Multiscale Modeling &Simulation, Volume 19, Issue 1, Page 320-343, January 2021.
It is well known that the generalized Darcy law describing multiphase flow in porous media has some shortcomings. In particular, it cannot explain hysteresis effects in the capillary pressure--saturation curve which have been observed in measurements. In this work, we derive a numerically tractable micro-macro model including coupled generalized Darcy's laws that still includes the microscale dynamics which are responsible, e.g., for hysteresis effects. For this purpose, we extend the two-scale expansion approach of periodic homogenization to include different time scales which allows us to start from a fully instationary Navier--Stokes--Cahn--Hilliard model at the pore scale as microscale. Identifying and separating the time scales allows us to derive local fast scale equations describing the microscale dynamics and global slow-scale equations giving rise to the macroscopic Darcy law.

中文翻译：

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从孔隙到达西尺度的多孔介质中两相流的均质化：相场方法

多尺度建模与仿真，第 19 卷，第 1 期，第 320-343 页，2021 年 1 月。
众所周知，描述多孔介质中多相流动的广义达西定律存在一些缺陷。特别是，它无法解释测量中观察到的毛细管压力-饱和曲线中的滞后效应。在这项工作中，我们导出了一个数值上易于处理的微观宏观模型，包括耦合的广义达西定律，该模型仍然包括负责例如滞后效应的微观动力学。为此，我们将周期性均质化的两尺度扩展方法扩展到包括不同的时间尺度，这使我们能够从微观尺度的孔隙尺度上的完全静止 Navier--Stokes--Cahn--Hilliard 模型开始。