## 6 Simulation Illustrations
### Simulation of the Mud Intrusion Procedure in a Mountain Tunnel  
During the excavation of mountain tunnels, mud inrushes may occur, as shown in Figure 11a. The mechanism of mud inrush in mountain tunnels remains unclear due to challenges in accurately assessing geological conditions. A two-dimensional fluid-solid coupling numerical model with dimensions of 5 cm in length and 4 cm in height is used to replicate the process of tunnel mud inrush, as shown in Figure 11b. Particle aggregates within this range were generated to mimic the formation process through self-weight deposition, equilibrium, and bonding, with particle sizes ranging from 0.06 to 0.1 cm. A total of 720 particles were randomly generated within the box. The left boundary features a water flow inlet and pressure boundary while employing a specific density value and non-equilibrium rebound method to simulate water pressure within the formation. The upper boundary is open, and the lower boundary is a no-slip boundary. The tunnel face located on the right boundary serves as a mudslide outlet. It should be noted that, in order to enhance the computational efficiency of coupled LBM-DEM simulation, the particle size in the numerical model is larger than that of typical mud bodies in practical engineering. When the particles and flow field are stable, the calculation stops. The detailed physical and mechanical parameters of fluid and solid particles are the same as in Table 1, except for the relaxation parameter \(\tau\) = 0.504. Figure 10: The flow velocity and particle position of the water continuously vary along with the increase of hydraulic gradient when solid particles in karst pipelines are washed away by the water flow. **(a)** The variation rule of the Darcy velocity of water in accordance with the hydraulic gradient. **(b)** The distribution pattern of the particle positions at the upper and lower boundaries in the case of water flow erosion. Figure 9: A series of snapshots from a simulation of karst piping flow with the coupled LBM-DEM. Throughout the simulation process, data such as particle distribution, inter-particle bonding distribution, flow velocity, water pressure, and mud-burst volume are recorded every 4000 time steps. The damage caused by tunnel mud burs is a result of the combined effects of various factors, including the soil medium properties, water pressure dynamics, and excavation activities. It is only under specific conditions that these three factors combine to cause mud-burst disasters in tunnels. Numerical calculations are solely used for simulating the phenomenon of mud inrush, which differs somewhat from actual engineering scenarios involving mud inrush. Figure 12 illustrates the process of tunnel mud inrush, which can be categorized into three stages: initiation, acceleration, and stability. During the initiation stage, as particles near the mud burst gradually enter, the upper particles collapse downward due to the loss of effective support, leading to the formation of a free surface that inclines towards the mud burst and creates a wider water passage. Under the erosion of water flow, a large number of particles migrate along the tunnel's palm face towards the direction of mud burst, resulting in a continuous and rapid increase in the amount of mud burst. In the accelerating stage, there is an increase in bonding strength between particles, causing rapid collapse of the tunnel face and scouring of many particles with a maximum distance between them exceeding 12 cm. The time for the acceleration stage becomes shorter with a slightly smaller rate of increase. At the stable stage, the free surface has essentially reached the lower edge of the mud mouth, and most of the water flows near the right boundary and exits through the tunnel palm face. The water flow on the free surface weakens, causing a slowdown in the growth of the tunnel palm face, which ultimately remains unchanged. At this point, both the particles and the flow field have stabilized, entering a steady state.