## 3 Results and Discussion
### tortuosity-Minimum Throat Length Relationship  
Figures 13 and 14 present the relationship between tortuosity and minimum throat length for sandstone and carbonate rocks, respectively. The tortuosity values of sandstone samples analyzed using Equation (17) decrease as the minimum throat length increases. This trend is particularly evident in both the streamline and Dijkstra methods, where the tortuosity values remain relatively stable and approximately linear, indicating a consistent flow path complexity relative to pore constrictions. This behavior suggests that in sandstone formations, larger minimum throat lengths correspond to simpler, less tortuous flow paths, consistent with the expected pore geometry where wider throats facilitate more direct fluid transport. The observed linearity in tortuosity for both the streamline and Dijkstra methods further supports their robustness in capturing the geometric constraints imposed by throat sizes in porous sandstone. The best fitting performance was obtained using Equation (17), with an average R\({}^{2}\) of 0.328, while the lowest was found in the Dijkstra method, with an average R\({}^{2}\) of 0.022. For the Dijkstra method, the lowest R\({}^{2}\) was observed in Bandera Gray (0.001), and the highest in Berea (0.130), whereas for the streamline method, the lowest R\({}^{2}\) values were recorded in Bandera Gray and Kirby, while the highest was found in Castlegate (0.090). For Equation (17), the lowest R\({}^{2}\) was observed in Bentheimer (0.196), and the highest in Buff Berea (0.553). Conversely, carbonate rocks (Figure 14) exhibited an inverse relationship, where higher tortuosity values tend to correspond with smaller minimum throat lengths, suggesting that narrow pore throats significantly contribute to the increased complexity of flow paths. Such behavior likely results from the inherently heterogeneous and anisotropic pore structures characteristic of carbonate formations, where restricted flow through smaller throats forces fluids to navigate more winding and tortuous pathways. Consequently, tortuosity increases as minimum throat size decreases, reflecting greater flow path complexity and resistance. The best fitting performance was observed with Equation (17), yielding an average R\({}^{2}\) of 0.554, while the lowest was obtained using the Dijkstra method, with an average R\({}^{2}\) of 0.041. For Dijkstra, the lowest R\({}^{2}\) was recorded in sample SW08, and the highest in SW04 (0.221), with the R\({}^{2}\) values ranging from 0.004 (SW08) to 0.289 (SW09). For Equation (17), the highest R\({}^{2}\) was found in SW01, and the lowest in SW03 (0.030). Figure 14: tortuosity-minimum throat length relationship of carbonate samples (**a**) SW01, (**b**) SW02, (**c**) SW03, (**d**) SW04, (**e**) SW05, (**f**) SW06, (**g**) SW07, (**h**) SW08, (i) SW09, and (j) SW10. Figure 13: tortuosity-minimum throat length relationship of sandstone samples (**a**) Bandera Brown, (**b**) Bandera Gray, (**c**) Buff Berea, (**d**) Bentheimer, (**e**) Berea, (**f**) Berea Sister Gray, (**g**) Berea Upper Gray, (**h**) Castlegate, (**i**) Kirby, and (**j**) Leopard. However, notable deviations from this general trend were observed in the streamline-derived tortuosity for samples SW05 and SW09, as well as in the Dijkstra-based tortuosity for sample SW07, where an increase in tortuosity coincides with an increase in minimum throat length. This behavior may be attributed to distinct pore-scale features or heterogeneities present in these specific samples, such as poorly connected throats or nonuniform throat size distributions. Such characteristics can result in more tortuous flow paths despite the presence of larger minimum throat dimensions, underscoring the complex and non-linear relationship between geometric parameters and effective flow tortuosity in heterogeneous carbonate rocks.