# Face Stability Analysis for Tunnels under Steady Unsaturated Seepage and Inhomogeneity Conditions
## 2 Derivation of Total Cohesion  
A consolidated expression for effective stress was proposed by [PERSON] and [PERSON] (2004) [17] as follows:  
\[\sigma^{\prime}=\sigma-\text{u}_{\text{a}}-\sigma^{\text{s}} \tag{1}\]  
where \(\sigma^{\prime}\) is the effective stress, \(\sigma\) is the total stress, \(\text{u}_{\text{a}}\) is the pore air pressure and \(\sigma^{\text{s}}\) is the suction stress. Following standard practice, positive values are assigned to normal compressive stress. [PERSON] et al. (2010) [14] presented the expression for \(\sigma^{\rm s}\) as follows:  
\[\sigma^{\rm s}=-(\rm u_{a}-\rm u_{w}) \tag{2a}\] \[\sigma^{\rm s}=-\frac{(\rm u_{a}-\rm u_{w})}{\left\{1+\left[\alpha(\rm u_{a}- \rm u_{w})\right]^{n}\right\}^{(n-1)/n}} \tag{2b}\]  
where \(\rm u_{w}\) is the pore water pressure and the values of \(\alpha\) and \(\rm n\) are determined by the different types of soil, where \(\alpha\) is the reciprocal of soil entry pressure, taking values between 0.001 kPa\({}^{-1}\) and 0.5 kPa\({}^{-1}\) and \(\rm n\) is a parameter reflecting the distribution of soil pores; [PERSON] et al. (2015) [15] pointed out that \(\rm n\) values should be in the range of 1.1-8.5. It is worth mentioning that Equation (2a) applies to the case when \((\rm u_{a}-\rm u_{w})\leq 0\) and Equation (2b) applies to the case when \((\rm u_{a}-\rm u_{w})>0\); this means that Equation (2a) applies to soils that are fully saturated while Equation (2b) applies to soils that are unsaturated, so Equation (2b) was used for the following derivation. Expressions describing the substrate suction were derived by [PERSON] et al. (2015) [15] and [PERSON] (2005) [18] based on [PERSON]'s law, in combination with the case for \(\rm z=0\) when \((\rm u_{a}-\rm u_{w})=0\), which takes the following form:  
\[(\rm u_{a}-\rm u_{w})=-\frac{1}{\alpha}\rm ln\left[\left(1+\frac{q}{\rm k_{s} }\right)e^{-\gamma_{w}\alpha z}-\frac{q}{\rm k_{s}}\right] \tag{3}\]  
where \(\rm q\) is the rate of steady unsaturated seepage, with infiltration for \(\rm q<0\) and evaporation for \(\rm q>0\), and \(\rm k_{s}\) is saturated hydraulic conductivity. The simultaneous equation for Equations (2b) and (3) is as follows:  
\[\sigma^{\rm s}=\frac{1}{\alpha}\frac{\rm ln\left[\left(1+\frac{q}{\rm k_{s}} \right)e^{-\gamma_{w}\alpha z}-\frac{q}{\rm k_{s}}\right]}{\left(1+\left\{- \rm ln\left[\left(1+\frac{q}{\rm k_{s}}\right)e^{-\gamma_{w}\alpha z}-\frac{ q}{\rm k_{s}}\right]\right\}^{n}\right)^{(n-1)/n}} \tag{4}\]  
On the basis of an extended form derived from the Mohr-Coulomb criterion ([PERSON] et al., 2015) [15], which describes the strength and the variation of unsaturated soils, the parameter called apparent cohesion was applied so as to obtain the shear strength component due to unsaturated seepage effects. The cohesion of unsaturated soils consists of an apparent cohesion (\(\rm c_{app}\)) and an effective cohesion (\(\rm c^{\prime}\)), expressed as follows:  
\[\rm c_{app}=-\sigma^{\rm s}tan\varphi^{\prime} \tag{5}\]  
\[\rm c=c^{\prime}+c_{app} \tag{6}\]  
For the case of inhomogeneous soils, an inhomogeneous cohesion model proposed by [PERSON] et al. (2008) [16] and [PERSON] (2016) [19] was employed, and so as to be consistent with the expression of the apparent cohesion, the model is rewritten as a function of \(\rm z\), which takes the following form:  
\[\rm c(z)=\left\{ \begin{matrix}n_{0}c+\frac{c-n_{0}c}{H}(C+D+z_{0}-z)&(C+D+z_{0 }-H)<z\leq(C+D+z_{0})\\ n_{2}c-\frac{n_{0}c-n_{0}c}{C+D-H}(z-z_{0})&z_{0}\leq z\leq(C+D+z_{0}-H)\\ \end{matrix} \right.