Permeability fluctuations in heterogeneous networks with different dimensionality and topology

The purpose of this work was to relate the spatial fluctuations and scaling properties of the transport properties of porous rocks to their underlying pore geometry. Our approach was to numerically simulate flow through networks of pipes with randomly prescribed radii. The permeability k and inverse formation factor 1/ F were calculated in a large number of network realizations of varying size and degree of heterogeneity (i.e., the width of the pipe radius distribution). We generally observed a large decrease of the ensemble arithmetic averages of k and 1/ F with increasing network size (i.e., negative scale effect). Conversely, the ensemble geometric averages showed a moderate positive scale effect in three‐dimensional simple cubic networks. We also found that in networks smaller than 32 × 32 or 10 × 10 × 10, the ensemble standard deviations of k and 1/ F had a power law dependence on network size (defined as the total number of pipes) with an exponent α varying from −0.5 in homogeneous networks to large negative values depending on lattice topology in highly heterogeneous ones (−α increased with increasing lattice connectiveness, i.e., with coordination number). Thus at small scales the network transport properties were characterized by a nonuniversal power law scaling. At larger scales we observed a transition to a presumably “universal” power law scaling with an exponent equal to −0.5 independently on the degree of heterogeneity, dimensionality and lattice topology. Comparing our results to published experimental data, we found a good agreement, except in cases where we suspect that the small‐scale measurements suffered a significant bias (indicated by non‐nested distributions at increasing scales). We speculate that the strong positive scale effect generally observed in nature is also caused by sampling bias at small scales.