Analytical solutions of geohydrological problems

PREFACE 1. Design and scope of the book Analytical solutions of geohydrological problems consist of mathematical formu...

Author: G. A. Bruggeman

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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


(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)


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.


(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)


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.


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 .

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 ~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 /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 0,

0

forx / 0,

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

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)


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)


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 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)


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)


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)


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)


¢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)


123.57 (D) As 123.56 with

(x)

q q(x)-

b

0

forO<x forx>b.

O,

f12X2

4t )"

Ox

0

for t -- O, fort >0.

(124a)


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


(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

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 (~-~)


(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


(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.

--

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

+ Jt]

(D)


~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)


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}.


(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)


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


(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)


•

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]

~ B"

O 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~ 0.

(215b)


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)


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).


(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)


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,


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


(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.


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,


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


(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.


(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


(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,


(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)


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 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


(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.


(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)


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)


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


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)


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)


~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.


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.


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)


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


(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


,

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.

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)


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)


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)


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


(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 (


(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)


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)


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)


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)


•

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.

[, 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)


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


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


(0,

216

(242)


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 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 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 0 together with y > 0, forx 0,

~o(0, y, t) = h

for y 0,

~o(0, y ) = 0

fory~~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

for0~0.

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 (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~/a,

See 323.06 with m -- ~1

and

~o( e ~ )

for 0 ~ 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


(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 0, 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


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)


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,


(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 t B.

with

tanh2( )


(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,

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 ~~ 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 ~ (L - B),

w lm),

m~

~° (c~, z) - o,

h [arcsintanh m120 + }I K(m)

(355b)


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 ) ,


(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)

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

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

~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 (L+B),

~(o, z) = h

for (d - l) < x < (d + l),

~(oo, 0x

009 - - ( o , z) = o Ox

for0~ 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


(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

==~ 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~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


(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,

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)


[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.

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)


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~ O,

a(o)

-

o.

~o(~, t) - O,

(410a)


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)


~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

/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)


~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~ 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 /,

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)


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 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)


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

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)


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


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)


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~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)


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,


(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


(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


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)


383

533.02 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)


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) ~


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)


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)


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)


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"


(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


(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)


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


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,

for0o 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) v ap~.~u! mOld

408


(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"


(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


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)


~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)


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


) (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,

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


(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) 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~ 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.


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.


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.


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)


--

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)


(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.

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

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~ 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 , 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 ~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.


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 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 ~ / - .

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 (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


(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


(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)


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


(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,


(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)


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


(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. •

.


(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)


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.


(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


(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


(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)


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,


(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)


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.


(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)


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> 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 )


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)


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)


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,


(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)


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


(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)


(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)


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


(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


(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.


(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


(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


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)


(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


(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


(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)


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)


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


(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:


( 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


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 )


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)


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.


(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)


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)


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


(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)


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)


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,


(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.


(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 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)


(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


(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)


(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


(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:


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


(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)


(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)


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)


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


(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


(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)


(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


(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)


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)


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


(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)


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.


(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


(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)


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


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

'


(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


(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


--

(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)


(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 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~ 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)


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~-+


(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


(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.


(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


(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 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


(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)


(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)


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


(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 )


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)


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)


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 )


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)


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.


(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)


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 b, yields two solutions

(2.2.5-4)


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.


(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)}


(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


(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 )


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)


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)


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)


(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)


(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)


¢ =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


(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)


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)


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.


(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~ 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


(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)


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)


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


(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


(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.


(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


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 0 and y > 0, forx 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)


~=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)


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,

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( )


(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

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)


(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


(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)


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


(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)


(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


(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


(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)


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)


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

~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)


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)


(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)


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)


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

,-


(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;


(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)


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)

"


(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)


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)


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)


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 ~,


(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)


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)


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)


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)


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


(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 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+


(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)


(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

~.
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)


921

The end result is, if # > 0: f0 ~ -f1 Ju (at) cos(bt) dt _

~cos

1(

/zarcsin

a

a

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 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

(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.

F r o m e q u a t i o n (158): b

L

oo 1 J1 (at)

sin(bt) dt -

forb a

and arcsin (

fo ~ 1 Jo(at)

sin(bt) dt -

b

)

forb a. 2

F r o m e q u a t i o n (159):

1

forb a

(166)

a

1 ~/b 2 - a 2

and

1

forb a.


(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 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)


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)


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 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 b.

(3.3.5)


929

from which 1

m

f0 °~ cos(at) Yo(bt) dt

forb - 3 1 and 0 < b - 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)


(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 / 3 > 0. For x = 1 we have from equation (25):

-

x t ) -~ dt

(25)


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

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