The Rayleigh-Taylor instability for the Musket free boundary problem

TL;DR

This paper investigates the Rayleigh-Taylor instability within the Musket free boundary problem, analyzing the behavior of two immiscible fluids in porous media through numerical simulations and homogenized models.

Contribution

It introduces a detailed analysis of free boundary problems in porous media, deriving homogenized models for rigid and elastic skeletons, and explores the transition from free boundary to mushy region models.

Findings

Homogenized models for rigid and elastic skeletons derived.

Free boundary persists in elastic skeleton model, mushy region in rigid skeleton.

Numerical simulations validate theoretical models.

Abstract

The present paper is devoted to the joint motion of two immiscible incompressible liquids in porous media. The liquids have different densities and initially separated by a surface of strong discontinuity (free boundary). We discuss the results of numerical simulations for exact free boundary problems on the microscopic level for the absolutely rigid solid skeleton and for the elastic solid skeleton of different geometries. The problems have a natural small parameter, which is the ratio of average pore size to the size of the domain in consideration. The formal limits as $\epsilon\searrow 0$ results homogenized models, which are the Muskat problem in the case of the absolutely rigid solid skeleton, and the viscoelastic Muskat problem in the case of the elastic solid skeleton. The last model preserves a free boundary during the motion, while in the first model instead of the free boundary appears a mushy region, occupied by a mixture of two fluids.