Analysis of Control Volume Heterogeneous Multiscale Methods for Single Phase Flow in Porous Media

Abstract

The standard approximation for the flow-pressure relationship in porous media is Darcy's law that was originally derived for infiltration of water in fine homogeneous sands. Ever since there have been numerous attempts to generalize it for handling more complex flows. Those include upscaling of standard continuum mechanics flow equations from the fine scale. In this work we present a heterogeneous multiscale method that utilizes fine scale information directly to solve problems for general single phase flow on the Darcy scale. On the coarse scale it only assumes mathematically justified conservation of mass on control volumes, that is, no phenomenological Darcy-type relationship for velocity is presumed. The fluid fluxes are instead provided by a fine scale Navier--Stokes mixed finite element solver. This work also considers several choices of quadrature for data estimation in the multiscale method and compares them. We prove that for an essentially linear regime, when the fine scale is governed by Stokes flow, our method converges to a rigorously derived homogenization solution---Darcy's law. Moreover the method gives the flexibility to solve problems with faster nonlinear flow regimes that is important in a number of applications, such as flows that may occur near wells and in fractured regions in subsurface. Those flows are also common for industrial and near surface porous media. The numerical examples presented in the paper verify the estimate and emphasize the importance of good data estimation.