Steady flow toward wells in heterogeneous formations: Mean head and equivalent conductivity

We consider steady flow of water in a confined aquifer toward a fully penetrating well of radius r(w) (Figure 1). The hydraulic conductivity K is modeled as a three-dimensional stationary random space function. The two-point covariance of Y = In (K/K-G) is of axisymmetric anisotropy, with I and I-v, the horizontal and vertical integral scales, respectively, and K-G, the geometric mean of K. Unlike previous studies which assumed constant flux, the well boundary condition is of given constant head (Figure 1). The aim of the study is to derive the mean head [H] and the mean specific discharge [q] as functions of the radial coordinate v and of the parameters sigma(Y)(2), e = I-v/I and r(w)/l. An approximate solution is obtained at first-order in sigma(Y)(2), by replacing the well by a line source of strength proportional to K and by assuming ergodicity, i.e., equivalence between (H) over bar, (q) over bar, space averages over the vertical, and [H], [q], ensemble means. An equivalent conductivity K-eq is defined as the fictitious one of a homogeneous aquifer which conveys the same discharge Q as the actual one, for the given head H-w in the well and a given head H in a piezometer at distance r from the well. This definition corresponds to the transmissivity determined in a pumping test by an observer that measures H-w, (H) over bar, and Q. The main result of the study is the relationship (19) K-eq = K-A(1 - lambda) + K(efu)lambda, where K-A is the conductivity arithmetic mean and K-efu is the effective conductivity for mean uniform flow in the horizontal direction in the same aquifer. The weight coefficient lambda < 1 is derived explicitly in terms of two quadratures and is a function of e, r(w)/I, and r/l. Hence K-eq, unlike K-efu, is not a property of the medium solely. For r(w)/I < 0.2 and for r/I > 10, lambda has the simple approximate expression lambda* = in (r/I) ln (r/r(w)). Near the well, lambda congruent to 0 and K-eq congruent to K-A, which is easily understood, since for r(w)/I << 1 the formation. behaves locally like a stratified one. In contrast, far from the well lambda congruent to 1 and K-eq congruent to K-efu, the flow being slowly varying there. Since K-A > K-efu, our result indicates that the transmissivity is overestimated in a pumping test in a steady state and it decreases with the distance from the well. However, the difference between K-A and K-efu is small for highly anisotropic formations for which e << 1. A nonlocal effective conductivity, which depends only on the heterogeneous structure, is derived in Appendix A along the lines of Indelman and Abramovich [1994].