Fractal Joint Signatures

Measures of many geophysical phenomena are fractal. One of these is fracture spacing along a rock face. For the most part, the only parameter that has been extracted from such a fractal data set is its dimension. However, fractals which are strictly self-similar can be succinctly described mathematically as an Iterated Function System (IFS). In this case the parameters of the functions of an IFS constitute a “signature” of the fractal. In this paper we introduce the possibility that fracture spacing data can be modeled as IFS fractal sets. The fracture spacing data sets examined in this study covered approximately 3 orders of magnitude in scale, 0.002 to 20 meters. They were approximately self similar, typically over 2 or more orders of magnitude, as shown by inter-fracture spacing plots and by a standard box counting dimension calculation. Not surprisingly, no data set was perfectly self-similar. We develop a technique for fitting IFS fractals to spacing data by using the criteria that the fit is a minimum in the Hausdorff distance. The technique found IFS fractals that reproduce the principal characteristics of the data sets of the study, but not the exact positions of the fractures. There remain open questions associated with the method: baseline, robustness and of course, physical meaning of the resulting fractal signature.