## 5 Numerical Modeling
### Slope Characterization
The rheological properties of the rock mass were estimated through the equation of [PERSON] et al. [17], as it nowadays represents the only available equivalent continuum approach for the evaluation of time-dependent behavior of a jointed material starting from the characteristic of the rock matrix and discontinuities. For jointed limestones, the joint dip was considered equal to the dip of each region, while for the cataclastic zone the dip was taken as always equal to 60\({}^{\circ}\). In the upper limestone sequence (Figure 7), an average bedding spacing of 50 cm was assumed, while in the lower limestone sequence a greater thickness of 150 cm (equal to the upper limit of the chosen approach) was considered. In cataclastic zones, the considered joint spacing was always of 15 cm, resulting from geomechanical surveyings, while in the HJL the spacing was considered as 10 cm. The time-dependent behavior of the mass was simulated through the rheological model of [PERSON] (Figure 8). The viscosity of intact limestones was attributed based on the literature data for similar rock masses [17; 20] and assumed to be about 2.00 \(\times\) 10\({}^{19}\) Pa\(\cdot\)s. Instead, the viscosity of the rock matrix in the HJL at the base of the Upper Cretaceous limestones was determined through a sensitivity analysis in a numerical model, by varying the parameters along the slope as a function of [PERSON] et al. 's [17] equation. This viscosity value, of about 2.00 \(\times\) 10\({}^{17}\) Pa\(\cdot\)s, is lower with respect to the one attributed to rock mass. The viscosity for the Kelvin rheological element referring to the jointed rock was considered equal to the parameter determined with the described equation, while the viscosity of the Maxwell rheological element was assumed to be one order of magnitude lower with respect to the previous one. The Bulk modulus and the shear modulus for the Maxwell rheological element derive directly from the Young modulus calculated through the approach of  
Figure 8: [PERSON] visco-plastic rheological model and related time-dependent behavior in relation to the considered parameters. [PERSON] et al. [43], while the shear modulus for the Kelvin component was considered 1.2 times higher than the Maxwell rheological element. In the upper portion of the slope, joint conditions were also considered in the numerical models. Regarding the joint strength, the parameters used in the Barton-Bandis model [54] were determined through the geomechanical survey conducted on the slope involved in the gravitational deformation. In particular, the \(JCS\) was calculated by using the Schmidt hammer rebound on discontinuity walls, while \(JRC\) was determined through an in situ test on joints. \(R\) and \(r\) numbers were calculated by using the same survey on both unweathered and weathered joint walls. All these parameters were determined separately for the jointed mass and the HJL. The geometrical characteristic of the discontinuities set derives from a geomechanical survey and represents the mean values of dips and orientations of fractures inside the deforming rock mass. One set, relative to bedding planes, was considered for the HJL; two sets, relative to high angle joints connected with the tectonic setting of the slope, were used for the Upper Cretaceous limestone sequence. The discontinuity behavior was considered only for the shallowest regions, while it was not taken into account for the deepest regions (where the role of discontinuity is less important for the development of DSGSD) and for cataclastic zones (where the jointing conditions do not allow to assume the control of one or few joint sets).