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Dissolution of carbonate stones caused by CO2 pollutant: an erosion model

This paper presents a new mathematical model and a corresponding numerical algorithm based on finite difference and level-set methods to simulate the erosion of carbonate stones caused by carbonic acid dissolution within a Darcy flow framework, with results consistent with existing literature.

Original authors: Elishan Christian Braun, Gabriella Bretti, Samuele Ferri, Maria Laura Santarelli, Matteo Semplice

Published 2026-04-01

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CC0 1.0

Original authors: Elishan Christian Braun, Gabriella Bretti, Samuele Ferri, Maria Laura Santarelli, Matteo Semplice

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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

Imagine ancient stone monuments—like the marble altar in the picture from the paper—as giant, porous sponges. They aren't just solid blocks; they are full of tiny, invisible holes (pores) that let air and water sneak inside.

This paper is about building a digital crystal ball to predict how these sponges slowly dissolve and crumble away because of pollution.

Here is the story of the paper, broken down into simple parts:

1. The Villain: The "Acid Rain" in a Bottle

You know how rain can be slightly acidic? In the modern world, the air is full of Carbon Dioxide (CO2) from cars and factories. When this CO2 mixes with moisture in the air (or rain), it turns into carbonic acid.

Think of this acid as a tiny, invisible army of termites. When they land on a stone monument, they don't just sit on the surface; they seep into the tiny pores of the stone. Once inside, they start eating the "glue" that holds the stone's crystals together. Over centuries, this turns solid marble into soft, crumbly dust.

2. The Problem: We Need a Better Map

Scientists have tried to guess how fast this happens using simple formulas. But those formulas are like using a flat map to navigate a mountain range—they give you an average idea but miss the details. They don't tell you exactly how the damage spreads inside the stone, or how the shape of the stone changes over time.

The authors of this paper wanted to build a 3D simulation (a video game engine for stone decay) that is much more accurate.

3. The Solution: A "Digital Stone" in a Computer

The team created a mathematical model that acts like a virtual stone. They fed it real-world rules:

The Sponge Rule: How water moves through the tiny holes (Darcy flow).
The Chemical Rule: How the acid reacts with the stone to dissolve it.
The Eating Rule: As the stone dissolves, the holes get bigger, letting more acid in, which makes the stone dissolve faster. It's a vicious cycle.

They used a clever trick called the Level-Set Method. Imagine drawing the outline of a stone on a piece of paper. As the "acid" eats the edge, the line moves inward. This method allows the computer to track that moving line perfectly, even if the stone has weird shapes or curves, without the computer getting confused.

4. The Experiment: Running the Simulation

The researchers ran their model on a computer to see what happens over time. They tested two different ways of describing how water moves through the stone (a "symmetric" way and a more complex "asymmetric" way).

What they found:

The "Slow Burn": In normal conditions (like a city park), the stone erodes very slowly. The computer showed the acid front moving inward, dissolving the stone layer by layer. The results matched real-world data from old tombstones, proving their model works.
The "Catastrophe": They also tested a "worst-case scenario" where the stone is completely soaked in water (like being underwater). In this case, the acid rushed in, and the stone dissolved much faster. This helps them understand how extreme weather events could damage heritage sites.

5. Why This Matters

Think of this model as a weather forecast for stone damage.

For Museums: If a museum knows the local pollution levels and humidity, they can use this model to predict: "If we don't fix the roof, this statue will lose 1 millimeter of detail in 50 years."
For Conservationists: It helps them decide where to spend money. Should they protect the statue from rain? Or from high CO2 levels? The model can simulate different scenarios to find the best protection strategy.

The Bottom Line

This paper gives us a new, high-tech tool to watch ancient stones "age" in fast-forward inside a computer. By understanding exactly how pollution eats away at stone, we can better protect our history before it turns to dust.

In short: They built a video game where the goal isn't to win, but to save history by understanding exactly how the "acid rain" villain destroys the stone castle.

1. Problem Statement

The paper addresses the chemical degradation of carbonate stone artifacts (such as limestone and marble) caused by environmental pollutants, specifically carbon dioxide ( $CO_2$ ).

Context: Ancient buildings and sculptures are weathering at accelerated rates due to increased atmospheric $CO_2$ and acid rain (pH < 5).
Mechanism: $CO_2$ dissolves in water to form carbonic acid ( $H_2CO_3$ ), which penetrates the porous matrix of the stone. This acid reacts with calcium carbonate ( $CaCO_3$ ), causing dissolution and a subsequent increase in porosity until the material structure fails (erosion).
Limitations of Existing Models: Current statistical tools (e.g., the Lipfert erosion index) provide average estimations based on precipitation and pollutant deposition but fail to account for:
- Specific material properties (chemical homogeneity, porosity).
- Time evolution of degradation.
- Local microclimatic phenomena.
- The specific dynamics of $CO_2$ penetration.

2. Methodology

The authors propose a nonlinear reaction-transport system governed by Darcy flow in porous media to model the erosion process.

A. Mathematical Model

The system couples three primary variables:

Water Saturation ( $\theta$ ): The volume fraction of water in the pores.
Carbonic Acid Concentration ( $c_a$ ): The concentration of dissolved acid.
Porosity ( $n$ ): The evolving porosity of the stone.

The governing equations (System 3) include:

Moisture Transport: A nonlinear diffusion equation based on the Richards equation, utilizing a function $B(\theta/n)$ to relate capillary pressure and moisture content.
Acid Transport: A convection-diffusion-reaction equation describing the movement of carbonic acid and its consumption via chemical reaction.
Porosity Evolution: An ordinary differential equation where porosity increases as the acid reacts with the carbonate matrix ( $CaCO_3 + H_2CO_3 \rightarrow Ca^{2+} + 2HCO_3^-$ ).
Boundary Conditions: Coupled with Neumann/Dirichlet conditions representing the exchange of humidity and acid between the ambient air and the stone surface.

Absorption Functions: The authors define two formulations for the capillary pressure/moisture relationship:

Symmetric ( $B'$ ): A simpler 3-parameter model.
Asymmetric ( $B'_{kP}$ ): A more complex model separating permeability and capillary pressure, allowing for richer physical description (calibrated against experimental data).

B. Numerical Algorithm

To handle the moving boundary caused by material loss (erosion), the authors developed a hybrid numerical scheme:

Domain Discretization: Finite Difference Method (FDM) on a uniform Cartesian grid.
Level-Set Method: Used to track the moving interface (erosion front). The physical domain is embedded in a larger grid, and a level-set function $\phi(x) = n(x) - n_{max}$ defines the boundary.
Ghost Point Method: To handle boundaries that do not align with the grid, "ghost points" are projected onto the physical boundary along the normal vector. Boundary conditions are enforced via Lagrange interpolation (biquadratic) on these ghost points.
Time Integration: Implicit Euler method is used to allow for larger time steps and avoid stability constraints.

3. Key Contributions

Novel Mathematical Framework: Introduction of a coupled reaction-transport model specifically tailored for $CO_2$ -induced dissolution in carbonate stones, moving beyond statistical averages to mechanistic physics.
Robust Numerical Scheme: Development of a Level-Set Ghost-Point Finite Difference algorithm capable of simulating complex 2D geometries and tracking the erosion front without remeshing.
Parameter Calibration: The model parameters (reaction constants, diffusion coefficients, porosity thresholds) are calibrated against literature data regarding recession rates of tombstones in urban vs. suburban environments.
Flexibility: The framework allows for the integration of real-time environmental data (temperature, humidity, $CO_2$ levels) to simulate specific site conditions.

4. Results

The authors performed numerical tests in 1D and 2D to validate the model.

1D Simulations:
- Showed that both symmetric and asymmetric absorption functions yield nearly identical results for humidity and erosion front positions under standard conditions.
- The system reaches a steady state for humidity and acid concentration, while porosity increases until material loss occurs.
- Erosion Rate: The model predicted a recession rate of approximately 2.36 $\times 10^{-6}$ cm/h (approx. 2 mm/100 years) for suburban conditions, aligning with literature values.
2D Simulations:
- Demonstrated the model's ability to handle complex geometries (e.g., a rectangular stone sample).
- Visualized the evolution of the erosion front over one year, showing realistic surface recession.
Catastrophic Scenario:
- Simulated a "submerged" condition (full saturation).
- Results showed rapid water penetration but limited acid accumulation due to consumption by reaction, resulting in a significantly higher erosion slope (8.25 $\times 10^{-4}$ cm/h).
Convergence: The numerical scheme demonstrated a global order of accuracy of 1, consistent with the first-order time and space discretization used.

5. Significance and Future Work

Cultural Heritage Conservation: The model provides a predictive tool to correlate pollution levels with material degradation, enabling site-specific conservation strategies for monuments (e.g., the Ostia Antica Archaeological Park).
Scalability: While currently tested in 1D and 2D, the algorithm is designed to be easily extended to 3D.
Future Directions:
- Calibration of parameters ( $K_w, K_a, K_n, n_{max}$ ) using in-situ sensor data.
- Extension to 3D simulations.
- Incorporation of gravity effects (capillary rise) and freeze-thaw cycles.
- Modeling the interplay between erosion and crust formation (deposition).
- Increasing the order of the numerical scheme (e.g., Crank-Nicolson) for higher accuracy.

In conclusion, the paper presents a rigorous, physics-based approach to modeling stone erosion that bridges the gap between chemical reaction kinetics and macroscopic material degradation, offering a promising tool for the preservation of historical artifacts.