We perform a numerical study of random walks (RWs) on 3D lattices with random and periodic distributions of blocked sites, analyze the dependence of effective diffusion coefficients $D_E$ on the effective porosity $\epsilon$, and show applications to diffusive solute transport in some porous materials. In the low-porosity regime of randomly disordered lattices (typically $0.03\lesssim \epsilon \lesssim 0.2$), where the total porosity exceeds $\epsilon$ by more than $1\%$, reliable estimates of $D_E/D_0$ are obtained by assuming that the mean square displacement is a sum of normal and anomalous contributions ($D_0$ is the free diffusion value). An extended form of the Archie's law, $D_E/D_0\approx A{\epsilon }^m$, is obtained with an uncommonly large exponent $m\approx 5$ and $A>1$, in agreement with the universal value $m=4.9\pm 0.3$ predicted by scaling arguments. The fit of diffusion data in a silty clay loam soil gives $m\approx 4.8$ and $A>1$, confirming the absence of long-range correlations in that pore system. This approach advances over previous studies that related $D_E/D_0$ to the total porosity and required an estimation of percolation thresholds. In periodic lattices with narrow planar gaps and the same low-porosity range, fits with $m\approx 1$ and $A<1$ are explained by the long-range correlations of the (nontortuous) pore systems. Similar values of $m$ and $A$ are obtained in ion diffusion and conductivity data in bentonite and granite samples, which may be a consequence of fractures that cross their pore systems. In the high-porosity regimes of the same lattices and of deposits produced by a growth model ($\epsilon \ge 0.8$), random walk values of $D_E/D_0$ are significantly smaller than the values of the diffusion equation [obtained in effective medium approximations (EMA) of Bruggeman or Maxwell]. This occurs because RWs do not reach the diffusion limit in lengths near the lattice constant, which is the order of magnitude of the widths of blocked domains; however, those EMA are recovered for RWs on lattices with obstacle sizes much larger than the lattice constant. This analysis suggests deviations from the EMA whenever RW models are used to model Knudsen diffusion. Finally, from intermediate to high porosities of randomly disordered lattices ($\epsilon \gtrsim 0.3$), fits of the original Archie's law $D_E/D_0={\epsilon }^m$ give $2<m<3$, which is the range obtained in several materials and suggests that their pore organization may be random at long distances.

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• Brownian motion
• Diffusion
• Fracture
• Growth
• Percolation theory
• Random walks
• Porous media
• Monte Carlo methods
• Scaling methods