A technique for revealing scale-dependent patterns in fracture spacing data

Data on fracture spacing along scan lines have been widely analyzed for the purposes of characterization. Most of these studies, however, either consider the cumulative frequency of spacing data without regard to the actual sequence of the spacing values or compute an average spacing that may not work for clustered fractures. The coefficient of variation parameter is often used to differentiate between clustered, random, and anticlustered fractures in a scan line but does not address the issue of scale-dependent variations in spacing. Lacunarity is a parameter that has been previously used for delineating scale-dependent clustering in fracture networks with similar fractal dimensions. This technique has the further capability of identifying scales at which different patterns emerge within the same data set. Lacunarity can also delineate possible fractal behavior. This paper tests the ability of lacunarity to find patterns (fractal/uniform/random) within synthetic and natural fracture clusters. A set of four model scan lines (uniformly spaced fractures, periodically spaced fracture clusters, fractal fracture clusters, and random fractures) was considered. The first derivative of the lacunarity curves of these models was used to find the intercluster distance and organization of fractures within the clusters. The same technique was then applied to a set of two natural fracture scan line data, one with fracture clusters with fractal organization within and the other with randomly spaced fractures. It was found that the proposed technique could discriminate between the random and clustered patterns, find the intercluster distance, and identify the type of spatial organization within the clusters.