Analysis of the Stability of Explicit Reservoir Simulation Calculations

Abstract

Abstract Von Neumann analysis is the most commonly used method for determining stability of an explicit-in-time finite difference method. Watts and Ramé (1999) describe a different approach based on the error growth matrix for the finite difference method. In this approach, called herein the matrix method, stability is determined by computing the eigenvalues of the error growth matrix. An advantage of the matrix method is it is it deals rigorously with heterogeneities. However, in a practical application, the growth matrix is large, making the cost of computing the eigenvalues prohibitive. Watts and Ramé deal with this through a local calculation. This paper extends the work of Watts and Ramé, making it possible to consider separately the stability impact of individual components of the flow equation. It also includes detailed comparisons with the results of Von Neumann analysis. This work’s primary application relates to research. It can be used to rigorously determine whether a particular explicit calculation is stable. This can lead to an improved understanding of the stability behavior of various computational methods and of various methods for predicting stability. In addition, the local form of it can be used to determine the stable timestep size in an explicit method or to determine what degree of implicitness is needed by individual blocks in an AIM calculation. It is concluded that (1) the most accurate way to determine stability of an explicit-in-time reservoir simulation calculation is the matrix method; (2) the matrix method is computationally expensive and impractical to use in large, field-scale models; (3) the local version of the matrix method offers promise as a way to extend the matrix method to larger models and to use it to determine which blocks to treat implicitly in an AIM application; and (4) the boundary conditions required by the von Neumann method do not occur in actual applications. The approach described here provides the researcher with a way to develop a better understanding of how individual components of the transport calculation affect stability, and it may provide a better way to determine stability in a practical setting.