Universal scaling laws for the activation of pre-existing natural fractures

The activation of sparsely distributed weak planes (fractures) in low permeability rocks by injecting a pressurized fluid creates flow paths that facilitate extraction of geothermal energy from hot dry rocks and gas from shale. Using a large-cell renormalization-group approach, we show that the process of forming percolating paths of activated fractures can be viewed as a critical phenomenon similar to the dilute n-vector model under the following assumptions: 1) a fracture is activated, following Mohr’s criterion, when the fluid pressure reaches a critical value that depends on its orientation with respect to the in-situ stress field; 2) the viscous pressure drop is negligible compared to the variability in the critical pressures; and 3) the lengths of the natural fractures are all of the same order of magnitude. Similar to the n-vector model, a crossover occurs as the linear dimension of the cluster of activated fractures becomes comparable to the correlation length of the natural fractures. The crossover exponent is found to be equal to 1 and the critical exponent ν is equal to the critical exponent of regular percolation. Based on the results, scaling laws relating the cluster radius with the transient injection pressure are provided.