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 σ² 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 σ² has been derived exactly for the three-dimensional flow. A conjecture has been made in the past on K_eff for any σ², but it was not yet proved exactly. This study derived the exact nonlinear correction, i.e. the term O(σ⁴) 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.