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

Abstract : 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 non negligible but remains small. It is equal to 0.09 for sparse networks and 0.14 for dense networks representing respectively 4.5% and 7% of the full possible range of \omega values. Power-averaging is not strictly valid but gives an estimate of upscaling at a few percents. \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 neighbour by points (effective coordination number) within the following structure (backbone).