Coats, K.H., Member AIME, The U. of Texas, Austin, Texas
Abstract
This paper describes a generalized analysis for calculating three-phase, three-dimensional flow in reservoirs. The analysis handles pressure maintenance type problems where fluid compressibility effects are negligible. A separate analysis for depletion type problems is described in another paper. The calculations consist of numerical, simultaneous solution of the three-flow equations using the iterative alternating direction technique of Douglas and Rachford. The mathematical details are fully described in the Appendix. The analysis is a computerized mathematical model that accounts for gravity, and capillary and viscous forces, and allows arbitrary reservoir heterogeneity, geometry, well locations and rates. A unique aspect of the analysis is the simultaneous solution of only as many difference equations in each grid block of the reservoir as there are mobile phases present. Thus, while the analysis handles phases present. Thus, while the analysis handles three-phase flow, the efficiency of the calculations (in a typical problem where three phases actually coexist only in a minor portion of the reservoir) is four to eight times greater than that of an analysis solving three equations in every block. The program may be applied to two-phase flow problems and to one-, two- or three-dimensional flow problems with negligible loss in efficiency, compared to programs specifically written for these sub cases. This paper also describes several applications of the analysis which illustrate some effects of gravitational and capillary forces in waterflooding of a heterogeneous reservoir. Another application indicates the utility of the program in simulating the fillup stage of water injection into a reservoir containing an initial free gas phase. Computer times and costs for the applications performed are given to indicate the current expense performed are given to indicate the current expense of three-dimensional, three-phase reservoir simulation.
Introduction
Under pressure maintenance by water and/or gas injection, fluid compressibility effects are generally negligible in producing operations. Although gas compressibility may be appreciable, the maintenance of pressure results in negligible time variation of gas density. In addition, the spatial variation of gas density is usually small in relation to the gas density itself. Producing schemes of pattern or flank waterflood and/or crestal gas injection, therefore, may be simulated with an analysis which presumes fluid incompressibility. presumes fluid incompressibility. The computing efficiency of a numerical model for simulating incompressible fluid flow is as much as 50 percent greater than that of a compressible flow model. Therefore, an analysis for numerically simulating three-dimensional flow of three incompressible, immiscible phases was developed and programmed. programmed.
THE MODEL
The equations describing three-phase, incompressible flow are the continuity equation and Darcy's law for each flowing phase. Combining these equations and introducing capillary pressures gives the three flow equations:
................ (1a)
............... (1b)
SPEJ
p. 331
Linear stability of finite difference equations for three-dimensional flow problems