A quasi-incompressible Cahn-Hilliard-Darcy model for two immiscible fluids in porous media

Abstract

We derive a quasi-incompressible Cahn-Hilliard-Darcy phase-field model (qCHD) with the logarithmic Flory-Huggins free energy density function for two-phase flows in porous media. The model satisfies an energy-dissipation law. In the formal sharp interface limit, the qCHD model gives rise to the classical Muskat's problem. By exploiting estimates of the pressure from Darcy's equations, we establish global existence of weak solutions in both 2D and 3D to the qCHD model.