2 Numerical Simulation of Reactive Flow using
SHEMAT
– Upgrade of the Chemical Reaction Modules Jörn Bartels, Michael Kühn, and Christoph Clauser
2.2 Governing Equations
2.2.5 Chemical Reactions
Chemical reactions between reservoir rock and fluid as well as reactive transport
can be simulated by SHEMAT. The current version offers the user two choices for
calculating the activity-concentration relationships underlying the mineral reaction
processes. They are based on either:
• Pitzer’s equations (best suited for concentrated brines) or
• Debye-Hückel’s theory (useful for more dilute solutions).
Pitzer’s approach
SHEMAT’s first chemical speciation module CHEMEQ is a modification of the
geochemical modeling code, PHRQPITZ (Plummer et al. 1988). It permits calculations of geochemical reactions in brines and other highly concentrated electrolyte solutions using the Pitzer virial-coefficient approach for activity-coefficient
corrections. Reaction-modeling capabilities include calculation of aqueous speciation and mineral-saturation as well as mineral solubility.
The results computed for each aqueous solution include the osmotic coefficient,
water activity, mineral saturation indices, mean activity coefficients, total activity
coefficients, and scale-dependent values of pH, activities of individual ions, and
activity coefficients of individual ions. A data base of Pitzer interaction parameters is provided for the system Na-K-Ca-Mg-Sr-Ba-Si-H-Cl-SO4-OH-(HCO3-CO3CO2)-H2O which is valid in the temperature range 0 – 150 °C; for the calcitecarbonic acid system set in parentheses the data base is valid in the temperature
range 0 – 90 °C.
PHRQPITZ was developed by Plummer et al. (1988) from the aqueous ionpairing model PHREEQE (Parkhurst et al. 1980) replacing the Debye-Hückel by
the Pitzer virial-coefficient approach (Pitzer 1973, Pitzer and Mayorga 1973,
Pitzer and Mayorga 1974, Pitzer and Kim 1974, Pitzer 1975).
The Pitzer treatment of the aqueous model is based largely on the equations as
presented by Harvie and Weare (1980) and Harvie et al. (1984). An expanded data
base of Pitzer interaction parameters is taken from He and Morse (1993) which is
6
Jörn Bartels, Michael Kühn, and Christoph Clauser
identical to the partially validated data base of Harvie and others (1984) for the
system Na-K-Ca-Mg-Sr-Ba-Si-H-Cl-SO4-OH-HCO3-CO3-CO2-H2O at 25°C, and
extended to higher temperatures.
The code was modified to include the calculation of temperature dependent
Pitzer coefficients. CHEMEQ uses the Newton-Raphson approach to solve a set of
algebraic equations by generating successively improved estimates of the molalities and activity coefficients of the aqueous species.
The temperature range for equilibria in the PHRQPITZ.DAT file is variable
and is generally 0 – 150 °C. However, the NaCl system is valid to 250 °C, while
the carbonate system is reliable only to 90° C.
The temperature dependence of the solubility of several of the minerals in
PHRQPITZ.DAT is not known and large errors could result if calculations are
made at temperatures other than 25 °C for these minerals. The minerals which are
tested for the temperature range 0 – 150 °C are the chlorides halite (NaCl) and
sylvite (KCl), the sulfates mirabilite (Na2SO4)*10H2O, arcanite (K2SO4), glaserite
(NaK3(SO4)2), anhydrite (CaSO4), gypsum (CaSO4*2H2O), glauberite
(Na2Ca(SO4)2), labile salt (Na4Ca(SO4)3*2H2O, syngenite (K2Ca(SO4)2*H2O, Celestite (SrSO4), and barite (BaSO4) as well as quartz and silica (SiO2). The phases
tested for the temperature 0 – 90 °C are calcite (CaCO3) and carbon dioxide
(CO2).
Debye Hückel’s Theory
To extend the applicability of SHEMAT with a second chemical speciation module, for example to enable re-engineering ore deposits or simulate permeable reactive barriers, the program has now been interfaced additionally to the chemical
code PHREEQC (Parkhurst and Appelo 1999). PHREEQC is based on the DebyeHückel ion association approach. Compared to PHRQPITZ and due to the considerably larger thermodynamic data set for the chemistry of dilute aqueous solutions, it allows to study chemical reactions between much more elements.
PHREEQC is a versatile computer program for simulating chemical reactions
in aqueous solutions, both natural and anthropogenically altered waters. So far, the
interaction between aqueous solutions and minerals in the recent interface to
SHEMAT is based exclusively on thermodynamic equilibrium. However,
PHREEQC itself provides many more chemical reaction features, such as solid solutions, exchanger and sorption surfaces, and isotopic fractionation. Many of these
are planned to be activated later. In contrast to CHEMEQ, the distribution of redox elements among their valence states is also taken into account now. The reaction calculations aim for a system equilibrium rather than merely aqueous equilibrium. System equilibrium is attained by distributing all of the moles of each
element in the system among the aqueous phase and the pure mineral phases.
PHREEQC is a general geochemical program and can be applied to many hydrogeochemical environments. However, there are several limitations. Most importantly and in contrast to PHRQPITZ which is based on Pitzer’s ion-interaction
approach, PHREEQC uses the Debye-Hückel ion-association expressions to address non-ideal aqueous solutions. The extended Debye-Hückel aqueous model
2 Numerical Simulation of Reactive Flow using SHEMAT
– Upgrade of the Chemical Reaction Modules 7
applied here, includes additional ionic strength terms and is therefore adequate not
only at low ionic strength but also in a range up to seawater salinities. Nevertheless, we recommend to employ CHEMEQ and Pitzer’s approach as described
above in the seawater salinity range and above. The other limitation of the aqueous model is a lack of internal consistency in the data base. The data base distributed with the code, chemeqdh.dat, is consistent with the aqueous model of
WATEQ4F (Ball and Nordstrom, 1991) and can be edited. However, careful selection of aqueous species and thermodynamic data is required by the user.
For more detailed information, such as basic equations of the constitutive laws,
readers are referred to the manual of PHREEQC (Parkhurst and Appelo 1999).
Reaction Kinetics
Since chemical reaction kinetics has not yet been tested in combination with Debye-Hückel’s approach this feature is only available in combination with Pitzer’s
approach at present. For a single mineral species, i, which is composed of two
ions, the change of concentration by precipitation or dissolution in a time interval
Δt is given by:
(
)
⎧ E ⎫ cion (t) − ceq (t)
Δcmin, i = Δt A react r0 exp ⎨− act ⎬ ⋅
,
kB T ⎭
ceq (t)
⎩
thermal activation
(2.32)
supersaturation
where Areact=Atotal × xmin, i is the reactive surface; Atotal is the total specific internal
surface in m2 m-3; xmin, i=0…1 is the percentage of the total surface covered by the
mineral i; Δcmin, i is the change in concentration [mmol L-1] of the mineral i; cion(t)
and ceq are the concentration and the equilibrium concentration [mmol L-1] of the
ion species of the ion-pair forming mineral i; r0 is the reaction rate [mol s-1 m-2];
Eact is the activation energy for the molecular surface processes associated with
precipitation; kB ist Boltzmann’s constant. The user has the choice to apply 1st order or nth order precipitation and dissolution reactions. Following Eq. (2.32), in the
first case only one of the ions constituting the mineral phase (the one occurring
earliest in the data set) is taken into account to determine the distance from equilibrium. This rate law is valid for reactions with equimolar abundance of aqueous
species in solution. In the second case, the calculations account for all ions of the
mineral phase by replacing the single supersaturation factor in Eq. (2.32) by the
product of the individual supersaturation factors of all ions of the mineral. This
rate law is recommended for large differences in the species concentrations of the
solution. Further, the defined internal surface area can be switched on and off in
the equations for the rate law “dependent or independent on surface”. In some
cases, precipitation reactions can be assumed to be independent of the surface
area. Table 2.1 lists the storage allocated in SHEMAT to the kinetic parameters
Areact, xmin, Eact, and r0.
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Jörn Bartels, Michael Kühn, and Christoph Clauser
Table 2.1. Input structure of kinetic parameters
parameter
Atotal(x,y,z)
xmin(i)
Eact
r0
dimension
imax×jmax×kmax
20
20
20
type
spatial distribution
for every mineral species i, independent of position
for every mineral species i, independent of position
for every mineral species i, independent of position
Generally, the chemical reaction calculation requires most of the computation
time. In general it is carried out for every single block. In cases where
(1) pressure, temperature and concentration have not changed since the last time
step, or (2) the time step is very small because of stability criteria defined outside
of the geochemical calculation, the required computation time can be significantly
cut down by reducing either the number of blocks per time step where calculations
are performed or the frequency for calculating new equilibrium concentrations.
Different options are available:
1. Set the frequency of transport steps per chemical reaction modeling step equal
to values greater than 1 (default: 1)
2. for the calculation of chemical equilibrium: Set a threshold for temperature and
concentration changes ΔT and Δc at a node (in %, after the heat and species
transport step) required for activating a new calculation of all equilibrium concentrations in this time step (defaults: ΔT=10-2 K; Δc=10-3 mmol L-1).
3. additionally for the calculations of reaction kinetics:
• if temperature and concentration changes do not exceed the thresholds ΔT
and Δc explained above, the concentration threshold Δc of the equilibrium
calculation is also applied to the change in mineral content at this node
when deciding which concentrations need to be calculated again for this
time step;
• If temperature and concentration changes do not exceed the thresholds ΔT
and Δc explained above, and if there was no mineral left at this node after
the previous time step, concentrations are not calculated again at this
node.
2.3 Numerical Techniques
2.3.6 Process Coupling
Non-linear Chemical Reactions
The chemical reaction models implemented in SHEMAT calculate the equilibrium concentration of each dissolved species and the amount of precipitation and
dissolution depending on the current concentration of each component. The first
chemical reaction module of SHEMAT is a modification of the geochemical simulation code, PHRQPITZ (Plummer et al. 1988) and is described in detail in Chapter 4.1. It calculates geochemical reactions in brines and other highly concentrated
2 Numerical Simulation of Reactive Flow using SHEMAT
– Upgrade of the Chemical Reaction Modules 9
electrolyte solutions using the Pitzer virial coefficient approach for the correction
of activity coefficients. The Pitzer coefficients depend, in turn, again on the concentrations of the components. The original PHRQPITZ code was extended to calculate temperature dependent Pitzer coefficients. The second chemical module of
SHEMAT is based on the program code PHREEQC (Parkhurst and Appelo 1999),
which is based on the Debye-Hückel ion-association approach. Both modules use
the Newton-Raphson approach to solve a coupled set of algebraic equations by
generating successively improved estimates for the concentrations of the involved
species. In combination with Pitzer’s approach (CHEMEQ / PHRQPITZ), reaction
kinetics can be taken into account for a number of minerals, if dissolution or precipitation of the minerals occurs too slowly and therefore chemical equilibrium is
not established instantaneously. The kinetic reaction rate depends linearly on the
concentrations of the reacting ions and on the concentration dependent supersaturation.
2.4 Input / Output
2.4.3 Input File
Scalar Chemical Reaction Parameters
FRACEXP
Three different fractal coefficients need to be specified. The first
coefficient accounts for the coupling of porosity and permeability in the porosity range 0 – 1 %, the second in the porosity
range 1 – 10 % and the third in the porosity range 10 – 100 %.
The fractal coefficient for the 0 – 1 % porosity range is commonly equal to 1.
IZV
one-dimensional integer vector containing all information required for the call of the chemical reaction subroutine
CHEMEQ:
IZV(1):
IZV(2):
IZV(3):
IZV(4):
IZV(5):
IZV(6):
number of dissolved chemical species (including pH)
number of minerals in reaction
charge balancing, 0=imbalance, 1=via pH,
2=via ionic elements
reaction modeling, 0=no reaction, 5=mineral
equilibria
not active
number of cation for charge balancing (if
IZV(3)=2)
10
Jörn Bartels, Michael Kühn, and Christoph Clauser
IZV(7):
number of anion for charge balancing (if
IZV(3)=2)
IZV(8 to 7+ANZCHEM): ion numbers from PITZER.DAT
IZV(8+ANZCHEM to 8+ANZCHEM+ANZMIN): mineral
numbers from data set PHRQPITZ.DAT
IZV(90):
Calculated redox state from previous time step
to be used as first estimate in actual time step
(0: yes, 1: no)
IZV(91):
output file dhcall.out listing all chemical
reaction calculations performed for one time
step, PHREEQC only (0: no, 1: yes)
IZV(92):
inactive
IZV(93):
defines for which minerals the kinetic approach
is to be used for dissolution / precipitation (binary coding)
IZV(94):
kinetic rate law (0: 1st order incl. reactive surface; 1: 1st order, independent of reactive surface; 2: nth order including reactive surface; 3:
nth order, independent of reactive surface)
IZV(95):
Ion activity approach used for chemical reaction calculations (0: Pitzer, 1: Debye-Hückel)
IZV(96-97): temperature threshold in % with IZV(96) ×
10-IZV(97)
IZV(98-99): concentration threshold in % with IZV(98) ×
10-IZV(99)
IZV(100): frequency of chemical reaction modeling
(1: always, 2: every 2nd time step, 3: …..)
AO
Dimensionless scalar vector (dimension: 20) containing the fraction of internal reactive surface occupied by each mineral (to be
multiplied with the total measured surface at each node,
AREAKT)
EACT
Activation energy (dimension: 20) for the thermally activated
molecular reaction at the pore surface where the dissolved molecule is incorporated into the crystal structure (i.e. surface diffusion)
RATE
Kinetic reaction rate factor (measured or estimated, dimension:
20), containing all unknown influences on the reaction rate (except the proportion of the reactive surface, the thermal activation
and distance from equilibrium state which are all accounted for
in the kinetic equation used here)
References
11
References
Ball JW, Nordstrom DK (1991) WATEQ4F--User's manual with revised thermodynamic
data base and test cases for calculating speciation of major, trace and redox elements in
natural waters: U.S. Geological Survey Open-File Report 90-129, 185 p, Reston Va.
Harvie CE, Moller N, Weare JH (1984): The prediction of mineral solubilities in natural
waters: The Na-K-Mg-Ca-H-Cl-SO4-OH-HCO3-CO3-CO2-H2O system to high ionic
strengths at 25 °C. Geochim Cosmochim Acta 48: 723-751
Harvie CE, Weare JH (1980) The prediction of mineral solubilities in natural waters: The
Na-K-Mg-Ca-Cl-SO4-H2O system from zero to high concentrations at 25°C. Geochim
Cosmochim Acta 44: 981-997
He S, Morse JW (1993) The carbonic acid system and calcite solubility in aqueous Na-KCa-Mg-Cl-SO4 solutions from 0-90 °C. Geochim Cosmochim Acta 57: 3533-3554
Parkhurst DL, Appelo CAJ (1999) User's guide to PHREEQC (version 2) -A computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations: U.S. Geological Survey Water-Resources Investigations Report 994259, Reston Va.
Parkhurst DL, Thorstenson DC, Plummer LN (1980) PHREEQE - a computer program for
geochemical calculations. U.S. Geological Survey Water-Resources Investigations Report 80-96, Reston Va. (Revised and reprinted 1990)
Pitzer KS (1973) Thermodynamics of electrolytes. 1. Theoretical basis and general equations. J. Physical Chemistry 77: 268-277
Pitzer KS (1975) Thermodynamics of electrolytes. V. Effects of higher-order electrostatic
terms. Journal of Solution Chemistry 4(3): 249-265
Pitzer KS, Kim JJ (1974) Thermodynamics of electrolytes. IV. Activity and osmotic coefficient for mixed electrolytes. Journal of American Chemical Society 96(18): 5701-5707
Pitzer KS, Mayorga G (1973) Thermodynamics of electrolytes. II. Activity and osmotic coefficient for strong electrolytes with one or both ions univalent. Journal of Physical
Chemistry 77(19): 2300-2308
Pitzer KS, Mayorga G (1974) Thermodynamics of electrolytes. III. Activity and osmotic
coefficients for 2-2 electrolytes. Journal of Solution Chemistry 3(7): 539-546
Plummer LN, Parkhurst DL, Fleming GW, Dunkle SA (1988) A computer program incorporating Pitzer’s equations for calculation of geochemical reactions in brines: U.S.
Geological Survey Water Resources Investigations Report 88-4153, Reston Va.