## 6 Simulation Illustrations  
### Simulation of the Water Flow Process and the Movement of Particles in a Karst Pipeline Flow  
In this subsection, the coupled LBM-DEM is utilized to model the particle flow in karst pipelines within mountain tunnels. The primary challenge arises from the need for more analytical solutions for estimating the forces exerted on particles by fluid flow containing a large number of particles. In a fluid-filled system comprising numerous particles, each submerged particle is subject to not only the hydrodynamic forces and torques exerted by the surrounding fluid field but also engages in intricate interactions with its neighboring particles. This system holds significant importance in civil engineering. The selection of interactions between a smaller number of individual particles and fluids within simple geometric shapes represents an initial step in the progressive exploration of such systems. For instance, the Xiaozhai Tunnel is a Class I high-risk tunnel on the China Yunnan-Guangxi Railway (from Yunnan Province to Guangxi Province). It has a total length of 6700 m and a maximum burial depth of 270 m. The survey area mainly consists of four lithologies: Quaternary silty clay, Triassic sandstone mixed with shale and mudstone, Devonian limestone mixed with dolomite, and Cambrian dolomitic limestone. The regional geological structure is relatively complex, with multiple faults developed. The rock mass is strongly influenced by its structure, with highly developed joints and fissures. The groundwater types in the tunnel area are mainly divided into clastic rock fissure water and carbonate rock karst water. Advanced geological prediction during tunnel excavation is essential for high-risk railway tunnels. It primarily involves the elastic wave reflection method and electromagnetic wave reflection method [37]. During the construction of the Xiaozhai Tunnel, Tunnel Seismic Prediction (TSP) was utilized for advanced geological prediction for the 100 m ahead of the section DK412+026\(\sim\)DK411+926 of the Xiaozhai Tunnel. It was found that from the chainage DK411+950 on the palm face, the longitudinal and transverse wave velocities of the rock mass ahead decreased, while the Poisson's ratio increased; the dynamic elastic modulus decreased with decreasing chainage, as shown in Figure 7. Subsequent excavation revealed a karst pipeline in the middle and lower part of the face, which experienced water with mud gushing, as shown in Figure 8a. In response to this phenomenon, a 2D simulation model of karst piping flow using coupled LBM-DEM was presented to verify if it accurately describes the linear relationship between erosion rate and fluid shear stress. A 2D simulation model of a karst pipeline flow, measuring 7 cm in length and 4 cm in width, is established in Figure 8b. Figure 6: Simulating incompressible flow around a cylinder using FEM. Figure 8: Water inrush and the 2D numerical model at the tunnel face of Xiaozhai Tunnel. (**a**) Water mixed with mud gushing out from a karst pipeline in the middle and lower portion of the tunnel face of Xiaozhai Tunnel. (**b**) A two-dimensional particle flow model with a length of 7 cm and a width of 4 cm. Figure 7: The TSP prediction results regarding the first 100 m of the section DK412+026 to DK411+926 of the Xiaozhai Tunnel encompass the rock longitudinal and transverse wave velocities, Poisson’s ratio, and dynamic elastic modulus. Beginning from the chainage DK411+950, water mixed with mud is gushing out from a karst pipeline in the middle and lower portion of the tunnel face. This system serves as a simplified model that disregards gravity effects and is spatially discretized utilizing a square lattice to facilitate the implementation of the LBM algorithm, with voids being filled with water. One hundred particles, randomly generated in a two-dimensional box, have an average radius of 0.155 cm, maximum radius of 0.206 cm, and minimum radius of 0.105 cm, as shown in Figure 9a.