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

This study validates a modified normalized Kruskal-Wallis fractal dimension approach as a robust and highly consistent method for characterizing pore-structure heterogeneity in Khuff carbonate reservoirs, demonstrating a strong positive correlation between fractal dimension and permeability while confirming its applicability across diverse porous materials.

Original authors: Prof. Khalid Elyas Mohamed Elameen AlKhidir

Published 2026-07-13

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CC BY 4.0

Original authors: Prof. Khalid Elyas Mohamed Elameen AlKhidir

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the inside of a rock not as a solid block, but as a giant, microscopic city made entirely of tunnels and caves. In the Khuff Formation of central Saudi Arabia, scientists are trying to map this hidden city to understand how fluids (like oil or water) move through it. The problem? These cities are messy. Some have wide, straight highways; others are a tangled web of tiny, winding alleys. This messiness is called "pore-network heterogeneity."

To measure this mess, the researchers, led by Prof. Khalid Elyas Mohamed Elameen AlKhidir, tried two different ways to count the complexity of these underground mazes.

The Two Map-Making Tools

The first tool is the "old reliable" method. It looks at the size of the tunnels (pore radius) and how much water is stuck in them. It's like measuring the width of every street in the city to guess how crowded it is.

The second tool is the new invention: the "Modified Normalized Kruskal-Wallis" (MNKW) index. Think of this as a new, high-tech scanner that doesn't just measure street widths, but looks at the pattern of how water fills the city. It uses a statistical trick to see how the water spreads out.

The big question was: Does this new scanner give the same answer as the old ruler?

The Verdict: They Are Almost Twins

The answer is a resounding "yes." When the team tested 26 different rock samples, the two methods produced results that were nearly identical. If you plotted the results on a graph, the line connecting them was almost perfectly straight, with a slope of 0.99531 and a match score (R²) of 0.9999986. That is a level of agreement so high it's practically a perfect mirror image.

However, the paper does point out a tiny, systematic quirk. While the two methods agree almost perfectly, the new MNKW scanner gets slightly more sensitive as the rock gets more complex. It's like two people counting the same crowd: they agree on the total number, but the person with the new scanner starts to notice a few extra people in the very back rows when the crowd gets huge. This "proportional bias" means the difference between the two methods grows just a tiny bit as the rock's complexity increases, but the overall agreement remains excellent.

What the Numbers Tell Us

The rocks they studied were from the Khuff Formation, a Permo–Triassic carbonate reservoir. Here is what they found:

Porosity (how much empty space exists): Ranged from 2.269% to 11.253%.
Permeability (how easily fluids flow): Ranged from 0.264 to 3.445 mD (millidarcies).
Fractal Dimension (the "messiness" score): This score, which they call Df, ranged from approximately 2.36 to 2.86.

The study found a clear pattern: the messier the rock (higher Df), the better the fluids could flow (higher permeability).

A "low mess" sample had a Df of about 2.36.
An "average mess" sample had a Df of about 2.72.
A "high mess" sample had a Df of about 2.86.

Interestingly, the messier the rock, the better the math worked. The "low mess" sample had a match score of about 0.826, while the "high mess" sample scored a near-perfect 0.9997. This suggests that fractal math is especially good at describing very complex, tangled pore systems.

What This Means (and What It Doesn't)

The paper confirms that the new MNKW method is a robust, highly consistent way to characterize these rocks. It suggests that fractal dimension is a powerful tool for describing the architecture of tight carbonate reservoirs, potentially offering better insights than just looking at porosity alone.

The authors are careful to note that while the agreement is "near-perfect," the methods aren't strictly interchangeable without accounting for that tiny proportional bias in highly complex systems. They also suggest this method could be useful for other materials like ceramics, catalysts, and nanomaterials, but they stop short of claiming it solves every problem in reservoir engineering.

In short, the researchers have validated a new, statistical "ruler" that measures the complexity of underground rock cities just as well as the traditional one, proving that even in the tiniest, most tangled tunnels, math can find a perfect pattern.

Technical Summary: Validation of a Modified Normalized Kruskal-Wallis Fractal Dimension for Khuff Carbonate Reservoirs

Problem Statement
Characterizing the heterogeneity of porous materials, particularly in carbonate reservoirs, remains a significant challenge in reservoir engineering. While fractal geometry has been established as a quantitative tool for describing pore-network complexity and predicting reservoir quality, existing methodologies often rely on conventional pore-radius scaling derived from capillary-pressure data. The study addresses the need to validate alternative statistical approaches that can robustly characterize pore-structure heterogeneity in tight carbonate formations, specifically the Khuff Formation in central Saudi Arabia, where pore-network complexity varies significantly.

Methodology
The research utilized carbonate samples collected from surface exposures of the Khartam Reservoirs within the Permo–Triassic Khuff Formation. The study employed two independent fractal approaches to analyze pore-network heterogeneity using capillary-pressure data:

Conventional Approach: This method utilized the relationship between the log-normalized pore radius ( $\log(R/R_{max})$ ) and the log of water saturation ( $\log_{10}(S_w)$ ).
Proposed Approach: This method employed a statistical parameter termed the Modified Normalized Kruskal–Wallis (MNKW) index, plotting $\log(MNKW)$ against $\log_{10}(S_w)$ .

For both methods, a second-order polynomial regression was fitted to the data. The fractal dimension ( $D_f$ ) was calculated using the linear coefficient ( $b$ ) of the fitted quadratic expression according to the equation $D_f = 2 + b$ . The study also analyzed the relationship between the calculated fractal dimensions and petrophysical properties, including porosity (ranging from 2.269% to 11.253%) and permeability (ranging from 0.264 to 3.445 mD).

To validate the new approach, the authors performed:

Regression Analysis: Comparing 26 paired $D_f$ values from both methods.
Bland–Altman Analysis: Assessing the bias and limits of agreement between the two techniques.
Correlation Analysis: Examining the relationship between $D_f$ , permeability, and the coefficient of determination ( $R^2$ ) of the fitted models.

Key Contributions
The primary contribution of this work is the introduction and validation of the Modified Normalized Kruskal–Wallis (MNKW) fractal approach as a statistically equivalent alternative to the conventional pore-radius scaling method. The study demonstrates that the MNKW parameter, which incorporates cumulative distribution characteristics of the capillary-pressure curve, effectively captures pore-structure complexity in carbonate reservoirs.

Results

Fractal Dimension Range: Calculated fractal dimensions ranged from approximately 2.36 to 2.86.
Methodological Agreement: Regression analysis of the 26 paired $D_f$ values revealed near-perfect agreement between the MNKW and conventional methods, yielding a slope of 0.99531, an intercept of 0.00641, and an $R^2$ of 0.9999986.
Bias Analysis: Bland–Altman analysis identified a small mean bias (~0.0063 $D_f$ units) and narrow limits of agreement. However, a strong proportional bias was detected, indicating that the difference between methods increases systematically as the fractal dimension increases. This suggests the MNKW approach may be slightly more sensitive to pore-system complexity in highly heterogeneous systems.
Relationship with Reservoir Quality: Fractal dimensions showed a strong positive association with permeability and pore-network connectivity. Samples with higher fractal dimensions (e.g., $D_f \approx 2.86$ ) exhibited stronger fractal scaling behavior, indicated by higher coefficients of determination ( $R^2 \approx 0.9997$ ), compared to samples with lower dimensions ( $D_f \approx 2.36$ , $R^2 \approx 0.826$ ).

Significance and Claims
The paper concludes that fractal dimension serves as an effective quantitative descriptor of pore-structure heterogeneity in tight Khuff carbonate reservoirs. The study claims that the MNKW method provides a robust and highly consistent characterization of pore systems, offering results that are virtually identical to the established conventional method.

The authors assert that while conventional petrophysical parameters like porosity alone may not adequately capture pore-network complexity, the fractal dimension correlates strongly with fluid-flow capacity, suggesting that pore-network connectivity exerts a dominant control on permeability in these samples. Furthermore, the paper posits that this methodology has potential applicability beyond carbonate reservoirs, extending to other porous materials such as sandstones, ceramics, catalysts, nanomaterials, and engineered porous media. The study emphasizes that the two methods are not strictly interchangeable across the entire $D_f$ range without accounting for the observed proportional bias, but the overall agreement confirms the validity of the MNKW approach for pore-structure characterization.

Technical Summary: Validation of a Modified Normalized Kruskal-Wallis Fractal Dimension for Khuff Carbonate Reservoirs

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