On measuring the fractal anisotropy of 2-D geometrical sets: Application to the spatial distribution of fractures

This work addresses a methodology on how to estimate and correlate the fractal anisotropy of a 2-D spatial distribution of segments with structural geometrical parameters and the relative dating of fracture sets. Tectonic characterisation of fracture traces can be performed to classify fractures into families, according to the tectonic event. To analyse their fractal anisotropy, several Cantorian profiles were constructed from the two-dimensional intersection between oriented transects and the distribution of segments in the plane. We applied this methodology to the spatial distribution of fault and fracture traces interpreted from digital terrain models and aerial photography. To obtain an appropriate mass number in the resulting Cantor set, parallel transects were arranged in a sequential pattern until a unique transect was obtained in each direction. When this procedure was applied to fracture maps filtered by stress fields (dynamical map), the fractal dimension D0 described an ellipse in a polar plot with the short axis as the minimum value (DHmin) and the long axis as the maximum value (DHmax). The fractal ellipse orientation was compared with the orientation of the horizontal projection of the main axes of the stress tensor: σHmax, and σHmin. From this comparison, a spatial perpendicular relationship between the axes of the ellipse of fractal spatial anisotropy of fractures associated with a particular tectonic event, and the axes of the stress tensor that produced it, was established. Furthermore, when this methodology was applied to a local whole fault trace map, the dynamical evolution of different fracture patterns (as a relative dating) were able to be estimated. The variation in the 1-D fractal value for neighbouring profiles in the fractal ellipse allowed azimuthal filters to be defined. Variation of the D0 value for consecutive profiles greater than 20% was considered as the borderline between different fracture sets. Whether the most recent stress field generated both new fracture sets and reactivated ancient fracture sets, the geometrical complexity increased as did the fractal value. Thus, fractal sets with lower fractal dimensions were interpreted as older than those with the highest values. The filtered fault families were in general agreement with those from the dynamical maps.