Steady Flow Toward Wells in Heterogeneous Formations: Mean Head and Equivalent Conductivity

Abstract

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 υ , 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 r and of the parameters σ y 2 , e = I / I υ and r w / I . An approximate solution is obtained at first‐order in σ y 2 , by replacing the well by a line source of strength proportional to K and by assuming ergodicity, i.e., equivalence between , , 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 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 , , and Q . The main result of the study is the relationship (19) K eq = K A (1 − λ) + K efu λ, 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 λ < 1 is derived explicitly in terms of two quadratures and is a function of e , r w / I and r / I . 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, λ has the simple approximate expression λ* = ln ( r / I )/ In ) r / r w ). Near the well, λ ≅ 0 and K eq ≅ 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 λ ≅ 1 and K eq ≅ 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].