Issue Paper/
Are Water Tables a Subdued Replica
of the Topography?
by Henk M. Haitjema1 and Sherry Mitchell-Bruker2
Abstract
The water table in unconfined aquifers is often believed to be a subdued replica of the topography or land
surface. However, this assumption has not been widely tested and in some cases has been found to be in error. An
analysis of ground water rise in regional unconfined aquifers, using both a two-dimensional boundary element
model and a one-dimensional Dupuit-Forchheimer model, reveals the conditions under which the water table does
or does not closely follow the topography. A simple decision criterion is presented to estimate in advance under
which conditions the water table is expected to be largely unrelated to the topography and under which conditions
the topography controls the position of the water table.
Introduction
When formulating mathematical solutions to ground
water flow problems in unconfined aquifers, the a priori
unknown water table is treated as a ‘‘free surface,’’ its location depending on the solution itself. Early computational
work often focused on local flow through or underneath
dams and near stream or canal boundaries (Muskat 1937;
Polubarinova-Kochina 1952, 1962; Harr 1962; Davis and
De Wiest 1966). The closed-form analytic solutions to
flow for these cases were developed by use of the hodograph method, an advanced conformal mapping technique
to treat the a priori unknown upper aquifer boundary. Dupuit (1863) was probably the first to address problems of
truly regional ground water flow. He simplified the problem by ignoring vertical flow, thus reducing the problem to
that of one- or two-dimensional horizontal flow. Later,
Forchheimer (1886) presented a differential equation for
the water table elevation in unconfined aquifers, referring
to Dupuit’s approximation. Hubbert (1940), using a flow
net construction, was one of the first who provided a conceptual ground water flow model for flow between
1Corresponding author: SPEA 439, Indiana University,
Bloomington, IN 47405; (812) 855-0563; fax (812) 855-7802;
[email protected]
2Everglades National Park, 950 North Krome Avenue,
Homestead, FL 33030; (305) 224-4286; fax (305) 224-4147;
[email protected]
Received August 2004, accepted January 2005.
Copyright ª 2005 National Ground Water Association.
doi: 10.1111/j.1745-6584.2005.00090.x
(parallel) streams that included resistance to vertical flow.
Toth (1963) also offered a regional ground water flow
solution that included resistance to vertical flow, but he
assumed the position of the water table in advance: a subdued replica of the topography. While the water table in
his example was observed to, at least approximately,
follow the topography, in many other settings the water
table and the surface topography appear poorly correlated
if not unrelated (Desbarats et al. 2002; Moore and
Thompson 1996; Blaskova et al. 2002; Shaman et al.
2002). Desbarats et al. (2002) suggested that the concept of
a correlation between the water table and the topography
may be valid locally but not universally. The question then
is under what circumstances does the topography control
the water table and under what circumstances does it not.
In this paper, we explore the different conditions that
affect the position of the water table and its possible interaction with the topography. Whether using a twodimensional boundary element model or a one-dimensional
Dupuit-Forchheimer model, we arrive at the same conclusions regarding the conditions under which the water table
will or will not interact with the topography. Finally, we
offer criteria for when the water table is controlled by the
surface topography and when it is mostly independent of
the topography and controlled by hydrological features.
Regional Ground Water Flow
The theories of Dupuit-Forchheimer and Toth are
seemingly conflicting conceptual models of regional
Vol. 43, No. 6—GROUND WATER—November–December 2005 (pages 781–786)
781
ground water flow. Toth (1963) introduced the concept of
nested regional, intermediate, and local flow cells, with
both horizontal and vertical components of flow being
important. Dupuit and Forchheimer proposed to ignore
vertical flow on a regional scale (Dupuit 1863; Forchheimer 1886). While in a modern interpretation of the
Dupuit-Forchheimer approximation vertical flow is allowed but resistance to vertical flow is ignored (Kirkham
1967; Strack 1984), there is no opportunity for the formation of nested flow cells such as presented by Toth. We
argue here that both models of regional flow can be expected to occur in nature but they apply to very different
hydrogeological conditions. Moreover, we demonstrate
that the Dupuit-Forchheimer approximation is often valid
in areas with properties typical of ‘‘usable aquifers,’’ that
is, aquifers from which adequate quantities of water can be
pumped for domestic, industrial, or public use.
Toth’s Model of Three-Dimensional Flow Cells
Toth developed his theory of regional flow while
studying the hummocky terrain of Alberta, Canada. This
region is known to contain areas of deep aquifers of lowpermeability shales and sandstone overlain by a relatively
thin layer of higher permeability sands (Ophori and Toth
1989). This stratigraphy accommodates large infiltration
rates in the surface sands that provide continuous recharge
to the underlying low-permeable aquifer. Toth was interested in deep geologic formations, so the aquifers that he
modeled were deep relative to the distance between drainage basins. In one of his examples, the aquifer depth is
3048 m (10,000 feet) while the distance between the
drainage divide and the stream is 6096 m (20,000 feet). In
order for the water table to rise to the highest point in this
theoretical watershed, the ratio of recharge over hydraulic
conductivity must be greater than 0.2 (Mitchell-Bruker
1993). For example, assuming a recharge rate somewhere
between 25.4 mm/year (1 inch/year) and 508 mm/year (20
inches/year) implies that the hydraulic conductivity of the
basin must be somewhere between 0.0003 m/d (0.001 ft/
d) and 0.007 m/d (0.023 ft/d). A recharge over hydraulic
conductivity ratio of 0.2 is quite high but could possibly
occur when the low-permeable basin is overlain by a layer
of much more permeable sand, as is the case in Toth’s
study area (Ophori and Toth 1989). The more permeable
sands could store much of the precipitation, rather than
letting it run off overland, and cause a sustained infiltration into the underlying basin.
In models where the water table is specified (e.g., Toth
1963), the distribution of recharge and discharge follows
from the solution. An alternative approach, which is more
commonly used in ground water modeling, is to apply
recharge to the aquifer top and only specify the head where
streams interact with the aquifer. In Figures 1 and 2,
a steady-state, two-dimensional boundary element model is
used to compare these two approaches (Mitchell-Bruker
1993). In this model, linear strength line sink functions
(Strack 1989) are used to create either recharge- or headspecified boundaries to the flow regime. The basin is
delineated by no-flow boundaries on the sides and bottoms, which are obtained by image line sinks (Haitjema
1995). In the recharge-specified simulation, the recharge
rate is specified for the line sinks and the water table is
obtained from the solution (Figure 1a). In the headspecified simulations (Figure 2), the water table is specified and the recharge distribution is obtained from the
solution. Figure 1a represents the case where the ratio
of recharge over hydraulic conductivity is relatively low
(R/k ¼ 1/12). In this situation, the recharge is equal to
25.4 mm/year (1 inch/year) throughout the watershed and
the water table does not reach the land surface, except all
the way to the left where it equals the water level in
a drainage stream. In the higher regions of the watershed,
the water table is far below the land surface. Figure 1b
represents the case where the ratio of recharge to hydraulic conductivity is much higher (R/k ¼ 5/12). In this situation, the water table rises to intersect the land surface well
Figure 1. Flow simulation of theoretical aquifer using recharge-specified boundary conditions: (a) recharge ¼ 25.4 mm/year,
hydraulic conductivity ¼ 0.3 m/d; (b) recharge ¼ 126 mm/year, hydraulic conductivity ¼ 0.3 m/d (after Mitchell-Bruker
1993).
782
H.M. Haitjema, S. Mitchell-Bruker GROUND WATER 43, no. 6: 781–786
Figure 2. Flow simulation of theoretical aquifer using head-specified boundary conditions: (a) 1000 feet aquifer depth, (b)
10,000 feet aquifer depth (after Mitchell-Bruker 1993).
before it links up with the left-hand drainage feature,
forming a local flow cell near the bottom of the watershed. The land surface near the bottom of the watershed
limits the water table rise, which is modeled by replacing
the discharge-specified line sinks in that area by headspecified ones with the head equal to the land surface.
These head-specified line sinks become discharge features as is seen from the flow lines in the left-hand part of
the watershed in Figure 1b. Further to the right, the water
table rises along with the ground surface; however, the
distance between the water table and the ground surface
increases as the land surface rises. When measuring the
water table in Figure 1b at several locations in the watershed, it appears as if the water table is a subdued replica
of the land surface, at least in a general sense. However, if
a model is developed where the water table is everywhere
equal to the land surface, a very different flow system is
obtained (Figure 2a). In this simulation, numerous local
flow cells are formed with zones of significant vertical
flow. If the same head-specified upper boundary conditions (water table equal to the land surface) are applied
to a deeper aquifer, as in Figure 2b, a system of local,
intermediate, and regional flow is obtained. The last Figure 2b is similar to those presented by Toth (1963).
Water Table in a Dupuit-Forchheimer Model
The analysis presented previously is based on a twodimensional cross-sectional flow model, thus including
resistance to vertical flow. Shallow regional aquifers,
however, have been modeled successfully by use of
the Dupuit-Forchheimer approximation, which ignores
resistance to vertical flow. Such a model, however, leads
to the same conclusions regarding the interaction of the
water table and the topography as arrived at previously.
The water table rise in a Dupuit-Forchheimer model of
the theoretical watershed in Figure 1 can be described
with a simple formula as shown subsequently. In Figure 3,
areal recharge due to precipitation generates flow toward
head-specified boundaries. If Figure 3 is interpreted as
representing one-dimensional flow between two parallel
streams, the mounding h [m], midway between the
streams, follows from (Haitjema 1995)
h ¼
RL2
8T
ð1Þ
where R [m/d] is the areal recharge rate, L [m] the distance between the streams, and T [m2/d] the average
Figure 3. Ground water mounding due to recharge may
interact with the topography.
H.M. Haitjema, S. Mitchell-Bruker GROUND WATER 43, no. 6: 781–786
783
aquifer transmissivity. Figure 1 may be seen as the left
half of Figure 3. Figure 3 may also be interpreted as flow
in a circular island with diameter L [m], in which case the
mounding in the center of the island becomes:
h ¼
RL2
16T
ð2Þ
The average transmissivity T may be written as the
aquifer hydraulic conductivity k [m/d] times the average
aquifer thickness H [m], hence:
T ¼ kH
ð3Þ
With Equation 3, both Equations 1 and 2 may be rewritten as:
h ¼
RL2
mkH
ð4Þ
where m is either 8 or 16, depending on the flow problem
being one-dimensional or radial symmetric, respectively.
Equation 4 illustrates that the water table rise is very sensitive to the distance L between hydrological boundaries
(surface waters). Given a particular aquifer geometry and
assuming a constant saturated thickness H, however,
the water table rise depends only on the ratio of recharge
over hydraulic conductivity, R/k, which is dimensionless.
Under unconfined flow conditions, the average saturated
thickness H will increase with a rising water table and the
response to R/k becomes nonlinear.
The result for the mounding at the center of the aquifer in Equation 4 is obtained by ignoring resistance to
vertical flow in the aquifer, which constitutes the DupuitForchheimer approximation. The validity of this approximation depends on the aquifer geometry and, to a lesser
extent, on the degree of vertical anisotropy. The DupuitForchheimer approximation is discussed in most standard
texts on ground water flow (e.g., Freeze and Cherry
1979; Bear 1972; Harr 1962). A detailed discussion
on the Dupuit-Forchheimer approximation is provided
by Kirkham (1967). As a rule of thumb, the DupuitForchheimer approximation is acceptable when the distance between hydrological boundaries is large compared
to the ‘‘effective’’ aquifer thickness:
rffiffiffiffiffi
kh
L
H
kv
In Figure 4 a cross section over a regional aquifer
is sketched with a water table in response to aquifer
recharge for two different cases. In Figure 4a, a high
recharge rate in combination with a low aquifer hydraulic
conductivity would lead to the dashed water table if the
aquifer would extend that high, while in Figure 4b a relatively low recharge rate and high aquifer hydraulic conductivity results in a water table that barely touches the terrain
surface. In fact, what matters is not the absolute values of
the recharge and the conductivity but the dimensionless
ratio of R/k (Equation 4). In Figure 4a the water table can
be characterized as ‘‘topography controlled,’’ while in Figure 4b there is essentially no correlation between the
topography and the water table. We demonstrate that the
nature of the water table (topography controlled or not) depends on the recharge, the aquifer transmissivity, the aquifer geometry, and to some degree the topography itself.
We introduce three dimensionless ratios to characterize the ground water flow regime: (1) The recharge over
hydraulic conductivity, R/k; (2) the distance between
hydrological boundaries over the saturated aquifer thickness, L /H; and (3) the distance between hydrological
boundaries over the maximum terrain rise, L /d. The maximum terrain rise d is defined here as the largest distance
between the (average) surface water elevation and the
terrain surface, as measured at the highest terrain elevation (Figure 3). The maximum ground water mounding
h in an aquifer may be estimated by the following
dimensionless relationship in terms of the previously
defined ratios:
rffiffiffiffiffi h 1 R L L
kh
¼
H
ð6Þ
L
d
mk Hd
kv
ð5Þ
where the square root term scales the aquifer thickness H
to account for the difference in horizontal conductivity kh
and vertical conductivity kv. In practice, it is often sufficient for L to be five times larger than the effective aquifer thickness (square root term times H ). In the event of
an anisotropic aquifer, the hydraulic conductivity k in
Equation 4 must be interpreted as the horizontal hydraulic
conductivity kh. However, anisotropy does affect ground
water mounding indirectly since it may invalidate the use
of the Dupuit-Forchheimer approximation by violating
Equation 5. Under these circumstances, the ground water
mounding predicted by Equation 4 should be considered
an underestimation.
784
Topography Vs. Recharge-Controlled
Water Tables
Figure 4. (a) Low-permeable aquifer with topography-controlled water table; (b) highly permeable aquifer with
a recharge-controlled water table (after Haitjema 1995).
H.M. Haitjema, S. Mitchell-Bruker GROUND WATER 43, no. 6: 781–786
The term h/d is the ratio of the maximum mounding over the maximum terrain rise, both measured with
respect to the surface water elevations of the surrounding
hydrological boundaries. Equation 6 is obtained from
Equation 4, whereby k must be interpreted as the horizontal hydraulic conductivity. The integer constant m in
Equation 6 depends on the type of flow. For one-dimensional flow between two parallel surface water boundaries
m ¼ 8, while for radial flow in an aquifer of diameter L,
m ¼ 16. In practice, therefore, m is expected to be somewhere in between these values:
8 m 16
ð7Þ
Equation 6 is obtained by use of the DupuitForchheimer approximation, thus the condition that the
distance between boundary conditions must be large compared to the aquifer thickness (Equation 5).
If the ratio h/d is larger than 1, the water table will
likely intersect the terrain surface at one or more locations. For values of h=d 1; the situation depicted in
Figure 4a occurs where the topography essentially controls the water table location. For this case, the recharge
rate R must be interpreted as the potential recharge rate
since the actual recharge rate will vary over the terrain,
being positive along the hillslopes and negative at local
topographic lows. Conversely, when h=d 1 the situation in Figure 4b occurs where the water table is controlled by the recharge rate (and other aquifer parameters)
and essentially unrelated to the topography. For the sake
of brevity, we will refer to the situation in Figure 4b as
that of a ‘‘recharge-controlled’’ water table. Equation 6, of
course, does not offer a clean cut between the two cases
in Figure 4. If the maximum ground water mounding is
about the same as the maximum terrain rise, h=d ffi 1;
the ground water flow regime is neither entirely recharge
controlled nor entirely topography controlled. Yet, as
a general rule of thumb the following may be stated:
RL2
> 1 water table is topography controlled
mkHd
RL2
<1
mkHd
water table is recharge controlled
ð8Þ
When reviewing the case presented by Toth (1963),
we find that the distance between the regional water
divide (on the right-hand side of his basin) and the primary discharge area on the left-hand side of the basin is
6096 m (20,000 feet). Assuming that the regional water
divide is midway between two (regional) drainages, the
distance L for his case is 12,192 m (40,000 feet). With a
basin depth H of 3048 m (10,000 feet), we find that L /H ¼
4. Estimating the maximum terrain rise d (difference
between the terrain elevation on the right and the left
of the basin) to be 30.48 m (100 feet), we find that
L/d ¼ 400. Finally, from the maximum recharge over
conductivity ratio that is consistent with the mounding on
the right-hand side of the basin, we observed earlier that
R/k ¼ 0.2. This maximum recharge over conductivity
ratio (R/k ¼ 0.2) is interpreted as belonging to the
‘‘potential recharge rate.’’ With these numbers, and m ¼
8, the dimensionless factor in Equation 8 is equal to 40,
which explains the topography-controlled water table in
Toth’s study. In fact, the use of the Dupuit-Forchheimer
approximation in this assessment is suspect for this
case with L /H ¼ 4, so that in view of the actual resistance to vertical flow, the water table mounding is likely
greater than predicted by Equation 6, further ensuring
a topography-controlled water table.
For the case discussed previously, the factor R/k is
rather large, R/k ¼ 0.2. Most productive aquifers (from
which usable quantities of water can be pumped) are
moderately to highly permeable with recharge over conductivity ratios in the range of 1026 to 1023 (at least 2 orders of magnitude lower than for Toth’s problem). With
these lower R/k ratios, the water table tends to be more
often recharge controlled. To illustrate this, consider the
following example: The distance L ¼ 5000 m, the thickness H ¼ 50 m, the hydraulic conductivity k ¼ 50 m/d,
the recharge rate R ¼ 25.6 mm/year (10 inch/year), which
is 0.0000701 m/d. With these data and assuming a maximum terrain rise d ¼ 10 m, the dimensionless factor in
Equation 8 becomes 0.0088. This is 2 orders of magnitude smaller than 1; hence, the water table will almost
certainly not reach the terrain surface and is thus unrelated to the topography.
Topography-controlled water tables are most likely
to occur in low-permeable sediments, which often are not
thought of as ‘‘aquifers,’’ at least not aquifers from which
usable quantities of water can be pumped. They can be
expected in systems where, due to circumstances, relatively high recharge rates are realized. These high
recharge rates may be due to wetland conditions, where
ponded water can augment aquifer recharge beyond that
derived from normal rainfall. In situations where a lowpermeable formation is overlain by a (thin) layer of more
permeable soils, a kind of ‘‘subsurface ponding’’ may
occur resulting in relatively high recharge rates in the
underlying low-permeable formation. In theory, the
recharge over conductivity ratio can be R/k ¼ 1 for incipient ponding or slightly higher for true ponding. Additionally, if the terrain surface is relatively flat or the water
table is generally shallow, the factor L /d may be large as
well (small d ), further increasing the potential for topographic control of the water table.
It should be pointed out that even if the water table
is topography controlled, it does not automatically follow that the water table must be locally parallel to the
terrain surface. In other words, even though the water
table is found to be a ‘‘subdued replica of the topography,’’ the local hydraulic gradient may be quite different
from the slope of the local terrain surface. If the water
table is not known in advance of modeling, it is not
advisable to specify a water table based on interpolation
of a sparse set of measured water table elevations, much
less on an assumed water table. When the head is specified instead of recharge rates, the recharge is not constrained to realistic rates and, if the water table is not
represented accurately, the solution produces erratic and
erroneous recharges and discharges (Stoerz and Bradbury
1989).
H.M. Haitjema, S. Mitchell-Bruker GROUND WATER 43, no. 6: 781–786
785
Conclusions
Acknowledgments
Consistent with Toth’s original analysis, topography
can significantly constrain ground water mounding. This
type of topography-controlled water table most likely
occurs in relatively low-permeable and/or anisotropic
aquifers subjected to unusually high (in view of the low
permeability) areal recharge rates or in nearly flat
terrain. However, most aquifers that are developed for
industrial or public water supply are relatively permeable
and show moderate to small degrees of ground water
mounding in response to average recharge. Such productive aquifers are generally recharge controlled. Recharge
and conductivity, however, are not the only parameters
that determine the difference between topography- or
recharge-controlled water tables. As is seen from Equation 6, the aquifer thickness and distance between surface
water features are also important. Consequently, shallow
aquifers in flat or gently rolling terrain may exhibit a relatively low R/k ratio and still exhibit a water table that
seems a subdued replica of the terrain surface. In fact, in
a very general sense the water table and the topography
are always related. After all, water table lows occur at
surface waters, which in turn occur in topographically
low areas.
We offer a simple dimensionless decision criterion to
assess the likelihood for either topography-controlled or
recharge-controlled water tables.
The authors thank Randy Hunt of the USGS in Madison, Wisconsin, for his valuable suggestions regarding this
article. We also thank the reviewers, George Hornberger
and Ken Bradbury for their constructive remarks.
RL2
>1
mkHd
RL2
<1
mkHd
topography controlled
recharge controlled
where R [m/d] is the average annual recharge rate, L [m]
the average distance between surface waters, m [2] a factor between 8 and 16 for aquifers that are strip like or circular in shape, respectively. The term k [m/d] is the
(horizontal) aquifer hydraulic conductivity, H [m] the aquifer thickness, and d [m] the maximum distance between
the average surface water levels and the terrain elevation.
These parameters are indicated in Figure 3. The criterion
has been developed for Dupuit-Forchheimer flow conditions, which generally occur when the distance between
surface waters is large compared to the aquifer thickness.
In case the Dupuit-Forchheimer approximation is not valid,
some additional water table rise must be anticipated, somewhat increasing the chance for topography-controlled
water table conditions when the factor in Equation 8 is
near unity. Recharge-controlled water tables tend to be
dominated by horizontal flow with mostly regional flow
cells, while topography-controlled water tables tend to
form local flow cells. Deep flow systems with topographycontrolled water tables tend to form local and regional flow
systems as pointed out by Toth (1963). In most cases
where the water table is topography controlled, it still is
likely not a truly subdued replica of the topography in that
the local hydraulic gradient is not correlated with the local
slope of the terrain surface.
786
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