Modeling chemical reactions in porous media: a review

The protection and conservation of cultural goods from natural stones, as, for example, historical monuments made of sandstone, subject to weathering, can happen by filling the pores by a water–lime mixture. As illustrated in Fig. 1a, the mixture penetrates into the near surface pore structure of the stone and the calcium hydroxide reacts with carbon dioxide and builds calcium carbonate and water. Thus, a solid layer is built in the pore space which strengthens the bonds lost due to weathering. For results of an experimental simulation of the process with corresponding pictures, see, e.g., [50]. One real example is the Königin–Victoria–Denkmal (Queen Victoria Memorial) in the wineyards of Hochheim in Germany which was build in 1854 in memory of the visit of the British queen Victoria regarding her journey along the Rhine river in 1845. In 2019, the heavily weathered monument was under reconstruction. The conservation has been finished in 2020 so that the visit of the queen 175 years ago could be celebrated with the tudor turret in old splendor.

A second example where chemical reactions appear in porous media is permeable reactive barriers. As illustrated in Fig. 1b, these are constructions built of permeable reactive materials either in the ground or in flowing waters which are created in the flow paths of contaminated (ground)water. Due to reactive processes between the pollutants in the water and the material of the retention barrier, a remediation of the groundwater takes place. For an overview on developments of the last centuries, see [95].

It is worth mentioning that in various fields examples of reactive porous media occur. In biology, they appear, e.g., in tissues, bioreactors and biofilms, and in classical chemical and petroleum engineering, e.g., in packed bed reactors or oil recovery with miscible fluids.

These are examples of processes and materials where chemical reactions in porous media take place. This article aims to summarize different approaches which were intended to describe such or similar processes by models which stem from different backgrounds. It must be pointed out that not physical processes like adsorption are of interest but processes which lead to the chemical transformation of one set of chemical substances to another. Adsorption–diffusion processes have been studied by¹ B. Albers, e.g., in [1] or [3].

One branch of the porous media theory has developed from the mixture theory (see Subsection 2.1) introduced by C.A. Truesdell in the 1960s [120]. In [35], R.M. Bowen initially investigated the mixture theory and extended it subsequently to porous media [36, 37]. He also was one of the first researchers who dealt with chemically reacting mixtures [33, 34] even though at that time subject of consideration was miscible mixtures of fluids and not immiscible mixtures mostly considered in porous media theories. The development of thermodynamic mixture theories started by the discoveries of A. Fick [56] who postulated a linear relation between the diffusion flux and the concentration gradient. This is known as Fick’s law of diffusion, and it is analogous to other relationships discovered at the same epoch which became important in the meantime as, for example, Darcy’s law of hydraulic flow (will be discussed later), Ohm’s law of charge transport or Fourier’s law of heat transfer. Already in 1871, J. Stefan [117] generalized Fick’s law to more than two constituents. Starting with C. Eckart [54], when linear irreversible thermodynamics was created, Fick’s law was generalized and the diffusion flux was considered to be a linear combination of the gradients of temperature and chemical potentials (see [93]). Although working rather well for many applications this form of constitutive relation for the diffusion flux leads to a parabolic diffusion equation and thereby predicts infinite speeds of propagation of disturbances in concentration. This is known as the paradox of diffusion and shown in [93]. As pointed out in [14] by R.J. Atkin and R.E. Craine due to successive revisions of ideas and a certain element of controversy, the subsequent literature on the developments of the mixture theory is not easy to follow. However, hereinafter it is tried to summarize the most important developments, especially those which are important for the description of chemical reactions within continuum thermodynamic theories. For a general overview, it is referred to [14] where both chemically reacting and non-chemically reacting continua are considered or [13] where mixtures of fluids, mixtures of solids and mixtures of an elastic solid and a viscous fluid are discussed. Furthermore, many classical textbooks on thermodynamics, as, for example, de Groot/Mazur [47], Gurtin et al. [67], Müller [93], Passman et al. [99], Prigogine [102], Raats [103], Silhavy [114], Slattery et al. [115], Truesdell [119] or Woods [131], provide a thermodynamically rigorous framework for the understanding how reactive mixtures should be modeled on the continuum level.

It should be mentioned that even for problems without chemical reactions difficulties regarding the application of mixture theories to porous media have been addressed, e.g., by W. Noll who already in the abstract of [96] notes that for him even the notion mixture theory is misleading, and thereupon, he classifies the large literature on the subject as mostly defective. Works by R. Sampaio, M.E. Gurtin, M.L. Oliver and especially W.O. Williams (see [124] and references therein) he considers sound but still lacking a precise conceptual foundation. The just mentioned researchers worked in the aftermath on the improvement in the theory, especially on the elimination of problems as, for example, the formulation of the stress tensor in mixtures. For the same purpose, subsequently, also the Thermodynamically Constrained Averaging Theory by W.G. Gray and C.T. Miller has been developed [64, 65]. In [64], also chemical reactions are addressed.

Another important branch of porous media theory goes back to M.A. Biot. In the beginning, he dealt with classical geotechnical problems like consolidation [27], afterward—mainly intended for the investigation of elastic wave propagation—he extended his considerations to the description of two-component porous media, i.e., fluid saturated solids [28]. Later on, Biot investigated also chemical reactions [26].

For a historical review of the formulation of porous media theories, see, e.g., de Boer and Ehlers [46].

In a series of papers Hassanizadeh and Gray [60, 68‐70] (extended in 1990 by [71]) describe the modeling of porous media by use of the averaging technique. Interestingly, they discern three different types of models: (1) descriptive models of empirical nature, (2) continuum theories of mixtures and (3) the averaging technique. However, in my opinion, the use of the averaging theory is only one possibility to extend mixture theory in order to describe porous media. Only the combination of (2) and (3) enables the description of porous media. Another possibility of extension, often used by mathematicians, would be the use of homogenization techniques (see, e.g., [76]). However, in most theories of porous materials macroscopic models are related to microscopic relations by means of volume averaging (see, e.g., [129] for two-component media or [8] for three-component media). In (9), e.g., is shown, in which way the macroscopic mass densities are obtained from their microscopic, measurable counterparts, the true mass densities, by consideration of their portion of the volume. In Bowen’s works macro-scale forms of the field equations are postulated. In the fourth paper, mentioned before, Gray also considers phase changes. Admittedly, nine restrictive assumptions are made which limit the applicability strongly.

In two further papers [60, 61], Gray introduced another concept in order to incorporate microscopic effects into macroscopic models. He explains that traditionally, porosity and fluid saturations, concepts that do not exist at the micro-scale, are included in macro-scale porous media theories to account for the presence of multiple phases at a point in a macro-scale continuum. However, he points out that these additional variables sometimes are insufficient to account for all important micro-scale processes that influence macro-scale behavior. Because of the dynamic motion of the fluids, many configurations and distributions of the fluids are possible for a given saturation. This point has been investigated also in Albers [9, 10] where the well known effect that the capillary pressure curve of partially saturated soils exhibits a hysteresis has been accounted for. For the same degree of saturation it has different values depending on the initial state of the soil, thus for drying of a wet soil or wetting of a dry soil. The influence of these different values of the capillary pressure on the propagation of sound waves has been studied by use of a linear hyperbolic model. Even if the model does not contain a hysteresis operator, the effect of hysteresis in the capillary pressure curve is accounted for. Gray pursues another approach and includes interfacial effects. In addition to porosity and saturation, specific interfacial area, the amount of interfacial area between two phases per unit volume of the system, is introduced as a macro-scale independent variable and this variable was used in some studies of mass transfer among phases of a porous medium (e.g., [21, 22] or [63]). Another approach on macroscopic multi-phase flow with diffusion and chemical reactions not based on the mixture theory after Truesdell but incorporating moving interfaces is by C. M. Marle [90]. He starts from the equations valid at the pore level and takes their mean values around each point to obtain macroscopic equations.

In the PhD thesis by F. Brunner [39], a mathematical model including three phases (gas, liquid and solid) consisting of multiple components, which may participate in chemical reactions (equilibrium or kinetic reactions, homogeneous or heterogeneous) is considered. Moreover, the transfer of mass across the phases is taken into account. The reduction scheme of Kräutle, Knabner and Hoffmann [75] is applied to this model, which means that linear combinations of equations are taken to eliminate the equilibrium reaction rates and that a linear variable transformation of the original variables is performed. The timescales at which chemical reactions take place typically vary over several orders of magnitude. Therefore, besides kinetically controlled reactions, also equilibrium reactions are considered in the work of Brunner which are running so fast that a state of local equilibrium can be assumed.

One example dealing with a similar topic as the initially mentioned conservation of buildings by carbonization is presented by Saetta, Schrefler and Vitaliani in [108, 109] and Creazza et al. [45]. The governing equations of moisture, heat and carbon dioxide flows through concrete within the framework of a distributed parameter model are described. A numerical procedure based on the finite element method (for basics see, e.g., [111]) is developed to solve the set of equations and to investigate the influence of relative humidity and temperature. The amount of calcium carbonate that forms in a unit of time depends on the degree of carbonation, i.e., the availability of calcium hydroxide, the temperature, the carbon dioxide concentration and the relative humidity in the pore structure of the concrete. Another model of hygro-thermal behavior of concrete at high temperature with thermo-chemical and mechanical material degradation is introduced in [58] by Gawin, Pesavento and Schrefler. According to Bažant (e.g., [18] or newer publications [17, 19]), the formulation of drying coupled with heat transfer is developed in terms of relative humidity rather than water content and the diffusivities of water and carbon dioxide are assumed to be strongly dependent on pore humidity, temperature and also on the degree of hydration of concrete. The authors realized that the use of evaporable water content as basic variable involved certain errors since Fick’s law in terms of the evaporable water content could not be applied to obtain the differential equation governing the drying phenomenon when hydration proceeds. The authors state as reason that the porosity becomes non-uniform in time and equal values of evaporable water content do not correspond to equal value of pore humidity in the same points. Another work in which mass transport accompanied by chemical reactions in porous media is studied according to Fick’s law and the so-called dusty-gas model is by Veldsink et al. [122]. A further work based on mixture theory and dealing with chemical reactions and diffusion in concrete is by A. J. Vromans et al. [123]. The model describing the corrosion of concrete with sulfuric acid is applied to the chemical reactions which transforms slaked lime and sulfuric acid into gypsum releasing water. A similar topic is dealt with in van Balen and van Gemert [121] in which is shown how the carbonation process in lime mortar is influenced by the diffusion of carbon dioxide into the mortar pore system, by the kinetics of the lime carbonation reaction and by the drying and wetting process in the mortar. Another report on experimental results of \(\mathrm {CaCO_{3}}\) precipitation kinetics can be found in [107] by Roques and Girou. Similar studies by Houst and Wittman are on the influence of porosity and water content on the diffusivity of \(\mathrm {CO_{2}}\) and \(\mathrm {O_{2}}\) through hydrated cement paste [77] and on depth profiles of carbonates formed during natural carbonation [78]. The article [48] by Despotou et al. deals with reaction mechanisms in mortars and the diffusion and reaction speeds during the carbonation process. An also rather experimental investigation of the physicochemical characteristics of ancient mortars with comparison to a reaction-diffusion model by Zouridakis at al. appeared in [132]. A slightly different reaction involving also sulfur is studied in [30] by Böhm et al. where the corrosion in a sewer pipe is modeled as a moving-boundary system. By the same authors also appeared a strategy for predicting the penetration of carbonation reaction fronts in concrete [94].

A couple of references deal distinctively with geochemical reactions. One of them by Gatica et al. [57] is of mathematical nature and contains nonlinear reaction terms (see 2. in Subsection 2.2). It studies the interaction between chemical reaction and natural convection in porous media. A global driving force for natural convection are thermal or concentration gradients applied to the boundaries of the system. As changes in the density lead to natural convection, chemical reactions can provide a driving force for secondary flows. The critical value for the onset of instabilities is found by performing a linear stability analysis. A similar stability analysis has been performed by Albers [4] showing the stability behavior of flows within the above-mentioned adsorption/diffusion model for porous materials [1, 3]. A comprehensive book on geochemical reaction modeling is by C. M. Bethke [25]. He presents a variety of model types of which each imposes on the system some variant of equilibrium but differs from others in the manner in which mass and heat transfer are specified. He mentions closed-system models in which no mass transfer occurs and titration models where one or more reactants are gradually added to the equilibrium system, as might occur as the grains in a rock gradually react with a pore fluid. Further types are fugacity models which are applied to geochemical processes in which a fluid remains in contact with a gaseous phase or kinetic reaction models. These are such in which the rates at which minerals dissolve into or precipitate from the equilibrium system are set by kinetic rate laws. Moreover, local equilibrium models, as for example a flow-through reaction path, are mentioned. In such a path, the model isolates from the system minerals that form over the course of the calculation, preventing them from reacting further. The last type of models are continuum models. Bethke describes them as a natural marriage of local equilibrium and kinetic models used to describe the mass transport in hydrology and various fields of engineering. As an example of a model which predicts the distribution in space and time of the chemical reactions that occur along a groundwater flow path he mentions the book by J. Bear [20]. For formulations either for systems involving dissolution and precipitation of minerals or those accounting for surface adsorption reactions he refers to publications of P.C. Lichtner [87, 88]. Lichtner’s works point to a model based on the reaction path approach developed for geochemical systems by H.C. Helgeson [72, 73]. In this method an initial assemblage of minerals reacts irreversibly with an aqueous solution. A further work of Helgeson together with W.M. Murphy [74] directly focuses on the influence of the surface area in geochemical processes. The dependence of adsorption-diffusion processes on this parameter and on bulk and surface permeabilities has been investigated also by Albers in [2]. Lichtner [87] points out that the rates of reversible heterogeneous reactions are independent of the surface area. He differs between reactions which can be considered locally in equilibrium and are reversible and others which require kinetic description and are irreversible. Setting up his equations he also follows J. Bear [20]. The model of Lichtner doubtlessly makes acceptance of several restrictive adoptions (e.g., the immobility of the solid, the restriction to Darcy’s law etc.), but it is one of the few models which is complete and also verified to really describe simultaneous chemical reactions and mass transport in hydrothermal systems. Thus, another reference to this model is quoted [86] and also data of a work by C.I. Steefel et al. which shows the outcome of the model [116]. In [103] P.A.C. Raats reflects on applications of the theory of mixtures in soil physics. He points out that each of the phases of a porous medium is itself a mixture. The solid phase is a mixture of numerous inorganic and organic constituents, many of which are positively or negatively charged. Then there is the gaseous phase whose composition may deviate from that of the atmosphere above the soil surface. Moreover, there is the aqueous phase which is a dilute aqueous solution, balancing the charges of the solid phase and being strongly buffered by the solid and gaseous phases. For the roots of plants the soil is a resilient provider of nutrients and receiver of excrements. Therefore, for many purposes it is useful to regard the root system itself as an additional phase of the mixture.

Also in application to phase transitions models for porous media stemming from mixture theory are used, e.g., the above-mentioned paper by Ricken et al. [106] on concentration driven phase transitions with application to methane oxidation in landfill cover layers or a paper by Bluhm et al. [29] on phase transitions under freezing-thawing load in porous media.

It got obvious that a number of scientists dealt with the problem of chemical reactions in mixtures and porous media. In order to formulate a general model, it is necessary to regard the governing rules stemming from thermodynamics. However, there are still several open questions even if the topic is relevant already for a long time and many works to it appeared already fifty years ago. One of this open questions is the temperature: do the constituents of a mixture possess a common temperature or do the constituents have own different temperatures which have to be considered as partial temperatures? Another open question is the exploitation of the second law of thermodynamics. While there is consensus for mixtures of fluids that a couple of methods are applicable, for mixtures containing solids not all methods are accepted. Especially the often used method after Coleman and Noll has been shown not to be entirely correct for solid–fluid mixtures. Nevertheless, applications of this method to such mixtures yielded not totally absurd results. In this case, one can easily circumvent the problems without additional expense in using the method after Liu for which the correctness has been proved also for solid–fluid mixtures. More difficult is the choice of the form of the chemical potential. There exist several suggestions for its introduction and it is by no means clear which of the formulations is correct and yields reasonable results. All the approaches shown in this paper have been introduced, in fact, without verification on a practical example. This is different for the models shown in the last subsection of the paper. These models are all created for a special application, and it was accepted that these models are partly strongly restricted. If one focuses on one special process, this is fair enough, but if one wants to describe several occurrences with one model—as, for example, both the above-mentioned carbonation process and also chemical reactions occurring in reaction walls—it is necessary to formulate it more generally. Thus, in future work a model describing both diffusion and chemical reactions on the basis of the mixture theory will be developed.