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. [ABSTRACT FROM AUTHOR]
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