time. This equation is valid for 0 t Tp, in which Tp is the period of growth
194
Cohesive Sediments in Open Channels
of the viscous sublayer. Thus, according to the outlined model, the instantaneous shear stress at any point of the bed varies from a theoretically infinite value to a minimum µ(U vb / πν Tp ) with an average value given by Equation 6.9. In a turbulent flow, Uvb is expected to vary randomly with time. To proceed with the development of a sediment-flow interaction model, one has to consider the microscopic nature of the bed surface. Macroscopically, that surface appears and functions as hydraulically smooth. Microscopically, however, it presents irregularities on the order of magnitude of individual fine particles, flocs, and aggregates. The size of water molecules is 2–3 Å, whereas the surface irregularities of the bed are on the order of a few microns to few tens of microns. Hence, fine particles and even more so flocs and higher order aggregates act as large protrusions, which part of the time are completely embedded in the viscous sublayer and for the rest of the time are within the turbulent flow field. For the modeling of the erosion process, a law of variation of the forces exerted of the surface protrusions has to be formulated. In their classical work on the statistical variation of uplift forces, L, on semispherical elements, Einstein and El Samni found that the variation of these forces on semi-spherical roughness elements follows closely the normal distribution law with a time mean value of L and standard deviation L/o [27]. The dimensionless parameter o is a constant to be determined experimentally and which Einstein and El Samni found for their particular experiments to be equal to 2.75. Assuming next that the same law of statistical variation applies to the instantaneous bed shear stress, o, the frequency distribution function of the latter has the following form:
f ( o )
2 ( ) 1 o exp − o 2 o 2π o 2 o o
(6.11)
The hydrodynamic forces on the surface elements include both drag and lift components. Each protruding element is subjected to bending, in addition to direct tension. Bending can induce tensile stresses at the joins much higher than the ones caused by the direct tension. All these effects can be represented by the instantaneous shear stress, o, which should then be viewed as the integral expression of all drag forces per unit area, that is,
o
N
∑ Fi i1
(6.12)
195
Chapter | 6 Erosion of Cohesive Soils
where N is the number of fine particles and/or aggregates per unit surface area of the bed. o is next expressed in the form
1 o o o ∗ o
(6.13)
in which * is a random variable with a normal distribution N(0,1), that is with mean zero and standard deviation equal to 1 and –ο /ο is the standard deviation of ο. If the force per unit area is assumed distributed equally to all surface particles and/or aggregates, then the maximum tensile stress on each unit can be written as
σmax
k o N
(6.14)
in which k is a dimensionless factor accounting for bending. It is, furthermore, assumed that the spread of the distribution of the interaggregate cohesive force, Fc, is small enough in comparison to the spread of o, so that it can be considered as constant. Then, if ce is the average erosive resistance per unit area, Fc can be expressed by
Fc
k ce N
(6.15)
in which k is a proportionality factor. The condition for erosion of a particle then becomes
k o σmax ke o 1 Fc k ce ce
(6.16)
where ke is an overall proportionality factor. The rate at which particles and aggregates will be eroded per unit time is proportional to the units available and to the probability, Pr, of such a unit to be eroded during a unit time. That probability is
k Pr Probability e o 1 ce
(6.17)
The number of particles and/or aggregates available per unit area of the bed 2 is 1/A1da , in which da is the average diameter of the eroded aggregates and A1 is a surface shape factor. Next, for the removal of a bed surface unit, a force needs to act for a certain time, t (ο ) ; that is, the impulse of the acting force is important.
196
Cohesive Sediments in Open Channels
Pr can also be interpreted as the fraction of time for which Equation 6.16 holds. Therefore, at one and the same point, Pr /t (o ) units will be eroded per unit time; hence, the number of particles eroded per unit area will be
dN 1 Pr dt A1da2 t (ο )
(6.18)
When Equation 6.13 is introduced into Equation 6.17, the probability obtains the form
1 ce Pr Pr obability ∗ o ke o o
(6.19)
The instantaneous shear stress, o, also can take negative values. This means that it can act in a direction opposite to that of the flow. Since its effect on erosion depends only on its absolute value, we take the square terms on the right side of Equation 6.19, i.e.,
2 2 ce2 1 k Pr Pr obability e o 1 P ∗ 2 o ce (ke o o )
(6.20)
or
ce ce 1 Pr 1 Pr obability ∗ ke o o o ke o o
(6.21)
or ce 1 2 1 ke o o o Pr 1 ∫ ce 1 exp d 2 2π − ke o o o
(6.22)
The weight of each particle is equal to A2 da3 s , in which s is the average unit weight of the clay minerals and/or the aggregates and A2 is a volume shape factor. The erosion rate, E, can be also written in the following form:
Ad 1 E 3 (a ) s 1 o 2π t
∫
ce 1 ke o o o ce 1 ke o o o
2 exp d 2
(6.23)
197
Chapter | 6 Erosion of Cohesive Soils
In both Equations 6.22 and Equation 6.23, is a dummy variable and A3 A2/A1. Equation 6.23 gives the erosion rates in terms of weight of eroded material per unit bed area and unit time. Because it has too many unknown constants, some of them have to be assumed. The coefficient o was assumed equal to 2.75, that is, the same as in Einstein and El Samni’s equations for rough boundaries [27]. The remaining constants can be evaluated for each particular case by fitting Equation 6.23 to the plot of the measured erosion rates versus the (t ) bed shear stresses. More specifically, the ratio (A3 da s )/t o is a function of the soil properties and can be evaluated for each particular type of sediment. The data for Series I and II are plotted in Figure 6.7 together with a plot of Equation 6.23 for Series I after the constants have been properly calibrated. In Figure 6.8 the same data have been plotted in a dimensionless form together with the average curve of similar data on kaolinite clay obtained at a later study by Mehta and Partheniades (Chapter 7) [84, 100] with abscissa o∗ o (k/ce ) and ordinate E ∗ Et (o ) /A3 da s . All the data coincide reasonably well on one and the same curve. It is observed that, for a shear stress between 0.5 and 1.5 Nm2 for Series I and from 1.5 to 3.0 Nm2 for Series II, the curve can be approximated by a straight line. This led certain researchers and engineers involved with estuarine sedimentation problems to generalize this straight line relationship in their mathematical models. The author wishes to caution against such a generalization. First, even 0.007
Rate of erosion, grams/cm2/hour
0.006 0.005
Computed from Equation 23
0.004 Series I
0.003 0.002
Series II 0.001 0
0
0.5
1.0
1.5
2.0
2.5
Average bottom shear stress, N/m2 Figure 6.7 Shear stress rate of erosion for dense bed of San Francisco Bay mud [102; 105, Fig. 7; 100].
198
Cohesive Sediments in Open Channels
0.8 E *� t τo�A d γ E 3 a 3
Dimensionless erosion rate
1.0
0.6 0.4
Rotating channel Series I, Partheniades (1965)
0.2 0
Series II, Partheniades (1965) 0
0.5
1.0 1.5 2.0 2.5 3.0 3.5 Dimensionless bed shear stress τb*�(k/c)τb
4.0
Figure 6.8 Dimensionless shear stress erosion rates relationships [100, Fig. 19].
for beds of uniform consistency, there are still zones of substantial deviation from the straight line. Second, and most important, cohesive beds deposited directly from suspension display a drastic variation of density and erosive resistance with depth, as the experiments of Series III have shown. This subject will be discussed in more detail in Chapter 7 on deposition and resuspension. The independence of the critical stress and of the erosion rates from the density and the strength of the bed needs some qualification. In Section 3.6.2, the four states through which a cohesive bed undergoes after deposition were presented. The loosest state, A, was the one immediately after deposition; state D represents the point at which the average interparticle distance reaches the order of that distance within the flocs. From state A to state D the surface structure does not change materially; so, as long as the flow does not induce mass shear stresses exceeding the shear strength of a finite section of the bed, there is no reason for the erosion conditions to be different. What happens upon further consolidation beyond state D is open to question. It seems logical to expect an increase of the resistance to erosion beyond the latter state; however, this should be a subject for future research. Much later, Lambermont and Lebon studied the problem of cohesive sediment erosion starting from the sediment continuity principle [69]. They accepted Migniot’s principle that the shear stress for which erosion begins is uniquely related to the yield strength of the eroding upper layer and that the latter is related to the sediment dry density, sb, by the equation
m y nsb
(6.24)
in which y is expressed in Nm2 and s in gL1. According to Migniot, m has a constant value of 5, but the value of n varies from 1012 to 1015, depending on the particular sediment. Lambermont and Lebon derived experimentally the
199
Chapter | 6 Erosion of Cohesive Soils
following relations for the critical friction velocity acting on the sediment layer in terms of the yield strength of the surface layer:
u*c 1/4 for y 15 dynes/cm 2 y
(6.25a)
and (6.25b) u*c 0.5 1/2 for y 15 dynes/cm 2 y where the friction velocity is in cm s1 and the yield strength in dynes/cm2. In SI units the coefficients of the second-hand members become 0.0178 and 0.016, respectively, with u* in ms1 and y in Nm2. Although the relationship of the minimum eroding bed stress to the yield strength of the sediment suspension seems to contradict the previously stated conclusion, in Equation 6.25a and Equation 6.25b, y is the yield strength corresponding to the upper layer subject to erosion. This layer is expected to have the lowest density and, therefore, its yield strength may be related to its erosive resistance. Moreover, the instantaneous physicochemical composition of the top bed layer that determines the yield strength also determines the erodibility of the same layer, so that these two variables can be correlated. The result of this study was an equation for the sediment flux from the bed involving a large number of constants. The following is the simplest form of this equation when the deflocculated particle diameter is very small:
(
1/Bi
J i Ai 1/Bi i
)
1/2
for i (i 1.2)
(6.26)
where i is the bed shear stress for zero flux given by i sv Ei1 i (6.27) where sv is the density of sediment component in the viscous sublayer and “Ei and Bi are constants appearing in the relations between erosion shear stress and solid particle density.” The subscript i indicates the density of the eroding upper layer taken equal to 1.2. The constant Ai in Equation 6.26 is given by B
Ai
Dv o νv1 2 Ei1
(6.28)
Bi
where o is a constant, Dv is the diffusion coefficient in the viscous sublayer, and v is the fluid density in the sublayer. The equations developed by Lambermont and Lebon are based on a quite elaborate theoretical analysis involving many assumptions and containing several constants that have to be evaluated by experiments. After evaluating these constants with the author’s experimental results, they gave the following equations
200
Cohesive Sediments in Open Channels
for the sediment erosion flux J1, which were found in close agreement with the curves of Figure 6.7 and Figure 6.8:
J1 1.7 106 ( 0.34)
(
1
2
for 0.34 Nm2 1.8 Nm2
)
J 2 19.9 106 2 1.12 1
1
2
for 1.48 Nm2
(6.29a) (6.29b)
The subscripts 1 and 2 indicate the two zones of the curves in Figure 6.7 and Figure 6.8. The diffusion coefficient Dv for region 2 was found to have a value of Dv2 370 1011, which, according to the investigators, is 100 times larger than for Series I of the experiments reported by the author. The outlined work by Lambermont and Lebon constitutes another approach to the erosion process of cohesive sediments, which seems to be very interesting from the standpoint of fluid mechanics. A more complete presentation of their analysis would be beyond the objectives of this book. However, readers involved in cohesive sediment research may be interested to study in more detail the analysis in question.
6.4 Summary and concluding remarks This chapter was confined to the erosion of two cases of cohesive soils of uniform physicochemical properties: a constant density bed and a deposited bed at its lowest density. The experiments on the first bed clearly indicated constant erosion rates independent of the suspended sediment concentration under constant bed shear stresses. The second bed displayed a stratified structure with erosive strength increasing with depth of erosion. Very limited deposition experiments suggested that the fraction of total sediment deposited under a given bed shear stress is independent of the suspended sediment concentration at the beginning of the run. This observation led to the next phase of research on cohesive sediment deposition by the author and coworkers. The equation for the erosion rates does contain a number of constants collected into two distinct groups representing the sediment properties and the flow variables. These groups can be readily evaluated on the basis of laboratory experiments and field measurements. It is also noteworthy that they fit the data by Lambermont and Lebon obtained by a different approach. Contrary to some early intuitive expectations, it has been shown that none of the common soil parameters can be used as a criterion for erodibility. In addition, there are strong indications that the mechanism of soil erosion by flowing water is drastically different to that of failure through a mass shear. In fact, the eroding stresses are by orders of magnitude smaller than the macroscopic strength of the soil. Tests of a small scale, such as rotating cylindrical specimen and jets impinging on small clay samples, are not recommended. In such tests, the detailed near bed flow structure is drastically different from the actual conditions in an open
Chapter | 6 Erosion of Cohesive Soils
201
channel. Likewise, the Bingham yield strength of clay suspensions cannot be used in general as an erosion criterion. That strength depends on the sediment concentration, whereas the stress corresponding to the initiation of erosion is independent of the latter. However, if it is used, it has to be for a bed density about equal to that of the upper few millimeters of the bed. It was demonstrated that a cohesive bed is very sensitive to minute changes of environmental parameters, such as water quality and temperature. For this reason, it is recommended that erosion experiments be conducted under strict control of all pertinent physicochemical parameters of the system. The bed should be as representative as possible to the in situ soil and should be allowed to set for sufficient time in the flume prior to testing. The water quality in the flume should be close to that of the natural water. The critical stress for the initiation of scouring can be significantly increased and the erosion rates substantially reduced by certain chemicals dissolved in the water and by covering the cohesive bed surface with granular material. Such improvements of the erosive resistance of cohesive soils should also be the subject of future applied research. To avoid chemical interaction of water and sediment with the material of the experimental flume, the latter should be made of or covered with an inert material, such as glass, fiberglass, or Plexiglas.
Chapter 7
Deposition and Resuspension of Cohesive Soils
7.1 Deposition of cohesive sediments 7.1.1 Early Experiments and Preliminary Conclusions Since the hydrodynamic processes of deposition and of resuspension of cohesive sediments beds are interrelated, we considered it more appropriate to treat the resuspension in the same chapter with deposition rather than in Chapter 6 on erosion. In that chapter, only some pilot experiments on erosion of deposited beds were presented. In fact, it was demonstrated that, unlike in uniform consistency dense beds, in flocculated deposited beds the erosive resistance increases and the rates of erosion decrease with increasing depth of scouring. The first systematic studies on the deposition of cohesive sediments were conducted by Krone for San Francisco Bay mud [62–64]. First, he found that, for constant initial sediment concentration, substantial flocculation occurs for a salinity of 1,000 ppm and that the median settling velocity attains a maximum from a salinity of 5,000 ppm for 120 ppm sediment concentration to 15,000 ppm for a sediment concentration of 1,000 ppm. For quiescent unhindered settling, and for sediment concentrations below 10 g L1 or 10,000 ppm at the optimum salinity, Krone found that the median settling velocities, wsm, follow the 4/3 law:
w sm 8C 4/3
(7.1)
where 8 is a dimensional constant. For hindered settling, which, according to Krone, starts for sediment concentrations above 10 gL1 or 10,000 ppm, his experiments seemed to fit Bosworth’s equation [7]: V ft Cohesive Sediments in Open Channels Copyright © 2009
Vf ∞
1
kf t
(7.2)
203
Cohesive Sediments in Open Channels
204
Suspended solids concentration, (g/L)
100 100-ft flume Redwood city sediment Salinity, 17 g/L
V Average velocity
V 0.5 V
0 ft/sec
0.4
4
10 1
V
V
V
0.
0
20
40
60
80
7
0.
20
10 2
0.3
25
100
120
140
160
180
200
Time, (hr) Figure 7.1 Deposition rates from flowing water [62, Fig. 13].
where Vft is the flocculent volume in time t, Vf is the flocculent volume at an infinite time, and kf is a constant with dimensions of time given as 0.95 of the time required for the flocculent volume to become equal to 2Vf . Subsequent experiments by Krone with running water in an open flume are shown in Figure 7.1. At that time, it was concluded that for a suspended sediment concentration below 0.3 gL1, the latter decreases logarithmically with time. However, the semilogarithmic relationship seems to follow a straight line only for velocities below 0.37 ft.s1, and the concentration eventually appears to approach zero. For velocities higher than 0.37 ft/sec, the time concentration line seems to have a distinct curvature and to approach asymptotically a constant value different from zero. The straight semilogarithmic portion of the curves can be expressed by the equation
C exp(kt t ) Co
(7.3)
where kt is a constant with dimensions of 1/time (t1), and Co is the initial sediment concentration in the flume. The dependence of kt on the bed shear stress was determined by plotting the relative rate of sediment deposited per hour versus the latter. The results are shown in Figure 7.2. It was concluded that there is no deposition for shear stresses higher than 0.6 dynes/cm2 and that at zero flow the deposition rates are 0.039 h1. One is reminded that these early conclusions were for sediment concentrations below 0.3 gL1 and for the special type of sediment used.
Chapter | 7 Deposition and Resuspension of Cohesive Soils
205
Fraction of suspended sediment deposited per hour
0.04 100-ft flume 30-cm depth Redwood city sediment mean particle size 2.5� C/Co � exp [�ktt]
0.03
�4
10 kt � 6.6 � y o
(1�1.67 τb) [cgs]
0.02
0.01
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Shear on bottom sediment, (τb), dynes. cm�2 Figure 7.2 Relation of sediment deposition rates and bed shear stress [62, Fig. 14].
Krone developed the following theory for deposition of cohesive sediments at concentrations below 0.3 gL1. He first introduced the concept of the probability, Pr, of a settling unit sticking to the bed. When one considers a control volume of suspension of unit area and height yo equal to the depth of flow in the flume, the sediment continuity principle gives
P dC ma nws r dt yo
(7.4a)
In Equation 7.4a, ma is the mass of the settling unit of suspension, n is the number of units per unit volume, and ws is the settling velocity. Obviously, man is the sediment concentration, C, at time t, and manvs is the mass flux of settling units approaching the bed. Equation 7.4a reduces to
P dC Cws r dt yo
(7.4b)
Cohesive Sediments in Open Channels
206
whose integration gives
P w t C exp r s Co yo
(7.5)
From Figure 7.2, ws was estimated to be about 6.6 104 cms1 and Pr 1 b /0.6 where b is the bed shear stress. Equation 7.5 involves some simplifying assumptions and is based on limited experimental information. The collision frequency was assumed to be independent of the flow velocity, and the apparent settling velocity independent of sediment concentration. Next, it was concluded from Figure 7.2 that no deposition takes place for bed shear stresses higher than 0.6 dynes/cm2 and that below that value all suspended sediment eventually deposits. However, in Figure 7.1 the concentration for the higher velocities appears to tend to a finite value rather than to zero. Second, it is an oversimplification to consider the same settling velocities and other aggregate properties for all suspended matter. This was clearly shown in Chapter 3 and Chapter 4. For sediment concentrations between 300 ppm and 10,000 ppm, Krone’s experiments suggested a logarithmic relationship of the form [62]
log C k4 log t constant
(7.6)
where k4 was found to be approximated by the equation k4
103 1 b yo dc
(7.7)
In this last expression, yo is the depth of flow, b is the average bed shear stress, and dc is the critical shear stress for complete deposition, found to be 0.78 dynes/cm2. This higher value of the latter, in comparison to that for concentrations below 300 ppm, was attributed to the formation of larger aggregates due to a higher collision frequency. For sediment concentrations above 10,000 ppm, Krone found a logarithmic law similar to Equation 7.6 but with an empirical constant k5. The following explanation was given by Krone for the three different deposition laws. For the lowest concentration range up to 300 ppm, the various order aggregates settle almost independently without much mutual interference. For the intermediate range between 300 ppm and 10,000 ppm, the increased collision frequency results in larger aggregates with higher settling rates and higher shear strength, so that not only are they able to deposit faster, but they are also able to develop stronger bonds with the bed material. Finally, for concentrations above 10,000 ppm, it was assumed that the sediment suspension takes the form of a continuous network, which Krone called fluid mud, settling more slowly than in the precious case with water percolating upward through the spaces of the network. The last mode of settling was defined as hindered settling.
Chapter | 7 Deposition and Resuspension of Cohesive Soils
207
Krone also conducted a deposition test in which he labeled part of the suspended sediment with radioactive gold-198. He observed that the rates of decrease of radioactivity were higher than the rates of sediment deposition. At that time, he attributed this difference to an interchange between suspended and bed aggregates deposited earlier from suspension. More recent work by the author, described in Chapter 6, contradicted this conclusion. The author conducted his first preliminary deposition experiments in conjunction with his erosion research discussed in Chapter 6. These experiments followed Series I and are shown in Figure 7.3 and Figure 7.4 [105]. The first substantial deposition occurred in run number 15 at a velocity of 0.81 ft.s1 or 0.25 m s1 corresponding to an approximate shear stress of 1.10 dynes/cm2. Run 15 displayed an initially rapid deposition at gradually diminishing rates, leveling off after several hours to a constant value of about 6.2 g L1. In runs 16 and 17 at velocities of 0.65 and 0.58 ft.s1, respectively, deposition was relatively small but still with an obvious tendency to reach asymptotically a constant finite value. That steady-state constant concentration, defined as equilibrium concentration, cannot be explained by the concept of an equal rate of deposition and resuspension of aggregates, as originally thought, because the possibility of such a simultaneous deposition and resuspension was precluded in the erosion experiments presented in Chapter 6. It seems to be rather indicative of the strength of the aggregates remaining in suspension. In addition, the equilibrium concentration
Cohesive soils
Suspended sediment concentration (g/L)
12
Run 12 Run 13
11 10
Run 14
9 Note: Points represent average concentration per day
8 7
Run 15 6 5
Equilibrium concentration 0
10
20
30
40
50
60
70
80
Time after start of run (hr) Figure 7.3 Deposition of sediment in a flume at high velocities [105, Fig. 5].
90
100
Cohesive Sediments in Open Channels
208
Suspended sediment concentration (g/L)
7 Run 16 Run 17
6 5
Note: Points of run 16 represent average concerntration per day
4 3
Run 1
8
2 Run
1 0
0
10
19
20
30
40
50
60
70
80
90
100
Time after start of run (hr) Figure 7.4 Deposition of sediment in a flume at low velocities [105, Fig. 6].
in run 15 at a velocity of 0.18 ft.s1 was about 55% of the highest concentration at the end of run 12 under a velocity of 1.92 ft.s1. This concentration is quite close to the 60% clay content of the natural sediment; therefore, it appeared at that time as if the deposited material was primarily the silt content. Moreover, one should note that little deposition occurred during runs 16 and 17. It took a reduction of velocity from 0.55 to 0.47 ft.s1 to reach the next stage of rapid deposition. The question was raised at that time as to the nature of the equilibrium concentration and the variables controlling it. A first tentative answer to that question was obtained by comparing the ratio [C f (1.34) C f (0.65)] / C f (1.34) , where the subscript f indicates final concentration at the end of the run under the velocity in parentheses, for two sets of runs at high and low concentrations. This ratio was found to be 0.36 for the high concentrations and 0.37 for the low ones. This was the first indication that the equilibrium concentration is independent, at least up to a certain concentration limit, of the absolute value of the initial concentration and appears to depend on the flow conditions.
7.1.2 D etailed Studies on Deposition. Part A: The Degree of Deposition The following detailed phase of investigation of the depositional behavior of cohesive sediment first aimed at elucidating the nature of the equilibrium concentration and next the variables that control it. The depositional behavior of
Chapter | 7 Deposition and Resuspension of Cohesive Soils
209
(a)
(b)
Figure 7.5a Annular channel and ring. Plan view [81, 82, Fig. 3.1.10a].
Figure 7.5b Annular channel and ring. Side view [81, 82, Fig. 3.1.9].
cohesive sediments is determined by the properties of the aggregates that, in turn, depend on both the physicochemical properties of the sediment and the flow conditions, and in particular on the flow microstructure. As discussed in Chapter 4, the flow microstructure is determined by the boundary shear; therefore, an experimental apparatus for a reliable study of the depositional behavior of cohesive sediments should have a uniform and readily determinable boundary shear all over the flow area. Clearly, a conventional laboratory flume does not satisfy these conditions because the water-sediment mixture has to go through the return pipe, in which the flow is drastically different than that in the flume itself. It was necessary, therefore, to design and build special research equipment to satisfy these conditions. Such equipment was developed first at the Massachusetts Institute of Technology (MIT) in 1963, where the first detailed studies on deposition were conducted, from 1963 to 1968 [31, 100, 109, 114, 118]. A larger but similar experimental setup with improved instrumentation was later developed at the University of Florida for the more extensive research on deposition of cohesive sediments, which took place from 1968 to 1983 [81–86, 94, 95, 154, 155]. The last version of this research equipment is pictured in Figure 7.5a and Figure 7.5b and outlined schematically in Figure 7.6. The equipment consists of an annular channel with a mean diameter of 60 in. or 1.525 m and a width of 4 in. or 10.2 cm. The maximum depth of the channel containing the sedimentwater mixture is 18 in. or 45.8 cm. A vertically movable annular ring of a diameter slightly less than that of the channel is positioned within the latter and placed in contact with the water surface. The ring is supported by four blades fixed to a collar that could be moved vertically on the supporting shaft so that it could readily be positioned at any desirable location in the channel. The two units are
Cohesive Sediments in Open Channels
210
Top horizontal support beam L-shaped steel angle
Triangular steel plate
Bearing for inner shaft Slipring and brush block assembly
Height adjustment mechanism Refilling well
Inner shaft
Arm supporting the ring Strain gauges Ring suspending blade Refilling funnel Fiberglass channel
Plexiglass ring
Chip board
Horizontal support beams
Sample tab Sample bottle
Drain Suspension collector
Fiberglass stiffener
Triangular plates supporting outer shaft
Turntable Outer shaft
Supporting leg Triangular steel corner
Driving motors
Figure 7.6 Schematic outline of annular channel and ring system [81, 82, Fig. 3.1.3].
driven by independent mechanisms in either direction. Their simultaneous rotation in two opposite directions generates a turbulent uniform flow field free of floc-disrupting elements, such as blades, return pipes, and diffusers, which are present in conventional laboratory flumes. The metallic blades, which supported the ring, were properly instrumented with strain gauges so that the shear stress exerted on the ring could be measured directly and from the latter the average bed shear stress could be evaluated. The details of this experimental system, of the calibration, and the operation can be found in Refs. [81, 82]. The effect of the rotation-induced secondary currents on the deposition has been eliminated by a proper selection of the speeds of the channel and of the ring so that the sediment deposits uniformly across the channel bed. A set of operational curves has thus been obtained for a wide variety of depths. At any such operational speed, the bed shear stress across the channel measured with a Preston tube was found to be uniform. The flow structure within the channel and its resemblance to that in an open channel was investigated prior to the beginning of the deposition experiments. Figure 7.7 shows four typical velocity profiles in the middle portion of the annular channel displaying two wall layers of
Chapter | 7 Deposition and Resuspension of Cohesive Soils
211
6
5
Y (in.)
4
3 Symbol 2
1
0
∆v (cm/sec) 25.1 38.1 60.5 74.2
0
10
20
30
40
50
60
70
80
U (cm/sec) Figure 7.7 Characteristic velocity profiles in the annular channel [81, 82, Fig. 4.1.3].
high shear rates and a core segment of near constant velocity and near homogeneous turbulence. A plot of the near-bed velocity profiles suggested a logarithmic velocity distribution of the form
u y 1 log u* yo′ κ
(7.8)
where u* is the friction velocity at the bed; κ is a constant of about 0.40, as in pipes and open channels; and yo′ is the virtual origin of the profiles, that is. the distance at which u 0 and which was found to be equal to about 0.021 in. or 0.5 mm. This means that in the neighborhood of the bed, the flow structure closely resembles that in an open channel. In the main core of the flow field and sufficiently far from the surfaces of the channel and the ring, the velocity remains almost constant with negligible gradients. Experiments for incipient motion of plastic beads gave results quite close to Shield’s diagram, particularly for low frictional Froude number [84]. The first experiments at MIT were concentrated on the equilibrium concentration and the controlling factors. Kaolinite clay in distilled water was used with a grain size distribution as shown in Figure 7.8. Figure 7.9 shows a concentration-time plot for constant flow conditions but with variable initial sediment concentration,
Cohesive Sediments in Open Channels
212
Mit classification
Sand Coarse Medium
Fine
Silt Coarse Medium
Fine
Clay Coarse Medium Fine
100
Percent finer by weight
90 80 70 60 50 40 30 20 10 0 10
1.0
0.1
0.01
0.001
0.0001
Diameter in mm Hydrometer analysis
Commercial average
Figure 7.8 Grain size distribution of peerless no. 2 kaolinite clay [31, 109, 114].
Co [31, 109, 114, 115]. In all runs, after a short period of deposition, the suspended sediment concentration reached a constant value, Ceq, defined again as equilibrium concentration. The ratio, Ceq/Co, defined as the relative equilibrium concentration, for a range of Co from about 1000 ppm to about 16,000 ppm, remained practically constant and independent of Co. Thus, the earlier indication that a given flow can sustain in suspension a constant fraction of suspended cohesive sediment regardless of the value of the initial concentration was confirmed. The grain size distribution of the suspended sediment at equilibrium concentration, shown in Figure 7.10, led to an interesting conclusion. The overall sediment size is clearly finer; however, the percentage of the silt size remains approximately the same before and after deposition. That means that it is the finer portion of the sediment that forms sufficiently large and resistant aggregates to be able to permanently settle on the bed [31, 109, 114]. The functional relationship of the relative equilibrium concentration to the flow conditions was investigated next. After a few trials, it was concluded that the relative equilibrium concentration, Ceq /Co, is best correlated to the average shear stress over the rotating channel boundary. This stress was evaluated from the measured shear stress on the ring. The latter was first correlated with the differential velocity, V, between the channel and the ring by the relationship
r 0.0017(V )2
(7.9)
where r is in dynes/cm2 and V in cm s1. V is proportional to the relative angular velocity of the two rotating components, , which is equal to the
Chapter | 7 Deposition and Resuspension of Cohesive Soils
213
30
20
Series IV Runs 1–6 Depth � 16 cm Velocity � 81.0 cm/sec.
10
Run 6
Ceq /Co � 0.528
Run 5
Ceq /Co � 0.540
Run 4
Ceq /Co � 0.568
Run 3
Ceq /Co � 0.540
Run 2
Ceq /Co � 0.498
Run 1
Ceq /Co � 0.540
8
Suspended sediment concentration vs. time for various initial concentrations
Suspended sediment concentration in (PPT)
6
4 3
2
1 0.8 0.6
0.4 0.3
0.2
.1
0
3
6
9
12
15
18
21
24
27
30
33
Time after start in hours Figure 7.9 Variation of suspended kaolinite clay for various values of Co [31, 109, 114].
absolute summation of the angular velocity of the two components. The average shear stress along the channel boundary, cav, is given by
cav
br b 2d
(7.10)
Cohesive Sediments in Open Channels
214
100
Percent finer by weight
90 Depth � 8 cm. �w � 17.71 RPM Co � 14,862 ppm Ceq � 5,710 ppm
80 70 60 50 40 30 20 10 0 10
1.0
0.1 0.01 Diameter in mm
0.001
0.0001
Figure 7.10 Grain size distribution of kaolinite clay at equilibrium concentration of Figure 6.10 [31, 109, 114].
Equilibrium concentration in ppm 10 3
12 11 10 9 8
Channel only rotating
7 6 5 4
Depth 8 cm. Depth 12 cm. Depth 16 cm. Co 8,020 ppm
3 2 1 0
Right only rotating
0
100
200 (∆�)2 1 2 (d/b)
300
400
in (RPM)2
Figure 7.11 Ceq bed shear stress relationship [31, 109, 114, 115, Fig. 32].
where b is the channel width and d is the depth of flow. Thus, the average shear stress on the channel boundary is proportional to ( )2 / (1 2 d/b) . A plot of the relative equilibrium concentration, Ceq/Co, versus this last parameter is shown in Figure 7.11 for four channel depths but a constant initial concentration. Thus, the first early phase of research led to three important conclusions: (a) the percentage of a given cohesive sediment that a particular flow is able to maintain in suspension is independent of the initial concentration; (b) the bed
C/CO
Chapter | 7 Deposition and Resuspension of Cohesive Soils
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
215
Symbol Depth (in.) Co(ppm) ∆V (cm/sec) 6 1005 0.0 6 1005 41.4 6 1005 49.4 6 1005 55.8 6 1005 59.8 6 1005 63.0 6 1005 70.0 6 1005 84.0
0
1
2
3
4
5
6
7
8
9
10 11 t (hr)
12
13
14
15
16
17
τb (dynes/cm2) 0.0 1.57 1.92 2.21 2.39 2.54 2.91 3.55
18
19
20
Figure 7.12 Variation of relative suspended concentration for kaolinite in distilled water [81, 82, Fig. 4.2.1].
shear stress of the channel is the flow parameter which determines the degrees of retention and deposition; and (c) there is a minimum bed stress below which all suspended sediment deposits. The more extensive and detailed experimental phase at the University of Florida led to the general formulation of the latter functional relationship for a wide variety of sediment-water systems and to equations for the deposition rates of the depositable portion of the total sediment. Figure 7.12 shows the results of 8 runs for a depth of 6 in.; a constant initial concentration, Co, of 1005 ppm; and a bed shear stress ranging from zero to 3.55 dynes/cm2 for kaolinite clay in distilled water. It was observed that complete deposition takes place for a bed stress between 1.57 and 1.92 dynes/cm2 and that Ceq is quickly attained. The bed shear stress, b, was directly measured for each operational speed by means of an instrumented false bottom. The following correlation between b and the average flow velocity, uav, was found:
0.94 b 0.126uav
(7.11)
In Figure 7.13, the relative equilibrium concentration, Ceq/Co, was plotted versus the bed shear stress for the indicated ranges of initial sediment concentration, Co, and the indicated depths of flow. The scattering of the data points appears to be random, but they clearly define a mean curve, thus suggesting, like the aforementioned MIT experiments, a unique dependence of the relative equilibrium concentration or the degree of retention on the bed shear stress [81, 82]. The curve intersects the b axis at a distinct point, indicated as bmin, and which for the present case has the value of 1.80 dynes/cm2. One should note that the ratio (Co Ceq)/Co represents the portion of the total sediment that deposits permanently on the bed and may be defined as the degree of deposition. An analytical expression was subsequently sought for the relative equilibrium concentration or the degree of retention. In Figure 7.14, the degree of retention, * Ceq Ceq / Co , and the dimensionless bed shear stress in excess of the minimum one, b* 1 ( b b min ) / b min , are plotted on normal probability-logarithmic paper with the shear stress as an abscissa and the degree of retention as the
Cohesive Sediments in Open Channels
216
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
Caq Co
Symbol Depth (in.) CO(ppm) Range 6 9 12 13
0
1
2
3
4
5
6
7
8
1078–25000 1058–24200 1135–25930 7680–16900
9
10
11
τb (dynes/cm2)
Figure 7.13 Relation of relative equilibrium concentration, Ceq /Co, with bed shear stress, b, for kaolinite in distilled water [81, 82, Fig. 4.2.2, 83]. 99 98 95
Ceq* in %
90
C*eq�
1 2π
ya
�e
2
–ω /2
dω, ya�
1 σ1
log [(τb*�1)/(τb*�1)50]
2 σ1�0.49, (τb*�1)50�0.72, τbmin(dynes/cm ) � 1.80
80 70 60 50 40 30 20 Symbol Depth (in.) Co range(ppm)
10
6 9 12 13
5 2 1 0.01
0.1
1080–25000 1060–24210 1150–25930 7680–16900
1
10
�b*�1 * Figure 7.14 Log-normal plot of degree of retention, Ceq , against the shear stress parameter * 1 [81, 82, Fig. 4.2.3, 83]. b
ordinate. The data points fall randomly on a straight line, which can be described by the expression * Ceq
ya
2 exp d 2 2π ∞
(7.12)
( * 1) 1 log * b 1 ( b 1)50
(7.13)
1
∫
where ya
Chapter | 7 Deposition and Resuspension of Cohesive Soils
99 98 95
Ceq* in %
90
217
Symbol Reference τb*–1 (dynes/cm2) (τb*–1)50 Kaolinite in 0.72 1.80 distilled water 0.82 Series C 1.19
80 70 60 50 40 30 20
Series D
0.96
1.18
10 5 2 1 0.01
0.1
1
10
�b*�1 *
Figure 7.15 Log-normal plot of Ceq versus b* 1 for San Francisco Bay mud and 50%–50% mixture of San Francisco mud and kaolinite in salt water [81, 82, Fig. 4.3.4].
In Equation 7.13, the term ( b* 1)50 is the mean of the logarithmic-normal relation described by Equation 7.12, 1 is the corresponding standard deviation, and is a dummy variable. The straight line was fitted by the method of least squares. The corresponding means are shown in the figure legend, while the standard deviation, 1, was found to be equal to 0.49 for all tests involving kaolinite in distilled water. The ** * degree of deposition is Ceq 1 Ceq . To investigate the effect of the physicochemical properties of the sedimentwater system on the deposition parameters, ( b∗ 1)50 and 1, two additional types of sediment were studied with salt water at about ocean salinity of 34,000 ppm. The first, Series D, was the same mud from the San Francisco Bay described in Chapter 6, and the second, Series C, was a mixture of 50% mud and 50% kaolinite clay. The results are shown in Figure 7.15. The test results of both Series C and D again appear to define a linear log-normal relationship, like the kaolinite clay in distilled water. These two straight lines are parallel to the original line for the first test, suggesting the same logarithmic standard deviation 1 5 0.49. Only the mean values of the shear stress parameter vary, being 0.82 for Series C and 1.18 for Series D. In Figure 7.16, the data for Figure 7.15 were plotted with the abscissa divided by the value of the shear stress parameter at relative equilibrium concentration or * degree of retention of 50%, ( b 1)50 . The results of Series A for kaolinite in distilled water at a depth of 6 in. and an initial concentration Co 6,140 ppm and of Series B for kaolinite clay in salt water at a depth of 9 in. and an initial concentration of 10,270 ppm are included. Finally, results are shown from earlier
Cohesive Sediments in Open Channels
218
Ceq* in %
99 98
Symbol
Reference τbmin (dynes/cm2)
(τb*–1)50
95
Series A
1.50
0.58
90
Series B
1.50
0.58
Series C
1.19
0.88
Series D
0.96
1.18
Partheniades et al.
0.40
0.92
Partheniades Rosillon and volkenborn
0.65 0.75
0.80 1.50
80 70 60 50 40 30 20 10 5 2 1 0.01
0.1
1
10
(τb*–1)/(τb*–1)50 Figure 7.16 Log-normal plot of [81, 82, Fig. 4.2.5].
* Ceq
against the bed stress parameter
( b* 1) /( b* 1)50
studies by Partheniades and coworkers [31, 109, 114] in the rotating channel at MIT, a single experimental point from the flume experiments on the San Francisco Bay mud by the author discussed in Chapter 6 [102, 104, 105], and the results by Rosillon and Volkenborn for flume studies of silty clay mud from Maracaibo Bay in Venezuela [124]. All these data for drastically different claywater systems fall on the same straight line described by Equation 7.12 and Equation 7.13 with comparatively little random scattering. The basic parameters bmin and ( b* 1)50 are shown in the insert. Figure 7.15 and Figure 7.16 lead to the very important conclusion that the degrees of retention and deposition under a given bed shear stress can be determined by the minimum bed shear stress, min, and the parameter ( b* 1)50. The standard deviation 1 has been shown to be constant and equal to 0.49. These first two parameters represent the gross hydraulic properties of the sediment-water system and can readily be determined in any laboratory with annular channel-ring equipment or any other suitable experimental setup that provides a uniform flow field without floc-disrupting elements. The equation for the degrees of retention and deposition can be further simplified if a relation can be found linking the parameter ( b* 1)50 to the minimum bed shear stress, min, which can be more readily evaluated. In Figure 7.17, the parameters 1/b50 at 50% degree of deposition and of ( b* 1)50 are plotted against bmin for the indicated types of sediment and water quality. These last two parameters are related by the following simple equation:
b 50 ( b* 1)50 1 b min
(7.14)
Chapter | 7 Deposition and Resuspension of Cohesive Soils
219
2
(τb*1)50 4e1.27 τbmin
(τb*1)50, (1/τb50)
1 0.9 0.8 0.7
(1/τb50) 0.65e0.30 τbmin
0.6 0.5 0.4 0.3
0.2
0.4
Symbol
Material
Series A,B
Kaolinite 50% kaolinite and 50% bay mud
C
Bay mud
D
Maracaibo sediment
0.6
0.8
1.0 τbmin
1.2
1.4
1.6
(dynes/cm2)
Figure 7.17 Relationships of depositional parameters [81, 82, Fig. 4.2.6].
There seems to be good correlation expressed by 1 and
b 50
0.65exp(0.30 b min )
( b* − 1)50 4exp(1.27 b min )
(7.15)
(7.16)
At this point, it is appropriate to look at the physical significance of the three parameters that determine the degrees of retention and deposition. In Section 4.4.1, the maximum aggregate size was related to the local rate of energy dissipation either per unit mass, ε, or per unit volume, ε (Equation 4.101 through Equation 4.106). That section also showed that the highest rate of energy dissipation takes place in the immediate neighborhood of the bed and, specifically, within the viscous sublayer, where it exceeds the average rate by three orders of magnitude. It was assumed that a cohesive bed behaves as hydraulically smooth. In open channels
Cohesive Sediments in Open Channels
220
the rate of energy dissipation within the near-bed viscous sublayer, εv, is given by Equation 4.72. For the inertia subrange the same energy, εt, is expressed by Equation 4.71, while the average energy dissipation over the entire channel, εav, is given by Equation 4.75. All these energies are related to the friction velocity, which, in turn, is related to the bed shear stress by the relationship u* b / . Experimental results by Kusuda et al., discussed in Section 4.4.2 [65], led to the conclusion that in a turbulent shear flow, the aggregates are composed of smaller and denser units formed under the highest shear rates that the sediment suspension was subjected to. In the studies with the rotating channel-ring system both at MIT [31, 109, 114, 118] and at the University of Florida [81, 82], the cohesive sediment was subjected to the highest shear rate permitted by the setup prior to each run. After deposition, the same sediment was resuspended under the same highest shear rates. It is reasonable, therefore, to assume that in every run, the basic unit of the aggregates is the one corresponding to these resuspension shear rates. Considering that the highest disruptive stresses are in the immediate neighborhood of the bed, it follows that the basic aggregate building unit is the primary floc, which, according to Table 4.2 in Section 4.4.1, has the highest density and strength. This is indeed what the experiments by Kusuda et al. [65] have shown. The time average bed shear stress for an open channel with a high width to depth ratio was given in terms of u* by Equation 4.68b in Section 4.3.1. However, the instantaneous value of that stress varies due to the imposed turbulence outside the viscous sublayer. The thickness of the latter is neither constant nor free from eddies, but it fluctuates instead from near zero to a maximum value. At this point, the viscous sublayer becomes unstable and disintegrates with turbulent eddies almost reaching the bed, while the bed stress attains its highest instantaneous value. From then on, viscous stresses slow down layer after layer, thus increasing the viscous sublayer thickness until the latter reaches its instability limit and the whole process is repeated all over again. As explained in Section 6.3, this periodic building and collapsing of the viscous sublayer was first demonstrated experimentally in 1958 by Einstein and Li [28], who derived the following equation for one-dimensional flow:
b
U vb πν t
(7.17)
where Uvb is the turbulent velocity right outside the sublayer, which reaches the bed instantaneously; and ν are the dynamic and kinematic viscosities, respectively; and t is the time from the moment of the collapse of the sublayer. Equation 7.17 suggests instantaneously infinite shear stress values; in reality, however, there must be a finite upper stress limit. Aggregates of various orders are being formed in the main flow far from the bed, where the shear stresses are low. These aggregates settle to the bed, where they are gradually subjected to increasing disruptive stresses. Depending on their physical
Chapter | 7 Deposition and Resuspension of Cohesive Soils
221
properties and, in particular, on their density and strength, some of these aggregates will be broken up and disintegrated before reaching the bed. Others will be entrained by turbulent diffusion back into the main flow before contacting the bed. Finally, those with the highest strength and settling velocity will overcome the nearbed upward turbulent velocities, will contact the cohesive bed, and will become part of it. It was shown in Chapter 6 and will be shown in Section 7.3 in the studies of resuspension of deposited cohesive sediments that the critical bed shear stresses for resuspension are considerably higher than the stresses under which suspended sediment deposits. The detail mechanism of the interaction between the settling sediment and the bed will be treated in Section 7.2. At this point, the discussion is limited to the general flow pattern in the immediate neighborhood of the bed. The quasi-periodic building up and disintegrating of the viscous sublayer generates a highly complex three-dimensional flow pattern in the immediate neighborhood of the bed. Kline et al. [58] demonstrated experimentally the development of three-dimensional low-speed streaks near the bed that interact with the flow in the outer layer. These streaks eventually break up, generating eddies that transport low momentum fluid to the outer region and high momentum fluid to the inner one. This transfer materializes as bursts and injections generating transverse vertical velocities, v, which are associated with high longitudinal velocities, u, when directed toward the bed and with low u when moving away from the bed. Grass measured actual instantaneous velocity profiles in both the longitudinal and transverse directions and confirmed that inrush toward the wall is associated with high momentum fluid and, therefore, with high wall stresses, and vice versa [38]. Similar ejections and inrushes have been observed from the surfaces of roughness elements of rough boundaries. On the basis of the outlined near-bed flow structure, the following hydrodynamic process takes place between the depositing sediment and the bed. Settling large aggregates formed in the far-bed zone will encounter the high disruptive near-bed stresses and gradients. Their approach velocity will be about equal to the summation of their own settling velocity and that of the transverse currents. If vsa is the approach velocity of an aggregate and tsa is the settling time for the aggregate to travel through the near-bed zone before reaching the bed, two situations may happen. If tsa is larger than the time of downward transverse velocities, the aggregate will be re-entrained back into the far-bed zone. If, on the other hand, tsa is smaller than the time of the downward currents, it will reach the surface of the bed. In the latter case, if the aggregates will be able to resist the highest stresses there, it will develop bonds with the bed and will form part of it. In the opposite case, it will be broken up into smaller units, which will be re-entrained into the far-bed zone, where they will start combining with other units to form lager aggregates, and the same process will continue. Eventually, all aggregates and flocs with sufficient settling velocity and adequate shear strength will deposit permanently on the bed, while the remaining units will be in a continuous process of aggregation and disaggregation.
Cohesive Sediments in Open Channels
222
The time history of the median aggregate size in the far-bed zone was investigated by Krishnappan and the author at the Canadian Center for Inland Waters in Burlington, Ontario, in 1991. Figure 7.18a and Figure 7.18b show two typical examples of this unpublished work. The experiments were conducted in a rotating annular channel-ring system similar to the one described in Section 7.1.2 but larger and equipped with a Malvern particle size analyzer for the measurement of the aggregate sizes. Kaolinite in distilled water was used as the sedimentwater system. The case in Figure 7.18a represents a high degree of deposition corresponding to a relatively low bed shear stress; and that in Figure 7.18b a low degree of deposition under a relatively high bed stress. In both cases, the median aggregate size increases with time as the suspended sediment concentration decreases and the depositable portion of the suspended sediment settles on the bed. The median aggregate size ranges from 12 to 35 m with aggregates as large as 200 m. These observations suggest a gradual coalescence of smaller flocs into larger aggregates as the process of deposition continues, but with strength, settling velocity, and density insufficient for them to reach the bed and
Median floc size (micron)
Concentration (ppm)
1200 1000 800 600 400 200
0
200
400
30
20
0
600
0
Time (minutes)
200
400
600
Time (minutes)
(a) 40 Median floc size (micron)
Concentration (ppm)
800
600
400
200
0
200
400
Time (minutes)
600
30
20
0
0
100
200
300
Time (minutes)
(b)
Figure 7.18 Variation of concentration and of median aggregate size of Kaolinite clay in distilled water [unpublished data]. (a) 0.70 degree of deposition; (b) 0.10 degree of deposition.
Chapter | 7 Deposition and Resuspension of Cohesive Soils
223
become permanently attached to it. The aggregates will eventually reach a quasi steady-state corresponding to the far-bed flow structure. These preliminary observations are in agreement with experimental results of Burban et al. [8] and Tsai et al. [147] shown in Figure 4.17 through Figure 4.20. They also confirm the aggregate structure as molded by the nearbed and far-bed flow-induced shear stresses analyzed in Chapter 4 and specifically in Section 4.3. The presented experimental evidence and the near-bed flow structure lead to the conclusion that the gross time average bed shear stress is the hydrodynamic parameter, which determines the part of the suspended sediment capable of depositing and, therefore, the degrees of retention and deposition. The same results suggest that all the complex physicochemical properties of the sediment-water system, which determine the aggregate properties controlling the degree of deposition, can be represented by two parameters: bmin, that is, the stress below which all sediment deposits, while above it some sediment remains in suspension; and ( b / b min 1)50 ( b* 1)50 . Both parameters can be readily determined, as already pointed out, by simple deposition experiments of the particular sedimentwater system in a devise similar to the annular channel-ring system or in any other suitable experimental setup with uniform bed stresses and flow structure throughout the flow field. We can add that in view of the tentative relations between these two parameters, bmin may be proven the only controlling parameter, while the second can be evaluated by means of Equation 7.15 and Equation 7.16. However, the validity of these equations has to be confirmed with more experimental research and properly modified if they are to be used with sufficient degree of confidence. For now, it is better to evaluate both parameters independently. Because bmin was shown to be representative of the physicochemical properties of the sediment-water system, an attempt was made to correlate it with some parameter indicative of these properties. The cation exchange capacity (CEC) was chosen as a correlation parameter. Such a correlation is shown in Figure 7.19 for both bmin and ( b* − 1)50 for the five different sediments and sediment mixtures indicated in the insert [81, 82]. One is reminded that Series A is for distilled water and Series B is for salt water. The CEC for the 50/50 mixture of kaolinite and San Francisco Bay mud was taken as the mean of the CEC of the constituent clays. The limited number of data points clearly indicates an increase of bmin with increasing CEC. This trend can be explained by the fact that the physicochemical interparticle forces, which control the aggregate strength, increase with the CEC of the sediment, thus leading to the formation of larger and more stress-resistant aggregates. Therefore, higher stresses are required to * keep all the sediment in suspension. The reduction of ( b − 1)50 with increasing CEC is also consistent with Equation 7.15. One should note that the correlation formulas presented in Figure 7.19 are at this point only indicative of trends and are based on a limited number of data. One should also remember that the CEC is a property of the sediment only and not of the sediment-water system. Therefore, these relations should not be used for quantitative purposes.
Cohesive Sediments in Open Channels
224
2.5 Symbol
Material
Series A,B
Kaolinite
(τb*1)50, τbmin (dynes/cm2)
2.0
1.5
50% kaolinite and 50% bay mud
C
Bay mud
D
τbmin 0.01 (CEC) 0.7
Maracaibo sediment
1.0
0.5 (τb*1)50 0.0114 (CEC) 1.45 0
0
10
20 30 40 50 60 CEC (milliequivalents per hundred grams)
Figure 7.19 Relationships between bmin, [81, 82, Fig. 4.2.7].
( b* − 1)50 ,
70
80
and cation exchange capacity (CEC)
7.1.3 D etailed Studies on Deposition. Part B: The Rates of Deposition The equations, principles, and criteria developed so far are for an estimate of the depositable portion of cohesive sediment in suspension under given bed shear stresses. For a reliable mathematical model for the prediction of shoaling, the rates under which the depositable sediment will deposit may be equally important. This was the second part of the detailed investigation by Mehta and the author [81, 82]. The portion of sediment that eventually deposits can be represented by Co Ceq. The fraction of that sediment deposited at time t can be represented by C∗
Co C Co Ceq
(7.18)
Likewise, the fraction of the depositable suspended sediment remaining in suspension at time t can be represented by C ∗∗
C Ceq Co Ceq
(7.19)
Chapter | 7 Deposition and Resuspension of Cohesive Soils
99 98 95
C*
1 2π
Tc
�e
ω2/2
dω,
∞
Tc
225
1 log (t/t50) σ1
C * in %
90 80 70 60 50 40 30 20
Symbol Material Co(ppm) Depth (cm) τb* t50(min) Kaolinite in distilled water
10 5 2 1 0.05 0.1
0.5
1150
31
1.70
3.4
1150
31
0.80
10.0
1150
31
1.11
6.7
1150
31
1.45
4.6
1060
23
1.54
3.3
1
5
10
50
t/t50 Figure 7.20 Concentration-time relationships for depositable kaolinite clay in distilled water [81, 82, Fig. 4.2.9].
In Equation 7.18 and Equation 7.19, C is the concentration at time t; the other two terms were identified earlier. Obviously, C ∗∗ 1 C ∗
(7.20)
The study on deposition rates was conducted by plotting C* in percent as an ordinate versus time nondimensionalized by t50, that is, the time required from the beginning of deposition for C* to become 50%. In Figure 7.20 a set of data for kaolinite in distilled water and for low Co is shown on a log-normal plot. The range of depth and relative bed shear stress, b∗ b / b min, are shown in the insert together with the corresponding values of t50. In all runs except one, d∗ was larger than unity. The data points fall very closely on a straight line, even for b bmin, described by the logarithmic-normal law indicated in the insert: C∗
1
Tc
2 ∫ exp ( /2)d 2π ∞
(7.21)
where Tc
t 1 log t50 2
(7.22)
In Equation 7.22, 2 is the geometric standard deviation, t50 is the geometric mean of the log-normal equation, and is a dummy variable.
Cohesive Sediments in Open Channels
226
Similar experimental results are shown in Figure 7.21 but for high initial concentrations and for two different depths. b was always larger than bmin. A similar plot is shown in Figure 7.22 for both high and low initial concentrations. As in Figure 7.20, all data points fall very closely on the same straight line described by Equation 7.21 and Equation 7.22. This close agreement suggests 99 98
Symbol Material Co(ppm) Depth(in.) �b*
95
Kaolinite in distiled water
C * in %
90 80 70 60 50 40 30 20
5280 5280 5280 5370 5370 5370
12 12 12 9 9 9
t50(min)
1.89 1.48 1.23 1.86 1.17 1.58
5.5 6.5 13.6 4.0 8.8 4.5
10 5 2 1 0.01
0.05 0.1
0.5 t / t50
1
5
10
50
Figure 7.21 Concentration-time relations for depositable kaolinite clay in distilled water [81, 82, Fig. 4.2.12]. 99 98 95
C* in %
90 80 70 60 50 40 30 20
Symbol Material Co(ppm) Depth (in.)
10
Kaolinite in distilled water
5 2 1 0.05 0.1
0.5
1
4920 10460 15790 1020 1020
6 6 6 6 6
5
10
�b*
t50(min)
1.60 1.60 1.59 1.58 1.23
8.4 8.8 6.9 3.1 8.2
50
t / t50 Figure 7.22 Concentration-time relations for depositable kaolinite clay in distilled water [81, 82, Fig. 4.2.15].
Chapter | 7 Deposition and Resuspension of Cohesive Soils
227
that only two parameters are needed to describe the time deposition rates: t50 and 2. Additional results have been obtained for higher values of b* , the following four of which are shown here. In Figure 7.23 and Figure 7.24, results for Co 13,760 – 18,1380 ppm and 24,210 – 25,000 ppm, respectively, are shown. In all the preceding cases, there is some deviation from the average straight line at the beginning of the deposition process due primarily to the sensitivity 99 98 95
C * in %
90 80 70 60 50 40 30 20
Symbol Material Co (ppm) Depth (in.)
10
Koalinite in distilled water
5 2 1 0.05 0.1
0.5
18380 18383 18380 13760 15300
1
9 9 9 9 12
5
�b*
t50(min)
2.07 1.56 2.32 2.07 2.63
2.2 10.4 2.3 3.2 1.5
10
50
t / t50 Figure 7.23 Concentration-time relations for depositable kaolinite clay in distilled water [81, 82, Fig. 4.2.20]. 99 98 95
C* in %
90 80 70 60 50 40 30 20
Symbol Material Co(ppm) Depth(in.) Kaolinite in distilled water
10 5 2 1 0.05 0.1
0.5
1
5 t / t50
10
24210 24210 24210 24210 25000 25000
9 9 9 9 6 6
50
�b*
t50(min)
2.91 2.73 2.41 3.08 3.18 1.85
2.1 2.5 3.9 0.6 0.6 8.3
100
500
Figure 7.24 Concentration-time relations for kaolinite clay in distilled water [81, 82, Fig. 4.2.23].
Cohesive Sediments in Open Channels
228
of the abscissa to concentration changes at that time. The sampling time was about 10–15 s, while the zone of maximum deviation was for t/t50 below 0.1–0.15, which corresponds to a range of t50 of 0.3–0.5 min or 20 to 30 s. This is the order of magnitude of the sampling time. In addition, as shown in Figure 7.1, Figure 7.3, Figure 7.4, and Figure 7.9, the time rates of change of suspended sediment concentration are highest at the beginning of deposition. These two factors contribute to the slight scattering at the initial stage of each deposition test. The deviation at the initial stage is more pronounced for bed shear stresses below bmin, as shown in Figure 7.25 and Figure 7.26. This deviation should be 99 98 95
C* in %
90 80 70 60 50 40 30 20 10 5 2 1 0.05 0.1
Symbol Material Co(ppm) Depth (in.) Kaolinite 9650 in distilled 1060 water
0.5
1
6 9
5
�b*
t50(min)
0.91 0.74
2.3 7.8
10
50
t / t50
Figure 7.25 Concentration-time relationships for kaolinite clay in distilled water [81, 82, Fig. 4.2.25].
99 98 95
C * in %
90 80 70 60 50 40 30 20 10
Symbol Material Co (ppm) Depth (in.) Kaolinite in distilled water
5 2 1 0.05 0.1
0.5
1
21830 1060 7680 5370
12 9 13 9
5 t / t50
10
�b*
t50(min)
0.54 0.44 0.61 0.90
3.3 5.2 5.6 4.7
50
Figure 7.26 Concentration-time relationships for kaolinite clay in distilled water [81, 82, Fig. 4.2.26].
Chapter | 7 Deposition and Resuspension of Cohesive Soils
229
expected because the lower the stress below the minimum, the wider the spectrum of aggregates and of the settling velocities and the higher the initial deposition rates. Equation 7.21 and Equation 7.22 can also be written in the forms C∗
T 1 1 erf c 2 2 2
(7.23)
T 1 erfc c 2 2
(7.24)
and consequently, C ∗∗ By definition T T erfc c 1 erf c 2 2
(7.25)
From Equation 7.21 and Equation 7.22 or, alternatively, from Equation 7.24 and Equation 7.25, the time rates of change of C* and of C** can readily be derived. These equations are
T 2 1 dC ∗ 0.434 exp c 2 t dt 2π2
(7.26)
T 2 1 dC ∗∗ 0.434 exp c 2 t dt 2π2
(7.27)
and
From Equation 7.26 and Equation 7.27, the following expression can be derived for the time rates of deposition in a channel of a depth h with steady uniform flow and uniform distribution of the suspended sediment: h
T 2 0.434 dC exp c (Co Ceq ) h 2 dt 2π2 t
(7.28)
Equation 7.28, however, is valid for a nondispersive system without longitudinal gradients of suspended sediment concentration. Similar log-normal relationships have also been found valid for kaolinite in salt water, as shown in Figure 7.27, as well as for other types of sediment-water systems. Figure 7.28 shows a plot of test results by Rosillon and Volkenborn for sediment
Cohesive Sediments in Open Channels
230
99 98 95
C*
1 2π
Tc
�e
ω2/2
∞
dω, Tc
1 log (t/t50) σ2
C * in %
90 80 70 60 50 40 30 20 Symbol Material Co(ppm) Depth(in.)
10 5 2 1 0.01
0.05 0.1
Kaolinite in 6140 salt water 6140 6140
6 6 6
0.5 1 t / t50
5
�b*
t50(min)
1.11 2.55 1.81
29.7 1.6 20.3
10
50
Figure 7.27 Concentration-time relations for kaolinite clay in salt water [81, 82, Fig. 4.2.39]. 99 98 95
C*
1 2π
Tc
�e
ω2/2
dω, Tc
∞
1 log (t/t50) σ2
C * in %
90 80 70 60 50 40 30 20 10
Symbol Material Co(ppm) Depth(in.)
5 2 1 0.2
Maracaibo 2108 sediment 2088
0.5
1
5
10 t / t50
20 20
�b*
t50(min)
0.50 0.14
55 12
50
100
200
Figure 7.28 Concentration-time relations for Maracaibo sediment in salt water [81, 82, Fig. 4.2.44].
from the Maracaibo Bay in Venezuela in an open flume [124]. Figure 7.29 shows a similar plot of data from three experiments with San Francisco Bay mud conducted at the University of California in an open flume by Krone [62] and by the author is his first studies on erosion and deposition of cohesive soils [102, 105]. Deviations from Equation 7.21 and Equation 7.22 started appearing for initial concentrations above 20,000 ppm, suggesting hindered settling and a possible effect on viscosity and on bed shear stresses. Such high concentrations, however, are very rare and should be a special subject for research.
Chapter | 7 Deposition and Resuspension of Cohesive Soils
99 98 95
C*
1 2π
Tc
�e ∞
dω, Tc
1 log (t/t50) σ2
Symbol
Material
ω2/2
231
C * in %
90 80 70 60 50 40 30 20 10 5 2 1 0.05 0.1
Co(ppm) Depth (cm.) �b* t50(min)
Bay mud 8450 (Partheniades) 1920 Bay mud (Krone)
0.5
1
21000
5
6 6
2.54 0.59
600 420
12
0.48
354
10
50
t / t50 Figure 7.29 Concentration-time relations for San Francisco Bay mud in a flume with salt water [81, 82, Fig. 4.2.49].
For ordinary cases of sediment-carrying channels and in estuaries, the suspended sediment concentrations are of the order of a few hundred or, at most, a few thousand parts per million. For these cases, the preceding equations may be accepted as valid and reliable, verified for clay suspensions of various mineralogical compositions and with both distilled and salt water. It is finally observed that, for the Maracaibo and the San Francisco Bay mud sediments, the t50 is substantially higher than for the kaolinite clay. Since the experiments for the last two cases were conducted in open flumes with recirculating water through return pipes and diffusers, the sediment suspensions were subjected to considerably higher stresses and to higher intensity of turbulent agitation in comparison to those generated by the flow in the open channel. These higher stresses lead to aggregate sizes and settling velocities considerably lower than the ones developed in the rotating annular channel-ring system, in which the bed shear stresses and the turbulence parameters are constant at every cross-section. This is an additional indication of the necessity to use experimental setups with constant bed stresses and flow structure for reliable relationships describing the deposition rates. The parameters t50 and 2 have to be evaluated in terms of the bed shear stresses and possibly Co. Some limited information obtained to-date is shown in Figure 7.30a and Figure 7.30b for t50 and 2 with b∗ and for four different depths. This subject should be investigated in more detail. These figures lead to the following general conclusions: 1. The mean deposition time, t50, increases first with b* reaching a peak for values of the latter between 1 and 1.25. From then on, it decreases with b*. 2. For the same b*, t50 seems to increase with increasing depth; there is, however, some overlapping.
Cohesive Sediments in Open Channels
232
1.5
log t50
1.0
0.5
0.0
0.5
0
0.5
1.0
1.5
2.0
2.5
3.0
�b*
(a) 1.5
2
1.0
Symbol Depth (in.) Co(ppm) 6 6 9 12 13
0.5
0
0
0.5
1.0
1.5 �b*
(b) Figure 7.30 Variation of t50 and 2 with
b∗
2.0
4920 6310 5370 5280 7680
2.5
3.0
[81, 82, Fig. 4.2.34].
3. The standard deviation, 2, increases initially with increasing b∗ reaching a peak for values of the latter between 1 and 1.5, almost like t50. 4. 2 seems to be very little affected by the flow depth; the effect of the depth, however, is not clear. These four observed trends can be explained by the following simplified model. We recall first that t50 is representative of the mean settling time of the suspended aggregates, while 2 is a measure of the spread of the settling time of the aggregate population. If h is the depth of the channel, then indicating the
Chapter | 7 Deposition and Resuspension of Cohesive Soils
233
f(log ts)
ts50
log ts
�2 Figure 7.31 Schematic probability density function for t50 [100, Fig. 32, 101, Fig. 12].
near-bed settling velocity of the depositable aggregates by Ua, the settling time can be represented by
ts
h Ua
(7.29)
Figure 7.31 shows a schematic probability density function, f(logts), plotted against the logarithm of the settling time. One can observe first that at the lower end of the bed shear stresses and as the latter become smaller and smaller, the flow-induced aggregate disrupting forces continuously diminish, becoming totally ineffective below a certain point. Therefore, the smaller aggregates will more and more readily join together and with larger units to form larger aggregates, thus increasing the average aggregate size and settling velocity and decreasing the average settling time t50. The dominant factors at this stage are the frequency of interaggregate collisions and the settling-induced drag forces. Because of the elimination of the finer portion of the aggregate population, the aggregate size distribution will shift to a coarser range with higher settling velocities. With increasing bed shear stresses beyond a certain point, more and more aggregates are eliminated from the permanently depositable population, starting from the smaller units and advancing gradually to the larger ones, thus making the average size of the depositable population gradually larger and larger. At the same time, the turbulence intensity and the coefficient of eddy viscosity increase, thus retarding the eventual deposition of the depositable part of the suspended aggregates on the bed. In fact, both Ua and ts in Equation 7.29 are apparent values of the turbulence structure. An increase of the apparent settling time is reflected in the shifting of logts to the right of Figure 7.31. It follows that the settling time and the rates of deposition are subjected with increasing bed shear stresses to two opposing effects: first, an increase of the sizes and of the settling velocities of the depositable aggregates in quiescent water and, second, an increase of the turbulent diffusion opposing the settling of aggregates and decreasing the net settling velocities. The results of Figure 7.30 suggest that, initially, the diffusion effect appears to be the dominant. However, the quiescent
234
Cohesive Sediments in Open Channels
settling velocity of the aggregates increases in proportion to the third power of the aggregate size, whereas the hydrodynamic drag due to the vertical components of the turbulent velocity fluctuations are proportional to the second power of the latter. The dominance of one of these two factors will determine the changes of both t50 and 2 with b. A critical value of the latter must exist below which the diffusion effect is dominant and above which the submerged weight of the aggregates predominates. According to Figure 7.30, this critical value of b ranges between bmin and about 1.25bmin. Regarding the standard deviation, 2, as long as b bmin, that is, as long as all suspended units deposit, the spread of the floc size distribution, for the stated reasons, is expected to decrease with decreasing b. This decrease is reflected in a decrease of 2. For values of b above bmin, the relative equilibrium concentration increases and the degree of deposition decreases. The distribution of the depositable aggregate population shifts to the coarser side with increasing b. Figure 7.30 suggests that, at least in most cases, this shifting is accompanied by a narrower spread of the aggregate size distribution and smaller values of 2. The critical point for the beginning of the spread decrease seems to be around 1.5bmin. Finally, the initial sediment concentration, Co, seems to have an effect on t50, which for similar depths and bed stresses seems to decrease with increasing Co. This is to be expected, since the frequency of collisions increases with concentration. However, the range of Co is too small for definite conclusions.
7.1.4 Variation of Depositional Parameters as the Sediment Sorts during Deposition in Open Conduits The parameters bmin, 1, and ( b∗ 1)50 , which appear in Equation 7.12 and Equation 7.13 for the degree of deposition, and the parameters t50 and 2, which enter in Equation 7.21 through Equation 7.28 describing the rates of deposition, are integral manifestations of the physicochemical properties of the sedimentwater system. Therefore, they maintain a constant value only if these properties do not change. The experimental system used in the research was nondispersive. That is, at any time, the sediment composition and properties were identical in every cross-section. In natural conduits, however, such as rivers and estuaries, a sediment sorting takes place during the phase of deposition, because the faster settling aggregates deposit first and the finest with the lowest settling velocities deposit last. It is expected, therefore, that the aggregate density and strength of the deposited sediment will vary along the shoaling reaches of the channel and/or estuary, and this variation will be reflected in the values of the previously mentioned parameters. The change of the depositional parameters as the sediment disperses and deposits in open channels was investigated by Dixit, Mehta, and the author [16]. A straight 100 m long flume in the Waterways Experiment Station in Vicksburg, Mississippi, was used with kaolinite clay and natural cohesive mud from Atchafalaya Bay, Louisiana, as sediment. The grain size distribution for both
Chapter | 7 Deposition and Resuspension of Cohesive Soils
Sand
Percent finer by weight
100
Medium
235
Silt Fine
Coarse
Medium
Clay Fine
80
60
40 Kaolinite Atchafalaya mud 20
0.4
0.2
0.1
0.06 0.04
0.02
0.01 0.006
0.002
0.001
Grain size (mm) Figure 7.32 Grain size distribution of kaolinite clay and Atchafalaya mud used in the 100 m flume [16, Fig. 2.4].
sediments is shown in Figure 7.32. The following was the experimental procedure: The sediment-water mixture at predetermined concentrations was introduced from a head tank to the upstream end of the channel and was allowed to deposit under specific velocities and bed stresses. At the end of the channel, the water with the remaining suspended sediment settled in a downstream tank. The deposited sediment was subsequently collected from specified reaches ranging from 6.1 to 18.3 m of the bed and were forwarded to the Coastal Engineering laboratory of University of Florida in Gainesville for testing in the same experimental setup and with the same methodology described in Section 7.1.2. The experimental and procedural details can be found in Ref. [16]. Four series of experiments were conducted, three with kaolinite clay and the fourth with Atchafalaya Bay mud. The nominal flow rates, initial concentrations, and other data pertaining to the deposition in the flume are shown in Table 7.1. The water temperature for the kaolinite sediment ranged from 23°C to 26°C and that for the Atchafalaya mud was 12°C. About the same temperatures were kept for the sediment slurry. The depth of flow at the end of the channel was kept at approximately 15.2 cm. After sufficient sediment was deposited in the open flume, samples were collected from the designated sections of the bed and were forwarded to the Coastal Engineering Laboratory of the University of Florida in order to determine their depositional properties in the rotating channel-ring system previously used. In the
Cohesive Sediments in Open Channels
236
Table 7.1 Data related to the sediment deposition in the 100 meter flume in Vicksburg, Mississippi [16, Table 2.2] Series No.
Sediment Type
Nominal Initial Conc. (ppm)
Nominal Flow Rate (m3/s)
Injection Duration (hr)
1
kaolinite
10,000
0.0076
1.00
2
kaolinite
10,000
0.0052
1.50
3
kaolinite
5,000
0.0034
3.00
4
Atchafalaya mud
2,000
0.0017
1.25
case of Series 1, 2, and 3 with kaolinite clay, samples were taken so that the bed of the flume was divided into six or seven segments. The selection was guided by the necessity of securing the desirable suspended sediment concentration in the annular flume, which ranged between 2 and 5 g L1. In the case of the fourth series with the Atchafalaya Bay mud, samples from only three reaches were collected up to 18.3 m from the upstream end of the flume. Table 7.2 lists the test conditions in the annular flume for all collected samples. One should note that the upper shear stress limit of 0.85 Nm2 was selected on the basis of previous experiments by Mehta and the author [81, 82], according to which no noticeable deposition took place for shear stresses above that limit. In Table 7.3, the test results for Series 1 are summarized as an example. The parameter ( b∗ 1)50 remained constant for about the middle of the channel, and from then on, it increased gradually by 60% of its initial value. b50, on the other hand, decreased slowly with distance from the upstream end of the flume but by only 13% of its highest value, so it can be considered as a fairly stable parameter. ∗ for Q 0.00757 m3s1 and The relative equilibrium concentration Ceq 2 b 0.3 Nm is plotted in Figure 7.33 versus the dimensionless variable ( b* − 1)/( b* − 1)50 as in Figure 7.16, but on a log-log graph for all segments of the flume [16,100]. All data in Figure 7.33 fall quite closely on the same straight line, suggesting ∗ 90%: the following power law for Ceq
( ∗ 1) 3 ∗ Ceq 50 ∗ b ( ) − 1 b 50
(7.30)
where the exponent 3 has the value of 0.861 for all reaches. For the second series with Q 0.0052 m3s1 and the corresponding b 0.21 Nm2, 3 has the value of 0.578, and for Series 3 with Q 0.0034 m3s1 and the corresponding b 0.094 Nm2, it has the value of 0.509. Finally, for Series 4 with Atchafalaya Bay mud, with a discharge of 0.0017 m3s1 and a bed shear stress of 0.023 Nm2, the value of 3 was 1.43.
Chapter | 7 Deposition and Resuspension of Cohesive Soils
237
Table 7.2 Test conditions in the annular flume-ring system for samples collected in the Vicksburg open flume [16, Table 2.3] Range of b (Nm2)
Range of Co (g/1)
0–12.2 12.2–24.4 24.4–36.6 36.6–48.8 48.8–61.0 61.0–73.2 73.2–91.5
0.25–0.85 0.25–0.85 0.25–0.85 0.25–0.85 0.25–0.85 0.25–0.85 0.25–0.85
2.95–3.12 1.79–1.81 3.45–3.58 2.42–3.12 2.09–2.16 1.73–1.78 2.63–2.71
1 2 3 4 5 6
0–18.3 18.3–30.5 30.5–44.2 44.2–67.1 67.1–77.0 77.0–93.0
0.20–0.85 0.20–0.85 0.175–0.85 0.175–0.85 0.125–0.85 0.15–0.85
2.97–2.99 4.60–4.78 4.51–4.53 3.61–3.67 2.51–2.55 3.13–3.18
kaolinite
1 2 3 4 5 6
0–15.2 15.2–27.4 27.4–39.6 39.6–54.9 54.9–70.1 70.1–91.5
0.15–0.85 0.15–0.85 0.15–0.85 0.15–0.85 0.15–0.85 0.15–0.85
2.67–2.73 4.07–4.15 4.25–4.34 3.72–3.90 3.11–3.13 2.61–2.66
Atchafalaya Mud
1 2 3
0–6.1 6.1–12.2 12.2–18.3
0.15–0.85 0.15–0.85 0.15–0.85
4.89–5.19 2.11–2.12 1.06–1.08
Series No.
Sediment type
Sample group number
Reach along 100 m flume (m)
1
kaolinite
1 2 3 4 5 6 7
2
kaolinite
3
4
Equation 7.30 differs from Equation 7.12 and Equation 7.13, which describe the relative equilibrium concentration for the entire population of a wide variety of sediment-water systems. Earlier log-normal plots of the data from the various reaches of the flume did not give consistent results. There is no explanation for this difference at this time other than a reflection of skewness in the distribution of aggregate properties in the various reaches as a result of the preferential settling. This aspect requires further study. The deposition rates follow Equation 7.23 through Equation 7.25 with very little scattering as for the entire sediment population. Figure 7.34 through Figure 7.37 show the time deposition plots for runs 1, 2, 3 and 4, respectively, for samples taken from the middle portion of the open channel but with (t / t50 )1/2 as the abscissa [16, 100]. The parameters t50 and 2 for Series 1 are shown in Table 7.3. With the exception of the first reach, t50 increases with distance along the channel, indicating diminishing settling velocities and size of deposited aggregates
238
Table 7.3 Properties of kaolinite clay deposited in an open flume under a discharge of 0.007 m3/sec and a bed shear stress of 0.294 N/m3 [16, Table 3.2] 0–12.2
12.2–24.2
24.2–36.6
36.6–48.8
48.8–61.0
61.0–73.2
73.2–91.5
bmin (Nm )
0.260
0.250
0.250
0.266
0.200
0.185
0.180
( b∗
0.90
0.88
0.90
1.02
1.30
1.35
1.40
b50(Nm2)
0.494
0.470
0.457
0.460
0.460
0.435
0.432
t50 (min)
1.0–1.7
0.8–2.0
0.4–2.0
1.5–8.0
3.8–10.5
4.0–11.3
4.0–11
t50av
4.0
1.4
1.2
4.75
7.15
7.65
7.50
2
2 2av
− 1)50
.249–.945 0.597
.301–.514 0.408
.505–.736 0.621
.286–.824 0.555
.409–.637 0.523
.426–.700 0.563
.420–.720 0.570
Cohesive Sediments in Open Channels
Reach (m)
Chapter | 7 Deposition and Resuspension of Cohesive Soils
239
100.
Ceq* 0.5[(τb*1)/(τb*1)50]σ3 Ceq* %
σ3 0.86 10.
Series 1 Reach (m) 0 – 12.2 12.2 – 24.4 24.4 – 36.6 36.6 – 48.8 48.8 – 61.0 61.0 – 73.2 73.2 – 91.5
1.0 0.1
1.0
10.
(τb*1) / (τb*1)50
C * (%)
Figure 7.33 Relative equilibrium concentration for the various reaches of kaolinite clay deposited in an open flume for Q 0.0076 m3/sec and o 0.294 Nm2 [16, Fig. 3.15, 100, 101]. 99.9 99.5 Series 1 Reach : 36.6 m�48.8 m 98 95 90 80 70 50 30 20 10 5 2 0.01 0.1
Symbol
Co t50 σ τb (Nm�2) (g/l) (min) 2 0.25 0.30 0.35 0.40
1.0
10.0
3.117 2.901 2.935 2.415
100
8.0 5.6 3.7 3.4
0.781 0.793 0.824 0.616
1000
(t/t50)1/σ2 Figure 7.34 Time-deposition plot for series 1 [16, Fig. 3.22a, 100].
downstream. The larger values in the first reach were attributed to an increased degree of turbulence there due to the entrance effect. There was little variation in the parameter 2 with an average value of 0.754. The Atchafalaya Bay mud displayed two values of t50 in the first reach. The difference was attributed to the 15% sand content.
Cohesive Sediments in Open Channels
C*(%)
240
99.9 99.5 Series 3 Reach: 39.6 m�54.9 m 98 95 90 80 70 50 30 20 10 5 2 0.01 0.1
Symbol
Co t50 σ τb (Nm�2) (g/l) (min) 2 0.15 0.20 0.25 0.40
1.0
10.0
3.731 3.722 3.752 3.897
30.0 18.5 20.0 16.5
0.997 0.664 0.658 0.923
100
1000
(t/t50)1/σ2
C* (%)
Figure 7.35 Time-deposition plot for series 2 [16, Fig. 3.35a].
99.9 99.5 Series 2 Reach: 30.5 m�44.2 m 98 95 90 80 70 50 30 20 10 5 2 0.01 0.1
Symbol
Co t50 σ τb (Nm�2) (g/l) (min) 2 0.50 0.60 0.75 0.85
1.0
10.0
4.515 4.520 4.531 4.511
10.5 8.0 5.5 3.0
0.55 0.52 0.52 0.48
100
1000
(t/t50)1/σ2
C*(%)
Figure 7.36 Time-deposition plot for series 3 [16, Fig. 3.28b].
99.9 99.5 Series 4 Reach: 6.1–12.2 m 98 95 90 80 70 50 30 20 10 5 2 0.01 0.1
Symbol
Co t50 σ τb (Nm�2) (g/l) (min) 2 0.50 0.60 0.70 0.85
1.0
10.0 (t/t50)1/σ2
Figure 7.37 Time-deposition plot for series 4 [16, Fig. 3.39b].
2.119 2.108 2.112 2.103
100
5.1 5.0 4.8 4.6
0.566 0.555 0.521 0.519
1000
Log t50
Chapter | 7 Deposition and Resuspension of Cohesive Soils
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
�0.2
0.0
min
0
σ2
Series 1 Reach: 12.2–24.4 m �2 �0.2 �bmin� 0.250 Nm
Series 1 Reach: 0–12.2 m �b �0.260 Nm�2
0.0
1.0
2.0
3.0
4.0
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
241
0
1.0
2.0
3.0
4.0
�b* Figure 7.38 Variation of t50 and 2 with [16, Fig. 3.43].
0.0
0
1.0
2.0
3.0
4.0
0
1.0
2.0
3.0
4.0
�b*
b∗ , Series 1 for sediment from first reach of the flume
The variation of t50 and of 2 will be more meaningful in their relation to the bed shear stresses and to the distance along the straight flume in which the sediment was deposited. In Figure 7.38, both parameters are plotted against the dimensionless bed stress, b∗ b / b min , in the rotating channel for the first two reaches of the open flume for Series 1. A similar plot is shown in Figure 7.39 for the same series for two central reaches of the flume. The trends in both figures are quite similar. Both t50 and 2 vary with the bed stress although the first more strongly than the second. This is indicative of a rapid initial reduction of the settling time and an equally rapid increase of the settling rates with relative bed stress. A drop in the spread up to a bed stress equal to twice the minimum is also observed. Beyond that point the rates of decrease of both parameters become much slower. These trends are more pronounced for sediment from the first reaches of the flume. For the central reaches, the initial drop of both parameters is substantially lower, reaching their quasi-constant values with smaller fluctuations. Similar trends have been observed for the sediments from the other three series in Ref. [16]. Figure 7.40 shows the variation of the same parameters with distance along the open flume for Series 1 and for the stresses shown in the inserts. One can observe from Figure 7.39 that t50 decreases with increasing bed stress. The same parameter reaches a minimum at a distance of approximately 20–40 m from the upstream end of the channel and from then on increases, reaching a relatively constant value at distances in excess of 50–60 m. The value of the standard
Cohesive Sediments in Open Channels
242
1.0
1.0
0.8
0.8
Log t50
0.6
0.6
0.4
0.4
0.2
0.2
0.0
Series 1 Reach: 24.4–36.6 m �b =0.250 Nm�2
�0.2 �0.4
�2
�0.2
min
0
1.0
2.0
3.0
4.0
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0
1.0
2.0
Series 1 Reach: 36.6–488 m �b � 0.226 Nm�2
0.0
3.0
4.0
0.0
min
0
1.0
2.0
0
1.0
2.0
�b*
Series 1
0.4
0.4
0.2
0.2
0.0
0.0
�0.2
�0.2
Log t50
0.6
�2
0
20
40
60
80
�0.4
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0
20
40
4.0
60
Distance along flume (m)
80
�b � 0.70 Nm�2
0.8
0.6
�0.4
3.0
b∗ , Series 1 for sediment from the middle reach of the
1.0
�b � 0.65 Nm�2
0.8
4.0
�b*
Figure 7.39 Variation of t50 and 2 with flume [16, Fig. 3.44].
1.0
3.0
0.0
Series 1
0
20
40
60
80
0
20
40
60
80
Distance along flume (m)
Figure 7.40 Variation of the deposition parameters t50 and 2 along an open flume [16, Fig. 3.65].
Chapter | 7 Deposition and Resuspension of Cohesive Soils
243
deviation, 2, also decreases with increasing bed shear stress in the rotating channel; however, it appears to be fairly constant along the straight flume. As pointed out earlier, the size, density, strength, and settling velocities of the aggregates that eventually settle on the bed increase with increasing bed shear stress in the testing rotating channel. Therefore, t50, which is a measure of the settling time, should be expected to decrease, as indeed was the case. The initial dip, however, suggests that in the first 20–30 m, the aggregate properties, as reflected in t50, decrease rapidly in the downstream direction before starting to increase. In interpreting the observed variation of t50 and 2, one has to take into account the flocculation and deposition condition in the 100 m open flume system. The sediment enters the flume in its most dispersed state dictated by the high shear rates in the return pipe and the high turbulence level generated by the diffuser and the entrance grid. Both effects result in an initial low apparent settling velocity and an apparent high t50. Shortly after the entrance, aggregation starts, while at the same time the initial high background turbulence dies. The largest and denser aggregates with the higher shear strength deposit at high rates, as reflected in the diminishing value of t50, within these first 30–50 m. A rapid increase of t50 follows beyond that point, indicative of diminishing settling velocities at an equally rapid rate. After a distance of 50–60 m, t50 appears to be reaching a nearly constant value, suggesting a more or less constant distribution of aggregate properties. Apart from some initial fluctuations, the standard deviation, 2, appears to be rather constant with distance increasing with decreasing b. The effect of the mean residence time, which is the average time it takes a segment of water-sediment mixture to travel through the channel, has to be considered. For Series 1, 2, 3, and 4, the corresponding residence time was 9 min, 13 min, 18 min, and 34 min. Therefore, part of the depositable material could not actually deposit. There is some more detailed discussion of this aspect in Ref. [16]. Finally, the relationship of bmin, which is the parameter representative of the strength of the depositable aggregates, with the bed shear stress, o, exerted by the flow in an open channel and with the distance along the latter was investigated. The first three series with kaolinite sediment shown in Table 7.1 deposited in the 100 m open flume were used for this purpose. The bed shear stresses, o, during deposition corresponding to the three indicated flow rates, are 0.1, 0.2, and 0.3 Nm2 [16, 100]. The collected bed samples were tested in the rotating channel-ring system at the University of Florida in exactly the same way as the previous ones. The results are shown in Figure 7.41. One can observe that for every reach of the channel, there is an increase in bmin with ο; and vice versa, for every bed stress during deposition, there is a decrease in bmin with distance. One can conclude that the sediment properties depend strongly on the flow conditions and, more specifically, on the bed shear stress during deposition.
Cohesive Sediments in Open Channels
244 0.4 Reach (m) 0–15 15–27 27–45 45–60 60–73 73–90
τbmin(Nm�2)
0.3
0.2
0.1
0
0.1
0.2
0.3
Shear stress τ0(Nm�2) Figure 7.41 Variation of bmin with bed stress, o, during deposition in an open flume [100, Fig. 37, 101, Fig. 17].
7.2 Hydrodynamic interaction of suspended aggregates with the deposited bed Before going into the study of the resuspension process of deposited aggregates, we should present a model for the interaction between the settling suspended aggregates and the deposited bed explaining the observed depositional phenomena for two reasons. First, certain depositional aspects, and in particular the independence of the equilibrium concentration from the initial one, seem to be quite different from the behavior of coarse cohesionless sediments. This requires some explanation. Second, the resuspension processes, which will be presented and discussed in the following section, will be better understood and connected with the depositional behavior after this model is presented. The independence of the concentration in suspension and of transport rates of fine sediment from the flow variables has long been known. Einstein [25] and Einstein et al. [26] called that part of sediment load, which bears no apparent relationship to the water flow, wash load, since it just washes through the channel. Guided by the mechanical composition of bed material and suspended sediment, the same investigators established the grain diameter, of which 10% by weight of the bed mixture is finer as the boundary between wash load and bed material load. The difference between these two types of sediment loads was attributed initially to the fact that the sediment-carrying capacity of the flow for that particular sediment size was not saturated. Indeed, according to Einstein’s theory, the bed load function represents the sediment saturation limit for a particular sediment size. It follows that, for saturation to occur, the sediment supply rates must
Chapter | 7 Deposition and Resuspension of Cohesive Soils
Q1
Q2
245
Q3
Sediment transport capacity Qs or sediment supply rate Q�s
Increasing water discharge Q
Qs vs Ds curves for various discharges
Q�s vs Ds curve
Ds1*
Ds2*
Ds3*
Sediment diameter Ds Figure 7.42 Qualitative sediment transport and supply relationships [126].
be at least equal to the sediment transport capacity of the flow; otherwise, the flow ceases to be the controlling parameter, and the sediment transport rates are determined by the supply rates. Figure 7.42 shows a qualitative relationship among sediment transport rates, Qs; sediment supply rates, Qs ; and sediment diameter, Ds; for three different water discharges, Q1, Q2, and Q3 [126]. In the same figure, the sediment supply rates are shown plotted against the sediment size. Indicating by Dsi∗ the sediment diameter corresponding to the point of intersection of the Qs Ds and Qs − Ds curves for each discharge, Qi, for i equal to 1, 2 and, 3, investigators reached the following conclusion. Considering Q1, ∗ for example, then to the left of Ds1 the sediment transport-carrying capacity of the flow exceeds the supply rate, whereas the opposite is true to the right of that point. This observation led Shen to define Ds∗ as the threshold diameter for the wash load because for Ds Ds∗ the sediment transport rate is controlled by the supply rate [126]. If this definition is adopted, then the threshold diameter cannot possibly be constant but should increase with increasing flow rates. Then a particular composition of coarse sediment with a representative sediment size Ds could behave either as a wash load or as a bed material load, depending on the saturation of the transport capacity of sediment of size different from Ds. Indeed, the availability and transport of the latter could saturate the overall sediment transport capacity of the flow. Depending on the particular conditions,
246
Cohesive Sediments in Open Channels
therefore, coarse sand and gravel could fall under the definition of the wash load. Nevertheless, the material found to behave as a wash load falls invariably within the silt and clay domain with a representative limit of 0.06 mm, which is near the boundary between silt and fine sand. It is, in fact, below that limit that the effects of Brownian motion and flocculation start becoming apparent. The explanation of this apparent drastic difference in hydrodynamic behavior between these two sets of sediment loads has to be sought in the nature of the erosion-resisting forces and in the assumptions involved in the description of the flow-induced forces responsible for sediment pickup and erosion. The first are generated by gravity, mechanical friction, and interlocking between the bed particles and the physicochemical attraction between fine particles. The latter are the combined effect of drag and lift forces on individual particles and/or particle aggregates. These forces are fundamentally the same for all kinds of sediment. According to Einstein’s original approach, a near-bed sediment grain in suspension will deposit if the instantaneous hydrodynamic lift force, L, exerted by the flow on the grain, is less than the submerged weight of the particle, Wb, i.e., if Wb /L 1. Likewise, a bed particle will be entrained from the bed to the main flow if L exceeds the summation of all the forces resisting erosion. For coarse sediment, Einstein considered the submerged weight of the sediment as the main resisting force, while the interparticle friction and interlocking forces were represented by a general parameter, defined as a hiding factor [25]. According to the experimental studies of Einstein and El Samni [27], the statistical distribution of the flow-induced drag and lift forces on coarse grains follows very closely the normal law. This law was used in the development of Einstein’s bed load function. For the lift forces, L, that distribution was expressed as
L L (1 )
(7.31)
in which L is the mean value of L, and is a dimensionless random variable of normal distribution with mean zero and standard deviation o, which was found for semicircular spheres to be equal to 1/2.75. According to that assumption, however, both lift and drag forces could instantaneously reach infinite values so that coarse gravel and large boulders could eventually move. That is contrary to observations and to intuitive expectations. Therefore, a modification of the stated original assumptions about the statistical distribution of the drag and lift forces and a unified hydrodynamic model are needed to explain the observed behavior of both coarse and cohesive sediments. That model is presented here [113]. The statistical distribution of the combined hydrodynamic forces was assumed to be quasi-normal but with an upper and a lower bound, as shown in Figure 7.43. The interparticle and interaggregate forces were added to the weight, friction, and interlocking of the bed units. The parameter in Equation 7.31 was considered bounded with upper and lower limits, u and , respectively, corresponding to similar limits of the lift forces. The probability distribution and the probability density curves are qualitatively shown
Chapter | 7 Deposition and Resuspension of Cohesive Soils
247
in Figure 7.43, and the acting and resisting forces on bed units are shown in Figure 7.44. The resisting forces consist of (a) the submerged weight, Wb, of the gains, flocs, and higher order aggregates; (b) the forces exerted among the bed units due to friction and inerlocking, and which, like in Einstein’s bed load theory, can be represented by a factor ; and (c) by the net interaggregate attractive forces represented by a force, Fa. Therefore, the total erosion-resisting forces on an aggregate should be equal to Wb Fa. The condition for erosion then becomes Wb Fa Wb Fa 1 L L (1 )
(7.32)
The probability of erosion, Pe, then becomes
W Fa Pe Pr b 1 L
Wb Fa 1 Pr 1 L
(7.33)
Normal distribution Modified normal distribution
Pr Pr ’ P’ r
p , p’ r
r
P 1 r Pr Prob
L – 1 � L
p’ r
0
�l
�
�u
�
Figure 7.43 Statistical distribution of lift forces [113].
Flow
L
Fa
Fa
Wb Figure 7.44 Forces due to friction, interlocking, and cohesive attraction on bed units [113].
Cohesive Sediments in Open Channels
248
For a suspended unit to be deposited the submerged weight, Wb, must exceed the instantaneous uplift force, L. The probability of deposition, therefore, takes the following form:
W Pd Pr b 1 L
(7.34)
In both Equation 7.32 and Equation 7.34, Pr means probability of. The condition for erosion and deposition and of the type of sediment transport depends on the relative position of the hydrodynamic lift force, L, with respect to Wb and Wb Fa, since Wb Fa Wb. We distinguish, therefore, the following three general cases: 1. Wb L Wb Fa. In this case the lift force is too large for deposition but too small for erosion. Therefore, the sediment washes through the channel without any resuspension and without depositing any sediment on the bed. The suspended sediment behaves as a wash load. 2. L Wb. The lift forces are low enough to allow deposition, but resuspension cannot take place. This is the case of deposition of suspended cohesive sediment under sufficiently low stresses. 3. L Wb Fa. Only erosion can take place because L is sufficiently high for resuspension but too high for deposition to take place. The erosion of dense and deposited cohesive sediments, described and discussed in Chapter 6, falls in this category. Now, the uplift force, L, which, as pointed out earlier, is rather a measure of the overall effect of both lift and drag hydrodynamic forces, varies statistically between very wide limits. If a normal distribution is assumed, then these limits are and respectively, which, as stated, are unrealistic. We define instead a quasi-normal distribution of L with an upper limit Lu and a lower limit L . It follows that the limits and u of the random variable in Equation 7.31 and in Figure 7.43 can be defined from by the simple relations
L 1 L
(7.35a)
u
Lu 1 L
(7.35b)
and
We introduce next the following dimensionless variables, e and d, for the resisting forces and for the submerged weight of the sedimentation units, respectively:
e
Wb Fa 1 L
(7.36)
Chapter | 7 Deposition and Resuspension of Cohesive Soils
249
and
d
Wb 1 L
(7.37)
The probability Equation 7.33 and Equation 7.34 of erosion and deposition, respectively, can be written in the form
Pe Pr { e } 1 Pr { e }
(7.38)
Pd Pr { d }
(7.39)
and
Thus, e and d define the sediment conditions for erosion and deposition, respectively, whereas and u indicate the lowest and highest values of the acting hydrodynamic forces. The behavior of a sediment load as wash load or as a bed material load and the conditions for only erosion, only deposition, simultaneous erosion and deposition, or neither erosion nor deposition depend on the relative magnitude of the intervals u − and e d and on the relative location of these four points. There are two main cases: Case I for e d u and Case II for e d u . In each case, five different subclasses may exist, according to the relative location of the four characteristic values of . Each of these cases and subclasses will be discussed individually. Case I: e d u . In Figure 7.45a the five possible relative locations, a b c d e, of the characteristic points are shown that can be included into the following three subclasses. Subclass 1, indicated by I1, includes positions a and b, in both of which Pd 0, since in both cases is or can be smaller than d, but it can never be higher than e so that Pe 0. Therefore, in this subclass only, deposition can possibly take place at rates depending on the relative location of the characteristic points. Subclass 2, noted by I2, is limited to arrangement c of the characteristic points. In this arrangement, both probabilities Pe and Pd are finite since the interval e d is contained within the interval of fluctuation of . Therefore, exceeds at times e, while at other times it can be smaller than d. Simultaneous deposition and resuspension are thus possible in this case, leading eventually to the sediment transport-carrying capacity of the flow when the rates of deposition and erosion become equal. The sediment is said to possess a bed load function because its rates of transport are determined by both the sediment composition and by the flow variables. Subclass 3 is noted as I3. The last two arrangements d and e of the characteristic values of belong to this subclass. In both subclasses, the lowest value of is higher than the characteristic value, d, for deposition. At the same time, the highest possible values of are higher than the critical value for erosion. This
Cohesive Sediments in Open Channels
250
(a) �I (b)
�u
�I
(c)
�d
�I
(d )
�d
�I
�e
�I
�d
�d
�u
�e
�e �e
�u
�e
�u
(e) �d
(a) (a)
�u
�u
�I
(b)
�I
(c)
�d
(d )
�d
�d
�I �I
�u �u
�e
�d
�e �e
�e
�u
(e) (b)
�d
�e
�I
�u
Figure 7.45 (a) Location of characteristic points for e d u , Case I; (b) location of characteristic points for e d u , Case II [113, Fig. 7].
means that the flow-induced forces are always too high to allow deposition, but over certain periods of time, they may become sufficiently high to erode original or deposited bed particles and/or aggregates. Therefore, erosion and/or resuspension will take place without simultaneous deposition either at constant rates for uniform bed or at gradually diminishing rates for a nonuniform bed deposited from suspension. The erosion of dense cohesive and lowest density deposited beds, described in Chapter 6, belongs to this subclass. Case II: e d u . The five subclasses of this case are shown schematically in Figure 7.45b as a b c d e. Subclass 1 is symbolized as II1. The relative locations of a and b belong to this subclass. In both locations can attain values smaller than d but never higher than e; therefore, the hydrodynamic forces will permit deposition for part of the time but never erosion exactly, as in subclass I1. The deposition of cohesive sediments belongs to this subclass. Subclass 2 represents location c only and is symbolized by II2. Here, the interval u falls within the interval e d. This means that the range of the acting near-bed forces is smaller than the difference between the deposition and erosion limits. The forces controlling erosion and deposition are never high
Chapter | 7 Deposition and Resuspension of Cohesive Soils
251
enough to erode and never small enough to allow deposition; therefore, the sediment washes through without leaving any traces in the bed, thus fitting the exact concept of the wash load. Subclass 3, the final subclass, symbolized by II3, encompasses the relative locations of d and e, in both of which can attain values higher than e but never lower than d. The reason for this is that the interval e d cannot be contained within the smaller interval u . This process represents erosion without simultaneous deposition, which is the case for cohesive sediment erosion of dense uniform and nonuniform deposited bed, discussed in Chapter 6. In summary, these six subclasses of arrangements of characteristic points fall into the following four modes of sedimentation: A. Pure deposition without simultaneous erosion. This mode includes subclasses I1 and II1. B. Pure erosion without simultaneous deposition. This mode includes subclasses I3 and II3. C. Simultaneous erosion and deposition. The only subclass in this mode is I2, which applies strictly to coarse sediment. If there are sufficient sediment supply rates and sufficient time for the near-bed sediment to reach the concentration level for saturation, the rates of erosion and deposition will become equal, and a bed load equation will exist. D. Neither erosion nor deposition. This mode contains only subclass II2 and could be called neutrality. The sediment just washes through without leaving any traces on the bed, thus fitting the definition of the wash load [25]. It can be thus concluded that the two extreme cases of either bed-material load or wash load are represented by modes C and D or by the corresponding subclasses I2 and II2. Mode A represents only the case of only deposition and mode B only the case of only erosion and/or resuspension. Finally, a look at the relative magnitude of e d and u is required. Because the standard deviation of has been found to be constant for coarse sediment, it seems reasonable to assume that the latter interval is also fairly constant. It is the first interval that is expected to vary. This interval can be represented by the equation
or by
e d
e d
( 1)Wb Fa L
9 ( 1)10 g(s w )Ds b
(7.40a)
10 Fa b Ds2
(7.40b)
Here, b is the bed shear stress due to the resistance of the bed grains and/ or aggregates, Ds is a representative grain or aggregate size, and 9 and 10 are dimensionless proportionality factors. For sediment coarser than silt, for which
252
Cohesive Sediments in Open Channels
the cohesive forces Fa are virtually zero, the difference e d decreases with decreasing Ds. In addition, the friction and interlocking factor is not expected to be much larger than unity because most of the entrained coarse grains simply rest on top of others with minimal interaction among them. Therefore, e d should approach zero, and as long as Wb lies between L( 1) and L (u 1) , the sediment will have a bed load function. This is the reason why Einstein’s bed load function is valid only for alluvial channels in equilibrium with sandy and gravelly beds. For sediment finer than 0.06 mm, which falls into the category of silt and clay, the factor Fa / Ds2 in Equation 7.40a and Equation 7.40b becomes very large in comparison to g(s w )Ds / b in the same equation and, in general but not unexceptionally, increases rapidly with decreasing Ds. Moreover, for fine sediment the settling and eroding units are the flocs and the floc aggregates whose submerged density is very low. The second term on the right side thus becomes dominant and may exceed the span of the flow-induced forces on the bed units. It is not surprising, therefore, that suspended sediment fitting the observed behavior of the wash load always falls within the silt and clay range. It has thus been shown that there are two basic physical processes in the transport of sediment to its final destination: either by a continuous exchange between bed and suspended units or by washing through the channel. Each process can be logically deduced as a special case of the developed generalized model. The wash load will eventually deposit in areas where conditions of subclass II1 and subclass II2 prevail, as, for example, in estuaries, at rates depending on the bed shear stresses according to Section 8.1. A more thorough discussion of the presented model is given by the author in Ref. [113].
7.3 Resuspension of deposited cohesive sediments 7.3.1 Introductory Remarks Some preliminary experimental results on the erosion of cohesive sediments deposited directly from suspension in flowing waters were reported in Section 6.3. These studies revealed that deposited cohesive sediment beds are eroded at continuously diminishing rates tending, under certain conditions, asymptotically to zero. In contrast, uniform beds of similar consistency to the deposited ones erode at constant rates. The difference between the two erosion processes was attributed to the stratification of the deposited bed with respect to the size, density, and strength. The stratification was linked to the variation of these three properties of the constituent aggregates. Indeed, it was shown in Section 4.4, and in particular in Section 4.4.1 and Section 4.4.3, that there is a strong interconnection between strength and density of the aggregates. A theory for the rates of erosion of cohesive soils of uniform consistency was presented in Section 6.3, and Equation 6.23 was derived. In this section the results and conclusions of the fundamental research on the resuspension of
Chapter | 7 Deposition and Resuspension of Cohesive Soils
253
deposited beds and of the factors affecting the process will be presented and discussed. The studies to be presented here had the double objective of first deriving an expression for the rates of resuspension and, second, of evaluating the various factors affecting these rates. The rotating annular channel-ring system, described in Section 7.1, was used for most of the experiments; however, a number of tests were conducted in a straight open flume 18 m long, 0.6 m wide, and 0.9 m deep. Kaolinite clay was used for all experiments with both distilled and salt water at ocean salinity. Since both systems were self-enclosed, the amount of resuspension was evaluated by means of the suspended sediment concentration. First, an expression was derived for the variation of the sediment concentration, C, with time. A similar relation was next sought for the variation of the bed density with the depth. And finally, an equation was developed for the rates of erosion in terms of a dimensionless excess bed shear stress over the erosive shear strength of the bed at any depth of erosion.
7.3.2 Fundamental Considerations The rate of erosion, E, expressed as the mass of sediment eroded per unit bed area and unit time, is related through the sediment continuity principle to the time rate of the increase of the suspended sediment concentration, C, by the equation [85]
Eh
dC dt
(7.41)
Similarly, the time rate of change of the depth of erosion, dz/dt, is given by the following relationship in terms of the rate dC/dt, the local dry density of the bed, sb(z) at an elevation z, and the total depth of flow, h:
( z ) dz dC sb dt h dt
(7.42)
From these last two relations, the rate of erosion can be determined if we know the readily determinable rate of increase of the suspended sediment concentration, dC/dt, and sb(z). The determination of the latter was the subject of a separate research effort. The hydraulic parameter controlling resuspension is expected to be the bed shear stress, b, as in the cases of erosion of uniform consistency beds and in the cases of deposition. As indicated in Section 6.3, a cohesive bed possesses a certain erosive strength, s, which is equal by definition to the minimum or critical bed shear stress, c, for the initiation of erosion. One is reminded that the nature of the erosive strength of a cohesive bed is different from the mass shear strength of the soil and that the latter exceeds the former by two to three orders of magnitude. In the first preliminary studies on resuspension, reported and discussed in Chapter 6, C increased initially rapidly but at gradually diminishing rates reaching, under certain conditions, a constant value, Cs, at which point any further erosion stops.
254
Cohesive Sediments in Open Channels
It can then be assumed that at that stage, the bed shear stress is equal to the erosive strength of the bed, i.e., s b c. The latter can be readily determined by controlled tests under continuous varying bed shear stresses until the suspended sediment concentration reaches a nearly constant value. The following factors may affect the erosive resistance of a cohesive bed and were investigated: (a) the conditions of bed preparation; (b) the consolidation time, that is, the time the deposited bed was left in a quiescent state before the start of erosion; and (c) the salinity of the eroding water.
7.3.3 Experimental Results The experimental studies consisted of four consecutive series corresponding to four specific steps. The objective of the first series was to explore once more the difference in erosive behavior between a naturally deposited bed and one artificially placed at an average density comparable to the first one. The second series attempted to derive an expression for the eroded sediment in terms of the suspended sediment concentration versus time. In that series the erosion process under a step-varying stress was first explored as a simulation of the natural erosion process of deposited mud in estuaries discussed in Chapter 8. Next, the variation of the erosive strength and the density of the bed with depth were explored under various conditions. Finally, in the fourth series, an expression for the density variation of the bed was first derived, followed by an equation for the rates of resuspension in terms of a nondimensional shear stress step. These steps were the difference between the eroding stress and the erosive strength of the bed at various depths of scouring. The experiments of the first three series were conducted in the rotating annular channel-ring system described in section 7.1.2, while the fourth series utilized the self-enclosed open flume with recirculating water [85, 94, 154, 155].
7.3.3.1 Results of Series 1 In Series 1, the erosive behavior was investigated for two beds of comparable average density but one deposited from suspension under a shear stress slightly below the bmin and the second of uniform density close to that of the deposited bed, artificially prepared and placed inside the rotating channel. Figure 7.46 shows the time variation of the suspended sediment concentration as a fraction of the total dry weight of the bed material for the deposited bed subjected to a flow-induced shear stress b 0.207 Nm2. This observation is in agreement with the resuspension experimental results by Krone [62] and by the author [102, 105] discussed in Section 6.3, with beds composed of flocculated silty-clay from the San Francisco Bay. One is reminded that the dominant clay mineral in the latter sediment was montmorillonite with some illite, whereas in the studies in question, it was kaolinite. One can observe that the concentration-time curve starts at a high slope indicative of relatively high initial resuspension rates. The slope of the curve and the erosion rate then diminish rapidly with the suspended
Relative suspended sediment concentration
Chapter | 7 Deposition and Resuspension of Cohesive Soils
255
0.12 0.10 0.08 0.06 0.04 �b � 0.207 Nm�2
0.02 0
0
2
4
6
8
10
12
14
16
18
20
22
24
Time (hrs)
Suspended sediment concentration in ppm
Figure 7.46 Variation of relative suspended sediment concentration for deposited kaolinite bed under b 0.207 Nm22 [85, Fig. 3].
600 500 400 300 200 100 0
�b � 0.413 Nm�2 0
4
8
12
16
20
24 28 32 Time (hrs)
36
40
44
48
Figure 7.47 Variation of suspended sediment concentration for uniform kaolinite bed eroded under a bed shear stress of b 0.413 Nm2 [85, Fig. 4].
sediment concentration reaching asymptotically a constant value, Cs. The latter will be used later as the reference parameter for resuspension. In Figure 7.47 similar test results are plotted for the artificially prepared kaolinite bed at a density close to that of the deposited bed under a shear stress of 0.413 Nm2. It is observed that the concentration time line is straight, suggesting a constant erosion rate, as in the original experiments with a dense bed of San Francisco Bay mud presented in Section 6.3 [102, 105]. Similar results were obtained for bed shear stresses of 0.445 and 0.483 Nm2. These last two experiments confirm the earlier preliminary conclusions that deposited beds are stratified with respect to density and erosive strength and that the stratification is caused by the difference in settling velocities, strengths, and densities of the various order aggregates rather than by mechanical consolidation after deposition.
7.3.3.2 Results of Series 2 In the Series 2 experiments, kaolinite clay in distilled water and in salt water was used in the annular flume-ring system, and the stratified beds were prepared
Cohesive Sediments in Open Channels
256
14
12 Shear stress τb 0.43 Nm�2 0.34 Nm�2 0.24 Nm�2 0.15 Nm�2
Concentration (g/L)
10
8
6
4
2
0
0
40
80
120
160
200
Time (hrs) Figure 7.48 Variation of suspended sediment concentration for deposited kaolinite bed in distilled water [85, Fig. 5, 154, Fig. 3, 155].
as those in the previous series [85]. Four resuspension tests with distilled water under bed shear stresses of 0.15, 0.24, 0.34, and 0.43 Nm2 are shown in Figure 7.48. The concentration time curves display the same general trend as the one in Figure 7.46. That is, they all start with a relatively steep slope, indicative of high erosion rates, becoming gradually flatter and flatter with suspended sediment concentration tending asymptotically to a value Cs for shear stresses below a certain limit. For the three lower stresses, that limit appears to have been reached at around 200 h of testing. For the highest bed stress of 0.43 Nm2, C seems to continue increasing with time, suggesting that the bed level with strength equal to the applied stress had not yet been reached. In order to analyze the test results and derive an expression for the amount of erosion in terms of the suspended sediment concentration, C, the asymptotic limit, Cs, was selected as the extrapolated value of C for 200 h. The ratio (Cs C)/Cs for all five tests of Figure 7.48 was then plotted on a logarithmic scale against the time, t, in arithmetic scale as in Figure 7.49. is an empirical
Chapter | 7 Deposition and Resuspension of Cohesive Soils
257
Dimensionless concentration (C–Cs)/Cs
1.0 0.5
�b (Nm�2)
Cs (gl�1)
� (hr�1)
0.15 0.24 0.34 0.43
0.82 3.8 6.5 14.2
0.017 0.024 0.017 0.016
0.1 C–Cs
0.05
0.01
Cs
0
e �t
1
2
3
4
5
Dimensionless time �t Figure 7.49 Variation of normalized concentration [85, Fig. 6].
Cs C / Cs
versus normalized time t
coefficient selected so that the data points fall on a straight line. This line is described by the equation
C Cs (1 e t )
(7.43)
where was found to vary within the narrow range of 0.013 to 0.028 h1. Equation 7.43 was also verified for kaolinite in salt water and for natural sediment from the Fernandina Beach in Florida composed primarily of montmorillonite, kaolinite, dolomite, and some illite and quartz [85, 154, 155]. The variation of the asymptotic value Cs of C with b in five experiments presented in Figure 7.48 suggests an increase of the erosive strength of the bed with depth as erosion proceeds, as qualitatively described in Figure 7.50, where z’ indicates the elevation below the original bed surface. Five characteristic stresses are indicated: bo, bA, bB, bC, and bD. For b bo, no erosion takes place. The other four characteristic stresses correspond to ultimate bed elevations with respect to the bed surface: zA, zB, zC, and zD. The curve starts at a low slope, increasing rapidly after bB but increasing slowly up to bB. Beyond that point, erosion will continue until two things may happen: either the erosion will be deep enough for the flow-induced shear stresses to reach the maximum bed strength, or the total cohesive layer gets eroded. sm is the highest erosive strength of the bed. Next, integration of Equation 7.42 yields
C
1 z sb ( z)dz h ∫0
(7.44)
Cohesive Sediments in Open Channels
258
�c
0
�bo
�bA
�bB �ch
�bC
�b
Depth below surface, (Z' )
ZA ZB Zch
Zc
Figure 7.50 Qualitative relationship between bed strength, s, and eroded depth, z’ [85, Fig. 7, 95].
�sm
Bed shear stress
Phase I
Phase II
Phase III
�m
�d1 �d2 Tm
Td1
Td2
Pre-erosion stress history
�b1 Tdc
T1
�b2 T2
�b3
T3
�b4
T4
�b5
T5
Time
Resuspension
Figure 7.51 Schematic variation of bed shear stress, b, in Series 3 and 4 [85, Fig. 8].
7.3.3.3 Results of Series 3 As stated earlier, the objective of the Series 3 phase was to observe the erosive behavior of a deposited cohesive bed under a shear stress increasing by finite steps. The effect of consolidation time on erosion was also investigated in this phase. The experiments were conducted in the annular flume-ring system using kaolinite in salt water. The stepwise stress increase was selected as an approximate digital simulation of the rising tide and is shown schematically in Figure 7.51, in which three phases of stress history are distinguished: Phase I represents the mixing stage, during which sediment at a concentration Co is mixed at a shear stress, m, equal to the highest possible stress of the system, over a time period Tm.
n
Chapter | 7 Deposition and Resuspension of Cohesive Soils
259
In Phase II, the shear stress is reduced to a value d1 over a time period Td1 sufficiently low to allow deposition. If the latter stress exceeds bmin, then some sediment will stay in suspension, in which case the deposition stress is lowered to d2 bmin over a period Td2 for complete deposition. The last part of Phase II is the consolidation time, Tdc, during which the deposited sediment is allowed to consolidate under zero flow and under its own submerged weight and its internal unbalanced physicochemical forces. These first two phases constitute the pre-resuspension history of the deposited bed characterized by the preceding mentioned variables. The effect of consolidation time, Tdc, on resuspension was investigated by changing Tdc while maintaining all the other variables constant. n The last phase, III, is the resuspension phase in which the deposited bed is subjected to erosion under a series of shear stresses b1, b2, b3, and so on applied over corresponding time intervals or time steps T1, T2, T3, and so on. Based on the results of Figure 7.48, a maximum erosion time of 200 h was established for all experiments. For low enough stresses, the concentration at that time was practically close to Cs. For higher stresses, though, there was still measurable erosion even after 200 h. However, for an approximate estimate, it was assumed that at 200 h the bed strength was close enough to the applied stress. n
In the analysis of the experimental data, the normalized bed stresses were used rather than the actual ones. For every shear stress, the corresponding normalized stress was defined as the ratio bi ( bi1 bi ) / bi corresponding to time interval Ti, where bi is the bed shear stress in the ith step. That is, bi is the difference between the new stress and the previous one, which also represents the strength of the bed at the end of the previous step. This parameter was selected because it is the excess stress over and above the bed strength that causes erosion. The bed shear stresses were selected in such a way that all values of bi were equal. The results of a particular experimental series for kaolinite with salt water are shown in Figure 7.52 as concentration-time relationships for the indicated values of the bed shear stress. Similar relationships were found for other sediment-water systems [85, 154, 155]. Values of the various parameters selected for this particular test are given in Figure 7.52. The shear stress d1 was selected less than the bmin (0.15 Nm2). In each time step i, the rate of erosion decreased from an initial high value, as was the case of the first tests shown in Figure 7.48. However, in the first seven steps, i.e., for values of i 1 to 7 (Type 1), the final suspended sediment concentration appears to have been close to its ultimate value of Cs. In contrast, for the last two steps, 8 and 9 (Type 2), there was still noticeable concentration increase and erosion rates after the end of the run. In Figure 7.53, the concentration C(Ti) at the end of the test period is plotted against the bed shear stress, bi, for each step and for three consolidation times, Tdc, of 24, 40, and 135 h [85, 154, 155]. For all three cases, the C(Ti)-bi relationships
Cohesive Sediments in Open Channels
260
Time (hrs)
Suspended sediment concentration (g/L)
7
5
6
7
Co � 44.1 gl�1 h � 30.5 cm �m � 0.9 Nm�2 Tm � 24 hrs �d 1 � 0.015 Nm�2 Td 1 � Tdc �d 2 � 0 Nm�2 Td 2 � 0 hrs Tdc � 40 hrs ��bi � 0.2 Ti � 1 hr
6 5 4 3
8
9
10
�b9�0.432 Nm�2
�b8�0.36 Nm�2
�b7�0.30 Nm�2
2
�2
�b6�0.25 Nm
1 0
�b3�0.145 Nm
�b2�0.12 Nm�2
�b1�0.10 Nm�2
1
2
�2
3
�b4�0.175 Nm�2
4
�b5�0.21 Nm�2
5
Time (hrs) Figure 7.52 Suspended sediment concentration-time curves for kaolinite in salt water for stepvarying stresses according to Figure 7.51 [85, Fig. 9].
consist of two straight lines intersecting at a point at which the bed shear stress is indicated by a characteristic value, bch. When bi exceeds bch, C(Ti) increases at a higher rate with bi than in the cases in which bi is less than bch [85]. The value of bch increases with the consolidation time, Tdc. Since the submerged weight of the deposited sediment is very low, in fact slightly higher than that of the water, it appears that this increase in shear strength is due to readjustment of the deposited aggregates by unbalanced interparticle and interaggregate forces rather than to mechanical consolidation. Moreover, for the same b, the ultimate concentration and, therefore, the ultimate amount of erosion decrease with increasing consolidation time. This aspect will be discussed in more detail later. Referring to Figure 7.50, the stresses bA and bB are both smaller than bch, while bC and bD exceed bch. Therefore, the value of bch must lie somewhere between bB and bC, and the corresponding depth of erosion must be between zB and zC. For type 1 profiles, the eroded bed strength at the end of the stress step was equal to applied shear stress, bi. No such equality should be expected for the type 2 profiles because substantial erosion was still taking place at the end of the time step. For these profiles, the bed strength at the end of the stress step is expected to be lower than the applied stress. The data of Figure 7.52 and Figure 7.53 were used to determine the variation of bed density and bed shear strength with depth. Strictly speaking, only the type 1 data can be utilized for
Chapter | 7 Deposition and Resuspension of Cohesive Soils
Suspension concentration C (g/L)
8
Expt
�d Nm 2
Tdc hrs
17 18 19
0.050 0.015 0
24 40 135
261
�m 0.9 Nm 2 Tm 24 hrs ( �)ex 0.2
6
4
2
0
�bch 0
0.4 0.2 Bed shear stress bi (Nm 2)
0.6
Figure 7.53 Relationships between concentration, C(Ti), at the end of run versus bed shear stress, bi, for kaolinite in salt water for three values of consolidation time, Tcd [85, 94, Fig. 85].
this purpose because only in these data is the condition s bi satisfied at the end of each time step. If the density distribution with depth is first determined, the depth of erosion for any particular stress step can be evaluated by means of Equation 7.42. This was the next objective of Series 3 [85, 94]. The distribution of the bed density below the bed surface was investigated after the end of step II (Figure 7.51) for various Tdc values just prior to the resuspension phase. The bed was first deposited in the rotating channel under various low shear stresses (d 0.015, 0.05 Nm2, etc.) and allowed in a quiescent state for a prescribed time. A bed sample was then taken with the following equipment and technique, especially developed for this purpose [94]. The apparatus consisted of
Cohesive Sediments in Open Channels
262
2cm dia. plastic tube 15 cm dia. plexiglass cylinder 15 cm
2.5 cm dia. metal tube Annular space for mixture of alcohol and dry ice Sediment
Metal plate
Porcelein dish
Filled with ice cubes
Piston with screw rod
Figure 7.54 Density measurement apparatus for deposited cohesive sediment beds [94, Fig. 31].
a 2.5 cm diameter metal tube 15 cm high, which was placed concentrically inside a circular Plexiglas cylinder with a diameter of 15 cm and a height of 15 cm and having a sealed bottom. This apparatus and the one for the sample extraction are shown in Figure 7.54. At the center of the bottom plate, there was a hole for the watertight placement of the metal 2.5 cm diameter tube. The annular space between the external wall of the metal tube and the internal wall of the 15 cm cylinder was used for the placement of the alcohol and dry ice to freeze the soil sample prior to testing. The latter was taken by vertically inserting into the deposited bed a transparent plastic tube 2 cm in diameter all the way to the bottom of the bed. The Plexiglas cylinder was then lowered vertically into the channel in a way that the 2 cm plastic tube was inserted symmetrically through the metal 2.5 cm tube. The annular space between the Plexiglas cylinder and the metal tube was then filled with a mixture of commercial grade denatured alcohol and dry ice. This mixture froze the sediment sample in less than 30 min.The plastic sampling tube was then removed and placed horizontally, covered with ice cubes to keep it frozen. To determine the density distribution with depth, the investigators used the indicated piston, which was activated by a threaded rod. The latter was used to push an approximately 5 mm portion of the frozen core sample out of the plastic sampler. This portion was then pressed against a vertical plate, was melted, and collected in a porcelain dish placed below the sample portion. The density of each portion was subsequently determined from the known volume and weight. To account for the sample expansion due to freezing, the frozen sediment core was divided into 10 equal parts, and the density of each layer thus determined was considered to correspond to the 10 parts of the actual bed. The elevation of the bed corresponding to each particular measured density was the elevation of the center of the sample. Figure 7.55 and Figure 7.56 present the distributions of density and erosive shear strength, respectively, for the series of resuspension tests shown in Figure 7.51.
Chapter | 7 Deposition and Resuspension of Cohesive Soils
263
Density �sb (g/cm3)
Depth z' (cm)
0
0
100
200
300
400
1 2
Figure 7.55 Distribution of bed density, sb, with depth, z’, for the conditions of Figure 7.51 [85, Fig. 11a].
3
Erosive shear strength �s (Nm�2) 0
0
0.1
0.2
0.3
Depth z' (cm)
1 2 3 4 5
Figure 7.56 Distribution of bed erosive strength, s, with depth, z’, for conditions of Figure 7.52 [85, Fig. 11b].
There are some preliminary albeit qualitative conclusions from the presented data. First, the surface strength of about 0.075 Nm2 corresponds to the minimum erosive shear stress. Second, there is a sharp increase of the density from about 150 g L1 at the surface to about 350 g L1 at a depth of approximately 2.5 cm. As pointed out earlier, this density increase can be attributed to the higher density of the deposited aggregates rather than to any mechanical consolidation effect. Finally, the shear strength distribution follows the density pattern to about 1.5 cm depth. From then on, while the density distribution seems to follow the same rate of increase with depth, the shear strength increases at a rate much slower than the rate of increase of the density. In fact, it appears to tend to a constant value even at higher densities. This observation leads to the tentative conclusion that the cohesive strength of the deposited aggregates has an upper limit even if their density becomes higher.
7.3.3.4 Results of Series 4 As stated earlier, the main objective of these series was to derive a semi-empirical expression for the rates of erosion in terms of the bed shear stress in excess of the erosive strength of the bed and to explore the effect of the consolidation time on these rates. The experiments of Series 4 were similar to those conducted under Series 3. Kaolinite clay was used with tap water in the straight recirculating flume [85].
Cohesive Sediments in Open Channels
264
The following range of the controlling parameters was used: Co 21.7–24.1 g L1; depth of flow, h 23 cm; mixing stress, m 0.46 Nm2; mixing time, Tm 4 h; first deposition stress, d1 0.026 Nm2 (less than the minimum shear stress of 0.18 Nm2) for a duration of Td1 12 h, second deposition stress, d2 0 for a duration, Td2 1.00 h. Nine tests were conducted with the following values of the consolidation time, Tdc: 2, 5, 11, 24, 48, 72, 96, 144, and 240 h. The first step was to derive an expression for the variation of the bed density with depth. The bed density, sb, was normalized by its average value, sb , over the entire bed and was plotted versus the relative distance from the original bed surface, z/Hb, where z Hb z, where Hb is the bed thickness and z is the elevation above the channel bottom. The plots are shown in Figure 7.57a and Figure 7.57b, from which, after a Tdc of 48 h, the distribution of sb seemed to be adequately described for the indicated range of z/H by the following expression: z ξ sb ς H b sb
(7.45)
where and ξ are experimental coefficients found for these experiments to be 0.794 and 0.228, respectively. The values of Hb, Tcd, and sb are shown in the insert of the figures. An important observation is that the average density increases and the bed thickness decreases with increasing consolidation time.
0.5
0.5
Hb
0.1
0.05 0.3
z' z
z /Hb
1.0
z /Hb
1.0
Bed
Tdc Hb sb (hr) (cm) (g cm�3)
Tdc Hb sb (hr) (cm) (g cm�3) 2 5 11 24
4.45 4.29 3.81 3.49
0.5
0.1
0.210 0.203 0.252 0.270
1.0
2.0
0.05 0.3
sb / sb (a)
48 72 96 144 240
3.33 3.18 3.02 2.86 2.50
0.5
0.292 0.304 0.270 0.325 0.336
1.0 sb / sb
2.0
(b)
Figure 7.57 Normalized bed density variation with depth below bed surface for kaolinite in tap water for various consolidation times [85, Fig. 12]. (a) Tdc 48 hours; (b) Tdc 48 hours.
Chapter | 7 Deposition and Resuspension of Cohesive Soils
265
The former specifically increases from 0.210 g cm3 for Tdc 2 h to 0.336 g cm3 for Tdc 240 h, that is, an increase of 60% of the original density. This is a very significant increase, which, for the reasons cited earlier, should be primarily attributed to the action of the unbalanced internal physicochemical forces. Equation 7.45 was found to be in agreement with reanalyzed data of Owen [91] and Thorn and Parsons [143]. For the derivation of the expression for the erosion rates, the data were analyzed in the following four steps. It was assumed first that, at the end of each time step of type 1 profiles, the bed shear stress, b, was equal to the erosive strength of the bed, s. The latter is equal to the critical applied shear stress, c, for the initiation of erosion at that particular level. This equality was concluded earlier on the basis of the results of Series 3, displayed in Figure 7.51 and Figure 7.52. Next, based on this last assumption, the variation of c with depth z below the bed surface was obtained as in Figure 7.56. And in the third step, the relationship between the rate of resuspension, E, and the normalized excess bed shear stress, (b c)/c, was empirically derived from the suspended sediment-time variation for each time step of the type 1 profiles. The procedure was to first calculate E from Equation 7.41 for each measured value of dC/dt. The erosion depth, z, was estimated from Figure 7.48 and from Equation 7.42, which, for a finite difference of time, t, gives z
h(C ) sb ( z)
(7.46)
where C is the change of suspended sediment concentration over a time t, h is the depth of flow, and sb ( z) is the average bed density over the depth z. Since the latter is not known in advance, it has to be estimated by iteration. The logarithm of the ratio of E/Eo for type 1 profiles was plotted against the corresponding values of the normalized excess bed shear stress, as defined earlier, multiplied by a coefficient 8. The coefficients Eo and 8 were selected so that the data fall on a straight line. Figure 7.58 through Figure 7.60 show such plots for three different consolidation times, Tdc, of 24, 48, and 240 h. In the fourth and final step, it was assumed that whatever expression for the rates of erosion was to be formulated, it should be applicable over the entire depth of erosion. The time concentration data of type 2 profiles were then used for this purpose. Representative depth-mean values of the coefficients Eo and 8 of the rate of erosion, previously derived on the basis of type 1 profiles, were selected to determine the variation of c with depth, z, below the depth, where this information could not be obtained from the type 1 profiles. The following expression for the erosion rates was then formulated:
c E exp 8 b Eo c
(7.47)
2.5 E exp 8 b c c Eo
Log (E/E0)
2.0
1.5 Series 4 Tdc 24 hr
1.0
Time b E0 104 d8 step (Nm 2) (g cm 2 min 1) 1 2 3 4 5
0.5
0
0
0.5
1.0
0.037 0.060 0.086 0.129 0.188
1.5 b c α8 � � c
5.6 15.0 10.8 12.1 11.1
0.38 0.80 1.30 1.21 1.53
2.0
2.5
Figure 7.58 Variation of erosion rates with normalized bed stress for kaolinite in tap water and for Tdc 24 hours [85, Fig. 13]. 2.5 E c exp 8 b c Eo
Log (E/E0)
2.0
1.5
Series 4 Tdc 48 hr
1.0
Time b E0 104 d8 step (Nm 2) (g cm 2 min 1) 1 2 3 4
0.5
0
0
0.5
1.0
1.5 b c α8 � � c
0.030 0.050 0.087 0.183
6.6 8.5 5.5 6.7
2.0
0.47 0.73 0.43 0.48
2.5
Figure 7.59 Variation of erosion rates with normalized bed stress for kaolinite in tap water and for Tdc 48 hours [85, Fig, 14].
Chapter | 7 Deposition and Resuspension of Cohesive Soils
267
2.5
log (E/E0)
E0
�
exp �8
�b – �c �c
�
1.5
Series 4 Tdc 240 hr Time τb step (Nm 2) 1
1.0
1 2 3 4 5 6 7
0.5
0 0
0.5
1.0
0.027 0.057 0.091 0.139 0.202 0.298 0.453
1.5 2.0 � �c �8 � b � �c
α
∝
E
2.0
E0 104 (g cm 2min 1)
8.9 6.1 19.8 6.1 11.2 10.8 12.4
0.48 0.92 2.26 4.03 1.38 1.59 4.23
2.5
3.0
Figure 7.60 Variation of erosion rates with normalized bed stress for kaolinite in tap water and for Tdc 240 hours [85, Fig. 15].
The erosive strength, c, is a function of the depth below the initial bed surface, and it increases with time approaching the applied stress, b, so that (b c)/c and E also decrease with time. The dimensional coefficient Eo is an empirical reference value. Both Eo and 8 were evaluated by fitting Equation 7.47 with experimental results and were found to vary with the bed shear stress b. In Figure 7.58, the value of 8 ranged from 5.6 to 15 and Eo from 0.38 104 to 1.5 104 g cm2min1. Comparative ranges were obtained for consolidation times of 48 and 240 h, shown in Figure 7.59 and Figure 7.60. One is reminded that the data of the last three figures were based on experiments conducted in the open straight flume. The data for Figure 7.52, obtained under Series 3 from experiments in the rotating channel, were reanalyzed and plotted on Figure 7.61 for comparison. The range of both Eo and 8 was narrower than in the experiments of Series 4. Specifically, Eo ranged from 0.04 104 to 0.22 104 g cm2min1, while 8 varied from 5.9 to 8.4. It was found that in each test both coefficients appeared to vary somewhat but in a systematic manner with the time step and, therefore, with the depth below the original bed surface. In general it was noted that both Eo and 8 were getting independent of the depth z for depths larger than a few millimeters. The average values, Eo and 8 , of the same coefficients were plotted against the consolidation time, Tcd, in Figure 7.62 and Figure 7.63 with results of Series 3 with kaolinite in salt water and of Series 4 with kaolinite in tap water.
Cohesive Sediments in Open Channels
268
2.0
E E0
�
exp �8
�b – �c �c
�
log (E/E0)
1.5
1.0 Series 3 Tdc 40 hr
1 2 3 4 5
0 0
0.5
1.0
0.100 0.120 0.145 0.175 0.210
α
∝
Time τb step (Nm 2) 1
0.5
E0 104 (g cm 2min 1)
59 55 55 55 84
1.5 2.0 � –� �8 � b� c � c
0.04 0.25 0.30 0.27 0.22
2.5
Figure 7.61 Variation of erosion rates with normalized bed stress for the data of Figure 7.52 [85, Fig. 16].
In Figure 7.62 and Figure 7.63, one can observe that both Eo and 8 seem to be independent of the consolidation time, Tcd, and dependent on the properties of the sediment water system. In both figures the average values of Eo and 8 over each consolidation time seem to vary randomly about a constant value for the entire range of Tcd. For kaolinite in salt water, the approximate average value of Eo is 4.0 105 g cm2min1 and of 8 is 5.9. For kaolinite in tap water, the same coefficients have values of 7.9 105 g cm2min1 and 9.3, respectively. The mean values of Eo and 8 over the entire range of Tdc for kaolinite in salt water are higher than for kaolinite in tap water. According to Equation 7.47, higher values of these coefficients mean higher erosion rates for the same relative excess bed shear stress. Therefore, kaolinite beds deposited in salt water get resuspended at higher rates than similar beds deposited in tap water for the same stresses. This difference in resuspension rates is indicative of a lower cohesive bond between deposited aggregates as well as among individual units within the aggregates. This trend appears to be consistent with flocculation characteristics of kaolinite. One is reminded that the latter flocculates predominantly by edge-to-face attraction and that such an attraction is stronger in water with the minimum amount of salt in solution.
Chapter | 7 Deposition and Resuspension of Cohesive Soils
269
Kaolinite in tap water
Eo (g cm�2 min�1)
Kaolinite in salt water 10�4
7.9 � 10�5 4.0 � 10�5 10�5
1
5
10
50
100
300
Tdc(hrs) Figure 7.62 Variation of average values of Eo with Tcd for kaolinite in salt and tap water [85, Fig. 17].
22 Kaolinite in tap water Kaolinite in salt water
18
a8
14
10 9.3 6 5.9 2 1
5
10
50
100
300
Tdc(hrs) Figure 7.63 Variation of average values of 8 with Tcd for kaolinite in salt and tap water [85, Fig. 18].
An example of the distribution of bed shear strength with depth below the original bed surface and consolidation time is shown in Figure 7.64 for kaolinite in tap water for the four consolidation times listed in the insert. It is first observed that the minimum value of c at the bed surface is constant and independent of the consolidation time, Tdc. The bed strength increases with depth, as
Cohesive Sediments in Open Channels
270 0
0.4
z�(cm)
0.8
1.2
1.6 Symbol Tdc(hrs) 2 11 48 144
2.0
2.4
0
0.2
0.4
0.6
�s(z)(Nm�2) Figure 7.64 Variation of bed strength, s, with depth and consolidation time for kaolinite in tap water [85, Fig. 19].
already discussed and shown in Figure 7.50 and Figure 7.56, for all consolidation times, while for the same depth, the bed strength, s, which is equal to c, increases with Tdc. The latter increase is far more pronounced during the shorter consolidation times converging asymptotically to a constant value after 48 h of consolidation. It was found that there is negligible strength increase after 240 h. On the basis of the experimental information presented, Equation 7.47 should be viewed only as a guideline and a basis for modeling the resuspension process in real-life cases. The form of that equation has been predicted in previous studies from a reinterpretation of the rate process theory of chemical reactions [92, 120]. This theory involves the activation energy concept according to which a threshold barrier must be crossed for the conversion of reactants to products. Paaswell, in fact, noted that Equation 7.47 helps in understanding the process of erosion of cohesive soils as an internal energy/external energy system, with E considered as a measure of the work required to be done on the system for erosion [92]. According to Paaswell, the external energy is reflected in the bed shear stress, b, while c or its equivalent, s, and 8 and Eo represent measures of the internal energy and, therefore, of the resistance of the deposited sediment bed to erosion.
Chapter | 7 Deposition and Resuspension of Cohesive Soils
271
The presented variation of bed density with depth due to aggregate segregation during settling was demonstrated by Torf et al. from experiments of mixtures of kaolinite clay with sand in settling columns [146]. A common feature in their experimental data was the observed segregation whereby sand had fallen to the bottom layer through the matrix of clay. The experiments also revealed a critical content of sand above which no further consolidation takes place. That critical limit was found to be 30% of sand by weight. The following are their main conclusions: 1. The structure of a bed formed by deposition of a mixture of cohesive sediment and sand depends on the input concentration and rate of supply; the sand content; and on sedimentological, physicochemical, and biological properties of the clay component. Of particular interest is the observation that the matrix of a gel of a dry density of 85 kg m3 is generally able to support further material without falling through, that is, without any collapsing of its structure and a new equilibrium position at higher density. 2. Segregation takes place under both low and high sand content. Most important, segregation can also occur for zero sand content due to the differential settling of higher density aggregates. This observation is in complete agreement with the presented results and theory in this chapter. 3. For segregated beds, the sand accumulates in a layer, while the upper portion of the deposited bed is sand free. The density of the latter was found to increase with sand content of the input mixture to a maximum density when the sand content varies between 18% and 32%.
7.4 Summary and closing comments This chapter concentrated on the deposition-resuspension-transport processes of cohesive sediments. These processes are of particular interest to navigable channels in tidal estuaries, in which shoaling occurs over continuous tidal cycles of deposition and resuspension. These shoaling processes will be discussed in Chapter 8, together with the presentation of some actual case histories of salinity intrusion and shoaling in estuaries. The fundamental studies presented revealed many significant aspects and important parameters that control deposition and resuspension in relation to the physical properties of the aggregates and led to a model explaining the observed phenomena. These parameters and flow variables can readily be determined in a laboratory with the proper apparatus. Such an apparatus, which can be easily built, was also described, and it is recommended, with whatever modifications and/or improvements may be considered for future studies anywhere. The following are the most important conclusions of the presented studies: 1. In both erosion and deposition, the bed shear stress, b, is the controlling hydraulic variable. 2. There exists a minimum value, bmin, of b below which all suspended sediment eventually deposits. bmin was shown to be representative of the physicochemical properties of the sediment-water system.
272
Cohesive Sediments in Open Channels
3. In contrast to earlier concepts, a given flow can still maintain part of the total suspended sediment in suspension for b bmin. The maintained sediment can be represented in an enclosed system by the equilibrium concentration, * Ceq. The relative equilibrium concentration, Ceq Ceq / Co , where Co is the ** * initial concentration, or the degree of deposition, Ceq , is uniquely 1 Ceq determined for a particular sediment-water system by the nondimensional bed shear stress b* 1 , where b* b / b min . Moreover, it was shown that ( b* 1) /( b* 1)50 , where the denominator is the value of the nominator for which 50% of the total sediment deposits, is the parameter that determines the degree of deposition or retention for any sediment-water system. Therefore, this last parameter represents the physicochemical properties of the system. An analytic expression was developed linking the degree of deposition and retention to the last parameter for a wide variety of sediment-water systems. 4. The deposition rates of the depositable portion of the suspended sediment, represented by C * (C Ceq ) /(Co Ceq ) , was closely related to the parameter t/to, where to is the time required for the deposition of 50% of the depositable sediment. An analytical expression for this relationship was obtained that led to an equation for the deposition rates. 5. The depositional and resuspension properties of cohesive sediments deposited in open channels vary with distance and with bed shear stress under which the sediment was deposited as a result of aggregate sorting. One is reminded that the aggregate properties are controlled by the bed shear stresses and by the near-bed flow structure. The same variables control the deposition and resuspension processes. 6. A hydrodynamic model was developed for the interaction between the bed and the settling sediment that explained the observed phenomena, and linked transport processes of both coarse and cohesive sediments. 7. The sorting of aggregates during deposition results in a stratified bed with respect to density and erosive strength. This fact was confirmed by experimental studies of clay and sand mixture deposited in settling columns [145, 146]. The resuspension rates diminish with time, and they may even stop altogether if, at a certain depth, the erosive strength exceeds the applied stress. The bed strength increases first rapidly and then more and more slowly with time after deposition. This observation is also consistent with the recent results by Torf et al. [146] that displayed comparatively small density variations beyond a certain depth. 8. Experimental equations have been developed for the vertical distribution of the bed density and for the rates of resuspension in terms of a nondimensional shear stress. At present time, these expressions should be used only as guidelines for rational estimates of the shoaling zones and for the general transport process of cohesive sediment in estuaries and other open channels. 9. Although the to-date knowledge on the dynamics of cohesive sediments leads to a good understanding of the erosion and deposition processes and provides rational criteria for actual estimates, it is by no means complete.
Chapter | 7 Deposition and Resuspension of Cohesive Soils
273
The effects of the bed stress during deposition on the density and the shear strength of the bed have to be investigated in more detail. The same is true for the relationship of the parameters into the equations for the degree and rates of deposition and resuspension. The variation of these last parameters over the deposition-resuspension-transport cycles as the sediment deposits along the channel is another very important aspect. The properties and the controlling parameters of the sediment deposited at a particular location may differ drastically from those of the total sediment introduced at the channel entrance. One is advised at this stage that such properties be evaluated for samples along the deposition areas. Finally, the relation between density and erosive bed strength has to be studied extensively. In Chapter 6, it was shown that for densities ranging from those of recently deposited sediments to those corresponding to the liquid limit, there is no correlation between density and erosive resistance, in spite of the tremendous density effect on the macroscopic mass shear strength of the soil. The explanation for that difference was given in Section 6.3. Systematic research is needed for medium- and highdensity cohesive soils formed with continuous varying overburden pressure in order to determine the relation between density and erosive strength for clay deposits subjected to higher degrees of consolidation.
Chapter 8
Engineering Applications of Cohesive Sediment Dynamics
8.1 Areas of application Problems related to cohesive sediments can be classified in the following three general categories of hydraulic engineering: 1. Design of open channels safe against scouring and deposition. Control of deposition in closed conduits is also a related problem. 2. Prediction and control of shoaling in navigable coastal and estuarine waterways for maintaining desirable navigable depths. 3. Control of environmental pollution. The problems in the first category were addressed to a limited extent in Chapter 6 in the discussion of erosion of such sediments and channel stability. In view of the experimental studies presented and discussed in Chapter 6 and Chapter 7, the stability of channels against scouring and the avoidance of shoaling due to transported fines constitute, with exception of special cases, two separate problems. Both criteria for stability depend on the flow-induced shear stresses on the bed and/or the banks. In scouring, we are interested in the highest boundary stresses, whereas in deposition the controlling bed stresses are the lowest expected. As we have seen, there is a substantial gap between these two critical forces for the same sediment. It is in the second category where the erosional and depositional behavior of cohesive sediments is particularly applicable. In fact, the need for a rational control of shoaling in estuarial navigable waterways was one of the primary motivations for most of the recent fundamental and applied research work on the subject. For this reason special emphasis will be placed on this category in the present chapter. The third category is so broad and so diversified that it should be the object of another special book. In addition to sediment dynamics, it involves biology, Cohesive Sediments in Open Channels Copyright © 2009
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marine life, and interaction of cohesive sediments with suspended and/or dissolved pollutants. To some extent, this interaction is expected to depend on the flow parameters. There is very limited information to date on this kind of interaction. This subject will be discussed only briefly with some suggested guidelines for control and for future research.
8.2 Design of stable channels 8.2.1 Design for Safety Against Scouring The complexities in establishing a representative parameter for the erosive strength of a cohesive soil were extensively presented in Chapter 6. In addition, in the discussion of the physicochemical aspects of these soils, it became appa rent that several factors may drastically affect the erosive resistance of a soil of a particular consistency. Chemical composition of the water, desiccation, compaction, and disturbances of various kinds are some of these factors. We should stress again that the macroscopic shear strength of a cohesive soil is by no means representative of its erosive strength. Relatively recent low-strength clay deposits may possess substantial erosive resistance, whereas highly dense consolidated clay samples, even shale samples with very high macroscopic strength, may completely disintegrate when immersed in water. We can state, therefore, that at present time no specific laws can be given in terms of the measurable physical, chemical, and mineralogical parameters and that every cohesive soil has to be treated as a special case. However, we can give a number of guidelines as to how to proceed and what to look for in order to reach a reasonable and conservative estimate of the erosive strength of particular cohesive soil: 1. The flow field of the testing apparatus should be similar to that in an open channel. That means it has to be either an open relatively long flume or a rotating annular channel-ring system similar to that described in Chapter 7. Because most open laboratory flumes are with recirculating water, one should take into consideration that the flow through the return pipe, the pump, and the diffuser is expected to affect the overall flow structure in the main channel. For practical purposes, it appears that an open channel with recirculating water of sufficient length, of the order of 15 to 20 m, would be satisfactory for an approximate evaluation. 2. The soil to be tested should occupy the entire bed of the open channel or of the rotating channel-ring system. Small soil samples may give misleading results since the frictional nonuniformity of the bed will alter the distribution of the flow-induced bed stresses. 3. The criterion of the critical stress should be decided on. According to the material presented in Chapter 6, soil erodibility is characterized by the minimum stress for the initiation of erosion and the erosion rates as functions of the applied bed shear stresses. Of course, it would be most desirable not to have any scouring at all. However, the rates of scouring and the possibility of
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the eventual termination of erosion may reveal important aspects of the soil in question. 4. As far as possible, the soil to be tested should be undisturbed. This is rather difficult considering the large volumes of samples to form the bed of a long open flume. It is recommended that remolded samples are allowed to stay under quiescent water for at least 1 week prior to testing and that the water quality should be very similar to that of the water of the actual channel. The results to be thus obtained will be on the conservative side. A complete physical, chemical, and mineralogical analysis including the Atterberg limits should be performed to fully identify the soil even if these variables do not by themselves determine the soil erodibility. 5. The range of the flow conditions may be of importance. The maximum expec ted and/or design flow should be considered in establishing the highest stress. Water level variations may expose a substantial part of the channel banks to desiccation that would normally significantly affect the erosive resistance. In any case, it is recommended that the channel bed and banks be monitored closely for at least several months after the beginning of its operation. 6. If the soil does not seem to have sufficient erosive strength or if after a while it shows signs of deterioration, measures of stabilization should be considered. There are several methods to stabilize a weak clay soil, the discussion even of the most important of which would be beyond the scope of this book. Torf found an increase of the erosive resistance of cohesive soils with sand mixtures [145]. She also reported results of other investigators indicating a similar effect of sand content. For conveyance and irrigation channels with cohesive boundaries, the first studies on erosion suggested two simple ways for increasing their erosive strength: (a) spraying sand and/or mixing sand and fine gravel and (b) applying cementing agents such as iron oxides [75]. The simplest way to do this would be to dissolve any iron oxide salts or any other stabilizing chemicals into the water inside the channel and let it stay there for at least a few weeks. There are several other ways to be explored for channel stabilization, such as plants and mixing of 2 to 3 inches of topsoil with various cementing materials and sand. Such mixtures should be individually tested for their resistance to erosion before application. For large conveyance channels and/or natural waterways, the protection of the bed and banks may require more elaborate protective measures, such as concrete covering, rip-rap, and wire meshes. The selection of the most suitable protective measure depends on the size and the importance of the channel, on the soil properties, and on the potential economic loss in case of severe scouring. This subject is very broad, it has been extensively studied for decades, and interested persons can find related information in several textbooks and handbooks. The most classical and most thorough of the books is by A. Schoklitsch titled Tratado de Arquitecture Hidraulica (Hydraulic Structures) [125]. Another good reference text is the handbook Bank and Shore Protection in California Highway Practice prepared by the Division of Highways, Department of Public Works of the State of California in 1970 [14].
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8.2.2 Design for Safety Against Deposition Channels can be designed for water conveyance and drainage, including irrigation canals. Only the conveyed water can introduce sediment in channels of the first category since no other water is supposed to be allowed to enter from the surroundings. In contrast, drainage canals which may drain large areas may be fed with a considerable sediment load, part of which may deposit inside the canal, thus requiring periodic cleaning. The sediment load in water conveyance canals may frequently fall in the fine category. The source of the water supply is either from a reservoir or from a groundwater aquifer or directly from a river. In the first two cases, the water is practically free of sediment. In the third case, the water intake will be at the upper part of the river where the sediment load is predominantly fine. The nature of the sediment load in drainage canals is more difficult to predict. It may originate from various sources, and its rates and composition depend on the rainfall intensity. Periodic cleaning may be required in case the highest flow rates are unable to erode and transport the deposited sediment. The design should start with the establishment of the minimum bed shear stress, bd, for the beginning of deposition. According to the information presented in Chapter 7, an open flume would not give representative estimates of either bd or the minimum bed shear stress, bmin, or the deposition rates. A rotating channelring system similar to the one outlined in Chapter 7 should be desirable, if available. However, if an open channel has to be used, the water with the sediment should be introduced at the upstream end of the channel and collected in a basin at the downstream end. It would be best to maintain the minimum bed shear stress above 4bmin to 5bmin. The highest stress should also be considered in case of cohesive bed and banks to avoid erosion. In cases of variation of the flow rates, the erosive characteristics of the deposited sediment have then to be determined to make sure that the latter will be resuspended at the higher flow rates.
8.3 Shoaling in estuaries A rational model for the description and prediction of shoaling in tidal estuaries has to make use of both the depositional and erosional properties of the sediment. That sediment consists predominantly of silt and clay, and it originates either from the land or from the sea. The coarse portion sediments transported by the river deposit within the upper parts of the estuary, where the flow velocities and the associated bed stresses become sufficiently low. Only the finer part settles in the lower estuarine reaches. Ocean-borne sediment can be sandy or fine, and in both cases it is set in motion by wave action. The sandy sediment comes from beaches and is transported along the coastline by littoral currents. It can enter the estuarine zone by tides and waves but predominantly by density currents, to be discussed in the following section. As soon as it enters the calmer
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waters of the estuary, it deposits near the ocean entrance. The fine ocean-borne sediment comes from the bay, where it has been deposited during the geological history of the system. It is suspended by wave-induced stresses, transported by the near-bottom density currents, and deposited either in zones with bed stresses sufficiently low to allow deposition or in the zones of near-zero near-bed velo cities averaged per tidal cycle. In either case, the process of sediment transport and its eventual accumulation in the shoaling zones consists of continuous cycles of deposition and resuspension. The sediment deposits during the periods of low water slack and gets resuspended partially or completely during the time of high velocities and shear stresses. As already shown in Chapter 6 and Chapter 7, the stresses for resuspension are higher that the stresses for complete deposition. Shoaling then, which consists of a gradual accumulation of sediment, takes place when in certain zones the amount of sediment resuspended is less than the amount of sediment deposited during a tidal cycle. One should also consider that during slack periods, the deposited sediment is subjected to various degrees of consolidation and, therefore, to various degrees of increase of its erosive strength. In this section, the hydrodynamic processes of the transport, deposition, and resuspension are first outlined, followed by a number of illustrative case histories.
8.3.1 Fine Sediment Transport Processes in Estuaries Chapter 4 and Chapter 7 showed that flocculation, erosion, and deposition of cohesive sediments are controlled by the near-bed flow structure. The far-bed flow structure, specifically the far-bed turbulence microstructure, provides the necessary mechanism for flocculation and controls the rates of the latter and the properties of the various order agglomerates. The generation of bed shear stresses and the overall microstructure within the near-bed zone in tidal estuaries is far more complicated than in regular open channels carrying pure water of uniform density. In the absence of tides or in cases of very low tidal range in comparison to the fresh water discharge, salt water from the ocean of higher density enters into the lower part of the estuary in the form of a wedge known as a salinity wedge [56, 110]. A stable interface separates this wedge from the fresh water flow above it. Such stratified estuaries are very rarely encountered, the Mississippi estuary being one of the closest examples. Most of the actual estuaries are subjected to substantial tidal variations, which induce the necessary shear stresses to break the interface and cause mixing of the lower salt water with the fresh water of the river. This mixing leads to the formation of a quasi steady-state vertical salinity distribution varying from a maximum near the bottom and tapering off to either a fresh water stage at some distance above the bottom or to a minimum near the surface. In the first case the estuary is called partially mixed, while in the second it is defined as well mixed or simply mixed. In a fully stratified or quasi-stratified estuary, the fresh water flow over the wedge induces shear stresses at the interface, thus forcing part of the salt water within the
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upper part of the wedge to flow toward the ocean. To compensate this loss, salt water flows continuously within the lower part of the wedge. This near-bed density flow is responsible for both the transport of ocean-borne sediments and for exerting shear stresses on the channel bed. As the latter diminish in the upstream direction, a point is reached at which they drop below the stresses required for the sediment to stay in suspension. Sediment then deposits around that point, thus forming the shoaling zones. In mixed and partially mixed estuaries, the tides are the ones that generate essentially the bed shear stresses, whereas the near-bed flow, averaged over a tidal cycle, provides the necessary mechanism for the net sediment transport. In both cases, there is a vertical distribution of the salinity and, consequently, of density, which drastically affects the dispersive and transport properties of the system. The distinction lies in the degree of the vertical salinity gradients. Moreover, for the same estuary, the type of stratification may change with the tidal phase and with the seasonal variations of the fresh water discharge. Figure 8.1 shows an example of vertical salinity and sediment concentration profiles for the Maracaibo estuary in Venezuela for November 1964 and for low and high tides. The difference between the two tidal cases can easily be distinguished [106]. In the partially mixed case, the vertical salinity distribution is characterized by (a) a lower zone of high salinity but with low gradients; (b) an upper zone of low salinity also with low vertical gradients; and (c) an intermediate zone of relatively sharp gradients connecting the two first zones. In the fully mixed case, these three zones still
1.0 Station E-33 Time 10:00 am ho � 45 ft.
0.8
y ho
Tidal height in ft.
November 16, 1964 1.0
0.8
2
6 12 18 24 0.6 Time (h) y ho
0.6 s
0.4
Station T-46 Time 3:30 pm ho � 45 ft.
4
0.4
C
ho
0.2
s
y
C
0.2
0
5
10 15 20 25 30
0
5
10 15 20 25 30
Salinity s in g L�1 0
0.2
0.4
0.6
0
0.2
0.4
0.6
Sediment concentration C in g/L Figure 8.1 Salinity and sediment concentration profiles in the Maracaibo estuary, Venezuela [106, Fig. 18].
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Chapter | 8 Engineering Applications of Cohesive Sediment Dynamics
Tidal height in ft.
exist, but the transition from one to the other is more gradual and less distinguishable. Also the salinity difference between the near-bed and the near-surface zones is much smaller than in the partially mixed case. In addition, the degree of stratification varies for the same estuary with the tidal phase and with time of the year. High tides and/or low fresh water discharges contribute to an increase of mixing and vice versa. Thus, during wet seasons and/or low neap tides, the estuary may well approach the stratified condition, while the same estuary may approach the fully mixed stage during dry seasons and at spring tides. The vertical density variation considerably affects the velocity distribution. Instantaneous velocity profiles vary continuously from totally upstream to totally downstream. At certain times there may even be a velocity reversal with an upstream flow near the bed and a downstream flow within the upper part of the section. Such a reversal is shown in Figure 8.2 for the Maracaibo estuary in Venezuela. The impact of such velocity distribution on the bed shear stresses is obvious. Integration of the velocity profiles over a tidal period results in a net upstream near-bottom flow and a net downstream flow in the higher zone (Figure 8.3). The first may thus transport sediment and whatever other suspended and/or dissolved matter in an upstream direction. The magnitude and intensity of the nearbed flow gradually diminishes in the upstream direction so that at some point it becomes zero. In that case, the entire velocity profile is directed downstream. This point is known as the null point [42, 47, 110]. In Figure 8.1, the distribution of suspended sediment is also shown by a dotted line. The sediment appears to be concentrated well within the lower zone of relatively high salinity. Suspended sediment cannot advance upstream of the
1.0
0.8 y ho
4
1.0
2 0
0.6
8
16 Time in h
24
0.8
0.4
0.2
0.40 Lake
0
y ho
0.6
ho 50 ft. y
0.4 Station T-50 Time 2.30 pm
0.40 Gulf
0.80
0.2
Station T-50 Time 7.30 am Velocity u in m/s
0.80
0.4
0
Lake
Figure 8.2 Instantaneous velocity distributions in the Maracaibo estuary, Venezuela [106, Fig. 23].
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Station E-33 1.0
Station T-50
ho
0.8
0.8
y
0.6
y ho
1.0
ho 45 ft.
u 0.0172 m/s
y ho
u 0.1700 m/s
0.6
0.4
0.4
0.2
0.2
1.0
3.0
1.0
Lake
Gulf
2.0
Velocity u in m/s
1.0 Lake
Figure 8.3 Average velocity profiles over a tidal cycle in the Maracaibo estuary, Venezuela [106, Fig. 26].
a
b Estuary
Ocean y
c Null pt.
Vo
River
ub x
�u ( ) �y
�u (a) �y
�u ( ) �y
Figure 8.4 Characteristic time average velocity profiles in an idealized estuary [100, Fig. 45].
null point. The latter is, therefore, very important since it marks the area around which sediment deposits, provided that the tide-induced stresses are sufficiently high for the transport of sediment to that point. Otherwise, deposition may occur either upstream or downstream of the null point, depending on whether the sediment is river borne or ocean borne. The subject of density stratification in tidal estuaries is too broad to be fully presented in this book, but it has been treated elsewhere. A concise treatment can be found in Chapter 11 by the author in the third volume of Handbook of Coastal and Ocean Engineering [110], while special aspects are treated in the cited references. A simplified distribution of a time average velocity profile in an idealized estuary is shown in Figure 8.4. In section a, there is a net upstream near-bed flow; in section b, the net flow near the bed is zero, marking the null
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point; and in section c, the flow is downstream throughout the section. The nearbed velocity gradient, du dy, is positive in the first section, zero in the second, and negative in the third. According to Figure 8.1, the suspended sediment is concentrated in both partially mixed and almost mixed estuaries within the near-bed zone of the highest salinity. As shown in Ref. [110], the interfacial zone in stratified estuaries and the intermediate zone of sharp density gradients in partially mixed estuaries suppress turbulence and, therefore, the rate of the vertical mass transfer through that zone is reduced. The near-bed shear stresses vary in both magnitude and direction with the tide, but they are expected to follow a quasi-harmonic variation with time. Figure 8.5 shows an idealized schematic variation of tidal velocities and bed shear stresses in a canal subjected to harmonic tides. The following zones can be distinguished: between time t 0, when u 0 and b 0, and the time at which b bmin for deposition, as defined earlier, almost the entire suspended sediment will deposit, with the possible exception of some nonflocculated particles and/or organic matter such as algae. That period is preceded and followed by a period during which bmin b bd, where bd is the bed shear stress below which noticeable deposition starts taking place. According to Chapter 7, bd ranges approximately from 4bmin to 5bmin. Some deposition does take place during that period at rates given by Equation 7.28 of Chapter 7. The third significant time interval is the one for which the bed shear stress ranges between bd and c, which is the minimum shear stress for the initiation of resuspension. During that third interval of the tidal phase, neither deposition nor resuspension take place, and the suspended sediment is being transported as a wash load. When the velocity increases to the point that the bed shear stress exceeds the critical value for resuspension, c, erosion of the previously deposited sediment begins. This fourth period is centered about the time of highest flood and ebb velocities. As soon as the bed stresses fall below c, resuspension ceases until they reach the value of bd for the beginning of deposition. We distinguish thus three main periods in a tidal cycle: (a) the deposition period centered about the high and low water slacks; (b) the resuspension period centered about the highest flood and ebb currents; and (c) the intermediate periods with neither deposition nor resuspension with wash load type sediment transport. One should note that, according to Chapter 7, the deposition rates attain their highest values for bed stresses below bmin. It is expected, therefore, that the bulk of shoaling will take place during that period with some deposits under stresses between bmin and bd. The shoaling process, therefore, in tidal estuaries takes place through a series of deposition-resuspension cycles with the resuspended sediment transported during each tidal cycle by a distance ubT, where ub is the near-bed velocity averaged over a tidal cycle and T is the tidal period. Continuous addition of sediment and shoal formation will take place either in estuarine sections where the tideinduced bed stresses are inadequate to erode all the sediment deposited during the previous deposition phase or within the neighborhood of the null point where the
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Cohesive Sediments in Open Channels
B A
T
T /2
O
Complete deposition
Partial deposition
Erosion & deposition
Erosion
No or partial deposition
Complete deposition
Partial deposition
Erosion & deposition
Erosion
3T /4
No or partial deposition
Complete deposition
Mean flow velocity
Datum
Bed shear stress b
c
bd bmin O bmin
T /2
T
T /4 3T /4
bd
c
Figure 8.5 Periods of deposition and resuspension within a tidal cycle [82, Fig. E.1; 109].
near-bed average velocity is about zero. In this way, more and more sediment will be added in every tidal cycle. The maximum tidal velocities and the associated bed shear stresses diminish in the upstream direction. If these stresses reach the critical low value before the null point, shoaling will occur, in which case the shoaling is classified as tide-controlled shoaling in contrast to the salinity intrusion-controlled shoaling, which takes place around the null point [119]. In estuaries with tide-controlled shoaling, river-borne fine sediment is expected to
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deposit upstream of the null point, whereas ocean-borne sediment will deposit downstream of the latter point. In salinity intrusion-controlled sedimentation, the main shoaling zones will be around the null point whether the sediment originates from the land or from the ocean. We should point out that salinity intrusioncontrolled shoaling is the dominant case. Regardless of the shoaling classification, the location of the null point is of paramount importance not only from the sedimentation standpoint but also for the overall salinity distribution and the environmental control of the system. A concise treatment of these subjects has been presented by the author in [110, 111], with references to many publications and reports on special aspects of the subject. The meaning of the null point becomes clearer in the idealized estuary depicted in Figure 8.4. The location of the null point is a function of the tides; the ocean salinity, which ranges within narrow limits; the geometry; and the fresh water discharge. It is also shown that the nondimensional near-bed velocity, ub / Vo, at a location x is a function of the densimetric Froude number, Fx9 , defined as Fx9
Vo ρ gho ρ
(8.1)
x
where (ρ / ρ) x is the relative density difference between the freshwater and the average salt water density at section x within the region of salinity intrusion, Vo is the average freshwater velocity, and ho is the channel depth below the mean water surface [46, 110]. An experimental relationship between Fx9 and ub / Vo was obtained by Harleman and Ippen on the basis of laboratory test results in a long open flume in the Waterways Experiment Station of the U.S. Corps of Engineers in Vicksburg, Mississippi. This relationship is shown in Figure 8.6 [46]. An extrapolation of this relationship to zero velocity gives the value of the densimetric Froude number corresponding to the null point from which the relative density at that point can be determined from Equation 8.1. The salinity at any station x can be evaluated from the relative density by the relationship
ρ 0.7 ρ 1000 s x x
(8.2)
where the salinity sx is expressed in parts per thousand. The location of the null point in the estuary can be determined from the salinity at that point by a method developed by Harleman and Abraham [44] and summarized by the author in Ref. [110]. The location of the null point after certain engineering operations, such as deepening and freshwater withdrawal, is critical not only in estuaries subjected
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5.0 15 4.0 Test 15 3.0 ub vo
16 14
15 14
2.0
16 4
1.0
14
4 10
11 15
0
�1.0 0.01
0.03
0.05 Fx' �
0.1
0.3
0.5
1.0
vo
�����
x
gho
Figure 8.6 Relation of near-bed velocity with densimetric Froude number in a tidal channel [46, Fig. 6].
to salinity intrusion-controlled shoaling, although this is the most common case, but also in cases where the shoaling is determined by the tide-induced bed shear stresses. Indeed, a change of the location of the null point will also indicate the change of the location of the critical zones after the completion of the planned operations. According to Harleman and Ippen [46], the dimensionless correlation curve of ub / Vo versus Fx9 appears to be unique for a given estuary and not significantly affected by changes in the freshwater discharge, tidal amplitude, and the depth. It follows that both Fx9 and sn/so at the null point will be sufficiently close to the values determined by the original data. Therefore, all that is needed for an estimate of the new location of the null point is the sx /so versus the distance x curve for the new values of the controlling data. For the determination of this distribution, the reader is referred to Ref. [106] for the necessary field measurements and procedures, coupled possibly with model studies involving salinity and density simulation. As an example, an increase of the estuarine depth, ho, will decrease the densimetric Froude number, as defined by Equation 8.1. According to Figure 8.6, the Froude number at the null point, Fn9 , has a constant value for a particular estuary. Therefore, if that value is to remain constant, the relative density at the null point, (ρρ / )n , has to decrease. Such a decrease will push the null point further upstream. The same null point displacement will take
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place if the average freshwater velocity, Vo, is reduced due to either withdrawal or construction of a dam on the contributing river or rivers. We should point out that small changes in estuary geometry and/or freshwater discharge may result in substantial upstream advancement of the null point as well as in an increase of the total volume of transported sediment. Likewise, the same changes will move the zone or zones of critical bed stresses for deposition further upstream with similar consequences. In either case, the zones of heavy deposition may move in areas with harbor installations from where it will be much more difficult and expensive to remove, or in ecologically sensitive zones. This has been the case in several major estuaries, examples of which are the Savanna, Delaware, London, and Maracaibo estuaries to be discussed in Section 8.4. In the shoaling zones, whether tide-controlled or salinity intrusion-controlled, sediment accumulates continuously over cycle after cycle. The deposited sediment is initially in the loose flocculated state A discussed in Section 3.6.2. As more and more sediment accumulates, consolidation takes place. The bonds between the higher order aggregates are broken, higher density aggregates are pressed more closely together, and the overall density gradually increases with the sediment going eventually through stages B and C to stage D, with a density of about the density of the first order flocs. Any further sediment accumulation and increase of the overburden pressure will result in a reduction of the pore space within the original flocs. A typical profile then in a shoaling zone will range gradually from highly flocculated recent deposits of minimum density and of extremely low macroscopic shear strength to denser, lower deposits of substantial dry density and measurable macroscopic shear strength. The deposited sediment will be gradually eroded during the phases of increasing bed stresses as soon as the latter exceed the erosive strength of the surface layers. As shown in Chapter 7, the erosive strength of cohesive sediments deposited from suspension increases with depth. Whether the entire sediment volume or part of it will erode depends first on whether the flow-induced shear stresses exceed the maximum erosive strength of the bed and, second, on the erosion rates during each time interval of the acting stresses. As long as the bed deposited during the deposition cycle can be completely resuspended, there will be no net deposition, and these particular zones will remain free of shoaling. Otherwise, there will be a net accumulation of sediment over every tidal cycle, with an eventual consolidation. There is a controversy considering the upper sediment layer of the lowest density, which should be addressed at this point. That layer, which appears as a fuzzy zone in an echo-sounding record, has been referred to by some engineers as fluff or fluid mud, as if it possesses different properties from those of the underlying sediment. Moreover, it was speculated that the fluid mud is transported as a form of bed load in a kind of creeping motion. According to the fundamental research results presented and discussed in Chapter 6 and Chapter 7, this speculation is incorrect. It has been well demonstrated in all laboratory experiments that the so-called fluff, in spite of its very low density, has a substantial erosive resistance.
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Moreover, there have not been any laboratory or field measurements indicating that such a creeping motion actually exists. As demonstrated earlier, cohesive sediment deposits are transported in tidal estuaries by resuspension and deposition, and that sediment can be moved only after having been resuspended. The misconception of a bed load type motion of the fluid mud is possibly due to the occasionally observed high concentration near the bed, primarily caused by high vertical salinity and density gradients. However, such concentrations are suspensions of individual units, not in a continuous network of aggregates. The modeling of fine cohesive sediments in estuaries first requires a reliable and sufficiently detailed analytical description of the flow field and of the appropriate formulation of the source and sink functions for the sediment as they appear in Equation 4.10 through Equation 4.13 of Chapter 4. Analytical expressions for the sediment source (resuspension) and sink (deposition) were developed and presented in Chapter 7. This section is limited to the presentation and discussion of estuarine shoaling processes without entering into the mathematical modeling techniques and with only some general suggestions as to how to apply the current framework to the prediction of shoaling. For comprehensive information on the subject, the reader is referred to the work of Ariathurai [2], Ariathurai and Krone [3], and Hayter [49]. In using the equations for deposition and resuspension, one has to remember that these equations were experimentally developed in nondispersive flow systems, such as the rotating annular channel-ring laboratory equipment. In a dispersive system, there is a segregation of the originally introduced sediment as it is transported and deposited along the estuarine canal. Therefore, the pro perties of the deposited sediment and the associated parameters that enter into the equations for deposition and resuspension are also expected to vary with both the distance from the sediment source and the time. These variations make the formulation of the sediment transport problem in tidal estuaries a very difficult task. In its most general form, the problem may be stated as follows: For given sediment inflow at a specific section of the estuary, determine the location and extent of the shoaling zones as well as the shoaling rates. Considering the deposition phase, the dominant parameters developed in Chapter 7, namely bmin, ( b* 1)50, y, 2, and t50, vary as the sediment deposits during the slack times. The following is an illustration of an elementary way to proceed in modeling deposition. First, the total deposition time can be divided into increments t in which the average cross-sectional flow velocity is assumed constant and equal to that at the corresponding time interval. The variation of the depositional para meters of the segregated sediment have to be determined next in the laboratory, according to the procedures of Chapter 7. These procedures are both lengthy and expensive. However, in view of the limited knowledge on the sediment supply rates and of the various mineralogical and physicochemical properties of the sediment, an excessively rigorous and detailed modeling of the shoaling process would not be normally justified. The modeler or the hydraulic engineer in charge
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should exercise judgment as to the desirable degree of detail based on a representative cross-section of the basic erosional and depositional parameters of the sediment. According to Chapter 7, the parameters of the segregated sediment vary along the channel, and they differ from those of the bulk sediment. The simplest approach would be the adaptation of a single set of parameters either equal to those of the original sediment or to some average value of the segregated sediment. The deposition rates can be simulated on the basis of Equation 7.28 of Chapter 7. However, in order to estimate the shoaling rates, that is, the rate of increase of the level of the bed surface, we require an equation relating the gross volume of the deposited sediment to its dry density. Such a relationship can readily be derived by simple laboratory tests. In modeling the resuspension phase, one is reminded again that Equation 6.23 of Chapter 6 is strictly valid for sediment beds of uniform consistency only. As shown in Section 7.3, for the resuspension rates from a naturally deposited bed, the sediment source function should be derived from Equation 7.44 through Equation 7.47 of the same section. These rates are functions of the bed shear stresses, which vary continuously with time, and they cease when the latter fall below the point of the minimum shear stress for resuspension. The following simple application on the deposition in a straight rectangular channel is presented only for the purpose of an illustration of the application of the existing current knowledge. The starting point is Equation 7.28 of Section 7.1.3, i.e.,
T 2 1 dC 0.434 * Co (1 Ceq ) exp c 2 t dt 2π2
(8.3)
The tidal motion is next approximated by a sequence of time steps of steady uniform flow. The channel is also divided in a number of segments of length x, as in Figure 8.7. For a given bed shear stress, b, the dimensionless equilib* rium concentration, Ceq , is constant. In a closed nondispersive flow system, as the described experimental annular channel-ring system, the original sediment concentration, Co, is also a constant parameter representing the total original sediment concentration in suspension just before the beginning of the deposition process. However, in a dispersive system, such as in an open natural channel, Co cannot be taken as a constant quantity any more because, even in the absence of deposition, dispersion will cause a reduction of Co as the sediment travels downstream. The sediment transport process can be better visualized if the dispersion term in the one-dimensional sediment transport Equation 4.13 of Chapter 4 is omitted, that is, if the equation is reduced to the simplified form
∂C ∂C V F (x, C , t ) ∂t ∂x
(8.4)
290
Cohesive Sediments in Open Channels
qs
V �x
�x
�x
�x
�x
y0
F �x
�x
�x
�x
�x
Discharge along channel (a) C Suspended sediment concentration
Suspended sediment concentration
C0
C1 C2
Concentration-time curve �t � time step
C3 C4
0
�t
2�t
3�t
4�t
5�t
6�t 7�t 8�t 9�t Time of deposition
(b) Figure 8.7 Simplified modeling of fine sediment transport in a straight rectangular channel [100, Fig. 46].
Next, a time step, t, and a length, x Vt, are selected. If sediment is introduced at x 0 at a rate qs and if the sediment is assumed to be completely mixed over the channel depth, ho, its concentration at x 0 will be
Co
qs bhoV w
(8.5)
where Co is the weight of sediment per unit volume of water, b is the width of the rectangular channel, V is the average velocity, and w is the unit weight of the water.
Chapter | 8 Engineering Applications of Cohesive Sediment Dynamics
291
The sediment concentration given by Equation 8.5 corresponds to the initial concentration in a closed nondispersive flow system. In the absence of dispersion, the sediment will be transported as a plug flow. That sediment plug will remain in the first segment x for a time t x/V during which the concentration will be reduced to C1, according to the deposition curve of Figure 8.7. The average weight of sediment deposited in the first segment of depth ho per unit length will thus be
Fd1
(Co C1 ) bho w t
(8.6)
Following the same reasoning, the weight of sediment deposited per unit length in the ith segment will be
Fdi
(Ci1 Ci ) bho w t
(8.7)
This procedure can be continued to the time at which the concentration C becomes either zero or constant and equal to the equilibrium concentration, Ceq. As stated, this example is too oversimplified and has been presented only for demonstration purposes. The following would be a more detailed and realistic approach also for a two-dimensional channel utilizing Equation 4.10 of Chapter 4: ∂C ∂C ∂C ∂ ∂C ∂ ∂C Fe (x, y, C , t ) Fd (x, y, C , t ) v +u Dy Dx ∂t ∂y ∂x ∂y ∂x ∂x ∂y (8.8) In Equation 8.8 and in Figure 8.7, x indicates the longitudinal direction, y is the horizontal direction normal to x, and the various other symbols were identified in Section 4.1 Again, the following is only an example of a possible procedure. The suspended sediment at a concentration Co at the segment x or xy of its injection is first treated as a conservative constituent by the appropriate one- and two-dimensional dispersion equations under the assumption of no deposition. A longitudinal time distribution of Co is thus obtained. Next, Equation 8.8 is used for the two-dimensional case and Equation 8.9 for the one-dimensional case. In each segment or rectangular element and at each time step, the values of the corresponding Co are obtained from the same equations under the assumption of zero source and/or sink functions. Equation 8.9 is the same as Equation 4.13 of Chapter 4 reproduced here:
∂C ∂V ∂ ∂C V Fe (x, C , t ) Fd (x, C , t ) Dx ∂t ∂x ∂x ∂x
where again the symbols are identified in Chapter 4.
(8.9)
292
Cohesive Sediments in Open Channels
In the resuspension phase, one has to definitely consider the highly stratified nature of the deposit. The resuspension rates can by no means be taken as constant, as assumed in some earlier studies. Indeed, the erosion-time relationships vary according to Figure 7.47 of Section 7.3, which means that the resuspension rates diminish rapidly with time and, in fact, they may cease altogether after a depth of erosion has been reached. As in the deposition phase, modeling of the resuspension process requires the introduction of a step function for the erosion-time modeling. In that function, the bed shear stress, b, is taken as constant over any time increment, t. The best approach would be to use the procedures and equations developed in Section 7.3.3 and specifically Equation 7.47 repeated here:
c E exp 8 b Eo c
(8.10)
In Equation 8.10, c is the critical or erosive strength of the bed, which changes with elavation, and 8 and Eo are experimental parameters. All these three factors require extensive experimentation in line with the principles of Section 7.3. In addition, they may change from cycle to cycle. Therefore, their application to real estuaries requires a great deal of judgment. The two experimental parameters could also be adjusted by field measurements in the prototype prior to the initiation of the engineering operations and the resulting changes of the estuarine regime. This could be accomplished through comparison of actual erosion rates in the shoaling zones of the estuary with Equation 8.10 and a continuous adjustment of the three parameters. At the present time, the derived laws of deposition and resuspension can be used at least as guidelines in the formulation of the sediment source and sink functions in the appropriate equations with the controlling parameters calibrated against field data. The modeling of estuarine shoaling also raises the question of the possibility of a combination of physical model testing, laboratory work on sediment properties, and analytical modeling. Since the dominant flow parameters controlling deposition and resuspension are the bed shear stress and the associated near-bed flow structure, the primary objective of physical model study could be the location of the zones of critical stresses for the new regime after the completion of the engineering operations. The highly complex estuarine flow pattern, particularly as complicated by the density stratification, makes an analytical prediction of such stresses extremely difficult if not impossible. Moreover, there is no way of simulating the cohesive sediment properties in a physical model. Laboratory models with simulation of salinity and density distributions can then be employed to determine the range of a parameter representative of the bed shear stresses and the near-bed flow. A convenient parameter could be a near-bed velocity such that the bed stresses could reasonably be assumed as proportional to the square of that velocity. A model calibration against field data will relate the critical near-bed velocities to the shoaling zones. The model then will reveal,
Chapter | 8 Engineering Applications of Cohesive Sediment Dynamics
293
after the introduction of the planned changes, the new critical zones in which the bed stresses will be similar to the ones in the original regime. The degrees of deposition and resuspension can then be estimated analytically using the appropriate relations of Chapter 7. In summary, a physical model can be used for the prediction of the near-bed flow pattern and of the potential shoaling zones, followed by the analytical estimates of shoaling. The latter should be based on the analytical application of the experimental relationships describing the degrees and rates of deposition and resuspension.
8.4 Illustrative case histories Since shoaling in estuaries constitutes one of the most important applications of cohesive sediment dynamics, this chapter will conclude with a brief description of five actual cases of estuaries subjected to heavy shoaling from fine cohesive sediments. These estuaries are the Savannah estuary in Georgia, USA; the Delaware estuary in Delaware, USA; the River Thames estuary in London, U.K.; the Maracaibo estuary in Maracaibo, Venezuela; and the San Francisco Bay estuary in California, USA. These examples were selected because of the special aspects of each one of them and demonstrate the effect of the various factors on shoaling.
8.4.1 The Savannah Estuary The Savannah estuary is one of the major and most important estuaries of the United States, located at the boundary between Georgia and South Carolina. Figure 8.8 shows the lower 23.3 miles (37.5 kilometers) of the river and estuary plus 7.5 miles (12 kilometers) of a bar channel between deep water in the Atlantic Ocean and the entrance jetties [46–48, 127]. The Savannah Harbor serves a large area in the southeast of the country, and its entrance is located at about 32o 029 latitude and 80o 559 longitude. The city of Savannah is located 16.1 miles (26 km) upstream from the entrance to the harbor. A navigation channel maintained by the U.S. Corps of Engineers and indicated by a dotted line serves the estuary and harbor. The channel extends another 7.2 miles (11.5 km) upstream from the city, and its locations refer to 1,000 ft. stations throughout the harbor. The port of Savannah is used for importing petroleum products, gypsum rock, raw sugar, fertilizer materials, as well as several other products. It also exports many commodities and, in particular, wood pulp, paperboard, naval stores, and iron and steel scraps [48]. As an example of the sea traffic volume, it is enough to mention that during 1962 4.5 million tonnes of water-borne commerce was carried to and from the harbor by 1,611 ships entering and leaving. According to the 1953–54 project, the navigation channel consisted of the three segments. The first 9.7 mile (15.6 km) segment started from the 36 ft. (11 m) depth contour at a depth of 36 ft. and a width of 500 ft. (153 m), which is from the deep water to station 190. The following 18.8 miles (30.2 km) of the channel to station 94 had a depth of 34 ft. (10.4 m) and a width of 400 ft. (122 m). The last section had
294
Cohesive Sediments in Open Channels
ATLANTIC OCEAN 200
TH CHA N E L N
160
N
O R
N
ON
T
OUT H CH A NN E L
SAVANNAH BEACH
S
BACK RIVER 140
FR
180
120
100 RIV
ER
SAVANNAH
Figure 8.8 The Savannah estuary and harbor [127, Fig. 1].
a length of by 2.5 miles (4.0 km), a depth of 30 ft. (9.2 m), and a width of 200 ft. (61 m), extending to the upstream limit of the project. Four turning basins constructed at selected locations are maintained to the project depth in the adjacent channels. The upper end of the harbor is connected with a 9 ft. (2.7 m) channel to the Savannah River, which extends about 200 miles (320 km) to the city of Augusta, Georgia. The 12 ft. (3.6 m) Atlantic Intracoastal Waterway crosses the harbor about 8 miles (13 km) below the city of Savannah. The tides in the Savannah estuary are semidiurnal with a tidal range of about 6.8 ft. (2.1 m) at the mouth and 7.4 ft. (2.3 m) at the city of Savannah. The normal range of tides in the Savannah harbor is about 9 ft. (2.7 m) for the spring tides and 5 ft. (1.5 m) for the neap tides. At that time, the highest current velocities in the navigable channels were of the order of 4 ft.s1 (1.2 m s1) on flood tides and 5 ft.s1 (1.5 m s1) on ebb tides. The Savannah River, which carries a heavy silt load, has its source in the Blue Ridge mountains about 400 miles (640 km) above the harbor and has a drainage area of 10,579 square miles (27,300 square km). The tides can reach the river to a distance of about 50 miles (80 km) upstream from the estuary mouth. There are two major reservoirs significantly affecting the river discharge located 238 miles (382 km) and 305 miles (490 km) above the estuary mouth. The normal fresh water inflow to the estuarine system is about 7,000 ft3s1 (200 m3s1). Because of the flow regulation by the two reservoirs, 16,000 ft3s1 (454 m3s1) and 5,580 ft3s1 (165 m3s1) are considered to be the normal high and low flows, respectively, expected during the year. The average annual shoaling rate was estimated in 1965 to about 7,00,000 cu yd, just about the same number in m3. The greatest concentration occurs in the 3.6 mile (5.8 km) reach above the city and the industrial areas where over half of the shoaling takes place. The annual shoaling in the bar channel amounts to 700,000 cu yd. The shoaling material is almost entirely fine with 22% colloids.
295
Chapter | 8 Engineering Applications of Cohesive Sediment Dynamics
20
120
140
160
180
75
Shoaling 10
50 Predominance
0
Flood Ebb predominance of bottom flow in percent of total
Distribution of shoaling in percent of total
100-ft. stations in Savannah Harbor 100
25
Figure 8.9 Annual shoaling in the Savannah channel before and after deepening (1953–54 project) [127, Fig. 3].
No information on the mineralogical composition of the sediment was provided. Figure 8.9 shows the approximate distribution of shoaling after the deepening of the channel according to the 1953–54 project. The channel had been maintained by the district Corps of Engineers through a pipeline dredge at an annual cost in 1964 of over $1,000,000. The studies of the Corps of Engineers indicated that the major portion of the shoaling material is supplied by the Savannah River with substantial amounts deriving from the ocean and runback from the spoil areas [48]. Only a negligible amount of sediment was attributed to industrial wastes. The average density of the bulk dredged material was estimated to about 20% of solids by weight (1,144 g L1). Of the solid portion, 78% was estimated to be suspended sediment, which is eventually deposited when the velocities were sufficiently reduced, and 22% colloidal clay and organic particles flocculated by the electrochemical action on contact with sea water and deposited in the harbor. This was the prevailing point of view at that time. However, as shown in Chapter 3, colloidal clay in suspension does not need ocean water salinity to flocculate, and most clays can readily flocculate even in river water. Their deposition in the Savannah harbor is most likely due to the lower bed stresses there. The vertical salinity profile is characterized, like in any partially mixed estuary, by a substantial difference between surface and bottom salinities and by a relatively thin zone of high salinity and density gradients a few feet above the channel bottom. The time history of shoaling and the sediment distribution along the estuary are of particular importance in understanding the shoaling processes and the interaction among sediments, freshwater discharge, salinity, and tides. The history of the channel depth is shown in Figure 8.10 and that of the annual shoaling rates is described in Figure 8.11. The four time periods in Figure 8.10 are the representative ones for the channel conditions with no changes immediately before or during these time periods.
296
Cohesive Sediments in Open Channels
100-ft. stations in Savannah Harbor 200 Quarantine
180
End of jetties
160
20
1923.1925 1931.1932 1939.1944
30
30
1953.1954
40
Depth in feet below MLW
20
140 Head of South channel
Depth in feet below MLW
120 Bull street
S.A.L bridge
100 Creosote wharf
80
40
8
8
7
7 Entire Harbor
6
6
5
5
4
4 Stations 107 to 134 (27,000 ft.)
3
3 Stations 166 to 205 (39,000 ft.) 2
2
1
1
0 1923–1925
1931–1932
1939–1944
Annual average shoaling rates in million cubic yards
Annual average shoaling rates in million cubic yards
Figure 8.10 Time history of the navigable channel in Savannah estuary [127, Fig. 4].
0 1953–1954
Figure 8.11 Time history of shoaling in the navigable channel in the Savannah estuary [127, Fig. 5].
Chapter | 8 Engineering Applications of Cohesive Sediment Dynamics
297
The two reaches in Figure 8.11 indicate the shoaling rates in the upper third (between stations 107 to 134) and the lower third (between stations 166 to 205) of the channel for the four channel conditions described in Figure 8.10. One can readily observe that the annual rates for the downstream one-third of the natural harbor decreased from about 2.1 million cu yd per year in the first period to near zero in the last one. In contrast, in the upper third of the channel, the shoaling rates increased from near zero in the first period to about 4.5 million cu yd in the last one. The annual shoaling rate for the entire estuary also increased very rapidly with the deepening of the navigable channel. However, unlike what happened in the upper third of the channel, the rates of the shoaling increase appeared to diminish. This leveling off of the total shoaling volume was interpreted at that time as an indication that for the final channel depth all the river-borne sediment is trapped, whereas in the earlier stages a substantial portion of the total sediment load was eventually washed through the estuary to the ocean. This time history of shoaling clearly demonstrates the upstream migration of the major shoaling zones as the navigable channel was progressively deepened, extended, or both. This migration is the result of a corresponding upstream advancement of the null point with the deepening of the navigable channel. The relationship between the location of the null point and the zones of heavy shoaling can be better demonstrated in Figure 8.9 by the near-bed flow predominance curve. This curve indicates the percentage of time of the near-bed ebb and flood currents. The point at which the line crosses the 50% mark is indicative of the location of the null point. For a concise treatment of the subject, the reader is referred to Ref. [46]. The history of shoaling and the migration of shoaling zones with the engineering operations suggest that the Savannah estuary is subjected to salinity intrusioncontrolled shoaling [47, 119]. The deepening of the navigable channel has thus a double harmful effect. It increases the total shoaling rates, and at the same time it shifts the zones of heavy shoaling to areas, including the Savannah City Harbor, where dredging is more difficult and expensive and where the distances to the disposal cites are longer. Of the total shoaling, 35% occurred between stations 120 and 130, that is, within a zone of 2 miles (3.2 km), while 69% of the total sediment volume settled within a 6 mile (9.7 km) zone centered around the first zone. That means that more than two-thirds of the total annual shoaling occurred in about one-fifth of the 31 mile (50 km) length of the navigable channel. Harleman and Ippen applied Equation 8.1 and Figure 8.6 to actual estuaries [46]. In Figure 8.12, the ratio ub / Vo is shown plotted against the densimetric Froude number, as given by Equation 8.1, for two river discharges, one for 7,000 ft3s1 (200 m3s1) and another for 16,000 ft3s1 (458 m3 s1). The results follow the trend of Figure 8.6 for laboratory data. It appears that the correlation is unique for practical purposes and independent of the fresh water discharge. A similar correlation can be derived for any estuary provided that sufficient field data are available. ub is the average value of ub per tidal cycle. Relationships like the one in Figure 8.12 can be used for the prediction of future changes of the location of the shoaling zones caused by changes in either the geometry
298
Cohesive Sediments in Open Channels
4.0 STA 9A
3.0
173 8A
STA 163 153 Savannah
2.0 6A
ub vo
143 173
1.0
Delaware
163
130
153
5A
147
123
0
Qf � 7,000 ft 3 s�1 Qf � 16,000 ft 3 s�1
�1.0 0.01
0.05 Fx' �
0.1
0.5
1.0
vo
�����
x
gho
Figure 8.12 Correlation of the near-bottom velocities with the densimetric Froude number for the Savannah estuary [46, Fig. 9].
of the channel or in freshwater discharge or both. Such an extrapolation assumes the dimensionless correlation is not measurably affected by changes of the regime of the estuary. This assumption seems reasonable as long as the changes are not very drastic and that the new estuarine regime is not very far from the original regime. According to Figure 8.12, the value of Fn9 at the null point, for zero ub, is about equal to 0.25. It follows then from Equation 8.1 that the relative density at the null point is
ρ 1 V2 9 2 o ρ (Fn ) gho n
(8.11)
The salinity at the null point can readily evaluated by means of Equation 8.2, which yields
sn 1000 ρ so (0.7)so ρ n
(8.12)
Equation 8.12 combined with Equation 8.1 gives
sn Vo2 1, 000 so (0.7)(so )(Fn9 )2 gho
(8.13)
Chapter | 8 Engineering Applications of Cohesive Sediment Dynamics
299
in which the subscript n indicates conditions at the null point, so is the ocean salinity, and Vo is the average velocity over the cross-section of the channel. From Equation 8.12 and Equation 8.13 and the assumed constancy of Fn9, the location of the new null point, and consequently the location of the new zones of heavy shoaling, can be estimated after the planned geometric and/or hydrologic changes. The new salinity and density distribution has first to be known for the same future conditions. For the one-dimensional approximation, the reader is referred to Ref. [46]. For more complicated situations, model studies can be conducted with the model verified against existing field data. Once the future salinity distribution is known, the new null point can be located along the estuary as the distance at which sn / so has the value given by Equation 8.13 for the densimetric Froude number obtained for the original situation. The previous discussion tacitly assumed that the entire fresh water discharge takes place through the navigable channel. However, generally, this is not true, because the estuary is crossed by a number of secondary channels and may have more than one tidal outlet. An estimate is then needed for the percentage of the total fresh water diverted through the main channel within which the shoaling does take place. For the existing situation, such an estimate can be obtained through direct field measurements, while for the anticipated new geometry and/or flow conditions, the same percentage has to be determined either by physical or by mathematical model studies. For example, in the case of the Savannah estuary, it was estimated that 75% of the total fresh water flows through the navigable channel. A series of model studies were conducted by the Corps of Engineers in order to predict the future shoaling situation after the planned operations [48, 127]. The model, which also involved salinity distribution, was constructed on a horizontal scale of 1:800 and a vertical scale of 1:80, that is, with a distortion ratio of 10. Hydraulic and salinity verification was based on prototype measurements made in 1950 and 1951. Shoaling verification was based on the quantitative distribution for the years 1953–54, which was the prevailing conditions for the existing (at that time) 34 ft project depth. The predominantly cohesive sediment was simulated by a material of low density readily movable by the near-bottom flow, which, as it has already been made clear, does not represent the actual hydrodynamic behavior of the real sediment. This was done long before the presented advances in cohesive sediment dynamics. It would be beyond the objective of this book to discus all the results of this model study. The case of this estuary, as well as of the other four, is presented rather as a demonstration of the relationship between deposition and the near-bed flow regime. The details of this study can be found in the cited reference. Several model studies were conducted for various channel alignments with and without retaining basin and for the effect of fresh water diversion to increase its outflow through the navigable channel. Plans for the latter were successful from the shoaling standpoint, but they were found to create pollution problems due to the reduced water volumes available for flushing in the other areas. For this reason, any future diversion plans were abandoned. The tests indicated
300
4000 5 Prototype
140 River
120
Entrance channel 300 ft wide 34 ft deep Sediment basin 600 ft wide 40 ft deep
13
0
Model
Elements of plain phase Vlll scheme 6
South 150
Test 4 Scales in feet 0 4000 0 5
el Chann
Navigation channel 34 ft deep North chan nel
160
8000 10
Cohesive Sediments in Open Channels
No front
ver
S.A.L
Tide gate 500 ft wide
Back ri
Canal 300ft. wide � 15 ft deep Side opening 100 ft wide
N
U.s. HWY. no 174
100
R,R
50
Figure 8.13 Proposed alignment and sedimentation basin for the Savannah estuary [48, Fig. 9].
that a sediment basin was effective in reducing the volume of shoaling material in the navigable channel. The same tests led to the conclusion that such a sedimentation basin constructed at the Back River could reduce the total shoaling quantity in the navigable channel by trapping 60% of the sediment. A sediment basin in the
Chapter | 8 Engineering Applications of Cohesive Sediment Dynamics
301
Back River with an opening and closing gate to allow water flow during the ebb periods was then proposed as the best solution. This plan is shown in Figure 8.13.
8.4.2 The Delaware River Estuary The Delaware River estuary with the typical funnel shape, outlined in Figure 8.14, is another case of a partially mixed estuary with salinity intrusion-controlled shoaling. Its description is about the same as presented by the author in Ref. [100] in a concise form. A more detailed description is given in Ref. [153]. The estuary is located between the states of New Jersey and Delaware at its mouth and for a distance of 80 miles (128 km) upstream, and between New Jersey and Pennsylvania for the remainder of its course to the head of the tide. The total estuary length is 133 miles (214 km). The 100 mile (161 km) long natural channel from the ocean to Philadelphia near the right bank had a depth of 17 ft. (5.2 m), becoming even smaller above Philadelphia. This same channel had been deepened by 1955 to 40 ft. (12 m) from the ocean to Philadelphia and to 25 ft. (7.6 m) upstream of Philadelphia. There also have been plans to extend the 40 ft. depth to within 8 miles (13 km) from the head of the tides and to provide a 35 ft. (11 m) depth in most of the remaining 8 mile (13 km) reach. The mean fresh water discharge varied from 11,699 ft3s1 (330 m3s1) at the head to 20,000 ft3s1 (570 m3s1) at the mouth of the estuary; however, these discharges fluctuate within very wide limits. For example, at the head of the tides, the fresh water discharge ranges from 1,200 ft3s1 (34 m3s1) during periods of severe draught to an estimated 300,000 ft3s1 (8,500 m3s1) at very wet periods. The mean tidal range in the Atlantic Ocean is 5 ft. (1.5 m). The sediment discharge into the estuary, which is contributed by 70% of its total drainage area, was estimated to about 6,000,000 yd3 year1. Like in the Savannah estuary, the shoaling zones were correlated to the null point, which means that the shoaling in the Delaware estuary is also salinity intrusion- controlled. To establish this correlation, the near-bed velocities, ub, were measured at a distance of 0.25 m above the bed and were averaged over a tidal cycle at five locations: 4A, 5A, 6A, 8A, and 9A.The locations of these points are shown in Figure 8.15 measured downstream from Allegheny Avenue in Philadelphia in 1,000 ft. intervals. For the 10.4 mile (16.7 km) reach between stations 4A and 5A, the channel depth was 21 ft. (6.4 m), and a fresh water discharge of about 28,200 ft3s1 (800 m3s1) was considered. This discharge corresponds to an average fresh water velocity, Vo, of 0.067 ft.s1 (2 cm s1) at station 4A and to 0.039 ft.s1 (1.2 cm s1) at station 9A. The average salinity varied from 10.3 ppt (parts per thousand) at station 4A to 13 ppt at station 9A. The correlation of ub with the densimetric Froude number, Fn9, is shown in Figure 8.12 together with the similar correlation for the Savannah estuary. The null point falls near station 275, which is about 52 miles (84 km) downstream of Philadelphia. This station is adjacent to the Artificial Island that has historically been the region of heavy shoaling. Detailed information can be found in Ref. [153].
302
Cohesive Sediments in Open Channels
juh
uvlk
ill
PENNSYLVANIA
TRENTON 130
Norristown
Ri
Lancaster
120
ve
r
ER
V RI
PHILADELPHIA S NN HA UE SQ
100
CAMDEN
80
ER RIV
EL D
A
AW
AR
E
WILMINGTON PA MD
EY
60
RS JE W
A
E AN
NE
M
DHESARDEL CANAL
OC
R
Atlantic City
A
ATL
AN
L
TIC
Y
40
DELAWARE
CHESAPEA
MD
DEL
KE BAY
N D
20 DELAWARE BAY
COYER
CAPE WAY CAPE HENLOPEN
STATUTE MILES ABOVE MOUTH
SCALE OF STATUTE MILES 8
0
8
16
24
32
Figure 8.14 Outline of the Delaware estuary [153, Fig. 1].
3
ub vo
1
EBB
Flood
4
2
9A 8A
Bottom velcoity
Artificial island
6A 5A
0
4A
�1 �2 �3 330
310 290 1000-ft. channel stations
270
Figure 8.15 Variation of near-bed velocity along the Delaware estuary channel [46, Fig. 10].
40
Chapter | 8 Engineering Applications of Cohesive Sediment Dynamics
303
Mud reaches
2
beam Southend N Rainbow Crook 10 miles Cold harbour 14 6 Barking 8 point 10 34 Lo reach Woolwich ng 4 16 18 re King George V reach Tilbury ac h 22 dock entrance tidel basin R. Darent 20 Key 30 24 Low water line 28 26 Broadness Shipping channel Gravegend reach Areas of accretion of mud Scale
London bridge
R Leo
R.roading Ra R ilw rea ay ions ch Gall ch 12 a re
Numbers in river are mileages from London bridge
Miles 0
1
2
3
4
5 Miles
Figure 8.16 Outline of River Thames estuary [15, 103, Fig. 1].
8.4.3 The River Thames Estuary The River Thames estuary in the U.K. is shown in Figure 8.16. It presents an interesting and special sedimentation case described in detail by the Hydraulic Research Station in the U.K. [15, 54] and in summary by the author [100, 103]. It was estimated that 3,00,000 cu yd per year were dredged from the tidal reaches of the estuary at the location of both the navigable channel and the docks. By far the greatest part of the dredged material was dumped in the Black Deep, located more than 20 miles (32 km) seaward from the eastern end of the estuary. The total dredged sediment volume was composed of silty clay with 20% to 35% of humified organic matter. The mechanical composition was described by only an average grain diameter of 45 m without any indication of the mineralogical characteristics. The sediment was found to possess a high surface physicochemical activity, which together with the humic material created a strong tendency for flocculation. The following are the two major shoaling areas: 1. The Tilbury tidal basin together with the adjacent docks about 26 miles (42 km) east of the London Bridge and within the Gravesend Reach shown in Figure 8.17. 2. The Royal Albert dock basin located from 8 to 10 miles (13 to 16 km) east of the London Bridge and outlined in Figure 8.18. Both zones lie at the upstream part of zones of mud accretion. The 8 mile (13 km) reach between 10 and 18 miles (17–30 km) east of the London Bridge, known also as Mud Reach, contains three segments: the Gallions Reach, the Barking Reach, and the Halfway Reach. The Tilbury tidal basin is subjected to heavy shoaling requiring, according to 1953 estimates, dredging of 300,000–400,000 cu yd annually in order to maintain a 24 ft. (7.3 m) depth below M.S.L. The deposited sediment was entering
304
Cohesive Sediments in Open Channels
Requires very frequent dredging. N
East branch dock
Requires frequent dredging.
Centre branch dock
Requires moderate dredging.
West branch dock
Seldom requires dredging. Never requires dredging.
Feet 0
500
Scale 1000 1500
2000 Feet
in
New lock entrance
Ma
ck
do
Old lock entrance
Impounding pump outlet Dry d Small ock dry dock Large dry dock Impounding pump intake Tidal basin
Landing stage
River jetty River Thames Plan of Tilbury docks showing areas where dredging is required
Figure 8.17 Outline of the shoaling zones in Tilbury docks of the London estuary [15, 103, Fig. 3].
ed us ot ent n ck es Lo t pr Royal Victoria a Dock Western entrance
sb
Lo ga ent we llio ran r ns ce Re ac h
Bug
Intake entrance Impounding station Royal Albert Dock Basin Royal Ge orge V D ock
y’
s
N
Re
Woolwich R
ach
each
Feet
0
Scale 4000
8000
Feet
Figure 8.18 The Royal Albert docks of the London estuary [15, 103, Fig. 4].
through the entrances connecting the Tilbury basin with the locks shown in Figure 8.17. Hydraulic model studies indicated that there is a net near-bottom flow of water loaded heavily with sediment into the area of the docks. Indeed, field measurements disclosed a high near-bed concentration of sediment ranging from 20,000 ppm to 60,000 ppm, reaching a peak of 130,000 ppm. This is indeed a very high density of the order of recent cohesive sediment deposits. For this
Chapter | 8 Engineering Applications of Cohesive Sediment Dynamics
305
reason it was at that time classified as fluid mud or fluff, while the denser bed below it, which forms a distinct boundary in the sounding chart of echo-sound measurements, was termed consolidated mud. The average concentration of suspended sediment above the fluid mud was of the order of 1,000 ppm. The shoaling zones and pattern in the Tilbury are outlined in Figure 8.17. The hypothesis at that time was that the near-bed flow of high concentration fluid induced a flow of the fluid mud into the Tilbury Basin which eventually consolidated in the tranquil zones behind the docks. As mentioned earlier, this speculation was perhaps due, at least partially, to an analogy with the bed load of coarse-grained sediment in rivers and other open channels. However, the process by which the fluid mud is transported was not described because no velocity measurements could be taken within the fluid mud zone. In view of the later advances in cohesive sediment dynamics discussed in Chapter 6 and Chapter 7, such a creeping near-bed sediment motion with the fluid mud being dragged is unlikely. According to Section 8.3.1, the flow of the mud indeed takes place not as a coherent thick fluid of high viscosity, but through a series of deposition resuspension cycles being in suspension while in the process of transportation. During sediment sampling, it is very difficult to distinguish sediment transport within the high concentration zone near the bed subjected to resuspension and transport within the still unconsolidated low density but not eroding layer below the first. As a result, patches of mud appear to an observer as advancing by a certain length during a tidal cycle. However, this advancement takes place after the sediment in the zone defined as fluid mud is eroded and placed in suspension. The study did not include any salinity distribution data. However, from the vertical description of the distribution of the sediment concentration and particularly from the fact that the near-bed water carries a much higher sediment concentration than the upper layers, it appears that the estuary is of the partially mixed type. Indeed, the suspended sediment in the water impounded into the Tilbury Docks by twin-propeller pumps installed between the Tilbury Basin and the Tilbury Docks amounted to just 15% of the total material dredged per year. In contrast, the sediment entering through the old entrances (Figure 8.17) amounted to about 80% of the total deposits, whereas the sediment inflow through the new entrance was found to be negligible. Dredging records indicated that the total known mud volume removed from the estuary and disposed near its mouth or in the Black Deep are twice as high as the estimated sediment inflow into the system accounted for from the tributaries, domestic and industrial sewage, and various other land sources. It was concluded, therefore, that the dredged mud not accounted for must be coming from the deep sea, being transported upstream to the shoaling zones around the null point by the net near-bed salinity and density currents. The shoaling of the Mud Reaches is most intensive during the summer. By contrast, during periods of winter floods, when the river discharge may exceed 4,500 million gallons per day (200 m3s1) for one or more weeks, the mud previously deposited in the Mud Reaches was eroded. That eroded sediment was
306
Cohesive Sediments in Open Channels
deposited in the lower Gravesend Reach, from where it had to be removed by dredging. These observations led to the conclusion that the shoaling in the Thames estuary is salinity intrusion controlled. During periods of low fresh water flows, the null point appears to be located within the Mud Reaches in the upper part of the estuary, whereas during periods of high flows, it is displaced downstream within the Gravesend Reach. Part of the dredged sediment disposed in the Black Deep does not remain there. Instead, it is transported up the river and is redeposited within the shoaling zones. This conclusion was furthermore confirmed by field studies involving sediment tracing with grains of radioactive glass as a tracer.
8.4.4 The Maracaibo Estuary The shoaling in the Savannah, Delaware, and London estuaries is salinity intrusion controlled because the shoaling zones were centered at the null point. This is indeed the case for most estuaries. However, as pointed out in Section 8.3.1, in some estuaries the bed shear stresses may drop below the critical value necessary to resuspend the particular deposited sediment downstream of the null point for ocean-borne sediment or upstream of the latter in the case the sediment originates predominantly from the land. The Maracaibo estuary in Venezuela is a striking example of tide-controlled shoaling [45, 100, 106]. The estuary is indeed a very special case. It connects the Gulf of Venezuela to a large fresh water lake by the Straits of Maracaibo and the shallow Tablazo Bay. The entire system of the lake and the connecting bay and channel are outlined in Figure 8.19. The estuary proper, together with its shoaling zones and tidal entrance to the Gulf of Venezuela, are shown in more detail in Figure 8.20. The lake has an oval shape with a north-south length of about 150 km, an eastwest width of approximately 110 km, and together with its marginal swamps it covers an area of about 18,000 km2. The Straits of Maracaibo and the shallow Tablazo Bay both have a total surface area of about 1,100 km2. The entire lake-estuary system drains 17 hydrologic basins with a widely varying rainfall. For example, the highest mean monthly rainfall recorded at the center of River Catatumbo basin was 500 mm in October and 200 mm in February. By far the highest part of rainfall drains directly into Lake Maracaibo proper. The average annual fresh water discharge volume, based on hydrologic data and analyses, was estimated to be about 50 109 m3, which is equivalent to an approximate uniform discharge of 56,000 ft3.s1 (1,600 m3s1). The straits of Maracaibo have a length of approximately 40 km and a width varying from about 18 km to 7 km, as shown in Figure 8.20. A natural channel with a width of about 3,000 ft. (1,000 m) and a depth varying between 18 and 58 ft. (12–18 m) runs through these straits. The distance between the northern end of the straits and the Gulf of Venezuela is about 24 km through the Tablazo Bay. The depth of that bay ranges from 2 to 15 ft. (0.6–4.6 m) except within some natural channels, where it can reach, and it may even exceed, 20 ft. (6 m).
Chapter | 8 Engineering Applications of Cohesive Sediment Dynamics
307
ISLANDS LOS MONJES
LA GUAJIRA PARAGUANA
Castilletes CO VE LUM NE B ZU IA EL A
Cajóro CALABOZO GULF OF VENEZUELA BAY da
FALCON
tari a Cap
TABLAZO BAY
MARACAIBO
ojó Bor
Ó
a
rid
za Za
N
LAKE MARACAIBO VENEZUELA
Approximate scale 1:2300,000 Figure 8.19 Outline of the Maracaibo estuary together with Lake Maracaibo [100, 106, Fig. 1, 115].
The environment of the entire estuarine-lake system had maintained a quasi steady-state regime with respect to salinity distribution and sediment motion with only seasonal and annual variations. Until 1938, navigation between the gulf and Lake Maracaibo took place only through the natural channels and particularly through the San Carlos-Zapara entrance. That entrance was very unstable and was shifting continuously. That shifting caused gradual changes of the depth of the entrance, which diminished from 13 ft. (4 m) in 1920 to 9.5 ft. (3 m) in 1928, thus limiting the draft of the boats to 3 m. The necessity for stabilization of the entrance and the need to provide for the increasing boat size prompted a navigation improvement plan. That plan started in 1938 with the deepening of the natural channel, indicated in Figure 8.20 as Canal Larrazabal, from its natural depth to 19 ft. (5.8 m). That channel could easily be maintained with only little dredging. Shortly after, however, the development of the oil industry and the rapidly increasing size of the
308
Cohesive Sediments in Open Channels
GULF OF VENEZUELA km 0 RIO SAN CA RL LIMON OS
km 10 N B 26
CANAL LARRAZABAL
ZAPARA E 33
km 10 S T 46 T 50 T 55
PESCADEROS TABLAZO BAY km 20 857 PUNTA PALMAS km 30 S N
B 69 MARACAIBO km 40 S km 50 S 0
10 KM.
20 B 89 km 60 S PUNTA ICOTEA
LAKE MARACAIBO Figure 8.20 Outline of the Maracaibo estuary and the navigable channel [106, Fig. 2; 100, 115].
oil tankers led to further deepening and to a realignment of the navigable channel. In 1947, the government of Venezuela initiated a new project for deepening the channel to 35 ft. (10.7 m) and in 1952 established the National Institute of Waterways (Instituto Nacional de Canalizaciones, or INC), which took charge of the dredging operations. These operations started on April 28, 1953, within the Tablazo Bay with the interior channel and ended on December 7, 1954. The dredging of the exterior portion of the channel, that is, in the Gulf of Venezuela, began on November 28, 1954, and was completed together with the 3.2 km breakwater at the western tip of the island of Zapara on December 8, 1956. Because of the increasing volume and displacement of the oil tankers, in 1953 the Venezuelan Institute of the Waterways decided to increase the depth of the
Chapter | 8 Engineering Applications of Cohesive Sediment Dynamics
309
navigable channel to a minimum of 45 ft. (13.7 m) below the lowest sea level. This deepening started in February 1960 and was completed in 1963, increasing the length of the channel from 40 km to 100 km.The width of the exterior portion of the channel is 1000 ft. and that of the interior portion is 800. ft with very mild slopes. The dredging and the shoaling situation to 1963 was the following. Since 1960, the dredging of the channel had been carried out mainly by an agitation dredge, Julia, with a capacity of 6,500 m3 of solid material per hour. A smaller pipeline dredge with a capacity of 650 m3 of solid material per hour was also used for assistance in special locations. Heavy shoaling started with deepening of the channel to 35 ft. and worsened even more when the depth was increased to 45 ft. to a point requiring almost continuous operation of both dredges. The shoaling distribution is shown in Figure 8.21a and Figure 8.21b. Both shoaling records for the 45 ft. channel were based on soundings right after each dredging and just prior to the following dredging [100, 106]. The subscripts N and S stand for north and south, respectively. The distances refer to the zero point in the island of Zapara (Figure 8.20). According to Figure 8.21a, the shoaling essentially takes place within the first 40–50 km of the channel with very little shoaling beyond 50 km. Two zones of heavy shoaling can be distinguished: 1. The first 10 km of the interior channel from the entrance to Tablazo Bay to the southern tip of the island of Pescaderos. 2. The 15 km segment between kilometer 20 south to kilometer 35 south (Figure 8.20). The sediment in the first shoaling zone was composed of 50% to 65% fine sand, 20% to 35% silt, and 5% to 15% clay. In the second shoaling zone, the sediment was found to be much finer of the sandy clay loam type, with an average composition of 30% clay, 50% silt, and 15% fine sand. Moreover, it was found that the latter has about the same mechanical composition of the sediment dredged from the Gulf of Venezuela. The observed rapid increase of the shoaling rates after the deepening of the navigable channel to 35 ft. (11 m) coincided with an increase of the salinity in Lake Maracaibo proper, as shown in Figure 8.22. Specifically, from 1957 to 1962 the chlorinity at a reference point in the epilimnion of the lake increased from 500 ppm to 2,200 ppm, and by 1966 it reached the value of 5,000 ppm. The salt composition of the lake was the same as that of the gulf water. This rapid salinity increase in the lake with the deepening of the navigable channel is the result of net upstream near-bed inflows into the lake. The rates of these inflows increase with increasing depth of the channel [100, 106]. These preliminary records of shoaling and salinity led to the following early conclusions: 1. The shoaling is tide controlled since the null point is well beyond the interior shoaling zone. In fact, there is not even a null point to speak of, because the salt water enters into the lake.
Rate of sedimentation in feet per day
310
Cohesive Sediments in Open Channels
Note : Values based on soundings from 1959 until November,1963
0.015' 0.010' 0.005' 0 �0.005' 30N
20N
10N
0
10S
20S 30S 40S Distance in km
50S
60S
70S
80S
Rate of sedimentation in feet per day
(a) 0.020' 0.015' 0.010' 0.005' 0' �0.005' 30N
0' Note : Values based on interval of 300 days between soundings without dredging
20N
10N
0 10S 20S Distance in km
30S
40S
(b) Figure 8.21 Rates of shoaling in the Maracaibo navigable channel; (a) Shoaling for years 1959–63; (b) Shoaling for the year 1964 [100, 106, Fig. 5; 115].
2. With the deepening of the channel, the shoaling zones advanced upstream since the tidal velocities, and consequently the bed shear stresses, increase with increasing depth at every distance. 3. The coarser material deposited in the first shoaling zone appears to come from the intense littoral movement across the coastline of the Gulf of Venezuela. This material possesses higher settling velocities and higher bmin, thus settling in a zone of relatively high bed stresses in comparison to those in the second shoaling zone. The depositing material in the latter zone, which, as indicated earlier, has a composition similar to that in the Gulf of Venezuela, seemed to originate from the Gulf after being scoured by wave
311
Chapter | 8 Engineering Applications of Cohesive Sediment Dynamics
Depth � 20 ft.
Constr. of 35 ft. channel
1500
1000
Constr. of 45 ft. channel 1955
1950
1945
0
1940
500
1960
Chlorinity in PPM
2000
Figure 8.22 Chlorinity variation in the epilimnion of Lake Maracaibo [100, 106, Fig. 7].
action and settled into the channel. That material possesses lower settling velocities and lower bmin than the coarser sediment in the first shoaling zone, thus being able to be transported further upstream, where the flow-induced bed shear stresses become sufficiently low. 4. The sediment concentration in the water in the Tablazo Bay during and after dredging was found to be near zero with only floating algae. This precluded some earlier suspicions that the shoaling was caused by the return of dredge spoils discharged into the Tablazo Bay. To confirm these preliminary conclusions and to obtain data for quantitative estimates of shoaling, M.I.T. and the I.N.C. jointly initiated a 3-year program of field studies in 1964. The program consisted of continuous measurements of velocities, salinities, and sediment concentrations at selected points in Tablazo Bay and in the Gulf of Venezuela. The first phase of this program, with the author’s participation, went through 1966. As shown in Figure 8.1, the vertical salinity profiles are characteristic of a partially mixed estuary whereby a lower zone of high salinity is separated from an intermediate layer of relatively high salinity gradients overlain from an upper zone of almost constant to near zero salinity. The same figure shows the vertical distribution of the suspended sediment concentration. Almost the entire suspended sediment is well contained within the lower zone of high salinity with only some traces of fine sediment and some algae in the upper zone. As explained in Section 8.3.1 and shown in Figure 8.3, the vertical distribution of the time averaged over a tidal cycle velocity profile is characterized by a net flow toward the ocean in its upper part and a net flow toward the lake
312
Cohesive Sediments in Open Channels
Station E-33 1.0
February 1965
Station T-50
u 0.0535
1.0
0.8
0.8 ho
y ho
y
0.6
0.6 u 0.0607 ho 45 ft.
0.4
0.4
0.2
1.0 Lake
y ho
0.2
1.0
2.0
2.0
1.0
u (m/s)
Gulf
Lake
1.0 Gulf
Figure 8.23 Average velocity profiles in the Maracaibo channel in Venezuela [106, Fig. 24].
(upstream) in its lower part. A similar distribution is shown in Figure 8.23 at the indicated two stations for February 1965, which was in the dry season. However, in tidal estuaries the instantaneous velocity profiles may be directed all the way downstream or upstream, depending on the tidal phase. Figure 8.2 shows an example of such an instantaneous velocity distribution at Station T-50. This is near the entrance to the second shoaling zone. The near-bed velocity profiles in Figure 8.23 and the fact the even at the entrance to Lake Maracaibo the net upstream flow still persists confirm directly that the shoaling is tide-controlled. The time average velocity, u ( y), over a tidal period T at a distance y from the bottom of the channel is given by the equation
u ( y)
∫t
t T
u( y)dy
(8.14)
The net fresh water discharge per unit width of the channel will be the integral of u ( y) over the entire average channel depth, yo, according to the equation
qf
ho
∫0
u ( y )dy
(8.15)
The time and depth average velocity, u, will them be u
qf ho
(8.16)
313
Chapter | 8 Engineering Applications of Cohesive Sediment Dynamics
Station E-33
�2.4
Gulf
�1.6
�0.8
0
8
10
12
14
16
18
Time
0.8
1.6
Lake
Sediment discharge rate qs in lb of dry material per foot width per second
Station T-50
2.4 Figure 8.24 Rates of sediment transport in the Maracaibo channel in Venezuela [100, 106, Fig. 27; 115].
The vertical distribution of the instantaneous sediment transport rate is given by qs ( y) u( y)C ( y)
(8.17)
An example of the vertical distribution of sediment concentration is given in Figure 8.1. Similar concentration profiles have been encountered for other times and sections. These profiles disclosed a well-defined zone of relatively high turbidity near the bottom of the channel, which extended to at least Punta Palmas. The thickness of that zone was found to vary from 2 to 15 ft. (0.60–4.6 m), decreasing from the ocean entrance toward the lake. An example of sediment discharge is given in Figure 8.24 for the two indicated stations, E-33 and T-50, located immediately downstream of each of the two shoaling zones for November 16, 1964. Similar distributions were found for different times. One can readily observe that the net sediment transport rate over the tidal period is directed toward the lake in an upstream direction [106]. Above the lower zone of high turbidity, the sediment concentration drops rapidly to very low values due predominantly to algae and to some organic matter. The total sediment transport rate, qs , per unit width of channel over the entire depth and over a tidal cycle was obtained by numerical integration of the following equation:
qs
ho
∫0 ∫t
t T
u( y,t )C ( y,t )dtdy
(8.18)
314
Cohesive Sediments in Open Channels
Table 8.1 Average velocities and suspended sediment transport rates over a tidal cycle [106] Date
Station E-33
Station T-50
m/sec
qs b/ft-sec
m/sec
qs b/ft-sec
2-25-65
.0535
10.2x102
.0607
37.3x102
3-17-65
.1010
8.8
.1080
18.7
4-14-65
.0172
12.7
.1700
30.9
u
u
5-20-65
–
1.3
–
–
6-8-65
–
2.7
–
9.1
NOTE: The positive sign indicates direction toward the Lake
These sediment transport rates are given in Table 8.1 in the indicated English units for stations E-33 and T-50 and the indicated dates. The origin of the sediment deposited in the two shoaling areas and the rates were thus determined by direct field measurements without any kind of tracer. Such tracers were used in the Thames River estuary and the San Francisco Bay estuary. This direct approach in evaluating sediment sources and motion is preferable to sediment labeling from the standpoints of reliability, economy, and environmental protection. It is, therefore, recommended wherever and whenever possible. From Table 8.1, it appears that 0.200 lb.ft2s1 is a rough but reasonably representative value of the average sediment discharge rate per unit width of the channel entering the second shoaling zone. According to Figure 8.21b, the average daily accumulation of sediment is of the order of 0.008 ft., and that accumulation takes place over an approximate length of about 20 km or 66,000 ft. Therefore the average daily rate of sediment accumulation in terms of dry weight of sediment per square foot is
86400qs 86400(0.2) 0.262 lb.ft 2day 1 66000 66000
(8.19)
which is equivalent to 1.278 kg.m2. That deposit corresponds, as indicated, to a daily shoaling depth of 0.008 ft. Therefore, the dry density of the deposited sediment was estimated to be about 33 lb.ft3, which is equivalent to 530 kg m3, and to a water content of about 150% of the dry weight of the sediment. Both numbers are comparable to the density of the mud of the San Francisco Bay. The dry density of that sediment was found
Chapter | 8 Engineering Applications of Cohesive Sediment Dynamics
315
to be about 40 lb.ft3 with a corresponding water content of about 110%. One should note, though, that the latter had been settled in place for considerable time and sampled within a depth of a few feet, whereas the former was evaluated from recently deposited surface samples. In any way, both values fall within the range of dry densities and water contents of recent deposits of cohesive sediments. To summarize, grain size distribution analyses, sediment accumulation records, and direct evaluation of sediment transport rates based on measured velocity and sediment concentration profiles led to the conclusion that the shoaling material comes from the Gulf of Venezuela, and it is transported and deposited in the two shoaling zones by the density-generated near-bed currents. We should finally add that the concentration of suspended sediments in the shallow zones of the Tablazo Bay surrounding the navigable channel were found to be very low, except very near the entrances, and of the order of 10 ppm. They seemed to be primarily composed of algae, some organic material, and some nonsettling small flocs, as within the upper water layer in the navigable channel. As pointed out earlier, the Maracaibo estuary case with its tide-controlled shoaling is rather rare and due primarily to the geomorphology of the estuarine system. There is specifically a significant dissipation of tidal energy there due primarily to the sudden expansion at the Zapara entrance and to the shallow water depth in the Tablazo Bay. For example, the tidal height diminishes from 4.6 ft. (1.4 m) at the Zapara entrance to 2.3 ft. (0.70 m) at Punta Palmas and to about 8 in. (0.20 m) near the lake. This would account, at least partially, for the low bed shear stresses in the navigable channel and for the tide-controlled nature of shoaling there.
8.4.5 The San Francisco Bay Estuary The San Francisco Bay with its tributary rivers is another special case of estuary with respect to its shoaling problems and processes and to its geology and hydrographic characteristics. It belongs to the class of estuaries built by tectonic processes [63]. It is one of the most important estuaries of the world, with very large population areas and industrial cites around it. Its scenic beauty and the recreational potentialities are additional assets of the estuary. The estuary is outlined in Figure 8.25. It consists of a broad and shallow bay communicating with the Pacific Ocean through a tidal inlet at Golden Gate. There are two major rivers, Sacramento and San Joaquin, that discharge into the bay through the smaller Suisun Bay and the shallow San Pablo Bay. In addition, there is a number of minor rivers and creeks discharging all around the perimeter of the estuary, the most important of which is the Napa River discharging north of the city of Vallejo. There is a network of navigable channels to and within the bay for sea traffic. These channels are being maintained by dredging at a high annual cost. For example, in 1964 the dredging cost exceeded $2 million. One of the worst areas for sediment deposition was the lower reach of the Carquinez Straits between Mare Island and the city of Vallejo. The Napa River discharges through these straits. There are several harbor installations in there, and the
316
Cohesive Sediments in Open Channels
Figure 8.25 Outline of the San Francisco estuary [63; 103, Fig. 5].
Chapter | 8 Engineering Applications of Cohesive Sediment Dynamics
317
transported sediment deposits right behind them, requiring difficult and expensive dredging [63]. The sediment in the San Francisco Bay is fine of the silty-clay type. The same sediment was used in the first experiments by the author on erosion and deposition of cohesive soils presented in Section 6.3. The sediment deposited in the shoaling zone of the Carquinez Strait was very similar to that of the San Francisco Bay and not to the sediment transported by the Napa River, although the latter was in substantial quantities. Moreover, it is noteworthy that shoaling occurred predominantly during the summer months, when the bay water contains the lowest sediment concentration and the rivers carry practically sediment-free water. It was, therefore, concluded at that time that sources other than the Napa River should be accounted for the shoaling in the Carquinez Strait. In 1955, the need to identify these sediment sources led to an extensive field and laboratory research project. This project was conducted by the University of California at Berkeley and supported by the U.S. Corps of Engineers. Radioactive gold, Au198, was selected and used as a sediment tracer, particularly because the high degree of surface electrochemical activity of the bay sediment made attachment of the gold molecules on the clay particles much more effective. We must point out that, in tracing, it is assumed that the hydrodynamic behavior of the labeled sediment, as it affects the formation and the properties of aggregates and the erosive and depositional characteristics, is the same as that of the unlabeled sediment. There is, however, some question about such an assumption because the surface attachment on any molecules or particles on the fine sediment surfaces is bound to have some effect on the surface forces of the latter and on their efficiency to coalesce with other units. The activated sediment was released at the beginning of the flood tide in the edge of the Mare Island (or Carquinez Straits), and the radioactivity was monitored. The tracer was found almost fully deposited in the straits during the first tidal cycle. No significant tracer movement was detected during the first four days. The radioactivity had its highest value behind piles and other marine structures. This sediment distribution was attributed to the high degree of turbulence in the wake of the marine structures [63, 103]. The investigators also claimed that the bulk of the shoaling material enters the straits in suspension. It was concluded that the observed sediment motion was caused by wind-generated waves in the San Pablo Bay. Because of the low depth of the bay, the waves exert sufficiently high shear stresses on the cohesive bed to cause substantial resuspension of bed sediment. The resuspended sediment is then transported by the near-bed currents and deposited in the shoaling areas. These currents are generated by the hydrodynamic interaction between the fresh water of the Napa River and the salt water of the bay. No profiles of salinity and velocity ware determined in that study.
8.4.6 Closing Remarks As we pointed out in the beginning of this section, the five case histories of estuarine shoaling were selected and presented for a demonstration of the various processes of
318
Cohesive Sediments in Open Channels
cohesive sediment transport rather than for the general hydrodynamics of estuaries. The latter is indeed a very broad subject treated extensively in various publications. For a concise treatment of the subject, readers are referred to Refs. [100, 110, 111]. Each one of the presented five cases is characterized by special hydraulic, hydrologic, and geomorphologic features and by distinct engineering requirements. In all of them, the upstream advancement of the shoaling zones and the increase of the shoaling rates with deepening of the navigation channel were quite clear. In the cases of the Savannah and Delaware estuaries, the sediment appeared to be predominantly land borne, and the shoaling zones were related to the null point. In the River Thames estuary, the bulk of the sediment was coming from the sea and particularly from the disposal area, and the shoaling zones were also related to the null point. The sediment motion was determined through the detection of granulated radioactive glass with hydrodynamic properties drastically different from those of the actual fine sediment. The sediment motion in the San Francisco Bay estuary was studied in the same way. Such tracing may very well disclose the general trend of sediment motion, but it can provide very little information as to the actual shoaling volumes and rates of deposition. In contrast, the field research in the Maracaibo estuary with tide-controlled shoaling first demonstrated the sediment sorting during its transport from direct analysis of sediment deposited in each of the two shoaling zones and, second, it led to a quantitative estimate of the shoaling rates. Therefore, such detailed measurements are recommended in cases where quantitative estimates are required or even desirable. Moreover, quantitative information regarding shoaling volumes and shoaling rates may be very useful in mathematical models, either physical or mathematical, or a combination of both for verification and predictions. One of the worst effects of deepening of navigable channels is the shifting of the shoaling zones in harbors and inside various other marine structures. Such installations were originally planned in a way to stay well beyond any zone of heavy shoaling. Unfortunately, in most cases, the planning was made before the developments in estuarine hydrodynamics and with little knowledge of the relationships among sediment properties, tides, channel geometry, and density currents. Sediment dredging from any area with harbors and/or marine structures can be much more expensive than dredging in unobstructed segments of the channel. Therefore, any plan for future changes in the geometry of an estuary should take the necessary steps to avoid unwanted sediment deposits inside the areas of harbors and other marine installations.
8.4.7 Applications to Estuarine Modeling In the past, physical models were used for the prediction of the effect of changes in an estuarine regime on salinity distribution and shoaling. Considering the strong coupling of density underflows and sedimentation, models with salinity have mostly been used. Since the advent of high-power computers, however, there is more and more reliance on mathematical models. There are advantages
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and disadvantages in both kinds of modeling. Physical models, particularly in a scale large enough to simulate a natural system on the order of magnitude and of the complexity of estuaries, are very expensive to build and even more so to verify and to operate. They have the great advantage of direct simulation and of providing a visual picture of the physical behavior of the system. Mathematical models are reliable if the fundamental behavior equations of the natural system are known. Even so, the physical models have the advantage of accounting for phenomena which could not be easily simulated in mathematical models, such as the effect of secondary currents on the dispersive characteristics of the system. In physical models involving prediction of new shoaling sites and rates due to planned engineering operations, the prevailing technique was to simulate the actual cohesive sediment with some lightweight coarse material, such a coal and various plastics, which can be moved readily by the model flows. Like the radioactive tracing techniques applied in the River Thames and San Francisco estuaries, that kind of model sediment may give some qualitative idea as to the trends of the future shoaling situation, but it is unlikely to lead to any quantitative predictions. The reliability of the model results, therefore, diminishes with increasing changes of the estuarine regime from its original state. The equations governing the deposition and resuspension of cohesive sediments presented in Chapters 6 and Chapter 7 can be used for an improved prediction of shoaling patterns and rates with either physical or mathematical models. At first glance, the application of these equations may appear very difficult. Cohesive sediments are dispersed during their transport, deposition, and resuspension, and in the process their properties may change significantly. Nevertheless, there are ways of using fruitfully the new knowledge for the improvement of the shoaling predictions. A successful incorporation of the sediment behavior equations into modeling can be achieved by using a salinity model, mathematical or physical, for the prediction of the salinity distribution in the estuarine channels and the new location of the null point, if any. The physical model with salinity simulation can be used for the prediction of the salinity and velocity regimes and specifically for the location of the null point. If the shoaling is salinity intrusion-controlled, then the new shoaling zone will be around the null point. In the case of tide-controlled shoaling, the model can be used to determine the segment or segments of the estuarine canal or canals with near-bed velocities and shear stresses similar to those in the original system. The rates and volumes of shoaling can then be estimated through the application of the appropriate equations of Chapter 7. Mathematical models could also provide information about the new estuarine regime. The choice depends on the complexity of the system, the desired degree of details, on the availability of modeling space and facilities and on the overall cost.
8.5 Control of environmental pollution Finally, we should add a few comments on the effect of sediment on the marine life and on the general environmental quality of estuaries. Changes of sediment
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distribution due to dredging and to various other solid waste disposals may have several unfavorable environmental impacts, of which two are the most obvious: excessive deposition on the feeding grounds and an increase in turbidity to a point prohibiting sufficient light for the development of plankton, one of the basic nutrients for marine life. Estuaries are highly complex natural systems with unique coastal features and with distinct physicochemical, biological, geomorphologic, hydraulic, and hydrologic characteristics [111]. These special estuarine properties are generated by the hydrodynamic interaction of geometry, tides, fresh water discharge sediment, and salinity. These components are subjected to a very wide range of variation so that the overall estuarine regime also varies very widely daily and seasonally. Throughout its geological history, an estuary reaches a quasi steady-state regime, still with high short-term variations, but with very small long-term changes on a human scale, except, of course, for changes due to isolated events, such as earthquakes and tsunami waves. A wide range of marine species is developed in every estuary adapted to the local environmental conditions and their variation. In addition to their rich marine life, estuaries possess several other advantages, such as abundant supplies of fresh water, productive agricultural land, access to navigation, favorable harbor sites, and recreational aspects. Large urban and industrial centers have been developed around estuarine areas attracted by these and other beneficial aspects. Such developments required encroachment on the estuarine environment and specifically withdrawal of fresh water, filling and dredging operations, and disposal of domestic and industrial wastes. In addition to sediment pollution, mentioned earlier, changes of the salinity and temperature distribution are expected to have a significant impact on the marine life [98]. Interaction of suspended cohesive sediments with organic matter and various dissolved chemicals from industrial and domestic sewers may become a major problem. The very same physicochemical forces acting on the surfaces of the fine particles, which are responsible for flocculation and which were discussed in Chapter 2, and Chapter 3 may also attract other suspended and/or organic and mineral matter and in particular toxic chemicals. Adsorption of such matters may affect the mutual attraction and flocculation of fine particles and of small flocs into larger aggregates. Very little is known at present time about the interaction of fines with organic matter and dissolved metals in estuarine waters, particularly the effect of flow-induced stresses on the rate of adsorption, other than mere indications. On the basis of laboratory experiments, Krank claimed that there are organic and inorganic proportions for optimum flocculation [59]. According to her, 65% to 75% organic matter by volume indicates that the component in excess of that value has been preferentially exported from the estuary. Krank also pointed that small marine organisms themselves may repackage particulate matter into regularly shaped, relatively compact fecal pellets containing various proportions of organic and inorganic matter depending on environment and species. Other sources also cited by Krank indicate a well documented role of organic coatings, bacteria and extracellular strands and monofilaments in
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aggregating particles or reinforcing other binding mechanisms. Earlier laboratory studies by Krank and Milligan demonstrated increased settling rates with organic matter [59, 60]. In spite of the absence of any quantitative information regarding the effect of organic matter and microorganisms on flocculation, it is clear that suspended fines do attract organic matter and toxic substances. Therefore, cohesive sediment beds formed by settling units exposed to such matter may contain a high quantity of toxic ingredients. As long as they stay in the bed, there is little danger for the marine life. Dredging operations, however, normally result in suspension of very large volume of bed sediment, which in turn may enter into the digestive system of the fishes and other marine organisms with harmful consequences. Before one starts any extensive dredging and/or filling operations, it is recommended that one conduct thorough chemical and biological analyses of the bed sediment to be dredged and, in case of detection of objectionable contamination, to avoid as far as possible any dumping of sediment into the estuarine waters.
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114. Partheniades, E., and Kennedy, J. F. “Depositional Behavior of Fine Sediments in a Turbulent Fluid Motion,” Proceedings, 10th Conference on Coastal Engineering, Tokyo, Japan, Vol. II, Chapter 41, pp. 707–724, September, 1966. 115. Partheniades, E., and Kennedy, J. F. “Sources and Motion of Sediment Deposited in the Maracaibo Channel, Venezuela,” (Spanish), Memoirs, Second Latin American Hydraulic Congress, Caracas, Venezuela, pp. 261–288, July 11–16, 1966. 116. Partheniades, E., and Paaswell, R. E. “Erosion of Cohesive Soils and Channel Stabilization. Part I: State of Knowledge,” Civil Engineering Report No. 19, Department of Civil Engineering, State University of New York, Buffalo, NY, October, 1968. 117. Partheniades, E., and Paaswell, R. E. “Erodibility of Channels with Cohesive Boundary,” Journal of the Hydraulics Division, ASCE, Vol. 96, No. HY 3, Proc. Paper 7156, pp. 755–771, March, 1970. 118. Partheniades, E., Cross, R. H., and Ayora, A. “Further Results on the Deposition of Cohesive Sediments,” Proceedings, 11th Conference on Coastal Engineering, London, England, Vol. II, Chapter 47, pp. 723–742, September, 1968. 119. Partheniades, E. Discussion of “Salinity Intrusion Effects in Estuarine Shoaling,” by Harleman, D. R. F., and Ippen, A. T. (Proc. Paper No 6340, January 1966), Journal of the Hydraulics Division, ASCE, Vol. 96, No. HY1, pp. 264–269, January, 1970. 120. Peele, T. C. “The Relation of Certain Physical Characteristics to the Erodibility of Soils,” Proceedings, Soil Science Society of America, Vol. 2, 1937. 121. Quinn, M. J. “A Scanning Electron Microscope Study of the Microstructure of Dispersed Kaolinite Clay Taken Out of Suspension,” M.S. Thesis, Department of Geology, University of Florida, Gainesville, FL, 1980. 122. Rectoric, R. J. “Critical Shear Stresses in Cohesive Soils,” M.S. Thesis, Department of Agricultural Engineering, Texas A & M University, College Station, TX, January, 1964. 123. Rosenquist, I. “Mechanical Properties of Soil-Water Systems,” Transactions, ASCE, Vol. 126, Part I, pp. 745–767, 1961. 124. Rosillon, R., and Volkenborn, C. “Sedimentation of Cohesive Material in Salt Water,” (Spanish), Diploma Thesis, University of Zulia, Maracaibo, Venezuela, November, 1964. 125. Schoklitsch, A. Tratado de Arquitecture Hidraulica, Second Edition, Vol. I, Editorial Gustavo Gill, S. A., 1961. 126. Shen, H. W. “Wash Load and Bed Load,” River Mechanics (Edited and Published by H. W. Shen), Water Resources Publications, Fort Collins, CO, Vol. I, Chapter II, 1971. 127. Simmons, H. B. “Channel Depth as a Factor of Estuarine Sedimentation,” Technical Bulletin No. 8, Committee on Tidal Hydraulics, U.S. Army, Corps of Engineers, March, 1965. 128. Smerdon, E. T., and Beasley, R. P. “The Tractive Force Theory Applied to Stability of Open Channels in Cohesive Soils,” Research Bulletin 715, University of Missouri, College of Agriculture, Agricultural Experiment Station, Vol. 13, pp. 441–448, October, 1959. 129. Smith, D. K. W. “The Strength of Aggregates Formed in Flocculation,” Chemical Engineering Science, Vol. 33, Pergamon Press Ltd, pp. 1631–1636, 1978. 130. Sündborg, A. The River Klarelven, A Study of Fluvial Processes, Geografiska Annalen, Stockholm, Sweden, 1968. 131. Swift, D. I., and Friedlander, S. K. “The Coagulation of Hydrosols by Brownian Motion and Laminar Shear Flow,” Journal of Colloid Science, Vol. 19, pp. 621–647, 1964.
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132. Tambo, N., and Hozumi, H. “Physical Aspects of Flocculation Process II. Contact Flocculation,” Water Research, Vol. 13, pp. 441–448, 1979. 133. Tambo, N., and Hozumi, H. “Physical Characteristics of Flocs II. Strength of Flocs,” Water Research, Vol. 13, pp. 421–427, 1979. 134. Tambo, N., and Watanabe, Y. “Physical Characteristics of Flocs I. The Floc Density Function and Aluminum Flocs,” Water Research, Vol. 13, pp. 409–419, 1979. 135. Tambo, N., and Watanabe, Y. “Physical Aspect of Flocculation Process I. Fundamental Treatise,” Water Research, Vol. 13, pp. 429–439, 1979. 136. Task Committee on Erosion of Cohesive Materials: Committee on Sedimentation “Abstracted Bibliography on Erosion of Cohesive Materials,” Journal of Hydraulic Division, ASCE, Vol. 92, No. HY2, Proc. Paper 4746, pp. 243–268, March, 1966. 137. Taylor, A. W. “Ion Exchange Phenomena,” Transactions, ASCE, Vol. 126, Part I, pp. 718–728, 1961. 138. Taylor, G. I. “Some Recent Developments on the Study of Turbulence,” Proceedings, 5th International Congress for Applied Mechanics, New York, p. 294, 1938. See also: Statistical Theory of Turbulence, V, “Effect of Turbulence of Boundary Layer,” Proceedings, Royal Society of London A, Vol. 156, pp. 307–317, 1936. 139. Tennekes, H., and Lumley, J. L. A First Course in Turbulence, MIT Press, 1972. 140. Terzaghi, K., and Freolich, U. K. Theorie der Setzung von Tonschichten, Vienna, Deuticke, 1925. 141. Thomas, D. G. “Turbulent Disruption of Flocs in Small Particle Size Suspensions,” Journal of American Institute of Chemical Engineers, Vol. 10, No. 4, pp. 517–523, 1964. 142. Thorn M. F. C., and Parsons, J. G. “Properties of Grangemouth Mud,” Report No. EX 781, Hydraulics Research Station, Wallingford, England, July, 1977. 143. Thorn, M. F. C., and Parsons, J. G. “Erosion of Cohesive Soils in Estuaries: An Engineering Guide,” Proceedings, Third International Symposium on Dredging Technology, Paper F1, March, 1980. 144. Tomi, D. T., and Bagster, D. F. “The Behavior of Aggregates in Stirred Vessels Part I—Theoretical Considerations,” Transactions, Institute of Chemical Engineers, London, Vol. 56, pp. 1–8, 1978. 145. Torf, H. “Erosion of Mud/Sand Mixtures,” Doctoral Dissertation, Hydraulics Laboratory, Catholic University of Leuven, Belgium, October, 1985. 146. Torf, H., Mitchener, H., Huysentruyt, H., and Toorman, E. “Settling and Consolidation of Mud/Sand Mixtures,” Coastal Engineering, Vol. 29, No. 1–2, pp. 27–45, 1996. 147. Tsai, C. T., Iacobellis, S., and Lick, W. “Flocculation of Fine-Grained Sediments due to Uniform Shear Stress,” Great Lakes Research, Vol. 13, No. 2, pp. 135–146, 1987. 148. U.S. Department of Interior, Bureau of Reclamation. “Interim Report on Channel Stability of Natural and Artificial Drainage Ways in Republican, Loup and Sioux River Areas, Nebraska and Iowa,” September, 1953. 149. Van Leussen, W. “Aggregation of Particles, Settling Velocity of Mud Flocs. A Review,” International Symposium of Physical Processes in Estuaries, Springer-Verlag, NY, pp. 347–403, 1988. 150. Van Olphen, H. An Introduction to Clay Colloid Chemistry, Interscience Publications, 1963. 151. Verwey, E. J. W., and Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids, Elsevier Publishing Co., 1948. 152. Vold, M. J. “Computer Simulation of Floc Formation in a Colloidal Suspension,” Journal of Colloid Science, Vol. 18, pp. 684–695, 1963.
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153. Wicker, C. F. “The Prototype and Model Delaware Estuary,” Proceedings, 6th Congress of IAHR, Vol. 1, Paper No. A12, Hague, Netherlands, 1955. 154. Yeh, H. Y. “Resuspension Properties of Flow Deposited Cohesive Sediment Beds,” M.S. Thesis, University of Florida, Gainesville, FL, 1979. 155. Yeh, H. W. “Resuspension Properties of Flow Deposited Cohesive Sediment Beds,” Proceedings, 4th Conference on Ocean Engineering in Republic of China, pp. 127–146, September, 1980.
List of Symbols
a aB A Å Aa Ab Ai An Ash ALT A1, A2, A3 b B Bi c cc ce ch cv C Ca CA, CB CD Ceq * Cep ** Ceq Cf (V)
Acceleration in general; side dimensions Acceleration due to Brownian motion Cross-sectional area; parameter defined by Equation 4.82 Angstrom units Constant in van der Waals attraction potential (Equation 3.32) Dimensionless coagulation constant in Brownian motion in Equation 4.134 Parameter in Equation 6.26 through Equation 6.28 Net cross-sectional area of aggregates Dimensionless coagulation constant for shear flow in Equation 4.135 ALT ratio Surface and volume shape factors Channel width; side dimension Binding force within an aggregate Constant in Equation 6.23 and Equation 6.27 Side dimension Proportionality constant in Equation 5.3 Average erosive resistance per unit bed area Cohesion Volume of suspension in Equation 4.127 Suspended sediment concentration in general; symbol of Coulomb units Coefficient in Equation 4.159 Concentrations of species A, B (Equation 3.35) Drag coefficient Equilibrium concentration Degree of retention = Ceq/Co * Degree of deposition = 1 2 Ceq Final concentration under a velocity V
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Ci Co Cs Cv CV’ Cvf Cvp C* C** CEC CS CV d da da dac df di, dj dmax dp dpi, dpj d1 D Dbr Dc Dij Dr Ds Ds* Dti Dx Dtx Dty Dtz Dv Dy e eq esu
List of Symbols
Concentration of i order aggregates Initial suspended sediment concentration Equilibrium concentration for resuspension Volume concentration of suspended sediment Cv /n Volume concentration of flocs and particles Volume concentration of mineral particles Instantaneous relative degree of retention equal to C* (Co C)/(Co Ceq) Instantaneous relative degree of deposition equal to 1 C* Cation exchange capacity Control surface Control volume Half distance between two particle surfaces; depth of rotating channel Aggregate diameter in general Dimensionless value of da Characteristic floc aggregate size in Equation 4.161 Diameter of microflocs in Equation 4.107 Aggregate diameters of order i and j, Highest aggregate diameter Representative particle diameter; original particle or floc diameter in Equation 4.147 Particle diameters of i and j order Diameter of primary particles Dial reading; molecular diffusion coefficient Brownian diffusion coefficient Diameter of capillary tube Diffusion coefficient of i particles in j direction Relative dielectric constant with respect to that in a vacuum Sediment diameter Threshold sediment diameter between wash load and bed material load Turbulent diffusion coefficient in i direction Dispersion coefficient in x (flow) direction Turbulent diffusion coefficient in x direction Turbulent diffusion coefficient in y direction Turbulent diffusion coefficient in z direction Diffusion coefficient in viscous sublayer Dispersion coefficient in y direction Basis of Naparian Logarithms; electronic charge equal to 16 1020 Coulombs or 4.8 1010 esu; porosity Equivalent Electrostatic units
List of Symbols
E (Eb)o (Es)o Ei Eo Eo Et E* fp F Fa Fb Fd Fe Fi Fij Fv Fn Fx g G h hB ho hs H Hb Hij i iˆ j ˆj J J1, J2 Jy k kˆ ke
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Rate of erosion and resuspension Total rate of borrowed energy of the flow Total rate of energy loss by friction Constant in Equation 6.27 and Equation 6.28 Total energy loss; reference coefficient for resuspension rates in Equation 7.47 Average value of Eo over entire range of consolidation time Tdc Total transmitted energy Dimensionless form of erosion rates Flux of particles across control surface SV Force in general Net interaggregate force; Faraday No 96.5; average erosion resisting force per aggregate Binding force in Equation 4.160 Sediment sink function Electric force between two particles; Sediment source function Total erosion resisting force Wb Fa; drag force on bed particles and/or aggregates in Equation 6.12 Generalized frequency of collision between i order and j order suspended units Volume flux in Equations 4.134, 4.135, 4.136 Densimetric Froude number at null point Densimetric Froude number at x Gravity acceleration Velocity gradient, ∂u/∂x, in y direction normal to flow Total energy head; height and depth in general Value of h to overcome Bingham yield strength Mean water depth in estuaries; average head in viscometer Distance above and below the interface within which the shear stress is less than y Height Bed thickness Collision frequency due to differential settling Index from 1 to 3 Unit vector in x direction Index from 1 to 3 Unit vector in y direction Joule unit Sediment fluxes in Equations 6.29a and 6.29b Sediment flux Boltzmann constant equal to 1.38 1023 JK1; selectivity constant in Equation 3.1 Unit vector in the z direction Overall proportionality constant in Equation 6.16 and Equation 6.17
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kf ki kt k, k, ke k4 K Kc1 Kij Kbrij Kdij Kr Ksij Ky L L Lu L m meq mV M ML Mij Mo Mp ma Mpt n n, n na n n f ni, nj
List of Symbols
Constant with dimensions t1(Equation 7.2) Slope of semilogarithmic lines in Figure 5.6 Constant with dimensions (time)1 in Equation 7.3 Dimensionless coefficients in Equations 6.14, 6.15, 6.16 Coefficient in Equation 7.6 defined in Equation 7.7 Degrees Kelvin; cation exchange equilibrium coefficient; parameter in Equation 4.133; cation exchange coefficient (Equation 3.35) Factor defined by Equation 4.109 defined by Equation 4.110 Collision frequency factor Value of Kij of Brownian motion (Table 4.1) Value of Kij for differential settling (Table 4.1) Constant exponent defining aggregate density (Equation 4.94) Value of Kij for velocity gradients (Table 4.1) Coefficient in Equation 5.19 Reference length for turbulence microscale; length of capillary tube in Equation 5.4 Lift force on bed units and settling units; liter Lower limit of L Upper limit of L Average value of L Mass; flocculation time; exponent in Equation 6.24; meter; coefficient in Equation 4.120 Milliequivalent Millivolts Mass of a particle in Equation 2.3; mole; Molarity; total number of collisions per Unit time of a particle (Equation 4.18) Molality Collision frequency due to Brownian motion between i and j order aggregates Dipole moment in Equation 2.5; total number of contacts per second and per unit volume (Equation 4.20) Number of collisions per unit time for uniform particles due to Brownian motion Mass of settling units Total number of contacts per unit volume and time Integer in general; Manning’s coefficient; ion concentration; number of suspended particles per unit volume; coefficient in Equation 6.24 Local concentrations of negative and positive ions, respectively Concentration of grown aggregates in Equation 4.107 Dimensionless value of n Flocculation concentration; microfloc number per unit volume in contact flocculation number of i and j order sediment units per unit volume of suspension
List of Symbols
no nR nv n* n N Na Ngrad Nijt Ntaf Nr Nij Nturb p pae pae1 ph pr pre pr ppm py P Pd Pe Pr Pr PI PR q qf qs qs(y) qs qx
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Original particle concentration per unit volume of suspension Particle concentration of order R Number of particles per cm3 larger than a given volume v Ion concentration in equilibrium liquid Total number of aggregates per unit volume in Equation 4.127 Number of units per unit surface of bed; maximum number of electrons in a shell; normality; number of revolutions per minute in a rotating cylinder viscometer Avogadro’s number equal to 6.02 1024 ions/mole Rate of collisions of uniform units due to velocity gradients Total turbulent collision frequency in a flocculator between i and j order aggregates in general and in a flocculator Collision frequency between microflocs and large aggregates (Equation 4.107) Rate of rotation of flocculation blades Collision frequency of i and j order aggregates Value of Nijt for constant diameter aggregates Pressure in general; mean collision factor; probability of pa, paa, paaa flocs, first and second order aggregates Aggregate porosity Limiting value of aggregates porosity Confining normal pressure on soil sample Probability density function; proportion of all near-bed settling units that reach and settle on the bed Repulsive force of interaction per unit area of two flat double layers in Equation 3.29 Probability density function for modified normal distribution Parts per million Pressure necessary to overcome yield strength Mean collision agglomeration factor in Equation 4.108; symbol for probability Probability of deposition Probability of erosion Probability of (probability function) Probability function for modified normal distribution Plasticity index Prandtl number Flow rate per unit width Net fresh water discharge per unit width in a tidal estuary Rate of sediment inflow at x 0 Vertical instantaneous distribution of sediment transport per unit width and depth in an estuary at a distance y from the bed Total sediment transport rate per unit width of an estuarial channel Flow rate per unit width in x direction
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qy Q Q Qs Qs r ras rc rm R R* R50ult Rb Re Rec Rp Rpc Rij s sn so sx S SAR Se Sh Sm Sv t tc ts tsa t (o ) t50 T Tc Tdi Tdc Ti Tm To
List of Symbols
Flow rate per unit width in the y direction Flow discharge; electric charge Electric charge Sediment transport capacity Sediment supply rate Distance from a particle surface; radius; radial coordinate; distance between two molecules Radius of largest aggregate in a shear flow Radius of a pipe Radius at which εo εs Order of aggregates; radius of units in general; Non dimensional value of R (Figure 4.13, Figure 4.15 and Figure 4.16) Value of R for the ultimate curve corresponding to 50% of volume accumulation Hydraulic radius pertaining to the bed friction Reynolds’s number Critical Reynolds number Particle radius Radius of influence (or coagulation) for uniform particles Radius of influence between i and j order particles Second; salinity Salinity at null point Ocean salinity Salinity at a distance x from estuary entrance Highest order of aggregates; symbol for surface Sodium adsorption ratio Slope of energy grade line Shear strength Highest order of aggregates for dmax Vane shear strength of a soil Time Coagulation time Settling time in Equation 7.29 Settling time through near bed zone Impulse time Time for 50 percent deposition of depositable sediment Absolute temperature; tidal period; time period Parameter in Equation 7.21 defined in Equation 7.22 Time of deposition under a bed stress bi Consolidation time Time interval of application of bed stress bi Sediment mixing time Torque
List of Symbols
Tr Tp u uo u u ub u−b unb up ur uav unB u u* u*c Ua Uvb v vc vo v vv vrx vsa vλ vλοο vm V V Va Va VA Vaf Vf VF VFd VF Vo Vp Vs Vs VT
339
Torque Period of growth and disintegration of viscous sublayer Velocity component in x direction Reference velocity u at center of an aggregate Time-averaged over a tidal cycle of u Depth averaged value of u− Near bed average per tidal cycle velocity in estuaries Average value of ub over a tidal cycle Brownian motion velocity normal to the surface of particles (Equation 2.2) Particle velocity Tangential velocity; relative velocity Average velocity of channel and ring Normal Brownian motion velocity Instantaneous turbulent velocity in x direction Friction velocity Critical friction velocity Near bed settling velocity of aggregates Instantaneous turbulent velocity near viscous sublayer Velocity component in y direction Critical velocity Flow velocity in a viscometer Instantaneous turbulent velocity in y direction Fluid velocity in a viscometer Relative velocity of two aggregates in flow direction x Approach velocity of an aggregate to bed Eddy velocities corresponding to eddy length scale Velocity microscale Average velocity in a viscometer tube Volts; average velocity; total volume of aggregates Velocity vector Aggregate volume; total volume of suspended sediment Dimensionless value of Va Attractive potential for a single particles Total aggregate volume in Equation 4.137 and Equation 4.138 Fluid volume in viscometer Free energy Free energy per cm2 of single double layer at a distance d Free energy per cm2 of single particle at infinity Average per tidal cycle fresh water velocity in estuaries Volume of solids; total volume of suspended particles Volume of solids within an aggregate Dimensionless value of Vs Water volume it flocculation tank
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Vft Vf VA VR Vt Vw w ws wsi wsm wsij Wb x x xc xo y ya yd yM yo y yo z z za, zb, zd zo
List of Symbols
Flocculent volume at time t (Equation 7.2) Flocculent volume at time (Equation 7.2) van der Waals attractive potential energy between two interacting particles Repulsive potential energy Total potential VA VR Water volume in general and within aggregates and in viscometers Velocity component in z direction Settling velocity in general Settling velocity of i order aggregates Median settling velocity Difference in settling velocity between an i and a j order unit Submerged weight of aggregates Coordinate axis; horizontal distance; distance from a particle surface Distance of energy hump from particle surface Measure of double layer thickness (Eq. 3.15) Reference location Coordinate horizontal axis normal to x; vertical distance; dimensionless variable in Equation 3.8 Parameter for the degree of retention in Equation 7.13 Value of y at mid distance d between two particles corresponding to ϕd (Equation 3.24) Depth below water surface at which εb εs’ Depth of flow; distance above the bed of an underflow at which 0 (Figure 5.3) Distance below water surface Virtual distance from bed at which u 0 Vertical coordinate axis; dimensionless variable in Equation 3.8; elevation of bed surface Distance from surface of deposited bed Characteristic distances from surface of deposited bed Bed elevation
Greek Symbols
c i k o R v w 1 8 8
k 1 s w s
Angle; stability factor; ratio of water volume within aggregates to solid volume; coefficient in Equation 4.85; constant in aggregate density function (Equation 4.94) Factor of collision effectiveness Dimensionless parameters (i from 1 to 7. 9, 10) Empirical coefficient in Equation 6.1 Initial aggregation factor in Equation 4.122; constant factor in Equation 6.28 Dimensionless factor in equation for the rate of generation of aggregates of various orders in Equation 4.122 Velocity head correction coefficient Ratio of immobilized water volume within aggregates in Equation 4.145 through Equation 4.147 Parameter defined by Equation 4.98 Dimensionless coefficient in Equation 7.1 and Equation 7.47 Average value of 8 over entire range of consolidation time Tdc Angle; dimensionless factor; ratio of collision rates due to velocity gradients versus those due to Brownian motion; exponent in Equation 4.161; experimental correction coefficient in Equation 4.80; factor in Equation 4.86; hiding factor for bed particles; empirical constant; empirical exponent in Equation 7.43 Empirical exponent in Equation 6.1 Ratio of collision rates due to velocity gradients to those due to differential settling Angle; unit weight; exponent in Equation 5.159; Dimensionless parameter in Equation 3.23 Average unit weight of clay minerals and/or aggregates in Equation 6.23 Unit weight of water Density of electric charge on a particle surface Plate thickness of clay particles Stern layer thickness
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t V f F i, j bi ~ bi U w wsij e eav ed ei eo εo ev εs εb εi εt η ηd ηe η ηo ηu η* ϑ o B
Greek Symbols
Thickness of viscous sublayer Indicates small finite differences Density difference Velocity difference between rotating annular channel and ring Net acting force per unit area of aggregates due to turbulence Force over aggregate area Density difference between water and aggregates of order i and j, respectively Bed shear stress step equal to (bi1 bi) Dimensionless value of bi equal to (bi1 bi)/bi Change of average velocity over a length scale l Differential settling velocity of particles Differential change of settling velocity difference between i and j order particles Local rate of energy dissipation per unit mass Average rate of energy dissipation per unit mass Dielectric constant Local turbulent energy dissipation per unit mass in inertia sublayer Dielectric constant of vacuum equal to 8.8542 1012C2J1m1; effective rate of energy dissipation per unit mass Effective energy dissipation per unit volume Rate of energy dissipation per unit mass in viscous sublayer Local energy dissipation per unit volume Rate of local energy withdrawal per unit volume Rate of local energy dissipated per unit volume Rate of energy transmitted per unit volume Experimental coefficient in density distribution Equation 7.45 Numerical exponent in Equation 4.122; indicates vh∞ / in Equation 4.131; Statistical parameter for distribution of forces on bed and suspended aggregates; coefficient of rigidity in Equation 5.3 Dimensional parameter in Equation 7.37 defining conditions for deposition Dimensional parameter in Equation 7.36 defining conditions for erosion Lower bound of Dimensionless constant parameter in statistical of local bed stresses Upper bound of Dimensionless statistical variable Angle in general Dimensionless variable in Equations 3.8 and 3.9; Karman’s constant in Equation 4.68a Constant in Equation 3.31; eddy scale in general Eddy microscale Dynamic viscosity in general Dynamic viscosity of a Bingham fluid
Greek Symbols
m s V a b bi bl e f i, j p s sb sb sv v w a max 1 2 3 as b b′ b* b bc bd bch bmin
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Dynamic viscosity of any liquid Micron Dynamic viscosity of suspension Microvolt Kinematic viscosity Dimensional variable in Equation 3.8; exponent in Equation 4.125; experimental exponent in Equation 7.45 Water density; density in general aggregate density Buoyant density of aggregates (Figure 4.8) Buoyant density of order i aggregates Buoyant density of primary fine particles Local electric energy charge in Equation 3.3 Floc density Density of i and j order particles Density of any liquid Density of solids Density of solid particles; density of suspension in Equation 5.5 Dry density of bed sediment Average value of sb Density of sediment suspension in viscous sublayer in Equation 6.27 Fluid density in the viscous sublayer Water density Total electric charge per unit area of particle surface; inter-aggregate cohesive force Pseudo-surface tension of aggregates Maximum tensile stress on aggregates Standard deviation for the degree of retention; electric charge in Stern layer Geometric standard deviation for the rates of deposition; electric charge in the Stern layer Exponent in Equation 7.30 Shear stress or shear strength in general Time averaged value of in turbulent flows Aggregate strength Bed shear stress Bed shear stress due to the resistance of bed grains and aggregates Dimensionless value of b in Figure 6.8; ratio of b/ bmin Average bed shear stress erosion and resuspension Critical boundary shear stress in Equation 6.5 Bed shear stress below which sediment deposition starts Characteristic value of b in Figure 7.53 Minimum bed shear stress at which sediment can be held in suspension and below which all suspended sediment deposits
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bo b50 b′ bc bi c cav B ch d dc i m o o o* r s sm w y h as ϕd s v d o δ λ Ψ1 i
Greek Symbols
Bed shear stress for initiation of resuspension in figure 7.50 Value of b for 50 percent deposition of depositable sediment Bed shear stress due to surface friction Critical boundary shear stress Bed shear stress during deposition in ith step Critical bed shear stress for erosion and resuspension Average shear stress over boundary of rotating annular channel-ring system Bingham yield strength Shear stress in rotating annular channel Deposition shear stress in general Critical bed stress for complete deposition in Equation 7.7 Bed shear stress for zero sediment flux in Equation 6.26; shear stress at the interface of an underflow in Figure 5.3 Bed shear stress during mixing time Local instantaneous shear stress in bed; bed shear stress in open laboratory channel Average value of o Dimensionless value of o in Figure 6.8 Shear stress on rotating ring Erosive shear strength of bed Highest erosive strength of a deposited bed in Figure 7.50 Wall shear stress Yield strength of sediment suspension Atomic valence Kolmogorov’s velocity microscale Angle of internal friction Total volume fraction of solids within the aggregate in Equation 4.146 Dimensionless value of d Volume fraction of aggregates in suspension in Equation 4.143 and in Equation 4.144 Total volume fraction of aggregates in Equation 4.129 Electric potential at a distance x from the particle surface Electric potential at midpoint between two particles Electric potential at particle surface Stern electric potential Kolmogorov’s time microscale Dimensionless function of η in Equation 4.130 Angular velocity; dummy variable Algebraic rate of increase of i order aggregates within control volume
About the Author Emmanuel Partheniades received his first degree in Civil Engineering from the Technical University of Athens, Greece in 1952, his MS degree in 1955 from the University of California at Berkeley, and his Ph.D. in 1962 in Hydraulic Engineering also from the University of California at Berkeley. From 1952 to 1954, and from 1955 to 1959, Professor Partheniades worked for various consulting firms both in Greece and in the San Francisco Bay Area in structures, irrigation, and soil Mechanics before pursuing an academic career in Hydraulic Engineering. He began his teaching career at the California State University at San Jose (1962–1963), which was followed by a three-year post-doctoral fellowship and assistant professorship at the Massachusetts Institute of Technology in Cambridge, Massachusetts (1963–1966). After a three year appointment at the Department of Civil Engineering at the State University of New York at Buffalo, he joined the faculty at the University of Florida in Gainesville as a Professor in the Department of Coastal and Oceanographic Engineering. In 1974 he became a faculty member in the Department of Engineering Science, which is now part of the Department of Mechanical and Aerospace Engineering, from where he retired in 1997 and where he is currently a Professor Emeritus. From 1973 to 1983 he held the position of Chair of Hydraulic Structures in the School of Engineering of the University of Thessaloniki in Greece while continuing his research at the University of Florida, where he returned in 1983. In all these universities he has taught many undergraduate and graduate courses in introductory and advanced fluid mechanics and hydraulics, sediment transport, hydraulic models, estuarine hydrodynamics, waves, coastal and harbor engineering, water supply engineering, and river engineering. His research interests and activities lie primarily in the areas of the hydraulics of cohesive sediments and estuary hydrodynamics, specifically in shoaling processes and salinity intrusion. He has also conducted extensive research in stratified flows and to a lesser extent in turbulence and jet dynamics He is author of three books in Greek and the author of chapters of three other books in English. He is also author and co-author of over 60 papers, conference presentations in proceedings, and technical reports. He has consulted for the United Nations in India and Argentina and participated as invited lecturer in a number of international short courses and Cohesive Sediments in Open Channels Copyright © 2009
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About the Author
seminars. In addition to his professorship in Greece, his foreign activities include a three year involvement in Venezuela from 1963–1966 while at MIT in the investigation of the hydraulic aspects of the Maracaibo Estuary and in Paraguay in 1967, where he consulted at the University of Asuncion. He is a member of the American Society of Civil Engineers, the Technical Chamber of Greece and an International estuary study group.
Author Index
A
Abdel-Rahman, N. M., 5, 180 Abraham, G., 285 Allen, F. H., 303 Allen, W., 107, 108, 109, 110 Anderson, A. G., 1, 244 Ariathurai, R., 95, 288 Ayora, A., 10, 209, 220
B
Bagster, D. F., 149 Bakhmeteff, B. A., 107, 108, 109, 110 Barkley, W. A., 151 Beasley, R. P., 5, 179, 180, 182 Berghager, D., 5, 180 Bingham, E. C., 155, 159 Bolt, G. H., 75 Bosworth, R. C. L., 203 Burban, P. Y., 124, 135, 138, 223
C
Camp, T. R., 116, 132 Casagrande, A., 82 Clark, M. M., 149 Cohen, J. M., 146 Cross, R. H., 10, 209, 220
D
Delichatsios, A Department of Public Works, State of California., 277 Department of Scientific and Industrial Research., 303, 304 Dixit, J. G., 6, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243 Dobbins, W. E., 92, 93 Du Boys, M. P., 4, 175 Du Buat, L. G., 4, 174, 175 Dunn, I. S., 5, 179
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E
Einstein, A., 156 Einstein, H. A., 1, 93, 114, 155, 156, 157, 158, 159, 160, 166, 173, 193, 194, 197, 220, 244, 246, 251 El Samni, S. A., 194, 197, 246 Espey, W. H., Jr., 5, 180 Etchceverry, B. A., 4, 174, 175 Etter, R. J., 6, 10, 92, 209, 212, 213, 214, 218, 220, 284
F
Flaxman, E. M., 5, 178 Fortier, S., 4, 174, 176 Freolich, U. K., 79, 82 Friedlander, S. K., 134, 141, 142, 143 Fuchs, N. A., 117, 120
G
Glasgow, L. A., 150, 151 Gouy-Chapman, 50, 59, 75, 76, 77 Graf, W. H., 3, 175 Grass, A. J., 221 Grissinger, E. H., 5, 182
H
Hannah, S. A., 146 Harleman, D. R. F., 182, 281, 285, 286, 293, 297, 298, 299, 302 Harris, J. W., 293, 295, 299, 301 Hayter, E. J., 95, 288 Hess, F. C., 15, 16, 19 Hinze, J. Q., 107, 112, 114 Hogg, R., 151 Hoyer, R. P., 6, 10, 92, 209, 212, 213, 214, 218, 220, 284 Hozumi, H., 119, 126, 127, 140 Hsu, J. P., 150 Hunt, J. R., 135, 143, 145, 146, 148, 149, 151
347
348
Hunter, R. J., 151 Huysentruyt, H., 271, 272 Hwaya, Y., 116, 124, 138, 139, 140, 151, 152, 220
I
Iacobellis, S., 124, 135, 136, 137, 138, 223 Inglis, C. C., 303 Ippen, A. T., 281, 285, 286, 293, 297, 298, 299, 302
J
Jenkins, D., 148 Johnson, J. W., 1, 244
K
Kaufman, W. J., 148 Kennedy, J. F., 6, 10, 92, 209, 212, 213, 214, 218, 220, 284, 308, 313 Kennedy, R. G., 4, 176 Klimpel, R. C., 151 Kline, S. J., 221 Koga, K., 116, 124, 138, 139, 140, 151, 152, 220 Krank, K., 320, 321 Krone, R. B., 6, 10, 93, 95, 119, 121, 122, 151, 152, 155, 156, 160, 161, 163, 164, 165, 166, 167, 168, 170, 173, 183, 184, 185, 191, 203, 204, 205, 206, 230, 254, 288, 315, 316, 317 Kusuda, T., 116, 124, 138, 139, 140, 151, 152, 220
L
Ladd, C. C., 5, 180 Lamb, H., 107 Lambe, T. W., 81, 82 Lambermont, J., 198 Lane, E. W., 4, 174, 175 Langmuir, I., 67, 85 Lebon, G., 198 Leentvaar, J., 150 Leliavsky, S., 4, 175, 176 Levich, V. G., 95, 99, 107, 115, 116, 117 Li, H., 114, 193, 220 Lick, J., 124, 135, 138, 223 Lick, W., 124, 135, 136, 137, 138, 223 Liou, Y. D., 277 Low, P. F., 79 Lumley, J. L., 107, 112, 113, 114
Author Index
M
MacLaughlin, R. T., 89 Manley, R., St. J., 141 Martin, A. S., 79 Mash, F. D., Jr., 5, 180 Mason, S. G., 141 Matsuo, T., 149 McAnnaly, W., 6, 10, 209 Mehta, A. J., 6, 10, 197, 209, 210, 211, 215, 216, 217, 218, 219, 220, 223, 224, 225, 226, 227, 228, 230, 231, 232, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 253, 254, 255, 256, 257, 258, 259, 260, 261, 263, 264, 266, 267, 268, 269, 270 Milligan, T. J., 321 Mirtskhulava, Ts. E., 181 Mitchell, J. K., 15, 25, 26, 27, 28, 29, 30, 31, 34, 35, 36, 37, 38, 40, 42, 44, 48, 49, 50, 51, 54, 78, 79, 80, 81 Mitchener, H., 271, 272 Moon, R. H., 83 Moore, W. L., 5, 180
O
Overbeek, J. Th. G., 59, 60, 61, 65, 66, 69, 73, 74, 77 Owen, M. W., 265
P
Paaswell, R. E., 4, 12, 174, 176, 177, 270 Parchure, T. M., 6, 209, 254, 261, 262 Parker, D. S., 148, 149 Parsons, J. G., 265 Partheniades, E., 4, 6, 10, 12, 24, 25, 28, 29, 30, 33, 50, 72, 87, 88, 92, 93, 95, 165, 174, 176, 177, 183, 185, 186, 187, 188, 189, 190, 191, 197, 198, 207, 208, 209, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 223, 224, 225, 226, 227, 228, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 246, 247, 250, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 263, 264, 266, 267, 268, 269, 270, 280, 281, 282, 283, 284, 285, 286, 290, 297, 300, 303, 304, 306, 307, 308, 309, 310, 311, 312, 313, 314, 316, 317, 318, 320 Peele, T. C., 270 Poisson, S. D., 51 Pollock, R. J., 151 Probstein, R. F., 151
349
Author Index
Q
Quinn, M. J., 82, 83, 84, 85, 86
R
Reblun, M., 150 Rectoric, R. J., 180 Reynolds, W. C., 221 Robeck, G. G., 146 Rosenquist, I., 79 Rosillon, R., 218, 230 Runstadler, P. W., 221
S
Schoklitsch, A., 4, 175, 277 Schraub, F. A., 221 Scobey, F. G., 4, 174, 176 Shen. H. W., 245 Simmons, H. B., 293, 294, 295, 296, 299 Smerdon, E. T., 5, 179, 180, 182 Smith, D. K. W., 150 Sundborg, A., 5, 178 Swift, D. I., 134, 141, 142, 143
T
Tambo, N., 117, 119, 122, 123, 124, 126, 127, 129, 130, 131, 132, 133, 134, 135, 140, 148, 151 Task Committee on Erosion of Cohesive Materials, 5 Taylor, G. I., 107 Tennekes, H., 107, 112, 113, 114 Terzaghi, K., 79, 82 Thomas, D. G., 146
Thorn, M. F. C., 265 Tomi, D. T., 149 Toorman, E., 271, 272 Torf, H., 271, 272, 277 Tsai, C. T., 124, 135, 136, 137, 138, 223
U
Unno, H., 149 U.S. Department of Interior, 5, 178
V
Van Leussen, W., 103, 105 Van Olphen, H., 12, 13, 25, 28, 29, 31, 32, 36, 39, 49, 50, 55, 56, 59, 61, 63, 64, 65, 67, 68, 69, 70, 71, 75, 76, 77, 85 Verwey, E. J. W., 59, 60, 61, 65, 66, 69, 73, 74, 77 Vold, M. J., 146 Volkenborn, C., 218, 230
W
Watanabe, Y., 117, 122, 123, 124, 129, 130, 131, 132, 133, 134, 135, 140, 148, 151 Whitman, R. V., 82 Wicker, C. F., 300, 302, 303
Y
Yeh, H. W., 209, 254, 257, 259 Yeh, H. Y., 209, 254, 257, 259
Yoruzu, H., 116, 124, 138, 139, 140, 151, 152, 220
Subject Index 1:1 layer minerals, 31 2:1 layer minerals, 31
A
Acceleration, 13, 73, 127 Activity, physicochemical, 41, 303 Aerosol behavior, 141 Aggregate properties: flow variable control, 118–28 normalized volume distribution, 142, 145, 146 quasi-steady-state distribution, 2, 128–40 research work, 140–52 Aggregates: counterion exchange, 75 density, 87, 104, 118, 120, 122, 124, 138, 139, 152, 162, 234 density function, 123, 124 disaggregation, 149, 221 double layer, 49 floc fabric, 81–8 formation, 2, 85, 106, 140, 146, 152 generation rate, 128–9 hydrodynamic interaction with deposited beds, 244–52 internal structure, 81–8 sensitivity, 80–1 size distribution, 2, 118, 134, 135, 143, 233 strength, 124–5, 149, 223 thixotropy, 80–1 turbulent flow distribution, 146 water phase, 78–80 Aging effect, 183 Allophane, 42 ALT ratio, 122, 123, 124, 132 Aluminum octahedral units, 28, 30 Amorphous minerals, 18, 42 Annular channel and ring system, 209, 210, 222, 223, 231, 253, 289 Annular flume and ring system, 237, 255–6, 258 Antecedent water content, 182–3 Artificially compacted bed erosion, 180 Cohesive Sediments in Open Channels Copyright © 2009
Atchafalaya Bay mud, 235, 236, 239 Atomic nature, 2 Atomic number, 16 Atomic structure, 15, 16 Atomic weight, 16 Attapulgite, 41 Atterberg limits, 6, 11, 178, 179
B
Barking Reach, 303 Basic clay minerals, 25–42 Bed density distribution, 261–2, 263 Bed load function, 93, 173, 184, 244, 246, 249, 252 Bed preparation conditions, 254 Bed shear stresses: consolidated cohesive soil erosion, 174 degree of deposition, 222, 273 deposited sediment resuspension, 252 deposition in open channels, 109, 175, 220, 235 rates of deposition, 273 shoaling in estuaries, 278–93 soft cohesive sediment deposit erosion, 183–200 Bed strength, 269–70 Bentonite, 38, 39, 43 Beryl, 25 Binary collision, 151 Bingham yield strength, 160, 201 Biological aggregates, 151 Biotite, 27, 40, 41 Black Deep, 303, 305, 306 Bohr model, 15 Boltzmann distribution, 50 Bonding mechanisms, 14 interatomic bonds, 14–22 primary bonds, 14–22 secondary bonds, 22–4 Boston blue clay, 180 Bravais classification, 25, 26 Brownian diffusion coefficient, 95, 99, 100
351
352 Brownian motion, 13, 72, 73, 95–100, 103, 105, 141, 143, 144, 146 Brucite, 25, 30, 35, 37, 41 Building blocks, 28, 30, 31, 32, 140 Buoyancy, 123, 128, 149
C
Calcite, 25 Canal Larrazabal, 307 Capacity of double layer, 55 Capillary viscometer, 159, 161, 163, 164, 165, 169 Cation bonds, 23–4, 33 Cation exchange capacity (CEC), 36, 40, 41, 48, 57, 75, 165, 223, 224 degree of deposition, 223 fine particles and aggregation, 75 isomorphous substitution, 36, 48 kaolinite structural units, 33 suspensions, 223 CEC, see Cation exchange capacity Cementation, 24, 80 Characteristic length, double layer, 53 Charge, double layer, 47–9 Chemical cementation, 24 Chemical weathering, 43 Chemicals and pollution, 320 Chemistry concepts and definitions, 20–2 Chlorites, 41 Classes, of sediments, 1–2 Clay deposit formation, 43–5 Clay micelle, 49, 50 see also Double layer Clay minerals, origin and occurrence, 43–5 Clay packets, 83, 84 Clay particles: forces, 47 mineralogy, 24–42 nature of, 24–42 orientation, 45, 182 Clay solution, 13 Clay suspension, 13, 201 Coagulation, 73, 95, 130, 138 Coagulation time, 72, 99 Coarse grains, 2, 246, 252, 305 Coastal navigable waterways, 275 Cohesive sediment deposits: floc fabric, 81–8 general properties, 11–14 hydrodynamic interactions, 244–52 internal structure, 81–8 microstructure, 87–8 resuspension, 252–71
Subject Index
suspended aggregates, 244–52 Cohesive sediment dynamics, 275 Cohesive sediment suspensions: basic properties, 11–14, 156–69 Bingham yield strength, 155, 160, 169, 201 cation exchange capacity, 165 cylinder viscometer, 165–6 density, 155, 156, 162, 169 estuarine muds, 162 property evaluation methods, 156–69 rheological properties, 155 salinity, 160, 161, 163, 164 shear strength, 155, 164 shear stresses, 157, 158 viscosity, 155, 156, 169 yield stresses, 155, 156, 160, 169 Cohesive soil erosion: consolidated soil, 174–83 soft sediment deposits, 183–200 Collisions: agglomeration factor, 127, 128, 130, 131 Brownian motion, 95–100 differential settling, 103–6 frequency, 2, 98, 103 turbulent flows, 116–18 velocity gradients, 100–3 Colloidal solution, 13 Colloidal suspension, 47, 81, 146 Concentration distribution, 21–2 chemistry concepts and definitions, 20–2 Maracaibo Estuary shoaling, 280 rates of deposition, 93, 225 shoaling in estuaries, 280 yield shear strength, 155 Concentration-time relationships, 225, 226, 227, 228, 231, 260 Consolidated cohesive soils, 174–83 Consolidation time, 254, 258, 260, 264, 267, 268, 269, 270 Constant charge density, 55–6 Constant electrolyte concentration layers, 63 Constant surface potential, 56, 62 Contact flocculation, 126, 127 Couette type flocculators, 146 Counterions, 49, 50, 75, 76 Covalence/covalent bonds, 18–20 Critical bed shear stress, 5, 174, 183, 221 Critical boundary stresses for erosion, 178, 180 Critical velocity, 5, 174, 178 Crustal movements, 44 Crystal structure, 28, 32, 39 Cubic system, 25 Cylinder viscometer, 165–6, 167, 168
Subject Index
D
Definite proportions, law of, 20 Deflocculated sols, 14 Degree of deposition, 208–24 Degree of retention, 215, 216, 217 Delaware estuary, 293, 300, 302 Densimetric Froude number, 285, 286, 298 Density: aggregate properties, 141, 171 cohesive sediment suspensions, 14 deposited cohesive sediment resuspensions, 252 deposition in open conduits, 234 quasi-steady-state aggregate distribution, 128 stratification, 282, 292 Deposited cohesive sediments: floc fabric, 81–8 general properties, 11–14 hydrodynamic interaction of suspended aggregates, 244–52 internal structure, 81–8 microstructure, 87–8 resuspension, 252–71 Deposition: degree of deposition, 208–24 deposited beds, 244 geological cycle, 45 hydrodynamic interaction of suspended aggregates, 244–52 open channel design, 275 parameter relationships, 272 phases, 287 rates of deposition, 224 shoaling in estuaries, 278 soft cohesive sediment erosion, 183–200 Development of cohesive sediment behavior, 3–6 Differential settling, 95, 103–6 Diffused ionic layer, 50 Diffusion equation, 94 Dioctahedral structure, 30 Dipole moment, 19 Disintegration, 33, 38, 89, 150, 180, 193 Dispersion, 13, 82, 94, 151, 289, 291 Distilled water, 14, 71, 211, 215, 216, 217, 222, 225, 226, 227, 228, 255–6 Distinction of sediments, 1–3 Double layer: charge, 47–9 constant charge density, 55–6 constant surface potential, 56, 62 electric charge, 47–9 electric force, 49 Gouy-Chapman theory, 59
353 interaction between flat surfaces, 59–75 surfaces of constant charge density, 55–6 surfaces of constant potential, 55 theoretical formulation, 49–59 Drag forces, 233, 246 Dredging, 305, 308, 309, 318, 320, 321 Driller’s mud, 38–9
E
Eddies, 111, 112, 113, 114, 117, 119, 220, 221 Effective density, 123, 124 Einstein’s approach, 244, 246 Electric charges, 47–9, 55 Electric force, 49 Electric potential, 51, 54, 55, 59, 60 Electrochemical nature, 47 Electronic energy, 16 Electrons, 15, 16, 17, 18, 19, 23 Electrophoresis, 47 Electrovalent (ionic) bonds, 17–18 Emulsions, 142, 143, 147 Energy: dissipation, 108, 112, 113, 114, 115–16, 126, 127, 152 double layer concentrations, 59, 64 hump, 71, 72, 73 turbulent flows, 114, 151 Engineering applications, 275 Environmental pollution, 275, 319–21 Equations, hydrodynamic transport processes, 89–95 Equilibrium concentrations, 187, 207, 212, 214, 216, 236, 239, 272 Equivalent weight, 20–1 Erosion: cohesive soils, 173 consolidated cohesive soils, 174–83 deposited cohesive sediment resuspension, 221, 252–71 particles from aggregates, 148 rates, 116, 171, 185, 203, 252–3, 263, 265 soft cohesive sediment deposits, 183–200 Estuaries: cohesive sediment suspensions, 155 fine sediment transport, 7, 114, 279–93 Maracaibo Estuary, 306–15 mud, 163 River Delaware Estuary, 300–3 River Thames Estuary, 303–6 San Francisco Bay Estuary, 315–17 sediment transport, 173, 279 shoaling, 7, 10, 71, 183, 278–93 Evaluation methods, 156–69
Subject Index
354
F
Fabric, 45, 80, 81, 84, 183 Failure, 11, 24, 121, 175, 179, 180, 181, 193 Ferric oxide, 150 Fine particles and aggregates: counterion exchange, 75 sensitivity, 80–1 thixotropy, 80–1 water phase, 78–80 Fine sediments: shoaling in estuaries, 278 transport in estuaries, 279–93 Flat double layers: constant surface potential, 62 energy interactions, 59–62 force interactions, 59–62 interaction, 59–62 potential energy, 64–5, 68–70 Flocculation: aerosol behavior, 141 aggregates, 140–52 collision rates in turbulent flows, 116 concentration, 74 distinction of sediments, 2 double layer, 47, 49 dynamics, 95–106 flocculation concentration, 74 flow variable control, 124, 126, 127 hydrodynamic transport, 89, 91 mean deposition time, 231 mineralogy, 14 particle arrangement, 81–7 quasi-steady-state aggregate distribution, 130, 132, 134, 137, 138, 139 standard deviation, 243 total potential energy, 70–5 Flocculation process, 134, 140, 141, 142, 153, 193 Flocculent content, 156, 157 Flocs: aggregate density, 104, 121, 124, 135 distinction of sediments, 2 fabric, 81–8 microflocs, 126, 127 particle arrangement, 81–7 strength, 146 Flow-induced shear stresses, 4, 45, 84, 106, 155, 173, 177, 179, 287 Flow variable control, 4, 118–28 Fluff, 287, 305 Fluid mud, 206, 287, 288, 305 Forces: clay particles, 47
distinction of sediments, 2 flat double layer interaction, 59–64 friction, 2, 246, 247 hydrodynamic interactions, 244 interaggregate cohesive force, 195 lift, 181, 246, 247, 248 resisting, 14, 246, 247 suspended aggregates, 149, 244 tractive, 4, 5, 175, 178 Van der Waals, 23, 33, 68 Formation, clay deposits, 43–5 Free energy, 64, 65 Fresh water flocculation, 84 Friction forces, 2, 246, 247 Friction velocity, 113, 115, 199, 220 Frictional resistance, 107, 108 Froude number, 285, 286, 297, 298, 299, 302
G
Gallions Reach, 303 Gapon equation, 49 Gegenions, double layer, 49 Gelation, 85–6 General properties, 11–14 Generation rates, 128–9 Geological cycle, 43, 44–5 Georgia, 293, 294 Gibbsite, 27, 35, 37, 39 Gold, 317 Golden Gate, 315 Gouy-Chapman theory, 50, 59, 75–8 Gouy layer, 50 Grain size distribution, 1, 2, 212, 214, 234–5, 315 Gravesend Reach, 306 Grown flocs, 126 Gulf of Venezuela, 306, 308, 309, 310, 311, 315
H
Halfway Reach, 303 Halloysite, 33, 36, 37 Hexagonal system, 25 Hindered settling, 91, 147, 203, 206, 230 Hydration, 78 Hydraulic behavior, 3, 6, 8, 9, 156 Hydraulic engineering applications, 275 Hydrodynamic interaction, 244–52 Hydrodynamic transport, 89 flocculation dynamics/processes, 95–106 governing equations, 89, 319 Hydrogen bonds, 23, 33, 78–9, 80
Subject Index
I
Illites, 39, 40, 41, 183, 257 Imogolite, 41 Impulse-momentum equation, 13 Inert elements, 17 Inertia sublayer, 114 Interaction: flat double layers, 59–75 hydrodynamic, 244–52 potential energy of interaction, 64–5, 68–70 suspended aggregates, 244–52 two flat double layers, 59–75 Interaggregate bonds, 171 Interaggregate cohesive force, 195 Internal structure, 81–8 Ionic agglomerates, 18, 19 Ionic bonds, 17–18, 19 Ionic distribution, 50, 52 Ions, 49, 68, 76 exchangeable octahedral units, 28, 30 Iron oxides, 24, 80, 190 Isometric systems, 25 Isomorphous substitution, 32, 33, 36, 48
K
Kaolinite, 27, 28, 33–4, 35, 36–7, 41, 48, 140, 235, 237, 238, 239, 253, 255–6, 257, 260, 263, 264, 266, 267, 268, 269, 270 degree of deposition, 208–24 deposited cohesive sediment resuspension, 252–71 deposition in open channels, 234 normalized volume distribution, 145 rates of deposition, 224–34 structural units, 33 Kennedy’s regime theory, 4, 175, 176 Krone’s deposition theory, 203
L
Lake Maracaibo, 306, 307, 309, 311, 312 Laminar sublayers, 193 Lattice of the crystal, 25 Lattice points, 25 Law of definite proportions, 20 Lift forces, 181, 246, 247, 248 Liquefaction, 80
M
Magnesium octahedral units, 28, 30 Maracaibo Estuary, 280, 281, 282, 293, 306–15, 318 Marine organisms, 320, 321
355 Maximum aggregate size, 89, 116, 124, 126, 128–40 Mean deposition time, 231 Mechanical composition, 189, 244, 303 Micas, 39, 40, 41 Microflocs, 126, 127 Microorganisms, 321 Microscales, 112, 113, 115, 117, 149, 151 Microstructure, 9, 10, 82, 87–8, 193, 209, 279 Migration, shoaling zones, 297 Mineralogy: clay particles, 24–42 Molality, 22 Molarity, 22 Mole, 20, 21 Monoclinic system, 26–7 Montmorillonite structural unit, 34–42 Moscovites, 39–40 Mud Reach, 303, 305–6
N
Nature of clay particles, 24–42 Navigable waterways, 1, 6, 183, 275 Net sediment discharge, 1 Neutrons, 15, 16, 20 Newtonian fluids, 155–6, 157 Noncrystalline materials, 42 Nonexpending three-layer clays, 39 Non-salt flocculation, 82 Normality, 22 Normalized aggregate size variations, 135 Normalized volume distributions, 145 Null point, 281, 283–4, 285, 286, 287, 297, 298, 299, 306, 309, 318, 319
O
Occurrence, clay minerals, 43–5 Octahedral units, 30 1:1 layer minerals, 31 Open channel design, 275, 276–8 Open conduits, 234–44 Open flume erosion experiments, 179 Organic matter, 80, 184, 185, 303, 320–1 Organisms, 321 Origin, clay minerals, 43–5 Orthorhombic system, 26 Oswald viscometer, 159 Oxides, 21, 42
P
Pacific Ocean, 315 Parallel flat plates, 66
Subject Index
356 Particles: arrangements, 81–7 coagulation, 73 from aggregates, 148 total potential energy, 70–5 see also Clay particles; Fine particles and aggregates Peptized sol, 14 Peptizing ions, 49 Percent of composition, 22 Phlogopite, 40 Photography, 150, 181 Physicochemical forces, 2, 3, 14, 130, 182, 320 Physicochemical properties, 11, 106, 200, 217 Plasticity, 11 Pollution, environmental, 275, 319–21 Porosity, 86, 120, 151, 152 Port of Savannah, 293 Potential constant, 62 Potential energy, 56, 64–5, 68–75 Power laws, 184, 236 Preferential adsorption, 48–9 Pressure, 22, 43, 67, 87, 157, 160, 287 Primary bonds, 14–22 Probability: density function, 233 deposition, 248 erosion, 247 Properties: cohesive sediment deposits, 11–14 cohesive sediment suspensions, 11–14, 155 evaluation methods, 156–69 physicochemical properties, 7, 11 turbulent flows, 106–18 see also Aggregate properties Protons, 15, 16, 20, 23, 78
Q
Quartz, 25, 151 Quasi-periodic built-up, 221 Quasi-steady-state distribution, 128–40 Quick clays, 80, 81
R
Radioactive gold, 207, 317 Radius of coagulation, 96 Rare earth elements, 17 Rates: deposition, 3, 10, 93, 94, 173, 224–34, 273, 318 energy dissipation, 112, 113, 115–16, 126, 127, 149, 151, 152, 193, 219–20 erosion, 197, 253, 259
Regime theories, 4, 175 Relative density, 141, 285, 286, 298 Relative equilibrium concentration, 212, 214, 215, 216, 217, 234, 236, 237, 239, 272 Remolded clay samples, 179 Repulsion, 47, 77 Repulsive potential, 66, 67, 71 Resistance to erosion, 84, 176, 178, 182, 183, 185, 188, 190, 192, 198, 277 Resisting forces, 14, 246, 247 Resuspension, 203, 252 Reynolds stresses, 110 Rheology: aggregate distribution, 128–40 aggregates, 146 cohesive sediment suspensions, 121, 146, 155 flow variable control, 4, 118–28 properties, 148, 151 turbulent flows, 146 River Delaware Estuary, 293, 300, 302 River Thames Estuary, 303–6 Rock cycle, 43 Rock weathering, 43, 83 Rotation-induced secondary currents, 210 Royal Albert Docks (London), 303, 304
S
Sacramento, 315 Safety against deposition, 278 Safety against scouring, 276–7 Salinity: cohesive sediment suspensions, 160, 161, 164 estuaries, 71 Maracaibo Estuary, 280, 306, 307, 309, 311 Savannah Estuary, 295, 297, 298, 299 shoaling, 71, 164, 285 wedge, 279 Salt water, 82, 84, 85, 86, 170, 217, 229, 230, 231, 258, 260, 261, 268, 279, 280 San Carlos-Zapara entrance, 307 San Francisco Bay Estuary, 293, 314, 315–17, 318 San Francisco Bay mud, 160, 161, 163, 173, 184, 197, 203, 217, 218, 223, 230, 231, 255 San Joaquin, 315 Saponite, 37, 39 Savannah Estuary, 293–300, 301, 302 Savannah River, 294, 295 Scale of eddy motion, 111 Schultze-Hardy rule, 74 Scouring, 10, 43, 173, 174, 176, 179, 185, 187, 188, 191, 192, 201, 275, 276–7
Subject Index
Secondary bonds, 22–4 Sediment importance, 1, 155 Sediment sink function, 94, 95, 288, 292 Sediment source function, 94, 95, 288, 289, 292, 317 Sediment transport, 8, 89, 93, 94, 245, 279, 289–90, 313, 314 Self-preservation aggregate size distribution, 142 Sensitivity, 80–1 Serpentine group, 35, 37 Settling suspended aggregates, 244 Settling time, 221, 232, 233, 241, 243 Settling velocities, 90 deposition, 2, 104, 176 flow variable control, 118–28 open conduits, 234, 235 quasi-steady-state aggregate distribution, 128 Severe failure, 180 Shear: aggregate properties, 141, 143, 144 deposition, 209, 210 flow rates, 100, 157, 159, 186, 278 resistance, 4, 175 strength, 11, 28, 80, 118, 121–2, 155, 159, 163, 164, 170–1, 179, 181, 185, 253, 260, 263, 269, 273 Shear stresses: aggregate size distribution, 135, 143, 233 cohesive sediment suspensions, 11, 155 consolidated cohesive soil erosion, 174–83 degree of deposition, 208–24 deposited cohesive sediment resuspension, 252–71 deposition in open channels, 109, 175, 220, 234, 235 flow variable control, 118–28 rates of deposition, 224–34 shoaling in estuaries, 278 soft cohesive sediment erosion, 183–200 turbulent flows, 106–16 Shells, 15, 16, 17, 18 Shoaling: estuaries, 275, 278–93 navigable waterways, 275 rates of deposition, 93, 224 River Delaware Estuary, 300 River Thames Estuary, 303–6 salinity: intrusion-controlled, 284, 285, 286, 287, 300, 319 tide-controlled, 284, 287, 306, 312, 315, 318, 319
357 San Francisco Bay Estuary, 315, 317 Savannah Estuary, 293–300 silica tetrahedral units, 28–30 Simple elements, 17 Size distribution: aggregates, 133, 134, 135, 141, 142, 143 degree of deposition, 1–2 deposition in open channels, 5, 231 distinction of sediments, 1–3 flow variable control, 118–28 quasi-steady-state distribution, 128–40 Slacking, 180 Smectite structural unit, 37–9 Soft cohesive sediment deposits, 183–200 Sol, 13, 14, 47 Solutes, 22 Solutions, 21–2 Solvent, 21 Specific area, clay minerals, 36 Sphere of influence, 97, 98, 101, 102, 104 Stability, 1, 4, 70, 173, 174, 175, 181, 275 Standard deviation, 194, 195, 217, 218, 232, 234, 243, 251 Steady-state constant concentration, 207 Stern layer, 75–8 Stern Potential, 76 Stirred vessels, 149 Stokes equation, 104 Straits of Maracaibo, 306 Strength: aggregates, 119, 153, 156, 169, 207 bed strength, 180, 257, 258, 259, 260, 269–70 Bingham yield, 160, 163, 201 deposited cohesive sediment suspensions, 11 flocs, 119, 146 shear, 11, 12, 80, 118, 121–2, 146, 155, 164, 165, 170–1, 179, 181, 185, 189, 243, 260, 263, 269, 273, 287 soft cohesive sediment deposits, 185, 186, 188–9, 192, 193, 198, 199 yield strength, 81, 86, 155–6, 157, 160, 163, 198–9 Stresses: critical, 179, 180, 181, 198, 201, 292 Reynolds, 110 tensile, 194, 195 time-averaged, 193 turbulent flows, 106–16 yield, 149 see also Bed shear stresses; Shear stresses Structure, see Internal structure; Microstructure Submerged weight of sedimentation unit, 248 Surface activity, 24, 36, 37
Subject Index
358 Surface forces, 2, 192, 317 Surfaces of constant charge density, 55–6 Surfaces of constant potential, 55 Suspended aggregates, 244–52 Suspended sediment concentration, 14, 84, 184, 185, 186, 187, 190, 191, 192, 231, 253, 254, 255, 256, 260 Suspensions: basic properties, 156 general properties, 11 photography, 150 rheological properties, 155
T
Tablazo Bay, 306, 308, 311, 315 Tactoids, 85 Tap water, 263, 264, 266, 267, 268, 269, 270 Temperature, 3, 43, 201, 235, 320 Tensile stress, 194, 195 Tetragonal system, 26 Tetrahedral units, 28–9 Theoretical formulation, double layer, 49–59 Thickness, double layer, 53, 55, 56, 59 Thixotropy, 80–1, 86 Three-layer minerals, 31, 32 Tidal motion, 289 Tilbury Basin/Docks, 304, 305 Time averaged shear stress, 107, 193, 220, 223 Time concentration relationships, 204 Time-deposition plots, 239, 240 Time history, 222, 295, 297 Time variations, 136, 184, 254–5 Tractive forces, 4, 5, 175, 178 Transport, 44 fine sediment, 7, 114, 279–93 hydrodynamic, 89 rates, 1, 2, 244, 245, 313, 314, 315 shoaling in estuaries, 278–9 Triclinic system, 27 Trioctahedral structure, 30 Turbulent flows: aggregate distribution, 118 collision rates, 116–18 fundamental properties, 106 parameters, 106–16 rheological behavior, 146 stresses, 106–16 velocity, 109, 111 2:1 layer minerals, 31 Two flat double layer interaction, 59–75
Two-layer minerals, 31, 32 Two particle potential energy, 68, 69, 70–5
U
U.K. (England), 303 Uniform dense beds, 187, 189 Unit cells, 25, 28, 30–1, 35 Universal equilibrium theory, 112
V
Valance number, 17 Valence, 17, 48, 74 Van der Waals bonds/forces, 23, 33, 36, 68, 71 Velocity: critical, 174, 176, 178 degree of deposition, 210–13, 220–2 flow variable control, 118–28 friction, 108, 113, 115, 199 gradients, 95, 100–3, 104, 105, 106, 109, 114, 115, 116, 152, 155, 283 Maracaibo Estuary, 281, 282, 311–12, 315 Savannah Estuary shoaling, 294, 295, 298 shoaling in estuaries, 278–93, 281 soft cohesive sediment deposit erosion, 184, 186–200 turbulent flows, 106–18 see also Settling velocities Venezuela, 280, 281, 282, 306, 308, 312, 313 Vermiculities, 39, 40, 41 Vertical salinity, 279, 280, 295, 311 Viscosity, 79, 80, 86, 112, 157, 161, 169 Viscous sublayers, 114, 117, 193, 194, 220, 221 Viscous subrange, 113, 116 Volume, 96, 105, 125, 130, 141, 142, 145, 147, 148, 319
W
Wash load, 244, 245, 246, 249, 252 Water phase, 78–80 Waterways Experiment Station, 234, 285 Waves, 8, 43, 278, 317, 320 Weathering, 43, 44, 80, 83 Weight per unit volume, 21 Wind generated waves, 317
Y
Yield strength, 81, 86, 155, 156, 157, 163
Yield stresses, 147, 148, 149, 153
