## 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 *t*_{D} = *Tt*/*Sr*^{2} 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 log*t*_{D} 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 within which heads are functions of time but head gradients are time invariant. This quasi-steady state region expands as . The head *h*_{τ} at its outer boundary *r*_{τ} 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 *h*_{τ} at *r*_{τ}. 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.