Generalized nonequilibrium capillary relations for two-phase flow through heterogeneous media

Abstract

For two-phase flow in porous media, the natural medium heterogeneity necessarily gives rise to capillary nonequilibrium effects. The relaxation to the equilibrium is a slow process which should be introduced in macroscopic flow models. Many nonequilibrium models are based on a phenomenological approach. At the same time there exists a rigorous mathematical way to develop the nonequilibrium equations. Its formalism, developed by Bourgeat and Panfilov [Computational Geosciences 2, 191 (1998)], is based on the homogenization of the microscale flow equations over medium heterogeneities. In contrast with the mentioned paper, in which the case of a sufficiently fast relaxation was analyzed, we consider the case of long relaxation, which leads to the appearance of long-term memory on the macroscale. Due to coupling between the nonlinearity and nonlocality in time, the macroscopic model remains, however, incompletely homogenized, in the general case. At the same time, frequently only the relationship for the nonequilibrium capillary pressure is of interest for applications. In the present paper, we obtain such an exact relationship in two different independent forms for the case of long-term memory. This relationship is more general than that obtained by Bourgeat and Panfilov. In addition, we prove the comparison theorem which determines the upper and lower bounds for the macroscopic model. These bounds represent linear flow models, which are completely homogenized. The results obtained are illustrated by numerical simulations.

A fully homogenized model for incompressible two-phase flow in double porosity media

“…Let us point out that the matrix-fracture source terms Q δ w , Q δ n of the system (1), given in an implicit form by (8), involve the function S δ m which is a solution of the local boundary value problem (16), which is coupled with the global problem (1)-(4) through its boundary condition. This feature of the system (1)- (8), (16) is captured by the concept introduced in [5]: the homogenized system of equations is said to be fully homogenized if it does not involve the unknown functions which are defined as the solutions of the coupled local problems. The global δ-problem (1)-(8), (16) is not fully homogenized in the said sense.…”

Approximating the vanishing capillarity limit of two‐phase flow in multi‐dimensional heterogeneous porous medium

“…As explained for example in [3,4,15], rock heterogeneities strongly influence the behavior of the flow, since a singular effect can be induced at the rock discontinuities. Therefore our purpose is to consider the capillarity only at the rock discontinuity, and to neglect it within the homogeneous domain where it plays a minor role.…”

Homogenized Model of Two-Phase Flow with Local Nonequilibrium in Double Porosity Media

“…The two-phase flow in double porosity media at local equilibrium is well studied; see, for example, [7][8][9]. It was shown that, in such systems, despite the local equilibrium, another nonequilibrium phenomenon appears on the macroscale, which is caused by the high delay in capillary redistribution of the fluids between blocks and fractures.…”