The arrest and recession dynamics of a deflating hydraulic fracture in a permeable elastic medium in a state of plane strain

We investigate the deflation dynamics of a fluid-driven fracture in a permeable elastic medium in a state of plane strain after fluid injection has ceased. Depending on the leak-off characteristics of the porous medium and the volume of injected fluid retained in the fracture at the time of shut-in, the fracture may start to recede almost immediately or continue to propagate till it arrests when the stress intensity factor drops below the fracture toughness. While occupying the arrest footprint, the fracture continues to deflate while the stress intensity factor decreases due to fluid loss to the porous medium. When the stress intensity factor drops to zero, the fracture starts the process of recession, which continues until it finally collapses. To establish a rigorous numerical scheme to explore the deflation dynamics of plane strain hydraulic fractures, we use recently established vertex and multiscale tip asymptotes for arrested and receding hydraulic fractures (Peirce and Detournay, 2022), including the r-vertex linear tip aperture asymptote wˆ∼xˆ for a receding hydraulic fracture and the stationary g-vertex asymptote wˆ∼xˆ^3/4. Numerical experiments demonstrate that the multiscale asymptotes are required in order to achieve solutions that remain smooth through the arrest–recession transition point. In contrast, numerical solutions, obtained by only using vertex solutions to model the arrest and recession, exhibit jump discontinuities through this transition point. However, once the transients from these jump discontinuities have decayed the numerical schemes that use vertex and multiscale asymptotes yield almost identical solutions. A scaling analysis shows the existence of asymptotic power law behaviour for various quantities, such as the arrest time, in terms of two new dimensionless parameters. Finally, numerical solutions explore the dependence on the two dimensionless parameters of the arrest time, the duration of the arrest period, and the duration of the recession phase and confirm their asymptotic power law behaviours.