Resistivity logging in a multi-scale medium with lognormal conductivity

We derived equations of effective conductivity coefficients to estimate mean current density and its variance in resistivity logging problems for isotropic multi-scale porous media with lognormal conductivity distribution. The equations have quite a simple form if the medium parameters satisfy refined Kolmogorov’s similarity hypothesis. The results of the refined perturbation theory obtained using subgrid modeling are compared with the results of numerical simulation and the conventional perturbation theory.