IMPROVED FRACTAL ANALYSIS OF FRACTURE NETWORKS

Fractals are irregular entities that show self-similarity over a wide range of scales and can be quantified by the fractal dimension, D. Fractal analysis can help in understanding the scaling characteristics of fracture networks. However, it is important that the D-values of such networks be properly evaluated. The box-counting algorithm is a popular technique for characterizing fracture networks as fractals and estimating their D values. If this analysis yields a power law distribution given by N = k r ^-D , where N is the number of boxes containing one or more fractures and r is the box size, the network is considered to be fractal. However, workers have used different versions of this algorithm over the years and are divided on their opinion about issues like estimating the ‘correct' D-value or whether a fracture network is indeed fractal. For instance, a closer look at the N vs. r plots for a set of previously published fracture trace maps show that such distributions do not follow power law scaling. In the present work, a model network with a known theoretical fractal dimension, D_theory, is used to develop a method for the proper evaluation of such fracture patterns, and for computing unbiased D-values. The model is constructed using hierarchical fracture networks (iterations) made up of line segments in 2-dimensions. It is observed on the box counting curves that, if instead of the entire range of points, the ones for which r > r_min (smallest element) are only considered, the estimated D-value closely approximates D_theory. But if the points for which r < r_min are also included, the distributions are no longer power laws. Because determining the smallest fracture element can be challenging for natural patterns a method for finding a proxy value, the estimated cut-off, r_cut-off, is also devised. Using this parameter instead of the r_min also returned satisfactory values of D for the model. Unbiased fractal dimensions for a suite of fracture maps that have previously been evaluated for their fractal nature have been successfully computed using the improved method. The resulting D values will be compared with the original estimates.