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arxiv: 2411.11129 · v1 · pith:W6NX2M4Bnew · submitted 2024-11-17 · 🧮 math.NA · cs.NA

Mathematical modelling of the water absorption properties for historical lime-based mortars from Catania (Sicily, Italy)

Gabriella Bretti , Cristina M. Belfiore This is my paper

Pith reviewed 2026-05-23 16:51 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords lime-based mortarswater absorptioncapillary pressurepermeabilityimbibitionretention curvemathematical modellingvolcanic aggregates
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The pith

A mathematical model with saturation-dependent capillary pressure and permeability, calibrated to imbibition data, reproduces the retention curves of two historic lime mortars from Catania.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to model how water moves through lime-based mortars made with ghiara and azolo volcanic aggregates by expressing capillary pressure and permeability as functions of the saturation level inside the material. The specific forms and parameters of those functions are obtained by fitting a numerical algorithm to experimental imbibition measurements on samples of each mortar. Once calibrated, the model generates a retention curve through simulation that is then compared directly with laboratory data. The approach reproduces the main observed features of water absorption for both aggregate types. This matters because moisture transport controls the durability of historic masonry exposed to rain and humidity.

Core claim

The authors formulate a mathematical model in which capillary pressure and permeability are written as functions of saturation; the concrete expressions and their coefficients are fixed by calibrating the numerical scheme against imbibition experiments performed on the two mortars. Validation consists of comparing the retention curve produced by the calibrated simulation with the curve measured in the laboratory. The comparison shows that the model reproduces the principal features of the experimentally observed absorption behavior for both the ghiara and azolo materials.

What carries the argument

Calibration of saturation-dependent expressions for capillary pressure and permeability against imbibition data to produce matching retention curves.

If this is right

  • The retention curve for each mortar can be obtained from simulation once the imbibition data are available.
  • The same calibration procedure works for both volcanic-aggregate mortars examined.
  • Direct laboratory measurement of the retention curve is not required once the imbibition-based calibration has been performed.
  • The numerical algorithm supplies a practical way to predict the main moisture-transport characteristics of these historic materials.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could lower the amount of material that must be sampled from protected historic structures.
  • Analogous calibration steps might be applied to other porous building materials whose imbibition behavior is easier to measure than their full retention curves.
  • The calibrated functions could serve as input for larger-scale simulations of moisture movement through entire walls or buildings.
  • If the functional forms prove transferable, the approach could support comparative studies of mortars from different historic sites.

Load-bearing premise

Capillary pressure and permeability can be written as functions of saturation whose shapes and parameters are fixed solely by fitting to imbibition measurements on these two mortars.

What would settle it

If the retention curve generated by the calibrated simulation deviates substantially from the laboratory-measured retention curve for either the ghiara or the azolo mortar, the claim that the model reproduces the main experimental features would be refuted.

Figures

Figures reproduced from arXiv: 2411.11129 by Cristina M. Belfiore, Gabriella Bretti.

Figure 1
Figure 1. Figure 1: Laboratory experiment for the measurement of water absorption for azolo and ghiara mortars. Laplace equation: (3) P i MIP = Tw cos(θw) THg| cos(θHg)| P i Hg, with i = 1, . . . , NMIP the number of MIP measurements, Tw and THg the surface tensions of water and mercury, respectively (Tw = 0.073N/m and THg = 0.489N/m); and θw and θHg the contact angles of water and mercury, respectively, θw = 0◦ and θHg = 130… view at source ↗
Figure 2
Figure 2. Figure 2: Left panel: Plot of functions B(s) and B′ (s) defined in (11) and obtained for sR = 0.1, sS = 0.9 and D =5e-03. Right Panel: Plot of functions B(s) and B′ (s) defined in (18) and obtained for sR = 0.1, sS = 0.9, α = 0.25, c =1.35e+05, Ks = 8e-10,γ = 1.45 with DkP =5e-03. 2.5. A new formulation of absorption function B′ for Darcy’s law. A slightly different formulation of function B in Darcy’s law (5) that … view at source ↗
Figure 3
Figure 3. Figure 3: Plot of permeability function (13) for Ks =1e-10 and of capillary function (14) for fixed c =1e+06, assuming different values of α. Then, keeping the same framework introduced in the previous paragraph we consider B′ as a compactly supported function in [sR, sS]. We introduce this new formulation using (13) and (15) in (5) so that we get the following expression for the derivative ∂sBkP called for simplici… view at source ↗
Figure 4
Figure 4. Figure 4: Left panel: Plots of the imbibition curve for ghiara mortar obtained numerically by the mathematical model (6)-(8)-(9)-(18) (green line) VS experimental data (red diamonds). Right panel: Plot of B′ function (18) for ghiara mortar (blue line) [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Left panel: Plot of the imbibition curve for azolo mortar obtained numerically by the mathematical model (6)-(8)-(9)-(18) (green line) VS experimental data (red di￾amonds). Right panel: Plot of B′ function (18) for azolo mortar (blue line) [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Left Panel: Plot of water retention curve for ghiara mortar. Experimental data (blue dotted line) VS curve profiles obtained numerically by the two models for capillary pressure Pc(s), reported in (14) for α = 0.25, c = 1.4e+06 (orange line). Right Panel: Plot of water reten￾tion curve for azolo mortar. Experimental data (blue dot￾ted line) VS curve profiles obtained numerically by the two models for capil… view at source ↗
Figure 7
Figure 7. Figure 7: Left Panel: Plot of the water saturation as a function of length at different times during imbibition ob￾tained numerically for ghiara mortar. Right Panel: Plot of the water saturation as a function of length at different times during imbibition obtained numerically for azolo mortar. 3.3. Sensitivity analysis. In the present Section we perform a sensitivity analysis that explores the space of parameter val… view at source ↗
Figure 8
Figure 8. Figure 8: Local sensitivity analysis with respect to model parameters for ghiara mortar: plots of the error functional E2 [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Local sensitivity analysis with respect to model parameters for azolo mortar: plots of the error functional E2. Conversely, from 10 to 90 mins, the slope of imbibition curve of azolo mortar appears to be slightly steepest than that of ghiara mortar. This unlike behavior at different times can be justified by the fact that at the beginning of the test the ghiara mortar tends to absorb water more quickly due… view at source ↗
read the original abstract

In this paper we propose a mathematical model of the capillary and permeability properties of lime-based mortars from the historic built heritage of Catania (Sicily, Italy) produced by using two different types of volcanic aggregate, i.e. ghiara and azolo. In order to find a formulation for the capillary pressure and the permeability as functions of the saturation level inside the porous medium we calibrate the numerical algorithm against imbibition data. The validation of the mathematical model was done by comparing the experimental retention curve with the one obtained by the simulation algorithm. Indeed, with the proposed approach it was possible to reproduce the main features of the experimentally observed phenomenon for both materials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a mathematical model for the capillary pressure and permeability properties of historical lime-based mortars from Catania produced with ghiara and azolo volcanic aggregates. The model expresses capillary pressure and permeability as functions of saturation, with parameters calibrated against imbibition data; validation consists of comparing the simulated retention curve to experimental data, with the claim that main observed features are reproduced for both materials.

Significance. If the calibration is robust and the validation independent, the approach could offer a practical method for predicting moisture transport in heritage mortars by combining limited experimental data with numerical modeling, which is relevant for conservation applications. The work applies standard inverse modeling techniques to a real-world heritage context.

major comments (2)
  1. [Abstract] Abstract: the validation compares a simulated retention curve (obtained after fitting p_c(S) and k(S) to imbibition data) to experimental retention data. Because the retention curve is the direct experimental counterpart of the capillary pressure function p_c(S), this step primarily checks consistency of the fitted p_c(S) rather than testing an independent prediction of absorption behavior.
  2. [Abstract] Abstract: no functional forms, governing equations, or error metrics (e.g., RMSE or R² for the imbibition fit or retention-curve comparison) are supplied, nor is it stated whether the retention-curve data were withheld from the calibration; without these details the claim that “main features” are reproduced cannot be assessed quantitatively.
minor comments (2)
  1. The manuscript should include the explicit expressions chosen for p_c(S) and k(S) together with the numerical scheme used to solve the resulting PDE system.
  2. Provide the raw imbibition and retention data (or at least tabulated values) and the fitting procedure so that the calibration can be reproduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each point below and will revise the abstract to improve clarity and provide the requested quantitative details.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the validation compares a simulated retention curve (obtained after fitting p_c(S) and k(S) to imbibition data) to experimental retention data. Because the retention curve is the direct experimental counterpart of the capillary pressure function p_c(S), this step primarily checks consistency of the fitted p_c(S) rather than testing an independent prediction of absorption behavior.

    Authors: We agree that the retention-curve comparison primarily validates the p_c(S) function obtained via inverse modeling of the imbibition data. However, the retention measurements constitute an independent experimental dataset not used in the calibration, providing a cross-check between dynamic imbibition and static retention experiments. The calibration simultaneously determines both p_c(S) and k(S) through numerical solution of the flow equations, so the retention comparison adds value beyond pure consistency. We will revise the abstract to explicitly state that retention data were withheld from calibration and to describe the validation more precisely. revision: yes

  2. Referee: [Abstract] Abstract: no functional forms, governing equations, or error metrics (e.g., RMSE or R² for the imbibition fit or retention-curve comparison) are supplied, nor is it stated whether the retention-curve data were withheld from the calibration; without these details the claim that “main features” are reproduced cannot be assessed quantitatively.

    Authors: We accept that the abstract is too brief and omits these specifics. In the revised version we will add: (i) the governing equation (unsaturated flow model), (ii) the chosen functional forms for p_c(S) and k(S), and (iii) quantitative error metrics (RMSE or equivalent) for both the imbibition calibration and the retention-curve comparison. We will also state explicitly that retention data were withheld from the fitting procedure. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard calibration followed by independent validation

full rationale

The paper calibrates functional forms and parameters for capillary pressure and permeability versus saturation to imbibition data, then validates by comparing the resulting simulation output against a separately measured experimental retention curve. This is a standard inverse problem workflow with forward prediction on an independent dataset, not a reduction of any claimed prediction to the fitting inputs by construction. No self-definitional steps, fitted-input predictions, or self-citation chains appear in the described chain.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Only abstract available; model rests on fitting saturation-dependent functions to imbibition data and standard assumptions of porous-media flow.

free parameters (1)
  • parameters defining capillary pressure and permeability versus saturation
    Explicitly calibrated against imbibition data for each aggregate.
axioms (1)
  • domain assumption Standard continuum assumptions for unsaturated flow in porous media (Darcy-type relations with saturation-dependent coefficients)
    Required to formulate the numerical algorithm for capillary and permeability properties.

pith-pipeline@v0.9.0 · 5641 in / 1188 out tokens · 40977 ms · 2026-05-23T16:51:50.352054+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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

    math.NA 2025-06 unverdicted novelty 3.0

    Experimental imbibition data on four porous materials are pre-processed with a monotonicity-preserving fit and used to calibrate a PDE model of capillary absorption.

Reference graph

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