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o = 0.6, (a-b) = ?0.002, and dc = 10 ?m. These values fall within the range of frictional parameters measured in laboratory tests on fault samples collected in carbonate rocks (Carpenter et al., 2014 , 2016 ). ?o is the friction coefficient at a reference slip velocity (vo). The parameter a quantifies the direct effect of a change in slip velocity. The parameter b describes the effect of the state variable (here we use the “aging law”; Dieterich, 1979 ). The characteristic slip distance, dc, governs the evolution of the state variable (?). For fault models with constant friction, we assume a static value (?s) of 0.6.
Figures 2c and 2d show how the development of the fluid pressure along the fault varies as a function of the permeability enhancement factor and the associated hydraulic aperture. Models indicate that the magnitude and distribution of the steady state overpressure as well as the size of the pressurized area depend strongly on the permeability change. For a constant permeability model (case k/ko = 1), the pressure perturbation is poorly prozerounced. The highest pressure and sharpest pressure gradients are located close to the injection. For models with changing permeability during slip, the size of the pressurized zone grows significantly with the fault permeability enhancement. Models show that the higher the permeability increase, the greater is the pressurized area (Figure 2c). The permeability increase is higher near the injection and decreases at larger distances (Figure 2d). We define the pressurized length of the fault as the distance from the injection to the limit of the pressurized zone where the fluid pressure is zero (this distance is then normalized by the fault length). Given the applied injection pressure and the resulting calculated fault deformation, the pressurized length reaches a maximum normalized value of 0.2635.
Our model results also indicate that the permeability evolution affects both the maximum diffusion length and the size of the slip zone (Figures 3a and 3b). The extent of the slip zone is defined as the distance between the injection and the limit of the slipping patch where the slip is zero. Figures 3a and 3b suggest that the larger the increase in fault permeability, the larger is the extent of the slip zone. When a sufficient portion of the fault is pressurized and weakened, fault slip accelerates (i.e., slow stick-slip, Figure S1 in the supporting information), and a step-like increase in the length of the slip zone occurs. The most pronounced difference between the slip and pressure fronts occurs for the higher, more critical, initial stress ratio (?o/?no = 0.47). For this case, results highlight that all simulations including permeability changes (k/ko > 1) show that the growth of fault slip outpaces the growing fluid pressure front. For a less critical initial stress ratio (?o/?no = 0.388), results show that the growth of fault slip outpaces the growing fluid pressure front for permeability changes (k/ko) greater than 10. For the highest permeability enhancement (k/ko = 60), the size of the slip zone is about 1.74 and 3.23 greater than the size of the pressurized zone, respectively, for initial stress ratios of 0.388 and 0.47 (Figure 4).
In Figures 3c and 3d, we compare the slip length as a function of the length of the pressurized zone for different stress ratio (?o/?no) and different friction laws for the case with the highest permeability increase (k/ko = 60). We find that the growth of slip is affected both by the background stress and frictional weakening. For example, a fault with higher background stress (?o/?no = 0.47) can produce larger slip growth (Figure 3c). Reducing the shear stress delays the timing of the slip front outpacing the fluid diffusion front (734 s in Figure 3d) and decreases the maximal size of slip area, whereas increased shear stress leads to earlier onset (471 s in Figure 3d) and a larger slip zone. The effect of fault friction is also illustrated in Figures 3c and 3d. Fault frictional weakening using the rate-and-state friction law influences the temporal evolution of the slipping area and may jak sprawdziД‡, kto ciД™ lubi w little armenia bez pЕ‚acenia produce larger ruptures. This is expected because friction weakening leads to reduced fault strength with sequences of accelerated and increased slip, while constant friction tends to stabilize fault strength resulting in less pronounced slip (Figure S1).