Self-similar fault slip in response to fluid injection

There is scientific and industrial interest in understanding how geologic faults respond to transient sources of fluid. Natural artificial can elevate pore fluid pressure on the fault frictional interface, which may induce slip. We consider a simple boundary value problem provide an elementary model physical process benchmark for numerical solution procedures. examine slip that interface two elastic half-spaces. Injection modeled as line source at constant assumed diffuse along interface. The resulting integro-differential equation governing slip, has single dimensionless parameter. expansion self-similar rupture front propagates factor $\lambda$ diffusive lengthscale $\sqrt{\alpha t}$. identify asymptotic regimes corresponding being small or large perform perturbation each limit. For $\lambda$, regime so-called critically stressed fault, layer emerges lengthscale, lags far behind front. demonstrate higher-order matched asymptotics equation, doing so, we derive multipole capture successive orders influence outer driving force relative crack dimensions. Asymptotic expansions are compared accurate solutions full problem, tabulated high precision.