Analytical solutions of geohydrological problems

PREFACE

1. Design and scope of the book Analytical solutions of geohydrological problems consist of mathematical formulae by which certain state variables such as heads and drawdowns of groundwater, concentrations of a solute, etc., are given as functions of space and time, and in which geohydrological parameters and geographical boundaries related to the problem have been incorporated at the same time. As compared to digital results of computer models, these formulae are very attractive, because they enable us to calculate all kinds of information related to heads, velocities, streamlines, traveling times of water particles, spreading of solutes, etc., by means of implementing mathematical manipulations such as differentiation and integration. Though, in general, complicated problems are not accessible for analytical solution methods, it almost always proves possible, after more or less strong simplification, to describe the problem by an analytical formula, thus giving a valuable insight in to the problem. On the other hand, analytical solutions may be also applied to rather intricate problems, as the potential for a successful numerical treatment of complicated formulae has considerably increased with the use of modem computers. We may find a further application of analytical formulae in the analysis of pumping tests and in the use in certain computer models, which are based on analytical formulae. All analytical formulae have been obtained by solving linear differential equations with initial and/or boundary values. About twenty five to thirty years ago, when my interest in analytical solutions arose, only a small number of them were available, notably solutions of ordinary differential equations. Though, at that time, solution methods for partial differential equations were sufficiently known and, for instance, in the theory of conduction of heat applied frequently, they were strangely enough not used for geohydrological problems. Then, after my attention has been drawn to the Laplace transformation method for solving partial differential equations and after having studied the existing and new integral transformation techniques, in the course of time the number of possibilities for me to find new analytical solutions increased considerably. The consequence was that publishing of the new results separately in scientific periodicals was put aside and, thus, the idea was born to collect the solutions and record them in a book. As a result of a large number of solutions, the classification of the problems according to certain characteristics had to be set up, while it also became necessary to design a kind of search-route in order to find the solution of a problem in the book more easily. The latter resulted in the use of determination tables and multiple choice questions similar to flora for the determination of plants. Composing of this determination book took much more time than it was anticipated as the number of problems to which I could find analytical solutions grew

viii

Preface

obviously due to the fact that I became increasingly skillful in applying various transformation techniques. Every new solution became, as it were, the startingpoint for a number of other solutions and I found it a challenge to continue on solving new problems analytically. However, a line had to be drawn somewhere in order to waste no more time and to keep the size of the book within limits. This line was drawn, somewhat arbitrarily, at about 1100 problems; the numbering of the problems, however, is such that new solutions, if desired, may be easily inserted without influencing other numbers. Another consequence of the obtained routine in solving partial differential equations was that I hardly found time to study literature as it proved less time consuming for me to solve a problem than to undertake a laborious search for an eventual solution of that problem in the world literature. Therefore, among the many solutions in this book, there will undoubtedly be solutions which were published somewhere before, but escaped my attention. To those people concerned I offer, in advance, my sincere apologies and invite them herewith to send me their reactions. Once the size and contents of the book were fixed, the question arose if, instead of giving only the base solutions of the various problems, it were also desirable to mention and explain the solution methods by which the solutions had been obtained. From several sides it was insisted on this completion because many users of the analytical formulae were not only interested in their digital conversion, but also in the way they had been realized. Therefore, the chapter "Analytical solution methods" was added. However, in order to understand the explanation of these methods, good knowledge of basic mathematical principles of geohydrology is essential, with the result that the chapter "Basic principles for saturated groundwater flow" was also written and included in the book. The derivation of several differential equations for the state variables such as head, drawdown, pressure, value of the stream function, concentration of a solute, etc., commands a central position in this chapter. Finally, the chapter "Functions", in which a number of mathematical functions frequently used in the solutions, completes the book. All this resulted in the division of the book into two parts: Part A "Solutions" comprising the determination book with the analytical solutions, and Part B "Mathematical tools" consisting of three chapters that describe basic principles, solution methods, and mathematical functions, respectively.

2. Acknowledgements I am indebted to Mr. G. Santing who taught me the first principles of geohydrology and to Professor G. de Josselin de Jong, who drew may attention to the Laplace integral transformation method and, thus, laid the foundation for my plan to write this book. I am also grateful to the two institutes, at which I worked: the National Institute of Public Health and Environment (RIVM), and its predecessor the Government Institute for Drinking Water Supply, that they gave me the opportunity to undertake scientific work, with minimal interruption by the usual reorganizations that took

Preface

ix

place. Especially my thanks are due to Professor C. van den Akker, chief of the Geohydrological section of the RIVM at the time just before my retirement, for his support with regard to the realization of this work. In the case of an analytical treatise such as this, it is probably in vain to hope that no mistakes, clerical or other, have remained undetected; nevertheless, the number of such mistakes has been considerably diminished by the criticism and the vigilance of my ex-colleague Mr. E.J.M. Veling whose labour to remove errors and obscurities has been of the greatest value, for which I thank him sincerely. Last, but not least, I wish to compliment the Lithuanian-Netherlands joint venture VTEX, bureau for typesetting services in Vilnius, on the admirable way in which they succeeded in transforming a handwritten manuscript of great mathematical complexity with more than a thousand illustrations drawn by hand into a conveniently arranged typewritten copy.

LITERATURE The following books have been studied:

Mathematical books Abramowitz, M. and Stegun, I.A. (1965). Handbook of Mathematical Functions. Dover, New York. Churchill, R.V. (1972). Operational Mathematics. McGraw-Hill Kogakusha, Tokyo. Gradshteym, I.S. and Ryzhik, I.M. (1965). Tables of Integrals, Series and Products. Academic Press, New York. Kreyszig, E. (1968). Advanced Engineering Mathematics. Wiley, New York.

Geohydrological books Bear, J. (1972). Dynamics of Fluids in Porous Media. Elsevier, New York. Polubarinova-Kochina, P.Ya. (1962). Theory of Groundwater Movement. Princeton Univ. Press, Princeton, NJ.

Special subjects Bessel-functions McLachlan, N.W. (1934). Bessel Functions for Engineers. Clarendon Press, Oxford. Watson, G.N. (1966). A Treatise on the Theory of Bessel Functions. Cambridge Univ. Press, Cambridge, UK. Solutions in heat conduction Carlslaw, H.S. and Jaeger, J.C. (1959). Conduction of Heat in Solids. Clarendon Press, Oxford.

X

Preface

Rotational flow Rouse, H. (1938). Fluid Mechanics for Hydraulic Engineers. McGraw-Hill, New York.

Multi-layer systems Hemker, C.J. and Maas, C. (1981). Unsteady Flow to Wells in Layered and Fissured Aquifer Systems, Journal of Hydrology, Vol. 90.

G.A. Bruggeman

Introduction

7

Introduction to Part A

For finding the relevant formulae in the labyrinth of available analytical solutions of geohydrological problems (about 1100), a logical classification of the various aspects of these problems is crucial. Since many ways of classification are feasible, however, the search for the class containing the problem studied is not straightforward. Thus, marking a route leading to the solution of a problem will be helpful. This idea is worked out in a series of multiple choice questions like in a flora, in use for the determination of plants. Question groups are formed and numbered. On the basis of an affirmative answer to a question, the other questions in the group can be skipped and one is referred to the next relevant group by a number. This will eventually lead to a detailed specification of the problem and a reference to its analytical solution. In general, each question is followed by a short explanation since the properties of the geohydrological problems considered were not defined uniquely by the questions. Sometimes a sketch is given as well. All solutions relate to groundwater problems in the saturated zone. The main classification used reads: A. B. C. D. E.

Phreatic groundwater Confined groundwater Multi-layer systems Dispersion Density flow (sharp interface)

10-100 100-700

700-800 800-900 900-1000

The bulk of analytical solutions are in class B. Therefore, this class is subdivided into six subclasses each characterized by the type of differential equation needed for description: BI. One-dimensional flow BII. Two-dimensional radial-symmetric flow B III. General two-dimensional flow BIV. Three-dimensional spherical flow BV. Three-dimensional axial-symmetric flow BVI. General three-dimensional flow

100-200 200-300 300-400 400-500 500-600 600-700

The numbers on the right side refer to the numbered solutions. The determination book begins with an orientation table for the main classification which leads to one of the above mentioned classes. Each class or subclass, in general, has been subdivided into groups, based on certain characteristics of those groups. For instance, subclass BV for three-dimensional axial-symmetric flow consists of six groups BV-1 to BV-6, using differences in boundary conditions. BV-1 corresponds with the solutions having 51 as initial digits (i.e. 510-519), BV-2 with the solutions numbered 520-529 etc. In general, each class or subclass has its own orientation table which leads to the groups, which on their turn may be divided into solution groups, having numbers consisting of three digitals. So, group BV-3

8

Introduction

has four solution groups, numbered 532-535. Finally the problems the solutions of which are looked for have as basic number the number of the solution group together with their own serial number; for example 533.01,533.02, etc. A problem is always described by a name, a sketch, its boundary conditions and if relevant an initial condition. The differential equation, governing the problem is almost always given for the whole solution group. The solution of a problem, the begining of which is denoted by an asterisk (.), is given by a potential function and if possible also as a stream function. Directly behind the number of the problem the solution method, applied to that problem, is given in code between ( ) . A review of the abbreviations of the solution methods used for this code is given hereafter. Abbreviations of solution methods

Abbr.

Solution Method

C D F FC FCR FH FHR FS FSR H HO I IC IFR IH IS J L M MI MS P PF

Conformal Transformation Direct Integration Forchheimer Approximation Finite Cosine Transformation Finite Cosine Resistance Transformation Finite Hankel Transformation Finite Hankel Resistance Transformation Finite Sine Transformation Finite Sine Resistance Transformation Herzberg Approximation Hodograph Method Method of Images Infinite Cosine Transformation Infinite Fourier Resistance Transformation Infinite Hankel Transformation Infinite Sine Transformation Joukowski Transformation Laplace Transformation Matrix Analysis Method of Impulses Method of Simplification Product Solutions (method o f . . . ) Periodic Flow Solutions Quadratic Transformation Superposition Schwarz-Christoffel Transformation Separation of Variables Variation of Parameters

Q S SC SV VP

8

Introduction

has four solution groups, numbered 532-535. Finally the problems the solutions of which are looked for have as basic number the number of the solution group together with their own serial number; for example 533.01,533.02, etc. A problem is always described by a name, a sketch, its boundary conditions and if relevant an initial condition. The differential equation, governing the problem is almost always given for the whole solution group. The solution of a problem, the begining of which is denoted by an asterisk (.), is given by a potential function and if possible also as a stream function. Directly behind the number of the problem the solution method, applied to that problem, is given in code between ( ) . A review of the abbreviations of the solution methods used for this code is given hereafter. Abbreviations of solution methods

Abbr.

Solution Method

C D F FC FCR FH FHR FS FSR H HO I IC IFR IH IS J L M MI MS P PF

Conformal Transformation Direct Integration Forchheimer Approximation Finite Cosine Transformation Finite Cosine Resistance Transformation Finite Hankel Transformation Finite Hankel Resistance Transformation Finite Sine Transformation Finite Sine Resistance Transformation Herzberg Approximation Hodograph Method Method of Images Infinite Cosine Transformation Infinite Fourier Resistance Transformation Infinite Hankel Transformation Infinite Sine Transformation Joukowski Transformation Laplace Transformation Matrix Analysis Method of Impulses Method of Simplification Product Solutions (method o f . . . ) Periodic Flow Solutions Quadratic Transformation Superposition Schwarz-Christoffel Transformation Separation of Variables Variation of Parameters

Q S SC SV VP

0

~ o o~

~ 0

.~~

~

.~ "In o~

-'~

i°

~g 0 rm

~o

~. o

r~

0

0

~

.~ 2

~,

o

0

o~

0 T//o

t-i

~

Classification of groundwater in the saturated zone

0:I

0

e r~ ~I

o~ 0

0

0

o~

/

o

0

ell

r~

0

0 0

9

10

(1-3)

Main orientation table

MAIN ORIENTATION TABLE

1. Groundwater flow in the saturated zone

a. Is inhomogeneous groundwater involved in your problem?

go to 2

Groundwater is considered as inhomogeneous if the concentration of some solute in the groundwater is not the same everywhere in the field or with time. Also problems concerning several kinds of groundwater, each of them of the same composition, but mutually different, belong to this category.

b. The groundwater in your problem may be considered as homogeneous

go to 5

Groundwater may be looked upon as homogeneous if there is no variation of a solute concentration with space and time at all, or if the variation is small and may be neglected. 2. Inhomogeneous groundwater

a. Are you interested in the groundwater flow together with the transport of the solute ?

go to 3

If the problem is a quality problem rather than a quantity problem then information as to the behaviour of the solutes, i.e. the concentrations as functions of space and time as a result of the groundwater flow, is necessary. This will happen for instance in many salt-fresh water problems and in groundwater pollution.

b. Are you only interested in the groundwater flow?

go to 4

In certain cases the behaviour of the solutes is less important than the influence of the inhomogeneity of the groundwater on the flow pattern, as is, for instance, the case with two kinds of water separated by a sharp interface. The problem then should be treated as a quantity problem. 3. Solute transport

a. Do you want to take into account only the convective transport of the solutes and neglect dispersion?

go to 4

As soon as transport of a solute in groundwater occurs, the phenomenon of dispersion is encountered. In general, taking into account the dispersion, will complicate analytical calculations considerably. For that reason it may be neglected in the first instance to receive a rough insight in the transport problem. Ignoring dispersion means that the assumption is made that a certain groundwater particle always maintains the same solute concentration (neglecting adsorption, solution and chemical reactions). The method of

Main orientation table

(3-5)

11

moving fronts is now available for the calculation of the transport of the solutes. b. The dispersion must be taken into account

go to 800 (D)

For a more rigorous approach of the transport problem the dispersion should be taken into account. In general, the differential equation with boundary values for the dispersion can only be solved if the flow pattern is known or simultaneously with the differential equation for the flow.

4. Only convective solute transport a. Will the varying solute concentrations result in a varying density ?

go to 900 (E)

Varying density of the groundwater with space and time may effect the flow considerably; in that case the flow is referred to as density flow. The density variation may be continuous or discrete. Continuous variation of the density occurs for instance in the transition zone between fresh and salt water, or in a heavily polluted area of the groundwater. A discrete variation happens if two kinds of groundwater with different densities (f.i. salt and fresh water) are assumed to be separated by a sharp interface.

b. No density variation owing to varying solute concentrations

go to 5

If there is no density variation as a result of varying solute concentrations, or if the (small) variations hardly affect the flow pattern, the groundwater may be considered as homogeneous for solving the flow problem.

5. Homogeneous groundwater a. Is phreatic groundwater involved in your problem?

go to 6

Phreatic groundwater is open to the atmosphere and contains the phreatic surface or water table, where the pressure equals the atmospheric pressure. In general, a phreatic aquifer is not covered by layers with big resistance to groundwater flow.

b. The groundwater is confined or semi-confined

go to 7

Confined groundwater flows in an aquifer that is bounded by impermeable layers. The piezometric head in every point in the aquifer lies above the upper side of the aquifer, which is called a confined aquifer. If at least one of the sides of the aquifer is bounded by a semi-permeable layer, the aquifer is called semi-confined.

12

(6, 7)

Main orientation table

6. Phreatic water

a. Are the expected fluctuations of the water table small compared with the thickness of the phreatic aquifer?

go to 7

In contrast with a confined aquifer, a phreatic aquifer has a thickness, that depends on the height of the (fluctuating) water table. If the fluctuations are small (for instance, less than 10%) compared with the average height of the water table, the phreatic aquifer may be considered as being confined and be treated as such. Afterwards a check must take place whether the assumption of small fluctuations was acceptable.

b. The water table fluctuations cannot be neglected compared with the thickness of the aquifer

go to 10 (A)

The thickness of the phreatic aquifer is a function of the height of the water table. The boundary condition along the phreatic surface is much more complicated than that in a confined aquifer and in many cases the differential equation differs from that of confined water. For these reasons special calculation methods must be applied for solving phreatic problems. 7. Confined and semi-confined groundwater

a. Does your groundwater flow system consist of more than one layer?

go to 700 (C)

A groundwater flow system will be referred to here as multi-layer system if it consists of more than one aquifer separated by one or more semi-permeable layers and if it is assumed that in one or more of the aquifers only horizontal flow takes place and in the semi-permeable layers only vertical flow. Then the differential equations for each aquifer also contain state variables (heads or drawdowns) of adjacent aquifers, thus making a simultaneous treatment of the differential equations necessary.

b. There is only one layer

go to 8

By "one layer" must be understood one aquifer in which the groundwater flow takes place. One or two semi-permeable layers may be included, provided that the surroundings (adjacent aquifer or surface water) have fixed water level (head) fluctuations. Also aquifers that consist of more layers of different permeability belong to this category. In general, only one differential equation with boundary values have to be solved, or if more equations are necessary as in the case of different permeabilities, they need not be solved simultaneously.

Main orientation table

(8)

13

8. One-layer systems a. Has your problem a general three-dimensional character?

go to 600 (BVI)

Here and in the sequel the word "dimensional" is related only to the dimensions of space, not to that of time. If the flow has components in all three coordinate directions x, y and z and thus the variables of state, such as the piezometric head, the pressure, the velocity components are all functions of x, y and z, then the flow is supposed to have a general three-dimensional character, provided that no axis or point of symmetry is present and that the flow is not a simple superposition of two or more symmetrical flow patterns. For instance, three point sources in an infinite field yield a general three-dimensional flow, but is the superposition of three spherical-symmetric cases. Concerning the calculation of the drawdowns it therefore belongs to the category of 8c, but in view of calculations of travelling times of water particles, for instance, it must be considered as general three-dimensional. A differential equation for general three-dimensional flow is

02(t9

02999 02(t9

F (x , y, z)

Ss 0999

This is the three-dimensional differential equation for the potential head 93 for non-steady flow of homogeneous groundwater through homogeneous and isotropic ground. This differential equation forms the basis of all analytical solutions. In special cases heterogeneity of the ground can be taken into account if it has a discontinuous character. b. Is your problem three-dimensional with axial symmetry?

go to 500 (BV)

If three-dimensional flow is symmetric with respect to an axis, which means that in all planes, passing through this axis, the same flow pattern presents itself, the flow is referred to as axial-symmetric. The several variables of state are now functions of r and z if the z-axis is the symmetry axis and r equals v/X 2 -+- y2. The general differential equation for the head or drawdown in axial-symmetric flow runs as follows: 0299

1

0(t9

Or---T + -r -gr +

02(19

F (r, z)

2 + ~ X

Ss 093 = X

at

e. Is your problem three-dimensional with spherical symmetry?

go to 400 (BIV)

Spherical symmetry in three-dimensional flow occurs if the flow is symmetric with respect to a point, which means that all straight lines, passing through that point, possess the same distribution of the variables of state. If the point

14

(8, 9)

Main orientation table

of symmetry is chosen as the origin of the coordinate system the variables of state are only functions of one variable p - x/x 2 + y2 + z2. The general differential equation runs: (head or drawdown) (P02=- -Jr- 2 O~p -~- ~F ( p ) -- Ss a~p

7V

K

d. The flow may be described with less than three dimensions

go to 9

9. One- and two-dimensional flow a. The flow is general two-dimensional

go to 300 (BIII)

The flow has components in two directions and will be referred to as general two-dimensional if no point of symmetry exists and if the flow is not a simple superposition of two or more symmetrical cases. The independent variables are chosen x and y in the case of mere horizontal flow and x and z for two-dimensional flow in vertical cross-sections of the aquifer. Differential equations"

02(.t9

02(t9

F (x , y)

Ox----g+ ~y2 +

K

Ss O~p = K at

and 0299 _

02(/9

OX 2 ~_ ~

_qt_

F(x, Z) Ss O~0 K = K at

b. The flow is two-dimensional with radial symmetry

go to 200 (BII)

If there consists a point of symmetry in two-dimensional flow, which means that along all straight lines passing through that point the variables of state have the same distribution, the flow is called radial flow. If the point of symmetry is chosen as the origin of the coordinate system, the variables of state are only functions of r where r - - V/X 2 -I- y2 or r -- ~/x 2 + Z 2. Differential equation for the head or drawdown: 02q9 ~__10~p - + -F(r) ~ =

Ss 0~o

---g Or - r -~r

K at

K

c. The flow is one-dimensional

go to 100 (B I)

The flow is unidirectional and can be described by one variable of space which will be x if horizontal flow takes place and z in the case of vertical flow. Differential equation for the drawdown: 02~o

F(x) .

OX 2

Ss O~p

~

K

K at

Orientation table A

(10, 11)

15

A. PHREATIC FLOW ORIENTATION TABLE A

10. Phreatic water

a. Is it allowed to assume only horizontal flow?

go to 11

As exact water table calculations in general are rather complicated and also not always lead to solutions, it is worth while to look for approximate solutions, at the same time maintaining the character of phreatic flow. The latter condition means that the flow profile is not constant but depends on the height of the water table (see comments of 6). Considerable simplifications of the calculations can be obtained by assuming only horizontal flow, which in general will be allowed if the vertical boundaries of the flow pattern are fully or almost fully penetrating the phreatic aquifer, for example:

~Q

b. Also vertical flow has to be taken into account

q¢=

go to 12 (A1)

If it is not allowed to neglect the vertical flow, for instance, in the case of flow to a horizontal drain (see example), then exact phreatic solutions have to be determined.

11. Horizontal flow

a. The flow is one-dimensional

go to 20 (A2)

See comments of 9c.

b. The flow is two-dimensional with radial symmetry

go to 30 (A3)

See comments of 9b.

c. The flow is general two-dimensional See comments of 9a.

go to 40 (A4)

16

( 12)

Phreatic flow

12 A-1. EXACT SOLUTIONS IN PHREATIC FLOW Only very few exact solutions for groundwater flow in phreatic aquifers are available and they are limited to two-dimensional steady flow, in general. Almost all solutions, given in the sequel, are obtained by the method of conformal mapping and therefore will satisfy the Laplace differential equation: 02(/9 OX 2

02(/9

+

= 0.

OZ2

12.01 (HO,Q)

z ~___

Flow of phreatic groundwater through a broad dam towards a horizontal drainage area. q9 = q)(x, z) = head.

phr. line

99(x,0)=0

,-o ............................~

X

--0

or

forx ~<0, 099 ~(x,0)=0 Oz

forx > 0 .

Phreatic line (surface): (/gph --" Zph, 1/rph --- q.

withS2--¢+iO

S2--ff2Kq( -- K f / 9 - ~ / ~ 1

( x 2 + z 2) 1/2 + x

q(q

andg-x+iz,

}1/2, ~P - f f ~ / ( x2+ z2)1/2 _ X }

1/2

)

Zph-- "~ -~- +2Xph •

12.02 (SC)

Inundation of a semi-infinite field and infiltration along a finite vertical screen of length 1. q9 = qg(x, z) = head.

q) Z

h

99(x,O)=h

x l

d

aq)(O z ) - O Ox

'

forx ~ 0 , for - I
Phreatic line: q)ph = Zph, ~ph -- O.

Exact solutions

X?--iK

( 12) h+d

~ / w - l + Kh

h ( - -- arccosh w

h +d/w

(= q5 + i~), _ 1

(=x+iz),

7/"

l--

(h+d) 2-

Jr arccos {

--~

17

2h / zr(h + d)/"

Z p h - - h - ( h + d) cosh(JrXph). 2h

12.03 {SC} cp z

Horizontal drain with discharge q [L2T -1] in an "infinitely" thick phreatic aquifer. 99 -

Dhr.line

q9(x, z) - head. Drain at the origin.

,h q¢=

099 ~ ( 0 , Z) -- 0

K

Ox

for Z --/=0.

Phreatic line: (/gph- Zph, lCrph- ~.

X2-- q l n w + K h , 2rr q ln2)~/w + 1 + ih, 7rK

(--i q--~ln(~/w+l+l)-i(h+ rrK

Xph- (h + rcK q l n 2 ) ¢ e x p { 2rcK(zph-h)]q q arccosexp[ rrK

( --x +iz.

-1

7rK(Zph- h)} q

12.04 (HO,SC)

~o

Horizontal drain at the bottom of a finite phreatic aquifer, cp(x, z) - head.

phr. line J

Drain with discharge q [LZT -1] at the origin. m

v

x q

GOa__(0 A., ' Z) -- 0 Ox

for Z :/: O,

~(x,0)--0

for x :/:0.

Oz

18

(12)

Phreatic flow

Phreatic line" ~0ph- Zph 1 / r p h - q2" (=q~+i~p),

-- q l n s i n h w + K h yf

2q ~-~22nB2nw2n+l(2n)V(2n+l) 2 ( (-

7rZKn= 0

.

+--zr h

q ) zrKe w

(:x+iz),

Bn -- Bernoulli number, e - 0.693147180546. q Zph --

zrK

In cosh t + h

t >/0 (parameter),

2(

2q 2 2n (2 2n -- 1) B2n t2n+ 1 -1- __ h X p h - jr2K n=l (2n)!(2n + 1) 7r

q e)t. rrK

12.os (sc)

Horizontal drain in an "infinitely" thick phreatic aquifer with precipitation, q9 =

,~,~ ,~,~,~,~,~,~,~,~+,~,,~,,~,~p

99(x, z) = head. Drains with discharge q at (+2nb, O) (n = 0, 1, 2 . . . . ) with q = 2pb. Original water table at z = h.

~

co0__L,(0, z) - 0

= O~J

Ox

b

~(b,

Ox

z) - O,

qg(x, --cx~) -- h,

P h r e a t i c line: ~Oph -- Zph, 1/rph(X,

£2 - P--b-bIn w -

rc

Z) --

K > p.

p(b - x).

2Kpb In (~/w + 1 + x/e + 1) r r ( K - p)

2p2b In (~/w + 1 + 1) + K H , +Jr(K-p) b 2pb ( = i - - l n ( w - e) + i In (~/w + 1 + 1) Jr z r ( K - p) 2Kb -i In ( 4 w + 1 -+-4e+ 1 ) + b + i H , :rr(K - p) H

zr(K - p)

In(4)

Jr(K - p)

ln(" +l 1)

~/e + 1 + 1 '

for z :/: 0,

Exact solutions

(12)

19

e from

2Kpb re(K-p)

h -- P---~blne7r

+

2p2b zr(K - p)

Zph --

rr(K-

In (x/e + 1 + 1)4- K H .

In

p)

Xph - - 7/"( K -

In (2~/e + 1)

+ H,

t + e

arctan

p)

2pb arctan (~/t - 1 ). 7 r ( K - p)

+ 1

12.06 (SC)

Z

9=H

q q

l

~

,~o=H

,

I

l I

I

I

2b

]

q

i

2B

r

~o= H

---- •

I

9=H q

2b

-~

-i

Combination of infiltration canals and drains in an "infinitely" thick aquifer.

, qb=KH

!#=q/2

=0

Drains with discharge q and equal distances 2b midway the infiltration canals with widths 2B and levels H. Phreatic line: ~ p h - Zph ' 1 / r p h - q2' ~0(x,H)=H for(b-B) ~<x ~
7,=0 b

O(b, z) --0,

7*(O, z) --

q

for z > 0,

0

for z < 0,

2

S2-- q l n ( ~ / w + a - ~ a ) + K H 2Jr ,/ w + a + ,/a

~(x,-oo)

(-- ~b + i ~ ) ,

- O.

20

Phreatic flow

(12)

_ (~/w+a.+__~_a~ i K ~ - - q In Jr

+

~/w+ 1 + 1 }

Kb

(~/d + l + ~/w + 1) In

( ( -- x + i z ) ,

~/d-+- 1 - ~ / w - + - 1

a from: b-

B -- - - arctan ~/a - 1 tanh

Jr

JrK

2Kb

lna)}]

KH

arctan ~/a - 1,

d from: ~/d+l-coth

Xph

-

Jr ( K H

2Kb

- - q} . 2 jlrn a )

-2b - arctan v / t - 1 Jr d + 1

JrK

q Zph -- 2jr K

arctan ~/t - 1,

~/-a + ~/a -- t

Potential at infinity" ¢(x

12.07

'

-ec) - KH -

/d + a + 4 rd )

q In

2jr

~/d + a -

(C)

zI i 4,,~,1,,1,,[,4,,I,,1, ,I,4,,[,,1,,l,,1,,[,,[,4,,1,4,,i,,1,,i,,~,1,~,1,,1, P I

= p(B-x) i

~ ~

Phreatic flow in a dam with width 2B and constant precipitation p. Outflow to a horizontal drainage area. Phreatic line:

7~= o x

,

~2 + Kp(2 _ 2 K p B ( -- O,

2 P Zph -- -~-Xph(2B -- Xph) @

( -- x + iz. p2B2

K(K-p)

qgph = Zph,

O(B,z)=0,

1/rph -- p ( B - x),

¢(x,0)=0

forx~<0,

~p(x,0)=0

/'orx >~0.

Exact solutions

(12)

21

12.08 (D)

~

~ •

•

•

~

~

o

~

~

~

P

•

•

•

~

Constant precipitation on a phreatic aquifer which is separated from an underlying aquifer with a constant head h by a semi-permeable layer. Steady state.

•

""H

".'"

dqo Kz -dz - p -

(p(O)-h c

(/gph - - Z p h .

~

~

c

~o=h P ~o(z) -- -;-;-. z + cp + h,

Zph - -

K z ( c p + h)

H

--

(Kz > p).

Kz - p

12.09 (D) Non-steady state of 12.08: Zph --

,

t

f(t)

or

t

Initial value: Zph(0)

-- g(Zph).

--

h.

m

Kz - p

H

-

-

Zph

with K z ( c p + h) H

_._.

ne --effective porosity or specific yield.

Kz - p

12.10
H

Zph(t). Zph

~///////////~

gph(0)

--

U//////////2 c

~o=h

t --

he{

K---~z (h + cKz) In

(H h) Zph -- h

-+- H - Zph

}

•

H,

gph(OO)

-- h.

22

Phreatic flow

(20, 21)

20 A-2.

ONE-DIMENSIONAL HORIZONTAL PHREATIC FLOW

a. Is the phreatic aquifer in your problem confined?

go to 21

If the groundwater flow in the aquifer is confined at the upper side by a water table and at the lower side by an impermeable layer, we may speak of a confined phreatic aquifer.

b. The phreatic aquifer of your problem is semi-confined

go to 22

Example of a leaky phreatic aquifer: The phreatic aquifer is confined at the upper side by a water table and at the bottom by a semi-permeable layer. It is assumed that the leakage of water through the semi-permeable layer L --does not affect the piezometric head C in the underlying aquifer (due, for instance, to big transmissivity of that aquifer).

21. Confined phreatic aquifers with horizontal one-dimensional flow Subdivision: a. Problems without precipitation or infiltration. b. Problems with precipitation or infiltration.

a. Problems without precipitation

d(h~) dx

- O,

h = head.

21.01 (ED) Flow through an infinite strip bounded at both sides by open water, h = h(x) = head.

h(O) = H,

b

h2 - H2

_

x

_i(i-i

_

h(b) = L.

One-dimensional flow

(21 a)

23

21.02 (ED) to

Flow through an infinite strip bounded at one side by open water and at the other side by open water with entrance resistance, h - h ( x ) head. h(O)-

dh --(b)dx

H,

b h2 _ H2 -- ~ x{H 2-h2(b)}

-

h(b)-

L

Kw

with

b L + v/b2L2 + K w H 2 ( 2 b + K w ) h(b) -2b + K'w 21.03 (ED) =~q

.

Y

Abstraction of water from an infinite well-screen in the middle of an infinite strip of width 2b and open water boundaries at both sides, h -h (x) = head. h(0)-

888888g~

h(2b)-

H,

dh h(b)--;-(b) -fix

b H

2K

for 0 ~< x ~
q K

~ X

h2=

q

H 2 -- q ( 2 b - x) K

f o r b ~ < x ~<2b.

21.04 (ED) =~q i

m

i

i | ! i i i

Abstraction of water from an infinite well-screen in the middle of an infinite strip of width 2b and open water boundaries with entrance resistance at both sides, h = h (x) = head. dh h"0' dx "0'

gg88~

h 2 _ h2(b)-

1

q (b-x)

1

h(b) - -~ H + -2v / H 2 - 2 q w

with H 2 >/2qw.

Minus sign: unstable situation. If H 2 < 2 q w no steady state possible.

q 2K'

dh dx

(b)

H - h(b) Kw

24

Phreatic flow

(21 b)

b. Problems with precipitation d (hdh p =0. dx -~x ) + -K 21.11 (ED) Flow, caused by precipitation through an infinite strip of width 2b, bounded at both sides by open water with equal level, h = h(x) = head.

dh dx (0) -- O,

h(b)

H.

b P 2 - x2). h 2 -- H 2 q-- -~(b

,

21.12 (ED} ,1,,1,,I,,1,,1,,1,,1,,1,,i,,1,,I,,i,,1,,1,,1,,1,,1,,i,P

Flow, caused by precipitation through an infinite strip of width 2b, bounded at both sides by open water with equal entrance resistance and level. h -- h (x) = head.

dh dx(0)

0,

dh dx(b)

h(b)- H Kw

b P 2 -h 2 -- h2(b) -+- -~-(b

,

x2),

h(b)-

1 + -~ 1 ~/H2 + 4 p b w . -~H

21.13 (ED)

H

H

Flow, caused by precipitation, in an infinite strip of width b, bounded at one side by open water and at the other side by open water with the same level and entrance resistance, h = h (x) = head.

dh dx (b)

h(O) - H,

b

px x { h2 (b) - H 2 } h 2 -- H 2 + -~-(b - x) + -~

with

h(b) -- ~/b2H2 + (2b + K w ) ( K w H 2 - pb2w) + b H 2b-+- K w

h(b) - H Kw

25

(22a)

One-dimensional flow

22. Leaky phreatic aquifers with horizontal one-dimensional flow Subdivision" a. Problems without precipitation. b. Problems with precipitation. a. Problems without precipitation

d(hdh ) H-h =0. dx ~-x -5 K-------~

22.01 (ED) Semi-infinite field with open water boundary at x - O. D r a w d o w n of the surface water level.

h ( O ) - L,

h ( o o ) - H.

~///~////////////////////~ ~ c x -- ~ / 3 K c { F ( h ) - F(L)}

v/3H -5 ~/2u -5 H

In

F ( u ) -- ~

with

- ~/2u -5 H.

v / 3 H - ~/2u -5 H

22.02 (ED) Semi-infinite field with open water boundary with entrance resistance at x - O. Drawdown of the surface water level to L.

W

dh K dx (0)

C

F(h(0))}

x -- ~/3Kc{F(h)-

h(0)- L w

h ((x~)

H.

with h(0) from

K w v / 2 h 3 ( O ) - 3Hh2(0) + H 3 -- h ( 0 ) { h ( 0 ) - L } ~ ' 3 K c .

F(u) as in 22.01.

22.03 (ED) q - - - - 7

.

ii

.

.

.

.

.

.

.

.

.

.

Abstraction of water from an infinite well-screen in an infinite leaky phreatic aquifer, h - h ( x ) -

.

head.

/-/

dh K h (0) -7dx C

q -- 2 '

h ( o o ) - H.

26

(22a, b)

x -- ~ / 3 K c { F ( h ) -

F(h(O))}

2h3(0) - 3Hh2(0) + H 3

Phreatic flow

with h(O) from

3cq 2 = 0 .

4K

F(u) as in 22.01. b. Problems with precipitation

d (hdh) dx

h-H Kc

-~x

P --0. + K

22.11 (ED) ~ , ~ ~ $ ~ 4 ~ ~ ~--.

.

.

.

.

.

.

.

.

.

.

.

P

Flow in a semi-infinite leaky phreatic aquifer towards an open water boundary, caused by precipitation, h - h (x ) - head.

" pc .

.

.

.

h(O)-

H

~~///////////////////~ x--

H,

h ( o o ) -- H + p c .

c

3v/'~7{F ( h ) - F ( H ) }.

F ( u ) as in 22.01 with H + p c for H.

22.12 (ED)

.... 22-~--- 2 H

pc

Flow, caused by precipitation, in a leaky phreatic aquifer towards an open water boundary with entrance resistance, h h (x) - head.

h

dh dx (0) x -- ~ / 3 K c { F ( h ) -

F(h(0))}

h(O)--

Kw

H '

h(cx~) - H + pc.

with h(0) from"

K w v / 2 h 3 ( O ) - 3(H + pc)h2(O) + ( H + p c ) 3 -- h(0){h(0) - H } ~ / 3 K c . F ( u ) as in 22.01 with H + p c for H.

Radial-symmetric flow

(30a)

27

30 A-3. T W O - D I M E N S I O N A L R A D I A L - S Y M M E T R I C H O R I Z O N T A L P H R E A T I C F L O W Subdivision: a. P r o b l e m s without precipitation. b. P r o b l e m s with precipitation.

a. Problems without precipitation d (rhdh

d-7

--~r) - O.

30.01 (ED)

K

Flow through a circular dam with inner radius r l and outer radius r2. Open water boundaries, h = h ( r ) = head, r l ~< r ~< r2.

7

h

r

h(rl) = H,

g88~ rl

r2

h(r2) = L.

ln(~) h 2 _ H 2 _ (H 2 - L 2 ) ~ r2 " In ( r , )

.

30.02 (ED) Flow through a circular dam with innerand outer radius r l and r2, respectively. Open water boundaries with entrance resistance, h -- h(r) -- head,

W2

7

N831ggS" r .... rl

H - hi Wl

rl <~r<~r2.

NfeN

h2-L

dh = - K d-T (rl),

dh ---

w K ~

dr

1132

(r2).

ln(~)

h2 - h21- (h21- h 2 ) ~

In ( rr-~l)'

hi and h2 from

H-hi Wl

h2_h 2 In (r~)

K

2hlrl

and

h2-L 1132

h2_h 2

K

ln( )

2h2r2

28

Phreatic flow

(30a, b)

30.03 (ED)

a

Well in the centre o f a circular island with open water b o u n d a r y , h - h (r) - head. lim ( r h d h ) r-, O

-gr

Q

h ( R ) - H.

- - + 2 ~----~ '

R •

h 2 -- H 2 - - ~ Q In R JrK r ~ o

30.04 (ED)

a

Well in the centre o f a circular island; open water b o u n d a r y with entrance resistance, h - h ( r ) head.

lim rh

--

r-~O

27rK

dh H - h(R) K -d-7 ( R ) -w

R ,

h 2 -- h 2 ( R ) -

Q zrK

R In - -

with

r

2Qw

1 H ± -~1 ~/H2 h ( R ) -- -~

Q<<.

zc R H 2

7rR '

2w

Minus sign" unstable situation. If Q > :rrRH2 2w no steady state possible.

b. P r o b l e m s with precipitation

l d ( dr hh )

r dr

~

+~

p K

-0.

3O.ll (ED) Flow, c a u s e d by precipitation t h r o u g h a circular island with open water b o u n d a r y , h - h (r) head. dh r- (d0 )

,,

P

R ,

h2

_ H 2 -4- ~P- ( R 2 - r2).

- O,

h ( R ) - H.

Radial-symmetric flow

(30b)

29

30.12 (ED) 4,,~4r4,4,,~4,,~,~,~,1,,~,,~,,~ P

Flow, caused by precipitation through a circular island; open water b o u n d a r y with entrance resistance, h - h (r) - head. dh d-7(0) - 0,

dh h(R)- H d-7 (R) - Kw

9

R h 2 _ h2(R

+

P

R 2 - r 2)

1 H + -~ 1 VH 2 + 2PRw with h ( R ) -- -~

30.13 (ED) Flow in a circular d a m with inner radius r l and outer radius r2, caused by precipitation. Open water boundaries, h - h ( r ) - head, r l <~ r ~< r2.

;/

h

h ( r l ) - h(r2) - H. r2

h2 _ H 2 _

p (r2 2K

r2 ) +

p ln r - l n r l 2-K (r2 - r12) lnr2 - l n r l

30

(40-42a)

Phreatic flow

40 A-4. GENERAL TWO-DIMENSIONAL HORIZONTAL PHREATIC FLOW

a. Is precipitation involved in your problem?

go to 48

As it is assumed that only horizontal flow takes place, the vertical precipitation (or infiltration) must be uniformly distributed over the thickness of the aquifer, thus represented as an additional term in the differential equation.

b. No precipitation or infiltration

go to 41

The differential equation for the head h becomes:

0 (h~x)+

Ox

0 (hOh ) - 0 .

7y

Introduction of the "Forchheimer" potential CPF-- ~I Kh 2 yields 02q~F Ox 2

+

02q~F Oy 2

- 0,

4~F thus satisfying the Laplace differential equation. So 4~F is a harmonic function and the real part of a complex function S2F = 4~F + i T~F. The stream function ~PF is the conjugate function of q~F. 41. Problems without precipitation

In view of the similarity of the differential equations for the potentials for confined and phreatic flow, the solution of a problem in phreatic flow will be principally the same as the solution of the corresponding problem in confined flow. Only the boundary conditions will cause some differences. In the following, all solutions for phreatic flow will be compared with the solutions of the corresponding problems in confined flow, making use of general rules for this correspondence which are valid for special groups of problems.

a. Are wells or well-screens involved in your problem?

go to 47

As it is assumed that only horizontal flow takes place, the wells or wellscreens must be totally penetrating, which means in a phreatic aquifer, that the filter lengths must reach from the water table to the bottom of the aquifer.

b. No wells or well-screens

go to 42

42. Problems without wells or well-screens

a. Is uniform flow involved in your problem?

go to 46

Uniform flow is defined as flow in a fixed direction. Unlike uniform flow in

General two-dimensionalflow

(42a-44)

31

a confined aquifer, in a phreatic aquifer the discharge is constant and not the groundwater velocity:

Oh qx -- -- Kh ~ -- q -- constant. Ox b. Flow to and from equipotential planes

go to 43

An equipotential plane may be represented by a more or less fully penetrating canal or water basin.

43. Equipotential planes

a. Is phreatic flow taking place between two planes with different levels? For these problems steady state solutions are available.

go to 44

b. The equipotential plane or planes has or have a given level which differs from the level of the surroundings go to 45 These problems are mostly non-steady, but, in general, tend to a quasi-steady state, where the stream function is defined uniquely, but the potential function contains an arbitrary constant.

44. Equipotential planes with different levels

0 (h~x)+

0 (hOh

7y Ty) - 0 .

For this group of problems the solutions for phreatic aquifers are derived directly from those for confined aquifers by applying the following procedure.

Procedure 1 If both S2 and z are given functions of a complex parameter w: 1 H 2 - ,f2F and h by H2-L2 Replace $2 by 5K 2 , the function z the same.

h=L

z(w) remaining

44.01 ( F ) ( c f 315.01)

Flow between two semi-infinite parallel equipotential planes with different levels, h(x, y) = head.

h { LH .

.

.

.

Derivation from 315.01 according to procedure 1.

fory=0andx>x0, fory=y0andx >0.

32

Phreaticflow

(44)

315.02)

44.02 ( F ) ( c f

Flow between two semi-infinite equipotential planes, making an other, where 0 < ol < 1.

H h--

L

angle arc with

each

for y = 0 and x > xo, for y = x tan(ctzr) + yo and x > O.

Derivation from 315.02 according to procedure 1.

315.03)

44.03 (F)(cf.

Flow between two each other.

H h--

L

semi-infinite equipotential planes

which are

perpendicular

to

fory=0andx>xo, for x = 0 and y > yo > 0.

Derivation from 315.03 according to procedure 1.

323.01)

44.04 ( F ) ( c f

infinite and

Flow between an other.

h-

H L

a

for y - O, fory-yoandx

semi-infinite equipotential plane, parallel

to each

>0.

Derivation from 323.01 according to procedure 1.

323.02)

44.05 ( F ) ( c f

Flow between an infinite and a semi-infinite c~zr with each other, where 0 < c~ < 1.

h_{

H L

equipotential plane

making an

angle

for y = 0, for y = xtan(otJr) + Yo and x > 0 (yo > 0).

Derivation from 323.02 according to procedure 1.

44.06 (F)(cf 323.03) Flow between two

semi-infinite equipotential planes parallel

opposite directions.

h-{

H L

fory=0andx fory=yoandx

>xo, <0(yo>0).

Derivation from 323.03 according to procedure 1.

to each other, but in

(44)

General two-dimensional flow

33

44.07 ( E D ) ( c f 335.01)

Yi

Two open boundaries perpendicular to each other with different levels, h(x, y) = head.

h=L

h(O, y ) = L,

h=H

h(x, O ) = H.

x

1 K S2F - - K H 2 -Jr---(H 2 - L2)i In z.

2

rr

44.08 (F} (cf 335.04) Flow between a straight open boundary and a circular basin with different levels. h = h(x, y) = head.

I

h(0, y) -- H,

.2j x

h ( ~ , y) = h ( x , cx~) -- H, h(xo, Y o ) - - L

1 S2F = - K H 2 2

f o r ( x o - a ) 2 + y 2 < ~ R 2.

K ( H 2 - L 2) i n ( z + ~/a2 -- R 2 ) .

Z - ~/a 2 - R 2

2arccosh(~)

Total discharge: z r K ( H 2 -- L 2) IQI =

arccosh(~)

"

44.09 ( F ) ( c f 335.05) Flow between a infinite and a semi-infinite equipotential plane which are perpendicular to each other. h (x, y) = head.

Y h=L ~Yo

h(0, y ) = L

fory>yo,

h=H X

K

1 K H 2 -+- __(H2

12F- ~

n

Oh

~ ( 0 , y) -- 0

h(x O) - H '

'

(_)

- L2)iarcsin h z Yo "

OX

for y < Yo.

34

(44)

Phreatic flow

44.10 ( F ) ( c f 355.01) Two straight open boundaries, parallel to each other, kept at high level. A third straight open boundary, perpendicular to them kept at low level, h(x, y) -- head. h(x, - b ) - h(x, b) - H.

h(O, y) -- L,

Derivation from 355.01 according to procedure 1. 44.11 (F)(cf. 355.03) As 44.10, but one of the parallel boundaries given a low level, the remaining other two boundaries kept at high level, h - h(x, y) - head. h(O, y) - H,

h(x,-b)-

H,

h(x, b ) -

L.

Derivation from 355.03 according to procedure 1. 44.12 (F)(cf. 355.07) A finite open boundary with low level at equal distance of two parallel infinite open boundaries with high level, h - h(x, y) - head.

for - B <<.x <~ B,

h(x, O) -- L,

h(x, - b ) - h ( x , b ) - H.

Derivation from 355.07 according to procedure 1. 44.13 ( F ) ( c f 364.01) Rectangular strip with 4 open water boundaries. One boundary kept at low level, the others at high level. h~':L

h = h(x, y) = head.

h=H a

h ( - a , y) = L,

x

h(a, y) = H,

h(x, - b ) = h(x, b) = H. -b

_1KH2 _ S2F-- 2 44.14

(F)(cf.

2K( H2 - L2) zr

~ m=l,3,...

(--1) (m-l)/2 sinh{ m~(a-zl2b } m sinh~ mrfa b )

364.02)

As 44.13 but with two opposite boundaries with low levels, the other two boundaries kept at high level, h - h(x, y) - head. h ( - a , y) - h(a, y) - L,

h(x,-b)-

h(x, b ) -

Derivation from 364.02 according to procedure 1.

H.

General two-dimensional flow

(44, 45)

35

44.15 (F)(cf 364.03) As 44.13 but with two adjacent boundaries kept at low level, the other two boundaries kept at high level, h(x, y) = head.

h ( - a , y) -- L,

h(a, y) - H,

h ( x , - b ) = L,

h(x, b) -- H.

Derivation from 364.03 according to procedure 1.

45. Equipotential planes with a level that differs from that of the surroundings O (hOh

ax

-gy-gy

=0.

For this group of problems the solutions for phreatic aquifers are derived directly from those for confined aquifers by applying the following procedure.

Procedure 2 Replace £2 by ~1 K H

2

- $2F and the arbitrary constant c by 21 H

2

- ~1 L 2 .

45.01 (F) (cf 315.12, 323.12 and 334.22)

y

Flow towards two semi-infinite equipotential planes, making an angle of ~Jr with each other, because of sudden lowering the head to L, keeping a head H at a reference point P ( R , loeyr).

P OlY¢

otrr~......~

h:L x

1

S2F -- - K H 2 - 1 K ( H Z 2 Z=x+iy,

I-2F = dpF + i TtF,

45.02 (F) (cf. 315.14) As 45.01 with oe - 32" 45.03 (F} (cf. 323.14) As 45.01 with c~ = 2. 45.04 ( F ) ( c f 334.03) 1 As 45.01 with oe - 7"

tz)-~

1+i

1

dpF "- - K h 2 2

,

36

Phreatic flow

(45)

45.05 (F) (cf 323.15) Flow outside an equipotential plane of parabolic shape and constant level L. Point P(R, 0) reference point for h - H. h - L for y2 _ 4p(p - x). Derivation from 323.15 according to procedure 2. 45.06 ( F ) ( c f 334.23) Flow inside an equipotential plane of parabolic shape and constant level h -- L. Point P(R, 0) reference point for h -- H. h - L for y2 _ 4p(p + x). Derivation from 334.23 according to procedure 2. 45.07 ( F ) ( c f 334.24) Flow towards an equipotential plane of orthogonal hyperbolic shape with constant level h - L. Point P (g1R ~/-2, g1R~/~) reference point for h - H. h - L for x y

g1 a2. Derivation from 334.24 according to procedure 2. 45.08 ( F ) ( c f 334.25) Flow towards an equipotential plane of elliptical shape with constant level h -- L. Point P (R, 0) reference point for h = H.

h--L

x2 y2 for ~-7 + ~-5- - 1.

Derivation from 334.25 according to procedure 2. Total discharge: Replace a by Q. 45.09 ( F ) ( c f 334.26) Flow to afinite equipotential plane of length 2a. Point P(R, 0) reference point for h - - H . h -- L for y -- 0 and - a < x < a . Derivation from 334.26 according to procedure 2. Total discharge" Replace a by Q. 45.10 ( F ) ( c f 354.03)

h=L

L

P(R,O)

Two straight open boundaries, parallel to each other and a third straight open boundary perpendicular to them. The surface water is given a constant low level, keeping a high level at a reference point P. h = h(x, y) = head.

-b

h=L

h(O, y) = h(x, - b ) = h(x, b) = L, h(R, O) = H.

(45, 46a)

General two-dimensional flow

1

S-2F-- -~K

H2

1

+ -~K(H 2

37

_ L2 { sinh(rrz

) sinh(~rR - 1

o

46. O b s t a c l e s in u n i f o r m f l o w

O(hOh

o-;

O(hOh ) --0.

+ Ty

7y

For this group of problems the solutions for phreatic aquifers are derived directly from those for confined aquifers by applying the following procedure.

Procedure 3 1 H 2 -,f2v and q by q. Replace $2 by 2K

Subdivision: a. Obstacles, acting as equipotential planes. b. Obstacles, acting as impermeable planes.

a. Obstacles, acting as equipotential planes 46.01 (F) (cf 324.01)

q

Finite equipotential plane with constant level in uniform flow of strength q, making an angle of oesr with the direction of the equipotential plane. h--Hfory=0and-b<x
h=H x

1KH2 -- q { + V/Z2 -- b 2 cos(an') - iz sin(an') }. $2F- -~ Total discharge through the plane" Q - 2bq. 46.02 (F)(cf 336.02) As 46.01 with ot - 0 . 46.03 (F)(cf 324.02)

Elliptical basin with constant level H in uniform flow with discharge q, making an angle c~rr with one of the main axis of the ellipse. Derivation from 324.02 according to procedure 3. 46.04 (F)(cf 336.03) As 46.03 with o~ - 0 .

38

Phreatic flow

(46a, b)

46.05 (F) (cf 336.01)

h=H

yT

Circular basin with constant level H in uniform flow with discharge q.

---~q

h - H

for x 2 + y2 _ R 2.

R

*

~¢2F-- ~1K H2

q ( -z~- -)

.

Total discharge through the basin:

Q - 4qR. x

b. Obstacles, acting as impermeable planes 46.11 ( F ) ( c f 324.11)

Finite impermeable plane in uniform flow, with discharge q, making an angle art with the

q

plane. -

x

Oh Oy

--(x,

o) -

o

for - b < x


OF-- -~IKH2 -- q{ 4-iv/z 2 -- b2 sin(otyr) + z cos(otyr)}

46.12 (F) (cf. 336.12)

Y

---> q h=H

2b

Finite impermeable screen perpendicular to uniform flow with discharge q.

Oh ox(O'Y)

~=0

h--H

0

forly] < b ,

for lYl > b.

x

1

h=H

I2F = - K H 2 + q V/z2 -k-b2 2

General two-dimensionalflow

(46b, 47a)

39

46.13 (F)(cf 324.12)

Impermeable screen of elliptical shape in uniform flow, with discharge q, making an angle otrc with one of the main axis.

y2

X2

~PF--0

for ~-7+~-7-- 1.

Derivation from 324.12 according to procedure 3. 46.14 (F)(cf 336.13) As 46.13 with o~ = 0. 46.15 (F) (cf 336.11)

Yl

--'q

h=H

,

Circular impermeable screen in uniform flow. ~F -- 0

for x 2 + y2 _ R 2,

h--H

for x - - 0 and lYl > R.

='R

I

1 K H 2 - q(z + R2) =

T

"

I

I

,

R

47. Wells and well-screens in phreatic aquifers

7x)+ryO (h Ohy)-O

O (hOh

Subdivision: a. Steady flow. b. Quasi-steady flow.

a. Steady flow to and from wells and well-screens For this group of problems the solutions for phreatic aquifers are derived directly from those for confined aquifers by applying the following procedure.

40

(47a)

Phreatic flow

Procedure 4

Replace X2 by g1 K H 2 -

X2F, ~ by Q and ~ by q.

47.01 (F) ( c f 3 3 7 . 0 1 ) Flow between a well and two semi-infinite equipotential planes, making an angle otrr with each other, h - h (r, 0).

ro P

h (r, 0) - h (r, oezr) - H. h=H

1

O

2

2zr

S2F -- - K H 2 z--re

i°

x

zo -- ro eiO°

~

Well with discharge Q at P(ro, 0o), ro :fi 0, 0 < 0o < otzr. -1/~

zll~

In

--

~0

--

~0

z l /ot

_ l /ot

zo -- ro e-iO°

•

47.02 ( F ) ( c f 3 3 7 . 0 2 ) 1 As 47.01 with o~ -- ~.

47.03 (F) As 47.01 with ot - 1.

1KH 2 S-2F- ~ .Oo ........................

h=H

47.04 (F)(cf. 3 1 6 . 0 2 ) 3 As 47.01 with ot - ~. 47.05 ( F ) ( c f 3 1 6 . 0 3 ) As 47.01 with ot - 2. 47.06 ( F ) ( c f 3 2 5 . 0 6 ) As 47.05 with 0 o - zr.

x

Q In (z - z'o - 2--~

z

zo Z0

41

(47a)

General two-dimensional flow

47.07 (F) (cf 316. 04)

.?'~._d ~

~

Y

"~& ~

r°~

Flow between a well with discharge Q and two finite equipotential planes of different lengths making an angle o~zr with each other.

:=~ Q

h

-H

..........

X

h (r, 0) - head.

h(r,O)-H

f o r 0 ~ < r ~
h(r,~zr)-H

f o r 0 ~ < r ~
Well located at P(ro, 0o).

Q In 27r

S-2F -- - K H 2 2

,(.,. ,) ,j

w(z) -- -~ z

+ -~ + -~ (zl/~ + a

s -- a 1/~ + b 1/~

47.08

(F)

41wol w _ s s ~wo w - wo

d - a 1/~ - b 1/~

) with

- bl/~),

.

(cf 316.05)

As 47.07 with a - b. 47.09 ( F ) ( c f 316.06) 1 As 47.08 with ot -- 2.

47.10 ( F ) ( c f 317.07) As 47.08 with o~ - 1. 47.11 ( F ) ( c f 316.08) Flow between a well with discharge Q and an equipotential plane of elliptical shape. Well located at P(xo, yo).

x2 h--H

y2

for ~-5 + ~ - S - 1.

Derivation from 316.08 according to procedure 4.

42

(47a)

47.12 ( F ) ( c f

Phreatic flow

325.01) Flow between a well and a circular basin with constant level. Well with discharge Q located at

Y

P(l, 0). h=H l

forx2+y2=R

2, l > R.

x R2

S2F -- ~1 K H2 -- Q In ( / 2zr

47.13 { F ) ( c f

• z--T-) z -- 1

325.04)

Flow between a well and an equipotential plane of parabolic shape. Well with constant discharge located at P(x0, Y0). h = H for y2 = 4p(x + p). Derivation from 325.04 according to procedure 4. 47.14 (F)

(cf 337.04) Flow between a well with constant discharge and an equipotential plane of equilateral hyperbolic shape. Well located at P(xo, Yo).

P o=~Q

h-

H

for

x y - ~a 2.

1

S2F -- ~ K H 2

47.15 (F) Y

~ In 2zr

( z 2 - ~ z - 2 i a 2) Z 2 -- 2 2

(cf 337.12) Well with discharge Q in the centre of a circular row of n recharge wells, regularly distributed along the circle, each with recharge a .

~'~

x

47.16 (F)

Q

1H2Q(zn+Rn - ~ In 27rn

12F -- ~ K

) zn

(cf 337.61)

Discharge well and recharge well of equal strength in uniform flow, angle otzc with the connecting line of the two wells. Derivation from 337.61 according to procedure 4.

making an

(47a)

General two-dimensional flow

43

47.17 (F) (cf 356.05) Infinite row of wells of equal strengths and at equal mutual distances along a straight line parallel to a straight open boundary with constant level H. h -- h(x, y) -- head. h(0, y) = H. Wells with discharge Q each at (b, ± h a ) with n = 0, 1, 2 , . . . . Derivation from 356.05 according to procedure 4.

47.18 ( F ) ( c f 356.12) Single well between two parallel open water boundaries with constant levels H, h = h (x, y) = head. h ( x , O ) = h ( x , b ) = H. Well with discharge Q at (0, yo).

Yo : : ~ Q ....... b .

.

.

.

1KH212F- ~

v

.

X

i

sinh{ Q In 2zr sinh{

2b } 1 7r(z+iyf)) 7r(z;byo)} "

47.19 ( F ) ( c f 365.01)

Yl Q

x

Well with excentricity d in a circular island, consisting of a phreatic aquifer with constant level H at the circular boundary, h - h(x, y) - h e a d . h (xo, Yo) - H for x 2 + y2 _ R 2. Well with discharge Q at (d, 0).

,QF -- -~K H2 -

. 2re

R2

In

•

z- d

47.20 (F) (cf 365.02) As 47.19 but with a circular row of n wells inside the circular island. Discharge of the wells is Q. Derivation from 365.02 according to procedure 4. /2

47.21 (F) (cf 365.03) Single well in a rectangular strip with open water boundaries at all sides with constant level H. h -h(x, y) - head. h(O, y) - h(2a, y) - h(x,O) h(x, 2 b ) - H. Well with discharge Q at (x0, y0).

2b Yo

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

Xo

~-

2a

x

44

(47a, b)

1

S2F -- - K H 2 2

Q

+ ~'~- Z /~/"-'-- OO

Phreatic flow

Esin'

Q

'

~(z-zo+4na) 4b } sinh { rc(z+zo+4na)4b } ] In Esinh{ rr(z+~f)+4na)4b} '

2rr

sinh {

n=-c~a

z -- x + i y, zo -- xo + i yo, zo -- x o - i yo.

47.22 (F) (cf 365.04) As 47.21 with the well at the centre of the strip, which is taken as the origin. h(a, y ) = h(x, b ) = H. Well with discharge Q at (0, 0). Derivation from 365.04 according to procedure 4. 47.23 (F) (cf 365.07) Infinite row o f wells o f equal strengths and at equal mutual distances along a straight line midway between two parallel open boundaries with constant levels H, h = h(x, y) = head. h ( - a , y) = h(a, y) = H. Wells with discharge Q each at (0, 2nb) with n -- 0, 4-1, +2, . . . .

Derivation from 365.07 according to procedure 4. b. Quasi-steady flow to and from wells and well-screens

For this group of problems the solutions for phreatic aquifers are derived directly from those for confined aquifers by applying the following procedure. Procedure 5

Replace £2 by ,f2F, a by - Q and ~ by - q , with c a real constant. 47.31 (F) (cf 337.11) Flow to a well situated near two semi-infinite impermeable planes, making an angle arc with each other, h = h (r, 0). Well with discharge Q, situated at P(ro, 0o), 0 < Oo < arc. 7rF(r, 0) = const.,

Q l/c~ ,QF -- c + ~ In { (z

47.32 (F) (cf 337.13) 1 As 47.31 with ot - ~.

zl/•o

)(z

1/or

-1/or }

2;0

)

•

~F(r, otzr) = const.

General two-dimensional flow

(47b)

45

47.33 (F) As 47.31 with o~ -- 1.

Q

P o=~Q

S2F -- c -+- ~ - I n

{(z -

zo)(z - zo)}.

x

47.34 (F}

(cf 316.22)

3 As 47.31 with ct -- 7"

47.35 (F)

(cf 316.23)

As 47.31 with c~ -- 2. 47.36 (F) F l o w to

(cf 316.24)

a well with d i s c h a r g e angle olzr with each

m a k i n g an

~ F ( r , 0) - const. 7tl~(r, otzr) -- const.

Q, situated near other and having

two finite impermeable screens, different lengths, h - h(r, 0).

f o r 0 ~
Well at P(ro, 0o). D e r i v a t i o n f r o m 3 1 6 . 2 4 a c c o r d i n g to p r o c e d u r e 5. 47.37 {F)

(cf 316.25)

As 4 7 . 3 6 with a - b. 47.38 (F)

(cf 316.26) P o=~Q

As 47.37 with o~ -

1 7"

Q

{]wol(W-Wo)(w-b4w°)}

12F -- c + ~ - In

41w°12 w0

1 2

47.39 (F)

(cf 316.27)

As 47.37 with ot -- 1.

1 4 + g,/z -b4.

w

,

46

Phreatic flow

(47b)

47.40 (F) (cf 316.28) Y

Flow to a well of discharge Q, situated near an impermeable obstacle of elliptical shape.

P

.~Q x2

~PF -- const, -a

x

y2

for ~-7 + ~-7 -- 1.

Well at P(xo, yo).

(a+b)2wo

S2F -- c + ~ - In

41w012 tOO

w(z)

-

1 +-~1 4Z 2 -~z

,

W

a2 + b2 .

47.41 (F) (cf. 325.21) Flow around a circular impermeable screen to a well with discharge Q, located at e(l, 0). 7iF = const,

for

X2

--I--y2 ._ R 2, l > R.

Derivation from 325.21 according to procedure 5. 47.42 IF) (cf. 325.24)

YT ,(x0yo,

Impermeable screen of parabolic shape and a well with discharge Q, located at P(x0, yo). OF = const,

for y2 = 4p(x + p).

-p

Q [sinh {~(~-4~/"~4~) } cosh{

S2F--c+~ln

47.43 (F) (cf. 337.15)

Impermeable screen of equilateral hyperbolic shape and a well with discharge Q, 1 2. located at P(x0, y0). ~PF --const. for x y - ~a Derivation from 337.15 according to procedure 5.

General two-dimensional flow

(47b)

47

47.44 (F) (cf 337.41) Finite well-screen with constant level, perpendicular to a straight impermeable boundary

(')

h(O, y ) - -

(1)

for a - -~l < y < a +-~l ,

r

x

Oh Ox (0, y)

(1)

for0~
0

Q •

~F

--

c-~-

~

2zr

(a+~

andy

ll) .

Z,2 -+-a 2 -+_ 1/2) arccosh al "

47.45 (F) (cf 337.42) Finite well-screen in an infinite field. Constant level at the screen surface 2l

~Q

h(O, y ) -

x

Oh ~xO( 0 ' y) - 0 •

for - l ~ y<~l, for y ~< --l and y >~ I.

S2F-- c + - - Q arcsinh(/), 7/" 1 Jo(lot)e_X~ sin(yot) dot. ~ F - - --jr Q f0 °° ot

47.46 (F) (cf 337.43)

Q

21

Finite well-screen in an infinite field. Discharge uniformly distributed along the well-screen. Oh (x 0 ) hoy

Q 2Kl 0

f o r - / < ~ x ~l.

Q s2F - c + ~ / { ( z -t-I)ln(z + I) - (z - l ) l n ( z - l ) } , ~PF- 2Jrl a fo ~ ~1 sin(lot) sin(xot)e -lyI~ dot.

48

(47b, 48)

Phreatic flow

47.47 (F) (cf 325.51)

Well with discharge Q in uniform flow with strength q. Derivation from 325.51 according to procedure 5. 47.48 (F) (cf 316.51)

Well-screen of length 21 with discharge Q in uniform flow with strength q. Derivation from 316.51 according to procedure 5. 47.49 (F) (cf 356.01)

Infinite row of wells of equal strengths and at equal mutual distances along a straight line. Quasi-steady state: h = h(x, y). Wells with discharge Q each, located at (0, ±na) with n = 0, 1, 2 . . . . .

-=~Q

--.~a ~a

X2F- c

+ --Q-QIn sinh( zrz ) 27r -a- •

~Q

~ a T -+=~a

48. Problems with precipitation

3 (hOh

3 (hOh

ox

P

-0.

For this group of problems the solutions for phreatic aquifers are derived directly from those for confined aquifers by applying the following procedure.

Procedure 6 Replace ~p by h - H and D by h+/-/ 2 " 48.01 (F) (cf 354.06)

Constant precipitation on a phreatic aquifer bounded by three fully penetrating straight canals with fixed level H. Two canals are parallel to each other, the third one is perpendicular to the first two. h - h(x, y) - head. h(O, y) - h ( x , - b ) - h(x, b ) - H. Derivation from 354.06 according to procedure 6.

General two-dimensional flow

(48)

49

48.02 (F) (cf 363.05)

Rectangular strip with constant precipitation and fixed level H of the surface water at the boundaries.

b

h - h(x, y) - head. -a

h ( - a , y) - h(a, y) - H, h(x, - b ) - h(x, b) - H. -b h 2 -- H 2 -Jr- ~P- ( b 2 __ y2 ) ,1,4,,1,,~,1,,I,,1,,1,,1,,1 P

I

~

32pb 2 at3 K

( _ 1)(m-I)/2

Z

m3

m = l , 3 ....

cosh(mnx -SV-J cosh(mTra] 2b

mzry COS

2b )"

]

2a

48.03 (F) (cf 363.08)

Constant precipitation on an island of elliptical shape with a fixed level at the elliptical boundary.

Y b

h - h(x, y) - head. -a

a

h(xo, y0) -- H

h 2 --H2+m

-b

I

~~;~;~P

I

for a-2

p

+j

a2b 2

K a2 + b2 y2

--1.

Orientation table BI

(100)

BI. O N E - D I M E N S I O N A L

51

GROUNDWATER

FLOW

ORIENTATIONTABLE BI 100. One-dimensional groundwater flow a. Does the f l o w field theoretically extend to direction o f the flow ? A flow field, flow, means x = - o o to metry exists

infinity in the go to 110 (BI- 1)

extending to infinity in one-dimensional that the flow variable x varies from x = + o o in such a way that no symon that interval.

--oo < x < +oo

go to 120 (BI-2)

b. The flow is semi-infinite

//--/

Semi-infinite flow in a direction is defined either as infinite flow with a line of symmetry, or as flow being present only at one side of a boundary. The variable x varies from x - 0 to x - +cx~.

O <<.x < + oo

go to 130 (BI-3)

e. The f l o w is finite

b

x

Finite flow in a direction is here defined as flow between two boundaries, situated at a finite distance of each other; generally the boundaries are parallel to each other and perpendicular to the direction of the flow.

O <~ x <<.b

d. The f l o w can be considered as a combination of the forementioned cases b and c

go to 140 (BI-4)

In general, all cases that do not fit properly in the fore-mentioned categories, have been classified under this section. An important group of problems, belonging to this categy, are those which involve discontinuities in one or more of the ground parameters.

52

(100d)

One-dimensional groundwaterflow

Example:

K1D1 K ~ ~

Parallel flow towards an open boundary (canal with fixed level) through a semi-infinite field with a strip of different transmissivity of the aquifer (KD) and different resistance of the covering layer (c). The flow is a combination of flow in a semi-infinite ' field and flow in a finite field.

Flow in an infinite field 110 BI-1.

(110, 111)

53

O N E - D I M E N S I O N A L G R O U N D W A T E R F L O W IN AN I N F I N I T E F I E L D

a. Has the soil discrete inhomogeneities in one or more of the ground parameters ?

I

h

C1//~//~J~_~~/~//~////~I/////////T/////" C2 +K1DI+ +

go to 140 (BI-4)

An example of a discrete inhomogeneity is flow through a so-called "Dutch polder profile" with different K D- and/or c-values.

xK2D2~,

b. The soil is assumed to be homogeneous

go to 111

111. Homogeneous soils

a. Is your problem a so-called initial value problem?

go to 112

If in an infinite field the groundwater head initially is a given function of x, the heads will diminish in a fixed manner, owing to gravity, whereas the groundwater will move horizontally in lateral direction, until finally all heads will be equal. This is a so-called initial value problem. The main objective is the determination of the depletion function, that is the function that shows the variation of the heads with time everywhere in the field.

b. Is horizontal flow in an aquifer caused by vertical infiltration on a given area ?

go to 113

The vertical infiltration, which may be natural (precipitation) or artificial, is supposed to be equally distributed over the thickness of the aquifer, so that no vertical flow takes place.

c. Flow to or from polders and drained lakes

go to 114

Polders and drained lakes consist of leaky aquifers, the leakage being controlled by the polder level in the top semi-permeable layer. It is assumed that only horizontal flow takes place in the aquifer and only vertical flow through the semi-permeable layer as a consequence of different polder levels and surface water levels. 112. Depletion functions for one-dimensional flow in an infinite field Subdivision: a. Confined aquifers. b. Leaky aquifers.

54

(112a, b)

One-dimensional groundwater flow

a. Confined aquifers

0299 __ f12 099

f12 __

S

-57'

,fTo

112.01 (L,VP) Depletion function for groundwater heads in a confined aquifer, starting from an arbitrary function. ~0 = 9(x, t) = head. Initial value: 9(x, 0 ) = F(x).

~o

J x KD' S__+

~M~eeeeee~ qg(x ' t ) -- 2 ~fl- - [ f _+oooo F(xo) exp { - ~7-(x /~2 - xo) 2 } dxo. 112.02 /L)

Depletion function for groundwater heads in a confined aquifer, starting from a constant head h for x positive and zero head for x negative. ~p - 9(x, t) - head. Initial value: ~0(x 0) ' qg(x,t)--~

l+erf

2~/t)

_ Jh, | 0

forx > 0 , forx < 0 .

"

h Steady state" cp -- 7"

b. Leaky aquifers 0 2 q)

q9 __ f12 0(.19

igx2

~2

1 _

57'

?7

_

1 _

f12L2

.

~

cS '

t~ 2

~ 2 __ K D c

'

-

-

-

S KD

112.11 (L,VP) Depletion function for groundwater heads in a semiconfined aquifer, starting from an arbitrary function. 9 - ~o(x, t) - head. Initial value: qg(x, 0 ) - F(x).

(112b, 113a)

Flow in an infinite field

•

o0 F(xo) exp

cp(x, t) -- 2 x / ~

- r/t - ~ - ( x

55

-

xo) 2

dxo.

112.12 (L) Depletion function for groundwater heads in a semi-confined aquifer, starting with a constant head h in the region where x is positive and zero head f o r x negative. ~o = ~o(x, t) = head. Initial value: {h ~o(x, 0) -

*

~o(x t) -- -~e -ot 1 + erf '

0

forx > 0 , forx < 0 .

] 2~/t/

"

113. One-dimensional flow in an infinite field, caused by vertical infiltration

Subdivision: a. Confined aquifers. b. Leaky aquifers. a. Confined aquifers 02~

t p(x, t) _ f12 "~O'l".

Ox 2

KD

Ot

113.01 (MI) The vertical infiltration (for example, precipitation that joins the groundwater) is an arbitrary function

~~

~ - ~ ~o(x, O

~

x,t)

o f x and t. p - - p ( x , t).

KD, S

X

•

l

~p(x, t) -- 2 f l K D x / ~

f_+°~fo'P(Xo, to) exp { oo

x/t - to

fl2(x - x0)2

4(t - to)

} dx0 dto.

113.02 (MI) The precipitation is an arbitrary function o f time and is equally distributed over the region where x is positive and is zero for x negative.

P--

p(t) 0

forx > 0 , for x < O.

56

One-dimensional groundwater flow

(113a)

~o(x, t) -

mfo

p(to){1 + e f t (

_fix_ } dto. \ 2 ~ / t - to

113.03 (S) As 113.02 but with p(t) = p = const. p 0

P-

cp(x, t)

--

for x > 0, forx<0.

pt ---ff { l - 2iZerfc( flx ] } \2~7]

forx >0,

2pts ieerfc/~ _ ~_~)fix

forx <0.

113.04 (D)

]

Precipitation (infiltration) equally distributed over the whole region and an arbitrary function of the time, p = p(t) for - o o < x < +oo.

~o(t)

$$$$$$$ I 4,$$$$$$$ 0

p(t)

l f o t p(to) dto. 9(t) -- -~

x

113.05 (D) As 113.04 but with p(t) = p = const., p = p for - o o ~o(t)

=

< x < +oo.

pt --. S

113.06 (D) As 113.04 but with p(t) = p sin(wt) for - oo < x < oo. ~o(t)--~{l+sin

Jr

,

2yr

T 113.07 (MI) As 113.01 but with p (x, t) = p (x) = an arbitrary function of x.

~o(x, t) -

47 /r += p (x0) ierfc ( fl Ix - x0] 2~/7 ) dx0. fl K D J_oo

Flow in an infinite field

(113b)

57

b. Leaky aquifers 0 2 ~0

q (x , t)

(t9

OX 2

)v 2

77'

K D

/7

_

1 fl2)v2

=

1

cS '

)2

_

KDc,

f12 __

S

-- K D "

113.11 (MI) The artificial infiltration q into the infinite leaky aquifer is an arbitrary function of x and t, q q(x,t).

go(x,t) /////////////////////////////////~ C

~ q ( KD, S

x

, t x

)

f+oo fot q (xo, to)

1

cp(x, t) -- 2~KD~/-~

x exp { - r/(t - to) -

fl2(X --X0) 2 } 4(t - to) dxo dto.

113.12 (MI) The artificial infiltration is an arbitrary function of time and is equally distributed over the region where x is positive and is zero for x negative.

q--

cp(x, t ) -

l fo' q(to)exp { -

-~

q(t) 0

forx >0, forx <0.

r / ( t - to)}{1 + erf(2~/' _ to) } dto.

113.13 (MI) As 113.12 but with q(t) - q -- const.

q--

q

for x > 0,

0

forx <0.

q

cp(x , t ) -

e- ot

- ~ S P ( -}-~ ~sq { P (

for x

erfc

2)~'x x / ~ ) - e - ° t e r f c (

-

~x

>

O,

f o r x -~ O.

58

(113b)

One-dimensional groundwater flow

113.14 (D) Steady state of 1 13.1 3.

(2)

I qc 1 - le--~ ~0(x) -

for x /> 0,

1 x ~qceZ

for x ~< 0.

113.15 (D) Infiltration equally distributed over the whole region and an arbitrary function o f time. q - q(t) for - c ~ < x < + c ~ .

I~o(O //'////////////////////////////////5

C

KD, S

q(t)

$$$$$~$$$$$$$$$$ 0

1/ot

~p(t) -- ~

x

q(t - to)e -~t° dto.

113.16 (L} As 113.15 but with q(t) - q - consc cp(t) -- qc(1 -- e-nt). Steady state: q9 - q c.

113.17 (L) As 1 13.15 but with q(t) - q sin(00t) for -cx~ < x < ~ . 00

00

q 77 sin(00t) - /.]2 _[_ 002 cos(00t) + ~ e - ° t qg(t) -- ~ { r] 2 "[- 002 /72 --[- 00 2 00--

2Jr T,

1 ~ 7 - - c~ So

113.18 (PF) Steady oscillation of 113.17.

•

¢(t)-

q {

r/

~ 772 + co2

sin(00t)- /72 -'[- 0) 2 cos(00t) .

113.19 (MI) As 113.11 but with q ( x , t) -- q(x), and arbitrary function of x.

•

qg(x,t)--

2KD

f_+oo o0

q (xo) Pconj

(,x-xo, 2~

'

dxo.

}'

(1 13b, 1 14)

Flow in an infinite field

59

113.20 (MI)

Steady state of 113.19. X •

f+oo

99(x)- 2KD

oo

q (x0) exp ( -

Ix - xol] dxo. X /

114. One-dimensional flow through infinite aquifers to or from polders and drained lakes 0299 h(t) --99 2099 OX 2 { ,~2 --" fl O---t

114.01 (L)

Non-steady flow through an aquifer, caused by draining a lake by means of lowering the water level according to an arbitrary function of time h(t). Drawdown 99 = 99(x, t).

099

KD, S

t) -

Ox x

h(O)

-- O,

399 Ox

t) -

99(0, t)

99(x, t) - rl fot h(t - r)e-Or { 1 - ~1 e r f c ( 2 ~ r ) } dr,

r/-

o,

1 -- ~99(oo, t). z 1

114.02 (L) "Sudden" lowering of the water level. As 1 14.01 but with h(t) - h - constant.

•

o(x,t)

h(1- e_,?t)- 2h{p(~X, ~/_~)- e_nterfc(2~t)

f o r x >/ 0,

~{p(_ ~ h x, w/-~-)- e_n,erfc( _ 2,/3~t)}

for x <~ 0,

-

114.03 (D)

Steady state of 1 14.02.

99(x) --

/

h h ~

h

m

2 x OX

o

_x x

for x >~ O, for x ~< O.

60

(120-122)

120 BI-2.

One-dimensional groundwater flow

ONE-DIMENSIONAL GROUNDWATER FLOW IN A SEMI-INFINITE FIELD

a. Has the soil discrete inhomogeneities in x-direction? Y

h .A Cl

U///////////~ ////////////////C 2 K1D1 K2D2

go to 140 (BI-4)

An example of flow through inhomogeneous soil is flow through a leaky aquifer with different K D - and c-values.

b. The soil is assumed to be homogeneous

go to 121

121. Homogeneous soils a. Is a periodic function involved in one of the boundary conditions of your problem ?

go to 128

In general, periodic functions occur in the form of sine- or cosine functions, for instance seasonal fluctuations of the precipitation or tidal fluctuations of surface water. Groundwater flow that more or less depends on such periodic fluctuations is referred to as periodic flow. This kind of groundwater flow can be solved in a particular manner and therefore will be treated separately. If a boundary condition contains a general function of the time it sometimes is useful to develop the function in a Fourier series (if possible) and derive the solution of the problem from a more simple periodic function.

b. The flow is non-periodic

go to 122

122. Non-periodic flow a. Is the boundary condition at x -- 0 a given head or drawdown?

go to 123

In most cases the given head or drawdown will be caused by surface water (straight river or canal that fully penetrates the aquifer) which level may be an arbitrary function of the time:

~o(0, t) - F(t). There is no entrance resistance between the aquifer and the surface water. In many problems F(t) will be a constant or zero; ~0(0, t) - h or ~0(0, t) - 0.

b. Is the boundary condition at x - 0 a given flux?

go to 124

A given flux will occur if a given amount of water is withdrawn from the aquifer by means ef a fully penetrating infinite well-screen located at x - 0.

Flow in a semi-infinite field

(122b, c)

61

Also a very long row of wells at short distances of each other can serve as such a well-screen. The discharge may be an arbitrary function of the time q ( t ) and the boundary condition will be

Ocp -~(o,

Ox

q (t) t) -

2KD

e. Is the boundary condition at x = 0 a combination of head and flux?

go to 125

In this category two kinds of boundary conditions may be distinguished:

1 o Entrance resistance between the aquifer and surface water; in general: OCP(o, t) -- F ( t ) - cp(O, t) -Kox w

~----~¢-,l,-,1,-,!, 1,-,!,

F({)~KD, S

KD, S x ,

There always exists a jump between the groundwater head at x = 0 and the surface water.

2 ° Surface water storage. The surface water level is no longer a given function of time (or a constant or zero) but depends on the outflow (inflow) of the groundwater from the aquifer, owing to limited dimensions of the water course (ditch or canal with width b) b O~P(0, t) - KDO~°(O, t). Ot Ox

123. G i v e n h e a d or d r a w d o w n at x -- 0 for o n e - d i m e n s i o n a l s e m i - i n f i n i t e f l o w

Subdivision: a. Confined aquifers with variable head at x - 0. b. Confined aquifers with given initial head and zero head at x -- 0. c. Confined aquifers with precipitation and zero head at x - 0. d. Leaky aquifers with variable head at x - - 0 . e. Leaky aquifers with given initial head and zero head at x - 0. f. Leaky aquifers with vertical infiltration and zero head at x -- 0.

62

(123a)

One-dimensional groundwater flow

a. Confined aquifers with variable head at x = 0 02q 9 __ f12 0q9

Ox 2

f12 __

-~'

S KD "

123.01 (L) The surface water level rises according to an arbitrary function of time. q)(x, t) --- head.

i

q)(x, 0) = 0 ,

KD,S

X

qg(c~, t) = 0 ,

99(0, t) = F(t)

~o(x,t)=~

2L(

with F(0) = 0.

f12X2 4~ 2 ) e -x2 d)..

F t

247

123.02 (L) F(t) i

Sudden rise of the surface water level which is kept constant thereafter, qg(x, t) = head.

h

¢p(x, 0) = 0,

q)(oo, t) -- 0,

q)(O, t) -- F(t) -- I O

[h q)(x, t ) = h

fort--0, for t > 0.

e r f c ( 2 @ t ).

123.03 (L) Linear rise of the surface water level, q)(x, t) = head.

q)(x, 0) = 0, 99(0, t) = a t ,

q)(x,t)-

q)(cx~, t) -- 0, a [LT-1].

4at i2erfc(fl~-~~).

123.04 (L) Parabolic rise of the surface water level, qg(x, t) = head.

q~(x, O) = O,

99(oo, t) = O,

~p(O, t) -- F(t) = av/t,

a [LT-1/2].

(123a)

Flow in a semi-infinite field

63

q)(x, t ) - a~/~ierfc(2@tt). Quasi-steady flow: for large values of the time the flux becomes constant.

1 q -- - ~ a ~ K D ~ .

123.05 (L)

Rise of the surface water level according to F(t) - a : / : -- ~o(0, t) (n -- 0, 1, 2, ...). ~o(x, t) -- head. q)(x, O) --0,

q)(cx~, t) --0. g/

qg(x, t)- aF(1 + ~)(4t) n/2 inerfc(/3x Special cases" 123.02, 123.03, 123.04. 123.06 (L) Exponentially increasing rise of the surface water lev-

F(t)

el. qg(x, t) - head. ~o(x, O) - 0 , -a

~o(c~, t) - 0 ,

q)(O, t) - F(t) = a(e bt - 1)

(a and b positive).

1 /3x ae bt P(-~flxv/-b, ~ / ~ ) - a erfc( ~ - ~ .)

qg(x, t ) -

123.07 (L) Exponentially decreasing rise of the surface water lev-

el. qg(x, t) - head. ~o(x, O) - 0 ,

~o(ee, t) - 0 ,

t

~o(0, t) - f ( t ) - h(1 - e -bt)

/

--

erfc

) - he- t

i

(h and b positive).

64

(123b)

One-dimensional groundwaterflow

b. Confined aquifers with given initial head and zero head at x - 0

02(/9

0(/9

O x 2 __ f12

-~'

f12

S KD

123.11 (IS,VP)

__ ~ f ( x )

Initial condition (t - 0 ) an arbitrary function of x. Surface water maintained at zero level.

99(x, t) - head.

KD,S

99(0, t) --0.

q)(x, O) -- f (x),

x

Depletion function: qg(xt)

flfo~[

f12

,82

123.12 (L) As 123.11. f (x) - h - const. h I f(x) •

I

ira.

~o(x,t)-herf(~-~x~) \2~/t /

(see 123.02).

x

123.13 (L) As 123.11.

f(x) I

b

f(x)--

ira,

{h 0

forO<<.x
for x >b.

x

~o(x,t)--~h i 2erf(fl/)\2~7 -erf//5 (x2~/Y-b) ] - e r f { ¢1(x2~/7 + b) }1 " 123.14 (L)

As 123.11. f (x) - h + ax. •

~o(x, t ) - h e f t ( f l ~ )

Steady state: qo(x) - ax.

"+" a x .

] dxo

(123c)

Flow in a semi-infinite field

65

c. C o n f i n e d aquifers with p r e c i p i t a t i o n a n d zero h e a d at x - 0 02(p

p ( x , t) = f12 Oq9 t

OX 2

K D

0-7"

123.21 (MI} A r b i t r a r y precipitation as a function of x and t. The p l a n e at x - 0 kept at zero head. (p(x, t) - head.

~o=0 p(x,t) KD, S

p -- p ( x , t),

x

~o(x, O) -- O,

¢(o,t)-o.

1

qg( x , t) -- 2 fl K D ~/-~

fo~/o '

p(x0, t -- to)

/~2

[oxp/ ~o~X xo,2/

-oxp{-~o(X +~o)2} ~o~xo 123.22 (L} As 123.21 but p ( x , t) - p ( t ) , an arbitrary f u n c t i o n o f time.

(,~)

'fo ~

p ( t - to) erf 24q_~ dto.

(p(x, t) -- -~

123.23 (L) As 123.22 but p ( t ) - p - const.

~(x, ~)- ~~t{,

~x / 4i~orfc(~)

123.24 (MI) As 123.21 but p (x, t) - p (x) an arbitrary f u n c t i o n o f x. •

(p(x,t) --

~fo ~

[

~

~o, ierfc/~+~o,] 2,/7

123.25 (MI) As 123.21 but

(x)

i~llIll~

p(x ,

b

t)-- / p [0

for0<x b.

"

66

One-dimensionalgroundwater flow

(123c, d)

pt [ 1 - 4 i 2 e r f c ( ~ -fix ~ ) + 2 i2erfc{ fl(b2 ~+ x) } -2i2erfc{ fl(b2~/7 - x ) }] *

~O(X,t) -"

for 0 < x < b ,

pt [ fix (x + b) ~ -4i2erfc(~-~)-t-2i2erfc{fl~ } +2 i2erfc[ f l (2~/7 /-b)}]

for x > b.

d. Leaky aquifers with variable head at x = 0 02~ 8x 2

q9 __ f12 099

-~

--

~2

-57,

f12 --

X D c ,

S --

1

K D

'

/7 --

fl2~2

1 --

--cS

123.31 (L)

Drawdown of the surface water level an arbitrary function of the time. q9 = ~o(x, t) = drawdown. ~V////////////////////////,//

¢

KD, S x

2 f ~x °° 24~

F(t

~o(x, O) =0,

~o(c~, t) = 0 ,

~o(0, t) = F(t)

with F(0) = 0.

f12X2

4c~2 ) exp ( - or2

,V J,,

X2

4,k2ot2 )dc~ •

123.32 (L)

Sudden drawdown of the surface water level, which is kept constant thereafter, q9 --- ¢p(x, t) -- drawdown. F(t)

~o(x, 0) --0,

(x

~o(x, t) - hP -~, 123.33 (D)

Steady state of 123.32. •

X

qg(x) -- h e - r .

~o(c~, t) = 0 ,

99(0, t) -- F(t) -- [0 / h .

fort--0, for t > 0.

(123d)

Flow in a semi-infinite field

67

123.34 (L) drawdown of the surface water level. ~0(x, t) - drawdown.

Linear

F(t)

~p(x, O) - - 0 ,

~p(cx~, t) - - 0 ,

~p(O, t) -- a t ,

qg(x, t) -- a t P

,

+

(~

aflZx,kPcon j

a [LT-1].

x ,

.

123.35 (L) -a

E x p o n e n t i a l l y i n c r e a s i n g drawdown of the surface wa-

ter level, q9 - qg(x, t) - drawdown.

FCt)-; qg(x, t) -- aebt p

~o(x, O) - 0 , ~p(O, t) -

~p(cx~, t) - 0 ,

F ( t ) -- a ( e bt - 1)

-~ flxx//O + b, x//(rl + b ) t

- ae

(a and b positive).

-f-f,

.

123.36 (L)

,j

E x p o n e n t i a l l y d e c r e a s i n g drawdown of the surface wa-

\

\

ter level. ~o(x, t) - drawdown. qg(x, O) - 0 ,

h

. . . . .

~p(O, t) x

~o(cx~, t) - 0 ,

F ( t ) -- h(1 - e -bt)

(h and b positive).

1

123.37 (D) S t e a d y state of 123.36. x

•

qg(x) -- h e - ~ .

e. L e a k y a q u i f e r s with g i v e n initial h e a d a n d zero h e a d at x - 0 02q9 Ox 2

q9 __ 1~2 0q9 ~2 --~'

i~ 2 __

S ) 2 __ K D c rl KD '

1 2)~2

1 ~o

cS

68

(123e)

One-dimensional groundwater flow

123.41 (VP)

[

Initial condition (t - O ) an arbitrary function of x. Surface water maintained at zero level.

~~-f(x)

q9 - ~o(x, t) - head. c

,S

~o(0, t) --0.

qg(x, O) -- f (x),

Depletion function:

qg(x' t)_ 2 ~fl e_rltfo~f ( x o ) [ exp [ - -~-7 f12(x -

xo)2 /

__ f12

-exp[-~(x+xo)2}]dxo

.

123.42 (L) As 123.41. f (x) -- h - const.

f(x)

•

~o(x, t) - he -oteft(

t~x

k

123.43 (L) As 123.41.

f(x)

h I

f(x)--

b

0

f o r O < x b.

x

h tE

247 }]

~o(x, t ) - ~e -~ 2erf/fix )\2~/t -eft[/~(x2:b) } _ erf[/~(x + b) 123.44 (L)

f(x)

As 123.41. f (x ) = h + ax.

/

.

qg(x,t)-

e-"t {h erf(2-~tt)+ax].

f. Leaky aquifers with vertical infiltration and zero head at x - 0 02~o

~o

OX2

~2 ~

q(x, t) = ~2 O~o KD

-~ '

[32 __

S )2 = K Dc, KD'

77=

1 fl2~2"

(123f)

Flow in a semi-infinite field

69

123.51 (MI) Arbitrary vertical infiltration as a function of x and t. The plane at x = 0 kept at zero head. qg(x, t) = head.

q9-- 0 ~//////////////__ ¢

~

q(x,t) q = q ( x , t),

99(x, O) = O,

~o(o, t) = o . 1 (p(x, t) -- 2 fi K D v/-~

/o~fot

i / ~

q (xo, t - to) e_Oto exp ~

-e×p{- ~o(1+~o~/

-

(x

-- X0) 2

-~o

j

dto d/o.

(L)

123.52

As 123.51 with q ( x , t) - q ( t ) an arbitrary f u n c t i o n o f the time. .

qg(x, t) -- -~

q ( t - to)e-'Tt°erf 2 n / ~

dto.

(L)

123.53

As 123.52 with q ( t ) = q - const. •

~o(x,t)--qc

123.54

1 - e -~terf

-P

~-~,~

.

(D)

Steady state of 123.53. •

¢p(x) - q c ( 1 - e-X/X).

123.55 (MI) As 123.51 but q (x, t) = q (x), an arbitrary f u n c t i o n o f x.

,

~fo ~

99(x, t) -- 2 K D

q(xo)

{ (~+~o Pconi

2X

' "~

ttxx°~t}dxo -Pconj

2k

'

123.56 (MI) Steady state of 123.55. •

~o(x)-

~fo~

2KD

q(xo)

/ ( ,x_xo,)_oxp( exp

-

x+x°t}dxo

70

(123f, 124a)

One-dimensional groundwater flow

123.57 (D) As 123.56 with

(x)

q q(x)-

b

0

forO<x forx>b.


x

Steady state qc

x

-x

-~-12-- 2exp ( - - ~ ) - exp ( +exp •

~(x)

-

-

forO<x

qc

x

-~-1 - 2exp ( +exp

b)v


x -b

~ ) + exp (

( - x~

)

~. )

forx > b .

124. G i v e n flux at x - 0 for o n e - d i m e n s i o n a l semi-infinite flow

Subdivision: a. Confined aquifers with variable flux at x = 0. b. Confined aquifers with given initial head and zero flux at x = 0. c. Confined aquifers with precipitation and zero flux at x = 0. d. Leaky aquifers with variable flux at x - 0 . e. Leaky aquifers with given initial head and zero flux at x = 0. f. Leaky aquifers with vertical infiltration and zero flux at x = 0. a. Confined aquifers with variable f l u x at x - 0

02~ = f12 O{p Ox 2

07 '

f12 __

S KD

124.01 /L)

Instantaneous abstraction Pi [ L2] of water from a fully penetrating plane source (discharge impulse from

a well-screen). ~o(x, t) - drawdown. x,~

S

~0(x, O) -- 0 Pi -- qidt

cp(x, t) -- 2 f l K D ~

99(o0, t) -- O, qi 2KD

Oqo (0, t) -

'

exp (

for x > O,

f12X2

4t )"

Ox

0

for t -- O, fort >0.

(124a)

Flow in a semi-infinite field

71

124.02 (MI) Continuous abstraction of water from a plane source. The discharge is an arbitrary function o f time.

qg(x, t) = drawdown. x

KD, S

qg(x, 0) --0, 0~o ~(O,t) Ox

q)(x, t) -- 2 K D ~ / - ~

¢i t

q t

(p(oo, t) --0,

q (t) = --~. 2KD

f12X2 '~e_Ot2 do/

40/2 ,]

0/2 •

2~/7

124.03 (L) Sudden discharge, which is maintained constant thereafter. As 124.02 except q ( t ) = q.

q(t)

•

go(x,t) --

qx/ff ierfc(/3 Ix I

124.04 (L) Linear increase of the discharge. q ( t ) = at, a [LZT-2].

~o(x, t) = 4at~/~

As 124.02 except

i3erfc(/3 Ix I"~

27J

124.05 (L) Parabolic increase of the discharge. q ( t ) - ax/7, a [LZT-3/2]. iP

t

a ~/~t iZerfc ( fl Ix l

~o(x, t) - fiKD

2-~)

As 124.02 but

72

(124a, b)

One-dimensionalgroundwater flow

124.06 (L) Increase of the discharge according to q(t) = at n/2 (n = 0, 1, 2 , . . . ) .

•

qg(x,t) =

at(1 + n/2) ( 4 t ) ~ -t in+lerfc( fllxl 2flKD

Special cases: 124.03, 124.04 and 124.05. 124.07 (MI) As 124.02 but the discharge an exponentially increas-

ing function of time. q (t) - qe ~2t .

_-q t

q ~e°t2t

1

pconj (-~otflx, o l ~ ) .

qg(x, t) -- - 2c~fl K D

124.08 (MI} As 124.02 but the discharge an exponentially decreas-

\ \

ing function of time. q(t) - qe -~2t.

\ \ \

-q. t

qg(x t ) '

q~e-~2tiPconj(~iflotlxl iota/"D. 2flotKD

b. Confined aquifers with given initial head and zero flux at x = 0 0 2 ~9 __ i~ 2 0(t9 OX 2

f12 __

--~ '

--

KD

124.11 IS) Initial condition (t = 0) an arbitrary function of x. Plane at x = 0 impermeable. ~o(x, t) = head.

f(x)

99(x, 0) = f ( x ) ,

0q9 ~(0,

Ox

t) -- 0.

Depletion function:

i xo,

]

dxo

Flow in a semi-infinite field

(124b, c)

73

124.12 As 124.11 with f (x) -- h -- const. •

~o(x, t) -- h.

124.13 (S) As 124.11 with

f(x) I b

h 0

f(x)-

for0<x b.

x

~p(x, t ) - - ~

erf

2v/}_

+eft

2v/7

.

124.14 (L) As 124.11 with f (x ) - h + ax. •

e(x,t)

2a~/7 fi i e r f c ( f l ~ )

-

+ h +ax.

c. Confined aquifers with precipitation and zero flux at x - 0

02~0 Ox 2

t

p ( x , t) __ f12 0(/9 KD --~-'

f12 __

S

K-D"

124.21 (S)

~-

p(x,t)

Arbitrary precipitation as a function of x and t. The plane at x -- 0 impermeable. 99(x, t) = head. p -- p(x, t),

~p(x, O) = O,

~ ( o , t) =o. Ox

1

cp(x, t) -- 2 f l K D v / _ ~

fo /ot xo t to, [oxp{

~2 +xo~2}1~o~o +oxp{- 7o~

_/~2

xo,2/

74

One-dimensionalgroundwater flow

(124c)

124.22 (S)

As 124.21 but

-b

lllli

llJ

0

b

for - b < x < b, for Ix l > b

with p(t) an arbitrary function of the time.

t ~o(x, t) - ~-~

p(t) 0

p(x, t) -

[ p(t - to) erf

2~/7o

÷ erf

/ ~(b+x)}] 2~/~

dto.

124.23 (S)

p [ p(x,O

As 124.22 but p(t) = p = const.

~llilllJ~IillllJJ -b

0

_ b

x

pt I ~ (b -- Ix 1) --S- L1 - 2iZerfc{ 2~/t } --2i2erfc{/~(b+l/I) ]247 ]

qg(x, t) --

for - b < x


pt 1P(Ixl- b) --S L2 iZerfc[ } 2~v/'/ - 2 i2erfc{ fl(lxl ] 2 4+7b) }

for

Ix l > b.

124.24 (S)

As 124.21, p(x, t) = p(x) = arbitrary function of x.

qg(x, t) -- flKD

~s }+ierfc{fllx+xol ~ 1 ] dx°

p(xo)[ierfc{~lx-xol

d. Leaky aquifers with given flux at x = 0 02q 9

q9 __ /~2 0(/9

OX 2

~,2

--~'

,~2 --

K Dc,

f12 =

S 1 1 K D' r l - ~2~2 -- CS ~ o

(124d)

Flow in a semi-infinite field

75

124.31 (L)

~////////////////////~ ]i

Instantaneous abstraction pi [ L2] of water from a fully penetrating plane source (discharge impulse from a well-screen). ~p(x, t ) - drawdown.

C

KD, S

qg(x,0)--0

q)(c~, t) -- O,

Oq9 -ff-fx ( O, t ) -

Pi -- qi dt,

for x > 0, [ [

qi 2K D

0 Pi exp ( -- ot ~o(x , t) -- 2 fi K D v/-~7

for t -- 0, for t > 0.

f12X2 4t )"

124.32 (MI) q(t)

~//////////////////~

Continuous abstraction of water from a plane source. The discharge is an arbitrary function of the time. qg(x, t) - drawdown.

C

0~o ~(O, Ox

t) --

q(t) 2KD

,

q)(x 0 ) - - 0 ' '

q)(oo, t) - - 0 .

~o(x, t ) -

2gDx/~

q t

f12x2 4ot2 ) exp ( -ot2

247

124.33

(L) Sudden discharge which is kept constant thereafter. As 124.32 with q ( t ) - q -- const, for t > 0.

q(t)

t q)~ •

qg(x,t) --

X

2KD

124.34 (D) Steady state of 124.33.

•

X2 )dot 4)~2ot2 ot2"

~o(x)

B

qJk e 2KD

~

_~

;~.

,,)

76

One-dimensional groundwater flow

(124d, e)

124.35 (L)

Linear increase of the discharge. As 124.32. q(t) = at, a [LZT-2].

•

~o(x,t)

=

- ~

a~ lot Pconj (~-~, ~ )

dto.

2KD

124.36 (L) As 124.32 but the discharge an exponentially increas-

ing function of the time. q(t) - qe ~2t.

s

t ~o(x,

t)

=

-

q

2 fl K D v/ rl + c~2

ea2t Pconj[ _~1 fl x v/ O +

}

~ 2 , ~/ t ( rl + ot 2) .

124.37 (L) As 124.32 but the discharge an exponentially decreas-

\ \

ing function of the time q(t) - qe -a2t.

\ \ \

t

*

cp(x, t) -

_

q

2flKD~/rl

-

o[ 2

e-aZt Pconj{ 21f l x ~ / 1 7

-

}

012' v / t ( o -- °12) "

e. Leaky aquifers with given initial head and zero flux at x = 0

02q9

(/9 .__ f12 0q9

OX2 ~2

--~'

)2

__

KDc

f12 _ '

-

S KD'

1

~ -

_

1

fl2)~2 - c--S"

124.41 (S) f(x)

q9

C

Initial condition (t = 0) an arbitrary function o f x. Plane at x = 0 impermeable. ~o(x, t) = head. qg(x, O) -- f (x),

0go ~(0, Ox

t) -- O.

(124e, f)

Flow in a semi-infinite field

77

/~2

e p{

qg(x , t) - 2"---~ fl e -r/t fo e<)

f12

(x-xo )2 J )2 ] )d o

124.42 (D} As 124.41 but f (x) - h - const. •

~o(x, t) -- he -~t.

124.43 (S) As 124.41.

f(x) h

h f(x)-

-b

0

b

0

for0<x b .

x

h ~o(x, t) -- - e -rtt [erf[fl(b2v/7+x)}+erf{fl(b-x)2~/7 ] "] 2

124.44 (L) As 124.41 with f (x) - h + a x .

xt'-e t{2a ierfct ,

fl

-~

+h+ax.

I

f. L e a k y aquifers with vertical infiltration a n d zero f l u x at x - 0 0 2 ~o Ox 2

~p )~2

q ( x , t) K D

= f12 0(t9 m, Ot

~,2

-- K D c ,

f12

=

S KD

,

fl2~2"

124.51 (S) A r b i t r a r y infiltration as a f u n c t i o n o f x a n d t. Imp e r m e a b l e p l a n e at x - 0. ~o - ~o(x, t) - head. /~//////////////¢

~x,t)

q -- q ( x , t),

qg(x, O) -- O,

~(0, Ox

t) -- O.

78

One-dimensionalgroundwaterflow

(124f)

1 qo(x , t) -- 2 fl K D #/_~

jo~fot q(xo,~ t - tO)e_Ot° [ exp ~- ~o,~ (x

-- X0) 2

j

+ exp { - ~o(1 +xo)2 } dto d/o.

124.52

(S>

As 124.51 but

I q(t) q(x, t) - 0 -b

0

arbitrary function of t.

b x

~o(x, t) -- ~-~

for-b<x b, and q(t) an

q(t - to)e -rlt° erf fl(b2~fi.o-x) + erf fl(b2~i.o+x)

dto.

124.53 (S) As 124.52 but q (t) = q - const.

q T q(x, t)

~illlilmlillli~ 0

-b

b

qc[

/~x2x

-T 2 - P

~ '

) - P (~+x~ / 2x

'

for - b < x < b , •

~ o ( x , t) -

qc[ (,x,_~ 2)~

,x,+~ ) '

2X

'

+e_,terf{fl(x-b) (x b) ] 2,k } - e-~/erf{/3 +

124.54

(D)

Steady state of 124.53.

e-~cos~(~)l qc sinh (~t e- ,x x qc{, -

~x,-

b

X

for-b<x


forx < - b a n d x

>b.

for Ix] > b.

Flow in a semi-infinite field

124.55

(124f- 126a)

79

(S)

As 124.51 but q(x, t) = q(x), an arbitrary function o f x . •

qg(x,t) --

2KD

q(xo)

{ txxo t Pconj

2)~

'

-+ Pc°nj ( x -+- xO2k, , ~'~-7)}dxo. 124.56

(S)

Steady state of 124.55. ~o(x)- 2KD

q(xo)

exp

-

Ix-xol)

+exp(

x +)~x 0 ) } dx0.

125. O n e - d i m e n s i o n a l s e m i - i n f i n i t e flow with c o m b i n e d b o u n d a r y value at x = 0

a. Is the combined boundary condition at x = 0 caused by entrance resistance ?

go to 126

See comments of 122c. b. Is the combined boundary condition at x = 0 caused by open water storage ?

go to 127

See comments of 122c. 126. O n e - d i m e n s i o n a l s e m i - i n f i n i t e flow with e n t r a n c e r e s i s t a n c e at x = 0

Subdivision: a. Confined aquifers with variable head at x = 0. b. Confined aquifers with given initial head and zero head at x -- 0. c. Confined aquifers with precipitation and zero head at x = 0. d. Leaky aquifers with variable head at x --- 0. e. Leaky aquifers with given initial head and zero head at x = 0. f. Leaky aquifers with vertical infiltration and zero head at x --- 0. a. Confined aquifers with variable head at x = 0 O2 q9 --- f12 0(t9 OX 2 -~' 126.01

~2 =

KD

(L)

The surface water level varies according to an arbitrary function of the time. ~0 = 99(x, t) = head. ~o(x, O) = 0 , w

x KD'S

~o(~, t) = 0 ,

KO~°(O~ t) -- F(t) - ~o(0, t) Ox ' w

F(O) = O.

80

(126a)

/ot ~t-,o, {1,~exp(-4~o )-,~(~x,~,~o) 1 } d,o

~x,,,-~ with 0

--

f1212w2

One-dimensionalgroundwaterflow

resistance parameter.

=

126.02 (L)

F(t)

Sudden rise of the surface water level, which is kept constant thereafter. ~o(x, t ) = head. As 126.01 but ~ g 0~°(0 t) -- h - ~p(0, t) 0X

'

1/3

~o(x, t ) - h erfc(2-~t) - h 126.03 (L)

Linear rise of the surface water level. As 126.01 but F(t) -- at, a [LT-1]. t

a{

~o(x,t)--~ 40 t i2erfc (2@t) - 2~0-} ierfc (2@t) + erfc(2-~t ) - R(~flx~/O, ~O-t)}. 126.04
Parabolic rise of the surface water level. As 126.01 with F(t)= a~/'{, a [LT-1/2].

~o(x, t) =

a,/7

2~-0 {2~--Tierfc(2~t)- erfc(2-~tt)-+-R(~ ~x v/-ff' ~ - Q }"

126.05 (L)

Exponentially increasing rise of the surface water level. As 126.01 with F ( t ) - he ~2t.

h t

( 126a, b)

Flow in a semi-infinite field

q)(x , t) - h J

4b.

1 , R(-~x~

,/7

[

R(_I 2(4~- + or)

81

,F)

0

~XO/, --(YW/~) -- t~ -- 0l 2

Confined aquifers with given initial head and zero surface water head and entrance resistance O2q3 __ fl2099

OqX2

[~2 __

-'~'- '

KD

126.11 (MS,L) Initial condition (t - O ) an arbitrary function of x. Surface water maintained at zero level.

f(x)

~o-0

99(x, t) - head.

jx w

99(x, t) --

f(xo)

O~o

q)(x, O) -- f (x),

KD, S

2~7

exp

- - ( 0 , t) -Ox

- ~-7(x - xo

/~ /~2 ÷ 2 ~ - 7 exp { - ~ 7 - ( x -+" X0) 2 }

-

1 + xo).,,/-ff, l---~-R{-~fl(x

Kw

V/-O--7] ] dxo.

(L)

126.12

As 126.11 with f ( x ) -- h -- const. 99(x, t ) -

h erf(2-~t ) -+-hR(~flx~/-ff, ~ff-O"

126.13 (L) As 126.1 1 with

f(x)

h I

b

~

f(x)--

0

for0<x b .

x

fix ,)- ~h i 2erf(~ t erf{~(x2~'b

} erf{~(x2,~} + b)

-2.{~.~ ' + ~.~ ~ } + 2 . ( ~' x ~ ~ t]

~o(0, t) Kw

82

(126b, c)

One-dimensional groundwater flow

126.14 (L)

h7 -

As 126.11 with f (x ) = h + ax.

~o(x, t)- h{erf(\2~7//3x'~+R(~flx~/"O, ~ ' ~ ) } + a K w { erf ( 2v't flx ]'~

R ( ~ flx~@, ~ 7 ) } +

ax.

c. Confined aquifers with precipitation; entrance resistance at x = O; zero surface water level 02q9

~

p(x, t) _ f12 O~o,

=

KD

126.21 (MS,L) Arbitrary precipitation as a function of x and t. Surface water maintained at zero

~o= 0 ~ ~ o

level. ~o(x, t) = head. p(x,t) to

•

KD, S

acp

p = p(x, t),

~ ( 0 , t) = Ox

x

~/o~/o ~p(xo, t - to) [ 2x/-~ ~~

¢p(x, t) -- f l g D

- ~/-OR{-~fl(x + xo)x/~, ~/-~} dtod/o. 126.2,2 (MS,L) As 126.21 with p(x, t) = p(t), an arbitrary function of time. ¢p(x, t ) -

~1 ~p ( t -

'1

to) eft(2~/~) + R (~flxx/~, x ~ o )

I

,

Kw

(~2~xxo,2,) 4to

exp -

+ exp ( - fi2 (x 4to + x°)2) )}

*

~o(0, t)

dto.

(126c, d)

Flow in a semi-infinite field

83

126.23 (MS,L)

Tp(x,t)

As 126.21 with

p(x , t) - I P(t) I

0

for0<x b

while p(t) is an arbitrary function of time.

cp(x, t) -- ~-~

p(t - to) erf

+ 2erf( fix

2~/~ 1

+ 2R(-~flx~,/O, ~ ~ o

- erfi

2~/7o

+

dto.

126.24 (MS,L) As 126.21 with p(x, t) = p(x), an arbitrary function ofx.

•

q)(x, t) -- flKD

p(xo) ~/~ierfc fllx2~-xol - ~/Tierfc fl(x2~/)_ + xo)

+_~erfc{fl(x+xo)

}-

1

,m} ] d/o

1

126.25 (MS,L) As 126.24 but p(x) = p = const.

•

~o(x, t)_~s{Ot_4otiZerfc(~)+2~-f[ierfc(_

__

- e r f c ( fix

_~)fix

2flxv/-ff, V-f[)

d. Leaky aquifers with entrance resistance at x = O; variable surface water level 0 2~

(t9

Ox2

X2 -- fl -57'

2 0 ~0

f12

~2

- KDc,

S 1 1 = KD' r / - flzx2 =~'cS

126.31 (L)

F(t) ~

. w

c KD, S

Rise of the surface water level according to an arbitrary function of time. cp = ~o(x, t) = head.

,p(x, O) =0, ~o(~, t) =0, Ocp F(t) - cp(O, t) - ~(o,t) = Ox Kw

F(O) -- O.

84

(126d, e)

One-dimensional groundwater flow f12X2

•

go(x, t) -- ~

fo'

F(t - to)e -~¢°

1 - ~/-~R (-~ ~x ~ ,

exp

1'

(

4,o )

v/-ff-~o) } dto.

126.32 (L) F(t)

As 126.31 with F(t) = h = const. (Sudden rise of the surface water level, which is kept constant thereafter.) go = qg(x, t) = head.

go(x, t) -- he-Ot {

2 ( v @ - ~/-~) 1 ~(_ ~ ~ -~) + 2(~/-~- + v/~)

_ ~o O-r/ 126.33

~-)}

R ( 1~ , / ~ ,

(D)

Steady state of 126.32. •

h)~ x ~o(x) - ~ e - r . K w + )~

e. Leaky aquifers with entrance resistance at x -- O; given initial head 0299 Ox 2

(.t9 __ 32 Og) )v 2 --~'

32 __

S

~2 __ K Dc, rl --

KD'

1 1 fi2)~2 -- cS"

126.41 (MS,L) ~

go=O

Initial condition (t = 0) an arbitrary function of x. Surface water maintained at zero level. go = go(x, t) = head.

~ f ( x )

~o

w

V(x, O) = f (x),

KD, S x

~(~,,~- fo~

Ogo ~(0, Ox

t) =

~(~o~[ ~ ,3~ ~xp { - ~3 2( ~ - xo~ / 3

3 (x + xo)2}

+ 2~/~ exp 1 - ~

- / 7 7'

~(~ + xo~,/-L ,/b7 }] e-~'dxo.

go(O, t) Kw

(126e, f)

Flow in a semi-infinite field

85

126.42 (MS,L) As 126.41 with f(x) = h = const. +

126.43 (MS,L) As 126.41.

f(x)

{h

i

f (x) -

i_

b

0

f o r 0 < x b.

x

h e-"' I 2 e r f ( 2/3x ~o(x, t ) - -2 - ~ ) - e r f { fl (x2~/7b) } _ erf{/~ (x2v~ } +b) -2R{~

1

126.44 <MS,L) As 126.41 with f (x ) = h + ax.

~p(x, t) - e -nt

(h + a K w ) e r f

-~

+(h-aKw)R

-~flx~fff , ~

+ax

.

f. Leaky aquifers with entrance resistance at x = 0 and vertical infiltration 02(t9

(t9

OX 2

~2

q (x , t ) t

K D

Oq9 = f12

--~'

f12

S _~.

KD'

~ 2 __ K D c .

126.51 <MS,L)

Y/////////////////////////Z ¢

~

as a f u n c t i o n o f at x = 0. Zero surface water level. ~p -- ~p(x, t) = head.

Arbitrary vertical infiltration x and t. Entrance resistance

~o- 0 ~ ~ ' ~ o ~(x,t)

q -- q ( x , t)

O__~ '

~(x, O) - O,

Ox ( O ' t ) -

~p(O, t) Kw

~ ( ~ , t) - O.

86

•

(126f, 127a)

One-dimensional groundwater flow fl2(x - x o ) 2

1

1 fo fo '

~o(x, t) -- f l K D

1

/~2(x+xo) 2 + exp

-

)

j] d,odxo

4to

126.52 (MS,L) As 126.51 with q(x, t) = q(t), an arbitrary function of the time.

*

qg(x, t ) -

"S1

t q ( t - ,o)e -nt° e r f ( 2 ~ / ~ ) q- R

flx~/-ff, v/-~o)

dto.

126.53 (D)

Constant vertical infiltration. Entrance resistance at x = 0. Zero surface water level, q9 = qg(x) = head. Steady state.

y/////////////////////_//_///C

~i$$,1,,1,,1,,1,$,1,,1,$$$q

dgo go(O) dx (0) - K - - - ~ '

dgo d---;(oo)- O.

x

(

q)(x) -- qc 1 - & + K w

"

127. One-dimensional semi-infinite groundwater flow with open water storage Subdivision: a. Confined aquifers with given initial head. b. Confined aquifers with precipitation. c. Confined aquifers with abstraction from the open water.

a. Confined aquifers with given initial head 02q9 __. f12 0~0 Ox 2

"~

'

f12 =

S KD"

127.01 (MS,L)

f(x)

Initial condition (t = O) an arbitrary function of x. q) = ~o(x, t) = head. ~o(x,

x

0)-

/O / f(x)

forx--O, forx >0,

KD, S b 0~° 0~°(0 t). 8--7(0, t) - K D ox '

Flow in a semi-infinite field

~o(x,t)--

foo

(127a, b)

[ fl

fl2(X __ X0)2

f(x) 2~.~.7 exp [ -

4t

fl 2~

87

]

fl2(X + X0)2 ] exp / -

4t

S 1 + ~R{ ~(x + xo~, ,m] ] d/o with a

-

127.02

fl2(KD)2 b2 --" storage parameter. (MS,L)

As 127.01 with f (x) - h - const.

{

1

~o(x, t) -- h 1 - R(-~flx~f~, 127.03 (MS,L) As 127.01 with

f(x)

l q)(x,t)-~

f(x)--

x hi

eft{

{h 0

f o r 0 < x I.

fl(x + l) erf[ fl(x - l) 2~q 1 2,,/7 /

+ 2R {~fl(x + l)~f~-,

1

~-~)

b. Confined aquifers with precipitation 02q9 p(x, t) __ ~2 099 OX2 ~ K D --~'

~2_ KD

127.11 (MS,L) Arbitrary precipitation as a function of x and t. ~o - ~o(x, t) - head.

p(x,t)

~o(x, 0) - 0,

b~-t (0 , t ) - KD 0~° a x (0,

t).

88

(127b) ¢p(x, t) --

fl K, D

- exp

p(xo,

fo /o '

One-dimensional groundwater flow

1 -#2(x t - to) ~ 2 ~ [ exp 4to

1

- # 2 ( x + xo) 2

xo) 2

]

4to

127.12 (MS,L> As 127.11 with p(x, •

an arbitrary function

t) - p(t),

lf0t

tp(x, t) -- -S

{ (1

of the time.

)/ dto.

p(t - to) 1 - R -~[3x~r~, ~

127.13 (MS,L)

l p(x,t)

As 127.11 with p(x, t) -

x

I

p(t)

~o(x, t ) -

'fo t p ( t -

-~

to)

p(t) forO <x /.

0

is an arbitrary function

+ x)- / [erf{ ~(124~_~

of t.

erf{ fl(12s/~ 1 +x)

, * ~l ~~(x.,~~, ~ / - ~(~x~, ~ t ] ~,o. 127.14 (MS,L) As 127.11 with p(x, •

t) -

~o(x, t) - f l K D

p(x), an arbitrary function p(xo)

of x.

~/Tierfc ¢~lx2v~-xol + ~/Tierfc ¢l(x2~/t + xo)

1 /~(x + x o ) /* 1 ~/ 1 ~ ( x . x o ~ , ~er~c/ 2s/7 ~

~/~1"] dxo.

//

127.15 (MS,L) As 127.14 with p (x) - p - const. *

¢p(x, t) -- ~-~ /~x + erfc(~-~) /~x - R ( ~1f l x ~ , Yr'-t) }. P { ¢7t -- 2s/~ierfc (~-~)

Flow in a semi-infinite field

(127c, 128a)

89

c. Abstraction from open water with storage for confined aquifers __ fi2 0(49

02(/9

f12 __

S

127.21 (L) Discharge an arbitrary function of t. ~o = qg(x, t) = drawdown.

]l~ ;~q(t)

l

~(x, 0) : 0 ,

bl

qg(~, t) - 0 ,

q(t) -- b-~(O, t) - K D o x (0, t).

KD, S x

lfo

qg(x, t) -- ~

q (t - to) R ~ fix v/~, ~

)

dto.

127.22 (L) As 127.21 with q(t) : q : const. ,

p(x, t ) -

q { 2 ~ ' ~ ierfc (2--~t ) - e r f c ( - ~ )fix ff~ + R ( - ~ f l x ~1/ ~ ,

~/-~) } .

128. One-dimensional periodic groundwater flow in semi-infinite aquifers Subdivision: a. Tidal fluctuations of the open water. b. Periodic abstractions. c. Periodic precipitation or infiltration.

a. Tidal fluctuations of the open water 128.01 (PF) Confined aquifer with open boundary at x - 0. Open water fluctuation according to a sine function. ~0 = ~o(x, t) = head = steady oscillation. 99(0, t) -- h sin(cot),

(p(x, t) -- h Im[e -/~x~/~'~+i~°t] - he - J x sin(cot - co'x), (2) - T

'

~p(c~, t) - - 0 .

90

One-dimensional groundwater flow

(128a)

128.02 {PF) Confined afuifer with entrance resistance at x -O. Open water fluctuation a sine function. ~0(x, t) = steady oscillation.

H

w

qg(cx), t) = 0 ,

KD, S x

-- K o~p h sin(cot) - qg(O, t) o x ( O , t ) -to

[e-~/~+iwt ]

q g ( x , t ) - h~/-fflm

t~7-~ +

e --O)l X

sin

=

cot - co'x - arctan ( ~ / ~ +

/~__~) ,

v/co + # + ~/2coO

1

m

f12K2w2

[T-l]

•

128.03 (PF) Leaky aquifer with open boundary at x - O. Open water level fluctuation according to a sine function, q) - ~p(x, t) - head - steady oscillation.

V////////////////////////~ C

I

KD, S

x

9(0, t) -- h sin(cot),

~0(c~, t) - 0 .

~o(x, t) -- h Im[e -~x~/74-77+i°Jt] -- he -eax s i n ( c o t - flbx),

1 x/O + ico -- a + ib

and

r# - -

1

1~2~2

--

cS

~

o

128.04 (PF)

V////////////////////////// C

~_..

KD, S

Leaky aquifer with entrance resistance at x - O. Open water level fluctuation according to a sine function. ~0 - qg(x, t) - steady oscillation of the head. K Og (O t) -- h sin(cot) -
'

~o(c~, t) - 0 .

W

Flow in a semi-infinite field

(128a, b)

91

e-[3x~//-~+io)+icot ]

q)(x, t) -- h4/-O Im I ~ - ~ i w ;

~-_1

e - ~ax

b

sin{cot-flbx-arctan(a

/ ( a --t- w/~0-) 2 + b 2

1

v/rl + im -- a + ib,

1 /7-

f12K2w2,

1

fl2)~2 -- c S

b. Periodic abstractions 128.11

(PF)

Confined aquifer with groundwater abstraction by a well-screen. Discharge according to a sine function. 99 - ~0(x, t) - steady oscillation. Oq)

q sin(wt)

~(0,

t) -- - - ~ ,

Ox

2KD

q g ( ~ , t) - - 0 .

e - [3x ~/[-~+ i cot

q

qg(x, t) -- 2 f l K D q 2~/(D,/-g e- J x

128.12

sin (cot

-

co'x

4

(PF)

vqsin(s ~////////////////////~

Leaky aquifer with groundwater abstraction by a well-screen. Discharge according to a sine function. q9 - qg(x, t) - steady oscillation.

C

Oq9

q sin(wt)

~(0,

t) -- - ~ ,

Ox

q ~ I m ~p(x, t) -- 2 f l K D q

~ p ( ~ , t) - - 0 .

-~ ~-7-£~~+io~,]

[e

~ / r / + iw

e -/~"x

2 f l K D ~/a e + b e v/rl + iw -- a + ib,

2KD

sin c o t - f l b x - arctan a

1

1

fl2~2

cS

'

92

(128b, c)

One-dimensional groundwater flow

128.13 (PF)

II

Abstraction from open water with storage for a confined aquifer. Discharge according to a sine function. ~o = ~o(x, t) = steady oscillation.

>qsin(~t) .

.

.

.

.

.

.

q sin(cot) -- b--~ (0, t) - K D O--~-9(O, t), Ox ~p(cx~, t) = 0 . --o.)tx C

~o(x, t) -- g Im ~-~(~7-~ + ~/~-)

ico(co + ~ + ~/2coo')

x sin { c o t - c o ' x - arctan( 2zr T'

co

co,

fl ~ '

~ ,/;+472

)

-4

7r}

'

fl 2( K D ) 2 b2

c. Periodic precipitation or infiltration 128.21 (PF)

tp = 0 ~ ' _ p $ $ $ $ $ $ $ x

Confined aquifer with open boundary at x = 0. Precipitation fluctuations according to a sine function. ~p -- ~o(x, t) steady oscillation.

psin(rot)

KD, S

a2~0 i- p sin(cot) = t~ 3~o -~x~

~o(x , t ) -

~p Im[1L i--w (1 _ e_t~x,4G~)eiO,,], ~p(0, , ) P -

(.O - -

2Jr

~ ,

sin w t -

-

t" 2 - -

at

0.

sin w t - c o x

~oS

09

K D

2

'

, #g/~~ m

T 128.22 (PF)

As 128.21 but with entrance resistance at X--0.

~o= 0 ~ ~ $ $ $ $ $ $ $ ~w

x

KD, S

psin(wt)

K 0___~_~ ¢p(O, t) Ox (O't) -w

Flow in a semi-infinite field

go(x t ) - - p I m '

S

(128c)

[l( m

1-

93

~/-Oe-/3x + ~/.~ ) eiWt]

i(0

P

~os

sin c o t -

-

Y

,Ao+ o + 2,/Y~

x sin lcot - w'x - arctan( 4c-d + -~-~) - -~ } , _

.

_

f12K2w2"

128.23

(PF)

Leaky aquifer with open boundary at x =

~

0. Vertical infiltration a sine function of the time. go = go(x, t) = steady oscillation.

qsin(wt)

02(/9 0X 2

go q sin((0t) = f12 ago ~2 ~ KD 0---t-'

go(O, t) = 0 .

go(x, t) -- ~q Im

I rl +1 i(0( 1

S V/r] 2 --[- (0 2

-

e

[sia/-t-

arctan(~) ]-e

x sin [(0t- flbx -arctan(-~)

128.24

,

-¢~ax

x//~ + i(0 -- a + i b.

(PF)

Leaky aquifer with entrance resistance at

-- -- ~~/////////////

C

N$$$,[.$,[.$,[.$$,[.,[. qsin(wt) Iw x

x = 0. Fluctuation of the vertical infiltration according to a sine function, go -go(x, t ) = steady oscillation.

KD, S 0299

q sin(cot) = f12 0(/9

(/9

KD K a_fv ~o(o t) ox (O' t) -- ~ ' Ox 2

i~,2

~

Ot

94

(128c)

~o(x, t) -- ~ Im

1-

r / + ico

One-dimensional groundwater flow

~77g77~,~e i°)t o9

~/-~e -flax

b ] a+~)/

- arctan(~

xsin{cot-flbx-arctan(c°)

\0 ~/o + ico - a + ib.

128.25

-

~

(PF)

,~ 1

Confined aquifer with open water storage. Fluctuation of the precipitation according to a sine function, q) = qg(x, t) -- steady oscillation.

- - - - - - - - P sin (coo

0299 Ox 2 t

p sin(wt) 0q) KD = f12 Ot'

b Oq) 0-'7 (0, t) -- K D Ocp Ox (0, t).

[11ico

qg(x, t) -- ~ Im

p _1 sin cot = -S co

xsinl ,

~7-~( /~-~ + ~/r~-)

2

IeiOOt

V/CO(co + a + ~/2wa)

Flow in a finite field

(130-132a)

95

130 BI-3. ONE-DIMENSIONAL GROUNDWATER FLOW IN A FINITE FIELD

a. Has the soil discrete inhomogeneities in x-direction? P _

_

go to 140 (BI-4)

An example of flow through an inhomogeneous medium is flow caused by precipitation on an infinite strip with different K D-values.

go to 131

b. The soil is assumed to be homogeneous

131. Homogeneous soils a. Is a periodic function involved in one of the boundary conditions of your problem?

go to 139

In general, periodic functions occur in the form of sine- or cosine functions, for instance, seasonal fluctuations of the precipitation or tidal fluctuations of surface water. Groundwater flow that more or less depends on such periodic fluctuations is referred to as periodic flow. If a boundary condition contains a general function of the time, it sometimes is useful to develop the function in a Fourier series (if possible) and derive the solution of the problem from a more simple periodic function. go to 132

b. The flow is non-periodic

132. Non-periodic flow a. The boundary condition at x : b is a given head or drawdown and at x = 0 a zero flux

go to 133

All solutions of one-dimensional flow in a finite field, given in the sequel relate to an interval of x between 0 and b. However, they may be also valid for vertical one-dimensional flow in an interval 0 ~< z ~< D for instance:

-IF(t) IF( t)

b In most cases the given head or drawdown will be caused by surface water (straight river or canal that fully penetrates the aquifer or a lake or sea on top of an aquifer) which level may be an arbitrary function of the time: ~o(b, t) -- F(t).

96

(132a-d)

One-dimensional groundwater flow

In many problems F ( t ) will be a constant or zero: go(b, t) = h or go(b, t) -- 0. A zero flux at x -- 0 means that the solution of the problem is valid for a strip with width 2b and symmetrical conditions. There is no entrance resistance between the aquifer and the open water. b.

There is a g i v e n h e a d or d r a w d o w n at x -

b and

a

zero h e a d at

x-O

go to 134 In this case a strip with width b will be considered with non-symmetric boundary values.

-

x

b c. The b o u n d a r y c o n d i t i o n at x -

b is a given f l u x a n d at x -

zero f l u x

.

.

0 a

go to 135 .

.

.

(t)

.

b

A given flux means that a given amount of water is withdrawn from the aquifer by means of a fully penetrating infinite well-screen, located at x = b. Also a very long row of wells at short distances of each other can serve as such a well-screen.

The discharge may be an arbitrary function of time q ( t ) and the boundary condition will be

a--~-'(b, t) Ox

--

q(t) 2KD

A zero flux at x - 0 means that also at x with the same discharge q ( t ) . d. G i v e n f l u x f o r x - b a n d zero h e a d at x - 0

_~(t) ~

go to 136

Also at x - 2b the head will be zero and the solution is that for an infinite fully penetrating well-screen in the middle of a strip with width 2b. D

b

- b groundwater is abstracted

(132e-133a)

Flow in a finite field

97

There is a c o m b i n e d b o u n d a r y condition at x = b, c a u s e d by

go to 137

e n t r a n c e resistance

In general, the boundary condition will be: O~p -~(b, Ox

t) --

~p(b, t) - F ( t ) Kw

while in many cases F ( t ) = 0. There always exists a j u m p between the g r o u n d w a t e r head at x = b and the surface water. The b o u n d a r y condition at x = 0 may be either 99(0, t ) - 0

or

aOx (o ' t) - o.

f. There is a c o m b i n e d b o u n d a r y condition at x = b, c a u s e d by surface

go to 138

water storage

I

h

b

~o0___L(b, , t) -Ox

a

The surface water level at x -- b is no longer a given function of time (or a constant or zero) but depends on the outflow (inflow) of the g r o u n d w a t e r from (into) the aquifer, owing to limited dimensions of the water course (ditch or canal with width a)

a 0~p (b t). K D Ot '

The b o u n d a r y condition at x = 0 may be either ~0(0, t ) - 0

133.

One-dimensional

or

a--EE(o Ox ' t) - o.

finite flow; g i v e n h e a d or d r a w d o w n at x -- b a n d z e r o

f l u x at x = 0

Subdivision: a. Confined aquifers with variable head at x - b. b. Confined aquifers with zero head at x - b. c. Leaky aquifers; given head or d r a w d o w n at x = b. a. C o n f i n e d aquifers with variable h e a d at x = b

02q 9

20(/9

ax2

77'

f12 __

S

KD

98

(133a)

One-dimensional groundwater flow

133.01 (L,FS)

~

The surface water level rises according to an arbitrary function of time. cp(x, t) - head.

]F(t) xKO,

~o(x, o) - o,

S

with F(O) - O,

qg(b, t) - F(t) b Oqg(O, t) - O. Ox OK)

•

~o(x,t) = Jr ~7~(_1)~(2n + 1)cos {(2n + 1)7rx ] f12b2 2b n--0

x

exp -

(t-- r)] F(r) dr.

2/3b

133.02 (FS) Sudden rise h of the surface constant thereafter, q9(x, t) -

F(t)

h

water level, which is kept head.

!a

•

~o(x, t) -- h - - -

Ox (0, t) -- O,

~p(x, O) -- O,

¢p(b,t)--F(t)=

l0 fort--0, / h for t > 0.

4h 1 " 2 f i b }2t] jr- ~ ( - 1 ) ~ ~ 2 n + l c°s{ (2n+l)rrx2b }exp [ -{ (2n+l)rr n--O

133.03 (FS) Linear rise

of the surface water level. ~o(x, t) - head.

~o(x, o) - o,

a--~ (o, t) - o Ox

cp(b, t) - F(t) - at. ~o(x, t) -- a t - aft 2 (b 2 - x 2) +

2

[

× exp -

2fib

16afl2b 2 ~

Jr 3

2t ] .

n=0

(-1)n

(2n + 1)

cos { (2n +2bl)Zrx }

Flow in a finite field

(1 33a, b)

99

133.04 (FS) Exponentially increasing rise of the surface water level. 99(x, t) = head. qg(x, O) = 0 ,

/ I I

099 ~ ( 0 , t) --0,

Ox qg(b, t)-- F(t)--h(e a t - 1) q9(x, t) -- h { e at

+

cosh(flx

/-d)

1]

_

cosh(eb,/a)

4ah k

(-1)n

(h and a positive).

c o s {(2n+l)nx 2b } exp

[_ { (2n + 1)Tr 2] 2fib

Jr n=O 2n + 1 a + { .(2n+l)n2flb . }2

} t .

133.05 (FS) Exponentially decreasing rise of the surface water level. 99(x, t) = head.

/[

~o(x o) - o,

f

'

o--E(o, t) - o, Ox

~o(b, t) -- F(t) = h(1 - e-at). q g ( x , t ) - h { 1 - e - a t c}cos(fib/-~) °s(~x~) + 4ah ~

( - 1 ) " c o s { (2n+l)rrx. 2b } exp [ - { (2n 1)rr2flb + } 2t]" Z.., 2n + 1 { (2n+l)n" 2

7l" n=O

2fib } -- a

b. Confined afuifers with zero head at x -- b 133.11 (FS)

Initial head an arbitrary function of x. Zero head at x = b. Zero flux at x -- O. 99(x, t) = head. ~p(x, O) = f (x),

~(b,t)

=o.

!0 ox(O,t) - 0 ,

100

(133b)

V(x, t) -- -£ Z

n=O

c°s

2n+l xj exp

One-dimensional groundwater flow

-

2b

t 2fib

x fo h f (xo) cos [ ( 2 n + 2b1)Zrxo ] dxo. 133.12 (FS)

f(x)

As 133.11 with f ( x ) -

[

h-

const. (see 133.02).

r

b

x

q)(x, t ) _ __4h~ (--1) n c°s[ (2n+l)zrx [ (2n+l)zr 2t] • 2b ]exp - 1 2fib ] 7r n - - - 0 2 n + l 133.13 (FS)

f(x)

As 133.11. Parabolic initial value: h

f ( x ) - ~--~(b2 - x2).

b q)(x, t) --

32h ~

(-1) n (2n + 1)Jrx ]exp[-{ cos 2b (2n + 1)3

Yr 3

(2n+l)rr2flb ]2It.

n=O

133.14 (FS)

As 133.11. Initial value a cosine"

f(x)

f (x) - h cos (_~._)Jr x . b

~o(x,t)-hcos

~

exP\4fl2b2 .

133.15 ( L ) Arbitrary precipitation as a function of time. Ze-

~ p ( t )

ro flux at x - 0, zero head at x - b. q) ~p(x, t) - head.

0299 ~ p(t) =320q9 b

Ox 2

KD

a-~-v(o, t) - o,

Ox

0--7'

g)(x, O) -- O,

v(t, t) - o.

(133b, c)

Flow in a finite field

4 £ ~o(x, t) - ~

101

(-1)" { ( 2 n + 1)7rx} 2n + ] cos 2b

n=O

x

exp

-

(t - r ) ] p ( r ) dr.

2/~b

133.16 (L) As 133.15 with p(t) = p = const. •

P

~p(x, t) -- 2 K D

(b2

x 2)

16pb 2 £

(-1)n

cos { (2n +2bl)Zrx }

rc3KD n=o (2n + 1) 3

x exp

[ / 2n÷l /2 ] -

2/3b

t .

133.17 (D) Steady state of 13 3.16.

•

P

~o(x) -- 2 K D

(b2 __ x2).

e. Leaky aquifers; given head or drawdown at x -- b and zero flux at x = 0 02q 9

(/9 .._ /~2 0(/9

Ox 2

~2

fi2 _

~2

--~'

--

KDc,

S

1

-- K D '

1

rl - - fl2~2 = c----S"

133.21 (L,FS) The surface water level rises according to an arbitrary function of time. ~o(x, t) = head. ~p(x, 0) = 0, with F(0) = 0, b

Jr°° ~p(x, t ) -

)n

f12b2 E ( - 1

(2n + 1)cos

n---0

x

fo' [ / 2n+ exp

-

4f12b 2

~o(b, t) -- F(t) 099 ~ ( 0 , t) = 0. Ox

{ (2n + 1)zrx } 2b

102

(133c)

One-dimensional groundwaterflow

133.22 (FS) F(t)

•

Sudden rise of the surface water level, which is kept constant thereafter. ~o - ~o(x, t) - head.

~o(x, t) -- h

Oq9(0, t) -- 0, 0x

~o(x 0) -- 0,

~o(b,t)- F(t) =

0 h

oo (-1)n(2n + 1) cos { (2n +2bl)Zrx ] b - 4n'h Z (2n + 1)27r2 + 4bZ/,L 2 cosh(~) n--O cosh(~) ~

x exp

133.23

fort--O, fort>O.

--

{ (2n + 1)27"t'2 4/~2b 2

+ Jt]

(D)

Steady state of 123.22. •

~o(x) -- h

133.24

cosh( ) ~ cosh( ) b

°

(FS)

Initial head an arbitrary function of x. Zero head at x - b. Zero flux at x - 0. ~o - ~o(x, t) - head. ~o(x O ) - f (x)

c

'

ogO__.L(O _~ t ) - - 0 '

OX

'

'

~o(b, t) - 0 .

~o(x, t) -- ~ ~ cos n=O

2b

exp

-

4/32b 2

fo b f (X0) COS { (2n + 1)7rx0 2b ] dxo. 133.25 (FS)

f(x)

As 133.24 with f (x) - h - const.

x

+

Flow in a finite field

(133c)

103

£ (-1) n cos { (2n +2bl)rrx } 7l" n=O 2n + 1 [--{ (2n q- 1)27r2 ] x exp 4f12b2 q- r/}t .

4h go(x, t) -- ~

133.26 (L) Arbitrary vertical infiltration as a function of time. Zero flux at x = O. Zero head a t x -- b. go = go(x, t) = head. xKD, S ]

0299 OX 2

99 )v 2

b

q(t) KD

aOgo x (0, t) -- O,

go(b, t) --0.

4 k (-1)~ cos 2n -t- 1 n=O

{ (2n +2bl)Zrx }

~o(x, o) - o,

go(x, t) - 7-S

t

x

~_

fo

exp

_ { (2n +

1)27/'2

4f12b 2

--

fl2 Ogo i at

+ ??}(t- r ) ] q ( r ) d r .

133.27 (L) As 133.26 with q(t) = q = const. { cosh(~) } go(x, t) -- qc 1 b cosh(2) × exp

-

133.28 (L) Steady state of 133.27. cosh(~) cosh(~)

4f12b 2

4q

0° ( - 1 ) n cos { (2n+,)zrx 2b } ~ (2n + 1){ (2,+1)2rr2 zrKD .=0 4b2 + ~} q- r/ t .

104

(134a)

One-dimensional groundwater flow

134. O n e - d i m e n s i o n a l finite flOW; given h e a d or d r a w d o w n at x -- b a n d zero h e a d at x -- 0

Subdivision: a. Confined aquifers. b. Leaky aquifers. a. Confined aquifers

02(t9 0(t9 OX2 __ f12__~,

fi 2

S KD

134.01 (L,FS) The surface water level falls according to an arbitrary function of time. ~o - ~p(x, t) - drawdown.

-

KD, S

x

•

2re

cp(x,t) --

~

f12b2 Z ( n= 1

134.02

cp(x, 0) -- 0,

qg(0, t) -- 0,

qg(b, t) -- F(t)

with F(0) -- 0.

1)n+ l n sin ( n z c x ] f0 t exp 1 - (~-~-) nTr 2 ( t - v ) } F ( z ) d v \b/ J

(FS)

Sudden fall of the surface water level, which is kept constant thereafter, q0(x, t) - drawdown.

h

F(t)

~p(x, 0 ) - 0 ,

~0(0, t ) - 0 ,

~o(b t ) - - F(t) -- I O f o r t - - 0 , ' | h fort>0.

•

hx

cp(x,t) --

134.03

b

2h ~ 7l"

(D)

Steady state of 134.02.

hx ~(x)

-

--.

b

n= 1

(-1) n + n

1

sin

\~b/

t}.

Flow in a finite field

(134a)

105

134.04 (FS) Linear rise of the surface water level. ~o(x, t) -- head.

~o(x, o) = o ,

~o(o, t ) = o ,

~p(b, t) = F(t) -- at. •

~(x,t)

=

axt b

aft 2x (62 _ x 2) 6b

2afl2b 2 ~ 7r3

(-

1)n+l n3

in(- )exp(

)

nygx

n= 1

134.05 (FS) Exponentially increasing rise of the surface water level. ~o(x, t) = head. qg(x, O) - - 0 ,

~ I~ i -h

i

'

~o(0, t) - - 0 ,

~p(b, t ) = F ( t ) = h(e a t - 1)

sinh(flb~)

(a and h positive).

b

+ rr ,~=l n( nz~r2~-7-~-+ a) sin

b/

exp

f1262

.

134.06 (FS) Exponentially decreasing drawdown of the surface wa-

\ \\

ter level. ~o(x, t) -- drawdown. t

~o(x,O) = 0 , h

.......

~o(O,t) = 0 ,

99(b, t) = F(t) = h(1 - e -at)

x _ e_,t sin(/~x ~/-d) } qg(x, t) - h ~ sin(flb~)

2ah ~ .ql_

( - 1 ) n+l (nrcx) [n27r2 -- a) sin - 7 exp Jr n=l n ~~-7-~

(a and h positive).

(--n27r2t) f12 b 2

.

106

(1 34a)

One-dimensional groundwater flow

134.07 (FS) Initial head an arbitrary function of x. Zero head at x = 0 and x = b. qg(x, t) = head.

99(x, 0) = f ( x ) ,

99(0, t) = 99(b, t) - - 0 .

88gerg83988

OO

nTrxo ) 2 Z sin ( n n ' x ) (--n27r2t) fob f (xo) sin b dx0. 99(x, t) -- ~ n=l b / exp f12b2

134.08 (PS)

f(x)

l /

As 134.07 with f (x) = ax a linear function of x.

i

,

I I

b 2ab ~ •

~ o ( x , t) =

7/"

n-- 1

( - 1)n+l n

nyrx .f12b2 ) sin ( - - - f f - - ) e x p ( -n27r2t

134.09 (D) x

Variable thickness of the aquifer. Zero drawdown at x = 0. Drawdown h at x = b. Steady state. D(x) an arbitrary function of x for 0 ~< x ~< b, all values of D(x) being positive and finite.

b dot

•

~o(x) =

h fo D(ot) fob dot D(ot)

discharge q =

Kh f~

dot D (ot)

b. Leaky aquifers 0299 OX 2

99 = ~2 a~o -g, )v 2

)v2 -- K Dc, f12 =

S KD'

1 1 ~ -- flz)v2 -- cS

Flow in a finite field

(134b)

107

134.11 (L,FS) The surface water level falls according to an arbitrary function of time. go - go(x, t) - drawdown.

V//////////////////////A C

~o(x, o ) - o ,

~o(o, t ) - o ,

go(b, t) -- F(t)

b OO

27r

1),+ ,

go(x, t) -- flZb2 Z ( tl--

×

with F(O) - O.

(nTrx] n sin \ b /

1

exp

--

f12b2 + ~ ( t -

r)

F(T)dr.

134.12 (FS) Sudden fall of the surface water level, which is kept constant thereafter, go(x, t) - drawdown. F(t)

~o(x, 0 ) - 0 , go(b t ) '

go(x, t) -- h

sinh(~) ~ -2zrh Z sinh(~) n=

~o(0, t ) - 0 ,

-- F(t) -- I O [ h

fort--0, fort>0.

nZTr2 -t- b2/)~ 2 sin \

b

/ exp

--

flZb2 +r I t

.

1

134.13 (D)

Steady state of 134.12.

•

g0(x) -- h

sinh(~) ~ sinh(-~)"

134.14 (FS) Initial head an arbitrary function of x. Zero head at x - 0 and x - b. cp(x, t) - head. m

_ _

(//////////////////////A C

b

go(x, O) - f (x),

~o(o, t) -

~o(b, t) -

O.

108

(134b)

2 °°

(/l:r/'x]


{

sin --b--/exp

n--1

x

f (xo) sin

One-dimensional groundwater flow

(n2:r/"2 -

f12b2

+ tt}

b

134.15 (FS} I

f(x)

As 134.14 with f (x ) -

ax a linear f u n c t i o n of x.

1)n+l

2 f12b2

, I I

b 2ab ~ •

(~

qo(x,t) -n=

n

1

rt~x

sin (---if-) exp { - ( n27r -k r/)t }.

134.16 (L,FS) Vertical infiltration an arbitrary f u n c t i o n o f space and time. Zero head at x - 0 and x - b.

~o -
q(x,t)

02(t9 (t9 q (x , t ) O~O OX 2 -- )v2 [ K D = t~la2 at 9

~

~o(x, O) - 0 ,

b

~o(x, t) -- ~ ~

n--1

sin k--if-

exp

go(0, t) -- go(b, t) -- 0.

-- f12b2 -+- rI

(t -- to)

dto dxo. × q (xo, to) sin nrrxo) b 134.17 (L,FS) As 134.16 with q ( x , t ) 4

c¢


q(t), an arbitrary function o f time only. 1

2n + 1

sin { (2n +b 1)Trx ]

?l~O

×

fo/exp { -- ( n2 f12b22 -+- ~7)

}

(t - to) q(to) dto.

134.18 (L,FS} As 134.16 with q ( x , t) - q(x), an arbitrary function o f space only.

Flow in a finite field

,

qg(x,t) --

(134b, 135 a)

109

/,/27/.2 } 2b ~ 1 sin \(nrcx~b/ 1 - e x p { - ( f i 2 b 2 +ri)t} K D Z narra + b2/)~2 n=

×fo b

1

q(xo) sin (nrcxo b ) dxo.

1 3 4 . 1 9 (FS)

Steady state of 134.18.

•

2b

oo

e(x) - K D Z/,/27/,2 n--1

1

_.1_ b 2/)V2 sin

(nrcx'] fob (nrcxO)dxo" -if- / q (xo) sin \ b

135. O n e - d i m e n s i o n a l finite flow; given flux at x - b a nd zero flux at x - 0

Subdivision" a. Confined aquifers. b. Leaky aquifers.

a. Confined aquifers 02~ ._ f12 Oq9 OX 2 -'~'

f12 __

KD

135.01 (L,FC)

_~

(

, . . . . . t)

Fully penetrating infinite well-screen at x -- b. Groundwater abstraction an arbitrary function of time. ~p(x, t) - drawdown. Oqo Ox

~o(x, O) - o ,

Oq9 Ox (b't)

b

- - ( o , t) - o,

q(t) 2KD

1 for q ( r ) dz" ~p(x, t) -- 2bS (t - r) }q(r) dr. +

135.02

-7

n=l

-

(FC}

As 135.01 with q(t) - q - const.

qt qg(x, t) -- 2bS

q ~(3x2 12bKD

71"2KD n--1

-- b 2)

n-------T--cos \ b

exp

\ fib / t .

110

(135a)

One-dimensional groundwater flow

135.03 (FC) Initial head an arbitrary function o f x. Zero flux at x = 0 and x = b. Depletion function qg(x, t) = head.

0~°(0 t)-- 0q9

cp(x, O) - f (x),

Ox

'

-~x (b' t) - O"

b

•

rp(x, t) -- ~

f (xo) dxo

+ cos( --if-- )exp{

\fib/

t

f (x0) cos (n:rrx° b ) dxo.

n=l

135.04 (D) Steady state

•

of 135.03.

ifoh

tp(x)-

-~

f(xo)

d x 0 - const.

(no flow).

135.05 (FC) As 135.03 with f (x) a linear function of x: f (x) = ax.

I

I I I

b

1

qg(x, t) - - a b 2

~ 1 {(2n+ 1)7rx } Jr 2 n~o (2n + 1)2 cos b

4ab

xexp{-(

(2n+ 1)2jr2) } f12b2

135.06 (D) Steady state

of 135.05.

1

~o(x) --- - a b

2

= const.

(no flow).

t .

(135a, b)

Flow in a finite field

111

135.07 (D} Arbitrary precipitation as a function o f time. Zero

flux at x = 0 and x = b. ~p = 99(t) = head.

(t)

f12 0(t9 _ p ( t ) at -- K D '

cp(O) -- O.

b •

~o(t) -

135.08

~

p(to)

dto.

(D)

As 135.07 with p ( t ) = p = const. pt

e(t) =

- - .

S

b. Leaky aquifers 0299

q9 = fi 2 0(t9

Ox 2

),,2

-~

)v 2 - -

K Dc,

t~ 2 - -

S

1

-- K D '

'

rl -

--

1

fl2)v 2 - - c S "

135.11 (L,FC)

F=~q(t)

Fully penetrating infinite well-screen at x = b. Groundwater abstraction an arbitrary function of time. ~p(x, t) = drawdown, qg(x, 0) = 0, 0~o ~(0, Ox

~

0~o ~(b, Ox

t) - - 0 ,

t) --

q (t) 2KD

,,~

b

.

q~(x, t)

--

e-'7{t-r)q(r) dr + ~

2bS

(-1

cos

---if--

n2~2

t

×fo

n=l

exp{ -- (f1262

+ r/)(t- r)}q(r)dr.

135.12 (L,FC) As 135.11 with q ( t ) = q = const. q)~ cosh(~) qg(x, t ) -

2 K D sinh(-~)

qb ~

qc 2b (-1) n

+ K-D ,~=l n27t "2 -t-- b2

-rlt

cos( exp ,

e/,/2~2

}

112

(13 5b, 136a)

One-dimensionalgroundwaterflow

135.13 (D)

Steady state of 135.12. q)~ cosh(~) ~o(x)- 2KD sinh(b) "

•

136. O n e - d i m e n s i o n a l finite flow; g i v e n flux at x = b a n d z e r o h e a d at x = 0

Subdivision: a. Confined aquifers. b. Leaky aquifers.

a. Confined aquifers 02q9 __ [32 0~0 OX 2 --~'

f12 = KD

136.01 (L,FC) Fully penetrating infinite well-screen at x = b. Groundwater abstraction according to an arbitrary function of time. ~p(x, t) -- drawdown.

x KD'S I "

b

~o(x, o) = o ,

"

0__~

~o(o, t) = o , q(t)

Ox (b, t) -- 2KD 1 °° )n {(2n+l)Jrx} ~p(x t ) - - ~ Z ( - 1 sin ' 2b

n=0 f0t exp { - ( (2/'2fib /-'1-1)7"C) 2( t x

136.02

/

r) q(r) dr.

(FC}

As 136.01 with q (t) = q = const. qx

4bq k (-1)" sin { (2n +2bl)rrx } D n=o (2n + 1)2

cp(x, t) -- 2KD

× exp 136.03 (D)

Steady state of 136.02.

7/" 2 K

-

(2n + 1)27l"2 } 4f12b 2 t .

(1 36a, b)

Flow in a finite field

113

for0 ~<x ~
2KD qx - x) q)(x) -- i q(2b

forb<~x <~2b.

2KD b. Leaky aquifers

0299

(/9

OX 2

)~2 --- fl - ~ '

20q9

~2 __ K D c ,

f12 __

-

S

1

KD'

1

~ -- fl2)~2 = c---S"

136.11 (L,FC) __ ~

~

_

/

Fully penetrating infinite well-screen at x - b. Groundwater abstraction according to an arbitrary function of time. qg(x, t) - drawdown.

q (t)

g/'/////////////////////A"//////. C

KD, S

~p(x, O)--0,

X

Oq) Ox (b't)

1 c~ q)(x,t)--~~(-1

),,

sin

~o(0, t)--0, q(t) 2KD

[ (2n + 1)zrx ] 2b

n--0

[ [ (2n -+-1)27r2+ 77]( t -

f0 t

exp -

4f12b2

r)]q(r)dr.

136.12 (FC)

As 136.11 with q(t) - q - const. ~o(x ' t)

q~ sinh(~) b 2 K D cosh(2)

[ /

× exp -

136.13 (D)

Steady state of 136.12.

*

q~ sinh(~) qo(x) -- 2 K D cosh(~)"

4bq ~ KD ~

(-1) n

n=0

4f12b2

(2n + i)2~ 2 +

/]

+ r/ t .

sin { (2n +2bl)Zrx } -4b2 L2

114

(137a)

137. O n e - d i m e n s i o n a l

Subdivision: a. Given b. Given c. Given d. Given

One-dimensional groundwater flow

finite flow; e n t r a n c e r e s i s t a n c e at x - b

flux at x - 0; confined aquifers. flux at x = 0; leaky aquifers. head at x = 0; confined aquifers. head at x -- 0; leaky aquifers.

a. Given flux at x - O; confined aquifers

0299

0(49

O x 2 __ f12

--~'

f12

S KD

137.01 (L,FCR) ..........

The surface water level rises according to an arbitrary function of time. q9(x, t) - head.

IF(t)

w

~(x, o ) - o , Oq) ~(b, Ox

b 2 •

~o(x,t)

--

~

fin

f12b2 Z

sin or,,

_E , n=0 1 + ot2+e 2

t) -

Oq9

~(o, t)-o,

Ox F(t) - q)(b, t) Kw

cos(nx fo/ {

with Otn being the roots of c~ tan ot - e -

---if-)

exp

,

F(0) --0.

2 ( t - r ) IF ( r ) d r

%

f12b2

b Kw"

137.02 (FCR) F(t)

Sudden rise h of the surface water level, which is kept constant thereafter. As 137.01 with F(t) - { O h

~p(x, t) -- h - 2h

sin ot,z

fort-0, for t > 0. cos (otnx

oe2t

n-o c~,z(1 -+- ot2We e 2)

137.03 (FCR)

Linear rise of the surface water level. 99(x, t) - head.

~

IP,

~o(x, o) - 0. As 137.01 with F(t) - at.

(137a)

Flow in a finite field

115

aft.~.--(b2 2 - x 2)

qg(x, t) -- at - a f i 2 b K w

oo

sin(ten)

+ 2afi2b2 Z

(

(_~)

3(1 + e ) cos n=0 tXn ot2+e2

exp

c~2t) "

/~2b2

.

137.04 (FCR) F(t)

..--

Exponentially increasing rise of the surface water level. As 137.01 with F ( t ) = h(e a t - 1) (h and a posi-

/

tive).

/ /

t

~0(x t) -- h{

eat c°sh(flxv/-d) K w f i ~ d s i n h ( f l b ~ ) 4- c o s h ( f l b ~ )

'

oo

-,}

sin cz,, 8 Otn ( Ot2 4- fl 2b 2a ) (1 4- ~2+E2)

4- 2 h a fl 2b 2 Z n--0

× cos (-~) exp (

c~t

f12b2)"

137.05 (FCR)

h

.

.

.

.

.

.

.

.

.

Exponentially decreasing rise of the surface water level. As 137.01 with F ( t ) = h ( 1 - e-"t).

.

//

~0(x, t) -- h{1 -

e - ' t cos(~x,/a)

}

cos(flb~/~) - K w f l ~ f ~ sin(flbv/-a) oo

4- 2haflZb 2 Z

sin Otn 8 flZbZa)( 1 4- ,,2+~2)

( o~2

n=O £gn

× cos (-~) exp (

°12t

f12b2)"

137.06 (FCR) Initial head an arbitrary function of x. Zero flux

at x = 0. 99(x, t) = head. xKD, b

S

w

Ocp ~(0, Ox

~o(x, O) = f (x),

a--E(b t) 'Ox

'

~o(b, t) Kw

t) -- O,

116

(137a)

~o(x , t)

_ 2~

cos (e_~)

exp ( -

7

n--O 1 + tz2A_82

One-dimensional groundwater flow

%t ) 2 fob f (xo) cos (CenX° b )dxo. ~2b2

137.07 (FCR) As 137.06 with f ( x ) = h = const. sin~n ~o(x, t) -- 2h ~ e ) COS n=o otn (1 + ~2+82

exp ( --

,4,

•

137.08 (L,FCR) Precipitation as an arbitrary function of time. Zero

f [~ . . . . . .

flux at x - 0. 99(x, t) -- head.

$$,1,,1,,1,,1,,1,,1,,1,,Wp(t)

02~0 OX 2

b

(t) KD

~ P

._ f12

O q9 at'

O(P(O t) -- O, Ox '

sin

99(X, t) -- ~2 ~°0 Or,, ,,=0 or,,(1 7 ~2+E2) cos e

cp(x, O) - o ,

O~°(b t) -Ox '

~p(b , t ) Kw

(~_.X)fot exp { __ o/2 (t __ r) }p ( r ) d r . f12b2

137.09 (FCR) As 137.08 with p(t) = p = const. •

P

~o(x, t) -- 2 K D

(b 2 -- X 2) -~- p b w D

2pb2 oo

sin oen

K D ~n=00/n3 ( i 7 ot2+e ~ 2 ) cos ( ~ ) e x p

~2t f12b2 ) .

(

137.10 (D) Steady state of 137.09. •

P (b 2 -- X 2) -]- p b w (p(x)- 2KD D

137.11 (L,FCR) q(t)

~

KD, S

Fully penetrating infinite well-screen at x - O. Groundwater abstraction an arbitrary function of time. qg(x, t) - drawdown.

~(x o ) - o, '

x

°--E(b, t) Ox

a~p

--(o t)-

Ox 99(b, t) ~ o Kw

'

q (t) 2KD'

Flow in a finite field

(1 37a, b)

1 ~

cos~, Otn bX ) fo +

=

!

0I 2

I

f12 b2

exp / -

ot2+e 2

1 17

(t -- r) } q ( r ) dr.

137.12
•

99(x,t) =

q(b-x)

t

2KD

2

%tt ) qb~~o c°s (~'~-~----) ( f12 b2 " K D = ol2(1 + ot2+e2) exp -

qw 2D

137.13 (D)

Steady state of 137.12. q(b-x) •

qg(x) =

qw

2KD

2D"

b. Given flux at x = O; leaky aquifers

0299

(t9

OX 2

/~2 - - /~ - ' ~ - '

20q 9

S

i~2 --

KD '

)2

KDc, rl-

1 fl2)~2

1 =

m

cS

137.21
F(t)

The surface water rises according to an arbitrary

function of time. q9(x, t) = head. 099

~o(x, o) =o,

Ox F(t) - 99(b, t) -Kw

099 ~(b,t) Ox

cos( ) fo'e'P{ (flZb2< -k- rl) } 2

•

99(x,t) =

~ ( o , t) =o,

oo oe,~ sin c~n

n=O

+e 2

( t - r)

with Otn being the roots of c~ tan c~ - e -

F(r)dr

b

Kw"

137.22 (FCR)

F(t)

Sudden rise of the surface water level, which is kept constant thereafter. As 137.21 with F(t)

-- / O

!h

fort <0, for t > O.

118

One-dimensional groundwater flow

(137b)

~p(x, t) --

h~. cosh(~) b + ,k cosh(b) K w sinh(~-) cx~

~n sin Otn

- 2h ~

b2

cos

e

)exp

f12b2 -k- ~)t

n=O ( 0/2 -+- ~--~')(1 + ot2q_e2)

137.23 (D)

Steady state of 137.22. ,kh cosh(~) •

o(x) =

K w sinh(b) + )~ cosh(b) "

137.24 (FCR)

Initial head an arbitrary function of x. Flux is zero at x = 0. qg(x, t) = head. 099

qg(x, O) -- f (x), x

x7

(°' t) - o,

w O~p ~ ( b , t) -Ox

b ~o(x , t)

2 k

cos

-;) (etnX

qg(b, t) Kw

exp{- ( c~2

f12b2 +

n=O 1 + c~2_+_e2

r/)}fo h f t

(x0) cos (°tnX0 b )dxo.

137.25 (FCR) As 137.24 with f ( x ) = h = const.

sin ot,~

go(x, t) -- 2h ~ ~ ) cos(--ff-/exp n=0 an (1 + ~2+~2

- ( f 1 2°tn b 2 + O)t .

137.26 (L,FCR)

Arbitrary vertical infiltration as a function of time. Zero flux at x = 0. ¢p(x, t ) = head.

~1,$,1,,I,$$$$$$~1 ~(t) KD, S b

Iw

0299

~0

OX 2

)~2

q(t) __ f12 099 I KD 0--7'

099(0 t) -- 0 Ox ' '

0~o Ox (b, t) -

qg(X O) -- O, ' ~o(b, t) K------~"

(137b)

Flow in a finite field

119

2 ~ sin O/n ~o(x, t ) - ~ _ ~ ( 1 + ~+~) cos \--b---)

exp -(flZb 2

137.27 (FCR) As 137.26 with q(t) = q = const.

,

~o(x, t)-2qbZ~KD= Oln(Oln~---J-sinc~nc°s(z-~)+ ~)(1. . . .+ ~+E2~) 1--exp

-- ClUb2

137.28 (D)

Steady state of 137.27. [ ~p(x)--qc 1 -

,

~ cosh(~) ] Kw sinh(~) + ,k cosh(~)

137.29 (L,FCR)

Fully penetrating infinite well-screen at x = O. Groundwater abstraction an arbitrary function of time. 99(x, t) = drawdown. b////////////////////AS~¢ KD, S ~, to

!o Ox (0, t) -

~o(x, O) - 0 ,

O~°(b, t) -Ox

~o(x t)

1 cos k"-if") t ~ = o 1 + ~ + ~ [2~ fo E

~o(b,t) Kw ~

o

expl_ ( f12b22 + r / ) ( t -

r)}q(z)dr.

137.30 (FCR) As 137.29 with q(t) = q = const.

qb ,

qg(x,t) --

oo

cos (~nx

__

E

%2+e

)

,_oxp{_ ( f12b22

137.31 (D)

Steady state of 137.30. q cosh { l ( b - x ) } --~-K--~sinh{z1 ( b - x ) } qg(x) -- 2KD ' sinh(b) + ~ cosh(~) )~ _

q(t) 2KD '

b

120

(137c)

One-dimensional groundwater flow

e. Given head at x = O; confined aquifers 02(/9

0~

Ox2 __ f120 7 '

f12

S

KD

137.41 (L,FSR) The surface water level falls according to an arbitrary function of time. Zero drawdown at x = 0. ~o(x, t) = drawdown. ~o(x, 0) --0,

~p(0, t) = 0 ,

O~P(b, t) -- F ( t ) - 99(b, t)

b

Ox •

~o(x,t)

2

=

~

_OtnCOSC~n

"-'7--a~2. "~_.~sin e

f12b2 Z l+n=l

Kw

(Ol,nX)fot --b---/

with or,, being the roots of ot cot ct = - e =

{ exp

F(0) -- 0. '

2

]

- fl2b2(t-r)F(r)dr

Kw"

137.42 (FSR) 7

As 137.41 with 0 F ( t ) --

F(t) •

~o(x,t) =

hx

+2hZ

h

fort<0, for t > 0 (const.).

oo

2

cos Otn (CtnX) (%t) 8 ) sin - - ~ exp f1262 n=l ot~(1 + ot2+e2

Kw+b

137.43 (D) Steady state of 137.42.

•

~o(x) =

hx Kw+b

137.44 (FSR) Initial head an arbitrary function of x. Zero

head at x = 0. qg(x, t) = head.

m~

~

n

m

~o(x, O) = f (x). 099

rp(b, t)

Ox

Kw

~ ( b , t) =

~(o, t) = o ,

b ~o(x, t)

2 ~-,°° sin(Z~x) ~

n-- 1

1 +

ot2+e 2

exp

(

f12b2

)

f (xo) sin (°6'x° b )dxo.

(137c)

Flow in a finite field

121

137.45 (FSR) As 137.44 with f ( x ) -- h -- const.

f(x)

I

b

x

~o(x, t) -- 2h

c~

(1-cosc~,,)

n=l oen(1 +

( /~2b2)" °l'2't

sin ( - ~ ) e x p

E

137.46
b ~o(x, t) -- 2ab

n= 1

s i n ~ , , - o6, coso6, sin ( ~ _ ~ ) exp ( o/2(1 _+_ e )

n

f12b2)"

ot2+e2

137.47 (L,FSR) Precipitation an arbitrary f u n c t i o n o f time. Ze-

ro head at x - 0 . ~,[, ,[,,l, ,[,,[,,[,,[,$ ,[~ P ( t)

~p(x, t) - head.

p ( t ) __ f12 0(t9 Ox 2 t K D Ot '

0 2 (t9

~o(x, O) - o , b

/)~o

~(b,t) Ox

2 ~

1 - cos oc,t

~(o, t) - o , ~o(b, t)

- - ~ . Kw

s i n ( ~ ~ -~) fot e x p { - f l 2 b 2c~2 (t-r)}P(r)dr.

137.48 (FSR) As 137.47 with p ( t ) - p •

~p(x t ) - '

px 2KD

const.

b-x+ 1 +e

2p b2 ~ 1 _--.cos o6___5, Z.., K D ,~=1 " %+~

sin( )exp(

f12b2).

122

(137c, d)

One-dimensional groundwater flow

137.49 (D) Steady state of 137.48.

•

99(x)-

b)

px (b - x + 2KD l+e

137.50 {L,FSR) Rise o f the surface water level at x = 0 an arbitrary f u n c t i o n o f time. 99(x, t) = head. 99(x, O) -- O,

99(b,

a-Y-~(b t) Ox

99(0, t) -- F ( t ) ,

'

t)

Kw

b •

2 ~ f1262

99(x, t ) -

n= 1

1+

sin (ct.x \ b )

Lt e x p { -

ot2+e 2

2

o6'

f12b2 (t - z') } f(z') dr.

137.51 (FSR) As 137.50 with F ( t ) = h = const, for t > 0. 99(x, t) -- h(1 -

x

)-2hZ

sin ( ~ _ x ) exp ( _

°t2t f12b2 ) "

n=l Otn(1 + ct2+e 2 )

Kw + b

137.52 (D) Steady state of 137.5 1.

(

99(x) -- h 1 -

x)

Kw + b

"

d. Given h e a d at x = O; leaky aquifers 0299 OX 2

99 __ f12 099 ~2 --~'

[32 __ S )2 -- K D '

1

KDc

1

77-- fl2~2 = ~ " ' cS

137.61 (L,FSR)

__ _ _ ~ ~ l

F(t)

V/////////////////////A

C

The surface water level at x --- b rises according to an arbitrary f u n c t i o n o f time. 99(x, t) head. Zero head at x = 0.

~o(x, o) = o , b

~o(o, t) = o ,

~90___:_(, b t) -- F ( t ) - 99(b, t) Ox ' Kw '

F(O) -- O.

(137d)

Flow in a finite field

•

2

~o(x, t) -

--0/n COS 0/n

f1262 n=l

x

123

s i n ( ~ -~)

1 "-t- °t2+62

lotexp { - ( ~ + r / ) ( t 0/n fleb2

r) } F(z") dr

with 0/n being the roots of 0 / c o t 0 / - - e . . . . Kw" 137.62 (FSR) As 137.61 with F ( t ) -- h - const. O0

--0/n COS 0/n

~o(x, t) -- 2h Z b2 e n=l (°/2 --t- ~ ) ( 1 -t- ~2+e2)

sin

~nx]

t b j[lexp/(

2 0/ n 77) f12b2 nt_

'Jl

137.63 (D) S t e a d y state of 137.62.

h,k sinh(~) ,

~o(x) -

b

K w cosh(~) + I. sinh(~)

137.64 (FSR) Initial h e a d an arbitrary f u n c t i o n o f x.

Zero

head at x - O. ~p(x, t) - head. U//////////////~

v ( o , t) - o ,

qg(x, O) -- f (x),

VA C

o~o

~(b, Ox

~o(b, t)

t) -

Kw

b

2 ~ sin (j~'x S ,~ ~o(x, t) - ~ ,,=~ 1 + ~ + ~

exp{ t

fl Z b 2 -t-

t

}fo

f (xo) sin (0/"x° b )dxo.

137.65 (FSR) As 137.64 with f (x) - h - const. •

~o(x, t) -- 2h

(1 - cos o/n) n=l 0/n(1 -t- ot2+82 e )

sin

(~texP{ t °'~ + ~It} f12b2

124

(137d)

One-dimensional groundwater flow

137.66 (FSR) Initial head a linear function of x. As 137.64 with f (x) = ax. cx~ sin ot n -- o~n c o s ot n

qg(x, t) -- 2ab ) n=l

C~2(1 ÷

sin\---ff-)exp

e

-(f12b2 c~'Z + r/)t

137.67 (L,FSR) Vertical infiltration an arbitrary function of time. Zero head at x = O. qg(x, t) = head. c

~ $ ,1,$ ,1,,[,,1,~,,~4 ~ q ( t ) x

KD, S ~ w ~

q9

OX 2

)~2 ~ K D

~o(x, o) =o, b

~°(b Ox

2~ 1 - cos otn qg(x, t) -- S ~'n=lo~,,(1 + e x

exp

q(t) __ f12 O~ Ot

02q9

'

t) -

~o(o, t) =o, qg(b, t) Kw

OlnX

sin\---if--)

fin

,-2-2 + r / ) ( t - r)}q(r) dr.

-

137.68 (FSR)

As 137.67 with q(t) = q = const. •

qg(x,t) --

1 - cosoln

2qb 2 ~ K D

__ Oln (Ot.12._~ b_~)( l + o~2h_82)

[

]

137.69 (FSR,D) Steady state of 137.68. qg(x) - 2qb2 ~'~

1 --bzCOSC~, ~

sin (ot, x/]

= q c [ 1 - Kw c°sh(b~---~x) + Jk s i n h ( ~ )

+)~ sinh(~) } K w cosh(~) + ~ sinh(~)

(137d, 138a)

Flow in a finite field

125

137.70 (L,FSR) Rise of the surface water level at x = 0 an arbitrary function of time. q9 = qg(x, t) = head. q)(x, O) = O,

aq) ~(b,

b

~o(x t)

Ox

oo

2

f12b2 x

_

exp

t) =

qg(O, t) = F(t),

~p(b, t) ~ . Kw

ofn sin ( - ~ ) E 1 -~- ot2+e2 ofn2 { t f12b2 + -

r / ) ( t - r ) } F ( r ) dr.

-

137.71 (FSR) As 137.70 with F(t) = h = const. OO

q0(x, t) -- 2h Z

b 2 oft'

n = l (of2 _+_ ~ ) ( 1 +

8

o f n x , ~ [ l _ e x p { _ ( of2 r l ) } ] sin ( - - ~ / fl2b 2 + t .

137.72 (D) Steady state of 137.71.

•

~o(x) = h

K w cosh { 21(b -- x)} + )~ sinh{ 2l(b - x)} b + )~ sinh(~) K w cosh(2)

138. O n e - d i m e n s i o n a l finite flow; o p e n w a t e r s t o r a g e at x - b

Subdivision: a. Given flux at x -- 0; confined aquifers. b. Given head (drawdown) at x - 0 ; confined aquifers. c. Leaky aquifers.

a.

Given flux at x = O; confined aquifers 02q 9 __ [~ 2 0(t9 OX 2 --~'

fi 2

__ --

KD

126

(138a)

One-dimensional groundwater flow

138.01 (L) Initial head an arbitrary function of x. Zero flux at x = O. ~o(x, t) = head.

b •

go(x, O) -- f (x),

Ox (0, t) -- O,

ox(b,t) -

(b,t).

K D Ot

a

~o(x, t) -- b(1 + 0)

foh

f ( x o ) dxo

(O/nX0 l+C°S(~"x]bO j exp ( - c~t ~o b f12b2 ) f ( / o ) cos \ b ) d / o n=l Otn2+02

+b2~

with c~n being the roots of c~ cot ot = - 0 =

bS

138.02 (L) As 138.01 with f ( x ) = h = const.

•

~o(x,t) =

ht~ 1+0

c~ +2h~z...,

,

sinoe,+o

n=l ffn[l

fin2+02)

cos

2

(~,x)

exp

b ]

(%t) - ~ . flZb2

138.03 (L) •

__!

Precipitation as an arbitrary function of time. Zero flux at x = O. ~o(x, t) = head.

02~o ~ p(t) = f12 0~° Ox 2 KD 0-7' b

•

~oO___: (O_, t) -- O,

a

~o(x,t) = S(1 -+- 0)

2 ~

Ox

/o t p ( r )

As 138.03 with p(t) -- p = const.

Ox

'

Oq)

a

ogO___: (b_, t) -

K D Ot

(b,t).

dr

sin c~,~

138.04 (L)

'

~o(x, O) -- O,

( ffnX ) ~0 t cosy

2(t

-

-

r)}p(r)dr.

Flow in a finite field

•

cp(x,t) =

(138a)

pot S(1 + O)

p

(0 + 3)b 2 - 3(0 + 1)x 2

6KD

(O + 1) 2

2pb2 ~n~1

KD

127

sino~no

)cos(C~nX]exP(b '

_ Ol3n(i7 ot2+02

,:4t f12b2 )"

138.05 (L)

Abstraction from the open water an arbitrary function of time. Zero flux at x = 0. 9(x, t) = drawdown.

q(t)

a~o 0-7 (o, t) - o,

~o(x, o) - o,

KDO__c_. t) + a (b, t) (t) Ox ' --~ --q ' q [L2T-1].

,

1 /ot q ( r ) dr

2 oo

~o(x, t) = a(1 + O)

x

for { exp

sin oe,

a ~ C~n(1 + o ) n=l ot2+O2

COS

(>)

2

f12b2 (t

- r) q ( r ) dr.

138.06 (L)

As 138.05 with q(t) -- q - const.

•

~p(x,t) =

qt a(1 + O)

qfl2 (0 + 3)b 2 - 3(0 + 1)x 2 6a

(O + 1) 2

0,2,)

+2qfl2b 2 c ~ sin~, (c~,,x] ( a ~oP(1T o cos exp - f12b2 " n--1 ot2q_O2) b ]

138.07 (L) q(t)

Fully penetrating infinite well-screen at x -- 0. Groundwater abstraction an arbitrary function of time. ~0(x, t) = drawdown.

___

~o(x o) - o, ,

Ocp

b

a

- - ( b , t) = Ox

o!

ox(O, t) -

a Ocp (b, t). K D Ot

q(t) 2KD '

128

One-dimensional groundwater flow

(138a, b)

1

9 ( x , t) -- 2a(1 + O)

Jotq ( r ) d r

l c°s °nX'fot / --b--J

exp

0 2 = 1 .qt_ Ot2+O

+ -~

ot~ (t - r )

-- f12b2

/

(r) dr.

q

138.08 (L) As 138.07 with q(t) = q = const.

q

•~

qg(x,t)-- 2 K D

1

1+0

qb

oo

O ( b - X) 2

[KDt [

+

a

bl~(1) -1- 3) 1 +b-x

2b

cos (~_~)

-

6(1+0)

/

~2t

K D n~l o[2 {-1 7 ct2--'~;2)exp (

f12b2 )"

b. Given head (drawdown) at x = O; confined aquifers

02~

0~

OX2 __ f12 - ~ '

f12

S KD

138.11 (L) Sudden drawdown h at x -- b; zero drawdown at x = 0. 9 ( x , t) = drawdown. h ~o(x, o) -

b ~o(x, t) -- 2h

Z

for x = b, for 0 ~< x < b , O~p a 09 -~-x (b t) . . . . (b, t).

~o(O, t) = o ,

a CK)

o

K D at

'

COSOfn

o n-O Oen(1 + ~2+02)

sin

OlnX

)exp(

f12b2

)

with Oen being the roots of oe tan oe - 0 - bS. a

138.12 (L) Arbitrary rise of the surface water level at x = 0 as an arbitrary function of time. ~o(x, t) = head.

F(t) I

~o(x, O) =

xKD, S b

O~P(b t ) Ox ' a

o,

99(0, t) = F(t), a 09 (b, t), K D Ot

F(O) = O.

Flow in a finite field

129

(138b)

2

,

°in (t -- r) } F(r) dr f12b2

2 ~~o °l'~ o sin (cznx'] - ~ / fot exp { f12b2 = 1 -1I- ot2q_O2

,

qg(x t ) -

138.13 (L) As 138.12 with F(t) - h *

const.

sin (~nX] ( b 0J exp = oen(1 + ot2bO2)

~o(x ' t ) - - h - 2 h ~

2t) f12b2 "

138.14 (L) Initial head an arbitrary function of x. Zero head at x - 0. 99(x, t) - head. ~o(0, t) - - 0 ,

cp(x, O) -- f (x), 399 ~(b, 3x b

a 3~p (b, t). K D at

a

_ 2~ •

t) -

'

sin (z_~) 0

qg(x t)

b n=O 1 + o~2+02

exp

(

f12b2

) fob

f (xo) sin (°enx°b )dxo.

138.15 (L) As 138.14 with f (x) -- h - const.

f(x)

b

x

1 -- cos c~n ~o(x, t) -- 2h Z o n=O oln(1 + ot2_+_02)

sin (- )exp (

]~2b2).

138.16 (L)

~

As 138.14 with f (x) - a x linear initial value.

./ab b

x O0

•

~p(x, t) -- 2ab ~

sin an -_.c~n_g__cosc~n

OtnX] exp ( sin ( b ¢

°t'z't

flZb2)

130

(138b)

One-dimensional groundwater flow

138.17 (L)

~

Precipitation an arbitrary function of time. Zero head at x - 0. 99(x, t) - head.

mm

l~$,l,$,l,$$$$~ p(t)

0299 OX 2

KD, S

b 2@

20(t9 - - fl ~

~o(0, t) --0,

a

e(x ' t)

p(t) KD

[

1-coso~n

~ 0 ~.=o ~,,(1 + ~.~+o~)

qg(X O) - - 0 Ot ' ' ' 0~p a 099 ~ ( b , t) -(b, t). Ox K D at

sin(°enX \ b ) lot e x p { -

%2 (t - r) } p ( r ) d r . f12b2

138.18 (L) As 138.17 with p(t) - p - const. 12) ~o(x , t ) - -~D ( b X - -~x 2p b2 ~

1 -- c os ~__5_n

K D n=O u3( 1-t- 20_2)

~nx'~ exp ( sin ( b /

°~n2t f12b2)

o~n +O

138.19 (D) Steady state of 138.18.

b

x

-

x2)

138.20 (L) q(t)

Abstraction from the open Water an arbitrary function of time. Zero drawdown at x - 0. qg(x, t) - drawdown.

q)(x, 0 ) - - 0 ,

b

q9 099 KDO---c- (b, t) 4- a (b, t) (t) Ox -07 --q ' q [L2T-1].

a 2

oo Z

qg(0, t) --0 ,

COS O/n

~o(x, t) -- -a n=O Oen(1 q- ~2+02") 0

sin ( ~ x \b

)fo texp / - 22°2(t - r) / q ( r ) d r .

(138b, c)

Flow in a finite field

131

138.21 (L) As 138.20 with q (t) = q = const. 2bq o° sin 0/n sin (0/nx ] (0/2t) K D Zn=o 0/~(1 + 0,2+0 o 2) eib2 --if'-/exp

qx •

~o(x,t) =

KD

.

138.22 (D) Steady state of 138.21.

bl

"~..q__x

•

e(x)

=

qx KD

c. Leaky aquifers 0299

99 __ f12 099

Ox 2

)~2

-~'

)v 2 - - K Dc,

S

f12 __

1

K D'

--

rl -

1

/~2)v2

--

c S

138.31 (L) ~o

h

Sudden drawdown h of the open water level at x = b. Zero flux at x = O. q)(x, t) = drawdown.

a! b

Ox

a OO

q)(x,

t)

0)-- [0 /h

forO~<x < b , for x -- b.

(0, t) -- O,

KDT-(b,ox

qg(x,

KD, S

2h

0//,/ COS /t --b--') °lnx exp(

S--' n=O

0/n

(1) + 2) cos0/n

o~o

Ol2t

t~zb2

r/t)

b2

--

(0/2 + 7z

a~o(b, t) "

t) -- - a 0-7

--

1)) sin 0/,,

with 0/n being the roots of (0l 2 .qt_ ~2)COS 0/ qt_ l)0/sin 0/ --O, 1) -- bS.a 138.32 (L)

jjjjjjjjjjjjjjjj! x

Sudden drawdown h of the open water level at x = b. Z e r o d r a w d o w n a t x = O. qg(x,t) =

drawdown. ~0(x 0 ) '

KD, S

a

~o(o, t) -

[0 | h o,

forO<~x < b , forx-b, 099

7(b Ox

t) '

a

099

K D Ot

(b, t).

132

(138c)

O~nsin ,--Y-)exp{-(~-2-U + r/)t}

2h V"

~o(x, t)

One-dimensional groundwaterflow

n--l~0/n(1) + 2) sin 0/n +

(0/,2 + 7b2 --

1)) COS0/n

with 0/n being the roots of (0[ 2 + b_~)sin 0/-- 00/coso~ --0.

138.33 (L)

IV_ II

Abstraction of water from the storage basin. Discharge an arbitrary function of time. Zero

q(t)

flux at x = 0. ~p(x, t) = drawdown.

~o(x, o) =o,

xKD, S b

Ox Oq9 KD O~p(b t) + a (b, t) 0--~ , -~

a

oo ~o ( x

'

t) -

099

--(o, t) =o, q

•

Oln X

cos

-

(t) -

-v-,

(

b2

a ~=o ~ ( 0 4- 2) c o s ~ - (~,2 4- ~ - O) s i n ~ x

f 0 t exp [(2

--

0/n + rl) ( t f12b2

}

r) q ( r ) d r

b2

with ot~ being the roots of (0/2 _[_ ~-~')COS0/ -[--tgO/sinot -- O.

138.34 (L) As 138.33 with q(t) - q •

qg(x, t) =

-- const.

q~. cosh(~)

KD sinh(~) 2q0b c~

0/nCOS(~'Xhexp{--( ~2 '

+

K D ~n=0 (0/2 + ~b2) { 0 / , ( 0 + 2)COS0tn- (0/2 + ~b2 _ 01 sin0/n}

138.35 (D)

Steady state of 138.34. qg cosh(~) •

~o(x)

=

KD sinh(~) "

(138c)

Flow in a finite field

133

138.36 (L) ~ q(t)

Abstraction o f water f r o m the storage basin. Discharge an arbitrary f u n c t i o n o f time. Ze-

ro drawdown at x = 0. qg(x, t) = drawdown. q)(x, 0) = 0 ,

KD, S X

099

~o(0, t) = 0 , Oq9

K D -ff-ffx ( b , t) + a --~ ( b , t) -- q ( t ) .

b

a

2 ~ oln sin (~"x ~p(x t ) - - - ~ - - " b ) , b2 a ~ = ~,, (0 + 2) sin ol~ + (or2 + 7: - 0) cos ol,~

× exp{-(

fo'

--2-2

-t-

e)(t-- )}q (r) dr "C

b2

with oln being the roots of (~2 _3t_~.g)sinc~- 0ol cosol --O.

138.37 (L) As 138.36 with q ( t ) = q = const. qX sinh(~) ,

cp(x,t) =

K D cosh (-~)

2q 0b ~ KD

or,, sin ( --y-) OlnX e x p { - ~ , 1"~ -ot~2 + r/),} b2

b2

= (a2 + 7:)1ol,,(0 + 2) sin~n + (a 2 + 7 - O) cosoln}

138.38 (D) Steady state of 138.37.

q)~ sinh(~) •

,#(x)

=

b K D cosh(2)

"

139. One-dimensional periodic groundwater flow in a finite field Subdivision: a. Periodic fluctuations of the open water level. b. Periodic abstractions. c. Periodic precipitation or infiltration.

134

(139a)

One-dimensional groundwater flow

a. Periodic fluctuations of the open water level

02q9

--2 099 for confined aquifers,

OX 2 -- ~ --57 0293

Ox 2

93 __ f12 093

)~2

f12 =

-07 for leaky aquifers,

1 S ) 2 _ K Dc, 17 KD' fl2~2

1 cS

139.01 (PF) _

_

Confined aquifer with open boundary at x -

b. Open water level fluctuation according to a sine function. 93 - 93(x, t) - steady oscillation. Zero flux at x - O.

m

2h

93(b, t) - h sin(cot).

ox(O,t) --0, 93(x, t) - h Im

h(flx t~7-~)

[cos

]

e i~°t

cosh(flb v'7-~)

+ COS(2cotX) cosh(2co'x! sin(cot + O) = h v cosh(2co'b) + cos(2co'b) with 0 - arctan{tanh(co'x)tan(co'x)} - arctan{tanh(co'b)tan(co'b)}, co m 2zr

co,_ V/

T '

JrS

KDT

139.02 (PF) As 139.01 but for a leaky aquifer.

[cosh(flx~/rl +

ico) eiCOt]

93(x, t) - h Im cosh(flb~/rl + ico)

"

139.03 (PF} ~ - - I 2 h

KD, S x

Confined aquifer with open water boundaries at x - 0 and x - b. Level fluctuation at x - b according to a sine function. ~0 - ~p(x, t) - steady oscillation.

o(o, t) - o ,

93(b, t) - h sin(cot).

Flow in a finite field

135

(139a)

~o(x, t) -- h lm[ sinh(flx ~/-~) ] sinh (fl b ~/~) ei~°t

~

cosh(2co'x) - cos(2co'x)

= h

sin(cot + 0)

cosh(2co'b) - cos(2co'b)

with O - arctan{coth(co'x)tan(co'x)}

- arctan{coth(co'b)tan(co'b)}.

139.04 (PF) As 139.03 but for a

leaky aquifer.

~o(x, t) -- h I m [

sinh(flx ~/r#

+ i co) eiOJt ] .

sinh(flb~/r# + ico)

d

139.05 (PF)

Confined aquifer with entrance resistance at x - b. O p e n water level fluctuation a sine function.

Zero flux at x - 0. ~p -
~o(x, t) -- h Im

0~p

0~o

0-7(0, t) -- O,

-~x (b, t) -

eiOJtJ

cosh(flxw/~ ) fiKw~

sinh(flbv/~)

-~o(b,t) Kw

h sin(cot)

+ cosh(flb-v/~)

139.06 (PF) As 139.05 but for a

~o(x, t) -- h I m

leaky aquifer. c o s h ( f l x ~ / r / + ico)

flKw~/r# + ico s i n h ( f l b ~ / r / + ico) + c o s h ( f l b ~ / r / + ico)

eiC°tI .

139.07 (PF)

L•-

-I2h

Confined aquifer with entrance resistance at x b. Zero head at x - 0. Level fluctuation at x - b a sine function. 99 - 99(x, t) - steady oscillation.

x. KD, S H w

~o(o, t) - o ,

a--E(b t)Ox '

-cp(b,t) Kw

h sin(cot)

136

One-dimensional groundwater flow

(139a)

~o(x, t) -- h I m [ fl K w ~

sinh(flx~7~) eiOOtJ cosh(flb t~7-~) + sinh(flb~7-~) "

139.08 (PF) As 139.07 but for a leaky aquifer. -- h I m [

~o(x, t)

s i n h ( f l x ~ / r / + ico)

LflKw~/rl + iwcosh(flb~/rl + ico) +

s i n h ( f l b ~ / r / + ico)

ei°OtI .

139.09 (PF)

Confined aquifer with entrance resistance at x - b. Open water boundary at x - 0. Level fluctuation at x - 0 a sine function. oscillation.

KD, S ~ w b

~o -

~o(x, t) -

Ox (b, t)

99(0, t) - h sin(cot),

steady

o(b, t) Kw

Num. eitOtl with

~p(x, t) -- h Im Denom--------~

flKw~-~-~wcosh{fl(b - x) ~/~w} + sinh{fl(b - x ) ~ } ,

Num.-

flKw~cosh(flb~w)

Denom.-

+ sinh(flb t~/~w).

139.10 (PF) As 139.09 but for a leaky aquifer. Num. ] ~o(x t) -- h Im ~ e i~°t ' Oenom.

with

fl K w v/ 0 + i co c o s h { fl ( b - x ) v/ rl + i co} + sinh{fl(b - x ) v / r / + ico},

Num.-

flKwv/rl + icocosh(flbv/rl + ico) + s i n h ( f l b v / r / + ico).

Denom.139.11 (PF)

2hIL

_

Confined aquifer with open water storage at x = b. Open water boundary at x - 0 with level fluctuation according to a sine function, qg(x, t) -- steady oscillation.

x

KD, S qg(0, t) = h sin(cot),

b

a

099 ~(b,

Ox

t) =

a

099

K D Ot

(b, t).

q9 =

Flow in a finite field

(139a, b)

~ Num. e go(x, t) -- h Im Denom. Num.-

ic°t]

137

with

fl K D c o s h { fi ( b - x ) ~ / ~ } + a ~

Denom.-

s i n h { fl ( b - x)~7-~},

f l K D cosh(fibv/7-~) + a ~ / ~ sinh(fibv/7-~).

b. Periodic abstractions

02q9 __ f12 0(t9 Ox 2

--~

02(t9

go

OX 2

f12

for confined aquifers,

20(t 9

)v2 --- fl --~ S _

_

KD'

for leaky aquifers,

~2 __ K Dc, 77

1

-

-

fi

2)v2

1

cS

139.21 (PF) ~ ~ _

L

_q_qsin(cot)

Confined aquifer with fully penetrating infinite well-screen at x -- b. Discharge of groundwater according to a sine function. Zero flux at x = 0. go = go(x, t) -- steady oscillation.

b q go(x, t) -- ~2 f iIK Dm

139.22

Goa__:_,(o ' t) - o,

a-E~ (b t) -

Ox

Ox

cosh(flx~/7-~) ~sinh(flb~_~)

eiCOtl '

co-

'

q sin(cot) 2KD

2re T

(PF}

As 139.21 but for a leaky aquifer. q

go(x, t) -- ~2 f lIK Dm

ico) eir°t ]" ~/r/+ ico sinh(fibx/r/+ ico) cosh(flx~/r/+

139.23 (PF) _ _ ~ s i n ( w t )

KD, S

x

Confined aquifer with fully penetrating infinite well-screen at x = b. Discharge of groundwater according to a sine function. Zero drawdown at x -- 0. go(x, t) -- steady oscillation. go(0, t) -- 0,

b

G0O_.__(b :_, t) -Ox '

q sin(cot) 2KD

138

(139b)

One-dimensional groundwater flow

q sinh(flx ~7-~) ] cosh(flbx/~) eiwt " q)(x, t) -- 2 f l K D Im ~

139.24 (PF) As 139.23 but for a leaky aquifer.

q sinh(flx~/r# + ico) eiWt ]" qg(x, t) -- 2flKD Im ~/r/+ icocosh(flb~/~ + ico) 139.25 (PF) Confined aquifer with entrance resistance at x = b. Fully penetrating infinite well-screen at x = 0. Groundwater discharge according to a sine function. KD, S w ~ x

~oa( ° t) - q sin(co/) Ox ' 2KD ' ~O~°(b t) qg(b,t) Ox ' Kw

b

~

q ,m[ ~um" e i°~t] q)(x, t) - 2/3 K D Denom.

o

with

N u m . - flKm~/~wcosh{fl(b - x)~i-~w} + sinh{fl(b - x ) ~ / ~ } , D e n o m . - fl K wico sinh(flb~/~w) + V ~ w c o s h ( f l b ~ ) .

139.26 (PF) As 139.25 but for a leaky aquifer.

q ,m[NUm e i°Jt] Denom.

?9(x, t) - 2flKD

with

N u m . - flKwv/i 7 + icocosh{fl(b - x)x/~ + ico} + sinh{fl(b - x)v/r / + ico}, D e n o m . - flKw(rl + ico) sinh(flb~/r/ + ico) + v/rl + icocosh(flbv/~ + ico). 139.27 (PF) 4, qsin(wt)

Confined aquifer with abstraction from open water storage at x = b. Zero flux at x = 0. Discharge according to a sine function. q9--~o(x, t) = steady oscillation.

aZ(o, t) - o, ax

b

a

K D a~o (b, t) + a Ta~o T ( b , t) - q sin(cot). Ox ot

Flow in a finite field

(139b)

139

cosh(flx,v/_~_._~)eioot qg(x, t) -- q Im

]

fl K D v/-{--~sinh(flbv/7-~) + a i co c o s h ( fl b v/7--~)

139.28 (PF} As 139.27 but for a leaky aquifer.

[ ,

qg(x, t) -- q Im

c°sh(fix~/rl+ic°)ei°)t ] flKD~/ri + ic0 sinh(flb~/r/+ ico) + aicocosh(fl~/~ + ico)

139.29 (PF)

Confined aquifer with open water storage at x = b. Fully penetrating infinite well-screen at x -- 0 with groundwater discharge according to a sine function. ~p = q9(x, t) = steady oscillation. Ii Ii

x

Oq9

b

~(0,

a

q sin(wt)

t) = -

Ox Ocp ~(b, Ox

~

2KD a 099 (b, t). K D at

t) =

[Num.] Im ~ e ia't cp(x, t) -- 2fl g------D Denom. q

with

f l K D cosh{fl(b - x) z~/7-~} + a ~ / ~ sinh{fl(b - x)~/Tco},

Num.-

Denom. -- fl K D ~

+ aico cosh(flbv/7-~).

sinh(flb~)

139.30 (PF) As 139.29 but for a leaky aquifer. q

(p(x, t) - 2fiK---~

Im

1

~

Num. ei~Otl

~/rl + iw Denom.

with

flKDv/rl + ico cosh{fl(b - x)v/~ + ico}

Num.-

+ aim sinh{fi(b - x ) v / r / + ico}, flKDv/rl + ico sinh(flbv/r/+ ico) + aicocosh(flbv/rl + ico).

Denom.139.31 (PF)

qsin(wt)

Confined aquifer with abstraction from open water storage at x = b. Zero drawdown at x = 0. Discharge according to a sine function. oscillation. ~o(x, o) - 0 ,

b

a

~0 = ~p(x, t) = steady

140

(139b, c)

K D Oq9(b t) + a

Ox

'

(b, t)

~

sin(cot)

--q

One-dimensional groundwater flow

"

sinh(flx x/~) qg(x, t) -- q Im f l K D ~ ~ c o s h ( f l b ~ ~ )

+ aico s i n h ( f l b ~ )

eiWt ]"

139.32 (PF) As 139.31 but for a leaky aquifer. •

qg(x,t)--qIm[

, sinh(flx'~/17+ico)ei°Jt ] flKD~/rl + icocosh(flb~/o + ico) + aicosinh(flb~/o + ico) "

c. Periodic precipitation or infiltration

02~0 sin(cot) 0q9 OX2 ~ P __ [32 for confined aquifers,

-~

KD

02(,/9 OX2 p2_

139.41

q9 p sin(cot) ~2 t KD

S KD '

_/~2 0(/9 -5-7 for leaky aquifers,

) 2 _ K Dc

1

'

1

7/- /~2~,2 "-" m.

cS

(PF)

~,1,,1, $,1,$,i, $ $,1,! p sin(cot)

Confined aquifer with open boundary at x = b. Zero flux at x = 0. Fluctuation of the precipitation

a sine function, q9 = qg(x, t) = steady oscillation. 0~°(0 t) -- O,

Ox '

b ~°(x1t ) - P-I Sm [ '

139.42

~ico 1 {

cosh(flx~/~) } ] cosh(flb t~7-w) ei°Jt'

~o(b, t) -- O.

~O--"

(PF)

As 139.41 but for a leaky aquifer. •

q [ 1 /1-

~o(x, t) - ~ Im

r#+im

cosh(flx~/r/+ ico) e iogt]. cosh(flb~/r/+ ico)

2~ T

n °

Flow in a finite field

(139c)

141

139.43 (PF)

Confined aquifer with entrance resistance at x = b. Zero flux at x = 0. Fluctuation of the precipitation a sine function, q9 = ~0(x, t) -- steady oscillation.

x ~ , [,1,,,i,$,1,$ ,1,,1,,!,~ ; s;n(wt) KD, S I w

Ocp ~(0, Ox

b qg(x t ) - - p l m

'

S

ieit{ ~ ico

1-

O~o ~(b, Ox

t) = 0 ,

qg(b, t) Kw

t) =

cosh(flx ~ / ~ ) } ]

fl Kw~7--~ sinh(flb~-~) + cosh(flb~7-~)

"

139.44 (PF) As 139.43 but for a leaky aquifer. •

~o(x, t) - ~ Im

q I rl + ico {1 cosh(flx ~/~/+ico) e/cot}]. flKw~/rl + icosinh(flb~/~l + ico) ÷ cosh(flb~/~/+ ico)

p

139.45 (PF)

Confined aquifer with entrance resistance at x = b. Zero head at x = 0. Fluctuation of the precipitation according to a sine function, q9 -qg(x, t) = steady oscillation.

l ~ ,I,,[,,l,,l, ,[,,l, ,l,,l,~l p si n (w t ) xKD, S

w

~o(x,t)-- s I m

1

Num.

7-£~(1 - Denom.)ei~°t

flKw~cosh{fl(b-

Num.-

a~o ~(b,t) Ox

~o(0, t) - - 0 ,

b

1

with

x)~7--~}

+ sinh{fl(b - x ) ~ / ~ } + sinh(fix~7-~), Denom.-

flKw~-~t~cosh(flb~-~) + sinh(fib ~-~w).

139.46 (PF) As 139.45 but for a leaky aquifer. •

q [, +i~o

qg(x, t) -- ~ I m

1-

De-~-om.

e i°)t

1

with

=

~o(b, t) Kw

142

One-dimensional groundwater flow

(139c)

flKwv/rl + icocosh{fl(b - x)x/r I + ioo}

Num.-

+ sinh{fl(b - x)v/rl + ico} + sinh(flxv/o + ico), Denom.-

flKwv/~ + icocosh(flbv/~ + ico) + sinh(/3bv/r / + ico).

139.47 (PF)

Confined aquifer with open water storage at x = 0. Zero flux at x = b. Fluctuation of the precipitation $$$$,b$$$~

b

according to a sine function. ~p = ~o(x, t) -- steady oscillation.

psin(wt)

Oq)(O, t) -- 0 Ox

a

qg(x t) - p I m [ { ' S

1 ico

K D ~ x (b, t) - - a a----~(b,t).

acosh(flx~7--~) }1 sinh(flb~T-~) + aico cosh(flbx/7-w) ei~°t "

~ K D~

139.48 (PF) As 139.47 but for a leaky aquifer. •

q)(x,t)--

Im

[ ( o+ico

N u m . - aicocosh(flxx/~ Denom.-

Num. 1 - Denom. )

eiWt]

with

+ ico),

flKDv/r I + ico sinh(flbx/r / + ico) + aicocosh(flbv/r I + ico).

139.49 (PF)

sin(0 b

~p(x t ) '

a

Confined aquifer with open water storage at x = b. Zero head at x = 0. Fluctuation of the precipitation according to a sine function. ~o -~o(x, t) = steady oscillation.

go(O, t) -- O,

Num. ~ I m ~ 1 - Denom.) ico

eiwt1

with

099 7 - ( b t) Ox '

a Oqo (b, t). K D at

Flow in a finite field

143

(139c)

flKDcosh{fl(b - x) z,v~} 4- a ~ s i n h { f l ( b

Num.-

- x),,~}

+ a~-ww sinh(flx ~[ww), Denom. - fl K D c o s h ( f l b , ~ )

+ a~

sinh(flb~7-~).

139.50 (PF) As 139.49 but for a leaky aquifer. ~o(x, t) --

~ E ~ (~um) Im

77 + i co

1-

Denom.

]

e i~°t

with

flKDv/~ + icocosh{fl(b - x)v/rl + ico}

Num.-

+ aicosinh{fl(b - x)v/rl + Denom.-

i~o] + aicosinh(flxv/rl

+ ico),

flKDv/~ + icocosh(flbv/~ + ico) + aicosinh(flbv/~ + ico).

144

(140a)

One-dimensional groundwater flow

140 BI-4. DISCONTINUITIES IN ONE-DIMENSIONAL GROUNDWATER FLOW Subdivision: a. Discontinuities in the hydrological system. b. Discontinuities in the ground parameters. In both cases the solution of the problem may be obtained by solving separately the continuous parts of the system, or those parts where the ground parameters are continuous, each with (an) arbitrary constant(s) and then determining these constants by applying the law of continuity at the location(s) of the abrupt change(s). There are numerous examples of both kinds of discontinuities and they may also occur together; here only some practical problems will be given. a. Discontinuity in the hydrological system

140.01 (L) Combination of finite and semi-infinite onedimensional groundwater flow, caused by a changing water level of a river (canal) in a

"///////z "//////)///////)////////////////// I

BiBl. '-- --I~ i

I I

--', xl

I

polder area. River level an arbitrary function of the time. Constant polder level -- reference level for the head h(O) = O.

KD, S

O <<.x <~ B"

x>~B"

99 -- 991 (X, t), 32991 OX 2

99 -- 992(X, t),

h(t) -99] = f12 099191 ~2 Ot' 0991

~ol (x, O) - O,

--~-x (o, t) - o .

02992

992 __ f12 0992

OX 2

~2

992(x, 0) - 0,

Ot ' 992(00, t) -- 0.

Continuity conditions" q91(n, t) -- 992(B,/),

0 99---~-l( B t ) Ox '

0992

-~x ( 8 , t)'

) 2 __ K Dc, f12 __

S 1 1 K D ' r l - ~2)~2 -- ~cSo

(140a)

Discontinuities

~1 (X, t) -- ~

j0 t h(t

- r)e -~r dr

11'

- -2 rl

~o2(x, t) -- - ~

145

I /

h(t - r)e -~r erfc /~(B2~r_-x)

h(t-

r)e - ~

erfc /~(x2,/~.-B)

}

+ erfc

(B2~/~_ )

- erfc fl(x2~_ + B)

dr, dr.

140.02 (L)

As 140.01 but with a sudden rise ho of the river level which is kept constant thereafter. h(t) -

0 ho

fort-0, fort > 0 .

991(x, t) -- ho - -~hoe -~t erf

lh o P 2

+ erf

2)~ '

Vr~ + P

1 [ fl(x-B) rp2(x,t)---~hoe -'Tt eft{ 2~/7

+~ho P ~ ~

}-eft{

,~

-P

2)~ ' fi(x+B) 2~/7

~~

' }

,~/~

]

.

140.03 (D)

Steady state of 140.02.

1-e

*

rpl(x)--ho

*

~o2(x) -- ho sinh

cosh ~

,

e - r.

140.04 (PF)

As 140.01 with tidal fluctuation of the river level, according to h(t) = ho sin(cot) 27r with c o - T"

[ho

]

~Ol(x, t) -- Im )~2M2 { 1 - c o s h ( M x ) e - M B } e i~°t , ho (p2(X, t) -- Im )~2M 2 sinh ( M B )e-

Mx

+ioot]

with M -- V/~ + i/32co.

146

(140a)

One-dimensional groundwater flow

140.05 (L) ,,

Confined finite aquifer of width 2b between two parallel straight surface water boundaries with etrance resistance. At x = i a well-screens are placed with discharge q [LZT -1] (a < b).

b

a

0~<x <~a:

a ~<x ~
99 "-- 991 (X, t),

991 (X, O) "~ O,

02q 91 __ f12 0991 Ox 2 Ot '

fl -- K D '

0991

~(0, Ox

2 __

99 = 992(X, t),

S

02992 = / ~ 2 0992 OX 2 0---7-' 0992

t) --0.

~(b,t) Ox

992(b, t)

= - - ~ Kw

Continuity conditions:

'

Ox

991(x,t)----~-

0992 ox

0 99-~1(a t ) -

991(a t) -- 992(a, t),

1+

-7--(a, t) --

'

q KD

Kw

2q b , ~ [cos (b~a~C~n) + c,--7~sin (b~a~ot,z)] (a~2 + 8 + 8 2) COSO/n

K D Z..,n=l X COS (

992(x,t)---D--

rOlnX) - p (

1+

2 ont

fi2b2)' Kw

cos( ~na [ (b-x -r) cos K D = (ot~ + e + e 2) cost~n b 2qb

c~

+ ~ , , sin

b

~,,

exp

-fl2b---7

b with ot,z being the roots of ot tan ct - e - Kw"

140.06 (D) Steady state of 140.05.

992(X, O) -- O,

Discontinuities

(140a, b)

qw(1 q- b -a) Kw b qm( - x ) .

(constant drawdown),

991(x)----~~o2(x)- --~

1 q-

147

Kw

140.07 (D) Leaky aquifer (polder), separated by an impermeable dike of width d from a totally penetrating river or canal with open boundary. Sudden drawdown of the river level with h.

~/////////////////~c x

-1

KD

x >~a"

O<.x~a" d2991 (/9 -- (/91(X),

dx 2

=0,

(/9 -- q92(X), (/92(00) -- 0,

q91(0) -- h.

d2q92 dx 2

q92 ~2 = 0,

~2 _ KDc.

Continuity conditions: d991 ~dx (d)-

(/91(d) -- 992(d),

dq92 (d) Tx

h ~ x , d+X hX ~-d ~o2 - ~ e x . d+X

q)l

-

h-

-

b. Discontinuous ground parameters 140.11 (D) Confined finite aquifer with two different transmissivities between two parallel surface water boundaries one open and one with entrance resistance. Constant precipitation.

p

a

O<<.x<~a"

a ~x

~0 -- ~01(X), dqOl ~ ( o )

dx

991(0) -

K1 w

d2qOl

p

dx 2

K1D

~
= 0,

d2~o2 ~o

~o2(x),

~o2(b) - O.

~

dx:

--

P t- KzD

-~

O

'

148

(140b)

One-dimensional groundwater flow

Continuity conditions" d(pl d~o2 K1-d-~-x (a) -- K2-~-x (a).

(/91 (a) -- ~o2(a),

P

( 2xnt-K1 w ) X+KlW a a + Kl W -- x 2 -t- qga a + K l W p b-x with ~2 K(2 xD - a ) ( b - x ) + ~Oa ~ b -- a

(/9I(x) -- 2K1D q92(X) p

a2)K1 w + ab(b

(b 2 -

-

a)

99a -- 2D K1 ( b - a) + K2(a + K1 w)

140.12

z (/91 (a) -- cp2(a).

(D) I

As 140.01 with different resistances of the semipermeable layers. Constant river level h.

i

t p = 0 Ih- - 4 -I- -

~o=0

~/////~V/////Z/////~ ~//////////~//////~ !Cl

'~-

J

B ! B

.',--

C2

-'

i

KD

X

!

~

x>~ B"

O<.x<.B" d2q)l

(/9 -- (/91 (X),

h - ~Ol = 0

dx 2 ~

X21

~o - q92(x) '

' ~o2(~)

dgOl

~(o)

-

dx

o.

(/91 (B) -- go2(B),

d991 (B) -- d(p2 (B) d---x~ " h,k lCOSh(~l-i )

(/91 ( x ) -- h -

B -+- ~'1 c o s h ( ~ ) ' Xasinh(~) x-B

h~.2 sinh(BX1)e •

~o2 ( x )

-

992

dx 2

X22

O,

~21 -- K D c l , ~ 2 _ KDc2.

Continuity conditions:

*

-

d2q92

x2

X2 sinh(B) + ~'1 c o s h ( ~ ) "

= 0,

Orientation table BII

(200a-c)

BII. T W O - D I M E N S I O N A L WATER FLOW

RADIAL-SYMMETRIC

149

GROUND-

ORIENTATIONTABLEBII 200. Two-dimensional radial-symmetric groundwater flow a. Does the f l o w field theoretically extend to infinity in r-direction ?

go to 210 (BII-1)

The radius r varies from zero to infinity or from almost zero to infinity in case of a pumping well or horizontal drain with small radius. O~
or withrw =

b. There is only groundwater flow outside a circular cylinder

well radius.

go to 220 (BII-2)

There is no flow in the region 0 < r < R and there will be some boundary value for r -- R with respect to flow outside the cylinder. In many cases the region inside the cylinder will contain surface water; the flow then is called flow to a circular pond or basin.

R <~ r < cx~.

c. There is only groundwater flow inside a circular cylinder

go to 230 (BII-3)

There is no flow outside the cylinder with radius R and there will be some boundary value for r = R with respect to the flow inside the cylinder. If the region r > R contains surface water the flow is referred to as flow in a circular island.

O <~ r <~ R. Because of symmetry the flux at r -- 0 is always zero.

150

Radial-symmetric groundwaterflow

(200d)

d. The f l o w can be considered as a combination of some of the foregoing cases

go to 240 (BII-4)

The main problems, belonging to this category are: 1° Problems with a discontinuity in one or more of the ground parameters at r=R. 2 ° A pumping well at the centre of a circular island. 3 ° Circular pumping screens. 4 ° Circular polder surrounded by a polder with different level. Examples.

R

i gil I

0

2°

i

-T- I Ih ~ R

~

~

c

I

I

KD, S

4°

3°

| | ill

(210-212a)

Flow in an infinite field

151

210 BII-1. TWO-DIMENSIONAL RADIAL-SYMMETRIC GROUNDWATER FLOW IN AN INFINITE FIELD

a. Has the soil discrete inhomogeneities in one or more of the ground parameters ?

go to 240 (BII-4)

An example of a discrete inhomogeneity is a pumping well at the centre of a circular region with a K D-value that differs from that of the surrounding area.

b. The soil is assumed to be homogeneous

go to 211

211. H o m o g e n e o u s soils

a. Is a line source (pumping well or drain) involved in your problem?

go to 215

Radial-symmetric flow is only possible if the line sourCe is situated at the origin of the coordinate system. In the case of a pumping well in an aquifer, the well must be fully penetrating in order to maintain horizontal radial flow. A horizontal drain at the top of a very thick non-leaky aquifer has the same solution as a drain in the middle of a thick aquifer, and therefore will be considered as a line source in an infinite field. drain

"~1 ~q

b. No line source

go to 212

212. P r o b l e m s w i t h o u t line source

a. Is your problem a so called initial value problem?

go to 213

If in an infinite field the groundwater head initially is a given function of r only, the heads will diminish in a fixed manner, owing to gravity, whereas the groundwater will move horizontally in radial direction, until finally all heads will be equal. This is a so called initial value problem. The main objective is the determination of the depletion function, that is the function that shows the variation of the heads with time and with the distance from the origin of the coordinate system.

152

(212b, 213a)

Radial-symmetricgroundwaterflow

b. Is horizontal radial f l o w in an aquifer caused by vertical infiltration on a circular area ?

go to 214

The vertical infiltration, which may be natural (precipitation) or artificial is supposed to be equally distributed over the thickness of the aquifer, so that no vertical flow takes place. Infiltration by means of a line source is excluded here (see 215). 213. Depletion functions for radial-symmetric flow in an infinite field

Subdivision: a. Confined aquifers. b. Leaky aquifers. a. Confined aquifers

02f/9

1 Oq9

f12 0(/9

Or 2 ~ r Or --

--~'

/~2

S

_ KD

213.01 (MI)

KD, S

Depletion function for groundwater heads in a confined aquifer, starting from an arbitrary function. ~o qg(r, t) - head. Initial value" ~o(r, 0 ) - F ( r ) . "

r

f12 foo

f12

F(ro) exp [ - -~-7(r2 + rg)}/o(

qg(r, t) -- - ~ --

fo°° fo °°

F(ro)otJo(rot~)Jo(rot) exp

(

fl2rr 0 2t /ro dr0

- ~---~0/2 d~rodro.

t

)

213.o2 (MI) ~0=0

~o=0 r

~o=0

0

Depletion function for groundwater heads in a confined aquifer, starting from a constant h e a d h f o r r < R and zero h e a d f o r r > R. q9 -~0(r, t) -- head. Initial value" ~o(r,O)- / h !0

forO~ R.

Flow in an infinite field

(213a, b)

/j2R2

f12 r2

q)(r, t) -- h exp (

4t

4, )fo

Io (~r v ~ ) e - r °

f0 OOJl(Rot)Jo(rot)exp (

= hR

153

- -~t

dro

ot2) dot.

Steady state: q9 = O. b. Leaky aquifers 0299 Or 2

.

1 O~o r Or .

.

.

.

q9 _. f12 0(t9 )~2 -57'

~-

1 1 )2__ K Dc. ~2~2 = c~'

213.11 (MI) Depletion function for groundwater heads in a leaky aquifer, starting from an arbitrary function of r. q9 q9(r, t) = head. Initial value: ~o(r, 0 ) = F(r).

~////////////~ ~///////////~ c KD

f12 f0oo F(r0) exp { - r / t - -4-7(r f12 2 + r 2 ) } l o ( fl2rrO q g ( r , t ) - 27 2t )ro dro = e -or

~

F(ro)otJo(root)Jo(rot)exp

--

~-2ot2 rodotdro,

213.12 (MI}

:

Depletion function for groundwater heads in a leaky aquifer, starting from a constant head h for r < R and zero head elsewhere, q) = qg(r, t) - head. Initial value:

o

qg(r, 0 ) -

h 0

for0~ R.

f12R2

~p(r, t) -- h exp ( - r/t

-" hRe -ot

fo

f12 r2 4t 4t ) fo I°(flr~

Jl(Rot)Jo(rot) exp

)e-r°dr°

(_' ~..~ot2 ) dot.

154

(214a)

Radial-symmetric groundwater flow

214. Radial-symmetric flow in an infinite field caused by vertical infiltration Subdivision: a. Confined aquifers. b. Leaky aquifers.

a. Confined aquifers ~ _1

0293

'Or --2-

093 _.}_ ~p(r, t) _ f12 093 r -~r KD -57'

S

f12 __

KD

214.01 (MI) The vertical infiltration (for example, precipitation that joins the groundwater) is an

arbitrary function of r and t. p -- p(r, t).

KD, S

1

,o.

93(r, t) -- 2 / ( D

to

exp - ~ o (r2+r2)}I°(

fl2rro

2t0 )rodrodto.

214.02 (MI) p=0

The precipitation is an arbitrary function of time and is equally distributed over the region, where r < R and is zero for r > R.

p=o

p=

P--

p=0

93(r, t) -- -~ .

fo /o t p(t

p(t) 0

for0~ R.

- to)J1 (R~)Jo(rot) exp

,o2)

- ~

dto dc~.

214.03 (MI) As 214.02 but with p(t) -- p - const.

P--

•

p 0

for0~R.

93(r,t) -- K D

-~ J1 (Rt~)Jo(rt~) 1 - exp

- ~-2

dot.

155

(214b)

Flow in an infinite field

b. Leaky aquifers 1 099 r Or

0299 Or 2

99 ~2

q(r, t) __ f12 099 -57, KD

/7-

1 1 ~ 2 __ K D c f12Z2 -- c---S'

214.11 (MI) The vertical infiltration q into the infinite leaky aquifer is an arbitrary function of r and t. Semi-permeable layer at the bottom of the aquifer. Constant level under this layer assumed (for instance, very permeable aquifer), q = q (r, t).

q(r,t)

KD, S -- r z////////////////////~ U//////~////////////A c tp=0

•

go(r,t) --

1 fo°°fotq(ro, t-to)

2KD

to

f12

exp { - ~t0- ~0 (r2 +

r~,}

{

fl2rro ~ × Iolv 2to / r o d r o d t o .

214.12 (IH) The vertical infiltration is an arbitrary function of time and is equally distributed over the region where r < R and is zero for r>R.

q(t) ++~+++++, KD, S

R <

q-

_-

r

q(t)

for0~
0

for r > R.

< R,

"¢I'/////////t'/////A U///A//////~'///////A c ~o=0 go(r, t) --

fo fo

q(t - to)J1 (got)Jo(rot) exp

- ~to - __~ot2 dto dot.

214.13 (IH) As 214.12 but with q(t) - q - const.

q-- Jq

/

0

f o r 0 ~ < r < R, for r > R.

KD

~ _+_ ot2 J1 (Rot)Jo(rot)

1 - exp

- r/t

__ 2

dot.

156

(214b, 215a)

Radial-symmetric groundwater flow

214.14 (IH)

Steady state of 214.13.

q)(r )

_

R qc-K---- ~ ~ / \ /r q R ~ I I | -| ~ I K/k ° | - ~k / KD

R~'KI( )Io( -~r )1

{

forO~ R.

215. Radial flow to a line source (pumping well or drain)

Subdivision: a. Confined aquifers. b. Leaky aquifers. a. Confined aquifers 02q9 1 O~ f12 Oq9 Or 2 ] r Or -Ot

The following formulas hold for a fully penetrating well (abstraction well or infiltration well) in an infinite aquifer with finite thickness D.

KD ', S t J !

r

-

s.-

They are also valid for an infinitely long drain at the top of an "infinitely" thick confined aquifer, provided that Q [L3T -1] and S (dimensionless) are replaced by 2q [LZT -1] and Ss [L-I], respectively. Then f12_ ~D for the well and f12_ ySs for the drain. 215.01 (L}

Instantaneous abstraction Pi [L 3] of water from a line source (discharge impulse), q)(r, t) = drawdown. KD i S

r

qg(r, 0) -- 0

Qi lim(r-~-r~)r-+O

•

qg(r,t) --

0 2yr K D

Pi 4rc K Dt

exp (

for t -- O, for t > O.

f12r2 4t )"

for r > 0,

Pi -- Q i dt.

qg(cxz, t) = 0,

(215a)

Flow in an infinite field

157

215.02 (MI)

22L_

Continuous abstraction of water from a line source. The discharge is an arbitrary function of t. cp(r, t) = drawdown.

~o(r, 0) = 0 , (Oq)) r -+O -~r

lim r

•

qg(r,t) =

'

4rcKD

2r2 4t

qg(cx~, t) = 0 , --

Q(t) . 2 zr K D

f12r2 ) 1 e - x dx 4x x

Q tt

o

215.03 (L) As 215.02 but with constant discharge Q(t) = Qo.

Q(t)

ao

•

Qo E1 qg(r, t) - 47r K----D

(fl 2r2 ) 4t

(Theiss).

t 215.04

(D)

Quasi-steady state of 215.03.

•

I2 - - c

215.05

Q 2zrD

lnz

with I2 = 4) + i dl , z = x + i y.

(L)

As 215.02 but with Q a linear function of the time. Q(t) = at, a [L3T-2].

Q(t)

*

at{ ~2r2 99(r, t) -- 4zr K------D E l ( 4t ) - E 2 (

f12r2 } 4t ) "

t

215.06

(L)

As 215.02 but with Q an exponentially increasing function of the time Q(t)Q° U

Qo e°t2t.

,, t

99(r, t) =

Qo 4rr K D

eot2 t

[ 2'1 exp ( - x --

JO

x

Qoe t~2t ~2f12r2 \ ! dx -- ~ W ( ot2 t , crfl r ) ] 4x 4rc K D

158

(215a, b)

Radial-symmetric groundwaterflow

(L)

215.07

As 215.02 but with Q an exponentially decreasing function of the time

a(0

Qo~..~~_~_

Ooe-U2t

• 215.08

Qoe -~2t.

Q(t)-

qg(r, t) - ~ W ( - ot2t, iot~r) 4Jr K D

(PF)

As 215.02 but with a periodic discharge

Q(t)

l

Q(t) - Qo sin(cot),

Qo

~Im{ei°)t qg(r, t) -- 2rr K D

Ko(flr~)

Qo

}

sin {cot + q~o(flr~/~) } with

~No(flr~) 2rr K D

No(z) = v/{ker(z)}2 + {kei(z)} 2,

k e r ( z ) - Re{Ko(zV~-)},

27r co - ~ . T

q~o(z)- arctan{ ker(z)kei(z)}

kei(z)= Im{K0(z4~-)}.

b. Leaky aquifers

02q9

1 Oq9

Or 2 { r Or

q9 f12 099 )~2 = -0-7'

)2 = K Dc,

1 1 7/ -- fl2~2 = ~cS.

215.11 (L)

~o=0

~ Pi

Instantaneous abstraction Pi [L 3] of water from a line source (discharge impulse), q)(r, t) - drawdown.

~//////~/////////////~ KD

IS ,

q)(r, 0) -- 0 for r > 0,

r

lim(r O~° r-+O ~ r ) - •

~p(r, t)

--

q)(oo, t) -- 0,

Pi = Qi dt,

i

"iexp(_.,

4rr K Dt

f12r2 4t

).

Qi { - 2zr~ K D 0

fort--O, for t > 0.

(215b)

Flow in an infinite field

159

215.12 (MI) Continuous abstraction of water from a line source. The discharge is an arbitrary f u n c t i o n o f t. q)(r, t) -

drawdown.

c

KD

S

q)(r, 0) - 0 ,

r

(Oq))

lim r r40 ~

1 ---------~fot ~o(r, t) - 4zr K'

,

qg(oo, t) - 0 , --

O ( t - to) exp ( - rlto -

Q(t)

2Zr K D

.

f12r2 dto 4to)-~-o"

215.13 (L) As 215.12 but with constant discharge Q ( t ) - Qo - const. ,

Qo W ot qg(r, t) -- 4zr g-----~ '

(Hantush).

215.14 (D) Steady state of 215.13.

(r)

,

Qo Ko q)(r) -- 27r K--------D ~

(de Glee)

215.15 (MI) As 215.12 but with Q a linear f u n c t i o n o f the time Q ( t ) - at, a [L3T-2].

,

at q)(r,t) -- ~ W 4Jr K D

(_~) r/t,

_

a 4zrKD

fo t exp ( - r/to - / ~2r2 4to ) dto.

215.16 (MI) As 215.12 but with Q an exponentially increasing function of the time Q ( t ) - Qoe ~2t. Qo e°12t

•

q)(r,t) - ~ W { ( r / + 4rr K D

ot2)t, f l r v / o + 012}.

160

(215b)

Radial-symmetric groundwater flow

215.17 (MI) As 215.12 but with Q an exponentially decreasing function of the time Q(t) - Qo e-°~2t.

a(t)

l ~._________ Qo~~. •

Qo e-~2t qg(r,t) = ~ W{(r/-cr2)t, flrv/rl -or2}. 47r K D

215.18 (PF) As 215.12 but with a periodic discharge

Q(t)

Q(t) = Qo sin(cot),

Qo

t

qg(r, t) = ~ I m { e 4rc K D

27r co = ~ . T

i°)t Ko(flrv/ico + rl) }.

Flow outside a circular cylinder

(220-222a)

161

220 BII-2. TWO-DIMENSIONAL RADIAL-SYMMETRIC GROUNDWATER FLOW OUTSIDE A CIRCULAR CYLINDER

a. Has the soil discrete inhomogeneities in one or more of the ground parameters ?

An example of a discrete inhomogeneity is an aquifer round a circular basin for which holds that

I J

KzD2

i

go to 240 (BII-4)

K D -- I K1D1 [ K2D2

for R ~< r ~< ro, for r >/ro.

R

r0 b. The soil is assumed to be homogeneous

go to 221

221. H o m o g e n e o u s soils

a. Is a periodic function involved in one of the boundary conditions of your problem?

go to 228

In general, periodic functions occur in the form of sine- or cosine functions, for instance, seasonal fluctuations of the precipitation or tidal fluctuations. Groundwater flow that more or less depends on such periodic fluctuations is referred to as periodic flow. This kind of groundwater flow is solved in a particular manner. If a boundary condition contains a general function of the time it sometimes is useful to develop the function in a Fourier series (if possible) and derive the solution of the problem from a more simple periodic function. b. The flow is non-periodic

go to 222

222. N o n - p e r i o d i c flow

a. Is the boundary condition at r -- R a given head or drawdown ?

go to 223

In most cases the given head or drawdown will be caused by surface water (circular pond or basin that fully penetrates the aquifer) which level may be an arbitrary function of the time ~o(R, t) -- F ( t ) .

There is no entrance resistance between the aquifer and the surface water. In many problems F ( t ) will be a constant or zero ~o(R,t)=h

or

~o(R,t)=0.

162

(222b, c)

Radial-symmetric groundwater flow

b. Is the boundary condition at r = R a given flux?

go to 224

A given flux will occur if a given amount of groundwater is withdrawn from the aquifer by means of a fully penetrating circular well-screen, located at r -- R, provided that the inside of the screen is impermeable to water in order to realize a unilateral discharge of the aquifer.

i

rq

I

I

I

!

I

r

:KD, S

The discharge may be an arbitrary function of the time, a constant or zero (impermeable wall or plane of symmetry). The general boundary condition is

0~o Q(t) - K D - z - - ( R , t) -- q(t) --

R

.

or

c. Is the boundary condition at r flux?

R a combination of head and go to 225

In this Category two kinds of boundary conditions may be distinguished: 1° Entrance resistance between the aquifer and surface water. There always exists a jump between the groundwater head at r - R and the level of the surface water. The general boundary condition is

i

~ w

KD, S ~

g

O~(R t) -- F(t) - cp(R, t) Or ' w

R

2 ° Surface water storage. In radial flow towards a circular basin practically always the storage of the open water plays a role, owing to the finite dimensions of the basin. The surface water level is no longer a given function of time (or a constant, or zero) but depends on the outflow (inflow) of water from or to the aquifer thus on its turn influencing the magnitude of the outflow (inflow). I

In the case of given initial open water level the boundary condition is

D

I I

R 2 a (R t)

1

I i

KD, S r

R

Ot

'

Oq9 = 2 7 r R K D - ~ r ( R , t).

Flow outside a circular cylinder

(222c, 223a)

163 In the case of abstraction of a given amount of water Q ( t ) from the basin the boundary condition becomes: Q ( t ) - rrR 2 0 q ) ( R t) Ot ' Oq9 - 2 r r R K D - ~ r (R, t).

R

223. G i v e n h e a d or d r a w d o w n at r -

R for radial flow o u t s i d e a circular c y l i n d e r

Subdivision: a. Confined aquifers. b. Leaky aquifers. a. Confined aquifers 02(t9 Or 2

t

10q) __ f12 0(/9

f12 __

-57'

r Or

S

gb

223.01 (L)

[ ------~ :I Iv(0

The surface water level sinks according to an arbitrary function of time. Open boundary at r - R. q9 - q)(r, t) - drawdown. ~o(r, 0) - - 0 ,

~o((x), t) - - 0 ,

~o(R, t) -- F ( t ) ,

F ( 0 ) -- 0.

R

u2to

•

~o(r,t) --

h(u,r) -

7rf12R2

u h ( u , r ) F ( t - to) exp

Jo(u)Yo(~u) - Yo(.) Jo(~u)

J~(u) + rg(u)

- f12 R 2 ,] dto du

= "head" function.

223.02 (L) As 223.01 with F ( t ) - h - const.

•

q)(r,t)--h

{

1---

72"

-h(u,r) U

exp

( u2t) } -

fl2R 2 du .

with

164

(223a)

Radial-symmetric groundwater flow

223.03 (L) As 223.01 with F (t) - at, a linear function of the time.

1

•

~p(r, t) -- at -

2afl2 R 2 fo ~ -~3 1 h (u , r ) U2t

F(t)

223.04 (L) As 223.01 with F(t) -- he ~2t, an exponentially increasing function of the time.

t

F(t) rp(r, t) - he a2t Ko(flrot) Ko(flRot)

2h fo c¢ u ( u2t ) rr ct2fl2R2 .if_ l/2h(u' r) exp -- ~f12R2 du.

223.05 (L) As 223.01 with F(t) -- he -~2t, an exponentially decreasing function of the time. /

F(t)

~o(x, t)

~2h e -

~2 t

7l"

f0 O°

U2

_ otzflgR2 h(u, r) 1 - exp otzt

flgR2

du.

223.06 (L)

Initial h e a d head.

p,

itp=0 I I

i J

KD, S r

=

D_

R

g)(r,O)--

~o(R, t) - o .

h - const, q9 - ~p(r, t) -

h 0

f o r r > R, forr--R,

Flow outside a circular cylinder

1

go(r, t) -- ~

- h ( u , r) exp

Jr

223.07

(223a)

u

t-

~~)du u2t

(L) r)

'

Initial head f (r) - h + a In ( ~ ) , a logarithmic function of r. go - go(r, t) -

h

I

igo=0/

i

i

165

head.

/

/

forr>

f(r) 0

go(r, O) --

R,

for r -- R,

go(R,t) - 0 .

R

u2t

go(r, t) -- ~

- h ( u , r) exp

yg

u

~~) d~+aln (~)

223.08 (L) Precipitation an arbitrary function of t.

i9= 0 I

oq2go

1 ago

p ( t ) _ f12 Ogo

~r-ffr ~ 7¢76 R go(r, 0) -- 0,

go(R, t) -- 0,

0go (c~, t) - 0 0-7 "

2fo~ftlh(u-,r)p(t-to)exp(-~ U2to

go(r, t) -- - ~

u

/~2 R 2 }

223.09 (L/ As 223.08 with p ( t ) -

,

7rgD

p-

const.

- ~ h ( u , r)

{1I-

u2t exp

-

dto du.

-57"

166

Radial-symmetric groundwater flow

(223b)

b. Leaky aquifers 0299 Or 2

1 099 r Or

~.2

__ f12 Oq9

f12 __

-~'

1

~ 2 = K Dc, 77 -- ~ . KD ' cS S

223.21 (L)

i

F(t)

ii

r

'

The surface water level sinks according to an arbitrary function of the time. Open boundary at r - R. q9 = q)(r, t) - drawdown.

'q)

~/////////////////2 I KD,s

~,

qg(r, 0) = 0 ,

R

~o(R, t) = F(t), t

•

qg(r, t) -- Yrfl 2 R 2

uh(u, r ) F ( t - to) exp ( - r/to

~p(c~, t) - - 0 ,

F(O) -- O.

u2'°)

/32R 2 dto du.

223.22 (L) As 223.21 with F(t) = h = const.

~o(r,t) -- hKo(C£)

•

Ko( )

2h fo ~ R2 U2 -t-- ~ •

h (u, r) exp ( - r/t

u2t) 132R2 du.

223.23 (D) Steady state of 223.22.

Ko( )

•

~p(r) = h ~

Ko( )

223.24 (L) i i

Initial head = h = const, q) = qg(r, t) = head.

h

!

~//////~.

c qg(r,

0)-- [h /0

~o(R,t) = o . R

f o r r > R, for r - R,

Flow outside a circular cylinder

q9(r, t) -- ~2he

71"

223.25

(223b)

167

.2t )

-ot f o ~ -l h (u, r) exp (

f12 R 2

U

du.

(L)

h~ ~ Z : _ _

f (r) - h + a In ( ~ ) , a logarithmic function of r. q) = ~o(r, t) = Initial head

f ( r)

~0= 0 ,/ I / ~P2/ / / / / / / / / / / / / / / / / / / 2 c /

head.

~o(r, 0 ) - -

f(r) 0

for r > R, forr--R,

qg(R, t) = 0 . R qg(r, t) -- ~2he -~t 7/"

223.26

fo l

- h ( u , r) exp U

(

U2I)

(-~)

fl2 R 2 du + ae -~t ln

.

(L)

Lowering of the polder level around a circular basin according to an arbitrary function of the time h = h (t).

i~o=O

~///////////////////2

i

c

O2q9 + _ 1 O~p + h(t) -- (t9 _ fi2 0~0 Or2 r ~ ~,2 at ~o(r, O) = 0 , ~p(R,t) = 0 ,

KD, S

i r

R

~ (cx~, t) = 0 . Or

~o(r, t) -- ~2rlfooO°fo t -lh(u , r)h(t - to) e x p ( - rlto

7r

223.27

dto du.

(L)

As 223.26 with

,

u

u2to ) /~2R2

h(t) -- h -- const. "Sudden" lowering of the polder level.

~p(r, t) -- h

223.28

{ K1 ° ( ~ ) } Ko(-£)

(D)

Steady state of 223.27. ~o(r) -- h { 1 -

Ko( ) } K0( ) "

u2t )

- ~----5 -e-~t

Jr

fO ~ h ( u ' r ) R2 exp ( -f12 R 2 du. U( u2"+" 7 )

168

Radial-symmetric groundwater flow

(223b, 224a)

223.29 (L)

Vertical infiltration an arbitrary function of t.

I

~///////////////~

I I

c

KD, S

I

i

go t q (t) -_ f12 Ogo )~2 K D Ot ' go(R, t) --0,

02go t 10go

I

Or 2 r Or go(r, O) --0,

r

~oO__(=_cx~, , t) -- O. Or

2 fo~fot -h1(U,u

go(r, t) -- - ~

r)q(t - to) exp

(-

r/to

u2to) dto du.

/52R2

223.30 (L) As 223.29 with q(t) - q - const.

Ko(~) } 2R2q go(r, t) -- qc 1 - Ko(-~) -- ~~ KeD

_rlt foCX~ h(u, r)

R2 exp u(u2 + ~_)

(

223.31 (D)

Steady state of 223.30. go(r) -- qc 1

Ko( ) } Ko( )

224. Given flux at r - R for radial flow outside a circular cylinder Subdivision" a. Confined aquifers. b. Leaky aquifers.

a. Confined aquifers 02go

1 099

-~7~

~ r- -~7 -

f12 ago -57'

f12 __

KD

u2t ) fl2R 2 du.

Flow outside a circular cylinder

(224a)

169

224.01 (L)

I i

Instantaneous abstraction pi [L 2] of g r o u n d w a t e r from a circular well-screen at r - R with a unilateral discharge, q9 = qg(r, t) -- d r a w d o w n .

KD, S

r

qg(r, 0) - - 0

R

qg(c~, t) -- O,

qi KD

a~°(R t ) Or '

--

Pi -- qi dt,

for t -- 0,

0

2pi L ~ ¢p(r, t) -- Jr R S

f(u,r)

for r > 0,

fort >0.

,2t )

( f (u, r) exp

- f12R2

J1 (u)YO(RU) -- Y1 (R)JO(RR)

j2(u) + r2(u)

with

du

-- ' f l u x " function.

224.02 (L) !

I i

~:~ q(t)

r

Continuous abstraction of g r o u n d w a t e r from a circular well-screen at r - R with a unilateral discharge. The discharge is an arbitrary f u n c t i o n of t. ~p - qg(x, t) drawdown.

KD, S

qg(r, O) - 0 ,

qg(cx~, t) - - 0 ,

a~(l~ t)Or ' •

cp(r,t) --

2fo fo '

7r R S

I q ( t~~ KD "

u2'° ) f (u, r ) q ( t - to) exp ( -

fl2R2

dt0 du.

2 2 4 . 0 3 (L) As 224.02 with q ( t ) - q - const.

•

2qRfo l- ~ f ( u ,

qg(r,t) -- r r K D

r)

{(

1 - exp

u2, } fl2R2 )

du.

170

(224a, b)

224.04

Radial-symmetric groundwater flow

(L)

As 224.02 with q(t) - at, a linear function of the time.

.2, ] •

go(r.t)-- rrKD

224.05

f(u.r)

u2

u4

1-exp

- ~

2R2)/d.

(L)

As 224.02 with q(t) - qe ~2t, an exponentially increasing function of the time.

qe ~2t Ko(~r~) go(r, t) -- ~KDt~ " KI(flR~)

•

224.06

2q R fo°° f (u, r) exp( rrKD ctzfl2R2 -k- u 2

u2t ) flZR2

du.

(L)

As 224.02 with q(t) - qe -~2t, an exponentially decreasing function of the time.

• go(r,t)2qR -~2tfo~u2 _f(u,r) { -- ~ e rrKD ot2fl2R 2 1 -

(

exp ot2t

u2t )} du .

f12R2

b. Leaky aquifers go

02('°t 10go Or 2 r Or

i~,2

- - fl

20go --~ '

f12 =

S

, ~2

KD

-- K D c rl -- 1 fl2~2 '

1

cS

224.11 (L)

Instantaneous abstraction pi

!

_

U///Y//////////~, I I

r. ~

[L 2] of groundwater from a circular well-screen at r - R with a unilateral discharge. ~0 - ~0(r, t) - drawdown.

••W//////////////" KD. S

go(r, 0) -- 0

I

go(oo, t) - - 0 ,

R

qi

O---~( R , t) --

Or

go(r,t) - - ~ e2pi rc RS

KD

0

for t - 0

Pi -- qi dt.

fort >0,

_.,f °of (u

r)exp

(

u2t ) flZR2 du.

forr

>0,

Flow outside a circular cylinder

(224b, 225a, b)

171

224.12 (L) As 224.11 but continuous abstraction at r = R an arbitrary function of t.

Oq) ~rO( R '

•

t) --

qg(r,t) = rcRS

q(t) KD'

q)(r, O) -- O,

fo /o t f (u

q)(oo, t) -- O.

U2to )

r)q(t - to) exp

- r/t0

flZR2

dto du.

224.13 (L) As 224.12 with q(t) = q = const.

•

~p(r,t) =

q)~Ko(~) R K D K I ( -Z )

2qR fo °° f ( u , r ) rc K D

....

R2

U 2 _.~._ .7

( exp

- Ot

u2t ) f12R2 du.

224.14 (D)

Steady state of 224.13. q)~ *

Ko(~)

99(r) --

225. Radial flow outside a circular cylinder with c o m b i n e d b o u n d a r y value at

r:R a. Is the combined boundary condition at r = R caused by entrance resistance ?

go to 226

See c o m m e n t s of 222c.

b. Is the combined boundary condition at r = R caused by open water storage?

go to 227

See c o m m e n t s of 222c.

226. C o m b i n e d b o u n d a r y condition at r = R for radial flow outside a circular cylinder. E n t r a n c e resistance at r - R Subdivision: a. Confined aquifers. b. Leaky aquifers.

172

(226a)

Radial-symmetric groundwater flow

a. Confined aquifers

0299 57

1 0go f12 099 r ar

~-

-

/~2

-~'

S -

K-D •

226.01 (L) The surface water level varies according to an arbitrary function o f the time. go = qg(r, t) = drawdown.

I i:h(t)I !

w

go(r, 0) = 0 ,

go(oo, t) - - 0 ,

- K O--~(R t) -Or

•

go(r,t)l ( u , r ) --

2 zrfl2R2

fo~fo t u l ( u , r ) h ( t - t o ) e x p

h(t)

'

(u2to) f12R2

go(R, t) w "

dtodu

with

{Jo}ro(-~u) - {to
{Jo(u) q- t~UJl(U)} 2 q- {Yo(u) -+-t~uYl(u)}

-- "leakage" function,

m

Kw R

226.02 (L) As 226.01 with h(t) -- h - const. •

u2t du. f12 R 2 )

2h fo~ ll(u' u r) exp ( -

go(r, t) -- h

226.03 {L) .,,

~,

As 226.01 with h(t) - at, a linear function o f the time. •

go(r, t) - - a t -

x

h(t)

{

2aflZ R 2 l e o 1 Jr - ~ l ( u , r)

1 -exp

Jo

} /32R2

du.

226.04 {L) As 226.01 with h(t) - he ~2t, an exponentially increasing function o f the time.

h(t)

Flow outside a circular cylinder

(226a)

173

he ~2t Ko(flra) ,

qg(r,t) --

Ko(flRoe) + f l K w a K 1 (tiRol)

u2t )

2h fo°t ul(u, r) ( 7l" U2 ~ ~z-fl2.R 2 exp - f12R2

du.

226.05 (L) As 226.01 with h(t) - he -~2t, an exponentially

decreasing function of the time. /

/

h(t) 2h

gO(X, t) -- ~e-jr

ot2t [ o o U l(u,r){1J0 u2 _ o/2fl2 R 2

u2t)} exp (ot2t

flZR2

du.

226.06 (L)

Initial h e a d -

h - const, q) - q)(r, t) -

head.

---[ii@. . --0 . . . h-t

q)(r, 0 ) - h

f o r r > R,

KTr (R't)-

w'

R

2h fo '-ul ( u ,

qg(r, t) -- ~ Jr

r) exp

(-

u2t ) f12 R 2 du.

226.07 (L)

,

~

hi//

f(r~

Initial head a logarithmic function of r, f(r) - h +aln(-~).

1

i

~p(r, O) -- f (r),

/

i / i

/

w~

KD, S

r

/,;-ffr(R,t) --

R

2 yr

~o(r, t) -- --(h - atg)

fo

l l ( u , r) exp u

(-

U2t)

f12 R2

~'

du +at9 + a l n

.

174

Radial-symmetric groundwater flow

(226a, b)

226.08 (L) Precipitation an arbitrary function of t.

i J tp = 0

i

02~o 1 099 p(t) 32 099 Or----g + - r-~r + ~K D = - Ot , qg(r, O) = O,

Ig444++++,444+++~+p(0

'

KD, S

r

=

=

R K a ~ ° ( R t) _ ~,qg(R' t) Or ' w

~0~°(oo, t) - O. Or

~o(r, t) = - ~2 f o e ° l o t

u2t \

lul ( u , r ) p ( t - to) exp (

132R 2 ,]

dto du.

226.09 (L) As 226.08 with p(t) - p - const.

qg(r,t) -- 2 p R 2 f0 °° zr K D

•

l(u,r)

1-exp

-

f12

S

f12 R2

du.

b. Leaky aquifers 02q9 Or 2

1 Oq9 r Or

q3 __. ~2 0(t9 ~2 -57,

= KD '

)2=KDc,

1 1 r / = fl2~2 = ~ "

cS

226.21 /L)

.... ]il ~I--~(t) ~//////////////~cp w :=~ i i

w

i .

r

c

The surface water level sinks according to an arbitrary function of the time. Combined boundary value (entrance resistance) at r = R. ~o = ~o(r, t) = drawdown.

~ KD, S IIA

qg(r, O) - - 0 ,

~p(oo, t) - - 0 ,

R _KO_._:_, Or

'

t) = h(t) - ~o(R, t) w '

2fo /o t ul(u,

qg(r, t) -- :rrfl2R 2

h(0) - 0.

r)h(t - to) exp

- ~lto

U2to

f12 R 2 ,J dt0 du.

Flow outside a circular cylinder 226.22

(226b)

175

(L)

As 226.21 with h(t) - h - const.

,

hKo(~)

cp(r, t) --

226.23

2h f o ~ Ul(u' r) Jr

Ko(R) + -K~KI(R)

/2 2 ; -

(-

R2 e x p

Ot

u2t f12R 2 } du.

(D)

Steady state of 226.22. hKo(~) ,

~o(r)

226.24

--

R K 0 ( R ) + - ~ K1 (~-)

°

(L) <x

Initial h e a d - h - const. ~o -- ~o(r, t) head.

I

~//////////////////~

i i

c

~o(r, O) -- h,

w

i

~o(R,t)

r

KTr (R,t) -

R

cp(r, t) -- 2he-~t Jr

fo

- l ( u , r) exp u

f12 R 2 )

du.

(L)

226.25

Initial head f (r) - h + a In ( ~ ) a logarithmic function of r. ~o - ~o(r, t) --

h

~~ o = O / ', /

head.

~////////////////////~ c

I

i

(

u2t

/

~o(r, O) - f (r),

w

0~o (R t) K 0--7 '

4

~o(R, t) w

R

2

cp(r, t) -- - - ( h - a O ) e

fo l-l(U,u

r)exp

t-f12R2 u2t)du+ae -nt {if)} 0+ln

~

.

176

(226b)

Radial-symmetric groundwater flow

226.26 (L)

~=o i I i

(t) ~

c 029 t - -1 O~o -Jr- h ( t ) ----T Or - r -~r ~2 9(r, O) = O,

w

K O-2-~(R t) _ ~,~°(R' t) Or ' w

9

-_

f12

O~p Ot '

~0~°(c~, t) - O. Or

cp(r,t) -- mjr 2r/foe°lot --I(U, 1 u

•

Lowering of the polder level around a circular basin according to an arbitrary function of the time.

r)h(t -- to) exp ( -- ~to

u 2 t ° ) dto du. f12R2

226.27 (L) As 226.26 with h(t) = h = const. Sudden lowering of the polder level.

Ko( )

~o(r, t ) - h ] -- K0(R ) -{- ~-~K1 (R) _

2hR 2

~ e - O

~x2

t

[oo

Jo

}

l(u r) u(u2 + 7R 2) exp(

u2t fl2R2 ) du.

226.28 (D) Steady state of 226.27.

~p(r, t) -- h { 1 -

Ko( ) Xo( ) +

}

226.29 (L) I

Vertical infiltration an arbitrary function of t.

i I , ~=0

i

~

! i

02(t9 1 09 Or 2 t r Or

c ~,****+*****$,4~4 q(t) r

KD, S

go(r, O) -- O, K O-2-~(R t) = Or '

R

~(r, t) -- ~-~ 2

~o q(t) 20q9 ;~,2 t- K D -- [3 Ot

fo /o -u1l ( u ,

r)q(t - to) exp

- r/to

O~°(cx~,t) -- O, Or 9 ( R , t) ~

o

w

U2to )

flZR 2 dtodu.

Flow outside a circular cylinder

(226b, 227a)

177

226.30 (L)

As 226.29 with q(t) -- q - const.

93(r, t ) -

x0(~)

qc 1 -

Ko(-~) + ~-~K1 (R)

2R2q

-~t

}

, l(u r)

foo~

u2t

exp (

f12 R 2 )

du.

226.31 (D)

Steady state of 226.30.

Ko(~) } Ko ( R ) ._t_ff_~_K1 ( R ) "

93(r) -- qc { 1 -

227. Radial flow outside a circular cylinder with open water storage

Subdivision: a. Confined aquifers. b. Leaky aquifers.

a. Confined aquifers 0293 1 093 i~ 2 093 ----~ 0 r -t- r Or = --~'

f12

=

KD

227.01 (L)

Groundwater flow to a circular storage basin, caused by sudden lowering of the surface water level with an amount h.

i--_ I~ I

i

I

KD, S

r

93(r, 0 ) - -

R

yl"R 2 0 93 t (

,

t)-

099 2 rr R K D -~r ( R , t ) .

2hfo~ 93(r, t) -- ~Jr

Sc(U, r) exp

a~(~, Or

( u 2 t )f12R2 du

h 0

for r = R, f o r r > R,

t) - o,

with

Sc(U,r) - {uYo(u) - 2SYI(u)}Jo(R u) - { u J o ( u ) - 2SJI(u)}YO(R u) {uJo(u) - 2SJl(u)} 2 + {uYo(u) - 2SYI(u)} 2 -- storage function for confined aquifers.

178

Radial-symmetric groundwater flow

(227a)

227.02 (L)

I

-

Precipitation an arbitrary function of the time.

-

I

i

,

02q9 ~"-1 0~o -4. p(t) . . _ . ~2 Oq9 ---2 Or - r ~r KD at

1~4,,1,,1,4,,1,4,,1,,1,4,1.,1,,1,4,,1,,1,,1, p(t)

i rl~O,S

qo(r, O) -- O,

R

~p(r,

aq9 (oo, t) -- 0 Or

7r R 2 0 ~o( R t) - 2 zr R K D O--~ 0-7 ' Or (R, t).

t) -- -~I f t p(to)

dto-

2fo /ot

-zrS -

(

Sc(U, r ) p ( t - t o ) e x p

u2t0 -

f12 R 2 J

dt0 du.

227.03 (L) As 227.02 with p(t) - p = const.

.

2 "2fo SC Ur'{.

S

227.04

7/"S

" - -u' - ~ -

ex, (

- fl 2 R-'----7

d u.

(L)

F ~`t'

Abstraction of surface water from a circular storage basin an arbitrary function of the time. ~p - qg(x, t) - drawdown. ~p(r, 0) - 0 ,

•9 ,

~p(c~, t) - - 0 ,

~-

R

Q(t) - Jr R 2 099 Oq9 O----;(R, t) - 2rc R K D - ~ r (R , t).

•

2/o /ot

~o(r,t) -- 7rZR2

Sc(U, r ) Q ( t - to) exp (

U2to -

f12 R 2 ]

227.05 (L) As 227.04 with Q(t) - Q - const.

.

~ o ( r , t ) - 2 f l 27/-2 Q

fo°° 7Sc(U, 1 r) { 1 -- exp (

u2t)} f12 R2

du.

dto du.

Flow outside a circular cylinder

(227b)

179

b. Leaky aquifers 02(t9 .

Or 2

1 Oq9 r Or .

.

.

.

q9 ._ f12 -07' ~2

S

f12 __ -- K D '

1

)2 = K Dc, 17 = ~ . cS

227.11 (L)

i

Groundwater flow to a circular storage basin, caused by sudden lowering of the surface water level with an amount h.

h

,

~/~//////////////////////~c

h 0

~o(r, 0 ) - -

for r -- R, forr >R,

7r R2 -~r Oq) (R, t) -- 2JrRKD-~r Oq9 (R, t), R

q)(oo, t) --0.

2hfo

~o(r, t) -- ~

7l"

sl(u, r) exp

t

u2t - r/t

f12 R 2 ,)

du

with

A (u) Jo(-~u) - B(u) Yo(-~u) SI(U , r)

A2(u ) -+- B2(u )

U

= storage function for leaky aquifers,

R2 A(u) -- (u2 + --~)Yo(u) - 2SuYl(U), B(u) -- (u2 -lt- -~2 )Jo(u) - 2SuJl(U).

227.12 (L)

=~ Q (t)

~

t#=o

~/~////////////////////2 c

Abstraction of surface water from a circular storage basin an arbitrary function of the time. q) - q)(x, t) - drawdown. q)(r, 0) - - 0 ,

qg(oo, t) - - 0 ,

Q(t) - JrR 2 Oqg(R t) Ot ' R

•

- 2rrRKDOq)(R t). Or '

2/o /o t Sl(U, r ) Q ( t

~o(r, t) -- 7r2R 2

u2to - to)exp

- r/to

/~ 2 R 2 ]

dto du.

180

(227b, 228a)

Radial-symmetric groundwater flow

227.13 (L) As 227.12 but constant abstraction Q. ~o(r, t) --

o

Ko( )

" R R 2zrKD ~- K1 (~-)

2/32Q f ~ sz(u,r) m _. R 2 U 2 nt-

7/" 2

(

exp

-

r/t

U2t du. f12 R 2 }

227.14 (D)

Steady state of 227.13.

o ~o(r)

x0( )

o

2rrKD

R T K1 (~-) R

"

228. Two-dimensional radial periodic groundwater flow outside a circular cylinder Subdivision" a. Periodic fluctuations of the open water level. b. Periodic abstractions. c. Periodic precipitation or infiltration.

a. Periodic fluctuations of the open water level 228.01 (PF)

Confined aquifer with open boundary at r - R. Open water fluctuation according to a sine function. ~0 -- ~o(r, t) -- steady oscillation. KD, S

r q

~o(R, t) - h sin(cot),

~o(oo, t) - 0 .

D

R

qg(r, t) - him

[ K o ( f l r ~ ) ei~°t]

= h ~No o ~(fir - ~ -~/-~) ~ - ) sin {cot + ~bo(fir v/~) - ( fq~o iR

No (x) -- v/ker 2 (x) + kei 2 (x),

,/~) }

~bo(x)- arctan{ kei(x) } ker(x) '

with

2yr CO

m

T

Flow outside a circular cylinder 228.02

(228a)

181

(PF)

Confined aquifer with entrance resistance t

' I

at r - R. Open water fluctuation a sine function. 99 -- qg(r, t) -- steady oscillation.

2h

I

! FWHKo,s

~p(cxz, t) - - 0 , h sin(cot) - q g ( R , t )

- K-~r (R, t) --

~o(r, t) -- h Im

228.03

Ko(flr~) Ko(~Rv/~)

11)

eiO~t]

+ Kw~/~K1

(~R~/~)

(PF)

Leaky aquifer with open boundary at r =

i 2h i

--

R. Open water level fluctuation according to a sine function, q9 - qg(r, t) steady oscillation.

c

i

q)(R, t) -- h sin(cot), q)(oe, t) - 0 .

R

~o(r, t) -- h Im

228.04

[ K o ( f l r ~ / o + ico)

Ko ( fl R ~/r/ + i co)

]

e i~°t

1

'

r]-

1

fl2)~2

--

cS

(PF)

Leaky aquifer with entrance resistance at

i "///////////////////////////2 c

r = R. Open water level fluctuation according to a sine function, q9 = q)(r, t) -steady oscillation.

KD,S Oq9 (R t) -

- KO--.7

h sin(cot) - qg(R, t)

,

W

g)(cx~, t) - - 0 .

Ko(~r~/rl+ico)

~o(r, t) -- h Im

K o ( fl R ~

) + K w fl ~/ rl + i cog l ( fi R ~/ ~ + i co)

e imt

] "

(228b)

182

Radial-symmetric groundwater flow

b. Periodic abstractions 228.11 (PF) .

Confined aquifer with groundwater abstraction by a circular well-screen with unilateral discharge at r = R according to a sine function. ~o--qg(r, t) = steady oscillation.

~ q sin(cot)

I I

0~p q sin(cot) ~(R,t) = - ~ , Or KD qg(oo, t) - - 0 .

R

•

qg(r,t) --

fl K D

~

K l ( fl R .v/-{-'~) e i '°t "

228.12 (PF) r

q sin (wt)

/

I

U///////////////////d///////////////////////////~ c i i i ~[', KD, S

ogO__(_:_,R t) - q Or ' go(oo, t) - - 0 .

R

•

¢p(r,t) --

Leaky aquifer with groundwater abstraction by a circular well-screen with unilateral discharge at r = R according to a sine function, q9 -- ~o(r, t) = steady oscillation.

q

Ko(~r~/rl -Jr ico) eiOOt] dO -% i coK1 ( fl R ~/ o + i co) ,

~f lIKmD

sin(cot)

KD

1

1

rl -- /32X2 = c-S"

228.13 (PF) Q sin(cot)

Confined aquifer with abstraction from open water with storage. Discharge according to a sine function. ~0 -- ~p(r, t) = steady oscillation.

r I

i

KD, S

Q sin(cot) - 7t'R2 O--~(R t) Ot '

R

¢p(r, t) -- ~ - ~ I m

- 2zrRKDOCP(R t). Or '

[

ioJKo(~Rq/~

2KD + ----U-~~-~K1 ( ~ R ~ / ~ )

eit] "

183

(228b, c)

Flow outside a circular cylinder

228.14 (PF) Q sin (wt)

r

Leaky aquifer with abstraction from open water with storage. Discharge according to a sine function. 99 - 99(r, t) -- steady oscillation.

~////////////////////////////~ c

Q sin(cot) -- zr R 2 099

O--7(R,t) o¢

R

- 2rcRKD-~r (R, t).

?r tc [ 99(r, t) -- ----~Im

K ° ( fl r K ~/Di co + rl) e iC° ] icoKo(~R~/o + ico) + T ~ / O + icoK1 (flR~/~ + ico)

c. Periodic precipitation or infiltration

228.21 (PF) ~o=0

tiiii+iiiii ~p .

.

.

sin (cot)

.

KD, S

Confined aquifer with open boundary at r -- R. Precipitation fluctuation according to a sine function. 99 = 99(r, t) - steady oscillation.

r

R

0299 Or---T

-F

1 099 p sin(cot) = r -~r + KD

-

f12 099 -~'

99(R, t) - 0 .

99(r, t) = P l m [ --~1{1 S kico Ko(fiR~/~)

"

228.22 (PF)

~ ~ ~ ~ p KD, S

sin(wt)

Confined aquifer with entrance resistance at r - R. Fluctuation of the precipitation a sine function. 9 9 - 99(r, t ) - steady oscillation. 0299 Or 2

t

1 099 r Or

+ p sin(cot) ___ f12 0_._~ KD Ot K ~99 99(R t) riT ( R' t ) - ~ ' 11) .

184

(228c)

Radial-symmetric groundwater flow

~o(r, t) -- -zIm ~ 1ico Ko ( fl R ~ / ~ ) + K w fl ~ / ~ K1 ( fl R ~

"

228.23 (PF) I

i i '

~////////////////////////////~ c ! ~ ~ ~ ~ ~ q sin(cot)

irl*O'S

Leaky aquifer with open boundary at r = R. Fluctuation of the infiltration according to a sine function, q9 = qg(r, t) = steady oscillation.

R

1 O~o

0299

Or 2 ~ r Or

~o(r,t)

~o ~2 ~

sIm[

f12 O~

q sin(cot) --

KD

at '

~o(R, t) = 0 .

1

1 { rl + i co

Ko(flR~/~I +-i~)

eiwt

'

--

fl2~2

1 ~

~ °

--

cS

228.24 (PF)

'

~///////////////////////////~

c

:

~ ~ 1 ; ~ ~ ~

q sin(cot)

Leaky aquifer with entrance resistance at r -- R. Fluctuation of the infiltration according to a sine function. ~o = ~o(r, t) = steady oscillation.

R

•

02q9

] 1 099

Or 2

r OF

~o(r,t) --

~p [ q sin(cot) _--/ 3 2 ~Oq9 )~2 KD Ot ' Im

rl + i co

K ~O~o (R Or

t)'

~o(R ' t) w

1

Ko(flR~/rl + iw) + Kwfl~/rl + icoK1 (flR~/rl + ico)

"

Z28.25 (PF)

liiii+iiiiiii;i sin,, R

Confined aquifer with open water storage at r -- R. Fluctuation of the precipitation according to a sine function. ~o = ~o(r, t) = steady oscillation.

Flow outside a circular cylinder

0299 ~---1099 +

(228c)

p sin(cot) __

----T Or - r -~r

/~2 099

KD

099 (R t) --~7(R, t) -- 2JrRKD 0-7 ' "

Jr R 2 099

at

ot, r,~'¢-t-~ t K , (~ R ~--w ) Jeit1"

1

99(r, t) -- s I m I { i co

1 85

i coK o ( ~ R ~/~--o~i ~- ~

228.26 (PF)

Leaky aquifer with open water storage at r = R. Fluctuation of the infiltration according to a sine q sin (cot)

function. 99 = 99(r, t) = steady oscillation.

R

0299

1 099 Or 2 t r Or

99 q sin(cot) )---~-+- K D =

q 99(r, t) -

[ ei°°t

-~Im

rl + iw

f12 099 --~ ,

7(R2099 (R t ) Ot

099 2JrRKD-~r(R

'

{ 1

i coK o ( fl R ~/ rl + i co) + 2~D ¢~~/ ~ + i coK1 ( fl R ~/ ~ + i co)

t).

186

(230-232a, b)

230 BII-3.

Radial-symmetricgroundwaterflow

TWO-DIMENSIONAL RADIAL-SYMMETRIC GROUNDWATER FLOW INSIDE

A CIRCULAR CYLINDER a. Has the soil discrete inhomogeneities as to one or more of the ground parameters ?

C1

C2

~

go to 240 (BII-4)

An example of flow through an inhomogeneous medium is flow caused by vertical infiltration of water on a circular island with different K D- and/or c-values.

b. The soil is assumed to be homogeneous

go to 231

231. Homogeneous soils a. Is a periodic function involved in one o f the boundary conditions o f your problem?

go to 238

In general, periodic functions are sine- or cosine functions, for instance, seasonal fluctuations of the precipitation or tidal fluctuations. Groundwater flow that more or less depends on such periodic fluctuations is referred to as periodic flow. If a boundary condition contains a general function of the time, it sometimes is useful to develop the function in a Fourier series (if possible) and derive the solution of the problem from a more simple periodic function. b. The flow is non-periodic

go to 232

232. Non-periodic flow a. The boundary condition at r -- R is a given head or drawdown

go to 233

In most cases the given head or drawdown will be caused by surface water outside a circular island, which level may be an arbitrary function of the time q)(R, t) - F(t).

In many problems F ( t ) will be a constant or zero: qg(R, t) = h, or qg(R, t) = 0. The flux at r - 0 is always zero. There is no entrance resistance between the aquifer and the open water. b. The boundary condition at r -- R is a given flux

go to 234

A given flux will occur if a given amount of groundwater is withdrawn from the aquifer by means of a fully penetrating circular well-screen, located at r = R, provided that the outside of the screen is impermeable to water in order to realize a unilateral discharge of the aquifer (no flow outside r -- R).

(232b, c)

Flow inside a circular cylinder q

187

The discharge may be an arbitrary function of the time, a constant or zero. The general boundary condition is

Oqg (R t) -- q ( t ) -- Q ( t ) K D o--7 ' 27r R c. The boundary condition at r -- R is a combination o f head and f l u x

go to 235

In this category two kinds of boundary conditions may be distinguished: 1 ° Entrance resistance between the aquifer and surface water. There always exists a jump between the groundwater head at r = R and the surface water level. In general, the boundary condition will be:

O~p -~(l¢, Or

t) =

~o(R, t) - F ( t ) Kw

while in many cases F ( t ) = O. R

2 ° Surface water storage. In this case the circular island is surrounded by a circular canal (ditch) with finite dimensions, such that the storage in the canal plays a role. The surface water level at r = R therefore is no longer a given function of time (or a constant or zero) but depends on the outflow (inflow) of the groundwater from (into) the aquifer, owing to the limited dimensions of the water course. In the case of given initial open water level the boundary condition is:

KDO__:__. Or

t)--a~(R '

099

Ot

t). '

In the case of abstraction of a given amount of water q ( t ) from the canal the boundary condition becomes: R

a

q(t)-

K D Oq) 099 O-T(R, t) + a-f-~(R, t).

233. G i v e n h e a d or d r a w d o w n at r = R for r a d i a l f l o w i n s i d e a c i r c u l a r c y l i n d e r

Subdivision: a. Confined aquifers with variable head at r -- R. b. Confined aquifers with zero head at r -- R. c. Leaky aquifers with given head at r -- R.

188

(233a)

Radial-symmetric groundwater flow

a. Confined aquifers with variable h e a d at r = R

02(/9 1 O~ /32 oq~o ~r ~ t r- 0 r -~'

132

S -- K-D •

233.01 (L,FH)

F(t F

The surface water level rises according to an arbitrary f u n c t i o n o f time. ~o(r, t) -- head. ~o(r, O) = O,

~o(R, t) = F ( t ) ,

F(O) -- O,

0~o

~(o,t) -o.

•

qg(r,t) --

OtnJo(-~)ft f12R 2 ~= J1 (fin) 2

2(t - z') e x p { - c ~ f l 2 R2 } F ( r ) dr

o~

with Otn being the roots of Jo(ct) = O. 233.02

(L,FI-I) Sudden rise h of the surface water level, which is kept constant thereafter. ~o(r, t) = head.

F(t)

O~° (0, t) -- O, ~o(r, O) -- O, Or ~o(R, 0) -- F ( t ) -- [ 0 f o r t - - O ,

/ h

OO

for t > 0.

2t

99(r, t) -- h - 2h ~ J°(~-~) n=0 o~,,J1 (C~n)

233.03 (L,FI-I) Linear rise of the surface water level, cp(r, t) = head.

F(t)

a~o 0--7(o, t) - o,

~o(r, 0) -- 0,

qg(R, t) = F ( t ) - at.

qg(r, t) - - a t

afl2(R2 - r2) + 2af12R2 Z tl "-O

c~3jl (Otn)

xp( ,2R2)"

(233a, b)

F l o w inside a c i r c u l a r c y l i n d e r

189

233.04 ( L , ~ ) Exponentially increasing rise of the surface water level. ~9(r, t) - head.

F(t)

0q) ~(O,t)--O, Or

~(r,O)--O, qg(R, t ) -

I i I

-h

F(t)-

h(e a t - l)

(h and a positive).

q)(r, t) -- h e at

lo(~r

-1

}

Io ( fl R ~l~ )

~

JO(~.r

--y- ) exp ( +2ah ~ °'} ,,=o ( + a ) oln J , ( oen)

°e2t

f12R2)"

233.05 ( L , ~ ) Exponentially decreasing rise of the surface water level, qg(r, t) - head.

F(t)

q)(r, O) - O, J

Or (0, t) - O,

(p(R, t) -- F(t) -- h(1 -- e -at)

/

(h and a positive).

} go(r, t) -- h 1 - e -at Jo(flr~/'-d) Jo (/~/¢,/'d)

~

jO('~,,r --r)

-t-2ah ~ [ °~2 _ ,,=o ~e~ R~

a)otnJ1 (otn) exp (-

2

' )"

f12R2

b. Confined aquifers with zero head at r - R 233.11 ( b---I~)

Initial head an arbitrary function of r. Zero head at r - R. Zero flux at r - 0. q)(r, t) = head. qg(r, 0) -- f (r),

a~o (o, t) - o, 0-7

q)(R, t) - - 0 .

R 2 ~Jo(

qo(r, t) -- ~

Otnr -r)

n=o J2(°tn)

2,

fo"

- - - ~ ) dro.

190

Radial-symmetric groundwater flow

(233b)

233.12 (FH) As 233.11 with f ( r ) = h = const. (see 233.02)

f(r) h

00 jo(Otnr] --y-j q9(r, t) = 2h E t~n J1 (Otn)

r

Rr

n=0

exp(/ 2R2).

233.13 (FH) As 233.11. Parabolic initial value h

f ( r ) = ~---~(R2 - r2). O O

•

L

r

Rr

¢p(r,t) -- 4h Z

J2(an)J°('q--~-) e x p (

n=0

f12R2)"

233.14 (FH) As 233.11. Initial value a Besselfunction (uor

f ( r ) = hJo\ R ] *

Rr

(cro -- 2.4048255577).

~p(r,t)-hJo(-~)exp(

f12R2 ).

233.15 (L,FH)

Arbitrary precipitation as a function of time. Zem

p(t)

ro flux at r = 0, zero head at r -- R. qg(r, t) = head.

02(/9 1 O~o p(t) f12 O~p Or----~ + -r -~r + K D = ~Ot, ~o(r, 0) = 0 , ~o(R,t) = 0 , 6o/9__2.,(0,t) = 0. Or

2 ~-~ J0(-~)fo t

q9(r, t) = ~ n=O OtnJ1 (0/n)

exp

[

2 f12R2(t an

r) ] p ( r ) dr

•

233.16 (L,FH) As 233.15 with p(t) = p = const. *

qg(r,t)--

P

4KD

(R 2 - r 2)

T J exp K D n=O °tn3J1 (C~n)

-

f12R2)"

q9 =

(233b, c)

Flow inside a circular cylinder

191

233.17 (D) Steady state of 233.16.

,

qg(r) -- P - - - ~ (R 2 - r2). 4KD

e. Leaky aquifers with given h e a d at r - R 02(t9 Or 2

~p _ f12 0(t9 -57' )v2

1 Ocp t

r Or

S

~2

-

KDc

'

f12

--

1

K D'

rl

-

-

fl2~2

1 --

-

cS

233.21 (L,FH) The surface water level rises according to an arbitrary f u n c t i o n o f time. ~o(r, t) = head.

I

F(f)

~p(r, 0) = 0 ,

C

cp(R, t ) =

F(t)

with F ( 0 ) =

0,

®a--(o, t) - o. Or

•

~o(r,t) --

.._~_) t 2 CX~OlnJO(°tnr fo exp

f12R2 E

-

J1 (Otn)

n--O

% +~ fi2R2

(t-r)

F ( r ) dr.

233.22 (L,FH) S u d d e n rise of the surface water level, which is kept constant thereafter. ~o - ~o(r, t) - head.

F(t)

~(0, Or

t) -- O,

q)(r, O) -- O,

t

! 0

qg(R t) -- F ( t ) -- I '

I h

fort <0, for t > 0.

oo ,

rp(r, t) -- h ~

- 2h E

anJo(

otnr -y-

)

n=O (Ol2 Jr- -~2 )Jl(Oln) 233.23 (D) Steady state of 233.22.

•

/o(1)

~o(r) -- h ~

exp

_ [

2

%

\ f12 R2

t/

192

(233c)

Radial-symmetric groundwater flow

233.24 (FH) Initial head an arbitrary function of r. Zero flux at r - 0. Zero head at r - R. q)(r, t) -- head.

r

qg(r, 0) - f (r),

C

"//////////////////////////~

a--f-~(o t) - o . Or

KD, S

2 ~ qg(r, t) -- ~

Jo(Z~-~)

n=0 J2(°tn)

q)(R, t) -- 0,

exp

'

{(2 -

% -t-rl)t}folCrof(ro)Jo(~n~)dro. /~2R2

233.25 (FH) As 233.24 with f (r) - h - const.

f(r)

T •

Ir

im

~o(r, t) - 2h ~

c~J1 (c~)

ex { (°:

fl2R 2 +

n--0

t .

233.26 (L,FH)

Arbitrary vertical infiltration as a function of time. --=

Zero flux a t r qg(r, t) -- head.

C

~++++++++++++++++~ q(t)

02q)Or 2 ~ 1 099 r Or go(r, O) -- O,

R

-

R.

99 ~ q(t) = f12~Oq9 ~2 K D Ot ' Og°(O, t) -- 0 Or

go(R, t) - - 0 .

2 ~ Jo(Otnr ( . . _fO i ft_ ) { q9(r, t) -

0. Zero head a t r

~ ~=o ot~ J1 (o~)

exp

2 -

f12R2 +r/ %

)

} (t-r)

q ( r ) dr.

233.27 (L,FH) As 233.26 with q(t) - q - const.

go(r, t) -- qc 1-- Io(~__~) Io(-~)

KD

=

O/n(O/2 _.]_ ~R.2) J l ( O / n )

exp

-

2R2

q) -

(233c, 234a)

Flow inside a circular cylinder

193

233.28 (D) Steady state of 233.27. •

go(r) -- qc ! 1 [

io/ , } io(~)

•

233.29 (L,FH) Circular island with polder level varying according to an arbitrary function of time. go(r, t) = drawdown.

~~~~c

02(t9 1 ago h ( t ) -- (.t9 f12 Ogo ~2 Ot ----7 Or + -r ~r + = h(O) = O, go(r, O) = O,

0go

~(0, Or

R

go(r,t) -- 20 E Jo( -g- ) n=00/n J1 (°/n)

exp

t) -- O,

{ (2 -

~o(R t) -- O.

% + r / ) ( t - r ) } h ( r ) dr. f12 R 2

233.30 (L,FH) As 233.29 with h(t) = h = const. .

go(r,t)--hll-I°(~r)} Io(-~)

233.31

2hR2 oo jo(_q_~_) - ~ ~ 2 n~O R2 = O/n (%2 -'k ~-y)J1 (Otn)

{2

exp ( f12R2 ~n +~)t

(D)

Steady state of 233.30.

to(~) 234. G i v e n flux at r = R for r a d i a l flow inside a c i r c u l a r c y l i n d e r

Subdivision: a. Confined aquifers. b. Leaky aquifers. a. Confined aquifers 02go

~+;

10go __ f12 0(t9

5r

-0-7'

f12 __

KD"

}

194

Radial-symmetric groundwater flow

(234a)

234.01 (L) q(t)

Fully penetrating circular well-screen at r = R. Unilateral discharge of the aquifer. Groundwater abstraction an arbitrary function of time. ¢p(r, t) = drawdown. qg(r, O) -- O,

oga___L(0 ~ t) -- O,

Or

'

O_.~_~ q (t ) or (R' t) -- K D ~o(r, t) -- ~2f0t q(v) dr 2

kfootl J° (~-¢)

2

exp

C~n ( t - r ) }

+ ~ ,,=~ Jo(c~)

f12R2

q

(r) dv

with an being the roots of J1 (or) = 0 (oto = 0 excluded).

234.02 (L) As 234.01 with q(t) = q = const.

2qt qg(r, t) -- - ~ +

q

2qR KD

(2r2 - R2 )

4RKD

Jo(~) °t2Jo(°tn)

n=l

exp

(°~2t) flZR2

"

234.03 (FHR)

Initial head an arbitrary function of r. Zero flux at r = 0 and r = R. Depletion function ¢p(r, t) = head.

qg(r, O) = f (r),

- - ( o , t) - ~O~°(R, t) -- 0

Or

R ~o(r, t) -- - 2 ~f

Rro f

(ro) dro

2 k Jo(~)

+ ~ .=1 Jo~(~")

ot2t ro f (ro) Jo ( ---~-) °tnr° dro. f12R2)fo R

exp (

Flow inside a circular cylinder

(234a, b)

195

234.04 (FHR)

Steady state of 234.03. ,

fo"

qg(r) -- -RE

r o f (ro) dro -- const.

(no flow).

234.05 (D) Precipitation as an arbitrary function of time. Zero flux at r - 0 and r = R. q9 = ~o(t) = head.

lllllllllilillp(t)

i~ 2 0(t9 _

5t -

S

p(t) KD '

0~o

r

~o(0) -- O,

Oq9

0--7(0, t) - -o-r-r(R, t) - O.

1/0'

R

~o(t) - ~

p(to) dto.

234.06 (D) As 234.05 with p(t) = p -- const.

pt ~o(t) - -~-. b. Leaky aquifers 0 2 rp Or 2

1 0(t9 ~ r Or

S

(t9 __ f12 O q9 )~2

~2 =

"~'

1

K D ' ) 2 __ K D c ,

1

rl - - /~2)~2 = --'cS

234.11 (L) Fully penetrating circular well-screen at r = R. Unilateral discharge of the aquifer. Groundwater abstraction an arbitrary function of time.

q(t)

qg(r, t) = drawdown.

"////////////////////////////~

KD, S

c

(p(r, 0) = 0 ,

! 1

r

099 -rO( R '

~(0, Or

t) = 0 ,

q (t) t) --

KD

R go(r, t) -- ~

e -rt(t-r~ q ( r ) d r

--R-)

+ ~

,,=1

JO('~n)

exp

flZR2 +17 ( t - r )

q(r)dr.

196

(234b-236a)

Radial-symmetric groundwater flow

234.12 (L) As 234.11 with q(t) = q = const. •

qg(r, t) -- q,k lo(C£)

2qCe_ot

2qR ~ KD E

xp{

Jo(Z-~) R2

f12 R 2 -+-

t .

234.13 (D)

Steady state of 234.12. qX Io(C£) •

~o(r) =

235.

Radial flow inside a circular cylinder with c o m b i n e d b o u n d a r y value at

KDII(R)"

r:R a. If the combined boundary condition at r = R is caused by entrance resistance

go to 236

See comments of 232c.

b. If the combined boundary condition at r = R is caused by open water storage

go to 237

See comments of 232c. 236. Radial flow inside a circular cylinder; entrance resistance at r = R

Subdivision: a. Confined aquifers. b. Leaky aquifers.

a. Confined aquifers

0299

1 099

Or 2 -+-_r 5r _

f12 099 --~'

f12

S _ I(-D .

236.01 (L,FHR)

F(t) ,i

The surface water level rises according to an arbitrary function of time. qg(r, t) = head.

qg(r, O) -- O, O~a ~r-O( R ' R

t) --

Oqg(O, t) -- O, Or F(t) - go(R, t) Kw

F(O) -- O.

,

197

(236a)

Flow inside a circular cylinder

2 ~cx) 8012 f12R2 ~ (0/2 q_ 82)Jo(0/n)

cp(r,t) =

2 f12R2 (t - r) F(r) dr

× Jo(~~--~) fot exp { with 0/n being the roots of o/J1 (0/) -

R

8J0(0/), 8 -

Kw"

236.02 (L,FHR) Sudden rise of the surface water level, which is kept constant thereafter. As 236.01 with

F(t)

F(t) OO

cp(r, t) -- h - 2h Z

n=0

_,[0 I

fort < 0 , for t > 0.

h

0/nr] 0/2t Jo(--~-/exp ( f 1 2 R 2) (0/2 + e2)Jo(0/n)

8

•

236.03 (L,FHR) Linear rise of the surface water level. As 236.01 with F(t) = at.

lafi2RK w _ ~afi2(R 2 - r 2) 2 e (~nr] (0/2t) + 2af12R2 ~n=00/2(0/2 + 82)Jo(0/n) Jo ----~-/exp - ~f12R2 .

,

qo(x,t) = a t

m

236.04 (FHR) Initial head an arbitrary function of r. qg(r, t) = head. o90..__2.,(0' t) -- O,

qg(r, O) = f (r),

{

0q9

~(R,t)

Or

-

2 cx~ 0/,,2 Jo (z~) exp( ~o(r, t) - ~ n ~° (0/2 + ez)j2(0/n )

ar

qg(r, t) Kw

f12R2

)fo RroI

ro''o( ) dro

198

(236a, b)

Radial-symmetric groundwater flow

236.05 (FHR) As 236.04 with f ( r ) = h = const.

o0

8 Jo ( anr__f)f_

01n2t f12R2 ).

*

~p(r, t) -- 2h Z (012 + 82)J0(01n) e x p ( n--0 236.06 (MI)

Precipitation an arbitrary function of time. ~o(r, t) = head. 02q9 1 099 p(t) f12 0~o ---T Or + -r -~r A- ~KD = ~t

99(r, 0) - 0,

099 q)(R, t) Or (R ' t) -- - ~ K .w

a~° (o, t) - o,

Or

R

2 •

qg(r, t) --

Z (012g-.Jo(~' 82)Jo(01n) 2°°n0 )_+_ l o t {exp

}

f12R2 (t - r) p ( r ) dr.

236.07 (MI) As 236.06 with p(t) = p = const. •

~o(r, t) --

P (R2 4KD

r2)q p R w 2D

2 p R 2 o0

2t)

eJo(~-)

KD ~n--0 012(012+ 82)Jo(01n)

exp ( -

f12R2

236.08 (D) Steady state of 236.07. •

2 - r 2) -~- p R w 2D

~o(r) -- P ~ ( R 4KD

b. Leaky aquifers 02~o

1 0q9

99 __ ~2 0~0

Or 2

r Or

~2

~2 __ S

-57'

~2 __ K Dc, rl

KD '

--

1

1

fl2~2 -- --'cS

236.11 (L,FHR)

~o

FF(t)

The surface water level rises according to an arbitrary function of time. q9(r, t) = head. ~o(r, O) - O,

R

t2

O~o ~r ( R

t)'

-7-(O, t) -- 0,

or

F(O) - - 0 .

F(t) - ~o(R t)

Kw

"

(236b)

Flow inside a circular cylinder

•

99(r,t) =

/~2R2 = (13/2_~_E2)J0(Oln )

199

lot {(

+ r/) ( t - r) } F ( r ) dr.

236.12 (L,FHR) Sudden rise of the surface water level, which is kept constant thereafter. As 236.11 with

F(t)

F(t)

~o(r, t)

2h

_ [0 Ih

o~ X-"

~,~Jo(~) .~ ~=o (~ + ~ ( ~ + ~)Jo(~)

fort <0, for t > 0.

[, ox~/ ( f12~ ~)t/] R2 +

236.13 (D) Steady state of 236.12. •

~o(r) -- h

~o(~) K w I , ( - ~ ) + )~Io(-~)"

236.14 (FFIR) Initial head an arbitrary function of r. qg(r, t) = head. 99(r, O) -- f (r),

m

Oq~(R t) --~r ' 4

0--7(0, t) -- O,

qg(r,t) Kw

---

R oo

qg(r, t ) -

2 _rlt ~--~e ~

2

n--O

x exp

ot n r

~XnJO(--y-) (or2 + e2)j2(an )

( f12o~tR 2 )fo Rr o f (ro)Jo(~n-~R0 ) dro.

236.15 (FHR) As 236.14 with f (r) = h = const. •

~ 99(r, t) -- 2he-Ot~__~ nO--

~;o(~.,~)

(~2 + E2) Jo(o/n)

~x~(~~~) 2t

200

(236b)

Radial-symmetric groundwater flow

236.16 (MI) Vertical infiltration an arbitrary function of time. 99 - 99(r, t) - head.

0299 ~ 1 O~o r Or

c q(t) r ~ ~ ~ 8 S~ ~ 5~ w,

__ 4 q ( t ) -KD 09 ~o(R,--(0' t) --

~p

99(r' 0) -- 0' O~° (R t) -Or '

R

2 °0 8Jo(-q-ff-) fOt ~p(r, t) -- ~ ~~o= (c~2+ 82)J0(°/n) exp

~

[~2

099 Ot '

q(0)--0,

,

Kw

{

--

(Or2)

}

)}2R2 -+-/7 ( t - r) q ( r ) d r .

236.17 (MI) As 236.16 with q(t) = q = const.

•

-- 2 R D Z n~O

~o(r, t)

(~-

e Jo ( -~- ) ~2)'~7R"2 j o ( ~ . )

1-exp

-

°in

+ ~ t

236.18 {D) Steady state of 236.17. ~o(r) -- qc { 1 -

Xlo(~) R

KwI1 (-2) +

}

R " )~Io(-2)

236.19 (MI) Circular island with polder level, varying according to an arbitrary function of time. ~o(r, t) - head.

h(t)

02(t9 c

1 099

~ ~ -

+

~ ~e ~

a-Z~(R t) Or

q)(r, t) - 17E

n--0

-k--J (0t2 + 82)Jo(otn)

2 099

,

h(O) = o,

t)

Kw

'

exp

-

0~o ~ ( 0 t) - 0 Or '

qg(r, O) -- O,

R

h(t) - ~o

-

% + 11 (t - r) h(r) dr. f12R2

Flow inside a circular cylinder

201

(236b, 237a)

236.20 (MI)

As 236.19 with h(t) -- h = const.

2hR 2°° •

go(r, t ) -

)v2

8Jo(°~nr]

[

{

n~O = ( 0/2 nt- 82)(0/2 --k-J -4- -'~2)Jo(0/n) 1 - e x p

-

236.21 (D)

Steady state of 236.20. ,

go(r) -- h

1-

KwlI(-~) + LIo(-~)

237. Radial flow inside a circular cylinder; open water storage at r = R

Subdivision: a. Confined aquifers. b. Leaky aquifers.

a. Confined aquifers oq2go 1 0(/9 f12 ago Or 2 + - r 57 37'

fi2

S _ KD .

237.01 (L)

Initial head an arbitrary function of r. Zero flux at r = O. ~o(r, t) = head.

go(x, O) = f (r), 099 XDur(R,t)--a

Ogo

~ ( 0 , t) = O, Or 099 (R t) a-7 ' "

a

.

20

go(r,t) = R2(O + 2) x exp

fo R

20 2 oo

Jo( ~nr]--k-/

rof(ro) dro + - - ~ E (02 + 20 + ~2)Jg(0/n) n=l

flZR2

rof(ro)Jo

dr0

with 0/n being the roots of 0/J0(0/) + 0 J1 (0/) - O, 0 - Rs. a 237.02 (L) As 237.01 with f ( r ) = h = const.

202

(237a) O~

*

99(r,t) --

Oh 0 + 2

Radial-symmetric groundwater flow

(<') f12R2 .

[ Oln F

Jo, , ) exp (02 + 20 + ~)J~(otn)

20h Z

n=l

237.03 (L) Precipitation an q)(r, t) - head.

arbitrary

02(.t9 10q)

f12099

p(t) Or 2 + - r-fir _~_ ~XD =

p(t)

function

-if'

of

time.

99(r, O) -- O,

a--E(o t) -o Or ' --

099 K D Oq) (R, t) -- - a ~ ( R Or Ot

a

*

t). '

R f t p ( r ) dr

qg(r,t) -- a(O + 2)

2R

Y~. Cxz

jo (~.~)

-"" (0 2 + 20 + ot2)Jo(otn) a n=l

lot

exp

{

2 f12R2 (t -- r) p ( r ) dr.

237.04 (L) As 237.03 with p ( t ) - p •

~o(r,t)

const.

Jo(~)

pRt -

2 0 R Z P Z c~2(02 + 20 + c~2)Jo(c~n) KD n=l

-

a(O + 2)

{ 1-exp ( %2t }

f12R2 ) "

237.05 (L) >q(t)

Abstraction f r o m the open water an arbitrary function o f time. ~o(r, t) - drawdown. ~o(r, O) - O,

0--~ (0, t) - 0 Or

K D O ~ ° ( R t) + aO~°(R t) -- q ( t ) Or ' ot ' ' q [L2T-1]. a *

2lot

~o(r,t) -- a(O + 2) 20

+ ~

~

~

a n=l

q ( r ) dr

jo(~nr...~_)

(02 + 2~ + ~.2)jo(~.)

fo { t

~xp

2 f12R2 (t %

- r ) } q ( r ) dr

(237a, b)

Flow inside a circular cylinder

203

237.06 (L) As 237.05 with q ( t ) -- q -- const. •

~o(r,t) --

2qt

2q RO 2

a(O + 2)

+ ~

KD

, nr

Jo~--g-)

~

7---]=0/2(02 -1--20 + a2)Jo(0/n)

{ 1 - e x p t-

~

f12 R2

"

b. Leaky aquifers 02([3 1 0(t9 Or 2 ~ r Or

(t9 2 0(t9 ,~2 -- fl --~'

1:12 __

S

~

~2

KD '

1

1

rl -- fl2)v 2

-- K D c ,

cS

~ o

237.11 (L)

----____vl

Sudden d r a w d o w n h of the open water level. qg(r, t) - drawdown.

J0

~///////////////////~

f o r 0 ~ < r < R, for r -- R.

q)(r, 0) -- / h

c

KD, S

aZ(o, t) - o Or

r

OqO(R t ) - - - - a 099 (R t)

R

K D o--7

a oo

~o(r, t) -- 2h Z

n=O

'

--~

0/nr

Jo(~.)

(0 + 2) Jg(0/n) + 0 J2(0/n)

'

{

Jo \ - - ~ - ) exp

"

2

-(

R2

%

fi2 R2

RS

with 0/. being the roots of (0/2 + ~-)Jo(0/) + 00/J1(0/) - O , 0 - -77-," 237.12 (L) ~,q(t)

Abstraction o f water f r o m the storage basin an arbitrary function o f time. qg(r, t) -

drawdown.

"~'] "

qo(r, O) -- 0 r

a~o (o, t) - o

0--7

'

c

K D Oq9 (R t) + a ar

'

q [L2T-1].

a

R 2 ~ qo(r, t) -- - E

a

x

Jo(0/n)JO(~nr'-~)

= (0 + 2)Jg(0/n) + OJ2(0/n)

fo t

{(2

exp

--

0/n

f12R2

(R t) -- q ( t ) -57

'

'

204

Radial-symmetric groundwater flow

(237b, 238a)

237.13 (L) As 237.12 with q(t) = q = const.

•

cp(r,t) =

q L Io (-~)

2q O R oo

R

KDII(-£) xexp

KD

J° (°ln ) J° (anr'-'R--)

Z

R2

n=O (C~n2+ ~T){( 0 + 2)J~(otn)+ OJ2(oen)}

fin

-

/32R2 +7/ t .

237.14 (D)

Steady state of 237.13. •

q)~Io(~) KDII(R)"

cp(r) =

238. Two-dimensional radial periodic groundwater flow inside a circular cylinder Subdivision: a. Periodic fluctuations of the open water level. b. Periodic abstractions. c. Periodic precipitation or infiltration.

a. Periodic fluctuations of the open water level 1 099 _ f12 099 r -Or '~

for confined aquifers,

0299 ~- -

Or 2

1 09

02~° t Or 2 r Or

¢p

~.2

-- fl

,k2 -- K Dc, i~2 --

2 09 -~

for leaky aquifers,

S 2re 1 co, rl--~. KD ' T cS

238.01 (PF)

22h

Confined aquifer with open boundary at r = R. Open water fluctuation according to a sine function. q9 = cp(r, t) = steady oscillation.

°!(o t) - o Or

'

cp(R, t) = h sin(cot).

gdSg~

cp(r, t) -- h Im[ I o ( ~ r ~ )

eiCOt]

Mo(flr~-w) = h Moo(-fiR'S-) sin {cot + Oo(~r~'-w) - 0o(/3R~/-w) } Mo (x) -- v/ber 2 (x) + bei 2 (x),

Oo(x) - arctan { ber(x)bei (x) ].

with

Flow inside a circular cylinder

(238a, b)

205

238.02 (PF) ~

Leaky aquifer with open boundary at r = R. Open

m

water fluctuation according to a sine function, q9 -~o(r, t ) - steady oscillation.

",///////////////////////A C

KD, S 0q9 (0 t) -- 0, -~r '

r

ggg88~

q)

(R t) -- h sin(cot) ' "

~ o ( r , t ) - h i m [ IO(~r~/o+ico) ] Io ( fi R ~/ o + i --~)e i~°t "

R 238.03 (PF)

Confined aquifer with entrance resistance at r - R. Open water fluctuation according to a sine function. ~ o - ~o(r, t ) - steady oscillation.

2h

aq9 (0, t) -- 0,

Or

Oq) h sin(cot) - ~p(R, t) ~rO( R ' t) -Kw "

R

~p(r, t) -- h Im I

I°([3r ~/-~) ] I1 (-fiR - ~ ~ ) ; Io ( ~ R ~/~) eic°t "

fl K w ~

238.04 (PF)

Leaky aquifer with entrance resistance at r - R. Open water fluctuation according to a sine function. ~0 - ~0(r, t) - steady oscillation.

~/////////////////////~~_hc

099 ~(o,t) -o, 0q9

R

~(R,

Or

q0(r, t) -- h I m [

t) -

-qo(R,t) Kw

h sin(cot)

Io(fir~/o + ico) e i wt ] . Kwfi~/O + icoI1 (fiR~/~ + ico) + Io(flR~/o 4 ico)

b. Periodic abstractions 0 2 (,fl

1 0 ~0 _ f12 O q9

57

r

-Or

1 0~o 02_~o -t Or2 r Or )2 _ K Dc,

-57 for coiffed aquifers,

~p 2 099 'k2 -- fi --~ for leaky aquifers, f12 =

S KD'

2rr T

CO--~,

17 --

1 -~

.

206

Radial-symmetricgroundwaterflow

(238b)

238.11 (PF)

Confined aquifer with fully penetrating circular well-screen at r = R. Unilateral discharge of the aquifer. Groundwater abstraction a sine

=} q sin(cot)

function. 9 = qg(r, t) = steady oscillation. 09

099 (R t) -- q sin(cot)

~r (O, t) =0,

07

'

KD

R

*

Im[ Io(flr~'~) 9(r,t) = q ~ fl K D ~/~Ii (fl R ~

e imt].

238.12 (PF)

Leaky aquifer with fully penetrating circular well-screen at r = R. Unilateral discharge of the aquifer. Groundwater abstraction a sine

=} q sin (cot) c

s

*

99(r,t) --

function, q9 = 9(r, t) = steady oscillation.

i

0 ! ( 0 t) -- 0,

0!

Or

Or (R, t) --

'

Io(~r~/o+ico)

q

flKD Im ~/r/+ i"coll(flRx/?? + ico)

q sin(cot)

KD

eiwtI"

238.13 (PF)

,q" sin(cot)

Confined aquifer with abstraction from open water with storage. Discharge according to a sine function. 99 = ~0(r, t) = steady oscillation. 6o0__(~.,0, t) -- O, Or a~o ~q9 KD-~r (R, t) + a - ~ ( R , t)

E R

a

= q sin(cot). 9(r, t) = q Im

[

Io(flr~ ~KD~I1

(flR~

+ aicolo(flR~)

eiWt].

(238b, c)

Flow inside a circular cylinder

207

238.14 (PF)

Leaky aquifer with abstraction from open water with storage. Discharge according to a sine function. ~o -~p(r, t) -- steady oscillation.

~qsin(wt)

l ...... ////////////////////////////// C KD, S

O~P(o t) - - 0 , Or '

r

0~o KDO---z-" t) + a (R t) Or ' -~ ' = q sin(cot).

R

a

qg(r, t) -- q I m [

Io(flr~/o + ico) eiWt I . fl K D ~/ o + i coI1 ( fl R ~/ o + i co) + a i colo ( fl R ,,/ rl + i co)

c. Periodic precipitation or infiltration

0299

1 099

-~;~ ~ - r -~r +

p sin(cot) /32 0q9 = KD -57 for confined aquifers,

02~o 1 0~o Or 2 -~ r Or

q9 q sin(cot) 0~o ~2 ~ KD __ f12 --57 for leaky aquifers,

f12

S _

_

KD '

) 2 __ K D c ,

Jr 1 co -- ~ , rI -- ~ . 2T cS

238.21 (PF)

llllllllll+liil IPsi'

~sin(wt)

Confined aquifer with open boundary at r - R. Fluctuation of the precipitation a sine function. ~o -- qg(r, t) -- steady oscillation. ®~--' (o, t) - o,

Or

~p(r, t ) -

¢(R

p ~Im[~{1

t) - o.

Io(flr~/~) ]. -- ~ o ~ R ~ } ei~°t

238.22 (PF)

,sin(wt) ~siI

Leaky aquifer with open boundary at r - R. Fluctuation of the vertical infiltration according to a sine function. ~o - ~p(r, t) - steady oscillation.

0q9 ~(0, Or R

t)-0,

~(R, t) - 0 .

208

(238c)

q [ 1 { 1 --

q)(r, t) -- ~ Im

r/+iw

Radial-symmetric groundwater flow

lo ( fl r ~/ o -t-ico) e iO)t] Io ( fl R ~/O + i"-w) ] "

238.23 (PF)

Confined aquifer with entrance resistance at r R. Fluctuation of the precipitation a sine function. q 9 - qg(r, t) - steady oscillation.

psin(wt)

q)(r,t)--Plm[

1[

°!(o, t) - o, Or

°--~ (R, t) Or

fl K w ~ / ~ I 1 (fl R ~ )

+ Io (fl R ~ / ~ )

~ ~e ~ t)

Kw

eiwt

"

238.24 (PF)

Im KD, S

~ qwsin(wt)

Leaky aquifer with entrance resistance at r R. Fluctuation of the vertical infiltration a sine function, q 9 - qo(r, t) - steady oscillation.

0----~-~ (0 t) -- 0 Or

'

'

Oq9 q)(R t) Or (R ' t) -- ___.___L~' Kw

R qIm[ 1 {1 q)(r, t) -- ~ ~7+ ico

lo(flr ~/rl+ico ) } ei~Ot] fl K w ~/ ~7+ i w I1 ( fl R ~/ r? + i w ) + Io ( fl R ff rl + i w ) " 238.25 (D)

Confined aquifer with closed boundary at r R. Fluctuation of the precipitation a sine funcpsin(oJt)

KD, S r

tion. q ) - qo(r, t ) -

steady oscillation.

°--' (o, t) - o, Or

~O~(R, t) - O. 8r

~o(t)-~

1-t-sin w t - ~

.

Flow inside a circular cylinder

(238c)

209

238.26 (D) Leaky aquifer with closed boundary at r -- R. Fluctuation of the vertical infiltration a sine function. ~o = ~o(r, t) = steady oscillation.

~//////////////////////~ C ~$$$$$~;$~$$$$$~ qsin(wt)

O~o ~(0, Or

o,s

0q) ~(R, Or

t) = 0 ,

t) = 0 .

. . .

R

q { // sin(cot) + //2 _.[_ 092 sin c o t - ~ ~p(t) -- ~ //2 _+_ O92

.

238.27 (PF)

1

Confined aquifer with open water storage at r -R. Fluctuation of the precipitation a sine function. ~o = ~o(r, t) = steady oscillation.

)

O{p (R, t) -- - a ~ ( R K D 0-7 Ot

R ~o(r, t) - p

~(0,

Or

a

S

'

t)

'

t) = 0 .

Im[{ 1

ico

f i K D ~ / ~ I 1 ( f i R ~ / ~ ) + aicolo(flR~v/-~)

}eitl

"

238.28 (PF)

~////////////////// //~

Leaky aquifer with open water storage at r = R. Fluctuation of the vertical infiltration a sine function. 99 = ~o(r, t) = steady oscillation.

- - ( 0 , t) = 0 , Or

R

099 KD~(R,t) Or

a

~p(r, t) -- q Im S

E' / / / + ico

O~O -- - a - - (R,t). at

1

fl K D ~/ o + i w I1 ~ R ~ ~ 7-i-£) 7+a Tcolo ( fl R ~/ // + i ~ )

"

210

(240-241)

Radial-symmetric groundwater flow

240 BII-4. C O M B I N A T I O N S IN R A D I A L - S Y M M E T R I C GROUNDWATER F L O W

a. Is a pumping well at the centre of a circular island involved in your problem ?

go to 241

In this case one-dimensional radial-symmetric groundwater flow is only possible if the well is fully penetrating the aquifer and is located at the origin of the coordinate system. The aquifer may be confined or leaky. b. Is a circular well-screen involved in your problem?

go to 242

The well-screen must be fully penetrating the aquifer and the centre of the screen must coincide with the origin of the coordinate system. A row of fully penetrating wells regularly distributed over a circle, may be approximated by a circular well-screen, if the mutual distances between the wells are relatively small. c. Is the hydrological system of your problem discontinuous?

go to 243

A hydrological system may be considered as discontinuous if it must be described by more than one differential equation with initial condition and boundary conditions. The solution of the problem may be obtained by solving separately the boundary value problems of the continuous parts of the system, each with (an) arbitrary constant(s) and then determining these constants by applying the law of continuity at the location(s) of the abrupt change(s). d. Has your problem a discontinuity in one or more of the ground parameters?

go to 244

Here, as under c, the solution of the problem may be obtained by solving separately those parts of the system, for which the groundparameters are continuous, each with arbitrary constants and then determining these constants by applying the law of continuity at the suitable locations. 241. Well at the centre of a circular island

Subdivision: a. Confined aquifers. b. Leaky aquifers. a. Confined aquifers 0299

1 Oq)

Or 2

r

132 =

Or S KD"

=

_57

0

(non-steady state), (steady state),

Combinations of foregoing cases

(241 a)

211

241.01 (L,FH) Pumping well at the centre of a circular island with open vertical boundary at r = R. Discharge an arbitrary function of the time. go = go(r, t) = drawdown.

=0

i

i: i R go(r, t) --

go(R,t) - - 0 ,

lim ( r O_~rr) - - -

Q(t) 27r K D '

r~O

R

_~) ,

go(r, 0) - - 0 ,

Q(0)-0.

t

Q(t - r) e x p

rrRzs n=o J12(o~)

fl°t2R2 dr

with ctn being the roots of Jo(o~) = 0. 241.02 (L,FH) As 241.01 with Qo -- const.

Q(t)-

0 Qo

go(r, t) -- 2rrKD

fort>0, for t = 0.

In( R )

0o

~

a°(~"~--r) e x p (

tl

)

f12R2 "

241.03 (D)

Steady state of 241.02.

Qo go(r) -- 2rr K D

'nt t

241.04 (L,FH) As 241.01 with Q(t) a linear function of the time. Q(t) = at, a [L3T-2].

Q(t)

at go(r, t) -- 27r K D

a 2R2 Ot nr { --k-J

r r g D n=o c~n4 j ~ ( ~

1 -exp

t 2} -

Otnt ) fi2R2 "

212

Radial-symmetric groundwaterflow

(241a)

241.05

(PF)

As 241.01 with a periodic discharge. Q(t) = 27r Qo sin(cot), co - T"

(t)

Qo I m [ K o ( f l r ~ ) q)(r, t) -- 2rrKD

] eiCOt

Ko(fiRx/-[-~)lo(firx/~)_]o_~R~i

Qo[

2rrKD

No(fir~'-~) sin {wt + ~bo(fir~/-~)}

_ Uo(~R,~)Mo(~r~)

Mo(~R,/-g)

sin {~o,+ ~o(~R,~) + Oo(~r,~) -Oo(~R~)}]

with

Io(x~/~l) - ber(x) + ibei(x) - Mo(x)e iO°(x), Ko(xx/-[) - ker(x) + ikei(x) -- No(x)e i4)(x), Mo (x) -- C b e r 2 (x) + bei 2 (x),

0o (x) -- arctan { ber(x)bei (x) },

No(x) -- Cker2(x) + kei2(x),

~bo(x) - arctan { ker(x)kei (x) ].

241.06 (L,FHR) Pumping well at the centre of a circular island with a closed vertical boundary at r = R (f.i. a closed building pit). Constant discharge. 99 -q)(r, t) = drawdown.

F~Qo

~o(r, o) - o , R

•

q)(r,t) =

lim r r--,o -~r

R

Qot rc R2S

+

o~ o r ( R , t ) --0, --

Qo ~_~ 1 - exp JrKD ,,=1 0/212 noo (~n)

with o6, being the roots of J1 (or) - - 0 (or - - 0 excluded).

2rrgD

(

.

~2R2)}

(24 la, b)

Combinations of foregoing cases

213

241.07 (L,FHR}

w/ Z

Pumping well at the centre of a circular island with entrance resistance at the vertical boundary at r = R, for instance, a circular building pit with a leaky screen. Constant discharge, q9 = qg(r, t) = drawdown.

!

S

w

~o(r, O) = O,

R

R ~ ( R ~t)

Oqg (R t) -Or '

Kw

lim r '

Q

r--+O

0~p) -~r

---

Qo

.

2rrKD

Kw

~ { ~o(r, t) -- 27c K D

In ( R ) + ---~}

(

1 J0(V) Jr K D .=o ~ ~ e x p

2t) f12R2

with c~,, being the roots of or J1 (or) -- 8Jo(ot) and e =

R Kw"

241.08 (D) Steady state off 241.07.

Q

Kto

~ {

In ( R ) + --R--}.

99(r)- 27rKD b. Leaky aquifers 0 2~0 Or 2

1 OqO r Or

S

f12 __

99

[0

~2

)v2

KD'

_ [ f12 -57 a~

(non-steady state), (steady state),

-- K D c .

241.21 (L,FHR) =e~Q ( t)

~o=0

U////////////////~ U////////////////~ c 113

KD

'

'

S

~w

'

r

~ R

Pumping well at the centre of a circular island with entrance resistance at the vertical boundaries (r = R). Discharge an arbitrary function o f the time. ~o = 99(r, t) -- drawdown. 0qg(R, t) qg(R, t) -~r - K----~'

~o(r, 0) -- O, (0~o)

lim r

r -+O

-~r

--

Q(t) 2 rc K D

.

214

•

Radial-symmetric groundwater flow

(241 b) 1

~o(r,t) =

otn jo _~__)

1) /

2

t

-4-~--~ r

dr

n--O R with oen being the roots of orJ1 (or) - eJo(ot) and e - 2-d~"

241.22 (L,FHR) As 241.21 with Qo-const. 0

Q(t) •

¢p(r,t) =

Qo ~ zrR2KD

=

fort>O, fort-0.

~ Otn (oe~ + e2) [~2n

Jo(~) 1 ) Jg(oen)

\ R 2 -~-k

I t c h : \flzK 2 +~-~l),j] .

x [.1-expt241.23 (D)

Steady state of 241.22. Qo ~o( r ) - 2 zr K D

{ (r) Ko

-~

+

KwKI(R) -~,Ko(R) K w l I ( R ) + ),Io(R)

241.24 (L,FH)

!

i

''

KD

" ''

R

•

Pumping well at the centre of a circular island with open vertical boundary at r = R. Constant discharge. ~p = ~o(r, t) = drawdown.

=*ao

~o=0

cp(r,t) =

S r

~p(r, O) = O,

qg(R, t) -- O,

(0~o) lim r -r ~ O -~r

Qo . 2 rc K D

R Q° n ~° J° (~-~) JrR2KD = ~R2[~2+ ~)J?(oen)

[

1-exp-

with u,~ being the roots of Jo(ot) = 0. 241.25 (D)

Steady state of 241.24.

•

Qo q g ( r ) - 2rcKD

go

Ko( R )

-~

r

}

io(~-~,o(~).

/ <

1)/]

c~n f12R2 + ~-~

(241b, 242)

Combinations of foregoing cases

215

241.26 (D)

Pumping well at the centre of a circular island with a closed vertical boundary at

~ao

~o=O

C

r - R. Constant discharge. ~o - ~o(r, t) -

drawdown. I R

•

lim r

R

r~ O

099) --~r

Qo

qg(r,t) --

Oqg (R t) -- O, 0--7 '

qg(r, O) -- O,

r

exp

Qo . 2 ~r K D -

with an being the roots of J1 (o~) - - 0 (~ - - 0 included). 241.27 (D) Steady state of 241.26.

r

Qo

~p(r, t) -- ~2 r{rKK oD( ~ ) +

•

gl(R) r el(R ) I 0 ( ~ ) } .

242. Circular well-screens 02P

-t-

Or 2

1 099 r Or

__ f12

Oq9 -~'

i~ 2 --

S KD

242.01 (L)

=:~Pi

=~ Pi

[ I I

NNNI

~8888g

I I

',

KD

i I

R (o0._2_,(0 Or '

Instantaneous abstraction of groundwater (discharge impulse) from a circular well-screen in a confined aquifer. ~o = qg(r, t) - drawdown. ~o(r, O) - - 0 ,

S

qg(oo, t) - - 0 ,

I"

R

t) = O,

8qg(R t)left_ Oq9

a---r

'

~ r (R' t)right --

{~i D

Pi -- qi dt.

~ peixRp q)(r, t) -- 2-KOt

{-- f12(r2+ R2) } I 0 ( ~ 2 r R ] 4t 2t - / "

for t - 0 , fort>0,

216

(242)

Radial-symmetric groundwater flow

242.02 (MI)

~ q(t)

=~q(t)

~88~88

i

i

i i i ! i

Continuous abstraction of groundwater from a circular well-screen in a confined aquifer. The discharge is an arbitrary function of t.

KD

S

0,

, ( r , O) -

i

r

i

R

--0,

99(c~, t) --0, a99 ( R t ) l e f t - 099

R

99(r, t ) -

a! or(O,t)

Or

2KD

'

q(t)

-~r (R, t)right -- K D

f12 (r2 + R2) } ( 2r Rx ) dx 4x I0 r2 + R2 e-X x "

2(r2+R2)q t 4t

242.03 (MI) As 242.02 with constant discharge qo.

99(r,t)-- qoR f oo ( 2rRx ) dx 2KD 2(r2+R2)Io r2 .~ R2 e-X~'x 4t

242.04 (MI) qo --I

=~ qo

I 19=0

~////~U////////////~////////////~ V//,/AI KD I S , r

|

[

R

As 242.03 but for a leaky aquifer. 99(r, t) - drawdown.

0299 t 1 099 Or2 r Or ~2__

S

K D ' ~,2

99(r, O) -- 0 '

R

KDc. 099

ra~(°' t) - o,

099(R t)left- 099 qo Or ' -~r ( R , t)right -- K D

99(c~, t) - 0 ,

~)

12r" )

2(r2+R2)Io r2 + R2 exp

r 2 -at_R2) d__x_x -x

4t

242.05
Steady state of 242.04.

/ qo"

~(r) -

99 _-- f12 099 Ot '

~2

for 0 ~< r ~< R,

K-D

2-5

f o r r ~> R.

-

4~.2x

x "

/9

217

(242, 243)

Combinations of foregoing cases

242.06 (MS) Instantaneous abstraction of ground:ff Pi I

I

w~

,'

I

KD

,

I

S

to

'

water from a circular w e l l - s c r e e n in a circular island with entrance resistance at the vertical b o u n d a r y of the confined aquifer. ~p - cp(r, t) drawdown. qg(r, 0) -- O,

ro

ro

R

O--~

R

Ocp { qi ~O~Por(ro, t)left - -~r (r0, t)right -KD

, t) = _cp(R,~,Kwt)

0

fort--0

'

for t > 0,

Pi -- qi dt. 2piE 0

,

qg(r,t)--

R2S

jo(Un_~_R ) j o ( a n r

=

(

2 )

j2(an)+j2(otn)

R

with oen being the roots of o/J1 (o/) -- 8Jo(ol), 8 -- Kw"

243. Discontinuous hydrological systems 243.01 (L,IH) I I

h(t) i . . . .

~o - 0

i. . . . . I I

U/////~Y/~ U////////////////2 "#/////////////////~ c I

i i, R

KD, S r~

C o m b i n a t i o n of g r o u n d w a t e r flow inside a circular cylinder and outside a circular cylinder, caused by a circular storage basin in open water on top of

a leaky aquifer. The head in the basin is an arbitrary function of the time. q9 = cp(r, t) = head.

R

O<~r<~R"

r>~R"

cp -- cpl (r, t),

~p -- ~02(r, t),

02(/91

02(/92 ] 1 0 (/92 Or 2 r Or

1 0(/91

h(t) - 991 ___ f12 O~91 )v2 -O - -rT + -r - ~ - r + at' OCpl ~(0, t) - - 0 . cpl (r, O) - - 0 , Or

~02(r, O) - O,

(/92 ---fl 2 0 (/92 )v2 Ot ~2(oo, t) - O.

218

(243)

Radial-symmetric groundwater flow

Continuity conditions: qgl(R

t)-

'

992(R

S

~ 2 __

'

t)

0qgl (R t ) -

'

Or

'

) 2 __ K D c

KD'

0q92

~(

h (O) - 0 '

"

R foC~

cp(r, t) -- --~

"R't''

J1 (Rot)Jo(rot)

fOt

h ( t - r) exp

{- (ot~+ 1) r }dr dot

-- q91(r, t) -- q92(r, t). 243.02 (L,IH) As 243.01 but with a s u d d e n rise ho of the water level in the basin which is k e p t c o n s t a n t thereafter.

ho h(t) -

•

0

qg(r,t) --

fort-0, fort>0.

)~2

Jl(Rot)Jo(rot)

= (/91 (r,

1-exp

--(~-~-+~--~)t]

ot2+~

t) -- (/92 (r, t).

243.03 (L,IH) S t e a d y state of 243.02. cpl ( r ) -- h o - -h-°fR- K1

q92(r) --

R

)~

(R)

(r)

-~ Io -f , r

243.04 (PF) I

i i~=0

2ho

I I

U/////////~

~//////)/////////~ "~/////////////////~ I

KD, S

I

i

R

r

R

c

Ci r c u l a r storage basin with c o n s t a n t level in open s u r f a c e w a t e r with tidal f l u c t u a t i o n of its level on topof a leaky aquifer, q9 = ~p(r, t) = head.

Combinations of foregoing cases

(243)

219

O<~r <~ R:

r>~R"

99 -- 991 (r,

99 -- 992(r, t),

t),

02991]

1 0991

Or 2

r Or

Ot

S ) 2 _ K Dc, f12 = KD'

-

(/92

= f12 0(/92

at

992 (r, 0) -- 0, 0(t92

0(t91 0---7(0, t) -- O.

991(F, O) -- O,

1 0992 + ho sin(cot) )~2 r -5-7r

02992 ~ + Or 2

991 )~2 = f12 0991 '

~(oo, Or

t) --0,

27r

T

Continuity conditions" (t91 ( R ,

3991 Or

0(/92 (R ' t) "

~(R,t)--d~

t) -- 992(R, t),

ho [R h0 [1- (/92(r, t) -- ~-~ Im { -p2

991(r, t) -- ~-~ Im -- K1 (pR)Io(pr)ei~°t], P

with p 2 _

R I l ( p R ) K o ( p r ) }ei°)t 1 P

~2-k-ifl2co.

243.05 (D)

WlI

Flow through a circular dam with innerand outer radius rl and r2 respectively. Open water boundaries with entrance resistance. 99 = 99(r) = drawdown. Different level inside and outside the dam.

KD

W2

d299 dr 2

NSgggg8

rl r

r2 d~ h -99(rl) - K d--r-(rl) 1/)1

99(r) -- h

d~ - K d-7 (r2) -

In ( ~ ) + Km____~2 r2 Kw I

Kw2 • r2

99(r2) 11)2

1 d99 --0, r dr

220

Radial-symmetricgroundwaterflow

(243, 244)

243.06 (D) [~

,

~ v.-q 92

Circular basin in a polder area (leaky aquifer). Impermeable dam of width r2 - rl around the basin. Sudden drawdown of the level of the basin.

rl r

r2 r />r2"

rl <~r <~r2: ~0 - -

q91

(r),

d2991 1 d~pl dr 2 t r dr : 0,

qgl ( r l ) -- h.

¢p -- 992(r), ~02(O0 ) -- 0,

d2~o2

1 d~o2

~P2

dr 2

r dr

~2

~ 2 __

--0,

K Dc.

Continuity conditions: q91 (r2) -- ¢p2(r2),

*

gO1 (r)

-- h

*

~o2(r) -- h

dcpl ~dr( r 2 ) -

d~02 -dT(r2).

XKo(~) + r2K1 ( ~ ) I n ( ~ )

)~Ko(~) + r2K, ( ~ ) I n ( ~ ) ' )~Ko(~) XKo(~') + r2K, (~z)In (rr-Zl)"

244. Discontinuous ground parameters

244.01 (D)

Fully penetrating pumping well in a confined aquifer at the centre of a cylindrical region with a permeability that differs from that of the surroundings. ¢p -- ~o(r) = drawdown. Quasi-steady state

ro

ro

Combinations of foregoing cases

(244)

O<.r<~ro"

r ~>ro"

(p -- ~ol (r), lim

dqgl)

r

r-, O

221

d 2 (/91

1 dqol

dr 2

r dr

d2~02

~0 -- q02(r),

Q

---

-&r

-- 0,

dr 2

t

1 d~02 r dr

--0,

q92((X)) -- 0.

2 7r K1D

C o n t i n u i t y conditions: qgl

(ro)

--

(/92

d~Ol dq~2 K1-dT-r (ro) -- K2-d~r (ro).

(ro);

991(r)--c+27rD

o(,

1

K1

K2

Q

In ro -

In r,

27r K1 D

Q

*

q92(r) -- c - ~ lnr. 2Jr K2 D

244.02

(D)

Circular basin w h i c h level differs f r o m that of the surrounding polder. Different semipermeable layers for basin and polder.

~o=0 I I

NNNNN Cl

R

":

KD

D

O <~ r <~ R"

r >~R"

(/9 -- (/91(r) -- head. d2q01 dr 2

dqOl

~(o)

dr

+-

--

1 dcpl

h - (/91

r dr

)2

=0,

(/9 2

d 2 (/92 dr 2

(r) -- head.

t

(p2(oo)-

- o.

1 dq02

(/92

r dr

X2

0.

)v2 -- K D c l ,

C o n t i n u i t y conditions:

qgl (R) -- (p2(R),

d(,Ol

dqo2

~(R)----~r(R). dr

)~IKI(~)Io(~)

(/91(r) -- h - h

R

)~110(~l ) g l (~2) -1t- ~211 (~11)go(~2) (p2(r) -- h

=0,

)~2 I1 (~1) go (xz-~) R R " )~II0(x~-)K1 (~2) + )~211 (~()Ko(F22)

X22- KDc2.

222

(244)

244.03

Radial-symmetric groundwater flow

(D)

a ¢1

C2

U//////////////////////~ g,

Fully penetrating pumping well in a leaky aquifer at the centre of a cylindrical region with ground constants that differ from the surroundings.

D

R

r>.R:

O~r<~R" q9 - - ~1

d2~l dr 2

~0 - - ~2

(r) = drawdown. 1 dcpl r dr

~ol =0, ,k2

)2 _ K1Dcl.

lim

(r) -- drawdown.

( dq~,~

r --+O

d2~o2 dr 2

r

=-

-~r ]

1 d~o2 r dr

Q 2 zr K1D

.

~o2 =0,. ~.22

)2 _ K2Dc2, ¢p2(00) - 0.

Continuity conditions: d~ol ~1 ( R ) "-- q 9 2 ( R ) ,

,o/ t

~ol(r)--cPRio(~) + ~

*

Ko(~)

~o2(r) = ~oRK o ( ~ )

d~o2

K~--~7-r(R) -/¢2-~r (R).

{Ko (r~ t -

Io(~)

while

Q Xo(~) ~" = 2 ~ R ~,,~ (~)Ko(~)~., + ~,,o(~)x, (~)~.~

r}

Orientation table Bill

(301-302a)

BIII. G E N E R A L T W O - D I M E N S I O N A L

223

GROUNDWATER

FLOW

ORIENTATIONTABLEBIII 300. General two-dimensional groundwater flow a. Is the f l o w finite in at least one o f the two directions ?

go to 302

General two-dimensional flow has components in two directions and has no point of symmetry as in radial flow. Finite flow in a direction is here defined as flow between two boundaries, situated at a finite distance of each other; generally the boundaries are parallel to each other and perpendicular to the considered direction. An example of finite flow is flow with a vertical component in a horizontal aquifer of finite thickness (0 <~ z <~ D). b. There is finite f l o w in neither direction

go to 301

The flow field always extends to infinity along both coordinates, either at one side (negative or positive) or at both sides.

301. Infinite and semi-infinite flow a. Does the f l o w field theoretically extend to infinity at both sides o f each o f the two coordinates?

go to 310 (BIII-1)

A flow field, extending to infinity at both sides of, for instance, the xcoordinate, means that the flow variables vary from x - - e ~ to x + e~ in such a way that no symmetry exists on that interval. If the two-dimensional flow in this way extends to infinity in both directions, we can speak of an infinite field. Flow in an infinite field is, in general, limited to horizontal f l o w in an aquifer of finite thickness.

-- 00 < x < nt-cx3,

--o,z < y < +cx~

b. The f l o w is semi-infinite in one direction and infinite in the other one

go to 320 (BIII-2)

Semi-infinite flow in a direction is defined either as infinite flow with a line of symmetry, or as flow being present only at one side of a boundary. If in the other direction the flow is infinite without line of symmetry or boundary,

224

Two-dimensional groundwaterflow

(301-302a)

it is referred to as flow in an infinite half plane. Besides horizontal flow in x- and y-direction also flow in a vertical cross section of a very thick aquifer (supposed to be "infinitely" thick) may belong to this category.

O <~ x < +c~, - c~ < y < q-cx:~, or -- ~

<

x

<

~OO,

0~
c. The flow is semi-infinite in both directions

go to 330 (BIII-3)

In both directions a line of symmetry or a boundary exists; the flow is referred to as flow in a quarter of a plane. This plane may be horizontal, for instance, if mere horizontal flow takes place in aquifers with finite thickness, or vertical in the case of flow in a vertical cross section of an "infinitely" thick aquifer. 0 ~ < x < +c~,

~

Y"

/

= I

x

"

x

0 ~< y < +c~, or 0 ~ < x < +c~,

Z

0~
302. Finite flow

a. The flow is finite in one direction and infinite in the other one

go to 340 (BIII-4)

In this case we can speak offlow in an infinite strip, which may take place in parallel horizontal planes in finite aquifers without vertical flow component, or in parallel vertical cross sections of finite aquifers with a vertical flow component.

a

O<.x~a,

D

x

-c~
or

-cx~<x

<+c~,

O<~z<~D

Orientation table Bill

(302b-d)

225

b. The flow is finite in one direction and semi-infinite for the other one

go to 350 (BIII-5)

There are two parallel boundaries perpendicular to one direction and one boundary or line of symmetry perpendicular to the other direction, both in horizontal planes and in vertical planes, the latter representing flow in vertical cross sections of a semi-infinite aquifer with a finite thickness. This flow is referred to as flow through a semi-infinite strip.

O<.x<~a, x

0 <~ y < +(x~, or

D

Y.

0~<x < e e ,

O~z~D I

a

x

e. The flow is finite in both directions

go to 360 (BIII-6)

This flow, either in horizontal or in vertical planes will be referred to as flow

in a finite strip.

O<~x<~a, a

X

O<~y<~b, or

D

O<~x<.a, O~z~D

a

x

d. The flow can be considered as a combination of some of the fore-mentioned cases a, b and c

go to 370 (BIII-7)

In general, all cases that do not fit properly in the fore-mentioned categories, have been classified under this section. An important group of problems, belonging to this category, are those which involve discontinuities in one or more of the ground parameters.

226

Two-dimensionalgroundwaterflow

(302d)

Example.

! ./~" ,,\

\

\ •

/

/

/

//~/I///I//~///5~//~i"6

u

/ /la\~o\

\

infinite strip and flow in a quarter of a plane.

Two pumping wells at (b; 0) and ( - b ; 0) with discharge Q in an infinite field with a strip of different permeability of the aquifer (K D) and different resistance of the covering layer (c). As the coordinate axes are axes of symmetry, the problem is a combination of flow in a semi-

(310-312a)

Flow in an infinite field

310 BIII-1.

227

G E N E R A L T W O - D I M E N S I O N A L G R O U N D W A T E R F L O W IN AN I N F I N I T E FIELD

a°

Has the soil discrete inhomogeneities in x- and~or y- direction?

go to 370 (BIII-7)

An example of a discrete inhomogeneity is uniform flow in a confined aquifer with different permeabilities inside and outside a region of elliptical shape. Y

(open water) o r K2 - 0 (impermeable region), there is no groundwater flow at all in that region; in those cases the soil will be considered homogeneous, with a boundary of elliptical shape (--+ b). If K2 -- oo

b. The soil is assumed to be homogeneous

go to 311

Also anisotropic homogeneous soils belong to this category, provided that the main directions of the anisotropy coincide with the coordinate directions. Solutions for anisotropic cases can easily be derived from analytical solutions for isotropic problems. 311. Homogeneous soils a. Are wells or well-screens involved in your problem?

go to 316

As flow in an infinite field is limited to horizontal flow in an aquifer of finite thickness, the wells and well-screens must be fully penetrating the aquifer. b. No wells or well-screens

go to 312

312. Problems without wells or well-screens a. Is your problem a so called initial value problem?

go to 313

If in an infinite field the groundwater head initially is a given function of x and y, the heads will diminish in a fixed manner owing to gravity, whereas the groundwater will move horizontally in lateral direction, until finally all heads will be equal. This is a so called initial value problem. The main objective is the determination of the depletion function, that is the function that shows the variation of the heads with time, everywhere in the field.

228

(312b-313a)

Two-dimensionalgroundwater flow

b. Is horizontal flow in an aquifer caused by vertical infiltration on a given area ?

go to 314

The vertical infiltration, which may be natural (precipitation) or artificial, is supposed to be equally distributed over the thickness of the aquifer, so that no vertical flow takes place. Point- and line-infiltrations, caused by wells and well-screens, respectively, are excluded here.

c. Flow towards and from straight open boundaries

go to 315

Cases, that do not fit properly in the categories a and b belong to this section. Mostly they involve flow between open boundaries of fixed length and position.

313. Depletion functions for general two-dimensional flow in an infinite field Subdivision: a. Confined aquifers. b. Leaky aquifers.

a. Confined aquifers 02q9

O2 0y 2

KD"

--~"

Initial value" ¢p(x, y, 0) - F(x, y). 313.01 (MI) Depletion function for groundwater heads in a confined aquifer, starting from an arbitrary function. ~o - ~0(x, y, t) - head. Initial value:

.~Yx

KD, S ~o(x, y, O) -- F(x, y).

Y

~o(x, y, t)

l1

-

-

-~

~

F(xo, Yo)

/32 x exp [ - ~-]-{(x - xo) 2 + (Y -- Yo)2 }] dxo d Yo. d

x

(313a, b)

Flow in an infinite field

229

313.02 (MI) Depletion function for groundwater heads in a confined aquifer, starting from a c o n s t a n t h e a d h in the f i r s t q u a d r a n t and zero head everywhere else. ~0 = ~p(x, y, t) = head. Initial value:

~o=0 x

¢p=O

¢p=O h

~o(x, y , O ) -

0

for x > 0 together with y > 0, for x < 0 and y < 0.

h S t e a d y state: qo -- -g.

313.03 (MI) Depletion function for heads in a confined aquifer, starting from a c o n s t a n t h e a d h in the w e d g e between y -- 0 and y -- m x and zero head everywhere else. ~0 = ~o(x, y, t) = head. Initial value:

_

x

~o=0

~o=0 h

~o(x, y , O ) -

for0 m x .

h erf(fix,__) }erf( fly ~{1+ 2~/t 2-~)

~o(x, y, t ) -

hfTl

2~"

x e-°°2erf - ~ ( Y

24'7

S t e a d y state: ~p -

0

- m x ) - moo

I

dco.

~h • -SY

b. L e a k y a q u i f e r s 02(,t9

02(t9 .-{-

OX 2

. - -

Oy 2

q9 __ fi 2 0 q9 i~.2 -~'

Initial value: ~0(x, y, 0) - G ( x , y).

f12 __

1

S ~ 2 __ K D c rl -/~2)v2 K D ' '

1 ~ °

cS

230

(313b, 314)

Two-dimensional groundwater flow

313.11 (MI)

//'///1//////////////////////////////~

~

y

KD, S

Depletion function for groundwater heads in a semi-confined aquifer, starting from an arbitrary function. ~o = ~o(x, y, t) = head. Initial value:

~X

~o(x, y, O) -- G(x, y).

~0(x, y, t) =

4rr t

G (xo, yo)

~

~2 ×

+ (y - yo) 2 }] dxo dyo.

313.12 (MI)

~p=0 X

~o=0

Depletion function for groundwater heads in a semiconfined aquifer, starting with a constant head h in the first quadrant and zero head everywhere else. ~o = ~o(x, y, t) = head. Initial value:

99=0 ~o(x, y,O)--

h

0

for x > 0 together with y > 0, for x < 0 and y < 0.

¢p(x, y , t ) - - ~e-~t -~) }• h { 1 + erf(2_~t)} { 1 + erf( ~fly

313.13 (MI)

The wedge of 313.03 for a semi-confined aquifer.

~o(x, y, t) -- e -or x solution of 313.03. 314. General two-dimensional flow in an infinite field, caused by vertical infiltration

Subdivision: a. Confined aquifers. b. Leaky aquifers.

Flow in an infinite field

(314a)

231

a. Confined aquifers

0299

0299

p(x, y, t) __ f12 099 -57' KD Ox-----g+ ~Oy 2 +

314.01

f12 __

S

/~b

(MI)

The vertical infiltration (for example, precipitation that joins the groundwater) is an arbitrary function of x, y and t. p - p(x, y, t).

/

.~Yx

KD, S

~o(x, y, t) --

1

4rr K D

oo

oo

t - to

f12

× exp [ - - 4 ( t -

314.02

to){(x - - X O ) 2 - + - ( y - yo)2}] dxodyodto.

(MI)

The precipitation is an arbitrary function of time and is equally distributed over the first quadrant and is zero everywhere else:

~x

p(t) P--

314.03

for x > 0 together with y > 0, for x < 0 and y < 0.

'Ji tp(to) / l + e r f < ~ )}{ l + e r f (~y)}

cp(x y t) -- - ~ '

0

'

2 ~ / t - to

2 ~ / t - to

dto.

(MI)

As 314.02 except p - constant in the first quadrant:

P-

p 0

for x > 0 together with y > 0, forx <0andy <0.

foil+ erf,,( 2~/t~--to )//,+erft 2~/t~Y--to ~/d,o

~o(x, y, t) -- &~4a

232

Two-dimensionalgroundwater flow

(314b)

b. Leaky aquifers 0299 OX2

0299 Oy2

)2

q(x, y,t) KD

__ ~ 2 0~0

-~'

f12 __ S ,k2 -- K Dc. KD'

314.11 (MI) Artificial infiltration q into an infinite semi-confined aquifer with thickness D is an arbitrary function of x, y and t. q - q (x, y, t). /////. "/////////////////////////////'~ C

~

,

y

,

t

)

~o,Isyx 1

*

q)(x, y , t) - -

~4rc K D f +~ f +~ ~ fo

f12

t-to

fl2~2

x exp

q (xo, _ yo, to

4(t

to)

{(x - xo) 2 + (y - yo)2}] dxodyodto.

314.12 (MI) The artificial infiltration is an arbitrary function of time and is equally distributed over the first quadrant and is zero everywhere else. I

x

for x > 0 together with y > 0, for x < 0 and y < 0.

q(t)

q--

0

1/o'

cp(x, y, t) -- --~

q(to) exp

('-'o)/ fl2~2

1 + erf 2~/t - to

flY ) }dto. x {1-1-erf(2~/t _ to 314.13 (MI) As 314.12 except q - const."

q- {qo y

for x > 0 together with y > 0, forx<0andy<0.

t, -

f0 t exp(-

z"

erf( Y

(3 14b, 3 15a)

Flow in an infinite field

233

314.14 (MI) Steady state of 3 14.1 3.

0299 OX 2

0299 Oy 2

t

99(x, y ) -

99 )v2

÷ KqD --0.

qCfo~e-~ll+erf(

T

x )}{l+erf(

Y )}

2 A.v/--d

2 )~. ~

d ol .

315. General two-dimensional flow in an infinite field; flow to and from equipotential planes 0299 0299 Ox----S + ~03,2 = 0

Subdivision: a. Flow between straight open boundaries. b. Flow towards straight open boundaries, caused by sudden lowering of the open water level. a. Flow between straight open boundaries

315.01 (SC) Flow between two semi-infinite parallel equipotentialplanes. 99(x, y) = drawdown.

iiiiiiii " =iiilhiiiiii iiiiill; ..........~ - o

xo-~"

*

Z=

99--

0 h

f o r y - - 0 , x >~x0, f o r y - - y 0 , x >~0.

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

I2 Yo _1e2irr(1- x~h) + (77 -- 1)eiTr(1-K~h) + irrr/~--~ zrr/ 2

~+~]

with r/from: 1

rl

2~7

2

xo In r / = z r ~ , yo

z = x + i y , I2 = q5 + i ~ ,

~b= K99.

1

234

Two-dimensional groundwater flow

(315a)

315.02 (SC) F l o w b e t w e e n two semi-infinite equipotential planes, m a k i n g an angle ofzr with e a c h other, w h e r e 0 < of < 1. q)(x, y) - d r a w d o w n .

Yo

Xo

q)=0 0 h

~o--

fory--0, x />xo, for y -- x tan(ofyr) + Yo, x >~ 0, Yo > 0.

of(1 - of)(2 - of) {cot(yrof) 4- i} Z -- YO (2 - of)~ - o f x

1 2-of

W2_Ot [ r / - 1 W 1-oe 4- -~ w - ~ ) - Yo cot(yrof) 1 -of of

with rl from: --of/7 2-°e 4- ( 2 - of)r/1-~ _-- __x°sin(zrof) + cos(zrof), (2 - o f ) r / - o f Yo

Z -- x 4- iy, w -- e 2Jri(1- x~h), $2 -- ~b 4- i ¢ .

315.03 (SC) F l o w b e t w e e n two semi-infinite equipotential planes w h i c h are perpendicular to e a c h other, qg(x, y) - d r a w d o w n .

q)-

,

Zm

with w r/~

{

3yo i 2 w v / - w 4- 2(0 - 1)v/-w 44 ( 3 0 - 1) e iJr(1- K~'g) and rl from:

4- 3 x° Yo

xo = 0 . Yo

0

for y - 0, x > xo,

h

for x - 0, y > Yo > 0.

2,}

Flow in an infinite field

(315b)

235

b. Flow towards straight open boundaries, caused by lowering of the open water level 315.11 (FS,SV}

Sudden lowering of the head with h in two semi-infinite equipotential planes, making an angle of otjr with each other, where 1 < ot < 2. Non-steady state with ~o(r, O, t ) = drawdown.

Y or Jr

.....

-a

99(r, O, t) = (p(r, c~Jr, t) = h

1 O~p _[_ m1 ~02(t9 02~ ~__ -Or - T - r -~r

= f12 Oq9

r 2 002

~p(r, O, t) = h

(r >1 0),

at

oejr Z n=O

{

sin (2n + 1)

0//0

J(.~)

(ru)exp

-

t

~-~u 2 --'u

315.12 (C) p

Y

Quasi-steady state of 315.11. Point P ( R , ~oelJr) reference point for 99 ((#p 0). Constant head c for the equipotential planes. a"2-gc

315.13 (FS,SV) As 315.11 with o~ - 3_ 2"

.

.

.

.

.

.

ib-t,

qo = h

315.14 (C) Quasi-steady state of 315.13. As 315.12 with o ~ - 32"

~o(x, 0, t) = h

for x ~> 0,

~o(0, y, t) = h

for y <~ 0.

1+i

~

.

236

(316a)

Two-dimensional groundwater flow

316. General two-dimensional flow in an infinite field to wells or well-screens

0299 ~ 02~0 OX----2 - -~y2 -- O.

Subdivision: a. Wells b. Wells c. Wells d. Wells

and and and and

equipotential impermeable equipotential well-screens

planes. planes. planes with entrance resistance. in uniform flow.

a. Wells and equipotential planes 316.01 (SC)

Flow between a well with discharge Q and two

\

semi-infinite equipotential planes, making an

angle of an" with each other, where 1 < ot ~< 2. q9(r, 0) = ~o(x, y) = drawdown, P (ro, 0o).

P

Oo

Z

~o(r, O) = qg(r, ~zr) -- 0

......

........... x-

_...~Q in ( z l / , .(2 -- 27r D Q

r 1/or C O S

=1/~)

Z0

zl/~

_1/~

,

~0

ln{(U-Uo)2+(v+vo)

¢ -- Kq9 -- 4zr--~ U ~

(r >~ 0).

(u

--

,

--

13 m

U0) 2 +

r

2}

(13 - -

sin

l)0) 2

'

,

{ 2 o u-uo,

}

Q arctan , ~P -- 2:rr--~ ( u - U0) 2 "l- 132 __ 13~)

uo - u(ro, 0o)

vo - v(ro, 0o).

316.02 (SC) 3 As 316.01 with o t - ~.

P~Q J 3~

2# \

• ......~b=t) ....

~o=0

x

~o(x,0)=0

forx/>0,

~o(0, y ) = 0

fory~<0.

S-2--

Q I n ( z2/3 ~

2re D

n

_-2/3) _2/3 '

~;0

k'z2/3 - ~0

etc.

(316a)

Flow in an infinite field

237

316.03 (Q) As 316.01 with o~ - 2.

Y

~0(x,0)-0

2zr

Q~ in (~/-z I-2 - 2zr---~ / ~ -- ~~ /) ~,

,

x-

forx>~0.

etc.

316.04 (J) Flow between a well with discharge Q and two finite equipotential p l a n e s of different lengths, making an arbitrary angle of c~rr with each other, where 0 < o~ ~< 27r. qg(r, 0) = ~o(x, y) = drawdown.

P~Q "~ V ........

_

rb

X

qg(r, otzr) - - 0

for0~
qg(r, 0 ) - - 0

for 0 ~ r ~
P (ro, 0o) -- P (xo, Yo).

,

I2--

S

Q 27rD in

-o

1 ( 1/c~ d) 1 w(z~- ~ z +-~ + ~ V/(zl/oe -1--a l / ~ ) ( z l / ~ wo

--

w(zo),

z -- x +

iy

--

re i°

d - a 1/~ - b 1/~

s -- a 1/~ nt- b 1/~

zo

_ bl/~),

-- xo +

o

316.05 (J) As 316.04 but with a - b.

•

2zrQDln(

I21

w(z)- ~z

1

21wOI

b l /°t

b l / a W -- W 02-~Iw O )

wo - w(zo)

iyo

--

roe i°°

238 316.06

(316a) (J)

1 As 316.05 but with ot - g.

P~Q

99(x,0)=0 qg(0, y ) = 0 b •

Two-dimensional groundwater flow

for0~x~b, for0~
x bEt°°) Q in (21%ylw - 21wol

I2=

2zrD 1 2

w ~ w0

1

W(Z) - -~Z nt- -~V/Z 4 - b 4,

wo = w ( z o ) .

316.07 (J) As 316.05 but with c~ = 1.

Y

~a

qg(x,0)=0

for - b ~ x <. b.

,f2___QQ ln(21b-~I w - 21wol bw°) 2zr D w - wo ' -b

b

316.08

1

x

w(z) -

1

z: z ± - ~ , / z 2 - 1)2,

wo -

w(zo).

(J)

Y

P (Xo, Yo)

Equipotential plane of elliptical shape

and a well with discharge Q, not situated on one of the two axes of the ellips, qg(x, y) = drawdown, P (x0, Y0). x2

--a

a

x

-b 21wol

a+b

Q l n ( , - S ~ w - 2 - ~ o l w°) 27r D w - wo ~ - ~ z i ~1

1 V/z2 - a2 + b2 ,

wo = w ( z o ) .

cp--O

y2

for ~-~ + ~ -- 1.

(316b)

Flow in an infinite field

239

b. Wells a n d i m p e r m e a b l e p l a n e s

316.21 (SC> Flow to a well with discharge Q along two semi-infinite i m p e r m e a b l e planes, make an angle c~zr with each other, where 1 < ot ~< 2. qg(r, 0) = drawdown, P(ro, 0o).

Y P

y

x

~ ( r , 0) = const, for r / > 0, (r, olrr) = const, for r >/0.

S2 - c z -- re i°

Q In { (z 1/~

2zrD

zo -- ro ei°°

_l/or 1/ct =1/o~ - ~ o >(z -~o

>}

zo -- ro e-i°°

~

•

316.22 (SC>

3 As 316.21 with c ~ - ~.

Yl

~p-const.

fory-0,

x >/0and

x--0,

Q •

,(2 - c

2:rD

y~<0.

_2/3 In {(z 2/3 - ~o )

× (Z2/3

316.23 (Q> As 316.21 with ~ - 2. --const. •

x

Stagnation point:

1 1 Zs - ~xo + ~lzoi.

I2 -- c

fory--0,

x >/0.

Q ln{(4c~_ 2zrD ×

-2/3

-~o

/~)

>}.

240

Two-dimensional groundwater flow

(316b)

316.24 (J) Flow to a well with discharge Q, situated near

P~Q

"" ']- b

S2 -- c -

=

two finite impermeable screens, m a k i n g an arbitrary angle of oen" with each other, and having different lengths. 0 < c~ <<.2zr. q)(r, O) = drawdown, P (ro, 0o).

x~

2rr----D Q In{ Iw°l ( w -

gr (r, O) -- const.

f o r 0 ~
(r, ctyr) -- const.

f o r 0 ~< r ~
w°)(w-

161w°12 '2w° )

'

d ) -4- -~1 V/ ( z l / °t nt- a 1/a) ( Z 1~or - - b 1lot ), W ( Z ) - -~1 (zl/Ot -1- "~ w o - - w(zo),

z - - r e i°

s -- a 1/~ -I- b 1/~

z o - - roe i°°

d - a 1/~ - b 1/~

o

Stagnation points: Ws -- - 4 - ~ Wo. 21wol

316.25

(J)

As 316.24 but with a - b. ,p(r, 0) -- const. for 0 ~< r ~
•

_

.Q--c

Q___Q.ln { IWol ( w 2zr D

w°)(w-

wo

h2/=~°°) 2

w

'

1 1 w ( z ) - -~z 1~or+ -~ ~//Z2/Ot__ b2/a 316.26

(J) P

:,Q

As 316.25 but with oe ~r--const.

1 2"

forx--0, y--0,

0~
and

Flow in an infinite field

(316b)

_Q__Qln{IWol(W-wo)(w-b4wo *

S2=c

2zr D

w0

~(z)- ~1 z2 +71 V,/Z 4 _

241

41wol 2

)]

w

'

b4"

316.27 (J) P

As 316.25 but with c~ = 1. ~Q

¢=const.

b

-b

fory=0,

-b~<x

~
x

~ l n / ~o ~ ~o,~ ;4lw012 ~2~°) / S2 - - c

2zr D

wo

w

'

1 1 ~(z)- ~z+ ~/z 2- b2

316.28 (J} Flow to a well with discharge Q, situated near an impermeable obstacle of elliptical shape and not lying on one of the axes. qg(x, y) = drawdown, P(x0, y0). x2

@ = const.

a

-b

Q ,n['Wo'~w- wo,{w_~'l+h'2wO}] 4[woI 2 S2 - c -

2zr------D

1 ~/z2

1

w ( z ) -- -~z + -~

wo

-

a2

+

w b2

Stagnation points: 1

Zs -- + - - ( a u o

Iwol

+ ibvo).

,

'

wo -- w(zo) -- uo + ivo.

y2

for ~-7 + ~-7 -

1.

242

(316c)

Two-dimensional groundwater flow

e. Wells a n d e q u i p o t e n t i a l p l a n e s with entrance resistance

316.41 (SC)

Groundwater flow between a well with discharge Q and two straight semi-infinite surface water b o u n d a r i e s with entrance resistance w, making an angle of oeJr with each other, 1 < ol<2.

zo. =~ Q

of It"

,'5

#

S

w

I l l l l , l l l l l l l l l l l l l ~

.......;;'"iS.......~-

Q {ln( zl/~ -1/~) ( Z 1/c~ 1/a 4- 2 exp --

,

S2-

2zr D

z - r e i°,

) E1 (z 1/or

iKw

z 1/~ _ Zo

zo -- roe iO°,

=l/or

z0

z o

=l/or

iKw

zo -- ro e-i°°.

316.42 (SC) 3 As 316.41 with o t - ~.

Zoo =:~Q

Z2/3

•

£2--

Z,|,|,~,,|,,,,,,,| .... ~ " - " 0 . . . . . . . .

x-

~z2/3 _ z o

+ ~-~ exp t

w-" q)=0

_2/3

z;0

=2/3

Q

z

-2/3

,n( z2/3 2rr D IZ2/3 x E1

t

'~

)

iK w =2/3

i Kw

"

316.43 (SC) As 316.41 with ot - 2. zoo=~ Q

•

,5,..... w . . . . . . . . . .

27r

7"~o'"-'6......

~?

S2-

Q in (~¢/z - ~ o 2zrD ~/~- ~/~) +~-~exp(

i Kw

iKw

/

/

/

Flow in an infinite field

(316d)

243

d. Wells and well-screens in uniform flow 316.51

(S)

Well-screen of length 2l with total discharge Q in a confined aquifer with thickness D, in which a uniform flow with strength q takes place. Quasi-steady state, q9 = qg(x, y) = drawdown.

q

X

...... ]l

-IF---~

x s2 ( ~ )

- -q~ z e -iotrr L)

Q

° Z ( x , o) -

D

Oy

4Kl 0

Q if2

--

C

4zc Dl

~<x ~
for Ix[ > 1.

a

{(z + l ) l n ( z + l) - (z - l ) l n ( z

Stagnation points:

Zs

for-l

, coth(27' c

)

- l)} + i--ze

D

--io~zr

(320-322a)

244

Two-dimensionalgroundwaterflow

320 BIII-2. GENERAL TWO-DIMENSIONAL GROUNDWATER FLOW IN AN INFINITE HALF PLANE a°

Has the soil discrete inhomogeneities in x- and~or y-direction ?

go to 370 (BIII-7)

An example of a discrete inhomogeneity is a groundwater abstraction from a confined aquifer with different permeabilities separated by a straight plane.

b. The soil is assumed to be homogeneous

go to 321

Also anisotropic homogeneous soils belong to this category, provided that the main directions of the anisotropy coincide with the coordinate directions. Solutions for anisotropic cases can easily be derived from analytical solutions for isotropic problems. 321. Homogeneous soils a. Are wells or well-screens involved in your problem?

go to 325

As flow in an infinite half plane almost always is limited to horizontal flow in an aquifer of finite thickness, the wells or well-screens must be fully penetrating the aquifer. b. No wells or well-screens

go to 322

322. Problems without wells or well-screens a. Is uniform flow involved in your problem?

y -.--.--~q

pond

go to 324 Uniform flow is defined as flow in a fixed direction with everywhere the same constant velocity. Disturbance of this flow pattern may take place by means of obstacles of different shape and character, for instance, finite impermeable screens or circular ponds. It is assumed that

Flow in an infinite half plane

(322a, b, 323a)

245

the uniform flow is undisturbed at infinity (in practice a certain distance from the obstacle). In this chapter only obstacles in uniform flow, that cause two-dimensional flow in an infinite half plane are treated. b. Flow to and f r o m equipotential planes

go to 323

Either flow between two equipotential planes with different (constant) head, or flow to a single plane, caused by lowering of the head of that plane, is regarded. In the latter case a so called quasi-steady state takes place, where the stream function is defined uniquely, but the potential function contains an arbitrary constant. In practice an equipotential plane may be represented by a more or less fully penetrating canal with a fixed water level.

323. Flow to and from equipotentiai planes Subdivision" a. Flow between straight open boundaries. b. Flow to open boundaries which levels differ from the surrounding groundwater levels. a. Flow between straight open boundaries

02q3

02(t9

Ox---7 + -ay2 - 0

0299 or

0 2~

+

-2Yz - 0 .

323.01 (SC) Flow between an infinite and a semiinfinite equipotential plane, which are parallel to each other, qg(x, y) -drawdown.

y

iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii_ iii yo

rp=0

x

q)--

0 h

for y -- O, fory--yo andx >0.

yo(

#2 e-i~ x~ - 1 z---irr--Jr Kh X? - d# + i ~ , z--x +iy,

4,- K~o.

323.02 (SC) Flow between an infinite and a semiinfinite equipotential plane, making an angle o~rc with each other, where 0 < o l < 1.

Y

¢p=O

x

246

(323a)

0 h

qg--

Two-dimensional groundwater flow

for y -- 0, fory-xtan(otzr)+yoandx>/0.

z -- yo{cot(zrot)+ i}{otw 1-~ + (1 - c t ) w -~ } - yo cot(zrot)

z -- x + iy,

w -- e rri(1-~hh),

with

~ -- Kgo + i~r.

323.03 (SC)

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Flow between two semi,infinite equipotential planes, parallel to each other, but in opposite directions. ~o(x, y) = drawdown, S = point of symmetry.

I;0 tp=0

Xo

,

for y -- 0 and x / > xo, for y - yo and x ~< 0 (Yo > 0).

0 h

~o-Z m

x

{

Yo rr(1 - a )

a}

w + (1 - a ) l n w

+ -to

yol+a

rrl-a

t- iyo

with w - e irr(1-~hh) and a from 2yo a + 1 X0~

zr a - 1

Yo Jr

In a.

323.04 (SC) Recharge- and discharge canal of different widths and with different levels on top of a confined aquifer which is "infinitely" thick.

~o=0

a -l-z L

a~o ( - ~ ,

oz

o) - a~o

(x, oo) = 0

I2 -- K(m) m=

b

~o(x O) '

7 x ( ~ , o) - o,

'

~(x, Oz

o)-

o

forx <-a,

,/arcsin,/: I /

L ( L + a + b) (L + a ) ( L + b)'

_ ]0 |h

L(( -t- a) m

K(ml) q -- K h K ( m )

for-a~<x~
(L+b).

(-- ~b-+- iap), (( -- x + iz),

ml=l--m.

Flow in an infinite half plane

(323a)

247

Drawdown at infinity: ~o(c~, o ) -

~o(-~,

o) -

~o(x, o o ) -

h F(arcsin¢ L+b ] ) K(m----~ R +a +b m .

323.05 (SC) As 323.04 but with one

~o=0

canal "infinitely" wide.

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

1.

.

l

-z

~o(-~,

.

X

.

L

b

z) - ~o(c~, z) - 0,

--(x,

for0<x

o) - o

Oz

£2--~F

m-


arcsin

K(m) L L +b'

0 h

qg(x, 0) -

for x ~< 0, for L ~< x ~< (L + b),

> (L+b),

~o(x, oo) - O.

m ,

b ml = L + b '

K(ml)

q - K h K(m) ~

Drawdown at infinity: ~o(cxz)- 0. 323.06 (SC) q

=

l

a

also on top of the aquifer.

z) = 0 ,

~o ( ~ , z) - o,

Ox Oz

X

=h

--(-c~, Ox

--(x,

=

b

co) = O,

~--Kh

Infiltration canal of finite width and constant level on top of an "infinitely" thick aquifer in combination with a well-screen with constant drawdown

Kh K(m)

~p(O,z)-h ~p(x, 0) - 0 099 --(x, 0z

forO~
o) - 0

- ~F(arcsinwlm

for x < a and for x > (a + b).

)

with

248

Two-dimensional groundwater flow

(323a, b)

c~ m

+l+-

a

K(ml)

m -- -~

a +b

q - K h ~ K(m) '

+l+a+b

ml--l--m.

Drawdown at infinity:

h [ar sin{v/(a + b)

qg(c~) -- h - K(m-----~

+ 1

a+1 /I mj

323.07 (SC)

q

As 323.06 but with an "in-

finitely" wide canal.

hi ~88888888~88

i

a

--

X

:~¢p = h

99(-00, z) -- q)(cx~, z) -- 0, ~0(x,0)--0

q)(0, z) -- h 0rp(x ' 0 ) - 0 0---z

forx>/a,

See 323.06 with m -- ~1

and

~o( e ~ )

for 0 <~ z ~< l, forx~
q)(x ' c ~ ) - 0 .

- - O.

b. Flow to open boundaries which levels differ from that of the surrounding groundwater 323.11 (FS,SV)

Sudden lowering of the head with h in two semi-infinite equipotential planes,

Y

making an angle of otzr with each o t h er, where g1 < oe < 1. Non steady state, q)(r, 0, t) = drawdown.

¢p= h

x q)(r, 0, t) = qg(r, otzr, t) = h, O2q 9 [ _ _1

0~o

"--T Or - r -~r

.

qg(r, 0, t) -- h

ot:r ~ n--0

[

- - 1 - -1- ~02(/9 - - f l

r 2 002

2 0(t9 .

at

t 2) du sin (2n + 1) O] fo °° J(2~l ) (ru) exp (--fl--Su Ol

U

Flow in an infinite half plane

(323b)

249

323.12 (C) Y ~°'~c

p

Quasi-steady state of 323.11. Point P(R, late) reference point for ,p

R

(99p = 0). Quasi constant drawdown c at the equipotential planes.

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

q~=C

~9(r, 0) -- ~p(r, otrr) -- c.

X

S?-Kc

1+i~

.

323.13 (FS,SV) As 323.11 with o~ -- 2. ~p(x,0, t ) = h 27r~..,

P

%

•

f o r x >~0.

R

~-;;::-:~ ..........x

cp(r, 0, t) = h

2hjr ~

/

°Jfo

sin (2n + 1)~

t

J(~+½) (ru)exp -~--~-u

n=O

2)6. ~

°

b/

323.14 (C)

Quasi-steady state of 323.13. 99(x,0)=c

f o r x ~>0.

cil+i 323.15 (Q) Groundwater flow outside an equipotential plane of parabolic shape and constant drawdown c. Point P (R, 0) reference point for ~0 -- 0. Quasisteady state.

"[ ~ x R

Kc •

s~

-

(,/-~- ~).

q9 = c

for y2 __ 4p(p - x).

250

Two-dimensionalgroundwaterflow

(324a)

324. Obstacles in uniform flow

02q9 029 i =0. Oy2 Ox2 Subdivision: a. Obstacles, acting as equipotential planes. b. Obstacles, acting as impermeable planes.

a. Obstacles, acting as equipotential planes 324.01

(J)

q

Finite equipotential plane in uniform flow of strength q, making an angle otzr with the hori1 3 zontal axis. (or :/: 0, ~, 1, ~). 9(x, 0) -- 0

q9 0

}ze-

if2 (OO)

-b

for Ix l < b, iot:rr

.

b I 2 = ~ q { 4- x/z 2 - b 2 cos(an') - iz sin(otzr) }. Stagnation points: Zs = +b sin(otzr). Total discharge through the plane: Q = 2bq.

324.02 (J)

Y b

q / 4 azr

.

Elliptical basin with zero level in uniform flow of strength q, making an 1 angle otzr with the x-axis. (a # 0, ~, 1,3).

I2(oo)-- ~qz e -

--a

iotyr

x2 9-0

I2=

q

D(a

b)

[{a cos(c~zr) + ib sin(otzr)} (-t-x/z 2 - a 2 + b 2)

- {bcos(otzr)+ iasin(otzr)}z]. Stagnation points: Zs - 4- {a sin(otzr) - ib cos(otzr) }. Total discharge through the basin: Q = 2q(a + b).

y2

for ~--~+ ~--~ - 1.

-b •

.

(324b)

Flow in an infinite half plane

251

b. Obstacles, acting as impermeable planes

324.11 (J) Finite impermeable plane in uniform flow of strength q, making an angle c~zr with the x-axis.

q

1 3 (c~ # 0, ~, 1, ~).

~ ( x , 0) - const. 7,

const b

-b

for Ix l < b.

S'2(Cx:~) -- q e -i~rr . D

x

$2 - q { z cos(otrr) -+-iv/z 2 - b2 sin(olTr)}. D Stagnation points: Zs - + b cos(~Tr). 324.12 (J) q

Y b

Impermeable screen of elliptical shape in uniform flow of strength q, making an angle c~zr with the x-axis. (or :/:

/ 4 c~zr

1 3 0, ~, l, ~).

a

--a

x2

;

~p-const.

for~+~-

S2(cx~)- -q~ z e D

iot:rr

y2

1.

.

-b

•

12-

q D(a-b)

[{a cos(~:r) + ib sin(an')}z

- {b cos(o~zr) ÷ ia sin(are) }(-I-v/z 2 - a 2 ÷ b2)]. Stagnation points" Zs - -+-{a cos(uzr) + ib sin(otTr) }. 325. General two-dimensional flow in an infinite half plane to wells or wellscreens

Unsteady flow: 0299

02q 9 __ f12 099

L-~t ay2

-57,

fi2 __

S

KD

25 2

(3 25 a)

Two-dimensional groundwater flow

Steady flow: 0293

OX 2

t

0293

Oy 2

Subdivision: a. Wells b. Wells c. Wells d. Wells

-0.

and and and and

equipotential planes. impermeable planes. equipotential planes with entrance resistance. well-screens in uniform flow.

a. Wells a n d equipotential planes

325.01 (I) Flow between a well with discharge Q and a circular p o n d with fixed constant level.

Q l

q9 - 0

f o r x 2 -+- y2 _ R 2 R2

-x S 2 - - 2rrD

R z-I

325.02 (FC) As 325.01 but for a semi-confined (leaky) aquifer. Differential equation for polar coordinates" 1 099 1 0293 Or----g-F -r -~r -+ r 2 O0 2

0299

93(R, 0) - - 0 ,

93(oo, 0) - 0 ,

Q

lim (p 099 p-+ o

93 ~2 = 0 '

)v2

-- K D c , 093 (r, O) -- 093 07 -ff~ (r, Jr) -- O,

f o r p2 _ r 2 + l 2 _ 21r c o s 0.

-~p ) - - 2 rr K D

Q 93(r, 0) -- 2 r r K D with p -- ~/r 2 -Jr-l 2 - 2lr cos 0.

1/---OO

r

cos(.0,

}

forr >~ R

Flow in an infinite half plane

(325a)

253

325.03 (J) Equipotential plane of elliptical shape and a well with discharge Q, situated on one of the axis.

Y b

-"5

--a

Q

_;7 a fi(xo, O)

See 316.08 with zo = xo and Iwol = wo.

-b

325.04 (Q) Y

Equipotential plane of parabolic shape and a well with discharge Q not situated in the axis of the parabola. Origin = focus of the parabola, q9 -- 0 for y2 __ 4p(x + p).

Q

o~:~

~

•

,f2 =

Q 2zrD

In

0

[cosh{ ~ ('/~44rfiw/~) ] ] sinh{

rr(vff-,Fd)4q~} "

325.05 (SC) Y

Flow between a well with discharge Q and two semi-infinite equipotential planes making an angle of o~Tr with each other.

Q

.e P (Xo, Yo) qg=0

See 316.01 with 21 < ot < 1.

x

go(r, O) = rp(r, c~:rr) = O,

r ~> O.

254

Two-dimensional groundwater flow

(325a, b)

325.06 (Q) Flow between a well with discharge Q and a semi-infinite equipotential plane, the well being in line with that plane. P(xo, 0) (xo here negative).

P (Xo, 0) ~o=0

a

X

~o(x,O)-O •

I2-

Q

forx>/0. ln(~/~+/I~

D

-

i

'] l,/TET/ "

325.07 (J) Flow between a well with discharge Q and a finite equipotentialplane, q9 = 0 for y = 0 and - b <<.x ~ b, P(O, Yo).

P(O, Y0)

See 316.07 with wo - w(iyo). . 0

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

b

-b

325.08 (J) Flow between a well with discharge Q and a finite equipotential plane, ~o = 0 for y = 0 a n d - b ~< x ~< b, P (xo, 0).

a ~o o

b P (Xo, O)

-b

See 316.07 with wo - w(xo).

_¢ X

b. Wells and impermeable planes 325.21 (J)

Y

Flow round a circular impermeable screen to a well with discharge Q. ~p - const. for x 2 + y2 _ R 2.

Quasi-steady state.

2zrD

,n{Z

R2

z

/

(325b)

Flow in an infinite half plane

255

-- solution for three wells, one recharge well in the origin and two discharge wells in (l, 0) and (@, 0), respectively, all with strength Q.

325.22 (FC) As 325.21 but for a s e m i - c o n f i n e d (leaky) aquifer. Differential equation for polar coordinates: 02q9 10q9 1 02q9 Or----T + -r -~r q r 2 O02

O ! ( R o) - o Or

'

)2 -- K D c

~oo__(, ~ , o) - o '

(0~o) lim p-~p p-+0

(t9 i~2 = O,

Or

~(r, O0

'

Q

O) -- -:-7(r, 7r) -- O,

Off

for p2 _ r 2 -k- 12 -- 21r cos 0.

-- - 2rr K D

Q

~Ko(1V/r

2 q-l 2 -

q)(r, O) -- 2 r r K D

-JcOO

21r c o s 0 )

l

R

nZ

r

gn(_£){In+l(_~) .qt_- g l n ( ~ ) } K , ( ) c o s ( n 0 )

+

K n + I ( R ) __ n)~

2rc K D

R

325.23 {J) Well in P (xo, 0) with discharge Q near an impermeable screen of elliptical shape. P situated on one o f the axis of the ellipse.

Y b

jJa

--a

Q Fr(xo, o) x

-b X2

gr = const.

I2=c w(z) -

y2

for ~-2 + ~-2 - 1.

,n2 o 1 + -~ 1 v/z2 _ a2 + b2, -~z

Stagnation points: Zs -- + a .

(a+b) 2

4o/1

wo = w ( x o ) .

(325b)

256

Two-dimensionalgroundwaterflow

325.24 (Q) Y

S'~ m C

2zrD

Impermeable screen of parabolical shape and a well with discharge Q, not situated on the axis of the parabola. Origin = focus of the parabola. ~p = const, for y2 = 4p (x + p).

'n [sinh/7r (~/z4~/_fi-~/~) ] c°sh { 7r (~/~4~/-fi-~/~-~) }]"

325.25 (SC)

a

.e P (x0, Y0)

Flow to a well with discharge Q near two semiinfinite impermeable planes, making an angle of otzr with each other. ap (r, 0) -- ~ (r, otzr) -- const.

~

x

See 316.21 with ~1 < c~ < 1. 325.26 (J) Flow to a well with discharge Q near a finite impermeable screen, P(O, yo). P (0, Yo) -const.

-b

b

See 316.27 with wo - w(iyo).

for y - 0

and Ix l < b.

(325c)

Flow in an infinite half plane

257

e. Wells and equipotential planes with entrance resistance

325.41 (SC) Groundwater flow between a well with discharge Q and two straight semi-infinite surface water boundaries with entrance resistance, w, making an angle of arc with each other. ~1< c ~ < l .

Y l Zo°/~ Q

. . . . .

(p(r, 0) = 99(r, c~rc) -- 0

. . . . . . . . . . .

- J l l l l l l l l~l l l~l l l0l l l l l l l l l ~X

(r >/0).

See 316.41 with 71 < c~ < 1.

325.42 (SC) Groundwater flow between a well with discharge Q and a semi-infinite surface water boundary with entrance resistance at both sides, the well being in line with the boundary. ~p=0

W

~o=0

X0

•

>/0.

x

Q {In (~/-z + i I~x--ol

Y2=

fory=0andx

2rc--D

~/~ + i I~x--ol)E1 (v/-z + i I~/]~)}

x/~-i [~~)+2exp(

i Kw

i Kw

"

d. Wells and well-screens in uniform flow

325.51 (S) y

ap=0

Well with discharge Q [L3T -1] in a uniformflow of strength q [L2T-1]. Quasi-steady state, q9 = qg(x, y) =

q

drawdown. qz

Q

~

D

S

Oq9

~(x,0)=0

forx¢0,

Oy

r O(p

12=c - ~

Q

2reD

q

ln z -+- --~ z.

Hydrological screen: ( 2rrq

x -- y c o t \ ~ y ] . ~

]

_

Q

-~r Jr~0 - 2rc K-------'~ (r -- x/x 2 + y2).

Stagnation point:

Q Zs = 2rcq"

(325c)

258

Two-dimensionalgroundwater flow

325.52 (S)

Well-screen of length 21 with total discharge Q in uniform flow of strength q in a direction parallel with the screen. Quasi-steady state. ¢p = qg(x, y) = drawdown.

y

l

~=0

a~ - l ~- . . . . -. x

- _qz.

~(~)

D

J Q

Oq)(x, O) -ay

for Ixl ~ III,

-t-4K i 0

for Ixl ~ I11.

q

a .(2 - -

c --

4rcDl

{(z + l)ln(z + l) - (z - l ) l n ( z

-/)} + =z. 13

Hydrological screen:

Q[2 (x+l)2+y2 4zrl y l n { ( x - l ) 2 + y 2 } + ( x + l ) a r c t a n ( -(x-l)arctan(xY-l)]

Y

-qY"

Stagnation point: zs--lcothk

( 2rrql

Q ....).

325.53 (S)

Well-screen of length 21 with total discharge Q in uniform flow of strength q, in a direction perpendicular to the screen. Quasi-steady state. ¢p = ¢p(x, y) = drawdown.

y

Ttq

S••,• . . . . . . Jl

=0

q ~N

x

a

q±

~(x

,

o)

-

I

q

O

... for Ixl <~ Ill,

4KI

for Ix] >/II].

S2(c~) -

°

---tz.

D

(325c)

Flow in an infinite half plane

•

$2--c

Q 47r Dl

259

{(z + l)ln(z + l) - (z -l)ln(z - / ) }

q

~

.

l Z .

Hydrological screen: Q[~

4zrl

(x+l)2+y2

yln{( x _/)2 + y 2 } - ( x

+(x-l)

arctan(Xy

)

+ 1)arctan( x---~---)+l

+ qx - O.

Stagnation point: Zs -- il cot( 2rrq/ Q)-

325.54 (S) Two wells with equal discharge in uniform flow perpendicular to the connecting line of the wells. Quasisteady state. ~0 = qg(x, y) -- draw-

"•Y q ____+

down. d

Q ln(z $ 2 = c - ~2JrD

qz , S 2 ( o o ) - -.~

S

2+d 2)+ q

~z.

Stagnation points: Q Zs = 2zrq

+

Q 2

)_d2

Oqg(x, O) -- O. Oy

260

(330-332)

Two-dimensional groundwater flow

3 3 0 BIII-3. GENERAL TWO-DIMENSIONAL GROUNDWATER FLOW IN AN INFINITE QUARTER OF A PLANE

a. Has the soil discrete inhomogeneities in x- and y-direction ?

go to 370 (BIII-7)

An example of a discrete inhomogeneity is a circular region with a permeability that differs from the surroundings, in uniform flow.

b. The soil is assumed to be homogeneous

go to 331

Also anisotropic homogeneous soils belong to this category, provided that the main directions of the anisotropy coincide with the coordinate directions. Solutions for anisotropic cases can easily be derived from analytical solutions for isotropic problems. 331. H o m o g e n e o u s soils

a. Are wells or well-screens involved in your problem?

go to 337

Groundwater flow in an infinite quarter of a plane can be both horizontal and vertical. Horizontal flow towards wells or well-screens takes place in aquifers with finite thickness, the discharge Q being a total quantity, whereas vertical flow in a quarter of a plane can be realised by abstraction of water from a drain with infinite length in the direction perpendicular to the vertical plane. In the latter case the discharge is measured by quantity per length. b. No wells or well-screens

go to 332

332. P r o b l e m s w i t h o u t wells or well-screens

a. Is uniform flow involved in your problem?

go to 336

See comments of 322a. In this paragraph only obstacles in uniform flow, that cause two-dimensional flow in an infinite quarter of a plane are treated. b. Flow to and from equipotential planes

go to 333

In practice an equipotential plane may be represented by a more or less fully penetrating canal in relation to horizontal flow, or the bottom of a lake or wide river in relation to flow in a vertical plane. Also polders with given level on top of an "infinitely" thick leaky aquifer and straight surface water boundaries with entrance resistance may act as equipotential planes.

Flow in an infinite quarter of a plane (333, 334a)

261

333. Equipotential planes a. Is groundwater flow taking place between two planes with different levels ?

go to 335

For these problems steady state solutions are available; in some cases also the non-steady state solutions are given. b. The equipotential planes have equal levels

go to 334

These problems are mostly non-steady but, in general, tend to a quasi-steady state, where the stream function is defined uniquely, but the potential function contains an arbitrary constant.

334. Equipotential planes with equal levels Unsteady flow:

0299 0299 __ /~2 099 -~ ~ t Oy 2 --~ ,

1~2 __

S KD "

Steady flow" 0299 0299 t =0. OX2 Oy2 Subdivision: a. Equipotential planes perpendicular to each other. b. Equipotential planes not perpendicular to each other and of different shapes. a. Equipotential planes perpendicular to each other 334.01 (P,L) The surface water level is an arbitrary function of the time the same for both boundaries. 99(x, y, t) drawdown. F(t) ~ ( x , y, o) - o,

99(0, y, t) -- 99(x, O, t) -- F(t) ..........

..........

x

with F(0) - 0.

099(c~, y t) -- 0q9 Ox ' ~y (X, C~, t) -- O. qg(x, y, t ) -

F(t)-

fo

--~(t-

r)erf

262

(334a)

334.02

Two-dimensional groundwaterflow

(P)

As 334.01 with F(t) -- h = const. (sudden drawdown of the water level which is kept constant thereafter). qo(x y , t ) - - h { 1 '

erf(2--~tt) erf( flY '~ } \2~/

"

334.03 (Q)

~O=C

~'-

P

1 Quasi-steady state of 334.02. Point P(~R~/2; ±R~/2) reference point for ~o (~o - 0). Constant 2 head c for the equipotential planes.

S2--Kc I............

334.04

1+i

~

.

............ *

(EL)

Constant drawdown h of the water level of two open boundaries perpendicular to each other for a leaky aquifer. Steady state, qg(x, y) - drawdown.

~o=h

0 2~o

~p=h

x

OX2

[

0 2~p

cp

02 2

~2

--0,

)2__ KDc.

~p(O, y) -- ~o(x, O) -- h,

hi -~//////////////////////////

O!(cx~, y) - =0~o -(x, Ox oy

-

o

KD, S

x qg(x, y) - h

fo{o~

x

}

1 -erf(2,k~)erf(

y 2)~V@) e-~ d°t.

334.05 (P,L) Confined aquifer with open boundaries at x - 0 and y - O. Constant precipitation.

~o=0

02(/9 _~_ ~0299 _~_ ~P Ox 2

~p=O

x

Oy2

~o(x, y, o) - o,

KD

_ f12 0(/9 -~'

Flow in an infinite quarter of a plane

(334a)

263

qg(O, y, t) -- qg(x, O, t) -- O, 0go 099 - - ( o o , y, t) -- 7 - ( x , oo, t) -- O.

--~$$,[.$$$,[,$,[,$$$p I

x

OX

KD, S

Oy

#x ) e f t ( 2x/~_ #y ) dr. qg(x , y , t) -- ~P fot erf( 2~/~

334.06 (P,L) As 334.05 but a leaky aquifer. Steady state.

c

02~0

02(/9

t

Ox2

Oy2

(/9

q

t

~2

KD

=0,

k 2 -- KDc.

q)(O, y) -- q)(x, O) -- O, Oq)(oo, y) -- - : - - ( x , Ox oy

(

qg(x, y ) = qc

erf 2 k ~ ) e f t (

o o ) - o.

y . ) e -~ dot. 2k~/ot

334.07 (P) y

J

~=h

Cl-- ~ = 0

--

~=0

-]IIIIIIIIIIIIIIII~ ¢2 X

Confined aquifer; two surface water boundaries with entrance resistance perpendicular to each other. 99(x, y, t) = head. Surface water level = 0. Initial head: ~p(x, y, 0) = h, K Oq)(O, y, t) -- q)(O, y, t) OX

C1

a 2 (oo, y, t) - 0, Ox

K Oq)(x, O, t) -- 99(x, O, t) Oy c2 --(x,

Oy

c~, t) -

O.

q~(x, y,t) - h { e r f ( - - ~ ~ ) + R ( ~ f l X V / ~ l , v / ~ l t ) l x {erf(2@t) + R(~fly

with

264

Two-dimensionalgroundwaterflow

(334a, b)

1 l~ 1 ---

i~2K2c 2 '

1 t} 2

~2K2cI

"

b. Equipotential planes not perpendicular to each other, or of different shapes 334.21 (FS,SV)

Sudden lowering of the head with h in two semiinfinite equipotential planes, making an angle of

1

o~zr with each other, where 0 < c~ ~< ~. Non-steady state with go(r, 0, t) - drawdown.

qg=h

0299_ 10go 1 02go Or 2 ~_ _ _~_ m ~ ~. f12 0(t9

x

r -~r

r 2 002

Ot

O0

•

go(r, O, t) -- h

4h Z s i n { ( 2 n + l ) O ]-

fo°° J 2~+1)(ru) exp ( t- ) d u u 2

334.22 (SC)

~ P

Quasi-steady state of 334.21. Point P(R, ½o~rc) reference point for go (99 - - 0 ) . Constant head c at the equipotential planes.

-.-go = 0

s2--gc qg=c

/

l+/

~

,,- }

.

x

334.23 (Q,SC)

j/"

Groundwater flow inside an equipotential plane of parabolic shape and constant drawdown c. Point P (R, 0) reference point for q9 (go = 0); go = c for y2 = 4p(p + x).

Quasi-steady state.

I2 -- K c { 1 -

cosht t } cosh(~

)

(334b)

Flow in an infinite quarter of a plane

265

334.24 (Q)

Quasi-steady flow to an equipotential plane of orthogonal hyperbolic shape and constant drawdown 1 2 rr c. q9 - c for xy - ga . Point P(R, 7) reference point for q9 (q) - 0). •

,(2-- R2 _ a2

1 + i ~-

.

334.25 (J)

Quasi-steady flow to an equipotential plane of elliptic shape and constant drawdown c. b

~

f

x2

99--c

~o-0 a! x

R

y2

for ~ - g + ~ - g - - 1.

Point P(R, 0) reference point for q9 (q) - 0).

arccosh(~) - arccosh(~)

£2--Kc

f - v/a2 - b 2, a > b. 1 In ("+h ~7-h ]J _ arccosh(7)R

Total discharge:

2rc K Dc

Q

a r c c o s h ( ~• ) -

1

In (a+b]" cT-bJ

334.26 (J)

Quasi-steady flow to a finite equipotential plane of length 2a. Point P(R, 0) reference point for 93 (9~ -

l .......

I.............. ......

Total discharge"

Q

2rc K Dc arccosh(E)"

x

•

o).

~2-Kc{1-

arccosh(z) } arccosh(~, ) "

266

Two-dimensional groundwater flow

(335a)

335. Equipotential planes with different levels Subdivision: a. Flow in horizontal planes. b. Flow in vertical planes.

a. Flow in horizontal planes Unsteady flow:

02q9 OX21

02~ oqy2

f12 0(,0 "-

f12 ._ K D

"~-'

Steady flow"

02q9 Ox 2

t

0299 Oy2

--0,

z--x

+iy.

335.01 (D) Two open boundaries, perpendicular to each other, with a constant drawdown h at one boundary. Steady state ~0 = qg(x, y) = drawdown. ~p(O, y) = h,

~o(x, O) -- O,

0~P (c~, y) --

•

2Kh

,(2 -- - ~ i

ay

Ox

....... -(o.=. 6 ..........

-

o.

lnz.

335.02 (L,IS)

Non-steady state of 335.01. Additional condition ¢p(x, y, 0) - 0 for x > 0.

q9(x, y, t) -- h erfc

- ~7l"

.

X2 + Ot2 exp

-

4t

335.03 (ISR) Two boundaries with entrance resistances of ¢1 and c2 respectively, perpendicular to each other, with a constant drawdown h at one boundary.

Y2' cl-- ~ = h "

Steady state, q9 = ~o(x, y).

~=0 C2

X

- K c l = - ( O , y) - h Ox

~o(0, y),

(335a)

Flow in an infinite quarter of a plane

Ogo K c 2 = - ( x , o) - e ( x , o), oy

2h [ ~ ~o(x, y) -- m 7r Jo

267

Oq9(oo, Y) _ 0q9 Ox 5-y-y(x' ~ ) - o.

e -x~ { sin(yot) + Kc2ot cos(yot)} ot(1 + KClot)(1 + K2c2ot 2) dot.

335.04 (I)

Groundwater flow between a straight open boundary and a circular basin with a constant drawdown h. Steady state. q9 = ~o(x, y).

Y

qg(O, y) = O, q)(oo, y) = qg(x, 00) = O, q)(xo, Yo) = h

for (xo - a 2) 4- y2 ~< R 2.

tj h •

ln { ZW~/a2- R2 }.

Kh

S2=

arccosh(~)

Z-

~/a 2 -

R2

Total discharge: 2rc K Dh

Q

arccosh(~)"

335.05 (SC) Y q)= h

Flow between an infinite and a semi-infinite equipotential plane which are perpendicular to each other. Steady state, q9 = qg(x, y).

I

qg(0, y ) = h

fory>/y0,

qg(oo, y) -- 0,

~oO__c(0, _. Y ) _ 0 Ox :¢

ff2--m

for y < Yo,

2Kh jr i arcsinh(~o)"

v ( x , o) = o,

°~° (x, ~ ) - o.

Oy

268

Two-dimensionalgroundwater flow

(335b)

b. Flow in vertical planes

02q 9

O2q9

I f12 0(t9 for unsteady flow,

- I f12__

Ss

KD'

-57

for steady flow,

ff -- x + iz.

335.11 (D>

....

Two semi-infinite reservoirs separated by an impermeable screen on top of an aquifer of great thickness. Rise H of one of the levels. Steady state, q9 = ~0(x, z) = head.

IH ~o=0

.. .............................. ......................... -_.-_......-.....,............................... .................... ~ X

~0(x

&,K

.

S2=-

KH

iln(,

0)'

[0 |H

forx>0, forx <0,

(--x+iz.

cf. 335.01.

7l"

335.12 (L,IS)

Non-steady state of 335.11. Additional initial condition: ~o(x, z, 0) - 0.

•

q)(x,z,t)=

2

erfc( >

rc z

x

x2 + °e2 exp

/ 2(x2+ 2'J 4t

335.13 (IS> Two semi-infinite polders separated by an impermeable small dike, on top of a very thick leaky aquifer. Rise H of one of the polder levels. Steady state, q) = q)(x, z) = head.

x

forx <0,

8q9 (x O) -- { qg(x, O) - H X c a--7 ' ~(x, o> •

Y2=

KHi[ln( 7r

q)(x, z) - ~ arctan

7r

f o r x > 0.

+ exp ( _

+

rr

_~c)El( -

i(

1 + Kcot

sin(xot)e -z~ dot.

Flow in an infinite quarter of a plane

(335b)

269

335.14 (L,IFR) Non-steady state of 335.13. Additional initial condition" ~o(x, z, 0) - 0.

sin(zct) + Kcot cos(zot) } Jr Jo

dot.

0/(1 -+- K2c20t 2)

335.15 (IC) Two semi-infinite reservoirs separated by an impermeable dam of width 2B on top of a very thick aquifer. Rise H of one of the levels. Steady state. 99 - q)(x, z) - head.

W=

........

x

~ o ( ~ , z) -

for0~<x
Oqg (x, O) -- O Oz KH

KH

2

Jr

m

H qg(x, z) -- 2 H

.

335.16

2

arcsin -~ ,

q)(x,0)--0

o,

~o(o, z) -

2

f o r x > B.

( -- x ÷ i z.

Harcsin{V/(X+B)2+Z2-v/(x-B)2+z2 Jr 2B .

H

-=,

}

H

. .f0 ~ 1 Jo(B~) sin(xol)e -z~ dot. Jr 19l

(IS)

Groundwater flow between two semi-infinite reservoirs around a screen o f length l on top of a very thick aquifer. Rise H of one of the levels. Steady state. 99 -- 99(x, z).

q)=0

It

x

K

o~

ox(O,z)--O

for0~
~o(x, 0) - 0,

°--~ (x, ~ ) Oz

~o(oo, z) - O,

- o.

~p(O, z) - 0

for z > l,

270

(335b)

(1)

I2----iarcsinh

,

Two-dimensional groundwater flow

( -- x + i z.

H{Jx2-k-(z-}-l)2-Jx2+(z-l)2} ~o(x, z) - - - arcsin Jr

21

H fo ~ 1 Jo(loe)e_X~ sin(zoo)dc~. 7l"

0/

(IC)(S)

335.17

L i:

A channel with width 2B separated by an impermeable screen from a semi-infinite reservoir on top of an "infinitely" thick aquifer. "Sudden" rise H of the channel level. Steady state. ~o = qg(x, z) = head.

¢p=O

~p(x,0)-

a~

qg(x, oo) -- O,

Ox (0, z) -- O,

KH

,

12 -- - - i

In

zr

335.18

H 0

qg(oo, z) -- O,

for0<~x B,

( -- x + iz.

((+B) (-B

(L,IC)

Non-steady state of 335.17. q9 = qg(x, z, t) = head. Additional condition: ~o(x, z, O) = O. *

99(x, z, t) -- 2 H fo°° -1 sin(Bol) cos(xoe)P (-~zot ot47 ) do~.

335.19

(S)

....

?=_o

;//////, ~ / / / / / / / / /!/ / / / / / / / / . Z ,

C

A channel with width 2B separated by an impermeable screen from a semi-infinite reservoir, on top of a leaky, "infinitely" thick aquifer. Sudden rise H of the channel level.

x B

_~

z

H + K c ~ ( x , o) for 0 ~< x < B, ~o(x, O) -

O~o Kc-~z (x, O)

~o(oo, z) - o,

~o 0-7 (0, z) - 0,

f o r x > B. ~o(oo, z) = 0 .

Flow in an infinite quarter of a plane (335b, 336a)

£2-- K H i V

•

~ +BB ) + e x p ( ( + B

In((

(t + B

iKc )El("KC)

-

•

336.

271

)

"Kc

Obstacles in uniform flow 02q9 Jr- --02q9 = 0 Ox 2

Oy 2

"

Subdivision" a. Obstacles, acting as equipotential planes. b. Obstacles, acting as impermeable planes. a. Obstacles, acting as equipotential planes

336.01 (J} Circular basin with constant level in uniform flow with strength q. 99 - 0 for x 2 + y2 ~< R 2.

Y

•

,(2--q(

R2) z

R

x

•

Stagnation points" Zs - +JR.

Hydrological screen" r -- R

1 + cos 0 sin 0

(polar coordinates).

Total discharge through the basin: Q = 4qR. 336.02 (J/ Finite equipotential plane parallel to uniform

flOW. ___.}q

-+Bq V / z 2 _ b 2 b Hydrological screen: y2 ._

b2x 2 x 2 + b 2"

Stagnation points" Zs - 4-0. Total discharge: Q - 2qb.

272

(336a, b)

Two-dimensional groundwater flow

336.03 (J)

Y

Basin of elliptical shape with constant level in uniform flow in the direction of the long axis.

=:~q

a

x

q ( + ax/z a - c 2 - bz). £2 -- D(a - b)

Stagnation points: Zs - +ib. Total discharge: Q - 2q(a + b). Hydrological screen" + a I m [ ~ z 2 - c i] - by - c 2. b. Obstacles, acting as impermeable planes

336.11 (J) Circular impermeable screen in uniform flow.

Y ==~q

I 2 - - ~ q ( z +~ )

.

Stagnation points" Zs - -t-R.

336.12 (J) Y

==~q

b ~=0

-b

Finite impermeable screen perpendicular to the uniform flow

cna__L(0,Y) _ 0 Ox

for lyl < b,

~0(0, y ) - 0

forlyl>b.

s? - + q , / z :

+ b2.

Stagnation points: Zs - 0.

273

Flow in an infinite quarter of a plane (336b, 337)

336.13 (J)

Impermeable screen of elliptical shape in uniform flow parallel to the long axis.

Y

•

~

a

x

I2-

q (az 4- bv/z 2 - c2). D(a - b )

Stagnation points: Zs -- -+-a.

337. General two-dimensional flow in an infinite quarter of a plane towards wells and well-screens

02q) _Jr- 0299 _ { f12-099 ~ OX2 f12 =

OY2

0

for unsteady flow, for steady flow,

KD

Many problems, concerning a well or a well-screen in an infinite quarter of a plane, may be solved by superposition of the drawdowns or heads, caused by so-called "image" wells, the place and character (abstraction or infiltration) of these wells depending on the location of the original well with regard to the boundaries and the character of the boundaries (zero head or zero flux). In general, all kinds of solutions for a single well in an infinite field (see 215) may be applied, thus giving, for instance, steady and non-steady solutions for a single well in an infinite quarter of a plane for both confined and leaky aquifers. Not all these solutions will be given in this paragraph; it will suffice to refer only to the possibility of applying the method of images. An exception will be made for wells or well-screens in uniform flow. A well-screen may be also considered as the approximation of an infinitely long row of partially penetrating line-wells, regularly distributed along a straight line and at small mutual distances. Vertical infiltration of water, uniformly distributed over the area of an infinitely long strip with finite width may be considered as a horizontal well-screen. Subdivision: a. Wells and equipotential planes. b. Wells and impermeable planes. c. Wells and equipotential planes with entrance resistance. d. Well-screens. e. Wells and well-screens in uniform flow.

274

(337a)

Two-dimensional groundwater flow

a. Wells and equipotential planes 337.01 (SC) Flow between a well and two semi-infinite equipotential planes, making an angle of otrc with each 1 other. 0 < o~ < g.

4

qg--qg(r,O),

0/5 ========================================== ~-

rp=0

x

0 < ~ 0 <~c~Jr,

qg(r, O) -- qg(r, otn') -- O,

~o(c~,o) - o ,

Q in zo(ro, 0o),

0 < 0o < c~zr.

ro ~= O,

:l/u)

,

S2--

zl/~ -- ;d0 zl/~ _1/~ ' ;d o

Q In 2rrD

z - - r e i°, zo -- ro ei°°, zo -- ro e-i°°.

337.02 (S) Flow between a well and two semi-infinite equipotential planes, perpendicular to each

q)=0

~Q m

P(Zo) *I',NQ

......

-zp

other.

I

I I

I

I

~o=Oi

I

I I .........

q9 -- qg(r, 0),

7r

0 ~< 0 <~ ~ ,

x

I

I

~o(~, o) - o ,

q)(r, O ) - qg(r, 2 ) - - 0 ,

Z0

--Z0

7l"

Qin

,

P(ro, 0o),

roe0,

0<0o

< ~.

Q ln(Z2-~'2'~ s2 -- 2rrD ~ - zg,]

= solution for one well with 3 image wells: two discharge wells at zo and - z o and two recharge wells at 2'0 and -z'o.

,r/ q~ -- 4zr---D

x + xo = + o , -

xo = +

+ yo =il

L {(x - xo) 2 + (y - yo) 2 }{ (x + xo) 2 + (y + yo) 2 }J '

Q {arctan(2-Y°)+arctan(y+y°) ~P - 2re D + xo x - xo - arctan

y - Y°, - arctan x xo

+ xo

(337a, b)

Flow in an infinite quarter of a plane

275

337.03 (Q) Equipotential plane of parabolic shape and a well situated on the axis o f the parabola. Focus of the parabola in the origin. 99 -- 0 for

Y

y2 _ 4 p ( x + p).

2p

•

V

~

Q lncoth{zr(~/-~- v/S) ]. 27rD 4V/-fi

£2--

a

337.04 (Q) Equipotential plane of equilateral hyperbolic shape and a well with discharge Q. 1 2 ~ o - - 0 f o r x y - - ~a . •

I2--

~2 _ 2ia 2

~ln 27rD

(z2 z 2

-z 2

)"

b. Wells and impermeable planes

337.11 (SC)

YT?

Groundwater flow towards a well situated near two semi-infinite impermeable planes, making 1 an angle of otrc with each other. 0 < ot < ~.

¢ P(zo)

Quasi-steady state.

~p(r, 0) -- const., •

Q

,(2 -- c

2rr D

In { (Z 1/~

_l/ot,

- z° )

x (z 1lot

337.12 (SC,S)

Y

~p(r, c~7r) -- const.

=1/o~ - z o )}.

Well with discharge Q in the centre of a circular row of n recharge wells, regularly distributed

Q

X

along the circle, each with recharge Q. Special case of 337.11 with c~ - 2, r0 - R, ,r combined with a discharge well in the 0o _ 7' origin. •

S2--

Q 2re Dn

i n ( z n - Rn) zn

(337b)

276

Two-dimensionalgroundwaterflow

337.13 (S) Flow to a well, situated near two semi-infinite impermeable planes, perpendicular to each other. Quasi-steady state.

- -~ zo I I I

I I I

--Z0~- -

Qg

~

I I

7r - (0, y) - const.,

x

,

O(x, 0) -- const.

In {(z 2 - z2)(z 2 - 2,2)/

S'2--c

27rD

__:~o ~Q

- solution for one well with 3 image wells, all discharge wells, respectively, in zo, -zo, zo and -z0. ~b - c

Q [ln{(x-xo) 47rD

2 + (y _ yo)2} + In {(x + xo) 2 + (y - yo) 2 }

+ In {(x + xo) 2 + (y + yo) 2 } + In {(x - xo) 2 + (y + yo)2}], - ~ +{ a r c2t a n7(

r xyD- xo y°)

+arctan

(y+Yo)

x+xo

arctan(YY °+- ) Xxo + arctan(Y + Yo) } x-xo "

337.14 (Q)

>

Impermeable screen of parabolic shape and a well, situated on the axis of the parabola. Focus of the parabola in the origin gr -- const. for y2 ._ 4 p ( x + p).

a ,

Q

s? - c -

2Jr----D

In sinh{ r r ( ~ / z - ~/~)} 2,/-fi

"

337.15 (Q) Impermeable screen of orthogonal hyperbolic shape and a well with discharge Q. ~p - const, for xy-

1 2 ~a .

Q

,

~(2--c

Stagnation point: Zs2 __ X2 - y2 + i a 2 •

- ~ In {(z 2 - zZ)(z 2 - 2,2 - 2ia 2) }" 27rD

Flow in an infinite quarter of a plane

(337c)

277

e. Wells and equipotential planes with entrance resistance 337.21 (IC,MS)

Y qg=0

Groundwater flow from a straight surface water boundary with entrance resistance, towards an abstraction well.

w

Oq9 Ox (0, y)

qg(0, y) Kw '

q)(e~, y) -- O,

Oq)(x, O) _ O (x # a), Oy ~o(x, ~ )

- O,

lira r --r--.o 2 Jr K D r 2 -- (x - a ) 2 ÷ y2.

Hw Q

z+a)+ .... a

~2zrD ln(z_

for

Q (z+a z+a ~--~exp k K w ) E l ( K w ) '

Q ln{ (x + a ) 2 + Y 2 } 4JrKD (x - a ) 2 ÷ y2

q)(x, y) --

÷

w wote -(x+a),~ cos(yot) do~. Jr KQ D fo ~ 1 +K K

337.22 (L,IC,MS)

Non-steady state of 337.21. ~o(x, y, 0) - 0.

•

qg(x, y, t) --

Q 4jr X D

([,'

-~

{(x - a

+ 2jr K D

Q f o ~ E(x,

E(x, t, ot) --

y.] [,.~-7{(X+

+ }

-- E 1

t, or) cos(yc~) dot

with

~q7 +-~} ~47 ,I 2,/7

Kw e (x+")~ erfc {/3(x + a) 1 - Kwc~ 247 ÷

Kw e-(X +a)c~ erfc {/~ (x + a) 1 +Kwc~

.Kw - 1xerfc{

K2t020/2

(x+a) exp

Kw

~/7 /~(x+a)} Kwfi + 2~ "

exp

K2w2 f12

t

+ y',J)

278

Two-dimensionalgroundwater flow

(337c)

337.23 (IC,MS)

a :=~

I

Problem 337.21 for a leaky aquifer.

f "////////////~ ///////////////~ ¢

~

0 2 (p

q9

Ox 2

Oy2

~2

"-- O,

)2 _ K D c .

a~o (0, y) -- ~o(0, y) O---x K----w-'

!

'!

02q9

KD

qg( c~, y) -- O,

E a

0--~(x,0) = 0 lim r

r~O

~

~o(x, c~) - 0,

forx Ca,

Oy

for r 2 -- ( x - a) 2 -k- y2.

--

2rr K D Q

qg(x, y) -- 2zc K D

I '

Q

+

Ko{~/(x -a)2+

f~

-

rcK D Jo

+ a) 2 + y3] y3 / - Ko{ ~x/(X 1

Kw

.e

-(x+a)

2+ ...~.

]

cos(yot) dc~.

l + Kw~ot2 + - ~

337.24 (SC) Groundwater flow between a well and two

straight semi-infinite surface water boundaries with entrance resistance, making an angle of 1

t~zr with each other. 0 < ot < 7. 1

See 316.41 with 0 < ot < ~. . . . . . . . . . . . . . .

337.25 (Q)

YI

~p=0

w

I...... ,

S2-

Groundwater flow between a well and two semi-infinite straight surface water boundaries with entrance resistance, perpendicular to each other.

o/TQ

tp=0

x

÷2exp(iKw

iKw

Flow in an infinite quarter of a plane

(337c, d)

279

337.26 (S) Groundwater flow to a well from two straight semi-infinite surface water boundaries perpendicular to each other. One open boundary and one with entrance resistance.

Y ~o=0

w

P(Zo) ,~ Q I I

I

•

..................... J ........................... .................................................

12-

x

~o=0',

Q ln(~ 2zr---D

-~'2 - z2 )

I

Q {

+-~-D

ZO

z+z'o

(z+z'o'~

e x p ( Kw )El - exp

Kw /

(z+zo , (z+zo } • /E Kw

Kw

= solution for a discharge well in zo and a recharge well in z'o in combination with a straight boundary with entrance resistance.

337.27 (S) Groundwater flow towards a well situated near two straight semi-infinite boundaries, perpendicular to each other. One impermeable boundary and one open boundary with entrance re-

Y

w

~o=0

P(zo) ~22 Q '

sistance.

I I

'

D,

~ 8 ~ ~ ~ 8 ~j x

•

,f2 -

Q

In

2reD

'~ Q

(z - zo)(z - zo)

Q {

z-o

+~--~

| (Z ~- ~70)(Z~- Z0) ]

(z + zo

exp\ Kw

z + zo

) E l ( ......Kw )

+ xp(ZK+zo (z+zo } w )El K w / -- solution for a discharge well in z0 and a discharge well in z'0 in combination with a straight boundary with entrance resistance.

d. Well-screens 337.41 (Q) =~ q

ai

~p=c

i

X

K

Finite well-screen in a very thick confined aquifer Constant level at the screen surface. Discharge q [LZT-]]. Quasi-steady state. 99 - qg(x, z) - drawdown. ~o(0, z) -- c

1

for ( a - ~ l )


1/),

280

(337d)

-x(a° '

z) - o

1/) a n d z > ( a + 1 ) ~ 21 ,

for 0 ~< z < ( a -

~o(c~, z) - 0 ,

Two-dimensional groundwater flow

Oqg(x, O) -- O, Oz

~O(x, cxz) -- O,

( -- x + i z.

1 2

arccosh ( (2 + a 2 + ~/ ). 2zr al

S2--Kc-~ 337.42 (Q) =~ q

x ~p=c

l

K

Finite well-screen on top of a very thick confined aquifer. Discharge q [L 2 T - 1 ]. Constant level at the screen surface. Quasi-steady state. 99 - qg(x, z) drawdown. qg(0, z ) - c

for0~
0--~(0 z ) - 0 0x

~o(c~, z) - 0 ,

a--£(x, o) - o, az

forz>l,

'

~(x, c~) - o.

S2 = K c - - q arcsinh (/(__) . 7t"

~p(x, z) -- -qrr fo ~ _otlJo(iot)e_X~ sin(zot)dot 1 = solution 337.41 with a - ~l.

337.43 (IC) q x

l

Finite well-screen on top of a very thick confined aquifer. Discharge uniformly distributed along the well-screen. Quasi-steady state, q9 - ~o(x, z) - drawdown.

Ss, K

Oqg(O, z) -Ox

q 2Kl 0

for z > / .

~o(c~, z) - o, ~o(x, c~) - O, I2 -- c ~r(x,z)--

for 0 < z < / ,

~ ( x , o) = o, 0z

( - x + i z.

2zrl q {(i( + l ) l n ( i ( + l) - (i( - l ) l n ( i ( - / ) } rrl q fo ~ ~1 sin(/ot)e -x~ sin(zot) dot.

Flow in an infinite quarter of a plane

(337d)

281

337.44 (L,IC) Non-steady state of 337.43. 99 -- 99(x, z, t). Initial value: 99(x, z, 0) = 0. •

qg(x,z,t) =

JrqKI f0~ ~1 sin(lot) Pconj (c~x ~ , c ~fl)

cos(zoe) dol.

337.45 (IFR) =~ q C///////////////× //////,////////////_

Finite well-screen in a very thick leaky aquifer. Discharge uniformly distributed along the wellscreen. Steady state. 99 = q)(x, z) = drawdown.

1 Ss, K

' 2Kl

a-E(°'axz) - /

K c a~p O---z(x . 0) - .

2q rc Kl

fO °

( a + - ~l l ) '


( 1 )

forO~
~o(c~, z) =0,

~p(x, z) --

for ( a - ~ l )

andz>

a - -~t

q)(x. 0)

( 1 )

a + -~l ,

99(x ~) - 0 .

.

F ( a ) e -x~ { sin(zot) + Kcc~ cos(zol) } dot

~(x, z) -- --77 2q fo ~ F(oe)e-X~ { cos(zoe) - Kcoe sin(zoe) } do~ with sin (/~){ sin(ac~) + Kcc~ cos(ac~)} F(c~) --

or2(1 ÷ K2c2ot 2)

"

337.46 (L,IFR) Non-steady state of 337.45. 99 = 99(x, z, t). Initial value: 99(x, z, 0) = 0.

•

qg(x, z, t) --

2q fo~ F(ol) Pconj (xc~ ~'

rc Kl

F(ol) = (see 337.45).

2

/3 / { sin(zot) + K c a c o s ( z a ) } dol,

282

Two-dimensional groundwater flow

(337d)

337.47 (S)

Finite well-screen with total discharge Q fully penetrating a leaky aquifer.

21 ]~. . . . .

0299

0299

Ox 2

0y2

99

~2 _

"-- O, )2

KDc.

Discharge uniformly distributed along the wellscreen.

~_a ,////////~9//////////~ c

Oy

iii III III III I I

I

I

for-/<x

0


forx <-landx

>l.

Q

I

I I J I I

QD 4l K

099 (x, 0) -

•

',II

99(x, y) --

4rc K DI

f

+t

1

l

337.48 (IC)

Vertical infiltration uniformly distributed over an infinite strip of width 2B into a very thick aquifer. Quasi-steady state. 99 - 99(x, z) - head.

q ~,1,,!,,i,,1,, B

--

B

X

a 2 ( o z) - o, Ox

a ~ ( x , o) Oz n -- c

-~ 0 q

q

for 0 ~< x ~< B,

'

~o(~, z) - o.

~0(x, ~ ) - O.

f o r x >~ B,

{(( + B ) l n ( ( + B) - (( - B) lfl(( - B)}

7/"

(( -- x + i z),

zr f o ~ ~-~ 7r(x,z) - 2q 1 sin(Bot) sin(xot)e -z~ dot.

337.49 (IC)

......... ,!,4,,1,,[,,1,4,,I,,1,4,.

Vertical infiltration uniformly distributed over an infinite strip of width 2B isolated from surrounding surface water, into a very thick aquifer. Steady state. 99 = 99(x, z) = head.

B -z B

099 Ox

~(0,

z) - 0 ,

~o(x, c~) - O,

~o(c~, z) - 0 ,

Flow in an infinite quarter of a plane Oq) - - ( x , 0) Oz

q K

(337d, e)

for 0 ~< x ~< B,

•

~ ( ( ) -- q ( i ( -+- v/B 2

•

,#(x, z) -

•

p ( x , z) -- qB

-

~.2),

283

q)(x, 0) - 0

f o r x >~ B.

~ -- X + iz,

-q~B f0 °° 1 J1 (Bct) cos(xot)e -z~ dc~

fo

- J l ( B o t ) sin(xot)e -z~ dot. 0[,

e. Wells and well-screens in uniform flow 337.61 (S}

Discharge well and recharge well of equal strength in uniform flow, making an arbitrary angle with the connecting line of the two wells.

¢ ¢

%

X

2l

•

Q ln(Z +l)

~2-- 2JrD

z-l

q

+-~ze-'

~zr

Stagnation points: Zs - -+-~/l2 -t-

Ql eic~rr. rrq

337.62 (S)

Y

r

Discharge well and recharge well of equal strength in uniform flow, parallel to the connecting line of the two wells, the discharge well situated downstream. As 337.61 with oe - 0.

Sx • Hydrological screen" X

2

_

12 _ y2 -k- 2ly cot( 2rrqy

o)"

Stagnation points:

Ql rrq

X s -- -'l-~/l 2 4- - -

YS - - 0 .

I2--

q Q ln( z+l ] 2reD \ +-~z.

284

(340-342)

340 BIII-4.

GENERAL TWO-DIMENSIONAL

Two-dimensional groundwater flow

F L O W IN A N I N F I N I T E S T R I P

a. Has the soil discrete inhomogeneities in x- and~or y- (z-) direction?

h z C1///'////////////////////~

K1D1

"/////////////////////.

C2 _

I

I

K2D2

x

//////////////////~

go to 370 (BIII-7)

An example of discrete inhomogeneities in this category is groundwater flow between two polders with different levels and different aquifer properties.

;////////////////////~Z b. The soil is assumed to be homogeneous

go to 341

Also anisotropic homogeneous soils belong to this category, provided that the main directions of the anisotropy coincide with the coordinate directions. Solutions for anisotropic cases can easily be derived from analytical solutions for corresponding isotropic problems.

341. Homogeneous soils a. Are drains or wells involved in your problem?

go to 343

Groundwater flow in an infinite strip can take place both in horizontal planes and in vertical planes. Flow to or from drains occurs in finite aquifers with a vertical flow component, the drains being infinitely long, whereas flow to wells takes place in finite aquifers with only horizontal flow, the wells being fully penetrating. b. Flow to and from equipotential planes

go to 342

In practice an equipotential plane may be represented by a more or less fully penetrating canal in relation to horizontal flow, or the bottom of a canal, lake or wide river in relation to flow in vertical planes. Sometimes an infinitely long row of partially penetrating line wells, regularly distributed along a straight line and at small mutual distances will be approximated by an infinite partially penetrating well-screen with an average drawdown, dependent on the total discharge, this screen thus acting as a partially penetrating vertical equipotential plane.

342. Flow to and from equipotential planes 02~o

0299

Ox 2

Oz2

-0.

(342)

Flow in an infinite strip

342.01

~

285

(SC)

:

qg=0 "

Infiltration canal with constant level on top of a confined aquifer with unilateral discharge q [L2 T - 1]. Steady state.

.......... -

2b

-

~

~p = ~o(x, z) = drawdown.

~(-~, Ox

O_~_~( ~ , z ) _ q Ox K D'

~(x,O)=O Oz ,

for - b

~o(x, O) -- 0

forx < - b a n d x

12 - 2q arctanh(x/~)

z) = 0 ,

<<.x <<.b,

~ ( x , D) --0. 0z

>b,

with

7~

err¢/D _ cJrb/D

w=

eTr¢/D _ e - r r b / D

,

( =x +iz.

2q arccoth(e~b/o). Drawdown at infinity" 99(-c~, 0 ) - ~-~ 342.02 (SC)

As 342.01 but with an "infinitely" wide

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

canal.

i' °i

~o(-oo, z) = 0 ,

..........

~=0

x D :=~q

~(~,z)

=

Ox

KD'

~88~888~88~t~88 ~ Z

~o(x,0)=0 •

099 ~(x,0)=0

forx<~0,

Oz

.f2 -- -2q - arccosh(eTr¢/2D),

forx > 0 ,

Oq) ~(x, Oz

D) = O.

( -- x + iz.

342.03 (SC)

4

E.... 1

--=

= 4

Recharge- and discharge canal of different widths

h •

L

x

and with different levels on top of a confined aquifer. Steady state. ~o = ~o(x, z) = drawdown.

286

(342)

ax ( - ~ '

qg(x 0 ) - - l 0 ' [h

z) - 7xx(~, z) - O,

co0__,(x, o) - o

~oO___(x ~_, ' D) -- O, Oz Kh

m)

K(m)

zrL

zra

D'

a--

D'

(L+b).

( -- X + iz,

zrb

Jr(

D'

r, = D '

b-

0<x

with

(e T M -- 1) (e ¢+a _ e a) to -- to(() -- (eL+ a _ ea)(e¢+a _ 1)' L-

f o r - a ~ < x ~<0, forL~<x<~(L+b),

forx <-a,

Oz

n -- ~ F ( a r c s i n 4 r w [

Two-dimensional groundwater flow

1 m--,

e - - t o ( L + b).

Drawdown at infinity:

(arcsin

h

¢(c~, o) =

/eL+a+b ea

Ve

K(m)

h

~o(-c~, o) =

K(m)

( ~/ e L + b - - 1 J F arcsin e L + a + b - 1

m) ) m

,

K(ml) Iql = K h ~

K(m) '

ml=l--m. 342.04

(SC)

~o=0

As 342.03 but with one canal "infinitely" wide. ~o(-c~, z) = 0 ,

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

1~

_~-

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

_~

~(~,

0x

~o(x

0)-/0 '

~(x, Oz

•

|h o) - o

z) = 0 ,

forx~<0, forL<~x~<(L+b), for 0 < x < L a n d x > ( L + b ) ,

Kh

f2 = K (m-----~V (arcsin ~/-w I m), exp (~-oL)sinh ( )2D rr¢ w -- w(() = sinh(~.oL) exp (~_~_o¢), K(ml) Jq[ = K h ~ . K(m)

O ! ( x , D) -- O. Oz

(=x+iz,

1 w ( L 4- B) ' ml -- 1 - m , m --

(342)

Flow in an infinite strip

287

Drawdown at infinity:

h [arccosexp/- Jr(L + b) I[ m.1

g~(c~, 0) -- K (m)

342.05 (SC)

~=0

L

2B

]D

Flow from an infiltration canal with constant level on top of a confined aquifer towards a partially penetrating well-screen at the bottom o f the aquifer with discharge q, the wellscreen acting as an equipotential plane with constant drawdown.

x

J

~o=h z

~ ( - c ~ , z) - ~-~-x ~ (c~, z) - 0, 0---~

qg(0, z) - h

~oO___=(x' D) -- 0, Oz

for x < (L - B) and for x > (L + B),

~o(x,0)=0

~(x,

0) -- 0

for (D - l) ~< z <~ D,

Oz

for(L-B)

~<x ~ < ( L + B ) .

Kh $-2 = K h - ~ F(arcsin w I m), K(m)

w=cwl(~),

w 1(() - ~/1 + tan2 (~D/) tanh2 (~D~) + tan (:r/~-~) tanh (~-D~) , C --

Wl

(B - L),

1

m -- b2,

b

CWl

(B + L),

ml

=

1 - m.

Drawdown at infinity: h

qg( cx~, O ) -

h-

{arc

K ( m----~

343. Drains in confined aquifers 82~o 8x 2

82~o +

8z 2

=0.

( rrl

cos

~O(--C~, O) = h -- ~ F ( a r c s i n d K(m)

l m),

--~

d = c

'

1 + sin (~-o/) '

Iql- K h ~

K(m)

288

(343)

343.01

Two-dimensional groundwater flow

(S)

qap=0 S

-

D =~qo

=:~

Drain with discharge q on top of a confined aquifer with uniform flow of strength q0 (positive in positive x-direction). Discharge chosen pos1 itive, q0 < ~q.

q

lz •

q In s i n h ( z r ( ) qo( 2zr 2-D + D '

$2--c

( --x +iz.

Hydrological screen: 2D X

m

arctanh

tan("z re) } tan (~q°z ~ k

qD !

Stagnation point: XS ~

Jr

arctanh(~qo' q)

ZS m

0.

343.02 (S) q D ~r ~- O

~

Drain with recharge q on top of a confined aquifer with unilateral discharge q. Steady state: ¢p(x, z) - head.

x

lim ( r. 0~r~) -r~O

q with r -- X//x2 + Z2 zrg ' Oq) q z) - o, = - ( o o , z) Ox KD'

~(-~,

OrP(x, O) - 0 0z S2 -- - q

for x 4: 0,

In (1 - errC/D),

~oa_.:_(x, , D) - O. Oz ( -x

+iz.

343.03 (SC) ¢p=0

Groundwater flow between an infiltration canal with constant level on top of a confined aquifer and a drain with discharge q. 0q9 0q) a x ( - C ~ , z) - ~ ( ~ ,

z) - o,

(343)

Flow in an infinite strip

lim

o)

r

r-+0

q

--

~r

289

with r - V/(X -

x o ) 2 n t- ( z -

zo) 2

n'K

~o(x, 0) - 0

for - b

093 ~(x, Oz

<~ x <~ b,

o) - o

for x < - b and x > b,

agO_._t(x , ' D) -- O. Oz

q arccoth ¢ w ~o

q arccoth./w

£2

7'/"

V w0

7"/"

with

eZr¢/D _ ezrb/D ezr¢/D _ e-zrb/D ,

w(¢)

-

wo -

w(¢o),

-wo - - w ( ( o ) ,

( -

x 4- i z ,

(o - - x o + i z o .

Drawdown at infinity:

t2 }' { (ercb/D -+-Uo)2+ S~2 q 99(--0o, 0) -- 2zr K In (e~b/O _ u~)2 + q 99(00, 0) -- 2zr K

v / - ~ - u'o +iv'o,

ln{(l+u'°'2+ l 12

(1

-

u ; ) 2 -}-

343.04 (SC)

~o=0

As 343.03 but with zo - 0 and xo > b. •

,.(2- 2qzr arccoth ~ u ~ ' w -- w ( ( ) -- (see 343.03), uo -

w(xo).

Drawdown at infinity" ~o(-oo, o)

-

2q arctanh{

zrK

exp (-~)

}

~o(,:x::,, 0) -

'

2q n'K

arctanh ,v"-uo.

343.05 (SC)

As 343 but with an "infinitely" wide canal. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

'Xo

i

~o(-c~, z) - 0 ,

~o(x,O)--O

~oa__,( ~ , z) - o

0x for x ~< O,

290

(343)

O~P(x, D) -- O

O~P (x, O) -- O

Oz

Oz

'

3q9) lim r -r-+O -~r

q zrK

Two-dimensional groundwater f l o w

forx/>O

with r -- I~" - ~'ol.

S 2 - qarccoth / w + qarccoth~/~_ ° 2l"

V WO

w ( ~ ) -- 1 - e - ~ ¢ / ° -- x + i z ,

with

7[ wo -- w ( ~ o ) ,

w o -- w(~o)

~o - xo + izo.

Drawdown at infinity: ~ lqn { ~b(oo, 0) - 2zrK

t2

(1 -+-u~)2-+- vO }t2 (1 - u ~ ) 2 + v o

with

~ -

u'o + i v ' o.

Flow in semi-infinite strip

(350-352)

291

350 BIII-5. GENERAL TWO-DIMENSIONAL GROUNDWATER FLOW IN A SEMI-INFINITE STRIP

a. Has the soil discrete inhomogeneities in x- and~or y- (z-) direction?

¢1///~, ~///////////////~/ / / / / / / / / / / / / / / / / / / / / , B

C2

x

go to 370 (BIII-7)

An example of discrete inhomogeneities in this category is groundwater flow between two polders with different levels and different aquifer properties.

KID1

b. The soil is assumed to be homogeneous

go to 351

Also anisotropic homogeneous soils belong to this category, provided that the main directions of the anisotropy coincide with the coordinate directions. Solutions for anisotropic cases can easily be derived from analytical solutions for corresponding isotropic problems. 351. H o m o g e n e o u s soils

a. Are drains, wells or well-screens involved in your problem?

go to 356

Groundwater flow in a semi-infinite strip can take place both in horizontal planes and in vertical planes. Flow to or from drains occurs in finite aquifers with a vertical flow component, the drains being infinitely long, whereas flow to wells takes place in finite aquifers with only horizontal flow, the wells being fully penetrating. Well-screens or plane sources may be considered as to be composed of an infinite number of line sources (drains or wells), integrated over the length of the screen, in such a way that the discharge (recharge) is equally distributed over that length. Sometimes well-screens will be considered as equipotential planes with a given head or drawdown the same all over its surface. Then go to 353. b. No drains, wells or well-screens

go to 352

352. P r o b l e m s without drains, wells or well-screens

a. Is uniform flow involved in your problem?

go to 357

See comments of 322a. In this paragraph only obstacles in uniform flow, that cause also vertical velocity components, are treated, which means that the flow pattern is shown in vertical cross sections of confined aquifers. b. Flow to and from equipotential planes

go to 353

In practice an equipotential plane may be represented by a more or less fully penetrating canal in relation to horizontal flow, or the bottom of a canal, lake

292

(352b, 354a)

Two-dimensional groundwater flow

or wide river in relation to flow in a vertical plane. In both cases boundaries with a fixed head or drawdown and with entrance resistance are included. Also well-screens, considered as equipotential planes belong to this caregory (see comments of 35 l a).

353. Equipotential planes a. Is groundwater flow taking place between planes with different levels ?

go to 355

For these problems steady state solutions are available; in some cases also the non-steady state solution is given.

b. The equipotential planes have equal levels

go to 354

These problems are mostly non-steady, but, in general, tend to a quasi-steady state with an arbitrary constant in the solution for the potential function.

354. Equipotential planes with equal levels Subdivision: a. Flow in horizontal planes b. Flow in vertical planes.

a. Flow in horizontal planes Non-steady flow:

0299

0299 __ f12 0~0 --~ ,

Ox-'---" ~ -t- Oy 2

f12 __

S KD"

Steady flow: 02(.t9

Ox2

02~p

t

Oy2

--0.

354.01 (P,L) Y

.....~o~t)

x-

Two straight open boundaries, parallel to each other and a third straight open boundary perpendicular to them. The surface water level is an arbitrary function of time and the same for all three boundaries. ~0 = ~o(x, y, t) = drawdown.

-b

~0(x, y, 0) -- 0, ~p(0, y, t) -- ~0(x, b, t) = ~0(x, - b , t) - F(t)

a~(oo, y, t) - 0. Ox i

x

KD, S

with F(0) -- 0,

Flow in semi-infinite strip

(354a)

293

4 fotdF (t- r)erf qg(x, y, t) -- F(t) - --

•

Jr

cxa

G(y t) ,

m-1

~

--

~

(-1)

2

cos

m

m = l , 3 ....

(mzry) 2b

(fiX) G(y, 2~/r-

exp

r)dr

(m27r2t) 4f12b 2

with

.

354.02 (P) As 354.01 with F(t) -- h -- const. (sudden drawdown of the water level). •

~o(x, y, t ) -

with G ( y , t )

/~X\ 4h --erf/'-z------|G(y t) Jr \2v/71 '

h-

according to 354.01.

354.03 <SC) Y

Quasi-steady state of 354.02. Point P(R, O) reference point for q9 (~0 = 0). Constant drawdown c for the equipotential planes.

~o=c

• -b

S2--Kc

~o=c

/ 1 - sinhtt / sinh (~--~)

z = x + i y, I2 = 4~ + i ~, ~ b = K c p .

354.04
Leaky aquifer with open boundaries at x - 0 and

~o=h

at y - + b .

Constant drawdown of the surface water levels, the same for all three boundaries. Steady state. ~o(x, y) - drawdown.

~o= h -b

02~0

0299

q9

Ox 2

Oy2

)~2

--- O,

)v2 - -

K Dc.

~o(0, y) -- qg(x, b) -- q g ( x , - b ) -- h,

hi--~/////////////////////////~ c

- - ( c x z , y) - 0.

Ox

KD (tC68g&~

4hfo~ Jr

~o(x, y) - h

F(y ot) ,

erf (

cx)

--

~

m = l , 3 ....

x 2)~

m-I

(-1) m

2

cos

) F(y, ot) dot

(mrry) 2b

exp

(

with m27r2~.2ot

-

4b 2

294

Two-dimensional groundwater flow

(354a)

354.05 (P,L) Y

tp=0

As 354.01, but with zero surface water level at the three boundaries and with constant precipitation.

=======================

02.____ 99 _~ ~0299 _[.. P

tp=0 -b

Oy2

OX 2

~"::

......

__ f12 099

KD

f12 ._

--~'

S

KD '

99(x , y, t) -- head. 099 (cx~, y, t) -- 0. 0x

99(0, y, t) -- 99(x, b, t) -- 99(x, - b , t) - O,

4Plot (fix) 99(x, y, t) -- ~-~ erf 247

a(y, r) dr

with G(y, r) according to G(y, t) with t - r (see 354.01) or m--1

99(x y, t) -- P (b 2 _ y2) ' 2KD --e

m2~2' erf( fix 4'2b2 ~)

16pb2 Jr3KD }

~°° z...,

m3

re=l,3 ....

{ (mx P

4b ' 2 f i b

(mJry cos \ 2b )"

354.06 (P) (FS) Steady state of 354.05. m-1

•

P (b2-y2) 99(x, y) -- 2 K D

16pb2 Jr3K D

(-1) 2 m~x (raTty) m3 e 2b cos 2b

k

o

re=l,3 ....

354.07 (P,L) As 354.04 (leaky aquifer), but with zero surface water level at the three boundaries and with constant vertical infiltration q. Steady state: q9 - 99(x, y) - head. 02(t9 [ 0299 Ox 2 Oy2

99 i q = O, )~2 K D

~2 _ K D c ,

99(0, y) -- 99(x, b) - 99(x,-b) - O ,

qg(x, y) - 4qc Jr

(

erf 2 ; xk ~

with F(y, ot) from 354.04.

Oq) ~ ( c x z , y) - O. Ox

) F(y, ot) dot

Flow in semi-infinite strip 354.08

(354a, b)

295

(P)

Confined aquifer with straight surface water boundaries at x = 0 with entrance resistance Cl and at y = +b with entrance resistance c2. Initial head -h. 99(x, y, t) = head. 99(x, y, 0) = h.

99=0

Y

.I~I~I~I~',~',~I~~',I',', C2

99=hfor t = 0

o=0i_ el

X

C2 -b ,,,,,,,,,~,,,,~,,,,,,,,,, ,,,,,,,,,,,,,,,,,,,,,,,,,,

~o="0

O__.~ K Ox (0, y, t)

99(0, y, t) --

099 ~(cx~, y, t) -- O, Ox 099 99(x,-b,t)

Cl

- K 099(x, b, t) -- 99(x, b, t) Oy c2

KTy(x,-b,t)-

C2

99(x, y,t) --2h{erf(~~-~) + R ( ~ f l x ~ / - O , ~ / ~ ) } c~ ×Z

sin Otn

(~nY C°Sk b )exp (

n = 0 0 t n ( 1 q'1

with 0 -

f12K2c2

a n d o~,,

being the roots of ot tan ot

-

e -

2 °tnt ) flZb2 b

Kc 2 .

b. Flow in vertical planes Non-steady flow: 0299 0299 099 ----'-~ Ox _~_ Oz 2 = f12 --~ '

Ss f12

= inK.

Steady flow: 0299 0299 OX---2 + ~ -- O. 354.11

(SC)

Drain with discharge q and finite width on top of a confined aquifer of finite thickness. Quasi-steady state. 'q"....~ ap=-~

~ K

D

x

ox (O, z) - 0 , 99(x,0)=c

qg(c~, z) - 0 , for0~<x~
296

(354b, 355a)

•

O~°(x, D) -- O.

forx >~ B,

a-YE(x, o) - o Oz

Oz

sinh (~--~)

q

Two-dimensionalgroundwater flow

, f 2 = K c - -- arccosh { 7r sinh(~B2-~) '

(=x+iz.

354.12 (SC) System of parallel drains of equal widths and at equal distances on top of an infinitely thick

2B 2b

aquifer. Discharge q per drain. Quasisteady state.

O~P(b z ) - O ,

a~(o, z) - o, Ox

Ox

~o(x, O) = c

*

~0(x, oo) = O,

'

for 0 ~< x ~< B,

~(x, Oz

o) = o

for B ~<x ~
{ sin (~¢ }

~-D) ~ 2 = K c - q arccosh Jr sin zr B "

355. Equipotential planes with different levels

Subdivision: a. Flow in horizontal planes. b. Flow in vertical planes. c. Well-screens considered as equipotential planes. a. Flow in horizontal planes

Unsteady flow: 02q 9

02f/9 __ f12 0q9

OX2 I Oy2

f12 __

--~'

Steady flow: O2~p + O2cP =0, Oy2 Ox 2

z = x +iy.

KD

Flow in semi-infinite strip

(355a)

297

355.01 (FS) Y

Two straight open boundaries, parallel to each other, kept at zero level. A third straight open boundary, perpendicular to them with sudden constant draw-

qg-O

KD, S

dowtl.

Steady state.

qg= h -b

99(0, y) -- h,

Oq9 ~(c~, Ox

y) --O,

~o(x, - b ) - ~o(x, b) - 0.

•

$2 --

4Kh

arctan

(e_Trz/2h).

7/"

355.02 (L,FS) Non-steady state of 355.01. Additional condition: qg(x, y, 0) - 0. O(3

4h ~o(x, y, t) -- - 7r

m=l,3

(-1)

m-I

2

....

m TrSX

P

m

4b '

m rc ~rf

)c°s(

m rcy

)

355.03 (FS) Y

As 355.01 but of which the level of one of the parallel boundaries is given a sudden constant drawdown, the levels of the remaining two boundaries being kept zero. Steady state.

~p=h KD, S

~o-0 -b

•

:.....::::~..:.........:....... ~0=o

S-2-- - ~ i l n

i s i n h -~-

~o(0, y) -- 0,

~ ( c x ~ , y) -- 0. Ox

~o(x,-b) -0,

~o(x, b) - h.

+1

.

355.04 (L,FS) Non-steady state of 355.03. Additional condition" qg(x, y, 0) - 0. •

cp(x, y, t) --

h(y + b) 2b

n=l

n27r2t } nrc(y + b) 4fi2b2) sin{ 2b }"

298

Two-dimensional groundwater flow

(355a)

355.05 (FCR) Y

Confined aquifer with straight surface water boundaries with entrance resistances. Boundary at x -0 with entrance resistance Cl and sudden constant drawdown of the water level. Boundaries at y -- + b with entrance resistance c2 and zero level. Steady

~o=0

b : .II',',[',',',~',',',',',',I',',',',',',I',',',',',',',~ C2

~o=h

KD, S ¢1 C2

-b

state.

~o=0

h

- KOq)(O y ) OX

'

Cl

~o(x, b_______~)

-- K Oqg(x, b) Oy •

~o(0, Y)

C2

'

sin ot~

~o(x, y) -- 2hb y ~ n=O

Oq9 ~(Cx), y) -- O, OX Oq) ~o(x, - b ) x-:-(~, -b) oy C2

%~+~) + e

Otn(KClOtn - k - b ) ( 1

with C~n being the roots of a tan c~ - e =

~.x

(~.y)

g cos\

b

b

Kc 2 •

355.06 (L,FCR)

Non-steady state of 355.05 with Cl - 0 (open boundary at x - 0). •

q ) ( x , y , t ) - - Z h ~ on=O tn(l+

sin c~n ( f i x otn ~/7 e ) P 2~/7' ~ '~ ~2+~2 /cos

any -i-)"

355.07 (SC) ............. _bt. y ........ ~ . - f l

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

IIIIIIIIII IIIIIIU ........i.......i - ~ ...... ; ~ h

.........~

x

A finite open boundary with constant drawdown at equal distance of two parallel infinite open boundaries with zero level. 099(0 y) - - 0 OX

99(cx~, y) -- 0,

'

..........." b ~o(x,0)-h

for0~<x~
~--Kh+i~

(~)

tanh JrB

Oy

Kh K(ml) {F(arcsin w [m) - K(m)}

tanh(~) tO n

~o0___~(_,x , O ) _ 0

m11mandm '

f o r x >t B.

with

tanh2( )

Flow in semi-infinite strip

(355a, b)

299

Total flow to the finite boundary [L2T-1] • Iql--4Kh~

K(m) K(ml)

b. Flow in vertical planes Unsteady flow: 02q 9 02q 9 __ f12 0(/9 OX 2 t Og 2 -~'-'

f12 __ Ss "~-.

Steady flow" 02(/9 OX 2

+

02q9

. . . . O,

( --x +iz.

OZ 2

355.11 (SC)

Two semi-infinite reservoirs separated by an impermeable screen on top of an aquifer of finite thickness. Constant drawdown of one of the reservoir levels. Steady state. ~o -~o(x, z).

~0=0

hi .....

h

K, Ss

~o(o, z) - ~,

~o(oo, z) - o.

Z

~o(x,0)-,

h 0

forx <0, forx >0,

O~p ~(x,D) Oz

-0.

(cf. 355.01).

,(2 -- ~ i In coth 7/"

355.12 (L,FS)

Non-steady state of 355.11 (cf. 355.02). Additional initial condition: ~o(x, z, 0) - 0. ~o(x, z, t)

_

2h

x--,°°

~

~

Jr

z...,

m=l,3 ....

1 --P

m

{ m rr x m T r S t ] sin (mTrz ~ ~4D ' 2flD / \ 2D )"

355.13 (FCR) As 355.11 but for a leaky aquifer (two polders with different level) and upward direction of z. Steady state. 2"////////////////////~ "////////////// ///////// ¢

D K,

Ss

h ~o(o, z) - ~ ,

~o(oo, z) - o,

300

Two-dimensionalgroundwater flow

(355b)

099 OZ

~(x,

3~o ~(x,D) Oz

0) - - 0 , oo

99(x, z ) -

-- (

~o(x, D) ~ Kc

ff n X

h Z sin Otn exp 2 ~ __-if-) COS ( n-0 o6,(1 + ~2+E2)

OlnZ

D with o~,t being the roots of ot tan o~ - e - x-T"

355.14 (L,FCR)

Non-steady state of 355.13. Additional initial condition: ¢p(x, z, 0) = 0.

p( flx ot~~

sin Otn ~o(x, z, t) -- h Z n = 0 ° h t ( l + ot2We e 2)

,

COS

ol,,z

--D--

"

355.15 (FCR)

Leaky aquifer with surface water boundary with entrance resistance. Two-dimensional steady flow (cf. 126.33). Constant drawdown of the open water lev-

-I-

h

¢

el.

3~o Ox

~(o,

Oq9 ~(x,D) 3z •

go(x, D)

= - ~ ,

Kc

~p(x, z) -- 2hD Z

z) = -

h - qg(O, z) , Kw

~o(c~, z) = o ,

aq9

--(x, 0)=0.

3z

s i n ot,,

e-~"x/D cos (~--~)

n=O Otn(Kwotn + D)(1 +

E

D with c~n being the roots of o~tan c~ - e - K---~" 1 (cf. 355.05). For w - - 0 - - + 355.13 with h -+ ~h

355.16 (SC)

Two reservoirs of different level, separated by an impermeable dam, on top of a confined aquifer of finite thickness. Steady state. qg(x, z) = drawdown. 99(0, z) ---- O,

qg(c~, z) -- h,

Flow in semi-infinite strip 0go ~(x,0)--0 0z

(355b)

f o r 0 ~ < x ~
301

Ogo ~(x, Oz

forx >~ b,

go(x,0)=h

D) = 0.

Kh S2 = ~ F ( a r c s i n w l m ) , ( = x + iz, g(m) rrg tanh(~-g) (n'b) w -m - tanh 2 tanh(g-~)~rb ' ~-D • Total discharge: K(ml) q -- K h ~K (,m )

ml -- 1 - m.

355.17 (SC)

Two reservoirs of different level, separated by an impermeable screen on top of a confined aquifer of finite thickness with a partially penetrating screen. Steady state, go = go(x, z) = drawdown. K

D

~Ogo (0, Ox

-

z) = 0

go(0, z ) = 0

go(cxz, z) = h,

go(x, O) = h,

Kh ,

.(2 =

~

F (arcsin

K(m)

w I m),

Ogo ~(x, Oz

f o r 0 ~
D) =- O.

1 +

zr~ tanh2 (~-~) tan2 (~-~)

,

m --

sin 2

(7rl) 2-D •

Total discharge: K(ml) q = K h ~ K(m) 355.18

(SC)

Two reservoirs of different level, separated by an impermeable dam on top of a confined aquifer of finite thickness with a partially penetrating screen. Steady state, go(x, z) = drawdown. l

_

_

099 ~ ( 0 , z) = 0 Ox ~o(~, z) = h, z

99(0, z ) = 0

f o r 0 ~ < z ~
for/~
(355b)

302

Oq9 OZ

~(x,

for0 ~< x ~< b,

o) = 0

Two-dimensionalgroundwaterflow

qg(x,0)--h

forx >~b

3qg(x, D) -- O. Oz

'

W

Kh F (arcsin w [ m), K(m)

(=x+iz,

~ t a n 2 ( ~-~) Jrl + tanh2 (~-~-0) tan 2 ( yg) rrl + tank2( g-g rrb ) '

m--l-

cos c°sh2 (~b)rrb'

ml--l-m.

Total discharge: K(ml)

q = K h ~ K(m)

355.19 (SC) .....

Drainage canal on top of a confined aquifer of finite thickness, near an open boundary. Constant drawdown of the water level in the canal. Steady state.

h___

qg(x, z) = drawdown. ~(o, z) - o, Z

~oo__z(_,x

, o) -

o

for0~<x ~< ( L - B )

Oz

q)(x, O) = h

for(L-B)~<x

Kh K(m)

S2 = ~ F ( a r c s i n W--

tanh(~-o~)

tanh{ ,~(L-B)2D } '

~< ( L + B ) ,

3qg(x, D) -- O. Oz

tanh2 { 7r(L-B) }2D tanh2 { rr(L+B)2D}

Total discharge: K(ml)

q = K h ~ K(m)

Drawdown at infinity: ~0(c~, z) =

Oz

and x /> (L - B),

w lm),

m~

~° (c~, z) - o,

h [arcsintanh m120 + }I K(m)

(355b)

Flow in semi-infinite strip

303

355.20 (S)

I~

.

A semi-infinite reservoir and a finite reservoir with width 2B separated by an impermeable screen on top of an aquifer of finite thickness. The reservoir levels are different. Steady state.

X

B

D

99(x,0)--

N888888

099 ~(x,D) Oz

$ 2 - - K h i In

•

Oq) ~(O,z)--0, Ox

--0,

h 0

f o r 0 ~ < x < B, f o r x > B,

~o(c~, z) - 0 .

[tanh{ 4-~(~" + B)] ] Jr " tanh{ ~-~(~" - B)}

355.21 (S) Non-steady state of 355.20. Additional initial condition" qg(x, z, 0) - 0.

l[,Im

2h ~o(x, z, t) -- ~ 7"( m=l,3

P

....

m

x - "' 4D

mrc(x + B) 4D

I ' 2flD

] (m.z

2/3D ] sin\2D)"

355.22 (S) ___

Y/////)

As 355.20 but for a leaky aquifer (two polders with different level) and upward direction of the positive z-axis. Steady state.

99=0

~////////////////////~

¢

0q)

~ ( 0 , z) - 0 ,

~o(c~, z) - 0 ,

0x

- h + qg(x, D) 0~° (x, D) -Oz

Kc _ ~o(x, D) Kc

for 0 ~< x < B,

~oo__,(x, o) - o. f o r x > B,

Oz

304

(355b, c)

~o(x z ) ,

Two-dimensional groundwater flow

(X)

(~,,z) sin ctn sinh(~--~-~-) _ _~.x _ ~ ---'7-- e O COS k--D--n=o ct,,(1 ~2+e2)

2h ~

D with C~n being the roots of ot tan ot - e, e - r--~"

355.23 (S) Non-steady state of 355.22.

Additional initial condition: ~o(x, z, 0) = 0. oo ~o(x, Z, t) -- h

sin otn

P { (x - B),~. ,~.4-i 2b

n=00~n(1 + ot2+e e 2)

-h~oo

sin Otn

n=O C~n(1 + ~2-t-e2)

'

fib

(,~.z ]cos\--~'-)

p[~.(x + B) , ,~.47 ]c°s(T,~.z).

e. Well-screens as equipotential planes

Steady flow: 32~o 32~-t=0, OX2 OZ2

(=x+iz

355.31 (SC)

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Partially penetrating well-screen in a finite aquifer under an infinitely large reservoir with a discharge

.

q and a constant drawdown h. ......... "d" ................................... -

o

~o(x O) -- O, '

co3_~.,( x , D) - 0 3z

'

~o(~, z) = 0 , for(d-l)

~o(0, z) -- h

° ! ( o , z) - o

for0<~z < (d-l)

3x

•

S2=Kh

- ~

Kh

K(m)

sin 2 { rr(d-l) }2D m --

< z < (d4-1),

sin 2 { rr(d+t)2D} '

and(d-t-l)
F(arcsin ~/-w I m),

K(ml)

q = 2 K h ~ K(m) '

w--

cot 2 { rr~D/) } + coth2 (~-D ¢) cot 2 { rr(d+l) rr( 2o } + c°th2(Y~)

ml-- 1--m.

'

(355c)

Flow in semi-infinite strip

355.32

305

(SC>

q ..................... ~ = 0

{ 2l]

qg=h

Partially penetrating well-screen in a confined aquifer under the middle o f a canal. Constant drawdown at the screen.

99(x,0)--0

ID

--(x,0)=0 Oz

399 =--(x, D) -- O, oz

f o r 0 ~ < x < B, f o r x > B,

99(0, z ) = h for(d-l)
,f2=(see355.31)

with

cot 2 { rr(d+l) rr B 2D } ÷ coth2(2-D) cot 2 { r r (20 d - l ) } ÷ c°th2 (.~_~.)rr B "

m=

Drawdown at infinity:

h {arcsin 1 >l m }

~o( cx~ , z ) -- h - K ( m------~ u -355.33

sin 2 { rr(~+l) 2D }

sin 2 { 7 r 2~ (d-l) }

'

K(ml) q -- 2 K h ~

with

m l -- 1 - m.

X(m)

(SC>

~0=0 ...................... £ 8 8 8 8 8 8 8 g

=~ q ~

.....g.-.0.. .....~ _

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

2B

2B

d K

~

L 21 I- q g - h

X

D

Partially penetrating well-screen in a confined aquifer, located symmetrically under two canals of equal width. Constant drawdown at the screen.

z ~o(x, o) = o

for ( L - B )

--(x, Oz

for0<~x < (L-B)

o) = o

<x < (L÷B), 0O9 ~(x,D) Oz

andx > (L+B),

~(o, z) = h

for (d - l) < x < (d + l),

~(oo, 0x

009 - - ( o , z) = o Ox

for0~

and ( d + l )

Kh F (arcsin ~/-w [m), S2 = K h - ~ K(m)

z) = 0,

=0,

306

(355c) c~cot2{ 7r(d+l) rr~ 2D }-+- fl coth2(2-D) cot 2 { 7r(d+l) Jr( 2D } + c°th2(g-g)

W --

m m

Two-dimensional groundwater f l o w

cot 2 { rr(d+l) 2D }

+ c°th2{

rr(L+B) 2D }

ot COt2{Jr(d+/) } 2 D -+- 13coth2{ ~r(L+B)2D}

with

rr(d-l)2D ] cot2 { 7r(d+/) 2D } + coth:{ Jr(~)B) } c o t 2{ rr(d+l)2D } c°t2{ 7r(a-/) } 2 D + c°th2{ ~r(L-s)2n} ' cot 2 { 7r(d+l) K(ml) 2D }+c°th2{ :rr(L-B) 2D } q -- 2 K h ~ c o t 2 { re(d-l) re(L-B) ' K(m) aD }+c°tha{ 2D } c o t 2{

0/

--

'

ml--l--m.

Drawdown at infinity:

h {arcsin( 1

q)(oo, z) -- h -

with

X(m-----S

u - o t c o s 2 { :rr(d -t-l) 7r(d nt-/)} 2D } +flsin2 { 2D " 355.34 (SC) q ,1= .

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"

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2B

go=0 :~888888~ ...................... N N N _ '¢=h2B -x

.

"

l]

L

K

As 355.33 but with the wellof the aquifer.

s c r e e n on top

~o(0, z ) - h for0~
D

Ox

£2 -- K h -

~K

Kh (m) \

F(arcsin ~-w I m),

/

cot2( gg)+coth2{ ~l 7r(LDB) 2 ] W--

7rl c°t2 (g-B)+

K(ml) q -- 2 K h

~

K (,

)m '"

c°th2( )2D ~r¢ m 1 --

cot2 (~_D t ) + cotha { ~r(LaD +z) } m --

cot2 (~D/ ) + coth 2 { 7r(L-B) '2D }

1 -- m .

Drawdown at infinity"

h {arcsin( ~ 1 )1 m }

~o((x:), z) -- h - K (m----~

with

]2o/

(355c, 356)

Flow in semi-infinite strip

307

355.35 (SC)

=~q

As 355.33 but with the wellscreen at the bottom of the aquifer.

(/9={'} .........................

2B L -----"

K

D

i~o=h

for ( D - l )

99(0, z ) - - h

° ! ( o , z) - o Ox

Y2 -- K h -

< z <. D,

for0 ~ z < Kh K (m----7F

(D-l).

(arcsin v/-w ] m),

cot 2( ~--~) yrl + tanh2(g-~) yr¢ tO m

rrl + tanh2{ yr(LB) } ' c°t2(g-~) 2D K(ml)

q -- 2 K h ~ K(m) '

m l -- 1 -

m

m

COt2(~-D/)+ tanh2{ Jr(L-B~2D}

7rl tanh2{ 7r(L+B) }' c°t2 (gB)+ 2D

m.

Drawdown at infinity:

h arcsin( { 1 })lm

K(m) mul

with

:sin2(~-D/)[c°t2(~-D/)+tanh2{ 7r(L+B) ] 2 D ]. "

356. General two-dimensional flow in a semi-infinite strip towards (from) wells, drains and well-screens

02990X 2

0299 OY2

{ f12 099 for non-steady flow, 0 for steady flow in horizontal planes,

S

fl2m.

KD

0299 0299 OX2 ~- OZ2

{ f12 099 for non-steady flow, 0 for stready flow in vertical planes,

f12 __ S___ss. K

Many problems, concerning wells, drains or well-screens in a semi-infinite strip may be solved by means of superposition of the heads or drawdowns, caused by so called "image" wells (drains, well-screens). As always two

308

Two-dimensional groundwater flow

(356a)

parallel boundaries occur in a semi-infinite strip, an infinite number of images are needed, which in most cases may lead to solutions in the form of infinite series. Only a restricted number of these solutions will be given in this section, and where it is possible the infinite series will be replaced by transcendental functions. Many solutions for horizontal flow towards wells have their counterparts in vertical flow towards drains which are almost identical; in such cases either the "well" solution or the "drain' solution will be given, dependent on practical considerations. A well-screen may be also considered as the approximation of an infinitely long row of partially penetrating line-wells, regularly distributed along a straight line and at small mutual distances. Vertical infiltration of water, equally distributed over the area of an infinitely long strip with finite width may be considered as a horizontal well-screen. For drains in uniform flow go to 357. Subdivision: a. Wells and drains. b. Well-screens. c. Infiltration strips.

a. Wells and drains 356.01 (FC) Y

Infinite row of wells of equal strengths and at equal mutual distances along a straight line and fully penetrating a confined aquifer. Quasi-steady state, q) =

~Q

q)(x, y). lim

r~O

~Q ~Q

r

Ocp) Q ---~r 2 rc K D

o9a__:_(0 _, ' Y) _ 0

for y ¢ 0

Ox Ocp ~(cxz, y)-Ox

Q 2aDK '

0~0 (x, 0) -- 0q0

K

~Q

Oy

Q

S2 = c

!

D

a

-if-fly(X,-~)

Q 27rD

for r 2

=0,

sinh( Z)

=

x 2 --t-- y2,

z=x+iy.

(356a)

Flow in semi-infinite strip

309

4~ - c - - ~l nQ{ s i n h 2 ( - ~ ) + s i n 2 ( ~ - - ~ ) } 4 z r D Qx ~C

m

2aD

n=

(2nzry)

2nTrx)

~ 27rD ~ £1oxp _n t

COS a

a

1

----~Q arctan c o t h ( - - ~ - ) t a n ( - ~ ) 2zrD

~y 2a D

~ £ ' enx p t 2re D n=

2nJrX)sin(2nzrY). a a

1

356.02 (L,FC) Non-steady state of 356.01. 99 - 99(x, y, t).

Additional initial condition" 99(x, y, 0) - 0. •

O~/7 ierfc(fl~--~~)

99(x, y, t) -- f l a K D

Q

~1 (2nrry) (nrcx -- COS Pconj ~ , 2 r r K D n=l n a a

2nTrv/[]

fl--a /"

356.03 (S)

Q

Infinite row of wells as in 356.01, but for a leaky aquifer. Steady state, q9 - 99(x, y).

~/~/////~//////////~

c

0299

0299

(t9

Ox 2

Oy2

~2

- - O,

ili QD 99 -- 2rrK

x

)2

_

+~ 1 Z Ko{ -£ v/x 2 + ( y + n a) 2 } . tZ--- -- CX)

356.04 (S) Non-steady state of 356.03. 99 - 99(x, y, t).

Additional initial condition: 99(x, y, 0) - 0. •

Q + ~ { ,--~', -f

99(x y, t) -- 4 r c K D

?l -~--- (X3

]

+ (y + na)2 "

KDc.

310

Two-dimensionalgroundwater flow

(356a)

356.05 (S)

Y

Infinite row of wells of equal strengths and at equal mutual distances along a straight line parallel to a straight canal and fully penetrating a confined aquifer. Steady state, q9 = q)(x, y).

~a

a

~a

[sinh{ rr(Tb) } ]

2

Q In

S2 - 2rrD

*

~a

sinh{ ~(z~-b~}

withz-x+iy

-- solution for an infinite row of positive wells together with an infinite row of negative wells. Drawdown at infinity:

~a

~o(c~, y ) - a

Ob aKD

356.06 (SC)(FC)

Infinite drain in a confined aquifer. Quasi-steady state. ~o - ~o(x, z ) . KD, Ss [ D ~ ~o0___2,(0,z ) - 0 0x

q

Z

°--:-' ( ~ , z) -

Ox

lim ( r 0-~r~) r-~O

q 2zr K

for

O~o (x o) = o~o

Oz

for z ¢ zo,

'

-~z (X, D) -- O, q

S2 - c

r 2 - - x 2 -t- (Z - - Z0) 2,

( - - x +iz.

In { c o s h ( - - ~ ) -

2zr

2KD '

(rCzo ) , cos\--D--

OO

•

¢(x,z)

-

qz 2D

q n ~ l - cos ( nrczo) exp ( 7r = n \ D

356.07 (L,FC)

Non-steady state of 356.06. ~0 = ~o(x, z, t). Additional initial condition: ~o(x, z, 0) - 0.

)sin(

n:rfx

(356a)

Flow in semi-infinite strip

311

~ o ( x , z , t ) -- q~/7 ierfc( fix

•

q

S-" 1 zn =- .l "n

wK

356.08

(nsrzo)(nrcz)

-- COS

COS

D

--D

Pconj

(nsrx n~_xlT~ \2D

,

riD /

•

(FCR)

Infinite drain in a leaky aquifer.

WlIIII~ "//////////////////// C

zoI

q

•

I

Steady state,

q9 =

~p(x, z) = drawdown. (z-direction chosen upwards and x-axis chosen along the bottom of the aquifer for computational reasons.)

x ~

0q9 (0, z) -- 0

for z ¢ z0,

go(oo, z) -- 0

0x

lim (r 099 r-+0 -g-r) --

a-5-~(x, Oz

2zrK

.

- - d x 2 --t-- (Z - - Z0) 2,

09a_.' _L(x, D) -- -

o) - o,

~o(x, z)

for r

Oz

~o(x, D) Kc

q ~ o 1COS(~n--~D~)(t~nZ] (ot,~x) . . . cos exp K = ot,~ 1 -4-~2+~2 ---D-/ --D

D with Otn being the roots of a~ tan oe - e - K-7"

356.09 (L,FCR) Non-steady state of 356.08. ~o - ~p(x, z, t).

Additional initial condition: ~p(x, z, 0) - 0. •

c p ( x , z , t ) --

1 cos q ZC~n 1+ 7 cos K n=0 ol2+E2

.47

(---D/Pc°nJ(2o' fiB )"

356.10 (SC)(FS)

....... t

sT iTx Z

Infinite drain in an aquifer under open water. Steady state. ?9 - qg(x, z).

099 ~(0, 0x

Z) -- 0

for Z ~ Z0,

~o(c~, Z) -- 0,

312

(356a) q 2Jrg

r 0(/9) -

lim

r-+0

~

q)(x O) -- 0 '

S2

1

for r -- 4 X 2 -'l- (Z -- Z0) 2,

Oq)(x, D) -- O, Oz

'

~2q

( -- x -t- iz.

}

q sin tTfi-) --i arctan ~r¢ :rr sinh(g-g)

~o(x, z) -

Two-dimensional groundwater flow

"

m1 s i n (

m~zo] exp (

m~x

m~z

2--D)sin(2D)"

2D /

re=l,3 ....

356.11 (L,FS) Non-steady state of 356.10. ~o - ~o(x, z, t). Additional initial condition ~o(x, z, 0) - 0. •

OO

2q

g ) ( x , z , t ) --

zrK

F~

re=l,3 ....

1 sin (mrrzo mzrz m \ 2D ) s i n ( 2 D )

x Pconj ( mzcx mzr S t ] \ 4 D ' 2 ~ D /"

356.12 (S)(FS) Y ~=0

y°I

Single well with discharge Q in an infinite strip with open water boundaries at both sides. Steady state, q9 ~p(x, y).

0~°(0, y) -- 0

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

tp=O

0x

x

lim (r Oq9] _

Q - 2rr K D qg(x, O) -- ~o(x, b) - O.

q)(oo y) -- 0,

for r -- ~ / x 2 -[- (y - yo) 2,

r--+O,, --~F]

sinh{ ....rr(z+iYO) ] 2b }

Q ,(2 - 2 z r D

for y -# Yo,

In sinh{ ~(z2/Yo) } ,

z-x+iy.

~(x y~- ~ ~ !sin (~;YO)exp(_~x)sin (~Y~ rrKD

n=l

n

b

b /"

356.13 (L,FS) Non-steady state of 356.12. q) - q)(x, y, t). Additional initial condition" ~0(x, y, 0) - 0.

•

q)(x, y, t) --

Q ~ _lsin(n~yo)sin(n~Y~,con~(n~xn~47 rcKD n b b / 2b' 3b )" n=l

(356a, b)

Flow in semi-infinite strip

313

356.14 (SC)

¢p=O B

--

"

Infinite drain in a confined aquifer under the axis o f a canal with fixed level. Steady state. 99 - 99(x, z) - draw-

x

down. ~o(x,0)-0

399

- - ( x , o) = o Oz

~(~,

Ox

forx > B,

z) = 0 ,

~(x,D) Oz

l i ~ (r-~-r~) --

--0,

q 27rK

for0~<x < B ,

for z ¢ zo,

~ ( o , z) - o Ox

f o r r - - V/x 2 --[- (Z -- gO) 2.

with

w ~ - q-Jr arcc°thv/w~ -t- qJr arcc°thv / No

exp (-~-) -exp(JrS--b-) w(;) -

wo -- w ( i z o ) , -wo -- w ( - i z o ) ,

exp (-~-) -exp(-rrB--D-) '

~ -- x -t- iz.

Drawdown at infinity: ,2

q

;12] ~o(c~, z) -- 2rr K In { ((11 +-- u~)2+ /g;) 2 -[-

with ~ - -

I

!

u o + iv o.

b. Well-screens

356.21 (L,FC)

Partially penetrating well-screen N o n - s t e a d y state, cp - 99(x, z, t).

in a confined aquifer.

r

x

'{2l

~o(x, z, 0) - 0, KD, S O~o (x O, t) -

O--z

'

O~o

-~7 (x, D, t) -- O,

Z

0

a-Y-~ x ( z, t) - {

q 41K

( p ( ~ , z, t) - - 0 .

for0~
and(d+l)

314 q~/~•

Two-dimensionalgroundwaterflow

(356b)

~o(x, z, t ) -

S. - 2 sin

/ ~ x \

ierfc|_~}

\2~/t / flKD qD ~ S n (nrrx nrc~/7) (nrcz) 2rc2KI n=l ~--~Pconj\ 2D ~ / cos "-5-

(nJrl

with

n:rrd].

356.22
Steady state •

stream function.

of 356.21 for the

¢ ( x , z) =

qz

qD ~ S n

2D

2zr 21

(

~ exp

nzrX)sin(nrcz ~ D

k--D--/"

n=l

356.23
Steady state. ~o= ~o(x, z).

/////////~//27////////////////C i

0~o 0---~(0,z) --

{0

for 0 ~< z < (d - l) and (d + I) < z < D,

q 4Kl

o~ Oz (x ' o) - o,

~o(~, z~ - o,

qD~ ~o(x, z ) -

for (d - l) < z < (d + l),

ff-gl

l__ ~

Sn

1+

n=

~

0q9

~o(x, D)

~(x,D) = - ~ Oz KC exp(-

c~nx)

(c~nz

with

D /cos\-~)

ct2+e 2

S~ - 2 sin (°tnl'~ cos (ot,~d~ \D/ \D/ D and Ctn being the roots of ot tan tz - e - ~-~c"

356.24 (L,FCR)

Non-steady state

of 356.23. q9 = qg(x, z, t). Additional initial condition: cp(x, z, 0) - 0. qD

•

qg(x , z , t)

--

2gl

1

s.

-- e COS 2_, m n=0 c~ 1 + ~-~+,2

Oln Z

~n47)

Oln X '

flD

"

(356b, c)

Flow in semi-infinite strip

315

356.25 (FS) Partially penetrating well-screen in an aquifer under open water. Steady state: ~o - cp(x, z).

~o=0 m

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

X

'I 2l KD, S Z

for0~
"~° az (°' z) - {o

q 4KI

~o(c~, z) - o , ~(x,z)

-

for(d-l)

0~o ~(x,D) Oz

Sm ~--~exp(

Z m=l,3



~o(x, 0) -- 0, 2q D rc2 Kl

and(d+l)

--0.

mrc x mTr z 2D ) s i n ( 2 D )

with

.....

Sm -- 2 sin (mrcl mrrd ~--5) sin ( 2D )" 356.26 (L,FS)

Non-steady state of 356.25. q9 - ~o(x, z, t). Additional initial condition: 99(x, z, 0) - 0.

•

~p(x, z, t) --

Sm

2q D rc2 K1

(mrcx

mrcz

~--S sin ( 2D )Pconj m=l,3

mzrx/~]

,

5gJ

•

.....

c. Infiltration strips

356.31 (L,IC) B

Uniformly distributed infiltration of water from an infinite strip into a confined aquifer. Non-steady state. ~p - cp(x, z, t).

B

P KD, S

D

x

i-

!a

~o(x, z, O) - O,

ox(O,z,t) -0,

~o(cx~, z, t) --0, Z

p

~(x, Oz

D, t) -- O,

a~(x, 0, t) Oz

-~

0

for Ix l < B, for Ix[ > B.

316

Two-dimensionalgroundwaterflow

(356c)

~p(x,z,t)-

2p f / ~ sin(nc~) cos(xot)E(ot, z, t) dct zrK J0 or2

E(ot, z t) -- - Z '

ot(2nD + z) ~ / [

Pconj

2

n----0 -t- Pconj

with

'

fl

{a~(2nD + 2D - z ) otv'7}] 2 ' fl "

356.32 (IC)

Steady state of 356.31 for the stream function. f0 ~ sin(Bc~_______~) 7r(x, z) -- 2__p_p sin(xot) sinh{ (D - z)ot } dot. Jr ol2 sinh(Dot)

•

356.33

(S) B

KD

Vertical infiltration from an infinite strip combined with an infinite drain, lying perpendicularly under and parallel to the axis of the strip. Steady state only if recharge = discharge: q =

B

,,I,,I,,I,$P~~~

zoI

2pB. Combination of 356.06 and 356.32.

1]d

q fo~C°S(X°t)[sin(B°t)c°sh{( D-z)°t }

~o(x, z ) -

Jr K

ot2

q ~1 OO

K

n=

1

-cos n

D

B sinh (Dot)

(nrczo) D

exp

(nrcx) D

COS

(rt~z

-5-),

7t(x, z) - ~p(x, z) of 356.32 - ~p(x, z) of 356.06.

356.34

(FC)

2B

2B

2B P y

x

b z

K

Flow in semi-infinite strip

(356c)

317

A row of infiltration strips with width 2B and at mutual distance 2b on top of an infinitely thick aquifer. Infiltration uniformly distributed.

0~

0~o

Ox (0, z) -- "~x (b, z) -- O, o~o (~, o o ) -

°~ (x 0 ) -

- ~P

for 0 ~< x < B '

O---z '

0

for B < x ~< b,

- P-£

Oz

Kb

Steady state for the stream function:

2pb

pB 7~(x, z) -

x +

b

~ 1 sin

nrr B nrrx ] n b ) sin ( - - - ~ - / e x p ( -

n=l

b

356.35 (S) =

2B

2B

:__

2B

2B

~$$,~, , ~ , ~ , ~ ~ r

X

b

_ q~

2b

I

A row of infiltration strips as in 356.34 in combination with a row of drains at mutual distance 2b and at depth a, the drains lying midway between the infiltration strips. Steady state only if recharge - discharge" 2 p B -- q.

q

(a

qb ~ ~

z) +

1

sm

/nrcB\

cos

/nzrx\

exp

q ,~ (-1) n [ nJra\ [nzrx~ ,[nJrz\ - - - ,) ~ - exp ! - - - 7 - / c o s / - - 7 - 1 cosn/--7--/ 7r~ n ', v / ', o / ', v / ~o(x, z) --

qb

for 0 ~< z ~< a, ~ 1 . [nTrB\ {nrcx\ [ sin cos exp

t, b )

OO

n

t,--b--)

nTrz\

t,---f-)

( 1) nrca nTrx nzrz [ - -7) \ - ~ -q V-' , , ~ - -~ - - cosh [l~---ff--)~ cos [~--ff--)\ exp ~, for z >~ a.

1"

nrcz]

3 18

(357)

Two-dimensionalgroundwaterflow

357. Drains in uniform flow in aquifers of finite thickness 02~ OX2

+

029 OZ2

= O.

357.01 (S) b

b

q

q

~88888~ x

Recharge drain and discharge drain with equal strength q in a confined aquifer with uniform flow of strength qo. Discharge q chosen positive, qo positive in x-direction.

I 2 = q In

2D } + q0 sinh{ ~r~Db) } D('

(=x+iz.

Hydrological screen:

[ tan(~-~) ] + q 0 _ 0 ! arctan tan (~-~) q - - - arctan tanh{ Jr(~ob) } "D Jr z tanh { a'(x +b) 2D } ~Z

"

Stagnation points: D { (Jr~_) q (Jr_~_)} Xs = + - - arccosh cosh + - - sinh , 7r qo

Zs -" 0.

357.02 (S)

S <

Two-dimensional doublet on top of an aquifer of finite thickness in a uniform stream of strength q0. Strength of the doublet = c [L3T-1].

S -

==,qo

•

c c o t h ( r r ( ] + qo

J-2---~

k2D/

-D('

(-x+iz.

(3 57)

Flow in semi-infinite strip

319

Hydrological screen: c sin ( ~ )

cosh( y t ' x -)- cos

qoz

Stagnation points: X s - 4 - - -2D arcsinh Jr

V/ rrc , 2qoD

Zs-0.

c the hydrological screen becomes a circle with radius R For small values of qo ~q°o with very good approximation 357.03 (SC>

==~ qo

A partially penetrating impermeable screen on top of a confined aquifer with uniform flow. 99 - qg(x, z) - draw-

q0

~=p0

down.

~v ox(O,z)--O

Z

~o(0, z ) - - 0 099 qo - - ( o o , z) Ox KD '

2qo

forl < z ~ < D ,

0O9 099 - - ( x , O) -- - ~ ( x , D) -- O. Oz oz

cosh(~) arccosh

for0~
yrl

( --x +iz.

'

cos(~)

357.04 (SC> 2b

Equipotential plane of finite width on top of a confined aquifer with uniform flow.

screen

>qo

~ K

~o(O, z) - o , o ~ ( ~ , z) - qo 8x KD'

~ ( x , o) - 0

forO <~x ~< b,

3go

--(x,

3z

o) - o

forx > b,

8q) - - ( x , D) --0. 8z

(357)

320

•

S2= 2qo arccosh cosh(~-B) 7r c o s h (~--~)rr b

'

(=x+iz.

Hydrological screen: Z

__2Darctan {

sinh(Jrb)sinh(~--~D) } ~ - B ~sinh2 (zrb rrx ~-B) + sinh2(~-~)

Stagnation points: ys = 4-0. Discharge through the equipotential plane: q

_2q0 ~ arctan{sinh~~b ~~ }

Two-dimensionalgroundwaterflow

(360-362a)

Flow in a finite strip

360 BIII-6.

321

G E N E R A L T W O - D I M E N S I O N A L GROUNDWATER F L O W IN A F I N I T E STRIP

a. Has the soil discrete inhomogeneities in x- and~or y- (z-) direction?

Tz

/

¢~;¢~¢~~p

!

go to 370 (BIII-7)

An example of discrete inhomogeneities in this category is rainfall on a dam between two parallel open boundaries, the aquifer consisting of soil layers of different transmissivity.

b. The soil is assumed to be homogeneous

go to 361

Also anisotropic homogeneous soils belong to this category, provided that the main directions of the anisotropy coincide with the coordinate directions. Solutions for anisotropic cases can easily be derived from analytical solutions for corresponding isotropic problems.

361. Homogeneous soils a. Are drains, wells or well-screens involved in your problem?

go to 365

Groundwater flow in a finite strip can take place both in horizontal planes and in vertical planes. Flow to or from drains occurs in aquifers with a finite thickness in which horizontal and vertical flow takes place, the drains being infinitely long, whereas flow to wells occurs in finite aquifers with only horizontal flow, the wells being fully penetrating. b. Flow to and from equipotential planes without wells or drains

go to 362

In practice an equipotential plane may be represented by a more or less fully penetrating canal in relation to horizontal flow, or the bottom of a canal, lake or wide river in relation to flow in a vertical plane. In both cases boundaries with a fixed head or drawdown and with entrance resistance are included.

362. Equipotential planes a°

Is groundwater flow taking place between planes with different levels ?

go to 364

For these problems steady state solutions are available; in some cases also the non-steady state solution is given. Sometimes, in the case of a so-called mixed boundary, where one part of the boundary is an equipotential plane with a fixed potential and the other part of the boundary has a given flux (zero included), then for computational reasons the equipotential part may be approximated by an equally distributed fixed flux. Afterwards the average head or drawdown along this part of the boundary must be calculated and set equal to the originally given head.

322

(362b, 363a)

Two-dimensional groundwater flow

b. The equipotential planes have equal levels

go to 363

These problems are mostly non-steady, but may tend to a quasi-steady state with an arbitrary constant.

363. Equipotential planes with equal levels Subdivision" a. Flow in horizontal planes. b. Flow in vertical planes. a. Flow in horizontal planes Unsteady flow: 0299

0299

fl2099

OxZl Oy 2 =

S

f 1 2--_ _K"

"-~'

Steady flow"

0299 Ox 2

0299 t 0y 2 - 0 .

363.01 (P,L) Rectangular strip with open boundaries. The surface water level is an arbitrary function of the time and the same for all four boundaries. 99 - 99(x, y, t) - drawdown.

Y b B

99(x, y, 0) -- 0, 99(-a, y, t) -- 99(a, y, t) = 99(x, - b , t)

-b

-- 99(x, b, t) - F(t) with F(0) - 0. •

99(x, y, t) -- H ( t )

16 fo t -d~H- ( t - z ) F ( x , z ) G ( y , r ) d r ~-~

O0 F ( x , t) -n=l,3 .... t3~

re=l,3 ....

n-1

n27r 2

cos

n

with

~

exp

and

4f12a 2

m-1

m

cos

k 2b

exp

4f12b 2

.

363.02 (P) As 363.01 with H ( t ) - h - const. (sudden drawdown of the water level).

Flow in a finite strip

•

79(x, y, t) -- h

(363a)

323

16h ~ - F ( x , t ) G ( y , t) 7r

with F ( x , t) and G ( y , t) according to 363.01. 363.03 (EL)

Rectangular strip with a leaky aquifer and open boundaries with constant drawdown of the surface water levels, the same for all four boundaries. Steady state" 79 - 79(x, y) - drawdown.

b

-a ............................... l a x

02(/9 OX 2

-b

I

3279

79

Oy2

~2

--0,

~2 _ K D c ,

79(-a, y) -- 79(a, y) -- h,

Ih !jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj

7 9 ( x , - b ) - 79(x, b) - h.

C

•

1-6h ~ fo ~

79(x, y) -- h

2a

F ( x , ol)

x G ( y , ot)e -°t du

with n-1

(-1)T

F ( x , or) -n=l,3 .... O0

G ( y , ot) -re=l,3 ....

n

(-1) m

m-1

2

c°S(:a,eXp -( 55--a )~

and

cos ~ )exp{ -(2~

363.04 (P,L) Rectangular strip with constant precipitation and zero surface water level at the boundaries. 79(x, y, t) - head.

b

P . a x

0 2-----~ -Jr- ~0279 ntOX 2 Oy2

¢p(x, y, O) - O,

-b

P

KD

= ~ 2079

3t

324

Two-dimensional groundwater flow

(363a)

~o(--a, y, t) = ~p(a, y, t) = O,

•

~o(x, y, t) --

16p fot

cp(x, - b , t) = cp(x, b, t) = O.

F(x r)G(y, r) dr

7r2S

with F and G according to 363.01 with t = r.

363.05 (FS) Steady state of 363.04. ~0(x, y) -

~ ( Pb

2 - y2)

2KD

m-1 Z

2b J

m3

j r 3 K D re=l,3 ....

( ) mzry

cosh(mzra ~ cos 2b J

2b

363.06 (P,L) As 363.05 but for a leaky aquifer with constant vertical infiltration q.

0 2cP I OX 2

02

cp

Oy2

cp(x, y) -- 16qc 7.c2

•

cp t

~,2

q = O, KD

fo ~ F(x,

)2 _ K D c.

ol)G(y, ot)e -~ dol

with F ( x , o~) and G(y, ol) according to 363.03. 363.07 (P)

................... iiiiiiiiii

iiiiiiiiiii

,~,,

C2 C]

' " ' " ' " ' " ' " ' " 'cl "I

--a

Rectangular strip with zero surface water levels at the boundaries. Entrance resistance Cl at the bounadries x -- - a and x -- a and entrance resistance c2 at the boundaries y = - b and y = b. Initial head = h. ~p(x, y, t) = head.

c2

~p(x, y, 0) = h,

I I I I ', : ', ', ', ', ~ ', ~ ', ', ', ', ', 1 ', ', ', '0 '01 '0 '01 '0 ', ', ', ', 1111 ~ 111 ~,

-b

K

coO.__2,(_a' Y, t) Ox

OCP(x, - b , t) -Koy

~p(-a, y, t) Cl ~o(x,-b,t)

C2

KO___:__, Ox

X

(x,b,t

Y, t) --

cp(a, y, t) Cl

-

cp(x, b, t) C2

(363a, b)

Flow in a finite strip

in which

q)(x, y, t) -- 4 h S n ( x , t ) S m ( y , t)

sin c~,,

~-,~

Sn(x, t) --

n'--O O/n(1 + Ctn281 2) +e 1

cos (ctnx ] exp ( _ a /

with an being the roots of ot tan ot - 81 sin Otm Sm (y, t) -- E

e2

m=O O/m(1 + otm 2 +822)

with

O/m

325

a

--

Kc I ,

0tZt

cos(°tmY)exp( \ b

being the roots of ct tan ot - E2

f12a2)

f12b2)

a

--

Kc 2 .

363.08 (D) Island of elliptical shape with constant precipitation. Zero surface water level.

Y b "---

g">O

0299-Jr- 0299 OX 2 Oy 2

--a

P KD

a cp(xo, Yo) -- 0

!a -b

Ox (0, y) -- O,

,xy,

a2 2{ (x2 y2)}

,

2KDaZ+b2

1-

--~-+--~

--- O,

xg y2 for ~-g + ~-g - 1,

o__f_~(x, 0) - 0. Oy

.

b. F l o w in vertical p l a n e s Steady flow: 0299 OX 2

0299 t

--0,

OZ2

( -- x + iz, l-2 -- ~ + i Tr, dp -- K qg.

363.11 (FC)

q .

.

.

.

.

Infinite strip with constant precipitation. Aquifer of finite thickness with horizontal and vertical flow component, q) = q)(x, z) = head.

.

x

- - ( o , z) = o ,

~o(b, z) = 0 ,

Ox

Oqg(x, O) Oz

P K'

O~P(x, D) -- O. Oz

Two-dimensional groundwater flow

(363b)

326

P

x?(~) - - ~ ( b 2

_ (2

(i

2pD ~

D

) + P ~ + T)

1 cosh(-~)

nyrb " .-, n 2 cosh (--e-)

7l"2

363.12 (FC)

.......

As 363.11 but with entrance resistances w at the surface water boundaries, q9 - ~o(x, z) head.

t .....

,,[,,1,,I,,[,,[,,1,,I,,I,,I,,I,,I, +.,], p

0~o (0 z) -- 0 Ox ' '

b

•

]z

b

O--~(x, O) --

--

Oz

pbKw p ~-t-~--~(b2-(

s2 (~-) -

2p D ~

2)

1

7t'2 n=l

+p

(

i( + ~

O~o(b, z) - - ~o(b, z) Ox Kw p K'

600__(x, _:_, D) -- O. Oz

D)

cosh[ n~r

-~ nrrKWDsinh(nrrb-D-) -+- c°sh(nzrb--ff-)"

363.13 (FS)

,,

_1~

_

Infinite strip between two open surface water boundaries. Partial constant infiltration over a strip of width 2b0, symmetrically with regard to the z-axis, q9 = qg(x, z) = head.

X

Oq) Ox

~ ( O , z) - o , q

a~(x,

o) -

Oz

•

S2(()-

f o r 0 ~ < x
-~

0

Oqg(x, D) -- O. Oz

for bo < x ~< b,

8qbyg2 Zc~

~o(b, z) - 0 ,

sin (mrrb° \ 2b ) cosh{ m r r (D+i() } 2b m2

m=l,3 ....

sinh(m~rD k 2b )

363.14 (FCR) As 363.13 but with entrance resistances w at the surface water boundaries. ~o - qg(x, z) head.

~o=0 -

g888~

I

-------~, bo bo w ~.

W

teD

~

},

x

a!(o, z) - o, Ox

~a~(b z) Ox

'

~o ( b ~ Z)

Kw

(363b, 364a)

Flow in a finite strip

o ~ (x, o) Oz

X2(()

q - ~-

forO~<x
0

forbo < x ~
327

~(x,D)--0. Oz

X--" sin (~n@) cosh { an (D+ib ¢ ) } Z~cz2(l+ E ) sinh(-~) n=O ct2+e 2

2qb

b

with an being the roots of u tan o~ - e - Kw"

364. Equipotential planes with different levels Subdivision: a. Flow in horizontal planes. b. Flow in vertical planes. a. Flow in horizontal p l a n e s

Steady flow" 02q9 OX 2

02(t9 +

364.01 (FS)

z -- x + i y, I2 -- c/) + i ~ , dp - K ~o.

= O,

Oy 2

yI

R e c t a n g u l a r strip with 4 open water boundaries. Sudden d r a w d o w n of the water level at one boundary, the water at the other boundaries kept at zero level. ~o =

b

~o= O--------

qg(x, y) = drawdown. qg(-a, y) = h, q)(a, y) = q g ( x , - b ) = q)(x, b) = O.

-b

•

S2(z) -

m-1

(a--z)

(-- 1 ) - 7 sinh{ mrr2b

4hK 7f

m

m = l , 3 ....

}

s i n h )~,m:rra / b'

364.02 (FS) R e c t a n g u l a r strip with 4 open water boundaries. S u d d e n d r a w d o w n of the water level at two opposite boundaries, the surface water kept at zero level at the two other boundaries, q9 = qg(x, y) =drawdown.

b

~=09 = h I

~o = h --a

a

~o=0 .

-b

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

I

x

qg(--a, y) = qg(a, y) = h, ~ o ( x , - b ) = q0(x, b) = 0 .

328

(364a, b) m -1

Se(z)

-

( m :rrz

( - 1) - ~ cosh~ - ~ )

4hK •

Two-dimensionalgroundwater flow

7/"

m

m=l,3 ....

c o s h ( -%--/ mzra ] "

364.03 (S) Rectangular strip with 4 open water boundaries. Sudden drawdown of the water level at two adjacent boundaries, the surface water kept at zero level at the two other boundaries. 99 = 99(x, y) = drawdown.

b ================================

-a

~o

!. . . .

.

.

.

~o=0

h

. - . .h . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

............................... la .................................

-b O0

4hK Se(z)

99(-a, y) -- h,

99(a, y) -- O,

99(x, - b ) : h,

99(x, b) -- O.

1) m~...~Il sinh{ m:rr(a--z) 2b }

(-

-

m

m=l,3 ....

sinh{ mTr(b+iz) ] +

s i n h ( ~ --~)

sinh(mTrh-7)

"

364.04 Rectangular strip with entrance resistances at the boundaries. b

See 364.12. qg=h

¢1

c1 :=hx

,,|~,,,,l,,,,,,,,,,,,, . . . . . . . . . . . . . . . . .

tp=O

, , , , , , , , ~ , , , , , , , , , , , , ,

-b

b. Flow in vertical planes Steady flow" 02(/9 ~ 0299

Ox-----~ - ~z 2 --0, ( -- x + i z,

S2 - ¢ + i ~ ,

¢-

K~p.

364.11 (FCR)

.

Leaky aquifer with two parallel open water boundaries. Constant drawdown of the surface water level. 99 - 99(x, z) - drawdown.

z I tp=O~

V////////////~////////////A c h

099 ~ ( 0 , z) -- 0, Ox x

b

-I-

b

99(b,z)-h,

(364b)

Flow in a finite strip

O(P (x, O) -- O, Oz

329

Oq°(x, D) -- _(p(x, D_~) Oz Kc

x-axis along the impermeable base of the aquifer; z-axis in upward direction for computational reasons. oo

S 2 ( ( ) - 2h ~ sin olne cosh(-~S-) n:O Oln(1 + ~2+e2)c°sh(-q-~) o~n being the roots of o~tan o~

with

D Kc"

364.12 (FCR) ~o=0

As 364.11 but with entrance resistances at the surface water boundaries.

z'

--~ V///////////~

"H

~ ( o , z) = 0 ,

X

~(x, Oz

"/////////////A C

~

b

b

099

899 (b, z) -- h - ~o(b, z)

Ox

Ox Oq) ~(x,D) Oz

o) = O,

Kw cp(x, D) -- - - ~ Kc

oo

sin~nE c°sh(Z~) n=O otn (1 + ~+E 2) K WOrn sinh(-q-~) + D c°sh( '~'ho,]

S2(() -- 2 h D ~

D with O~n being the roots of ot tan o~ - 2--Sc"

364.13 (FCR) Z

~ ....

//A/////////////////Z/1

Leaky aquifer with two parallel partially penetrating canals. Constant drawdown h of the canal levels. 99 - 99(x, z) - drawdown.

lh

s

0q)

~(o, z)-0, Ox

-

Oq)-(x, D) --

x ~o(b, z) -

'

Ox(b,z) -

Oz

q)(x, D)

- K-----U-'

0

°---~(b z ) - 0 0x

0z

099

~(x, 0)-0,

for ( D - d) < z ~< D, / This boundary condition replaced by" for0~
{o q

K

!

forO<~z < ( D - d ) , for ( D - d )


330

Two-dimensional groundwater flow

(364b)

~2 (~ )

2q D

Z°° s i n a n - s i n { n=0

k'-D-']~,

~"(D-d) } cosh/'°tn(] "

a2+e2]

D with an being the roots of o~tan c~ - e - X---~" Average drawdown at x - b, ( D - d) < z ~< D"

2qD2~

[ s i n a n - s i n { ~ " ( D -od ) } coth( °tnb

hay -Kd

n=O

ot3 (1 -t- ,~2+e2)

364.14 (FC)

[ r;7

go=0 2L ///////////////////////////¢

C ~////////////////////Z

~o=0

Infiltration of surface water into a leaky aquifer by means of infinitely long ditches with widths 2B at equal distances 2L of each other. The ditches hard! ly penetrate the aquifer. Under the semi-permeable x layer at the bottom of the aquifer the ground water head is assumed to remain constant (~0 = 0). The D total discharge q [LZT -1] per ditch is caused by a 7-/-/% V///// C rise h of the water level. The problem is schematized to a rectangular part of a leaky aquifer with the discharge q equally distributed over the width of the ditch. ~o - ~p(x, z) - head.

°!(o z ) - o~o Ox

'

- - ~ "L' z-

q 2B K

O~°(x, O) 3z

O ~

for 0 4 x < B, f o r B < x ~ L,

0

~ ( ( ) --

~

(Kc -+- D + i() 4- zr2B

3~p (x, D) . . . 3z

-~5 sin

L

n=l

nJr cosh{nzr(D+i¢) } _[_ 1 sinh{nrr(D+i()

-£ F (n, () --

t~

-nTr sinh ( -nTr r )o+

L

~c I~ 1 cosh: n~ro ~ ~c ~ L :

, "

.

~o(x, D)

Kc

F(n, ()

with

Flow in a finite strip

(364b, 365)

331

Average head (level) of the ditches:

q ( D) qL 2 ~ hav--~--~ c - t - ~ - ÷ 7r3BZK n -

1 (nrcB) L ~- sin 2 F(n ' 0). 1

365. General two-dimensional flow in a finite strip towards (from) wells and drains 02(t9 -

-

-~-

OX 2

OZcp -

-

OX 2

02(t9 Oy 2

-~-

-0

for steady flow in horizontal planes, z - x ÷ i y,

-0

for steady flow in vertical planes, ~ - x ÷ i z.

02q9 OZ 2

Many problems concerning wells or drains in a finite strip may be solved by means of superposition of the heads or drawdowns, caused by so-called "image" wells (drains). As always two pairs of parallel boundaries occur in a finite strip and for each pair an infinite number of images is needed, the solution, in general, may be obtained in the form of double infinite series. In many cases the solution for the finite strip is an infinite series of the corresponding solutions for a semi-infinite strip, as given in 356. Only a restricted number of these solutions will be given in this section. Many solutions for horizontal flow towards wells have their counterparts in vertical flow towards drains, which are almost identical; in such cases either the "well" solution or the "drain" solution will be given, dependent on practical considerations.

365.ol (s) Well with excentricity d in a circular island consisting of a confined aquifer with constant zero level at the circular boundary. ~0 - ~0(x, y) drawdown. X

q9(xo, Yo) -- 0 0q9

--(x,

Oy

,n(dR zz - d T) ' 27rD

0) - 0.

R2

,

12-

z--x

+iy.

for x 2 + y2 _ R 2,

332

(365)

Two-dimensional groundwater f l o w

365.02 (SC,S)

YT

of n w e l l s regularly distributed over the circle with radius r0, located symmetrically ins i d e a c i r c u l a r i s l a n d with radius R. Confined aquifer with zero drawdown at the boundary. Discharge of the wells - Q. n Circular row

:~888

~o=0

---~Q l n { ( R ) 2rr D n

Zn

r0

o

n

--

365.03 (S) with discharge Q in a r e c t a n g u l a r strip with o p e n w a t e r boundaries at all sides. Single well

2b Yo ------]....... --99=0

1

S t e a d y state, q9 - qg(x, y ) .

~a I I

~p(0, y) -- ~o(2a, y) -- 0,

I [

I :::::::::::::::::::::::::::::::::::

Xo

2a x

~o(x, 0) -- ~p(x, 2b) - 0.

Well with discharge Q at (xo, yo).

•

12--

ln[

2zrD n-'--O0

sinh{ ~(z-~O+4na)} sinh { zr(z-zo+4na)4b

2zrD Z In R=--(X)

} sinh{ :rr(z+zo+4na)4b } ] sinh{ 7r(z+zo+4na)4b } '

z -- x + i y , zo -- xo + i yo, zo -- xo - i yo,

= solution deduced from 356.12 (semi-infinite strip) by superposition.

Flow in a finite strip

(365)

333

365.04 (FS) As 356.03 with the well at the centre of the strip, which is taken as the origin. q) = q)(x, y) = drawdown.

Yt ...........................

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

e..-..O.

.........

..............................~ ............................... ~o=0 Q a x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

099 Ox

~(0,

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

y) = 0

for y :/= O,

q)(a, y) = O,

Oqg(x, 0) - 0 Oy

for x -¢ 0,

q)(x, b) - 0.

Well with discharge Q [L3T -1] at (0, 0).

O

S2(z) -

7/'O

~ ~ m--l,3,..,

1 sinh{ ~-~a(b + iz)} m cosh(-~-ab )

z=x+iy.

365.05 (S)

--L

d X

Single drain in a confined aquifer between two parallel open boundaries. 99 = rp(x, z) = drawdown. 99(0, z) = rp(2b, z) = 0,

z

O~o(x 0) - O~o o~ ' ~(x,

2b

D) - 0.

Drain with discharge q [L2T -1] at (x0, z0). Solution to obtain from 356.06 or 356.12 by means of superposition.

365.o6 (FS) As 365.05 with the drain midway between the open boundaries on the bottom of the aquifer. ~o = qg(x, z) = drawdown.

J

0q) ~(0,

b

b

z)=0

Ox O99 ~(x,0)=0 Oz

forz¢O,

~(b,z) =0,

forx-¢0,

0q9 ~z(X, D) - - 0 .

334

(365)

Two-dimensional groundwater flow

Drain with discharge q [L2T -1] at (0, 0). 2q x?(~-)--Jr

~ ~ m--l,3 ....

1 cosh{ ~-~(D + i()} m

( --x +iz.

sink (~O-b °)

365.07 (FS)

Infinite row of wells of equal strengths and at equal mutual distances along a straight line midway between two parallel open boundaries and fully penetrating a confined aquifer, q9 = q)(x, y) = drawdown. b 888888~~

x

a

~a

a

0q9 (x, 0) -- 0 0y

for x :fi 0,

~oa__:_,(x ' b) - o.

Oy

S2(z) -

Q yrD

oo ~ m--1,3 ....

1 cosh{ ~i-a(b -+- iz)} -m sinh( m~rb ~ )

(S)

Single drain in a leaky aquifer between two parallel open boundaries. ~o - (p(x, z) drawdown.

V//////////////////////////~ ¢

i . qlio "

~o(a, y) -- O,

z--x +iy.

~Z

Z0

for y :/: 0,

Wells with discharge Q [L3T -1] at (0, +2nb).

•

365.08

~o0.__:_(0, _, y) -- 0 Ox

2b

~o(0, z) - ~o(2b, z) - 0, .

x

0~o (x , 0) - 0,

Drain with discharge q [L2T -1] at (x0, z0). Solution to obtain from 356.08 by superposition.

0q9 oz

-z-(x,

D ) --

-

~o(x, D) ~ Kc

Flow in a finite strip

(365)

335

365.09 (FCR)

¢'////////////~///////////~

As 356.08 with the drain midway between the open boundaries on the bottom of the aquifer. ~p = ~o(x, z) = drawdown. Drain with discharge q at (0, 0).

c

q b

b

° ! ( o , z) - o

for z # O,

~o(b, z) = 0 ,

- - ( x , o) = o

for x # 0,

° ! ( x , D) - _ ~o(x, D)

Ox 099 Oz

Oz

Kc

sinh{ ~~"( b - ~)}

(=x+iz,

I2(() -- q Zn:O Otn cosh(.q_~) (1 + ~2+E2) 'E D with Otn being the roots of o~tan t~ - e - K---~. "

365.10 (FCR)

L ~

Single drain on the bottom of a confined aquifer midway between two parallel open boundaries with entrance resistances, q9 = qg(x, z) = drawdown. Drain with discharge q at (0, 0).

q)=0

•

x

b

b

~o(b, z)

Oq) --(0, z)=0 Ox

forz#0,

Oq) --(b,z)=-~ Ox

O~p(x, 0) -- 0 Oz

for x # 0,

8__~(x, D) - 0. az

Kw

OO

S2(() -- q Z

cosh{ ~ ( D + i()} E

n=O o t n s i n h ( - ~ ) ( 1 + ~2+~2 ) b

with ctn being the roots of ot tan o~ - Kw"

(=x+iz,

336

(365)

Two-dimensional groundwater f l o w

365.11 (FCR) Single drain on the b o t t o m of a leaky aquifer m i d w a y between two p a r a l l e l open boundaries

Z///////////~U//////////~C 2 g

with e n t r a n c e resistances. ~o -- 9 ( x , z ) = drawdown. Drain with discharge q at (0, 0).

D~ ~q

b

b

o(b, z)

Ox(0'z)-0

forz#0,

Ox (b, z) -

~(x,0) 0z

for x :riO,

~(x,D) Oz

=0

O~o

= -

K------~

~o(x, D) ~

Kc

Drain with discharge q [L2T -1] at (0, 0).

oo a(()

D sinh{ o,,, (b - () } + K w cosh{ ~"

-- q Z (1 + n=O

e ) { D c o s h ( - ~ ) + Kwo6~ s i n h ( - ~ ) }

ot2+e2

D with an being the roots of at tan ot - e - V-de or

oo ~O/n s i n h { ~ ( D + i()} + K c c o s h { ~ ( D "Q(() = q Z E ){bcosh(O~D n=o (1 + %2+e2 ~)

+ i()}

+ Kcoen sinh(~_~_D)}

with at,, being the roots of ot tan ~ - e - K-~7, ( -- x + i z.

(370)

Combinations of foregoing cases

337

370 BIII-7. COMBINATIONS IN GENERAL TWO-DIMENSIONAL GROUNDWATER FLOW

See comments of 302 d. The solutions of the problems with discontinuous groundparameters are obtained by solving separately the parts of the hydrological system for which the ground parameters are continuous, each with arbitrary constants and then determining these constants by applying the law of continuity at the location of the abrupt change. There are numerous examples of problems with discontinuous parameters; here only some solutions of practical problems will be given. 370.01 (IC) Fully penetrating pumping well in a leaky aquifer with different transmissivities of the aquifer and different resistances of the semi-permeable layer at both sides of a straight line. Pumping well at P(a, 0). @ = @(x, y) = drawdown.

K2D2, C2 p/~ Q a

K1 D1, Cl

T J

_1

I I

X ~ O,

Q

I I I

02(/91

=~

OX 2 c2

( t 9 - (t91(X, y)"

02(/91 Oy 2

991 = O, )~

@l ( - - 0 0 , y) -- O,

K2D2

0(t91 --(X,

Oy

(/91(X, 00) -- O. x />0,

02(/92 OX 2

@ -- @2(x, y)"

+

02(/92 Oy2

(/92

~2

= O,

lim (r 0(/92 r~0

)2 _ K2D2c2,

Q

- - ~ r ) -- -- 27t'K2D2

(/92(00, y) -- 0,

for r - v/(x - a ) 2 -+- y2,

@2(X, OO) -- 0,

0(/92 --(X,

Oy

0) -- 0

for x --/: a.

Continuity conditions: (/91(0, y) -- (/92(0, y),

'

os0

-- --zr

0(/91

0(/92

I(1D~-~x (O, y) - I(2D2-STx (O, y).

A(0/) exp x

0/2 -~- )2

)2 _ K1DlCl, O) -- O,

338

(370)

Two-dimensional groundwater flow

[1

1

Q D2 Ko{ ~2 V/(x _ a ) 2 + y 2 ] _ Ko{ ~_72V/(X + a ) 2 + y2 ~P2(x, y) - 2st K2

+Oio --st

A(ot) exp

exp (-- a ? 2 A(ot)

- x

/]

2 + )~" cos(yot) dot with

-+- ~ )

--

K1Dl?2 --I----lk~l--}l K2D2v/ot2 --t---.~2 370.02 (IC)

~Q

S p e c i a l case of 370.01 with K1 D1 -- K2 D2 and a - O. ) 2 _ K D c l , )~2 _ K D c 2 .

-- K D

¢1///////////////" ~/;/////////////~/C2 KD

KD

~01(X, y) -y) --

go2(x,

B(ot)

Q

L ~ B(ot) exp ( ~x o t 2 + ~--x ~1) cos(yot) dot,

Q

So xp(-- X

zc K D

2 __[_,,~-'g"

B(ot)

rc K D

with

--

72-t- ~12 -I- /~2-~- ~22 Special result for x - 0 "

~oy)_ '

~

'/'

~(c~- ~)y 1

-

~2K1

(y) l (y)} 722 - 7, K1 Z

-

Q/o ~

rc K D

B(ot) cos(yot) dot.

370.03 (IC) As 370.01 but for a

f

c2 -

\\

Q

KID

KzD

confined

c~. Q u a s i - s t e a d y

state.

aquifer, cl -

Combinations of foregoing cases

(370)

Q *

(/91 (x, y) -- c -

*

(/92(X , y) -- C--

ln{(a-x)2+y2},

2Jr(K1D -k- K2D)

Q

l n { ( x + a ) e + y e}

2Jr(K1D + K2D)

Q 4jr K2 D2

+

339

In { (x -k-a) 2 nt- y2 (x - a) 2 -I- y2 }"

qgl(X, y) represents radial flow towards P(a, 0) with a discharge Q and average transmissivity. Total discharge descended from the K1 D-area: Q1 -

K1

~ Q . K1 + K2

370.04 (FC) Fully penetrating pumping well in a confined aquifer with different transmissivities inside and outside a circular cylinder. Quasi-steady state, q9 -qg(x, y) = drawdown. qg(-oo, y) -- qg(oo, y) -- 0,

K2D

~o(x, c~) - O,

2D l

Oq9 . r~O Or lim r .

•

.

Q 2JrKzD .

Oq9 Oy

for r - v/(x - l) 2 + y2.

K*Q

lnl-

Q

~(x,

2Jr(KiD -t- KzD)

0) - 0,

ln{(x-/)Z+y2}

qgl (x, y) - c -

2JrKzD

q92(x, y) - c -

Q In {(x - l)2 + y2} + 4JrK2D 4JrK2D

K*Q • with K * .__

K1 - K2 K1 + K2

The flow q)l (x, y) inside the cylinder represents radial flow towards P(l, 0) with a discharge Q and average transmissivity. Total discharge through the cylinder:

2

K1

(

R

)

Qc - - ~ Q arctan . . Jr K1 d- K2 ~/12 R2

(370)

340

Two-dimensional

groundwater flow

370.05 (SV) Uniform flow in a confined aquifer with different transmissivities inside and outside a circular cylinder. Steady state, q9 = ~o(r, 0) - drawdown.

f L

A_ /I

I I I/

10(p

02(t9

\/Ri\jy

Or 2 -+- _ r

-~r

1 02 -Jr- _ _ q9 _ O, r 2 002 --

~o=0 q

~p r, -~

qg( oo, O) - ~K2rD cos 0,

--0,

2q ~o(r, O) -

~(r,

O0

r cos 0

K1D + K2D

q{ r

r

O) - - 7 7 ( r , O(4

y r ) - - O.

for 0 ~< r <~ R,

2 K1 + K2

1 )}cos0 K2

forr~>R.

The flow inside the cylinder is uniform and parallel to the uniform flow outside the cylinder. 4qK 1R Total discharge through the cylinder: Qc - /qT-~"

370.06 (J)

b

q

v

x

Uniform flow in a confined aquifer with different transmissivities inside and outside a cylinder of elliptical shape, the direction of the flow making an angle o~7c with the positive xaxis. Steady state. (p = (p(x, y) = drawdown.

KzD

Inside the ellipse" ~0 - rpl (x, y), ~Pl(0, 0) - 0. Outside the ellipse: (p - - (fl2(x, y), q92(-{'-(X), y)

- - q92(x , - 4 - 0 0 ) - -

q {x cos(an') + y sin(otzr)}. K2D

q { cos(otn') sin(otzr) } S21 (z) - -~k(a + b) -i z, a + kb b + ka

q[

I

}1 1 522(Z) -- -~ ze -i°~rr + ab(a + b)(1 - k) cos(oln') + isin(°~rr) ~ a+kb b+ka m

Combinations of foregoing cases

(370)

341

K1 with w -- z 4- V/z 2 - (a 2 -- b2), k -- -XS2' z -- x + iy, S'21 -- K1991 -t- i~rl, ff22 -K2q92 + i7r2. The flow inside the elliptical cylinder is uniform, making an angle flrr with the positive x-axis, while

a+kb

tan(flzr) --

tan(otzr).

b+ka

Total discharge through the cylinder"

2qk(a Qc = (a + kb)(b+ +b)ka) V/ a2(a + kb)2 sin2(°tzr) + b2(b + ka)2 COS2(0/7r)"

370.07 (FCR) Groundwater flow between two polders with different levels and different aquifer properties, q) - q)(x, z) - drawdown. Cl z

C2

(/9 -- q91(X, y)"

x ~ 0, 02991 OX 2

02(/91 -- 0, OZ 2

-~-

X

Oz

o) -

0(t91

x >/0,

~(x,D)Oz

q) -- q92(x, y)"

02(/92

02992

OX 2

OZ2

0(/92 ~(x, Oz

0) -- 0,

= 0,

qgZ((X),Z) -- h, 0(/92 ~(x, Oz

D) --

h - q92(x , D) K2c2

Continuity conditions" (/91 (0, z) -- (/92(0, z),

0(/91 0@2 K1--~-x (0, x) -- Ka--~x (0, x).

Also fo D (t91(0, Z) dz - fo

(/92(0, Z) dz

and

~o D 3~pl

-g2x(o , z) dz - Ka

D 0~02

L

--~-x (0, z) dz,

(/91(--OO, Z) -- 0,

O,

-

(t91(x , D)

~ KlCl

342

(370)

Two-dimensional groundwater flow

leading to approximate solutions: c~ ~ol (x, z) "~ 2hK2 Z

n--0

Otn sin 2 fin E3[3n(otnK 1 -!- flnK2) sinc~n O~3

~02(X, Z) "~ h - 2hK1

n--0

370.08

' nZ] cos (-o-,,

otn sin fin exp(-fl"X'~c°s(-~)D/ E3fln(otnK1 q- flnK2)

with Otn being the roots of c~ tan ot - e and E ~ -

exp

-

D and fin the roots offltanfl Klcl

-- ~ =

1 + ~+~2" (FS)

D1

K1D1

D2

KED2

z

Constant precipitation on an infinite dam between two parallel open boundaries, the confined aquifer consisting of two horizontal layers with different transmissivities, q9 = q9(x, z) - head.

2b

0 ~< z ~< D1, 02q91 OX2

I

99 - -

991 (X, Z)"

02(/91 =0, OZ2

0991 OZ

~(x,

0) -

D1 ~< z ~< (D1 q- D2),

q9 -- @2(X, Z)"

02q92 OX2

~(x,

02(/92 =0, OZ2

0992 Oz

991(0, Z) -- 991(2b, z) -- 0.

K1

992(0, Z) -- q92(2b, z) = 0.

D1 if- D2) -- 0,

Continuity conditions: 0991 (X D1)

q91(x, D1) -- q92(x, D1), 8pb *

@1 (X, Z) --

K1 - ~ z

sin [mrrx

o0

~2

o0 Z m=l,3 ....

8pb

Km

0992

-- K2 ~

(x, D1).

[mrtz

m= 1,3,... m EKm cosh 2 ( mTr2bDl) 8pb

~o2(x, z ) -

'

\ 2 b ) cosh,-%-)

~

q- 7r 2 K1

•

D K2c2

sin (mrrx mrr (D1 - Z)} k 2b ) sinh { Tb-

m 2 cosh [ m~r D 1 ]

\ 2b 1

oo

sin (myrx ~ cosh{ mzr (Ol -'1"-0 2 -- Z)} 2b / ~yr2 ~ m 2 Km cosh(m2 DI ) cosh(myrDT~ ) re=l,3 .... 2b ,

rc D ( mrr D2 2b" ) + K z t a n h \ 2 b )" 2b ....

- - K1 t a n h ( m r r u 1

\

with

Orientation table BIV

(400a-c)

343

BIV. THREE-DIMENSIONAL SPHERICAL SYMMETRY ORIENTATION TABLE BIV

400. Three-dimensional spherical-symmetric groundwater flow

a. Does the flow field theoretically extend to infinity in p-direction ?

go to 410 (BIV-1)

The radius p varies from zero to infinity or from almost zero to infinity in case of a point well or a point source with small radius. 0~
or

pw~
with Pw = well radius.

b. There is only groundwater flow outside a sphere

\

\ \

y

go to 420 (BIV-2)

There is no flow in the region 0 < r < P0 and there will be some boundary value for p --p0 with respect to flow outside the sphere. The region inside the sphere will contain free water (without soil).

po <<.p < cx~.

e. There is only groundwater flow inside a sphere There are no practical examples in groundwater flow for this category, in contrast, for instance, with conduction of heat in solids. Therefore we shall restrict ourselves to the problems belonging to a and b.

344

(410a)

Three-dimensional spherical symmetry

410 BIV-1. THREE-DIMENSIONAL SPHERICAL-SYMMETRIC GROUNDWATERFLOW IN AN INFINITE FIELD Most problems in this category relate to groundwater abstraction from a point well with a very small diameter. Also spherical initial value problems (see comments of 212a) belong to this category; though they have no practical meaning for groundwater flow, they are of value for the spread of a spherical pollution by diffusion. Subdivision: a. Point well in an infinite field, b. Spherical depletion functions. a. Point well in an infinite field 02q9

~-~Op2

2 Oq9 __ f120@ p Op 0-7'

/~2 __ as

410.01 (L) I

Instantaneous abstraction Pi [L3] of groundwater from a point well (discharge impulse). ~o - ~o(p, t) - drawdown.

Pi

s,,K

qg(p,O)--O

for p > O,

qg(cx~,t)-O,

P

lim (p20~o) P~

--

~P

~o(p, t) --

{

Qi 4rr K

0 fl Pi 8Kzrt~

fort-O,

Pi -- Qi dt.

for t > 0, exp

f12p2) 4t

410.02 (MI) Q(t)

Continuous abstraction of water from a point well according to an arbitrary function of t. q9 - qg(p, t) - drawdown. p

lim (p2 ~--~) -

p--+0

~o(p, O) - 0 Q(t) 4zc K '

for p > O,

a(o)

-

o.

~o(~, t) - O,

(410a)

Flow in an infinite field

q)(P, t) --

410.03

-z~t~ ~/Jr p

Q t 2,//

f12p2 ,~e_k2 dk 4)~2 }

(L)

As 410.02 but with constant discharge: Qo.

Q(t)

Qo

•

410.04

345

Qo

~p(p, t) - 47r K p

Q(t) --

erfc( tip

(D)

Steady state of 410.03. •

~o(p) =

410.05

Qo 47r K p

(L)

As 410.02 but with a linear function of the time for the discharge: Q(t) -- at, a [L3T-2]. (p(p, t) -- ~ Jr K p

iZcrfc 2~/7

t 410.06


As 410.02 but with the discharge a parabolic function of the time: Q(t) - - a q r [ , a [L3T-3/2].

Q(t)

~o(p, t) - 4 ~ / ~ K p ierfc

410.07

(L)

As 410.02 but the discharge according to the function: Q(t) = at n/2 (n = 0, 1 , 2 , . . . ) .

•

,;(;,t)-

4~-Kl,

Special cases: 410.02, 410.05, 410.06.

Y~

346

(410a)

410.08

Three-dimensionalspherical symmetry

(L)

As 410.02 but with an exponentially increasing discharge: Q(t) - a(e b t - 1) (a and b positive).

Q(t)

f I

-a

4JrKpa { e b t p ( l p ~ / ~ , ~ c - ~ ) _ e r f c ( 2 ~ t ) } "

~o(p, t ) -

410.09

(L)

As 410.02 but with an exponentially decreasing discharge: Q ( t ) - Qo(1 - e -bt) (Qo and b positive).

Q(t) Qo

/ I

/ /

•

~o(p, t) --

410.10

Qo

flP )-e-btp(-~iflp~/-b,i

47r Kp

.

(D)

Steady state of 410.09. •

~o(p) -

410.11

Qo

4rc Kp

(PF)

As 410.02 but with a periodic discharge: Q(t) 27r Qo sin(wt), w - -7-"

Q(t)

......

IOo I

•

~o(p, t) - ~ Qo e

4rc Kp

t

co'-fi~°V/ ~.--

-~o'psin(wt - w' p ) ,

Flow in an infinite field

(410b)

347

b. Depletion functions for spherical diffusion

02C Op2

2 Oc p Op

. m

~

~

1 Oc Dd Ot ~

c -- concentration [ML-3], Dd --coefficient of diffusion [L2T-1].

410.21 (MI)

Depletion function for a solute concentration in an infinite aquifer, starting from an arbitrary spherical initial concentration c(p, O) = F(p).

F(p)

Oc ~ ( O , t ) --0, Op c(c~, t) = 0 .

,

c(p, t) -- 2p~/rCDdt

f0~ poF(po) [ exp / - ,~4Ddt~o~~

- - e x p { - ( p + p] 4° D ) 2d t }- dpo.

410.22 (MI) As 410.21 with F(p) -- co

c(p, O) -CoI F(p)

1

p+R~

c,:,o, ~- ~. co erf(2,~

co~ +

V~

0

OC

~(O, t) --0, Op

1

p-R

~ - ~- coerf(2,~ 4Ddt

c(cx~, t) --0.

)

1 [~xp/_ ~,o+ R~2 p

f o r 0 ~ < p < R, for p > R.

}--exp{--

~,o_ R~2j]. 4Oat

348

Three-dimensional spherical symmetry

(420a)

420 BIV-2.

THREE-DIMENSIONAL SPHERICAL-SYMMETRIC GROUNDWATER FLOW OUTSIDE A SPHERE

0299 Op2

t

2 099 p Op

/~2 099 --

-~'

f12

Ss ___

K

All problems in this category are related to groundwater flow towards an abstraction well with a spherical well-screen of radius Po. Subdivision: a. Given head or drawdown at p - Po, b. Given flux at p - Po, c. A combination of given head and flux at p - Po.

a. Given head or drawdown at p - Po

420.01 (L)

F(t) I . . . .

Spherical well-screen in an infinite aquifer. The water level inside the screen sinks according to an arbitrary function of time. Open boundary at p = p0. ¢P = ¢p(p, t) = drawdown.

'ft ~. . . .

I

¢p(p, O) = 0,

'\

/

¢P(Po, t) = F(t),

/ P

2 pop f~ ~ /7~

go(p, t) --

(p-po)

F[t--

~p(oo, t) = 0,

F(O) -- O.

/~2(p __ p0)2 4)~2 ]e-X2 d)~.

247

420.02 (L) As 420.01 with constant drawdown. F(t) -- h.

'F(t)

¢p(p, t) - po___h_herfc[/3(p2v/7p Po) }.

420.03 (D) Steady state of 420.02. ¢p(p) = poh P

(420a)

Flow outside a sphere

349

420.04 (L) As 420.01 with linear drawdown. F ( t ) -- at, a [ L T - 1 ] . •

~o(p, t ) -

4at P°izerfc[ ~(p - P0)]

p

/

2v/7

l"

420.05 (L) As 420.01 with a p a r a b o l i c drawdown. F ( t ) -- av/-i,

~F(t)

a [LT-1/2].

~p(p,t) --a~/-~P° ierfc{ fl(p - p°) ] p

2,,/7

"

420.06 (L) As 420.01 with a drawdown according to the function. F ( t ) - at n/e (n -- 0, 1 . . . . ). •

~p(p, t) -- a F

1 + -~ (4t) n / 2 m inerfc

p

2v/7

/

Special cases: 420.02, 420.04 and 420.05.

420.07 (L) As 420.01 with exponentially increasing drawdown of the water level of the well-screen. F ( t ) - a ( e b t - 1) (a and b positive).

t

I -a

P

~/3(p - po)~/-b, 4Cb-t r - a -

p

erfc

247

420.08 (L) F(t)

/

J

I

As 420.01 with exponentially decreasing drawdown of the well-screen level. F ( t ) -- h(1 - e -bt) (h and b positive).

(420a, b)

350

Three-dimensional spherical symmetry

[erfc{ f l ( P2~/~ -Po) 1 ~o(p,t) = hpo p ]-e-btPl-2i~(p-P°)~/b'i~/~}

•

420.09

]

"

(D)

Steady state of 420.08. •

~o(p) =

hpo P

420.10

(PF)

As 420.01 with a periodic drawdown of the wellscreen level. F(t) -- h sin(cot).

F(t)

¢p(p , t) - - ~ epoh -oo'(p-po) sin {cot - co' ( P - P o ) }

P

t

with c o -

2rr T

and w'

- flv/~ •

b. Given flux at p = Po 420.21

(L)

Continuous abstraction of groundwater according to an arbitrary function of t from a spherical well-screen at p = Po.

:~ Q(t)

~o = ~(p, t) = drawdown. _

[

'

~o(p,O)=O

-

for P > P o ,

~o(c~,t)=O,

Oq9 Q(t) Op (Po, t) - -~'47rKp~ \

/ .._...

,

Q¢,

qg(p,t) =

exp,

(p

1R(P-P-------~° ~/~)] dto.

/~Po 420.22

2po ' ,,6po

(L)

As 420.21 with constant discharge Q(t) = Qo. ,

~o(p,t)

=

Po ( . - R - ~p 2po ' t~po) "

Qo [erfc{/3(p-p°) /

4zr Kp

2~/7

/

(420b, c)

Flow outside a sphere

351

420.23 (D) Steady state of 420.22. •

~(p)

=

Qo 4yr K p

(p >/po).

420.24 (PF)

As 420.21 with a periodic discharge. 27r Qo sin(cot), c o - --f-.

Q(t)

•

~(p, t ) -

Q(t) --

Qo Im[ e-H(P-P°)4'-~ eiWt]" K' 47r -------~ 1 -!-flpo4'Tw

t

e. Combination of given head and flux at p = po

420.31 (L) Spherical well-screen with entrance (outflow) resistance in an infinite field. The open water level inside the screen sinks according to an arbitrary function of .

.

.

.

.

time. q9 = ~o(p, t) = drawdown.

. 1

~o(p, 0) = 0 ,

w

qg(cx:), t) --0,

- K o¢p F(t) - qg(po, t) Op (Po, t) -w ,

Po q9(p, t) -- fl K w p

fo

F(t - to)

[

~

1

exp

/

-

- ~/-O R { fl ( p - 2p ° ) 4/-~ , ~ ~ o } ] dto

F(O) -- O.

~2(p _ po)2

4to

with

o _ (P° + K w ) 2 flpoKw [T-l]"

420.32 (L) 'F(t)

As 420.31 with constant drawdown. F(t) = h. .

qg(p, t ) -

hp2 [erfc{ fl(P - Po) } p(po + K w ) 24/[

,~}]

- R { ~ ( p - p°)~-~ 2

'

"

352

(420c)

Three-dimensional spherical symmetry

420.33 (D)

Steady state of 420.32. •

~o(p)

-

P(Po + Kw)

420.34

Spherical well-screen with open water storage in the bore hole with radius rw in an infinite aquifer. Sudden lowering of the open water level in the bore hole with an amount h. q9(p, t) = drawdown.

hi I w

.

_

-a t-

.

.

.

.

~- ~ 2 r w

qg(p,O)\

/

h 0

forp-p0, forp>p0,

rc r 2 --~(Po, 0 cp Oq9(PO, t). t) -- 4zrKp~-~r

¢p(p, t) - hP°Im[zR { 1 f l ( p - p o ) z , zw/t} ]

yp

z - - x d-iy,

¢p(oo, t ) - O .

~

1

1/4c

x--~/3c, y-~w~o

_

with

fl2c2 , C - - 4Kp 2

Orientation table BV

(500a-c)

BV. T H R E E - D I M E N S I O N A L

353

AXIAL SYMMETRY

ORIENTATIONTABLEBV 500. Three-dimensional axial-symmetric groundwater flow a. Does the flow field theoretically extend to infinity in the direction of the r-axis ?

go to 501

The great majority of axial-symmetric flow fields have a vertical axis of symmetry, in the sequel chosen as z-axis, while the r-axis will represent the horizontal direction, with

r = v/x2-k- y 2. If r varies from zero to infinity, we may speak of flow in a horizontally

infinite field: 0~
b. There is only groundwater flow inside a circular cylinder

go to 540 (BV-4)

There is no flow outside the cylinder with radius R and there will be some boundary value for r -- R with respect to the flow inside the cylinder. This flow is referred to as flow in a circular island:

O~r<~R.

e. There is only groundwater flow outside a circular cylinder

r

There is no flow in the region 0 < r < R and there will be some boundary value for r -- R with respect to the flow outside the cylinder. In most cases the region inside the cylinder will contain surface water; for that reason the flow is called flow to a circular pond or basin:

R<.r
go tO 550 (BV-5)

354

(500d, 501 a, b)

Three-dimensional axial symmetry

d. The flow can be considered as a combination of some of the foregoing cases a, b and c

go to 560 (BV-6)

The following two kinds of problems belong to this category. (a) There is a discontinuity in one or more of the ground parameters at r -- R. (b) A part of the flow takes place inside and/or outside a circular cylinder, for instance, over the distance 0 ~< z ~< D, whereas for the other part of the flow (z > 0) r varies from zero to infinity.

R

D

Kz

K1

5 0 1 . I n f i n i t e field 0 ~< r < c ~

a. The flow field also extends to infinity in the direction of the axis of symmetry: infinite field

go to 510 (BV- 1)

There is no plane of symmetry perpendicular to the axis of revolution and no boundary at a finite value of this axis. O~


or x

--CX:) < x < Cx:),

0 < ~ r
b' The flow field is semi-infinite

go to 520 (BV-2)

The plane z - 0 (or x -- 0) is a plane of symmetry, or acts as a boundary, outside which no flow takes place.

r

O~
O~
I

0~
0~<x
Flow in an infinite field

(501 c, d)

355

c. The flow field has finite dimensions in the direction of the axis of revolution

r

D

go to 530 (BV-3)

All aquifers with a finite thickness which extend horizontally to infinity belong to this category, provided that axial symmetry exists

rZ

0~
d. Combinations of some of the cases a, b, c

K1 K2

z K3

O<<.z<~D

go to 560 (BV-6)

This category includes discrete inhomogeneities in the direction of the axis of symmetry.

356

(510, 511 a)

Three-dimensionalaxial symmetry

5 1 0 B V - 1 . THREE-DIMENSIONAL AXIAL-SYMMETRIC GROUNDWATER FLOW IN AN INFINITE FIELD

a. Has the soil discrete inhomogeneities in r- and~or z-direction ? go to 560 (BV-6) An example of discrete inhomogeneities in an infinite field is a cylindrical salt dome, surrounded by horizontal layers with different permeabilities.

b. The soil is assumed to be homogeneous

go to 511

Also anisotropic homogeneous soils belong to this category, provided that the main directions of the anisotropy coincide with the r- and z-direction. Solutions for anisotropic cases can easily be derived from analytic solutions for isotropic problems.

511. Homogeneous soils a. Is uniform flow involved in your problem?

go to 513

A point well in uniform flow, for instance, yields an axialsymmetric flow pattern and so also do finite and infinite line wells in the direction of the uniform flow. The solutions ~--- hydr. screen for all these cases can be written down immediately as the superposition of the solution for the uniform flow and those of the separate cases. So they have as such not a "real" axialsymmetric character. More important however is the analytical solution for the "hydrological screen" that is formed by various means of abstraction in uniform flow. The hydrological screen in axial-symmetric flow is a body of revolution that separates the original uniform flow from the injected or abstracted groundwater. Uniform flow is mainly horizontal. For that reason we choose the x-axis as the axis of revolution, while the r-axis will be the vertical axis.

Flow in an infinite field

(511 b, 512)

357 go to 512

b. No uniform flow

Non-symmetric configurations of several kinds of injection wells and abstraction wells belong to this Q category, provided that the axial-symmetric character of the flow is maintained. In general, the solu' =~ Q2 tions will be the superposition of simpler ones. For . that reason only a few examples will be given. An example is a finite line well and a point well along z and on the axis of revolution. If the wells and/or well-screens have opposite signs and are of unequal strength, a hydrological screen will be present, that represents a body of revolution inside which only groundwater will flow to or from the dischargeor recharge, well with the smallest strength. 1~]

~

|

512. Wells and well-screens without uniform flow 02(/9

1 099

02~ +

-0 '

02~° F--1 O~o + 029 - 0 Or--T- r-~r ~ • 512.01 (S)

Point source and point sink of unequal strength in an infinite field. Steady state. q9 = qg(x,r) = drawdown. Point source with discharge Q1 in (l, 0). Point sink with recharge Q2 in ( - l , 0). Q1 and Q2 positive numbers, Q1 < Q2.

rT " P2 7/ Q2

x~ / Q1

1(o, 02),

~o(x, r) -- 4JrK

•

x

¢(x,r)

-

-

Pl

Q2(x + l) 2p2

pl - 4 ( x - 1 ) 2 W r 2, P2 Q1 (x - l) P2-- 4 ( x + / ) 2 + r 2pl

Stagnation point S: XS =

~ / ~ + ~/Q1

rs - 0 .

4-0S-

Hydrological screen:

o1(1 x_,) -o2( Pl

x+l). P2

2.

35 8

(512, 513)

Three-dimensional axial symmetry

512.02 {S) Point source and finite line sink of unequal strength in an infinite field. Steady state. 9 = ~o(r, z) = drawdown.

Q2 21

Point source with discharge Q1 in (0, a). Line sink along the z-axis with recharge Q2 from z - - l to z - l. Q1 and Q2 positive numbers, Q1 < Q2.

Q1 1

~o(r, z) =

Q1

4zr K v / r 2 + (z - a) 2 8rc K l

~(r,z) - -

{ arcsin/Z

+ l ) r - a r c s i n h ( Z - /' ) }r

Ql(z -a)

02 + -~{ ~ r 2 + (z + / ) 2 - v/r 2 + (z - / ) 2 }.

2x/r 2 + ( z - a) 2

Stagnation point S" Zs from

Q1 (Zs -- a ) 2

=

Q2

rs - O.

Z 2 -- 12,

Hydrological screen: ap(r, z) --

Q2 -

Q1

513. Various means of abstraction or injection in uniform flow 029

~__10q9 + 829

"---7 Or - r -~r

Ox 2

-0.

513.01 (S) Point well in uniform f l o w in an infinite field. Steady state. ¢p -

~q P

q9(x, r) - drawdown. O!(x,O)--O Or

for x -7/=0,

~p(-cx~, r) -- 9 ( c ~ , r) qx =

-

(513)

Flow in an infinite field

Q

l i m [ p 20g° ] ;---,0 -~p (X, r) --

qx ~o(x , r) -- --;-;- +

4zrK

-

with p -

X//X2 nt- r 2.

Q

4 ~ K ~/x 2 -Jr-r 2

Qx

[~(x, r) I

359

2~/x2

q-

r2

Q

+ 7rqr 2

Stagnation point S"

Xs - -

~Q

4zrq ,

rs - - O.

Hydrological screen in polar coordinates"

Q

/9 2 - -

2Jrq(1 + c o s O )

with p -- v/x 2 -t- r 2 and 0 - arctan

(r)

.

513.02 (S) Finite drain with discharge Q in uniform f l o w with strength q parallel to the drain. Steady state. q9 = q) (x, r) = drawdown.

--,q

~p(-oo, r) -- q)(oo, r)

C

x

Y

Q r 099

-g-r]r-,O

-

0

qx = ~o(x, o o ) -

Q 47r K l

forx <-l for - I

<x

andx >l, <1.

, _ ~Q {arcsinh(X+l)_arcsinh(X-/)}_k___qx ~o(x r) 87r K l r r

[~(x,r)l-

~Q- { v / r 2 + (x + l )2 - v/r 2 + ( x - / ) 2 } + n"q r 2

Hydrological screen" 7: -- 0. Stagnation point:

Xs --

K '

~/ 2 + 47r----q' Q

rs m 0°

Q 2

K'

360

Three-dimensional axial symmetry

(513)

513.03 (S)

~0

r '[R~

Circular disk with discharge Q in uniform flow with strength q, perpendicular to the disk. Steady state. 99 -- ~p(x, r) = drawdown.

~'q

~p(-c~, r) -- ~p(c~, r)

y

qx

x

= ~(x,

0--~(0 r ) Ox '

for0
I -K q -C- 2Jr R2 K


Oq9 Or

--(x,

q

f o r r > R,

K

o) -

c¢) -

-=,

o.

Q o0 1 J1 (Rot)Jo(rot)e -Ixl~ dot + --~, qx qg(x , r) -- 2 rr K R fo -ot

x

- - e -Ixl~ dot - 7rqr 2 + S(x) ~ ( x , r) -- ~Qr fo ~ ot1 Jl (Rot)Jl ( r o t )Ixl Q 0

S(x) -

with

forx <0, forx >0.

Hydrological screen" ~r -- 0. Stagnation point: R(qscr- q) r S --

x S ~

O~

~/q (2qscr- q) with qscr --

q (screen)

--

Q 27rR 2 •

513.04 (S) ~0

rl

Circular drain with discharge in uniform flow with strength perpendicular to the plane the drain. Steady state, q9 qg(x, r) = drawdown.

~,q

s2 x

a qg(-cx~, r) -- qg(cxz, r) -- qg(x, c~) --

qx K'

--(x,

Or

o) - o.

Q q, of =

(513)

Orientation table B V

•

, ~p(x r )

•

1 Qr ~ ( x , r ) -- -~

J o ( R o e ) J o ( r o e ) e -Ixl~ doe + qKx 4reQ K f o °°

Q S(x) -

361

0

f0 °°

J o ( R o e ) J : ( r o e ) ~xxl e -

Ixloedoe - 7rqr 2 -t- S ( x )

forx <0, forx >0.

Q > : 3 ~/~ then Stagnation points" If 4,~R2q x

Xs from

(X2 + R2)3/2 =

47rq

Q ,

rs -- O.

with

362

(520, 521)

520 BV-2.

THREE-DIMENSIONAL SEMI-INFINITE

Three-dimensional axial symmetry

AXIAL-SYMMETRIC

GROUNDWATER

FLOW IN A

FIELD

a. Has the soil discrete inhomogeneities in r- and~or z-direction ? go to 560 (BV-6) Inhomogeneity in z-direction: r

K1D1

K2D2 z K3D3

b. The soil is assumed to be homogeneous

go to 521

Also anisotropic homogeneous soils belong to this category, provided that the main directions of the anisotropy coincide with the r- and z-direction. Solutions for anisotropic cases can easily be derived from analytical solutions for isotropic problems.

521. Homogeneous soils a°

There are discontinuous boundary values along the z-axis and continuous boundary values along the r-axis ~a

The z-axis is taken here, as

r

almost always, as the axis of

I i ............ r

r

go to 522

revolution. Point wells and

~a

line wells, located along and on the z-axis cause discontinuities in the boundary value, as to the flux.

bo Discontinuous boundary values along the r-axis and continuous ones along the z-axis

Q ......K........... ~

r

=

R Z

r

go to 523

Flow in a semi-infinite field

363

(521b-522a)

The z-axis is taken as symmetry axis. Discontinuities along the r-axis may be caused by circular disks, with given drawdown or flux or by circular drains with given discharge. go to 524

c. Uniform f l o w

----> q

Uniform flow is mainly horizontal. For that reason the x-axis has been chosen as the axis of symmetry. The configuration of injection wells and abstraction wells or other obstacles must be such, that also the plane x = 0 is a plane of symmetry.

x

-----~ q

r

522. Discontinuous boundary values along the z-axis

02q9

----T Or + _

1 099 _+_~0299 _ { f12~0q) r -~r

Oz 2

0

(non-steady state), (steady state).

Subdivision: a. Finite line wells. b. Point wells. a. Finite line wells

522.01 (L,IC)

==~P i

Instantaneous abstraction Pi [L3] of water from a finite line well on top of an infinitely thick confined aquifer. Discharge uniformly distributed along the well-screen.

r l

) -- { 2rcKl Qi lim ( r O._~rr r--+0 0

z Oqg(r, 0 t) -- 0 Oz .

q)(r,

Z, t)

.

q)(oo, z, t) -- 0 .

_ 8zrKltPi e x p ( _

.

for O <<.z <<.l and t -- O, for t > 0 and/or z >/l,

qg(r, c~, t) - O,

'~ +l) f12r____224t } [ eft|fl(z |

Pi -- Qi dt.

/ ere{ ' (z -

247

l) ]

}"

522.02 (MI) Continuous abstraction of groundwater from the line well of 522.01. The discharge is an arbitrary function o f t. ~o(r, z, t) - d r a w d o w n . ~ ( r , z, O) -

O,

Q(O) -

O.

364

(522a)

•

~o(r,z,t) --

Three-dimensional axial symmetry

f12r2 { 8zrKI

2r2 4t

-erf

4y ) erf( z +/~/~)r

Q t

(z I

e_y~. y

r

522.03 (L,IC)

~a

Continuous constant abstraction Q of water from a finite line well on top of a confined aquifer of infinite thickness. Discharge uniformly distributed along the well-screen.

8888 |

r

,

l

i

lim ( r a_~r~) -- { r-+O 0 ~oO_.(r :_., ' O, t) -- 0 Oz qg(r, M(u,

Z, t)

qg(c~, z, t) - - 0 ,

{ (f12r2

Q M 8rr K-----~

4t

2rrQKI

for0<~z~ l,

qg( r, cx~, t) -- O,

z-k-l)(f12r2 -M '

r

z-l)} 4t

'

~o(r, z, O) - O.

with

r

ot) -- fu ~ _1e_Yerf(ot~/~) dy. Y

522.04 (IC) Steady state *

of 522.03.

qg(r,z) --

O f/c~ _1 sin(lot)Ko(rot) cos(zot) dot

jr2 K l Jo

ot

{arcsin(z÷' ) - a r c s i n h ( Z -'/ )r} r

4~Kl

2Qr f/~ _1 sin(lot)K1 (rot) sin(zot) dot ]~(r, z)l- 7rl Jo ot /')

_

I~ {x/r2 + (z +/)2 _ x/r 2 + (z -/)2 / 21 522.05 (IFR)

=~Q c

Y///× I//////,~//////~

Finite line well in an "infinitely" thick semi-confined aquifer. Discharge uniformly distributed along the well-screen.

~o(r, z) - drawdown.

(522a)

Flow in a semi-infinite field

I 2rcKI Q lim[ Oq9 ] r-+0 r -~-r (r, Z) -- 0 Oq9 ~(r, Oz

¢p(oo, z) - - 0 ,

O) --

365 1

fora

1

2 1 < z < a +-~l,

for 0 <~ z < a

qg(r, 0) Kc '

1

1

2 l andz >a+-~l,

~p(r, oo) - O.

Steady state: 2Q Jr 2 K l

•

~o(r, z ) - -

•

i~/(r, z) I _ F(ol) --

f(o~)Ko(rol){ sin(zol) + gcol cos(zol) } dol, F ( o l ) K l ( r o l ) { K c o l sin(zc~) -- cos(zoe)} doe,

4Qr zrl

sin ( ~ ) { sin(aol)+ Kcol cos(aol)} or(1%- K 2 c 2 0 / 2 )

"

522.06 (L,IFR) Non-steady state of 522.05, ~o(r, z, 0) - 0. •

~o(r,z,t) - 7r

)

F ( o l ) W ( \ °~2t f12 ' roe { sin(zoe) + gcoe cos(zu) } d~,

l

F(ot) -- (see 522.05). 522.07 (IC)

m

r

r

l

Finite line well on top of an infinitely thick confined aquifer. Discharge not uniformly distributed along the well-screen. Approximate constant level at the screen. Steady state. ~p = ~o(r, z) = drawdown.

lim r-+0

go(oo, z) -- O,

~(r, az

~o(r, z) --

fo °

Q 7r2 K

= 2zrK~-TF

f01r ~0~ dz -Or O) -- O,

a

r--+0

~p(r, oo) -- O.

Jo(lu) Ko(rot) cos(zot) du

~,~,1;

21wl

( 0.~r~) -

lim r

27r K '

with

w -- z 2 + (r + il) 2 -- u + iv, Ig/(r, z)[ - 2Qrrc fo °° Jo(lot) K1 (rot) sin(zoo) dc~.

0

forz >/,

366

(522a, b)

Three-dimensional axial s y m m e t r y

522.08 (L,IC} Non-steady state of 522.07, ~o(r, z, O) - O.

rp(r, z, t) -- 2zr2K

Q f0 ~

Jo(Io{)W(\ °{2t f12'

rot) cos(zo{) do{

b. Point wells along the axis of revolution

Many axial-symmetric problems, concerning point wells in a semi-infinite field may be solved by superposition of the drawdowns or heads, caused by so called "image" wells, the place and character (abstraction or infiltration) of these wells depending on the place of the original well with regard to the boundaries and the character of the boundaries (zero head or zero flux). In the following, some examples which are of some importance for the determination of ground constants, will be given. 522.21 <S) 888~

Instantaneous abstraction Pi [L3] (discharge impuls) of water from a point well in a very thick confined aquifer. q9 - qg(r, z, t) - drawdown.

r

r a

Pi~= Z

Qi

for t - O ,

lim (P2 ~p~) -- { 0 4n'K p-->O Pi -- Qi dt,

cp(r, z, t) --

p2 _ r 2 q_ ( Z - a) 2,

fort>O,

~(r,O,t) Oz

--0,

go(c~, z, t) --0,

qg(r, cxz, t) -- O.

/~2 /~2 flPi exp [---~{r2-k-(z-a)g}]-k-exp [ - ~ 7 { r 2 + (z-+-a)2}] 8Jr K t 4~ff-i

522.22 <S) As 522.21, but with a continuous constant discharge Q. Q Ierfc{2-@v/r2 + ( z - a) 2}

•

~o(r, z, t) - 47r----Kk

7 (z - a) 2

erfc{ 2--~v/r2 + (z + a)2}l

+

v/r 2 + (z + a) 2

522.23 (S) Steady state of 522.22.

1 •

1

}

~o(r, z) -- 47r----K x/r 2 + (z - a) 2 + x/r 2 -t- (z + a) 2 '

(522b, 523)

Flow in a semi-infinite field

Q{

z -a

~p(r, z)l -- -~- ~/r2 + (z -- a) 2

+

367

z+a }. x/r 2 + (z + a) 2

522.24 (IH)

7///, / / ' / / / / / / / / / / Z / ' / / / / / Z

Constant discharge of groundwater from a point well in a Very thick leaky aquifer. Steady state, cp - ~p(r, z) drawdown.

C

r

~a

lim (p2 0_~p~) -p~0

,o~__:_,(o, z) - o Or

~p(c~, z) - 0 ,

~o(r, z) --

O---~-~(r, O) -Oz

forz C a ,

Q 4rrK

with p -

x/r 2 + ( Z - a) 2,

q)(r, O) Kc

qg(r, c~) -- O.

4Jr K +

1

1

}

v/r 2 + (z - a ) 2

~/r 2 + (z + a) 2

Q fo ~ Kcot (rot)e_(Z+.)~ 2rr K 1 + K cot Jo dot,

Q{ + Qr

z+a

z -a

~(r, z) -- ---~- x//r 2 + (Z -- a) 2 fo °°

}

x//r 2 -+- (Z + a) 2

K cot J1 (rot)e -(z+a)~ dot. 1 + K cot

522.25 (L,IFR)

Non-steady state of 522.24, ~p(r, z, 0) - 0.

•

Q /'~ ~o(r, z, t) - 2zr2K Jo

sin(aot) + Kcot cos(aot) ot2t rot) • W\--~-, g2c2ot 2 + 1

× { sin(zot) + Kcot cos(zot)} dot.

523. Axial-symmetric flow in a semi-infinite field with discontinuous boundary values along the r-axis, 0 ~< r < c~; 0 <~ z < c~

02q9

1 0q9

0299

r ~-57r ~ 2z5 --

{ /~2 0q9 (non-steady state),

0

(steady state).

(523)

368

Three'dimensional axial symmetry

523.01 (IH) h-

Circular pond in a semi-infinite field with drawdown h of the water level. Steady state. ¢p -

q) 0

¢p(r, z) - drawdown. ...... E--:...... '-~ • . .

r

0q9

li'i:

~(0,

Or

z) - 0 ,

~o(c~, z) - 0 .

Mixed boundary value: ¢p(r, 0 ) - h

0¢p(r, 0 ) _ 0

f o r 0 < ~ r < R,

Oz

forr > R

¢p(r, e~) -- 0.

~o(r, z) -- ~2h f o ~ -1 sin(Rot)Jo(rot)e -z~ dot Jr

ot

2harcsin{V/Z2+(r+R)2-~z2+(r-R)2 Jr 2r lp(r, z) - Q {rfoC~lsin(Rot)Jl(rot)e-Z~dot-~ ot R

Q -4KhR

-

} '

1}

2 {v/(re + ze - Re)e + 4Rez2 - (re + za - Re)}

with

--total discharge to the pond.

523.02 (IH) Vertical infiltration from a circular pond at the horizontal

q 4+4

r •

•

R

•

•

•

•

boundary in a semi-infinite

Steady state. ¢p - ¢p(r, z) - head.

r

Oq9 Or

•

~(O, z) - 0 ,

~o(ee, z) - 0 ,

Z

q Oqg(r,O)_

-

for 0 ~< r < R,

Oz

0

f o r r > R,

q)(r, oo) -- 0.

qR fo °° --d 1 Jl(Rot)Jo(rot)e_Z~ dot, qg( r, z) -- --~ ~p(r, z) -- 2jrqRr

fo

-Jl(Rot)Jl(rot)e -z~ dot. ot

field.

(523)

Flow in a semi-infinite field

369

523.03 (L,IH) Non-steady state of 523.02. Additional initial value cp(r, z, 0) --0. qo(r, z , t )

1 (R~)Jo(rol)E(z, t, ~) dol, -qR ~ f o ~ --J1 c~

--

E ( z tf° t') -l- e - -zZ ~ e' r f c ( 2~/7

o~~/7) _ eZ~erfc (flz___~+ ~ f l t )

= _2Pconj(C~Z c~~/7) 2 ' 523.04 (IH)

---1 ....

.-777.....-..

Circular reservoir, isolated from surrounding surface water on top of a semi-infinite field, with drawdown of its water level. Steady state. ~0(r, z) = drawdown.

cp=0 .

.

-:Z: ......: - - - 2 : - i - - - :

.

.

.

.

.

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

r

~ ( 0 , z) - 0 , Or

qg(r, 0) --

{h 0

qg(r, z) -- hR

for0~ R,

fo 0 J 1

~(r, z) -- 27rKhRr

~o(00, z) - 0 ,

qg(r, ee) - 0.

(Rol)Jo(rol) e-z~ dol,

jo

Jl(RC~)Jl(rot)e -z~ dot - 4 K h ~ / ~ r g(m4) - E(m4) m

and

m

v / ( R q- r ) 2 + z 2 _ x//(R - r ) 2 + z 2

523.05 (L,IH) Non-steady state of 523.04. Additional initial value

99(r,z, 0)--

99(r, z, t) -- hR

h 0

forz-0and0~0.

Jl(Rol)Jo(rol)P


-

'

fi

dc~.

370

(523)

523.06

Three-dimensional axial symmetry

(IH)

=°°

°

°

Circular reservoir isolated from surrounding surf a c e water on top of an infinitely thick leaky aquifer. D r a w d o w n of the water level in the reservoir. Steady state, qg(r, z) = drawdown.

r

a ! ( o z) - o Or ' '

h - c p ( r , 0)

i

O~° (r, O) -Oz

~(c~, z) - o

for0<~r < R ,

Kc

qg(r, oo) -- O.

~o(r, O)

for r >~ R,

Kc

qg(r, z) -- hR fI c¢ J1 (Rot) J o ( r a ) e -z~ dct,

•

1 +otKc

Jo •

~ ( r , z) - 2 r c K h R r

f0 ~ J1 (Rot)J1 (rot) e -z~ dot. 1 + otKc

(L,IH)

523.07

N o n - s t e a d y state of 523.06. Additional initial value qg@,z, 0) - 0 . •

hR qg(r, z, t) -- --f-

J1 (Rc~)Jo(rot)E(a, z, t) do~ with

e~Z

E(ot, z, t)

e r f c ( f l ~ + --°e~7)+

1 - Kcot

fi

1-

2

1-

523.08

K2c2ot 2

exp

e -~z

erfc( flz

1 + Kcot

+

K2c2c~ 2

f12K2c2

t ) e r f c ( ~/7

flz

+ 27)"

(IH)

¢p=0 q

Circular cylinder, isolated from surrounding surf a c e water on top of a semi-infinite aquifer with vertical infiltration into the aquifer.

r

R

0q9

a r ( 0 , z) - 0,

~o(~, z) - 0,

qg(r, oo) = 0, O~° (r, O) -Oz

q K

for 0 ~< r < R,

qg(r, 0 ) - - 0

forr /> R.

(523)

Flow in a semi-infinite field

•

qg(r,z) -

371

2q [ ~

sin(Rot) - Rot cos(Rot)

r r K J0

ot2

Jo(rot)e -z~ dot,

f0 °° sin(Rot) - Rot cos(Rot) J1 (rot)e -z~ dot. ot2

(r, z) -- 4 q r

523.09 (IH)

Q --+.

5

Circular drain with total discharge Q on top of a very thick confined aquifer. Steady state. ~p ~0(r, z) - drawdown.

r

llz"

099

~ ( 0 , z) - 0 ,

~o(oo, z) - 0 ,

Or

lim 0 ! ( r , 0) -e--,o 0z 0___~_p(r, 0) -- 0 Oz

forR-e
4rr K Re

qg(r, oo) -- O.

for r :/= R,

Q/o

cp(r, z) -- 27r K

< R+s,

Jo ( R ot ) Jo ( r ot ) e - z ~ dot 4rR

Q 1 K rr2K v/(r + R)2 + Z 2

(r + R

)2

}

+

Z2

with K ( m ) -- complete elliptic integral of the first kind, d/(r, z) -- Q r

fo °

Jo(Rot)J1 (rot)e -z~ dot.

523.10 (IH) q r

]_

~ 2B ~

r

Vertical infiltration from a circular canal with width 2B into a very thick confined aquifer. Steady state. 99 - qg(r, z) - head.

z Oq)

~(O,z) - o ,

(p(oo, Z) -- 0,

Or

q 0---~-~(r, O) -0z

-0

for ( R - B ) for0~
qg(r, oo) -- 0,
< (R+B),

< (R-B)

qg( r, z) -- --K q fo °° F ( ot ) Jo ( r ot )e -z~ dot,

andr > (R+B).

372

(523)

~(r, z) -- 2rrq r

Three-dimensional axial symmetry

F(ot)J1 (rot)e -z~ dot

f0 x3

1

F(ot) -- - [ ( R + B ) J , { ( R + B)ot} - ( R

with

- B ) J I { ( R - B)ot}].

o/

523.11 (L,IH) Non-steady state of 523.10, q)(r, z, 0) = 0. •

F (ot) Jo(rot) Pconj motz ot

q)(r,z,t) =

Kq

f0 ~

( f l2~' )

with F(ot) from 523.10.

523.12 (S) Circular well-screen of radius R and length l and a total discharge of Q = 2rcRq [L3T -1] in a semi-

I I

!1

R

infinite field.

r

l

Oq9

aq)

{

-~r (R' Z)]left -- [-~r (R' Z)]right Oq) ~(0, Or

z) = O,

~p(cx~, z) = O,

-

Q

02rcKRI Oq9 ~(r, Oz

f o r 0 ~ z 1, O) = O,

q)(r, ¢x~) = O.

Q fo ~ 1 sinh(iot)Jo(Rot)Jo(rot)e_Z~ dot qg(r, z) -- 2re KI ot 1 sinh(iot)Jo(Rot)Jl(rot)e_ZU dot. ~ (r, z ) - -Qr T fo ~ -~

523.13 (L,S) Non-steady state of 523.12, ~o(r, z, O) = O.

•

q)(r,z,t) =

Q fo ~ _1 Jo(Rot)Jo(rot){F(z - 1, t) - F(z + 1, t)} dot 4 rr K I ot

with F ( z , t) -- P

otz ot~/[ )

-~,

fl

-

c~2t\ (7_~). flz exp ( _ _~)erfc

Flow in a semi-infinitefield

(523, 524)

373

523.14 (IFR)

~a

Circular well-screen of radius R and length l with a total discharge Q in a semi-infinite leaky aquifer.

C

Steady state.

r

R ,o0__:_,( ° z) - o

~o(~, z) - 0,

Or '

ml

[a-~-ro(R, z) ]left

-

Q _

2zrKRl 0

Z']ri ht for ( , a - ~1I) < z < forO~
0q) g)(r, 0) ~ ( r , 0) = ~ ,

Oz

1

l),

andz>

(1) a+-~l ,

qg(r, cx~) = 0.

Kc

2Q

a--~l

(a+

f o ~ -1 sin (Iot)F(a'ot)K°(Rot)io(rot)F(z ot -21 + K2c2ot 2

ot)dot'

•

q)(r, z) -

*

i~p(r, z) I _ 4Qrjrlf o ~ (1/ osin t ) _ o-2t F(a, ot)Ko(Rot)l + K 2c2ot 2 I,(rot)G(z, ot) dot

rc2Kl

for 0 ~< r < R,

and

•

2Q f o ~ l - sin (Iot)F(a, ot)Io(Rot) qg(r, z) - rc2Kl ot -2- 1 -k- K2c2ot 2 Ko(rot)F(z, ot) dot,

*

I~p(r, z) I _ 4Qrrclf o ~ (1/ osin e ) _ o-2t F(a, ot)Io(Rot)l + K 2c2ot 2 Kl(rot)G(z, ot) dot forr > R

with F(z, ot) = sin(zot) + Kcot cos(zot) and G(z, ot) = Kcot sin(zot) - cos(zot). 524. Uniform flow with wells or obstacles

02~0 1 099 0299 =0. Or----T + -r -~r + OX 2

374

524.01

Three-dimensional axial symmetry

(524)

(S)

Point source and point sink of equal strength in uniform flow, parallel to the connecting line of the two wells, the point source situated downstream.

a

a

Q {

qx go(x, r) - -~- + 4zr K ~ ( x r) -- --zrqr 2 '

1

1 x/(x - / ) 2 + r 2

Q{ -- -2-

v/(x d- I)2 + r 2 '

x+,

x - l x/(x - - / ) 2 -k- r 2

}

V/(x+/)2+r

} 2 "

Stagnation points: Xs from (x 2 - t 2)2

- - ~QlX ~ 0 , zrq

r S ~0.

Hydrological screen: 7r = 0. 524.02

(S)

Impermeable sphere of radius R in uniform flow of strength q. qx + -qR go(x, r) -- --~ - ~ 3 x(r 2 + x 2) -3/2

~

R

x

~ ( x , r) -- 7rqr 2 { R 3 (x 2 -+- r2) -3/2 - 1 }.

(530, 531)

Flow in a finite field

530 BV-3.

375

THREE-DIMENSIONAL AXIAL-SYMMETRIC GROUNDWATER FLOW IN A F I N I T E F I E L D

a. Has the soil discrete inhomogeneities in r- and~or z-direction ? go to 560 (BV-6)

Inhomogeneity in r-direction" Cl

c2

m

go to 531

b. The soil is assumed to be homogeneous

Also anisotropic homogeneous soils belong to this category, provided that the main directions of the anisotropy coincide with the r- and z-direction. Solutions for anisotropic cases can easily be derived from analytical solutions for isotropic problems.

531. Homogeneous soils a°

There are discontinuous boundary values along the z-axis and continuous ones along the r-axis

go to 532

The z-axis is taken here as the axis of revolution. Point wells and partially penetrating line wells, located along and on the z-axis cause discontinuities in the value of the flux along that axis.

~Q c

,/'//~

~////////~/////////////////~ r

1I

Z

bD Discontinuous boundary values along the r-axis and continuous ones along the z-axis

go to 533

The z-axis is taken as axis of symmetry. Discontinuities along the r-axis may be caused by circular disks, with given drawdown or given flux, or by circular screens, drains or canals with given discharge or recharge.

376

(531 b-532)

-h _

Three-dimensionalaxial symmetry

a

r

r

R

D

K Z

c. Combinations of the boundary values mentioned under a and b

go to 534

There are many possibilities to combine certain discontinuous boundary values, mentioned under a and b and, in general, most of these combined problems may be solved by superposition of the separate solutions, given under a or b, especially those, concerning steady and non-steady states.

/I

=~Q

q ,~+~~ r

=~a

°

K

;

q ~ B :

r

lI

Z

Z

The only solutions that will be given here, are some combinations of problems of which the separate solutions cannot reach a steady state for the head or the drawdown, as we often encounter if water is abstracted or infiltrated from or into a confined aquifer. However, together a recharge problem and a discharge problem may yield a steady state, if the recharge equals the discharge. Two examples are shown here. d. Both the boundary values along the r-axis and the boundary values along the z-axis are continuous

PQ ~///////////////////~c

!i

Iro

For instance, a fully penetrating well in a leaky aquifer, considering also the vertical velocity components of the flow.

532. Discontinuous boundary values along the z-axis 1 Ogo

02~0 Or 2

+-

r~r

O2~p

+

~z~

/ - /

f12 O~O (non-steady state), 0

go to 535

-57

(steady state).

(532a)

Flow in a finite field

377

Subdivision: a. Partially penetrating line wells. b. Point wells. a. Partially penetrating line wells

532.01 (L,FC)

~Q r

a

l

Partially penetrating line well with constant discharge in a confined aquifer. ~o = qg(r, z, t) - drawdown.

, qg(r, z, O) --0,

qg(cx~, z, t) --0,

O~(r, D ' t) -- O, ~Oz ( r , O, t) -- -Oz

NSgS8

Q lim(099) r-~-r --

2:rrKL

qo(r, z, t) --

1 for O~
0

r--~O

(a+~',)

1

for ( a - ~ I ) < z <

(a+~

1/) < z<~ D.

Q

El(~2r 2 Q ~-~1 (Trnl~ 7rna~ 4t ) + s i n cos( D / rc2K I n \-~ / 4rc K D n-- 1 × W

t

D

fl2D 2'

COS

-D

"

532.02 (FC) (S) Steady state of 532.01 for the stream function ~.

ez

4er

1

[~(r, z ) l - -D- + - 7

n-I

~l)

X sin (-~-~-/

(~a)K ( O

\

/

1

O /

sin k,--D--/

r + z+2nD-a+-~l 21 n=-oo -~r2+(z+2nD-a

-

r2+

z+2nD+a

11)2 + ( r 2 +

2

21 l

"

(

')

z+2nD-+-a+-~l

37 8

(5 32a)

Three-dimensional axial symmetry

532.03 (L,FS) Partially penetrating line well with constant discharge in an aquifer under open water. .............................

~,, li--

-

lim(099) r--+0 r0-r-r -•

q) = qg(r, z, t) = drawdown.

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

a

~o(r,z,t) =

I O

qg(cxz, z, t) =0,

qg(r, 0, t) = 0,

O99 ~ ( r , D, t) -- 0, Oz

for ( a - ~1/) < z < ( a + 1/),

27rKl

0

qg(r, z, O) =0,

for0~
Z mlsin(-4-D-)sin ( 2D )

20

rc2 Kl

m=1,3,5 ....

( rr2m2t 7rmr)sin(mrcz ) x W 4/32D2, 2D 2D " 532.04

(FS)(S)

Steady state of 532.03. oo

•

qg(r,z) = 7r4Q 2Kl

Z

1 sin(mrrl~

m=1,3,5 ....

m

\ 4D / sin

(mrca ) K° (m r) sin (m z)) 2D

2D

2D

( - 1)n arcsinh z + 2n Dr- a + g l

= 4Jr Kl ~ n~--OO

-

(z + 2nD - a - ~1 l)

arcsinh

r ll

+ arcsinh •

~ ( r , z) =

~

Trl

~

1

(-1)n

tl--'-- OO

r

}

mrrl

--sin(m - ~ ) s i n (

mrra

(mrrr

mrrz

2D ) K I \ 2D ) c o s ( 2D )

m=1,3,5 ....

= 25 Z

(z + 2nD + a + ~1 l)

,

r

8Or

- arcsinh

r2+ z + 2 n D - a + - ~ l

'G>2>

soa

~

~+~,~o

u;o O=U

I

-lOt, =1< ~''>~1 u;o 0 = u

I~

oX

lX~z

= (z 'a)o~

* ,

W~) u!s "~0"~

( dUTo ,~,0

,;0;9'

al"

a~ ~-I

-~

a-

.aX -6- =

jo

rams ~FvmS

a3 uel ~o jo sloo.~ aql ~u!aq u~o ql!,~

~u70~ 0 --'u

( ~ ) ~o~ (r~) ~:~

lXaZ 0

= (~ 'z 'a)o~

,

ze

oaX

-

l-(+v

'-0~z{?

= (~ ' G

pu~

li-V

'0-

(~ 'z 'e~)o~

<J 'o '~) e~--

'0-

o

>z>O~O J

(0 'z '.0o~

~,.~ I _(~d~ d) O+--d

lX aZE 0

'(1I + v)> z > (17Z; v) ~oj d

S

•( s u o s s o ~ ISUO!lmndtuoa .toj ~oj!nbs oql jo tuolloq oql ffUOle uosoqa s!x~-.~ pus sp~sA~dn uosoqa uo!laa.I!p-2) UA~OpA~s.Ip = (t 'z 'a)o~ -- d~ "aa~nbv ~va I ~ u! o~.t~qas!p

lumsuoa qlIA~ lla~

(IX

',_]j

a ~////////////////Z////~ 0

am.1 gul.malauad dllm.lat~d

•

l-+V+GUg+

z +z4

-

+;

v-Gug+z

+z

l__V+GU.~+z 1

6L~

+ -

pld~.fdll.u~fv u! ~Old

('eE~g)

380

(532b)

Three-dimensionalaxial symmetry

b. Point wells 532.11 (L,FC) 888~

r

r

a

a~,

Point well with constant discharge in a confined aquifer, q9 - qg(r, z, t) - drawdown. lim (p2 0qg) _ Q p~o -~p -4zrK

KD, S

with p -

Cr 2 + ( z - a) 2,

Z

a!(O, z, t) -- O forz-Ca, Or qg(c~, z, t) --0,

~o(r, z, O) --0,

099 (r, 0---z

O, t) --

0~o -0-z-z(r, D, t) -- 0.

Q El(fl 2r2 ) ~o(r, z, t) -- 4zr K D 4t Q ~ (nrra] (n2rr2t nrcr) (nzrz) cos \--5- / W cos . + 2rcKD n=l f12D2' D 532.12 (FC)(S) Steady state of 532.11 for the stream function. Qz 2Qr ~ [~(r, z ) l - --~ + "-D-- ~ c o s

nrca (nTrr] (nTrz] (---D--)K1 k--D--/sin \---~-/

n=l

z+2nD+a }. =--Q ~ { z+2nD-a + 2 n - - ' - - O O x/r 2 + (z + 2 n D - a) 2 x//r2 + (z + 2nD + a) 2 532.13 (L,FS) Point well with constant discharge in an aquifer under open water. ~o - ~o(r, z, t) drawdown. ~a

KD, S

qg(r, z, O) --0, ~o(r, O, t) --0,

z

~p(cx), z, t) --0,

Flow in a finite field

(532b)

Q

lim (p2 0qg) p--+0

Oq9 --(r,D,t) Oz

with p - 4 r 2 -k- ( z -

4rrK

"~p

0q9 --(O,z,t)-O Or

--0,

Q

381

a) 2,

forzCa.

oo

mrCr)sin(mrcz ) 2D

m~a

~p(r, z, t) -- 27r K D

sin(2D)W(m2zr2t Z 4f12D 2 ' 2 D m=1,3,5 ....

532.14 (FS)(S) Steady state of 532.14. - qg(r, z) --

oo

Q rr K D

mrc a mrc r sin(2D)K°(2D)sin(mrcz2D)

m=1,3,5 ....

Q

+~

Z

(--1)n

{

4rrK n=-~

1

4 r 2 + (Z + 2n D -

a) 2

1

j

v/ r 2 -k- ( z -+- 2 n D -+- a) 2 '

•

I~(r, z ) l -

oo

2Qr D

mrc a mrr r sin(2D)Kl(2D)C°s(mrrz

2D )"

m=1,3,5 ....

532.15 (L,FCR)

~///////////////////~

c

a

Point well with constant discharge in a leaky aquifer, q9 = cp(r, z, t) = drawdown. (zdirection upwards; r-axis along the bottom of the aquifer.) 99(r, z, O) - - 0 ,

~o(c~, z, t) - - 0 ,

r

lim (p2 0~°) _ Q p~o ~p " 4zr K

O~P(O, z t) -- O -~r '

with p - 4 r 2 -+- ( z -

0~o ~o(r, D, t) - - ( r , D, t) -- Oz Kc

forz:/:a,

Q ~p(r, z, t) -- 27r K D

O~°(r, O, t) -- O, Oz

a) 2,

cos(

) w

cos

= 1 + ot2+e2

fiZ D2 ' O D

with O~n being the roots of ot tan ot - e - 27"

---O-/

382

Three-dimensional axial symmetry

(532b, 533a)

532.16

(FCR)

Steady state of 532.15. cos •

~o(r,z)

Ko

~-

--

Jr K D n=O

(otnr

cos

~2+E2

T), otnZ

I (r, z ) l - 2Qr E°°l + c°s( o / Kl(otnr~Dlsin\--D-

•

n=O

c¢2+e2

5 3 3 . D i s c o n t i n u o u s b o u n d a r y v a l u e s a l o n g the r - a x i s

02~

10q9

O r 2 nt- _

02~

if_

{ f12 O~

_

-~

0Z2

r -~r

0

(non-steady state), (steady state).

Subdivision: a. No line well along the z-axis. b. Fully penetrating line well along the z-axis.

a. No line well along the z-axis 533.01 (L,IH)

q

Vertical infiltration from a circular pond into a confined aquifer. ~o - ~o(r, z, t) --- head.

y =

~

r

R

~o(r, z, 0) - 0,

KD, S

3~° (0, z, t) - 0 Or

qg(cxz, z, t) - 0 , Z

q

3--~(r, 0 t) -

-K"

Oz

0

'

for 0 ~< r < R,

a!(r,D t)-O. 3z

f o r r > R,

~o(r, z, t) -- --~ qR fo°° ot1 J1 (Rot)Jo(rot)E(ot, z, t) dot with ot (2 n D q- z) otv/-[ E(ot, z , t ) - -

Z

n--0

Pconj

+ Pconj

2

'

/3

{ ot(2nD + 2D - z) 2 '

fl

"

(533a)

Flow in a finite field

383

533.02
sinh { (D - z)ot } dot. sinh(Dot)

~(r, z) -- 27rqRr fo ~ -Jl(Rot)Jl(rot) 1 ot

533.03 (IH)

:

Circular reservoir, isolated from surrounding surface water on top of an aquifer of finite thickness. Drawdown of the water level of the reservoir. Steady state.

.-_-_-.-.--.---.--.-:::.-.-_-:. ;

0q9 Or

~ ( o , z) =o,

~p(r, 0 ) - -

h 0

for0~ R,

~p(r, z) -- hR

fo °°

0q9 ~(r,D) Oz

J1 (Rot)Jo(rot)

~(r, z) -- 2 z r g h R r

fo

~(c~, z) =o,

=0.

cosh { (D - z)ot } dot, cosh(Dot)

Jl(Rot)Jl(rot)

sinh { (D - z)ot } dot. cosh(Dot)

533.04 (IH)

[

Circular polder with a water level that differs from the surrounding polder level on top of an aquifer of finite thickness. Steady state. ~p = qg(r, z) = drawdown.

c

099 Or

~ ( 0 , z) =0,

OCP(r, O) -Oz

h-~p(r,O) Kc q)(r, 0) Kc

q)(r, z) - hR

f0 ~

forO~
~o(~, z) =0,

a~o '

for r > R,

(r, D)

O.

Oz

cosh {(D - z)ot} dot, ot g c sinh(Dot)

J1 (Rot)Jo(rot)cosh(Dot) +

~(r, z) -- 27rKhRr

fo ~

sinh {(D - z)ot } dol. Jl(Rot)Jl(rot)cosh(Dot) + otKc sinh(Dot)

384

(533a)

Three-dimensional axial symmetry

533.05 (IH) a _

Circular drain with constant total discharge Q on top of a confined aquifer Quasi-steady state. ~0 = ~p(r, z) = drawdown.

y

r =

m.

R

KD

° ! ( o , z) - o, Or

~(c¢, z) - o,

Z

lim OCP(r, O) -- Q----~ ~ o Oz 4zr K Re

for(R-e)


OCP(r, D) -- O. Oz

Oqg (r, O) -- O f o r r # R , Oz

Steady state for the stream function:

I~(r, z)l -- Qr

f0 °

Jo(Rot)Jl(rot)

sinh {(D - z)ot } dot. sinh(Du)

533.06 (IH) q

~

~ I~

4

~

~ ~ "2B"

_ r

Vertical infiltration from a circular canal into a confined aquifer. Quasi-steady state. °=-' (o z) - o, Or '

q

0--~ ( r, O) -az

- -K 0

for(R-B)

~(c¢, z) - o,


for0<~r < (R-B)

andr > (R+B),

¢aO___(r, ~, D ) -- O.

Oz Steady state for the stream function:

lap(r, z)l -- 27rqr

F(ot)J1 (rot)

f0 X)

sinh {(D - z)ot} du sinh(Dot)

1 F ( u ) -- - [ ( R + B ) J I { ( R + B)u} - ( R ot

with

- B ) J , { ( R - B)u}].

Flow in a finite field

(533a)

385

533.07 (S)(FC) Circular well-screen with radius R and length 1 on top of a confined aquifer. Quasi-steady state, q) = qg(r, z) = drawdown.

I .....!11 / I_

r

R

KD

-

~Z(o z) - o , Or '

-~-r (R' Z)]right -

02rcKRI

~o(,:x:,, z) = 0 ,

for0<~z < I , for z > l,

099(r, D) -- O. Oz

~o0_2(_,r, O) -- O, Oz

Steady state for the stream function:

]~(r'z)] - TQr f0 °° 1 sinh(/ot) Jo(Rot)J1 (rot)

sinh { ( D - z)ot } doe, sinh(Dot)

or

•

~(r,z) =

2Qr ~ 1 (nTrl~ nrcR nrcr --~sin \ - - ~ - / K o ( )I1()sin() -if5n=l ,, for 0 ~< r <~ R, Q

Qz D

2QF rrl

n=l

nzrz b-

1 nrcl] nrCR]Kl( nTrr ~ sin (nrcz ) n Sin(--D--/I°( D / D / \---Df o r r ~> R.

533.08 (FCR)

~Q "/////////////~, "////////////////////,2, c

Partially penetrating circular well-screen at the bottom of a leaky aquifer. Steady state. q) = ¢p(r, z) = drawdown. (z-direction upwards; r-axis along the bottom of the aquifer.)

KD .

.

.

.

.

.

.

.

.

°--E(o, z) - o, Or

-n!

r

v(oe, z) - o,

386

(533a, b)

[-~-r ] 0q9 (e z) right

~r(R,z) left

,

Oq9 ~(r,D) Oz

"co --z-' (r, O) -- O, Oz

Three-dimensional axial symmetry

{ 2rcKRI Q 0

--

forO~ l,

~p(r, D) = - - ~ Kc

sin( -b~"/~ T +]' "" ~r"'0~it'a"R ~,-b-) I0( otnr]D] cos \__/~.)(OtnZ Jr K l

n=oOln

a n -1-

forO~
go(r, z) -Jr K l

n=oOtn

sin(an/~, [anR~ (otnr i+--~ Ko \ D

OtnZ'~ )cos (T '

a n +e

forr > R, 1 sin( --ffan/~l ,r"x o [~,---ff~nR ~]

2Qr~ l

i+-~2

n=oOln

( Oln r

Otn Z ]

I1 k--D-)sin (--~-/

a n q-

for 0 ~ r < R,

~ ( r , z ) -a

2Qr ~

I

l sin [anl~,l

__ n=0 C~n

[anR~ K l ( ° t n r

~-bL-_/ "o k-y- ! +$2+~

OtnZ]

)sin(--D- /

\ D forr > R

D with Un being the roots of ot tan ot - e, e - x--7."

b. Fully penetrating line well along the z-axis

533.21 (IH)

=,a

IR ,

Vertical infiltration from a circular p o n d into a confined aquifer, combined with a fully penetrating line well along the z-axis. ~o = ~o(r, z) = head. Steady state only if

r

recharge = discharge: Q = rc RZq.

K D

lim r r--~O

q 0__~_~(r, 0) --

- K

Oz

0

for 0 ~< r < R, forr > R,

Q fo

~o( r , z) -- 2 rr K D

ot

Oqg) -~r

Q 27r K D '

~0(co, z) =0,

Oq)(r, D) -- O. Oz

Jo(ra) [ ~ - J 1 (Rc~)

cosh { (D - z)ot } - lldc~ , sinh(Du)

I~(r, z ) [ -

387

(533b)

Flow in a finite field

Qz + -R--2Qr fo °° 1 --D--

J1

(Rol)J1 (rot)

sinh {(D -- z)c~} d~. sinh(D~)

533.22 (IH)

~Q ~8888~

Constant infiltration from a circular drain on top of a confined aquifer, combined with a fully penetrating line well along the z-axis. ~p = qg(r, z) = head. Steady state only if recharge -- discharge = Q.

Q

!

-

y

r

-', R

"

KD

Oqg) lim r r-+O ~r

Z

lim O~° (r, O) --

~o

Oz

for(R-e)

4re K R e

~(r, 0)--0 Oz

q)(r, z) --

Q 27r K D

IVy(r, z)l- --D-oz +

DJo(Rot)

Qr

fo

~(oo, z) - 0 ,


Ogo ~(r,D) Oz

for r # R,

Q 2rc K D '

--0.

cosh {(D - z)c~ } sinh(Dol)

Jo(Rot)Jl(rot)

1 - - ] Jo(ra) da, ot

sinh {(D - z)ol} dol. sinh(Dot)

533.23 (IH) ~Q

q =

i

Vertical infiltration from a circular canal into a confined aquifer in combination with a fully penetrating line well along the z-axis. cp = qg(r, z) = head. Steady state only if recharge = discharge; Q -- 47r B Rq.

=

2B

r

|

R

"

lim r r~O

KD

27r K D ' q

aq)(r, 0 ) _ az

o

~o(r, z) -- 27rKD

~oO__(r, _:, D) - O. Oz

~o(oe, z) - 0 ,

for(R-B)


for0<~r < (R-B)

2BR

andr > (R+B).

sinh(Dol)

1 1 Jo(rot) dol,

Ol

388

(533b, 534)

sinh {(D - z)ot} Sl(r~)dot sinh(Dcz)

Qz + 2BR Qr fo ~ f(ot) I~(r, z)l - --ff

1

F(ot) -- -[(R +

B)JI{(R

+

B)cz}

Three-dimensional axial symmetry

-(R

- B)Jl{(R

-

with

B)ot}].

o/

533.24 (IH)

4

q',

:'

iLl'

i I

,

Constant infiltration from a circular wellscreen on top of a confined aquifer, in combination with a fully penetrating line well along the z-axis, q9 = ~0(r, z) = head. Steady state only if recharge = discharge --- Q.

KD

i

o

! i

lim r r-+O

O~o [O~o z)] [ "~F (R' left -

{

-0--r(R' Z)]right -

600__z(_,r, O) -- 0

Oz

Q 02yrKRI

2Jr K D '

for 0 ~< z < I , for z > I,

coO_z(r, ._, D) -- O. Oz

cosh {(D - z)ot } ~o(r, z) -- 2 z r K D

~o(~, z) - 0 ,

--i- sinh(l~)Jo(Rot)

Qz + TQr fo ~ I~(r' z)l - --b-

sinh(Dol)

1 sinh(l~)Jo(Rot)J](rol)

- 1J l- Jo(rot) dot, Ol

sinh {(D - z)ot } dot. sinh(Dol)

534. Combination of discontinuous boundary values along the z-axis and the r-axis oZfP }---1 3~o + 32q9 --0 ar---Y- r-g-r ~ "

534.01 (S)

i

. _

a

i

r R

Vertical infiltration from a circular pond into a confined aquifer, combined with a partially penetrating well along the z-axis. qo = qg(r, z) = head. Steady state only if recharge = discharge: Q = Jr RZq.

~o(c~, z) - 0 , z

(5 34)

Flow in a finite field

lim

Oqg)

r

r-+O

0qg(r, O) -Oz


~Q-K 2 l

11) for ( a - ~

0

forO<<.z< ( a - - ~ l )

--

~

389

-K-q 0

1

for 0 ~< r < R for r > R,

099

~(r,D) Oz

qg(r, z) -- 2 zrQK D f o ~ ot1 Jo(rot) [ ~

J1 (Rot)

nzrl]

2Q

7rgKl

1 sin ( 2 D c o s

n=l

1

and ( a + - ~ ! )

n


--0.

cosh{(D-z)ot} _ 1 sinh(Dot)

dot

J

(nrca] nrrr] nrrz] K0( cos (--D--! \--D-/ D / '

O(r, z) -- O(r, z) of 533.02- ~(r, z) of 532.02. 534.02 (S)

As 534.01 but with a point well in (0, a). qg(r, z )

-

Q f 0 ° ° l - Jo(rot) [ 2D J1 (Rot) ot --R

2 rr K D

OO

Q

~KD

~ cos n=l

(nrca) --5-

K0

(nrcr] ~ D :

cosh {(D - z)~} - 1]dot sinh(Dot) cos

(nrrz]

,,---if-:'

~(r, z) - ~(r, z) of 533.02- ~(r, z) of 532.12. 534.03 (S)

~Q ~Q ,ggggggggggga~gg~~~~_ r

aT

R

l

|

KD

Constant infiltration from a circular drain on top of a confined aquifer, in combination with a partially penetrating line well along the zaxis. q9 = ~0(r, z) -- head. Steady state only if recharge -- discharge =

Q.

| L

~o(ee, z) - 0 ,

~388~

z

Q lim ( ar ~ o )

2rcKl

r--+O

0

~

for (a - ~1,)

< z < (a + ',),

forO~
and a + - ~ l

< z ~ D,

390

(534)

lim 0~° (r, 0) -~-,o Oz

for(R-e)

4re K Re

~o0__:(r, _, 0) -- 0 for r # R Oz

Three-dimensional axial symmetry


0~o(r, D) -- 0. '

Oz DJo(Rot)cosh {(D - z)c~}

Q ~o(r, z) -- 2JrKD

sinh(Dot)

2Q ~ n=l

-

1 Jo(r~)dc~

nrcr , )cos (nrca] T (--5)cos(-nrcz ) -if-'

1 sin (nrcl n

ap(r, z) -- ap(r, z) of 533.05- ap(r, z) of 532.02. 534.04 (S)

As 534.03 but with a point well in (0, a).

Q fo [

~o(r, z) -- 2rcKD

DJo(Rot)

OO

Q

:rKD

•

Z cos n=l

cosh {(D - z)ot } sinh(Dc~)

1 ] Jo(rt~) dc~ 0t

(nrca] (n:rr] (nzrz) Ko cos \ D / \ D / --5 '

ap(r, z) - ap(r, z) of 533.05- ~p(r, z) of 532.12.

534.05 (S)

~Q

q r

2B l

r

R

KD

Vertical infiltration from a circular canal into a confined aquifer, in combination with a partially penetrating line well along the zaxis. ~0 = ~0(r, z) = head. Steady state only if recharge = discharge: Q = 4Jr B Rq.

~o(c~, z) - 0 ,

Z

for ( a - ~l / ) < z < ~i~(r~r~)-

0 q

( a + 1/),

f o r 0 ~ < z < ( a - ~ l l)and (a + l / ) < z < ~ D ,

0qg(r, 0)--

-K

for(R-B)


Oz

0

for0~
andr > ( R + B ) ,

~oO_(r _z_,. ' D) -- O. Oz

Flow in a finite field

(534, 535)

Q

391

fo~[DF(ot) c o s h { ( D - z ) or} 2BR sinh(Dc~)

~o(r, z) -- 2rrKD

2Q ~ yr2Kl

_l

1 ] Jo(re~)dot

Ol

sin (nrrl ~ cos ( n y r a ] K 0 ( n y r r \2D/ ~ D ¢

n=l

-g- ) cos ( -nyrz

with

1 F(c~) -- - [ ( R + B)J,{(R + B)ol} - ( R - B ) J , { ( R - B)c~}], o/ ~t(r, z) -- 7t(r, z) of 533.06 - ~(r, z) of 532.02.

534.06 (S) As 534.05 but with a point well in (0, a).

Q fo~[DF(ot) c o s h { ( D - z ) Or} ~o(r, z) -- 2rrKD 2BR sinh(Dc~)

1 ] Jo(rc~) dot Ol

OO

~cos(nyra__D__)K0(nyrr] (nrcz] rcKD \ - - D - - / c ° s k--D--/ n--1 Q

with F(ct) as in 534.05, 7t(r, z) -- ~(r, z) of 533.06 - ~(r, z) of 532.12.

535. Continuous boundary values both along the r-axis and along the z-axis

0(/9 02~0 + _1. 0q9 _~_ 02(/9 __ { /~2 --~

Or2

r -~r

Oz 2

0

(non-steady state), (steady state).

535.01 (L,FS)

=,a .

.~

.

.

.

.

.

.

.

.

Fully penetrating line well with a constant discharge in an aquifer under open water. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

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

q9 - ~o(r, z, t) - drawdown.

r

~p(r, z, O) --0, ( O~o)

KD, S

lim r r~0 -~r

----

cp(r, O, t) --0, Q

(X)

qg(cx~, z, t) --0, Q 2Zr K D '

coO_(r, __t, D, t) -- O. Oz

m r)sin(m2D z)

1 / m27r2t W qg(r, z, t) -- Jr2 KD Z m 4f12D 2 ' 2D m=l,3 ....

392

Three-dimensionalaxial symmetry

(535)

535.02 (S) (FS) Steady state of 535.01. *

~o(r,z) --

2Q

_1 Ko

rc2 KD re=l,3

~(r,z) -

4Qr reD

m

2D

(mrcz)

sin

2D

--]-oo

O 2rc K D •

....

(mrcr)

(-1)"arcsinh(z + 2nD) r

n----oo

Z m=l,3

....

_1 K1 (mrcr) cos (mzrz) m 2D 2D

-~-(x)

Q ~ D

(-1)nv/r 2 -4-(z -t- 2nD) 2.

535.03 (L,FCR)

Za C

Fully penetrating line well with constant discharge in a leaky aquifer. ~p = ~o(r, z, t) = drawdown. (z-direction chosen upwards and r-axis chosen along the bottom of the aquifer for computational reasons.) ~o(r, z, 0) = 0,

KD, S

lim r

~z~ coO__(_r_r,~. D, t) -- _ 09(r, D, t) Oz Kc Q

~o(r, z, t ) -

o0

2rrKD ~

---

r--+0

r

2zr K D '

qg(c~, z, t) = 0 , 6o0___(~,r, O, t) -- O. Oz sin(ctn)

( 2 t ont Otnr) cos (OtnZ) k--D-Oln + sin(o6~)cos(ol~) W f12D2, D

n=O D

with C~n being the roots of c~tan ot - K-T"

535.04 (FCR) Steady state of 535.03. •

~o(r,z)

Q --

[~(r, z ) l -

~-~

rcKD ~n

u

2Qr ~ D

sin(un) an + sin(ctn) cos(c6z)

sin(an) = otn + sin(o6~) cos(c~n)

Otnr

Otnr

otnz ~ ,

sin

(540, 541)

Flow inside a circular cylinder

393

540 BV-4. T H R E E - D I M E N S I O N A L A X I A L - S Y M M E T R I C GROUNDWATER F L O W INSIDE A CIRCULAR CYLINDER

a. Has the soil discrete inhomogeneities in r- and~or z-direction ? go to 560 (BV-6) An example of discrete inhomogeneities in the soil is a layered aquifer with a discontinuous variation of the transmissivity of the layers.

K1D1 K2D2

K3D3 ~:~ R

z

b. The soil is (assumed to be) homogeneous

go to 541

Also anisotropic homogeneous soils belong to this category, provided that the main directions of the anisotropy coincide with the r- and z-direction. Solutions for anisotropic cases can easily be derived from analytical solutions for corresponding isotropic problems (Part B, Section 2.3.4). Practically all groundwater flow problems concerning flow inside a circular cylinder relate to flow in an aquifer with finite thickness and generally is referred to as flow within a circular island. For that reason solutions for flow problems, concerning circular cylinders with flow fields that extend to infinity in vertical direction, will not be given here, as being of no practical use. 541. Axial-symmetric flow inside a circular island

~x~

~ D

r

w1

R

tance w [T].

K ~

z

The boundary at r - R, extending from z - 0 until z - D will be considered as semi-permeable with a resis-

0~o ~

~(R,

Or

h - qo(R z) z) -

Kw

'

This case is the generalisation of the two cases w = 0 and w tends to infinity, respectively open and closed boundary. Outside this boundary (r > R) a fixed water level is assumed, generally taken as the reference level for the groundwater heads o r - drawdowns (~0 = 0), but sometimes given as a sudden rise or drawdown h of the surface water level.

394

(541-542a)

Three-dimensional axial symmetry

In general, the lower boundary of the aquifer will assumed to be impermeable; hence Oqg(r, D) -- O. Oz

These two boundary values being given, the flow is further governed by the values along the boundaries r - 0 and z = 0. a. The boundary values along r = 0 and z = 0 are continuous

go to 542

Continuity along the z-axis (r = 0) or in the upper plane of the aquifer (z = 0) means that along these boundaries no jumps will occur, either in the values of the head or drawdown qg, or in its derivative, the flux. In many cases these variables have constant values (zero included). go to 543

b. There are discontinuities in some boundary values

As soon as abstractions or injections are involved in the groundwater flow, jumps in the boundary values may occur, for instance, in the case of partially penetrating wells or ditches, or infiltration via circular disks, etc.

542. Continuous boundary values 02q )

10q)

~_ _ .

---~ Or - r -~r

~_ ~ozq) __ OZ2

[ ~e -Oq) ~

(non-steady state), (steady state).

0

--

Subdivision" a. No line well along the z-axis. b. Fully penetrating line well along the z-axis. a. No line well along the z-axis

542.01 (FS) ~0=0

KD R

Circular island with resistance at the vertical boundary and open horizontal upper boundary. Sudden fall of the surface water. Steady state, q9 - q)(r, z) = drawdown. O~o (0 z) -- 0

Or

Z

rp(r, 0) - 0,

ogO_Z(r , ' D) -- O. Oz

'

0~o '

r(R ' z ) -

h - qg(R, z) Kw

(542a)

Flow inside a circular cylinder

4h •

q)(r,z) --

oo Z

Jr

i0 \ .~.~_ m Jrr ]

1

mzrR

]~(r,z)[--8Khr

Z

mzrR] '

)+Io(

m rrr

1

23

/

[ m rrz

11 (TD-) cos ~,TD-)

m Kw

m - - l , 3 ....

m zrz sin [~,Tg)

mrrR

m Kw-5-ff-ll(

m - - l , 3 ....

oo

.

395

mrr

I1

{mrrR'~

mrrR

"

+ Io(T6-)

542.02 (L,FHR) Non-steady state of 542.01. (X)

Ejo(Otnr~

(c~ + e2)Jo(o6,) E(otn, m, z, t),

~o(r, z, t) -- h - 2h

e --

n--0

with o~,~ being the roots of

'

'

and with

or J 1 ( o / ) - - e J o ( o t )

E(o~n, m z t) -- ~ - ~ ( - 1 ) m p

Otn(2mD + z) 2R

m--0

+ Z(_l)m

P

Otn'v/7 '

fiR

an(2mD + 2D2R

m=0

Kw'

z)

C~n~/~ ' fiR "

542.03 (FS) As 542.01 but with open vertical boundary. Steady state, qg@, z) - drawdown.

9=0 ,_-_-_--.--.----_-.-:_--.--::.-.-.--_--.-.-.-.-.-~w::

r

6o0__..2,(0, z) -- 0

S, KD

Or

R

qo(R z) -- h '

q)(r, O) - - 0 ,

'

'

- - ( r , D) - - 0 . Oz

Z

4h 99(r, z) -- ~ 7r

•

oo

myrr

1 Io~,Tb-) -

m = l , 3 ....

Ig:(r,z)l-Sghr

(mrrz]

[ m:rr R

m Io ~--g-ff-) -

Z m = l , 3 ....

sin

2D ,/'

mrrr] [ mTrz ). _1 I I ( -Tfi-: m IO(m~n~ COS 2 D ~ !

542.04 (L,FH) Non-steady state of 542.03. O0

•

Jo(Otnr_._.R_)

~o(r, z, t) -- h - 2h ~_~ E (Oln, m, z, t) n=O OtnJ1 (Otn)

with Otn being the roots of Jo(oe) - 0 and E(ol,,, m, z, t) according to 542.02.

396

(542a)

Three-dimensional axial symmetry

542.05 (FC)

]~ = 0

Equally distributed constant precipitation on a circular island with resistance at the vertical boundary. Steady state. ~p(r, z) = head.

M

r

& KD

•

w

qg(r,z) =

pRw

+

2D

Oq)

Oq9

- - ( o , z) = o , Or Oq9 ~(r, Oz

Or

P K'

O) -

P (R 2 _ r 2q_2z 2 ) 4KD

=

-

~

Kw

Oqg(r, D) -- O. Oz

_

[nTgz

1

2pO

cp(R, z)

~ ( R , z )

Io (~-c) cos ,---ff- )

i

yc2K

1 n2

n=

nJr --ffgwI

( z)

[~p(r, z ) [ - prcr 2 1 - -~

•

[nzrR 1\ D

n=l

n2

'

nzr r ?ITgZ I1 (-b--) sin (~,--b)

__1

4pDr Jr

nrrR ) -qt- 10( "-D-)

nrr

g w I l ( ' -nzrR 5 - ) + I0( n rDr R]]

"

542.06 (L,FHR)

Non-steady state of 542.05 with ~p(r, z, O) = O.

•

qg(r, z t) -- 2 p R

'

K

~ n~o

sJo(

t~n r

--8--

)

= c ~ ( ~ + e2)jo(c~)

E (an, m, z, t)

R with otn being the roots of ctJ1 (or) -- e Jo(ot), e -- -K--ww' and

E(an m, z t) -- - E Pconj ~n(2mD + z) t~nv/-[ ' ' 2R ' fiR m--O - Z

Pconj Otn(2mD + 2D - z) O t n ~ 2R ' fiR m----0

"

542.07 (FC)

~o=0

As 542.05, but with open vertical boundary. Steady state, qg(r, z) = head.

P 0q9

s

1 gdSgS~

e(R,z) =0,

- - ( o , z) = o , Or Oq9 i(r, Oz

O) =

P K'

~oO__(r, ~. D) -- O. Oz

Flow inside a circular cylinder P q)(r,z) = ~ ( R 4KD

•

(542a)

397

P ( z - D-~-) 2 - r 2 + 2z 2) - -~

2pD ~

1 Io(~~-) n'2K n=l ~ I°(nTrR-E-)

IO(r,z)l - pzrr2(1--~)

k----b--/'

4pDr ~

Z

l I1( n:rrr]D, sin (nrcz]

n=l

?12 IO(---6-) nTcR

D:

542.08 (L,FH)

Non-steady state of 542.07 with rp(r, z, 0) -- 0.

2pR ~ •

~o(r, z , t ) - -

Jo( °tnr

--if-) E (oln, m, z, t ) c~2 J1 (Oln)

K m--0

with oln being the roots of Jo(ol) = 0, and E(oln, m, z, t) according to 542.06. 542.09 (FCR) Z

Circular island with resistance at the vertical boundary and a semi-permeable layer at the horizontal upper boundary. Sudden fall of the surface water level. Steady state, q9 = qg(r, z) = drawdown. (z-axis cho-

qg=0

KD

sen upwards and r-axis chosen along the bottom of the aquifer.)

w

N388

0~o Or

Oq) ~(R,z) Or

~ ( 0 , z) =0,

Oq9 Oz

~(r,

0) = 0 ,

Oq9 ~(r,D) Oz

z)[- 4zrghr

Z n=0

h - q)(R, z) Kw

~o(r, D) = - ~ Kc

Io(~"r~

sin ot~ qg(r, z) - 2h Z ~ ) n=0 an (1 + ~2+62

I~(r,

=

~n Kw -ff I1 (unR

sin ol~ otn(1 + E ) o~2+~2

with C~n being the roots of c~ tan ot - e -

D

K---S."

I1 ( --D-) otnr sin ( --O-~nz~] Kw -~ 11( - ~ ) + Io(~---~)

398

(542a, b)

Three-dimensional axial symmetry

542.10 (FCR)

Z

As 542.09 but with open vertical boundary. Steady state, q9 - qg(r, z) - drawdown.

~o=0

399(0 z) -- 0, Or ' 3q9 ~(r,O) --0, 3z

"///////////////////////~ KD

c R

¢p(R z) -- h ' ' 099 ~o(r, D) ~ ( r , D ) -- - ~ Or Kc

D,

r

oo ~o(r, z) - 2h Z

[ OlnZ

sin c6,

Io(Z-~) cos ~-b- )

io( )

n=00tn(1 -k- o~2-k-e2 e )

17'(r, z ) l -

4yrKhr ~

sin~n Otn(1 + E )

n=0

I1 ( -D--] °tnr ~ sin ~--ff-) ( OtnZ

oe2+e2

D " with C~n being the roots of ot tan c~ - e - K---S, b. Fully penetrating line well along the z-axis

542.21 (FHR) Fully penetrating line well along the axis of a circular island with resistance at the vertical boundary and open horizontal upper boundary. Steady state. qg(r, z) = drawdown.

=~Q rp=0

;"M-

i .............................

r

lim (r 099 Q r--+O -~F ) -- -- 2 zr K D

"i";; ................ i

n

,

N

o~o ~ ( R , z )

Or

~(R, z) -

-

~

99(r, 0) - - 0 ,

Kw

099 ~(r, Oz

D) - - 0 .

cosh { C t n ( D - z) ~KD Qz

[~(r, z ) l -

= (or2 + 82)J~(oln) 2 Q r 00

--5 + --~

jl (otnr]

= (ot~ + e2)j~(ot,z)

cosh

(°tnRD ]

sinh {an~(D - z)}

cosh ( ~)an D

R with C~n being the roots of or J1 (or) - eJo(ot) and e - xw"

Flow inside a circular cylinder

(542b)

399

542.22 (FH) ~Q "r

As 542.21 but with open vertical boundary. Steady state, qo(r, z) = drawdown.

~o=0

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

r

i gD

lim

r-+O

:

R

r

Ogo) Q ---~r 2 yr K D '

go(r, O) -- O,

,n(")

~o(r, z) -- 2rrK------D

Qz

r

Q ~Jo(

e(R,z)

=0,

690__L(r, _, D) -- O. Oz Otnr -r)

cosh { -k~" (D - Z) }

~KD ,,_-o~J1~ ( ~

cosh ( ~ 1

~" Jl (q-ff- ) sinh { -T(D - z)}

2 Qr ~

IVy(r,z)l- --~ + --5-~=0 ~2j2(~)

cosh [~,'~"RD)

with c~, being the roots of Jo(ot) = O. 542.23 (FI-IR) As 542.21 but with closed vertical boundary. Steady state, qg(r, t) = drawdown. lim r r--+O ~

~p(r, O) -- O,

qg(r, z) --

27r K D '

-D

Qr 2 D - z + R2 " D

~(R,z) Or

=0,

Oqg(r, D) = O. Oz

Q 2Dz -- Z2 Q Jo~--U) rcKD 2R 2 + yrKD n=l Ocn2 Jg (Otn) _ Qz

[~(r, z)l

----

°~" -- Z) } ] 1 - c°sh { --R-(D cosh (Z~__D) '

2Qr ~ Jl(---~-/~nr] . sinh {o~,,_K(D - z)} +- - - D z__, 2jz(otn ) cosh ( z ~ ) n=l O/n

with a,z being the roots of J1 (or) --- 0 (ot = 0 excluded).

542.24 (SV)

~Q p i

i Kv

qg=0

Fully penetrating line well along the axis of a circular island with closed vertical boundary and equally distributed precipitation at the upper horizontal boundary. Steady state only if Q=JrRZp

or

qo(r, z) = drawdown.

p=

Q 7rR 2"

400

Three-dimensional axial symmetry

(542b)

lim

r~O

r

o)

-~r

Q

---

~----~-~(r, O) -- P

Oz

O~p

2rcKD'

Or (R, z) -- 0,

~----~-~(r, D) -- 0.

K'

Oz

R2P l n ( R)_ _ ~ P (R2_F2)+P( z2) cp(r, z) -- 2 K D r 4KD -~ z - ~ ,

Q{ r2

I~(r, z)[ - ~

~-5-(D - z) + z

]

with Q -- yrRep.

542.25 (FHR)

a

I

Fully penetrating line well along the axis of a circular island with resistance at the vertical boundary and a semi-permeable layer at the upper horizontal boundary. Steady state. ~o(r, z) = drawdown.

~o=0

U/~U/////////////////////A r ,, c 113 ,~ KD ,

,//

,

,,:

( Ocp)

lim r

r--+0

R

~

---

Q 2zrKD '

Ir Z

o~ ~(R' rO

~(R,z) z) -- --

OCP(r, 0) -- qg(r, 0____~) Oz Kc '

K------~-'

,rz,

Oqg(r, D) -- O. Oz

I'n( r + Q

c~

jo(Otnr] --R-!

JrKD

:

(Ol2n_[_ 82)Jg(oln)

Qz [~(r, z ) [ -

~

2Qr oo 4- --D--- ~

n=O

c°sh { ° i n ( D - z )

}

an Kc sinh (~-~) + cosh (~-~1 -~

jl (o~nr,~ --k--,

.

(~2 4- e2)Jg(c~n)

sinh {o~,~(D - z)}

Kc 7~" sinh (~-~) + cosh (~-~) R

with otn being the roots of or J1 (or) - eJo(ot) and e - K--~" 542.26

(FH)

~a

As 542.25 but with open vertical boundary. Steady state, qg(r, z) - drawdown.

~//2, ~/////////////////////~ r I KD

Ng88

'Z

c

j

lim

r~O

r

-- -

0 2zr K D

Oq)(r, O) -- q)(r, 0)) Oz Kc '

,

~o(R

'

z) -- 0,

Oqg(r, D) -- O. Oz

(542b, 543)

Flow inside a circular cylinder

401

Q in( R ) qg(r, z) -- 2Jr K-----D

o

~

Jo(~)

cosh { ~- (D - z ) } J r K D n=O Otn2 J? (Ogn) K c ~ sinh ( - ~ ) + cosh ( OlnD'~'---~] O[n

2Qr ~-~ Jl( °tnr]---~]

Qz

I~(r, z)l - --5 +

D

n=O o~J~(otn)

sinh {~" T (D - z)}

xc-~ sinh,, RD ) + cosh (~) ~n

( ~n

with Oln being the roots of J0(ol) = O. 542.27 (FHR)

__l

As 542.25 but with closed vertical boundary. Steady state. ~o(r, z) = drawdown.

!

•

( a~p) lim r ----

r

~o(r, z) --

r--+O ~ 2Jr K D ' Oq) qg(r, O) O~o ~ ( r , O) = ~ , ~(r, Oz Kc Oz

Or

D) = O.

2R 2

Qr 2

c°sh{ ~"T(D - z)} ] K c T sinh (~nn --k-) + c ° s h (anD , R ) ' 0ln

Jr K D n=l Ol2J g (Oln) Qz

acp

--(R, z) =o,

2Dz -- Z2

Q Qc R2 -k- JrKD Jr

o 2 ,oil)[, IT'(r, z ) l -

Q

D - z

-D-- + R2 " O

+

2Qr ~

D

~)

J l ( ~nr

n=l 0/n2 Jg (C~n)

sinh { T~n(D -- Z) } Ofn

K c --y sinh ( - ~ ) 4- cosh ( - ~ )

with Otn being the roots of J1 (or) : 0 (o/ -- 0 excluded).

543. Boundary values with discontinuities 0299 1 0~0 02(/9 Or---T + -r-57r-5-Sz~ + -- 0

Subdivision: a. Discontinuous boundary values along the r-axis. b. Discontinuous boundary values along the z-axis.

(543a)

402

Three-dimensional axial symmetry

a. Discontinuous boundary values along the r-axis

543.01 (FHR) q

~

Circular island with resistance at the vertical boundary and vertical infiltration from a circular disc at the upper horizontal boundary. Steady state.

9=0

I ro _1

.N

12

°~N

'R

;

9 (r, z) -- head. °-~ ( o , z) - o

Or

q ---K 0

O~° (r, O) -Oz

*

9 ( r , z ) --

I~(r,

for0
z)l-

J l ( ~n--R°) Jo ( unr--f)f-

n~O (0/2 -~- E2)Jg(0/n)

Or

09 ~(r,D) Oz

forr0 < r <~ R,

2 q ro ~

g

< ro,

O-£~ ( R z) -

'

'

@ ( R ~ Z)

Kw

--0.

cosh { -k~" (D - z) } sinh (--T-) ~"o

cx~ J1 ( ~ ) S l (E~.E) 4rcqror Z (0/,2 + e2)jg(0/n) n=O

sinh {~" (D - z)} sinh ( - ~ )

R with 0/n being the roots of o/J1 (0/) -- 8J0(0/) and e -- Kw" 543.02 (FH)

q ~ 44,~444,

9=0

I_~. [

]r

q

0~o (r, O) -Oz

09 ~rO( 0 '

~o(r,z)

z) -- O,

for0~
--K 0

J1 (~n~ R ) Jo (°tnr-ff)

cosh {(D--k~" - Z)}

-R---O

cx~ an r0 -glgr(r, z)i -- 4zcqror Z J1 ( )J1 ( ~ ) n=O 0/n

with o6, being the roots of Jo(u) - O .

9(R, z) -- O,

~oO__(r _t, ' D) - O. Oz

forr0 < r ~
2qro ~ •

As 543.01, but with open vertical boundary. Steady state, q)(r, z) - head.

sinh ( - ~ ) sinh { -fi~" (D - z) } sinh (o~___~)

Flow inside a circular cylinder

(543a)

403

543.03 (FHR, S)

~Q

As 543.01, but with closed vertical boundary in combination with a discharge by a fully penetrating line well along the z-axis. 99(r, z) = drawdown.

q

Steady state only if r e c h a r g e - - d i s c h a r g e .

( O~o) Q lim r ---r--+O ~ 2rr K D '

~8g O~° (r, O) _ Oz *

go(r,z) =

q -~ 0

for 0 ~< r < r0,

Oq)

~(r,D) Oz

forr0 < r <<. R,

Q

1D2

7 r K D R 2 16

=o,

=0.

1 2 1 + 8 ro - ~ ( D - z) 2 }

Jo,=)

2qro J l ( ~ r=° ) c o s h {Otn -~(D-z)}

o

n=l of2 j 2 ( ofn ) 7r K D Qz

099 Or

~(R,z)

Qr 2

K

sinh ( --V-)an o

1

J'

z

I~(r, z)l -- -b- + - T ( 1 - ~ ) OO

+ 4:rrqror ~

J1 (~nr°)J1 (Olnr~ R -~!

sinh { -~(D ~" - z) } sinh ( ~ --~)

with of,z being the roots of

J1 (of) - - 0 (of - - 0

excluded) and Q - rcrZq.

543.04 (FHR)

-o

iro:

r

_0go (r, 0) -- {,im(~o Oz

Circular island with resistance at the vertical boundary and abstraction from a circular drain at the upper horizontal boundary. Steady state, q)(r, z) = drawdown.

0 )

47r Kero

0

°E(o, z) - o,

o~o

Or

-~r ( R , z) -- -

for (ro - e) < r < (ro -+- e), f o r r --/: ro,

~oO__(r ~. ' D) -- O. Oz

a~ (D - z) } O --fr- ) cosh { -k~ X R ,,=o ( ~ + ~2)J~ (~n) sinh (g-~) OO

[~o(r, z)l-

0gn r

~(1¢, z) K--~

404

(543a)

•

2 Qr ~

I~(r, z ) ] -

R

=

with ot,~ being the roots of

Three-dimensional axial symmetry

OlnJo (~n~R ) Jl ( °tnr ]

sinh {~"~(D - z)}

( ~ + e2)jg(~,,)

sinh ( - ~ )

o~J1 ( ~ ) -- 6J0(o/)

R and e - Kw"

543.05 (FH) As 543.04, but with open vertical boundary. Steady state, qg(r, z) - drawdown. r

--(O,z) - o ,

~(R,z) - o ,

Or

~oO__(r, _L, D) -- O, Oz

.

0q9 Oz (r' 0) -- { ~--->o lim(-

)

for ( r o - e) < r < (ro -k- e),

47r K ero

f o r r :/: ro .

0

cosh {~"(D - z)}

Q

•

cp(r,z) --

Jr g R .=o

sinh ( - ~ )

or. J~ (o~,,) Otn r

•

[~p(r,z)12Qr

--fi- )

Jo(~)Jl(

sinh { T~n(D - z) } sinh ( - ~ )

with an being the roots of Jo(u) - 0. 543.06 (FHR,S)

~Q 7

1, ro

Q ~

As 543.04 but with closed vertical boundary, combined with a discharge by a fully penetrating line well along the z-axis. ~p(r, z) = drawdown. Steady

r

state only if recharge = discharge.

i

' '

R

KD

lim r

r~ O

0qg(r, 0 ) _ Oz ¢p(r, z) --

{lim( +Q ) E~o 47rk~ro 0

+ ~

-~r

----

Q 2 rc K D '

for(ro-e)
a--' (R z) - o, Or

'

~099 (r,D) Oz

for r :/: ro ,

Q {2D 2 + 3r 2 - 6 ( D 12RZD

.

O~o)

,ot t [ l

,,=1 ~ n J 2 ( ~ )

Dctn

z) 2 }

,ot tcosh/ ~ ( D R sinh (z_~)

z)}

] '

--0.

Flow inside a circular cylinder Qz

(543a, b)

Qr 2

405

z

I (r,z)l - -ff +--F. (1--6) 2Qr ~-~

+ -s

Jo(~n~R)Jl(~n?)

=,

with oen being the roots of

Z 70 (-Z

J1 (o/) - - 0 (o/ - -

sinh {~-(D - z)} sinh (~)~Zn D

0 excluded).

b. Discontinuous boundary values along the z-axis 543.21 (FCR)

Z

Partially penetrating well in the centre of a circular island, with resistance at the vertical boundary and semi-permeable layer at the upper horizontal boundary. Steady state. 99(r, z) = drawdown.

tp=0

C U//2~/////////////////////~ liC L~o

"Ia

R

KD t

~&a ~ ~

w g&N r Q

_ lim {r Oq),~ ]

r--+O

2:rrKl

-gr

0

099 99(R, z) ~ ( R , z ) -- - ~ , Or Kw 9o(r,z)

for (a

1/)
(a+

- -2

1

I)

-2 ' ( _1 ) ( 1 ) forO~
099 99(r, D) ~ ( r , D ) -- - ~ . Oz Kc

Q °c [ (o6,r)(olnr~} (oenz) 7rKI ~ S(°en) A(oln, w)Io --5 + KO ---~-/ cos - - 5 ,

n--0

{ I ~S(°ln)A(°ln'W)II( Otnr --~-)

]gr(r,z)lgQr°°

n-0

with

S(oln) -

2 cos (~-~) sin (-~-~ 2) +

8

,

A(oe,~,w) --

OlnZ ) -- K1 (~-~) } (sin,--D-KWognK 1 (ot_~) __ OKo(U__n_~) K woln i 1 ( ff_n~) _qt_O lo ( ff_.~ )

D with Otn being the roots of c~ tan c~ - e - x-7" 543.22 (FCR) As 543.21, but with open vertical boundary"

Iz II!]4~,a RKD

99(R, z) - O (w - O). Steady state. 99(r,z) - drawdown. •

1 r

~o(r,z) -- (see 543.21),

406

(543 b)

Three-dimensionalaxial symmetry

with

lTt(r, z)l -- (see 543.21) KO ( otn R

D ]

A(otn, O ) - - io(~n6R-----~ •

543.23 (FCR) As 543.21, but with closed vertical boundary" ~Or( R z) - 0 (w - oo).

~Q

KD

~ •

•

~o(r,z) -- (see 543.21),

•

lap(r, z)l -- (see 543.21)

A(otn, oo) --

with

K1 (°t--~)

I1 ( otn R_._)5_

543.24 (FC)

z

Partially penetrating well in the centre of a circular island, with resistance at the vertical boundary and with open upper horizontal boundary. Steady state.

tp=0

KD

~!

~p(r, z) = drawdown.

li~,!-,=~Q w (Ta R g888~

~ggg~gSg88~N~ ~ r

(8~o) lim r r-+0

I O

~

_e(R, z)

O~°(r, O) - O, 8z

Kw

qg(r, z) -- Q rcKl

( a + 1/),

for0~
0

a!(R, z) -

Or

for (\ a - ~l l ) < z <

2zrKl

l l ) < z ~
~o(r, D) -- O.

{ (mrcr)(mrcr)] S(m) A(m, w)Io 2D + Ko 2D

(mrcz) cos

2D

'

re=l,3 ....

lap(r, z)l -- 2Qrl

~°°

mzr r mzr r } mrr z S(m) { A(m, w)Ii\ 2D ) - K I ( 2 D ) sin ( 2 D )

re=l,3 ....

with

S(m) --

4 cos (mrra2D]~ sin m~

mrrl (-o-)

[mrrR]

[mrrR]

Kw" ~ KI ~-5-B-j - DKo~ 20 J A(m, w) CmrrR'~ Kw 2 11CmrrR'~ ~, 2D J + Dlo~--gg-j

,~ ~ >

•o -

( , ;~; +

(a

'~)~-~ ze

v) pu~ (l T~ -

'0-

|--H

= I(~'~)'~1 0<3 [--"H

O0

~-d + ~ z~ (0 ' - 0 ~ d)e

u~

o

~_mX

= (2 '.0o~

= (2 '8)

o/

,,) > ~ o , o ~

>

.,t 888 II

•uaxopaxeJp- (2' '.Od~ "alms dpvalS "fiJepunoq imuoz~.ao~1 aaddn pasola ql!A~ pue ,(J~punoq IVm.laaa oql 1~ aouvlg~.gaa ql!~ 'pu~Is! J~I -noJ[.3 1~jo OJlUO3 Oq] HI iio~ gul.lvJlauad dllm.lar~d

8 o=o~ (3a)

• (~-~) tI (/~

q~!a

= ( ~ 'uO V

,

'(17~;'~17~ oos) - (z '.0o~

*

(~,~:'~; o o s ) -

(z '-04

•(cx: - a ) 0 -

L~:'f:ts

-/gggs~

(z 8 ) ~

0=tb

• (_%)oz

2

(Dd/ 9Z'£17~;

(~)o:~ ql!,~

= (o ' u O v

d~X~

('¢Z:'t~'¢~;o o s ) - (z '-04

* (/X

'(t,Z;'t~17~; o o s ) - (z 'J)~

,

vI , hr

"uaxopax~Jp = (z'.Oo~ "awls ,(pva2S " ( 0 - o n ) 0 (z'8)~ •fiJ~punoq IVm.laaa uado ql!ax mq '17~'£17~ s v

Off"-

0=o~

2 $~,'£,t'$

LO~¢

(q£~¢~)

ldpu~.lda .mlnan.:> v ap~.~u! mOld

408

Three-dimensional axial symmetry

(543 b)

with { n rr a

S(n) =

2c°s~,o)sin(

mr l

)Sff

A(n, w) =

rtTr

KwnTrK 1( -nrrR - 5 - ) - DKo ( . ~ ) K w n r c I1

nzr R + DIo (--D--) nrr R (--b--)

"

543.28 (FC)

_•••• a

As 543.27, but with open vertical boundary: 9 ( R , z ) = 0 (w = 0). Steady state. ~p(r,z) = drawdown.

=0

R

r

•

~o(r,z) = (see 543.27)

•

~p(r, z) = (see 543.27)

with w = 0, with

Ko ( nrrR.._).f_ ~

A(n, O) -- -

I0(nTrR----ff-) "

543.29 (FCR)

Tz

I

9=0 c

Point well at the axis of symmetry of a circular island with resistance at the vertical boundary and semi-permeable layer at the upper horizontal boundary. Steady state, qg(r, z) = drawdown.

U/A z//////////////////////A

~

~

~ r

09 lim ('°2 -~p) -

Q 4n'K

p--*0

O!(O,z)--O Or

with p -- ~ r 2 -+- ( z -

forz#a,

Q

q)(r,z) =

,

[~r(r, z ) [ -

Or

'

~ ~R ~ Z)

Kw

O--~(r, D) - _ 9 ( r , D____~) Oz Kc

aCP(r, O) -- O, Oz

•

a ~ ( n z) -

a ) 2,

zrKD n=O D

~S'(°tn)

)+ A(°tnW)II( D '

~r~

T,

\--D-(anr

~z~ cos (W,' --D-)°lnZ~ with

n=O S t (otn) =

COS (k ana '~ 8

1 + 0t2+82 D and A(czn, w) - (see 546.21) with Oen being the roots of ol tan ol - e - K---7"

Flow inside a circular cylinder

(543b)

409

These solutions have been derived from those of 543.21 by letting l approach zero"

lim I2 sin ( o,nl] ] ~ 2D

]

1 _

In all problems 543.21-543.28 the partially penetrating line well may be replaced by a point well yielding new solutions, which can be found by taking the limit for l --+ 0.

(550, 551 )

410

Three-dimensional axial symmetry

550 BV-5. THREE-DIMENSIONAL AXIAL-SYMMETRIC GROUNDWATER FLOW OUTSIDE A CIRCULAR CYLINDER

a. Has the soil discrete inhomogeneities in r- and~or z-direction ? go to 560 (BV-6) An example of this category of problems is a layered aquifer with a discontinuous variation of the transmissivity of the layers.

~///////////////////////////////,¢ c KID1

R

K2D2

b. The soil is (assumed to be) homogeneous

go to 551

Also anisotropic homogeneous soils belong to this category, provided that the main directions of the anisotropy coincide with the r- and z-direction. Solutions for anisotropic cases can easily be derived from analytical solutions for corresponding isotropic problems (part B, Section 2.3.4). Practically all groundwater flow problems concerning flow outside a circular cylinder relate to flow in an aquifer with finite thickness and generally is referred to as flow outside a fully penetrating circular pond (basin). For that reason solutions for flow problems, concerning circular cylinders with surrounding flow fields that extend to infinity in vertical direction, will not be given here, as being of no practical use.

551. Axial-symmetric flow outside a circular basin 1 8~p

82~ =0.

~r---~ - r ~ 551.01 (FS) ~0=0

r

Circular pond with resistance at the vertical boundary and open horizontal upper boundary of the surrounding aquifer. Sudden fall of the surface water. Steady state. ~o(r, z) = drawdown.

~q9 ~ ( R , z )

=

Or ~o(c~, z) = 0 ,

-

h - qg(R, z) Kw

(55 l)

Flow outside a circular cylinder

~o(r, O) -- O,

*

(p(r,z) --

60a__~(r, , D ) -- O.

Oz

4h

mrr r

~ ~

]O(r, z ) l -

mrr z

1 Ko ~Tb-) sin (-Tff j -mTr K1 (mzrR m K w -Uff [mrrR] 7r T d - ) + Kot 2D ] re=l,3 .... O0

•

411

8Khr

[ mJrr

Z m=l,3 ....

mrrz ~

1 Kl~ 2D ) COS ( T D - ] mTg K1 (mTrR] m K w 7-ff [mrrR] 2D I -~- K O t - ' ~ - j

551.02 (FS) ~o=0

As 551.01, but with open vertical boundary.

h!] ............................. -

q)(R, z) -- h,

~o(oo, z) -- O,

~o(r, O) -- 0

aq) (r, D) -- O. Oz

r l .............................

r

'

myrr

4h

1 Ko~,TB-) (mrrz) q)(r, z) -m Kogm~R~ sin 7[ \ 2D ] 2D re=l,3 .... [ mTfr

•

~ .~.

]~(r,z)l--8Khr

K1I ~ , ~ ] _ cos(mTrg) [m~rR]

re=l,3 .... m K o ~ z - f j

2D

551.03 (FCR) Circular pond with resistance at the vertical boundary and leaky surrounding aquifer. Sudden fall of the surface water level. Steady state, qg(r, z) -- drawdown.

tp=0 ~//////////////////~ c

o~9 N888

Or

Oq9 --(r,O) Oz

-

-

Kw

~o(oo, z) - 0 ,

r

Oq) --(r,O) -0, Oz

h - ,9(n, z)

~ ( n , z )

R

- -

q)(r, D) ~ Kc

sin ~,, ~o(r, z) -- 2h ~ n=O C~,,(1 + ot2+a¢2)

K,,,

0/y/

K,

c°s ( -e) Xo(

)

')

412

(551)

Three-dimensional axial symmetry

K1 ( --E-) Otnr sin ( ~ z h -b-J K w -.~. Kl (-q-~ ) -]- Ko ( °tnR

sin C~n

0l n

n=0

c~2+e2 D r-~."

with an being the roots of ot tan ot - e 551.04 (FCR) Z

As 551.03, but with open vertical boundary. Steady state. ~p - qg(r, z) - drawdown.

tp=0

m _

cp(R, z) - h,

~///////////////,~ c

OCP(r, O) -- 0 Oz '

KD

R

oo

qg(r'z)--2hZ

z)[- 4 y r K h r

Xo( )

ot2+e 2 )

co

Z n=0

O~P(r, D) -- _qg(r, D_____~) Oz Kc

K0 (~n r

sinan (1+ e

n=00~n

]~(r,

q~(cx~, z) - O,

K1 (otnr -b-) sin (z-~)

sin c~n

Otn(1 +

e

)

ot2+e 2

with Otn being the roots of ot tan c~ -- e -

D K---~."

Ko(anR] D I

Flow with discontinuous parameters

(560-561)

413

560 BV-6. THREE-DIMENSIONAL AXIAL-SYMMETRIC GROUNDWATER FLOW WITH DISCONTINUOUS GROUND PARAMETERS

a. The soil has discrete inhomogeneities in r-direction

go to 561

See comments of 500d and 530a. b. The soil has discrete inhomogeneities in z-direction

go to 562

See comments of 520a, 540a and 550a.

561. Discontinuous ground parameters in r-direction 561.01 (FC) Equally distributed constant precipitation on a circular island with two different transmissivities in r-direction. Steady state, q)(r, z) head.

P

----7

(/9-

0 2 q)i 1 0(t9i 02q3i Or-----T + -r -~r + Oz 2 = 0 (i -- 1, 2), Oq)l ~(0, Z) -- 0, q)2(R, z) -- 0, Or

R1

R2

Oq)i ~(r,O) OZ

P -- - ~ , Ki

Oqgi ~(r,D) OZ

P (R2-r q91(r, z) -- ~ gl

2

¢p2(r, Z) --

--0.

+ R~ - R1~ +

X~

~

E ~ltr,.)cos n=

1

2 ~ (n:rrz] P (R 2 - r 2) + ~ q32(r, n)cos 4 K2 D --D \--D-- / ' n=

(ol(r, ~) --

{~01 f o r 0 ~< r ~< R1, q)2 for R1 ~< r ~< R2,

A(n)

Io~'r

U(n----~

pD 2

"

--D-J + n27r2 K1 nrrr)

N(n)

---D-/

+

A(n) -- pKz(K1 - K2) KI \

+ 11

nzrR1 D

1

C(n) N(n) D

,o(

nrcr] D ] D

Ko D

nrrR1

pD 2

+ n27r2K2

)

--6-'

414

(561)

{

C(n)=-PK1

Three-dimensionalaxial symmetry

) (rlTr:R1)

i1( nTrR1 + K1Ko ( rtyFRl D D)

KzIo\(nJrDR1 K1 k D

+(K2-K1)II(nT~R1)K°(

n~R2 O ) '

n27r2

[ (nzrR1 ( nJr R2 -~ K1K2 (K2 -- K1)lo nrrDR1) i 1 D )KO, D )

N(n) --

(nzr R2 { nTr R1 nzr R1 +/ok D )Kz/o( D )KI( D ) + K1K°( nrrR1D )II(nrrR1D

//]

561.02 (FS)

go=Oi Q ........................

K2

1--1 i |

!

!

!

r

K1 ' K1 !

t

Kz

D

Fully penetrating line well with a constant discharge in an aquifer with two different transmissivities in r-direction, under open water, go - go(r, z) = drawdown. Steady state.

R '1' R !

i

go--

gol for 0 <~ r ~< R, 92 forr t> R,

029i 1 39i 029i = 0 (i = 1, 2), Or 2 + - r -~r + Oz2 0gol) = Q (/92(00, Z) -- 0, lim r r--,O ~ 2Jr K1D ' ~Ogoi ( r , D ) --0. goi (r, O) = O, Oz

91 (r, z) =

2Q 7rZK1D +

E°0

n=l,3 ....

_1{ Ko (nzrr) -~ n

nrrR (K1 - K2)Ko~,( nrr 2DR ]I K 1( "~-D-) R ] + Kz lo ( "Tb-) nr~R K1 [,'Tfi nrrR-,] K1 K° (n2-~) I1 [, n~ 20,

( nrrr ) } (nrr z ] x / o k 2D / sin \ - - ~ - / , O0 (mrr] 4Q ml Ko ~TO: sin (n~zTff) go2(r, Z) -- :rr3R Z nrrR )I1 ( 2mr0R, ~ + Kzlo('~-ff', mrn ~K1 ('~--) nrrR " n=,,3 .... n 2 " K1K o [,20

Flow with discontinuous parameters

(561)

415

561.03 (FCR)

Circular basin which level differs from the surrounding polder level. Different ground parameters for the basin and the polder. Steady state. ~o = ~o(r, z) = drawdown.

C1///"///////~ ~////////A, ~//////////////////////// C2 KzD

g2D

q91 f o r 0 ~ < r ~< R, q92 f o r r ~> R,

(/9-im

R

02q9i ~__1 0(t9i + 02q9i "---Tar r--~r 0~Ol ~(O,z) -o, Or

Oqgi Oz

(r, O) -

Oz2

=0

(i-

1,2),

K1Cl ~Oqgl (r, D) -- -qgl (r, D),

q92(cx:), z) -- h,

0~02 (r, D) -- h - q92(r , D). K2c2 -~z

O,

Continuity conditions: @1 (R, z) -- q)2(R, z),

0(/91

0~2

K~ --ff-r(R, z) - th --ff-r(R, z).

Also

f0 ° g)l(R, K1

/o

z) dz --

f0 ° q)2(R, z) dz

D 0(/91 (R z) dz -- K2

-~r

'

/o

-L-r

and (R, z) dz,

leading to the approximate solutions:

Oln sin2 fin K1 (P_~)

([91(r, Z) "~ 2h K2 Z

n--0

fin sin O/n G'F"(~nl ~n)

,o( nr)cos,T)

~ oen sin f l n l l ( - ~ ) ( _ ~ ) ( f l n z ) q92(r, z ) ~ h - 2 h K 1 E flnEfl-~i~- ~ KO cos n---0

F(otn, f l n ) - o t n K 1 K o ( ~ ) I I (

---5/,

°[nR~ D / + flnK2Io( ° lDn R/~ K I ( ~ )

D and fl~ the roots of fl tan fl -- ~ with or,, being the roots of ot tan c~ - s - K-~ and E B -

1 + B~+~a"

D K2c2

416

Three-dimensional axial symmetry

(562)

562. Discontinuous ground parameters in z-direction 562.01

(SV) z=d0=0 i

---r

!

K~

al~::

D1

K2

a2~ii

DE

z=dl

d2

ai --,

bi

Di

ai~

gi

tpi di

Ki

Di

di

dn-2 i"

Qn_l¢=

!

,

Kn-1

Dn-1

d~_l

:

K~

Dn

d~

i!

Qn¢=

Three-dimensional axial-symmetric groundwater flow in a multi-layered infinite aquifer, towards wells located in a vertical line. The wells are partially penetrating each layer and have arbitrary but constant discharges. Steady state.

di-1 ~ z ~ di

(i-- 1,2 .... ,n),

q9i --" qgi(r, z) "- drawdown. ~2~i

-Or -T-

~---1 Oq9i -~- ~~2~i -- 0,

r Tr

0z 2 _

( Oqgi~ lim r r--~O Or / ~i ((X:), Z) --" 0.

I

0

ai

2rrKi(bi - ai)

for (di-1 -]- ai) <~ z <~ (di-1 -t- bi), for di-1 <<.z <~ (di-1 + ai) and (di-1 + bi) <<.z <~ di,

Boundary conditions along z = di (i = 0, 1, 2 . . . . . n): Z -- do - - 0 " Otlqgl(r, 0) -q-/~1 0@1 (r, 0) - 0 (Ogl,/~1, O/n and fin are constants, zero included),

(562)

Flow with discontinuous parameters

z -- dn" Otnqgn(r, dn) + fin --~z(r, dn) -- O, Z -- di (i -- 1, 2 . . . . , n - 1)

qgi(r, di) - qOgi(r) (unknown beforehand)

(ctl and ill, ctn and 13,, not equal to zero at the same time) (Also Otl and otn, in order to guarantee a steady state.) O0

q9i (r, z) -- ~

*

Gi (k, z ) K o ( X ~ r )

(i -- 1, 2 . . . . , n),

k=l O/1 sin(X~z) - )~kfll cos(XkZ)

G1 (k, z) -- gl (k)

Gi (k, z) --

O/1 sin(XkD1) -- Xkfll cos(~.kD1) '

gi-1 (k) sin{,kk(di - z)} ÷ gi(k) sin{,kk(z -- di-1)} sin(,kk Di )

(i - 2 , 3

..... n-l), C~n sin{Xk(dn - z)} - Xk/Sn cos{Xk(dn - z)}

Gn (k, z) = gn-1 (k)

gi (k) -- gi -- Pi

ol,~ sin(~.kD,,) -- )~,/~,~ cos(,k~Dn)

n-1

E~=I (Ui + Ui+l)p 2 + Ei"=-12ViPi-lPi

Qi 2rcKi(bi - ai)

qi -

n-1 ~-'~i=1 (qi Wi -k- qi + l Xi + l ) Pi

(pl - 1) (i - 1, 2 , . . . , n - 1),

1 sin()~k Di) cos(,kk Di) } Ui -- sin2(X~Di)

Ti

V/

Wi

Xi

~,kDi

2

{ sin(Xk Di)

sin2 (~.~Di) ~"

2

~,kDi

-- COS(~,k Di ) },

~.kDi sin(,kkDi) { cos(X~ai) - COS()~kbi)},

T, )~kO i sin()~kDi) [ cos {~,k(Oi -- bi) } - cos {~,k(Oi -- ai) }],

Ti - K i D i ,

(i - 2, 3 , . . . , n -

1),

T1 { ot 2 + / 3 2l~'k2 .q_ /~2X2 _ or2 . sin(,kkD1) cos(~kD1) U1 -- N2 2 2 XkD1

-- 0gl/~ 1 sin2(XkD1) },

417

418

(562)

Tn Un -- N 7

2

otn + flnXk + 2

Three-dimensional axial symmetry

sin(XkDn) cos(XkDn)

2

~kDn

- otnfln sin2(~.kDn) [ /

with N1 - Otl

sin(XkD1)

--

Xkfll

cos(~kD1),

Nn -- otn sin(XkDn) - ~kfl,, cos(XkDn),

T1 [ C~l { cos(~ka~) -- cos(~kbl)} W1 -- ~ l l

~,k D1

D1

{ sin(Xkbl) -- sin(Xgal)}],

~ n ' n { COS(,kk Dn - Lkbn) - COS(~,k Dn - ~kan) }

Xn --

+ -~n { sin(XkDn -- Xkbn) - sin()~kDn -- )~kan)} •

)~k (k -- 1, 2 . . . . ), should be solved from the goniometric equation: B1 + B2 -A2 0 D =

0 0

-A2 0 B2 "-t-B3 -A3

0

-Ai

0

0

Bi -t- Bi+l -Ai+l

-An-2 0

0 0

Bn-2 + Bn-1 -An-1

-An_ 1 Bn- 1 -4c-Bn

in which Ki Ai "- sin(,kk D i ) ' B1 --

K1

Bi--

Ki cos(Xk Di ) sin(~.k Di)

Otl c o s ( X k D 1 ) -+- Xkfll sin(LkD1) 0/1

sin(,kkD1)

-- )~kfll COS(~.kD1) '

Otn COS(~.kDn) + Xkfln Bn = Kn

(i -- 2, 3 , . . . , n -

sin(XkDn)

Otn sin(~.kDn) -- ~,kfln COS(~.kDn) '

Pi (i -- 2, 3, ..., n-- 1) should be solved from the n - 2 equations:

1),

=0

Flow with discontinuous parameters

(562)

419

-A2P2

= -(B1 +B2)

(B2 -k- B3)P2 - A3P3

= A2

=0

-A3P2 + (B3 + B4)P3 - A4P4

=0

-AiPi-1 -k (Bi q- Bi+l)Pi - Ai+lPi+l

-An-2Pn-3 nt-(Bn-2-k- Bn-1)Pn-2- An_l Pn_l = 0

562.02 (SV) do=0 U///7//~Uf/f////J/f//////J/f//K{ D ,_ d 1 r .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

K2D2 d2

K3D3

~g88g&g8~ z (/91 (r,

d

thick semi-permeable layer with relatively small resistance, considered as a threelayered aquifer with an impermeable base and a zero drawdown on the top. Fully penetrating well in the second layer. Special case of 562.01 with: n -- 3, Q 1 -- 0, Q2 -- Q, Q3 = 0, a2 -- 0, b2 -- D2, /~1 = 0 ,

3

z) -- Z

Two-layered leaky aquifer with a relatively

gl

k=l

sin()~kz) sin(XkD1)

O/n - - 0 .

Ko(Xkr),

oo

992(r, Z) -- Z k--1

gl sin {Xk(d2 - z)} + g2 sin {Xk(Z -- dl)} Ko(Xkr), sin(~,kD2)

oo cos {Xk(d3 - z)} Ko(Zkr), ~03(r, z) -- Z g2 cos(XkD3) k=l

K1 D1

{1 2

U1 -- sin2(XkD1)

K2 D2

{1

U2 - sin2 (2.k D2)

2

1 sin(,kkD1) cos(~,kD1) /

2

~,kD1

~'

1 sin()vkD2) COS()~kD2)

2

)~kD2

/ I'

K3 D3 [ 1 1 sin(XkD3) cos(,kkD3) / U3 = COS2(/~'kD3) I + - 2 ~kD3 I' K2 D2

{ sin()vk D2)

V2 -- sin2 ()~kD2)

cos()~kD2) } '

)~gD2 K2 D2 { 1 - cos(Xk D2)}

W 1 -- 0,

W 2 --

,kk D2 sin()~k D2)

-- X2,

X3 - 0,

420

Three-dimensional axial symmetry

(562)

(X2 + P2 W2)q2 gl

(U 1 -~- U2) + p 2 ( U 2 -3I- U3) + p 2 V 2

B1 + B 2 g2 -- P 2 g l --

g 1,

A2

K2 A2 --

B1 = K1

sin ()~ D2)' COS()~kD2)

B2=K2

Q

q2

2zr K2 D2

COS()~kD1) sin (~.kD1 ) ' sin (&~D3)

B3 = - K3

sin (~.kD2)'

cos (~.kD3)'

L~ (k = 1, 2 , . . . ) from: O

B1 -+- B2 -A2

-A2 = 0 B2 -at- B3

or

KI K2 cot()~D1) cot()~D2) - K1 K3 cot(~.D1) tan(*D3)

-- K2K3 cot(~.D2) tan(~.D3) = K2.

562.03 (SV) 9=0

Three-dimensional axial-symmetric groundwater flow in a multi-layered aquifer in a circular island, surrounded by surface water in which a sudden drawdown h takes place. Steady state.

-Ih Z=0

K1D1

dl

d i - 1 <~ z ~ di

di-1

KiDi

t di

0299i 10q9 i 02q3i Or 2 -+- -r ~ + OZ2 = O,

I 1 I

Oftgi

~(0, Or R

z) -- O,

qgi(R, z) = h.

Boundary conditions along z 562.01. O0

q9i (r, z) -- h ~

k=l

Gi (k, z) lo(~.kr____.__~)

•

Io(,kkR)

with Gi (k, z) as in 562.01 provided that

qi = h,

(i -- 1,2 . . . . ,n),

ai = 0

and

bi = Di

(i = 1, 2 , . . . , n).

di:

see

421

(562)

Flow with discontinuous parameters

562.04 (SV) ~o=0 ~-z=O K1D1

dl di-1

di

r

Three-dimensional axial-symmetric ground-water flow in a multi-layered aquifer outside a circular basin with surface water in which a sudden drawdown h takes place. Steady state.

di-~ <. z <. di KiDi

02q9i

10q3 i

(i-

1,2 . . . . . n),

02q9i

----T Or + -r -~r + ~OZ 2 = O, dn-1

99i(R, z) - h,

K~Dn

qgi(00, Z) - - 0 .

Boundary conditions along z - di" see 562.01.

R

oo

q9i (r, z) -- h ~

Gi (k, z)

k=l

with Gi(k, z) as in 562.03.

Ko(~r) Ko()~kR)

Orientation table BVI

(600)

BVI. T H R E E - D I M E N S I O N A L

ORIENTATION

423

GROUNDWATER FLOW

BVI

TABLE

600. General three-dimensional groundwater flow a. Does the flow field theoretically extend to infinity in all three directions ?

go to 610 (BVI- 1)

General three-dimensional flow has components in three directions and has no point of symmetry as in spherical flow or a line of symmetry as in axialsymmetric flow. A flow field extending to infinity, for instance along the x-coordinate, means that the flow variables vary from x -- - c ~ till x -- +cx~ (infinite field) or from x = 0 till +cx~ (semi-infinite field).

b. The flow is finite in at least one of the three directions f

/

/ / f

/ k

/

:X

._,

D J 7 f f f_

, 7 _.t._ ~ , .

go to 620 (BVI-2)

Finite flow in a direction is here defined as flow between two boundaries, situated at a finite distance from each other, in almost all practical cases representing the top and the bottom of an aquifer. These boundaries are horizontal planes perpendicular to the vertical axis, chosen as z-axis.

424

Three-dimensional groundwater flow

(610)

610 BVI-1.

GENERAL THREE-DIMENSIONAL

G R O U N D W A T E R F L O W IN A

S E M I - I N F I N I T E O R I N F I N I T E FIELD

0299

0299

ax--5 + ~

0299

2 0q)

SS

-07' ~

+ az 2 = / ~

= -K "

610.01 (MI)

Infiltration according to an arbitrary function of x, y and t into an infinitely thick confined aquifer at z = 0.

o

/

/

/t

/

/ "/ X /

99 = 99(x, y, z, t) = head,

Z

99(x, y, z, O) = O,

Y

99(-cx:~, y, z, t) = 99(cx:~, y, z, t) = O,

099(x y O, t) -- q(x, y, t) O--z ' ' K '

99(x, - c ~ , z, t) --- 99(x, cxz, z, t) -- O,

99(x, y, cxz, t) -- O. f1202

fot f_+~ yo,3/2 to) e ~ f_+~ c~ q(t(xo, - to)

99(x, y, z, t) -- 4Kzr ~

with p~) - (x

- x 0 ) 2 --I- (y -

4(t -to)

dxo dyo dto

yo) 2 -+- z 2.

610.02 (MI) As 610.01, but with a constant infiltration q, regularly distributed over a rectangular area with sides 2a and 2b.

q -b ,,__a_~_/__a_~_~, -a//

,/ //a ~_:l__+~b _tz_,__~ ,,a

x

99 = 99(x, y, z, t) = head,

x

99(x, y, z, O) = O, 6o0__z_,(0,y ,

z, t) -- 0

99(cx~, y, z, t) = O,

O!(x, O,z,t) --0, Oy

99(x, cx~, z, t) -- 0,

Ox

q

0 99 z (X y , O, t ) -- {-0 K

for Ixl < a and lYl < b,

99(x, y, oo, t) -- 0.

for Ix l > a or lYl > b,

q

/1

erf ~ f l ( x + a ) r

99(x, y, z, t) -- 2 f l K ~

×[

1

/ erf/ 1 y

-erf

/1

~fl(x-a)v

j]

] _l~2zzv2dv

Semi-infinite and infinite field

(61 O)

425

610.03 (MI)(IC,IC)

Steady state

of 610.02.

q[ b+y, arcsinh{v/(b +a y)2x + I

qg(x, y, z) -- 27r K

Z2

{ + (a - x) arcsinh

b+y } v/(a _ x) 2 + z 2

+ (a + x) arcsinh

/ {

+ (b - y) arcsinh

{ v/(b _a-~y)2 + z 2 }

+ (a - x) arcsinh

{

+ (b - y) arcsinh

{ a - q - x } v/( b _ y)2 + z 2

+ (a + x) arcsinh

/

+ (b + y) arcsinh

- z arctan {

{ -zarctan

{ - z arctan

{ -zarctan

Zv/(a

v/(b + y)2 + z 2

v/(a + x) 2 + z 2

v/( a _ x) 2 + z 2

v/( a + x) 2 + z 2

} } / /

(a-x)(b+y)} - - x ) 2 @ (b -4-

y)2 + Z 2

(a+x)(b+y) Zv/(a+x) 2 + ( b + y ) 2 + z (a-x)(b-y) Zv/( a _ x) 2 + (b -

} 2

} y)2 + z 2

(a+x)(b-y) zv/(a÷x) 2 + ( b - y ) 2 + z

}] 2

"

610.04 <S)

n horizontal drains of length 2R, regularly distributed over a circular disk of radius R in the plane z = 0, in an infinite field, each drain System of

with discharge Q. ~o -- ~o(r, 0, z, t) = drawdown in cylindrical coordinates:

f

X

0299_ 1 099 1 Or2 ~- --Jr r -~r r2

02

q9

O0 2

0

2q)

[ OZ 2

= f12 0 ~0 Ot

426

(61 O)

Q 16ar K R

q)(r, O, z, t) =

m--0

R Xm J]

(mar) 0

Ym -t- z 2) R W x m } 4t ' v/y2 + z 2

M

f12( Ym2 + z2) ' v/y 2 -t- Z 2 4t

+ M

Xm--rcos

Three-dimensional groundwater flow

,

(mar)

ym--rsin

0

.

H

610.05

with

/1

(MI)

Depletion function for a solute concentration in an infinite aquifer, starting from an arbitrary three-dimensional initial concentration c(x, y, z, O) = F ( x , y, z). 02C

02C

02C

1 OC

OX---5 + -Oy - 2 + -OZ - 2 = Dd Ot

c = concentration [ML-3], Dd = coefficient of diffusion. •

c ( x , y, z, t) =

8 ar D dt ~/ ar D d t

x exp( with pg - ( x 610.06

0o

~

oo

F (xo, Yo, zo)

p2 ) d x o d y o d z o 4Ddt

- xo) 2 -~- (y - yo) 2 + (z - zo) 2.

(MI)

As 610.05 with

CO

F(x, y,z)

/COo

I

[

..Jl -

--'-

-

-I- .

/ /

.

.

.

.

.

C

r. /

I

/x

lZ

1{

c~x y z t,- ~co e r f (x+a 2~,

~{~rfIY+~ z q-c ~ _

o r f (x2 ~al I

Y~ z--c

.

for Ixl < a,

lyl < b, Izl < c for Ixl > a, lyl > b, Izl > c

(620)

Finite field

427

620 BVI-2. GENERAL THREE-DIMENSIONAL GROUNDWATER FLOW IN A FINITE

FIELD 02q 9 _ 02q 9 02(t9 /~2 0(/9 Ox 2 t- - - -tay 2 ag 2 -07 02(/9 Or---T

+

1 Oq0 1 02(/9 -r -~r -t r 2 002

F

(Cartesian coordinates),

02(/9 Og2

f12 099 --

/~2 __ Ss

K-'

(cylindrical coordinates),

p2 _ r 2 _+_r 2 _ 2ror cos 0.

In this category there are no exact analytical solutions available for problems with so-called mixed boundary values (see comments of 362a), in contrast with problems in general two-dimensional flow with mixed boundary values, which may be solved by means of the method of conformal transformations. In that case approximate solutions may be obtained by replacing the equipotential part of the boundary by an equally distributed fixed flux and calculating afterwards the average head along this part of the boundary. Most of the remaining problems can be solved by means of superposition of known solutions for separate problems in other categories. There are many possibilities to combine such separate problems but here only some examples of recharge = discharge problems will be given (see comments of 531c). 620.01 (FC,IH,S)

~Q

q l_

I

x

a

Vertical infiltration from a circular p o n d into a confined aquifer, combined with a partially penetrating well along an axis, parallel to the z-axis, q9 = qg(r, 0, z) = drawdown. Steady state only if recharge -- discharge: Q = Jr RZq.

~gSg8889888g~

8k~ rz

i

O~P(r, O, O) _ 8z

q -~

forO~
0

forr>~ R,

- - ( r , O, D) - - 0 , 8z

lim (p O~p

2rr KI

~-~o V ) -

0

OqO (r, O, Z) -- a(p

a---ff

for ( a -

~1/) < z <

~o(c~, O, z) - - 0 ,

( a + 1/),

forO~
-if-d@, Jr, z) -- O.

(620)

428

Three-dimensional groundwater flow

For 0 ~< r ~
•

Q

ln( R )

+ 2zrK-------D

Q

~o(r, 0, z) -- ~Ow(r, 0, z) - q92(r, z) -k-

2Q ~-~ l sin (nzrl )

~Ow(r, 0, z) -- JrZgl

n

\-~

Q

2Jr KD

D2) •

- Dz +

(r)

For r ~> R" •

1

(lz2

- zrKDR2 -~

In p

with

nzca

--a-), nTrz

\ D /cos

COS

n--1 (/91 (r,

z) --

2Q ~ 1 (nzrR) (nzrr) (nzcz) - K1 Io cos 7r2gR n \ D ~ ---D- ' n=l

q92(r, z ) - -

620.02

2Q

_111

rr2K R

n=l

n

D

Ko

cos

---~

.

(S)

Non-steady state of 620.01. Initial value ~p(r, 0, z, t) - 0, q and Q independent. ~o(r, 0, z, t) -- ~p(r, 0, z, t) of 532.01 - qg(r, z, t) of 533.01 with p - ~/r 2 + r g -2ror cos 0 instead of r in 532.01. 620.03

(FC,IC,S>

~a

q gS~ ~

b

x

i i

~-----~"

b

i i

' i i i

D

i i i

Xo Ng8

:

i i

Z

Vertical infiltration from an infinite strip into a confined aquifer, in combination with an infinite row of fully penetrating wells of equal strengths and at equal mutual distances along a straight line parallel to the strip. ~0 -- ~0(x, y, z) = drawdown. Steady state only if recharge = discharge or Q = 4abq. q

I

° ! ( x , y, O) _ Oz

I

;~a I

,

,/7 I

Q x

-~ 0

for Ix] < b, for Ix] > b,

O~°(x, y, D) -- 0 Oz

I

--(-c~, I

2a

i I

Xo

ii

Ox

y, z) -- -2- (¢~, y, z) -- O, Ox

Finite field

(620)

0__~(x, m a , z) -- 0

Oy

for m -- 0, + 1,-t-2 . . . . , if x :/: xo for y =

Wells with discharge Q at x -- xo, y =

q) ( , y,

_.x

z

) --

Q

429

~

2rcKD

2ma.

1 exp{ -

n

2ma.

nrc(x-xo)} a

cos

(nrcy] a /

n=l

Q foC~[sin(bot)cosh{(D-z) ot} 2rca K b sinh(Dot)

1 J cos(xot) dot. D 0l 2

620.04 (S)

Non-steady state

of 620.03. Initial value: ~p(x, y, z, O) -- O. ~p(x, y, z, t) = qg(x, y, t) of 356.02 - ~o(x, z, t) of 356.31

if x is replaced by x - x o , a by 2a in 356.02, B by b and p by q in 356.31. q and Q independent.

(700-701 a)

Orientation table C

C. M U L T I - L A Y E R

431

SYSTEMS

ORIENTATION TABLE C

700. Groundwater flow in multi-layer systems go to 701

a. Is the multi-layer system continuous ?

A multi-layer system here is referred to as continuous, if the aquifers and semi-permeable layers that form the ~/////////////////////¢1 system are continuous all over the '3' K1D1, S1 ~///////////////////~ C2 flow field, that is to say if each aquifer i ir and each semi-permeable layer has constant properties between the boundaries of the flow field under consid~/////////////////////~ Cn eration. The hydrological constants K.D.,S. thus only differ mutually. An example of a continuous multi-layer system problem is radial flow towards a well situated in the upper aquifer of a n-layer system, extending to infinity. ,

,

'

I

I I I

b. There are discontinuities in the system ----+--

----

Cl

!

KlOlSl

?///2"//////////////////////////////////////,4 C2 i

i

x K2D2, $2

q

=

B

go to 750 (C5) In general a multi-layer system becomes discontinuous, if one or more aquifers or semi-permeable layers (not all) are interrupted, for instance by a partially penetrating river of width 2B and with a level that differs from the original level of the system. In such a case the system is not the same everywhere in the flow field.

701. Continuous systems a. Has the groundwater flow a disz~inct vertical component in one or more o f the aquifers ?

go to 740 (C4)

If an aquifer of the system is relatively thick and a boundary value in that aquifer only holds along a small part of the aquifer thickness, the vertical flow may not be neglected.

432

(701 b, 702)

b. Only horizontal flow

Multi-layer systems

go to 702

702. Horizontal groundwater flow a. The flow is one-dimensional

go to 710 (C 1)

The flow is unidirectional and can be described by one variable of space, which will be x.

b. The flow is two-dimensional with radial symmetry

go to 720 (C2)

Radial flow is two-dimensional flow with a point of symmetry that is chosen as the origin of the coordinate system. The flow can be described by one variable of space, which will be r - v/x 2 + y2. c. The flow is general two-dimensional

go to 730 (C3)

The flow has components in two horizontal directions and has no point of symmetry. The independent variables are chosen x and y.

One-dimensional flow

710 C-1.

(710)

ONE-DIMENSIONAL

433

F L O W IN C O N T I N U O U S M U L T I - L A Y E R S Y S T E M S

02 0 Ox q~ - Aq~ + B ~-~ ~0

for n o n - s t e a d y flow,

d2

dx 2 ~0 -- A~o

for steady flow,

a l l + a12

--a12

--a22 A

m

a22 -k- a23

0

...

...

0

•

o

•

o

--a23

0 •

o

0

...

0

1 ~ , aii -- Ki Di ci

0

0

...... t~2

o

ann -t- an n+ 1 Si

i

Ki Di ci + 1

Ki Di '

991 ,

•

0

•

--ann

q92

m

•

......

1 aii+l ~

jq2

,~ B

0

o

.

•

......

j~2

q9n

t'/

T w o - l a y e r s o l u t i o n for f ( A ) "

f(A)

--

Pl -- P2

a22Y2 f (Pl) -- a22Yl f (P2) --a22 f (Pl) -k- a:z2f (p2)

--al2 f (Pl) -+- al2 f (P2) --a22Yl f (Pl) -+- az2Y2f (P2)

with p l and p2 the roots o f p2 _ ( a l l + a12 -+- a22 -k- a 2 3 ) P -k- a l i a 2 2 -k- a l i a 2 3 -+- a 1 2 a 2 3 - - 0 and yl and y2 the roots o f a22Y 2 -- (all

-+- a12 - - a22 - - a 2 3 ) Y - - a12 - - 0.

Also a22Yl,2 -- all

-+-a12 - - P l , 2 - - - - ( a 2 2 + a23 - - P 2 , 1 ) -

Subdivision: a. F l o w in an infinite field. b. F l o w in a semi-infinite field. c. F l o w in a finite field (strip).

)

434

(710a, b)

Multi-layer systems

a. Flow in an infinite field

710.01 (M) n-layer system under open water with sudden rise h of its level 9 = q~(t). Only vertical flow through the semi-pervious layers.

:////: "//////////////////////////////////////:ci

KID1, $1 c2

KED~ S2

dq~

C3

dt

+ A9 - h

with Ki Oi in A replaced by Si, Cn

KnDn, Sn

Cn+l

~o(0)-0

~=0 *

and

h-

0 o

q~(t) -- (I - e - A t ) A - l h .

710.02 (M) Steady state of 710.01. ,

¢p - A - l h .

b. Flow in a semi-infinite field

710.11 (M)

h I "//////////////////////////////////////.4 Cl

All aquifers with open boundary. Sudden drawdown of the surface water level, which is kept constant thereafter. 9 - ~o(x, t) - drawdown.

K1Db Sl c2

KED2, $2

tp(x, O) - - 0 ,

C3

h

9(O,t) - h - I

...... >

°"

tp(oo, t) - - 0 , h .

,

h

..... C n + I

X

•

q~(x , t) -- e_X4Xh - -2 fo °° a sin(xot )e_tB-l(A+ot2i) (A + ot21) -1 h dot. 7Z"

710.12 (M) Steady state of 710.1 J.

*

9 ( x ) = e -x4X h.

One-dimensional flow

710.13

(710b)

435

(M)

~///////////////////¢

C1

KID1

C2

K2D2

All aquifers with open boundary. The plane at x = 0 kept at zero head. Constant vertical infiltration q into the u p p e r aquifer. Steady state, tp = tp (x) = head.

C3 d 2

dx 2 ~ - A ~ - q,

>

~ ( 0 ) - 0,

Cn

K~D,,

Cn+l

d dx 9 ( o o ) - 0,

q

0

~

°

o tp(x) -- (I - c - x ' d A ) A - l q .

710.14 (M)

=~Zq

Fully p e n e t r a t i n g well-screens in all aquifers at x - O. Sudden discharges which are kept constant thereafter, tp tp(x, t) - d r a w d o w n .

V//z ~//////////////////////////////////////>CI

K1D1, S1

I

9 ( x , 0) -- 0,

C2

~::~q2

K2D2, 82

i

9 ( o o , t) -- 0,

C3 Pl

0 ~ q~(0, t) -- --p, 0x Cn %,

i

Cn+l

~o=0

X

2fo

7/"

(M)

Steady state of 710.14. •

--

P2

Pn qi

Pi -- 2Ki Di

~o(x, t) -- e - X d X A - 1 / 2 p - _

710.15

p

-I

p.

cos(xot)e-tB-l(A+a2I) (A + o t 2 I ) - l p dt~.

•

436

Multi-layer systems

(710b)

710.16 (M} All aquifers with open boundary with entrance resistance. Sudden drawdown of the surface water level which is kept constant thereafter. ~o = tp(x) = drawdown. Steady

hll ~~////////////////////// ~ Wl

K1D1

W2

K2D2

¢1 C2 C3

I I

state. 9(oo) = O. d d-~ 9(0) - - H { h - 9 ( 0 ) } ,

i I I

X

c.

Wn

KnD~

Cn+l 771

1

rli--~ , Ki toi

O

.

0

h h

r/2

H--

h

m

•

°

°°°

O

.

.

.

.

.

.

r/n

Hh. 710.17 (M)

~//////////////////~ Cl ~,1,,i, ,1,,I,,,1,,I, ,I, ,I, ,I, ,I, 4, ,1,,1,,1,4,,1,,1,,,I,,1,,1,? q 1101 K1D1 c2

~ w2

K2D2

All aquifers with open boundary with entrance resistance. Constant vertical infiltration in the upper aquifer Zero head at x - 0. Steady state. ~o = ~o(x) = head. d2 dx 2 9 -- A~p - q,

C3

X 0

K.D.

~Wn

d dx ~o(0)

n

m

Cn

q

Cn+l

~

.

~

o

H~p(O),

r/1

0

0

r}2

° o •

0

rli-o

O

.

.

.

.

° .

rln

Ki wi

d d-~9(c~) -- O.

(710b, c)

One-dimensional f l o w

437

~p(x) = {I -- e - X ' / X ( ~ / ~ q- H ) - I H } A - l q .

710.18 (M} Vertical infiltration into the upper

aquifer with I

~////////////////////

,I,,1,,1,,1,,I,,1,,1,,1,~,1,,1,,1,,I,,I,,1, ,1,,1, q

K1DI i KzD2 I ~1i

q(x)-

c2 C3

I

b

q

Cl

b

d2

X

d x 2 ~o - -

err K.D.

!I

forlxl b.

Flux - 0 for x = 0. S t e a d y state. - ~o(x) = head.

i i

0

A 9 - q(x),

Cn+l __._q__

q(x) -- q --

for Ix l < b,

,~ u

0

¢p(x) -

for Ix l > b.

{I - cosh (x w/-A)e-bV~}A -1

for Ix l < b,

e_X,/X sinh ( b ~ / A ) A - l q

for Ix[ > b.

e. F l o w in a f i n i t e f i e l d (strip)

710.21 (M} G i v e n d r a w d o w n h for x -

b.

0 for

Flux -

x - O. S t e a d y state. ~o - ~o(x) - drawdown.

h h

Cl

d w-~o(o)

X I.

i I I

~

.

.

.

K2 2

li c; I J

.

b

i I I

KnDn cn+l ~

-

o,

~o(b) -

h -

•

ax

~p(x) -- cosh ( x ~ / A ) c o s h -1 (b~/A)h.

.

438

(710c)

Multi-layer systems

710.22 (M) G i v e n d r a w d o w n h for x - b. Zero d r a w d o w n at x - O. S t e a d y state. 9 - tp(x) - drawdown. -~

~P

h

h h

Cl ~o(0) - - 0 ,

> K2D2 x I I

h

•

I I

J

9(b) - h -

I c3 iI

9(x) -- sinh (x~/A)sinh -1 (b~/A)h.

KnDn cn+l ~

710.23 (M) Zero h e a d at x - b. Zero f l u x at x - O . C o n s t a n t vertical infiltration. Steady state. ~o - q~(x) -

head.

////////////////////////////4, Cl q KID1 : C2 K2D2 C3 x

d2

d

dx 2 ~ - Atp - q ,

dx tp(O)

O,

Kbo

___q__

I

q~(b) - 0 ,

q-o

tp(x) -- {I - cosh (x~/A)cosh -1 (bVrA)}A-lq. 710.24 (M) =~Zq

=~Zq

cl et

9"///, ~ / / / / / / / / / / / / / / / / / / / /

",=:~ql

~

K1D1 c2

"~qz

~

:2D2 X

I I L-

•]

~,=:~ql

I:1~ q 2

Fully penetrating well-screens in all aquifers at x = b, - b , 3b, - 3 b etc. C o n s t a n t but different discharges qi. Flux = 0 f o r x = O. S t e a d y state. 9 = 9 ( x ) =

drawdown.

C3 iI _1

b Pl d

~o(0) - 0,

dx 9(b)

qi

p2

d

p

PiPn

2KiDi

(710c)

One-dimensional flow

(p(x)

439

cosh (x~//A)sinh -1 (b~/A) (~/A) -1 p.

710.25 (M) =~Zq

Fully penetrating well-screen in all aquifers in the middle of a strip with width 2b. C o n s t a n t but different dis-

"///////////////////////~ ///////////////////////A Cl |

charges qi.

=~ql

K1D1

|

C2

= 2b.

= 0 for

S t e a d y state.

~o = ~o(x) = drawdown.

=~ q2

K2D2

Drawdown

x = 0 andx C3

j-

x -

o,

Pl

d

_

dx (p(b)

_

P2

p

_

.

with Pi

qi

2 K i Di

Pn

tp(x)

sinh ( x V ~ ) c o s h -1 (b~/A) (~/-A) -1 p.

710.26 (M) All aquifers with entrance resistances 11)i to open water with a c o n s t a n t drawdown. Zero f l u x at x O. S t e a d y state. ~o - q~(x) - drawdown. /////////////////////////////~

el

K1D1

wl ~

K2D2

11°2

d ~o(0) -- 0, d-~

C2 .

d

d-~ ~o(b) -- H { h - ~o(b)},

C3 x I..

'

b

771 . . . . . . 0 r/2 n

0 0

1

n

~

•

0

"'.

......

"

r/~

'

rli - -

~

lxre"i

h-

h h .

°

Wi

h

9(x) -- cosh ( x V ~ ) { H - l ~ / A sinh (by/A) + cosh (b~/A)}-lh.

440

(710c)

Multi-layer systems

710.27 (M)

- m

All aquifers with e n t r a n c e r e s i s t a n c e 113i to open water at x -- b. C o n s t a n t drawdown of the open water level. Z e r o d r a w d o w n at x = O. S t e a d y state, fp = 9 ( x ) = drawdown.

go

C1

,~

KID1

wl~

K2D2

w 2 ~ c2 c3

'

b

~1 0 H-

d x ¢p(b)

m

H{h

m

¢p(b)}

'

O

...

0

/72

h

0

.

.

. •

0

d m

¢p(0) - - 0,

h ,

h-

.

.

o

......

r/n

h

~p(x) -- sinh (x~f'A) {H-l~f-Acosh (b~/A) + sinh ( b ~ / A ) } - l h .

710.28 (M) All aquifers with e n t r a n c e r e s i s t a n c e w i to open water at x -- b. Vertical infiltration q in the upper aquifer (q = const.). F l u x = 0 f o r x = O. S t e a d y state. ~p = ~p(x) = head.

Cl

q

K1D1

Wl ~

. K2D2

w2

--

d2

C2

dx 2 ¢P -- A 9 - q,

C3

d dx ¢p(0) - 0 ,

-

171

......

0

0

r12

0

d dx 9(b)

-

-Hop(b)

K qlOl 1

_

0

O

•

0

"'.

......

"

tin

'

rli

--

lxt" i W i

q

"

0

~p(x) -- [I - cosh (x~/A) {H-1 ~/"A sinh (b~/A) + cosh ( b ~ / A ) } - l ] A - l q .

Radial flow

720 C-2.

(720a)

441

T W O - D I M E N S I O N A L R A D I A L F L O W IN C O N T I N U O U S M U L T I - L A Y E R SYSTEMS 02 1 O O Or 2 q~ -Jr- -r -OTrtp -- A 9 + B -~7 9

d2 1 d dr 2 q~ + -rdTr 9 - At#

for non-steady flow

for steady flow,

9, A a n d B as i n 7 1 0 . For the general two-layers solution for f (A) see 710. Subdivision: a. Radial flow in an infinite field. b. Radial flow outside a circular cylinder. c. Radial flow inside a circular cylinder. d. Combinations of the foregoing cases a, b and c. a. Radial f l o w in an infinite field

720.01 (M) Vertical infiltration q(r) into the upper aquifer with

P"/////////////////////~~

C

,~4,,1,¢4,,[-¢+, ,,1,,1,,1,,1,,1,,1,,1,,1,q K1D1

K2D2

1

q(r)

C2 C3

_,[q

f o r 0 ~ < r < R, for r > R.

I0

Steady state, q~ = q~(r) = head. d2 1 d dr 2 ~0 + -rd-Tr ~0 - A 9 - q ( r )

C/,l

K.D.

d --~(0) dr

Cn+l

=0,

q

o(r) -' 1 \ /

for 0 ~< r < R,

q --

d d--7 q~(oo) - O.

o

0

forr>

R,

q~(r) -- { {I - R x / A K 1 ( R x / A ) I o ( r x / A ) } A - l q R I1 ( R ~/'A) K0 (r x/A) (~/A) - l q

for 0 ~< r < R, f o r r > R.

442

(720a, b)

Multi-layer systems

720.02 (M) =~ZQ

Fully penetrating wells in all aquifers at r - O. Constant but different discharges Qi. tp - 9(r, t) - drawdown.

Y///~ ~ / / / / / / / / / / / / / / / / / / / / /

Q1

Cl

K1D1,S1

9(r, O) -- O,

'

9(cx~, t) - 0 ,

C2

' ~ Q2

K2Dz, $2

,

ql q2

C3

r

>

lim r r-+0

,

q

~rr ¢p

__

qn

Cn

, 'l::~Qn

Qi qi -- 2re Ki Di

KnDn, Sn

' '

¢n+l

¢p(r, t) -- Ko(r~/A)q

-

.

--q'

olJo(rot)e -B-l(A+°t2I)/t (A + or21)-1 q dc~.

-

f o °°

720.03 (M) Steady state of 720.02.

•

¢p(r)- Ko(r~)q.

b. Radial flow outside a circular cylinder

720.11 (M)

Cl

K1D1 C2

K2D2

All aquifers with open boundary. Constant drawdown of the open water level. tp -- 9(r) -- drawdown. Steady state.

C3

h h h __

R

o

I Cn

K,D, Cn+l

¢p(r) -- Ko(r ~/-A) K o 1(R~/A)h.

~o(oo) - 0 ,

¢p(R) - h ,

h

o

Radial flow

720.12

(720b)

(M)

Constant lowering o f the polder level

Ih

I

a r o u n d a c i r c u l a r basin.

I 1

//'////////////////////~ C1

I

i

443

-

R.

9

--

9@)

Zero drawdown --

drawdown.

Steady state.

KID1

r

at r

C2 K2D2

I

i

1 d

d2

C3

d r 2 ~0 + -rd-rr 9 -- A(~o - h)

I

i

R Cn

I I

Cn+l

h 0

d

9 ( R ) -- O,

d---~9(00) - O,

h -

•

o

o

¢p(r) -- {I -- K o 1 (R~/-A)Ko(r~/A)}h.

720.13

(M) All aquifers w i t h open boundary and zero

head at r = R. ~///////////////////////5 Cl 4,4,44444444~,44 q

1 •

Steady state.

KID1 K2D2

c2 C3

d2 1 d d r 2 ~0 + - r ~ r r ~0 -- A~o - q,

i r I

R

i i

K.D.

Cn

Cn+l

d 9(R)

-- O,

d--7 9(cx~) -

0 O,

q -

o

o

tp(r)-

{I-

Constant infiltration q

into the u p p e r aquifer, q~ = q~(r) = head.

Kol(R~/A)Ko(r~/-A)}A-'q.

.

444

(720b)

Multi-layer systems

7 2 0 . 1 4 (M)

i

Fully penetrating circular well-screens with unilateral discharges qi in all aquifers at r = R. ¢p = tp(r) = d r a w d o w n . S t e a d y state.

r Cl

i i i

r

i

R

J

d

c2 --1

d-7 tp(R) -- - p ,

Ca

P --

Cn

K,,D,,

!=* q,,

,

qi Pi -- Ki D-----~"

,

Pn

Cn+l

I

Pl P2 .

¢p(oo) -- O,

tp(r) -- Ko(r~/r-A) K [ -1 ( R ~ / A ) ( ~ / A ) - l p .

7 2 0 . 1 5 (M) i

g4

liI

All aquifers with entrance resistances Wi to open water with a c o n s t a n t d r a w d o w n

-Ih--

i

~/////////////////////////

,

r

i I

"

H

Wl

~/A ~ w2

h at r = R. S t e a d y state.

Cl

tp = tp(r) = d r a w d o w n .

K1D1 d d--~ ¢p(R) -- - H { h

C2 C3

K2D2

- ¢p(R)},

~o(~) - o , R i i

~w.

K.D.

c. Cn+l

l

n

......

0

772

0

• 0

... ......

.

1

,

On

tp(r) - K o ( r ~ / - A ) { H - I ~ / - A K I

Yli - -

h-

h h.

Ki W i ' h

(R~/-A) + K o ( R ~ " A ) } - l h .

.

Radial flow

(720b)

445

720.16 (M) All aquifers with entrance resistances to open water with zero d r a w d o w n at r = R. Constant lowering of the p o l d e r level around the circular basin. ¢p = ¢p(r) = drawdown. Steady state.

~//////i~ h//////////////~ C1

r..

KID1

1102

C2

K2D2

¢3

d2 1 d dr 2 ~o + -r~rr ~p -- A((p - h)

I I I

R

[ I

IC.D.

a

/71

......

0

772

m

0 "..

......

(p(r)-

[I-

d ~rr ( p ( o o ) - - 0 ,

Cn+l h 0.

0

•

0

d ~rr (p(R) -- Hop(R),

fit

.

,

rli--

1 Kiwi'

h--

~.

.

0

Ko(r~f-A){H-'~/-AK,(R~"-A)+

Ko(R~/A)}-I]h.

720.17 (M) All aquifers with entrance resistances to open water with zero head at r = R. Constant infiltration q into the upper aquifer. (p = (p(r) = head. Steady state.

I

i

4,4,4,4,,1, q KID1 c2 K2D2

I

i

r

i

>

Wl ~ W2

I I

d2

C3

dr 2 (P + -rd-Tr (p - A(p - q,

I I

I

i~

d drOP(R) -- H(p(R),

---!

R

i

I

I

i

i

i

~wn

K~D~

......

a

m

•

~o(r)-

......

[I-

Cn+l K qlD1

0

...

•

d ~rr (p(oo) -- 0,

Cn

0

1 7 0rl2 1 0

1 d

1

,

~i -- "~'itUi

rl,~

Ko(rx/~){H-]~/-AKI(Rx/A)+

0 q --

"

"

0

Ko(R~/A)}-I]A-lq.

446

Multi-layer systems

(720c)

c. Radial f l o w inside a circular cylinder 720.21

(M)

Th ~/////////////////////~

Given drawdown h for r - R. All aquifers with open boundary to the surface water. Flux - 0 at r - 0. tp - ~o(r) - d r a w d o w n . Steady state.

Cl

K1D1 K2D2

C2 C3

d d r ~o(O) -- O,

h h q~(R) -- h,

h

•

__

•

h •

R K.D.

tp(r)-

Io(rV/-A)Iol(R~)h.

Cn+l~ 720.22

(M)

Cl q

KID1 K2D2

C2 C3

All aquifers with open boundary to the surface water with zero head a t r R. Flux - O at r -- O. Constant vertical infiltration into the u p p e r aquifer. ~o - ~o(r) - head. Steady state. d2 1 d d r 2 ~o -k- -r~rr ~o - A~o - q , __q__

d d r ~o(O) -- O,

~o(R) -- O,

q

__

o

Cn+l~ 9 ( r ) -- {I -- Io(r~/-A)Io 1( R ~ / - A ) } A - ' q . 720.23

(M)

--___%

All aquifers with entrance resistance

~7//////////////////////, K1D1 K2D2

Cl

Wl

w2

C2 C3

"I I I I

i

R KnD.

Wi

to o p e n

w a t e r with a constant drawdown. Zero f l u x at r 0. tp - tp(r) - d r a w d o w n . Steady state.

I

11)nC + I ~ ~ ' Cn

d dr

d

~o(0) -- O,

d r ~o(R) - H { h - ~o(R)},

Radial flow

H

(720c, d)

71

O

0

112

--

O 0

.

...

0

*

.•.

447

•

......

h h

1 ,

~

7i

~

•

h

K i wi

o

h

7n

~ o ( r ) - Io(rV/-A){H-I~/-AII(R~/-A)+ Io(RV/A)} -1 h •

7 2 0 . 2 4 (M)

Y//////////////////////A

K1D1 KzD2

All aquifers with entrance resistance w i to open water with zero head at r - R. Zero flux at r -- O. Constant vertical infiltration into the upper aquifer. ~o - ~o(r) - head. Steady state.

C1

Wl ~ c2

d2 1 d dr 2 tp + -r~rr tp - Atp - q,

W2 ~ C3 I

r

I I

~"

d

R

d

dr~O(O) - 0,

I I

~r ~o(R) - -He(R),

I

K.D.

w, ~ Cn Cn+l ~88&g8~

H-

0

/72 •

...

0

•

......

1 ,

7i

--

~xv i

0

Wi

q

-

"

•

73

0

[ I - - Io(r~/-A){H-lv/-AII(Rv/--A)+ I o ( R ~ / ~ ) } - l ] A - l q .

~o(r)-

d. Combinations of the foregoing cases a, b and c 720.31

(M)

_ h

C1 C2 C3 I I

1 i

r

i i

,

Fully penetrating wells in all aquifers at r - 0. Constant, but different discharges Qi. Open water boundary at r - R with zero d r a w d o w n . q~ - ~o(r) - drawdown. Steady state.

r--~0

~rr tp

i i

R

I'~an g~o~

i cn Cn+1

ql

(d) lira r

q2 -

-q

-

qn Qi

qi -- 2rrKi Oi'

~o(R) -- O.

448

Multi-layer systems

(720d)

~ ( r ) -- { K o ( r ~ / A ) - Io(r~/-A)Iol(Rv/-A)Ko(R~/-~) }q.

720.32 (M)

=~ ZQ

Fully penetrating wells in all aquifers at r = 0. Constant but different discharges Qi. Closed boundary (flux = 0) at r -- R. ¢p = ¢p(r, t) = drawdown.

Steady state. ~=~Q1

K1D1

=~ Q2

K2D2

C2

(d) lim r

C3

r ---+0

~ r ¢p

ql --

--q

--

q2

--

"

'

qn qi --

K.v. , ~

Cn+ 1

q~(r)-

{Ko(r~/~)+

Qi 2Jr Ki D i '

d

-T-cp(R) -- O. or

Io(r~-A)I~I (R~/-A)KI (R~/-A) }q •

720.33 (M) Fully penetrating wells in all aquifers at r 0 with constant, but different discharges Qi. Boundaries with entrance resistances wi to open water at r - R. ¢p -- ¢p(r) - drawdown. Steady

=~EQ

~

,

!

Cl

state.

(d)

lim r r---~ 0 ~r ~o

C3

r

d d-7 q~(R) - - H i p ( R ) ,

j-

R

i '~ a. ! I

-q,

1,;.19. w . ~

Cn

Cn+1

~1

0

0

172

...

0 "

a

•

0

¢p(r)-

1

17i

ql q2

Kiwi

O.o

......

[Ko(r~JA)+

tin

Qi qi - 2zr Ki Di

qn

Io(r~/--A){~/-AIl(R~/-~) + H I o ( R ~ / ~ ) } -1

× {~/AK1 ( R ~ / A ) - H K o ( R ~ / A ) } ] q .

Radial flow

(720d)

449

720.34 (M)

Zq

All aquifers with fully penetrating cylindrical well-screens with radius R and constant but different discharges qi. Zero flux ¢1 ¢2

:

q2

atr-O. d

d right

c3 ql

r

=q-R

KnD,,

' i=~qn

i

¢n Cn+ 1

{ RKo(Rv/-A)Io(r4r-A)q go(r)-- Rlo(Rx/~)Ko(rx/~)q

q2 o

qn qi qi -- Ki Di'

d d--7go(O) -- O.

for 0 ~< r ~ R, for r ~> R.

720.35 (M) b

Circular polder with radius R, surrounded by a polder with different level, or iso-

. . . . . .

~/~//////////////////~ I

c2

I

,

K2D2

I I

I

¢1

K1D1

!

c3

r

lated reservoir in open water, drawdown. Steady state.

d2 1 d dr 2 go + go - A{go -- h ( r ) } rdrr d dr ~(0) -

i

R

K.D. h 0 °

h ( r ) --

~(~)

¢n Cn+l

for 0 <~ r < R,

f o r r > R.

Rv/--AK1 (RV/-A)lo(rv/-A) }h RV/_~II(Rx/~)Ko(r~/_A) h {I -

go(r)-

O,

0

0

go(r) =

for 0 ~< r ~ R, f o r r >~ R.

- O,

450

(730)

730 C.3.

Multi-layer systems

GENERAL TWO-DIMENSIONAL FLOW IN CONTINUOUS MULTI-LAYER SYSTEMS 02

02

OX 2 (,,0 -~- ~Oy2 ~0 - -

A tp

for steady flow with q~ and A as in 710. For the general two-layer solution for f (A) see 710. 730.01 (M) Vertical, constant infiltration q, equally distributed over the first quadrant. Zero infiltration everywhere else. ~/////////////, ~ / / / / / / / / / / / / / / / / / / ~ Cl ~~,,I,,~+,l,~ ,l,,~,l,,l,+~ ,~~,,~,~ q KID1

q q(x,y)--

¢2 K2D2

0

C3

Steady state, qo - qo(x, y) - head.

X I I I

02 Cn

02

Ox 2 q9 -Jr- 7 2 2 ~0 - -

Cn+l

;

x

q--

forx > 0 a n d y > 0 , forx < 0 o r y < 0 .

0

~o(-oo, y) - 0,

~(x,-~)

o

1 q~(x, y) - ~ x

e -~ I + erf

{I + e r f ( 2 ~Y

x 2~/~ ~/~ qdol.

-0.

A tp - q ,

(730)

Radial flow

451

730.02 (M) ¢p=0 ////////////////////////////////////////

61

KID1 C2

Multi-layer system with two open boundaries, perpendicular to each other. Constant drawdown h of the surface water level. ¢p = ¢p(x, y) - drawdown. Steady state.

K2D2

h h

¢3 h

~

h

Cn Cn+l

.

¢p(O, y) - ¢p(x, O) -- h,

0 Ox ~p(oo, y)

0 -~y ¢p(x, oo) --O.

• ¢p(x, y) ~o=O

~p= h

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

_

fo { xerf(

:.-:.-_._-..-_-eeeee_-e_-e_-

t

e-~ I - e r f ( 2 x

~p=h

y

C A ) / h do~

2,F

"

730.03 (M) Multi-layer system with two open boundaries, perpendicular to each

¢p=0 Y//////////////////////////////////////~CI

other. Constant infiltration q. ¢p =

~p(x, y) = head. Steady state.

KID1

¢2

02

K2D2 ¢3

02

OX 2 ~0 + ~Oy2 ¢p -- A~p -- q _.._L__

Klol

Cn

K.D.

q

~

.

~

Cn+l

o

tp(O, y) = ¢p(x, O) -- O, Y O Ox tp(oo, y)

¢p=0

q

¢p=0

x

0 -~y tp(x, oo) --O.

452

(730)

/o erf (x24/_~ 4rA )erf

tp(x, y) --

Multi-layer systems

2,f~

qe -~ do~.

730.04 (M)

i

ZQ

Straight surface water boundary with entrance resistance. Abstraction wells at (a, 0), fully penetrating all aquifers with constant but different discharges.

y//////////////////,, y,//////////////////////// Cl 1/) 1

K1D1

11)2

K2D2 I

' ~ QI i i ' =~ Q2

tp = tp(x, y) = drawdown.

c2 c3

i J

0

a

I I

Ox

X

~o(0, y) - H~o(O, y),

9(cxz, y) = O, Cn

' =~Q.

~D.

- - ~ o ( x , o) = o,

Cn+l

I I

0y

~o(x, ~ ) = O, lim r---~0

r

~o - - - q ~rr

for r -- g/(x - a ) 2 + y2,

1/)i

ZQ

~

x a

171

0

0

~12

...

0

1 ,

n •

0

--

~ ,

Ki lwi

,Oo

......

r/i

r/,,

ql

q2 q

Qi qi -- 2re Ki Di

qn

~o(x, y ) -

{ Ko(~/(x - a ) 2 +

+2

y2~/'~)_ Ko(~/(x + a ) 2 + y 2 ~ / ~ ) ] q

(n + ,/A + ~ i ) - '

e-("+x~4~+~iqcos(y~)d~.

'

( ~,.+~,,+~,,/~_o

z~o + zzo + z~o/~zzo + ~oz~o

= (-)3 7:~o+ ~7:~+ 7:l+~p ~_0 -- o;0_;9"UgD

q~!~ ~op (~o£)soo (lo)ar ×

++~,.+ ~._~ ++++,.):f.+.,. tO

+

[/~, + ~(- +,)/(+~- + +,+/'o, /

~£ + z ( v - ~c) A

/

(

'zop (~o£)so3 ( ~ ) d

zZvtb

zzv + z~v uI

= (£ '~c)zo~

,

x

)++~,- :f ,+,-oo

/

{~x+~(v-~)}(~v+~v)

0~

~tv~b

~£ + z ( v - ~c) uI f £ + z(v + x')/

+ = (£ ' x ) ~

,

zZo[b

I

tO ~" i,

£

I I

.(0

0

O) cx~

(o)

, L(/I)/g~ =~b ~0

-H

(..)

-- b

z~v-

z~v

~;3 ~ / / ' / / / / ~ / / / / / / / y / / / / / ¢

,o,,

-- V

........

rJ o ~!

" 0 - ~0 ' o o - ~m ' 0 - tm ' o o - o 'oo = t:) 'E: - u ql!~ 170"O~L jo a s w lm.aads

<~) so'ocL ~ ~17

(O~L)

'~°lf lv2.PV~t

454

(730)

Multi-layer systems

730.06 (M) Finite well-screens, with lengths 2l and discharges Oi, fully penetrating the sep-

arate layers of a multi-layer system. (i 1,2 . . . . . n).

Cl

Q1

0 Ox (p(O, y) - - 0 •

i !

I

12/

I

(p(c~, y) = O,

.

I I

0 ON ~o(x, O)

I

Cn

I[lllllllllifllll e. Y ,-

,

for y -¢ 0

C3

I ½

Cn+l

for Ix l > l, for Ix l < / ,

~o(x, c~) - O, ql

/~ZQ

q2

C . . . . . . . .

{0 -q

.I

.~

21

q --

Qi

.

,

qi -- 4 1 K i D i

qn

lf+l

(p(x, y) -- -7r

Ko {v/(x - xo) 2 + y2 ~/~}q dxo.

l

730.07 =, z a Y ~Y'////////////////Z, f f / / / / / / / / / / / / / / / / / / / / / Z 4 c l K1D1

!~ Q1 !

|

', =* Q.2

c2

, ,

C3

=~ Qi

'

==~ Q i

,i

._~ Qi "--'T

a

a

K2D2 I I I I I I

,

x

a

~ = ~ Q~ Cn

a

Cn+l

~~:~

m_~=:~ Qi

Infinite row of wells, in each layer of equal strengths and at equal mutual distances along a straight line and fully penetrating each layer. Steady state. (p - (p(x, y) -

drawdown. lim r r~O ~ r ~p = - q

for

r

-- ~//x 2 --[- ( y -+- n a ) 2,

(730)

Radial flow

O

Ox

¢p(O, y) - - 0

for y # na,

¢p(cx~, y) --O,

ql q

m

q2

Qi qi -- 2rr Ki Di

qn

0 (.a)

@ with n - O,-1-1, ±2, + 3 , . . . .

Oy~° x, T

q-O0

~0(X, y) -- E K0{V/x2 q- (y + n a ) 2 ~ } q" --(X)

455

456

Multi-layer systems

(740)

7 4 0 C - 4 . MULTI-LAYER SYSTEMS WITH A VERTICAL FLOW COMPONENT IN ONE OR M O R E OF THE AQUIFERS

In the aquifers for which the vertical flow component should not be neglected, the leakage from or to adjacent aquifers must be presented as a boundary condition, while in the aquifers with only horizontal flow, the leakage occurs as an injection term in the differential equation (see comments of 7a and 7b). Especially for more than two aquifers, the solutions will become very complicated and in many cases the problems may be better solved by considering the multi-layer system, consisting of aquifers and semi-permeable layers, as one aquifer consisting of many layers with different transmissivities (the semipermeable layers included), in which both horizontal and vertical flow takes place (cf. 562). Therefore only two examples, conceming two-layer systems with a vertical flow component in one of the aquifers, will be given here. 740.01 (IH,FCR) z

Two-layer system with a fully penetrating well in the upper aquifer and a partially penetrating well in the lower aquifer. qgl = g)l(r) = drawdown, (/92 -- q)2(r, Z) -drawdown. Steady state.

tp=0

//////////~ ~ / / / / / / / / / / / / / / / / / / / ~ Cl ~1 i >=}Ol K1D1 //////////~~//////////////////////C2 ,

,

i

i

r

d2~ol 1 d~ol t dr 2 r dr

( a l l nt- al2)q)l q- a12q92(r, D) = 0,

02(/92 --[--1 0(492- [ - 02q92 ~ --0, Or2 r Or OZ2 all = ~ , K1Dlcl

a12 =

K1DlC2'

(dqgl) Ol lim r ---- ~ , r~0 -d-7-r/ 2yrK1D1

lim(r

r-+0

-~F)

--

992(00, Z) = 0,

/

a22 ---

K2D2c2

~0~(c~) = 0 ,

Q2 2Zr K2 (b -- a) 0

fora
0q92 ~(r, Oz

O) = O,

0~02 ~(r, Oz

D2) -- -{~02(r, D2) - ~01(r)}a22.

(740)

Vertical flow component Q1 (t91 (r) -- 2zr K1 D1 K o ( r x / a l l

-1- a12)

coJo(rw)

2a12 fo ~

457

~,

Qn(oe~) + P(co) COS~n

all if-a12 -f-09 2 ~n=O

(z)2 -I- ~n2 02

cos(~.)

x

s do), 1 + ~2+E2

2/o°°

~02(r, z) -- -~2

COS (OlnZ___~2) Qn(oln) -t- P(co) COSOZn do) ~2 1 + ot2+e 2 n=0 092 -[- D---~

coJo(roo) Z

with Otn being the roots of o~tan oe - e - D2a22

Qn(oe~)- 2zrg2(b - a) ot,~

a12+°)2

al 1+a12+ o92

-~-2 ) - s i n

and

\ D2 )

Q1 a22 P(co) -- 2zrK1D1 all + a12 -k- 002.

740.02 (IC,IC) z

qg=O

Y/////////////////,4 Y'////////////////////////Cl

K1D1 ///////////////////.4. 9"////////////////////////C 2

02(/91 OX 2 O@l

~(0, Ox

t

Three-dimensional two-layer system with a horizontal finite drain of length 2L parallel to the x-axis through the point (0, 0, a) (a < D2). Steady state. ~ol -- ~ol (x, y) = drawdown. (t92 -- ~2(X, y, Z) = drawdown.

02(/91

Oy 2

(all nL al2)q01 -k- a12992(x, y, D2) -- 0,

y) -- O,

0(t91 ~ ( X , O) -- O,

(t91 ( (x~, y) -- O,

Oy

(t91 ( X , (30) -- O .

02(/92 02(/92 02(/92 -Ox - T -t- Oy 2 + OZ 2 = 0 , 0(.t92

~(O, Ox 0(t92

~(x, Oy

y, z) -- O

but not for (0, 0, a),

O, z) - 0

but not f o r z - a ,

(/92(00, y, z) -- O,

-L <x
~o2(x, ~ , z) - O,

458

(740)

Q lim (r 0992~ -r-+O - ~ f / 0q92 Oz

- 4rr K2 L 0 0992 ~(x, Oz

(x, y, O) = O,

1 all ~ ~ , K1 D1Cl

for

Ixl < L with r - x//y 2 --k (z - a ) 2,

for

Ix l > L,

y, D2) -- -a22{(,02(x, y, D2) - q91(x, y)}, 1

a12 =

and

K1 D1 c2

4 fo Jo

(/91(X, y) -- ~-5

Multi-layer systems

a22 --

1

K2 D2c2

~1 (C~, CO)COS(XOt) Cos(yco) dc~ do,

~02(X, y, Z) = q92a(X, y, Z) + q)2b(X, y, Z),

O2axy, z,- 87rK2L [arcsinh{ '

L-,-x }

x/y 2 4- (z - a) 2

+ arcsinh

g)2b(X, Y, Z) -- ~-5 4 fo~fo~ (~I(Ct O)) __ ,

~/y2 + ( z - a) 2

'

~b2b(Ot, co, z) cos(xot) cos(yco) dc~ do),

a12 all 4-al2 + y2

{Qsin(Lo~)l_(o_a)× } 8K2Lot

• -- e y

+ ~2b(D2) ,

y ~ ~¢/0/2 q-O92, y cosh{y (D2 - z) } -+- p sinh{y (D2 - z) }

(P2b(O/, 09, Z) -- Q sin(Lc~) e_,~× 8 Kz Lot y

y sinh(y D2) -+- p cosh(y D2)

y cosh(y z)

+

y sinh(y D2) -4- p cosh(y D2)

p = a22

all -4- Y

2

all -t--a12 4- y2

and

with

Q sin(Lot) | p q

~

m

8KzLot

[y

e-(D-a)Y _4_e-aY }.

(750)

D i s c o n t i n u o u s sy s t e m s

750 C-5.

DISCONTINUOUS

459

MULTI-LAYER SYSTEMS

For the c o n t i n u o u s multi-layer systems, described in C-I, C-2 and C-3 the problems may be solved by means of matrix analysis, making use of one system-matrix for the entire flow field. In the case of d i s c o n t i n u o u s multilayer systems, there are different system-matrices each referring to a continuous part of the field. The solutions may be obtained by solving separately the sub-systems with some unknowns and next by applying the law of continuity between two adjacent sub-systems. There are numerous examples of discontinuous multi-layer problems which may be solved in this way: here we will restrict ourselves to problems in which the upper aquifer partly has been replaced by a fully penetrating open water body (river, canal, lake, circular basin), thus giving there a systemmatrix that differs from the main matrix by the values of the elements of the upper row, which become all zero, because open water may be considered as having a transmissivity K D = ee (infinite) (see comments and figure of 700b). The main matrix A remains the same as in 710, 720 and 730: all

4-a12 --a22

A

_._.

--a12

0

0

+

--a23

a22

a23

0

0

0

.

.

•

.

0

--a33

a33 + a34

--a34

0

0

0

0

1 with aii - - Ki Dici ,

ai

i+1

m

...

...

--ann

ann+ann+l

1

Ki Dici+l

The system-matrix for the system in which the upper aquifer has been replaced by open water, becomes: 0 --a22 A 0 --

0 a22

+ a23

0

0

--a23

0

. •

0

--a33

a33 -+- a34

--a34

0

0

0

0

...

...

0

-an,,

0

an,, + a n n + l

Here it is supposed that the elements aii and ai i+1 are the same for the main matrix and the reduced matrix, which however will not be a necessity for solving the problems. Subdivision: a. One-dimensional flow. b. Radial-symmetric flow. c. Two-dimensional flow.

460

(750a)

Multi-layer systems

a. O n e - d i m e n s i o n a l f l o w 750.01 ----+

.

(M} .

.

Multi-layer system with a canal o f width 2B f u l l y penetrating the upper aquifer without entrance resistance. Constant drawd o w n h of the surface water level.

.

i

"////////////////////////5 C1

KID1

i

V///~//////////////, /////////////////////////

C2

0 4 x 4 B (left side --+/):

K2D2

d2 dx 2 ~Ol -- Al ~Ol,

C3

X ~v

d

B

-~°1(°) xd

Cn

K.D.

- O,

e ~ ( B ) -- h.

Cn+l

x >~ B (right side--+ r):

X m B"

d2 dx 2 ~r - Arq)r,

d d Tl -~x ¢RI ( B ) -- T r -~x ~or ( B ).

¢#r(O0) -- O,

~r(B) --h.

At - Ao,

Ar - A,

c¢

0

o

T2

Tt-

.

...

0

.

Tr-

, "

0

h --

......

Tn h ~02 91

0

T2

--

hn •

9 l ( X ) - cosh ( x ~ ) c o s h

•

tpr(X) -- e -(x-B)vr-~" h

...

0 Ti -- Ki D i ,

Ooo

0

h ,

0

°

h2 .

7"1

......

T,,

~Ol

.

92 ,

~Or

--

~on

9n

-1 ( B x / ~ t ) " h,

with h from T / x / ~ / t a n h ( B x / ~ / ) h -- - T r x ~ r

h.

This vector equation corresponds to n equations in the n - 1 unknowns h2, h3, . . . , hn. The first equation yields oo x 0 - indefinite at the left side and equals the discharge from the upper aquifer into the canal at the right side.

(750a)

Discontinuous systems

750.02

461

(M) As 750.01 but with entrance resistance w between the upper aquifer and the open water.

i V//////////////////////~ C1 ~Kl/D// I////.//////) C2 K2D2

h h2 .

ht --

c3

x

hi h2

,

hr --

hn

•

•

hn

B Cn

Cn+l

~I (X) -- cosh (x v/AI) ~Or(X)

cosh -1

(B ~/A l).

h,,

exp {-(x - B ) ~ r } " hr

--

with hl and hr from T i V/ A l t a n h ( B v/--~l ) h l -

- Tr V/ A r . h r .

This vector equation represents n equations in the n unknowns hi, h2 . . . . . hn. The first equation yields cx~ x 0 - indefinite at the left side and must be replaced in this case by

Ol //)

-(h - h 1)~ 750.03

(M) As 750.01 with two l a y e r s n -

(o o)

I

!~

~////////////////////////¢1

I

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

K,o,

--a22

~///////////////////// ,

02

[

i i

x

~

d

i

N

Tl-~l

Ar -- ( a l l + a12 \ --a22

K2D2

~

--

i

0

(h) ~12

0)

T2

'

'

Tr t~r --

0

( rl) ~r2

022

'

/

,

0)

T2

'

h --

th) he

.

--a12 t a22 '

2, c3 - cx~.

462

Multi-layer systems

(750a)

~12(X) - -

h -- (h - - h 2 )

cosh(x a~-d~) cosh(B a~h-~) '

qgr I a n d (Pr2 f r o m

qgrl + Ulq3r2 -- (h + u l h 2 ) e -(x-B)4~ff(, q)rl + H2@r2 -- (h + u 2 h 2 ) e -(x-B)~/-~,

h a/h-~tanh(B a ~ - ~ ) + ,/-PS-,/-P7 h h2 --

Ul-U2

a ~ - ~ t a n h ( B v / - d ~ ) + Ul~-fff--U2~l'-~ u 1--u 2

with Pl and p2 being the roots of p 2 _ ( a l l + a12 + a 2 2 ) P + a l l a 2 2 - - 0

and a 2 2 u l , 2 - - a l l + a12 - - Pl,2.

7 5 0 . 0 4 (M)

Multi-layer system with constant infiltration q into an earthen dam of width 2 B , the dam consisting of one layer.

9=0 I .

C///~//////////////, Cl q i K1D1 y///7////////////// Y//////////////////////// C2

0 ~< x ~< B (left side --+ l): d2 dx 2 ~ l - - A t ~ l - q,

I

K2D2 I

C3

I

d d x ~p/(0)

~Ol(B)

0

h.

B Cn

,

K.D.

Cn+l

I

x >~ B (right s i d e - + r): d2 dx 2 ~Or -- Arq~r ,

X

m

B"

d d Tl -~x ~l ( B ) -- Tr -~x ~pr ( B ) .

~Or(0o) -- O,

~r(B) --h.

At - A,

Ar - Ao,

T1

0

...

0

TI--

0

T2

•

0

0

0

T2

...

0 Ti -- Ki Di,

Tr •

Ooo

......

oe

o

o

T~

0

Ooo

......

r~

Discontinuous

(750a)

systems

463

q

K IoD1 q

991

0

0

992

992

h2

m

.

,

~1

--

.

0

,

(~r

--

.

99~

h

,

~

,

•

99~

q~,(x) -- A~lq -- cosh (x ~ )

cosh -1 (B ~ )

(AT'1 q - h ) ,

q~r ( X ) - - e - (x - B ) ~/-~rr • h

with h from Ttx/~l tanh(Bx/~t)(A~-lq - h) -- T r x ~ r h.

The first equation at the left side of this set of equations yields c~ x 0 - indefinite. 750.05 (M) Confined multi-layer system with two layers. As 750.04 with n - 2, cl - cxz, C3 m (X3. U///2"~///////////////////~ I

C2

A, -- ( a12

I

i

X

I

B

K~D2

i

Ar (0

--a22

~Ol l (X ) --

-ale)

--a22

0)

a22

(o)

~Or - -

'

992

rl(x) + al20(x)

99/2(x) --

a 12 -Jr-a22

r/(x) -- a12h2 -k- a 2 2 P

,

q --

,

a22

KIoD1

,

tp/-- (991) , 992

h -

h2

17( x ) - a 2 2 0 ( x )

a 12 -+- a22 (B 2 _ x2).

2K1D1

O(x) -

p K1Dl (al2 -k- a22)

99r2(X) -- h 2 e - ( x - B )

h2-

Ev/~

-

} c o s h ( x x / a l 2 -k- a22) h2 c o s h ( B ~ / a l 2 -k- a22)

K1 D1 (a12 -Jr-a22)'

with

p tanh(B~/al2 %- a22) q- B ~ / a l 2 -+- a22 K1D1 + K2D2 a22 tanh(B~/a12 + a22) -k- ~/a22(a12 ~- a22)

"

464

Multi-layer systems

(750b)

b. Radial-symmetric flow 750.11 (M) Multi-layer system with a circular open water basin of radius R fully penetrating the first layer. Constant drawdown h of the surface water level.

I-h"////////////////////////~ C1 K1D1 I y///)'/////////////A ///////////////////////// C2 I

0 ~< r ~< R (left side --+ 1):

K2D2 i

d2 1 d dr 2 q/l -q- r dr q/l - Atcpl,

C3

r

d d---; qll(0) -- O,

I i~

,

--

R

~Ol( g ) -- h.

¢n

Cn+l r >/ R (right side--+ r):

F

m

R"

d

d2 1 d dr 2 qlr-k-F'~rq~r--Ar~Or'

d

Tt ~ ~l(R) -- Tr ~ ¢pr(R).

q)r(R) - h.

~Or(00) - - 0 ,

Al, Ar, Tl, Tr, qll, q/r and h as in 750.01.

q~l(r ) - lo (r v/-All ) l o l ( R vF-All) h, ~r(r)-

Ko(rv/-Arr)Kol(Rx/Ar)h

with h from

Tlv/-~lI1 ( R x / ~ / ) I o

1(Rx/~/)h

-- _TrV/-~rK1 ( R ~ r ) K o

1( R x / A r ) h .

The first equation of the latter vector equation yields at the left side c~ x 0 indefinite. 750.12 (M)

.

.

.

.

iz-I

~//////////////////////~ C1 w~ K1D]

~ / / / / / / / / / / / / / / / /I / / / / / i

I[ I I

As 750.11 but with entrance resistance w between the upper aquifer and the open water. h h2

C2 hl --

K2D2 • ¢3

.

h.

hi

h2 ,

hr -

•

hn

•

(750b)

Discontinuous systems

*

¢pl(r) -- I o ( r v / A l l ) I o l ( R V A l ) h l ,

*

~Or(r)- K o ( r x / ~ r ) K o l ( R d A r ) h r

465

hr from:

with hi and

TIv/-~lI1

(Rx~/)Io 1(Rv/~/)h/ -- - T r x ~ r K 1 (Rx~r)Ko 1( R d A r ) h r .

The first equation of this set of equations with unknowns hi, the left side oe x 0 - indefinite and must be replaced by

-(h-hi) 750.13

D1 w

~ o

(M)

~o=0

As 750.12 with two layers: n -- 2, c3 -- c~.

i

mM ,<,D,

Ar -- (all ~- a12 --a12 ) \ --a22 a22 '

C2

I

i

KzD2

I

T

I

(OO Tt-

0

0 ) ( T T2

'

--

q)12

'

1 0)

Tr--

( h )

*

--a22 a22 ' /

¢'1112"~11/111//I//111////,,~ I

o)

Cl

Y/III/I/III/II//I/II////¢

i

~0l

h2, ..., hn yields at

q~r - -

~p/2(r) -- h - (h - h2)

(~Orl) q)r2

0

T2

,

'

hl - -

(h) h2

,

hr

_ hi) ( h2

"

lo ( r x/-d~) Io(Rx/-d-~) '

~Pr1 and qgr2 from qgrl -+- Ulq)r2 -- (hl -+- ulh2)

Ko (r x f - ~ ) Ko(Rx/~)

and

g)rl -Jr-U2qgr2 --

(hl + u2h2)

Ko (r x/~2) Ko(Rx/~)

'

hi and h2 from hi

a22ul f (Pl) nt- a22u2f (P2) } -- P2 = hg(a:22)

Pl

Pl -- P2

466

Multi-layer systems

(750b)

and

h l [ K l W a 2 2 { u 2 f ( P l ) - u , f ( p 2 ) } + p, - P2] + h2KlWa12{f(p2) - f ( P l ) } = (Pl - p2)h in which

g(a22) -- ~/-d~I1 (R a~/-d~) Io(n ~4~5S)

f ( P ) _ ~/-fi K1 ( R q/-fi)

Ko(R4-~)

Pl and P2 are the roots of p2 _ (all + a12 -+- a22)P + alla22 -- 0 while a22ul,2 -all + a12 -- Pl,2. 750.14

(M)

Multi-layer system with a circular open water basin of radius R, fully penetrating the first layer. Fully penetrating wells in all aquifers at r = 0 under the basin, with constant but different discharges Qi.

9=0 "/////////////////////////Cl

ZQ ~ K1D1 ~////////////////////~

Y///

' i i

C2

I

~Q1

K2D2

0 ~< r ~< R (left side --+ l)"

!

C3 r

d2 1 d dr2 ~l -t- r dr tpl - AltPl,

i

i

R

K.D. r >~ R

(right side --+

lim

C n+l

r ---~O

r):

r -

d2 1 d dr 2 ¢ffr -1" --r -~r ~Or = A r ~or , ¢ffr ( g ) - - h ,

(d) r

~Ol - - - q ,

-~f

R"

d

T l --~r r q~l ( R

d ) =

Tr ~

¢ffr ( g ) .

~Or ( O0 ) - - 0 .

Al, Ar, Tl and Tr as in 750.01.

~Ol

0

qgl

~02

(/92

- -

.

~Or - -

,

~On

.

9n

0 h2 ,

h --

.

,

hn

0

q2 q

-

.

,

qi

-

Qi 2zr Ki D------S"

qn ~l(r)

Ko(r~/)q

+ Io(rv/-All)Iol(R~l){ h-

Ko(R~/)q},

~Ol(R) - -

h.

Discontinuous systems

*

(750b)

467

tpr(r) -- Ko(rv/-~r)Kol(RVAr)h

with h from: 1

TI[~-

Iol(RVAI)q - ~ l I , ( R x / ~ l ) I o l ( R v ~ l ) h }

--- Tr~rrK1

1 (Rx/A-Tr)h.

(R~r)Ko

The first equation of the latter vector equation yields at the left side oo × 0 indefinite. 750.15

(M}

Q2

Confined multi-layer system with two layers. As 750.14 with n - 2, cl - co, c 3 -- oo.

E-0

,, (0--a22 a220 t

¢1=oo

--

K1D1

,

¢p/--

(0) ~12

7//////////////////////////////////////9 ¢2 ', K2D2 r I I

3

~-

I

=

c3=00

_--!

R

Ar-

q --

•

( a12 --a22 (0) q2

~r2

h_(0) h2

qg12(r) --

q2Ko(r~/-d-27) +

qgrl(r)_

h2a12 a12 -+- a22

{1

h2

{

a12-t-a22

' Q2

'

qgr2(r) --

with

~Or-- ((#rl)

-a12) a22 '

'

q2 - 27r K2 D2

{h2 -

Io (r ~d-~ ) q2Ko(R~dT-2 ) } io(Rx/_d~ ) ,

Ko(r~/a_12-i-a22) } Ko(R~/al2 -+-a22 ) ' Ko(rx/a12-k-a22)}

a 12 q-- a22

KooiR~72 -~-a22)

h2 from: I1 (R a~/h-~) +

h2 "~/a22 Io(R~/~)

a22

~/a]2 +a22

Drawdown at infinity"

(/grl(OO) -- ~Or2(OO) --

h2a12 a12 -+- a22

K1 (R~.al2 + a22 ) } _ Ko(R~/al2 q-a22 )

q2 RIo(R~-~)"

,

468

Multi-layer systems

(750b)

750.16 (M)

¢p=O Cl

i=~ Q1

i i i

KID]

'

;,'7/2 "//////////////, ////////////////////////P C2 i

!~ Q2

K2D2

i !

¢3

r i ,_

Multi-layer system with the upper aquifer being a circular island with radius R. Fully penetrating wells in all aquifers at the centre of the island, with constant but different discharges Qi. Differential equations and boundary values as in 750.14, but with Al -- A, Ar = A0, T1

0

o

T2

0

......

...

0

L

i

R

Tz-

C n

K.D.

¢n+l

c~

0

...

o

T2

0

......

0

T,,

ql

o

q2

qm

Tr

Qi qi -- 2zr Ki Di '

0 992

~01

~o2 ~01

--

q,

T,

•

,

~ r

~On

0 h2 h

--

~

h,,

~n

Solutions -- solutions of 750.14. 750.17 (M) ~o=0

As 750.16 with constant infiltration q into the upper aquifer instead of wells at the centre.

/////~//////////////, el

q

0 ~< r ~< R (left side -+ 1)"

! K]D] V///)/////////////A 7//////////////////////// C2 I I

K2D2

d2 1 d dr2 ~Ol -~- r cir ~Ol - At q~l - - q,

I

C3

I

i

r

I

i

d

d---~~l(O) -- O,

~v

R

K~D,,

Cn Cn+l

~ l ( R ) -- h.

Discontinuous systems

(750b, c)

r ~> R (right side--+ r)" d2

r--R: d d Tl ~r ~ot(R) -- Tr -~r ~or(R).

1 d

dr 2 ~ ° r - k - r - ' ~ r ~ ° r - - A r ~ ° r '

q~r(R) -- h,

469

~Or(O0) -- O.

Al, Ar, Tt, Tr, ~ol, ~0r and h as in 750.16;

•

q

°

0

*

~Ol(r) -- Al-lq - I o ( r v / - ~ l ) I o l ( R x / ~ l ) { A ; l q - h},

*

q~r(r)- K o ( r ~ r ) K o l ( R v / - ~ r ) h ,

with h from Tt~A-~lI1 ( R ~ l ) I -'- T r ~ r g l

o' (R~/) (gV/~r)go

{ A / ' q - h} 1 (Rv/-~-r)h.

The first equation of the latter vector equation yields at the right side oo x 0 indefinite.

c. Two-dimensional flow

750.21 (M) Multi-layer system with a semi-infinite open water body (lake), fully penetrating the upper aquifer, with entrance resistance. Fully penetrating wells in all aquifers at (a, 0), with constant but different discharges Qi.

9=0 Qy////////////~////////. E cl

w

Y/////////////////S

KzD2

•y///////////~)?/w///~ c2

Q2~: i

x ~< 0 (left side --+/):

C3

X

O2

02

0 X 2 ~Ol -~- ~Oy2 ¢Pl -- AI ¢Pl,

KnDn

Q~,~, t

•.

Cn

9t (-cx~, y) = O,

~ot(0, y) = hi (y),

Cn+1

0 Oy q~t(x, O)

q~t(x, oo)

XQ a

x

O,

O.

(750c)

470

x >~ 0 (right

side -+

Multi-layer systems

r):

x=O:

O2 02 OX2 ~r -}- ~y2 ~0r -- Ar ~0r, lim

r-+O

r

--

~ ~r

3

3

Tl~x ~l(O, y) -- Tr ff-X'X~r(O, y).

for r - ~/(x - a ) 2 + y2,

--q

9r(0, y) = hr(y), 0 3y ~0r(X, 0) 0,

~0r( ~ , y) "- 0, ~0r(X, OO)

0.

AI, Ar, TI and Tr as in 750.01. 0

qgl

92

(/92 ~Ol - -

.

,

,

~Or

--

qgn

~on

0 hi

h2

---

.

hr

~

hi hE

--

•

hn

q

-

~

,

hn

ai

q3li = qgli(X,

qi -- 2zr Ki Di ' hli -- hli (Y),

hri -- hri

ql q2

qn Y),

(tgri = qgri (X,

y),

(y).

Y) -- _2 fo ~ ex~/Al+Ot2i~iI(ot) cos(yot) dot,

~Ol(X,

7~

~Or(X,

y ) = {Ko(~/(x - a ) 2 + y2. v/-~r)_ Ko(~(x + a) 2 + y2" ~ r ) } q + - -2

fo

e-X~'/A~+~2Ilar(ot) cos(yot) dot

with ~ll and l~lr from

Tlx/Al + ot21l?l/(ot) - Tr {~e-a~/Ar+°12Iq -- x/Ar + ot211?lr(ot)}.

The first equation of this set of equations with unknowns hi, h2,..., h,, yields at the left side oo x 0 = indefinite and must be replaced by D1

]~l

with

113 o /,2 .

/;,,

,

f,2 .

A,,

and

fzi (ot) -

f0 °

hi (y) cos(uy) dy.

Orientation table D

(800)

471

D. DISPERSION ORIENTATION TABLE D 800. G r o u n d w a t e r flow with dispersion of a solute

a. Does the dispersion of the solute takes place in uniform flow ?

go to 810 (D 1)

Uniform flow is defined as flow in a fixed direction with everywhere the same constant velocity. It is assumed that the uniform flow will not be disturbed by the dispersion of the solute.

b. The groundwater flow in which the dispersion takes place is radial-symmetric See comments of 9b.

go to 820 (D2)

472

(810-812)

Dispersion

810 D-1. DISPERSION IN U N I F O R M FLOW

a. Has the dispersion a one-dimensional character?

go to 811

• Though the groundwater flow is one-dimensional, the dispersion in general has a three-dimensional character. Only if the solute concentration is uniformly distributed over a cross-section perpendicular to the flow direction, we have to do with one-dimensional dispersion. In this case there is no transversal dispersion and only the longitudinal dispersion coefficient DL must be considered.

b. The dispersion is two-dimensional

go to 815

The practical cases in this category are limited to injection of a solute (tracer) into an aquifer with uniform flow by means of a fully penetrating well. The dispersion is anisotropic, as also transversal dispersion takes place.

c. The dispersion is three-dimensional

go to 816

Injection of a solute by means of a point well into uniform flow causes spreading of the solute in three directions: one is the flow direction and the other two perpendicular to it, yielding an axial-symmetric concentration distribution of the solute, as the coefficient of transversal dispersion D-r is the same for the two directions perpendicular to the flow direction.

811. One-dimensional dispersion in uniform flow a°

Does the area where dispersion occurs extend to infinity in the direction of the flow ?

go to 812

A concentration distribution of a solute extending to infinity in uniform flow, means that the concentration variable x varies from x = -cxz till x -- +cx~ in such a way that no plane of symmetry exists on that interval.

b. The concentration distribution of the solid is semi-infinite

go to 813

Semi-infinite dispersion in a direction is defined as dispersion which takes place only at one side of a boundary, the concentration variable x varying only from x - - 0 till x = +cxz.

c. The dispersion is finite

go to 814

Finite dispersion in a direction is here defined as dispersion in the direction of the uniform flow between two boundaries, situated at a finite distance of each other, parallel to each other and perpendicular to the flow direction.

812. One-dimensional infinite dispersion in uniform flow O2C OC DL ~ - - u Ox

OC Ot

(812)

Uniform flow

473

with DL -- longitudinal dispersion coefficient, u - average real velocity of the groundwater.

812.01 (L) Solute (tracer) mass impuls of strength m i and initial concentration Ci in an infinite field with uniform flow. c - c ( x , t).

C Ci --------~ U

c(x,O)-

X

ci

for x - 0 ,

0

forx--#0,

c ( - o o , t) - c(+cx3, t) - 0. mi

--

Ci

dx - mass per fluid area [ML -2] of the plane source.

mi exp{ (x -- ut)2 } c ( x , t) -- 2~/rC DLt -4DLt ' •

f(x,t)

--

mi (x + ut) e x p { 4t ~/zr DL t

(x - ut) 2 4DLt }"

812.02 (MI) Arbitrary initial concentration Ci ( X ) as function of x only, in an infinite field with uniform flow. c - c(x, t).

c

) :- Ci

c(x, O) - ci (x).

r

dxo

x

1 c ( x , t) -

2~/rC DLt

ci (xo) exp

~

i,xxo.,)2 -

I dxo,

4DLt

Oc

f (x, t) -- u c ( x , t) -- DL 7 ( x , ox

t).

812.03 (MI) From 812.02 with CO

m

-c0

co ----% U

1 ci(x) -

forx <0,

5 co

forx --0,

0

f o r x > O.

474

(812, 813a)

1

Dispersion

(x-u,)

c(x, t) -- -~ co erfc 2 D~/-D~ '

1 x 2 u t ' ] Co~/DL f (x, t) -- -~ cou erfc(2 D ~ - . ~ / +

(x -- ut) 2

/"

812.04 (MI)

c

From 812.02 with

Co ~ 1

~Co

/ CO for Ix l < a, 0 for lx i > a, Ci (x) -co for I/I - a 2

----->u Co

o

+a

--a

c(x, t) --

x

-2-c°{ (x+a-ut)__ erf 2 Dv/_D_~L t

- erf

(x-a-ut)}

Oc f (x, t) -- uc(x, t) -- DL -~x (X, t).

813. One-dimensional semi-infinite dispersion in uniform flow

Subdivision: a. Given solute concentration at x - 0. b. Given solute flux at x -- 0. a. Given concentration at x = 0 32C OC DL ~ - - u 3x

OC

3t

813.01 {L) c(x,O

co(O = c(O,O

~,u

The concentration at x = 0 an arbitrary function of the time co(t), c = c(x, t).

t

c(x, O) = 0

fort > 0 ,

co(0) = 0.

c(cx~, t) = 0 ,

c(x, t) =

c(O,t)=co(t)

forx > 0 ,

x-~/Tr DL

ux f ~ co(,47

X20t 2

4 DL

U2

4 DLot2 ]

dot.

(813a)

Uniform flow

475

813.02 (L)

From 813.01 with

Co(t) Co /~/~//~/~~/~/

c(O, t) --

co(t)-co 0

fort > 0 , for t - - 0

t

= sudden rise of the concentration which is kept constant thereafter.

c(x t) -- co exp '

P

, 4DL 2~D7

1 { UX)erfc(X+Ut ) x-ut]}, = ~ c o exp(~LL 2 D~/-D-~ + e r f c ( 2 D.v/-~TLt/ P -- polder function. 813.03 (L)

From 813.01 with c(0, t) - co(t) - at - linear rise of the concentration.

co(t)

t

tuft{ t4DL ' 2~-D--LL + u tux 4DL

c(x, t) -- a exp ~

tP

-

Pconj

,

2~D-7/

1 UX)erfc(X +ut,~ erfc(X-Ut,~ = -~at {exp (D--EL 2 D~--~Lt/+ 2 D~/~Lt/}

813.04 (PF)

C0

-----

Periodic fluctuation of the concentration at x - 0 according to a sine function. From 813.01 with c(O, t) - co(t) - co sin(cot)

--C O

2rr and T - period. with c o - -y-c(x, t) - steady oscillation of the concentration.

476

Dispersion

(813a, b)

c(x, t)

bx

x

co exp

--

-

2 L)

with a 2 - b 2 --/,/2 and ab -- 2coDE (a > b; a > u; a > 0; b > 0).

b. Given flux at x - 0 of u Ox

a2f

DL ~

of Ot

of Ox

Oc at

813.11 {L)

(t)

Solute flux impuls at x - 0 with strength mi and initial flux f/, giving a concentration distribution c(x, t) in a semi-infinite field

f(x,0 -

-------~u

with uniform flow.

lot c(x, 0) - 0

/

f(O, t)

for x > 0,

c(cx~, t) - 0,

uc(O, t) -- DL -~x (O, t) -- fi

fort--0,

0

fort-ri0,

mi -- fi dt -- mass per fluid area.

•

mix exp{ (X -- Ut)2 } f ( x , t) -- 2t~/zrDLt -- 4DLt '

•

c(x,t) -

mi ~/yr DL t

exp { - (X4DLt--ut)2 }

ux

miu

x

).

2DL exp (~-~L) erfc ( 2 D~77Lt

813.12 (MI) f(x,

The flux at x - 0 is an arbitrary function of the time.

t) -'---'% U

t f--

f (x, O) - O.

f(x,t),

f(x,t)

--

f (c~, t) - - 0 .

/o(0) - 0,

f (O, t) -- fo(t), x

exp ~/~ DL

ux

F ot, ) xot 47

x20/2 4DE

u2 4 DL0/2 ,]

da.

(813b, 814)

Uniform flow

477

813.13 (L) From 813.12" Constant flux fo at x - 0.

fo(O

fo

l

f ( x , t) -- f0exp

•

P

'24-~

.

814. One-dimensional finite dispersion in uniform flow

O2C DE ~

OC

OC

- - u Ox

Ot

o

814.01 (L)

The concentration at x - 0 is an arbitrary

c(x,t)

Co(t) = c(O,t)

"''~

.

function of the time. Zero concentration at

c=O

U

x -

0

t

c(x,O)-O

forx > 0 ,

with An

t) -

--

c(O,t)-co(t)

L.

c -

c(x,

t).

x

fort > 0 ,

co(0) - 0.

c(L, t) -- O, c(x,

L

2rr DL e 2---b' ttX-~L~@o ( - - 1 ) n + l n sin { rt~(L - x ) } fO t e -An(t-r) c0(r) dr L2 n=l L

n2rt 2 D L u2 L2 _qt 4 D L "

814.02 (L)

From814.01 with

coIco(0

co(t)-co c(O, t) -

t

0

-- sudden rise of the concentration which is kept constant thereafter.

exp( )-exp( ) c(x,

t) -

co

fort>0, fort-0

exp(_~L)_l

( ux -

L2 exp 2D

4D

(x)

xZ n= 1

with An from 814.01.

(-1)n+ln L2 ) A2 sin {nTr (LL - x) } exp ( _ n2rC2DLt

Dispersion

(814, 815)

478 814.03 (D)

Steady state of 814.02. exp (-~L) - exp (~-~L) c(x) -

co

exp (-~L) - 1

f (x) -- uco

e x p ( ~ L)

= const.

exp (-~L) - 1 814.04 (L,FSR) co(t) =

c(O,t)

c(x,O

""~

t

c(x, 0) - - 0 Oc ox(L,t)

0

L

for x > O, c(O, t) - co(t)

O,

Oc ~=0

U

co(0)

~

0

The concentration at x = 0 is an arbitrary function of the time. Zero concentration gradient at x - L.

x

c = c ( x , t).

for t > 0,

.

2D ~ ( anX f t c(x t) - --~ e~°-£L an sin k L ) Jo e -Bn (t-~) co(r) dr ux Z ' 1 + o~2 +e 2 n= 1 Lu and B , , 2OL

with an being the roots of a cot or = - e -

or2DL

/2

u2

-~ 4OL"

814.05 (L,FSR)

Sudden rise of the concentration, which is kept constant thereafter

coTco(0

c(O, t) -

co(t)-co 0

t ,,x oo an sin(~nx c(x, t) -- c o - 2coe 2°-''~ ~ L ] e-Bnt

n = l 012 -Jr-g 2 + 8

with c~,~ and Bn from 814.04. 815. Two-dimensional dispersion in uniform flow O2C

O2C

OC

DL ~x 2 + DT ~y 2 " /,/ Ox

0C

8t

fort>0, for t - 0 .

(815)

Uniform flow

479

DE -- longitudinal dispersion coefficient, DT = transversal dispersion coefficient, u -- average real velocity of the groundwater.

815.01 (L) Solute mass line impuls of strength mi and initial concentration ci in an infinite field with uniform flow in an aquifer of finite thickness. mi -- 7r(dr)2ci -- mass per fluid length of the line source [ M L - 1 ] . c - c(x, y, t).

--------~ u

I

I

x

c(x, y , O ) - -

Ir•dY

ci 0

at the origin, anywhere else,

c ( - c x z , y, t) - c(c~, y, t) - c(x, -cx~, t) = c(x, + c ~ , t) -- O.

r

|

mi

c ( x , y , t ) --

exp

4re ~f DL DT t

i

(x -

ut) 2

y2

I

4DLt

4DTt }"

815.02 (MI) Arbitrary initial concentration Ci ( X , y) as function of x and y in an infinite field with uniform flow. c - c(x, y, t).

C

~

/

c(x, y,O) = ci(x,y)

c(x, y, O)

---~U

- - ci ( x ,

y).

/ •

c ( x , y, t) --

1 47r ~/DE DT t × exp [ [

~

~

ci (xo, Yo) exp

/-

xo 4 DE t

yo) 2 /

(y

4DTt

| dxo dy0.

815.03 (MI) c

From 815.02 with co Y

Ci (X, y) -- /

0

---~u x

for - c x ~ < x ~ < 0 and - b ~< y ~< b, anywhere else.

/

480

•

(815, 816)

c(xy '

'

Dispersion

x-ut]{erf(Y+b]_ t ) - - c ° erfc(~ --D~/-D~Lt/ 2-~/

ere( y

4

-b

)}

2 D~/~-

"

815.04 (MI)

c

From 815.02 with co for - a ~ < x ~
Ci(X, Y ) ---"")"U

/-b co{ ( x + a - - u t ) c(x, y, t) - -~ erf 2 D~-~

- erf

(x--a-ut)} 2 Dv/-~-TL t

× {erf( y + b y-b 2 D ~ D - ~ ) - e r e ( 2 D.v/-D-~)}" 816. T h r e e - d i m e n s i o n a l dispersion in uniform flow

O2C DL ~

(02¢

O2C)

+ DTkOy 2 + ~

OC

OC

- u o----x-- 0--7'

DL -- longitudinal dispersion coefficient, DT -- transversal dispersion coefficient, u -- average real velocity of the groundwater. 816.01 (L)

Solute mass point impuls of strenght mi and initial concentration ci in an infinite field with uniform flow.

Z

===~u

4

x

Mi

mi -- ~ :rt'(dp) 3ci -- ~ " ne

[M]

with ne - effective porosity and Mi - mass of injected solute, c - c(x, y, z, t). ¢i

c(x, y , z , 0 ) - -

0

at the origin, anywhere else,

c ( - c ~ , y, z, t) - c(cxz, y, z, t) - c(x, -cx~, z, t) - c(x, c~, z, t) -- c(x, y, -c~, t) - c(x, y, cxz, t) - O.

(816)

Uniform flow

exp{

mi •

c(x, y,z, t)--

8rrDTt~/rrDLt

481 Y2+Z2}

(x--ut)2

--

4DLt

--

4DTt

"

816.02 (MI)

as a function of x, y and z in an

Arbitrary initial concentration C i ( X , y, z) f i e l d with uniform flow. c = c ( x , y, z, t). c ( x , y, z, O) = ci (x, y, z).

l •

c ( x , y , z, t) =

8rC DTt ~/Tr DL t

× exp

{ -

oo

oo

c~

(x - xo - u t ) 2 4DLt

x exp { - (z - zo) 2 4DTt

Ci (Xo, YO, ZO)

2

_

}exp{ -(y4DTty°)

}

} dxodyodzo.

816.03 (MI)

From 816.02 with ci (x, y, z) -

{co 0

for - a ~<x ~
co{

c ( x , y, z, t) -- -~

erf

2 D/_D~Lt - e r f

x {erf(Y+b x {erf( z + c

z

(x-a-ut l 2 D~-~Lt

-b

}

-

}

2 o,/-~t) •

-c~
infinite

482

(820)

820 D-2.

Dispersion

DISPERSIONIN RADIAL-SYMMETRICFLOW

We only consider here radial-symmetric dispersion in radial-symmetric flow, which means that the dispersion is caused by injection of a solute into a fully penetrating well that recharges water into a confined aquifer. The differential equation for the dispersion then becomes"

02C

OC

OC

a L G e r 2 - G or

r ~ Ot

with aL "- longitudinal dispersivity [L]. G - 2rrHne Q [L2T -1 ], ne -- effective porosity, H -- thickness of the aquifer.

820.01 (L)

colc

|

Injection of a solute with constant concentration co at r -

|

O. c - c ( r , t ) .

t_ r

t limc(r,t)--c0

r---+0

c(r,O)=O

u(r) I

fort >0,

c(r, t) -- co - ~3c0 exp

2rr

Ai()~- ~

forr >0,

c(c~, t) - - 0 .

(2@L) f0 c~ F ( r , ~ ) e x p (

- G)~-3/2 t )dX 8a 2

)~-l/2)Bi(~.)- Ai()QBi()~- ~

F (r, ~.) --

with

~--1/2)

Ai 2 ()Q -t- Bi 2 (,k)

Ai(x) and Bi(x) are Airy functions.

820.02 (L) Solute f l u x impuls at r - 0 with strength Mi and initial flux Fi. c -- c(r, t).

L[-

¢=

H

u(r) Idt

f

lim c(r, t) -- aL

r--+O

u(O r

Oc "~r

(r, t)

}

--

Mi Q dt'

c(r,O)--O c(cx~, t) - - 0 ,

Mi - Fi dt.

fort >0,

Radial-symmetric flow

*

(820)

483

c(r,t) -- 2Mi exp ( ~ a L ) fo ~ BuAi(pu) - AuBi(pu)

~--6

~ + B~u

Au -- Ai(Puo) + 2aLhul/3Ai'(Puo),

Ai'(x) --

dAi dx

Bu -- Bi(Puo) + 2aLhul/3Bi'(PuO),

Bi'(x) -

dx

Pu - - g u - 2 / 3

-- hrul/3

1 _4/3G2/3 g - - -~ a L

,

Puo - - g u - 2 / 3

h - aL1/3G -1/3

'

dBi '

e -ut

du

with

Orientation table E E. S H A R P

INTERFACE,

(900-901)

DENSITY

485

FLOW

ORIENTATION TABLE E 900. Interface flow a. Is it allowed to assume only horizontal flow?

go to 901

Interface flow will be understood here as flow of fresh water along a sharp interface that completely separates the fresh water body from underlying salt water, the latter generally assumed to be immobile. As exact calculations concerning interface flow, in general, are rather complicated and also not always lead to solutions, it is worth while to look for approximate solutions, at the same time maintaining the character of interface flow. The latter condition means that the flow profile is not constant but depends on the shape of the interface and eventually on the height of the water table. Considerable simplifications of the calculations can be obtained by assuming only horizontal flow, which, in general, will be allowed if the corresponding flow problem without interface also is based on horizontal flow only. b. Also vertical flow has to be taken into account

go to 910 (E-l)

If it is not allowed to neglect the vertical flow, for instance, if also in the corresponding flow problem without interface vertical flow is taken into account, then exact interface solutions have to be determined.

901. Horizontal flow a. The flow is one-dimensional

go to 920 (E-2)

See comments of 9c.

b. The flow is two-dimensional with radial symmetry

go to 930 (E-3)

See comments of 9b.

c. The flow is general two-dimensional See comments of 9a.

go to 940 (E-4)

486

Sharp interface, density flow

(910)

910 E-1.

EXACT SOLUTIONS IN INTERFACE FLOW

Only very few exact solutions for groundwater flow in aquifers with an interface are available and they are limited to two-dimensional steady flow, in general. Almost all solutions, given in the sequel, are obtained by the method of conformal mapping and therefore will satisfy the Laplace differential equation 02~

F 02q9

ax-

=0.

Interface oarameter: ct -- p s - pf K

pf

910.01 (HO,Q)

.........~

........

¢ ---o

Very thick confined aquifer in a semiinfinite field. Flow of a quantity q of fresh water towards a horizontal drainage area in the sea. ~o = ~0(x, z) = head.

x

~o(x,0) = 0

for x <~ 0,

0!(x,0)-0 Oz

forx>10.

Z I n t e r f a c e : (/9i :

q,

olzi, ~i -" Kqgi, l[ti :

Y2 = x/2otKq(

with ,f2 = q~ + i ap and ( = x + i z.

Interface:

Z2 -

-- H 2.

q (-~+2xi) otK

910.02 (HO,SC)

........ _e_a0 .......

As 910.01 but with a confined aquifer with finite thickness D. tp = ~o(x, z) = head.

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

7z=:o

H

x

D

q¢--

q~(x,O)=O

forx~O,

0~O(x,0)_0 Oz O!(x, D) - O. Oz

forx/>0,

Exact solutions

(910)

Interface"

(tgi - - O l Z i ,

~)i - -

K~oi,

~"

q,

lPi - -

Y2 -- q arccosh( 1 - c w ) - o k +

487

iTr,

w--c

arccosh u du

~ = q~/c 2 - 1 f w 7c2c~K

(u -

=x+iz

c)Ju 2 - 1

D with c - cosh(TrotK-7). Interface:

q ~/c 2 Xi

--"

Zi =

1f t

--

7r2o~K

arccoshu du

J1 (u

-at- C ) ~ / / , / 2 - -

q arccosh(1 + c t ) Jro~K t + c q4c

X0--

2

--

1 f+l

7r20tK

J-1

1

(t >/ 1) -- parameter,

-- X0,

with

arccos u du (u - c)~/1 - c 2

910.03 (C)

~o(x, o)

Very thick confined aquifer with infiltration in a finite field between two parallel open sea water boundaries with zero level. Distance between the b o u n d a r i e s - 2b. ~o(x, z) - head.

r

~(o z) -o, Ox

Interface:

q9 i -

otzi,

¢i

-

K~oi,

° ~ ( x , o) Oz

K

~o(x, O) - 0

f o r x >~ b.

ff/i - - O.

X22 + ol K p ( 2 - 2ip~2( - p ( p + otK)b 2 = O, ~--¢+i~,

-x

+iz.

Interface:

z2i -

Pb 2 otK

p x 2 _ H 2. p + otK

'

for 0 ~< x ~
488

Sharp interface, density flow

(910)

910.04 (HO,SC)

Two fresh water reservoirs separated by an impervious screen (dam) on top of an aquifer with an interface between fresh and salt water, lying originally at a depth d. The level of one of the reservoirs is lowered with h < otd. q) = q)(x, z) = drawdown.

~o=0

~I I I I I I'

I

......

q)(x 0 ) '

/_interface

IIIIIIIIIIIIIIIIIIIIIIIIII

Z

ot(d

Interface: q3i - -

-

zi),

~)i =

Kq)i,

l0 [ h

forx~<0, forx~>0.

lPi --" O.

For h < ~d: ( 7rS-2 ) 2h { (zr~)} 2d In tan + In 2 cos --~ 2Kh 7rot 2Kh

(-

,f2 = ~ + i~p,

-

iS2

+id,

( =x+iz.

Interface: X i

--

~

7r

In tan

+

2h

zrot

In 2 cos

.

2h

Critical situation:

h = otd - - - - - %-%-;.

(the interface makes contact with the bottom of the fresh water reservoir; no inflection point).

I I I I I I I I I I I I I I I I I I l [ l l l l l l l l l l l l l l

I I I I I I I I I I I I I I I I I I I I I I I l l l l l l l l l l

910.05 (HO,SC)

_~_ . . . . .

qff"

f,_

salt water x

72=0 ~ Ho

H

Horizontal drain on top of a very thick confined aquifer. The original horizontal interface has been lifted to a distance H0 from the drain, q) = qg(x, z) = head. Drain with discharge q at (0, 0). 0q)

~(0,

z) = 0

Ox Oq9 (x, 0) = 0 Oz

for z : / : 0 ,

for x --/: 0.

Exact solutions

(910)

489

Interface: q) i

q

Ol Z i 4 - fi s

q~i - - Kqgi,

Pf

with ~Os - salt water head - const., Ps (t9i ( 0 )

- - (t90 - - o l g O 4 - - - ( t 9 s. Pf

X2 -- q In sinh w + K~oo (= q5 + i~p), Jr 2q ~ 2 2nB2nw 2n+l 2 { ( = Jr2a g "---'2 n=o (z)!(z'"n'""n + 1) + -Jr kH°

q e)w

Bn -- Bernoulli number, e - 0.693147180546. q Zi ---

lncosht+Ho--H,

JraK

2q Xi - -

k

Interface"

parameter t >/ 0,

22n(22n - 1 ) B2nt2n+l

Jr2o¢K

(--x4-iz)

Jra K

(2n) .V(2n 4- 1)

2( 4- --Jr Ho

q ) J r a K e t.

n=l

910.06 (HO,SC)

Horizontal drain at a distance d under open water in a very thick aquifer with an interface, originally horizontal at a depth h. q)(x,z)= head. qg(x, 0 ) = 0. Drain

q)=O

W

...............t ..................x

with discharge q at (0, d). Interface: l~ri~

~i

---

-a(h

q 2"

2haK (rounded top of the interface)" If q < ---~£-2--

q ln (~/w + a + ~/a + l )

2Jr

~w +a-

2h ~ arccot Jr

( -- ~ i

~/a + 1

~

q

i In

Jra K

-- ~b + i~p,

(~/w + a + ~/a + l ) 1 + 4r-w '

a from 2h d

m

--

arctan ~ -

Jr

In ~/a + 1. Jra K

Interface" 2h Xi--

7/

arctanh

q Jra K

arctan ~/7, r

-

zi),

~i

"-

Kqgi,

490

(910)

]a-t

q h -~arctanh zr ot K

Zi - -

=H,

+1

Sharp interface, density flow

O~
Critical situation" 2h~K

q-

,/-d

(sharp top of the interface). 910.07 (D) Non-steady one-dimensional density flow through a semi-permeable layer. The interface, lying originally along the bottom of the semi-permeable layer, moves upwards because of a lowering of the fresh water head in the aquifer, with h. zi = f (t) or t = g(zi). Initial value: zi (0) = O.

Z

Ih salt water head

D illllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll

•

"el

t--

otKfM

hl(h

M D -t- (1 - M ) - In ot {

h - o~zi

- (1 - M)zi

]

.

Velocity of the interface: KfM(h

Vz~(t)

-

M-

-

olzi)

(1 - M ) z i -1- M D '

/zf-

/zf _ mobility ratio, /Zs viscosity of fresh water,

/Zs -

viscosity of salt water,

?,f -

specific weight of fresh water,

?'s -

specific weight of salt water,

Kf-

kvf

-- fresh water permeability of the semi-permeable layer,

/Zf

k-

intrinsic permeablity, V s -° - V f

~

~

Vf Steady state

zi=-

h o/

h if D > / - .

o/

One-dimensional flow 920 E-2.

(920-921)

ONE-DIMENSIONAL

HORIZONTAL

491

INTERFACE FLOW

go to 921

a. Does the interface lie in a confined aquifer? ......

The aquifer in which the interface lies is confined by an impermeable layer at the upper side and also if the aquifer is not assumed to be "infinitely" deep at the bottom. The fresh water flows according to the shape of the piezometric head which is lying above the upper side of the aquifer.

~ _ ~ a d

.L__...._~ ..... salt

go to 922

b. The interface lies in a phreatic aquifer

h

water table

The fresh water flows between a water table and an interface in a phreatic aquifer, which may be assumed to be "infinitely" deep.

H fresh

salt

c. The interface lies in a semi-confined aquifer (leaky aquifer)

5////////, ~ / / / / / / / / / / / / / / / / / / / / ~

/,

fresh/

The aquifer in which the interface lies is confined at the upper side by a semipermeable layer, above which a constant water level is maintained, often serving as the reference level for the piezometric head.

C

Jill

H

go to 923

...... salt

921. Interface in a confined aquifer with horizontal one-dimensional flow o/

dx

+ otK - 0,

m

Ps - - P f

Pf

H -- depth of the interface [L], p -- infiltration or precipitation [LT -1] which may be zero, Ps -- density of salt water [ML-3], pf -- density of fresh water.

492

(921)

Sharp interface, density flow

921.01 (H,D)

x

~

Coastal aquifer of "infinitely" thickness between two open parallel sea water boundaries with infiltration

sea

p$$,W$$ g,l,,l, $ $ $ $,1, $,1,,I,$ , 1 , , 1 , , 1 ; , ~ - ~

fresh / :

dH ~(0) dx

salt

=0,

H(B) =0.

H2-

B

P (B 2 -- x 2 ) , otK ~o = o~H.

__

"-'rrm

921.02 (H,D)

~

As 921.01 but with an aquifer with thickness D such that H(xo) -- D with xo > 0.

sea

d~o ~(0) dx

salt

P

B

_ ~

--0,

~o - 2 K D

._.....~

H(B) =0. (X~ -- X 2) + 0¢ D

for 0 ~< x ~< xo, X0

.

H 2-

qg=otH

P (B2 - x2)' ~---K

x°2 -- B2

ot K D 2 P

'

f o r x o ~ < x <~B,

D
921.03 (H,D) q sea ............./ , ~ ~ ~ l ~ + ~ a : ~ ~ ~ ;

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

H "%~r~-" B

Coastal aquifer of infinitely thickness between two open parallel sea water boundaries with infiltration. Well-screen with discharge q [LZT -1] at x -- xo, with [xo[ < B, fully penetrating the fresh water body. H(-B)

.i Xo ~B

salt

=0,

H(B) =0.

(92 l, 922)

One-dimensional flow

H 2 :

P

(B 2 -

(e

x 2) -

c~K H 2 :

P

.(B 2 _ x 2 ) _

o~K 99 =

921.04

xo)q

ol K B (e + x o ) q ol K B

493

(B + x)

for - B

(B - x )

f o r x o ~ < x <~B,

~<x ~<xo,

(q < p B ) .

olH

(H,D)

Coastal confined aquifer with thickness D, such that H ( x o ) = D if xo > 0, the aquifer lying between two parallel open boundaries, a salt water boundary and a fresh water boundary. Fresh water flow caused by constant high fresh water level.

sea

H

Xo

B

99(0) = 990,

99(xo) = o~D,

990 - otD

~

99=99o-

x

H ( x o ) = D,

H ( B ) -- O.

f o r 0 ~ < x ~<xo,

X0

99=c~H H2 =

f o r x o ~ < x ~
990 - c~D xo -- 2990 - c~D"

Total discharge: KD q -- --2-:-~(299o - o~D). Z15

922. I n t e r f a c e in a p h r e a t i c a q u i f e r with h o r i z o n t a l o n e - d i m e n s i o n a l flow

d(dH) dx

~x

p + c~(1 + u ) K - 0 ,

0/ - -

Ps -- Pf

Pf

H = depth of the interface [L], p = precipitation or infiltration [ L T -1] which may be zero, ps -- density of salt water [ML-3], pf = density of fresh water.

494

Sharp interface, density flow

(922)

922.01 (H,ED) Coastal aquifer of infinitely thickness between two open parallel sea water boundaries with precipitation.

p $,[,~$,[,$$$,1,,1,,l,$,1,$$$,1,,1,$$,1,

dH

~(0) dx ,

~_.~

H2=

H(B) --0.

P (B 2 - x 2 ) , or(1 + o~)K

h--otH.

D,

B

--0,

B

922.02

(H,ED)

As 922.01 but with an aquifer with thickness D, such that H(xo) -- D (xo > 0). m

m

dh dx (0) B

m

0

~

H (B)

m

0.

_~

Xo

P (x 2 (D + h) 2 -- (1 + ~)2D2 + ~H2= X2

n

P (B 2 - x 2) u(1 + a ) K

B2 - or(1 + ct)KD 2 P '

x 2)

for 0 ~< x ~< xo,

for xo ~< x ~< B,

D < B~

h - all,

P

V(z(1 + ¢z)K"

922.03 (H,ED) q P

Coastal aquifer of infinitely thickness between two parallel open sea water boundaries with precipitation. Well-screen with discharge q [LZT-1] at x - xo (Ixo[ < B), fully penetrating the fresh water body.

H(-B) -0, B

..... ~ X° B

H(B) --0.

One-dimensional flow

,

(922, 923)

P

H2=

(B - xo)q

( 9 2 - - X 2) --

0/(1 + 0/)K 9 2 --

495

0/(1 -t- 0/)KB (B 2 -x

2)

0/(1 + 0/)K

(B + xo)q _

0/(1 + 0/)KB

(B+x)

for - B < ~ x

~<xo,

(B-x)

f o r x o ~ < x ~
h--0/H. 922.04 (H,ED) Coastal phreatic aquifer with thickness D such that H(xo) = D (xo > 0), the aquifer lying between two parallel open boundaries, a salt water boundary and afresh water boundary. Fresh water flow caused by a raise of the fresh water level. h(O) - ho, Xo

H ( B ) -- O.

B

(D+h) 2- (D+ho) 2-x{(D+ho)

2 - (1 + 0/)2D2}

x0

H2 __ B - x D2 B - xo

for xo ~< x ~< 0,

for 0 ~< x ~< xo,

h - O/H,

(D + ho) 2 - (1 + 0/)20 2 x0-- B

0/(1 + 0/)D 2 + (D + ho) 2 - (1 + 0/)2D2"

Total discharge" q -- ~K- {0/(1 + 0/)3 2 + (D + ho) 2 - (1 + 0/2)D2}.

923. Interface in a leaky aquifer with horizontal one-dimensional flow 923.01 (H,D) [KX

Semi-infinite very thick leaky aquifer with open sea water boundary. Raise of the polder level with ~Oo.

sea

....................... ~///////////////////////////////k

C X

Ho

d(HdH ) Ho- H dx -~x + Kc

O,

H - depth of the interface, H(O) - O, H(cx~) - head.

Ho -

~o

496

Sharp interface, density flow

(923)

x -- ~/3Kc{F(H) - F(O)} F(u) -- / _ ~ ln ( q)o Ho-- - - ,

with

34~-~°+ ~/2u + H° ) _ ~/2u + Ho, 3 ~ ~ - ~/2u + Ho

~o-o~H.

Ol

Total discharge:

~/

Kqgo ]q (O) i -- ~oo 3otc

923.02 (H,D}

sea

- i /t. / / ~ / / / / / / / / / / / / / / / / / / / / /

Semi-infinite very thick leaky aquifer with open sea water boundary, with infiltration.

C

1,1,,[',~,['4,P x

d(d.) dx

H

H(O) --0,

rmm-rr x -- ~/3Kc{F(H) - F(O)},

q9 -- otH. pc

F(u) as in 923.01 with Ho replaced by -g-. Total discharge:

Iq(0) i - - p c

~pK 3-d"

.

-di-x - T / c + ~/( - 0 , dH -7-(0o) --0. dx

Radial-symmetric flow

930 E-3.

(930a)

TWO-DIMENSIONAL

497

RADIAL-SYMMETRIC

HORIZONTAL

INTERFACE

FLOW

Subdivision: a. Confined aquifers. See comments 920a. b. Phreatic aquifers. See comments 920b. a. Confined aquifers

,d( r H d.)

r dr

~

+

p otK

--0

'

O/ =

Ps -- P f

Pf

H = depth of the interface [L], p = infiltration or precipitation [LT-1].

930.01 (H,D) sea

Circular island with infiltration. Very thick confined aquifer with sea water boundaries.

P dH ~(0) dr ,

H 2 =

=0,

H(R) =0.

P---~-(R 2 _ r2),

2c~K ~o = oeH.

930.02 (H,D)

....... t

~

sea

As 930.01 but with an aquifer with thickness D such that H(ro) = D

(r0 > 0). dcP (0) - O, dr

r0

R P ~o= ~ ( r o 4KD ~p=uH

2 - r 2) q-otD f o r r o ~
f o r O ~ < r ~
and

H ( R ) - O.

498

Sharp interface, density flow

(930a)

.

H 2=

930.03

P (R 2 _ r 2 ) , 2oeK

r2-

2°e K D2 , p

R 2_ ~

D
P 2otK

(H,D)

a

Circular island with infiltration. Discharge well at the centre. Very thick confined aquifer with sea water b o u n d aries.

sea ,!,,[, ,I,,i,,1,~ ~ ,I,,!,,!, ,!, ,[,,!;,1,,1,,I,,!,,!,,I,,!, i

J~

i

|

I

i

-

r

lim r --+ O

Ir

---

Q

2 Jrot K H

,

H ( R ) --0.

R H 2 = ~ (PR 2oeK

930.04

2 - r 2)

Q In ( R ) , zrc~K

9-oeH.

(H,D) Ring-shaped dam in the sea on a very thick confined aquifer with infiltration. Inner radius rl and outer radius r2. H ( r l ) -- 0,

r1 5/

H ( r 2 ) -- 0.

r2

,

H 2P (r2_r2)ln(~) -- 2oeK In (rr-~l)

930.05 i

i i fresh 4

_ _

se___aa

i

i

ro _

Ring-shaped dam round afresh water basin in the sea. Aquifer with thickness D, such that H(ro) -- D (rl < ro < r2). Constant high fresh water level. Inner radius rl, outer radius

~o(rl) -- qgo,

i

i i

9 -- oeH. '

r2.

i i

2)

(H,D)

i _i

P (r2_r 2o/g

r2

H (r2) -- 0.

Radial-symmetric flow

(930a, b)

499

In r - In r l q) -- 9 9 0 - ( q 9 o - olD) c# -- o t H •

H2

for r l ~< r ~< ro,

for ro ~< r ~< r2,

In r2 - In r

--

In ro - In r l

0 2,

In ro -

2(q9o - c~D) lnr2 + otD l n r l

In r2 - In ro

2~oo - oeD

Total discharge: 7r KD(2qgo - otD)

Q

In r 2

--

In r l

b. Phreatic aquifers rH

r dr

--dTr

+

or(1 + oe)K

=0

'

oe -

Ps -- Pf

Pf

H -- d e p t h o f the interface [L], p - p r e c i p i t a t i o n or infiltration [ L T - I ] . 930.11

(H,ED)

Circular island with precipitation. Very thick phreatic aquifer with sea

p ,!,,1,,1,$$,i,,1,,i,,1,,1,,,i,,1,,1,,i,,1,,1,$,!,,[,,1,$ -.------ ~ _

water b o u n d a r i e s .

dH

~ ( 0 ) --0,

H(R) --0.

dr

,

H2=

P

(R 2 -

r 2)

2or(1 + ot)K

h--~H.

rrrrr vrrrrvm~,, ..... R

930.12 p

(H,ED)

~$~$~~$~~~$~

As 930.11 but with an aquifer with thickness D such that H(ro) -- D

r____

(ro > 0). dh d--~(O) - O,

NNNN

ro

(h+D)

R

a-

P (rg_r ~-~

2) + ( l + c ~ )

2D 2

for0~
~
H ( R ) -- O.

500

Sharp interface, density flow

(930b)

h--c~H H 2 --

forro~
~/ D < R

r2 _ R2

2or(1 -t- ot) K D 2 P

P 2c~(1 + o t ) K

930.13 (H,ED)

Circular island with precipitation. Discharge well at the centre. Very thick phreatic aquifer with sea water boundaries.

,1,~,i,,1,,1,,I,,1,,!,,I,,I,,1,,1,,1,,I,,I,,1,,!,,1,,1,,!,,1,p

a r,

!:

sea --

;

lim r ro0

:!

d(H + h) ] dr

Q

'"

H

2rcotK(H + h)' H(R) --0.

I-

,

In ( R ) , P (R 2 _ r 2) Q 2or(1 + ot)K zrc~(1 + c~)K

H 2 --

h--otH.

930.14 (H,ED)

Ring-shaped dam in the sea on a very thick phreatic aquifer with precipitation. Inner radius rl and outer radius r2.

,1,,1,$$,1,,1,,I,,I,,1,,1,$~,,i,$,i,$,1,,I, 4,,1,$ p

.

se)

H ( r l ) - 0,

r

H(r2) - 0.

rl

- -

r2

In r H 2 __

In r l

2cg(1 + c~)K (r22 - r~)In r2 - I n rl

-(r2-

r2)},

h--otH.

Radial-symmetric flow

(930b)

501

930.15 (H,ED)

r

Ring-shaped dam r o u n d afresh water basin in the sea. Phreatic aquifer with finite thickness. Constant high fresh water level.

Fho

rl

H (ro) = D

_

with

r l < ro < r2, h ( r l ) = ho, r2

(h+D)

2-

(ho+D) x

h = oeH

2-

{(ho-+-D) 2-

In r2 - In r l In ro - In r l

(1 + o l ) 2D 2}

for r l ~< r ~< ro,

for ro ~< r ~< r2,

H2 =

In r2 - In r D2 ' In r2 - In ro

or(1 + ol)D2 l n r l + { ( h o + D) 2 - (1 + ol)2D 2} lnr2 In ro =

or(1 + ot)O 2 -+- (ho + D ) 2 - (1 + ff)2D2

Total discharge:

Q = zrK

oe(1 + ot)D 2 -+- (ho + D) 2 - (1 + ot)2D 2 In r2 - In r l

H (r2) -- 0.

502

Sharp interface, density flow

(940a)

940 E-4. G E N E R A L T W O - D I M E N S I O N A L H O R I Z O N T A L I N T E R F A C E F L O W

O (HOH

O (HOH

ax

P K,

=0

with K* = otK for confined aquifers and K* = c~(1 + c~)K for phreatic aquifers,

o/ "--

Ps - - Pf

Pf H = depth of the interface [L], p = precipitation or infiltration [LT-1]. Introduction of the "Herzberg" potential ~u - ~1 K* H 2 yields

°q2~I-I +

°q2~bH -4-

OX 2

p -- 0.

Oy 2

This differential equation is very similar to the corresponding equation for steady two-dimensional flow in confined aquifers without interface and the solutions for ~bn are almost identical to those for the head or drawdown ~o. Therefore, in the following, all solutions for interface flow will be compared with the corresponding solutions for problems without interface, q~H is the real part of the complex function I2n. No distinction will be made between confined and phreatic aquifers, as in all the here given problems the solutions for both categories are identical, apart from the above given values for K*. In all problems infiltration or precipitation is present. Subdivision: a. Precipitation or infiltration without wells. b. Precipitation or infiltration with wells.

a. Precipitation or infiltration without wells For this group of problems the solutions for interface aquifers are derived directly from those for confined aquifers without interface by replacing ~0 by q~H and ~

by p.

General two-dimensional flow

(940a)

503

354.06)

940.01 ( H ) ( c f

Two straight open boundaries, parallel to each other and a third straight open boundary, perpendicular to them, with zero sea water level. Phreatic (or confined) very thick aquifer with precipitation (or infiltration).

T 4,H=0 X I

-b

I~

$$$$~$~$$$$$$$$$$$$p _ - - - - -

X

H

q~H(0, y) = 0,

O¢H

~(cx~, Ox

y) = 0,

~bH(X, - b ) = q~H(X, b) = 0. Derivation from 354.06: P __ y2 ~bH(X, Y) -- -~(b 2 )

H2=

2

16pb 2 7t-3

m-1

~ ~ m = l , 3 ....

(-1) 2 ~ m e 3-

2~-- cos

(m y) 2b

'

t~H.

K*

Phreatic aquifer: K* = oe(1 + ol)K, h = o~H. Confined aquifer: K* = oeK, cp -- ot H.

363.05)

940.02 ( H ) ( c f

Rectangular strip with constant infiltration (precipitation) and zero sea water level at the boundaries. Confined (phreatic) very thick aquifer.

b D

~ s ( - - a , y) = q~H(a, y) = 0, ~bH(X, - b ) -- q~H(X, b) = 0. -b

504

Sharp interface, density flow

(940a)

Derivation f r o m 363.05" 16pb 2

q~n(X, y) -- ~P ( b 2 -- y2)

o0

m-1

( - 1) -T- c o s h ( ~ )

~3

m3

r e = l , 3 ....

c°sh/mJra \ 2b)

cos

(m y) 2b

'

2 H 2 _ K,~H. Confined aquifer: K* -- c~K, q9 -- ot H . Phreatic aquifer: K* -- or(1 + or)K, h - olH. 940.03 ( H ) ( c f

363.08) Very thick phreatic (confined) aquifer in an island of elliptical shape in the sea, with precipitation. Zero sea water level at the boundaries.

OqbH

-a

~(0, Ox

a

y) -- O,

~4)H

~(x,

o) - 0,

Oy

-b

~bH(X0, Y0) -- 0

,[,,[,,l,,[,,l,,b,l,,l,,[,,I,,l,,l,$,[,$,l,,l,$$$$ p h

sea

sea

for

Xo

a-5-

Derivation f r o m 363.08:

p

a2b 2

X2

22

2 H 2 _ K,~bH. C o n f i n e d aquifer: K* -- ct K, ~p -- otH. Phreatic aquifer: K* -- o ~ ( l + o l ) K , h -- olH.

yo

-1.

General two-dimensional flow

(940b)

505

b. Precipitation or infiltration with wells For this group of problems the solutions for interface aquifers are derived directly from those for confined aquifers without interface by replacing ~o by 4~H, ~ by p and 22-5 by Q. 940.11 (H)(cf. 356.12)

Very thick coastal aquifer, confined or phreatic, between two open parallel sea water boundaries with precipitation. Single well with discharge Q [L3T -1] at (x0, y0), fully penetrating the fresh water body.

___oOQ

Yo

I I

-a

Xo

a

x

a

~bH(--a, y) = ~bH(a, y) = 0,

p ,1,4, 4,,~,1,~ 4, ~,$ 4, ~,,~$ 4, 4,,1,,~$ 4,,~

(a¢.)

lim r r--+0 - ~ r

-

O 27r

for

r -- v/(x - x0) 2 + (y - y0) 2.

X0

a

a

Derivation from 356.12"

4,.(x,y)-

~P (a 2 -

x 2

)

Q In 47r

[cosh, Y2aY°'} + c o s { x+x0, 2, } ] cosh { 7r(y-Y°) 2a }--COS{ rr(x-xo) 2a } '

2 H 2 -- K,q~H. Confined aquifer: K* - ot K, ~o - c~H. Phreatic aquifer" K* -- ol(1 + or) K, h - olH. 940.12 (H) (cf 354.06 and 356.12)

Very thick coastal aquifer, confined or phreatic, between two open boundaries parallel to each other and a third open boundary, perpendicular to them with precipitation and a discharge well at (x0, Y0). Zero sea water level.

Y Yo I I I I I

XO a

qSH(--a, y) = q~H(a, y) = 0, CH(X, O) -- O,

506

Sharp interface, density flow

(940b)

lim (r 0~H r---~0 -'~-r ) --

Q

27/"

for r - v/(x - x 0 ) 2 -~- (y - y0) 2.

Derivation from 354.06 and 356.12" ~bH(X, y) -- P (a 2 -- x 2) .-

Q m 47r

Q

In

In

16pa 2

00

7/-3

Z m=l,3 ....

(-1)

m-1 2

m3

cos

(2a)e--

mTry 2a

m~x

cosh{rr(-2aYg)}-I-cos { Jr(x+xg) 2a } ] cosh{ :rr(YaYO)} -- cos { rr(x-xO)2a} cosh{Zr(Y+Yo) } nt-cos {rr(x+x9 2a 2a } rr(Y+YO) rr(x-xo) 2a }--COS{ 2a } '

cosh{

2 H 2 _ K,~H. Confined aquifer: K* - otK, q9 - a H. Phreatic aquifer: K* -- or(1 + or)K, h - c~H. 940.13 (H) (cf 365.01 and 233.17)

Well with excentricity d in a circular island with precipitation. Very thick confined or phreatic aquifer

Y

with zero sea water level at the circular boundary. 4~H(Xo, YO) -- 0

x 2 + y2 __ R 2,

OCH ~(x,

Oy

lim ( r 0~bH

r--+0 " ~ F ) --

Q

o) - o,

for r - ~/(x - d ) 2 nt- y2.

2Jr

Derivation from 365.01 and 233.17" 4~H -- ~P { R 2 - ( x 2 + y 2 ) }

{d 2 Q In 4zr ~-5

( x _ _ R2_d.. 2) +y2}

2 H 2 _ K,@~. Confined aquifer: K* -- otK, ~o -- otH. Phreatic aquifer: K* -- or(1 + u ) K , h - otH.

(x-d)

2+y2

'

for

(940b)

General two-dimensional f l o w

507

940.14 (H) (cf 365.02 and 233.17)

Circular row of n wells regularly distributed over a circle with radius r0 located symmetrically inside a circular island with radius R. Confined or phreatic very thick aquifer with zero sea water level at the circular boundary and with precipitation. Discharge of each well - Q. n Derivation from 365.02 and 233.17:

*

o

S'2H -- 27rn P

H 2 --

(r t" Zn

r 0n

'

R 2 _ (x 2 + y2)} _ Re[S2H],

2 K* q~H-

Confined aquifer: K* -- c~K, q9 -- u H. Phreatic aquifer: K* -- ol(1 + ol)K, h -- o~H. 940.15 (H} (cf 365.03 and 363.05)

Single well in a rectangular strip with open sea water boundaries at all sides. Very thick confined or phreatic aquifer with precipitation. Derivation from 365.03 and 363.05 by means of superposition. 940.16 (H) (cf 365.04 and 363.05) As 940.15 with the well at the centre of the strip, which is taken as the origin. Well with discharge Q at (0, 0).

b

oQ -a

O~H

P

~(0, Ox

ax

~ H ( a , y ) -- O,

...............................l -b

a~H

~(x,0) Oy

--0

4~=0

for x :/=0,

y) -- 0

qSH(X, b) -- 0.

Derivation from 365.04 and 363.05.

for y -¢ O,

508

Sharp interface, densityflow

(940b)

940.17 (H) (cf 365.07 and 133.17)

Q¢= 2b

Q~

r

Infinite row of wells of equal strengths and at equal mutual distances along a straight line midway between two parallel open sea water boundaries. Very thick confined or phreatic aquifer with precipitation.

x ~bH(--a, y) -- q~H(a, y) -- 0.

~Q

Wells with discharge Q at (0, +2nb).

Q~,

$,~$,i,$$,1,$,1,,!,,1,,~$4,,1,,!,4,,1,$$$ p

a

a

Derivation from 365.07 and 133.17"

~ . - - ~P

(a 2

-x

2)

_

Re[[2H]

with

£2H--Q-7/"

k

1 cosh{ ~ ( b + iz)} m

sinh(mTrh-NVa)

2

H 2 -- K,~bH.

re=l,3 ....

Confined aquifer: K* -- otK, ~o - c~H. Phreatic aquifer: K* -- o~(1 + or)K, h - o~H.

The continuum approach

(1.1.1)

515

I. B A S I C P R I N C I P L E S F O R S A T U R A T E D GROUNDWATER FLOW

1.1. T H E C O N T I N U U M A P P R O A C H TO GROUNDWATER F L O W

1.1.1. Molecular and microscopic level The continuum approach is treated extensively by Bear in his book "Dynamics of Fluids in Porous Media". A short abstract of his vision necessary in understanding some basic principles of groundwater flow, will be given here. Groundwater flow means that a fluid, water, moves through a porous medium, the ground. In fact, a fluid is composed of a large number of molecules, possibly combined with molecules of solutes. All these molecules move around, colliding with each other and with the solid walls of the soil particles. Since it is impossible to describe the flow of fluids in general (not only through porous media) mathematically, by following the motion of each molecule separately and combining all these motions, here we abandon the molecular level of treatment and try to describe the flow of groundwater at a coarser level, the microscopic level. For this purpose, we introduce the concept of a water particle, an elementary volume, composed of a great number of molecules. Its size must be much larger than the mean free path of a single molecule, such that a meaningful value for the density (mass divided by volume) of the water particle can be defined; it should however, be smaller than the pores and grains of the ground. The entire space between the grains is now filled with these water particles: instead of a large collection of molecules, moving in a vacuum, we consider a continuum, the groundwater, in each point of which a value can be assigned for the density, being the density of the water particle for which this point is the centroid. This density is a continuous function of space, which means that for any two close points P and P' the relation

p(P) = lim p(P') pI_+ p

(1)

holds. Flow of groundwater is now considered as the movement of water particles through the pores of the ground, instead of the arbitrary, uncontrollable movements of the molecules. This continuum approach seems to be evident as we actually see water as a continuous fluid, rather than as a collection of moving molecules; however, to explain certain phenomena such as diffusion, dispersion or heat flow, we have to revert to the molecular level. During its movement through the pores, a water particle exchanges continuously molecules with neighbouring particles, due to molecular diffusion. If we were to

516

(1.1.2)

Basic principles

label a cloud of water molecules, forming a particle, we should notice that, after a certain time interval, even if there was no flow, the labelled molecules would have spread and occupied a larger volume around the centroid of the initial cloud; this volume however now contains more molecules than the original particle, as molecules of other particles are now also comprised in it. The limitations of size of a particle mean that after a certain time interval a new particle must be defined for the same centroid, so that it has the same number of molecules (or mass) as the original one. Each time the end of the path of a former particle becomes the centroid of a new particle. By following this procedure, a continuous path of a water particle is obtained. Such a particle always contains the same number of molecules, although these molecules are continuously changing. 1.1.2. Macroscopic

level

Even on this microscopic level, which is a higher level than the molecular, it is impossible to give an exact mathematical description of the very complicated motion of a single water particle through the heterogeneous system of pores and grains. For this reason, in order to describe the flow of groundwater in a proper way, we have again to pass to a coarser level of averaging, the macroscopic level; this is again a continuum approach, but on a higher level. We now define a so-called representative elementary volume (REV) consisting of a large number of particles which, on the one hand, compared with the pore size, is large enough to assign a meaningful value for the porosity of the element (a number of pores, more or less, ought not to influence the porosity appreciably). On the other hand, a REV must be much smaller than the dimensions of the flow field. The entire space of the flow field is assumed to be filled with such elementary volumes. Instead of a grain skeleton with pores through which water flows, we now consider a fictitious continuum in any point of which we may assign values of any property, obtained by averaging those values over the REV for which that point is the centroid. If properties of the ground, such as porosity, intrinsic permeability, or pressure of the solid matrix are involved, the entire flow field is assumed to be a continuum, consisting only of ground, whereas in the case of groundwater properties like density, viscosity, water pressure and flow velocity the entire field is assumed to be filled by water. So at the same time we can use more than one continuum, which will overlap each other, each representing one phase (solid, first fluid, second fluid, etc.) and filling the entire porous-medium domain. Following this procedure, we define the velocity of the groundwater at a point at a fixed time as the average of the velocities of all the water particles in the REV, having that point as a centroid. In general, the velocities of the single particles differ both in magnitude and in direction. The averaging procedure has been performed by taking the integral of the product of the volumes d Vw of the individual particles and their velocity vectors over the total water volume A Vw of the REV, divided by that volume: u --

~ AVw

udVw. Vw

(2)

The continuum approach

(1.1.2)

517

This average velocity vector u is called the average real velocity of the REV. If we divide by the total volume A V of the REV instead of by the water volume A Vw, the specific discharge vector v is defined. v--

1 L VwudVw

and as

AVw--neAV,

(3)

AV

the relation between both is: V = neU,

(4)

where ne - effective porosity. If temporal fluctuations take place in the local velocities, an average over time must be performed, such as: v -

AtA V

t

Vw

u dVw dt.

(5)

In close relation to velocity vectors in a flow field, the concepts of streamlines and pathlines may be defined. A streamline is an oriented curve, for which it holds that at every point on it the velocity vector is tangent to the curve at that point. For non-steady flow, the streamlines change continuously; in that case the pathline of a REV is the limit of the small distances that the REV has traveled during short time intervals along the instantaneous streamlines related to those intervals. In steady flow streamlines and pathlines are permanent and coincide, while in unsteady flow they may be distinct. By means of the continuum approach thus, the very complicated flow of the single water particles through the pores is replaced by a movement of small water elements, composed of groups of particles, along pathlines, such that the centroid of an element remains on the pathline, while the velocity vectors of the composing water particles of the element all have the same magnitude and the same direction. In a permanent situation, flow takes place through stream tubes, consisting of successive REV's, having axes that coincide with streamlines; the cross-section dA is the same of that of a single REV. As all velocity vectors of the water particles of a stream tube are assumed to be parallel to the axis of the tube, there can be no exchange of particles with neighbouring stream tubes. The discharge through such a representative elementary stream tube therefore remains constant: dQ = Iv] dA = const.,

(6)

whereas the specific discharge v and the cross section dA may vary continuously, inversely proportionally to each other.

518

( 1.1.3)

Basic principles

1.1.3. Dispersion .~p.. •

/ / /

./

,/

./

/ / 4"

//

I ~ " '" " ~

VB

In accordance with this concept, a cloud of labelled particles of a REV with centroid in point A (Fig. 1) and mean velocity VA, will arrive after some time at point B only slightly deformed, due to the small difference in direction and magnitude between VA and YR. Thus, in theory, no spreading of the water particles occurs in the flow direction.

Fig. 1. Flow through a stream tube.

streamtube

Fig. 2.

Flow in a stream tube (detail).

In practice however, a number of particles in A travel along different paths between the several grains, lying within the stream tube, towards B and therefore need different travel times to reach B (Fig. 2). Moreover, even along a fixed path, the velocities of the water particles in a pore channel vary (Fig. 3) causing different arrival times. The result is that the tracer gradually spreads in the direction of the flow, occupying an increasing portion of the stream tube. This phenomenon is called hydrodynamic dispersion and in particular, since in theory and also in general in practice the water particles of a stream tube, do not leave the stream tube, only the longitudinal dispersion here has been explained by the irregularity of the pore system through which flow takes place. Actually, experiments show that after injection of a tracer in flowing groundwater, spreading of the tracer also takes place in a direction perpendicular to the flow direction; this transversal dispersion cannot be explained in a proper way from flow through the heterogeneously distributed pores, since exchange of water particles in a transversal direction does not take place. However, as has been explained before, by molecular diffusion, molecules may pass from one particle to neighbouring ones

The continuum approach

( 1.1.3)

519

L

Fig. 3.

Flow in a pore channel.

C

® D

Fig. 4.

Exchange of water molecules.

and thus an exchange of molecules through neighbouring streamlines (tubes) occurs. In the case of the presence of a concentration gradient this exchange results in a transport of the tracer perpendicular to the streamlines. The combination of the diffusion and the groundwater flow through the irregular pore system, cause a considerable spread of the tracer molecules in transversal directions, which is stronger than if pure diffusion only had acted on the water particles. This is illustrated by the idealized pore system in Fig. 4 where a molecule, originally in a particle at point A may arrive, dependent of the number of transfers to neighbouring pathlines and stream tubes at B, C, D or E. An important aspect is that the distance between B and E is of the order of magnitude of a REV, whereas spread, caused by diffusion only, is of the order

520

Basic principles

(1.1.4)

of magnitude of a water particle. Thus, the effect of diffusion is considerably strengthened by the microscopic heterogeneity of the soil. Summarizing, it can be stated that the longitudinal dispersion is mainly caused by flow through the irregular pore system (longitudinal diffusion also takes place, but is less important), whereas transversal dispersion is, in the first instance, caused by the process of diffusion and intensified by the irregular pores and grains through which the tracer is transported. So longitudinal dispersion can be explained from the microscopic level, whereas for understanding the transversal dispersion we even have to go back to the molecular level.

1.1.4. Mathematical description of geohydrological problems The continuum approach enables us to assign values of the velocity and density of a fluid (groundwater) and in the case of transport of a solute, also its concentration, to any point of the flow field. These values may be expressed by piecewise continuous functions of space and time, for instance: v -- v(x, y, z, t),

p = p(x, y, z, t)

c = c(x, y, z, t)

(concentration of a solute),

(density).

From these variables of the flow, continuity equations can be derived, by application of the conservation principle for mass or volume; these equations, which will be treated in Section 1.4 are partial differential equations with mass flux or specific discharge as dependent variables. Since these equations do not contain any information regarding the properties of the soil and the groundwater, they are insufficient to yield solutions of groundwater problems, unless further equations are supplied. The necessary additional information usually takes the form of relationships between fluxes and driving forces. Such relationships or physical laws in reality hold for the microscopic or even the molecular level. As we are unable to express them mathematically for their real level, we use the continuum approach to describe the relationships in a simplified way. The result is, in general, an equation of motion with one or more coefficients or parameters, which may be called "coefficients of ignorance", behind which is hidden the impossibility of describing the phenomenon at its real level. An example of such a relationship is Darcy's law where the driving force is the head or pressure, the gradient of which is related to the specific discharge by means of the conductivity coefficient. Some equations of motion are discussed in Section 1.2. Although we may submit the parameters to theoretical considerations by describing them in an analytical or statistical manner in order to determine their relation with the soil and fluid properties, it is impossible to derive absolute values for the parameters in this way. In order to find absolute values for the parameters experiments, either in the field or in the laboratory, are always necessary. Parameters and variables will be discussed in Section 1.3. The combination of continuity equations with equations of motion yields differential equations in which water and soil properties are now incorporated and thus describes the groundwater flow in a proper way. In Section 1.4 a number of

(1.1.4)

The continuum approach

52 1

I I

H

i~

~ . . . . .

~

?.

898888{

Ro

Fig. 5. Circular dike. differential equations will be derived. Solutions of the differential equations can be obtained with the aid of initial and boundary values (Section 1.5). Three main methods of the solution of the differential equations together with the initial and boundary conditions can be distinguished: analytical methods, numerical methods and use of analogons. This work is only dedicated to analytical methods. Summarizing, the successive main steps necessary to solve a geohydrological problem will be illustrated by a simple example, concerning a circular dike with inner radius Ri and outer radius R0 (see Fig. 5). This dike, which is schematized to an aquifer of thickness D and covered with an impermeable layer encloses a circular basin with a level H and a radius Ri. Outside the basin a level h (< H) is maintained. Groundwater flow takes place through the permeable part of the dike from the high level H inside to the low level h outside. Given the dimensions of the dike and the permeability K of the aquifer, what will be the total discharge Q [L3T -1] through the dike and in which way will the piezometric head ~0 of the groundwater (see Section 1.2.2) vary? 1. Replace the arbitrary moving water molecules by water particles that travel along fixed paths through the pores of the soil. 2. Replace the heterogeneous soil by a fictitious continuum, at each point of which water and soil properties may be assigned, obtained by averaging these properties over representative elementary volumes (REV's). In this example, the specific discharge of the groundwater can be defined as a continuous function of the distance from the origin, the centre of the circular basin: v = v(r) with Ri <~ r <~ Ro. 3. This value for v serves as the base for the continuity equation with the aid of the law of conservation of mass (here volume as well, as the groundwater is assumed to be homogeneous). 27rrDv -- const. -- Q.

522

(1.1.4)

Basic principles

4. The continuum approach is also used by deriving the relationship between flux and driving force (here head qg) which results in an equation of motion, containing one or more parameters of the ground or groundwater. Here the equation of motion is Darcy's law, which gives the relation between velocity and piezometric head (see Section 1.2.3): v --

-K

d--~

dr with q) = qg(r) and K = conductivity coefficient. 5. Combining the continuity equation with the equation of motion yields the differential equation (here a very simple one):

Q - -2re K Dr 60d__z..," dr

6. The boundary values of the flow problem are and

q~(Ri) = H

qg(R0) = h.

7. The solution of the differential equation with boundary values can be found as follows: dq)

--

-

Q

dr

~,

q) -- -

2rr K D r

Q 27r K D

In r + const.,

Q

In Ri + const., qg(Ri) = H - - - ~ 2rc K D from which Q

const.-- H +

2rr K D

In Ri

and Q

qg(Ro) -- h - H -

In( r )

"

o ,n (";). '

2zrK-------D

which leads to Q--2JrKD~

H-h

In(R°) Ri

(r)

and

qg(r)-H-(H-h)~

In Ti

In(R°) Ri

the required solutions of the problem (note that qg(r) is independent of K!).

(1.2.1)

Equations of motion

523

1.2. EQUATIONS OF MOTION

1.2.1. Stream functions

Making use of the continuum approach, a groundwater flow in a field can be described by means of a vector function for the average real velocity of the groundwater u or for the specific discharge or volume flux v (Section 1.1.2) with the mutual relation v = neU, in which ne = effective porosity. According to the assumptions of the continuum approach the functions u and v are continuous functions of space and time in the most general case: u--u(x,

v - v(x, y, z, t).

y,z,t),

The components of these vector functions in the coordinate directions are also functions of space and time, for example: vz -- vz(x, y, z, t), etc.

Ux -- Ux(X, y, z, t),

The small groundwater elements (REV's) assumed in the continuum approach move along pathlines according to the equation of motion: ds-luldt

or

ds-/I.v__.=dt He

in which ds represents a small distance traveled by a REV along the instantaneous streamline during the small time interval dt. lul and Ivl are absolute values (magnitudes) of the real velocity and specific discharge, respectively. The components of the equation of motion in the three coordinate directions become: dx

dy

Vx -- ne d t

Vy

ne dt

dz and

vz

nedt

from which: dx Vx (x, y, z, t)

-

dy vy (x, y, z, t)

=

dz

dt = --. vz (x, y, z, t) ne

(1)

(The effective porosity ne will in general also be a function of space.) The analytical solution of the three differential equations represented by (1) is only possible for very simple cases, because generally vx, vv and vz are complicated functions of x, y, z, and t. Theoretically, however, solution of (1) by integration yields three mutually independent functions of x, y, z and t with integration constants" f l (x, y, z, t, or) - O, f 3 ( x , y, z, t, )/) -- O.

f 2 ( x , y, z, t, fi) - O,

(2)

524

( 1.2.1 )

Basic principles

Each of these three functions with parameters represents a (time-dependent) set of surfaces; the mutual intersection of the three surfaces which arise by assigning fixed values ct0, fl0 and Y0 to ct, fl and y define the position of a REV at the time t. Together, the equations (2) define the pathlines of the several water elements. If the vector field is considered at a fixed moment to, the time as an independent variable disappears in (1) and the equations of motion become: dx

dy

dz

=

Vx (x, y, z, to)

=

l)y (x, y, z, to)

dto =

~ .

Vz (x, y, z, to)

(3)

He

The solution of (3) can theoretically be written down by means of two functions of x, y and z with two integration constants: fl (X, y, Z, to, ct) -- 0

and

(4)

f2(x, y, z, to, fl) -- O.

These two functions with parameters represent two families of surfaces, of which the mutual lines of intersection form the instantaneous set of streamlines at the time t--to. In the case of steady flow, the velocity vector v is only a function of space; pathlines and streamlines then coincide and the equations of motion become: dx _ Vx (x, y, z) --

l)y

dy = dz (X, y, Z) Vz (X, y, Z)

( _ d~e) '

(5)

from which can be derived two families of surfaces" f l ( x , y, z, ct) -- O

and

f2(x, y, z, fl) - O,

(6)

representing a permanent set of streamlines. Solution of (6) for ct and fl gives: ct--ct(x,y,z)

and

fi-fl(x,y,z).

(7)

The two families of surfaces then may be considered as the so-called level surfaces of the scalar functions o~ and ft. As the mutual lines of intersection of these surfaces represent a set of streamlines, these surfaces are called stream surfaces and the functions ct and fl stream functions. Along a fixed streamline, ct and fl have constant values ct0 and fl0. The velocity vector v at a point P has the direction of the tangent to the streamline (oe0, fl0) through P; as the streamline lies on the stream surface c~0 through P, the tangent lies in the plane through P tangent to the stream surface ct0 and thus perpendicular to the normal of ct0 at the point P. This normal can, according to a theorem of vector analysis, be represented by grad ct or Vct, which is a vector with components Oct ~, Ox

Oct Oy

and

Oct Oz

,(z{? . .x~?.

. xe z e ) a _ ~ca

. ( x e *e ge ~°e ,( a~e zC ~Je " e

2e

x~ \

/ a -- Za / \

/a

-

-

ge ,oe /

~a

qa!qA~ tUmJ

(o~)

,(e

ze

ge

ge

dC

2e

f.

~t

~oC

~oc

xe

~e xe

(7 'X 'X')O - - A

Joc !

Jo ~/A x ~OA(Z 'cO 'x)a -- A "luapuadap aatds ~IltJaUa8 s! qa!qA~, fil!leuo!lJodoJd Jo JolaeJ t ql!A¢ gA x ~oA lanpoJd JOlaaA aql 01 ItUO!lJodoJd s! A ltql S,~OIIOJ 1! (8) moJq 2:C za + d c '~a

(6)

o'0-

g-7

7e

ze x3e

-x~ - )Ca

+ ~e

de ~oe

xC x3e

--Za +--~a

+--Xa i0

(8)

'0 -

~/A" A

'0 -

~°A" A

:uo!lt110u lOlaaA uI "~/A put A aO=l p!lti S[ amts aql put atlna!puachad £Ilenlnm alt snql a~A put A SIOlaaA aqJ~ "(I ":~!d aas) suo!laaI!p altu!pIoo3 aaIql aql u! •aU!ltUea.ns pue saaejms tuea:I1S

"I "~t!~l

(o~/,o~,)

gZ g

(V'~'I)

uo!lou, fo suo!wnb3

526

(1.2.1)

Basic principles

This result could have been obtained also by solving the equations (9) for ~ and Vy 13z

By choosing other parameters ot' and/3' instead of ot and/3, such that both are functions of ot' and fl', i.e., ot - u (ot',/3') and fl - fl (u', fl'), the same set of streamlines will be obtained as given by equation (6), however as the intersection of two other sets of surfaces u' - ot'(x, y, z) and fl' - fl'(x, y, z). Repeated changes of ot and/3 with small, but constant values dot and d/~, for instance, successively ot, ot + dot, ot + 2 dot etc., therefore give a sequence of streamlines different from that with changes of ot' and 13' with equal values dot' and dfl'. Compare, for instance, in two-dimensional radial flow, the set of streamlines y - c x with parameter c that can also be represented by y - x tan 0 with parameter 0; stepwise variation of c gives a sequence of streamlines different from that with a similar variation of 0. Moreover, grad c or Vc differs from V0: Vc-

y 1 x ~--~i4-j'x

y V0--~ix 2+y2

x +x 2+y2j"

Now we shall try to determine stream functions (which are parameters of families of stream surfaces) gr and r/ in such a way that increase or decrease of ~p with equal values d~p, 2 d~p, etc., and also stepwise variation of r/with values dr/, yield s t r e a m t u b e s which all transport the same quantity of groundwater d Q, in agreement with the conception of stream tubes given in Section 1.1.2. Such a stream tube, for instance, is composed of the four stream surfaces 7t, ~P + d~p, r/and ,7 + dr/(Fig. 2). If a cross-section of the stream tube perpendicular to the streamlines with area dS is considered, we may write: d Q = Ivl dS = Ivl da db sin y and according to (10): v-

c(V~ x v~)

(ap + d~, 7"/+ dr/) (~ + d~, r/) ~ / I ' (~, r/) /~b -~ v,/ -, (ap, 71+ do)

~+d~k

Fig. 2. Stream tube with cross section.

VO v

vr/

(1.2.1)

Equations of motion

527

from which, according to the definition of a vector product: Ivl -

sin y

clV~llV~l

and so dQ -

clV~llVol da

db sin 2 g.

Now IV7~l --

d~

and

da

IVr/I - n

dr/ db

resulting in" d Q - c sin 2 y d ~ dr/. In general, c and 7 are functions of x, y and z; so if it is required that through every stream tube (dgr, dr/) d Q has the same value, then c and sin y must be constants. We c h o o s e c - 1 a n d T - T 7r which gives: dQ - dgt dr/,

(11)

while Vgt and Vrl are orthogonal vectors: V~p. V r / -

0.

(12)

At the same time, the important relation between the specific discharge and the two stream functions that define a set of stream tubes with equal discharges has been derived: v-

Vrl × VTr,

or

Vx = 3y Oz

3z Oy

'

vv = Oz Ox - ~Ox~ Oz

(13) V z = 3x 3y

3y 3x

In cylindrical coordinates r, 0 and z, (13) may be written as"

Ur

l) z

1 Or~ Oap

1 or~ oV~

r O00z

r Oz 0 0 '

r Or 30

r 30 Or

--"

----

UO

o~ o~

o,7 oV~

Oz Or

Or Oz

n

(14)

528

(1.2.1)

Basic principles

In t w o - d i m e n s i o n a l flow, for instance, horizontal flow in planes, perpendicular to the z-axis, one of the two families of stream surfaces represents these planes, i.e., 3~ 3 77 77 -- z. In equation (13) ;5; and G then vanish and ~3 r/ equals 1, from which Vz - 0 and

Vx =

o7, -~,

o7, v~ = - - .

Oy

"

(15)

Ox

In two-dimensional flow there is only one stream function left, ~p (dimension [LZT-1]), from which can be derived the vector field for the specific discharges, by differentiation according to (15). If we take d r / - 1, then for two-dimensional flow (11 ) becomes" dq - d~,

(16)

in which q = discharge per unit length [L2T -1] (in this case per unit of depth). In a x i a l - s y m m e t r i c a l f l o w , the flow takes place in a number of planes all going through a common straight line, the axis of symmetry (mostly the z-axis); in each of these planes the stream pattern is identical. Now one of the two families of stream surfaces, for instance ~, becomes a set of surfaces of revolution round the axis of symmetry: ~ - ~p(r, z) with r ~/X 2 + y2; the other is the set of planes through the axis of symmetry r/-- cO with c a constant which has to be determined afterwards. From (14) it follows, as -

Or/

=~

Or

Or/ 3z

=0

Or/

and

~C"

30

cog, V0

~

O,

Vr

~

-

- - ~ ,

cog, V z --

r 3z

r Or

If we define d Q as the total discharge between the surfaces of revolution belonging to 7t and ~p + d~p, it can be shown, that c 2 ~1 has to be chosen such as to ensure that: dQ - d g t .

(17)

The relationship between the function 7t (dimension [L3T-1]) and the specific discharge in this case thus becomes

10~ Vr-

~, 2rrr Oz

10~ Vz -

~ ~ . 27rr Or

(18)

In general, it can be stated that if in a flow field a permanent set of streamlines exists, stream functions can be defined from which the components of the specific discharges may be derived by differentiating them in the direction perpendicular to that component. In three-dimensional flow, there are two stream functions, both with dimension [L3T-1]; in two-dimensional flow there is only one stream function with dimension [LZT-1].

(1.2.2)

Equations of motion

529

1.2.2. Pressure and piezometric head Under natural conditions a porous-medium volume, such as Pz an aquifer or ground layer, rzy which contains stagnant or moving groundwater, is sub\ jected to internal and external forces which will cause P stresses, both in the groundwater and in the solid matrix of the ground. Applying the continuum approach rxz to groundwater flow (Section 1.1), we may assign values of these stresses in any point of the field under consideration, which have been obtained by averaging them over the REV Fig. 3. Stress types in a cube. for which that point is the centroid. The thus obtained stresses are continuous functions of x, y, z and t. In general, stresses in the g r o u n d w a t e r consist of normal stresses and shear stresses, as can be seen from Fig. 3, in which a stress state is represented in three mutually perpendicular planes in the immediate vicinity of a point P (taken as the origin of a coordinate system). In this figure Pz, for instance, represents the normal stress acting in the z-direction, normal to the X O Y-plane, while, for instance, rxz represents the shear stress in the plane normal to the x-axis acting in z-direction. There are two principal differences with regard to the effects of a stress state on a solid and on a fluid. Firstly, a fluid cannot undergo tension; therefore normal stresses in fluids are always pressures. Secondly, a solid in a state of shear will deform elastically, the magnitude of deformation being in direct proportion to the shearing force, whereas if a fluid is subjected to shear, it will also deform, but continuously so long as the shearing force is applied; the magnitude of the force now governs not the magnitude, but the rate of deformation. The proportionality factor is the viscosity /z (mu, see Section 1.3.1-3) from which the internal shear stresses of the fluid (not to be confused with shear stresses between fluid and soil grains) may be written as: X

N

\

N

\

N

N

N

Ovv 3v~] "Cx y -- "Cy x -- lZ ( --~x + -~y / , 3vz Ovv~ r y z -- "Cz y -- lZ ( ~ -k- -~z. / ,

rzx -- rxz -- l z ( .--~z + --~x Ovz

(19)

530

Basic principles

(1.2.2)

(For the determination of angular deformation see Section 1.4.4: rotational flow.) These equations show that shear stresses only exist in the presence of variable velocities, which means that fluids at rest or fluids with uniform flow have no shear stresses. In general, both the variations of the velocity in groundwater flow and the viscosity of water are small. For this reason the shear stresses in water can be neglected compared with the normal stresses (pressures). The magnitude of the shear stress in the neighbourhood of a well, pumping a discharge Q out of a confined aquifer, may serve as an example: ~ Q In r 2 7cK D

~o(r) - c -

vx--KOqg-Ox -

OVx Oy

or

qg(x, y) - c -

Q x 2reD x 2 + y2 '

Q xy Jr D (x 2 + y2)2

Q ln(x 2 -k- y2 ), 4 7r K D

Oq9 _ Vy - K Oy -

Q y 2reD x 2 + y2 '

Ovv Ox

from which

75xy - -

"gyx - -

~ k,

3vv ) Oy + ~ -

2Qlz xy 7rD (x 2 + y 2 ) 2

in which ~p -- drawdown, Q = constant discharge of the pumping well, K D -- transmissivity of the aquifer, r -- distance to the pumping well. The shear stress is maximal on the line y = x and then becomes: Q/z Txy

--

27r D x 2"

Water at 10 °C has a viscosity of 1.31 cP (centipoise) 1 31 x 10 -3 Pa.s (Pascal second) •

from w h i c h / z -

1.31 x 10 -2 P (poise) -

1.31x 10-3 Pa.d (Pascal day) 24 x 3600

1 5162 x 10 -8 Pa.d "

1.5162 x 10 -8 Nd/m 2.

Take, for instance, Q -

100 m3/h - 2400 m3/day, D -- 20 m and x - 0.5 m,

then r-

1.1583 x 10 -6 N/m 2.

C o m p a r e d with the pressure exerted by, for instance, a 1 m high column of water, i.e., "--, 10.000 N / m 2, it will be clear that the shear stresses may be neglected with respect to pressures.

(1.2.2)

Equations of motion

531

This implies that for flowing groundwater Pascal's Law holds with a very good approximation: The water pressure in a point

ct z

is independent o f the direction o f the normal o f the plane on which that pressure acts. This can be shown as follows (Fig. 4):

Az

P

Y

The forces acting on an infinitely small fluid particle O A B C are in equilibrium with each other (accelerations are neglected in groundwater flow). In the y-direction py x 0 surface triangle O AC -- p cos 13x surface triangle OAC with fl = the angle between the normal of the plane A B C and the yaxis. Now surface triangle O AC -- surface Fig. 4. Pascal's law. triangle A B C x cos fl, from which py = p and also px = py = Pz = P (the weight of the element is infinitely small, of the third order, and can be neglected). In all directions at a point, thus, the water pressure is the same, so that p is a scalar (and not a vector or a tensor). In general, the pressure in flowing groundwater is a continuous function of space and time p = p ( x , y, z, t).

In groundwater at rest, the water pressure increases linearly with depth as a consequence of the weight of the overlying water. In the presence of a water table (or phreatic surface - surface of the groundwater open to the atmosphere), the pressure at a depth h [L] below that level equals the atmospheric pressure in addition to the pressure exerted by the weight of the overlying column of water with length h: Pt = Pa + p g h = pa + ?'h

(20)

in which pt = total water pressure [ M L - 1 T -2] or [FL-2], Pa -- atmospheric pressure [idem], p = density of the water [ML-3], g = gravity acceleration [LT-2], y = p g = specific weight of the water [ M L - Z T -2] or [FL-3]. The water pressure is usually considered with respect to the atmospheric pressure and is written as water pressure p = p t - Pa. Then (20) becomes: p = p g h = )/h.

(21)

532

(1.2.2)

Basic principles

In the sequel p will be generally used instead of P t - Pa; only if the atmospheric pressure varies in a distinct way must the total pressure be considered. From (21) it follows that, if the density of the water is constant, the pressure in groundwater at rest increases linearly with depth, the specific weight y being the factor of proportionality. Then in general, if the positive z-axis is directed upwards, along a vertical the following relation holds: p(z) -- - y z + const.,

or

p(z)

~

-t- z - const.,

(22)

Y from which 0p

~

~y•

0z Pressure distributions according to (22) are called hydrostatic pressure distributions; these distributions hold not only for fluids at rest but also for fluids that flow purely horizontally. In the latter case, equation (22) is valid in any vertical line, but the "constant" term is variable with space. Even if the specific weight is variable, for instance, groundwater of which the chlorine content increases with depth, the pressure distribution is called hydrostatic if 8p 8z

= -y(z)

(23)

and p -- --

fz

y(z) dz.

The term P

P

Y

Pg

is called the pressure head, independent of the pressure distribution, and can be defined as follows" The pressure head in a fixed point of a flow field is the elevation to which water of the same density as the water that is present at the point in question, rises above that point, if a small screen with connected pipe (piezometer) has been placed in that point. In the case of variable atmospheric pressure, p must be replaced by p t - Pa and the pressure head becomes P t - Pa Y The term z in equation (22) is called the elevation head; the sum of pressure head and elevation head is the piezometric head ~o if the latter is measured with respect to the plane z - 0" q) -- z

+_P. Y

(24)

Equations of motion

( 1.2.2)

533

Sometimes, the piezometric head is measured from a level other than z = 0; if additionally the atmospheric pressure is variable, then the generalization of equation (24) becomes (Fig. 5):

P__ Y

99 - - c q- Z q- P t Z

P a.

(25)

0

C Z -" --C

Fig. 5. Pressure distribution in groundwater.

From (22) and (24) it follows that, if the pressure distribution is hydrostatic, the piezometric head is the same for each point of a vertical line (this rule does not hold for a fluid with variable density in the z-direction).

In an aquifer containing a free water level, the so-called water table or phreatic surface, where the pressure equals the atmospheric pressure, the piezometric head coincides with that surface if the groundwater is at rest or flowing horizontally. Such an aquifer is called a phreatic aquifer; in general, it is not covered by layers with considerable resistance to groundwater flow. In a phreatic aquifer, a change of piezometric head (water level) is mainly accompanied by a change in storage capacity (see Section 1.3.5). Aquifers bounded by one or more impermeable or semi-permeable layers are called confined or semi-confined aquifers, respectively. The piezometric head in every point of the aquifer lies above the lower side of the covering layer. The locus of the piezometric heads in a number of distinct points is a fictitious surface, the piezometric surface. If this surface lies above the ground surface we may refer to the groundwater as artesian groundwater. Besides water pressures, also stresses in the grain skeleton (solid matrix) will be created by internal or external forces acting on an aquifer. Any horizontal plane passing through the ground, will intersect both solid grains and void spaces. Equilibrium of forces acting on this plane requires that the sum of the atmospheric burden and the weight of the overlying (wet) ground is compensated by the sum of the water pressure and the pressure in the solid matrix, multiplied by the area in question: Pa + Crg = Pt + O's,

(26)

in which Crg = ground stress and Os -- stress of the solid matrix. Equation (26) is only valid for vertical ground and solid matrix stresses; distinct from water pressures, stresses in the solid matrix depend on the direction under consideration as a result of the presence of shear stresses in the ground massif.

534

( 1.2.2)

Basic principles

ground surface -i

i

eati-

s

ace

. . . . .

piezometric surface phreatic aquifer with fresh water (Y1)

~\\~.\\~\\\\\\\" \ semi-permeable layer ~\\\\\\\\\\\\\"

leaky aquifer with salt water (Y2)

impermeable base

Fig. 6. Soil stress diagram. Figure 6 may serve as an example of a stress diagram in which the different kinds of vertical stresses in a vertical cross-section over a ground system, containing two aquifers, separated by a semi-permeable layer and filled with two kinds of water (fresh and salt), are represented. The atmospheric pressure is assumed constant and is omitted in the figure; in this case the reduced relation (26) is sufficient: ~rg = p + as.

(27)

1.2.3. Darcy's law In general, groundwater flows very slowly and also variations of the velocity with space and time are small. Therefore, acceleration is supposed to be absent in groundwater flow and the forces acting on an elementary volume must be in equi-

librium with each other. In order to investigate the relationship between these forces and the magnitude and direction of the flow, consider a REV with cylindrical shape (Fig. 7), arbitrarily oriented in s-direction in the flow field. The elemental area and the elemental length, normal to the area, are AA and As, respectively. The forces acting on the water in this REV are the pressure force Fp, the weight Fw and the shearing force Fs, that is the force exerted on the water by the solid skeleton of the ground.

Equations of motion

p

a.p 1 -

Fig. 7.

( 1.2.3)

535

~.."

Forces acting on an elementary volume.

The components of these forces in s-direction must also be in equilibrium with each other and they become successively" a. Pressure force

Fp--

Op 1 -As-p - ~Os2

p + O----~-~As n A A

or

Op Fp -- - - n A A As O---s in which p - water pressure, ;-57.,,- pressure gradient in s-direction, and n - porosity of the ground, i.e., that part of the soil which is filled with groundwater. b. Weight Fw In general, the density p of the groundwater varies with space and a density gradient ~2 ds in s-direction will be present. As 7' - Pg the specific weight y then varies fly 1

~y 1

along As from y -- ~ ~ As till y + ;-57,,.~ As. The average along As equals y as can be seen from:

gav --

2

= 7'

from which

OZ

o Fw -- - - n A A A s y cosc~ -- - - n A A A s y ~Os

536

( 1.2.3)

Basic principles

c. Shearing force Fs

The shearing force is caused by the shear stresses exerted on the moving groundwater by the solid matrix of the ground. The total surface AAs of the grains, in contact with the water, within the volume A V is proportional to the volume of the solids A Vs (see definition of specific surface in Section 1.3.2-3) and thus proportional to AV: AAs = cl A V in which cl = constant of proportionality. Assume that r with dimension [ M L - 1 T -2] or [FL -2] represents the shear stress, which is called the external shear stress, as distinct from the internal shear stress between the water particles (caused by the viscosity of the water), then: f s =

-c

17 5 A A A s .

The minus sign means that the shearing force is opposite to the direction of flow. From physical considerations it may be concluded that the external stress r is dependent on the geometry o f the pores and the grains, the velocity o f the groundwater and the internal shear stresses in the fluid and consequently of the viscosity o f the fluid.

By means of the zr-theorem in dimensional analysis, the relation between these parameters can be found: Jr

- - ~IZ

x

v

u;,a

z

in which Jr = dimensionless number [0], r = shear stress [FL-2], /z = viscosity of the groundwater [ F L - 2 T ] , a = arbitrary length [L], Us = component in s-direction of the real velocity [LT-1]. Jr = [ F ° L ° r °] = [ F L - Z ] [ F L - a T ] X [ L T - 1 ] y [ L ] z from which F---~ x + 1 - - 0

/

L~-2x+y+z-2--0

x--1 -+ y - - - 1

T-+x-y--O

z-1.

So Jr = constant

ra =

c2

--

or

/3,t/e l) s Z" --" C 2 ~ ,

lZUs

a

in which ne -" effective porosity and Vs = specific discharge (see Section 1.1.2). If c3 = clc2 [L-l], then the shearing force becomes: ].L/'/e U s

Fs = --c3 ~

A A As.

The equilibrium equation of the three forces Fp + Fw + Fs = 0 gives: Op Oz lZne__Vs ) _ O. - A A A s n -~s + n Y -~s + c 3 ~ a

Equations of motion

(1.2.3)

537

Solving for vs"

an l) S

(Op

--

+

Y

Oz)

"

If the part of the factor before the brackets, that is related to the ground, is separated and defined as the coefficient of intrinsic permeability of the porous medium: an

k-

[L

2]

(see Section 1.3.2-4),

(28)

C3ne

then the expression for Vs becomes"

k (Op l) S

Oz

m

.

As the s-direction was chosen arbitrarily, this equation may be generalised to all directions and it then becomes in vector notation v -- - -

k (Vp + ?, Vz) /z

(29)

or

Vx =

k Op # 3x '

k Op v v = - -# yO ,

k ( Op ) # -~z + )/

vz -

(V stands for gradient). Equation (29) represents the general differential form of Darcy's law, which is valid for heterogeneous isotropic ground with heterogeneous fluid, which means that the parameters k, /, and ?' = pg are variable and are functions of x, y and z. (For anisotropic ground, the permeability of which is also dependent on the direction, see Section 1.3.3.) If the density of the water (and so also the specific weight) is constant, equation (29) may be written as v-----V(p+?,z)

or

v--

V

+z

"

£t/ + z equals the piezometric head ~p if z - 0 is the plane of reference (see Section 1.2.2, equation (24)) while ky

= K [LT -~]

(30)

= the so-called Darcy permeability coefficient or conductivity coefficient. So in the case of constant density, equation (29) is simplified to Darcy's law in its well-known form:

538

Basic principles

(1.2.3) v = -KV~p

Vx

--

or

- K O~O ~, Ox

vv

-- -K~,

0(p Oy

Vz -

-K~.

O~o Oz

(31)

In words: the specific discharge vector of a homogeneous fluid (constant density) flowing through a porous medium is proportional to the negative gradient of the piezometric head. In this equation K, in general, is a function of space: K -- K(x, y, z) (heterogeneous ground). If also K is a constant (homogeneous ground and water) then the specific discharge vector or velocity vector equals the negative gradient of a scalar function q~ = K~o, which is characteristic for potential flow (irrotational flow); q~ is called the potentional function. v -- -V4~.

(32)

The piezometric head ~0 is usually also denoted by the word "potential" and lines of equal head are called equipotential lines, which is not quite correct. From equations (31) and (32) it follows that if the density of the groundwater is invariable, the velocity vector is proportional to the negative gradient of the piezometric head ~p and even equal to the negative gradient of a scalar function 4~ (= Krp) if K is constant. The importance of equation (31) lies in the fact that the piezometric head can be measured directly in standpipes of piezometers. From the general form of Darcy's law (equation (29)), it is impossible to derive a similar relation between the velocity vector and the piezometric head; for instance, if we put again ~o - ~ + z then O~o

Ox

=

1 Op

p

Oy

y2 Ox

y Ox

etc., and as Vx =

k

Op

I~ Ox

it follows that Vx is not proportional to ~Dx and a real potential function does not exist. All attempts to introduce substitutive heads (point-water head, fresh-water head, environmental head, true environmental head) in case of variable densities, have, in general, little sense and lead to unnecessary complications, because of the fact that the velocities cannot be derived directly from these heads. Only if the variation of the density is due merely to pressure changes: y -- y (p), a kind of potential function can be derived. If we put in equation (29) Vp Vq)* = ~ - t Y

Vz,

then .¢

m

k× (vp +vz) /z

y

-- -KVq)*

with y -- y (p),

(1.2.3)

Equations of motion

539

while dp

,

d~o* = ~ -t- dz Y(P)

and

~o

fP -- Jp

du ~

-

o y(u) ~-z.

This is " H u b b e r t ' s f o r c e potential", indeed a function from which the velocity vector can be derived directly, but on the other hand, not a function that can be measured directly in the field, as ~0" is not a head. (~0" =/: ~0.) Use of this potential therefore is unpractical. In the case of variable densities of the groundwater, for instance, a brackish water zone between fresh and salt groundwater, it is much more convenient to make use of pressures, and if possible of stream functions (see Sections 1.4.3 and 1.4.4). For fresh-salt groundwater problems, measured piezometric heads are only of importance if the piezometer pipe is filled with water with the same density as that which is present at the screen location and if that density is known; conversion to pressure then easily can take place according to P ~o----+z Y

or

p--y(~p-z).

For anisotropic ground (in which the intrinsic permeability varies with direction; see Section 1.3.3) the foregoing derivation of Darcy's law only holds if the main directions of the anisotropy coincide with the coordinate axes. Equations (29) then become:

vx =

kx Op I~ Ox

,

kv Op vv . . . . , I~ Oy

vz --

kz ( Op

"

I~ ~

+ y

) '

(33)

in which kx, kv and kz represent the permeability coefficients in x-, y- and zdirections, respectively; in general, they are functions of space (for heterogeneous anisotropic grounds). A derivation of the generalization of Darcy's law for anisotropic grounds will be given in Section 1.3.3. If the fluid is homogeneous (constant y) the equations (33) become: O~p

Vx -- - K x -2--, ox

O~0

vv - - K , , = - , Oy

v,"

0~0

-Kz~. Oz

(34)

The derivation of Darcy's law, given here, is a generalization of Darcy's onedimensional experiments to three-dimensional flow. In 1856, Henry Darcy investigated the flow of water through vertical homogeneous sand filters as shown in Fig. 8.

540

( 1.2.3)

Basic principles

From his experiments, he concluded that the rate of flow (volume per unit time) Q is proportional to the constant cross-sectional area A, proportional to ( h i - h2) and inversely proportional to the length L. When combined, these conclusions lead to the famous Darcy formula

U 6 6

-II

Pl

m

t'

I

,-7

I

ii ~~screen m _

hi

I_

P2

ii!l." _" . . . .

.....

Zl

I

S.anc

Y

/

KA(hl -h2) iQi -

screen

z;I ////////.

~/////////.d,

where K is a coefficient of proportionality. The lengths

Y/////..4

h l and h 2 (measured with respect to some arbitrary horizontal datum level) represent the piezometric heads and h l - h 2 (01 - ( 0 2 is the difference in piezometric head across the filter of length L. Now ~o1~o2 may be interpreted as the hydraulic gradient in z-direction: ~dz Fig. 8. D a r c y ' s law; vertical column,

((01 > (02) If we define the specific discharge "O in z-direction as "Oz Q (direction of negative z-axis), we find a form of Darcy's law in agreement with formula (31)" •

"Oz

~

A

_ _ K d _ _~

dz

Figure 9 shows how Darcy's law may be extended to flow through an inclined homogeneous ground column (direction s): IQI

K A ((01 L

-

(02)

,

"Os = -

K ((0/91 - L

(0./92)

= -K~

d(0 ds

with Pl (/91 - - Z l -I- ~

and

q92 - - z2 nt

Y

P2

g

It is important to note that these equations state that the flow takes place from a higher piezometric head to a lower one and not from a higher to a lower pressure.

As shown in Fig. 8 ' p_l < p2 i.e ., the flow in this case is in the direction of y y' increasing pressure and decreasing head. It is only in the special case of horizontal flow, i.e., z l - - z2 that we may write K(pl•O h

~

~

yL

P2)

K dp T ds

Equations of motion

( 1.2.3)

541

U _ ~

~.-.

[

___

~

I

_

P2 El

tp2 Z2

area A

I-I

datum level -"h

T

Fig. 9.

Darcy's law" oblique column.

In the vertical flow shown in Fig. 8, we have (with

Uz

KL 7((/91

--

=-K

--

(1+

(/92)

pl

_

--

LK( Zl "-[-- Pl Y

--"T-

p2

)

---K

×L

Z2

Z l --Z2

=

L):

7)--7(

P2

K

Pl - P2

)

1+ ~-~

in agreement with equation (29). From many experiments it appears that Darcy's law, which specifies a linear relationship between the specific discharge v and the gradient of the piezometric head V~0, is no longer always valid if the specific discharge increases. The so-called Reynolds number is now used therefore as a criterion Re =

pvd [0], /.t

(35)

in which v = specific discharge [LT-1], d = mean grain diameter [L], p = fluid density [ML-3], /z = viscosity [ML-1 T - 1]. In the range from Re -- 1 to 10 there is a transition from laminar flow to turbulent flow. Almost all kinds of groundwater flow possess Reynolds numbers < 1, so that Darcy's law is applicable; only in the immediate vicinity of screens of pumping wells, where gradients are high, may Re be greater than 1.

542

(1.2.4)

Basic principles

1.2.4. Relations between stream functions and potential functions

For three-dimensional steady flow of a homogeneous fluid (constant p) through homogeneous isotropic ground, the relation between the potential function and the stream functions can be derived directly from the equations (32) of Section 1.2.3 and (13) of Section 1.2.1: V4~ = Vgt

x

Vr/,

(36)

in which q~ = Kq9 with q9 = piezometric head and K = coefficient of conductivity. The three vectors are mutually orthogonal; each of these vectors is perpendicular to its level surface, so that these level surfaces, i.e., the two families of stream surfaces ~p = const, and 1/ = const, and the equipotential surfaces ~b = const, are also mutually perpendicular. The streamlines, being the intersection lines of the two sets of stream surfaces, are perpendicular to the equipotential surfaces. In two-dimensional steady flow of homogeneous groundwater through isotropic homogeneous ground, the relation between potential function and stream function becomes, after combining (15) of Section 1.2.1 with (32) of Section 1.2.3, 0___~_~= 0lp OX Oy

and

0__L~b = _ 0__L" ~k Oy OX

(37)

These are the well-known Cauchy-Riemann conditions. They have great practical importance for solving geohydrological problems, especially in two-dimensional steady flow, as we shall see in Section 2.2.5. In this case, the streamlines (curves ~p = const.) are perpendicular to the curves ~ = const. (equipotential lines). If heterogeneous isotropic ground is considered, for which K = K(x, y, z) holds, equation (36) becomes: KVq) = V~p x Vr/

(38)

and for two dimensions: K0qgOx

0gr Oy

and

K0qg-Oy

~/r0__c_,. Ox

(39)

Notwithstanding the fact that v is not equal to Vqg, this vector is perpendicular, everywhere in the field, to the surfaces of equal piezometric heads (not equipotential surfaces) because v is proportional to Vqg. In the two-dimensional case, the streamlines (curves ~p = const.) are perpendicular to the lines of equal piezometric heads (curves q9 = const.) or isohypses. For anisotropic ground, if the main directions of the anisotropy coincide with the coordinate directions, equations (34) of Section 1.2.3 hold: for instance, for two-dimensional flow we obtain: 0rp _ 0gr Kx Ox -- Oy

and

Kv0q9 0gr " -~y - - O---x"

(40)

(1.2.4)

Equations of motion

543

Though the piezometric head q) is defined as a function of x, y and z, which means that also surfaces of equal piezometric head (isohypses in two-dimensional flow) exist, the streamlines are no longer perpendicular to those surfaces (lines) because v is no longer proportional to Vcp. Combination of: Vr -- - K ~

099 Or

and

vz -- - K ~

0q) 3z

with equation (18) of Section 1.2.1 gives, for axial-symmetric flow,

3~ Or

=

10~p

and

2rrr Oz

O~ 3z

2rcr Or

(41)

or

399 K - - -Or

1 3~p 2rcr 3z

a~

and

K~

Oz

-

1 07, 2rcr Or'

(42)

for homogeneous and heterogeneous isotropic ground, respectively. As already pointed out above, in the case of inhomogeneous fluid (variable density p) it is of no use to introduce isohypses or equipotential lines (-surfaces) because the velocity vector is not proportional to the gradient of the piezometric head. The relationships between the velocity vectors and the stream functions, however, (formulas (13), (14), (15) and (18) of Section 1.2.1) do not contain the variable p (as a part of the conductivity K) and therefore also are valid for inhomogeneous fluid flow: the stream functions no longer have a direct relationship with potentials or piezometric heads, but with pressures:

(Vp + × vz)

×

(43)

In two-dimensional flow, a plot of equipotential lines or isohypses and streamlines is called a f l o w net. Figure 10 shows a portion of a flow net with two stream tubes in a homogeneous isotropic medium, for which holds that equipotential lines and streamlines are mutually orthogonal. It is customary to draw the flow net such that the difference Aq9 between any two adjacent isohypses is constant. The difference A ~ between any two adjacent streamlines is also constant and is equal to the discharge Aq (equation (16) of Section 1.2.1). Because streamlines behave as impervious boundaries of the stream tube, we may write: A q -- K A n ~

399 Os

-- A ~ -

An

~Adp As

SO

An

A~p

As

A4~

(44)

544

(1.2.5)

Basic principles

¢=

Fig. 10.

Flow net.

Hence, in a homogeneous medium the ratio A,, of the sides of the rectangles must remain constant throughout the flow net and equals the ratio of the small increments of stream- and potential functions. If A ~ -- A¢ is chosen, An = As everywhere and the flow net becomes a net of curvilinear squares. This conception of a flow net consisting of squares, affords a means of determining the flow field graphically by sketching in the flow net by eye.

1.2.5. Equations of motion for diffusion and dispersion The concentration c of a solute in groundwater is expressed in solute mass per unit volume of the water and has the same dimension as the density p [ML-3]. In general, c is, according to the continuum approach, a continuous function of space and time: c = c(x, y, z, t). Phenomena that influence the transport of a solute and thus cause changes in the concentration, are, in addition to the groundwater flow (convective transport), diffusion, dispersion, adsorption, radioactive decay, solution, ion exchange, chemical reactions and others. In Section 1.1.3 it was stated that the phenomenon of dispersion, that means the spreading of water particles and water molecules in the direction of the flow and perpendicular to it, was caused by flow of water through the irregular pore system and by molecular diffusion. The diffusion causes spreading of molecules and thus changes in concentration of the solute, if differences in concentration are present. The driving force behind the transport of molecules is the concentration gradient in a chosen direction. Experiments have shown that, though diffusion is a molecular phenomenon, it may be described on a coarser level, the microscopic level, by means of Fick's law, in which a linear relationship between the specific mass flux of the solute and the concentration gradient is supposed.

Oc fs -- --Dd-=--. Os

(45)

Equations of motion

(1.2.5)

545

The negative sign means that the transport of the molecules takes place in the direction of decreasing concentration. The dimension of G ~t~ - [ML -4] • the mass flux F of a solute is generally expressed as mass per unit time [MT-1], whereas the specific massflux f is defined as the mass flux per fluid area of a cross-section of a ground volume [ML-ZT-1]. The relation between the two is: f =

F neA '

(46)

in which ne -- effective porosity [0] and A --= area of the cross section [L2]. Comparison of the dimensions in equation (45) yields the dimension of the coefficient of diffusion Dd: [L2 T - 1]. Diffusion also occurs in stagnant water (only if a concentration gradient is present) and so is not committed to groundwater flow; however, the effect of diffusion in the direction perpendicular to the flow direction is strengthened considerably as a result of flow through the heterogeneous pore system, as we have seen in Section 1.1.3. This intensified diffusion was called transversal dispersion and now we assume that the latter may also be described by Fick's law, but with a much larger coefficient, the coefficient of transversal dispersion: DT [L2T -1 ] Oc fn -- - - O T ~ n n ,

(47)

in which fn - specific mass flux in the n-direction, perpendicular to the direction 0c of flow and ~ - gradient of the solute concentration in the n-direction. In the flow direction, longitudinal dispersion takes place, which is mainly caused by the irregularity of the pore system and the solid matrix of the ground. Although diffusion plays a role of minor importance in longitudinal dispersion, the longitudinal dispersive transport of the solute is also described by means of Fick's law, this time with a coefficient of longitudinal dispersion DL [L 2T-1], which has the same character as a diffusion coefficient, but is much larger. The longitudinal dispersive flux is always accompanied by mass flux caused by the flowing water, which equals the average real velocity multiplied by the concentration c: uc (convective transport). The total specific flux in the flow direction thus becomes: f s -- u c -

Oc DL--.

Os

(48)

Although dispersion is caused by physical processes that take place on a microscopic level or even on a molecular level (transversal dispersion as a result of intensified diffusion), we make use of the continuum approach to describe the dispersive transport mathematically on a coarser level, the macroscopic level, with the aid of the equations of motion (47) and (48) with pertaining parameters.

546

(1.3.1-1 )

Basic principles

1.3. PARAMETERS AND VARIABLES

1.3.1. G r o u n d w a t e r parameters

1. Density The density of a gas, fluid or solid is defined as the mass of the substance per unit volume [ML-3]; The density of groundwater p is a function of the pressure, the concentration of solutes and the temperature: p - p(p, c, T) in the most general case. The variation of the density with temperature is small and --------xb may be neglected with respect to common groundwater flow, 15 though the specific character T in °C of that variation (a density max10 imum at 4°C, see Fig. 1) is of extraordinary importance for the life of organisms in water (ice is lighter than water). ~8 999 /1000 With the exception of those casp in kg/m3 es where considerable differences of temperature are introduced in groundwater intentionally (injection of cooling waFig. 1. Relation between the density of water and temperature. ter, subsurface heat storage, etc.), the groundwater temperature normally fluctuates slightly around about l0 °C; in the following the density of groundwater will be considered as independent of temperature, unless otherwise mentioned. In general, the variation of the density with the concentration of a solute is linear, for instance:

9!

p - po = a ( c - co)

(1)

in which P0 and co are reference values for the density and the concentration, respectively, while a is constant. For water with increasing chlorine contents (varying from fresh via brackish to salt water) and a temperature of 10 °C, the following relation holds (see Fig. 2): P-

P0 = 1.4075c

with po = 999.7 kg/m 3 and co chosen zero. For instance, c = 10 000 mg/1 = l0 kg/m 3 C1 p = 999.7 + 10 x 1.4075 = 1013.78 kg/m 3.

(2)

Parameters and variables

(1.3.1- l)

547

1030

J

1020

xx'/ /

1010

1000 5000

10000

15000

20000

Cl-contents (c) in mg/1 (= 10-3 kg/m 3) Fig. 2.

Relation between the density and the Cl-contents for water of 10°C.

A property of the groundwater, that is very closely related to the density, is the specific weight, commonly given the symbol g

(3)

× = pg,

g = gravity acceleration [LT-2]. The dimension of specific weight is [ML-2T -2] or [FL-3]; the acceleration of gravity varies slightly with situation on the earth, from 9.78 m/sec 2 at the poles, via 9.813 m/sec 2 in Europe up to 9.83 m/sec 2 at the equator.

2. Compressibility of water. Modulus of elasticity Like solids and gases, fluids are compressible; the density p is a function of water pressure p: ,o = p(p) which may be derived as follows. In a small volume A V of the soil, the change of the total groundwater pressure Pt (Section 1.2.2, formula (20)) is proportional to the relative change of water volume

AVw d(A Vw) dpt = - E w e .

/XVw

(4)

548

(1.3.1-2)

Basic principles

In this relation pt - - total water pressure [ F L - 2 ] and Ew -- modulus of elasticity of water, which has the same dimension as stress or pressure [ F L - 2 ] . The negative sign means that an increase of pressure gives a decrease of volume and inversely. Conservation of mass p A Vw of the water in the water volume A Vw gives d(pAVw)-O

or

AVwdp+pd(AVw)-O,

from which d(A Vw)

dp =

.

ZXVw

(5)

p

Combination of (4) with (5) yields: dpt-

Ew d__L.o

(6)

P Integration of (6) gives the desired relation pt P - P0 exp

- p0)

Ew

(7)

'

in which P0 and p0 are reference values for the density and the pressure, respectively. If the atmospheric pressure is constant, then p0 is mostly replaced by Pa and P t - P0 becomes P t - P a - - P (Section 1.2.2) and P0 --- Pa - - density at atmospheric pressure. In this c a s e dpt can also be written as dp. As the pressure p in the most general case is a function of space and time" p = p(x, y, z, t) it follows that also the density p = p(x, y, z, t). The derivatives of p with regard to space and time may now be expressed as the derivatives of the pressure, according to (6) or (7):

Op Ox Op -- =

Ot

dp Op dp Ox dp Op dp at

p Op Ew Ox' p Op =

Ew at'

(8) etc.

The modulus of elasticity of water amounts to 2 x 10 9 N/m 2 -- 2 x 10 9 Pa (Pascal). Steel has a modulus of elasticity of about 200 x 1 0 9 Pa, so that water is about a hundred times more compressible than steel. Compared with gases, water is about 104 less compressible.

3. Viscosity A solid in a state of shear will deform elastically, the magnitude of deformation being in direct proportion to the shearing force. A fluid subjected to shear, however, will also deform, but in this case continuously so long as the shearing force is applied; the magnitude of the force now

Parameters and variables

(1.3.1-3)

549

Vy + -~yy d y

-~y~dy dt dy

OVy dx dt

Ox

lly

, v~+aV-~dx Ox

Vx

dx Fig. 3.

Shear stresses in a horizontal plane.

governs not the magnitude, but the rate of deformation. The majority of fluids, water included, are so-called Newtonian fluids, which have the property that shear stresses, if present, are proportional to the angular deformation per time unit. In a horizontal plane, perpendicular to the z-direction, the angular deformation equals Ol)y

Oldx

(--~-x + --~-y) d t as can be derived from Fig. 3, where a~y dx dt Ovv ax = " dt dx Ox

tanda-daand tan dfl - dfl -

0 l)x

Oy

dt

and so the rate of deformation equals

da - dfl 3 vv OVx = + dt 3x -~y

(see Section 1.4.4)

The proportionality factor is called the dynamic viscosity and is denoted b y / , . The relation between the shear stress and the rate of deformation then becomes:

rxv

•

--

rvx

.

--

l~

( 3 Vy

3 Vx

--~-x + --~y )

(see also Section 1.2.2, formula (19)).

(9)

550

Basic principles

(1.3.1-3)

100 T in °C

50

0.5

1.0

1.5

2.0

Fig. 4. Relation between viscosity and temperature for pure water. As the viscosity/z is directly proportional to the shear stress (dimension [ F L - 2 ] ) and inversely proportional to the rate of deformation (dimension [T-l]), the dimension of the viscosity becomes [FL-ZT]. The magnitude of the viscosity is related according to the International System 1 Nsec of Units (SI) to a special unit the Poise (P)= 126 Pa.sec (pascalsecond) - 10 m2 •

Pressure hardly affects the viscosity of a fluid, except for very high values. On the other hand, the temperature has much more influence. In Fig. 4 the relation between the temperature and the viscosity is given for pure water from which it can be seen that an increase of temperature causes decrease of the viscosity: the water becomes "thinner" and the water particles move more readily along each other. In passing it must be noted that the viscosity of a gas increases if the temperature rises, as distinct from the viscosity of a fluid. Both water and air have relatively low viscosities, compared, for instance, with glycerine, which is roughly a thousand times as viscous as water. Solutes in water will increase the viscosity but not spectacularly, as can be seen from the curve in Fig. 5 that represents the relation between viscosity and chlorine contents of water at 10 °C. Therefore, in fresh-salt water problems the viscosity may be considered as a constant parameter. In hydrodynamics the so-called kinematic viscosity v is frequently used; it equals the dynamic viscosity divided by the density of the fluid /z v = --. P

(10)

The dimension of v is [L2T -1] and the unit is the Stokes = 1 cm2/sec. In geohydrology the use of the kinematic viscosity should be discouraged, because, for

Parameters and variables

(1.3.2-1)

551

1.4

1.35

1.3 Cl-contents in mg/1

0 Fig. 5.

I

f

I

I

5 000

10 000

15 000

20 000

Relation between viscosity and chlorine content for water at 10 °C.

instance, in fresh-salt groundwater problems or other flows with different densities a more or less constant value to the viscosity v can no longer be assigned. 1.3.2. Ground

parameters

1. Statistical description of the ground For different purposes, some description of the geometric properties of the soil, such as size and shape of the grains and the pores, is needed. Because such a description is impossible in a deterministic form owing to the complexity of the material, one resorts to defining a number of macroscopic properties, such as, for example, porosity and specific surface (cf. 2 and 3 of this section), which have to be determined empirically. However, such macroscopic properties need a sound theoretical basis in agreement with physical laws, in which a statistical description of the soil will be a valuable device. Soils may be devided into consolidated and unconsolidated conglomerates. Unconsolidated ground is composed of loose particles (grains) between which pores are present, for instance, sand or clay. Consolidated ground is more or less a solid with cavities and pores, e.g., sandstone. Unconsolidated ground may be described by the distribution of the soil particles according to their size. However, the size of a grain is not unique (purely spherical and cubic particles excepted) and cannot be described by means of one single parameter. In general, the "dimension" of a ground particle depends on the method of measurement. Well known methods of obtaining grain distributions of soils are sieve analysis for particles bigger than approximately 0.06 mm (60/zm) and hydrometric analysis for smaller particles. Nowadays, more and more apparatures are being used, with which the particles can be counted, according to their size, by means of laser beams.

552

Basic principles

(1.3.2-1 )

100

,WofUL

wei % through I the 50

t&]h I I t n sieve rivqx d

I

1

clay

50

0 I

100 1000

4000

].

10

100

-- grain diameter in n~icrometer /

gravel 2 000

coarse sand

|

/

200

clay

silt

fine sand 20

2

Fig. 6. Sieve curves and classification of soils. If sieve analysis is applied to a soil sample, the sample must be dried and made free from silt before it is placed on the upper sieve of a set of sieves, with apertures of successive diameters, from coarse to fine. After shaking the set of sieves simultaneously the weight percentages of the material that falls through each sieve and that remaining on each sieve are determined, thereby defining the dimension of the grains by their position between two successive sieve diameters. In hydrometric analysis, the rate of sedimentation of the particles in water is compared with that of small spheres of fixed diameters; in the case of equal rates the soil particles are considered to have the same diameter as the correspondent spheres. The obtained results may be shown by means of a particle-distribution diagram or sieve curve (see Fig. 6). The horizontal axis of the diagram gives, on a logarithmic scale, the dimensions of the particles, while the weight percentage of the material that each sieve passes, can be read from the vertical axis. The measured points (one for each sieve diameter and each hydrometric determination) will then be connected by a smooth curve, which represents the sieve curve for the ground sample. These diagrams may be used for statistical operations such as the determination of the mean grain diameter and the distribution of the grains and they are particularly important devices for the determination of permeabilities of ground samples in laboratories and of the specific surface (cf. 3 and 4 of this section). Unconsolidated grounds include several soil types, classified according to their particle diameters, such as clay, silt, fine sand, coarse sand and gravel. The classification laid down by the International Society of Soil Science is given at the foot of Fig. 6.

(1.3.2-2)

Parameters and variables

553

Unlike unconsolidated soils consolidated soils such as sandstone have to be classified according to the distribution of the pore dimensions, which turns out to be much more difficult than the determination of the particle distribution. 2. P o r o s i t y

The porosity or the v o l u m e t r i c p o r o s i t y n, a macroscopic property of the porous medium, is defined as the ratio of the pore volume to the total volume of a certain amount of ground

n --

Avp AV

=

/xv~ 1 -

~ .

(11)

AV

In this relation A Vp represents the volume of the pores and A Vs the volume of the soil grains in the total volume A V. In the saturated zone of the ground the volume of the groundwater A Vw equals the pore volume (apart from a slight number of air bubbles) and (11) becomes: n-

AVw AV

= 1-~.

AV~

(12)

AV

The porosity is dimensionless [0]" usually it is expressed as a percentage. The so-called a r e a l p o r o s i t y nA is defined in saturated ground as the ratio of the area of the pores in a cross-section to the total area of that cross-section

na-

AAw AA

-- 1

AAs

(13)

AA '

(Ap -- Aw in the saturated zone). The relation between n A and n can be determined by considering a REV of cylindrical shape with length L, volume A V and cross-section area AA with A V = L A A . The pore volume in the REV becomes:

A V w -- n A V

--

½~ A A w d s

--

½LnaAA

ds -- A A j - ½ L n a

ds"

In general, n A is variable along s" if we define a mean porosity /~A as

1[ ~a -

nA ds

-{ j _ ½ L

then n A V -- A A L f i A nA -- n.

from which (14)

554

(1.3.2-2)

Basic principles

isolated pore dead end pores

Fig. 7.

Different pores.

As we can see from Section 1.1, the continuum approach that we use in our study of groundwater flow is based on different levels of averaging. So we need not distinguish between tiA and h a ; therefore in the following we suppose that (15)

nA -- n.

On the other hand, a distinction must be made between different types of porosity, if we consider the purpose to which the porosity has to be applied. For instance, if a consolidated soil, apart from mutually connected pores also contains a number of isolated pores or a number of so-called "dead end" pores, which are not isolated, but in which groundwater stagnates (see Fig. 7), then the total amount of water will no longer take part in the flow, but only an "effective" part of it, with a volume (A Vw)e (,< A Vw). The effective porosity n e is then defined as

(AVw)e

ne ---- ~

.

AV

(16)

Also in unconsolidated soils a portion of the water volume usually does not participate in the flow: for instance, groundwater that is linked with the soil skeleton by absorption forces; here, too, the concept of effective porosity holds. However, it must be kept in mind that the concept "effective" only means effective with regard to an object, for instance, with regard to flow, as was discussed here. For problems concerning diffusion, dispersion, adsorption and others also stagnant water is of importance and the effective porosity for such cases may be different from that for flow. The porosity of unconsolidated grounds depends on the structure of the ground (the packing of the grains), the size distribution and the spatial distribution of the grains and their shape, while the porosity of a consolidated ground depends greatly on its state of cementation. Especially the grain-size distribution may influence the porosity strongly, because small particles may partly fill up the pores between the larger particles, thus

Parameters and variables

(1.3.2-3)

555

Table 1. Porosity of soil types. Designation of soil

Porosity

Gravel Sands Sandy loam Loam Clayey soil Peaty soil

0.30-0.40 0.30-0.45 0.35-0.45 0.35-0.50 0.40-0.55 0.60-0.80

decreasing the porosity. A survey of the porosity for some soil types is given in Table 1.

3. Specific surface The specific surface 0 of an unconsolidated soil is defined as the total area AAs of the surface of the grains that makes contact with the groundwater inside a unit volume AV of the ground, divided by the volume of the grains AVs AAs O = ~. ZXVs

(17)

The dimension of specific surfaces is [L-l]. This concept is very important for groundwater flow, because the magnitude of the specific surface is directly proportional to the shearing force which the solid matrix exerts on the moving groundwater. The shearing force is opposite to the flow direction and therefore offers resistance to the groundwater flow thus influencing the flow capability considerably (see Section 1.2.3). As the flow capability is directly related to the permeability of the ground, it is obvious that much research has been done in order to relate the permeability of the ground directly to the O-number (see 4 of this section). The specific surface also plays a role in problems concerning absorption of solutes in groundwater to the soil particles, for instance, clogging phenomena round screens of pumping wells. For consolidated grounds the specific surface is defined as the area of the surface of the solid matrix that forms the boundary of the pores, divided by the volume of the solid matrix. The specific surface of a fictitious material, consisting of equal small spheres with diameter d, is called the specific standard surface Os: ~d 2

6

Os = l:rrd3 6

d

(18)

From the particle distribution diagram (sieve curve) of a soil sample the specific surface Osa can be determined as follows.

556

(1.3.2-3)

Basic principles

On each sieve with aperture diameter Di (i = 1, 2, . . . , n) of a set of n sieves, a certain amount of grains remains of which the weight has been determined, say wi. We suppose that these grains have an average diameter di equal to the geometric mean of the diameters of the two adjacent sieves: di - ~ / D i - 1 D i (the sieve curve is considered to be constructed on a half logarithmic scale by straight lines between the sieve diameters). We assume further that each fraction consists of identical spheres with diameter di, belonging to that fraction, giving mi spheres in a fraction i, where wi mi = !~d3i y s 6 with ys - specific weight of the grain material. The specific surface of the sample now becomes: n

n

wi

n

Y-~i=I mirrd~ _-- 6 Zi=ln d-7 ---- 6 Z we Osa = n l zrd3 ~-'~'i= 11)i Gdi Z i = I mi g . 1 i=1 R

in which G - - Z i = I w i - - total weight of the sample. wi - Wpi - the weight percentage of the fraction i and We set W Z

?l

1/)pi __ 1

dh

i=1 di

and we call dh [L] the harmonic mean dialneter with respect to the weight of the sample. We have thus found: 6 Osa = d----h with WP------L-~- tOP2

1

dl

l/)Pm

(19) '

Wpi are the weight percentages which may be read directly from the sieve diagram (y~ Wpi = 1). From equation (19) we conclude that the specific surface of a soil sample is inversely proportional to its harmonic weighted mean diameter. In the literature much use is made of a specific surface that is defined as the total surface of the particles in a sample, divided by the total surface of the particles in a fictitious sample of the same weight but consisting of uniform spherical particles of 1 cm diameter. This specific surface, also called the U-number, is dimensionless and its relation to the specific surface O is: 0 U

__

_ _

Os

Od -

-

6 '

according to (18) with d -

1 cm,

Parameters and variables

(1.3.2-4)

557

which means that the U-number is the ratio of the specific surface of a sample to the specific standard surface. The choice of a fixed diameter of 1 cm for the spheres means that U - o , always bearing in mind that O has to be expressed in cm -1, whereas the specific surface O has a certain value, independent of the dimension that has been measured (cm -1 or m - l ) • the specific surface U on the contrary is bound to the chosen unit, here the centimeter. To avoid confusion it is better to work with O than with U. This also holds for experimental formulas, developed for the determination of U-numbers in which numbers occur that are not dimensionless.

4. Intrinsic permeability In the theoretical derivation of Darcy's law in Section 1.2.3 a value of an

k-

C3ne

was found in which a was an arbitrary length and c3 = ClC2 had been set with C2 as a dimensionless constant and cl the ratio of the grain surface AAs to the volume A V of the ground unit. According to equation (17), the relation between Cl and the specific surface O becomes: AA C1

__

s

Av

= (1 -- n)

AA~

Av~

-- (1 - n)O

and with equation (19)" 6(1 - - n ) Cl --"

dh

Taking for the arbitrary length a the harmonic weighted mean diameter dh and putting the dimensionless c o n s t a n t 6he(inn)c2 -- C, the intrinsic permeability may be written as

k-

cd 2 [L2].

(20)

In general, the constant c will be a function of the porosity of the soil (which already found its expression in the factor h e ( l"- n ) ) and of the shape of the grains, while dh represents the grain distribution. The general form for k, therefore, is" k - fl (s) f2 (n)d~

(21 )

(according to Bear in "Dynamics of Fluids in Porous Media", p. 134). In this expression s is a dimensionless parameter related to the shape of the grains (or the pores); fl (s) is called the shape factor and fz(n) the porosity factor.

558

(1.3.2-4)

Basic principles

A particular form of this general formula is the formula of Kozeny-Carman, well known to soil scientists" 1 n3 1 1 n3 k - ~ -(1- - - ~ n) ---5 0 = 180 (1 - n) 2 d2"

(22)

Here n3

l

fl (s) -

180

and

f2 (n) -

(1 - n) 2"

This is an example of a formula which is suitable for determining the k-value by means of laboratory tests (sieve analysis, etc.). For sands with little or no silt contents, good results may be obtained in this way; however, silt contents of some importance will already give a strong disturbance of the results, because the silt will have been removed (washed out) before the sieve analysis and so will not count in the dh determination, whereas in reality the permeability of the ground strongly depends on the silt contents. The value of c in equation (20) or f l ( s ) f l ( n ) in equation (21) generally have to be examined experimentally either in the laboratory by means of permeameter tests on soil samples or in the field by means of pumping tests. Both tests yield values for the Darcy permeability K -- ~p--~g(see Section 1.2.3, formula (30)) and so pg and /z must be known numerically in order to get k. Field tests for the determination of the permeability give the most reliable results, because soil samples will always be more or less disturbed and in permeameter tests only the vertical permeability is measured, and this can differ considerably from the horizontal one. Pumping tests will be discussed more extensively in Section 1.3.6.

5. Soil compressibility and modulus of elasticity If the solid stress as, also called grain stress (see Section 1.2.2, equation (26)), in a point in a grain skeleton increases, the result will be a compression of the grain skeleton caused both by a change of the vertical dimension Az of a volume element A V -- A x A y A z and a change of the porosity n; the changes in the horizontal dimensions Ax and Ay of A V are neglected. The relation between the changes of the grain stress and of the vertical dimension of the volume element can be written as

das - - E g

d(Az) Az

(23)

Eg -- modulus of elasticity of the grain skeleton with dimension [ML-1T -2] or [FL-2], the same as that of stress. The negative sign means that increase of stress results in decrease of volume and inversely. The modulus of elasticity of the grain skeleton Eg must be distinguished from the modulus of elasticity of the solid material Es, which is very much higher.

(1.3.2-5)

Parameters and variables

559

The values of Eg lie approximately between 10 6 and 10 7 Pa (N/m 2) for clay and between 107 and 108 Pa for sand at depths up to ca 500 m. The values given in the literature, however, diverge rather considerably. As discussed earlier, the equilibrium of stresses acting on a plane lying in a horizontal cross-section of the ground can be written as: Pa + o'g - Pt + O's (equation (26) of Section 1.2.2). If it is assumed that the atmospheric pressure Pa and the total vertical ground stress o'g do not change, then p t n t- Os = constant and as p = p t - Pa = water pressure with respect to the atmospheric pressure, we find: das - - d p ,

(24)

which means that in geohydrological processes decrease of the water pressure goes together with increase of the solid stress and inversely if the external forces remain constant. From the equations (23) and (24) it follows that: dp-

d(Az) Eg~. Az

(25)

The change in the volume element AV does not result in a change in the grain volume A Vs, as the compressibility of the individual grains may be neglected, compared with the change of the total volume (Es >> Eg); therefore, A Vs = (1 - n ) A x A y A z -- constant. Differentiation gives (with Ax and Ay assumed to be constant and Az and n variable)" d(AVs) -- {(1 - n ) d ( A z ) -

Azdn}AxAy

--0

or

d(Az) dn - (1 - n ) ~ Az With equation (25) this gives" 1

dn -- ~ ( 1 Eg

- n) dp.

(26)

Integration of this expression yields" n-- 1-(1-n0)

exp

(P) -~g

(27)

,

where no -- porosity at p - - 0 (atmospheric pressure). As the pressure p in the most general case is a continuous function of space and time p - p ( x , y, z, t), according to the continuum approach for groundwater flow this also holds for the porosity: n = n(x, y, z, t). The derivatives of n with regard to space and time can now be expressed in the derivatives of the pressure, according to (26) or (27), for instance, On to= =

On Op _ ot -

1 -(1gE

Op

etc.

(28)

~tu/~I L'666 dl!suap aql pue (I'E'I uo!laaS jo 17 "ff!~I) da I E'I st. qa!q~ jo/~]ISO:3SI.A aql Do 0I le Jale~punoa~ ql!m tunipatu snoaod e ut dep aad tu I jo ,{l!A!lanpuo3 V •,{ep/tu 1798 - aas/tua I s[ aol:3eJ uoIsaaAUOa aq,L "puoaas ,tad ,talauqmao aajaad '.taAan~oq 'SlS!lUa!as I!OS :dt~p ,tad saalaut s! sls!~oloap~ q ,{q pasn dlluanbaa J lSOLU S! leql dl!A!lanpuoa jo l!un aq,L •aaeld a~tel uo!leaedas s!ql plnoqs ~Olj ieai1aaA paaunouoad jo asea aql u! ,{iuo "G pue X uaa~laq apetu s! uoIleae -das ou slsal pla~J pue suo!lmndLuoa iea!~OlO.Xp,~qoa$ jo 1.xed aaleaa:~ aql u! asneaaq aalatueaed aql jo aalaeaeqa aql saq!aasap .xallaq qa!q~ 'Z LIIeaaua~ 'Ioq tu'(s auo ,(q paleu~!sap ualjo s[ dl!A!SS!tUsueJ1 aq,L ".tale~punoa$ l!tusueJ1 ol aajinbe aql jo al!iiqe aql saz!aalaeaeqa 11 "[~-ZzT] uo!suatu!p ql[n~ G X .xaj!nbe ue jo &t.at.sst.tusuoal aql ,qdtu!s ao &~.a~.ss~.utsumlfo luat.obffaoo aql patuaal s[ aaj!nbe aql jo uo!laas Iea!laaA e u! sa!l!A!lanpuoa a!IneJp,~q aql jo anleA aSe.xaae aql pue aaj!nbe aql jo (7 ssau -:,Ia!ql aql jo lanpoJd aql 'ssau:,Ia!ql u[ tuJoj!un patunsse aq ,~etu leql sJaj!nbe u I •tunt.patu snoaod aql jo ~/,(l!i!qeatu.xad a!su!alu! aql ol pal!tu!I aq plnoqs 1! lnq 'dl!a!lanpuoa aql aoj pasn dlluanbaJ J s! &!I!qeatuaad atueu aq,L "~7"~'I uo!laaS u! passnas!p uaaq seq 'punoa~ aql jo sa!laadoad uo spuadap ,quo qa!qt~ '[zT] ~/al!I!q eamaad aq,L "I'~'I uoIlaaS u! I!map u! paq!aasap uaaq seq qa!qA~ ' [ Z z - T d ] r/,{a!soas!A aql pue [~-Td] a a - x lq$!aA~ a~iaads aql ao [~;_7/A/] d al!suap aql a.m uo!lelaa s!ql u! sa!laadoad pInlj aq,L

z~ (6Z)

"~

rl =

= X

:((00 uo!lenba '~:'Z'I uo!~azS) elntuaoj aqa tuoa=I peza aq uea se 'p!nlj aq~ jo pue n,po

jo

o. oao a . o . , o q

puoaop

,I

-

,~l!un aA!lgffatl aql s! lua!pe.xff a!IneJpA q aql jo apnl!uffetu 0ql j! 'aale~punoaff aql jo a~Jeqas!p atj!aads aql slenba X '~eI s,,~aae(I ol ~ut.pJoaalz 'Jo 'Jalt~A~ptlno.I~ 1Jodsueal ol tun!patu snoaod e jo dl!I!qe aql saz!aalaeJeqa qa!q~ '[~-Z7] &IaOlaA jo uo!suatn -!p aql ql!~ (saauelstunaa!3 a!doJlos! u[) .mleaSe st. 11 "~O~.a~.lanpuooaql ',qdtu!s ao '&~.a~.lonpuoa a~.lno.tpdt1 fo luat.abgfaoo aql PalIea st. (E'g'I uo!laaS u! (I~) uo!l -enba "ja) ~e I s,,qaae(I aoj uotssaadxa aql u! X dl!IeUO!laodoad jo luelsuoa aq,L &t.at.lonpuoo o,.doalos I "[ s:lua!aUjao:~ h~:l!A!:lanpuo~ "~'~'[ •pzaolsaa ,~IInJ oq lou II!aX rams IeU!~!ao oql 'ssoaoad oql ~u[ddms amje leql sueotu qa!qn~ 'a!lseia dilazjaad tuoplas s! 'IIa~ ~u!dtund e jo sueatu Lq ameaxpunoa~ jo ieaxeapql!ax Lq 'aaumsu! aoj 'pasnea uolaD~IS u!ea~ e jo uo!letuaoj~p ~ql asnea~q ~a~q paq!aasap ueql pmeaIId -tuoa aaotu ,qqeaapIsuoa s! ssaaoad zql 'dl!IeZa u I "uo2.wHlosuoo PaIIea s! sa~ueqa aansszad jo linsaa e se uolzD>IS u!ea~ pmeanles e jo uo!letuaojzp jo ssaaoad aq,L

sHdpu,.dd a.wt~/

( [-¢" ~'I)

09g

Parameters and variables

(1.3.3-1 )

561

(Fig. 1 of Section 1.3.1) corresponds to a permeability k of the porous m e d i u m of 1.31 x 10 .3

m2

or

24 x 3600 x 999.7 x 9.813 K -- 1 m/day is equivalent to k -

1.55 x 10 .8 cm 2 -

1.55 # m 2 for pure ground-

water at 10 °C. In the oil industry the unit darcy for the permeability k is in use, which is based on the general form of Darcy's formula for horizontal flow Q

k 0p

A

#Ox

from which 1 darcy-

1 cm3/sec per c m 2 × 1 centipoise 1 atm. per cm

;

(30)

in words" a porous m e d i u m has a permeability of 1 darcy if a fluid with a viscosity of 1 centipoise flows through it with a discharge of 1 cm3/sec over a cross-section of 1 cm 2 under a pressure gradient of 1 arm per cm. 1 atmosphere is 10.1325 N/cm 2 and 1 c P - 10 .7 N s e c / c m 2, so that

1 darcy-

1 cm/sec x 10 .7 N s e c / c m 2

= 9.869 x 10 .9 cm 2 - 0 . 9 8 6 9 / ~ m

2

10.1325 N/cm 3

which corresponds with a conductivity of 0.64 m/day for water at 10 °C. A global survey of some unconsolidated soil types and their conductivities for pure groundwater at 10 °C is given in Table 2. If the permeability k of the ground is space dependent: k = k(x, y, z) then the porous m e d i u m is called inhomogeneous or heterogeneous, contrary to a h o m o g e neous porous m e d i u m with more or less the same k everywhere. If in a fixed point the permeability k varies with the direction, the m e d i u m is called anisotropic, unlike Table 2. Global ground conductivities for water at 10°C. Designation of soil Peat Clay Peaty sand Loamy sand Fine sand Middle fine sand Coarse sand Gravel

Conductivity (m/day) 0. I-5 1 0 - 6 - 1 0 -3

3 x 1 0 - 2 - 1 0 -2 6 x 10-4-4 x 10-4 1-10 10-50 50-200 > 200

562

(1.3.3-2)

Basic principles

m

Fig. 8.

Anisotropic ground.

an i s o t r o p i c medium where the variation of k is independent of the direction. As K is directly related to k, we may also distinguish between isotropic and anisotropic conductivity. 2. A n i s o t r o p i c

conductivity

Anisotropic grounds are of considerable importance in practice, especially those that consist of layers deposited in the past (sediments). If this have been taken place horizontally, the conductivity in the horizontal direction mostly exceeds the conduction in vertical direction. In general, one distinguishes in anisotropic ground three main directions with different p e r m e a b i l i t i e s , which are mutually perpendicular, for instance, ks in s-direction, kn in n-direction and km in m-direction (see Fig. 8). As the permeability in a point is no longer a constant, but depends on the direction, Darcy's law according to (29) or (31) of Section 1.2.3 no longer holds. The specific discharge vector is no longer proportional to the gradient of the piezometric head for homogeneous groundwater (constant p), so that v an V~o, in general, d o n o t c o i n c i d e (cf. equation (31) of Section 1.2.3). In anisotropic ground the ratio of the components of v and the corresponding components of V~0 along arbitrary directions are variable, contrary to isotropic ground where that ratio is always the same ( - K for homogeneous groundwater). In the following derivation of Darcy's law for anisotropic groundwater flow, we at first assume that the ground is homogeneous and anisotropic and the water also homogeneous, which gives the opportunity to consider the c o n d u c t i v i t i e s in three main directions K,., Kn and Kin. Based on physical considerations, it is assumed that along the main directions Darcy's law, in its simple form, holds: 099 Vs -- - K s -~s ,

099 vn -

- K,, -~n ,

0~0 vm -

- Km ~ L .

(31)

Parameters and variables Table

(1.3.3-2)

563

3.

S

H

m,

X

Olsx

Otnx

Otmx

Y

Otsy

Otny

~my

Z

Orsz

Otnz

Olmz

To find expressions for Darcy's law in three arbitrary directions x, y and z, which are also perpendicular to each other, we suppose that tese directions make angles with the main directions s, n and m, according to the scheme in Table 3. By means of these angles the components Vx, Vy and vz of the vector v can be expressed in the components of v in the main directions vs, v,, and Vm" Vx --

U s C O S Olsx ~

l) n C O S Olnx -J- 1)m C O S t Y m x ,

U v --

U s C O S O/sv -t- Un C O S Olny n t- Um C O S Olmy ,

Uz - -

U s C O S Olsz n t- U n C O S Olnz n t- Um C O S Olmz.

(32)

In a similar way, the components of Vq9 along the main directions can be expressed in the components of Vq) along x, y and z"

Oq) Os

--

090 COS 3x

Otsx -J- ~

099 On

--

~

090 COS 3x

Otnx -}- ~

Oq9 Om

--

--

O~o

3x

099 COS 3y

Otsv q -

090 COS 3y

Ofnv q - - -

099 C O S Olmx q -

~

Oy

099 COS Oz

C(~,z,

09o COS 3z

O/nz ,

--

(33)

099 C O S O/my n t- ~

3z

COS Olmz.

If we substitute these values in the equations (31), we get values for Vs, Vn and which, substituted in (32), yield values tbr v,, vv and vz; for instance, for Vx"

Z cos ol,.x + 3099 0q) COSOr,z) - Vx -- K,. cos ot,.x( 03x y cos c~,.v q- O---z

'09o

3~o

Oq)

+ Kn COSOtnx ~X COSO~nx -t. Oy. COS -t OZ C O S . Oenv . 099 ..~ -k- Km cosoln~x 03x

COS ~mx

3y

q- ~

) Olnz

O99

3z

C O S Olmv q - - -

) C O S O/mz

Vm

564

(1.3.3-2)

Basic principles

or rearranged:

-Vx -- (Ks COS 2 Olsx -Jr-Kn c o s 2 ~nx -~- Km c o s 2 0 l m x ) - ~0(/9 X Oq)

-1t- (Ks cos Ogsxcos Ogsy-[- Kn cos Ognxcos Ogny_qt_Km cos C~mxcos O/my)~y aq~

-1t- ( K s c o s of,.x c o s Ofsz -[- K n c o s Ognx c o s Olnz -[- g m c o s Olmx c o s Olmz) -~z"

The terms between the brackets all consist of constants and so their sums can be denoted by constants, successively, for instance, by Kxx, Kxy and Kxz. The same procedure holds for vv and vz; therefore the general form of Darcy's law for anisotropic and homogeneous ground and homogeneous groundwater can be written as:

Vx -- - K x x Oq9 Ox

Kxv Oq) " Oy

Kxz Oq9 "-~z'

0q9

Kvv 0q0

Kyz Oq) 0--7'

Vz -- - Kzx 099 Ox

0q9 Kzv " -Oy

099 Kzz '-Oz

Vy -- - K y x Ox

"

Oy

(34)

With the aid of the so-called summation convention of Einstein the equations (34) can be written in compact form: 0q)

(i, j -- x, y, z).

l)i --" --Ki.j"y-:'.

Oj

(35)

This convention states that if in a product an index occurs more than once (here j), this product should be replaced by a series of products according to the values that index may adopt (here x, y, z). The constants Kij also can be written in short notation as functions of K,., K,, and Kin:

Ki.j : Kh cos

Olhi

COSOlh.j

(i, j = X, y, Z, h = s, n, m),

(36)

for instance, i = z, j = x:

Kzx = Ks cos Olsz cos C~sx + Kn cos o~,,zcos ~,,x + Km cos Olmz COS Olmx. From (36) it immediately follows that Ki.j -= Kji. The nine K-values in (34) or (35) thus are reduced to six: Kxx, Kvv, Kzz, Kxv, K~z and Kvz. Together they constitute a symmetric tensor, the conductivity tensor. •

.

Parameters and variables

(1.3.3-2)

565

For non-homogeneous fluids (variable p) a similar derivation for Darcy's law holds. In index notation (31) becomes:

kh ( Op Uh--

OZ ) q-y

Formula (32) becomes vi

( h - - s n m) ,

=

1)h COS O/hi

Op Op 0 h = Oj cos Oth.j and

,

.

(i = x, y, z), while

Oz Oz 0 h = ~Oj cos Othj (j = x , y , z ) .

Combination of these 4 equations yields:

Ui - -

@ y ~

kij (Op Ui =

"

with kij = kh cos

--~ y

COS O~hi COS

Oz)

Olhi COS Olh.j

C~h.i

(i , j -- x ,

or

z)

(37)

y , ,

(h = s, n, m), for instance:

kxxOpkxvOpkxz(Op Vx = - ~ " lz Ox lz Oy lz

+ ~') -~z

etc

Oz and Oz are z e r o). Ox Oy The nine k-values together form the (intrinsic) permeability tensor of the anisotropic soil. From (31) it follows that as soon as the head gradient in one of the main directions, for instance, the m-direction becomes zero ( ~Om - 0), the flow becomes two-dimensional (only flow in s- and n-directions). In anisotropic flow, however, if in an arbitrary direction, not coinciding with a main direction the head gradient vanishes, for instance, ~ - 0, then Vz does not vanish as can be read from equation (34). In order to obtain two-dimensional anisotropic flow both 57z and vz must equal zero, which implies that still three different conductivity constants Kxx, Kxy and Kyy are left. In practical problems concerning anisotropic groundwater flow the coordinate system xyz will be chosen along the main directions of the anisotropy. As we saw, the discharge vector v does not coincide with the gradient of the piezometric head Vq9 in anisotropic flow, as a result of which streamlines and equipotential lines (isohypses) are not perpendicular, but, in general, make angles with each other. In order to obtain information about these angles we introduce the concept

566

Basic principles

(1.3.3-2)

of directional conductivity, which is defined as the ratio of the velocity component along a chosen direction to the negative head gradient in that direction, i.e., Kx

-~.

~s as

In the same way, the directional conductivity Kv in the direction v of the velocity vector v is defined, and thus

Ivl - -Kv ~O---z-".

(38)

Ov

We suppose the coordinate axes x, y and z to be the main directions of the anisotropy from which

~x -- - K ~ ~Ox,

vv -- - K v

-

- 5y

~z - - K z -0-2"

and

The piezometric head ~0 is a scalar function of x, y and z and according to a theorem of vector calculus that the directional derivative of a function in a chosen direction equals the scalar product of the gradient of that scalar function and the unit vector in that direction, it follows that O~o v ~ = ~.V~o

o~v

Ivl

and from (38):

Ko Oqo O~p Oqo ,vl (Vx +vv + Vz )

Ivl --

which becomes with vx - - Kx ~~)x ' etc." 1¥12

-

2

Vx

+

Ko-Kx

vv2

Vz2

(39)

+KT

From (39) it can be seen that K~ is not a constant, but a state variable, depending on x, y and z. If we assume in the origin a position vector r parallel and proportional to v, with components x0, y0 and z0 and length V'R--~, i.e., ¥ m

r = ~/Kv Ivl then

[r] - - ~Kv,

xo - - ~~

Vx,

Ivi

Yo

--

~/Kv v v ,

~

Ivl

zo

--

Vz

Ivl

(1.3.3-2)

Parameters and variables

567

and so according to equation (39): xg + Yg z2 Kx K-~-v+ ~ - l K z which means that the endpoint of r lies on the ellipsoide: x2 . .

.

y2 .

Z2

.

1

(40)

K x ~- -~v -~ K z

with axes ~/~x, x2

y2

---iKx

Ky

dgy a n d ,v/-K~z. For two dimensions (40) becomes = 1

an ellipse: (41)

which has been drawn in Fig. 9. (Here a horizontal and a vertical direction have been chosen, because in practical problems, especially in sediments, the horizontal conductivity may exceed the vertical conductivity considerably.) z

r t

ol i--.

4Vf X v

- )

x

The ellipsoid (or ellipse in two dimensions) represents the locus of the end points of the position vectors r, which are parallel to v and having lengths ~/Ko, thus giving directly in each direction the magnitude of the directional conductivity, associated with the velocity vector, having this direction (see O P = ~/Ko in Fig. 9).

Fig. 9. Conductivityellipse.

Moreover, the picture of the ellipsoid (ellipse) enables us to determine the direction of the velocity vector (streamline) if the direction of the head gradient is given and inversely. This is based on the fact that equipotential surfaces (-lines) and streamlines are each others' conjugates in the ellipsoid (ellipse) which means, for instance, in Fig. 9, that the plane (line) through P tangent to the ellipsoid (ellipse) is perpendicular to the head gradient and therefore parallel to the plane (line) through O, tangent to the equipotential surface (line) in O. This can be shown as follows: Point P (x0, y0, z0) lies on the ellipsoid, so

568

(1.3.3-2)

Basic principles

x2 t- yg z2 "~'x- ~-~-v+----1;Kz

(a)

O P has the direction of v, hence v_.j_x= v_L = v__z. xo yo zo

(b)

The vector perpendicular to the plane through P tangent to the ellipsoid is the gradient of a function f, which is obtained by writing the equation of the ellipsoid in the form: X2

y2

Z2

f (x' Y' Z) -- -Kx + -~v + K---~- 1 = 0 .

The gradient of f in point P becomes Vf(P)-

(2xo 2y0 2z0) K x ' K v ' Kz "

(c)

The head gradient can be written as: 3rp 3~o O~o

vx

vv

Vz

(d)

Comparing (b), (c) and (d) with each other it follows that V~0 and V f are mutually parallel and also the planes to which they are perpendicular. From the conductivity ellipsoid it can be seen at once that if in a point the discharge vector v has the direction of one of the main directions of the conductivity tensor, the streamline through that point is perpendicular to the equipotential surface in that point. 3. Resistance to flow in layered soils

The inverse of the conductivity K is called resistivity r 1

r = -K

(42)

which is a measure of the resistance to flow that is offered by the grain particles. It is of importance in problems concerning flow in layered aquifers and flow through semi-permeable layers.

Parameters and variables

(1.3.3-3)

KID1 K2D2

giDi

~ qi

569

Consider an aquifer, consisting of horizontal layers, each having a constant thickness Di and constant conductivity Ki (i = 1 , 2 , . . . , n). See Fig. 10. We assume in the first place uniform horizontalflow through the aquifer, caused by different heads (difference H) at a distance L from each other, the heads being the same in a vertical. As the n is the same for hydraulic gradient Zall layers, the specific discharge qi for each layer thus becomes:

D

H qi -- Ki Di Kn-lDn-1

L

according to Darcy's law. The total discharge q [L2T -1] becomes"

K~Dn

Fig. 10. Parallel flow in a layered aquifer.

H q -- - ; - ( K 1 D1 + K2D2 + ' . " + KnD,,) L,

or H

tl

(a)

q---EEl, i=1

where T,. -- KiDi is the transmissivity of a layer. The same discharge q would be produced in a uniform soil (no layers) of the n if it had same total thickness D -- D1 + D2 + . . . q- D,z with the same gradient Z, the equivalent conductivity Khor" q-

H KhorD--. L

(b)

From (a) and (b) it follows that:

ghor-

ZI'= 1 r i

T

2_.~-'i=1Di

D'

(43)

in which T - total transmissivity - K1 D1 + K2D2 -Jr-... + K,,D,,. If K - K(z) varies continuously with the vertical z, then Khor - - -D-

=o K ( z ) dz.

(44)

570

(1.3.3-3)

t

~°o = H

I

I

. . . . . .

Basic principles

¢po=H

i

K1D1 K2D2 iI iI

KiDi / iI

D

/i

I

Kn-lDn-1

i

K,,D,,

i I

•

A .

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

•

.

t

¢p,, - 0 Fig. 11.

Normal flow in a layered aquifer.

Secondly, we suppose uniform v e r t i c a l f l o w through the same layered aquifer caused by different heads at the top and at the bottom of the aquifer (see Fig. 11). In this case the specific discharge v is the same for all layers because of continuity. If the successive heads at the bottom of each particular layer are ~oi and qgi-1 -qgi Hi, then =

/4, H,, H1 = K 2 H 2 v -- K1 D----~ D2 -- Ki---~i -- Kn--~n [ L T - 1 ] . From this the heads Hi can be determined as functions of v: D1 H1 -- -z v,

Hi-

Di ~v Ki

-- r i D i v -- Riv,

with ri - resistivity and Ri -- ri Di is the resistance to vertical f l o w of a layer with conductivity Ki and thickness Di Oi

R,- ~

IT].

(45)

As

H -- H1 + H2 + ' "

+ H,z --

Hi -i=1

~v i=1

-- v

Ri, i=1

Parameters and variables

(1.3.3-3)

571

the vertical flux v can be represented by the total head difference H, divided by the some of the resistances of the different layers: H v =

~

f

.

'x

tcj

R,.

The same flux v would be transmitted in a uniform soil (no layers) of the same H total thickness D -- D1 4- D2 4---. 4- Dn and the same total hydraulic gradient ~if it had the equivalent conductivity Kvert: H V-

(d)

Kvert m .

D

(c) combined with (d) gives: Ein= 1 Di D ,~ = -Kvert -- Y-~i=l R i R

(46)

in which R = total resistance to vertical flow of the aquifer. If r = r(z) varies continuously with the vertical z, then 1

to-

=

Kvert

1 fD

D

dz

.

(47)

=0 K(z)

It may be shown that for any layered system where the conductivity Ki is not always the same everywhere Khor > Kvert-

The expressions for Khor and Kvert in the equations (43) and (46) are useful for giving a rough insight into the anisotropy of an aquifer if the soil samples of the several layers, obtained from a boring, have been analysed by means of sieve analysis (Section 1.3.2-1). The determinations of Kho,- and Kvert described here, have their analogies in the coupling of electrical resistances in parallel and in series, respectively. In problems concerning flow in leaky aquifers, the assumption is generally made that only horizontal flow takes place in the aquifers and only vertical flow in the semi-permeable layers. For the aquifers then, the transmissivity T -- K D is domiD' with D' - thickness nant and for the semi-permeable layers the resistance R - 2v, of the layer and K' the conductivity of the layer, usually denoted by the letter c: D

!

K' which is called the leakage factor [T] of the aquifer.

572

Basic principles

(1.3.4)

1.3.4. Coefficients of dispersion For an extensive discussion of dispersion and the dispersion coefficients the book by Bear "Dynamics of Fluids in Porous Media" is recommended. Here we start from the simplifying assumption that both the longitudinal and the transversal coefficient of dispersion are directly proportional to the magnitude of the real velocity of the groundwater, which assumption has been rendered acceptable by means of statistical considerations and laboratory tests. In Section 1.1.3 it has been shown that longitudinal dispersion differs from transversal dispersion; laboratory tests have demonstrated that the former (in the direction of the flow) is larger than the latter (perpendicular to the flow direction). The proportionality factor is called dispersivity (dimension [L]) and is denoted by the letter a

DL = aLu,

DT = aTu.

(48)

The two dispersivity parameters may be considered as characteristic lengths of the porous medium with regard to dispersion. Because of the difference occurring between the dispersivity in the direction of the flow and that perpendicular to it, the dispersion is always anisotropic. We assume that the coefficients of dispersion may be written as a dispersion tensor, like stresses in a body or conductivities in anisotropic ground. One of the main directions of the dispersion tensor in a fixed point of the flow field, coincides with the discharge vector v in that point; the other two are perpendicular to the flow direction and may be chosen arbitrarily, because the transversal dispersion is assumed the same in all directions perpendicular to the flow directions. In general, as a consequence of the variable velocity vector, the main directions of dispersion vary from point to point. If we suppose the main directions of the dispersion in a point along the mutual orthogonal directions u, v and w, in which v represents the flow direction in that point, then we may derive the transport equations for the dispersion in three arbitrary mutual orthogonal directions from the equations for the main directions. The procedure is almost similar to that of the anisotropic conductivity (see Section 1.3.3-2) with K replaced by D, ~0 by c and the directions s, n and m by u, v and w, respectively. According to (47) and (48) of Section 1.2.5, the following equations for the flux in the three main directions u, v and w hold:

Oc fu---Du~u,

fv-uc-D~-~v

Oc

Oc and

fw--Dw~

Ow'

(49)

in which Dv = DL - the longitudinal and Du = Dw = DT = the transversal dispersion coefficient. In index notation (cf. equation (35)) equation (49) may be written as Oc

fh -- UhC- Dh-;-;-.

017

(h -- u, v, w),

(50)

in which it should be kept in mind that uv = u and u. = uw = 0. The fluxes in idirection (i = x, y, z) can be composed from the fluxes in h-direction (h = u, v, w) as follows

Parameters and variables

(1.3.4)

573

f i - - f h COS Olhi

(51)

(compare formulas (32) of Section 1.3.3), for instance,

fx -- fu cos C~ux + fv cos oevx + fw cos Oewx,

etc.

The concentration gradients in h-direction may be expressed by means of those in j-direction (j - x, y, z)

Oc Oh

Oc

= ~ cos Oeh.i,

(52)

Oj

for instance,

Oc Ov

Oc -- ~

Ox

Oc COS Olvx -Jr- ~

Oy

Oc COS O/vv -~-

"

Oz

COS Olvz.

Combination of (50), (51) and (52) yields" OC ft" - - blh COSOlhiC -- D h 7-: COS Othi COS Othj.

oj

As Uu cos oeux + uv cos oevx + Uw cos oewx = u cos oevx = Ux, because uv = u and Uu -- uw -- 0, we may write shortly ui for Uh cos Othi. Analogously to the coefficients Kij in formula (36) of Section 1.3.3, we may write Dij for Dh cos Othi COSOthj. The transport equations of a solute in three arbitrary mutual orthogonal directions thus become: OC

fi -- u i c -

Dij-2= oj

(i, j -- x, y , z )

(53)

or written at length"

Oc UxC- DxxSx

Oc DxySy

Oc f "v -- uvc - Dyx -OX

Oc Dyy o-gy

fx

0c

ac

fz -- UzC - Dz~ -Ox

" oTy Dz.v

Oc O x z-~z'

Oc D Y ZOz --' ac

Dzz. . OzT-"

From Di.j -- Dh COSOfhi COSOfh.j it follows that Di.j - D.ji, the nine D-values in (53) thus are reduced to six: Dxx, Dyy, Dzz , Oxv, Dxz and Dy z. Together they constitute the dispersion tensor Di.j (i, j - x, y, z) or the symmetrical square matrix:

D--

Dxx Dxv

Dxv Dvv

Dxz ) Dvz •

(54)

574

(1.3.4)

Basic principles

As Dxx -- Du cos 20tux + Do cos 20tvx + Dw cos 2 C~wx, we find, if we introduce the dispersivity with Do - at.u and D . -- Dw - aTu" Dxx - a T u ( COS20tux + cos 20twx) + at.u cos 20tox

and as cos 2 aux + cos 20tox + cos 2 C~wx -- 1" Dxx -- aTu (1 -- cos 20tox) + aLU COS20tox.

Finally with Ux -- u cos Oeox" 2

Dxx -- aTu + (aL --aT)U---~x • u

Further w e have Dxy -- Du cos ~ux c o s Oluy At- Do cos ~vx c o s Olvy _4_ Dw cos Otwx c o s Olwy = aTu(COS Otux COS O/uy -'[- COS Otwx COS Otwy) -[- aLU COS Ofvx COS Otoy;

cos Otux, cos Oeox and cos C~wx are the c o m p o n e n t s of the unit vector along the x-axis in the directions u, v and w and also cos O~uy, cos C~oy and cos Otwy, the c o m p o n e n t s o f the unit vector along the y-axis in those directions. Thus, the scalar p r o d u c t of these vectors cos Otux cos Otuy + cos Otox cos Otoy + cos C~wxcos Oewy -- 0 b e c a u s e of the o r t h o g o n a l i t y of these vectors. T h e r e f o r e or with

D x y - - ( a L - - a T ) U COS Otvx COS Olvy ,

cosotvx=m

Ux u

uv

and

cosotvv=-="

u

UxUy Dxy = (aL -- a T ) ~

In a similar w a y we can derive expressions for the other dispersion coefficients; with u -- real velocity vector with c o m p o n e n t s Ux, Uy and Uz and u - lul, the six e x p r e s s i o n s for D become" 2

2

Dxx -- aTu + (at. -- aT)U--y-x,

Dyy -- aTu + (aL - - a T ) uv

u

u

2

UxUv

Dzz -- aTu + (at. -- aT)m,Uz

Dxv. -- (aL -- aT)

Dxz - - ( a L -

D y z -- ( a L -

u

aT) uxu----'-Zz -- Dzx, u

u

" = Dvx,

aT) uyuz u

-- D z y .

In index notation U i Uj D i j - - a T u S i j + (at. - a T ) ~

(i, j = x , y , z ) ,

~i.j -

1 0

(55) for i -- j , f o r / - ¢ j.

( K r o n e c k e r delta.)

(1.3.5-1)

Parameters and variables

575

Some special cases are: 1° u parallel to the X O Y plane: Uz --O, Dxz and Dyz vanish and Dzz : aTU. 2 ° u coincides with one of the axes, for instance, the x-axis: Ux = u and Uy = Uz = 0 gives: Dxx = aTu + (aL -- aV)U = aLU, Dyy = Dzz = aTu while Dii = 0 (i 5/= j). The coordinate axes are the main directions in this case.

1.3.5. Storage coefficients 1. Elastic and phreatic storage In general, a change of the piezometric head in an aquifer will result in a change of the amount of water that is stored in that aquifer. Lowering of the head diminishes the amount of stored water which means that water is withdrawn whereas an increase of the head results in a supply of water to the aquifer. So the conclusion is that an aquifer has a certain storage capability or, as it is shortly (but less correctly) called, storage. This concept of storage is principally different for a phreatic aquifer, as compared with a confined aquifer, as will be shown in the following. If in a confined or semi-confined aquifer the head is reduced, for instance, as a result of withdrawing groundwater by means of a pumping well or of flow to a reduced surface water level, also the water pressures will be diminished.

dp = 7 d~o,

(56)

which follows from (24) of Section 1.2.2: (the atmospheric pressure assumed to be constant) q)=z+-.

P ?,

Now in the first place the decrease of water pressure will lead to an expansion of the groundwater to the extent permitted by its elasticity, reducing its density according to (6) of Section 1.3.1

1

dp -- ~ p dp. /Zw Secondly, if it is assumed that the atmospheric pressure Pa and the total vertical ground stress Crg do not change, the solid stress Ors will increase according to dos = - d p (formula (24) of Section 1.3.2). The soil is compressed and the porosity reduces according to (26) of Section 1.3.2 dn=

1

~(1-n) Eg

dp.

These two processes lead to a change of the groundwater mass AMw in a REV with fixed dimensions Ax, Ay and Az: d(AMw) = d ( p n A x A y A z ) = A V ( p d n + n dp), which becomes

576

(1.3.5-1 )

Basic principles

( 1 - n) pn = { p Eg -Jr--~w } d P.

d(AMw)

AV

The amount of released water per unit volume of the ground is thus given by

(,-.

AV

=

Eg

+

dp

and from (56):

(1-n

d(A Vw) AV

= y

n_ff__)&p.

E g + Ew

If we put

1-n Ss- y

Eg

n) + ~w '

(57)

then d(A Vw) AV

= Ss d~.

(58)

The coefficient Ss is called the specific storage of the aquifer. It has the dimension [L -1] and can be defined as the volume of water which a unit volume of the aquifer releases from storage because of expansion of water and compression of the aquifer under a unit decline in the average head within the unit volume of the aquifer. If the head increases with a unit length, Ss is the volume of water which is stored in a unit volume of the ground as a result of compression of the groundwater and expansion of the solid matrix. For aquifers with mainly horizontal flow, the coefficient of specific storage Ss is usually replaced by the storage coefficient S which is defined as the volume of water that is released from (is stored in) a vertical column of an aquifer of unit crosssectional area and length, equal to the thickness D of the aquifer, if the average head within this column decreases (increases) by a unit length. S is dimensionless [0]. For a confined aquifer it is usually assumed that

S-

DSs,

(59)

with D = thickness of the aquifer [L]. This is not quite correct as there is a small variation of D with time because of the compression (expansion) of the aquifer. In practice, however, the variation of D is of the same order as the error made in measuring or estimating the thickness of the aquifer. In an aquifer with phreatic water, also called a water-table aquifer, the volume of water released from or taken into storage, in response to a change in head, is mostly due to dewatering or refilling of the zone through which the water table moves. The volume of water released from or taken into storage in a water column with cross-section AA -- A x A y amounts to:

d V -- A Ane dqg,

Parameters and variables

(1.3.5-2)

577

in which ne = effective porosity (see Section 1.3.2-2). As the length of the column here equals d~o, the phreatic storage coefficient Sph, also called the specific yield, becomes, according to the definition for the storage coefficient, equal to the effective porosity: Sph -- ne [0].

(60)

The elastic storage coefficient in phreatic water is given by S = Ssh(t), in which h(t) represents the height of the water table above the base of the aquifer, which is a function of time. The elastic storage coefficient in phreatic water is, however, many times smaller than the effective porosity and can be neglected.

2. Barometric sensitivity In a confined or semi-confined aquifer, the head fluctuates with changes in the atmospheric pressure. This can be shown by using the following arguments. Equilibrium of vertical stresses in a point of a confined aquifer gives (Section 1.2.2, equation (26)) ag = P t - - P a + a s . Variable atmospheric pressure Pa and constant total ground stress ag yields dpa = dpt + do's,

(a)

or in words: increase of atmospheric pressure is partly compensated by an increase of the total water pressure and partly by an increase of the grain stress. So dpt < dpa and as dp -- d p t - dpa the water pressure dp becomes negative. As dp = y d~0 it follows that rise of the barometric height is accompanied by a decrease of the piezometric head and vice versa, in confined or semi-confined aquifers. In a phreatic aquifer, changes in the atmospheric pressure will be totally compensated by the total water pressure; dpa = dpt (dots = 0) so that dp = 0: the water table remains unchanged. The ratio of the change in head of a confined aquifer to the corresponding change of atmospheric pressure, expressed in lengths of water columns is called the barometric sensitivity Bs: Bs =

d~p dp___~a=

y dcp dpa [0].

(b)

y

The minus sign means that increase of Pa gives a decrease of qg. In the case of variable atmospheric pressure, equation (56) becomes d p t - dpa -- y d~0 and according to (a) y d~0 -- - d a s whence do" s

Bs = dpt + das"

(c)

578

(1.3.5-2)

Basic principles

The mass of groundwater AMw in a fixed volume element A V does not change, because no water is discharged from or recharged into it. Hence d(AMw) - A V ( p d n + n dp) - 0, 1

(d) 1

dn - ge--(1 - n ) ( d p t - dpa) - -E--(lg 1 dpt so that ggw np d p t dp = ~-w-wP

dpt

1 -

das

n

n

-- /'l)do's,

p(1-n) do's, from which Eg

Ew Eg

From (c) it follows that tl

Ew Bs

---

i'l

Ew t

1-n Eg

or with (51): nF Bs

--

E--w _

-- Ss --

dq) dpa"

y

(61)

3. Tidal and phreatic sensitivity If a completely confined aquifer is situated under open water whose level is subject to tidal fluctuations, the head of the groundwater in this aquifer will be influenced by the variations in level of the surface water, because of pressure propagation. The tidal sensitivity Ts is defined as the ratio of the change of the head in the aquifer to the corresponding change of the level h of the overlying surface water

dq9 Ts = d---h"

(a)

The tidal fluctuations in the open water can be considered as a fluctuating live load with regard to the underlying aquifer; the instantaneous equilibrium at a point in the aquifer becomes ?' h + Og - p --[-as, and as ag is supposed to remain unchanged: g dh - dp + das. With dp = g dcp, (a) becomes"

Ts=

dp y dh'

(b)

(1.3.5-3)

Parameters and variables

579

and with (b): dp Ts -- d p + dtrs"

(c)

The mass of groundwater AMw in a fixed volume element remains unchanged, because no water is discharged or recharged, so that d,,s d-e--P--- --;7, 1-n ew E---~-'as in the case of barometric efficiency. Then (c) becomes:

TS

1-n Eg n 1-n Ew t Eg

or with (57)"

Ts --

Ss

=

dh

.

(62)

If, for instance, the tide may be described by h(t) - ho sin(cot), then the fluctuations in the aquifer become qg(t) -- Tsho sin(cot).

Although the tidal sensitivity of an aquifer is so called after the tidal fluctuations of open water levels, it should be remarked that any movement of the open water level will influence the heads in the underlying aquifer in like manner. It is of interest to observe that comparison of (61) with (62) yields: Bs + Ts -- 1.

(63)

From this relation and the definition of Ss (equation (57)) it can be concluded that the percentage of storage attributable to expansion of water is equal to Bs and that attributable to compression of the aquifer is equal to Ts. If a completely confined aquifer is situated beneath an aquifer with a water table (phreatic aquifer), a change of the water table will be accompanied by a change of the piezometric head in the underlying aquifer as a consequence of pressure propagation. The phreatic sensitivity Ps is defined as the ratio of the change of the head in the confined aquifer to the corresponding change of the water table hph in the overlying phreatic aquifer d~o Ps = dhph

580

(1.3.6)

Basic principles

A rise of the phreatic surface, for instance, one caused by precipitation, can be considered as an increase of the live load acting upon the confined aquifer, which equals the increase of the weight of a water table column neyhph. The instantaneous equilibrium at a point in the confined aquifer gives: ne y hph ÷ Crg = p + Ors

and with Crg = constant, ney dhph - dp + dos. In a similar way as has been done for the tidal sensitivity, we may obtain a relation between the phreatic sensitivity and some ground and groundwater constants:

ney( 1-n Ps =

Ss

=

dhph

.

(64)

If we put ne = 1, we obtain formula (62) for open water, as should be expected.

1.3.6. Field tests (general description) Numerical values for the ground parameters are indispensable tools for solving geohydrological problems. Principally there are two methods of determining these values: laboratory tests and field tests. Although various laboratory techniques are available, the numerical values obtained therefrom are not fully reliable for hydrological purposes, due to the fact that the soil samples which are to be tested are not subject to the same conditions as those in the field. In the following, only field experiments for determination of ground parameters will be discussed. Field tests, in general, comprise the measuring of state variables, such as heads, water tables, velocities, discharges and concentrations of solutes at several locations in the field and at different times. These measurements may be performed on "natural" groundwater flow or more usually on flow artificially evoked by means of abstraction of groundwater via pumping wells, the so-called pumping tests. Generally, pumping tests are more suitable for determinations of ground parameters than the measuring of existing groundwater flow, because pumping affords more pronounced variations in heads and velocities than natural flow, which leads to relatively smaller errors in measurement. On the other hand, in many cases it is very important to measure and analyse the existing flow regime in order to eliminate its influence on the observed data during the pumping test. Essential for the determination of ground parameters by means of field tests is that the natural or artificial flow pattern is described by a proper analytical formula, mostly in the form of a head or drawdown as a function of space and time and of the ground parameters, the latter occurring in the formula as unknowns. The analysis, in short, of the data, collected during field tests, consists of comparing these data with the theoretical values, derived from the analytical solution, by attaching numerical values to the ground parameters. Those numerical values that give the best fit of the measured state variables with the theoretical ones, are the required numerical values.

Parameters and variables

(1.3.6)

581

This comparison usually is done by means of the method of curve fitting, in which the measured values, collected in the data curve in a specific way, are compared with the theoretical curve, the so-called type curve. Nowadays computer programs for analysing field test data are also available; such applications may be summarized under the name of digital curve fitting, although however, graphical curve fitting gives a more allover view of the deviations of the determined numerical values of the parameters with respect to the real values. The method of analysis suitable for any particular problem depends, among other factors, on the type of flow, the geometry of the flow system, the range of time within which the observed data fall, and the distribution in time and distance of the collected data. For a detailed study of pumping test analyses the reader should consult existing textbooks. Here we shall limit ourselves to giving some guidelines at the end of this section. Concerning the practical performance, the pumping tests may be divided into two main groups: the multi-filter tests and the single-filter tests. To the multifilter tests belong the classical pumping tests, in which, in general, one pumping well is installed with a filter that fully penetrates the aquifer (or aquifers) that is (are) to be tested, while a large number of observation wells are placed in all directions at varying distances from the well to be discharged. During the period that water is abstracted from the well with a constant discharge, the water levels in the observation wells are observed frequently and compared with their initial values, thus measuring the drawdowns. If a steady state is possible, the pumping test will go on until this state is approximately reached; after the shutdown of the well, a recovery test will usually be made in which no pumping takes place and the rises of level in the various observation wells are measured. The classical pumping tests are based on merely horizontal flow in the aquifers and merely vertical flow through the semi-permeable layers, thus giving information about the transmissivity and the storage capability of the aquifers and about the resistance of the semi-permeable layers, all averaged over the area that is influenced by the pumping. However, as soon as vertical flow plays a role in any geohydrological problem, for instance, upconing of salt or brackish water into a fresh-water region, the eventual anisotropy of the aquifer has to be taken into account and the tests should be adapted accordingly. This means that the tests must be designed in such a way that besides horizontal flow also a distinct vertical flow component is introduced during the pumping, usually obtained by making use of partially penetrating well filters. In order to obtain data concerning the vertical flow in such special pumping tests, the installation of several observation wells in a vertical line is necessary, the differences in head of which may give an impression of the vertical conductivity. If the average numerical values of the ground constants that can be obtained by means of classical pumping tests are not sufficient because of distinct heterogeneities of the aquifer, then more test have to be performed in the same region. In that case, local values of the parameters may be required in a more simple and less expensive way by means of single-filter tests, in which short tests are carried out on a single well, without additional observation wells. The single-well tests can be divided into

582

(1.3.6)

Basic principles

tests on line wells and tests on point wells. Line well tests are applied to existing pumping wells, which generally are equipped with more or less fully penetrating filters, and therefore may give information about the horizontal conductivity only. Point well tests make use of a very small filter with length about the same as the filter diameter, thus causing approximately three dimensional spherical flow. These tests are applied during the boring of a well in order to get information with regard to the variation of the conductivity in verticals. In analysing single-well tests the resistance of the filter, the skin effect, should always be taken into account. During pumping tests, especially tests with a relatively long duration (days or weeks) external influences may disturb the measured data series and even disturb the whole test if it were not possible to eliminate them. The most frequently occurring external influences come from tidal fluctuations of the heads, barometric fluctuations and seasonal fluctuations, caused by precipitation and evaporation, all three naturally influencing the heads and from artificial hydrological activities such as groundwater abstractions or recharges for drinking water purposes and others. As the analysis of pumping test data is fully dependent on that part of the measured values that is influenced only by the pumping, it is obvious that methods must be found to eliminate other influences. Thus, a good knowledge of the existing nonsteady flow regime is essential. This knowledge can be obtained by performing field tests before the start of the pumping test, comprising tidal, barometric and other observations in each of the observation wells, installed for the pumping test, trying to correlate the measured values with those of a particular observation well, situated outside the zone of influence of the pumping well. The data, gathered from this control well, during the pumping test may then serve, by means of the earlier found correlations, to determine the external influences in the observation wells in use during the test and to eliminate them. A number of guidelines that may be useful for the design and analysis of a pumping test will be given in the following. These guidelines will be coupled with the problem in Fig. 12 in which a thick leaky aquifer has to be tested in order to find the ground variables. 1. The aquifer may be anisotropic; so a partially penetrating pumping well and observation wells at different heights in the aquifer must be used, for instance, as shown in Fig. 12. 2. Find out if external influences may disturb the test and if so develop methods to eliminate them. 3. If a steady state is possible, try to find an analytical formula for the drawdown as a function of r and z because of pumping with a constant discharge Q. (See part A of this book: solutions.) 4. Try also to find an analytical formula for the non-steady state, certainly in the case that no steady state can be reached (for instance if c is infinitely large). But even if a steady state is possible, a non-steady solution is important because most observations will take place in the non-stationary phase.

(1.3.6)

Parameters and variables

583

~

Q

U//~/////////////////~/////////~

~llllllll/lllllll/llll, J

'i!

s

:8 Ss

D

11

12

Xh

:9

21 31

1

N88888983~ Z

Fig. 12.

Situation of pumping well and observation wells.

5. Begin the analysis with the steady state, if present, because one ground parameter, the storage coefficient S or Ss is absent and the formula is simpler. A disadvantage is that it may take a very long time before a steady state can be reached (several weeks). 6. The formulas will mostly be strongly simplified by choosing special values for the independent variables; besides t -- e<~, also r -- 0 and z -- 0 give simpler results and thus easier analysis of the data. For this reason try to place a number of observation filters in the axis of the pumping filter (r - 0) and also directly under the semi-permeable layer (z = 0) (see Fig. 12). 7. If for a certain non-steady problem no exact solution can be found because of a difficult boundary value (phreatic surface or a sharp interface between fresh and salt water), then replace the flow system by a simpler one of which the solution is known and limit the analysis of the observations to those filters which do not lie in the neighbourhood of the boundary. Consider also the first part of the time-drawdown curves, until a certain time criterion before which the boundary under consideration has not yet or hardly influenced the drawdowns. 8. Rewrite the analytical solutions in such a way, that the different constants and variables in the formulas are grouped in dimensionless parameters and construct so-called type curves on logarithmic scale. Draw the measured data curves on the same scale and determine, by means of curve fitting, the required values of the ground parameters.

584

Basic principles

(1.4.1)

1.4. C O N T I N U I T Y EQUATIONS AND D I F F E R E N T I A L EQUATIONS

1.4.1. General continuity equation (without dispersion) A fundamental law in physics is the Law of Mass Conservation. This law, applied to flow of a fluid through a porous medium means that the mass flux of the fluid which enters or leaves a fixed volume element must be equal to the increase or decrease of the amount of mass in the unit volume per unit time. The volume element is called the control volume and has an invariable fixed form and position in the flow field although the quantity and the character of matter within the element may vary with time. This is termed the method of Euler, in which the behaviour of the fluid particles is studied at fixed points in the flow field, unlike the method of Lagrange, in which the particles are followed on their ways along the stream tubes (-lines).

AA~xA::

pv ~

O(pv~)_

pry + Oy zxy .

Ay

x/ I

The control volume may be a REV of arbitrary shape; here is chosen for a small box with sides Ax, Ay and Az parallel to the coordinate axes x, y and z, respectively (see Fig. 1). The following holds for inhomogeneous groundwater flowing through heterogeneous ground.

Fig. 1. Control volume.

At the left side of the box, groundwater enters the box in the y-direction with a specific discharge Vy and a density p. During a time interval At, this yields a quantity of water mass: pvvAxAzAt. At the right side, a quantity

{pry + ~I(pvy) A y } A x A z A t is leaving the box. In total, the mass of groundwaely ter that enters the box in the y-direction will be ~(p~y~ ~)y A x A y A z A t In the x- and z-directions, we find in a similar way the amounts of mass of groundwater entering the box, o(p~X)AxAyAzAt and ~(pOZ)AxAyAzAt respectively. The change of the mass of the groundwater AMw in AV is ~(~tMw)At; AMw -- npAxAyAz with n = porosity. This mass change must, according to the law of mass conservation, be equal to the total mass that enters the control volume. So the equation for mass conservation- also called the continuity equation- becomes:

O(pVx) + Ox

O(pvy)4 o(pVz) ~ O(pn) = 0

Oy Oz O(pn) div(pv) + = 0. Ot

Ot

or

(1)

Differential equations

( 1.4.2-1 )

585

The second equation is in vector notation with div = divergence. In the most general case, Vx, Vy, vz, ,o and n are functions of x, y, z and t. Equation (1) then can be developed to:

Ovv Ovz] Op Op Op + + V X -~x + V V -~y + V z --O z Ox Ty Oz / Op On _ 0 + + a-7 +

(2)

or

Op OH p d i v v + v. Vp + n - ~ + p r o = 0 . at

In the foregoing derivation account has not been taken of the mass conservation of an eventual solute in the groundwater with the accompanying phenomenon of dispersion. This will be discussed in Section 1.4.5.

1.4.2. Differential equations for homogeneous water without dispersion 1. Non-steady flow of compressible groundwater through a compressible porous medium As already has been noted in Section 1.1.4, the general continuity equation which has been derived in the previous section does not contain information regarding the properties of the soil and the groundwater and as such is insufficient to yield solutions of groundwater problems. Combination of the continuity equation with equations of motion, such as Darcy's law or Fick's law, yield differential equations for the state variables, such as pressure, head and solute concentration, in which water and soil properties are incorporated. These differential equations describe the groundwater flow and the transport of solutes in such a way that with the help of initial and boundary conditions the state variables may be solved as functions of the coordinates. In this section the dispersion will be left out of consideration and will be postponed until Section 1.4.5. Here homogeneous groundwater will be dealt with which means that the density p does not vary owing to concentration variations, but only owing to pressure variations. Combination of the continuity equation (2) of Section 1.4.1 with Darcy's law in the form of (33) of Section 1.2.3 gives

0

kxOp

0

kvOp

0

kzOp

0

kz~

kzOp lz Oz

kx OP OP I~ Ox Ox The

derivative,~

nf

r~ a n d

kv Op Op I~ Oy Oy n

with

kz ( OP ) OP OP On # ~ + p g -~z + n -~ + p Ot

rocn~et

tn

r

~

7

~nr]

t

con

ho

o.

ovnroccor]

;n

tho

586

Basic principles

(1.4.2-1 )

3p p 3p = , etc. 3x Ew 3x

On 1 - n Op = at Eg Ot

and

The differential equation thus, after dividing by p, becomes:

3 kx 3p -Jr-kx

3 kv Op

1 (OP) 2

3 (k~Op) O (kz) kz 1 3p + -~z -~z + Pg -~z 7 + Pg lz Uw aZ

kv 1 (OP) 2

Ix-~w -~x + -~ -ETw -~y 1--n n )3p -

kz 1 (OP) 2 + --~ -ETw -~z

kz 1 0 P + p g lZ ew Oz

(3)

57

This is the general differential equation for the water pressure p for non-steady flow of homogeneous, compressible groundwater through compressible, heterogeneous and anisotropic ground. The main directions of the anisotropy are supposed to coincide with the directions of the coordinate axes; otherwise, the complex form of Darcy's law of (37) in Section 1.3.3 must be applied in the derivation of the differential equation. If the ground is homogeneous and anisotropic, then kx, ky and kz become constants and can be put before the differential characters in (3) and the term pg~( ) vanishes if/z is constant. In the case of homogeneous and isotropic ground and constant viscosity lz of the groundwater, equation (3) becomes:

k ( 0 2p + 0 2p + 0 2p) + k 1 { (

2

k 1 3p 3p + 2pg - I = S~-57 tx Ew 3z

Op 2

3p)2 (4)

or

_k7 2

IX

kl klOp 3p (V p)2 -l- 2pg = s; 7 ; lZ ~w lZ Ew Oz

P -1- - -

with S~-- 1-nEg -[ /~w'n

k 1 3p in (4) and the terms ~k w1 (Vp) 2 are very Generally the gravity term 2pgT-Ugw~ 1 . neglecting these small, compared with iuV 2 p ' especially owing to the factor E-Tw' terms, there remains:

32p

32p

02p

/~(1--n n ) 3p a x--Y + -jy 2 + ---y a z = -; e + 57

(5)

It should be kept in mind that in the formulas (3), (4) and (5) the density p and the n porosity n are functions of p and so also of x, y, z and t. So ~1-n + 2-gw - Sst is not

(1.4.2-2)

Differential equations

587

a constant strictly speaking. For practical computations, however, an average value is taken for n, for instance, no -- porosity at p = 0 (atmospheric pressure), and S~ is considered a constant. As has been discussed in Section 1.2.3 it has little sense to replace p in the differential equations, shown up till now by some kind of head or by Hubbert's force potential.

2. Non-steady flow of incompressible groundwater in a compressible porous medium If we compare the given values for the moduli of elasticity for soil and water Eg and Ew with each other (Ew - 2 x 10 9 Pa and Eg - 106-10 8 Pa) it appears that the modulus of elasticity of unconsolidated grounds like sand and clay is much smaller than that of water (by more than 50 times). For unconsolidated grounds the compressibility of water may be neglected and the density of the groundwater is supposed to be a constant. In the continuity equation (2) of Section 1.4.1, the terms v. V p and n ~ disappear and the equation is reduced to: div v ~

OH at

- 0.

(6)

The density p is constant and so Darcy's law according to (31) of Section 1.2.3 can be applied to equation (6) 099

Vx---Kx-~x,

vv---Kv--

O99

Oy

and

0~0

Vz--Kz-07,

while

On 1 - n Op 1 - n 099 = = pg-Eg Ot Eg Ot Ot which yields the general differential equation for the piezometric head cp for nonsteady flow of homogeneous incompressible groundwater through compressible, heterogeneous and anisotropic ground:

O (K~O~°

O (KzO~°

-- Ss~

Ot'

(7)

with Ss - specific storage - pg(1-n) e-----7---, a reduced form of the specific storage of (57) of Section 1.3.5 (without ~w"). If the main directions of the anisotropy are not parallel with the coordinate axes, the equation (7) becomes (in index notation, see (35) and (36) in Section 1.3.3-2):

a--Tz Ki.j

- Ss-~

(i, j - x, y, z).

(8)

If the ground is homogeneous and anisotropic, then (7) becomes"

02(/9

02(/9

Kx ~-5-x2 + K v ~ •

02

099

Oy2 + KzOZ ~p 2 = S s ~Or'

(9)

588

(1.4.2-3)

Basic principles

and for homogeneous, isotropic ground: 02q9

02q9

a---5 +

02(t9 +

OZ2

Ss 099 =

.

K at

(lo)

In all these equations q9 - ~ + z. The differential equations (9) and (10) serve as the starting point for the greater number of the analytical solutions for geohydrological problems, while (7), which contains variable conductivity coefficients, is the basis for various numerical solutions. If the modulus of elasticity of water does not differ much from that of the soil, which case may occur in semi-consolidated soils, it is not allowable to neglect Ew compared to Eg in the expression for S~ in equation (4). However, the terms with Ew in the left-hand side of equation (4) are mostly in fact neglected, giving formula (5), from which for homogeneous and isotropic ground and homogeneous compressible groundwater, equation (10) can be derived, provided that Ss - (~x-n + g_~w)pg .n

3. Flow through an incompressible porous medium. Steady flow In hardly compressible or incompressible soils, such as consolidated soils like sand1 can be neglected. The equations (3) and (4) remain the same, provided stone, ~g that S ~ - ggw' whereas equation (5) becomes" VZp

.

.

lz n Op . . . k Ew Ot

(11)

These differential equations are in use in the oil industry; the ground is considered incompressible but the fluid compressible. For steady flow, the terms with ~ap or ~a~o vanish everywhere in the differential equations as a result of the fact that both the density p and the porosity n are supposed to be constant, so that the term ~a(pn) in the general continuity equation for homogeneous groundwater becomes zero. As soon as both the ground and the groundwater are assumed to be incompressible and the groundwater homogeneous, then a steady state may be reached, which means that the groundwater flow will be

independent of time. The steady state is always the end phase of a non-steady state (theoretically after an infinitely long time), in which ground and groundwater are no longer compressed or decompressed. On the other hand, not each non-steady flow will become a steady one after an infinitely long time; it depends especially on the possibility of receiving water from or supplying water to the direct surroundings of the flow field. The continuity equation for steady flow becomes (cf. equation (1)): divv = 0.

(12)

The differential equations for steady flow can be obtained by putting ~at - 0 in the equations (7), (8) and (9). For heterogeneous anisotropic ground (from equation (8)) ~

a gi.i ai

--0

(i,j

~

x y,z)

'

o

(13)

Differential equations

( 1.4.2-3)

589

For homogeneous anisotropic ground (from equation (13))

0299 -- 0

Kij Oi Oj

(i j -- x, y, z).

(14)

With the main directions parallel to the coordinate axes:

02(t9 2 +

02(t9 • Oy2

02(t9 OZ 2

= 0

(15)

and finally for homogeneous, isotropic ground the well-known differential equation

of Laplace: 0299 -~- 02~0 -+- 02(/9 -- 0 OX 2 Oy 2 OZ2

(16)

or div(V~0) = 0, or V2~o = 0. Equation (16) does not contain the conductivity of the ground, which means that if the conductivity K also is absent in the boundary values, the variation of the piezometric head is only governed by geometrical conditions and is independent of the soil type.

4. Differential equations in other coordinate systems All differential equations dealt with up to now were related to Cartesian coordinates x, y and z. Transformation to other coordinate systems takes place by means of transformation formulas which may be applied to all the foregoing differential equations; for convenience, we shall restrict ourselves to transformation of equation (10):

02~0 02(/9 0299 __ Ss 099 Ox-"-'~-{- 7 7 2 -Jr- 0Z 2 -- K Ot

Spherical coordinates r, 0 and r/ are related to

Z

Cartesian coordinates x, y and z, as follows (Fig. 2): P (r, O, 7"/) x = r cos 0 sin rl,

y = r sin 0 sin rl,

z = r cos r/ and inversely:

r - v/x2 + y 2 +z2 CX 2 + y2

Fig. 2. Spherical coordinates. r / - arctan (

0 - arctan( )

)"

590

(1.4.2-4)

Basic principles

By applying the chain rule for differentiating repeatedly, for instance, ~ax - - - - O~O;O"x' jr O-I'r a0 aO ax ' etc. we can transform equation (10) to (see mathematical textbooks): 1 { 0 (099) ~rr r2 -~r

--g r

1 0 (Oqg) sin/7 -t sin 77 0rl ~

q

1 02 } q9 __ f12 Op sin2r/00 2 07

(17)

with

f12 __ Ss. K If the boundary values in a problem are independent of one or more of the independent variables, then the terms in the differential equation that are related to those variables may be omitted. If, for instance, the problem is s p h e r i c a l - s y m m e t r i c , then the head q9 is independent of 0 and 77 and the terms with 0 and r/ in (17) can be deleted. The differential equation reduces to:

1{0

r.~ _~r ( r 2

O~o) }

_

f12

099

(18)

__

at

and for the steady state: 02q9 Or

-I-

2 099 r Or

= 0.

(19)

Other important coordinates are c y l i n d r i c a l coordinates (Fig. 3): x - r cos0,

P (r, 0, z)

y - r sin0,

ZmZ

and inversely" r -- J x 2 Z ~

-k- y 2,

0 - arctan(Y),

Z.

Fig. 3. Cylindrical coordinates. The differential equation (10) becomes: 0299

1 099

1 0299

0299

Or---T + -r -~r -+ r 2 00 2 ~- ~

f12 0(/9 --

~Ot .

(20)

For a x i a l - s y m m e t r i c problems the term with 0 vanishes and the differential equation becomes:

0299

1 099

0299

Or 2 at- -r -~r -4- ~OZ2 --

f12 0(/9 ~at.

(21)

Differential equations

(1.4.2-5)

591

If the flow is two-dimensional, the flow has components in only two directions and the derivations of the dependent variable with respect to the third direction can be omitted. If the latter is the z-direction, the cylindrical coordinates are replaced by the well-known polar coordinates with x - r cos 0,

r -- V/X2 @ y2,

y - - r s i n O,

0 - arctan(Y). x

The equations (20) and (21) become respectively: __ f12 099 02q9 ] - - -1 099 -'[. . 1 . 0299 . ---~ Or - r -~r r 2 O02 Ot'

(22)

0299 -F 1 0q) -_ f12 ~099 .

(23)

Or2

r Or

at

The last equation represents the differential equation for radial-symmetric cases.

5. Differential equations for the stream function 7r In stead of describing groundwater flow problems by means of differential equations for the dependent variables p or qg, in certain cases it is useful to describe a steady flow problem by means of the stream function 7r, as 7r can also be related (like 99) to the groundwater velocities (see, e.g., the formulas (15) and ( 1 8 ) o f Section 1.2.1). In this paragraph we shall restrict ourselves to the derivation of differential equations for the stream function for permanent two-dimensional flow of homogeneous groundwater and for axial-symmetric steady flow. Stream functions related to inhomogeneous groundwater are treated in Section 1.4.4 under rotational flow. The relations between the stream function and the potential function in twodimensional flow are according to equation (37) of Section 1.2.4

a4~ aT~ = Ox Oy

and

04, = Oy

a~v Ox

for homogeneous isotropic ground function, while grad 4~ or Vq~ is a vector analysis it is known that if function, its curl is the zero vector. OUy

0l) x

Ox

Oy

(~b = K~0 with K constant). ~b is a scalar vector which equals - v (Darcy's law). From a vector function is the gradient of a scalar So curl v = 0 means curl(V~b) = 0 or:

= 0,

which also follows immediately from v~ Oq2q~

axay

(24) a~ Ox and v v -

,J4~ as

i~v

Oq2q~

=

ayax

if it is assumed that the second derivatives of ~0 exist and are continuous.

(25)

592

(1.4.2-5)

Basic principles

Combination of Vx - _ a_q~_ Ov and Vy - ~Bx with equation (24) gives the differential equation for ~k for homogeneous and isotropic ground a2~

021//t

OX 2

Oy 2

= 0

(26)

or V21/t - - 0 (Laplace differential equation). If the ground is heterogeneous and isotropic, then K - K(x, y, z) is not a constant and Darcy's law takes the form v - - K ( x , y, z)Vq9 or Vq) - - ~ v which means that 399

3x

Vx

=

3q)

and

K

3y

=

vv

K

and from equation (25): 1) x

m

0

1)v

0

10~

Oy and thus" 0

10~

If the ground is anisotropic Darcy's law becomes vx - - K x ~Bx and 1)y - - - K y a-~#y ' if the main directions of the anisotropy are parallel to the coordinate axes"

399 Ox

Vx Kx'

3q) Oy

vv Ky

and again with equation (25): 0

Vv

o-;(-

0

-

Vx

(-

a~, and 1)y -- ~ax, we find the differential equation for ~p for and with Vx = -;57 heterogeneous and anisotropic ground (main directions parallel to the coordinate axes)"

a

1 a7,

a

la7,

If the ground is homogeneous and anisotropic, Kx and Kv are constants and the differential equation becomes: 021/f

Kx ~ x 2 + Kv.

021/f Oy2

= 0•

(29)

Also for the general case of anisotropy in which the main directions of the anisotropy do not coincide with the x- and y-axis, a differential equation may be derived for

(1.4.2-5)

Differential equations

593

the stream function. In this case Darcy's law for two-dimensional flow follows from the equations (34) of Section 1.3.3 Vx -- - K x x

Kxv~, " Oy

Ox

O~o vv -- - Kxv -Ox . . . .

fix

(3o)

099

Kvv

;

Oy

~y solved from these equations gives"

and

~

and

099 = Kxyuy - - Kyyvx Ox Kxx Kyy - K2y

at#

Kxy Vx - Kxx Vy

Oy

K.

-

K2

"

Again with 02(t9

=

OxOy

02(t9

Ool/f

and

OyOx

Vx =

and vv =

Oy

"

a~v Ox

the differential equation for this case becomes: 'Tx + K x v ,,Ty Kxx Kyy - K2v

Ox

a~ a~ • " Kxx Kyy - K2y

+ ~

- O.

(31)

For h o m o g e n e o u s anisotropic ground, all K-values become constants and we get:

021//`

021/f

K x x - ~ x 2 + 2Kxv "

OxOy

021//" + Kvv

" Oy 2

= 0

(32) "

It should be noted that if Kxv - O, the conductivity coefficients Kxx and Kyy become the coefficients Kx and Ky in the main directions x and y respectively, which means that equation (28) may be derived directly from equation (31) and equation (29) from equation (32) by putting Kxy = O, Kxx = Kx and Kyy = Kv. Also, equation (27) can be obtained from equation (28) by putting K -- Kx -- Ky (switching from anisotropic to isotropic ground) and by putting K = const, in equation (27), we obtain equation (26) (homogeneous ground as a particular case of heterogeneous ground). For a x i a l - s y m m e t r i c f l o w through homogeneous, anisotropic ground, Darcy's law can be written as: 099 Vr -- - - K r

07

0~0 and

vz -

-Kz

Oz

if it is supposed that the main directions of the anisotropy are horizontal ( K , Ky - K r ) and vertical (Kz). The discharge vector v is also related to the stream function (cf. equation (18) of Section 1.2.1)"

1 0~v I)r

~-

2Jrr Oz

and

Vz =

1 a~ 2rrr Or

594

(1.4.2-5)

Basic principles

As O 099

02(t9

O Oq)

(Tz)-

Or

0

r (_

OZ

a ( l a O +) 1

1027r

Kz

r Or2

Kr 02~ Or 2

~Z.)

O(1

O---r Kzr Or

or, introducing the stream function gr:

3~)

-~z Kzr 3z

1 Oft ) +

- 0

1 102~r Kr r Oz2

r 2 Or

)

1 OTr + Kz r Or

it follows that

OrOz

021P

from which = 0

and so:

= O.

(33)

" 3z 2

For isotropic homogeneous ground the differential equation becomes:

021p

10l~r

Or2

r Or

k

021If = O. 3Z 2

(34)

6. Injection term in the differential equation The general continuity equation (1) in Section 1.4.1 has been obtained by applying the law of conservation of mass to flow which enters and leaves a control volume. The total mass of water that entered the control volume was put equal to this increase of mass in the control volume. If also during the considered time interval At an amount of water F per volume unit A V should have been generated within the control volume, the continuity equation would have become: O(pn) - d i v ( p v ) + p F -- ~ . Ot

(35)

In general, F will be a function of x, y, z and t: F = F(x, y, z, t) and it has the dimension [T-l]; F is positive in the case of injection and negative in the case of abstraction. The injection term F will occur in all the differential equations that can be derived from equation (35), for instance, after adding a term with F equation (10) becomes: 0299 ]-- ~0299q-- 02(/9 _ __F _-- Ss 0q) . -Ox - - s - Oy 2 ~z 2 -} K K Ot

(36)

This differential equation (and simplifications of it for less than three dimensions or for steady flow) is frequently used for describing problems concerning so-called semi-confined or leaky aquifers. Although the exchange of water through a semipermeable layer principally has a vertical character and thus the leakage should have to be presented as a boundary condition at the top of the aquifer, it is assumed that this leakage is hypothetically generated in the aquifer, uniformly distributed over the

Differential equations

(1.4.2-6)

595

thickness D thereof. In this way, the vertical component of the flow is replaced by an injection term, while the groundwater movement is assumed to be horizontal but now in a confined aquifer. The intention of this device is obvious: the dimensions of the problem are reduced by one, thus simplifying the solution considerably, while this idealization of the flow system does not affect the flow pattern in the aquifer inadmissably, since the main aquifer is much more transmissive than the semipermeable layer. If the head above the semi-permeable layer is ~00 = qg0(x, y, z, t), the leakage may be simplified by ~°°c~° and uniformly distributed over the thickness D, it becomes: F -- ~°0-~° with c - leakage factor (Section 1 3.3-3). In polder Dc ' areas 9o is generally taken as zero. Then equation (36) is reduced to

02(/9

02@

(/9

Ox 2 ~ Oy2

Ss 0(/9

K D c = K Ot .

(37)

1.4.3. Differential equations for non-homogeneous groundwater, without dispersion. Density flow If the density p of the groundwater is dependent in addition to pressure also on the concentration of a solute therein, which is variable with space and time, then we have to do with inhomogeneous groundwater and the flow is denoted as density flow. So if c = c(x, y, z, t) then also p = p(x, y, z, t). In general, the density of compressible water will then be a function both of pressure and of solute concentration: p = p (p, c) (variation with temperature neglected; see Section 1.3.1-1). The gradient of p: Vp may then be written as: Vp -

m

Op

ap Vp + ~ Vc. Op Oc

(38)

In the following the density p is assumed to be dependent only on the concentration c and no longer on p, though the derivation of the differential equations will take place in a similar way as in Section 1.4.2, in the case that p = p (p, c), by replacing Vp in equation (2) of Section 1.4.1 by the more complicated expression (38). As the water is now supposed to be incompressible, an increase of the solute concentration in the control volume only results in a change of the density and not of the volume of the groundwater. Therefore, it is allowable to apply the continuity equation for the volume, which means: div v J On _ 0 , at

Ov~ ~+ ax

or

Ovv

+

Ovz

On +~-0. at

(39)

The continuity equation (2) of Section 1.4.1 thus may divided into two equations, one for the groundwater (equation (39)) and one for the solute: v . Vp +

Op n ~

- 0

or

v~

Op

+ vv

Op

+ vz

Op

Op + n ~ = O.

(40)

596

Basic principles

(1.4.3)

The density p -- p(c) is a function of the concentration of a solute (for instance, chlorine content in salt-fresh water problems) so, as the density is not a constant, the general form of Darcy's law must be chosen:

k Op Vx = - - - - ~ , I~ Ox

k Op vv = - - - - ~ , # Oy

vz --

k ( Op ) + pg . # -~z

Equation (39) then becomes for homogeneous isotropic, compressible ground, with

y - pg: 02p 02p 02p iz 1 - n Op Ox----~ -F ~y2 + Oz---T= k Eg Ot

07' Oz

(41)

If the ground is incompressible the term with Eg vanishes. Together with equation (40), evaluated in a similar way, the flow of incompressible groundwater of

variable density through a homogeneous, isotropic, incompressible porous medium can now be described by the following two differential equations

02p 02p 02p Oy Ox----~ -F ~Oy2 + ~z 2 + Oz -- 0 k

OpOy

.

and (42)

OpOy ~ Op Oy t-), Oy ) - n ~ Oy

ayay

az az

at'

or in vector notation: Oy _ 0

V=p+ 0-2

and

k

Vp.V f+f

Oy)

Oy

-n~.

0t

These two differential equations for three-dimensional density flow are mutually dependent and have to be solved simultaneously for p(x, y, z) and y(x, y, z, t). Although both water and soil are assumed to be incompressible, the flow is nevertheless non-steady as a result of the varying density of the groundwater. Only if v . V p -- 0 (see equation (40)) can steady density flow occur, so if the streamlines are everywhere tangent to the planes p = const. For anisotropic ground Darcy's law runs according to (37) of Section 1.3.3 with

i,j=x,y,z: ki.i ( Op Vi

Oz )

--

These values, combined with (39) and (40) yield two differential equations for three-dimensional density flow through anisotropic ground (here not written down). As soon as transport of a solute in groundwater takes place, as in the case of the here above described density flow, we encounter the phenomenon of dispersion. The differential equations that describe this transport may be considered as additional to the foregoing equations and will be derived in Section 1.4.5.

Differential equations

(1.4.3)

597

As has already been remarked in connection with the derivation of Darcy's law (Section 1.2.3) there is no sense in introducing some kind of potential function instead of pressure as dependent variable, because in density flow the velocity vector is no longer simply proportional to the gradient of a potential function. In some cases, especially in two-dimensional problems, it is worthwhile to introduce the stream function in density flow. Since there is a close relation between it and the rotation of the flow, the differential equations for the stream function ~k concerning inhomogeneous groundwater will be studied in Section 1.4.4: rotational flow.

1.4.4. Rotational flow If, during flow through a porous medium, water elements are also subjected to rotation as well as to translation, we may speak of rotational flow. The following, which is taken from H. Rouse's book "Fluid Mechanics for Hydraulic Engineers", will make this clear. Consider an element of cubic form in the field (Fig. 4) having small but finite sides dx, dy and dz. At the point (x, y, z) the velocity vector v has the three components vx, Vv and vz. Since the flow varies with distance in each of the coordinate directions, at every corner of the cubic it will vary by amounts that depend upon the length of the sides and the gradients of each velocity component. For the sake of simplicity, we shall follow the changes in one face of the cube, later extending the relationships to the other faces. For instance, in Fig. 5 the velocity components in the x- and y-directions will be considered. Neglecting for a moment the change with time at a fixed point in space (steady flow) and omitting derivatives of the second order, it is seen from the illustration that during a small time increment dt the actual motions at the several corners of the face must be different. Hence, not only will the face be moved through space, but it must at

l)y-Jr- ~y dy

1)yy!,~Z~Vx ~ dZ 011

d

_

Vx

" dx Vz + -~z dz

)¢ Fig. 4. Variation in velocity in a volume element.

OVxdx + Ox

598

(1.4.4)

1)y -k-

vy + ~ y dy OVx Vx + -'d-yd y

Basic principles

~ y dy +

-Ydx

X

~x+ Ox

dy 1)y

+ OyaY

OV_~dx OX

1)y+

Vx+ b d x

Fig. 5. Velocity components in the x-y-plane. the same time suffer a change in its original form. There are four essential types of movement which the face may undergo: a translation of the quadrangle in the x- and y-directions, a change in the length of each pair of parallel sides (linear deformation), a change in each of the four corner angles (angular deformation), and a turning movement in one direction or the other (rotation). Each of these four essential types of displacement is shown schematically in Fig. 6. During the time increment dt the magnitude of the translation in the two directions will be represented by the distances Vx dt and l)y dr. The magnitude of the linear deformation will be given by the difference between the distances moved by

O•y

dy dt

. . . . I

I

I I

.......

li --

aVXdx dt

vydt

vxdt

translation Ov x .y dt. ~-~d

linear deformation

~.~ ~

x , y dt - OV ]~-~a .i

I

II

--1

II

""Ib -dfl

~

.-

angular deformation

"

\

\\\ i\

-" Ib at

\\

dot)

rotation

Fig. 6. Schematic representation of the essential types of movement in the x-y-plane.

Differential equations

(1.4.4)

599

each pair of opposite sides:

3Vx dx dt 3x

and

Ol)y

3y

dydt.

The change in the right angle at the point (x, y) will depend upon the angular movements dot and d/3 of the two sides dx and dy. Since over a very short time these angular increments will be small, they may be put equal to their respective tangents; selecting the counterclockwise direction as positive:

dot-

'ova.,dx dt 3vv ;~x = • dt dx 3x

0 Vx

and

~ dy dt • = dy

dfl--

OVx dt. Oy

In general, the angular changes are composed of two types of movement: rotation and angular deformation. The rotation of the face is defined as the total angle through which the face is rotated; this will be the average value of dot and d/3

Rotation-

dot + dfl l(Ovv 2 = 2 Ox

Ovx)dt, Oy

whereas Angular deformation - dot -- dfl -- /{Ovy \

+ 3y/

dt.

Owing to the finite dimensions of the original box the values just given for each type of displacement are not exact, derivatives of second order and higher having been omitted. But if the sides of the box are now assumed to become infinitesimal, with all comers then approaching the point (x, y, z) as a limit, each type of movement may be expressed as a rate of change with time. At the given point the rate of translation in the three coordinate directions will be simply the three velocity components Vx, Vy and Vz, or in vector notation: rate of translation = v.

(43)

The rate of linear deformation in each of the coordinate directions will be

Ol)y dy

3Vx dx, 3x

and

3y

OVz dz. 3z

(44)

Applying the continuity principle for a constant volume of the box, it will be clear that d iv v -

Ov~

Ovv

OVz _ O.

3---~ + -~y + 3z

600

Basic principles

(1.4.4)

The rate of angular deformation at a given point in planes normal to each of the three coordinate directions will be:

OVz Ovv ~ ----=, 3y 3z

OVx Oz

~ + ~

Ova 3x

and

Ovv OVx " +~. Ox Oy

(45)

In this paragraph special attention will be given to the rotation of the groundwater which is a type of displacement, allied to translation, since both denote movement of the water element without change of its form. Thus the rate of rotation will also be a vector with components in the three coordinate directions and equal to the angular velocity to or vorticity (usually = 2o9) with dimension [T-l].

l ( Ovz

OUy)

l(OVy

OVx)

C°z -- 2

Ox

l ( Ovx COy--~ Oz

Opz ) Ox ' (46)

Oy '

or in vector notation: 1 = - curl v. 2

The rotation vector ro at a point in the flow field is perpendicular to the infinitesimal plane in which the rotation takes place. A positive value of the vector or of one of its components means a counterclockwise rotation in the plane perpendicular to it. The magnitude of the vector gives an indication of the strength of the rotation, i.e. the angular velocity. So long as the rotation vector re has a finite value, the flow is characterized as rotational flow. If each component of this vector is reduced to zero, the flow will become irrotational. If the flow is a so-called potential flow, which means that the velocity vector is the gradient of a scalar quantity, the potential 4~ -- K cp with K constant (see Section 1.2.3), the flow is irrotational, which can be seen immediately when applying the values for the velocities

Vx -- - K ~

0q9

0~o

vv - - K ~ " Oy

Ox '

and

vz - - K ~

0~o

Oz

to the equations (46) with curl v = 0, for instance,

Ovv Ox •

K 02(D O1)x =

-

~

OyOx

=

Oy

0299 =

-

K

~

OxOy

etc.

If K is not constant, as is the case in anisotropic and/or heterogeneous soils, the flow becomes rotational (exceptions will be mentioned later in this section). In these cases the heads will nevertheless also be called potentials though properly speaking

Differential equations

(1.4.4)

601

this is not quite correct because it is not potential flow but rather rotational flow that takes place. Considering the general expression for the conductivity k?"

K--

(see equation (29) of Section 1.3.3),

= K(x,y,z)

t* two particular cases may be distinguished which are essential to rotational flow. In the first place we suppose that k is a function of x, y and z while ?' - Yo and /x - / x 0 are constants (homogeneous groundwater, heterogeneous ground). Darcy's law v -- - ~ ~ ( V P + ?'Vz) becomes v - K V ~ o with K -- k--D-,*o= K ( x , y , z ) . According to (46) we find

Ovz "0y

2Wx

02(/9

"0vv "0z

~

`0K `0~o

~ K - -

"0y"0z

"OK "0~

"OK "0~

"0z "0y

"0y "0z

"0y "0z

~K

`02(/9

`0K "0~o 4-~~ "0y"0z "0z "0y

Similar expressions can be found for coy and coz. So the rotation of h o m o g e n e o u s g r o u n d w a t e r in heterogeneous isotropic ground can be expressed mathematically as"

2COx-

20.)z

•

OK "0~o `0z `0y

OK `0q) "0y "0z '

"OK "0q)

OK "09)

"0y "0x

"0x Oy

OK `0~o 2COy = "0x "0z

OK `0~o "0z "0x '

or

2 0 ) - curl v -

V~o x V K.

(47)

These formulas may be extended for anisotropic heterogeneous ground; then also terms with ~~)x~)v ' etc. appear. As can be seen from equation (47), by way of exception the fl0w can become irrotational, if curl v = Vqo x V K = 0; this means, if grad 99 and grad K are parallel, for instance, vertical flow through a horizontally layered aquifer. Secondly, it is assumed that ?' is a function of x, y and z, while ~- - c o n s t . - /,~. Darcy's law becomes u ~,o

Vx=

#0 Ox

_ "ova

,

vv=

"0vv

--

#0 "0Y

_~k°

vz=

+?"

#0

"02p +

"07') + k0`02 p

"0y"0z

-~y

ko "0?"

"0Y

Oz -

2co,, =

OVx "0z

`0Vz _ k o `02p k o ( `02p "0?') k o "0?' "0x - #---05xOz ~- -no - axaz + -a-7 - ~o ax '

0 vv

0 vx

"0x

Oy

ko 02p =

#o "0x "0v

ko t

#o "0y"0z

=

)

2oox -

2COz --

#o

(

and

"02D

#o "0x "0y

--0.

#o Oy

and

602

(1.4.4)

Basic principles

Thus, rotation caused by flow of heterogeneous groundwater (with constant viscosity) through a homogeneous isotropic soil can be described by: ko OF 2cox = - ~ ~ , lzo Oy

2coy =

ko 07,

and

IXo Ox

2coz - 0 (48)

ko 2w--curlv--~VzxV?'. /z0

or

From these equations it appears that flow of groundwater with a variable density (for instance, salt-fresh water flow) through a homogeneous, isotropic aquifer is rotational flow with a horizontal rotation vector (wz = 0): there is only rotation in vertical planes, though the flow may be three-dimensional. The flow is irrotational, if g is a function of z only, as can be seen immediately from the equations (48). Like in irrotational flow (Section 1.4.2-5) also in rotational flow stream functions may be defined. In the following the relation between the stream function and rotation of inhomogeneous groundwater in homogeneous ground will be studied both for two-dimensional and for axial-symmetric flow. Consider two-dimensional flow in vertical cross-sections of a homogeneous isotropic aquifer with water of variable density (for instance, groundwater with variable chlorine contents). The groundwater velocity and the density are functions of x and z, so that the rotation takes place entirely in the x z-plane as can be seen from eqs. (48). COx and Wz are zero and only COy has a finite value. The total vector of the rotation is perpendicular to the xz-plane and has the value: kOy 2co -- -- ~ . i~ Ox

(49)

Again from the continuity equation ~~}Vx + ~~)vz - 0 the function ~p can be defined such that

Vx --

Oz

-

and

vz =

Ox

The vorticity, expressed in the function ~p now becomes" 2co-

O1)x

O l)z

OZ

OX

=

021[r

O2 ~J

OZ 2

OX 2

=

-- 2 l[t

and thus we have found the important relation between the stream function 7t(x, z) and the vorticity co" 021p OX 2

t

021p OZ 2

=

k 07' t~ OX

= -2co.

(50)

This differential equation is a so-called Poisson equation; it states that at every point (x0, z0) of the flow field, where the gradient of the fluid density has a horizontal

Differential equations

(1.4.4)

603

component, the flow contains a vorticity 2co(x0, z0). As has already been pointed out in Section 1.4.3, density flow will, in general, be non-steady, though ground and water are assumed to be incompressible. Equation (50), which is the result of the combination div v -- 0 and curl v = 2o~, still has to be completed by (see equation (40) of Section 1.4.3) v. Vy + n ~

Oy =0 Ot

which in terms of 7r becomes:

oO o×

oO o×

Ox Oz

Oz Ox

oy

t--n~ =0. at

(51)

Steady flow is possible as an exception, if v . V y, -- 0; this means, if the discharge vector is perpendicular to the density gradient everywhere in the field: the streamlines then coincide with the lines p -- constant. In general, the vorticity is defined as the limit of the circulation around the enclosing curve of a region, divided by the area of that region, if that area approaches zero. Circulation F (dimension [LZT-1]) should be understood as the line integral of the tangential velocity component around any closed curve (see Fig. 7).

Fig, 7. Circulation. I-'A -- (~ VL dL

JL

and

lim FA -- 209.

(52)

A~0 A

This relationship will be apparent after considering the circulation around the rectangle in the xy-plane shown in Fig. 5. If the elementary rectangle is sufficiently small, the line integral of tangential velocity along each side will equal the average velocity between the two corners times the length of the side. Writing the circulation in counterclockwise direction, we obtain: AFxy--

(

Xdx Vx-4--~1 0 VOx

( _-

OVx

Vx + -~y

OVy dx dy Ox

)

Ovv low ) dx + (Vy + Ox" dx ~ 2 0 y " dy dy

d y -4-

l OVX dx ) d x -~ Ox

OVx dx dy Oy

-

2~o z

( W - k - -l o w " dy ) dy " 2 Oy

dx dy.

604

Basic principles

(1.4.4)

As the area of the surface approaches zero in the limiting case, the resulting expression for circulation per unit area will be the component of the vorticity vector perpendicular to that area: lim

AI"xy

dx dy--+0 dx dy

Ov v

OVx Oy

-- 2coz -: " Ox

The differential equation for the stream function ~p for density flow through anisotropic soils can be found as follows. Consider twodimensional flow in a vertical plane (Vy -- 0) and an anisotropy with main directions k,. and k,, which do not coincide with the coordinate axes x and z (see Fig. 8). The permeability perpendicular to the vertical plane is also ks. In practice, abstraction of groundwater by means of a partially penetrating "well-screen"

Fig. 8. Two-dimensional anisotropic density flow.

(long series of wells, close upon each other) from an obliquely layered aquifer, may be schematized to this case. Darcy's law here becomes (equation (37) of Section 1.3.3)

Vx = -

kxx Op lZ Ox

kxz ( O_~z ) " --]- Y , lz

kxz Op " lz Ox

Vz -

kzz ( Op lz Oz + Y

Solving ~~P and ~az from these equations we get:

o0,_ =

/x

Ox

D

with D -

Op Ox

(-kzzvx + kxzvz) "

and

oO..e_ , =

Oz

k x x k z z - k2z " As V x - - ~ - ~~ z and v z -d

(kzz a 7~

a

and

xzV,)- ×

~~Jx ' this becomes:

ap -

+

D

az -

~

a¢

a~/

-ox

If we assume that the second derivatives of p exist and are continuous, we obtain from ~jZp ~ ~)2p . ~x~Jz ~ ~)Z~X

lz { kzz 02 llr

02 lit

IZ

02 lp

02 ~ ~

0y

OX 2

kxz. OxOz }

Ox

Differential equations

(1.4.4)

605

Hence, the differential equation in ~ for two-dimensional density flow through anisotropic homogeneous ground becomes"

02fir 02fir 0 2~ k x x k zz - k 2 0}1 O. kxx-~x 2 + 2kxz~.OxOz + kzz..-~z2 + tx xz Ox =

(53)

If the main directions of the anisotropy are parallel to the coordinate axes, then kxx - kx, kzz -- kz and kxz - O, from which"

02~P 02~ kxkz OV kx--~x2+kz-~z2+ =0. lz Ox

(54)

•

For isotropic ground kx = kz = k and equation (54) is replaced by equation (50). The differential equation for the stream function ~p for density flow through an anisotropic porous medium can be found in a similar way if the flow is axialsymmetric.

I

/

r

.i j |

Consider groundwater abstraction from a horizontally layered aquifer by means of a partially penetrating well. We take the z-axis as the axis of symmetry and the r-axis as horizontal. The main directions of the anisotropy are parallel to the coordinate directions (other directions are, due to the symmetry, unrealistic) with permeabilities kr and kz.

~r

i

Fig. 9. Axial-symmetric anisotropic density flow. Darcy's law here becomes"

kr Op Or ~

lz Or

k z { Op

and

vz --

and

Op = Oz

lZ ~,-~z

+Y),

from which

Op Or

=

-

-

lz Vr kr -

lZVz kz

yo

With the equations (18) of Section 1.2.1 becomes Op Or

=

At

10~

27rkr r Oz

and

Op Oz.

=

Vr

At

--

10~

2rrkz r Or

1 ~~--~. and Vz -- ~ 1 ~_g~. 2rrr ~r' this

yo

606

Basic principles

( 1.4.4)

As 82 P "--" ~)2p

OrOz

Oz~)r' we have

/~

1 02~

/~

=

2rckr r Oz 2

1

021//

/Z

2zrkz r Or 2

10~

09/

2rckz r 2 Or

Or

or

021P

kr Olp

kr Or 2

r Or

021~r

t- kz

~

2rckrkz Oy

+ ----=r~ -- O, 13, Or

(55)

the differential equation in 7: for axial-symmetric density flow through anisotropic homogeneous ground. In isotropic ground, with kr - kz - k, this becomes:

oq21/r

1 01//

Or 2

r Or

I-

021p ~

2Jrk 09/

(56)

+ ~ r ~ --0. lZ Or

1.4.5. Differential equations for dispersion

fy___~/

A~: f y + ~ y A y

Ay

We apply the law of mass conservation to the specific mass flux f (Section 1.2.5) of a solute in groundwater with a concentration c(x, y, z , t ) through a control volume (Fig. 10).

ira,

x / /

Y

Fig. 10. Control volume.

During a short time At the total mass of solute that enters the volume is

0fz

Ox

neAx A y Az At - - - neAx A y Az At -- - - n e A x A y A z A t Oy Oz

with ne - effective porosity. The total mass that enters the control volume must be equal to the amount of mass that is stored in the elemental volume, either by increase of the concentration of the solute in the water or by means of adsorption of the solute on the soil particles:

Oc OA -ff-~neAxAyAzAt + -oT(1 - n e ) A x A y A z A t . A is the mass of the amount of solute that is adsorbed on the soil particles per volume of the grains; A has the dimension of concentration [ML-3].

(1.4.5)

Differential equations

607

The equation o f continuity f o r the mass f l u x of a solute in groundwater thus becomes"

aL

af~

oL

ac

1 -neOA

5x

ne

Ot

= 0

or

Oc 1 -- ne OA divf + -57 -~ ne at = 0 .

(57)

In the literature concerning adsorption, many kinds of relations between the concentration of a solute c and the adsorbed amount per grain volume A are proposed of which the most frequently occurring is a linear one: (58)

A -- ac,

with a a dimensionless constant. Equation (57) then becomes"

OC

div f + Rd - ~ -- O,

(59)

with Rd -- 1 + a 1-n_____~e. tie Rd is called the retardation f a c t o r because o f adsorption, as adsorption causes an effect that corresponds with a reduction of the mass flux" 1

Rd

div f +

Oc Ot

-0.

We leave adsorption out of further consideration (A -- 0 gives a -- 0 and Rd -- 1) and take as the basic equation of continuity" div f ~

OC Ot

- O.

(60)

If needed, the adsorption may be taken into account again by dividing each flux term by a retardation factor. As has been discussed in Section 1.2.5 the transport of a solute comprises dispersive transport and convective transport, the latter, in a fixed direction, being equal to the solute concentration multiplied by the average real velocity component in that direction. Because of the anisotropy of the dispersion (Section 1.3.4) with one main direction along the velocity vector and of the variation with respect to space and time of this vector, the general equation of motion for the mass flux f, as represented by equation (53) of Section 1.3.4, has to be applied:

OC

fi -- U i C - Di.jT. OJ

(i, j -- x, y, Z)

combined with equation (55) for Dii"

UiU.j

Dij - aTu~ij -+- (aL - - a T ) ~ (i, j -- x, y , z ) ,

~i.i -- {

for i -- j, for i :/: j.

608

(1.4.5)

Basic principles

These equations, combined with the continuity equation (60) yield the general differential equation f o r the convective and dispersive transport of a solute in groundwater:

(

0 Dij

Ui

Oi

oc

--

Oi

oc

(61)

(i, j -- X y, Z)

Ot

or written at length: 0

(

5-;

Dxx

Oc

~

+ Dxv

Oc

.Ty

o(

of

+ Dxz

Oc)

.Tz

of

+ ~z D ~ z ~ + Z ) y ~

+

0 (

7y

of)

Oc

~

+ Dvv

of

of

Dxy

Oc

..Ty

+ Dvz

Oc)

.7z

of

of

+ Z)z~.~ - .xS-;x - U y ~ - ~z Oz - 5 7

The sum OUx OUy OUz c--~x + c--~y + c Oz

that also appears in combining the equations of motion with the equation of continuity, can be put equal to zero, as it is assumed that the groundwater flow is steady, which implies that div v - 0 and div u - 0. The dispersion coefficients Dij are functions of the real velocities (see Section 1.3.4, equation (55)), and the two constants DE and DT, the dispersivity coefficients. In the case of uniform flow, for instance, in x-direction, Ux - u - const. and Uy -- Uz - O. For Dij we find: Dxx -- aLu, Dyy -- aTu, Dzz -- aTu, Dxy -- DXZ -- Dyz --O. The differential equation then becomes"

02C

(02C

aLU-~x 2 + aTu ~Oy2 +

02¢~ 0¢ -u ~ OZ2,] Ox

OC -- m Ot.

(62)

This differential equation describes a point injection or a point pollution in uniform

flOW. Although the flow is one-dimensional, the dispersion is three-dimensional, because transport of the solute perpendicular to the flow direction takes also place. If a pollution is uniformly distributed along a plane, perpendicular to the direction of the uniform flow the dispersion becomes also one-dimensional, with one coefficient of dispersion D -- aLu while aT = 0

02C

~OC_

D-~x2 - U ox

__Oc. Ot

(63)

The derivation of the differential equation for the dispersion in cylindrical and polar coordinates may be performed in the usual way, making use of the well-known

(1.4.5)

Differential equations

~ .-.-

609

(r + dr) dO

A Fig. 11. Control v o l u m e in cylindrical coordinates.

relations between Cartesian coordinates and cylindrical coordinates and between the velocities in x-, y- and z-directions and those in r-, 0- and z-directions: r--

3 3x

v/x2 + y 2,

= cos 0

0 -- arctan(xY-),

O

sin0 O

Or

r

U x - - b/r COS 0 - -

U0

O

O cos0 O = sin 0 - - -t 3y Or r O0'

30'

O,

sin

uv

--

Ur

s i n 0 + uo c o s O.

This requires much complicated mathematical analysis; therefore it is much easier to depart from the elementary control volume for cylindrical coordinates (Fig. 11) with dimensions dr, dz, r dO and (r + dr) dO. The specific flux in the r-direction is fr perpendicular to the elementary area a.fr dr to the elementary area (r 4-dr)dO dz, so in the r-direction r dO dz and fr q- ;-dT-r the flux that enters the control volume amounts to

(

frr dO dz - \ fr + --

--fr

dr dO dz

\

0fr

77

dr) (r + dr) dO dz

air -

3r

r

dr dO dz,

in which the term - ~af (dr)2 dO dz has been neglected, compared to the other terms (infinitely small of higher order). The flux that enters in the direction perpendicular to r and in the z-direction becomes, respectively" 3fo dO dr dz O0

and

3fz dz dr r dO. Oz

The law of conservation of mass then yields: --fr

dr dO dz

-

aL

r ~ Or

dr dO dz

-

afo 30

or:

_f r_f _ air + _ l afo + afz + - -ac r

-~r

r-~

-~Z

Ot

-0.

aL dr dO dz - r

Oc

dr dO dz -- r m dr dO dz at Oz

610

Basic principles

(1.4.5)

Thus the continuity equation for the dispersion incylindrical coordinates becomes:

1 O (rfr) + -1 Ofo + Ofz 4- -Oc - - 0. 70-7 r-~ -~z Ot

(64)

The dispersion tensor for the three mutually perpendicular directions r, 0 and z, all three different from the direction of the velocity vector, in matrix notation is:

Drr Dro Drz

D(r, O, z) --

Dro Doo Doz

Drz ) Doz

(65)

Dzz

with Dij

(aL -- aT) uiuj u

= a T u ~ i . j -k-

(i, j -- r, O, z),

(66)

u2 and Doz -- (aL -- aT) uouz for instance, D r r - - a T u -Jr- ( a L - - aT)-u-u The specific fluxes in the three directions are:

Oc fr -- UrC- Drr-~r Oc fo -- u o c - Dro-~r Oc fz -- UzC- Drz-or

10c Oc Dro - m - Drz m , r O0 Oz 10c Oc Doo-~ - Doz~, r O0 Oz 1 Oc Oc Doz-r ~30 - Dzz~...Oz

(67)

Combination of (64) and (67) yields the differential equation for the dispersion in

cylindrical coordinates l O ( r D r r Oc r Or ~r

l a(

-Jr- D r o

ac

Oc -Jr-r D r z O~zz)

1

ac

ac)

+ -r-~ Dro -~r + -r Doo -~ + Doz -~z

(68)

O( Oc 1 Oc O_~zz) + -~Z Drz-~r + -Doz-~r + Dzz Oc Ur Or

uo Oc r O0

Oc Oz

UZ~

Oc Ot

--- ~ .

In this derivation the differentiation of the convective terms gave:

lO lO 0 ~ ( r u r c ) + -r-~ (uoc) + -~z (UzC) r-O

r

1 0 = c

-

r-~r

10uo ( r U r ) -l-

7--~-

-Jr-

OUz I "I ~

-It- u r

Oc ~r

uo Oc r-~

+ ~

Oc Oz

n{- U z ~

(1.4.5)

Differential equations

611

and as steady flow was assumed, the expression between the accolades became zero. For axial-symmetric flow, uo - 0 and u 2 __ Ur2+U 2. Then Doo -- aTu, DrO -- Doz ---0 and Drr Drz and Dzz. remain unchanged. Equation (68) becomes (au _ 0)" ,

1 O( r Or

r Drr

Oc -~r

O

+ r Drz

Oc) -~z

Oc

1

+

7

Oc

aTu Oc

O2c

002 Oc

Oc

(69)

@ "~z (Orz-~F @ Ozz-~z ) - "r'~F -- Hz -Oz - -Ot. This may be the differential equation for a point pollution in an axial-symmetric f l o w field (caused, for instance, by abstraction of groundwater from a partially penetrating well in an infinite aquifer). If also the concentration distribution of the solute is axial-symmetric: c = c(r, z), for instance, as a result of injection into a partially penetrating well, the term with 02c disappears from equation (69). For mere radial flow, also uz = 0 and ur = u(r) while Drr = aLu, Drz = 0 and Dzz - - a T u . Equation (69) then becomes:

aL 0 ( O C ) r Or

ur

-~r

+ aTu

( 1 02c -~-~

+

02c~ OC OC -- u - - -- ~ OZ2 ] Or at'

(70)

with u = u(r). This differential equation holds for a point pollution in radialflow, caused, for instance, by water abstraction by means of a fully penetrating well. Injection of a solute into a fully penetrating well gives a radial concentration distribution in a radial flow field; in equation (70) the terms with ~ij2c and i)2c disappear. In radial flow, the velocity of the groundwater is inversely proportional to the distance r from the well: ur =

Q 27r Dne

= G --constant,

with D -- thickness of the aquifer. Then the differential equation becomes:

02C

OC

OC (71)

a L G -~r 2 -- GO-7 = r - - . Ot

In the most general situation the density and the viscosity of the groundwater will be influenced by the differences in concentration and we will have to deal with an inhomogeneous fluid with variable p and #. If the groundwater is assumed to be incompressible, the continuity equation remains valid and becomes (see Section 1.4.3, equation (42)):

02p + 02p + 02p

ax2

ay---7

Op

+

-°'

which shows that the pressure distribution and so also the flow pattern depends on p. This differential equation has to be solved simultaneously with the differential

612

(1.4.5)

Basic principles

equation (61) in which c must be replaced by its function of p: c = c(p). In the case of salt-fresh water flow, the density can be expressed as a linear function of the chlorine contents c (see Section 1.3.1, equation (1)): p - po = a ( c -

co),

from which Vp = aVc. The two mutually dependent differential equations then become: V2p -k- g Op O---~- 0

and

O---i

+ ~lZne V p

V p + go ~

= ~ . (72)

With the exception of the dispersion terms this couple of differential equations is the same as (42) of Section 1.4.3. If the density and viscosity of the groundwater are not influenced by varying concentrations of a solute, we may speak of an ideal tracer (for instance, a dye) and also if the differences in concentration are small, their influence on the density may in most cases be neglected (brackish water problems); as in the equation div v = 0 the term ~Op disappears, the flow pattern is independent of the density and can be solved separately, after which the values found for Ux, Uy and uz may be used in the differential equation for the dispersion.

Initial and boundary conditions

( 1.5.1-1 )

6 13

1.5. INITIAL AND BOUNDARY CONDITIONS

1.5.1. Linear conditions

1. General The differential equations for the state variables head or water table qg, pressure p, stream function ~, specific weight y and solute concentration c, developed in the previous sections are, in general, partial, linear differential equations of the second order. This means that more than one independent variable (x, y, t, etc.) is present, that the dependent variables and their derivatives are of first degree (not ~o2 or ( ~~)x) 2 etc.) and that the highest derivative in the equation is of second order. The general solution of a second-order ordinary differential equation theoretically contains a number of constants, dependent on the order of the derivatives with respect to the variable. Thus, additional information has to be obtained in order to get a uniquely determined solution and this may happen if the conditions at the boundaries of the flow system are known, i.e., the boundary conditions and in case of the time variable also if the initial condition is given. The derivatives with respect to the space variables x, y, z, r, 0 and r/occur as second-order derivatives in the partial differential equations and, therefore, each space variable requires two given values at both sides of its interval (negative and positive infinity included), whereas the derivatives with respect to time in the equations always are of the first order, so that only one condition for the time variable is necessary. The boundary conditions may be divided into linear and non-linear conditions depending on the character of the boundaries. If the boundaries are geometrically fixed and are known, the state variables or their derivatives that appear in the expressions for the boundary conditions are of the first degree or linear. These boundary conditions will be discussed in this paragraph. Sometimes the boundary of the flow region is unknown beforehand and depends on the flow pattern, still to be determined, as in the case of a phreatic surface or an interface between two fluids of different density. The conditions then become much more complicated and turn out to become quadratic as to the first derivatives. These non-linear boundary conditions are the subject of Section 1.5.2. Also the boundary conditions necessary for solving dispersion problems will be treated separately under 6 of this section, because of their specific character. 2. Initial values In order to solve a problem of non-steady flow, the distribution of the state variable under consideration at a certain time must be known and is mostly given at t = 0. Then, for instance, if a three-dimensional non-steady differential equation for the piezometric head is given, which means that ~0 = ~p(x, y, z, t), then ~o(x, y, z, 0) must be a known function and represent the initial value of ~0 or initial condition for ~p. As this function contains three independent variables (x, y and z), it is not possible to give a geometrical interpretation of this function. At each point of the flow field, fixed by the coordinates x, y and z the head ~0 is known at the time t = 0. In many groundwater flow systems the head is constant for each vertical and 99

6 14

Basic principles

(1.5.1-2)

is independent of z and a function of x, y and t: q9 = qg(x, y, t). The initial value qg(x, y, 0) may then be represented geometrically by a surface. If q)(x, y, z, 0) is a continuous function, the analytical solution of the problem will tend to this function if t approaches zero. If the initial distribution is discontinuous in regions in the flow field, theoretically the discontinuity must disappear in an infinitely short time and in this case the solution will converge to that part of the initial distribution that was continuous. The following example may illustrate this (see Fig. 1). An arbitrary continuous head-distribution as a function only of x: q)(x, 0) = F(x) in an infinite confined aquifer yields one-dimensional flow of which the solution is a function of x and t (see 112.01 of Part

S

888~$x~x~

A):

KD, S

,1..-

x

Fig. 1. Arbitrary initial function in one-dimensional flow.

~o( x t ) -

~

r l +~

'

2~-7

f12

F(xo)exp{ - -~(x-xo)2}dxo.

(1)

J_

Putting /~(x - xo)

247 we find F ( x + ~ S - Z ) e -x2 d;k

~o(x, t) - ~

and letting t approach zero this becomes" ,

~o(x, O)- ---~F(x) v Jr

e -x2 d ~ -

F ( x ) e r f ( e o ) - F(x),

oo

which is indeed the initial continuous head distribution. If for instance, F(x)--

h 0

F(x)

for-b<x <+b, forx <-bandx >+b

(see Fig. 2), the solution for the depletion function becomes"

h q)(x, t) -- ~ erf

2~/7

+ erf

~(b-x)

is discontinuous,

Initial and boundary conditions ,

(1.5.1-2)

615

q9 I

I I

h

'-b

I~0=0

KD, S x

~88888888888888~

Fig. 2.

Discontinuous initial function in one-dimensional flow.

(see 124.13 of Part A). Letting t approach zero, we find for Ixl < b ~0(x, 0) = h--2 erf(e~) -- h and for Ix l > b ° 2

~(x, 0) - h (erf(oo) - erf(oo)) - O, both in accordance with the continuous parts of the original initial head distribution. Besides this, however, for x - ± b we find from equation (2): qg(-+-b, t) - h e r f ( ~ t ) which becomes if t --+ 0: lim qg(-l-b t) t--+0

'

h 2"

The solution (2) thus, if t tends to zero, converges to the continuous function as shown in Fig. 3, in which the original continuous parts are now connected by a continuous curve.

h

b Fig. 3. Continuous curve at x - b.

h

l

dx

Fig. 4. Head impulse in a plane.

If we let b in Fig. 2 approach a very small value, ~1 dx , the initial value becomes an impulse function (see Fig. 4) for the head h, which may be written as

~0(x, 0 ) = h~(x),

(3)

616

(1.5.1-3)

Basic principles

in which

S(x)--

1 0

if x -- 0, ifx#0.

(Kronecker delta)

The impulse functions are very useful, because they enable us to find the solutions of a large number of problems, that can be written down immediately from first principles, as will be explained in Section 2.3.1-3. In the example in Fig. 4, the impulse function has been applied to a plane, perpendicular to the x-axis, but impulses can also be given in points, lines, spherical surfaces, cylindrical surfaces, etc. 3. Open boundaries

Pervious or semi-pervious boundaries such as bottoms and banks of rivers, canals, lakes, or well-screens, in general, boundaries which permit the outflow of groundwater into open water or conversely the inflow of open water into the ground, are called open boundaries. There may be distinguished by three kinds of conditions at open boundaries: a. K n o w n h e a d distribution

At each point of the boundary surface, the head distribution is known at any time, which means that ~0b = ~0b(x, y, Z, t) is a known function of space and time, in general. An additional condition is the equation of the boundary surface Zb -- Zb(X, y). If this function for Zb is substituted in ~0h(x, y, z, t), qgb becomes a function of x, y and t and the boundary condition is given as a known function ~ob - ~Ob(X, y, t) and then represents the level distribution of the open water at the boundary surface at any time. If the boundary is a body of surface water whose volume is so large that its level is uniform and independent of changes in groundwater flow, the head distribution everywhere at the boundary is uniform in position; it may be either constant or variable with time. For instance, a lake with a constant level gives ~Pb = known constant, which becomes 99b = 0 if the level is taken as reference level, and a sea with a tide: ~0b gOb(t) - - known function of the time. In both cases the boundary surface is an equipotential surface with the streamlines perpendicular to it in isotropic ground. In steady two-dimensional isotropic flow, the boundary condition for the stream function gr in the case of a given constant head at the open boundary becomes -

a~vh an

=0

if n is the direction normal to the boundary surface at a point. If the water pressure p is the dependent variable, as is usual in density flow, the boundary conditions may be obtained from those of the head by putting

p-×(e-z).

Initial and boundary conditions

(1.5.1-3)

617

b. Known flow distribution At each point of the boundary surface the velocity 099/, Vsh -- - K , . - ~ - s (x, y, z, t) is a given function of x, y, z and t while s is a given direction. Between z and x and y the equation for the boundary surface zh -- b(x, y) holds, which after substitution yields: 099/, v,+ - - K,. ~ (x, y, t) - known function. If a constant discharge is abstracted from a pumping well and the total amount of water is supposed to be uniformly distributed over the well-screen, the function a3--°-h may become constant and s becomes the direction n perpendicular to the wellOs screen: 099/, v,~h - - K ~ 0n = constant. If the discharge Q = Q(t) is a known function of time, the boundary value may also become a known function of time: ,,.h -

099/, -K.--~(t).

The boundary values for the stream function may be derived from the well-known relations between stream functions and potential functions (see Section 1.2.4). For instance, in two-dimensional flow to a partially penetrating, infinitely long wellscreen, the boundary condition at the screen for the drawdown ~o = ~o(x, z) is: 0q0 ~(0, Ox

q

z) =

2KI

and thus ' as KAY-aO. ;Ix m Oz

alp

q

~(O,z)

=

Oz

2l'

the condition for the stream function ~ -- ~p(x, z). Axial-symmetric flow to a partially penetrating well yields, in a similar way, the condition for ~p at the wellscreen (~p = ~ ( r , z)):

0¢ ~(0, Oz

Q z) =

l'

derived from K 0qo _

Q

Or

2rc lr

618

(1.5.1-3)

Basic principles

with the relation Kay_ Or

10~p 2rrr Oz

e. Known combination of head and flux Principally, two kinds of boundary conditions may be distinguished in this category: 1° Entrance resistance due to a semipervious layer of low conductivity and small thickness between the ground and open water. The flux through that layer will, according to Darcy's law, be proportional to the difference in heads on both sides of the resistance layer, divided by the thickness, the coefficient of proportionality being the conductivity of the layer Vnb =

gOlb - - ~02b

w

with w =

D' K"

in which ~01his a known function of space and time representing the open-water level, and ~P2hthe still unknown value of the head at the inside of the boundary. D' and K' are the thickness and the conductivity of the semipervious layer, respectively, and n -- direction perpendicular to the boundary. Because of continuity, the flux in the porous medium at the inside of the boundary must be equal to the flux through that boundary, whence the boundary condition becomes: 0 ~02b

0

991b - - 9 2 b

w is called the resistance to inflow of surface water into the ground (or conversely resistance to outflow of groundwater into open water). There is always a jump between the groundwater head at the inside of the boundary and the open water level at the outside, as distinct from boundaries without resistance, where the groundwater head and open-water level at the boundary are equal. Open water storage. If the open water body, into which outflow of groundwater through a fixed boundary takes place, is relatively small and the water level is not maintained artificially at a prescribed function of time, the level will fluctuate in a manner that depends on the outflow or inflow of water from or to the aquifer, thus in its turn influencing the magnitude of the outflow or inflow. The amount of water that enters the open-water body per unit of time is the rise a~0h of the water level per time unit ~ times a surface area, As depending on the dimensions of the finite water boundary. This amount is equal to the volume of groundwater that leaves the ground body" the groundwater velocity Kn a~oh perpendicular to the boundary, times the area Ah of the boundary surface. So the condition becomes: As Oq)h = A h K Ocph at n On

Initial and boundary conditions

( 1.5.1-4, 5)

619

For instance, groundwater flow in an aquifer with thickness D towards a circular storage basin with radius R needs a boundary condition at the cylindrical boundary: rr R 2 3 qo

t -(OR ' ~

399

t) -- 2rcRKD~(R,or

t),

where q9 = qg(r, t) is a function of r and t. In the case of abstraction of surface water with a discharge Q = Q ( t ) , the boundary condition becomes:

Q ( t ) -- JrR 2 3~P (R t) -- 2 7 r R K D O ~ ° ( R t). Ot ' Or '

4. I m p e r v i o u s b o u n d a r i e s Impervious boundaries are boundaries that do not allow groundwater pass them and they may be thick claylayers bounding a confined aquifer, or artificial screens, impermeable to groundwater round building pits, etc. In general, such a boundary is fixed in space. The flow is tangent to it, so there is no normal flow component: Oqgh

5; = 0 , and the boundary condition for the head becomes ~3~ob - 0 . The boundary surface is a stream surface, from which: ~h = constant.

5. C o m m o n b o u n d a r i e s Frequently, two adjacent aquifers have uniform but different hydraulic properties. Consider two soils with different conductivities K' and K, with K' > K. At every point of the boundary the elevation and the water pressure must be the same for both sides of ! ! the boundary" z h - zh, Ph - Ph and also as ~o - z + 2_. pg the heads must be the same:

--

K'~ K Vn

/

Fig. 5. Boundary between different soils.

99;, - ~Ph if the groundwater is homogeneous ( p ' - p). As 99;, - ~0h, then the derivatives

in the directions of the boundary surface must also be the same: ~

,

v,.--g

, oG Os

and

v,--K~,

aeh 3s

so

v',

v,

~=--. K' K

=

as • (a)

According to the continuity principle, the velocities normal to the interface at both sides of it, must be equal: vnf - v,,, tan 0 - - - ,Vs Un

tan0'=

v,' Un'

and so

tan 0' _ v,.' tan 0 v,.

620

(1.5.1-6)

Basic principles

and with (a): tan 01

K' ~--

tan 0

~

o

K

Upon passing from the pervious (K') to the less pervious (K) soil, the streamlines intersecting the boundary are refracted towards the normal, and conversely. When K' is much larger than K (K' >> K) then 0' >> 0 and the refracted streamline is almost normal in the less pervious soil. This is an indication that the assumption of merely horizontal flow in aquifers and merely vertical flow in the semi-permeable layers is generally allowable for leaky aquifers. 6. Boundary conditions f o r solute concentrations The specific character of the boundary conditions for solute concentrations originates from the convective terms in the transport equations of a solute (Section 1.3.4, equation (53)), if dispersion is studied in combination with groundwater flow. If dispersion takes place in groundwater at rest (only diffusion), the initial condition and the boundary conditions are similar to those for groundwater heads. This also holds for dispersion in flowing groundwater as far as the dispersion takes place perpendicular to the flow direction and even in the direction of the flow for boundary conditions with a known concentration distribution. Only if the flux distribution along a boundary is given, the condition to match differs from the boundary condition for a similar flow problem without dispersion. This difference may become clear by discussing the following examples where it is assumed that dispersion of a solute takes place in uniform flow with a constant real velocity u of the groundwater in x-direction. The concentration of the solute is then, in general, a function of x, y, z and t" c - c(x, y, z, t). In that case the groundwater flow is one-dimensional, but the dispersion three-dimensional, but if we assume that the solute concentration is uniformly distributed over a cross-section perpendicular to the x-axis (flow through a tube) we have to do also with one-dimensional dispersion" c - c(x, t). A given concentration as a known function of the time, for instance, at the boundary x - 0 will result in the boundary condition: c(O, t) - c ( t )

in which c(t) also may be a constant (zero included). This boundary condition corresponds to an open boundary with known head distribution, discussed in 3a of this section and is as such not characteristic for dispersion boundary conditions. If, on the other hand, a flux distribution at x -- 0 is given as a known function of the time: f (O, t) - f (t)

we must, if we want to express this conditions in terms of concentration and its derivatives, rewrite this in Oc

uc(O, t) -- D T - ( O , t) -- f (t) Ox

(4)

Initial and boundary conditions

(1.5.2-1 )

621

as f (x, t) - uc(x, t) - DE '~c ,-F;x(x , t) (see Section 1.2.5, equation (48)). This condition differs from the open boundary with known flow distribution discussed in 3b of this section, owing to the convective term uc. We see that in equation (4) not only the derivative of the concentration in x-direction, but also the concentration itself occurs. The function f ( t ) may also be a constant f0 or equal to zero. 1.5.2. N o n - l i n e a r

conditions

1. Phreatic surface The phreatic surface or water table (see Section 1.2.2) represents the upper boundary of a phreatic aquifer, if flow in the capillary zone is negligible. In general, it may be considered as a moving front, open to the atmosphere. So Pph -- 0. As, in general, P ~o(x, y, z, t) -- z -t- --, ?' the condition at the phreatic surface becomes: (a)

qgph( X , y, Zph, t) -- Zph Z p h - - Zph(X, Y, t) -- equation /'or the phreatic surface. arbitrary moving front is

or

If the equation of an

Zs -- Zs(X, y, t), then differentiating to t yields" dzs

dt

3zs dx 3zs dy 3zs t t --. Ox dt Oy dt Ot

=

dx __ _ K T;' ~)~0 vv - ne ddvt As Vx - n e ~ can be written as"

Oq)s Ozs

+

(b)

--

=

K ~;~o and vz -- n e ~ dz

Oq)s Ozs

Oq),.

ne OZs

3y 3y

Oz

K 3t

Ox 3x

__

_ K ~;)z' equation (b)

and, in particular, for the phreatic surface: 0(Pph OZph

3X

3X

t

0@ph0Zph

3y

3y

0q)ph

ne

0Zph

OZ

K

Ot

(c)

Differentiation of (a) with respect to x and y gives: 0qgPh t 0 ~0ph 0 Zph -- 0 Zph Ox Oz Ox Ox

and

0 qgph

3y

q

0 (/gph 0 Zph

3Z

3y

OZph

3y

622

(1.5.2-1 )

Basic principles

from which 3~Oph

OZph ~

Ox

1 --

0 Z ph

and

"3x

0y

3 tpph

3z

3tPph 3x

=

(d)

3qgph

1

Differentiating (a) with respect to t gives" O~0ph

0 q~ph 0 Zph

Ot

Oz

OZph

dt '

Ot

from which

OZph Ot

=

3tPph Ot . 1 - 0~Oph

(e)

3z

Replacing 3Zph 0x' 3Zph 0v and ~3t in (c) by the expressions (d) and (e) gives the differential equation for the phreatic surface in non-steady flow" 0 ~0ph 2

0 ~ph

2

0 ~0ph ) 2

099ph _

ne 0 ~ph

This expression can be understood as the (non-linear) boundary condition along the phreatic surface, whose shape is, however, unknown beforehand. For steady flow, the term with t disappears and equation (5) may be written 2 2 2 Oxp h -11-Vyph + Pzph + K VzPh - - 0 ,

(6)

which is the steady phreatic boundary condition written as a relation between the velocity components at every point on the phreatic surface, which is now fixed in space, but still unknown. The steady water table is a stream surface; so the total 2 2 2 2 velocity Vph at each point of this surface is tangent to it and Vph -- Vxp h + Vyph + Vzp h. From equation (6) it follows that Vzph is negative, so the phreatic surface has a downward slope. Also Uph < K as V2p h - - K v z p h and l)ph > IVzph]. A phreatic boundary with accretion is a boundary where water is added vertically from above (precipitation or artificial recharge) or removed by evapotranspiration. If the accretion is denoted by A [LT-1], it may be incorporated in the differential equation for the phreatic surface by simply adding the amount A to 1)zph. In equation (c) the term 0~0ph _ !1)

Oz

then becomes

K

Vz + A K

and yA must be added to the right side of equation (c). The resultant boundary condition f o r a water table with accretion then becomes: 0 ~0ph 2

0 qgph '~ 2

0 ~ph

2

A

O

A ne 0~ph q- __ K

Ot

K

(7)

(1.5.2-1 )

Initial and boundary conditions

623

and for steady flow: 2 2 2 l)xp h ~- Pyph -q- Uzph + ( K - A ) vzp h -

(8)

K A.

If the positive z-axis is directed upwards, as is assumed here, for downward accretion like precipitation A -- - p and for evaporation A -- E. It can easily be shown that in anisotropic soils of (Kx, Ky, Kz) in the principal directions the boundary condition (8) becomes" Kx ( 0 q)ph 2

-- -- n e

OCflph

Ot

0 ~Pph 2

0 (flph '~ 2

0 (/gph

+ A

and with A - - 0 and/or ~ equations may follow.

(9)

-0

the anisotropic versions of the previously derived

Apart from the non-linear conditions for the phreatic surface, by which the shape and position of that surface is determined, there are linear conditions along the phreatic surface, for instance, for the pressure Pph = Patm and for the head ~Oph= Zph, as we have seen, both conditions also valid for non-steady flow and for phreatic flow with accretion. Moreover, the boundary condition for the stream function ~ along the phreatic surface for steady flow without accretion is 1/fph --" constant, also for three-dimensional flow. Also in two-dimensional flow with accretion the boundary condition along the phreatic line for the stream function is a linear condition as can be shown as follows. The vertical accretion A is supposed to be constant and to be distributed uniformly in horizontal direction. Then the discharge q (or recharge, dependent on the sign of A), caused by the accretion A must satisfy the differential quotient: dq dx if the x-axis is chosen as the horizontal axis. In Section 1.2.1 we found for twodimensional flow dq - d ~ (equation (16)) and so we have d~ph _ A and thus dx

lpp h -- Ax + c

--

'

(10)

in which c is a constant that must be determined from the choice of the streamline - 0 and from geometrical data. Equation (10) also holds for the case that no accretion takes place (A - 0 ) ; then the phreatic line is a streamline and ~ - constant.

624

(1.5.2-2)

Basic principles

The phreatic surface almost always ends at the downstream face, not at the water surI \ \ face, but some distance above it, thus crei I i ating a zone where water seeps out of the seepage ground into the air, hence the name seepface age face. An example of a seepage face i----is given in Fig. 6 where intersections of i some potential- and stream surfaces with i a vertical plane are given for radial flow of I rw phreatic groundwater to a discharging well with diameter 2rw. A steady state is supposed, so that the water table is a stream surface and at the same time the groundF i g . 6. S e e p a g e f a c e f o r a w e l l . water level in the aquifer. This stream surface meets the cylindrical screen at a level that is higher than the open-water level in the well itself. The zone at the face of the well between these two levels is the seepage face, which is neither a stream surface nor an equipotential surface but forms a transition zone between the stream surface of the sloping water table and the vertical equipotential surface of the well below its level. The existence of a seepage face can be proved with the aid of the hodograph concept (see under 3 of this section). The hodograph also will show that the phreatic level must be tangential to the seepage face. I

i i i

2. Interface between fluids of different density Consider a sharp interface between two liquids of different densities, here chosen as fresh water (index f) which overlies salt water (index s). It is assumed that the interface completely separates the two fluids and that therefore each region has its own differential equation which is independent of the differential equation of the other region. At every point of the interface (index i) both elevation Zi and pressure Pi must be the same in both sides"' so (flif - - Zi '1t- Pi and q)is Zi -l-" Pi in which yf }Is yf = pfg = specific weight of fresh water and Ys = Psg = specific weight of salt -

-

water, with Ys > Yr. Elimination of Pi yields: yf(q)if

-- Zi)

(a)

-- Ys(q)is -- Zi)

at the interface. In the first place, we suppose that the salt water is immobile and only the flesh water flows. Then qgis -- constant and from (a) it follows that: qgif - -

--q)is

Yf

Ysqgis

- -

C

- -

gf

~Oi(X,

-1--

1

-

--

Zi,

Yf

constant and if we take y,

Z i ) --" C - - OlZi

o~ -

×s-×f Yf

and

(Dif

is replaced by q9i, we get (b)

(1.5.2-2)

Initial and boundary conditions

625

with Zi -- Zi(X, y) -- equation for the steady interface. Differentiating with respect to t of zi - zi(x, y) yields: dzi

Ozi dx

=

dt

t

Ox dt

3zi dy Oy dt

d x _ - K ~ax' etc." which becomes with vx - ne-d--F Oqgi OZi Ox Ox

t

Oqgi OZi

Oqgi

Oy Oy

Oz

(c)

=0.

Differentiating (b) with respect to x and y gives'

Ox

O(/gi OZi 0Zi --"-~~ Oz Ox Ox

Oq3i

Oqgi OZi

Oy t

Oz Oy

Oqgi t

~JZi

;Jcpi aZi =_ ;~_.....~__x Ox ~ + a~

which gives

3z

OZi

3q~i

OZi

c~ Oy --+ Oy

3-7

ot 4 a~0i az

(d)

3Zi

Replacing W and ~ in equation (c) by these equations yields the differential equation for the interface between two fluids of different density, with the underlying fluid at rest:

(099i 2

v)+(

099i 2

-Uy

)+

(099i ~ 2

az J

+o099i -o -aT

(11)

This expression can be understood as the (non-linear) boundary condition along the interface, the shape of which is, however, unknown. In terms of velocities, equation (11) can be written as:

Ux2 i _}_ Uy2 i _qt_Uz2i

--otK

Uzi

--0.

(12)

The steady interface is a stream surface and so ~i - constant

(13)

and the total velocity vi in each point of this surface is tangent to it and v2 = Vxi2 -Jr-Vyi2 + Uzi.2 From (12) it follows that vzi is always positive, so the interface has an upward slope. Also vi < o~K as v 2 - otKvzi and vi > vzi. Secondly, we assume simultaneous flow of both fluids; so also the salt water is moving. The interface i now becomes a moving front zi - zi(x, y, t). At every point on the moving front the following expression holds: dzi dt

Ozi dx -

3x dt

3zi dy t

3y dt

3zi t

3t

626 As

Basic principles

(1.5.2-2)

dx_ dt - -

v_,rd" ne '

ddty _- - v~,_j.f and ne

dz_ dt m

Vz__. f_at f the fresh water side of the aquifer and ne

dx _ onxe__' _~ dy _ Oys and d_z _ one~.__~sat the salt water side, we find two differential dt - dt ~ ne dt m equations for the moving interface: OZi .......

Ox OZi ,,

Ox

OZi Vxf + "2"--l)vf

oy

OZi Vzf -at- n e ~ -- 0

--

•

and

Ùt

(14)

OZi OZi Vxs + ~ Vvs -- Vzs + ne --0.

oy

Ot

Subtraction of these two equations gives a first relation between the differences of the velocities in the three coordinate directions on both sides of the interface: OZi OZi (Uxf -- Uxs) ~ -]" (Pyf -- Uys) -~y -- (l)zf -- l)zs) -- O.

(e)

Differentiation of the boundary condition (a): Yf(fflif --

Zi)

--

Ys ( ~ i s

--

Zi)

with respect to x gives: ( Oq3if O(tgif OZi Yf -~X ~ OZ OX

OZi ) ( O~is OX -- Ys ~ - ~

O(tgis OZi OZ OX

OZi )

(f)

OX "

Within each region on both sides of the interface separately, Darcy's law for potential flow is applicable, however, with different permeabilities: Oqgif __ Vxf "-- -- K f Ox --

kTf O~if Iz Ox

and

i)qgis

Vxs - - K s Ox -

kVs Oqgis

lz Ox

etc. assuming / Z f --" /Zs (equal viscosities). These values substituted in equation (f) gives the second relation between the differences of the Darcy velocities on both sides of the interface: 1)xf - l)xs +

(l)zf-

OZi Uzs) "~x =

k(ys-

~/f)

iz

OZi

(g)

Ox

In the same way, differentiating (a) with respect to y we obtain the third relation.

Oz~ l)yf-

Pys +

(l)zf-

Vzs)

k(ys - ?'f) Ozi

Oy --

tx

(h)

o Oy

~

From the three equations (e), (g) and (h) the three velocity differences dx, dy and

dz can easily be solved: dx=

K* Ozi D Ox'

K* Ozi dr. = D Oy

and

dz-.

K* ~(D-1)

(15)

(1.5.2-2)

Initial and boundary conditions

if dx -- Vxf

-- Vxs, dy --

627

Vvs , dz -- Vzf - Vzs,

vvf -

K*----(ys/z - ?'f) and

D--l+\(-~-x/

+\(-~-y/ "

These velocity differences may be considered as the components of a vector di (dx, dy, dz) at a point on the interface. The absolute value of the velocity difference v e c t o r

is

K*22 { \-if--x/

Ozi~2}

K .2

=

K*2

K*2

D2 { D - I + ( D - 1 )

2}---D-2-(D-I+D

2-2D+1)

= K. 2 D- 1 D ' so, with equation (15):

Idi 12

(16)

- - d 2 - - g * d z - - d 2 .qt._d 2 _.jr_d 2.

The vector ni, normal to the interface at a point of the interface is, if we write the interface as ~. - zi - zi (x, y ) - 0: ni - g r a d ~ . - V f i -

(

Oz~

Oz--2 1'~.

Ox '

03' '

]

It can easily be seen that di and ni are orthogonal vectors as the scalar product equals zero: di . n i -

0.

This means that the velocity difference vector at a point on an abrupt interface between two homogeneous fluids of different density is directed along the interface (lies in the plane tangent to the interface in that point). It follows that in the direction normal to the interface, the velocity difference equals zero and the flow is continuous in that direction, as might be expected. The boundary conditions along the interface of two fluids which flow simultaneously are given by the two equations (14) or by one of the equations (14) together with the equations (15). A steady state of the interface can be reached as soon as the heaviest fluid has become immobile, in which case vxs, Vys and Vzs are zero and dx becomes vxi, etc. Equation (16) will then be transformed into equation (12). 3. The h o d o g r a p h

If the velocity vector at a point P of a flow field is plotted from an origin with coordinate axes Vx, vy and vz the endpoint Ph is called the h o d o g r a p h r e p r e s e n t a t i o n of P.

628

(1.5.2-3)

Basic principles

1)z

Qh

X Fig. 7.

1)x

Hodograph of a streamline in two-dimensional flow.

When all points of a flow domain are plotted in this way, a hodograph domain is obtained and, for instance, the points on a stream surface have their images on a hodograph surface as can be seen in Fig. 7, where the velocity vectors at two points P and Q on a streamline in a vertical x z-plane are plotted from the origin in a V x V z - p l a n e . Hodograph mapping is especially very useful in two-dimensional flow with respect to the boundaries of the flow field; in particular the boundaries with non-linear conditions, like the phreatic line and the interface between two fluids of different density. Although the exact position and shape of these boundaries are unknown, its hodograph is known, because they appear as circles through the origin of the hodograph plane, as can be seen from the equations (6) and (12). The relation between the velocities for the phreatic line is v x2 + v z2 -t- K Vz -- 0 which represents a 1 circle with centre (0, - 1 K ) and radius gK in the V x v z - p l a n e . The interface between moving fresh water and immobile salt water is represented in the hodograph plane by the circle v x2 + v z2 - o l K v z -- 0 (see Fig. 8). Hodograph mapping may assist in solving the flow problem analytically, or at least in giving better understanding of flow in complex domains with non-linear boundary conditions. The following example may illustrate this. Consider an earth dam lying on an impermeable underground, with different water levels at both sides (see Fig. 9). Point A and point F are s t a g n a t i o n p o i n t s with zero velocity, so their hodographs are at the origin. However, A F is a streamline and between A and F the velocities have positive values, with a maximum somewhere in G (unknown). The hodograph representation becomes the double line F h G h A h . The upstream boundary A B is an open-water boundary and the velocities in the dam are perpendicular to it, increasing from zero at A to a value VB at B which is determined by the point Bh in the hodograph plane which is the point of intersection of the line A h B h 2_ A B with the hodograph circle, that corresponds to the phreatic boundary B CD. The phreatic line ends in D and the part DE is the seepage face. The head q) in the dam is a function of x and z: q9 = qg(x, z). At the seepage/'ace, where p = 0,

Initial and boundary conditions

(1.5.2-3)

629

Vz interface

oeK

13x

phreatic line

mi

-K

Fig. 8.

Hodograph circles for phreatic line and interface.

_

Z

I

~'/////////////2/2/2/2/2/2/2/2/S~ ~ ~ ~ ~/////72~

A

G

Vz

E/E/

F

....

/

//

,i! \ \\

-½K ~i I

-) 2K

Fig. 9.

e /I/

/ ii

~ --'- i /

Hodograph of an earth dam.

y////////////,4

F

630

(1.5.2-3)

Basic principles

q)(x, z) - z. The gradient of ~o in the direction s of the seepage face becomes d~o - -

Orp dx =

ds Ux

-

O~odz +

- - cos fl +

dz =

Ox ds

8z ds 1)z

K

- -

ds

or

sin fl - - sin fl

from which Vz -- Vx c o t f l - K,

which in the hodograph is a straight line, perpendicular to the downstream slope of the dam and going through the point ( 0 , - K ) . The intersection of this line with the circle yields the point Dh. At point B the velocity is perpendicular to A B and has the magnitude K cos ~ as can be seen immediately from the hodograph. At the other end of the water table, at point D, the velocity is parallel to the seepage face (AhDh d_ DhEh) and is equal to K sin 13. Judging from the nearly parallel directions of VB and VD, a reflection point C must be present somewhere on the water table, where the velocity reaches its minimum; the representation of the phreatic line thus becomes the portion BhChDh on the circle, with C D double. Point E is a cavitation point where the velocity theoretically is infinite, so its hodograph is the infinite end of the seepage line DhEh. The hodograph of the downstream water boundary E F is the normal to that boundary through the origin; it is therefore parallel to the hodograph of the seepage face. It may be shown that the shaded area in the hodograph plane corresponds to the shaded area in the physical plane. For a more detailed treatment of the hodograph plane, the book by Bear, Zaslawsky and Irmay "Physical Principles of Water Percolation and Seepage" published by UNESCO in 1968, is recommended.

(2.1.1)

Ordinary differential equations

II. A N A L Y T I C A L

SOLUTION

631

METHODS

2.1 ORDINARY DIFFERENTIAL EQUATIONS

2.1.1. Direct integration Ordinary differential equations in groundwater flow only occur in steady onedimensional flow, in steady radial-symmetric flow and in more-dimensional and also in non-steady flow if the concerning partial differential equations have been reduced to ordinary equations by means of one or more integral transformations. They are always linear and almost always of the second order and may be homogeneous or non-homogeneous (see Section 2.1.2). For direct integration it is necessary that the differential equation is homogeneous or eventually non-homogeneous, but then with a constant value for the term that makes the equation non-homogeneous. Example 1. One-dimensional steady flow in a leaky aquifer with vertical infiltration. Differential equation: d2(,o dx 2

q9 q -t)~2 K D

- - O,

~2 __ K D c .

General solution" x

x

q)(x) -- Aler + B l e - r + q c

or

(x) + B 2 c o s h (x)~ + q c .

qg(x)-A2sinh ~

The first form is suitable for a semi-infinite field and the second for a finite field. This solution has been obtained by first solving the corresponding homogeneous differential equation d299

q9

dx 2

~2

=0,

and next adding the particular solution qX 2 ~o --

to it.

KD

= qc

632

Analytical solution methods

(2.1.1)

In the case of a semi-infinite field with zero head at x - 0 (see Fig. 1), the boundary values become"

/------]-99 7///////////////////////////~

99(0)- 0

and

d99 1 x = A1 - - e x dx )~

Fig. 1. Vertical infiltration in a leaky aquifer.

d99 (cx)) -- 0, d-~

--

1 x B1 e-r" ~ '

~dx( o o ) -- 0 gives A1 -- 0 while 99(0) -- 0 gives B1 -- - q c . becomes (see 123.54, Part A):

The solution thus

~o - qc(1 - e - ~ ) .

E x a m p l e 2. Radial-symmetric steady flow in a leaky aquifer. Differential equation: d299 1 d99 J dr 2 r dr

99 ---0, I. 2

)2 - K D c .

This is a homogeneous equation and is known as the modified Bessel differential equation of zero order (see Section 3.3.2) with the general solution r

r

99(r) -- AIo(-£) + B K o ( - ~ ) , in which I0 and K0 are modified Bessel functions of zero order and of the first and second kind, respectively.

iih~,I __

Radial flow towards a circular basin the water level of which sinks by an amount h gives a boundary value for the drawdown at infinity of zero and at the border of h. As 10(cx)) = oo, then A = 0 while 99(R) - h - B K0 (~). The solution thus becomes:

~///////////////~99 c

I I

i

KD

4

R

X0( ) x0(r)

Fig. 2. Radial flow towards a circular basin.

~o(r) - h ~ R "

(See Fig. 2 and 223.23, Part A.) E x a m p l e 3. The ordinary differential equations obtained from partial differential equations by means of integral transformations are almost always of the form: d2cpt dx 2

N99t + M -- 0

or

d299t

d-7 q

1 d99t

r dr

N~t + M -- 0,

(2.1.2)

Ordinary differential equations

633

where 99t = transformed head or drawdown and N and M are functions independent of opt or x or r with N > 0. The general solutions then become respectively (see Examples 1 and 2): M (t9t - A e x "/W + Be - x "/~ - 4 - m N

and

(/9 t -

M AIo(r x / N ) + B Ko(r v/-N) 4- - N

2.1.2. Variation of parameters We consider an ordinary differential equation of the second order, which is nonhomogeneous and linear: d2y dx 2

dy

-F f ( x ) - - : - + g ( x ) y -- h ( x ) ,

(1)

assuming that f , g and h are continuous on an open interval I. We shall obtain a particular solution of (1) by using the method of variation of parameters as follows. The corresponding homogeneous equation d2y dx 2

dy

(2)

-F f ( x ) - : - + g ( x ) y -- O,

has a general solution yh (x) on I, which is of the form yh(X) "- ClYl(X) Jr- c2Y2(X)

with Cl and c2 as constants. The method consists in replacing ¢1 and c2 by functions of x: A(x) and B(x) to be determined such that the resulting function yp(X) = A ( x ) y l ( x ) q- B ( x ) y 2 ( x )

(3)

is a particular solution of (1). By differentiating (3) we obtain dyp dA dyl dB dy2 dx = dx y l -+- A--~x + ~-x Y2 + B dx

We assume that we can determine A and B such that dA dB dx Y 1 + -dTx Y2

(4)

o. d Vp

This reduces the expression for ~ dyp = A d y l + B dy2 dx dx dx

to the form:

(5)

634

Analytical solution methods

(2.1.2)

By differentiating this expression, we obtain

d2yp

dA dyl d2yl dB dy2 d2y___~2 dx2 = d--~-d--x-4- a d - ~ -t dx dx l- B dx2"

(6)

By substituting equations (3), (5) and (6) into equation (1) and collecting terms containing A and terms containing B, we readily obtain d2yl

dyl

(d2y2

dy2

A(-d----xZ 4- f--~---xx 4-gyl) 4-B\--d-----xZ 4- f---~-xx -tI

gy2)

dA dyl dB dy2 ~ =h. dx dx dx dx

Since yl and Y2 are solutions of the homogeneous equation (2), this reduces to dA dyl dB dy2 I =h. dx dx dx dx Equation (4) is" dA

dB

dx Y 1 + -d-SY2

o.

This is a system of two linear algebraic equations for the unknown functions dB The solution is obtained by Cramer's rule" and 7~-" A

dy2

Yl

Y2

dyl

dy2

dx

dx

--

Yl-~x-

d__~A dx

dyl --

W,

Y2 d x

the Wronskian of Yl and Y2.

AA--

0

Y2

h(x)

dr2

/

-- -y2h(x),

AB--

Yl dyl -a-Z

0 -- ylh(x). h(x)

We find

dA dx

y2h(x)

- - - ~ W

and

dB dx

ylh(x)

= ~ . W

(7)

Clearly, W :/: 0, because the functions yl and Y2 constitute a fundamental system. By integrating equation (7) we have f y2h(x)

a(x) -- - ] J

W

dx

and

B(x) -- f ylh(x) ~7 dx.

(8)

These integrals exist because h(x) is continuous. By substituting these expressions for A and B into (3), we obtain the desired particular solution of equation (1)" W dx. yp(X) -- --Yl f y2h W dx 4- y2 f ylh

(9)

Ordinary differential equations

(2.1.2)

635

The general solution of equation (1) then becomes" Y--Yh +Yp, which can readily be verified by substituting this function in (1). If the constants of integration in equation (9) are left arbitrary, then equation (9) represents the general solution of equation (1). Then the result can be written as: y -- A ( X ) y l ( x ) + B(x)y2(x) with dA dx

y2h(x) W

'

~

dB dx

ylh(x) W

( 1O)

and dy2 dyl W - - yl-~x- -- Y2 dx

Example 4. In an infinite field the groundwater head initially is a given function of x (see Fig. 3), for a confined aquifer with transmissivity K D and storage coefficient S. Owing to the gravity, the heads will diminish in a fixed manner, whereas the groundwater will move horizontally in a lateral direction until finally all heads will be equal.

KD, S x

Fig. 3. Given initial function in a confined aquifer. The main objective of this initial value problem is the determination of the depletion function, that is the function that shows the variation of the heads with time everywhere in the field. If the initial function is chosen arbitrarily, the boundary value problem can be written down as: 02(t9 __ f12 0(/9 with f12 = S Ox 2 at KD ' q)(x, O) -- f (x),

qg(-oo, t) - 99(00, t) - 0.

Laplace transformation with respect to t gives (see Section 2.2.2-1): d2q3 dx 2

~(-oo,

s) -

C,(oo, s) -

O.

636

(2.1.2)

Analytical solution methods

The differential equation is an ordinary one and non-homogeneous:

d2q3 dx 2

fl 2 S ~) - - - - fl 2 f ( x ) "

The solutions of the homogeneous differential equation e -~x'F. The Wronskian becomes"

dq)2

w - ,~1 ~d-x , ~ 2 ~

dq31 -d--~-

t

In this case h(x) - - f i 2 f ( x ) , dA

dx

=

fl2 f (x)e-l~x'fi

-2fi,~

are

q)l

-

e ~x~ and

q~2 --

e~ X4-i ( - fl ~/7) e - ~X~ - e - ~X'/-i fl V~et~ X~ = - 2 fl ~/7.

so, according to equation (10)" and

dB

dx

=

fl2 f (x)ePX'fi

-2fi~

and A --

2~/7

f ( x ) e -[~x'/-i dx ÷ Cl,

B -- 2~/-s

f (x)e t~x~ dx ÷

C2.

The general solution is q5 -- A(x)q31 ÷ B(x)q52 -- A ( x ) e z x ~ + B ( x ) e -zx'/i. As qS(cx), s) - 0, it follows that A (co) -- 0, and from ~5(-oo, s) - 0 it follows that B(-cx)) must be zero. Suitable expressions for A ( x ) and B ( x ) then become: fl f f e~X°~ B (x) -- 2~/7 ~ f (x0) dx0

A (x) -- 2~/_~

f (xo)e -~x°'F dx0,

fl qS(x, s) -- 2V/7

e~ f (x0)e -~(x° -x),/-i dx0 ÷ ~ -fl~

and f (xo)e -~(x-x°)4~ dxo,

or 99(x, s) -- 2~-s

_

~

~

f+~

f (xo)e -[~lx-x°l'/~ dx0.

This is the solution of the transformed boundary value problem. Inverse Laplace transform leads to the desired solution (see Section 2.2.2-3b, equation (41)) q0(x t ) - - f l f+oo ' 2v'~7 oo f (x0) exp (see 1 12.01 of Part A).

{ fi2(X_X0)2} -

4t

dx0

Ordinary differential equations

(2.1.3)

637

2.1.3. Use of matrix functions for solving problems in multi-layer systems A groundwater flow system will be referred to here as a multi-layer system or multiple-aquifer system, if it consists of more than one aquifer, separated by one or more semi-permeable layers and if it is assumed that in the aquifers only horizontal flow takes place and in the semi-permeable layers only vertical flow. Then the differential equations for each aquifer also contain state variables (heads or drawdowns) of adjacent aquifers. In that case, a simultaneous treatment of the set of differential equations, one for each aquifer, is necessary, which is a condition for applying the solution method with use of matrix functions.

~EQ

Consider the geohydrological scheme of Fig. 4, where a n-layer system consists of y//# ~ / / / / / / / / / / / / / / / / / / / / Cl n aquifers of transmissivity T1 = K1D1, i T2 -- K2 D2 . . . . , Tn = Kn Dn and n + '.~ Q1 qgl T1 = KI D1 t 1 semi-permeable layers with resistances m ~///////////////////// C2 Cl, c2 . . . . , Cn+l in days. It is supposed r that above the cl-layer and below the cn+li > I layer a fixed water level is maintained, I ~///////////////////// which is chosen to be the reference level. Further, the assumptions are made that on~////////,///////////// Ci+l ly horizontal flow takes place in the aquifers and only vertical flow in the semipermeable layers. The amounts of water passing the semi-permeable layers depend I ~//////////////////// cn on the difference in head on both sides of i]",,,~Qn ~Pn Tn = KnDn the layers and on the value of the resistance c, and are assumed to be distributed ~//////////////////// Cn+l equally over the thickness of the aquifers, ~n+a - - 0 thus occurring as separate terms in the Fig. 4. n-layer system with wells. differential equations for the aquifers (see Section 1.4.2-6, equation (37)). These terms, for instance, for aquifer 1 become: ~oo=0

991 - - q90

K1DlCl

991 - - q92 -k-

K1Dlc2

=

(all

@ a12)~Ol - - a12992,

a s qg0 -

0,

and for aquifer i"

qgi -- qgi- 1 + qgi -- qgi+1 = KiDici KiDici+l 1

1

--aii~Oi-1 Jr- (aii -Jr-ai(i+l))qgi -- ai(i+l)qgi+l,

if we put ~ - - aii, Tici+l = ai(i+l), etc. As an example of flow in a multiple-aquifer system, consider groundwater flow in the system, caused by withdrawal of water with a fully penetrating line well

638

Analytical solution methods

(2.1.3)

with screens in all aquifers and discharges Q l, Q2, . . . , Q,,, as given in Fig. 4. The (ordinary) differential equations for this system for steady flow are:

d2~l

1 d~l

dr 2

r dr

d2~2

1 d~2

dr 2

r dr

= (all -4- a12)~l -- a12992, -- -a22~Ol + (a22 -+- a23)q92 -- a23993, (11)

d2~oi

1 dqpi

dr 2

r dr

d2tpn

-- --aii99i-1 + (aii + ai(i+l))qgi -- ai(i+l)qgi+l,

1 dqgn

dr 2

r dr

-- --ann~On-1 -4- (ann + an(n+l))~On,

and the boundary values: qgi(c~) -- 0,

lim r~0

( dqgi] r

--

dr ]

Oi

= -qi

2zr~

(i - 1 2 . . . . n). ' '

Using matrix notation, this system of n simultaneous differential equations can be written in c o m p a c t form: Vr2tp -- Atp,

(d)

¢p(c¢) - 0,

lim r ~ r-+0

- --q'

(12)

in which V2 =

d2

!

dr 2

1 d

,

r dr

--a12 a22 -I- a23

f a l l -+- a12 --a22

0

0

--a23 .

A

m

0

°°°

--aii

aii -4- ai(i+l)

--ai(i+l)

°

o

.

,

q--

O

qn

ann -f- an(n+l) j

0 0

ql q2

~o2 ~o--

--ann

°°°

•

o

o

The system matrix A is a square n x n matrix; tp, q and 0 are column vectors, each with n elements; Atp is also a column vector with n elements.

Ordinary differential equations

(2.1.3)

639

To solve the vector equation vZcp - Acp, we consider the corresponding single differential equation d2q9 dr 2

1 dq) _I-

-- pq),

r dr

with the general solution:

q) -- aKo(r~/-fi) + blo(rv/-p)

(a and b constants).

Now suppose ¢p -- c{aKo(rv/-fi) + blo(r~/-fi)} -- cq) is a solution of equation (12), with Cl c2

Cn

is a column vector with n constants. As V 2~o -- cV rq) 2 - cpq9 , and also length:

(all + al2)Cl

-

a12c2

2

Acqg, we find: Ac - pc, or written at

V r ~o -

-- pCl

-- a22cl -at- (a22 -+ a23)c2 -- a23c3 -- pc2

(13) -- annCn-1

n t-

(ann

+ an(n+l))Cn

-- pCn.

The vector c = 0 is a solution of this set of equations for any value of p. A value of p for which equation (13) has a solution c -¢ 0 is called an eigenvalue of the matrix A. The corresponding solutions c are called eigenvectors of A. To determine the eigenvalues and eigenvectors, we write equation (13) as (all -

+ a12 -

a22cl

p)cl

-

a12c2

-k- ( a 2 2 q-- a 2 3 -

z

p)c2

-

a23c3

0

-- 0

(14)

-- annCn-1 + (ann -k- an(n+l) -- p)Cn

-- O.

This h o m o g e n e o u s set of linear equations has a non-trivial solution if the determinant of the coefficients is zero" all

+

a12 -

ma22

p

--a12

a22 + a23 - p

• • •

0

--a23

D ( p ) -0 -- 0,

--ann

a n n n t- a n ( n + 1) - - P

(~5)

(2.1.3)

640

Analytical solution m e t h o d s

D(p) -- d e t ( A - pl), where I is the n-rowed unit matrix •

0 I

1

• ° •

0

...

0

Ooo

"

m °

0

...

•

1

By developing D(p) we obtain a polynomial of n-th degree in p. So the eigenvalues of the square matrix A are the roots of the equation (15). In general, there are n solutions p l , p2, . . . , p,,. Once the eigenvalues have been calculated, the corresponding eigenvectors can be determined from the system (14). We have thus found that a solution for ~oi of Vr2q~ = A~p is: q~i -- ei {ai

Ko(r~/-~) 4- bi I o ( r x / ~ ) },

where ei is the eigenvector c i of the matrix A, corresponding to the eigenvalue Pi. There are n solutions, so n

~0 -- E ei { ai K o (r ~/~t ) 4- bi lo (r ~PT) }. i=1

With ~o(c~) - 0 ,

all constants bi become zero and so

n

q~ - - E ei ai K o (r x / ~ l ), i=l

or written at length •

991 -- ellal K o ( r ~ - { ) + e21a2Ko(r~/~) 4 - . . . 4- enlanKo(r~/~n), q92 -- el2al Ko(r ~ff{) 4- e22a2Ko(r ~ff2) 4 - " " 4- en2an Ko(r ~/~n), (16)

gOn - elnalKo(r~ff~) + e2na2Ko(r~/~) + ' "

+ ennanKo(r~n).

In matrix notation equation (16) can be written as: (17)

tp - - E F a ,

where the matrix E is composed of the eigenvectors ei as follows" ell

e21

•••

enl

el2

e22

...

en2

E - (elez-..en) •

eln

• •o

e2n

•••

enn

Ordinary differential equations

(2.1.3)

641

and is called the eigenmatrix of A,

0

K0 ( ; x / ~ )

0

Ko(r~) F

--function diagonal matrbc,

m

0

Ko (r x/~-~n)

and al a

m

al

- column vector of constants.

an

That equation (17) is identical to equation (16) can easily be proved by applying the rules for matrix multiplication. The constants vector a can be solved with the help of the boundary conditions for the well-screens"

lim(rd)

m

_q,

r--+0

as follows: Differentiation with respect to r of equation (16) gives" d r d---~cp - - -

e i a i r ~-fi-[ K 1 ( r ~fi-[t ) , i=1

,im(rd.)

r-+O

f/

-- -- E

eiai

--

-Ea

-- -q,

i=1

from which a - E - l q , where E -1 - inverse eigenmatrix of A. The end solution of the problem then becomes with equation (17)" ~o(r) -- E F ( r ) E - l q ,

(18)

which means that, in matrix notation, the solution consists of a function diagonal matrix, premultiplied by the eigenmatrix of the system matrix, post-multiplied by the inverse eigenmatrix, delivering a new matrix and this matrix multiplied by the column vector of the discharges. The diagonal matrix F contains the same function as will be found in solving the problem for a single leaky aquifer, but now with the eigenvalues of the system matrix as arguments. The question suggests itself whether this is a general rule

(2.1.3)

642

Analytical solution methods

and not only a result of this particular problem. Let us consider an arbitrary square matrix of the n-th order, and its eigenmatrix C:

M

mll

m12

...

mln

m21

m22

.. •

m2n

~

o

,

,

C-(Cl

•

C2 . . .

Cn),

°o

mnl

mn2

•••

mnn

with C i eigenvectors of M. Then M C - (Me1 Me2 . . . Men). As ci is an eigenvector of M corresponding to the eigenvalue ,ki, it is clear that M C - (~1Cl ,k2c2 . . . )~,c,), from which M C - CD, with D the diagonal matrix of the eigenvalues of M:

O

)~1

0

...

0

0

)~2

...

0

Oo•

•

...

,kn

•

0

0

•

Postmultiplying both sides with C -1 we obtain (19)

M - CDC-1.

So an arbitrary square matrix can be written as the diagonal matrix of its eigenvalues, premultiplied by its eigenmatrix and postmultiplied by its inverse eigenmatrix. For the matrix A of the geohydrological system, this means that A - E D E -1. The matrix A 2 then can be written as: A 2 --

E D E - 1 E D E -1

_ EDZE -1

with 0 2 the diagonal matrix of ,k2. This result can easily be extended to yield: A m = EDmE -1

and to the power series f ( A ) f (A) - E

~ Em=l O/mAm;

°lmEDmE-1 - E

m=l

Otm D m

E -1

m=l

Hence: (20)

f ( A ) - E F E -~ with 0

o

.

•

f (,k2) F

~

0 0

o

0

0

o

o

o

s(L)

Ordinary differential equations

(2.1.3)

643

We have thus obtained the important result that an arbitrary function of the arbitrary square matrix A can be written as the product of three matrices, provided that the function f ( A ) can be developed in a power series. The product consists of a diagonal matrix, containing the functions f()~l), f().2), . . . , f ()~n) in which

A has been replaced by its eigenvalues, premultiplied and postmultiplied by the eigenmatrix and inverse eigenmatrix of A, respectively. Now we may write the solution (18) as: ¢p(r) - K o ( r x / A ) q

with q i -

Qi 2rc----~i"

(21)

This result generalizes the well-known formula for steady flow towards a fully penetrating well in a leaky aquifer: Q ~o(r) -- ~ K o ( r ~ / - a l l ) 2 r c K D

1 with a l l -- ~ . K D c

(22)

(See 215.14, Part A.) Many properties of matrix functions are similar to the properties of analogue scalar functions and this enables us to solve systems of simultaneous differential equations in a way quite similar to solving a single differential equation, as may be shown by the solution with use of the matrix functions of the foregoing problem.

Example 5. dZcP ~ = Acp, dr 2 r dr

¢p(oo) -- 0,

(d.)

lim r r-+O ~ r

--q"

General solution: ¢p -- Ko(r~/A)a + Io(r~/A)b with a and b as column vectors containing constants. As ¢p(e~) = 0, it follows that b = 0, dcp = - x / A K 1 (r x/A)a, dr

lim r x/AK, (r v/-A) - I, r-+O

SO

lim

= -a = -q,

r--+0

and therefore ¢p(r) - K0(r~/A)q

(see 720.03, Part A).

Although the use of matrix functions in multiple-aquifer systems is very convenient and enables us to solve rather complicated problems in an easy way, leading in most cases to simple analytical formulas, the analytical evaluation of the ob-

644

Analytical solution methods

(2.1.3)

tained matrix functions will become very complicated if more than two aquifers are involved. However, nowadays effective methods are available to evaluate matrix functions numerically by means of computer programs. For a two-layer system it is possible to derive analytical expressions for an arbitrary function of the matrix that describes that system. Consider the function f ( A ) , where

al )

--a22

a22 -t- a23

(see A in equation (12) with n - 2). According to equation (20) we have to develop

f(A)--E(f(pl) 0

0 )E-1 f(P2) '

and we start with the determination of the eigenvalues pl and p2. tion (15) we obtain

D(p) --

all -t- a12 -- p --a22

--a12 a22 + a23 -- p

From equa-

-- 0,

which means that pl and p2 are the roots of the quadratic equation" p2 _ (all -+- a12 ~- a22 + a23)P -+- alia22 + alia23 -+- a12a23 -- 0. For p -

Pl the system (14) becomes" (all -t- a12 -- pl)Cl -- a12c2 -- 0, --a22cl + (a22 + a23 -- pl)c2 -- 0,

from which Cl

a12

c2

all + a12 -- Pl

and an eigenvector el belonging to Pl becomes: el-(

a12

).

a l l -+- a12 -- Pl

In a similar manner we find: e2--(

a12 ). all + al2 -- P2

The eigenmatrix E of A then becomes: E--(

012 all + a12 -- Pl

a12 ) all -k- a12 -- P2 "

(23)

Ordinarydifferentialequations E

Det

= al2(all

+ a12 -

E-l--

P2) - a l 2 ( a l l

-- P 2 )

As f(A) - E (f(Pl) \ 0

--(all

Pl) -- al2(Pl

-

P2), and thus

2_al2)

+ a12 -- P l )

0)

f(P2)

645

+ a12 -

ta,l+al2_

1 al2(Pl

(2.1.3)

a12

"

E-

we find, after some mathematical manipulations, taking into account that from equation (23) it follows that P l + P2 = a l l + a12 + a22 + a23

and

P i p 2 - - a l i a 2 2 + a l i a 2 3 + a12a23,

that f(A) = Pl

-- P2

(24) X ( (all +al2--P2).f(Pl)--(all +al2--Pl)f(p2)

\

--a22 {f ( P l ) - f (P2)}

--a12 {f(Pl ) - f (P2) } (a22 +a23 -- P2)f (Pl)-- (a22 +a23 -- P l ) f (P2)

/

E x a m p l e 6. Example 4 for 2 layers • 9 ( r ) - Ko(rv/A)q, if q - (q~) (pumping in the upper aquifer): ~p(r) =

ql ((all+ P l -- P2

a12-

p2)Ko(r~-7)-

(all-Jr-a12--a22 { Ko (r ~P-7) -- Ko (r ~ / ~ ) }

pl)Ko(r~-~)) "

E x a m p l e 7.

d ///////////////////'//////////////////,//,/////~ C1

T1 ~//////////////////////,/~

Another example of a two-layer system is a fully penetrating canal or river with a level different from the reference level of the system (for instance, the polder level; see Fig. 5). d2

C2

dx 2 ~o -- Ag,

r2

9(0) - h,

~o(~) =o,

Nergg Fig. 5. Fully penetrating canal in a 2layer system.

q~-

, q92

A--

(all +a12 a12) a23--0, h --a22

a22

'

"

646

Analytical solution methods

(2.1.3)

The solution is" ~o(x) - e-X4Ah

(see 710.12, Part A)

which, according to equation (24) can be written:

--

(qgl) ~02

-

h ((all--p2)e-X4~-(--(all_Pl)e-X~) Pl -- P2 --P2 e-x4Ti + Pl e - x 4 ~

with Pl and p2 the roots of the quadratic equation" p2 _ (all -k- a12 -k- a22)P -+- alla22 -- O. R e m a r k 1. In the foregoing examples the matrix ~/A occurs, and although ~/A cannot be developed in a series, it is still allowable to write ~/A - EA D(v/-~/)EA 1, because ~/~/~-

EAD(~/)EA1EAD(v/-~/)EA 1 -- EAD(v/-~/)D(v/-~/)EA 1 = EAD(Xi)EA 1 -- A,

so indeed ~ is a matrix, that multiplied by itself yields A. ~ is not unique, as can be shown, for example, for a (2 x 2) matrix:

0

,

0 0 -

0

-4

0) S

l

0

because all 4 solutions, if multiplied by themselves, yield A. The square root of a (3 x 3) matrix has even eight solutions, four of them only differing in sign from the other four. In general, the square root of a n x n matrix has 2n solutions. It can be shown that the system matrix A of the multiple-aquifer system has real and positive eigenvalues, so that there is always a solution of ~/A with real and positive eigenvalues too, the only solution that satisfies the boundary conditions. R e m a r k 2. Matrix multiplication is commutative if the factors involved are functions

of the same matrix f (A)g(A) = g(A) f (A), because f ( A ) g ( A ) - EAD{ f (~,i) }EA1EAD{g(~,i) }EA 1

: EAD{f(Xi)g(Xi)}EA 1 = EAD{g(Xi)f(Li)}EA 1 -- g(A) f (A).

(2.2.1)

Partial differential equations

647

2.2. PARTIAL DIFFERENTIAL EQUATIONS

2.2.1. Separation of variables If two or more independent variables occur in the differential equation to solve, for instance, x and t, r and t or r, z and t, we have to do with partial differential equations, and unlike ordinary differential equations, a general solution with a number of constants does not exist. In many cases, however, it is possible to reduce the partial differential equation to two ordinary ones, by applying the m e t h o d o f separating variables.

Consider, for instance, the partial differential equations for one-dimensional and for radial symmetric non-steady flow:

02(/9

and

02cP = f12 O~p

1 0(/9

f12 0(/9

r Or

Ot

We assume that solutions of these equations are such that the independent variables occur separately in them: qg(x, t) = X ( x ) T ( t ) and ~0(r, t) = R ( r ) T ( t ) where X, R and T are functions of x, r and t only. Differentiation of ~0(x, t) and qg(r, t) twice with respect to x and r, respectively, and once with respect to t, yields: d2X

dT

and

- - fl 2 X ~

T dx2

dt

T

d2R

T dR

dT

-~. . . . ~r 2 r dr

fl ZR ~ dt

Deviding these expressions by T X and TR, respectively, we obtain 1

d2X

X dx 2

=

f12 dT

1 d2R

and

T dt

R dr 2

1

t

dR

=

rR dr

f12 d T T dt

The expressions on the left side of these equations depend only on x or r and on the right side only on t. Hence, both sides of the equations must be equal to a constant, for instance, p. Thus,

1 d 2X X dx2

=

/~2 dT

= p

T dt

1 d 2R

and

R dr2

~

1

dR

rR dr

=

f12 dT T dt

=p,

from which three ordinary differential equations are obtained: d2X dx 2

p X - O,

d2R

1 dR

dT

p

dt

f12 T

-- 0

and (1)

dr 2

t

r dr

pR

-0.

In these equations p is still arbitrary. The solution of these equations depends on the initial and boundary conditions of the problem, which will be made clear by the following examples.

648

(2.2.1)

Analytical solution methods

E x a m p l e 1. In an infinite confined aquifer, the initial head of the groundwater is a given function of x: ~o(x, 0) = f ( x ) . This is the same problem as in Example 4 of Section 2.1.2; the objective is to determine the depletion function. The boundary value problem can be written as:

02q 9 __ f12 0(19 OX2 Ot '

qP(X, O ) - f (x),

~0(-cxz, t) = 99(cx~, t) = 0 .

We assume ~p(x, t) = X ( x ) T ( t ) and, according to equation (1), we find d2X dx 2

pX-O

and

If we take p positive, say p

dT dt

p

j~2 T

-- 0.

the general solution for X becomes

- - 0/2,

o~2

X (x) = c le ux + c2e -~x,

and for T:

T (t) -- c3e~'Tt

and for ~p(x, t) = X ( x ) T ( t ) : (p(x, t ) -

~t

(Ae ~x + Be-~X)e~

.

Because increasing values of x and t lead to increasing exponential functions and thus to increasing values of the head, this solution has no physical meaning. Therefore we choose p negative, say p = _0/2, and then we find a solution: ~o(x, t) -- {A cos(o/x) + B sin(o/x)} exp

- ~-~t .

Since A and B are arbitrary, we may consider these quantities as functions of 0/ and write A = A(0/) and B = B(0/). Since the differential equation is linear and homogeneous, the superposition principle is valid (Section 2.3.1) and the function ~o(x, t) -

f0

{a(0/) cos(o/x) + B(0/) sin(o/x)} exp

(

)

- ~2 t do/

is then a solution, provided this integral exists and can be differentiated twice with respect to x and once with respect to t. According to the initial condition ~o(x, 0) = f ( x ) , it follows that oo

f ( x ) --

L

{A(0/) cos(o/x) + B(0/) sin(o/x)} do/.

This is a Fourier integral representation for f (x) (see Section 2.2.3-1, equation (60)), whence A(0/) -- --

f ( k ) cos(0/)~) dk

and

i f +~ f(~.) sin(oeX) d~,

B(0/) -- --

7Z"

cx3

(2.2.1)

Partial differential equations

649

SO

~o(x, t) -- --

?

f()~) { cos(o/x)cos(o/)~) + sin(o/x) sin(o/~.) } d)~

Jr

cx~ o/2

x exp ( - fi-:ft) do/

= --1 fo~f_~-,c f (~.) cos(o/x - o/)~) exp ( - o/2 t ) d)~ do/. Jr

~

Assuming that we may invert the order of integration, we obtain:

i f +°°f()~) fo °~ cos(o/x -

qg(x, t) -- -Jr

o/)~) exp ( -

o 2)

do/d),..

cxD

The inner integral can be evaluated by the use of the formula f0 ° e -a~2 cos(2bo/) do/-- ~1 ~

e b,,2

(a > 0),

to obtain by differentiating both sides of equation (21) of Section 3.1.1 with respect to x, giving

fl2(X --)v) 2

/~ [ + ~

_

qg(x, t)

2 ~ - - [ J_oo

f ( L ) exp

--

4t

} d)~ (see 112.01 of Part A).

Example 2. In a circular island with radius R the initial head is a constant amount h higher than the surrounding surface water level. There is a resistance w against outflowing groundwater at the boundary r -- R. The aquifer is assumed to be confined. The objective is to determine the depletion curve as a function of space r and time t: ~o = ~0(r, t) = head.

R Fig. 1. Depletion function for a circular island with entrance resistance. 2----~-~ Oq -t-

Or 2 09o

~(o,

Or

1 099 _- -/ 3 2 099 r ar at' t) = o ,

099 --(R,t) Or

=

~o(r, t) Kw

99(r, 0) = h.

650

Analytical solution methods

(2.2.1)

Application of separation of variables gives, according to (1): d 2R dr2

1 dR

t

r dr

pR = 0

with ~p(r, t) = R(r)T(t). solution qg(r, t ) -

and

dT

p

dt

f12

T =0

If we choose p positive, say p

=

C~2, we find a general

{AKo(oer) + BIo(o~r) } exp (~2 t).

This solution has no physical meaning because increasing values of time lead to increasing heads. Therefore, we take p negative, say p -- -or 2, finding as a general solution:

01"2t)" qg(r, t) -- {AJo(otr) + BYo(otr)} exp ( - ~ , 0~o 0---7-= { _ AotJ1 (oer)

-

B o l Y 1 (fir)}

exp

--

~--~t .

~Or( 0 , t) - - 0 and as Y1 (0) - - o o B must be zero, so °12t)

~p(r, t) -- AJo(c~r) exp ( - ~-g .

__ 0~o (R t) -- -AOtJl (cZR) exp ( Or

'

-

-~

°t2t) ,

t)---KwA Jo(c~R)exp(-~-~

from which

¢zJl (ot R ) = ~ Jo (oeR ), Kw ocn from which ot - W with ol,, being a root of OtnJl(Oln) - 8Jo(Otn), with e -There are infinitely many roots oln (n - 0, 1 ' 2, " " .) and thus ~o(r, t) may be Kw" represented by

oo

~o(r, t) -- Z

(olnr) (ot2t) enJo\---R- exp f12R2 ,

n=0

according to the superposition principle (see Section 2.3.1), provided that, according to the initial value qg(r, 0) = h, the constant h may be represented by

ot2r~ c~lr'] h -- A1Jo (--~--) -J- A 2Jo (--R- j

+-.-

if

oe. J1 (Otn) -- 8 Jo (Otn).

In Section 2.2.4-3 it will be shown that under certain conditions an arbitrary function can be represented by a so-called Fourier-Bessel series:

Partial differential equations

( nx)

C¢3

f (x) -- ~

(2.2.1)

cnJo ,,--~-/

651

with

n--O

2

2 o~n cn = a2 (or2 + 82)jg(O/n )

fo

( OlnX

x f (x) 3"o

', a

) dx,

with o6, being the roots of OgnJl(Ogn) -- 8Jo(ogn). (See equations (123) and (124) of Section 2.2.4.) oo ffnr ~ So, if h - ~--~n=oAn Jo ( ~ , then 2

06,

2

An = R2 (ol2 + e2) j2 (Oen)

°lnr dr. fo R hrJo( ---~-)

Otnr. The integral in this expression becomes, with u - --y-

h fo R rJo (O~nr) ~ dr-

hR2[uJl(u)]Un hR2foUn °e2 u Jo ( u ) d u - oe---Tn o hR 2

J1 (Otn) -

h R 2 e 3'o(O~n)

Oln

Thus, the final solution becomes" oo e J0(~nr---R--) 99(r, t) -- 2h S (oe2 + e2)J0(oln) exp

2t ) f12R2

R ,

e-- K w

n--O

(see 236.05, Part A) with otn being the roots of OlnJ 1 (Ogn) -- 6 J o ( o l n ) . In general, the method of separation of variables, applied to practical problems, leads to Fourier and Fourier-Bessel integrals and series, as has been shown in the foregoing examples. The method, therefore, forms an inportant part of the development of the theory of Fourier and Hankel transformations as will be shown in the Sections 2.2.3 and 2.2.4. As soon as knowledge about these transformation methods has been acquired application of them to practical problems leads to much easier and quicker solutions than the method of separation of variables.

2.2.2. Laplace transformation

1. Fundamental properties The integral transformation techniques are based on the fact that, by means of a skillful substitution in integral form, one of the dimensions of the differential equation for the groundwater flow can be eliminated, and by continuing this procedure the partial differential equation can be reduced to an ordinary differential equation or even to a common algebraic equation, which usually will not yield any difficulties in their solution with help of the likewise transformed boundary values. The solutions obtained in this way must undergo a reverse or some reverse transformations in order to obtain the desired solution of the problem.

(2.2.2-1)

652

Analytical solution methods

The Laplace transformation generally is used to eliminate the time variable; so it is most frequently applied to non-steady problems. There are many textbooks on operational methods in which the Laplace transformation is thoroughly treated; here we will restrict ourselves to developing the Laplace transformation method briefly, giving a number of theorems and examples, necessary to understand the way in which most of the unsteady problems in this book have been solved. Let F(t) be a given function which is defined for all positive values of t. We multiply F(t) by e -st and integrate with respect to t from zero to infinity. Then, if the resulting integral exists, it is a function of s, say F(s) and is independent of t" F ( s ) --

f0

L{F}.

e-StF(t) d t -

(2)

This operation on the function F(t) is called the Laplace transformation of F(t), and the new function F(s) is called the Laplace transform of F(t), which will usually be indicated by L{F} or by means of a bar: F(s) or shortly F. Furthermore, the original function F(t) is called the inverse transform or inverse of F(s) and will be denoted by L -1 {F}, that is

F(t) -- L -1 {F}. In geohydrological problems we generally wish to find a solution for the head or drawdown qg. Besides a function of t, if the problem is unsteady, q9 may be also a function of the space variables x, y, z, r; during the Laplace transformation with respect to t, however, they will be considered as constants. In the following we will assume that q9 is a function of x and t and ~3 or L {~0} a function of x and s" q~ -- ~o(x, t),

q3 -- qg(x, s)

L(cp(x, t)} --

e -st~p(x, t) dt - q3(x, s)

so

and

L -l{q3(x, s)} - ~o(x, t).

T h e o r e m 1 (Linearity theorem). The Laplace transformation is a linear operation,

that is, for any functions f (t) and g(t) whose Laplace transforms exist and any constants a and b, we have L { a f ( t ) + bg(t)} - a L { f } + bL{g} - a f + b~, for, by definition, L { a f ( t ) + bg(t)} -

-

-

e-St{af(t) + bg(t)} dt a

/o

e - ' t f (t) dt + b

=af(s)+b~(s).

fo

e -st g(t) dt

(3)

(2.2.2-1 )

Partial differential equations

Theorem

65 3

2 (Transforms of derivatives). I f 99 = 99(x, t), then L ~

--

--

fo fo

e - s t ~ dt 8t

ji

e -st d 9 9 - [99e-St]0 -

= -99(x, O) + s

99 de -st

99e-st dt.

So,

{~}

L -~

- s~(x, s) - e(x, o),

(4)

or in words: the Laplace transform o f the first derivative o f a function is the Laplace transform o f the f u n c t i o n multiplied by s minus the initial value o f the function (if t is the time variable).

By applying (4) to the second-order derivative ~~)t2 , we obtain:

1" 0299 L|

099

099 099

= s[sL{99} - 99(x, 0)] - -~-(x, 0),

or L { 02(/9

-5-ff } - s

2@

099

(x, s) - s~o(x, o) - -57(x, o),

and, in general: 0n99 } __ S n n-1 99(X, O) -- S n - 2 0 9 9 ( X , O) . . . . L --~ ~)(x, s) - s Ot

- 0~-~99(x O) Otn-1

'

(5)

•

Transform of the derivative with respect to other variables becomes:

fo ~ -- ~O f ~ Ox

fo ~ e-St 99 dt - ~O~ (x, Ox

s)

654

(2.2.2-1)

Analytical solution methods

and also

L[ 02¢p~} -- aX 202 ~°°

e-'e~o dt -

a2q9 OX2 "

(6)

Theorem 3 (Transforms of integrals). L

/fot

f(r)dr

J

--f(s).

S

(7)

dg

Take g(t) -- fo f (r) dr, then 77 - f (t) and according to (4): L { f ( t ) } -- L { -dT} dg - s L { g ( t ) } - g(O). Clearly g(O) - 0 and therefore L {g(t) } -- { L {f (t) }. Thus also

ll

s, J - fo

qg(x, r) dr.

(8)

Theorem 4 (First shifting theorem). If L{qg(x, t)} = ~(x, s), then L { e atqg(x, t)} -- ~(x, s - a),

(9)

because L { e at

~(x, l)}

e at~o(x, t)e -st dt

--

fo ° qg(x, t)e -(s-a)t dt -- ~(x, s - a);

--

that is, the substitution o f s - a f o r s in the transform corresponds to the multiplication o f the original function by e at (shifting along the s-axis).

Theorem 5 (Second shifting theorem). If q3(x, s) = L{qg(x, t)}, then multiplying q5 by e -tos, where to is some positive constant, we have e-t°S~(x, s) --

•

/o

e-S(r+t°)qg(x, r) dr.

Substituting t = r + to, we obtain: e-t°S~(x, s) --

f?o

e -stqg(x, t -- to) dt.

Partial differential equations

(2.2.2-1)

655

f(t)

Ua(t)

/ a

f ( t - a)

f(t-

a)Ua(t )

I

a

I

i

"-"---'----"

/

[ i

t

a

t

Fig. 2. Shifting along the t-axis and unit step function. We need to write this integral as an integral from 0 to oe. For this purpose we may replace ~p(x, t - to) by the function, which is zero on the interval 0 ~< t < to, and is equal to qg(x, t - to), when t > to. The function

qg(x, t - to)Uto(t) --

0

for t < to,

qg(x

,t-t0)

fort >to

has the desired property, where Uto (t) is the unit step function, defined as follows"

b/to (t)

-

l0 /1

whent to.

Therefore

e-tos(9(x, s) -- L{(p(x, t - to)Uto(t)},

(10)

that is, the multiplication of the transform of a function by e -tos yields the Laplace transform of the original function, shifted along the t-axis over an amount to and multiplied by the unit step function Uto(t). An illustrative example of a function f ( t - a ) U a ( t ) , where f ( t ) is an arbitrary function of (t), which is also defined for t < 0, is shown in Fig. 2. Theorem

6

(Convolution theorem). If the convolution of two functions F(t) and

G(t) is denoted by F(t) • G(t) and is defined by F(t) • G(t) -

f0 t F ( r ) G ( t -

r)dr,

(11)

656

(2.2.2-1)

Analytical solution methods

the Laplace transform o f this function o f t is the product of the transforms o f the functions: (12)

L { F ( t ) • G(t) } -- -ff(s)-G(s), which can be made plausible as follows (no proof)" -ff (s)-G(s) -- -if(s)

f000

G ( r ) e -st dr --

f000

G ( r ) F ( s ) e -st dr.

From Theorem 5 we know that F ( s ) e -st -- L { F ( t - r)ur(t)}. It will be convenient to define F ( t ) to be zero when t < 0; then F ( t - r) will be zero for t < r and the unit step-function ur(t) can be omitted. Thus,

f0

-ff(s)G(s) --

G(r)

f0

e -st F ( t - r) dt dr.

Here the boundaries of the first integral refer to r, whereas the boundaries of the second integral relate to t. We assume that the order of integration can be interchanged, finding: -F(s)--G(s) --

f0 f0 e -st

G ( r ) F ( t - r) dr dt

in which the boundaries of the second integral relate to r. However, F ( t - r) is zero for r > t, so the boundary oo may be replaced by t and we have

-ff (s)-G(s) -- L

{/ot

G ( r ) F ( t - r) dr

}

--L{F,G}.

It can easily be shown that fo G ( r ) F ( t - r) dr - fo F ( t - r ) G ( r ) dr, which means that the convolution operation is commutative. T h e o r e m 7 (Derivatives of transforms). ~ ( x , s) -Og) 5s -

02( Os 29

f0 ° e_St

_

e-Stq)(x, t) dt,

(-t~(~,

fo c~ e-Stt2 q)(x,

t) dt - L { - t ~ ( x , t~ },

t) dt -- L {t2qo(x, t)},

(13)

(2.2.2-2)

Partial differential equations

657

so that, in general:

Osn

= L{(-t)"~o(x, t)}.

(14)

Theorem 8 (Integral of a transform). Consider the transform F ( p ) (Laplace transform with letter p instead of s)

fr__F ( p )

dp -

fr fOCXaF(t)e -pt dt dp

foC~OF(t) fr

-

e -pt dp dt

-- f o ° ° f ( t ) ( - ~ ) ( e - r t - e - ' ~ t ) d t . Letting r approach infinity, we have: F ( p ) dp -

provided that limt_+0 ( - ~ )

t

e

dt - L

t

'

(15)

exists (not proved here).

2. Some examples of Laplace transforms The transform of a constant c may be derived directly from the definition of the Laplace transform (2)"

L{c}--c

fo

e-stdt---

e -st o o _ _

]

S

0

(16)

S

1 and t h u s L { 1 } - - 7" Also, by direct derivation"

L {t/~} --

fo °

Substitute x - s t , L,

t/~e-st dt.

then

,!1 -

S

Sk+l

f cx~Xke-x dx

1

= (by definition) --z-rzF(k + 1) S,~-v,

(gamma function, see Section 3.4.1).

So,

L{t k} -

F ( k + l)

sk+l

(k > - 1 )

(17)

658

Analytical solution methods

(2.2.2-2)

For k - m (m integer and positive) m!

L {t m }

--

sm+l

(18)

,

2 L{t 2} -- ~ ,

1 -~,

L{t}-

1

L{1} -- - , s

k-

~,

(19)

s~/s- = 2s4rJ -'

1

L {eat } - -

{1}

L

fo

e ate -st

1

F(½) - ~

--

,/7

dt

fo

--

•

(20)

e -(s-a)t

[e_(S_a)t] ~

with a a

dt

constant.

s-a

Thus, 1

L {e at } --

(21)

s--a 1 _ - a t } and with Theorem 1 and equation (21): L{sinh(at)} - L{~1 e a t - ~c

1

1

1

1

L{sinh(at)}--~s_a-2s+a

a

-- s 2 - a 2"

(22)

Also 1

L{cosh(at)}

--

1

2s - a

1 -i-

1

.2 s. +. a.

s S2

-

(23) a2.

Substituting a - ib in equations (22), (23) we find: b L{sin(bt)} -- s2 + b2

L{cos(bt)} -

F (t) d2F dt 2

s 2 s +

- - c o s 2 (at),

and

(24)

(25)

b 2"

dF dt

-- - 2 a sin(at) cos(at),

= 2a2{ s i n 2 ( a t ) _ cos2(at)} -- 2a2{1 - 2 c o s 2 ( a t ) } - 2a2{1 - 2F(t)}.

(2.2.2-2)

Partial differential equations

659

According to Theorem 2" L{ d 2 F -~

and as F ( O ) -

dF

-- s F ( 0 )

] -- s 2 F ( s )

(o),

1 and -d-/ dF (0) -- 0 we find:

s2F(s)

- s -- 2 a 2 { 1 s

2if(s) }.

Solving for F ( s ) yields s 2 ~- 2 a 2 L{ cos2(at)}

-S ( S 2 nt-

(26)

4a2) "

L{ sin 2(at)} -- L{1 - cos 2(at)} - L{1} - L{ cos 2(at)} 1

s 2 + 2a 2

2a 2

S

S (S 2 + 4 a 2)

s (s 2 -1-- 4 a 2)

(27)

L{t sin(cot)}. If we put F ( t ) --sin(cot), then according to Theorem 7

dF(s) L { t F ( t ) } -- - ~ ; ds L{sin(cot)}-

F(s)-

CO S 2 _.]._CO2

(see equation (24))

and thus L {t sin(cot) } --

Also with F ( t ) L

t

m

F ( p ) --

2cos (28)

(S 2 _Jr_0)2)2"

sin(cot)"

i 1 --

/~(p) dp

according to Theorem 8;

09

p2 -k- o92

(see equation (24))

and thus p2 -k- 092 --

arctan co

.,.

=

2

arctan

- arctan

;

SO

L

t

-- arctan

.

(29)

660

(2.2.2-3)

Analytical solution methods

The Laplace transformation of an impulse function (see Section 1.5.1-2, Fig. 4 and equation (3)) can be derived from the second shifting theorem (Theorem 5 of Section 2.2.2-1). For instance, a discharge impuls I ( Q ) = Q i a ( t - to)

is defined as an instantaneous abstraction of a finite quantity of water Pi [L 3] from a groundwater body with a discharge Qi [L 3T-l] during a very short time dt. The delta function means that 1 6(t - to) -

I(Q)

for t -- to, f o r t -/=to.

0

(Kronecker delta)

We write Pi - Qi A t and thus Qi with Pi a fixed quantity (see Fig. 3) and

At

ai

I(Q)--

Qiuto(t)-

Qiuto+zxt(t),

where u is the unit step function (see Theorem 5). Then according to Theorem 5, equation (10) we have

to to+At Fig. 3. Discharge impulse.

Qie-StO _ Qi e-s(to+At) = Pie_St ° 1 - e -Ats L{I(Q)}

=

s

s

Ats

and thus lim L { I ( Q ) }

= Pie -st°.

At--+0

(30)

3. I n v e r s e t r a n s f o r m s

By means of Laplace transformation, applied to a differential equation for nonsteady groundwater flow, with boundary values and an initial value, a new differential equation is obtained, in which the time variable has been eliminated, as follows from Theorem 2, applied to the non-steady one-dimensional differential equation, 02~0 __ f12 0(/9 OX 2

Ot

which gives, after Laplace transformation: d2q5 dx 2

t~2{s,~- ~(x, o))=o,

Partial differential equations

(2.2.2-3)

661

in which 93 - q3(x, s) is a function independent of time. The solution of this ordinary differential equation, called the subsidiary equation, together with the also transformed boundary values for x, is a function of the Laplace parameter s. In order to obtain the desired solution of the problem, the transformed solution q3(x, s) which has been found must undergo a reverse Laplace transformation (see, for instance, Example 4 of Section 2.1.2).

a. Application of the inversion theorem There are a number of ways to find the inverse transform of a given function of s. In the first place there are explicit formulas for the inverse transform L -1 {F(s)} as is the case for the transform L{F(t)}. The most useful of these formulas involves an integral in the complex plane, where s is a complex variable, which is stated in

Theorem 9 (Laplace inversion theorem).

f y+iot

F(t)-

L-I{F(s)}-

1 lim eSt-ff(s) ds. 27ri o~-+oo,,×_io,

(31)

The integration takes place in the complex plane, with respect to s - x + i y, along the straight line x - ?, from - o o to +oo; 9/ is so large that all singularities of F(s) lie to the left of the line x - 9/ (see Fig. 5). There are conditions on F(s) or F(t) for the validity of equation (31), but neither these nor the proof of the theorem will be considered here, which implies that it should be verified subsequently whether the solution that was found by means of this theorem satisfies the differential equation and the initial and boundary conditions of the problem. Dependent on the character of the complex function F(s), there are two standard methods of procedure to put the line integral of equation (31) in real form. Two examples of geohydrological problems, the solutions of which have been obtained using each of these procedures, follow below.

Example 3. N~

b

a

Fig. 4. Strip with open water storage.

One-dimensional finite flow in a strip with width 2b. Zero flux at x = 0 and open water storage at x - b for a confined aquifer (Fig. 4). Sudden drawdown h of the surface water level.

Analytical solution methods

(2.2.2-3)

662

Differential equation for one-dimensional non-steady flow" Oq2~0 ._ f12 a~O

ax 2

at '

where 9 - 9 ( x , t) - drawdown and f12 = KD" s The initial condition is: ~p(x t ) = {0 [ ' [ h

f o r 0 <~x < b , forx >b.

Atx-0, ~Ox( 0 ' t ) - 0 . The condition for open water storage at x - b becomes (see Section 1.5.1-3c2°):

K D ~--~-~

O9 (b t)

Ox (b, t) - - a 0-7

'

"

Laplace transformation with respect to t applied to the differential equation with initial and boundary conditions, yields (see Theorem 2 of Section 2.2.2-1): d2~

fl2Sq3 --" 0,

dx 2 dq5 x- (-bd'

d~

~(0

S) -- 0,

dx a s) = -~{sdp(b,KD

' s) - h},

with q3 = q3(x, s). The general solution of this ordinary differential equation is q3 -- A cosh(flx ~/~) + B sinh(flx ~/s). As ~dx( 0 ' s) = 0, it follows that B - - 0 dq5 (b, s) = Aft sinh(fib~7) = dx

a

K D { As cosh(flb ~/7) -- h },

and thus as

A fl ~ sinh(flb ~/7) + K--D cosh(flbx/~)

ah = KD

The transformed solution then becomes: h ~(x,s)

=

cosh(flx ~Cs)

-

S flKa D" sinh(flb,f~) ~

+ cosh(/3 b Vts)

Application of the inversion theorem gives, according to equation (31): 1

lira

f y+iot

e st~(x, s) ds.

(32)

Partial differential equations

(2.2.2-3)

663

The integrand eStq)(x, s) is a single-valued function of s, because all roots of s vanish, as can easily be shown by using the series for the hyperbolic functions: Z2

coshz=

Z4

1+~.

Z3

+~.v + " "

and

sinhz-z+~.v

Z5

+~+'''"

The points for which the integrand becomes infinite are s - 0 and the zeros of the denominator: f l K D sinh(flb4ff)

a

+ cosh(flb,v/7) - O,

47

fib,v'7 c o t h ( f l b ~ ) --

-

~2 K Db a

or

bS

As coth z -- i cot(iz), these zeros are the roots c~,, (n - 1, 2 , . . . ) of bS withO--~,

OenCOtOen----O

O/n

a

- - i f l b ~ d T , so that ~

/~b and Sn --

C

+Or

R

B

o,2

iol n

=

V

So 1

-- Ol

A

f12 b2 "

Thus, so = 0 and s~ are single poles of the integrand, lying on the origin and on the negative x-axis. Now consider the line integral as a part of a closed curve which should not pass through any of the poles, such as, for instance, the circle A B C in Fig. 5. It follows from Cauchy's integral theorem that the integral along the closed curve A B C A is equal to 2zri times the sum of the residues of the integrand at its poles within the contour. It can be shown that as the radius of the large circle tends to infinity, the integral over the arc C B A tends to zero. Thus, at the limit, the line integral of equation (32) is equal to 2zri times the sum of the residues of its integrand at the poles s - 0 and

Fig. 5. Contour for a single-valued function. Sn --- -

The residue at s - 0 Res [e st q3(x s--O

°'2 ( n - - 1 2,

f12 b2

,

becomes" h

h

-

fl K D ~ a

b

.qt_ 1

0 + 1

..

.)

.

664

Analytical solution methods

(2.2.2-3)

f (s) with If we write e st q3(x, s) as heSt sg(s)

f (s) -- cosh(flxv~)

and

g(s) --

fl K D sinh(flb~/7) a

47

+ cosh (13b x/~-),

then the residue at s - Sn becomes:

Res s=s,,

he m f (s) ] S~sg(S)

,,'=s,,

ot2

n With Sn =-~2"/,2 and ~ - - t ~ - - - f iol f we find

f(sn) _ c o s h l f l x (

}_tiff) iOtn __COS( b°lnX~/,

fl K D 1 sinh (flb ~/7) + fl2 K Db cosh (/3 b V/'s-) a 2s ~/7 2as

d d--Tg ( s ) --

fib + ~ - ~ sinh (,6 b,~/J-),

d cls

s.-:-g(s.)

-

-'

1 fl2KDb flKD sinh(-io6,) + ~ cosh(-o6,) otn 2a a - 2 i~--T

io.

-~)

sinh(-iOtn)

0 0 1 sinc~n+ cosoen-sinotn 2a~n 22 c~" " fin

Using these results and remembering that sin o~, =

cos oln, we finally have from

equation (32)

oo c ° s ( ~ n-X ~be x p] (

go(x, t) -- 1 +-----~+ 2hO Z

n=l

(0 +

l) 2 -+-Ot 2)

with otn as the roots of ol,~cotun - - 0 - - ~

~°t2t )

COSO~n

(n-

1, 2 . . . . ).

(33)

Partial differential equations

(2.2.2-3)

665 The graphical construction of c,,, is shown in Fig. 6. If t tends to infinity, the series in equation (33) vanish and a steady state is reached with

Y y=t

/

zr/2

Ot1 / z r

3zr/2 i

O~2

2zr h qg(x, oo) - hs = 1 + 0 '

;

which means that the amount of groundwater released from the aquifer equals the amount of water responsible for the rise h - hs of the open water Fig. 6. Roots of otn cot ogn a level in the storage basin: bsh,, - a(h-hs). Solution (33) has been used for finding the depletion function for an initial head h, given in 138.02 of Part A by putting the head ~o'(x, t) - h-qo(x, t) and by modifying --

hS

0 (O~,2 + l) 2 --t- l))COSOI n

Example

sin c~,,

to

.,,(1 +

for practical reasons.

L9

4.

i~~ __h[~///////////////////~., q9 ~ c

R Fig. 7. Leaky aquifer around a circular basin.

Radial flow in a leaky aquifer towards a circular basin with radius R in which the surface water level is "suddenly" lowered by an amount h. Above the semipermeable layer, a constant level is maintained, which serves as a reference level for the drawdown qo(x, t). The differential equation for non-steady radial

flow in leaky aquifers becomes (see Section 1.4.2-6): 02@

1 8~0

Or2

t r Or

99 ~2 =

f12 0(/9 Ot

The initial condition is ~p(r, O) --

! 0 / h

f o r r > R, for r - - R.

with ~2 _

K Dc

and f12 =

KD

666

Analytical solution methods

(2.2.2-3)

The boundary conditions are cp(R, t) -- h and ~p(cx~, t) - 0. Laplace transformation with respect to t gives:

d293 t-

dr 2

r dr

fl2s Jr-

~-~

qg-O,

~(R

s)-

'

-

93(oo, s ) - - 0

s'

with the solution (see Example 2 of Section 2.1.1):

h Ko(flr~/s + ~2-~) ~ ( r , s ) --

(34)

-

s

+

We consider the function -if(r, s)

-

Ko(flr~/~) KO (flRvC~)" The inverse transform of this function

yields, according to equation (31),

f y+iot e st Ko(flr~/C#) 1 lim ds. F(r, t) -- L -1 {F(r, s)} -- 2yri ~--,~ ay-iot K o ( fl R v/-# )

F +or

Now consider this line integral as a part of a closed curve which does not pass through the singular points of the integrand. These singular points are the zeros of Ko(~R~/-#) and the point s = 0. The integrand is doublevalued, owing to the occurrence of ~/J, except at the point s = 0, which therefore is a so-called branch point.

1

jJ¢

We choose a contour such that the value of the contour integral, according to the theorem of Cauchy, becomes zero, which means that the

A -o~

singularities must lie outside this contour.

The contour A B C D E F

of

Fig. 8 satisfies this condition, as it is known from the theory of Bessel Fig. 8. Contour around a branch point.

functions that no zeros of Ko(fl R ~/'#) are enclosed by this contour. In the limiting case, c~ and Pl tend to infin-

ity and P0 approaches zero. Thus we may write, according to Cauchy's theorem:

Partial differential equations

2rci F(r, t) -- lim +

(2.2.2-3)

F

lim Pl -+oo p0-+0

+

lim

f f (r' s)eSt ds -

667

Pl-+cxzlimfa B

-ff (r' s)eSt ds

££-

P0 --+0

lim

£-

E

Pl -+oo

F

P0 -+0 pl -+oo

F(r, s)e st ds

F(r, s)e st ds + lim

C

F(r, s)e st ds +

D

F ( r , s)e st ds.

It can be shown that the integrals along the parts A B and E F of the large circle become zero if pl tends to infinity. Along the small circle s - poe i° and ds -

i poe i° dO, ~

- ~p-oe½i° and thus f

lim /

if(r, s)e st dt

PO--+0 J C D

= lim P0 -+0

yr

F

exp () . t p 0 i°. e

n"

Ko(flr~-~e½ i°) i io) ipo ei° dO - O, Ko(flR~e7

Ko(ax) =1. x-+O Ko(bx)

as lim

Along B C is s -- pe-~ri= - p

and ds - - d p ; ~ / 7 -

~I-pe -1iTr -

-i~/-~. We have

from equations (79), (74) of Section 3.3.3" Jr

Ko(flr ~/7) -- K o ( - i f l r v/--fi) -- i -~ { Jo(flr x/-fi) + i Yo(flr v/-fi) }, and so F ( r , s)e st ds - - f ~ e -pt Jo(firv/-fi) + iYo(flr~/-fi)

lim

p,~oo

po-+O

c

Jo(~Rq/-P) + iYo(flRv/-fi) •

Along D E we find, with s - pe '~r - - p , ds - - d p ,

dp.

(35)

1.

v/}- - ~/-~e7 'Jr - i~'-~ and

from equation (84) of Section 3.3.3:

Xo(flr

lim ;o -+°

pl -+oo

~/7)

--

Ko(ifir ~/-p)

D -F (r, E

-

s ) e 't d s - - -

-i

Jr { Jo (fir q/-fi ) - i Yo ( fi r ,g/-fi) }, -~

fo °°

e -p'

Jo(flr ~f-d) - i Yo(flr~Ffi) dp.

Jo (fl R V/-~) - i Yo (fl R ~/~)

(36)

668

Analytical solution methods

(2.2.2-3)

Thus, only the line integrals along the negative real axis contribute to the end value of the line integral along AF. Together they yield ((35) + (36)):

BC+DE -if(r, s)e st ds = 2i fo ~ J°(flRv/-fi)Y°(~r~/-fi)- Jo(~r~-fi)Yo(~R~/~) pt Jg (fl R ~/-fi) + r~ (fl R ~-fi) e- dp , as can easily be verified. Substituting u -/~Rff-~, we find for the inverse transform of F(r, s)"

F(r, t ) -

rcflaR2

fo

-

J~(u) + Yg(u)

exp

(

u2t flZ R2 ) u du

or shortly

F(r, t) -- 7rf12R2

h(u, r) exp

as the inverse transform of F(r, s)

--

f12R2 u du, Ko(flr~ )

Ko(flR~/~ ) .

(37)

Then, according to the first shifting

1. theorem (Theorem 4, equation (9) of Section 2.2.2-1), we find, with 77 - ~21)2 = c-g" g -1 {F(r, s + r/)} --

F(r, t)e -~t

and from equation (34), applying Theorem 3: u2c

rp(r, t) -- rrfl2R2

)

flZR2

h(u, r) exp

r/r u du dr.

As exp

/~2 R 2

r/'r dr --

1 - exp

u2

f12R2

f12 R 2

r/t

,

7]

we find

¢p(r, t) -- ~Jr

{

R------Th(u, r) 1 - exp u2+~

(

/~2R 2

r/t)} du.

As

u2t Ko(~R~/s + r/)

YrflZR 2

h(u, r) exp

f12 R2

r/t)u du

(38)

Partial differential equations

(2.2.2-3)

669

is, by definition of the Laplace transform:

Ko(fir~/s+rl) Ko(flR~/s + rl)

2

-- 7rfl2R 2

2 7rf12R2

foO°fo °°

h(u, r) exp

fo ~ h(u, r)

(

u2t

f12R2

) rlt - st u du dt

u du

u2 . fl2 R--"7-- nt- 7] -]- S

Letting s approach zero, we have:

Ko(C£) _ _2 fo ~ ~ h (uu ,R2 r)du.

Ko(-~)

rc

bt 2 -Jr- --~

The right side of this equation equals the first term of equation (38). Thus, the end solution of our problem may be written as"

~o(r, t) -- h K ° ( - ~ ) 2 h f o ° ° U Ko(R)

n"

R------yh(u,r) exp (

U2 -]- 7

u2t ]~2R2

~/t) du

(39)

with

h(u r) ,

-

Jo(u)YO(R u) -- Yo(u)Jo(R u) Jg(u) + y2(u)

1 and

/7-

1

fl2)v 2 ---" c S "

The first term of equation (39) represents the steady state solution (see 223.22 and 223.23 of Part A).

b. Solutions obtained without use of the Inversion Theorem Though all Laplace transformed functions F(s) can be inversely transformed to the original function F(t) by means of the inversion theorem, in general, it is a laborious undertaking, which requires a considerable knowledge of complex analysis, as is shown in the foregoing Examples 3 and 4. Therefore, methods have been developed to obtain reverse transforms in an easier way, generally starting from already known transforms, collected in a table of transforms and derived from these by means of the various theorems, discussed in Section 2.2.2-1. For instance, the function F ( s ) - e -h'/;: which, together with related functions, occurs frequently in one-dimensional problems in semi-infinite aquifers, can be transformed back to the original function by constructing a linear differential equation for F(s) and next by determining the inverse differential equation for F(t) and solving for F(t), as follows" T -- e_k,/;:,

dF _ ds

__k -kvq, 2~/s e

from which dZF 2 dF ds 7 -~ s ds

k 2 -s F -- 0.

d2ff ds 2

k__ -kvG _jr_k2 e-kv~ 4s x/s e 4s

670

(2.2.2-3)

Analytical solution methods

The inverse Laplace transform yields (see Theorems 6 and 7): 4t2F - 2

fot

r F ( r ) dr - k 2

fot

F ( r ) dr - O.

Differentiation with respect to t gives" d F - + 8t F - 2t F - k2 F -- 0 4t2--~

4t 2

or

dF dt

+ (6t - k 2) F -- 0,

from which k2 (c--constant).

F(t)--ct-3/2exp(---~)

ds

2~/~e

- L{-tF(t)}

-- L

- c t -1/2 exp

~

(Theorem 7)

or k2 L-1l!e -k~} --c't-1/2exp(---~) 47 1

1

c!

and f o r k - 0 "

(see equation (20))

from which !

c --

1

and

c

k --

2v~

Thus we have found: L-l{e-k47}-

k 2t~

k2 exp ( - - ~--~)

(k

>

O)

(40)

and 1

1 e-k47

}

exp(-~--~)

(also k - 0).

(41)

Example 5. One-dimensional groundwater flow in a semi-infinite leaky aquifer towards open water without entrance resistance. The drawdown of the surface water level is an arbitrary f u n c t i o n of the time F ( t ) , with F(0) = 0; ~0(x, t) = drawdown. Fig. 9. Leaky aquifer with open boundary.

(2.2.2-3)

Partial differential equations

0299

99t9 __ f12 099

OX 2

~2

~ 2 __ K D c ,

-~'

e ( x , o) - o,

671

~2 __

KD'

99(oo, t) - 0 .

99(0, t) -- F ( t ) ,

Laplace transformation with respect to t gives"

d2q3

( 2S

dx 2

fl

1) ÷ -2~ 0 -- O,

~ ( 0 , s) -- F ( s ) ,

~ ( o o , s) -- 0

m with the solution ~ ( x , s) - F ( s ) e x p ( - f l x ~ / s

+ rl), in which 7/--

fl21).2 - -

1

7s"

If we write C,(s) -- e x p ( - f l x ~/s + rl), then according to the convolution theorem (Theorem 6, equations (11) and (12)):

99(x, t) -- L -1 {q3(x, s)} -- F ( t ) • G ( t ) -- f0 t F ( t - r ) G ( r ) dr. F ( t ) is arbitrary, but G ( t ) can be obtained from equation (40), using Theorem 4:

~x

G ( t ) --

2t ~ - 7

/

f12X2

k

4t

exp /

rlt),

and thus

99(x, t) --

X/o'

2~/~-

Substituting 0/2 __ --

F ( t - r ) r -3/2 exp

f12x2 4r ' from which r

--

t

4r

fl4ot2 2x2

,

2

)

Or dr.

and r -3/2 dr -

- 2 d ( r - 1/2) _

- 2 d ( ~-7) 2~ -- _ 4 doe, while the boundaries 0 and t of the integral are replaced by ~x cx~ and 7-~' respectively, 99(x, t) can be written: I

99(x

,

t) -- ~ _

F t

40/2

exp

-- 0/2

X2 4~.20/2 doe.

(42)

2,,/7 (See Part A, 123.31.) If a s u d d e n d r a w d o w n h takes place which is kept constant thereafter, then -- 0 for t -- 0 and - h for t > 0 while F ( s ) - h__. s Equation (42) then becomes" F(t)

99(x,t) -- ~

exp 2,/7

-

x2) 4)~20/2 do/.

672

Analytical solution methods

(2.2.2-3)

This integral can be evaluated by making use of the solution of the following indefinite integral:

f exp(-a2x2 - bx-4)dx =

# 4a

{ e2ab erf(ax

+ -b) + x

e-2aherf( a x b-- - ) } + x

constant,

which can easily be verified by differentiating the right side of the equation. We find:

~o(x, t) -- ~h{ e~x erfc (fl~

+

~)

x

(fix

+ e-~ erfc 2~/7

~)}

(43)

or qg(x, t) -- hP(~, ~-~) - polder function (see Section 3.1.2 and Part A, 123.32) 1 We have also found the inverse Laplace transformation" with r / - fl21~2 = ~"~.

~ ,lies ~~ / -,

(~ ,~)

(44)

If we consider a gradual drawdown of the surface water level, for instance, according to a linear function F(t) = at we have, from equation (42):

2a qg(x, , ) -

~

F

{t

x Iv 247

2at ff~

-~_

f12X2 4ot2 )exp (_ot2

exp ( - ot2

X2 4,k2ot2)dot

x 2

4~2ot2)dot

247

afl2x2 fz~Xt exp ( _

ote

X2 )dot 4)~2ot2 ~'"

The first integral equals at P (~, ~ ) and the second integral can be evaluated, by substituting c o - - ~ , 1 to lafl2x)~Pconj(~,~-i) (conjugate polder function). Thus: ~o(x, t ) - - a t P

1

,

(x~

+ =a~2xXPconj

2

4~

)

(Part A, 123.34).

(45)

For this problem we could have started directly from the transformed solution qS(x, s) with F(s) -- ~ (equation (18) with k - 1) finding the inverse transform:

~ {~

/ t'(~

~ ,con~(~

,~)

~46,

(2.2.2-3)

Partial differential equations Example

673

6.

Continuous abstraction of groundwater from a leaky infinite aquifer by means of a fully penetrating line source. The discharge is an arbitrary function o f t. ~0(r, t) = drawdown. See Fig. 10.

c

Fig. 10. Line well in a leaky aquifer.

0299

1 0(t9

(t9

Or 2 ~ r Or

2 0(t9

)v 2-

)v2 = fl --~'

qg(r, O) -- O,

99(00, t) -- O,

K Dc, f12 ._ Oqo)

lim r r -+O -~r

---

KD' Q(t) 2 rc K D

Laplace transformation with respect to t gives" d293

ldq3

dr 2 ~

(

1) fl2s --~

r dr

~a(oo, s) - 0 ,

q9 - 0,

m

lim r

r -+ O

---

-~r

2 7r K D

with solution: ~(r, s ) with 1 " 1 -

Q(s) ~Ko(flr~/s 27r K D

1 fl2~2

1 c"-S"

_ --

(47)

+ 71)

The inverse transform may be obtained if the inverse

transform of Ko(k,,/7) is known. In Section 3.3.2 the inverse Laplace transform L -1 {(~/7)~Kv (a~/}-)} -a~ 1 e x p ( - @ ) has been derived in a similar way as has (2t)v+ been done for L -1 {ekv~s} (equation 40). So 1

k2

5 oxp(- at),

(48)

Now, applying the Theorems 4 and 6 to q3(r, s) we have: ~p(r, t) --

1

47rKD

fot

Q(t-

r)exp

(

-

f12r2

4r

- ~r

)dr -r- ,

and with c o - r/r" ~o(r, t) - 4 r r K D

(see Part A, 215.12).

Q(t - - ) e x p r/

- co

4~.2co

09

(49)

674

Analytical solution methods

(2.2.2-3)

If a sudden abstraction Qo takes place, which is kept constant thereafter, then Q(t)-Ofort-OandQ0 f o r t > 0, while ( ~ ( s ) - a0.s Equation (49) then becomes:

QOforlt exp ( - 09

qg(r, t) -- 4zrKD

r2 ) ~de° , 4X2co co

(50)

or

Qo W r/t qg(r, t) - 4zr K-------D ' (well-function of Hantush, Section 3.2.2 and Part A, 215.13). We have also found the invers transform: _

1

L_l{ 1 Ko(k~/s + ~)} -- ~ W (r/t, k~/-~).

(51)

S

If a sudden abstraction Q0 is followed by an exponentially increasing function of the time: Q(t) Q0e ~2t from which ~)(s) - Q0 qS(r,s) then becomes"

Qo Ko(flr~/s + rl) and q3(r, s) -- 2re K D q3(r, s

+ 0/2) - -

s

--

0l2

Oo Ko(flrv/s + ot2 + ~) -- L {e -~2t qg(r, t)}

2rcKD

and thus Qo eo~2t ~ W { ( o l

~o(r, t) -- 4rrKD

2.2.3. Fourier

2 -°t-

rl)t, flrx/ot 2 -at- r/} (Part A, 215.16).

(52)

transformations

1. Fourier series and integrals Practically every periodic function occurring in geohydrological problems can be represented by a trigonometric series, also called a Fourier series, for instance, an arbitrary function f(x) with period 21"

f (x) -- E

anCOS --'~-/ + bn sin \---i-- /

n=0 (x)

~?/X

oo

22"X

- - a o + E a n c o s ( - - - ~ - ) + ~ bn sin (--~-) n=l

= ao 4- a l cos --~

n=l

q- a2 cos \

1

4- a3 cos \ - - 7 / q - . . .

rex (2rex (3rex +blsin(-T-)+b2sin\ l )+bgsin\---~)+....

(53)

Partial differential equations

(2.2.3- l)

675

The determination of the coefficients an and bn is done by integrating both sides of (53) with respect to x between the boundaries -1 and +l. We first determine a0 by integrating both sides from -1 to +l"

f+'

f+'

l f(x) dx-ao

l dx +

f+' {an (.,~x l

~-"

COS

(~]]

1 ) +bnsin\--T- /

dx

n=l

The first term on the right equals 2a0/, whereas all the other integrals are zero, as can be readily seen by performing the integrations. Hence we find for a0

,f+'t

ao -- -~

f (x) dx.

(54)

Next we multiply (53) by cos(~-~), where m is any fixed positive integer, and then integrate from - l to +l, finding:

t

f (x) cos

1

dx

---/-) + b,~sin (.~] \---~-/}] f7' [a0+ /a cos (.~

COS

myrx

1 ) dx.

n=l

The first term on the right side f ~ l a0 c o s ( m- rTc x - ) d x - 0

t

bn sin

1 ]

cos

1

again and also

dx - 0

because sin(nJrx -7-) is an odd function, cos(m~x ~ dx an even function and their product l / an odd function, yielding zero if integrated between boundaries which are at the same distance from the origin at both sides of the origin. Further:

(nyrx )cos( mrcx )dx cos,,,-5I

f+l

{~x 1an f+'l 1 f_+, --[--(m + n) } dx + -~ l

2an

cos

cos --~(m - n) dx.

The first integral on the right side again equals zero, whereas the second integral is zero if m, -y: n and equals anl if m - n. Thus we find for an"

if_ +l n~x~ l f (x) cos ( - - - ~ / d x .

an -- -[

(55)

676

Analytical solution methods

(2.2.3-1)

By applying a similar procedure for the determination of bn in (53) (multiplying by sin(mJrx - 7 - J~ and integrating from - l to -t-l) we find:

bn

--

1 f+l

7

l

nzrx

(56)

f (x) sin (----[--) dx.

If f ( x ) is an even function, we have

1 fol f (x) dx,

(t/WX)

a n - 72 fol f (x) cos --7-- dx

ao -- 7

and

bn - O,

so that f ( x ) can be represented by the cosine series:

0(3

nTrx,~.

f (x) -- ao + ~-~an COS(--7-- /

(57)

n---1

If f (x) is an odd function, we find a sine series

oo

nyrx

sin (-7-)

(58)

n=l with

bn --

72f0'

f (x) sin \ - - ~ / d x .

If a function f (x) is given only over the interval between x = 0 and x = l, both the cosine series and the sine series will represent this function in the interval 0 < x < l. Outside this interval the cosine series (57) will represent the even periodic extension, or continuation of f ( x ) , having the period 2l, and the sine series (58) will represent the odd periodic continuation of f ( x ) with the period 21 (see Fig. 11). The series (57) and (58) with coefficients are called half-range expansions of the given function f (x). They will have important applications in connection with the solution of partial differential equations, especially if periodic functions are involved, but also in finite regions, where parts of non-periodic functions may be considered as parts of periodic functions. Since, of course many practical problems do not involve periodic functions or finite regions, it is desirable to generalize the method of Fourier series to include non-periodic functions as well. in general, if we have a periodic function fT (x) of period T and let T approach infinity, then the resulting function f ( x ) is no longer periodic. The Fourier series of f T (x ) is

f r (x ) -- ao + Z

n--1

an COS

(2 nx) T

+ b,, sin

(2 nx)j T

'

(2.2.3-1)

Partial differential equations

677

f(x)

i

jl

given function f(x) it, X

even continuation of f(x) I

J

f

-2l

I I i i

-l

2l I

f(x)

odd continuation of f(x) I

/

J

Y

-l

-21

21 J

Fig. 11.

Even

and odd

continuation

of a periodic

X

i i

x

f

function.

in which

if,

ao -- -~

T

T

r f r (x) dx 2 T

b~ -- T

f r (x) cos

a~ - ~

'

r fr (x) sin

T

dx

T

2

and (59)

dx.

2

If we use the short notation

Otn =

2rrn T ' then

27r(n + 1) O / n + 1 __ Of n m

AIy

2zrn

m

2jr --

T

T

T

2

and

--

T

Ac~ =

Jr

and thus,

if= T

fT (X) -- ~

oo

T

cos(olnX) Aol

r fT (X) dx + -2

ofT (X) COS(OtnX) dx

Jr

n=l

2

T

+ --1 ~~ sin(olnx) Aot f ~ 22"

n=l

fT (X) sin(olnX) dx. T 2

678

(2.2.3-1)

Analytical solution methods

We now let T approach infinity; then T1 approaches zero and so also the first term 2rr of f T ( x ) . Furthermore, Aot -- T approaches zero and it seems plausible that the infinite series in (59) becomes an integral from 0 to c~, which represents f ( x ) , namely,

lf0~ {A(ot) cos(xot) 4- B(ot) sin(xot) } dot,

f ( x ) -- --

(60)

Jr

f ( x ) cos(otx) dx

A(ot) --

and

B(ot) --

f (x) sin(otx) dx.

¢x:)

This is a representation of f (x) by a so-called Fourier integral. It is clear that this approach merely suggests the representation (60), but by no means establishes it; in fact, there are some conditions for the validity of (60), of which the most important is that the integral

F

~ If(x)l dx

exists.

(x)

The simplifications for even and odd functions are quite similar to those in the case of a Fourier series representation of f ( x ) . If f ( x ) is an even function, then B(ot) = 0 in (60) and

If °~A (ot) cos(xot) dot

f (x) -- Jr

with A(ot) -- 2

f

ox:)

f (x) cos(otx) dx.

(61)

If f ( x ) is an odd function, then A(ot) = 0, and B(ot) sin(xot) dot

f (x) -- -Jr

with Oo

B(ot) -- 2

L

f (x) sin(otx) dx.

(62)

2. Finite Fourier transformations a. Finite Fourier sine transformation

By this transformation the operation is understood in which the unknown function , l ) and the product is integrated with respect to x from 0 q)(x) is multiplied by sin ~mrx

Partial differential equations

(2.2.3-2)

679

to l, thus obtaining a new function, denoted as Sn{{p(x)} or as 93(n), which is independent of x" q~(n) -- fo l {p(x) sin (nrcX 1 ) dx

(n-l,2,

..).

(63)

For instance, the finite sine transform of a constant c is: g3(n) -- c

fo l sin (I12TX)dx _ nTr Cl {l l /

--C0S(/'/~)};

so

Sn{c} --

Sn {x } -

cl nTl"

{1 - (-1)n}.

f0' x sin

--~~

nTr

l

(64)

'fo'

/dx-

XCOS

riTZ"

l

o

+ ~ nzr

+ nZ:rr2

l n~

x d cos cos

sin

-T-]

dx

l / o

Thus" 12

Sn{x} -- ~ ( - 1 )

"+l.

riTZ"

(65)

The main property from which the finite Fourier sine transformation derives its value for solving certain partial differential equations, is, that it reduces the differentialquotient of the second order, ~Ox2 ' to the transformed function itself, as can be shown as follows"

Snl 0293

ff~x2} -

102(t9

-~-Tx2s i n (

I /

sin(T) d(~° ~xx )"

dx

Partial integrating gives:

£n{O2q 9

099

~x2}-[-~-xSin( =

1

117~X~ l

IITg fol

cos

d(p,

l

/]o

l /

l

as the first term on the right side becomes zero. yields"

F}-

,

°c°S,,Tj o (T) -

(IITgX 0~9 dx

cos\

/ )0x

Once more partial integration

~osin\

1 )dx.

680

(2.2.3-2)

Analytical solution methods

The second integral equals the transformed function q3(n). Hence:

Sn { 0299

-

n~

+ -7- {99(0)- (- 1)nq:,(1)}.

-

(66)

Thus, for finite regions along a space variable x, y or z, the finite Fourier sine transformation deserves consideration if the heads (or drawdowns) on both sides of the interval are given. According to the Fourier analysis developed in the preceeding section, the function qg(x) can be represented in the interval from 0 to l by (see equation (58)): oo

~p(x)--~bnsin(

l

/'

n--1

2 l mrx 2 2 in which bn -- 7 fo qg(x)sin(--T-)dx - 7Sn{qg} - 793. Thus, the inverse transformation, turns out to be very simple:

S/1 { ~ ( ~ ) } I ~ (X) l

2 ~ 7 Z n--1

nrcx

~(~)sin( - U )

(67)

Example 7.

.

.

.

.

One-dimensional finite flow through a confined aquifer in a strip with at one side a constant surface water level and at the other side a sudden fall of the surface water level, which is thereafter kept constant, qo(x, t) = drawdown. See Fig. 12.

i;

.

Fig. 12. Strip with open boundaries. 02q 9

Ox 2 -- fl

209 O'---t

--0, ~o(0 t) '

with f12 __

S

K D ' qg(x, O) - O, 9(b t) - J O ' [ h

fort-O, fort>O.

A finite Fourier sine transformation with respect to x gives, according to (66)"

-

nTr)2

-~

~

nrch

~---(-1

)n

--

fl2dq3

d-t-'

q3(n,O)-O.

Partial differential equations

(2.2.3-2)

681

The transformed solution becomes"

~(n' t) - hb (-1)n exP

- ( n2~t }) - ~

-n---~hb(-1)n"

(68)

The inverse transform of -,,--Y hh (_l)n - ~TY(-1 hb2 )n+l becomes ~ as is given by (65). So with (67) we have:

hx qg(x, t)

-

b

2h ~-~ ( - 1 ) n+l 7r n=l

sin

n

(nTrx](n27r2t) f12b2 . b / exp

(69)

The first term on the right side represents the steady state as would be expected (see 134.02 and 134.03 of Part A).

b. Finite Fourier cosine transformation This transformation consists of multiplying the unknown function qg(x) by c o s ( ~ -~-) and integrating the product with respect to x between the boundaries 0 and l, yielding a new function, denoted by C,,{qg(x)} or ~(n), which is independent of x" q3(n) --

~o(x) cos --7-- dx

(n - 1,2 . . . . ).

(70)

Examples:

(nrcx ) cl n yrs x t c o s \ I d x - nzr sin \ 1

[

C.{c}-ct

)1' .

0'

SO

C.{c} - o .

(71)

Cn{x} - £ l

(nrcx )

1

xcos\

1

dx-

nzr

,[ x sin ,2[

l

/ 0

nzr

nzr

///221.2 COS \ T

'

fo

xdsin sin

\

---7--

I

dx

0

12

.2~2 { cos(.~)

- 1};

SO

C.{x} -

12

(72)

682

Analytical solution methods

(2.2.3-2)

The main property of the finite Fourier cosine transformation may be derived as follows:

(/'/7I'X ~

fo I cos (F/TFX 0(/9 \-'T-)d(-~x )

cos \ - - i - - t dx -

--

I / o + -7-

~x cos

sin \ ~

] dq9

Oqg(1)cos(nrr) _ Oq) nzr [ (nzrx]] l O--~~xx (0) + -l- ~osin \ - 7 - - 7 o 12

~pcos ---7-- dx;

SO Cn{ 02(/9

/'/27/'2 12 q3 +

_~x2 } _

(_l)n ~O~o O~o(0). x ( 1 ) - ~x

(73)

Like the sine transformation, also the finite Fourier cosine transformation reduces the differential quotient of the second order to the transformed function itself, together with boundary values at both sides of the interval. As these boundary values concern the fluxes, it is obvious that the finite Fourier cosine transformation should be applied to finite regions, if the derivatives are given at both sides of the interval. The function ~p(x) can be represented over the interval from 0 to l by (see equation (57))

(n x)

oo cp(x) -- ao + Z

an cos --7--

n=l

with

1/o1

ao -- 7

and

qg(x)dx

an -- -[

q)(x) cos - - ~

dx,

or

2 2~ an -- 7 C n { 9 } - 7q)(n)

and

1 a 0 - 793(0),

in which q~(0) represents the value of q3 for n - 0. Thus, the inverse transformation becomes:

1 c; 1

-

-

7

2 ~ (o) + 7

(nrrx) cos

-7-

•

(74)

n---1

Example 8. Non-steady one-dimensional flow in a confinied aquifer, caused by constant groundwater abstraction by means of fully penetrating well-screens at mutual distances 2b; q)(x, t) - drawdown (see Fig. 13).

(2.2.3-2)

Partial differential equations

683

=~q

=*q

I

KD, S

X

b

iI

b

2b

Fig. 13. Abstraction from well-screens.

Only the interval between x -- 0 and x = b has to be considered and qg(x) there will be represented by the cosine series (cf. Fig. 10), because of the even periodic extension. The boundary value problem can be written down as: 02q 9 __ f12 0(t9

f12 __

57'

S XD '

Oq)

a--f-~ Ox (o ' t) - o ,

q)(x, O) - O,

q

Ox ( b ' t )

2KD

Finite Fourier cosine transformation with respect to x gives, according to (73): -

/'/7r) 2 q __ f12 dq~ -b--- q~+(-1)"2K----D -d-t-'

~3(n,0)-0

with the transformed solution: O(n t) -- (-- 1)n b2q ' 2n27r 2KD

1 -- exp

t f12b2

(75) •

The inverse transform needs the value q3(0, t) (see equation (74)). Letting n 2 - m approach zero in (75) we find, after applying once the rule of l'H6pital: (o(0, t) -

qt 2bS'

cp(x, t) -

2bS

so that ÷

E Yr2 K D ,,=l

n2

cos

---if--/

1 - exp

flZb2 t

(76)

(see 135.02 of Part A). e. Finite Fourier resistance transformations In the previous examples we have seen that one-dimensional problems in finite regions, concerning partial differential equations, can be simplified considerably by means of finite Fourier transformations, the choice between sine and cosine

(2.2.3-2)

684

Analytical solution methods

transformation depending on the character of the boundary conditions, as the sine transformation is related to given heads and the cosine transformation to given fluxes at both sides of the finite region. The question arises whether for more complicated boundary conditions and for various combinations thereof at both sides of the interval, generalized Fourier transforms are also available, leading to simplifications in a similar way. Indeed, a general theory has been developed in which special series as solutions of a generalized differential equation, satisfy conditions of the form 099 c199 q- c2-~x - 0

at x - a

and

Oq9

c3q9-Jr-c4 Ox = 0

at x -- b

for the interval a ~< x ~< b. Here C1, C2, C3 and C4 are given as constants, at least one in each condition being different from zero (Sturm-Liouville problem). This general theory will not be discussed here; only some problems that may be solved by means of Fourier transformations will be described. As to the boundary conditions, we will restrict ourselves to the interval 0 ~< x ~ b with either 99(0) -- 0 or ~0x (0) - 0 and 399

CliO(b) + c2-2--(b) -- O. Ox

In practical problems the latter condition occurs if there is a certain resistance against outflow of groundwater from an aquifer into open water, or inversely inflow of surface water into an aquifer (see Section 1.5.1-3c1°). If on the other side of the finite region the head is given we may apply the so-called finite Fourier sine "resistance" transformation as can be shown by the following example.

Example 9.

•

In a finite confined aquifer the initial head is an arbitrary function of x. There is a constant zero head at x - 0 and an entrance resistance w at x - b. See Fig. 14. The boundary value problem for the head ~p(x, t) becomes: 0299 __ f12 0(/9

f12 __

-dT'

S xo

b Fig. 14. Strip with entrance resistance.

q)(x, O) - f (x),

99(0, t) - O,

aqg(b, t) 3x

qg(b, t) Kw

Separation of variables (see Section 2.2.1) yields a general solution ~o(x, t) -- {A c o s ( p x ) + B sin(px)} exp ( -

PZt)

Y

'

Partial differential equations

(2.2.3-2)

685

From ~o(0, t) - - 0 we may conclude that A - - 0 .

ox(b't)--Bpc°s(pb)exp

-~--~t

p2t)

- - ~ es ixnp((p- b~)- ~ K w

or

pbcos(pb) -

Kw

sin(pb),

if we assume B -fi 0. Thus, p must satisfy, with p - g, c~ cot or - - e O/X

~0(x, t ) -

B sin ( T - )

b ,and xw

ot2t exp (

flZb2 )"

As there are infinitely many solutions of c~ - Oln (n -- 1, 2 , . . . ) , assuming or0 -- 0, ~o(x, t) may be represented by

99(x, t) - Z

Bn sin

exp

flZb2

(77)

,

n=l

provided that, according to the initial condition ~o(x, 0) - f ( x ) , an arbitrary function can be represented by the series" OO

f (x) -- Z

Bn sin (°lnx--ff-)

(78)

n=l

b ( n - - 1 ' 2 , . . )" " with o~n being the roots of otn cot otn - - e Kw To find Bn we multiply both sides of equation (78) by sin( --F-) ~mx and integrate with respect to x from 0 to b; O/m being also a root of ot cot ot - - e (m - 1, 2 . . . . ).

fo b f

OlmX OlnX~ . sin (---b-- / dx f0 hsin \---if--)

(OlmX)dx -- Zn=ln n b

(x) sin \

For m ¢ n we have

fo b s i n

(\ Olmx b ) s i n ( - - -°l'nX ff--)dx

'f0 [ /"

-- ~

cos

J /"

~(oen - otto) - cos

1 b - ~ sin(oln - Olm) 2 otn - C~m 1 b ( sin otn cos O/m - 2 c~,~ - o/m 1 b ( sin ~xn cos Otm m

2 O/n - -

O/m

1

~(Otn + O~m)

b

2 C~n + elm sin

dx

sin(or. + elm)

O~m COS O/n)

sin Olm cos c~n)

(79)

686

Analytical solution methods

(2.2.3-2)

1

b

2

+

an

( sin an

c o s a m --[-- sin a m COS a n )

am

b 0/2 __ 0l 2

(a m sin or, COS a

m -- a n

sin

a m COS a n ) .

N o w an cos an - - e sin an and a m COS a m - - --8 sin a m . This substituted in the last expression between brackets yields zero. So we have found that

fb if an and

sin

(amX) n --if-- sin (~,X)dx_Oform~ b / are roots of a cot a -- - e . If m -- n we find:

a m

fobsin 2 (\ anX ) dx -- 2 b

1 -- cos

1 (

--~b

_1 2b

1

(1-

1 an

dx - - b - - - sin(2an) 2 4ct,

b

sinancosan

an cot ~n a 2 + a 2 cotZan

)

-

1 ( 2

b 1

)- (

1

cota.

)

an 1 + cot2an

_1b 1 + 2 a 2 + e2

)

Thus, the right-hand side of (79) is reduced to the single term, belonging to m - n, whence we find for B,"

Bn

=-

2

1

- ~ b 1 + ot2+e2

~obf ( x ) s i n

(anX]dx. b /

This value for Bn, substituted in equation (77), gives the solution of the problem. N o w we define the finite Fourier sine resistance transformation of a function, that function multiplied by sin(Z-~), .where a , (n - 1, 2 , . . . ) are the roots of a c o t a - e , and integrated with respect to x from 0 to b, thus yielding a function, which is independent of x

Snr t . .,[~(xI,~ -- ~(n) --

fb ~(x)

sin

(anY)/dx

(80)

\ b

Kw" b

with an cot an - - e It is then shown that this transformation reduces the differential quotient of the second order to the transformed function itself and the boundary value at x - O, using the boundary condition at x - b, concerning the entrance resistance.

f0h 2 --

-~x sin

( nX o-

f0h OnX -~

cos \ b

d9

Partial differential equations

--

(2.2.3-2)

[ ~(19 (OlnX) -~x sin

687

Oln

b /

(OgnX)]b __ Ogn 2fob

g~ocos --if-

099

O/n

o

Otn

~-

(OgnX)

qgsin --ff-/dx

Ot2

= sin Oen-0-Tx( b ) - --b-v(b)COSOen + -b-V(0)- ~-~q~(n). As ~(b)ax - - Kw ' we have, with ten cos oe,~+ ~ sinoe,, - 0 , 2 C~n ,.,

Snr { ~x2

O~n

- -~o(,,)

+ -b--~o(o).

(81)

According to the definition (80) for this transformation, we have found

2

Bn = -b

f(n)

e_

1 ÷ ~+e2

and, referring to equation (78), we may represent an arbitrary function

2~

_f(n~

f ( x ) - ~ n=o 1 + ~2n27_e2

f(x) by

sin (---~-/,

(82)

where f (x) sin (~nX --U)

f ( n ) - fo

dx

and otn are the roots of Otn cot o6~ = - e . Equation (82) represents the

inversefinite

Fourier sine resistance transformation. We can now solve the problem of Example 9 very quickly: the transformed differential equation with transformed initial condition becomes: /~2 dq~

2

b

-d7 + ~ o ( n )

- 0,

~(,,, 0) -

b )dxo fo f (x0) sin (~nX°

with the solution q3(n, t) -- exp

fleZ;b2

f (x0) sin ( 06, b x0 )dxo,

and the inverse transform, according to equation (82), is: OQ

2 ~(x, t) - g ~

n--0

Oln X

sin(s ) exp ( 1+ ot2+8 2

2t fobf (xo)sin (°lnX° b )dxo

%

f12b2)

with Oen being the roots of c~cot c~ = - e - - K - V h (see 137.44 of Part A).

(83)

Analytical solution methods

(2.2.3-2)

688

If the flux at x -- 0 is given, application of the finite Fourier cosine resistance transformation is suitable. This transformation arises if a function of x is multiplied by cos (~"x --y-), and integrated with respect to x from 0 to b, thus yielding a function which is independent of x

Cnr{99(x)} -- ~o(n) -- fo b 99(x) cos (ku nbx ) dx

(84)

b In a similar way as where ot~ (n - 0, 1, 2 , . . . ) are the roots of o~ tan o~ - e - K--w" has been done for the sine resistance transformation, it can easily be shown that

Cnr { 0299

°12

099

~7x2} - - ~ ( , , )

- ~(0),

(85)

and that the inverse finite Fourier cosine resistance transformation is given by

2 oo

q3(n)

C~r 1{q3(n)} -- 99(x) -- ~ ~~o- 1 + =2+e 2 c o s (

with Otn being the roots of ot tan o~ - e -

b /

(86)

b

K--7"

Example 10. Axial-symmetric groundwater flow in a leaky aquifer, towards a partially penetrating well with discharge Q. The drawdown is a function of r and z in the steady state: 99 = 99(r, z). The boundary value problem is given by (Fig. 15):

Z

:,a

99=0

/////.,¢ ~ / / / / / / / / / / / / / / / / / / / , 4

Ir-|

C

S,K D

b

0299 ~---1 099 -Jr- 0299 --0 -Or -z-

O--~( r, O) -- 0 Oz

Fig. 15. Partially penetrating well in a leaky aquifer.

r- 0 -Yr-r)

m

'

'

O--~( r, D) -- - ~99(r, D) Kc Oz

for 0 <~ z ~< a and for b ~< z ~< D,

0 lim ( r 099

r-fir

m

Q

2rcK(b - a)

for a ~< z ~< b, while 99(o0, z) = 0.

We apply the cosine resistance transformation with respect to z over the interval 0 4 z ~< D, as for z - 0 ~ - 0. According to equation (85) the transformed az

differential equation becomes

1 d~

d2q3 t dr 2 r dr

2

oln O2q3 - O.

Partial differential equations

(2.2.3-2)

689

The transform of the discontinuous boundary condition along the z-axis (r can be written as:

lim

r

r--+0

57

cos

--D--/

dz - -

_-_

Q

COS

27cK(b - a)

Q

_D{

27r K (b - a) Otn

O)

( OtnZ dz otnb

t~na

sin ( - - - ~ - ) - s i n ( - - D - ) }

D = - Qn ~ • O/n

The boundary condition qg(oo, z) - 0 becomes q3(cx~, n) - O. The solution of this ordinary differential equation is

D (Otnr) (o(r, n) -- Q n Oln - - Ko --D--

(87)

and the inverse transform according to equation (86):

go(r,

z) --

Q

~

with c~,, (n -- 0, 1, 2 , . . . ) Part A).

1 sin

m = ot,~

7 r K ( b - a)

( - ~ ) -sin (~"]--b---, ~ cos 1 + ~2+e2

being the roots of o~tano~ -

e -

OfnZ ~

ofnr

-6D

(cf.

532.06 of

3. Infinite Fourier transformations a. Infinite Fourier sine transformation The operation in which an unknown function ~p(x) is multiplied by sin(ctx) and the product is integrated with respect to x from zero to infinity, is called the infinite Fourier sine transformation. This transformation yields a new function which depends on ot and is independent of x

-

- f0

~o(x) sin(otx) dx.

For instance, S{e -~x } - f o e-ex sin(olx) dx -

(89)

k2+ot2 • (Laplace integral, see equa-

tion (24) of Section 2.2.2-2.) An infinite Fourier transformation only has significance if f0~ I~p(x)[ dx exists. For instance, the transform of a constant has no sense. As

690

Analytical solution methods

(2.2.3-3)

sin(ox d( t

fo

---

~ x sin(otx) o - ot

cos(otx) dcp

--

~ 99 sin(otx) - otq9cos(otx) 0 -

l

q9 sin(otx) dx

__ __ot2~ _~_otqg(0), if both q9 and ~Ox are zero at infinity, S [ 02q9

} _ _~2~ + ,~o(o),

(90)

provided that 99(o0) - ~ ( o 0 ) - 0. This means that the infinite Fourier sine transformation reduces the differential quotient of the second order to the transformed function itself and the value of the function at x - 0, and thus should be applied to problems in semi-infinite regions with given head (or drawdown) at x - 0, and both zero head (drawdown) and zero flux at x - o0. The Fourier analysis of nonperiodic functions shows that an odd function ~p(x) over the interval from zero to infinity, may be represented by (see equation (62)):

if0

~0(x) -- -Jr

B(ot) sin(xot) dot,

with B(ot) -- 2 f o qg(x) sin(otx) dx - 2~b(ot), from which it follows that the inverse infinite Fourier sine transform can be written as"

~(x~- s-l{~(~}-

2

~(ot) sin(xot) doe.

(91)

E x a m p l e 11.

~o= H

~°=0

c K

lz Fig. 16. Leaky aquifer with different water lev-

els on top.

Two semi-infinite lakes or polders separated by an impermeable screen (small dike) on top of a very thick leaky aquifer. Steady groundwater flow takes place, caused by a difference in level between the two lakes (polders). The flow is symmetrical with respect to the z-axis, which is taken as positive downwards; so we need only to consider the infinite quarter of a plane between the positive parts of the x- and z-axis (see Fig. 16).

(2.2.3-3)

Partial differential equations

691

The boundary value problem becomes: 0299

0299

H (/9(0, Z) - - T '

"} oqg2 - - O,

OX 2

a~o (x 0) _ ~,~9(x' 0) K O--z

'

~o(c~, z) - 0 ,

~a~°(x, co) - 0.

c

Oz

Infinite Fourier sine transformation with respect to x gives (see equation (90)):

d2rp

2~

dz 2

Hoe

c~

+

K c d ~ (ol O) -- ~(cg, O)

__ O,

2

dz

and

d~ (or, oo) -- 0 dz

'

H o with the general solution ~b - A e -'~z + B e '~z -J 2t~,

~~bd_._5_=

- A ot e - ~ z +

B oee ~ z

and

dz from which B -

~d---L(c~,o o ) dz

0,

O;

d~b

K c - 7 - ( o ~ , O) -- - K c o t A az

-- ~(c~, O) - A q-

H

or

H

A-

1

2oe 1 + K cot

Thus we have the transformed solution:

^

H(

qg(c~, z) -- ~ 1

__

1

1-

e -~ +Kcoe

1

Kc ' l+Kcot

N o w ot(l+Kcot) - - ot to equation (91):

~o(x, z) -- -H- J i ° °

)

.

(92)

and thus the inverse transformation becomes, according

-1 sin(xol) dc~ - -H- f o ° ° -1e -z~ sin(xoe) doe

Jr

19l

Jr

+ -H- fo ~ ~ K c Jr

1 + Kcot

og

e -z~ sin (x c~) do~.

The second integral in this expression is the Laplace transform of I13/ sin(xoe) (see equation (29)) and is equal to arctan(z), x which becomes T 7r for z - 0 ; so the first integral equals 2 and together the first and second intgerals become"

-{Jr

2

arctan

-

- - - arctan Jr x

z

"

Thus we find" ~o(x,z)-~arctan Jr

x

Kc

+ Jr

sin(xoe)e -z~ dc~,

(93)

1 +Kcoe

the first term being the solution for a confined aquifer (c - 0). (See 335.13, Part A.)

692

(2.2.3-3)

Analytical solution methods

b. Infinite Fourier cosine transformation By this integral transformation the operation is understood in which an unknown function q)(x) is multiplied by cos(oex) and the product is integrated with respect

to x from zero to infinity, yielding a new function, independent of x:

c{~0(x)} - ~(~) - fo ~ ~o(x) cos(olx) dx.

(94)

For instance, {

1

}

C k2 q- x2

f0C~c°s(°lx)~

--

k 2 -t- x 2 dx

_Jr ~k ~e-2k

The main property of the infinite Fourier cosine transformation is that it reduces the differential quotient of the second order of the differential equation to the transformed function itself and the derivative of the function at x = 0:

C { 32q9 0(/9 ~ x 2 } - -ty2~ - ~X (0),

(95)

provided that ~o(c~) - ~3x( c ~ ) - 0, as can easily be derived in a similar way as has been done for the infinite sine transformation. From equation (95) we may conclude that the infinite cosine transformation should be applied to problems in semi-infinite regions with given flux at x = 0 and both zero head (drawdown) and zero flux at x = co. An even function ~p(x) over the interval from zero to infinity may, according to (61), be represented by

~0(x) - -I f Jr

c~ A (or) cos(xot) dot

with A(a) - 2 f ~ ~o(x)cos(~x)dx - 2q3(ot), from which it immediately follows that the inverse Fourier cosine transform can be written down as:

~(X)- C -1 {(p(Ol)}_ __jfor2c~ ~b(oe) cos(xoe) doe.

(96)

Partial differential equations

(2.2.3-3)

693

E x a m p l e 12 (see Fig. 17).

z"/ / / / _

a/

/~/~ K1DI ~/

2D2/////

=~Q

Cl/////////////////,;'//,

K2D2

K1D1 x

a

Abstraction of groundwater by a fully penetrating well from a leaky aquifer that has a discrete inhomogeneity, caused by different transmissivities of the aquifer and different resistances of the semipermeable layers at both sides of a straight line (for instance, a geological fault), this line not passing through the well. We choose the y-axis along the straight line and the x-axis through the well. The problem then becomes symmetrical with respect to the xaxis. Along the boundary x -- 0 we assume ~01(0, y) = ~02(0, y) = ~0g, a still unknown function of y, where ~01(x, y) represents the drawdown in the aquifer at the left side of the boundary (x < 0) and ~02(x, y) that for x > 0.

,, I I I

I I I

Fig. 17. Well in a leaky aquifer with discrete inhomogeneity along a straight line.

As usual in problems with discrete inhomogeneities, we consider each homogeneous part separately and try to find a solution for each part. These solutions contain common boundary values, which are still unknown and can be solved by applying the laws of continuity, as follows: l°x~
Ox 2

-+-

0 2991

991

Oy2

~2

~o~( - o o , y) - O, ~ol (x, ~ ) - O.

-0

with ~2 _ K1D1c1, 0o1(0, y) - qOg(y),

~(x,

Oy

O) - 0

and

Analytical solution methods

(2.2.3-3)

694

Owing to the boundary condition ~Oy (x ' 0) - 0 we apply the infinite Fourier cosine transformation with respect to y, giving according to equation (95):

d2~l dx 2

(

0l2 -Jr-

~..~)" q)l - 0,

(Pl (-oe, 0/) - 0 and ~bl(0, 0/) -- ~g,

with the solution: (pl(X, 0 / ) - q3gexp (x

(97)

o/2-1- ~-~1).

2 ° x >~ 0: We consider the flow pattern at the right side of the y-axis as the result of the superposition of two simpler flow fields, caused by" (a) (b)

abstraction of Q by the well, but with an open boundary along the y-axis and a drawdown q)g along the y-axis.

Case (a) gives, with )~2

9a

_

Q

2Jr K2 D2

_

K2D2c2"

[ Ko{_£Tx/(x_a)2+y2}-Ko{-.~2x/(x-k-a)2-k-y2 1 1 ] }

and as

f0 c~ Ko(p,/a~ + y2)cos(~y)dy- ~Jr exp(--av/0/2 + p2) , / ~ + P~

(a >/0)

(see equation (217) in Section 3.3.5), we find for the transformed solution ~a"

~(x, c~) -

Q 4K2D2

{ (

1 exp 2 + x7

-Ix -al

-exp(-[x+al~/ot2+~~2)

*)

2 "-I- ~--~2

}.

Case b can be solved in a similar way as q91"

(pb(x,0/) -- ~geXp(--x~/ot2q--~2), and thus ,~ 2 ( x , o,) = ¢,,, + ,~ h .

(98)

(2.2.3-3)

Partial differential equations

695

The continuity at x = 0 is responsible for the equation: d~l K1D1--~x

(0, ot)

-

-

K2 D2 -~"~x' d~2 x(O, ot) ,

and by differentiating the expressions in (97) and (98) and putting x - 0: 1 -- "~Q exp 2 _~_ )'-~1

K1 D1 ~g

- a

2 _1_ ~)~I

- q3gK2 D2

2 --l- )~I'

from which Q

exp(-a?2+~

2) (99)

g)g (ot) - '2 K1D l ? 2 _.l_.__lflq._ K 2D 2v/ ot2 .+__122 The inverse transformation yields, with equation (96): exp ( - a ? 2 + ~ ) q)g(Y)- --Jr Q fo ~

cos(yot) dot,

K1Ol~ot2-~- ~11 -t-K2O2?2nt- ~2

exp (x~ot2 + ~ - a ? 2 + 1~) 991(x, y) - --n" Q fo °°

cos(yot) dot

(x negative)

(100)

and

Q [=o/1, +a~2+ y2}] ~2(x. y~- 2.=2z~2 g,/u-a~2+ y2j -,~o{g,/(x +~/o" exp/ (x+a,(~+~/~ -(see Part A, 370.01).

cos(yot) dot

696

Analytical solution methods

(2.2.3-3)

Remark. If we replace the leaky aquifers in Example 12 (Fig. 17) by confined ones and maintain the different transmissivities, we find a very interesting and unexpected result for the flow pattern in the unpumped aquifer. The flow now is no longer steady, but quasi-steady, which means that the groundwater velocities are uniquely defined everywhere in the field, as distinct from the potential function which contains an arbitrary constant. This can be shown by letting Cl and c2 approach infinity, 1/Xl and 1/~.2 thus becoming zero in the expression for qgl (x, y) in equation (100), which yields the integral

fo ~ e -(a-x)°l o{

cos(yot) dot

(x negative)

which does not converge for the lower boundary. However, if we consider the derivatives of qgl with respect to x and y, we find, with ~1 - ~: - cx~:

0(/91 _-Ox

Q 7r(K1D1 + K2D2)

f0 °

Q

e -~(a-x) cos(yot) dot

a-x

zr(K1 D1 + K2D2) ( a - X) 2 -4- y2 and =

Oy

-o

zr(K1D1 + K2D2) Q

f0

e -~"-x) sin(yot) dot y

zr(K1 D1 + KzD2) ( a - x) 2 q-- y2 (Laplace integrals, see equations (24) and (25)). Both equations, integrated, give:

O 991(x' Y) -- -2re(KID1 + K2D2)

In {(a - x)2 + y2} + c

(x negative).

Because the quasi-steady drawdown, caused by abstraction of a fully penetrating well in an infinite confined aquifer, situated in the origin is determined by"

~o(r)

--

Q c - 2zr K D

= c - ~ l n ( xQ 4Jr K D

lnr--qg(x y ) - - c '

Q 2rr K D

In x/x 2 -q- y2

2 + y2 ),

it follows that the flow pattern at the left side of the y-axis in our problem can be considered as caused by a pumping well at the point (a, 0) in a confined aquifer with a transmissivity that equals the average value of the two given values K1 D1 and K2D2 (see Fig. 18, where K1 D1 < K2D2). So in the left half plane, where the unpumped aquifer lies, the streamlines are straight lines and the equipotential lines are circles with their centre at the well location.

Partial differential equations

(2.2.3-3)

697

Y

T \\

K2D2 "\\

[_

I-

\

o

___

Fig. 18. Well in a confined aquifer with different transmissivities on both sides of a straight line.

e. Infinite Fourier resistance transformation This transformation may be derived in a somewhat artificial way from the finite Fourier sine or cosine resistance transformation (see Section 2.2.3-2c), as follows. Consider the problem of Example 9 (Fig. 14) where the initial value is given by equation (78): oo

f(x)--~B,,sin(~-)

with

n-- 1

B,,

and

c~,, c o t

--

-2 1 ~ fo h f ( x ) sin ( _ ~ ) b 1 + oe2+~z

oe,, - -

--e

--

dx

b K w •

In order to find solutions for a semi-infinite aquifer as drawn in Fig. 19, we have to shift the origin to the point x -- b in Fig. 14, convert the x-direction and let b approach infinity. Then x has to be replaced by b - x and thus sin( --y-) '~'x becomes sin (oe,,

Semi-infinite field with entrance resistance. Fig. 19.

Oln x

cos oe,, sin -\(ot,,X__b_)"

698

(2.2.3-3)

Analytical solution methods

If we write F ( x ) for f ( x ) and ot for ~ , we have b

sin otn Kw sin C~n = - K wot cos otn,

or

otn COS otn --"

and thus s i n ( ~ -~) becomes - c o s o t n { s i n ( o t x ) + Kwot cos(otx)}, from which 2

COS 2 ot n

F(x)--Z~I+

{ sin(otx) + Kwot cos(otx)}A(ot)

7

n=0

ot2+s 2

with A(ot) - f bo F(x){sin(otx) + Kwot cos(otx)} dx. As otn are the points of intersection of the curve yl -- tan or,, and the straight Kw line through the origin y2 - ---y-otn, it can easily be shown that if b approaches infinity, C~n approaches nzr (n = 0, 1, 2 , . . . ) (cf. Fig. 6). We introduce Aot with Aot -- (,+l),rh ,,~rh ---- _,rh and replace the factor ~2 in F(x) by -2Aot 7 - , and evaluate COS 2 otn

1+

(otn2 -'1" t32) COS 2 otn

+

~

t3 2

/32 s i n 2 otn -1-" 132 COS 2 otn

+

2+

+

b2

132

+

+

1

ot 2 K 2 w 2 nL b 2 nt- b K w

K2w2ot2

gto b °

q-- 1 q

Now we have 2 ~ F ( x ) - - - - yt" 2. n=0 A(ot) --

sin(xot) + Kwot cos(xot) . k-~ ,A(ot) Aot KZw2ot2

fo b F(x){sin(otx)

q - 1 --1

with

b

+ gwot cos(otx)} dx

and

ot - -b--. otn

Now letting b approach infinity and putting dot for Aot, it seems plausible that the infinite series for F(x) becomes an integral from zero to infinity, which represents F(x): 2 t "~ sin(xot) + Kwotcos(xot) F ( x ) - -- Jo At(or) K2w2ot 2 dot Jr 1+ A'(ot) -

with

(101)

F(x){sin(otx) + gwot cos(otx)} dx.

Now if we define the infinite Fourier resistance transformation as Fr{qg(x)} -- qb(ot) -

f0 °

qo(x){sin(otx) + Kwot cos(otx)} dx,

(102)

Partial differential equations

(2.2.3-3)

699

the inverse transformation becomes, with equation (101):

Kwol cos(xc~) f0 oK)q3(o~) sin(xu)1 ++ K2to2o/2 dol.

(/9(X) -- Fr-l{(p(of)} __ __2 7/"

(103)

The transform of ~i)x2 becomes: Fr{O2(/9

~x2} - foCX~ 0299 {sin(o~x)+Kwotcos(otx)Idx -

{ sin(o,x) + K,,,o, cos(,,,.,,:)} d

=

{ sin(oex)+ Kw~ cos(otx)} 0 { cos(otx) - Kwot sin(otx) } d~o

- c~ :

in( x) + K

cos(

x)}

-- 0e~0{COS(OtX) -- K w u sin(c~x)} o _--

_~O~°(O)Kwa Ox

+ a~o(O) -

C~2~

ol2q),

provided that ~o(oo)

-

7--(00)

-

ox

o.

With the boundary condition K ~a~o( 0 ) Fr{

Oq2~ 572} - - ~ '

~w , we find (104)

provided that ~o(oo) -

-7(00)

ox

-

-

o.

Example 13. Q 5/////2 /////~/~///x~/////////////////////////>c j.

r

a

l z Fig. 20. Finite line well in a thick leaky aquifer.

Abstraction of groundwater by a finite line well from a very thick leaky aquifer; the discharge Q is uniformly distributed along the well-screen. The drawdown in the steady state is a function of r and z: ~0 = ~0(r, z) (see Fig. 20).

700

(2.2.3-3)

Analytical solution methods

02(t9 ~- --10q) -+- 0299 --0

----g Or - r -~r

Oz 2

[r Oq) ~ _

r-+olimIV -~-r/

0

' 2re Kl

I °

(1) (1) for a - -~l < z < a +-~l , (1) (1) for0~ a + -~l ,

Ocp(r, 0) -- ~p(r,0) 0-7 K-----~'

~o(c~, z) - 0 ,

099 ¢p(r, oo) -- -~-z(r, oo) --0.

We apply the infinite Fourier resistance transformation with respect to z. The differential equation is transformed into d2~ dr 2

}

1 d~b - - O f 2 ~ r dr

"-- O,

while Kcot cos(otz)} dz

Q [a+ll{sin(otz)+ 2rr g l Ja-½l

lim ( r d-~r~) -

r--+O

Q [_cos(~z)+Kcusin(~z)]a+i; 2rr K l ~ --

[ Kcoesin{u.(a+ Q 2rr K l u 1l

1l) } - K c o e sin {ol(a - 11) }

-2

/

cos a(a

Q {Kcot cos(aot)sin ( 2 ) + zr K l ot

Q

-2

1 sin(aot)sin ( ~ ) }

sin (l~_ ] F ( a , ~),

rr K l ot

\21

if we use a short notation for F(z, u) - sin(zot) + K c u cos(zoo). The transformed solution is ~b(r, or) --

Q zr K l u

sin

('2)

F(a,u)Ko(r~),

(105)

and the reverse transform according to equation (103): fo °° sin ( ~ ) F ( a , ~) q)(r, z) -- 7r2Q 2K l -£-(] ~ -~ ~-~2-ot-~i K o ( r ~ ) F ( z , ~ ) d u , with F(z, c~) = sin(zoe) + Kcot cos(zot). (See Part A, 522.05.)

(106)

Partial differential equations

(22.4-1)

701

2.2.4. Hankel transformations

1. Orthogonal functions The Fourier analysis and the closely connected Fourier transformations of the preceding section are based on the fact that for the determination of the Fourier coefficients a,, and bn the property has been used that integrals of the form

f_,

sin

l

)cos( \

1

)dx

and

1

f_'

cos (---7-- / cos (

t

m.. /

)dx

become zero (the second integral for m :/= n). The set of functions cos( n~x ~] (n - 1 ' 2 ' ' " .) thus has the property that if two l distinct functions of this set are multiplied by each other and the product is integrated over the interval - l ~< x ~< l, the result becomes zero. Such a set of functions is called an orthogonal set of functions. A set of functions g! (x), g2(x), g 3 ( x ) , . . , is said to be orthogonal over a finite interval a ~< x ~< b if

f~

b gm (X)gn(X) dx -- 0

for m # n.

For m - n the integral f,h g2 (x) dx will generally not be equal to zero. The positive square root of this last expression is called the norm of gm(X) and is denoted by Ilgmll:

g2 (x) dx - IIgm II.

(107)

Clearly, an orthogonal set gl, g2, . . . on an interval a ~< x ~< b whose functions have norm 1 satisfies the relations

gm (x) gn (x) dx --

0

when m # n, m = l, 2, .. .,

1

whenm=n,n--

1,2 . . . . .

Such a set is called an orthonormal set of functions on the interval a ~< x ~ b. Obviously, from an orthogonal set we may obtain an orthonormal set by dividing each function by its norm. The set of functions gm(x) = sin(mx), namely sinx, sin(2x), sin(3x) . . . . may serve as an example of an orthogonal set on the interval - J r ~< x ~< Jr, because

f_

Usin(mx) sin(nx) dx - 0

for m :/: n.

71"

The norm Ilgmll equals ~

as Ilgmll 2 -- f_~ s i n 2 ( m x ) d x - rr.

702

(2.2.4-1)

Analytical solution methods

Hence, the corresponding orthonormal set consists of the functions sinx

sin(2x)

sin(3x) ,

. . . . .

Because of the fact that the possibility of representing an arbitrary function f ( x ) by a Fourier series is based on the orthogonality of the goniometric functions (see Section 2.2.3-1) on a certain interval, it is tempting to represent given functions in terms of any other orthogonal set gn (x) in the form oo

f (x) -- E

Cngn(X) -- Clgl (x) + c2g2(x) + ' " ,

(108)

n=l and determine the coefficients cn in a similar way as for the Fourier series, as follows: Multiply both sides of equation (108) by gm(X) and integrate over the interval a ~< x ~< b on which the functions are orthogonal:

b

L

c~ Lb f (x ) gm (X ) dx -- E Cn gn (x ) gm (X ) dx .

n=l

If n ~ m, all integrals on the right-hand side become zero except that only the integral for n = m does not, but becomes IIg,,IIz. So the formula for the determination of the constants is:

1yah

c,, = ilgnll2

f(x)gn(x) dx.

(109)

If the series (108) converges and represents f ( x ) , it is called a generalized Fourier series of f ( x ) , and its coefficients are called Fourier constants of that series, represented by equation (109). Some important sets of real functions occurring in applications are not orthogonal but have the property that, for some function p(x),

L

b p(X)gm(X)gn(X) dx -- O,

when m ¢- n.

(11 O)

Such a set is then said to be orthogonal with respect to the weight function p(x) on the interval a ~< x <~ b. The norm of gm is now defined as

I[gm I[ = ~fa b p (x) g2m (x) dx,

(111)

(2.2.4-1)

Partial differential equations

703

and if the norm of each function gm is 1, the set is said to be orthonormal on that interval with respect to p ( x ) . If we set hm -- X/c-figm, then equation (11 l) becomes

f

b h m ( x ) h n ( x ) dx - 0

(m ¢ n),

that is, the functions hm form an orthogonal set in the usual sense. 2. Series a n d integrals o f Bessel f u n c t i o n s

Consider the differential equations (Bessel's equations, see Section 3.3.1): d2y 1 dy dx 2 ~--x d-ffx + k 2 y - 0 '

and

d2y 1 dy dx 2 + x-d-fix- + 12y - 0; it follows that u -- J o ( k x ) is a solution of the first and v = Jo(lx) a solution of the second equation, in which J o ( k x ) and Jo(lx) are Bessel functions of the first kind and order zero (Section 3.3.1); so it is allowed to write d2u 1 du dx 2 q- x-~x- q- k2u - 0

and

d2v 1 dv ~ -k- x~xx- -}- 1 2 v - 0

or

1 d (du) x if- k2u -- 0 x dx ~xx

and

1 d (dr) 12 -~ x + v =0. x dx ~xx

Multiplying the first equation by x v and the second by x u and subtracting the second equation from the first, we find d(xdU d(xdV Vdx ~ x ) - U~x-x d-x-x) + ( k 2 - 12)xuv - O. Integration with respect to x gives

f u xd -- f ud(x-dyx)-f dv

_

dv = x u dx

As

do du f x~

f

d. }- y dx .-~x dv

dv du - x v x dx dx ~

-- f x ~do ~du dx -- f x ~du dr, we have

(k 2 - I 2)

f

x u v dx - x

-dx

(X~x)

v~

.

_

o

704

(2.2.4-2)

Analytical solution methods

Inserting u = Jo(kx) and v = Jo(Ix) in this expression, we get (k 2 _ / 2 ) / xJo(kx)Jo(lx) dx = x { - lJo(kx)J1 (lx) + kJo(lx)J1 (kx) }, or

f xJo(kx)Jo(lx) dx = k 2 x_ l 2 {kJo(lx)J1 (kx) - IJo(kx)J1 (tx) }

(112)

It should be noted that the general formula

/ xJv(kx)J~(lx) dx -

x {kJv(Ix)Jo+l(kx) -IJv(kx)Jo+l(Ix)} k 2 _ 12

(113)

may be derived in a similar manner. In particular it follows from equation (112) that the definite integral over the interval 0 ~< x ~< a becomes

fo

a x Jo(kx)Jo(lx) dx -

a

k 2 _

l2

{kJo(la)Jl(ka) -lJo(ka)Jl(la)}

•

Now set ka = 0/1 and la - 0/2 and assume that 0/1 and o/2 are two roots of the equation J0(c~) = 0; then Jo(la) = 0 and Jo(ka) = 0 and thus,

a / fo°xJo ( aa' X )/j o ( 0/2X)dx

=0.

(114)

From this equation it can be stated that, according to equation (110), the set of functions J0 (~"x ---y-) (n = 0, 1, 2, ...), is orthogonal with respect to the weight function p(x) = x if 0/n (n = 0, 1, 2 . . . . ) are the roots of the equation J0(0/) = 0. 0 When k - l, the result in equation (112) assumes the indeterminate form +. By using l'H6pital's theorem it can be shown that

f x j 2 ( k x ) d x - 2 x e { J ~ ( k x ) + j2(kx)},

(115)

from which f oa x j2 (kx) dx - ~la2jZ(ka) , so that the norm of Jo( ~"x a ,] has now been determined (see equation (111)): a

1]J°( o/n' ) a~x I 1 - ~

Jl(~n).

(116)

Using the results (114) and (116) we may conclude, referring to Section 2.2.4-1, equations (108) and (109), that under certain conditions an arbitrary function f ( x )

Partial differential equations

(2.2.4-2)

705

may be represented by a so-called Fourier-Bessel series of the form

oo

OllX

c0.0 v ) +

n----O Cn ----

'

aZJ2(oln)

c1Jo( Ol2x) a

+...

1o"x f (x ) Jo (o.x) dx, \ a

with (117)

if Ofn (rt "--0, 1, 2 , . . . ) are the roots of Jo(ot~) = 0. The Fourier-Bessel series are powerful tools in treating geohydrological problems involving radial and axial-symmetric flow, especially in finite regions, as they are the basis for the finite Hankel transformations, to be discussed in the next section (Section 2.2.4-3). Since, of course, many practical problems in radial- or axial-symmetric flow involve semi-infinite instead of finite regions, it is desirable to know whether an arbitrary function can also be represented by an infinite integral of one or more Bessel functions. Now, indeed, it can be proved that, under certain conditions, a function f ( r ) can be represented by the so-called Hankel integral

f (r) -- fo ~

otA (ot) J,(rot) dot,

in which

A(ot) --

f0 °

r f (r)J~(otr) dr.

(118)

The proof of this statement will not be given here; instead, the following derivation would make it acceptable. Integrals of the type

f txJu(at)J~(bt) dt have been investigated by Weber and Schafheitlin, and they found that discontinuities occur in these integrals at a = b for certain relations between X, lZ and v. A well known Weber-Schafheitlin integral is (see Section 3.3.5, equation (156)):

{ bu-1

oo f0

Jtz(at)Jtz-l(bt) dt -

a~ 0

for 0 < b < a (/1. > 0) forb>a >0.

From this we have

r ~ fo ° Jv(r~)J~_l(Xot)d~

-

-

JX ~-l I0

for0<Xr >0.

706

(2.2.4-2)

Analytical solution methods

Both sides of this equation, multiplied by )~f(~.) and integrated with respect to )~ between zero and infinity, yields

r ~ f0 cx~Jv(r(~)

f0 cxz)~f()~)Jv-l(~)~) d)~ do~ -- fo r )~f()~)

d)~.

Now, differentiating both sides with respect to r and making use of the general d differentiation formula for the Bessel functions T;{x~Jv(x)} -- x~Jv_l(x) (equation ( 2 2 ) o f Section 3.3.1), we find r~

f0

f0

otJv_l (rc~)

or w i t h / z - - v -

f (r) --

Lf(~) Jv-l(a)~)d~ dc~- r V f ( r )

1

fo

otJ~(rot)

fo

~.f (~.) Ju (c~.) d)~ do~,

which is identical with equation (118).

3. Finite Hankel transformations Considering the representation of a function by a Fourier-Bessel series as given in equation (117), we may understand the definite integral in the expression for cn as an operation performed on that function and may call this operation the finite Hankel transformation. As we want to apply this transformation to the solution of problems in radial flow or axial-symmetric flow, we replace the function f ( x ) in (117) by qg(r). The finite Hankel transformation is then the operation in which the unknown function qg@) is multiplied by the factor r Jo( --R--) ~"r and the product is integrated with respect to r over the interval 0 ~< r ~< R, thus obtaining a new function, denoted as Hn {qg(r)} or as q3(n), which is independent of r" Hn{~o(r)} -- ~(n) -- fo R rqg(r)Jo( °enr R '~ / dr

(119)

with c~n (n = 0, 1, 2 , . . . ) being the roots of J0(ot,) = 0. From equation (117) we can immediately determine the inverse transformation

2~

q3(n) Otnr ~ J0 \--~--), ~ o ( r ) - H n - - ' l ~ ( n ) } - R--2-n=o J~(o~n)

(120)

again, with or,, being the roots of Jo(a,) = O. The main property from which the finite Hankel transformation derives its value

d299 in solving radial and axial-symmetric problems, is, that it reduces the terms ~-7~-+

Partial differential equations

(2.2.4-3)

707

1 ~ which always occur in the differential equations, governing these problems to rdr' the transformed function itself, as can be shown as follows:

~ + r ~ r ] - ~ r~r ~r) --

_ _ _d ( r - ~ ) d r /o Rr J o ( °tnr]l R }rdr

(Otnr dqgin ---[Jo\ R )rdrJ0-

_

r ) d (r~rr) f0R Jo (otn \--R-

fo R dq9 otnr l rd-rrdJ°( R ]

dq9 R Oln fo R Otnr] + ~ rJl (---~-j d~ -[~0(~O~nr]~r ~r]0

--

,~)r - --~

h-Tr+ --Y-~, ~-T qgd r J l \

R

)~]0

"

Now

Otnr d---r ---~-J1

R ]

(otnr)Otn R Jo\ R --~-

o~n2 ( ~ ) R---~Jo

r

(equation (22), Section 3.3.1), whence we have, as J0(c~n) - 0 , d299 1 d99 0q9 c~n Hn -~r 2 + -- O~n J1( otn ) 99( R ) - l i m r ~b(n) r--&r r~O -~r ~ "

{

]

(

)

(121)

Hence, application of the finite Hankel transformation is useful in radial or axialsymmetric problems involving wells with given discharge at r - 0 and with a given head (drawdown) distribution at r = R.

Example 14.

7 Q

i

r KD, S

Fig. 21. Well in a circular island.

Non-steady flow of groundwater through a confined aquifer caused by abstraction of a constant discharge Q by a fully penetrating well at the centre of a circular island. ~p - qg(r, t) drawdown (see Fig. 21).

708

Analytical solution methods

(2.2.4-3)

02(t9 1 0(t9 __ f12099 (p(r, 0) - - 0 ,

S

(p(R,t) --0,

lim r, = r~O ( a ~ r ) 2rc K D

for t > O.

Finite Hankel transformation with respect to r gives 2

Q

Otn

2rc K D

f12

R 2 (~ - -

~b(n, 0) -- 0,

dt '

an ordinary differential equation with initial value, and solution

o R2{1 -- exp

~ ( n , t) -- 2 7r K D ot---~n

t

fl2R2

"

The inverse Hankel transformation immediately yields qg(r,t) --

Q

k

{anr{

Jo~--k-)

1 -exp

(

2t ) }

%

Now

jO(--.-) °r n=0 is the Fourier-Bessel representation, over the interval 0 ~< r ~< R, of the function 1 in(rE ) as is shown in Section 3.3.4, equation (115); so we get

2

(p(r, t ) -

Q In 2rc K------D

/r"--)

k

Q J° ( z - ~ ) e x p 7r K D n=O °tz J~(°tn)

/ o t) flZR2,

where an are the roots of Jo(~n) - 0 (see 241.02, Part A). If t tends to infinity the series vanishes and the well known solution for the steady state is obtained, given by the first term in the expression for (p(r, t). The Hankel transformation technique, though in a somewhat modified way, enables us to solve also problems in radial flow with more complicated boundary values. Like the Fourier transformation, the Hankel transformation can be extended to the so-called finite Hankel resistance transformation, which should be applied in particular to problems involving entrance resistance against outflow of groundwater from an aquifer into open water, or inversely, inflow of surface water into an aquifer (see Section 1.5.1-3c1°). We will derive the necessary equations for this transformation, starting from the following theoretical example.

Partial differential equations

(2.2.4-3)

709

Example 15.

t!

~

In a circular island, with radius R, the initial head in the confined aquifer is an arbitrary function of r. There is no flux at r - 0 and entrance resistance w at r -- R (Fig. 22). The boundary value problem for the head ~p(r, t) can be written as

m

02~0 Or 2

R

10q 9 ~

r-Or

S )

2099( ._ fl

f12 _

-~

-- K D

'

Fig. 22. Circular island with entrance resistance. 0__~

~p(r, O) -- f (r),

099

Or (0, t) -- O,

K~(R, Or

t) --

~o(r, t) w

To solve the differential equation we apply the method of separation of variables (see Section 2.2.1), which yields a general solution

(- y,). O/2

+

From ~Or (0, t) -- 0 we may conclude that B -- O, so

o/2

99(r, t) -- AJo(o/r) exp ( - ~-f

t)

The boundary value at r - R gives:

Kw~(R Or

'

t) -- - K w A o / J l

(o/R) exp

-

-y

t

-- - A J o ( o / R ) exp

-

-y

t

,

from which K wo/J1 (o/R) -- J0 (o/R). Writing o/,, for o/R and e for X--8-y,we have found that o / - W has to satisfy the relation

0tn

~ . J1 ( ~ )

- ~ Jo (~,,),

while o/2 t

o/n r

q)(r't)--AJokR)exp

(

f12R2 ) •

As there are infinitely many solutions of otn (n represented by

0, 1, 2 . . . . ), ~o(r, t) may be

OO q)(r,t)--ZAnJ0 n=0

R lexp f12 R 2

'

(122)

710

Analytical solution methods

(2.2.4-3)

according to the superposition principle (see Section 2.3.1-1), provided that, according to the initial condition ~0(r, 0) - f(r), an arbitrary function can be represented by the series oo

f (r) " ~

(123)

AnJo(~-~- )

n--0 R

with 0/n being the roots of 0/,,J1 (0/,,) - e Jo(0/~), where e - y ¥ . Now,

as

r {aJo(br)Jl(ar) - bJo(ar)Jl(br)} f r Jo(ar) Jo(br) dr -- a 2 -b2 (see equation 112), over the interval 0 <~ r ~< R we have: R

fo

l~ r Jo(ar) Jo(br) dr = a2 _ b2 {aJo(bR)Jl(aR) - bJo(aR)Jl(bR)}.

If we assume that

0/1

--

a R and

0/2

--

b R are roots of the equation

0~n J1 (0/n) - -

e J0 (0/,), we find 0/1

E

aJo(bR)J1 (ag) -- --~ J0(0/2)J1 (0/1) -- ~ J0(0/2) Jo(0/1), and also

bJo(an)J~(bg)-

~Jo(O~)J~(o~2)-~Jo(O~)Jo(a2),

O/2

8

and therefore

fo R r JOk--~--)Jo(~--~) (0/lr dr - O . From this result it follows that, according to equation (114), the set of functions J0(-q-~) is orthogonal with respect to the weight function p(r) - r if 0/n (n 0, 1, 2 , . . . ) are the roots of the equation 0/n J1 (o/n) - e J0(0/n). ( Otmr , Multiplication of both sides of expression (123) by rJo~--~-) in which am is also a root of 0/J1 (0/) - eJ0(a), and integration with respect to r from 0 to R, gives

f0 r f (r)Jo

dr - ~ n--O

A.

f0 rJo('--~-/Jo o.r

dr

(m -- 0, 1, 2 , . . . ) . All terms of the series with m -¢- n become zero and only the term m - n is left, i.e."

An

fo"

r J~

dr.

(2.2.4-3)

Partial differential equations

711

From (115) it follows that fo R r J~ \( -Otnr - ~ ) dr - !R'{g(~n> 2

#(~.>}

+

So, we find 2 1 fo R rf(r)Jo (otnr) dr An = ~R 2 JZ(c~n) + JZ(t~n) \ R

(124)

with C~nthe roots of OlnJ1(Oln) -- EJo(oln). This value of An substituted in (122) gives the solution of the problem. We now consider the definite integral in the expression for An, replacing f (r) by qg(r), as the finite Hankel resistance transformation Olnr ) Hnr{q)(r)} -- ~(n) -- fo R rq)(r)Jo\ R dr

(125)

with o~. being the roots of OtnJ1 (O/n) -- E J0 (O/n).

From equations (122) and (123) we get the inverse transformation" ~o(r)-

2

Hn~l{~(n)}-

~

Otnr

6(n)

~ Z n--O

jo2(O~n)-I-J~(otn)Y0\ R )"

The transform of ~dr 2 + 1r ~ becomes

~r {'d(rd.} r~r Tr)-

foRr'0(,.r~r ".r ld(rd~r) dr

(~.r (rd~ -/o . '°,,) d

.,)r

]o- fo"d~ r--~rdJ°\C"F R /

=[jo(Otnr dq)lR Otnfor \ R )rdr_lo + R r J l ( -T)d~ Otnr = [ jo ( Unr , d q) C~nr ( otnr R --~-)r--~r + --~-q)Jl\ R )]0 Otn~'oRRq) d{ r

J, (-~) }

d~o

= Yo(otn)R-~r(R ) + Oln(tg(R)Yl(Oln) -

lim

(d~)~.2

r-~0 r ~r

-

~

~b(n)

(126)

712

Analytical solution methods

(2.2.4-3)

d~o (R) -- - / ¢~o(R) The boundary condition for the resistance at r - R, namely ?-7 w ' makes the first two terms in the last equation (setting e - K---~)equal to -eJo(0/n)q)(R) + 0~nJ1 (0/n)qg(R), and therefore equal to zero if 0/,, are the roots of 0/,, J1 (0/,,) - eJo(a,,). Thus, we have found:

Hnr{d2q)

ldq9

~r2 + --r_~_r } _ _ r--+0 li m

(rdq 9 ~r

)_

2 0/n ,, -~q)(n).

(127)

The problem of Example 15 can now be solved very quickly. The transformed differential equation becomes: 2 R2 q3 =

d~b dt

~nr]

~(n, O) - f R rf (r)Jo(---ff-/ dr - f (n)

with

the solution of which is

o~2t

e(.,

The reverse transformation then gives, according to equation (126)

2 oo

2t

f(n)

~o( r, t) -- - ~ n ~ ° j g ( 0/n ) + J ~ ( 0/n) J 0 ( - ~ )

exp (

flZ'R2)

with f(n) - fog rf(r)Jo(f-~-)dr and 0/n being the roots of 0/,,Jl(0/n) - ~J0(0/n), where e - ~ (see Part A, problem 236.04).

4. Infinite Hankel transformation The operation in which an unknown function ¢p(r) is multiplied by the factor r Jo(0/r) and the product is integrated with respect to r from zero to infinity, is called the infinte Hankel transformation. This transformation yields a new function, which is independent of r and will be denoted as H{qg(r)} or shortly as ~b(0/)" q3(0/) -- H{go(r)} --

f0 °

(128)

rqg(r)Jo(0/r) dr,

Jo(c~r) in this expression is the Bessel function of the first kind and of zero order. The infinite Hankel transform of r1e-Cr' for instance, becomes lTe-CrI -- fo~ e-Cr Jo(0/r) dr -

40/2 -Jr-c2

(Laplace integral).

It is easy to show that, if the function qg(r) satisfies the condition that both q) and d__e dr vanish for r --+ oo H{d2g° I&P ~ r 2 -+- -d--r-r }r

(r dq9 - - r~olim ~ - r ) - 0/2~(0/).

(129)

Partial differential equations

(2.2.4-4)

713

Hence, application of the infinite Hankel transformation is possible for radial and axial-symmetric problems, for which the horizontal groundwater velocity at r - 0 is given and the head (or drawdown) and velocity at infinity can be assumed to be zero. Now, from equation (118) it follows that, according to equation (128), for v - 0 A(ot) can be replaced by f(ot) and so the inverse infinite Hankel transformation becomes simply" 99(r) -- H-l{~b(c~)}- fo ~ ot~(c~) Jo(rc~) dot,

(130)

Example 16. I I

I I

I I

I I

I

I

Consider vertical infiltration of water with a constant velocity q [L T - l ] , uniformly distributed over a circular area with radius R, into an assumed semi-infinite aquifer (see Fig. 23). This infiltration will cause an axialsymmetric rise of the original piezometric head, which after some time becomes steady and will then be a function of r and z: q9 = 99(r, z) = head.

I

I I I I I

1" r

z

Fig. 23. Verticial infiltration on a circular area. This boundary value problem can be translated mathematically as follows: 0299 1 399 0299 Or 2 i f - -r -~r q- ~3z 2 - - 0 ,

qg(c~,z)--0

oo(r,

-

o,

and also

099 ~(~ Or

'

q

a---V(r, o) Oz

z)-0,

0

099 ~(0 Or

'

z)-0,

for 0 ~ r < R forr>

R.

Infinite Hankel transformation with respect to r of the differential equation becomes d2~ dz 2

o/2~ - 0

(see equation (129))

714

(2.2.4-4)

Analytical solution methods

and transformation of the discontinuous boundary condition yields R

d(~ (ol, O) - fo ~ oqq9 d--z -~-z(r, O)r Jo(ar) dr = - qK [Jo r Jo(otr) dr =

qR

- ~ J1 (orR), K~

whereupon the transformed boundary value problem can be written as d2(p dz 2

c~2~b -- 0

q3(c~, oo) -- 0

'

and

qR

"~d--E~(c~, 0) - - ~ J 1 dz Kot

(otR).

The solution of this ordinary differential equation is ff(~, z) =

q R J1 (orR)e -~z Kot2

Inverse infinite Hankel transformation gives the desired solution q R fo c¢ 1 Jl (ROt)Jo(rot)e_Z~ dot go(r, z) -- --~

(see Part A, 523.02). This solution can be evaluated in infinite series and thus calculated for every value of r and z. Along the coordinate axes the integral function reduces to transcendental functions, for example, along the z-axis: q R fo ~ -g1 Jl(R(z)e -z~ d(z ~ ( o, z) - -~-

K

(~//R 2 +

Z2 -

Z)

(Laplace integral),

and along the r-axis the Weber-Schafheitlin integral (see Section 3.3.5, equation (170)):

q) ( r, O) --

q R f0 c¢ 1 Jl(R~)Jo(rot) dol -- 2 q R E ( r 2 ) --K --d 7r K -~

2q (r 2 -- R2) K (R -

7rKr

2)

-rT

+ 2qr E (R -~

2)

-rT

for 0 ~< r < R, f o r r > R,

in which K (z) and E(z) represent complete elliptic integrals of the first and second kinds, respectively (see Section 3.5).

2.2.5. Conformal transformation 1. Complex analytic functions

If x and y are real variables, then z -- x + i y is said to be a complex variable. Consider two complex variables z = x + iy and w = u + iv, and suppose that

Partial differential equations

(2.2.5-1)

715

a relation is given such that to each value in some region of the complex z-plane there corresponds one or more values of w in a well-defined manner. Then w is said to be a complex function of z, defined in that region, and we write

w--f

(z)

w - w(z).

or simply

If to each z in that region there corresponds only one value of w = f ( z ) , then the function f ( z ) is said to be single-valued. A function which is not single-valued, is called multi-valued. Since w depends on z -- x + iy and w = u + iv, it is clear that, in general, u depends on x and y, and so does v. We may, therefore, write

(131)

w -- f ( z ) -- u(x, y) ÷ iv(x, y),

and this shows that a complex function f ( z ) is equivalent to two real functions u(x, y) and v(x, y). It is known from complex analysis that a number of basic concepts, such as limits, continuity, derivatives, etc., concerning complex functions are quite similar to those in real calculus. So, for instance, a complex function f (z) is said to be differentiable at a point z = z0 if the limit df f'(z0) -- -=--(z0) -- lim f(z0 + Az) -- f(z0) Az-+0 AZ dz

exists.

(132)

A basic statement in complex analysis is the definition of analyticity of a function. A function f ( z ) is called analytic at a point z -- z0 if it is defined, and has a derivative, at every point in some neighbourhood of z0.

B

z+Az

AYt=iA

x

Consider the function w = f ( z ) in equation (131) to be analytic in a domain D of the zplane. Then, by definition, f ( z ) has a derivative according to equation (132), everywhere in D, where Az may approach zero along any path. We may set Az = Ax + i A y . Choosing path A in Fig. 24, we let Ay approach zero first and then Ax -+ 0. After Ay becomes zero, Az - Ax and by equation (132)

Fig. 24. Paths in the z-plane.

f ' ( z ) -- lim

u(x ÷ Ax, y) + i v(x + Ax, y) - {u(x, y) ÷ iv(x, y)} Ax

Ax-+0

=

lim

Ax-+0

u (x + Ax, y) -- u (x, y) Ax

+

i

lim Ax---,0

v(x + Ax, y) - v(x, y) Ax

716

(2.2.5-1 )

Analytical solution methods

Since f'(z) exists, the last two real limits exist. They are the partial derivatives of u and v with respect to x. Hence, f'(z) can be written

dw Ou Ov = +i~. dz Ox Ox

f'(z)-

(133)

Similarly, if we choose path B in Fig. 24, we let Ax approach zero first and then Ay --+ 0. After Ax becomes zero, Az -- i Ay and f'(z)--

lim

Ay--+O

u(x, y + Ay) -- u(x, y) v(x, y + Ay) -- v(x, y) + i lim iAy Ay--+O iA y

that is,

dw

f ' (z) . . . .

dz

Ou

Ov

= -i ~ + -0y 0y

(134)

because 71 -_ - i • The existence of f'(z) thus implies the existence of the four partial derivatives in (133) and (134). By equating the real and the imaginary parts of the right-hand sides of (133) and (134) we obtain

Ou Ov =~ Ox Oy

and

Ou Oy

~=-~.

Ov Ox

(135)

These basic relations between the partial derivatives of the real and imaginary parts of a complex analytic function are called the Cauchy-Riemann equations. These equations are fundamental because they are not only necessary but also sufficient for a function to be analytic, which can be stated by the following T h e o r e m 10. If two real functions u(x, y) and v(x, y) of two real variables x and y

have continuous first partial derivatives that satisfy the Cauchy-Riemann equations in some domain D, then the comnplex function f (z) = w = u(x, y) + iv(x, y) is analytic in D. The proof of this theorem will not be given here. It can also be proved that the derivative of an analytic function f ( z ) is itself analytic, so that u(x, y) and v(x, y) will have continuous partial derivatives of all orders. In particular, the mixed second derivatives of these functions will be equal:

02u OxOy

=

02u OyOx

and

02v OxOy

=

02v OyOx

Differentiating the Cauchy-Riemann equations of equation (135), we thus obtain

02U

021)

02U

02V

OX2

OxOy'

Oy2

OxOy'

02U

OxOy

02V =

Oy2'

OZu and

OxOy

02l) =

OX2"

(2.2.5-1)

Partial differential equations

717

This yield the following important result: Theorem 11. The real part and the imaginary part of a complex function f (z) u(x, y ) + i v ( x , y) that is analytic in a domain D are solutions of Laplace's equation, for two dimensions, 02u 02u V 2 u - - Ox 2 + ~ - 0 Oy2

021)

and

V2v-~+~-0, OX 2

021)

Oy2

in D and have continuous second partial derivatives in D.

A solution of Laplace's equation, having continuous second-order partial derivatives is called a harmonic function. Hence, the real and imaginary parts of an analytic function are harmonic functions. This is one of the main reasons for the great practical importance of complex analysis for solving geohydrological problems, because we have found in Section 1.2.4, equation (37) that the potential function ok(x, y) and the stream function ~ ( x , y) in steady two-dimensional groundwater flow satisfy the Cauchy-Riemann conditions, thus being the real part and imaginary part, successively, of a complex analytic function, that we shall denote by f2: f2 -- q~ + i~,

(136)

f2 is called the complex potential function for the flow. The (real) potential function 4~( - K~p) and the stream function 7r are, according to Theorem 11, harmonic and are thus solutions of Laplace's equation as has been shown already in Section 1.4.2-3, equation (16) (for the three-dimensional case) and Section 1.4.2-5, equation (26). Let us consider two-dimensional horizontal steady groundwater flow in a confined aquifer, which means that the motion of the groundwater is the same in all planes parallel to the x y-plane, the velocity being parallel to that plane. It then suffices to consider the motion of the fluid in the x y-plane. If we put z = x + i y then the xy-plane is assumed to be the complex z-plane and f2 becomes a function of z: or shortly

~2 -- F(z),

,f2 - ,f2(z).

From equations (133) and (134) it follows that

dz

=

~

Ox

+

i ~

Ox

-

-i

Oy

I

Oy

and also with equation (135): d~Q

O@

dz

Ox

.o4,

t ~

Oy

- - - Vx +

ivy,

(137)

718

(2.2.5-1)

Analytical solution methods

which gives the relation between the complex potential function and the D a r c y velocities in the two coordinate directions (cf. equation (32) of Section 1.2.3). If one of the three functions I2, q~ and 7r is known, the other two are determined and, in general, can be found easily.

Example 17. Radial flow of groundwater in a confined aquifer towards a fully penetrating well with a constant discharge Q. At a distance R from the well the drawdown is assumed to be zero. After some time a steady state will be reached with a drawdown s of the original head as a function of r in the domain 0 ~ r ~< R (see Fig. 25).

KD

R Fig. 25. Pumping well in a

circular island.

At a distance r (< R) from the well the velocity vector is directed towards the well and has the magnitude l)r = + K - -

ds dr'

in which the positive sign means that the velocity has the direction of the increasing drawdown, whereas Darcy's law requires the negative sign in the case of variable heads. The law of conservation of mass (volume, if the groundwater is homogeneous) applied to a circular cylinder with radius r and thickness D, gives ds

Q = - 2 r c r Dvr = - 2 r r K D r m dr

(negative sign, because the flow direction is negative and the discharge is a positive number). Q dr ds = - ~ - -

2rr K D r '

from which

For r - R the drawdown s - 0 ' so s q~ = - Ks becomes Q ln(R ) ~ b - 27rD

s - Q

2rrKD

~

Q

2rr K D

ln(~)

"

In r + constant.

Then the potential function

which is negative for r < R.

In Cartesian coordinates we have ~ b - 4-Y-B Q ln( x2+v2 R2 )' as r -- v/X ~ + y2.

(2.2.5-1 )

Partial differential equations

719

To find the stream function, we make use of the Cauchy-Riemann conditions" Od?

Q

Ox

2x

_-- 0gr

from which

~P -

Oy

4reD x 2 4- y2

Qx f dy 27c D J X 2 + y2

or

~P -- ~2zrQD arc t an ( Y ) 4- f (x ' Q Ox

=

v2 2reD 1 + x~

---~

4-f'(x)--

so

f'(x)-O

y

2Jr D x 2 4- y2

+ f ' ( x ) --

o4) Oy

Now

0.__.~¢= Q y 0y 2zr D x 2 + y2'

and

f (x) - c (constant),

from which it follows that Q a r c t a n ( y ) + c. ~P-- 2zrD x If we choose gt -- 0 along the x-axis (y - 0), we have c - 0, and ~p becomes"

~--

Q arctan( y)_ . 2zrD x

Now, from equation (136), we get

2rrD

R

) + i arctan(Y)

'

from which t'2 -- 2zrD as In z - In Izl + i arg z. The expression for 4, determines a set of equipotential lines, which are, in this case a set of circles with the well location as a centre: X 2 4- y2 _

R2

e x p (4zrDq~\ Q )'

and with parameter ¢ (negative or zero). The streamlines are straight lines through the origin with parameter Or.

y -- x tan (2zr D~p

Q ).

720

(2.2.5-2)

Analytical solution methods

2. Conformal mapping A continuous real function y -- f ( x ) of a real variable x can be exhibited graphically by plotting a curve in the Cartesian xy-plane; this curve is called the graph of the function. In the case of a complex function w -- w(z) -- w(x + iy) =

u(x, y) + iv(x, y), the situation is more complicated, because each of the complex variables w and z is represented by the points in the complex plane. This suggests the use of two separate complex planes for the two variables: one the z-plane, in which the point z = x + iy is to be plotted, and the other the w-plane, in which the corresponding point w = u + iv is to be plotted. In this way, the function w = w(z) defines a correspondence between points of these two planes. This correspondence is called a mapping or transformation of points in the z-plane onto points in the w-plane, and we say that w(z) maps its domain of definition in the z-plane onto its range of values in the w-plane. The point w0 = w(zo) corresponding to a point z0 is called the image point. If z moves along some curve C and w(z) is continuous (not a constant), the corresponding point w - w(z) will, in general, travel along a curve C* in the w-plane. This curve is then called the image of the curve C, and the word "image" applies also to regions or other point sets. To investigate the specific properties of a mapping defined by a given analytic function w(z), we may consider the images of the straight lines x - const, and y = const, in the w-plane. Another possibility is the study of the images of the circles ]zl = constant and the straight lines through the origin. Conversely, we may consider the curves defined by u(x, y) -- constant and v(x, y) = constant in the z-plane. These curves are called the level curves of u and v. An important geometrical property of the mappings defined by analytic functions is their conformality. A mapping in the plane is said to be angle-preserving, or conformal, if it preserves angles between oriented curves in magnitude as well as in sense; that is, the images of any two such curves, make the same angle of intersection as the curves, both in magnitude and direction, which leads to the following theorem: T h e o r e m 12. The mapping defined by an analytic function w -- f (z) is conformal,

except at points where the derivative -dZz dw is zero. To prove this theorem we represent a curve C in the complex z-plane in the parametric form

z --z(t)--x(t)+iy(t), where t is a real parameter.

Partial differential equations

(2.2.5-2)

721

The tangent to C at a point z0 - z(to) is defined as the limiting position of the straight line through z0 and another point Zl - z(to + At) as zl approaches z0 along C, that is, as At --+ 0. So, the tangent to C at z0 is represented by dz (to) - - dx dv d-T d"t(t0) + i-d7 (t0), and the angle between this tangent and the positive x-axis dz is arg ~/-(to). Consider now the mapping given by an analytic function w - w ( z ) defined in a domain containing the curve C. Then the image of C under this mapping is a curve C* in the w-plane represented by w = w(t) = w{z(t)} = u(t) + iv(t).

The point zo -- z(to) corresponds to the point w(to) of C* , and -d-F dw (t0) represents the tangent to C* at this point. Now, by the chain rule, dw

dw dz

dt

dz dt

o

do) hence, if dw (Zo) -¢ 0, we see that -d-F(t0) -¢ 0 and C* has a unique tangent at

dw

w(to), making an angle arg-dy(to) with the positive u-axis. Since the argument of

a product of two complex numbers equals the sum of the arguments of the factors, we have dw dw dz arg ~ (to) -- arg -7-(zo) + arg -7-(to). 0Z

at

Thus, under the mapping the tangent to C at z0 is rotated through the angle between the two tangents dw dz dw arg ~ (t0) - arg ~- (t0) - arg ~(zo). dz Since the expression on the right is independent of the choice of C, we see that this angle is independent of C, that is, the transformation w -- f ( z ) rotates the tangents of all the curves through z0 through the same angle arg -d?-z dw (Z0). Hence, two curves through zo whose tangents form a certain angle at z0 are mapped onto curves, the tangents of which form the same angle in both sense and magnitude at the image point w0. If we apply the foregoing considerations to the complex potential function ,(2 (z) in (136) we see that this function maps the relevant domain in the z-plane onto a domain in the S2-plane in a conformal way, that is, at every point in the domain where -d2-z dS? --/: 0, the angle between the images of two curves is the same as the angle between the curves, both in sense and in magnitude. In particular, the curves defined by ~b -- const, and ~ = const, in the S-2-plane, which are perpendicular to each other, are the images of the potential lines and streamlines in the z-plane, which are theretore also perpendicular to each other, as already has been proved in another way (see Section 1.2.4).

722

(2.2.5-3)

Analytical solution methods

The practical importance of conformal mapping for solving two-dimensional steady groundwater flow results from the fact that the total flow region in the physical p la n e (the z-plane) can be mapped onto a flow region in another plane without losing its character, as it can easily be proved that harmonic functions remain harmonic under a change of variables arising from a conformal transformation. Consequently, if it is required to find a solution of Laplace's equation of two independent variables in a given region D with known boundary conditions, it may be possible to find a conformal mapping which transforms D into some simpler region D*, for instance, a half plane or a circular disk. Then we may solve Laplace's equation subject to the transformed boundary conditions in D*. The resulting solution when carried back to D by the inverse transformation will be the solution of the original problem. The following special mappings, the quadratic transformation and the Joukowski transformation, will make this clear. 3. The quadratic transformation

An important transformation, frequently applied in potential flow, is the quadratic mapping 11) ~

(138)

Z 2.

If we switch to polar coordinates, we have Z -- re iO

and

w -- R e i~°

and thus

R e i~° -- r2e 2i0.

Then R - - r 2 and q) - 20 and.we see that circles r - r0 - constant are mapped onto circles R = r 2 -- constant, and rays 0 - 00 - constant onto rays q9 20o - constant. In particular, the positive real axis (0 - 0) is mapped onto the positive real axis in the w-plane, and the positive imaginary axis (0 - 2) in the z-plane is mapped onto the negative real axis in the w-plane. So the first quadrant 0<~0~<~-rr is mapped upon the entire upper half of the w-plane (Fig. 26). ly

v

z-plane

w-plane o

0 = 67.5 ° ~

0=45°

X

Fig. 26. Mapping w - -

z 2.

1 4

1

21 4

4 u

Partial

differential

(2.2.5-3)

equations

As the right angles between circles and rays in the z-plane remain right between their images in the w-plane, we see that the mapping is conformal, dw dw where at points for which -Uz -¢ 0 holds. As -~z - 2z, we see that at the (z -- 0) this derivative becomes zero and the mapping is not conformal: the are doubled there in this case. In rectangular coordinates the transformation w - z 2 becomes u + i 1) - -

x 2 -

y2 -k- 2 i x y

y2

and

723

angles everyorigin angles

and therefore U -- X 2 --

v -- 2 x y .

We see that the level curves of u and v are equilateral hyperbolas with the lines y = i x and the coordinate axes for asymptotes, while these curves are the orthogonal trajectories of each other. Another property of this mapping is that every point w ( u o , vo) is the image of two points z(xo, Yo) and z ( - x o , - y o ) in the z-plane. Finally, we determine the images of the straight lines x -- const, and y - const. The line x -- c has the image U - - C2 _

V -- 2cy.

y2,

Eliminating y from these equations, we find l) 2 - - 4 C 2 ( C 2 - - U ) .

This is a parabola with focus at the origin and opening to the left. Similarly, the image of a line y - c can be represented in the form U2 - - 4 C 2 ( C 2 + U ) ,

a parabola with focus at the origin and opening to the right. E x a m p l e 18. Y,l ~

z-plane

'"

P(zo)

l a/"~O

,

F i g . 27. ary.

W e l l , near a h y p e r b o l i c b o u n d -

Abstraction of groundwater by a fully penetrating well from a confined aquifer in the neighbourhood of a fully penetrating river bend. The shape of this bend can be approximated by an equilateral hyperbola, while the water level is assumed to be constant.

724

(2.2.5-3)

Analytical solution methods

We choose the asymptotes of the hyperbola as the coordinate axes; the equation of the hyperbola then becomes: xy -

~ a 2.

We consider the drawdown ~p(x, y) as a function of x and y caused by abstraction of a discharge Q in P ( x o , yo) and set ~0 - 0 at the hyperbolic boundary. To solve this problem, i.e., to find the flow pattern (streamlines and equipotential lines), we consider the physical xy-plane as the complex z-plane and try to find the complex potential £2(z) which level curves determine the required stream and potential functions of the flow. One level curve is already known: the equipotential line q5 - K99 - 0, which coincide with the hyperbola. Now from the mapping of w - z 2 (see the example in this section), we know that the level curves u and v are equilateral hyperbolas, their images in the w-plane being straight lines. So it is obvious to apply this transformation to our problem. Then the first quadrant of the physical z-plane is mapped upon the upper half w-plane and as v - 2 x y the hyperbolic boundary x y - - gal2 is mapped upon the straight line v - a 2 (Fig. 28) The image of point P ( z o ) , the location of the well, becomes P ( w o ) where wo -- uo + i vo -- x~ - y~ + 2ixoYo.

For further simplification, we apply a second mapping ¢ - w - ia 2, where ~" - rl + i 0, which consists of a t r a n s l a t i o n of the w-plane through a distance a 2 in the direction of the negative v-axis, yielding the g-plane with the straight boundary v - a 2 now becoming the 0-axis (0 - 0) of this plane (Fig. 28). We see that the original problem, to be solved in the z-plane, has been reduced to a much simpler problem in the g-plane by means of a transformation according to the function (z) = z 2 - ia 2.

The solution for the drawdown q)(rl, O) in the g-plane is the well-known solution for a well near a straight boundary with constant level, or also a positive well

1)0

I [

w-plane •

A

P(Wo)

?

I I IA

. . . . . .

a2

g-plane _

P(~o)

o0

?

I I

I t

I

t

I ,

Uo

Fig. 28.

w-plane and ~'-plane.

L

O0

Partial differential equations

(2.2.5-3)

725

(abstraction) at the point ~'o -- r/o +iOo and a negative well (infiltration) of the same strength at the point ~o - r/o - i 0o, that is

m

___.__~Q In { ( r / - / 7 0 ) 2 ÷ ( t~ + 0 0 ) 2 } 4rcKD ( r / - r/o) 2 + (0 - 0o) 2

(see Example 5 of Section 2.3.1-2). It can easily be shown that q~ - Kq9 is the real part of the complex potential function

~" -

~'o

S 2 ( ~ - ) - 2rrD As ~" -- z 2 - ia 2, ~0 - z 2 - ia 2 - x 2 - y2 + i ( 2 x o Y o _ a 2) and thus ~0 xg - z~ - i(2xoYo - a 2) -- z.~)÷ ia 2, we find for the inverse transform

O In ( z2 _z2 _ _z 2ia 2)

(see Part A, 337.04).

~2 (z) - 2reD

The real part of ,f2 is the potential function qS:

Q In { (x2- y 2

q~(x, y) --

4zrD

X2 + y2)2 ÷ 4 ( x y + XOYO- a2) 2 }

(X 2 __ y2 __ X 2 ÷ y 2 ) 2 ÷

4(xy

-

xoYo) 2 -

'

and the imaginary part the stream function ~p:

Er2 xy+xoyoa2,

~p(x, y) --

12xyxoyo, j]

Q arctan - arctan D X 2 __ y 2 _ _ X02 ÷ yo2 X 2 __ y 2 _ _ X02 ÷ yo2

2zr

•

Example 19.

.~. . . .

-_ _ _ - . . . - _ - _ _ - _ - _ - - . . _ . _ . - _ - _ . . _ . _ . . , _ , _ . _ . . . _ . . . . . .

_._._.._

,,.~

x

A

B

R

Fig. 29. Flow to a semi-infinite canal.

C

A

Flow in a confined aquifer towards a semi-infinite fully penetrating straight canal with constant drawdown h. The drawdown at a point C ( R , O ) is kept at zero (reference point). The x-axis is an axis of symmetry of which the negative part is a potential line (4~ = - K h ) and the positive part a streamline, chosen ~p -- 0 (Fig. 29).

(2.2.5-3)

726

Analytical solution methods

The complex potential plane 12 is shown in Fig. 30. Now we have to find the function S2 (z) that maps the z-plane onto the I2-plane. We see that the negative x-axis is mapped onto the line ~b - K h and the point B(0, 0) onto the point B * ( - K h , 0). The image of the posi* * B C tive x-axis becomes the 4~-axis between - K h and +oo of the O-plane. The A* 4' -Kh point c(R, 0) becomes the origin of the S-2-plane. If we translate the gt-axis Fig. 30. O-plane. through a distance K h to the left by means of the transformation w - S2 + K h, we see that the relation between the z-plane and the w-plane is the reverse of the mapping w - z 2 of Example 18, that is

ai

im

Z

aw 2

--

or

z

--

a(I2

+

Kh) 2

in which a is some complex constant. R and (point C*), so a - /~2h2

For z -- R (point C) we have ,f2 - 0

S-2- K h ( £

-1)

is the required solution of the problem in complex form (see Part A, 323.14). If we switch to polar coordinates, we have I2

= ~{

cos ( 0 ) + i sin ( 0 ) } - 1

Kh

and if we introduce the dimensionless variables S2 - Kh' s2 q~ -- K h '

-

ap ~ P K h' --

r -

r R'

x-

x -R

y and

y -

R

,

we find

rcos2(°/

r - ~(1 + cosO)-

~p___2= r s i n 2 ( 0 ) - ~r ( ~ - c o s o )

-

~l(v/X2 + y_2 + x)

~1V/X_ ( ~ + y~ -

and

~).

From this we may easily derive the equation for the set of streamlines with parameter ~, y2 = 4ap2(~2 + x), which are parabolas with focus at the origin and opening

(2.2.5-4)

Partial differential equations

727

to the right and the equation for the set of equipotential lines with parameter ¢ y2 __ 4(¢ + 1)2{(¢ + 1) 2 - x}, which are parabolas with focus at the origin and opening to the left. 4. The Joukowski transformation The mapping defined by R2

w -- z + - -

(139)

is called the Joukowski transformation, which is important for solving special problems in potential flow. Since dw dz

R2

-~-l

(z + R ) ( z -

--

Z2

R)

Z2

the mapping is conformal except at the points z - R and z - - R where d__~_w dz becomes zero. These points correspond to w - 2R and w - - 2 R , respectively. From (139) we find z 2 - w z + R 2 - 0 or 1

(140)

1 v/w2 _ 4R 2

To any value w there correspond two values of z, except to w = 2R and w = - 2 R for which there is only one image in the z-plane (R and - R , respectively). Hence, the points w -- + 2 R are branch points in the w-plane. Consequently, (140) maps the z-plane onto a two-sheeted Riemann surface, the two sheets being connected crosswise from w -- - 2 R to w -- 2R (Fig. 31), and this mapping is one to one. (Here it is assumed that the reader is more or less familiar with the representation of many-valued functions by Riemann surfaces.)

--

w-plane

z-plane b I f /

(

\\

/

/

/ -2R ,a I \ \

x \

/ \

/ J ..._.

_....

Fig. 31. The Joukowski transformation.

\ \ 2R ! _ /a u / / J

728

(2.2.5-4)

Analytical solution m e t h o d s

If we set z -- r e i°, then w -- u + i 1) -- r e i° + ~R2e_iO, from which it follows that

and

u-(r-t-R~Zr)COsO

v-(r-R~2r)

sin0

and also u2

1)2

a--7 + ~

- 1,

a--r+~

R2

where

and

b-

ir - ~ . R21

F

F

The circles r -- constant are thus mapped onto ellipses whose principal axes lie in the u and v axes and have the lengths 2a and 2b, respectively. Since a 2 - b 2 -- 4, independent of r, these ellipses are confocal, with foci at w = - 2 R and w = 2R. The circle r = R maps onto the line segment from w -- - 2 R to w = 2R. The circles r -- ot R and r - _R (o~ --fi 1) are mapped onto the same ellipse in the w-plane, 0t corresponding to the two sheets of the Riemann surface. Hence, the interior of the circle r -- R corresponds to one sheet, and the exterior to the other. Furthermore, from the expressions for u and v we obtain /,/2

1)2

cos 2 0

sin 2 0

= 4R 2 u2

l) 2

4R 2 cos 2 0

4R 2 sin 2 0

or

---

l.

The lines 0 -- constant are thus mapped onto the hyperbolas which are the orthogonal trajectories of those ellipses as they have the same foci as the ellipses (w - - 2 R and w - 2R). The direction of the asymptotes is defined by 2R sin 0 t a n c~ -

4

-

~

=

+tan

0

2R c o s 0 and thus ¢z - 4-0 which means that at infinity in the w-plane the image of a line in the z-plane through the origin has the same direction as that line. This is important in connection with problems involving uniform flow, as a Joukowski transformation does not change the direction of the uniform flow. E x a m p l e 20. A circular impermeable screen (for instance, around a building pit) is placed in a confined aquifer in which a uniform flow takes place. What will be the effect of this circular obstacle upon the flow pattern?

Partial differential equations

(2.2.5-4)

Y

729

We choose the positive x-axis in the flow direction and apply the Joukowski mapping R2

w--zq-~ Z

which maps the circle onto a line segment along the u axis, that is in the direction of the uniform flow, this direction being the same in both planes, as we have seen. So this line segment in the w-plane does not offer any resistance to the flow and the flow is purely uniFig. 32. Circular i m p e r m e a b l e screen in uniform flow. form. If the thickness of the aquifer is D and the strength of the uniform flow q (dimension L2T-1), the drawdown in the w-plane will be q5 - Kq9 -- qu, assuming zero drawdown along the v-axis. Then the complex potential a2(w) will become: a2 - ~Dw. The inverse transformation yields (see Part A, 336.11) x

from which the potential function and the stream function can be derived easily. In polar coordinates" ~b- ~

r +

cos 0

and

---~ r-~

.2)

sin0,

r

and in Cartesian coordinatesqx

q~---~-(l+x2

R 2

+y2)

and

qY

O----~-(1-x2

R 2

+y2).

According to equation (137) the total velocity at a point in the flow field is v -- I da2

q

R2

1'-71

At the point A(0, R) in the z-plane, Z A - - i R and thus VA = --~, which means that the velocity at A is twice the velocity of the uniform flow, independent of the radius R! Example 21. A fully penetrating water basin of elliptical shape in a confined aquifer in which uniform flow takes place. The flow makes an arbitrary angle otrr with one of the principal axes of the ellipse and has the strength q [L2T -1] (see Fig. 33). The equation of the ellipse in the z-plane is x2 y2 a--~, -~-1.

730

(2.2.5-4)

Analytical solution m e t h o d s

Y

q

ya .

.

.

.

x i

'''I

Fig. 33. Elliptical water basin in uniform flow.

Foci F1 and F2 in (c, 0) and ( - c , 0) with c 2 = a 2 - b 2. We know that the Joukowski transformation maps circles of one plane onto ellipses in another plane and inversely. As the problem of a circular basin in uniform flow is easier to solve than that of an elliptical one, we look for a Joukowski transformation that maps the ellipse in the z-plane onto a circle in the w-plane, say with radius P0. Considering the inverse mapping, there is one circle with radius R in the w-plane which is mapped onto the line segment from z = - c to z = c of the z-plane. So c corresponds to 2R (compare Fig. 31 with Fig. 33) and according to (139) we apply the transformation c2

z - w-F - - . 4w

(a) c2

We set w - p e in, then z - x + i y - p e in + -g-floe-in. from which c2

(,o + V)cos. x2

and

y--

( c2) p-~p

sinr/

or

y2

+

2

=1.

This ellipse represents the ellipse from

x2.-+7 ~v2-

c2

c2

a2 -- (Po + ~po)

1 for p - P0 where P0 has to be solved

and

which, if a > b, yields two solutions

(2.2.5-4)

Partial differential equations

1

and

po -- -~ (a + b)

731

1

P0 -- ;-(a - b). z

So, by means of the transformation 1v/z2

1

c2

the ellipse x2

y2

a--~- +

-1

in the z-plane is mapped onto two circles in the w-plane, a large circle with radius P 0 - gl(a + b) and a small circle with radius P0 - 51(a - b)" Now, the mapping according to the positive sign in the expression for w, that is 1

1

1/3 -- ~z -1t- ~v'/z 2 - c 2, transforms the right half of the ellipse (x > 0) into the right half of the large circle, and the left half of the ellipse (x < 0) into the left half of the small circle. For the mapping w - g1 z - ~ l ~ / Z 2 - C2 the inverse holds: the left side of the ellipse maps onto the left side of the large circle and the right side of the ellipse onto the right side of the small circle. So the required transformation of the outer region of the ellipse, the ellipse itself included, into the outer region of the circle with radius Po - ~l(a -+- b) this circle included, needs the mapping according to 1

w -- - z -+- sign x 2

v/z 2 -

c2

"

u

(b) Now we have to solve the problem in the w-plane, that is a circular basin in uniform flow (Fig. 34). The radius of the circle is P 0 - ~l(a + b) while the direction of the uniform flow remains unchanged under the Joukowski transformation, as has been proved before.

Fig. 34. w-plane.

In the first place we apply the transformation //31 ~ w e

-init

which means that the w-plane is rotated clockwise through an angle 7ra~, as IOle irJl -- ,oe i(rl-rr°~)

and thus

r/1 - r / - n'oe.

Analytical solution methods

(2.2.5-4)

732

In the wl-plane the circle remains the same but the uniform flow is parallel to the ul-axis. Secondly, we apply the Joukowski transformation of Example 20, with R - i po, that is

1)2 2Po

W2 -- Wl -- ~ ,

Wl

U2

-2po

which maps the circle onto a line segment along the v2-axis (Fig. 35) as can easily be shown. This line segment with assumed zero potential does not constitute any obstruction to the uniform flow, as it is perpendicular to the direction of the flow. So the solution in the wz-plane is simply q

Fig. 35. w2-plane.

S"2 -- ~ w2,

D

like as in Example 20. The solution in the wl-plane then becomes

_(W l - ~

q S2--D

Wl

and in the w-plane

q(we-iOlrr

P2eiOerr)

(c)

The end solution is equation (c) with w according to equation (b). If we write C2

C2 q- 4 p ~ e 2i~7r )

a"2 -- q e -i~Tr w -+-

D

4w

4w

c2

and write z - w + ~ according to equation (a) we find with equation (b)" q e -ic~rr { Z --

C2 k- (a -Jr b ) 2 e 2i°trr

/

2(z + sign x N/z 2 - c '2) ---- qe-i~ZrD { z - c 2 + (a + b)2e 2i~rr (z - sign x ~ z 2 2c 2

c2)}

Partial differential equations

(2.2.5-4)

7 33

or

S2 -- q I c2e-i~zr -- (a 4-b)2e ic~zr 2c 2 D I

+

c2e -i°err -Jr-(a +

b ) 2 e i°tn"

sign xv/z 2 - C2}.

2c 2

Now c2e -i~Tr + (a +

b ) 2 e i°trr

1

-- ~{ cos(c~zr)- i sin(oen')}

2c 2

a+b

+ 2(a - b) { c°s(crzr) + i sin(otzr/} a a -b

cos(otzr) +

b a -b

i sin(oe:rr),

and in the same way

c2e -i°~zr -- (a + b)2e i°~zr 2c 2

a

a -- b

cos (otzr)

a-b

i sin(c~zr).

The end result therefore is

S2 (z) - D(a q- b) [{a cos(crzr) + ib sin(crzr)} signxv/z 2 - c 2 - {b cos(c~zr) + ia sin(crzr)}z] (see 324.02 of Part A). Some special characteristics of the flow field will be considered in the following.

dX2dz

D(aq- b) {a cos(crJr) + ib sin(otzr)}

signxz C2

~/Z 2 __

-- {b cos(otzr) + ia sin(otrr)}] -- -Vx + ivy with Vx and l)y being the velocities the field where both these velocities are the stagnation points, for which

in the coordinate directions. At the points in are zero, there is no flow at all. These points dg2 - 0 holds, that is -d-rz

z2{a cos(~zr) + ib sin(otzr)} 2 - (z 2 - c2){b cos(crrr) + ia sin(otrr) }2 which can be evaluated to Zs -- + {a sin(c~zr) - ib cos(aTr) }.

734

Analytical solution methods

(2.2.5-4)

As Xs -- ± a sin(otzr) and ys -- Tb cos(otzr), we see that the stagnation points lie on x2

v2

the ellipse ~ + ~ = 1. It will be interesting to know the amount of groundwater that enters and leaves the elliptical basin as a result of the interrupted uniform flow. For that purpose, we determine the values of the stream function ~ ( x , y) along the circumference of the ellipse. The easiest way to do this is to make use of the parameter representation of the ellipse: Ze -- a cos t 4- i b sin t,

where

Xe --

a cos t and Ye - - b sin t.

This expression for Z e , substituted in I2 (z), gives the values of the complex potential function ff2e along the edge of the basin. As Ze2 _ C 2 - a 2 cos 2 t + 2iab sin t cos t b 2 sin 2 t - a 2 + b 2 - b 2 cos 2 t 4- 2iab sin t cos t - a 2 sin 2 t - (b cos t 4- ia sin t) 2, we have v/Ze2 - c 2

4-(b cos t + ia sin t) or

--

~/

(b Z 2 -- C 2 --

4-

Xe

a ) + i~ye

•

The parametric representation of the ellipse applied to the mapping according to (b) yields 2We -- a cos t + ib sin t 4- (b cos t 4- ia sin t) = (a 4- b)(cos t 4- i sin t), from which we may conclude that the positive sign here determines the mapping of the total ellipse in the z-plane onto the total large circle with radius l(a + b ) in the w-plane and the negative sign the mapping onto the small circle with radius L2 ( a - b) " So we have to substitute x/z 2 - c 2 - aXe b + i ag Ye in the expression for

S?(z) q D(a-b)

~e-2e - -

-

[{a cos(otyr)+ ib sin(c~yr)}

(b

a

aXe +i-~ye

)

{b cos(otn')+ ia sin(~yr)}(/e +/Ye)]

or

S'2e--" q SO ~e

--0

(a + b)i {

Xe

a

Ye } sin(oeyr) + --b-cos(olrr)

(as it ought to be) and

a P e = ~ q (a + b)[ -b-Yecos(otyr)

-

- - sin(otyr) }. Xea

-

~be - t - i l ~ e ,

(2.2.5- 5 )

Partial differential equations

735

The values of lpe at the stagnation points $1 and $2 for which zs -- +{a sin(otrr) i b cos(otn') } become lfirel -- -~q(a + b)

and

1/re2 = --~q (a + b),

respectively. Now the streamlines towards and from the points $1 and $2 are the outermost streamlines that reach the basin (see Fig. 33); the total flow through the basin is therefore determined by the difference between the two values of the stream function: Qe - 2 q ( a + b)

which is independent of the direction of the uniform flow! 5. The S c h w a r z - C h r i s t o f f e l transformation

In the previous section the solution of geohydrological problems by means of conformal mapping required a foreknowledge of the mapping properties of certain complex functions, such as the quadratic function and the Joukowski function. A more systematical way of finding a suitable function that maps the physical plane onto the S2-plane is the method, discovered independently by two German mathematicians, Schwarz and Christoffel, which will be discussed in the following. Consider a region in the z-plane bounded by a given polygon with vertices z l, z2, . . . , z,,-1, zn (Fig. 36). Now we try to find a function that maps this region onto the entire upper half of the w-plane; this means that the circumference of the polygon must coincide with the real axis u. The vertices zi (i - 1, 2 . . . . , n) have images ui (i -- 1, 2 , . . . , n)

\\ \\

Z2 c

t/

I t

/

tO

}

-I /'/n

//1 p

Fig. 36. Schwarz-Christoffel transformation.

*

U2

t

I

u3

u4

I ¸

Un_ 1 p*

U

736

(2.2.5-5)

Analytical solution methods

for instance, and one point P (here chosen between Z n - 1 and zn) becomes the point P* at infinity of the u-axis. It is assumed, as it were, that the polygon has been cut at P and stretched along the real axis of the w-plane from minus infinity to plus infinity. Suppose that this transformation can be realized by means of the function z F ( w ) (it turned out that the determination of this function is easier than that of the inverse function w = f ( z ) ) , then F ( w ) should be analytical in the whole region, eventually with the exception of a limited number of isolated points (poles) where dz = F ' (w) has no finite value. a~ The image of w in z is conformal, which means that angles between curves in the w-plane remain the same as the angles between their images in the z-plane, both in sense and in magnitude, except at points where ~d z - 0 . However, as we see, at the vertices zi and ui, the angles between the adjacent sides of the polygon in the z-plane differ from the corresponding angles in the w-plane, the latter all having the magnitude 0 or rr. So the transformation z = F ( w ) is not conformal at dz - oo there. The form of the derivative the vertices, which means that ~dz _ 0 or T~ of the function thus may be: dz

-- F ' ( w ) -- c l ( w

-

ul)al(w

-

u2)a2(w

-

u3) a3"'"

dw X (W--Un_

1) an-1 ( W - - U n ) an .

(141)

dz = 0 for w -- un (analytical, not conformal) and if a,, < 0, then If an > 0, then T~ ddz __w_w= C~ for w -- Un (not analytical, pole). Now we let a point z travel along the sides of the polygon, its image w then following the real u-axis. Suppose that z lies between z l and z2 (Fig. 36) on the line segment ZlZ2 with parameter representation z(t) = t ÷ i(tanOlt + c). The direction of the tangent at a point z to the curve z(t) is determined by a r g ( ~ ) (see Theorem 12, Section 2.2.5-2), which in this case is equal to the direction of the straight line itself: dz = 1 + / t a n 01 dt

arg(d~t ) - arctan(tan 01) - 01.

dz As long as z is situated between Z l and za, the value of arg(7?) remains constant and is equal to the angle between the straight line through Zl and z2 and the positive

x-axis. The image point w of z shows a similar behaviour: as long as w finds itself between Ul and U2 the value of arg( -aS-) dw remains constant and is equal to zero. Now z -

F ( w ) and

dz

dz dw

dt

dw dt

(2.2.5-5)

Partial differential equations

737

and from this a r g ( - ~ - ~ ) - arg(dd-@Zw)+ a r g ( - ~ )

-arg(

dz

-01,

if z lies between z l and Z2. As soon as the travelling point z passes Z2 in order to pursue its way along the line segment z2z3 the value of arg(-~) jumps by an amount or2, whereas arg(-~) dz remains zero, so that arg(T~ ) is increased by an amount or2 as well. From equation (141) we have arg ( ff----~zw) -- arg C1

-3t.

alarg(w

-+- an-1 arg(w

-

-

Ul) @

a2 arg(w

U n _ l ) -Jr- an

-

u 2 ) -~-

a3 arg(w -- u3) + . . -

arg(w - Un).

Now, the argument of the difference of two complex numbers wl and W2 equals the angle between the straight line that connects the points wl and w2 and the positive real axis (see Fig. 37). So if w lies on the real axis between u l and u2 (see Fig. 36) then a r g ( w - u l ) = 0, a r g ( w - u 2 ) = Jr, a r g ( w - u3) = Jr . . . . . a r g ( w - Un-1) = Jr and arg(w - un) = 0. From thiswe find

1)

W2

W 1 -- 1132

0 U F i g . 37. A r g u m e n t o f the d i f f e r e n c e of two complex numbers.

arg

-- arg cl + azzr + a3zr + .-. + an_lzr

if w lies between u l and

U2,

and

arg(ff~Zw) - arg cl + a3rc + a47r 4- ...-4- an_ire if w moves between U2 and u3. Hence, we may conclude that arg( T-~) dz remains constant, if z moves between two vertices of the polygon and decreases by an amount airr as soon as it passes a vertex. Compared with the increase of Ofi we found before, we have Ol i - - --7rai and thus dz

dw

=

Cl(//) -

,~1

Ul)--Y-(w

-

~2 u 2 ) -'5-- • • • ( t o -

Un)

Otn

Jr - - Cl

i~i

(113 -

c~i

ui)--Y

i=1

from which

clf

n

°ti

1--I(w - ui) - ~ d w + i--1

C 2.

(142)

Analytical solution methods

(2.2.5-5)

738

This result is called the transformation function of Schwarz-Christoffel. That arg(dd~zw) also remains constant if w moves from Un-1 to u, via the point P* at infinity, can be shown as follows: If w moves between u,-1 and + o e arg(dd-~zw)-arg Cl, and if w moves between - o e and u, arg (dd--~Zw) - =

arg

cl

arg Cl

+ a l : r r -+- a 2 7 r -k- . . -

tyl

~

o l 2 -1- . . . .

+

a._lzr

O/n-1

-

O/n

- arg c1 - 2zr - arg C l .

The transformation (142) maps the real u-axis onto a polygon P' of which the sides make angles O~i with each other. As long as the real constants ui (i = 1, 2 . . . . , n) and the complex constants Cl and c2 are arbitrary, the polygon P' will not coincide with the given polygon P in the z-plane and will not even be similar (equal form) to P. The constants ui are responsible for the similarity of the two polygons P' and P and it is easy to see that if two polygons with n vertices have the same angles, they become similar if n - 2 connected sides of P' have a common ratio to the corresponding sides of P; this condition is expressed by means of n - 3 equations in the n real unknowns ui. Thus, three of the numbers ui can be chosen arbitrarily, the point at infinity included; the remaining n - 3 unknowns can then be determined in the transformation of the u-axis onto the polygon P' which is similar to the given polygon P. Now the size and position of P' still have to be adjusted to match those of P by introducing the appropriate constants cl and c2, as multiplication of a mapping function by a complex number causes a rotation combined with a dilatation or contraction and addition of a complex number to a mapping function gives a translation. If one point is chosen as the point at infinity, for instance, u,, = cx~, then the factor w - u, vanishes in the Schwarz-Christoffel equation ( w - u, becomes 1) as can be seen as follows: equation (141) could also have been written as dz - - C 3 ( l / ) w /,tl ) a l ( W m

.

b. / 2 ). a 2 .

.

(1

~)""

dw with c3 instead of C l . If u. --> oe, then -~ --+ 0 and (1 - ~)~. --~ 1. Un Un The Schwarz-Christoffel transformation is a powerful method for solving geohydrological problems concerning flow fields of which the boundaries are straight lines, making fixed angles with each other, thus representing the contour of a polygon. An additional advantage of the Schwarz-Christoffel mapping is that also degenerated polygons, such as semi-infinite or infinite strips, are mapped onto the real u-axis. By means of the Schwarz-Christoffel transformation, we may find z as a function of w, but also the I2-plane can be mapped onto the w-plane by means of a Schwarz-Christoffel transformation, finding I2 as a function of w as well. Elimination of w from z = z ( w ) and I2 = I2(w) yields the required solution I2 = S-2(z)

Partial differential equations

(2.2.5-5)

739

or the inverse solution z -- z(Y2). The following examples may give an impression of the valuable features of the Schwarz-Christoffel transformation for solving geohydrological problems. E x a m p l e 22.

~o=0

Two semi-infinite reservoirs separated by an impermeable screen lie on top of an aquifer with transmissivity KD. The reservoirs have different levels, causing a steady flow of groundwater through the aquifer (Fig. 38). The differential equation with boundary values for the drawdown ~0 = ~0(x, z) can be written

rp=H

as:

Fig. 38. Two semi-infinite reservoirs with different levels.

0299

02(t9 -

H -0

OX 2 1"- OZ 2 -o(x, o ) -

o,

~9(O,z) -

'

7'

~o(~,z)

-0

'

Oq°(x ' D ) - O

(positive z-axis chosen downwards; aquifer thickness D). We now introduce the complex physical plane ( - x + i z. I I

Jr

i

- 7 .... _ v _

T

'

O

t I I

Jr

1/,=0 B Z

I I

i/

Fig. 39. Physical plane.

Jr at A is twice an amount 7, so o/A - - Yr.

m

2

t

2

The flow domain is bounded by a semiinfinite strip which may be considered as a degenerated polygon with three vertices A, B and C, one vertex lying at infinity (Fig. 39). The angles between the sides of the polygon at B and C are ~2 ' whereas the jump

740

(2.2.5-5)

Analytical solution methods

¢ =IKH

-(

) "

(ap = 0)

8

[

c

I

T

1

-1

l

+1

¢=0

i g

1)

Fig. 40.

w-plane.

Applying the Schwarz-Christoffel transformation (142) we have

d~

dw

=

A(w

1 b/A)-I(w

-

--

UB)-2(N

1 -- blc)-2

with A being an arbitrary complex constant. We choose UA -- co, UB -- --1 and uc -- + 1 (see Fig. 40) with which we find d~" dw

~ 1 = A(w + 1 ) - 7 ( w - 1)-7

and thus

~" -- A

f

dw ~/W 2-

1

+B

with B also an arbitrary complex constant. ~" - A arccoshw + B

or

~" - (A1 + ia2) ln(w + v/w 2

i) + B1 + iB2.

The point C in the g-plane, for which ~'c - 0, has its image in the w-plane, for which wc - 1, so that 0 - (A1 + i A2) In 1 + B1 + i B2, from which B1 - B2 - 0. Point B" (B -- iD and w8 - - 1 whence iD -- (A1 +iAz)iJr, from which A1 - ¥D and A 2 -

0. The Schwarz-Christoffel transformation thus becomes

~ ' - - - aDr c c o s h w 7r

or

w-cosh

(Tr()D --=- .

Partial d i f f e r e n t i a l e q u a t i o n s

(2.2.5-5)

741

The groundwater flow takes place from the top of the aquifer (horizontal boundary with KH q) = 0) towards the line B C (vertical boundary with drawdown 99 - 1H) with a streamline ~ - 0 along the bottom of the aquifer, whereas at the point C the stream function. theoretically tends to infinity. In the 22-plane, 7~ IC therefore, the flow domain is a semi-infinite strip with C being the point at infinity of Fig. 41. ~-plane. the imaginary axis ~ (see Fig. 41) and A B (~ - 0 ) along the real axis qS. Now we also map the 22-plane onto the w-plane by means of the SchwarzChristoffel transformation A

B

r

d22

1

= A(w

dw

or, as

1

U A ) - ~ ( W -- U B ) - ~ (W -

-

UA - - CX~, UB

d22

-- --1 and

uc --

Uc) -1

+1,

A

dw

(w-

1)~/W + 1

and thus 22--A

f

dw (w-1)~/w+l

+B

or

22 -- (A1 ÷ iA2) In

dw+ 1 - ,/5)

~ww ÷ 1 ÷ ~

+ B1 ÷ iB2.

Point A" WA - - CX~, 22A - - 0 - - (A1 + i A 2 ) In 1 + B1 + and B2 - - 0. Point B" WB - - - - 1 ' 22B - - -5-KH = (A1 ÷ i A 2 ) l n ( - 1 ) which A1 - - 0 and A 2 KH Hence 2~ • 22

2Jr

-i In

from which B1 -- 0

-- (A1 ÷ i A z ) i J r ,

from

~/w + [ +

With w -cosh(--~-) we have ~ the end result becomes: - ~2jr i In

iB2,

+ 1

g-g) ~ - 1 cosh(2-5) + 1

with 22 - ~b + i ~ and

~ -

x + iz

- v /cosh(--~) or

÷ 1

- v / 2cosh2(~-5)

,f2 - ~K H i In coth ( zr~",~ Jr \4DI

(see Part A, 355.1 1).

and thus

742

Analytical solution methods

(2.2.5-5)

Equating the real and imaginary parts of this equation we find, making use of some standard formulas of complex analysis,

4~ (x

'

z) -

{

KH sin(~) - - - - arctan ~x Jr sinh(~-fi)

/

and ~rz / cos(~-$) ~p(x, z) - -

7r

arctanh

cosh(~--~) "

E x a m p l e 23. Horizontal flow between an infinite and a semi-infinite fully penetrating canal which are parallel to each other in a confined aquifer. The canals have different levels. The differential equation with boundary values for the drawdown q) = ~o(x, y) can be written as qg(X, 0) = 0,

02(/9 ] 02q9 = 0, OX2 Oy 2

qg(X, b) = h

0q° (+oo, y) = 0,

qg(-cx~, y) = 0,

for x >~ 0,

q)(x, cxz) -- 0.

Ox

The physical plane is shown in Fig. 42; we see that the infinite canal A B with drawdown q9 = 0 loses water at one side, which flows horizontally through the aquifer and reaches the semi-infinite canal with drawdown h on both sides. The contour of the degenerated polygon becomes A B D A with angles O / a - - 2 7 " f , O/B - - 7"f, c~o -- - z r and the Schwarz-Christoffel transformation gives:

7/"

2

i iI

i I

dp = Kh .

.

.

.

.

.

.

.

.

.

- - - -

I

--

i

.

.

.

.

.

,,

t

ap=0

I

C A

5/"

A .

.

i

4>=0

B

x

~ 2

Fig. 42. Flow between an infinite and a semi-infinite canal.

--

(2.2.5-5)

Partial differential equations

743

dz

dw

A(w

=

-

UA)-2(W

w e take U A - - (:X), U B - - 0

dz dw

w-

- - UD) +1.

-- UB)-I(w

a n d u D - - 1.

1

- - A ~

w

and z - A ( w -

l n w ) + B. Point D: wo = 1, zo = i b - - A + B, from which B -- i b - A .

-e* -c* •

I

E

C

Point C: w c -- real and negative (between A and B on the u-axis), for instance, w c = - c * (c* > 0).

1 I

B

u

D

zc i b-

A

= 0 = A(-c*

-

ln c* -

A , from which A -- L and c* +

l n c * + 1 = 0, B = i b - A so that

Fig. 43. w-plane.

in-) +

= ib-~

~'

b z - --(w - lnw - 1 + in). 7/"

A

E

If we choose the streamline ~ - 0 through point D, coming from a still unknown point E on the infinite canal, the flow domain in the ,f2-plane becomes an infinite strip (Fig. 44). The Schwarz-Christoffel transformation that maps the S2-plane onto the wplane satisfies

A

~=0

D Kh

dS2 ¢

A(tO

dw UA

-- UA)-I(to

-- UB) -1

(X) and UB - - 0 .

--

Hence

B

B

dS2

A

dw

to

Fig. 44. S2-plane.

Point D" w o - - 1, I2o -- K h -- B .

and

I-2 -- A l n w

+ B.

744

(2.2.5-5)

Analytical solution methods

Point E: wE = real and negative (between A and C on the u-axis), for instance, WE = --e* (e* > O) X?E = 0 = A lne* + iArr + K h from which e* = 1 and A=i~;

Kh 7r

hence Kh Y2 -- ~ ( 7 r

+ i In w)

or w - e irr(1 - K-' s?h) -- _e_iTrs? g---hSubstituting this value of w in the expression for z as a function of w, we find the end result z---with

b (n'X2 i rr Kh

z = x + iy

e

_i 7rS2

-~-1

)

(see Part A, 323.01),

and ~2 = ~b + i~p.

2.2.6. Successive transformations

In problems involving several independent variables, a great economy in calculations can be achieved by making integral transforms successively with regard to those variables. Either a Laplace transformation may be used first to remove the timevariable, followed by other integral transforms on the space-variables, or successive integral transforms may be used on the space-variables. This method of successive transformations makes it possible to reduce a partial differential equation to an algebraic equation in the multiply transformed dependent variable, which can be solved directly. The obtained solution will now still have to be subjected to a number of inverse transformations in order to get the desired solution. The several transformations and inverse transformations may be undertaken in different sequences, yielding solutions that may differ considerably from each other, dependent on the way in which the solution has been obtained. However, as the solution of a well-posed boundary value problem is unique, the obtained solutions must be various forms of the same result. In this way, alternative expressions for the same solution are obtained, a fact that may be very useful for further evaluation of the solution. To illustrate this method of successive transformations we consider the following examples.

Partial differential equations

(2.2.6)

745

Example 24. Cf. Example l0 of Section 2.2.3-2 (Fig. 15). Non-steady, threedimensional axial-symmetric flow of groundwater, caused by abstraction of a constant discharge Q from a leaky aquifer by means of a partially penetrating well. The boundary value problem, becomes, with 99 - ~o(r, z, t) - drawdown: 02(t9 1 O q) Or 2 q_ _r -~r q

02(t9

0(t9 07~2 _ fi2 a t '

99(r, z, O) - 0

OCP(r, O, t) - O, Oz

'

099 lim(r ) -- F ( z ) - r-+o -~r

O--~(r, D t) Oz '

{0

-

~o(r, D, t) Kc

for0~
-27rK(b-a)

f o r a ~
q ) ( ~ , z, t) - - 0 .

The solution of this boundary value problem will be shown step by step, while interchanging the steps will yield a number of interesting relations between the obtained results. First step. Laplace transformation with respect to the time-variable t yields, according to Theorem 2 of Section 2.2.2-1 and equation (16): 02@ 10q~ 02@ ~)r 2 _+_ _r -OTr q_ ~ z 2 _ f12 s (# -- O ,

a--~-~(r, 0, s) - 0 Oz

a-~ (r, D s) - _ ~ ( r , D, s) '

(O~)

lira r r~ O

--

-~r

- ( r, z , s ) , (p- -- ~o

Oz

F(Z) s

=

'

Kc

'

forO~
{ 0

Q

1

-- 2 rc K ( b -- a) s

for a <~ z <~ b.

¢(c~, z, s) - 0 . Finite Fourier cosine resistance transformation with respect to the space variable z gives, according to (84) and (85):

S e c o n d step.

d2~ l d~ dr 2 t r dr

o~2 Dn2( ~ _ f i 2 S ~ ) __ O,

~- -- ~p(r, (P °/n' s) '

' i mo( r -dTr d ~ ) - - f 0 D F ( z ) c ° s ( --~-) r-~ °enz dz

= -

-

2 7 r K ( b - a) s

cos

\--D

dz - - ~ S n

SOen

OllZ a with P _ 2Jr/((h-a) Q and Sn - s i n ( ~ -) - sin(--b--), ~(oo, otn, s) D the roots of ot tan oe -- c = x---7"

0, while Oen are

746

Analytical solution methods

(2.2.6)

Third step. Infinite Hankel transformation, finally, with respect to the space variable r yields according to equation (129) with co instead of or" ^

PD& --

2

^

^

CO2~_ ~C~n ~ _

fl2S~ -- 0 "

S Oln

This is an algebraic equation with the solution"

PD&

^

~0(oo, ~ ,

s) =

(a)

~2

"

s~.(co: + ~ + ~:s)

Expression (a) represents the threefold transformed solution of the problem, which requires three inverse transformations in order to obtain the end solution q9 as a function of the variables r, z and t. The result (a) is independent of the order of transformations; so, for instance, if we denote the Laplace, Fourier and Hankel transformations shortly by their first letter, the sequences L FH, L HF, F LH, F H L , H L F and H F L all give the same result (a), as can easily be verified. Now, the inverse transformations will be given in the same sequence LaplaceFourier-Hankel as the transforms.

Fourth step. We apply first the inverse Laplace transformation to equation (a). Setting 2 M _ co2 _l_ °t._.E. n

D2

for a moment, we have:

, --

s ( m + fl2S)

1(, --

m

s

,) m

'

s + -fl

and thus (see equations (19) and (21))" Mt

~(co an t ) ' '

PDSn c~nM ( 1 - e

e2

).

(b)

Fifth step. The inverse finite Fourier cosine resistance transformation applied to expression (b), gives, according to equation (86)

~(co, z, t) = 2P

~ Sn cos (1-e " n=00en M 1 + ~2+62

-

t~2)

(c)

Partial differential equations

(2.2.6)

747

Sixth step. The last step consists of the inverse infinite Hankel transformation of (c), which becomes, according to equation (130):

(p(r, z, t) -

Q

~

rcK(b - a)

f0

D,

(OlnZ]

sin ( - q - ~ ) - s i n ( ° ' ' a l 8

Oen(1 -31- o~2q_t¢2)

=

O)2 --i-- ~2

{, .xp(-

COS --D--]

a'2+~ } f12

(d)

t) Jo(rw)dw,

provided, that it is allowed to interchange the summation and integration. With 06, as the roots of ot tan ot - s - ~D and f12 - ~KSD ' the expression (d) represents the end result. It can readily be verified that also the six possibilities of the inverse transform sequences all lead to the same result (d). So it may be stated that, in general, if we apply successive transformations and inverse transformations to all the independent variables of a differential equation and its boundary values, the end result will be

independent of the order of the transforms and inverse transforms. If we apply successive transformations to all but one independent variables instead of to all of them, an ordinary differential equation will then be left, which generally may be solved for the untransformed variable. Application of the still necessary inverse transformations will give the end result, which may be different from the result found before. For instance, in our example, if we should have omitted the third step and have solved the ordinary differential equation with boundary values for ~(r, otn, s), after the second step, we would have found: (cf. Example 6 in Section 2.2.2-3):

((°2)

.... ~(r, an , s) -- PDSn Ko r

- ~n + fl2s

(e)

SOl n

Inverse Laplace transformation of (e) becomes (see equation (51)):

P D S,~ q3(r, Ogn, t) -- ~ W Or'n

( °tzv

olnr

,',2,--2'

'

(f)

and finally inverse Fourier transformation, according to (86) gives _

qg(r, z ' t)

Q

~

2rcK(b - a) n~0 =

sin ( - ~ ) - sin (~-~-~) otn(1 "_ql_ot2q_82) --7-.--

co. ) (<, o.r)

× COS \ - - 5

W

fl202,

(g)

[)

with o~n being the roots of ot tan c~ - s - ~D and W - well-function (see Part A, 532.05).

748

Analytical solution methods

(2.2.6)

Here we have found an alternative expression for the end solution of the problem. Compared with solution (d), it is obvious that we have also found the equality:

f0 °

002 + 00 p2 { 1 - exp ( -

002/32+P2t)

jJo(r00)do) _l_~Wk_~,pr).( p2t

(h)

If we had first applied the inverse Hankel transformation to equation (a) during the fourth step, we would have found:

¢)(r, Otn, s) -- PDSn fo ~ S Oln

~co 002 + ~

Jo(r00) do).

(i)

_it_ f12 S

The expressions (e) and (i) for ~ must be equal and thus we have found that, in general,

f0 c~z002_+_ 09 p2 Jo(r00) do) - Ko(pr),

(j)

which may also be proved in other ways (equation (214) of Section 3.3.5). The alternative expression (g) for the end solution has the advantage, in this case, compared with expression (d), that it has a simpler composition, as the well-function, though defined as an integral (see equation (50)) can be evaluated numerically somewhat more easily than the integral of equation (d). Thus, it can be stated that, in general, it is worthwhile to investigate alternative solutions for a problem. Remark. The steady state of the problem may be derived from equation (d) with t --+ c~. According to equation (j) it becomes:

q)(r,z)-

Q

.K(b-a

oo

~~0 =

Otn a

sin(~)-sin( E

) (OtnZ --if) cos - 5 )

(Olnr

)(k)

If the well is fully penetrating, we get, with a = 0, b = D and e = O~ntan Otn: .

. . . cos Ko Jr K D = otn + sin Otncos Oln \--D \--D

(1)

D with O~n being the roots of ot tan ot - e - K--S" This formula represents the exact axial-symmetric solution for the abstraction of groundwater with a constant discharge from a leaky aquifer by means of a fully penetrating well. Compared with the "classical" solution of de Glee: Q Ko qg(r, z) = 2zrK------D

(r)

(m)

Partial differential equations

(2.2.6)

749

with )~ = ~ / K D c , in which it is assumed that only horizontal flow of groundwater takes place and that the leakage through the semi-permeable layer is uniformly distributed over the thickness D of the aquifer (cf. Section 1.4.2-6), we see that in formula (1) also vertical flow has been taken into account. Solution (m) is valid only for large values of the leakage factor c, as can be shown as follows. For large values of c, the product K c becomes large with respect to the thickness D D of the aquifer and thus e - 27~ becomes small. Now, if we consider the roots o~,, (n = 0, 1, 2, 3 , . . . ) of c~ tan ol = e as the points of intersection of the two curves yl - t a n c ~

and

y2--,

8 o/

a tangent and a hyperbola successively (see Fig. 45), we see that for small values of e the values of oe all tend to rr, 2rr, 3~ . . . . except the first one which approaches the value x/7. Because of the factor sin otn, all terms of the series in expression (1), for which n > 0 disappear. As or0 is small, sin a~0 may be replaced by o~0, cos oe0 by 1 and cos(~-~-j) also by 1, so that the only term of the series that is left, becomes: r

1

r

So with )~ - ~ / K D c , the result is expression (m).

- i! ,/2 /

I

I

I

i Ol

Fig. 45. Roots of ~ tan ct = e

2

2 2

750

Analytical solution methods

(2.2.6)

Example 25 (see Fig. 12). We consider the problem of Example 7 of Section 2.2.3-2 and apply first the Laplace transformation with respect to t, which leads to: d2q3

fl2sq5= 0,

q3(0 s) = 0,

dx 2

q3(b, s) = h

'

s

(a)

according to Theorem 2 of Section 2.2.2-1 and equation (16). Secondly, finite Fourier sine transformation with respect to x gives, according to equation (66): -

q5 -

(-1

b

0

s

with the solution: ~(n,s) =

(_ 1)n+l nrr h b7

(b)

~s + (,¢)2.

If we write ~ ( n , s ) = ( - - 1 ) n+l

1 )

hb (1 /'/~

s+~-~ n2rr2

s

'

the inverse Laplace transformation becomes (see eqs. (19) and (21))

O(n' t) -- (-1)n+l hb { 1 - eXP

n2rr2t ) } f12b2

,

(c)

which corresponds to equation (68). Inverse Fourier transformation then gives equation (69). If we solve the differential equation with boundary values (a) directly, we find h sinh (fix ~/s-) q~(x, s) -- s sinh(flb4~ ) w

,

(d)

First inverse Fourier transformation, applied to solution (b), also gives an expression for q3(x, s):

~(x, s ~ -

2Zrhbzs k. - , ~2s(-1)n+ln (nrcx] + ()'~'2 sin ~ b J "

(e~

The right sides of both (d) and (e) must be equal, thus, in general, with a = fl~/s: cx)

a2b2 nt- n27r 2 sin n=l

(n x) --if--

sinh a -- ~

sinh(ab)

for - b < x < b.

(f)

Partial differential equations

(2.2.6)

751

The inverse Laplace transformation of (d) can be performed in two different ways. First, we apply the inversion theorem (Theorem 9 of Section 2.2.2-3a, equation (31)). We find: y +iot

99(x, t ) -

1 lim e stqS(x, s) ds. 2zri ~-,c~ ~,×-i~

(g)

The integrand eSt,(x, s) is a single-valued function of s, because all roots vanish, as can easily be shown by using the series representation for the hyperbolic sine function: Z3

Z5

sinhz -- z + ~ + ~ + . . . The points for which the integrand becomes infinite, are s - 0 and the zeros of sinh(flbv/7 ). sinh(flb~/7) - - i s i n ( i / 3 b ~ ) - 0, from which i f l b ~ n - nrc (n - 1 . .2. . .) and so sn - _(,-r)2 ~ . Thus, so and s,, are single poles of the integrand, lying in the origin and on the negative x-axis. Now, as in Example 3 of Section 2.2.2-3, we consider the line integral (g) as a part of a closed curve which should not pass through any of the poles, such as the circle A B C of Fig. 5. The solution then will be the sum of the residues at the poles. The residue at s - 0 becomes Res [e stqS(x s)] - [se stqS(x, s)]s= 0 = s=O

'

hx

-b

and Res [eSt~b(x s)] -- ~ ,,,=s,,

'

sinh (/3x x/~-)

n=, = Z

s ~ sinh(flb~/7) h exp{-(~_~.)2

he s' sinh (fix x/s-) ] s=s.

-- Z

n=,

~,~b

~

t} sinh(n-'~-i !

,,'~ 2"-7-COS h(,,,~) "-7-

n= 1

= h ~c~ - 2 i sin/'~Jrx _ T _ ) e x p { _ ( ~ nJr ) 2t_" } n=l

--inrc cos(nrr)

Thus, the end solution is: 2h ~

hx

o(x, t ) - T +

-

-

Y/"

n--I

(-1) n

- -ns i n ( t-/ ~;X) e x p{-

as has already been found (equation (69)).

//7/" 2

cosh(/3bxF)J.,,=s,

752

(2.2.6)

Analytical solution methods

Secondly, we may write equation (d) as" h

h e~ x ~ - e - ~ x ~ ~(x,s)

-

-

e-~x~

s et3b~ (1 - e-Z/~b4g:)"

s e~b,/-~ - e-~b,/-~

Now, as

e~ x ~ -

1 is the sum of the geometric series O0

1 + y + y2 + . . . .

~

yn,

if lYl < 1,

n=O

we have for positive s" OO

~ ( x , s) - -h{ e_~(b_x),/-~ - e - ~ ( h + ~ }

S

e_2n~bvG

n=0

or h

h oo

OO

Cp(x, S) -- -- Z s

e-Cl"F{(2n+l)b-x}

-- ~ S

n---0

e -~'fd{(2n+l)b+x}

(i)

n--0

The inverse Laplace transformation can be found if the general inverse transform

L,/se1

-k./~

v/

is known. Now, from equation (40) we know that k k2 g-l{e-k4~:}_ 2 t ~ / ~ exp ( - - 4-t).

So, applying Theorem 3 of Section 2.2.2-1, we see that L_l{le_k,fi}_ s

k

2~/~-

f0 t -3/2 k2 ~_~ r exp ( - ~--r-r)dr -

2 K247

e -~2 dot

= erfc(2~/7 ( - complementary error function, see Section 3.1.1). The inverse transform of (i) then becomes" O(3

n--0 - h ~ e ~r f c n--0

(j) [ fl~{(2n+l)b+x}].

(2.2.6)

Partial differential equations

753

Compared with equation (h), this equation can be considered as an alternative solution of the problem. Remark. If we consider the same problem of Example 7, Section 2.2.3-2, but for a leaky aquifer and we assume that only the differential equation changes, while the initial and boundary conditions remain the same, we have: 02(/9

(/9

OX 2

)v 2 = fl

2 0q)

q)(x 0) - - 0 . .

with/32 -

Ot

S

and

)v 2 - - K D c

-- KD

99(0 t ) - - 0 . .

and

'

qg(b t ) -

l0 | h

fort-0, fort>0.

Laplace transformation with respect to t and finite Fourier sine transformation with respect to x gives:

b s

'

with the solution: }(n,s)

-

( - 1)n-+-I nwr h b s (~-)2-ql-/~2S--~- ~2

If we set r/-- fll~2.. = ~1 it is easy to verify that after inverse Laplace transformation, we find

n 2 7 r 2 + flZb27]

-

flab2

+ rl t

,

and after inverse Fourier transformation: 99(x, t) - 27rh Z n ( - 1 ) n + l sin ~.-b---) 1 - exp n=l ; 2 7 2 ~ fl2b2r]

-

n f1262 +/7 t

.

Now we can make use of the equality found before, as given by equation (f) to evaluate the series related to the unit term between the brackets: sinh(~) oo ( _ 1),,+, n sin (--if-) exp g)(x, t) -- h ~ - 2zrh ~ 11271. 2 /~2 sinh (-~) n=l -11- ~2

-

+ f12b2

t rl

"

The first term at the right side of this expression represents the steady state of the groundwater flow, which also follows after solving the differential equation for the steady flow of this problem (without ~a~o ).

(2.2.7)

754

2.2.7. Table

Survey

of integral

Analytical solution m e t h o d s

transformations

1

Transform

Laplace

Main property

Invers transform

L{9(t)} = qS(s)

(Section 2.2..2) = f o qO(t)e-St dt

a by means of operations •-

s~(s)

and from tabels.

~(o)

b by integration in the complex plane.

Fourier

s.{~o(x)} = ~(n)

A. Finite

= fob ~o(x)sin (mrXb *~ dx

1. Sine

,

s,{a->-} = (~)~(~)

s2J{(o(n)}

dx 2

--~ ~ - { ~ ( 0 )

= ~o(x)

2 X-'-,oo

~enX

• -- ~/--.n=l Vat )

sin

lnzrx

k---b--/

'

(-1)n~(b)}

(Section 2.2.3-2a)

2. Cosine (Section

G {~o(x)} = ~ ( n ) [117rX

- fo~ ~o(x) cos ~ ~, ) dx

d29

=

C n l {q~(n) } -- qg(x)

(torT) 2 ¢(n)

+ ( - 1y ~dx (b)

2.2.3-2b)

3. Sine

C.{d-V}

S n r {q)(X)} = q~(n)

(~.x b /~ dx

S

nr

=

(Section

with ogn being the roots of ogcotog

"-

s~rl {q3(n) } = qo(x).

J d~9 I. I d~2 I

Resistance

fo~ ~o(x)sin

+2 EL~ ~(~ cos (.~x_;_~

a_,e dx (0)

~; r"~ (n) + T~"9 9(0)

" a n +e

=

-e

~b

-

2.2.3-2c)

4. Cosine

Cnr {qg(x) } -- q3(n)

c2} {~(n)} = ~0(x)

Cnr { d--~dx 2 }

Resistance

- fo~ ~o(x)cos ( - ~ ) 4 2

(Section

with ogn being the roots of og tan og = e -

"--

~qS(n)

dd-~x(0)

' oln +e

gi~ h

2.2.3-2c)

Fourier

s{~0(x)} = ¢(c~)

B. Infinite

= f o ~p(x) sin(ogx) dx

1. Sine

dx 2

--

= --~2~a(~) + ~0(0)

,, • "-- ~2 f0c~ ~0(=) sln(xog) dog

,

and q) (cx~)

•- ~ (~) = o

(Section 2.2.3-3a)

2. Cosine

c{~o(x)} = ~ ( ~ )

c{~} dx 2

(Section

= fo~ ~o(x) cos(ogx) dx

= -~(~

I

~(o~

c -~1¢(~)1- - ~(x) = ~2 f o ~(~

cos(x,~ a~

• and ~o(oo) -

2.2.3-3b)

~dx (oc,) = 0 3. Resistance

F,.{~o(x)} = ~ ( ~ )

(Section

= f o ~o(~){ sin(~)

2.2.3-3c)

+ K woe cos(ogx) } dx

d2cp

Fr{ d__~ } __ _og2¢(og) and q)(cxz) • -- dd2 x (00) -- 0

,,

FI.--1 {q3(og)} ._ 2 fo

1+~u12~2 ' { sm(xog)

+ K wog cos(xog) } dog

.,

Partial differential equations

(2.2.7)

755

T a b l e 1 (Continued) Transform

Hankel

Hn{~p(r)} -- ~b(n)

A. Finite

= f o r ~ ( r ) J o ( " "Rr ]~ dr

1. Jo

with otn being the roots

(Section

of Jo(ot) -- 0

Main property d2~o 1 d~o

Hn { ~r 2 + r -d7 } •2 ^

-

Invers transform

H n 1 {~b(n)} = ~o(r) ^

,-

:7

=o

+otn J1 (Oen)~o(R) limr-+O (r d~o

aT)

2.2.4-3)

2. Resistance

Hnr {Cp(r) } = ~ ( n )

(Section

.__

2.2.4-3)

with otn being the roots

[ Otnr f o r~o(r)Jo,--R--) dr

H . {dr~2 + 7~~r } "--

4~(n) R l i m r ~ 0 (r dd~r)

Hn-1 {~b(n)} = ~o(r) I_

_2 p o e . 93(n) R2 Z-.,n=l j,~(oen)+j2(oe,,)

.

Jo ( -a',r r)

of o~J 1 (or) -- eJ0(ot) • and e --

R Kw

B. Infinite

H{~o(r)} = q3(ot)

(Section

= fo

2.2.4-4)

rcp(r)Jo(otr) dr

d2~o 1 d~o H{ d--~ + 7?-7}

-

2^ ~ ~o(~)

H - l { ~ ( o t ) } = 99(r)

. - fo~o ~o(ot)Jo(rot) ^ dot d~o

limr-+0 (r aT) and ~p(oo) - ~ (~) = o

.

756

(2.3.1-1)

Analytical solution methods

2.3. SOLUTIONS, DERIVED F R O M KNOWN SOLUTIONS

2.3.1. Superposition 1. Principles Although the most general differential equations for groundwater flow, developed in Section 1.4.2 (equations (3) and (4)), are non-linear because of the terms with (Op~2 (~.¥ ~)p)2 a n d (~p~2 ~ . j , the simplified differential equations that are used to solve ,,0x j , geohydrological problems analytically, such as the equations (9) and (10) of Section 1.4.2-2, are all linear. This means that the state variables, mostly the head or drawdown ~0 and its derivatives with respect to space and time, that appear in the differential equation, are of the first degree. Consider, for instance, the partial differential equation 02q9

O2q9 ax---5 + Oy 2

~

__ /~2 0q9

x--5 -

(1)

a---7'

with )v2 -- K D c and f12 __ ~ which is used for non-steady horizontal flow in a KD' leaky isotropic aquifer. This is a linear differential equation, which is homogeneous moreover (see Section 2.1.2, equation (2)). Now it is easy to show that if ~ol, ~02, . . . , ~0n are all solutions of equation (1), independently of each other, a new function q9 -- Cl~01 at- C2q92 -at - . . . -{- Cn~0n

can be composed which will be a solution of equation (1) as well, if cl, C2, . - ' , are constants. This is a basic property which may be characterized as follows"

Cn

Theorem 1. If a solution of a homogeneous linear differential equation is multiplied by any constant, the resulting function is also a solution. If two or more solutions are added, the resulting function is also a solution of that equation. This theorem is called the superposition principle or linearity principle and is of great practical and theoretical importance, because it enables us to generate more complicated solutions from simpler ones, but it must be kept in mind that it does not hold for non-homogeneous linear equations or non-linear equations. In the previous sections we have met already with some applications of this theorem. In Section 2.2.1, Example 2, we found the solution o/r

(-

0t2t -

-

-

f12R2)

of the differential equation 02([3 0r2t

10q3

r Or --

f12 Oq9 ~at,

with ol being a root of the equation otJl(~) - sJo(~) - 0 (A, R, ~, ~ and s are all constants). As there are infinitely many roots o~ - o~,~ (n - 0 , 1,2, 3 . . . . ) there are

Solutions, derivedfrom known solutions (2.3.1- l)

757

also infinitely many solutions qgn(r, t), which may be taken together in an infinite series, according to the superposition theorem, yielding the new solution:

oo

(Oenr)

qg(r,t)--ZAnJo

--~

(ol2t)

exp

(2)

fl2R 2 '

n---0

provided that this series and also the series that result from differentiating equation (2) twice with respect to r and once with respect to t, all converge. In Example 1 of Section 2.2.1 we have the solution 99(x, t) -- {A(ol) cos(czx) 4- B(cz) sin(olx)} exp ( - -c~2t -~) of the differential equation ~~)x2 - f12~ with ot and A(ot) and B(ot) being arbitrary ~}t ' constants. Then according to the superposition principle, also the integral function ~o(x t) -'

f0

{A(o~) cos(o~x) + B(o~) sin(o~x)} exp

(

- °t2t] do~

(3)

/~2 ,]

is then a solution of the differential equation, provided that this integral exists and can be differentiated twice with respect to x and once with respect to t. So far, we considered superposition of more or less arbitrary solutions of a linear homogeneous differential equation. However, for the application of the superposition principle to solutions of boundary value problems it is not sufficient that the solutions satisfy the differential equation (which must be linear and homogeneous). In that case, also the boundary values play a role. It will be clear that the following theorem must hold: Theorem 2. The superposition principle for solutions of boundary value problems

is valid if the separate problems are governed by a linear homogeneous differential equation and if the resulting solution satisfies that equation and also boundary conditions and eventually the initial condition. Example 1. Q

h

~o=0

[

d

:,1 KD, S

Y Q

,/? -

x

Consider a fully penetrating well with constant discharge Q in a semi-infinite confined aquifer at a distance d from a fully penetrating straight open water body. The original surface water level is equal to the original groundwater level and is supposed to be the reference level for the head 99 (~o = 0). At the time t -- 0, the surface water rises by an amount h and the discharge of the well is started (see Fig. 1). The head ~p is a function of x, y and t: ~0 = 99(x, y, t) and the boundary value problem can be written as: 0299 0299 ._ f12 099 -Ox - 3 -t- Oy2 0---7

Fig. 1. Well near a straight open boundary.

with/32 ---

S K D'

758

Analytical solution m e t h o d s

(2.3.1-1)

q)(x,y 0 ) - - {0 ' h lim r r~ O

forx >0, for x - - O ,

= -~r

Oqg(c~,y,t)=O, q)(O, y , t) -- h,

Ox

for r - xf(x - d ) 2 + y2, 2 rr g D

O--~ ( x , O, t) -- O, Oy

- - ( x , -t- c ~ , t) Oy

O.

This problem includes a combination of one-dimensional flow from an open water boundary and abstraction of water from a well. To solve this problem it is therefore obvious to apply the superposition principle in one way or another, as the differential equation is linear and homogeneous. First, we solve the separate problems: a. Non-steady one-dimensional flow of groundwater from a straight open water boundary through a confined aquifer (see Fig. 2). 02(t9 __ f12 0(/9 ax---g --

fi2

-~'

S -- K D '

Fig.2. One-dimensionalflow from an open water boundary.

qO(X t) -- head, '

{0

for x > O,

h

for x -- O,

qg(x O) -' !a

~o(0, t) -- h,

Ox ( o o , t) -- O.

Laplace transformation with respect to t gives (equations (4), (6) and (16) of Section 2.2.2)" d2q3

fl2sq) -- 0,

q)(0, s) -

dx 2

h

dq5

-,

--(oo,

s

dx

s) - 0

with the solution: qS(x, s) =

h -e

s

The inverse transform of e -t~x47 is, according to equation (40) of Section 2.2.2: /~x

/~2x2

2t~Texp(

4t )

and thus, from equation (8) of Section 2.2.2:

hflx qg(x, t) -- 24%-

.~-3/2 exp

fo'

(

- f12x2) d.c. 4r

Solutions, derived from known solutions

(2.3.1-1)

759

With the substitution 0/2 -- f12x2 4r ' we find Z"-3/2 dr -- --2

dz " - 1 / 2 ~-~- - 2

d( 20/'] -\fix/

40/

_ m

~x

dot

fly giving and the new boundaries of the integral: oo and g-~, e -a~2 d o / - h erfc( fix

q)a(X, t) -- h - - ~

(a)

2,/7

(see Part A, 123.02). b. Non-steady radial flow of groundwater in a confined aquifer towards a fully penetrating well with a constant discharge (see Fig. 3). 02(/9

Fig. 3. Radial flow towards a well.

1 0(t9

Or---T -F r Or --

f12 0(/9 -57'

99(r, t) -- drawdown, lim r r--+0

---

2= K D

f12

S -- K D ,

~p(r, O) - O, ~0(oo, t) --0.

Laplace transformation with respect to t gives (see equations (4), (6) and (16) of Section 2.2.2): d2q3

1 dq3

dr 2

r dr dq3)

lim r

r--+O

~

/~2S q) - - 0 , ----

Q 2rc K Ds

~(oo, s) - 0

with the solution: ~ Ks o ( f l r x / ~ . ~(r, s) -- 2 7 r Q KD

The inverse transform of Ko(flr~/7) is, according to equation (48) of Section 2.2.2:

1 2t exp (

f12r2 4t )

and thus, from equation (8) of Section 2.2.2"

760

Analytical solution methods

(2.3.1-1)

99(r, t)

Q

-

4rc K D

fo t exp ( f 1 2 r 2 )

-4r.

With the substitution u -

f12r2 4r

dT

~.

r

we find or _

du and the new boundaries of u

flZr2 the integral: oo and --W-, giving

qgb(r, t)

--

QfOO 47rKD

2r2

e -u

a7

du ~

U

Q

(f12r2)

4rrKD

4t

-- ~ E 1

(b)

(see Part A, 215.03), the solution for a well located in the origin of the xy-plane. The solution for a well in the point x --- a, y -- 0 becomes, if q) is considered as the head:

-Q

(pc(x'y't)--47rK'---'-~

E1

[ e2 {(x 47

a) 2

+

y2

] }.

(c)

Both solutions (a) and (c) for qg~ and qgc, respectively, are solutions of the differential equation

02(/9

02q9

0(/9

OX----~ -F Oy2 __ f12 at and according to the superposition theorem also qg(x, y, t) = 99a+~0c is a solution of that equation. However, as qga(0, y, t) = h and

{f12

Q E1 (a 2 -+- y qgc(0, y, t) -- - 47rK--------~ ~

2)}

the boundary value for x = 0 is not satisfied, as qg(0, y, t) = q)a(0, y, t ) + qgc(0, y, t) :/: h. So q0(x, y, t) = q9a --i-qgc is not a solution of the boundary value problem.

a

0

.e

Fig. 4. Well near a straight open water boundary.

Obviously, in order to find the right solution we must add to solution (a) not the solution of a well in an infinite field, but the solution for a well near a straight open water boundary, as given in Fig. 4, which in this case will be (q9 is head):

Solutions, derived from known solutions (2.3. l- l)

(#d(X, y,

t)

761

f12

= ------~Q E1 [-~-- { (x -ql-a)2 -l- 22}] 47c K D

Q

4rr K--------~

(d)

E1

4t-

{(x - a )

2

y2 +

}

(well and image well, see sub 2 of this section). The right solution then becomes: ~o(x, y, t) = ~0a(x, t) + 99d(x, y, t), which satisfies both the differential equation and the boundary conditions and initial condition of the posed problem. Remark. A steady state will be reached if t approaches infinity, i.e.: q)(x, y) = h +

Q

ln { (X - a)2 + Y 2 } (x-k-a) 2 + y 2 " 4rr K D

From this example we may learn that if a solution of a boundary value problem is known, a second solution of the differential equation may be superposed upon the first one, but only if it does not affect the initial value and the boundary values belonging to the first solution and inversely if the first solution does not influence the initial value and the boundary values of the second solution. For instance, in this example, solution (Pc did affect the boundary value h of solution ~0a for x - 0, while solution 99d did not, because ~0d(0, y, t) = 0. On the other hand, the solution 99a does not affect the boundary value at the well-screen of solution qgc: the total discharge remains Q, because the total flux entering and leaving the well-screen, caused by the one-dimensional flow described by ~0a, is zero. This last property, arising from the law of continuity, leads to the following theorem: Theorem 3. In a certain groundwater flow field the superposition method is valid for a finite number of all kinds of abstraction means or infiltration means, such as wells, well-screens, drains, infiltration canals and others, provided that the concerning boundary conditions relate to the total fluxes only and, .for instance, not to potentials (heads or drawdowns) or combinations of fluxes and potentials. Thus, for instance, the flow field caused by an arbitrary number of wells or drains or combinations of them with fixed discharges (which may be constant or given functions of the time), can be determined by adding the flow fields caused by the separate wells or drains. Example 2. A simple example of application of this theorem is a well field consisting of n wells with yields Q1, Q 2 , . . . , Q,. The drawdown at a certain point

762

(2.3.1-1)

Analytical solution methods

and at a certain time, caused by the i th well may be expressed as q9i (X,

Qi

y, t) -- 2rr K----~~ (x, y, t)

(i -- 1, 2 . . . . , n)

(4)

in which ~ ( x , y, t) is a dimensionless function, depending on the location of the point, the time passed since the beginning of the discharge, the location of the i th well (xi, yi) and on the characteristics of the aquifer and the well-screen. For a fully penetrating well in a confined aquifer, for instance, j~(x, y, t) becomes: j~(x, y, t) -- ~E1 -~-{(x

-xi)

2 -at-

(y

-

yi) 2

,

with El(z) being the exponential integral (Part A, 215.03). The total drawdown, according to the superposition principle then becomes: 1

n

~p(x, y, t) -- 2rcKD E

Qifi(x , y, t).

(5)

i=1

It is obvious that the superposition principle, applied to well fields, will simplify calculations on groundwater intake areas and on drainage of building pits, considerably. If, instead of the discharges, the drawdowns at the well-screens are kept constant, the principle of superposition according to equation (5) is no longer valid because the drawdown at a well-screen is influenced by the pumping of the other wells. Then, before formula (5) can be used the yields must be computed from the given drawdowns. In Theorem 3 it is stated that application of the superposition method is limited to a finite number of abstraction means, but in most cases the theorem is also valid for an infinite number; however, there are exceptions and one of these will be discussed under sub 2 of this section. Also an infinite number of infinitely small discharges may be superposed, which is shown in the following example.

Example 3.

l /,-

z0l

l Q

P

The drawdown, caused by abstraction of a discharge Q from a line well of length l on top of a semiinfinite aquifer (Fig. 5) can be determined by integrating the drawdown caused by a point well along the length of the well. The drawdown caused by a point well in the origin is equal to Q

Z Fig. 5. Line well in a semiinfinite field.

~p -

4rr Kp

Q =

4rr K ~/r 2 +

(6) Z2

(2.3.1-1)

Solutions, derived from known solutions

and for a well in point P(O ' zo) with strength

dq)-

763

2__QQ. 21

Q dzo 4re K l v l r 2 + (z

go) 2

-

Integration from -1 to +l yields for the total drawdown"

Q f+] Q

99(r' z) -

dzo

, v/r2 _+_(g ZOi2 and so

99 -- 4 r c K l

{ arcsinh( l + z

4rr K------1

r

arcsinh( I - z

)+

)}"

r

(7)

This is the solution for the drawdown if the discharge is uniformly distributed along the well-screen (Part A, 522.04). The drawdown distribution along the well-screen is not uniform in this case. An important application of Theorem 2 is the m e t h o d o f simplification. This method implies that the solution of a problem, for instance, the solution for the drawdown 99 is supposed to be the sum of two other solutions 991 and 992 of which 991 is the already known solution of a simplified problem, related to the problem to be solved, and a still unknown function 992 which must be determined making use of the condition that 991 + 992 has to satisfy the differential equation and the boundary conditions and initial condition.

Example 4.

I

Groundwater flow in a confined aquifer from a straight surface water boundary with entrance resistance, towards a fully penetrating line well with discharge Q. The differential equation with boundary values for the drawdown 99 = 99(x, y) becomes (see Fig. 6):

P X

Q

=~

~o=0

0299 /..1~p

i

i w

a

.+

Ox--7 + ~

- O,

a~

~(O, y)

--(0, Ox

J

'

0299

y) = ~ , Kw

~0(oo, y) = 0,

KD

Fig. 6. Well near a boundary with entrance resistance. ~90_~,( x , O) _ O Oy

lim ( 0r 9 9 ) - r ---~ O

-~F

f o r x ~>0 a n d x 7/= a , Q 2 rr K D

for r2

--

(X - - a )

y ( x , c~ ) - O ,

2 -at-

v2 .

764

Analytical solution methods

(2.3.1-1)

Now, assume that the solution 99(x, y) - 991(x, y ) + 99:(x, y), where 991 is the solution for a well near an open boundary (without entrance resistance) and 992 a still unknown function, dependent on K w . The solution for 991, the simplified solution, is"

Q

ln{ (x -+- a)2 -+- Y2 } (x - a ) 2 + y2

991(x, y) -- 47cKD

(8)

(well and image well, see sub 2 of this section). As both 99 and 991 are solutions of the differential equation 0299 t 0299 = 0, OX 2 Oy 2 992 is also a solution of this equation. The boundary values for (/92 follow from:

992 -- (./9- 991 0992 Ox

099 Ox

and

0992 099 = Ox Ox

0991 Ox

Q [ 2(x + a) / 4re K D (x -+- a) 2 + y2

or 2(x - a ) / (X - - a ) 2 -k- y2

099 Q 4a 0 99---~2 x ( 0 y) - -~x ( 0 ' y) - 4 rc K D a 2 -4- y2 991(0, y) + 992(0, y) Kw

992(0, y) Kw

99(0, y) Kw

Q

a

r c K D a 2 -4- y2

Qa r r K D ( a 2 + y2)

Q a J r K D a: + y : '

992(00, y) - 99((x~, y ) 0(/92 _ 0(/9 Oy -- Oy

and so

991((x~, y) - 0 -

0(t91 _ 099 Oy -- Oy

0 ' O,

___~Q { 2y 47r K D (x -t- a) 2 + y2

2y (x - - a ) 2 --t- y2

!

and so °~--22(x, o) - -099 0-T(x,0)-0-0 Oy lim r r--+O -~r

-lim

r~O

r

Or

forxCa,

r

-Or

---

2rr K D - ( -

for r 2 -- (x - a ) 2 + y2, from which T~)~°2(X ' 0) -- 0 ~o(x, oo) - ~01(x, oo) - 0 - 0 - 0.

Q

2rcKD

also for x

-

)-o

a, 992(x co) -

Solutions, derived from known solutions (2.3.1-1)

765

The boundary value problem for 992(X, y) thus becomes: 02(/92 02992 OX2 ] Oy 2 = O,

992(0, Y)

0992 Ox (0, y) --

992(00, y) -- O,

0992 (X, O) -- 0 Oy '

Kw

Q

a

7 r K D a 2 4- y 2 '

(D2(X, 00) -- 0

"

Owing to the boundary condition ~0V( x ' 0) - 0 we apply the infinite Fourier cosine transformation with respect t o y (see Section 2.2.3-3), giving according to equations (94) and (95): d2 (~2 _ 0/2 (P2 - 0 dx 2 d(~2 dx

~(0,

~) =

~b2(oo, or) - O,

(~2 (0, oe) Kw

Qa

JrKD

f0 ~ cos(oey) a 2+y2 dy-

~2(0, oe)

Q

Kw

2KD

_e-aO~

with the solution: (~2(X, oe) __

Q Kw e-(X+a)~ 2 K D 1 + Kwoe

From equation (96) we find the inverse transform: Q

(#2(X, y) -- r r K D

fo o°

Kw e -(x+")~ cos(yoe) doe 1 + Kwoe

(9)

and finally 99(x, y) - q91(x, y) + 992(X, y) (equations (8) and (9)). Remark. As cos(yoe) - Re[e-iy ~] is, with z - x + iy, [ f o °° Kw e -(z+a)~doe] q)2(x, y) - r r KQD Re 1 + Kwoe

and thus S'22 -- ~2 q - i ~2 --

_

Q [~ reD Jo

Q exp(Z + a

-- - ~

Kw e -(z+a)°t doe 1 + Kwoe

z +a,]

....K t o ) E l ( K 1 1 )

,I

[substitute u - (z+a)(x--!y+oe)] with E1 - exponential integral and a"22the complex potential function (see Section 2.2.5-1, equation (136)). The end solution, written as a complex function thus becomes: Q

z

2 r r D { l n ( z - a+ (Part A, 337.21).

a

)+2exp\

(z+a z+a K w ) E 1( )}

(10)

766

Analytical solution methods

(2.3.1-2)

2. Method of images A very important application of the superposition principle is the so-called method of images for solving geohydrological problems involving straight or circular boundaries with prescribed conditions. Instead of theoretical discussions, we shall show the principles of this method by giving some examples. E x a m p l e 5.

A fully penetrating line well in an aquifer near a fully penetrating straight surface water body (channel, river, lake, etc.) with a constant water level (see Fig. 7). P e The straight open water boundary has been -'hQ x Q~,a a -tchosen along the y-axis; the groundwater a =~ level in the aquifer is initially the same as , ' . ~o=0 the surface water level and assumed to be zero (q9 = 0). Pumping of groundwater with a discharge Q from the line well at P (a, 0) yields a steady flow of water from KD the surface water body through the porous medium towards the well, causing a drawdown qg(x, y) everywhere in the half plane Fig. 7. Well near an open boundary. (x > 0, --c~ < y < +cx~) except at x = 0. If we imagine an infinite aquifer without surface water boundary and another line well at point P ' ( - a , 0) with a recharge (negative discharge) of strength Q equal to the discharge at P (a, 0), then the y-axis (x = 0) is the locus of the points that have equal distances to P and P'. Such a point undergoes influences from both wells that are equal but of opposite sign, resulting, according to the superposition principle, in an unchanged head (q9 = 0). So the problem of a well in a half plane near a straight open boundary can be replaced by a discharge- and a recharge well of equal strength in an infinite field. The solution for the drawdown caused by the discharge well at P (a, 0) in an infinite field is: l

I I I

•" - - . . ~

_

q)=c-

2rc K D

lnr

with r - v/(x

a)2 + y2,

and for the drawdown (negative) caused by the recharge well at P ' ( - a , 0) in an infinite field: (,o' = c' +

Q In r' 2~rK D

with r ' = x//(x + a) 2 + y2.

Solutions, derived from known solutions

(2.3.1-2)

767

Making use of the principle of superposition, we find for the solution of a discharge well and a recharge well of equal strength in an infinite field: Q l n { ( x + a ) 2 + y2} 99 -- 4zrK--------D (x - a ) 2 -k- y2 '

(1 1)

as c + c' = 0 from the condition ~o(0, y) = 0. If we define a discharge well as a positive well, giving a positive d r a w d o w n and a recharge well as a negative well, causing a rise of the groundwater heads (negative drawdown), we have seen that the combination of a positive and a negative well in an infinite field has the same effect as a positive well near a straight open boundary in a semi-infinite field. (Here, and in the sequel we assume all wells and image wells to have the same strength.) On the other hand, two positive wells or two negative wells in an infinite field is the substitute for a well near a straight impervious boundary in a semi-infinite field, as can be deduced from symmetrical arguments and also derived mathematically by differentiating the total drawdown

~o -

Q In [{(x - a) 2 + y2}{(x + a) 2 + y2}] + c 4 7r K D

(12)

(for this case c + c' --/=0): Oq) _ Ox -

Q

{

2(x+a) } a) 2 + y2 -+- (x -nt- a) 2 -+- y2

2(x - a)

47rK-------D ( x -

and putting x -- 0, from which a~0(0, y) -- 0. This means that no flux of water takes place through the boundary along x = 0. In general, all kind of solutions for a single well in an infinite field, such as solutions for steady or non-steady flow in confined or leaky aquifers or multilayer systems, may be used for practising the method of images. An example for non-steady flow has already been given in sub 1 of this section (equation (d); see Fig. 4). E x a m p l e 6. As in Example 5 (Fig. 7), but for a leaky aquifer, the solution becomes:

~o -

2 rc K D

in which r -- v/(x - a) 2 + y2, r' -- v/(x + a) 2 + y2, )~ _ ~/~2Dc -- leakage factor and c -- resistance against vertical flow of the semi-permeable layer.

(2.3.1-2)

768

Analytical solution methods

Example 7. Y

,,~ Q!-_)._

P(xo, Yo) 1

i

Iy°....Xo2 Q(-)

I

Flow towards a discharge well in an infinite quarter of a plane bounded by two straight boundaries, perpendicular to each other; one of them is an open water boundary, the other is impervious.

I

;,,'~Q(+)

Fig. 8. Well near an open and an impervious boundary. The differential equation with boundary values becomes, with q9 -

qg(x, y),

0<~x < c~ and 0 ~< y < c~" 02~0

0299

OX----5+----Oy2 = 0 ,

°--~(x, Oy

o) - o,

~o(0, y) - 0, ~o(x, ~ )

(0~0) Q lim r = r~0 -~r 2zr K D '

qg(c~, y) - 0,

- 0,

with r - ~/(x - x 0 ) 2 -1- (y - Y0)2

It is not difficult to see that this problem of one well in a quarter of a plane may be replaced by a system of four wells in an infinite plane; namely, the original well together with three image wells, the place and character (positive or negative) of the latter depending on the location of the original well with regard to the boundaries and the character of the boundaries (open or closed). In the present case, the image wells are a positive well at (xo,-yo) and two negative wells at (-xo, yo) and ( - x o , - y o ) , respectively. The solution for x >~ 0, y >/0 thus becomes" Q l n [ { ( x + xo)2 + ( Y - Y°)2}{(x + x°)2 + (Y + Y°)2}] q9 - 4rrK------D {(x - xo) 2 + (y - yo)2}{(x - xo) 2 + (Y + Yo)2} "

(14)

Example 8.

Fig. 9. Drain in a confined aquifer.

An infinite drain with discharge q [L2T -1] on top of a confined aquifer with thickness D (Fig. 9). In order to make use of the method of images, we have to realize that two parallel boundaries need an infinite number of image wells or drains, as is shown in Fig. 10, where the original aquifer first has been replaced by an aquifer of thickness 2D and a drain in the middle with discharge 2q.

Solutions, derived from known solutians

(2.3.1-2)

As the boundaries are impermeable, the image drains must be positive (discharge drains) and be located at the points (0, + 2 n D ) with n = 0, 1, 2 . . . . . The solution for the drawdown caused by a drain with discharge 2q in an infinite field is, if the drain is situated in the origin:

2q z = -6D 2q -4D +

2q -2D .,~ 2q

+

769

2D % 2q

q~ -- c -

4D % 2q

= c

6D 2q 8D % 2q Fig. 10. Infinite n u m b e r of positive drains in an infinite field.

q ~ lnr JrK q ln(x 2 + 2JrK

Z2

).

For a two-dimensional steady or quasi-steady problem it is, in general, convenient to work with the complex potential solutions because of the simpler expressions for the latter. Then for one drain in an infinite field we have (cf. Example 17 of Section 2.2.5-1): q £2 - c - - - I n ( .

The physical plane is now considered as the complex plane ( - x + i z and the image drains are located at the points ( - + 2 i n D (n - 1, 2 , . . . ) . Now, according to the superposition principle, the drain together with its image drains should give: ,(2 -- co

q In ( + cl - q ln(( - 2i D) + Jr

Jr

C2 - - q Jr

In(( + 2i D)

+ c3 - q ln(~" - 4 i D ) + c4 - q ln(~" + 4 i D ) + . . Jr Jr OQ

--c

- q In ~ - q ~ 7/"

(X3

ln(( - 2 i n D ) -

Jr

q E

ln(( + 2 i n D )

or

Jr n-- 1

n-- 1

0(3

S2-c

- q In ( - q ~ Jr

In(( 2 + 4n2D2).

Jr n--1

Unfortunately, the series in this expression diverges, as the general term does not approach zero with increasing n; so it was not permitted to apply the superposition theorem in this case.

(2.3.1-2)

770

Analytical solution methods

However, if we apply this theorem to the groundwater velocities, caused by each of the drains, as expressed in the relation (see equation (137) of Section 2.2.5-1): dX?

d~

= - Vx + i vz,

we find: dX?

q(1

d~ =

rc - ~ + ~ - 2 i D

q

1

1

1

1

+ ~'+2iD +~-4iO

+~+4iD

) +""

+

Jr

=

~-2 ÷ 4n2D 2

"

As

coth(Jvx) -

1 2x ~ 1 7//,/ ÷ --7/" k~l= x ÷ 2 k 2 '

we find d£2 _ d~" -

q coth(Zr~') 2D 2--D '

as can easily be verified. The solution for the total complex function, which is also the solution for the posed problem, thus becomes" X2 - c

7l"

In sinh

(15) rr~

(Part A, 356.01) from which follows 4) - R e [ ~ ] - cl - ~ In Isinh(g-5)l if c = Cl ÷ ic2, and as 4) - K~0 we find for the drawdown: ~o(x, z) -- c - 2~r----K

2-D

÷ sin2 2-D

"

(16)

With some slight adaptations this formula is also valid for an "infinite" row of wells of equal strengths Q and at equal distances d from each other in a confined aquifer of thickness D: ~0(x, y) - c - 47r K D

-d-

+ sin2 --d-

"

(17)

These solutions are important for finding solutions for more complicated problems concerning drains and wells between open and closed boundaries, for instance, the following example. * Formula 1.421-4 of Tables of Integrals, Series and Products of Gradshteyn and Ryzhik.

Solutions, derived from known solutions

(2.3.1-2)

771

E x a m p l e 9.

A discharge well in a confined aquifer of thickness D between two parallel open boundaries. We suppose a -¢ g1b which implies the use of a more complicated system of image wells in order to replace the flow field shown in Fig. 11 by an infinite field.

............. Y I ..................................

a I I ............1................................. ................

x

Fig. 11. Well between parallel open boundaries.

a4b+a 4b-a

"~a2b+a 2b - a

This system is shown in Fig. 12 and consists of an infinite number of positive wells situated at the points (0, a -t- 2rib) (n - 0, 1, 2 . . . . ) and an infinite number of negative wells located at the points ( 0 , - a - t - 2 n b ) (n - 1, 2 . . . . ). Both the positive and the negative wells have mutual distances of 2b. If we apply the formula for a row of wells (see Example 8) we find for the positive wells: q)l(X, Y ) - - C l -

)------~Q 4 r Inc [sinh2 K D( -zrx ~

:...... :......:........-...--.-.-...~-.-.-.....,.,.,.

+ sin2 \(7r(Y2bS-a))] -a

x

Q

and for the negative wells" 992(X, y) -- C2 _Jr_~4 r rQK D ln[sinh2 ( -~-) 7rx

~-2b + a

+ s i n 2 ( 7 r ( y]+2 ab) )

-2b -a

m

and together: Q ~-4b + a

-4b -a

Fig. 12. Infinite number of positive and neg-

ative wells.

Q q) -- 4rr K D In

[sinh2(~_~) q_ sin 2 (~r(v+a) ] sinh2(~-~) + sin 2 (Tr(2~-a))

(Part A, 356.12) as cl +

C2

--

(18)

0 because q9 -- 0 for y -- 0.

Analytical solution methods

(2.3.1-2)

772

Remark.

The problem of Example 9 can also be solved by means of a finite Fourier sine transformation with respect to y (Section 2.2.3-2a). To make the boundary value at the location of the drain suitable for this transformation, we consider the discharge Q of the line drain as the limit of a discharge g1 Q at both sides of an infinite screen with length l, if l approaches zero (see Fig. 13).

............. Y.l ................................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

liill~aQ .llb ................] ....I............................... x

Fig. 13. Well-screen between two parallel open boundaries.

The boundary value along the y-axis for the well-screen becomes" for 0 ~< y < a and for (a + l) < y ~< b,

0 Oq°(O y) --

Q

Ox '

2KDI

for a < y < (a + 1).

Finite sine transformation gives: dq3(0 , n )

dx

_

Q 2KDI

f a + t sin ( ~ )

dy.

Letting l approach zero, we have" lim dq3 t~0 d--x-x(0, n) -

Q I f~:'+t sin ("~---Y-v)dY ] _- 0 - i n d e f i n i t e . 2KD 1 t--,0 0

Applying the rule of l'H6pital, by differentiating the numerator and the denominator with respect to l, we find lim dq3 t-+o ~x-x(0'

Q [sinlnyr(a+l)]] n) -- 2 K----D b t-+o

Q sin(nzra) 2K D --if-

o

The transformed differential equation becomes: d2q3 dx 2

(~_)2q3 -- 0

and as q3(oo, n) - 0 we find for the transformed solution: nyra Qb 1 (o(x, n) - 2rrK D n sin (---b--) exp (

nTrx]

- ---~-/

and thus"

sin( nrr T )a sin( nrry )exp( n=l

nrrx ). b

(19)

Solutions, derived from known solutions

A s s i n z l s i n z2 -

~1 {COS(Z 1 -

COS(Z 1 -3t- Z2)} , a n d

Z2) -

CXD

E

773

(2.3.1-2)

Cx3

le-,Lx n cos(ny) -- Re Z

n-- 1

_n1 e-nZ.

with z - x + i y,

n-- 1

and OO

E n--1

1-un -- - - l n ( 1 -- u), Ft

we can show after some mathematical exertion, that equation (18) is another form of equation (19)*. E x a m p l e 10.

Groundwater flow between a straight open / \ boundary and a circu/ N / \ / \ lar basin with a conI I stant drawdown h in ', " d / a confined aquifer. X / N / x / -x .. Both boundaries are penetrating the aquifer fully (see Fig. 14). At first sight, this problem of steady groundwater flow should be solved by adding to the solution of flow to a circular basin in an infinite field the solution of flow from a circuFig. 14. Circular basin near a straight open water boundary. lar image basin with rise h of the level at the other side of the straight boundary, according to the method of images. Y

- KL/

h

k=

However, on further consideration, we may notice that the solution of the image basin will affect the values of the drawdown inside the real basin, which are supposed to be constant. So a direct application of the method of images is impossible in this case. On the other hand, if we consider the solution of a well and an image

* Formula 1.462 of Tables of Integrals, Series and Products of Gradshteyn and Ryzhik.

774

(2.3.1-2)

Analytical solution methods

well of Example 5, we see that we can write solution (11) as

4rc K Dq9 = In { (x + a) 2 + y2 (x - a ) 2 + y2 }" Q Setting

A

-

exp(4Zr

K QD~o),

(20)

we have A ( x - a ) 2 -k- Ay 2 -- (x -k-a) 2 -+- y2 or x2(A -

1) + yZ(A - 1) + a2(A - 1) - 2ax(A + 1) - 0, giving" A+I (x - a ~ - - i )

2

+

4a2A

y2 --

(A - 1)2'

the equation for a set of circles with centres on the positive x-axis. The parameter of this set is A and as A is dependent on qg, we see that the set of circles represents the equipotential lines, belonging to the flow pattern of the well and image well at (a, 0) and ( - a , 0), respectively. Now we assume that the circumference of the circular basin of our problem coincides with one of these equipotential circles; the problem is thus then brought back to that of a positive well and negative image well, with still unknown location. But from equation (20) it follows that for (x - d) 2 q- y2 __ R 2, the equation of the circular basin, the equipotential q9 must be equal to the drawdown h: (x+a) 2+R 2- (x-d) 2 = In { (1 - a ) 2 + R 2 _ ( x _ d ) 2 }

4rc K Dh

Q

or

4:rrK Dh = ln{ 21(d + a) + a 2 + R 2 - d 2 } 2x(d - a) + a 2 + R 2 - d 2 " Q As the left side of this equation is a constant, the right side must be independent of x from which it follows that a 2 - d 2 - R 2 and

47r K Dh Q

= In

d + ~/d 2 - R 2 d - ~/d 2 - R 2

or Q

u

2Jr K Dh arccosh(-~)"

The solution for the drawdown qg(x, y) becomes:

I ~o--

h 2 arccosh(-~) h

(Part A, 335.04).

In

{ (x +

~/d 2 - R2) 2 -+- y2

for

(x - ~/d2 - R'~-+- y2 } for

(x - d )

2 + y2 > / R 2,

(x - d )

2 + y2 <~ R 2

(21)

Solutions, derived from known solutions (2.3.1-3)

775

3. Discharge impulses A discharge impulse is an instantaneous abstraction of a finite quantity of water from a groundwater body, for instance, a discharge Qi during an infinitely short time dt from a line well in an aquifer, yielding a total abstraction Pi = Qi dt. A recharge impulse is an instantaneous injection of a finite quantity of water Pi into a groundwater body, that is, a recharge Qi during an infinitely short time dt. The dimension of Pi is [L3]. A discharge impulse causes a non-steady flow field, starting from an initial drawdown distribution ~0 = 0 at all points of the field with the exception of the impulse location where the drawdown becomes infinitely large. In course of time the drawdowns will gradually spread, originating from the impulse centre, reach a maximum and then decrease to zero if the time tends to infinity. The next example will make this clear. Example

11.

Instantaneous abstraction Pi [L 3] of groundwater from a fully penetrating line well in a confined aquifer. The drawdown is a function of r and t: ~0 = ~0(r, t).

KD]S.

r

Fig. 15. Discharge impulse in a line well. The boundary value problem can be written as:

0299

1 0(/9

/~2 0(/9

f12

-

~p(r, 0) = 0

(O~p)

lim r r-+0 -~r

-~' {__

'

KD

for r > 0, -

S -

r -Or

cp(oo, t) = 0,

Qi

fort-0,

27r K D

0

Pi -- Qi dt.

for t > 0,

Laplace transformation with respect to t gives:

d299 dr 2

t

1 d~ r dr

flesq9 "-0,

~)(OO, S) = 0 ,

lim

(d~a) r ---

r-+O --dTr

Pi 2rc K D

(cf. equation (30) for to = 0 of Section 2.2.2-2) with the solution:

Pi Ko (fi r v/7) . ~(r, s) -- 27rKD The inverse Laplace transformation becomes (cf. equation (48) of Section 2.2.2-3):

Pi e x p ( 99(r, t) -- 47r K D------7 (Part A, 215.01).

f12r2 ) 4t

(22)

776

Analytical solution methods

(2.3.1-3)

For t = 0 ~0(r, 0) = 0 for every value of r with the exception of r = 0 where the drawdown ~0(0, 0) becomes infinitely large. A control on the correctness of equation (22) is that at every time t > 0 the total quantity of groundwater released from storage must be equal to Pi. The total quantity of water d V, released from the aquifer between the two cylinders with length D (thickness of the aquifer) and radii r and r + dr, respectively, at the time t, is:

Pi S exp dV - 27rr dr Scp - 2 K Dt

f12r2 ) r dr 4t

and thus

Pi S 2KDt

V=

f12r2

exp (

4t ) r d r - - P i [ e x p (

f0 °°

f12r2 4t ) ] 0 - - P i .

For a fixed distance r0 from the line well, the drawdown as a function of the time, the time-drawdown line, follows from equation (22) by putting r - r0. The curve begins with qg(r0, 0) - 0, reaches a maximum at t - tmax and tends to zero again for t approaching infinity. The maximum can be determined, if we set u -

O~p Ot

=

-

Pi 4zr K Dt 2

e -u

+

~

Pi e

4re K Dt

-u-

u t

=

-

Pi

- - e - "

4rr K Dt 2

(1 -

f12r2 4t

'

from

u),

which becomes zero for u - 1' from which tmax -- f12r2 Sr2 and 99max(r0, tmax) -4 --" 4xo Pi e-1 this result being independent of the transmissivity K D So if we perform ~r2S , a pumping test, by giving a fully penetrating line well in a confined aquifer a discharge impulse Pi m 3 of water and measuring the values, qgmax and tmax of the time-drawdown curve for an observation well at distance r0 from the pumping well, we can directly determine the values for S and KD. A first application of the impulse function(s) in combination with the superposition principle is the solution of initial value problems. An example of an initial value problem has been given in Example 4 of Section 2.1.2, concerning one-dimensional flow. In the following example a two-dimensional initial value problem will be discussed. Example

12.

Z

~o=0

~888~ y

KD, S '

X

Fig. 16. Two-dimensional initial val-

ue problem.

In an infinite field the groundwater head in a confined aquifer with transmissivity K D and storage coefficient S is initially an arbitrary function of x and y (see Fig. 16). Owing to the gravity, the heads will decrease and the groundwater will move horizontally with velocity components in x- and y-direction until all heads are equal. These movements are governed by the depletion function, which is the function that shows the variation of the heads with time everywhere in the field.

Solutions, derivedfrom known solutions

(2.3. I-3)

777

In order to determine this depletion function, we may regard the original head

~(x, y, O) = F(x, y) as being due to infinitely many recharge impulses at t = 0. From the definition of the storage coefficient S (Section 1.3.5-1) it follows that at the vertical through the point (x0, y0) at t = 0 an amount of water is stored equal to F(xo, yo)S dx0 dy0. This small quantity of water, considered as due to a small recharge impulse dPi gives, according to equation (22), a head distribution dqg(x, y, t) -

F (xo, Yo)S dxo dyo

47r K Dt

f12

exp[-- --~{(X --Xo)2 + (y-- yo)2}].

The total head distribution at any time and location, the depletion function, then becomes: ~o(x, y, t) --

f12 f+oof+oo F (xo, yo)

4rrt

oo

x exp [ (see 313.01, Part A).

~o=0

~o=h

~o=0

~o=0

(23)

oo fi2 ~-7 {(x --XO)2--~ - ( y -

20)2}] dxo dyo

A special case of this example is the depletion function, starting from a constant head h in the first quadrant and zero head everywhere else (Fig. 17).

Fig. 17. Constant head in the first quadrant as initial value. Here

~9(x, y, O) -- F(x, y) --

h

0

for x > 0 and y > 0, forx <0or y <0.

Equation (23) then becomes: q)(x, y, t ) -

f12h fOOOexp{

~-

f12

-- - ~ ( X --XO) 2 } dxo

/ '2 x

As

exp

/

- -4-7(y - yo) 2 dyo

= -e -c°2 do). h -.f~ x~ e -°t2 doe - ~ 7/" 2v/t 247

fz ~00 e -a~2 doe --

f O e -°~2 doe + ~0 -c~ e -°t2 doe z

q7 erfz + ,/7 erf(-oo)

= - ~

2

2

-- - ~ ( 1

+ erfz),

778

Analytical solution methods

(2.3.1-3)

we find: ~o(x, y, t) -- ~h{ 1 + erf (fix_ 2~/t) }{ 1 + erf (2@t) }

(24)

(Part A, 313.02).

A second application of the impulse function(s) in combination with the superposition principle is the integration of impulse functions with respect to time, yielding continuous functions such as continuous discharges or recharges at wells or drains. Example 13. As an example we choose the line well of Example 11, now with a discharge Q(t) which is an arbitrary function of time. We regard this discharge as due to an infinite number of successive discharge impulses, starting at t = 0. At the time to, the discharge impulse Q(to)dt0 takes place which gives at the time t (> to) a drawdown according to equation (22):

Q(to)dto e x p { f12r2 } dgo = 4rrKD(t - to) 4(t - t0--------~' and in total:

t ~p(r, t) -- 4rrKD

l 2r j

dto

Q(to) exp - 4(t -----to) t - t o f12r2

which by substituting ) ~ - 4(t-to) can be written as

l f~ ( ~o(r, t) -- 41rKD 2r2 Q t

flere)le-Xd) ~ 4X

~.

(25)

4t

(Part A, 215.02). As the Exponential Integral is defined as: El(Z)

-

f

e~ e-)~

)~ d)~ (see Section 3.2.1),

z

we find for a constant discharge Q0:

Qo E l ( fl 2r2 ) ~p(r, t) - 41r K--------D

4t

(26)

(cf. Part A, 215.03) and for a discharge that increases linearly with the time Q(t) = At (A is a constant):

A f~ ( ~o(r, t) - 4~rKD 2r2 t 4t

/32r2)le-)~ 41. X d~,

Solutions, derived from known solutions (2.3.1-3) and with

779

p2r2

u =

o

4t

~o(r,t)_

At fu°° ( ~ ) _xd)~ 4Jr K D 1e

and as E,(z) is defined as E, (z) -- z'-1 fz ~ --e 1 )~n we find

~p(r, t) =

At{

-)" d~.

f12r2 }

f12r2

4rrK--------~ E l ( 4t ) - E 2 (

(27)

41 )

(Part A, 215.04). A third application of the impulse function(s) in combination with the superposition principle is the possibility to write down directly the solution for the heads caused by vertical infiltration (precipitation or artificial recharge) into an aquifer, which will become clear from the following example. Example

14.

I rP

q(t)

~sxa~$,1,,1,,1, $ , b , l , , l , ~

1,

b

]

b

-b

ii K D ' S

Vertical infiltration over an infinite strip of width 2b into a confined aquifer in which the flow is supposed to be merely horizontal. The infiltration velocity q [LT -1] is an arbitrary function of the time and equally distributed over the area of the strip. The mathematical translation of this onedimensional problem is:

02(t9 099 OX 2 ~ q(t) K D = fl 2 --~,

+b x

~p = ~o(x, t),

f12

-- K D ' ~p(x, O) = O,

Fig. 18. Infinite strip with vertical infiltration.

a---~(o, t) - o

Ox q-

q(t) 0

'

~ o ( i o o t) - o,

'

for-b<x b.

In order to solve this problem, we may regard the vertical infiltration as an infinite number of successive recharge impulses in line wells everywhere in the infiltration area. For this reason, we consider the flow field as two-dimensional for a moment

(2.3.1-3)

780

Analytical solution methods

and find that at the time to at the point (xo, yo) a recharge impulse takes place equal to d Pi -

q (to)

dxo dyo dto,

which will give at the time t (> to) and at the point (x, y) a rise of the head, according to equation (22)" d~o-

q (to) dxo d yo dto 4re K D ( t - to)

/~2 /] F exp L - 4 ( t - to){(x - x o ) 2 + ( y - yo)2lj.

All impulses together form the vertical infiltration and will give a total head:

~o(x, y, t) --

1 f_+bf_+c~Lt q(t°) 4rcKD

x exp

b

~

f12

F

L- 4 ( t -

t -- to

to){(x- xo)2 + ( y -

/"] yo)2/j dxo dyo dto.

Now

f12 (y _ yo)2 [

4(t - to)

/ dyo

may be evaluated with the substitution f i ( y - Yo)

2~/t - to to

2~/t - to [ + ° ° e -~2 dol = - ~ ~ / t

- to { - erf(-oo) + erf(oo) }

J- ~ 2~/-~-~/t - t o

/3 while

f +bexp { - p2 (x _ xo)2 I dxo = b

4(t -- to)

and g 2 where g l - 2~(x+b) ~

-

2~/t - to [g2 e_~2 dol,

fl(x-b) and thus equals 2,/~-~'

~~'-~ '° [ere/~x 2. +~/,o -ere/~x 2~ -~/] - 'o

J~

781

Solutions, derived from known solutions (2.3.1-3)

So if we set r = t - to the end result becomes: ~p(x, t) - - ~

q(t - r) erf

2~/~-

- erf

2,/7

dr

(28)

(Part A, 124.22). For q(t) -- q as a constant, we have to solve integrals of the type / - f0t e r f ( 2 ~ r ) dr

k (k >/0). (see Section 3.1.1, equation (22)), which equals t{1 - 4i2erfc(~--~)} Due to the condition k >~ 0, we must split up the solution (28) for q(t) = q -- const into a solution for - b ~< x ~< b and one for x < - b and x > b, as follows: 99(xt) -

'

qt

fl(b - I x l )

qt

[2i2erfc{ f i ( ]2~/7 x l - b) }-

- ~ - [ 1 - 2i2erfc{

2~/t

} -2i2

erfc{ ~(b + Ixl) 2x/7 }]

2i2erfc{

fl(Ix] Jr- b) 247 }]

for Ix] < b, for Ixl > b

(Part A, 124.23), with i2erfcz - f ~ i e r f c t d t and ierfct - ftC~erfcudu (Section 3.1.1). So far we have considered discharge- and recharge impulses in line wells that fully penetrated confined aquifers, extending to infinity. But also impulses in point wells, well-screens, cylindrical surfaces, disks and others may be performed and not only in infinite confined aquifers. The solutions for the heads or drawdowns of these impulses may be determined in a similar way as has been done for the impulse in the line well (Example 11 of this section). From these solutions, the solutions of a great many problems can also be written down immediately in the form of definite integrals. 4. Hydrological screens

A very important application of the superposition principle is the creation of socalled hydrological screens; these are fictitious screens through which either no water particles pass or flow takes place only in one (desired) direction. Hydrological screens that are impermeable for flow of groundwater can be obtained by discharging or recharging groundwater from or into aquifers in which a natural, more or less uniform, flow takes place. Example 15. A simple example of a hydrological screen arises if a fully penetrating line well with a constant discharge Q [L3T -1] is placed in an infinite confined aquifer which is subject to a uniform flow with strength q [L2T -1] (Fig. 19). The flow is two-dimensional and quasi-steady and is the superposition of the flow field caused by a single well in an infinite field and the uniform flow.

782

(2.3.1-4)

Analytical solution methods

~=0

q

.q +

a 71=-~-fi S

x

q

q

q

Fig. 19. Well in uniform flow. The solution, written as a complex potential thus becomes: 2zrQD l n z + ~qz

$2 - c

(Part A, 325.51)

(29)

with z - x + i y dX2 dz

Q = Vx - i v v

"

=

1

q

~_ m . 2rr D z D

If both Vx and Vy are zero, then ~dI2 - 0

-

(

Q

ys -

Zs - 2zr q

0, Xs -

Q)

27r q

and we find the stagnation point

.

The potential function q~ - K qg, with ~o = drawdown, becomes Q

~b - c -

4rr---D

l n ( x 2 q_ y 2 ) q_ DX

(a)

and the stream function:

--

Q arctan(Y) + D y ' 2zrD

(b)

if we choose the positive x-axis for ap - 0. The set of streamlines with parameter ~ can be represented by x--ycot\

(2~rq

2rrD

Q y---~lp

)

.

(c)

Solutions, derived from known solutions (2.3.1-4)

783

In polar coordinates the stream function becomes: q ~(r, 0) - -D- r sin 0

QO 2arD ,

(d)

and the set of streamlines with parameter ~p may be represented by:

r=

2rcD~ + QO 2zrqsin0 '

(e)

which means that every value of ~p represents a streamline, for which r is a function of 0. The values of ~ vary from - ~ to +e~; the values of ~ for which r = 0 follow from equations (d) or (e): ~(o,o)

=

QO 2rrD

The streamlines belonging to of the well. As 0 varies from R D ' the end-values included, their way from left to right. flowing towards the well is streamlines ~ - 0 and ~ p from equation (e): QO r

-~

2rrq sin 0 Q(O - 2zr) 2zr q sin 0

these values all pass through the origin, the location 0 to 27r, we see that the streamlines with values 0 to all reach the well, while all other streamlines pursue This means that the total quantity of groundwater separated from the rest of the groundwater by the D" Q The equations for these two streamlines follow

for 0 ~< 0 ~< zr, (f) for Jr ~<0~<2rr

and in Cartesian coordinates from equation (c): x - - y cot(2rCqy)

(30)

y.

As y approaches zero, we have

{

y

x -- lim ~,, v--,0 tan( Q , )

} -- -0 -- indefinite. 0

Applying the rule of l'H6pital, we find x - 2-~q, the stagnation point. The equation (30) is the equation for the hydrological screen, a straight cylinder with length D and with the shape of a hyperbola, symmetrical with respect to the xaxis, going through the stagnation point and approaching asymptotically the straight lines y - + ~q. This fictitious screen may be considered as an impermeable obstacle that separates the groundwater flowing to the well from the original uniform flow.

784

(2.3.1-4)

A n a l y t i c a l solution m e t h o d s

The practical importance of such a screen for p o l l u t e d a r e a s is obvious, because a pollution inside the screen cannot escape to the stream field outside the screen and thus always reaches the abstraction well. Only the phenomenon of transversal dispersion of the contaminants may cause a slight deviation of the solute from the theoretical streamlines; so the hydrological screen has to designed amply around the polluted area in order to isolate that area from the surrounding flow field. This example of an o p e n h y d r o l o g i c a l s c r e e n has the disadvantage that because of the quasi-steady character of the flow field, the drawdowns may become unacceptably large. An improvement therefore will be the addition of a recharge well of the same strength as the discharge well as shown in the following example. Example 16. A discharge well and a recharge well of equal strength Q, fully penetrating a confined aquifer in which a uniform flow of strength q takes place. The mutual distance between the two wells is 21. The wells are situated such that their line of connection is parallel to the direction of the uniform flow and the recharge well lies upstream of the discharge well (Fig. 20). Again applying the principle of superposition, we find for the complex potential:

Q ln(Zz +l)l

I2 - 2zr----D

+ D z'

(31)

with z - x + i y dS2 _ dz

-

Ql

1

Jr D z 2 - 1 2

q + -D -

vx - i vy"

The location of the stagnation points follows from Vx Z2 .

.i2_~. . Q t zrq

-

v,,r -

x 2 - y2 + 2 i x y .

hydrological screen

q

q

2,. s_ I I I I I_

r

i

.---.--------~"~

I

I

_

21

I

i

Fig. 20. Positiveand negative well in uniform flow.

0

dS2 - O" and thus -dT-z

Solutions, derived from known solutions

(2.3.1-4)

785

Equating the real and imaginary parts of both sides, we find for the stagnation points:

X s --

--t-

2

+ ~ rrq

and

Ys - 0.

If we choose gr = 0 for y = 0 and Ix l > 1, the stream function becomes:

Q larct

-- 2zrD

n(Y)-ar

x+l

tan(Y)}+ q x 1 DY'

(a)

which can also be written as: x 2 _

12 _

y 2 _+_ 21y

27rq cot(---~-y,,

2rrD Q ~).

(b)

For ~p -- 0 and ~ -- --QD ' we can derive from equation (b) the equation for the hydrological screen: 2zrq)

x 2 _ l 2 _ y2 + 2 l y cot ---~--y ,

(32)

a straight cylinder with the shape of an ellipse and passing through the stagnation points. As distinct from Example 15, we have to do here with a c l o s e d h y d r o l o g i c a l s c r e e n which has the great advantage that inside such a screen a steady groundwater flow takes place that hardly effects the groundwater heads outside it. Moreover, it is possible to wash out a pollution in the aquifer inside the cylinder with clean water and thus purify the contaminated soil and groundwater to a certain degree. The case of one abstraction well and one injection well may easily be extended to more wells in different places, thus allowing of creating any desirable shape of the hydrological screen, which can be seen in Fig. 21 where five injection wells have been placed opposite one abFig. 21. One discharge well with 5 recharge wells in uniform straction well. But in flow. order to obtain a steady state inside a closed hydrological screen, one must take care that:

786 a. b. c.

(2.3.1-4)

Analytical

solution

methods

the total discharge of the positive wells is equal to the total recharge of the negative wells, the configuration of the wells is symmetrical with respect to a line (chosen as x-axis) parallel to the uniform flow, the discharge well(s) lie(s) downstream of the recharge well(s).

In this way, taking into account that a slight transversal dispersion may occur, any pollution of the underground in uniform flow can be isolated by a suitable location and strength of positive and negative wells, while at the same time the underground is sanitated in situ. If no appreciable natural groundwater flow takes place, it is impossible to create impermeable hydrological screens by means of positive and negative wells. However, it is possible in that case to construct closed fictitious screens that allow water particles to pass only in one direction, namely to the inside, thus representing a water r e p e l l i n g s c r e e n for the groundwater inside that screen, together with an eventual pollution. Example 17 will make this clear.

Example 17. y

A confined aquifer with fully penetrating wells. One well with discharge Q in the centre of a circular row of n recharge wells, regularly distributed along the circumference of the circle, each with recharge Q. Suppose the recharge wells are located at the points Zm (m -- 0, 1, 2 . . . . . n - 1) and the positive x-axis has been chosen through the point z0. The complex potential caused by one recharge well at the point Zm in an infinite field has the value

Q

Zo

x

Fig. 22. Discharge well in the centre of a circular row of n recharge wells.

Q ,.~"-2m --- Cm @

27r D n

ln(z

- - Zm)

and for all recharge wells together, according to the superposition principle: Q 2re D n

The points

Q

m=0

(m -- 0 , 1 , 2 . . . . , n -

R n e i2rrm =

R n.

Thus we have £2 =

In { ( z - z o ) ( z - z l )

2rrm

Re i-7-

0 as

(Zm) n =

2zr D n

. . . (z-zn-l)

}+c.

are regularly distributed along the circumference of the circle with

Zm

radius R; so Zm Z n -- R n =

,,- 1 ~'-~ l n ( z - - Z m ) + C --

Q 2rr D n

ln(z" - R") + c

1). Now Zm are the roots of

(2.3.1-4)

Solutions, derived from known solutions

787

and together with the discharge well:

Q

In (z" - R n

~2 -- 2rr D n

z '~

(33)

)"

Written in polar coordinates, with z - re i°, equation (33) becomes" Q ln{1-(R)ne-i"°} a'2 -- 27rD-----n = ~Q

In [ 1 - ( R ) ~ { c o s ( n 0 ) - i

27r D n

sin(n0)} 1.

A s q ~ - K 9 9 - - 4~rDn 0 lnll -- ( E ) n e- i,,o 12, we find for the drawdown:

{ (R) -

Q In 1 + ~o( r, O) - 4 7r K D--------~

2n

r

-2(R)n

cos(n0) }.

0~o

Q

- 2 n R 2 n r -2n-1 + 2 n R n r n-1 cos(n0)

Or

4rr K D n

1 + ( R ) 2n -- 2(~)" cos(n0)

and thus

Vr(r,O) -

099

K~

Or

=

Q

1

(r, - ( r

cos( 0

2 7 r D r 1 + ( E ) 2 , , - 2(R],~ 7-, cos(n0)

represents the velocity of the groundwater in the direction of the origin. For r -- R this velocity becomes Q v r ( R , O) -

-27cDR

1 - cos(n0) 2 - 2 cos(n0)

47r R D '

provided that cos(n0) -J: 1 or 0 -J: 2mn'. ?l This surprising result means that, with the exception of the locations of the recharge wells, everywhere on the circumference of the recharge circle the velocity of the groundwater is a constant, directed towards the origin, independent of 0 or norK!

So groundwater inside the circular cylinder with radius R and length D cannot leave the cylinder and a pollution may thus be isolated and washed out. This cylinder may be considered as a water repelling hydrological screen for the water inside the screen.

788

(2.3.2)

Analytical solution methods

2.3.2. Product solutions

Besides solutions obtained by adding solutions of simpler problems as has been discussed in the previous section, solutions may also be constructed by multiplying solutions of simpler problems. These so-called product solutions are, in general, solutions of non-steady problems with two or three space variables and are composed of the solutions of related problems in one-dimensional flow. Consider the homogeneous differential equation for three-dimensional non-steady flow of groundwater through homogeneous isotropic ground (equation (10) of Section 1.4.2-2):

0299 -1-0299 nt-- 02 q9- fi2 099 Ox 2 OY 2 OZ 2 --~,

f12 with

Ss --K

(34)

and the three differential equations for one-dimensional non-steady flow in the same ground in the three coordinate directions:

02(/91 ___ f12 0(t91 OX 2

Ot '

02(t92 __ fi2 0(/92 Oy 2

at '

02(/93 ._ f12 0993 OZ 2

at

then, if ~o:(x,t), ~02(y,t) and ~p3(z,t) are successively solutions of these onedimensional differential equations, their product 99(x, y, z, t) -- q91 (x, t)q92(y, t)tp3(z, t) is a solution of the three-dimensional differential equation (34), which can be shown simply by writing:

02(/9 OX 2 - -

02(/91 q92993 OX 2 '

02(/9 Oy2 - -

02(/92 991993 Oy2 '

02(p 02 2

02q03 --- 991992 OZ 2

and

099 0991 0992 0(/93 0"--'7"= (/92(/93'~ -Jr- (/91gO3-'~ -1- (/91992 Ot and substituting these expressions in equation (34). Similar considerations are the basis for product solutions in three-dimensional axial-symmetric flow: if qgl (r, t) is a solution of the differential equation O2qgl ~ 1 O(pl = /~2 0991 Or 2

r Or

Ot

for a radial-symmetric problem, and 992(2, t) a solution of the one-dimensional equation

~2~2 __ f12 ~992 OZ2 bS '

Solutions, derived from known solutions

(2.3.2)

789

then ~o(r, z, t) = 991(r, t)992(Z , t) is a solution of the differential equation 02(/9 1 0(/9 0299 /~2 099 Or---T - t - -r -Sir + ~OZ2 = at for axial-symmetric flow, as can easily be verified. In order to apply this method of product solutions to boundary value problems, we have to investigate the types of initial conditions and boundary conditions that will make such an application possible. It will be clear that in the three-dimensional case, if the initial condition qg(x, y, z, 0) = F(x, y, z) should be equal to the product of the initial conditions of the one-dimensional problems, this is only possible if, theoretically, F ( x , y, z) = f ( x ) g ( y ) h ( z ) . In practice, f (x), g(y) and h(z) will almost always be constants. The boundaries along which the boundary conditions are given should be straight lines (planes) perpendicular to the coordinate directions, for instance, along x = a or y - - 0 , etc., and must have the general form A1 ~---fx + Bl~O - - 0

for x -- xo,

A2-~y -+- B 2 q 0 - 0

for y = Yo,

A3-7- + B3~o - - 0 oz

for z = zo,

(35)

where A,, and Bn (n = l, 2, 3) are constants, either of which may be zero, so that the cases of zero head and zero flux are included, while x0, y0 and z0 may vary from zero to plus or minus infinity. Now we may use the solutions for the related one-dimensional problems with similar boundary conditions as factors for the product solution. If the boundary condition along x = x0 for the one-dimensional case is 0991 A1-2--(x0, t) + Bl~Ol (x0, t) -- 0, Ox

the condition along x = x0 for the three-dimensional problem becomes, with ~p = ~p1992~P3: A1 ~ x (xo, y, z, t) + Bl~O(Xo, y, z, t) -- Al~P2~o3--~-x (x 0, t) -+- B1~02~03~01(xo, t) -- 992993 A1--2---(x0, t) nt- Ol(fll(XO, t)

as it should be.

-- 0

790

(2.3.2)

Analytical solution methods

Similar considerations hold for the other coordinate directions.

Example 18.

~p=h

~o=0

~o=0

x

Two-dimensional flow of groundwater in a confined aquifer towards two straight open boundaries, perpendicular to each other, starting from an initial head h, compared with the zero head along the boundaries which is kept constant (Fig. 23). The boundary value problem can be written as:

02(/9

~5

g----W- hi

0299 _ f12 099

ax---~ + ~y~ rp(x,y,0)-h

~888~

x •

KD, S

qg(O, y, t) -- O,

Fig. 23. Two-dimensional flow

qg(x, O, t) - - 0 ,

to open boundaries.

with f12 _

0-7 forx >0,

S

- KD' y>0,

Oqg(~, Y, t) -- O, Ox

e)a___~(x, , cxz, t ) - O. Oy

The boundary conditions are in agreement with equations (35) as for x0 - 0 A1 = 0, for xo -- cxz B1 -- 0, for Y0 - 0 A2 - 0 and for y0 - e~ B2 -- 0, while also the initial condition may be considered as the product of f ( x ) and g(y) with f ( x ) - ~ and g ( y ) - ~/-h, for instance. The related one-dimensional solution in the x-direction has to satisfy:

02(/91 __ f12 0f/91 Ox 2

~01(0, t ) - 0

8t

and

991(x, 0) - v/-h, O~l.(c~,t)-0, Ox

and in the y-direction:

02(/92 = fl20q 92 Oy 2 0--t-'

~02(y, 0) -- v/h,

0(/92 q)2(0, t) - 0

and

oY (c~, t) - O.

From these conditions it follows that the method of product solutions is applicable. The solution for the head q91(x, t) can be obtained from equation (a) of Example 1 of Section 2.3.1-1 which is the solution for the same one-dimensional problem but with zero initial head and head h along the open boundaries; so

erfc

ere( )

Solutions, derived from known solutions

(2.3.2)

791

In a similar way, we find q92(y t ) -

4'-herf(flY)

'

2,/7"

Thus the solution of the two-dimensional problem becomes" q)(x y t ) , , (Part A, 334.02).

h eft( fix \ \ ) e f t ( f l y ) k2v/7 k2v/7

(36)

Example 19. Y

Two-dimensional flow of groundwater in a confined aquifer towards three straight surface water boundaries with entrance resistances, two of them parallel to each other with entrance resistance c2 and the third one perpendicular to them with entrance resistance cl. The initial head is h compared with the zero head along the boundaries, which is kept constant.

~o=0

dll'll~llIl',',',l',',l',Ill;II',ll',l',

C2

~o=h for t = O ~o=0

Cl

X C2

-b

Fig. 24. T w o - d i m e n s i o n a l flow in a semi-infinite strip.

The differential equation with boundary values and initial value is, for the head ~p -- ~p(x, y, t)" 02q 9 OX 2 1

02q 9 = fi20(t9 fi 2 __ Oy 2 --~' -- K D '

99(x,y,0)-h

forx >0and

K 099(0, y, t) -- 99(0, y' t), OX

Cl

K 099(x, b, t) - q)(x, b, t) Oy c2 '

-b


099(oo, y, t) -- 0, OX

K OqO(x, - b , t) -- q)(x, - b , t) Oy c2

The boundary conditions may be compared with the equations (35). For x - 0 we have A1 - K and B1 -- Cl1 . For x - ec B 1 - 0 . __1 Fory-bA2-KandB2c2" Fory---b A2-K andB21 C2 " As the boundaries are straight planes perpendicular to the coordinate directions, and also the boundary conditions and the initial condition are in agreement with the demands underlying the application of product solutions, we may write the solution as"

q)(x, y, t) -- 991(x, t)q)2(y, t),

(37)

792

(2.3.2)

Analytical solution methods

in which 991(x, t) is the solution of the one-dimensional boundary value problem: 32991

23991

3X 2 = fl

K

3"7'

991(X, O) -- h l,

0991 991 (0, t) (0, t) = ~ , OX Cl

0991 OX

~(oo,

t) = 0 ,

and 992(y, t) is the solution of the one-dimensional boundary value problem: 02q 92 __ f12 0992 Oy 2 oq--7-'

-

K

992(Y, O) -- h2,

0992 (b, t) _ ~,992(b' t) Oy c2

K 0992 . . . .( b, t) - 992(-b, t) Oy c2

In order to determine 991(x, t) we apply the Laplace transformation with respect to t, giving: d2q31

fl2(Sqgl -- hi) -- 0,

dx 2

dq51

K-T-(0, s) -tax

qgl (0, S) Cl

dq~l

~ ( o o , s) =0, dx

with the solution qgl(X, S ) -

hi (1 s

e-t~x'fi

)

1 + Xc~,/-i

"

1 If we define the resistance parameter as 0 -- f12K2c2 [T- 1 ], we find from tables for

Laplace transforms, the inverse transform 991(X, t) -- hi e r f\(2f ~l x/ ] + h 1R ( ~ f l X ~ ,

(38)

(Part A, 126.12), where the resistance function R is defined as R(x

-- e2X+y2erfC(y ) + y

(see Section 3.1.3).

For the solution of 992(y, t) we apply the finite Fourier cosine resistance transformation (see Section 2.2.3-2, the equations (84), (85) and (86)), with q32 -- q~2(n, t)" 2 dq~2 O/n f12 b2 q~2 -dt '

q32(n, 0) -- h2 foot, cos ( °lbn Y /'] d y --

h2b O/n

sin Otn,

Solutions, derived from known solutions

(2.3.3)

793

with the solution:

h2b

(

q32(n, t) -- ~ot,, sin c~,, exp

°t2t ),z flZb2

and the inverse transform: oo

_v) (

sin oe,,

~02(y, t) -- 2h2 ~ (1 -+- ,,e, cos (or,, y n--o Oln ol2q-g2)

f12b2 )

(39)

b with c~,, being the roots of oe tan oe = e = Kc 2• Now, with h lh2 : h (for instance, h l = h2 -- v/h ) the solution of our twodimensional problem is equation (37) in combination with equation (38) and equation (39) (Part A, 354.08). 2.3.3. Periodic flow solutions

Periodic functions are of great practical importance for the solution of geohydrological problems because they not only serve as the basis for the Fourier transformations (Section 2.2.3) but in many cases they also allow a general function to be written as a Fourier series, consisting of sine- and/or cosine functions, thus reducing the solution of a problem to a much simpler one. In many groundwater problems a periodic function of the time is involved in one of the boundary conditions, for instance, seasonal fluctuations of the precipitation or tidal fluctuations of surface water. As we have seen in Section 2.2.3-1, periodic functions may be represented by series of sine functions and cosine functions; so it will be sufficient to investigate the influence of a simple sine function (or cosine function) of the time on the behaviour of the groundwater flow, that under such circumstances is referred to as periodic flow. In the following we will make use of the general trigonometric function

f (t) = A sin(cot + 3)

27/"

with c o - ~ T'

(40)

in which A -- the amplitude, T = the period, and 3 - an arbitrary constant. Theoretically, solutions for periodic flow problems may be obtained from solutions involving arbitrary functions of the time by substituting a sine function of the time. In most cases, this will yield complicated expressions with definite integrals that can hardly or not at all be evaluated to simpler results. However, there are methods of solving these problems in a particular manner, which will be the subject of this section. To show the method, we start with a simple example.

794

Analytical solution methods

(2.3.3)

Example 20.

2h:

One-dimensional groundwater flow in a confined semi-infinite aquifer towards open water, without entrance resistance. There are tidal fluctuations of the open water according to the trigonometric function.

KD, S

x

F ( t ) - h sin(cot + 8)

Tidal fluctuations along a half-plane.

with c o -

Fig. 25.

27/" T'

in which h = amplitude, T -- period and 8 an arbitrary constant (Fig. 25). The drawdown or head qg(x, t) is a function of the distance from the open boundary and of the time and can be solved from"

0299 __ f12 0(/9 with 1~2= S Ox 2

at

99(x, 0) -- 0

KD '

for x > O,

99(0, t) -

F ( t ) -- h sin(cot + 8),

~o(oo, t) - - 0 . We first solve the problem, assuming F ( t ) to be an arbitrary function of time. Laplace transformation gives"

d2q3 dx 2

f12

sq3 - O,

(p(O, s) - F(s),

q3(oo, s) -- O,

with the solution q3(x, s) - F(s)e - ~ * ~ .

(a)

According to the convolution theorem (Theorem 6 of Section 2.2.2-1) and equation (40) of Section 2.2.2-3, the inverse Laplace transformation yields:

~p(x, t) -

flXfot

2~.~

With the substitution ot2

gO(X, t) -- ~2

F(t-

_ -

-

f12x2 47: '

ft~x F ( t 2qri

r)exp

(f12x2) 4r

dT

r~r-

for x > 0.

(b)

we find

fl2X2)4ot 2 e -~2 dot

for x > 0,

the solution for an arbitrary fluctuation of the open water with time.

(c)

Solutions, derived from known solutions

(2.3.3)

795

Secondly, we replace F ( t ) by h sin(cot + 3), giving ~p(x, t) -

--~

sin co t

+ 3 e -~2 dc~.

(d)

f (oe) dot,

(e)

5-Z Now,

/~x

f(oe) doe -247

/0

f(oe) dot --

/0

f12x2

with f(c~) - sin{co(t - 4¥gY) + ~}e -~2. The second integral on the right-hand side of equation (e) is a transient disturbance caused by starting the oscillation of the surface water at time t - 0; it dies away as t increases, leaving the first integral as the solution for large t:

v/~

sin

l(

co t

40/2

+ ~

/

e -a2 d~.

(f)

As sin 0 is the imaginary part of e iO, we have ~o(x, t) -- Im

2h ei(O~t+3)

exp

- - 13l2 - -

icofl2x2 40e-------5--- dc~,

which can be evaluated, with the help of Section 2.2.2-3 if we set there in equation ( 4 3 ) ) 2 1 andt-oo, as -

-

iwfl2

~p(x, t) -- Im[hei(°~t+~)e-t~x'/7-d] or, as ~ -

~~/2(1 + i )

99(x, t) -- Im[hei(~°t+~)-½45/3x~(l+i)]. So, with c o ' - fix/-~, we find

~o(x, t) - he - J x sin(cot - c o ' x + 3),

(41)

which means that the head (drawdown) 99 at every fixed distance x - x0 is represented by a steady oscillation with period T 2__~and amplitude he-"/x0 and with a time lag co'x0 with respect to the surface water. The solution (d) is the non-steady solution of the problem if the oscillation of the surface water starts at the time t - 0; it tends to the "steady" solution (t) or (41) with increasing values of t. This solution is a special steady state solution because it still contains the time variable and the storage coefficient S. So the head ~o keeps -

796

Analytical solution methods

(2.3.3)

changing with time but has become periodical with the same period T as that of the surface water movement, but with an amplitude and a difference in phase that are both functions of the distance only. It can easily be verified that equation (41) satisfies the differential equation 0299 __ ~}x2 - -

f120~0 and the boundary conditions for x - 0 and x - oo. Of course, it no at

longer satisfies the initial condition qg(x, 0) - 0 for x > 0. It would be convenient to find a direct method to solve the steady oscillation represented by equation (41); therefore we make use of this result and seek for a solution that has the general form: q0(x, t) -- f (x) sin {cot + ~ - g (x) },

(g)

in which f ( x ) and g(x) are functions of x only and must be determined such that f ( 0 ) - h, g(0) - 0 and f ( o o ) - 0. Now ~p(x, t) may be considered as the imaginary part of a complex function 4~(x, t) as follows: ~0(x, t) - Im[4~(x, t)] - Im[f(x)e i{rot+a-g(x)}]

or

~o(x, t ) - Im[F(x)e i(°~t+a)] with F(x) - f (x)e -ig(x),

(h)

where F(x) is a complex function of x only. As a2~° ax2 - Im[ a2~1 ax2 j and ~a~0 _ im[a~l at J' we have the differential equation for qS: a2~

= /~2 atp

OX 2

at

which becomes, with equation (h)"

02 Fei(O)t+~) __ fl2icoF(x)ei(°)t+~) Ox 2 02F OX 2

or

ifl2coF - 0 .

~0(0, t) -- h sin(cot + ~), but also ~o(0, t) -- Im[~b(O, t)] -- Im[F(O)e i(°)t+~)] - F(O) sin(cot + ~) from which F ( 0 ) - h. In a similar way we find F(oo) - - 0 . Thus, the differential equation with boundary values for the complex variable F as a function of x, becomes" d2F dx 2

i fi ZcoF - 0 ,

F (0) - h, F (oo) - O,

(2.3.3)

Solutions, derived from known solutions

797

with solution: F ( x ) -- he -flx#7-d = he -°/x(l+i)

(i)

if c o ' - flv/~. Compared with equation (h), we see that and

f (x) - he -°Ix

g ( x ) - +co'x.

These values, substituted in equation (g) give the result (41), obtained before. The function F ( x ) has been solved from a differential equation with boundary values similar to the Laplace transformed function qS(x, s) in equation (a). So the solution (i) can be derived immediately from solution (a) by replacing F(s) and s by h and ico, respectively. Then F ( x ) must be multiplied by e i(c°t+6) to find ~b(x, t), the complex function the imaginary part of which equals the function ~0(x, t) that was looked for. In this way, also more complicated problems concerning periodic flow may be solved by making use of the Laplace transforms of already known solutions for non-periodic flow, as we shall see in the following examples. Example

~

21.

$

Periodic precipitation on a circular island, surrounded by open water with a constant level without entrance resistance. Assumed horizontal flow in a confined aquifer. We first consider the case of a precipitation being an arbitrary function of the time p ( t ) with p(0) = 0 (Fig. 26).

$ $~ p(t]

m

.

.

.

.

"r

= 0c

.

KD, s

I

Fig. 26. Precipitation on a cir-

cular island. We have" 02(/9 1 099 p (t ) 2 0(t9 Or 2 +--¢-~ =/3 ~ r -~r K D Ot

~p(r, O) -- O,

f12

with

-8--;-r(0, t) -- 0

and

S

=

K D '

¢p(R, O) - O.

Laplace transformation with respect to t gives" d2q3 1 dq3 •dr 2 t

/?(s) flZsq) @

r dr

KD

- 0

d (o s)-o,

'

dr

'

s) - 0 ,

with the solution:

1 99(r, s) -- S

} s

Io(~R~/-s)

"

(a)

798

Analytical solution methods

(2.3.3)

The inverse Laplace transformation can be obtained by means of the Laplace Inversion Theorem (Theorem 9 of Section 2.2.2-3), but is rather complicated; it is easier to apply a finite Hankel transformation with respect to r to the differential equation with boundary values of which equation (a) was a solution, making use of the equations (119) and (121) of Section 2.2.4-3:

2

13(s) R 2

~" ~ - ~ 2 s ~ 4 J~ (~.) - 0, R2 K D O/n an algebraic equation with the solution

~(n, s) --

/3(s)

R2 J1 (o/n)

~--7

.

(b)

fl2KD s-+- flZR2 Inverse Hankel transformation yields, according to equation (120): 2 oo s) -

-(s) p

._0s+

Jo(°tnr~

TJ

(c)

O/n Jl (o/n) '

with O/n being the roots of J0(o/) = 0. Inverse Laplace transformation gives the end solution:

T ) f0 ' 2~ J0(c~nr qg(r, t) -- ~ n=0 O/nJ1 (O/n) exp {

2 f12R2( t - r ) } p ( r ) d r ,

(42)

with O/n being the roots of J0(o/) = 0 (Part A, 233.15). Secondly, we consider periodic precipitation and substitute p sin(mr + 5) for p ( r ) with p as a constant, in equation (42) and let t approach infinity in order to find the steady oscillation of the groundwater head which we may expect. The evaluation of the combined series and integral in equation (42) becomes rather complicated; therefore, like in Example 20, we assume a steady state of the form:

qg(r, t) -- f ( r ) sin{cot + ~ - g(r)},

(d)

in which f ( r ) and g(r) are functions of r only and rp(r, t) is considered as the imaginary part of 4) (r, t): ~o(r, t) -- Im[q~(r, t)] -- Im[f(r)ei{a't+a-g(r)l],

or

~o(r, t) = Im[F(r)e i(°°t+a)] with (r) - f (r)e -ig(r). F(r) is a complex function of r only.

(e)

Solutions, derived from known solutions

(2.3.3)

799

The differential equation for 99" 0299

1 Oq9

p sin(cot + 3)

Or 2 4-_r -~r 4-

0q)

= ~2 at

KD

leads to"

[O2~b

10q5

Im 7r2-r 24--r-~-r 4- KPD ei(Oot+a)]- fi2im[0q~__~_] from which d 2F 1 dF dr 2 t r dr while -dT-r dF (0) F(r)-

--

f12

P

icoF 4- ~ = 0, KD

0 and F ( R ) -- 0, with the solution

~S

1-

Io(~n47-£)

I

(f)

This solution is identical to the solution (a) if we replace there fi(s) by p, s by ico and q3(r, s) by F(r), in a similar way to that found in the previous example (equation (i) compared with equation (a)). The solution for the steady oscillations becomes, if we combine the equations (e) and (f)" q)(r, t) -- = I m - - 1 ico

Io(fl r ~ ) ] Io(~ R ,/~) ! e~(~'+~

(43)

(Part A, 238.21). Making use of the Kelvin functions ber(x) and bei(x) as the real and imaginary parts of Io(x~/7) (Section 3.3.3) Io(xv/Tl) -- ber(x) + i bei(x) - Mo(x)e iOo(x) with Mo (x) -- v/ber 2 (x) 4- bei 2 (x)

and

00 (x) - arctan [ ~ ] , b(x) ei

we find" ~o(r, t) -

p sin cot + coS

rr } (44)

p Mo (fl r.v/-~) sin cot + 3 + 0o (fl r x/~) - 0o (fl R VG) - -}- .

~oS Mo(~/~/-£)

R e m a r k . In practice, a periodic precipitation will generally be composed of a constant precipitation and a variation more or less according to a sine function, for

800

Analytical solution methods

(2.3.3)

instance: p(t) -- Po + P sin(cot + ~), in which po and p are constants with po ~> P. In that case, the steady state solution for a constant precipitation on a circular island P0 (R 2 _ r 2) ~p(r)- 4KD

(45)

must be added to equation (44). Solution (45) may be obtained by direct solution of the differential equation for the steady state: 1 dq0

po

dZg° + -~r -t -- 0 dr 2 r KD

with ~o(R) -- 0

and

d~o

-a-~-r(0) - 0,

namely:

1 d [r d~p

d (r dqo por d---~ --~r ) - - K D

po -~,

r-dr

-gr ) - dcp po r2 r~ +Cl, dr 2KD po r2

qo --

4K D

dqo ~ = dr

por 2KD

{ Cl , r

+ cl In r + c2. poR 2

From ~ ( 0 ) -- 0 it follows that cl - 0 and from ~o(R) - 0 it follows that c2 = 4/(D, which leads to equation (45). This result may also be derived from equation (42) by replacing p ( r ) by po and letting t approach infinity. Then, as

Po

2

fot /

f12R2

exp

= poexp ( ~ - ~ ~ 2 ) _ [12R 2po exp --

fot

exp (/32R2] dr

- -----

Ot2

f12 R2

-- /~2R2p0 -012

1-exp(/32R2)

exp

- 1 \ f12 R 2 1

,

p(r, t) -- 2p°R2 J°(~-~) 1 - exp K D n=O or3 J1 (O/n) with an being the roots of Jo(o~) - 0 . 2poR2 ~ KD

Jo (~-~)

n=O ~3 J1 (~n)

f12R2

The first term

,

(g)

Solutions, derived from known soluticns

(2.3.3)

801

in this equation can be evaluated by comparing the two solutions (a) and (c) for 93(r, s) from which it follows that

io(flrv/7 ) }

ls 1 - I0(-flR-~

oo

2~ 0

must be equal to

_~)

Jo ( °tnr ~2

= oln(s + - ~ ) J l ( c ~ ) or, in general, if we set a -- fl V/s-:

oo

jo(~_~_)

1

,,=0 Otn (or2 -4- a 2 R e) J1 (Otn) = -2a- -2-R~

{

Io(ar) }

(h)

1 - Io(aR)

In particular, for a - - 0 we have oo

n--O

go(Unr ] R !

R2

~3nJ1 (~n) --

r2

(46)

8R 2

which can easily be verified by developing the modified Bessel functions in the right-hand side of equation (h) in infinite series according to

lz 2 (¼z2)2 I o ( z ) - 1 -4-~

9- (2!) 2 -4- (3!) 2

•

°

°

°

So equation (g) becomes" P0

(R 2 _ r 2)

~o(r, t) -- 4K----D

2po R2

KD

J0(~RZ) exp

,,=o c~ J1 (c~)

("2')

f12R2 ,

(47)

with c~,~ being the roots of J0(ot) - 0 . The first term of this solution represents the steady state which we found before. From the Examples 20 and 21, we may conclude that the following theorem holds: Theorem 4. If the solution of a periodic flow problem tends to a steady oscil-

lation, it can be derived from the Laplace transform of the solution of a similar problem involving an arbitrary function G(t) of the time instead of the sine function A sin(cot + ~), according to the following rules: 1° Determine the function F, which is a complex function of space only, from the Laplace transformed function ~, which is a function of the same space variable(s) and s, by replacing the transformed arbitrary function G(s) by the amplitude A of the sine function and s by i co. 2 ° Multiply F by e i(~°t+a) to find the complex function qb, which is a function of space and time.

0

0

Analytical solution methods

(2.3.3)

802

Take the imaginary part of ¢ which equals the solution q) (head or drawdown) of the problem. Verify whether q) satisfies the differential equation and the boundary values (not the initial value) in order to be sure that a steady oscillation can be reached, a condition for the validity of this theorem.

Example 22. We take Example 9 of Section 2.3.1-2, a discharge well between two parallel open boundaries (Fig. 11), and suppose that the discharge is an arbitrary function of time: Q - Q(t) with Q ( 0 ) - 0. Then we have:

02(/9 t 02q9 -- fl2Oq ~ ) Oy2

OX 2

with

f12 -

S

at

lim ( r Orp) = - Q(t) r-+0 -~r 27r K D ~0(cx~, y, t) -- 0

,

~o(x, y, 0) - 0

KD

and

'

for r -- V/X2 + ( y - a) 2, qg(x, 0, t) - ~0(x, b, t) - 0.

Finite Fourier sine transformation with respect to y gives (see remark concerning Fig. 13): OX 2

T

(0 --

0(o (0, n t) 0-7 '

-~ ,

~O(X , 11,

O(t) sin (nrra,~ 2KD --7/

--

and

q3(c~, n, t) - 0.

Laplace transformation with respect to t yields:

d2~ dx 2

{(nTr) 2 } '-if- q-flgs ~ = 0 '

dq3(O,n S ) d---x '

Q(s) sin nrra

and

,}(oo, ,,, s ) = o , with the solution

(o(x n s ) '

'

Q(s) sin

--2K-----D

--b----/

M.

with M n -

+

Inverse Fourier transformation gives:

O_.(s) ~

nrra']

qS(x, y, s) - b / ~ D ~ s i n ( - - - b - / s i n \

n=l

(arty b )

e -M~x Me

(a)

which still needs an inverse Laplace transformation in order to get the solution for the drawdown q) as a function of space and time, caused by an arbitrary timedependent discharge. In the context of this section we shall not perform this transformation but make use of equation (a) for the determination of the steady oscillations of the drawdown, caused by a periodic abstraction, for instance:

(2.3.3)

Solutions, derived from known solutions

803

Q(t) = Qo + Q sin(cot + 3) 27r with co -- -7and Q0 ~> Q, which describes a discharge composed of a constant part and a part that varies according to a goniometric function. The solution is the superposition of the solution for a constant abstraction given in Example 9 (equation (18)) and the solution for a sinusoidal abstraction. Now the latter may be written down immediately from equation (a) according to Theorem 4:

q)(x, y, t) =

Q bKD

I

nrra nzry Im e i(wt+3) ~-'~sin --if-- sin - - ~

e-xv/(-cff--)2+ifl 2w ]

( ) ( ) V/(t--~) 2 q- ifl2co

n=l

'

which may be evaluated, if we put V/( ~.)2 n~r nt_ ico - pn + iq, to"

qg(x, y, t) --

(nrca ~ ( n r c y ) e -~xp" Q Z sin \--if--/sin \ - 7 v/P 2 ÷ q2 b f l K D n=l

x sin {c o t - g x q n -

(48)

arctan(q--L) + 3 } .

Pn

Verification, of whether this solution satisfies the differential equation and boundary values may be most easily performed by investigating whether the function cx~

(nrca] F(x, y) = Q Z s i n \ - 7 / s i n b K D n=l

(nrcy] e

-x

v/(--if-)2+i[32co nTr

b ] v/(n~_)2_t_ ig2co

is a solution of 02F OX 2

-t--

02F Oy 2

fl z i coF = O,

with F(x,O) = F ( x , b ) = 0 and F(oo, y) = 0, which can be easily shown. As qg(x, t) - Im[F(x)ei(~°t+a)], it follows that qg(x, t) is a solution of the posed boundary problem. 2.3.4. Solutions in anisotropic soils

The general three-dimensional differential equation for non-steady flow of homogeneous groundwater through homogeneous anisotropic ground in which the main directions of the anisotropy are parallel with the chosen coordinate directions, has been given in Section 1.4.2-2, equation (9):

02(,0

0299

02

099

K x - ~ x2 + K "v ~Oy 2 + Kz ----~-~ Og 2 = S s ~a t

(49)

804

(2.3.4)

Analytical solution methods

If the main directions of the anisotropy (as is assumed here) are horizontal (x and y) and vertical (z), also horizontal and vertical boundary conditions, which are present or will be assumed to be present in almost all geohydrological problems suitable for analytical solutions, can easily be expressed in terms of the different conductivities. For instance, flow through a semi-permeable horizontal layer often serves as a boundary condition for problems in leaky aquifers, and may be described as" Vz - - Kz Oq) = Oz

A~ c

for anisotropic soils, where c = the leakage factor (cf. Section 1.3.3-3). Problems in such anisotropic soils may be solved in a similar manner to that which has been employed in the previous examples for isotropic soils, as the following example may show. Example

_

23.

....

Two-dimensional steady flow through an anisotropic leaky aquifer from a fixed water level above the semi-permeable layer towards a vertical surface water boundary with entrance resistance. The differential equation with boundary values for the drawdown q9 = qg(x, z) becomes:

e - 0

~'////////////////////////,//C

0299

Fig. 27. Anisotropicleaky aquifer with entrance resistance. Oq9

02(/9

K~~x2 + Kz OZ 2 = 0

h -99(0, z)

-

K~ 7x(O,

-

I
z) -

099

w

,

q)(x, D) , c

~o(~, z) -

099 (x,0) Oz

~

_

O,

0. _

Finite Fourier cosine resistance transformation with respect to z (see Section 2.2.32c) gives, with q 3 - q3(x, otn)" d2q3 Kx dx 2

2 Kz °tn -~--~q3 -- 0,

with c~n being the roots of ot tan c~ - e - K o__0__ zc' dO - K x -7-(0, Otn) (IX

hD

(o(0, o~,~) sin(otn) -

-

~

tOOln

,

q3(oe, Otn) - O. W

The solution is, with ~ 2 _ K_...Lz. Kx

~(x, ~ ) -

h D 2 sin (Otn)

u~nx e

Otn( D + I~ Kx wetn )

o

Solutions, derived from known solutions

(2.3.4)

805

Inverse finite Fourier cosine resistance transformation yields the end solution for the anisotropic case" I~C~nX

99(x, z) -- 2 h D

Z

sin(ot~)e

D E

n=O otn(D + lzKxwotn)(1 +

ol.z] cos (--D--/'

(50)

with oe,~ being the roots of ot tan oe - e - Kzc D " The solution for the isotropic case follows immediately from equation (50) by putting Kx -- Kz -- K and t h u s / z - 1" sin(otn)e

OlnX

z~ ~o(x, z) -- 2 h D Z n=o C~n(D + Kwotn)(1 +

E

COS(0/nZ ~--5)

(51)

D" (Part A, 355.15), with or,, being the roots of ot tan ot - e - K---~. Inversely, deriving the anisotropic solution (50) from the isotropic solution (51) is less simple but is of more importance, because most solutions of geohydrological problems concerning two-dimensional or three-dimensional flow through aquifers are available for the isotropic cases. In order to find a general method to perform such derivations, we must compare relations between variables in isotropic flow with corresponding relations in anisotropic flow. For that reason we consider the isotropic case as a model and the anisotropic case as the prototype and try to find the several scale factors underlying the similitude of model and prototype. The general differential equation for the anisotropic case (prototype) is equation (49) and for the isotropic case (model): 0299m

0299m

O2@m

0~m

Km Ox2 +Kin Oy~m + K i n 0~m =Ssm'0~m'

(52)

in which the index m refers to the model. We rewrite the equations (49) and (52) as"

Kx 0299 K y 0299 0299 Ss 0~9 t t = Kz OX 2 Kz Oy2 OZ 2 Kz at

0299m 0299m 0299m

and

(a)

Ssm 099m

Ox-----~ -~- Oy2 + Oz2 = Km Otto

(b)

and set /z 2 -

Kz

K~

and

va = Kz. Kv

(53)

806

(2.3.4)

Analytical solution methods

As the piezometric head 99 is defined as q) - z + ~ , it is obvious to choose the same scale for q) and z and even the scale factor equal to 1 in order to ensure that the definition for the piezometric head remains unchanged. So q)m -- q9

and

Zm -- z.

(c)

Then, as the corresponding terms of the differential equations (a) and (b) should be the same, we find: Xm-/zx

and

Ym

vy

(d)

and

Ss

Ssm Kmtm

=

Kzt

•

(e)

The discharge equations for the prototype are:

Oq)

0q) IdQy[ - Ky- Z- dx dz

[dQx[ - KX-~x dy dz,

oy

and

099 Id Qz I = Kz -~z dx dy, and for the model: 0q)m Id Qxm[ -- Km ~ dym dzm,

060~ IdQvm[ = Km -T''' dxm dzm " 0Yrn

and

0qgm [dQzml - Km~zm dxm dym. If we choose, in general, scale factor 1 for Q, we have

Q x m - Qx,

Q y m - Qy,

Q z m - Qz,

(f)

and according to equations (d): 0p

0q)m

0(,/9

0~0m

099

0q)m

099

Kx ~x d y dz - Km~x m dym dzm - K m - ~ x V d y dz, K y -ff-fy dx d z - Km ~y m-

0q) dxm dzm - Km v~y/z dx dz,

and

099

gz -~z dx d y - Kin-g--- dxm d ym - Km-2--/z dx v d y, oz OZm from which 1

Km -- Iz Kx __ _v Ky -KZ~ v IX tzv

(g)

Solutions, derived from known solutions w/ ~Kz a n d v -

or w i t h # -

(2.3.4)

807

W/ K~--7, Kz . (h)

Km -- v / K x K y .

The value Km = /zv K---zzfrom equation (g) substituted in equation (e) gives" Ssm

Ssm#

Kmtm

P

Ss

Kztm

Kzt

For reasons which will become clear in the following, we choose: SS m - -

(i)

SS

and thus"

0)

tm -- l z v t .

The t r a n s p o r t e q u a t i o n s for the prototype are (see Section 1.2.1)" dx

090

dy

ne d--7 = - g x -~x ,

Oqo

ne d--7 = - K v" Oy '

dz

Oqo

n e - - = - Kz - - , dt Oz

and for the model: dxm nem dtm

dym __

099m Km OXm

dzm

0(/gm

nero dtm

Km OZm

nemdt m

3q)m -K'n 5~m '

in which ne -- effective porosity (Section 1.3.2-2). Comparison of the corresponding equations and substituting the already obtained scale factors according to the equations (c), (d), (g) and (j), yields" lzdx nem~ -lzvdt

lZ v

Kx~

Oq)

#Ox '

vdy v Oqo nem~ -- - - - K v lz v dt lZ r o y dz n e m ~

Kz 09o ~-

lz vd t

lZ V Oz '

from which nero -- he, f r o m w h i c h nero - he,

from which nero -- he,

all three equations yielding the same scale for ne, determining for practical reasons also the same scale factor 1 for the volumetric porosity n (Section 1.3.2-2) nem -- ne

and

nm -- n.

(k)

808

(2.3.4)

Analytical solution methods

From these equations the scale factors for the velocities in the directions of the coordinates also follow: 1)x l)xm = -

v

,

I)ym--

Uy ----:-

/z

and

l)zm--

1)z ~.

/zv

(1)

The scale factor for the volume of a part of the ground follows from the volume d V of a small box with sides dx, dy and dz" dV - dx dy dz and dVm - dxm dym dzm /~ dx v dy dz, namely (m)

Vm - i~ v V.

Also, the scale factors for the water volume Vw and the soil volume Vs, which follow respectively from Vw = n V, Vwm - - nm Vm and Vs = (1 - n) V, Vsm - - ( | - n)m Vm, are equal to/zv, because nm -- n (equation (k)). So the condition V = Vw + Vs from the prototype also holds for the model: Vm - - Vwm -~- Vsm.

Now we see that if we had chosen Ssm :~ Ss instead of Ssm -- Ss (equation (i)), for instance, tm - t ' yielding Ssm - % # v ' we would have found nm - ~ #v instead of t/m - - t/.

Then, though the scale factor for V would have remained unchanged (equation (m)), the scale factors for Vw and Vs would have become different from the scale factor for V and thus Vwm + Vsm ~ Vm, which is not logical. The boundary conditions along planes to which the main directions of the anisotropy are parallel (here horizontal and vertical planes); in particular boundary conditions involving entrance resistances (Section 1.5.1-3cl °) may serve to find scale factors for the latter. For instance, along a horizontal semi-permeable layer with resistance c, the boundary condition for the prototype is: _KzOq) = Aq9 Oz c

and for the model: Oqgm

A(flm

- K m OZm --

As K m - - #-'-~, Kz (tim

Cm

- - (/9

and Zm - z, according to the equations (g) and (c), respec-

tively, we have Kz Oq)

Aq)

ILl) OZ

Cm

from which it follows that Cm- ~vc.

(n)

Solutions, derived from known solutions

(2.3.4)

809

Along a vertical surface water boundary with entrance resistance, the boundary condition is" 099

Acp --

-X~ ~

099 or

w~

-

Kv

Oy

A~p --

Wy

for the prototype and for the model: AC, Om - - K m ~0(Pm X m __ m tOln

and

O~pm

- Km -0Y-ram -

A~om tOm

or according to (g), (c) and (d): # Kx 099

A99

V

tom

#OX

and

v #

O(# roy

Wm

from which tom -- Utox -- #toy.

(0)

From Ss -- (,-~g 1-,, + -~w)pg n (Section 1.3.5, equation (57)), it follows that, because both Ss and n remain the same for the prototype and the model (equations (i) and (k)), also the scale factors for Eg, Ew, /9 and g should be chosen equal to 1. So, Egm-

Ewm-

Eg,

Pm - - P ,

Ew,

(p)

(q)

Ym - - P m g m - - Y.

As M - p V, the scale factor for the mass becomes, according to the equations (m) and (q)"

Mm -- # v M .

(r)

From the definition for the piezometric head 99 1 for the pressure Pm -- P.

z +

, we find the scale factor

(s)

The relations between the piezometric head and the stream function ~ are for axialsymmetric flow for the prototype

Kr ~

--

2rcr

Oz

and for the model 099m | Ogtm (see Section 1 . 2 . 4 ) Km 0r-ran -- 27rrm OZm

810

(2.3.4)

Analytical solution methods

As for axial-symmetric flow with the z-axis as axis of revolution Kr - - K x - - K y we have from equation (h)" K m - Kr and from

V

~

I~

rm -- V//./,2x2 "-l- v 2 y 2 -- lzr, so that ~

KF

099

Olpm

1

---"

lzOr

2rclzr Oz

from which for axial-symmetric flow

~m -- ~ [L3T-1] •

(t)

The relations between the piezometric head and the stream function ~p are for twodimensional flow in vertical planes for the prototype

0q)m = 0ff~m and for the model: Km 0~m OZm from which

0~0 _ 0~p Kx-~x-az

0q)

Kx lzox

01pm

=

Oz

,

as Kv

-

-

Kx and thus v - #.

This means that for two-dimensional flow in vertical planes

(u)

~rm -- % [L2T-1]. In two-dimensional flow in horizontal planes we have

099

0~r

Kx-~x - 5z

0~0m and

Km

OXm

0~m =

Oym

for the prototype and the model respectively, from which ~--'gx Oq9 _

O~m

v

roy

lzOx

So for two-dimensional flow in horizontal planes we find l~rm -- ~

[L2T-1].

(v)

The knowledge of the various scale factors obtained for the variables and parameters in the relation between isotropic and anisotropic flow of groundwater, enables us to transform known solutions for groundwater flow in isotropic soils into desired solutions for analogous flow in anisotropic soils, which can be recorded in the following theorem.

Solutions, derived from known solutions

(2.3.4)

81 1

Theorem 5. An analytical solution for flow of homogeneous groundwater through an anisotropic homogeneous medium of which the main directions of the anisotropy are parallel to the planes subjected to boundary conditions, can be derived directly from the analytical solution for the analogous problem in an isotropic medium, according to the following rules.

1° Keep the state variables ~o (piezometric head or water table), p (pressure) and Q (total discharge) unchanged. 2 ° Multiply the space variables x, y and z and also constant dimensions in the directions of x, y and z by #, v and 1, respectively, in which ~ Kz

~ Kz

(z remains unchanged).

3 ° Multiply the time variable t by #v. 4 ° Replace K by v/KxKy. 5 ° Keep Ss (specific storage coefficient), the effective porosity ne and the porosity n unchanged. 6 ° Multiply the resistance in a horizontal layer c by #v and the resistance w in a vertical plane by v and Wy in a vertical plane by lz. 7 ° Keep the axial-symmetric stream function ~ [L3T -1] and the two-dimensional stream function ~ [LZT -1] in horizontal planes unchanged and multiply the 1 two-dimensional stream function ~ [LZT -1] in vertical planes by 7" 8 ° Multiply the velocities Vx, vy and vz in the coordinate directions [LT -1] by ± p~ Lu and --~, respectively. This theorem, applied to the isotropic solution (51) of Example 23, yields the anisotropic solution (50), which can easily be verified, as in two-dimensional anisotropic flow K. = Ky and thus # = v and #v = #2; then x has to be replaced by ttx, K by Kx ' w by /zw and c by #2c, yielding K x bDt 2 c ~_ KD for E, the other zc parameters (99, h, D, c~,, and z) remaining unchanged. Example 24. For the derivation of the analytical solution of a non-steady threedimensional groundwater flow problem in an anisotropic medium, we choose Example 24 of Section 2.2.6 involving axial-symmetric flow, caused by abstraction of a constant discharge from a leaky aquifer by means of a partially penetrating well. The solution for isotropic conditions is given there by equation (g) and becomes, if we transform qg(r, z, t) into ~0(x, y, z, t) with r - v/X 2 + y2:

Q

O0

sin ( - ~ ) -

--b-J sin (°tna~

qg(x, y, z, t) -- 2rcK(b - a) =

(54) ( ~n z X COS

ot2K t Otnv/Xe + ye } D

812

Analytical solution methods

(2.3.4)

D with O~n being the roots of o~tan o~ - e - K--~." In order to find the solution for the anisotropic case we must, according to the transformation Theorem 5, replace K by v/Kx Ky (rule 4), t b y / x v t (rule 3), x and y b y / x x and vy, respectively (rule 2) and c b y / ~ v c (rule 6), the other parameters remaining unchanged (rules 1, 2 and 5). So

D

g__+

--

v / K x KyiJ, v c

D

v/X2 _+_ y2 ~

V//j2X 2 nt_ v2y2

Kzc '

Oln2Kt

2 Otn~KxKylzVt

Ss D 2

Ss D 2

2

Oln Kzt Ss D 2 '

and thus the anisotropic solution becomes for 3 dimensions"

~(x y, z, t) -'

Q

~~0 sin (z_~k) _ sine(~""ID'

2rcv/KxKy(b - a) =

Otn(1 -+- ~2+e2)

(55)

OlnZ )

{ Oln2Kzt u"v/lz2x2+v2y2] x c o s \ - - D - - w SsD2, D ' D

with Oen being the roots of ot tan ot - e - 2-7zC" For axial-symmetric flow we have Kx - Ky -- Kr and v - / , ; and V///~2x2 --~ v2 y 2 -- l Z v / X 2 -at- y2 -- lzr.

so v/K~ Ky -- Kr

Example 25. q

~,4&$, R

s~

R Kr

Fig. 28. Vertical infiltration from a circular area.

Vertical infiltration from a circular area (see Fig. 28) on top of a confined aquifer of "infinite" thickness. The infiltration rate q [L T -1 ] is uniformly distributed over the circular area with radius R, so that the total recharge Q = 7r R2q. The horizontal conductivity is the same in all directions (Kx = Ky = Kr) and differs from the vertical conductivity. So the flow is axial-symmetric. The solution for the isotropic case (see Part A, 523.02) for the steady state is:

1 J1 (Rot) Jo(rol)e -z~ du, ~o(r, z) -- ~q R fo ~ --£

(56)

~(r, z) - 27rqRr fo ~ -1 J1 (Rot)J1 (rot)e -z~ dot, Ol

(57)

and

Solutions, derived from known solutions

(2.3.4)

813

and for the non-steady state:

qg(r,z,t)- qRKfo ee -oeljl(Roe)J°(roe)Pc°nj( oez2'oeS'] doe'/3 ]

(58)

where J0 and J1 are Bessel functions of the first kind and order zero and one, respectively, and Pconj is the conjugate polder function (Section 3.1.2). The solutions (56), (57) and (58) can easily be derived by means of a Hankel transformation and a Laplace transformation (Sections 2.2.4-4 and 2.2.2-3). The transformation of these solutions to solutions for axial-symmetric anisotropy can be performed by replacing K by Kr (rule 4 of Theorem 5 with Kx -- Kv = Kr), q by ~ (rule 8 for a velocity in vertical direction), R and r by # R and #r, respectively (rule 2, because v = #) and t by /,2t (rule 3), the other parameters remaining constant (qg, z, ~P, Ss, according to the rules 1, 2, 7 and 5, respectively). Then 7,/7 = K ~ s b e c o m e s C K r u Ss 2 t " = CKzt TS~' and the solutions for the analogous problem in axial-symmetric anisotropic ground for the steady state become: 99(r, Z) --

qR f ~

1 - J1 (#Roe) Jo(#roe)e -z~ doe,

(59)

#Kr J0 oe

(60)

~/(r, z) - 2rcqRr fo ~ -1J l ( l z R o e ) J l ( l z r o e ) e -z~ doe,

oe

and for the non-steady state:

~p( r, z, t) -

# KR

oo oel J l ( # R oe) Jo ( # r oe) Pc onj ~

oeV--~s

doe.

(61)

Up till now we have considered anisotropic soils of which the main directions of the anisotropy were horizontal and vertical while the boundary conditions were also given in horizontal and vertical planes. If, however, the boundary planes are not perpendicular to the main directions of the anisotropy, it is no longer possible to derive simple rules for the transformation of solutions of problems on isotropic soils to solutions of problems on anisotropic soils.

2.3.5. Approximate solutions for phreatic flow and for density flow with interface 1. Phreatic f l o w Phreatic flow means groundwater flow in a phreatic aquifer, that is, an aquifer that contains a free water level, the so-called water table or phreatic surface, where the water pressure equals the atmospheric pressure (see Section 1.2.2). Unlike a confined aquifer, a phreatic aquifer has a thickness that depends on the height of the (fluctuating) water table the shape of which is unknown beforehand.

814

(2.3.5-1)

Analytical solution methods

As exact solutions for phreatic flow cannot always be found easily because of the complicated boundary conditions along the phreatic surface (see Section 1.5.2-1), it is worthwhile to look for approximate solutions, at the same time maintaining the character of phreatic flow, that is, taking into account the variable thickness of the aquifer. A considerable simplification of the calculations on phreatic flow can be obtained by assuming only horizontal flow, which, in general, will be allowable if the boundary values along vertical planes do not or hardly not contain vertical flow components. In Fig. 29 some examples are given of flow through phreatic aquifers. In Figs 29a and 29b the vertical boundaries relate to fully penetrating surface water bodies, with or without entrance resistance and to a fully penetrating well-screen, all boundaries giving only horizontal flow. Here the assumption of only horizontal flow in the whole aquifer will be a good approximation of the exact phreatic flow. In Fig. 29c a horizontal drain at the bottom of a phreatic aquifer causes vertical flow along the vertical plane of symmetry through the drain; here the assumption of only horizontal flow is not allowable in the neighbourhood of the drain. This assumption was first suggested by Dupuit for a particular case and later on generalized by Forchheimer. Only horizontal flow means that in a vertical plane all points have the same piezometric head q9 which in phreatic flow coincides with the water table and in the sequel will be denoted by h. Darcy's law, which gives the specific discharges in x=~q _

_

I

to

a

Fig. 29. Examples of phreatic flow.

b

Solutions, derived from known solutions (2.3.5- l)

815

and y-directions, now becomes" Vx

-

Oh - K o x~ ,

Vy

--

-

K

~

Oh , Oy

(a)

while the total discharge over a vertical becomes: Oh qx -- - K h o x

Oh -Khoy

qy

(b)

The continuity equation in this case becomes" Oqx

Oqv Oh + -7-- + Sph-7-7 --0, 0---7oy

(62)

in which the term Sph ,-57 oh describes the amount of water that will be taken into storage if the water table rises or will be released from storage if the water table falls. Here Sph is the phreatic storage coefficient or specific yield (dimension [0]), in contrast with the elastic storage coefficient S which originates from the compressibility of the ground and the groundwater (see Section 1.3.5-!). As the latter is very much smaller than the specific yield: S << Sph, the elastic storage coefficient has been neglected and omitted in equation (62). As O q x --- K h O 2 heax o-x- - = - K (Oh)2~xx and

O2h ( o')h'] 2 Oqv = - K h ~ K . ! o y\ Oy2 Oy

the differential equation (62) appears to be non-linear. However, if we consider the steady state (ah _ 0) and write 1 oq(~Kh 2)

qx -- -

and

Ox

0(1Kh 2) qy -

-

Oy

,

equation (62) becomes"

02(1Kh 2) OX 2

O 2 ( 2 K h 2)

+

Oy 2

= 0

(63)

1 and this is a linear differential equation in ~Kh 2° We set

1

CDF-- -2 K h 2

(64)

and call it the "Forchheimer" potential, which has the dimension of a discharge [L3T-1]. Equation (63) now becomes"

02~F O2q~F OX 2

+

Oy 2

= o.

(65)

816

(2.3.5-1 )

Analytical solution methods

The Forchheimer potential thus is a solution of the two-dimensional Laplace differential equation and therefore the real part of a complex function S2F, with S2F -- ~OF+ i~F

(66)

(see Section 2.2.5-1). The stream function OF is the conjugate function of q~F and has also the dimension of a discharge [L3T-1]. The Cauchy-Riemann conditions are also valid:

a4~F

ag~F

a~F

Ox

Oy '

Oy

a~PF with Ox

OdpF

Oh = Kh~--hvx--qx Ox Ox OC/)F Oh = Kh~--hvy--qy. Oy Oy

(67)

and

(c)

As a consequence of the similitude of the complex Forchheimer function I2F with the complex function I2 for confined aquifers, the solutions according to the Forchheimer approximation for steady two-dimensional flow in phreatic aquifers are identical to the solutions for analogous problems in confined aquifers, apart from some adaptations in relation to the boundary values. Phreatic flow with accretion (see Section 1.5.2-1), for instance, constant precipitation p [LT-1], may also be approximated in this way, the precipitation not acting as a boundary condition with a vertical flow component, but as a quantity, hypothetically generated in the aquifer and uniformly distributed over the height h of the water table, in order to maintain the assumption of horizontal flow only. The precipitation now appears as a separate term in the differential equation, as can be shown as follows.

I ~_

P

I

[

'~

hreatic surface

I I

qx

qx +

I I - -f-fl-~

/

,

/

~dy .....

dx

Fig. 30. Phreatic surface with precipitation.

Considering a vertical column with small cross-section dx dy and average height h in a phreatic aquifer with precipitation p (see Fig. 30) and applying the law of continuity, we see that per unit of time at the left side an amount of water vxh dy -- qx dy enters the column and at the right side an amount of water equal to Oqxx dx) dy leaves the col(q x + "g£' umn, the difference ~~)x dx dy being the total amount of water that leaves the column in

Solutions, derived from known solutions (2.3.5- l)

817

x-direction per unit of time. Similarly, the total amount of water that leaves the iJqr column per unit of time in y-direction is T dy dx. Together they must be equal to the replenishment of the column per unit of time by the precipitation, namely p dx dy. So we have: Oqx Ox

Oqy =p Oy

or with equations (c):

O2~F -~-02q~F - ~ - p - 0. Ox 2

(68)

Oy 2

This differential equation is very similar to the corresponding equation for steady two-dimensional flow in confined aquifers with precipitation:

0299

02(./9

p

---~ Ox + ~Oy2 + K D = 0.

(d)

The solutions for qSF are almost identical to those for q9 and can easily be derived from the latter. For two-dimensional radial-symmetric flow, the continuity equation becomes (see Fig. 31):

P

dQr - 27cr drp. r

Qr+dQr

(e)

With dh Qr - 27rrhvr - - 2 7 c r h K ~ dr

dr

Fig.31. Radial phreatic flow with precipitation.

and with q~F -- ~1 Kh2 , we have Qr - -27cr ~ 1 d ( d~bF] r + -0 r dr --~-r/ P

or

and thus, according to equation (e):

d2~bF l d~bF ~ + +p-0 dr 2 r -~r

'

(69)

the differential equation for two-dimensional radial-symmetric steady flow in a phreatic aquifer, according to the Forchheimer approach, with or without precipitation (p = 0 in the latter case). Also this differential equation has its counterpart for groundwater flow in confined aquifers, namely:

d2~o

1 d~o

p

-dr - T + -r -d-~r + K D = 0.

(f)

All solutions of the linear differential equations (68) and (69) and also solutions of particular cases of these equations like p = 0, or qSF independent on y (onedimensional flow), are almost identical to solutions of corresponding problems in confined aquifers and may be derived, according to the following theorem:

818

Analytical solution methods

(2.3.5-1)

Theorem

6. The approximate solution dpFfor a phreatic groundwater flow problem,

for which one of the differential equations (68) and (69) holds, particular cases included, may be found fromthe solution q9for the analogous problem in a confined aquifer, as follows: 1° Replace in the expression for ~p: q9 - A f ( x , y ) + B or cp -- A f ( x ) + B or q9 -- A f (r) + B the known constants A and B by the unknown constants cl and c2 (also if B - 0). 2 ° Determine the constants c l and c2 by means of the differential equation and

boundary conditions belonging to the phreatic flow problem. Some examples may serve for the application of this theorem. Example

26.

A circular basin with radius R and constant level H is lying in a phreatic aquifer in which uniform flow takes place with a discharge q [L 2T -1] (Fig. 32). In a confined aquifer the solution for the complex

Y ----->, q

h=H

R

potential S2 is:

q (z _ R 2

s2--

phreatic surface

-7)

(Part A, 336.01)

and for the drawdown ~o:

H

qx ~0---~(1-x2+y2 Fig. 32. Circular basin and uniform flow in a phreatic aquifer. According to Theorem 6, the solution ~PF becomes:

R2 ) 1Kh2 ~F -- ClX 1 -- X2 + y2 q-C2 -- ~ • Now h -- H for x 2 + y2 _ R 2, so c2 -- ~1 K H 2 and 0~bF _

qx --

Cl -Jr- Cl R 2

OX

y2 x 2 (X2 _+_y2)2' _

while limx~oo qx - q, so cl - - q and thus

epF---

1KH2

R2)

--qx 1-- X2_{_ y2 (

"

R2 )"

Solutions, derived from known solutions (2.3.5- l)

819

The stream function becomes:

!/fF

-

R2 - q y ( 1 + X2 + y 2 )

and the complex potential function: 1KH2

,.~'F-- ~ z

-q

(

z-~

R2) 27

,

and can easily be verified. Remark. Uniform flow in a phreatic aquifer means that the discharge q is constant everywhere; as q - - K h ~ ,dh it follows that the velocity Vx - - K ~ dh is not constant.

Example 27. Constant rainfall p [LT-1] on a rectangular island with a phreatic aquifer and surrounded by surface water with a constant level (Fig. 33). For the analogous problem with a confined aquifer the solution for the head ~o can be found by means of a finite Fourier sine transformation with respect to y (Part A, 363.05):

======================================================================

t -

~o=H

aye

Fig. 33. Rainfall on a rectangular phreatic aquifer.

4b2p ~o(x, y) -- 7r3E D

Z

1 --

m=l,3 ....

(mna cosh~--~)

~

sin

(m.,) . b

According to Theorem 6, the solution for q~F becomes:

[

q~F(x, y ) - - c, m=,,3 ~ .... 1 -

cosh(m~a )_Ty_

cm yt +c2.

m33 sin \ b

(f)

820

(2.3.5-1)

Analytical solution methods

Now h -- H and thus qSF -- 21 K H 2 for x -- 0 and x -- a and for y -- 0 and

1 H 2. ~p(x, y) is a solution of the differential y -- b, from which we find c2 - 2K equation (d) and q~F(X, y) must satisfy the differential equation (68), from which it follows directly that cl -

1

qSF(X, y) -- -~ K

H2

4b2p

rr3 " So the phreatic solution becomes:

1

+ -~ py(b - y)

4b2 p

oo m = l , 3 ....

1 cosh{ ~--~( a - 2x)} (tarry) m3 cosh(mzra~ sin 2b ] b

as

~ Z

1 ~

sin

(mzry) 7r 3 b - -~y(b-

y),

m = l , 3 ....

which may be verified by applying the finite Fourier sine transformation (Section 2.2.3-2a) with respect to y from 0 to b to the right side of this equation.

E x a m p l e 28. Constant rainfall on a circular island with a phreatic aquifer, surrounded by surface water with a constant level and with entrance resistance along the boundaries (Fig. 34). The solution for the analogous problem with a confined aquifer is (Part A, 236.08):

,,+++++++,+,~+4g~

R Fig. 34.

p (R 2 _ r 2 ) q p R w ~ 0 ( r ) - 4K---~ - 2---D--

Rainfall on a circular phreat-

(g)

ic aquifer with entrance resistance. According to Theorem 6, this solution can be written as" qg(r) - Ar 2 + B, with p A --

4KD

pR 2 and

B --

4KD

pRw q-

2D '

the solution for ~bF(r) thus becoming; ~bF(r) -- Cl r2 -+-C2. If we set at r -- R the (unknown) head h ( R ) - hR, we have

1

qSF(R)- = K h 2" -- Cl R 2 --]'-C2, Z from which

C2-

~1 K h ~

- c l R 2 and thus ~bF(r) -- ~I K h 2

must satisfy the differential equation (69)" d~F _ 2c 1r and

--

c 1(R

dz~bF - dr 2

--

2 -- r 2 ).

2C 1

,

q~F

and with

Solutions, derived from known solutions

821

(2.3.5-2)

equation (69): 2cl + 2c! + p -- 0, it follows that

C 1

--

--~ and thus

p q~F(r) -- ~1 K h 2 + ~-(R 2 - r2).

The boundary value at r - R gives v(R) dh hR - H K d--~-(R) w

(h)

--

h R-H W

or

and

dqSF dh h R (h R - H ) d-T -(R) - KhR--dTr (R) - - - w From equation (h) we have d a - ~ ( R ) - - P R , so that he(hR - H)

w

=

pR

or

2

h 2 - HhR

pRw

2

= 0

'

from which h R -- ~! H + ~1VH2 + 2 p R w " this expression substituted in equation (h) gives the end solution. With w = 0 we find hR = H and qSF(R) = H. 2. Interface flow Interface flow will be understood here as flow of fresh water along a sharp interface that completely separates the fresh-water body from underlying immobile salt water. Like phreatic flow also for interface flow it holds that exact analytical solutions cannot easily be found because of the complicated boundary conditions along the interface (see Section 1.5.2-2). Therefore it is obvious to apply some simplifications of the conditions for interface flow in order to obtain (approximate) solutions in a more easy way. The main assumptions are:

1° 2° 3° 4°

....

Sharp interface between fresh and salt water. No flow of the salt water. Only horizontal flow of the fresh water. Neglection of the dimensions of the region where fresh water flows into the sea.

_71_H__~ ~-----~~~~

P ~ x ~ c e~ ~,,..~xgtex ~.-,-

sea kzvel

•

..

Fig. 35. Phreatic aquifer with interface.

Calculations based on these assumptions were first made by Herzberg and later on independently by BadonGhyben. We may distinguish between calculations for phreatic conditions and for confined conditions. In a phreatic aquifer near the sea with a sharp interface, the pressures at a point P of the interface must be the same for the fresh water and for the salt water (Fig. 35):

822

Analytical solution methods

(2.3.5-2)

ysH = yf(H + h) or h = otH with ot -- Y s - Y f = interface parameter Yf

(70)

with ys and yf being the specific weights of salt and fresh water, respectively (Section 1.3.1.- 1). Darcy's law gives the specific discharges in x- and y-directions for an isotropic medium: Oh Vx -- - K o x

and

Oh Vy -- - K ~ . 3y

The discharges over a vertical become, as a consequence of the assumption of only horizontal flow: qx -- - K ( H

Oh -+-h)-~x,

qy - - K ( H

Oh -Jr-h) Oy

and with equation (70): OH qx = --Kol(1 + ~ ) H Ox

OH -Kol(1 + ~ ) H Oy

qy

We set ~H -- --1 Koe(1 + oe)H2 2

(71)

and call it the "Herzberg potential", which has the dimension of a discharge [L3T-1]. Now we have

qx --

0~H Ox

and

qy--

0~H [L 2T -1 ]. Oy

The continuity equation with precipitation becomes, for two-dimensional flow: Oqx ~ Oqy Oh OH 0---~" - "~y -'1- Sph-~ "-[-"tle--'~- -- P,

and thus the differential equation 02~H 02~H q---[--p OX2 Oy2

_

_

OH

(OtSph+ ne) at

(72)

Solutions, derived f r o m known solutions

(2.3.5-2)

823

with Sph -- specific yield and ne - effective porosity. The term Sph ~;)h describes the amount of water that will be taken into storage if the phreatic level rises, and the #H term ne ,-57- describes the amount of fresh water that replaces the salt water if the interface goes down. For radial-symmetric flow, the general differential equation for the Herzberg potential becomes: 02q~n 1 0q~n Or 2 ~ r Or ~- p

fresh

H

--

( O f S p h -nt-

0H ne) 0--7"

(73)

In a c o n f i n e d a q u i f e r near the sea with a sharp interface and sea level hs, we find for the pressure ps of the salt water at a point P on the interface ps = ys(H + hs) (see Fig. 36), with H being the depth of the interface below the upper edge of the aquifer.

/ ~i:lll.

P

Fig. 36. Confined aquifer with an interface. The salt water level in point A is hs and the corresponding fresh water level there is h f - ys ×f h s. Now if we take the fresh water level hf as reference level for the piezometric head h, we find for the fresh water pressure at P: pf -- yf(H + hf + h). Now again, Ps = Pf, so ys(H + hs) = Ff(H + hf + h), from which, with Fshs = yfhf, we have FsH = Ff(H + h) and thus h=o~H

withot=

ys-yf

(see equation (70))

Vf the same result as for the phreatic aquifer. The discharge over a vertical, however, is different from that for phreatic conditions because of the reduced thickness of that part of the aquifer that contributes to the flow of the fresh water (H instead of H+h): Oh qx -- - K H ~ Ox '

Oh qv -- - K H ~ " Oy

824

(2.3.5-2)

Analytical solution methods

and with h --o~H" Oh qx -- - K ° l H ox

OqbH Ox '

Oh qy -- - K o l H oy

aeH Oy

if we set 1

(74)

C])H-- - Kol H 2. 2

Now the differential equations (72) and (73) are with Sph -- 0 also valid for confined aquifers, provided that equation (74) is used instead of equation (71). These equations are non-linear, but the steady state versions (a/-/ -57 _ 0) are linear differential equations: 02t~H 02t~H Ox----5- 4- OYz + p -- O,

(75)

d2t~n [- -1 d~H 4- p -- 0, ----5dr - r - ~ - r

(76)

with CH -- 1Kct(1 4- ot)H 2 for a phreatic aquifer and Cn - 1KotH2 for a confined aquifer. These equations are very similar to the corresponding equations for confined aquifers without interface (see equations (d) and (f) of Section 2.3.5-1) and therefore all solutions of the equations (75) and (76) and also solutions of particular cases of these equations, like p - 0, or ~n independent of y (one-dimensional flow) are almost identical to solutions of corresponding problems in confined aquifers without interface and may be derived, according to the following theorem: Theorem 7. The approximate solution CH for a groundwater flow problem involving a steady sharp interface between two fluids of different density, for which one of the differential equations (75) or (76) holds, particular cases included, may be found from the solution cp for the analogous problem in a confined aquifer without interface, as follows:

1° Replace in the solution for 99" ~o -- A f (x, y) 4- B

or

cp -- A f (x) 4- B

or

~p -- A f (r) 4- B

the known constants A and B by the unknown constants cl and ¢2 (also if 0

B --0). Determine the constants Cl and c2 by means of the differential equation and boundary conditions belonging to the interface problem.

Solutions, derived from known solutions

Example

(2.3.5-2)

825

29.

A long earthen dam in the sea with precipitation p and discharge Q from a well at the location (x0, y0). The aquifer is assumed to be phreatic and of sufficient thickness to guarantee the development of a complete interface (Fig. 37). The differential equation for the Herzberg potential ~bH is equation (75) and the boundary conditions:

Y0 I I i

x0 B

B

Q

e vel

p ,[,,l,,l,,l,,l,,I,,l,,l,,l,,l,,l,~$~,l,$,l,,[;,l,

sea_ _

!

- _

~H(B, y)

2

-- ~H(--B, y) -- O,

Fig. 37. Earthen dam in the sea with precipitation and discharge from a well.

lim (r o~H r--+0

Q

"--~-r ) --

2rr

for r -- v/(X -- x0) 2 ~- (y -- y0) 2.

The solution for the analogous problem in a confined aquifer without interface for the head 99 is" P

(B 2 - x 2) -

go(x, y) -- 2K---D

Q

I cosh{

} + cos

2B

}

]

4zr K D In cosh{rr{~iv0,} - cos {rr(2-~x0) } '

where q9 is a solution of the differential equation" 02('/9-~ 0299 -t

Ox 2

Oy 2

P

KD

=0

with boundary value for the well" 0qg) Q lim r --r-+0 -~r 2rrKD

for r -- v/(x - xo) 2 + (y - yo) 2.

Comparison with the differential equation (75) and the boundary value for the well in this problem immediately gives the solution for qSH by replacing ~ by p and

826

(2.3.5-2)

A n a l y t i c a l solution m e t h o d s

Q by Q. With qSn -- $K0/(1 1 KD + 0/) H 2, we have the solution for H"

H 2 __

P (B 2 _ 0/(1 + 0/)K

x 2)

c°sh{ zr(v-y°) }'2B o ,n[ cosh { rr(Y-YO) 2:fro/(1 + 0/)K

"-I'-COS { n'(x+x0)2B} ] 2B } -- COS { n'(2;0) }

and h 2 - 0/2 H 2. The smallest value of H will be at the well-screen, for instance, with good approximation at the point (xo, yo + rw) with rw - well radius:

P

(B2

x 2) _

0/(1 + 0/) K

cosh,, ~ j + cos ( ~ )

Q

2~0/(1 + 0/) K In

cosh(nrw ] B _TTj_ 1

this expression determining the relation between the precipitation p and the discharge Q if the admissible upconing of the salt water is given. In practice, H is much larger than h; for instance, water of the North Sea has an average Cl-content of 18000 mg/1 and according to Fig. 2 of Section 1.3.1-1 a density of 1025 kg/m 3. Then, as y = p g we find 0/ --

)/s -- Yf yf

=

Ps -- Pf pf

=

1025 -- 1000 1000

= 0.025

and thus H = 40h. In contrast with phreatic water without interface as has been discussed in the first part of this section, the thickness of the fresh water body, responsible for the flow of the groundwater varies considerably and even becomes zero at the outflow into the sea, while also the outflow opening is reduced to zero. In this context, especially the assumptions 3 ° and 4 ° namely mere horizontal flow and neglect of the outflow opening become questionable, the more the flow approaches the outflow region. For this reason, it is worthwhile to compare a Herzberg approximation with an exact solution and investigate for which regions this approximation is still admissible. This will be done in the following example. Example 30. A long earthen dam in the sea of width 2b considered as a confined very thick aquifer, with constant infiltration p (Fig. 38). First we determine the e x a c t s o l u t i o n of the problem, which must satisfy the two-dimensional Laplace equation 8299 0x 2

t

0299 0z 2

=0.

We choose the x-axis along the upper edge of the aquifer and the positive z-axis downwards, the origin lying in the centre of the dam. If we set ~l - x + i z, then 4~ - K~0 is the real part of a complex function S2 - q~ + i gt which is a function

Solutions, derived from known solutions

(2.3.5-2)

827

¢ 5 ¢ ¢

B . . . . . . .

p$$,1,$,1,$ W,4,,1, 4, 4,,1,1,~4 4,,l,,b,l,,l,,~ .................... ~=o fresh D

(¢//

/

Vz

Fig. 38. Earthen dam in the sea with precipitation and exact interface" hodograph of the flow field. of rl = x 4- i z, where ~, the stream function, is the imaginary part of a'2 (see Section 2.2.5-1). The general solution X2 = £2(r/) cannot always be found as an explicit function of r/and therefore we write F(£2, r / ) = 0 .

(a)

If we consider the real and imaginary parts of equation (a) we may write F(£2, 77) = f (ck, ~r, x, z) 4- i g ( ~ , ~, x, z) = 0

(b)

and we have both f@5, O , x , z ) = O

and

g(qS,~,x,z)=O.

(c)

Now we try to find functions f and g which satisfy the boundary conditions. The boundary conditions are (cf. Section 1.5.2-1, equation (10)): Along the x-axis (z = 0) gr = p x for 0 ~< x ~< b (along D C ) and q5 = 0 for x >~ b (along C B , point B still being unknown). Along the z-axis (x = 0) fr = 0 (along D A ) . Along the interface, which shape is still unknown, 05 = ~ K z and gr = 0. The function px(qb - olKz) 4- Oh(qb, d/, x, z) = 0

(d)

with h being an arbitrary function of qS, gr, x and z (h --/: 0), with ~ - 0 already satisfies the boundary values along D A (x = 0) and along the interface (q~ = olKz). For z = 0 we have ~p = p x and q5 = 0, so that if we substitute z = 0 in equation (d) we should get: ( p x - ~)q5 = 0, for instance, h(d,b, ~, x, z) -- zk(q5, ~, x, z) - q5.

828

(2.3.5-2)

Analytical solution methods

Equation (d) then becomes: p x ( ~ - o~z,:z) + ~ {z~(~, ~ , x, z) - ~} - o

(e)

or

(f)

~c/) + ozpKxz - pxqb - grzk(dp, ~p, x, z) = O.

All boundary values of the problem are satisfied by this equation and if it corresponds with f = 0 or g -- 0 (equations (c)), the real and imaginary parts of a complex function S2(r/), then the latter is the solution of the problem. 1 2 Now 7r~b is the imaginary part of ~S2 , as £2 - q~+igr and if2 2 - - ~bz-~rz-k-2ioS~. Also ~ p K x z is the imaginary part of lc~Kprl 2 and if we set k - - p we have -pxc]) + pzgr as the imaginary part of - i p £ 2 ~ , as - i p I 2 r l = -ip{(dp + i ~ ) ( x + iz)} = -ip{(dpx - ~ z ) + i(grx + qSz)}. The complex solution thus becomes: 1

2

1

S22 + =otKprl 2 2

ip.f-2rl + c

(g)

0

with c as a real constant. The real part of equation (g) becomes: 1

~(q~2

_7~2

1

) + " ~ K p ( x 2 -- Z2) + p(~px + qbZ) + c -- O.

(h)

The head go(x, O) along C D can be determined by substituting z = 0 and 7r = p x in (h): 1 2 - -~p2x2 1 -~4) + ~1 X p x 2 + 1 2

1

2

~4, + ~ p ( p + ~ / ~ ) ~

p 2X 2 + c - o,

or

+c-O.

For x = b go(x, O) and also q~(x, O) = O, from which 1

(i)

c -- - - ~ p ( p -t- o t g ) b 2.

So the end solution F(S-2, r/) -- 0 becomes: 1 $2 2 -~- ~otKptl2 2

ipX2rl

1 -~p(p + otK) b2 - O.

(77)

The real part f(4~, ~, x, z) = 0 becomes 1

1

1

~(4~2 _ ~p2) + _~c~Kp(x 2 _ z 2) + p(grx + dpz) - ~ p ( p + ~ K ) b 2 -- O, (78)

Solutions, derived from known solutions (2.3.5-2)

829

and the imaginary part g(¢, ~p, x, z) = O: ¢ ~ + 0/pKxz - pCx + p O z = O.

(79)

If we substitute ~k -- 0 and ¢ -- 0/Kz in equation (78), we find the equation for the interface: 1 °l 2 K 2 z 2 nt-

-2

1

2

1

0/Kpx2

1

- -0/2Kpz2 + 0/KPz2 - -2p ( p + 0/K)b2 -- 0

or

0/K(0/K + p ) z 2 + 0/Kpx 2 - p(0/K + p)b 2 - 0

and thus P b2

7..,2

p

-- 0/K

x 2 -- H 2

p + oeK

"

(80)

This line intersects the x-axis at B where z = 0, from which

XB - -

For x - 0

+ 0/ K

b,/ V

~ I(

"

(J )

we have the maximum value of H: !

Hmax - - b /

P

V0/K"

(k)

The interface line is an ellipse for which the long axis is represented by (j) and the short axis by (k). The variation of the piezometric head ~Pi along the interface can be derived directly from equation (80), as (/9i - - c ~ H . The piezometric head ~p(x, 0) along C D becomes, with z - 0 and ~p - p x substituted in equation (h): ~2

+

p(p

+ 0/K)x 2 _

p ( p + 0/K)b2 _ 0

and thus, with ¢ = K~p(x, 0): (/92 (X, 0) =

p ( p + 0/K) K2 ( b2 -- X2).

(81)

The maximum head ~o(0, 0) is b (p(O, O) - -7; v / p ( p + 0/K). lk

(1)

Equation (81) is an ellipse with the long axis b and short axis represented by equation (1).

830

(2.3.5-2)

Analytical solution methods

The velocities everywhere in the flow field can be obtained by differentiating (78) and (79) with respect to x and z; for instance, differentiation of (79) with respect to x gives: Og

O~

O~

~4)

o7,

+ 4 -g2x 4- otp K z - pc/) - p X -~x 4- p z .-~x - 0

Ox -

and w i t h v x = - o x -~ --

_

~Oz and V z -

~4)

_ Oz -

( p x - ~/)Vx + ( p z + ~)Vz + p ( o t K z -

~x we find: ~ ) = O,

(m)

( p z + ~)Vx + (7z -- pX)Vz -- p ( o t K x + ap) = O,

(n)

Bg

and in the same way 57 = 0 yields

giving two equations in Vx and Vz from which Vx and Vz are determined as functions of ~b, 7r, x and z and thus as functions of x and z. Differentiation with respect to x and z of the real part, f (q~, gr, x, z) = O, of the solution gives also the results represented by (m) and (n). Especially the velocities along the boundaries of the flow field can easily be obtained from (m) and (n). For instance, along the interface we have gr - 0 and = o t K z , which substituted in these equations gives: pXVx + ( p + o t K ) z v z = 0

and

( p + o t K ) z v x - pXVz - p o t K x = 0 .

Elimination of ( p 4 - o t K ) z from these two equations gives 2

2

vx + v z + o t K v z - - O ,

(o)

which is the boundary condition along the interface, representing the circle in the hodograph representation (see Fig. 38). Along A D the velocities become, with ~ - 0 and x - 0 from equation (n) v~ = 0 and from equation (m): Vz =

p(4, - o~Kz) pz+¢

•

(p)

Point D (z - 0) gives Vz - p and point A (4) - o t K z ) gives Vz - 0 (stagnation point because also Vx = 0). Along D C we have z = 0 and ~ = p x . From equation (m) it follows that Vz = p and from equation (n) p(p + otK)x Vx =

.

(q)

(2.3.5-2)

Solutions, derived from known solutions

831

Point D (x -- 0) gives Vx -- 0 and point C (x - b, q~ - 0) gives Vx - cx~. Along C B ( z - 0, ¢ - 0) the velocities become vx - 0 and p(aKx VZ

+ ~)

---

(r)

--px

Point C (x - b, ~ - p b ) gives Vz - cx~ (cavitation point because also Vx - e~) and point B (gr - - 0 ) gives Vz -- - o t K . Now, applying the Herzberg approximation for a confined aquifer with precipitation, we must solve the differential equation d2q~H dx 2

+p--0,

with q~H - ~1 K o t H 2, dq~H (0) -- 0 dx

and

CH(b) -

0,

with the solution P 2 - x 2), ~H -- -~-(b

from which

H 2 = P._~_(b 2 X2). otK -

(82)

This equation represents an ellipse with long axis b and short axis b,/ff-~. If we compare this result with the exact solution of the interface,V~'"according to equation (80), we see that in both cases the interface is an ellipse, having a short axis of the same length (cf. equation (k)) and a long axis, the length of which differs by an amount b~P

+ c~K

o~K

-b

(cf. equation (j)).

p is, in general, very small compared with c~K. For instance, p - 0.002 m/day, ot - 0 . 0 2 5 and K - 10 m/day; then of K V/P + = 1.004 otK

and the difference in length of the long axis of the two ellipses, which corresponds with the dimension of the outflow opening in the exact solution, becomes very small, in this case 0.004b, which means that if b - 500 m the outflow opening is only 2 m. So in this particular case the Herzberg solution is an extremely good approximation of the exact shape of the interface.

832

(2.3.6-i)

2.3.6.

The

reciprocity

Analytical solution methods

principle

1. Description and mathematical proof 8. The drawdown of the piezometric head at an arbitrary point A in a

Theorem

heterogeneous anisotropic porous medium with homogeneous groundwater, caused by pumping with a discharge Q in another arbitrary point B at the time t after the beginning of the pumping is equal to the drawdown in B as the result of pumping in A with the same discharge Q and after the same time t. This theorem describes the so-called reciprocity principle and is valid for , J , , e%~ I I • "° ", " " I homogeneous groundwater in heterogeneous anisotropic soils of whatever complex composition (Fig. 39). (In the sequel we take it for granted eB ,' I that by "groundwater" is meant "homogeneous groundwater".) In order to prove this principle, we '- S~_ ~. , . , ,s ~, .... I make use of the divergence theorem ' " I • % 8 • • of Gauss, applied to a closed groundwater body with volume V and suri o s ~/ face A in which a vector field for the clay specific discharge or bulk velocity v J • • P has been generated. This means, in l oA • " , general, that at every point of the groundwater body the velocities Vx, Vy and Vz in the directions of the coordinate axis x, y and z, respecFig. 39. Cross section of anisotropic heterogeneous tively, are functions of x, y, z and soil. t. Then, according to the divergence theorem, we have at the time t, per unit of time: I

I

s

,,

S

%

e

I

'~

~

p

fffdivvdV-ffAo, dA, OVx

OVy

(83

OVz

in which div v - -~x + ~ ÷ -~z and v, - v . n is the component of the vector v in the direction of the outer normal of the surface A. If we apply equation (83) to steady flow, we find, according to the continuity principle: d i v v - 0 (Section 1.4.2-3, equation (12)), and thus

f fa Vn dA -- O,

(84)

Solutions, derivedfrom knownsolutions (2.3.6-1)

833

which means that the total amount of groundwater that flows across the surface A equals zero, provided that no sources or sinks exist within the body, enclosed by A. A second application of the divergence theorem leads to Green's formulas for groundwater flow, as follows: Let f and g be scalar functions of x, y and z and let the vector v in equation (83) be related to f and g by v-fgradg-

f Vg.

Then div v - div(fVg)

Of ag O2g Of Og ~-f Oeg + Of Og + f O2g 0-7 0--7+ f ~ ~ Oy Oy ---~ ay ~ ~z 022 = fV2g + V f . Vg and vn--v.n--(fVg).n--f(Vg-n)-According to equation (83) we find

ag f On.

Green'sfirst formula:

Og dA. fff
(85)

By interchanging f and g, we obtain a similar formula. On subtracting this formula from equation (85) we find Green's second formula:

f f f(fV2g-gV2f)dV-f

Og

Of

(86)

If both f and g satisfy Laplace's differential equations g 2 f - 0 and V2g - O, respectively, we have:

Og Of f £ (f-~n-g-~n) dA --0.

(87)

In particular, if the functions 99a(x, y, z) and 99t,(x, y, z) both represent the piezometric head of a groundwater flow and they satisfy the differential equations vZq)a -- 0 and vZqgb --0 respectively, equation (87) becomes

ff (

g)a On

991,~

dA -- 0

This equation serves as the basis of the proof of the reciprocity principle.

(88)

(2.3.6-1)

834

Analytical solution methods

First Theorem 8 will be proved for s t e a d y f l o w . Let the points A and B belong to a closed groundwater Qa V body with volume V, whose boundary is a piecewise smooth orientable surn zI B[] face A (Fig. 40). x/ Assume that pumping from a A with a discharge Qa gives a steady drawdown distribution ~0a and a bulk velocity v, and pumping from B with a discharge Q b causes a steady drawFig. 40. Abstraction of water from A and B. down distribution ~0h and a bulk velocity vb in that groundwater body; ~Pa, ~Ph and the components of Va and vb (Vxa, Vya, Vza, Vxh, vyh and Vzh) are functions of x, y and z. Qb

=~

If the soil is isotropic the velocity vectors are proportional to the negative gradients of the piezometric heads (Section 1.2.3, equation (31)): Va = - K g ~ o a

and

Vb = - K V ~ o h ,

and if besides the soil is homogeneous with K = constant, the functions ~p, and ~Ph will satisfy the differential equations of Laplace. In that case equation (88) is valid. It is important to know if equation (88) also holds for flow through a n i s o t r o p i c h e t e r o g e n e o u s ground. Then the bulk velocity is no longer proportional to the gradient of the piezometric head as its components are defined by

Vx = - Kx -~x ,

vy -

- K y -~y

and

vz - - K z -~z ,

in which K x , Ky and Kz are different functions of x, y and z. Then Green's second formula (equation (86)) is not valid and we need an extension thereof to show that also for anisotropic ground a generalization of formula (88) holds. For that purpose we compose the vector U - - ~OaVb -- qgbVa

and apply Gauss' divergence theorem thereto. Then formula (83) takes the form:

fff

Solutions, derived from known solutions (2.3.6- l)

835

in which Van and vhn are the components of va and vh in the direction of the outer normal of A. Now div(99aVb)

--

(/9 a

div vh + V~o..vh

and

div(99bVa) -- q)h d i v v . + Vq)b'Va, and on subtracting these equations and setting div v,, - 0 and div vh - 0 as the flow is steady, we find div(q)aVb) -- div(~OhVa) = Vq)a "Vb -- VqOb "Va

3~pa

Oq)a

099h

099b

Furthermore, Vx,, - - K x ~ax and vxh - - K x ;-57-x and thus 3q)a a--7- v x h

3q)h 09% 399b t- Kx 3q)h 399a -~-x v x" - - K x -~x a x -~x a x = 0

and also 3q)a 3y Vyb

099h ~Vva

-- O

and

O~Oa 099h ~3z vzh - -~z Vza -- O.

So the volume integral of equation (89) becomes zero and we have found an extension of Green's theorem for anisotropic soils with steady flow:

f

(90)

f a (99,, Vhn -- 99h Van) d A -- O.

A condition for equation (90) is, that no sources or sinks exist within the body enclosed by A. So we have to choose for A the surface indicated by Fig. 41, where A = A' 4- AA 4- AB in which A' is the outer boundary of the groundwater body and A A and A B are the surfaces of the filters used by pumping in A and B. In practice, there is always a surface A' (finite or infinite) where either the head or the flux or a combination thereof is equal to zero: a~

+ fly,,

-

0

along A' in which a and fl are constants, including zero, but not at the same time. Fig. 41. Outer and inner boundaries of a groundwater body.

836

Analyticalsolutionmethods

(2.3.6-1)

Then, along A'

fi qga V b n - - (tgb V a n m

fi

__ ~ Ua n 12bn _4_ ~ Vbn V a n ~ Ol Ol

0

and thus the surfaces A A and A B are left in the double integral of equation (90) which now becomes:

ffaA(~O~,vb~--~OhV~'~)dA+ffa8(qga Vbn

- - qgb l ) a n )

d A = O,

which, of course, is also valid for drawdowns instead of heads. Now we assume that pumping from A causes a drawdown ~0aa at A A and a drawdown ~oha at A B and also that pumping from B gives a drawdown qgbh at A B and a drawdown 99ab at A A, the first index of qOi.j referring to the location of the drawdown and the second to the location of the well. Then we have:

°a,,ffavbdA--°"hffA

A vhndA B

(91)

0. B

f

fA A Dan dA is the velocity, perpendicular to the surface A A and caused by abstraction of water from filter A, integrated over the surface AA and thus is equal to the discharge Qa, while in the same way

f fasvhndA-- Qh. f

fA A l)bn dA is the velocity, perpendicular to the surface A A, integrated over A A, caused by abstraction of water, not from filter A but from filter B and therefore equals zero, because the total flux entering and leaving the well-screen at A as a consequence of the flow field described by ~0h, equals zero. In the same way

f fA8 VandA -- O. So equation (91) is reduced to: - - qgab (,Oab ~

Qa + qgba

qOba Q b

-'- O ,

or if

Qa = Qb (92)

by which the reciprocity principle is proved for steady flow. Secondly, the theorem will be proved for unsteadyflow. ~0,,, ~oh, va, vh and the components of va and vh are now also functions of the time t and the divergence

Solutions, derived from known solutions

(2.3.6- l)

837

of the bulk velocity is not equal to zero as a result of the compressibility of the ground (Section 1.4.2-2, equations (6) and (7))"

Oq),, div v,, - - Ss 0-t

and

Oqgh div vh - - Ss 3---7'

(a)

where Ss is the specific storage (in general a function of x, y and z). The volume integral in the divergence theorem of Gauss applied to the time-dependent vector U --

q)a Vb - - qOb V a

now does not vanish but becomes equal to

The dependence of time can be eliminated by applying the Laplace integral transformation to the functions that depend on t (Section 2.2.2). Instead of the vector u - q)aVb --(pbVa, which depends on x, y, z and t, we now start from the vector V --

q)a Vb - - (#b V a ,

which depends on x, y, z and s. In this case e -stq).(t) dt

qSa --

and

% -

e - ' t Va(t) dt,

etc.

It must be noted that 9 -# fi because the Laplace transform of a product of two functions of time is not equal to the product of the transforms of the separate functions (Section 2.2.2-1, Theorem 6). Applying the divergence theorem to the vector ~, we find:

dA. Now div(q3a~h) = q3a div~h + Vq3,. vh, div(q3h%) = q3h d i v % + Vq3h" %, and on subtracting these equations: div ~ = ~b, div vh - q3h div % + Vq3a • vh - Vq3h • %.

OG , div ~', =

3x

+

Ovv,, OVz,, ( Ov~,, ,~ O-y + = 3z 3x /

= (div Va) - - S s \-=--t,ot/ according to equation (a).

!

(b)

838

Analytical solution methods

(2.3.6-1)

As (~-g) = S~a -qga(0) (Section 2.2.1-1 Theorem 2) and the drawdown g)a at at the time t - 0 is equal to zero, we find div~'a - -Sssq)a and in the same way div~,b -- - & s O b . Then, the first two terms on the right of equation (b) become q3a div g'h - q3b div re,, - - S s s ~ a ~ b + Sssqghqga - O,

Vq)a" Vb -- Vq)b "Va -

--

aqga OX

'~)xb

aqga aqga -. -~- --~Z Uzb -~y Uvb

nt-

-

aq3b _ .

aqSb _

. . . 3x Vxa

OqSh _

. . 3y vya

OZ

l)za .

°af-fi x and if K~ is independent of time:

Uxa -- - K x

aqSa _

(O~-[~a

Vxa -

\ ax

-- - Kx

and also

a¢h f)xb -- -- Kx

ax

and thus O~)a

a-Tf'x

Oq9b _

-T2x " * ' - - K x

a~a a~h t- K xa ah ~ ~a ao = O.

5x

ax

ax ax

A similar derivation may be performed for the coordinate directions y and z, from which it follows that also the second two terms of the right side of equation (b) become equal to zero, and thus div ~' = O, which means that also the triple integral of the divergence theorem equals zero and consequently also the double integral:

f

f A (~af)bn -- ~hf)an) d A -- O.

(93)

We assume that also Qa and Qb are functions of time and their Laplace transforms are ~)a and Qb. Then we may consider qSa and Va as the drawdown and the bulk velocity caused by an abstraction ~)a; the same is valid for the functions with index b. Now mathematical analysis of equation (93) in a similar way as has been done for steady flow, and if also the boundary condition olqo + flVn - 0 has been transformed to otq3 + firOn -- O, will result in: Q a q g a b - Qb~ba

for every value of the parameter s. If Qa is the same function of s as Ob, q~ab will be the same function of s as ~h,,, and inasmuch as the inverse Laplace transform of a function is essentially unique,

Solutions, derived from known solutions (2.3.6- l)

839

we can conclude that q)ah is the same function of t as qgba, if Qa is the same function of t as Qb. The reciprocity principle enables us to derive a drawdown at a certain point A, caused by abstracting water from B, if the drawdown at B as a result of pumping water from A, is known. As such it is the theoretical representative of the category of solutions, derived from known solutions, which are the subject and title of this Section 2.3. The practical applications are, in general, limited to checking existing or knewly found analytical solutions, especially those concerning layered soils and multilayered systems. Example 31. An aquifer, infinitely extended under open water, consists of two horizontal layers with different transmissivities K1 D1 and KzD2 (Fig. 42). From each layer an arbitrary amount of groundwater is abstracted by means of a partially penetrating well. The wells are located in a vertical chosen as the axis of revolution for the axial-symmetric flow. Each layer has its own differential equation with boundary values for the steady drawdowns (/91 (r, z) and 992(r, z). For 0 ~< z ~< D1 we have:

02991 1 0991 02991 -----5Or + -r ~ + 0 Z 2 "-- 0,

~01 -- q91(r, z)

~=0

al i

K1

|

Q1¢= ' '

bl

D1

A

q)g(r) a2

b2 B'

K2

;ol(r, z)

ro

-IA' I I I I I I I

ro

r ~ i

i

'' ==~a2 D2 i

go2(r, Z)

|

Fig. 42. Layered aquifer under open water and with partially penetrating wells.

840

Analytical solution methods

(2.3.6-1)

lim [r

0qgl (r, z ) ] -

Ol {

r--,o -~r

-

0

2:7rK1 (bl - a l )

for al < z < bl, for0~
andbl
(t91 (OO, Z) -- O.

~Pl(r, 0) -- 0, (/91(r, D1) -- ~Og(r) a still unknown function. In the lower layer (D1 ~< z ~< (D1 -q- D2)) we find: 02q92

1 0(/92

-Or - 5 - + -r ~

02992 q)2 -- q92(r, Z),

+ ~oz 2 = 0 ,

Q2 lim r

r--,0 -~r

(r, z)

--

2zr K2 (b2 - a2)

0

for (D1 + a2) < z < (D1 + b2), for D1 <~ z < (D1 + a2) and (D1 -+-b2) < z ~< (D1 + D2),

(/92(00, Z) -- O,

q92(r, D1) -- qgg(r),

O~p2 ~(r,

D1 + D2) -- 0.

Oz

For the solution of this problem, we apply the method of separation of variables (see Section 2.2.1). We assume that ~pl(r,z)= FI(r)GI(z) and ~02(r, z ) = F2(r)G2(z), in which F1 (r) and F2(r) are functions of r only, and G1 (z) and G2(z) are functions of z only. Then we have for i -- 1, 2:

d2Fi Gi dFi d2Gi Gi-=--z-.-t " ~-Fi " =0 dr z r dr dz 2 or

1 d2Fi

1 d2Gi

1 dFi

"F//( dr 2 + r - - d T ) -

Gi dz 2 "

The expression on the left side of this equation depends only on r and on the right side only on z. Hence, both sides must be equal to the same constant, for instance, o~2, from which two ordinary differential equations are obtained: d 2 Fi 1 dFi dr 2 t r dr

ot2 Fi -- 0

and

d2Gi

~ + ot2Gi - O, dz 2 •

with the general solutions:

Fi(r)

-- ¢ilKo(otir)

Gi(z)

- - ci3

-t- c i 2 I o ( o l i r )

sin(otiz) -+- ¢i4

and

COS(0tiZ).

Solutions, derivedfrom known solutions (2.3.6-1) The boundary condition 99i(oo, z) = ci2 = O. With Ai - - C i l C i 3 and Bi =

~.(oo)Gi(z) CilCi4,

841

-- 0 gives

F,(oo)

=

0

and thus

we find:

(a)

qgi(r, Z) -- {Ai sin(o~iz) -q- Bi COS(CeiZ)}Ko(ceir). For z = D1 the functions q)l and 992 become: q)l (r, D1) -- {A1 sin(o
and

992(r, D1) -- {A2 sin(o<2D1) ÷ B2 cos(o~2D1) }Ko(o<2r). Both expressions must be equal to the same function g)g(r) of r, from which it follows that o
G1 = A1 sin(o
The boundary conditions along z = 0, z = D1 and z = D1 -+- D2 become, for G1 and G2:

GI(O) --O,

GI(D1) -- Gz(D1) -- g,

dG2 ~(D1 dz

-nt- D2) - - 0 ,

from which G1 -- g

G2=g

sin(o
and

(b)

cos{or(D1 + D2 -- Z)}

(c)

cos(o/D2)

In these expressions, the constants g and c~ still have to be determined. The continuity condition along the boundary z = D1 becomes: 0(/91 0(/92 X l - ~ Z (D1) -- K 2 - ~ g (D1), dG1 (D1) -- K2

or

dG2 --(D1),

(d)

and thus, by differentiating the equations (b) and (c): K1 cot(o
(e)

842

Analytical solution methods

(2.3.6-1)

This expression is a goniometric equation for oe and so oe has infinitely many solutions C~k (k -- 1, 2, 3 , . . . ) . Hence, the equations (a) become" sin(otkZ) Ko(otkr)

cpl(r, z) -- ~

k--1

and

(f)

g(k) sin(otkD1)

cx)

C_OS{Otk(D1 + D2 - z)! Ko(otkr). cos(otkD2)

~p2(r, z) -- Z g(k) k=l

(g)

The function g(k) is a function of Oek and may be determined with the help of the boundary values at r - 0. We now write the functions G1 and G2 from equations (b) and (c) for a special value of oe, for instance, Olk, as G I (k, z) and Gz(k, z). Then by differentiating equation (f) with respect to r we have: oo

a~o---~l(r, z) -- - Z G1 (k, z)otkK1 (akr), Or k=l and thus oo

lim r r-+O

-

- ~

G1

as lim

,

{xK,(x)}- 1;

x-+O

k=l

also oo

lim ( r. = - ~ G2(k , z) r--+0 --~-r ] k=l The boundary conditions at r - 0 therefore become: c~ al(k,

Z) --

k=l

sin(otkZ)

~

g(k)sin(oekD1)

k=l

(h)

/

Q1

=

oo

oo

for0~
andal
cos{oek(D1 + D2 - Z)}

G2(k, z) - ~ k--1

for al < Z < bl,

2rr K1 (bl - a l ) 0

g(k)

k=l

cos(otkD2)

(i) for (D1 + a) < z < (D1 + b2), =

2rr K2 (b2 - a2) 0

for D1 <~ z < (D1 + a 2 ) and (D1 + b2) < z <~ (D1 + D2).

From the equations (h) and (i) we have to solve g(k), which can be done as follows. Besides the function Gl(k, z), we consider the function Gl(l, z) in which Olk of

Solutions, derived from known solutions (2.3.6-1)

843

G1 (k, z) has been replaced by another root of equation (e), namely oel. The functions a l (k, z) and a l (1, z) satisfy the differential equations: d2G1 (k, z) dz 2 + ot~G1 (k, z) - 0

d2G1 (1 z) + c~2Gl(l, z) - 0. dz 2 '

and

Multiply the first equation by G j(1, z) and the second by G l(k, z), subtract the second from the first and integrate with respect to z between z = 0 and z = D~: _

--

2 f0D1G1 (k, z)G1 (l, z) dz f D1

Gl(l,

d2Gl(k ' dz2

Z)

z)

dz -

fD,

(k, z) } -- fo D1 Gl(l, z) d { dGl-d~ --

Gl(l,z)

Gl(k

'

z)

d2Gl(l z) ' dz dz2

foD,Gl(k,z) d{ dG 1(/, Z) } dz

dGl(k, z) _ G l(k z) dGl(I,z)] D1

dz

dz

[ DI dG1 (k, z) dG1 (1, z) J0

dz

fO D1 dGl(/, z) dGl(k, z) dz. dz dz

dz +

dz

o

The last two terms cancel each other, so that

-

2 f0D1Gl(k,z)Gl(l,z)dz (J)

dGl(k, = g(l) dz

D 1 ) -- g ( k ) d G 1 (1 D )

----~Z- '

1 ,

as G1 (k, 0) -- G1 (l, 0) -- 0 and G1 (1, D1) = g(l) and G1 (k, D1) = g(k). In a similar way, for the second layer we find:

_

fo °1+ 2 G2(k,z)G2(l,z)dz 1

dG2(k,D)+ = - g (I) ---~-z 1

g

(k) dG2(l --dz

(k) D1) '

"

If we now multiply equation (j) by K1 and equation (k) by K2 and add both equations, we find, making use of the continuity equation (d), the important result:

K1fo l Gl(k,z)Gl(1, z) dz + K2 fD ol+D2 G2(k, z)G2(I, z) dz =- 0, 1

provided that k =/: 1.

(1)

844

Analytical solution methods

(2.3.6-1)

Now we multiply both sides of equation (h) with K1GI(I, z) and integrate with respect to z between z = 0 and z --- D1, yielding

K1

foD1G1 (l, z) ~

Q1

Gl(k, z) dz - 2yr(bl - al)

f~bl Gl(l,z)dz. 1

(m)

k=l

Next, we multiply both sides of equation (i) by K2G2(I, z) and integrate with respect to z over the interval D1 <~ z ~< (D1 + D2): +D2 G2(l, z) K2 fo D1

Gz(k, z) dz

f D1 +b2 / G2 (l, z) dz. (n) 27r(b2 - a2) j Dl+a2

Q2

-

k=l

1

Now, addition of the equations (m) and (n) gives an infinite series, for which all the terms with k --/: l vanish, according to equation (1) and only the terms with k - l are left. Hence equation (m) + equation (n) gives: K1

foD1G 2(k, z) dz +

K2

fOO1+D2 G 2 (k, z) dz 1

=

(o)

,

Q1 2yr(bl - a l )

G1 (1 z) dz + 2yr(b2 - a2) d Dl+a 2

1

G2(l, z) dz.

Now, substituting the expressions for G1 and G2 from equations (b) and (c) in equation (o) gives: g2

foD1 sinZ(°lz)

K1

dz + Kzg 2 fD°l+DZ c°s2{~(Ol -t- O2 -- Z)} dz 1 COS2 (0l D2)

sin 2 (ol D1)

Qlg

--

[bl

2yr(bl - al) Jal

Q2g

sin(olz) dz sin(olD1)

f D1Wb2cos{or(D1 +

+ 2yv(b2 - a2) ]dDl+a 2

D 2 - z)}

dz,

cos(otD2)

which may be evaluated to: g2 {

K2 D2 K1 D1 _ K1 cot(olD1) + + K2 tan(or D2) 2 sin2 (olD1) 2or 2 cos2(olD2) 2~

Q1

cos(oral) -- cos(otbl)

-- g 2yr(bl - al) +

}

Q2 2Jr(b2 - a2)

ol sin(czD1) sin(otD2 - ota2) - sin(olD2 -- orb2)/" ot cos(olD2)

The second and fourth terms on the left side cancel each other according to equation (e), and we may solve for g. Substituting the value for g in the equations (f) and (g), we find the final solutions.

Solutions, derived from known solutions

(2.3.6-1)

oo Wl + W2 sin(otkZ) Ko (Otkr ) q91(r, z) -- ~ ~ + U2 sin(~k D1)

845

and

k=l

992(r' g ) -

c~ Wl + W2 COS{Olk(D1 + D2 -- Z)} Ko(oekr), ~ ~11 + U2 cos(c~kD2) k=l

(94)

with

U1W1 =

W2

K1D1

sin 2 (C~kD1)

,

U2-

K2D2

cos 2 (O/kD2)'

Q1 cos(otkal) -- cos(otkbl) xr~(bl - al) sin(~kD1)

and

Q2

sin(c~kD2 -- otka2) -- sin(c~kD2 -- otkb2)

:rrOlk (b2 -- a2)

cos(otkD2)

with O~k being the roots of tan(otD1)tan(otD2) -- L K2 according to equation (e) (Part A, 562.01). We may now verify these solutions by applying the reciprocity principle for the two locations A and B (see Fig. 42). Therefore, we must modify the solutions ~01 and q92 for partially penetrating wells into solutions for point wells. A point well at point A(0, bl) may be considered as the limit of the partially penetrating well with length ll = bl - a l if 11 approaches zero. Then W1 becomes:

W1 "--

Q1 lim [cos{oek(bl- l l ) } - cos(otkbl)-[_ Q1 sin(c~kbl) 7rc~k sin(otkD1) lifO [ ll J Jr sin(oekD1)

So, if we abstract only an amount of water from point A(0, bl), we find with Q1 - Q and Q2 - 0 (so also W2 - 0) from equation (94) the drawdown at the point B(ro, D1 + a2)"

Q w-~°° 7r

k=l

1

sin(~kbl) cos{c~k(D2 -- a2)}

UI + U 2

sin(ak D1) cos(¢k D2)

Ko(akro).

(p)

Now, pumping in B(ro, Ol + a2) with an amount Q will give a drawdown ~P,,b at point A(0, bl), which is the same drawdown at point A'(ro, bl) as a result of pumping in B'(0, D1 + a2) with a discharge Q, because of symmetry. A point well at B'(0, D1 + a2) may be considered as the limit of the partially penetrating well with length 12 -- b 2 - a2 if 12 approaches zero. Then W2 becomes

846

(2.3.6-2)

W2

[sin{otk(D2-a2)}-sin{otl~(D2-a2-12)}]

lim

Q2

Analytical solution methods

YrOCkCOS(OtkD2) /2 --+0

12

Q2cos{~k(D2 --a2)} :It COS(Otk D2)

Pumping an amount of water only from point B'(0, D1 + a2) we find, with Q1 - 0 and Q2 - Q from equation (94), the drawdown in the point A'(ro, bl)" Q ~ qga'b' -- ~

1

COS{Otk(D2 -- a2)} sin(otkbl)

COS(OtkD2)sin(olkD1)

' ~ U1 --I-- U2 k=l

Ko(u~ro).

(q)

Comparing equation (p) with equation (q), we see that qgatb, - - q g b a , and as ~o,,,b, -~Oab, we have found 9ab -- ~Oba, the reciprocity of discharges and drawdowns for the points A and B in Fig. 42. 2. The reciprocity principle in multi-layer systems A multi-layer system has been described in Section 2.1.3 and consists of more than one aquifer, separated by one or more semi-permeable layers, while the flow in the aquifers is assumed to be merely horizontal and the flow through the semi-permeable layers merely vertical (see Fig. 4 of Section 2.1.3). The principle difference with a layered soil, as discussed in the previous section, where for each layer the drawdown has its own (three-dimensional) differential equation with boundary values, is that, in a multi-layer system, the (two-dimensional) differential equation for the drawdown of each separate aquifer also contains terms referring to drawdowns in adjacent aquifers. As the vertical flow in the aquifers is neglected, the pumping takes place by means of wells that fully penetrate the distinct aquifers; then, the reciprocity principle refers to groundwater abstractions from line wells instead of point wells and to drawdowns being the same along a vertical in an aquifer.

Qb :=~

Qa

Cl

i

K1D1

B'

c2 K2D2 c3 r I

A '11

K3D3

I I I

Fig. 43. Three-layer system.

B • c4

For instance, pumping an amount Q. from A in aquifer 1 gives a drawdown qgba in B in aquifer 3, which must be equal to the drawdown qgab i n A as a result of pumping an amount Qb from B, if Q. = Qb, as is shown schematically in the three-layer system of Fig. 43. Suppose that r is the horizontal distance between A and B, then (Pb., the drawdown in B, is a function of r, if the location of the well in A is taken as the origin and qg.b the same function of r if the origin is chosen in the vertical through B.

Solutions, derived f r o m k n o w n solutions

(2.3.6-2)

847

From the supposed homogeneity of the layers, it follows that pumping an amount Qh from A' in aquifer 3, lying in the same vertical as A, will give a drawdown ~0h,,,, in B' in aquifer 1 in the same vertical as B, which is equal to 99,h, the drawdown in A caused by pumping in B. So, q)h',' = ~0,h and as q)c,h = qghc, we have q)h',' = qgh,, which means that pumping in aquifer 1 from a vertical gives a drawdown in aquifer 3 at a horizontal distance r, which is equal to the drawdown at the same distance in aquifer 1, caused by pumping from aquifer 3 in the same vertical. Because r may be chosen arbitrary, the drawdown line as a function of r in aquifer 3, caused by pumping Q1 in aquifer 1, namely q)31(r) is equal to q)13(r), the distance-drawdown line in aquifer 1 as a result of pumping Q3 in aquifer 3 by means of a well in the same vertical as the well in aquifer 1, provided that Q1 = Q3. In order to check the reciprocity principle also for multi-layer systems, we consider the n-layer system of Fig. 4, Section 2.1.3, where we have to show that pumping Qi from aquifer i gives a drawdown q).ji(r) in aquifer j, which is equal to the drawdown qgij(r) in aquifer i, caused by pumping Qj from aquifer j, the wells being located in the same vertical, if Qi -- Qj.

za

Instead of checking the reciprocity 9=0

y/////////~//////////////////////////////////////////~ Cl principle in the general n-layer sys,

1

=~ Q 1

q91

tem by means of the solution

T1 =

U//'////////////////////////////////////////////////~////,~

ili

C2

¢ p ( r ) - Ko(rx/A)q,

I I I I

2

=¢" Q2

Ill

3

~°2

T2 = K2D2

r

I I I I I I

, l:::~ 0 3

which has been derived in Section 2.1.3, and in order to avoid very inq)3

tricate matrix analysis, we will con-

T3 = K3D3

I I

(.U////////J/.)//////////////////////////////////////////~ I

¢4

Fig. 44, and consider groundwater

I I

4

= , ~' Q4

q94

sider a four-layer system as given in

T4 = K4D4

flow, caused by abstraction of wa-

I

ter by means of fully penetrating line

I

c,

~o - 0

wells with discharges Qi (i = 1, 2, 3,4).

Fig. 44. Four-layer system.

The ordinary differential equations for steady flow are, if we put 1 -- aii Tt ci

and

1 Ti ci + l

= ai(i+l),

848

Analytical solution methods

(2.3.6-2)

with i -

1--4 for Ti and i -

d2rpl

1 dq)l

dr 2

r dr

d2q92

+-

r dr

d2q)3

1 dq)3

~-~ dr 2

d2q94

dr 2

Ci"

-- ( a l l -t- al2)fPl -- a12q)2,

1 dq92

dr 2

1-5 for

r dr

= --a22q)l -k- (a22 -t- a23)q92 -- a23993,

(a) -- --a33q)2 -+- (a33 -i- a34)q93 -- a34q)4,

1 dq94 ~ -- -a44993 -at- (a44 -k-a45)q94,

r dr

and the boundary conditions are: lim

(Pi ( 0 0 ) -- O,

( dqoi~ r

r--+0 "~f, /

Qi

-

=

27rTi

-qi

(i -- 1 2 3, 4). ' '

For the solution of this boundary value problem, we apply the infinite Hankel transformation (Section 2.2.4-4). Then we have d2~0i 1 d99i H { --~-r2 + r - ~ - r ] - - l i mr----~0 in which ~)i(0/) -- f~x~

(rd~Oi "] _ --~-r/

r~oi(r)Jo(0/r)dr,

0/2~i (0/),

and use this expression to reduce the differ-

ential equations (a) to four algebraic equations in g3i (oe)" ( a l l + a12 -at- 0/2)(p1 -- a12932 -- ql, -- a22~1 + (a22 + a23 + 0/2)~2 -- a23933 -- q2,

(b)

--a33(P2 nt- (a33 "at- a34 "-i-0/2)(p3 --a34(P4 -- q3, --a44(P3 -t- (a44 q - a 4 5 + 0/2)(p4 -- q4.

The solution of these 4 linear equations for the 4 unknowns ~i (i - 1, 2, 3, 4) can be written in the form: ~ b l - D1 -D-' in which D and

O

~b2-

Di

D2

D

,

~b3-

D3

D

and

D4

~b4- ~ , D

(c)

are determinants of the 4th order:

all + a12 + 0/2 --a22 0 0

--a12 a22 -k- a23 + 0/2 --a33 0

0

0

--a23 a33 -at- a34 -k- 0/2 --a44

0 --a34 a44 nt- a45 --t- 0/2

Solutions, derived from known solutions

(2.3.6-2)

849

and D1, D2, D3 and D4 are derived from D, by replacing the elements of the columns 1, 2, 3 and 4 by the elements of the column matrix

q2

, respectively.

q3 q4

For instance, D3 becomes"

all + a12 nt- 0/2

--a12

ql

0

--a22

a22 nt- a23 nt- 0/2

q2

0

0 0

--a33 0

q3 q4

--a34 a44 -+-a45 -t-0/2

D3--

Now, if we p u m p only in aquifer 1 with a discharge Q, we have

Q ql

--

2:rr T1

,

q2--q3

--q4--0

and D3 becomes"

all + a12 nt- 0/2

--a12

ql

0

--a22

a22 + a23 nt- 0(2

0

0

0 0

--a33 0

0 0

--a34 a44 q- a45 + 0/2

D3--

--a22

a22 + a23 -t-- 0/2

0

0 0

--a33 0

--a34 a44 q- a45 -t- 0/2

=ql

= qla22a33(a44 -+- a45 -+- 0/2),

and thus

D3 q~ ~31 -- - ~ = --~a22a33(a44 -+- a45 -+- 0/2).

If we p u m p only in aquifer 3 with a discharge Q, we have

Q q3 -- 27r T~'

q l - q2 - q4 - 0 ,

(d)

Analytical solution m e t h o d s

(2.3.6-2)

850

and D1 becomes" 0

--a12 a22 + a23 + or2

q3

--a33

--a23 a33 + a34 -t- ot 2

0

0

-a44

0 D1 --

0

0

0

--a12

0

--a23

= q3 a22 + a23 + ot 0

0 --a34 a44 -t- a45 -k- ol 2

--a44

0 a44+a45+ot

= q 3 a 12a23 (a44 -Jr- a45 nt- t3/2),

and thus q313

(e)

D1 q3 -- - - a 1 2 a 2 3 (a44 + a45 -+- 19/2) • D D

-- ~

Comparing solution (d) with solution (e) we see that Q

1

1

O q3a 12a23 -- ~

q 1a22a33 - ~27rT1 T2c2 T3c3

1

1

T1 c2 T2c3

and thus ~31 ~ (P13"

As the inverse Hankel transform is essentially unique, we also have q931 - ~013. In the same way, we may easily find that ~012 - ~021, (/914 - - 9)41, (/923 - - q932' q924 - - q942 and (/934 - - q943• For non-steady flow the differential equations (a) Si ~)~Pi have to be modified by adding the -terms T , ~ (i - 1, 2, 3, 4) to the right sides. i The initial values are assumed to be zero: qgi(r, 0) - 0. Now, if we apply the infinite Hankel transformation with respect to r and the Laplace transformation with respect to t, we find a set of 4 algebraic equations for the 4 unknown functions qSi(o~, s), which is similar to the equations (b), provided that tY2 in each equation is replaced qi ( i - 1 ' 2, 3 ' 4) " by o/2 -+-/~2s, with/32 - ~Si and qi by .-7. The results (d) and (e) for steady flow now become, respectively" ql

~31-

- - ~ a 2 2 a 3 3 ( a 4 4 + a45-t-ol 2 -t- fl2s)

-

q3

a, a 3(a44 +

+

and

+

from which it follows that ~31 - ~13 and thus as the inverse Hankel and Laplace transforms are unique: q931 -q913. Also for other values of n, the reciprocity principle can be demonstrated in a similar manner for a n-layer system, using determinants for solving the differential equations transformed to algebraic equations, the order of the determinants being equal to n.

Solutions, derived from known solutions (2.3.6-2)

851

Example 32. In Fig. 45a a double-layer system is given, consisting of two leaky aquifers, separated and confined by 3 semiprevious layers. Both aquifers will be drained in succession by means of a steadily discharging pumping well that is screened only throughout the 11 aquifer in question. In the same way that has been used I I f I for a 4-layer system, we will deter,, S1, T1 mine the various drawdown distribu'5/'///J/J////////////////////////////////////////////////~ C2 tions, in the non-steady state, by applying Laplace and Hankel transfor$2, T2 mations with respect to t and r successively. First we consider Fig. 45a c where pumping with a discharge Q1 ~o=0 takes place in the upper aquifer, yielding a drawdown line ~oll (r, t) in the Fig. 45a. Double-layer system. Pumping the upupper aquifer and a drawdown line per aquifer. ~021(r, t) in the lower aquifer. The doubly transformed functions ~11 (ol, s) and ~21 (or, s) may then be solved from the two algebraic linear equations (cf. equations (b) for the 4-layer system)" Q1

=>

~r

( a l l -[--a12 -+-og 2 -+- f l 2 S ) ~ l l -

ql

a12~21 -- - - , s

(f)

--a22~11 + (a22 + a23 _+_ o/2 .qt_/~2s)~21 _ 0, DI D2 with the solutions" ~ll -- -D-a n d ~21 "- --D-, in which

a l l -+- a12 -+- Ot2 + O

--a22 -

fl2S

--a12 a22 -+- a23 + Ot2 -+- fl2S

( a l l 9- a12 -+- ot 2 -+- f 1 2 s ) ( a 2 2 + a23 q- ot 2 --[- f l 2 s ) -- a12a22,

or

D

-- ¢s 2

-Jr-(aot 2 -k d)s + Og4 -Jr-bot 2 -+- e

a - - fl 2 -k- fl 2, 2

with

b - - a 11 -+- a 12 + a22 nt- a23 ,

d -- (all +- a l Z ) f l 22 -k (a22 -+- a23)fi~

and

c - - f121 f12, 2

e -- a l l a l 2 -Jr- a12a23 -+- a l l a 2 3 .

(g)

852

(2.3.6-2)

Analytical solution methods

Further, D1--

q_.l. s

--a12

0

a22 -nt- a23 nt- 012 + fl 2s

all + a12 + 0/2 -t-- fl2s

q_l

--a22

0

D2-

_ q__}_l(a22 + a23 -k- 0/2 + / ~ 2 s )

and

S

ql - --a22. s

First we apply the inverse Laplace transformation to ~11 and ~21. Therefore, we write D as D -- cs 2 + (a0/2 -% d ) s + 0/4 + b0/2 jr_ e -- c(s + Xl)(S --I-x2),

in which a0/2 + d xl -t- x2 -

0/4 + b0/2 + e and

c

X lX2 -

(h) •

c

Obviously, X l and x2 are the roots of the quadratic equation c x 2 - (a0/2 -t-

d)x +

0/4 -t- b0/2 -t- e

-

0

(i)

(Xl and x2 are both positive, because all constants a, b, c, d and e are positive). Now ql (a22 nt- a23 -Jr-0/2 %. fl2s) ~11 (0/, S) =

and

CS(S + Xl)(S "-{-X2)

(j)

qla22

,~~ 2 1 ,~ o ~ , s ~ -

CS(S nt- Xl)(S "-i-X2)

The inverse Laplace transform of the function

1 f(s)

_

-- (s -+- X l ) ( S + x2)

1(1

1)

Xl - x2 s --1-x2

s --i- Xl

is 1

F ( t ) -- ~ ( e

Xl while L - l { Isf ( s ) } 1

XlX2

1 +

-x2t - e-Xlt)

(Section 2.2.2-2, equation (21)),

x2

-

1

-

f0 (e-X2r - - e - X l r ) d r ,

~1-~

X2

_Xl t - ~ Xl _x2t) e Xl -- X2 Xl -- X2 e

which can be evaluated to (Section 2.2.2-1, equation (8)).

Consequently, for the inverse Laplace transform of ~11 and ~21 we obtain: @11(0/, l) -- q1(a22 + a23 + 0/2) ( 1 + ~ - -X2 e CXlX2 _%

q lfl2 C(Xl --X2)

Xl - - X 2 (e-Xate-Xlt)

-xlt -- ~ Xl e Xl - - X 2

-x2t)

Solutions, derived from known solutions

(2.3.6-2)

853

and @21 (oe, t) -- qla22

1 -+- ~ X2 e

CXlX2

_Xl t

-- ~ Xel

Xl --X2

-x2t )

.

Xl --X 2

Inverse Hankel transformation gives the final solutionq-2-1L °c oe(a22 nt- a23 -4- 0l 2)

9911 (r, t) --

C

XlX 2

x

l+~--e X1

-C

X2 ~

- x 1t

-

X2

~ X1

Jo(oer) X1

XlX 2

Q2

] doe

-xlt -- e -x2t) doe,

~

X2 e

X 1 ~ X2

_Xl t

-

(95) Xl

---~~e

X 1 ~ X2

_x2t' ~

]

d~.

Secondly, we will derive the drawdown lines 9912(r, t) and (/922(r, t) in the upper and lower aquifer respectively, caused by p u m p i n g with a discharge Q2 in the lower aquifer (see Fig. 45b).

=~

r

X2

~Jo(oer)(e X 1 ~ X2

~o21(r, t) -- qla22 f0 ec oe Jo(roe) ( 1 + C

_x2 t

e

~

~o=0

S1, T1 5"//,,'2Y//////////////////////////////////////////////ZC 2

82, T2

~o=0 Fig. 45b. Double-layer system. Pumping the lower aquifer.

The doubly transformed f u n c t i o n s ~12 a n d ~22 can then be solved from the linear equations( a l l ~L_ a , 2 + 0l 2 J r - / ~ s ) ~ 1 2

-- a12~2 2 -- 0

- - a 2 2 ~ 1 2 -+- (a22 -~- a23 -~- 0/2 _4_ f i 2 s ) ~ 2 2 _

q2. s

and

(k)

The determinant D of this set of equations is the same as the determinant of the equations (f) and is given by equation (h): D - c(s + Xl)(S + X2),

854

(2.3.6-2)

Analytical solution methods

so q2a12

g912(0/, s) --

and

CS(S + X1)(S -at- X2)

O)

q2(all Jr- a12 -+- 0/2 _+. fl2s)

^

q~22(0/, S) --

CS(S "nt- X1)(S -'l- X2)

The inverse Laplace and Hankel transforms will yield, in a similar way to that in the foregoing problem:

rplz(r, t ) -

q2a12 fo ~ ~0/ ¢

XlX2

-xlt - ~ Xl - x 2 t ) du, Jo(r0/) ( 1 + ~ X2 e e Xl X2 Xl -- X2

q922(r , t) -- qZc f0 ~ 0/(all +Xlx2al2nt- 0/2) Jo(r0/) X2

x

-Xlt

1+ ~ e Xl

~

qzf12 fo ~ ~ ¢

X1

-

x2

Xl

0/

_x2 t

~ e

--

X2

] dot

(96)

Jo(r0/) (e -x't - e -x2t) d0/.

Xl -- X2

According to the reciprocity principle, q921 in equation (95) should be equal to (/912 in equation (96) if Q1 = Q2 and indeed the right-hand sides of both equations have the same integral form and differ only with respect to the constants before the integral signs. But if we assume that Q1 = Q2 = Q, we find q2a12 c

Q 1 1 2zr T2 T1c2 c

Q 1 1 2zr T1 T2c2 c

qla22 c

and consequently q912(r, t) - q)21(r, t) if Q1 - Q2 for all values of r and t. The steady state can be obtained by letting t approach to infinity in the solutions (95) and (96), which gives, with C X l X 2 - ot4 nt_ b0/2 + e (equation (h))"

qgll (r) -- ql

f0 o° 0/(a22 -+- a23 -+- 0/2) Jo(r0/) do~, 0/4 + b0/2 + e

(/921(r) -- qla22

~

oo

u Jo(r0/) du, ot4 q._ b0/2 + e

(m)

0/ + e Jo(r0/) du, q)12(r) -- q2a12 ~0 °° 0/4 + b0/2

~22 (r) -- q2

fo ~ 0/(all q- a12 -at- 0/2) Jo(r0/) du. 0/4 -Jr-b0/2 -}- e

Putting 0/4 -~- b0/2 -Jr- e -- (0/2 q_ yl)(0/2 -k- Y2), where Yl + Y2 - b and Yly2 - e and thus yl and y2 are the roots of the quadratic equation y2 _ by + e -- O,

(n)

Solutions, derived from known solutions (2.3.6-2)

855

we find"

1

1

1

0/4 4- b0/2 4- e

(0/2 4- yl)(0/2 4- Y2)

~yl ( - 0 y2 / 2

a22 4- a23 4- 0/2

0/4 4- b0/2 4- e

l (a22 4-a23 -- Yl Y l - Y2 0/2 4- Y l

a l l 4- a12 4- 0/2

1

0/4 4- b0/2 4- e

Yl - Y2

[all 4- a12 -- Yl 0/2 4- Yl

1)

1 4- Yl

0/2 4- Y2 '

a22 4-a23 - Y2) 0/2 4- Y2

and

a l l 4- a12 -- Y2'~. 0/2 4- Y2

)

Substituting these equalities in the equations (m), integrals of the form

4- k2 J0(r0/)do/ f0 °° 0/2 0/ occur, which m a y be replaced by Ko(kr) (Section 3.3-5, equation (214)). Thus, the equations (m) become"

(/911

ql ~{(a22 Y2 -- Yl

4- a23 - y l ) K o ( r ~ ) -- (a22 4- a23 -- y2)Ko(r~iC~)},

q)21

q912 --

(97)

q l a22 { Ko (r ~y-~) - Ko (r ~ / ~ ) }, Y2 -- Yl

q2a12

Y2 - Yl q2 q922 -- ~ { ( a l l y2 -- Yl

{ K0 (r ~y]-) - Ko (r ~y-~) }, 4- a12 -- y l ) K o ( r ~ )

(98)

-- (all 4- a 1 2 - y 2 ) K o ( r ~ / ~ ) } . Solution (97) is the s a m e as that found in Section 2.1.3, E x a m p l e 6, as Yl 4- y2 b -- a l l 4- a l2 4- a22 4- a23 and thus a22 + a23 -- Yl -- - - ( a l l -4- a12 -- Y2) a22 4- a23 -- Y2 -- - - ( a l l + a12 -- Yl).

and

Error functions and related functions

(3.1.1)

857

III. F U N C T I O N S 3.1. E R R O R F U N C T I O N S A N D R E L A T E D F U N C T I O N S

3.1.1. Error function X2 •

The function y -- e is an even function, whose graph, the so-called bell-shaped curve, is frequently used in the field of probability and statistics and forms there the basis of the normal distribution. Its integral is called the error function and is defined as: erf z -

x 2/ ~ foZ e -t2 dt.

(1)

The factor ~2 has been chosen because f o e

_t 2

dt - - g1 ,/-if and thus

(2)

lim erfz = 1.

z---~oo

Now the complementary error function, defined as erfc z = 1 - erf z,

(3)

thus becomes: erfc z -- ~ 2

f z cx~ _t 2

e

dt.

(4)

We write the integral of erfc z as ierfc z and define: ierfc z --

erfc(t) dt, f z x3

and n times repeated: inerfcz -- fzz ~ in- 1 erfc(t) dt

(n -

1, 2 , . . . )

(5)

with i °erfc z = erfc z. These repeated integrals of the complementary error function satisfy the recur-

rence relation: z in_l e r f c z +

i"erfcz-

/7

with i l erfc z -- ~ -

2

e -z2

.

~

1 in " -2erfcz

(n-

1 2 '

'"

..)

(6)

85 8

(3.1.1)

Functions

This relation can easily be verified by differentiating both sides of equation (6) with respect to z, making use of the derivative of i"erfc z, following from equation (5):

d inerfcz

-i~-lerfcz

~

( n - O, 1 2, ~

•

.

.)

(7)

o

dz Also d

d (erf z) dz

2

(erfc z) -

dz

e -z2

~

(8) "

A series representation for erf z may be found by integrating the Maclaurin series of e -z2" erfz -- ~2

foZ e -t2 dt

2 (z = ~

foZ ( 1 -

t4 t 2 -k 2!

t3!6

t----

)

dt

Z7

z3 z5 -~+~2!5

2 ~ erfz --

- ~2

3!7 t - " ' )

or

(-1)nz 2n+l

(9)

~/~ n=0 n!(Zn + 1)

Some special values follow from the foregoing formulas: 1 2 erf(O) - 0 from equation (1) or equation (9), erfc(O) - 1, ierfc(O) - g . ~ ~1,

1lt~,u):2er~"m'_ ~1 erfc(O)

1 etc., and, in general: - ~,

1

i"erfc(0)-

2.r(~'~ + 1)

erf(c~) = 1,

(n---I,0,1,2

. . . . ),

(11)

erfc(c~) = 0,

erf(-z)--erf(z),

erfc(-z)-

e r f ( - c ~ ) = - 1,

e r f c ( - ~ ) = 2.

(10)

1 + erf.(z),

(12)

(13)

The error functions (repeated integrals of complementary error functions included) occur frequently in solutions of non-steady groundwater flow in confined aquifers, especially in infinite or semi-infinite fields, where the argument of the error function takes the form 247" For instance, erf(z-~t) is a solution of the differential equation

02q9 __ f12 0q9 Ox2

Ot

with f12_

S KD '

as may easily be shown by applying equation (8).

Errorfunctions and relatedfunctions

(3.1.1)

859

So the solution for non-steady one-dimensional flow from an open straight boundary, caused by a sudden rise h of the surface water level, which is subsequently kept constant, is 99(x, t ) - - h erfc(2@t )

(Part A, 123.02).

We may verify that q)(x, 0) = 0 for x > 0, ~o(0, t) = h for t > 0 and = 0 for t = 0 and ~o(oo, t) = 0. A derivation of this solution has been given in Section 2.3.1-1, Example l a, by means of the Laplace transformation method, which resulted in the Laplace transform of the complementory error function with the argument 2~/7 ~ • L - l { 1} e-k'F s

--erfc (2--~)

and inversely:

L{erfc(~)}-Some other As

foOOerfc(+) e-'tdt-le-~'/7"s

inverse Laplace transforms, leading 1

L 1{7)-

1

to error functions follow here.

(see equation (20) of Section 2.2.2),

then L-l{

1 } _ e-at ,/s + a

(Section 2.2.2, equation (9)).

According to equation (8) of Section 2.2.2, we have

,,{ s ~/s, + a } Setting a t

=

/o, x/~1 r e -"r dr.

0/2, we may modify the integral in fo "/St e -~e d0/,

1 V/~ ~2

from which it follows that 1

L-l { s~/s + a }

--

1 erf(~)

(14)

860

(3.1.1)

Functions

and inversely: fo ~ erf(~a-t)e -st dt - ~

(15)

s~/s + a

Replacing s by s - a in

f(s)

of equation (15) means multiplying ea t erf (~'a-~)

L-1l (s _ 1a)~/-~ ] -- --~1

F(t)

by e at. (16)

As

s --a

=

1

t

~

a ,q/}(s - a )

we find from equation (16): s - a

] -- ~ 1

t- ~/de a t e r f ( ~ ) .

(17)

Also 1

~/'j - ~

~-(~cj- + ~/-~)

1

~/}-(s - a)

s - a

~/J-(s - a ) '

and thus, according to equation (16)" 1

(18)

L - l [ ,~cj-(,q/'}-+ ~/~)] -- eat e r f c ( ' ~ ) " From

~/'-s + ~/'d

s - a

s - a

s - a

we find from equation (17)" 1 1 L-1{%/~_{_%~] -- %/~

4rdeat erfc (~aT).

(19)

Integrating both sides of equation (14) with respect to k over the interval from k to infinity, we find"

L-

1

dk

--

Cx:)

k_~)

erfc\2

,-

Error functions and related functions

(3.1.1)

861

and thus:

1 e_~

(k__~

L-{~--~

} --2v~ierfc 2~/7)"

This procedure, repeated n times, yields the general formula: k L -1 , [ s - ~ l + ~ )-ek ~ ] -(4t)~"i~erfc(~--~]\/

(n - O , 1,2 . . . . ).

(20)

If we consider again the differential equation with the initial value and boundary values with the solution 99(x ' t) - h effc( 2v'~~x ) in this paragraph and we now solve the problem by means of an infinite Fourier sine transformation (Section 2.2.3-3a) with respect to x, we get the solution qg(x, t) - h

- sin(xot) exp Jr

which must be equal to h erfc(z~t). representation 1 sin(xot) exp

-

t dot,

u

-

~

So, we have found the following integral

t dot - -- erf 2

(21)

"

This equality may also be established by replacing sin(xot) by its Maclaurin series and then integrating this term by the term, resulting in the infinite series, given in equation (9). The integral

I--foterf(

2 ~k - ) dr'

which we encountered in Example 14 of Section 2.3.1-3, can be evaluated by taking the Laplace transform:

L{I} -- - L erf

(equation (8), Section 2.2.2)

S

---L s

1-erfc

2~/7

. . . . s s

s

e-k~

-

s2

s ~ ek4~'

according to equation (14). Now, with the aid of equation (20) with n = 2, we find

f0 t

4i r ct )/ k

862

Functions

(3.1.1)

In some problems, concerning staedy and quasi-steady two-dimensional flow, the integral I --

erf(aw)e -b2°fi do

fo ° occurs, which can be evaluated by differentiating I with respect to the parameter a" dI _ da

2

d

f0

e -azm2 •

e-bZw20) do) --

1

1

f0

e_ (a2+b2)o)2

do)2

~"~-(a 2 -k- b2) '

and thus

I m As I - 0

,f

a 2 da + b 2 = b ~ /|~ a r c t a n ( a~) + c .

for a = 0, we get c - 0, from which (23)

fo ~ erf(aco)e -b2°92 do) - b---~ 1 arctan ( a~) .

3.1.2. Polder function

In a great number of problems concerning non-steady flow in leaky aquifers, a certain combination of error functions appears, which may be called the Polder function, after the areas where a more or less constant level is maintained above the leaky aquifer. We define the polder function as:

l e 2 x e r f cy( X+) lYe - z+X e-~r f c ( X - y ) \Y , P (x, y ) -- -~

(24)

and the conjugate polder functions as" X

X

Pconj (x, Y) - ~1e2XerfC(y + Y) - ~1e-2XerfC(y - Y)"

(25)

Closely related to the polder functions is the indefinite integral: b2

f e x p ( - - a 2 t 2 - -~) dt (26) ~A- eZaberf(at + b ) + e _ 2 a b e r f ( a t _ t ) 4a

+constant;

Error functions and related functions

(3.1.2)

863

by differentiating the right side of this equation, its correctness can easily be verified. From equation (26) we may derive the following integral representations for the polder functions: exp

- a2t 2 -

dt - -a P ab, -z '

and

2 f °°exp(-a2t2 b~_~)dt_ 1 Pconj(ab _)b -

V

b

'z'

(27)

(28)

by first substituting u - 71 in the integral of equation (28) and next substituting the integral boundaries z, c~, 0 and z1 in the right side of equation (26). The derivatives of the polder functions with respect to x and y become, with the aid of equations (24) and (25): OP =2Peon j _ 2 1( Ox ~/-~" y

2x

OP Oy ' = x~~ y2 exp

x2 y2

( x2

)

(29)

y2)

y2

0 Pconj __ 2 P, Ox 0 Pconj _ 2 Oy ~/-~ (

y2

(30) ' (31)

x2 y2

y2).

(32)

Some special values follow from the foregoing formulas: P(0, y ) = 1

and

P(x 0 ) - - ~0 ' | 1 P(x, c x z ) = e -2x

P(cx~,y)=0

forx >0, forx--0,

(33)

f o r x >~0,

Pconj(O, Y) -- - e r f y Pconj (x, O) - 0

fory/>0,

Pconj(C~, y) - - 0

for y ~> 0,

Pconj(x, oe) - - e -2x

for x/> 0.

and

and

(34)

Further, by adding and subtracting the equations (24) and (25) we have:

(x ) and (x) e-Z~erfc Y - y •

P(x, y) + Pconj(X, Y) - e2Xerfc Y + y P(x, y) - Pconj(X, y) -

(35)

The polder functions occur frequently in solutions for non-steady flow in leaky aquifers, especially in infinite or semi-infinite fields, where the arguments x and y usually take the forms ~ and ~ with ~ 2 __ K Dc a n d f12 _ KSD "

864

(3.1.2)

Functions

x~ ) is a solution of the differential equation:

For instance, P ( ~ ,

ax 2

at '

~2

as has been shown in Example 5 of Section 2.2.2-3 where we found the solution (43) for one-dimensional groundwater flow in a semi-infinite leaky aquifer towards open water in which a sudden drawdown takes place. This solution was obtained by means of the Laplace transformation method, which resulted in the Laplace transform of the polder function: L-'{1 e-k47~]-

p(k~20,~

)

(36)

s

and inversely

fo~

p

(~~~) ,

e -st

dt -- 1 e_k~/T-s-~ s

Some other inverse Laplace transforms leading to polder functions follow below.

s ~/s + ~ e

/

~

, ~#~ ,

(37)

which can be shown directly by differentiating both sides with respect to k, making use of equation (36) and equation (31). Replacing s by s - r/in equation (36), we find" ------ e

e ot p

, ~

.

(38)

s--r/

If we differentiate the Laplace transform of the polder function in equation (36) with respect to s, we get: d 1 e_k~ d---~(s

) _

e-k,/7~

1 e_k47-~

-~

-

2S ~/S + r/

Applying the inverse Laplace transform to both sides of this equation, and with the aid of Theorem 7 and equation (13) of Section 2.2.2-1 and equation (37) of this section, we find:

~-1~1~e-~~/_,,(k As

1 __1(1 1) s(s+o)

~ s

s+o

2

,,~) + 2,~'con~( ,,~)

~9,

(3.1.3)

Error functions and related functions

865

we have from equation (36) and equation (14): L - 1{ r/-------~l s(s + e_k,/Tg-¢} -- ~ P (

, ~-~-7)-- ~ e - y t e r f c ( + ) .

(40)

3.1.3. Resistance function

Besides the polder function, another function related to the error function also plays an important role in analytical solutions of certain geohydrological problems, especially those involving entrance resistances (Section 1.5.1-3c). We therefore call this function the Resistance function and define it as: R ( x y, )

-- e2x+yZerfC(y + y . )

(41)

The relation to the polder functions follows directly from equations (35): R(x, y) - e y2 { P ( x , y) q- Pconj (x, y) }

and (42)

R ( - x , - y ) -- e y2 {P(x, y) - Pconj(x, y)}.

Special values are: R(0, y) -- ey2erfcy R(x, 0) = 0,

and

R(oo, y) -- 0,

R(x, oo) = O.

(43)

The resistance function is important in solutions for non-steady flow in semi-infinite areas bounded by open water, if the boundary condition there is a combination of head andflux (Section 1.5.1-3). In the case of entrance resistance, the arguments x respectively, in w h i c h f12 _- KSD and y usually take the forms" ~1 flx~/-O and ~ and the 1 resistance parameter 0 = f12 K2w2 [T-l]'

where w = entrance resistance ([T]). If the combined boundary condition refers to open water storage (Section 1.5.1-3b), the arguments of the resistance function become 1 fix v / 7 and .,/77 in which the fl2(KD)2 storage parameter a =

b2

[z-~],

where b = width of the outflow canal [L]. At first sight it may appear somewhat strange that also solutions for problems concerning open water storage are based on the resistance function R; however, the gradual rise of the level of the surface water actually represents a sort of resistance,

866

(3.1.3)

Functions

which may be shown by comparing 0 with a" substituting for the entrance resistance b w in the resistance parameter 0 the value 3-Y [T], which may be considered as a substitute resistance for the open water storage, we again find the storage parameter. The resistance function for open water storage R(7l f l x ~ , ~ ) is a solution of the differential equation

02q9 _. f12 099 at '

OX 2

and also satisfies the boundary condition b O~p (0, t ) = K D 0_.~

at

ax (o, t);

this may be verified by performing the necessary differentiations, but may also be established by solving the boundary value problem of 127.02 of Part A, with an initial drawdown h of the surface water, instead of an initial head h of the groundwater. We then have

02(/9 _. f12 0(/9

[h

07'

OX 2'

qg(c~, t) -- 0

~o(x, 0) -

b Oq9 (0, t) -at

and

for

0

x --0,

for x > 0, K D Oq9

ax (o, t).

Laplace transformation with respect to t gives:

d2~ dx 2

d~

~2sq~ - 0 ,

bsq3(O, s) - b h -- K D - z - ( O , s ) , Clx

q3(c~, s) - 0

with the solution" he-~X~ ¢(x,s)

=

47(47

+ 47)

Now we have to find the inverse Laplace transform of this expression, or e-kv G L -1

47 v7 +

} = F(t).

Therefore we write:

1

~ - - ~/'a +

- a)

1 s - a

Then we have: F(t)

-

L -1

- L -1 s - a

~/s(s - a)

"

Error functions and related functions

(3.1.3)

867

Replacing s by s + a means multiplication of F(t) by e -at, so

e -atF(t)

--

L -1

-

L -1

s

s~/s + a

'

or with the equations (36) and (37):

e-atF(t)- P

,~

-k-Pconj

,~

,

and thus, according to equation (42): F(t) - R(½ kv/-d, ~/-~), which means that L -1

--R

~/}-(~/7+ v/-~)

2 '

~-t

(44)

'

and the solution of our boundary value problem is:

'

2

'

Other inverse Laplace transforms, leading to the resistance function will follow here. Differentiating both sides of equation (44) with respect to k readily yields: -~} ~ g-1 { e,~/~-_~_

--

k2 ~2 ~_Texp(-~)-~R(k 1

,~/~).

(45)

The inverse Laplace transform of s(~+,/S) e-kV(;: may be obtained by writing

and using the inverse Laplace transforms of the equations (14) and (44): L-1 s(~/7 + ~/-d)

~

erfc(~-~)-

~

R(

, ~/~),

(46)

which corresponds with the solution for non-steady one-dimensional flow towards a straight canal with entrance resistance to and a sudden drawdown h of the surface water level (problem 126.02, Part A): ~0(x t ) - h e r f c (

fix

flx~/-O ~ / ~ )

,

2

,

In general, we have:

e-k~ f (s ) -

( ,/7

( 4 7 + ,/-& =

{

1 ( ,/7

1

} e -~

868

Functions

(3.1.3)

which leads to a recurrence relation between the Laplace transforms of the resistance function:

1 { e -k~

fn(S)---"~ fo(s)

=

fn--1 (S) } (n -- 1, 2,...)

(~f'~)n

with

e-k4~ ,/7+ ~ '

and from equation (45):

1

Fo(t) = ~ e x p

k2 (-- ~ - / ) - ~/dR( k~/-d2 ' ~a-D

n--l:

e-k4 ~: fl(s)

=

1 {e-k4 ~

47(,/7+ ~ ) = ~

,/7

} /0(s),

and thus: 1 {e-k4~s} F1 (t) -- - ~ L -1 ~/7

1

--~ Fo (t ) .

From equation (41) of Section 2.2.2 we have L -1

e k,F } 1 k2 ~ -- ~ / ~ exp ( - ~-~),

from which it follows that

in agreement with equation (44). n--2: e-k4 ~ f2(s)--

s(47+ 4~)

=~

1 {e-k4 ~ ~--fl(s)

~

}

s

;

so from equation (14) we find

1

k

Fz(t)--~erfc(~)-~R(

1

k~/~ ~-7) 2 ,

(equation (46)).

(47)

(3.1.4)

Error functions and related functions

869

n=3: e -k'/5

/3(s) = ,,/7(,/7 +,/a)

_

1 [ e -~,/7

/2(s)};

,/a / s,/7

from equation (20)

~(t)-

k ierfc ( ~ - ~ ) - - al e r f c ( ~ -k~ )

2

1 +-Ra

(k2v/-d, ~ )

,

(48)

and so on.

3.1.4. The M-function The M-function was introduced by Hantush and defined as: M ( u , ~) --

f

~ 1

- e - Y e r f ( o l ~ ) dy.

(49)

Y

This function occurs in non-steady state solutions for axial-symmetric flow in confined aquifers towards partially penetrating wells, the discharges of which are assumed to be uniformly distributed along the well-screens. For instance, continuous constant abstraction Q of water from a finite line well with length l on top of a confined aquifer of infinite thickness (522.03, part A) causes a drawdown distribution

{

qg(r, z, t) -- 87rQK-----7 M u, f12r2 u - - 4t

and

r

-M

( z_,)} u,

r

with

(50)

f12= Ss K'

which has been obtained as follows: The boundary value problem can be written as: 0299 1 099 0299 099 0t..___ ~ _~__r ~ _qL_ OZ2 __ /~2 Ot ' qg(oo, z, t) = 0 ,

lim (r 0~o r --+ O

-~r ) --

99 -- (p(r, z, t),

{

Q

99(r, z, O) = 0 ,

for 0 ~< z < l ,

0 2re K1

for z > 1, ~(r,O,t) OZ

=0,

~(r, oo, t) = O.

870

(3.1.4)

Functions

Laplace transformation with respect to t and infinite Fourier cosine transformation with respect to z yields (see survey of integral transforms, Section 2.2.7): d2~

t

dr 2

lim

r ---~O

ld~

(ot2 +/52s)0 _ 0,

~ - ~(r, ot, s),

r dr

(d~) r

- - ~

~

Q

2 rc g l s ot

,, qS(oo, ot, s) - 0

sin(lot),

with the solution" Q ~ ( r , ot, s) -

27r K l s o t

sin(lot)Ko(rv/ot 2 +

fl2s).

Inverse infinite Fourier cosine transformation gives" ~(r,z,s)

=

Q

jr 2 K l

fo o° _1 sin(lot)cos(zot) . _1 Ko(rx/ot 2 + fl2S) dot. ot

s

From equation (51) of Section 2.2.2 we know that the inverse Laplace transform - { ( ~/ot2 )} 1 ( ot2 ) L 1 1 Ko fir +s W ¢,otr . So q)(r, z, t) -

Q

c~ _1 sin(lot) cos(zot)W

2rc2KI

ot

t, otr dot

is the solution of the problem. Herein the well-function W is defined as"

c~2t

ot2 otr) 1 W(~--~,, _ -- f0 -7 x exp ( - /

ot2r2 41 ) d x

(see Section 3.2.2). The obtained solution 9(r, z, t) may be evaluated, bearing in mind that 2 sin(lot) cos(zot) - sin(zot +lot)--sin(zol--la), by evaluating two integrals of the type:

u2t I --

/0 ot

- sin(zot)

f0

-

X

exp(-x

ot2r2 41 )d/dot.

To get rid of the integration variable ot in the upper boundary of the second integral, we substitute y I =

fo

ot2r 2 4x, which yields:

- sin(zol) ot

exp(-- y

ot2r2 4y ) dydot.

(3.1.4)

Error functions and related functions

871

Now we interchange the order of integration and find with u I--

le_y y

/o

~lsin(zc~)exp cz

f12r2

4t "

r.)

-~ot 4y

2 dc~dy,

and from equation (21) of Section 3.1. l" I----

2

- e -yerf y r

.

dy - - - ~ M

(uz, -)

r

.

Replacing z by z + l and z - l successively, we readily obtain solution (50). Some special values follow from the foregoing formulas" From equation (49) we have: M ( u , -c~) -- - M ( u , M ( u , O) -- O,

(51)

~),

M ( u , c~) --

(52)

- e - y dy - E1 (z) Y

= exponential integral (see Section 3.2.1), M(0, or) -- f0 ° -1e -y e r f ( o t ~ ) dy, Y which is the Laplace transform

1

L[terf(ot~7) ]

for s - - 1 .

As L { e r f ( o t ~ ) } --

(equation (15)), $ 4 S -lt- Ol2

if follows by the integration theorem of Laplace transforms (Theorem 8 of Section 2.2.2-1) that

M(0, or) -- ot

=-ln

as 1 S~/S qt-Ol 2

-

[ In 4S q-Of2 --Of] °° ~/s +c~ 2 +oe

1

~/1 + c ~ 2 - ~ = 21n(~ +v/1 +~2),

or

~/1 +or 2 +or M (0, or) - 2 arcsinh ol.

(53)

This special value for the M-function leads, with t - oo and thus u - 0, to the steady state solution of the partially penetrating well on top of an infinitely deep

872

Functions

(3.1.4)

aquifer, substituting u = 0 in equation (50):

Q

{arcsinh(Z+l)-arcsinh(Z-l)}

q)(r, z) - 4zr K----~

r

r

(54)

(cf. equation (7) of Section 2.3.1). The drawdown distribution along the z-axis (r -- 0) for z > l, follows for the steady state from equation (54) by writing

cp(r, z)

--

I z+l v/(Z+/]2 /

~ lO n

-7-,

7+

4rc Kl

+1

Zr___.l/_~_ ~/(.~)2 + 1

Q___~ln { Z + l + x/(z + l)2 + r2 } 4zr KI z - l + ~/(z - / ) 2 + r 2 ' and thus Q

q)(O, z) -- 4zr Kl

ln(Z+l)

forz>/

z- l

Substitution of r = 0 in the non-steady formula, however, gives M(0, oo) and is senseless. Therefore, we apply the method of superposition as given in Example 3 of Section 2.3.1-1 for the non-steady state. The drawdown caused by a point well in the origin is equal to:

Q erfc(fl~) ~o(p, t) - 4zrK-----7

~o(r, z, t) -

Q

4rr K %/r 2 -Jr-Z 2

(Part A, 410.03)

or

erfc ( fl---~-x/r2 + z 2)

247

and by a point well in point P (0 ' zo) with strength 2Q. 21 d99 =

Q dzo

4zcKlx/r 2 + (z - z0) 2

erfc{ ~ fl - ~ v/r 2 -+- (z - z0) 2 }

Integration from - l to +l yields for the total drawdown:

qg(r, z, t)

--

4re Kl

l C r 2 -~- (Z -- Z0) 2"

x erfc{ + x / r Z

+ (z - zo)Z} dzo,

(55)

Errorfunctions and relatedfunctions

(3.1.4)

873

an alternative solution, c o m p a r e d with equation (50). Now along the z-axis (r = 0) for z > l we find: dzo

99(O,z, t) =

Z -- ZO

-----~Q

4rcK1

{ln( z+l)-fbh2

lerf(u)du}-

z-1

with

1 IA

/3

b , - ---~ (z + l)

and

2~/t

b2 -- 7---7~ (z - 1 ) .

2~/t

According to equation (9),

2 ~

1

(-1)nu 2n

- erf(u) = b/

n=O n?(2n + 1)'

from which 2

,t qg(0, z, t) -- K-----~ 47ro {lnt ~22

n + l - b 22n+1 } )

~(-1)n(b~

n!(2n -4- 1) 2

n=0

"

A series representation for the M-function is obtained by writing (cf. equation (53)): M(u, o~) -- 2 arcsinh o~ - f0 u -1 e - Y e r f ( o ~ )

dy

Y

= 2 arcsinh c~ - 2

-u 1 - e-X2erf (c~x) dx.

f0"/

X

1 2 1 ~ (ml)mx2m - e -x erf(c~x) -- X X m--0 mr.

=

2 ~ ~

(bl)n(otx)2n+l

n=0

nV(2n. + 1)

20t ~ ~ ( - 1)m+nol2nx2m+2n F.Z mVnV(2n +

m=0 n=0

f0 ~/-d -1 e - x 2erf(c~x) dx -X

" "

2olx/~ oo ~

1)

,

so

( _ l )m +n ol2n um +n

x~~ m=0 Z ~ m?n'(Zn + 1)(2m + 2n + 1) n=0

If we put k = m + n, then k varies from 0 to e~ and m = k - n, so n varies from 0 to k. The end result is, therefore, the product of a finite and an infinite series:

4~v~ ~

M (u, c~) = 2 arcsinh ot k

M~(~)

~-

o~2n

i

Z ~ (2n 4- 1 ) n V ( k - n)V n--0

"

(-u) ~

~=o 2k +-----i-M~(oe)

"

with

(56)

874 3.2.

(3.2.1)

Functions

E X P O N E N T I A L I N T E G R A L AND R E L A T E D F U N C T I O N S

3.2.1. Exponential integral Definitions 1

- et

E1 ( z ) - - f z ~

t dt -

f~

-1 t

e-Zt dt

(1)

= exponential integral of order one. (2)

(n - 0, 1, 2 , . . . )

E n ( z ) -- f l ~ tn1 e_Zt dt

= exponential integral of order n.

No(Z) --

f

~

e -zt at - - -

1 [e_Zt

]1~ - -1 e-z

Z

(3)

Z

Derivative and indefinite integral: dEn(z)

f

dz

f

~

t e_Zt dt tn

- - E n - l (Z)

(4)

(n -- 1, 2 , . . . ) .

(5)

(n -- 0, l, 2 . . . . ).

E n ( z ) dz - - E n + I (z) + c

R e c u r r e n c e relation O0

1

En+l(Z)- f

e

-zt

1 f~

dt - - ----n J1

n

n

e -zt dt-"

lde-Z,_ l_z Z/l l_ztd, '

tn

n

n

tn

and so"

1 -{e -z g/

-zG z }

(n - 1, 2 , . . . ) .

(6)

Series representation

Integration of the Maclaurin series of 71 e - t term by term between the boundaries z 1 which, when integrated, yields and cx~ encounters difficulties with the first term 7, dE1 In t and so gives no significant value at infinity. Therefore, we develop ~ in an

(3.2.1)

Exponential integral

875

infinite series and then integrate term by term: dE1 = _ _ e1- Z dz

= _ ~v-~ .( - - 1 ) n z n - 1

z

~

1

=

n!

n--O

~

Z

(--1)Z n-1 n!

n= 1

c~ ( _ 1)~z ~ El(Z) -- - lnz - E n " n! + c . n= 1

It can be proved that c - Euler's constant y with negative sign and thus we have

E1 (Z) -- - - l n z -

E n--1

(_l)~z ~ - Y n n!

(7)

1,2,...),

(8)

with y - 0.5772156649 . . . . S p e c i a l values are: E n ( c ~ ) -- O

(n-O,

En (0) --

t -n dt -

~ -n+l

1

=

fort/

-n+l

1

> 1,

SO

En(0)--

(n--2,3 t/

m

1

..).

(9)

~ "

The exponential integrals occur in solutions for n o n - s t e a d y radial flow of groundwater in a c o n f i n e d aquifer towards a f u l l y p e n e t r a t i n g well, as has been shown in E x a m p l e l b of Section 2.3.1-1, where such a well had a constant discharge Q. It was shown there that the exponential function El(~2~2) was a solution of the 4t differential equation 02~

}

Or 2

1 0cp _ f12 0~o r Or

Ot

which solution was obtained by means of the Laplace transformation method, yielding an inverse L a p l a c e transform, that led to the exponential integral:

1

k2

-

S

and inversely:

L{ ,t tJ

f0 ,t tc

s

•

876

(3.2.1)

Functions

In a number of solutions for steady flow in t w o - d i m e n s i o n a l problems concerning straight surface water boundaries with entrance resistance, the exponential integral also plays a role, particularly as part of a complex potential function X2 (Section 2.2.5-1, equation (136)) and multiplied by the exponential function with the same argument: F ( z ) -- e a z E l ( a z )

with z - x + i y. We write F ( z ) --

f

~ 1 z

- e -t+az dt t

and substitute t - az + p z , from which we find the integral representation: fo ~

e-ZP d p -- e az E1 (az) a+p

(11)

.

As e -zp - - e -(x+iy)p - - e - X p { c o s ( y p ) - i s i n ( y p ) } , the real part of F ( z ) becomes: Re[eaz(E1 ( a z ) ] - fo o° e-xp cos(yp) dp, a+p

and the imaginary part: Im[e az E1 (az)] -- -- f0 ~ e-Xp sin(yp) dp. a+p

An example has been given in Example 11 of Section 2.2.3-3, where a solution for steady flow between two lakes with different levels on top of an infinitely deep leaky aquifer (resistance c) was obtained by means of an infinite Fourier sine transformation (equation (93)). In this solution for the head ~o(x, z), the integral fo ~

Kc

1 +Kca

sin(xot)e -z~ dot

is, according to equation (11), equal to [ (Y)(Y)] Im exp ~cc E1 ~ c

'

if y -- x + i z, or equal to -iy

1

(3.2.2)

Exponential integral

877

as, in general, 4~ = Kq9 is considered as the real part of a complex function ,(2. Laplace transforms leading to exponential integrals may easily be derived from equation (11) and the definition of the exponential integral of order n (equation (2)). Replacing z by s and p by t in equation (11) we immediately find: L

{1} t+a

(12)

-- easEl(as)

and, in general,

/ 1J/0 e

L (t + a) n

(t + a) n dt,

which gives, after substituting t + a = pa and using equation (2): 1 ]_ L (t + a) n

3.2.2. Hantush's

e as an_l En(as)

(a > 0) (n = 0, 1, 2 , . . . ) .

(13)

well function

Definition

W(r, p) --

-exp x

- x

dx

4x

and also ' if we substitute y - 24x with u W ( r , p ) - - f ~ l -ye x p ( - y -

-

(14)

p2

T~r,

p2) dy.

(15

Equation (15) is the definition of the well function given by Hantush. In this work the definition according to equation (14) has been chosen as the standard form, because of the separation of the variables r and p, which in equation (15) are combined in u. However, for special cases it is convenient to use equation (15). Series representation We expand the factor e x p ( - ~ ) in equation (15) in its Maclaurin series and interchange the order of summation and integration: W(g,

/o) - -

(-1

_ e_y

Y

-E n=O

n=0

n

dy

n! y - n - 1e-y dy.

n!

878

(3.2.2)

Functions

and with y - ux"

°° (__l)n()2n foe x - n n! 2P u-" n=0

w(r,p)-~

1e-UX d x ,

and thus with the aid of equation (2) and as

( 2 ) 2nu-n-(p2]n-rn'\4u,1

OO

(16)

.gn En+l

~

n--0

rt!

-~t'

while 1

(n -- 1, 2 , . . . )

En+l(U) = - { e -u -uEn(u)} 1l

according to equation (6).

Special values W ( - r , ip) -

f0-r -x1e x p ( p- 2x) +

1 - fu ° yeXp (-

dx

a-Tx

/92 dy y + ~y)

oo 72n

p2

(17) n=O

W(oo, p) --

f0

- exp

x

(

- x

4x

p2 at dx - f0 ° t1 exp ( - -~-7)e-St

for s -- 1, which means that W(oo, p) equals the Laplace transform

1

p2)

L { t exp ( - ~ x - x } -- 2K0(p~/7) (see Section 2.2.2, equation (47)) for s - 1; so W(oo, p) - 2K0(p),

(18)

w(o, p) -o,

(19)

W(r, ~ ) - O.

In some problems the arguments r and p are

p2 U

~

~

4v

f12r2 4t

r respectively. Then and ~,

Exponential integral

(3.2.2)

879

which is independent of X; so, if ,k tends to infinity then p approaches zero and we have W(r, O) - fu °~ -1 e -y dy - E1 (u). Y An interesting result follows if r equation (15) we have ,P)--fp~

lexp(-Y

y

(20) p2

~; then u -- ~ becomes u -

- P2)dy

_--f o ~ Yl e x p ( _ y _ ~ y ) _ f ; / 2 1a0 = w(oo, p)-

w

~ and from

-~,p

yeXp(-

y-

p2

~ y ) dy

,

according to equation (14). So 2W(~, p) - W(c~, p), and thus, with equation (18): -

The well functions play a role in solutions for non-steady flow of groundwater towards wells in confined and leaky aquifers; in general, the wells should be point wells or partially penetrating line wells. Only in the case of a leaky aquifer will a fully penetrating well yield the well function as the solution, as has been shown in Section 2.2.2-3, Example 6, where it was found that: Q0

q)(r, t) -- 4rcKD

w r/t

with 1 / - ~1 , as a result of the

L-l{1Ko(k~/ sS

inverse Laplace transform"

+ 7/)] - -~1 w ( ~ t ,

k,/-~).

(21)

880

(3.3.1)

Functions

3.3. B E S S E L F U N C T I O N S A N D R E L A T E D F U N C T I O N S

3.3.1. Bessel functions J and Y The ordinary differential equation dZY 1 - -l d y -dx -~x~x

1)2 )

+ ( 1--

--0

~

(1)

y

is called B e s s e l ' s differential equation. The parameter v may be any arbitrary number, real or complex. The solutions y l ( x ) and yz(x) of equation (1) are called Bessel f u n c t i o n s .

To solve equation (1) we write d2y + . _lA dy = dx 2 x dx

1 d ( d xy ) x dx

~xx

and assume that a solution can be represented by an infinite power series of the form: oo

y -- x v (ao + a l x nt- a2x n--0

Then

1 d(dy) x dx

x -~x

- Z

an(n

+ 1))2xn+V-2

n=0

as may easily be shown. These expressions substituted in equation (1) yields, after dividing by x ~" oo

oo

co

n=O

n=O

n=O

This equation is valid if the sum of the coefficients of each power of x is equal to zero, which gives for the coefficients of x -2" aov 2 - aov 2 -- O, of x

-

1. a l ( v -t- 1)2 _ a l i)2 - - O,

o f x °" a 2 ( v

-Jr- 2 ) 2 -k- ao -- a2 v2 - - O,

of x 1" a3(v -Jr-3) 2 q-- al -- a3 v2 -- O, of X 2" a4(v nt- 4) 2 nt- a2 -- a4 v2 -- O, of X 3" a s ( v + 5) 2 + a3 - asv 2 -- O, of x 4" a6(v -+- 6) 2 q- a4 - a6 v2 -- 0

(3.3. l)

Bessel functions and related functions

881

and so on. It follows that a0 may be chosen arbitrary, that a l - 0 and consequently a3 - O, a5 - O, . . . , azn+l - 0,

ao a2 -- --4(V + 1)'

a4

a2

ao

8(1) + 2)

4. 8(v ÷ 1)(1) + 2)

a4 a 6

a2n

- -

_

ao

12(v + 3)

~ .

4 . 8 . 1 2 ( v + 1)(1) + 2)(v + 3)'

a0(-1) n 22"n !(v + 1)(v + 2 ) . . - ( v + n)

Now we may write the solution y ( x ) as:

{

(2 x)

y -- aox v 1 -

(½*)4

1!(1)+1) +2!(v+1)(1)+2)

(+ X) 6 3t(v + 1)(v + 2)(v + 3)

+-.-}.

The denominators in this expression are related to gamma functions, because (v + 1)(v + 2 ) . . - ( v + n) --

F(1)+n+

l)

r ( v + 1)

as will be shown in Section 3.4.1, equation (5). With a0 chosen as 1 ao-

2 ~ F ( v + 1)'

the solution for y is called the Bessel function of the f i r s t k i n d a n d order v and is denoted by J ~ ( x ) , and we have:

oo ( - 1 ) " ( ~ 1 X) 2n+v Jr(x) -- Z

n--0

n ! F ( n + v + 1)

t ){ 1

-

-~x

v

1

F ( v + 1) -

(+ X) 2

,>4

(~

X

l ! F ( v + 2) + 2 ! F ( v + 3) + " "

/

(2) "

Replacing 1) by -1), we find:

oo ( _ l ) n ( l x ) 2 n - v

(3) n=O Since Bessel's equation (1) involves !)2, the functions J~ and J_~ are solutions of the equation for the same v. If v is not an integer, they are linearly independent,

882

(3.3.1)

Functions

because the quotient of the first terms ,

(g /-'(v -+- 1)

and

/ - ' ( - v -+- 1)

of equation (2) and equation (3), respectively, is not a constant or equal to zero or infinity. This yields the following result: If v is not an integer, the general solution of Bessel's differential equation (1) is

y(x) = A Jr(x) + BJ_v(x).

(4)

Integer values of v are frequently denoted by n and Bessel's differential equation of integer order becomes: d2y ldy ( n 2) m--~+-x~xx dx + 1-~-~ y--O.

(5)

Replacing the term number n in equation (2) by m and the order number v by n, we have f r o m / - ' (m + n + 1) -- (m 4- n)! (see Section 3.4.1):

J~(x)

cx~ ( _ l ) m ( . ~ X 1) X--"

A...,

m--O

2m+n

mV(m + n)) "

which is a solution of equation (5). J_,,(x) is also a solution of equation (5) dependent on Jn(x), as can be shown as follows: If we replace n by - n in equation (6), we get the factor (m - n) ? = F (m in the denominator and because F ( ~ ) = -t-c~ for ~ - - 0 , - 1 , - 2 . . . . , all m = 0, 1, 2 . . . . , ( n - 1) become zero and the series (6) begins with m yielding: (_l)m(2

J-n(X) -- Z

(6)

"

but is n + 1) terms -- n,

x) 2m-n

m!(m -- n)!

m=n

Setting m -

n = k, we have

c~ ~=0

)2k+n (k + n)!k!

and comparing this series with the series (6) we see that

J-n(x) = (-1)'~Jn(x),

(7)

which proves that for integer v = n, the Bessel functions J,z (x) and J-n (X) are linearly dependent and do not form a fundamental system of solutions of equation (5).

(3.3.1)

Bessel functions and related functions

883

We shall now obtain a second independent solution, starting with the case n - 0. Bessel's equation of zero o r d e r becomes: d2y ~ - -1 dy + y - - O -dx --5- x ~

(8)

and a first solution yl follows from equation (6) with n - 0 "

cxa 1 X) 2m Yl - J o ( x ) - ~ ( - 1 ) m ( 2 m=0

(m!)2

(9)

"

A second solution Y2 which is independent of Yl is y2 - yl" u(x), with u ( x ) a still unknown function of x, not equal to a constant. dy2 du dyl dx = yl ~xx + u dx

d 2 Y2

and

d2u du dyl d2yl dx 2 = yl ~ + 2 -dx - dx + u dx 2 .

These expressions, substituted in equation (8), give:

d2u

(dyl

Yl d---~+

yl)du

2--~-x + - -x

(d2yl

1 dyl d - 7 +-x--~-x +

d--xx+

As Yl is a solution of equation (8), we find, with p dp (dyl Yl) Yl ~--~-x-[- 2--~x - [ - -x- P - 0 ,

dp ~+2 P

or

yl)u -- O. du°

dyl

dx +m_0 Yl x

with the solution p

m

du

c

dx

xy 2

and with c being an arbitrary constant.

y2 __ { Jo(x )}2 _

,2(

--

t2

{ ( 12 ) 1 --

q-

~ x

1

(gX) 4 -~ (2!) 2

{ ,

(2!)2

So y2 is of the form: y2

_ 1 + A2 x2 --}-A4 x4 nt- A6 x6 Jr-...

and x y 2 -- x + A e x 3 + A 4 x 5 -+-... ,

2

l X)6 o

(3!) 2

o

o

884

Functions

(3.3.1)

while, with c -

1"

du d--7 = p -

1

1

xy 2

= - + B lX + B3x 3 + Bsx 5 + . . . x

and thus" 1

2

1

1

u(x) -- lnx + ~ BlX +-~ B3x 4 +-~ BSx 6 + ' ' ' . Now Y2 -- Yl • u(x), so the second solution y2 is of the form" Yl ln x + b2x2 -at- b4x4 -Jr-b6x6 -+-"'.

Y2-

If we substitute this expression for y2 and its derivatives in equation (8), we find

1 d(dy2) x

x dx

-~x

or

+y2--0,

2 dyl (d2yl 1 dyl ) x d---7- + 'd-TTx 2 + - x -~-x + y l In oo

(x)

q.Z(2m)2b2mx2m-2nr-Eb2m x2m - 0 . m=l

m=l

The term with ln x disappears because yl satisfies equation (8), and from equation (9) we may derive ~2 dyl d~" Then the equation which enables us to solve the constants b2m becomes"

oo (_l)m(lx) 2m-2 -

oo

oo

-+-E (2m)2b2mx2m-2 q- E b2mx2m --O.

-

m=l

m=l

m=l

Now again, the sum of the coefficients of each power of x is equal to zero from which we may solve" (~)2 b2 --

,

1

( )1 41+$1 b4 - - ~ (2!)2,

(~)61+~+~ b6 - (3!)2

1

etc '

"

So

Y2 -- Yl l n x +

~x

(2!) 2

1+ ~

q- (3!) 2

~

-~

•

This second solution is called the Bessel function of the second kind and order zero and is denoted by Yo(x)" oo

Y2 -- g 0 ( x ) -

Jo(x)lnx - E m=l

1x)2m (

(-1)m (2 2 (m!)

)

1 + ~1 + 51 + " "

+ m1 " (10)

(3.3.1)

Bessel functions and related functions

885

Now we may write the complete solution of equation (8) as: (11)

y(x) -- A1 Jo(x) + B1 Yo(x).

In most works on Bessel functions another solution, due to Weber and denoted by Yo(x), is used, which will be followed in this work. So the general solution of the differential equation of Bessel of order zero (equation (8)) is:

y(x) -- AJo(x) + BYo(x) Yo(x)---

In

7r

x

with

(12)

+ y Jo(x)

(m!) 2

m---1

1 1 l+~+~+...+--m

•

1) '

(13)

in which y -- Euler's constant = 0.5772 . . . . Solution (12) is identical with solution Jr BI and A - A 1 - 7 2 (ln 2-- ?' ) B 1. (11) provided that the arbitrary constants are B - ~In a similar way, we may obtain the second solution of Bessel's equation of integer order n, Yn(x) and the complete solution of equation (5) is:

y(x) -- AJn(x) + BYn(x).

(14)

J,(x) has been given in equation (6) and Y,,(x) may be obtained by substituting Y2 - - J n ( x ) l n x

+ b2x2

-Jr- b 4 x 4 -k- " "

in equation (5), similarly to the determination of Yo(x). The series representation of Yn(x) is rather complicated and will not be given here because the Bessel functions of the second kind, used in this book, are limited to Y0, Y1 and Y2, which may be defined in another way. As we have seen, the second solution of Bessel's equation (1) has been defined differently, depending on whether the order v is integral or not. To provide uniformity, it is desirable to adopt a form of the second solution which is valid for all values of the order. For this reason, a standard second solution Y,,(x) has been introduced, defined for all v by the formula: Y~ (x) =

c o s ( v J r ) J ~ ( x ) - J_~(x) sin(vrr)

and

Yn(x) = lim Y~ (x). ~,,

(15)

For noninteger order v, the function Y~(x) is evidently a solution of Bessel's equation because J~(x) and J_~(x) are solutions of that equation and as they are linearly independent, also J~(x) and Y~(x) are linearly independent. Furthermore, it can be proved that the limit in equation (15) exists and Y,,(x) is the second solution of equation (5). Here we will show that the previously found series for Yo(x)

886

(3.3.1)

Functions

(equation (13)) equals the limit of Y~(x) if v approaches zero. Substituting v = 0 in the expression for Y~(x), both the numerator and the denominator tend to zero, so that we must apply the rule of l'H6pital and we have: Y0(x) -- lim

,-+0

OJ-v(X)

cos(vrr) og,(x) _ Jr sin(vrc)J~(x) 0v

Ov o

Jr c o s ( v z r )

From equation (2) we can derive the derivative of Jr(x) with respect to v, making use of the standard derivatives:

__ d (az) _ aZ lna dz

and

d{ 1 } ~

F(z)

-

F'(z)_

~(z)

F2(z)-

F(z)

(psi function, see Section 3.4.1). cx~

OJr(x) _ Jr(x)In ( 1 ) Ov ~x

1 X) 2m+v

(--l)m(~"

-- Z

m=0

7r(m + v + 1).

m.lF(m + v + 1)

Equation (3) gives: oJ_.(x)

Ov

=-J_

(1)

(x)ln ~ x

1

+

,~2m-v

~ (--1)m(2.x]7_/m!F(m -1) lp(m m=O

- v + 1)

and thus

v,,t' "~ 1 lim {OJ~,(x)

.,,x,

Jr ~-+o

OJ_v(x)}

Ov

2j0Cx ) In ( ~ x ) -

Ov

2 ~

(-1)m(1x)2m

-

tp(m+ 1).

m--0 In Section 3.4.1 it will be shown that

1

1

1

~p(m + 1) -- -?' + 1 + ~ + ~ + - - . +--m and thus Y0(x), here obtained as limiting value of Y~(x), corresponds with Yo(x) of equation (13). This result means that the most general solution of Bessel's equation (1) for all values of v is y ( x ) -- A J r ( x ) + BYv(x),

with Jo and Y~ according to equations (2) and (15), respectively.

(16)

(3.3.1)

Bessel functions and related functions

887

Some relations of the Bessel functions with circular functions follow by inserting v -- ~1 and v -- - 7 1 in the series for J~(x). From equation (2) it follows that ~(-1)n(½x)

J1/2(x) --

2n+l

-?zi-F(n 7 3_~

n--O

~{

--

1

(lx)2

]-,(3)

r(~)

t

1

(:X) 4

/

2!/"( 7 )

This can be evaluated to (see Section 3.4.1):

¢-~X(

X3X5

J1/2(x)-

X

) ¢ 2

3! t 5!

reX sinx.

In like manner, it can be shown that

lxv/2x

~COSX,

and from equation (15) COS (2)Sl/2(x) - J-1/2(x) sin ( 2 ) - --J-1/2(X)

Y1/2(x) -

and

Y- 1/2 (x) = J1/2 (x).

Summing up, we have

sinx,

J1/2(x) --

¢5

J-1/2(x) --

cosx,

Y1/2(X) -- --

Y-1/2(x) --

2

~ cos x, 7rx

2 7rx

sinx.

(17) (18)

Some special values of the arguments give the following results: J n ( - x ) is obtained by multiplying the series for Jn(x) (equation (6)) by ( - 1)2m+n = ( _ l)n, SO that Jn(-x) = (-1)nJn(x),

(19)

for instance, J o ( - x ) = Jo(x),

J1 (--x ) = - J1 (x ).

J. (o) - / o

/

1

for all values of n --/:0, for n-= 0; so J 0 ( 0 ) = 1,

r.(o) = -e~.

(20)

888

Functions

(3.3.1)

and for all n" Jn(oo)

-

0

and

Yn(oo) -

(21)

O.

Derivatives and recurrence formulae of Bessel functions Using equation (2) we have: ( _ l ) m ( + x) 2m+2v . 2 v

xVJv(x) -- Z

m t F ( m + v + l)

m--O

Then, as F(m + v + 1) -- (m + v)F(m + v)" oo

1 x ) 2 m + 2v-1 . 2 v . 1

d {xVjv(x) } _ y ~ ( - 1 ) m 2 ( m + v)(~ dx m!F(m + v + 1)

2

m-'O

o0 ( _ l ) m ( l x ) 2 m + v - 1

= xV ~

m t F ( m + v)

m=0

which means that

d {x~Jv(x) } _X~Jv_t(x)" dx

(22)

Also oo m--O

(-1) m

1 x ) 2m . 2-v

m t F ( m - + v ; 1)

2--P the first term being the constant F(v+l)"

d { x -v J v ( X ) } _ Z

dx

cx~

So

1 ( __ 1)m 2m(.ffx) 2m-1 2- v l

m=l

m t F ( m + v + 1)

the numbering of the terms beginning with m - 1. Now m

mt

1

(m-

1)t

and setting n - m - 1, we get cxz

dx

t£--O

1 X) 2n+l

ntF(n + v + 2)

d {x_ v J,,(x)} - - x -~ gv+l ( x ) . dx

or

(23)

(3.3.1)

Bessel functions and related functions

889

Performing the differentiations in the equations (22) and (23), we have 1)X v-1 J r ( x ) "31-X v d Jr

(x) ---X J v - l ( X ) dx

and

- 1)X_V_ 1 j v ( x ) "{- X-- v d Jr (x) = - x - v J , , + l ( x ) , dx

from which the following recurrence f o r m u l a e can be obtained: dJ~(x) dx d Jr(x) dx

v Jr(x) = J.-l(x)

~

(24)

,

x

-- - Jv+ l (x ) -~-

1)Jr(x)

(25)

x

Adding these equations we find: d Jr(x) dx

1 2 J,-1 (x)

1 ~ J,+l (x)

(26)

and subtracting 21)

J r - 1 ( x ) @ Jv+l ( x ) -- ~ x

Jr(x).

(27)

The corresponding formulae for Y. can be obtained from the definition for Y,,, given in equation (15). For instance" cos(vrr) d sin(vrr) dx

dx

1 d sin(vrr) dx

cos(vrr) sin(v~)

1 sin(vzr)

~xVJv-l(X)

+ ~x"J-(,,-1)(x)

from equations (22) and (23). From equation (15) we find cos {rc(v -- 1 ) } J , _ l ( X ) - J-(,-1)(x) F._l(x)

-

sin {Tr(v - 1)} - cos(vrr ) Jr-1 (X) -- J-(v- l) (x) sin(urr) from which it follows that d

dx In like manner, we find __d { x - " r , , ( x ) } dx

-

-x-"z,,+l

(~)

1'

890

Functions

(3.3.1)

These equations are identical to the corresponding equations for J~ (x) (equations (22) and (23)) and as the recurrence formulae for J~(x) are based on these formulae, we may conclude that the derivatives and recurrence formulae for Y~(x) can be obtained by replacing J by Y in the equations (22) to (27). Indefinite integration of the equations (22) and (23) yields:

f f

x v j v _ l (X ) dx - xUJv(x) + c,

f x~Yv_l(x) dx - x~Y~(x) -t- c,

(28)

x - v Jr+ 1(x) dx -- - x -~'J.(x) + c,

(29)

f x-VYvq_l (X) dx -- - x -~Y~(x)+c.

The Wronski determinant, in short referred to as the Wronskian of the general solutions J~ and Y~, can be written as

J~ Wx{J.,Y

} -

Y~

dY~

dJ.__..~v d Y._.~v - J r dx dx

dx

dJ~

Yv d x '

and is a criterion for testing the linear dependence (W~ - 0) and independence (W~ ¢ 0 ) o f the solutions J~(x) and Y~(x). To evaluate Wx we write equation (1) as

1 d (xdY (1 xV-~22) y -x dx d--x'x) + -

0,

replace y by Yv and multiply by J~, yielding a first equation; next we replace y by Jv and multiply by Y~, giving a second equation and subtract the second equation from the first. Then, we get

d (xdYv' ]

d (xdJv

-0.

Integration with respect to x gives

f

-dT)-fr d

~ d(x dYv

(x d Jr

or by partial integration, dYv

x Jr dx

f

dY~ d Jr dx - x Y~ d Jr +

x dx dx

giving dYv

x Jr dx

d Jr

xYv dx = c - const.

~

f

x d Jr dYv dx - c,

dx dx

Bessel functions and related functions

(3.3.1)

891

So, w~{J~,r~}

C -.

-

X

The constant c can be determined by evaluating Wx with the aid of equation (15)"

dY~ J~ dx

dJ. 1 (j_ Y~ dx = sin(v Jr)

dJ. dx

j

dJ_v ) dx "

Assuming x very small, we may approximate the four functions by considering only the first terms of their series: ,

J~(x) ~

J_~(x)

F ( v -t- 1)' v--

' (~i x ) 2F(v) '

dJ~(x) dx

and

f'(1 - v)' (21 X ) - V - 1

dJ-v (x) dx

2F(-v)

Thus, for small x

'

21 / - ' ( 1 - v' ) 2 F ( v )

Wx ~ sin(vzr)

x

'

F(v + 1)2/-'(-v)

/

As

F(1-v)F(v)--

sin(vzr)

(see Section 3.4.1) and replacing v by - v also F(1 + v)F(-v)

-- -

7/"

. sln(vzr) '

we find Wx ~ 7r--7 2 for small x " However, we found for all x W~- - x' c from which we may conclude that after multiplication of the series all terms vanish exept the first ones, and that c - 2. Furthermore it is easy to show that, using equation (25) 7/"

dYv Jv dx

d J,, Yv dx = J"+ l Y~, - J~ Y~,+l ,

which gives the final result" 2 w ~ { J ~ , r,,} -

-J~(x)r,,+~(x)

+ Y,,(x)J,,+~(x) ~X

(30)

Examples for small orders"

dJo(x) dx

= - J 1 (x),

dro dx

= -Yl(x),

(31)

892

(3.3.1)

f x Jo(x) dx - x J1 ( X )

d J1 ( x )

dx

-I- C,

1 J~ (x), x

= Jo(x)

Functions

f xYo(x) dx - xYI(x) + c,

(32)

dY1 (x) = to(x) dx

(33)

2 Jo(x) + J z ( x ) -

J3/2(x)

1 r~ (x), x

2

r0(x) + rz(x) -

- Jl(X), x

1 -- - J1/2(x) x

-

- rl (x), x

(34)

J-1/2(x),

according to equation (27) and from equations (17) and (18):

Jw (

-- ~ / ~ x ( ~T-M cosx

)

x

Y3/2(x)--

,/ xt

sinx +

cos/ x

(35)

.

5 7 We may continue in this fashion with v - q-~, +~, etc., and conclude that for these values of v the Bessel functions Jv and Yv are elementary functions. From the series for Yo(x) given in form (13) we now can derive the series for Y1(x) by means of equation (31)"

Y1(x) = --

In

2 ~ +-zr

m=l

2 x

+ Y

J1 ( x )

(-1)m( 1 X) 2m-1 ( m - 1)!m!

TrY

Jo(x)

(36)

('

1

'

1

+m)

From equation (30) we have" 2

w~{Jo, Y 0 } -

Jl ( X ) r o ( x ) - J o ( x ) r ~ ( x ) - ~ .

(37)

~x

The Bessel functions J and Y occur frequently in geohydrological solutions concerning radial-symmetric or axial-symmetric flow, mostly in combination with other functions. In particular, the J0-function plays an important role in the solution of these problems, because it serves as the base of the Hankel transformations (Section 2.2.4).

Bessel functions and relatedfunctions

(3.3.2)

893

3.3.2. Modified Bessel functions I and K

If in Bessel's differential equation (1) x is replaced by ix, we must write 71 ~dv for dv and d2y d2y dx --d--~ for d--~' which results in Bessel's modified differential equation: ldy dZY ~---5 dx - x dx

(

1+

!)2 ) y

-- 0.

(38)

Equation (38) can be solved in a way analogue to the solution of equation (1), which yields the modified Bessel functions of the first kind and order v, denoted by

Iv(x): cxz

Iv(x) -- Z

( .~1X) 2m + v

m!F(.m + v + 1)

(39)

m--O

Replacing v b y - v ,

we find 1 X) 2m-v

l_v(x) m--O

m!F(m - v + 1)

(40)

Iv(x) and I_v(x) are linearly independent if v is not an integer and then constitute the general solution to equation (38): y(x) = Air(x) + Bl_v(x).

(41)

If v is integer, then 1

In(x)- ~

x) 2m+n

(42)

m!(m + n)!

m=O

is a solution of ldy dZY } dx 2 x dx

(

1+

n2)

y - 0.

(43)

I-n(X) is also a solution of equation (43) but is dependent on In(x), because I-n (X) = In (X),

(44)

obtained in a similar manner as equation (7). So In (x) and I_n(x) do not form a fundamental system of solutions and we need a second solution for integer values n of v. This solution can be obtained in a fashion similar to the solution Y0 in

894

(3.3.2)

Functions

the previous section, by assuming y2 = yl" u(x) as the solution of the modified differential equation of zero order d2y dx 2

1 dy !

y -

x dx

O.

(45)

The result is the modified Besselfunction of the second kind and order zero, denoted by K0(x):

~ (+x)2m( l l <9O

m=l

(mr)2

1 +2+3+'"+m

(46)

1) "

So the general solution of equation (45) is (47)

y ( x ) = AIo(x) + B Ko(x)

and the general solution of equation (43): (48)

y ( x ) -- AIn (x) + B Kn (x).

Also for the modified differential equation of general order v (equation (38)), it is desirable to adopt a form of the second solution which is valid for all values of the order. Therefore the standard solution K~(x) has been introduced, defined for all v: Jr I - v ( X ) - Iv(x) Kv (x) -- ~ sin(vzr) '

Kn(x) - lim Kv(x). v-+n

(49)

The last formula may be verified in a manner like to that used for Y,(x) in Section 3.3.1. Now the most general solution of the modified Bessel's equation for all values of v (equation (38)) is: y ( x ) -- A i r ( x ) + BK~(x).

(50)

From equation (49) we have directly: Kv(x) = K_v(X).

(51)

Bessel functions and related functions

(3.3.2)

895

The relations of the modified Bessel functions with the hyperbolic functions follow by inserting v - g1 or v -- - g 1 in the series for Iv(x), given in equation (39)"

oo I1/2 (X) --

(~

m=o m ! F ( m -t- 3)

(l)½{

l r(~)

(1 x)l/2 ( r(3)

I1/2(x)

X2

~ x+ Jrx

--

Jrx

+

~

(1X)4 2!r(7)

+

X4

1+2--~+

=

2

(lx)2 l!r(~)

+

)

2.3.4.5 +...

~

'

} +...

or

sinh x,

(52)

and also

I-1/2(X)

2

--

Jrx

__ Jr I - 1 / 2 ( X )

K1/2(x)

(53)

coshx,

-~

-- I 1 / 2 ( X )

sin (g1 Jr)

__ ¢

Jr

~x (coshx

sinhx)

from which (54) Some special values of the arguments give the following results:

(55)

In(-X) = (-1)nln(x), for instance: Io(--X)

(0) _

-- I o ( x ) ,

J0 / 1

I1 ( - - x )

= --I1 ( x ) ,

for all values of n --/: 0, for n - 0; so I 0 ( 0 ) -

1,

K.(O) = -~.

(56)

For all values of n:

In (e~) = cx~ and

Kn (oe) = 0.

(57)

896

Functions

(3.3.2)

The derivatives and indefinite integrals of Iv and Kv closely resemble those of J~ and Yv, but differ with respect to the sign here and there and are also mutually different. They can be obtained in a similar manner: d

d

{x v Iv(x) } - x vIv-1 (x),

{X v K v ( x ) } -- --X v K v - l ( x ) ,

dx

dx

d

d {x-Vlv(x)} - x-Vlv+l(X), dx

dx

{x-VKv(x)} - _x-VKv+l(X),

(58)

(59)

f xVlv_l(X) dx -- xVlv(x) + c, (60)

f xVKv_l(X) dx - -xVKv(x) + c,

f x-Vlv+l(X) dx - x-Vlv(x) + c, (61)

f x-VKv+l(x) dx = -x-VKv(x) + c. The recurrence formulae become: dlv(x) = Iv-1 (x) dx

1)Iv(x), x

dlv(x) = Iv+l (x) + ~1)Iv(x) , dx x

__

vKv(x)

dKv(x) _ - K v - l ( x ) - ~ , dx x

(62)

dK~,(x) = --Kv+l (x) + ~1)K~ (x ) , dx x

(63)

dlv(x) 1 1 dx = ~ I v - l ( X ) + ~ Iv+l(X), (64) dKv(x) _

dx

1

1 Kv-l (X) - -2 Kv+l (X)

2

Z

21)

Iv-1 (x) -- Iv+l (x) = - - Iv(x), x

(65) 2v

-

Kv-

1 (X) -'1- Kv+ t ( x ) = - - K v ( x ) . x

(3.3.2)

Bessel functions and related functions

897

The Wronskian for Iv and Kv becomes dKv(x)

Wx {Iv(x), Kv(x)} -- I v ( x ) ~

dlv(x) -

dx

X~(x)

= --Iv(x)Kv+l(X)-

dx

(66)

K v ( x ) I v + s ( x ) --

Some examples for small orders are: dlo(x) dx

=

dKo(x) dx

Is ( x ) ,

-- -Ks

x Io(x) dx - x I1 (x) nt- c,

dI1 (x) dx

1 -- Io(x)

x

f

(67)

(x),

dK1 (x) Ii (x),

(68)

x K o ( x ) dx - - x K1 (x) ÷ c,

dx

1 = -Ko(x)

x

Kl (x)

(69)

'

Wx { Io(x), K o ( x ) } - - I o ( x ) K s (x) - Is ( x ) K o ( x ) --

Kl(x)--

{ (') / In -~ x

+),

(70)

1 Is (x) + - Io(x) x

(71) -E

m--0

m ! ( m + 1), 1 + ~ + 5 + ' " +

.

Laplace transformations leading to K - f u n c t i o n s In certain non-steady problems, we need the inverse Laplace transform of f (s ) - ( x/~ ) VKv (a x/~ ) ,

say L -1 {f(s)} -- F ( t ) . Consider the function y d__y_y_ dx - -

v -x

Kv-1

x V K ~ ( x ) . From equation (58) it follows that

d2y (x);

dx 2 --

d v- x V dx { Kv-1 (x) } - I)x 1Kv_ 1(x)

From equation (63) we find d2y v{ v-1 ] dx 2 -- - x - K v ( x ) ~- ~ Kv-1 (x) - vx v-l K v _ l (x) X d2y = x V K v ( x ) + (1 - 2 v ) x V - l K v _ l ( X ).

dx 2

or

o

898

Functions

(3.3.2) dv

d2v

From the expressions for y, ~ and d-~ we can compose the differential equation d2y ~- (1 - 2v) dy -dx 5~xx

y - O,

(a)

of which y - x v K~ (x) is a solution. Now assume x -- a~/7, then y - a ~f ( s ) , which together with its derivatives substituted in equation (a) leads to the differential equation d2f 4(1 - v) d f 4----=_ 4ds 2 s ds

a2 s

f --0,

(b)

of which f ( s ) = ( 4 ~ ) V K v ( a ~ ) is a solution. Inverse Laplace transformation of this equation yields (see Theorems 7 and 8 of Section 2.2.2-1)"

4t2F(t) - 4(1 - v)

fot

r F ( r ) dr - a 2

fot

F ( r ) dr - 0.

Differentiating with respect to t gives: 4t 2 --~ d F + 8tF - 4(1 - v)tF - a2F -- O, dF. = ( a 2 F 4t 2

or

1 + V)dt, t

with the solution" a2

F(t) -- ct-(v+l) exp ( -

-~).

To calculate c we write:

f(s) = (~7)VKv(a~/7)- L{F(t)} (c)

= c

fo

t-(v+ t/exp

(

st) dt

4t

and let s approach to zero. Now from equations (49), (39) and (40) we may derive, for small values of x"

rr

Jr

1 X) -v

(7

Kv(x) ~" 2 sin(vrr) l_~(x) - 2sin(vTr) F(1 - v)' and as r(v)r(1

-

v) -

sin(vrr)

Bessel functions and related functions

(3.3.2)

899

(Section 3.4.1), we have 1

1

Kv(a~/~) ~ -~ F(v)(-~

/

-v

and thus

2~-la-~F(v).

lim {(~/7)~K~(av/7)}s--->0

(d)

The integral in equation (c) becomes, if s - 0 and with the substitution u equal to"

C fo °

71

a2u du - c22Va-ZVI-'(v) blv-1 exp ( - ---4-)

(according to the definition of the gamma function). Compared with equation (d), we find c - 2 -(v+l)av and thus

(a2)

L -l{(x/7)~K~(aqq)} -- (2ti;+ 1 exp

.

(72)

a2 ~-;).

(73)

- ~

For instance, |

a 2

L-l{Ko(a~'s)} -- - ~ e x p ( -

--~),

a L-l{,~fsKl(aV~)} -- - ~ e x p ( - -

a2 --d-[),

and so on. As K_~,(a~/7) -- K~,(a,,~), we have also

1

L-1 { (~/~-)v Kv(a~ r~) } --

(2t) -l+v

a---------y~exp( -

For instance,

1

1

-exp a

a2 )

The modified Bessel functions I and K are of considerable importance for solutions of radial-symmetric or axial-symmetric groundwater flow. In particular, they appear in pure form as solutions of two-dimensional radial-symmetric flow in leaky aquifers, for which the differential equation runs (see Example 2 of Section 2.1.1): d2~o 1 d~o t dr 2 r dr

99 )v 2

=0

900

Functions

(3.3.3)

with the solutions r qg(r) -- A I o ( ~ ) 4- BKo(-~),

as can easily be derived from equation (45). Special forms of this differential equation also appear as equations obtained from partial differential equations by means of integral transformations (Example 3 of Section 2.1.1 and Example 6 of Section 2.2.2-3).

3.3.3. Complex functions In some geohydrological problems there is a practical need for solutions of Bessel's equation which are complex for real values of x. For this reason, the so-called Bessel functions of the third kind have been introduced: H{ 1)(x) -- Jv (x) 4- i Yv(x)

/_/(2)(x)

and

-- Jv (x) - i Yv(x).

(74)

It is obvious that H{t)(x) and H(2)(x) satisfy Bessel's equation (1) and since Jv (x) and Yv(x) satisfy the same recurrence formulae and derivatives as given in the equations (22) to (27), in which the functions enter linearly, and since the functions of the third kind are linear functions (with constant coefficients)of J~(x) and Yv(x), it follows that these same recurrence formulae are satisfied by functions of the third kind. There are many other relations between the three kinds of Bessel functions and the two kinds of modified Bessel functions in the complex plane. In fact, every Bessel function of the second or third kind and every modified Bessel function of the first or second kind may be expressed in terms of the functions J~ and J_~ of the first kind, as follows. Yv and Y_~ as functions of J~ and J_~ follow from their definitions according to equation (15). cos(wr)J.(x) Y~(x)

-

-

J_~(x)

sin(vzr)

Iv(x) -- i-~Jv(iX)

and

J~(x) ,

r_~(x)

-

-

cos(v:r)J_~(x) sin(vzr)

I-v - i v j - ~ ( i x )

(75)

follow immediately from their series representations (equations (2), (3), (39) and (40)), and from equation (49) we have:

Jr i v J_~(ix) - i -~ J~(ix) = K_~(x), Kv(x)- ~ sin(vTr) H(1)(x) = Jr(x) + iYv(x) - Jr(x) + i J_v (x)

-e-iVnJv(x)

isin(vrr)

cos(vzr)J~(x)- J_.(x) sin(vrr)

(76)

(3.3.3)

Bessel functions and related functions

901

Now, as e -ivrr -- i -2v, we may write" -/v 1) (X)

ivJ_v(x)-i-vJv(x

i -v-1

)

sin(vn.) H (1) ( x ) - - i v-1

(77)

i v J_v(x) - i -v J+v(x)

--V

sin(vn.)

from which H_(1) (x) -- i 2vH (1) (x). In the same way, we have H(2)(x)

i v-1. iv J r ( x ) - i - v J - v ( x ) sin(vTr)

-

_/_(2) ( X ) - - i -v-

1 .

and (78)

ivJv(x)-i-vJ_v(x)

V

sin(vTr)

and H (2) ( x ) -- i -2v H(_2)v ( x ) . On the other hand, it is possible to express all functions in terms of the functions g (1) and/_/(1~_, in terms of H(~2)(x) and H(~3)(x) and so on. For instance, a relation connecting Kv to H~(1) follows from equations (76) and (77): Kv(x)

-

1

- n.iv+l tl~"(1)(ix) -

2

K_v(x).

(79)

Some s p e c i a l v a l u e s of the a r g u m e n t s leading to complex functions follow here. J o ( i x ) -- I o ( x ) ,

J1 ( i x ) - i l l ( x ) .

(80)

From equation (74) we have Yo(ix) -

i J o ( i x ) - i H0(1) ( i x )

and with equations (79) and (80): Yo(ix) -

Ko(x) + i Io(x).

(8 l)

Differentiating with respect to x gives: 2 Y1 ( i x ) -- - I 1 ( x ) + -- i K1 ( x ) . 7/"

(82)

From the series representation for Y o ( x ) given in equation (13), and with l n ( - 21 x) ln( 1 x) + in', we immediately find: Yo(-x)

-- Y o ( x ) + 2i J o ( x ) .

(83)

902

Functions

(3.3.3)

Equation (79) gives:

1 zri H(o1) ( - x ) -- 12 j r i l J o ( _ x ) + i Y o ( _ x ) Ko(ix) -- -~

}

and with equation (83):

Ko(ix) -- - - ~ { Yo(x) + i Jo(x) } and

(84)

K1 (ix) -- - ~ { J1 (x) - i Yl(X) }.

(85)

Replacing x by ix in equation (84) yields

K o ( - x ) -- - - ~ { Y o ( i x ) + iJo(ix)}, which becomes, with equations (80) and (81):

K o ( - x ) -- Ko(x) - ire Io(x).

(86)

This result can also easily be obtained by inserting - x for x in the series for Ko(x), given in equation (46). The Bessel function of the third kind H~-1) for the values 1 and 2 of n and the real argument, giving according to equation (79) the functions Ko(+ix) and K1 (+ix), occurs in problems concerning non-steady radial flow outside a circular cylinder, in particular, if the solution is to be obtained by means of a Laplace transformation with respect to t (see Example 4, Section 2.2.2-3) and the inverse transformation takes place in the complex plane. A similar problem arises in non-steady radial-symmetric flow with dispersion where a so-called Airy function with complex argument must be evaluated. This function is a solution of the Airy differential equation: d2u dp 2

pu - O.

(87)

The connection of this equation with Bessel's equation can be obtained by substituting u - y~/-fi

and

giving: dZY t 1 dy dx 2 x dx

2 x -- "~ p3/2,

(1) 1+

y-0

~

(88) '

a modified Bessel's equation of order one third, with the solution (equation (41))

y = A I - 1 / 3 ( x ) + BI1/3(x),

Bessel functions and related functions

(3.3.3)

903

and thus 2 2 u -- AV/-pI_I/3(~ p3/2) 4- B~/-plI/3(~ ,03/2).

(89)

From the series representations for Iv and I_~ we find:

31/3

3 2/3

2

~f(p)

and

x/~I1/3 (~ p 3 / 2 ) _ F(1)

g(p)

with

f(P)--l+~.t

1.4.7 1"4p6 +-~.v + 9! 2 p4 2 . 5 p7 + 2 . 5 . 8 10!

1 p3

p9 _at_...

and

p10 + . . . .

(90)

So the general solution of equation (87) can be written as"

u -- A l f ( p ) ÷ Big(p). For real positive values of p the values of f and g will increase with increasing p and tend to infinity if p becomes increasingly larger. To avoid this we set, analogously to cosh x and sinh x in relation to e x and e -x"

F(p) -- cl f (p) - c2g(p)

and

G(p) - cx f (p) + c2g(p)

and consider

u - otF(p) + fiG(p) as the general solution of equation (87). The constants Cl, c2, ol and fl may be chosen arbitrarily; for computational reasons F(p) and ~/3G(p) have been chosen as standard solutions of the Airy differential equation (87) and called Airy functions, denoted by Ai(p) and B i(p). The constants cl and c2 have been chosen such that 1

2

c2f (p) --

~1 ~#'pI1/3(-~2 p3/2),

SO 3-2/3 C1 -"

3-~/3

,

C2 - -

Ai(p) -- c l f ( p ) - c2g(p)

,

and

(91)

Bi(p) -- x / 3 { c l f ( p ) 4- c2g(p)},

while f ( p ) and g(p) are given by equation (90).

904

Functions

(3.3.3)

We may also write: Ai(p)

1

-- -~~{I_l/3(x) - I1/3(x) } (92)

1

B i ( p ) - --~~/-fi{ I-1/3 (x) -b I1/3 (x) } with x = 52 p3/2 , or according to the definition of K~ (equation (49)): with x -- ~2 io3/2.

K1/3(x)

Ai(p) - --

(93)

The close connections of the Airy functions to the modified Bessel functions, given in the equations (92) and (93) are a guarantee that many properties of the Airy functions are similar to those of the modified Bessel functions. For instance, if we write

1(3)1/31/3 x K1/a(x),

Ai(p) -

we have, according to equation (58): dAi

dAi

dx

dp -

dx

dp

1(3)1/31/3

jr ~/~ -2

x

K-2/3 (x) . p

1/2

and as Kv -- K_~: dAi _ dp -

P K2/3(x) zr~'~

'

and making use of equations (58) and (59): dBi

p

dR = V~ { I-2/3 (x) + 12/3(x) }.

(94)

In the already mentioned radial-symmetric dispersion problem, which gave rise to the treatment of Airy functions, we encounter the complex argument p exp(~ rri) for the Ai-function, which can be evaluated as follows. If in equation (92) we replace p by p exp( 3 zri), the argument x becomes xe ~i = - x , and thus:

Ai(pe ~;¢i) - -~rpe½rCi{I-1/3(-x)- I1/3(-x)} = ~1 ~//pe~1 ~-i{e-~ Trii_l/3(x) _ e½ ~rii1/3(x)} ---- 1 x/'-Pe~ - rri 3 {(

2ix/~) I-1/3(x)- ( l + i ~I 1) / 3 ( x ) 2

}'

905

(3.3.3)

Bessel functions and related functions

in which the evaluation of the /-functions with negative argument follow from their series representations. Now, according to the definitions of the Airy functions (equation (92)), we have: A i ( p e 2~ri) a i ( p e - ~ ~i)

--

1 e 7' ~ i { a i ( p ) - i B i ( p ) }

and also

2

(95)

1 - ~ r i { a i ( p ) + iBi(p)} 2

~ e

o

This example of a complex argument of a Bessel function is, in practice, an exception rather than a rule and in most cases we have to do with real arguments. Therefore, all the theory of the Bessel functions in the previous paragraphs was based on the real argument x. However, all properties, definitions and derivations are also valid for a complex variable z, which occurs in certain problems of groundwater flow, in particular problems concerning so-called periodic flow (see Section 2.3.3), where we encounter the differential equation d2y

dx 2

1 dy t

x dx

iy

= 0,

(96)

with the solution

y -- A I o ( x ~ t ) + BKo(x~/-[),

(97)

in which x is a real number. Now, as

Io(x)

-- 1 +

(1 X) 2 (21 X) 4 (21 X) 6 (l !)2 + (2 !)2 + (3 !)2

+

(21 X) 8 +..., (4 !)2

we have

(1) (1 !)2

1 x) 4

Io(x~/Tz)- 1 + i "~x'2

(2!)2

(½X) 6

i

(3!)2

+

(~l X)8 (4!)2

. . .

and if we separate the terms with i from the terms without i, we find the socalled Kelvin functions of the first kind, denoted by ber x and bei x as the real and imaginary part, respectively, of Io(xv/-{) • I0 (x v~-) -- ber x + i bei x

12 (1 X) + (2!) 2 (4!) 2 (6!) 2 ( ) )6 lx)IO ~x (3 !)2 ~- (5!)2 1

b e r x - 1- (½ x)4 ~ (~ X)

bei x -

with

(98)

8

...

and (99)

906

(3.3.3)

Functions

or c~

c~

1

berx - Z

(-1)~(4 x:)2" (2n)!(2n) !

n--0

and

beix -

(__l)n/1

2x2n+l

tax ) (2n + ])T.(~n + 1)'

n=0

If we write (1

)

~ (~lX)2n (n!)2 O ( n + 1),

Ko(x) -- -- ln -~ x Io(x) + Z n=O

in which 1

~(n-t- 1 ) - - y

1

+ 1 q-~-+- . . . . n

(psi-function, Section 3.4.1) we obtain, as 1

In (~ x~/i) --In ( ~ x ) +

~zri,

Ko (x ~/i) = ker x + i kei x

(1) ~x

kerx=-ln

cx~

+ ~

n--O

kei x - -

+ ~

n---0

(100)

1

berx+~zrbeix

(-1)n(a1 x a) 2n (2.n')i(2n)!

(1) In ~ x oo

with

~p(2n + l)

and (101)

1 4 zrber x

bei x

( - 1 ) " ( 1 x 2)2n+1 (2n + 1)!(2n + 1)!

~(2n + 2),

where ker x and kei x are Kelvin functions of the second kind. It is often advantageous to express the functions Io(x v/-{) and Ko(x ~/-{) in terms of modulus and phase. Thus we have lo(x ~c[) _ Mo(x)eiOo(x)

and

(102) (103)

Ko (x'v/-i) -- No (x )e i4~°(x)

with Mo(x) -- v/ber 2 x + bei 2 x,

0o (x) - arctan ( berx)bei x

No (x) -- q/~ker2 x + kei 2 x,

(keix) 4~0(x) - arctan k e r x "

and

Bessel functions and related functions

(3.3.4)

907

In practical geohydrological problems the ber- and bei-functions occur in solutions for periodic flow inside a circular cylinder and ker- and kei-functions in solutions for periodic flow outside a circular cylinder, including periodic flow to or from wells. The general functions Iv (x ~/i-) and K, (x ~/i-) lead to the Kelvin functions ber~ (x), bei,(x), kerr(x) and kei~(x) and to M~(x), N~,(x), O~(x) and q~,(x), but these will not be discussed here. Most of the properties of these general Kelvin functions can be derived from the properties of the corresponding Bessel functions.

3.3.4. Series of Bessel functions The generating function for the Bessel functions of the first kind is

1 xt -~

Fg--exp

21 x) t

or

if t :/: 0,

{ l + ~ x't + ~ .

,(,

{ I. . x. . × 1 2 t

- 1(lx)2_ ~ 2t

Fg--

-~xt

+~

(104)

,(,

-~xt """ I

l(lx)3 ~ 2t

/ +""

"

Performing the multiplication, we see that the coefficients of t o (constants) yield the series t o --+ 1 -

( 1 )2 1 4 (~X) ~ x -~ (2!) 2

(~lX)6 (3!) 2 Jr- . . . .

Jo(x).

The coefficients of t give:

,

lx)5

1

t -+-x 2

1!2!

t

(½x)

2!3!

3!4!

F- . . . .

J1 ( x ) .

In general, J,, (x) is the coefficient of t n. The coefficient of t -1 becomes"

1 2

x+

(½x) 3 1!2!

-

(~1 x) 5 2!3!

+

(2 x) 7 3!4!

....

=-Jl(x)-

J-l(X)

and, in general, J-n(X) - (--1)n Jn(x) is the coefficient of t -n. It follows that

Fg -- Jo(x) -4- tJl(x) -4- t2jz(x) +...-4- t-l J_l(X) -+-t-2J_2(x) + . . . , or (x)

e2

--

r Jn (x ) tt-"-- CX3

(n -- integer)

(lo5)

908

(3.3.4)

If we substitute t

--

Functions

ei° we obtain:

eylx(t-1) --exp {l_~x (e iO - e -iO }) -- e ixsinO and from equation (105) e ixsinO --- J o ( x ) + { J1 (x) eiO -t- J-1 (x) e-iO } -}- { J2(x)e 2i0 -}- J - 2 ( x ) e -2i0 } -}-... , which becomes with J_~ (x) - (-- 1)n j~ (x): sinO q- 2J2(x)cos(20) + 2iJ3(x) sin(30)

e ixsinO = J o ( x ) q- 2 i J l ( X )

q- 2J4(x)cos(40) + . - . . Separating the real and imaginary parts at both sides of this equation, we find oo

cos(x sin0) -- Jo(x) + 2 ~

Jzn(x) cos(2nO),

n=l sin(x sin0) -- 2 Z

(106)

J2n+l(X)sin {(2n + 1)0}.

n--0

From these two formulae we can derive numerous other functions, expressed in series of Bessel functions by giving various values to 0, directly, or after differentiating several times with respect to 0. Some examples follow here: (ND

0 - 0

gives

1 - J0 (x) 4- 2 ~

Jzn (x),

(107)

n=l Jr

0 -- -2

co

gives

cosx - Jo(x) + 2 ~ ( - 1 ) n J z n ( x )

and

n= 1 (9O

sinx - 2 E ( - 1 ) n h ~ + l ( x ) . n=O

Differentiation of sin(x sin 0) with respect to 0 gives" d d---~{ sin(x sin0)} - cos(x sin 0)x cos0 oo

= 2 E J2n+l(x)(2n + 1)cos {(2n + 1)0} n--0

(108)

Bessel functions and related functions

(3.3.4)

909

and with 0 - 0 ÷ 1)J2n+l(X).

x -- 2 Z ( 2 n

(109)

n---0

In a similar way, we may develop the generating function for the modified Bessel function of the first kind: (x)

(110) n - - ' - - CX:)

and derive associated series with the substitution t - e iO, like

e x cosO= Io(x) + 2 ~

In (x) cos(nO),

(111)

n=l

which becomes for 0 - 0 : oo

e x -- Io(x) + 2 Z

In (x).

(112)

n=l

Other formulae may be easily obtained from the formulae for the Jn-series by writing ix for x and Jn (ix) - i n I (x). In Section 2.2.4-2 it was stated, that under certain conditions an arbitrary function f ( r ) may be represented by a so-called Fourier-Bessel series of the form: oo

f ( r ) -- Z C n J °

----~/

with

(113)

n--O

2~oR

rf(r)Jo(-~--~--) dr,

while an (n - 0, 1, 2 , . . . ) are the roots of Jo(o~n) - 0 (cf. equation (117) of Section 2.2.4-2). For instance, if f ( r ) -- c - const, we have

foRr f (r)Jo

dr - c

/oRr Jo

Otnr '] R /dr-

cR 2 Oln

Jl(C~n)

(formula (28) of Section 3.3.1). So,

Cn =

2c c~ J1 ( ~ )

and n--O

~ J1 ( ~ )

=-

1

2

(0 < r < R).

(114)

910

Functions

(3.3.4)

If f ( r ) = In ( ~ ) t h e integral in the expression for cn becomes: I=

fo R rln ( R ) J°\--R-)dr--ot-~ (otnr R2fol¢ l n ( R- ) -d J{otnr -R l r R2[~.ff_

= ct-~

°tnr]] e

In (rR---)J1( R i o

(~)}

R2 fo otnr (otnr ) dr + ot---~n T R2

As limr->0[r In r] --0, the first term disappears and the integral I becomes -7 and 2

c~

Thus

jo(Otnr (g) -g-) _ l l n

(0 < r >~ R).

(115)

n=0

In this way, many functions may be expressed as Fourier-Bessel series, but, in general, require rather laborious calculations. However, sometimes Fourier-Bessel series occur as alternative expressions if successive transformations of variables (Section 2.2.6) have taken place. This was shown in Example 21 of Section 2.3.3 by equation (h):

n--0

-fi-) _ 1 Io(ar) Otn(Ot2 -k- a2R2)Jl(Otn) -- ''--''~ 2a2R 1 - Io(aR)

(116)

with otn being the roots of J0(c~) = 0. In particular, if a = 0: for0~r~
cx~ jo(Otnr] '-R-! _ R2 _ r2

ZU~Jl(C~n)-

8R 2

(117)

n=0

Differentiating equation (116) with respect to r gives:

J1 ( __~_ nr n--0

(or2 + a 2R 2) J1 (ot,)

1 11 (ar) 2aR Io(aR)'

(118)

and with a = 0:

oo J1 ( --g-) unr _ 1

r

(119)

Z n=o un2J1 (c~) - 4 R

If we multiply both sides of equation (118) by r and we differentiate with respect to r, we easily obtain:

1 Io(ar) --2 Io(aR) + a2R2)j,( .) n--0

(0 <~r <~R).

(120)

Bessel functions and related functions

(3.3.4)

911

With a - 0 we find equation (114) again. In all these formulae 0/n is a root of J0(0/) - 0. If a is replaced by ib in the equations (116), (118) and (120), we find J-functions instead of/-functions at the right side. Another category of Fourier-Bessel functions arises if in equation (113) the 0/,, are the roots of 0/J1(0/) - 8Jo(0/), in which e is a constant (see Section 2.2.4-3). Then, according to equation (126) of that section, we have Fourier-Bessel series of the form:

CX3 f(r)--Zc'Jo( n--O

o/'r ---~-)

with

(121)

1 fo R r f (r ) Jo (0/nr) dr, Cn = ~2 R2 J g ( 0/n ) -4- J 2 ( 0/n ) \ --'Rwith 0/, being the roots of 0/J1(0/) - 8Jo(0/). An example, obtained also as an alternative expression from successive transformations is"

~ (0/2n + e2)(0/Zn + aZRZ)Jo(0/') "=0 elo(ar) }. 2a2R 2 a R I l ( a R ) + elo(aR)

1{1_

(122)

This formula is a generalization of formula (116), which can be derived from this formula by letting e approach infinity, setting in the left side of this equation 8 N

0/'J1(0/')

J0(ot.) and thus the factor (ot2+e28)jo(otn)in the series of equation (122) becomes

0/nJl(0/n)

ot2J?(otn) { 0/2"31-

j2(otn )

Jl(0/n)

] Jg (0/')

and next, as

0/nJl(0/') Jo(c~.)

-

by setting J0(0/n) equal to zero. At the same time the condition that 0/" must be the roots of J0(e,) - 0 is satisfied. By differentiating equation (1.22) with respect to r, we find

oo

n---0

0/n J1 ( otRn r J

(0/2 + 82)(0/2 _[_ a 2 R 2 ) J o ( 0 / n )

=

1 Ii.ar.( ) 2aR a R l l ( a R ) + elo(aR)'

(123)

912

Functions

(3.3.5)

and if a -- 0

J1 ( "-'R nr

c~

?-

) = 4Re"

+

n--0

(124)

Going on in this way, a lot of other equalities may be obtained; they will not be given here but are left to the reader to find. The Fourier-Bessel series play an important role in solving problems concerning radial flow inside a circular cylinder, by means of the finite Hankel transformations (Section 2.2.4-3).

3.3.5. Integrals of Bessei functions

Laplace integrals of Bessel functions By these integrals we will understand Bessel functions with various arguments in combination with powers of t, multiplied by e -or and integrated with respect to t from zero to infinity. A general example is

I --

txJu(at)e -c' d t -

L{tXJu(at)}

f0 CK)

with c instead of s, and a > 0 and c > 0. We develop Ju in its series according to equation (2) of this section. I - - fo ~ t ~"~ ( - 1 )n (3l at)t *+2n e -ct dt n=0 n!F(l~ + n + 1)

= Z

n--O

a).+2, f0

t ~'+#+2n • e - c t d$.

n!F(Ix + n + 1) "

The last integral represents a gamma function (Section 3.4.1). So

I -- Z

n--0

( - I ) " ( l a ) u+2" F(/, + X + 2n + I) n!F(l~ + n + 1) " c/z+)'+2n+l

provided t h a t / , + )~ + 2n + 1 > 0, or/~ + )~ ÷ 1 > 0. As the hypergeometric series

F(el, fl; V; x) --

F(y) ~ F(~ + n)r(fl + n) . x_~_ ~ F(~)F(fl) ,,=o F(y + n) n!

(see Section 3.4.2) and /-' (/~ ÷ X ÷ 1 ÷ 2n) =

+2n, r(n+

( 2 r n+

Bessel functions and related functions

(3.3.5)

913

(duplication formula, Section 3.4.1, equation (11)), we find:

fo

OGtz Ju (at) e-ct dt

cl*+~+~ F (/z + 1)

(125)

(

2

'

2

• #+

1;-

,

i f c > a > 0, /z +)~ + 1 > 0 and # -J: - 1 , - 2 , . . . . With )v = - 1 , we find from equation (125):

t J• (at)e-Ct dt -

c" F (l~ + 1)

1 o As F ( p , p + g, 2 p + 1; x) -- 22p(1 + ~/1 /z F (/~), we find

fOG ]

t Ju (at) e - c ' dt -

2'

,/x+

1,-

.

X) -2p (Section 3.4.2) and F ( / z + 1) -

1 (~/a2-+-c2--c)

#

2

a

bt

( # > 0).

(126)

The restriction c > a has disappeared. F o r / z = 1 we have OG 1 t J1 ( a t ) e -ct dt -

f0

1 - ( v / a 2 + c 2 - c).

a

(127)

The case ~ -- 0 and thus/~ > - 1 in equation (125) gives the integral f ~ J u ( a t ) e -Ct dt as a h y p e r g e o m e t r i c function, but can also be derived by differentiating equation (126) with respect to c:

fo

OGJu (at) e-Ct dt --

1

~/hJ' + 6.2

a

F o r / z = 0 we have

fo

OGJ°(at)e-Ct dt =

(129)

~/a 2 ' ._1_c 2'

and f o r / ~ -

1:

OG J1 ( a t ) e -ct dt -

~/a2 + c2 -- c a ~ / a '2 + C 2

f0

"

(130)

If we differentiate equation (129) with respect to c, we have:

f

OG t J o ( a t ) e -ct dt

-

-

C ( a2 ÷ c2)~/a 2 ÷ c2 ,

(131)

914

(3.3.5)

Functions

and with respect to a:

f0 °

(a2 4- c 2 ) ~ / a 2 4- C2"

t J1 ( a t ) e -ct dt -

(132)

a

With the relation I u ( a t ) = i - u J u ( i a t ) , a similar set of formulae may be derived for integrals of the modified Bessel function I: In equation (125) only - 7a2 changes in 7" a2 Equation (126) becomes fo °°1

1 ( c - ~ / c 2 - a2) ~t

7 I~, ( a t ) e -c't dt -

-l*

(133)

a

with/z > 0 and c > a. /x -- 1 in equation (133) gives f0 ~ 71 I1 ( a t ) e -ct dt - -1( c -

x/Ic 2-

a

a 2)

(c > a).

(134)

From equation (128): fo °° I~ ( a t ) e -ct dt

--

% / c 2 1_

--

~/C2

a 2 ( c - - ~ / c 2a- - a 2 )

#

(c > a).

(135)

For/z = 0 we have fo ° I o ( a t ) e -ct dt

1_ a 2

(c > a)

(136)

and for/x = 1: f 0 ~ I1 ( a t ) e - c t d t - - C a--~ / c~/C2 2- --a 2a 2

(c > a)

(137)

Also: o ~ t I o ( a t ) e -ct dt -

(C 2 -- a 2 ) ~ ' c 2 _ a 2

(c>a),

and

(138)

a

fo ~ t 11 ( a t ) e -ct dt --

(c 2 - a2)~/c 2 - a 2

(c > a).

Interesting results follow if c -- a; then the hypergeometric series F(/x+)~+l 2

'

/x+L+2 2

;/,+

) 1; 1

(139)

Bessel functions and related functions

(3.3.5)

915

which occurs in the expression for

fo ~ t z I u (at)e -"t dt, only converges if /z+l -

/z +,L + 1 2

-

/z +,L + 2 2

or

> 0

1 2

~.<

From the integrals shown in the equations (133) to (139), only those in equations (133) and (134) satisfy this condition, and thus

fo ~ -2I 1 u(at)e-"dt-/-

1

(/z>O),

/z

and

(1 40)

f0 ~ t1 11 (at)e -at dt - 1.

From the definition of Y~ (equation (15)) and of K~ (equation (49)) the integrals

fo c~ tx Ylz (at) e-ct dt

and

fo ~ t x Klz (at)e -ct dt

may easily be evaluated from equation (125). As J~ and J_~ and also I N and I_~ are involved, the conditions besides c > a > 0 and/z :/: - 1 , - 2 . . . . become / z + ) ~ + 1 > 0 and - / z + , L + 1 > 0, from which - ( , L + 1) < /z < ,L+ 1, or I. + 1 > I/xl; so, for I. = - 1 these integrals do not exist. If ~. = 0, we have, from equation (128):

o ~ Yu (at) e-ct dt (141) a

/

sin(/zzr)~/a 2 + c 2

(llzJ < 1).

For # - 0 the right side of equation (141) becomes indefinite and we must apply l'H6pital's rule, differentiating both the numerator and the denominator with respect to/z and next letting lZ approach zero. The result becomes: 2

fo ~ Yo(at)e -ct dt -- 7r.~/a 2 -[--c 2

in (~/a2--~-C2--C) a

"

(142)

916

Functions

(3.3.5)

In a similar way, we may write:

fo ~ K~ (at)e -ct dt (143) 2

(Iz~l < 1)

sin(/zzr)~/c 2 - a 2

and letting/x approach zero, we find"

fo ~ Ko(at)e -ct dt -

2~/c 21 _ a2

~/c 2 -- a 2

In ( c + ~/c 2 - a 2 C -- ~/C 2 -- a 2 )

in(c+~/c2-a2) a

_-- arccosh(c)

forc>a

~/C 2 -- a 2

If c -- a this expression becomes indefinite, but after applying l'H6pital's rule, we find 1a f o r c - a .

Ifc
i arccos ( c )

arccos ( c ) m

i ~/a 2

~/a 2

__ ¢2'

__ C 2

which also becomes ± if c - a. The end result is: a

arccosh(c)

for c > a,

%/c 2 -- a 2

fo ~ Ko(at)e -ct dt -

1 a

arccos (a)

for c - a,

(144)

for c < a.

~/a 2 -- c 2

If we consider f o Ko(at)e-Ct dt as a function of c with a - constant, we see that the formula of the function is discontinuous but the function itself is continuous. 1

3

5

It must be remarked that if we choose f o r / z the numbers -t-~, ± ~ , :t:~, . . . we obtain expressions for infinite integrals of circular functions, for instance" (from equation (141) and with the aid of equation (17)):

fo ~

Y1/2(at)e -ct dt - -

~2-~a foCX~l --~ cos(at)e -ct dt

- -

(4a21-¢2-1-c) 1/2 al/Z~/a 2 -t- C2

(3.3.5)

Bessel functions and related functions

917

from which f 0 e~

1

~(%/a

cos(at)e -ct dt --

2 ÷ c 2 ÷ c)1/2

~/a 2 + c 2

The integral

i-fo

tz Ju (2~-a-7) e-ct dt

may be evaluated by developing Ju (2v/~) in its series according to equation (2)" tz

( - 1 (,~)2n+#

f0 ~ ~n=0

)" + lz + 1) n!F(n

I --

e -ct dt

_-- ~ ( - 1)"an+-~ tn+-~+Xe-Ct dt. n=O n!F(n + # + 1) f0 °°

The last integral is a gamma function if @ + i. + 1 > 0. So oo

/1

I-- Z a-f(-a) n F ( n + ~ + , k + 1) n=o n!F(n + # + 1) cn+~+)~+l

If we take ~ . - ~, we get ag

(_a)

c-~--" ~n--0

n! "

from which e

fo (at-)

g Jtz (2~-7) e-ct dt -

_a c

c#+l'

(145)

and

(146)

and, in particular,

f

~ Jo (2 ~ )

a

e -ct dt --

e-,C

Va

J1 (2~/~)e - a dt

e

a

c

(147)

-

C2 "

Replacing a by - b , we easily find b

/0

d,

e c

c#+l

"

(148)

918

(3.3.5)

Functions

A number of infinite integrals, here given are also valid for c = 0. From equation (126) 1 -tx

fora >0,

0

for a - O.

f o ~ l t Jl (at) dt - {10

fora >0, for a = 0, for a < 0.

fo c~ t1 Jlz (at) dt -

/x>O,

(149)

From equation (127)

(15o)

From equation (128): fo °~ J u ( a t ) d t - - - a1

(a > 0 , I~ > - 1 ) .

(151)

From equation ( 141) fo °° Yu(at) d t -

cos(/zrr) - 1 = - - 1 tan ( ~ ) a sin(/xzr) a

(a > 0

I~l < 1).

(152)

From equation (143): fo ~

Jr i lz - ( - i ) tz _ zr(eitZ~ - e-ilz~) _ Jr sin (/x ~-) 2ia sin(/zzr) - a sin(/xzr)

K~(at) dt -- -~ ia sin(/xzr) -

So 2"t"

fo ~

K u ( a t ) dt -

Jr

2a cos (/z 5)

(a > 0, I~1 < 1)

(153)

Weber-Schafheitlin integrals In certain problems it is necessary to use infinite integrals involving products of two Bessel functions multiplied by a power, for instance:

I =

f0 X)txJu(at)J~(bt)dt.

This integral is known as the Weber-Schafheitlin integral. The condition for convergence of this integral is - ( / x + v + 1) < ~ < 1. The evaluation of this integral and its convergence condition will not be given here; it will suffice to state that there are two solutions for this integral in the form of hypergeometric series: one

Bessel functions and related functions

(3.3.5)

919

for a > b and one for a < b, while for certain combinations of the values of/z, v and ~ the integrals are discontinuous for a = b. These solutions are: 2Zb u

/-, (#+v+~.+ 12

aV+Z+l

)

F(v + 1 ) F ( U - " ~ ~+1)

x F / l z + v +)~ + 1

i

2

'

v - / z +,k + 1 b2\ 2 " v + l ; a -2

)

for a > b,

fo ~ tx Ju (at) J~ (bt) dt -

(154)

/-, (#+v+~.+l 2 ) /-'(/Z -t- 1)/-'( v-lz-z+12 )

2Za ~

bu+~.+l

x F/lZ~ + v + ~

+ 1 , /z - v

2

+~. + 1 ; / z + --a2\) 2 1;b2

f o r a < b, with-(/x+v+l)<)~< 1. For many combinations o f / z , v and )~, the hypergeometric series may be expressed in transcendental functions. The hypergeometric series F(ot,/3, y; x) reduces to unity if c~ = 0 or/3 = 0 (see Section 3.4.2). So if X = - 1 a n d / z = v, both hypergeometric functions in equation (154) are equal to 1, because 13 = 0, from which we find, if/z > 0: 1

o~ f0

dt (at) (bt) Ju Jlz t

b

2-~%( a ) 1 a u

forb
(155) for b > a.

The integral is continuous for a = b. With )~ = 0 and # = v - 1 the h.g. function for a < b in equation (154) reduces to 1, while in the expression for a > b

F( lz-v

-)~+2 1 ) - F ( 0 ) - - I - c x ~ '

which means that the integral becomes zero for a > b. We find if v > 0:

o

Jv-l (at)Jv(bt) dt

/°

for b < a,

a v-1 b~

(156)

for b > a.

The integral is discontinuous for a = b. If one of the Bessel functions has the order + ~1 it is reduced to a circular function (see equations (17) and (18)), for instance, if v - - g 1 and )~ - - g ,1 we have

1

--~ J~ (at)

j2- ~

cos(bt) dt -

j2f0 -~ l t Jtz (at)

cos(bt) dt.

920

(3.3.5)

Functions

According to equation (154), we will find rather complicated expressions for this integral. Therefore, we consider the integral f0 ~ 71 J l z ( a t ) e - C t dt of equation (126) and substitute i b for c. We then find

/o

fo fo _---l(~/a2-b2-ib) t Ju ( a t ) cos(bt) dt - i

t Ju ( a t ) e-ibt dt -

tz

tx

t Ju ( a t ) sin(bt) dt

ifa >b.

a

To find the real and imaginary parts of the right side of this equation, we assume z - ~/a 2 - b 2 - ib - r e iO, in which r 2 - Izl 2 - a 2 - b 2 + b 2 - a 2 and

0-arctan

-b (~fa 2 _ b 2 ) -

b - arcsin ( a ) .

So

{ / /

z u - - a t Z e i t *° --atZcos

Ixarcsin(b)

-ialZsin

Ixarcsin(a )

and thus f0 ~ t1 J~ ( a t ) cos(bt) dt - - cos IX arcsin

a

f o r a > b.

At the same time, we found f0 ~ t1 Jr* ( a t ) sin(bt)

~t----sin ,arcsin -

for a > b.

IX

Forb >a l(~/a2-b2-ib) IX

tz

a

~ {/(,~2 _a2_ --~

a

1 (

a

-- IX b +

l( -"

-

~,}._ ~(~-4~2-a~ a

']lZe_itz ~

x/b 2 - a ~/

a

-

IX b +

;

~/b2 - a 2

).{

COS

(, ~)_ / sin(, ~)

( i)"

(3.3.5)

Bessel functions and related functions

921

The end result is, if # > 0: f0 ~ -f1 Ju (at) cos(bt) dt _

~cos

1(

/zarcsin

a

a

forb
(157)

forb >a

and if # > -1" f0 ° -f1 J~ (at) sin(bt) dt 1

b

~ sin {/x arcsin ( a ) } 1(

for b < a,

a

-~ b + ~/b2 _ a 2 ) s i n ( # 2 )

(158)

for b > a.

The criteria for ht follow from the criterion of equation (154): - ( # + v + 1) < ,k < 1 1 Both integrals are with v - - ~1 for the cosine and v - ~1 for the sine and ~. - - ~. continuous for a -- b. From equation (128) we can derive, in a similar way: if# >-1"

o ° J~ (at) cos(bt) dt cos {# arcsin (~,) } for b < a,

x / a 2 -- b 2

a ~'

(,/b:

sin (# y7/")

+ ,/b:- a:)"

(159)

forb >a

and if # > -2" sin {# arcsin (~) } ~/a 2 -- b 2

tbrb
fo °° Ju (at) sin(bt) dt -

(160) cos

for b > a.

Both the integrals are discontinuous for a - b. Special values for the orders # and v of the Bessel functions in the formulae (155) to (160) lead to a number equations that frequently occur in analytical solutions"

922

Functions

(3.3.5)

F r o m equation (155):

dt

Jl(at)Jl(bt)t

f0

--

b 2a

forb
-a

for b > a

(161)

'

2b F r o m e q u a t i o n (156):

f0 ~

Jo(at) Jl (bt) dt -

0

for b < a,

1 2b

for

_1 b

forb > a

b--a,

(162)

F r o m equation (157):

f0

_1 J1 (at)

c o s ( b t ) dt -

forb
a 0

t

(163)

for b > a.

F r o m e q u a t i o n (158): b

L

oo 1 J1 (at)

sin(bt) dt -

forb
a

t

a b -t- ~/b 2 - a 2

forb >a

and arcsin (

fo ~ 1 Jo(at)

sin(bt) dt -

b

)

forb
a

t

for b > a. 2

F r o m e q u a t i o n (159):

1

forb
a

f0

J1 (at)

c o s ( b t ) dt -

b e t w e e n - and - c~

for b = a,

1 a

forb >a

(166)

a

1 ~/b 2 - a 2

and

1

forb
~/a 2 -- b 2

fo ~ Jo(at)

c o s ( b t ) dt -

(167)

b e t w e e n + co and 0

for b = a,

0

for b > a.

Bessel functions and related functions

(3.3.5)

923

F r o m equation (160): b

for b < a,

a ~/a 2 -- b 2 fO c~ J1 (at) sin(bt) dt --

(168)

between + o e a n d 0

forb=a,

0

forb >a

0 between0and

for b < a, forb=0,

and

fo ~ Jo(at) sin(bt) dt -

+oo

(169)

1

for b > a.

x/b 2 -- a 2

A n u m b e r of W e b e r - S c h a f h e i t l i n integrals can be expressed as elliptic integrals. For instance, with X = - 1 , # -- 0 and v = 1, we get from equation (154)

~l

~aaF

fo

forb
~,~;2;~--~ 1

F ~, - g ; 1. ;5

for b > a.

If we consider the integral representation for the hypergeometric function

F(y) fo 1 t ~ - l ( 1 - t ) × - ~ - l ( 1 -- xt) -~ dt F ( f l ) F ( y - fl)

F(ot,/3; V; x) --

1 1 Y - 2 and (see Section 3.4.2), and we substitute ot - 3,/3 - 3,

F

(11

~, ~ ; 2 ; x

f 1

F(2)

-- Jr

_4f01 {_ Jr

- t, we find

)

r(½)r(~)

4 fol~

172

l-r/2 1 -- X/72

,

1

1

t - ~ ( 1 - t ~ ( 1 - x t ~ - ~ at

dr/

4f01 --jr

1

v/(1 - r/2)(1 - xr/2)

2

l-r/

d77

V/(1 -- r/2)(1

-- xr/2)

1 1 - (1 - X r / 2 ) ] do x ~ ( 1 - r/2)(1 - xr/2) 1 v/1 - xr/2 /

Jr -- Jr

x x

v/(1 - 772)(1 - x r/2)

K ( X ) + x- E ( x )

x V/1 _ r/2

J

do

924

(3.3.5)

Functions

in which K(x) and E(x) represent the complete elliptic integrals of the first and second kind, respectively (see Section 3.5). In a similar way we have

(

1

F

1. lx"

2'2'

'

) 2--

Jr

~

E(x)

o

The final result becomes:

fo ~ t1 Jo(at)Jl(bt) dt b

1 1

2a / b 2 \ 2 =-~-bE~-~)--~ab(a

-F /~

2

b2 - b2)K (~-~)

1 1 a2\ 2 /a2\ -~ , -~;1; -~ ) = -~ E ~-~ )

(170) forb < a , for b > a.

2 • In a similar way The integral is continuous for a - b and has then the value hwe find

fo °° J1 (at) J1(bt) a

dt

1 3 a2 { a2 a2 } F ( ~ , ~; 2; ~ - ~ ) - 2rra K (~--~)- E (~--~)

for a < b, (171)

~(~ ~ ~;~ ) - ~ ~ (~)- ~ (~)

fora > b.

The integral is e~ for a = b and, therefore, discontinuous for a = b.

fo

°° Jo(at)Jo(bt) dt

1F( _

a

11

~)

~'~'"1"' ~

--

1(11 a 2) ~ F ~, ~, 1, ~-~

and = c~ for a = b (discontinuity).

~ K (~)

m

~ a

2 (a 2) ~--~K

forb < a , forb > a

(172)

(3.3.5)

Bessel functions and related functions

925

An integral, related to the Weber-Schafheitlin type is

fo ~ tx Ju (at) K~(bt) dt

2X-lag

F (~z+v+X+l 2 ) F ( u - v+x+ 1 F (/z + 1)

b z+u+l

(173)

× F ( / z + v +,k + 1 /z - v +)~ + 1 a 2) 2 ' 2 ;/Z + 1; -~-~for b > a >~ 0 and - / z + I v l - 1 < ~ (~ unrestricted). If in a hypergeometric function g - fl, its value becomes: F(o~, fl; fl; x) -(1 - x) -~ (see Section 3.4.2). So if in equation (173) /z-v+)~+l

=/Z+I,

or

,% -- /z + v + l,

we easily find

fo

~ t u+v+l J u ( a t ) K v ( b t ) dt -

(2a)U(2b)~F(/z + v + 1) (a 2 -+- b2)#+v+l

(174)

if/z>

[1)[

--

I)

-1,

a >~0andb >0.

As both the expression and the convergence condition in equation (173) do not change if we replace v by - v (K~(x) -- K_~(x)), we also find

fo

~ t u-v+l J u ( a t ) K ~ ( b t ) dt -

if/z> For/Z-

[vl + v

( 2 a ) U ( 2 b ) - ~ F ( / z - v + 1) (a 2 -+- b2)lz-v+l

(175) -1,

a>~0andb>0.

v we find

/0

t J u ( a t ) K u ( b t ) dt -

if/z>-l,

-~ (a).

a2 + b 2

,

(176)

a ~>0andb>0.

Many special results follow from the formulae (173) to (176), among which also the 1 cases for which lZ - + g , yielding combinations of the K-functions with a sine or a cosine. The latter may also be obtained by using equation (143) or equation (144) by replacing c by i b. A number of special cases follow below:

926

(3.3.5)

Functions

From equation (174): and

fo ~ t 2 J o ( a t ) K l ( b t ) dt - (a 2 -q2bb2)2

(177) (178)

fo ~ t 2 J1 (at) go(bt) dt - (a 2 +2a b2)2. From equation (176): fo °° t Jl ( a t ) K l (bt) dt - -~ a a2 _+_ 1 b2

(179)

and

(180)

fo ~ t Jo(at) Ko(bt) dt -- a 2 -at1 b2 . 1 From equation (175) f o r / ~ - ~, v - 1"

fo ~ sin(at)K1 (bt) dt

zra 1 2b Via 2 + b 2

(181)

,,

-

From equation (174) f o r / z - - 7 1, v - 0 : f0 c~ cos(at)Ko(bt) d t -

7r ~/a2 1+ b 2 -~

(182)

From equation (144) we have fo ~ Ko(bt)e -iat dt - arccos (i b) ~/a2 + b:~

fo

arcsinh(b ) Jr 1 --i %/a 2 + b 2 2 ~/a2-+-b 2

Ko(bt){ cos(at) - i sin(at)} dt.

So it follows that f0 e~ sin(at)Ko(bt) dt -

arcsinh(b )

(183)

~/a 2 + b 2

Integrating equation (182) with respect to a, we find foC~l sin(at)Ko(bt) dt _ J r arcsinh(b)

t

2

.

(184)

(3.3.5)

Bessel functions and related functions

927

From equation (173) we have for ~. = 0:

fo

eCJ1 (at) K1 (bt) dt

jra ( 1 3 a 2) 4b 2- F ~, ~; 2;-~--5 jra. b (~ 1. a2 ) 4b 2 Via 2 + b 2 F , ~ , 2 ; a 2 + b 2

(see Section 3.4.2).

We found above (see the derivation of equation (170)) that 1

1

)

4

F~, ~ ; 2 ; x -- ~ { E ( x ) - ( 1 - x ) K ( x ) } , jrX

from which

fo

CX~j1(at) K1 (bt) dt 1

jra

1

a2

)

4b~/a 2 q- b 2 F ~ , ~ , 2 ; a 2 + b

2

~/a2+b2

b

(

ab

a2

_

a 2 + b2

(~85) a2

a~/a 2 + b 2 K(a2+b2)"

Also from equation (173) fo ~

J1 ( a t ) g o ( b t ) d t -

~ F 1, 1; 2 ; a ( a~) 1 a 2 + b2 = +~aa ln( b 2 )

(186)

as easily may be derived from equation (25) of Section 3.4.2 and which result can also be obtained by integrating equation (179) with respect to b, and

fo

J o ( a t ) K o ( b t ) dt -- - -

(

1 1 " l" - a 2

b

(1

2---b ~/a2 + b 2 F -2'

) 1 a2 ) ~; 1; a2 + b2 ,

and thus

f

1

~ J o ( a t ) Ko(bt) dt -

1

a2

2x/a 2 --t-b 2 F ~, ~, 1; aZ_k_b2

) (187)

a2+b 2 ] ~ / a 2 -qt- b2

928

(3.3.5)

Functions

Another integral, related to the Weber-Schafheitlin integral is

fo

~ t Z Ju (at) Yv(bt) dt,

which can be split up into two other integrals by means of the definition for Y~, given in equation (15). The condition for convergence becomes - # + I v l - 1 < )~ < 1. Of interest are some cases, for )~ -- ~1 or )~ - - ~ 1 and /x ----1--1 in which Y~(bt) is combined with sin(at) or cos(at). The evaluation of these integrals can be most easily performed by making use of the equations (141) or (142), for instance:

I - fo ~ Yo(bt)e -ct dt - Jr ~/b22 + C2

In ( ~/b2 ÷b c2 - c ) .

We replace c by i a and find: In ( ~/b2 - b a2 - i a ) I --

fo °

Yo(bt)e -iat dt -

fora
Jr ~/b 2 - a 2 In ( i ~/a 2 - b b2 - ia )

f o r a > b.

ijr ~/a 2 - b 2 As In (v/b 2 - a 2

ia)--ln[v/b

( ~/b 2a- a 2 )

2-a 2-ial-iarctan

-- In b - i arcsin

(;)

,

and ln(ix/a 2 - b 2 - i a )

- In { - i ( a -

- 4 a 2 - b2)}

ln(a - x/a 2

-

b 2)

-

i ~

2' we have 2i

Jr%/b 2 - a I-1

~/a 2 -- 62

I-fo

a 2 arcsin ( ~ )

fora
2i (aJr ~/a 2 _ b2 In

cos(at) Yo(bt) dt - i

~/a 2 b

_b2)

sin(at) Yo(bt) dt,

f0 °

f o r a > b.

(3.3.5)

Bessel functions and related functions

929

from which 1

m

f0 °~ cos(at) Yo(bt) dt

forb
"~/a2 -- b2 between - c~ and 0 0

--

(188) for b = a, for b > a,

f0 ~ sin(at)Yo(bt) dt 2

In ( a - ~/a2b - b2 )

Jr ~/a 2 - 62 --

forb
between

and oe

forb =a,

Jra

2 Jr ~/b 2 - a 2

a

arcsin ( ~ )

for b > a.

Integration of equation (188) with respect to a gives: -ln(a

fo ~ 1t sin(at)Yo(bt) dt -

+

~/a 2 -

- - 1 lna 2 0

b 2)

forb
(190)

for b > a.

Differentiation of equation (190) with respect to b yields: a-

fo ~ sin(at) Y1 (bt) dt -

~/a2 - b 2

-

b~/a2 - b2 between - e ~ a n d O 0

forb
(191)

forb-a, for b > a.

From the equations (84), (162) and (186) the following integral may be deduced:

( a2)

fo

J1 (at) Yo(bt) dt -

1 In 1 Jra b-~ +cx~ 1 In

Jra

-1

fora b.

The integral formula, known as Weber's first exponential integral I --

fo °

tz Ju (at) e-c2t2 dt,

(192)

930

Functions

(3.3.5)

which sometimes occurs in the analysis of geohydrological problems, differs from those considered earlier in this section by containing the square of the variable in the exponential function (t 2 instead of t). To secure convergence at the origin, it must be supposed that/z + ~ + 1 > 0. Expanding Ju(at) in powers of t and integrating term-by-term we find:

O0 n=O

(--1)n(1 a)lZ+2n

dr.

n!F(Iz + n + 1)ao

With t 2 - u the integral becomes: 1fc~ 2 J0

--

U

½x+½r~+n-½e-C2U a n

m

£(n -t- lz+2+l )

if/z -t- )~ -4- 1 > 0.

2cU+~+l+2n

Now, if we choose k - / z + 1 the gamma functions in the series for I disappear and the series reduces to an exponential function: f0 °° tlz+lJlz(at)e-C2t2 dt -- (2c2)1z+ alZ 1 exp ( - ~a2c2 )

(/z > - 1 ) .

(193)

In particular, w i t h / z - O, we have:

f0 ~ t Jo(at)e -c2t2 dt -- ~c12 e 4c2. a2 If we take Z - - ~

I

m

1

and/z-

(194)

~1 in I, we find

~ / -~a 2 f0w1t sin(at)e-C2t2 dt,

an integral that we met with before in equation (21) of Section 3.1.1 as an integral representation of the error function. It is easy to show that in this case the series for I are closely related to the series for the error function, as given in equation (9) of Section 3.1.1. With Z = 0 and lZ = 2v it can be shown that f0 °° J 2 v ( a t ) e - C 2 t 2 d t - - ~ / ~ I v ( ~ - ~ ) e x p ( - - ~ c 2a) 2 2c

(v>

2) 1

for instance, by means of series multiplication. In particular, we have:

(195)

Bessel functions and related functions

(3.3.5)

931

It is not difficult to find, with the help of the equations (15) and (49), the similar integrals with Y2~ and K2v which are convergent for 2v + 1 > 0 and - 2 v + 1 > 0, 1 thus for Ivl < 3" For that reason we give the solution for v - 0 as:

oC~Yo(at)e -c2t2 dt -

and

fo

C~Ko(at)e -c'2t2 dt -

(197)

(198)

Closely related to Weber's first exponential integral is the equality

f0 e~ J o ( a o t ) ( 1 - e - ~ 2 t ) d°~ = 1 E1 (~-i) 02 ot

2

(199)

which follows immediately from equation (194) if we differentiate both sides of equation (199) with respect to t. An important formula, due to Gegenbauer, is

fo zr Cv(~/a 2 + b 2 - 2abcosO) (sin0)2v dO 1 (a 2 -+- b 2 - 2ab cos 0) ~ v ( 1) ( ~ ) C u ( a ) J u ( b ) =2vF v+~ F .a. ~ .. bv

(200)

with v > - 3 1 and 0 < b <~ a, in which C~ is a cylinder function viz. J~, Y~, H~(1) or H~(2). From this formula many other formulae of practical interest may be deduced, for instance, for v - - 0 :

f0 rr J0(v/a 2 -4- 6 2 -- 2abcosO)dO -- rrJo(a)Jo(b),

(201)

f0 rr Yo(~//a 2 + 6 2 --

(202)

2abcosO)dO

-

zrYo(a)Jo(b),

fo rr Ho(1)(~/a 2 + b 2 - 2abcosO)dO - ~rH(ol~(a)Jo(b).

From K o ( z ) -

(203)

~1 rci H~o1)(iz) it follows that

f0 rr Ko(v/a 2 + b 2 - 2abcosO)dO - 7rKo(a)lo(b).

(204)

932

Functions

(3.3.5)

From Weber's first exponential integral in combination with Gegenbauer's integral Weber's second exponential integral can be obtained, if v > -1"

fo c~tJv(at)Jv(bt)e-C2t2 dt - ~cz 1 lv(~cz ab ) exp ( -

a 2 + b 2) 4c 2 .

(205)

We will show the deduction of this formula for v - 0, starting from equation (194), written, with a slight modification, as"

fo ~ otJo (pot)e_tOt2 d o t - 2t1 e ,02 4t and assuming p2 _ r 2 + R 2 _ 2rR cos 0. Now we integrate both sides of this equation with respect to 0 from 0 to 7r, giving

fO cx~ote -t~ 2 fo zr J0(otx/r 2 -q- R 2 - 2rR cos0) dO dot

1 -"-

~

e

r2+R2 4t fo g e-~ cosOdO.

2t The left side becomes rcf~ote and the right side is equal to Jr exp 2t

I0

-t°~2Jo(rot)Jo(Rot)

,

dot, according to equation (201),

because

4t

f0 rr e x cos0 dO - Jr Io(x). The latter integral representation for Io(x) follows from equation (111), as, after integration with respect to 0 between the limits 0 and rr, all terms containing cos(n0) vanish. So we have found

fo °° otJo(rot)Jo(Rot)e-t~2 dot - 2t1 Io -r ~R )

r 2 -k- R 2 ) exp ( -

4t

'

(206)

which is equation (205) for v - 0 . An interesting result is obtained if we apply the Laplace transformation with respect to t to both sides of equation (206). As

1

L {e-tOt2 } - s _it_ot2, the left side of equation (206)becomes

fo cx~S +ot ot2 Jo(rot)Jo(Rot) dot,

Bessel functions and related functions

(3.3.5)

933

and the right side:

fo °° ~71 I 0 ( -rR~ )

exp ( -

r 2 + R2"~e_St dt 4t ) /.2

R2 |e-St\ /

1 foJrfo~ 2t1 exp ( p-2 )47 e-st dt dO ~-

with

- - Jr

1 ~

exp

(rRcosO 2t

+ 4t

lf0~f07r

dO dt

,o 2 - - r 2 -+- R 2 -

2rR

cos 0.

1 exp(- 7/-) p2 The infinite integral represents the Laplace transform of ~7 and equals Ko(Pv/7) (cf. equation (72)). Because, according to equation (204),

1

/0

Ko (p ~/-J-) dO -

{

Xo (r ~/-J-)Io (R ~/-~) f o r r ~> R, f o r r ~< R, Xo ( n ,/7) Io ( r

we have found:

fo

S --]- Ot 2

Jo(rot)Jo(Rc~) dot -

{

f o r r ~< R,

lo(R~/s)Ko(r~/-s)

f o r r ~> R.

(207)

This integral is a particular case of the more general integral formula involving a product of Bessel functions, known as Hankel's integral:

f o ° e X px-2l j_jr_ # (p2 a x ) { ( 1cos ) -~ fro Jv(bx) -k- sin ( 1~) rr0 Y,(bx) } dx (208)

= -pp-2Iu(ap)gv(bp)

with 0 - p + / x -

v

i f 0 < a ~ 0 a n d [ v ] - # < p . For practical geohydrological problems the case 0 = 2 in this equation is of much interest. Then we have

fo ° X 2X p-1 -Jr- p2 J u ( a x ) J v ( b x ) d x -

pP-Zlu(ap)Kv(bp)

(209)

with the conditions: 0 < a ~< b, p > 0 and I v ] - # < p; with p -- 2, # -- 0, v = 0 we find equation (207). Many other integral formulae can be deduced from this equation following from certain combinations of p, # and v. A case which deserves special mention is p = 1 , # = 1, v = 0 :

I --

fo ~

X 2

+1 p2 Jl(ax)Jo(bx)dx - -1 Ii(ap)Ko(bp) P

for a <~ b.

934

Functions

(3.3.5)

If in this case, a >~ b we must write this integral as

I -- f0 ° 1 p 2 Jo(bx)Jl(ax) dx X 2 %_

with a ~> b,

but now, a s / z = 0 and v = 1, the condition lv ] - / * < p is not satisfied and it is not allowable to apply formula (209). However, if we write 1

1 x 2 + p2

X2 + p2

p2

X2

X2 %- p2

we obtain two integrals for I:

l fo°°Jo (bx) J1 (ax) dx

I -- - ~

y1 f0 °° X 2 X +2 p2 Jo(bx)Jl(ax) dx.

The first integral is the discontinuous Weber-Schafheitlin integral of equation (162) 1 for a > b. The second integral is a Hankel and equals zero for a < b and - 7, integral according to equation (208) with 0 --- 2, p = 3 , / , = 0 and v = 1, yielding:

f0 ° X 2 x%-2 p2 J o ( b x ) J l ( a x ) d x - p l o ( b p ) K l ( a p )

fora >b,

and from equation (207) with p = 3,/z = 1, v = 0 and thus 0 -- 4:

fo 00x 2 x+2 p2 J1 (ax)Jo(bx) dx -

- p I 1 (ap)Ko(bp)

This integral is also discontinuous for a = b.

f o r a < b.

Together the two discontinuous

integrals give: 1

- 11 (ap)Ko(bp) Jl (ax)Jo(bx) dx fo ° X2 "a1t- p--------~

for a <~ b,

P

(210) 1

ap 2

1

p Io(bp)Kl(ap)

for a >/b.

As the determinant of Wronski for I0 and K0 gives, according to equation (70): 1

Io(ap)Kl(ap) + I1 (ap)Ko(ap) = ~ , ap we notice that for a -- b the integral of equation (210) is continuous and equals there 71 11 (ap) Ko(ap) For /, -- 0 equation (208) is also valid for a -- 0 and a number of Hankel integrals involving one Bessel function can be deduced, for instance:

(3.3.5)

Bessel functions and related functions

p=3,

v = 1, 0 = 2 :

f

o cxa~ J lx( 2b x ) x 2 + p2

p=3,

v=0,

(211)

dx = p K l ( b p ) ,

0=3:

~0 °° x 2 x+2 p2 p=2,

935

(212)

Yo(bx)dx - pKo(bp),

v = 1, 0 = 1: (213)

fo ° x 2 + X p2 Y1 (bx) dx = - K 1 (bp),

p=2,

v=0,

0=2:

f0 cx)X 2 -JrX p2 p=

1, v = 0 ,

fo

°°X

(214)

J o ( b x ) d x - Ko(bp),

0 = 1:

1 2 _]._ p2 Y o ( b x ) d x -

1 P Ko(bp).

(215)

The integral o~Jlz (ax) Ko(bv/x 2 -+- y2)x/Z+l dx

--aU (

y

%/a2 -]- b 2

)u+l

(216)

K#+I (yv/a 2 -Jr-b2)

with a and b positive and /.t > - 1 is called Sonine's integral and is of some importance in geohydrological problems. It may be evaluated as follows. Replace b by x/7 and consider the function I ( s ) --

f0 °

J u ( a x ) K o ( x / T v / x e + ye)xu+l dx

as the Laplace transform of a function I(t). Then, according to equation (72), we get 1

I(t)--L

l { I ( s ) } - - m2te

4t

xU+l Ju(ax) exp

--

dx,

936

Functions

(3.3.5)

which is Weber's first exponential integral in equation (193). So

22 I (t) -- a lz(2t) tz exp ( - at 2 - - - ] k 4t ] and with the aid of equation (72) and the first shifting theorem (Theorem 4 of Section 2.2.2-1) we find

L{I(t)}--I(s)-at~(

Y ~/s 4-a 2

) tL+' Klz+l(YV/S 4- a2).

For s -- b 2 this is the result stated in equation (216). In particular, for tt - - 3 1 we find, with

J-1/2(z)-

~z

COSZ and

K1/2(z)- ~ z e -z"

fo

~COS(ax)Ko(bv/x 2 4- y2) dx

7r 1 _v~/aZ+h2 2 ~/a 2 4- b 2

~

e

-

o

(217)

Differentiation with respect to y gives:

fo ~ cos(ax) K1 (bx/x 2 + y2) dx x/x2 + y2

--

~

:rre

2by

-y

~/aZ+h2.

(218)

Most integrals of Bessel functions which have been discussed up till now occur in solutions and derivatives of solutions for axial-symmetric groundwater flow in a semi-infinite field (Section 523 of Part A), and have been deduced by means of an infinite Hankel transformation (Section 2.2.4-4). The infinite integral

h(u, r)uexp

~2R 2 du

with

Jo (u ) Yo (-~ u ) - Yo (r ) Jo (-~ u ) h ( u , r) -

(219)

+

is one of a category of integrals of which the Laplace transforms lead to quotients of modified Bessel functions of the second kind, for instance, in this case:

Ko(~R~/'s)

7rfl2R2

f0

u2t h(u, r)u exp

f12 R 2 ,]

du.

(220)

These integrals are of practical importance in solutions for two-dimensional radialsymmetric groundwater flow outside a circular cylinder. Equation (220) has been derived in Section 2.2.2-3, Example 4 by means of the Laplace inversion theorem (Theorem 9).

Bessel functions and relatedfunctions

(3.3.5)

937

From equation (220) we may deduce:

L_I{1 Ko(flr~/'7) } _ s Ko(flRx/7)

2

yrfl2R 2

fotL °°

h(u, r)u exp

(

u2T ) f12R2 du dr (a)

_-- 2jrfo °°h(u, r) { 1 -

exp

(

d/gU flu2t)} 2R2 m.

By definition of the Laplace transform, it follows also from equation (220):

K o(fl r ~/7) 7rf12R2 2fo°°fo°°

Ko( R,/7)

h(u, r)u exp (

-

U2t fl~-~2 )e-Stu du dt (b)

2 fo °° h(u, r) f12R2udu - --Jr S qt_ u 2" Letting now s approach zero, we find

f

oo

h(u,

du

7/"

r ) ~ = -u 2

(221)

(h(u, r) according to (219)). This integral represents the first term in the integral in equation (a), so that we find

s Ko

2f°°l-h(u 7/" U

R,/7)

, r) exp (

u2t f12 R 2 J du.

(222)

Many other Laplace transforms may be deduced from equation (220), for instance (see Section 2.2.2-3, Example 4): L - l { 1_

Ko(flrX/s+~)} + ,7) (223)

Ko(flr ~l-~) Ko(flR~/-~)

2 fo~° u ( Jr rlfl2R2Jr-U2 h(u, r)exp - tit

u2t ) f12R2 du

with h(u, r) according to equation (219). Also by means of application of the Laplace inversion theorem in the complex plane, similar to the transform of equation (220), the following Laplace transform may be obtained: L_l{

Ko(flr~lff ) } 2 fo°° ( .~l~K l (fl R ~lT) -- 7rfl R f (u , r) exp

u2t)

f12R2 du

with (224)

f(u,r) =

938

Functions

(3.3.5)

By differentiation with respect to r, we find:

{

L_ 1 Kl (flr~/7) K1 ( f l R ~ )

g(u, r) --

/

_

2 3Tfl2R2

to"

ug(u r) exp '

J, (u)Y, (-~ u)- Y, (u)J, (-~ u) j2(u ) + y2(u )

u2t flZR2 ) du

with (225)

.

As

Kl(flr~/~) 2fo°°fo°° ( KI(flR~/~) -- 7rfl2R 2 ug(u,r)exp

7rf12R2 2 f0 ~ ~-z-~+s u2 u g(u, r) du, lim [ Kl(flrqrs) ]

R

s-~0 K I ( f l R ~ )

r

u2t f12R2

) st dudt

and

we find for s ~ 0: °°I

lrR

u

2r

L

- g(u, r) du - ~ .

(226)

(g(u, r) according to equation (225)). Remark. The result _n for the limit follows from limx~o[xK1 (x)] -- 1, which may r deduced from the formula for Kl(X) for small values of x (cf. equation (71)).

(3.4.1)

Gamma and hypergeometric functions

939

3.4. GAMMA FUNCTION AND HYPERGEOMETRIC FUNCTION 3.4.1. G a m m a function and related functions Definition F(x)

--

t X - l e - t dt

(x > 0).

(1)

Writing x 4- 1 in place of x we get: F ( x 4- 1) --

/o

tXe - t at -

-

/o

t x de -t -

[-tXe-t]o

+

/o

e - t x t x-1 dt

and thus F(x 4-1)--xF(x)

(x > 0 ) .

(2)

Also F(x)

-- (x -

1 ) F ( x - 1) -- (x - 1)(x - 2 ) F ( x - 2)

= (x - 1)(x - 2)(x - 3 ) . . . (x - n ) F ( x - n)

(x > n).

(3)

Consequently, F (x 4- l) is sometimes called the f a c t o r i a l f u n c t i o n and is written x !. If x is a positive integer n, it follows from equation (2) that F ( 1 4- n ) -- n ( n - 1)(n - 2 ) . - - 3 . 2 . 1

-- n!.

(4)

Also from equation (2)" F ( x + n ) -- (x 4- n -

(5)

1)(x 4- n - 2 ) . . . (x 4- 2)(x 4- 1 ) x F ( x ) .

The integral representation (1) is meaningful only when x is positive. Now we may write equation (5) as F(x)

=

F(x +n) x ( x + 1)(x + 2 ) - . - ( x

+n-

1)

(x :/: 0 , - 1 , - 2 , . . . )

(6)

and use it for defining the g a m m a function for negative values of x (-7(= 0, - 1, - 2 , . . . ) , choosing for n the smallest integer such that x 4-n > 0. Then the equations (1) and (6) together give a definition of F (x) for all x n o t e q u a l to z e r o o r a n e g a t i v e integer. For x = 0 , - 1 , - 2 , . . . the g a m m a function has asymptotes and equals 4-00. The equations (2) and (3) are now also valid for all x -J: 0 , - 1 , - 2 , . . . and from equation (3) it follows that

f(x)

F ( x - n) --

=

(x - 1)(x - 2 ) . . . (x - n)

(-1)hE(x)

( - x 4- 1 ) ( - x 4- 2 ) - . . ( - x 4- n)

and because from equation (5) we have" F(-x

+ n + 1) -- ( - x

4- n ) ( - x

4- n -

1 ) . - . ( - x 4- 1 ) F ( - x

4- 1),

940

Functions

(3.4.1)

we find:

r(x)r(1 - x )

F ( x - n) = ( - 1 ) n

(7)

(x :/: 0, - 1, - 2 , . . . ) .

F ( 1 - x + n)

The product of two g a m m a functions F ( p ) and F(q) can be written as

F ( p ) F ( q ) --

fo

x p - l e -x dx

--

/o

yq-le-y dy

x P - l y q - l e -(x+y) dx dy

fo fo

= 4

x2p-ly2q-le -(xa+y2) dx dy,

where x and y in the double integral have been replaced by X2 and 2 2. Now, if we introduce polar coordinates x = r cos 0 and y --- r sin 0, we get:

F ( p ) F ( q ) -- 4

fo ~

rZP+Zq-le -r2 dr

f~12 (COsO)ZP-l(sinO) 2q-1 dO. dO

The first integral on the right is :1 F ( p + q) which follows immediately if we substitute t - r 2. Furthermore, by setting t d t = - 2 sin 0 cos 0 dO and, therefore,

r(p)r(q) = r(p + q) . fo I tP-l(1

cos20 we have 1 -

t -

sin 2 0,

-- t) q-1 dt.

The integral in this expression is defined as the Beta function:

1

B(x, y) --

f0

t x - l ( 1 - t) y-1 dt

(x > 0, y > 0),

(8)

while we found the relation between this function and the g a m m a function: B ( x , y) --

r(x)r(y) r(x + y)

(9)

= B ( y , x).

Moreover, we found another expression for B(x, y):

B(x, y) -- 2 f0 zr/2 (sinO)Zx-l(cosO) 2y-1 dO. For y = x we find

B(x, x) -- 2

frr/2

(sin 0 cos

0) 2x-1

dO

dO = 21-2x. 2

(sin p) 2x-1 d~o - 21-2XB x,

(10)

Gamma and hypergeometric functions

(3.4.1)

941

and thus according to equation (9)

r ( x ) r ( x ) = 2,_~ r(x)r(½)

V(x + ~)

r'(2x)

I

'

from which the duplication formula may be deduced: 22x-1

/-' ( 2 x ) -

/

/-' (x)/-" ~x +

1

(11)

/

as F ( 21) -- 4%- (see equation (13)). Fory-- 1-x, with0<x < 1 becausey >0andx

B(x, 1 - x) --/-'(x)/-'(1 - x) -- 2

f

> 0 , wefind

~/2

(tan0) 2x-1 dO

dO

and as (tanO) u d O -

fo ~'~

2sec

_~

~/x (~)

forO<#<

1,

we have

F ( x ) F ( 1 - x) =

sin(rrx)

( 0 < x < 1),

(12)

which is called the reflection formula. It can be shown that by means of analytic continuation formula (12) is valid for all values of x not equal to zero or a positive or negative integer. From the preceding formulae we may deduce the values of the gamma function for some special values of the argument. From equation (12) we have 1 ~(~)-,~

and from equation (2):

3 14~-, s 3 3F ~(~)-~ ~(~)~3 ~(~)-~

'

and so on. Also from equation (12) we find:

and thus

~t-~t--2,~

(13)

942

(3.4.1)

Functions

From equation (6): (3)

F

_ F(-3+

-~

F

2) _ F ( ½ )

~

,

~

-~(-~)

=4

-~,/~,

(_ 75) _ _ ~i!--,_~,3~ ~ + 3)~, -- rtl)l~ _- __8~,~5

and so on. The logarithmic derivative of the g a m m a function is called the Psi Function, and is defined as: d

~(x)- ~{ lnr(x)}-

1

d

(14)

r(x) dx r(x).

F r o m the recurrence formula (2) for the g a m m a function the recurrence formula for the ap-function can easily be deduced:

1

~p(x + 1) - ~ ( x ) + - . x

(15)

The psi function is defined for all x --/: 0, - 1 , - 2 , . . . . It follows from the definition (1) of the g a m m a function that dF(x) _

~x

t x-1 ln te -¢ dt

and, therefore, ap (1) -

In te - t dt.

To compute this integral we evaluate I =

In te - t dt

and we then let s approach zero: I - -

In t de -t - - [ In t e - t ] ~ +

- e -t dt - In se -~ + E1 (s) t

(see Section 3.2.1, equation (1)). According to the series representation (7) of the exponential integral, we find

~-,°° (_l)~e~

I - e-~ lne - lne - )L _ ~

n_~

nn!

)/

(16)

and with s -+ 0: ~p (1) = - ~,, where y = 0 . 5 7 7 2 1 5 6 6 5 . . . = Euler's constant, the value of which has been computed in a different way.

Gamma and hypergeometric functions

(3.4.2)

943

Now we are able to compute the psi function for positive integers with the aid of equation (15):

1 1 ~(m+l)----y'+l-4-~+~+---+--

1

(m -- 1, 2, 3 . . . . ).

(17)

m

3.4.2. H y p e r g e o m e t r i c function The differential equation d2y dy x(1-X)~x 2 +{g-(c~+fl+l)X}~xx-Otfly--0,

(18)

where or, fl and V are constants, is known as Gauss's hypergeometric equation. This equation can be solved in the same way as Bessel's equation (1) in Section 3.3.1 by assuming the solution y in the form of a power series with undetermined coefficients" oo

y

-- x v (ao

+

a l x -4- a2 x 2 + ' '

anxn+V.

") -- Z n=0

Then

dy 0° v)xn+V-1 dx = Z an (n +

and

n--O

d2y c~ dx 2 = ~ an(n nt- v)(n + v -

1)x n+v-2.

n--0

These expressions, inserted in equation (18) yield, after dividing by x ~" oo

o~3

a.(,, + ~ ( , , + ~ -

l~x "-~ - ~

n--0

a.(,, + ~)(,, + ~ -

l~x"

n=O oo

-+- ) / ~

oo

an(n + v)x n - l -

oo

(or -t- ~ + 1 ) ~

n=0

an(n + v)x n -oe~ ~

n=0

any ~ -- O.

n=0

This equation is valid if the sum of the coefficients of each power of x is equal to zero, which gives for the coefficient of x - 1. a0v (v - 1) + va0v - 0, from which it follows that v - 0 or v - 1 - ?', while a0 may be chosen arbitrarily. If we choose v - 0 and a0 - 1, it is easy to show that a solution y of equation (18) becomes: y-l+~x+ +

~fi

~(~ + 1)/~(¢~ + 1)

)'

?'(V + 1)

x2 .-2!

or(or + 1)(or + 2)fl(fl + 1)(/3 + 2)

X •

}, ( 9 / + 1)(?, + 2)

(19)

3

~

3~

+

o

o

o

o

944

Functions

(3.4.2)

This series is called the hypergeometric series and its sum by F(ot, 13; y; x) and called the hypergeometric function. As, according to equation (6) of Section 3.4.1, oe(ot -+- 1)(or -q- 2 ) . . . (o~ + n - 1) =

y(x) is usually denoted

F(oe + n )

r(~)

we may write the hypergeometric function also as:

y = F(oe, ~; y; x) =

/-'(y)

~/-'(or

+ n)F(/3 + n) . x'__~ ~"

.=0

r ( × + n)

(20)

The series converges for [x[ < 1 and for x - 1 it converges if g - o l - 13 > 0. From equation (19) it can be seen that for g = - n (n = 0, 1, 2 , . . . ) the hypergeometric series is not defined (provided that ot or fl is not zero or a negative integer). From equation (19) it also follows that the series reduces to a polynomial of degree n in x when ot or/~ is equal to - n (n = 0, 1, 2 , . . . ) and y is not equal to zero or a negative integer. For instance: F(0, fl; T; x) = 1,

(21)

F ( - 1 , 13; y; x) = 1 - - - x .

(22)

From equation (20) it can immediately be seen that (23)

F(cg, 13; y; x) -- F(/~, o~; y; x).

Differentiation with respect to x yields: _ _ _ { d yotfl dx - y

l+(°t+l)(fl+l)x+(°~+l)(°t+2)(fl+l)(fi+2)x2--y + 1 (y + 1)(y + 2)

} 2! ~-""

and thus:

__dF ( o t , / 3-"°yt '¢,x3), dx

--F(u+l, y

fi-+-l;y-+-l;x)

(24)

An important integral representation of the hypergeometric function, of which many special cases may be deduced, is F(o~, fl; y,• x) -

r(y) -

~)

fo ~t ~ - l ( 1

-

t)×-/~-l(1

with y > / 3 > 0. For x = 1 we have from equation (25):

-

x t ) -~ dt

(25)

Gamma and hypergeometric functions

F(oe, fl; y; 1) =

(3.4.2)

[ 1 t~-l( 1 _

/-,(y)

r(/~)r(×

945

t) y-/~-~-I

- ~) Jo

r(×) V(~)r(×

- ~)

v(×)r(~)r(× r(~)r(×

B(/~, × - ~ - c ~ ) -

-

dt

~ -

~)

-

~)

~)r(×

in which B(x, y) is the Beta function (Section 3.4.1, equations (8) and (9)), and thus F(oe, fl; ?,; 1) -

r(y)v(× - ~ - ~ ) F(V -oe)F(y - fl)'

(26)

provided that y :/: 0 , - 1 , - 2 . . . . and y - f i - ot > 0. From equation (25) a number of transformation formulae for F(ot, fl; y ; x ) can be derived by changing the integration variable t in the integral. For instance, if we set u = 1 - t we easily find: F(ot, fl; y" x) -- (1 - x)-~F(ot, y - fl; y;

x

)

x--1

(27) '

which formula is specially suited for negative arguments, as we have already seen 3. a2 when deriving equation (185) of Section 3.3.5, where F(2, ~, 2 ; - ~ ) was transformed into b

F(1

~ / a 2 --}-b 2

~,

1 ~" 2;

a2

)

a 2 -[- b2

Many elementary functions may be expressed as hypergeometric functions for special cases of the constants c~, fl and y. For y = fl we have from equation (19) (not from equation (25)!): x2

x3

F(ol, fi;/3; x) -- 1 + c~x + Ol(Ol+ 1)-Z-: + Ol(Ol + 1)(Or + 2 ) = + . . . Z~

3[

which represents the binomial series (1 - x ) -~ and thus F(c~, fl; fl; x) = (1 - x) -~.

(28)

From equation (25) we have: F(1, 1; 2; x)

--

f01

dt 1 -xt

and thus F(1, 1; 2; x) =

ln(1 - x). x

(29)

946

3.5.

Functions

(3.5)

ELLIPTICINTEGRALS

The elliptic integral of the first kind is defined as fo e

dO x/1 - sin 2 ot sin 2 0

F(~olm) --

(1)

If we substitute t = sin 0, we have dt - cos 0 dO, or dt

dO-

dt

=

cos 0

=

v/1 - sin 2 0

dt ~/1 - t 2

The integration limits b e c o m e 0 and sin ~0 and we find, with x sin 2 c~, another form for the elliptic integral.

F(cplm) --

fo x

sin q9 and m -

dt . V/(1 _ t2)( 1 _ mt2),

(2)

m is called the parameter and ol the modular angle, while ~0 - amplitude - arcsin x. The elliptic integral of the second kind is defined as E(~plm) --

E(~p[m) --

If we set rp

_

x/1 - sin 2 ce sin 2 0 dO

fo ~°

or

(3)

fo x ~ 1 - mt 2 dt. 1 - t2

(4)

7r and so x - 1 in the integrals (1) to (4) the integrals are said to be -f

complete elliptic integrals and are designated as follows:

dO

fo 7r/2

K(m)-E(m) --

f01

~/1 - m sin 2 0

f0 zr/2q/1

dt ~/(1 - t2)(1 - mt 2)

11- m- t 2t dt. - m sin 2 0 dO - f 0 1 ~ ---------7

and

(5)

(6)

The relation of K(m) with the hypergeometric function follows from the integral 1 1 representation of the latter (equation (25)) by setting u - g, /3 - g, y - 1 and x - m and substituting t - u2; then the integral becomes equal to the integral (5), whence zr(11

K(m)-~-F

~,~;1;m

)

.

(7)

(3.5)

Elliptic integrals

With

o~-

- g , 1 fl -

g1

zr

E ( m ) -- -~ F

and

(

y

947

-- 1, we find

1 1 l'm)

2'-2;

(8)

"

The equations (7) and (8) are the series representations for the complete elliptic integrals; they may be written at length, according to equation (19):

1{1

K(m)--~zr

l+(~)

m-t-

~

m-

~

-t- 2 4 6

(9)

(Iml < 1),

1{1 --~--

2 4 6

5 ....

/

(10)

(Iml < 1). In computations with elliptic integrals, the complete integrals with the c o m p l e m e n t a r y p a r a m e t e r m l = 1 - m frequently occur; they are defined as: K(ml)-

F

(2)

and

Iml

E(ml)-

E ( 2 l m l ).

(11)

A number of s p e c i a l c a s e s of the elliptic integrals follows if we assert special values to the parameter and/or the amplitude. From equations (1) and (3) it can immediately be seen that a negative amplitude gives: F(-cplm) = - F ( q g l m ) ,

E(-~plm) = -E(qglm).

(12)

An amplitude of any magnitude q9 + nzr (n = 0, 4-1,-t-2 . . . . ) can be evaluated as follows: dO

F ( cp + n :rr [m ) -- f ~o+ n Tr

v/1 - m sin 2 0

dO

If we substitute 0 - / 3 + nsr, we have dO - dfl, sin 2 0 - sin 2(fi + n:r) - sin 2 fl, while the integration limits become - n z r and 99, respectively. So we have f_~

d~

= f]~

d/~

F ( q) + n Tr lm ) -

,~~r V/1 - m sin 2 fl

,~r V/1 -

m s i n 2 fl

o +

dfl v/1 - m sin 2 fl

948

Functions

(3.5)

T h e s e c o n d integral equals F(~olm) and as the integrand of the first integral is a positive periodic function with period Jr and has a vertical axis of s y m m e t r y in the points fl - 4 - ~ (n - 0, 1, 2 . . . . ), the first integral m a y be replaced by

2n

f Jrl 2 a0

d fi

= 2nK(m).

X/1 -- m sin 2 fl

Thus we f o u n d

F(~o + nzrlm) -- 2 n K ( m ) + F(~olm)

(n - 0, + 1 , 4-2 . . . . ).

(13)

O t h e r special cases are: F(go[O) - go,

E(~o[O) - ~p,

F(goll) - a r c t a n h x ,

7/"

E(~pI1) - x,

(15)

7/"

K (0) -- ~-, K(1)-

(14)

E (0) -- ~ ,

+oo,

E(1)-

(16) 1,

(17)

An i m a g i n a r y amplitude i~o gives, if we substitute 0 - ifl" t~o

F(i~plm) --

/o

dO

-

V/1 - m sin 2 0

Now, with sinh fl --tanc~, or f l 1

d~-

i

~/1 + tan 2 a

q/1 4- m sinh 2 fi

arcsinh(tano~) and

dol

•

fo

da

COS20l

=

COS O/

we find

f arctan(sinh~0)

da

F(icp]m) - i so

cos a ~/i + m tan 2 el

w h i c h can be evaluated to

f arctan(sinh~o)

dt~ with m 1 -- 1 - m

F(i~olm) - i ao or

v/1 - m l sin 2 o~

F(i~olm) -- i F { arctan(sinh~o)[ml}.

(18)

List of symbols

949

LIST OF SYMBOLS (The numbers refer to the pages on which the symbols are defined) Latin-letter symbols A A Ai(p) aL aT B

B

B(x, y) Bi(p) Bn B~ b

ber(x) bei(x)

c{~9(x)} c-l{~(~)} cn{~o(x)} Cnl {q~(n)}

Cnr{(#(x)}

C2rl{~(n)} c D

D Dd DL DT d d dh E

N-1

En(z) E(~plm)

E(m) Eg

system matrix for multi-layer systems 638 surface (area) [L 2] 545 Airy function 903 longitudinal dispersivity [L] 572 transversal dispersivity [L] 572 diagonal matrix for f12 433 width of half a canal, river etc. [L] 19 Beta function 940 Airy function 903 Bernoulli number 18, 489 barometric sensitivity [0] 577 distance between two straight boundaries [L] 95 Kelvin function: real part of I0(oe~/i-) 905 Kelvin function: imaginary part of Io(x~/-{) 905 infinite Fourier cosine transform 692 inverse infinite Fourier cosine transform 692 finite Fourier cosine transform 681 inverse finite Fourier cosine transform 682 finite Fourier cosine resistance transform 688 inverse finite Fourier cosine resistance transform 688 leakage factor [T ] 571 concentration of a solute [ML -3] 544 diagonal matrix 642 dispersion matrix 573, 610 thickness of an aquifer [L] 560 coefficient of diffusion [L2T -1] 545 coefficient of longitudinal dispersion [L2T -1] 545 coefficient of transversal dispersion [L2T -1] 545 differential operator for ordinary differential quotients 631 diameter [L] 556 harmonic mean diameter [L] 556 eigenmatrix of A 640 inverse eigenmatrix of A 641 exponential integral of order n (n = 1,2 . . . . ) 874 elliptic integral of the second kind 946 complete elliptic integral of the second kind 946 modulus of elasticity of the grain skeleton [FL -2] 558

950

List of symbols

Ew

modulus of elasticity of water [FL -2] 548 error function 857 complementory error function 857 function diagonal matrix 641 force in dimensional analysis [F] 536 mass flux [MT -1] 545 elliptic integral of the first kind 946 hypergeometric function 944 infinite Fourier resistance transform 698 inverse infinite Fourier resistance transform 699 specific mass flux [ML-2T -1] 545 flux function 169 goniometric functions in layered aquifers (i -- 1, 2 , . . . ) 842 gravity acceleration [LT -2] 547 diagonal matrix for r]i 436 interface depth [L] 822 high level in phreatic flow [L] 22 Bessel functions of the third kind and arbitrary order (m = 1, 2) 900

erf(x) erfc(x) F

F F(q)lm) F(ot, fi; ?,; x)

Fr{~O(x)} Frl{¢(o')} f

f(u,r) Gi(k,z) g H

H

n(m)(x) n(m) (x) H{~p(r)}

H-l{~(ot)} Hn{qg(r)}

nn- 1{~ (rt) }

nnr{(fl(r)} H~rl{qgF(n)} h I

Iv(x)

I.(x) Im i inerfc(x)

J.(x) J.(x) J K KD Kv(x) Kn(x) K(m)

Bessel functions of the third kind and integer order (m = 1, 2) 900 infinite Hankel transform 712 inverse infinite Hankel transform 713 finite Hankel transform 704 inverse finite Hankel transform 704 finite Hankel resistance transform 711 inverse finite Hankel resistance transform 711 phreatic head (water table) [L] 22 surface water level [L] 59 unit matrix 640 modified Bessel function of the first kind and arbitrary order 893 modified Bessel function of the first kind and integer order 893 imaginary part of a complex function 717 imaginary unit = ~ / ~ 714 as index: integer (i - 1, 2 , . . . , n) repeated integral of the complementary error function (n = 1, 2 , . . . ) 857 Bessel function of the first kind and arbitrary order 881 Bessel function of the first kind and integer order 882 as index: integer (j = 1, 2 , . . . , n) coefficient of conductivity [LT- 1] 560 transmissivity of an aquifer [LZT -1] 560 modified Bessel function of the second kind and arbitrary order 894 modified Bessel function of the second kind and integer order 894 complete elliptic integral of the first kind 946

List of symbols k ker(x) kei(x) L L{F(t)} L-I{F(s)} t

l(u,r) M M(u, ol) Mo(x) m ml

N No(x) n nA 12e

0 Os P Pa

Pi P~ P(x, y)

Pconj(x, y) P Pa Pt Pi

ph

Q Qi

q

qi

R

R(x, y)

951 coefficient of intrinsic permeability [L 2] 537 modified Kelvin function: real part of Ko(x~/-{) 906 modified Kelvin function: imaginary part of Ko(xv/-{) 906 low level in phreatic flow [L] 22 length in dimensional analysis [L] 536 Laplace transform 652 inverse Laplace transform 652 length of a screen or filter [L] 16 leakage function 172 mass in dimensional analysis [M] 544 M-function 869 modulus of Io(xv/-{) 906 parameter in elliptic integrals 946 complementary parameter in elliptic integrals 947 Newton [F] 530 modulus of Ko(x ~/-{) 906 volumetric porosity [0] 553 areal porosity [0] 553 effective porosity [0] 554 specific surface [L -1] 555 specific standard surface [L -1] 555 poise [FL-ZT] 530 Pascal [FL -2] 530 instantaneous abstraction from a line well [L 3] 775 phreatic sensitivity [0] 579 polder function 862 conjugate polder function 862 precipitation [L T- 1 ] 816 pressure [FL -2] 531 atmospheric pressure [FL -2] 531 total pressure [FL -2] 531 eigenvalues of the system matrix A (i -- 1, 2, ..., n) 639 instantaneous abstraction from a well-screen [L 2] 70 as index: phreatic discharge from a line well [L3T -1] 530 total discharge through an area [L3T -1] 521 discharge impulse from a line well [L3T -1] 775 infiltration [LT -1] 779 strength of uniform flow [LeT -1] 729 discharge from a drain [LZT -1] 17 discharge impulse from a well-screen [LZT -1] 70 radius [L] 649 resistance function 865

952

Re

]"w

S

Ss Sph s{~o(x)} S - 1{q3(o/) }

s.{~o(x)} syl{(o(n)} Snr{qg(x)}

Snrl {q~(n) } s Sc(U, r)

Sl(U, r) T T

U u

u

Uto (t) V v v

W(x, y)

W (Yl, Y2) to Wp X

r.(x) r.(x) y

List of symbols

Reynolds number 541 real part of a complex function 717 polar-, cylindrical-, spherical coordinate [L] 591,590, 589 well radius [L] 149 storage coefficient [0] 576 specific storage coefficient [L-l] 576 phreatic storage coefficient [0] 577 infinite Fourier sine transform 689 inverse infinite Fourier sine transform 690 finite Fourier sine transform 679 inverse finite Fourier sine transform 680 finite Fourier sine resistance transform 686 inverse finite Fourier sine resistance transform 687 Laplace parameter 661 storage function for confined aquifers 177 storage function for leaky aquifers 179 diagonal matrix for the transmissivity 460 time in dimensional analysis [T] 536 period in periodic functions [T] 676 transmissivity [LZT -1] 560 time variable [T] 517 parameter 720 U-number [0] 556 real velocity vector [L T - 1] 517 real part of complex number w 714 unit step function 655 volume [L 3] 517 velocity vector [L T - 1] 517 imaginary part of complex number w 714 well function 877 Wronskian 634 complex number (= u + i v) 714 entrance resistance IT] 618 weight percentage in a sieve diagram 556 Cartesian horizontal coordinate [L] 589 real part of complex number z 714 Bessel function of the second kind and arbitrary order 885 Bessel function of the second kind and integer order 885 Cartesian horizontal coordinate [L] 589 imaginary part of complex number z 714 Cartesian vertical coordinate [L] 589 complex number (= x + iy) 714

95 3

List of symbols

Greek-letter symbols

Oln

F r(x)

g

A ~(x - xo)

lTi

0 Oo(x)

0

)~

Jr

p Pf

Ps Pw o"

interface parameter [0] 822 parameter of hypergeometric functions 943 modular angle of elliptic integrals 946 roots of a goniometric equation (n --0, 1, 2 . . . . ) 663 roots of a Bessel equation (n = 0, 1, 2 , . . . ) 650 non-steady flow parameter [L-1T 1/2] parameter of hypergeometric functions 943 circulation [LZT-1] 603 gamma function 939 Euler's constant 875 specific weight [FL -3] 547 parameter of hypergeometric functions 943 difference operator 804 small value 516 Kronecker delta 660 resistance parameter [0] 685,710 complex number in vertical flow (= x ÷ iz) 16 non-steady leakage parameter [T -1] 54 spherical coordinate (angle) 589 stream function in general three-dimensional flow [0] 526 resistance parameter in multi-layer systems [L -1] 436 polar-, cylindrical-, spherical coordinate (angle) 591,590, 589 phase of Io(x~/i) 906 resistance parameter [T -1] 90, 351,865 storage parameter [0] 126, 201,663 leakage parameter [L] 54 eigenvalues of a matrix 642 dynamic viscosity [FL-ZT] 549 anisotropy parameter [0] 805 as index: arbitrary order of a Bessel function 912 kinematic viscosity [LZT-1] 550 as index: arbitrary order of a Bessel function 880 product sign 737 pi-theorem 536 density [ML -3] 546 spherical coordinate [L] 343 density of fresh water [ML -3] 826 density of salt water [ML -3] 826 well radius [L] 343 summation sign 674 storage parameter [T -1] 92, 865

954 O'g O"s 75

~F epH ~o(x) q)

lPF

~H ~(x) £2 £2F £2H 60

O9 1

List of symbols

ground stress [FL -2] 533 solid matrix stress [FL -2] 533 shear stress [FL -2] 536 integration variable in convolution integrals 655 potential function [L2T -1] 538 Forchheimer potential function [L3T -1] 815 Herzberg potential function [L3T -1] 822 phase of Ko(x~/7) 906 head or drawdown [L] 532 amplitude of elliptic integrals 946 stream function in two-dimensional flow [L2T -1] 528 stream function in axial-symmetric flow [L2T -1] 528 Forchheimer stream function [L3T -1] 816 Herzberg stream function [L3T -1] 822 psi-function 942 complex potential function (= 4~+ i 7r) 717 complex Forchheimer potential function (= ~bF+ i~PF) 816 complex Herzberg potential function (= q~H+ i Tell) 822 vorticity IT -1] 600 period parameter [T -1] 793 phase parameter [L -1] 795 Other symbols

8 V Vn t

differential operator for partial differential equations 647 gradient 537 Laplacian operator 588, 638 factorial symbol 939

Index

955

Index (The numbers refer to the pages on which the subjects are defined and/or treated, or first mentioned)

A Accretion 622 Adsorption 606 Airy differential equation 902 functions 903-905 Alternative expressions 744 Amplitude 793, 946 Analogons 521 Analytical methods 631-855 solutions 7-508 Analytic function 714-719 Angular deformation 598 velocity 600 Anisotropic conductivity 562-568 dispersion 572 Ground, soil 803-813 Areal porosity 553 Artesian groundwater 533 Atmospheric pressure 531 Axial-symmetric flow 353-422, 590 B

Badon-Ghyben (approximation) 821 Barometric sensitivity 577-578 Bernoulli numbers 18, 489 Bessel differential equation 880, 893 functions 880-938 Beta function 940 Binomial series 945 Boundary conditions 613-630 Branch point 666 C Cartesian coordinates 589 Cauchy-Riemann (equations) 542, 716 Cavitation point 630 Chlorine (contents) 546

Circulation 603 Classical pumping tests 581 Closed hydrological screen 785 Coefficients (geohydrological-) 520 Common boundaries 619 Complementary error function 857 parameter 947 Complete elliptic integrals 946 Complex potential function 717 Compressibility of soil 558-560 of water 547-548 Concentration (of a solute) 544 Conductivity 550-568 Confined aquifer 533 Conformal mapping 720-722 transformation 714-744 Conjugate Polder function 862 Conservation principle 520 Consolidated grounds 551 Consolidation 560 Continuity equation 584-585 Continuous discharge (recharge) 778 Continuum 515-522 Control volume 584 Convective solute transport 545, 607 Convolution theorem 655-656 Correlation 582 Cramer's rule 634 Curve fitting 581 Cylinder functions 931 Cylindrical coordinates 590 D

Darcy's law 534-541 Darcy (unit) 561 Data curve 581 Deformation (rate of-) 529 Density 546-547 Density flow 485-508, 595-597, 821-831

Index

956

Depletion function 635 Diagonal matrix 642 Differential equations 585-612 Diffusion 544-545 Dimensional analysis 536 Direct integration 631-633 Directional conductivity 566 Discharge impulse 775-781 Discontinuous parameters 144 systems 144, 459-470 Dispersion 471-484, 518-520, 544-545, 572-575, 606-612 Dispersivity 572 Divergence theorem 832 Duplication formula 941 Dupuit (approximation) 814 Dutch polder profile 53 Dynamic viscosity 549

G Gamma function 939-943 Gauss' divergence theorem 832 hypergeometric function 943-945 Gegenbauer's integral 931 Generalized Fourier series 702 Generating function 907 Geohydrology 7-855 de Glee 748 Grain stress 533 Green's first formula 833 second formula 833 Ground 7-855 Ground parameters 551-560 Groundwater 7-855 Groundwater parameters 546-551 H

E

Effective porosity 554 Eigen matrix 641 values 639 vectors 639 Einstein ('s summation convention) 564 Elasticity of a grain skeleton 558 of water 548 Elastic storage 575-580 Elevation head 532 Elliptic integrals 946-948 Entrance resistance 618 Equipotential line 542 surface 542 Error function 857-862 Euler 's method 584 's constant 875 Even function 676 Exponential integral 874-877 F

Factorial function 939 Fick's law 544 Field tests 580-583 Flow net 543 Flux function 169 Forchheimer potential 30, 815 Fourier integrals 674-678 series 674-678 transformations 674-700 Fourier-Bessel series 705 Function diagonal matrix 641

Half-range expansion 676 Hankel integral 933 transformations 701-714 Hantush ('well function) 877-879 Harmonic function 717 weighted mean diameter 556 Head (piezometric) 532 Head function 163 Herzberg potential 502, 821 Heterogeneous fluid 595 ground 586 H-functions (Bessel) 900 Hodograph 627-630 Homogeneous differential equation 633 fluid 585 ground 586 l'H6pital ('s theorem) 772 Hubbert ('s force potential) 539 Hydrological screen 781-787 Hydrometric analysis 551 Hydrostatic pressure 532 Hypergeometric function 943-945 I

Ideal tracer 612 /-function (Bessel) 893 Image (point, curve, region) 720 Images (method of-) 766-774 Impervious boundary 619 Impulse function 615, 660 Impulses (method of-) 775-781 Incomplete elliptic integrals 946 Infiltration 779 Initial value 613-616

Index

957

Initial value problem 635, 776 Injection term 594-595 Instantaneous discharge (recharge) 775 Interface 624-627 flow 485-508, 821-831 parameter 822 Intrinsic permeability 557-558 Inverse eigenmatrix 641 Inversion theorem (Laplace) 661-669 Irrotational flow 600 Isohypse 542 Isotropic conductivity 560-562 ground, soil 805 J

J-function (Bessel) 881 Joukowski transformation 727-735 K

Kelvin functions 905-907 K-function (Bessel) 894 Kinematic viscosity 550 Kozeny-Carman formula 558 Kronecker delta 616, 660 L Laboratory tests 580 Lagrange (method of-) 584 Laminar flow 541 Laplace differential equation 589 transformation 651-674 Layered aquifer 568-571, 839-846 Leakage factor 571 function 172 Leaky aquifer 594 Level curve 720 surface 524 Linear boundary conditions 613-621 deformation 598 differential equation 756 Linearity theorem 652 Line well tests 582 Longitudinal dispersion 518 M

Matrix function 637 Macroscopic level 516 Mass conservation 584 flux 545 M-function 869-873 Microscopic level 515 Mixed boundary values 321 Model 805

Modified Bessel functions 893-900 Bessel differential equation 893 Modular angle 946 Modulus of elasticity of ground 558 of water 548 Molecular diffusion 515 level 515 Motion (equation of-) 523-545 Multi-filter tests 581 Multi-layer system 431-470, 637-646, 846-855 N Newtonian fluids 549 Non-homogeneous differential equation 633 Non-linear boundary conditions 621-630 Non steady flow 586 Norm (of an orthogonal function) 701 Normal stress 529 Numerical (methods of solution) 521 O Obstacles (in uniform flow) 250-251,728-735 Odd function 676 One-dimensional groundwater flow 51-148 Open boundary 616-619 hydrological screen 784 water storage 618 Ordinary differential equation 631-646 Orthogonal function 701-703 Orthonormal function 701 Oscillation 795 P

Parameters 546-551 Parameter (in elliptic integrals) 946 Partial differential equations 647-755 Particle 515 Particular solution 633 Pascal (law of-) 531 Path (of a water particle) 516 Pathline 517, 524 Period 793 Periodic flow 793-803 function 674 Permeability 537 Permeameter test 558 Phase 796 Phreatic aquifer 533 flow 15-49, 813-821 sensitivity 579 storage 577 surface 533, 621-624 Physical plane 722

Index

958

Piezometric head 532 surface 533 Pi-theorem 536 Point well test 582 Poisson ('s equation) 602 Polar coordinates 591 Polder function 862-865 Pole 663 Porosity 553-555 Porosity factor 557 Potential flow 538, 600 function 538 Precipitation 816 Pressure 529-534 force 535 head 532 Product solutions 788-793 Prototype 805 Psi function 942 Pumping tests 580 Q Quadratic transformation 722-727 Quarter of a plane 224 Quasi-steady state 696 R

Radial-symmetric flow 149-222, 591 Real velocity 517 Recharge impulse 775 Reciprocity principle 832-855 Recovery test 581 Recurrence relations Bessel functions 889, 896, 900 error function 857 exponential integral 874 gamma function 937 psi function 942 resistance function 868 Refraction of streamlines 620 Reflection formula 941 point 630 Repeated integral 857 Representative elementary volume 516 Residues 663 Resistance function 865-869 parameter 865 Resistivity 568 Retardation factor 607 Reynold ('s number) 541 Riemann surface 727 Rotation 598 Rotational flow 597-606

s

Saturated zone 553 Scale factor 805 Schwarz-Christoffel transformation 735-744 Seepage face 624 Separation of variables 647-651 Shape factor 557 Shearing force 536 Shifting theorem (first and second) 654 Sieve analysis 551 curve 552 Simplification (method of-) 763 Single-filter tests 581 Skin effect 582 Solute 544 Sonine's integral 935 Special pumping tests 581 Specific discharge 517 mass flux 545 standard surface 555 storage 576 surface 555-557 weight 547 yield 577 Spherical coordinates 589 Spherical-symmetric flow 343-352, 590 Stagnation point 628 Statistical description of soils 551-553 Steady flow 588-589 Stokes 550 Storage function 177, 179 parameter 865 Stream function 523-528, 542-544, 591-594, 602-606 surface 524 tube 526 Streamline 517, 524 Stress 529 Stress diagram 534 Sturm-Liouville 684 Successive transformations 744-753 Summation convention 564 Superposition 756-787 Survey of integral transforms 754-755 System matrix 638 T Tensor conductivity- 564 dispersion- 572 permeability- 565 Three-dimensional groundwater flow 423-429

Index Tidal fluctuations 793 sensitivity 578-580 Time lag 795 Tracer 518 Translation 598 Transmissivity 560 Transversal dispersion 518 Travelling time 13 Turbulent flow 541 Two-dimensional groundwater flow 223-342 Type curve 581 U Unconsolidated grounds 551 Uniform flow 608 Unilateral discharge 162 Unit matrix 640 Unit step function 655 U-number 556 V Variation of parameters 633-636 Velocity vector 538 Velocity difference vector 627

959

Viscosity 548-551 Volume flux 523 Volumetric porosity 553 Vorticity 600 W

Water repelling screen 786 table 813 Weber-Schafheitlin integrals 918 Weber's first exponential integral 929 second exponential integral 932 Wedge 229 Weight 535 Weight function 702 Well function 877 Wronskian 634, 890 Y Y-function (Bessel) 885