Solving three-dimensional problems of two-phase filtration using averaging

Abstract

The present work examines a method of solving three-dimensional problems of two-phase filtration based on averaging the equations and introducing functions reflecting the layering of the flow in collectors inhomogeneous over the thickness. These functions are constructed from the calculation of the two-dimensional flow in the plane of the vertical cross section of the bed. This approach is generalized to the case of the displacement of petroleum by aqueous solutions of chemical reagents. Inhomogeneous problems of multiphase filtration may only be solved numerically. However, in the case of three-dimensional flow, even the use of an effective difference scheme is beset with considerable difficulties, dueprimarily to the increased requirements for memory and speed of operation of the computer. In [1] a principle of approximate integration of the filtration equations in a thin inclined bed was proposed. Assuming a hydrostatic law of vertical pressure distribution, the equations were averaged, and successive solution of problems of ever smaller dimensionality was carried out, not with the initial curves of phase permeability, but with curves of averaged phase permeability. Subsequently, there was further development of this principle of capillary-gravitational equilibrium, and others, to deal with the case of flooding of an inhomogeneously laminar bed [2, 3]. A significantly different approach to the determination of the auxiliary functions, not involving assumptions to as the relations between the viscous drag and the capillary and gravitational forces, is to use the solution of the two-dimensional problem of the replacement of petroleum by water in the plane of the vertical cross section of the bed.