Bioremediation Journal, 8(1–2):47–64, 2004Copyright ©c 2004 Taylor and Francis Inc.ISSN: 1040-8371DOI: 10.1080/10889860490453177

Modeling of DNAPL-Dissolution,Rate-Limited Sorption and Biodegradation

Reactions in Groundwater Systems

T. Prabhakar ClementDepartment of Civil Engineering,Auburn University, Auburn,Alabama, USA

Tirtha R. GautamDepartment of Civil Engineering,Auburn University, Auburn,Alabama, USA; Center for WaterResearch, University of WesternAustralia, Perth, Australia

Kang Kun LeeSchool of Earth andEnvironmental Science, SeoulNational University, Korea

Michael J. TruexPacific Northwest NationalLaboratory, Richland,Washington, USA

Greg B. DavisCSIRO Land and Water,Perth, Australia

ABSTRACT This article presents an approach for modeling the disso-lution process of single component dense non-aqueous phase liquids(DNAPL), such as tetrachloroethene and trichloroethene, in a biolog-ically reactive porous medium. In the proposed approach, the overalltransport processes are conceptualized as three distinct reactions. Firstly,the dissolution (or dissolving) process of a residual DNAPL source zoneis conceptualized as a mass-transfer limited reaction. Secondly, the con-taminants dissolved from the DNAPL source are allowed to partitionbetween sediment and water phases through a rate-limited sorption reac-tion. Finally, the contaminants in the solid and liquid phases are allowedto degrade by a set of kinetic-limited biological reactions. Although allof these three reaction processes have been researched in the past, lit-tle progress has been made towards understanding the combined effectsof these processes. This work provides a rigorous mathematical modelfor describing the coupled effects of these three fundamental reactivetransport mechanisms. The model equations are then solved using thegeneral-purpose reactive transport code RT3D (Clement, 1997).

KEYWORDS NAPL dissolution, rate limited sorption, biodegradation, solute trans-port, chlorinated solvent

INTRODUCTION

Chlorinated solvents (e.g., tetrachloroethene [PCE] and trichloro-ethene [TCE]) are commonly observed soil and groundwater contam-inants (Wiedemeier et al., 1999). Because of their physical properties,PCE and TCE can contaminate the environment as non-aqueous phaseliquids (NAPL). Since chlorinated compounds are denser than water theyare further classified as DNAPL. If sufficient contaminant mass is dis-charged at a hazardous waste site, the DNAPL products can rapidly mi-grate into the deeper regions of the saturated groundwater zone, contam-inating large volumes of the aquifer. Within this contaminated region,

Address correspondence toT. Prabhakar Clement, Department ofCivil Engineering, Auburn University,Auburn, AL 36849, USA.E-mail: clement@eng.auburn.edu

47

FIGURE 1 Conceptual representation of various rate-limited reactive transport processes considered in the numerical model.

the DNAPL will eventually be trapped in the form ofblobs. These trapped DNAPL-contaminated regionscan act as long-term residual sources.

Figure 1 is a conceptual model that illustratesvarious contaminant transport processes occurringwithin a DNAPL contaminated aquifer. The im-portant fate-and-transport processes identified inthe conceptual model include: (1) DNAPL dissolu-tion processes, which will control the rate at whichthe contaminant mass dissolves from the trappedDNAPL into the surrounding groundwater; (2) sorp-tion processes, which will control the rate at whichthe dissolved contaminants will partition on to thesurrounding soil material; and (3) biodegradation re-action processes, which will control the rate of con-taminant transformation. At several contaminatedsites, these physical, chemical, and biological pro-cesses can be mediated by kinetically-controlled,rate-limiting mechanisms.

Designing a remediation system for treating thegroundwater systems contaminated with DNAPLproducts would first require a thorough understand-ing of the dissolution kinetics of the DNAPL source.DNAPL dissolution is a complex process that canbe influenced by several factors. Miller et al. (1990)concluded that the inter-phase mass transfer rate ofcontaminants from an NAPL source depends on atleast ten non-dimensional parameters, which are inturn functions of several other flow and transport pa-rameters. Unfortunately, the experimental researchrequired to quantify the system in terms of all ofthese parameters has not been accomplished even

under ideal laboratory conditions. Therefore, in mostpractical field problems, if an NAPL source is knownor suspected to be present in the system, a partition-ing relationship is often used to model the source re-gion having concentrations close to the saturations(e.g., Schafer and Therrien, 1995) or by fixing thesource zone concentrations at those levels observedin the field (e.g., Burnell, 2002). Others have alsoused a set of imaginary injection wells to recreate ob-served source concentration levels (e.g., Rifai et al.,1988). However, none of these fixed source-zonemodels may be valid for sources that are expectedto deplete considerably over the simulation period.Furthermore, field studies have shown that the dis-solved phase concentrations near an NAPL sourcezone can be significantly lower than the expectedsolubility level (e.g., Clement et al., 2002), therebyindicating that the dissolution of NAPL is often arate-limited exchange process.

As illustrated in Figure 1, the dissolved contam-inant mass transferred from a DNAPL source zonemay partition between the sediment and water phasesdue to sorption reactions. Sorption is often mod-eled using the retardation approach, which assumesan equilibrium relationship between solid and liquidphases. The retardation approach for modeling sorp-tion may provide an adequate description for mod-eling transport in slow natural groundwater flow sys-tems. However, when considerable external pumpingand injection stresses are imposed on the system (e.g.,using a pump- and-treat system), then the system maydiverge from the equilibrium transport conditions

48 CLEMENT ET AL.

(Harvey et al., 1994). Under these conditions, non-equilibrium effects such as the plume tailing effect(where low persistent contaminant levels are ob-served at the pumping well for long periods) andthe rebounding effect (where a “remediated” aquifershows considerable increase in contaminant concen-tration levels when the pumps are shut down) arecommonly observed. These non-equilibrium effectscannot be simulated using retardation models. Theycan, however, be modeled using mass-transfer limitedsorption models, where the contaminant mass ex-change between the soil and groundwater is assumedto be a rate-limited kinetic process (e.g., Clementet al., 1998; Haggerty and Gorelick, 1994; Zhang andBrusseau, 1999).

Recent field studies have shown that natural mi-crobial populations present in groundwater aquifershave the ability to degrade several anthropogenicchemicals including the chlorinated solvents (e.g.,McCarty, 1997; Lu et al., 1999; Clement et al.,2000). These observations have led to the develop-ment of various types of bioremediation technolo-gies for cleaning contaminated groundwater systems.Application of bioremediation processes to reme-diate contaminated sites includes two distinct ap-proaches: the passive bioremediation approach andthe active bioremediation approach. The passive ap-proach is also known as natural attenuation or mon-itored natural attenuation. The natural attenuationapproach is a long-term management strategy, whichrelies on the natural assimilative capacity of the sys-tem to control contaminant migration and supportsite-specific remediation goals. The active bioremedi-ation approach, on the other hand, is an acceleratedcleanup strategy which is usually accomplished byenhancing the activities of an indigenous or a non-indigenous microbial population within the contam-inated region (e.g., Truex, 1995; Duba et al., 1996).Both passive and active bioremediation systems aremediated by rate-limited microbial degradation reac-tions (Hooker et al., 1998; Clement et al., 2000).

Determining the feasibility of a bioremediationsystem at chlorinated solvent-contaminated siteswith complex hydrogeological conditions will re-quire simulation of a set of coupled rate-limitedprocesses that describe the effects of DNAPL-dissolution, sorption and biodegradation reactions.Although all of these three rate-limited processeshave been researched in the past, little progress hasbeen made towards understanding the combined ef-

fects of these processes. The objective of this workis to develop a mathematical model for predict-ing the combined effects of NAPL-dissolution, rate-limited sorption, and biological reactions in a sat-urated porous medium. The coupled model equa-tions are solved by the reactive transport code RT3D(Clement, 1997; Clement et al., 1998). The modelwas used to simulate a test problem to demon-strate the use of the approach for describing thebio-attenuation patterns of contaminants emanatingfrom a residual DNAPL source. A detailed simula-tion analysis was also completed to quantify the ef-fects of parameter variations.

MODEL DEVELOPMENT

In this section, a kinetic modeling framework is de-veloped to describe the fate and transport processesoccurring at a site contaminated with PCE-DNAPLproducts. PCE is selected as the DNAPL specie be-cause it is a widely used industrial solvent; further,there are multiple reported cases of soil and ground-water contamination problems associated with PCEspills (Wiedemeier et al., 1999). Also, the physi-cal and chemical properties of PCE are well under-stood since it is a commonly used DNAPL speciesin laboratory-scale column experiments (e.g., Brad-ford and Abriola, 2001). PCE can be biologicallydechlorinated under anaerobic conditions via the fol-lowing sequential reaction chain (McCarty, 1997):PCE → Trichloroethene (TCE) → Dichloroethene(DCE) → Vinyl Chloride (VC) → Ethene. PCE andits biotransformation products are regulated con-taminant species; therefore, remediation of PCE-contaminated sites requires careful consideration ofthe fate of PCE and all its degradation products.

The details of various reactive transport pro-cesses considered in this study are illustrated inFigure 1. As shown in the figure, three coupled re-action mechanisms, which include DNAPL dissolu-tion, rate-limited sorption, and biodegradation, areconsidered. The following three sections summarizethe mathematical descriptions used for modelingDNAPL dissolution, rate-limited sorption, and bi-ological processes, respectively.

Modeling of DNAPL Dissolutionand Solute Transport Processes

A detailed review of the NAPL dissolution andrelated laboratory studies can be found in Miller et al.

MODELING OF BIOREMEDIATION SYSTEMS 49

(1998). In several DNAPL modeling studies, a quasi-steady state form of Fick’s linear dissolution modelis used to describe the contaminant mass exchangefrom the NAPL phase to the aqueous phase (e.g.,Miller et al., 1990; Powers et al., 1994; Saenton et al.,2002). Using the linear dissolution model, the fateand transport of a contaminant plume emanatingfrom an aquifer contaminated with residual DNAPLproducts can be written as:

∂C∂t

= ∂

∂xi

(Dij

∂C∂xj

)− ∂

∂xi(viC) + kLa(C∗ − C) ± F

(1)

where C is the aqueous-phase concentration of theDNAPL contaminant [ML−3], C∗ is equilibriumaqueous phase concentration (the solubility limit)of the contaminant [ML−3], D is the hydrodynamicdispersion coefficient [L2T−1], v is the pore watervelocity [LT−1], kLa is the DNAPL dissolution rateconstant [T−1], and the factor F represents all otherphysical, bio/geo-chemical reactions [ML−3T−1]. Inthis work, DNAPL is assumed to be trapped in thesubsurface as an immobile phase, and the influenceof the trapped DNAPL on aquifer porosity and hy-draulic conductivity are assumed to be negligible.These assumptions are valid at very low DNAPL sat-uration levels (Zhu and Sykes, 2000).

The changes in the DNAPL mass associated withthe contaminated sediment can be modeled by usingthe following equation:

ρ

φ

∂CN

∂t= −kLa(C∗ − C) (2)

where ρ is the dry bulk density of the soil matrix[ML−3]; φ is the porosity; and CN is the mass frac-tion of the residual DNAPL in the sediment [massof trapped NAPL per unit dry mass of porous media,MM−1]. Note, traditionally, the presence of NAPLin a porous medium is represented by its volumet-ric fraction θN . The value of DNAPL-mass fractionCN can be computed from the value of θN using therelationship:

CN = ρNθN

ρ(3)

where ρN is the DNAPL density [ML−3].We used the following empirical expression for

computing the DNAPL dissolution rate constant kLa

(see Appendix A):

kLa = kmaxLa

(CN

CoN

)β

(4)

where kmaxLa is the initial maximum dissolution rate

constant, which can be calculated based on the prop-erties of the porous medium (Appendix A), β is anempirical constant and Co

N is the initial mass fractionof the residual DNAPL in the sediment. Equation (4)is a scaling-type model for estimating the value ofkLa for different amounts of residual DNAPL. Notewhen β = 1, the model reduces to a simple expressionthat scales the initial kLa value based on the fractionof DNAPL concentration remaining in the system.Powers et al. (1994) presented a detailed dissolutionmodel that has a format similar to (4). In Appendix A,we review the details of the Powers et al. (1994) workand relate Equation (3) to the Powers’ model. The ap-pendix also provides expressions for computing thevalues of the model parameters kmax

La and β.It is important to note that the Powers et al.

(1994) model was originally developed based on dataobtained from one-dimensional experiments con-ducted using homogeneous sandy soils. This simplemodel will not be directly applicable for hetero-geneous multi-dimensional flow domains. Othershave attempted to extend this model for use in het-erogeneous systems involving non-uniform DNAPLsources. For example, Zhang and Brusseau (1999)identified one of the Powers’ model parametersas a calibration parameter and used the calibratedmodel to predict the regional-scale behavior of atrichloroethene plume at a Superfund site in Tucson,Arizona. Brusseau et al. (2002) used a simple ratioapproach to estimate the discrete dissolution val-ues for a three-dimensional grid using the Powers’model based values estimated from local-scale col-umn experiments. The scaling approach given in theappendix is similar to the Brusseau et al. (2002) ap-proach. However, it is important to note that fieldheterogeneity and NAPL source geometry can greatlyinfluence the dissolution behavior, especially in amulti-dimensional aquifer where transient flow by-passing effects can be dominant (Saenton et al., 2002;Brusseau et al., 2000). Further, DNAPL dissolutionprocesses can also be affected by the biological pro-cesses active near the NAPL source (Seagren et al.,1994; Yang and McCarty, 2000; Chu et al., 2003).The Powers’ dissolution model used in this work isa simplified approach, and it ignores several issuesincluding multi-dimensional by-pass flows, soil het-erogeneities, source heterogeneities, and biologicaleffects of NAPL dissolution. The overall modelingframework described, however, is a general approach

50 CLEMENT ET AL.

and can be easily adapted for more complex situa-tions by replacing the Powers’ approach with an ap-propriate dissolution model.

Modeling of Rate-Limited Sorptionand Solute Transport Processes

When the partitioning of dissolved contaminantsbetween the solid and liquid phases is assumed to berate-limited, the contaminant concentration levels inboth solid and liquid phases must be simulated bythe model. Following Haggerty and Gorelick’s (1994)approach, the fate and transport of a sorbing solutein aqueous and soil phases can be predicted usingthe following transport equations:

∂C∂t

+ ρ

φ

∂C̃∂t

= ∂

∂xi

(Dij

∂C∂xj

)− ∂

∂xj(νiC) ± F (5)

ρ

φ

∂C̃∂t

= ξ

(C − C̃

λ

)(6)

where C is the concentration of the contaminantin the aqueous phase [ML−3]; C̃ is the concentra-tion of the contaminant in the sorbed phase [massof sorbed contaminant per unit dry mass of porousmedia, MM−1]; ξ is the mass-transfer rate param-eter [T−1]; λ is the linear partitioning coefficient[L3M−1]; and the factor F represents all possibleaqueous-phase physical, bio/geo-chemical reactions[ML−3T−1]. It can be shown that the above non-equilibrium model relaxes to an equivalent equilib-rium retardation model formulation when the valueof ξ is large, and relaxes to non-sorbing tracer modelwhen the value of ξ is small (Clement et al., 1998).

Modeling of CoupledDNAPL-Dissolution, Rate-LimitedSorption, and Biological Reaction

Processes

In this section we couple the rate-limited disso-lution and sorption models discussed above with arate-limited biodegradation reaction model to pre-dict the overall fate and transport of the dissolvedPCE plume originating from a PCE-DNAPL source.The biodegradation model assumes that PCE will bedegraded by a uniformly distributed stable micro-bial population present in the aquifer. The modelalso assumes biodegradation by anaerobic reduc-tive dechlorination processes yielding the follow-

ing sequential degradation pathway: PCE → TCE →DCE → VC. The “true” reaction mechanisms ofmicrobially-mediated reductive dechlorination pro-cesses are indeed complex and will be influenced byseveral biogeochemical factors (Maymo-Gatell et al.,1997). For instance, the rate of biodegradation wouldstrongly depend on the types and number of mi-croorganisms present in the subsurface, and the typesand amounts of the electron donors available forsupporting the dechlorination reaction. However, inmost practical, field-scale applications, chlorinatedsolvent degradation steps can be conceptualized aslumped first-order decay reactions (e.g., Wiedemeieret al., 1999; Clement et al., 2000). Assuming first or-der, sequential decay kinetics for describing the PCEbiodegradation process, the governing equations forthe coupled reactive transport system can be writtenas:

∂[PCE]∂t

+ ρ

φ

∂[PC̃E]∂t

= ∂

∂xi

(Dij

∂[PCE]∂xj

)− ∂

∂xi(νi[PCE]) + qs

φ[PCE]s

+ kLa([PCE]∗− [PCE]) − kaPce[PCE] − ρ

φks

Pce[PC̃E]

(7)

∂[TCE]∂

ρ

φ

∂[T C̃E]∂t

= ∂

∂xi

(Dij

∂[TCE]∂xj

)− ∂

∂xi(νi[TCE]) + qs

φ[TCE]s

+ YTce/Pce kaPCE[PCE] − ka

Tce[TCE]

+ ρ

φYTce/Pce ks

Pce[PC̃E] − ρ

φks

Tce[T C̃E] (8)

∂[DCE]∂t

+ ρ

φ

∂[DC̃E]∂t

= ∂

∂xi

(Dij

∂[DCE]∂xj

)− ∂

∂xj(νi[DCE]) + qs

φ[DCE]s

+ YDce/Tce kaTce[TCE] − ka

Dce[DCE]

+ ρ

φYDce/Tce ks

Tce[T C̃E] − ρ

φks

Dce[DC̃E] (9)

∂[VC]∂t

+ ρ

φ

∂[VC]∂t

= ∂

∂xi

(Dij

∂[VC]∂xj

)− ∂

∂xj(νi[VC]) + qs

φ[VC]s

+ YV c/Dce kaDce[DCE] − ka

V c[VC]

+ ρ

φYV c/Dec ks

Dec[DC̃E] − ρ

φks

V c[V C̃] (10)

ρ

φ

d[PCENAPL]dt

= −kLa([PCE]∗ − [PCE]) (11)

MODELING OF BIOREMEDIATION SYSTEMS 51

ρ

φ

∂[PC̃E]∂t

= ξPce

([PCE] − [PC̃E]

λPce

)− ρ

φks

Pce[PC̃E]

(12)

ρ

φ

∂[T C̃E]∂t

= ξTce

([TCE] − [T C̃E]

λTce

)+ ρ

φYT ce/Pceks

Pce[PC̃E]

− ρ

φks

Pce[T C̃E] (13)

ρ

φ

∂[DC̃E]∂t

= ξDce

([DCE] − [DC̃E]

λDce

)+ ρ

φYDce/T ce ks

Tce[T C̃E]

− ρ

φks

Dce[DC̃E] (14)

ρ

φ

∂[V C̃]∂t

= ξVc

([VC] − [V C̃]

λVc

)+ ρ

φYV c/Dceks

Dce[DC̃E]

− ρ

φks

Vc[V C̃] (15)

where [PCE], [TCE], [DCE], and [VC] representaqueous phase concentrations of the contaminantspecies [ML−3]; [PCENAPL] is the mass fractionof residual PCE NAPL [MM−1]; [PC̃E], [T C̃E],[DC̃E] and [V C̃] represent sorbed phase concentra-tions of the contaminants [MM−1]; [PCE]s, [TCE]s,[DCE]s, and [VC]s represent the source/sink con-centrations of various contaminant species [ML−3];[PCE∗] represents the solubility limit of PCE[ML−3]; ka

Pce, kaTce, ka

Dce, and kaVc are the first-order aque-

ous phase degradation rate constants [T−1]; ksPce, ks

Tce,ka

Dce, and ksVc are the first-order sorbed phase degrada-

tion rate constants [T−1]; ξPce, ξTce, ξDce, and ξVc arethe first-order sorption mass transfer rate constantsof various contaminant species [T−1]; λPce, λTce, λDce,and λVc are the first-order linear partitioning coeffi-cients of various contaminant species [L3M−1]; andY represents various stoichiometric yield values. Thevalues of Y can be calculated from the reaction sto-ichiometry and the molecular weights; for example,degradation of one mole of PCE would yield onemole of TCE, therefore YTce/ Pce = molecular weightof TCE/molecular weight of PCE (131.4/165.8 =0.79). Based on similar calculations, the values ofYDce/Tce and YVc/Dce can be calculated as 0.74 and 0.64,respectively.

It should be noted that the above model assumesthat degradation reactions in the aqueous phase

would release daughter products into the aqueousphase, and degradation reactions in the sorbed phasewould release daughter products into the sorbedphase. Biodegradation in the solid phase is includedin the model as a means to simulate the bioavail-ability of sorbed-phase contaminants. Conceptually,biodegradation in the sorbed phase assumes thatmicroorganisms are in contact with the sediment-bound contaminants and can mediate the release anddegradation of the bound contaminant mass. Novaket al. (1995) reviewed several experimental studieswhere addition of soil amendments has been shownto simulate microbial activity and helped increasethe release and degradation rates of bound residues.While it is difficult to directly quantify the effec-tive rates of such solid phase biodegradation reactionrates, inclusion of this mechanism in our theoreticalmodel allowed investigation of the impacts of vari-ous types of bioavailability conditions on the overallfate and transport. Several scenarios related to thebioavailability effects are discussed in the simulationanalysis section of this work.

Equations (7–15) provide the complete set ofmathematical equations for describing PCENAPL dis-solution coupled with rate-limited mass transfer andbiodegradation reactions. The model equations, to-gether with appropriate initial and boundary condi-tions, were solved using the general-purpose reactivetransport code RT3D (Clement, 1997; Clement et al.,1998; Clement et al., 2000). The RT3D code uses theoperator-splitting (OS) technique which allows repre-sentation of various reaction frameworks (Clement,1997). Using the OS technique, the reaction kineticsin Equations (7–15) can be separated from the trans-port equations to form the following reaction kineticframework:

d[PCE]dt

= ktLa([PCE]∗− [PCE]) − ka

Pce[PCE]

− ξPce

([PCE] − [PC̃E]

λPceE

)(16)

d[TCE]dt

= YT ce/PcekcPce[PCE] − ka

Tce[T CE]

− ξTce

([TCE] − [T C̃E]

λTce

)(17)

d[DCE]dt

= YDce/T cekcTce[TCE] − ka

Dce[DCE]

− ξDce

([DCE] − [DC̃E]

λDce

)(18)

52 CLEMENT ET AL.

d[VC]dt

= YV c/DcekcDce[DCE] − ka

V c[VC]

−ξVc

([VC] − [V C̃]

λVc

)(19)

d[PCENAPL]dt

= −φktLa

ρ([PCE]∗ − [PCE]) (20)

d[PC̃E]dt

= φξPce

ρ

([PCE] − [PC̃E]

λPce

)

− ksPce[PC̃E] (21)

d[T C̃E]dt

= φξTce

ρ

([TCE] − [T C̃E]

λTce

)

+ YTce/PceksPce[PC̃E] − ks

Pce[T C̃E]

(22)

d[DC̃E]dt

= φξDce

ρ

([DCE] − [DC̃E]

λDceE

)

+YDce/TceksTce[T C̃E] − ks

Dce[DC̃E]

(23)

d[V C̃]dt

= φξVc

ρ

([VC] − [V C̃]

λVc

)

+ YVc/DceksDce[DC̃E] − ks

V c[V C̃]

(24)

In the RT3D model, a subroutine, known as theRT3D reaction package, is used to define the govern-ing reaction kinetics. The reaction kinetics describedby Equations (16–24) were programmed as a user-defined reaction package using four mobile compo-nents to represent the aqueous phase contaminantsand five immobile components to represent the solidphase contaminants.

MODEL TESTING

Analytical solutions are not available for the multi-species transport equations developed in this work,even for simple one-dimensional cases. Therefore,the new reaction package was first tested against asingle-species analytical solution involving source de-cay, sorption, and biological degradation processes.Later a more complex multi-species case was solvedto verify the mass balance characteristics of the nu-merical solution.

Comparison Againstan Analytical Solution

The objective of this test is to verify the workingsof the new RT3D user-defined reaction package. Thetest problem considers one-dimensional transport of

a contaminant degrading via first-order reaction in asorbing porous medium. The contaminant source isassumed to degrade with time. The governing equa-tion for this problem can be written as:

R∂C∂t

= D∂2C∂x2

− ν∂C∂x

− kC (25)

The initial and boundary conditions for this problemare:

C(x, 0) = 0

C(0, t) = Cs exp(−µt) (26)∂C∂x

(∞, t) = 0

where R is the retardation factor; D is the disper-sion coefficient [L2T−1]; k is first-order degradationrate constant [T−1]; µ is rate constant of the expo-nentially decaying source [T−1]; C is the contam-inant concentration [ML−3]; and Cs is the initialmaximum contaminant source concentration level atthe boundary [ML−3]. Analytical solution to Equa-tions (24) and (25) is given in van Genuchten andAlves (1982).

The numerical modeling framework developed inthis study was used to solve the analytical prob-lem described above. In the numerical model, theDNAPL source was assumed to yield an initial sourceconcentration (Cs) of 1 mg/L, and this concentra-tion was allowed to decay exponentially. The sorp-tion processes were modeled under two extreme masstransfer conditions: firstly, a large mass transfer rateconstant (ξ ) of 1.5 day−1 was used to model the equi-librium retarded plume condition, and later a verysmall mass transfer rate constant of 0.000015 day−1

was used to model the no sorption (or tracer) condi-tion. The value of the first-order linear partitioningcoefficients λ was assumed to be 0.1875 L/kg (whichwould yield a retardation coefficient of R = 2 un-der high mass transfer conditions), bulk density ofthe soil is 1.6 kg/L, porosity is 0.3, the degradationrate constant k is 0.01 day−1, the source decay con-stant µ is 0.005 day−1, transport velocity is 1 m/day,and dispersion coefficient is 0.5 m2/day. The columnwas discretized using 50 uniform grid cells. Figure 2compares the breakthroughs obtained from the ana-lytical and numerical models at x = 50 m, and theconcentration profiles observed along the column af-ter 50 days. Under both tracer transport and equilib-rium sorption conditions, the result obtained fromthe numerical model closely matches the analyticalsolution.

MODELING OF BIOREMEDIATION SYSTEMS 53

FIGURE 2 Comparison of numerical and analytical results (a) breakthrough curves at x = 50 m (b) Concentration profile in thecolumn after 50 days of transport.

Mass Balance Analysis

The hypothetical test problem considered for themass balance study is a 510 m long one-dimensionalaquifer. The aquifer is divided into 51 cells of di-mensions 10 m × 10 m × 10 m. This problem isa one-dimensional version of a validation problemdiscussed in the MT3D manual (Zheng, 1990). Themodel parameters and other numerical parametersassumed in our simulation were similar to those usedby Zheng (1990). The aquifer is assumed to be con-fined with constant head boundaries at either end ofthe flow domain and no flow boundaries at the topand bottom of the domain. The hydraulic gradientacross the aquifer is 1/500, hydraulic conductivity is50 m/day and porosity is 0.3; these parameters yieldan effective transport velocity of 0.33 m/day. The

longitudinal dispersivity is assumed to be 10 m andthe ratio of transverse to longitudinal dispersivity isassumed to be 0.3. Other transport and reaction pa-rameters assumed are given in Table 1.

To simulate the source zone, 20,000 kg of PCE-NAPL was instantaneously discharged into the gridcell centered at x = 155 m resulting in the initialDNAPL mass fraction (CN ) of 0.0125 mg/mg (orthe volumetric NAPL content of 0.0123, yielding4% residual NAPL saturation). The DNAPL fractionsat all other nodes are assumed to be zero. The ini-tial aqueous and sorbed-phase concentrations of allother species are also assumed to be zero.

The objective of the first simulation experimentis to verify the code performance by testing themass balance characteristics of the model. In thistest, the fate and transport of PCE emanating from

54 CLEMENT ET AL.

TABLE 1 Summary of model parameters

Model parameters Values Reference

PCE solubility, PCE∗ (at 20◦C) 203 mg/L Bradford and Abriola (2001)Dynamic viscosity of water, µw 0.89 centi poise Bradford and Abriola (2001)Uniformity, Uin 1.21 Bradford and Abriola (2001)Median grain size, d50 0.036 cm Bradford and Abriola (2001)Diameter of mean grain size, dM 0.05 cm Powers et al. (1994)PCE diffusivity, Dm 6.56 × 10−6 cm2/sec Bradford and Abriola (2001)Water density, ρw 1.0 kg/L AssumedNAPL PCE density, ρm 1.623 kg/L Bradford and Abriola (2001)Bulk density of porous media, ρb 1.6 kg/L AssumedPartitioning coefficient for PCE, λPce 1.0 kg/L (6.3) Clement et al. (2002)

(Retardation factor for PCE, RPce)Partitioning coefficient for TCE, λTce 0.57 L/kg (4.0) Clement et al. (2002)

(Retardation factor for TCE, RTce)Partitioning coefficient for DCE, λDce 0.25 L/kg (2.4) Clement et al. (2002)

(Retardation factor for DCE, RDce)Partitioning coefficient for VC, λVc 0.05 L/kg (1.3) Clement et al. (2002)

(Retardation factor for VC, RVc)Initial dissolution rate constant for PCE 19 day−1 Calculated based on assumed

NAPL, kmaxLa porous media properties

Empirical constant, β 1 Assumed

the DNAPL source was simulated for two years. Ashort two-year simulation period was selected in thistest to avoid the advective loss of contaminant massthrough the downstream boundary of the column.Also, in order to close complete mass balance withinthe column, the degradation rate of VC and thesolid-phase biodegradation rates of all the specieswere set to zero. The aqueous phase biodegradationrates of PCE, TCE, and DCE were set to 0.02 day−1,0.01 day−1, and 0.002 day−1, respectively. The val-ues of the sorption mass transfer rate constants (ξ )for all the species were set to 0.15 day−1. The simu-lated aqueous phase concentration profiles of all thespecies at the end of the two-year period are presentedin Figure 3a.

Mass balance calculations indicated that themodel conserved the total mass. At the end ofthe two-year simulation period, the amount ofDNAPL remaining in the NAPL-contaminated nodewas 1.0101 × 10−2 mg/mg, which is equivalent to16,160 kg of DNAPL. Therefore, the remaining 3,840kg of PCE must have dissolved and either partitionedinto the solid phase or degraded. The total PCEequivalent mass of all the contaminants species (i.e.,the area below all the concentration profiles multi-plied by their respective yield and retardation values)is estimated to be 3,880 kg, which is close to the totalmass of PCE dissolved into the system. The overallmass balance error at the end of the two-year simula-tion period is about 0.2% of the initial DNAPL mass.

Figure 3b shows the breakthrough curves of var-ious dissolved species at the 20th cell (40 m down-stream from the DNAPL source) after 18 years ofsimulation. It is interesting to note that the concen-tration of the dissolution product (PCE ) reaches aquasisteady state condition within two years and re-mains steady until the DNAPL mass is considerablydepleted. All other daughter products also appearto reach a similar quasisteady state condition withinthree years. Depletion of the DNAPL source zonestarts to impact the breakthrough concentration pro-files after 12 years, when the concentration profilesappear to relax into a transient mode.

MODEL SIMULATIONS RESULTS

Obtaining precise values for all the parametersused in the proposed model is a difficult task. There-fore, several sets of simulations were completed toprovide a basic understanding of the model be-havior under various parameter conditions. The re-sults are summarized by quantifying the response ofthe model to systematic variations in the values ofbiodegradation rates, DNAPL dissolution rates, andsorption mass transfer rates.

Model Response to Variationsin Biodegradation Rates

In the first set of simulations, biodegradation rateconstants were varied. Simulations were completed

MODELING OF BIOREMEDIATION SYSTEMS 55

FIGURE 3 Predicted concentration distribution of PCE, TCE, DCE, and VC (a) Profile along the column after two years(b) Breakthrough curve at x = 195 m.

with aqueous phase biodegradation rate constantsas 0.02 day−1, 0.01 day−1, 0.002 day−1, and 0.001day−1 for PCE, TCE, DCE, and VC, respectively.These assumed values of degradation rate constantsare within the range of values reported in the liter-ature (Wiedemeier et al., 1999). Biodegradation wasassumed to occur in the solid-phase with rates iden-tical to those assumed for the aqueous phase (i.e.,complete bioavailability is assumed). In order to as-sess the sensitivity, two sets of simulations were com-pleted by perturbing the base level biodegradationrates of PCE (both aqueous and soil phase rates) byan order of magnitude. In all of these simulations, theDNAPL dissolution rate parameter, kmax

La , was fixedat 19 day−1, and the values of sorption mass transferrate constant, ξ , were fixed at 0.008 day−1 for all the

species. Other flow and transport parameters usedwere identical to those used in the previous massbalance example problem.

Figure 4 shows the results of the model responseto variation in PCE biodegradation rate constant atthe end of 5 years. The results indicate that whenthe biodegradation rate for PCE is high, PCE andTCE concentrations decrease rapidly yielding highconcentrations of DCE and VC. It should be notedthat although higher biodegradation rate for PCE re-sult in rapid removal of PCE and TCE contaminants,the reactions lead to accumulation of daughter prod-ucts, which could pose several environmental con-cerns. For example, decay products, such as VC, aremore toxic than its parent products TCE and PCE.Therefore, the performance of a chlorinated solvent

56 CLEMENT ET AL.

FIG

UR

E4

Mod

elpr

edic

tions

for

vario

usPC

Ebi

odeg

rada

tion

rate

s(t

=5

year

s).

57

bioremediation system should always be carefullyscreened for conditions that could possibly lead tothe accumulation of more toxic daughter products.

Model Response to Variationsin DNAPL Dissolution Rates

The response of the model to variations in theDNAPL dissolution rate was studied by perturbingthe rate constant kmax

La . Since the overall DNAPLdissolution rate (given by Equation [4]) varies lin-early with the value of kmax

La , varying this parame-ter by a factor will vary the overall DNAPL dis-solution rate by a similar factor over the entiresimulation period. The variations in the DNAPLdissolution rate was quantified under the follow-ing two bioavailability conditions: in the first case,biodegradation was assumed to occur both in aque-ous and solid phases (complete bioavailability con-dition), and in the second case biodegradation wasassumed to occur only in the aqueous phase (i.e.,direct solid-phase bioavailability was assumed to benegligible). Three simulations were completed underthe two assumed bioavailability conditions with thevalues of maximum DNAPL dissolution rate con-stant kmax

La = 19, 1.9, and 0.19 day−1. The values ofξ were fixed at 0.008 day−1. The values of aqueousphase biodegradation rates were fixed at the baselinevalues used in section 4.1. Whenever degradation wasassumed to occur in the solid phase, the rates were as-sumed to be identical to aqueous phase degradationrates.

Figure 5 summarizes the result for the model re-sponse to variation in DNAPL dissolution rate con-stant. Figures 5a and 5b show the concentration dis-tribution of PCE and VC, respectively, at the endof the five-year simulation period when biodegrada-tion is assumed to occur in both phases. The simula-tion results showed that the PCE contamination wasconcentrated to a small region close to the DNAPLsource but its immediate daughter product TCE (re-sults not shown) was spread over a wider region. Simi-larly, the DCE concentration profile (not shown) waswider than the TCE profile, and finally the VC con-centration profile (shown in Figure 5b) was widerthan the DCE profile. Thus, contamination by adaughter species tends to impact a larger zone whencompared to its immediate parent species. Increasesin dissolution rate lead to wider concentration pro-files and higher concentration peaks.

Figures 5c and 5d show the results for the secondset of simulations where biodegradation is consid-ered to occur only in the aqueous phase. The ob-served trends in the predicted concentration profileswere similar to those observed in the previous setof simulations. However, the total amount of PCEmass in the column was relatively high when solidphase biodegradation processes were ignored. Also,the model predicted VC profiles were observed to berelatively insensitive above a critical DNAPL dissolu-tion rate constant of 1.9 day−1. For PCE, the value ofcritical limiting rate constant was slightly higher. Theoverall dissolved concentration level of PCE appearsto control the contaminant dissolution process whenthe dissolution rate constant was above the criticalvalue.

Comparison of the results shown in Figures 5aand 5c (or 5b and 5d) indicates that the predictedaqueous phase contaminant concentration levels aremarginally lower for the case when degradation wasassumed to occur in both phases. This is because,when complete bioavailability is assumed, the con-taminant mass is consumed on the solid phase lead-ing to lower solid phase concentration levels. This in-creases the concentration gradient (the driving forcefor mass transfer reactions) towards the solid phaseand facilitates increased aqueous phase contaminantmass partitioning onto the solid phase, thus resultingin lower aqueous phase concentration levels. There-fore, for a given solubility limit, aqueous phase con-centrations can be expected to be at a lower levelwhen total bioavailability conditions are assumed.

Model Response to Variationsin Sorption Mass Transfer Rates

In this set of simulations the response of the modelto variations in the sorption mass transfer rate con-stant ξ was analyzed. The values of degradation rateconstants were fixed similar to the baseline valuesused in previous simulations. When biodegradationwas assumed to occur in both liquid and solid phases,the solid phase biodegradation rates were assumed tobe identical to the aqueous phase rates. Three simula-tions were completed by setting the ξ values of all thespecies at 0.15, 0.008, and 0.00015 day−1, which arethe range of ξ values used in Clement et al. (1998).The value of DNAPL dissolution rate parameter kmax

Lawas set at 19 day−1, which was the calculated rate us-ing the assumed porous media properties.

58 CLEMENT ET AL.

FIG

UR

E5

Mod

elpr

edic

tions

for

vario

usD

NA

PLdi

ssol

utio

nra

tes:

(a)

and

(b)

biod

egra

datio

nis

assu

med

tooc

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ueou

san

dso

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ases

;(c

)an

d(d

)bi

odeg

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tion

isas

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occu

ron

lyin

the

aque

ous

phas

e(t

=5

year

s).

59

FIG

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E6

Mod

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for

vario

usso

rptio

nm

ass

tran

sfer

rate

cons

tant

s(t

=5

year

s).

60

The predicted aqueous phase concentration pro-files after 5 years, assuming biodegradation to occurin both solid and aqueous phases, are summarizedin Figure 6. When sorption mass-transfer rate washigh, biodegradation can destroy contaminants inboth solid and aqueous phases. Therefore, the to-tal amount of PCE and TCE mass in the columnwas minimum when ξ = 0.15 day−1. Since the as-sumed biodegradation rates of DCE and VC are rel-atively less than the PCE and TCE degradation rates,the column tends to accumulate large amounts ofDCE and VC when the value of ξ is high. Unfor-tunately, it is difficult to observe further definitetrends from the results. The overall mass transferprocess seems to be controlled by complex non-linear exchange mechanisms. Therefore, the simu-lation experiments were re-run for 25 years and a

FIGURE 7 Total solvent mass removed by biodegradation under various values of ξ (a) biodegradation is assumed to occur inboth aqueous and solid phases; (b) biodegradation is assumed to occur only in the aqueous phase.

detailed mass balance analysis was completed to in-terpret the predicted aqueous and sorbed phase con-centration profiles. Figure 7a shows the cumulativemass removed by the biodegradation activity withinthe column over the 25-year simulation period. Themass removed due to biodegradation was computedby deducting the following two components fromthe initial solvent mass: (1) the mass of various sol-vent species (in PCE-equivalent mass units) remain-ing within the aquifer column, and (2) the mass ofvarious solvent species flushed through the down-gradient boundary. It can be observed from Figure 7athat the cumulative contaminant mass destroyed dueto biodegradation increases rapidly with time andreaches a steady level around 15 years, after whichthe DNAPL source was totally depleted. The totalmass destroyed due to biodegradation is high when

MODELING OF BIOREMEDIATION SYSTEMS 61

the ξ value is high. Note for the low value of ξ , thepartitioning rate of contaminants to the solid phaseis small and, therefore, under this condition, most ofthe contaminant mass was advected and lost throughthe right boundary. This can be clearly seen in Figures6c and 6d where the predicted aqueous phase con-centrations of DCE and VC are considerably high,closer to the right boundary, when ξ values are small.

Nonlinear trends, similar to those shown inFigure 6, were also observed for the second set ofsimulations where the solid phase biodegradationwas assumed to be negligible (results not shown).Figure 7b shows the mass balance results for thesesimulations. The mass balance profiles presented inthe figure are relatively insensitive to changes in ξ

values; this is, in part, due to the fact that the totalmass removed via biodegradation was considerablylow in this case when compared to the mass removalresults predicted for the complete bioavailabilitycase.

SUMMARY AND CONCLUSIONS

This article presents a mathematical frameworkfor coupling DNAPL dissolution processes with rate-limited sorption and biodegradation processes. Thecoupled model equations were implemented in auser-defined reaction package and solved by the reac-tive transport code RT3D. The reaction package wasfirst tested by solving a reactive transport problemfor which an analytical solution is available. Numer-ical performance of the model was further validatedthrough the mass balance analysis of the simulationresults. Application of the model was illustrated byusing a hypothetical case study.

A detailed simulation analysis was completed toexplore the effects of changes in various reaction pa-rameters used in the model. The analysis indicatedthat the sequential PCE dechlorination process couldlead to excessive accumulation of certain degradationdaughter products. Therefore, chlorinated solventbioremediation systems should always be carefullymonitored for conditions that could possibly lead toaccumulation of more toxic degradation byproducts,such as vinyl chloride. Comparison of the simulationresult for different bioavailability conditions indi-cated that the predicted aqueous phase contaminantconcentration levels are relatively low under com-plete bioavailability conditions. This is because whenthe solid-phase degradation (or complete bioavail-

ability) was allowed in the model, more contaminantmass partitioned to the soil phase (to compensatefor the decayed soil phase mass) and this partition-ing process led to lower aqueous phase concentrationlevels. Therefore, for a given solubility limit, aqueousphase concentrations can be expected to be at lowerlevels when complete bioavailability conditions areassumed. Predictive simulations for the variations inthe rate-limited sorption process rate showed thatthe mass transfer process assumed in the model ap-pears to be mediated by complex non-linear massexchange mechanisms. Varying the value of mass-transfer rate constant yielded concentration profilesand mass-balance results that help demonstrate thesenonlinear effects.

This study provides a general framework for mod-eling DNAPL dissolution processes coupled to rate-limited sorption processes in a biologically reactiveporous medium. The framework is a useful tool fordeveloping a better understanding of the combinedeffects of physical, chemical and biological reactivetransport mechanisms in groundwater systems. Theproposed framework can serve as a basis for develop-ing rational design methods for quantifying the ef-fectiveness of natural attenuation and/or enhancedbioremediation systems at DNAPL-contaminatedfield sites.

ACKNOWLEDGEMENTS

This study was completed at Auburn Universityand was, in part, funded by a project from the SeoulNational University funded by the Frontier Projectof the Korea Ministry of Science and Technology.Additional funding for Dr. Gautam was providedby Battelle Pacific Northwest National Laboratorythrough the Australian Research Council’s SPIRTGrant program.

REFERENCESBradford, S. A., and L. M. Abriola. 2001. Dissolution of residual

tetrachloroethylene in fractional wettability porous media:Incorporation of interfacial area estimates. Water Resour.Res. 37:1183–1195.

Brusseau, M. L., N. T. Nelson, M. Oostrom, Z. Zhang, G. R.Johnson, and T. W. Wietsma. 2000. Influence of hetero-geneity and sampling method on aqueous concentrationsassociated with NAPL dissolution. Environ. Sci. Technol.34:3657–3664.

Brusseau, M. L., Z. Zhang, N. T. Nelson, R. B. Cain, G. R.Tick, and M. Oostrom. 2002. Dissolution of nonuniformly

62 CLEMENT ET AL.

distributed immiscible liquid: Intermediate scale experi-ments and mathematical modeling. Environ. Sci. Technol.36:1033–1041.

Burnell, D. K. 2002. A groundwater flow and solute transportmodel of sequential biodegradation of multiple chlorinatedsolvents in the surficial aquifer, Palm Bay. Florida, PhD dis-sertation, Georgia Institute of Technology.

Chu, M., P. K. Kitanidis, and P. L. McCarty. 2003. Effects ofbiomass accumulation on microbially enhanced dissolutionof a PCE pool: A numerical simulation. Journal of Contami-nant Hydrology 65:79–100.

Clement, T. P. 1997. RT3D—A modular computer code forsimulating reactive multi-species transport in 3-dimensionalgroundwater aquifers. Battelle Pacific Northwest NationalLaboratory, PNNL-SA-28967, http://bioprocess.pnl.gov/rt3d.htm

Clement, T. P., C. D. Johnson, Y. Sun, G. M. Klecka, andC. Bartlett. 2000. Natural attenuation of chlorinated ethanecompounds: Model development and field-scale applicationat the dover site. J. Contam. Hydrol. 42:113–140.

Clement, T. P., M. J. Truex, and P. Lee. 2002. A case study fordemonstrating the application of USEPA’s monitored naturalattenuation screening protocol at a hazardous waste site.J. Contam. Hydrol. 59:133–162.

Clement, T. P., Y. Sun, B. S. Hooker, and J. N. Petersen. 1998.Modeling multi-species reactive transport in groundwateraquifers. Groundwater Monit. Rem. 18:79–92.

Duba, A. G., K. J. Jackson, M. C. Jovanovich, R. B. Knapp, andR. T. Taylor. 1996. TCE remediation using in situ resting-state bioaugmentation. Environ. Sc. Technol. 30:1982–1989.

Haggerty, R., and S. M. Gorelick. 1994. Design of multiplecontaminant remediation: Sensitivity to rate-Limited masstransfer. Water Resour. Res. 30:435–446.

Harvey, C. F., R. Haggerty, and S. M. Gorelick. 1994. Aquiferremediation: A method for estimating mass transfer ratecoefficients and an evaluation of pulsed pumping. WaterResour. Res. 30:1979–1991.

Hooker, B. S., R. S. Skeen, M. J. Truex, C. D. Johnson, B. M.Peyton, and D. B. Anderson. 1998. In situ bioremediationof carbon tetrachloride: Field test results. BioremediationJournal 3:181–193.

Lu, G., T. P. Clement, C. Zheng, and T. H. Wiedemeier. 1999.Natural attenuation of BTEX compounds: Model develop-ment and field-scale application. Ground Water 37:707–717.

Maymo-Gatell, X., Y. Chien, J. M. Gossett, and S. H. Zinder.1997. Isolation of a bacterium that reductively dechlorinatestetrachloroethene to ethene. Science 276:1568–1571.

McCarty, P. L. 1997. Breathing with chlorinated solvents. Sci-ence 276:1521–1522.

Miller, C. T., G. Christakos, P. T. Imhoff, J. F. McBride, J. A. Pedit,and J. A. Trangenstein. 1998. Multiphase flow and transportmodeling in heterogenous porous media: Challenges andapproaches. Adv. Water Resour. 21:77–120.

Miller, C. T., M. M. Poirier-McNeill, and A. S. Mayer. 1990. Dis-solution of trapped non-aqueous phase liquids: Mass trans-fer characteristics. Water Resour. Res. 26:2783–2796.

Novak, J. M., K. Jayachandran, T. B. Moorman, and J. B. Weber.1995. Sorption and binding of organic compounds in soilsand their relation to bioavailability. In: Bioremediation: Sci-ence and Application, pp. 13–31. SSSA Special Publication43, Soil Science Society of America, Madison, WI.

Powers, S. E., L. M. Abriola, and W. J. Weber Jr. 1994. An exper-imental investigation for non aqueous phase liquid dissolu-tion in saturated subsurface systems: Transient mass transferrates. Water Resour. Res. 30:321–332.

Rifai, H. S., P. B. Bedient, J. T. Wilson, and J. M. Armstrong.1998. Biodegradation modeling at aviation fuel spill site.Journal of Environmental Engineering 144:1007–1029.

Saenton, S., T. H. Illangasekare, K. Soga, and T. A. Saba.2002. Effect of source zone heterogeneity on surfactant-enhanced NAPL dissolution and resulting remediation endpoints. J. Contam. Hydrol. 59:27–44.

Schafer, W., and R. Therrien. 1995. Simulating transport andremoval of xylene during remediation of a sandy aquifer. J.Contam. Hydrol. 19:205–236.

Seagren, E. A., B. E. Rittmann, and A. J. Valocchi. 1994. Quan-titative evaluation of the enhancement of NAPL-pool disso-lution by flushing and biodegradation. Environ. Sci. Technol.28:833–839.

Truex, M. J. 1995. In situ bioremediation destroys carbon tetra-chloride in hanford aquifer. Environmental Solutions 8:30–33.

Van Genuchten, M. T., and W. J. Alves. 1982. Analytical Solu-tions of the One Dimensional Convective-Dispersive SoluteTransport Equation, Tech. Bull. 1661, US Dep. Of Agric.,Riverside, CA.

Wiedemeier, T. H., H. S. Rifai, J. T. Wilson, and C. Newell. 1999.Natural attenuation of fuels and chlorinated solvents in thesubsurface, John Wiley, New York.

Yang, Y., and P. L. McCarty. 2000. Biologically enhanced dis-solution of tetrachloroethene DNAPL. Environ. Sci. Technol.34:2978–2984.

Zhang, Z., and M. L. Brusseau. 1999. Non ideal transport ofreactive solutes in heterogeneous porous media: 5. Simu-lating regional-scale behavior of a trichloroethene plumeduring pump-and-treat remediation. Water Resources Re-search 35:2921–2935.

Zheng, C., 1990. MT3D: A modular three-dimensional trans-port model for simulation of advection, dispersion andchemical reactions of contaminants in groundwater sys-tems. USEPA report.

Zhu, J., and J. F. Sykes. 2000. The influence of NAPL dissolu-tion characteristics on field-scale contaminant transport insubsurface. J. Contam. Hydrol. 41:133–154.

APPENDIX A: ESTIMATIONOF DNAPL DISSOLUTION

RATE CONSTANT

In this appendix, we modify the model by Powerset al. (1994) to develop a simple scaling-type expres-sion for predicting the transient variations in DNAPLdissolution rates. The first order dissolution rate con-stant, kLa, can be computed using the following em-pirical model (Powers et al., 1994):

Sh = 4.13R0.598e

(d50

dm

)0.673

U 0.369in

(θN

θNi

)β

(A.1)

where Sh[= kLad250

Dm] is the modified Sherwood num-

ber; Re[= νρwd50µw

] is the Reynolds number; d50 is the

MODELING OF BIOREMEDIATION SYSTEMS 63

median grain diameter of the porous medium mate-rial [L]; Dm is the molecular diffusivity of the con-taminant in water [L2T−1]. dm is the medium graindiameter [L]; Uin is the uniformity index (Uin =d60/d10) [dimensionless]; θN is the volume fractionof the DNAPL [dimensionless]; θNi is the initialvalue of θN ; ρw is the density of the water [ML−3]; vis the pore water velocity [LT−1]; µw is the dynamicviscosity of water [ML−1T−1]; and β is an empiricalconstant.

The volumetric fraction of DNAPL, θN , can becomputed from the relation:

θN = ρCN

ρN(A.2)

where CN is the residual DNAPL mass fraction in thesoil [mass of NAPL per unit mass of dry soil, MM−1];ρNapl is the DNAPL density [ML−3]; and ρ is thedry bulk density of the porous medium [ML−3].

Using Equations (A.1) and (A.3), we can write thefollowing general expression for the mass transfer rateconstant:

kLa = 4.13R0.598e

(d50

dm

)0.673

U 0.369in

(Dm

d250

)(CN

CoN

)β

(A.3)

where CoN is the residual DNAPL mass fraction

when θN = θNi. Note the term (CN/CoN )β in Equa-

tion (A.3), which is equivalent to (θN/θNi)β in Equa-tion (A.1), accounts for the transient changes in themass transfer rate due to changes in the interfacial

surface area. The constant β can be computed fromthe empirical equation (Powers et al., 1994):

β = 0.518 + 0.114(

d50

dm

)+ 0.10Uin (A.4)

By combining all the terms containing the physi-cal properties of the porous medium, we can rewriteEquation (A.3) as:

kLa = kmaxLa

(CN

CoN

)β

(A.5)

Where,

kmaxLa = 4.13R0.598

e

(d50

dm

)0.673

U 0.369in

(Dm

d250

).

Note kmaxLa is an intrinsic mass transfer property of

a porous medium. Using the soil properties givenin Table 1 (for the transport problem described insection 4.1), the kmax

La value for our test problem wasestimated to be 19 d−1. Similarly, the value of β wasestimated as 0.81 for the given properties. Typicalvalues of β for idealized DNAPL geometries rangefrom 0.5 to 1.0 (Powers et al., 1994). However, thevalues of β measured in the experimental columns ofPowers et al. (1994) range from 0.75 to 0.96. We stud-ied the sensitivity of dissolution rate to changes in β

values and found that the rate is relatively insensitiveto β for the range of NAPL fractions considered inthis study.

64 CLEMENT ET AL.