## 4 Conclusions  
This study explored the use of convolutional neural network (CNN) models to estimate essential physical parameters--permeability, surface roughness, and mean aperture--of three-dimensional fractures. The dataset included 3878 synthetic fracture samples for training, 970 for testing, and 93 real fracture samples for evaluation. Among the five models tested--CNN-1, CNN-2, and three pre-trained models (DenseNet201, VGG16, and Xception)--CNN-2 demonstrated the highest performances when estimating synthetic fractures. CNN-2 achieving superior R\({}^{2}\) values and a lower MAPE compared to the other models. For permeability estimation, DenseNet201 had an R\({}^{2}\) of 0.980 and an MAPE of 8.597%, with CNN-1 being close behind but exhibiting slightly higher error rates. CNN-2 was the most accurate for surface roughness, showing an R\({}^{2}\) of 0.962 and an MAPE of 1.470%, followed by VGG16. In terms of the mean aperture, CNN-2 again led with an R\({}^{2}\) of 0.997 and an MAPE of 0.941%, outperforming other models. However, when tested on real fracture datasets, the models' performance dropped, indicating challenges in generalizing when applying synthetic data to real-world geometries. This decline was most noticeable for extreme values, where the accuracy of predictions suffered. CNN-1 and CNN-2, in particular, demonstrated the weakest generalization capabilities, suggesting the need for further improvements in their models' architectures. Pre-trained models, such as DenseNet201 and VGG16, demonstrated generalization capabilities when faced with real fracture data and outperformed CNN-1 and CNN-2.  
To address these limitations, future research should focus on training models with real fracture data to better capture the complexities of actual samples. Implementing data augmentation strategies to mimic the variability of real fractures and employing physics-informed CNN architectures that incorporate domain-specific knowledge could enhance predictive performance. These advancements would help bridge the gap between synthetic training data and real-world scenarios, enabling more accurate predictions and promoting wider use in geosciences and reservoir engineering.  
The authors' responsibilities were as follows: [PERSON] and I.A.D. were responsible for the concept of this study; [PERSON], [PERSON] and A.A. were responsible for the conduct of this study, samples, and clinical data collection; [PERSON] was responsible for writing and editing this manuscript; [PERSON], [PERSON], [PERSON] and I.A.D. were responsible for sample analysis and data analysis. All authors have read and approved the published version of the manuscript.  
This research was financially supported by the Ministry of Education, Culture, Research, and Technology, Directorate of Research, Technology, and Community Service, Directorate General of Higher Education, Research, and Technology under the Fundamental Research and Master's Thesis Research Scheme (Contract No. 074/E5/PG.02.00.PL/2024). Additional funding was provided by Hibah Riset Universitats Padjadjaran under the Riset Kompetensi Dosen Unpad scheme (Contract No. 1948/UN6.3.1/PT.00/2024).  
Not applicable.  
All authors of this work concur with this submission. The data presented have not been previously reported, nor are they under consideration for publication elsewhere.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
The authors acknowledge the Department of Geophysics Universitas Padjadjaran supercomputing resources \"RockExplorer\", which were made available for conducting the research reported in this paper.  
The authors declare no conflicts of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l l} CNN & Convolutional neural network \\ MAPE & Mean absolute percentage error \\ fBm & Fractal Brownian motion \\ LBM & Lattice Boltzmann method \\ \end{tabular}