Allen-Cahn and Cahn-Hilliard-like equations for dissipative dynamics of saturated porous media

TL;DR

This paper derives Allen-Cahn and Cahn-Hilliard-like equations to model the dissipative dynamics of saturated porous media, explaining interface formation during solid-fluid segregation under different dissipation mechanisms.

Contribution

It establishes a theoretical framework linking dissipation mechanisms to specific evolution equations for porous media, advancing understanding of phase segregation processes.

Findings

Dissipation via Darcy law leads to Cahn-Hilliard-like dynamics.

Stokes dissipation results in Allen-Cahn-like evolution.

The model describes interface formation between fluid-rich and fluid-poor phases.

Abstract

We consider a saturated porous medium in the regime of solid-fluid segregation under an applied pressure on the solid constituent. We prove that, depending on the dissipation mechanism, the dynamics is described either by a Cahn-Hilliard or by an Allen-Cahn-like equation. More precisely, when the dissipation is modeled via the Darcy law we find that, for small deformation of the solid and small variations of the fluid density, the evolution equation is very similar to the Cahn-Hilliard equation. On the other hand, when only the Stokes dissipation term is considered, we find that the evolution is governed by an Allen-Cahn-like equation. We use this theory to describe the formation of interfaces inside porous media. We consider a recently developed model proposed to study the solid-liquid segregation in consolidation and we are able to fully describe the formation of an interface between the fluid-rich and the fluid-poor phase.