Recent developments on relationships between the equivalent permeability and fractal dimension of two-dimensional rock fracture networks

Over the last 30 years, discrete fracture network (DFN) modeling has been increasingly utilized in various practical applications in geoengineering and geosciences. The geometric properties such as the aperture, length, orientation, and connectivity of fractures in the networks significantly influence the permeability of the fractured rock masses. Two key issues include determining the distributions of these geometric properties and establishing relationships between permeability and the geometric properties of DFNs. Previous studies have shown that both single fractures and complex fracture networks exhibit fractal properties, and recent studies have established analytical and/or empirical expressions between the permeability and fractal dimension of the fracture networks. In this study, we review the fractal properties of rock fractures and fracture networks and their correlations with the permeability of DFNs. Analytical, numerical, and analytical-numerical solutions for permeability are individually reviewed, and the simplifications used in each model are extensively discussed. Moreover, ways to determine and improve the mathematical relationships between the permeability and fractal dimension of DFNs in future studies are specifically noted. This work provides a reference for engineers and hydrogeologists who use fractal methods, especially beginners who are interested in predicting the permeability of fractured rock masses.