## 3 Methods
### Hydro-Mechanical Coupling Concept in PFC2D  
In order to simulate the pore pressure variations induced by the dynamic geomechanical processes, i.e., fault dynamic rupture, which is a hydro-mechanical coupled processes, we developed a pore-pipe network model and implemented it in the 2D particle assembly. Figure 5a shows the 2D version of the pore-pipe network model. The blue polygons are the fluid domains constructed by lines linked to adjacent particles at their centers. The \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Normal stiffness & \(3\times 10^{8}\) (Pa/m) & Tensile strength & \(1\times 10^{6}\) (Pa) \\ Shear stiffness & \(5\times 10^{7}\) (Pa/m) & Friction angle & 30 (\({}^{\circ}\)) \\ Friction coefficient & 0.5 & Dilation angle & 3 (\({}^{\circ}\)) \\ Cohesion & \(5\times 10^{6}\) (Pa) & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Smooth joint contact model parameters. All parameter values were best estimated, based on study region lithology. Figure 4: Faults represented by smooth joint contact model and the orientations of the in situ horizontal stresses shown by the stress tensor. Black star is the hypocenter of the 2016 GJEQ. short red lines are the pipes for the fluid flow based on [PERSON]'s law and Poiseuille equation, which assumes fluid from one domain to a nearby one in the form of laminar flow. Each cycle of hydro-mechanical coupling in the pipe network model can be divided into three steps. Firstly, the volumetric flow rate in each pipe is calculated by the hydraulic aperture and fluid pressure gradient. Secondly, the fluid pressure in each fluid domain is updated considering the fluid volume increment. Lastly, fluid pressure is exerted on solid particles by an additional force, as shown in Figure 5b. Figure 6 presents the calculation steps of the hydro-mechanical coupling processes. In this algorithm, the left part is the DEM modeling in PFC2D for the solid deformation and movement, while the right part is the fluid flow modeling based on FVM. FVM for fluid flow has the following two steps: constructing a linear system based on the continuity equation of a Stokes flow in each fluid domain and solving the linear system in Python to obtain the pore pressure field. During the coupling process, the local conductance of each pipe and the volume increment in each fluid domain are calculated from the DEM particle system. The pore pressure includes the hydrostatic pressure and the losses of piezometric pressure due to fluid flow. The mathematical formulation of the pore-pipe network model can be referred to in [23]. Figure 5: (**a**) Pore–pipe network implemented in a bonded particle assembly and (**b**) description of how pore pressure exerts force (multiplication of pore pressure \(P_{f}\), length \(l_{i}\), and model’s out-of-plane direction thickness \(t\)) to the surrounding particles (hydro-mechanical coupled effect). Figure 6: Algorithm of hydro-mechanical coupling processes in PFC2D modeling (a solid line arrow means iterative process within a single code, while a broken line arrow means iterative communications between different codes). The coupled process calculation begins with particle movement initiated by fault rupture. The equation of motion governs particle movement, where translational and rotational forces determine how particles move and rotate in response to external forces. \[F_{i}=m\left(\ddot{x}_{i}-\ddot{\bar{g}}_{i}\right) \tag{4}\]  
\[M_{i}=I\ddot{\theta} \tag{5}\]  
As the simulation progresses, particle velocity and displacement are updated, and particles come into contact with each other. The forces at these contacts are calculated using the force-displacement law, which considers normal and shear stiffnesses, to model how forces evolve at particle interfaces. This part of the calculation is handled by a Discrete Element Method (DEM) loop, implemented through PFC2D.