## 2 Methodology
### Analysis of Instability around the Contour of an Underground Exacuation Leading to Overbreak  
The deterministic method is based on the theory of block stability in jointed rock mass [15; 16]. Block theory is based on a geometric analysis of the possible occurrence of \"key\" blocks of a certain size and position in relation to the geometry of the excavation, and it considers three spatial dimensions. A key block can be defined as a block for which there is the possibility of movement towards the contour of the tunnel excavation, and which is unstable if not supported. The dip, strike, and azimuth of discontinuities are recorded at each step of tunnel excavation during geological mapping, along with other significant joint characteristics from the perspective of block theory analysis. Once available, these data can be effectively used to conduct block analysis for each tunnel cross-section during the tunnel drive. In order to apply block theory, it is necessary to isolate an intersection of at least three sets of joints in the rock mass. The joint sets present along the unsupported tunnel section form wedges in the shape of a tetrahedron, the dimensions and position of which depend on the orientation and spatial location of joints and the orientation and geometry of the tunnel contour. Depending on the shear strength parameters along the discontinuities, the factors of safety against failure can be determined for each wedge. In the scenario in which four or more sets of joints are present in the rock mass, it is possible to analyse combinations of any three sets and thus determine the most critical wedge. The analysis is usually repeated with different combinations of joint sets so that all potentially unstable blocks around the contour can be duly assessed. The dimensions of the wedges depend on the persistence of the joints and the length of the unsupported excavation. In the case of persistent joints and an infinite excavation length, the critical size of the block is limited by the key block size. The key block size can be calculated from the geometry of the largest pyramidal surface in the intersection of sets of joints and the contour of excavation. To obtain realistic dimensions of wedges for each analysed tunnel section, the length of the unsupported excavation is taken to coincide with the length of the tunnel excavation step. The persistence of the joints is interpreted on the basis of geological mapping, which is usually based on the rock mass rating (RMR) classification initially proposed by [PERSON] [17; 18]. Three-dimensional block theory is based on vector analysis, in which the stability of the wedge around the contour of the underground excavation is determined following several predetermined steps that are integrated in the UnWedge software package (Rocccience, Inc., Toronto, ON, Canada). In the first step, the geometry and position of the wedge are determined, as illustrated in Figure 1, so that the volume, surface area of each side, and vectors normal to each plane of the wedge can be determined. Once all forces acting on the wedge are determined, the results of active (driving) and passive (resisting) forces can be evaluated. Based on the equilibrium of active and passive forces, the direction of sliding of the wedge as well as the normal forces acting on the sides of the wedge are determined. The resulting active force vector (A) is obtained as  
\[\mathbf{A}=\mathbf{W}+\mathbf{C}+\mathbf{X}+\mathbf{U} \tag{1}\]  
where \(\mathbf{W}\) is the weight vector of the wedge, \(\mathbf{C}\) is the weight vector of the shotcrete lining, \(\mathbf{X}\) is the vector of active pressure on the wedge, and \(\mathbf{U}\) is the water pressure vector. The resulting passive vector (\(\mathbf{P}\)) is obtained by the addition of component vectors:  
\[\mathbf{P}=\mathbf{H}+\mathbf{Y}+\mathbf{B} \tag{2}\]  
where \(\mathbf{H}\) is the shear resistance vector of the shotcrete, \(\mathbf{Y}\) is the passive resistance vector, and \(\mathbf{B}\) is the vector of the resultant force from the anchor, if used.