## 3 Friction Experiments of Granitoid Fault
### Velocity-Stepping Experiment
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Gauge/Rock** & **Temperature** & \begin{tabular}{c} **Effective** \\ **Normal Stress** \\ \(\overline{\sigma}\) \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Pore Fluid** \\ **Pressure** \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Slip Rate** \\ **(\(\mu\)m/s)** \\ \end{tabular} & \
\begin{tabular}{c} **Shear** \\ **Displacement** \\ **(mm)** \\ \end{tabular} &
\begin{tabular}{c} **Experiment** \\ **Type** \\ \end{tabular} & **References** \\
**Sample** & \begin{tabular}{c} **\(\text{(}^{\circ}\text{C}\text{)}\)** \\ \end{tabular} & \
\begin{tabular}{c} **\(5\text{-}40\)** \\ \end{tabular} &
\begin{tabular}{c} minor water \\ \end{tabular} & 0.1–30 & \(\sim\)2.5 or \(\sim\)112 & VS & [45] \\ \hline
**Nateral granite** &
\begin{tabular}{c} 25–600 \\ \end{tabular} & 200 & 30 & 0.04–0.2–1 & \(\sim\)4 & VS & [68] \\ \hline
**Nested granite** &
\begin{tabular}{c} 25–250 \\ \end{tabular} & CP: 20 & 10 & 0.1 & \(\sim\)5 & HSH & [69] \\ \hline
**Nateral granitiod** &
\begin{tabular}{c} 20–650 \\ \end{tabular} & 100 & 100 & 0.1–100 & -15 & CV & [41] \\ \hline
**Nateral granite** & \begin{tabular}{c} 25–600 \\ \end{tabular} & 100, 200 & 30,100 & \
\begin{tabular}{c} 0.04–0.2–1 and \\ \(1\text{-}3\text{-}10\text{-}30\text{-}100\) \\ \end{tabular} &
\begin{tabular}{c} -25 \\ \end{tabular} & VS and CV & [9] \\ \hline \end{tabular} Note. VS = velocity stepping, CV = constant velocity, SHS = slide-hold-slide, CP = confining pressure.  
\end{table} 
Table 1: _Cont._On the contrary, three \(T\) regimes were observed under hydrothermal conditions for granitic gouge and rock: (a) (Regime I) velocity-strengthening (V-S) at a low \(T\) of 25-100 \({}^{\circ}\)C; (b) (Regime II) velocity-weakening (V-W) at an intermediate \(T\) of 150-300 \({}^{\circ}\)C; (c) (Regime III) velocity-strengthening (V-S) at a high \(T\) of 350-600 \({}^{\circ}\)C. Notably, the boundaries of these \(T\) ranges may shift toward a higher \(T\) with different \(V\) and \(P_{t}\). [PERSON] et al. [37] found that at 350 \({}^{\circ}\)C, lower \(V\) (0.01-0.1 \(\upmu\)m/s) transition from the V-W regime to a high-\(T\), V-S regime earlier than higher \(V\) (0.1-1 \(\upmu\)m/s). A similar trend is observed in the natural granite gouge [9]. They conducted friction experiments over a wider \(V\) range of 1-100 \(\upmu\)m/s and found that the V-W regime occurs at 150-300 \({}^{\circ}\)C for low \(V\) = 1-3 \(\upmu\)m/s, while it extends to 100-450 \({}^{\circ}\)C as the \(V\) increases to 30-100 \(\upmu\)m/s (Figure 2). In addition to \(T\) and \(V\), pore pressure (\(P_{t}\)) and shear displacement influence the occurrence of the three regimes. For example, increasing \(P_{t}\) shifts all three regimes toward lower \(T\), particularly narrowing the V-W regime to within 100-350 \({}^{\circ}\)C. This behavior has also been observed in phyllosilicate and quartz-rich fault gouges [71]. Regarding shear displacement, large cumulative slips can promote an earlier transition from the V-W regime to a high-\(T\), V-S regime [9]. Note that in tests conducted over a narrow range of \(V\), the (\(a-b\)) values may capture only two of the three frictional regimes [40; 68; 72]. Recent microstructural studies suggest that the transitions among the three frictional regimes are primarily governed by underlying microphysical mechanisms, particularly the intergranular flow and deformation of fine-grained gouge facilitated by intergranular pressure solutions, often accompanied by healing processes, such as cavitation-creep behavior, and CNS models [42; 73; 74; 75]. Under hydrothermal conditions, the strain rate of the fine grains is strongly dependent on temperature (\(T\)), differential stress (\(\sigma\)), and grain size (\(d\)).