A New Iterative Method For Solving Reservoir Simulation Equations

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Abstract
Recently Poussin and Watts have presented some interesting iterative methods of solving matrix equations resulting from the finite-difference approximation of partial differential equations. These methods are however, not competitive with the strongly implicit (SIP) method of Stone for highly anisotropic and heterogeneous reservoirs. 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. Next we present Watts’ method in the notation of our paper. Following this, the new method is developed. Computational 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 usually faced with the problem of solving equations of the parabolic type, (Equation Available In Full Paper) where n is the direction normal to the bounding surface. In spite of the fact that computers are getting bigger and faster and the cost per computation is being reduced the cost of reservoir simulation in many important cases is still too high and cannot be justified, Efforts are being made to reduce the cost of reservoir simulation. Along with improvements in computers, it is also necessary that more efficient methods be developed for the solution of reservoir simulation equations. This is indeed the case and many new methods are continually appearing in the literature. Many useful techniques are often buried in mathematical details which the engineer usually tries to avoid. In another paper(1), we have given mathematical details of a new method for solving the matrix equations resulting from the finite-difference approximations of Equations (1) and (2), In addition to the consideration of the boundary condition given by Equation (3), we have also discussed other boundary conditions in that paper. Our objective here is to present important practical results without giving any mathematical details which are not necessary as far as applications 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 Equation n) and then show how the same method can be applied to Equation (2). Many important theoretical results not presented here are contained in the companion paper(1), and some of the practical considerations 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 shown in Figure L As usual, we divide the domain of interest into grid blocks and locate a grid point inside each grid block.

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