Solution of the Immiscible Fluid Flow Simulation Equations

Abstract

American Institute of Mining, Metallurgical and Petroleum Engineers, Inc. Abstract This paper presents the methods used to solve the finite difference equations which we developed in a companion paper (1). Various possible methods of solution are discussed. Experience has narrowed the numb of suitable numerical methods that are practical to three: Gauss elimination, successive practical to three: Gauss elimination, successive overrelaxation, and the iterative alternating direction implicit process. The final sections of the paper are devoted to a presentation of computational technique which are vital to actual use of each of the above-mentioned methods. FINITE DIFFERENCE EQUATIONS, THE MATRIX AND DEFINITIONS The final finite difference equation for pressure developed in Reference (1) is: pressure developed in Reference (1) is: ..........................................(1) All the terms are defined in the paper. Here, however, we have dropped the subscript denoting the pressure, p, as an oil pressure. Further breakdown requires definition of the numerical solution to be used. This paper describes the breakdown and solution processes most often used in the MUFFS program. Sufficient detail is given so that computer programming can be done. Contrary to popular opinion, economic simulation has been found to require the development of several solution methods, rather than relying on a single one. This requires that the computer subprogram for generating coefficients (A's and O's) be written as a distinct, separate entity to supply the coefficients in Equation (1). Furthermore, it is necessary to be able to obtain these coefficients automatically in column-by-column, row-by-raw, or point-by-point form, in any order required by a point-by-point form, in any order required by a numerical solution. Columns, rows, and points refer to the columns, rows, and points of the finite difference grid. A program that can generate coefficients in several forms is a simple but important concept, for it allows the easy insertion and modification of experimental methods. The computing inefficiencies that may be incurred within a general coefficient generator are small in comparison to the computing time saved by using the fastest of several solution techniques.