Three-dimensional fluid-driven stable frictional ruptures

Understanding the interplay between fluid flow and frictional slip on geologic structural discontinuities (fractures and faults) is important to a number of industrial applications (deep geothermal energy, CO2 geological storage) and natural phenomena (fluid-driven seismic swarms and aftershocks, slow slip events). In this work, we build over a previous 2-D plane-strain model [1] to investigate the growth of a fluid-driven frictional shear crack propagating in mixed mode (II+III) on a planar fault interface that separates two identical half-spaces of a linearly elastic 3-D solid. The fault interface is characterized by a shear strength equal to the product of a constant friction coefficient (a condition that guarantees stable slip) and the local effective normal stress. Fluid is injected into the fault from a point source at a constant volume rate. We demonstrate by dimensional analysis that the spatiotemporal evolution of fault slip is self-similar. Analytical solutions are derivedfor circular ruptures which occur in the limit of a Poisson’s ratio ν = 0, while the more generalcase in which the rupture shape is unknown (ν ̸= 0) is solved numerically. For ν = 0, the rupture 4αt is the nominal position of the fluid pressure front, with α the fault hydraulic diffusivity, and λ the so-called 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 limit- ing cases of critically stressed faults (λ ≫ 1, T ≪ 1) and marginally pressurized faults (λ ≪ 1, T ≫ 1), respectively. Moreover, we show that the evolution of the rupture area is independent of the Poisson’s ratio and grows simply as Ar = 4παλ2t. Asymptotic closed-form expressions for the evolution of the self-similar quasi-elliptical crack fronts are also provided in both limiting cases (see [2] for further details). Since a constant friction coefficient corresponds to a fault interface with zero fracture energy, we explore the effect of non-negligible fracture energy on the rupture dynamics by solving the 3-D crack energy balance of a fluid-driven frictional rupture with constant fracture energy Gc. It is shown that the self-similar property of the solution is broken at an initial stage that is controlled by the presence of a finite fracture energy Gc. However, the self-similarity of the response is finally recovered at late times as the effect of Gc becomes negligible. The transition between the early-time and late-time propagation regimes is captured by a dimensionless fracture energy which is decreasing with time.