## 3 Results and Discussion
### Limitations and Perspective  
The evaluation of tortuosity across diverse rock types--specifically sandstone and carbonate--demonstrates the crucial influence of pore structure heterogeneity on fluid flow behavior. Carbonate rocks, with their complex and irregular pore geometries, exhibit higher and more variable tortuosity than the relatively homogeneous sandstones. The streamline-based tortuosity from LBM simulations provides the most physically representative results among the four methods assessed. However, the Dijkstra-based geometric and empirical models, particularly Equation (17), offer a practical balance between accuracy and computational efficiency when microstructural parameters are well-integrated.  
While an inverse relationship between porosity and tortuosity was consistent, deviations in flow tortuosity highlight the limitations of porosity as a sole predictor of transport behavior. Comparisons with the literature confirm the reliability of the derived values and underline the importance of 3D modeling in capturing realistic flow dynamics. These findings emphasize the need for structure-aware, multi-parametric models for accurate tortuosity estimation in complex porous media, with important implications for reservoir characterization.  
This study, while comprehensive in scope, has several limitations that merit consideration. The dataset, although diverse, comprises a fixed number of digital samples comprising ten types of sandstone and carbonate rocks sourced from publicly available micro-CT repositories. While widely used in pore-scale research, these samples may not fully represent the broad variability found in natural formations across different depositional environments or diagenetic histories. Additionally, flow simulations are limited to single-phase, incompressible conditions using the Lattice Boltzmann Method. Realistic reservoir conditions involving multiphase flow, capillary effects, and wettability were not considered, which may restrict the broader applicability of the findings.  
The empirical models rely on geometric features, including effective porosity, coordination number, and minimum throat length, derived from image segmentation, which is sensitive to voxel resolution and thresholding. Although the original 10003 voxel datasets were downsampled to 2563 to reduce computational load, the representative elementary volume (REV) condition was maintained to preserve relevant pore-scale features and statistical validity.  
In this study, LBM simulations were conducted on 2563 voxel volumes over 30,000 time steps per sample, across 1963 rock samples. All simulations were performed on the RockExplorer supercomputing cluster at the Department of Geophysics, Universitas Padjadjaran, equipped with an Intel(r) Xeon(r) Gold 5118 CPU @ 2.30 GHz (48 cores) and 128 GB RAM. The simulations utilized MPI-based parallelization to efficiently distribute computational loads. Each simulation required approximately 1.5 to 2 h of runtime and consumed around 4 to 6 GB of RAM, enabling an average throughput of 10-15 samples per day.  
Finally, the assumption of isotropy in tortuosity estimation does not capture directional anisotropy, especially in layered or fractured carbonate rocks. Future studies should consider anisotropic flow, multiphase conditions, and data-driven approaches. Hybrid models that integrate geometric, streamline, and machine learning methods trained on high-resolution, multi-lithology datasets could improve prediction accuracy. Incorporating reactive transport and validating simulations with microfluidic or core flooding experiments would further enhance model robustness and applicability.