## 3 Numerical Example
### Granular Dam Break Simulation  
Simulations of dam breaking are widely regarded as a benchmark when testing numerical simulation methods [37; 38; 39]. We simulated a three-dimensional granular version of the dam breaking problem, configured as a box-shaped pile of granular mass initially at rest behind a wall that collapsed with the sudden emergence of a rectangular breach (Figure 7). In the simulation, the calculation domain was 200 m \(\times\) 200 m and the dimensions of the granular pile were 200 m (width) \(\times\) 100 m (length) \(\times\) 10 m (height). We placed the midpoint of the breach at \(y=100\) m and varied the width of the rectangular breach (B) from 5 m to 10 m, 15 m, 20 m, 35 m, 50 m, and 70 m in separate simulations. The angles of internal friction and coefficient of basal friction were set at \(\varphi=40^{\circ}\) and \(\delta=30^{\circ}\), respectively. The earth pressure coefficients were calculated using (10a,b) for the simulations. The volumetric flow rate per unit width \(Q\) (i.e., \(\sqrt{\overline{u}^{2}+\overline{v}^{2}}\cdot h\cdot dy\), we used a unit width of \(dy=5\) m in the simulation), the velocity in the \(x\)-direction \(u\), and the granular depth \(h\) at \(x=105\) m and \(y=100\) m were tracked (Figure 7, red point) during the dam breaking simulations for different breach widths (Figures 8-10). From Figures 8 and 9, the flow rate and the velocity in the \(x\)-direction with different breach widths all reached large values (\(Q>100\) m\({}^{3}\)/s and \(u>1.4\) m/s) within 3 s, and then decreased rapidly within 10 s. The average values and the fluctuations of the relatively steady flow rate and the velocity after 10 s decreased with the breach width. Figure 10 plots the flow depths at \(x=105\) m and \(y=100\) m with time for various breach widths. The depth and its variation decreased with the breach width as well. Larger fluctuations in the flow rate, the velocity, and the depth indicated that the granular mass collapsed more discontinuously with the decrease in breach width. Larger average values for the flow rate and the velocity were likely because  
Figure 7: Schematic of the box-shaped pile of granular mass before collapse through a rectangular breach into a horizontal runout plane (unit: m). more grains flowed over the tracked point, and the duration of dam break process increased with the decrease in breach width. In the governing equation (i.e., Equations (1) and (2a-e)) used to describe the granular motion, the flow of granules was mainly balanced by the flux vectors and the source terms. Specifically speaking, particles started to move intensively when the depth gradient of the granular flow in the flux vector term exceeded the friction in the source term. The fluctuations in the flow rate, the velocity, and the depth in Figures 8-10 indicate that the granular mass collapsed discontinuously. To be specific, the motion of grains remained negligible when the depth gradient of the granular flow was smaller than the friction. As the flow surface gradually grew steeper with the accumulation of grains, an avalanche suddenly occurred when the depth gradient term exceeded the friction term. After the avalanche was terminated, a new balanced state was reached, which is represented by the plateaus in Figures 8-10. Figure 8: Changes in volumetric flow rate per unit width at the track point marked in Figure 7 over time for different breach widths. Figure 9: Plots of tracked velocity in the \(x\)-direction with different breach widths during dam breaking. According to the results presented in Figure 10, the depth of the mass collapse satisfied a regression model  
\[h=h_{\max}\,\left(1-e^{-t/\tau}\right), \tag{11}\]  
where \(h_{\max}\) is a limiting value of the depth and \(\tau\) is a \"relaxation\" parameter.