## 3 Results and Discussion
### Model Testing Using Real Fractures
DenseNet201 and CNN-1 performed the worst, with significant scatter in their predictions and low R\({}^{2}\) values (0.195 and 0.154, respectively). This drop in performance could be attributed to these fractures having a more complex and non-uniform geometry, differing from the synthetic data used for training. The performance of various models in predicting the Hurst exponent of shale samples is depicted in Figure 18, with detailed metrics provided in Table 12. DenseNet201 was the most accurate model, with minimal deviation from the true values and the highest R\({}^{2}\) value, indicating strong predictive performance. CNN-2 followed as the second-best model, performing well but showing slightly larger prediction errors than DenseNet201. VGG16 performed reasonably, while Xception had a broader spread of points, signalling larger prediction errors. CNN-1 had the weakest performance, with points furthest from the diagonal and the lowest R\({}^{2}\) value, indicating less accurate predictions compared to the other models. Similarly to the andesite fracture dataset, the models performed poorly in capturing the variations in surface roughness from shale data, which led to relatively low R\({}^{2}\) values. This can be attributed to the small range of values in the shale fracture compared to the broader range in the training dataset. However, the models performed reasonably well when evaluated using the MAPE measure, with DenseNet201 having an MAPE of 4.493%. This result shows that the roughness of shale fractures is more similar to the training dataset, leading to more accurate predictions by the model when compared to the andesite fracture. Figure 16: Scatter plot of the predicted mean aperture versus the actual mean aperture from the an-desite fracture dataset: (**a**) Xception, (**b**) DenseNet201, (**c**) VGG16, (**d**) CNN-1, and (**e**) CNN-2 models. Figure 19 and Table 13 showcase the performance of different CNN models in predicting the mean aperture of shale samples. Among all models, VGG16 demonstrated the best performance, with its data points being tightly clustered around the diagonal and obtaining the highest R\({}^{2}\) value, having the most accurate predictions. Following VGG16, Xception showed strong performance, with a narrow spread of points and competitive accuracy. DenseNet201 was next, exhibiting slightly larger deviations but still delivering reliable results. CNN-1 performed reasonably well but had a wider spread of data points, indicating more variability in predictions compared to VGG16, Xception, and DenseNet201. CNN-2 had the weakest performance, with points furthest from the diagonal and the lowest R\({}^{2}\) value, suggesting less accurate predictions. \begin{table}
\begin{tabular}{c c c} \hline
**Model** & **R\({}^{2}\)** & **MAPE** \\ \hline Xception & 0.473 & 51.414\% \\ DenseNet201 & 0.195 & 42.712\% \\ VGG16 & 0.393 & 38.881\% \\ CNN-1 & 0.154 & 37.581\% \\ CNN-2 & 0.258 & 35.666\% \\ \hline \end{tabular}
\end{table} 
Table 11: Model performance in estimating the permeability of shale oil reservoir fracture. Figure 17: Scatter plot of the predicted permeability versus the actual permeability using a shale oil reservoir fracture dataset: (**a**) Xception, (**b**) DenseNet201, (**c**) VGG16, (**d**) CNN-1, and (**e**) CNN-2 models. The evaluation of transfer learning models for predicting permeability, the Hurst exponent and surface roughness of andesite and shale samples has revealed key insights. This analysis suggests that models struggled with the complexity inherent in real shale fracture geometries, which differ significantly from the synthetic data used in training. These findings emphasize the critical need to incorporate real fracture data during the training process. Additionally, exploring hybrid models that integrate physics-based methodologies with data-driven approaches could further enhance the predictive accuracy and generalizability of machine learning models for characterizing shale surface roughness and the Hurst exponent.