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

We consider a saturated porous medium in the solid–fluid segregation regime under the effect of an external pressure applied 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, provided the solid deformation and the fluid density variations are small, the evolution equation is very similar to the Cahn–Hilliard one. 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 give a complete description of the formation of an interface between the fluid-rich and the fluid-poor phase.