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Two-Phase Inertial Flow in Homogeneous Porous Media: A Theoretical Derivation of a Macroscopic Model

Authors:

Didier Lasseux Azita Ahmadi Ali Akbar Abbasian Arani

Institution:

(1) TREFLE – UMR CNRS 8508 – University Bordeaux 1, Esplanade des Arts et Metiers, Talence, 33405, France

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 $\nabla \langle p_{\alpha}\rangle^{\alpha}$ to the filtration or Darcy velocity $\langle {\mathbf{v}}_{\alpha}\rangle$ in a coupled nonlinear form explicitly given by

$\langle {\mathbf{v}}_{\alpha}\rangle =-\frac{{\mathbf{K}}_{\alpha \alpha}^{{\mathbf{\ast}}}}{\mu _{\alpha}} \,\cdot\, (\nabla \langle p_{\alpha}\rangle^{\alpha}-\rho _{\alpha}{\mathbf{g}})-{\mathbf{F}}_{\alpha \alpha} \,\cdot\, \langle {\mathbf{v}}_{\alpha} \rangle -\frac{{\mathbf{K}}_{\alpha \kappa}^{{\mathbf{\ast}}}}{\mu_{\kappa}} \,\cdot\, (\nabla \langle p_{\kappa}\rangle^{\kappa}-\rho_{\kappa}{\mathbf{g}})-{\mathbf{F}}_{\alpha \kappa}\,\cdot\, \langle {\mathbf{v}}_{\kappa}\rangle \qquad\alpha ,\kappa =\beta, \gamma \quad\alpha \neq \kappa$

or equivalently

$\langle {\mathbf{v}}_{\alpha}\rangle =-\frac{{\mathbf{K}}_{\alpha}}{\mu _{\alpha}}\,\cdot\, (\nabla \langle p_{\alpha} \rangle^{\alpha}-\rho _{\alpha}{\mathbf{g}})-{\mathbf{F}}_{\alpha \alpha} \,\cdot\, \langle {\mathbf{v}}_{\alpha}\rangle +\,{\mathbf{K}}_{\alpha \kappa}\,\cdot\, \langle {\mathbf{v}}_{\kappa}\rangle - {\mathbf{F}}_{\alpha \kappa} \,\cdot\, \langle {\mathbf{v}}_{\kappa}\rangle\qquad \alpha ,\kappa =\beta ,\gamma \quad\alpha \neq \kappa$

In these equations, ${\mathbf{F}}_{\alpha \alpha}$ and ${\mathbf{F}}_{\alpha \kappa}$ are the inertial and coupling inertial correction tensors that are functions of flow-rates. The dominant and coupling permeability tensors ${\mathbf{K}}_{\alpha \alpha}^{{\mathbf{\ast}}}$ and ${\mathbf{K}}_{\alpha \kappa}^{{\mathbf{\ast}}}$ and the permeability and viscous drag tensors ${\mathbf{K}}_{\alpha}$ and ${\mathbf{K}}_{\alpha \kappa}$ 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.

Keywords:

Homogeneous porous media Two-phase flow Inertial or non-Darcian flow Up-scaling Volume averaging

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