Estimating Reaction Rate Coefficients Within aTravel-Time Modeling Frameworkby R. Gong1, C. Lu1, W.-M. Wu2, H. Cheng2, B. Gu3, D. Watson3, P.M. Jardine3, S.C. Brooks3, C.S. Criddle2,P.K. Kitanidis2, and J. Luo1,4

AbstractA generalized, efficient, and practical approach based on the travel-time modeling framework is developed

to estimate in situ reaction rate coefficients for groundwater remediation in heterogeneous aquifers. The requiredinformation for this approach can be obtained by conducting tracer tests with injection of a mixture of conservativeand reactive tracers and measurements of both breakthrough curves (BTCs). The conservative BTC is used toinfer the travel-time distribution from the injection point to the observation point. For advection-dominant reactivetransport with well-mixed reactive species and a constant travel-time distribution, the reactive BTC is obtainedby integrating the solutions to advective-reactive transport over the entire travel-time distribution, and then isused in optimization to determine the in situ reaction rate coefficients. By directly working on the conservativeand reactive BTCs, this approach avoids costly aquifer characterization and improves the estimation for transportin heterogeneous aquifers which may not be sufficiently described by traditional mechanistic transport modelswith constant transport parameters. Simplified schemes are proposed for reactive transport with zero-, first-, nth-order, and Michaelis-Menten reactions. The proposed approach is validated by a reactive transport case in atwo-dimensional synthetic heterogeneous aquifer and a field-scale bioremediation experiment conducted at OakRidge, Tennessee. The field application indicates that ethanol degradation for U(VI)-bioremediation is betterapproximated by zero-order reaction kinetics than first-order reaction kinetics.

IntroductionEstimation of contaminant degradation rates is one

of the primary tasks for in situ remediation of contami-nated groundwater. Tracer tests designed for quantifyingin situ reaction rate coefficients often involve the injection

1School of Civil and Environmental Engineering, GeorgiaInstitute of Technology, Atlanta, GA 30332-0355.

2Department of Civil and Environmental Engineering, StanfordUniversity, Stanford, CA 94305-4020.

3Environmental Science Division, Oak Ridge NationalLaboratory, Oak Ridge, TN 37831-6038.

4Corresponding author: School of Civil and EnvironmentalEngineering, Georgia Institute of Technology, Atlanta, GA 30332-0355; (404) 385 6390; fax: (404) 385-1131; jianluo@ce.gatech.edu

Received April 2009, accepted January 2010.Copyright © 2010 The Author(s)Journal compilation © 2010 National Ground Water Association.doi: 10.1111/j.1745-6584.2010.00683.x

of a mixture of conservative and reactive tracers, in whichthe conservative tracer serves as a control (e.g., Reinhardet al. 1997; Schreiber and Bahr 2002; Luo et al. 2006a).By analyzing the breakthrough curves (BTCs) of bothconservative and reactive tracers measured at the sameobservation points, one can estimate transport parametersand reaction rate coefficients. A common approach is tofit the BTCs jointly or subsequently with one- or mul-tidimensional transport models, which generally includethe description of advection, dispersion, sorption, masstransfer, reaction kinetics, etc. (Toride et al. 1993). Dueto insufficient aquifer characterization, transport modelsoften assume uniform, constant transport parameters, andthe reaction kinetics are simplified to zero- or first-orderreactions (Haggerty et al. 1998; Snodgrass and Kitanidis1998). Although the estimated reaction rate coefficientsare lumped parameters, they are still useful to helppractitioners to predict contaminant degradation rates, to

NGWA.org Vol. 49, No. 2–GROUND WATER–March-April 2011 (pages 209–218) 209

approximate effective consumption rates of injected sub-strates and other chemical compounds, and to optimizeexperimental operations. In general, the development ofsuch a mechanistic transport model for field-scale appli-cations requires a sufficient characterization of aquiferheterogeneity, which is costly and may not be available(Semprini and McCarty 1991).

Alternative approaches to avoid aquifer characteriza-tion have been developed to estimate in situ reaction ratecoefficients. The method of temporal moments is an effi-cient one, but it is only available for first-order reactionsand is also associated with a mechanistic transport modelwhich defines the relationships among transport param-eters, reaction rate coefficients and temporal moments(e.g., Kreft and Zuber 1978; Das and Kluitenberg 1996;Luo et al. 2008). The methods developed by Snodgrassand Kitanidis (1998) for estimating zero- and first-orderreaction rates are efficient for directly working on the con-servative and reactive BTCs rather than fitting transportmodels. Thus, no information about aquifer heterogene-ity is required. Haggerty et al. (1998) presented a similarmethod to determine first-order reaction rate coefficients.These methods were developed for “push-pull” tests andhave been successfully applied in practice (e.g., Istok et al.2001; Hageman et al. 2004). Within the context of trans-fer function and travel-time approach, Luo et al. (2006a)and Fienen et al. (2006) applied the method of para-metric and nonparametric transfer functions to estimatefirst-order reaction rate coefficients, respectively. In theirwork, transfer functions of a conservative tracer were usedto infer travel-time distributions from an injection pointto observation points, and first-order reactions occurredthroughout all travel times. Luo et al. (2008) also investi-gated the effects of nonequilibrium mass transfer on esti-mating first-order reaction rate coefficients. The methodsmentioned above are more advantageous than the classi-cal method of fitting mechanistic transport models due tothe minimization of efforts for aquifer characterizations.However, most of the approaches were developed only forsimplified reaction kinetics, such as zero- and first-orderreactions, which may have limited predictive applicabil-ity for complicated reaction kinetics, such as the microbialreaction kinetics (Bekins et al. 1998).

The main objective of the present study is to developan efficient and practical approach to infer more gen-eral reaction rate coefficients in heterogeneous aquifersfor a tracer test with the measurements of BTCs of bothconservative and reactive tracers. The method derived isbased on the travel-time modeling framework (Simmonset al. 1995; Ginn et al. 1995; Cirpka and Kitanidis 2000a;Luo and Cirpka 2008), which allows one to avoid thecharacterization of aquifer heterogeneity by using the con-servative BTC to infer the travel-time probability densityfunction (PDF) and assuming reactions are associatedwith travel times. Simplified approaches are developedfor specific reaction kinetics, including zero-, first-, nth-order reactions, and the Michaelis-Menten reaction, whichare the reaction kinetics mostly assumed in contami-nant transport and remediation practice. Both synthetic

and field cases are presented to validate the developedapproach.

Mathematical Models

Travel-Time Based Transport ModelFollowing the methodology of Simmons et al. (1995),

the travel-time based transport model for advection-dominated conservative transport in an individual streamtube is described by:

∂ct

∂t+ ∂ct

∂τ= 0 (1)

subject to the initial and boundary conditions:

ct (τ, t = 0) = 0, ct (τ = 0, t) = ct0(t) (2)

where ct is the concentration of a conservative tracer, t istime, ct0(t) represents the concentration profile of injectedtracer at the injection location xin, and τ is the travel-timecoordinate along a streamline.

The concentration BTC measured at an observationpoint is considered as the weighted average of anensemble of noninteracting stream tubes passing throughthe observation point. The weights of these stream tubesare described by a travel-time PDF. By integrating thesolutions in all stream tubes over the entire travel-timePDF, one can evaluate the BTCs at an observation point.

ct (xout, t) =∫ ∞

0ct (τ, t)f (τ |xin, xout)dτ (3)

where xout is the location of the observation point,and f (τ |xin, xout) represents the travel-time PDF at xout,describing the distribution of stream tubes.

Equation (1) has the solution:

ct (τ, t) = ct0(t − τ ) (4)

which indicates that the concentration of a conservativetracer in a stream tube is equal to a shifted inputconcentration by its travel time τ , that is, the streamtube acts as a plug-flow reactor. Thus, the BTC ofa conservative tracer at the observation point xout isgiven by:

ct (xout, t) =∫ ∞

0ct0(t − τ)f (τ |xin, xout)dτ (5)

which is the well-known convolution principle for time-invariant linear systems (e.g., Jury 1982). ct0 is theinput function, and the travel-time PDF f (τ |xin, xout)

is the transfer function, which is the response functioncorresponding to a unit impulse input function. The travel-time based model is distinguished from the classicalmechanistic transport model by the assumption that eachstream tube is physically independent, and both diffusionand pore-scale dispersion are described by the travel-timedistribution, but neglected along the streamline (Jury

210 R. Gong et al. GROUND WATER 49, no. 2: 209–218 NGWA.org

and Fluhler 1992). Thus, the travel-time based model isparticularly effective for advection-dominated transport,that is, the travel-time distribution is primarily caused byheterogeneous advection.

The conservative tracer BTC corresponding to a unitimpulse input may be considered as the travel-time PDFfrom the tracer injection point to the observation point. Forother input functions, the travel-time PDF can be obtainedby deconvoluting the conservative BTC or assuming aparametric distribution function, such as inverse Gaussian(Rao et al. 1981, Cirpka and Kitanidis 2000b), lognormal(Simmons 1982), or gamma distributions (Loaciga 2004;Luo et al. 2006a). The distribution of τ is specific to theinflow boundary, the flow field, the aquifer heterogeneityand transport processes. For example, in a dipole flowfield, the distributed travel times of a conservative tracerfrom the injection well to the extraction well are primarilycaused by nonuniform advection (Luo et al. 2007), whilethe travel-time distribution at an observation point withsmall sampling volumes is primarily controlled by localdispersion or mass transfer (Cirpka and Kitanidis 2000b;Luo and Cirpka 2008).

For advection-dominated reactive transport, theaquifer is assumed chemically homogeneous, that is,the reaction rate coefficient is not a function of spatialcoordinates, and the reactive tracer experiences the samephysical transport processes as the conservative tracer.Similarly, the travel-time based transport model for areactive tracer in a stream tube can be written as:

∂cr

∂t+ ∂cr

∂τ= r(τ, t, cr) (6)

subject to initial and boundary conditions:

cr(τ, 0) = 0, cr(τ = 0, t) = cr0(t) (7)

where cr is the concentration of a reactive tracer, cr0(t)

represents the concentration profile of injected reactivetracer at the injection point xin, and r(τ, t, cr) representsthe reactive rate. In this study, we assume that the reac-tion rate depends only on the species concentrations, thetime, and the travel time, rather than the spatial coordi-nates. The reaction rate can also involve more than onechemical species and sorption, such as bioreactive systemsinvolving electron donor, electron acceptor, and biomass,in which transport models need to be developed for eachspecies (Ginn 2001; Cirpka and Kitanidis 2000a). Chem-ical heterogeneity may also be included provided withthe relationship between travel times and spatial coordi-nates (Crane and Blunt 1999). However, if the travel-timePDF is the only known information at the observationpoint, the assumption of chemical homogeneity is neces-sary for applying this model because of insufficient infor-mation to establish the relationship between travel timesand spatial coordinates. More importantly, the travel-timemodel described by Equation (6) assumes that all reactivespecies are completely mixed during transport or premixedbefore injection into the subsurface. This assumptionis particularly valid for engineered remediation systems

involving injection and extraction wells, where advec-tion is the dominant transport process due to enhancedgradients and chemical species are premixed before injec-tion into pumping wells (Cirpka and Kitanidis 2000a).For mixing controlled by transport processes, such as dis-persion and mass transfer, additional mixing descriptionsmust be included (Cirpka and Kitanidis 2000b; Ginn et al.2001; Luo and Cirpka 2008).

The average BTC of a reactive tracer at the observa-tion point xout can be evaluated by:

cr(xout, t) =∫ ∞

0cr(τ, t)f (τ |xin, xout)dτ (8)

Equation (6) is a first-order semilinear equation andcan be solved using the method of characteristics.Simmons et al. (1995) summarized the solution approachpropagating from boundary and initial conditions. Tofacilitate its application, we present several solutions forreaction kinetics often considered in practical applications.

Table 1 summarizes the solutions to Equation (6)for zero-, fist-, nth-order, and Michaelis-Menten reactionkinetics. k0, k1, and kn are the reaction rate constants.For reactions with exponent less than 1, the solutionmay yield negative concentrations after certain periods.Thus, the solution must be enforced to be nonnegative.For the Michaelis-Menten reaction kinetics, λ is a rateconstant, and K is the concentration giving one-half themaximum rate. The Michaelis-Menten reaction kineticsmay be used to describe bioreaction kinetics with con-stant biomass and without limitations from other reactivespecies. When K � cr , the reaction rate is approximatelyzero order and when K � cr , it becomes first order. W

represents the Lambert W-function, also called the omegafunction (Corless et al. 1996), which is the inverse func-tion of F(W) = WeW .

Reaction Kinetics EstimationThe proposed estimation approach based on travel-

time modeling framework is similar to the general methodof fitting BTCs of conservative and reactive tracers byone- or multidimensional mechanistic transport models.However, the transport model here is described by thetravel-time PDF instead of by the mechanistic transportprocesses. The general inversion algorithm for estimatingreaction rate coefficients can be summarized as follows:

Step 1 Estimating travel-time PDF f (τ |xin, xout) from theinjection point xin to the observation point xout

based on the BTC of a conservative tracer. Fora unit impulse input, the conservative BTC isthe travel-time PDF. For other input modes, thetravel-time PDF can be obtained by deconvolutingthe conservative BTC (Luo et al. 2006a; Fienenet al. 2006).

Step 2 Discretizing the travel-time PDF into a number ofnoninteracting stream tubes with assigned traveltimes and weights.

NGWA.org R. Gong et al. GROUND WATER 49, no. 2: 209–218 211

Table 1Reaction Kinetics and Solutions in an Individual Stream Tube

Reaction Reaction Rate r(τ , t , cr ) Solution

Zero order −k0

cr (τ , t) =

⎧⎪⎪⎨⎪⎪⎩

−k0τ + cr0(t − τ ), τ ≤ cr0(t − τ )

k0

0, τ >cr0(t − τ )

k0

First order −k1cr cr (τ , t) = cr0(t − τ ) exp(−k1τ )

cr (τ , t) =[−knτ (1 − n) + c1−n

r0 (t − τ )] 1

1−n , n > 1

nth order −kn cnr

cr (τ , t) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

[−k0τ (1 − n) + c1−n

r0 (t − τ )] 1

1−n , τ ≤ c1−nr0 (t − τ )

k0(1 − n)

0, τ >c1−n

r0 (t − τ )

k0(1 − n)

, 0 < n < 1

Michaelis-Menten −λ crcr +K cr (τ , t) = KW

{cr0(t−τ )

K exp(

−λK τ + cr0(t−τ )

K

)}

Step 3 Making initial guess of the reaction rate coeffi-cients and solving the travel-time based reactivetransport model to obtain cr(τ, t).

Step 4 Evaluating the weighted average concentrations ofreactive tracers at xout, that is, the reactive BTC,by integrating cr(τ, t) over the travel-time PDF.

Step 5 Minimizing the mean squared error between themodel simulation and the measurements by opti-mizing the parameters of reaction rate coefficients(software such as PEST).

As discussed in the previous section, there are sev-eral assumptions to be satisfied in order to apply thealgorithm presented above: (1) reactive species are wellmixed during transport and (2) reactive tracer experiencesthe same physical transport processes as the conservativetracer. The advantages of the proposed algorithm include:(1) travel-time distributions (both parametric and nonpara-metric) have the flexibility to capture anomalous concen-tration measurements, such as heavily tailing, multimodalconcentrations, which may not be described by classicalmechanistic transport models, in the absence of thoroughcharacterizations of aquifer heterogeneity in a spatiallyexplicit form; (2) travel-time distributions can be derivedbased on concentration BTCs, which can be convenientlyevaluated through tracer tests (Simmons 1982; Simmonset al. 1995); and (3) transport in a travel-time domainbecomes one-dimensional (1D) with a uniform “velocity”whereas traditional spatial models describe transport ina multidimensional domain with potentially highly vari-able velocity (Simmons et al. 1995; Crane and Blunt1999; Cirpka and Kitanidis 2000a). In fact, provided thatthe reaction rate coefficients are not spatial functions,only one 1D advective-reactive transport equation, that is,Equation (6), needs to be solved (Nauman and Buffham

1983; Cirpka and Kitanidis 2000a). In addition, if theshape of the travel-time PDF is known, for example, aparametric distribution with unknown parameters, decon-volution of the conservative BTC in Step 1 can be avoidedfor the estimation of the distribution parameters and reac-tion rate coefficients by jointly fitting the conservativeand reactive BTCs. The inversion algorithm can be mod-ified to:

Step 1 Making initial guess of distribution parameters andreaction rate coefficients.

Step 2 Calculating the travel-time PDF f (τ |xin, xout)

based on the known distribution function andestimated parameters from Step 1.

Step 3 Discretizing the travel-time PDF into a number ofnoninteracting stream tubes with assigned traveltimes and weights.

Step 4 Solving the travel-time based reactive transportmodel to obtain cr(τ, t).

Step 5 Evaluating the weighted average concentrationsof both conservative and reactive tracers at xout,that is, the conservative and reactive BTCs, usingEquations (5) and (8), respectively.

Step 6 Minimizing the mean squared error between themodel simulations and the conservative and reac-tive tracer measurements by optimizing the param-eters of distribution function and reaction ratecoefficients.

Simplified Algorithm for Specific ReactionKinetics

The algorithm presented above is general and validfor an arbitrary r(τ, t, cr). In the following, we will

212 R. Gong et al. GROUND WATER 49, no. 2: 209–218 NGWA.org

present simplified algorithms for the four specific casessummarized in Table 1. Numerical cases are presented inthe supporting information to demonstrate their applica-tions. Assume that the conservative and reactive tracersare well mixed before injection and the input mixtureconcentrations of conservative and reactive tracers havethe relationship: cr0(t)/ct0(t) = α, where α is a scalingfactor.

Zero-Order ReactionBecause of the nonnegative constraint, the solution

in Table 1 needs to be evaluated individually for eachstream tube for an arbitrary input. Thus, the generalalgorithm summarized in the previous section should beapplied to estimate the zero-order reaction rate coeffi-cient. However, for an impulse input function, simplifiedapproaches can be developed. For an impulse input, thatis, cr0(t) = cr0δ(t) and ct0(t) = ct0δ(t), where δ is thedelta function, one can observe nonnegative measure-ments for the reactive BTC until T = cr0/k0. Thus, thesimplest method to estimate k0 is k0 = cr0/T . In addi-tion, substituting the impulse inputs and the solution inTable 1 into Equations (5) and (8) yields the conservativeand reactive BTCs:

ct (xout, t) =∫ ∞

0ct0δ(t − τ )f (τ )dτ = ct0f (t) (9a)

cr(xout, t) =∫ ∞

0[cr0 − k0τ ]δ(t − τ )f (τ )dτ

= (cr0 − k0t)f (t), t < cr0/k0 (9b)

Thus, the relationship between ct and cr can beexpressed as:

ct0cr(xout, t)

ct (xout, t)− cr0 = −k0t (10a)

or

k0 = − d

dt

[cr(xout, t)

f (t)

](10b)

The scaled concentration difference ct0cr(xout, t)/

ct (xout, t) − cr0 or the normalized concentration cr(xout, t)/

f (t) is a linear function of time. By plotting the scaledconcentration difference or the normalized concentrationvs. time, one can approximate k0 by the slope. Note thelinear relationship predicted by Equation (10) is differentfrom the result derived by Snodgrass and Kitanidis (1998)due to the different ways to treat the travel-time distribu-tion. The latter considered the travel-time distribution asa result of dispersion, while Equation (10) is based on theintegration of a number of advective stream tubes. Numer-ical cases are presented in the supporting information todemonstrate the application of Equation (10).

First-Order ReactionSubstituting the solution to the first-order reaction in

Table 1 into Equation (8) yields:

cr(xout, t) = α

∫ ∞

0ct0(t − τ ) exp(−k1τ )f (τ |xin, xout)dτ

(11)

Reactive transport with a first-order reaction can beconsidered as a linear system with a transfer functionexp(−k1τ )f (τ |xin, xout). Thus, for a unit impulse input,the reaction rate constant can be evaluated directly usingthe BTCs:

k1 = − d

dtln

cr(xout, t)

ct (xout, t)(12)

For arbitrary inputs, both conservative and reactiveBTCs can be deconvoluted to obtain the transfer functions,which can subsequently be substituted into Equation (12)to estimate the first-order reaction rate constant. Bothparametric and nonparametric methods are available toobtain the transfer functions (Luo et al. 2006a; Fienenet al. 2006).

nth-Order ReactionFor nth-order nonlinear reaction (n �= 1), the general

algorithm should be applied to estimate kn. It should benoticed that the solution in Table 1 always gives non-negative concentrations for n > 1, while the nonnegativeconstraint needs to be included for 0 < n < 1. If we char-acterize the transport from xin to xout as a stream tubewith a mean travel time 〈τ 〉, the solution in Table 1 canbe written as:

cr(xout, t) = [−kn〈τ 〉(1 − n) + α1−nc1−nt0 (t − 〈τ 〉)] 1

1−n

(13)

Thus, we have:

kn = α1−nc1−nt (xout, t) − c1−n

r (xout, t)

〈τ 〉(1 − n)(14)

Equation (14) is valid for the case without thenonnegative constraint, that is, n > 1. The numeratorin Equation (14) may also be considered as a scaledconcentration difference between the conservative andreactive tracers for an nth-order reaction. Equation (14)indicates that the scaled concentration difference isconstant at an observation point with a constant traveltime. In addition, if the concentration profile is available atdifferent locations along a stream tube, that is, at differenttravel times, the scaled concentration difference at thesame measurement time should be linearly dependent onthe travel time with the slope kn, that is,

kn = 1

1 − n

d[α1−nc1−nt (xout, t) − c1−n

r (xout, t)]

d〈τ 〉 (15)

NGWA.org R. Gong et al. GROUND WATER 49, no. 2: 209–218 213

Michaelis-Menten Reaction KineticsSimilarly, the general algorithm can be applied to

estimate the reaction rate constant λ for a Michaelis-Menten reaction kinetics. By approximating the transportas 1D advective transport with mean travel time 〈τ 〉, thesolution in Table 1 can be written as:

cr(xout, t)

K+ ln

cr(xout, t)

K

=[

αct (xout, t)

K+ ln

αct (xout, t)

K

]− λ

K〈τ 〉 (16)

Thus,

K(ln cr(xout, t) − ln[αct (xout, t)])

= −cr(xout, t) + αct (xout, t) − λ〈τ 〉 (17)

The rate constant λ can then be evaluated by:

λ =[αct (xout, t) − cr(xout, t) + K ln αct (xout, t)

−K ln cr(xout, t)]

〈τ 〉 (18)

In addition, Equation (17) represents a linear functionwith variables −cr(xout, t) + αct (xout, t) and ln cr(xout, t)

− ln[αct (xout, t)]. Thus, for estimating both K and λ, onemay obtain:

K = − d[cr(xout, t) − αct (xout, t)]

d(ln cr(xout, t) − ln[αct (xout, t)])(19)

λ =[αct (xout, t) + K ln αct (xout, t)]−[cr(xout, t) + K ln cr(xout, t)]

〈τ 〉 (20)

Moreover, if the BTCs at different observation pointsare available, that is, with different travel times, λ may beestimated by:

λ =d[αct (xout, t) − cr(xout, t) + K ln αct (xout, t)

−K ln cr(xout, t)]

d〈τ 〉 (21)

Note that the simplified algorithms need the estima-tion of the mean travel time 〈τ 〉, which is dependent uponthe complete measurements of the conservative BTC (Luoet al. 2006b).

ApplicationsBoth numerical and field cases are presented to

demonstrate the applications of the proposed algorithmin estimating reaction rate coefficients. In the supportinginformation, we present a bioreactive transport system in a2D heterogeneous aquifer. The reactive kinetics involvesthe growth and decay of biomass. Transport equations aresolved using numerical methods based on spatial coordi-nates to generate BTCs of both conservative and reactivetracers, which are then assumed to be the only informa-tion we have for estimating the reaction rate coefficients.In the following, we apply the proposed approach to anengineered in situ bioremediation experiment of U(VI)-contaminated groundwater implemented in an inter-wellsystem at Oak Ridge, Tennessee, for estimating zero- andfirst-order reaction rate coefficients of injected substrates.

Field ApplicationA multiple-well system (Figure 1) was installed to

investigate the in situ bioremediation feasibility of U(VI)-contaminated groundwater at the Field Research Center

Figure 1. The multiple-well system installed at Oak Ridge, Tennessee, for in situ remediation of U(VI)-contaminatedgroundwater. The well system includes two injection (FW024 and FW104) and two extraction wells (FW026 and FW103),and three MLS wells. The dashed lines with arrows indicating flow directions are the streamlines within the nested inner cell,and the solid lines are the streamlines in the outer flow cell.

214 R. Gong et al. GROUND WATER 49, no. 2: 209–218 NGWA.org

of the U.S. Department of Energy Environmental Reme-diation Science Program at the Oak Ridge Reservationin Oak Ridge, Tennessee (Luo et al. 2006c). The result-ing flow field is divided into a nested inner recirculationcell, a transition zone between the inner and outer wellpairs, and an outer protection cell that separates the nestedinner cell from exterior regional flow (Luo et al. 2006c).In situ bioremediation experiments were conducted in thenested inner cell by intermittently injecting ethanol as anelectron donor to enhance microbial activity in the subsur-face leading to biochemical reduction and immobilizationof uranium within the aquifer (Wu et al. 2006a, 2006b,2007). Tracer tests were conducted in the inner cell byinjecting a mixture of bromide and ethanol to evaluatethe consumption rate of injected electron donors. Sam-ples for the measurement of bromide and ethanol werewithdrawn at the injection well FW104, selected multi-level sampling (MLS) well ports and the extraction wellsFW026 and FW103. More details about the tracer tests,well pumping manipulation, and analytical methods forthe analysis of the collected samples have been describedelsewhere (Gu et al. 2005; Wu et al. 2006a; Fienen et al.2006; Luo et al. 2007, 2008).

Figure 2 shows the measured input concentrations ofbromide and ethanol at the injection well, FW104. Dur-ing the period of injection of the mixture of bromideand ethanol, no large concentration deviation (see normal-ized concentrations shown in Figure 2b) was observed atthe injection well because tracer injection dominated themeasurements of concentrations. The gradual increase inconcentrations instead of a step function was observed atthe injection well mainly due to the incomplete mixingin the well and slow release from the well to the aquifer.After the injection was terminated, ethanol concentrationsdecay faster than bromide due to the biodegradation, that

is, a much longer concentration tail was observed forbromide.

We apply the general algorithm summarized in theprevious section to estimate the ethanol consumption ratesat MLS wells. The bromide BTCs are first fitted byassuming inverse Gaussian travel-time PDFs, which aresubsequently accepted by travel-time based models witheither zero- or first-order reaction kinetics to fit the ethanolBTCs. Ethanol is assumed not to undergo retardation(Tomich et al. 1973) so that ethanol and bromide have thesame travel-time PDFs. The goodness of fit of simulatedresults to the measured ethanol data is quantified by theroot mean square error (RMSE):

RMSE ={

1

nEtOH − m

×[nEtOH∑

i=1

(cmodel(EtOHi ) − cmeas(EtOHi ))2]}1/2

(22)

where nEtOH is the number of measurement points ofethanol concentrations; m is the number of fitting coef-ficients, which is equal to 1 for fitting zero- or first-order reaction rate coefficients; and cmodel and cmeas arethe normalized modeled and measured concentrations,respectively.

Figure 3 shows the bromide and ethanol measure-ments and model fitted results at two MLS wells.Figure 3b shows the estimated travel-time distributions,which give good fitting results for the bromide BTCs atboth MLS wells (Figure 3a). The estimated travel-timedistributions are then used by the travel-time based modelwith zero- and first-order reactions to fit the ethanol BTCs(Figure 3c and d). Both zero- and first-order reaction

0 10 20 30 40 50 60 70 80 90 1000

100

200

300

400

Time [h]

c [m

g/L]

BrEthanol

0 10 20 30 40 50 60 70 80 90 1000

0.5

1

Time [h]

c [m

g/L]

BrEthanol

(a) Measured concentrations

(b) Normalized concentrations

Figure 2. Input bromide and ethanol concentrations measured at the injection well FW104. (a) Measurements and(b) concentrations normalized by the maximum injection concentrations.

NGWA.org R. Gong et al. GROUND WATER 49, no. 2: 209–218 215

0 10 20 30 40 500

0.1

0.2

0.3

0.4

Time [h]

f(τ)

0 20 40 60 80 1000

50

100

150

200

250

Time [h]

c Br [m

g/L]

0 5 10 15 200

50

100

150

Time [h]

c EtO

H [m

g/L]

0 5 10 15 200

50

100

150

Time [h]

c EtO

H [m

g/L]

FW1012: measurementsFW1012: modelFW1023: measurementsFW1023: model

FW1012FW1023

FW1012: measurementsFW1012: modelFW1023: measurementsFW1023: model

FW1012: measurementsFW1012: modelFW1023: measurementsFW1023: model

(a) Bromide fitting (b) Estimated travel timedistributions

(c) Zero order reaction (d) First order reaction

Figure 3. Measured and fitted bromide and ethanol BTCs at two MLS wells (FW101-2 and FW102-3). “-2” and “-3” arethe sampling port numbers, representing depth 45 and 40 feet, respectively. (a) Bromide BTCs and fitting; (b) estimatedtravel-time distributions; (c) ethanol BTC fitting by zero-order reaction kinetics; and (d) ethanol BTC fitting by first-orderreaction kinetics.

Table 2Estimated Parameters of Travel-Time PDFs, Zero- and First-Order Reaction Rate Coefficient, and RMSE

MLS Wells Mean Travel Time (h) Variance (h2) k0 (mg/L/h) k1(1/h) RMSE

FW101-2 6.3 44.1 10.5 0.08 7.3 (for k0)12.9 (for k1)

FW102-3 8.0 291 14.5 0.11 6.0 (for k0)8.5 (for k1)

kinetics give good fitting results. However, the zero-orderkinetics show much better fitting for the BTC tails.

Table 2 summarizes the estimated parameters and theRMSEs for zero- and first-order reaction kinetics. Thefirst-order reaction rate coefficients are consistent to pre-vious findings by deconvoluting the BTCs (Fienen et al.2006; Luo et al. 2006a). Estimating first-order reactionrate coefficients is simpler than estimating zero-order reac-tion kinetics because first-order reaction kinetics is linearand zero order is nonlinear. Also, the first-order reactionrate coefficient can be used to evaluate the half life of thereactants, which is popular in groundwater remediationapplications because it gives practitioners the intuitive ofhow fast the reactants decay. However, RMSEs indicatethat the fitting performance of zero-order reaction kinet-ics is better than that of first-order reaction kinetics forthe present case. This finding is consistent to the fact that

the half saturation coefficient for ethanol is small and theMichaelis-Menten reaction for ethanol degradation maybe approximated by zero-order reaction kinetics (Nymanet al. 2007).

Summary and ConclusionsAn efficient, practical method within a travel-time

modeling framework is presented to estimate in situreaction rate coefficients for groundwater remediation.Within the travel-time modeling framework, multidimen-sional transport on spatial coordinates can be describedby transport in noninteracting stream tubes with assignedtravel times and weights according to the travel-time PDF,which may be conveniently evaluated from conservativetracer BTCs by applying deconvolution techniques. Sim-plified schemes are developed for zero-, first-, nth-order

216 R. Gong et al. GROUND WATER 49, no. 2: 209–218 NGWA.org

reactions and the Michaelis-Menten reaction. For first-order reactions, the reactive transport is a linear time-invariant system for steady-state flow fields. Thus, thefirst-order reaction rate coefficient is the slope of the curveof − ln(cr/ct ) vs. time for impulse response functions. Forzero-order reactions, the scaled concentration differencect0cr(xout, t)/ct (xout, t) − cr0 or the normalized concen-tration cr(xout, t)/f (t), where f is the travel-time distri-bution or the transfer function, is a linear function of time.For nth-order reactions with n > 1, the scaled concen-tration differences become α1−nc1−n

t (x, t) − c1−nr (x, t),

which are constant for advection-dominant problems. ForMichaelis-Menten reactions in advection-dominant trans-port, one can conveniently obtain the reaction constant λ

by plotting [αct (xout, t) − cr(xout, t) + K ln αct(xout, t) −K ln cr(xout, t)] vs. time. These schemes can also serveto determine the reaction type, that is, one may treat themeasurements following the derived methods to determinewhich reaction type fits best. For joint estimation of thereaction order and rate coefficient, the general algorithmneeds to be applied.

The developed approach avoids the need for thor-oughly characterizing aquifer heterogeneities and per-forming multidimensional advective-dispersive-reactive(ADR) simulations by directly working on the BTCs ofboth conservative and reactive tracers, this method. Therequired information for applying this method can beobtained by conducting a tracer test with injection of bothconservative and reactive tracers and taking measurementsat both injection and observation wells. In addition, forlong-term in situ experiments, flow fields may be influ-enced by ongoing reactions, which may make it question-able for applying mechanistic transport models based oninitial aquifer characterizations over the long-term period(Luo et al. 2008). By contrast, the proposed method basedon travel-time distributions, which can be updated bytracer tests, offers more flexibility to estimate reactionrates for a long-term reactive system. The assumptionof the proposed method is that both tracers experiencethe same advection-dominated transport processes exceptthe reactive one undergoes reactions which are associatedwith travel times. For reactive transport controlled mix-ing processes, that is, dispersion and kinetic mass transfer,more sophisticated travel-time models need to be devel-oped to incorporate the description of mixing processes(Luo and Cirpka 2008). In addition, the development ofthe general algorithms and simplified methods for sim-ple reaction kinetics lead to another important researchtopic, that is, under what condition can complex reactionkinetics be described by simple kinetics, such as zero- andfirst-order reactions. This research is under way.

AcknowledgmentsThis research was sponsored by the U.S. Department

of Energy (DOE) Office of Science, Biological andEnvironmental Research, as part of the Integrated FieldResearch Challenge (IFRC) at Oak Ridge NationalLaboratory (ORNL). ORNL is managed by UT-Battelle,

LLC, for DOE under Contract DE-AC05-00OR22725.The authors thank Dr. O.A. Cirpka for providing theoriginal streamline-based numerical codes, Dr. M. Fienenand Ms. T. Mehlhorn for analytical support. We alsothank Dr. L. Semprini, Dr. M. Riley, and an anonymousreviewer for their constructive comments on this work.

Supporting InformationSupporting Information include numerical cases for

the developed simplified algorithms and a bioreactivetransport case in a 2D heterogeneous medium.

Figure S1. BTCs of conservative and reactive tracersfor a zero-order reaction constant k0 = 2 mg/L-day withrespect to impulse inputs.

Figure S2. Estimation of the zero-order reaction rateconstant. (a) Scaled concentration difference proposed bySnodgrass and Kitanidis (1998); and (b) scaled concen-tration developed in the present work.

Figure S3. (a) BTCs of conservative and reactivetracers for a first-order reaction constant k0 = 1/daywith respect to Dirac inputs. (b) Parameter estimation byplotting − ln(cr/ct) versus time.

Figure S4. (a) BTCs of conservative and reactivetracers for a second-order reaction constant k2 = 0.1L/mg/day with respect to a step-like input. (b) Parameterestimation by plotting 1/cr-1/ct versus time.

Figure S5. Estimation of the second-order reactionrate constant versus the Peclet number.

Figure S6. (a) BTCs of conservative and reactivetracers for a Michaelis-Menten reaction constant λ =10/day with respect to a step-like input. (b) Parameterestimation.

Figure S7. Estimation of the reaction rate constant lfor a Michaelis-Menten reaction versus the Peclet number.

Figure S8. 2D Heterogeneous hydraulic conductivityfield (log10K), contour lines of hydraulic heads (dashedlines), and streamlines (solid lines).

Figure S9. (a) Travel-time PDF at the outflowboundary and the parametrization by 1-D AD model.(b) Reactive BTC and the fitting results of 1-D ADRmodel and travel-time model.

Please note: Wiley-Blackwell are not responsible forthe content or functionality of any supporting materialssupplied by the authors. Any queries (other than missingmaterial) should be directed to the corresponding authorfor the article.

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