A Two-scale Operator Splitting for Flow Problems in Porous Media

Operator splitting time discretization techniques are getting more and more popular for the numerical simulation of non-linear problems in porous media. The main idea is to split complex operators in evolution equations into simpler ones which are successively solved in each time step. For porous media flow a natural splitting is given by the diffusive and the advective part of the flux. Then for each part, optimal solvers for the particular evolution equation can be used, i.e., a parabolic solver for the diffusion equation and a hyperbolic solver (e.g., characteristics methods) for the advection equation. In this talk, we first present an operator splitting discretization using an implicit finite volume scheme for the diffusive part and a semi-Lagrangian method for the advective part. Hence, the computational cost of the semi-Lagrangian method can be neglected compared to the one of the implicit finite volume scheme. In the second part, we introduce a two-scale operator splitting method where the diffusion part is solved on a coarse grid while the advection part is considered on a fine grid. This is motivated by the fact that the diffusion and the advection act on different scales. Then, the resulting operator splitting discretization is equivalent in performance to a fast hyberbolic solver. In numerical examples, our proposed methods are compared with standard implicit finite volume methods. We will demonstrate that by using the operator splitting time discretization, the number of iterations in the implicit finite volume scheme may be significantly reduced.Moreover, for the two-scale method, it turns out that despite of the poor approximation in the diffusive part, the solution is still of equal quality compared to that of standard methods on fine grids. To summarize, operator splitting techniques provide a very flexible and powerful framework which makes it quite attractive for an efficient solution strategy for flow problems in porous media.