The Sunset Solution: Closure on Leaking Proppant

Recent analytic work has established the tip asymptotics for receding hydraulic fractures that close due to leak-off to the surrounding permeable rock. This asymptotic result has enabled the development of rigorous numerical schemes to explore the recession dynamics of hydraulic fractures that develop in a state of plane strain and radially symmetrically. These detailed studies enabled the identification of the so-called Sunset Solution, which is a similarity solution that emerges close to the ultimate collapse of the fracture. Asymptotic analysis has established that the existence of the sunset solution is due to a fundamental decoupling of the kinematics from the dynamics in the governing equations, which leads to a robust way to measure the Carter leak-off coefficient from the rate of change in the fracture aperture at the wellbore. These results were established assuming that no fluid is leaking from the closed section of the fracture. In this paper, we explore the viability of the procedure to estimate the Carter leak-off coefficient using the sunset solution when the fracture is assumed to close on proppant that is filled with fluid upon closure and which continues to leak off and exchange fluid with the open portions of the fracture. In particular, we consider radially symmetric geometries and geometries that have significant symmetry-breaking changes to the fracture free boundary induced by jump discontinuities in the confining stress field and the elastic moduli between sedimentary layers.