On the connectivity of three-dimensional fault networks

Natural fault networks involve a very broad range of fault lengths, modeled in general by a power law length distribution, n(l) similar to alpha l(-a). Such a scaling law does not allow to define any a priori pertinent scale of observation for hydraulic field experiments in fractured media. To investigate the relative effects of faults depending on their length, we undertake in the spirit of percolation theory a theoretical and numerical study of the connectivity of three-dimensional fault networks following power law length distributions. We first establish the correct analytical expression of a percolation parameter p, which describes the connectivity of the system. The parameter p is found to be dependent on the third moment of the length distribution for fault planes. It allows us to identify different regimes of connectivity depending on a, the exponent of the fault length distribution. The geometrical properties of the infinite cluster, which partly control transport properties, are also established at the percolation threshold. For natural fault networks, our theoretical analysis suggests that faults larger than a critical length scale may form a well-connected network, while smaller faults may be not connected on average. Tl-lis result, which implies an increase of the connectivity with scale, is consistent with scaling effects observed on permeability measurements.