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Dimensional effects on breakthrough behavior and pore-scale residence time distributions

This study utilizes Lattice Boltzmann simulations to demonstrate that while 2D pore-scale models systematically underestimate 3D permeability due to reduced out-of-plane connectivity—a discrepancy best predicted by the Euler characteristic—the impact of dimensionality on transport metrics like breakthrough curves and residence times varies significantly depending on the specific pore geometry and porosity of the medium.

Original authors: Shahram Asgari, Christof Meile

Published 2026-10-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shahram Asgari, Christof Meile

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

Water and dissolved substances move through the hidden, intricate world of soil, rock, and engineered materials in ways that are often impossible to see. This movement, known as solute transport, is critical for everything from cleaning up contaminated groundwater to storing carbon deep underground. To understand how these fluids navigate the tiny, winding tunnels between grains of sand or fibers of a filter, scientists build computer models that mimic the flow at the scale of individual pores. For decades, a practical dilemma has faced researchers: while three-dimensional models offer the most realistic picture of these complex networks, they demand immense computing power. Two-dimensional slices, which are essentially flat cross-sections of the material, are far cheaper and faster to run, leading many to use them as stand-ins for the full three-dimensional reality. The question has long been whether this shortcut sacrifices too much accuracy to be useful.

A team of researchers at the University of Georgia set out to test exactly how much these flat models miss. They created digital versions of three distinct types of porous materials: one made of tangled fibers, another resembling packed spheres like a pile of marbles, and a third mimicking the random, clumpy structure of natural geological formations. For each material, they built both a full three-dimensional block and a series of two-dimensional slices cut from it. Using a powerful simulation method that tracks the movement of fluid particles and dissolved substances, they compared how water and tracers moved through the flat slices versus the full blocks. They looked specifically at how easily the fluid could pass through the material, how long it took for a substance to travel from one side to the other, and how much the substance spread out along the way.

The results confirmed that the flat models consistently underestimate how easily fluid can flow through the material. In the three-dimensional blocks, fluid has the freedom to move around obstacles in every direction, finding shortcuts that simply do not exist in a flat slice. When the researchers looked at the two-dimensional slices, they found that the fluid was forced into longer, more winding paths because it could not bypass obstacles by moving "up" or "down" out of the plane. This effect was most dramatic in materials with low porosity, where the available space for flow is already tight. In these cases, the flat models predicted permeability values that were nearly ten times lower than the true three-dimensional reality for fibrous materials, and about four times lower for the geological-like structures. The researchers found that this discrepancy is not just a matter of how much empty space exists, but rather a fundamental loss of connection. By flattening the world, the models sever the vital out-of-plane links that allow fluid to weave through the network efficiently.

To predict when a flat model might be a safe substitute, the team turned to a mathematical way of counting the connections within the pore network. They used a measure that essentially counts the number of separate pockets, loops, and enclosed spaces in the structure. They discovered that when the flat slice and the full block had very different counts of these connections, the flow predictions diverged sharply. This topological measure acts as a quick diagnostic tool: if a flat slice captures the same pattern of loops and connections as the full block, it is likely to give an accurate picture of the flow. If the counts differ significantly, the flat model is missing critical pathways, and a full three-dimensional simulation is necessary.

Beyond just how fast water flows, the study examined how long it takes for a substance to travel through the material and how much it spreads out. The behavior of these transport metrics depended heavily on the type of material. In the fibrous medium, the difference between the flat and full models remained steady regardless of how permeable the material was, likely because the flow is dominated by a few main channels that look similar in both dimensions. In the geological-like structures, the gap between the two models shrank as the material became more permeable, suggesting that in looser, more open materials, the flat slices capture the flow paths well enough. However, in the packed sphere models, the flat slices sometimes predicted faster travel times and sometimes slower ones, with no clear pattern, indicating that the specific arrangement of spheres in a single slice can be misleading.

The researchers also tested whether the chemical nature of the substance being tracked changed these results. They compared a conservative tracer, which simply moves with the water without reacting, against a reactive tracer that interacts with the material as it moves. Surprisingly, the type of tracer made very little difference to the overall comparison. Whether the substance was inert or reactive, the errors introduced by using a flat model were governed primarily by the shape and connectivity of the pores, not by the chemistry of the fluid. This suggests that the structural limitations of two-dimensional models are a fundamental geometric issue, not a chemical one.

Ultimately, the study provides a clear guide for scientists and engineers who must choose between speed and accuracy. While two-dimensional models are useful for quick estimates, they systematically miss the ease of flow found in three-dimensional reality because they cannot account for the extra routes available in the third dimension. The research shows that by checking a simple measure of how the pores are connected, one can determine before running a simulation whether a flat slice is a faithful proxy or a misleading simplification. This insight allows for more efficient use of computing resources, ensuring that the complex, three-dimensional nature of the natural world is respected when it matters most.

Technical Summary: Dimensional Effects on Breakthrough Behavior and Pore-Scale Residence Time Distributions

Problem Statement
Two-dimensional (2D) pore-scale models are frequently employed as computationally efficient proxies for three-dimensional (3D) simulations in the study of solute transport within porous media. However, 2D and 3D porous structures exhibit fundamentally different flow and transport behaviors due to differences in connectivity, tortuosity, and mixing regimes. While 3D simulations offer the most realistic representation of pore networks, they are computationally intensive. Consequently, researchers often rely on 2D slices derived from micro-CT images or thin sections. The central problem addressed by this study is the extent to which these 2D analogues can accurately predict 3D permeability, breakthrough curves, and fluid age distributions, and whether specific topological or geometric indicators can determine when a 2D model is a suitable substitute for a full 3D simulation.

Methodology
The authors utilized Lattice Boltzmann (LB) simulations to resolve Navier–Stokes flow and advection–diffusion–reaction (ADR) equations on discrete voxel grids. The study focused on three distinct types of synthetic porous media:

Fibrous: Representing biological tissues and engineered materials, generated by placing cylindrical fibers.
Gaussian: Resembling geological materials like soil aggregates and sandstone, generated via a filtered random field.
Overlapping Spheres: Representing granular materials and packed beds, constructed by randomly placing spheres.

For each medium type, 3D computational domains (100 × 100 × 100 voxels) were generated at low, medium, and high porosities (0.4, 0.6, and 0.8). 2D domains were extracted as xz-plane slices from these 3D structures, ensuring flow percolation.

The simulations compared:

Permeability: Calculated via Darcy's law for both 2D slices and their 3D parent domains.
Transport Metrics: Flux-weighted breakthrough curves were analyzed to derive the first temporal moment ( $M_1$ , mean arrival time) and the coefficient of variation ( $CV$ , breakthrough spreading).
Tracer Types: Both conservative (non-reactive) and reactive tracers (representing fluid age with a first-order reaction rate) were simulated under advection-dominated conditions (Péclet numbers of 4.1, 12.5, and 37.5).
Topological Analysis: The pore-normalized Euler characteristic ( $\tilde{\chi}$ ), derived from persistent homology (Betti numbers $\beta_0, \beta_1, \beta_2$ ), was computed to quantify pore-network connectivity as a geometry-only metric.

Key Results

Systematic Underestimation of Permeability:
2D simulations systematically underestimated the true 3D permeability across all media types and porosities. The magnitude of this discrepancy was most pronounced at low porosities. For instance, in the Fibrous medium at $\phi=0.4$ , 2D permeability was nearly 10 times lower than 3D values. This bias was attributed to the loss of out-of-plane connectivity in 2D slices, which increases effective tortuosity and reduces the fraction of pore space contributing to active flow. The Kozeny–Carman relationship confirmed that porosity differences alone could not explain this discrepancy; the topological disconnect was the primary driver.
Role of the Euler Characteristic ( $\tilde{\chi}$ ):
The pore-normalized Euler characteristic, a topological measure of connectivity, largely tracked the difference between 2D and 3D permeability. Negative $\tilde{\chi}$ values (indicating well-connected networks dominated by loops) correlated with higher permeabilities and smaller 2D–3D discrepancies, while positive $\tilde{\chi}$ values (fragmented networks) correlated with lower permeabilities and larger discrepancies. The authors propose $\tilde{\chi}$ as a low-cost, geometry-only proxy to screen 2D slices before running flow simulations.
Medium-Specific Transport Trends:
Dimensional effects on transport metrics ( $M_1$ and $CV$ ) varied significantly by medium type:
- Fibrous Medium: The relative difference between 2D and 3D mean travel time and breakthrough spreading remained almost constant across the permeability range. This is attributed to flow organizing into dominant channels that are preserved in 2D.
- Gaussian Medium: The 2D–3D differences decreased as permeability increased. At low permeability, narrow throats restrict 2D flow but can be bypassed via out-of-plane routes in 3D; as permeability rises, these constraints relax, reducing the dimensional gap.
- Overlapping Spheres Medium: Differences in mean arrival time were large and bidirectional (some 2D slices overpredicted, others underpredicted 3D results) with no clear trend relative to permeability. This reflects the complex interplay of large, interconnected pores and the specific porosity of individual 2D slices.
Impact of Tracer Reactivity:
Conservative and reactive tracers exhibited similar dimensional biases. While reactive tracers showed a systematic 20–40% larger relative difference in mean travel time ( $M_1$ ) compared to conservative tracers, the overall pattern was governed by pore geometry rather than tracer reactivity. Breakthrough spreading ( $CV$ ) showed more variability, with the Gaussian medium showing a distinct bias where conservative tracers exhibited larger 2D–3D differences than reactive ones.

Significance and Claims
The paper concludes that pore geometry and connectivity are the primary governors of dimensional bias in pore-scale modeling, with tracer reactivity playing a secondary role. The study demonstrates that while 2D models are computationally efficient, they introduce systematic errors in permeability and transport predictions that cannot be corrected solely by matching porosity.

The authors claim that the pore-normalized Euler characteristic ( $\tilde{\chi}$ ) offers a practical, pre-simulation screening tool. By comparing the $\tilde{\chi}$ of candidate 2D slices against the 3D parent, researchers can identify slices that preserve the topological connectivity of the 3D system. If the $\tilde{\chi}$ values agree in sign and magnitude, a 2D slice may serve as a valid proxy; otherwise, a full 3D simulation is required. This approach suggests that when constructing 2D micromodels (e.g., microfluidic devices) to represent 3D systems, matching pore-scale connectivity via topological descriptors is as critical as matching porosity or surface area.

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