## 2 Materials and Methods
### Breach Evolution Model
#### 2.2.4. Breach Lateral Evolution Module  
Under the lateral erosion of the outburst flood, the dam breach will become wider. It is currently assumed in most models that the lateral erosion rate is twice that of the vertical erosion rate during the dam break [48; 49]. A linear extension model for calculating the lateral development of the breach is provided, while the initial movement conditions of sediment particles and the difference in slope are ignored. Therefore, the simulated lateral erosion of the breach does not fully match the actual situation. In order to solve this problem, a method is proposed for calculating the width of the breach by deducing the lateral evolution formula through the theoretical analysis of the initial movement of sediment in the dam slope. The forces acting on sediment particles in the slope mainly include particle gravity component \(Gsin\theta\), the combined drag force \(\mathbf{F_{d}}\), and the critical drag force \(\mathbf{F_{d}^{\prime}}\), as shown in Figure 2.  
The combined drag force of dragging \(\mathbf{F_{d}}\) on the side slope of breach and the underwater gravity along the slope is calculated as follows:  
\[\mathbf{F_{d}}=\sqrt{\left(\mathbf{G}sin\theta\right)^{2}+\mathbf{F_{d}^{\prime}}^{2}}. \tag{17}\]  
The initial motion condition of sediment particles on the breach side slope:  
\[\sqrt{\left(\mathbf{G}sin\theta\right)^{2}+\mathbf{F_{d}^{\prime}}^{2}}=tan\varphi( \mathbf{G}cos\theta-\mathbf{F_{L}}-\mathbf{F_{s}}). \tag{18}\]From the Equation (18), the critical drag force \(\mathbf{F_{d}^{\prime}}\) of sediment particles on the breach side slope can be expressed as the following:  
\[\mathbf{F_{d}^{\prime}}=\sqrt{tan^{2}\varphi(\mathbf{G}cos\theta-\mathbf{F_{L}}-\mathbf{F_{s}})^{ 2}-\left(\mathbf{G}sin\theta\right)^{2}}. \tag{19}\]  
The change rate of sediment erosion \(\frac{\Delta E_{r}}{\Delta t}\) on the breach side slope is calculated as the following:  
\[\frac{\Delta E_{r}}{\Delta t}=\frac{\sqrt{tan^{2}\varphi(\mathbf{G}cos\theta-\mathbf{F _{L}}-\mathbf{F_{s}})^{2}-\left(\mathbf{G}sin\theta\right)^{2}}}{\frac{\pi D^{2}}{4} \cdot\rho}. \tag{20}\]  
The initial sediment erosion rate \(E_{s}\) on the breach side slope can be expressed by:  
\[E_{s}=\frac{\Delta E_{r}}{\Delta t}.\tau_{c}=\frac{\sqrt{tan^{2}\varphi(\mathbf{ G}cos\theta-\mathbf{F_{L}}-\mathbf{F_{s}})^{2}-\left(\mathbf{G}sin\theta\right)^{2}}}{ \frac{\pi D^{2}}{4}\cdot\rho}.\tau_{c}. \tag{21}\]  
In the scouring process of dam break, the erosion rate of the breach side slope varies with shear stress \(\tau\) of the water flow. Thus, the actual widened erosion area per unit time \(E_{b}\) can be simulated by the following formula [50]:  
\[E_{b}=\frac{E_{s}}{\gamma_{s}}.\frac{\tau-\tau_{c}}{\tau_{c}}, \tag{22}\]  
substituting Equations (21) into (22):  
\[E_{b}=\frac{\sqrt{tan^{2}\varphi(\mathbf{G}cos\theta-\mathbf{F_{L}}-\mathbf{F_{s}})^{2}- \left(\mathbf{G}sin\theta\right)^{2}}}{\frac{\pi D^{2}}{4}\cdot\rho\cdot\gamma_{s }}(\tau-\tau_{c}). \tag{23}\]  
The increase in breach bottom width within the time step is expressed as:  
\[\Delta B_{i}=\frac{C_{b}\cdot E_{b}\cdot\Delta t}{L}, \tag{24}\]  
substituting Equations (23) into (24):  
\[\Delta B_{i}=\frac{4C_{b}\sqrt{tan^{2}\varphi(\mathbf{G}cos\theta-\mathbf{F_{L}}-\mathbf{ F_{s}})^{2}-\left(\mathbf{G}sin\theta\right)^{2}}}{\pi D^{2}\cdot\rho\cdot\gamma_{s }\cdot L}(\tau_{i}-\tau_{c})\Delta t, \tag{25}\]  
where \(L\) is the length of flow chute, \(C_{b}\) is the coefficient of side slope, if only one side is eroded, \(C_{b}\) equals to 1.0. For a breach with erosion of both sides \(C_{b}\) equals 2.0.  
Figure 2: Diagram of forces acting on a particle in the slope.