A Non Uniform Algebraic Dynamic Multilevel for Unstructured Meshes for the numerical Simulation of Fluid Flow in Heterogeneous Porous Media

Abstract

In the present work, we introduce, for the first time in the literature, the Non-Uniform Algebraic Dynamic Multilevel with Unstructured Meshes (NU-ADM-UM) method, which extends the original Non-Uniform Algebraic Dynamic Multilevel method to the simulation of single-phase and two-phase flows on unstructured meshes. In this new method, the employed multiscale transfer operators are the Algebraic Multiscale Solver for General Unstructured Grids (AMS-U) and the Multiscale Restriction Smoothed Basis (MsRSB). To construct the coarse meshes (primal and dual), we adopted the procedure proposed by AMS-U method. The dual mesh obtained via this approach was then used to define the support and boundary regions required by the MsRSB method. The mathematical model results in a system solved using an IMPES (Implicit Pressure Explicit Saturation) strategy, in which pressure is solved implicitly and saturation explicitly. For the pressure field, we adopted a finite volume formulation of the MPFA-D (Multi-Point Flux Approximation with a Diamond scheme) type, with weights computed using the Global Least Squares (GLS) method. This formulation can handle non-K-orthogonal meshes and yields highly accurate solutions. The saturation field is solved using a first-order upwind method. The results show that the mesh refinement parameters of NU-ADM-UM successfully preserved, at the fine scale, the regions that would otherwise produce spurious pressure oscillations. Moreover, the method was able to follow the saturation front in two-phase flow problems, delivering results very close to those obtained by solving the problem on the full fine scale, but with a reduced number of active control volumes throughout the simulation and, consequently, a reduction in total simulation time.