Micro-scale diffuse interface modeling with regularized logarithmic potential in porous media

This paper presents a diffuse interface model for the evolution of two incompressible fluids within a porous medium. The model is governed by the Cahn–Hilliard equations, which incorporate a Flory–Huggins logarithmic potential, and is coupled with the evolutionary Stokes equations at the pore scale through a surface tension term. A family of regular potentials $\{ F_{\delta }\} $ that approximate the singular logarithmic potential F is introduced. The existence of solutions for the pore-scale problem with the regular potential $F_{\delta }$ is established using Galerkin’s method. Subsequently, the two-scale convergence approach and the unfolding operator technique are applied to derive the homogenized model from the microscopic description. In the homogenized model equations, associated with the regularized potentials $\{ F_{\delta }\} $, the limit as $\delta \to 0 $ is taken to recover the upscaled model associated with the original logarithmic functional.