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

Fluid Dynamics;Geophysics