Influence of Pore Pressure Diffusion on Propagation of KGD Fracture

This paper reports a novel framework for modeling toughness-dominated KGD fracture propagation in permeable rocks, which accounts for 2D pore pressure diffusion. The model introduces a critical pre-breakdown injection phase with two distinct non-propagating stages: (i) a first stage where the crack pressure builds from in-situ pore pressure to compressive stress with the fracture remaining closed, and (ii) a second stage where the fracture opens until breakdown. Three dimensionless parameters govern the system: initial fracture length γ0, injection rate I, and rock property φ, defined as the product of stiffness and storage. The analysis reveals that large γ0 and I promote 1D diffusion during injection, while smaller values induce 2D diffusion. There is a critical threshold γ0I2 ≃ 0.4 below which propagation becomes unattainable without an infinite injection rate. Post-breakdown, the fracture potentially evolves through a toughness-dominated K-vertex at intermediate times when I is large, and consistently reaches a leak-off-controlled "2D K-vertex" at large times, when the fracture length still behaves as τ1/2 with radial leak-off, but the prefactor is smaller than at the K-vertex. For small I, there is no linear flow and the fracture propagates under a regime of pseudo-steady diffusion. These findings, particularly the intermediate- and large-time diffusion behaviors, redefine the limits of Carter's law for a KGD fracture.