We study the homogenization of a microscopic Stokes–Cahn–Hilliard system in a periodically perforated porous medium with logarithmic free energy. The model describes an incompressible two-phase flow with diffuse interfaces inside pore-scale geometries and incorporates capillary effects through the chemical potential. We prove that the microscopic system converges to a Darcy–Korteweg type law coupled with a generalized Richards-type equation. A central consequence of the analysis is that the resulting macroscopic law generally cannot be reduced to the classical unsaturated Darcy law. In particular, the effective driving mechanism is not, in general, governed by a scalar gradient structure, reflecting the influence of interfacial energy and microscopic phase configurations.
This is a collaborative effort with Lei RONG.