Quantitative evaluation of the percolation threshold and backbone fraction of networks of transverse isotropically oriented cracks

Electrical conductivity is the crucial transport property that allows us to infer the amount of conductive material, such as aqueous fluids, in subsurface rocks. Electrical conductivity strongly depends on the backbone fraction, that is, the volume fraction of the fractures contributing to the transport of electric charge. Therefore, to quantify the amount of conductive material and fracture configurations from the observed rock conductivity, it is vital to know the backbone fraction and percolation threshold at which a non-zero backbone fraction is initiated. However, because the backbone fraction and percolation threshold have rarely been modeled and are often assumed in most studies, I quantitatively evaluated them for circular cracks with transversely isotropic orientation using numerical simulations. I performed 500 simulations for individual cases with different stochastic distributions of crack size and orientation to obtain the statistical properties of the percolation threshold and backbone fraction for each case. The simulation results confirmed the quasi-invariance of the percolation threshold, even for significantly different stochastic distributions of crack size and anisotropy. The threshold was around 0.22–0.25 in terms of the crack density parameter. In addition, it was inferred that the backbone fraction, as well as the percolation threshold, does not depend on direction even when the crack orientation has significant anisotropy. The simulated backbone fraction increased sharply near the percolation threshold, but its rate of increase gradually decreased as the distance from the threshold increased. The power-law exponent of the backbone fraction near the percolation threshold was close to unity, especially when the probability density of the smallest cracks was significantly higher than that of the largest cracks. The simulation results showed a significant dependence of the backbone fraction on the assumption of crack aspect ratio, suggesting that information on the stochastic distribution of the crack aspect ratio is crucial for accurately estimating the backbone fraction.