## 5 Numerical Modeling of the Vertical Section of the Cava Madre Cavern  
The numerical modeling was carried out using the Flac2D code, considering the vertical marble bank affected by the exploitation together with other gneiss and mica schist banks located to its right and left. The primary goal of the numerical modeling is to confirm the stability conditions of the walls and the stress level induced in the structural elements that contrast the walls because the cavern has a tendency to grow in height and the major thrusts coming from the rock are directed horizontally [4,20]. Furthermore, in the future, the necessity of extracting good quality marble blocks pushes towards a further deepening of the foot of the cavern in the entrance area, where the mining activities are currently concentrated. For this reason, the two-dimensional numerical modeling was focused on the entrance area, in a vertical section transversal to the axis of the cavern located approximately in correspondence with the new central concrete strut beam. The numerical model developed, with the geomechanical parameters calibrated by the back-analysis procedure, turns out to be a powerful tool for predicting the development of stresses and deformations in the rock mass surrounding the cave and inside the support structures and rock reinforcement. It is able to predict the static behavior of the cave as the extraction of marble blocks from the foot of the cave progresses in the near future. The correct simulation of the succession of excavation phases is a fundamental step to allow the model to adhere as realistically as possible to the actual evolution of excavations in the past and also for what is expected in the future. Due to a lack of comprehensive historical data, it was not possible to replicate the removal of every block from the cavern's opening; instead, a simplified version of the cavern's mining history up to the present was created, based on the primary excavation stages that are described here:  
Phase 0--Initial lithostatic condition, with the insertion of the lithostatic stress state (vertical and horizontal in situ stress);  
Phase 1--Initial cavity geometry: extraction activities up to the 1950s is simulated in a single phase;  
Phase 2--Construction of the old seven concrete strut-beams;  
Phase 3--Simulation of further mining activities;  
Phase 4--Demolition of some old strut beams (the outermost ones) and remodeling of the right wall of the cavern with the enlargement of the cavern in the entrance zone from 12 to 22 m;  
Phase 5--Insertion of the new three concrete strut beams, more massive (section of 6 m \(\times\) 2.5 m) than the previous ones and located at a lower level;Phase 6--Lowering the foot of the cavern (bench 0) and partial underlaying of the supports of the new strut-beams on the right wall (activity in progress in the entrance area). Additional support systems, such as long rebars and cable tendons, are also taken into account in order to maintain adequate confinement against rock boundary displacement [21; 22; 23]. The end of Phase 6 corresponds to the current condition of the cavern. The subsequent phases simulate the excavation activities and support structures of the side walls, which are hypothesized for the future in order to be able to continue the mining operations downwards through the excavation of a further bench (bench 1) at the foot of the cavern:  
Phase 7--Completion of bench 0 by exhausting the marble extraction at the current level at the foot of the cavern;  
Phase 8--Creation of a second row of strut beams (4 m \(\times\) 4 m sections) resting at the foot of the cavern;  
Phase 9--Opening of bench 1: repetition of the extraction and wall support techniques described in phases 6 and 7, in order to correctly carry out the excavations and underlaying of the supports on the right wall of the existing strut beams.