Stereological analysis of fractal fracture networks

We assess the stereological rules for fractal fracture networks, that is, networks whose fracture‐to‐fracture correlation is scale‐dependent with a noninteger fractal dimension D. We first develop the general expression of the probability of intersection p(l, l′) between two populations of fractures of length l and l′. We then derive the stereological function that gives the fracture distribution seen in 2‐D outcrops or 1‐D scan lines for an original three‐dimensional (3‐D) distribution. The case of a fractal fracture network with a power law length distribution, whose exponent a is independent of D, is particularly developed, but the results can, however, be extended to any other length distributions. The analytical results were tested using a numerical model that generates 3‐D discrete fractal fracture networks. The corresponding 1‐D and 2‐D length distributions are still described by a power law with exponents a1‐D and a2‐D that are related to the original 3‐D exponent by a3‐D = a1‐D + 2 and a3‐D = a2‐D + 1, respectively, regardless of the fractal dimension. The density distributions of fractures in two or one dimensions remain fractal but with a dimension that depends on both the original 3‐D distribution and the power law length exponent a. The fractal dimension of 2‐D or 3‐D fracture networks cannot be directly inferred from one‐dimensional scan‐line data sets unless a is known. We found a good adequacy between our predictions and measurements made on a few natural data sets. We propose also an original method for measuring the fractal dimension from the variations of the average number of fracture intersections per fracture.