## 4 Verification and Analysis of Solution
### Consolidation Analyses
Furthermore, the figure highlights that \(N_{\sigma}\) exerts a more substantial effect on \(U_{\rm s}\) compared to \(U_{\rm p}\), signifying a greater influence of \(m_{\rm v}\) on \(U_{\rm s}\) relative to \(U_{\rm p}\). When parameters are held constant, the average settlement-defined consolidation degree (\(U_{\rm s}\)) consistently surpasses the average pore water pressure-defined consolidation degree (\(U_{\rm p}\)), underlining the superior speed of soil settlement compared to the dissipation rate of excess pore water pressure. In the case of \(c_{ci}/c_{ki}=1\), the values of \(k_{vi}\) and \(m_{vi}\) exhibit synchronous decrease as \(N_{\sigma}\) increases. Figure 10a focuses on analyzing the influence of \(N_{\sigma}\) values on the average settlement-defined consolidation degree (\(U_{\rm s}\)). The graph illustrates that the average consolidation degree \(U_{\rm s}\), determined by settlement under continuous drainage boundary conditions, rises as the \(N_{\sigma}\) value increases. While the variations in the \(N_{\sigma}\) value, as observed in [PERSON] et al. 's (2002) [6] solution based on Terzaghi drainage boundaries, have no discernible effect on the Us consolidation curve. This disparity could stem from the circumstance of \(c_{ci}/c_{ki}=1\), where under continuous drainage conditions, the influence of the compression coefficient \(m_{vi}\) on soil settlement surpasses that of the permeability coefficient \(k_{vi}\). In contrast, under the traditional Terzaghi boundary condition, the influence of compression coefficient \(m_{vi}\) on soil settlement is similar to that of permeability coefficient \(k_{vi}\), which means that \(U_{\rm s}\) does not follow \(N_{\sigma}\) change. Figure 10b shows the effect of \(N_{\sigma}\) value on the average degree of consolidation \(U_{\rm p}\) defined by pore water pressure. It is evident that both the continuous drainage boundary solution and [PERSON] et al. [6]'s solution depict a decrease in \(U_{\rm p}\) as the \(N_{\sigma}\) value increases. This means that when calculating the average degree of consolidation according to pore water pressure, the influence of permeability coefficient \(k_{vi}\) on soil consolidation is greater than that of compression coefficient \(m_{vi}\) on soil consolidation. In addition, the disparities between the \(U_{\rm p}\) curves of various \(N_{\sigma}\) values in [PERSON] et al. [6]'s solution are substantial, whereas the variations in \(U_{\rm p}\) curves under the continuous drainage boundary solution are comparatively minor, which indicates that the influence of permeability coefficient \(k_{vi}\) under the Terzaghi boundary on \(U_{\rm p}\) outweighs that of \(k_{vi}\) under the continuous drainage boundary. For cases where \(c_{ci}/c_{ki}>1\), it follows that \(k_{vi}\) experiences a greater reduction than \(m_{vi}\) for the same \(N_{\sigma}\) value, so the permeability coefficient \(k_{vi}\) exerts a more pronounced effect on the dissipation of pore water pressure. With \(c_{ci}/c_{ki}\) held constant at 1.5, Figure 11 undertakes an analysis of the \(N_{\sigma}\) impact on both \(U_{\rm s}\) and \(U_{\rm p}\). The diagram reveals that \(U_{\rm p}\) diminishes as the \(N_{\sigma}\) value increases, indicating that under conditions where \(c_{ci}/c_{ki}>1\), the controlling factor for \(U_{\rm p}\) is the permeability coefficient \(k_{vi}\). By contrast, the influence of \(N_{\sigma}\) on \(U_{\rm s}\) is more intricate. In the initial stages of consolidation, elevating the \(N_{\sigma}\) value corresponds to an increase in \(U_{\rm s}\). This behavior signifies that when assessing the degree of consolidation (\(U_{\rm s}\)) through settlement definition, the soil's initial settlement during the early phases of consolidation is predominantly governed by the compression coefficient \(m_{vi}\). Consequently, the degree of consolidation increases as \(N_{\sigma}\) grows.