Reservoir Simulation by Galerkin's Method
Authors: McMichael C.L. , Thomas G.W. 
Published: December 2007

Abstract
Multidimensional multiphase flow of compressible fluids is characterized by a set of nonlinear partial differential equations. Their solution is normally achieved numerically through the use of a finite-difference scheme. Not all reservoir problems, however, are readily amenable to such treatment and recently an alternate approach; Galerkin's method, has been employed. Until now, the application of this technique has been limited to one- and two-phase systems in at most two dimensions.
The primary thrust of this work was to investigate the feasibility of using Galerkin's method on three-phase, multidimensional, compressible flow problems. A reservoir simulation model that treats problems. A reservoir simulation model that treats the reservoir as a nonhomogeneous, irregularly bounded system has been developed. Since this work was primarily concerned with the feasibility of Galerkin simulation, no attempt was made to study a wide spectrum of reservoir problems. However, a few typical applications are presented and some of the results are compared with those derived from a finite-difference simulator.
This work shows that the use of Galerkin's method is feasible and that, in many cases, it results in solutions that are more realistic than those from a finite-difference model. There are, however, certain disadvantages. For example, the computational time and programming effort are usually in excess of programming effort are usually in excess of that required by a finite-difference scheme. Even so, it is felt that the potential of the technique is sufficient justification for this work and for a continuing effort to apply it to reservoir simulation problems.

Introduction
Within the last 20 years, the oil industry has experienced a rapid growth in reservoir engineering technology, stimulated by a desire to maximize recoveries from known reserves. It has been characterized by efforts to predict reservoir behavior in a more realistic manner than in the past. Thus the "tank" concept of a reservoir is now considered inadequate when one desires a model that will reflect the presence of wells and reservoir heterogeneities and that will also simulate unsteady-state flow behavior. This trend toward more realistic descriptions has culminated in treating coupled systems of nonlinear partial differential equations describing multiphase, multidimensional flow in porous media. In the late 1950's and early 1960's the mathematical apparatus was developed to solve such systems of equations. Primarily, the methods referred to rely upon Primarily, the methods referred to rely upon reducing the partial differential equations to algebraic systems by means of finite-difference approximations.
While some success has been enjoyed with these techniques, not all reservoir problems are readily amenable to such treatment. For example, those regimes that give rise to shock-front development or involve convective dispersion are usually poor candidates for standard finite-difference treatment. In such cases, one must resort to special computational techniques that frequently are impractical to implement in a field-scale simulator. Since finite - difference approaches involve discretizing both the dependent and the independent variables, only a discrete solution in time and space is obtained. This is frequently a disadvantage when, for example, one would like to predict accurately the behavior of bottom-hole well pressures with time. pressures with time. Recognition of the problems cited above has led to a search for alternative approaches. Among the more promising are protection methods and, in particular, Galerkin's method. With Galerkin's particular, Galerkin's method. With Galerkin's procedure, one obtains continuous solutions in procedure, one obtains continuous solutions in space much like an analytical expression. Furthermore, by appropriate selection of basis functions, one can even achieve continuity in the derivatives. Thus the method lends itself to problems where sharp gradients occur, either in problems where sharp gradients occur, either in saturation or in pressure.
The purpose of this work was to determine the feasibility of Galerkin's method in treating multidimensional, three-phase flow in nonhomogeneous, irregularly bounded systems containing multiple wells. The mathematical formulation for reducing the system of differential equations to a set of algebraic equations is presented.