## 4 Velocity Dependence Behavior Explained Using Microphysical Model  
To describe the frictional behaviour of granular fault materials, [PERSON], [PERSON], and [PERSON] developed a microphysical model (CNS) [73; 74; 75; 42]. This model attributes friction to two key factors: porosity and grain boundary friction. It captures how velocity-dependent transitions arise from the competition between porosity generation via granular flow and  
Figure 4: (**a**) Schematic diagram of restrengthening or secant friction evolution during slide-hold-slide (SHS) tests. (**b**) Diagram showing how secant friction (blue lines), tangent friction (red lines), and cohesion (y-intercept of red lines) are determined. Lower gray dashed line represents steady-state sliding stress conditions; upper black dashed line represents stress conditions at failure. Figure reproduced from [91] with permission. porosity reduction via time-dependent, thermally activated creep processes. In the CNS framework, the rates of change of shear stress (\(\tau\)), porosities of shear band (\(\varphi^{ib}\)), and bulk gauge (\(\varphi^{bulk}\)) are described by:  
\[V_{imp}-\dot{\tau}/K=L_{t}\Big{[}\lambda\dot{\gamma}^{sb}_{ps}+(1-\lambda)\dot{ \gamma}^{bulk}_{ps}\Big{]}+L_{t}\lambda\dot{\gamma}^{sb}_{gr} \tag{6}\]  
\[\tau=\overline{\sigma}\frac{\ddot{\mu}+tan\varphi}{1-\widetilde{\mu}tan\varphi} \tag{7}\]  
\[\dot{\varphi}^{sb}/\Big{(}1-\varphi^{sb}\Big{)}=tan\varphi\dot{\gamma}_{gr}- \dot{\varepsilon}^{sb}_{pl} \tag{8}\]  
\[\dot{\varphi}^{bulk}/\Big{(}1-\varphi^{bulk}\Big{)}=-\dot{\varepsilon}^{bulk}_ {pl} \tag{9}\]  
Here, \(V_{imp}\) represents the velocity imposed at the loading point, and the gouge layer has a thickness of \(L_{t}\), with a shear strain localizing within a fraction \(\lambda\) of that thickness. The dot notation indicates a time derivative. Granular flow (gr) is assumed to be confined to the localized shear zone with a width of \(L_{t}\)\(\lambda\), whereas intergranular plastic creep (pl) is considered to occur both inside this localized region and throughout the remaining bulk gouge material, covering a thickness of \(L_{t}(1-\lambda)\). The dilatancy angle is denoted by \(\varphi\). Superscripts \"sb\" and \"bulk\" refer to the shear band and surrounding bulk material, respectively. Shear strain and normal strain are indicated by \(\gamma\) and \(\varepsilon\). The frictional resistance at grain boundaries, represented by \(\widetilde{\mu}\), depends on the strain rate associated with the granular flow within the shear band. Figure 5 shows a schematic diagram outlining the three distinct temperature-dependent regimes in terms of the velocity-dependence parameter. This schematic links the apparent friction coefficient and its velocity dependence to the underlying micro-mechanical deformation mechanisms proposed by the model. In Regime I, deformation is characterized by significant grain size reduction and a cataclastic flow involving quartz, albite, and K-feldspar, without any microstructural evidence for the creep or mass transfer processes, consistent with model A in Figure 5. For Regime II, the microstructures indicate a more distributed deformation pattern with multiple sharp Y-shears, in contrast to the localization observed in Regimes I and III. While cataclastic processes remain active, features such as grain-to-grain indentation and grain truncation, particularly within ultrafine-grained zones, suggest that pressure solution mechanisms began to operate, likely triggered by grain size reduction [79]. This regime reflects the interplay between dilatant granular flow and solution-assisted deformation, in line with model B of Figure 5. In Regime III, the presence of active B-shear bands containing ultrafine, equant grains of quartz, albite, and K-feldspar--accompanied by intergranular pores or healed cavities with low overall porosity and lacking shape-preferred orientation (SPO)--indicates a shift towards grain boundary sliding assisted by solution-precipitation creep and pore healing [73].