A heterogeneous multiscale MPFA method for single-phase flows in porous media with inertial effects

Abstract

Upscaling of flow from pore to Darcy scale is a long-standing research field within flow in porous media. It is well known that non-linearities can occur in near-well regions and high-porosity or fractured media. At the same time, the upscaled non-linear effects associated with high flow rates are hard to quantify a priori with single-scale models. Advances in pore-scale imaging combined with increased computational have made flow simulations in small pore-scale domain feasible, but computations on domains larger than at most a few centimeters are still elusive. In this work, we present a multiscale simulation framework that automatically adapts to non-linear effects as they arise. We formulate a control volume heterogeneous multiscale method (CVHMM) by coupling of a Darcy-scale control volume method with a constitutive relation that is captured based on the fine-scale physics. While the CVHMM formulation works with arbitrary upscaled laws, we emphasize its ability to be applied in fully discrete multiscale context, in particular when a finite element solver is used for solving Navier-Stokes equations on the fine-scale pore geometry. Previous versions of CVHMM are consistent only when the coarse grid is aligned with the upscaled permeability. Herein, we generalize CVHMM by introducing a new coarse solver, thus significantly improving the applicability of the method. The presented method is applied to study flow in near-well regions, as well as media with fractures and irregular grain shapes. The examples show that the method successfully copes with general grids and pore geometries and handles flows with varying degree of non-linearities even outside the domain of applicability of classical upscaled models. In terms of computational efficiency, the method seamlessly localizes computations to regions where non-linear effects are important.