## 2 Materials and Methods
### Breach Evolution Model
#### 2.2.2 Starting Erosion Condition of the Breach  
The critical flow velocity formula, with and without seepage force, is derived by analyzing the force of a soil particle on the slope. The forces acting on a soil particle on the dam slope mainly include the soil particle gravity \(G\), the drag force \(F_{d}^{\prime}\), the uplift force \(F_{L}\) exerted by water flows, the interacting force \(F_{m}\), the friction force \(F_{f}\), and the seepage force \(F_{s}\). The force analysis is shown in Figure 1.  
Figure 1: Force analysis of soil particles.  
The particle gravity of the soil particle is expressed as follows [41]:  
\[\mathbf{G}=\frac{1}{6}(\gamma_{s}-\gamma_{w})\pi D^{3}, \tag{2}\]  
where \(\gamma_{s}\) is the specific weight of the soil particle, \(\gamma_{w}\) is the water weight; \(D\) is the diameter of the soil particle, represented by \(D_{50}\), the median particle size of the non-uniform mixed sediment.  
The uplift force calculation formula is as follows [42]:  
\[\mathbf{F_{L}}=C_{L}\frac{\pi D^{2}}{4}\frac{\rho v^{2}}{2}=\frac{\pi}{80g}\gamma_{ w}V^{2}D^{2}, \tag{3}\]  
where \(C_{L}\) is the coefficient of the uplift force and is usually equal to \(0.1\); \(g\) is the acceleration due to gravity, \(\rho\) is the density of water; and \(V\) is the velocity of flow.  
The drag force calculation formula is written as follows [42]:  
\[\mathbf{F_{d}}=C_{D}\frac{\pi D^{2}}{4}\frac{\rho v^{2}}{2}=\frac{\pi}{20g}\gamma_ {w}V^{2}D^{2}, \tag{4}\]  
where \(C_{D}\) is the coefficient of the drag force and usually takes the value of \(0.4\).  
The seepage force calculation formula is written as follows [43]:  
\[\mathbf{F_{s}}=\frac{\pi}{6(1-e)}i\gamma_{w}D^{3}, \tag{5}\]  
where \(i=\Delta H/L\), \(\Delta H\) is the difference of water level elevation between two points of the water level line, that is, the difference of water level between the unit before the dam and the unit behind the dam. \(L\) is the horizontal distance between these two points, which is the width of the dam; \(e\) is the porosity of the dam material.  
The interacting force calculation formula is written as follows [44]:  
\[\mathbf{F_{m}}=1.5M(\gamma_{s}-\gamma_{w})D^{3}, \tag{6}\]  
\[M=0.75-\frac{0.65}{2+C_{u}}, \tag{7}\]  
where \(C_{u}\) is the non-uniformity coefficient.  
According to the force analysis of the soil particles, it could be concluded that the friction force of the soil particle at its incipience when seepage force is considered can be written as follows:  
\[\mathbf{F_{f}}=tan\varphi(\mathbf{G}cos\theta+\mathbf{F_{m}}-\mathbf{F_{L}}-\mathbf{F_{s}}). \tag{8}\]  
when seepage force is not considered, the friction force is:  
\[\mathbf{F_{f}}=tan\varphi(\mathbf{G}cos\theta+\mathbf{F_{m}}-\mathbf{F_{L}}), \tag{9}\]  
where \(\varphi\) is the internal friction angle of soil material, \(\theta\) is the inclination of dam slope. The analysis of the forces acting on the soil particle reveal that the critical condition for incipient motion is as follows:  
\[\mathbf{F_{d}}+\mathbf{G}sin\theta=\mathbf{F_{f}}. \tag{10}\]  
Substituting Equations (3)-(6) and (8) into (10), the starting flow velocity considering seepage flow is obtained as follows:  
\[\mathbf{V_{c}}=\sqrt{\frac{120 gtan\varphi M(\gamma_{s}-\gamma_{w})D}{\pi\gamma_{ w}(4+tan\varphi)}+\frac{40g(tan\varphi cos\theta-sin\theta)(\gamma_{s}-\gamma_{w})D}{3 \gamma_{w}(4+tan\varphi)}-\frac{40\pi giDtan\varphi}{3(1-e)(4+tan\varphi)}}. \tag{11}\]Substituting Equations (3)-(6) and (9) into (10), the starting flow velocity without considering seepage flow is obtained as follows:  
\[\mathbf{V_{c}}=\sqrt{\frac{120 gtan\varphi M(\gamma_{s}-\gamma_{w})D}{\pi\gamma_{w}(4 +tan\varphi)}+\frac{40g(tan\varphi cos\theta-sin\theta)(\gamma_{s}-\gamma_{w})D }{3\gamma_{w}(4+tan\varphi)}}. \tag{12}\]