  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.
