a dupuit formulation for flow in layered, anisotropic aquifers

A Dupuit formulation for flow in layered, anisotropic aquifers

Mark Bakker a,*, Kick Hemker b

a Department of Biological and Agricultural Engineering, University of Georgia, Athens, GA 30602, USAb Faculty of Earth Sciences, Vrije Universiteit, De Boelelaan 1085, 1081 HV Amsterdam, The Netherlands

Received 1 January 2002; received in revised form 29 April 2002; accepted 4 July 2002

Abstract

A new theory is presented for groundwater flow in layered, anisotropic aquifers; flow must remain semi-confined. Two main

approximations are made: (1) the aquifer consists of a number of horizontal, homogeneous layers, each with its own anisotropic

transmissivity, and (2) the resistance to flow in the vertical direction is neglected. Analytic solutions are derived for the head and

horizontal flow in each layer by use of a comprehensive transmissivity tensor. The vertical component of flow at the layer interfaces

is computed analytically by vertical integration of the horizontal divergence. The theory is applied to both uniform flow and flow to

a well; solutions may be superimposed. Flow in layered, anisotropic aquifers is three-dimensional when the anisotropy between

layers differs. In the context of contaminant transport, the resulting three-dimensional flow field can be of great importance. Three-

dimensional flow lines become especially complicated near pumping wells. In a two-layer aquifer, the flow lines to a well may be

grouped into four bundles of spiraling flow lines, referred to as groundwater whirls. These whirls are bounded by two vertical planes

that intersect at the well; horizontal flow along these planes is radial. For a well in a uniform flow field, the complications of the

three-dimensional flow field are illustrated by the difficulties that are encountered in delineating the capture zone of the well.

� 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Layered aquifer; Anisotropy; Dupuit approximation; Capture zone

1. Introduction

In the modeling of groundwater flow, the resistance

to vertical flow may be neglected when flow is essentially

horizontal. This approximation is an alternative inter-pretation of the Dupuit approximation [12], which ne-

glects the vertical variation of the head. The advantage

of the alternative interpretation is that it allows for the

presence of vertical flow, because the vertical flow is not

anymore associated with a vertical variation of head [21,

p. 351;23;24, p. 36]. Forchheimer [13] applied the Dupuit

approximation to phreatic aquifers; Dietz [11] applied

the approximation to model interface flow in coastalaquifers. Several studies have explored the range of

practical applicability of Dupuit models through com-

parison of exact three-dimensional solutions to Dupuit

solutions; these include flow between ditches [4,17], in-

filtration at the top of the aquifer system [15,23], flow in

multi-aquifer systems [2,18], and interface flow [3,5].

When the Dupuit approximation is adopted, the pi-

ezometric head is a function of the horizontal x and y

coordinates, and does not vary in the vertical direction

within an aquifer. For vertically homogeneous aquifers,

this means that the horizontal flow is equally dividedover the aquifer thickness. The vertical component of

flow may be computed by considering three-dimensional

continuity of flow at any point in the aquifer [23]. Gi-

rinski [14] and Strack [24] adopted the Dupuit approx-

imation to develop a potential flow formulation for flow

in layered aquifers, where the hydraulic conductivity is

different but isotropic in each homogeneous layer. In

this formulation, the head gradient is the same in everylayer, but the velocities differ (because the hydraulic

conductivities and porosities of the layers differ). This

enables the study of advective transport of contaminants

in layered aquifers.

In this paper, the formulations of Girinski and Strack

will be extended to account for a different anisotropic

horizontal hydraulic conductivity in each layer. Both the

principal directions of the horizontal hydraulic con-ductivity and the principal values may differ in each

layer; flow must remain semi-confined. It will be shown

Advances in Water Resources 25 (2002) 747–754

www.elsevier.com/locate/advwatres

*Corresponding author.

E-mail addresses: [email protected] (M. Bakker), hemk@

geo.vu.nl (K. Hemker).

0309-1708/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0309-1708 (02 )00074-X

mail to: [email protected]

that the difference in anisotropy between layers usually

results in a vertical component of flow between layers.

The vertical flow between layers creates a kind of rota-

tional flow within the aquifer, as was observed from

numerical experiments by Hemker et al. [16]. It is real-ized that even though evidence of anisotropy may be

observed in the field (elliptical drawdown cones near

pumping wells, for example), measurement of differences

in anisotropy between layers in a layered aquifer will not

always be possible. When the measurement of anisot-

ropy in specific layers is difficult, the proposed formu-

lation may be used for hypothesis testing by changing

the anisotropy of different layers.Layered or stratified aquifers are discussed frequently

in the groundwater literature, as are aquifers with an

anisotropic transmissivity (e.g. [20,25,26]). Here, an

analysis of flow in an aquifer with both these conditions

is presented. Since this type of layered anisotropy is not

considered in earlier groundwater flow studies, the

geological circumstances that may lead to this type of

physical properties will be discussed briefly. There aremany texts that discuss the structure of sedimentary

deposits (e.g. [8,22]). With regard to layering, these texts

identify stratification or bedding as the most common

sedimentary structure (e.g. [8, p. 128]). The thickness of

these beds ranges from a centimeter to more than 1 m.

With regard to anisotropy, it is well known that an-

isotropy in the vertical plane often results from a series

of horizontal sediment layers. However, in most naturalenvironments, ripples, sand waves, and dunes are the

common type of bed forms, much more common than

planar beds [8, p. 141]. Within a bed, the transmissivity

will be anisotropic as a result of the internal sedimentary

structure of these ripples, waves or dunes, and their

preferential direction. The origin of anisotropy in con-

solidated rock aquifers is quite different. In a fractured

rock mass the principal direction of the hydraulic con-ductivity is parallel to preferentially aligned fractures

and fissures. Since the development of these fissures is

caused by effective stress, the principal direction will

usually be constant within a single lithological unit.

2. Mathematical formulation

A formulation will be derived for flow in layered

aquifers. The horizontal hydraulic conductivity may

differ between layers and may be anisotropic, but must

be piecewise homogeneous within a layer. It is empha-

sized that the presented theory is not applicable toaquifer systems consisting of aquifers separated by

aquitards. This paper deals with single, stratified aqui-

fers where all layers are fully connected. The resistance

to flow in the vertical direction is neglected for the

computation of the head and the horizontal components

of flow (the Dupuit approximation); the vertical com-

ponent of flow is computed from three-dimensional

continuity of flow. The formulation is presented for

steady-state flow conditions, but this is not a restriction

on the approach.

Consider an aquifer consisting of M homogeneouslayers and a Cartesian x, y, z coordinate system with the

z-axis pointing vertically upward. The horizontal hy-

draulic conductivity tensor Km (L/T) in layer m is sym-

metric. The major principal value is called k1;m and the

minor principal value k2;m; the major principal direction

makes an angle am with the positive x-direction

(�p=2 < am 6 p=2; Fig. 1). Each layer is homogeneous,

so that the transmissivity tensor Tm of layer m may beobtained as

Tm ¼ HmKm ð1Þ

where Hm is the thickness of layer m. Tm, which has

principal values T1;m ¼ Hmk1;m and T2;m ¼ Hmk2;m, may be

written in terms of the x, y coordinate system as

Tm ¼ Txx;m Txy;mTxy;m Tyy;m

� �ð2Þ

where (e.g. [7, p. 140–1])

Txx;m ¼ T1;m cos2 am þ T2;m sin2 am

Txy;m ¼ ðT1;m � T2;mÞ sin am cos am

Tyy;m ¼ T1;m sin2 am þ T2;m cos2 am

ð3Þ

The vertically integrated flow ~QQm (L2/T) in layer m is

defined as the horizontal specific discharge vector inte-

grated from the bottom to the top of layer m and may be

computed as

~QQm ¼ �Tm~rrh ð4Þ

where ~rrh is the gradient of the piezometric head h (L). Itis emphasized that the head gradient is the same in each

layer under the Dupuit approximation [24, p. 36].

The comprehensive discharge vector ~QQ (no index)

represents the vertically integrated flow of the entire

Fig. 1. Definition of angles.

748 M. Bakker, K. Hemker / Advances in Water Resources 25 (2002) 747–754

aquifer and is obtained through the summation of the

discharge vector of each layer

~QQ ¼XMm¼1

~QQm ð5Þ

Substitution of (4) for ~QQm with (2) in (5) gives

~QQ ¼ �T ~rrh ð6Þwhere T (L2/T) is the comprehensive transmissivitytensor

T ¼ Txx TxyTxy Tyy

� �¼

PMm¼1 Txx;m

PMm¼1 Txy;mPM

m¼1 Txy;mPM

m¼1 Tyy;m

� �ð7Þ

The comprehensive transmissivity tensor T is symmetric

because every transmissivity tensor Tm is symmetric. The

principal values T1 and T2 and the direction of maximum

transmissivity b of T may be determined from the well-

known formulas (e.g. [7])

tan 2b ¼ 2TxyTxx � Tyy

ð8Þ

T1;2 ¼Txx þ Tyy

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTxx � Tyy

2

� �2

þ T 2xy

sð9Þ

The x, y coordinate system is rotated over the angle bresulting in the new Cartesian �xx, �yy coordinate system,

such that the �xx-direction points in the direction of

maximum T (Fig. 1)

�xx ¼ x cos b þ y sin b

�yy ¼ �x sin b þ y cos bð10Þ

The �xx and �yy components of the comprehensive discharge

vector simplify to

Q�xx ¼ �T1oho�xx

Q�yy ¼ �T2oho�yy

ð11Þ

Continuity of flow may now be written, for steady-state

conditions, as

oQ�xx

o�xxþ oQ�yy

o�yy¼ N ð12Þ

where Nð�xx; �yyÞ (L/T) is the leakage into the aquifer.

Substitution of (11) for Q�xx and Q�yy in (12) gives

@

o�xxT1

oho�xx

� �þ @

o�yyT2

oho�yy

� �¼ �N ð13Þ

This differential equation may be transformed into

Poisson�s differential equation through a scaling of the

coordinate system, resulting in [6,19].

@2ho~xx2

þ @2ho~yy2

¼ �NeTT ð14Þ

where ~xx, ~yy are scaled coordinates defined as

~xx ¼ �xx

ffiffiffiffiffieTTT1

s~yy ¼ �yy

ffiffiffiffiffieTTT2

sð15Þ

and where eTT is the isotropic transmissivity in the scaled

domaineTT ¼ffiffiffiffiffiffiffiffiffiT1T2

pð16Þ

Eq. (14) is the standard differential equation for

steady-state flow with a leakage rate N into the aquifer.

A variety of methods is available to obtain analytic so-

lutions to this differential equation, depending on the

boundary conditions (e.g. [9,24]). For now it suffices to

assume that boundary conditions have been specified,

(14) has been solved, and the function hð~xx; ~yyÞ has beentransformed back to the �xx, �yy or x, y coordinate system.

Equations for uniform flow and wells will be presented

in the next section.

Once the head function is known, the discharge vec-

tor in layer m may be obtained with (4). The leakage Nm

(L/T) into layer m, defined as the sum of the leakage

through the bottom and top of layer m, is obtained by

taking the divergence of the discharge vector of layer m(note that the divergence in the x, y coordinate system

is equal to the divergence in the �xx, �yy coordinate system,

because the coordinate systems are related through

a rotation)

Nm ¼ ~rr ~QQm ð17ÞIf the vertical component of the specific discharge vector

(positive upward) at the interface between layers m and

m� 1 is called qz;m then

Nm ¼ qz;mþ1 � qz;m ð18ÞThe vertical component of the specific discharge varies

linearly in the vertical direction within a layer and is

continuous at the interface between layers. This is a

consequence of the Dupuit approximation [23]. Hence,

in a system with M layers, with the bottom of layer M

impermeable (qz;Mþ1 ¼ 0), the vertical flow at the inter-face between layers m� 1 and m may be computed as

qz;m ¼ �XMj¼m

Nj ð19Þ

where Nj is obtained from (17). It is important to note

that even when the layered aquifer is fully confined(N ¼ 0), the vertical flow between layers will not be zero

when the different layers of the aquifer have different

anisotropies. Furthermore, it may be observed from

(19), but also from (17) and (12), thatXMj¼1

Nj ¼ N ð20Þ

3. Uniform flow and wells

In this section, functions are presented for uniform

flow and flow to a well; solutions may be superimposed.

M. Bakker, K. Hemker / Advances in Water Resources 25 (2002) 747–754 749

Consider a uniform comprehensive flow with compo-

nents Qx0 in the positive x-direction and Qy0 in the

positive y-direction and no areal leakage into the aquifer

(N ¼ 0). The head as a function of x and y is presented

without derivation

h ¼ ðTyyQx0 � TxyQy0Þxþ ðTxxQy0 � TxyQx0ÞyT 2xy � TxxTyy

þ h0 ð21Þ

where h0 is the head at the origin. It may be verified that

this expression represents the given uniform flow field bysubstitution of (21) for h into (6). The flow is uniform in

each layer, but both the magnitude and direction will

differ, depending on the anisotropy of the layers. The

flow in each layer may be obtained by substitution of

(21) for h in (4), which gives

Qx;m

Qy;m

� �¼ �1

T 2xy � TxxTyy

Txx;m Txy;mTxy;m Tyy;m

� �TyyQx0 � TxyQy0

TxxQy0 � TxyQx0

� �ð22Þ

A comprehensive uniform flow does not create any

leakage between layers, as may be verified from substi-

tution of (22) for ~QQm into (17).Next, functions are derived for flow to a well with

discharge Q at the center of the ~xx, ~yy coordinate system;

the well screen penetrates the aquifer fully and there is

no areal leakage into the aquifer (N ¼ 0). The head

function in the ~xx, ~yy coordinate system is the Thiem

solution

h ¼ Q

4peTT lnð~xx2 þ ~yy2Þ þ h0 ð23Þ

where h0 is the head at a unit distance from the well in

the ~xx, ~yy system. The components of the head gradient in

the �xx, �yy coordinate system are

oho�xx

¼ Q2pT1

�xxq2

oho�yy

¼ Q2pT2

�yyq2

ð24Þ

where q2 ¼ eTT �xx2=T1 þ eTT �yy2=T2. The flow components in

layer m may be obtained by combination of (4) and (24),

which gives, after rearrangement of terms

Q�xx;m

Q�yy;m

� �¼ � Q

2pq2

Txx;m=T1 Txy;m=T2Txy;m=T1 Tyy;m=T2

� ��xx�yy

� �ð25Þ

where the components of transmissivity tensor m in the�xx, �yy coordinate system may be obtained with (3) when am

is replaced with am � b. It may be seen from (25) that

horizontal flow is, in general, not radial towards thewell. There are still two lines, however, along which the

horizontal flow is radial. These lines point in the direc-

tion of the eigenvectors of the tensor in (25). The leakage

into layer m may be obtained with (17), which gives,

after carrying out the differentiations and gathering

terms

Nm ¼ � Q2p

Txx;mT1

�"þ Tyy;m

T2

�1

q2

� Txx;m�xx2

T 21

þ Tyy;m�yy2

T 22

þ 2Txy;mxyT1T2

!2eTTq4

#ð26Þ

It is noted that the vertical leakage decreases as 1=q2,

while the horizontal components of the flow decrease as1=q (25).

To gain insight in Eqs. (25) and (26), consider an

aquifer consisting of two layers. The transmissivity

tensors of the two layers have the same major and minor

principal values, and are called s1 and s2, respectively.The principal directions of the two layers are normal to

each other so that the comprehensive transmissivity is

isotropic with major and minor components equal tos1 þ s2. Since the comprehensive transmissivity tensor is

isotropic, the analysis may be continued in the x, y

plane. If the major principal direction of layer 1 is in the

x-direction, then Txx;1 ¼ Tyy;2 ¼ s1 and Tyy;1 ¼ Txx;2 ¼ s2,and the off-diagonal terms are zero. Horizontal flow is

radial in the direction of the eigenvectors of the tensor in

(25); for this case those directions are the x and y axes.

The upward leakage at the interface between the twolayers may be written with the aid of (26) as

N1 ¼ qz;2 ¼Q2p

ðs1 � s2Þðs1 þ s2Þ

ðx2 � y2Þr4

ð27Þ

For this special case, the leakage between the two layers

equals zero along the lines x ¼ y and x ¼ �y. These twolines divide the domain in four quadrants. In the two

quadrants where jxj > jyj the leakage is upward, in the

other two quadrants the leakage is downward. Eq. (27)

may alternatively be written as

qz;2 ¼Q2p


cosð2hÞr2

ð28Þ

where h is defined as h ¼ arctanðy=xÞ. The total upwardleakage Qtot in the quadrant �p=46 h6 p=4 from thewell radius rw to a distance r from the well is obtained

from integration of (28) and gives

Qtot ¼Q2p


lnrrw

� �ð29Þ

4. Example 1

It is rather easy to create fascinatingly complicatedflow line patterns by specifying several wells in a layered

aquifer consisting of a number of layers with different

anisotropy ratios and directions. These three-dimen-

sional flow line patterns are very difficult to analyze.

Here, the discussion is limited to a rather simple two-

layer aquifer with one well and, in the next example, a


well in a uniform flow field. The top layer of the aquifer

consists of sandstone with an isotropic hydraulic con-

ductivity while the bottom layer consists of highly

fractured carbonate rock with an anisotropic hydraulicconductivity. It will be shown that flow lines to a well

screened in this aquifer are far from radial and that the

difference in anisotropy between the two layers has a

major effect on the capture zone envelope of the well.

Specific aquifer data used in both examples are shown in

Table 1.

A well with discharge Q is screened over the entire

aquifer thickness and is located at the origin of an x, ycoordinate system; flow in the area of interest is con-

fined. Flow lines are computed through numerical in-

tegration of the (analytic) specific discharge vector; a

predictor–corrector method is applied. Flow lines are

started at three locations: ðx; yÞ ¼ ð�100; 0Þ, ð�100;100Þ, and ð0; 100Þ, which are labeled A, B, and C, re-

spectively (Fig. 2). At each location six flow lines are

started at elevations z ¼ 2, 6, 10, 14, 18, 22 m. A flowline is represented by a thin line when it is in the top

layer and with a thick line in the bottom layer. It is

noted that the direction of the flow lines changes

abruptly when crossing the interface between the two

layers. The dotted lines represent the lines along which

the horizontal flow is radial in both layers (obtained by

evaluating the eigenvectors of the tensor in Eq. (25)).

The flow lines starting at point A are projected on avertical plane normal to the y-axis (bottom part of Fig.

2). A contour plot of the vertical flow at the interface

between the two layers is shown for a square area of 20

by 20 m around the well in Fig. 3; the shaded areas

represent upward leakage and the dotted lines the

directions in which horizontal flow is radial.

It may be seen from Fig. 2 that the anisotropy of the

bottom layer has a strong influence on the flow field.Flow lines in both layers deviate from the straight radial

lines that would be obtained in an aquifer with isotropic

layers. Flow lines in the bottom layer tend to deviate

towards the major principal direction of the hydraulic

conductivity in the bottom layer; flow lines in the top

layer deviate in the opposite direction. Flow lines tend to

spiral towards the well. This is best illustrated by con-

sidering flow line 3 in Fig. 2. A three-dimensional rep-resentation of flow line 3 is shown in Fig. 4; notice that

the y and z scales are exaggerated. The thick line is flow

Table 1

Aquifer data of examples 1 and 2

m (layer #) Hm (m) T1;m (m2/d) T2;m (m2/d) am

Specified

1 12 120 120 N/A

2 12 120 24 )30�

ComputedeTT ¼ 185:90 m2/d b ¼ �30�

Fig. 2. Flow lines to a well at ðx; yÞ ¼ ð0; 0Þ in a two-layer aquifer. Plan

view of all flow lines (top) and projection on a vertical plane of the flow

lines starting at A (bottom; vertical scale is exaggerated). Flow lines are

thin in top layer and thick in bottom layer.

Fig. 3. Contour plot of the vertical leakage at the interface between the

two layers around the well; shaded area indicates upward leakage.


line 3, the thin lines are projections on the x, y, and z

planes. Flow line 3 consists of three sections. The first

and third section are in the top layer, the second section

is in the bottom layer. It may be seen from Fig. 4 that

flow line 3 has the shape of a counter-clockwise spiral

when looking from point A in the direction of flow

towards the well.

5. Example 2

The previous example is modified by the addition

of a comprehensive uniform flow Qx0 in the positivex-direction; everything else remains the same. As was

stated previously, a comprehensive uniform flow means

that flow is uniform in each layer but both their mag-

nitudes and directions are different (22); the sum of the

two flows gives Qx0 in the positive x-direction. Evalua-

tion of (22) shows that the uniform flow in the top

aquifer makes an angle of 13.9� with the x-axis, and in

the bottom aquifer an angle of )19.1�. The flow in thetop aquifer is 36% larger than in the bottom aquifer.

The uniform flow solution is added to the well solu-

tion, where the relative strengths are such that

Qx0H=Q ¼ 0:24 (H is the total thickness of the aquifer).

It will be attempted to determine the boundary of that

part of the aquifer that is captured by the well. This

boundary is referred to as the capture zone envelope. In

a single-layer aquifer, the capture zone may be obtainedby starting flow lines on either side of the stagnation

point associated with the well, and tracing the flow lines

against the flow direction [1]. For the problem under

consideration there is also only one stagnation point

(the point on the x-axis where oh=ox ¼ oh=oy ¼ 0). Flow

lines are started on either side of the stagnation point at

the top of the aquifer and at the bottom of the aquifer

and are represented by the dashed lines in Fig. 5 (the

thin line is the top layer). As expected, away from the

well the envelopes consist of two straight parallel lines

corresponding to the direction of uniform flow in that

layer. Note that the confluence of these two envelopes

does not encompass the entire volume of aquifer that is

captured by the well. This is illustrated by starting fiveflow lines at different elevations from point D at ðx; yÞ ¼ð�200; 20Þ. These five flow lines are also shown in Fig. 5;

again a flow line is represented by a thin line when it is in

the top layer and by a thick line in the bottom layer. As

may be seen from Fig. 5, flow lines that are started near

the interface between the two layers (z ¼ 12) are cap-

tured by the well. This remains the case far upstream of

the well: streamlines starting between the two dashedenvelopes and near the interface still reach the well. For

example, a flow line starting at ðx; y; zÞ ¼ ð�1000; 20;12:1Þ reaches the well, but a flow line starting at

ð�1000; 20; 12:2Þ bypasses the well.

6. Discussion and conclusions

The research described in this paper was initiated

when it became clear from experiments with finite-ele-

ment models that flow lines in groundwater models

sometimes represent spiraling curves. These flow lines

were found to occur in layered aquifers that have dif-ferent anisotropic horizontal hydraulic conductivities in

adjacent layers. Bundles of such spiraling flow lines,

turning in the same direction, were referred to as

groundwater whirls [16]. Within a single model several

of these whirls may exist, on top and next to each other,

where adjacent whirls turn in opposite directions. In the

Fig. 5. Flow lines for a well in a uniform flow field in a two-layer

aquifer, starting at point D at different elevations z from the base.

Fig. 4. Three-dimensional representation of flow line 3 (thick line),

with projections on the x, y, and z planes (thin lines); y and z scales are

exaggerated.


context of transport modeling, these whirls can be of

great importance.

In this paper, a new theory was presented for

groundwater flow in layered anisotropic aquifers; flow

must remain semi-confined. Two main approximationswere made: (1) the aquifer consists of a number of hor-

izontal, homogeneous layers, each with its own aniso-

tropic transmissivity, and (2) the resistance to flow in the

vertical direction is neglected (the Dupuit approxima-

tion). Analytic solutions for head and horizontal flow in

each layer may be derived by use of a comprehensive

transmissivity tensor (7). The vertical component of

flow may be obtained analytically through vertical inte-gration of the horizontal divergence. Equations were

derived for steady-state uniform flow and flow to a well

in an aquifer with an arbitrary number of layers; solu-

tions may be superimposed. Flow lines were computed

through numerical integration of the specific discharge

vector.

Three-dimensional flow line patterns for flow to a

well can become quite complicated, even in a two-layeraquifer with only one anisotropic layer (Fig. 2). Eqs.

(25) and (26) show that both the horizontal and vertical

components of flow are directly proportional to the well

discharge. This implies that the flow line pattern is in-

dependent of the specific value of the discharge. From

previous numerical experiments [16] it is known that a

single well in a two-layer anisotropic aquifer produces

four bundles of spiraling flow lines. Here it was shownthat for a two-layer aquifer, the groundwater whirls are

bounded by two vertical planes intersecting in the well,

with directions given by the eigenvectors of the tensor in

(25); these planes are shown in Fig. 2 with dotted lines.

Horizontal flow along these planes is radial towards the

well. Flow lines starting from point A are part of a

counter-clockwise rotating whirl when looking in the

direction of flow; flow lines starting from B and C arepart of a clockwise whirl. It is noted that the two planes

along which horizontal flow is radial are, in general, not

normal to each other. They are normal to each other

for the presented example because one of the layers is

isotropic, so that the off-diagonal elements of the tensor

in (25) are zero. The results of Example 1 were com-

pared to results obtained with the finite element model

MicroFEM [10]. Results of the finite element modelshow small differences in heads between the layers (as

may be expected; in the finite element model there is no

vertical flow if there is no head difference). But the re-

sulting three-dimensional flow lines were virtually

identical to those presented in Fig. 2; small differ-

ences were only visible very close to the well (less than

5 m).

The theory presented in this paper may be applied tostudy patterns of spiraling flow lines to wells in aquifers

with more than two layers. Such flow fields can become

quite complicated and are beyond the scope of this

paper. The boundaries of whirls will be different in each

layer, depending on the transmissivity of the layer and

the comprehensive transmissivity tensor. In addition,

whirls may extend over any number of layers and two or

more smaller whirls may be enclosed by a larger one, allturning in the same direction. The actual configuration

of clockwise and counter-clockwise whirls depends on

how the anisotropy changes from the top to the bottom

of a layered aquifer.

Finally, it was shown that layered anisotropy has

a strong influence on the shape of the capture zone of

a well. Even for a two-layer aquifer consisting of one

isotropic layer and one anisotropic layer, the capturezones extended in different directions in the two layers.

Due to the vertical exchange of water between the layers

it is difficult to define the shape of the capture zone of

a well accurately (Fig. 5).

Acknowledgements

The authors thank B. Hunt, A. Kacimov, and twoanonymous reviewers for their comments and sugges-

tions.

References

[1] Bakker M, Strack ODL. Capture zone delineation in two-

dimensional groundwater flow models. Water Resourc Res

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a dupuit formulation for flow in layered, anisotropic aquifers

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