Random neighborhood graphs as models of fracture networks on rocks: Structural and dynamical analysis

We propose a new model to account for the main structural characteristics of rock fracture networks (RFNs). The model is based on a generalization of the random neighborhood graphs to consider fractures embedded into rectangular spaces. We study a series of 29 real-world RFNs and find the best fit with the random rectangular neighborhood graphs (RRNGs) proposed here. We show that this model captures most of the structural characteristics of the RFNs and allows a distinction between small and more spherical rocks and large and more elongated ones. We use a diffusion equation on the graphs in order to model diffusive processes taking place through the channels of the RFNs. We find a small set of structural parameters that highly correlates with the average diffusion time in the RFNs. We found analytically some bounds for the diameter and the algebraic connectivity of these graphs that allow to bound the diffusion time in these networks. We also show that the RRNGs can be used as a suitable model to replace the RFNs in the study of diffusion-like processes. Indeed, the diffusion time in RFNs can be predicted by using structural and dynamical parameters of the RRNGs. Finally, we also explore some potential extensions of our model to include variable fracture apertures, the possibility of long-range hops of the diffusive particles as a way to account for heterogeneities in the medium and possible superdiffusive processes, and the extension of the model to 3-dimensional space.