## 4 Verification and Analysis of Solution
### Consolidation Analyses  
Through an examination of the time-dependent consolidation characteristics or the depth-dependent pore pressure variations under different soil or soil layer properties, we can investigate the influence of various factors on the consolidation process. A steeper consolidation or pore pressure curve indicates a faster consolidation rate. Figure 3 illustrates the impact of the permeability coefficient ratio (denoted as '\(a\)') between the lower and upper soil layers on the average consolidation degree (\(U_{\text{s}}\)), with other variables remaining constant. The graph clearly demonstrates that an elevated value of '\(a\)' leads to an augmented soil consolidation rate. This trend indicates that, while maintaining consistent drainage conditions, the consolidation process accelerates when the permeability of the lower soil layer surpasses that of the upper soil layer. From the findings depicted in Figure 4, it is deduced that the soil's consolidation rate diminishes with the escalation of the compression coefficient ratio (\(b\)), which signifies the ratio between the compression coefficient of the lower soil layer and that of the upper soil layer. This observation underscores the fact that the soil tends to exhibit reduced consolidation tendencies when the compressibility of the lower soil layer surpasses that of the upper layer. Figure 3: Influence of the ratio of permeability coefficient \(a\) on \(U_{\text{s}}\) curves. Figure 2: Comparison of consolidation degree curves between numerical and analytical solutions [(23)]. Here, we are presented with a graph (Figure 5), under the conditions of \(a=2\) and \(b=0.5\), corresponding to a subsoil with higher permeability and lower compressibility compared to the upper soil. The diagram shows that the degree of consolidation of soil increases with the increase in value \(d\), indicating that increasing the thickness of soil layer with high permeability and low compressibility in double-layered soil can improve the rate of soil consolidation. In Figure 6, the effect of exchanging the interface parameters of the top and the bottom soil on the consolidation characteristics of the soil is analyzed. When both soil layers possess identical permeability and compressibility, and the compression index to penetration index ratio equals 1, the consolidation degree curve remains invariant upon interchanging the top and bottom interface parameter values. This observation underscores the fact that, in this case, the exchange of top and bottom interface parameters has no effect on the soil consolidation rate. However, when the compression index to penetration index ratio differs (\(c_{1}=0.5\), \(c_{2}=1\)), the consolidation degree curves resulting from reversing the top and bottom interface parameters exhibit substantial disparities. This indicates that when the properties of the upper and lower soil layers differ, the interface parameters significantly influences the consolidation rate. In engineering practice, it is crucial to take into account  
Figure 4: Impact of compression coefficient ratio (\(b\)) on consolidation curves (\(U_{\text{s}}\)). Figure 5: Impact of soil layer thickness ratio (\(d\)) on consolidation curves (\(U_{\text{s}}\)). the comprehensive consolidation characteristics resulting from the coupling of interface permeability and soil nonlinearity when making settlement predictions. Under the stipulated conditions of uniform interface parameters for the upper and lower soil layers, along with equal permeability and compressibility within these layers, Figure 7 scrutinizes the impact of the compression index to penetration index ratios (\(c_{1}\) for the upper layer and \(c_{2}\) for the lower layer) on the degree of soil consolidation. The graph illustrates a negative correlation between the compression index to penetration index ratio and the soil consolidation degree. This negative correlation indicates that an increased compression index to penetration index ratio leads to a reduction in the soil consolidation rate.