Upscaling multicomponent transport in porous media with a linear reversible heterogeneous reaction
Introduction
Multicomponent transport with reaction in porous media is a common phenomenon, which may occur in many scientific and engineering disciplines, including chemical reaction engineering, petroleum engineering, groundwater hydrology, etc. Many researchers have taken great interest in the experimental studies of mass transfer and reaction in porous media. However, the experimental results only focused on some specific reactive system or some specific porous media structure (Gibilaro and Waldram, 1981, Baiker et al., 1982, Suzuki and Smith, 1972). In recent years, due to the advances in computational capabilities, numerical simulation has been applied to help researchers to elucidate the complex multicomponent transport with reaction process in porous media. The need for a precise mathematical model of reactive transport in porous media which is the basis for numerical simulation has been recognized.
The modeling of reaction-diffusion phenomena in porous media is a multiscale problem, which can be described either by pore-scale or by Darcy-scale models. At pore scale, despite current advances in computational capabilities, it is still impractical to model this complex process by the direct numerical simulations, since the pore-scale simulations require the detailed knowledge of pore geometry that is seldom available, especially for the complex porous system. Nevertheless, macroscopic model regards the porous medium as an averaged continuum system and can be obtained via suitable upscaling method, thus overcoming the difficulties at the pore scale. The upscaling techniques include the method of moment (Brenner, 1980), the method of volume averaging (Whitaker, 1999) and its modifications, pore-network models (Prat, 2002) and the method of homogenization (Ene and Poliševski, 1987), and so on.
Upscaling mass transfer and reaction processes in porous media has attracted considerable attentions from researchers in the literature. Most upscaled models are expressed in terms of effective coefficients that can be used to relate microscopic (pore-scale) characteristics to the macroscopic (Darcy-scale) counterparts. To quote some examples: Dykaar and Kitanidis, 1996, Sharratt and Mann, 1987, Valdés-Parada and Álvarez-Ramírez, 2010 and Valdés-Parada et al. (2011) focused on the first-order irreversible reaction, and obtained the corresponding macroscopic models for the diffusion-reaction process in a porous medium. They found that the effective coefficients inside the upscaled models depend upon the nature and magnitude of the microscopic reaction rate as well as the essential geometrical structure of the solid matrix and the flow rate. Moreover, Wood et al. (2007) and Heße et al. (2009) considered a heterogeneous reaction with Michaelis-Menton type and Monod type kinetics, respectively. The kinetics equations are similar and can be given by . The macroscale transport equations for reactive chemical species were obtained. Dadvar and Sahimi (2007) used pore network and continuum models of porous media to estimate the effective diffusivities under reactive and nonreactive conditions, where the reactive conditions involved a second-order reaction and one governed by the Michaelis-Menten kinetics. The studied reaction kinetics equations were and , respectively. Porta et al. (2012) investigated reactive transport processes involving fast bimolecular homogeneous irreversible reaction occurring within a porous medium based on the method of volume averaging. Guo et al. (2015) used the method of volume averaging to achieve the upscaled model for the mass transport in porous media with a heterogeneous reaction at the fluid-solid interface, typical of dissolution problems. The studied reaction kinetic equation was . Lugo-Méndez et al. (2015) revisited the upscaling process of diffusive mass transfer of a solute undergoing a homogeneous reaction in porous media using the method of volume averaging. They explored a linearization approach for the purpose of solving the associated closure problem for the nonlinear reactions. Zhang et al. (2015) investigated the adsorption-diffusion process in the nanoporous media by means of homogenization theory. Santos-Sánchez et al. (2016) derived a mass equilibrium model to describe the diffusion and reaction processes in a cell cluster composed of different cell populations.
In the above-referenced works, several kinds of reactions were investigated, involving homogeneous and heterogeneous reactions. The involved reaction kinetic equations included , , , , and et al. However, in the chemical reaction engineering, there are two kinds of chemical reactions, namely irreversible and reversible reactions. Reversible reaction as a common reaction type was seldom investigated, except the work by Morales-Zárate et al., 2008, Battiato et al., 2009, Boso and Battiato, 2013 and Battiato and Tartakovsky (2011). Morales-Zárate et al. (2008) presented a macroscopic model for diffusion and chemical reaction in double emulsion systems using the method of volume averaging. In this three-phase system, an irreversible reaction takes place in the drops phase while a reversible reaction occurs in the membrane phase. Furthermore, passive diffusion is considered in the continuous external phase. Battiato et al. (2009) and Boso and Battiato (2013) established macroscopic reaction-diffusion equations for the multicomponent reactive transport in porous media involving both homogeneous (bio-) chemical reactions between species dissolved in the fluid phase, and heterogeneous reactions occurring at liquid-solid interface, . Battiato and Tartakovsky (2011) used the method of multiple-scale expansions to upscale a pore-scale advection-diffusion equation with a nonlinear heterogeneous reaction at the fluid-solid interface, .
Although these above-referenced works focused on the reversible reactions, the studied reaction types were different with the catalytic reaction which plays an important role in chemical engineering process. Firstly, many studies related to the reversible reactions focused on homogeneous reactions, such as Morales-Zárate et al. (2008). Nevertheless, in the catalytic reaction process, the reaction occurs at the fluid-solid interface. Secondly, the reversible heterogeneous chemical reactions investigated by Battiato et al., 2009, Battiato and Tartakovsky, 2011 and Boso and Battiato (2013) are only appropriate for specific processes, such as precipitation and dissolution process. The precipitation and dissolution rates can be simplified as and , respectively, with the equal reaction rate constants k. Nevertheless, the forward and backward reaction rate constants in the catalytic reaction process are unequal (Qiu et al., 2014, Huang et al., 2014, Wang et al., 2014), Thus, the problems investigated in the literature can not be used to describe catalytic reaction process. Furthermore, it has been found in the literature that pore-scale reaction rate has significant effect on the effective parameters in the macroscopic transport equations (Valdés-Parada and Álvarez-Ramírez, 2010, Valdés-Parada et al., 2011). To the best of our knowledge, however, for the heterogeneous reversible reaction with unequal forward and backward reaction rate constants, the influences of forward and backward reaction rate constants on effective parameters were not reported yet in the literature.
In this work, a simple and typical reversible heterogeneous reaction with unequal forward and backward reaction rate constants at the fluid-solid interface in porous media was investigated. We begin with the pore-scale description of multicomponent mass transport, interfacial mass transport and simple first-order reversible heterogeneous reaction, and then upscale to obtain the macroscopic mass balance equations via the method of volume averaging (Whitaker, 1999). Then, the influence of reaction rates on the effective parameters in the macroscopic equations was investigated. Eventually, the verification of the presented macroscale model was performed at eleven situations which may happen in the actual reactive process by comparing with the direct numerical simulations in the pore-scale mode. The obtained corresponding macroscopic governing equations are the powerful supplements for the mathematical models of reactive transport in porous media in the literature.
Section snippets
Pore scale problem
In this study, we consider a rigid and homogeneous porous medium (Ω), which is fully saturated with a fluid phase illustrated in Fig. 1. We identify the solid phase shown in Fig. 1 as the κ-phase and the liquid phase as the γ-phase.
In this work, we are interested in studying the mass transport and reaction through the system. For the chemical reaction engineering, the reaction kinetic equations have many expressions, such as , , , and other complex
Upscaling procedures
In this section, we proceed by employing the method of volume averaging (Whitaker, 1999) to derive the macro-scale equations. We consider the representative elementary volume V as illustrated in Fig. 1 whose characteristic radius is . And the averaging domain V contains the solid phase κ-phase and the liquid phase as the γ-phase, i.e., . Here, and are the domains occupied by the γ-phase and κ-phase, respectively. If a characteristic length-scale of the macroscopic domain Ω is
Verification of the effective parameters
It is interesting to note that the parameters in the closed forms of the mass balance equations given by Eqs. (37), (38) are dependent on the sub-pore scale diffusion coefficients, the reaction rate parameters and volume geometry. In this section, the effective medium coefficients were verified by the comparisons of the effective coefficients obtained from the first-order heterogeneous reaction in the literature.
Valdés-Parada et al. (2011) carried out the upscaling diffusion with first-order
Influence of reaction rates
The works in the literature concluded that the effective coefficients in the macroscopic model depend on the magnitude of microscopic reaction rate. Thus in this section, the influence of reaction rates on the effective parameters in the macroscopic equations was discussed with the existence of reversible reaction.
The closure problems were solved in the unit cell illustrated in Fig. 4 at . In this figure, the characteristic length scale associated with this cell is l = d, which is the
Comparisons of the macroscopic model with microscopic DNS calculations
About eleven situations may happen in the actual process, which are shown in Table 1. In this section, in order to ponder about the predictive capabilities of the present closed macroscale model, the upscaled and the microscale models were solved and the average concentration profiles were compared, considering these eleven situations.
Conclusion
In this paper, we have investigated a mass transport problem involving a linear first-order reversible heterogeneous reaction at the interface inside a homogeneous porous medium. The method of volume averaging was used to derive the corresponding average governing equations expressed in terms of effective medium coefficients, which can be obtained by solving the associated closure problems in representative unit cells.
The effective parameters were calculated at , where the process studied
Acknowledgements
The authors wish to acknowledge the financial support by the National Natural Science Foundation of China (Project No. 91534106 and No. 21506032). The authors want to express their gratitude to two anonymous reviewers, whose comments and suggestions helped improving the manuscript.
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