The goodness-of-fit of models for fracture trace length distribution: Does the power law provide a good fit?

Abstract

Fracture trace length distributions are often assumed to follow a power law, which implies that the distribution is scale-independent. The present study tests this assumption by evaluating the goodness-of-fit of three statistical models—the power law, piecewise power law, and lognormal distribution—upon a dataset of 57 trace maps that cover a range of fracture modes, host rock types, network scales, and topologies. The goodness-of-fit was assessed through the unbiased Kolmogorov-Smirnov (KS) test, which accounts for the fitting procedure and the degrees of freedom of each model. The results show that the power law provides a poor fit to trace length distributions, being rejected in 24 trace maps at a significance level of 0.05. In contrast, the piecewise power law and lognormal distribution demonstrated better fits across the fracture networks, with the piecewise power law performing the best overall. The poor fit of the power law can be attributed to mechanical and chemical controls on fracture growth, mainly fracture abutment, as well as stress shadowing and cementation, which affect growth rate at different length scales and result in scale-dependent trace length distributions. The consistent poor fit of the power law across various fracture networks suggests that these controls are prevalent in natural systems. While the power law remains a simple and effective model for trace length distribution, it should be recognized that it overlooks such controls that can influence the mechanical and hydraulic properties of fracture networks. Meanwhile, the fit of the piecewise power-law suggests the existence of a characteristic length where a transition in fracture growth occurs.