Validation Of a Modified Normalized Kruskal-Walis Fractal Dimension For Characterizing Pore Structure in Khuff Carbonate Reservoirs, Central Saudi Arabia

Abstract Carbonate samples were collected from surface exposures of the Khartam Reservoirs within the Permo–Triassic Khuff Formation of central Saudi Arabia. Porosity and permeability measurements were obtained for all samples, and pore-network heterogeneity was investigated using two independent fractal approaches derived from capillary-pressure data. The first method utilized the relationship between log-normalized pore radius, Log(R/Rmax), and Log10(water saturation), whereas the second employed Log(modified normalized Kruskal–Wallis parameter, MNKW) versus Log10(water saturation). In both methods, the fractal dimension was determined from the positive linear coefficient of the fitted quadratic expression according to Df = 2 + b. The calculated fractal dimensions ranged from approximately 2.36 to 2.86 and showed a strong positive association with permeability and pore-network connectivity. Porosity ranged from 2.269–11.253%, whereas permeability ranged from 0.264–3.445 mD. Representative samples with low, average, and high fractal dimensions yielded Df values of approximately 2.36, 2.72, and 2.86, respectively. Corresponding coefficients of determination increased from about 0.826 to 0.995 and 0.9997, indicating that fractal behavior becomes increasingly pronounced as pore-system complexity increases. Comparison of the two approaches demonstrated remarkable consistency. Regression analysis of 26 paired Df values produced a slope of 0.99531, an intercept of 0.00641, and R² = 0.9999986, confirming near-perfect agreement between methods. Bland–Altman analysis revealed a small mean bias (~ 0.0063 Df units) and narrow limits of agreement; however, a strong proportional bias was identified, indicating that differences between methods increase systematically with increasing Df. The results demonstrate that fractal dimension is an effective quantitative descriptor of pore-structure heterogeneity in tight carbonate reservoirs. Both approaches provide robust and highly consistent characterization of pore systems and may be extended to other porous materials, including nanomaterials, ceramics, catalysts, and engineered porous media.