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The effects of vertical barrier walls on the hydraulic control of contaminated groundwater
Advances in Water Resources 29 (2006) 89–98 www.elsevier.com/locate/advwatres
The effects of vertical barrier walls on the hydraulic control of contaminated groundwater Erik I. Anderson *, Elizabeth Mesa Department of Civil and Environmental Engineering, University of South Carolina, 300 Main Street, Columbia, SC 29208, United States Received 14 December 2004 Available online 11 July 2005
Abstract We present explicit analytic solutions describing the hydraulic head and discharge vector for two-dimensional, steady groundwater flow past an impermeable barrier embedded in a regional flow field. We use the solution to investigate the effects of open vertical barriers on the flow field; in particular, we examine the hydraulic containment of contaminant plumes or source zones by combination of a vertical barrier wall and extraction wells. We quantify the local reduction in discharge rates due to the barrier wall and the local increase in the size of the capture zone of an extraction well near an open, up-gradient barrier. We find that the combination of an open vertical barrier with down-gradient extraction wells can be very effective in decreasing the well discharge rate necessary to control a contaminant plume or source area. Design charts are presented for quantifying the effects of the barrier wall on the hydraulic control of the groundwater flow field and for estimating the jump in head across a barrier. The charts are appropriate for use in the preliminary design and cost estimating of remedial systems, and for the design of dewatering systems. Ó 2005 Published by Elsevier Ltd. Keywords: Groundwater; Analytic; Remediation; Vertical barriers; Conformal mapping; Capture zone
1. Introduction Vertical barrier walls are often used in conjunction with groundwater extraction wells as components of waste containment or pump and treat systems. The purpose of the barrier wall in a remedial design is to restrict the flow of uncontaminated groundwater onto a site and to limit the flow of contaminants off site. Barriers can also provide groundwater control during excavation of wastes or contaminated soil. Vertical barrier walls are commonly constructed of grout, slurry, and plastic or steel sheetpiling. The exact configuration and effects of a barrier wall depend upon site-specific conditions. However, a variety of texts [21,5,11,7] and USEPA documents [2,4], discuss *
Corresponding author. Fax: +1 803 777 0670. E-mail addresses: [email protected] (E.I. Anderson), mesae @engr.sc.edu (E. Mesa). 0309-1708/$ - see front matter Ó 2005 Published by Elsevier Ltd. doi:10.1016/j.advwatres.2005.05.005
qualitatively the use and placement of vertical barriers for control of contaminated groundwater. Rumer and Ryan [21], for example, discuss the placement of circumferential barriers used to completely enclose a site, and open barriers used for redirecting groundwater flow. The circumferential barrier design has been applied to many landfills and hazardous waste sites throughout the United States, including the well-documented Gilson Road Superfund Site in Nashua, New Hampshire [1,3]. A four foot wide slurry wall extending up to 100 ft in depth and 4000 ft long encloses that site. Open barrier designs commonly include up-gradient barriers with down-gradient extraction wells or downgradient barriers with up-gradient extraction wells. The desired effect of the open barrier is to minimize the discharge rate of wells needed for hydraulic control of a contaminant plume [27, p. 183]. Down-gradient barriers have been used at the Solvent Recovery Services of New England Superfund Site in Southington,
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Connecticut [1], and the Rocky Mountain Arsenal Site in Colorado [7]. The Connecticut site includes a wall of steel sheetpiling extending to bedrock with 11 extraction wells located up-gradient of the wall; the Colorado site includes a grout curtain with up-gradient extraction wells and down-gradient injection wells. Sites where upgradient barriers have been constructed include Necco Park Landfill in Niagara Falls, New York [28] and the LaBounty Landfill Superfund Site in Iowa [22]. A single line grout curtain was placed along three sides of the Necco Park industrial landfill to reduce the volume of underflow being removed by down-gradient extraction wells; a bentonite wall was placed up-gradient of the LaBounty Landfill for the purpose of dewatering submerged wastes and to reduce the groundwater discharge beneath the site. Much information about the geotechnical design and construction of vertical barrier walls is available [4,2,11,18,21]. However, little design guidance is available for evaluating the impacts of a barrier wall on groundwater flow. As discussed by Mitchell and Van Court [27] proper modeling of groundwater flow patterns and the influences of barrier and extraction well interaction is necessary to evaluate the effectiveness of any design. This is commonly achieved by developing numerical models of the local groundwater flow field. Hudak [14] used a numerical model to investigate the effects of a cutoff wall on monitoring well networks near landfills. Recently, Bayer et al. [6] presented results of a numerical study of the effects of different barrier configurations on groundwater flow; a numerical framework is presented for evaluating the effects of barrier geometry on the advective control of contaminant plumes. It is often necessary during the preliminary design phase to draw some conclusions about the effectiveness of a barrier wall on site conditions without the time and expense required to develop a detailed numerical
model of the local groundwater flow field. The purpose of this paper is to examine quantitatively, and in a general setting, the combined effects of extraction wells and open vertical barriers. We develop explicit analytical expressions for the hydraulic head and the discharge vector for general well/barrier systems embedded in a regional groundwater flow field. The results provide a tool for engineers, in the form of dimensionless plots, to help assess the effects of a barrier wall on the hydraulic control of contaminant plumes. In particular, this work helps to quantify the effects of the barrier wall on local advection of a plume, the decrease in the discharge rates required of extraction wells to control a plume after placement of a vertical barrier wall, and the down-gradient lowering of the hydraulic head caused by the barrier wall. In addition, the analytic solution can be applied to more general cases with barriers and multiple extraction and injection wells for more detailed design considerations. We formulate the groundwater flow problem in terms of a complex potential and solve the problem using conformal mapping and the method of images. References on the application of conformal mapping to solve problems of groundwater flow include [20,13,26,24].
2. Problem description We consider steady, two-dimensional flow in the complex z-plane (z = x + iy), as illustrated in Fig. 1a. A vertical barrier wall, lying along a circular arc of radius R and length Ra, is embedded in the infinite domain; the barrier wall is assumed to be impermeable. A fully penetrating well of discharge rate Q [L3/T] is located at z = zw, and a uniform discharge of strength Q0 [L2/T] and oriented at an angle b to the real axis is specified at infinity. We define a complex potential as an analytic function of z as
Fig. 1. (a) The physical plane (z-plane), (b) the auxiliary plane (v-plane), and (c) the upper-half plane (f-plane). In the physical plane, a well of discharge rate Q is located at z = zw.
E.I. Anderson, E. Mesa / Advances in Water Resources 29 (2006) 89–98
XðzÞ ¼ U þ iW
ð1Þ 3
The imaginary part of the complex potential, W [L /T], is the stream function; the real part, U [L3/T], is the discharge potential, and is related to the hydraulic head, h [L], in the aquifer as follows: 1 U ¼ kHh kH 2 2
ðh P H Þ
ð2Þ
1 U ¼ kh2 ðh < H Þ ð3Þ 2 where k [L/T] is the hydraulic conductivity of the aquifer and h is measured with respect to the aquifer base; H is the distance between the aquifer base and an overlying confining layer. Using (2) and (3) the complex potential may be applied to combined shallow confined and shallow unconfined flow systems [24]. We further define the complex discharge function, W(z), as W ðzÞ ¼
dX ¼ Qx iQy dz
ð4Þ
where Qx and Qy [L2/T] are the x- and y-components of the discharge vector (the depth-integrated specific discharge vector). We present the boundary conditions for the problem illustrated in Fig. 1 in terms of the complex potential and complex discharge. First, the condition along the impermeable barrier is specified z ¼ Rþ expðihÞ aþp ap 6h6 JðXÞ ¼ 0 for 2 2 z ¼ R expðihÞ ð5Þ +
where R and R are used to distinguish between the two sides of the wall, and where J indicates the imaginary part of the complex function. The condition at infinity may be expressed as follows: lim W ¼ Q0 expðibÞ
z!1
ð6Þ
The behavior near the discharge well may be expressed in the form of an expansion of X about z = zw as Q lnðz zw Þ þ f ðzÞ X¼ 2p
field. Historically, solutions to problems of potential flow past impermeable barriers embedded in a uniform regional flow field have been developed for the study of lift on aerofoils; the impermeable circular arc used here to represent the barrier wall is a special case of a Joukowski foil (see for example, [19,25]). In hydrodynamics, however, point sources/sinks inside the domain have no physical meaning; the solution presented here focuses on the interaction of the impermeable barrier and groundwater wells.
3. Solution by conformal mapping The solution to the boundary-value problem described in the previous section may be obtained by conformal mapping and the method of images: we map the physical plane (z) onto the upper-half plane (f), with the impermeable slot mapped to the entire real axis; singularities are added to the flow field in the f-plane to represent flow features in the physical plane, and the method of images is applied to satisfy the condition along the impermeable barrier. 3.1. The physical plane, the auxiliary plane, and the upper-half plane We map the z-plane onto the f-plane in two steps, first transforming it onto the auxiliary v-plane. The physical, or z-plane is shown in Fig. 1a. The impermeable boundary is represented by a slot with the shape of a circular arc; points of interest along the impermeable slot are labelled 1 through 4. The circle of radius R in the z-plane is mapped to a straight line in the auxiliary v-plane by a bilinear transformation; the impermeable slot lying along an arc of the circle in the z-plane is therefore mapped to a straight slot in the v-plane. The v-plane is shown in Fig. 1b. The mapping of z onto v is given by v¼
ð7Þ
iðz þ iRÞ ðz iRÞ
ð9Þ
The mapping may be inverted to give z(v)
where f(z) is a function that is analytic at z = zw. Finally, we require a reference point to make the complex potential unique Uðz0 Þ ¼ U0
91
ð8Þ
where z0 is the coordinate of the reference point and U0 is a known value of the discharge potential at the reference point. Many researchers have applied complex analysis to groundwater flow with multiple combinations of extraction and injection wells for capture zone analysis and remedial design [15,23,10,12,9,29,8,16]; here we include explicitly the effects of a vertical barrier wall on the flow
z¼
iRðv þ iÞ ðv iÞ
ð10Þ
Common points in the two planes are labelled along the impermeable boundary. From (9) we find that the point at infinity in the z-plane is mapped to v = i; the location of the well in the z-plane, zw, is mapped to the point vw in the v-plane. Both points are labelled in Fig. 1b. The dimensionless length L of the impermeable slot in the v-plane may be evaluated from (9) as follows: L ¼ vðz4 Þ vðz2 Þ ¼
2Rðz4 z2 Þ ðz4 iRÞðz2 iRÞ
ð11Þ
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where z4 and z2 are the complex coordinates of the corner points of the impermeable barrier in the physical plane (Fig. 1) h a þ p i z4 ¼ R exp i ð12Þ 2 h a pi ð13Þ z2 ¼ R exp i 2 After some algebra we find L¼
2 sinða=2Þ 1 þ cosða=2Þ
ð14Þ
Finally, the f-plane is mapped onto the auxiliary v-plane by a mapping of constant argument [24]. The f-plane is illustrated in Fig. 1c. The mapping is given by v¼
Lf ðf2 þ 1Þ
ð15Þ
The impermeable slot in the z- and v-planes is now mapped to the entire real axis of the f-plane. This mapping may also be inverted: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi L L L þ 1 þ1 ð16Þ f¼ 2v 2v 2v We chose the sign of the square root term such that the domains correspond to those shown in Fig. 1 and the arguments such that the branch cut lies along the slot in the v-plane: L L 0 6 arg 1 6 2p p 6 arg þ1 6p 2v 2v ð17Þ The mappings (15) and (16) are valid for 0 6 a < 2p. The location in the f-plane that the point z = 1 maps to is of importance in evaluating the complex potential. We refer to the point f(z = 1) as r; the value of r may be obtained from (16) and (9), using (17) pffiffiffiffiffiffiffiffiffiffiffiffiffi i r ¼ ðL þ L2 þ 4Þ 2
conditions along the impermeable boundary. We will consider the two flow features in the physical plane— the well and the uniform flow—separately. To represent the well located in the physical plane at z = zw (7), we place a sink in the f-plane at f = fw. To satisfy the condition along the impermeable slot in the physical plane (5), we image the sink about the real axis of the f-plane. A source exists at infinity in the physical plane, balancing the sink at zw. To produce the proper behavior in f we must place a source of strength equal to the sink at f = r; that source must also be imaged about the real axis to satisfy the condition along the impermeable slot. The portion of the complex potential reflecting the well and satisfying the condition along the impermeable slot is Q ðf fw Þðf fw Þ ln Xwell ¼ ð19Þ Þ 2p ðf rÞðf r The uniform flow specified at infinity in the physical plane (6) is represented in f by a dipole placed at f = r. The dipole is represented graphically in Fig. 2 by a source()/sink(+) pair. The dipole must also be imaged about the real axis as shown in the figure. The portion of the complex potential reflecting the uniform flow and satisfying the condition along the impermeable slot is ic s e eic Xuni ¼ þ ð20Þ Þ 2p ðf rÞ ðf r For now we let the strength and orientation of the dipole, s and c, remain arbitrary; later we will relate them to the boundary condition in the physical plane (6). The final expression for the complex potential is the sum of Xwell and Xuni
ð18Þ
We note that r is purely imaginary. It may also be of interest to note that another pole in the mapping z(f) exists when the sign of the square root term in (18) is changed. This pole, however, does not lie on the proper branch of the mapping function f(z) defined by (17) and therefore lies outside the problem domain. 3.2. The complex potential The complex potential may now be evaluated by placing singularities in the f-plane to represent the well and uniform flow specified in the physical plane. For each singularity placed in the f-plane we apply the method of images about the real axis to satisfy
Fig. 2. Orientation and imaging of the dipole about the impermeable boundary in the f-plane.
E.I. Anderson, E. Mesa / Advances in Water Resources 29 (2006) 89–98
Q ðf fw Þðf fw Þ ln X¼ Þ 2p ðf rÞðf r ic ic s e e þ þ þC Þ 2p ðf rÞ ðf r
lim W ¼ f!r
ð21Þ
where C is a real constant that may be evaluated from a reference point (8). The complex potential (21) is analytic at f = 1, as may be seen by expanding the expression about infinity. Additional sources or sinks, representing recharge or discharge wells in the physical plane may be included by superposition in the f-plane. A general expression for N sources and sinks of strength Qn and located in the upper-half plane at fw,n is given by N X ðf fw;n Þðf fw;n Þ Qn X¼ ln Þ ðf rÞðf r 2p n¼1 ic s e eic þ þ þC ð22Þ Þ 2p ðf rÞ ðf r
4RLQ0 pr2 r2 1
ð28Þ
c ¼ ðb þ pÞ
ð29Þ
s¼ and
3.5. Solution The explicit analytical solutions for both the complex potential and complex discharge in the physical plane are now complete; the solutions are given by (21) and (24) along with the mappings and parameters given by (9), (14), (16) through (18), (28), and (29). We present the complex potential in a more direct form here, by combining Eqs. (21), (28), and (29): X¼
The details are straightforward and are omitted here for brevity. The resulting expression for the complex discharge is ( 2 2 ðf rÞ ðf r1 Þ 2f ðfw þ fw Þ W ¼ Q 2 ðf fw Þðf fw Þ 4pRLðf 1Þ ) 2ðf2 þ r2 Þ cos c þ 4i fr sin c s ð24Þ 2 Þ2 ðf rÞ ðf r 3.4. Strength and orientation of the dipole The strength and orientation, s and c, of the dipole placed at f = r are related to the strength and orientation, Q0 and b, of the uniform flow specified at infinity in the physical plane. To evaluate the relationship, we apply the boundary condition (6): lim W ¼ Q0 expðibÞ
z!1
ð25Þ
We apply this condition to the complex discharge function (24), noting that for z ! 1, f ! r and we obtain " # 2 sðr r1 Þ 4r2 expðicÞ lim W ¼ ð26Þ f!r 4pRLðr2 1Þ ðr r Þ2 Recalling that r is purely imaginary (18), we note that Þ ¼ 2r, and the expression simplifies as ðr r
sðr2 1Þ sðr2 1Þ expðicÞ ¼ exp½iðp þ cÞ 2 4pRLr 4pRLr2 ð27Þ
By equating (25) and (27) we obtain
3.3. The complex discharge The complex discharge function W(z) may be evaluated from the complex potential (21) and the mappings (10) and (15) as
dX dX dv dz ¼ W ¼ ð23Þ dz df df dv
93
Q ðf fw Þðf fw Þ ln Þ 2p ðf rÞðf r iðbþpÞ 2 2RLQ0 r e eiðbþpÞ þ þ þC Þ r2 1 ðf rÞ ðf r
ð30Þ
where pffiffiffiffiffiffiffiffiffiffiffiffiffi i r ¼ ðL þ L2 þ 4Þ 2
and
L¼
2 sinða=2Þ 1 þ cosða=2Þ
ð31Þ
In addition, the intermediate mapping onto the v-plane can be eliminated to provide a direct mapping from the f-plane to the z-plane. We combine Eqs. (10) and (15) to obtain 2 f þ iLf þ 1 z ¼ iR 2 ð32Þ f iLf þ 1 It is necessary to use the inverse mapping f(z) to compute fw given zw, for example. In this case, it is useful to retain the intermediate mapping onto the v-plane; Eqs. (10), (16), (17) represent the most useable form of f(z). When evaluating the inverse mapping, care must be taken to choose the proper branch of the function defined by (17). Flow nets can be constructed from the solution by contouring the real and imaginary parts of X(z), corresponding to the discharge potential and stream function, respectively. A flow net obtained from the analytical solution is plotted in Fig. 3 to illustrate the general features of the solution; the solid lines are contours of the stream function and the dashed lines are contours of the discharge potential. The problem includes a vertical barrier wall of length Ra = Rp/2, and a uniform flow at infinity of strength Q0 with an orientation of b = 70°.
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Fig. 3. Example flow field: a = 0.5p and b = 0.389p. Each well has a discharge rate of Q = 2Q0R. The solid lines originating at the wells and extending to the top of the figure are branch cuts in the stream function.
Three extraction wells of discharge Q = 2Q0R are located down-gradient from the wall. In the figure, the contour interval for both the dimensionless discharge potential, DU/(Q0R), and the dimensionless stream function, DW/(Q0R), is 0.5. 4. Effects of the barrier wall on groundwater flow We first consider the effects of a barrier wall on groundwater flow without the presence of extraction wells. The open barrier creates two stagnation points in the flow field—one on the up-gradient side of the wall and one on the down-gradient side; associated with each stagnation point is a region of low discharge. When the wall is curved, a relatively large region of low discharge is created on the interior (the R side) of the wall in comparison to the exterior (R+) side. This zone of low discharge may be used, by itself, to slow the movement of contaminants in the aquifer. Fig. 4a shows the flow net for the case of a wall of length Ra where a = p embedded in a uniform regional flow of orientation b = ±p/2. The figure may be used to analyze flow in either the positive or negative y-direction. Fig. 4b shows contours of theffi magnitude of the discharge vector, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jW j ¼ Q2x þ Q2y , made dimensionless by dividing by the regional discharge, Q0; this is equivalent to the dimensionless advective speed within the flow field. The plot quantifies the change in advection in the vicinity of the wall. Inside the wall, a large zone of reduced speed exists, while outside the wall a smaller region of
reduced speed is seen. The effect of radius of curvature of the wall on the speed is illustrated in Fig. 4c: a wall of equal length to that in Fig. 4b is shown, but with twice the radius of curvature. Inside the wall the 0.2 contour covers a smaller area; at the 0.6 level, the areas enclosed are similar. The zone of reduced speed on the exterior side of the wall has increased in comparison to the previous case. In Fig. 4b and c, the contours labelled 1.0 represent the curves along which the advective speed is the same with or without the barrier wall present. From Fig. 4, we can make the following conclusions about the placement of a wall with respect to the location of a contaminant plume. For a wall placed up-gradient of a plume or contaminant source (plume on the interior side of the wall, with regional flow in the positive y-direction), the rate of advective transport can be reduced significantly; in addition, near the wall and for a short distance down-gradient, the direction of advection is laterally inward which will tend to reduce the plume width. This is illustrated in Fig. 4a by the dashed streamlines plotted near the wall on the left side of the figure. For a wall placed down-gradient of the plume (plume on the interior of the wall with regional flow in the negative y-direction), the rate of advective transport is again reduced significantly, and the reduction is identical to the previous case. In this case, however, advection will cause the plume to widen as the streamlines spread to pass the barrier wall. This effect may not be desirable in many applications.
5. Combined effects of a barrier wall and wells We now consider the effects of vertical barriers on the capture zone of groundwater wells. In general, an open barrier placed up-gradient from a well will widen locally the capture zone of the well and the well will control hydraulically a larger region than would a well of the same discharge without the barrier present. In general, a smaller discharge may be required to contain a plume when a barrier is included in the design. This is the primary benefit of including an open vertical barrier wall in a remedial design; reduction of discharges needed to control a plume may have large impacts on the operating cost over the life of a remedial system, and may justify the expense of installing the barrier wall. The analytic solutions presented provide general tools that allow different barrier geometries, direction of regional flow, and multiple recharge and discharge wells to be investigated. Here we consider a limited number of cases that may be useful for the preliminary design of a plume-control system: we consider a semicircular impermeable barrier (a = p), a uniform flow aligned with the y-axis (b = p/2), and two separate locations of a discharge well, zw = iR and zw = 0. We
E.I. Anderson, E. Mesa / Advances in Water Resources 29 (2006) 89–98
95
0.8
0.8 1.0
1.0 0.6
1.0
1.0
0.6
0.4 0.4
0.2
0.2
1.0
0.4
0.6
1.0
0.6
1.0
1.0
0.8
a
b
c
Fig. 4. Flow past an impermeable barrier: (a) flow net for a = p; (b) contours of dimensionless speed for a = p; and, (c) contours of dimensionless speed for a = p/2. In (a), extra streamline (dashed) are plotted on the left side of the figure to highlight the direction of groundwater flow in the vicinity of the wall; the contour interval between the dashed streamlines are unequal, decreasing from left to right.
2.0
Q/(2Q o R) 1.0
2 1 0.5 0.25 0.125 0.0625
0.0
y/R
consider the capture zones for the two cases for varying discharges of the wells. Figs. 5 and 6 are dimensionless plots of the capture zone envelopes for the two cases with the well operating under different discharge rates. The capture zone envelopes were obtained by contouring the stream function and selecting the dividing streamline. The dimensionless well discharge, Q/ (2RQ0), varies in each case from 0.0625 to 2.0 with the discharge doubling between adjacent capture zones. To judge the effectiveness of the barrier wall at reducing the well discharge required to control a plume, we compare the present solution with the type curves presented by Javandel and Tsang [15] for uniform flow and a well; the type curves are used often in the design of hydraulic containment systems. From that solution we find that a well of discharge rate Q in a regional uniform flow of strength Q0 has an ultimate capture zone width of Q/Q0 at an infinite distance up-gradient from
-1.0
-2.0
-3.0 -2.0
-1.0
0.0
1.0
2.0
3.0
x/R Fig. 6. Capture zone envelopes for a down-gradient well located at z = 0.
2.0
Q/(2Q o R) 1.0
2 1 0.5 0.25 0.125 0.0625
y/R
0.0
-1.0
-2.0
-3.0 -2.0
-1.0
0.0
1.0
2.0
3.0
x/R Fig. 5. Capture zone envelopes for a down-gradient well located at z = iR.
the well, and a width of Q/(2Q0)—half the ultimate width—at the location of the well. Therefore, if a plume of characteristic width 2R is to be controlled by a downgradient well, the well will have to pump at a rate between 2Q0R and 4Q0R. From Figs. 5 and 6 we see that a width of 2R can be captured with a significantly smaller discharge when a barrier wall is included in the design. For example, we observe in both figures that a discharge of Q/ (2Q0R) = 0.25 controls a significantly increased region of flow than would be contained without the wall. This is only 12.5–25% of the discharge required to contain a similarly sized area without a wall. For site-specific plume geometries and varying plume lengths, the method of overlaying the capture zones on an outline of the plume [15] can be applied readily to determine the required discharge. The dimensionless plots of Figs. 5 and 6 provide a simple tool for comparing alternative
E.I. Anderson, E. Mesa / Advances in Water Resources 29 (2006) 89–98 6 4 2
Φ /(QoR)
designs and estimating potential savings in operating costs due to the installation of a vertical barrier wall. In application, an estimate of Q0 must first be made based on observed heads in the aquifer and estimates of the hydraulic conductivity. An additional design concern for vertical barrier walls is the effect of the wall on the hydraulic head in the aquifer; if the water table is near the ground surface, the wall may cause springs or ponded water to appear up-gradient. Figs. 7 and 8 provide plots of the dimensionless discharge potential along the y-axis for each of the discharges shown in Figs. 5 and 6; expressions (2) and (3) may be used with Figs. 7 and 8 to calculate hydraulic head using site-specific data. In Fig. 7 we see that the jump in the value of the discharge potential across the barrier wall is within the range of 3 to 4Q0R for the range of discharges presented. In Fig. 8, with the well closer to the wall, the jump in the discharge potential increases up to 5Q0R. We include in both figures the potential for the case of no well and no barrier wall for reference. For each plotted solution, a reference point was chosen far down-gradient from the barrier, at z = i10R with a potential value of 10Q0R. For application to specific sites, a different reference point may be chosen. For example, a point of known head along a nearby stream may be used. The plots provided in Figs. 7 and 8 may be modified by calculating a new value of the constant C (21) and shifting the presented curves vertically. Finally, we consider the effects of a down-gradient barrier—a common configuration in practice—on the capture zone of an extraction well. We again consider the specific case where a = p and the location of the well is at zw = 0. In this case however, the direction of regional flow is reversed by specifying b = p/2. Capture zones for this case are presented in Fig. 9; note that the direction of regional flow is in the minus y-direction
0
Q/(2QoR) -2
2 1 0.5 0.25 0.125 0.0625
-4 -6 -8
-10 -5
-4
-3
-2
-1
0
1
2
3
4
5
y/R
Fig. 8. Potentials along the line x = 0 for a well located at z = 0. The straight line represents the potential without a well or barrier wall present.
3.0
2.0
1.0
Q/(2QoR)
y/R
96
2 1 0.5 0.25 0.125 0.0625
0.0
-1.0
-2.0 -2.0
-1.0
0.0
1.0
2.0
3.0
x/R Fig. 9. Capture zone envelopes for an up-gradient well located at z = 0. Note that the direction of flow in the far-field is from the top of the figure to the bottom.
6
in this example. In contrast to the previous case of an up-gradient barrier and a down-gradient well, the down-gradient barrier does not appear to be effective at increasing the width of a capture zone for a given well discharge.
4
Φ /(QoR)
2 0
Q/(2QoR) -2
2 1 0.5 0.25 0.125 0.0625
-4 -6
6. Conclusions and discussion
-8 -10 -5
-4
-3
-2
-1
0
1
2
3
4
5
y/R Fig. 7. Potentials along the line x = 0 for a well located at z = iR. The straight line represents the potential without a well or barrier wall present.
We have developed explicit analytical expressions for the hydraulic head and the discharge vector for steady, shallow groundwater flow past an impermeable barrier in the presence of multiple discharge and recharge wells. We have used the solution to draw some general conclusions about the effectiveness of open vertical barrier walls on the hydraulic control of contaminant plumes
E.I. Anderson, E. Mesa / Advances in Water Resources 29 (2006) 89–98
for open barrier configurations that are used often in practice. By itself an open barrier creates an up- and downgradient zone of low discharges, that can be used to immobilize or retard the advection of a contaminant. If the barrier is placed up-gradient of a plume, advection acts locally to shrink the plume width; if the barrier is placed down-gradient, advection acts to widen the plume as the groundwater flows around the barrier wall. The combination of an up-gradient barrier wall and one or more down-gradient wells can be very effective at reducing the well discharge necessary for hydraulic control of a plume; the barrier acts to widen the capture zone of the discharge well in the vicinity of the wall. In contrast, the combination of a down-gradient wall with an up-gradient extraction well has little useful effect on the capture zone of the well; the discharge necessary to control hydraulically a given plume is not reduced by the presence of the barrier wall. We have also quantified the effects of a barrier on heads, discharges and the capture zone of wells for special cases; we have provided tools in the form of dimensionless graphs to aid engineers in estimating the improved hydraulic performance of a remedial system when including a barrier wall, and for estimating the effects of a barrier on site conditions. We anticipate that the design charts presented here will be useful in feasibility studies, preliminary design, and cost estimating of hydraulic control systems, particularly in early phases when alternative designs are being considered. The results provide insight as to the reduction in well discharges that can be expected by placement of a vertical barrier wall. In some cases detailed numerical modeling may not be necessary.
Fig. 10. An example flow field showing an injection/extraction well pair operating within the area of hydraulic containment.
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The analytical model is quite general and can be used directly for more detailed design considerations and evaluating alternative remedial schemes that include barrier walls. An example is shown in Fig. 10 consisting of an injection/extraction well pair down-gradient from an open barrier wall and within the capture zone of a low discharge well used for gradient control. The solution could be used, for example, to determine residence times for an injection/extraction well system [16].
References [1] USEPA. Abstracts of remediation case studies, vol. 3. EPA 542R-98-010; 1998. [2] USEPA. Investigation of slurry cutoff wall design and construction methods for containing hazardous wastes. EPA/600/52-87/ 063; 1987. [3] USEPA. Construction quality control and post construction performance verification for the Gilson road hazardous waste site cutoff wall. EPA/600/2-87/065; 1987. [4] USEPA. Slurry trench construction for pollution migration control. Municipal Environmental Research Laboratory, Office of Research and Development; EPA-540/2-84-001; 1984. [5] National Academy of Sciences. Alternatives for ground water cleanup. Wasington, DC: National Academy Press; 1994. [6] Bayer P, Finkel M, Teutsch G. Combining pump-and-treat and physical barriers for contaminant plume control. Ground Water 2004;42(6):856–67. [7] Campbell DL, Quintrell WN. Cleanup strategies of Rocky Mountain Arsenal. In: Proceedings, 6th national conference on the management of uncontrolled hazardous waste sites, HMCRI, November 4–6, Washington, DC; 1985. p. 36–42. [8] Christ JA, Goltz MN. Containment of groundwater contamination plumes: minimizing draw-down by aligning capture wells parallel to regional flow. J Hydrol 2004;286:52–68. [9] Christ JA, Goltz MN. Hydraulic containment: analytical and semi-analytical models for capture zone curve delineation. J Hydrol 2002;262:224–44. [10] Christ JA, Goltz MN, Huang J. Development and application of an analytical model to aid design and implementation of in situ remediation technologies. J Contam Hydrol 1999;37:295– 317. [11] Davy DE, editor. Geotechnical practices for waste disposal. London, UK: Chapman and Hall; 1993. [12] Erdmann JB. On capture width and capture zone gaps in multiplewell systems. Ground Water 2000;38(4):497–504. [13] Harr ME. Groundwater and seepage. New York, NY: Dover; 1962. [14] Hudak PF. Locating groundwater monitoring wells near cutoff walls. Adv Environ Res 2001;5:23–9. [15] Javandel I, Tsang CF. Capture-zone type curves: a tool for aquifer cleanup. Ground Water 1986;24(5):616–25. [16] Luo J, Kitanidis PK. Fluid residence times within a recirculation zone created by an extraction–injection well pair. J Hydrol 2004;295:149–62. [18] Meegoda JN, Ezeldin AS, Fang HY, Inyang HI. Waste immobilization technologies. Pract Periodical Hazard Toxic Radioact Waste Manage 2003;7(1):46–58. [19] Milne-Thomson LM. Theoretical aerodynamics. New York, NY: Dover; 1958. [20] Polubarinova-Kochina PY. Theory of ground water movement [de Wiest JMR, Trans.]. Princeton, NJ: Princetion University Press; 1962 [from Russian].
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[21] Rumer RR, Ryan ME. Barrier containment technologies for environmental remediation applications. New York, NY: John Wiley and Sons, Inc.; 1995. [22] Sato C, Braithwaite DA, Protopapas A, Stewart PP. Hazardous waste containment with a bentonite cutoff wall. In: Grouting, soil improvement and geosynthetics, vol. 2. ASCE Geotechnical Special Publication No. 3; 1992. p. 1298–310. [23] Shan C. An analytical solution for the capture zone of two arbitrarily located wells. J Hydrol 1999;222:123–8. [24] Strack ODL. Groundwater mechanics. Englewood Cliffs, NJ: Prentice Hall; 1989. [25] Vallentine HR. Applied hydrodynamics. London: Butterworths; 1969.
[26] Verruijt. Theory of groundwater flow. New York, NY: Gordon and Breach Science Publishers; 1970. [27] Mitchell JK, van Court WAN. Barrier design and installation: Walls and covers. In: Ward CH, Cherry JA, Scalf MR, editors. Subsurface restoration. Chelsea, MI: Ann Arbor Press, Inc.; 1989. p. 175–95 [chapter 11]. [28] Weaver KD, Coad RM, McIntosh KR. Grouting for hazardous waste site remediation at Necco Park, Niagara Falls, New York. In: Grouting, soil improvement and geosynthetics, vol. 2. ASCE Geotechnical Special Publication No. 3; 1992. p. 1332–43. [29] Yeo IW, Lee KK. Analytical solution for arbitrarily located multiwells in an anisotropic homegeneous confined aquifer. Water Resour Res 2003(5):39. Article number 1133.
The effects of vertical barrier walls on the hydraulic control of contaminated groundwater Erik I. Anderson *, Elizabeth Mesa Department of Civil and Environmental Engineering, University of South Carolina, 300 Main Street, Columbia, SC 29208, United States Received 14 December 2004 Available online 11 July 2005
Abstract We present explicit analytic solutions describing the hydraulic head and discharge vector for two-dimensional, steady groundwater flow past an impermeable barrier embedded in a regional flow field. We use the solution to investigate the effects of open vertical barriers on the flow field; in particular, we examine the hydraulic containment of contaminant plumes or source zones by combination of a vertical barrier wall and extraction wells. We quantify the local reduction in discharge rates due to the barrier wall and the local increase in the size of the capture zone of an extraction well near an open, up-gradient barrier. We find that the combination of an open vertical barrier with down-gradient extraction wells can be very effective in decreasing the well discharge rate necessary to control a contaminant plume or source area. Design charts are presented for quantifying the effects of the barrier wall on the hydraulic control of the groundwater flow field and for estimating the jump in head across a barrier. The charts are appropriate for use in the preliminary design and cost estimating of remedial systems, and for the design of dewatering systems. Ó 2005 Published by Elsevier Ltd. Keywords: Groundwater; Analytic; Remediation; Vertical barriers; Conformal mapping; Capture zone
1. Introduction Vertical barrier walls are often used in conjunction with groundwater extraction wells as components of waste containment or pump and treat systems. The purpose of the barrier wall in a remedial design is to restrict the flow of uncontaminated groundwater onto a site and to limit the flow of contaminants off site. Barriers can also provide groundwater control during excavation of wastes or contaminated soil. Vertical barrier walls are commonly constructed of grout, slurry, and plastic or steel sheetpiling. The exact configuration and effects of a barrier wall depend upon site-specific conditions. However, a variety of texts [21,5,11,7] and USEPA documents [2,4], discuss *
Corresponding author. Fax: +1 803 777 0670. E-mail addresses: [email protected] (E.I. Anderson), mesae @engr.sc.edu (E. Mesa). 0309-1708/$ - see front matter Ó 2005 Published by Elsevier Ltd. doi:10.1016/j.advwatres.2005.05.005
qualitatively the use and placement of vertical barriers for control of contaminated groundwater. Rumer and Ryan [21], for example, discuss the placement of circumferential barriers used to completely enclose a site, and open barriers used for redirecting groundwater flow. The circumferential barrier design has been applied to many landfills and hazardous waste sites throughout the United States, including the well-documented Gilson Road Superfund Site in Nashua, New Hampshire [1,3]. A four foot wide slurry wall extending up to 100 ft in depth and 4000 ft long encloses that site. Open barrier designs commonly include up-gradient barriers with down-gradient extraction wells or downgradient barriers with up-gradient extraction wells. The desired effect of the open barrier is to minimize the discharge rate of wells needed for hydraulic control of a contaminant plume [27, p. 183]. Down-gradient barriers have been used at the Solvent Recovery Services of New England Superfund Site in Southington,
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Connecticut [1], and the Rocky Mountain Arsenal Site in Colorado [7]. The Connecticut site includes a wall of steel sheetpiling extending to bedrock with 11 extraction wells located up-gradient of the wall; the Colorado site includes a grout curtain with up-gradient extraction wells and down-gradient injection wells. Sites where upgradient barriers have been constructed include Necco Park Landfill in Niagara Falls, New York [28] and the LaBounty Landfill Superfund Site in Iowa [22]. A single line grout curtain was placed along three sides of the Necco Park industrial landfill to reduce the volume of underflow being removed by down-gradient extraction wells; a bentonite wall was placed up-gradient of the LaBounty Landfill for the purpose of dewatering submerged wastes and to reduce the groundwater discharge beneath the site. Much information about the geotechnical design and construction of vertical barrier walls is available [4,2,11,18,21]. However, little design guidance is available for evaluating the impacts of a barrier wall on groundwater flow. As discussed by Mitchell and Van Court [27] proper modeling of groundwater flow patterns and the influences of barrier and extraction well interaction is necessary to evaluate the effectiveness of any design. This is commonly achieved by developing numerical models of the local groundwater flow field. Hudak [14] used a numerical model to investigate the effects of a cutoff wall on monitoring well networks near landfills. Recently, Bayer et al. [6] presented results of a numerical study of the effects of different barrier configurations on groundwater flow; a numerical framework is presented for evaluating the effects of barrier geometry on the advective control of contaminant plumes. It is often necessary during the preliminary design phase to draw some conclusions about the effectiveness of a barrier wall on site conditions without the time and expense required to develop a detailed numerical
model of the local groundwater flow field. The purpose of this paper is to examine quantitatively, and in a general setting, the combined effects of extraction wells and open vertical barriers. We develop explicit analytical expressions for the hydraulic head and the discharge vector for general well/barrier systems embedded in a regional groundwater flow field. The results provide a tool for engineers, in the form of dimensionless plots, to help assess the effects of a barrier wall on the hydraulic control of contaminant plumes. In particular, this work helps to quantify the effects of the barrier wall on local advection of a plume, the decrease in the discharge rates required of extraction wells to control a plume after placement of a vertical barrier wall, and the down-gradient lowering of the hydraulic head caused by the barrier wall. In addition, the analytic solution can be applied to more general cases with barriers and multiple extraction and injection wells for more detailed design considerations. We formulate the groundwater flow problem in terms of a complex potential and solve the problem using conformal mapping and the method of images. References on the application of conformal mapping to solve problems of groundwater flow include [20,13,26,24].
2. Problem description We consider steady, two-dimensional flow in the complex z-plane (z = x + iy), as illustrated in Fig. 1a. A vertical barrier wall, lying along a circular arc of radius R and length Ra, is embedded in the infinite domain; the barrier wall is assumed to be impermeable. A fully penetrating well of discharge rate Q [L3/T] is located at z = zw, and a uniform discharge of strength Q0 [L2/T] and oriented at an angle b to the real axis is specified at infinity. We define a complex potential as an analytic function of z as
Fig. 1. (a) The physical plane (z-plane), (b) the auxiliary plane (v-plane), and (c) the upper-half plane (f-plane). In the physical plane, a well of discharge rate Q is located at z = zw.
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XðzÞ ¼ U þ iW
ð1Þ 3
The imaginary part of the complex potential, W [L /T], is the stream function; the real part, U [L3/T], is the discharge potential, and is related to the hydraulic head, h [L], in the aquifer as follows: 1 U ¼ kHh kH 2 2
ðh P H Þ
ð2Þ
1 U ¼ kh2 ðh < H Þ ð3Þ 2 where k [L/T] is the hydraulic conductivity of the aquifer and h is measured with respect to the aquifer base; H is the distance between the aquifer base and an overlying confining layer. Using (2) and (3) the complex potential may be applied to combined shallow confined and shallow unconfined flow systems [24]. We further define the complex discharge function, W(z), as W ðzÞ ¼
dX ¼ Qx iQy dz
ð4Þ
where Qx and Qy [L2/T] are the x- and y-components of the discharge vector (the depth-integrated specific discharge vector). We present the boundary conditions for the problem illustrated in Fig. 1 in terms of the complex potential and complex discharge. First, the condition along the impermeable barrier is specified z ¼ Rþ expðihÞ aþp ap 6h6 JðXÞ ¼ 0 for 2 2 z ¼ R expðihÞ ð5Þ +
where R and R are used to distinguish between the two sides of the wall, and where J indicates the imaginary part of the complex function. The condition at infinity may be expressed as follows: lim W ¼ Q0 expðibÞ
z!1
ð6Þ
The behavior near the discharge well may be expressed in the form of an expansion of X about z = zw as Q lnðz zw Þ þ f ðzÞ X¼ 2p
field. Historically, solutions to problems of potential flow past impermeable barriers embedded in a uniform regional flow field have been developed for the study of lift on aerofoils; the impermeable circular arc used here to represent the barrier wall is a special case of a Joukowski foil (see for example, [19,25]). In hydrodynamics, however, point sources/sinks inside the domain have no physical meaning; the solution presented here focuses on the interaction of the impermeable barrier and groundwater wells.
3. Solution by conformal mapping The solution to the boundary-value problem described in the previous section may be obtained by conformal mapping and the method of images: we map the physical plane (z) onto the upper-half plane (f), with the impermeable slot mapped to the entire real axis; singularities are added to the flow field in the f-plane to represent flow features in the physical plane, and the method of images is applied to satisfy the condition along the impermeable barrier. 3.1. The physical plane, the auxiliary plane, and the upper-half plane We map the z-plane onto the f-plane in two steps, first transforming it onto the auxiliary v-plane. The physical, or z-plane is shown in Fig. 1a. The impermeable boundary is represented by a slot with the shape of a circular arc; points of interest along the impermeable slot are labelled 1 through 4. The circle of radius R in the z-plane is mapped to a straight line in the auxiliary v-plane by a bilinear transformation; the impermeable slot lying along an arc of the circle in the z-plane is therefore mapped to a straight slot in the v-plane. The v-plane is shown in Fig. 1b. The mapping of z onto v is given by v¼
ð7Þ
iðz þ iRÞ ðz iRÞ
ð9Þ
The mapping may be inverted to give z(v)
where f(z) is a function that is analytic at z = zw. Finally, we require a reference point to make the complex potential unique Uðz0 Þ ¼ U0
91
ð8Þ
where z0 is the coordinate of the reference point and U0 is a known value of the discharge potential at the reference point. Many researchers have applied complex analysis to groundwater flow with multiple combinations of extraction and injection wells for capture zone analysis and remedial design [15,23,10,12,9,29,8,16]; here we include explicitly the effects of a vertical barrier wall on the flow
z¼
iRðv þ iÞ ðv iÞ
ð10Þ
Common points in the two planes are labelled along the impermeable boundary. From (9) we find that the point at infinity in the z-plane is mapped to v = i; the location of the well in the z-plane, zw, is mapped to the point vw in the v-plane. Both points are labelled in Fig. 1b. The dimensionless length L of the impermeable slot in the v-plane may be evaluated from (9) as follows: L ¼ vðz4 Þ vðz2 Þ ¼
2Rðz4 z2 Þ ðz4 iRÞðz2 iRÞ
ð11Þ
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where z4 and z2 are the complex coordinates of the corner points of the impermeable barrier in the physical plane (Fig. 1) h a þ p i z4 ¼ R exp i ð12Þ 2 h a pi ð13Þ z2 ¼ R exp i 2 After some algebra we find L¼
2 sinða=2Þ 1 þ cosða=2Þ
ð14Þ
Finally, the f-plane is mapped onto the auxiliary v-plane by a mapping of constant argument [24]. The f-plane is illustrated in Fig. 1c. The mapping is given by v¼
Lf ðf2 þ 1Þ
ð15Þ
The impermeable slot in the z- and v-planes is now mapped to the entire real axis of the f-plane. This mapping may also be inverted: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi L L L þ 1 þ1 ð16Þ f¼ 2v 2v 2v We chose the sign of the square root term such that the domains correspond to those shown in Fig. 1 and the arguments such that the branch cut lies along the slot in the v-plane: L L 0 6 arg 1 6 2p p 6 arg þ1 6p 2v 2v ð17Þ The mappings (15) and (16) are valid for 0 6 a < 2p. The location in the f-plane that the point z = 1 maps to is of importance in evaluating the complex potential. We refer to the point f(z = 1) as r; the value of r may be obtained from (16) and (9), using (17) pffiffiffiffiffiffiffiffiffiffiffiffiffi i r ¼ ðL þ L2 þ 4Þ 2
conditions along the impermeable boundary. We will consider the two flow features in the physical plane— the well and the uniform flow—separately. To represent the well located in the physical plane at z = zw (7), we place a sink in the f-plane at f = fw. To satisfy the condition along the impermeable slot in the physical plane (5), we image the sink about the real axis of the f-plane. A source exists at infinity in the physical plane, balancing the sink at zw. To produce the proper behavior in f we must place a source of strength equal to the sink at f = r; that source must also be imaged about the real axis to satisfy the condition along the impermeable slot. The portion of the complex potential reflecting the well and satisfying the condition along the impermeable slot is Q ðf fw Þðf fw Þ ln Xwell ¼ ð19Þ Þ 2p ðf rÞðf r The uniform flow specified at infinity in the physical plane (6) is represented in f by a dipole placed at f = r. The dipole is represented graphically in Fig. 2 by a source()/sink(+) pair. The dipole must also be imaged about the real axis as shown in the figure. The portion of the complex potential reflecting the uniform flow and satisfying the condition along the impermeable slot is ic s e eic Xuni ¼ þ ð20Þ Þ 2p ðf rÞ ðf r For now we let the strength and orientation of the dipole, s and c, remain arbitrary; later we will relate them to the boundary condition in the physical plane (6). The final expression for the complex potential is the sum of Xwell and Xuni
ð18Þ
We note that r is purely imaginary. It may also be of interest to note that another pole in the mapping z(f) exists when the sign of the square root term in (18) is changed. This pole, however, does not lie on the proper branch of the mapping function f(z) defined by (17) and therefore lies outside the problem domain. 3.2. The complex potential The complex potential may now be evaluated by placing singularities in the f-plane to represent the well and uniform flow specified in the physical plane. For each singularity placed in the f-plane we apply the method of images about the real axis to satisfy
Fig. 2. Orientation and imaging of the dipole about the impermeable boundary in the f-plane.
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Q ðf fw Þðf fw Þ ln X¼ Þ 2p ðf rÞðf r ic ic s e e þ þ þC Þ 2p ðf rÞ ðf r
lim W ¼ f!r
ð21Þ
where C is a real constant that may be evaluated from a reference point (8). The complex potential (21) is analytic at f = 1, as may be seen by expanding the expression about infinity. Additional sources or sinks, representing recharge or discharge wells in the physical plane may be included by superposition in the f-plane. A general expression for N sources and sinks of strength Qn and located in the upper-half plane at fw,n is given by N X ðf fw;n Þðf fw;n Þ Qn X¼ ln Þ ðf rÞðf r 2p n¼1 ic s e eic þ þ þC ð22Þ Þ 2p ðf rÞ ðf r
4RLQ0 pr2 r2 1
ð28Þ
c ¼ ðb þ pÞ
ð29Þ
s¼ and
3.5. Solution The explicit analytical solutions for both the complex potential and complex discharge in the physical plane are now complete; the solutions are given by (21) and (24) along with the mappings and parameters given by (9), (14), (16) through (18), (28), and (29). We present the complex potential in a more direct form here, by combining Eqs. (21), (28), and (29): X¼
The details are straightforward and are omitted here for brevity. The resulting expression for the complex discharge is ( 2 2 ðf rÞ ðf r1 Þ 2f ðfw þ fw Þ W ¼ Q 2 ðf fw Þðf fw Þ 4pRLðf 1Þ ) 2ðf2 þ r2 Þ cos c þ 4i fr sin c s ð24Þ 2 Þ2 ðf rÞ ðf r 3.4. Strength and orientation of the dipole The strength and orientation, s and c, of the dipole placed at f = r are related to the strength and orientation, Q0 and b, of the uniform flow specified at infinity in the physical plane. To evaluate the relationship, we apply the boundary condition (6): lim W ¼ Q0 expðibÞ
z!1
ð25Þ
We apply this condition to the complex discharge function (24), noting that for z ! 1, f ! r and we obtain " # 2 sðr r1 Þ 4r2 expðicÞ lim W ¼ ð26Þ f!r 4pRLðr2 1Þ ðr r Þ2 Recalling that r is purely imaginary (18), we note that Þ ¼ 2r, and the expression simplifies as ðr r
sðr2 1Þ sðr2 1Þ expðicÞ ¼ exp½iðp þ cÞ 2 4pRLr 4pRLr2 ð27Þ
By equating (25) and (27) we obtain
3.3. The complex discharge The complex discharge function W(z) may be evaluated from the complex potential (21) and the mappings (10) and (15) as
dX dX dv dz ¼ W ¼ ð23Þ dz df df dv
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Q ðf fw Þðf fw Þ ln Þ 2p ðf rÞðf r iðbþpÞ 2 2RLQ0 r e eiðbþpÞ þ þ þC Þ r2 1 ðf rÞ ðf r
ð30Þ
where pffiffiffiffiffiffiffiffiffiffiffiffiffi i r ¼ ðL þ L2 þ 4Þ 2
and
L¼
2 sinða=2Þ 1 þ cosða=2Þ
ð31Þ
In addition, the intermediate mapping onto the v-plane can be eliminated to provide a direct mapping from the f-plane to the z-plane. We combine Eqs. (10) and (15) to obtain 2 f þ iLf þ 1 z ¼ iR 2 ð32Þ f iLf þ 1 It is necessary to use the inverse mapping f(z) to compute fw given zw, for example. In this case, it is useful to retain the intermediate mapping onto the v-plane; Eqs. (10), (16), (17) represent the most useable form of f(z). When evaluating the inverse mapping, care must be taken to choose the proper branch of the function defined by (17). Flow nets can be constructed from the solution by contouring the real and imaginary parts of X(z), corresponding to the discharge potential and stream function, respectively. A flow net obtained from the analytical solution is plotted in Fig. 3 to illustrate the general features of the solution; the solid lines are contours of the stream function and the dashed lines are contours of the discharge potential. The problem includes a vertical barrier wall of length Ra = Rp/2, and a uniform flow at infinity of strength Q0 with an orientation of b = 70°.
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Fig. 3. Example flow field: a = 0.5p and b = 0.389p. Each well has a discharge rate of Q = 2Q0R. The solid lines originating at the wells and extending to the top of the figure are branch cuts in the stream function.
Three extraction wells of discharge Q = 2Q0R are located down-gradient from the wall. In the figure, the contour interval for both the dimensionless discharge potential, DU/(Q0R), and the dimensionless stream function, DW/(Q0R), is 0.5. 4. Effects of the barrier wall on groundwater flow We first consider the effects of a barrier wall on groundwater flow without the presence of extraction wells. The open barrier creates two stagnation points in the flow field—one on the up-gradient side of the wall and one on the down-gradient side; associated with each stagnation point is a region of low discharge. When the wall is curved, a relatively large region of low discharge is created on the interior (the R side) of the wall in comparison to the exterior (R+) side. This zone of low discharge may be used, by itself, to slow the movement of contaminants in the aquifer. Fig. 4a shows the flow net for the case of a wall of length Ra where a = p embedded in a uniform regional flow of orientation b = ±p/2. The figure may be used to analyze flow in either the positive or negative y-direction. Fig. 4b shows contours of theffi magnitude of the discharge vector, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jW j ¼ Q2x þ Q2y , made dimensionless by dividing by the regional discharge, Q0; this is equivalent to the dimensionless advective speed within the flow field. The plot quantifies the change in advection in the vicinity of the wall. Inside the wall, a large zone of reduced speed exists, while outside the wall a smaller region of
reduced speed is seen. The effect of radius of curvature of the wall on the speed is illustrated in Fig. 4c: a wall of equal length to that in Fig. 4b is shown, but with twice the radius of curvature. Inside the wall the 0.2 contour covers a smaller area; at the 0.6 level, the areas enclosed are similar. The zone of reduced speed on the exterior side of the wall has increased in comparison to the previous case. In Fig. 4b and c, the contours labelled 1.0 represent the curves along which the advective speed is the same with or without the barrier wall present. From Fig. 4, we can make the following conclusions about the placement of a wall with respect to the location of a contaminant plume. For a wall placed up-gradient of a plume or contaminant source (plume on the interior side of the wall, with regional flow in the positive y-direction), the rate of advective transport can be reduced significantly; in addition, near the wall and for a short distance down-gradient, the direction of advection is laterally inward which will tend to reduce the plume width. This is illustrated in Fig. 4a by the dashed streamlines plotted near the wall on the left side of the figure. For a wall placed down-gradient of the plume (plume on the interior of the wall with regional flow in the negative y-direction), the rate of advective transport is again reduced significantly, and the reduction is identical to the previous case. In this case, however, advection will cause the plume to widen as the streamlines spread to pass the barrier wall. This effect may not be desirable in many applications.
5. Combined effects of a barrier wall and wells We now consider the effects of vertical barriers on the capture zone of groundwater wells. In general, an open barrier placed up-gradient from a well will widen locally the capture zone of the well and the well will control hydraulically a larger region than would a well of the same discharge without the barrier present. In general, a smaller discharge may be required to contain a plume when a barrier is included in the design. This is the primary benefit of including an open vertical barrier wall in a remedial design; reduction of discharges needed to control a plume may have large impacts on the operating cost over the life of a remedial system, and may justify the expense of installing the barrier wall. The analytic solutions presented provide general tools that allow different barrier geometries, direction of regional flow, and multiple recharge and discharge wells to be investigated. Here we consider a limited number of cases that may be useful for the preliminary design of a plume-control system: we consider a semicircular impermeable barrier (a = p), a uniform flow aligned with the y-axis (b = p/2), and two separate locations of a discharge well, zw = iR and zw = 0. We
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95
0.8
0.8 1.0
1.0 0.6
1.0
1.0
0.6
0.4 0.4
0.2
0.2
1.0
0.4
0.6
1.0
0.6
1.0
1.0
0.8
a
b
c
Fig. 4. Flow past an impermeable barrier: (a) flow net for a = p; (b) contours of dimensionless speed for a = p; and, (c) contours of dimensionless speed for a = p/2. In (a), extra streamline (dashed) are plotted on the left side of the figure to highlight the direction of groundwater flow in the vicinity of the wall; the contour interval between the dashed streamlines are unequal, decreasing from left to right.
2.0
Q/(2Q o R) 1.0
2 1 0.5 0.25 0.125 0.0625
0.0
y/R
consider the capture zones for the two cases for varying discharges of the wells. Figs. 5 and 6 are dimensionless plots of the capture zone envelopes for the two cases with the well operating under different discharge rates. The capture zone envelopes were obtained by contouring the stream function and selecting the dividing streamline. The dimensionless well discharge, Q/ (2RQ0), varies in each case from 0.0625 to 2.0 with the discharge doubling between adjacent capture zones. To judge the effectiveness of the barrier wall at reducing the well discharge required to control a plume, we compare the present solution with the type curves presented by Javandel and Tsang [15] for uniform flow and a well; the type curves are used often in the design of hydraulic containment systems. From that solution we find that a well of discharge rate Q in a regional uniform flow of strength Q0 has an ultimate capture zone width of Q/Q0 at an infinite distance up-gradient from
-1.0
-2.0
-3.0 -2.0
-1.0
0.0
1.0
2.0
3.0
x/R Fig. 6. Capture zone envelopes for a down-gradient well located at z = 0.
2.0
Q/(2Q o R) 1.0
2 1 0.5 0.25 0.125 0.0625
y/R
0.0
-1.0
-2.0
-3.0 -2.0
-1.0
0.0
1.0
2.0
3.0
x/R Fig. 5. Capture zone envelopes for a down-gradient well located at z = iR.
the well, and a width of Q/(2Q0)—half the ultimate width—at the location of the well. Therefore, if a plume of characteristic width 2R is to be controlled by a downgradient well, the well will have to pump at a rate between 2Q0R and 4Q0R. From Figs. 5 and 6 we see that a width of 2R can be captured with a significantly smaller discharge when a barrier wall is included in the design. For example, we observe in both figures that a discharge of Q/ (2Q0R) = 0.25 controls a significantly increased region of flow than would be contained without the wall. This is only 12.5–25% of the discharge required to contain a similarly sized area without a wall. For site-specific plume geometries and varying plume lengths, the method of overlaying the capture zones on an outline of the plume [15] can be applied readily to determine the required discharge. The dimensionless plots of Figs. 5 and 6 provide a simple tool for comparing alternative
E.I. Anderson, E. Mesa / Advances in Water Resources 29 (2006) 89–98 6 4 2
Φ /(QoR)
designs and estimating potential savings in operating costs due to the installation of a vertical barrier wall. In application, an estimate of Q0 must first be made based on observed heads in the aquifer and estimates of the hydraulic conductivity. An additional design concern for vertical barrier walls is the effect of the wall on the hydraulic head in the aquifer; if the water table is near the ground surface, the wall may cause springs or ponded water to appear up-gradient. Figs. 7 and 8 provide plots of the dimensionless discharge potential along the y-axis for each of the discharges shown in Figs. 5 and 6; expressions (2) and (3) may be used with Figs. 7 and 8 to calculate hydraulic head using site-specific data. In Fig. 7 we see that the jump in the value of the discharge potential across the barrier wall is within the range of 3 to 4Q0R for the range of discharges presented. In Fig. 8, with the well closer to the wall, the jump in the discharge potential increases up to 5Q0R. We include in both figures the potential for the case of no well and no barrier wall for reference. For each plotted solution, a reference point was chosen far down-gradient from the barrier, at z = i10R with a potential value of 10Q0R. For application to specific sites, a different reference point may be chosen. For example, a point of known head along a nearby stream may be used. The plots provided in Figs. 7 and 8 may be modified by calculating a new value of the constant C (21) and shifting the presented curves vertically. Finally, we consider the effects of a down-gradient barrier—a common configuration in practice—on the capture zone of an extraction well. We again consider the specific case where a = p and the location of the well is at zw = 0. In this case however, the direction of regional flow is reversed by specifying b = p/2. Capture zones for this case are presented in Fig. 9; note that the direction of regional flow is in the minus y-direction
0
Q/(2QoR) -2
2 1 0.5 0.25 0.125 0.0625
-4 -6 -8
-10 -5
-4
-3
-2
-1
0
1
2
3
4
5
y/R
Fig. 8. Potentials along the line x = 0 for a well located at z = 0. The straight line represents the potential without a well or barrier wall present.
3.0
2.0
1.0
Q/(2QoR)
y/R
96
2 1 0.5 0.25 0.125 0.0625
0.0
-1.0
-2.0 -2.0
-1.0
0.0
1.0
2.0
3.0
x/R Fig. 9. Capture zone envelopes for an up-gradient well located at z = 0. Note that the direction of flow in the far-field is from the top of the figure to the bottom.
6
in this example. In contrast to the previous case of an up-gradient barrier and a down-gradient well, the down-gradient barrier does not appear to be effective at increasing the width of a capture zone for a given well discharge.
4
Φ /(QoR)
2 0
Q/(2QoR) -2
2 1 0.5 0.25 0.125 0.0625
-4 -6
6. Conclusions and discussion
-8 -10 -5
-4
-3
-2
-1
0
1
2
3
4
5
y/R Fig. 7. Potentials along the line x = 0 for a well located at z = iR. The straight line represents the potential without a well or barrier wall present.
We have developed explicit analytical expressions for the hydraulic head and the discharge vector for steady, shallow groundwater flow past an impermeable barrier in the presence of multiple discharge and recharge wells. We have used the solution to draw some general conclusions about the effectiveness of open vertical barrier walls on the hydraulic control of contaminant plumes
E.I. Anderson, E. Mesa / Advances in Water Resources 29 (2006) 89–98
for open barrier configurations that are used often in practice. By itself an open barrier creates an up- and downgradient zone of low discharges, that can be used to immobilize or retard the advection of a contaminant. If the barrier is placed up-gradient of a plume, advection acts locally to shrink the plume width; if the barrier is placed down-gradient, advection acts to widen the plume as the groundwater flows around the barrier wall. The combination of an up-gradient barrier wall and one or more down-gradient wells can be very effective at reducing the well discharge necessary for hydraulic control of a plume; the barrier acts to widen the capture zone of the discharge well in the vicinity of the wall. In contrast, the combination of a down-gradient wall with an up-gradient extraction well has little useful effect on the capture zone of the well; the discharge necessary to control hydraulically a given plume is not reduced by the presence of the barrier wall. We have also quantified the effects of a barrier on heads, discharges and the capture zone of wells for special cases; we have provided tools in the form of dimensionless graphs to aid engineers in estimating the improved hydraulic performance of a remedial system when including a barrier wall, and for estimating the effects of a barrier on site conditions. We anticipate that the design charts presented here will be useful in feasibility studies, preliminary design, and cost estimating of hydraulic control systems, particularly in early phases when alternative designs are being considered. The results provide insight as to the reduction in well discharges that can be expected by placement of a vertical barrier wall. In some cases detailed numerical modeling may not be necessary.
Fig. 10. An example flow field showing an injection/extraction well pair operating within the area of hydraulic containment.
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The analytical model is quite general and can be used directly for more detailed design considerations and evaluating alternative remedial schemes that include barrier walls. An example is shown in Fig. 10 consisting of an injection/extraction well pair down-gradient from an open barrier wall and within the capture zone of a low discharge well used for gradient control. The solution could be used, for example, to determine residence times for an injection/extraction well system [16].
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