Special Hydraulic Fractures: Cusps in a Hele-Shaw Cell and Dipoles during Extraction

We first study the dynamics of hydraulic fracturing of an elastic solid in a Hele-Shaw cell. Compared with standard hydraulic fractures in an infinite elastic bulk (e.g., Spence & Sharp, 1985), the viscous resistance mainly comes from the drag by the two parallel plates that forms the Hele-Shaw cell rather than by the fluid-solid interface. Such a feature leads to a different nonlinear differential-integral system that describes the coupled evolution of the fracture shape and pressure field. Our theory leads to hydraulic fractures of cusp shapes in the neighbourhood of the fracture tip, which is consistent with recent experimental observations. Accordingly, there exists no pressure singularity at the location of the fracture tip, which is also fundamentally different from our previous understandings of hydraulic fracturing of elastic solids.

We then show the dynamics of fracture deflation following hydraulic fracturing of an infinite elastic solid, with fluid removal from a narrow conduit at the centre. This process involves coupled lubricating flow and elastic deformation, now subject to appropriate descriptions of fluid removal through the conduit towards the ambient, driven by elastic stresses and extraction. When the influence of material toughness is negligible, the model predicts that the fracture thickness eventually approaches zero at the centre under extraction, while the fracture evolves into a self-similar shape of the dipole type that conserves the dipole moment. The fracture's front continues to elongate according to $x_f \propto t^{1/9}$, while the total fluid volume within a fracture decreases according to $V \propto t^{-1/9}$.