Validation of a Modified Normalized Spotted Hyena Optimization Fractal Dimension of Characterizing Pore Structure in Permo–Triassic Khuff Carbonate Reservoirs, Central Saudi Arabia

Abstract Characterization of pore-network heterogeneity is essential for understanding fluid-flow behavior and reservoir quality in carbonate formations. This study investigates the relationship between fractal dimension, pore structure, porosity, and permeability in carbonate samples collected from surface exposures of the Permo–Triassic Khuff Formation in central Saudi Arabia. Porosity and permeability measurements were obtained for all samples, and pore-network heterogeneity was quantified using capillary-pressure-derived fractal approaches. Fractal dimensions were initially determined using the conventional normalized pore-radius model based on the relationship between Log(R/Rmax) and Log10(water saturation). A second approach employed the modified normalized Kruskal–Wallis (MNKW) parameter, while a third method utilized Modified Normalized Spotted Hyena Optimization (MNSHO) to evaluate the robustness of optimization-based fractal characterization. Calculated fractal dimensions ranged from approximately 2.36 to 2.86, reflecting substantial variations in pore-system complexity and heterogeneity. Porosity values ranged from 2.269% to 11.253%, whereas permeability values ranged from 0.264 to 3.445 mD. Increasing fractal dimension was consistently associated with improved permeability and enhanced pore-network connectivity, indicating that fractal geometry effectively captures variations in reservoir quality. Regression analysis demonstrated near-perfect agreement between fractal dimensions obtained from the conventional Log(R/Rmax) method and the MNKW approach (R² = 0.9999986), confirming the validity of the modified statistical method. Similarly, comparison between the Log(R/Rmax) and MNSHO methods yielded an exceptionally strong positive correlation (R² ≈ 0.9999), demonstrating that optimization-based techniques can accurately reproduce conventional fractal-dimension estimates. Bland–Altman analysis further confirmed excellent agreement between the methods, although a small systematic bias was observed for the MNSHO approach. The narrow limits of agreement indicate that both MNKW and MNSHO provide reliable and reproducible estimates of pore-network heterogeneity. Comparison with recent carbonate-reservoir studies demonstrates that fractal dimension is strongly linked to pore-throat complexity, reservoir connectivity, and permeability behavior. The results establish fractal dimension as a robust quantitative descriptor of carbonate pore systems and demonstrate that both statistical and optimization-based approaches provide effective tools for characterizing heterogeneity in tight carbonate reservoirs. The proposed methodologies offer potential applications in reservoir characterization, digital-rock analysis, permeability prediction, and advanced evaluation of heterogeneous carbonate formations.