Hydrological Sciences—Journal—des Sciences Hydrologiques, 43(2) April 1998 999

Multicomponent groundwater transport with chemical equilibrium and kinetics: model development and evaluation

THOMAS WERNBERG Danish Hydraulic Institute, Agern Allé 5, DK-2970 Horsholm, Denmark

Abstract A multicomponent reactive transport model, MIN3KIN, has been devel­oped. MIN3KIN handles both multiple chemical equilibrium and multiple component chemical kinetic reactions. A two-dimensional transport code has been coupled with a chemical equilibrium code and a chemical kinetic subroutine. A finite element method is used to solve the transport equations. Different coupling methods between transport and equilibrium calculations are evaluated and chemical equilibrium may be solved sequentially or iteratively. MIN3KIN has been evaluated against an analytical solution and applications for different coupling schemes and temporal discretization. It is shown that the chemical kinetic flux should not exceed 10% of the mass depletion for a given time step. Systems with both chemical equilibrium and irreversible chemical kinetics reach a condition, where the concentration quotient does not correspond with the equilibrium constant. The model is finally applied to a two-dimensional case study to illustrate the presence of an intermediate aqueous state governed by transport and irreversible chemical kinetics.

Transport par l'eau souterraine de composés multiples avec prise en compte des équilibres chimiques et des cinétiques: développement et évaluation du modèle Résumé Un modèle de transport de divers composés réactifs, MIN3KIN, a été développé. MIN3KIN traite à la fois un ensemble d'équilibres chimiques et de cinétiques de réaction. Un code de transport bidimensionnel a été couplé à un code d'équilibre chimique et à un code de cinétique chimique. La méthode des éléments finis a été utilisée pour résoudre les équations de transport. Différentes méthodes de couplage des calculs du transport et des équilibres ont été évaluées, et l'équilibre chimique peut être obtenu séquentiellement ou iterativement. MIN3KIN a été évalué en le comparant à une solution analytique et pour différents schémas de couplage et de discrétisation temporelle. On a pu constater que le flux de la cinétique chimique ne devrait pas excéder 10% de la réduction de masse au cours d'un pas de temps. Les systèmes avec équilibres chimiques et cinétiques chimiques irréversibles atteignent un état où le rapport des concentrations ne correspond pas à la constante d'équilibre. Nous présentons enfin une étude de cas où le modèle est employé en vue d'illustrer l'existence d'un état aqueux intermédiaire gouverné par le transport et des cinétiques chimiques irréversibles.

INTRODUCTION

When a substance is transported through a water-saturated porous medium, it is sub­ject to chemical interactions with other substances either within pore water and/or on mineral grains. The type and duration of these chemical interactions are essential, as they control the development of the chemical composition of the groundwater. All substances dissolved in the aqueous phase are subject to transport. In the presence of

Open for discussion until I October 1998

300 Thomas Wernberg

solid phases, the groundwater composition will be influenced by the solid phase, and some chemical substances may be retarded. Front developments are commonly found in groundwater systems, where major changes in the chemical composition occur over a short distance. Some of these changes occur over a very short distance relative to the model scale and groundwater velocity, and the reactions are fast, resulting in the aqueous phase being under local chemical equilibrium. Other fronts develop over a longer distance, and with a slower chemical reaction rate, in the same order of magnitude as the time scale. In these situations, chemical kinetic reactions become important.

The time for chemical reactions to establish an equilibrium varies over several orders of magnitude. Some chemical substances may be assumed to be under chemical equilibrium, due to fast reaction rates compared to the transport time scale. Other chemical substances are assumed to be conservative, and will not interact with other substances in the aqueous solution due to very slow chemical reactions. Slow chemical kinetics reactions become important for chemical systems, where the chemical interactions are fast enough to change the composition of the water, but cannot be assumed to be under local chemical equilibrium. The chemical reactions assumed to be influenced by slow chemical kinetics depend highly on the nature of the problem and time scale used. Solid-water interfaces and electron transfers are generally referred to as slow chemical reactions, whereas all other chemical reactions within the porous medium are referred to as relatively fast chemical reactions.

The major types of chemical interactions in groundwater are: proton (acid-base) reactions, electron (redox) reactions, solid phase (precipitation-dissolution) reactions and ionic exchange-adsorption reactions. Apart from chemical reactions, other processes such as convection, dispersion and diffusion are considered in the reactive transport model.

Most multicomponent transport models are constructed under the assumption that water is under local equilibrium. These models usually include acid-base reactions, complexation, precipitation-dissolution, redox reactions and adsorption to solids with ionic exchange (Liu & Narasimhan, 1989). Walter et al. (1994) combined a transport model with the chemical equilibrium program, MINTEQ, by Allison et al. (1990), and compared it successfully to another existing multicomponent transport model, PHREEQM, by Appelo & Willemsen (1987), which is a one-dimensional mixing cell model. Engesgaard (1991) combined PHREEQE by Parkhurst et al. (1980) with a one-dimensional finite difference transport code and gives examples of the use of multicomponent equilibrium transport models including redox equilibria.

The solution of a combined transport and chemical reaction model may be found by various numerical methods. If the chemical interactions are fast compared to the velocity of the pore water, the numerical transport model may be split into two parts. One part solves the transport differential equations, and another part solves the chemical equilibrium nonlinear algebraic equations. The advantage of operator splitting is the ability to handle a large number of components. Solving for the transport equations and chemical equilibrium reactions by operator splitting reduces the computational effort in finding a solution. Therefore this method is preferred for solving large systems involving many chemical components and nodal points. Two-

Multicomponent groundwater transport with chemical equilibrium and kinetics 301

step models are presented by Kirkner et al. (1984), White et al. (1984), Cederberg et al. (1985), Narasimhan et al. (1986), Liu & Narasimhan (1989), Engesgaard, (1991) and Walter et al. (1994).

Other transport models with chemical kinetics have been developed, but lack of knowledge about chemical mechanisms and rates of kinetic reactions in nature limits the use of these models. Among the models, McNab & Narasimhan (1993), presented a sequential decay chain for multiple kinetic reactions. Large-scale transport models with reaction kinetics have also been presented. In the Fuhrberg case described by Frind et al. (1990), nitrate reduction was examined in a large-scale aquifer with a limited number of involved reactive components.

In some transport models with chemical kinetics dealing with natural aqueous environments, redox reactions are treated as Monod type chemical kinetics, since biodégradation of nutrients have been of interest (cf. Brun et al., 1993; Celia et al., 1989; Lensing et al, 1994; Widdowson et al., 1988). The Monod type chemical kinetics is based on an empirical approach, and it has proved to be an excellent tool for investigating microbial growth rates in the subsurface. Transport models with both chemical equilibrium and kinetic reactions are presented by Lensing et al. (1994), Zysset et al. (1994) and Brun et al. (1993). The latter presented a transport model which combines a sequential kinetic model to a chemical equilibrium part.

THE MIN3KIN MODEL

In this paper a new model is presented. This model has a capability to combine physical transport with both chemical equilibrium and chemical kinetic reactions. The chemical equilibrium code includes the following types of reactions: acid-base, complexation, precipitation-dissolution, oxidation-reduction and adsorption. In natural groundwater systems, cationic exchange, acid-base reactions and complexa­tion are usually assumed to be controlled by local chemical equilibrium. Slow chemical reactions, which usually include electron transfer and precipitation-dissolution, cannot always be assumed to be in chemical equilibrium. A kinetic approach is therefore suitable when modelling show chemical reactions.

The model is limited to two dimensions with steady state flow in a medium, which may be anisotropic and/or inhomogeneous. The sequence in which the transport and chemical reaction equations are solved is evaluated and compared with existing analytical and numerical models in order to achieve knowledge of mass balance and stability criteria. This model uses a new coupling method for the chemical kinetic differential equations, as they may be solved simultaneously with the transport equations. The chemical kinetics approach is based on thermodynamic reaction rate laws, where the reactive flux of a component is dependent on the activity of some species. In addition to the chemical kinetics approach, the chemical components may also be involved in chemical equilibrium reactions.

The model presented here is therefore an alternative to other reactive multi-component transport models which include chemical equilibrium and kinetics. This model handles both chemical reaction type and introduces a new coupling of

302 Thomas Wernberg

chemical kinetics. These features may contribute to the general improvement of future reactive transport models.

The transport part

The governing transport equations are the basic advection/diffusion differential equation. The groundwater velocity is obtained as input from an external groundwater flow model and is assumed constant in time. The change of concentration in time is (Frind, 1993):

dc: dci { dci | dc, ,,,

t-£{D"e£)-u&;+s! (1)

where c, is the aqueous concentration of the z'th component, xk and JC, are the spatial coordinates in two dimensions, Dkl is the diffusion-dispersion coefficient, U is the steady state velocity and is S,r is a source-sink term. This source-sink term includes external contribution, biological transformation and chemical reactions as described by Rubin (1983). The biological transformation is not included in the present model unless it is treated as chemical kinetic reactions. The transport equation in two dimensions becomes (Frind, 1993):

dc, d2ci d2c, d2Cj dc, dc, ., . . .

t - D~ i?+D* U+D' w-"-t-">t*sr + s (2)

S,equil = cp,(c,, ..., c„) and S.kmet,c = ^.(c,, ..., c„), where i = 1, ..., n is the number of chemical components and cp, and v|/,. are both nonlinear functions.

A two-dimensional finite element method is used to solve the transport equations. The finite element grid can consist of triangles or rectangles. The transport equations are solved using the Galerkin method (cf. Pinder & Gray, 1977; Frind, 1993).

The matrices are partitioned, the known concentrations (Dirichlet nodes, type 1 boundary and last time-step concentrations) are moved to the right-hand side of the matrix equation, and only unknown concentrations remain on the left-hand side of the matrix equation.

The chemical equilibrium part

The chemical equilibrium is solved by use of the chemical equilibrium code, MINEQL (Westall et al, 1976; Schecher & McAvoy, 1991). The composition of the chemical equilibrium solution is found by solving a set of nonlinear algebraic equations using Newton-Raphson iterations. The method is essentially the same as in other chemical equilibrium codes such as MINTEQA2 (Allison et al., 1990) and PHREEQE (Parkhurst et al., 1980). Differences between these chemical equilibrium codes are primarily the thermodynamic database and the activity coefficient correction methods. MINEQL uses the extended Debye-Huckel activity coefficient

Multicomponent groundwater transport with chemical equilibrium and kinetics 303

approximation, which is applicable for single charged ions for ionic strengths under 0.1 M. Equilibrium redox processes are formulated as half-cell reactions using the electron e~ as an aqueous component.

The chemical kinetics part

The equations for chemical kinetic reactions are added to the transport differential equations by using a thermodynamic approach. The reaction term, i|/(c), can be written in the general way:

w(c) reactions reactants ( .Q

(3)

where (dc/d£)Mn is the rate of change of concentration of component c due to kinetic reactions, k is a kinetic rate coefficient, s is a stoichiometric constant for the reaction in which component c participates and a is a constant.

The coupling between transport and chemical equilibrium

The coupling between the transport part and the chemical equilibrium part is done sequentially in two steps: one step considering the transport calculation and one step considering the calculation of the chemical equilibrium. First the transport of each component is calculated, solving the partial differential equations for a single time step. The chemical kinetic reactions are calculated during this step. After both transport and chemical kinetics are solved, the aqueous composition is not in chemical equilibrium:

At dCi

ÔX D.

k V

dCf -U-

dci

dx, - + vfo) (4)

where c* is an intermediate value for an aqueous solution without chemical equilibrium. The transport equations are solved using a time centred scheme. The chemical equilibrium is established at the end of each time step for each node by the chemical equilibrium code, MINEQL, where the new concentrations are found as:

c,'+A' - c, = cp,equ" (5)

where c'+A' is the new concentration and (p,.eqml are the changes in the aqueous solution from MINEQL to obtain chemical equilibrium.

The coupling with chemical kinetics

The differential equations of chemical kinetics are either solved separately or simultaneously with the transport equations. Using the latter method, the flux due to

304 Thomas Wernberg

chemical kinetic reactions is assumed to be constant during a single time step and solved within the transport matrix equations.

In the sequential method, chemical kinetic reactions are calculated explicitly. The kinetic flux is calculated either before or after the transport equations have been solved. By setting w = 1, the chemical kinetics are calculated before the transport equations are solved; with w — 0, the chemical kinetics are calculated after the transport equations have been solved. For w = 0.5, half of the chemical kinetics is calculated before the transport equations are solved, and the other half is calculated after the transport part.

By the sequential method, the change in concentration of a component is calculated as:

c';AI = c\ + Aty(c) (6)

where vj/(c) has been calculated in equation (3). After the chemical kinetics and transport have been solved, the chemical equilibrium is established at the end of the time step, and initialization of a new time step can take place. By including both chemical kinetics and equilibrium, it becomes possible to model slow chemical reactions and simultaneously control the general quality of water with fast secondary reactions. For example, an electron transferring reaction causes a production of hydrogen ions (slow reaction). These ions cause not only a decrease in pH, but also a change in the whole chemical balance (fast reactions), including an eventual dissolution of precipitated carbonates.

By using the simultaneous method, the chemical kinetics and transport differential equations are solved simultaneously. The kinetic weighting factor acts in a similar way to the transport weighting factor, w, i.e. for w = 1, the chemical kinetics are solved explicitly for d, and for w = 0 the chemical kinetics are solved for d+At. The chemical equilibrium equations are solved sequentially at the end of each time step with the simultaneous method.

The iterative method solves the kinetic/transport differential equations simul­taneously, and includes the chemical equilibrium calculations within the iteration loop.

Trial runs by MIN3KIN have shown that the sequential method is the fastest and most inaccurate method, whereas the iterative method requires more computational effort, but is more accurate. The simultaneous method is intermediate in accuracy and computational speed. The assessment of accuracy is based on model results compared with analytical transport solutions. For the different simulations, transport parameters such as temporal and spatial discretization were identical.

In all cases, the chemical equilibrium equations are solved for each nodal point, by calling the chemical equilibrium code MINEQL (Westall et al., 1976). With a sequential coupling between transport and chemical equilibrium, no iterations are performed. The simultaneous technique requires iterations in the transport step, which might introduce an error up to second order error in time in the transport solution alone with a centred weighting in time. By including the chemical equi­librium calculation within the iterative loop, the error in time is close to second order for the whole system when using the iterative method.

Multicomponent groundwater transport with chemical equilibrium and kinetics 305

Alternatively, an iterative coupling between transport, kinetics and chemical equilibrium could have been chosen, but for a transport and chemical equilibrium scheme, Walter et al. (1992) show that there is good agreement between the two solution techniques, although the CPU time used is remarkably reduced with the sequential method. In some cases, 95% of the CPU time is used to solve the chemical equilibrium. The reason for the high computational cost for solving the chemical equilibrium is the highly nonlinear algebraic equations, which are more complex than the linear transport differential equations and, furthermore, the number of nodes often exceeds the number of defined chemical components thereby requiring higher computational effort to solve the chemical equilibrium.

When the differential equations for the chemical kinetics and transport are solved simultaneously, it is possible to keep the error close to second order in time by centred time weighting and iterations. The iteration loop must be established, since all concentrations at the new time step must be known in order for the chemical kinetic flux to be calculated.

The differential equation for chemical kinetic flux is added to the transport differential equation. The equations are solved by creating a partitioned time weighted matrix equation obtained from the Galerkin finite element scheme (Frind, 1993):

1 E ( K ] + K > K ] + K ] ) + ^ K ] Uc}k+

^(1-S)([M«] + [M"] + [M^ + [MA']) + ^ [ M - ] { C K (7) At1

-({^L+(n+1 -Mj+o-^L+{FCLH*K) where [M] represent the matrices for dispersion, advection, decay, boundary entry and mass storage (in that order of appearance in equation (7)), {Fs} is the type 3 boundary flux vector, {Fc} is the type 1 boundary linking vector and {FK} is the kinetic flux vector; k is the old time step; k+l is the new time step; and s is a time weighting factor. For triangular elements {FK} is found as:

1

{FKr=V(c)fù i

(8)

i|/(c) is defined in equation (3), and A is the element area.

Numerical stability

No stability criteria for multicomponent transport with chemical equilibrium have yet been made (cf. Walter et al., 1994), but the restrictions on the variability of the dimensionless Peclet (Pe) and Courant (Co) numbers, for non-reactive flow with retardation, must be satisfied (Daus et al., 1985).

306 Thomas Wernberg

The discretization in time should principally cause only a finite change in the chemical composition, though be large enough to make a progression in time. Stability criteria for transport with chemical kinetic reactions are sparse in the litera­ture. Valocchi & Malmstead (1992) suggest that Mr < 0.1 for transport with first order decay, to obtain a solution with a low mass balance error. This is an aqueous concentration reduction of 10% for a single time step with explicit time weighting. For higher order kinetic reactions or multiple component kinetic reactions, this criterion cannot be used. Instead a criterion of the kinetic flux should be used, so the change in concentration and flux during a single time step is low. Satisfying the same criterion for a first order decay, a maximum of 10% of the mass is transformed, due to kinetic reactions, in a single time step. Therefore the discretization in time for an explicit scheme can be:

A? < 0.1 A (9) M

where c is the concentration and \\> is the chemical kinetic flux. This criterion should be applied to all components with a total negative flux to prevent a fast mass depletion.

EVALUATION AND DISCUSSION

MIN3KIN was evaluated against existing analytical solutions of different transport-related problems to show its flexibility and accuracy.

A sensitivity analysis of MÏN3KIN using different coupling methods was evaluated with transport of a reactive substance under first order decay. A 100 cm long column was divided into 20 elements. The velocity was 0.50 cm h"1 and the longitudinal dispersivity was 2.0 cm. The simulation time was 100 h and the number of time steps ranged from 10 to 250. The Peclet number was constant, Pe = 1.25, for all simulations, whereas the Courant number depended on the discretization in time, and had a value between 0.04 and 1.0. The decay was of first order, with a kinetic rate constant of 0.01 h4.

The simulation data were compared with the analytical solution for one dimensional advective-dispersive transport under first order decay (McNab & Narasimhan (1993): equation (16) and Fig. 1):

c if (x(v-U)) (Rx-Ut) (x(v+U)) (Rx+Ut)~\

where x is the spatial location, v is the velocity, t is the time, R is the retardation,

D is the dispersion, k is the first order decay constant and U = 4v2 +4DRk. The concentration discrepancy induced by the sequential method for different

time steps was compared (Fig. 2(a)). A centred weighting was chosen, wk = 0.5, i.e. half of the chemical kinetic decay was calculated before the transport step and the other half was calculated after the transport step. The error becomes lower as the

Multicompomnt groundwater transport with chemical equilibrium and kinetics 307

+ Sequential, w=0, dt=10.0 h x Sequential, w=1, dt=10.Q h o Iterative, w=1, dt=10 h - Analytical

50 Distance (cm)

100

Fig. 1 The numerical solution of transport under first order decay by MIN3KIN compared to the analytical solution for different chemical kinetic weighting (w = 0 and vc = 1) and coupling (sequential-iterative method). The temporal discretization is àt = 10 h.

0 . 0 5 ^ ! à. Sequential, w=0.5, dt=10 h I x Sequential, w=0.5, dt=2 h + Sequential, w=0.5, dt=0.4 h j

O Ô

P

1 <S -o.o

=2 0.00

. Sequential, w=0, dt=10 h xSequential, w=1, dt=10 h o Iterative. w=1, dt=10 h

50 100 0 50 Distance (cm) Distance (cm)

Fig. 2 The error introduced (a) at different timestepping (sequential method with centred weighting; discretization in time = 10, 2 and 0.4 h, respectively); and (b) at different weighting and coupling methods (as Fig. 1; temporal discretization, At = 10 h).

100

discretization in time is denser. At the large discretization in time, àt = 10 h, the kinetic flux is 10% of the total mass and the error introduced is significant. With denser discretization, the error is reduced.

The error introduced to the solution is more dependent on the coupling method (Fig. 2(b)). With a kinetic weighting factor of wk = 1, the chemical kinetic flux is calculated before the transport step and the error introduced is positive, i.e. the numeric solution tends to overshoot the analytical result. For wk = 0, where the chemical kinetic flux is calculated after the transport step, the numerical result is underestimated. In the iterative method, here calculated explicitly, an overall smaller error than in the sequential method is introduced, and a slight overestimation of the solution is observed.

The time discretization suggested by Valocchi & Malmstead (1992) is in this case

308 Thomas Wernberg

10 h and gives a 5% error. Shorter time steps are recommended, as the error introduced to the result becomes much smaller. The error made by the simultaneous method is generally much smaller than the error made by the sequential method.

Lensing et al. (1994) suggested that the chemical kinetic equations should be solved between the transport part and the chemical equilibrium (wk = 0.0, sequential method). This is not recommended, if relatively large time steps are used, as the kinetic flux is calculated after transport but before chemical equilibrium, where the aqueous solution has not been equilibrated. Buffering by solid phases may play a very important role in this situation.

The simultaneous method, where the chemical flux equations are calculated simultaneously with the physical transport equations, provides the best results in this case. The simultaneous method introduces only a minor error in time (close to second order) and shows the highest accuracy of the three methods.

Transport and chemical kinetics

MIN3KIN was compared with a transport/kinetic model by McNab & Narasimhan (1993). The following example presents a kinetically controlled decay chain involving three species, NH4

+ -» N02~ -» N03"", and the solution obtained by MIN3KIN is evaluated against Van Genuchten (1985).

The initial model parameters were a one-dimensional column, 200 cm in length, divided into 400 elements (802 nodes, since MIN3KIN was working in two dimensions), with a length of 0.5 cm and the same arbitrary element width. The simulation time was 200 h, and the discretization in time was 0.25 h, keeping both an acceptably low Peclet and Courant number, respectively. The active boundary was of Cauchy type (type 3). The numerical solution was compared with the analytical solution and shows a good agreement with the latter (Fig. 3).

0 100 200 Distance (cm)

Fig. 3 A kinetic chain reaction under transport: MIN3KIN (•) evaluated against Van Genuchten (1985) (-).

Multicomponent groundwater transport with chemical equilibrium and kinetics 309

Irreversible kinetics and fast chemical equilibrium

The following example shows how MIN3KIN is able to model both chemical equilibrium reactions and slow chemical kinetic reactions simultaneously. In a container with water, Fe2+ and a ligand (L), Fe2+ was oxidized to Fe3+ as oxygen was added to the system. As the oxidation continued, the equilibrium between Fe2+

and the ligand could not be kept due to a slow reaction rate, and a new equilibrium was established. This equilibrium is not equal to the thermodynamic equilibrium. The chemical reactions in the system are:

K Fe2++ L e> FeL2+ (11)

k Fe2+ ^ F e 3 + (12)

The reaction (11) is reversible, while reaction (12) is assumed to be irreversible. The oxidation rate of Fe2+ is assumed to be catalysed on the surface of ferrihydroxide and is found empirically to be proportional to [Fe2+], p02 and [OH]2 (Stumm & Morgan, 1981):

d[Fe3+| , n l , ,2 - i _ J = £2[Fe2+HOH-] -P02 (13)

where [Fe2+] and [OFT] are the concentrations of Fe2+ and OFT respectively and p02

is the partial pressure for oxygen. The coupled linear differential equations now read:

A\V 2+l

- ^ J - ^ = t1[FeL2+]~Â:+1[Fe2+]-[L]-/t2[Fe2+]-[OH-]2-^02

d ' l (14)

^ p = -£_,[FeL2+] + £+1[Fe2+]-[L]

Since the total mass in the system is preserved, [Fe]T is constant, and the oxidized iron is found as:

d[Fe3+] d[Fe2+] d[FeL2+]

~~~dT~ d/ d/ ( 1 5 )

Model simulations were performed with constant pH at 7, 8 and 8.5, respectively. All other initial parameters were kept constant. LT = 5.0 x 10"6 M, Fer = 10~5 M, kx = 1.0 s"1, k_x — 1.0 x 106 M"1 s"1. The stability constant for the iron-ligand reaction was assumed to be constant in all the simulations, K a = kAlkx = 106 M"1

(Pankow & Morgan, 1981). In this case the influence of different reaction rates, due to different pH values, is impressive (Figs 4(a)-4(c)). The activity of water was assumed to be constant (1) in all examples and no activity correction was made.

The concentration quotient, Q, is defined as:

310 Thomas Wernberg

<2-[FeL2+]

Fe2+]-[L] (16)

Q is an estimate of how close the system is to a thermodynamic chemical equilibrium. At thermodynamic equilibrium, Q = K^.

Note that the aqueous system is under a process towards chemical equilibrium and reaches a steady state condition, which is close to the thermodynamic equilibrium. At low pH, the oxidation of Fe2+ is slow compared to the exchange of iron between the ligand binding, and an equilibrium between Fe2+ and L can be established, causing Q = Zequil (Fig. 4(a)). At higher pH values, the oxidation rate of Fe2+ is fast compared to the iron-ligand reaction and Q •*• Kequil (Fig. 4(c)). The equilibrium established is not necessarily the same as the thermodynamic equi­librium. The MIN3KIN simulations were compared to Pankow & Morgan (1981), and the results are identical.

As long as an irreversible reaction is occurring, the stabilized steady state equilibrium does not have to be the same as the thermodynamic equilibrium (Pankow & Morgan, 1981). Some species in the aqueous solution reach a steady state equilibrium which does not correspond with the thermodynamic equilibrium.

The oxidation of Fe2+ by 02 changes the pH in the aqueous solution. A change in

(a)

1er6

10*

irvS

Fe(ll)

L

Fe(lin^-—"

pH = 7.0

20 40 Time (s)

pH = 8.5

20 _ , , 40 Time(s)

2.5

2.0

1.5 § 1

1.0

0.5 60

pH = 8.0

~ 10*

8 10-7,

I ^rr^fse—^r_ —

8 10-'

KHH

.z\ \ v pH

— - 8.5

6.5 10 20

Time (s) 30

Fig. 4 The development of the chemical kinetic system (a) at pH = 7; (b) at pH = 8; (c) at pH=8.5; and (d) where pH is allowed to change.

Multicomponent groundwater transport with chemical equilibrium and kinetics 311

pH affects the oxidation rate for Fe2+. Since the solubility of Fe3+ is very low in the model pH range, precipitation of ferric iron may be expected. Fe3+ precipitates by the following reaction:

2Fe2+ +40FT + | 0 2 +H 20 ^ >2Fe(OH)3(s) (17)

As ferrihydroxide precipitates, pH decreases due to removal of hydroxide ions from the aqueous solution. The rate of change for [OH ] due to oxidation of Fe2+ is:

d[OFT]

àt =^2/c2[Fe2+]-[OH"]2 -P02 (18)

In the simulations performed, the initial pH was 8.5 and other conditions were the same as in previous simulations. The precipitation of ferric hydroxide was assumed to be a fast reaction, and was treated as an equilibrium reaction. As ferrihydroxide precipitated, pH decreased fast, and the oxidation rate of Fe2+ became very low (Fig. 4(d)). The oxidation of Fe2+ occurred within the first few seconds, after a few time steps; pH fell to a value below 7 and no changes in chemical system were observed thereafter. This example shows how sensitive a multicomponent model may be in the choice of parameters and assumptions, as a change in one parameter can lead to different results.

Transport with chemical equilibrium and kinetics

The simulation of multicomponent transport under both chemical equilibrium and chemical kinetics are provided in this application, which illustrates the presence of an intermediate aqueous state during denitrification in an aquifer.

The groundwater flow was calculated for a 1000 m long vertical cross section starting near the water divide. The base of the aquifer was confined by an imper­meable clay formation; elsewhere the aquifer was unconfined. The hydraulic conductivity was 10"3 m s', the constant top boundary recharge was 180 mm year1

and the porosity 0.3 (Table 1). The left and bottom boundaries were impermeable, whereas groundwater could exist at the right boundary. The steady state velocity field for the aquifer was calculated by the finite element code FLONET (Nilson et al., 1993) and the streamlines are shown in Fig. 5.

A Cauchy type boundary condition was applied to the top, and all other boundaries were treated as Neumann zero flux boundaries. The upper part of the aquifer is contaminated with nitrate from extensive cultivation. Pyrite is abundant in

Q -9 0 200

I Pyrit Impermeable

400 600 800 Distance (m)

Fig. 5 Calculated streamlines for the aquifer application. The impermeable base of the aquifer is shown with black and the pyrite rich zone with a grey shade.

312 Thomas Wernberg

Table 1 Parameter values used in the aquifer model applications.

Discretization Value NX (elements in a row) NY (elements in a column) AX(m) AF(m) Af (d)

Aquifer parameters: Hydraulic conductivity (ms1) AL(m) AT(m) D* (m2 day4)

Stability criteria: Peclet number (Pe) Courant number (Co)

40 39 25 0.25 max. 30

1.0 x 10~3

20 0.01 8.6 x lO6

1.25 0.23

the aquifer between the base of the flow system and 7 m below the water table. Nitrate receives electrons from pyrite, depletes, and sulphate is produced. The reaction is assumed to occur in two steps, reactions (19) and (20), where nitrite appears as an intermediate species:

4 N O j + S 2 0 ^ + H 2 0 ^ 4 N O - + 2 S O ^ + 2 H + (19)

4FeS2(S) + 14N02 + 6H+ + 3H20 -» 4S20*_ + 4Fe(OH)3(S) + 7N2Oo te) (20)

The rates of the reactions are assumed to be sufficiently fast in the application and the reaction rate coefficients are selected ad hoc to prevent front smearing caused by chemical kinetics. The half-life of nitrite is 170 days in the presence of pyrite. The use of the chemical kinetic approach on the electron transferring reactions neglects a definition of the electron activity and it is possible to treat the reactions as irreversible.

The simulation was performed until a steady state condition was reached. The temporal discretization was variable, and maximized with agreement to both the Courant and kinetic mass flux criteria. The chemical parameters in the model are

Table 2 Chemical components and initial conditions in the aquifer simulation.

Component Lower aquifer (mol l"1) Upper aquifer and top boundary (mol 1"1)

Ca2+

co32-

F e 3 +

NO3-so4

2

N02-S203

2-pH

3.68 x lO3

3.35 x 10~6

5.69 x 1018

0 1.80 x lO3

1.0 x 10s

1.0 x 108

7.43

3.55 x 10~3

3.25 x 10" 5.69 x 10» 2.50 x lO3

0.55 x lO'3

1.0 x 108

1.0 x If/8

7.41

HC03- = CC

HSO4- = SO,

CaC03(aq) =

CaS04(aq) =

Ferrihydrite Calcite

)32- + H+

»2" + H+

Ca2+ + C03:

Ca2+ + S042"

Multicomponent groundwater transport with chemical equilibrium and kinetics 313

Table 3 Chemical equilibrium reactions and equilibrium constants for aqueous and solid phases for the aquifer model application.

Chemical reaction logK _ _ _ _ _ _ _ _ _ _

FeOH2+ = Fe3+ + 2H20 - 2H+ -2.77

FeOHj(aq) = Fe3+ + 3H20 - 3H+ -13.90

FeOH4~ = Fe3+ + 4H20 - 4H+ -21.00

CaHC03+ = C03

2- + H+ + Ca2+ 10.76

H2C03(aq) = C032~ + 2H+ 16.60

10.50

1.24

2.32

1.70

-3.04

7.95

Pyrite NA

Equilibrium constants from Schecher & McAvoy (1991). The chemical equilibrium constants have been correlated to a system of 10°C and an ionic strength of 0.016. NA: not applied.

Table 4 Rate coefficients and order of chemical reactions controlled by kinetics.

Kinetic rate Rate coefficient Rate order reaction

~1 lb3 mol'1 year1 [N03:][S203

2:] 2 4.0 year1 [N021[FeS2(s)]

Table 5 Mass balance matrix (sn in equation (3)) of chemical reactions controlled by kinetics.

Component Reaction 1 Reaction 2 __ +2 0 0 -2 -7 +2

shown in Tables 2-5. Three solid phases were considered in the model: calcite, ferrihydrite and pyrite. Pyrite is assumed to be present in an unlimited amount in the lower part of the aquifer, which gives a stationary pyrite front. Calcite and ferri­hydrite are assumed to be present throughout the model domain. Precipitation and dissolution of calcite buffer pH. Nitrate contaminated water enters the aquifer along the top boundary.

H+

Fe3+

NO/ so4

2-FeS2(s) N02-s2o3

2-

2 0 -4 +2 0 4 -1

314 Thomas Wernberg

Within a few years of simulation time, most aqueous components in the aquifer reached a steady state condition and outputs for major components after 30 years of simulation time are presented in Fig. 6. Fluctuations in the nitrite concentration at the nitrate front were observed until stabilizing at around 25 years. Nitrate depletes in the presence of thiosulphate, which is produced during the first reaction step (reaction (19)). During this reaction, nitrite is produced. Since nitrite is unstable in the presence of pyrite, nitrite depletes in the lower part of the aquifer but is stable above the pyrite front. Nitrite is present in a zone in the aquifer which is located slightly above the pyrite front. Since nitrite is produced in a limited zone in the aquifer, transport of nitrite perpendicular to the groundwater flow direction explains the presence of nitrite above the pyrite front. Reactions (19) and (20) have a stoichiometric imbalance with nitrate and nitrite relative to thiosulphate and a surplus of thiosulphate is produced during the overall nitrate depletion reaction. Thiosulphate is stable in the part of the aquifer without nitrate and is present in the lower part of the aquifer (Fig. 6). The transverse diffusion of thiosulphate above the pyrite front initiates nitrate depletion above the pyrite front. Reactions (19) and (20) thereby dominate in separate parts of the aquifer around the pyrite front. Reaction (20) produces hydrogen ions, which is revealed by a decrease in pH, dissolution of calcite and an increase in the alkalinity just above the pyrite front (Fig. 6). Since the aquifer is saturated with calcite, the decrease in pH is minimal due to calcite precipitation/dissolution. Reaction (20) consumes hydrogen ions and consequently

Nitrite {Û-Q.Û8 mM] Thiosulphate (0:0,1 j f _ l

Jierrihxdrite

5

0 200 400 600 800 0 200 400 600 800 Distance (m) Distance (m)

fig. 6 Concentration contours for nitrate, sulphate, nitrite, thiosulphate, pH, alkalinity, calcite and ferrihydrite respectively for the aquifer application. The concentrations of the solid phases are given as relative amounts. A dark shade represents high concentrations. (Nitrite appears as an intermediate species and is only present in a limited zone in the aquifer; the presence of an intermediate state with nitrite develops a double front, which is revealed on the concentration contours for pH, bicarbonate and amount of precipitated calcite).

Multicomponent groundwater transport with chemical equilibrium and kinetics 315

pH increases followed by precipitation of calcite and a decrease in the alkalinity. Reaction (20) is limited to the lower part of the aquifer and the effect on the carbonate equilibrium system is less distinct since the hydrogen production is countered by reaction (19) in the pyrite part of the aquifer as nitrate is transported across the pyrite front by transverse dispersion. The precipitation of ferrihydrite occurs in the lower part of the aquifer (Fig. 6). The presence of aqueous Fe2+ has not been considered in this model. The changes in alkalinity due to transverse dispersion of Fe2+ above the pyrite front followed by oxidation and precipitation as ferrihydrite may occur, but the concentration of Fe2+ below the pyrite front is limited to the order of 10"5 mol l"1 before the aquifer is supersaturated with siderite and the effect on the net changes to the carbonate equilibrium due to oxidation of Fe2+ is assumed to be minimal.

CONCLUSIONS

A model (MIN3KJN) with three numerical methods to solve transport with reaction has been presented. The chemical reactions are either assumed to occur as a chemical equilibrium reaction or to be controlled by slow kinetic reactions. The different numerical methods either solve the reactive transport as a three step sequential method, an iterative physical transport/chemical kinetics with a sequential equilibrium step or a fully iterative transport/kinetics/equilibrium procedure. The choice of method depends on computational effort vs accuracy for the problems formulated.

Whether the sequential kinetic/transport coupling or the simultaneous solving technique should be used, depends highly on the nature of the chemical system. With sharp fronts, the time weighted sequential solution technique might be useful, but not necessary optimal. As seen with MINTRAN, and other chemical equilibrium transport codes, sharp concentration gradients can be solved sequentially without loosing accuracy, and at the same time saving computational effort and calculation time.

The computational time is very different for the transport/kinetic part and the equilibrium part. Keeping the relative change in the concentration of a component, caused by chemical kinetic reactions, under 10% during a single time step, provides an acceptably low error. By controlling the discretization in time, the change in the concentration of an aqueous component can be kept at an acceptable value, and mass depletion can be avoided.

MIN3KIN is capable of simulating transport with both multicomponent kinetic reactions and chemical equilibrium reactions simultaneously. The model is able to simulate transport of substances under acid-base, oxidation-reduction, adsorption and precipitation-dissolution reactions. Chemical kinetic reactions may be included in the model to simulate slow chemical reactions.

Three examples of different problems were solved successfully; one that involved transport during a chemical kinetic controlled chain reaction, one with slow kinetic reactions and chemical equilibrium where transport was omitted, and finally a

316 Thomas Wernberg

complex example where substances under both chemical equilibrium and chemical kinetics were under transport. When transport is omitted, MIN3KIN can be used in investigations of complex chemical reaction systems including chemical kinetics.

The two-dimensional multicomponent application with chemical equilibrium and kinetics shows that it is possible to create an intermediate aqueous state by use of irreversible reactions, which also neglects a definition of the electron activity. The transverse dispersivity becomes an important parameter as the presence of the intermediate species depends on the transport perpendicular to the groundwater flow direction to zones in the aquifer where the intermediate species are stable.

Acknowledgements The Faculty of Science at the University of Aarhus, Denmark, sponsored this work. The author is grateful to Keld Romer Rasmussen, Department of Earth Sciences, University of Aarhus and Birgit Tejg Jensen, Department of Chemistry, University of Aarhus, for their guidance. Thanks are also due to Emil Frind and Uli Mayer, Centre for Groundwater Research, University of Waterloo, Ontario, Canada for guidance during the model development and to Peter Engesgaard, ISVA, Technical University of Denmark for the final comments.

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Received 9 September 1996; accepted 12 September 1997