Efficient FFT-based upscaling of the permeability of porous media discretized on uniform grids with estimation of RVE size

Abstract

An efficient numerical scheme at the crossroad of Fourier based methods and boundary element methods is derived to upscale the permeability of porous media from Stokes flow within the pore space. The method relies on a variational framework which involves trial force fields and the Green function. The discretization of the trial force fields is carried out on a uniform grid, which allows for the use of fast Fourier transforms to evaluate the convolution product arising from the Green function. A discretization of the Green function consistent with the variational framework is introduced to ensure the upper bound status of the homogenized permeability. The energy consistent discretization is compared to six alternative discretizations arising from truncation of Fourier series or from various finite difference schemes, in terms of accuracy of the homogenized permeability and of the local velocity fields as well as on the number of iterations required for convergence of the iterative solver. The method is combined to a strategy to estimate the Representative Volume Element (RVE) size at a target accuracy level from a single simulation on a large sample to study the permeability and associated RVE size for the Voronoi cell model, over the whole range of porosity.