Kinetic Approach to Model Reactive Transport and Mixed Salt Precipitation in a Coupled FreeFlow–PorousMedia System
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Abstract
Evaporative salinization of the agricultural soil is a chronic problem in arid, semiarid and coastal regions worldwide. As a shallow subsurface issue, it is strongly influenced by the freeflow (henceforth, FF) and porousmedia (henceforth, PM) flow and transport processes. In specific, it is mainly affected by the mass, momentum and energy exchange between the FF and the PM regions. Furthermore, salt precipitation in such systems is strongly determined by the interaction between different dissolved ionic species. In the scope of this work, we extend the REVscale coupled FF–PM model concept for salinization proposed by Jambhekar et al. (Transp Porous Media, 2015), to describe reactive transport of ionic species \(\text {Na}^+\), \(\text {Cl}^\) and \(\text {I}^\) and mixed salt precipitation (here, NaCl and NaI). In the first part of our numerical analysis, we illustrate precipitation behavior of NaCl using equilibrium and kinetic approaches. Both approaches are found to be in good agreement with experimental observations. However, in the literature it is often discussed that for mixed salt systems found in the nature, the equilibrium precipitation–dissolution assumption is not easily justifiable. Therefore, in the second part of our numerical analysis, we extend the kinetic approach to describe mixed salt precipitation in an NaCl–NaI system and compare it with the equilibrium precipitation approach. Our numerical analysis indicate that the simulation results for these approaches are very similar and analogous with the phenomenological explanations.
Keywords
Reactive transport Precipitation Dissolution Coupling Evaporation1 Introduction
The primary cause of the agricultural salinization is evaporative reduction of the irrigated water volume and resulting salt precipitation, as shown schematically in Fig. 1. The irrigation water normally taken from sources such as rivers, lakes or water reservoirs always contains some amount of dissolved salts. Thus, salinization to a certain extent is inevitable. However, encouraged excessive supplementary irrigation practices for ambitious crop yield and poor natural drainage of the irrigated fields risk acute soil salinization and therefore reduction in the crop yield in the long run. Therefore, in the last decades there is a developing research interest to understand the physical processes related to evaporationdriven soil salinization in the shallow subsurface.
As depicted in Fig. 1, during evaporative salinization dissolved salt transport is significantly influenced by the flow and transport processes in the PM and the FF. On the PM side dissolved salt transport is governed by capillary, viscous and gravitational forces and advective and diffusive processes. On the FF side, wind speed, temperature, humidity and solar radiation control evaporation strongly and therefore influence soil salinization.
For FF–PM interaction, evaporation takes place either directly in the waterfilled pores at the FF–PM interface (saturated case) or within the PM (unsaturated case), where the water vapor is transported to the FF by diffusion through the PM gas phase. Evaporation promotes salt accumulation and precipitation at the evaporation sites, which has the following implications: (a) decrease in the saturation vapor pressure of saline water due to an increase in the osmotic potential (salt concentration) at the evaporation site (Battistelli et al. 1997; Kelly and Selker 2001; Nachshon et al. 2011a) and (b) evolution of the pore geometry due to salt precipitation at the evaporation site. Therefore, evaporative salinization can influence the evaporation dynamics through change of the void space by salt precipitation and vapor pressure lowering.
Numerical modeling of saline water evaporation from a naturally existing PM system in contact with the FF is hardly addressed in the literature. In our previous work Jambhekar et al. (2015), we have extended the model concept proposed by Mosthaf et al. (2011) to describe evaporative salinization in a coupled nonisothermal compositional FF–PM system. In Jambhekar et al. (2015) we used a simplified or Layman’s approach to describe precipitation of a single dissolved salt (\(\text {NaCl}\)), considered to be one component.
However, in a natural shallow subsurface system (unsaturated zone), dissolved salts exist in the form of dissociated ionic species, e.g., \(\text {Na}^+\), \(\text {Cl}^\) and \(\text {I}^\) (Fig. 1). In such a system, one needs to describe reactive transport of individual ionic species. In addition, interactions between ions of different species and its influence on the precipitation–dissolution dynamics of various salts (e.g., NaCl and NaI in Fig. 1) must be accounted. Moreover, complexity of such systems may increase dramatically in the presence of flow and transport processes (Steefel and Cappellen 1990).
In this work, we present a new robust representative elementary volume (REV) scale approach for reactive transport and precipitation–dissolution in a drying PM system coupled with the FF. For reactive transport and precipitation, the transition from microscale to REV scale is schematically shown in Fig. 2. The new approach also accounts for interactions between different dissolved ionic species and mineral precipitates.
2 Objectives and Structure
 (1)
Development and implementation of an REVscale approach for reactive transport and ratecontrolled kinetic precipitation–dissolution processes.
 (2)
Numerical modeling of evaporationdriven reactive transport of \(\text {Na}^+\), \(\text {Cl}^\) and precipitation of \(\text {NaCl}\). The simulation results will be compared with the experimental data presented by Rad Norouzi et al. (2013).
 (3)
Calibration of the kinetic precipitation approach in the model against the experimental data.
 (4)
Analysis of the precipitation dynamics of \(\text {NaCl}\) and \(\text {NaI}\) for brinecontaining ionic species \(\text {Na}^+\), \(\text {Cl}^\) and \(\text {I}^\) with the new kinetic precipitation model.
3 Reactive Precipitation Approach
A reactive PM flow system is mainly driven by the feedback between precipitated salt and surrounding local composition of the fluid phase. As a feedback to variation of the fluid composition, solid salt aggregation is regulated through precipitation–dissolution processes (Steefel and Cappellen 1990). However, salt aggregation in a PM system may vary in space and time as illustrated schematically in Fig. 3. Here, at a given time (t), differing salt aggregates (shown in green) persist at different locations in the PM system, and in addition to this, solid salt present at any location (x) may vary with time. Therefore, the reaction taking place at a given location can potentially influence the reaction elsewhere in the flow field (Domenico and Schwartz 1990).
3.1 Ionic Activity
Activity coefficient parameters for existing ions for Debye–Hückle model and the Truesdell and Jhones model (Appelo and Postma 2005)
Variation of the activity coefficients (\(\gamma _i\)) for different ionic species involved in the scope of this work over the ionic strength is presented in Fig. 4. Here, the activity coefficients determined using the Truesdell and Jones model are represented by solid lines and the symbols correspond to the extended Debye–Hückle model. The extended Debye–Hückle model is well in accordance with the Truesdell and Jones model (Appelo and Postma 2005) within the applicability limit. Here, it is important to notice that the activity behavior for \(\text {Cl}^\) and \(\text {I}^\) are identical. Moreover, one must also note that, even if the Truesdell and Jones model is accurate up to \(I\le 2\), it is largely used for geochemical applications where ionic strength I can also be \(>\)10.0. Therefore, Fig. 4 depicts activity behavior up to \(I\le 10\).
In addition to above models, activity models like the Devies model (Appelo and Postma 2005) and the Pitzer and Kim model (1974) are also available. However, the Devies model is only applicable to the ionic strength of 0.5, and the Pitzer and Kim model is limited in terms of components that can be treated (Crowe and Langstaffe 1987). Moreover, ion paring values needed for the Pitzer and Kim formulation are not available at hand and very difficult to find for a multicomponent system. Thus, in line with most of the geochemical codes (e.g., TOUGH2 Xu et al. 2012), we use the Truesdell and Jones model.
3.2 Law of Mass Action
3.3 Equilibrium Versus Kinetic Reaction
The state of chemical equilibrium of a saline PM system describes its condition of maximum thermodynamic stability. At this state, there is no potential to alter the mass distribution between dissolved and precipitated salts. As the system moves away from this state, with the available chemical potential, it continuously attempts to attend chemical equilibrium through precipitation–dissolution processes.
The existence of equilibrium in such PM flow system is audited by the competition between the reaction and the transport processes (Appelo and Postma 2005). If the reactive processes are faster in relation to the species transport, local chemical equilibrium exists. Adversely, if the reaction processes are slower than the species transport, chemical nonequilibrium persists locally. Here, a kinetic formulation must be adopted for the reaction description (Domenico and Schwartz 1990).
3.4 Deviation from Equilibrium State
Moreover, for mixed salt systems found in the nature, interactions between different ionic species play a significant role in the salt precipitation dynamics. These ionic interactions can also potentially influence the precipitation of other salts in the system (Crowe and Langstaffe 1987). Furthermore, the precipitation–dissolution behavior of distinct salts present in such systems vary from each other. Therefore, for a mixed salt system, the equilibrium reaction assumption is not easily justifiable.
Here, \(\phi _\mathrm{S}\) is pore volume occupied by the precipitated salt, and \(\phi _0\) is the initial porosity. During evaporation, as the PM dries out, the available solid–liquid interfacial area will be reduced. Therefore, based on phenomenological explanations, the interfacial area is weighted with the liquidphase saturation \((S_w)\) (see Eq. 13). It is important to note that the factor 500 represents the specific solid surface of the porous medium and it strongly depends on the particle size distribution in the porous medium.
 (1)
If the \(K_{\mathrm{sp}_{n}}\) \(>\) \(K_{\mathrm{eq}_{n}}\) (i.e., \(\varOmega _n\) \(>\) 1.0), the reaction progresses from right to left (i.e., \(\text {NaCl}\) precipitates).
 (2)
If the \(K_{\mathrm{sp}_{n}}\) \(<\) \(K_{\mathrm{eq}_{n}}\) (i.e, \(\varOmega _n\) \(<\) 1.0), the reaction proceeds from left to right (i.e., \(\text {NaCl}\) dissolves).
 (3)
At equilibrium, the \(K_{\mathrm{sp}_{n}}\) equals \(K_{\mathrm{eq}_{n}}\) (i.e, \(\varOmega _n\) \(=\) 1.0) and the saturation index is \(\hbox {SI}_n = \log _{10}\;(\varOmega _n) = 0\).
4 Model Concept
The focus of this work is to couple the REVscale singlephase compositional nonisothermal FF and the multiphase compositional nonisothermal PM flow systems, for mixed salt precipitation near the FF–PM interface, as shown in Fig. 7. Here, in the FF subdomain Stokes flow is used, whereas in the PM subdomain Darcy’s flow is assumed.
Material laws and equation of state dependent on primary variables
Parameters  Equation of state  References 

Density  
Gas density \(\varrho _{\alpha }\)  \(\varrho _{\alpha } = f(\varrho ^{\kappa }, x^{\kappa }_{\alpha }, p_{\alpha }, T)\)  IAPWS (2009) 
Liquid density \(\varrho _\mathrm{l}\)  \(\varrho _\mathrm{l}= \varrho _\mathrm{w}+1000X_\mathrm{l}^\mathrm{s}\{0.668+0.44X_\mathrm{l}^\mathrm{s}\)  
\(+\,[300p2400pX_\mathrm{l}^\mathrm{s} + T(80+3T3300X_\mathrm{l}^\mathrm{s}\,13p \)  Batzle and Wang (1992)  
\(+\,47pX_\mathrm{l}^\mathrm{s})]\times 10^{6}\}\)  
Component water \(\varrho ^{\mathrm{w}}\)  Incompressible fluid  Reid et al. (1987) 
Component air \(\varrho ^{\mathrm{a}}\)  Ideal gas  Reid et al. (1987) 
Salt NaCl \(\varrho ^{\text {NaCl}}_\mathrm{S}\)  2165.0 \({\hbox {kg}}/{\mathrm{m}^3}\)  
Salt NaI \(\varrho ^{\text {NaI}}_\mathrm{S}\)  3670.0 \({\hbox {kg}}/{\mathrm{m}^3}\)  
Diffusion coefficient \(D_{\alpha ,\mathrm{pm}}\)  \(D_{\alpha ,\mathrm{pm}} = \tau \phi S_{\alpha } D_{\alpha }\)  Millington and Quirk (1961) 
Tortuosity \(\tau \)  \(\tau = \frac{(\phi S_{\alpha })^{7/3}}{\phi ^2}\)  Millington and Quirk (1961) 
Capillary pressure \(p_\mathrm{c}\)  van Genuchten model \(p_\mathrm{c} = f(s_\mathrm{l})\)  Genuchten (1980) 
Relative permeability \(k_{r \alpha }\)  van Genuchten model  Genuchten (1980) 
Eff. thermal conductivity \(\lambda _\mathrm{pm}\)  \(\lambda _\mathrm{pm}=\lambda _\mathrm{eff,g} + \sqrt{S_\mathrm{l}}(\lambda _\mathrm{eff,l}\lambda _\mathrm{eff,g})\)  Somerton et al. (1974) 
Internal energy \(u_{\alpha }\)  \(u_{\alpha }=h_{\alpha }p_{\alpha }/\varrho _{\alpha }\)  
Enthalpy h  
Gas phase  \(h_\mathrm{g} = x^\mathrm{w}_\mathrm{g} h_\mathrm{w} + x^\mathrm{a}_\mathrm{g} h_\mathrm{a}\)  
Liquid phase  \(h_\mathrm{l}(p_\mathrm{l}, T) = X^\mathrm{w}_\mathrm{l} h^\mathrm{w} + X^\mathrm{a}_\mathrm{l} h^\mathrm{a} + X^\mathrm{s}_\mathrm{l} (h^\mathrm{s}+ \varDelta h^\mathrm{s})\)  Michaelides (1981) 
Component air  \(h^\mathrm{a}=1005(T273.15\,\hbox {K})\)  IAPWS (2009) 
Component water  \(h^\mathrm{w}=f(p_{\alpha }, T)\)  IAPWS (2009) 
Component salt  \(h^\mathrm{s}=f(p_{\alpha }, T)\)  Michaelides (1981) 
Secondary variables  
Saturation  \(S_\mathrm{g} = 1S_\mathrm{l}\)  
Phase pressure  \(p_\mathrm{g} = p_\mathrm{c} + p_\mathrm{l}\)  Genuchten (1980) 
Mole fraction  \(\sum \nolimits _{\kappa } x^{\kappa }_{\alpha } = 1\)  
Air in liquid pahse  Henry’s law: \(x^\mathrm{a}_\mathrm{l}=p^\mathrm{a}_\mathrm{g}/H^\mathrm{a}_{\mathrm{gl}}\)  
Vapor in gas phase  \(x^\mathrm{w}_\mathrm{g}=p^\mathrm{w}_{\text {sat,Kelvin}}/p_\mathrm{g}\)  
Kelvin’s equation  \({p^\mathrm{w}_{\text {sat,Kelvin}}} = \text {exp}\left( \frac{\left( p_\mathrm{c} + \psi _\mathrm{s}\right) }{\varrho _{\mathrm{mol,l}}RT}\right) {p^\mathrm{w}_{\text {sat}}}(T)\)  Kelly and Selker (2001) 
Osmotic potential  \(\psi _\mathrm{s} =  RT \varrho _{\text {mol,l}} \text {ln}\left( x^{\mathrm{w}}_\mathrm{l}\right) \)  Kelly and Selker (2001) 
4.1 PorousMedia Submodel

Local thermodynamic (mechanical, thermal and chemical) equilibrium,

Rigid solid phase (solid matrix and precipitated salt) (subscript S),

Twophase flow consisting of a liquid phase (subscript l) and a gas phase (subscript g),

Gas phase is assumed to be ideal,

Slow flow velocities (\(Re \ll 1\)) allow usage of the multiphase Darcy’s law,

Binary diffusion is assumed in fluid phases, dispersion is neglected because of slow flow velocities and relatively high diffusion coefficient in the gas phase.
Porousmedia equations and associated primary variables
Balance equations  Primary variables 

Mass balance for component \(\kappa \,\in \{w, a, \text {Na}^+, \text {Cl}^, \text {I}^ \}\):  
\(\sum \nolimits _{\alpha \in \mathrm{\{l, g\}}}\frac{\partial \left( \phi \varrho _{\mathrm{mol},\alpha }x_\alpha ^\kappa S_\alpha \right) }{\partial t}+ \nabla \cdot \mathbf{F}^\kappa = q^\kappa ,\)  \(S_\mathrm{l}\) or \(x^{\kappa }_{\alpha }\) 
\(\mathbf{F}^\kappa = \sum \nolimits _{\alpha \in \{l, g\}} \left( \varrho _{\mathrm{mol},\alpha }{} \mathbf v _{\alpha }x_{\alpha }^{\kappa } D_{\alpha ,\textit{pm}}^{\kappa }\varrho _{\mathrm{mol},\alpha }\nabla x_{\alpha }^{\kappa }\right) ,\)  
\(q^{\kappa } = \left\{ \begin{array}{l} r_n \quad \forall \quad \kappa \in \left\{ \text {Na}^+,\text {Cl}^,\text {I}^\right\} \\ 0 \qquad \quad \text {else} \end{array}\right. \)  
Mass balance for gas phases  
\(\frac{\partial (\phi \varrho _\mathrm{g}S_\mathrm{g})}{\partial t} + \nabla \cdot (\varrho _\mathrm{g}\mathbf {v}_\mathrm{g}) = q_\mathrm{g},\)  \(p_\mathrm{g}\) 
Mass balance for solid salt  
\(\frac{\partial (\phi _{\mathrm{S}}^\mathrm{s}\varrho _{\mathrm{mol,S}}^\mathrm{s})}{\partial t}=\sum \nolimits _{\kappa }q^{\kappa }\)  \(\phi _\mathrm{S}^\mathrm{s}\) 
Darcy’s law  
\(\mathbf v _{\alpha } = \frac{k_{r\alpha }}{\mu _\alpha } \mathbf K \left( {\nabla }\, p_\alpha  \varrho _{\alpha } \mathbf {g} \right) \)  
Energy balance  
\(\left( 1  \phi _0 \right) \frac{\partial \left( \varrho _\mathrm{s} c_\mathrm{s} T\right) }{\partial t} + \frac{ \partial \left( \phi _\mathrm{S}^\mathrm{s} \varrho _\mathrm{S}^\mathrm{s} c_\mathrm{S}^\mathrm{s} T\right) }{\partial t} \)  T 
\(+ \sum \limits _{\alpha \in \{\mathrm{l,g}\}} \frac{ \partial \left( \phi \varrho _{\alpha } u_\alpha S_\alpha \right) }{\partial t} + \nabla \cdot \mathbf{F}_T = q_T,\)  
\(\mathbf{F}_T = \sum \limits _{\alpha \in \{\mathrm{l, g}\}} \varrho _{\alpha } h_\alpha \mathbf v _{\alpha }  \lambda _\mathrm{pm} {\nabla } \, T\) 

Two component mass balance equations for water w and dissolved salt s, where \(\phi \) is porosity, \(\varrho _{\hbox {mol},\alpha }\) is the molar density for phase \(\alpha \), \(x^{\kappa }_{\alpha }\) is the mole fraction of component \(\kappa \) in phase \(\alpha \), \(S_{\alpha }\) is the saturation of phase \(\alpha \), \(\mathbf{v}_{\alpha }\) is the velocity of phase \(\alpha \), \(D^{\kappa }_{\alpha , \mathrm{pm}}\) is the macroscopic diffusion coefficient for component \(\kappa \) in phase \(\alpha \), and \(q^{k}\) is a source/sink term for the component. The source/sink for dissolved salt (s) is given by Eq. 12.

A mass balance equation for the gas phase (g), where \(\varrho _\mathrm{g}\) is the density of the gas.

A mass conservation equation for precipitated solid salt, where \(\phi ^\mathrm{s}_\mathrm{S}\) is the pore volume fraction occupied by the precipitated solid salt.

Based on the assumption of local thermal equilibrium one energy balance equation accounts for energy conservation. Here, \(c_\mathrm{s}\) is the specific heat capacity of the solid phase, \(c^\mathrm{s}_\mathrm{S}\) is the specific heat capacity of the solid salt, \(u_{\alpha }\) is the internal energy of phase \(\alpha \), \(h_{\alpha }\) is the enthalpy of phase \(\alpha \), \(\lambda _\mathrm{pm}\) is the bulk thermal conductivity of the porous medium, and \(\lambda _\mathrm{eff,l}\) and \(\lambda _\mathrm{eff,g}\) are the effective thermal conductivity of the liquid and gassaturated porous medium, respectively (see Table 2).

The flow velocity within the porous medium is given by Darcy’s law, where \(\mathbf{K}\) is the intrinsic permeability tensor, \(p_{\alpha }\) is the phase pressure, \(k_{r\alpha }\) is the relative permeability, and \(\mu _{\alpha }\) is the dynamic viscosity for phase \(\alpha \). Moreover, changes in the intrinsic permeability of the porous medium caused by salt precipitation is taken into account using Kozeny–Carman relationship and Leverett scaling is used to update corresponding changes in the capillary pressure (see Jambhekar et al. 2015 for details).
Freeflow equations and associated primary variables
Balance equations  Primary variables 

Mass balance for water vapor (w)  
\(\frac{\partial \left( \varrho _{\mathrm{mol,g}}x_\mathrm{g}^\mathrm{w}\right) }{\partial t}+ \nabla \cdot \mathbf{F}^\mathrm{w} = q^\mathrm{w},\)  \(x^{\mathrm{w}}_\mathrm{g}\) 
\(\mathbf{F}^\mathrm{w} = \left( \varrho _\mathrm{mol,g}{} \mathbf v _\mathrm{g}x_\mathrm{g}^\mathrm{w} D_\mathrm{g}^\mathrm{w}\varrho _\mathrm{mol,g}\nabla x_\mathrm{g}^\mathrm{w}\right) \)  
Mass balance for gas phases  
\(\frac{\partial (\varrho _\mathrm{g,mol})}{\partial t} + \nabla \cdot {(\varrho _\mathrm{g}\mathbf {v}_\mathrm{g})} = q_\mathrm{g}\)  \(p_\mathrm{g}\) 
Momentum balance  
\(\frac{\partial (\varrho _\mathrm{g} \mathbf {v}_\mathrm{g})}{\partial t} + \nabla \cdot \left[ p_\mathrm{g}{} \mathbf{I}  \mu _\mathrm{g}(\nabla \mathbf{v}_\mathrm{g} + \nabla \mathbf{v}_\mathrm{g}^T)\right] = \varrho _\mathrm{g}\mathbf {g},\)  \(\mathbf {v}_x\),\(\mathbf {v}_y\) 
Energy balance  
\(\frac{ \partial \left( \varrho _\mathrm{g} u_\mathrm{g} \right) }{\partial t} + \nabla \cdot {(\varrho _\mathrm{g} h_\mathrm{g} \mathbf v _\mathrm{g}  \lambda _\mathrm{pm} {\nabla } \, T)} = q_T\)  T 
4.2 FreeFlow Submodel

laminar or creeping gasphase (g) flow,

gas phase is composed of components water (w) and air (a) (see Fig. 7).

negligible inertia forces for low Reynolds numbers,

binary diffusion in the gas phase.
4.3 Coupling Conditions
Suitable coupling conditions must be applied at the FF–PM interface to account for the underlying physics of the exchange process for mass, momentum and energy between the FF and the PM. Based on the phenomenological explanations, coupling conditions for compositional nonisothermal coupled FF–PM system are developed by Mosthaf et al. (2011). As Mosthaf et al. (2011) stated, these conditions are motivated from the porescale processes and are also valid on the REV scale. This is then extended by Jambhekar et al. (2015) to the salinization applications. For brevity we only list them in the following.
 (1)Continuity of normal stresses resulting in feasible jump in the gasphase pressure$$\begin{aligned} \mathbf{n}\cdot \left[ \left( \left( p_\mathrm{g} \mathbf{I}\mu _\mathrm{g}\left( \nabla \mathbf{v}_\mathrm{g} + \nabla \mathbf{v}_\mathrm{g}^\mathrm{T}\right) \right) \mathbf{n}\right) \right] ^{\text {ff}} = \left[ p_\mathrm{g}\right] ^{\text {pm}}. \end{aligned}$$(16)
 (2)Continuity of the mass fluxes normal to the interface$$\begin{aligned}{}[\varrho _\mathrm{g} \mathbf{v}_\mathrm{g} \cdot \mathbf{n}]^{\text {ff}} = [(\varrho _\mathrm{g} \mathbf{v}_\mathrm{g} + \varrho _\mathrm{l} \mathbf{v}_\mathrm{l})\cdot \mathbf{n}]^{\text {pm}}. \end{aligned}$$(17)
 (3)Beavers–Joseph–Saffman condition for the tangential component of the FF velocity (Beavers and Joseph 1967; Saffman 1971)where \(k_i=\mathbf{t_i\cdot (\text {K}t_i)}\) is the tangential component of the permeability tensor.$$\begin{aligned}&\left[ \left( v_\mathrm{g} + \frac{\sqrt{k_i}}{\alpha _{\text {BJ}}}\left( \nabla v_\mathrm{g} + \nabla v_\mathrm{g}^\mathrm{T} \right) \mathbf{n} \right) \cdot \mathbf{t}_i\right] ^{\text {ff}}=0, \nonumber \\&\quad i \in \left\{ 1,\ldots ,d1\right\} , \end{aligned}$$(18)
 (1)Continuity of temperature$$\begin{aligned}{}[T]^{\text {ff}}=[T]^{\text {pm}}. \end{aligned}$$(19)
 (2)Continuity of heat fluxes$$\begin{aligned}&\left[ \left( \varrho _\mathrm{g} h_\mathrm{g} \mathbf{v}_\mathrm{g} \lambda _\mathrm{g}\nabla T\right) \cdot \mathbf{n}\right] ^{\text {ff}} \nonumber \\&\quad =\left[ \left( \varrho _\mathrm{g} h_\mathrm{g} \mathbf{v}_\mathrm{g} + \varrho _\mathrm{l} h_\mathrm{l} \mathbf{v}_\mathrm{l}\lambda _{\text {pm}}\nabla T\right) \cdot \mathbf{n}\right] ^{\text {pm}}. \end{aligned}$$(20)
 (1)Continuity of mole fractions$$\begin{aligned} \left[ x^{\kappa }_\mathrm{g}\right] ^{\text {ff}} =\left[ x^{\kappa }_\mathrm{g} \right] ^{\text {pm}} \end{aligned}$$(21)
 (2)Continuity of component flux across the interface for water (w) and air (a)$$\begin{aligned}&\left[ \left( \varrho _\mathrm{g} \mathbf{v}_\mathrm{g} x^{\kappa }_\mathrm{g}  D_\mathrm{g}\varrho _\mathrm{g}\nabla x^{\kappa }_\mathrm{g}\right) \cdot \mathbf{n}\right] ^{\text {ff}} \nonumber \\&\quad = [(\varrho _\mathrm{g} \mathbf{v}_\mathrm{g} x^{\kappa }_\mathrm{g}D_{\mathrm{g,pm}}\varrho _\mathrm{g}\nabla x^{\kappa }_\mathrm{g} \nonumber \\&\qquad +\, \varrho _\mathrm{l} \mathbf{v}_\mathrm{l} x^{\kappa }_\mathrm{l}D_{\mathrm{l,pm}}\varrho _\mathrm{l}\nabla x^{\kappa }_\mathrm{l})]^{\text {pm}} \end{aligned}$$(22)
5 Numerical Model
A vertexcentered finitevolume method is used for space, and implicit Euler method is used for time discretization (Baber et al. 2012). The primary variables for the FF and PM subdomain are presented in Tables 3 and 4, and the switch criteria for existence of different phases are discussed in detail by Class et al. (2002). The unknown mole fractions are determined as secondary variables using the primary variabledependent constitutive relationships given in Table 2.
The developed coupled model concept is implemented in the numerical framework \(\text {DuMu}^x\), a free and opensource simulator for flow and transport processes in PM (Flemisch et al. 2011), which is based on the Distributed and Unified Numerics Environment (DUNE) (Bastian et al. 2008). The discritization and solution schemes used for this work are addressed in detail by Baber et al. (2012).
6 Experimental Setup
The experimental setup is shown in Fig. 8a, where the sand column of 0.011 m in diameter and 0.035 m in height is placed in the Xray chamber. The details of the experimental setup can be studied in detail in Rad Norouzi et al. (2013) and Jambhekar et al. (2015). In the following, we briefly discuss the experiments carried out by Rad Norouzi et al. (2013).
Initial conditions for the porous medium and the free flow used for experiments
Primary variable  Initial conditions 

Free flow  
Velocity (\(v_\mathrm{g}\))  0.01 m/s 
Humidity (\(x^w_\mathrm{g}\))  0.007925 (25 %) 
Temperature (T)  303.15 K 
Porous medium  
Saturation (\(S_l\))  98 % 
Salinity (molal (M))  3.5 mol/kg 
Solidity (\(\phi ^\mathrm{s}_\mathrm{S}\))  0.0 (–) 
Even though the evaporative demand was nearly constant during experiments, the relative humidity and temperature inside an Xray chamber could not be controlled precisely (Rad Norouzi et al. 2013). So approximate temperature and relative humidity are used for numerical simulations. To investigate the dependence of the salinization behavior on initial salt concentration, three rounds of evaporation experiments were conducted in Rad Norouzi et al. (2013) for sand columns saturated with 3.5, 4.0 and 6.0 M saline water solution.
7 Simulation Setup
Summary of initial, boundary and coupling conditions for the porousmedia and freeflow subdomains
Domain  Primary variable  Initial  Boundary  

Left  Right  Top  Bottom  
Porous medium  \(p_\mathrm{g}\)  1 bar  No flow  No flow  Coupling (Eq. 17)  No flow 
\(S_\mathrm{w}\) or \(x^\mathrm{w}_\mathrm{g}\)  0.98 (–)  Coupling (Eq. 22)  
\(x_\mathrm{w}^{\text {Na}^+}\)  0.0668 (–)  No flow  
\(x_\mathrm{w}^{\text {Cl}^}\)  0.1030 (–)  No flow  
\(\phi _\mathrm{S}^\mathrm{s}\)  0.0 (–)  No flow  
T  303.15 K  Coupling (Eq. 20)  
Free flow  \(v_x\)  0.01 m/s  0.01 m/s  Outflow  Slip  BJS condition (Eq. 18) 
\(v_y\)  0 m/s  0 m/s  0 m/s  Coupling (Eq. 16)  
\(p_\mathrm{g}\)  1 bar  Outflow  1 bar  Outflow  Outflow  
\(x^\mathrm{w}_\mathrm{g}\)  0.0079 (–)  Flux (\(q^\mathrm{w}_\mathrm{g}\))  Outflow  No flow  Coupling (Eq. 21)  
T  303.15 K  Flux (\(q_T\))  No flow  Coupling (Eq. 19) 
Porousmedia and freeflow properties for the reference experimental setup
Properties  Reference value  Unit 

Porous medium  
Porosity (\(\phi _0\))  0.37  (–) 
Permeability (\(K_0\)) (Shokri and Salvucci 2011)  \(6.37\times 10^{12}\)  \(\mathrm{m}^2\) 
Van Genuchten (\(\alpha \)) (Shokri and Salvucci 2011)  \(6.024\times 10^{4}\)  \({1}/{\mathrm{Pa}}\) 
Van Genuchten (n) (Shokri and Salvucci 2011)  12.18  (–) 
Thermal conductivity solid matrix (\(\lambda _\mathrm{S}\))  5.26  \({\hbox {W}}/\mathrm{mK}\) 
Specific heat capacity solid matrix (\(C_\mathrm{p}\))  830  \({\hbox {J}}/\mathrm{KgK}\) 
Solidsalt density (\(\varrho ^\mathrm{s}_\mathrm{S}\))  2165  \({\hbox {Kg}}\mathrm{m^3}\) 
Solidsalt specific heat capacity (\(C_{\mathrm{p,S}}\))  629  \({\hbox {J}}/\mathrm{KgK}\) 
Free flow  
Beaver–Joseph coefficient(\(\alpha _\mathrm{BJ}\))  1.0  (–) 
8 Numerical Experiments
In this section, the performance of the implemented model concept is tested by comparing the numerical results with the experimental data presented in Rad Norouzi et al. (2013). Here, one must notice that different from Jambhekar et al. (2015), as indicated in Table 3, in this work transport of dissolved ionic species is considered and precipitation–dissolution processes are accounted for by considering interionic interactions.
8.1 Equilibrium Approach
Figure 9 shows the grid convergence analysis for cumulative salt precipitation. In this work, we use three different grid configurations: in specific, \(10 \times 17 \), \(30 \times 51 \) and \(45 \times 71\). The grid convergence analysis highlights that the simulation result for cumulative salt precipitation are grid independent for resolutions higher than \(30 \times 51 \). Therefore, for further simulations we use \(30 \times 51 \) grid.
Guess values for the fitting parameters of kinetic precipitation–dissolution model given by Eq. 12
Parameters  Guess1  Guess2  Guess3  Guess4  Guess5  Guess6 

\(k_n\) (\({\hbox {mol}}/{\mathrm{m^2\,s}}\))  1.0  1e\(\)3  6.5e\(\)6  1.0  1e\(\)3  6.5e\(\)6 
\(\theta \) (–)  1.0  1.0  1.0  1.0  1.0  1.0 
\(\eta \) (–)  1.0  1.0  1.0  1.3  1.3  1.3 
As discussed above, the reactive transport and equilibrium precipitation approach convincingly predicts the cumulative NaCl precipitation and cumulative water loss during evaporative salinization. Moreover, the simulation results presented above differ marginally from the equilibrium approach discussed in Jambhekar et al. (2015). These difference are related to the differences in the model concept and the conservation equations used in the porousmedia subdomain models.
For the existence of multiple ionic species in a natural system, as discussed in Sect. 3.4, it is rather difficult to justify the assumption of equilibrium precipitation, and the generalized kinetic approach should be employed. Therefore, in the following, firstly, in Sect. 8.2, we employ and analyze the kinetic approach for NaCl precipitation dynamics, and later in Sect. 8.3, we apply it for the mixed NaCl–NaI precipitation and also compare the results with equilibrium approach.
8.2 Kinetic Approach
Therefore, in this section, we first calibrate the kinetic precipitation model against experimentally observed NaCl precipitation by Rad Norouzi et al. (2013) to find appropriate values of \(k_n\), \(\theta \) and \(\eta \). In the following, numerical simulations are performed using some guess values for \(k_n\), \(\theta \) and \(\eta \) given in Table 8 and inspired from the literature, e.g., Xu et al. (2012). As illustrated in Table 8, in this work, we use three different precipitation rate constants \(k_n\) in combination with two values for the fitting parameter \(\eta \). The value for parameter \(\theta \) is kept unity.
Simulation results for the kinetic precipitation approach using different combinations of \(k_n\) and \(\eta \) are presented in Fig. 11. Figure 11a displays that the cumulative water loss for all the simulations is excellent in agreement with the experimental observations. Figure 11b points out that the salt precipitation dynamics is significantly dominated by the kinetic precipitation rate constant \(k_n\), and moreover, the exponent \(\eta \) influences the shape of the plot.
8.3 Mixed Salt Precipitation
In this section, we present a numerical case study, where we apply the developed kinetic model for mixed salt precipitation. The twodimensional simulation setup for this case is shown in Fig. 12. The PM subdomain is 0.25 \(\times \) 0.25 m and the FF subdomain is \(0.4\,\times \,0.2\) m. For grid convergence analysis, three different configurations, namely 15 \(\times \) 26, 30 \(\times \) 51 and 35 \(\times \) 61, are used for the complete coupled FF–PM system with refinement at the FF–PM interface. The boundary conditions for the FF and PM subdomains and initial conditions for the FF subdomain are the same as in the earlier simulation setup, as discussed in Sect. 7. The initial conditions for the PM subdomain are discussed in the following.
Here, in the PM subdomain in addition to \(\text {Na}^+\) and \(\text {Cl}^\), reactive transport of ionic species \(\text {I}^\) is considered. Initially, the PM subdomain is assumed to be 98 % saturated with saline water containing dissolved NaCl and NaI, 1.0 M each; i.e., the initial condition of primary variables \(\text {Na}^+\), \(\text {Cl}^\) and \(\text {I}^\) is 2.0, 1.0 and 1.0 M, respectively. The initial condition for other primary variables in the PM subdomain are same as earlier simulation setup (see Sect. 7). In comparison with the earlier simulations in this case larger simulation domain and lower initial salt concentration are chosen: (1) to model evaporative salinization for longer durations and (2) to avoid complete clogging of the freeflow–porousmedia interface due to precipitated salt.
Guess values for the fitting parameters of kinetic precipitation–dissolution model (given by Eq. 12)
Parameters  NaCl–NaI 

\(k_n\)  1e\(\)3 
\(\theta \)  1.0 
\(\eta \)  1.3 
During the course of evaporative salinization, the water vapor concentration in the ambient FF gas phase along the FF–PM interface increases due to water vapor exchange with the PM. Consequently, dissolved salt is transported toward the FF–PM interface and accumulates at the evaporation front. As a result, the activity of each ionic species, namely \([\text {Na}^+]\), \([\text {Cl}^]\) and \([\text {I}^]\), increases at the evaporation front causing NaCl and NaI precipitation, provided that the kinetic saturation index for the corresponding salt species, i.e., \(\varOmega _{\text {NaCl}}\) and/or \(\varOmega _{\text {NaI}}\) exceeds unity.
However, based on the phenomenological explanations discussed by Nachshon et al. (2011a), delay in the initiation of NaI precipitation is expected. This delay can be clearly seen in Fig. 15a, b, taking into account that same precipitation rate constant and exponents are used for these cases. In Fig. 16, the saturation indices (\(\varOmega \)) are plotted along a vertical section at the top of the PM subdomain after 1, 7 and 14 days. As shown in Fig. 16, early outset of NaCl precipitation can also be predicted from the faster increase of \(\varOmega _{\text {NaCl}}\) in comparison with \(\varOmega _{\text {NaI}}\) locally along the PM depth.
From the above discussion, it is clear that the developed new model concept is capable to model ionic species transport and kinetic mixed salt precipitation in a coupled FF–PM system. For the mixed salt precipitation scenario, we initially assumed the same kinetic precipitation rate constant for NaCl and NaI. In the following, we want to perform a sensitivity analysis of NaI precipitation dynamics in a mixed salt system to its precipitation rate constant (\(k_{\text {NaI}}\)).
Figure 18 exhibits the cumulative solidity evolution for kinetic NaI precipitation rates: 1e\(\)3, 1e\(\)7 and 1e\(\)10. Here, likewise precipitation behavior is observed for \(k_{\text {NaI}}\) 1e\(\)3 and 1e\(\)7. However, the precipitation dynamics largely changes for \(k_{\text {NaI}}\) = 1e\(\)10. In this case, significant delay in NaI precipitation (approximately 1 day) is observed.
For a better prediction of the NaI precipitation rate constant, the new kinetic model must be validated against experiments. This rate constant can be further used in a mixed salt system, to acquire better understanding about the influence of interionic interactions on NaCl and NaI precipitation dynamics. Furthermore, the delay in the initiation of NaI precipitation can also be validated by comparison against NaCl–NaI precipitation experiments.
9 Summary and Outlook
A coupled FF–PM interaction model for evaporationdriven reactive transport and precipitation under both equilibrium and kinetic conditions is presented in the scope of this work. This new model is an extension of our previous work (Jambhekar et al. 2015). Nonisothermal compositional models are applied in the FF and PM subdomains and a simple interface, which cannot store mass, momentum and energy, is employed to account for the FF–PM interaction. The model is implemented in the opensource modeling framework \(\text {DuMu}^x\) (Flemisch et al. 2011), where the vertexcenteredfinitevolume (box) scheme is employed for spatial discretization and the implicit Euler method is used for time discretization.
The new concept is applied to model evaporative salinization under both equilibrium and kinetic conditions. As shown in Sect. 8.1, the numerical results for NaCl precipitation under equilibrium assumption are in accordance with experiments. However, for mixed salt precipitation in a natural system interionic interactions between different salt ions play a significant role in the precipitation dynamics. Therefore, both equilibrium and generalized kinetic precipitation–dissolution approach are employed and the comparison is discussed.
As discussed in Sect. 8.2, the kinetic precipitation approach is in line with the experimental observations for NaCl precipitation. However, the kinetic precipitation rate constant \(k_n\) and fitting parameters are usually not available at hand and must be determined explicitly for each salt type. Numerical simulations for NaCl–NaI precipitation are observed to be analogous with the phenomenological explanations. However, for better understanding, model validation with NaI precipitation experiments must be undertaken in the near future. This should also help to better predict the mixed salt (NaCl–NaI) precipitation dynamics.
Due to constraint of space, in this work, we are unable to address combined precipitation–dissolution processes in a mixed salt system. Salt dissolution processes in the shallow subsurface are rather complicated and are hardly addressed in the literature. Therefore, a detailed numerical modeling and analysis of salt dissolution processes are strongly recommended in the future.
The developed model concept can be applied to environmental, engineering and industrial applications, where mixed salt precipitation processes are significantly influenced by FF–PM interaction.
Notes
Acknowledgments
This work is supported by the German research foundation (DFG) under the framework of the International research and training group NUPUS (GRK 1398). We thank Dr. Mansoureh Norouzi Rad, School of Chemical Engineering and Analytical Sciences, University of Manchester and Saideep Pavuluri, Institute of Petroleum Engineering, HeriotWatt University, Edinburgh, for their valuable support. Dr. Nima Shokri acknowledges the funding by The Leverhulme Trust (RPG2014331) and the donors of the American Chemical Society Petroleum Research Fund for partial support of the experimental work (PRF No. 52054DNI6).
References
 Alazmi, B., Vafai, K.: Analysis of fluid flow and heat transfer interfacial conditions between a porous medium and a fluid layer. Int. J. Heat Mass Transf. 44(9), 1735–1749 (2001). doi: 10.1016/S00179310(00)002179 CrossRefGoogle Scholar
 Alazmi, B., Vafai, K.: Constant wall heat flux boundary conditions in porous media under local thermal nonequilibrium conditions. Int. J. Heat Mass Transf. 45(15), 3071–3087 (2002). doi: 10.1016/S00179310(02)000443 CrossRefGoogle Scholar
 Appelo, C.A.J., Postma, D.: Geochemistry, Groundwater and Pollution, 2nd edn. Balkema, Leiden (2005)CrossRefGoogle Scholar
 Baber, K., Mosthaf, K., Flemisch, B., Helmig, R., Müthing, S., Wohlmuth, B.: Numerical scheme for coupling twophase compositional porousmedia flow and onephase compositional free flow. IMA J. Appl. Math. 77(6), 887–909 (2012)Google Scholar
 Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klfkorn, R., Kornhuber, R., Ohlberger, M., Sander, O.: A generic grid interface for parallel and adaptive scientific computing. Part ii: implementation and tests in dune. Computing 82(2–3), 121–138 (2008). doi: 10.1007/s0060700800049 CrossRefGoogle Scholar
 Battistelli, A., Calore, C., Pruess, K.: The simulator tough2/ewasg for modelling geothermal reservoirs with brines and noncondensible gas. Geothermics 26(4), 437–464 (1997). doi: 10.1016/S03756505(97)000072 CrossRefGoogle Scholar
 Batzle, M.L., Wang, Z.: Seismic properties of pore fluids. Geophysics 57, 1396–1408 (1992). doi: 10.1190/1.1443207 CrossRefGoogle Scholar
 Beavers, G.S., Joseph, D.D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197–207 (1967). doi: 10.1017/S0022112067001375 CrossRefGoogle Scholar
 Class, H., Helmig, R., Bastian, P.: Numerical simulation of nonisothermal multiphase multicomponent processes in porous media.: 1. An efficient solution technique. Adv. Water Resour. 25(5), 533–550 (2002). doi: 10.1016/S03091708(02)000143 CrossRefGoogle Scholar
 Clement, T.P., Hooker, B.S., Skeen, R.S.: Macroscopic models for predicting changes in saturated porous media properties caused by microbial growth. Ground Water 34(5), 934–942 (1996). doi: 10.1111/j.17456584.1996.tb02088.x CrossRefGoogle Scholar
 Crowe, A., Langstaffe, F.J.: Extension of geochemical modelling techniques to brine: coupling of the pitzer equations with models. In: Proceedings of Solving Groundwater Problems with Models. National Water Well Association Conference, Denver, CO, 10–12 Feb 1987Google Scholar
 Domenico, P.A., Schwartz, F.W.: Physical and Chemical Hydrogeology. Wiley, New York (1990)Google Scholar
 Flemisch, B., Darcis, M., Erbertseder, K., Faigle, B., Lauser, A., Mosthaf, K., Müthing, S., Nuske, P., Tatomir, A., Wolff, M., Helmig, R.: DuMuX: DUNE for multiphase, component, scale, physics.., flow and transport in porous media. Adv. Water Resour. 34(9), 1102–1112 (2011). doi: 10.1016/j.advwatres.2011.03.007. New Computational Methods and Software ToolsCrossRefGoogle Scholar
 Fujimaki, H., Shimano, T., Inoue, M., Nakane, K.: Effect of a salt crust on evaporation from a bare saline soil. Vadose Zone J 5(4), 1246–1256 (2006). doi: 10.2136/vzj2005.0144 CrossRefGoogle Scholar
 Gamazo, P., Saaltink, M., Carrera, J., Slooten, L., Bea, S.: A consistent compositional formulation for multiphase reactive transport where chemistry affects hydrodynamics. Adv. Water Resour. 35, 83–93 (2012). doi: 10.1016/j.advwatres.2011.09.006 CrossRefGoogle Scholar
 IAPWS: Revised release on the IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use. http://www.iapws.org (2009)
 Jambhekar, V., Schröder, N., Helmig, R., Shokri, N.: Freeflowporousmedia coupling for evaporationdriven transport and precipitation of salt. Transp. Porous Med. 110, 251–280 (2015)Google Scholar
 Kelly, S.F., Selker, J.S.: Osmotically driven water vapor transport in unsaturated soils. Soil Sci. Soc. Am 65, 1634–1641 (2001)CrossRefGoogle Scholar
 Lasaga, A.C., Soler, J.M., Ganor, J., Burch, T.E., Nagy, K.L.: Chemical weathering rate laws and global geochemical cycles. Geochim. Cosmochim. Acta 58(10), 2361–2386 (1994). doi: 10.1016/00167037(94)900167 CrossRefGoogle Scholar
 Mayer, K.U., Frind, E.O., Blowes, D.W.: Multicomponent reactive transport modeling in variably saturated porous media using a generalized formulation for kinetically controlled reactions. Water Resour. Res. 38(9), 13–1–13–21 (2002). doi: 10.1029/2001WR000862 CrossRefGoogle Scholar
 Michaelides, E.: Thermodynamic properties of geothermal fluids. Geotherm. Res. Counc. Trans. 5, 361–364 (1981)Google Scholar
 Millington, R., Quirk, J.: Permeability of porous solids. Trans. Faraday Soc. 57, 1200–1207 (1961)CrossRefGoogle Scholar
 Mosthaf, K., Baber, K., Flemisch, B., Helmig, R., Leijnse, A., Rybak, I., Wohlmuth, B.: A coupling concept for twophase compositional porousmedium and singlephase compositional free flow. Water Resour. Res. (2011). doi: 10.1029/2011WR010685 Google Scholar
 Munns, D.R.: The Impact of Salinity Stress. CSIRO Division of Plant Industry, Canberra ACT (2011)Google Scholar
 Nachshon, U., Shahraeeni, E., Or, D., Dragila, M., Weisbrod, N.: Infrared thermography of evaporative fluxes and dynamics of salt deposition on heterogeneous porous surfaces. Water Resour. Res. (2011). doi: 10.1029/2011WR010776 Google Scholar
 Nachshon, U., Weisbrod, N., Dragila, M.I., Grader, A.: Combined evaporation and salt precipitation in homogeneous and heterogeneous porous media. Water Resour. Res. (2011). doi: 10.1029/2010WR009677 Google Scholar
 Norouzi Rad, M., Shokri, N.: Effects of grain angularity on nacl precipitation in porous media during evaporation. Water Resour. Res. 50(11), 9020–9030 (2014). doi: 10.1002/2014WR016125 CrossRefGoogle Scholar
 Pitzer, K., Kim, J.: Thermodynamics of electrolytes: 3 activity and osmotic corfficients for 2–2 electrolytes. J. Solut. Chem. 3, 539–546 (1974)CrossRefGoogle Scholar
 Rad, M.N., Shokri, N.: Nonlinear effects of salt concentrations on evaporation from porous media. Geophys. Res. Lett. (2012). doi: 10.1029/2011GL050763 Google Scholar
 Rad Norouzi, M., Shokri, N., Sahimi, M.: Porescale dynamics of salt precipitation in drying porous media. Phys. Rev. E 88(32404), 1–5 (2013)Google Scholar
 Reid, R.C., Prausnitz, J.M., Poling, B.E.: The Properties of Gases and Liquids. McGrawHill, Inc, New York (1987)Google Scholar
 Saffman, R.: On the boundary condition at the surface of the Porous medium. Stud. Appl. Math. 50, 93–101 (1971)CrossRefGoogle Scholar
 Shokri, N., Salvucci, G.: Evaporation from porous media in the presence of a water table. Vadose Zone J 10(4), 1309–1318 (2011). doi: 10.2136/vzj2011.0027 CrossRefGoogle Scholar
 Somerton, W.H., ElShaarani, A.H., Mobarak, S.M.: High temperature behavior of rocks associated with geothermal type reservoirs. In: SPE California Regional Meeting. Society of Petroleum Engineers, San Francisco, CA (1974). doi: 10.2118/4897MS
 Steefel, C.I., Cappellen, P.V.: A new kinetic approach to modeling water–rock interaction: the role of nucleation, precursors, and Ostwald ripening. Geochim. Cosmochim. Acta 54(10), 2657–2677 (1990). doi: 10.1016/00167037(90)900034 CrossRefGoogle Scholar
 Truesdell, A., Jones, B.: A Computer Program for Calculating Chemical Equilibrium of Natural Waters, vol. 73. U.S. Geological Survey, Washington, DC (1973)Google Scholar
 Van Genuchten, M.T.: A closedfrom equation for predicting the hydraulic conductivity of unstructured soil. Soil Sci. Soc. Am. J. 44, 892–898 (1980)CrossRefGoogle Scholar
 Xu, T., Spycher, N., Sonnenthal, E., Zheng, L., Pruess, K.: TOUGHREACT User’s Guide: A Simulation Program for NonIsothermal Multiphase Reactive Geochemical Transport in Variably Saturated Geologic Media, Version 2.0. Lawrence Berkeley National Laboratory, Berkeley, California, lbnldraft edn. (2012)Google Scholar