## 3 Results and Discussion
### Comparison of tortuosity Studies  
A comparative analysis with previous experimental and numerical investigations on similar porous materials was conducted to assess the validity of the tortuosity values obtained in this study (Table 3). [PERSON] et al. investigated 2D synthetic porous media using the lattice-gas automata method (FHP-III model) to examine the empirical relationship between tortuosity and porosity [45]. Analytical methods were found to be inadequate for capturing the complexity of fluid transport in such systems, prompting the use of direct numerical simulations. Their results indicated that the flow tortuosity varied between 1.1 and 1.7 within a porosity range of 0.4 to 0.9. There was a correlation between increasing tortuosity and decreasing porosity, highlighting the nontrivial nature of pore-scale connectivity in controlling \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Study** & **Input Data** & **Absolute** & **Effective** & **Geometric** & **Flow** & **Empirical Models** \\  & **Type** & **Porosity** & **Porosity** & **Tortuosity** & **Tortuosity** & **Empirical Models** \\ \hline [PERSON] et al. (1998) & Synthetic 2D & 0.4–0.9 & - & - & 1.1–1.7 & - \\ [PERSON] et al. (2024) [2] & Sandstone \& 0.07–0.37 & - & 1.00–1.80 & - & 1.00–7.00 \\ [PERSON] et al. (2024) [26] & Sandstone 2D & 0.12–0.40 & 0.06–0.325 & 1.38–2.2 & - & - \\ [PERSON] et al. (2025) [33] & Sandstone 3D & 0.10–0.43 & 0.05–0.35 & - & 2.25–3.75 & - \\  & Sandstone 3D & 0.11–0.29 & 0.08–0.22 & 1.29–1.75 & 1.18–2.61 & 2.47–4.42 (Equation (13)) \\ Present Study & \multirow{2}{*}{Carbonate 3D} & \multirow{2}{*}{0.11–0.34} & \multirow{2}{*}{0.03–0.30} & \multirow{2}{*}{1.00–3.19} & \multirow{2}{*}{1.00–3.68} & \multirow{2}{*}{1.59–3.47} (Equation (17)) \\  & & & & & & \\ \hline \hline \end{tabular} Note: [PERSON] et al. [45] and [PERSON] et al. [2] report absolute porosity values only; effective porosity was not provided in their studies. The present study distinguishes between sandstone (first value) and carbonate (second value) for all reported ranges.  
\end{table} 
Table 3: Comparison of tortuosity studies across various porous media and methods. Porosity values are explicitly categorized as either absolute or effective porosity, where reported. Flow tortuosity values are based on direct fluid simulations. Dijkstra tortuosity values are typically obtained from shortest path algorithms. transport behavior. [PERSON] et al. reported geometric tortuosity values between 1.00 and 1.80 for sandstone and carbonate samples within a porosity range of 0.07 to 0.37, whereas the empirical tortuosity values derived from correlation-based models showed a broader distribution from 1.00 to 7.00. This suggests that while empirical models are highly sensitive to variations in pore structure, they may produce significant deviations if not calibrated carefully to internal geometrical features. A recent study by [PERSON] et al. [26] analyzed 2D representations of sandstone and found geometric tortuosity ranging from 1.38 to 2.20 across effective porosities of 0.06 to 0.325. In a follow-up study conducted in 2025, the same group extended their investigation to 3D models (128 \(\times\) 128 \(\times\) 128 voxels) and reported flow tortuosity values ranging from 2.25 to 3.75 for effective porosities between 0.05 and 0.35. These findings support theoretical expectations, as the transition from two to three dimensions typically leads to greater complexity in flow paths due to enhanced connectivity and geometric constraints. In the present study, the tortuosity values observed for 3D sandstone samples with effective porosity from 0.08 to 0.22 were in good agreement with these prior reports: geometric tortuosity ranged from 1.29 to 1.75, flow tortuosity from 1.18 to 2.61, and empirical tortuosity estimates were between 2.47 to 4.42 for Equation (13) and 1.18 to 2.59 for Equation (17).