WATER RESOURCES RESEARCH, VOL. 40, W11507, doi:10.1029/2004WR003115, 2004
Dupuit-Forchheimer solutions for radial flow with linearly
varying hydraulic conductivity or thickness of aquifer
Nils-Otto Kitterød1
Department of Geoscience, University of Oslo, Oslo, Norway
Received 17 February 2004; revised 8 July 2004; accepted 9 August 2004; published 16 November 2004.
[1] This paper presents new analytical solutions for steady state piezometric heads in
phreatic and confined aquifers with radial flow. The new solutions utilize two constant
head boundaries and a hydraulic conductivity k or thickness of aquifer H that is a linear
function of radius r. Previous solutions applied constant k and H, and they were usually
based on one constant head boundary, which might be superimposed on a solution that
allowed a constant discharge or recharge at the center of the aquifer. The previous
solutions may be difficult to apply to real aquifers because constant k and H is usually an
oversimplification of nature. The Dupuit-Forchheimer assumption implies that vertical
head gradients are ignored. The magnitude of the error introduced by allowing H = H(r) is
INDEX TERMS: 1829 Hydrology: Groundwater hydrology;
evaluated numerically in this paper.
3210 Mathematical Geophysics: Modeling; 1832 Hydrology: Groundwater transport; KEYWORDS: analytical
modeling, Dupuit-Forchheimer, groundwater flow
Citation: Kitterød, N.-O. (2004), Dupuit-Forchheimer solutions for radial flow with linearly varying hydraulic conductivity or
thickness of aquifer, Water Resour. Res., 40, W11507, doi:10.1029/2004WR003115.
1. Introduction
[2] There is increased interest in applications of analytical modeling of subsurface flow and contaminant
problems [Wu and Pan, 2003; Yeo and Lee, 2003; Bakker
and Strack, 2002; Luther and Haitjema, 1998, 1999,
2000]. Innovations in finite mathematics and computer
technology have made numerical modeling a routinely
used tool in scientific computations as well as in practical
management and administration of water resources. However, the successful development of numerical technology
has not made analytical solutions redundant. On the
contrary, by abstracting nature into mathematically manageable quantities, analytical solutions are easy to implement in a spreadsheet or in a computer code. In this way,
analytical solutions offer direct insight into the physical
conditions that are important for specific flow problems.
Numerical simulations, on the other hand, require a
considerable amount of time to achieve similar evaluations. Observations indicate that groundwater flow in
delta structures may be simplified to radial flow due to
the axial symmetry of the aquifer. This simplified geometry is possible because a lot of prograding deltas are
radial structures where the river mouth is positioned on
the axis of symmetry. Work here expands previous radial
solutions of the Poisson’s equation by introducing two
constant head boundaries and letting the hydraulic conductivity or the thickness of the aquifer be a linear
function of radius (Figure 1).
1
Also at Norwegian Centre for Soil and Environmental Research, Ås,
Norway.
2. Discharge Potential Assuming Constant
Hydraulic Conductivity or Constant Aquifer
Thickness
[3] On a regional scale, where the ratios between vertical
and horizontal dimension are very small, the vertical resistance to groundwater flow may be neglected (the DupuitForchheimer assumption). The discharge potential F =
(dQ/dr) [Strack, 1984, 1989] of a circular structure (like
a delta or an island) may therefore be simplified to a
function of the radius r only [Haitjema, 1995]:
F¼
N 2
r R22 þ F2 ;
4
ð1Þ
where the constant head boundary is F2 at r = R2 and N is
the net infiltration to the groundwater. By the use of the
discharge potential as suggested by Strack [1989], equation
(1) is the solution for both a confined aquifer (F = k H f)
and a phreatic aquifer (F = (1/2) k f2), where k is hydraulic
conductivity, H is the thickness of a confined aquifer, and f
is the piezometric head, which is equal to the saturated
thickness of a phreatic aquifer.
[4] For geological reasons it may be convenient to introduce a constant head boundary F1 at r = R1. First we derive an
expression for the area between the inner boundary R1 and the
groundwater divide l. The water balance Q for steady state
flow implies that net precipitation N on the inner cylindrical
area R1 r l (see Figure 1 for boundary conditions):
Q ¼ Np l 2 r2 ;
ð2Þ
is equal to the radial flow Qr across the cylinder with radius
r:
Copyright 2004 by the American Geophysical Union.
0043-1397/04/2004WR003115$09.00
Qr ¼
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Q
:
2pr
ð3Þ
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KITTERØD: TECHNICAL NOTE
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which is the discharge potential for radial flow with
constant head boundary value at r = R1 and impervious
boundary at r = l.
[7] The same reasoning may be applied if l r R2 with
boundary value F2 at r = R2. The discharge potential Fb for
the outer cylindrical area is equal to
Nl 2
N 2
r
2
ln
Fb ¼ þ F2 :
r R2 þ
2
4
R2
ð7Þ
[8] At the groundwater divide (r = l in equations (6) and
(7)), there is no radial flow: dFa/dr = dFb/dr = 0, and the
discharge potential F = Fa = Fb, is used to form a closed
expression by elimination of l in equations (6) and (7):
F¼
N 2
ln r ln R2
r R21 F1
4
ln R2 ln R1
N 2
ln r ln R1
2
r R 2 F2
:
4
ln R2 ln R1
ð8Þ
Equation (8) is referred to as the simple ‘‘doughnut’’
equation.
[9] It is easy to see that equation (8) is consistent with
equation (1) because
lim
R1 !0
lim
Figure 1. Principal sketch of delta geometry. Figure 1a
shows the interface between the paleoglacier and the
sediments, radius to the inner and outer delta boundaries
(R1 and R2), and the location of the groundwater divide (l ).
Figure 1b is a radial cross section through the delta
indicating a confined aquifer with linearly decreasing
aquifer thickness with piezometric head f1 and f2 at R1
and R2. Figures 1c and 1d are modified from Gilbert [1890],
indicating the sediments at the glacier interface (Figure 1c)
and after withdrawal of the glacier (Figure 1d) and with a
phreatic groundwater table. If net recharge N > 0, the
groundwater divide (dF/dr = 0) is at r = l.
The flow direction is opposite of r, and thus Qr has to be
negative in this case.
[5] Combined with Darcy’s law of laminar flow,
Qr ¼ dFa
;
dr
ð4Þ
the water balance equations (2) and (3) yield
Fa ¼
Nl 2
N
ln r r2 þ Ca :
4
2
ð5Þ
[6] The boundary value Fa = F1 at r = R1 inserted in
equation (5) gives
Nl 2
N 2
r
ln
þ F1 ;
Fa ¼ r R21 þ
2
4
R1
ð6Þ
R1 !0
ln r ln R2
ln R2 ln R1
ln r ln R1
ln R2 ln R1
¼ 0;
ð9Þ
¼ 1:
ð10Þ
[10] As pointed out by an anonymous reviewer, the
existence of a groundwater divide l between R1 and R2 is
no longer necessary because l is eliminated in equation (8).
This implies that a (theoretical) groundwater divide may be
outside the aquifer. The groundwater divide (dF/dr of
equation (8)) is equal to
2
1
ðF1 F2 Þ þ R21 R22
N
2
l ¼
:
ln R1 ln R2
2
ð11Þ
3. Piezometric Head in a Confined or Phreatic
Aquifer Where Thickness or Hydraulic
Conductivity is a Linear Function of Radius
[11] To make the simple ‘‘doughnut’’ equation (8) more
realistic, we may let the thickness of the aquifer H or
hydraulic conductivity k be a linear function of radius r.
In this case the discharge potential, as defined above, does
not exist, and the equations for confined and phreatic
aquifers are more convenient to derive separately. Starting
with the confined aquifer, we let the thickness of the aquifer
H be given as a linear function of r:
H ðrÞ ¼ H1;2 a r R1;2 ;
where a = (H1 H2)/(R2 R1) (Figure 1).
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ð12Þ
KITTERØD: TECHNICAL NOTE
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[12] If a is small, i.e., H1 H2 R2 R1, we may
neglect the vertical component and solve the approximate
piezometric head as a one-dimensional ordinary partial
differential equation. Again we divide the flow equation
into two parts where the groundwater divide l is the no-flow
boundary. Balance of mass for steady state flow where R1 r l gives the expression
qr H ¼
N
2
l2
r :
r
ð13Þ
Darcy’s law, qr = k(dfa/dr), inserted into equation (13)
gives
dfa ¼
N l 2 r2
dr:
rH
2k
ð14Þ
If H = H(r) is a linear function of r as given in equation
(12), then two simple integrals have to be solved, namely,
Z
Z
1
1
H0 ar
þ c;
dr ¼ ln
rðH0 arÞ
H0
r
r
1
dr ¼ 2 ðar H0 þ H0 lnðH0 arÞÞ þ c;
H0 ar
a
N
Nl 2
N
A1 þ B1 þ f1 ;
ðr R 1 Þ 2ka
2k
2k
Table 1. Differences d Between Analytical and Numerical
Solutions (Equation (19))a
H1 = 100 m
H2kH1 m
100.0
200.0
300.0
400.0
500.0
750.0
1000.0
Max (d),
m
0.022798
0.136579
0.202831
0.249633
0.287973
0.368054
0.438660
0.042431
0.171761
0.247430
0.304703
0.355404
0.471294
0.590987
0.044346
0.182896
0.294309
0.378903
0.450699
0.604228
0.740620
0.011847
0.030846
0.042964
0.054727
0.066562
0.097509
0.130050
df
d A1 L2 A2 L1
¼
¼ 0;
dr dr
A1 A2
2
l ¼ H0
ð16Þ
ð17Þ
R2 R1
a
2k
2
þ Ha20 ln H
H1 þ N ðf1 f2 Þ
R1
2
ln H
H1 þ ln R2
ð20Þ
!
:
ð21Þ
[18] Next, we may develop the same kinds of equations
for phreatic aquifers where the hydraulic conductivity k is
given as
k ðrÞ ¼ k1;2 b r R1;2 ;
ð22Þ
where b = (k1 k2)/(R2 R1), similar to the linear equation
applied for a confined aquifer (Figure 1).
[19] Balance of mass and Darcy’s law yield
df2 ¼ N
2
l r2
dr;
rk
ð23Þ
with k given in equation (22). Imposing the boundary
conditions f = f1 at r = R1 and f = f2 at r = R2, the solution
of equation (23) is
ð18Þ
f2 ¼
where A2 = (1/H0)ln(R2H/H2r) and B2 = (H0/a )ln(H/H2),
where H is given in equation (12) for index 2.
[15] By eliminating the groundwater divide l in equations
(17) and (18), we get one expression for the piezometric
head:
A1 L2 A2 L1
;
A1 A2
0.013306
0.039585
0.072078
0.096665
0.117329
0.161174
0.200173
[17] The groundwater divide for R1 l R2 exists if the
derivative of equation (19) is df/dr = 0:
2
f¼
0.021947
0.110088
0.179970
0.235209
0.282543
0.382758
0.469550
which is equal to
where A1 = (1/H0)ln(R1H/H1r) and B1 = (H0/a )ln(H/H1),
where H is given in equation (12) for index 1.
[14] We find the piezometric head for the outer area l r R2 by the same token:
N
Nl 2
N
A2 þ B2 þ f2 ;
ðr R 2 Þ 2k
2ka
2k
Max (d), Mean (d), SD (d),
m
m
m
a
Numerical values in equation (19): k = 1.727901 105 m/s; N =
1.266730 108 m/s; R1 = 1000 m; R2 = 5100 m; f1 = f2 = 100 m. Here
d = jfanalytical fnumericalj.
2
fb ¼
H2 = 100 m
Mean (d), SD (d),
m
m
ð15Þ
where H0 = H1,2 + aR1,2, and c is the integral constant. H0 is
the (theoretical) aquifer thickness at r = 0.
[13] For the boundary condition f = f1 at r = R1, the
solution of equation (14) is
fa ¼
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a1 P2 a2 P1
;
a1 a2
ð24Þ
where
ð19Þ
where L1 = (N/2ka)(r R1) + (N/2k) B1 + f1, and L2 =
(N/2ka)(r R2) + (N/2k) B2 + f2. A1, A2, B1, and B2 are
defined as in equations (17) and (18).
[16] If a ! 0 (e.g., if the sloping bottom of the aquifer in
Figure 1 becomes horizontal), it is easy to verify that
equation (19) is consistent with equation (8) by power
expansion of ln(1 + (a(R1,2 r))/H1,2).
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N
ðr R1 Þ þ N b1 þ h21 ;
b
N
P2 ¼ ðr R2 Þ þ N b2 þ h22 ;
b
1
R1 k
a1 ¼ ln
;
k0
k r
1 1
R2 k
a2 ¼ ln
;
k0
k r
2 k0
k
b1 ¼ 2 ln
;
b
k1
k0
k
b2 ¼ 2 ln
;
b
k2
P1 ¼
ð25Þ
KITTERØD: TECHNICAL NOTE
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Figure 2. Three-dimensional numerical computation of heads (solid) and analytical (dashed) solution of
equation (19). In Figure 2a the outer boundary H2 is kept constant equal to 100 m while the inner
boundary H1 is stepwise increased: H1 = [1, 2, 3, 4, 5, 7.5, 10] 100 m (see label in Figure 2b). In
Figure 2c the opposite is performed: The inner boundary H1 is kept constant at 100 m while the outer
boundary H2 is increased in steps: H2 = [1, 2, 3, 4, 5, 7.5, 10] 100 m. Differences between the 3-D
numerical and the analytical solutions are shown in Figures 2b and 2d. Numerical results and values of
parameters in the equation are given in Table 1. See color version of this figure at back of this issue.
and where k0 = k1,2 + bR1,2 and k is given in equation (22)
with the corresponding indexes.
[20] Necessary conditions for a groundwater divide R1 l R2 is df2/dr = 0, which for an open aquifer is equal to:
2
l ¼ k0
R2 R1
b
!
þ bk02 ln kk21 þ N1 h21 h22
:
ln kk21 þ ln RR12
ð26Þ
4. Numerical Evaluation
[21] The Dupuit-Forchheimer assumption ignores the
vertical gradients in piezometric heads: @f/@z = 0. This
assumption is not valid close to the aquifer boundaries or in
the close vicinity of partially penetrating wells. Haitjema
[1987] performed a thorough numerical analysis of the
discrepancies between 3-D flow and the approximate
Dupuit-Forchheimer solutions. In the work by Haitjema
[1987] the recharge took place from a circular pond at the
center of an island, and he demonstrated that the 3-D effect
can be ignored at a distance 1 – 2 times the thickness H of an
isotropic
aquifer.
For an anisotropic aquifer the distance is
pffiffiffiffiffiffiffiffiffiffi
ffi
H kh =kv , where kh and kv are horizontal and vertical
hydraulic conductivity, respectively. Haitjema [1987] used a
constant thickness of the aquifer, but this rule of thumb may
not be valid when H (or k) is a variable in space. It is
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KITTERØD: TECHNICAL NOTE
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Figure 3. Piezometric head contours of analytical (solid) and 3-D numerical (dashed) solutions. The
contours in Figure 3a indicate that the numerical and analytical solutions are identical if the inner (H1)
and outer (H2) boundaries are equal to each other. In this case, H1 = H2 = 100 m. In Figure 3b, where H1 =
100 m and H2 = 10 H1, the maximum deviation is 0.59 m (see Table 1). See color version of this figure
at back of this issue.
therefore of interest to evaluate the additional error that is
introduced by allowing H (or k) to be a linear function of the
radial distance r from the axis of aquifer symmetry.
[22] Here the numerical evaluation was carried out using
MODFLOW with the PMWIN preprocessing and postprocessing [Chiang and Kinzelbach, 2001]. Only the solution
for a confined aquifer is included in these calculations. The
inner boundary R1 = 1000 m, and the outer boundary R2 =
5100 m. The boundary conditions were hydrostatic at R1
and R2: f1 = f2 = f = 100 m. First, H1 is kept constant equal
to 100 m, and H2 is increased in steps from 100 to 1000 m:
H2 = [100, 200, 300, 400, 500, 750, 1000] m. Second, H2 is
kept constant to 100 m and H1 is increased by the same
intervals. For H1 = H2 the analytical and the numerical
solutions gave identical results, exept for the numerical
noise. The numerical deviations between the analytical
and numerical solutions are given in Table 1 and presented
graphically in Figures 2 and 3. The deviations increase with
increasing differences between H1 and H2, but the maximum relative error, (fanal fnum)/f, is less than 1% even
for the most extreme case: H1,2/H2,1 = 10.
5. Conclusion
[23] Equations (19) and (24) are the Dupuit-Forchheimer
solutions for groundwater heads in confined and phreatic
aquifers where the geometry is simplified to a ‘‘doughnut’’
structure and the aquifer thickness or the hydraulic conductivity is expressed as a linear function of the radius. The
proposed extensions of the radial flow equations are consistent with previously published solutions where H or k are
constants in space. The error introduced by ignoring the
vertical flow component is not significant if j(H1 H2)/
(R2 R1)j < 1/5 (see Table 1). Examples where equations
(19) and (24) may be applied are delta structures that
comprise axial symmetry. It is important to keep in mind,
however, that the analytical solutions will not capture all the
details of a real aquifer because the purpose is to estimate
the regional trend of the average hydraulic parameters.
[24] Acknowledgment. I thank T. Lindstrøm for verifying the consistency between equations (19) and (8) and the two anonymous reviewers
for constructive comments.
References
Bakker, M., and O. D. L. Strack (2002), Analytic elements for multiaquifer
flow, J. Hydrol., 271, 129 – 199.
Chiang, W.-H., and W. Kinzelbach (2001), 3D-Groundwater Modeling With
PMWIN: A Simulation System for Modeling Groundwater Flow and
Pollution, 360 pp., Springer-Verlag, New York.
Gilbert, G. K. (1890), Lake Bonneville, Monogr. of the U.S. Geol. Surv.,
vol. 1, 438 pp.
Haitjema, H. M. (1987), Comparing a three-dimensional and a DupuitForchheimer solution for a circular recharge area in a confined aquifer,
J. Hydrol., 91, 87 – 101.
Haitjema, H. M. (1995), Analytic Element Modeling of Groundwater Flow,
394 pp., Academic, San Diego, Calif.
Luther, K., and H. M. Haitjema (1998), Numerical experiments on the
residence time distribution of heterogeneous groundwatersheds, J. Hydrol., 207, 1 – 17.
Luther, K., and H. M. Haitjema (1999), An analytic element solution to
unconfined flow near partially penetrating wells, J. Hydrol., 226, 197 – 203.
Luther, K., and H. M. Haitjema (2000), Approximate analytic solutions to
3D unconfined groundwater flow within regional 2D models, J. Hydrol.,
229, 101 – 117.
Strack, O. D. L. (1984), Three-dimensional streamlines in DupuitForcheimer models, Water Resour. Res., 20(7), 812 – 822.
Strack, O. D. L. (1989), Groundwater Mechanics, 732 pp., Prentice-Hall,
Upper Saddle River, N. J.
Wu, Y.-S., and L. Pan (2003), Special relative permeability functions with
analytical solutions for transient flow into unsaturated rock matrix, Water
Resour. Res., 39(4), 1104, doi:10.1029/2002WR001495.
Yeo, I. W., and K.-K. Lee (2003), Analytical solution for arbitrarily
located multiwells in an anisotropic homogeneous confined aquifer,
Water Resour. Res., 39(5), 1133, doi:10.1029/2003WR002047.
N.-O. Kitterød, Department of Geoscience, University of Oslo, P.O. Box
1047, Blindern, N-0316 Oslo, Norway. (nilsotto@geo.uio.no)
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KITTERØD: TECHNICAL NOTE
Figure 2. Three-dimensional numerical computation of heads (solid) and analytical (dashed) solution of
equation (19). In Figure 2a the outer boundary H2 is kept constant equal to 100 m while the inner
boundary H1 is stepwise increased: H1 = [1, 2, 3, 4, 5, 7.5, 10] 100 m (see label in Figure 2b). In
Figure 2c the opposite is performed: The inner boundary H1 is kept constant at 100 m while the outer
boundary H2 is increased in steps: H2 = [1, 2, 3, 4, 5, 7.5, 10] 100 m. Differences between the 3-D
numerical and the analytical solutions are shown in Figures 2b and 2d. Numerical results and values of
parameters in the equation are given in Table 1.
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KITTERØD: TECHNICAL NOTE
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Figure 3. Piezometric head contours of analytical (solid) and 3-D numerical (dashed) solutions. The
contours in Figure 3a indicate that the numerical and analytical solutions are identical if the inner (H1)
and outer (H2) boundaries are equal to each other. In this case, H1 = H2 = 100 m. In Figure 3b, where H1 =
100 m and H2 = 10 H1, the maximum deviation is 0.59 m (see Table 1).
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