Sunset similarity solution for a receding hydraulic fracture

This paper derives approximate ‘sunset’ similarity solutions for receding plane strain and radially symmetric hydraulic fractures in permeable elastic media close to the point of closure. Local analysis is used show that a fracture has linear aperture asymptote $\hat {w}\sim \hat {s}$ tip, where distance from front. Due regularity asymptote, it possible determine form power series expansions, which, integers $N\ge 2$ values radius decay exponent $\gamma =1/N$ , can be shown terminate yield polynomial degree $N$ . Of this countable infinity solutions, final profile as approaches closure associated with second-degree =1/2$ called sunset solution. For reverse time $t^{\prime }$ measured closure, solution characterized by $w\sim t^{\prime $R\sim 1/2}$ all admissible an attractor, }\rightarrow 0$ wide variety points parametric space. Using solution, estimate duration recession, assuming at start recession are given, how scales dimensionless shut-in parameter. As term responsible coupling force balance fluid conservation becomes subdominant other terms lubrication equation, which reduces local kinematic relation between decaying leak-off velocity. fundamental decoupling dynamics kinematics results being dependent on only single material parameter – namely coefficient. isolation coefficient opens possibility laboratory or field measurements.