Two-phase inertial flow in homogeneous porous media: A theoretical derivation of a macroscopic model

Abstract

The purpose of this article is to derive a macroscopic model for a certain class of inertial two-phase, incompressible, Newtonian fluid flow through homogenous porous media. Starting from the continuity and Navier–Stokes equations in each phase β and γ , the method of volume averaging is employed subjected to constraints that are explicitly provided to obtain the macroscopic mass and momentum balance equations. These constraints are on the length- and time-scales, as well as, on some quantities involving capillary, Weber and Reynolds numbers that define the class of two-phase flow under consideration. The resulting macroscopic momentum equation relates the phase-averaged pressure gradient ∇ pα α to the filtration or Darcy velocity vα in a coupled nonlinear form explicitly given by : (equations) In these equations, Fαα and Fακ are the inertial and coupling inertial correction tensors that are functions of flow-rates. The dominant and coupling permeability tensors K∗αα and K∗ακ and the permeability and viscous drag tensors Kα and Kακ are intrinsic and are those defined the conventional manner as in (Whitaker, Chem Eng Sci 49:765–780, 1994) and (Lasseux et al., Transport Porous Media 24(1):107–137, 1996). All these tensors can be determined from closure problems that are to be solved using a spatially periodic model of a porous medium. The practical procedure to compute these tensors is provided.