## 2 Materials and Methods
### Dataset Distribution  
The distributions of the fracture parameters are quite important for understanding the dataset structures. Figure 4 presents the histograms and box plots of the three primary features used for training data in this study.  
Figure 4 shows the distribution of dataset used for training the model. The histogram of permeability values shows a skewed distribution, with a higher frequency of samples in the lower permeability range (0-0.0002 mm\({}^{2}\)). The frequency decreases as permeability increases, with relatively fewer samples beyond 0.0008 mm\({}^{2}\). This distribution reflects the inherent variability of synthetic fractures, where lower permeability values are more common due to tighter fracture geometries. Despite this imbalance, the dataset still spans a wide range of permeability values, ensuring sufficient representation for model training. Surface roughness, represented by the Hurst exponent, exhibits a discrete distribution with samples concentrated at six specific values used on the data generation: 0.50, 0.55, 0.60, 0.65, 0.70, and 0.75. The uniformity of sample counts across these values demonstrates an equal representation of different roughness levels, ensuring no bias toward a particular roughness scale. The mean aperture values are distributed across fixed intervals, ranging from 10 to 30 lattice units (lu). The histogram shows an approximately uniform distribution, with similar sample counts across all intervals. This uniform representation provides balanced training data for predicting the mean aperture, ensuring the model learns effectively across the entire range of possible values.  
Figures 5 and 6 present the histograms and box plots of the three primary features from real rock fracture in this study.  
Figure 4: Histograms and box plots of the parameters in the training dataset: (**a**) permeability, (**b**) surface roughness, and (**c**) mean aperture.  
Figure 5: Histograms and box plots of the parameters in the andesite fracture dataset: (**a**) permeability, (**b**) surface roughness, and (**c**) mean aperture.  
Figure 6: Histograms and box plots of the parameters in the shale oil reservoir fracture dataset: (**a**) permeability, (**b**) surface roughness, and (**c**) mean aperture.  
Figure 5 (andesite) and Figure 6 (shale) reveal the distinct fracture properties of these two rock types. Andesite has significantly higher permeability, ranging from 0.0002 mm\({}^{2}\) to 0.001 mm\({}^{2}\), with most values falling between 0.0002 mm\({}^{2}\) and 0.0006 mm\({}^{2}\). In contrast, shale exhibits much lower permeability, predominantly between 0.0001 mm\({}^{2}\) and 0.0005 mm\({}^{2}\). The Hurst exponent for andesite, which clusters around 0.61-0.62, indicates a more organized and interconnected fracture network, while shale, with a lower Hurst exponent (0.58-0.60), has more random and less systematic fractures. Additionally, andesite has larger mean aperture sizes, ranging from 10 to 30 lu and peaking around 20 lu, with its aperture also having consistent patterns, whereas shale features have smaller apertures (12 to 28 lu) with greater variability. These findings suggest that andesite, with its higher permeability, larger apertures, and more structured fractures, is more favourable for fluid flow, whereas shale's lower permeability, smaller apertures, and less organized fractures make it a better barrier to fluid movement.