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Data-Informed Mathematical Characterization of Absorption Properties in Artificial and Natural Porous Materials

This contribution presents a combined experimental and mathematical modeling approach that encompasses a noise-reducing adaptation procedure and simulations of partial differential equations to characterize the water absorption properties of various porous materials for cultural heritage conservation.

Original authors: Elishan C. Braun, Gabriella Bretti, Melania Di Fazio, Laura Medeghini, Mario Pezzella

Published 2026-05-08

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CC BY 4.0

Original authors: Elishan C. Braun, Gabriella Bretti, Melania Di Fazio, Laura Medeghini, Mario Pezzella

Original paper licensed under CC BY 4.0 (http://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

Imagine you have a sponge, a piece of bread, and a rock. If you dip the bottom end of each of these objects into a cup of water, they all absorb the water, but they do so in very different ways. Some drink quickly, others slowly, and some stop drinking long before they are "full."

This article is like a detective story in which the authors want to find out exactly how various building materials (such as those used in ancient Roman ruins) absorb water. They aim to create a "mathematical recipe" that precisely predicts how much water a specific stone or brick will absorb at any given moment.

Here is the story of how they did it, broken down into simple steps:

1. The Problem: Why do old buildings get sick?

Imagine old buildings as living beings that can fall ill. One of the biggest "germs" is water. When rain or groundwater seeps into the tiny holes (pores) of stones and bricks, it causes trouble. It can promote the growth of salt crystals (like frost on a window), introduce pollutants, or allow mold to thrive.

To prevent this, scientists must understand precisely how these materials "drink" water. However, measurement is tricky because real data is messy, like a shaky video recording.

2. The Experiment: The "Thirst Test"

The researchers took samples of four different material types:

White Marble: A very dense, smooth stone (like a hard candy).
Travertine: An uneven, porous stone often used in fountains (like a stone sponge).
Wackestone: A muddy, fine-grained rock.
Mortar: The "glue" used to hold bricks together. They tested both ancient glue and modern, eco-friendly glue made from algae.

They dipped the bottom end of these samples into water and weighed them every few minutes. It is like watching a thirsty person drink through a straw and recording how much liquid was consumed every second.

3. The Mathematics: Building a "Digital Twin"

The authors did not just look at the numbers; they built a mathematical model. Imagine this model as a "digital twin" of the stone.

The Rules: They used a set of rules (equations) based on how water naturally flows through holes.
The Simulation: They ran a computer program that acted like the stone. The computer tried to guess how the water would move inside.
The Challenge: The computer must know specific settings to function correctly, such as: "How fast does the water move?" or "How full can the stone get before it stops?" These are the "knobs" on the machine.

4. Cleaning the Data: Smoothing the Shaky Video

Real experiments are never perfect. Sometimes the scale shakes, or water drips down the side, causing the data to look like a jagged, shaky line instead of a smooth curve.

The authors developed a special smoothing technique. Imagine you have a shaky video of a runner. You cannot simply delete the shaky parts, or you would lose the speed. Instead, they used a clever mathematical trick to redraw the line so that it moves smoothly upward (since water absorption should always increase, never decrease), while preserving the runner's true speed. This provided them with a clean, reliable picture of what actually happened.

5. Tuning the Machine: The "Swarm" Search

Now came the difficult part: finding the right settings for the computer model. They had to match the "digital twin" with the "real stone."

The Problem: There are millions of possible settings. If one tried them out individually, it would take forever.
The Solution: They used a method called Particle Swarm Optimization. Imagine a flock of birds searching for the best landing spot. Each bird tries a different spot. If a bird finds a good spot, the others fly there. If they find an even better spot, everyone adjusts their course.
The Multigrid Trick: To speed this up, they started with a coarse, low-resolution map (like looking at a map from high up in an airplane) to find the general area. Once they found the general area, they zoomed in with a high-resolution map (like walking on the ground) to find the exactly perfect spot.

6. The Results: A Perfect Match

Once they finished tuning the "knobs," the computer model matched the real experimental data almost perfectly.

For Marble and Wackestone: The model was a perfect match.
For Travertine: This stone was tricky because it has layers (like a sandwich). The water moved differently depending on whether it flowed along the layers or across them. The researchers realized they had to treat these two directions as different "types" of travertine to achieve an accurate match.
For Mortar: The model successfully predicted how the new, eco-friendly mortars absorbed water compared to the ancient Roman mortars.

The Conclusion

The authors created a reliable tool that combines real experiments with intelligent computer simulations.

They proved that by cleaning up messy data and using a "flock of birds" search algorithm to tune the mathematics, they can accurately predict how various building materials absorb water. This is a powerful tool for anyone trying to preserve old buildings, as it helps them understand how water will damage the structure over time and enables them to select the right materials for repairs.

Technical Summary: Data-Driven Mathematical Characterization of Absorption Properties in Artificial and Natural Porous Materials

Problem Statement
The decay of cultural heritage, particularly historical buildings and archaeological sites, is significantly driven by water ingress into porous natural stones and artificial geomaterials. Water acts as a carrier for reactive flows and triggers destructive processes such as salt crystallization, pollutant ingress, and biocolonization. Although capillary suction is a primary degradation mechanism, detailed mathematical characterizations are lacking for specific materials used in restoration, particularly eco-friendly new-generation mortars and certain natural stones like wackestone. Existing literature frequently describes these materials from a mineralogical-petrographic perspective and omits rigorous mathematical modeling to predict water uptake dynamics. This work addresses the need to integrate experimental data with mathematical modeling to reliably characterize the absorption properties of these materials.

Methodology
The authors propose a data-driven framework that combines laboratory experiments with a partial differential equation (PDE) model and numerical optimization.

Experimental Setup: Capillary water uptake tests were conducted on natural stones (Carrara marble, Tivoli travertine, wackestone) and artificial geomaterials (hydraulic mortars based on Roman recipes and air-lime mortars with organic/acrylic additives). Samples were dried, partially immersed in distilled water, and weighed at specific intervals to determine the mass of water absorbed per unit area ( $Q_{data}$ ).
Mathematical Modeling: The suction process is modeled using a one-dimensional Richards equation derived from Darcy's law. The model describes the evolution of fluid saturation $\theta(z,t)$ $θ (z, t)$ in a porous medium. A specific absorption function $B(s)$ $B (s)$ is defined, where $s$ $s$ represents the degree of saturation. This function is parameterized by:
- Residual saturation ( $s_R$ ) and maximum saturation ( $s_S$ ).
- A diffusion rate ( $D$ ).
- A water exchange rate ( $K_w$ ) at the interface.
- The material porosity ( $n_0$ ).
Numerical Simulation: The PDE is solved using a Method of Lines (MOL). Spatial derivatives are discretized using second-order central finite differences, and the resulting system of ordinary differential equations (ODE) is integrated using a second-order Heun approximation. This explicit scheme demonstrates quadratic convergence and computational efficiency compared to standard Forward-Time-Central-Space (FTCS) schemes.
Data Preprocessing: To address instrumental errors and non-monotonicities in raw experimental data (common in natural materials and fast-absorbing mortars), a difference reconstruction technique is applied that preserves monotonicity. This method formulates the data fitting problem as an unconstrained optimization of a function $w_f(t)$ within a space of smooth monotone functions using truncated Legendre expansions and regularization.
Parameter Calibration (Inverse Problem): Optimal model parameters are determined by minimizing a cost function that compares the simulated total fluid absorption ( $Q_{num}$ $Q_{n u m}$ ) with experimental/reconstructed data ( $Q_{data}$ $Q_{d a t a}$ ). The loss function combines:
- The squared relative Euclidean (SRE) error for pointwise deviations.
- The Dynamic Time Warping (DTW) error to account for temporal misalignments.
- A penalty term for deviations in the final state.
  Optimization is performed using a Particle Swarm Optimization (PSO) algorithm, extended by a multigrid strategy (refinement from coarse to fine) to handle nonlinearities and local minima.

Main Contributions

Integrated Framework: The work presents a novel workflow linking laboratory capillary absorption tests with a calibrated PDE model, specifically tailored for both natural stones and modern restoration mortars.
Monotonicity-Preserving Reconstruction: A new differential fitting technique is introduced for preprocessing experimental data. This method reduces noise and instrumental errors while strictly adhering to the physical constraint that water uptake must not decrease, without losing the information content of the original measurements.
Robust Optimization Strategy: The development of a PSO-based multigrid calibration approach enables efficient and robust estimation of model parameters ( $s_R, s_S, D, K_w, n_0$ ), even with complex, non-smooth objective functions.
Material Characterization: The study provides specific, calibrated parameters for a diverse group of materials, including wackestone and eco-friendly air-lime mortars (GS, GSN, GSP), which have not previously been mathematically characterized in this manner.

Results

Model Accuracy: The calibrated models successfully reproduce the suction dynamics for most materials. The combined use of SRE and DTW errors in the cost function effectively balances amplitude and shape differences, resulting in low objective function values across all tested materials.
Anisotropy in Travertine: A notable discrepancy was observed in travertine samples. By classifying samples based on the orientation of anisotropy planes (parallel vs. perpendicular to water flow), the authors achieved distinct parameter sets for each class. This underscores the model's sensitivity to structural anisotropy, which, although not explicitly modeled in the 1D PDE, is captured through parameter differentiation.
Performance: The MOL numerical scheme demonstrated superior performance and quadratic convergence compared to FTCS methods. The reconstruction procedure effectively smoothed experimental noise while preserving the physical monotonicity of the absorption curves.
Parameter Ranges: The study provided specific diffusion rates and saturation levels for each material. For instance, hydraulic mortars (OT) exhibited high porosity (33.5%–44.3%) and varying diffusion rates, while natural stones showed lower porosity and different saturation behaviors.

Significance and Claims
The authors claim that this work provides a realistic and reliable tool for characterizing the water uptake properties of construction materials relevant to cultural heritage. By moving beyond purely descriptive petrographic analysis to a quantitative mathematical approach, the study offers valuable insights into degradation mechanisms. The proposed methodology is presented as a versatile tool for investigating and preserving cultural heritage, capable of handling both natural and artificial geomaterials. The work modestly suggests that while the current 1D model is effective, future work could incorporate anisotropy and heterogeneous permeability to further refine the representation of complex materials like travertine. The overarching goal is the deployment of these techniques as diagnostic and predictive tools for real-world conservation efforts.