## 4 Engineering-Geological Model
### Rock Mass Rheological Behavior
\[\gamma_{m}=\gamma_{r}-\sum_{j=1}^{n}\frac{\gamma_{r}a_{j}{l_{m}}{l_{j}}^{2}}{V_{t} \left(s_{j}+a_{j}\right)}+\sum_{j=1}^{n}\frac{\gamma_{j}a_{j}{l_{m}}{l_{j}}^{2}} {V_{t}\left(s_{j}+a_{j}\right)} \tag{1}\]  
where \(\gamma_{r}\) is the unit weight of the intact rock, \(\gamma_{j}\) is the unit weight of the joint filling material in a particular joint set, \(a_{j}\) is the average opening of the joints belonging to a particular set, \(l_{m}\) is the average length of the considered mass, \(l_{j}\) is the average length of a certain joint set, \(s_{j}\) is the average spacing values of the discontinuities, \(V_{t}\) is the total volume of the considered mass, and '\(n\)' is the number of joint sets. For the fault zones corresponding to cataclastic rocks, the physical properties were attributed by considering a chaotic rock mass composed of rock blocks and a fine matrix. In this case, the unit weight \(\gamma_{m}\) of fault zones was calculated through the following relation:  
\[\gamma_{m}=\sum_{c=1}^{n}\ \gamma_{c}V_{c} \tag{2}\]  
where \(\gamma_{c}\) is the unit weight of different lithotypes, \(V_{c}\) is the volumetric ratio of each material, and '\(n\)' is the number of discrete materials constituting the mass. The rock mass strength was defined following the 2002 version Hoek-Brown failure criterion [45]. This approach allows the determination of the mechanical properties of a jointed rock mass considering the strength of rock matrix and the jointing conditions, as a function of the stress field existing within the slope. In this version, the criterion is expressed by the following equation:  
\[\sigma_{1}=\sigma_{3}+\sigma_{c}\bigg{(}m_{b}\frac{\sigma_{3}}{\sigma_{c}}+s \bigg{)}^{a} \tag{3}\]  
In which \(\sigma_{1}\) and \(\sigma_{3}\) are the principal failure stress, \(\sigma_{c}\) is the uniaxial compressive strength, and \(m_{b}\), \(s\), and \(a\) are empirical nondimensional constant related to the geological-structural setting of the rock mass. Specifically, \(m_{b}\) can be determined through the intact rock constant \(m_{i}\), the degree of disturbance suffered by the rock after an excavation \(D\), and the Geological Strength Index \(GSI\), a parameter indicative of the jointing state of the mass and the characteristics of the discontinuities [49]. The tensile strength of rock mass was determined by using the alternative relationship of Hoek-Martin [50]. In this case, starting from the uniaxial compressive strength of intact rock \(\sigma_{ci}\), the tensile strength of the material \(T\) can be calculated through the following equation:  
\[T=\frac{\sigma_{ci}}{8.62+0.7m_{i}} \tag{4}\]  
The elastic modulus of rock mass was estimated through the approach proposed by [PERSON] et al. [43], which considers not only many geomechanical parameters but also the stress field acting on the rock. This approach is based on the determination of the \(J_{f}\) parameter, the Joint Factor [50, 51, 52], which directly depends on the jointing condition of the rock mass and on the geometrical and mechanical characteristics of discontinuities. The parameter \(J_{f}\) is given by the relation:  
\[J_{f}=\frac{J_{p}}{n\ r} \tag{5}\]  
where \(J_{p}\) is the number of discontinuities per cubic meter of rock, \(n\) is a parameter related to the dip of the most frequent joint set, and \(r\) is an additional variable coefficient function of the uniaxial compressive strength of the intact rock [43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53]. The uniaxial compressive strength of the jointed rock \(\sigma_{cj}\) can be computed by the equation:  
\[\sigma_{cj}=\sigma_{ci}\bigg{[}0.04+0.89\ exp\bigg{(}-\frac{J_{f}}{161}\bigg{)} \bigg{]} \tag{6}\]in which \(\sigma_{cl}\) is the uniaxial compressive strength of the intact rock.