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Voronoi-entropy characterisation of geometric disorder in two-dimensional porous networks

This paper demonstrates that the Shannon entropy of the Voronoi cell-area distribution serves as a scalar, porosity-independent metric to quantify the geometric disorder in two-dimensional porous networks, thereby complementing traditional porosity and permeability measures by capturing how pore-space reorganization affects fluid transport.

Original authors: Vladivostok Suxo, Alexsandro Kirch, Caetano Rodrigues Miranda

Published 2026-08-24

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CC BY 4.0

Original authors: Vladivostok Suxo, Alexsandro Kirch, Caetano Rodrigues Miranda

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

Fluids moving through the ground—whether water seeping into an aquifer, oil drifting through rock, or ink soaking into paper—must navigate a hidden, twisting maze. The ease with which these liquids pass depends on two things: how much empty space exists within the material, and how that space is arranged. Scientists have long used two main tools to describe this underground world. The first is porosity, a simple measure of how much of the rock is empty space versus solid rock. The second is permeability, which describes how easily a fluid can flow through that space when pushed by pressure. While these two numbers are essential for engineers and geologists, they tell only part of the story. Two samples of rock can have the exact same amount of empty space, yet allow fluids to pass at vastly different speeds because the shape and connection of the pores differ. This hidden variation, known as geometric disorder, is difficult to measure but crucial for understanding how the Earth's subsurface works.

A team of researchers at the University of São Paulo set out to find a way to quantify this disorder without getting lost in the complexity of real rock. They created a simplified, two-dimensional model of a porous medium, imagining it as a flat sheet filled with circular solid obstacles, like coins scattered on a table. The spaces between these coins represent the pores through which fluid flows. To understand the arrangement, they used a mathematical technique called a Voronoi tessellation. This method divides the empty space into distinct regions, where each region belongs to the nearest solid obstacle. By measuring the area of these regions, the researchers could calculate a single number, known as Shannon entropy, which acts as a score for how disordered the arrangement is. A perfectly ordered grid of obstacles would have a score of zero, while a completely random scattering would have a higher score. They then used powerful computer simulations to watch how a fluid, specifically a salty water solution, moved through these different arrangements.

The researchers tested their model by changing one factor at a time while keeping the others constant. First, they looked at how the position of the obstacles affected the flow. They started with a neat, ordered grid and gradually scrambled the positions of the obstacles to increase the disorder. They found that as the arrangement became more chaotic, the path the fluid had to take became slightly more winding, a property called tortuosity. However, the overall ability of the fluid to pass through the material, the permeability, changed very little. Even when the obstacles were scattered randomly, the fluid found a way through almost as easily as it did in the ordered grid. This happened because, while disorder narrows some pathways, it simultaneously widens others, effectively balancing the flow.

Next, the team examined what happened when they changed the size of the obstacles or the total amount of empty space. When they kept the disorder level the same but made the obstacles larger, the permeability increased dramatically, rising nearly seven times. This occurred because larger obstacles meant fewer of them were needed to fill the space, which left wider gaps between them for the fluid to rush through. Similarly, when they increased the amount of empty space while keeping the obstacle size the same, the permeability jumped by more than seventy times. These results confirmed that the size of the gaps, or "throats," between the solid parts is the primary driver of how fast fluid moves. The degree of disorder, while it changes the shape of the maze, does not control the speed of the flow in the same way.

The study concludes that the new entropy score is a valuable tool, but not for predicting flow speed. Instead, it serves as a precise description of the geometry itself. It captures the local reorganization of the pore space, summarizing the complex mix of wide and narrow channels into a single variable. This allows scientists to distinguish between two rocks that look identical in terms of porosity and flow speed but have completely different internal structures. While the model used in this study is a simplified two-dimensional representation and does not yet account for the full complexity of three-dimensional rock or the interaction of multiple fluids, it successfully isolates the role of disorder. The work suggests that to fully understand fluid transport in the Earth, we need to measure not just how much space is there, but also the specific statistical signature of how that space is arranged.

Technical Summary: Voronoi-entropy characterisation of geometric disorder in two-dimensional porous networks

Problem Statement
Quantifying fluid transport in porous media remains a challenge because standard descriptors—porosity (void fraction) and permeability (ease of flow)—are insufficient. Samples with identical porosity can exhibit vastly different permeabilities due to variations in the geometric arrangement of pore spaces. While porosity measures how much space exists, it does not quantify how that space is arranged. Existing descriptors often fail to isolate geometric disorder independently of porosity or pore size. This study addresses the gap in characterizing this geometric disorder using a scalar metric derived from information theory, specifically testing whether the Shannon entropy of the Voronoi cell-area distribution can serve as a porosity-independent measure of disorder and how it correlates with single-phase flow properties.

Methodology
The authors employed an idealized two-dimensional model consisting of circular obstacles representing the solid phase of a porous medium. The study utilized a hierarchical approach combining molecular dynamics-derived fluid properties with pore-scale simulations:

Network Generation: Two-dimensional domains ( $5.68 \, \mu\text{m}$ lateral length) were populated with circular obstacles. The study systematically varied four factors: positional disorder, obstacle size, porosity, and the resulting Voronoi entropy. To isolate variables, the authors generated 1,000 independent realizations for each configuration, ranging from ordered square lattices to fully random placements, while enforcing a minimum surface-to-surface separation to ensure connectivity.
Voronoi Analysis: Geometric disorder was quantified using the Shannon entropy ( $H$ ) of the distribution of Voronoi cell areas. The Voronoi tessellation was constructed based on obstacle centers. The entropy was calculated by discretizing the distribution of scaled cell areas ( $A/\langle A \rangle$ ) into bins. This metric is distinct from topological Voronoi entropy (based on cell sides) as it is sensitive to the spread of cell sizes.
Flow Simulation: Single-phase flow was simulated using the Lattice-Boltzmann method (LBM) via the Taxila package. The fluid properties (density, kinematic viscosity) and fluid-solid interaction parameters (contact angle) were imported from prior molecular dynamics studies of an API brine in a montmorillonite matrix. Simulations were conducted in the creeping flow regime (Reynolds number $< 1$ ) to ensure linear response.
Transport Metrics: From the steady-state velocity fields, the authors extracted hydraulic tortuosity and Darcy permeability.

Key Contributions

Isolation of Geometric Disorder: The study successfully decoupled the effects of positional disorder, obstacle size, and porosity on transport properties, a separation difficult to achieve in natural rock samples.
Voronoi Cell-Area Entropy as a Descriptor: The paper introduces and validates the Voronoi cell-area Shannon entropy as a compact, scalar measure of geometric disorder that is computable solely from obstacle positions, independent of flow simulations.
Clarification of Permeability Drivers: The work demonstrates that while porosity and permeability are standard, they leave unmeasured the specific geometric reorganization of the pore space. The entropy metric captures this "local reorganization" where disorder widens some throats and narrows others at fixed porosity.

Results

Permeability Dependence: The bulk permeability was found to be governed primarily by the pore-throat scale. A single power law ( $k \propto w_{med}^{2.9}$ ) collapsed the data across variations in porosity and obstacle size.
Effect of Positional Disorder: At fixed porosity and obstacle size, increasing positional disorder (from ordered lattice to random) caused a slight increase in permeability (from $\approx 4.75$ to $5.01$ mD) and an increase in tortuosity (from $1.04$ to $1.16$). The permeability increase was attributed to the broadening of the throat-width distribution, where the emergence of preferential wider channels offsets the narrowing of others.
Effect of Entropy: When varying entropy alone within fully random networks (fixed porosity and obstacle size), no systematic trend in permeability or tortuosity was observed; variations remained within the statistical scatter of the configurations. This indicates that once the crystalline order is lost, further increases in entropy do not significantly alter transport properties.
Obstacle Size and Porosity: Increasing obstacle size (at fixed porosity) significantly increased permeability (nearly sevenfold) and tortuosity. Increasing porosity (at fixed obstacle size) caused a sharp rise in permeability (over seventyfold) and a decrease in tortuosity.
Entropy vs. Tortuosity: The entropy correlated with tortuosity during the transition from an ordered lattice to a disordered state. However, this correlation is partly an artifact of the ordered lattice's alignment with the flow direction, which minimizes path length. In fully disordered states, the correlation weakens.

Significance and Claims
The paper claims that the Voronoi cell-area Shannon entropy serves as a complementary descriptor rather than a direct predictor of transport.

Complementary Role: The entropy quantifies the geometric disorder that porosity and permeability leave unmeasured. It condenses the complexity of pore-space reorganization into a single variable.
Predictive Limitations: The study explicitly states that the entropy does not predict permeability better than the pore-throat scale. The weak dependence of permeability on entropy is consistent with critical-path transport theory, which suggests flow is controlled by the bottleneck scale (widest connected throats) rather than the global arrangement statistics.
Redundancy: The authors note that the entropy is mathematically redundant with the coefficient of variation (CV) of the cell areas for the Gamma-like distributions observed in these packings. Thus, it acts as a convenient single-number summary of the cell-area spread.
Scope: The conclusions are strictly limited to idealized, two-dimensional, single-phase creeping flow through circular obstacles. The authors do not claim these results directly apply to complex, three-dimensional, multiphase reservoir rocks without further validation, though the entropy could be computed from grain centroids in digital-rock workflows without full pore-network extraction.

In summary, the work establishes that while geometric disorder (measured by Voronoi entropy) alters the local topology of the pore space, the macroscopic permeability in this regime is dominated by the pore-throat scale. The entropy metric is valuable for characterizing the arrangement of the medium but does not replace porosity or permeability as a primary transport descriptor.

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