## 4 Summary and Conclusions  
The research presented in this paper was motivated by the lack of a methodology to determine the probability of overbreaks, which inevitably occur during drill-and-blast excavation in hard rock, and to quantify them. Overbreaks of up to several hundred meters regularly occur in hard rock tunnelling in low-confinement conditions and can severely undermine tunnel construction. Evaluating the causes of occurrence and quantifying overbreaks should be conducted as a valuable engineering practise during tunnel construction to optimise the excavation and avoid significant financial and time losses. The occurrence of overbreaks is typically attributed to poor drill-and-blast technology, which is inappropriately applied in given rock conditions. In mining engineering, the focus on drill-and-blast efficiency is related to good fragmentation, and overbreaks are of less interest because there is typically no need to install secondary lining. In civil engineering, this problem is reversed: even with completely adequate drill-and-blast technology, superfluous overbreaks can occur, significantly increasing the consumption of shotcrete for primary lining and cast-in-place concrete for secondary lining. Inevitable geological overbreaks are dictated by geological conditions, which are defined by the state of hard rock mass discontinuities, that is: (a) the spatial layout of sets of discontinuities relative to the direction of advancing tunnel and the shape of the tunnel contour and (b) the shear strength along the surface of discontinuities. Along elongated tunnels, the problem must be regarded as not only deterministic but also stochastic due to large quantities of relevant data coming from the natural environment. For this reason, the proposed methodology has two components: (a) deterministic evaluation of the typical types of overbreaks and their quantification and (b) probabilistic analysis to extrapolate the probability of failure established by the deterministic method along the tunnel route. The methodology works as a synergetic combination of deterministic and probabilistic components so that inevitable overbreaks can be quantified using statistical analysis, including an evaluation of the threshold value to distinguish between geological and technological causes (e.g., inadequate use of drill-and-blast technology). The application of the methodology, demonstrated on the example of an 8.1 km long tunnel section in hard rock, led to the following conclusions:  
* Using deterministic evaluation, the position and size of unstable wedges (for a safety factor of less than 1.0) was adequately defined. This analysis showed that the positions and sizes of the considered wedges were in good agreement with the contours of the geodetic measured overbreaks in analysed sections. * Probabilistic analysis was conducted in the same sections as the deterministic evaluation, and the results showed that the probability of failure is high and often exceeds 90%. * Based on the combination of deterministic and probabilistic analysis results, the \"stability\" criterion was defined: a wedge is stable if the safety factor is greater than 1.0 and probability of failure is less than 7%. Based on this criterion, it was possible to distinguish stable from unstable parts of an unsupported tunnel excavation along the tunnel route. \begin{table}
\begin{tabular}{c c c c c} \hline
**Threshold Value (m2)** & **Overall** & **\textless{1.6}** & **1.6-2.5** & **\textgreater{2.5}** \\ \hline Total overbreak area (m2) & 4703 & 416 & 1086 & 3201 \\ \hline Total overbreak volume (m3) & & 22,880 & & \\ \hline Technological overbreak volume (m3) & & 3322 & & \\ \hline Geological overbreak volume (m3) & & 19,558 & & \\ \hline \end{tabular}
\end{table} 
Table 4: Calculated technological and geological overbreak area and volume. * The analyses confirmed that the main cause of overbreaks was of a geological nature, i.e., unfavourable orientation of discontinuities and relatively low values of the shear strength parameters at relevant joints.