## 2 Materials and Methods
### Methods  
Experiments were performed on bare rock surfaces using a high-pressure and high-temperature triaxial testing apparatus [32]. The samples were jacketed in a hot shrinkage pipe with a wall thickness of 1 mm in the test carried out at room temperature or in an annealed copper tube with a wall thickness of 0.35 mm in the test carried out at 100 \({}^{\circ}\)C (Figure 2c). A detailed description of the sample assembly is presented in the study of [PERSON] et al. [32]. In the experiments, high-pressure gas was injected into the pressure vessel, forming a constant confining pressure (\(P_{c}\)). Axial deformation was applied with an electro-hydraulic servo-control system at 1 \(\upmu\)m/s to ensure stable slip conditions for the sample.  
The average overburden pressure gradient equals the lithostatic pressure gradient plus the fluid pressure gradient, which is 22.7 kPa/m [33]. Injection-induced earthquakes usually occur at the shallow sedimentary fault, typically at 3-5 km depths [8; 13; 15; 16; 34]. The lithostatic pressure, equaling the overburden pressure gradient minus the hydraulic pressure gradient, is around 50 MPa at this depth, so we ran experiments at 1 \(\upmu\)m/s initial loading rate and 50 MPa confining pressure (\(P_{c}\)) in this study (Table 2). A preset pore pressure (\(P_{p}\) = 1 MPa or 15 MPa) was injected into the sample when the fault reached the steady-state friction.  
In the applications associated with unconventional oil/gas resource development, pressurized water may be injected into a position connecting to nearby pre-existing faults or separated from nearby faults. Considering the difference between these situations, we performed the injection tests in blind and open borehole samples (Figure 2a,b). In the blind borehole tests, the 15 MPa pressurized water was injected into the blind borehole, separated by a 1 mm rick from the fault plane, to investigate the impacts of rock permeability on injection-induced earthquakes. In the open borehole tests, 1 MPa and 15 MPa pressurized water were injected into the fault successively to investigate the influence of fault deformation on injection-induced earthquakes.  
Figure 2: Sample configuration for an injection hole (**a**) reaching the position at 1 mm from the fault (blind borehole) and (**b**) the fault plane (open borehole). (**c**) Sketch of sample assembly.  
According to rate-and-state dependent friction (RSF) laws [35; 36; 37], friction slip stability is characterized by the velocity-dependent parameter \((a-b)=d\mu_{ss}/dln(v)\), where \(a\) and \(b\) are empirical constants in RSF laws, \(\mu_{ss}\) is the fault frictional strength in steady state, and \(v\) is sliding velocity. When \((a-b)>0\), the steady-state friction coefficient increases with the slip velocity, called velocity-strengthening, and slip is accommodated by stable sliding. On the contrary, when \((a-b)<0\), the friction coefficient decreases as the sliding velocity, called velocity-weakening, and the fault may experience dynamic instability, meeting the conditions for earthquake nucleation [35; 37]. To test the velocity dependence of friction, we carried out velocity stepping tests in three faults at different loading conditions, including dry and wet samples with 1 MPa and 15 MPa pressurized water, by switching the loading rate between 1 \(\upmu\)m/s and 0.2 \(\upmu\)m/s.  
Most experiments were carried out at room temperature. Given the critical roles of temperature on fault velocity-dependent properties, a limestone experiment was conducted at \(T=100\)\({}^{\circ}\)C for comparison. A thermocouple was used to regulate the test's temperature at \(T=100\)\({}^{\circ}\)C (Figure 2c). Table 2 lists all experiments performed in this study.