## 3 Results and Discussion
### Boxplot of tortuosity Values by Porosity Group (L1-L5)  
To evaluate the consistency of the relationship between porosity and tortuosity across different rock types, box plots in Figures 7 and 8 were employed to visualize the variation in tortuosity as a function of grouped porosity values. Porosity was discretized into five categorical groups, labeled L1 to L5, arranged from the lowest (L1) to the highest (L5) porosity interval, with each group representing 20% of the porosity distribution to balance statistical robustness and resolution, ensuring meaningful comparisons while covering the observed porosity spectrum. The five-bin grouping aligns with conventional quantile-based methods in porous media studies, enabling comparisons with prior research. Each boxplot displays tortuosity (\(y\)-axis) calculated by three methods: [PERSON]'s algorithm, streamline tortuosity, and an empirical model represented by Equation (17).  
Preliminary evaluations showed that the predictions significantly deviated from both simulation-based results and the observed empirical behavior. Equation (13), although widely cited, tends to under- or overestimate tortuosity under certain pore geometries, particularly in heterogeneous formations like carbonates; thus, it is less representative of actual tortuosity under the studied conditions and was therefore omitted from detailed interpretation. The selection of Dijkstra, streamline simulation, and Equation (17) ensures that structural and flow-related mechanisms are captured while maintaining a tractable  
Figure 6: tortuosity estimated values from Dijkstra, Equation (13) by [PERSON] et al. [30] and Equation (17) of the proposed percolation-based model are plotted against the streamline on the \(x\)-axis.  
and reliable empirical reference, providing a comprehensive framework for understanding tortuosity variations across lithologies and validating the robustness of porosity-torutosity trends in porous media.  
The \(x\)-axis categorizes samples by their porosity (L1-L5), revealing clear distinctions between sandstone and carbonate lithologies. Sandstone samples exhibit more homogeneous behavior, with relatively stable tortuosity values and consistent distributions across porosity groups. This homogeneity is evident in the tighter and more symmetric boxplot spreads, indicating uniform pore structures and flow pathways within sandstones. In contrast, carbonate samples display far greater variability in tortuosity values, especially for  
Figure 8: Tortuosity 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 7: Tortuosity 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.  
streamline tortuosity and Dijkstra tortuosity derived from simulations of actual rock structures. The widespread and irregular distribution of data points highlights the inherently heterogeneous pore geometry and complex flow networks typical of carbonates.  
Interestingly, tortuosity values estimated from the empirical model using Equation (17) tend to be more stable even for carbonates, suggesting that empirical correlations may smooth out microstructural heterogeneities that significantly affect simulation-based methods. These findings underscore the difference in pore-scale complexity between sandstones and carbonates and validate the use of porosity grouping as a useful framework for comparative analysis. The more homogeneous sandstone behavior contrasts with the heterogeneous carbonate response, providing important insight into how lithological variability influences tortuosity-porosity relationships and porous media flow physics.