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).

使用Cahn-Hilliard方程的一般化方法在孔隙尺度上模拟多孔介质中的两相流动已成为一种流行方法，因为它具有通过增加最小的复杂性来捕获界面效应的能力。在这里，我们使用提升的场方程并本着有理热力学的精神利用热力学的第二定律来开发一个框架，该框架在宏观上针对两个相恢复液相的Korteweg应力张量，推广达西定律并恢复经典Cahn-Hilliard方程。得出了在宏观尺度上三相的相应结果，并证明是理查兹方程的一般化，并给出了适当的简化假设，以恢复两相结果。适当简化结果可生成两个液相的孔尺度模型，并被证明可以推广Cueto-Felgueroso和Juanes和Boyer和Quintard等人的先前著作。等 还将结果与Cahn-Hilliard Brinkman方程进行了比较，其中指出，为了物理上一致，状态变量应表示物理量。将该公式与其他公式区分开的一个关键方面是，它捕获了三个界面（气-液，气-固和液-固）的不同能量，而没有在微观尺度上引入相应的量（界面张力，接触角等） ）。