modeling groundwater flow to elliptical lakes and through multi-aquifer elliptical inhomogeneities

Advances in Water Resources 27 (2004) 497–506

www.elsevier.com/locate/advwatres

Modeling groundwater flow to elliptical lakes and throughmulti-aquifer elliptical inhomogeneities

Mark Bakker

Department of Biological and Agricultural Engineering, University of Georgia, Athens, GA 30602, USA

Received 8 December 2003; received in revised form 11 February 2004; accepted 12 February 2004

Available online 25 March 2004

Abstract

Two new analytic element solutions are presented for steady flow problems with elliptical boundaries. The first solution concerns

groundwater flow to shallow elliptical lakes with leaky lake beds in a single-aquifer. The second solution concerns groundwater flow

through elliptical cylinder inhomogeneities in a multi-aquifer system. Both the transmissivity of each aquifer and the resistance of

each leaky layer may differ between the inside and the outside of an inhomogeneity. The elliptical inhomogeneity may be bounded

on top by a shallow elliptical lake with a leaky lake bed. Analytic element solutions are obtained for both problems through

separation of variables of the Laplace and modified-Helmholtz differential equations in elliptical coordinates. The resulting equa-

tions for the discharge potential consist of infinite sums of products of exponentials, trigonometric functions, and modified-Mathieu

functions. The series are truncated but still fulfill the differential equation exactly; boundary conditions are met approximately, but

up to machine accuracy provided enough terms are used. The head and flow may be computed analytically at any point in the

aquifer. Examples are given of uniform flow through an elliptical lake, a well pumping near two elliptical lakes, and uniform flow

through three elliptical inhomogeneities in a multi-aquifer system. Mathieu functions may be applied in a similar fashion to solve

other groundwater flow problems in semi-confined aquifers and leaky aquifer systems with elliptical internal or external bound-

aries.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Elliptical lake; Elliptical inhomogeneity; Groundwater–surface water interaction; Analytic element method

1. Introduction

The topic of this paper is the analytic element solu-

tion of groundwater flow problems in semi-confined

aquifers or leaky aquifer systems with elliptical internal

or external boundaries. Solutions are obtained through

separation of variables in elliptical coordinates, which

are known to lead to infinite series of Mathieu functions

[25]. This approach has become practical since the recent

development of general algorithms for the computationof Mathieu functions [2,3]. Analytic element solutions

for two specific problems will be presented: the inter-

action of groundwater with shallow elliptical lakes in

single-aquifer and multi-aquifer systems, and ground-

water flow through elliptical inhomogeneities in multi-

aquifer systems.

Lake–groundwater interaction has been studied

extensively in vertical cross-sections with numerical

E-mail address: [email protected] (M. Bakker).

0309-1708/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.advwatres.2004.02.015

models [4,26,28,35] and with analytic models [5,6].

Townley and Davidson [29] applied a boundary integralapproach to study horizontal plane flow to fully pene-

trating circular and elliptical lakes. A three-dimensional

finite-difference model was applied to study shallow

circular lakes by [30]. Several techniques have been

developed to model flow to lakes of which the overall

water budget is specified and the corresponding lake

stage is computed. These include the lake packages for

MODFLOW and for Gflow [16]. Exact solutions existfor horizontal uniform flow to circular and elliptical

lakes that penetrate the aquifer fully (e.g. [32]); these

solutions may be extended to include pumping wells

through conformal mapping and the method of images.

Kacimov [21] presented an exact solution for three-

dimensional flow to a hemispherical lake in an infinitely

deep aquifer. Bakker [8] presented an exact solution

for uniform flow to a shallow circular lake with a leakylake bed in a multi-aquifer system. In this paper, an

analytic element solution is presented for shallow

elliptical lakes in single- and multi-aquifer systems;

T T τ

c

h*

x

z

Fig. 1. Definition sketch: cross-section through a shallow lake.

498 M. Bakker / Advances in Water Resources 27 (2004) 497–506

either the lake stage or the net water budget of the lake

may be specified.

Many analytic element solutions exist for flow

through inhomogeneities. Inhomogeneities are defined

as bounded domains with different homogeneous aqui-fer properties on the inside and outside of the domain.

Solutions for steady-state flow in single aquifers include

flow through polygonal inhomogeneities [33], circular

inhomogeneities [11,31], and elliptical inhomogeneities

[27,32,34]. A solution for cylindrical inhomogeneities in

a multi-aquifer system was presented by [9]. Solutions

for three-dimensional flow through ellipsoidal inhomo-

geneities were developed by [13,17,24]. A transientsolution for circular inhomogeneities was presented by

[14]; they used analytic elements in the Laplace domain

and a numerical back-transformation. All these analytic

element solutions may be used to model inhomogenei-

ties that are accurate up to machine accuracy, provided

enough terms are used in the series expansion, or, in case

of a polygonal inhomogeneity, a polynomial of high

order is used for the line-elements [18]. Polygonalinhomogeneities are applied regularly to model regional

flow (e.g. [7,22]). Circular and ellipsoidal inhomogenei-

ties have been used to study effective aquifer and

transport properties of highly heterogeneous media in

infinite aquifers, e.g. [19,20]. In this paper, a new solu-

tion is presented for the modeling of steady flow through

elliptical cylinder inhomogeneities in multi-aquifer sys-

tems.

2. Single-aquifer flow

2.1. Problem statement

Consider steady-state, Dupuit–Forchheimer flow in a

piece-wise homogeneous, isotropic aquifer; the Dupuit–

Forchheimer approximation is interpreted to mean that

the resistance to vertical flow is neglected within an

aquifer (e.g. [32]). A Cartesian x; y; z coordinate system

is adopted. Flow in the aquifer is governed by Laplace’s

differential equation

o2Uox2

þ o2Uoy2

¼ r2U ¼ 0 ð1Þ

where the discharge potential U ½L3=T � is defined, for

confined flow or for linearized unconfined flow, as

U ¼ Th ð2Þ

where T ½L2=T � is the transmissivity of the aquifer and

h ½L� is the piezometric head.

The aquifer contains a shallow elliptical lake with a

fixed water level h�. At the bottom of the lake is a leaky

resistance layer with a resistance c ½T � to vertical flow

(Fig. 1). The lake is in hydraulic contact with the aquifer

over its entire area. Leakage through the lake bed is

approximated as vertical and may be computed as

qz ¼ ðh h�Þ=c ð3Þwhere qz ½L=T � is the vertical specific discharge through

the lake bed. Flow in the aquifer below the lake isgoverned by the modified-Helmholtz equation [32, Sec-

tion 14]

r2/ ¼ /=k2 ð4Þwhere the discharge potential below the lake is defined

as

/ ¼ sðh h�Þ ð5Þand where s is the aquifer transmissivity below the lake.

k ½L� is the leakage factor, defined as

k ¼ffiffiffiffiffisc

pð6Þ

The components of the comprehensive discharge vector,the vertically integrated specific discharge, may be ob-

tained from a discharge potential U (either U or /)through differentiation

Qx ¼ oUox

Qy ¼ oUoy

ð7Þ

An analytic element equation will be derived for an

elliptical lake in a general flow field. The lake may have

a net amount of groundwater inflow, referred to here as

the net discharge A0 ½L3=T � of the lake (A0 is positive if

the net amount of inflow is positive). The head is con-

tinuous everywhere in the aquifer. The flow component

tangential to the elliptical boundary is discontinuous if Tand s are not equal, but otherwise the flow is continuouseverywhere as well.

2.2. Elliptical coordinates

An analytic element solution is obtained through

separation of variables of the governing differentialequations in terms of elliptical coordinates. First, a local

X , Y coordinate system is defined as shown in Fig. 2.

The complex transformation from the x; y system to the

X , Y system is

W ¼ X þ iY ¼ ðw w0Þeih ð8Þwhere w ¼ xþ iy, w0 is the complex coordinate of the

center of the ellipse, and h is the inclination of the major

ψ = π/2 ψ = 0

ψ = –π/2 ψ = – π

ψ = π

η = η*

V

V Y

X

θ η = 0

x

y

Fig. 2. Elliptical coordinates g (dashed) and w (solid); are focal

points.

M. Bakker / Advances in Water Resources 27 (2004) 497–506 499

axis (Fig. 2). The elliptical coordinates g, w are defined

as [25, p. 17]

X ¼ d cosh g cosw Y ¼ d sinh g sinw ð9Þwhere 2d is the focal distance of the ellipse. The coor-

dinate g is constant along confocal ellipses and varies

from zero along the line connecting the foci to infinity at

infinity. The w coordinate varies from p to p, jumps

from w to þw across the line connecting the foci, and

is discontinuous by 2p along the negative X -axis to the

left of the left focus (Fig. 2). The boundary of the lake

corresponds to g ¼ g�, such that the length of the longsemi-axis is d cosh g� and the length of the short semi-

axis is d sinh g�. The two-dimensional Laplacian in terms

of elliptical coordinates is [25, p. 18]

r2U ¼ 1

d2ðcosh2 g cos2 wÞo2Uog2

�þ o2U

ow2

�ð10Þ

The elliptical coordinates are written as a complex

coordinate x ¼ g þ iw and the relationship (9) becomes

[34]

W ¼ d coshx ð11ÞA unique inverse transformation is

x ¼ g þ iw ¼ ln½W =d þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðW =d 1ÞðW =d þ 1Þ

p� ð12Þ

The correct jump in w, as described above, is obtainedwhen the arguments of ðW =d 1Þ and ðW =d þ 1Þ are

chosen ½0; p� for Y P 0 and ½p; 0� for Y 6 0 [32]. This

may be achieved in most programming languages when

the argument of the square root in (12) is programmed

as shown, rather than as ðW =dÞ2 1.

2.3. Analytic element equations

The solution for the discharge potential outside the

lake is obtained through separation of Laplace’s equa-

tion [25, p. 19]. One of the possible solutions is a linear

function of g, which may be written as

U ¼ A0

2pg ð13Þ

and represents an elliptical equipotential boundary with

a net discharge A0 in an infinite aquifer (e.g. [12, Eq.(334.25)]). A solution representing an elliptical lake with

zero net discharge in a general flow field is added to this

solution. The separated solution of Laplace’s equation

as a function of g and w is [25, p. 19]

U ¼ ðAeng þ BengÞ � ½C cosðnwÞ þ D sinðnwÞ� ð14Þ

where n is an integer separation variable and A, B, C, Dare constants. As this part of the solution represents a

lake with zero net inflow, it must vanish at infinity and

therefore cannot contain any exponentials with positive

powers. The general solution for the potential outside

the lake may be written as

U ¼ A0

2pg þ

X1n¼1

½An cosðnwÞ þ Bn sinðnwÞ�eng ð15Þ

where the coefficients An and Bn will be computed such

that the boundary conditions along the elliptical

boundary are met (Section 2.4). An expression for the

discharge vector outside the lake may be obtained from

differentiation of the potential as explained in the

Appendix A.

The solution for the discharge potential under the

lake is obtained through separation of the modified-Helmholtz equation in elliptical coordinates. Moon and

Spencer [25, p. 20], show that the general solution may

be written as (changing their result to the notation used

in this paper)

/ ¼ F ðgÞGðwÞ ð16Þ

where F fulfills

o2Fog2

a

�þ d2

2k2coshð2gÞ

�F ¼ 0 ð17Þ

and G fulfills

o2G

ow2þ a

�þ d2

2k2cosð2wÞ

�G ¼ 0 ð18Þ

where a is a separation constant. Eqs. (17) and (18) are

modified-Mathieu differential equations of the radial

and circumferential kind, respectively, and a is also re-

ferred to as the Mathieu characteristic number. Solu-tions to these differential equations are referred to as

modified-Mathieu functions. For a recent overview of

the work on Mathieu functions, see [2]. The literature on

Mathieu functions is somewhat confusing, as there

seems to be no agreement on the notation. Abramowitz

and Stegun [1] present a table with comparative nota-

tions between leading works. The practical evaluation of


Mathieu functions has become possible since the publi-

cation of algorithms and C++ subroutines for the

computation of Mathieu functions for orders up to 200

and values of d=k up to 800 by Alhargan [2,3]. As the

routines of Alhargan are used in this paper, the notationof Alhargan is adopted as well.

There are two kinds of solutions to the modified

circumferential Mathieu differential equation (18). Cir-

cumferential Mathieu functions of the first kind are

called Qonðd=k;wÞ and Qenðd=k;wÞ, where Qon are odd

functions of order n, and Qen are even functions; the odd

functions start at n ¼ 1, the even functions at n ¼ 0. An

example of functions of the first kind is shown in Fig. 3.Circumferential functions of the second kind are non-

periodic. As the second kind functions create disconti-

nuities in the flow domain (recall that w varies from pto p), they are not used here.

There are also two kinds of solutions to the modified

radial Mathieu differential equation (17). The first kind

even and odd functions of order n are Ienðd=k; gÞ and

–3 –2 –1 0 1 2 3

–1

0

1

ψ

Qe,

Qo

Qe

Qo

Fig. 3. First kind, third order circumferential modified-Mathieu

0 0.1 0.2 0.3 0.4 0.50

1

2

3

η

Io, I

e, K

o, K

e

5Ko

5Ke

Io

Ie

Io’,

Ie’,

Ko’

, Ke’

Fig. 4. Third order radial modified-Mathieu function

Ionðd=k; gÞ, respectively. The even and odd nomencla-

ture in the context of radial Mathieu functions merely

indicates that they are complimentary to the even and

odd circumferential Mathieu functions (i.e. QenIen con-

stitutes a solution to Helmholtz’s equation). The func-tions Ien are finite at g ¼ 0 and have a zero derivative

Ie0n ¼ dIen=dg at g ¼ 0; conversely, the functions Ion are

zero at g ¼ 0 and have a finite derivative at g ¼ 0. Both

Ion and Ien and their first derivatives approach infinity

for g approaching infinity (Fig. 4). The second kind even

and odd functions of order n are Kenðd=k; gÞ and

Konðd=k; gÞ, respectively. Both Kon and Ken and their

first derivatives are finite for g ¼ 0 and approach zerofor g approaching infinity (Fig. 4).

As w jumps by j2wj across the line connecting the

foci, the radial functions of the second kind (Ken and

Kon) cannot be used, as they would create a jump in

either the function or the flow across the line connecting

the foci. The general solution for the potential in the

aquifer below the lake may now be written as

–3 –2 –1 0 1 2 3

–4

–2

0

2

4

ψ

Qe’

, Qo’

Qe’

Qo’

functions (left) and their derivatives (right) for d=k ¼ 5.

0 0.1 0.2 0.3 0.4 0.5–15

–10

–5

0

5

10

15

20

η

5Ko’

5Ke’

Ie’

Io’

s (left) and their derivatives (right) for d=k ¼ 5.

A

A’

h=h*

h = h*

λ

H

Fig. 5. Example 1. A lake (gray area) in uniform flow. Head contours

(dashed) and pathlines (solid); vertical cross-section along A–A0 (bot-

tom figure).


/ ¼ a0Qe0ðd=k;wÞIe0ðd=k; gÞ

þX1n¼1

½anQenðd=k;wÞIenðd=k; gÞ

þ bnQonðd=k;wÞIonðd=k; gÞ� ð19Þ

where an and bn are coefficients that will be computed

such that the boundary conditions are met (Section 2.4).

It may be seen that the potential is continuous across the

line connecting the foci as follows. The functions Qen are

even and thus give the same value for w and w. Thefunctions Qon are odd and jump across the line con-

necting the foci, but since all Ion equal zero for g ¼ 0,the terms QonIon are again continuous. An expression

for the discharge vector in the aquifer under the lake

may be obtained from differentiation as is explained in

Appendix A, where it is also shown that the discharge

vector is continuous across g ¼ 0.

2.4. Implementation and solution of the coefficients

The discharge potential outside the lake is given by

(15) plus the discharge potential of any other featuressuch as uniform flow, wells, line-sinks, line-doublets, or

other elliptical lakes. The discharge potential for the

aquifer under the lake is given by (19). Implementation

of the discharge potential requires the truncation of the

series in (15) and (19). The truncated series still fulfill the

governing differential equations exactly (as every term in

the series does), and thus truncation affects only the

accuracy with which the boundary conditions are met;this effect is investigated in Section 2.5. Both the series in

(15) and (19) are truncated at N terms, so that each

series has 2N þ 1 unknown coefficients for a total of

4N þ 2 unknown coefficients.

Continuity of head in the aquifer from just outside

the lake to just below the lake may be written as

hþðg�;wÞ ¼ hðg�;wÞ ð20Þ

where the superscripts �)’ and �+’ indicate evaluation

just outside the lake and just below the lake, respec-

tively; recall that g ¼ g� is the boundary of the lake. Eq.

(20) may be written in terms of potentials, using (2) and

(5) as

sUðg�;wÞ T/ðg�;wÞ ¼ h� ð21Þ

This equation is applied at 2N þ 1 collocation points at

equal intervals Dw along the elliptical boundary of the

lake; this results in 2N þ 1 linear equations in An, Bn, an,and bn.

Continuity of flow in the aquifer normal to the

elliptical boundary below the lake may be written as

Qþg ðg�;wÞ Q

g ðg�;wÞ ¼ 0 ð22Þ

This equation is applied at the same 2N þ 1 collocation

points, also resulting in 2N þ 1 linear equations. The

resulting system of 4N þ 2 linear equations for 4N þ 2

unknown coefficients may be solved with a standard

method. It may be beneficial to apply boundary condi-

tions at more than 2N þ 1 collocation points and solve

the resulting overdetermined system in a least squaressense (e.g. [18]), although this was not done in the cur-

rent study.

Alternatively, the discharge A0 of the lake may be

specified and the water level h� may be treated as an

unknown. In this case, Eqs. (21) and (22) must be

reorganized such that terms containing the now known

value of A0 appear on the right-hand side of the equa-

tions and terms containing h� in (21) appear on the left-hand side of the equation.

2.5. Example 1: A lake in uniform flow

The objective of the first example is to demonstrate

the accuracy of the solution and to show how the

solution can be used to study lake–groundwater inter-

action. Consider an elliptical lake in a uniform flow field

of magnitude Qx0 ½L2=T � aligned in the positive x direc-

tion (from left to right in Fig. 5). The transmissivity T isthe same outside the lake and under the lake; the leakage

factor is k. The long semi-axis of the lake is 9k and the

short semi-axis is 3k; the long axis makes an angle of 30�with the x-axis. The net flow into the lake is zero and the

water level in the lake is h�. The solution outside the lake

consists of the potential for uniform flow [32] plus the

potential due to the lake (15); the potential under the

h=h*

h=h*

λ

Fig. 6. The limiting case of h ¼ h� touching the lake boundary; head

contours (dashed) and enveloping pathlines (solid).


lake consists of Eq. (19). The problem is solved using 16

terms in the series representations and the results are

shown in Fig. 5. The gray area represents the lake and

head contours are dashed; the contour interval is

20Qx0k=T . Note that the contour representing h� passesthrough the center of the lake. Groundwater flows into

the lake to the left of this contour, and out of the lake to

the right of this contour.

Pathlines are computed through numerical integra-

tion of the velocity field using a standard predictor–

corrector method (solid lines in Fig. 5). Fourteen

pathlines are started equally spaced at x ¼ 50k, and at

z ¼ 0:1H where H is the aquifer thickness. Eight path-lines end at the lake. The two thicker lines represent the

envelope of pathlines that either end in the lake, flow

under the lake, or originate at the lake. These pathlines

were started from the two points on the boundary of the

lake where Qg ¼ 0, and were traced both forward and

backward. To illustrate underflow, 9 pathlines were

started from the starting point of pathline A–A0. A

projection of these pathlines on the vertical x; z plane isshown in the bottom part of Fig. 5. The pathlines started

at z ¼ 0:05H and z ¼ 0:15H represent underflow.

To assess the accuracy of the solution between the

collocation points, the average and maximum absolute

difference in head jDhj and normal flow jDQgj across theelliptical boundary below the lake are computed at

8N þ 4 equally spaced points along the boundary. The

results for different values of N are reported in Table 1.All four quantities decrease with increasing N ; for

N ¼ 24, Dh and DQg reach machine accuracy for this

specific example. Experimentation has shown that the

number of terms in the series required for an accurate

solution is larger for more elongated lakes. For example,

when the long axis of the lake is doubled to 18k, anaccuracy of jDhjavg < 1� 1012 m is obtained with

N ¼ 29 terms, while the original problem with a longaxis of 9k required N ¼ 21 terms to reach the same

accuracy.

Next, the water level in the lake is lowered such that

the lake has a net inflow; the uniform flow remains

unchanged. The water level h� is lowered until the lake is

gaining water over its entire area. The limiting case oc-

curs when the minimum value of the aquifer head along

the boundary of the lake is h�. The limiting case isachieved iteratively with the bisection method and is

Table 1

Approximate error as a function of truncation N

N

8 12 1

jDhjavg (m) 1.06· 103 7.74· 107 2

jDhjmax (m) 3.90· 103 2.93· 106 5

jDQgjavg (m2/d) 2.80· 103 5.65· 106 8

jDQgjmax (m2/d) 1.57· 102 3.20· 105 4

shown in Fig. 6. Note that the head contour for h ¼ h�

now touches the boundary of the lake, and the head in

the aquifer below the lake is nowhere smaller than h�.The two thick pathlines in Fig. 6. represent the envelope

of pathlines that either end at the lake or flow below the

lake. Even though the lake is gaining water over its

entire area, there is still some underflow; pathlines that

represent underflow continue between the two envelop-ing pathlines on the right side of the lake. To arrive at

the situation where there is no underflow, the lake level

should be lowered further until the flow normal to the

boundary of the lake ðQgÞ is zero or towards the lake

everywhere.

2.6. Example 2: A well near two lakes

The purpose of the second example is to demonstrate

that the solution can be used to simulate flow to wells

near multiple lakes, and that capture zones of wells near

lakes can be quite complex. Consider two lakes in an

aquifer with a constant transmissivity and a leakage

factor k below the lakes (Fig. 7). Lake 1 has a long semi-axis of 10k and a short semi-axis of 6k. Lake 2 has a longsemi-axis of 15k and a short semi-axis of 3k. The water

level h�1 in lake 1 is higher than the level h�2 of lake 2.

Groundwater flows from infinity towards both lakes;

6 20 24

.22· 109 3.40· 1012 1.32· 1014

.37· 109 8.90· 1012 7.11· 1014

.97· 109 1.28· 1011 1.66· 1013

.83· 108 6.64· 1011 1.16· 1012

h = h1*

h = h2*

λ

Fig. 7. Example 2. A well near two lakes with h�1 > h�2; head contours

(dashed) and pathlines toward the well (solid).


there is no uniform flow. A well is situated at a small

distance from lake 2. The problem is solved with N ¼ 30

terms for lake 1 and N ¼ 40 terms for lake 2. This results

in a solution that is accurate up to machine accuracy for

lake 1, and along lake 2 the approximate error is

jDhjavg ¼ 2� 106 m. A more accurate solution may be

obtained by increasing the number of terms for lake 2,although this does not lead to visually different results.

The value of jDhjavg decreases by an order of magnitude

for every 8–9 additional terms. When N ¼ 92 terms are

used, jDhjavg < 1� 1012 m for this complicated case of

an elongated lake with a pumping well and another lake

nearby.

Contours of the head are shown in Fig. 7; the contour

interval is ðh�1 h�2Þ=2 and the gray areas are the lakes.The contours h ¼ h�1 and h ¼ h�2 are bold. Lake water

leaks from lake 1 into the aquifer inside the contour

h ¼ h�1. Lake water leaks from lake 2 into the aquifer

inside the contour h ¼ h�2. Twenty pathlines are started

equally spaced around the well and are traced against

the flow; pathlines are started at z ¼ 0:1H . Three path-

lines flow under lake 2; two of the three pathlines orig-

inate at lake 1, but one of the three pathlines flows frominfinity under a small section of lake 1 towards the well.

Note that if the latter three pathlines were started at a

higher elevation in the aquifer, they follow the same

horizontal path, but originate in lake 2.

C1

C2

C3

CM+1

T2

T1

TM

c1

τ1

τ2

τM

cM+1

c2

c3

x

z

y

Fig. 8. Problem definition of an elliptical inhomogeneity in a multi-

aquifer system.

3. Multi-aquifer flow

3.1. Problem description

The solution presented in the previous sections is

extended for use in multi-aquifer systems. Consider

steady-state flow in a multi-aquifer system consisting of

M aquifers and M leaky layers. Aquifers and leaky

layers are numbered from the top down, so that leaky

layer m is on top of aquifer m (Fig. 8); all aquifers and

leaky layers are homogeneous and isotropic, and flow is

confined everywhere. The Dupuit–Forchheimer approx-

imation is adopted for flow in aquifers; flow in leaky

layers is approximated as vertical. Flow in the aquifersystem is governed by the matrix differential equation

[9,10,12,15,23]

r2~U ¼ A~U ð23Þ

where ~U is the discharge potential vector with compo-

nents

Um ¼ Tmðhm h�Þ ð24Þ

where Tm is the transmissivity of aquifer m, hm is the head

in aquifer m, and h� is the water level in the lake on top

of the aquifer system; if such a lake is not present, h� iszero. A is the system matrix, a tri-diagonal M by Mmatrix with diagonal terms

Am;m ¼ 1

cmTmþ 1

cmþ1Tmð25Þ

and off-diagonal terms

Am;m1 ¼1

cmTm1

Am;mþ1 ¼1

cmþ1Tmþ1

ð26Þ

where cm is the resistance of leaky layer m. Eigenvalue mand corresponding eigenvector m of the system matrix

are called lm and~mm, respectively.For a semi-confined aquifer system (i.e. c1 is finite),

the general solution for the discharge potential may bewritten as [10,15]

~U ¼XMm¼1

um~mm ð27Þ

where umðx; yÞ fulfills

r2um ¼ um=k2m ð28Þ

and where leakage vector m is defined as

km ¼ 1=ffiffiffiffiffiffilm

p ð29Þ


For a confined aquifer system (i.e. c1 ¼ 1), the system

matrix has one zero eigenvalue, and the general solution

for the discharge potential becomes [10]

~U ¼ uL~T þXM1

m¼1

um~mm ð30Þ

where uLðx; yÞ fulfills Laplace’s equation (1), and ~T is a

vector with the transmissivities of the aquifers.

An analytic element equation will be derived for an

elliptical cylinder inhomogeneity. The transmissivities of

the aquifers and the resistances of the leaky layers maydiffer between the inside and the outside of the elliptical

cylinder; the elliptical cylinder may be bounded on top

by a lake with a fixed water level h� and a resistance c1 ofthe lake bed. The head is continuous in each aquifer, as

is the component of flow normal to the elliptical

boundary in each aquifer. A solution is again sought in

terms of elliptical coordinates g and w.

3.2. Analytic element equations

The leakage factors and eigenvectors of the system

matrix outside the inhomogeneity are called Km and ~V m,

respectively, such that the potential vector ~U outside the

inhomogeneity may be written, according to (30), as

~U ¼ UL~T þ

XM1

m¼1

Um~V m ð31Þ

where ~T is a vector with the transmissivities of the

aquifers outside the inhomogeneities, and UL is given by

(15)

UL ¼A0

2pg þ

X1n¼1

½An cosðnwÞ þ Bn sinðnwÞ�eng ð32Þ

The functions Um must vanish at infinity, and may be

represented by the series (15) but with the radial Mat-

hieu functions of the first kind replaced by functions of

the second kind, which vanish at infinity. This gives

Um ¼ C0;mQe0ðd=Km;wÞKe0ðd=Km; gÞ

þX1n¼1

½Cn;mQenðd=Km;wÞKenðd=Km; gÞ

þ Dn;mQonðd=Km;wÞKonðd=Km; gÞ� ð33Þ

where Cn;m and Dn;m are constants.The leakage factors and eigenvectors of the system

matrix inside the inhomogeneity are called km and ~vm,respectively. If the inhomogeneity is bounded on top by

a lake, the potential ~/ inside the inhomogeneity may be

written, according to (27), as

~/ ¼XMm¼1

/m~vm ð34Þ

otherwise, it becomes

~/ ¼ /L~s þXM1

m¼1

/m~vm ð35Þ

where ~s is a vector with the transmissivities inside the

inhomogeneity. In both cases, /m is given by (15)

/m ¼ c0;mQe0ðd=km;wÞIe0ðd=km; gÞ

þX1n¼1

½cn;mQenðd=km;wÞIenðd=km; gÞ

þ dn;mQonðd=km;wÞIonðd=km; gÞ� ð36Þwhere cn;m and dn;m are constants. The function /L con-

sists of a constant plus an infinite series of terms of the

form (14). As the potential inside the inhomogeneity

must remain continuous across the line connecting the

foci, exponential terms are grouped as follows [34]

/L ¼ a0 þX1n¼1

½an coshðngÞ cosðnwÞ þ bn sinhðngÞ sinðnwÞ�

ð37Þwhere an and bn are constants.

3.3. Implementation and solution

The discharge potential for an inhomogeneity in amulti-aquifer system was presented in the previous seven

equations. Implementation and solution of the coeffi-

cients is analogous to the solution for an elliptical lake in

a single-aquifer (Section 2.4). Infinite series are truncated

at N terms, and the potential inside and outside of the

inhomogeneity each have Mð2N þ 1Þ unknowns, whereM is the number of aquifers. The unknowns are com-

puted by requiring continuity of head and normal flow at2N þ 1 points along the inhomogeneity boundary in

each aquifer. The resulting system of Mð4N þ 2Þ linearequations may be solved with a standard method.

3.4. Example of multi-aquifer flow

To demonstrate the validity of the multi-aquifer

solution, consider three elliptical cylinder inhomogene-ities in a two-aquifer system. The transmissivity of both

aquifers is equal to T . Inside the inhomogeneity on the

left, the transmissivity in the top aquifer is 50T , while inthe two inhomogeneities on the right the transmissivity

in the bottom aquifer is 50T ; otherwise, the aquifer

properties are the same as outside the inhomogeneities.

The leakage factors are K and k outside and inside the

inhomogeneities, respectively. The leakage factors areshown graphically on the upper left-hand corner of Fig.

9a. A uniform flow is applied from left to right. The

problem is solved and contour lines of the head are

shown in Fig. 9a; solid lines represent the top aquifer

and dashed lines the bottom aquifer. Contour lines of

the vertical leakage between the two aquifers are shown

Λλ

(a)

(b)

Fig. 9. Multi-aquifer example of uniform flow through three elliptical

inhomogeneities: (a) head contours in top (solid) and bottom (dashed)

aquifers, shaded areas are elliptical inhomogeneities; (b) contours of

vertical leakage, shaded areas represent upward leakage.


in Fig. 9b; the shaded areas indicate upward leakage. Asthe inhomogeneity on the left has a much higher trans-

missivity in the top aquifer, the leakage on the left side

of the flow field is upward, with the highest value of the

leakage occurring near the left tip of the left ellipse. In

the middle section of the flow field, water starts to leak

to the bottom aquifer, where the transmissivity is higher

in the two right ellipses; the highest values of leakage

again occurs near the tips of the ellipses. On the rightside of the flow field the leakage is again upward so that

it can distribute equally over the two aquifers down-

stream of the inhomogeneities.

4. Summary and conclusions

Analytic element solutions were presented for flow to

elliptical lakes and elliptical inhomogeneities in multi-

aquifer systems. Both the transmissivity of each aquifer

and the resistance of each leaky layer may differ between

the inside and the outside of an inhomogeneity; an

inhomogeneity may be bounded on top by a lake with a

fixed stage and a leaky lake bed. Solutions were ob-tained through separation of variables and the use of

modified-Mathieu functions; the Mathieu functions

were computed with the algorithms of Alhargan [2,3].

Complicated flow fields of multiple elliptical lakes,

elliptical inhomogeneities, and wells were solved ana-

lytically. Mathieu functions may be applied to solve

other groundwater flow problems in semi-confined

aquifers and leaky aquifer systems with elliptical inter-nal or external boundaries; solutions for leaky aquifer

systems may also be used for the modeling of quasi-3D

flow in stratified aquifers. The solutions allow for the

analytic computation of head, flow and vertical leakage

at any point in the aquifer system.

Acknowledgements

The author thanks Ed Veling of Delft University of

Technology for suggesting the use of Mathieu functions

for the modeling of groundwater flow. The presented

solutions are implemented in the open-source computer

program TimML (www.engr.uga.edu/~mbakker/timml.

html). This research was funded in part by USDA grant2002-34393-12117.

Appendix A

The discharge vector is derived corresponding to a

discharge potential Uðg;wÞ (either U or /). The com-

ponents of the discharge vector (7) are written in com-

plex form (e.g. [32]) and derivatives of the

transformations (8) and (12) are applied to give

Qx iQy ¼� oU

ogþ i

oUow

�dxdW

dWdw

¼� oU

ogþ i

oUow

�eih

d sinhxðA:1Þ

where i is the imaginary unit. The derivatives of U and /with respect to g and w are elementary for the expo-

nential and trigonometric functions used in this paper

and are not presented here; algorithms and routines for

the derivatives of the modified-Mathieu functions arepresented in [2,3].

Next, it is shown that the discharge vector corre-

sponding to the potential (19) is continuous across the

line connecting the foci of the ellipse. It is sufficient to

show that the components QX and QY in the X , Ycoordinate system are continuous. Along the line

http://www.engr.uga.edu/~mbakker/timml.html

http://www.engr.uga.edu/~mbakker/timml.html


connecting the foci, g is zero, so that x ¼ iw, and (A.1)

simplifies to

QX iQY ¼� o/

ogþ i

o/ow

�1

di sinwðA:2Þ

and thus

QX ¼ 1

d sinwo/ow

QY ¼ 1

d sinwo/og

ðA:3Þ

o/=ow in (19) consists of a sum of terms Qo0nðwÞIonð0Þ

and Qe0nðwÞIenð0Þ. The former terms are zero asIonð0Þ ¼ 0. The latter terms result in a continuous QX

across g ¼ 0, because both Qe0n and sinðwÞ are odd

functions, and thus

Qe0nðwÞIenð0Þd sinw

¼ Qe0nðwÞIenð0Þd sinðwÞ ðA:4Þ

A similar argument holds for QY . o/=og consists of a

sum of terms QonðwÞIo0nð0Þ and QenðwÞIe0nð0Þ. The latter

terms are zero as Ie0nð0Þ ¼ 0. The former terms result in a

continuous QY across g ¼ 0, because both Qon and

sinðwÞ are odd functions, and thus

QonðwÞIo0nð0Þ

d sinw¼ QonðwÞIo0

nð0Þd sinðwÞ ðA:5Þ

The analysis in the previous four equations may be re-

peated to demonstrate that the discharge vector corre-

sponding to potential /L (37) is continuous across g ¼ 0.

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