Role of critical sets in the permeability of random fracture networks: A percolation scaling perspective

Fracture networks in the subsurface often comprise multiple fracture sets, introducing anisotropy and heterogeneity due to wide variations in fracture orientations, sizes, and conductivities. In this study, we develop a finite-size scaling (FSS) hypothesis, rooted in percolation theory, to highlight the role of critical sets—those that conduct the majority of flow—in controlling the permeability of random fracture networks. Definitions of the FSS input quantities are provided, incorporating nondimensionalizations to account for the complexity of the networks, based on the statistical properties of the fracture parameters across all sets. For cases where critical sets are present, we emphasize their dominant role and propose modified definitions based solely on the parameters of these critical sets, in line with the idea that the density of critical sets determines the magnitude of permeability. FSS results demonstrate that the hypothesis serves as a universal model for characterizing anisotropic permeability in complex fracture networks, provided that the influence of critical sets is properly addressed. This finding can be extended to other complex systems where conductive objects can be categorized into distinct sets.