Higher-order correction of effective permeability of heterogeneous isotropic formations of lognormal conductivity distribution

Abstract

Steady flow of an incompressible fluid takes place in a porous formation of spatially variable hydraulic conductivityK. The latter is regarded as a lognormal stationary random space function and Y=ln(K/K _G ), whereK _G is the geometric mean ofK, is characterized by its variance σ² and correlation scale I. Exact results are known for the effective conductivityK _eff in one- and two-dimensional flows. In contrast, only a first-order term in a perturbation expansion in σ² has been derived exactly for the three-dimensional flow. A conjecture has been made in the past onK _eff for any σ², but it was not yet proved exactly. This study derived the exact nonlinear correction, i.e. the termO(σ⁴) ofK _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 ofK _eff for in the three-dimensional case.