Large-time self-similar propagation of a toughness-dominated hydraulic fracture in a poroelastic medium

We present a theoretical framework for modelling a plane-strain hydraulic fracture propagating in a poroelastic rock in the toughness-dominated regime. The formulation explicitly incorporates two-dimensional (2-D) pore-pressure diffusion, thereby generalising the classical Carter leak-off model, which can be interpreted as the limiting case of one-dimensional (1-D) diffusion. The poroelastic response is captured by superposing pore pressure and backstress contributions from a spatial and temporal distribution of instantaneous point sources along the extending fracture. A scaling analysis reveals the existence of a class of large-time, self-similar solutions for which the fracture length grows as $\ell \sim t^{1/2}$ , with a prefactor function of a dimensionless injection rate $\mathcal{I}$ and a poroelastic stress coefficient $\eta$ . The injection rate $\mathcal{I}$ emerges as the dominant controlling parameter. Asymptotic analysis provides large-time closed-form solutions in the limits of both large and small $\mathcal{I}$ , which show excellent agreement with full numerical simulations. For large $\mathcal{I}$ , diffusion reduces to 1-D and the solution converges to the classical toughness- and leak-off-dominated solution governed by Carter’s law. For small $\mathcal{I}$ , fracture growth is instead controlled by pseudo-steady (2-D) diffusion. The transition from 2-D to 1-D diffusion is characterised by an increase in the fracture length prefactor and a reduction in leak-off. The poroelastic coefficient $\eta$ acts to shorten and narrow the fracture while increasing both leak-off and driving pressure. This framework delineates the transition between 2-D and 1-D diffusion and establishes quantitative conditions under which Carter’s law remains valid in the large-time limit.