Applied Geochemistry 55 (2015) 46–61
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Applied Geochemistry
journal homepage: www.elsevier.com/locate/apgeochem
A chemical kinetics algorithm for geochemical modelling ⇑ Allan M.M. Leal a, , Martin J. Blunt a, Tara C. LaForce b a Imperial College London, QCCSRC and Department of Earth Science and Engineering, London, UK b CSIRO, Earth Science and Resource Engineering, Australia article info abstract
Article history: A chemical kinetics algorithm is presented for geochemical applications. The algorithm is capable of han- Available online 12 October 2014 dling both equilibrium- and kinetically-controlled reactions in multiphase systems. The ordinary differ- ential equations (ODEs) are solved using an implicit multistep backward differentiation formula (BDF) algorithm to ensure efficiency and stability when integrating stiff ODEs. An adaptive control scheme of the time step is adopted to guarantee small steps in steeper regions and large steps in smoother regions of the integration. Analytical derivatives of the reaction rates and species activities are used to permit the use of larger time steps, and to increase the robustness of the calculations. The chemical equilibrium calculations are performed using a Gibbs free energy minimisation algorithm, which is based on a trust-region interior-point method adapted with a watchdog strategy that yields quadratic rates of con- vergence near the solution. The chemical kinetics algorithm is applied to geochemical problems relevant to carbon storage in saline aquifers. The calculations assume aqueous, gaseous and mineral phases, where the kinetics of the water–gas–rock interactions are investigated. The results allow us to estimate the time
frames at which brine of different salinities and supercritical CO2 attains equilibrium with a carbonate rock, as well as the amount of carbon dioxide trapped by solubility and mineralisation mechanisms. Ó 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/).
1. Introduction time-dependence, chemical kinetics consists of solving a system of ordinary differential equations, while chemical equilibrium Frequently geochemical investigations of a system assume requires only the solution of algebraic equations. chemical equilibrium conditions. Calculating the solubilities of Another complexity in geochemical kinetics is the broad differ- minerals and gases in aqueous solutions at different temperatures, ence of the speeds of the aqueous, gaseous and mineral reactions pressures, salinities, and other conditions require only equilibrium (Lasaga, 1998). Commonly, aqueous reactions proceed substan- calculations (Anderson and Crerar, 1993). Sometimes, however, tially faster than mineral reactions, with the former sometimes one might be interested in the time scales over which such pro- achieving equilibrium in microseconds, while the latter requiring cesses occur, and equilibrium calculations will not provide this. several days to many years. Therefore, this can result in an ineffi- Application of chemical kinetics theory is vital when the tran- cient numerical integration of the ordinary differential equations, sient chemical state of a system is important. This is useful, for requiring tiny time steps to ensure accuracy and stability. example, to analyse the temperature and pressure effects on the To address this problem, we consider the geochemical system time required for a mineral to equilibrate with a solution. In addi- to be in partial equilibrium (Helgeson, 1968; Helgeson et al., tion, it describes the water–gas–rock effects over time of a geochem- 1969, 1970). A system in partial equilibrium means that it is in ical process, such as the continuous consumption or production of equilibrium with respect to some reactions and out of equilibrium gases while minerals are reacting in an aqueous solution. with respect to others. For example, since aqueous and gaseous A more detailed modelling procedure has its consequent reactions are often considerably faster than mineral reactions, it complexities, however. Chemical kinetics calculations require seems plausible to assume they are in equilibrium at all times. more input data and models than chemical equilibrium calcula- As the mineral reactions proceed kinetically, the aqueous and gas- tions. For example, calculating the evolution of the system compo- eous reactions are constantly perturbed and then instantaneously sition demands rate laws of the reactions. In addition, due to its re-equilibrated. Note, however, that the partial equilibrium assumption is based on the relative speed of the reactions. There-
⇑ Corresponding author. fore, it is possible to assume a mineral reaction in equilibrium and E-mail addresses: [email protected] (A.M.M. Leal), [email protected]. an aqueous reaction out of equilibrium at any instant. uk (M.J. Blunt), [email protected] (T.C. LaForce). http://dx.doi.org/10.1016/j.apgeochem.2014.09.020 0883-2927/Ó 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). A.M.M. Leal et al. / Applied Geochemistry 55 (2015) 46–61 47
The partial equilibrium assumption simplifies the problem by of K-feldspar where the partial equilibrium assumption was replacing stiff differential equations by algebraic ones. These alge- adopted by considering the aqueous reactions in equilibrium. braic equations govern the equilibrium condition of those reac- However, their reaction path modelling was not based on kinetic tions assumed to be in equilibrium. As a result, the governing rate laws of the reactions, but on the use of a progress variable equations become a system of non-linear differential–algebraic to describe the compositional change of the system. Helgeson equations, with the constraint that mass of the chemical elements (1971) later on modelled the feldspar hydrolysis using a parabolic in the system must be conserved and charge balance of an electro- rate law. The simplistic rate was derived using Fick’s first law of lyte solution attained. diffusion on a one-dimensional problem of diffusion along the sur- In this work we present an algorithm for chemical kinetics cal- face layer of the mineral reacting with the aqueous solution. culation in multiphase geochemical systems. The method supports Helgeson and Murphy (1983) combined the rate laws proposed the mixing of reactions controlled by chemical equilibrium and in Aagaard and Helgeson (1982) with a numerical integration rou- kinetics. However, different from common practices in geochemis- tine to model irreversible reactions among minerals and aqueous try modelling, only the reactions assumed out of equilibrium (i.e., solutions. They again considered the hydrolysis of feldspar, with the kinetically-controlled reactions) and their rate laws are the possibility of precipitation of secondary minerals (muscovite, required in the calculations. This is because our approach does gibbsite, and kaolinite). The secondary minerals were assumed to not use a stoichiometric method (also known as law of mass-action precipitate under the partial equilibrium assumption. approach) for equilibrium calculations, which requires the equa- Although these preliminary works led by Helgeson 30–40 years tions of the equilibrium reactions and their equilibrium constants ago were the precursors of many others, they were always intended (Smith and Missen, 1982). for a specific system. There was no formalisation of geochemical The chemical equilibrium calculations are performed using our kinetics calculations for general multiphase systems with a mixing Gibbs free energy minimisation algorithm for multiphase and non- of equilibrium- and kinetically-controlled reactions. In addition, no ideal systems Leal et al. (2014). This method was specifically efficient methodology was discussed for the solution of the result- designed for applications that require sequential equilibrium ing system of differential–algebraic equations. Some of these gaps calculations, such as chemical kinetics and reactive transport mod- have been addressed, as we shall discuss next. However, we believe elling. The results shown there indicated quadratic rates of conver- that an efficient, general, and flexible algorithm has yet to be devel- gence near the solution by using a trust-region interior-point oped for chemical kinetics in geochemical modelling. method adapted with a watchdog strategy. Therefore, by using The following is a list of computer codes commonly used for the compositional state in a previous time step as the initial guess geochemical kinetics modelling: EQ6 (Wolery and Daveler, 1992), for the equilibrium calculation in a subsequent step, only a few PHREEQC (Parkhurst and Appelo, 1999, 2013), MINTEQA2 iterations should be necessary to solve the problem. (Allison and Kevin, 1991), CHESS (van der Lee and Windt, 2002), Integration of the ordinary differential equations is performed SOLMINEQ.88 (Kharaka et al., 1988), and The Geochemist’s Work- using an implicit multistep BDF algorithm (Ascher and Petzold, bench (Bethke, 2007). They calculate the evolution of systems as 1998). This algorithm is specially effective for stiff ODEs, which minerals kinetically dissolve or precipitate. In addition, The Geo- are characterised by solutions with rapid variations in some of chemist’s Workbench, as described in Bethke (2007), provides sup- the variables (Hairer and Wanner, 2010). Note that assuming the port for modelling redox reactions controlled by kinetics. fastest reactions in the system (e.g., aqueous reactions) to be in As discussed in Leal et al. (2013, 2014), these geochemical pack- equilibrium is not enough to prevent a stiff system of ordinary dif- ages adopt a stoichiometric approach for aqueous speciation calcu- ferential equations. Modelling chemical kinetics with only one lations. Their databases contain only the equilibrium constants of kinetically-controlled reaction is already susceptible to form a stiff the reactions, which are required for the solution of the system differential equation if the reaction causes fast variations in the of mass action equations. The chemical potentials of the species, system composition. on the other hand, are not available, which are needed to calculate Analytical derivatives of the rate laws and species activities are the Gibbs free energy of the system. Therefore, determining the used in the calculation. This is a consequence of our choice of an stable equilibrium phase assemblage of the system is a difficult implicit integration method, which requires the solution of a sys- task, since, given two or more states, it is not possible to determine tem of non-linear algebraic equations. The derivatives of the rate which one has the lowest Gibbs free energy. laws and the species activities are, therefore, necessary in the In addition, these geochemical codes use an incomplete Newton assembly of the Jacobian matrix of the right-hand side of the sys- scheme for aqueous speciation calculations. This approach was lar- tem of ODEs. Although the computation of these derivatives gely influenced by the algorithm of Morel and Morgan (1972), with requires some computational effort, it allows the use of larger time further improvements by Reed (1982). As argued in Leal et al. steps and increases the stability of the integration (Ascher and (2013, 2014), this practice results in slow rates of convergence in Petzold, 1998; Hairer and Wanner, 2010). the solution of the non-linear system of equations. This is because An adaptive control scheme of the time step is adopted in the the incomplete Newton’s method consists of combining Newton’s integration. This ensures small steps in steeper regions and large method with a successive substitution approach, preventing the steps in smoother regions. As a result, we achieve both accuracy calculation to converge at quadratic rates near the solution. and efficiency throughout the calculation. We have observed that Recently, Mironenko and Zolotov (2011) developed a computer this adaptive control is essential in geochemistry, since, for exam- code for modelling equilibrium-kinetics of water–rock interac- ple, minerals react very fast initially (a steep region), and then pro- tions. Instead of using a stoichiometric scheme for chemical equi- ceed very slowly (a smooth region) until equilibrium. If a constant librium calculations, they used the algorithm of de Capitani and time step is adopted, then it must be small enough to guarantee Brown (1987), which minimises the Gibbs free energy of the sys- that the integration is accurate and stable at the beginning of the tem using a convex simplex approach. The chemical equilibrium process. However, the required initial time step (e.g., 10 4 s) is usu- algorithm of Leal et al. (2014) used here, however, is capable of ally orders of magnitude smaller than what should be used near minimising non-convex objective functions using a trust-region equilibrium (e.g., days or years). interior-point method. In addition, the adaptive time step control The first work on computational reaction path modelling in we adopt in our work is based on numerical analysis, while their geochemistry can be tracked to Helgeson (1968) and Helgeson approach is based on a heuristic that aims to prevent strong et al. (1969). They presented a modelling example of the hydrolysis changes in pH between time steps. 48 A.M.M. Leal et al. / Applied Geochemistry 55 (2015) 46–61
The proposed chemical kinetics algorithm and the chemical 2.2. Partial equilibrium equilibrium method based on a Gibbs energy minimisation approach has been implemented in Reaktoro, a scientific library The reactions in geochemical systems proceed with different for computational geochemical modelling written in the C++ pro- speeds. Their time scales can differ from each other by several gramming language. The code is freely available at www.bitbuc- orders of magnitude, ranging from microseconds to years ket.org/reaktoro, where its licensing information can be found. (Lasaga, 1998). Langmuir (1996) presents a list containing some common geochemical reactions and their respective half-times.1 For example, the acid–base reaction involving only solutes: 2. Chemical kinetics
H CO aq Hþ HCO ; 2:6 In this section we present the governing equations that model 2 3ð Þ þ 3 ð Þ the compositional evolution of a chemical system subject to reac- 6 tions controlled by kinetics and equilibrium. The formulation has a half-time of about 10 s. The solute–water hydration assumes a closed-system for simplicity. Addition of source and sink reaction: contributions to the equations should, however, be straightfor- ward. Moreover, we assume that the kinetic processes occur in a CO2ðaqÞþH2OðlÞ H2CO3ðaqÞ; ð2:7Þ well-mixed batch reactor, which allows us to neglect any transport phenomena such as diffusion, convection and dispersion. However, has a half-time of 0.1 s. Compare these time scales with the half- the algorithm is designed to be coupled to a transport simulator in time of the mineral dissolution reaction: future work. þ 2þ CaCO3ðsÞþH Ca þ HCO3 ; ð2:8Þ 2.1. Governing equations which can be in the order of weeks at low temperatures. Consider the following linearly independent reactions taking These broad differences on the speed of the reactions pose sev- place in a chemical system: eral numerical complications. The ordinary differential Eqs. (2.4) are severely stiff, requiring appropriate methods for its integration. XN However, a carefully selected numerical method might still need 0 m a ðj ¼ 1; ...; MÞ; ð2:1Þ ji i tiny time steps in order to capture the kinetics of the fastest reac- i¼1 tion in the system. It is not optimal to use time steps in the order of where ai is the i-th chemical species; mji is the stoichiometric coef- microseconds when there are some reactions in the system requir- ficient of the i-th species in the j-th reaction; N is the number of ing years to achieve some progress, and where the application of chemical species; and M is the number of reactions. The stoichiom- interest has time scales of millennia, such as for carbon storage. etric coefficients in any reaction are assumed to be positive for Therefore, a simplification is necessary to allow larger time steps products and negative for reactants. for efficient integration and still provide accurate calculations. From the theory of chemical kinetics, it follows that the compo- It is plausible to assume partial equilibrium in some geochem- sitional evolution of a system is governed by the following ordin- ical processes (Helgeson, 1968; Helgeson et al., 1969, 1970). Con- ary differential equations: sider the dissolution of calcite in an aqueous solution given by reaction (2.8). Recall that the speed of the reactions involving only dni aqueous solutes are, in general, considerably faster than the speed ¼ f iðT; P; nÞði ¼ 1; ...; NÞ; ð2:2Þ dt of this reaction. Thus, it is reasonable to consider that the aqueous solutes are in equilibrium at all times during the process, while cal- where t is the time variable; ni is the number of moles of the i-th species; n 2 RN is the molar composition vector of the system; T cite is kinetically reacting, and thus out of equilibrium with them. and P are the given temperature and pressure of the system; and This assumption has also been adopted by Lichtner (1985), Steefel 2þN # and Cappellen (1990) and Steefel and Lasaga (1994). f i : R R is defined by: The partial equilibrium assumption eliminates the dependence XM of the calculations on the time scales of the fast reactions. Because f iðT; P; nÞ :¼ mjirjðT; P; nÞ; ð2:3Þ only the slow reactions are assumed to be controlled by kinetics, j¼1 while the fast reactions are controlled by equilibrium, the rate laws which accounts for the production and consumption of the i-th spe- of the latter are no longer necessary. Their equilibrium conditions cies in every reaction (2.1). The kinetic rate function of the j-th reac- are governed by algebraic constraints instead of differential ones. 2þN In addition, the partial equilibrium assumption simplifies the tion is denoted by rj : R # R. The convention adopted is that rj is positive when the reaction proceeds towards the products, and neg- modelling. Assuming all reactions in geochemical processes to be ative towards the reactants. controlled by kinetics can be a daunting task. For example, the rate The system of differential Eqs. (2.2) can be written in matrix law of every reaction would be necessary, which in general notation as: requires several temperature and pressure dependent parameters. Moreover, every heterogeneous reaction, including the gaseous– dn aqueous reactions, would require some reactive surface area ¼ fðnÞ; ð2:4Þ dt model, increasing the complexity of the modelling. Nevertheless, care must be taken not to assume partial equilib- N # N with f : R R defined by: rium inappropriately. Analysing the rates of all reactions occurring in the process is fundamental for identifying the fast and slow fðnÞ :¼ mT rðnÞ; ð2:5Þ reactions and guaranteeing some degree of accuracy. Sometimes, where m 2 RM N denotes the stoichiometric matrix of reactions however, accuracy can be compromised by modelling a reaction (2.1); and r : RN # RM the rate function of these reactions. with equilibrium control. Note that for convenience reasons we omitted the dependence on temperature and pressure from Eqs. (2.4) and (2.5). 1 The necessary time to consume half of the initial amount of a reactant. A.M.M. Leal et al. / Applied Geochemistry 55 (2015) 46–61 49
2.3. Partitioning Table 2.2 Partition of the chemical system H2O–CO2–Halite–Calcite–Magnesite–Dolomite in equilibrium and kinetic species. The reactions are classified into two groups: fast reactions and slow reactions. Following the discussion of partial equilibrium in Equilibrium species Kinetic species H2O(l) CaCO3 (s) the previous section, we will assume the slow reactions to be con- + H MgCO3(s) trolled by kinetics, and the fast reactions to be controlled by OH (CaMg)(CO3)2(s) equilibrium. HCO3 Define a kinetic reaction as a reaction controlled by kinetics, and 2 CO3 an equilibrium reaction as a reaction controlled by equilibrium. In Na+ addition, define the following terms: Cl Ca2+ Mg2+ equilibrium species is any species that participates in an equi- H2CO3 (aq)
librium reaction, CO2(aq) kinetic species is any species that participates in a kinetic reac- CaCO3 (aq) tion, but not in an equilibrium reaction, MgCO3(aq) CaCl (aq) inert species is a species that does not participate in an equilib- 2 CO2(g)
rium or kinetic reaction, H2O(g) NaCl(s) which will be useful in the formulation of the governing equations of chemical kinetics coupled with chemical equilibrium.
Let ae; ak, and ai denote the set of equilibrium, kinetic and inert species can be found in Table 2.2. Note that no inert species was species respectively. Moreover, let Ne; Nk, and Ni denote the assumed. respective number of equilibrium, kinetic, and inert species. From the previous definitions, it follows that: 2.4. Governing equations: revisited ae [ ak [ ai ¼ a; ð2:9Þ The formulation in Section 2.1 assumed that all reactions were and controlled by kinetics. In this section, however, we will separate reactions (2.1) in equilibrium and kinetic reactions as follows: a \ a \ a ¼ Ø; ð2:10Þ e k i XN 0 me a ðj ¼ 1; ...; M Þ; ð2:12Þ where a denotes the set of all species in the system. The set of equi- ji i e i¼1 librium species ae can be constructed by the union of the species participating in the equilibrium reactions. The set of kinetic species and ak, on the other hand, can be constructed using: XN k 0 mjiai ðj ¼ 1; ...; MkÞ; ð2:13Þ ak ¼ a ðae [ aiÞ; ð2:11Þ i¼1 which can be derived from conditions (2.9) and (2.10). e k where mji and mji are the stoichiometry of the i-th species in the j-th In order to elucidate the partitioning of the species in equilib- equilibrium and kinetic reactions respectively; and Me and Mk are rium, kinetic and inert species, consider the example chemical sys- the number of equilibrium and kinetic reactions in the system. Me N Mk N tem in Table 2.1. The reactions occurring in this system is listed in As before, let me 2 R and mk 2 R denote the stoichiome- Table 2.3. Note that for the modelling of the chemical kinetics of tric matrices of the equilibrium and kinetic reactions respectively. this system we assume the reactions involving aqueous and gas- From the partitioning discussion in Section 2.3, it follows that equi- eous species to be controlled by equilibrium. Also, due to the fast librium reactions only contain equilibrium species, while kinetic rates of dissolution and precipitation of mineral halite, its reaction reactions can include both equilibrium and kinetic species. There- Mk Ne Mk Nk is also assumed to be controlled by equilibrium. The reactions fore, we let mke 2 R and mkk 2 R denote the stoichiometric involving calcite, magnesite and dolomite were assumed to be con- matrices constructed from the columns of mk corresponding to trolled by kinetics, because their rates are not as fast as the others. equilibrium and kinetic species respectively. Based on our previous definitions, the equilibrium and kinetic Let us now formulate the mathematical equations for a general chemical kinetics problem coupled with equilibrium conditions. From Eq. (2.4), we can write the following governing equations Table 2.1 for the evolution of the molar abundance of the kinetic species: Description of the chemical system H2O–CO2–Halite–Calcite–Magnesite–Dolomite with their phases and respective chemical species. dn k ¼ f ðnÞ; ð2:14Þ Aqueous phase Gaseous phase dt k H O(l) CO (g) 2 2 N N + # k H H2O(g) with fk : R R defined by: OH Mineral phase #1 T fkðnÞ :¼ m rkðnÞ; ð2:15Þ HCO3 NaCl(s) (Halite) kk 2 Mineral phase #2 CO3 N # Mk + where rk : R R denotes the rate function of kinetic reactions Na CaCO3(s) (Calcite) Cl Mineral phase #3 (2.13). 2þ Although we can write a similar equation for the equilibrium Ca MgCO3ðsÞ (Magnesite) Mg2þ Mineral phase #4 species:
H2CO3ðaqÞðCaMgÞðCO3Þ2ðsÞ (Dolomite) dne CO2(aq) ¼ feðnÞ; ð2:16Þ CaCO3 (aq) dt MgCO3(aq) we cannot, as before, write an analytical expression for function CaCl2(aq) N Ne fe : R # R . Because of the equilibrium conditions imposed by 50 A.M.M. Leal et al. / Applied Geochemistry 55 (2015) 46–61
E Table 2.3 where be; bk 2 R are the molar abundance vectors of the chemical Description of the equilibrium and kinetic reactions in the chemical system of E Ne elements in the equilibrium and kinetic partitions; and We 2 R Table 2.1. E Nk and Wk 2 R are the formula matrices of the equilibrium and Equilibrium reactions Kinetic reactions kinetic species. H O(l) Hþ OH + þ2 2 þ CaCO3(s) + H HCO3 þ Ca From the principle of mass conservation, it follows that: þ + þ2 CO2(aq) + H2O(l) HCO3 þ H MgCO3(s) + H HCO3 þ Mg H CO (aq) HCO Hþ CaMg CO s 2Hþ db dbe dbk 2 3 3 þ ð Þð 3Þ2ð Þþ ¼ þ ¼ 0: ð2:20Þ þ2 þ2 dt dt dt 2HCO3 þ Ca þ Mg 2 þ CO3 þ H HCO3 By multiplying Eq. (2.14) by Wk we obtain: þ þ2 CaCO3ðaqÞþH HCO3 þ Ca 2 db MgCO ðaqÞþHþ HCO þ Mgþ k 3 3 ¼ WkfkðnÞ; ð2:21Þ þ2 dt CaCl2ðaqÞ Ca þ 2Cl
CO2ðgÞ CO2ðaqÞ which can be used to write the evolution of the molar abundance of H O(g) H O(l) 2 2 the elements in the equilibrium partition: NaCl(s) Na+ +Cl db e ¼ g ðnÞ; ð2:22Þ dt e the equilibrium reactions, their rates of production and consump- N # E where ge : R R is defined by: tion are, in fact, unknowns in the problem. Therefore, an alternative approach must be used to evolve the geðnÞ :¼ WkfkðnÞ: ð2:23Þ molar abundance of the species without requiring the produc- Therefore, combining Eqs. (2.14) and (2.22) we have the following tion/consumption rates of the equilibrium species fe. For this we system of ordinary differential equations: will rely on the principle of mass conservation, which allows us du to state that the mass that leaves or enters the kinetic partition ¼ wðuÞ; ð2:24Þ must, respectively, enter or leave the equilibrium partition. dt In Fig. 2.1 we illustrate the chemical system of Table 2.1, with where u : RD # RD denotes the unknown function to be integrated; its equilibrium and kinetic species. The figure shows the exchange and w : RD # RD the right-hand side function of the ordinary differ- of element atoms among the equilibrium and kinetic partitions. ential equation, both defined by: The fact that these atoms are preserved in the system will allow nk us to calculate the evolution of the molar abundance of the u :¼ ; ð2:25Þ elements in the equilibrium partition. As a result, the composition be of the equilibrium species ne can be calculated at any time by and solving an equilibrium problem using these elemental molar abundances. wðuÞ :¼ ArkðnÞ; ð2:26Þ E Let b 2 R denote the molar abundance vector of the chemical with the coefficient matrix A 2 RD Nk given by: elements in the system, and W 2 RE N the formula matrix of all "# species in the chemical system. The formula matrix W is defined mT A :¼ kk ; ð2:27Þ such that its ðj; iÞ-th entry, given by w , denotes the number of T ji Wkmkk atoms of the j-th element in the i-th species. Therefore, it follows that the molar abundance of the elements can be calculated using: where D :¼ Nk þ E. Therefore, Eqs. (2.25) and (2.26) govern the evolution of the b ¼ Wn: ð2:17Þ molar abundance of both kinetic species nk and chemical elements Similarly, we can write the following equations for the equilibrium in the equilibrium partition be. Observe, however, that the rate and kinetic partitions: function rk in Eq. (2.26) depends on the system composition n, which cannot be explicitly obtained from u. In what follows we b W n ; 2:18 e ¼ e e ð Þ will see how this problem can be resolved. and 2.5. Chemical equilibrium bk ¼ Wknk; ð2:19Þ To integrate Eqs. (2.25) and (2.26), it is necessary to construct a function u : RD # RN such that: n ¼ uðuÞ: ð2:28Þ Unfortunately an explicit expression for the function u is not avail- able due to the intricate dependence of n on u. In fact, all the com- plexity of this dependence lies in the calculation of the molar
abundance of the equilibrium species ne from the elemental molar
abundance be. This is because the molar abundance of the kinetic T T species nk can be explicitly obtained from u ¼ðu1; u2Þ ¼ðnk; beÞ using:
nk ¼ u1: ð2:29Þ In Section 2.2 we introduced the concept of partial equilibrium, and in Section 2.3 we formalised the concept of partitioning the species in a set of equilibrium and kinetic species. As a result, we assumed Fig. 2.1. Exchange of elements between the equilibrium and kinetic partitions for that the equilibrium species constitute a sub-system in which the chemical system in Table 2.2. chemical equilibrium is always attained. A.M.M. Leal et al. / Applied Geochemistry 55 (2015) 46–61 51
Therefore, from the principle of minimum Gibbs free energy, banded matrices, using direct or iterative methods for linear sys- the chemical equilibrium state of the equilibrium species can be tems. The algorithm uses a Adams–Moulton method for non-stiff calculated by solving the following constrained minimisation ODEs, and a backward differentiation formula (BDF) method for problem: stiff ones. In both cases, the step-size used in the numerical inte- 8 gration is variable, such that smaller steps are used in sharper > W n b < e e ¼ e regions, and larger steps in smoother regions. T min Geðne; T; P; nkÞ subject to ce ne ¼ 0 ; ð2:30Þ The solver CVODE calculates the solution at a new time step using ne :> ne P 0 an implicit scheme, requiring a system of non-linear algebraic equations to be solved. Two approaches are offered for solving where ce denotes the vector of electrical charges of the equilibrium these non-linear equations: Newton iteration and functional itera- Ne # species; and Ge : R R the Gibbs free energy function of the equi- tion. The former uses Newton’s method to solve the equations, librium partition, defined by: while the latter uses a successive-substitution method. As a result, T the functional iteration approach is advised to be used only for Geðne; T; P; nkÞ :¼ ne l ðne; T; P; nkÞ: ð2:31Þ e non-stiff ODEs (Hindmarsh et al., 2005). The chemical potential function of the equilibrium species The Newton iteration approach is adopted in this work. Ne # Ne le : R R is defined by: Although it is more suitable for stiff ODEs, resulting in more effi- cient calculations that use larger time steps, this approach l ðn ; T; P; n Þ :¼ l ðT; PÞþRT ln a ðn ; T; P; n Þ; ð2:32Þ e e k e e e k increases the level of complexity of the numerical integration. This 2 # Ne is because using Newton’s method requires the Jacobian function where le : R R is the standard chemical potential function of D D D Ne Ne J : # defined by: the equilibrium species; ae : R # R is the activity function of R R the equilibrium species; and R is the universal gas constant. More @wðuÞ discussion on the calculation of these thermodynamic quantities JðuÞ :¼ : ð2:33Þ @u can be found in Leal et al. (2013). Note that the solution of the min- imisation problem (2.30) and the evaluation of all functions in Eqs.
(2.31) and (2.32) assume T; P and nk as constant parameters. 2.7. Jacobian function In Leal et al. (2014) we presented a non-stoichiometric2 method for multiphase chemical equilibrium calculations that solves prob- Let us now present a methodology for the calculation of the lem (2.30). The method is based on an interior-point minimisation Jacobian function JðuÞ. Combining Eqs. (2.26) and (2.33), and algorithm capable of converging from arbitrary initial guesses. It applying the chain rule in the derivative term, results in: was shown to be able to capture efficiently and robustly any transi- @r @n tion in phase assemblage during the calculation. In addition, a per- J ¼ A k : ð2:34Þ formance assessment of the algorithm indicated quadratic rates of @n @u convergence near the solution, with sequential equilibrium calcula- The partial molar derivatives of the reaction rates @rk=@n can be tions converging in only a few iterations. Therefore, we adopted this obtained by differentiating the rate functions either analytically or chemical equilibrium method in this work. numerically. In this work we adopt an analytical approach. Alternatively, one can also use a stoichiometric method for the The calculation of the partial derivatives @n=@u is slightly more solution of the chemical equilibrium problem. In Leal et al. (2013) complicated. From the definition of u in Eq. (2.25), it follows that: we presented a multiphase chemical equilibrium method based hihi on the solution of a system of mass-action equations coupled @n @n @n @n @n @ne ¼ @n @b ¼ @n @n @b ; ð2:35Þ with general equilibrium constraints. Although this method was @u k e k e e also shown to be efficient, we identified that it is not as robust where @n=@ne and @n=@nk are constant matrices obtained by as the recent one in Leal et al. (2014) for capturing transitions N N extracting the columns of the identity matrix I 2 R correspond- in the phase assemblage (i.e., when phases are appearing and ing to the equilibrium and kinetic species respectively. The matrix disappearing). @ne=@be, on the other hand, needs more effort to be calculated, since Hence, the function u is defined as the solution of Eq. (2.29) and it depends on the equations governing the equilibrium state of the the Gibbs energy minimum problem (2.30). equilibrium species.
In order to calculate @ne=@be, let us write the Lagrange function 2.6. Numerical integration L of the minimisation problem (2.30):
T T T Several numerical methods exist in the literature for the inte- Lðne; ye; zeÞ :¼ ne leðneÞþðMene meÞ ye ne ze; ð2:36Þ gration of Eq. (2.24). Ascher and Petzold (1998), Hairer et al. (2008) and Hairer and Wanner (2010) present methods for stiff with Me and me denoting the mass-charge balance matrix and vector and non-stiff system of ordinary differential equations. From our respectively, defined as: discussions in the previous sections, however, a suitable method We be for stiff equations should be adopted because of the large differ- Me ¼ and me ¼ : ð2:37Þ cT 0 ences that can exist in the speeds of the kinetic reactions. e In this work we use the package CVODE (Cohen and Hindmarsh, Eþ1 Ne The vectors ye 2 R and ze 2 R are known as the Lagrange multi- 1996; Hindmarsh et al., 2005) for integration of the chemical kinet- pliers. In addition, let us write the gradient of the Lagrange function ics Eqs. (2.24). This solver is based on the well-known algorithm with respect to the molar abundance of the equilibrium species ne: VODE (Brown et al., 1989), coded in the programming language C, T with improved interface and added capability for dense and rnLðne; ye; zeÞ¼leðneÞþMe ye ze: ð2:38Þ
We refer the reader to Leal et al. (2014) for a more in-depth discus- 2 In Smith and Missen (1982) two approaches were shown for chemical equilib- sion about these equations. rium calculations: a stoichiometric and a non-stoichiometric. The former, also known as equilibrium constant method, solves a system of mass-action equations, while the Assume that ðne; ye; zeÞ is the solution of the minimisation prob- latter minimises the Gibbs free energy of the system, which we use here. lem (2.30). From optimisation theory (Nocedal and Wright, 1999), 52 A.M.M. Leal et al. / Applied Geochemistry 55 (2015) 46–61 it follows that the following first-order optimality conditions are 3. Rates of mineral reactions satisfied at ðne; ye; zeÞ: In order to model the kinetic dissolution and precipitation of T leðneÞþMe ye ze ¼ 0; ð2:39Þ minerals, kinetic rate laws for the mineral reactions are necessary. Mene me ¼ 0; ð2:40Þ By adapting the mineral rate laws presented in Lasaga (1981,
NeZeee ¼ 0; ð2:41Þ 1998), Aagaard and Helgeson (1982), Steefel and Cappellen (1990), Steefel and Lasaga (1994), Perkins et al. (1997), Palandri ne; ze P 0; ð2:42Þ and Kharaka (2004), we write the following general rate law for
Ne where Ne :¼ diagðneÞ; Ze :¼ diagðzeÞ and ee 2 R is the vector of crystal growth and mineral dissolution adopted in this work: ones. Applying the derivative operator @=@be in Eqs. (2.39)–(2.41) X yields: rmðT; P; nÞ :¼AmðnÞ Mm;iðT; P; nÞ; ð3:1Þ i
@ne T @ye @ze l ðn Þ þ M ¼ 0; ð2:43Þ 2þN r e e e where rm : R # R is the rate function of mineral m (in units of @be @be @be moles per unit time); Am is the surface area function of the mineral; @ne 0 Me Ie ¼ 0; ð2:44Þ and Mm;i is the i-th kinetic mechanism function of the mineral (in @be units of moles per unit surface area and unit time). This functional @n @z Z e þ N e ¼ 0; ð2:45Þ form of r allows us to model several kinetic mineral mechanisms e @b e @b m e e such as acid, neutral, base, carbonate, and so forth (Palandri and which can be simplified to: Kharaka, 2004). Estimating the evolution of the mineral surface area in geolog- 1 @ne T @ye ical formations is very difficult and one of the major source of rle þ Ne Ze þ Me ¼ 0; ð2:46Þ @be @be uncertainty in reaction path simulation. The process is so intricate @ne 0 that it is still not completely understood even in batch reactors Zhu Me ¼ Ie; ð2:47Þ @be (2009). In particular, there is the intricate phenomenon of crystal nucleation and growth, which Fritz and Noguera (2009) states that where I0 RðEþ1Þ E is defined as: e 2 its understanding is hindered by at least three factors: the change in aqueous composition due to parallel reactions such as the disso- 0 Ie I ¼ ; ð2:48Þ lution of primary minerals; the variable compositional state of the e 0 formed minerals (i.e., solid-solutions as opposed to pure minerals); E E with Ie 2 R denoting the identity matrix. and the coupling of the reactive processes with transport of the
In order to simplify Eqs. (2.46) and (2.47) even further, let Ke aqueous solutes. Besides all these complexities, however, simpli- denote a matrix whose columns form a basis of the kernel of Me, fied models for surface area have been introduced in the literature such that: to allow us to understand a little bit more about the water–rock- gas interactions. MeKe ¼ 0: ð2:49Þ Helgeson and Murphy (1983) and Helgeson et al. (1984) adopted a constant total surface for the reactant mineral. It seems Thus, multiplying Eq. (2.46) by KT yields: e to be a more common practice nowadays, however, to adopt mod-
@ne els that allow for the variation of the total surface area of the min- Ae ¼ Be; ð2:50Þ eral. We implemented two models for the evolution of mineral @be surface areas. The simplest one consists of assuming a constant where matrices Ae and Be are defined as: specific surface area rm for the mineral, which allows us to com- "# pute the surface area by: KT ðrl þ N 1Z Þ A :¼ e e e e ; ð2:51Þ e : n ; 3:2 Me Am ¼ mrm ð Þ and where nm is the number of moles of mineral m. The other model is based on the one used by PHREEQC (Parkhurst and Appelo, 2013): 0 Be :¼ 0 : ð2:52Þ g nm Ie : ; 3:3 Am ¼Am ð Þ nm Therefore, @ne=@be can be calculated by solving the general system of linear Eqs. (2.50), whose coefficient matrix Ae can be computed where Am is the initial surface area of the mineral; nm is the initial once the equilibrium state of the system has been found (i.e., once number of moles of the mineral; and g ¼ 2=3 for uniformly dissolv- ne; ye, and ze has been calculated). Note that the kernel matrix Ke ing cubes and spheres. should be computed only once in the beginning of the integration In order to model the kinetic dissolution of minerals, it may be for efficiency reasons. necessary to resort to several sources to collect data. In general, Note that for some chemical systems the charge-balance condi- each source will present a slightly different equation for the calcu- tion in the minimisation problem (2.30) may be unnecessary. This lation of the mineral rates. As a result, modelling these kinetic pro- is because the system does not have charged species or, a less obvi- cesses can be hampered by the necessity to handle a multitude of ous case, because the charge-balance condition is already implicitly rate equations. enforced by the mass–balance equations. One way to determine if In Palandri and Kharaka (2004), however, a general and semi- this happens is to check if the charge-balance equation is linearly empirical rate equation is presented for the calculation of mineral dependent of the mass–balance equations. In this case, the explicit rates, which is based on the one adopted by GAMSPATH (Perkins charge-balance condition should be removed, implying Me ¼ We et al., 1997). To achieve this uniformity, they analysed mineral dis- and me ¼ be. Our implementation of the algorithm performs this solution data from several sources, and used them to regress the check automatically. parameters of the general equation. Therefore, to take advantage A.M.M. Leal et al. / Applied Geochemistry 55 (2015) 46–61 53 of their large mineral kinetic database, we define the i-th mecha- The following activity and fugacity coefficient models were nism function Mm;i by: adopted in our calculations: M :¼ sgnð1 XÞj j1 Xpi jqi C ; ð3:4Þ m;i m;i m;i the HKF extended Debye–Hückel activity coefficient model for where X is the saturation index of the mineral; jm;i is the rate con- solvent water and ionic species, Helgeson and Kirkham stant of the mineral reaction (in units of moles per unit surface area (1974a,b, 1976), Helgeson et al. (1981); the Setschenow activity coefficient model for neutral aqueous and unit time); pi and qi are empirical exponents used to fit the rate law; and Cm;i is a function to model catalysts and inhibitors of the species other than CO2(aq); mineral reaction. The saturation index X of the mineral is defined the activity coefficient model of Duan and Sun (2003) for by: CO2(aq); the fugacity coefficient models of Spycher et al. (2003) for Q m X :¼ ; ð3:5Þ CO2(g) and H2O(g). Km where if the mineral reaction is written as: The standard chemical potentials li of the species were obtained using the equations of state of Helgeson and Kirkham (1974a), XN Helgeson et al. (1978), Tanger and Helgeson (1988), Shock and 0 m a ; ð3:6Þ i i Helgeson (1988) and Shock et al. (1992). For this, we used the i¼1 parameters of the database file slop98.dat from the software then Km is its equilibrium constant; and Q m is its reaction quotient, SUPCRT92 (Johnson et al., 1992) and the equation of state of defined by: Wagner and Pruss (2002) to calculate the density of water and YN its temperature and pressure derivatives. The equilibrium con- mi stants K of the mineral reactions (3.6) were calculated using the Q m :¼ ai : ð3:7Þ m i¼1 standard chemical potentials of the participating species.
The reaction rate constant jm;i in Eq. (3.4) depends on temperature. This dependence can be modelled via the Arrhenius equation 4.1. Kinetic modelling of CO injection into carbonate saline aquifers (Lasaga, 1998) as: 2 Em;i 1 1 Consider a subsurface fluid in equilibrium with a carbonate j :¼ j exp ; ð3:8Þ m;i m;i R T 298:15 rock. Assume that supercritical carbon dioxide is injected into this system with an amount large enough to saturate the fluid and pro- where j is the reaction rate constant at 25 °C; E is the activa- m;i m;i duce a supercritical CO -rich phase. In order to model and analyse tion energy; and R is the universal gas constant. 2 the water–gas–rock interactions produced by the gas injection, we Finally, the catalyst/inhibitor function Cm;i is defined as: Y Y use both our chemical kinetics and chemical equilibrium method- nj gg ologies. The entire modelling can be subdivided into three stages: Cm;i :¼ aj Pg ; ð3:9Þ j g Stage 1: calculation of the equilibrium state of the system com- where aj is the activity of the j-th species; Pg is the partial pressure prised of the subsurface fluid and the rock-forming minerals. of the g-th gaseous species; and nj and gg are the exponents of the Stage 2: calculation of the equilibrium state of the system com- catalysts, when positive, and inhibitors, when negative. Thus, acid prised of the injected supercritical carbon dioxide and the resul- mechanisms can be modelled by setting a non-zero value for nHþ , tant solution of the previous stage. while a carbonate mechanism would require g to be non-zero. CO2ðgÞ Stage 3: calculation of the transient state of the entire system A neutral mechanism, on the other hand, can be modelled by setting comprised of the rock-forming minerals, the subsurface fluid, both n and g to zero for all species. j g and the emerged CO2-rich phase. The parameters jm;i; Em;i; nj, and gg for several minerals can be found in Palandri and Kharaka (2004). The first stage ensures that before carbon dioxide is injected, both fluid and rock are in thermodynamic equilibrium. In other 4. Results and discussion words, it models the state of long residency time of the fluid in contact with the reservoir rock, as it would happen in a saline In this section we apply our proposed chemical kinetics algo- aquifer. rithm, coupled with chemical equilibrium, to a problem pertinent The second stage assumes that the injected carbon dioxide to carbon dioxide injection in saline aquifers. The injection of car- achieves equilibrium with the fluid considerably faster than the bon dioxide in saline aquifers perturbs the reservoir and initiates rock-forming minerals. This is a reasonable assumption since the several physical and chemical phenomena due to the interactions speed of mineral dissolution is in general slower than the one for of the injected gas with the resident fluid and the reservoir rock. gas dissolution. Hence, we neglect any amount of mineral that is The complexity of its computational modelling can be high when dissolved between the carbon dioxide injection and its equilibrium both transport processes (e.g., advection, diffusion, dispersion) with the brine. and chemical processes (e.g., mineral dissolution/precipitation, At the third stage, the rock-forming minerals are in disequilib- gas dissolution/exsolution, etc.) are considered (see Pruess et al., rium with the rest of the system. By using the chemical kinetics 2003; Kumar et al., 2004; Ennis-King and Paterson, 2005, 2007; methodology of the previous sections, we can calculate the tran- Xu et al., 2003, 2006; Obi and Blunt, 2006; Audigane et al., 2007). sient state of the entire system until it achieves equilibrium. Nevertheless, in order to apply the methodology presented in An important assumption in the previous modelling is that the previous sections, we neglect all transport processes in this pressure is kept constant at all stages. In other words, we assume study. As mentioned before, we assume that the kinetic process an expandable system that can accommodate the injected gas occurs in a well-mixed batch reactor, although our method is and permit an isobaric process. We remark, however, that support designed for eventual incorporation in a reactive transport for a constant volume system is a work in progress, which will simulator. allow us to model the pressure rise with gas injection. 54 A.M.M. Leal et al. / Applied Geochemistry 55 (2015) 46–61
4.1.1. Subsurface fluid and rock compositions Table 4.2 In our modelling, we considered a brine composition represen- The composition of a rock representative of a Qatari reservoir. tative of a Qatari subsurface fluid. Table 4.1 presents the analysis of Mineral Rock 1 (%) Rock 2 (%) the subsurface fluid of two Qatari reservoirs. There are three sam- Calcite 93.3 97.2 ples for each reservoir, from which we can observe large differ- Dolomite 5.2 0.0 ences in composition, even within the same reservoir. This lack Quartz 1.5 2.8 of homogeneity motivated us to choose samples 2 and 3 of Reser- Note: Composition in units of volume percent. voir B as the composition of the brines used in this study, which we will denote by Brine 1 and 2 respectively. Note that Brine 1 is the lower limit case in terms of concentrations of cations and anions, while Brine 2 is the upper limit case (about five times more con- Table 4.3 The chemical system for the representation of the subsurface fluid and rock of a centrated than Brine 1). We neglected the presence of the compo- Qatari reservoir. nents iron and barium in our calculations. The mineral composition of the rock chosen in this work is also Aqueous phase þ CO(aq) HCO KSO O (aq) 2 CaðHCO3Þ 3 4 2 S5O6 representative of a Qatari reservoir rock. In Table 4.2 we show the þ þ CaðHSiO3Þ CO2ðaqÞ HO2 MgðHCO3Þ OH SiO2(aq) volume percent of the minerals in two samples of the subsurface 2+ 2 þ 2 Ca CO HS MgðHSiO Þ S SO2(aq) rock. The volume composition was obtained by X-ray diffraction 3 3 2 CaCl+ H+ HS O Mg2+ 2 2 2 3 S2O3 SO3 analysis. + 2 CaCl (aq) H (aq) HS O MgCl 2 SO4 2 2 2 4 S2O4 CaCO (aq) H O(l) HSiO MgCO (aq) 2 þ 3 2 3 3 S2O5 SrðHCO3Þ + + 2 2+ CaOH H2O2ðaqÞ HSO3 MgOH S O Sr 4.1.2. Chemical system 2 6 2 + CaSO4(aq) H2S(aq) HSO4 MgSO4(aq) S O SrCl Given the composition of the subsurface fluid and rock, we need 2 8 + 2 Cl H2S2O3ðaqÞ HSO5 Na S SrCO3(aq) to define the multiphase chemical system to be used in our compu- 3 + 2 + ClO H2S2O4ðaqÞ K NaCl(aq) S O SrOH tational modelling. Using the database of Johnson et al. (1992), 3 6 2 ClO HClO2(aq) KOH(aq) NaSO4 S SUPCRT92, we detected all possible aqueous species that could 4 5 ClO HClO(aq) KHSO (aq) NaOH(aq) 2 3 4 S4O6 be present in the subsurface fluid. These are listed in Table 4.3, 2 ClO HCl(aq) KCl(aq) NaHSiO3(aq) S which also shows the assumed gaseous and mineral species. The 2 4 pure mineral phases composed of magnesite and halite are consid- Gaseous phase H O(g) CO (g) ered to capture eventual secondary mineral precipitation. 2 2 The chemical system in Table 4.3 contains several aqueous spe- Mineral phases Calcite Dolomite Quartz Magnesite Halite cies. In Table 4.4 we show the result of the equilibrium calculation at Stage 1, which corresponds to the equilibrium state of the sub- surface fluid (Brine 1) and carbonate rock (Rock 1) at 60 °C and 150 bar. Note that several aqueous species are present only at very 4.1.3. Kinetic rate models and parameters low concentrations. As a result, one can argue that many of these As mentioned before, we use the mineral rate parameters com- species could be potentially removed without compromising the piled by Palandri and Kharaka (2004). The data they present was accuracy of the calculation. This would also dramatically improve compiled from several sources, and it has been used extensively efficiency of the calculations, which is specially important for reac- in the literature for modelling carbon dioxide storage in saline tive flow simulations due to its large number of chemical equilib- aquifers, and the quantification of its trapping by mineral mecha- rium and kinetics calculations. nisms. In addition, it is also adopted by the TOUGHREACT simula- However, this must be done very carefully. For example, a spe- tor (see Gunter et al., 2004; Xu et al., 2006, 2007; André et al., cies that is present at low concentrations initially can later increase 2007; Gaus et al., 2008). considerably during the kinetic process. In this case, the calculation Table 4.5 shows the specific surface areas of the minerals con- of the evolution of a simplified system could be inaccurate or even trolled by kinetics at Stage 3 of our modelling problem. The choice become unstable. Therefore, the full chemical system described in of the specific surface area of calcite was motivated from the dis- Table 4.3 was used in this work. cussion in Schultz et al. (2013) and the value used in Garcia et al.
Table 4.1 The subsurface fluid composition of two Qatari reservoirs.
Composition Reservoir A Reservoir B Sample 1 Sample 2 Sample 3 Sample 1 Sample 2 Brine 1 Sample 3 Brine 2 Cations Na+ 8320 9930 14,177 14,204 7180 33,838 Ca2+ 1483 1200 3240 5109 1500 10,586 Mg2+ 316 425 624 1128 345 1851 K+ 1574 365 484 800 300 460 Sr2+ 40 63 98 159 48 299 Fe2+,Fe3+ < 1 – 0.4 2 –3 Ba2+ – < 0.05 – – 0.23 – Anions Cl 16,632 17,880 28,795 33,969 13,660 76,113 HCO3 315 475 277 258 390 170 2 1340 1230 1194 1147 1520 1015 SO4 2 00 0 0 32 0 CO3 OH 00 0 0 00
Note: Composition in units of mg/l, where l is volume of solvent in litres. The two samples labelled Brine 1 and Brine 2 are used for quantitative analysis. A.M.M. Leal et al. / Applied Geochemistry 55 (2015) 46–61 55
Table 4.4 The chemical state of the system at the end of Stage 1, assuming Brine 1 and Rock 1.
Species Amounta Activityb Activity coefficient Concentrationc
þ1 1 1 1 H2O(l) 5.55099 10 9.87507 10 9.87507 10 9.86559 10 Cl 3.72609 10 1 2.51946 10 1 6.76184 10 1 3.72600 10 1 + Na 2.98870 10 1 2.00642 10 1 6.71354 10 1 2.98862 10 1 2+ Ca 2.25051 10 2 4.26960 10 3 1.89722 10 1 2.25046 10 2 2+ Mg 2.11714 10 2 4.01451 10 3 1.89624 10 1 2.11709 10 2 NaCl(aq) 9.97974 10 3 1.09230 10 2 1.09455 10þ0 9.97950 10 3 2 SO4 9.27278 10 3 1.81520 10 3 1.95760 10 1 9.27255 10 3 + K 7.53420 10 3 5.05526 10 3 6.70991 10 1 7.53401 10 3 3 3 1 3 NaSO4 3.46048 10 2.33524 10 6.74849 10 3.46039 10 3 3 1 3 HCO3 2.89327 10 1.95119 10 6.74407 10 2.89319 10 3 3 þ0 3 MgSO4(aq) 1.58017 10 1.72953 10 1.09455 10 1.58013 10 + MgCl 1.30618 10 3 9.04922 10 4 6.92817 10 1 1.30615 10 3 3 3 þ0 3 CaSO4(aq) 1.25845 10 1.37740 10 1.09455 10 1.25842 10 + CaCl 1.25826 10 3 8.49569 10 4 6.75212 10 1 1.25823 10 3 4 3 þ0 4 CO2ðaqÞ 9.95103 10 1.08431 10 1.08967 10 9.95078 10 2+ Sr 5.03681 10 4 9.57751 10 5 1.90155 10 1 5.03668 10 4 4 4 þ0 4 SiO2(aq) 3.18428 10 3.48526 10 1.09455 10 3.18420 10 þ CaðHCO3Þ 1.75637 10 4 1.17862 10 4 6.71068 10 1 1.75633 10 4 þ 4 4 1 4 MgðHCO3Þ 1.63728 10 1.11277 10 6.79662 10 1.63724 10 4 5 1 4 KSO4 1.31253 10 8.92201 10 6.79774 10 1.31250 10 HS 6.73588 10 5 4.55233 10 5 6.75850 10 1 6.73571 10 5 5 5 þ0 5 H2S(aq) 5.78897 10 6.33615 10 1.09455 10 5.78883 10 5 5 þ0 5 CaCl2(aq) 5.39894 10 5.90926 10 1.09455 10 5.39881 10 + SrCl 3.14527 10 5 2.11459 10 5 6.72324 10 1 3.14519 10 5 þ 5 6 1 5 SrðHCO3Þ 1.30311 10 8.85998 10 6.79926 10 1.30308 10 6 6 þ0 6 CaCO3(aq) 9.01224 10 9.86409 10 1.09455 10 9.01202 10 6 6 þ0 6 Ca(CO3)(aq) 8.93584 10 9.78046 10 1.09455 10 8.93562 10 KCl(aq) 7.91279 10 6 8.66071 10 6 1.09455 10þ0 7.91259 10 6 6 6 þ0 6 NaHSiO3(aq) 3.58169 10 3.92024 10 1.09455 10 3.58161 10 6 6 þ0 6 MgCO3(aq) 2.81041 10 3.07605 10 1.09455 10 2.81034 10 6 6 þ0 6 Mg(CO3)(aq) 2.78540 10 3.04868 10 1.09455 10 2.78533 10 2 6 7 1 6 CO3 2.59866 10 5.05965 10 1.94708 10 2.59859 10 6 7 1 6 HSiO3 1.00650 10 6.83353 10 6.78958 10 1.00647 10 + MgOH 5.23607 10 7 3.62755 10 7 6.92817 10 1 5.23594 10 7 + H 4.92786 10 7 3.34180 10 7 6.78162 10 1 4.92773 10 7 OH 4.48929 10 7 3.08129 10 7 6.86382 10 1 4.48918 10 7 7 7 1 7 HSO4 2.23389 10 1.50760 10 6.74893 10 2.23383 10 8 8 þ0 8 SrCO3(aq) 7.78895 10 8.52517 10 1.09455 10 7.78875 10 8 8 þ0 8 Sr(CO3)(aq) 7.72124 10 8.45106 10 1.09455 10 7.72105 10 þ 8 8 1 8 MgðHSiO3Þ 7.64075 10 5.32874 10 6.97428 10 7.64056 10 + CaOH 6.92677 10 8 4.66881 10 8 6.74041 10 1 6.92659 10 8 þ 8 8 1 8 CaðHSiO3Þ 5.62231 10 3.81725 10 6.78965 10 5.62217 10 NaOH(aq) 3.56951 10 8 3.90690 10 8 1.09455 10þ0 3.56942 10 8 HCl(aq) 1.36065 10 8 1.48926 10 8 1.09455 10þ0 1.36062 10 8 2 9 9 1 9 S2O3 5.39452 10 1.06813 10 1.98007 10 5.39438 10 KOH(aq) 6.98844 10 10 7.64900 10 10 1.09455 10þ0 6.98827 10 10 + SrOH 6.38840 10 10 4.28878 10 10 6.71354 10 1 6.38824 10 10 10 10 þ0 10 H2(aq) 5.27009 10 5.76822 10 1.09455 10 5.26996 10 2 11 12 1 11 S2 3.27169 10 6.42550 10 1.96402 10 3.27161 10 12 12 1 12 HSO3 2.88534 10 1.95767 10 6.78504 10 2.88527 10 2 12 13 1 12 S3 1.42434 10 2.81914 10 1.97932 10 1.42430 10 2 12 13 1 12 SO3 1.27810 10 2.49241 10 1.95015 10 1.27807 10 13 13 þ0 13 KHSO4(aq) 6.21707 10 6.80472 10 1.09455 10 6.21692 10 14 14 1 14 HS2O3 4.38233 10 2.96525 10 6.76654 10 4.38222 10 2 14 15 1 14 S4 3.90984 10 7.82466 10 2.00132 10 3.90975 10 CO(aq) 1.02352 10 14 1.12027 10 14 1.09455 10þ0 1.02350 10 14 2 16 16 1 16 S5 6.79615 10 1.38004 10 2.03067 10 6.79598 10 17 16 þ0 17 SO2(aq) 9.61334 10 1.05220 10 1.09455 10 9.61310 10 20 20 þ0 20 H2S2O3ðaqÞ 6.50277 10 7.11741 10 1.09455 10 6.50260 10 2 24 25 1 24 S4O6 4.02922 10 8.85636 10 2.19809 10 4.02912 10 2 27 28 1 27 S2O4 1.94819 10 3.88538 10 1.99441 10 1.94814 10 2 28 29 1 28 S2O5 2.53174 10 5.06895 10 2.00221 10 2.53167 10 2 31 32 1 31 S2O6 1.83482 10 3.70240 10 2.01791 10 1.83477 10
(continued on next page) 56 A.M.M. Leal et al. / Applied Geochemistry 55 (2015) 46–61
Table 4.4 (continued)
Species Amounta Activityb Activity coefficient Concentrationc