Darcy's Law for Porous Media with Multiple Microstructures

TL;DR

This paper analyzes the homogenization of Stokes equations in perforated domains with multiple microstructures, demonstrating that the effective flow obeys Darcy's law with piecewise constant permeability, and providing sharp error estimates.

Contribution

It establishes the Darcy law for complex microstructured media and proves the continuity of pressure across interfaces using Tartar's method, with precise convergence error bounds.

Findings

Effective velocity and pressure follow Darcy's law with piecewise constant permeability.

Pressure is continuous across interfaces between subdomains.

Sharp error estimates for velocity and pressure convergence are derived.

Abstract

In this paper we study the homogenization of the Dirichlet problem for the Stokes equations in a perforated domain with multiple microstructures. First, under the assumption that the interface between subdomains is a union of Lipschitz surfaces, we show that the effective velocity and pressure are governed by a Darcy law, where the permeability matrix is piecewise constant. The key step is to prove that the effective pressure is continuous across the interface, using Tartar's method of test functions. Secondly, we establish the sharp error estimates for the convergence of the velocity and pressure, assuming the interface satisfies certain smoothness and geometric conditions. This is achieved by constructing two correctors. One of them is used to correct the discontinuity of the two-scale approximation on the interface, while the other is used to correct the discrepancy between boundary values of the solution and its approximation.