## 2 Problem Statement and Chosen Methodologies  
### Problem Statement  
The considered problem consists of analysing the stability at the long-term (100 years) of a circular tunnel accounting for the uncertainty of both the creep rock behaviors and the anisotropic initial stresses. The time-dependent behavior of the viscoelastic Burger rock is characterized by two springs \((G_{M},\ G_{K})\) and two dashpots \((\eta_{M},\ \eta_{K})\) of the connected Maxwell and Kelvin elements. The tunnel is supported by a system of double elastic liners which consists of an outer flexible liner and an inner concrete liner. The outer one is assumed to be firmly bonded with the rock mass and with the inner liner; both liners are installed at the same time \(t_{0}\). Figure 1 presents the considered tunnel constructed in the viscoelastic Burger rock with double supports.  
Whilst the deterministic analysis of deep tunnels constructed in viscoelastic materials has been largely performed [5; 6; 18; 19], the time-dependent behavior's uncertainty of rock masses has not yet received a lot of attention. Recently, [PERSON] et al. [17] investigated the stability of the tunnel supports using reliability analyses. Their stochastic analyses by MCS demonstrated the significant effects of Burger parameters' uncertainty on the stress state in liners and hence their stability at the long-term. More precisely, the \(Pf\) of these elements was estimated at 100 years using the following limit state function (LSF):  
\[\text{LSF}_{1}=\sigma_{e\_max}-\ \sigma_{e}(\mathbf{X}) \tag{1}\]  
where \(\sigma_{e\_max}\) is the allowable value of the equivalent stress \(\sigma_{e}(\mathbf{X})\) in the support element; this latter depends on the random input parameters gathered in the vector \(\mathbf{X}\). An analytical solution of this equivalent stress in each support element with respect to \(\mathbf{X}\) was presented in the work of [17]. However, this last contribution was limited to the case of isotropic stress state at far field and only the uncertainty of the rheological rock was considered (i.e., the vector of random variables consists of four parameters \(\mathbf{X}=[G_{M},\ G_{k},\ \eta_{M},\ \eta_{K}]\) in [17]).  
In comparison with [17], the uncertainty of the anisotropic stress state at far field is involved in the present study. Noting \(\lambda\) and \(P_{0}\) as the anisotropic ratio and the vertical principal value of stress state at far field, the vector of random variables is now characterized  
Figure 1: Deep tunnel with double liners in a viscoelastic Burger rock under anisotropic initial stresses.  
by six parameters \(\mathbf{X}=[G_{M},\ G_{k},\ \eta_{M},\ \eta_{K},\ \lambda,\ P_{0}]\). Furthermore, a volumetric strain based LSF for the outer flexible liner is proposed as follows:  
\[\text{LSF}_{2}=\epsilon_{v\_max}-\ \epsilon_{v}(\mathbf{X}) \tag{2}\]  
According to Equation (2), the elastic behavior of the outer flexible liner is limited to an allowable value \(\epsilon_{v\_max}\) of the volumetric strain before attaining the non-linear behavior stage due to the pore collapse mechanism.  
The calculation of the equivalent stress \(\sigma_{e}(\mathbf{X})\) in the concrete liner and the volumetric strain \(\epsilon_{v}(\mathbf{X})\) in the flexible liner is based on a semi-analytical solution presented in the work of [14]. To derive this solution of a circular tunnel under an anisotropic stress state of a viscoelastic Burger rock, [PERSON] et al. [14] combined the Laplace transform technique and the complex potential theory. More precisely, the authors expressed the complex potentials in the form of the truncated Laurent series whose coefficients are numerically determined from the inverse Laplace transform. For details of this solution, the interested readers can refer to [14].