## 2 Fundamental Problems  
A one-dimensional nonlinear consolidation analysis model of double-layered soil is shown in Figure 1, in which \(h_{i}\) is the thickness of the soil layer \(i\) (\(i=1\), \(2\)). Ground thickness is \(H=h_{1}+h_{2}\), \(q_{0}\) and represents the instantaneous load acting on the soil. The nonlinear consolidation behavior of soils is described by using the generally accepted nonlinear compression and permeability relationships proposed by [PERSON] and [PERSON] [12]:  
\[e_{i}-e_{0i}=c_{ci}\text{lg}\left(\frac{\sigma_{0}^{\prime}}{\sigma_{i}^{ \prime}}\right) \tag{1}\]  
\[e_{i}-e_{0i}=c_{ki}\text{lg}\left(\frac{k_{vi}}{k_{v0i}}\right) \tag{2}\]  
In the Equations (1) and (2), \(e_{i}\) and \(e_{0i}\) represent the pore ratio and the initial pore ratio, respectively; \(\sigma^{\prime}_{i}\) and \(\sigma^{\prime}{}_{0i}\) are the effective stress and the initial effective stress, respectively; \(k_{vi}\) and \(k_{v0i}\) are the permeability coefficient and the initial permeability coefficient, respectively; and \(c_{ci}\) and \(c_{ki}\) are the compression index and the penetration index respectively. It can be obtained from Equations (1) and (2):  
\[k_{vi}=k_{v0i}(\frac{\sigma_{0i}^{\prime}}{\sigma_{i}^{\prime}})^{\frac{c_{ki} }{k_{v}}} \tag{3}\]  
\[m_{vi}=-\frac{1}{1+e_{0i}}\frac{\partial e_{i}}{\partial\sigma_{i}^{\prime}}= \frac{m_{v0i}\sigma_{0i}^{\prime}}{\sigma_{i}^{\prime}} \tag{4}\]  
where \(m_{vi}\) and \(m_{v0i}\) are the compression coefficient and the initial compression coefficient, respectively. The \(m_{v0i}\) expression is as follows:  
Figure 1: Double-layered soil model. \[m_{\rm v0i}=\left.-\frac{1}{1+e_{0i}}\frac{\partial e_{i}}{\partial\sigma^{\prime} _{i}}\right|_{\sigma^{\prime}_{i}=\sigma^{\prime}_{0i}}=\frac{c_{ci}}{(1+e_{0i} )\sigma^{\prime}_{0i}\ln 10} \tag{5}\]  
The continuous equation for one-dimensional saturated soils is:  
\[\frac{\partial}{\partial z}\bigg{[}\frac{k_{\rm v0i}}{\gamma_{\rm w}}\frac{ \partial u_{i}}{\partial z}\bigg{]}=\frac{1}{1+e_{0i}}\frac{\partial e_{i}}{ \partial t} \tag{6}\]  
In the expression, \(t\) represents time. It is assumed that the applied load is an instantaneous uniform load \(q_{0}\) and that the initial effective stress is evenly distributed along the depth (i.e., \(\sigma^{\prime}_{0i}=\sigma^{\prime}_{0}\)). It can be derived from the effective stress principle:  
\[\sigma^{\prime}_{i}=q_{0}+\sigma^{\prime}_{0}-u_{i}=\sigma^{\prime}_{\rm f}-u _{i} \tag{7}\]  
In the equation, \(u_{i}\) is the excess static pore water pressure and \(\sigma^{\prime}_{\rm f}\) is the final effective stress. Under instantaneous load, the values are \(\sigma^{\prime}_{\rm f}=q_{0}+\sigma^{\prime}_{0}\). In combination with Equations (3)-(7), the one-dimensional nonlinear consolidation equation of soil mass can be obtained as follows:  
\[\frac{\partial}{\partial z}\Bigg{[}\frac{k_{\rm v0i}}{\gamma_{\rm w}}(\frac{ \sigma^{\prime}_{\rm f}-u_{i}}{\sigma^{\prime}_{0}})^{-\frac{c_{ci}}{\partial z }}\frac{\partial u}{\partial z}\Bigg{]}=\frac{c_{ci}}{(1+e_{0i})(\sigma^{ \prime}_{\rm f}-u_{i})\ln 10}\frac{\partial u_{i}}{\partial t} \tag{8}\]  
Define dimensionless parameter: initial permeability coefficient ratio of lower and upper soil layers \(a=\frac{k_{\rm v0i}}{K_{\rm v0i}}\), initial compression coefficient ratio of lower and upper soil layers \(b=\frac{m_{\rm v0i}}{m_{\rm v0i}}\), initial thickness ratio of lower and upper soil layers \(d=\frac{h_{2}}{h_{1}}\). Dimensionless pore water pressure \(\overline{u}_{i}=\frac{u_{i}}{\sigma^{\prime}_{0}}\), dimensionless depth \(Z=\frac{z}{H}\), dimensionless first layer depth \(Z_{\rm f}=\frac{h_{1}}{H}\), dimensionless effective stress \(N_{\sigma}=\frac{\sigma^{\prime}_{\rm f}}{\sigma^{\prime}_{0}}\).