IMPLICIT FLUX LIMITING SCHEMES FOR PETROLEUM RESERVOIR SIMULATION

Explicit total variation diminishing (TVD) numerical methods have been used in the past to give convergent, high order accurate solutions to hyperbolic conservation equations, such as those governing flow in oil reservoirs. To ensure stability there is a restriction on the size of time step that can be used. Many petroleum reservoir simulation problems have regions of fast flow away from sharp fronts, which means that this time step limitation makes explicit schemes less efficient than the best implicit methods. This work extends the theory of TVD schemes to both fully implicit and partially implicit methods. We use our theoretical results to construct schemes which are stable even for very large time steps. We show how to construct an adaptively implicit scheme which is nearly fully implicit in regions of fast flow, but which may be explicit at sharp fronts which are moving more slowly. In general these schemes are only first-order accurate in time overall, but locally may achieve second-order time accuracy. Results, presented for a one-dimensional Buckley-Leverett problem, demonstrate that these methods are more accurate than conventional implicit algorithms and more efficient than fully explicit methods, for which smaller time steps must be used. The theory is also extended to embrace mixed hyperbolic/parabolic (black oil) systems and example solutions to a radial flow equation are presented. In this case the time step is not limited by the high flow speeds at a small radius, as would be the case for an explicit solution. Moreover, the shock front is resolved more sharply than for a fully implicit method. © 1992 Academic Press, Inc. All rights reserved.