## 5 Discussion
### Adaptation from Block Area to Block Volume Analyses  
One major limitation of this study is that the analyses of the block distributions were done only two-dimensional. Within this framework, this limitation was accepted because, even in 2D, a high number of simulations were required to perform a reliable statistical analysis. 3D simulations would have been very time-consuming, and starting initially with a 2D approach allows for a first and crucial validation of the methodical approach. Conceptually, if a third fracture set is added, an estimation of the mean 3D block volume (\(V_{b}\) [m\({}^{3}\)]) on the basis of the 2D analyses can be done according to following general mathematical relation:  
\[V_{b}=A_{b}\cdot\frac{s_{3}}{\cos\gamma_{3D}} \tag{16}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Threshold Length** & **a** & **b** & **c** & **R\({}^{2}\)** \\ \hline
0.05 m & 0.360 & \(-\)0.668 & 1.005 & 0.972 \\
0.10 m & 0.924 & \(-\)0.842 & 1.026 & 0.980 \\
0.20 m & 2.575 & \(-\)1.115 & 1.089 & 0.982 \\
0.30 m & 4.823 & \(-\)1.263 & 1.134 & 0.984 \\
0.40 m & 6.443 & \(-\)1.259 & 1.149 & 0.985 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fitting parameters for Equation (15) of the power law in relationship to the threshold length. where \(A_{b}\) is the simulated mean block area considering non-persistent fractures [m\({}^{2}\)]; s\({}_{3}\) is the mean normal set spacing [m]; \(\gamma_{3D}\) is the angle between the mean normal to fracture set #3 and the normal vector to the \(A_{b}\)-plane for fracture sets #1 and #2. If the dip direction, \(\alpha_{d}\), and dip angle, \(\beta_{d}\), of a fracture set is known, the trend and plunge of its normal can be found from the following expressions:  
\[\alpha_{n}=\alpha_{d}\pm 180\text{ with }0\leq\alpha_{d}\leq 360 \tag{17}\]  
\[n=90-\beta_{d}\text{ with }0\leq\beta_{d}\leq 90 \tag{18}\]  
Accordingly, the normal vector to a plane (i.e., fracture set) is determined from the trend, \(\alpha_{n}\), and the plunge, \(\beta_{n}\). \[n=\left[ \begin{array}{c}n_{x}\\ n_{y}\\ n_{z}\end{array} \right]=\left[ \begin{array}{c}\cos\alpha_{n}\cos\beta_{n} \\ \sin\alpha_{n}\cos\beta_{n}\\ \sin\beta_{n}\end{array} \right] \tag{19}\]  
The orientation of the \(A_{b}\)-plane is clearly defined by the normal vector of fracture set #1 (\(n_{set1}\)) and fracture set #2 (\(n_{set2}\)). Furthermore, the normal vector, \(n_{Ab}\), to the \(A_{b}\)-plane is calculated by the cross product of the normal vectors of fracture sets #1 and #2. \[n_{Ab}=n_{set1}\times n_{set2} \tag{20}\]  
The angle, \(\gamma_{3D}\), between the normal vector of fracture set #3 and the normal vector to the \(A_{b}\)-plane is determined by the dot product:  
\[\cos\gamma_{3D}=\frac{n_{AB}\cdot n_{set3}}{|n_{AB}|\cdot|n_{set3}|} \tag{21}\]  
The limitations of this approach are manifold and comprise that fracture set #3 is fully persistent and that the 3D shape of the fractures, i.e., polygonal or circular, are not taken into account. Furthermore, scanline mapping or window sampling based on laser scanning or photogrammetric methods adds additional biases in the data set. These biases comprise that the observed fracture dimensions do not necessarily represent the actual fracture dimensions, given that they are locally obscured. The intact rock bridge failure that can be considered for fracture sets #1 and #2 cannot be considered for fracture set #3. The occurrence of more than three fracture sets cannot be considered as well. Besides that, the difference between the trace length in 2D and the areal extent of the fractures in 3D must be considered. For example, three orthogonal fracture sets with circular-shaped fractures, trace lengths and frequencies of 1 m and 1 m\({}^{-1}\), respectively, can form a closed block area of 1 m\({}^{2}\) but cannot form a block volume of 1 m\({}^{3}\). Consequently, the shape of the fractures must be known to be able to determine the IBSD in 3D.