The Multiscale Finite Volume Method on Unstructured Grids

Abstract

Abstract Finding a pressure solution for large-scale reservoirs that takes into account fine-scale heterogeneities can be very computationally intensive. One way of reducing the workload is to employ multiscale methods that capture local geological variations using a set of reusable basis functions. One of these methods, the multiscale finite-volume (MsFV) method is well studied for 2D Cartesian grids, but has not been implemented for stratigraphic and unstructured grids with faults in 3D. With reservoirs and other geological structures spanning several kilometers, running simulations on the meter scale can be prohibitively expensive in terms of time and hardware requirements. Multiscale methods are a possible solution to this problem, and extending the MsFV method to realistic grids is a step on the way towards fast and accurate solutions for large-scale reservoirs. We present a MsFV solver along with a coarse partitioning algorithm that can handle stratigraphic grids with faults and wells. The solver is an alternative to traditional upscaling methods, but can also be used for accelerating fine-scale simulations. Approximate solutions computed by the new MsFV solver are compared with fine-scale solutions computed by a standard two-point scheme for grids with realistic permeability and geometries. The results show that the MsFV method is suitable for solving realistic permeabilities, but can fail for highly anisotropic systems. The implementation is a suitable framework for further experimentation with partitioning algorithms and MsFV variants. To achieve better precision, the implementation can use the MsFV method as a preconditioner for Arnoldi iterations using GMRES, or for smoothing cycles using Dirichlet Multiplicative Schwarz (DMS).