This article investigates a space-time differential model related to the degradation of stone artifacts caused by exposure to air and atmospheric agents, which specifically lead to the accumulation of salt crystals in the material. A numerical method based on finite-element space discretization and implicit-explicit time marching is proposed as an extension of a one-dimensional finite-difference framework introduced in the literature. Within the same one-dimensional setting, a sensitivity analysis is performed, based on the techniques developed therein. They are also used as a comparison tool for the finite-element formulation, here introduced for more realistic simulations in higher space dimensions. Considerations about stability will be provided, together with an experimental convergence analysis highlighting the performance of the proposed approach. Numerical results in two and three space dimensions, obtained by an efficient code implementation, will be presented and discussed.
Lapideous materials used in cultural heritage can be regarded as porous media, permeable to damaging agents transported by moisture which may lead to structural deterioration. When exposed to environmental factors, stones undergo weathering processes driven by water penetration, whether from meteoric precipitation or groundwater infiltration. Numerous studies have investigated the degradation of porous building materials in heritage structures (see, for instance, [10,12] and references therein). Degradation of monumental stones arises from a combination of physical-mechanical, chemical and biological processes. Physical-mechanical effects include salt crystallization [14], erosion [20] and thermal expansion [29]. Chemical reactions, such as carbonation [8,16], sulfation [2,4], oxidation [28] and hydration [15,30], alter the material composition, producing clay, crusts that swell, soluble salts and shrinkage. In addition, the growth of biological entities, including plants [31], mosses, lichens and bacte-ria [32], contributes to further deterioration. The interplay of these effects may induce microstructural changes in the material, sustaining an ongoing decay. In this work, we focus on salt crystallization processes, which constitute one of the main causes of deterioration in porous materials. The phenomenon involves capillary penetration and internal reactions of salts dissolved in water flowing through the stone pores. Under certain conditions, the salts precipitate, forming crystals either on the surface (efflorescence) or within the pores (subflorescence). When crystallization occurs within the pores, it reduces the available pore volume, disrupts the liquid network and slows water transport, with consequent effects on the material deterioration. In a context where preventive strategies play a critical role, several differential models have been proposed to characterize and quantify the degradation, alongside numerical methods specifically developed for its simulation. These models account for water transport in the porous medium, incorporating Darcys law, the changes in porosity due to crystal blooming and the effects of protective treatments aimed at mitigating the damage [5,9]. Existence and uniqueness results for the model in [5], both without and with boundary conditions, are provided in [17,18]. However, despite their modeling potential, the existing mathematical frameworks are based on a one-dimensional spatial representation. Although capable of describing vertical rise in porous materials, these models find limited applicability in realistic case studies. The first contribution of this work consists in the extension the model of [5] to multidimensional domains, that provides a more detailed investigation of diffusion-driven phenomena. Furthermore, to address the challenges inherent in multidimensional simulations, a Finite Element Method (FEM) approach is employed. FEM provides a flexible and robust framework for the discretization of complex geometries and heterogeneous domains, ensuring accurate representation of the underlying physics. A comparison between FEM and traditional one-dimensional finite difference (FD) schemes, together with a discussion of their respective advantages and limitations, is reported in [34].
In this paper, we build upon the mathematical model introduced in [5], describing the coupled processes of moisture transport, salt migration and crystallization in porous materials exposed to saline water and atmospheric agents. The formulation is motivated by standard laboratory imbibition-drying experiments performed on stone-and brick-like specimens, where salt accumulation within the pore network leads to progressive porosity reduction and material degradation. The physical domain here considered is a bounded region Ω ⊂ R d , with d = 1, 2, 3, representing a prismatic porous specimen subjected to water absorption from the bottom surface and moisture exchange with the surrounding air at the exposed boundaries. The evolution of the system is studied over a finite time interval [0, T ], which may be decomposed into an imbibition phase followed by a drying phase in order to reproduce the experimental protocol. The model accounts for the interaction between a liquid phase carrying dissolved salt ions and a solid crystalline phase precipitating within the pore space. A key feature of the formulation is the explicit coupling between transport processes and microstructural evolution induced by salt crystallization.
The unknowns of the problem are four space-and time-dependent fields defined for (x, t) ∈ Ω × [0, T ]:
• θ l (x, t), the volumetric fraction of liquid water, representing the portion of the pore volume locally occupied by the liquid phase;
• c i (x, t), the concentration of dissolved salt ions transported by the liquid phase;
• c s (x, t), the concentration of crystallized salt deposited within the pore network;
• n(x, t), the porosity of the material, defined as t
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