Connectivity properties of two-dimensional fracture networks with stochastic fractal correlation

[1] We present a theoretical and numerical study of the connectivity of fracture networks with fractal correlations. In addition to length distribution, this spatial property observed on most fracture networks conveys long-range correlation that may be crucial on network connectivity. We especially focus on the model that comes out relevant to natural fracture network: a fractal density distribution for the fracture centers ( dimension D) and a power law distribution for the fracture lengths (exponent a, n(l) similar to l(-a)). Three different regimes of connectivity are identified depending on D and on a. For a < D + 1 the length distribution prevails against fractal correlation on network connectivity. In this case, global connectivity is mostly ruled by the largest fractures whose length is of the order of the system size, and thus the connectivity increases with scale. For a > D + 1 the connectivity is ruled by fractures much smaller than the system size with a strong control of spatial correlation; the connectivity now decreases with system scale. Finally, for the self-similar case ( a = D + 1), which corresponds to the transition between the two previous regimes, connectivity properties are scale invariant: percolation threshold corresponds to a critical fractal density and the average number of intersections per fracture at threshold is a scale invariant as well.