Implicit Flux Limiting Schemes for Petroleum Reservoir Simulation

Explicit total variation decreasing (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. In general these schemes are only first order accurate in time over all, but locally may achieve second order time accuracy. Results are presented for a one dimensional Buckley Leverett problem, which demonstrate that these methods are more accurate than conventional implicit algorithms and more efficient than explicit methods, where smaller time steps must be used. Results from black oil and compositional simulators are presented.