Hydraulic fractures of the dipole type subject to fluid withdrawal through a conduit

We study 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/suction. When the influence of material toughness is negligible, the dynamics is found to be governed by two dimensionless parameters that describe the relative influence of elasticity-driven backflow ( $\Pi _c$ ) and ambient-pressure-driven backflow ( $\Pi _e$ ), respectively. We also found that the fracture’s thickness eventually approaches zero at the centre, while the fracture evolves into a self-similar shape of the dipole type that conserves the dipole moment $M$ . The fracture’s front continues to elongate according to $x_f \propto t^{1/9}$ , while the total fluid volume within the fracture decreases according to $V \propto t^{-1/9}$ . The model and solutions might find use in practical problems to estimate the rate of backflow and effective permeability of a fractured reservoir once pressure is released.