Stereological analysis of fractal fracture networks

[1] 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.