# Temperature Effect on Stability of Tunnel Face Under Unsaturated Seepage Condition
## 2 Impact of Temperature on Unsaturated Seepage Soil  
The strength of unsaturated soil is affected by temperature conditions, mainly reflected in the changes in soil matric suction with temperature variations. [PERSON] and [PERSON] [24] proposed an expression for effective stress:  
\[\sigma^{\prime}=\sigma-u_{\text{a}}-\sigma^{\text{s}} \tag{1}\]  
where \(u_{\text{a}}\) is the pore-air pressure, \(\sigma^{\text{s}}\) is the suction stress, \(\sigma\) is the total stress, and \(\sigma^{\prime}\) is the effective stress. An equation for \(\sigma^{\text{s}}\) in unsaturated soil was proposed by [PERSON] et al. [25]; this work extends it into a form that is related to temperature:  
\[\sigma^{\text{s}}=-\frac{\psi}{\left[1+(\alpha\psi)^{n}\right]^{(n-1)/n}} \tag{2}\]  
where \(\psi\) is the matric suction under unsaturated soil condition with temperature effect; \(n\) and \(\alpha\) are parameters that describe the soil's pore distribution and the inverse of the soil's entry pressure, respectively. These two values are specific to the distinct characteristics of various soil types: the value of parameter \(\alpha\) varies within the range of 0.001 kPa\({}^{-1}\) to 0.5 kPa\({}^{-1}\), while the value of parameter \(n\) varies within the range of 1.1 to 8.5 ([PERSON] et al. [26]). Utilizing the effective stress attributed to matric suction and the conventional Mohr-Coulomb criterion for shear strength, the formula of unsaturated soil's shear strength is as follows:  
\[\tau=c^{\prime}+(\sigma-u_{\mathrm{a}}-\sigma^{\mathrm{s}})\mathrm{tan}\varphi^ {\prime} \tag{3}\]  
where \(c^{\prime}\) is the effective cohesion, \(\tau\) is the shear strength, and \(\varphi^{\prime}\) is the effective friction angle. Matric suction's contribution to shear strength is termed as apparent cohesion ([PERSON] et al. [25]; [PERSON] et al. [26]), and it can be expressed as  
\[c_{\mathrm{app}}=-\sigma^{\mathrm{s}}\mathrm{tan}\varphi^{\prime} \tag{4}\]  
Equations (1)-(4) are defined under general environmental conditions, and temperature's role in influencing matric suction can be formulated as follows:  
\[\psi=\psi_{\mathrm{Tr}}\left(\frac{\beta+T}{\beta_{\mathrm{Tr}}+T_{\mathrm{r} }}\right) \tag{5}\]  
where \(T\) is the temperature (as a variable in Kelvin). It should be clarified that for the sake of convenience in expression and intuitive display, although Kelvin is used as the unit of temperature in the calculations, Celsius will be used as the unit for analysis in the subsequent text. \(\psi_{\mathrm{Tr}}\) is the matric suction at the reference temperature \(T_{\mathrm{r}}\) (0 \({}^{\circ}\)C), and \(\beta_{\mathrm{Tr}_{\mathrm{r}}}\) represents the regression parameter at the reference temperature, depending on surface tension, specific immersion enthalpy, and the contact angle. In the formula, \(\beta\) and \(\beta_{\mathrm{Tr}_{\mathrm{r}}}\) can be calculated as follows:  
\[\beta=\frac{-\Delta h}{-\Delta h+(a+bT_{\mathrm{r}})\mathrm{cos}\alpha} \tag{6}\]  
\[\beta_{\mathrm{Tr}_{\mathrm{r}}}=-\frac{\Delta h_{\mathrm{Tr}_{\mathrm{r}}}}{ C_{1}} \tag{7}\]  
where \(\alpha\) is the temperature-dependent soil-water contact angle, and \(a\) and \(b\) are the fitting parameters that can be estimated as \(a=0.11766\) N/m, \(b=0.0001535\) N/(m\(\cdot\)K) ([PERSON] and [PERSON] [27]). \(\Delta h\) represents the immersion enthalpy per unit area, and the temperature-dependent expression for \(\Delta h\) is:  
\[\Delta h=\Delta h_{\mathrm{Tr}_{\mathrm{r}}}\left(\frac{1-T_{\mathrm{r}}}{1-T }\right)^{0.38} \tag{8}\]  
where \(\Delta h_{\mathrm{Tr}_{\mathrm{r}}}\) is the immersion enthalpy per unit area at the reference temperature. The soil-water contact angle \(\mathrm{cos}\alpha\) can be obtained from the following expression ([PERSON] and [PERSON] [28]):  
\[\mathrm{cos}\alpha=\frac{-\Delta h+TC_{1}}{a+bT} \tag{9}\]  
where \(C_{1}\) is a constant.