The effect of in-situ linear stress gradient on the frictional shear rupture growth in 2D

Deep heat mining requires activation of slip on pre-existing geological discontinuities and the creation of hydraulically conductive fracture networks. Fluid injection or diffusion of ground waters can rise the fluid pressure near pre-existing fractures and faults, which may induce frictional slip. The fracturing process depends strongly on the initial stress conditions and rupture planes orientation. It is known that vertical stress is varying linearly with depth whereas horizontal stresses are likely not to exhibit linear dependence. Nevertheless, within certain length scales, one may assume linear relations for all stress tensor components. In the previous study [1], it was shown that for a planar rupture which is propagating due to fluid injection under a constant overpressure in the absence of stress gradient, the solution is self-similar and depends only on one dimensionless parameter which determines two limiting regimes. The first so-called "critically-stressed limit” designates that the fault is initially close to failure, whereas the “marginally pressurized limit” represents the case when the fluid pressure is “just sufficient” to activate the fault. One of the main features of the solution in the uniform stress case is that the rupture tips are propagating symmetrically. In our work, we investigate how linear stress gradient acting initially on the fault affects the shear rupture growth, namely, how it breaks the symmetry of the rupture propagation. The problem couples quasi-static elastic equilibrium and fluid flow on the fault plane via a Coulomb shear failure criterion with a constant friction coefficient. From a scaling analysis, it is shown that the problem is governed by two dimensionless parameters, To (similar to the one found in [1]) and dimensionless time. Parameter To is the ratio between the initial distance to failure and the strength of injection [1] calculated at the injection point. To determines two propagation regimes similar to those found in [1] (critically stressed and marginally pressurized limits). Dimensionless time parameter determines symmetric and asymmetric propagation periods and encapsulates the information about stress-gradient values. At early times, the solution is similar to the homogeneous stress case and the rupture stays symmetrical. At times near the characteristic time of each regime, the non-uniform in-situ stress distribution makes the rupture to propagate asymmetrically. We investigate the transition time for each limiting regime and compare it with real field observations. Our solution can also provide a benchmark for numerical solvers.