Some Invariant Solutions to Two-Phase Fluid Displacement Problems Including Capillary Effect (includes associated papers 18744 and 19037 )

Abstract

Summary Analytical investigation of the two-phase displacement problem that considers capillary force is one of the most important and difficult problems in the theory of fluid flow through porous media. This paper presents some invariant solutions on the subject, including a self-similar solution for the axisymmetric case, a self-similar solution in a linear, isoclinal reservoir that considers the effects of gravity and capillary forces simultaneously, and a progressive wave solution corresponding to a stabilized zone of saturation distribution. The relations and differences amonga model that completely includes capillary effect,a model that considers capillary action partially and implicitly (the Buckley-Leverett model), anda model that completely ignores capillary effect are indicated and discussed. Introduction The capillary effect may be an important force in multiphase flow through porous media. It is included dually in the classic mathematical model of multiphase flow through porous media first proposed by Leverett:indirectly in terms of relative permeability and directly through capillary pressure, the local difference in pressure between two phases. The resultant differential equation is a secondorder nonlinear parabolic equation that is difficult to solve analytically even for the simple case of one-dimensional(ID), two-phase flow. Neglecting the capillary pressure, Buckley and Leverett reduced the equation to a quasilinear, hyperbolic equation of first order for which they obtained an analytic solution. This solution has become a base for engineering calculations in the secondary recovery of oil. However, the discontinuity inherent in the Buckley-Leverett solution bothered reservoir engineers for a long period of time. It was recognized afterward that this physical absurdity is just the result of neglecting the capillary pressure. Several invariant solutions of the unsteady-state two-phase flow equation obtained in the early 1960's taking the capillary pressure into consideration played an important role in this issue. These solutions, including a recent paper, are of great value both in revealing and clarifying the essential nature of two-phase flow in porous media and in evaluating the validity and accuracy of various approximate solutions, including numerical ones. This paper presents some invariant solutions described in Refs. 4, 6, and 7. In addition, a comprehensive discussion on the capillary effect on the displacement process is carried out.