## 5 Numerical Results  
The slurry particles and the tracers move together toward the drainage board filter membrane under the vacuum gradient. However, their displacements should not be the same due to the differences in physical properties such as particle size and density. As shown in Table 4, eight cases considering different combinations of the tracer density (\(\rho_{\rm t}\)), tracer size (\(d_{\rm t}\)), and initial void ratio (\(e_{0}\)) of the dredged slurry. As shown in Figure 6b, a statistical region of 25 \(\times\) 100 \(\times\) 85 \(\mu\)m in length \(\times\) width \(\times\) height was defined within the particle-fluid mixed region. The particles in the statistical region were sufficiently away from the PVD membrane and thus had large space for movement, which was beneficial to obtaining more particle movement information. Figure 8 compares the time variation of the ratio (\(D_{\rm t}/D_{\rm s}\)) between the statistical displacements of the tracers \(D_{\rm t}\) and the slurry particles \(D_{\rm s}\) for three cases of \(d_{\rm t}/d_{\rm s}=1.6\), 1.8, and 2.5, i.e., Cases 1-3 in Table 4. The mass density of the tracer and the initial void ratio of the slurry were taken as \(\rho_{\rm t}=2650\) kg/m\({}^{3}\) and \(e_{0}=3\), respectively. Obviously, better representativeness of the tracer for the slurry particles can be achieved when \(D_{\rm t}/D_{\rm s}\) is closer to 1. It can be seen from the figure that \(D_{\rm t}/D_{\rm s}\) tends to be stable after the previous fluctuations. The stable values of \(D_{\rm t}/D_{\rm s}\) in Cases 1-3 were 1.042, 1.089, and 1.137, respectively. The observations indicate that the representativeness of the tracer decreases when the diameter ratio between the tracer and the slurry particle (\(d_{\rm t}/d_{\rm s}\)) increases. Figure 8: Variations of displacement ratio \(D_{\rm t}/D_{\rm s}\) between tracer and slurry particle with respect to time t under conditions of \(\rho_{\rm t}=2650\) kg/m\({}^{3}\), \(e_{0}=3\) and \(d_{\rm t}/d_{\rm s}=1.6\), 1.8 and 2.5. By reducing the mass density \(\rho_{\rm t}\) of the tracer from 2650 kg/m\({}^{3}\) to 2000 kg/m\({}^{3}\) and keeping \(e_{0}=3\), Figure 9 compares the variation of \(D_{\rm t}/D_{\rm s}\) with \(t\) for the same diameter ratios \(d_{\rm t}/d_{\rm s}=1.6\), 1.8, and 2.5 (i.e., Cases 5-7 in Table 4). By comparing Figure 9 to Figure 8, it can be seen that the mass density of the tracer had little effect on the variation of \(D_{\rm t}/D_{\rm s}\) with time. However, the stable values of \(D_{\rm t}/D_{\rm s}\) were reduced, i.e., the reductions were about 1.4%, 3.5%, and 10%, respectively, for \(d_{\rm t}/d_{\rm s}=1.6\), 1.8, and 2.5. Figure 10 summarizes the variations of the stable values of \(D_{\rm t}/D_{\rm s}\) with \(d_{\rm t}/d_{\rm s}\) for the two mass densities of the tracer, i.e., \(\rho_{\rm t}=2650\) kg/m\({}^{3}\) and 2000 kg/m\({}^{3}\). It can be seen from the figure that the stable value of \(D_{\rm t}/D_{\rm s}\) increased with the mass density ratio \(\rho_{\rm t}/\rho_{\rm s}\), which means the representativeness of the tracer decreased. Among the Cases 1-3 and 5-7 considered in Figure 10, the tracer of diameter 18 \(\mu\)m and of mass density 2000 kg/m\({}^{3}\), i.e., Case 6, provided the best representativeness on the displacement response of the slurry particles. Figure 10: Variations of stable values of the displacement ratio \(D_{\rm t}/D_{\rm s}\) with respect to the tracer diameter dt for two mass densities of the tracer, \(\rho_{\rm t}=2000\) kg/m\({}^{3}\) and 2650 kg/m\({}^{3}\). Figure 9: Variations of displacement ratio \(D_{\rm t}/D_{\rm s}\) between tracer and slurry particle with respect to time t under conditions of \(\rho_{\rm t}=2000\) kg/m\({}^{3}\), \(e_{0}=3\), \(d_{\rm t}/d_{\rm s}=1.6\), 1.8, and 2.5. For slurries of different particle concentrations, the variations of \(D_{\rm t}/D_{\rm s}\) with \(t\) were compared in Figure 11a for two initial void ratios, i.e., \(e_{0}=2\) and \(3\).