Viscous pressure drop modulates the morphology of a network fractures activated by hydraulic stimulation

Convective transport in low permeability rocks can be enhanced by injection of a fluid to activate pre-existing weak planes (fractures) above a critical fluid pressure given by Mohr's criterion. Using a discrete fracture network (DFN) simulation and complementary averaged equation solutions for a highly heterogeneous rock, we show that the morphology and average transport properties of a cluster of activated fractures depend on the ratio, $F_{N}$, between the standard deviation of the critical pressures and the viscous pressure drop across a fracture. When $F_{N}$ \textless \textless 1, the cluster is well connected, and a linear diffusion equation can be used to describe the cluster's growth. When $F_{N\thinspace }$\textgreater \textgreater $R/l$ where $R$ is the cluster radius and $l$ is the fracture length, a fractal network is formed by an invasion percolation process. In the intermediate regime, 1\textless \textless $F_{N}$\textless \textless $R$/$l$, percolation theory relates the porosity and permeability of the network to the local pressure and an averaged fluid transport equation with pressure-dependent properties describes the cluster growth on length scales much larger than $l F_{N}$. The theory is also applicable to the displacement of a wetting fluid by a more viscous non-wetting fluid in a permeable rock with the capillary number replacing $F_{N}$ in the two-phase flow application.