# Face Stability Analysis for Tunnels under Steady Unsaturated Seepage and Inhomogeneity Conditions
###### Abstract  
In the field of tunnels, the stability of tunnel faces is generally considered in dry, saturated and homogeneous soils. However, the actual condition of some soils has been found to be inhomogeneous, with unsaturated seepage. In this paper, an analytical method is applied to estimate the safety factor when the supporting force at the tunnel face is zero under steady unsaturated seepage and inhomogeneous conditions. This method combines kinematic limit analysis techniques with strength reduction techniques; an efficient stress formulation utilizing suction stress is employed to determine the apparent cohesive force to obtain the solution of the steady unsaturated seepage problem, and indicators of soil inhomogeneity are attributed to the effect on cohesion. A 3D log-spiral collapse mechanism is used to find the zero supporting pressure and determine the safety factor through an iterative method. This paper analyzes the effect of variations in the unsaturated parameters, inhomogeneity parameters and tunnel dimensional parameters on the stability of the tunnel face. t 20242024149377 10.3390/app1420937 10.3390/app1420937 10.3390/app1420937  
tunnel face. A new discretization technique was proposed by [PERSON] et al. (2023) [10] for a three-dimensional (3D) tunnel face in weak strata with a random position in space. Many subsequent studies have been inspired by these fundamental failure mechanisms. Numerous researchers have explored kinematic limit analysis methods, with [PERSON] et al. (2004) [11] employing such an approach to assess the seismic and static stability of slopes. Integrating the suggested coefficients into the kinematic framework of limit analysis, [PERSON] and [PERSON] (2023) [12] crafted a three-dimensional (3D) failure mechanism capable of capturing the full spectrum of potential instability scenarios. Depending on the construction process, tunnels can be generally classified into two main types: closed-face and open-face. The primary distinction between the two types of tunnels is that face pressure generally exists for support on the face of closed-face tunnels, such as shield tunnels, which are often assessed with the necessary face pressures, while open-face tunnels do not need to be supported by face pressures, such as tunnels excavated with the New Austrian Tunneling Method, which are more appropriately assessed using a safety factor. This paper focuses exclusively on open-face tunnels to examine the safety factor under conditions of unsaturated seepage and soil inhomogeneity. In existing studies, two methods of safety factor calculation are proposed. The gravity increase method was used by [PERSON] et al. (2009) [13], with the safety factor defined as the ratio of the internal dissipated work rate to the external work rate. The strength reduction technique has been employed by many studies, defining the safety factor as the ratio of the initial soil strength to the soil strength at the ultimate collapse state. It has been shown that more conservative results can be produced by the strength reduction technique. Therefore, the strength reduction technique was employed in this paper. Steady unsaturated soils were modeled by defining a friction angle \(\varphi^{\text{b}}\), but it was experimentally demonstrated that \(\varphi^{\text{b}}\) varies with matric suction; however, assumptions about the matric suction distribution are inaccurate. A framework for the stress behavior of unsaturated soils based on a one-dimensional model of seepage and suction stress was established by [PERSON] et al. (2010) [14] and [PERSON] et al. (2015) [15]. However, the above studies were mostly used for the assessment of slope stability and calculation of soil pressure, and did not address the issue of tunnel face stability. Inhomogeneity refers to the variation in soil strength parameters with the change of three-dimensional coordinates, and this property is proved to be particularly obvious in the vertical direction.