## 1 Introduction
The effective stress \((\sigma^{\prime})\) owns a linear relationship with the total stress \((\sigma)\) and the coefficient of effective stress \((\chi)\) is set to reflect the degree to which the effective stress is affected by matric suction. To some extent, the dimensionless coefficient \(\chi\) takes values between 0 and 1. When \(\chi\) is 0, it corresponds to completely dry soil, but when \(\chi\) is 1, it is consistent with saturated soil. Regrettably, conducting a direct measurement of the parameter \(\chi\) appears to be unattainable. However, conducting shear strength tests on unsaturated soil specimens offers an alternative means to determine the value of \(\chi\) under extended Mohr-Coulomb criteria in unsaturated conditions. [PERSON] and [PERSON] (2013) [21] explored the relationship between matric suction and matric stress as a means to elucidate shear strength in unsaturated soils and the novel Mohr-Coulomb criteria are outlined as follows:  
\[\tau_{f}=c^{\prime}+(\sigma-u_{a})\tan\sigma^{\prime}+\chi(u_{a}-u_{w})\tan \sigma^{\prime} \tag{2}\]  
where \(\tau_{f}\) represents shear strength while \(c^{\prime}\) refers to the effective cohesion intercept and \(\sigma^{\prime}\) denotes the effective internal friction angle. Many scholars assessed the stability of geotechnical structures in unsaturated soils with the aid of the above extended criteria. [PERSON] and [PERSON] (2022) [22] focused on the three-dimensional stability of backfill in unsaturated soils. Nevertheless, unsaturated soils are often influenced by rainfall infiltration. This unsaturated flow alters the distribution of matric suction in the soil matrix, adding complexity to various stability analyses of unsaturated soils under seepage conditions. Various analytical formulas proposed by scholars aim to facilitate the determination of both the matric suction distribution and suction stress distribution. [PERSON] and [PERSON] (2006) [20] analyzed the effective expression to discuss the shear strength in unsaturated soils under seepage conditions. [PERSON] and [PERSON] (2013) [21] built a framework corresponding to the profile of the suction head in both a one-layer system and a two-layer system under the background of [PERSON]'s law. With these guiding tools, many studies related to the stability of structures under unsaturated flow conditions began to emerge. [PERSON] et al. (2020) [23] explored the potential failure of a circular tunnel face under steady-state unsaturated flow conditions, employing a discrete failure mechanism. In a parallel investigation, [PERSON] and [PERSON] (2022) [24] analyzed the stability of tunnel faces under unsaturated seepage conditions, taking into account the impact of fluctuating water levels. Expanding on these findings, a subsequent study by [PERSON] and [PERSON] (2020) [25] delved into three-dimensional stability assessments for a tunnel face reinforced by bolts in the presence of seepage flow conditions. Among the research that only considered the unsaturated flow condition in homogenous soils, all of the studies ignored the intricate heterogeneous stratigraphic condition. The soils that are spatially discretely distributed will exhibit discretely distributed matric suction as well. It is essential to examine the impact of spatially discretely distributed soil on the stability of a tunnel face under unsaturated flow conditions. This paper takes into account the inclined stratification characteristics in the spatial variability of soil properties, combining the distribution formula of stratified water heads to consider the scenario of vertical steady-state seepage. On this basis, the stability issue of a tunnel face in unsaturated, steady-state seepage conditions for inclined stratified soils is investigated. The characteristics of the spatially uneven distribution of the water head and soil stratification are considered.