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We examine the circular, self-similar expansion of frictional rupture due tofluid injected at a constant rate. In a prior problem [Viesca, 2021], theauthor examined rupture of a planar frictional fault driven by line source offluid injected at constant pressure. Here, fluid injection occurs at constantvolumetric rate at a point on the fault and fluid migration occurs within athin, relatively permeable layer containing and parallel to the fault plane.For the particular case when the Poisson ratio $\nu=0$, self-similarity of thefluid pressure profile implies that fault slip will also evolve in anaxisymmetric, self-similar manner, reducing the three-dimensional problem forthe spatiotemporal evolution of fault slip to a single self-similar dimension.The rupture radius grows as $\lambda \sqrt{4\alpha_{hy} t}$ , where $t$ is timesince the start of injection, $\alpha_{hy}$ is the hydraulic diffusivity of thepore fluid pressure, and $\lambda$ is a prefactor determined by a singleparameter, $T$, which depends on the pre-injection stress state and injectionconditions. The prefactor has the range $0<\lambda<\infty$, where the lower andupper limits of $\lambda$ correspond to, respectively marginal pressurizationof the fault and critically stressed conditions, in which the fault-resolvedshear stress is close to the pre-injection fault strength. In both limits of$\lambda$, we derive asymptotic solutions for the slip by perturbationexpansion, to arbitrary order. For the critically stressed limit, the solutionfor slip contains an interior boundary layer and an outer solution. We performa matched asymptotic expansion to provide a composite solution with uniformconvergence over the entire rupture. The leading order of these solutions wasrecently used by Saez et al. [2022] to verify three-dimensionalboundary-element solutions in the limit $\nu=0$.

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