## 2 Materials and Methods  
A high-resolution 3D dataset derived from the Brazilian pre-salt carbonate formation, including SW01, SW02, SW03, SW04, SW05, SW06, SW07, SW08, SW09, and SW10, was utilized to investigate pore-scale fluid dynamics in heterogeneous porous media. This geological system is widely recognized for its intricate pore architecture and extreme internal heterogeneity, effectively representing complex carbonate reservoirs. The original microstructural data acquired through X-ray micro-CT consisted of volumes measuring 1000 \(\times\) 1000 \(\times\) 1000 pixels and are publicly accessible via the Digital Porous Media ([[https://www.doi.org/10.17612/xr50-s717](https://www.doi.org/10.17612/xr50-s717)]([https://www.doi.org/10.17612/xr50-s717](https://www.doi.org/10.17612/xr50-s717))) [35]. Ten sandstone types were included for comparison: Bandera Brown, Bandera Gray, Buff Berea Upper Gray, Berea Sister Gray, Berea, Kirby, Leopard, Bentheimer, and Castlegate. These sandstones are well established in pore-scale studies due to their relatively homogeneous and predictable pore networks, and the samples are publicly accessible via the Digital Porous Media ([[https://doi.org/10.17612/f4h1-w124](https://doi.org/10.17612/f4h1-w124)]([https://doi.org/10.17612/f4h1-w124](https://doi.org/10.17612/f4h1-w124))) [36]. Both sandstone and carbonate data were accessed on 30 October 2024. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Study** & **Input Data** & **Geometric** & **Flow** & **Empirical** & **Limitation** \\  & **Type** & **Tortuosity** & **Tortuosity** & **Models** & **Limitation** \\ \hline [PERSON] et al. [28] & Synthetic 2D & No & Yes & Yes & Synthetic input data; no \\ [PERSON] et al. & Sandstone \& & & & geometric tortuosity; 2D input \\
[2] & Carbonate 3D & Yes & No & Yes & No flow tortuosity \\ [PERSON] et al. [26] & Sandstone 2D & Yes & No & No & No flow \& empirical models; \\ [PERSON] et al. [33] & Sandstone 3D & No & Yes & No & No geometric tortuosity \& empirical models \\ Present Study & Sandstone \& & & & & Other additional parameters \\ Carbonate 3D & Yes & Yes & Yes & Yes & are limited to \(\varphi_{eff}\), \(Z_{max}\), \(Z_{avg}\) and \(L_{tmin}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of tortuosity studies. The methodological flowchart in Figure 1 begins with segmentation employing a thresholding technique to convert grayscale images into binary format, thereby enabling the distinction between the matrix phase and the pore space within the rock structure. This thresholding method operates by setting a defined intensity cutoff value: pixels exhibiting intensity values exceeding this threshold are classified as pore space and assigned a white value (1 or 255), whereas pixels with intensity values at or below the threshold are designated as matrix and assigned a black value (0). By leveraging the contrast in intensity between these two phases, this approach constitutes a critical preprocessing step in image analysis for the quantitative characterization of porous media microstructures. The grayscale voxel intensity function \(\mathrm{g}(x,y,z)\) was obtained from 3D micro-CT scans of rock samples. A fixed global threshold of 128, corresponding to the midpoint of the 8-bit grayscale range (0-255), was used to segment pore space (binary value 1) from the solid matrix (value 0). This threshold was selected to ensure consistent segmentation between pore and solid phases, particularly in datasets with bimodal grayscale histograms. The use of a fixed global threshold reduces subjectivity, simplifies the segmentation process, and minimizes the influence of image noise. This approach enhances reproducibility and facilitates direct comparison across multiple samples with varying characteristics. Each volume was resampled from its original resolution (1000 \(\times\) 1000 \(\times\) 1000 voxels) to 256 \(\times\) 256 \(\times\) 256 to standardize input for simulation, preserving essential pore-scale features while reducing computational cost.