Fast Sequential Implicit Porous Media Flow Simulations Using Multiscale Finite Elements and Reordering of Cells for Solution of Nonlinear Transport Equation

It is demonstrated previously in the literature that multiscale methods can used to provide accurate highresolution velocity fields at a low computational cost. However, to achieve enhanced accuracy in flow simulations compared with a standard approach, the multiscale method must be accompanied by a transport solver that can account for the fine-scale structures of the velocity fields. In this paper, we use the standard implicit single-point upwind (SPU) finite-volume method for computing transport. This method requires that a nonlinear system is solved at each time-step. However, if we assume (as in streamline methods) that capillary forces can be disregarded and that gravity can be treated by operator splitting, and reordering the cells in an optimal way, the nonlinear systems in each implicit advective step can be solved on a cell-by-cell (or block-by-block) basis. This approach makes the standard SPU method at least as fast as a streamline method, even on geo-cellular models with multimillion cells, and alleviates many limitations that streamline methods have. In particular, the method is mass conservative, compressibility can be handled in a straightforward manner, and pressure can be updated frequently without severely influencing the computational efficiency. By combining this transport solver with a multiscale pressure solver, we obtain a very efficient solution method capable of direct simulation of geo-cellular models with multimillion cells within an acceptable time-frame on a single desktop computer.