## 3 Numerical Modeling
### Model Parameters  
Based on the geology investigation and drilling information, the surrounding rock at the tunnel portal is mainly composed of a clay layer with a thickness of about 1.0 m, strongly weathered sandstone with mudstone with a thickness of 25-30 m, and medium-weathered sandstone with mudstone. The rock mass at the portal section of the tunnel is relatively broken, showing a fragmented mosaic structure and poor stability, and is a grade V surrounding rock.  
The Mohr-coulomb elastic-plastic criterion was widely used in numerical computational studies of small clearances tunnels, but the damage problem of the surrounding rock was less considered. The generalized Hoek-Brown criterion better reflects the nonlinear characteristics of the rock mass than the former, and takes into account the weakening effect of plastic damage on the strength and stiffness of the rock mass, while the model can be used for the safety factor calculation in FLAC 3D. For the weak and broken rock mass structure, the geological strength index (GSI) surrounding the rock-rating system can better reflect the characteristics of the jointed rock mass [17; 18]. Therefore, according to the surrounding rock grade, weathering degree, and rock properties, the rock mass was simulated by the Hoek-Brown elastic-plastic model suitable for the brittle fracture of rock, given by Equation (1) [19; 20]:  
\[\sigma_{1}=\sigma_{3}+\sigma_{ci}(m_{b}\frac{\sigma_{3}}{\sigma_{ci}}+s)^{a} \tag{1}\]  
where \(\sigma_{1}\) and \(\sigma_{3}\) are the maximum and minimum principal stresses; \(\sigma_{ci}\) is the uniaxial compressive strength; and \(m_{b}\), \(s\), and \(a\) are rock mass material parameters, given Equations (2)-(4) [19; 20]:  
\[m_{b}=\exp(\frac{GSI-100}{28-14D})m_{i} \tag{2}\]  
\[s=\exp(\frac{GSI-100}{9-3D}) \tag{3}\]  
\[a=\frac{1}{2}+\frac{1}{6}\bigg{[}\exp(-\frac{GSI}{15})-\exp\bigg{(}-\frac{20}{ 3}\bigg{)}\bigg{]} \tag{4}\]  
where \(GSI\) is the geological strength index, indicating the geological conditions of the site; \(D\) is the excavation disturbance coefficient; and \(m_{i}\) is a rock constant, which comprehensively reflects the friction and rupture characteristics as well as compressive and tensile strengths of rock.  
The values of \(GSI\), \(m_{i}\), and \(D\) were selected based on a combination of existing studies [21; 22; 23; 24], in situ geological conditions, and indoor uniaxial tests. Based on the geological borehole exposures of YK29+261, the strongly weathered rock masses are broken along with the medium-weathered rock masses in elastic form, and for the method of quantifying the \(GSI\) system proposed by [PERSON] et al. [21], the \(GSI\) is taken as 25 and 40, respectively. The uniaxial compressive strengths of the strongly weathered and moderately weathered sandstone with mudstone were obtained from indoor tests as 8.5 MPa and 26.1 MPa, respectively. Combined with the lithologic, fracture characteristics and with reference to literature [22; 23], the \(m_{i}\) values for the strongly weathered and moderately weathered sandstone with mudstone were taken as 5 and 10, respectively. \(D\) depends on the disturbance of the rock mass by blast damage and stress relaxation; this tunnel area is mechanically excavated, with controlled pre-cracking or smooth blasting, as the value of 0.5 is taken [24]. The results are given in Table 1.  
The elastic modulus of the weakly jointed surrounding rock is determined on the basis of the method proposed by [PERSON] [25], given as Equation (5):  
\[E=100\bigg{(}\frac{1-D/2}{1+\exp[(75+25D-GSI)/11]}\bigg{)} \tag{5}\]Combined with the engineering geological survey report, Code for Design of Highway Tunnels (JTG 3370.1-2018) [26], and indoor geotechnical tests, the soil constitutive of the model was Mohr-Coulomb. The parameters of the tunnel model rock formations are listed in Table 1.