Self-similar Approximants of the Permeability in Heterogeneous Porous Media from Moment Equation Expansions

TL;DR

This paper introduces a self-similar functional renormalization method to accurately estimate permeability in heterogeneous porous media, improving upon traditional perturbative expansions especially for large heterogeneity levels.

Contribution

It develops a novel approach using self-similar approximants based on moment equation expansions to better predict permeability in highly heterogeneous media.

Findings

Method provides accurate estimates for large heterogeneity.

Comparison shows significant improvement over existing expansions.

Encouraging agreement with numerical simulations.

Abstract

We use a mathematical technique, the self-similar functional renormalization, to construct formulas for the average conductivity that apply for large heterogeneity, based on perturbative expansions in powers of a small parameter, usually the log-variance $\sigma_Y^2$ of the local conductivity. Using perturbation expansions up to third order and fourth order in $\sigma_Y^2$ obtained from the moment equation approach, we construct the general functional dependence of the transport variables in the regime where $\sigma_Y^2$ is of order 1 and larger than 1. Comparison with available numerical simulations give encouraging results and show that the proposed method provides significant improvements over available expansions.