The Mathematical Model of Non-Equilibrium Effects in Water-Oil Displacement

Abstract

Abstract Forced oil-water displacement and spontaneous countercurrent imbibition are crucial mechanisms of secondary oil recovery. The classical mathematical models of these phenomena are based on the fundamental assumption that in both these unsteady flows a local phase equilibrium is reached in the vicinity of every point. Thus, the water and oil flows are locally redistributed over their flow paths similarly to steady flows. This assumption allowed the investigators to further assume that the relative phase permeabilities and the capillary pressure are universal functions of the local water saturation, which can be obtained from steady-state flow experiments. The last assumption leads to a mathematical model consisting of a closed system of equations for fluid flow properties (velocity, pressure) and water saturation. This model is currently used as a basis for predictions of water-oil displacement with numerical simulations. However, at the water front in the water-oil displacement, as well as in capillary imbibition, the characteristic times of both processes are comparable with the times of redistribution of flow paths between oil and water. Therefore, the non-equilibrium effects should be taken into account. We present here a refined and extended mathematical model for the non-equilibrium two-phase (e. g., water-oil) flows. The basic problem formulation as well as the more specific equations are given, and the results of comparison with experiments are presented and discussed. 1. Introduction The problem of simultaneous flow of immiscible fluids in porous media, and, in particular, the problem of water-oil displacement, both forced and spontaneous, is fundamental to the modern simulations of transport in porous media. This problem is also important for engineering applications, especially in the mathematical simulation of the development of oil deposits. The classical model of simultaneous flow of immiscible fluids in porous media was constructed in late thirties-early forties by the distinguished American scientists and engineers M. Muskat and M. C. Leverett, and their associates. The model was based on the assumption of the local equilibrium, according to which the relative phase permeabilities and the capillary pressure can be expressed through the universal functions of the local saturation. The Muskat-Leverett theory was in the past and is nowadays of fundamental importance for the engineering practice of the development of oil deposits. Moreover, this theory leads to new mathematical problems involving specific instructive partial differential equations. It is interesting to note that some of these equations were independently introduced later as simplified model equations of gas dynamics. Gradually, however, it was recognized that the classical Muskat-Leverett model is not quite adequate, especially for many practically important flows. In particular, it seems to be inadequate for the capillary countercurrent imbibition of a porous block initially filled with oil, one of the basic processes involved in oil recovery, and for the even more important problem of flow near the water-oil displacement front. The usual argument in favor of the local equilibrium is based on the assumption that a representative sampling volume of the water-oil saturated porous medium has the size not too much exceeding the size of the porous channels. In fact, it happens that it is not always the case and the non-equilibrium effects are of importance. A model, which made it possible to take into account the non-equilibrium effects, was proposed and developed by the first author and his colleagues. This model was gradually corrected and modified. It was confirmed by laboratory and numerical experiments. In its turn, this model leads to non-traditional mathematical problems.