## 3 Three-Dimensional Finite Element Modeling
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multicolumn{2}{c}{**Value**} & \multicolumn{2}{c}{**Value**} & \multirow{2}{*}{**Method**} \\ \cline{2-2} \cline{4-7}  & **Soft Soil** & **Improved Soil** & & & **Soft Soil** & **Improved Soil** & **Method** \\ \hline \(c^{\prime}\) (RPa) & 13.1 & 350 & \begin{tabular}{c} Triaxial consolidated \\ and drained test \\ \end{tabular} & \(G_{0}\)\({}^{\prime\prime}\) (MPa) & 40.1 & 350 & \(G_{0}\)\({}^{\prime\prime}\) = (1.5–2.5)\(E_{\omega}\)\({}^{\prime\prime\prime}\)[32] \\ \hline \(\varphi^{\prime}\) (\(c^{\prime}\)) & 13.3 & 43 & \begin{tabular}{c} Triaxial consolidated \\ and drained test \\ \end{tabular} & \(P^{\prime\prime}\) (RPa) & 100 & 100 & \begin{tabular}{c} Triaxial manual \\ \end{tabular} \\ \hline \(\Psi\) (\(c^{\prime}\)) & 0.1 & 0.1 & \begin{tabular}{c} \(\Psi=0\) when \(\varphi^{\prime}<30^{\circ}\) \\ [33] \\ \end{tabular} & \(v_{\omega r}\) & 0.2 & 0.2 & \begin{tabular}{c} Triaxial manual \\ \end{tabular} \\ \hline \(K_{0}\) & 0.77 & 0.318 & \(K_{0}=1-\sin\varphi^{\prime}\)[34] & \(E_{\omega}\)\({}^{\prime\prime\prime}\) (MPa) & 16.03 & 200 & \(E_{\omega}\)\({}^{\prime\prime\prime}\) = (3–8)\(E_{\omega}\)\({}^{\prime\prime\prime}\)[35] \\ \hline \(R_{f}\) & 0.9 & 0.9 & \begin{tabular}{c} Triaxial manual \\ \end{tabular} & \(m\) & 0.8 & 0.1 & \begin{tabular}{c} Triaxial manual \\ \end{tabular} \\ \hline \(E_{\omega}\)\({}^{\prime\prime\prime}\) (MPa) & 2.29 & 30 & \
\begin{tabular}{c} \(E_{\omega}\)\({}^{\prime\prime\prime}\) = (0.7–1.0) \\ \(E_{\omega}\)\({}^{\prime\prime}\) = (35] \\ \end{tabular} & \(\gamma_{0.7}\) & \(1\times 10^{-4}\) & \(4\times 10^{-4}\) & \(\gamma_{0.7}\) = (1–4) \(\times 10^{-4}\)[32] \\ \hline \(E_{\omega}\)\({}^{\prime\prime\prime}\) (MPa) & 1.91 & 27 &
\begin{tabular}{c} \(E_{\omega}\)\({}^{\prime\prime}\) = 0.9\(E_{\omega}\)\({}^{\prime\prime}\)\({}^{\prime\prime}\) \\ \end{tabular} & \\ \hline \hline \end{tabular} (Note: \(c^{\prime}\): effective cohesion; \(\varphi^{\prime}\): effective friction angle; \(\Psi\): dilation angle; \(K_{0}\): lateral pressure coefficient; \(R_{f}\): failure ratio; \(E_{\omega}\)\({}^{\prime\prime}\): secant stiffness in standard drained triaxial test; \(E_{\omega}\)\({}^{\prime\prime\prime}\): tangent stiffness for primary odometer loading; \(G_{0}\)\({}^{\prime\prime}\): reference shear modulus at very small strains (\(c<10^{-6}\)); \(P^{\prime\prime}\): reference stiffness stress; \(v_{\omega r}\): [PERSON]’s ratio for unloading-reloading; \(E_{\omega}\)\({}^{\prime\prime\prime}\): unloading/reloading stiffness from drained triaxial test; \(m\): power for stress-level dependency of stiffness; \(\gamma_{0.7}\): shear strain corresponding to an initial shear modulus of 70%).  
\end{table} 
Table 1: Main parameters of the HSS model. Figure 2: Sketch of reinforcement strategies for paralleled twin tunnels: (**a**) RR; (**b**) LR; (**c**) TR; (**d**) CR. The tunnel construction process was modeled using a step-by-step approach. Each excavation step corresponds to an advancement of the tunnel face of 4 m, which is equal to the width of two lining rings [4]. A pressure equal to the initial ground horizontal stress was applied at the tunnel face [36]. The distribution of the jacking force, which acts on linings of the last ring, was assumed to be linear over the height of the tunnel [19; 20]. A radial pressure was used to simulate the grouting pressure, taking into account the effect of gravity, over a length of 4 m behind the shield. The procedures for excavation of paralleled twin tunnels are summarized as follows [37]:  
1. According to the lateral pressure coefficient \(K_{0}\), the initial boundary and stress conditions of the soil layer are established under the self-weight of soil. 2. The soft soil layer is reinforced and then the displacement caused by the reinforcement is reset. 3. The right tunnel is excavated, 4 m at each step, and relevant pressure is applied. 4.