📄 chemistry

Effective diffusion in sharp-front reactive transport experiments

This study presents a closed-form analytical solution for sharp-front reactive transport in porous media, demonstrating that the experimentally inferred effective diffusion coefficient fundamentally differs from the intrinsic molecular diffusion coefficient due to coupled advection-diffusion-thermodynamic effects, a framework validated through Portland cement carbonation experiments under various flow conditions.

Original authors: Viktoriya M. Yarushina, Reinier van Noort, Yury Y. Podladchikov, Yannik F. Schneider

Published 2026-09-01

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CC BY 4.0

Original authors: Viktoriya M. Yarushina, Reinier van Noort, Yury Y. Podladchikov, Yannik F. Schneider

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 underground, where rock and fluid meet, a silent chemical battle is constantly being waged. When water or gas rich in carbon dioxide flows through porous materials like cement or rock, it triggers a reaction that transforms the solid structure. This process does not happen evenly throughout the material; instead, it creates a distinct boundary, a sharp front, that moves inward, separating the altered, reacted zone from the untouched original material. This phenomenon is critical for the safety of geological carbon storage, where cement seals are used to trap greenhouse gases deep underground. If these seals degrade too quickly, the stored gas could escape. For decades, scientists have tried to measure how fast this front moves to predict the lifespan of these seals, often assuming that the speed is driven by how easily molecules drift through the tiny pores of the material, a process known as diffusion.

However, a new study by researchers at the Institute for Energy Technology and the University of Lausanne challenges the way we interpret these measurements. They found that the speed at which this reaction front travels is not a simple reflection of how fast molecules move through the fluid. Instead, the movement is governed by a complex interplay between the flow of the fluid, the diffusion of molecules, and a chemical "buffering" effect where the solid material absorbs the reacting substance. The researchers developed a new mathematical framework to describe this movement, showing that the "effective" speed measured in experiments is actually a compound result of the fluid's movement and the massive amount of chemical storage the solid material can hold. This distinction is vital because it explains why different experiments, even those using very different methods, often produce confusingly different numbers for how fast the cement degrades.

To test this theory, the team turned to Portland cement, the material used to seal wells for carbon storage. They subjected identical cement samples to two very different environments. In one set of experiments, the cement sat quietly in a container filled with either carbon-dioxide-saturated water or wet, supercritical carbon dioxide. This is known as a batch experiment, where there is no forced flow, and the fluid simply sits around the sample. In the second set, they forced the same fluids to flow through the cement samples under pressure, simulating a scenario where fluid is actively moving through a well. Conventional wisdom might suggest that the forced flow would push the reaction front much faster than the quiet, static conditions. Yet, when the researchers measured the depth of the carbonation after weeks and months, the results were startlingly similar. In both the quiet and the forced-flow scenarios, the reaction front moved at a rate that followed the same pattern: the distance it traveled increased with the square root of time. This pattern is the hallmark of a process controlled by diffusion, not by the speed of the flowing fluid.

The researchers used this surprising similarity to explain a long-standing puzzle in the field. When scientists measure how fast a reaction front moves in an experiment, they often calculate an "effective diffusion coefficient" and treat it as a fixed property of the material, like the density of a rock. The new study shows that this is a misunderstanding. The speed of the front is not determined solely by how fast carbon dioxide molecules can wiggle through the water in the pores. It is also determined by how much of that carbon dioxide the solid cement can absorb and store as it reacts. The cement acts like a sponge, soaking up the carbon dioxide and converting it into a new mineral. Because the solid holds so much more carbon than the fluid does, the front moves slowly, not because the fluid is sluggish, but because the reaction has to "fill up" the solid sponge before it can advance. The researchers introduced a new way of looking at this, using dimensionless numbers to describe the balance between the flow of the fluid and the chemical capacity of the solid. They found that even when fluid is forced through the cement, the chemical storage capacity is so large that the flow makes almost no difference to the speed of the front.

To confirm that the reaction front was indeed sharp and distinct, the team examined the cement samples under powerful microscopes and used infrared light to map the chemical changes. They looked at the boundary between the reacted and unreacted cement with extreme precision. The images revealed that the transition was incredibly sharp, occurring over a distance of only about 50 to 100 micrometers, which is roughly the width of a human hair. In this narrow zone, the chemical composition changed abruptly. The calcium silicate hydrate, the glue that holds the cement together, disappeared and was replaced by calcium carbonate. The researchers also saw that the calcium in the cement did not dissolve and wash away; instead, it stayed put, rearranging itself into the new mineral structure. This observation supports the idea that the front is a sharp interface where the solid phase acts as a stationary buffer, absorbing the incoming fluid and reacting locally.

The implications of these findings are significant for how we design and test materials for carbon storage. The study demonstrates that simply increasing the flow of fluid through a well will not necessarily speed up the degradation of the cement seal, because the process is dominated by the chemical storage capacity of the material rather than the speed of the fluid. This means that experiments conducted in quiet, static containers can be directly compared to those with forced flow, provided the chemical conditions are similar. The "effective" speed measured in the lab is not a fundamental property of the fluid's ability to move, but a result of the specific chemical dance between the fluid and the solid. By understanding this, scientists can better predict how long a cement seal will last in the harsh environment of a deep well, ensuring that the carbon dioxide remains safely trapped underground for thousands of years. The work clarifies that the movement of these chemical fronts is a story of storage and balance, not just of flow.

Technical Summary: Effective Diffusion in Sharp-Front Reactive Transport Experiments

Problem Statement
Reactive transport in porous materials, such as cement systems exposed to CO₂, frequently generates sharp alteration fronts separating reacted and unaltered regions. While these fronts are critical for understanding fluid-rock interactions in geological CO₂ storage and well integrity, their propagation rates are often interpreted using empirical diffusion laws (e.g., $x \propto \sqrt{t}$ ). A significant ambiguity exists in the literature regarding the "effective diffusion coefficient" ( $D_{eff}$ ) inferred from these experiments. Reported values vary widely and are often orders of magnitude smaller than intrinsic molecular diffusion coefficients, leading to confusion about whether these values represent true material properties or artifacts of coupled transport-reaction processes. Furthermore, it remains unclear how to consistently compare experiments conducted under different boundary conditions (e.g., batch vs. forced-flow) or how advection influences front propagation when chemical buffering is strong.

Methodology
The authors develop a closed-form analytical solution for advection-diffusion reactive transport featuring a moving sharp reaction front. The model assumes:

Mass Conservation: The transport of a reactive species is governed by advection and diffusion in the fluid phase, coupled with exchange with the solid phase via equilibrium partitioning.
Sharp Interface: A moving boundary (Stefan-type) separates reacted and unreacted zones, across which porosity, density, and concentrations exhibit finite jumps.
Rankine-Hugoniot Condition: The front velocity is derived from mass conservation at the interface, accounting for discontinuities in total concentration.

The solution introduces two key dimensionless numbers:

Péclet Number ( $Pe$ ): Controls the relative importance of advection versus diffusion.
Chemical Stefan Number ($Ste$): Controls the front-propagation rate by quantifying the ratio of the mobile concentration driving flux to the total species accumulation required across the front (incorporating solid-phase storage and concentration jumps).

The framework is validated and applied to experimental data from Portland cement samples exposed to CO₂-rich fluids under both batch and forced-flow conditions. The experiments utilized wet supercritical CO₂ and CO₂-saturated water. Characterization was performed using Scanning Electron Microscopy with Back-Scatter Electrons (SEM-BSE), Energy Dispersive Spectroscopy (EDS), and Fourier Transform Infrared (FTIR) microscopy to resolve front morphology and chemical gradients.

Key Contributions

Analytical Framework: The paper provides a Stefan-type analytical solution that explicitly accounts for discontinuities in porosity, density, and concentrations at a moving interface. This solution unifies diffusion- and advection-dominated regimes through a composite parameter $\Lambda = Pe / (Ste \cdot R)$ , where $R$ is the retardation factor.
Redefinition of Effective Diffusion: The study demonstrates that the "effective diffusion coefficient" inferred from front propagation is not an intrinsic material property of the pore fluid. Instead, it is a compound propagation coefficient dependent on the molecular diffusion coefficient ( $D_f$ ) scaled by the chemical Stefan number ( $D_{eff} = D_f \cdot Ste$ ).
Resolution of Discrepancies: The framework explains why experimentally inferred diffusion coefficients are often several orders of magnitude lower than molecular values: a small Stefan number implies that only a small fraction of the diffusive flux contributes to front advancement, with the majority consumed by solid-phase storage.
Experimental Validation: The authors show that despite fundamentally different exposure methodologies (batch vs. forced-flow), front propagation rates can be similar if the governing dimensionless parameters are comparable.

Results

Diffusion-Controlled Propagation: In both batch and forced-flow experiments involving Portland cement, the carbonation front propagation followed a square-root-of-time scaling ( $x \propto \sqrt{t}$ ). Calculated Péclet numbers were low ( $Pe \approx 0.2$ for wet supercritical CO₂ and lower for CO₂-saturated water), indicating that even under imposed flow, transport remained diffusion-dominated.
Effective vs. Intrinsic Coefficients: The inferred effective diffusion coefficients were $1.4 \times 10^{-11}$ m²/s (CO₂-saturated water) and $2.5 \times 10^{-11}$ m²/s (wet supercritical CO₂). These are orders of magnitude smaller than the intrinsic molecular diffusion of CO₂ in water ( $\sim 10^{-8}$ m²/s). The authors attribute this to small Stefan numbers ( $Ste \sim 10^{-3}$ ), reflecting the high carbonation capacity of the solid phase relative to the fluid concentration contrast.
Front Morphology: High-resolution SEM-BSE, EDS, and FTIR observations confirmed that reaction fronts are extremely sharp, with transition zones localized to micrometer scales (approx. 50–110 µm).
- FTIR analysis revealed that carbonate appearance and C-S-H gel disappearance occur over narrow, distinct zones.
- EDS profiles showed sharp transitions in Ca/Si ratios, supporting the assumption of a sharp interface and local equilibrium.
- In forced-flow experiments, a narrow enrichment zone of Ca/Si was observed immediately behind the front, but the overall front remained sharp, validating the immobile-calcium approximation in the low- $Pe$ regime.

Significance and Claims
The paper claims to provide a general, physically transparent basis for interpreting alteration-front propagation in reactive porous media. By distinguishing between intrinsic molecular diffusion and the effective front-propagation coefficient, the study resolves the apparent contradiction between widely varying experimental diffusion values and consistent molecular diffusion properties. The authors assert that their framework allows for the consistent comparison of experiments performed under different boundary conditions (batch vs. flow-through) and fluid compositions. The results support the applicability of sharp-interface, local-equilibrium formulations for modeling reactive transport in systems ranging from engineered cement barriers to natural fluid-rock interaction zones, provided the governing dimensionless numbers ( $Pe$ , $Ste$, and $\Lambda$ ) are properly characterized.

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