Derivation of generalized Cahn-Hilliard equation for two-phase flow in porous media using hybrid mixture theory

Abstract

Using generalizations of the Cahn-Hilliard equation for modeling two-phase flow in porous media at the pore scale has become popular due to its ability to capture interfacial effects by adding minimal complications. Here we use upscaled field equations and exploit the second law of thermodynamics in the spirit of rational thermodynamics to develop a framework that, for two phases at the macroscale, recovers the Korteweg stress tensor for the liquid phase, generalizes Darcy’s law, and recovers the classical Cahn-Hilliard equation. The corresponding results for three-phases at the macroscale are derived and are shown to be a generalization of Richards equation, and with appropriate simplifying assumptions are shown to recover the two-phase results. Simplifying the results appropriately produces a pore scale model for two liquid phases, and are shown to generalize previous works by Cueto-Felgueroso and Juanes and Boyer and Quintard et. al. The results are are also compared with the Cahn-Hilliard Brinkman equations, where it is noted that to be physically consistent the state variable should represent a physical quantity. One key aspect that distinguishes this formulation from others is that it captures the different energies of the three interfaces (gas-liquid, gas-solid, and liquid-solid) without introducing the corresponding quantities at the microscale (interfacial tension, contact angle, etc).