A Block Successive Over-Relaxation Method for Coupled Equations

Abstract

American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. Abstract The simultaneous solution method in reservoir simulation gives rise to a system of coupled linear equations. This paper shows that the system can be efficiently solved by a block successive over-relaxation method even for large 3-D, 3-phase models. Examples are given which include a large 3-D problem and a radial coning problem. problem and a radial coning problem Introduction In reservoir simulation problems involving flow in the vertical direction, it is desirable to formulate a semiimplicit or fully implicit finite difference model and simultaneously solve for as many unknowns as practical. This approach yields a very stable solution, and is most useful when the reservoir is highly heterogeneous and when the throughput across the grid blocks is high. Simultaneous solution also allows a convenient implementation of strongly coupled production and reservoir flow terms, which further contribute to stability. However, for easy reservoir simulation problems, the implicit pressure explicit saturation method is often more economical. The technique of solving for the unknowns simultaneously is known as the simultaneous solution method. There are, however, disadvantages in solving for several unknowns by this solution method. The number of linear equations to be solved is equal to the number of mesh points multiplied by the number of unknowns. For example, if a model has 1500 mesh points and 3 unknowns (oil pressure, points and 3 unknowns (oil pressure, gas saturation, and water saturation), there will be 4500 linear equations. A large system of this size precludes the use of direct methods because of excessive storage requirement, unless a special algorithm such as the storage management algorithm is available. For 3-D problems, direct methods also become rapidly uneconomical, as the number of mesh points increases. For example, the computational work for a cubic mesh with Nc3 mesh points is proportional to Nc6. Indeed, it is usually proportional to Nc6. Indeed, it is usually uneconomical to apply direct methods to simultaneous solution models, if the mesh exceeds a few hundred points. Iterative methods require less storage than direct methods and also less computational work if they converge rapidly.