The arrest and recession dynamics of a deflating rectangular hydraulic fracture in a permeable elastic medium

This paper considers the propagation, arrest and recession of a planar hydraulic fracture in a porous elastic medium whose footprint is constrained to a growing or shrinking rectangular region with a constant height. Hydraulic fractures with large aspect ratio rectangular footprints are frequently referred to as PKN fractures in recognition of the original researchers (Perkins & Kern 1961 J. Petrol. Tech. 13 , 937–949) and (Nordgren 1972 J. Petrol Technol. 1972 , 306–314) who first analyzed models of such fracture geometries. We investigate the one-dimensional non-local PKN approximation to a fully planar rectangular hydraulic fracture model in a three-dimensional elastic medium. By analysing the tip behaviour of the non-local PKN model, a transformation procedure is established to render the asymptotic equations for the dynamics of the steady semi-infinite PKN and plane strain models formally identical, which implies that all the existing multiscale plane strain asymptotes can be converted directly to the PKN case by making use of this transformation. Using this transformation, it is shown that the appropriate PKN asymptotes for the average aperture $\bar {w}$ with distance $\hat {x}$ to the fracture front are $\bar {w}\sim \hat {x}^{1/2},\ \hat {x}^{5/8}\ {\textrm{and}}\, \ \hat {x}^{2/3}$ in the toughness, leak-off and viscous modes of propagation, respectively; as well as the linear elastic fracture mechanics tip asymptote $\bar {w}\sim \hat {x}^{1/2}$ for arrest, which transitions to the linear asymptote tip $\bar {w}\sim \hat {x}$ for a fracture driven to recede due to fluid leak-off. Both the arrest and recession tip asymptotes share the intermediate leak-off asymptote $\bar {w}\sim \hat {x}^{3/4}$ . A scaling analysis yields the arrest time, length and aperture as functions of a dimensionless injection-cessation time $\omega$ . An asymptotic analysis of the non-local PKN model is used to establish the fundamental decoupling between dynamics and kinematics, which leads to the emergence of a similarity solution – termed the sunset solution – close to the time of collapse of the fracture. The multiscale PKN numerical solutions agree well with those for a fully planar multiscale rectangular hydraulic fracture model in a three-dimensional elastic medium. The scaling laws and the emergence of the sunset solution are confirmed by the PKN numerical model. The sunset solution also emerges in the fully planar numerical model and persists beyond the collapse time of the PKN model, by which time its footprints have separated from the upper and lower constraining sedimentary layer boundaries and have assumed self-similar elliptic shapes that shrink as they approach collapse.