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

Technical Summary: Validation of a Modified Normalized Spotted Hyena Optimization Fractal Dimension for Characterizing Pore Structure in Permo–Triassic Khuff Carbonate Reservoirs

Problem Statement
Carbonate reservoirs, such as the Permo–Triassic Khuff Formation in central Saudi Arabia, exhibit extreme pore-network heterogeneity driven by complex depositional and diagenetic histories. Unlike siliciclastic systems, these reservoirs contain multiscale pore networks (interparticle pores, vugs, fractures) that significantly influence fluid storage and flow. While fractal theory has been established as a framework for characterizing pore-space geometry and linking capillary pressure to permeability, there is a need for robust, alternative statistical formulations that can describe pore-network heterogeneity while preserving physical interpretability. Specifically, the applicability of non-parametric statistical descriptors and optimization-based techniques within a fractal framework for carbonate reservoirs requires comprehensive evaluation and validation against conventional methods.

Methodology
This study analyzed surface exposure samples from the Khartam Member of the Khuff Formation to evaluate pore-network heterogeneity using three distinct approaches to calculate fractal dimensions ($D_f$):

1. Conventional Normalized Pore-Radius Model: This baseline method utilized Mercury Injection Capillary Pressure (MICP) data. Pore-throat radii ($R$) were derived from capillary pressure ($P_c$) using the Washburn equation. A logarithmic relationship was established between normalized pore radius ($R/R_{max}$) and water saturation ($S_w$). A second-order polynomial regression ($Y = aX^2 + bX + c$) was fitted to the data, where the fractal dimension was calculated as $D_f = 2 + b$.
2. Modified Normalized Kruskal–Wallis (MNKW): An independent statistical descriptor was developed by normalizing the Kruskal–Wallis parameter derived from pore-throat distributions. Similar to the conventional method, a regression of $\log(MNKW)$ versus $\log_{10}(S_w)$ was performed to extract the coefficient $b$ and calculate $D_f$.
3. Modified Normalized Spotted Hyena Optimization (MNSHO): This optimization-based approach utilized Shannon entropy calculated from pore-throat size probability distributions. The entropy values were normalized to create the MNSHO parameter. A quadratic regression of $\log(MNSHO)$ versus $\log_{10}(S_w)$ was fitted to determine the fractal dimension.

All samples underwent standard petrophysical testing, including helium porosity and steady-state gas permeability measurements. The study employed regression analysis to correlate fractal dimensions with porosity and permeability, and Bland–Altman analysis to assess the agreement between the MNKW and MNSHO methods.

Key Results

• Fractal Dimension Range: Calculated fractal dimensions ranged from approximately 2.36 to 2.86. Lower values corresponded to homogeneous pore systems with limited connectivity, while higher values indicated complex, heterogeneous networks with improved fluid-flow pathways.
• Petrophysical Correlations: Porosity values ranged from 2.269% to 11.253%, and permeability ranged from 0.264 to 3.445 mD. A strong positive correlation was observed between fractal dimension and permeability. Samples with higher fractal dimensions consistently exhibited better permeability and pore-network connectivity, suggesting that fractal geometry captures flow-path continuity more effectively than porosity alone.
• Methodological Validation:
  • Regression Analysis: The MNKW approach showed near-perfect agreement with the conventional Log($R/R_{max}$) method ($R^2 = 0.9999986$). The MNSHO approach also demonstrated an exceptionally strong positive correlation with the conventional method ($R^2 \approx 0.9999$).
  • Bland–Altman Analysis: This analysis confirmed excellent agreement between the MNKW and MNSHO methods, with a small systematic bias observed for the MNSHO approach. The narrow limits of agreement indicated that both methods provide reliable and reproducible estimates of pore-network heterogeneity.

Key Contributions

• Introduction of MNKW and MNSHO: The study successfully introduced and validated the Modified Normalized Kruskal–Wallis (MNKW) and Modified Normalized Spotted Hyena Optimization (MNSHO) parameters as robust fractal descriptors for carbonate reservoirs.
• Validation of Optimization-Based Techniques: The research demonstrated that optimization-based techniques (MNSHO) can accurately reproduce conventional fractal dimension estimates derived from capillary pressure data.
• Quantification of Heterogeneity: The study established that fractal dimension serves as a comprehensive quantitative descriptor that integrates pore-size distribution, connectivity, and structural complexity, offering a more nuanced view of reservoir quality than porosity measurements alone.

Significance and Claims
The paper claims that fractal dimension is a robust quantitative descriptor of carbonate pore systems, effectively characterizing the heterogeneity of tight carbonate reservoirs. The results establish that both statistical (MNKW) and optimization-based (MNSHO) approaches are valid alternatives to conventional methods for evaluating pore-network architecture.

The authors assert that these methodologies offer potential applications in:

• Reservoir characterization and digital-rock analysis.
• Permeability prediction in heterogeneous carbonate formations.
• Advanced evaluation of reservoir quality where pore connectivity is a critical control on fluid flow.

The study concludes that the proposed frameworks provide reliable tools for assessing the Khuff Formation and other heterogeneous carbonate systems, confirming that increasing fractal dimension is a direct indicator of improved reservoir performance driven by enhanced pore-network connectivity and complexity.