## 1 Introduction
Thus, failure of tunnel supports can occur during their service life as a consequence of the time delay effect when the stress state exceeds the constituent materials strength of each liner, even if these liners were well designed to verify the stability condition at the short-term (e.g., at the first months/years after the excavation). The consideration of uncertainties of time-dependent behaviors for rock masses on the stability of deep tunnels was firstly conducted in the recent contribution of [17]. In their work, the time-dependent behavior of rock mass is characterized by the viscoelastic Burger model represented by four parameters: two springs and two dashpots of the connected Maxwell and Kelvin elements. The reliability analysis of a circular tunnel excavated in this kind of creep rock was undertaken using the direct MCS. More precisely, under the hydrostatic initial stress, the probabilistic response of the tunnel was assessed by MCS thanks to a developed closed-form solution. Their vast numerical investigations showed the important effects of Burger's parameters on the evolution in time of stress states and hence the stability of liners at long term of 100 years. In this work, the probabilistic analysis of a deep tunnel excavated in viscoelastic Burger rocks is reconsidered by accounting for the uncertainty of the initial stresses at far field and particularly of their anisotropy. For the sake of clarity, it is supposed that the position at depth \(H\) of deep tunnel is largely higher than the diameter of tunnel \(D\) (i.e., \(H/D>10\)) that the variation of initial stress state around tunnel can be neglected. In addition, the studied deep tunnel is supported by a system of double elastic liners: an outer flexible liner and an inner concrete liner; both are installed at the same time. Owing a low elastic modulus, the outer flexible liner can absorb the high evolution in time of the rock convergence and reduce the transmitted loading to the concrete liner. Furthermore, a firmly bonded assumption of the outer liner with the rock mass and the inner concrete layer is adopted. In comparison with the case of isotropic initial stresses as considered in [17], the behavior of the tunnel supports is more critical when the moment induced by anisotropic initial stress increases the concentration of compressive stress in liners as shown in the deterministic analyses performed by different authors [2,5,6,18,19]. From a statistical point of view, it is more rational to consider the anisotropy of the initial stresses and the related uncertainties compared to the assumption of isotropic initial stresses using constant parameters. The present study is among the first attempts to consider the uncertainty and anisotropy of the initial stress for the probabilistic assessment of a deep tunnel drilled in Burger rocks and supported by double liners. Additionally, a probabilistic analysis procedure for the above-mentioned problem is proposed in the paper. It combines an effective tunnel stability analysis tool with the subset simulation and the Sobol method. The procedure allows evaluating the reliability of a deep tunnel from different aspects at a first design stage, as different results can be provided by the procedure within a short computational time.