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