A 3-D Extension of the Multiscale Control Volume Method for the Simula- tion of the Steady-State Diffusion Equation

Resumo

The level of detail on modern geological models requires higher resolution grids that may render the simulation of multiphase flow in porous media intractable. Moreover, these models may comprise highly heteroge-neous media with phenomena taking place in different scales. The Multiscale Finite Volume (MsFV) method can tackle such issues by constructing a set of numerical operators that map quantities from the fine-scale domain to a coarser one where the initial problem can be solved at a lower computational cost and the solution mapped back to the original scale. Unlike more traditional techniques like homogenization and upscaling, the MsFV has the advantage of maintaining the coupling between the scales even when there is no clear scale separation. However, the MsFV formulation is limited to k-orthogonal grids since it uses a Two-point Flux Approximation (TPFA) method and employs an algorithm to generate the coarse meshes that is not capable of handling general geometries. The Multiscale Restriction Smoothed-Basis method (MsRSB) improves on the MsFV by introducing a new iterative procedure to find the multiscale operators and modifying the algorithm for the generation of the multiscale geometric entities to accommodate unstructured coarse grids, but is still limited to structured fine grids due to the TPFA discretization. Finally, the Multiscale Control Volume method (MsCV) replaces the TPFA by the Multipoint Flux
Approximation with a Diamond stencil (MPFA-D) scheme on the fine-scale while further enhancing the generation of the geometric entities to allow truly unstructured grids on the fine and coarse scales for two-dimensional simulation. In this work we propose an extension to three-dimensional geometries of both the MsCV and the algorithm to obtain the multiscale geometric entities based on the concept of background grid. We also modify the MPFA-D to use the very robust Generalised Least Squares (GLS) interpolation technique to obtain the required auxiliary nodal unknowns. We show that the 3-D MsCV method produces satisfactory results even for heterogeneous and highly anisotropic media, employing true unstructured grids on both scales to handle the simulation of the steady-state diffusion equation.