## 2 Problem Description and Governing Equations  
Figure 1 illustrates the schematic diagram of a soft clay foundation. \(H\) is the thickness of the clay layer; \(q_{0}\) is the instantaneous uniformly distributed load applied on the top surface of the clay layer; \(v\) depicts the seepage velocity; \(z\) and \(z_{0}\) represent the downward  
vertical coordinate originated from the top boundary and the depth of soil unit, respectively. In [PERSON]'s flow, the relationship between seepage velocity and the hydraulic gradient of soil is composed of the exponential segment for the conditions of the small hydraulic gradient, and the linear segment for the cases of the large hydraulic gradient. Therefore, the seepage law in soil can be expressed as [9]:  
\[v=\left\{ \begin{array}{l}k_{\mathrm{v}}i^{m}/(m_{1}^{im-1}),\;i<i_{1}\\ k_{\mathrm{v}}(i-i_{0}),\;i\geq i_{1}\end{array} \right., \tag{1}\]  
where \(k_{\mathrm{v}}\) denotes the permeability coefficient in the linear section, \(i\), \(i_{0}\) and \(i_{1}\) represent the hydraulic gradient, initial hydraulic gradient of seepage calculation in linear section and initial hydraulic gradient of linear seepage, respectively, and \(m\) is a constant determined by test. The expression of \(i_{0}\) can be written as:  
\[i_{0}=\frac{i_{1}(m-1)}{m}. \tag{2}\]  
Introducing the relationship between soil compression and seepage proposed by [PERSON] and [PERSON] [18], the nonlinear characteristics of soil can be obtained as:  
\[e-e_{0}=c_{\mathrm{c}}\mathrm{lg}\bigg{(}\frac{\sigma_{0}^{\prime}}{\sigma^{ \prime}}\bigg{)}, \tag{3}\]  
\[e-e_{0}=c_{\mathrm{k}}\mathrm{lg}\bigg{(}\frac{k_{\mathrm{v}}}{k_{\mathrm{v0} }}\bigg{)}, \tag{4}\]  
where \(e\) and \(e_{0}\) represent the void ratio and initial void ratio, respectively. \(\sigma^{\prime}\) and \(\sigma_{0}^{\prime}\) denote the effective stress and initial effective stress, respectively. \(k_{\mathrm{v}}\) and \(k_{\mathrm{v0}}\) are the permeability coefficient and initial permeability coefficient, respectively. \(c_{\mathrm{c}}\) and \(c_{\mathrm{k}}\) represent the compression index and permeability index, respectively. Combining Equation (3) with Equation (4) yields:  
\[k_{\mathrm{v}}=k_{\mathrm{v0}}(\frac{\sigma_{0}^{\prime}}{\sigma^{\prime}})^{ \frac{c_{\mathrm{c}}}{k_{\mathrm{v}}}}, \tag{5}\]  
\[m_{\mathrm{v}}=-\frac{1}{1+e_{0}}\frac{\partial e}{\partial\sigma^{\prime}}= \frac{m_{\mathrm{v0}}\sigma_{0}^{\prime}}{\sigma^{\prime}}, \tag{6}\]  
Figure 1: Schematic diagram of soft clay layer. where \(m_{\rm v}\) and \(m_{\rm v0}\) denote the compressibility and initial compressibility coefficients, respectively. The expression of \(m_{\rm v0}\) can be written as:  
\[m_{\rm v0}=-\frac{1}{1+e_{0}}\frac{\partial e}{\partial\sigma^{\prime}}\bigg{|} _{\sigma^{\prime}=\sigma_{0}^{\prime}}=\frac{c_{\rm c}}{(1+e_{0})\sigma_{0}^{ \prime}\ln 10}. \tag{7}\]  
Then, the consolidation coefficient of soil can be obtained as:  
\[c_{\rm v}=\frac{k_{\rm v}}{\gamma_{\rm w}m_{\rm v}}=c_{\rm v0}\bigg{(}\frac{ \sigma_{0}^{\prime}}{\sigma^{\prime}}\bigg{)}^{\frac{c_{\rm c}}{k_{\rm k}}-1}, \tag{8}\]  
where \(\gamma_{\rm w}\) denotes the unit weight of water. \(c_{\rm v0}\) represents the initial consolidation coefficient of soil and can be written as:  
\[c_{\rm v0}=\frac{k_{\rm v0}(1+e_{0})\sigma_{0}^{\prime}\ln 10}{\gamma_{\rm w}c_{ \rm c}}. \tag{9}\]  
According to the continuous condition of seepage in soil, the following equation can be obtained:  
\[\frac{\partial v}{\partial z}=-\frac{1}{1+e_{0}}\frac{\partial e}{\partial t}, \tag{10}\]  
where \(t\) is time. Assuming that the instantaneous uniformly distributed load and the initial effective stress are uniformly distributed along the depth, and introducing the effective stress principle, the following equation can be gained:  
\[e-e_{0}=c_{\mathrm{c}}\mathrm{lg}\bigg{(}\frac{\sigma_{0}^{\prime}}{\sigma^{ \prime}}\bigg{)}, \tag{3}\]0  
where \(u\) represents the excess pore water pressure.