Asymptotic solutions for self-similarly expanding fault slip induced by fluid injection at constant rate

We examine the circular, self-similar expansion of frictional rupture due to fluid injected at a constant rate. Fluid migrates within a thin permeable layer parallel to and containing the fault plane. When the Poisson ratio =0, self-similarity of the fluid pressure implies fault slip also evolves in an axisymmetric, self-similar manner, reducing the three-dimensional problem for the evolution of fault slip to a single self-similar dimension. The rupture radius grows as 4 ₇ₘ t, where t is time since the start of injection and ₇ₘ is the hydraulic diffusivity of the pore fluid pressure. The prefactor is determined by a single parameter, T, which depends on the pre-injection stress state and injection conditions. The prefactor has the range 0, the lower and upper limits of which correspond to marginal pressurisation of the fault and critically stressed conditions, in which the fault-resolved shear stress is close to the pre-injection fault strength. In both limits, we derive solutions for slip by perturbation expansion, to arbitrary order. In the marginally pressurised limit (0), the perturbation is regular and the series expansion is convergent. For the critically stressed limit (), the perturbation is singular, contains a boundary layer and an outer solution, and the series is divergent. In this case, we provide a composite solution with uniform convergence over the entire rupture using a matched asymptotic expansion. We provide error estimates of the asymptotic expansions in both limits and demonstrate optimal truncation of the singular perturbation in the critically stressed limit.