Two-phase flow in heterogeneous porous media III: Laboratory experiments for flow parallel to a stratified system

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Transport in Porous Media 5: 341-379. 1990. Kluwer Academic Publishers. Printed in the Netherlands.

© 1990

Two-Phase Flow in Heterogeneous Porous Media I: The Influence of Large Spatial and Temporal Gradients MICHEL QUINT ARD and STEPHEN WHITAKER * Laboratoire Energetique et Phenomenes de Transfert, VA CNRS 873, Ecole Nationale Superieure d'Arts et Metiers, 33405 Talence Cedex, France (Received: 28 February 1989, revised: 14 September 1989)

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 {3-phase that are given by a{E{3}* at

+ v. {(vll)} = 0,

{(VIl)} =

"'!"K$. (v{(p/3)/3}Il- P/3g) - u/3 a{E/3}* - U/3' v Il-Il at 1 1 1 - -.illl : vv{(p/3)Il}/3 - - ~1l:v/3 - -Il' 1l-/3 1l-/3 1l-/3 -

ah}* at

Here {(v/3)} represents the large-scale averaged velocity for the {3-phase, {Ed* represents the largescale volume fraction for the {3-phase and K$ represents the large-scale permeability tensor for the {3phase. We have considered only the case of the flow of two immiscible fluids, thus the large-scale equations for the 'Y-phase are identical in form to those for the {3-phase. The terms in the momentum equation involving a{ E/3}*lat and v iJ{ EIl}*/iJt result from the transient nature of the closure problem, while the terms containing vV{(p/3)Il}/3. v/3 and /3 are the results of nonlinear variations in the largescale 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 {pJc by { Pc } c -_

CP' ( {

Yfr

)/3}1l ' a{ E/3}* ) Ell }* ' ( Py _ Pil ) g, "{( v P/3 at ' etc ..

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 *Permanent address: Department of Chemical Engineering, University of California, Davis, CA 95616 U.S.A.

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concern. On the basis of the derived closure problems, estimates of ull' UIl , etc., are available and they clearly indicate that the nontraditional terms in the momentum equation can be discarded when IH