Three-dimensional steady state flow to a well in a randomly heterogeneous bounded aquifer

Authors


Abstract

[1] We consider flow in a confined aquifer of uniform thickness due to a well of zero radius that fully penetrates the aquifer and discharges at a constant rate. If the lateral extent of the aquifer is infinite, a steady state flow regime never develops. It is, however, well known that if the aquifer is additionally uniform, a quasi-steady state region extends from the well out to a cylindrical surface whose radius expands as the square root of time. On the expanding surface, head is uniform and time invariant. Inside this surface, head at any time is described by a steady state solution. A rigorous analysis of the analogous situation in a randomly heterogeneous aquifer would require the solution of a three-dimensional transient stochastic flow problem in an aquifer of infinite lateral extent. Here we take a different approach by developing a three-dimensional steady solution for mean flow to a well in a randomly heterogeneous aquifer with a cylindrical prescribed head boundary. In analogy to the uniform case we expect our solution to approximate a quasi-steady state region whose radius is initially small in comparison to the horizontal correlation scale of log conductivity but grows with time to become eventually much larger. We treat log conductivity as a statistically homogeneous random field characterized by a Gaussian spatial covariance function that may have different horizontal and vertical correlation scales. Our solution consists of analytical expressions for the ensemble mean and variance of head in the aquifer to second order in the standard deviation of log conductivity. It is based on recursive approximations of exact nonlocal moment equations that are free of distributional assumptions and so apply to both Gaussian and non-Gaussian log conductivity fields. The analytical solution is supported by numerical Monte Carlo simulations. It clarifies the manner in which relationships between the horizontal and vertical scales of the quasi-steady state region and those of statistical anisotropy impact the statistical moments of drawdown and the equivalent and apparent hydraulic conductivities of the aquifer. Both conductivities are shown to exhibit a scale effect by growing with distance from the well within a radius of one to two horizontal integral scales from it.

1. Introduction

[2] Flow to wells in randomly heterogeneous aquifers is of considerable theoretical and practical interest to groundwater hydrologists. Yet such flow is not well understood and the problem of characterizing nonuniform aquifers by means of pumping tests remains an open challenge.

[3] Previous theoretical analyses have shown that it is generally not possible to represent randomly heterogeneous aquifers under radial flow by a single effective hydraulic conductivity or transmissivity [Matheron, 1967; Dagan, 1989; Gomez-Hernandez and Gorelick, 1989; Ababou and Wood, 1990; Butler, 1991; Naff, 1991; Desbarats, 1992, 1994; Neuman and Orr, 1993; Oliver, 1993; Franzetti et al., 1996; Sánchez-Vila, 1997]. Most of these analyses were two-dimensional and focused largely on mean flow behavior. Early attempts to characterize uncertainty by considering the variance of radial flow variables were published by Shvidler [1964] and Dagan [1982]. A detailed review of this literature can be found in a recent paper by Riva et al. [2001]. The latter authors present two-dimensional analytical expressions for leading statistical moments (mean, variance and covariance) of hydraulic head and flux under steady state flow to a well in a bounded, randomly heterogeneous aquifer. Their solution is based on recursive local approximations of exact nonlocal moment equations developed by Neuman and Orr [1993] and Guadagnini and Neuman [1999a]. Both the exact moment equations and their approximations are free of distributional assumptions, rendering them applicable to both Gaussian and non-Gaussian log hydraulic conductivity fields.

[4] Three-dimensional analyses of steady state flow to a well in a randomly heterogeneous aquifer have been published by Naff [1991], Desbarats [1994], Indelman et al. [1996], and Fiori et al. [1998]. Desbarats dealt numerically with spatial flow moments in a finite domain. The other three analyses dealt mathematically with ensemble flow moments by considering the radius of the well to be small in comparison to the horizontal correlation scale of log hydraulic conductivity, so as to render it effectively zero; the lateral extent of the aquifer to be large in comparison to this scale, so as to render it effectively infinite; and the thickness of the aquifer to be large in comparison to the vertical correlation scale of log hydraulic conductivity, so as to effectively eliminate the influence of horizontal boundaries. To develop expressions for the variance and vertical integral scale of head fluctuations, Fiori et al. treated both the variance of log conductivity and the statistical anisotropy ratio between its vertical and horizontal correlation scales as small parameters. Naff considered volumetric discharge per unit length of the well to be a known constant, even though hydraulic conductivity and gradient vary randomly along the well. Indelman et al. and Fiori et al. treated head in the well as a finite deterministic quantity, even though head in a well of zero radius is theoretically infinite. More recently, Indelman [2001] developed steady state expressions (to first order in the log conductivity variance) for mean head in one-, two- and three-dimensional unbounded domains due to a concentrated unit source.

[5] In this paper we consider flow in a confined aquifer of uniform thickness due to a well of zero radius that fully penetrates the aquifer and discharges at a constant rate. If the aquifer has infinite lateral extent, a steady state flow regime never develops. If the aquifer is additionally uniform, head within it varies with time according to the Theis equation [Theis, 1935]. It is well known that when dimensionless time tD = Tt/Sr2 exceeds a certain value τ (where T is transmissivity, t is time, S is storativity, and r is radial distance from the pumping well), head according to the Theis solution varies linearly with logtD and its radial derivative is thus independent of time. At any t, a quasi-steady state region extends from the well out to a radial distance equation image within which heads are functions of time but head gradients are time invariant. This quasi-steady state region expands as equation image. The head hτ at its outer boundary rτ remains fixed. Thus, at any t, head within the quasi-steady state region is described by a steady state solution obtained upon prescribing a constant, time-invariant head hτ at rτ. It is this steady state solution that renders the well-known Thiem equation applicable to extensive aquifers.

[6] A rigorous analysis of the analogous situation in a randomly heterogeneous aquifer would require the solution of a three-dimensional transient stochastic flow problem in an aquifer of infinite lateral extent. In this paper we take a different approach by developing a three-dimensional steady solution for mean flow to a well in a randomly heterogeneous aquifer with a cylindrical prescribed head boundary. In analogy to the uniform case, we expect our solution to approximate a quasi-steady state region whose radius is initially small in comparison to the horizontal correlation scale of log conductivity but grows with time to become eventually much larger. Hence various ratios between this radius and correlation scale are of practical interest, and we investigate a range of them in this paper.

[7] We treat log conductivity as a statistically homogeneous random field characterized by a Gaussian spatial covariance function that may have different horizontal and vertical correlation scales. Pumping tests in both uniform and heterogeneous aquifers are more commonly conducted by controlling the total volumetric rate of discharge from the well than by controlling the head (as is often assumed in stochastic solutions); we therefore prescribe the total flow rate at the well deterministically. This does not prevent us from allowing the horizontal flux into the pumping well to vary randomly along it. Most aquifers are less than perfectly stratified and may thus exhibit vertical correlation scales of log hydraulic conductivity that are not negligible in comparison to aquifer thickness. This is especially true in the presence of vertical fractures that may render the vertical correlation scale comparable in magnitude to aquifer thickness. We therefore investigate a range of ratios between aquifer thickness and vertical correlation scale.

[8] Our solution consists of analytical expressions for the ensemble mean and variance of head in the aquifer to second order in the standard deviation of log conductivity. Like the two-dimensional analysis of Riva et al. [2001], our three-dimensional solution is based on recursive approximations of exact moment equations [Neuman and Orr, 1993; Guadagnini and Neuman, 1999a] that are free of distributional assumptions and are thus applicable to both Gaussian and non-Gaussian log hydraulic conductivity fields. We explore the effect of boundaries and statistical anisotropy on statistical moments of drawdown around the well and on equivalent and apparent hydraulic conductivities of the aquifer. To assess the accuracy of our analytical solution, we compare it to the results of three-dimensional numerical Monte Carlo simulations.

2. Statement of Problem

[9] Consider an aquifer confined between two horizontal no-flow boundaries separated by a vertical distance D. A well of zero radius fully penetrates the aquifer and discharges at a deterministically prescribed volumetric rate, QD. A constant head, HL, is prescribed deterministically on a vertical boundary of cylindrical shape located at a radial distance L from the well (representing the radial extent of a continuously expanding quasi-steady state domain). The natural logarithm Y(r) = ln K(r) of hydraulic conductivity K(r) forms a statistically homogeneous random function of the cylindrical space coordinate r = (r, θ, z)T where r is radial distance from the well, θ is horizontal angle measured relative to an arbitrary radius, z is elevation above the bottom of the aquifer, and T indicates transpose. Y(r) is characterized by a statistically anisotropic Gaussian spatial covariance function

equation image

with variance σY2, horizontal correlation scale λ, vertical correlation scale λz and horizontal to vertical anisotropy ratio e = λ/λz. Here

equation image

is dimensionless distance defined in terms of ξ = r/L, χ = z/D and ε = L/D.

[10] Flow in the aquifer is governed by the steady state continuity equation and Darcy's law

equation image

subject to the Neumann and Dirichlet boundary conditions

equation image

where ∇ is the gradient operator in cylindrical coordinates, q(r) is volumetric flux, f(r) is a source term and h(r) is head. The source is represented by

equation image

where qr(z) is radial flux into the well and δ(r) is the Dirac delta. Whereas HL and QD are deterministic the flux, source term and head are random due to the randomness of K.

[11] Let 〈K〉 be the constant ensemble mean (statistical expectation) of K(r) and K′(r) = K(r) − 〈K〉 a zero mean random fluctuation about this mean. In the absence of site specific information about the values of K(r), 〈K〉 represents the best available estimate of hydraulic conductivity in the aquifer and K′(r) represents the corresponding estimation error. By the same token, the ensemble mean quantities 〈h(r)〉 and 〈q(r)〉 represent optimum unbiased estimates of head and flux which are associated with zero mean prediction errors h′(r) = h(r) − 〈h(r)〉 and q′(r) = q(r) − 〈q(r)〉, respectively.

[12] Taking the ensemble average of (3)(5) yields the mean flow equations

equation image
equation image
equation image

where rf(r) = − 〈K′(r)∇h′(r)〉 is a “residual flux.” The latter can be expressed exactly in the form of a compact integrodifferential expression [Neuman and Orr, 1993], which renders (6) nonlocal and non-Darcian. This implies that one cannot define an effective hydraulic conductivity for mean flow conditions except in special cases. We shall see that in the case considered here, the exception does not apply and the best one can do is to define equivalent or apparent conductivities that vary not only with statistical properties of the aquifer but also from point to point within it.

[13] For simplicity we treat 〈qr〉 as being independent of z. We note that this does not prevent the random qr(z) to vary with elevation. It then follows from (8) that the mean source term is given deterministically by

equation image

[14] The residual flux depends on a stochastic Green's function whose moments cannot be evaluated exactly [Neuman and Orr, 1993]. To overcome this difficulty, we evaluate the nonlocal moment equations recursively [Guadagnini and Neuman, 1999a] by expanding them to second order in powers of σY. This nominally limits our results to mildly nonuniform aquifers. We note, however, that Guadagnini and Neuman [1999b] obtained good agreement between second-order finite elements approximations of two-dimensional nonlocal moment equations and numerical Monte Carlo results for at least σY2 = 4 under superimposed mean uniform and convergent steady state flows in a rectangular aquifer. Riva et al. [2001] obtained equally good agreement between second-order analytical solutions of two-dimensional nonlocal moment equations and numerical Monte Carlo results for at least σY2 = 1.5 under convergent steady state flow to a well in a bounded aquifer similar to that we consider here.

3. Mean Hydraulic Head

[15] Let 〈K[2](r)〉 = 〈K(0)(r)〉 + 〈K(1)(r)〉 + 〈K(2)(r)〉, 〈h[2](r)〉 = 〈h(0)(r)〉 + 〈h(1)(r)〉 + 〈h(2)(r)〉 and 〈q[2](r)〉 = 〈q(0)(r)〉 + 〈q(1)(r)〉 + 〈q(2)(r)〉 where superscripts in rectangular brackets indicate the order of expansion in σY and those in parentheses designate the order of its individual components. A corresponding set of mean recursive equations is given by (24)–(39) of Guadagnini and Neuman [1999a]. In our special case the zero-order approximation of mean head, 〈h(r)(0)〉, satisfies

equation image

subject to

equation image

where KG = eY is the geometric mean of K. Due to radial symmetry, the solution depends solely on the normalized radial coordinate ξ = r/L,

equation image

The zero-order solution coincides with the widely used deterministic Thiem equation. It represents two-dimensional radial flow in an aquifer with uniform, nonrandom transmissivity equal to TG = KGD.

[16] The first-order component of mean head vanishes. To evaluate the second-order component, 〈h(2)(r)〉, we start from the second-order approximation of (6) [Guadagnini and Neuman, 1999a]

equation image

Here rf(2) is given explicitly by

equation image

where Ω is the flow domain; CY(r, r′) is the covariance of Y; and G(r, r′) is a deterministic Green's function that satisfies (6) with the source term replaced by δ(rr′), subject to homogeneous boundary conditions. The Green's function is given by (A2)(A4) in Appendix A.

[17] Our treatment of 〈qr〉 as being independent of z (which has led to (9)) implies, by virtue of the first expression in (34) of Guadagnini and Neuman [1999a], that the flux predictor reduces to its zero-order approximation 〈q(r)〉 = 〈q(0)(r)〉. This in turn implies that 〈q(n)(r)〉 ≡ 0 for all n ≥ 1 (since 〈q(n)(r)〉 is proportional to σYn, their sum cannot be identically zero unless each individual term vanishes identically). Indeed, numerical analyses of flow to a well in a rectangular domain by Guadagnini and Neuman [1997, 1999a, 1999b] have demonstrated that 〈q(2)(r)〉 = 0. The same was found by Riva et al. [2001] for two-dimensional mean radial flow. It was concluded by these authors that whereas second order accuracy is important for the prediction of heads, it has no effect on the prediction of fluxes.

[18] Setting 〈q(2)〉 = 0 in (13) implies that 〈h(2)(r)〉 satisfies the differential equation

equation image

subject to homogeneous boundary conditions. We demonstrate in Appendix B that the tangential component of the second-order approximation of the residual flux, rfθ(2)(r), vanishes so that 〈h(2)〉 is independent of angle and depends solely on ξ and χ.

[19] Substituting (1)(2), (12) and (A2)(A4) into (14)(15) and integrating leads to the following expression for the second-order mean head component,

equation image

where Ij (j = 1, …, 6) are multidimensional integrals defined in Appendix C. The second-order approximation of mean head, 〈h[2]〉, is the sum of (12) and (16).

[20] The integrals Ii were evaluated at various dimensionless locations (ξ, χ) with the aid of Gaussian quadrature using 30 to 100 Gauss points, depending on the integral. For each point (ξ, χ) the computation took between 1 and 2 hours on a 800 MHz Pentium III processor (RAM being immaterial). We illustrate below the behavior of our solution for σY2 = 1 when L/D = 1 (the radial extent of the quasi-steady state region is equal to aquifer thickness) and L/D = 10 (the radial extent of this region exceeds aquifer thickness by a factor of 10).

[21] Figure 1 shows how the second-order dimensionless head correction {〈h(2)(ξ, χ)〉/[QDσY2/(2πDKG)]} varies with dimensionless elevation χ at ξ = 0.1, 0.4 and 0.8 when L/D = 1, L/λ = 10 and e = 1 (D/λ = Dz = 10). Dependence on χ, which is fully accounted for by the two- and three-dimensional integrals I5 and I6, respectively is evident as one approaches the well. The second-order correction is seen to form nonuniform symmetric vertical profiles across the aquifer (the zero-order mean head is of course uniform along the vertical). Even though the mean head varies vertically between the two horizontal no-flow boundaries, the mean flux remains uniform along the vertical. This is so because the mean flux depends not only on the mean hydraulic head gradient but also on the residual flux [Guadagnini and Neuman, 1999a, equation (35)].

Figure 1.

Second-order component of dimensionless mean head, {〈h(2)(ξ, χ)〉/[QDσY2/(2πDKG)]}, versus dimensionless elevation, χ, at ξ = 0.1, 0.4, and 0.8 when L/D = 1, L/λ = 10, and e = 1 (D/λ = Dz = 10).

[22] Figure 2 illustrates the effect of statistical anisotropy ratio e on the variation of second-order dimensionless mean drawdown with dimensionless radial distance ξ at χ = 0.5 (midway between the horizontal no-flow boundaries) when L/D = L/λ = 1. The same is illustrated for L/D = 1 and L/λ = 10 in Figure 3, L/D = 10 and L/λ = 1 in Figure 4, and L/D = L/λ = 10 in Figure 5. Included for comparison are the zero-order solution and a second-order solution obtained for two-dimensional flow by Riva et al. [2001]. The zero-order solution is the same for three- and two-dimensional flows and remains unaffected by variations in L/D, L/λ or e (as well as σY2). The second-order two-dimensional solution is sensitive to L/λ but not to L/D or e.

Figure 2.

Dimensionless mean drawdown versus dimensionless radial distance, ξ, for various values of the anisotropy ratio, e, at χ = 0.5 when L/D = L/λ = 1; σY2 = 1.

Figure 3.

Dimensionless mean drawdown versus dimensionless radial distance, ξ, for various values of the anisotropy ratio, e, at χ = 0.5 when L/D = 1 and L/λ = 10; σY2 = 1.

Figure 4.

Dimensionless mean drawdown versus dimensionless radial distance, ξ, for various values of the anisotropy ratio, e, at χ = 0.5 when L/D = 10 and L/λ = 1; σY2 = 1.

Figure 5.

Dimensionless mean drawdown versus dimensionless radial distance, ξ, for various values of the anisotropy ratio, e, at χ = 0.5 when L/D = L/λ = 10; σY2 = 1.

[23] In all four cases depicted in Figures 25, rendering the aquifer more stratified (by increasing e from 1 to 10) causes the (second-order three-dimensional) normalized drawdown and its horizontal gradient (except near the well) to diminish. This is so because resistance to horizontal flow becomes weaker and the aquifer is able to deliver a prescribed discharge rate to the well under a reduced hydraulic gradient. Later we will show that this is reflected in an increase in equivalent and apparent hydraulic conductivities with e.

[24] When Dz ≤ 1 (e = 1 in Figures 2 and 5, e = 1, 5 and 10 in Figure 4), the (second-order) three-dimensional solution approaches its two-dimensional equivalent. This happens because the vertical correlation scale of log hydraulic conductivity is large enough relative to the aquifer thickness for drawdown to equilibrate along the vertical. When additionally L= 10 (e = 1 in Figure 5) or larger, the zero-order solution yields a good approximation for the mean drawdown. The same happens to the two-dimensional solution regardless of e (Figures 3 and 5). In both cases the flow is virtually horizontal and, at distances that are large in comparison to λ, is known to be represented adequately by the same effective (geometric mean) transmissivity as that which controls mean uniform flow [Neuman and Orr, 1993]. This is also seen in the graphs of equivalent hydraulic conductivity presented in Figures 14 and 17 below.

4. Hydraulic Head Variance

[25] Whereas for purposes of solving the mean head equations it was enough for us to treat the mean flux along the borehole as being uniform, now we must adopt the stronger assumption that the flux itself is uniform and deterministic. Recursive expressions for the variance/covariance of head that include this case were derived by Guadagnini and Neuman [1999a, (40)–(44)]. To second (lowest) order of approximation, the head covariance Ch(rI, rII) between two points rI and rII in the aquifer satisfies

equation image

subject to the homogeneous boundary conditions

equation image

The solution of (17)(18) is

equation image

where ChK(2) is a second-order approximation of the cross-covariance between hydraulic head and conductivity. The latter is given by

equation image

[26] We note on the basis of (1)(2) and (A2)(A4) that, by virtue of radial symmetry, the log conductivity covariance and G depend only on the angular separation (θ − θ′) between points r and r′, and not on their absolute angles θ or θ′. Hence integration with respect to one of these angles is identical to integration with respect to their difference. It follows that all local moments (defined at a single point in space) are independent of angle and are solely functions of ξ and χ. Substituting (1)(2), (12) and (A2)(A4) into (19)(20) yields, after some manipulation, the following expression for the second-order head variance, σh2(2) = Ch(2)(rIrII),

equation image

where Iσ is the sum of multidimensional integrals defined in Appendix D. These were evaluated using Gaussian quadrature with 20 to 100 Gauss points, depending on the integral. For each point, the computation took between 1 and 3 hours on a 800 MHz Pentium III processor (RAM being immaterial).

[27] Like in the case of mean head, we illustrate the behavior of our solution for σY2 = 1. Figure 6 shows how the dimensionless head variance {σh2(2)(ξ, χ)/[QDσY/(4πDKG)]2} varies with dimensionless elevation χ at ξ = 0.4 and 0.8 in the previous case (L/D = 1, L/λ = 10 and e = 1). Included for comparison are two-dimensional solutions obtained for the same radial locations by Riva et al. [2001]. The variance is seen to form a symmetrically nonuniform vertical profile across the aquifer with a trough midway between the horizontal no-flow boundaries. It decreases and its profile becomes more uniform with distance from the well. The no-flow boundaries and the well are thus seen to have a negative impact on the predictability of head, which diminishes with distance from each of them. The three-dimensional head variance is always lower than its two-dimensional counterpart.

Figure 6.

Dimensionless head variance, {σh2(2)(ξ, χ)/[QDσY/(4πDKG)]2}, versus dimensionless elevation, χ, at ξ = 0.4 and 0.8 when L/D = 1, L/λ = 10, and e = 1 (D/λ = Dz = 10).

[28] Figure 7 illustrates the effect of statistical anisotropy ratio e on the variation of second-order dimensionless head variance with dimensionless radial distance ξ midway between the horizontal no-flow boundaries (at χ = 0.5) when L/D = L/λ = 1. The same is illustrated for L/D = 1 and L/λ = 10 in Figure 8, L/D = 10 and L/λ = 1 in Figure 9 and L/D = L/λ = 10 in Figure 10. Included for comparison is a second-order solution obtained for two-dimensional flow by Riva et al. [2001]. The latter is sensitive to L/λ but not to L/D or e.

Figure 7.

Dimensionless head variance, {σh2(2)(ξ, χ)/[QDσY/(4πDKG)]2}, versus normalized radial distance, ξ, for various values of the anisotropy ratio, e, at χ = 0.5 when L/D = L/λ = 1.

Figure 8.

Dimensionless head variance, {σh2(2)(ξ, χ)/[QDσY/(4πDKG)]2}, versus normalized radial distance, ξ, for various values of the anisotropy ratio, e, at χ = 0.5 when L/D = 1 and L/λ = 10.

Figure 9.

Dimensionless head variance, {σh2(2)(ξ, χ)/[QDσY/(4πDKG)]2}, versus normalized radial distance, ξ, for various values of the anisotropy ratio, e, at χ = 0.5 when L/D = 10 and L/λ = 1.

Figure 10.

Dimensionless head variance, {σh2(2)(ξ, χ)/[QDσY/(4πDKG)]2}, versus normalized radial distance, ξ, for various values of the anisotropy ratio, e, at χ = 0.5 when L/D = L/λ = 10.

[29] In all four cases depicted in Figures 710, rendering the aquifer more stratified (by increasing e from 1 through 5 to 10) causes the (second-order three-dimensional) normalized head variance and its horizontal rate of variation to diminish. We explained earlier why an increase in e causes resistance to horizontal flow and the hydraulic gradient to go down. The more uniform is the hydraulic head across the aquifer, the more closely is it controlled by the deterministic constant head boundary and the more reliably can it be predicted by the mean. This is reflected in a reduced head variance.

[30] When Dz ≤ 1 (e = 1 in Figures 7 and 10, e = 1, 5 and 10 in Figure 9), the three-dimensional normalized head variance approaches its two-dimensional equivalent. We explained earlier that here the vertical correlation scale of log hydraulic conductivity is large enough relative to the aquifer thickness for drawdown to equilibrate along the vertical, so that the flow becomes effectively horizontal as in the two-dimensional case.

[31] We end this section with an illustration in Figure 11 of how the standard deviation of head varies with dimensionless distance about the mean when L/D = L/λ = 10 for the isotropic and stratified cases.

Figure 11.

Dimensionless mean drawdown and its standard deviation versus normalized radial distance, ξ, for various values of the anisotropy ratio, e, at χ = 0.5 when L/D = L/λ = 10; σY2 = 1.

5. Comparison With Numerical Monte Carlo Simulations

[32] To gauge the accuracy of our analytical solution, we compared it against numerical Monte Carlo (MC) simulations, which we consider sufficiently accurate to serve as a standard of comparison. Simulations were conducted with the widely used and thoroughly tested finite difference code MODFLOW [McDonald and Harbaugh, 1988]. The code is widely used for this purpose [e.g., Ptak, 1996] (Groundwater Modeling System,http://www.scisoft-gms.com) and was found by Chin and Wang [1992] to reproduce accurately the mean and second moments of three-dimensional Eulerian velocities in statistically isotropic media for σY = 0.1. We simulated flow on a square grid measuring 20 × 20 arbitrary length units along the horizontal plane (consisting of 81 × 81 nodes spaced 0.25 units apart) and 2.5 length units along the vertical direction (11 nodes spaced 0.25 units apart).

[33] Pumping at the well was simulated using two different approaches. In the first approach, a volumetric pumping rate of QD = 50 (in arbitrary space-time units) was distributed uniformly along a vertical column of finite difference cells at the center of the grid, representing a fully penetrating well. In the second approach, the well cells were ascribed a relatively large hydraulic conductivity so as to allow rapid equilibration of heads along it, and a concentrated sink of strength QD = 50 was placed at the center of the well. In this way, horizontal fluxes along the well were free to adjust to the random vertical distribution of hydraulic conductivities around it while maintaining uniform head along the well.

[34] In both cases a cylindrical vertical boundary of radius L = 10 was defined about the well by designating all finite difference cells outside it as inactive. Head HL on the vertical cylindrical boundary was set equal to 500 length units. The upper and lower boundaries were made impermeable.

[35] A Gaussian sequential simulator, SGSIM [Deutsch and Journel, 1998], was used to generate random realizations of log hydraulic conductivity across the three-dimensional grid. Each realization constituted a sample from a multivariate Gaussian, statistically homogeneous Y field with an isotropic Gaussian covariance, mean 〈Y〉 = 0 (corresponding to KG = 1 in units of length per time), variance σY2 = 1, unit spatial horizontal correlation length λ, and anisotropy ratio e = 1. These correspond to L/D = 4, L/λ = 10 and e = 1. Stable values of sample mean and variance, and sample correlation functions that resemble closely their theoretical (Gaussian) counterparts, were obtained after 1000 Monte Carlo flow simulations. In both cases, the generation of random log hydraulic conductivity fields required about 2 hours on a 600 Pentium III (256 Mb RAM), while flow simulations took about 12 hours.

[36] Figure 12 compares our second-order analytical solution for dimensionless drawdown with the two types of Monte Carlo results at χ = 0.5 (midway between the horizontal no-flow boundaries). Our solution is seen to correspond more closely to Monte Carlo results obtained by allowing flux along the well to be nonuniform (diamonds) than to those obtained by imposing a uniform flux along the well (crosses). The difference between the two types of Monte Carlo results is small everywhere except near the well.

Figure 12.

Second-order dimensionless mean drawdown versus dimensionless radial distance, ξ, compared with numerical Monte Carlo results at χ = 0.5 when L/D = 4, L/λ = 10, e = 1, σY2 = 1.

[37] Figures 13 shows a similar comparison of analytical and Monte Carlo results for dimensionless head variance at χ = 0.5. Even though our analytical second moments represent their lowest possible order of approximation, they compare favourably with the Monte Carlo results. Our analytical solution for the head variance is based on the assumption of uniform flux distribution along the well. This is reflected in it being somewhat closer to Monte Carlo results obtained by imposing a uniform flux along the well (crosses) than to those obtained by allowing flux along the well to be nonuniform (diamonds), especially as one approaches the well and the constant head boundary. The difference between the two types of Monte Carlo results is, however, generally quite small. Obtaining a fourth order approximation for the head variance is probably not worth the extra computational effort.

Figure 13.

Second-order dimensionless head variance versus dimensionless radial distance, ξ, compared with numerical Monte Carlo results at χ = 0.5 when L/D = 4, L/λ = 10, e = 1, σY2 = 1.

6. Equivalent and Apparent Hydraulic Conductivities

[38] We define the second-order equivalent hydraulic conductivity Ke[2](ξ, χ) as the quantity one would obtain from the Thiem equation on the basis of second-order mean head,

equation image

On the other hand, we define the second-order apparent hydraulic conductivity Ka[2](ξ, χ) as the quantity that would yield a local form of Darcy's law in terms of second-order mean radial flux and hydraulic gradient,

equation image

[39] Figure 14 illustrates the effect of statistical anisotropy ratio e on the variation of Ke[2](ξ, χ)/KG with dimensionless radial distance midway between the horizontal no-flow boundaries (at χ = 0.5) when L/D = L/λ = 1. Figure 15 shows the same for Ke(2)(ξ, χ)/Ka(2)(ξ, χ). Analogous depictions are given in Figures 16 and 17 for the case where L/D = L/λ = 10. Included for comparison are corresponding quantities obtained from the two-dimensional solution of Riva et al. [2001]. The latter are sensitive to L/λ but not to L/D or e.

Figure 14.

Ke[2]/KG versus normalized distance, ξ, for various values of the anisotropy ratio, e, at χ = 0.5 when L/D = L/λ = 1; σY2 = 1.

Figure 15.

Ke[2]/Ka[2] versus normalized distance, ξ, for various values of the anisotropy ratio, e, at χ = 0.5 when L/D = L/λ = 1; σY2 = 1.

Figure 16.

Ke[2]/KG versus normalized distance, ξ, for various values of the anisotropy ratio, e, at χ = 0.5 when L/D = L/λ = 10; σY2 = 1.

Figure 17.

Ke[2]/Ka[2] versus normalized distance, ξ, for various values of the anisotropy ratio, e, at χ = 0.5 when L/D = L/λ = 10; σY2 = 1.

[40] In all four cases depicted in Figures 1417, rendering the aquifer more stratified (by increasing e from 1 to 10) causes the (second-order three-dimensional) equivalent and apparent hydraulic conductivities to increase. This has been anticipated by us earlier on the grounds that resistance to horizontal flow becomes weaker and the aquifer is able to deliver a prescribed discharge rate to the well under a reduced hydraulic gradient.

[41] When Dz ≤ 1 (e = 1 in Figures 1417), the three-dimensional equivalent and apparent conductivities are closed to their two-dimensional counterparts as obtained by Riva et al. [2001]. This is consistent with our earlier explanation that here the vertical correlation scale of log hydraulic conductivity is large enough relative to the aquifer thickness for drawdown to equilibrate along the vertical, so that the flow becomes effectively horizontal as in the two-dimensional case.

[42] Though Ke[2](ξ, χ) and Ka[2](ξ, χ) are generally not equal to each other, they tend not to differ significantly. Both generally vary with radial distance from the well. Both exhibit a scale effect by growing with distance from the well within a radius of one to two horizontal integral scales from it. At distances that exceed two horizontal integral scales from the well and the outer boundaries, the equivalent and apparent hydraulic conductivities stabilize. These stable values tend to KG as the flow becomes two dimensional and to the arithmetic mean as it becomes more stratified.

[43] As in the two-dimensional case of Riva et al. [2001], Ka[2](ξ, χ) tends to KH, the harmonic mean of K, close to the well for all L/D, L/λ and e; Ke[2](ξ, χ) is always larger than Ka[2](ξ, χ) especially for large anisotropy ratios e.

7. Conclusions

[44] A three-dimensional analytical solution was developed for mean steady state flow to a well of zero radius that fully penetrates a randomly heterogeneous confined aquifer and pumps at a constant rate. If the lateral extent of the aquifer is infinite, a steady state flow regime never develops. It is however well known that if the aquifer is additionally uniform, a quasi-steady state region extends from the well out to a cylindrical surface whose radius expands as the square root of time. On the expanding surface head is uniform and time-invariant. Inside this surface, head at any time is described by a steady state solution. A rigorous analysis of the analogous situation in a randomly heterogeneous aquifer would require the solution of a three-dimensional transient stochastic flow problem in an aquifer of infinite lateral extent. Our steady state solution constitutes an approximation of the corresponding quasi-steady state region whose radius is initially small in comparison to the horizontal correlation scale of log conductivity but grows with time to become eventually much larger.

[45] Our solution assumes that the logarithm of hydraulic conductivity forms a statistically homogeneous random field characterized by a Gaussian spatial covariance function that may have different horizontal and vertical correlation scales. It is based on recursive approximations of exact nonlocal moment equations that are free of distributional assumptions, and yields expressions for the ensemble mean and variance of hydraulic head.

[46] The mean head solution allows horizontal flux at the pumping well to vary randomly along it. The head variance solution requires that this flux be distributed uniformly along the well. Both solutions agree closely with corresponding Monte Carlo simulations, which in turn differ little from each other.

[47] The zero-order solution coincides with the widely used deterministic Thiem equation. It represents two-dimensional radial flow in an aquifer with uniform, nonrandom geometric mean hydraulic conductivity.

[48] The second-order head approximation forms symmetric nonuniform vertical profiles across the aquifer with reduced values near the no-flow boundaries. The profiles become progressively more uniform with distance from the well. The second-order mean flux remains uniform along the vertical.

[49] Increasing the anisotropy ratio e between horizontal and vertical log hydraulic conductivity correlation scales causes the second-order drawdown and its horizontal gradient (away from the well) to diminish. This is so because, as the aquifer becomes more stratified, resistance to horizontal flow weakens and the aquifer is able to deliver a prescribed discharge rate to the well under a reduced overall hydraulic gradient. This is reflected in an increase in equivalent and apparent hydraulic conductivities with e.

[50] When the vertical correlation scale of log hydraulic conductivity becomes as large as the aquifer thickness, drawdowns equilibrate along the vertical and the (second-order three-dimensional) solution reduces to its two-dimensional counterpart. When additionally the external radius of the aquifer becomes large relative to the horizontal correlation scale, λ, the mean drawdown is described quite accurately by its zero-order approximation. The same happens to the two-dimensional solution regardless of e. In both cases the flow is virtually horizontal and the three-dimensional equivalent and apparent conductivities coincide with their two-dimensional counterparts. At distances that are large in comparison to λ, both conductivities are represented adequately by the geometric mean, as under mean uniform flow.

[51] Second-order head variance forms symmetrically nonuniform vertical profiles across the aquifer with a trough midway between the horizontal no-flow boundaries. The variance decreases and its profile becomes more uniform with distance from the well. The well and no-flow boundaries thus have a negative impact on the predictability of head near them. Three-dimensional head variance is always smaller than its two-dimensional counterpart.

[52] Rendering the aquifer more stratified by increasing the anisotropy ratio e causes the second-order head variance and its horizontal rate of variation to diminish. It was concluded earlier that an increase in e causes resistance to horizontal flow and the hydraulic gradient to go down. The more uniform is the hydraulic head across the aquifer, the more closely is it controlled by the deterministic constant head boundary and the more reliably can it be predicted by the mean. This is reflected in a reduction in head variance.

[53] When the vertical correlation scale of log hydraulic conductivity is of the order of aquifer thickness, the three-dimensional head variance approaches its two-dimensional equivalent. This happens because drawdowns tend to equilibrate along the vertical and cause flow to be effectively horizontal, as in the two-dimensional case.

[54] Equivalent hydraulic conductivity is always larger than apparent conductivity though the two do not differ significantly from each other. Both conductivities exhibit a scale effect by growing with distance from the well within a radius of one to two horizontal correlation scales from it. At distances that exceed two horizontal correlation scales from the well and the outer boundaries, the equivalent and apparent hydraulic conductivities stabilize. These stable values tend to the geometric mean as the flow becomes two-dimensional and to the arithmetic mean as it becomes more stratified. As in the two-dimensional case, the apparent conductivity tends to the harmonic mean close to the well.

Appendix A

[55] The steady state Green's function is obtained from its transient equivalent G(r, r′, t) [Carslaw and Jaeger, 1959] by taking the normalized limit [Dagan, 1982]

equation image

This yields, upon considering Staff of Bateman Manuscript Project [1953, p. 104],

equation image

where

equation image
equation image

and Ii, Ki are modified Bessel functions.

Appendix B

[56] According to (14), the tangential component of the second-order approximation of the residual flux, rfθ(2)(r), is given by

equation image

Using (12) and introducing dimensionless coordinates we obtain

equation image

Integrating (B2) by parts with respect to ξ′ and recalling that the angular derivative of (A2) vanishes at ξ′ = 0, 1 yields

equation image

Introducing the derivative of (A2) with respect to θ and of (1) with respect to ξ′ into (B3) yields, after some manipulations,

equation image

Since

equation image

rfθ(2)(ξ, χ) vanishes.

Appendix C

[57] Integrals Ij (j = 1,…6) in (16) are given by

equation image
equation image
equation image
equation image
equation image
equation image

where α is defined by (A4), γim (with i = 0, n) is defined by (A3) and

equation image
equation image
equation image

Appendix D

[58] The integral expression Iσ in (21) is given by

equation image

where

equation image

the subscript i corresponding to subscripts 0, q or n in (D1) and

equation image
equation image
equation image
equation image
equation image
equation image
equation image

In (D5) and (D8), the subscript v corresponds to subscripts 0, n, or q while subscript j corresponds to subscripts m or p of (D1).

Acknowledgments

[59] This work was supported by the European Commission under contract EVK1-CT-1999-00041 (W-SAHARA - Stochastic Analysis of Wellhead Protection and Risk Assessment).

Ancillary