## 2 Materials and Methods  
### Materials  
The 2-D sandstone rock data from CT scans were downloaded from [[https://www.digitalrocksportal.org/projects](https://www.digitalrocksportal.org/projects)]([https://www.digitalrocksportal.org/projects](https://www.digitalrocksportal.org/projects)), accessed on 20 August 2024. Several types of sandstone, including Bandera, Berea, Bentheimer, Castlegate, and Leopard, are predominantly composed of quartz, with variations in feldspar content, clay minerals, and lithic fragments. Bandera and Bentheimer sandstones exhibit low anisotropy and a low degree of alteration, characterized by exceptionally clean quartz grains. Berea sandstone shows moderate anisotropy and low-to-moderate alteration, while Castlegate sandstone tends to be anisotropic with a moderate degree of alteration. Leopard sandstone is distinguished by its characteristic iron oxide staining and visual anisotropy, although its mechanical properties and fluid flow characteristics are generally isotropic. The rock data samples were originally 3-D, and were segmented and sliced into 2-D because sandstone rocks have homogeneous porosity so the porosity distribution is the same for each rock segment. The eight selected rocks were categorized based on their porosity and grain shapes. Digital Image Analysis was used to calculate the total number of rock pores against the total rock volume. The porosity range of the rock samples used was in the range of 0.21-0.29. This range corresponds to the porosity range of sandstone reservoir rocks. Figure 1 Illustrate the microstructural differences among the samples, with black areas representing the pore spaces and white areas representing the solid matrix. The visual comparison highlights the variation in porosity and pore connectivity across different sandstone types, which is essential for understanding their fluid flow properties. This set of images serves as the basis for analyzingthe interrelations among effective porosity, permeability, average throat size, and tortuosity in our 2-D study.  
The porous rock parameters were obtained in several stages, starting from data collection to the visualization of the data obtained, as shown in Figure 2.  
Figure 1: 2-D rock samples of sandstones: (**a**) Berea, (**b**) Bentheimer, (**c**) Leopard, (**d**) Bandera gray, (**e**) Castlegate, (**f**) Buff Berea, (**g**) Berea upper gray, and (**h**) Berea sister gray.  
Figure 2: Research methodology.  
Figure 2 illustrates the research methodology employed in this study, starting with the collection of 3-D X-ray micro-computed tomography sandstone data. The data undergo segmentation to identify different phases, resulting in a \(1000\times 1000\times 1000\) pixels of 3-D rock data segmentation. These 3-D data are then resampled and sliced to produce \(256\times 256\) pixel of 2-D binarized images for simulation input. The Lattice Boltzmann method (LBM) is used to simulate fluid velocity and pressure profiles, which are then segmented to produce average throat size and effective porosity. Permeability is calculated using the velocity and pressure data through [PERSON]'s equation. Additionally, the Kozeny-Carman permeability is obtained from image processing techniques, utilizing tortuosity, specific surface area, and absolute porosity. The data visualization and analysis stage includes comparisons and correlations among the various calculated parameters, culminating in a comprehensive understanding of fluid flow properties in the sandstone samples.