## 2 Discrete Fracture Network Models  
DFNs are synthetic stochastic models of naturally fractured rock masses that can be generated using data collected from various sources (e.g., 1D boreholes and 2D rock exposures). High-quality drill coring can obtain relatively undisturbed rock core samples. In contrast, observations on exposed rock faces at or near the project site have the advantage of allowing direct measurements of discontinuity orientation, spacing, and fracture length.  
DFN modeling has a broad range of applications in geomechanics due to its ability to accurately represent the complex geometry and connectivity of fracture networks within rock masses. Some key applications include slope stability analysis, rockburst and seismicity prediction in mining engineering, and tunnel design and stability and dam foundation analysis in civil engineering.  
[PERSON] et al. [15] introduced a quantitative classification system incorporating a DFN approach to better capture rock mass scale and anisotropic effects and improve the reflection of the natural fracture network's connectivity. [PERSON] et al. [16] combined DFN and machine learning to improve the estimation of GSI. [PERSON] al. [17] used DFN models to estimate volumetric fracture intensity from borehole data. [PERSON] et al. [18] utilized a rectangular joint model to enhance the precision of simulating fracture geometries and their spatial distribution. This improved the accuracy of predicting rock mass behavior under various conditions. [PERSON] et al. [19] developed a hybrid modeling method to capture the complex behavior of fluid flow and solute transport in fractured rock systems generated using DFNs in applications such as groundwater contamination. Other examples include the stability design of tunnels [20], open stope stability [21], and analysis of rock mass strength and fragmentation [22; 23].  
The typical process involved in the generation of a DFN model requires the definition of four primary fracture properties (Table 2), including (i) fracture orientation, (ii) fracture length, (iii) fracture terminations, and (iv) fracture intensity. The reader should refer to [PERSON] et al. [24] for a detailed discussion of the methodology required to generate calibrated DFN models.  
\begin{table}
\begin{tabular}{c c c} \hline
**Fracture Data** & **Source** & **DFN Input Data** \\ \hline Orientation & Boreholes, outcrops, tunnels & Orientation of fractures for every fracture set \\ \hline Length & Tunnels, outcrops, lineaments & Fracture radius distribution \\ \hline Terminations & Tunnels, outcrops, lineaments & Choice of the model hierarchy of the sets \\ \hline Intensity & Boreholes, scanlines (P\({}_{10}\)), outcrops (P\({}_{21}\)) & Fracture intensity (P\({}_{10}\) or P\({}_{32}\)) \\ \hline \end{tabular}
\end{table}
Table 2: Fracture data and derived input data for a DFN model from digitally and conventionally mapped data (modified from [25]).  
Typically, in the DFN community, fracture intensity is expressed concerning a unified system of fracture intensity measures that provide an easy framework to move between differing scales and dimensions [26]. Fracture intensity is referred to as \(P_{ij}\) intensity, where the subscript \(i\) refers to the dimensions of the sample and subscript \(j\) refers to the dimensions of measurement. In DFN modeling, fractures are represented by circular discs or polygons with \(n\) sides. Fracture length is a critical DFN input and a key parameter for sensitivity studies, as it significantly influences block size and fracture connectivity [27].