A fractal model for estimating the permeability of tortuous fracture networks with correlated fracture length and aperture

Many fractures are present in the crust and dominate fluid flow and mass transport. This study proposes a fractal model of permeability for fractured rock masses that includes fractal properties of both fracture networks and fracture surface tortuosity. Using this model, a mathematical expression is derived based on the traditional parallel-plate cubic law and fractal theory. This expression functions as the equivalent permeability of the tortuous fracture network in terms of the maximum fracture length l(max), the fractal dimension of the length distribution D-f, porosity phi, fracture orientation theta, and the proportionality coefficient between fracture length and aperture beta. The fractal scaling law of the fracture length distribution and fractal permeability model is verified by comparison with published studies and fluid dynamic computation, respectively. The results indicate that the deviation of permeability values predicted by the models that do or do not consider the fracture surface tortuosity are as large as three orders of magnitude, which emphasizes that the role of tortuosity should be considered to avoid the overestimation of permeability due to the smooth fracture surface assumption. Further analyses show that the permeability increases with increasing fractal dimension D-f, proportionality coefficient beta, maximum fracture length l(max), and effective porosity phi but decreases with increasing tortuosity dimension D-tf and orientation theta. The fractal dimension of the fracture length distribution D-f has the most significant influence on the permeability of the fracture network, followed by D-tf, beta, l(max), theta, and phi, sequentially.