## 3 Results and Discussion
### Tortuosity-Average Coordination Number Relationship  
Figures 11 and 12 illustrate the relationship between tortuosity and average coordination number \(Z_{agg}\) for sandstone and carbonate rocks, respectively. This plot was constructed because \(Z_{agg}\) is directly involved in one of the empirical equations used to estimate tortuosity. A higher \(Z_{agg}\) generally indicates better pore connectivity which may reduce the effective flow path length, thereby lowering tortuosity. Conversely, lower coordination numbers often correspond to more isolated or poorly connected pores, potentially increasing the tortuosity of the medium. A clear inverse correlation was observed in sandstones (Figure 11), particularly in the empirical tortuosity values derived from Equation (17). Overall, the best fit was achieved using Equation (17), with an average R\({}^{2}\) of 0.609, while the lowest was observed for the streamline method, at 0.148. For the Dijkstra method, the lowest R\({}^{2}\) occurred in Berea Sister Gray, and the highest in Castlegate (0.465). In comparison, the lowest R\({}^{2}\) was found in Buff Berea, and the highest in Berea Sister Gray (0.592) for the streamline-based tortuosity. The R\({}^{2}\) ranged from 0.159 (Berea) to 0.917 (Castl  
Figure 10: Tortuosity-porosity 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. As tortuosity increases, the average \(Z_{avg}\) decreases significantly, supporting the theoretical expectation that higher tortuosity--representing more convoluted flow paths--is associated with lower pore connectivity. This trend was also observed, though less distinctly, in both the streamline and Dijkstra-derived tortuosity values, which show a more gradual decline in \(Z_{avg}\) with increasing tortuosity. The consistency of this inverse relationship across methods, particularly evident in Equation (17), confirms the structural coherence of sandstone pore networks, where variations in local coordination are effectively captured by tortuosity metrics. We aim to validate the physical consistency of the empirical model by examining this relationship, further exploring how topological features of the pore structure influence fluid transport behavior. Figure 11: tortuosity-average coordination number relationship in 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. Figure 12: tortuosity-average coordination number relationship in carbonate samples (**a**) SW01, (**b**) SW02, (**c**) SW03, (**d**) SW04, (**e**) SW05, (**f**) SW06, (**g**) SW07, (**h**) SW08, (**i**) SW09, and (**j**) SW10. In contrast, carbonate rocks (Figure 12) exhibit a more complex and less predictable relationship. While the Dijkstra and Equation (17) tortuosity values demonstrate a marked inverse correlation with coordination number--similar to that observed in sandstones--the streamline-based tortuosity reveals a divergent pattern. Here, tortuosity and \(Z_{avg}\) appear to be positively correlated, indicating that samples with higher streamline tortuosity may also exhibit higher average connectivity. Overall, the best fitting performance was observed for the empirical method Equation (17), with an average Overall, the best fitting performance was observed for the empirical method Equation (17), with an average R\({}^{2}\) of 0.585, while the lowest was found in the streamline method, with an average R\({}^{2}\) of 0.205. For the Dijkstra method, the lowest R\({}^{2}\) was obtained for sample SW01 (0.016), and the highest for SW02 (0.616). In the streamline method, the lowest value occurred in SW03, while the highest was in SW08 (0.435). For the empirical model Equation (17), R\({}^{2}\) ranged from 0.092 in SW01 to 0.916 in SW10.