Scaling of Fracture Patterns in Three Deep Boreholes and Implications for Constraining Fractal Discrete Fracture Network Models

Abstract

We present a methodology for generating fractal fracture networks in one, two and three dimensions that respects the dual power-law model, in which the scaling characteristics are set by the two independent parameters: (1) the correlation dimension that pertains to separation of fracture centers, and (2) the length exponent that governs the distribution of fracture lengths. Synthetic fracture distributions were generated to evaluate the stereological relationships between the scaling parameters of 2D and 3D networks, and the scaling of fracture intersection points along a scanline through the network. The results showed that it is not possible to estimate the 2D and 3D fractal scaling parameters of the correlation dimension from the 1D correlation dimension of fracture spacing from scanlines through the network, even if the length exponent is known a priori. Synthetic 1D distributions of fracture spacing of known correlation dimension were used as a benchmark to test the consistency of estimates of fractal dimension derived from box-counting, two-point correlation, and power-law fitting. The results showed that the correlation dimension obtained from the two-point correlation method provided the most stable and reliable estimate of the fractal dimension of fractures on 1D scanlines or boreholes. Application of the two-point correlation function to the observed fracture distributions along three deep boreholes in crystalline rock at Basel, Switzerland and Soultz-sous-Forets, France showed that the distribution was fractal over more than two orders of magnitude in scale, and in all cases the fractal dimensions was in the range 0.86–0.88. Similar results were obtained for fracture sets of common orientation within the wells, although the fractal dimension ranged between 0.65 and 0.75. This constitutes strong evidence that fracturing in rock masses penetrated by the wells follows a fractal organization.