Correlation analysis of fracture arrangement in space

Abstract

We present new techniques that overcome limitations of standard approaches to documenting spatial arrangement. The new techniques directly quantify spatial arrangement by normalizing to expected values for randomly arranged fractures. The techniques differ in terms of computational intensity, robustness of results, ability to detect anti-correlation, and use of fracture size data. Variation of spatial arrangement across a broad range of length scales facilitates distinguishing clustered and periodic arrangements—opposite forms of organization—from random arrangements. Moreover, self-organized arrangements can be distinguished from arrangements due to extrinsic organization. Traditional techniques for analysis of fracture spacing are hamstrung because they account neither for the sequence of fracture spacings nor for possible coordination between fracture size and position, attributes accounted for by our methods. All of the new techniques reveal fractal clustering in a test case of veins, or cement-filled opening-mode fractures, in Pennsylvanian Marble Falls Limestone. The observed arrangement is readily distinguishable from random and periodic arrangements. Comparison of results that account for fracture size with results that ignore fracture size demonstrates that spatial arrangement is dominated by the sequence of fracture spacings, rather than coordination of fracture size with position. Fracture size and position are not completely independent in this example, however, because large fractures are more clustered than small fractures. Both spatial and size organization of veins here probably emerged from fracture interaction during growth. The new approaches described here, along with freely available software to implement the techniques, can be applied with effect to a wide range of structures, or indeed many other phenomena such as drilling response, where spatial heterogeneity is an issue.