Two-phase flow in heterogeneous porous media I: The influence of large spatial and temporal gradients期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Two-phase flow in heterogeneous porous media I: The influence of large spatial and temporal gradients

Authors:

Institution:

1. Laboratoire Energétique et Phénomènes de Transfert, UA CNRS 873, Ecole Nationale Supérieure d'Arts et Métiers, 33405, Talence Cedex, France

Abstract:

In order to capture the complexities of two-phase flow in heterogeneous porous media, we have used the method of large-scale averaging and spatially periodic models of the local heterogeneities. The analysis leads to the large-scale form of the momentum equations for the two immiscible fluids, a theoretical representation for the large-scale permeability tensor, and a dynamic, large-scale capillary pressure. The prediction of the permeability tensor and the dynamic capillary pressure requires the solution of a large-scale closure problem. In our initial study (Quintard and Whitaker, 1988), the solution to the closure problem was restricted to the quasi-steady condition and small spatial gradients. In this work, we have relaxed the constraint of small spatial gradients and developed a dynamic solution to the closure problem that takes into account some, but not all, of the transient effects that occur at the closure level. The analysis leads to continuity and momentum equations for the-phase that are given by

$\begin{gathered} \frac{{\partial \left\{ {\varepsilon _\beta } \right\}}}{{\partial t}} + \nabla \cdot \left\{ {\langle V_\beta \rangle } \right\} = 0, \hfill \\ \left\{ {\langle V_\beta \rangle } \right\} = - \frac{1}{{\mu _\beta }}K_\beta ^* \cdot \left( {\nabla \left\{ {\langle p_\beta \rangle ^\beta } \right\}^\beta - \rho _\beta g} \right) - u_\beta \frac{{\partial \left\{ {\varepsilon _\beta } \right\}}}{{\partial t}}^* - U_\beta \cdot \nabla \frac{{\partial \left\{ {\varepsilon _\beta } \right\}^* }}{{\partial t}} - \hfill \\ - \frac{1}{{\mu _\beta }}\mathcal{M}_\beta :\nabla \nabla \left\{ {\langle p_\beta \rangle ^\beta } \right\}^\beta - \frac{1}{{\mu _\beta }}\mathcal{R}_\beta :\nabla \Phi _\beta - \frac{1}{{\mu _\beta }}\Phi _\beta \hfill \\ \end{gathered}$

Keywords:	Two-phase heterogeneous media large-scale averaging dynamic effective properties
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