Sunset similarity solution for a receding hydraulic fracture

Abstract

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 to show that a receding hydraulic fracture has a linear aperture asymptote$\hat {w}\sim \hat {s}$in the fracture tip, where$\hat {s}$is the distance from the fracture front. Due to the regularity of the linear asymptote, it is possible to determine similarity solutions in the form of power series expansions, which, for integers$N\ge 2$and values of the radius decay exponent$\gamma =1/N$, can be shown to terminate to yield polynomial solutions for the fracture aperture of degree$N$. Of this countable infinity of polynomial solutions, the final aperture profile as the fracture approaches closure is associated with the second-degree polynomial with$\gamma =1/2$called the sunset solution. For the reverse time$t^{\prime }$measured from closure, the sunset solution is characterized by$w\sim t^{\prime }$and$R\sim t^{\prime 1/2}$. Of all the admissible polynomial similarity solutions, the sunset solution is shown to form an attractor, as$t^{\prime }\rightarrow 0$, for receding hydraulic fractures associated with a wide variety of points in parametric space. Using the sunset solution, it is possible to estimate the duration of recession, assuming that the fracture aperture and radius at the start of recession are given, and determine how it scales with a dimensionless shut-in parameter. As the fracture approaches closure, the term responsible for coupling the elastic force balance and fluid conservation becomes subdominant to the other terms in the lubrication equation, which reduces to a local kinematic relation between the decaying fracture aperture and the leak-off velocity. This fundamental decoupling of dynamics from kinematics results in the sunset solution being dependent on only a single material parameter – namely the leak-off coefficient. This isolation of the leak-off coefficient by the sunset solution opens the possibility to determine this parameter from laboratory or field measurements.