## 3 Results and Discussion
### Model Performance  
Five CNN models were trained to estimate the fracture physical parameters' permeability, surface roughness, and mean aperture using a dataset of 3878 training data, and they were then tested on 970 testing data. The model performance in estimating permeability is shown in Table 5 and Figure 11, which plots the predicted permeability against the permeability from the LBM simulation. \begin{table}
\begin{tabular}{c c} \hline \hline
**Models** & **Time to Train** \\ \hline Xception & 2 h 13 min \\ DenseNet201 & 1 h 56 min \\ VGG16 & 2 h 10 min \\ CNN-1 & 20 min \\ CNN-2 & 42 min \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Time to train the models. \begin{table}
\begin{tabular}{c c} \hline \hline
**MAPE (\%)** & **Interpretation** \\ \hline \(<\)10 & Highly accurate \\
10–20 & Good forecast \\
20–50 & Reasonable forecast \\ \(>\)50 & Inaccurate forecast \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Prediction accuracy based on MAPE [40]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **R\({}^{2}\)** & **MAPE** \\ \hline Xception & 0.839 & 20.966\% \\ DenseNet201 & 0.948 & 14.171\% \\ VGG16 & 0.938 & 14.507\% \\ CNN-1 & 0.964 & 14.146\% \\ CNN-2 & 0.989 & 8.597\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Model performance in estimating permeability. CNN-2 is the most accurate model with the smallest error (R\({}^{2}\) value of 0.989 and MAPE of 8.596%), followed by CNN-1, DenseNet201, and VGG16. The model with the largest error was Xception. The performance of each model in estimating the surface roughness is presented in Table 6 and Figure 12, which shows that CNN-2 has the smallest error compared to the other models with an R\({}^{2}\) value of 0.962 and MAPE of 1.470%. Therefore, it is highly accurate in estimating surface roughness. The second-best performing model was VGG16, followed by DenseNet201 and Xception, with CNN-1 having the largest error. The performance of each model in estimating the mean aperture is presented in Table 7 and Figure 13, showing that CNN-2 has the smallest error of all the models with an R\({}^{2}\) value of 0.997 and MAPE of 0.941%, indicating that its prediction is highly accurate. The second-best performing model is CNN-1, followed by the DenseNet201 and VGG16. Xception had the largest error of all models that were evaluated. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **R\({}^{2}\)** & **MAPE** \\ \hline Xception & 0.899 & 3.087\% \\ DenseNet201 & 0.909 & 2.916\% \\ VGG16 & 0.922 & 2.585\% \\ CNN-1 & 0.889 & 3.304\% \\ CNN-2 & 0.962 & 1.470\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Model performance in estimating surface roughness. Figure 11: Scatter plot of the predicted permeability versus the actual permeability of the (**a**) Xception, (**b**) DenseNet201, (**c**) VGG16, (**d**) CNN-1, and (**e**) CNN-2 models. The models performed best in estimating the mean aperture, possibly because the mean aperture can be observed more straightforwardly than other parameters; thus, it is easier to predict. Five CNN models, consisting of CNN-1, CNN-2, and pre-trained models VGG16, DenseNet201, and Xception, were used to estimate three physical parameters of rocks: permeability, surface roughness, and average aperture width of a three-dimensional fracture. The evaluation metrics results show that the models can predict the physical parameter values accurately. Based on Table 3, the mean aperture and surface roughness parameters can be predicted with small errors, where R\({}^{2}\) values are close to 1 and MAPE values are small. For permeability, only the CNN-2 model has very accurate predictions, with the other four models having MAPE values greater than 10%, indicating less accurate predictions. Based on R\({}^{2}\) and MAPE metrics, the results show that CNN-2 outperforms the other CNN models. Compared to CNN-1, CNN-2 has a more complex architecture that makes use of the RELU activation function and batch normalization.