Evaluation of the path integral for flow through random porous media

TL;DR

This paper develops a path integral approach to model flow through one-dimensional random porous media, using Monte Carlo methods to compute pressure distributions and validating against stochastic differential equations, with discussions on extending to higher dimensions.

Contribution

It introduces a novel path integral formulation for Darcy's equation with correlated lognormal permeability and demonstrates its effectiveness through Monte Carlo evaluation.

Findings

Pressure distributions match stochastic differential equation solutions.

Path integral approach effectively models flow in random media.

Extension to higher dimensions is feasible and discussed.

Abstract

We present a path integral formulation of Darcy's equation in one dimension with random permeability described by a correlated multi-variate lognormal distribution. This path integral is evaluated with the Markov chain Monte Carlo method to obtain pressure distributions, which are shown to agree with the solutions of the corresponding stochastic differential equation for Dirichlet and Neumann boundary conditions. The extension of our approach to flow through random media in two and three dimensions is discussed.