## 2 Materials and Methods
### Fracture Geometry Dataset  
The dataset consisted of synthetic three-dimensional singular fracture geometry. A singular fracture refers to an isolated fracture that is not interconnected to other fractures. This fracture was modelled by a three-dimensional channel formed between two parallel surface plates. This fracture is characterised by its geometric form with two values, the width of the aperture and its internal surface roughness of the fracture [4; 25]. The rough internal surface of the fracture could result in a distribution of aperture widths rather than a uniform aperture width. The roughness is frequently modelled as a stochastic function of a self-affine fractal. In this study, the surface roughness was described using the Hurst exponent, a parameter used to measure roughness by assessing the tendency of a time series or signal to follow a pattern. The height of the fracture surface is represented as a time series to describe its roughness and the Hurst exponent is related to the fractal dimension according to Equation (1) [6; 26]:  
\[D=3-H \tag{1}\]  
where \(D\) is the fractal dimension and \(H\) is the Hurst exponent. Roughness profiles can be generated using the fBm method, which has a certain Hurst exponent similar to the roughness of real fracture surfaces in rocks [25]. The fractures were generated using the fBm method with Hurst exponent values ranging from 0.5 to 0.75, reflecting those found in natural rock fractures [5] and the mean aperture width (10 to 30 voxels) [27]. The fracture is created by two identical rough surfaces generated using the fBm method that with separated with a specific aperture. The surface roughness and mean aperture values in the dataset are discrete due to data generation process. Specific Hurst exponent values (0.5, 0.55, 0.6, 0.65, 0.7, and 0.75) were used to define surface roughness, while mean aperture values were selected at fixed intervals. Despite that, the machine learning models would predict this as continuous regression targets. The dimensions of the generated fractures were \(200\times 200\times 200\) voxels. Each voxels was represented as a lattice unit (lu) that was used in the fluid simulation. This lattice unit also served as the unit for the mean aperture because of the difference in the resolution of different fractures used in this study. The geometries were converted to binary arrays, with 0 representing the fracture and 1 representing solids. Figure 1: The study flow of CNN modelling to estimate fracture physical parameters. Additional data used were real rock fracture samples for model validation from andesite rock in the geothermal region in Kyushu, Japan [28]. These samples were digital models of a fractured rock mass, mapped using a laser profilometer to obtain its fracture surface topography. The samples have dimensions of 200 \(\times\) 1800 pixels with a resolution of 0.1 mm, divided into upper and lower surface plates. Two mapped surface plates were paired and separated with an aperture to create a three-dimensional fracture model. The roughness has a fractal dimension of 2.4 consistent with typical rock surface roughness [29]. Both surface plates were combined into multiple three-dimensional binary geometries with dimensions of 200 \(\times\) 1800 \(\times\) 200 voxels, varying aperture width and a Hurst exponent value of 0.6. These samples were then divided into subsamples measuring 200 \(\times\) 200 \(\times\) 200 voxels to match the dimensions of the generated fractures (Figure 2). Other data that were used comprised hydraulic fracture samples from a shale oil reservoir in China [30]. The samples were obtained from original three-dimensional greyscale computed tomography images. The samples used consisted of the main fracture, aligned with its maximum effective subsurface stress direction.