The Sunset Solution: Closure on Leaking Proppant

Recent analytic work (Peirce and Detournay, 2022a) has established the tip asymptotics for receding hydraulic fractures that close due to leak-off to the permeable rock. This asymptotic result enabled the development of rigorous numerical schemes to explore the recession dynamics of hydraulic fractures that develop in a state of plane strain (Peirce and Detournay, 2022b) and radially symmetrically (Peirce, 2022). These detailed studies enabled the identification (Peirce and Detournay, 2022c) of the so-called Sunset Solution, which is a similarity solution that emerges close to the ultimate collapse of the fracture. The asymptotic analysis performed in (Peirce and Detournay, 2022c) establishes 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. 1 INTRODUCTION The diagnostic fracture injection test (DFIT) is used in petroleum engineering to assess the minimum in situ stress, Carter's leak-off coefficient, and other reservoir parameters. The test consists of monitoring the well pressure during injection of fluid over a time long enough to create a hydraulic fracture, as well as after shut-in of the well. Interpretation of the pressure decline that follows the end of pumping was pioneered by Nolte (1979, 1986). His original analysis relies on two main assumptions: (i) the fracture stiffness remains constant after shut-in until the fracture completely closes and (ii) Carter's leak-off coefficient CL is a constant. In Nolte's analysis, the coefficient CL is interpreted from the pressure decline following the sudden drop of pressure observed when the well is shut-in, by expressing the declining pressure as a linear function of G(Δt), which is a function of Δt - the time elapsed since shut-in divided by the shut-in time ts (Nolte, 1979, 1986).