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A Dupuit model of groundwater-surface water interaction
473
A Dupuit model of groundwater-surface water interaction Erik I. Anderson a ~Department of Civil and Environmental Engineering, University of South Carolina, 300 Main Street, Columbia, SC 29208 We consider two-dimensional groundwater flow in the vertical plane to a stream in direct contact with the top of a confined aquifer. By comparison of the exact solution with a Dupuit solution to the same problem, we find that global errors in head and discharge in the Dupuit model can be avoided by treating the fictitious resistance of the stream bed as an effective parameter and introducing a new parameter into the model. Although the choice of the new parameter is arbitrary, we choose to model the portion of the aquifer beneath the stream as an inhomogeneity, thus introducing a second hydraulic conductivity into the Dupuit model. This approach fits well into common AEM modeling practices. We evaluate dimensionless values of the effective parameters such that there are no global errors in head or discharge in the Dupuit model. We demonstrate that the values of the effective parameters are weakly dependent on the geometric parameter D/H, the ratio of the length of the stream bed and the thickness of the aquifer, and independent of the flow parameter Q1/Q~,the ratio of the left and right model discharges. These properties of the effective parameters provide some evidence that the effective values presented here may be applied, at least approximately, to more general groundwater flow problems; the results provide guidance for simulating groundwater-surface water interaction with Dupuit models. 1. I N T R O D U C T I O N
Surface water features play a prominent role in nearly every model of groundwater flow even when the details of the interaction are not the focus of the model. Surface water features are of primary importance as they represent real hydrologic boundaries. They also present difficulties for modelers; as discussed by Hunt et al. [1], the manner in which surface water features are represented in a groundwater flow model can have significant effects on the model results. In regional models, the Dupuit approximation is often made, and the full details of groundwater-surface water interaction at streams are difficult to include due to the size of the domain and the computational effort required for fine vertical discretization. Global errors in head and discharge can be introduced into a model when employing the Dupuit approximation in regions of local, concentrated vertical flow [2,3]. These global errors in the Dupuit model can result in biased estimates of aquifer properties when calibrating a model to observed head data. Kaleris [4] demonstrates, using a Dupuit model, that forecasting groundwater surface water interaction under flow conditions other
474 than those used for calibration will give erroneous results; this is especially true in cases where the resistance of the stream bed is small and the stream is narrow. As surface water bodies are a common feature of nearly all groundwater flow models, this limitation restricts the predictive capabilities of Dupuit-based flow models. In many cases, Dupuit models can be improved by defining effective aquifer properties that incorporate, in an approximate fashion, the head losses due to vertical flow that are neglected in the Dupuit formulation" Aravin and Numerov [5] describe the local head loss summation method for analyzing seepage beneath hydraulic structures; Streltsova [6] defines an additional seepage resistance to deal with vertical flow for several groundwater flow problems; Halec and Svec [7] describes the method of substitute lengths to approximate losses due to vertical recharge from a reservoir; Strack [8] uses an equivalent length of aquifer to include losses due to vertical flow to a partially penetrating ditch; Bakker [9] defines an effective resistance to vertical flow in a two layer Dupuit model of flow to a ditch. In each case, an effective property is defined and evaluated by comparison of a Dupuit solution to an exact (vertical plane) solution.
2. E R R O R S I N D U P U I T
MODELS
To examine errors that can be introduced in a model of groundwater surface water interaction by making the Dupuit approximation, we consider the problem shown in Figure 1.a flow in the vertical plane to a stream bed in direct contact with the aquifer. The aquifer is an infinite strip of height H [L] with impermeable bottom and top, except along the stream bed of width D [L], shown as the dashed line; the head is constant and equal to h* [L] along the stream bed. Discharges Q1 and Q,. [L2/T] are specified at the left and right infinite vertex, respectively. Figure 1.b and 1.c show two common representations of the stream used in regional groundwater flow models. In Figure 1.b, the stream is assumed to penetrate the aquifer fully, and the head h* is fixed on both sides of the stream. In Figure 1.c, the flow beneath the stream is modeled as semi-confined flow, being separated from the stream by a fictitious leaky stream bed. The parameter A [L] (the leakage factor) is related to the resistance of the stream bed, c [T], as A - (kHc) 1/2, where k [L/T] is the hydraulic conductivity of the aquifer. In the Dupuit model of Figure 1.c, we have also allowed for a second parameter, k* [L/T], the hydraulic conductivity of the aquifer beneath the stream. An analytical solution to the exact, 2D problem (Figure 1.a) may be found in [5] and solutions to the two Dupuit formulations (Figures 1.b and c) may be developed using standard techniques. In the Dupuit model of Figure 1.b, the resistance to vertical flow of the aquifer is entirely neglected by assuming full penetration of the stream. In Figure 1.c, we lump the true, distributed resistance to vertical flow of the aquifer at the boundary along the stream bed. Here the resistance of the stream bed represents an estimate of the average resistance to vertical flow of the aquifer; the aquifer beneath the stream is modeled as being semiconfined. There is little guidance for selecting the effective resistance of the stream bed when a stream is in direct contact with the aquifer. McDonald and Harbaugh [10] suggest treating the resistivity as an intrinsic property of the aquifer by using a value of one half the total resistance to flow of the aquifer (c - H/2k). This approach results in a dimensionless value of the leakage factor of A/H = 0.707. The parameter k*, shown
475
Figure 1. Flow to a stream intersecting the top of an aquifer. (a) Two dimensional flow in the vertical plane to a stream of width D, and two common representations of the stream in Dupuit models. (b) The stream fully penetrates the aquifer, and (c) the flow beneath the stream bed is modeled as being semiconfined with a fictitious resistance specified at the stream bed.
in Figure 1.c, has not been identified as an effective parameter in previous studies. We specify the discharges Q~ and Ql in all three models and evaluate heads to the left and right of the stream at a distance of 2H. We express the error in head on the right side of the stream as
error-- (hr)exact- (hr)Dupuit (hl)exact- h*
(1)
where h / i s the head calculated at a distance of 2H from the left edge of the stream and h~ represents the head calculated at a distance of 2H from the right edge of the stream. We normalize the error in head on the right by a measure of the change in head over the model on the left. We can evaluate the errors induced in a Dupuit model by specifying fully penetrating boundaries as in Figure 1.b, or by the common practice of specifying the leakage factor to be 0.707H in Figure 1.c, by comparing solutions from the exact and Dupuit problems. We specify Q1 and Q~ to be the same in both the exact and the two Dupuit models; in the second Dupuit model we specify , ~ / H - 0.707 and k / k * - 1.0. We then compare heads predicted by each model a short distance from the stream bed.
476
Figure 2. Error in head in the two Dupuit models of groundwater surface water interaction. To the left, the error in the fully penetrating model (Figure 1.b) and to the right, the error in the semiconfined model with intrinsic properties (Figure 1.c).
Figure 2 shows plots of the error in head predicted by the two Dupuit models; Figure 2.a shows the errors induced by making the stream fully penetrating and Figure 2.b shows errors induced by using intrinsic values of vertical resistivity. Figure 2 shows that, in both cases, the error in head is a function of the problem geometry (D/H) and a parameter reflecting the flow regime or flow field (Q~/Q1). This confirms the conclusion of Kaleris [4] that using a Dupuit model to forecast heads under changed flow conditions will lead to errors. The error in Figure 2.b ranges from approximately minus 10% (for D/H large) to values exceeding 100% (for D/H small); the error can be significant. Similar errors in head on the left side of the stream should be expected. In a typical groundwater flow model, neither the distribution of discharge, nor the hydraulic conductivity of the aquifer are known before calibration of the model. The errors shown in Figure 2 will result in biased estimates of aquifer parameters during calibration of a groundwater flow model; the correct values of k and @/Q1 in the Dupuit model will not reproduce the correct (exact) head distribution in the aquifer. The Dupuit model shown in Figure 1.c may be modified to produce the correct head and discharges a short distance from the stream by either defining an effective head at the stream, or a vertical resistance layer separating the stream from the aquifer. The effective head or resistivity may be evaluated, but the dependence of parameters on Q~/Q1 will remain: the solution will not be able to forecast results under conditions of changed flow regimes. 3. E F F E C T I V E
PROPERTIES
It may be shown, for this problem, that global error in heads and discharges can be eliminated by properly defining the effective parameters k* and k, illustrated in Figure 1.c. The relationship between the exact and Dupuit solutions may be evaluated analytically for this case, and values of effective parameters may be derived, following [11]. For this
477
Figure 3. Effective parameters for groundwater-surface water interaction.
problem the effective parameters may be expressed as
__ = __
/ I (1--71/2) 1/-1
A H
D H
k*-
Arc
arcosh
in ( ! - ~ )
(2)
In 1+,.)/1/2
in 2 ~ ( 1 - 7 )
-ln2
11+-71/2 71/2
1/2
(3)
where 7 is a function of D / H only. The appropriate values of the effective parameters, evaluated analytically, are plotted in Figure 3. Surprisingly, the introduction of two effective parameters in this problem eliminates any dependence on the flow parameter Q~/QI; the effective values may be applied to a model without prior knowledge of the distribution of flow from the right and left the values are independent of the flow field. Further, Figure 3 shows that the values of the effective parameters are only weakly dependent on geometry, being fairly constant over a wide range of stream widths, and that the effective resistance is much smaller than the intrinsic value of c k / H - 0.5 suggested by McDonald and Harbaugh [10]. 4.
COMPARISON
OF
FLOW
FIELDS
Figures 4 and 5 show comparisons of flow fields obtained from solutions to the exact problem and the Dupuit problem; the effective parameters used in the Dupuit solution are obtained from Figure 3. The specific problem considered has geometry and flow parameters of D / H - 1.5; Figure 4 shows the case where Q~/Q1 - - 0 . 5 and Figure 5 the case where Q ~ / Q 1 - +0.5. Figure 4.a shows the solution to the exact problem and Figure 4.b the Dupuit solution. In both, contours of head are shown as dashed lines and contours of stream function are
478
Figure 4. Comparison of flow fields for D / H - 1.5 and Q , . / Q , - -0.5" (a) the exact vertical plane solution showing head contours (dashed) and contours of the stream function (solid); (b) the Dupuit solution with the correct effective properties specified; (c) a comparison of heads obtained from the exact solution (solid) and the Dupuit solution (dashed); and (d), a comparison of the stream function from the exact (sdid) and Dupuit (dashed) solutions.
479
Figure 5. Comparison of flow fields for D / H - 1.5 and Q ~ / Q ~ - +0.5" (a) the exact vertical plane solution showing head contours (dashed) and contours of the stream function (solid); (b) the Dupuit solution with the correct effective properties specified; (c) a comparison of heads obtained from the exact solution (solid) and the Dupuit solution (dashed); and (d), a comparison of the stream function from the exact (solid) and Dupuit (dashed) solutions.
480 shown as solid lines; the dividing streamline is shown as a heavier line. Figure 4.c shows the exact and Dupuit head distributions overlaid; the solid lines represent the exact head and the dashed lines the Dupuit heads. Note that the heads converge a short distance from the stream on both sides. In a similar fashion, Figure 4.d shows the exact and Dupuit stream functions overlayed. Note here that the values of the stream function converge a short distance from the stream on both sides. Global errors in both head and discharge have been eliminated from the Dupuit solution by proper choice of the effective parameters. Figure 5 shows similar comparisons under changed flow conditions, emphasizing the fact that the results are independent of the flow regime. 5. D I S C U S S I O N
The independence of the values of the effective parameters on the flow conditions, and their weak dependence on geometry suggest that the effective properties may be applied to models of more general settings, where, for example, the stream width varies and the head in the stream varies. Application of these results in a general groundwater flow model requires simply modifying parameter values; no changes in modeling technique, approach, or the numerical method of solution are required. However, to be useful, these preliminary results must be verified in more general settings. REFERENCES
1. Hunt, R.J., Haitjema, H.M., Krohelski, J.T. and D.T. Feinstein, Simulating ground water-lake interactions: approaches and insight, Ground Water, 41(2), pp227-237, 2003. 2. Bear, J. and C. Braester, Flow from infiltration basins to drains and wells, Journal of the Hydraulics Division, ASCE, HY5, pp. 115-135, September, 1996. 3. Bakker, M., Simulating groundwater flow in multi-aquifer systems with analytical and numerical Dupuit models, Journal of Hydrology, 222, pp. 55-64, 1999. 4. Kaleris, V., Quantifying the exchange rate between groundwater and small streams, Journal of Hydraulic Research, 36(6), pp. 913-932, 1998. 5. Aravin, V.I. and S.N. Numerov, Theory of Fluid Flow in Undeformable Porous Media, Daniel Davy, New York, 1965. 6. Streltsova, T.D., Method of additional seepage resistances- theory and application, ASCE Journal of the Hydraulics Division, Vol. 100, HY8, pp. 1119-1131, 1974. 7. Halek, V. and J. Svec, Groundwater Hydraulics, Elsevier Scientific Publishing Co., Amsterdam, 1979. 8. Strack, O.D.L., Groundwater Mechanics, Prentice Hall, Englewood Cliffs, New Jersey, 1989. 9. Bakker, M., Transient Dupuit interface flow with partially penetrating features, Water Resources Research, 34(11), pp.2911-2918, 1998. 10. McDonald, M.G., and A.W. Harbaugh, A modular three-dimensional finite-difference ground-water flow model, Techniques of Water Resources Investigations, USGS, 1988. 11. Anderson, E.I., The effective resistance to vertical flow in Dupuit models, Advances in Water Resources, 26, pp.513-523, 2003.
A Dupuit model of groundwater-surface water interaction Erik I. Anderson a ~Department of Civil and Environmental Engineering, University of South Carolina, 300 Main Street, Columbia, SC 29208 We consider two-dimensional groundwater flow in the vertical plane to a stream in direct contact with the top of a confined aquifer. By comparison of the exact solution with a Dupuit solution to the same problem, we find that global errors in head and discharge in the Dupuit model can be avoided by treating the fictitious resistance of the stream bed as an effective parameter and introducing a new parameter into the model. Although the choice of the new parameter is arbitrary, we choose to model the portion of the aquifer beneath the stream as an inhomogeneity, thus introducing a second hydraulic conductivity into the Dupuit model. This approach fits well into common AEM modeling practices. We evaluate dimensionless values of the effective parameters such that there are no global errors in head or discharge in the Dupuit model. We demonstrate that the values of the effective parameters are weakly dependent on the geometric parameter D/H, the ratio of the length of the stream bed and the thickness of the aquifer, and independent of the flow parameter Q1/Q~,the ratio of the left and right model discharges. These properties of the effective parameters provide some evidence that the effective values presented here may be applied, at least approximately, to more general groundwater flow problems; the results provide guidance for simulating groundwater-surface water interaction with Dupuit models. 1. I N T R O D U C T I O N
Surface water features play a prominent role in nearly every model of groundwater flow even when the details of the interaction are not the focus of the model. Surface water features are of primary importance as they represent real hydrologic boundaries. They also present difficulties for modelers; as discussed by Hunt et al. [1], the manner in which surface water features are represented in a groundwater flow model can have significant effects on the model results. In regional models, the Dupuit approximation is often made, and the full details of groundwater-surface water interaction at streams are difficult to include due to the size of the domain and the computational effort required for fine vertical discretization. Global errors in head and discharge can be introduced into a model when employing the Dupuit approximation in regions of local, concentrated vertical flow [2,3]. These global errors in the Dupuit model can result in biased estimates of aquifer properties when calibrating a model to observed head data. Kaleris [4] demonstrates, using a Dupuit model, that forecasting groundwater surface water interaction under flow conditions other
474 than those used for calibration will give erroneous results; this is especially true in cases where the resistance of the stream bed is small and the stream is narrow. As surface water bodies are a common feature of nearly all groundwater flow models, this limitation restricts the predictive capabilities of Dupuit-based flow models. In many cases, Dupuit models can be improved by defining effective aquifer properties that incorporate, in an approximate fashion, the head losses due to vertical flow that are neglected in the Dupuit formulation" Aravin and Numerov [5] describe the local head loss summation method for analyzing seepage beneath hydraulic structures; Streltsova [6] defines an additional seepage resistance to deal with vertical flow for several groundwater flow problems; Halec and Svec [7] describes the method of substitute lengths to approximate losses due to vertical recharge from a reservoir; Strack [8] uses an equivalent length of aquifer to include losses due to vertical flow to a partially penetrating ditch; Bakker [9] defines an effective resistance to vertical flow in a two layer Dupuit model of flow to a ditch. In each case, an effective property is defined and evaluated by comparison of a Dupuit solution to an exact (vertical plane) solution.
2. E R R O R S I N D U P U I T
MODELS
To examine errors that can be introduced in a model of groundwater surface water interaction by making the Dupuit approximation, we consider the problem shown in Figure 1.a flow in the vertical plane to a stream bed in direct contact with the aquifer. The aquifer is an infinite strip of height H [L] with impermeable bottom and top, except along the stream bed of width D [L], shown as the dashed line; the head is constant and equal to h* [L] along the stream bed. Discharges Q1 and Q,. [L2/T] are specified at the left and right infinite vertex, respectively. Figure 1.b and 1.c show two common representations of the stream used in regional groundwater flow models. In Figure 1.b, the stream is assumed to penetrate the aquifer fully, and the head h* is fixed on both sides of the stream. In Figure 1.c, the flow beneath the stream is modeled as semi-confined flow, being separated from the stream by a fictitious leaky stream bed. The parameter A [L] (the leakage factor) is related to the resistance of the stream bed, c [T], as A - (kHc) 1/2, where k [L/T] is the hydraulic conductivity of the aquifer. In the Dupuit model of Figure 1.c, we have also allowed for a second parameter, k* [L/T], the hydraulic conductivity of the aquifer beneath the stream. An analytical solution to the exact, 2D problem (Figure 1.a) may be found in [5] and solutions to the two Dupuit formulations (Figures 1.b and c) may be developed using standard techniques. In the Dupuit model of Figure 1.b, the resistance to vertical flow of the aquifer is entirely neglected by assuming full penetration of the stream. In Figure 1.c, we lump the true, distributed resistance to vertical flow of the aquifer at the boundary along the stream bed. Here the resistance of the stream bed represents an estimate of the average resistance to vertical flow of the aquifer; the aquifer beneath the stream is modeled as being semiconfined. There is little guidance for selecting the effective resistance of the stream bed when a stream is in direct contact with the aquifer. McDonald and Harbaugh [10] suggest treating the resistivity as an intrinsic property of the aquifer by using a value of one half the total resistance to flow of the aquifer (c - H/2k). This approach results in a dimensionless value of the leakage factor of A/H = 0.707. The parameter k*, shown
475
Figure 1. Flow to a stream intersecting the top of an aquifer. (a) Two dimensional flow in the vertical plane to a stream of width D, and two common representations of the stream in Dupuit models. (b) The stream fully penetrates the aquifer, and (c) the flow beneath the stream bed is modeled as being semiconfined with a fictitious resistance specified at the stream bed.
in Figure 1.c, has not been identified as an effective parameter in previous studies. We specify the discharges Q~ and Ql in all three models and evaluate heads to the left and right of the stream at a distance of 2H. We express the error in head on the right side of the stream as
error-- (hr)exact- (hr)Dupuit (hl)exact- h*
(1)
where h / i s the head calculated at a distance of 2H from the left edge of the stream and h~ represents the head calculated at a distance of 2H from the right edge of the stream. We normalize the error in head on the right by a measure of the change in head over the model on the left. We can evaluate the errors induced in a Dupuit model by specifying fully penetrating boundaries as in Figure 1.b, or by the common practice of specifying the leakage factor to be 0.707H in Figure 1.c, by comparing solutions from the exact and Dupuit problems. We specify Q1 and Q~ to be the same in both the exact and the two Dupuit models; in the second Dupuit model we specify , ~ / H - 0.707 and k / k * - 1.0. We then compare heads predicted by each model a short distance from the stream bed.
476
Figure 2. Error in head in the two Dupuit models of groundwater surface water interaction. To the left, the error in the fully penetrating model (Figure 1.b) and to the right, the error in the semiconfined model with intrinsic properties (Figure 1.c).
Figure 2 shows plots of the error in head predicted by the two Dupuit models; Figure 2.a shows the errors induced by making the stream fully penetrating and Figure 2.b shows errors induced by using intrinsic values of vertical resistivity. Figure 2 shows that, in both cases, the error in head is a function of the problem geometry (D/H) and a parameter reflecting the flow regime or flow field (Q~/Q1). This confirms the conclusion of Kaleris [4] that using a Dupuit model to forecast heads under changed flow conditions will lead to errors. The error in Figure 2.b ranges from approximately minus 10% (for D/H large) to values exceeding 100% (for D/H small); the error can be significant. Similar errors in head on the left side of the stream should be expected. In a typical groundwater flow model, neither the distribution of discharge, nor the hydraulic conductivity of the aquifer are known before calibration of the model. The errors shown in Figure 2 will result in biased estimates of aquifer parameters during calibration of a groundwater flow model; the correct values of k and @/Q1 in the Dupuit model will not reproduce the correct (exact) head distribution in the aquifer. The Dupuit model shown in Figure 1.c may be modified to produce the correct head and discharges a short distance from the stream by either defining an effective head at the stream, or a vertical resistance layer separating the stream from the aquifer. The effective head or resistivity may be evaluated, but the dependence of parameters on Q~/Q1 will remain: the solution will not be able to forecast results under conditions of changed flow regimes. 3. E F F E C T I V E
PROPERTIES
It may be shown, for this problem, that global error in heads and discharges can be eliminated by properly defining the effective parameters k* and k, illustrated in Figure 1.c. The relationship between the exact and Dupuit solutions may be evaluated analytically for this case, and values of effective parameters may be derived, following [11]. For this
477
Figure 3. Effective parameters for groundwater-surface water interaction.
problem the effective parameters may be expressed as
__ = __
/ I (1--71/2) 1/-1
A H
D H
k*-
Arc
arcosh
in ( ! - ~ )
(2)
In 1+,.)/1/2
in 2 ~ ( 1 - 7 )
-ln2
11+-71/2 71/2
1/2
(3)
where 7 is a function of D / H only. The appropriate values of the effective parameters, evaluated analytically, are plotted in Figure 3. Surprisingly, the introduction of two effective parameters in this problem eliminates any dependence on the flow parameter Q~/QI; the effective values may be applied to a model without prior knowledge of the distribution of flow from the right and left the values are independent of the flow field. Further, Figure 3 shows that the values of the effective parameters are only weakly dependent on geometry, being fairly constant over a wide range of stream widths, and that the effective resistance is much smaller than the intrinsic value of c k / H - 0.5 suggested by McDonald and Harbaugh [10]. 4.
COMPARISON
OF
FLOW
FIELDS
Figures 4 and 5 show comparisons of flow fields obtained from solutions to the exact problem and the Dupuit problem; the effective parameters used in the Dupuit solution are obtained from Figure 3. The specific problem considered has geometry and flow parameters of D / H - 1.5; Figure 4 shows the case where Q~/Q1 - - 0 . 5 and Figure 5 the case where Q ~ / Q 1 - +0.5. Figure 4.a shows the solution to the exact problem and Figure 4.b the Dupuit solution. In both, contours of head are shown as dashed lines and contours of stream function are
478
Figure 4. Comparison of flow fields for D / H - 1.5 and Q , . / Q , - -0.5" (a) the exact vertical plane solution showing head contours (dashed) and contours of the stream function (solid); (b) the Dupuit solution with the correct effective properties specified; (c) a comparison of heads obtained from the exact solution (solid) and the Dupuit solution (dashed); and (d), a comparison of the stream function from the exact (sdid) and Dupuit (dashed) solutions.
479
Figure 5. Comparison of flow fields for D / H - 1.5 and Q ~ / Q ~ - +0.5" (a) the exact vertical plane solution showing head contours (dashed) and contours of the stream function (solid); (b) the Dupuit solution with the correct effective properties specified; (c) a comparison of heads obtained from the exact solution (solid) and the Dupuit solution (dashed); and (d), a comparison of the stream function from the exact (solid) and Dupuit (dashed) solutions.
480 shown as solid lines; the dividing streamline is shown as a heavier line. Figure 4.c shows the exact and Dupuit head distributions overlaid; the solid lines represent the exact head and the dashed lines the Dupuit heads. Note that the heads converge a short distance from the stream on both sides. In a similar fashion, Figure 4.d shows the exact and Dupuit stream functions overlayed. Note here that the values of the stream function converge a short distance from the stream on both sides. Global errors in both head and discharge have been eliminated from the Dupuit solution by proper choice of the effective parameters. Figure 5 shows similar comparisons under changed flow conditions, emphasizing the fact that the results are independent of the flow regime. 5. D I S C U S S I O N
The independence of the values of the effective parameters on the flow conditions, and their weak dependence on geometry suggest that the effective properties may be applied to models of more general settings, where, for example, the stream width varies and the head in the stream varies. Application of these results in a general groundwater flow model requires simply modifying parameter values; no changes in modeling technique, approach, or the numerical method of solution are required. However, to be useful, these preliminary results must be verified in more general settings. REFERENCES
1. Hunt, R.J., Haitjema, H.M., Krohelski, J.T. and D.T. Feinstein, Simulating ground water-lake interactions: approaches and insight, Ground Water, 41(2), pp227-237, 2003. 2. Bear, J. and C. Braester, Flow from infiltration basins to drains and wells, Journal of the Hydraulics Division, ASCE, HY5, pp. 115-135, September, 1996. 3. Bakker, M., Simulating groundwater flow in multi-aquifer systems with analytical and numerical Dupuit models, Journal of Hydrology, 222, pp. 55-64, 1999. 4. Kaleris, V., Quantifying the exchange rate between groundwater and small streams, Journal of Hydraulic Research, 36(6), pp. 913-932, 1998. 5. Aravin, V.I. and S.N. Numerov, Theory of Fluid Flow in Undeformable Porous Media, Daniel Davy, New York, 1965. 6. Streltsova, T.D., Method of additional seepage resistances- theory and application, ASCE Journal of the Hydraulics Division, Vol. 100, HY8, pp. 1119-1131, 1974. 7. Halek, V. and J. Svec, Groundwater Hydraulics, Elsevier Scientific Publishing Co., Amsterdam, 1979. 8. Strack, O.D.L., Groundwater Mechanics, Prentice Hall, Englewood Cliffs, New Jersey, 1989. 9. Bakker, M., Transient Dupuit interface flow with partially penetrating features, Water Resources Research, 34(11), pp.2911-2918, 1998. 10. McDonald, M.G., and A.W. Harbaugh, A modular three-dimensional finite-difference ground-water flow model, Techniques of Water Resources Investigations, USGS, 1988. 11. Anderson, E.I., The effective resistance to vertical flow in Dupuit models, Advances in Water Resources, 26, pp.513-523, 2003.