## 3 Results and Discussion
### Are  
Absolute Relative Error (ARE) as the primary metric for performance evaluation, representing the proportional deviation of estimated tortuosity from streamline-based reference values. This formulation ensures scale-independence and enables robust inter-method comparisons. Figure 15a presents the ARE for the sandstone samples, using streamline tortuosity obtained from streamline simulations as the ground truth for comparison.  
Equation (17) exhibits the highest predictive accuracy with an ARE of 13.91%, effectively capturing the dominant flow paths in sandstone, which typically feature a relatively homogeneous and well-connected pore structure. In contrast, the Dijkstra method and Equation (13) exhibit significantly larger discrepancies, with ARE values of 32.03% and 45.00%, respectively, suggesting that the assumptions or sensitivities underlying these models may not be well-suited to the simpler and more uniform sandstone geometry. Specifically, while the Dijkstra algorithm considers the shortest paths through the medium, it may overestimate or underestimate tortuosity in systems where flow is more uniformly distributed across the pore space. Meanwhile, Equation (13) relies solely on porosity, which likely limits its ability to capture the actual pore structure and connectivity. These findings confirm that Equation (17) offers a robust and efficient estimation of streamline tortuosity for relatively homogeneous porous media such as sandstone.  
Figure 15b shows that Equation (13) yields a very high ARE of 110.82% for carbonate rocks, indicating a substantial deviation from the streamline-derived tortuosity and a fundamental misrepresentation of the actual flow pathways. This underperformance likely stems from the model's limited ability to account for the intricate and highly heterogeneous pore systems typical of carbonate rocks.  
In contrast, Equation (17) and the Dijkstra algorithm exhibit superior performance, with AREs of 48.83% and 23.09%, respectively, attributed to their better adaptability to complex pore networks, where connectivity, tortuous paths, and dead-end pores are prevalent. [PERSON]'s algorithm traces the most efficient path from inlet to outlet within a voxelized pore space; thus, it is inherently sensitive to variations in pore geometry and spatial heterogeneity, making it more suitable for modeling flow in such systems.  
These results underline the importance of model selection based on lithological characteristics and pore architecture complexity. Taken together, the contrasting performances of the models across sandstone and carbonate samples indicate that no single model provides universally accurate tortuosity predictions across different rock types. While Equation (13) demonstrates satisfactory performance in systems characterized by relatively regular and continuous pore networks, such as high-porosity sandstones, its application to more complex and heterogeneous media, such as carbonate rocks, results in notable discrepancies. Its underlying assumptions rely on simplified geometric representations and thus are less applicable in formations with irregular throat distributions, isolated pore bodies, or anisotropic connectivity, thereby reducing its predictive accuracy in such environments.  
Figure 15: ARE of (**a**) sandstone and (**b**) carbonate samples.  
The findings highlight the necessity for adaptive modeling approaches that can accommodate structural variability. Analytical or semi-empirical models may suffice for systems with straightforward pore geometry, but path-based or topology-aware methods, such as [PERSON]'s algorithm, demonstrate significant advantages in rocks with complex microstructures.