dynamics of mixing and bimolecular reaction kinetics in aquifers

ORIGINAL PAPER

Dynamics of mixing and bimolecular reaction kinetics in aquifers

Jagadeesh Anmala • Vivek Kapoor

Published online: 29 December 2012

� Springer-Verlag Berlin Heidelberg 2012

Abstract Descriptions of chemical transformation kinet-

ics and hydrologic transport need to be coupled to understand

the composition of flowing waters. The coupling of a

bimolecular transformation reaction (reactant 1 ? reactant

2 ? product; rate r ¼ jc1c2) with spatially heterogeneous

subsurface flows is addressed here. The flow microstruc-

ture—that controls the spreading rate of solutes and the mean

reactant concentrations (C1, C2)—creates concentration

microstructure whose intensity is characterized by the vari-

ances r2c1

, r2c2

, and the cross-covariance c1c2. In addition to

the macroscopic overlap of the reactants, that is quantified by

the product of the mean concentrations that are routinely

modeled using effective dispersion coefficients Dij, the

concentration microstructure plays an important role in

determining reaction macro-kinetics as the mean reaction

rate is r ¼ jðC1C2 þ c1c2Þ. For initially non-overlapping

reactants c1c2 þ C1C2 ¼ 0 and r ¼ 0 under pure advection.

It is shown that due to the action of local dispersion, at large

time, the c1c2 budget is characterized by a balance between

its rate of production and dissipation, which results in

c1c2 � 2sDijðoC1

�oxiÞðoC2

�oxjÞ, where s is the dissipation

time-scale characteristic to the destruction of concentration

fluctuation by local dispersion. This results in r ¼ jeffC1C2,

where jeff � j½1þ 2sDijðolnC1

�oxiÞðolnC2

�oxjÞ�, which

accounts for the influence of concentration microstructure

and small-scale mixing on the macroscopic bimolecular

kinetics. The effective rate parameter jeff is greater than the

intrinsic rate constant j measured under well-mixed condi-

tions if the macroscopic concentration gradients have the

same sign (initially overlapping reactants). For the initially

non-overlapping reactants which result in macroscopic gra-

dients having opposite signs, jeff \j. The macroscopic

reactant concentration gradients, effective dispersion coef-

ficients, and the dissipation time-scale control the reaction

macro-kinetics, in addition to the intrinsic rate constant j and

the mean reactant concentrations. The formulation for

reaction macro-kinetics developed here helps explain

previously reported disparities between laboratory and field-

scale transformation rates and also provides a way to rep-

resent the influence of reactant concentration microstructure

in large-scale descriptions of reactive transport.

Keywords Mixing � Bimolecular reaction � Kinetics �Aquifers � Two-dimensional � Heterogeneous �Concentration microstructure � Transport �Concentration variance � Concentration cross-covariance �Production-dissipation balance � Macro dispersion

1 Introduction

In quantifying chemical transformation processes in the

field, fundamental questions arise regarding the coupling of

transport descriptions with the reaction kinetics. Due to the

practical interest in biodegradation in aquifers, some of

these questions are posed in the context of biologically

mediated reactions. For example Molz and Widdowson

(1988) observed the sharp gradients in the dissolved oxy-

gen levels in the vertical dimension and pointed out that

practical models that describe depth averaged concen-

trations based on effective dispersion coefficients are

J. Anmala (&)

Civil Environmental Group, Birla Institute of Technology,

Pilani, Hyderabad Campus, Hyderabad 500 078, AP, India

e-mail: [email protected];

[email protected]

V. Kapoor

Citigroup, 399 Park Avenue, New York, NY 10043, USA

123

Stoch Environ Res Risk Assess (2013) 27:1005–1020

DOI 10.1007/s00477-012-0679-5

incapable of incorporating the potentially important influ-

ence of the variable oxygen concentration on the overall

degradation rate. Semprini and McCarty (1991) found that

in order for their reactive transport model to correctly

predict the extent of biotransformation, the effective dis-

persion coefficients had to be reduced from the values that

are consistent with the spreading of a non-reactive tracer.

In line with these observations, Sturman et al. (1995) point

out that field degradation rates were reported to be 4–10

times less than the rates measured in the laboratory. These

departures are often attributed to mass-transfer limitations

which are pertinent to both abiotic and biotic chemical

transformation reactions. Friedly et al. (1995) discuss the

challenges in applying batch-scale parameters pertinent to

chromium reduction to describe chemical transformation in

the field. The focus of this work is in understanding mass

transfer limitations and their representation in large-scale

descriptions of multiple species undergoing chemical

transformations and transport.

The field observations and experiments reported above

are valuable as they document the challenges in field-scale

reactive transport modeling. They point to the need for

conducting experiments with comprehensive measure-

ments to test and refine modeling approaches. Carefully

controlled field experiments with three-dimensional moni-

toring of independently well-characterized multi-species

chemical transformation reactions have yet to be under-

taken. The complexity of real chemical transformation

reactions and their monitoring makes it challenging to

design a field experiment with independent documentation

of reaction kinetics, transport characteristics, and the cou-

pled reactive transport. Mass transfer limitations within the

pore-spaces can render the reaction rate-laws observed in a

well-mixed batch reactor incapable of representing the

reaction kinetics of fast multi-species reactions even over a

collection of pores, i.e., the Darcy scale, as shown in the

laboratory experiments of Kapoor et al. (1998) and Raje

and Kapoor (2000). Therefore it is not surprising that the

problem of representing chemical transformation kinetics

in field-scale models remains unsolved (Fig. 1). While

significant advances have been made in representing the

influences of small-scale heterogeneity on solute spreading

rates (e.g., Gelhar and Axness 1983) a commensurate

understanding of how mass-transfer limitations influence

large-scale chemical transformation kinetics is lacking.

Presently most numerical modeling applications of coupled

chemical transformation kinetics and transport are based

upon using chemical kinetic rate laws and parameters

Fig. 1 Upscaling of reaction

kinetics is addressed in this

work

1006 Stoch Environ Res Risk Assess (2013) 27:1005–1020

123

inferred under well mixed conditions. Chemical reactions

and solute transport are treated sequentially in these

numerical models, and the effects of small-scale (sub-grid)

heterogeneity are represented through ad-hoc parameter

adjustments that render the model descriptions of limited

scientific interest and engineering utility. Yet a host of

practically important contamination problems require

understanding chemical transformations in large hydro-

logic systems, and the coupling of biochemical processes

and hydrological processes is at the heart of understanding

the ecology of drainage-basins (Webb and Walling 1996).

The heterogeneous distribution of grain sizes in natural

systems in relation to the discrepancy of laboratory mea-

sured rate constants on uniform grain sizes and field

measurements was also addressed by Lichtner and

Tartakovsky (2003). Computational tools are introduced by

Srinivasan et al. (2007) to quantify model (structural) and

parametric uncertainties of geochemical reactions while

disregarding transport mechanisms including advection,

diffusion and hydrodynamic dispersion. Two-dimensional

pore-scale simulations were performed and transverse

dispersion coefficients were computed for conservative

tracers and reactive solutes to model mixing (Acharya et al.

2007). The issues related to upscaling of reactive transport

in heterogeneous media were examined by Luo et al.

(2008). Sufficient conditions were established by Battiato

et al. (2009) and Battiato and Tartakovsky (2011) under

which macroscopic reaction–diffusion equations provide

an adequate averaged description of pore-scale processes

by neglecting and including advection. Volume averaging

was employed by them as done by Wang and Kitanidis

(1999) in analyzing macrodispersion previously. An itera-

tive hybrid numerical method was developed by Battiato

et al. (2011) to incorporate pore-scale effects into contin-

uum models of reactive transport in porous and fractured

media. A brief review of the state of art discussion on

mixing, spreading and reaction in subsurface heteroge-

neous media is attempted by Dentz et al. (2011).

When reactions are slow compared to the molecular

diffusion time-scale characteristic to the pore-spaces the

well-mixed reaction laws and parameters can be effectively

applied to describe the averaged concentration over a pore

cross-section or a collection of pores. However reactions

that are slow at the Darcy-scale may not be slow in the

field. This is due to the fact that td ¼ 2V1aL= l1ð Þ2þh

2V1aL= l2ð Þ2��1and tr ¼ jCinitial½ ��1

(from Table 1). For

increasing heterogeneity (or a heterogeneous case) due to

‘‘scale effect’’ denominator of td can also increase giving

rise to varied possibilities (Kapoor et al. 1997). Owing to

the large spatial scales of heterogeneity in the field, the

time-scales characteristic to the dissipation of concentra-

tion fluctuations in the field can be much larger than the

molecular diffusion time-scale characteristic to the pore-

spaces. When the molecular diffusion time-scale charac-

teristic to the pore-spaces is smaller than the reaction

time-scale, the reaction kinetics determined in well-mixed

batch tests apply to the Darcy-scale, which enables directly

addressing the question of field scale mass-transfer limi-

tations and their influence of field-scale reaction kinetics

(Fig. 1), as is done here by analyzing concentration fluc-

tuation budgets and detailed numerical simulations for an

irreversible bimolecular reaction (Anmala 2000).

In Sect. 2 we formulate the problem of mixing between

two species and its connections with chemical transfor-

mations for an irreversible second order bimolecular

reaction. In Sect. 3 we describe the heterogeneous porous

media flow field used to examine the coupling of mixing

and reaction in this work. In Sect. 4 we focus on mixing

without reactions, and in Sect. 5 we focus on reactive

mixing and outline an approach for coupling reactions with

macroscopic transport descriptions. Section 6 provides a

summary of our key findings.

2 Role of mixing in controlling bimolecular reactions

We use an irreversible bimolecular reaction (reactant

1 ? reactant 2 ? product) to provide a specific example of

coupling chemical transformations kinetics with flow and

transport descriptions. For such a reaction the reactant con-

centrations (c1, c2) and product concentration (c3) follow

dc1dt¼ dc2

dt¼ � dc3

dt¼ �jc1c2 ð1Þ

under well mixed conditions, and j is the intrinsic rate

constant which has the units of cu-1 Time-1 where cu

stands for concentration units. The elementary non-

linearity in the reaction rate law (1) is sufficient to make

Table 1 Flow, transport, and reaction parameters

Porosity: n 0.3

eln K:KG (cm/day) 100

Hydraulic gradient magnitude: J 0.01

ln K std. dev: rln K 1.0

lnK heterogeneity scales: l1; l2(cm) 50, 25

Mean x1 velocity: V1 (cm/day) 3.77

Velocity std. dev: rv1;rv2

(cm/day) 2.66, 1.15

Spatial discretization: Dx1;Dx2 (cm) 1.22, 0.488

Local dispersivities: aL, aT (cm) 1, 1

Local dispersion time-scale:

td ¼ 2V1aL= l1ð Þ2þ2V1aT= l2ð Þ2h i�1

(days)66.3

Advection time-scale: ta ¼ l1=V1 (days) 13.3

Reaction time-scale: tr ¼ jcinitialð Þ�1(days) 0.19

Stoch Environ Res Risk Assess (2013) 27:1005–1020 1007

123

the coupling of reaction kinetics to aggregated flow and

transport descriptions non-trivial. The experimental

investigations of Kapoor et al. (1998) and Raje and

Kapoor (2000) with a bimolecular reaction in flows in a

tube and porous medium column, document the difficulty

in coupling such a reaction with flow and macroscopic

transport descriptions when the reaction is rapid compared

to the molecular diffusion time-scales characteristic to the

pore-spaces. When the reaction is slow compared to the

molecular diffusion time-scales, there is no problem in

appending the rate law to an aggregated transport description.

In the context of a porous medium even if the well-mixed rate

law can be trivially appended to a Darcy scale transport

description, the challenge remains in describing the influences

of geological heterogeneity induced flow microstructure on the

macroscopic reaction kinetics, that is addressed here. Therefore

we consider the concentrations of the bimolecular reaction

system undergoing advection and small-scale mixing in a

porous medium. With mi describing the seepage velocity, the

conservation of mass statements for the reactants and product

are (Kapoor et al. 1997)

ockotþ o

oxivick � dij

ockoxj

� �¼ �jc1c2 k ¼ 1; 2 ð2Þ

oc3otþ o

oxivic3 � dij

oc3oxj

� �¼ jc1c2 ð3Þ

The divergence-free seepage velocity mi is well known to

have a rich microstructure that is responsible for enhancing

the dispersive flux in the field, compared to laboratory-

scale homogeneous columns. The above continuum scale

transport equations rely on the hypothesis of well-mixing

within the pore and the micro-scale fluctuations exert non-

negligible effects up to the field scale representation of

transport (Kapoor and Gelhar 1994a, b, Kapoor et al.

1997). The small-scale mixing is parameterized by the

local dispersion coefficients dij that are routinely evaluated

in laboratory scale tracer tests. We treat the local dispersion

coefficients as constants (dependent on the mean flow) and

focus on describing the role of the velocity microstructure

and the resultant concentration fluctuations on reaction

kinetics. We decompose the concentration and velocity

into their ensemble mean and perturbations

vi ¼ Vi þ vi; ck ¼ Ck þ ck ð4Þ

Substituting these decompositions into the reactive

transport Eqs. (2) and (3) and taking expectations (which

are denoted by an overbar or capital letters) yields

oCk

otþ o

oxiViCk þ vick � dij

oCk

oxj

� �¼ �j C1C2 þ c1c2ð Þ

ð5Þ

oC3

otþ o

oxiViC3 þ vic3 � dij

oC3

oxj

� �¼ j C1C2 þ c1c2ð Þ ð6Þ

In field-scale applications the numerical grid resolution

scale is never sufficient to resolve all the details of the flow

field, hence, the modeled flow and concentration are (at best)

spatial averages of the actual flow and concentration.

When the spatial averaging occurs over scales that are

larger than the flow microstructure, the ensemble average

provides useful estimates of spatial averages. For modeling

the averaged concentration of non-reactive tracers the

dispersive flux needs to be modeled—that has been the

motivation of much theoretical research and carefully

controlled and intensely monitored field tracer tests that

have led to an understanding of the connection between

aquifer microstructure and field-scale solute dispersion. For

chemical transformation problems involving rate laws that

depend on the product of the concentrations, in addition to

the dispersive flux, the cross-correlation between reactant

concentrations, c1c2, needs to be evaluated, as it appears in

the reaction macro-kinetics in Eqs. (5) and (6). In the

numerical simulations of Kapoor et al. (1997) it was shown

that for initially non-overlapping reactants c1c2 can be

significantly negative and can appreciably slow down the

macro-kinetics. These simulations along with the empirical

observations that field-scale kinetics are slower than what is

suggested by laboratory scale experiments build a strong

case for the inclusion of reactant cross-correlations in large-

scale models of reactive transport where reaction rates are

determined by the product of reactant concentrations. The

recent works of Cirpka and Kitanidis (2000a, b) document

the issue based on analyzing temporal moments from

numerical simulations, and by developing an advective–

dispersive streamtube approach that incorporates the crucial

role of local dispersion on mixing and reactions. However

there are virtually no guidelines on how to represent this

cross-covariance between reactant concentration in a mean

concentration model, and the current applied modeling

practice ignores c1c2 altogether in the reaction term. This

modeling practice presumes that the well mixed rate law

directly applied to the mean concentration, and the extent of

reaction, and the rate of reaction, is then determined by the

large-scale mixing, which may be quantified by the

product of the mean concentrations C1C2. This work

extends the previous work by Kapoor et al. (1997) by

analyzing c1c2 and determining its relationship with

macroscopic variables.

Actual concentrations vary around the averaged values

and these fluctuations are subject to the physical–chemical

reality of a bimolecular reaction in a heterogeneous flow,

which can be expressed (for the reactants) by subtracting

Eqs. (5) from (2) to obtain


123

ock

otþ o

oxiVick þ vick � vick � dij

ock

oxj

� �¼ �vi

oCk

oxi

� j c1C2 þ C1c2 þ c1c2 � c1c2ð Þ k ¼ 1; 2

ð7Þ

This concentration perturbation equation will be used to

understand the dynamics of c1c2 in this work. We focus on

initially non-overlapping species undergoing spatially

heterogeneous advection and local dispersion and develop

an understanding of the nature of the cross-correlation

between the reactants. The overall configuration for the

numerical simulations reported here is shown in Fig. 2.

The spatially varying flow field used for a specific example

calculation is described next.

3 Flow in heterogeneous porous medium

The flux of fluid qi in the porous continuum (averaged over

many pores and solids) follows Darcy’s law, which for the

locally isotropic-, constant density-case is qi ¼ �K ohoxi

. K is

the hydraulic conductivity, and h the hydraulic head. The

conservation of the incompressible fluid in the incompress-

ible porous continuum requires r2hþr ln K � rh ¼ 0.

3.1 Hydraulic conductivity field

As K is positive and its logarithm appears naturally in the

flow problem, ln K has been represented as a random field in

previous studies on understanding the consequences of geo-

logical heterogeneity on flow and transport (e.g., Gelhar and

Axness 1983). Making the decomposition ln K ¼ ln KG þ y,

a description of the hydraulic conductivity up to second

statistical moments is made by specifying KG and the spatial

covariance function, RðnÞ ¼ yðxÞyðx�nÞ, of the lnK varia-

tions (assumed to be statistically stationary), or its spectrum

SðfÞ ¼R

ei2pf�nRðnÞdn. For the sake of illustration the

spectrum-covariance pair is taken as

Sðf1; f2Þ ¼r2

ln Kl1l2

2pl21f 2

1 þ l22f 22

� �2exp � l2

1f 21 þ l2

2f 22

� �� ð8Þ

Rðx1; x2Þ ¼r2

lnKl1l2

2pexp � px1

l1

2

� px2

l2

2" #

� 1� 2px1

l1

2

þ 1

2

px1

l1

4

�2px2

l2

2"

þ 1

2

px2

l2

4

þ px1

l1

2 px2

l2

2#

ð9Þ

At small frequencies f, S increases and achieves its

maximum value at fi ¼ 1=ðffiffiffi2p

liÞ and decreases thereafter—

qualitatively similar to some spectra inferred from

measurements (Gelhar 1993). The variance of ln K is

denoted by r2ln K and the correlation scale ki is the distance

along the axis xi at which the covariance falls to r2ln Ke�1:

Rðki; xj ¼ 0; j 6¼ iÞ ¼ r2ln Ke�1. For the chosen spectrum

ki � 0:1674li and the hydraulic conductivity covariance

function is shown in Fig. 3. Figure 4 shows the simulated

two-dimensional realization of the hydraulic conductivity

field in the visualization window (shown in Fig. 2), using an

FFT based algorithm (Dykaar and Kitanidis 1992). Table 1

lists the parameters of the hydraulic conductivity field.

3.2 Flow solution

Flow in a realization of the random hydraulic conductivity

(Fig. 4) is numerically simulated. The difference in the

Fig. 2 Domain, initial, and

boundary conditions, and

visualization window


123

hydraulic head specified at the two longitudinal ends of the

hydraulic conductivity field is JL, where L is the overall

length of the domain. In the simulation L ¼ 50l1. The

domain extends 20l2 in the vertical, and vertical averages

that span a large number of correlation scales are taken to

estimate ensemble averages. Such an approach enables

studying averages and fluctuations based on one realization

of the hydraulic conductivity field, as has been previously

done by Kapoor and Kitanidis (1997, 1998). No flux occurs

through the boundaries at x2 ¼ 0 and 20l2. The hydraulic

head solution was numerically evaluated using the suc-

cessive over relaxation method with centered and second

order finite-differences, and the over-relaxation parameter

was calculated based on the spectral radius of a Poisson

equation with the same discretization (Ames 1992). The

porosity of natural porous medium typically varies less

than its hydraulic conductivity and tends to be poorly

correlated with the hydraulic conductivity, and the porosity

variations themselves are not the primary mechanisms for

enhanced field-scale spreading (Gelhar 1993). Therefore

we focus on the dominant influence of a spatially

heterogeneous hydraulic conductivity field and calculate

the seepage velocity by dividing the specific discharge by

the porosity (n) which is assigned a constant value of 0.3.

The correlation coefficient between the numerically com-

puted ov1=ox1and ov2=ox2 is -0.96, therefore the numer-

ically computed flow fields are practically divergence free.

The numerical scheme converges when Dxi� 2=

maxðo ln K=oxiÞ, which ensures that the set of algebraic

equations resulting from discretization have a unique solution

(Ames 1992) and that the hydraulic conductivity field is well

resolved. The solution to the flow problem is shown in Figs. 5

and 6, and Table 1 lists the flow parameters.

Fig. 3 Covariance of ln K (l1 = 50 cm, l2 = 25 cm, rln K ¼ 1)

Fig. 4 Logarithm of hydraulic conductivity K (l1 = 50 cm,

l2 = 25 cm, rln K ¼ 1, KG = 100 cm/day)

Fig. 5 Hydraulic head h (cm)

Fig. 6 a Flow field—v1 (cm/d), b flow field—v2 (cm/d)


123

3.3 Pure advection case

It is easy to describe what occurs under advection alone.

The interfacial evolution is tracked by introducing particles

(say, around 30,000) uniformly over the interface and by

letting them experience the numerically calculated velocity

field. The interface gets stretched and increasingly distorted

with time (Anmala 2000). The mean concentrations of the

different species may give the appearance of overlap

between the species (Anmala 2000), however there is no

actual overlap of the different species. Under advection

alone the average of the product of the concentrations is

zero, which implies that the cross-covariance between the

reactant concentrations is equal to the negative of the

product of the mean concentrations

c1c2 ¼ 0) c1c2 ¼ �C1C2 ð10Þ

This stark picture of the pure advection case is important

to consider as it illustrates how even when an upscaled

(averaged) model does an excellent job of describing

the mean concentration field (using macrodispersion

coefficients that can be practically independent of local

dispersion coefficients at large Peclet numbers), the overlap

between the reactant implied by the mean concentrations can

be absolutely misleading. Under advection alone there is no

mixing of reactants despite the immense distortion of the

interface and the overlap of the mean concentrations

(Anmala 2000). We explore next how mixing occurs due

to local dispersion and how the lack of perfect mixing may be

expressed as a function of upscaled concentrations.

4 Dynamics of mixing of solutes in heterogeneous

porous media flows

The reactant concentration cross-covariance c1c2 can con-

trol the reaction macro-kinetics (Eqs. 5 and 6). We first

focus on how c1c2 can be related to macroscopic variables

in the absence of reaction (j ¼ 0). The equation governing

the cross-covariance is found by multiplying the c1 equa-

tion by c2 and the c2 equation by c1 and adding the resultant

equations and taking expectations to get

oc1c2

otþ o

oxiVic1c2 þ vic1c2 � dij

oc1c2

oxj

� �

¼ �vic2

oC1

oxi� vic1

oC2

oxi� 2dij

oc1

oxi

oc2

oxjð11Þ

The structure of the c1c2 equation is similar to the

variance equations which are found by multiplying the ck

equation by ck and taking expectations to obtain

or2ck

otþ o

oxiVir

2ckþ vic

2k � dij

or2ck

oxj

� �

¼ �2vickoCk

oxi� 2dij

ock

oxi

ock

oxjk ¼ 1; 2 ð12Þ

In (11) and (12), c1c2 and r2ck

undergo a mean advective

flux, a dispersive flux due to velocity fluctuations, and a

diffusive flux due to local dispersion. In addition to transport

of c1c2 and r2ck

, they are produced and dissipated, as indicated

by the terms on the right hand sides of Eqs. (11) and (12). If

the reactant concentrations were spatially uniform in the

mean and in the transverse directions, then heterogeneous

advection does not result in concentration fluctuations

because all the fluid elements are carrying the same

concentration—hence the role of the mean concentration

gradient in producing fluctuations. When the mean

concentration varies, say for a concrete example, in the x1

direction, then due to the differences in the velocities of the

fluid elements that were once carrying the same concentration

(same x1 location), concentration fluctuations will result.

However local mixing (local dispersion) transports reactants

from where they are at larger concentrations to where the

concentrations are less—hence countering the creation of

fluctuations due to the presence of a mean gradient and

velocity fluctuations.

In the numerically simulated flow field described in

Sect. 3 we simulate the transport of two initially non-

overlapping species that are adjacent in space, with the

overall configuration and initial and boundary conditions

depicted in Fig. 2 (by solving Eq. (2) with j = 0). The

concentrations are numerically simulated, using the stan-

dard explicit, centered, finite-difference method (Ames

1992). Initially species 2 was placed to the right of the

interface and species 1 was introduced uniformly to the left

of the interface. The local dispersion coefficients were

taken to be constants proportional to the mean velocity

(d11 ¼ aLV1, d22 ¼ aT V1) and the local dispersivities

ðaL; aTÞ and the local dispersion time-scale characteristic to

the scale of heterogeneity are given in Table 1. The small

grid size (Table 1) and adherence to the stability criterion

for an explicit scheme (Ames 1992; Kapoor et al. 1997)

assures resolution of diffusive fluxes and of the concen-

tration microstructure (including its gradient). The simu-

lated concentration fields are shown in Fig. 7. The

concentration contours show that at any cross-section

(fixed x1) larger than average values of c1 occur together

with smaller than average values of c2 and vice versa.

4.1 Production-dissipation balances for the variance

and the cross-covariance

To predict the concentration cross-covariance and con-

centration variance, the fluctuation budgets given by Eqs.

(11) and (12) need to be understood. It has been previously

shown by detailed numerical simulations that the dominant

balance governing the variance Eq. (12) is between its rate

of production and dissipation at large time (Kapoor and


123

Kitanidis 1998). On integrating the terms of the fluctuation

budgets in x1 we obtain global (integrated in x1) production

and dissipation rates and at large time the unsteady terms

become unimportant and the dominant balance is between

the rates of production and dissipation (Anmala 2000). The

flux terms integrate out to zero, so it is no surprise they do

not have a role to play in the global fluctuation budgets.

However on plotting the production and dissipation terms

as a function of space, we find that a production-dissipation

balance occurs even locally (Figs. 8, 9) after 0.2 td:

�vic2

oC1

oxi� vic1

oC2

oxi� 2dij

oc1

oxi

oc2

oxj� 0 ð13Þ

�2vickoCk

oxi� 2dij

ock

oxi

ock

oxj� 0 ð14Þ

The dominant production dissipation balance will not

hold at points where the production rate is zero and the

concentration variance and cross-covariance are not zero.

For example for a finite size input of a solute in a random

flow-field, at the center of mass the production rate is zero

and yet the concentration variance is not zero, therefore the

production dissipation balance for the variance takes place

at a distance away from the center of mass (Kapoor and

Gelhar 1994a, b; Kapoor and Kitanidis 1998). However the

singular points where the production dissipation balance

does not hold are points of small values for the concen-

tration fluctuations intensity because of the lack of any

fluctuation production at those points. For the configuration

examined here such singular points do not exist.

4.2 Macrodispersive fluxes of the mean, variance

and cross-covariance

It has been previously theoretically shown that the linear-

ized analysis of macrodispersive fluxes developed for the

mean concentration also apply to the concentration vari-

ance, and the attendant macrodispersion coefficients for the

mean and variance are identical (Kapoor and Gelhar 1994a,

Fig. 7 a Concentration fields for the nonreactive case (c1). b Con-

centration fields for the nonreactive case (c2)

Fig. 8 Rates of production and dissipation of r2ck

for the nonreactive

case

Fig. 9 Rates of production and dissipation of c1c2 for the nonreactive

case


123

b; Kapoor and Kitanidis 1997), and this prediction was

observed in detailed numerical simulations (Kapoor and

Kitanidis 1997, 1998). These analyses are easily extended

to the macrodispersive flux of the concentration cross-

covariance. Therefore the influence of heterogeneous

advection on the macroscopic fluxes of Ck, r2ck

, and c1c2

can be parameterized by

ckvi ¼ �Dkij

oCk

oxj; c2

kvi ¼ � ~Dkij

or2Ck

oxj; c1c2vi ¼ �D12

ij

oc1c2

oxj

ð15Þ

where

D1ij � ~D1

ij � D2ij � ~D2

ij � D12ij � Dij ð16Þ

The macrodispersion coefficients associated with the

linear gradient approximation of the dispersive flux (15)

were evaluated from our detailed numerical simulation by a

linear regression. In Fig. (10) it is shown that the dispersion

coefficients associated with all the dispersive fluxes are

close to each other and rapidly approach an asymptotic

level.

4.3 Dissipation time scale s

The time-scale associated with the destruction of concen-

tration variances (sk, k = 1, 2) and the concentration cross-

covariance (s12) may be defined as:

2dijock

oxi

ock

oxj¼ r2

c

skð17Þ

2dijoc1

oxi

oc2

oxj¼ c1c2

s12

ð18Þ

Based on the detailed numerical simulations the

averaged quantities involved in defining the dissipation

time-scales may be calculated and a global (integrated in

x1) estimate of these time-scales is found by global values

of the variances, cross-covariance and their dissipations.

These 3 dissipation time-scales are shown in Fig. 11, which

shows that

s1 � s2 � s12 � s at all times tð Þ ð19Þ

and at large time the dissipation time-scales appear to

approach an asymptotic level. The approach to an asymp-

totic level for the dissipation time-scale is slower than the

approach of the macrodispersion coefficients to their

asymptotic values (Fig. 10). This is consistent with the

prior results of Kapoor and Kitanidis (1998) for the con-

centration variance dissipation. The time-scale character-

istic to the dissipation of the cross-covariance is identical to

that associated with the dissipation of the concentration

variances.

Although the dissipation time-scales have been exam-

ined much less in the subsurface transport literature,

compared to macrodispersion coefficients, their cardinal

role in quantifying concentration fluctuations and dilution

has been recognized. The identification of the dissipation

time-scale in heterogeneous porous medium lead to the

prediction of the large-time demise of the intensity of

concentration fluctuations (rc=C) for a finite-sized impulse

input (Kapoor and Gelhar 1994a, b). The Cape Cod bro-

mide tracer data (Leblanc et al. 1991) exhibits this large

time demise of rc=C that implies s � 125 days at the Cape

Cod site (Kapoor and Gelhar 1994b). Subsequent numeri-

cal simulations have also documented the dissipation time

scale and the large-time demise of rc=C for a finite-sized

impulse input (Kapoor and Kitanidis 1998). For perfectlyFig. 10 Evolution of longitudinal dispersion coefficients for the

nonreactive case

Fig. 11 Evolution of dissipation time-scales of r2ck

and c1c2 for the

nonreactive case


123

stratified media that result in small-scale variation in flow

in a direction perpendicular to the flow direction, the close

proximity of an upper bound on the dissipation time-scale

and the numerically simulated values (Kapoor and Anmala

1998) provides a way to theoretically estimate dissipation

time-scales. These works have spurred the development of

new Lagrangian approaches that can also account for dis-

sipation (e.g., Pannone and Kitanidis 1999).

4.4 Simplified model for concentration cross-

covariance and variance

Using the production dissipation balance (Eqs. 13 and 14),

the Fickian representation of the dispersive flux (Eq. 15 for

the mean concentrations, and the definition of the dissi-

pation time-scale (Eqs. 17 and 18), we can write for the

concentration cross-covariance and variance

c1c2 � D1ij þ D2

ij

� s12

oC1

oxi

oC2

oxjð20Þ

r2ck� 2Dk

ijskoCk

oxi

oCk

oxjð21Þ

In (20) and (21) the macrodispersion coefficients (Dij)

and the dissipation time-scale (s) are macroscopic

constitutive parameters of the transport system. Using

these macro-constitutive parameters the concentration

cross-covariance and the variance can be related to the

macroscopic concentration gradients. In Fig. 12 it is shown

that the cross-covariance represented by Eq. (20)

compares very well with that calculated from the detailed

numerical simulations, which is expected because the

production-dissipation balance that is the basis of (20)

holds well (Fig. 9), and the macro-constitutive parameters

of macrodispersion coefficients and the dissipation time-

scale were evaluated from the detailed numerical

simulations (Figs. 10, 11). Similarly the variance model

(Eq. 21) also works very well (Fig. 13). Therefore if we

know the macro-constitutive parameters for the trans-

port system, Dij and s, we can relate the concentration

variances and the cross-covariance to the mean/upscaled

concentration field.

5 Bimolecular reaction kinetics in heterogeneous

porous media flows

The cross-covariance of reactants undergoing a bimolecu-

lar reaction and transport follows

oc1c2

otþ o

oxiVic1c2 þ vic1c2 � dij

oc1c2

oxj

� �¼

� vic2

oC1

oxi� vic1

oC2

oxi� 2dij

oc1

oxi

oc2

oxj

� j c1c2ðC1 þ C2Þ þ C1r2c2þ C2r

2c1þ c1c2

2 þ c21c2

h i

ð22Þ

The reactant variances follow

or2c1

otþ o

oxiVir

2c1þ vic

21 � dij

or2c1

oxj

� �¼ �2vic1

oC1

oxi

� 2dijoc1

oxi

oc1

oxj� 2j C1c1c2 þ C2r

2c1þ c2

1c2

h ið23Þ

or2c2

otþ o

oxiVir

2c2þ vic

22 � dij

or2c2

oxj

� �¼ �2vic2

oC2

oxi

� 2dijoc2

oxi

oc2

oxj� 2j C2c1c2 þ C1r

2c2þ c2

2c1

h ið24ÞFig. 12 Comparison of simulations and model of c1c2 for the

nonreactive case

Fig. 13 Comparison of detailed numerical simulation and model of

r2ck

(k = 1,2) for the nonreactive case


123

A detailed numerical simulation is performed (by

solving Eqs. 2 and 3 including reactions) with initial

conditions and transport parameters identical to the

simulation performed to study mixing without reactions in

the previous section (Table 1). The simulation reported here

incorporates the bimolecular reaction and the reaction is fast

relative to the local-dispersion time scale characteristic to

the scales of heterogeneity (Table 1). We focus on the fast

reaction case as slow reactions are trivially appended to

macroscopic transport models, because from the point of

view of the reaction process the transported reactants of slow

reactions are locally well mixed (Kapoor et al. 1997; Mo and

Friedly 2000). Considering fast reactions may be in contrast

with the well-mixed pore assumption. With the criterion that

the simulation time step should be much smaller than the

smallest possible reaction time-scale (tr=100), the transport

simulation procedure is easily extended to handle reactions

explicitly (Kapoor et al. 1997; Anmala and Kapoor 2012).

The simulated concentration fields are shown in Fig. 14. The

negative correlation between the reactants can be observed

from the concentration contours of Fig. 14. The main

distinction from the non-reactive case is the separation of the

reactants along the flow direction because of their

transformation into the product (Fig. 14). As a

consequence of the reaction, the mean concentration

gradients remain steep at large time whereas without

reactions the mean concentration gradients decreased with

time (Anmala 2000).

5.1 Production-dissipation balances for the variance

and the cross-covariance

We observed that notwithstanding rapid reaction, the bal-

ance between production and dissipation holds. This bal-

ance is shown for the reactant variances in Figs. 15a and b,

and for the cross-covariance in Fig. 16. The production-

dissipation balance takes place at t [ 0.2td, as it does for

the non-reactive variances and cross-covariance (Figs. 8,

9). The preservation of the production dissipation balance

leads to a major simplification in the reactant cross-

covariance and concentration variance budgets (Eqs. 22,

23, and 24) insofar as (13) and (14) hold notwithstanding

reaction. Accompanying the feature of production-dissi-

pation balance is the feature that the reaction terms in the

cross-covariance budget balance out the unsteady and the

transport terms.

5.2 Macrodispersive fluxes of the mean, variance

and cross covariance

Figure 17 shows the macrodispersion coefficients associ-

ated with the dispersive fluxes of the means and variance of

the reactants and the product, as well as the reactant

concentration cross-covariance. The bimolecular reaction

does not alter the macrodispersion coefficient associated

with the macrodispersive fluxes. Figure 17 can be com-

pared with Fig. 10 to find that the bimolecular reaction

does not alter the coefficients that much and all of them

quickly reach asymptotic values.

5.3 Dissipation time scale s

The time-scale characteristic to the dissipation of variances

of the reactants and the product, shown in Fig. 18, have the

same overall behavior and values as those for the non-

reactive case in Fig. 11. The dissipation time scales of

Fig. 14 a Concentration fields for the reactive case—c1. b Concen-

tration fields for the reactive case—c2. c Concentration fields for the

reactive case—c3


123

species variances and cross-covariance match with each

other for non-reactive case. This is not the case for reactive

case as can be seen from Fig. 18. The understanding of

evolution of dissipation time scales of species variances

and cross covariance is an important step towards the

understanding of upscaling of bimolecular reaction kinetics

in aquifers.

5.4 Simplified model for cross-covariance, reaction

macro-kinetics, and variance

The production-dissipation balance and the values of the

macrodispersion coefficients and the dissipation time-scale,

were not altered in the reactive case from the non-reactive

case, despite the concentration fields being significantly

different due to reaction. Therefore the relationship of the

concentration cross-covariance and variance to the mean

concentrations remains unchanged from that given by

Eqs. 20 and 21, as shown in Figs. 19 and 20. Setting D1ij �

D2ij � Dij and s1 � s2 � s12 � s, we can write (20) and

(21) as

c1c2 � 2sDijoC1

oxi

oC2

oxjð25Þ

r2ck� 2sDij

oCk

oxi

oCk

oxjð26Þ

The macroscopic reaction rate can therefore be written

in terms of macroscopic variables (Fig. 21):

Fig. 15 a Rates of production and dissipation of r2c1

for the reactive

case. b Rates of production and dissipation of r2c2

for the reactive case

Fig. 16 Rates of production and dissipation of c1c2 for the reactive

case

Fig. 17 Evolution of longitudinal dispersion coefficients for the

reactive case


123

j C1C2 þ c1c2ð Þ ¼ j 1þ c1c2

C1C2

� �C1C2

� j 1þ 2sDijo ln C1

oxi

o ln C2

oxj

� �C1C2 ð27Þ

The macroscopic reactive transport model describing the

mean concentrations becomes

oCk

otþ o

oxiViCk � ð dij þ DijÞ

oCk

oxj

� �¼ �jeff C1C2ð Þ

k ¼ 1; 2

ð28Þ

oC3

otþ o

oxiViC3 � ðdij þ DijÞ

oC3

oxj

� �¼ jeff C1C2ð Þ ð29Þ

where the ‘‘effective rate parameter’’ jeff is

jeff � j 1þ 2sDijo ln C1

oxi

o ln C2

oxj

� �ð30Þ

Note that jeff is not a constant—to the contrary it

depends on the macroscopic gradients of the macroscopic

reactant concentrations. Therefore we are not suggesting a

fixed value for the effective reaction rate parameter. The

effective reaction rate parameter can be less than the

intrinsic constant value for non-overlapping reactants

jeff \ j) due to the opposite signs of the gradients. The

effective reaction rate parameter can be greater than the

intrinsic rate constant value (jeff [ j) if the reactants are

initially completely overlapping. That the intrinsic rate

parameter is not directly applicable to describe reaction

macro-kinetics should come as no surprise considering that

the coefficients used to describe mixing are routinely

accepted to be scale-dependent, hence the distinction

between molecular diffusion coefficients, ‘‘effective

Fig. 18 Evolution of dissipation time-scales for the reactive case

Fig. 19 Comparison of simulations and model of c1c2 for the

reactive case

Fig. 20 a Comparison of detailed numerical simulation and model of

r2c1

for the reactive case. b Comparison of detailed numerical

simulation and model of r2c2

for the reactive case


123

diffusion coefficients’’, ‘‘local dispersion coefficients’’, and

‘‘macrodispersion coefficients’’.

Solving (28)–(29), given the macrodispersion coeffi-

cients and the dissipation time-scale is considerably sim-

pler than the detailed numerical simulations shown here.

Considering that in all applications macroscopic averages

of concentration are modeled, it is useful to develop a

macroscopic representation of reaction kinetics that

accounts for small-scale transport limitations (Eqs. 28, 29).

The dependence of the reaction macro-kinetics on the

macroscopic gradients is consistent with the dependence of

macrodispersive fluxes on macroscopic gradients that is

routinely accepted in applications and has been theoreti-

cally shown in the considerable body of work on dispersion

due to flow microstructure. It would be unseemly if the

reactant cross-covariance did not depend on the mean

concentration gradients and yet to the correlation of the

reactant concentrations with the velocity perturbations

were controlled by those gradients. This is in contrast with

the belief that reaction kinetics, no matter how non-linear

they may be, can be represented in macroscopic transport

models with rate laws of the same functional form as that

inferred under well-mixed conditions, with at most some

‘‘adjustment’’ of the rate parameters. We find that the

reaction macro-kinetics can be completely different from

the small-scale rate law and that the macroscopic reactant

concentration gradients control the reaction rates.

That the macrodispersion coefficient did not change not

withstanding reaction is also important to recognize,

because of the often expressed suspicion that macrodi-

spersion coefficients are ‘‘unable/unsuitable to model

reactive transport’’ because actual mixing is not quantified

by macrodispersion coefficients. We find that even though

macrodispersion coefficients may not describe actual

mixing and dilution, the macrodispersive fluxes of reac-

tants undergoing a bimolecular reaction can be quantified

using the same macrodispersion coefficients that describe

spreading of a non-reactive tracer. It is the reaction rate law

that needs to be modified to incorporate the influence of

small-scale mixing limitations on the macro-kinetics

(Fig. 21) and not the macroscopic representation of dis-

persive flux.

6 Summary

It has long been remarked that field-scale reaction rates are

considerably smaller than the ones inferred in the labora-

tory conditions. It is widely believed that this scale-effect is

due to the flow microstructure on account of the ubiquitous

spatial heterogeneity of aquifer materials. Non-reactive

tracer tests have carefully documented that the heteroge-

neity results in greater spreading rates in the field, and that

effect is routinely parameterized in transport models by the

use of macroscopic dispersion coefficients that are much

larger than their laboratory scale counterparts. Field-scale

numerical modeling efforts of reactive transport using

independently assessed macrodispersion coefficients (from

non-reactive tracer tests) and with the reaction kinetics

observed in careful laboratory tests have been known to

overstate the transformation rate in the field. This over-

prediction can be reduced/eliminated by decreasing the

dispersion coefficients with the reasoning that actual mix-

ing in aquifers is less than that suggested by the macrodi-

spersion coefficients, and/or, by decreasing the reaction

rate constant by presuming the field-scale constitutive

parameter is smaller than the one inferred in small-scale

laboratory experiments. Methods to a priori quantify the

dispersion coefficients and the reaction rate parameters that

will work in field-scale reactive transport models have not

been developed, therefore comparisons of model results

with field-scale measurements can become curve-fitting

exercises with little assurance of successful extrapolation

to different spatial–temporal scales. This unsatisfactory

state of affairs is because refinements to transport pro-

cesses, and less work has been done in coupling the two

processes at scales of practical interest. The practitioner

faced with assessing multi-species chemical transforma-

tions in an aquifer is likely to have some guidelines on the

choice of the dispersion coefficients, and possibly an

Fig. 21 Bimolecular reaction macro-kinetics


123

elaboration of the reaction kinetics in either well-mixed

small-scale microcosms using the aquifer material, or, in

small-scale column studies. There is little guidance on how

the reaction kinetic parameters and laws inferred in small-

scale experiments may need to be modified so that they will

work in field-scale modeling. In this work the problem of

upscaling multi-species reactions in heterogeneous sub-

surface flows is systematically addressed by analyzing

detailed numerical simulations and the budgets governing

the reactant concentration cross-covariance. Here is sum-

mary of the main findings of this work:

1. The macroscopic dispersive flux of solutes undergoing

aqueous phase bimolecular reactions can be described

in the same manner as that of a non-reactive solute. In

a linear gradient dependent relationship for the mac-

rodispersive flux the macrodispersion coefficient

remains unchanged on account of transformation

reactions.

2. The concentration variance and the cross-covariance

between the reactant concentrations also undergo a

macrodispersive flux that can be described a linear

gradient relationship with the macrodispersion coeffi-

cient being the same as that for the mean dispersive

flux of a non-reactive solute.

3. The dominant balance in the cross-covariance budget

is between its rates of production and dissipation of

concentration cross-covariance, with, and without

reactions. The same balance holds for the concentra-

tion variances.

4. The characteristic time-scale over which small-scale

mixing (local dispersion) dissipates the concentration

variance and the cross-covariance, which we refer to as

the ‘‘dissipation time scale’’, and has been previously

referred to as the ‘‘variance residence time’’, is

unaffected by reaction.

5. The macrodispersion coefficient and the dissipation

time scale appear to approach constant asymptotic

values, although the dissipation time scale approaches

the asymptotic constant level much slower than the

macrodispersion coefficient.

6. Exploring the dominant production dissipation balance

in the concentration cross-covariance budget yields a

simple formula for the concentration cross covariance

(Eq. 25). The macrodispersion coefficients, the dissi-

pation time-scale, and the product of the mean reactant

concentration gradients control the reactant concen-

tration cross-covariance. The influence of the flow

microstructure and small-scale mixing are present in

this model through the macrodispersion coefficient and

the dissipation time-scale. This formula for the con-

centration cross-covariance was shown to apply under

reactive and non-reactive conditions.

7. The simple result for the concentration cross-covari-

ance that reflects the production-dissipation balance

that characterizes second order fluctuation budgets can

be applied to upscale bimolecular reactions and results

in a macroscopic gradient dependent effective rate

parameter. The effective rate parameter can be larger

than its intrinsic value (inferred under well-mixed

conditions) if the macroscopic gradients have identical

signs (initially overlapping case) or smaller than the

intrinsic value if the macroscopic gradients have

opposite signs (initially non-overlapping reactants).

The effective reaction rate parameter for the bimolec-

ular reaction derived here provides a specific example

of how mixing in a heterogeneous environment

controls the reaction rates. The macro-kinetics them-

selves are dependent on the macroscopic concentration

gradient, because these gradients control the small-

scale concentration fluctuations of the reactants. It is

easy to incorporate the gradient dependent reaction

macro-kinetics in models that routinely evaluate the

mean concentration field and its gradients.

The dependence of the field-scale effective rate param-

eter on the dissipation time-scale underlines the critical

need to theoretically understand the relationship between

the dissipation time-scale and the flow microstructure in

order to couple biotic and abiotic chemical transformations

processes and hydrologic transport process. Not only does

the dissipation time-scale determine dilution and concen-

tration fluctuations, it is also needed to quantify mixing

between different solutes, which controls reactions.

Experimental techniques that could document the dissipa-

tion time-scale also need to be developed. Perhaps reac-

tive-transport experiments can also play a role in

quantifying the dissipation time-scale because of the sen-

sitivity of the mean concentrations to small-scale mixing.

A carefully controlled field experiment with independently

well characterized chemical transformation kinetics and

intense sampling is also needed to critically assess our

ability to couple chemical transformations processes with

hydrologic transport.

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