# Cnoidal Wave-Induced Residual Liquefaction in Loosely Deposited Seabed under Coastal Shallow Water
## 4 Result Analysis
### Residual Liquefaction  
There are two types of liquefaction mechanisms: momentary liquefaction and residual liquefaction. Momentary liquefaction can only occur in very dense seabed floors beneath wave troughs. Residual liquefaction generally occurs in loosely deposited seabed floors under cyclic loadings, such as wave loading and seismic loading. The occurrence of residual liquefaction is a fatal factor for the stability of offshore structures built on a loose seabed foundation. In this section, the characteristics of residual liquefaction and its process are both analyzed in the loose seabed floor. Figure 7: Stress–strain relationship at three typical buried depths on \(x=100\) m.  
The following stress-based criterion proposed by [PERSON] and [PERSON] (2018) [14] is adopted to judge the occurrence of residual liquefaction:  
\[L_{potential}=\frac{\left(I_{1}\right)_{d}}{-\left(I_{1}\right)_{0}+\alpha C}= \frac{\left(I_{1}\right)_{d}}{L_{r}}\geq\alpha_{r} \tag{15}\]  
where \(L_{potential}\) is called liquefaction potential. \(\left(I_{1}\right)_{d}=I_{1}-\left(I_{1}\right)_{0}\) is the cnoidal wave-induced dynamic mean effective stress, \(I_{1}\) is the current mean effective stress, and \(\left(I_{1}\right)_{0}\) is the initial mean effective stress. \(C\) is the cohesion of soil, and \(\alpha\) is a material parameter. \(L_{r}=-\left(I_{1}\right)_{0}+\alpha C\) is actually the term characterizing the liquefaction resistance of soil, in which the contribution of the cohesion of soil to enhance the resistance ability against liquefaction is considered. In this study, the seabed floor is set as cohesiveness sand soil, leading to \(C=0\) kPa. \(\alpha_{r}\in\left[0,\ 1\right]\) is a critical value to judge whether residual liquefaction has occurred or not, whose value is dependent on the properties of soil. [PERSON] (2004) [31] suggested that \(\alpha_{r}\) is in the range of 0.78-0.99. For Nevada sand related to the model parameters listed in Table 1, \(\alpha_{r}=0.86\) was proposed by [PERSON] et al. (2015) [32]. The distributions of \(L_{potential}\) along \(x=100\) m at four typical times are illustrated in Figure 8. It is observed that the liquefaction potential along \(x=100\) m is gradually increased over time under the action of the cnoidal wave. If \(\alpha_{r}=0.86\) is taken as the critical value for the occurrence of residual liquefaction, then we can find that the liquefaction depth is gradually increased from the upper part into the medium part of the seabed floor. At \(t=500\) s, the final depth of the liquefaction zone is about 14 m. In the lower seabed floor, there is no residual liquefaction occurring. This phenomenon can be more clearly observed in Figure 9. It is shown in Figure 9 that the liquefaction zone is gradually enlarging from the upper to the medium part of the seabed floor in the process of wave application. Finally, the frontier of the residual liquefaction zone reaches the position \(z=16\) m, indicating that the thickness of the liquefaction zone is 14 m.  
Figure 8: Vertical distribution of the liquefaction potential \(L_{potential}\) at typical times \(t/T=10\), 12.5, 37.5, and 62.5. The above analysis has more or less implied that the residual liquefaction first occurs in the upper part, and then gradually transmits toward the lower part of the seabed floor. Figure 10 exactly proves this progressive process. It is observed that the residual liquefaction first occurs at the seabed surface, and then the liquefaction zone gradually enlarges toward the bottom of the seabed floor. Finally, the frontier of the liquefaction zone reaches \(z=16\) m.  
Residual pore pressure is the direct driven factor for the residual liquefaction. Its distribution characteristics along \(x=100\) m at different times are demonstrated in Figure 11, where the distribution characteristics of the oscillatory pore pressure are also presented. It is observed that the residual pore pressure along the seabed depth is gradually increasing with time.