⚡ electrical engineering

Testing spatial Markovianity of breakthrough curves: a model-free criterion to discriminate heterogeneity-driven from mass-transfer- driven non-Fickian transport

This paper introduces a model-free criterion based on the self-convolution property of spatially Markovian transport to distinguish between heterogeneity-driven and mass-transfer-driven non-Fickian transport, demonstrating through simulations that combining this test with a flow-rate tail analysis can accurately classify transport regimes in multi-distance tracer tests.

Original authors: Thi Thanh Hoa Nguyen, Thi My Hong Tran, Song Giang Le

Published 2026-11-05

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CC BY 4.0

Original authors: Thi Thanh Hoa Nguyen, Thi My Hong Tran, Song Giang Le

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

When water moves through the ground, it rarely flows in a straight, uniform line. Instead, it navigates a labyrinth of soil and rock, encountering pockets of high resistance and channels of easy passage. This uneven journey creates a phenomenon scientists call non-Fickian transport. In a perfectly uniform world, a drop of dye injected into an aquifer would spread out smoothly, like ink in a glass of still water. But in the real, messy underground, the dye arrives in a chaotic pattern: some molecules race ahead through fast lanes, while others get stuck in slow zones, emerging much later than expected. This creates a "breakthrough curve," a graph showing how much dye arrives at a monitoring well over time. The problem is that this graph looks the same whether the delay is caused by the water taking a winding, fast path through a rocky maze or by the dye getting trapped in tiny, stagnant pockets of water that it slowly diffuses out of. For decades, researchers have struggled to tell these two very different physical causes apart using just a single measurement, often leading to models that fit the data perfectly but fail to predict what will happen further downstream.

A team of researchers set out to solve this puzzle by looking at the problem from a different angle. Instead of trying to guess which model fits a single curve, they asked a simple question: if you measure the dye at one distance, can you predict exactly what the curve will look like at twice that distance? They discovered that for many standard models of groundwater flow, the answer is yes. If the ground is uniform or if the trapping happens in a way that forgets the past, the curve at the second distance is simply a mathematical combination of the first curve with itself. However, they found that when the delay is caused by the complex, correlated structure of the rock itself—where fast paths lead to other fast paths and slow paths lead to other slow paths—this prediction breaks down. The curve at the second distance does not match the prediction; it arrives too early and spreads out differently.

To prove this, the scientists ran thousands of computer simulations of water moving through virtual underground fields. They created scenarios where the delay came purely from the rock's structure, scenarios where it came purely from trapping, and scenarios where both happened at once. In one striking test, they took a simulation of a highly heterogeneous field, where the rock varied wildly, and fitted a standard "trapping" model to the data collected at a specific distance. This model matched the data perfectly, looking like a flawless solution. But when they used that same model to predict the arrival of the dye at a distance four times further away, it failed miserably, missing the early arrivals and underestimating the late ones. This confirmed that a single measurement cannot distinguish between the two causes, no matter how well a model fits it.

The researchers then developed a two-step test to separate these causes, using data from two different distances and two different flow speeds. First, they compared the curve at a second distance against the mathematical prediction derived from the first. If the prediction failed, it signaled that the ground was structurally heterogeneous, with water remembering its path. If the prediction held true, the delay was likely due to simple trapping. Second, they slowed down the flow of water in their simulations. They found that if the delay was caused by diffusion into stagnant pockets, the tail of the curve would shrink significantly when the water moved slower. If the delay was caused by the rock structure, the curve would not change in the same way. By combining these two checks, the team could correctly identify the underlying mechanism in 86% of their simulated cases.

This approach offers a new way to understand the hidden world beneath our feet without needing to guess the right mathematical model. It suggests that by measuring tracer movement at multiple distances and under different flow conditions, scientists can finally tell whether the delays they see are due to the complex architecture of the underground or to the solute getting stuck in dead ends. The study, which relied entirely on computer simulations, provides a clear, model-free method to diagnose these transport problems. It turns the choice of a transport model from a guessing game into a testable hypothesis, allowing researchers to design better experiments and make more reliable predictions about how contaminants or nutrients will move through the environment. While the results are currently limited to simulated data, the method lays the groundwork for applying these rigorous tests to real-world field experiments, potentially transforming how we manage groundwater resources.

Technical Summary: Testing Spatial Markovianity of Breakthrough Curves

Problem Statement
Non-Fickian breakthrough curves (BTCs) in heterogeneous porous media are commonly observed, characterized by early arrivals and long late-time tails. These features typically arise from two distinct physical mechanisms: (1) spatially correlated velocity heterogeneity (creating persistent fast and slow pathways) and (2) mass transfer between mobile and immobile regions (trapping). A fundamental challenge in hydrogeology is that a single BTC cannot distinguish between these mechanisms. Macroscopic models such as the Advection-Dispersion Equation (ADE), Mobile-Immobile (MIM), Multirate Mass Transfer (MRMT), and Continuous Time Random Walk (CTRW) can all reproduce the same BTC shape with different parameters. Consequently, a good model fit does not reveal the underlying physical process, and model selection criteria (e.g., AIC) often fail to identify the correct mechanism, sometimes inferring mass transfer where only heterogeneity exists.

Methodology
The authors propose a model-free approach to discriminate between heterogeneity-driven and mass-transfer-driven transport by exploiting the spatial Markov property. The study utilizes two-dimensional particle-tracking simulations in heterogeneous hydraulic conductivity fields ( $\ln K$ ) with varying variance ( $\sigma^2_{\ln K}$ ) and capacity ratios ( $R$ ).

The methodology relies on two primary tests:

The Spatial Markov (Self-Convolution) Test: Based on the proposition that if transport is spatially Markovian and stationary at scale $d$ , the BTC at distance $2d$ must equal the self-convolution of the BTC at distance $d$ ( $f_{2d} = f_d * f_d$ ). This identity holds for ADE, MIM, MRMT, and uncorrelated CTRW models. Violations indicate spatially variable transport (velocity memory or non-stationarity). The test compares observed arrival times at $2d$ against the distribution generated by summing pairs of arrival times from $d$ .
The Flow-Rate (Exchange) Test: Based on the scaling behavior of diffusion-controlled exchange. If the mean flow velocity ( $U$ ) is reduced, advective travel times scale as $1/U$ , but diffusion-controlled exchange delays do not scale linearly. Consequently, the late-time tail of the BTC, when scaled by $U$ , should contract at lower flow rates. This signature is absent in pure advection through a fixed geometry.

The study evaluates these tests using:

Mechanism A: Resolved heterogeneity without exchange.
Mechanism B: A homogeneous medium with sub-grid exchange (calibrated to mimic Mechanism A).
Mechanism C: Heterogeneity combined with sub-grid exchange.
Data Types: Both particle arrival times (noise-free) and emulated concentration curves with Poisson counting noise, multiplicative Gaussian noise, and detection limits.

Key Contributions and Results

Equifinality of Single Curves: The study confirms that a single BTC cannot distinguish between heterogeneity and mass transfer. A MIM model calibrated on a highly heterogeneous field ( $\sigma^2_{\ln K} = 4$ ) reproduced the BTC over six orders of magnitude at the calibration distance ( $8\lambda$ ). However, this model failed to predict arrival times at larger distances ( $64\lambda$ ), underestimating the tail by a factor of two, while its fitted parameters (dispersivity and exchange rate) drifted significantly with distance.
Validity of the Self-Convolution Test: The self-convolution test successfully identified spatially correlated velocities. For heterogeneous fields, the test rejected the null hypothesis (Markovianity) with high power (up to 99% for $N=3000$ samples) because travel times over successive segments are positively correlated. In contrast, the test maintained a low false-alarm rate (2–7.5%) for homogeneous exchange-only media (Mechanism B), where the identity held.
Flow-Rate Signature: The flow-rate test effectively isolated exchange mechanisms. When flow velocity was reduced, the late-time tail of exchange-dominated systems contracted in scaled time, whereas pure heterogeneity systems showed minimal contraction.
Combined Classification: By applying both tests, the authors achieved an average correct classification rate of 86% for distinguishing between heterogeneity-only, exchange-only, and mixed regimes (with $N=3000$ $N = 3000$ samples).
- Limitations: Strong exchange ( $R=1$ ) can mask heterogeneity, reducing the power of the Markov test. Conversely, strong heterogeneity ( $\sigma^2_{\ln K} \ge 2$ ) can mimic exchange signatures due to diffusion into low-conductivity inclusions, leading to misclassification in some cases.
Robustness: The tests remained effective for noisy, censored concentration data, though power decreased with higher noise levels. The test was found to be conservative regarding false alarms.

Significance and Claims
The paper claims to provide a model-free criterion for identifying the physical mechanisms behind non-Fickian transport. Unlike previous approaches that rely on fitting specific models or comparing model complexities, this method tests a fundamental hypothesis about the spatial structure of transport:

It demonstrates that violations of the spatial Markov property (specifically the self-convolution identity) are a definitive signature of spatially variable transport (velocity memory or non-stationarity), rejecting the entire class of uncorrelated models (ADE, MIM, MRMT, uncorrelated CTRW).
It establishes that flow-rate scaling of the late-time tail is a distinct signature of diffusion-controlled mass transfer.
The combined approach supports the design of multi-distance, multi-rate tracer tests to identify transport mechanisms without the ambiguity of model calibration.

The authors conclude that while a single BTC is insufficient for mechanism identification, comparing curves at multiple distances and flow rates allows for the discrimination of heterogeneity and mass transfer in most practical scenarios, turning the choice of a macroscopic transport model into a testable hypothesis about the underlying physics.

Technical Summary: Testing Spatial Markovianity of Breakthrough Curves

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