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Validation Of a Modified Normalized Skillings-Mack Fractal Dimension For Characterizing Pore Structure in Permo-Triassic Khuff Carbonate Reservoirs , Central Saudi Arabia

This study validates the Modified Normalized Skillings-Mack fractal dimension as a robust and highly consistent alternative to the normalized pore-radius method for characterizing pore-network heterogeneity and predicting reservoir quality in Permo-Triassic Khuff carbonate reservoirs, demonstrating that higher fractal dimensions correlate with improved pore connectivity and permeability.

Original authors: Khalid Elyas Mohamed Elameen Alkhidir

Published 2026-08-14

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CC BY 4.0

Original authors: 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 ground beneath our feet as a giant, ancient sponge. Some of these sponges are made of sand, which is usually straightforward, but others are made of limestone and other carbonates. These carbonate rocks are the hidden treasure chests of the world's oil and gas, holding massive amounts of fuel. However, they are also the most complicated sponges imaginable. Instead of having neat, round holes like a kitchen sponge, their insides are a chaotic maze of tunnels, caves, and tiny cracks formed over millions of years by water, heat, and chemical changes.

To get oil out of these rocks, scientists need to understand how the fluid moves through this maze. They use a concept called "fractal geometry." Think of a fractal like a coastline: if you look at it from a satellite, it looks jagged; if you zoom in with a telescope, the jaggedness is still there, just smaller. This "self-similarity" means the rock's internal structure has a specific mathematical complexity. Scientists measure this complexity with a number called the "fractal dimension." A higher number means a more twisted, complex, and interconnected maze, which usually helps oil flow better. But measuring this is tricky, and scientists often use different mathematical tools to get the number, hoping they all tell the same story.

This study dives into a specific, ancient rock layer in central Saudi Arabia called the Khuff Formation, which dates back to the Permo-Triassic period (a time when dinosaurs were just starting to appear). The researchers wanted to test a new, fancy mathematical tool they invented, called the "Modified Normalized Skillings-Mack" (MNSM) method. They wanted to see if this new tool could measure the rock's complexity just as well as the old, trusted way of doing it. They took 26 rock samples from the surface, measured how much oil and water they could hold, and then used capillary pressure (the force that pushes fluids into tiny holes) to calculate the fractal dimension using both the new MNSM method and the traditional method.

The results were a perfect match. The paper found that the new MNSM method gave almost the exact same numbers as the old method. For the 26 samples, the fractal dimensions calculated by the new method ranged from 2.3584 to 2.8575, while the old method gave values from 2.3628 to 2.8647. The average for the new method was 2.7038, and for the old one, it was 2.7101. When the scientists compared the two sets of numbers, they found a correlation coefficient of 1.0, which is the highest possible score, meaning the two methods moved in perfect lockstep. They also found a "root mean square error" of just 0.0063, which is a tiny, tiny difference, and a statistical check called a Bland–Altman analysis showed that there was no systematic bias—neither method was consistently lying or exaggerating.

The study also confirmed a key rule of the road for these rocks: the higher the fractal dimension, the better the rock is at letting fluids flow. The samples with higher fractal dimensions (around 2.86) had better pore connectivity and higher permeability, meaning the oil could move through them more easily. The samples with lower fractal dimensions (around 2.36) were more restricted.

In short, the author suggests that this new MNSM method is a robust and reliable alternative for characterizing these tricky carbonate rocks. It doesn't replace the old way, but it offers a fresh statistical perspective that arrives at the same conclusion. The paper concludes that this approach provides a solid framework for evaluating how "messy" or "connected" the pore networks are, which helps in predicting how well a reservoir will perform. It's like discovering a new way to count the twists in a knot that gives you the exact same answer as the old way, but with a different kind of math, giving geologists another tool to understand the hidden highways of the Earth.

Technical Summary: Validation of a Modified Normalized Skillings-Mack Fractal Dimension for Khuff Carbonate Reservoirs

Problem Statement
Carbonate reservoirs, particularly the Permo-Triassic Khuff Formation in central Saudi Arabia, are characterized by complex, multiscale pore systems and significant heterogeneity. Traditional petrophysical parameters such as porosity and permeability often fail to adequately describe the geometric complexity and scale-invariant nature of these pore networks. While fractal analysis has emerged as a robust tool for quantifying pore-throat architecture and fluid-flow pathways in various reservoirs, there remains a need for further investigation into specific carbonate systems like the Khuff Formation. Specifically, the relationship between pore-network heterogeneity, fractal dimension, and reservoir quality in this formation has not been fully quantified using alternative statistical descriptors.

Methodology
The study utilized twenty-six carbonate core samples collected from surface exposures of the Khuff Formation. The samples underwent standard petrophysical characterization, yielding porosity values between 2.269% and 11.253% and permeability values between 0.264 and 3.445 mD. Capillary pressure data were transformed into pore-throat radius distributions using the Washburn equation.

Two independent fractal approaches were employed to characterize pore structure based on this capillary pressure data:

Conventional Normalized Pore-Radius Method: This method utilized the relationship between the logarithm of the normalized pore radius ( $\log_{10}(R/R_{max})$ ) and the logarithm of water saturation ( $\log_{10}(S_w)$ ).
Modified Normalized Skillings-Mack (MNSM) Method: A novel approach utilizing a statistically derived Modified Normalized Skillings-Mack parameter. The logarithm of the MNSM parameter was correlated with $\log_{10}(S_w)$ .

For both methods, the data were fitted to a quadratic equation of the form $Y = a[\log_{10}(S_w)]^2 + b[\log_{10}(S_w)] + c$ . The fractal dimension ( $D_f$ ) was calculated using the positive linear coefficient ( $b$ ) according to the relationship $D_f = 2 + b$ .

The consistency between the two methods was evaluated using linear regression (correlation coefficient $r$ and coefficient of determination $R^2$ ), Root Mean Square Error (RMSE), and Bland–Altman analysis to assess systematic bias and limits of agreement.

Key Results

Fractal Dimension Range: The calculated fractal dimensions ranged from 2.3584 to 2.8575 for the MNSM method and from 2.3628 to 2.8647 for the normalized pore-radius method. The mean values were 2.7038 and 2.7101, respectively.
Statistical Agreement: The comparison between the two methods demonstrated near-perfect agreement, with a correlation coefficient ( $r$ ) of 1.0 and a coefficient of determination ( $R^2$ ) of 1.0. The Root Mean Square Error (RMSE) was extremely low at 0.0063.
Bias Analysis: Bland–Altman analysis revealed a negligible mean bias (-0.0063) and narrow limits of agreement, confirming that there is no systematic overestimation or underestimation by the MNSM method compared to the conventional approach.
Reservoir Properties: The study observed that increasing fractal dimensions were associated with improved pore connectivity and higher permeability. Samples with low, intermediate, and high pore-network complexity yielded fractal dimensions of approximately 2.36, 2.72, and 2.86, respectively.

Significance and Claims
The paper claims that the Modified Normalized Skillings-Mack (MNSM) fractal dimension is a robust and reliable alternative parameter for characterizing pore structure in Khuff carbonate reservoirs. The near-perfect statistical agreement with the conventional normalized pore-radius method validates the MNSM approach as an effective framework for evaluating carbonate reservoir heterogeneity.

The author asserts that fractal geometry provides a quantitative measure of pore-network heterogeneity that complements traditional petrophysical measurements. Specifically, the study concludes that the MNSM method successfully reproduces the heterogeneity information captured by conventional capillary-pressure-based models while offering an independent statistical framework. This approach is presented as a potential tool for improved reservoir characterization and flow-property prediction, particularly in scenarios where direct pore-radius normalization may be challenging or where independent validation of fractal estimates is required. The study does not claim to introduce new physical laws but rather validates a statistical descriptor that aligns with established fractal theories in porous media.

Technical Summary: Validation of a Modified Normalized Skillings-Mack Fractal Dimension for Khuff Carbonate Reservoirs

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