IMPES Stability: The CFL Limit

Abstract

Summary This work considers cocurrent, 3D, single-phase miscible and two-phase immiscible, hyperbolic flow in a general grid, structured or unstructured. A given gridblock or control volume may have any number of neighbors. Heterogeneity, anisotropy, and viscous and gravity forces are included, while tensor considerations are neglected. The flow equations are discretized in space and time, with explicit composition and mobility used in the interblock flow terms [the IMPES (implicit pressure, explicit saturation/concentration) case]. Published stability analyses for this flow in a less general framework indicate that the CFL number must be < 1 or < 2 for stability. A recent paper reported stable 1- and 2D Buckley-Leverett two-phase simulations for CFL limits up to 2. A subsequent paper presented a stability analysis predicting a CFL limit of 2 for one of those simulations. This work gives a different reason for that stability up to a CFL limit of 2. This work shows that the eigenvalues of the stability matrix are equal to its diagonal entries which, for any ordering scheme, are in turn equal to 1-CFLi.This leads to a conclusion of an early paper that CFL < 1is required for nonoscillatory stability. This paper discusses cases in which larger CFL limits between 1 and 2 exhibit stability, the existence or absence of applicable theory in such cases, and the practical contribution of such larger CFL limits to increased model efficiency.