On Efficient Implicit Upwind Schemes

Although many advanced methods have been devised for the hyperbolic transport problems that arise in reservoir simulation, the most widely used method in commercial simulators is still the implicit upwind scheme. The robustness, simplicity and stability of this scheme makes it preferable to more sophisticated schemes for real reservoir models with large variations in flow speed and porosity. However, the efficiency of the implicit upwind scheme depends on the ability to solve large systems of nonlinear equations effectively. To advance the solution one time step, a large nonlinear system needs to be solved. This is a highly non-trivial matter and convergence is not guaranteed for any starting guess. This effectively imposes limitations on the practical magnitude of time steps as well as on the number of grid blocks that can be handled. In this paper, we present an idea that allows the implicit upwind scheme to become a highly efficient. Under mild assumptions, it is possible to compute a reordering of the equations that renders the system of nonlinear equations (block) lower triangular. Thus, the nonlinear system may be solved one (or a few) equations at a time, increasing the efficiency of the implicit upwind scheme by orders of magnitude. Similar ideas can also be used for high-order discontinuous Galerkin discretizations. To demonstrate the power of these ideas we show results and timings for incompressible and weakly compressible transport in real reservoir models.