## 1 Introduction
[PERSON] and [PERSON] (2019) [1] used 3D finite element simulations for numerical modeling to compare tunnel face stability in both homogeneous and layered soft ground. Their study provided insights into tunnel face behavior under varying soil conditions. [PERSON] et al. (2020) [12] employed the earth pressure balance method in a case study of the Dez-Ghomrood water transfer tunnel, focusing on non-homogeneous inclined soil layers. Their investigation emphasized the practical application of this method for assessing face stability. [PERSON] and [PERSON] (2020) [13] introduced a novel upper bound limit analysis method, dividing the failure mechanism into horizontal strips with small heights, aimed at assessing tunnel face stability in nonuniform soils. [PERSON] et al. (2023) [14] examined the influence of inclined soil stratification on tunnel face stability, particularly in shield tunneling contexts. These studies offer valuable insights into the complexities of face stability in inclined soils, providing methodologies and case studies to enhance understanding and guide future tunneling projects. More recently, [PERSON] and [PERSON] (2021) [15] applied a pseudodynamic approach to analyze the three-dimensional stability of tunnel faces, contributing a critical methodology for assessing dynamic stability in tunnel engineering. [PERSON] et al. (2023) [16] focused on the pseudodynamic analysis of 3D tunnel faces in inclined weak strata, further enhancing the understanding of tunnel stability in challenging ground conditions. In addition to soil stratification, natural soils typically contain water, and the interaction between soil and water is an important topic. In tunnel surface stability analysis, the literature often assumes mostly dry soil above the water table or fully saturated soil below it. However, in natural environments, soils are predominantly unsaturated due to factors such as ambient humidity, groundwater, and rainfall infiltration. The soil above the groundwater table is commonly referred to as unsaturated soil, as demonstrated by experimental studies ([PERSON] and [PERSON] (1988) [17]). Recognizing the unsaturated state of soils in natural conditions is critical for achieving a more precise and comprehensive understanding of tunnel stability. Understanding the shear strength of unsaturated soils is crucial for various geotechnical designs, and the soil-water characteristic curve is often used as the key parameter for estimating unsaturated shear strength. [PERSON] et al. (1996) [18] explored the correlation between unsaturated soil shear strength and the soil-water characteristic curve. Building on this, [PERSON] and [PERSON] (1998) [19] proposed a unique relationship to determine the shear strength of unsaturated soils. Existing models generally consider suction as an additional net normal stress component influenced by changes in matric suction. The shear strength of unsaturated soils has been a significant focus of research, especially in understanding their behavior under applied stress gradients. Unsaturated soils can be conceptualized as a two-phase mixture striving for equilibrium when subjected to external stress. The distinctiveness of their stress response lies in the influence of matric suction, which differs from saturated and dry soils. [PERSON] et al. (1996) [18] made pioneering contributions by incorporating matric suction into the Mohr-Coulomb failure criterion for unsaturated soils, suggesting a modified failure criterion that revealed a linear relationship between shear strength and matric suction. To bridge these perspectives, [PERSON] and [PERSON] (2004) [20] refined the understanding of effective stress in unsaturated soils, developing a new expression connecting matric suction to matric stress:  
\[\sigma^{\prime}=(\sigma-u_{a})+\chi(u_{a}-u_{w}) \tag{1}\]  
In this context, matric suction is characterized as the disparity between pore air pressure \((u_{a})\) and pore water pressure \((u_{w})\).