Fractal and multifractal characterization of stochastic fracture networks and real outcrops

Abstract

The fractal dimension and multifractal spectrum are widely used to characterize the complexity of natural fractures. However, systematic investigations, considering impacts of different fracture geometrical properties (fracture lengths, orientations, center positions) and system sizes, on the fractal and multifractal characterization of complex fracture networks are insufficient. Here, we utilize an in-house developed DFN modeling software, hatchfrac, to construct stochastic fracture networks with prescribed distributions and systematically study the impact of three geometrical properties of fractures and system sizes on the fractal and multifractal characterization. We calculate the single fractal dimension and multifractal spectrum with the box-counting method. The single fractal dimension, D, and the difference of singularity exponent, Δα, are used to represent the fractal and multifractal patterns, respectively. We find that fracture lengths, orientations and system sizes positively correlate with D and Δα, while the system size has the most significant impact among the four parameters. D is uncorrelated with fracture positions (FD), which means that a single fractal dimension cannot capture the complexity caused by clustering effects. However, Δα has a strong negative correlation with FD, implying that clustering effects make fracture networks more complex, and Δα can capture the difference. We also digitize 80 outcrop maps with a novel fracture detection algorithm and calculate their fractal dimension and multifractal spectrum. We find wide variations of D and Δα on those outcrop maps, even for outcrops at similar scales, indicating that a universal indicator for characterizing fracture networks at different scales or the same scale is almost impossible. D and Δα have negligible correlations with scales, supporting the self-similarity patterns of natural fracture networks.