## 2 Materials and Methods
### CNN Models  
The CNN is designed to process image or spatial datasets using three main components: convolutional layers, pooling layers, and fully connected layers. Convolutional layers were applying convolution operation to extract features using a filter or kernel. This kernel is matrix that could detect simple patterns or features such as edges or textures. This convolution operation generates a feature map that highlights those features to be processed by another layer. Pooling layers reduce the spatial dimension of the data while retaining most of their critical information. This is typically carried out using the operation of max pooling, where the layers take the maximum value from a small region of the feature map. These pooling layers also reduce the computational load and make the model more invariant to small distortions in the input data. Fully connected layers are artificial neural  
Figure 7: The process of creating a composite image as the input data. The green slice represents the \(xy\) plane, and the blue slice represents the \(zy\) plane. These slices are extracted, separated, and stacked into a composite image. networks that learn the relationship between the input and the output label, enabling the model to make predictions based on the features extracted from convolutional and pooling layers. Three pre-trained models using transfer learning and two custom-designed CNN models were used to estimate physical parameters of three-dimensional fractures. Each model's architecture was independent and trained for each of the three prediction targets: permeability, surface roughness, and mean aperture. This was completed because each physical parameter has unique characteristics and dependencies, which may require distinct feature representations. Training independent models allows each model to specialize in learning features relevant to its specific target without interference from unrelated features of other parameters. The three pre-trained models were available from the Keras library and included DenseNet201 [33], VGG16 [34], and Xception [35], which were previously trained using the ImageNet dataset, a large-scale image dataset containing 14 million labelled images across more than 20,000 categories. These models have been used in several studies for predicting physical parameters using a CNN for fractured and porous media [21, 36, 37]. The three pre-trained models were adapted with a modified final layer to change from networks that perform classification into regression functions to predict physical parameters. This enables us to make use of the initial layers of the network that perform generic feature extraction, while the final layers act as classifiers tailored to specific prediction tasks [38]. The new final layer was implemented with a global maxpooling layer, a flattening layer, a fully connected layer with 512 hidden units with ReLU activation, and a fully connected sigmoid layer for predicting physical parameters. The full description of the model architecture can be seen in Appendix A. An illustration of the modifications made to the pre-trained models is shown in Figure 8.  
In this research, two custom CNN models, CNN-1 and CNN-2, were designed and implemented to predict the physical properties of three-dimensional fractures. CNN-1 consists of three convolutional layers, each followed by a pooling layer to reduce feature dimensions connected to a fully connected layer and used linear activation functions (Figure 9). Figure 8: Pre-trained model modification. Figure 9: Illustration of the CNN-1 model. CNN-2 is a modified version of CNN-1, incorporating batch normalisation layers to improve its training speed and generalisation, and ReLU activation functions to introduce non-linearity in the model, thus improving the model performance for learning more complex patterns (Figure 10). Model training was performed using a dataset of composite images created from three-dimensional fractures as input that was split into training (80%) and testing (20%) data.