Two-phase flow in heterogeneous porous media II: Numerical experiments for flow perpendicular to a stratified system

Abstract

In this work, we make use of numerical experiments to explore our original theoretical analysis of two-phase flow in heterogeneous porous media (Quintard and Whitaker, 1988). The calculations were carried out with a two-region model of a stratified system, and the parameters were chosen be consistent with practical problems associated with groundwater flows and petroleum reservoir recovery processes. The comparison between theory (the large-scaled averaged equations) and experiment (numerical solution of the local volume averaged equations) has allowed us to identify conditions for which the quasi-static theory is acceptable and conditions for which a dynamic theory must be used. Byquasi-static we mean the following: (1) The local capillary pressure,everywhere in the averaging volume, can be set equal to the large-scale capillary pressure evaluated at the centroid of the averaging volume and (2) the large-scale capillary pressure is given by the difference between the large-scale pressures in the two immiscible phases, and is therefore independent of gravitational effects, flow effects and transient effects. Bydynamic, we simply mean a significant departure from the quasi-static condition, thus dynamic effects can be associated with gravitational effects, flow effects and transient effects. To be more precise about the quasi-static condition we need to refer to the relation between the local capillary pressure and the large-scale capillary pressure derived in Part I (Quintard and Whitaker, 1990). $$\begin{gathered} p_c \left| {_y = \left\{ {p_c } \right\}^c } \right|_x + \left( {\rho _\gamma - \rho _\beta } \right)g \cdot \left( {y - \left\{ y \right\}^c } \right) + \Omega _\gamma \cdot [\left( {y + b_\gamma } \right) - \left\{ {y + b_\gamma } \right\}^c ] - \Omega _\beta \cdot [\left( {y + b_\beta } \right) - \left\{ {y + b_\beta } \right\}^c ] + \hfill \\ + \tfrac{1}{2}\nabla \Omega _\gamma :[\left( {yy + D_\gamma } \right) - \left\{ {yy + D_\gamma } \right\}^c ] - \tfrac{1}{2}\nabla \Omega _\beta :[\left( {yy + D_\beta } \right) - \left\{ {yy + D_\beta } \right\}^c ] + \hfill \\ + [\left( {\mu _\gamma A_\gamma - \mu _\beta A_\beta } \right) - \left\{ {\mu _\gamma A_\gamma - \mu _\beta A_\beta } \right\}^c ]\frac{{\partial \left\{ { \in _\beta } \right\}*}}{{\partial t}} + \hfill \\ + [\left( {\mu _\gamma c_\gamma - \mu _\beta c_\beta } \right) - \left\{ {\mu _\gamma c_\gamma - \mu _\beta c_\beta } \right\}^c ] \cdot \nabla \frac{{\partial \left\{ { \in _\beta } \right\}*}}{{\partial t}} + \hfill \\ + \mu _\gamma (E_\gamma - \left\{ {E_\gamma } \right\}^c ):\nabla \Phi _\gamma - \mu _\beta (E_\beta - \left\{ {E_\beta } \right\}^c ):\nabla \Phi _{\beta \cdot } \hfill \\ \end{gathered} $$ Herep c ¦y represents the local capillary pressure evaluated at a positiony relative to the centroid of the large-scale averaging volume, and {p c }¦x represents the large-scale capillary pressure evaluated at the centroid. In addition to{p c } c being evaluated at the centroid, all averaged terms on the right-hand side of Equation (1) are evaluated at the centroid. We can now write the equations describing the quasi-static condition as $$\left\{ {p_c } \right\}^c = \left\{ {\left\langle {p_\gamma } \right\rangle ^\gamma } \right\}^\gamma - \left\{ {\left\langle {p_\beta } \right\rangle ^\beta } \right\}^\beta ,$$ , $$p_c \left| {_y = \left\{ {p_c } \right\}^c } \right|_{x \cdot } $$ , This means that the fluids within an averaging volume are distributed according to the capillary pressure-saturation relationwith the capillary pressure held constant. It also means that the large-scale capillary pressure is devoid of any dynamic effects. Both of these conditions represent approximations (see Section 6 in Part I) and one of our main objectives in this paper is to learn something about the efficacy of these approximations. As a secondary objective we want to explore the influence of dynamic effects in terms of our original theory. In that development only the first four terms on the right hand side of Equation (1) appeared in the representation for the local capillary pressure. However, those terms will provide an indication of the influence of dynamic effects on the large-scale capillary pressure and the large-scale permeability tensor, and that information provides valuable guidance for future studies based on the theory presented in Part I.