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

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.