Asymptotic series solution for plane poroelastic model with non-penetrating crack driven by hydraulic fracture

Abstract

A new class of coupled poroelastic problems describing fluid-driven cracks (called fractures) subjected to non-penetration conditions between opposite crack faces (fracture walls) is considered in the incremental form. The nonlinear crack problem for a plane isotropic setting in a two-phase medium is expressed in polar coordinates as a variational inequality with respect to the solid phase displacement and the pore pressure. Applying nonlinear methods, the asymptotic theory and Fourier analysis, a semi-analytic solution given as the power series in the sector of angle $2\pi$ is proven using rigorous expansions with respect to the distance to the crack-tip. Here no logarithmic terms occur in the asymptotic expansion. Consequently, a square-root singularity for the poroelastic medium with a non-penetrating crack is derived, and the integral formulas for calculating the corresponding stress intensity factors are obtained.