HIGHER-ORDER CORRECTION OF EFFECTIVE PERMEABILITY OF HETEROGENEOUS ISOTROPIC FORMATIONS OF LOGNORMAL CONDUCTIVITY DISTRIBUTION

Steady flow of an incompressible fluid takes place in a porous formation of spatially variable hydraulic conductivity K. The latter is regarded as a lognormal stationary random space function and Y = ln(K/K(G)), where K(G) is the geometric mean of K, is characterized by its variance sigma2 and correlation scale I. Exact results are known for the effective conductivity K(eff) in one- and two-dimensional flows. In contrast, only a first-order term in a perturbation expansion in sigma2 has been derived exactly for the three-dimensional flow. A conjecture has been made in the past on K(eff) for any sigma2, but it was not yet proved exactly. This study derived the exact nonlinear correction, i.e. the term O(sigma4) of K(eff), which is found to be the one resulting from the conjecture, strengthening the confidence in it. It is also shown that the self-consistent approximation leads to the exact results for one-dimensional and two-dimensional flows, but underestimates the nonlinear correction of K(eff) for in the three-dimensional case.