Supplemental Materials. Modeling Flow into Horizontal Wells in a Dupuit-Forchheimer Model

Transcription

1 Supplemental Materials Modeling Flow into Horizontal Wells in a Dupuit-Forchheimer Model Henk Haitjema, Sergey Kuzin, Vic Kelson, and Daniel Abrams August 8,

2 Original publication Modeling Flow into Horizontal Wells in a Dupuit-Forchheimer Model, Henk Haitjema, Sergey Kuzin, Vic Kelson, and Daniel Abrams, Ground Water, 2010, in press. Abstract Horizontal wells or radial collector wells are used in shallow aquifers to enhance water withdrawal rates. Groundwater flow patterns near these wells are three-dimensional, but difficult to represent in a 3D numerical model because of the high degree of grid refinement needed. However, for the purpose of designing water withdrawal systems it is sufficient to obtain the correct production rate of these wells for a given drawdown. We developed a Cauchy boundary condition along a horizontal well in a Dupuit-Forchheimer model. Such a steady state 2D model is not only useful for predicting groundwater withdrawal rates, but also for capture zone delineation in the context of source water protection. A comparison of our Dupuit-Forchheimer model for a radial collector well with a three-dimensional model yields a 5% lower production rate. Particular attention is given to horizontal wells that extend underneath a river. A comparison of our approach with a 3D solution for this case yields satisfactory results, at least for moderate to large river bottom resistances. 2

3 Summary of results In the next two sections the primary results from the original publication are summarized with references to appendices not included in the original publication. A third section contains a validation of the case for a horizontal well (lateral) underneath a river, which was also not included in the original publication. Horizontal well in a confined aquifer The drawdown (φ L φ w ) at a infinitely long horizontal well in a confined aquifer, where resistance to vertical flow is included (2D flow in the vertical plane), is given by, φ L φ w = σ 2πk ln 2 sin( π r wh ) sin(π h+rw/2 2 cosh(π L ) cos(π h ) H H ) H (1) where k and H are the hydraulic conductivity and thickness of the aquifer, respectively. The parameter r w [L] is the radius of the horizontal well, which must be small compared to H, for instance r w < 0.1H. In practice this means that aquifers should not be thinner than about 2 to 3 meters. The horizontal well is located at a distance h above the aquifer base and the head φ L occurs at a distance L from the well. The head in the horizontal well is φ w. The analysis that leads to (1) is similar to that in Strack (1989), page and provided in Appendix A. Comparison of the result in (1) with Dupuit-Forchheimer flow to a stream yields a fictitious stream resistance of c ls = w { ( ) ( π 2πk ln r w 4 sin sin π h + r )} w/2 2 H H The width w may be selected arbitrarily. With (2) applied as bottom resistance to a stream in a Dupuit-Forchheimer model, this stream may be interpreted as a horizontal well, yielding approximately the same production rate as a horizontal well in a three dimensional model. Horizontal well underneath a river with bottom resistance A solution for the case of an infinite long horizontal well (lateral) underneath an infinitely long (and wide) river is given by Bruggeman (1999, case on page 311), which leads to the following expression for the drawdown (φ 0 φ w ): φ 0 φ w = σ 1 k n=0 α n (2) cos ( ) h ( ) α n H h + r w 1 + ɛ cos α n (3) H α 2 n +ɛ2 where φ 0 is the river stage, φ w the head in the horizontal well, and where α n are the roots of: α tan α = ɛ (4) The parameter ɛ is defined as: ɛ = H k c b (5) 3

4 with c b the resistance to flow in the river bottom sediments, defined as: c b = δ k s (6) where δ [m] and k s are the thickness and the vertical hydraulic conductivity of these sediments, respectively. The roots of (4) are periodic and may be written as: α n = α + n π (7) where α is the principal value of (4). A graph for α as a function of ɛ is provided in Appendix B. A resistance to vertical flow in the horizontal plane c 2 is obtained as: c 2 = w k 1 n=0 α n cos ( ) h ( ) α n H h + r w 1 + ɛ cos α n (8) H α 2 n +ɛ2 The resistance c 2 includes the resistance to (a) vertical flow through the river bottom, (b) horizontal flow in the aquifer, and (c) vertical flow in the aquifer. When modeling a horizontal well underneath a river in a Dupuit-Forchheimer model we are already including the first two types of resistances. This resistance, therefore, must be subtracted from c 2 when defining the Cauchy boundary condition. This leads to the following resistance to be applied as bottom resistance in a Dupuit-Forchheimer model: c ls = c 2 λw 2kH (9) where λ is a characteristic leakage length defined as: λ = Some graphs for c 2 are provided in Appendix B. khc b (10) Validation of equation (9) The use of (9) may be validated by use of a high resolution MODFLOW model with a horizontal well (lateral) that extends underneath the entire width of a river and with no-flow boundaries coinciding with these river boundaries. In such a setup the flow is forced to be two-dimensional in the vertical plane, perpendicular to the well, anywhere underneath the river. This situation is consistent with the conditions for which (9) was developed. The river stage and head in the horizontal well are φ 0 = 30 meters and φ w = 20 meters, respectively. The radius of the horizontal well is r w = meters, while the aquifer thickness is H = 24 meters. The well is midway the aquifer (h/h = 0.5), which has a hydraulic conductivity of k = meters per day. Table 1 presents the total flow [m 3 /day] in the lateral obtained in three different ways. Firstly, Q exact is obtained by use of (3). Secondly, Q MF has been obtained from a cross-sectional MODFLOW model representing the aquifer underneath a stretch of the river 4

5 c b Q exact Q MF error Q GF error [days] [m 3 /day] [m 3 /day] [%] [m 3 /day] [%] , , , , , , , , , , , Table 1: Comparison of total inflows into a lateral underneath a stream for the case of a lateral that extends underneath the entire river that is bounded by no-flow boundaries to force flow perpendicular to the lateral. of length 480 meters with cells that are 0.1 meters on a side remote from the lateral. Near the lateral the cells have been refined down to meters on a side. Finally, Q GF has been obtained by use of the Dupuit-Forchheimer model GFLOW using a no-flow boundary along a 480 meter long stream stretch for values of c b that are between 0.01 and 1 day and a 1600 meter stream stretch for the case of c b = 10 [days]. In this manner we ensured that for all cases the stream on either side of the lateral is longer than 4λ thus capturing stream seepage sufficiently far away, see the original publication. The GFLOW model is a Dupuit- Forchheimer model that utilizes (9) with (8) and (10). No results are presented for the river bottom resistances of c b = 10, 000 [days] and c b = [days]. For the first case the length of the river would have to be about 20,000 meters in each direction from the lateral in order to capture enough of the river seepage, which is cumbersome to implement. For the latter case the river is completely separated from the aquifer, hence there would be no flow at all into the lateral. We also omitted the MODFLOW run for c b = 10 [days], which would require an unduly long river section in the high resolution MODFLOW model. The percent error in the column that follows Q MF is (Q MF Q exact )/Q exact 100%, while the values in the column following Q GF are defined as (Q GF Q exact )/Q exact 100%. It is seen that the MODFLOW cross-sectional model as well as the GFLOW model using the c ls values obtained from (9) are quite accurate. 5

6 Figure 1: A confined aquifer with a lateral at a distance h above its base (z-plane) is mapped onto a reference plane (ζ-plane). Appendix A: Derivation of equation (1) The analysis in this appendix is similar to that presented by Strack (1989) page Figure 1 depicts a vertical section over an aquifer with a small diameter lateral at a distance h above the aquifer base; the z-plane on the left. The upper half of the ζ-plane on the right in Figure 1 is a conformal map of this z-plane. The aquifer top and bottom, which form no-flow boundaries in the z-plane, are mapped along the real ξ-axis in the complex ζ-plane with the numbering along these boundaries detailing which points map where. The mapping function is given by Strack (1989) equation 32.30: ζ = e πz/h (11) where z = x + iy and ζ = ξ + iη, see Figure 1. The lateral with discharge σ [m 2 /day] is located at z w = ih and maps onto ζ w = exp(iπh/h). Half of the flow toward the lateral in the z-plane comes from + (between points 2a and 2b), which corresponds to infinity in the upper half-plane (ζ-plane). The other half of the flow comes from in the z-plane (between the points 4a and 4b), which corresponds to the origin in the ζ-plane. The latter water is supplied by a point source with strength σ/2 at ζ = 0 (not shown in Figure 1). To maintain the no-flow condition along the horizontal axis in the reference plane, η = 0, an image lateral at ζ w = exp( iπh/h) and an image of the source with strength σ/2 at ζ = 0 are added. The resulting complex potential Ω(ζ) is given by: where Ω 0 is an integration constant. Ω(ζ) = σ 2π ln (ζ ζ w)(ζ ζ w ) ζ 6 + Ω 0 (12)

7 The specific discharge potential Φ [m 2 /day], defined as the product of the hydraulic conductivity k and the head φ, Φ = kφ (13) is the real part of Ω (Strack 1989), so that with (11) and (12): { σ Φ(z) = R 2π ln (eπz/h e iπh/h )(e πz/h e iπh/h } ) + Φ e πz/h 0 (14) where Φ 0 is the real part of Ω 0. If the potential at some point z = L is given as Φ = Φ L expression (14) becomes: { σ Φ(z) = R 2π ln (eπz/h e iπh/h )(e πz/h e iπh/h )e πl/h } + Φ e πz/h (e πl/h e iπh/h )(e πl/h e iπh/h L (15) ) In (15) constants (independent of z) have been added to the argument of the logarithm such that it becomes 1 at z = L, so that Φ = Φ L at z = L. The potential inside (and just outside) the lateral is denoted by Φ w = kφ w, where φ w is the head inside the lateral. For small diameter laterals the head will be nearly constant along the perimeter of the lateral. In writing (12) it was tacitly assumed that the lateral could be seen as a point in the z-plane, mapping to a point in the ζ-plane. Since we will apply the analysis to laterals of finite diameter, the analysis presented here is approximate. The potential at a collocation point z w, selected on top of the lateral, z w = i(h + r w ) (16) is set to Φ w. Substituting (16) into (15), thus replacing z by z w, results in the following expression for the potential Φ w : h+rw σ (eiπ H e iπ h h+rw iπ H )(e H e iπ h H )e π L H Φ w = R ln h+rw 2π iπ e H (e π L H e iπ h H )(e π L H e iπ h H ) + Φ L (17) The argument of the logarithm in (17) may be rewritten as follows. Referring to the argument as A: ( ) ( A = eiπ h H e i π rw 2 H e i π rw 2 H e i π rw 2 H e iπ( h H + rw 2H ) e iπ( h H + rw )) 2H e i π rw 2 H e π L H ( e iπ( h H + rw H ) e π L H 1 e π( L H i h )) ( H 1 e π( L H +i h )) (18) H e π L H The last exponential term in the numerator cancels the last exponential term in the denominator. The two first exponential terms in the numerator with what is now the last term in the numerator cancel the first exponential term in the denominator. Rearranging terms leads to: ( ) ( e i π rw 2 H e i π rw 2 H e iπ( h H + rw 2H ) e iπ( h H + rw )) 2H A = ( π e 2 ( L H i h H ) e π 2 ( L H i h )) ( H e π 2 ( L H +i h H ) e π 2 ( L H +i h )) (19) H The denominator may be multiplied out and rearranged to give: A = ( e i π rw 2 H e i π 2 ( e π L H + e π L H ) ( rw H e iπ( h H + rw 2H ) e iπ( h H + rw ) ( ) (20) e iπ h H + e iπ h H 2H )) 7

8 Dividing the expressions between parentheses in the numerator by 2i and those in the denominator by 2 yields: A = 2 sin( π r wh ) sin(π h+rw/2 2 H )eiπ cosh(π L ) cos(π h ) (21) H H Replacing the argument of the logarithm in (17) by (21) and writing the expression as a drawdown using (13) yields: where use has been made of (13). φ 0 φ w = σ 2πk ln 2 sin( π r wh ) sin(π h+rw/2 2 cosh(π L ) cos(π h ) H H ) H (22) 8

9 Appendix B: Graphs for α and c 2. Evaluation of (8) is complicated by the infinite series. For the reader s convenience some results are presented in dimensionless form in Figure 3 where k c 2 /w (kc/w on the graph) is plotted versus ɛ for various values of r w /H (r/h on the graph) and h/h. The graph will only provide approximate results, however. For more accurate values of c the value of α may be read from the graph in Figure 2 and used with (7) in the infinite sum in (8) which may be aborted after sufficient accuracy is obtained. For further accuracy the implicit equation (4) may be evaluated instead of using the graph in Figure 2. For small values of ɛ it was found that several thousand Newton-Raphson iterations were required to obtain accurate values for α. Additional graphs for the resistance c 2, for various values of k, H, and stream bottom resistance c b follow in Figure 4 through Figure 10. The parameters in these graphs are presented in dimensionless form as kc /w, where c = c 2, as kc/h (= 1/ɛ), where c = c b, and as r/h, where r = r w. 9

10 Figure 2: Principal values for α as a function of ɛ to be used in equation (7). 10

11 Figure 3: Resistance c 2 due to 3D flow underneath a stream with a resistance layer is plotted as the dimensionless parameter k c/w (c = c 2 ) versus ɛ (= H/kc b ). The dotted lines refer to the case of h/h = 0.2, while the solid lines refer to the case h/h =

12 Figure 4: Resistance c 2 due to 3D flow underneath a stream with a resistance layer with resistance c = c b. The parameters are: hydraulic conductivity k, stream bottom resistance c = c b, aquifer thickness H, lateral radius r = r w, lateral elevation h, line-sink width w, and resistance c = c 2. The curves are for r/h =0.001, 0.002, 0.004, 0.006, 0.008, 0.01, 0.02, 0.04, 0.06, 0.08 and

13 Figure 5: Resistance c 2 due to 3D flow underneath a stream with a resistance layer with resistance c = c b. The parameters are: hydraulic conductivity k, stream bottom resistance c = c b, aquifer thickness H, lateral radius r = r w, lateral elevation h, line-sink width w, and resistance c = c 2. The curves are for r/h =0.001, 0.002, 0.004, 0.006, 0.008, 0.01, 0.02, 0.04, 0.06, 0.08 and

14 Figure 6: Resistance c 2 due to 3D flow underneath a stream with a resistance layer with resistance c = c b. The parameters are: hydraulic conductivity k, stream bottom resistance c = c b, aquifer thickness H, lateral radius r = r w, lateral elevation h, line-sink width w, and resistance c = c 2. The curves are for r/h =0.001, 0.002, 0.004, 0.006, 0.008, 0.01, 0.02, 0.04, 0.06, 0.08 and

15 Figure 7: Resistance c 2 due to 3D flow underneath a stream with a resistance layer with resistance c = c b. The parameters are: hydraulic conductivity k, stream bottom resistance c = c b, aquifer thickness H, lateral radius r = r w, lateral elevation h, line-sink width w, and resistance c = c 2. The curves are for r/h =0.001, 0.002, 0.004, 0.006, 0.008, 0.01, 0.02, 0.04, 0.06, 0.08 and

16 Figure 8: Resistance c 2 due to 3D flow underneath a stream with a resistance layer with resistance c = c b. The parameters are: hydraulic conductivity k, stream bottom resistance c = c b, aquifer thickness H, lateral radius r = r w, lateral elevation h, line-sink width w, and resistance c = c 2. The curves are for r/h =0.001, 0.002, 0.004, 0.006, 0.008, 0.01, 0.02, 0.04, 0.06, 0.08 and

17 Figure 9: Resistance c 2 due to 3D flow underneath a stream with a resistance layer with resistance c = c b. The parameters are: hydraulic conductivity k, stream bottom resistance c = c b, aquifer thickness H, lateral radius r = r w, lateral elevation h, line-sink width w, and resistance c = c 2. The curves are for r/h =0.001, 0.002, 0.004, 0.006, 0.008, 0.01, 0.02, 0.04, 0.06, 0.08 and

18 Figure 10: Resistance c 2 due to 3D flow underneath a stream with a resistance layer with resistance c = c b. The parameters are: hydraulic conductivity k, stream bottom resistance c = c b, aquifer thickness H, lateral radius r = r w, lateral elevation h, line-sink width w, and resistance c = c 2. The curves are for r/h =0.001, 0.002, 0.004, 0.006, 0.008, 0.01, 0.02, 0.04, 0.06, 0.08 and