Path Integral Renormalization of Flow through Random Porous Media

TL;DR

This paper develops a renormalization-group approach using path integrals to analyze flow in random porous media, providing a systematic way to understand how conductivity properties evolve across scales.

Contribution

It introduces a novel RG method based on path integrals for flow through random media, including exact solutions for effective conductivity at different scales.

Findings

Exact one-loop $eta$-functions for conductivity

Effective conductivity over larger length scales

Framework applicable to non-Gaussian and multiphase flows

Abstract

The path integral for Darcy's law with a stochastic conductivity, which characterizes flow through random porous media, is used as a basis for Wilson renormalization-group (RG) calculations in momentum space. A coarse graining procedure is implemented by integrating over infinitesimal shells of large momenta corresponding to the elimination of the small scale modes of the theory. The resulting one-loop $\beta$-functions are solved exactly to obtain an effective conductivity in a coarse grained theory over successively larger length scales. We first carry out a calculation with uncorrelated Gaussian conductivity fluctuations to illustrate the RG procedure before considering the effect of a finite correlation length of conductivity fluctuations. We conclude by discussing applications and extensions of our calculations, including comparisons with the numerical evaluation of path integrals, non-Gaussian fluctuations, and multiphase flow, for which the path integral formulation should prove particularly useful.