## 3 Results and Discussion  
### Tortuosity Distribution  
The tortuosity was evaluated as a key parameter for modeling fluid flow in carbonate and sandstone rocks, utilizing 1963 digital rock samples. Tortuosity was computed using three different approaches: Streamline tortuosity derived from streamlined paths obtained through LBM simulations and used as the reference, geometric tortuosity calculated using [PERSON]'s algorithm, and two empirical models based on macroscopic geometric parameters. The statistical analysis of tortuosity, as summarized in Figure 5, reveals the distinct behaviors of sandstone and carbonate rocks across Dijkstra, streamline, and empirical tortuosity measures. Distributions were fitted using the best-matching probability functions, gamma, log-normal, and normal, selected based on visual inspection and goodness-of-fit statistics (e.g., skewness, kurtosis, and standard deviation). The Dijkstra tortuosity of sandstone exhibited a gamma distribution with a mean of 1.50 and a low standard deviation of 0.08. The distribution shows near symmetry, with skewness of 0.22 and slight platykurtosis (kurtosis \(=-0.16\)), indicating a relatively homogeneous pore structure. The streamline tortuosity, obtained using the LBM approach, also follows a gamma distribution but with a mean value of 2.22 and a similarly narrow standard deviation of 0.11. Its skewness (0.17) and kurtosis (\(-0.04\)) suggest mild asymmetry but overall statistical normality. The empirical methods yielded more varied results. Tortuosity values calculated using Equation (13) conformed to a log-normal distribution (mean = 1.93, std = 0.22), showing slight positive skewness (0.14) and low kurtosis (0.20), while Equation (17) obtained a significantly higher mean tortuosity of 3.20 (std = 0.28), as well as log-normal distribution with marginal skewness (0.15) and kurtosis (0.17). These differences indicate that empirical estimators may over- or under-predict tortuosity depending on their sensitivity to pore geometry. Figure 5: Tortuosity distribution in (**a**) sandstone, (**b**) carbonate, and (**c**) mixed using the Geometric Tortuosity (Dijkstra), Flow Tortuosity (Streamline), and Empirical Equation (Equations (13) and (17)) methods. Across all tortuosity measures, Dijkstra tortuosity follows a log-normal distribution with a mean of 1.69 and a relatively high standard deviation of 0.24. Its strong positive skewness (1.84) and high kurtosis (6.38) reflect a long-tailed distribution indicative of significant variability in pore path lengths caused by irregular carbonate microstructures. Flow tortuosity modeled via gamma distribution has a mean value of 1.86 with higher variability (std = 0.43), moderate skewness (0.25), and near-normal kurtosis (0.17), suggesting a broader distribution of flow path lengths compared to sandstone. For empirical models, Equation (17) fitted a log-normal distribution with a mean of 2.59 (std = 0.28), skewness of 0.53, and moderate leptokurtosis (kurtosis = 2.61), while Equation (13) conforms to a normal distribution (mean = 3.74, std = 0.48) with notable negative skewness (\(-\)0.96) and slightly elevated kurtosis (0.93). The presence of both left and right-skewed distributions for carbonates highlights the complexity of estimating tortuosity in highly heterogeneous pore systems. When analyzing the combined dataset across all rock types, the aggregate Dijkstra tortuosity follows a log-normal distribution with a mean of 1.60 and a standard deviation of 0.20. However, its high skewness (2.43) and kurtosis (10.32) point to a strongly right-skewed, heavy-tailed distribution, reinforcing the structural diversity present in carbonate-dominated samples. In contrast, streamline tortuosity fitted a normal distribution (mean = 2.03, std = 0.36), with negative skewness (\(-\)0.74) and moderate kurtosis (0.89), suggesting a central tendency with moderate variation, likely due to the averaging effect across lithologies. The empirical tortuosities exhibit diverse behaviors.