## 4 Discussion  
### Overpressure in Fracture Reactivation  
The stimulation of triggering pre-existing fractures by cyclic water pressurization is a process of over-pressurization [6; 14; 15; 16; 17] and pore pressure diffusion [18; 19; 20; 21; 22; 23; 24] along the fracture. Each process promotes the reactivation of the pre-existing fault and induces seismicity. Overpressure is an especially crucial factor, as it controls the state of the fault. The combinations of the shear stress, normal stress, and water pressure of a fault are plotted in Figure 5, by modifying the graphical representation of Garagash [25] and Gischig [26]. According to the Mohr-Coulomb failure criterion, \(\tau\) is the initial shear stress on the fault and \(\tau_{p}\) is the shear strength acquired from the displacement-driven shear test. \(r_{c}\) is the ratio between shear stress (\(\tau\)) and shear strength (\(\tau_{p}\)), which is used to describe the critical state of the fault. The understress is evolved as \(1-r_{c}\). The fault is assumed to be critically stressed if the understress decreases to zero. On the contrary, the shear stress is 0 if the understress is equal to one. The increasing water pressure monitored by the system is defined as \(\Delta p\) and \(\Delta p\) and is normalized by the fault-effective normal stress. \(\alpha\)eff (\(\Delta p/\sigma_{\textit{neff}}\)) is defined to describe the overpressure of the fracture. The diagonal in Figure 8 is a limitation in the fracture failure, dividing the fault into two states: if \(\Delta p/\sigma_{n}<1-r_{c}\), the increased pressure and shear stress cannot induce slip; if \(\Delta p/\sigma_{n}>1-r_{c}\), the fault is transitioning from the stable state to unstable state and an aseismic slip or seismic slip might happen. In our test, the overpressure is a dynamic oscillating state rather than a quasi-static state. Due to the heterogeneity of the pressure distribution and low permeability, the change in outlet pressure is delayed compared to the inlet pressure. Therefore, the inlet pressure and outlet pressure corresponding to every slip are analyzed in Figure 8, showing that the unstable slip occurs with the accumulation of pressure. The whole process is divided into three stages according to the inlet pressure (6 MPa, 7 MPa and 8 MPa). For each stage, the overpressure increases in both the inlet and outlet pressure. Every initiation of reactivation of the fault requires a higher pressure in both the inlet and outlet. Therefore, a higher pressure along the fault surface is a necessary condition for inducing a dynamic slip. And when a new balance between the inlet and outlet pressure has formed, the pressure-controlled stick-slip will turn into a stable sliding state. This phenomenon is consistent with the experiment investigated by [PERSON] [24]. Finally, when the outlet and inlet pressure remains equal, the maximum injected energy by the rising water pressure remains constant and there is no more dynamic slip during pressure oscillation. Referring to the theory for describing the criterion of fault stability by combining the elastic dislocation theory with the rate-and-state-friction (RSF) constitutive, a relationship between fault stiffness (\(k_{c}\)) and effective normal stress (\(\sigma_{\textit{neff}}\)) is defined as Equation (5). \(D_{c}\) is the critical slip distance for the friction of the fracture surface changing from static friction to dynamical friction. Parameter \(a\) is the fracture friction at velocity \(V_{1}\). Parameter \(b\) is the fracture friction at velocity \(V_{2}\). \(\sigma_{\textit{neff}}\) is the effective normal stress on the fracture. All the parameters can be measured in the rate and state law friction test. \[k_{c}=\frac{\sigma_{\textit{neff}}(b-a)}{D_{c}} \tag{5}\]  
where \((b-a)/D_{c}\) is the friction weakening rate parameter [27; 28]. Equation (5) shows that increasing the pressure inside the fault can reduce \(k_{c}\) and promote stable sliding, rather than an earthquake slip. But seismological observations contrast with this prediction.