1
March 16, 2004
Application of the Dupuit-Forchheimer assumption for calculating
groundwater heads in Gilbert-type deltas
Nils-Otto Kitterød1
Department of Geoscience, University of Oslo, Norway
Abstract
This paper presents analytical solutions for a steady state piezometric head in phreatic and
confined aquifers with circular structures. The solutions utilize two constant head boundaries
and a hydraulic conductivity or thickness of aquifer that is a linear function of radius r. Previous
solutions are based on one constant head boundary, superimposed on a solution that allowed
constant discharge or recharge at the center of the circular aquifer. This primitive solution may
be difficult to apply to real aquifers because a constant transmissivity is usually an
over-simplification of nature. An important example is delta formations that prograde into a
sedimentation basin. These kinds of deposits are referred to as Gilbert type deltas. The
Dupuit-Forchheimer assumption implies that vertical head gradients are ignored. This may not
be valid close to the flow boundaries. Nevertheless, if the thickness H of a confined aquifer is
constant, the error is small at a distance equal to H from the boundaries. However in this case,
where H is a function of r, the vertical flow is not limited to the boundaries. The magnitude of
the error introduced by allowing H = H(r), is evaluated numerically in this paper. Furthermore,
the approximate analytical solutions are evaluated against observations of piezometric heads
from an ice-contact delta. This evaluation shows that the regional trend of the observations can
be reproduced.
2
1. Introduction
There is increased interest for 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, 2000;
1999; 1998]. Innovations in finite mathematics and
computer technology have made numerical modeling
to a routinely used tool in scientific computations as
well as practical management and administration of
water resources. However, the successful development
of numerical technology has not made analytical solutions redundant, rather the contrary. By abstracting
nature into mathematically manageable quantities,
analytical solutions offer direct insight into the physical conditions that are important for specific flow
problems. In this way analytical solutions offer an alternative to numerical simulations, which usually require a considerable amount of details to achive realistic simulations. Work reported here results from observations of groundwater heads from an ice-contact
Gilbert-type delta in Norway, the Gardermoen delta
[Tuttle, 1997; Tuttle et al, 1997]. These observations
clearly indicate that groundwater flow in this structure may be simplified to a one dimensional flow due
to the axial symmetry of the aquifer. Futhermore,
observations such as the groundwater divide and the
regional trend of piezometric heads may be calculated
by solving the Poisson’s Equation: ∇2 Φ = −N . This
simplified geometry is possible due to the fact that
a lot of prograding deltas are radial structures where
the river mouth is positioned on the the axis of symmetry. Deltas deposited close to a melting glacier are
classified as a special class of Gilbert deltas [Gilbert,
1890; Bogen, 1980; Reading, 1998]. The interface between the sandy foresets and the silty bottom sets
may be treated as constant head boundaries. In such
cases the groundwater divide l, is located between the
inner boundary at distance R1 and the outer boundary R2 as indicated in Fig. 1. Work here expands
simple 1d solutions of the Poisson’s Equation by introducing two constant head boundaries and let hydraulic conductivity or thickness of the aquifer be a
linear function of radius.
The groundwater resources in Gilbert-type deltas
constitute valuable resources. The groundwater in
this type of formations is usually robust with respect
to contamination, due to the good remediation capacity of the vadose zone. In addition there may
appear semi impervious barriers in the aquifer that
reduce the hazard of pumping contaminated water
from supply wells. The system of highly permeable
coarse grained channels (generated either by the sediment feeding rivers or turbidity flow) give the reservoirs exceptional hydraulic connectivity, suitable for
the extraction of large amounts of water. Where observations of groundwater heads are available, the analytical solution presented below may be used to estimate the ratio of the groundwater recharge N to the
hydraulic conductivity k or thickness of aquifer H as
a function of the distance r to the axis of symmetry.
This function: k(r) or H(r), is of great importance
both for location of wells for water extraction and
for estimation of average velocities of contaminants
transport in the groundwater.
2. Discharge potential assuming
constant hydraulic conductivity or
constant aquifer thickness
On a regional scale where the ratio between vertical and horizontal dimension are very small, the vertical resistance to groundwater flow may be neglected
(Dupuit-Forchheimer assumption). The discharge potential Φ = − dQ
dr [Strack 1988; Haitjema 1995] of a
circular structure (like a Gilbert-delta, or an island),
may therefore be simplified to a function of the radius
r only:
Φ = −
N 2
(r − R22 ) + Φ2 ,
4
(1)
where the constant head boundary is Φ2 at r = R2 ,
and N is the net infiltration to the groundwater. By
the use of the discharge potential as suggested by
[Strack, 1988], eq. 1 is the solution for both a confined aquifer (Φ = k H φ), and a phreatic aquifer
(Φ = 12 k h2 ), where k is hydraulic conductivity, H
is the thickness of a confined aquifer, φ is piezometric head, and h is the groundwater level in a phreatic
aquifer.
For geological reasons it may be convenient to introduce a constant head boundary Φ1 at r = R1 .
First we derive an expression for the area between
the groundwater divide l and the inner boundary R1 .
The water balance Q for steady state flow implies
that net precipitation N on the inner cylindrical area
l ≤ r < R1 :
Q = N π(l2 − r2 ),
(2)
is equal to the radial flow Qr across the cylinder with
radius r:
−Qr =
Q
.
2π r
(3)
3
The flow direction is opposite of r, thus Qr has to
be negative in this case.
Combined with Darcy’s law of laminar flow:
Qr = −
dΦa
,
dr
(4)
the water balance equations (2 and 3) yields:
Φa =
N l2
N 2
lnr −
r + Ca .
2
4
(5)
The boundary value: Φa = Φ1 at r = R1 , inserted
in 5 gives
N l2
r
N 2
r − R12 +
Φa = −
ln
+ Φ1 , (6)
4
2
R1
which is the discharge potential for radial flow with
constant head boundary value at r = R1 with impervious boundary at r = l.
The same reasoning may be applied if R2 ≤ r < l
with boundary value Φ2 at r = R2 . The discharge
potential Φb for the outer cylindrical area is equal to:
N l2
r
N 2
2
+ Φ2 . (7)
r − R2 +
ln
Φb = −
4
2
R2
The distance to the impervious boundary l in 6
and 7 is equivalent to the groundwater divide where:
dΦa
dΦb
dr = dr = 0. At r = l the discharge potential
Φ = Φa = Φb , may be derived to form a closed expression of Φ by elimination of l in 6 and 7:
lnr − lnR2
N 2
2
r − R1 − Φ1
Φ =
4
lnR2 − lnR1
N 2
lnr − lnR1
2
−
r − R2 − Φ2
.(8)
4
lnR2 − lnR1
Equation 8 is refered to as the simple ’doughnut’
equation.
It is easy to see that 8 is consistent to 1, because:
lnr − lnR2
= 0,
(9)
lim
R1 →0 lnR2 − lnR1
and
3. Piezometric head in a confined- or
a phreatic aquifer where thickness or
hydraulic conductivity is a linear
function of radius
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. For simplicity we develop the equations for confined and phreatic aquifers separately, starting with
the confined aquifer.
Let thickness of aquifer H be given as:
H(r) = H1 − a(r − R1 ),
H1 −H2
R2 −R1
H0 −H1
R1
(12)
H0 −H2
R2
=
=
(Fig. 1).
where a =
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 l ≤ r < R1 gives the expression:
N l2
−q H =
− r .
(13)
2
r
a
Darcy’s law: q = −k dφ
dr , inserted in 13 gives:
N l2 − r2
dr.
dφa =
2k
rH
(14)
If H = H(r) is a linear function of r as given in 12,
then two simple integrals have to be solved, namely:
Z
1
1
H0 − ar
dr = −
ln
+ c, (15)
r(H0 − ar)
H0
r
and
Z
1
r
dr = − 2 (ar − H0 + H0 ln(H0 − ar)) + c. (16)
H0 − ar
a
For the boundary condition φ = φ1 at r = R1 , the
solution of 14 is:
N
N l2
N
( r − R1 ) −
A1 +
B1 + φ1 , (17)
2ka
2k
2k
R1 H1 −aR1 (r−R1 )
1
where A1 = H1 +aR
and B1 =
ln
rH1
1
H1 +aR1
1)
.
ln H1 −a(r−R
a2
H1
φa =
lim
R1 →0
lnr − lnR1
lnR2 − lnR1
= 1.
(10)
The groundwater divide is:
2
l =
2
N
(Φ1 − Φ2 ) + 21 R12 − R22
,
lnR1 − lnR2
which is the derivative of 8 where r = l at
(11)
dΦ
dr
= 0.
We find the piezometric head for the outer area
R2 ≤ r < l by the same token:
φb =
N
N l2
N
( r − R2 ) −
A2 +
B2 + φ2 , (18)
2ka
2k
2k
4
1
2 (r−R2 )
where A2 = H2 +aR
ln R2 H2 −aR
rH2
2
H2 +aR2
2)
ln H2 −a(r−R
.
a2
H2
and B2 =
α2
=
By eliminating the groundwater divide l in 17 and
18 we get one expression for the piezometric head:
β1
=
β2
=
A 1 L2 − A 2 L1
φ =
,
A1 − A 2
(19)
R1 ) +
B1 + φ1 , and L2 =
where L1 =
N
(r
−
R
)
+
φ
.
A
, A2 B1 , and B2 is
2
2
1
2ka
defined as in 17 and 18.
it is easy
By power expansion of ln 1 + a(R−r)
H1,2
to verify that 19 is consistent with 8 if a → 0.
The groundwater divide at r = l is the derivative
of 19 where dφ
dr = 0:
dφ
d A 1 L2 − A 2 L1
= 0, (20)
=
dr
dr
A1 − A 2
which is equal to:
2
l = H0
R2 −R1
a
+
H0
H2
a2 ln H1
H2
+
ln H
1
2k
N
R1
ln R
2
+
(φ1 − φ2 )
!
(21)
Next we can develop the equation for a phreatic
aquifer where the hydraulic conductivity k is given
as:
k(r) = k1 − b(r − R1 ),
(22)
−k2
1
2
= k0R−k
= k0R−k
, similar to the
where b = Rk21 −R
1
1
2
linear equation applied for a confined aquifer (Fig. 1).
Balance of mass and Darcy’s law yield:
2
l − r2
2
dh = N
dr,
(23)
rk
with k given in 22. Imposing the boundary conditions
h = h1 at r = R1 and h = h2 at r = R2 the solution
of 23 is:
h2 =
α1 P2 − α 2 P1
,
α1 − α 2
(24)
where
P1
=
P2
=
α1
=
1
k2 R2 − bR2 (r − R2 )
ln
k2 + bR2
k2 r
k1 + bR1
k1 − b(r − R1 )
ln
,
b2
k1
k2 + bR2
k2 − b(r − R2 )
ln
2
b
k2
,(25)
The groundwater divide (r = l) for an open aquifer
N
2k
N
2ka (r −
N
+ 2k
B2
N
(r − R1 ) + N β1 + h21 ,
b
N
(r − R2 ) + N β2 + h22 ,
b
1
k1 R1 − bR1 (r − R1 )
,
ln
k1 + bR1
k1 r
is:
2
l = k0
R2 −R1
b
+
k0
k2
1
b2 ln k1 + N
R1
ln kk12 + ln R
2
h21 − h22
!
(26)
2
which is the derivative of 24 where dh
dr = 0.
Equations 19 and 24 is the analytical solutions
for groundwater heads in a confined and a phreatic
aquifer where the geometry is simplified to a ’doughnut’ structure, and the aquifer thickness or the hydraulic conductivity can be expressed as a linear function of radius.
4. Numerical evaluation
The Dupuit-Forchheimer assumption ignore the
vertical gradients in piezometric heads: ∂φ
∂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 3d flow and
the approximate Dupuit-Forchheimer solutions. In
Haitjema (1987) the recharge took place from a circular pond at the center of an island, and he demonstrated that the 3d effect can be ignored at a distance one to two times the thickness H of an isotropic
aquifer.
p For an anisotropic aquifer the distance is
∼ 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 is a variable in space. It is therefore of interest
to evaluate the additional error that is introduced by
areal recharge and by allowing H (or k) to be a linear
function of the distance from the center H = H(r)
given in 12.
Here, the numerical evaluation was carried out using MODFLOW with the PMWIN pre- and post processing (Chiang and Kinzelbach, 2001). Only the
solution for a confined aquifer is included in these
calculations. The boundary conditions were hydrostatic at R1 and R2 (i.e. constant head boundaries
φ1 = φ2 = φ = 100 m) for both the numerical
5
mean-square deviation D, between the observations
yi and the calculated groundwater heads y(xi ; p):
and the analytical model. In this way the numerical and the analytical solution should give identical
results. That is also true if H1 = H2 (exept for the
numerical noise), but as soon as H1 /H2 6= 1 an error is introduced due to the vertical flow component
that is ignored in the analytical solution (Fig. 2). It
should be noted however that the max. relative error:
(φanal − φnum )/φ, is less than 1% even for the most
extreme case: H1 /H2 = 10 or H2 /H1 = 10. Numerical values are listed in Tab. 1, and spatial deviations
are illustrated in Fig. 3.
where x is the location of the observation tubes,
i = 1, · · · , n and n is the number of observations.
Measured and calculated heads are shown in Fig. 5
together with the D function. Estimation results are
given in Tab. 2.
5. Experimental field data
6. Discussion
The Gardermoen aquifer is a superposition of two
main delta structures, the Trandum delta to the west
and the Helgebostad delta to the south (Fig. 4).
The axial symmetry of the deltas around the paleoportals is evident. Furthermore the radial structure of the paleo-river channels supports the assumption that the groundwater flow is radial. At Gardermoen the groundwater divide is more distant to
the inner than to the outer boundary, indicating decreasing transmissivity with increasing distance to
the paleo-portals. This is also the reason why the
main drainage of the aquifer goes to the north into
the river called Risa. Because the river is greatly
fed by groundwater, the response of precipitation
and snowmelt to runoff is dampend and the river
discharge is fairly constant throught the the year.
In the period of groundwater monitoring, the specific discharge to Risa was 1.095 mm/d, which correspond well with average values of precipitation and
estimated evapotranspiration. Thus a steady-state
recharge of N = 1.095 mm/d = 1.26673 × 10−8 m/s
was used for calculation of the groundwater heads.
The second most important parameter to specify is
the boundary condition. In this case the origo (or the
axis of symmetry) was located at UTM-East: 8400.0
m; UTM-North: 7150.0 m (UTM zone 32 V, EU89
datum). The distances to inner and outer boundaries
are: R1 = 336.0 m and R2 = 5100.0 m, with corresponding constant head 171.5 m and 185.0 m (Fig. 4).
A large number of groundwater levels were monitored at the Trandum delta (Engen, 1995, Fig. 4).
After proper choice of recharge and boundary conditions these observations are used for estimating k1
and k2 for Eq. 24 and H1 and H2 for Eq. 19. In the
latter case k is assumed to be the arithmetic average
of k1 and k2 . Optimal estimates are the parameter
values p = [k1 , k2 ; H1 , H2 ] that minimize the root-
If the ration between H1,2 /H2,1 < 5 − 10 the error introduced by the Dupuit-Forchheimer assumption may be ignored. However, it is important to
keep in mind that the analytical solution presented
in this paper will not capture all the details of a
real aquifer, because the purpose is to estimate average flow parameters. The main regional character
however, is reproduced. The proposed extension of
Dupuit-Forchheimer flow in circular structures is consistent to previous published solutions where H or k
is constant in space. This technical derivation of analytical solutions has potential applications:
n
D =
1/2
1X 2
yi − y 2 (xi ; p)
n i
(27)
Stochastic boundary values. The ravine processes
at the inner- and outer boundaries generates a
landscape that is quite irregular (Fig. 4). This
process may be mimicked either conditionally
or unconditionally by simulation of a stochastic
deviation from the constant values R1 and R2 .
This is possible by taking the radial angle ϕ into
account. If the flow component perpendicular
to the radial direction is not too large, it may
be possible to approximate this flow condition
in a 1d solution.
Bayesian estimation. Estimation of unknown parameters may be improved by combining geostatistics and Bayesian methods (Omre and Halvorsen,
1989). Goshu (2003) improved the estimation
of transmissivities and storativities in points
where only grainsize distribution is available by
combining Gustafson’s (1983; 1986) analytical
method for estimation of aquifer parameters by
Bayesian up-dateing of stochastic parameters by
conditioning on local pumping tests. The practical estimation procedure is connected to a numerical model for calculation of drawdowns. At
a regional scale the rivers may be considered
6
as pumping wells that extract water from the
aquifer. If the aquifer geometry can be simplified as discussed above, the numerical model
may be replaced by an analytical solution. This
will simplify the estimation procedure considerably.
Risk mapping. Wong (2003) and Wong et al. (2002)
have implemented fuzzy rule based methods for
spatial estimation of risk for groundwater contamination and hazards of transport. The motivation for fuzzy methods is the need for tools to
aid real time management. Inaccurate methods
that quantifies the uncertainties within specified
confidence intervals, are more valuable than exact solutions that are coming too late. Analytical approximations at hand may narrow confidence intervals and supplement fuzzy-rule based
methods.
Geochemical reactions. Weathering processes provide the geochemical background for adsorption
and bio-degradation of contaminants in the subsurface. Approximate vertical flow may be included in the 1d radial solution by mass-balance
considerations in the vertical direction (Strack,
1984; Haitjema, 1995). Given the mineralogical
composition of the deposit and chemical parameters of the precipitation, an approximate 3d
flow field may be used as a first order approximation for estimation of geochemical zoning of
the subsurface.
References
Bakker, M., O. D. L. Strack, Analytic elements for
multiaquifer flow, J. Hydrol., 271, 199-129, 2002
Bogen, J., Morphology and sedimentology of deltas
in fjord and fjord valley lakes. Sedim. Geol., 36,
254-267, 1983
Chiang, W.-H., W. Kinzelbach, 3D-groundwater modeling with PMWIN, ISBN 3-540-67744-5
Engen, T., Stochastic interpolation of groundwater
levels in an ice-contact delta at Gardermoen (in
Norwegian, Cand. Scient thesis, University of
Oslo, department of Geophysics), 1995
Gilbert G.K., Lake Bonneville, Mon. U.S. geol Surv,
1, 438 pp, 1890
Goshu, A.T., Bayesian Inversion and Geostatistical
Methods applied to some Groundwater Problems,
Ph.D thesis at Department of Mathematical Sciences Norwegian University of Science & Technology, Trondheim, Norway, ISBN 82-471-5632-6,
2003
Gustafson, G., Well system for heat storage and heat
extraction in aquifers, (In Swedish: Brunnsystem
för värmelagring och värmeutvinning i akviferer),
Byggforskningsrådet, R39, 1983
Gustafson, G., One-hole pumping tests in Swedish
glaciofluvial aquifers - prediction of transmissivity
and storage coefficients Nordic Hydrological Conference, Reykjavik, 1986
Haitjema, H. M. Analytic element modeling of groundwater flow, ISBN 0-12-316550-4, 1995
Haitjema, H. M., Comparing a three-dimensional
and a Dupuit-Forchheimer solution for a circular
recharge area in a confined aquifer, J. Hydrol., 91,
87-101, 1987
Luther, K., H. M. Haitjema, Approximate analytic solutions to 3D unconfined groundwater flow within
regional 2D models, J. Hydrol., 229, 101-117, 2000
Luther, K., H. M. Haitjema, An analytic element solution to unconfined flow near partially penetrating
wells, J. Hydrol., 226, 197-203, 1999
Luther, K., H. M. Haitjema, Numerical experiments
on the residence time distribution of heterogeneous
groundwatersheds, J. Hydrol., 207, 1-17, 1997
Omre, H., K. B. Halvorsen, The Bayesian bridge between simple and universal kriging, Math. Geol.,
21(7), 767-786, 1989
Reading, H. G., Sedimentary Environments: Processes, Facies and Stratigraphy, ISBN 0-632-036273, 1998
Strack, O. D. L., Three-dimensional streamlines in
Dupuit-Forcheimer models. Water Resour. Res.,
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20(7), 812-822, 1984
Strack, O. D. L., Groundwater Mechanics, ISBN 013-365412-5, 1988
Tuttle, K. J., Sedimentological and hydrogeological
characterisation of a raised icecontact delta - the
Preboreal deltacomplex at Gardermoen, southestern
Norway, Ph.D. thesis, Dept. of Geology, University
of Oslo, Nov. 1997
Tuttle, K. J., S. R. Østmo, B. G. Andersen, Quantitative study of the distributary braidplain of the
Preboreal ice-contact Gardermoen delta complex,
southeastern Norway, Boreas 26, 141-156, 1997
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mapping in the subsurface - the case of Gardermoen, Norway, PhD thesis at the University of
Oslo, Dept. of Geophysics, ISSN , 1501-7710,
No.251, 2003
Wong, W.K., I. Krasovskaia, L. Gottschalk, A. Bárdossy,
Risk mapping of groundwater contamination. In:
Hydrological Models for Environmental Management, edited by M.V. Bolgov, L. Gottschalk, I.
Krasovskaia, R.J. Moore NATO Science Series, 2.
Environmental Security 79:227-240, Kluwer Academic Publishers, Dordrecht. ISBN 1-4020-0910-0,
2002
Wu, Y.-S., L. Pan, Special relative permeability functions with analytical solutions for transient flow
into unsaturated rock matrix. Water Resour.
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doi:10.1029/2002WR002047, 2003
N.-O. Kitterød, University of Oslo Department of
Geoscience P.O.Box 1047, Blindern, N-0316 Oslo,
Norway. (e-mail: nilsotto@geo.uio.no)
1 And at The Norwegian Centre for Soil and Environmental
Research, Frederik A. Dahls vei 20, N-1432 s, Norway
This preprint was prepared with AGU’s LATEX macros
v5.01. File analytical˙gilbert formatted March 16, 2004.
8
Figure 1. Principal sketch of delta geometry: (a) indicates the interface between the paleo-glacier and the sediments;
radius to the inner and outer delta boundaries (R1 and R2 ), and the location of the groundwater divide (l); (b) is a radial
cross-section through the delta indicating a confined aquifer with linearly decreasing aquifer thickness with piezometric head
φ1 and φ2 at R1 and R2 ; (c) and (d) are modified from Gilbert (1890) indicating the sediments at the glacier interface (c)
and after withdrawal of the glacier (d) and with a phreatic groundwater table. If net recharge N > 0, the groundwater
divide ( dΦ
= 0) is at r = l.
dr
Figure 2. 3d-numerical computation of heads (solid) and analytical (dashed) solution of Eq. 19. In (a) 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, c.f.
label in (b). In (c) 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 3d-numerical and the analytical solutions
are shown in (b) and (d). Numerical results and values of parameter in the equation are given in Tab. 1.
Figure 3. Piezometric head contours of analytical (solid) and 3d-numerical (dashed) solutions. The contours in (a) indicate
that the numerical and the analytical solutions are identical if the inner- (H1 ), and the outer boundary (H2 ) are equal to
each other. In this case H1 = H2 = 100 m. In (b) where H1 = 100 m and H2 = 10 × H1 , the max. deviation is 0.59 m (cf.
Tab. 1).
Figure 4. The Gardermoen aquifer is a superposition of two deltas with paleo-portals at Trandum and Helgebostad. The
Trandum delta has the highest desity of groundwater observation wells (small circles) therefore this part of the aquifer is
used for the estimation of aquifer parameters as shown in Fig. 5.
Figure 5. Optimal parameters p= [k1 , k2 ] of Eq. 24 and p= [H1 , H2 ] of Eq. 19 are estimated by minimizing the rootmean-square deviation D given in Eq. 27. The groundwater heads corresponding to optimal p are indicated as crosses for
the phreatic aquifer condition and diamonds in the case of a confined aquifer. Observed heads are indicated as circles. The
shaded areas that envelopes the observations are the perturbation range of p: [2.2 × 10 −5 m/s ≤ k1 ≤ 4.1 × 10−5 m/s,
2.6 × 10−6 m/s ≤ k2 ≤ 9.4 × 10−6 m/s; 235 m ≤ H1 ≤ 401 m, 35 m ≤ H2 ≤ 103 m]. Optimal estimates are given in Tab. 2.
9
Table 1. Differences d between analytical- and numerical solution (eq. 19)
H1 = 100 m
H2 = 100 m
H2 kH1 m
mean(d) m
std(d) m
max(d) m
mean(d) m
std(d) m
max(d) m
100.0
200.0
300.0
400.0
500.0
750.0
1000.0
0.022798
0.136579
0.202831
0.249633
0.287973
0.368054
0.438660
0.011847
0.030846
0.042964
0.054727
0.066562
0.097509
0.130050
0.042431
0.171761
0.247430
0.304703
0.355404
0.471294
0.590987
0.021947
0.110088
0.179970
0.235209
0.282543
0.382758
0.469550
0.013306
0.039585
0.072078
0.096665
0.117329
0.161174
0.200173
0.044346
0.182896
0.294309
0.378903
0.450699
0.604228
0.740620
d = |φanalytical − φnumerical |
Numerical values in eq. 19: k = 1.727901 × 10−5 m/s; N = 1.266730 × 10−8 m/s; R1 = 1000m; R2 = 5100m;
φ1 = φ2 = 100m
10
Table 2. Estimation results based on observations from the Gardermoen aquifer (Norway) of confined- and phreatic aquifer conditions.
Confined (eq. 19)
parameter
H1
H2
a
dip
grw.divide
rms-error
values
301.86 m
66.19 m
4.947e-02
2.832◦
3220.7 m
13.84 m
Phreatic (eq. 24)
parameter
k1
k2
b
grw.divide
rms-error
values
2.886e-05 m/s
5.693e-06 m/s
4.864e-09
3238.2 m
14.30 m
Boundary conditions: Net infiltration N : 1.266730e-08 m/s (or 1.094455
mm/d); constant head h1 kφ1 : 171.50 m at R1 : 336.00 m; constant head
h2 kφ2 : 185.00 m at R2: 5100.00 m. In eq. 19: k = 12 (k1 + k2 ) = 1.727901e05 m/s. Location of origo (UTM-East, UTM-North): 8400.00 m 7150.00
m in UTM-zone: 32 V, EU89 datum.
φ
φ
Figure 1: Principal sketch of delta geometry: (a) indicates the interface between the paleoglacier and the sediments; radius to the inner and outer delta boundaries (R 1 and R2 ), and
the location of the groundwater divide (l); (b) is a radial cross-section through the delta
indicating a confined aquifer with linearly decreasing aquifer thickness with piezometric
head φ1 and φ2 at R1 and R2 ; (c) and (d) are modified from Gilbert (1890) indicating
the sediments at the glacier interface (c) and after withdrawal of the glacier (d) and with a
phreatic groundwater table. If net recharge N > 0, the groundwater divide ( dΦ
dr = 0) is at
r = l.
1
0.8
differences in heads between analytical and numerical solutions (m)
125
a
120
head (m)
115
110
105
100
95
500
1000
1500
2000
2500
3000
3500
radius (m)
4000
4500
5000
c
120
115
head (m)
0.5
0.4
0.3
H1=H2
H1=2*H2
H1=3*H2
H1=4*H2
H1=5*H2
H1=7.5*H2
H1=10*H2
0.2
0.1
0
1000
2000
3000
4000
radius (m)
5000
6000
7000
8000
0.8
differences in heads between analytical and numerical solutions (m)
125
110
105
100
95
500
0.6
0
5500
b
0.7
1000
1500
2000
2500
3000
3500
radius (m)
4000
4500
5000
0.6
0.5
0.4
0.3
H2=H1
H2=2*H1
H2=3*H1
H2=4*H1
H2=5*H1
H2=7.5*H1
H2=10*H1
0.2
0.1
0
5500
d
0.7
0
1000
2000
3000
4000
radius (m)
5000
6000
7000
8000
Figure 2: 3d-numerical computation of heads (solid) and analytical (dashed) solution of
Eq. 19. In (a) 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, c.f. label in (b).
In (c) 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 3d-numerical and the analytical solutions are shown in (b) and (d). Numerical
results and values of parameter in the equation are given in Tab. 1
2
H1 = H2
5000
102
104
115
10
2
10
4
10 6
8
11
0
b
3
11
0
0.
5
10
1.
103.5
5
104
10
10
4
2.
10
3.
5
5
1
4
10
10
10
2
1
10
10
1.5
2500
3.1
50
3
2
10
2000
0.5 101.5
120
10
023.5
110
2
10 104
2
1
10 1
1.5 0
2
10
5
12
3000
102
102.5
110023
.5
103.150
10
10
0
122
11
Y−northing (m)
10
12
2000
103
3500
104
122
2500
101
100
.5
101
.5
101
101.5102
102
102.5
120
3000
4500
4000
115
3500
101.5
a
1
10 06
110 8
100.5
100.5
101
115
1500
2000
2500
3000
X−easting (m)
3500
4000
102
101.5
1000
101
500
102
103.5
5000
2
2.5
1010
4500
103.5
4000
103
.5
3500
103
2500
3000
X−easting (m)
104
2000
4
1500
104
104
1000
3.5
2 .5
1
10
500
102
500
10
3
115
500
115
4
122
10
120
2
120
1000
1500 1010
4 4
1
1003.5
101101102 3102.
.5
5 10
1000
10
12
1
1010
8
106
106
108
110
1500
4500
101
106
108
110
4500
4000
H2 = 10*H1
102
104
Y−northing (m)
5000
5000
Figure 3: Piezometric head contours of analytical (solid) and 3d-numerical (dashed) solutions. The contours in (a) indicate that the numerical and the analytical solutions are
identical if the inner- (H1 ), and the outer boundary (H2 ) are equal to each other. In this
case H1 = H2 = 100 m. In (b) where H1 = 100 m and H2 = 10×H1, the max. deviation
is 0.59 m (cf. Tab.1).
3
Figure 4: The Gardermoen aquifer is a superposition of two deltas with paleo-portals
at Trandum and Helgebostad. The Trandum delta has the highest desity of groundwater
observation wells (small circles) therefore this part of the aquifer is used for the estimation
of aquifer parameters as shown in Fig. 5.
4
Figure 5: Optimal parameters p= [k1 , k2 ] of Eq. 24 and p= [H1 , H2 ] of Eq. 19 are
estimated by minimizing the root-mean-square deviation D given in Eq. 27. The groundwater heads corresponding to optimal p are indicated as crosses for the phreatic aquifer
condition and diamonds in the case of a confined aquifer. Observed heads are indicated as
circles. The shaded areas that envelopes the observations are the perturbation range of p:
[2.2 × 10−5 m/s ≤ k1 ≤ 4.1 × 10−5 m/s, 2.6 × 10−6 m/s ≤ k2 ≤ 9.4 × 10−6 m/s;
235 m ≤ H1 ≤ 401 m, 35 m ≤ H2 ≤ 103 m]. Optimal estimates are given in Tab.2.
5