Two-scale modeling in porous media: Relative permeability predictions

Abstract

We present a numerical analysis of fluid flow through a porous medium with two distinct characteristic scales. The system considered is a monodisperse matrix with porosity ϕ and permeability Kpm with an embedded second phase, characterized by a phase content or saturation s and phase length scales Lϕ and Ls. Both two- and three-dimensional simulations are performed to compute the mobile fluid phase relative permeability kr,m and its dependence on s and Kpm. The relative permeability is found to vary as a power law of saturation, with a quasilinear behavior for low permeability, and increasing values of the exponent as Kpm increases. For media with low permeability, the linearity of kr,m is attributed to the drag force, whereas for high Kpm, the decrease of kr,m with s is due primarily to viscous forces. An analytical model for kr,m is also presented to aid the interpretation and to corroborate the simulation results. In the second part, in order to elucidate the role of the length scales on kr,m, simulations explicitly resolving both porous media and second-phase scales are performed. The relative permeability is found to drop rapidly when both scales are of the same order (Ls≈Lϕ) or when Ls<Lϕ. Three regimes (Darcy, Brinkman, Stokes) are consequently identified based on the length scales.