Transient Propagation of a Toughness-Dominated Plane-Strain Hydraulic Fracture in a Diffusive Porous Medium

We develop a transient framework for modeling toughness-dominated plane-strain hydraulic fracture propagation in permeable formations that incorporates fully two-dimensional (2D) pore pressure diffusion. The formulation employs an impulse response approach to evaluate the leak-off volume through a diffusion kernel, enabling a consistent treatment of finite-time diffusion without relying on classical one-dimensional (1D) leak-off assumptions. Asymptotic limits of the kernel agree with full kernel numerical computations. A finite pre-propagation stage is included, consisting of pressure buildup in a closed fracture and subsequent opening to breakdown. The governing dimensionless parameters—the initial half-length γ ₀ γ 0 , the injection rate parameter I , and the storage parameter φ φ —control both the onset and evolution of propagation. A critical condition I²γ ₀ 1.2 I 2 γ 0 ≃ 1.2 is identified, below which propagation cannot occur without unbounded injection rate, emphasizing the need for a finite initial fracture length. A 2D leak-off-dominated regime is identified at large times of propagation, which shares the same square-root time dependence of length as the classical vertex, with a reduced prefactor dependent on I . During propagation, large I produces a pore pressure field concentrated near the fracture and leads to a regime similar to 1D leak-off before transitioning to a 2D leak-off-dominated regime. Small I results in immediate 2D diffusion and no passage through the storage-dominated toughness vertex. The model quantifies the transition between 1D and 2D diffusion and delineates the conditions under which Carter-type leak-off provides an adequate approximation. This framework offers a physically consistent extension of classical hydraulic fracture models to permeable formations.