Multidimensional transport for reservoir simulation is typically solved by applying 1D numerical methods in each spatial-coordinate direction. This approach is simple, but the disadvantage is that numerical errors become highly correlated with the underlying computational grid. In many real-field applications, this can result in strong sensitivity to grid design not only for the computed saturation/composition fields but also for critical integrated data such as breakthrough times. Therefore, to increase robustness of simulators, especially for adverse-mobility-ratio flows that arise in a variety of enhanced-oil-recovery (EOR) processes, it is of much interest to design truly multidimensional schemes for transport that remove, or at least strongly reduce, the sensitivity to grid design.
We present a new upstream-biased truly multidimensional family of schemes for multiphase transport capable of handling countercurrent flow arising from gravity. The proposed family of schemes has four attractive properties: applicability within a variety of simulation formulations with varying levels of implicitness, extensibility to general grid topologies, compatibility with any finite-volume flow discretization, and provable stability (monotonicity) for multiphase transport. The family is sufficiently expressive to include several previously developed multidimensional schemes, such as the narrow scheme, in a manner appropriate for general-purpose reservoir simulation.
A number of waterflooding problems in homogeneous and heterogeneous media demonstrate the robustness of the method as well as reduced transverse (cross-wind) diffusion and grid-orientation effects.