2-PHASE FLOW IN HETEROGENEOUS POROUS-MEDIA .1. THE INFLUENCE OF LARGE SPATIAL AND TEMPORAL GRADIENTS

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 β-phase that are given by {Mathematical expression}. Here {〈vβ〉} represents the large-scale averaged velocity for the β-phase, {εβ}* represents the largescale volume fraction for the β-phase and Kβ* represents the large-scale permeability tensor for the β-phase. We have considered only the case of the flow of two immiscible fluids, thus the large-scale equations for the γ-phase are identical in form to those for the β-phase. The terms in the momentum equation involving {Mathematical expression} and {Mathematical expression} result from the transient nature of the closure problem, while the terms containing {Mathematical expression} and Φβ 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 {Mathematical expression}. 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 of uβ, etc., are available and they clearly indicate that the nontraditional terms in the momentum equation can be discarded when lH ≪ℒ. Here lH is the characteristic length scale for the heterogeneities and ℒ is the characteristic length scale for the large-scale averaged quantities. When lH 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. © 1990 Kluwer Academic Publishers.