Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue5.2000:

Advances in Water Resources 23 (2000) 461±474

The method of images for leaky boundaries
Erik I. Anderson *
SEH, Inc., 421 Frenette Drive, Chippewa Falls, WI 54729, USA
Received 15 July 1999; accepted 20 October 1999

Abstract
An analytic solution is presented describing ¯ow to a drain in a semi-in®nite domain bounded by a leaky layer of constant
thickness. The solution is developed by applying the method of images to two parallel boundaries: an inhomogeneity boundary and
an equipotential boundary. It is then demonstrated that the solution for the problem with the leaky layer approximated by a leaky
boundary (a mixed boundary condition) may be obtained by allowing the thickness, h , and the hydraulic conductivity, k  , of the
leaky layer to vanish while holding the ratio h =k  constant. A method of images for leaky boundaries is proposed, in which a drain
is imaged with respect to a leaky boundary by an image drain and an image line dipole. The method of images for a leaky boundary
is applied to solve the problem of ¯ow to a horizontal drain in a semi-con®ned aquifer. Ó 2000 Elsevier Science Ltd. All rights
reserved.
Keywords: Groundwater; Analytic; Method of images; Leaky boundary; Semi-con®ned

1. Introduction
Groundwater ¯ow domains often contain thin layers
of low hydraulic conductivity that act as leaky barriers

to ¯ow. A typical example, a€ecting the interaction of
groundwater and surface water, is a low-permeability
layer of sediment or silt lining a streambed. The leaky
layer may have a major impact on both heads and discharges in an aquifer and cannot be ignored in the
mathematical formulation of the problem. Barriers are
often included in engineering design. Waste isolation is a
common example; engineered systems for waste isolation may include geomembranes, bentonite blankets,
grout curtains and slurry walls.
The ¯ow within a leaky layer is approximated often
as being one-dimensional and the physical layer is replaced by a mixed boundary condition. We refer to this
type of boundary as a leaky boundary. The e€ects of a
leaky boundary can be included in numerical studies in
an approximate fashion [12], but exact solutions have
proven dicult to obtain. Polubarinova-Kochina [13,
p. 376] presents a general approach for solving problems
with leaky boundaries by conformal mapping; Van der
Veer [18] applies the method to solve two problems. The

*

Fax: +1-715-720-6300.
E-mail address: eanderson@sehinc.com (E.I. Anderson).

approach is applicable to problems with horizontal
leaky boundaries, and vertical equipotential and impermeable boundaries. This is a signi®cant restriction on
the types of problems that may be investigated analytically; for example, the approach does not apply to
problems of ¯ow in semi-con®ned aquifers (i.e. aquifers
bounded on top by a leaky boundary and on the bottom
by an impermeable base). Van der Veer [19,20] presents
an approach for solving problems in semi-con®ned
aquifers based on the superposition of basic solutions.
However, the resulting solutions do not satisfy the
conditions along the leaky boundary; an explanation of
the error is provided herein. Bruggeman [4] presents
several solutions, expressed in Fourier series, to problems in semi-con®ned aquifers.
Problems of ¯ow in semi-con®ned aquifers are often
solved by using the Dupuit approximation. Use of the
Dupuit approximation results in a governing di€erential
equation which is often simpler to solve than the exact
equation. Strack [16] presents several cases. Dupuit solutions are widely used and generally accepted in engineering practice. Bear and Braester [3], however,

demonstrate that application of the Dupuit approximation to cases of semi-con®ned ¯ow may result in
signi®cant errors in discharge.
In this paper we develop a general approach, based
upon the method of images, for solving problems with
leaky boundaries. The approach may be applied to more

0309-1708/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 0 9 - 1 7 0 8 ( 9 9 ) 0 0 0 4 4 - 5

462

E.I. Anderson / Advances in Water Resources 23 (2000) 461±474

general problems than the approach presented by
Polubarinova-Kochina and is applicable to problems of
semi-con®ned ¯ow. The method of images is a classical
technique for solving boundary value problems; the
method is frequently used for solving problems governed by Laplace's equation, such as those encountered
in the ®elds of groundwater mechanics, hydrodynamics,
and electrostatics. Maxwell [9] cites Thomson [17] as the

originator of the method of electrical images.
In groundwater mechanics the method of images is
applied frequently to solve steady and transient problems in two and three dimensions. Strack [16] presents
several steady and transient solutions in two dimensions.
Hantush and Jacob [6] solve the problem of transient
¯ow to a well in a leaky strip by applying the method of
images; their solution is based on the Dupuit approximation. Muskat [11], Haitjema [5] and Steward [15],
solve various three-dimensional ¯ow problems; in each
case the author considers ¯ow in a con®ned aquifer to a
partially penetrating well. Shan et al. [14] applies the
method of images to model air ¯ow to a partially penetrating well; in this case the top of the aquifer is
modeled as a constant pressure boundary.
The method of images is applied to satisfy conditions
along equipotential and impermeable boundaries in
most cases. Muskat [11], following Maxwell [9], shows
that the method of images may also be applied to inhomogeneity boundaries. Fitts [8] presents an approximate approach, loosely based on the method of images,
for solving three-dimensional groundwater ¯ow problems in strati®ed aquifers.
In the following, we extend the method of images to
include leaky boundaries. First, we develop the method
of images for a leaky layer ± a thin layer of low hydraulic conductivity bounded on top by an equipotential. Second we demonstrate that the leaky boundary ± a

boundary of zero thickness and ®nite resistance ± is
a limiting case of the leaky layer. Next, we propose a
method of images for a leaky boundary. Finally, we
apply the method of images for a leaky boundary to
solve the problem of ¯ow to a horizontal drain in a semicon®ned aquifer.

2. Problem description
We consider steady, two-dimensional groundwater
¯ow to a drain in a semi-in®nite aquifer bounded by a
horizontal leaky layer of thickness h , as illustrated in
Fig. 1. We adopt a Cartesian coordinate system with the
y axis pointing vertically upward; the problem is formulated in terms of the complex coordinate z ˆ x ‡ iy.
The drain, represented by the dot in the ®gure, is located
at z ˆ zd . The boundary B divides the complex z plane
into two domains, D and D . The domain D contains
the aquifer and D contains the leaky layer. The upper

Fig. 1. De®nition sketch: ¯ow from a leaky layer to a drain.

boundary of the leaky layer is an equipotential of head

/0 , and is represented by the dashed line in the ®gure.
We de®ne a complex potential in each domain
z in D X ˆ U ‡ iW;
z in D X ˆ U ‡ iW ;

…1a†
…1b†

where the speci®c discharge potential is
U ˆ k/;




…2a†


U ˆk / ;

…2b†





W and W are stream functions and X and X are analytic functions of z; k and / are the hydraulic conductivity and head, respectively, in the aquifer; k  and / are
the hydraulic conductivity and head in the leaky layer,
respectively. The boundary B links the two domains
with the conditions that both the head and the normal
component of ¯ow are continuous across B. The ®rst
condition, expressed in terms of the two potentials is
z on B

Uˆ

k 
U:
k

…3†

The second condition, expressed in terms of the two
stream functions, is
z on B

W ˆ W :

…4†

These are the standard conditions that apply along the
boundary of an inhomogeneity (e.g. [16, p. 412]).
It is often assumed that the ¯ow in the leaky layer
may be approximated well as one-dimensional ¯ow. The
domain D is then eliminated from the problem and
replaced by a boundary condition on D
y ˆ 0;

qy ˆ

/ ÿ /0
;

c

…5†

where qy is the speci®c discharge in the y direction, /0
represents the ®xed head above the leaky layer and c is
the resistance of the leaky layer
cˆ

h
:
k

…6†

Eqs. (5) and (6) de®ne the leaky boundary. The leaky
boundary will provide a good approximation of the

E.I. Anderson / Advances in Water Resources 23 (2000) 461±474

e€ects of a leaky layer when the horizontal component
of ¯ow in the leaky layer is negligible.

3. The method of images for a leaky layer
We present an analytical solution describing ¯ow to a
drain near a leaky layer. The solution is derived by applying the method of images to both an equipotential
boundary and an inhomogeneity boundary. Numerous
authors apply the method of images to inhomogeneity
and impermeable boundaries to model strati®ed formations in the ®elds of groundwater mechanics and
geophysical prospecting. These papers are summarized
by Muskat [10,11] who cites Maxwell [9] as originally
solving an analogous problem by the method of images.
The method has not been applied previously to problems with leaky layers. Three basic solutions will be used
to determine the appropriate forms of the images about
each boundary.
The ®rst basic solution contains a drain in the lowerhalf plane with a horizontal equipotential located a
distance h above the real axis (Fig. 2(a)). The solution is
a classical application of the method of images (e.g. [16,
p. 44])
Xˆ

Q
Q
ln…z ÿ zd † ÿ
ln‰z ÿ …zd ‡ 2ih †Š ‡ U0 ;
2p
2p

…7†

where Q is the discharge of the drain and U0 is the value
of the horizontal equipotential.
Basic solutions 2 and 3 are developed from a single
classical solution consisting of a drain in an aquifer of
two hydraulic conductivities ([13, p. 372]). In both solutions, the lower-half plane is designated as D with a
hydraulic conductivity of k and the upper-half plane is
designated as D with a hydraulic conductivity of k  .
Basic solution 2 has a drain at z ˆ zd in D (Fig. 2(b))
which gives

Q
ln…z ÿ zd † ‡ C;
2p
Q
Q
k
ln…z ÿ zd † ‡ j
ln…z ÿ zd † ‡  C:
Xˆ
2p
2p
k
X ˆ …1 ÿ j†

463

…8a†
…8b†

The third basic solution contains a drain at z ˆ zd in D
(Fig. 2(c))
Q
Q
k
ln…z ÿ zd † ÿ j
ln…z ÿ zd † ‡ D;
k
2p
2p
Q
ln…z ÿ zd † ‡ D:
X ˆ …1 ‡ j†
2p
X ˆ

…9a†
…9b†

In both solutions
jˆ

k ÿ k
k ‡ k

…10†

and C and D are real constants.
To create a solution with a leaky layer from the three
basic solutions, we image drains about both the inhomogeneity boundary B and the equipotential boundary
in D (see Fig. 1). First consider a drain of discharge Q
at zd and the inhomogeneity boundary. The solutions in
D and D are given by (8a) and (8b). For now, we neglect the real constants, and write
Q
ln…z ÿ zd †;
2p
Q
Q
ln…z ÿ zd † ‡ j
ln…z ÿ zd †:
Xˆ
2p
2p
X ˆ …1 ÿ j†

…11a†
…11b†

These expressions include the e€ects of the drain at
z ˆ zd and satisfy the conditions (3) and (4) along the
inhomogeneity boundary. The boundary condition,
U ˆ U0 for z ˆ x ‡ ih is not satis®ed by (11a). We
create an equipotential of value U ˆ 0 for z ˆ x ‡ ih
by placing an image drain of opposite strength in D
with respect to the equipotential boundary. We see from
(7) that the complex coordinates of the image drain are
given by z ˆ zd ‡ 2ih . We add the drain to (11a) to
obtain

Fig. 2. The three basic problems: (a) ¯ow to a drain from a horizontal equipotential, and (b) and (c), ¯ow to a drain in an aquifer of two hydraulic
conductivities.

464

E.I. Anderson / Advances in Water Resources 23 (2000) 461±474

Q
ln…z ÿ zd †
2p
Q
ln‰z ÿ …zd ‡ 2ih †Š:
ÿ…1 ÿ j†
2p

X ˆ …1 ÿ j†

…12†

The real constant U0 may be added to the complex potential to satisfy the boundary condition; for now, we
neglect the constant, leaving the equipotential along
z ˆ x ‡ ih with the value U ˆ 0.
We have satis®ed the boundary condition U ˆ 0 for
z ˆ x ‡ ih , but the conditions (3) and (4) along the inhomogeneity boundary are now violated by addition of
the new drain to X . To correct this, both X and X must
be modi®ed: the new drain in D at z ˆ zd ‡ 2ih must be
imaged about the inhomogeneity boundary. We obtain
the complex coordinates and strength of the image drain
by inspection of (9a). From (9a), the image of a drain at
zd in D with respect to an inhomogeneity boundary
along the real axis is located at zd . In (12) the drain is
located at z ˆ zd ‡ 2ih so that the image drain must be
placed at z ˆ zd ÿ 2ih . We also see from (9a) that the
image of a drain of strength Q has a strength of ÿjQ. In
(12) the strength of the drain is ÿ…1 ÿ j†Q so the
strength of the image drain must be j…1 ÿ j†Q. We add
the image drain to (12) to obtain
Q
ln…z ÿ zd †
2p
Q
ln‰z ÿ …zd ‡ 2ih †Š
ÿ …1 ÿ j†
2p
Q
ln‰z ÿ …zd ÿ 2ih †Š;
‡ j…1 ÿ j†
2p

X ˆ …1 ÿ j†

…13†

X now contains the proper imaging across the inhomogeneity boundary, but inspection of (9b) shows that
we must add a new drain to X also. We see from (9b)
that the complex potential in D feels the e€ect of the
drain outside of D, and so a new drain must also be
included in the expression for X. We see from (9b) that
the new drain must be located at z ˆ zd ‡ 2ih and have
a strength of ÿ…1 ÿ j†…1 ‡ j†Q. We add this drain to
(11b) to obtain
Q
Q
ln…z ÿ zd † ‡ j
ln…z ÿ zd †
Xˆ
2p
2p
Q
ln‰z ÿ …zd ‡ 2ih †Š:
ÿ …1 ÿ j†…1 ‡ j†
2p

…14†

We have now taken three steps in the imaging process.
We began with the basic solutions (8a) and (8b) which
satisfy the conditions along the inhomogeneity boundary. In the second step we applied the method of images
in D to satisfy the equipotential boundary condition.
This violated the conditions along the inhomogeneity
boundary. In the third step, X and X were modi®ed to
satisfy again the conditions along the inhomogeneity
boundary. Once again, X contains a drain outside of D
which violates the conditions along the equipotential

boundary; we have returned to conditions similar to
those at the conclusion of the ®rst step. However, the
location of the last drain added, located at z ˆ zd ÿ 2ih ,
is farther from the equipotential boundary than the
original drain at z ˆ zd and the strength has changed
from …1 ÿ j†Q in (11a) to j…1 ÿ j†Q. We recall that j
equals …k ÿ k  †=…k ‡ k  † and conclude that j is less than
one for k  < k; the strength of the drain has decreased.
We have established a pattern in the imaging process;
repetition of the second and third steps described above
will result in drains of decreasing strength being added in
X and X that lie farther from the real axis than in previous steps. The imaging must be continued inde®nitely;
the ®nal expressions contain an in®nite number of drains
1
Q X
z ÿ zd ‡ 2ih n
‡ k  /0 ;
jn ln
X ˆ …1 ÿ j†
2p nˆ0
z ÿ zd ÿ 2ih …n ‡ 1†
…15a†

Xˆ

Q
Q
Q
ln…z ÿ zd † ‡ j
ln…z ÿ zd † ÿ …1 ÿ j2 †
2p
2p
2p
1
X
…15b†
jn ln‰z ÿ zd ÿ 2ih …n ‡ 1†Š ‡ k/0 ;

nˆ0

where the constants, k  /0 and k/0 , are evaluated from the
boundary conditions. Fig. 3(a) and (b) show the distribution of drains in X and X , respectively. The complex
potential X, valid in D, contains the actual drain at zd and
an in®nite sum of drains outside of D whose locations
approach in®nity and whose discharges decay exponentially with their distance from the real axis. The complex
potential X , valid in D , contains an in®nite sum of
drains outside of D and another in®nite sum of drains
inside of D , but outside the domain of interest. Anderson
[2] demonstrates that the expressions (15a) and (15b)
satisfy the boundary conditions exactly and that the in®nite series appearing in (15a) and (15b) converge.
4. Limiting case: the leaky boundary
Hantush [7], Van der Veer [18], and Bruggeman [4]
solved the problem discussed above with the leaky layer
replaced by the boundary condition
/ ÿ /0
:
c
The solution is

y ˆ 0;

Xˆ

qy ˆ



Q
z ÿ zd Q
i
ln
…z ÿ zd †
ÿ exp
z ÿ zd p
2p
ck


i
…z ÿ zd † ‡ k/0 ;
 E1
ck

…16†

…17†

where E1 …z† is the exponential integral [1] de®ned as
Z 1
exp…ÿt†
dt j arg…z†j < p:
…18†
E1 …z† ˆ
t
z

E.I. Anderson / Advances in Water Resources 23 (2000) 461±474

465

Fig. 3. The locations and strengths of drains appearing in: (a) X; (b) X .

We will show here that the solution (17) for a leaky
boundary is a limiting case of the solution (15b) for a
leaky layer obtained by letting the thickness and hydraulic conductivity of the leaky layer, h and k  , approach zero while keeping the ratio h =k  constant and
equal to c, the resistance of the leaky boundary.
The solution for X (15b), presented graphically in
Fig. 3(a), consists of drains of decreasing recharge distributed evenly along the line extending from
z ˆ zd ‡ 2ih to in®nity along the positive imaginary
axis. In the ®gure, the drains are represented by dots; the
discharge of each drain is displayed to the right of each
dot. The drains in the upper-half plane approach one
another as the parameter h is decreased. In the limit as
h and k  vanish while holding h =k  constant, we pass
from a discrete sum of drains to a continuous distribution of drains, or a line sink.
We rewrite the series in (15b) in a more convenient
form to aid in taking the limit
1
Q X
jn ln‰z ÿ zd ÿ 2ih …n ‡ 1†Š
…1 ÿ j2 †
2p nˆ0
1
X
rm
ln…z ÿ dm †Dn;
ˆ
2p
mˆ1

…19†

where

dm ˆ zd ‡ 2ih …n ‡ 1† ˆ zd ‡ 2imh :

…20†

We express the discharge of the nth, or …m ÿ 1†th, drain
as rmÿ1 Dn where
2

rm ˆ

Q 1ÿj m
j
2h j

…21†

and where Dn represents the distance between the mth
and …m ‡ 1†th drains and is equal to 2h . We may express the integer m in terms of the location of the mth
drain, dm . From (20) we obtain
m ˆ …dm ÿ zd †=2ih :
…22†
Substituting (22) into (21) we obtain the strength of the
mth drain as a function of its location
Q 1 ÿ j2 …dm ÿzd †=2ih
j
Dn
2h j




Q 1 ÿ j2
1
exp
z
†
ln
j
…d
ˆ 
ÿ
d Dn:
m
j
2h
2ih

rm Dn ˆ

…23†

We obtain the expression for the complex potential
upon substitution of (23) and (19) into (15b)
Q
Q
ln…z ÿ zd † ‡ j
ln…z ÿ zd †
Xˆ
2p
2p




1
Q 1 ÿ j2 X
1
ÿ
ln j …dm ÿ zd †
exp
4p h j mˆ1
2ih
 ln…z ÿ dm †Dn ‡ k/0 :

…24†

The coecients in (24) containing j may be expanded in
Taylor series about k  ˆ 0 as follows:
n
1 
X
k ÿ k
k
ˆ
1
‡
2
;
…25a†
ÿ
jˆ
k ‡ k
k
nˆ1
1   2n
1 ÿ j2
k X
k
ˆ4
;
j
k nˆ0 k
1   2n
k X
k
:
ln j ˆ ÿ2
k nˆ0 k

…25b†

…25c†

466

E.I. Anderson / Advances in Water Resources 23 (2000) 461±474

We substitute the above expansions into (24) to obtain
an expression valid for k  =k < 1 and h =k  ˆ c. The
complex potential may now be written as
"
n #
1 
X
Q
k
Q
ln…z ÿ zd † ‡ 1 ‡ 2
ÿ
Xˆ
k
2p
2p
nˆ1
"
#
1   2n
X
Q 1
k
 ln…z ÿ zd † ÿ
p ck nˆ0 k
#
"
1   2n
1
X
i X
k
…dm ÿ zd †
exp

ck nˆ0 k
mˆ1
 ln…z ÿ dm †Dn ‡ k/0 :

…26†

In the limit, for h and k  approaching zero with
h =k  ˆ c, the ®rst series appearing in (26) vanishes, the
second series and fourth series are reduced to single
terms, and the third series becomes an integral. The
expression for the complex potential becomes
lim X ˆ
 

h ;k !0
h =k  ˆc

Q
ln…z ÿ zd †…z ÿ zd †
2p


Z zd ‡i1
Q i
i
…d ÿ zd † ln…z ÿ d†dd
exp
‡
p ck zd
ck
‡ k/0 ;

…27†

where the complex number d denotes a point of the line
extending along the imaginary axis from zd to in®nity
and may be written in terms of n as
d ˆ zd ‡ in;

0 6 n 6 1:

…28†

We rewrite the integral in (27), making the following
change of variables:
i
…z ÿ zd †;
ck
i
D ˆ …d ÿ zd †:
ck
Zˆ

The second term in (27), written as F …z†, becomes

…29a†
…29b†



Z zd ‡i1
Q i
i
F …z† ˆ
…d ÿ zd † ln…z ÿ d† dd
exp
p ck zd
ck
 

Z ÿ1
Q
i
ˆ
dD
exp D ln…Z ÿ D† ‡ ln
p 0
ck
 
Z ÿ1
Q
Q
i
exp D ln…Z ÿ D† dD ‡ ln
: …30†
ˆ
p 0
p
ck
The integral represents a semi-in®nite line sink of exponentially decaying strength lying along the negative
real axis of the Z plane. The integral in (30) may be
integrated by parts to give
Z
Q
Q ÿ1 exp D
Q
dD ‡
F …z† ˆ ÿ ln Z ‡
p
p 0
ZÿD
p
 
i
 ln
;
…31a†
ck
where use is made of
lim exp D ln…ÿD† ˆ 0:

D!ÿ1

…31b†

The remaining integral represents a line dipole ([16, p.
291]) and may be evaluated as (see (18))
Z
Q ÿ1 exp D
Q
dD ˆ ÿ exp…Z†E1 …Z†:
…32†
p 0
ZÿD
p
The ®nal form of the complex potential is obtained from
(27), (29a), (29b), (31a), (31b), and (32) as


Q
z ÿ zd Q
i
ÿ exp
ln
…z ÿ zd †
Xˆ
z ÿ zd p
2p
ck


i
…z ÿ zd † ‡ k/0 ;
 E1
…33†
ck
which is identical to (17). The leaky boundary is a limiting case of the leaky layer, as asserted. The leaky
boundary represents exactly the e€ects of a membrane,
where the membrane has a ®nite resistance but no
thickness. Fig. 4 shows ¯ow nets for the case of a leaky
layer, developed from (15a) and (15b) and for the case of

Fig. 4. Contours of constant head (dashed) and stream function (solid) for ¯ow to a drain from: (a) a leaky layer with k=k  ˆ 10 and h =jzd j ˆ 0:4;
(b) a leaky boundary with ck=jzd j ˆ 4. The heavy solid lines originating at the drains are branch cuts in the stream function.

467

E.I. Anderson / Advances in Water Resources 23 (2000) 461±474

a leaky boundary, developed from (33). In Fig. 4(a), the
parameters of the problem are k=k  ˆ 10 and
h =jzd j ˆ 0:4; in Fig. 4(b), the parameter is ck=jzd j ˆ 4.
The ¯ow ®elds in the aquifer are nearly identical; the
horizontal component of ¯ow in the leaky layer is indeed negligible in this case.

5. The method of images for a leaky boundary
We applied the method of images earlier to both an
inhomogeneity boundary and an equipotential boundary to construct a leaky layer. In passing through the
limit for h and k  approaching zero while holding h =k 
constant, the two boundaries join to form a single leaky
boundary. We introduce in this section the method of
images for a leaky boundary; we examine the solution
(33) to determine the form of the image, re¯ected across
a leaky boundary, of a single drain.
Eq. (33) indicates that a drain beneath a leaky
boundary must be imaged across that boundary by a
recharge drain, with an extra term added; the extra term
has the functional form exp…Z†E1 …Z† and we recall from
(32) that the function represents a line dipole of exponentially decaying strength distribution lying along the
negative real axis of the Z plane. The function is ®nite,
but not analytic, at in®nity with branch points at Z ˆ 0
and Z ˆ ÿ1. The image drain in (33) at z ˆ zd is the
source for the discharge drain at z ˆ zd , while the line
dipole corrects the behavior of the complex potential
along the leaky boundary without a€ecting the net discharge in the aquifer.

This interpretation provides the basis of the method
of images for a leaky boundary. The image for a drain of
strength Q at zd with respect to a leaky boundary consists of a drain of strength ÿQ at zd , and a semi-in®nite
line dipole originating at zd and oriented normal to the
leaky boundary. The line dipole has an exponential
strength distribution which vanishes at in®nity.
We present in Fig. 5 the solution from Fig. 4(b)
continued in the upper-half plane. The real axis of the z
plane, corresponding to the leaky boundary, is shown as
the solid horizontal line. The heavy line originating at
the drain in the lower-half plane represents the branch
cut in the stream function created by the drain. The
heavy line originating at the image drain in the upperhalf plane represents the branch cut for the image drain
and the branch cut for the line dipole.

6. The image of a line dipole
The images, with respect to a leaky boundary, for
other features such as line sinks, dipoles, line dipoles and
line doublets, may be generated from the basic solution
(33) for a single drain beneath a leaky boundary. We
present without derivation the image, with respect to a
leaky boundary, for a line dipole of the form
Z ÿ1 nÿ1
Q …ÿ1†n
D exp D
dD; …n ˆ 1; 2; 3 . . .†:
Xld ˆ
p …n ÿ 1†! 0
ZÿD
…34†

We will use the results to solve the problem of ¯ow to a
horizontal drain in a semi-con®ned aquifer. We note,
that for n ˆ 1, (34) has the same form as the line dipole
appearing in (32). Eq. (34) may be integrated to express
the line dipole in terms of an exponential integral
Xld ˆ

Q
exp…Z†En …Z†;
p

…35†

where the function En …z† is related to E1 …z† by the following relationships [1]:
1
En‡1 …z† ˆ ‰ exp… ÿ z† ÿ zEn …z†Š
n

…n ˆ 1; 2; 3; . . .†;
…36a†

dEn …z†
ˆ ÿEnÿ1 …z†
dz
E0 …z† ˆ

Fig. 5. The method of images for a drain beneath a leaky boundary:
¯ow net for Fig. 4(b) continued into the upper-half plane.

exp…ÿz†
:
z

…n ˆ 1; 2; 3; . . .†;

…36b†

…36c†

The equality of (34) and (35) may be veri®ed, for n P 2,
by repeated substitution of (36a) into (35) to express
En …z† in terms of E1 …z†. The function obtained in this
manner, when expanded, may be shown to equal the
integral given in (34).

468

E.I. Anderson / Advances in Water Resources 23 (2000) 461±474

We place the line dipole along the negative imaginary
axis, originating at z ˆ zd by de®ning Z and D as
i
…z ÿ zd †;
ck
i
D ˆ ÿ …d ÿ zd †:
ck

Zˆÿ

The complex potential for the line dipole is

 

Q
i
i
Xld ˆ exp ÿ …z ÿ zd † En ÿ …z ÿ zd † :
p
ck
ck

…37a†
…37b†

…38†

The image of this line dipole, with respect to a leaky
boundary lying along the real axis, is given by

 

Q
i
i
…im†
…z ÿ zd † En
…z ÿ zd †
Xld ˆ exp
p
ck
ck


i
…z ÿ zd † :
…39†
ÿ 2En‡1
ck
We demonstrate that the function
…im†

X ˆ Xld ‡ Xld ‡ k/0

…40†

satis®es condition (16) along the leaky boundary. We
may express (16) in terms of the function L…z†, de®ned in
the following manner ([13, p. 376])
dX
;
…41†
dz
where ÿdX=dz is the complex speci®c discharge function
(e.g. [13, p. 36]) and is equal to qx ÿ iqy . The condition
(16) along the leaky boundary, y ˆ 0, may be expressed
as Re L…z† ˆ k/0 . We di€erentiate (40) and substitute
into (41) to obtain





Q
i
i
exp ÿ …z ÿ zd † Enÿ1 ÿ …z ÿ zd †
Lˆÿ
p
ck
ck




i
i
…z ÿ zd † Enÿ1
…z ÿ zd † ‡ 2
ÿ exp
ck
ck


 

Q
i
i
exp ÿ …z ÿ zd † En ÿ …z ÿ zd †

p
ck
ck
 


i
i
…42†
…z ÿ zd † En
…z ÿ zd † ‡ k/0 :
ÿ exp
ck
ck
L…z† ˆ X ‡ ick

For y ˆ 0 we have
Zˆÿ

i
…z ÿ zd †;
ck

i
…z ÿ zd †:
ck
Using the symmetry relationships [1]
Zˆ

…43a†
…43b†

exp…z† ˆ exp…z†;

…44a†

En …z† ˆ En …z†;

…44b†

we see that
Re f exp…z†En …z†g ˆ Re f exp…z†En …z†g

…45†

and therefore, from (42), for y ˆ 0, the real part of L…z†
is equal to k/0 . The condition along the leaky boundary
is satis®ed exactly.
As a demonstration, we apply (33) and (39) to identify the image, with respect to a leaky boundary, of a
drain and a semi-in®nite line dipole. The resulting solution will be used both to derive the solution for ¯ow to
a drain in a semi-con®ned aquifer and to identify the
error in the work by Van der Veer [19,20]. The leaky
boundary lies along the real axis of the complex plane;
the drain lies in the lower-half plane at z ˆ zd and the
line dipole originates at z ˆ zd and extends to
z ˆ zd ÿ i1. The portion of the complex potential associated with the drain is
Xdr ˆ

Q
ln…z ÿ zd †
2p

…46†

and the portion associated with the line dipole, obtained
from (35) with n ˆ 1, is given by

 

Q
i
i
…47†
Xld ˆ exp ÿ …z ÿ zd † E1 ÿ …z ÿ zd † :
p
ck
ck
The image of the drain in (46), obtained from (33), is
…im†

Q
ln…z ÿ zd †
2p
 


Q
i
i
…z ÿ zd † E1
…z ÿ zd †
ÿ exp
p
ck
ck

Xdr ˆ ÿ

…48†

and the image of the line dipole in (47), obtained from
(38) and (39), is given by

 

Q
i
i
…im†
…z ÿ zd † E1
…z ÿ zd †
Xld ˆ exp
p
ck
ck


i
…z ÿ zd † :
…49†
ÿ 2E2
ck
The complex potential containing Xdr and Xld and satisfying the condition (16) is given by
…im†

…im†

X ˆ Xdr ‡ Xld ‡ Xdr ‡ Xld ‡ k/0 :
We combine (46)±(49) to obtain


Q
z ÿ zd Q
i
ln
‡ exp ÿ …z ÿ zd †
Xˆ
z ÿ zd p
2p
ck




i
Q
i
…z ÿ zd †
 E1 ÿ …z ÿ zd † ÿ 2 exp
ck
p
ck


i
…z ÿ zd † ‡ k/0 :
 E2
ck

…50†

…51†

The image of the drain and the line dipole in the lowerhalf plane consists of a drain and a line dipole in the
upper-half plane. The image drain at z ˆ zd is opposite
in strength of the drain at z ˆ zd and the image line dipole originating at z ˆ zd and extending to z ˆ zd ‡ i1,
is of the form exp…Z†E2 …Z†. Van der Veer [19,20] suggests that the image of a drain and a line dipole about a
leaky boundary is simply a drain; the term of the form

E.I. Anderson / Advances in Water Resources 23 (2000) 461±474

exp…Z†E2 …Z† in (51) is neglected and therefore his solution does not satisfy the proper condition along the
leaky boundary.
Van der Veer's approach [19,20] is to superimpose
two basic solutions, each satisfying conditions along a
leaky boundary lying on the real axis, to satisfy conditions along an impermeable base. The ®rst basic solution
(17) describes ¯ow to a drain in the lower-half plane
from the leaky boundary; the second solution describes
¯ow to a drain in the upper-half plane from the leaky
boundary. The second solution is obtained from the ®rst
by a simple rotation of the domain. However, this
conformal transformation alters the form of the leakage
function, L…z† (41), by introducing a minus sign in front
of the term dX=dz. The two basic solutions may be superimposed, but the appropriate condition along the
leaky boundary will not be satis®ed, as demonstrated
above.

7. Example: ¯ow to a horizontal drain in a semi-con®ned
aquifer
We develop the solution to the problem illustrated in
Fig. 6 as an application of the method of images for a
leaky boundary. The ®gure shows an aquifer of ®nite
depth H, bounded on top by a leaky boundary of resistance c and on the bottom by an impermeable base. A
horizontal drain of discharge Q exists at z ˆ zd . We
apply the method of images about the two parallel
boundaries to obtain a solution. Bruggeman [4, p. 311]
solves the same problem using a di€erent approach.
We begin with the solution (33) for a drain beneath a
leaky boundary. The condition along the impermeable
base may be satis®ed by imaging the two drains and the
line dipole in (33) about z ˆ x ÿ iH . The images of a
drain and of a line dipole, with respect to an impermeable boundary, are known (e.g. [16]). The image of a
drain is a drain of the same discharge re¯ected across the
boundary; the image of a line dipole is a line dipole of
the same strength distribution re¯ected across the
boundary. We image the terms in (33) about the aquifer
base and obtain

Fig. 6. De®nition sketch: ¯ow to a drain in a semi-con®ned aquifer.

Xˆ

469


 

Q
z ÿ zd Q
i
i
ln
…z ÿ zd † E1
…z ÿ zd †
ÿ exp
z ÿ zd p
2p
ck
ck


Q
z ÿ zd ‡ 2iH Q
i
ÿ exp ÿ …z ÿ zd ‡ 2iH †
ln
‡
2p
z ÿ zd ‡ 2iH p
ck


i
 E1 ÿ …z ÿ zd ‡ 2iH † ‡ k/0 :
…52†
ck

The third and fourth terms in (52) are the images of the
®rst and second terms with respect to the aquifer base.
The condition along the leaky boundary, y ˆ 0, is now
violated; we may satisfy that condition by imaging the
two drains and the line dipole, the third and fourth
terms in (52), about y ˆ 0 using (33) and (51) to obtain
their images. We obtain

 

Q
z ÿ zd Q
i
i
ln
…z ÿ zd † E1
…z ÿ zd †
ÿ exp
Xˆ
2p
ck
ck
z ÿ zd p


Q
z ÿ zd ‡ 2iH Q
i
ÿ exp ÿ …z ÿ zd ‡ 2iH †
ln
‡
2p
z ÿ zd ‡ 2iH p
ck


i
Q
z ÿ zd ÿ 2iH
ln
 E1 ÿ …z ÿ zd ‡ 2iH † ‡
ck
2p
z ÿ zd ÿ 2iH

 

Q
i
i
…z ÿ zd ÿ 2iH † E1
…z ÿ zd ÿ 2iH †
ÿ exp
p
ck
ck


Q
i
…z ÿ zd ÿ 2iH †
‡ 2 exp
p
ck


i
…z ÿ zd ÿ 2iH † ‡ k/0 :
 E2
…53†
ck
Once again, we may satisfy the condition along the
aquifer base by imaging the new terms in (53) about
z ˆ x ÿ iH . The process must be continued inde®nitely
resulting in in®nite sums of drains, and line dipoles of
the form (38) of increasing order n. The process is tedious but straightforward. The resulting solution may
be expressed as
Xˆ

M
Q
z ÿ zd
Q X
m
ln
‡
…ÿ1†
z ÿ zd 2p mˆ1
2p


…z ÿ zd ÿ 2iHm† …z ÿ zd ‡ 2iHm†
 ln
…z ÿ zd ÿ 2iHm† …z ÿ zd ‡ 2iHm†


M ‡1
Q X
i
…z ÿ zd ÿ 2iH …m ÿ 1††
exp
‡
2p mˆ1
ck
(
)

m
X
i

…z ÿ zd ÿ 2iH …m ÿ 1††
am;n En
ck
nˆ1


M
Q X
i
exp ÿ …z ÿ zd ‡ 2iHm†
‡
2p mˆ1
ck
(
)

m
X
i

am;n En ÿ …z ÿ zd ‡ 2iHm†
ck
nˆ1


M
Q X
i
…z ÿ zd ÿ 2iHm†
exp
‡
2p mˆ1
ck

470

E.I. Anderson / Advances in Water Resources 23 (2000) 461±474



(

m
X

am;n En

nˆ1



i
…z ÿ zd ÿ 2iHm†
ck

)



M ÿ1
Q X
i
exp ÿ …z ÿ zd ‡ 2iH …m ‡ 1††
2p mˆ1
ck
(
)

m
X
i
am;n En ÿ …z ÿ zd ‡ 2iH …m ‡ 1††

ck
nˆ1
‡

‡ k/0 ;

…54†

where
am;1 ˆ



ÿ2
0

for m odd; m P 1;
for m even; m P 1;

a1;n ˆ 0 for n P 2;
am;n ˆ ÿ2amÿ1;nÿ1 ‡ amÿ1;n

…55†

for n P 2; m P 2:

The values of the coecient am;n for m 6 7 and n 6 7 are
given in Table 1. The expression (54) is written such that
for any ®nite M the condition along the leaky boundary
is satis®ed exactly and the condition along the impermeable base is satis®ed approximately. We demonstrate
in Appendix A that expression (54) satis®es exactly the
boundary conditions along the aquifer top for any M. M
must become in®nite to satisfy the condition along the
impermeable base; we o€er no proof that, for M ! 1,
the in®nite series of line dipoles converge. The proof is
dicult as each term of the series of line dipoles represents a semi-in®nite line integral which is ®nite but not
analytic at in®nity. As Table 1 shows, the coecients
am;n grow rapidly and alternate in sign. However, each
individual line dipole has only a local e€ect on the ¯ow
®eld and each successive line dipole in a series is located
farther from the domain of interest.
The ®rst two terms in (54), consisting of the drains,
converge to a known function for M ! 1
1
Q
z ÿ zd
Q X
m
ln
‡
…ÿ1†
2p
z ÿ zd 2p mˆ1


…z ÿ zd ÿ 2iHm† …z ÿ zd ‡ 2iHm†
 ln
…z ÿ zd ÿ 2iHm† …z ÿ zd ‡ 2iHm†
(
 p
)
…z ÿ zd †
tanh 4H
Q
 p
 :
ln
…56†
ˆ
2p
…z ÿ zd †
tanh 4H

This may be seen by writing the left-hand side of (56) as
the logarithm of an in®nite product and identifying
the in®nite product as the ratio of hyperbolic functions
([1, p. 85]) shown on the right-hand side of (56). By itself, (56) represents the solution for ¯ow from an equipotential to a drain in an aquifer of ®nite depth; the
upper boundary, lying along the real axis, is an equipotential and the lower boundary, at z ˆ x ÿ iH , is impermeable. The remaining terms in (54), consisting of
the in®nite sums of line dipoles, may be viewed as two
semi-in®nite line dipoles with discontinuous strength
distributions. One line dipole originates at zd and extends along the positive imaginary axis and the other
originates at zd ÿ 2iH and extends along the negative
imaginary axis. We interpret the solution obtained by
the method of images as consisting of a basic solution
(56), which re¯ects an equipotential boundary along the
real axis and an impermeable boundary for z ˆ x ÿ iH ,
and line dipoles which correct the behavior along the
leaky boundary without changing the net discharge into
the aquifer.
Fig. 7 shows a ¯ow ®eld for the case ck=H ˆ 2 and
jzd j=H ˆ 0:4. The ¯ow ®eld is obtained from (54) by
setting M equal to 4. The aquifer top and base are
shown as heavier lines than the streamlines. We have
extended the contours of head and streamfunction
across the impermeable base to demonstrate graphically
the approximated lower boundary condition; the condition is approximated well. The solution is quite accurate, although it was obtained with only a few terms;
the e€ect of the line dipoles on the area of interest
vanishes rapidly in this example.

Fig. 7. Flow net showing ¯ow in a semi-con®ned aquifer to a horizontal drain. The solution shown is obtained from (54) with M ˆ 4.

Table 1
Values of the coecient am;n for m 6 7 and n 6 7
am;n

am;n
1

2

3

4

5

6

7

1
2
3
4
5
6
7

ÿ2
0
ÿ2
0
ÿ2
0
ÿ2

0
4
4
8
8
12
12

0
0
ÿ8
ÿ16
ÿ32
ÿ48
ÿ72

0
0
0
16
48
112
208

0
0
0
0
ÿ32
ÿ128
ÿ352

0
0
0
0
0
64
320

0
0
0
0
0
0
ÿ128

471

E.I. Anderson / Advances in Water Resources 23 (2000) 461±474

8. Conclusions

and

A new approach for solving problems of groundwater
¯ow with leaky boundaries was presented. The approach is an extension of the classical method of images
commonly used to solve groundwater ¯ow problems.
The new approach may be applied to more general
problems than the approach presented by PolubarinovaKochina [13, p. 376]. The image for a drain beneath a
leaky boundary was presented as well as the images for a
speci®c class of line dipoles. The results were applied to
solve the problem of ¯ow in a semi-con®ned aquifer to a
horizontal drain. The results may be applied to obtain
the solutions to other groundwater problems and the
approach may be generalized further by considering
other boundary types.
The method of images for both leaky layers and leaky
boundaries was presented for steady, two-dimensional
groundwater ¯ow. The approach is general, however,
and may be applied to three-dimensional ¯ows. For
example, basic three-dimensional solutions exist for a
point sink beneath a planar discontinuity in hydraulic
conductivity and for a point sink beneath a planar
equipotential [9]. These solutions have forms similar to
the basic solutions (7), (8a), (8b), (9a), and (9b) used in
the present analysis. It is clear that the approach presented here could be used to generate the solution for a
point sink beneath a planar leaky boundary. In a similar
fashion, the approach is applicable to some transient
problems.

Xld ˆ

M ‡1
Q X
exp ‰Z1 …z; m ÿ 1†Š
2p mˆ1
(
)
m
X
am;n En ‰Z1 …z; m ÿ 1†Š

nˆ1

(
)
M
m
X
Q X
exp ‰Z2 …z; m†Š
am;n En ‰Z2 …z; m†Š
‡
2p mˆ1
nˆ1
(
)
M
m
X
Q X
exp ‰Z3 …z; m†Š
am;n En ‰Z3 …z; m†Š
‡
2p mˆ1
nˆ1
M ÿ1
Q X
exp ‰Z4 …z; m ‡ 1††Š
2p mˆ1
(
)
m
X
am;n En ‰Z4 …z; m ‡ 1††Š


‡

…A:3†

nˆ1

and where

i
…z ÿ zd ÿ 2iHm†;
…A:4†
ck
i
…A:5†
Z2 …z; m† ˆ ÿ …z ÿ zd ‡ 2iHm†;
ck
i
…A:6†
Z3 …z; m† ˆ …z ÿ zd ÿ 2iHm†;
ck
i
…A:7†
Z4 …z; m† ˆ ÿ …z ÿ zd ‡ 2iHm†:
ck
Similarly, we partition the leakage function L…z†, given
by (41), into a portion due to drains and a portion due
to line dipoles
Z1 …z; m† ˆ

L ˆ Ldr ‡ Lld ‡ k/0 ;

…A:8†

where

Acknowledgements
I wish to thank an anonymous referee for a particularly thorough review of the manuscript and for identifying Bruggeman [4] as a necessary reference.

Ldr ˆ Xdr ‡ ick

dXdr
dz

…A:9†

and
dXld
:
…A:10†
dz
Recall that the condition along the leaky boundary is
given by Re L ˆ k/0 for y ˆ 0.
We di€erentiate (A.2) and substitute into (A.9) to
obtain
Lld ˆ Xld ‡ ick

Appendix A
We verify that the solution (54) satis®es condition
(41) along the leaky boundary. We partition the potential (54) in the following manner:
X ˆ Xdr ‡ Xld ‡ k/0 ;

…A:1†

where
Xdr ˆ

M
Q
z ÿ zd
Q X
ln
…ÿ1†m
‡
z ÿ zd 2p mˆ1
2p


Z3 …z; m†Z2 …z; m†
 ln
Z4 …z; m†Z1 …z; m†



M
Q
z ÿ zd
Q X
Z3 …z; m†Z2 …z; m†
m
ln
‡
…ÿ1† ln
Ldr ˆ
z ÿ zd 2p mˆ1
2p
Z4 …z; m†Z1 …z; m†


M
Q
1
1
Q X
m
…ÿ1†
ÿ
ÿ
‡ ick
2p z ÿ zd z ÿ zd
2p mˆ1


1
1
1
1
ÿ
ÿ

‡
:
Z3 …z; m† Z4 …z; m† Z2 …z; m† Z1 …z; m†

…A:11†

…A:2†

We di€erentiate (A.3) and substitute into (A.10) to
obtain

472

E.I. Anderson / Advances in Water Resources 23 (2000) 461±474

M ‡1
Q X
Lld ˆ
exp‰Z1 …z; m ÿ 1†Š
2p mˆ1
(
)
m
X
am;n Enÿ1 ‰Z1 …z; m ÿ 1†Š



‡

nˆ1

M
Q X
exp‰Z2 …z; m†Š
‡
2p mˆ1
(
)
m
X
am;n ‰2En ‰Z2 …z; m†Š ÿ Enÿ1 ‰Z2 …z; m†ŠŠ

nˆ1

(
)
m
M
X
Q X
am;n Enÿ1 ‰Z3 …z; m†Š
exp‰Z3 …z; m†Š
‡
2p mˆ1
nˆ1
M ÿ1
Q X
exp‰Z4 …z; m ‡ 1†Š
2p mˆ1
(
m
X
am;n ‰2En ‰Z4 …z; m ‡ 1†Š


…A:16†

a1;1 ˆ ÿ2;

…A:17†

am;m‡1 ˆ 0

…A:18†

and
am‡1;1 ÿ am;1 ˆ



‡2
ÿ2

for m odd;
for m even:

…A:19†

Q
exp‰Z1 …z; 0†ŠE0 ‰Z1 …z; 0†Š
p

M
Q X
exp‰Z1 …z; m†Š
2p mˆ1
(
)
m
X
am;n ‰Enÿ1 ‰Z1 …z; m†Š ÿ 2En ‰Z1 …z; m†ŠŠ


‡
…A:12†

Q
exp‰Z1 …z; 0†ŠE0 ‰Z1 …z; 0†Š
ˆ a1;1
2p
(
)
M
m‡1
X
Q X
exp‰Z1 …z; m†Š
am‡1;n Enÿ1 ‰Z1 …z; m†Š :
‡
2p mˆ1
nˆ1

nˆ1

ÿ

F ˆ ick



…A:20†

M
Q 1
QX
1
Q
m
‡
ÿ
…ÿ1†
p z ÿ zd p mˆ1
Z1 …z; m† 2p

M
X

exp‰Z1 …z; m†Š

mˆ1

…A:14†

M
Q
Q X
exp‰Z1 …z; 0†ŠE0 ‰Z1 …z; 0†Š ‡
2p
2p mˆ1
(
m‡1
X
 exp‰Z1 …z; m†Š
‰am;n

M
QX
m
…ÿ1† exp‰Z1 …z; m†ŠE0 ‰Z1 …z; m†Š:
p mˆ1

Finally, we use (36c) to obtain

…A:13†

for n P 2

M
Q X
exp‰Z1 …z; m†Šf…am‡1;1 ÿ am;1 †E0 ‰Z1 …z; m†Š
2p mˆ1

We use (55) to evaluate the following coecients:

F ˆÿ

We modify the ®rst term in (A.12) into a more convenient form; we operate on the indices to represent
Z1 …z; m ÿ 1† as Z1 …z; m†
(
)
‡1
m
X
X
QM
exp‰Z1 …z; m ÿ 1†Š
am;n Enÿ1 ‰Z1 …z; m ÿ 1†Š
F ˆ
2p mˆ1
nˆ1

From (55) we obtain
am‡1;n ˆ am;n ÿ 2am;nÿ1
and we may write

am;n ‰Enÿ1 ‰Z1 …z; m†Š ÿ 2En ‰Z1 …z; m†ŠŠ

nˆ1

)

‡ am;m‡1 Em ‰Z1 …z; m†Šg:

nˆ1

ÿ Enÿ1 ‰Z4 …z; m ‡ 1†ŠŠ :

m
X

Substituting (A.18) and (A.19) into (A.16), we obtain

‡

)

(



(

m
X
nˆ1

)

am;n ‰Enÿ1 ‰Z1 …z; m†Š ÿ 2En ‰Z1 …z; m†ŠŠ :
…A:21†

F ˆ a1;1

We modify the third term in (A.12) in a similar fashion,
by operating on the indices to transform the terms
Z3 …z; m† into the form Z3 …z; m ‡ 1†. We obtain

nˆ2

ÿ 2am;nÿ1 ŠEnÿ1 ‰Z1 …z; m†Š
)
‡ am‡1;1 E0 ‰Z1 …z; m†Š

or
Q
exp‰Z1 …z; 0†ŠE0 ‰Z1 …z; 0†Š
2p
M
Q X
exp‰Z1 …z; m†Š
‡
2p mˆ1

F ˆ a1;1

…A:15†

(
)
m
M
M
X
Q X
QX
am;n Enÿ1 ‰Z3 …z; m†Š ˆ ‡
exp‰Z3 …z; m†Š
…ÿ1†m
2p mˆ1
p
nˆ1
mˆ1
M ÿ1
1
Q X
‡
exp‰Z3 …z; m ‡ 1†Š
Z3 …z; m† 2p mˆ1
(
)
m
X
am;n ‰Enÿ1 ‰Z3 …z; m ‡ 1†Š ÿ 2En ‰Z1 …z; m ‡ 1†ŠŠ :




nˆ1

…A:22†

E.I. Anderson / Advances in Water Resources 23 (2000) 461±474

We substitute (A.21) and (A.22) into (A.12) to obtain Lld


M
Q 1
QX
1
1
m
…ÿ1†
Lld ˆ ick
ÿ
ÿ
p z ÿ zd p mˆ1
Z1 …z; m† Z3 …z; m†
M
Q X
exp‰Z1 …z; m†Š
2p mˆ1
(
)
m
X
am;n ‰Enÿ1 ‰Z1 …z; m†Š ÿ 2En ‰Z1 …z; m†ŠŠ


‡

nˆ1

M
Q X
exp‰Z2 …z; m†Š
2p mˆ1
(
)
m
X
am;n ‰Enÿ1 ‰Z2 …z; m†Š ÿ 2En ‰Z2 …z; m†ŠŠ


ÿ

nˆ1

M ÿ1
Q X
exp‰Z3 …z; m ‡ 1†Š
2p mˆ1
(
)
m
X
am;n ‰Enÿ1 ‰Z3 …z; m ‡ 1†Š ÿ 2En ‰Z3 …z; m ‡ 1†ŠŠ


‡

473

We substitute (A.25) and (A.26) into (A.8) to verify the
boundary condition along the real axis



Q
1
1
for y ˆ 0; Re L ˆ Re ick
‡
2p z ÿ zd z ÿ zd
(


M
Q X
1
1
… ÿ 1†m
ÿ
‡ Re
2p mˆ1
Z2 …z; m† Z1 …z; m†

)
1
1
…A:27†
ÿ
‡ k/0 :
‡
Z3 …z; m† Z4 …z; m†
We use the following relationship to evaluate (A.27):


1 1
ˆ 0:
…A:28†
ÿ
Re
z z
The remaining terms in (A.27) vanish except for the
constant. We obtain Re L…x† ˆ k/0 ; the boundary condition is satis®ed exactly for any value of M.

nˆ1

M ÿ1
Q X
exp‰Z4 …z; m ‡ 1†Š
2p mˆ1
(
m
X
am;n ‰Enÿ1 ‰Z4 …z; m ‡ 1†Š


ÿ

References

nˆ1

)

ÿ 2En ‰Z4 …z; m ‡ 1†ŠŠ :

…A:23†

We evaluate the real part of Lld along the real axis
(y ˆ 0). Note that for y ˆ 0, from (A.4) to (A.7), we
have
Z1 ˆ Z2 ;

Z3 ˆ Z4 :

…A:24†

Using the symmetry relationship (45), we see that the
real parts of the terms in (A.23) containing the exponential integrals cancel and we have
(
M
Q 1
QX
… ÿ 1†m
ÿ
for y ˆ 0; Re Lld ˆ Re ick
p z ÿ zd p mˆ1

)
1
1
ÿ
:
…A:25†

Z1 …z; m† Z3 …z; m†
Next, we evaluate the real part of Ldr for z ˆ x. Using
(A.24) we see that the real part of the logarithmic terms
in (A.11) vanishes along the real axis and we have



Q
1
1
ÿ
for y ˆ 0; Re Ldr ˆ Re ick
2p z ÿ zd z ÿ zd
(

M
Q X
1
1
‡
… ÿ 1†m
ÿ Re
2p mˆ1
Z3 …z; m† Z4 …z; m†
)
1
1
ÿ
ÿ
:
…A:26†
Z2 …z; m† Z1 …z; m†

[1] Abramowitz M, Stegun, IA. Handbook of mathematical functions, New York: Dover, 1965.
[2] Anderson EI. Groundwater ¯ow with leaky boundaries, Ph.D.
Thesis, Department of Civil Engineering, University of Minnesota-Minneapolis, 1999.
[3] Bear J, Braester C. Flow from in®ltration basins to drains and
wells. In: Proceedings of the American Society of Civil Engineers,
J Hydra Div 1966;95(5):115±34.
[4] Bruggeman GA. Analytical solutions of geohydrological problems, developments in water science, vol. 46. Amsterdam: Elsevier,
1999.
[5] Haitjema HM. Modeling three-dimensional ¯ow in con®ned
aquifers by superposition of both two and three-dimensional
analytic functions. Water Resour Res 1985;21(10):1557±66.
[6] Hantush MS, Jacob CE. Non-steady Green's functions for an
in®nite strip of leaky aquifer. Trans Am Geophys Union
1955;36(1):101±12.
[7] Hantush MS. Wells near streams with semipervious beds.
J Geophys Res 1965;70(12):2829±31.
[8] Fitts CR. Simple analytic functions for modeling three-dimensional ¯ow in layered aquifers. Water Resour Res 1989;25(5):943±
8.
[9] Maxwell JC. A treatise on electricity and magnetism, vol. 1.
Oxford: Clarendon Press, 1873.
[10] Muskat M. Potential distribution about an electrode on the
surface of the earth. Physics 1933;4(4):129±47.
[11] Muskat M. The ¯ow of homogeneous ¯uids through porous
media. New York: McGraw-Hill, 1937.
[12] Nield SP, Townley LR, Barr AD. A framework for quantitive
analysis of surface water±groundwater interaction: ¯ow geometry
in a vertical section. Water Resour Res 1994;30(8):2461±75.
[13] Polubarinova-Kochina PY. Theory of ground water movement.
Princeton, NJ: Princeton University Press, 1962 (translated by De
Wiest, JMR).
[14] Shan C, Falta RW, Javandel I. Analytical solutions for steady
state gas ¯ow to a soil vapor extraction well. Water Resour Res
1992;28(4):1105±20.

474

E.I. Anderson / Advances in Water Resources 23 (2000) 461±474

[15] Steward DR. Three-dimensional analysis of the capture of
contaminated leachate by fully penetrating, partially penetrating,
and horizontal wells. Water Resour Res 1999;35(2):461±8.
[16] Strack ODL. Groundwater mechanics. Englewood Cli€s, NJ:
Prentice-Hall, 1989.
[17] Thomson W. On the mathematical theory of electricity in
equilibrium. The Cambridge and Dublin Math J 1848;III.

[18] Van der veer P. Exact solutions for two-dimensional groundwater
¯ow problems involving a semi-pervious boundary. J Hydro
1978;37:159±68.
[19] Van der veer P. Exact solutions for two-dimensional
groundwater ¯ow in a semicon®ned aquifer. J Hydro 1994;
156:91±9.
[20] Van der veer P. Corrigendum. J Hydro 1995;171:213±4.