Average steady flow toward a drain through a randomly heterogeneous porous formation

We consider the problem of steady pumping of water from a line drain on the surface of a wet ground. Unlike the classical formulation, which regards the conductivity parameter K as uniformly distributed in the domain, the problem here is solved within a stochastic framework in order to account for the irregular (random), and more realistic, spatial variability of K. Due to the linearity of the problem at stake, we focus on the derivation of the mean Green function G. This is computed by means of an asymptotic expansion. The fundamental result is an analytical (closed form) expression of G which generalizes the classical solution. Based on this, we develop an equivalent conductivity Keq which enables one to tackle the problem similarly to the classical one. In particular, it is shown that the equivalent conductivity grows monotonically with the radial distance r from the drain, and it lies within the range Keq(0) ≤ Keq(r) ≤ Keq(∞) < ∞.