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

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 theβ-phase that are given by $$\begin{gathered} \frac{{\partial \left\{ {\varepsilon _\beta } \right\}}}{{\partial t}} + \nabla \cdot \left\{ {\langle V_\beta \rangle } \right\} = 0, \hfill \\ \left\{ {\langle V_\beta \rangle } \right\} = - \frac{1}{{\mu _\beta }}K_\beta ^* \cdot \left( {\nabla \left\{ {\langle p_\beta \rangle ^\beta } \right\}^\beta - \rho _\beta g} \right) - u_\beta \frac{{\partial \left\{ {\varepsilon _\beta } \right\}}}{{\partial t}}^* - U_\beta \cdot \nabla \frac{{\partial \left\{ {\varepsilon _\beta } \right\}^* }}{{\partial t}} - \hfill \\ - \frac{1}{{\mu _\beta }}\mathcal{M}_\beta :\nabla \nabla \left\{ {\langle p_\beta \rangle ^\beta } \right\}^\beta - \frac{1}{{\mu _\beta }}\mathcal{R}_\beta :\nabla \Phi _\beta - \frac{1}{{\mu _\beta }}\Phi _\beta \hfill \\ \end{gathered} $$ . Here {〈vβ〉} represents the large-scale averaged velocity for theβ-phase, {eβ}* represents the largescale volume fraction for theβ-phase andKβ* represents the large-scale permeability tensor for theβ-phase. We have considered only the case of the flow of two immiscible fluids, thus the large-scale equations for theγ-phase are identical in form to those for theβ-phase. The terms in the momentum equation involving\({{\partial \left\{ {\varepsilon _\beta } \right\}^* } \mathord{\left/ {\vphantom {{\partial \left\{ {\varepsilon _\beta } \right\}^* } {\partial t}}} \right. \kern-\nulldelimiterspace} {\partial t}}\) and\({{\nabla \partial \left\{ {\varepsilon _\beta } \right\}^* } \mathord{\left/ {\vphantom {{\nabla \partial \left\{ {\varepsilon _\beta } \right\}^* } {\partial t}}} \right. \kern-\nulldelimiterspace} {\partial t}}\) result from the transient nature of the closure problem, while the terms containing\(\nabla \nabla \left\{ {\langle p_\beta \rangle ^\beta } \right\}^\beta ,\nabla \Phi _\beta \) andΦβ are the results of nonlinear variations in the large-scale field. All of the latter three terms are associated with second derivatives of the pressure and thus present certain unresolved mathematical problems. The situation concerning the large-scale capillary pressure is equally complex, and we indicate the functional dependence of {pc}c by $$\left\{ {p_c } \right\}^c = \mathcal{F}\left( {\partial \left\{ {\varepsilon _\beta } \right\}^* ,\left( {\rho _\gamma - \rho _\beta } \right)g,\nabla \left\{ {\langle p_\beta \rangle ^\beta } \right\}^\beta ,\frac{{\partial \left\{ {\varepsilon _\beta } \right\}^* }}{{\partial t}},etc.} \right)$$ . Because of the highly nonlinear nature of the capillary pressure-saturation relation, small causes can have significant effects, and the treatment of the large-scale capillary pressure is a matter of considerable concern. On the basis of the derived closure problems, estimates ofuβ, etc., are available and they clearly indicate that the nontraditional terms in the momentum equation can be discarded whenlH ≪ℒ. HerelH is the characteristic length scale for the heterogeneities and ℒ is the characteristic length scale for the large-scale averaged quantities. WhenlH is not small relative to ℒ, the nontraditional terms must be considered and nonperiodic boundary conditions must be developed for the closure problem. Detailed numerical studies presented in Part II (Quintard and Whitaker, 1990) and carefully documented experimental studies described in Part III (Berlin et al., 1990) provide further insight into the effects of large spatial and temporal gradients.