The Comparable Efficiency of Mathematical Techniques Used in Reservoir Simulation

Abstract

Abstract The results of this work compare the computing efficiency of several mathematical methods which can be used in solving the equations governing reservoir fluid flow. Two-dimensional, water-oil movement in a vertical cross-section and in two areal reservoirs are considered as examples. The fluids used are slightly compressible and both gravitational and viscous forces are considered. The mathematical techniques considered are successive line overrelaxation; block Jacobi, semi-iterative; iterative and noniterative alternating direction implicit; the strongly implicit procedure; symmetric successive line overrelaxation, semi-iterative, and line Saul'Ev. A new method is presented for comparing "equal quality" results. Basically, this involves the determination of a correlation of quality in a material balance-type term vs quality in solving the equations, the L2 norm. The conclusions of this work disagree with reports in the literature in some respects. These differences are attributed to the fact that a compressible fluid system was considered in this work. Results indicate that geometry and fluid properties affect the convergence rates of iterative techniques. Furthermore, noniterative techniques require extremely small time steps to produce the same quality solutions as iterative techniques produce. Introduction The use of reservoir simulators to predict field performance has become widely accepted by the petroleum industry in the past decade. Engineers use both iterative and noniterative techniques to obtain approximate solutions of linear equations which described fluid flow. Variational methods are presently being investigated. With the increased used of reservoir simulators, means of reducing the computer time required to make field studies are needed. Indeed, the amount of computer time required to solve a particular field problem may vary tenfold, depending only on the mathematical method used. This investigation is concerned with comparing the computational efficiency of solution techniques of finite difference approximations to the equations describing reservoir fluid flow. Iterative methods are discussed in detail and noniterative methods are briefly considered. Iterative procedures usually require a set of parameters to speed convergence to an acceptable parameters to speed convergence to an acceptable answer. These methods are of little value to the engineer unless a reliable procedure for choosing effective parameters is known. Where possible, this investigation makes use of possible, this investigation makes use of programmed algorithms to compute parameters programmed algorithms to compute parameters internally in the reservoir simulator.