## 2 Mechanical Analysis of Subsea Tunnels Considering Ground Reinforcement  
### Analytical Model  
This section is aimed at deriving surrounding rock displacement as well as the plastic zones of subsea tunnels considering ground reinforcement effect. It was assumed that the seepage of the seawater obeys [PERSON]'s law. Due to excavation, the seepage field changes within the range \(R_{0}\) (\(R_{0}\) = min [20\(r_{0}\), \(h_{0}\)], where \(h_{0}\) is the depth of the tunnel center below the sea level) due to the excavation [18; 19], and the seepage force at the boundary (\(R_{0}\)) is denoted by \(p_{a}\). For analytical modelling of the problem, a circular tunnel is considered within an infinite ground subjected to a far-field hydrostatic stress (see Figure 1). Both of the reinforced ground (RG) and the natural ground (NG) were assumed to behave like as elastic perfectly plastic materials whose behaviors obey the Mohr-Coulomb yielding criterion. Due to the difference in the mechanical parameters between the reinforced ground and the natural ground, as the stress is released, the plastic zone may be initiated within the reinforced ground or  
Figure 1: Curtain grouting for subsea tunnels. the natural ground. Under certain circumferences, the plastic zones could simultaneously appear in both zones and gradually expand outward. Therefore, six different configurations based on the development process of the plastic zone(s) were generated, as presented in Figure 2[20]. Each kind of configuration was likely to be the final state, so there existed in total eight transitions and eleven possible paths, as given in Table 1. Each path met certain qualifications, which can be illustrated by the unconfinement factor (\(\lambda\)) defined by the internal support pressure (\(p_{\mathrm{i}}\)), the in situ stress (\(p_{0}\)), and the water pressure (\(p_{\mathrm{e}}\)) as:  
\[\lambda=1-p_{\mathrm{i}}/\left(p_{0}+p_{\mathrm{e}}\right), \tag{1}\]  
As \(\lambda\) progresses from 0 to 1, additional plastic zone(s) appear, as demonstrated in Figure 3. Given that the transitions from CFG I to CFG II and that from CFG III to CFG IV are similar such that an additional plastic zone appears in the reinforced zone; therefore, these two paths can be represented by \begin{table}
\begin{tabular}{c c c} \hline \hline
**Processing Path** & **Development History** & **Critical Unconfinement** \\ \hline
1 & I & - \\
2 & I–II & \(\lambda_{12}^{*}\), \\
3 & I–II–IV & \(\lambda_{12}^{*}\), \(\lambda_{24}^{*}\) \\
4 & I–II–IV–VI & \(\lambda_{12}^{*}\), \(\lambda_{24}^{*}\), \(\lambda_{46}^{*}\) \\
5 & I–II–V & \(\lambda_{12}^{*}\), \(\lambda_{25}^{*}\) \\
6 & I–II–V–VI & \(\lambda_{12}^{*}\), \(\lambda_{25}^{*}\), \(\lambda_{56}^{*}\) \\
7 & I–III & \(\lambda_{13}^{*}\), \\
8 & I–III–IV & \(\lambda_{13}^{*}\), \(\lambda_{34}^{*}\) \\
9 & I-III–IV–VI & \(\lambda_{13}^{*}\), \(\lambda_{34}^{*}\), \(\lambda_{46}^{*}\) \\
10 & I–IV & \(\lambda_{14}^{*}\) \\
11 & I–IV–VI & \(\lambda_{14}^{*}\), \(\lambda_{46}^{*}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Evolution processes of the plastic zone(s). Figure 2: Schematic diagram of the development of plastic zone(s) (“CFG” stands for “configuration”). a single illustration (see Figure 3a), and the same story holds true for Figure 3c,d. The final state of the plastic zone(s) resulted from the superposition of the constitutive subgraphs in Figure 3. Taking path 4 in Table 1 as an illustrative example, when \(\lambda\) was smaller than \(\lambda^{*}_{12}\), both RG and NG remained elastic (CFG I). Once \(\lambda\) reached \(\lambda^{*}_{12}\), the plastic zone was initially developed inside the RG (CFG II). Once \(\lambda\) increased up to \(\lambda^{*}_{24}\), an additional plastic zone was initiated and developed inside the NG (CFG IV). Otherwise, when \(\lambda\) increased up to \(\lambda^{*}_{46}\), the plastic zone inside the RG finally expanded to occupy the entire RG (CFG VI).