Comparison of PKN asymptotes in porous elastic media

ABSTRACT: In a recent asymptotic analysis Peirce (2024) of the nonlocal PKN formulation Adachi and Peirce (2008) for a slender rectangular hydraulic fracture propagating in a permeable elastic medium, two distinct classes of tip asymptotes were identified. These two distinct classes of tip asymptote are the result of different limiting processes that were used to derive the asymptotes. The first class of asymptote, is established by a local analysis in the tip region of the nonlocal PKN model equations. The second class of asymptote, is established by combining the expression for the average fracture aperture and the appropriate plane strain asymptote for the desired mode of propagation. These asymptotes differ by from one another by between 4% - 9%. Since these two classes of limit asymptotes are equally valid, this paper aims to identify which asymptote yields a closer approximation to a fully planar model of a PKN fracture. The fully planar model is assumed to be endowed with the complete multiscale asymptote for a planar a hydraulic fracture in a 3D elastic medium that can propagate and arrest after shut-in, and which combines the competing processes of viscous dissipation, toughness energy release, and fluid leak-off.