Hydraulic fracturing is commonplace in the geo-industry, whether designed or unintended; e.g., stimulation of hydrocarbons reservoirs [1, 2], disposal of waste water , waterflooding operations , enhanced oil recovery by injection of CO2 , and preconditioning of rock mass in the mining industry [6, 7]. Nonetheless, modeling of hydraulic fracturing usually relies on oversimplified assumptions [1, 8]; in particular, fluid leak-off is often studied within Carter's model  that assumes that the transport of the filtrate and the porous fluid through the porous medium is one-dimensional. While this assumption is quite reasonable in the case of short treatments such as hydraulic fracturing of a hydrocarbons reservoir , it is unlikely to be applicable for injection operations over long periods of time.
This study is part of an ongoing effort to rigorously introduce large-scale 3D diffusion in a model of hydraulic fracture. The increase of pore pressure around the fracture caused by fluid leak-off from the fracture leads to an expansion of the porous medium. This expansion can be accounted for by the introduction of the so-called backstress [10, 11]. By definition, the backstress would be the stress induced across the fracture plane if the fracture were closed. Here we restrict our investigation to the toughness-dominated regime of propagation, for which the viscosity of the fluid is negligible. In other words we assume that the energy spent for hydraulic fracturing is mainly due to the rock damage rather than due to dissipation associated with viscous flow of the fracturing fluid. Setting the fracturing fluid viscosity to be equal to zero implies that the fluid pressure inside the fracture is uniform.