Relationship between box-counting fractal dimension and properties of fracture networks

Abstract

Due to the capacity to quantify the complexity of a fracture network, fractal dimension (D) is widely applied in analyzing fracture network-related issues, such as connectivity and permeability. While the relationship between D and individual properties of a fracture network has been extensively studied, D is influenced by a combination of multiple attributes of the fracture network. Therefore, this work utilizes multivariate analysis to establish an equation for predicting D, taking into account various properties of the fracture network, namely fracture length, number, and orientation. Monte Carlo simulation is employed to generate a substantial number of fracture network models. Subsequently, relationships between the fractal dimension (D) and various properties are derived for three types of fracture networks with (1) invariant fracture length and random orientation, (2) exponential fracture length and random orientation, and (3) exponential fracture length and von-Mises orientation. The initial analysis focuses on the simplest relationship, wherein the fundamental formula of fractional expression is determined. Then the second and third relationships are obtained through replacing the fixed parameter in the first relationship with the distribution parameters of fracture properties. Correlation analyses between the predicted D and the actual values reveal a remarkably high correlation (>0.99). To validate the established relationships, a fracture network obtained from geological outcrops is utilized. The results demonstrate the validity of the derived relationships. The utilization of these equations enhances the efficiency, practicality, and convenience of estimating fractal dimensions from fracture properties. As a result, the analysis of fracture network-related issues becomes more feasible and accessible.