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 $\nu =0$, 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 $\lambda \sqrt {4\alpha _{hy} t}$, where $t$ is time since the start of injection and $\alpha _{hy}$ is the hydraulic diffusivity of the pore fluid pressure. The prefactor $\lambda$ is determined by a single parameter, $T$, which depends on the pre-injection stress state and injection conditions. The prefactor has the range $0\lt \lambda \lt \infty$, 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 ($\lambda \rightarrow 0$), the perturbation is regular and the series expansion is convergent. For the critically stressed limit ($\lambda \rightarrow \infty$), 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.