On the definition of macroscale pressure for multiphase flow in porous media

We consider immiscible two-phase flow in porous media, starting with the Stokes equations. Our analysis leads to Darcy's law but with notable differences from the usual interpretation. The most immediate difference is the interpretation of macroscale pressure, which, contrary to previous derivations, does not equal the intrinsic phase average pressure. We recover the intrinsic average only when systematic subscale heterogeneities, in material properties or fluid distribution, are absent. Examples using capillary tube and dynamic pore network models are given. These results impact our understanding of multiphase flow and have a direct effect on numerical upscaling efforts, including calculations of continuum-scale flow parameters from pore-scale network models.