## 4 Engineering Application
### Model Calculation Results and Analysis  
(1) Porosity evolution. Figure 8 shows the results of porosity evolution contours. As depicted in Figure 8, porosity changes are insignificant near the exit and in the fault rupture zone when the evolution time is below 200 min. At 300 min, porosity changes dramatically near the exit zone, with a maximum porosity of approximately 0.55. This indicates more active erosion and considerable particle loss. By 400 min, the maximum porosity near the exit zone reaches close to 0.8, with drastic alterations in the fault fracture zone vicinity, suggesting considerable particle loss and unstable cavity formation. With continued evolution, porosity development increases between the exit and fault fracture zone, with regions exceeding 0.8 porosity transfixing rapidly, indicating the formation of the water inrush channel. (2) Pore water pressure and fluidized grain concentration evolution. Figure 9 shows the results of the pore water pressure and fluidized grain concentration at different times. When the evolution time is 350 min, more active erosion is observed near the exit zone. The maximum fluidized grain concentration at the exit zone is close to 0.35,  
Figure 11: Time variation in the average channel size. Figure 12: Temporal variation in the water flow rate and mud inflow rate at the outlet boundary of the tunnel face. the pore pressure around the model outlet decreases gradually with increasing evolution time, and the pore pressure gradually forms a low-pressure funnel area at the model outlet, which grows larger with increasing evolution time, as shown in Figure 9b,c. With significant particle loss, unstable cavities formed in the fault fracture zone. As the evolution progresses, the porosity between the exit and fault fracture zones quickly increases to 0.5 (Figure 9e). This indicates that all the movable particles have been eroded except for the large skeleton particles, and a water inrush channel has formed (see Figure 9f). 3. Analysis of the development of the water inrush channel and the water and mud inrush disaster. Soil with high porosity will induce the collapse and result in the formation of the water flow channel. To analyze the development of the water inrush channel, the zone with high porosity (>0.5) was further investigated, as shown in Figure 10. It can be observed that a porosity greater than 0.5 first appeared in the tunnel vault at t = 300 min, which inclined to the tunnel axis at an angle of approximately 45\({}^{\circ}\). As the evolution continued, the porosity development increased in the fault fracture zone (Figure 10b-d), and the region with a porosity exceeding 0.5 rapidly developed toward the fracture zone top, as depicted in Figure 10e,f, suggesting the establishment of a water inrush channel. Channel formation further exacerbates erosion development because of the elevated fluidized grain concentration and flow velocity. As depicted in Figure 10e,f, the channel width and height increase with continual erosion. To quantitatively evaluate the size of the water inrush channel, the porosity of the water inrush channel or cavity was set to greater than 0.7. Based on this criterion, the temporal evolution of the mean water inrush channel size was determined in Figure 11. From 0 to 350 min, the channel height and width are considerably small owing to the weak erosion, which is approximately zero. From 350 to 500 min, the water inrush channel develops rapidly, and the channel height increases quickly from 0 to approximately 50 m. The width of the channel increases rapidly from zero to approximately 25 m. After channel formation, its width remains stable, while its height increases quickly from 50 m to 62 m.  
Figure 12 illustrates the temporal fluctuations of the water flow rate and mud inflow rate at the outlet boundary of the tunnel face. Combining this analysis with Figure 11, we segment the development process of the water outburst channel into three phases: slow development, rapid development, and stable phases.