## 2 Methodology  
### Theoretical Derivation: Interfacial Force  
As shown in Figure 1, the initiation process of particles is a complex dynamic process including the gravity \(\mathbf{G}\), horizontal thrust force \(\mathbf{F_{ht}}\), uplift force \(\mathbf{F_{ul}}\), viscosity force between particles \(\mathbf{F_{\mu}}\) and many other forces [26]. Among them, the horizontal thrust \(\mathbf{F_{ul}}\) and uplift force \(\mathbf{F_{ul}}\) are the forces exerted by the fluid phase on the solid particles to be moved, which are also affected by the drag force \(\mathbf{F_{d}}\) and the generalized virtual mass force \(F_{vmg}\).  
[PERSON] [27] has given the expression of the generalized interfacial force.  
\[F_{0}=F_{d}+F_{vmg}=C_{d}\Big{(}u_{f}-u_{s}\Big{)}\Big{|}u_{f}-u_{s}\Big{|}+C_{ vmg}\frac{d}{dt}\Big{(}u_{f}-u_{s}\Big{)} \tag{1}\]  
where \(F_{0}\) is the interfacial force, \(C_{d}\) is the drag coefficient, \(C_{vmg}\) is the generalized virtual mass coefficient, \(u_{f}\) and \(u_{s}\) are the fluid and solid phase velocity, respectively. In Equation (1), the drag force term is square with the relative velocity between phases, and the second term is the generalized virtual mass force.  
When the relative velocity between the solid and liquid phases has an acceleration over time, the fluid around the particles will be affected by an additional force, which is related to the time derivative of the relative velocity.  
Based on previous research results [27; 28; 29], Equation (1) can be transformed into:  
\[\begin{array}{l}\frac{F_{0}}{B_{d}}=\frac{\alpha_{s}\sigma_{d}\big{(}\rho_{s }-\rho_{f}\big{)}\delta}{\big{|}\omega[PF(\mathbf{F_{\mu}})+(1-P)G(\mathbf{F_{ \mu}})]\big{|}}\big{(}\mathbf{u_{f}}-\mathbf{u_{s}}\big{)}|\mathbf{u_{f}}- \mathbf{u_{s}}|+\\ \frac{1}{2}\alpha_{s}\rho_{f}\Big{(}\frac{1+2\alpha_{s}}{\alpha_{f}}\Big{)} \Big{(}\Big{(}\frac{\partial\mathbf{u_{f}}}{\partial t}+\mathbf{u_{f}}\cdot \
abla\mathbf{u_{f}}\Big{)}-\Big{(}\frac{\partial\mathbf{u_{f}}}{\partial t}+ \mathbf{u_{s}}\cdot\
abla\mathbf{u_{s}}\Big{)}\Big{]}\end{array} \tag{2}\]  
Figure 1: Interfacial forces acting on a particle.  
where \(B_{d}\) denotes the particle volume, \(\alpha_{s}\) and \(\alpha_{f}\) denote the volume fractions for the solid and the fluid constituents, \(\rho_{s}\) and \(\rho_{f}\) denote the density of the solid and the fluid, \(\omega\) denotes the particle settling velocity through fluid, \(F(Re_{p})=\gamma(\alpha_{s}\alpha_{s})^{3}Re_{p}/180\) depends linearly on the particle Reynolds number \(Re_{p}\), density ratio \(\gamma\) and the cube power of the ratio \(\alpha_{g}\alpha_{s}\) between fluid and solid volume fractions. \(G(Re_{p})=\alpha_{f}^{M(Rep)-1}\) depends on the fluid volume fraction and the particle Reynolds number. For sediment particles in critical initiation state, \(\frac{\partial u_{s}}{\partial t}+u_{s}\ \
abla u_{s}\ =0\), when we focus on the instantaneous state of particle initiation, the derivative of liquid velocity to time \(\ \frac{\partial u_{s}}{\partial t}\), therefore, Equation (2) can be simplified as follows:  
\[\begin{array}{l}\frac{F_{0}}{B_{d}}=\frac{\alpha_{s}a_{f}\big{(}\rho_{s}- \rho_{f}\big{)}g}{\big{[}\omega[F(\mathbf{Re}_{p})+(1-P)G(\mathbf{Re}_{p})] \big{]}}(\mathbf{u_{f}}-\mathbf{u_{s}})|\mathbf{u_{f}}-\mathbf{u_{s}}|+\\ \frac{1}{2}\alpha_{s}\rho_{f}\Big{(}\frac{1+2\alpha_{s}}{\alpha_{f}}\Big{)} \mathbf{u_{f}}\cdot\
abla\mathbf{u_{f}}\end{array} \tag{3}\]