Three-dimensional fluid-driven stable frictional ruptures

We investigate the quasi-static growth of a fluid-driven frictional shear crack that propagates in mixed mode (II+III) on a planar fault interface that separates two identical half-spaces of a three-dimensional solid. The fault interface is characterized by a shear strength equal to the product of a constant friction coefficient and the local effective normal stress. Fluid is injected into the fault interface and two different injection scenarios are considered: injection at constant volume rate and injection at constant pressure. We derive analytical solutions for circular ruptures which occur in the limit of a Poisson's ratio ν=0 and solve numerically for the more general case in which the rupture shape is unknown (ν≠0). For an injection at constant volume rate, the fault slip growth is self-similar. The rupture radius (ν=0) expands as R(t)=λL(t), where L(t) is the nominal position of the fluid pressure front and λ is an amplification factor that is a known function of a unique dimensionless parameter T. The latter is defined as the ratio between the distance to failure under ambient conditions and the strength of the injection. Whenever λ>1, the rupture front outpaces the fluid pressure front. For ν≠0, the rupture shape is quasi-elliptical. The aspect ratio is upper and lower bounded by 1/(1-ν) and (3-ν)/(3-2ν), for the limiting cases of critically stressed faults (λ≫1, T≪1) and marginally pressurized faults (λ≪1, T≫1), respectively. Moreover, the evolution of the rupture area is independent of the Poisson's ratio and grows simply as Aᵣ(t)=4παλ²t, where α is the fault hydraulic diffusivity. For injection at constant pressure, the fault slip growth is not self-similar: the rupture front evolves at large times as ∝(αt)⁽¹⁻ᵀ⁾ᐟ² with T between 0 and 1. The frictional rupture moves at most diffusively (∝√(αt)) when the fault is critically stressed, but in general propagates slower than the fluid pressure front. Yet in some conditions, the rupture front outpaces the fluid pressure front. The latter will eventually catch the former if injection is sustained for a sufficient time. Our findings provide a basic understanding on how stable (aseismic) ruptures propagate in response to fluid injection in 3-D. Notably, since aseismic ruptures driven by injection at constant rate expands proportionally to the squared root of time, seismicity clouds that are commonly interpreted to be controlled by the direct effect of fluid pressure increase might be controlled by the stress transfer of a propagating aseismic rupture instead. We also demonstrate that the aseismic moment M₀ scales to the injected fluid volume V as M₀ ∝ V³ᐟ².