⚡ electrical engineering

Physically-Consistent Fracture Capillary Pressure Type Curves and Relative Permeability: A Unified Aperture-Based Model without Empirical Matching

This study presents a unified, physics-based framework for modeling fracture capillary pressure and relative permeability using a single interpretable aperture parameter, eliminating the need for empirical fitting while offering a robust, simulator-ready alternative to traditional oversimplified assumptions and costly experiments.

Original authors: Alireza Golsefatan, Seyed Reza Shadizadeh, Hassan Shokrollahzadeh Behbahani

Published 2026-08-28

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CC BY 4.0

Original authors: Alireza Golsefatan, Seyed Reza Shadizadeh, Hassan Shokrollahzadeh Behbahani

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

Deep beneath the Earth's surface, vast reservoirs of oil and gas often reside not in porous sandstone like a wet sponge, but trapped within a complex network of cracks and fissures. These fractured rocks present a unique challenge for engineers trying to predict how fluids move through them. In the world of reservoir engineering, two invisible forces dictate the flow of oil and water: capillary pressure and relative permeability. Capillary pressure is the force that holds fluids in place within tiny spaces, while relative permeability describes how easily one fluid can flow when another is already present. For decades, the standard approach to modeling these fractured systems has relied on a convenient but flawed assumption: that these cracks are so wide and uniform that capillary pressure is effectively zero, and that the fluids slide past each other in a perfectly linear, predictable way. This simplification has made calculations easier, but recent evidence suggests it is physically wrong, leading to significant errors in predicting how much oil can be recovered and how fast it will flow.

A team of researchers at the Petroleum University of Technology in Iran has developed a new, unified way to describe fluid flow in these fractured rocks that abandons these old shortcuts. Instead of guessing or fitting curves to experimental data with arbitrary numbers, they built a model that starts with the actual physical shape of the cracks. They recognized that no fracture is perfectly smooth or uniform; the gap between the rock walls, known as the aperture, varies wildly from one spot to another. By treating this variation as a specific statistical pattern, they created a framework that links the geometry of the crack directly to how fluids behave inside it. Their work provides a clear, physics-based method to calculate both the capillary pressure and the relative permeability of a fracture using just one key measurement: the standard deviation of the fracture aperture. This single number, which describes how much the crack width varies, replaces the need for multiple confusing fitting parameters that have plagued previous models.

The researchers began by conceptualizing a fracture not as a single flat gap, but as a network where the opening size changes constantly along its length and width. They determined that these variations follow a specific statistical distribution, similar to how heights vary in a large population. Using this distribution, they derived mathematical relationships that show exactly how water and oil distribute themselves within the crack. Smaller openings tend to hold onto the wetting fluid, like water, while larger openings allow the non-wetting fluid, like oil, to pass through. By integrating these physical realities, they generated a set of "type curves." These curves act as a universal map, allowing engineers to predict the behavior of a fracture simply by knowing how much its width varies. If the width is very consistent, the flow behaves one way; if the width varies significantly, the flow behaves differently. This approach eliminates the need for the old, oversimplified "X-shaped" curves that assumed fluids move in a straight line regardless of the crack's complexity.

To prove their model worked, the team tested it against a wide array of existing data, including both theoretical studies and real-world laboratory experiments. They compared their predictions against data from dozens of different studies, covering everything from smooth laboratory fractures to rough, naturally occurring rock surfaces. In cases where the physical properties of the fracture were known, their model matched the observed data with remarkable precision. In cases where the properties were unknown, they used a matching technique to find the single best-fit variation parameter, which then successfully predicted the fluid behavior across different conditions. The results were consistent: the new model could replicate complex experimental data with high accuracy, often outperforming older methods that relied on multiple empirical constants. Crucially, the model also captured how the fracture changes under pressure. As the rock is squeezed by the weight of the earth above, the fracture closes, and the variation in its width changes. The researchers showed that their model could track these mechanical changes, adjusting the flow predictions to reflect the new, tighter geometry of the crack.

The implications of this work extend beyond a simple improvement in calculation. By providing a model that relies on a single, physically meaningful parameter rather than a collection of fitting numbers, the researchers have offered a tool that is both simpler and more robust. This means that engineers can now integrate these more realistic flow descriptions directly into the large-scale computer simulations used to manage oil fields. Instead of relying on the outdated assumption that fractures are empty, open channels with no internal resistance, they can now simulate the intricate dance of fluids within the actual, uneven geometry of the rock. The study confirms that fracture capillary pressure is not zero and that relative permeability is not a simple straight line; these are complex, geometry-dependent phenomena that must be accounted for to understand the true potential of a reservoir. While the model was developed for single fractures, the authors suggest that this framework could be expanded to handle entire networks of cracks and even more complex three-phase flows, offering a clearer path to understanding the hidden dynamics of the Earth's fractured crust.

Technical Summary: Physically-Consistent Fracture Capillary Pressure Type Curves and Relative Permeability

Problem Statement
Accurate modeling of fluid flow in fractured reservoirs remains a significant challenge due to the complex geometry of fracture networks and intricate matrix-fracture interactions. While fracture capillary pressure ( $P_c$ ) and relative permeability ( $k_r$ ) are the fundamental constitutive relationships governing phase interference and displacement efficiency, traditional reservoir engineering practices often rely on oversimplified assumptions. Specifically, simulators frequently assume zero capillary pressure and linear (X-curve) relative permeability behavior for fractures. Recent experimental and modeling evidence suggests these assumptions lack a sound physical basis, potentially leading to significant errors in performance prediction. Furthermore, existing models often rely on multiple empirical "fitting parameters" that lack clear physical meaning or are case-specific, limiting their predictive capability across different fracture geometries. A critical gap exists in the direct link between fracture micro-structural characteristics (specifically aperture distribution) and these flow parameters.

Methodology
This study develops a unified, physics-based framework for fracture $P_c$ and $k_r$ models by integrating fracture aperture distributions with fundamental flow mechanics. The core methodology involves the following steps:

Statistical Characterization: The fracture system is conceptualized as a unit volume containing a fully connected network where individual fractures exhibit spatially varying apertures. The study employs a lognormal distribution to characterize fracture aperture, width, and length, and a normal distribution for fracture orientation. The standard deviation of the aperture ( $\sigma_b$ ) is identified as the primary governing parameter.
Phase Saturation Derivation: Mobile and immobile saturations for wetting and non-wetting phases are derived by calculating phase volumes within specific aperture ranges defined by capillary pressure thresholds. These ranges correspond to irreducible wetting phase, residual non-wetting phase, and total phase saturations.
Type Curve Development:
- Capillary Pressure: Dimensionless wetting phase saturation is formulated as a function of effective saturation and dimensionless capillary pressure ( $P_{cDf}$ ). By incorporating the lognormal aperture distribution, Drainage and Imbibition Fracture Capillary Pressure type curves are derived using error functions (erf). These curves depend on a single physical parameter, $\sigma_b$ , and dimensionless boundary pressures ( $P_{cDminf}$ , $P_{cDmaxf}$ ).
- Relative Permeability: The model derives fracture relative permeability by integrating the Hagen-Poiseuille equation for fluid flow in conduits with Darcy's law for porous media. The integration accounts for the aperture distribution and phase saturations across drainage and imbibition processes. Tortuosity coefficients are incorporated, with exponents determined through validation against experimental data.
Validation and Matching: The models are validated against a wide spectrum of theoretical frameworks (e.g., Pruess and Tsang, Zhou and Wheater, Liu et al.) and experimental datasets (e.g., Reitsma and Kueper, Huo et al., Chen, Diomampo).
- When $\sigma_b$ is known, the model generates curves directly for comparison.
- When $\sigma_b$ is unknown, a type curve matching procedure is employed. Experimental $P_c$ vs. saturation data is plotted on a log-log scale and matched against the developed dimensionless type curves to extract the optimal $\sigma_b$ and other parameters.
Quantitative Metrics: Model accuracy is evaluated using Mean Squared Error (MSE) and Average Absolute Relative Error (AARE).

Key Contributions

Unified Aperture-Based Model: The study presents a single framework that derives both fracture capillary pressure and relative permeability directly from the fracture aperture distribution, eliminating the need for arbitrary empirical matching coefficients.
Single Physical Parameter: The model relies on the fracture aperture standard deviation ( $\sigma_b$ ) as the sole input parameter. This parameter is physically interpretable and represents the geometric heterogeneity of the fracture, contrasting with traditional models that require multiple case-specific fitting parameters.
Type Curve Methodology: The introduction of fracture capillary pressure type curves allows for a dual application: (1) predicting $P_c$ and $k_r$ from known aperture distributions, and (2) characterizing fracture aperture distributions from experimental $P_c$ data via type curve matching.
Mathematical Simplicity and Robustness: The derived equations utilize standard statistical functions (error functions) and are designed for seamless integration into commercial reservoir simulators.

Results
The developed models demonstrated exceptional robustness and efficacy across diverse fracture types and flow conditions:

Capillary Pressure: The model successfully replicated theoretical models (e.g., Zhou and Wheater) with near-perfect alignment (MSE in the range of $10^{-9}$ to $10^{-10}$ ) and matched experimental data (e.g., Huo et al., Reitsma and Kueper) with low AARE values (typically < 20%). Notably, the model captured the physical trend where increasing effective stress leads to an increase in the optimized $\sigma_b$ , reflecting fracture closure and increased heterogeneity.
Relative Permeability: The model accurately predicted relative permeability curves for both drainage and imbibition processes. It showed strong agreement with theoretical limits (e.g., Brooks and Corey with $\lambda \to \infty$ ) and experimental datasets (e.g., Chen, Diomampo). The model successfully decoupled phase interference from secondary experimental variables, providing a more rigorous validation than traditional saturation-based plots.
Parameter Efficiency: In all validation cases, the model achieved comparable or superior accuracy to complex existing models while utilizing only the single parameter $\sigma_b$ .

Significance and Claims
The paper claims that the proposed framework offers a reliable, cost-effective alternative to costly and time-consuming laboratory experiments for determining fracture capillary pressure and relative permeability. By grounding the models in the physical reality of fracture aperture distributions rather than empirical curve-fitting, the study provides a physically consistent tool for reservoir engineers.

The authors emphasize that the mathematical structure of the models facilitates their direct implementation in commercial reservoir simulators, enabling more realistic large-scale simulations of fractured systems. The study concludes that fracture capillary pressure is neither zero nor constant, and fracture relative permeability deviates significantly from linear "X-curves"; these phenomena must be accounted for using aperture-based models to accurately capture phase interference. The work bridges the gap between complex analytical solutions and practical reservoir engineering applications, offering a standardized method to characterize fractured media without the ambiguity of arbitrary fitting parameters.

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