Use of power averaging for quantifying the influence of structure organization on permeability upscaling in on-lattice networks under mean parallel flow

We numerically assess the relevance of power averaging as a means for permeability upscaling on a variety of 2D and 3D, dense, and sparse on-lattice networks. The power average exponent omega determined on a realization basis converges with the system size within the range of scales explored for all cases. Power averaging is strictly valid only for the 2D dense square case for which omega is equal to 0 with a numerical precision of 0.01 both for the lognormal and log-uniform permeability distributions consistently with the theoretical proof of Matheron (1967). For all other cases, the variability of omega with the local permeability distribution variance sigma(2) is nonnegligible but remains small. It is equal to 0.09 for sparse networks and 0.14 for dense networks representing 4.5% and 7%, respectively, of the full possible range of omega values. Power averaging is not strictly valid but gives an estimate of upscaling at a few percent. Here omega depends slightly on the local permeability distribution shape beyond its variance but mostly on the morphological network structures. Most of the morphological control on upscaling for on-lattice structures is local and topological and can be explained by the dependence on the average number of neighbor by points (effective coordination number) within the flowing structure (backbone).