## 3 Ground Loss Effect Factors
### Displacement Solution  
Based on the Verruijt analytical formula [34; 35], considering quantify ground loss with non-equivalent radial displacement, [PERSON] [21] put forward a method for estimating stratum displacement caused by ground loss in undrained clays; the formula is as shown in Equation (2). \[\left\{ \begin{array}{c}S_{z}=R^{2}\bigg{\{}-\frac{z-H}{x^{2}+(z-H)^{2}}+(3-4 \mu)\frac{z+H}{x^{2}+(z+H)^{2}}-\frac{2z}{\left[x^{2}+(z+H)^{2}\right]}\bigg{\}} \times\varepsilon_{x,z}\\ S_{x}=-R^{2}x\bigg{\{}\frac{1}{x^{2}+(z-H)^{2}}+\frac{(3-4\mu_{x})}{x^{2}+(z+ H)^{2}}-\frac{4z(z+H)}{\left[x^{2}+(z+H)^{2}\right]^{2}}\bigg{\}}\times\varepsilon_{x,z} \end{array} \right. \tag{2}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Material** & \begin{tabular}{c} **Unit Weight** \\ **(kN/m\({}^{3}\))** \\ \end{tabular} & \begin{tabular}{c} **Thickness** \\ **(m)** \\ \end{tabular} & \begin{tabular}{c} **Poisson’s** \\ **Ratio** \\ \end{tabular} & \begin{tabular}{c} **Cohesion** \\ **(kPa)** \\ \end{tabular} & \
\begin{tabular}{c} **Friction** \\ **Angle** \\ \end{tabular} &
\begin{tabular}{c} **Elastic Modules** \\ **(MPa)** \\ \end{tabular} \\ \hline Topsoil & 20 & 2.1 & 0.3 & 45 & \(14^{\circ}\) & 5.94 \\ \hline Intermediary & & & & & & & \\ weathered mudstone & 23.4 & 37.9 & 0.17 & 300 & \(18^{\circ}\) & 180 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of soil. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & \begin{tabular}{c} **Shield Tunneling** \\ **Speed (mm/min)** \\ \end{tabular} & \begin{tabular}{c} **Rotation Speed of** \\ **Cutter Head (t/min)** \\ \end{tabular} & \begin{tabular}{c} **Cutter Opening** \\ **Ratio (\%)** \\ \end{tabular} & \begin{tabular}{c} **Closure** \\ **Number of** \\ **Cutters** \\ \end{tabular} & \begin{tabular}{c} **Diameter of Shield** \\ **(mm)** \\ \end{tabular} \\ \hline Value & 80 & \(3.55\) & 36 & 4 & 6280 \\ \hline \begin{tabular}{c} **Length of Main** \\ **Machine (mm)** \\ \end{tabular} & \begin{tabular}{c} **Diameter of** \\ **Segment (mm)** \\ \end{tabular} & \begin{tabular}{c} **Synchronous** \\ **Grouting Pressure** \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Strain Rate of** \\ **Grouting Layer (\%)** \\ \end{tabular} & \
\begin{tabular}{c} **Slip Softening** \\ **Coefficient** \\ \end{tabular} &
\begin{tabular}{c} **Interface Friction** \\ **Angle (\(\lx@notemark{\circ}\))** \\ \end{tabular} \\ \hline
8427 & 600 & 0.3 & 8.4 & 0.9 & 6.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Construction parameters of shield machine. where \(R\) is tunnel external radius; \(H\) is burial depth of tunnel center; \(\mu\) is Poisson's ratio of soil; \(\varepsilon_{x,z}\) is the parameter of non-equivalent radial displacement, fitting by exponential function based on measured data, which can be expressed as Equation (3). \[\varepsilon_{x,z}=\frac{4 gR+g^{2}}{4R^{2}}\exp\left\{-\left|\frac{1.38x^{2}}{ \left(H+R\right)^{2}}+\frac{0.69z^{2}}{H^{2}}\right|\right\} \tag{3}\]  
Considering that the displacement solution caused by ground loss derived by Loganathan is a two-dimensional solution, which is often a three-dimensional problem in actual engineering, it is necessary to extend the two-dimensional displacement solution to three-dimensional. The three-dimensional ground loss parameter \(\varepsilon_{x,z,y}\) can be extracted from the [PERSON]' solution [20]. [PERSON] deduced the expression of ground displacement caused by tunnel construction by using the mirror source-sink method. \[\left\{