## 4 Numerical Model
### Boundary Conditions  
The wall boundary conditions were adopted for the DEM domain. As for the CFD domain, on the top and the bottom surfaces of the model, the zero-gradient and the fixed-value boundary conditions, i.e., the patch boundary condition, were assigned to the fluid velocity and the pressure, respectively; on the front and back surfaces, the symmetrical boundary conditions were prescribed; for the right side surface that was tangential to the  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Parameter** & **Symbol** & \multicolumn{6}{c}{**Value**} \\ \hline \multicolumn{10}{c}{**Particle—PVD membrane**} \\ \hline Number & & & & & & & 336 & & & \\ Diameter/(m) & \(d\) & & & & & \(5\times 10^{-6}\) & & & \\ Density & \(\rho\) & & & & & 500 & & & \\ /(kg/m\({}^{3}\)) & & & & & & & & & \\ \hline \multicolumn{10}{c}{**Particle—Slurry particle**} \\ \hline  & & **Case1** & **Case2** & **Case3** & **Case4** & **Case5** & **Case6** & **Case7** & **Case8** \\ \hline Number & & & 1751 & 1340 & 1400 & 1776 & 1751 & 1340 & 1400 & 1776 \\ Diameter/(m) & \(d_{\text{s}}\) & & & & & \(1.0\times 10^{-5}\) & & & \\ Density & \(\rho_{\text{s}}\) & & & & & 2650 & & & \\ /(kg/m\({}^{3}\)) & & & & & & & & & \\ Void ratio & \(e_{0}\) & 3 & 3 & 3 & 2 & 3 & 3 & 3 & 2 \\ Slurry & \(h\) & \(3\times 10^{-4}\) & \(3\times 10^{-4}\) & \(3\times 10^{-4}\) & \(3\times 10^{-4}\) & \(3\times 10^{-4}\) & \(3\times 10^{-4}\) & \(3\times 10^{-4}\) & \(3\times 10^{-4}\) \\ Thickness/(m) & & & & & & & & & \\ \hline \multicolumn{10}{c}{**Particle—Tracer particle**} \\ \hline Number & & 20 & 12 & 5 & 12 & 20 & 12 & 5 & 12 \\ Diameter/(m) & \(d_{\text{s}}\) & 16 & 18 & 25 & 18 & 16 & 18 & 25 & 18 \\ Density & \(\rho_{\text{s}}\) & 2650 & 2650 & 2650 & 2650 & 2000 & 2000 & 2000 & 2000 \\ /(kg/m\({}^{3}\)) & & & & & & & & & \\ \hline \multicolumn{10}{c}{**Particle—DEM**} \\ \hline Stiffness & \(k_{n}\)/\(k_{t}\) & & & & & & 500/143 & & & \\ /(N/m) & & & & & & 20,000/10,000 & & & \\ Damping/(N/ms) & \(\gamma_{n}\)/\(\gamma_{t}\) & & & & & & & \\ Friction & \(\mu\) & & & & & 0.4 & & \\ coefficient & & & & & & & & \\ \hline \multicolumn{10}{c}{**Fluid— CFD**} \\ \hline Density/(kg/m\({}^{3}\)) & \(\rho_{f}\) & & & 1000 & & & & \\ Viscosity/(m\({}^{2}\)/s) & \(v\) & & & 1.0 \(\times 10^{-6}\) & & \\ Gravity/(m/s\({}^{2}\)) & \(g\) & & & & 9.8 & & \\ \hline \multicolumn{10}{c}{**Domain dimensions**} \\ \hline Height/(m) & \(H\) & & & \(4\times 10^{-4}\) & & \\ Width/(m) & \(W\) & & \(1\times 10^{-4}\) & & \\ Length/(m) & \(L\) & & & \(1\times 10^{-4}\) & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Modeling cases and parameters of dredged slurry and PIV tracers.  
tracers, the wall boundary condition was applied; and for the left side surface, the periodic boundary condition was specified. The above boundary conditions have been indicated in Figure 6b.  
In the engineering practice, the designed value of vacuum pressure \(P\) was generally 80 kPa, and the spacing \(s\) between PVDs was about 1.4 m. Then the maximum pressure gradient across the slurry can be estimated as \(i=P/(s/2)=114\) kPa/m. However, the actual pressure gradient is usually smaller than 114 kPa/m since the vacuum pressure at soil increases along with the consolidation process. In the numerical model, the thickness of the slurry and the pressure difference was set to 0.3 mm and 30 Pa, respectively. Then, the corresponding pressure gradient was 100 kPa/m, which was close to \(i=114\) kPa/m in practice. The pressure gradient was achieved by setting zero (\(P_{\rm a}=0\) Pa) and negative pressures (\(P_{\rm b}=-30\) Pa) to the top and bottom surfaces of the CFD domain, respectively.