An analytical approach to capture zone delineation for a well near a stream with a leaky layer

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1 Hydrological Sciences Journal ISSN: (Print) (Online) Journal homepage: An analytical approach to capture zone delineation for a well near a stream with a leaky layer Mahdi Asadi-Aghbolaghi, Gholam Reza Rakhshandehroo & Mazda Kompani- Zare To cite this article: Mahdi Asadi-Aghbolaghi, Gholam Reza Rakhshandehroo & Mazda Kompani- Zare (23) An analytical approach to capture zone delineation for a well near a stream with a leaky layer, Hydrological Sciences Journal, 58:8, , DOI:.8/ To link to this article: Published online: 3 Oct 23. Submit your article to this journal Article views: 27 View related articles Citing articles: 4 View citing articles Full Terms & Conditions of access and use can be found at

2 Hydrological Sciences Journal Journal des Sciences Hydrologiques, 58 (8) An analytical approach to capture zone delineation for a well near a stream with a leaky layer Mahdi Asadi-Aghbolaghi, Gholam Reza Rakhshandehroo 2 and Mazda Kompani-Zare 3,4 Faculty of Agriculture, Water Engineering Department, Shahrekord University, Shahrekord, Iran 2 Civil Engineering Department, School of Engineering, Shiraz University, Shiraz, Iran rakhshan@shirazu.ac.ir 3 Department of Desert Region Management, School of Agriculture, Shiraz University, Shiraz, Iran 4 Department of Geography and Environmental Management, University of Waterloo, Ontario N2L 3G, Canada Received 7 October 2; accepted 28 January 23; open for discussion until April 24 Editor D. Koutsoyiannis; Associate editor A. Koussis Citation Asadi-Aghbolaghi, M., Rakhshandehroo, G.R., and Kompani-Zare, M., 23. An analytical approach to capture zone delineation for a well near a stream with a leaky layer. Hydrological Sciences Journal, 58 (8), Abstract An analytical solution is developed to delineate the capture zone of a pumping well in an aquifer with a regional flow perpendicular to a stream, assuming a leaky layer between the stream and the aquifer. Three different scenarios are considered for different pumping rates. At low pumping rates, the capture zone boundary will be completely contained in the aquifer. At medium pumping rates, the tip of the capture zone boundary will intrude into the leaky layer. Under these two scenarios, all the pumped water is supplied from the regional groundwater flow in the aquifer. At high pumping rates, however, the capture zone boundary intersects the stream and pumped water is supplied from both the aquifer and the stream. The two critical pumping rates which separate these three scenarios, as well as the proportion of pumped water from the stream and the aquifer, are determined for different hydraulic settings. Key words groundwater regional flow; stream boundary; capture zone delineation; complex potential theory; leaky layer Une approche analytique pour délimiter la zone de captage d un puits près d une rivière sur couche perméable Résumé Nous avons développé une solution analytique pour délimiter la zone de captage d un puits de pompage dans un aquifère avec un écoulement régional perpendiculaire à une rivière, en supposant l existence une couche perméable entre la rivière et la nappe phréatique. Trois scénarios différents ont été envisagés pour différents débits de pompage. Pour de faibles débits de pompage, la limite de la zone de captage sera entièrement contenue dans l aquifère. Pour des débits de pompage moyens, la pointe de la limite de la zone de captage atteint la couche perméable. Dans ces deux scénarios, toute l eau pompée est fournie par l écoulement des eaux souterraines dans l aquifère régional. Pour des débits de pompage élevés, cependant, la limite de la zone de captage recoupe la rivière et l eau pompée est fournie à la fois par l aquifère et par la rivière Les deux débits de pompage critiques séparant ces trois scénarios et la proportion de l eau pompée dans la rivière et l aquifère ont été déterminés pour différents paramètres hydrauliques. Mots clefs écoulement régional des eaux souterraines; limite de cours d eau; délimitation de la zone de captage; zone aride; théorie du potentiel complexe; couche perméable INTRODUCTION Capture zone delineation for pumping wells in aquifers has been studied by many researchers (Grubb 993, Fienen et al. 25, Kompani-Zare et al. 25, Intaraprasong and Zhan 27). This topic may be considered as a critical subject in water resources management from different perspectives. From an environmental perspective, for example, the need for groundwater pollution prevention in areas with sources of heavy contamination requires a relatively accurate mathematical model of the capture zone 23 IAHS Press

3 84 Mahdi Asadi-Aghbolaghi et al. geometry in the affected aquifer. In the case where the capture zone of a pumping well interferes with surface water bodies in and around the aquifer, the source of the pumped water would be a vital subject from a resource management perspective. Analytical delineation of capture zones dates back to 946, when Muskat performed a thorough and detailed analysis of the problem using potential theory in a complex domain. Then, Dacosta and Bennett (96) delineated the capture zone analytically for two recharge and discharge wells with different regional flow angles with respect to the wells. Since then, many research studies have developed different mathematical schemes to investigate capture zone properties (e.g. Javandel and Tsang 986,Shafer 987, 996, Grubb 993, Faybishenko et al. 995, Shan 999, Christ and Goltz 22, 24). In particular, Javandel and Tsang (986), Faybishenko et al. (995), Shan (999), and Christ and Goltz (22, 24) delineated the capture zone for multiple vertical pumping wells placed at different angles to the groundwater regional flow. Different types of hydraulic boundaries influence groundwater flow in the aquifer, and more specifically, the pumping well capture zone. Theis (94) was among early researchers who considered a fully penetrating vertical well in a confined aquifer perfectly connected to a stream on one side. He utilized the concept of image well theory to incorporate groundwater stream interaction in his analytical derivations. Later, Glover and Balmer (954) generalized Theis s approach and obtained solutions to the problem based on a series of idealistic assumptions. Their method of solution was improved by Jenkins (968), who gave dimensionless tables and curves leading to a scheme for management of the water resources. Since then, the method of images has been used by researchers to satisfy various types of boundary conditions, such as no-flow and constant head boundaries (Zhan 999, Zhan and Cao 2, Kompani-Zare et al. 25). Groundwater flow domains often contain boundaries or inhomogeneities that act as leaky barriers to flow. Typical examples include problems of groundwater surface water interaction, in which an aquifer is separated from a river by a layer of silt (Anderson 23). In most real cases, a thin layer with low hydraulic conductivity along the streambed separates the aquifer from the stream (Anderson 2, 23). Mathematically, finding an analytical solution for this case is more complicated than the case with perfect hydraulic connection (Anderson 2). Hantush (965) developed an analytical solution for drawdown of a vertical pumping well near a stream with a semi-pervious layer between the stream and aquifer. His approach contained some limiting assumptions in the definition of the leaky layer. Anderson (2) obtained an analytical solution to the problem in a complex domain by dropping Hantush s assumption. Recent researchers who have investigated the interaction between groundwater and streams through assumption of a leaky layer between them include Hantush (25), Intaraprasong and Zhan (27), Ha et al. (27), Rushton (27), and Intaraprasong and Zhan (29). Some researchers have studied groundwater flow of a pumping well near a partially penetrating stream (Hunt 999, Zlotnik and Huang 999, Butler et al. 2, Bakker and Anderson 23). They ignored the vertical component of velocity and applied the Dupuit approximation. Hunt (999) developed an analytical solution to transient groundwater flow of a pumping well near a partially penetrating stream with a clogged streambed. He considered both sides of the stream in his solution. Bakker and Anderson (23) presented an explicit analytical solution for steady, two-dimensional (2D) groundwater flow to a well near a leaky streambed that penetrates the aquifer partially. They assumed that leakage from the stream is approximated as occurring along the centreline of the stream. In their setting, the problem domain is infinite and pumping on one side of the stream may induce flow to the other side. Analytical solutions have been presented for different engineering applications, with governing equations similar to groundwater flow equations (Cole and Yen 2, Lin2). Lin (2) presented analytical solutions of heat conduction for isotropic media with finite dimensions. He utilized a Fourier transform together with the image method to find solutions to a composite-layer medium. He assumed two different boundary types: thermal isolation (Neumann) and isothermal (Dirichlet), and expressed explicit full field solutions as simple closed-forms, which may be easily used in other engineering applications too. In this study, a steady-state analytical solution is developed to delineate capture zone for a fully penetrating well in an aquifer with regional flow perpendicular to a stream, on the assumption that there is a leaky layer between the stream and the aquifer. Complex potential theory and superposition law are used to obtain the analytical solutions to the problem. Three different scenarios are considered for different

4 An analytical approach to capture zone delineation for a well near a stream with a leaky layer 85 pumping rates. At low pumping rates, the capture zone boundary will be completely contained in the aquifer. At medium pumping rates, however, the tip of the capture zone boundary will intrude into the leaky layer. Under these first two scenarios, all the pumped water is supplied from the regional groundwater flow in the aquifer. However, at high pumping rates, the capture zone boundary intersects the stream and the pumped water is supplied from both the aquifer and the stream. The two critical pumping rates separating these three scenarios the proportion of pumped water from the stream and the aquifer, and the interval during which the water is gaining from the stream, in the third scenario are determined for different hydraulic settings. MODEL DESCRIPTION Figure illustrates a schematic plan view of a vertical pumping well in an aquifer that is separated from a stream boundary by a leaky layer. The origin of the coordinate system is located at the interface of the leaky layer and the aquifer. The stream and the leaky layer fully penetrate the aquifer and are parallel to the x-axis. The well is also fully penetrating and is located at (, a). The thickness of the leaky layer, h, is constant and groundwater regional flow, q, is perpendicular to the stream axis. Complex potential theory is adopted to investigate 2D steady-state groundwater flow in the aquifer and the leaky layer. The complex variable, z,isdefined as z = x + iy,wherei =, for the coordinate system shown in Fig.. The entire domain is subdivided into two domains, D and D, corresponding to two Fig.. Schematic view of a fully penetrating pumping well in an aquifer that is separated from a stream by a leaky layer. media, the aquifer and the leaky layer, with two different hydraulic conductivities, K and K.Twocomplex potentials for the domains are introduced as: Ω = Φ + iψ in D Ω = Φ + iψ in D (a) (b) where Ω refers to the complex potential, Φ to the potential function, and Ψ to the stream function, and superscript denotes the same parameters for the leaky layer. In a confined aquifer the potential function is associated with hydraulic head, φ, viaφ = KBφ + C C,inwhichB is the aquifer thickness and C C is an arbitrary constant. For an unconfined aquifer, Φ = /2Kφ 2 + C u,wherec u is another arbitrary constant (Strack 989). Along the stream boundary, hydraulic head is constant and, therefore, the potential function Φ, should be constant there. Along the leaky layer and aquifer interface, y =, named the inhomogeneity boundary, the hydraulic head and normal component of flow should be constant, and this can be expressed in terms of potential and stream functions as (Anderson 2): Φ = K K Φ and Ψ = Ψ at y = (2) SOLUTION TO THE PROBLEM An analytical solution is developed based on superposition of complex potentials for two components of the system to delineate the capture zone of a pumping well near a stream with a leaky layer. The first component is the well without any regional groundwater flow in the aquifer and the second is the regional flow. Both components have a stream with a leaky layer at their boundaries. The first component was presented by Anderson (2), who utilized three basic solutions to determine the appropriate forms of the images across each boundary. The first basic solution contains a drain near a horizontal equipotential, and the basic solutions (2) and (3) have been developed from a single classical solution consisting of a drain in an aquifer with two hydraulic conductivities (Polubarinova- Kochina 962). Anderson (2) establisheda pattern in the imaging process, and the final solution may be expressed as follows:

5 86 Mahdi Asadi-Aghbolaghi et al. Ω = ( κ) ln 2πB κ n z + ai + 2ihn z ai 2ih (n + ) + Φ (3a) Ω = ( κ) ln 2πB κ n z + ai + 2ihn z ai 2ih (n + ) + qiz + Φ (6a) Ω = ln (z + ai) + κ ln (z ai) 2πB 2πB ( κ 2) 2πB κ n ln [z ai 2ih (n + )] + Φ Φ = K Bφ for confined aquifer Φ = KBφ for confined aquifer Φ = / 2K φ 2 for unconfined aquifer Φ = / 2Kφ 2 for unconfined aquifer (3b) (4a) (4b) (4c) (4d) where φ is the constant hydraulic head evaluated from the boundary condition, κ = (K K )/(K + K ) = ( β)/( + β) and is the discharge of the pumping well. Anderson (999) demonstrates that expressions (3a) and (3b) satisfy the boundary conditions exactly and the infinite series appearing in (3a) and (3b) converge. If hydraulic head at the stream boundary is set equal to zero, then in equations (3a) and (3b) Φ = Φ =. For the second component, the regional groundwater flow, q, passes through both the aquifer and the leaky layer perpendicular to the stream boundary. Applying equation (2) across the inhomogeneity boundary, the complex potentials are obtained as: Ω = qiz + Φ Ω = qiz + Φ (5a) (5b) Again, if hydraulic head at the stream boundary (y = h) is set equal to zero, then in equations (5a) and (5b) one obtains Φ = qh, and Φ = K K qh, respectively. Combining equations (3) and (5) yields the final solution to the problem as: Ω = ln (z + ai) + κ ln (z ai) 2πB 2πB ( κ 2) 2πB κ n ln [z ai 2ih (n + )] + qiz + Φ (6b) where Φ and Φ are evaluated from boundary conditions. Following arguments presented for equations (3) and (5), if hydraulic head at the stream boundary is zero, then constants in equations (6a) and (6b) would take the forms Φ = qh and Φ = K K qh, respectively. These equations may be expressed in dimensionless form as: Ω D = ( κ) D κ n ln z D + i + 2ih D n z D i 2ih D (n + ) + iz D + Φ D Ω D = D ln (z D + i) + κ D ln (z D i) ( κ 2) D κ n ln [z D i 2ih D (n + )] + iz D + Φ D (7a) (7b) where the subscript D denotes the dimensionless terms and dimensionless parameters are defined as Ω D =Ω/aq, D = /2πBaq, = x/a, y D = y/a, z D = z/a, h D = h/a. The branch of functions Ω(z) and Ω (z) (showninfig. 3(a) (c)) isfixedinthezplane with the cut along the imaginary axis from the point i to the point ih D. The real and imaginary parts of Ω D are Φ D and Ψ D, respectively. These parts for domains D and D would be:

6 An analytical approach to capture zone delineation for a well near a stream with a leaky layer 87 Φ D = ( k) D κ n x 2 D + (y D + + 2h D n) 2 ln x 2 D + [y D 2h D (n + )] 2 y D + Φ D Ψ D = ( κ) D κ n [ tan y D + + 2h D n tan y D 2h D (n + ) Φ D = D ln [ 2 + (y D + ) 2] + κ D ln [ 2 + (y D ) 2] ( κ 2) D 2h D (n + )] 2} y D + Φ D ] + κ n ln { 2 + [y D Ψ D = D tan y D + + κ D tan y D ( κ 2) D y D 2h D (n + ) Stagnation point κ n tan + (8a) (8b) (8c) (8d) The stagnation point is a key point in delineating the capture zone of a well. The stagnation point exists on the streamline forming the capture zone boundary in a flow domain (Strack 989, Kompani-Zare et al. 25). In the vicinity of this point, the streamlines turn from parallel to the capture zone boundary to perpendicular to it. In the present case of a uniform flow and a single well, only one stagnation point exists on the capture zone boundary at low pumping rates. At the stagnation point, the flow velocity and hydraulic gradient are zero, i.e. Ψ D / = Ψ D / y D =. Differentiating the stream functions given by equations (8b) and (8d), with respect to and y D yields: Ψ D = ( κ) D κ n { y D + + 2h D n x 2 D + (y D + + 2h D n) 2 } y D 2h D (n + ) + x 2 D + [y D 2h D (n + )] 2 + Ψ D y D = ( κ) D κ n { x 2 D + (y D + + 2h D n) 2 } x 2 D + [y D 2h D (n + )] 2 Ψ D y D + = D x 2 D + (y D + ) 2 κ D y D 2 + (y D ) 2 + ( κ 2) D κ n y D 2h D (n + ) 2 + [y D 2h D (n + )] 2 + Ψ D y D = D 2 + (y D + ) 2 + κ D 2 + (y D ) 2 ( κ 2) D κ n 2 + [y D 2h D (n + )] 2 (9a) (9b) (9c) (9d) Based on the location of the stagnation point(s), equations (9a) to (9d) may be set equal to zero to find the stagnation point(s) coordinates, s and y Ds. However, the equations contain series which make explicit determination of their solution impossible. Hence, a numerical method is adopted to find the stagnation point coordinates. It is worth noting that the stagnation point may be located in the aquifer, in the leaky layer, or on the stream boundary owing to different pumping rate values.

7 88 Mahdi Asadi-Aghbolaghi et al. CAPTURE ZONE DELINEATION The capture zone boundary can be determined based on the stagnation point location. For the case in which a regional flow perpendicular to the stream is considered the capture zone boundary is symmetric relative to y-axis. Figure 2 depicts the capture zone boundary for the hydraulic configuration shown on Fig., where a low pumping rate of D =.4 is applied. In Fig. 2, h D =.4, β = K /K =.5, Φ = Φ = and the stagnation point is located at s = and y Ds =.547. Note that basic configurations are considered in this research where the well pumping rate, D, together with the thickness of leaky layer, h D,andthe ratio of hydraulic conductivity in the aquifer to that in the leaky layer, β, vary. However, parameters such as the aquifer hydraulic conductivity, regional flow rate and distance between the well and stream are held constant. Three scenarios for capture zone configuration For different D values, the capture zone configurations create three scenarios. At low pumping rates, water is withdrawn solely from the aquifer and the capture zone boundary is completely contained in the aquifer (Fig. 3(a)). For this scenario, the stream gains water from the aquifer, or the regional flow, along its entire length, the tip of the capture zone, or the stagnation point, is located inside the aquifer, and equations (9c) and (9d) may be set equal to zero to find its coordinates. From equation (9d) one gets y D Stream Boundary Branch Cut Leaky Layer Inhomogeneity Boundary Pumping Well Capture Zone Boundary Fig. 3. Streamlines (solid), iso-potential lines (dashed), capture zone boundary (solid bold) and branch cut (dashed bold) for h D =.4, β =.5, ΦD = Φ D =, and for different dimensionless pumping rates: (a) D =.4, (b) D =.8, and (c) D = Fig. 2. Capture zone boundary for D =.4, h D =.4 and β =.5. s =, meaning that the stagnation point is located on the symmetry axis. Substituting = inequation (9c), the ordinate of the stagnation point would be obtained. This scenario occurs when D is between zero and a critical value, DC,where DC may be viewed as the pumping rate at which the stagnation

8 An analytical approach to capture zone delineation for a well near a stream with a leaky layer 89 point would be located on the inhomogeneity boundary. As D increases, however, the capture zone boundary crosses the leaky layer, the stagnation point would be located inside the layer (Fig. 3(b)), and equations (9a) and (9b) may be used to obtain its coordinates. Setting equation (9b) equal to zero, is obtained to be zero, meaning that the stagnation point is again located on the symmetry axis. Substituting = in equation (9a) and setting it equal to zero, the ordinate of the stagnation point can be obtained. In this scenario, the stream water does not enter the well and pumped water is still supplied solely by the aquifer or the regional flow. The second scenario ends when D increases and reaches a value DC2,at which the capture zone touches the stream at only one point. This touched point with (,h D ) coordinates is the stagnation of the capture zone. The third scenario is for the case in which D is greater than DC2. In this case, the capture zone boundary crosses the stream at two distinct points and the pumped water from the well is supplied by both the aquifer and the stream (Fig. 3(c)). In this scenario, there would be two stagnation points at the stream boundary, and equations (9a) and (9b) may be used to obtain their coordinates. By setting equation (9b) equal to zero, one may obtain y D = h D, and by substituting y D = h D in equation (9a) and setting it equal to zero, two symmetrical abscissas are obtained for s of the stagnation points. For the third scenario, it is important to determine the proportions of pumping water gain from the stream. For good accuracy, Fig. 3(a) (c) has been drawn using n = in the series. It is worth noting that, from equations (8a) (8d) and Fig. 3(a) (c), one may conclude that far away from the well, iso-potential lines would be parallel to the stream boundary, groundwater flow direction would be perpendicular to it, and the specific discharge would be q. First critical pumping rate, DC As mentioned before, at D = DC the stagnation point is located on the inhomogeneity boundary. Therefore, to find DC asafunctionofh D and κ, the stagnation point should be placed at the origin of coordinates. / By substituting = in equation (9d), Ψ D yd would equate zero for all values of y D,h D and β. By setting = y D = in equation (9c) and making it equal to zero, the following equation will be determined: Ψ D = DC + κ DC ( κ 2) Therefore, DC = DC κ n + 2h D (n + ) + = κ + ( κ 2) κ n +2h D (n+) () () It must be noted that DC depends not only on the aquifer hydraulic conductivity, but also on the thickness and conductivity of the leaky layer, via h D and κ, respectively. In general, larger h or smaller K would result in greater DC, which make sense physically. In other words, higher pumping rates are required to place the stagnation point on the boundary when a less conductive or thicker leaky layer exists. Figure 4 shows DC as a function of β for different values of h D. As depicted in Fig. 4, DC increases significantly as β decreases and the changes are greater for smaller β values. For a given β value, DC increases for larger h D values. For the very small values of β, the leaky layer acts as a barrier for groundwater to reach the stream as a constant head boundary; a phenomenon that requires higher DC values. For the special case in which K K, β thenκ, meaning that the leaky layer no longer exists or, in other words, is considered as a part of the aquifer. Then, DC would only depend on h D : DC = + 2h D 2 + 2h D (2) Second critical pumping rate, DC2 As mentioned previously, at D < DC2 the capture zone boundary does not cross the stream, and at D > DC2 the capture zone boundary crosses the stream boundary. At D = DC2, the stagnation point lies on the stream boundary and, hence, s = andy Ds = h D. Therefore, Ψ D/ xd,and Ψ D/ yd should be equal to zero at (,h D )for D = DC2.However, Ψ D/ yd = at = for all values of β and h D (equation (9b)). Therefore, DC2 should be obtained by equation (9a) for = andy D = h D. Substituting coordinate (,h D ) in equation (9a):

9 82 Mahdi Asadi-Aghbolaghi et al. DC h D =. h D =.2 h D =.3 h D =.4.. β Fig. 4. DC vs β for different values of h D. ( κ) DC2 2κ n + h D + 2h D n + = (3) DC2 = + h D 2 (5) Setting h D in equation (5), one would get DC2 = /2. This pumping rate has been derived by researchers such as Strack (989), who did not consider the leaky layer at all. Water pumped from stream for D > DC2 When the pumping rate is greater than DC2, a certain portion of pumped water would be supplied by the stream, DR (to be determined). From water resources management and contaminant transport view points, it is vital to know what portion of the pumped water is actually withdrawn from the stream (Muskat 946). To calculate DR, it is essential to locate the two stagnation points on the stream boundary (Fig. 3(c)). For this purpose, equation (9a) should be set equal to zero at y D = h D, the stream boundary: and DC2 = ( κ) 2κ n +h D +2h D n (4) Ψ D = ( κ) D 2κ n + h D (2n + ) S 2 + [ + h D (2n + )] 2 + = (6) Figure 5 depicts DC2 with respect to β for different values of h D. As expected, DC2 is always greater than DC for all values of β and h D.However,comparing Figs 4 and 5 reveals that both DC and DC2 have similar increasing trends as β decreases and/or h D increases. For the special case in which K K, β and κ, then: DC h D =. h D =.2 h D =.3 h D = β Fig. 5. DC2 vs β for different values of h D. As seen in the equation, s depends upon D, β (via κ) andh D. Unfortunately, it is not possible to find an explicit analytical solution for equation (6) and numerical methods are implemented to solve it. As shown in Fig. 3(c), there are two stagnation points on the stream boundary, symmetrically located on both sides of the y-axis, with 2s distance between them. Water enters the aquifer from the stream along the interval between two stagnation points. The problem has a symmetric axis passing through the pumping well and y-axis. Monotonically increasing curves of s vs D for different values of h D and β are shown in Fig. 6(a) (d). In these figures, curves cross the horizontal axis, s =, where D = DC2. In these curves, for D values less than DC2, s also equals zero. As can be seen in Fig. 6(a) (d), for D > DC2 by increasing D,s also increases. However, the rate of increase in s with D is different at different β and h D values. For smaller values of D, the slope of the curve is increasing and the maximum slope occurs at s =. With increase in D, the curve slope decreases until it approaches the linear condition at high D values. The linear parts of the curves with smaller values of β that have higher slopes occur sooner. Also, the

10 An analytical approach to capture zone delineation for a well near a stream with a leaky layer 82 (a) s (c) s β =. β =.5 β =. β =.5 β = h D = D β =. β =.5 β =. β =.5 β = h D = D (b) s (d) s β =. β =.5 β =. β =.5 β = h D = D β =. β =.5 β =. β =.5 β = h D = D Fig. 6. s vs D for different values of β, and for h D equal to: (a)., (b).2, (c).3 and (d).4. curves with smaller β values cross the horizontal axis at higher D values. The higher linear part slope of the curves means that a longer interval on the stream boundary is required to supply the increase in D. Also, the higher D for s = in the curves with smaller β shows that, as the hydraulic conductivity of the leaky layer declines, larger D is needed for the capture zone to touch the stream boundary. The half portion of water pumped from the stream, R /2, may be calculated based on the difference in the stream function values for the streamline passing through the stagnation point, Ψ Ds,andthe streamline passing through = andy D = h D, Ψ Dt. To calculate the stream function for the streamline passing through (,h D ), the limits of Ψ D are calculated (equation (8b)) when + : lim +Ψ D = ( κ) D κ n π = π D (7) Therefore, f may be introduced as the proportion of pumped water from the stream to the total pumping water as: f = π D Ψ Ds π D (8) Figure 7(a) (d) shows fvs D for different values of β and h D. As expected, for the given values of β and h D,f increases with any increase in D ;however, the increase is not linear. The general trend (except in low conductive and thick leaky layers) is that the rate of increase in f is larger at low D and smaller at high D values. For the given D and h D values, f decreases with any decrease in β. It reflects the fact that the stream would make less contribution to the pumped water when the conductivity of the leaky layer decreases. For any D and β, f decreases as h D increases. This may be interpreted as thicker leaky layers lessening the stream contribution to the pumped water, which makes sense hydraulically. For low conductive (β =.) and thick leaky layers (h D.3), the stream contribution to pumped water (f ) becomes very small (<.3%) even for large values of D (Fig. 7(c) and (d)). CONCLUSIONS A steady-state analytical solution was developed to delineate the capture zone of a fully penetrating pumping well in an aquifer with regional flow perpendicular to a stream boundary, assuming a leaky layer between the stream and the aquifer. Complex potential theory and superposition law were used to obtain analytical solutions to the problem. Three different scenarios were considered for different pumping rates. At low pumping rates, the capture zone boundary was completely contained in the aquifer. At medium pumping rates, however, the tip of the capture zone boundary intruded into the leaky layer.

11 822 Mahdi Asadi-Aghbolaghi et al. (a) f (c) f β =. (b) h D =..8 β =. β =.5 h.7 D =.2 β =.5 β =. β =.5.6 β =. β =.5.5 β =.4 β = D D β =. h D =.3.8 β =.5 (d) β =. h D =.4.7 β =.5 β =. β =. β =.5 β =.5 β = β = D f f D Fig. 7. Variation of fvs D for different values of β, and for h D equal to: (a)., (b).2, (c).3 and (d).4. Under these two scenarios all of the pumped water was supplied from the regional groundwater flow in the aquifer. Finally, at higher pumping rates, the capture zone boundary intersected the stream and pumped water was supplied from both the aquifer and the stream. The two critical pumping rates which separate these three scenarios were obtained analytically. The results show that both critical pumping rates have similar trends and increase significantly as the conductivity of the leaky layer decreases and/or its thickness increases. Furthermore, in the third scenario, the contribution of the stream to the pumped water and the length of the segment through which water enters the aquifer from the stream were determined and investigated for different hydraulic settings. It was found that, for a given pumping rate, the proportion of pumping water supplied from the stream would decrease as the conductivity of the leaky layer decreases or its thickness increases. Acknowledgements The detailed and useful comments and suggestions of the reviewers, associate editor and co-editor of the journal regarding this manuscript are hereby acknowledged. REFERENCES Anderson, E.I., 999. Groundwater flow with leaky boundaries. Thesis (PhD). Department of Civil Engineering, University of Minnesota, Minneapolis. Anderson, E.I., 2. The method of images for leaky boundaries. Advances in Water Resources, 23 (5), Anderson, E.I., 23. An approximation for leaky boundaries in groundwater flow. Journal of Hydrology, 274 ( 4), Bakker, M. and Anderson, E.I., 23. Steady flow to a well near a stream with a leaky bed. Ground Water, 4 (6), Butler, J.J., Zlotnik, V.A., and Tsou, M., 2. Drawdown and stream depletion produced by pumping in the vicinity of a partially penetrating stream. Ground Water, 39(5), Christ, J.A. and Goltz, M.N., 22. Hydraulic containment: analytical and semi-analytical models for capture zone curve delineation. Journal of Hydrology, 262 ( 4), Christ, J.A. and Goltz, M.N., 24. Containment of groundwater contamination plumes: minimizing drawdown by aligning capture wellsparalleltoregional flow. Journal of Hydrology, 286 ( 4), Cole, K.D. and Yen, D.H.Y., 2. Green s functions, temperature, and heat flux in the rectangle. International Journal of Heat Mass Transfer, 44, Dacosta, J.A. and Bennett, R.R., 96. The pattern of flow in the vicinity of a recharging and discharging pair of wells in an aquifer having areal parallel flow. In: Subterranean waters (IAHS, IUGG General Assembly of Helsinki). Wallingford: IAHS Press, IAHS Publ. 52, Faybishenko, B.A., Javandel, I., and Witherspoon, P.A., 995. Hydrodynamics of the capture zone of a partially penetrating well in a confined aquifer. Water Resources Research, 3(4), Fienen, M.N., Luo, J., and Kitanidis, P.K., 25. Semi-analytical homogeneous anisotropic capture zone delineation. Journal of Hydrology, 32 ( 4), Glover, R.E. and Balmer, C.G., 954. River depletion resulting from pumping a well near a river. EOS, Transactions of the American Geophysical Union, 35, Grubb, S., 993. Analytical model for estimation of steady-state capture zones of pumping wells in confined and unconfined aquifers. Ground Water, 3 (), Ha, K., et al., 27. Estimation of layered aquifer diffusivity and river resistance using flood wave response model. Journal of Hydrology, 337 (3 4), Hantush, M.M., 25. Modeling stream aquifer interactions with linear response functions. Journal of Hydrology, 3 ( 4),

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