Directional Pair-Correlation Analysis of Fracture Networks

Fractures result from complex mechanical processes producing irregular, hierarchical, and correlated networks. The statistical analysis of such networks is an important step toward characterizing and modeling fractures. However, established exploratory statistics for the investigation and quantification of fracture networks use only first-order or mean-value characteristics such as the density or the length and orientation distributions of fractures, leaving much to be desired. Here, we present a second-order statistical theory to characterize the inner variability of fracture networks in 2D. We use ideas from marked point process theory treating the barycenters of fractures or fracture branches as points and fracture lengths and orientation as marks. The statistics are based on oriented distances between object centers, which are represented by pair-correlation and mark-correlation functions describing fracture network variability. The forms of the corresponding plots give information on the degree of randomness, the most frequent center-to-center distances, and possible local order, all with respect to fracture orientations. We demonstrate the application of these ideas by analyzing three fracture networks. First, we study a synthetic structure as a benchmark test of the methods under ideal conditions. Then, we focus on two previously characterized field exposures: a well-developed and highly connected network and a very irregular network with many small and isolated fractures. The correlation functions successfully characterize the different spatial arrangements of fractures in all three cases.

Automated summary

This paper proposes a second-order statistical theory to characterize the inner variability of fracture networks in 2D. The theory utilizes marked point process theory to treat fracture barycenters as points and fracture lengths and orientations as marks. It employs pair-correlation and mark-correlation functions to describe fracture network variability based on oriented distances between object centers. The application of these ideas on three fracture networks demonstrates their effectiveness.