## 2 Rock Mass Characteristics
### Estimation of Rock Mass Strength Properties  
Even the most sophisticated failure criteria, laboratory tests of intact rock, and rock mass classification systems have little practical value unless they can be used to estimate the representative mechanical properties of jointed rock masses [27]. Therefore, in the present study, the following strength degradation relations for undisturbed rock mass, proposed by [PERSON] et al. [28] and reiterated by [PERSON] and [PERSON] [29], were employed to determine the Hoek-Brown material constants \(m\), \(s\), and \(a\):  
\[\frac{m}{m_{i}}=\exp\left(\frac{GSI-100}{28}\right), \tag{5}\]  
\[s=\exp\left(\frac{GSI-100}{9}\right), \tag{6}\]  
\[a=\frac{1}{2}+\frac{1}{6}\cdot\left[\exp\left(\frac{-GSI}{15}\right)-\exp \left(-\frac{20}{3}\right)\right]. \tag{7}\]  
The recommended method for determining the Hoek-Brown material constant \(m_{i}\) is the triaxial compressive test of intact rock samples [30]. However, due to the lack of relevant data, a different approach had to be employed in the present study, based on the research conducted by [PERSON] [31], who provided the following relationship between the material constant \(m_{i}\) and intact strength parameters for sedimentary rocks:  
\[m_{i}=\exp\left[1.3\cdot\left(\frac{\sigma_{c,i}-2.5\cdot\sigma_{t,i}}{\sigma _{t,i}}\right)^{0.26}\right]. \tag{8}\]  
Equivalent Mohr-Coulomb strength parameters, found by fitting linear Coulomb failure envelope to the parabolic Hoek-Brown failure envelope, were calculated with use of the solution provided by [PERSON] et al. [28]:  
\[c^{eq}=\frac{\sigma_{c,i}\cdot\left[(1+2\cdot a)\cdot s+(1-a)\cdot m\cdot \sigma_{3n}\right]\cdot(s+m\cdot\sigma_{3n})^{a-1}}{(1+a)\cdot(2+a)\cdot\sqrt{1 +\frac{6\cdot a\cdot m\cdot(s+m\cdot\sigma_{3n})^{a-1}}{(1+a)\cdot(2+a)}}} \tag{9}\]  
\[\varphi^{eq}=\sin^{-1}\left[\frac{6\cdot a\cdot m\cdot(s+m\cdot\sigma_{3n})^ {a-1}}{2\cdot(1+a)\cdot(2+a)+6\cdot a\cdot m\cdot(s+m\cdot\sigma_{3n})^{a-1} }\right], \tag{10}\]  
where \(c^{eq}\) is the equivalent cohesion, \(\varphi^{eq}\) is the equivalent internal friction angle, \(m\), \(s\), and \(a\) are Hoek-Brown material constants for jointed rock mass and \(\sigma_{3 max}\) is the upper limit of the confining stress.  
In the present study, the peak strength properties of the jointed rock mass were calculated based on the Geological Strength Index (Table 3), while the post-peak properties were estimated using the Residual Geological Strength Index (Table 4).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**Hoek–Brown**} & \multicolumn{3}{c}{**Coulomb–Mohr**} \\ \cline{3-7}
**Lithology** & **Depth** & **m** & **s** & **a** & **c** & **\(\varphi\)** \\  & **(m)** & **m** & **s** & **a** & **(MPa)** & **(\({}^{\circ}\))** \\ \hline Medium-grained sandstone & 957.00–983.50 & 2.668 & 0.027 & 0.502 & 2.60 & 32.56 \\ Mixed-grained sandstone & 983.50–988.50 & 2.332 & 0.017 & 0.502 & 2.94 & 34.89 \\ Mixed-grained sandstone & 988.50–993.20 & 2.396 & 0.018 & 0.502 & 3.23 & 36.31 \\ Claystone & 993.20–995.70 & 1.025 & 0.002 & 0.504 & 1.58 & 24.52 \\ Mudstone & 995.70–999.20 & 1.434 & 0.005 & 0.503 & 2.64 & 33.20 \\ Fine-grained sandstone & 999.20–1012.20 & 1.570 & 0.005 & 0.503 & 2.59 & 33.17 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Peak strength properties of the rock masses in the area of Shaft II.