Reactive Transport in Porous Media 9781501509797, 9780939950423

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REVIEWS in MINERALOGY

Vol ume 34

REACTIVE T R A N S P O R T IN POROUS M E D I A Edited by

Peter C. Lichtner Center for Nuclear Waste Regulatory Analysis

Carl I. Steefel University of South Florida

Eric H. Oelkers Université Paul Sabatier Cover: Contour plot of computed mineral volume fraction and fluid streamlines (white with arrows indicating direction) after an elapsed time of 7,500 years. Fluid undersaturated with respect to the mineral is injected at the left, causing the mineral to dissolve and thus increasing the porosity and permeability in this region. Zones of red indicate regions where the mineral has been dissolved completely while the blue zones indicate regions where the initial mineral volume fraction is still present. Due to the reactive infiltration instability, the front propagates as fingers rather than as a planar front. (Lichtner, this volume, Figure 19, page 76).

Series Editor: Paul H. Ribbe Virginia

Department of Geological Sciences Polytechnic Institute 2 and carbonate mineral concentrations 415 Setup of the 1-D reactive transport simulations 416 Simulation results: Movement of the Fe(II)rich waters and of the MnC>2 dissolution front 418 Simulation results: Evolution of the low-pH waters 419 The effect of the initial carbonate to initial Mn02 ratio on the evolution of the low-pH waters 421 Influence of the aluminum mineral allowed to precipitate on the evolution of the low-pH waters 422

xii

Effects of the irreversible dissolution of Ca- and Mg-silicates on the evolution of low-pH Fe(II)-rich waters The effect of not allowing rhodochrosite precipitation The CO2 open system simulations The effect of longitudinal dispersion The influence of ion exchange and surface-complexation sorption processes Other minor effects on the evolution of the low-pH waters Comparison of the reactive transport simulation results with observations at the Pinal Creek site How to obtain U.S. Geological Survey computer codes and the PHREEQM code

Conclusions Acknowledgments References

xiii

424 425 427 427 428 430 431 436 433 436 436

Chapter 1 CONTINUUM FORMULATION OF MULTICOMPONENT-MULTIPHASE REACTIVE TRANSPORT Peter C. Lichtner Center for Nuclear Waste Regulatory Analyses San Antonio, Texas 78238 U.S.A. (lichtner@swri. edu) INTRODUCTION Quantitative models of physicochemical processes in the Earth's crust can be applied to a broad range of phenomena involving fluid-rock interaction and transport of aqueous, non-aqueous and gaseous species. These phenomena encompass environmental and industrial problems as well as fundamental geologic processes. Mathematically the reactive mass transport equations represent a moving boundary problem in which it is necessary to determine the regions of space the various phases in the system—solid, liquid and gas—occupy as functions of time. Computer models can provide, if not a direct quantitative description, at least a far better qualitative understanding of the geochemical and physical processes under investigation than might otherwise be possible. Further, reactive transport models can provide a useful tool to society for evaluating various environmental hazards. For example, models can be used to predict the rate of migration of a contaminant plume resulting from acid mine drainage and heavy metal mobilization. Quantitative models can also help design remedial strategies to clean up toxic waste sites, such as removal of volatile hydrocarbons by steam injection, and others. While the emphasis of this book is on modeling, it is essential that modeling be closely integrated with field and laboratory studies if the predicted results are to have anything to do with reality. Considerable progress has been made in the past few years in modeling reactive flow and transport in natural systems. Ever faster computers, including massively parallel machines, have enabled complex multicomponent chemical systems to be included in multi-dimensional simulations. New algorithms for high Peclet number flows have made it possible to reduce the effects of numerical dispersion. And growing thermodynamic and kinetic databases have provided fundamental data needed to model realistic geochemical systems. Nevertheless, there remain many challenges for an accurate or even semi-quantitative description of natural systems. Reaction mechanisms need to be understood first before they can be successfully incorporated into a reactive flow and transport model. A description of fluid-rock interaction requires knowledge of mineral surface area, a quantity that is difficult to measure and may change with time. Microbial systems are still poorly understood but may be an essential aspect of many geochemical systems. Initial and boundary conditions are often unknown and unknowable. And finally, a problem of fundamental importance is upscaling properties obtained from observations at the pore and molecular scales to the macro or continuum scale at which most reactive transport models are applicable. Natural porous media are undoubtedly heterogeneous at all scales. Yet they are often approximated as if they were homogeneous. Incorporating heterogeneous porous media into reactive transport models presents a uniquely challenging problem, both conceptually as well as computationally. 0275-0279/96/0034-0001 $05.00

2

Lichtner: Continuum Formulation of Reactive Transport

This chapter begins with a discussion of the continuum hypothesis on which conservation equations of mass and energy describing flow and transport in porous media are founded. A general conservation law is developed from which mass and energy conservation equations can be deduced. Reactive mass transport equations are derived for a multicomponent-multiphase system involving chemical reaction of minerals, aqueous and gaseous species for a system in local partial equilibrium. It is shown that by transforming the chemical reactions to canonical form, the number of unknowns in the mass transport equations can be greatly reduced by eliminating of rates corresponding to local equilibrium reactions. Concepts of local equilibrium, scaling relations, asymptotics, ghost zones and the quasi-stationary state approximation are introduced. An analytical solution for a single component system derived from the quasistationary state approximation is presented. The concept of equilibration length is illustrated. The question of charge balance in the aqueous phase is investigated for species-dependent aqueous diffusion coefficients and incorporating batch sorption models with transport. Finally several examples are presented using the computer code MULTIFLO, a multicomponentmultiphase reactive transport model. These include acid mine drainage, redox front migration, reactive transport in heterogeneous porous media, an example of reaction front instability, and a simple hydrothermal system. CONTINUUM HYPOTHESIS A rock mass consisting of aggregates of mineral grains and pore spaces or voids is referred to as a porous medium. An actual porous medium is a highly heterogeneous structure containing physical discontinuities marked by the boundaries of pore walls which separate the solid framework from the void space. Although it is possible in principle at least, to describe this system at the microscale of a pore, such a description rapidly becomes a hopeless task as the size of the system increases and many pore volumes become involved. It is therefore necessary to approximate the system by a more manageable one. One quantitative description of the transport of fluids and their interaction with rocks is based on a mathematical idealization of the real physical system referred to as a continuum. In this theory the actual discrete physical system, consisting of aggregates of mineral grains, interstitial pore spaces, and fractures, is replaced by a continuous system in which physical variables describing the system vary continuously in space. Allowance is made for the possibility of a discrete set of surfaces across which discontinuous changes in physical properties may occur. In this fictitious representation of the real physical system solids and fluids coexist simultaneously at each point in space. The continuum hypothesis asserts that a real physical system may be approximated by one in which the properties of the system vary sufficiently smoothly enabling the use of differential calculus to describe processes taking place in the system. The utility of the continuum hypothesis is that it enables changes in a system to be formulated in terms of partial differential equations. Real rocks, however, are far from continuous, and it must be understood at the outset that invoking the continuum hypothesis requires a major leap of faith that cannot be justified a posteriori. Certainly, the concept of a continuum applied to porous media is on a much less firm foundation compared to its use in other areas of physics. By way of contrast, in fluid mechanics a real fluid consisting of discrete molecules is replaced by a continuum in which fluid properties vary continuously throughout space. This is justified on the basis that any finite volume of fluid which characterizes changes in its macroscopic properties such as temperature and pressure, contains many molecules. For example, at standard conditions of 25°C and 1 bar, approximately 1022 water molecules occupy one cubic centimeter of liquid water. In the continuum representation of a porous medium, the physical variables describing the system, which are discontinuous at the microscale or pore scale as a consequence of the granular nature of rocks, are replaced by functions which are continuous at the macroscale. The

Lichtner: Continuum Formulation of Reactive Transport

3

rock matrix

>y v

R

E v = y

p

+ v

s

X Figure 1. Schematic diagram of continuum space.

value of each physical variable assigned to a point in continuum space is obtained by locally averaging the actual physical property over some representative elemental volume, referred to simply as a REV (Bear, 1972). The essential feature of a REV is that it characterizes the properties of the system locally. The dimensions of a REV are usually large compared to the grain size, but small compared to the characteristic length scale over which the quantities of interest change. The quantities describing a system at the macroscale include the temperature, pressure, fluid density, fluid saturation, average fluid flow velocities, concentration of minerals, gaseous and aqueous species, porosity, and permeability among others. Functions defined at each in point in space and time are referred to mathematically as fields. Thus, for example, the temperature T is represented by the scalar field T(r, t) where the vector r refers to a point in continuum space with spatial coordinates (x, y, z) (see Fig. 1), and t represents the time. Bold face type is used throughout to denote vectors in 3-space. The value of the temperature at the point r is obtained by averaging the temperature over a REV centered at r. Similarly the porosity, liquid saturation, mineral, aqueous and gaseous concentrations are represented by the scalar fields m(r, t),Cj(r, t), and C-(r, t), respectively, where superscripts I and g refer to liquid and gas phases, and subscripts i and m refer to the ith solute and mth mineral, respectively. The velocity field v(r, f) = (vx(r, t), vy(r, t), vz(r, t)) is an example of a vector field. Covering the entire system with a connected set of REVs provides a global description of the system. For some systems a typical size of a REV may be a hand sample collected by a geologist in the field. But a REV may also be much smaller than that, perhaps on the order of tens of mineral grains. For rocks exhibiting patterns such as a reaction halo surrounding a fracture, care must be taken to choose the size of the REV smaller than the structure being observed. A REV may not be taken too small, however, because then it no longer provides an average property of the rock-fluid system. There is no guarantee that a single set of REVs belonging to a single continuum is sufficient to characterize a rock. This is especially true of fractured rocks for which primary and secondary porosities corresponding to the matrix and fracture network can be defined. For such rocks at least two sets of REVs are needed, one for the fracture network and the other for the rock matrix. More generally a hierarchical porous medium may be required characterized by many sets of REVs, referred to as multiple interacting continua. It is important to realize that continuum theory can provide only a macroscale description of the properties of rocks and interstitial fluids and not a microscale, or pore scale, description. This is not to say that microscale properties are not important. In fact it is the microscale properties averaged over a REV which define the macroscale properties. However, usually such averages

4

Lichtner: Continuum Formulation of Reactive Transport

are too difficult to perform mathematically and we must let nature carry out the averaging process for us. We may therefore attempt to measure directly the macroscale properties of a rock, such as its porosity and permeability, thereby providing a phenomenological or empirical description. Even if it is not feasible to predict the values of the various parameters entering the continuum theory from fundamental principles, a phenomenological description can provide a first attempt to model such systems. Given an actual sample of a representative elemental volume for some porous medium, values of the various physical variables which describe the system may, in principle at least, be determined. The volume of the REV, VREV or simply V, is equal to the sum of the solid volume Vs and void, or pore volume, Vp at position r in continuum space and time t: VREV(r,t)

= Vs(r,t)

+ Vp(r,t).

(1)

As indicated, the REV volume may vary with position and even time. The total porosity (f> is defined as the fraction of volume of the rock made up of pore space or voids

where the change Stjr is presumed to occur over the time interval St. Consequently, the time rate of change of the number of moles of the ¡th species can be expressed as the sum of the reaction rates weighted by the stoichiometric coefficients i>,> 8m ^ -zr = ¿ > r / r , St 7=1

(52)

Lichtner: Continuum Formulation of Reactive

Transport

13

as follows by dividing both sides of Equation (50) by St. In an open, continuous, system the rate I r is referenced to a REV and represents the volumetric average rate in units of moles per unit volume per unit time. The reaction progress variable becomes the reaction progress density defined at each point in space. The change in the number of moles of some species contained in a REV can occur not only as a result of chemical reactions taking place in the system, but also because of the transport of matter into and out of the REV. In addition there may be other source/sink terms present. Denoting the contribution from external changes resulting from transport processes across the boundaries of the V by Seni, and that caused by internal changes resulting from chemical reactions by total change in the ¡th species during the time interval St may be expressed as the sum Sni = Seni + Sim,

(53)

with NR

Sim = St^virlr, r=1

(54)

and sem

(55)

= -StV-Ji.

The minus sign occurs because, by convention, the flux /, is taken as positive flowing out of the representative volume. The total change in the number of moles of the ¡th species can thus be expressed as NR

Sni = -StV

• Ji + St

Vhh•

(56)

r=l

Dividing by St and taking the limit St species

0, yields the mass conservation equations for aqueous

3 -(¿Cj+V./,-

Nr

= £vir/r.

(57)

r=l

01

This result is obtained by noting that Sni

3 ,

.

Comparing this equation with Equation (21), the source/sink term R, in the mass conservation equation is thus equal to NR

(59)

Ri = J^virIr. r= 1 For minerals the transport equations simplify to

£in which the flux term is absent.

-

r=1

14

Lichtner: Continuum Formulation of Reactive Transport

Reaction rates To solve the transport equations requires knowledge of the reaction rates for kinetically controlled reactions. Different forms of the rate law are required depending on the type of reaction, that is whether it is homogeneous or heterogeneous, and whether it is an elementary reaction or whether it represents an overall reaction or some specific reaction mechanism. Homogeneous reactions. The rate of an elementary kinetic homogeneous reaction of the general form N

0 ^

^Pv.yAi, ¡=1

(61)

is given by the difference between forward and backward reaction rates h = kf H (YiCi)~Vir - kbr inCiT" Vir0

,

(62)

where kr' b denote the forward and backward rate constants. At equilibrium the net rate vanishes, the forward and backward rates being equal, resulting in a relation between the equilibrium constant and the kinetic rate constants kf Kr = ^ where the ion activity product Qr is defined by

-

Qr,

N

Qr = I K ' ¡=1

(63)

m > 0, or if 0) occurs if Am < 0, and dissolution (7m < 0) if Am > 0. The reaction rate has units of moles per unit time per unit volume of bulk porous medium. Thus it represents an average rate taken over a REV. The rate law given by Equation (67a) should really be referred to as a pseudo-kinetic rate law. This is because it refers to the overall mineral precipitation/dissolution reaction, and generally does not describe the actual kinetic mechanism by which the mineral reacts. Nevertheless, it provides a useful form to describe departures from equilibrium and is certainly no worse than the assumption of local equilibrium. Far from equilibrium the expression for the reaction rate reduces to the forms Im — for Am

^-m ^m r K

(70)

0 corresponding to dissolution, and T l m\!RT m —_ h. ~ JA ^

IK

(71)

16

Lichtner: Continuum Formulation of Reactive Transport

for Am corresponding to precipitation. Close to equilibrium the rate becomes proportional to the chemical affinity according to the expression I = km

r K

(72)

RT'

valid for \Am/RT\ 1. There is an inherent asymmetry in the rate law regarding precipitation and dissolution that should be noted. According to Equation (71), the precipitation rate grows indefinitely as Am -> —oo, whereas according to Equation (70) the dissolution rate tends to a finite constant times the factor in square brackets as Am — o o . Of course physically the reaction rate cannot grow indefinitely, but must be limited by the rate of transport of reactants and products to the site where the reaction takes place. Note that under such farfrom-equilibrium conditions, although the rate is transport limited, the reaction is not in local chemical equilibrium. The temperature dependence of the kinetic rate constant may be calculated from the Arrhenius equation (Lasaga, 1981) km(T)

=

1 /l

C ^ g e x p

^ -

R \T

T0J

(73)

A£„

where k°m denotes the rate constant at To, A(T) represents a pre-exponential factor, and AEm denotes the activation energy. Surface area. One recurring difficulty in describing heterogeneous reactions involving minerals is characterizing the reacting mineral surface area. Consider, to be specific, an aggregate of cubical grains of dimension d. Different grain geometries result in different geometric factors. If the grains are assumed to be arranged in a evenly spaced, three-dimensional array with N grains contained in Vrev, a total volume b3 = V r e v / N is associated with each grain including pore space, and the grain number density t] is equal to 1 =

N

1

Vrev

b3'

(74)

The pore volume is equal to b3-d3

Vrev

= 1

b3

© -

= 1 -

nd\

(75)

The specific surface area s of the grains can be expressed in terms of the porosity either as 6 d2 ^

(76a)

V

for fixed grain size, or, alternatively, in terms of a fixed number density as 2

= 6^(1

-4>f'\

(76b)

Accordingly, a different dependence of the surface area on porosity is obtained depending on whether the grain number density r) is considered constant, or the grain size itself is fixed. For

'

1=1 it is possible to transform the reactions to the canonical form Nc

J2»jiAj ^ j=1

A-.

G = tfc + 1

AO,

(90)

20

Lichtner: Continuum Formulation

of Reactive

Transport

in which a single species, referred to as a secondary species, appears on the right-hand side with unit stoichiometric coefficient. The Nc species appearing on the left-hand side are primary species. The term primary species is used throughout for a system in local partial equilibrium, rather than the term component which is reserved for a closed system in thermodynamic equilibrium. There are an equal number of secondary species as there are reactions. The secondary species Ai serves to identify the ¡th reaction. The number of primary species is equal to the difference between the total number of species N involved in the reactions and the number of reactions NR: Nc = N-Nr.

(91)

The stoichiometric coefficients vji are related to the original stoichiometric coefficients vlr by the equation V^

, -k

0" = 1. . . . , N C )

where ( v _ 1 ) n represents the inverse of the submatrix vir where i runs over the secondary species. The submatrix v,> is a square, nonsingular matrix. This can always be accomplished by virtue of the fact that the reactions are presumed to form a linearly independent set. The canonical form defined by Equation (90) may be derived by noting that the chemical reactions given in the general form of Equation (89) can be rewritten as follows NC

N

Y^VjrAj+ y=l

£ virAi, l=Wc + l

(r = l, . . . , N r ) ,

(93)

where the first sum on the right-hand side is over primary species and the second sum over secondary species. The partitioning the reacting species into primary and secondary species is done selectively so that the submatrix vlr involved in the sum over secondary species is nonsingular. Hence it is possible to "solve" Equation (93) for the secondary species to give NR

Ai ^

-¿TvjrCv-^riAj,

(94)

r= 1

from which Equation (90) follows with v Jt defined by Equation (92). Because of its special form, reactions written in the form of Equation (90) are referred to as being in canonical form (Lichtner, 1985; Smith and Missen, 1982; Van Zeggeren and Storey, 1970). This terminology appears to have been first introduced by Aris and Mah (1963). The canonical form is characterized by associating a single secondary species with each reaction, with the remaining species belonging to a common subset of species which appear on the left-hand side of each reaction. Constructing the canonical form is equivalent to the Gauss-Jordan procedure used in linear algebra in which a matrix is transformed to an upper nonzero matrix and a lower unit matrix (see Steefel and MacQuarrie, this volume). It should be especially noted that the designation of a species as primary or secondary refers to the canonical form of the reaction in which the species occurs. Reactions written in canonical form are expressed as association reactions. Several examples may help clarify the the meaning of the canonical representation. Consider the reaction describing the formation of the aqueous species CaSO^. With primary species chosen as Ca 2 + and S O 2 - , and secondary species CaSO^, the reaction becomes C a 2 + + S° 0O44 - ^ =5= CaSO^. L1JU4

(95a)

Lichtner: Continuum Formulation of Reactive Transport

21

Equally possible is to chose as primary species Ca 2 + and the complex CaSO^. In this case SO? - becomes the secondary species. The reaction now has the form CaSC>4 — Ca 2 + ^

SO2-.

(95b)

Both forms lead to equivalent results. As a more complicated example, consider the precipitation or dissolution of goethite and iron speciation in the aqueous phase. Taking as primary species {Fe 2+ , H + , (h.^), H2O}, and secondary species {Fe 3+ , Fe(OH)^, FeOOH}, the reactions have the form Fe 2 + + H+ - J H 2 O

Jo 2 (g) -

Fe 3 + ,

(96a)

Fe 2 + - 3H+ +

402(g)

Fe(OH)",

(96b)

Fe 2 + - 2H+ + h2 +

1 402(g)

FeOOH.

(96c)

+

I °

An alternative set of primary species are {Fe 2+ , Fe 3 + , H + , H2O}, and secondary species {02(g), Fe(0H)4 , FeOOH}. This choice leads to the set of reactions 4Fe 2 + - 4Fe 3 + + 4H+ - 2 H 2 0 ^ Fe

3+

Fe

3+

0 2( g),

(97a)

- 4H+ + 4 H 2 0 ^

Fe(OH)",

(97b)

- 3H+ + 2 H 2 0 ^

FeOOH.

(97c)

Provided equilibrium can be assumed to hold between the redox couples F e 2 + - F e 3 + - 0 2 ( i ) , the two sets of reactions are equivalent. In the canonical representation, the charge conservation equation for a chemical reaction becomes a statement expressing the charge of the ¡'th secondary species in terms of the charges of the primary species. Thus Nc

j=1 with vji defined by Equation (92). Likewise mass conservation reads Nc

Mi = J2"J'MJ-

(99)

These relations should be compared with the general forms, Equations (43) and (42). The mass action equation corresponding to the i'th canonical reaction is given by

* -

^

with equilibrium constant Ki. The advantage of the canonical form is that this equation may be formally solved for the concentration of the secondary species C, in terms of the concentrations of primary species. For aqueous and gaseous species it follows that Nc

Q = (*)">£ n ( y , c , p . ;=1

(ioi)

22

Lichtner: Continuum Formulation of Reactive Transport

To obtain this expression, the activity has been replaced by the product of the activity coefficient and the molar concentration. Actually, because the activity coefficients also depend on the concentrations of the solute species, this relation only provides an implicit expression for the concentration. For stoichiometric mineral species, replacing the index i by m and noting that am = 1, the mass action equations become an inequality: Nc KmY[{YjCj)v,m j=i

< 1.

(102)

This relation provides a constraint among the primary species concentrations. Equality holds if the aqueous solution is saturated with respect to the mineral and thus the mineral is in local equilibrium with the solution. The inequality holds if the solution is undersaturated with respect to the mineral. The presence of an inequality for heterogeneous reactions reflects the moving boundary problem nature of the reactive mass transport equations. A simple relation exists between the equilibrium constants of the original reactions Kr and the reactions written in canonical form denoted by Ki. Taking logarithms of both sides of the mass action equation, Equation (100), yields N

'

In Ki = In a, — ^ v Jt In a j , ;=i NR

NC

= In a, + J2 J2 vjr(v~ls>ri In ah r=l j=1

(103)

where to obtain the second equality, vji is replaced by its definition given in Equation (92). Taking the logarithm of Equation (44) and separating terms for primary and secondary species results in the expression NC

N

InKr = ^ V j r l n a j + ^ v,>lna,-. i=Nc+1 j= 1

(104)

Multiplying this equation through by (v _ 1 ) r l and summing over all reactions yields NR

IX

1

)"'

1

"^

= Ina; + ^

r

NC

^

Vy^v" 1 ),,-lna 7 .

(105)

r=1 j=1

Comparing this relation with Equation (103), it follows that In Ki =

NR

r=1

In

(106)

K

(107)

or, alternatively Nr

Ki =

-i

r=l

yielding the desired relation. Thermodynamic databases. The relation between equilibrium constants obtained in Equation (106) or (107), is actually of direct practical use in transforming a particular thermodynamic

Lichtner: Continuum Formulation of Reactive Transport

23

database from one representation to another. Thermodynamic databases which store chemical reactions and their corresponding equilibrium constants, as well as other properties such as Debye-Huckel parameters, charge, molar volume, and gram-formula weight, store reactions in canonical form. The freedom of choice of primary species may be used to advantage by enabling the user of such a database in which a particular choice has been hardwired, to use any other set of species that may be more relevant to the problem at hand. This technique is used in several computer codes developed by the author (MPATH, MULTIFLO, GEM). These codes employ a modified form of the EQ3/6 database (Wolery et al., 1990). Actually the EQ3/6 database uses two sets of primary species resulting in two distinct canonical representations. One representation is referred to as the basis set and the other as the auxiliary basis. The EQ3/6 database employs a special canonical form using the species O2(g) as a primary species to describe redox reactions. There is nothing magical about using C ^ j and, in fact, in some cases it may be more convenient to use some other species. For example, under anaerobic conditions in which the oxygen concentration is just a few atoms per cubic centimeter, it may make little sense physically to use C>2(g) even though formally, of course, it must give equivalent results and serves to parameterize the redox state. The user may wish to know the total iron II and III in the system, in which case using the redox couple Fe 2 + -Fe 3 + as primary species provides this information directly. Furthermore, using 02(g) as primary species does not allow for easy decoupling of different redox couples. In this case a customized database may be necessary (Lichtner, 1995). For problems involving a high pH it may be more desirable to use OH~ rather than H + as primary species. Relation between source/sink terms. In the canonical representation the source/sink term for primary species becomes

N Rj = -

J2

(108)

VjJ"

i=Nc+1

and for secondary species simply Ri = Z,

(109)

where /, represents the reaction rate for the canonical form of the reaction as given by Equation (90). Substituting Equation (109) into Equation (108) leads to the following relation among the source/sink terms

N R

J +

"J i R i =

10

i=Nc+1

>

For kinetic reactions it may not be possible to associate /, with a kinetic rate law since the canonical form of the kinetic reaction may not represent the actual reaction mechanism. In such cases / r ke represents the more fundamental quantity. Local partial equilibrium For a general geochemical system in a state of local partial equilibrium the chemical reactions taking place may be divided into two sets: those which may be described by conditions of local equilibrium, and those which are kinetically controlled and require kinetic rate laws. The reactions taking place in the system can be written in the general form

0 ^

Nk ¡=1

(r = 1

Nmin

, -w

+

Nf

= m= 1

(137) r=1

as follows from the definition of L, given in Equation (86). Collecting terms yields the primary species transport equations

a

N'min „....

dt

m=1

N