The Impact of Pore‐Scale Flow Regimes on Upscaling of Immiscible Two‐Phase Flow in Porous Media

Abstract

AbstractEmpirical or theoretical extensions of Darcy's law for immiscible two‐phase flow have shown significant limitations in properly modeling the flow at the continuum scale. We tackle this problem by proposing a set of upscaled equations based on pore‐scale flow regimes, that is, the topology of flowing phases. The incompressible Navier‐Stokes equation is upscaled by means of multiple‐scale expansions and its closures derived from the mechanical energy balance for different flow regimes at the pore scale. We also derive the applicability conditions of the upscaled equations based on the order of magnitude of relevant dimensionless numbers, that is, Eotvos, Reynolds, capillary, Froude numbers, and the viscosity and density ratio of the system, as well as a set of closures valid for the basic flow regimes of low Eotvos number systems, that is, core‐annular and plug and drop traffic flows. We provide analytical expressions for the relative permeability of the wetting and nonwetting phases in different flow regimes and demonstrate that the effect of the flowing‐phases topology on the relative permeabilities is significant. Finally, we show that the classical two‐phase Darcy law is recovered for a limited range of operative conditions, while specific terms accounting for interfacial and wall interactions should be incorporated to accurately model ganglia or drop traffic flow.