Computation of the equivalent macroscopic permeability tensor of discrete networks with heterogeneous segment length

Flow in discrete networks can be observed in biological, geological, or technical systems. Often, the number of individual segments is very large. Therefore, discrete pressure and flow calculation in each single segment becomes very time consuming, and a continuum model becomes attractive. We apply a global upscaling method based on a spatial average to investigate a porous media network model with heterogeneous pore length distribution. For this purpose, the porous media was modeled by triangular networks. For such networks, we characterize the representative elementary volume (REV) size and show that using window sizes smaller than the REV requires an heterogeneous Darcian description. We find that the permeability has to be distributed according to a log-Gaussian distribution with variance and correlation length depending on the window size and the porous microstructure. Finally, we apply the procedure to an anisotropic regular network for which an analytical solution can be found and show that using this method leads to the correct analytical behavior.