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A NEW ITERATIVE METHOD FOR SOLVING RESERVOIR

SIMULATION EQUATIONS

K. AZIZ A. SETTARI

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JCPT72-01-04 A New Iterative Method for Solving Reservoir Simulation Equations KHALID AZIZ, Professor, Department of Chemical Engineering, and ANTONIN SETTARI, Graduate Student, Department of Mechanical Enginee ring, University of C algary, Calgary, Alta. ABSTRACT Recently, Poussin and Watts have presented some interesting iterative methods of solving matrix equations resulting from the finite-difference approximation of par- tial differential equations. These methods are, however, not competitive with the strongly implicit (SIP) method of Stone for highly anisotropic and heterogeneousreser- voirs. Our method is an extension of the method of Watts and it is competitive with SIP from a computational point of view; furthermore our method is easier to program. In this paper, we first formulate the matrix problem using the notation of Stone. N ext we present Watts' method in the notation of our paper. F ollowing this, the new method is developed. Computa tional algorithms for the implementation of both of these methods are also presented. Finally, we present numerical results and a comparison of several numerical methods. INTRODUCTION IN RESERVOIR SIMULATION and in many

other problems involving partial differential equations, we are usual- ly faced with the problem of solving equations of the parabolic type, Q(X,Y) + P(lx,y) at .......... (1) K. AZIZ A. SETTARI K. AZIZ is Professor of Chemical Engineering at The University of Calgary. He received his engineering educa- tion at The University of Michigan, University of Alberta and Rice University. He is author or co-author of ap- proximately 40 technical papers and one book. His re- search interests include reservoir simulation, multiphase flow in

pipelines, atmospheric pollution and heat transfer _ in porous media. A. SETTARI received his first degree from Technical University, Brno, in Czechoslovakia, and worked for three years in gas turbine research. He joined The University of Calgary in 1969 as a graduate student and is now enrolled in a Ph.D. program. 62 or elliptic type, (2) The boundary conditions for most reservoir simula- tion problems are of the Neumann type, i.e., ................. ........... (3) where n is the

direction normal to the bounding sur- face. In spite of the fact that computers are getting bigger and fast er and the cost per computation is being reduced, the cost of reservo ir simulation in many important cases is still too high and canno t be justified. Efforts are being made to reduce the cost of reservoir simu lation. Along with improvements in computers, it is also nec essary that more efficient methods be develope d for the solution of reservoir simulation equati ons. This is indeed the case and many new methods are continuall y appearing in the litera- ture. Many useful techniques areoften buried in mathematical details which the engineer usually tries to avoid. In another paper(1), we have given mathematical de- tails of a new method for solving the matrix equations resulting from the finite-difference approximations of Equations (1) and (2). In additio n to the considera- tion of the boundary condition given by Equation (3), We have also discussed other boundary condi- tions in that paper. Our objective here is to present important practical results, without giving any mathe- matical details which are not necessary as far as ap- plications of the method are

concerned. In order to keep our discussion brief and simple, here we will deal in detail with the solution of the parabolic Equa- tion (1) and then show how the same method can be applied to Equation (2). Many important theoretical result.@ not presented here are contained in the com- panion paper(1), and some of the practical considera- tions given here are not discussed elsewhere. DERIVATION OF MATRIX EQUATION The method to be presented is quite general and can be applied to irregularly shaped domains (like natural petroleum reservoirs) ; however, to illustrate the method we will

choose a simple example of a rectangular domain as show n in Figure 1. As usual, we divide the domain of interest into grid blocks and _ locate a grid poin t inside each grid block. The general form of the finite-difference equ ation for each grid point is the same for equally or unequally spaced gri d points; whereas the accuracy of the solution does depen d on the selection of grid blocks for the reservoir(2). Again for the sak e of simplicity, we will select grid blocks so that the distance betw een all grid points in any direction is the same. We emphasize that these restrictions are not

necessary for the ap- The Journal of Canadian Petroleum

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