prediction of mass-transfer coefficient for solute transport in porous media

Ž .Journal of Contaminant Hydrology 50 2001 1–19www.elsevier.comrlocaterjconhyd

Prediction of mass-transfer coefficient for solutetransport in porous media

Munjed A. Maraqa)

Department of CiÕil Engineering, College of Engineering, United Arab Emirates UniÕersity, P.O. Box 17555,Al-Ain, United Arab Emirates

Received 9 February 2000; received in revised form 2 January 2001; accepted 5 February 2001

Abstract

Several previously reported laboratory studies related to transport of solutes through packedcolumns were utilized to develop predictive relationships for mass-transfer rate coefficient. Thedata were classified into two groups: those obtained under rate-limited mass transfer between

Ž .mobile and immobile water regions physical nonequilibrium conditions , and those derived fromŽrate-limited mass transfer between instantaneous and slow sorption sites sorption nonequilibrium

.conditions . The mass-transfer coefficient in all these studies was obtained by fitting breakthroughdata to a transport model employing a first-order rate limitations with a AconstantB mass-transfercoefficient, independent of flow conditions. This study demonstrated that the mass-transfercoefficient in these models is dependent on system parameters including pore-water velocity,length-scale, retardation coefficient, and particle or aggregate size. Predictive relationships weredeveloped, through regression analysis, relating mass-transfer coefficient to residence time. Thedeveloped relationships adequately estimated previously reported field mass-transfer values.

w Ž .Successful simulations of field desorption data reported by Bahr J. Contam. Hydrol. 4 1989x205 further demonstrate the potential applicability of the developed relationships. q 2001

Published by Elsevier Science B.V.

Keywords: Mass transfer; Physical nonequilibrium; Sorption nonequilibrium; Velocity; Residence time

1. Introduction

Today, there is increasing awareness that mass-transfer limitation can have a signifi-cant impact on the mobility of contaminants in groundwater. This has been demonstrated

Žby several laboratory investigations Hutzler et al., 1986; Bouchard et al., 1988; Lee et

) Tel.: q971-3-7051524; fax: q971-3-623154.Ž .E-mail address: [email protected] M.A. Maraqa .

0169-7722r01r$ - see front matter q2001 Published by Elsevier Science B.V.Ž .PII: S0169-7722 01 00107-3

( )M.A. MaraqarJournal of Contaminant Hydrology 50 2001 1–192

.al., 1988; Brusseau, 1992; Brusseau et al., 1991a; Ptacek and Gillham, 1992 and fieldŽstudies Goltz and Roberts, 1986, 1988; Bahr, 1989; Bowman, 1989; Harmon et al.,

.1992; Pang and Close, 1999 .As accurate estimation of the nonequilibrium mass-transfer coefficients is critical,

several efforts have been directed towards characterizing mass transfer usinglaboratory-scale systems. The ability, however, to use laboratory-derived mass transfercoefficients for the prediction of nonequilibrium solute transport at the field-scale isconstrained by several factors. First, rate limitation, in most of the current nonequilib-rium transport models, has been simulated using a formulation that assumes a AconstantB

Žmass-transfer coefficient. Several studies Brusseau, 1992; Kookana et al., 1993; Zim-.merman, 1998; Maraqa et al., 1999 questioned the validity of this assumption and

suggested a correlation between mass-transfer coefficient and pore-water velocity.Second, rate limitations to solute transport in the field occur at large scales compared to

Žthose observed in the laboratory Goltz and Roberts, 1986; Quinodoz and Valocchi,.1993 , resulting in a field mass-transfer coefficient that is much lower than its labora-

tory-determined counterpart. Third, heterogeneity of hydrodynamic and sorption parame-ters in the field may affect transport more significantly than particle-scale limitationŽ . Ž .Harmon et al., 1992 . Quinodoz and Valocchi 1993 fit kinetic mass-transfer models todata from the Borden field experiment and obtained a field-scale mass-transfer rateparameter estimate that was more than an order of magnitude lower than that obtained

Ž .by Ball and Roberts 1991 during long-term laboratory adsorption studies.At comparable pore-water velocities, discrepancies between values of mass-transfer

coefficient for a certain solute exist even among laboratory studies conducted byŽ .different investigators. Brusseau et al. 1991a noticed a time-scale effect on sorption

Ž . Ž .mass-transfer coefficient, as did Ball and Roberts 1991 and Maraqa et al. 1999 . If itis true that mass-transfer coefficient is time dependent, then the coefficient is influencedby other system parameters including the extent of solute interaction with the medium,in addition to pore-water velocity and the length-scale. Hence, valid comparisonbetween values of mass-transfer coefficients reported by various investigators in thelaboratory will be obtained only when similar residence times are employed.

The primary objective of this study was to develop relationships between mass-trans-fer coefficient and system parameters using previously reported laboratory-derived data.

Ž .This study expands on the work of Maraqa et al. 1999 by considering additionalprevious studies, and explores the impact of other factors including particle size.Another objective was to determine the adequacy of the developed relationships inestimating field-mass transfer values and in simulating field transport data.

2. Model background

Nonequilibrium during transport of solutes in porous media has been categorized asŽeither transport-related or sorption-related. Transport nonequilibrium also called physi-

.cal nonequilibrium is caused by slow diffusion between mobile and immobile waterŽregions. These regions are commonly observed in aggregated soils van Genuchten and

. ŽWierenga, 1976; Nkedi-Kizza et al., 1983 , or under unsaturated flow conditions De

( )M.A. MaraqarJournal of Contaminant Hydrology 50 2001 1–19 3

.Smedt and Wierenga, 1979, 1984; De Smedt et al., 1986; Bond and Wierenga, 1990 , orin layered or otherwise heterogeneous groundwater systems. Sorption-related nonequi-

Ž .librium results from either slow chemical interaction van Genuchten et al., 1974 orŽ .slow intrasorbent diffusion Ball and Roberts, 1991 . In most of these models, the soil

matrix is conceptually divided into two types of sites; sorption is assumed to beinstantaneous for one type and rate-limited for the other type.

Solute transfer between mobilerimmobile water regions or instantaneousrrate-limitedsorption sites is commonly described by a first-order rate expression or by Fick’s law ifthe geometry of the porous matrix can be specified. Models that are based onwell-defined geometry are difficult to apply to actual field situations, as they require

Žinformation about the geometry of the structural units that are rarely available Fortin et.al., 1997 . Hence, the first-order rate formulation has been extensively used to model

underground contaminant transport.The dimensionless equations of the first-order, two-site nonequilibrium model for the

case of linear sorption are given as

EC) ES)

Õ L E2 C) EC)

obR q 1yb R s y 1Ž . Ž .2ET ET D EZEZ

ES)

) )1yb R sv C yS 2Ž . Ž . Ž .ET

where,

ZszrL 3aŽ .TsÕ trL 3bŽ .o

C)sCrC 3cŽ .o

1 S2)S s 3dŽ .

1yF K CŽ . D o

rK DRs1q 3eŽ .

u

uqFrK Dbs 3fŽ .

uqrK D

kLvs 1yb R 3gŽ . Ž .

Õo

where C is the aqueous solute concentration, C is the solute concentration in theo

influent solution, z is distance, L is the column length, t is time, Õ is the averageo

pore-water velocity, D is the hydrodynamic dispersion coefficient, S is the average2

sorbed concentration in the rate-limited domain, R is the retardation coefficient, K isD

the linear sorption distribution coefficient, r is the soil bulk density, u is the moisturecontent, F is the fraction of sorbent for which sorption is instantaneous, and k is thefirst-order mass-transfer coefficient. b and v are the dimensionless parameters thatspecify the degree of nonequilibrium in the system.


Although the physical and the two-site sorption nonequilibrium models have beenbuilt on physically distinct hypotheses, their mathematical formulations are similar for

Ž .the case of linear sorption Nkedi-Kizza et al., 1984 . Therefore, the system of equationsdescribing the first-order, physical nonequilibrium models is not presented here but has

Ž .been reported elsewhere Brusseau et al., 1989; Kookana et al., 1993 . The parameter vin the first-order, physical nonequilibrium model is defined as

aLvs 4Ž .

Õ um m

where Õ is the average pore-water velocity of the mobile phase, u is the moisturem m

content of the mobile region, and a is the first-order mass-transfer coefficient betweenŽ Ž ..the mobile and immobile water region analogous to k in 3g .

Ž .The mass-transfer coefficient a or k in the above models is assumed to be constant,independent of flow conditions. Several relationships have been developed relating a todiffusion coefficient and diffusion path length for a specified aggregate geometry. For

Ž .spherical aggregates, a is given as Harmon et al., 1992

15D up imas 5Ž .2a

where D is pore diffusivity coefficients, u is the immobile water fraction, and a isp im

the diffusion path length. Similarly, k is estimated for the case where sorptionŽ .nonequilibrium is due to intraorganic matter diffusion as Brusseau et al., 1991b

S Df OMks 6Ž .2a 1yFŽ .

where S is a shape factor and D is the effective diffusion coefficient in soil organicf OM

matter.

3. Data collection

Data were collected from 19 previously reported studies conducted in the laboratoryand five studies conducted in the field. The studies used one of the above two models to

Žestimate transport parameters. Eight laboratory studies Krupp and Elrick, 1968; vanGenuchten and Wierenga, 1977; Rao et al., 1980a,b; Nkedi-Kizza et al., 1983; De Smedtand Wierenga, 1984; De Smedt et al., 1986; Schulin et al., 1987; Seyfried and Rao,

. Ž .1987 were related to physical nonequilibrium Table 1 . The other laboratory studiesŽLee et al., 1988; Nkedi-Kizza et al., 1989; Brusseau, 1992; Brusseau et al., 1990;Ptacek and Gillham, 1992; Kookana et al., 1993; Piatt et al., 1996; Zimmerman, 1998;

. ŽMaraqa et al., 1999; Zhao et al., 1999 were related to sorption nonequilibrium Table.2 .

Ž .Unless indicated, packed soil or glass beads columns were injected with the targetŽ .solute and generated breakthrough curves BTCs were analyzed to determine the

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Table 1Review of some experimental investigations employing first-order physical nonequilibrium formulations

aInvestigator Porous medium Tracer Column Average Retardation Nb Ž . Ž .length cm velocity cmrh coefficientŽ .Description Size cm

Ž .Krupp and Elrick 1968 Unsaturated, 0.01 Cl 10 2.46–13.4 1.0 4nonaggregatedglass beads

3Ž .De Smedt et al. 1986 Unsaturated, 0.03 H O 100 6.98–14.33 1.0 32

nonaggregated, sand36De Smedt and Unsaturated, 0.01 Cl 30 0.88–40.42 1.0 6

Ž .Wierenga 1984 nonaggregated,glass beads

3van Genuchten and Saturated, 0.1, 0.31 H O 30 0.17–4.48 1.024–1.027 152Ž .Wierenga 1977 aggregated soil

36 3Ž .Rao et al. 1980a,b Saturated, 0.55, 0.75 Cl, H O 30 1.87–96.61 1.0 102

synthetic aggregates36 3Ž .Nkedi-Kizza et al. 1983 Saturated, 0.075, 0.15, 0.335 Cl, H O 5 0.46–8.72 0.81–1.49 342

aggregated soil3Ž .Seyfried and Rao 1987 Saturated, NA H O 12–15 4.74–54.4 1.12–1.17 72

aggregated soil3Ž .Schulin et al. 1987 Unsaturated, )2 H O, Br 42, 50 0.06–11.43 0.73–1.21 232

55% stones

Note: NA means not available.aN is the number of data points.bAverage particle or aggregate size employed.

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Table 2Review of some experimental investigations employing first-order sorption nonequilibrium formulations

a bInvestigator Soil organic Tracer Column Average Retardation NŽ . Ž . Ž .carbon % length cm velocity cmrh factors

Ž .Maraqa et al. 1999 0.7, 1.57, 2.25 BEN, DMP 30 0.62–36.47 1.61–4.03 18Ž .Brusseau 1992 0.02, 0.03, 0.1 XYL, NAP, PCE, DCB 6.7, 7 5–93 1.48–2.27 12Ž .Lee et al. 1988 0.025, 0.034 TCE, XYL 6, 8.8 24 1.61–2.65 4

Ž .Brusseau et al. 1991a 0.007, 0.025 TCM, TCE, PCE, XYL, 30 0.9 1.4–4.9 8DCB, NAP, BPN

cŽ .Nkedi-Kizza et al. 1989 0.2 ATR, DRN 5 5–6 1.16–4.37 7dŽ .Brusseau et al. 1990 0.39 BEN, CB, DCB, TCB 5.3 90 1.33–13.5 4eŽ .Kookana et al. 1993 0.6 SIM 30 1.9–37.2 3.66 4

Ž .Ptacek and Gillham 1992 0.019 BROM, CT, PCE, DCB, HAC 10 0.75, 12.08 1.08–4.1 20Ž .Zhao et al. 1999 0.03 CT 15 1.27 2.84 1

fŽ .Piatt et al. 1996 0.017 PNT 27.1, 27.6 27 2.64–16.71 4gŽ .Zimmerman 1998 0.05, 0.3, 1.0 NAP, XYL, BEN, TOL 15.3 1.7–93.2 1.05–19.28 34

a BEN: Benzene, DMP: Dimethylphthalate, XYL: Xylene, NAP: Naphthalene, PCE: Tetrachloroethene, TCE: Trichloroethene,TCM: Trichloromethane, CB:Chlorobenzene, DCB: 1,2 Dichlorobenzene or 1,4 Dichlorobenzene, TCB:Trichlorobenzene, BPN:Biphenyl, SIM:Simazine, ATR:Atrazine, DRN: Diuron, CT: Carbontetrachloride, BROM: Bromoform, HAC: Hexachlorobenzene, PNT: Phenantherene, PYN:Pyrene, TOL: Toluene.

bN is the number of data points.c Includes both aqueous phase and cosolvent data.d Values for the miscible displacement experiment. F values are not available.eb values were determined by independent means. Negative F values were not considered.f Naphthalene was excluded because its v values)4.g Twelve values out of 46 were excluded for which v-0.05 or )4. Transport parameters were obtained from the desorption part of the BTCs after equilibrating

the soil with the target compounds.


mass-transfer coefficient and other transport parameters. Both b and v were deter-mined by curve fitting technique. The length of the column, r, u and Õ were allo

determined by independent means. The dispersion coefficient in the studies related tophysical nonequilibrium was determined by curve fitting along with b and v, while thatin the case of sorption nonequilibrium was determined from BTCs generated fornonreactive tracers. Retardation coefficient was either determined independently or bycurve fitting along with b and v.

Ž .It has been reported that for cases wherein a system is near equilibrium v)4 orŽ .under extreme nonequilibrium v-0.05 , attainment of a unique solution by curve

Ž .fitting may prove difficult Brusseau et al., 1989 . Therefore, values of mass-transfercoefficient were not considered when v fell outside of this range. The only occurrence

Ž .of this end-member behavior was in the study of Piatt et al. 1996 and that ofŽ .Zimmerman 1998 , with a total of 14 data points were excluded for this reason.

Ž .Studies presented in this paper that are related to physical nonequilibrium Table 1Ž .are categorized into three groups: 1 those conducted under unsaturated conditions

Žusing fine, nonaggregated media Krupp and Elrick, 1968; De Smedt and Wierenga,. Ž .1984; De Smedt et al., 1986 ; 2 those conducted using saturated, aggregated media

Žvan Genuchten and Wierenga, 1977; Rao et al., 1980a,b; Nkedi-Kizza et al., 1983;. Ž .Seyfried and Rao, 1987 ; and 3 those conducted under unsaturated conditions using

Ž .stony soil Schulin et al., 1987 . The above characterization was solely based on theŽ .average particle or aggregate size employed in these studies Table 1 . The tracers used

Ž 3 . Ž .in all these studies Cl, Br and H O are believed to behave as conservative Rs1 or2

near conservative tracers. The length of the column employed in these studies rangesbetween 5 and 100 cm, while the pore-water velocity ranges between 0.17 and 96.6cmrh.

Eleven laboratory studies are utilized to investigate mass transfer of sorbing tracersŽ .Table 2 . These studies incorporate 20 organic compounds, most of them nonionic,including aromatic, aliphatic and polynuclear aromatic hydrocarbons. All these resultswere obtained by conducting saturated column experiments using fine, nonaggregated

Ž .media sand, sandy loam or silt loam . Therefore, physical nonequilibrium is not likelyto be present in these systems. The molecular weight of the various compounds ranged

Ž .from 78 for benzene to 284 for hexachlorobenzene. Soil organic carbon SOC reportedin these studies varied significantly and fell within 0.007–2.25%. The column lengthranged between 5 and 30 cm, pore-water velocity ranged between 0.6 and 93 cmrh, andretardation coefficient fell in the range of 1.1–19.3.

4. Discussion

4.1. Physical mass transfer

Ž .Fig. 1a plots the mass-transfer coefficient a obtained from the studies indicated inTable 1 versus pore-water velocity. The mass-transfer coefficient increases with the


Ž . Ž .Fig. 1. Effects of a pore-water velocity, and b residence time on mass-transfer coefficient between mobileand immobile water regions.


increase in pore water velocity. The best-fit relations between a and Õ for theoŽ 2 .categorized studies of Table 1 along with the coefficient of determination r are

Fine, nonaggregated media as0.034Õ0.85 r 2s0.57 7aŽ .o

Aggregated media as0.031Õ0.71 r 2s0.52 7bŽ .o

Stony media as0.00076Õ0.76 r 2s0.7. 7cŽ .o

Variation of mass transfer of nonsorbing solutes between mobile and immobile waterŽregions with pore-water velocity has been reported van Genuchten and Wierenga, 1977;

Nkedi-Kizza et al., 1983; De Smedt and Wierenga, 1984; De Smedt et al., 1986; Schulin.et al., 1987 . The increase in a with increasing pore-water velocity is attributed to

Žhigher mixing in the mobile phase at high pore-water velocities De Smedt and.Wierenga, 1984 or to shorter diffusion path lengths as a result of a decrease in the

Ž .amount of immobile water van Genuchten and Wierenga, 1977 .Fig. 1a shows that the values of a vary significantly at a given pore-water velocity.

These variations are possibly due to differences in the column length employed amongŽdifferent investigators, and the fact that some of these tracers did not behave ideally i.e.

.R/1 . To account for these variations, the values of a were plotted against residenceŽ . 2time Fig. 1b . The resulting best-fit relations along with r are

y0 .8 2Fine, nonaggregated media as0.95 LRrÕ r s0.91 8aŽ . Ž .o

y0 .88 2Aggregated media as0.26 LRrÕ r s0.77 8bŽ . Ž .o

y0 .73 2Stony media as0.012 LRrÕ r s0.73. 8cŽ . Ž .o

ŽImprovement in the fit when a was regressed on residence time compared to.pore-water velocity can be seen by inspecting the values of the coefficient of determina-

Ž 2 . Ž . Ž .tion r for Eqs. 7a – 8c . Significant improvement was gained with the studiesŽ Ž .conducted under unsaturated conditions using fine, nonaggregated media Eqs. 7a and

Ž ..8a . But little improvement was obtained with those conducted under unsaturatedŽ Ž . Ž ..conditions using stony soil Eqs. 7c and 8c due possibly to the limited number of

Ž . Ž .studies utilized for this case. Comparison between Eqs. 7b and 8b shows that theŽ .improvement in the fit with the studies conducted using aggregates Group 2 was not as

Ž .great as that gained with fine, nonaggregated media Group 1 . This may be due to theŽ . Ž .wide variations in the size of the aggregates associated with Eqs. 7b and 8b .

Fig. 1b revealed that, at comparable residence times, the mass-transfer coefficientbetween mobile and immobile water regions decrease as the particleraggregate size

Ž .increases. This is consistent with the observation of Rao et al. 1980a,b that theasymmetry in BTCs, for a given velocity, was greater for columns packed with largerporous spheres, suggesting a decrease in the mass-transfer coefficient as the aggregatesize increases.

Ž . Ž .Eqs. 8a – 8c indicate that the physical nonequilibrium mass-transfer coefficient isinversely proportional to residence time, and takes the form asmtyn, where t is ther r

residence time. Little variations in the value of n is observed among the three


Ž .categorized studies 0.73–0.88 . In fact, for a 95% confidence level, deviations in theŽ . Ž .value of n in Eqs. 8a – 8c were "0.113, "0.097, and "0.076, respectively.

Therefore, the value of n appears to be independent of particleraggregate size.However, the coefficient m decreases noticeably as the particleraggregate size in-

Ž . Ž .creases. Deviations in the value of m in Eqs. 8a – 8c , based on a 95% confidencelevel, were "0.101, "0.147 and "0.258, respectively.

4.2. Sorption mass transfer

Ž .Variation of k for the studies listed in Table 2 with pore-water velocity is presentedin Fig. 2a. As pore-water velocity increases, k increases, with the following best-fitrelation between the two parameters

Sorption nonequilibrium ks0.095Õ0.97 r 2s0.61. 9Ž .o

Similar to the physical nonequilibrium mass-transfer coefficient, high variations in kŽ .can be observed at comparable pore-water velocities Fig. 2a . To account for variations

in the length of the columns and the differences in retardation among the compounds,Ž . Ž .values of k were regressed on values of LRrÕ Fig. 2b to obtain Eq. 10o

y0 .93 2Sorption nonequilibrium ks1.72 LRrÕ r s0.84. 10Ž . Ž .o

Ž 2 .Comparison of the coefficients of determination for the regression on Õ r s0.61oŽ 2 .with that for the regression on residence time r s0.84 indicated that the residence

time was a better parameter for explaining variations in k than was pore-water velocityŽ .alone. Eq. 10 accounts for the dependence of k on K as reported by othersD

ŽKarickhoff and Morris, 1985; Brusseau and Rao, 1989; Brusseau et al., 1991a,b; Ball.and Roberts, 1991 but only as the latter affects residence time.

Ž .Attempts to improve Eq. 10 by considering differences in SOC, and some chemicalŽ Ž .characteristics of the organic compounds i.e. octanol–water partition coefficient K ,ow

.molecular weight, and aqueous diffusion coefficient were not successful. This wasŽ .y0.93accomplished by plotting kr1.72 LRrÕ for all values against the parametero

under consideration. In all cases, the value of r 2 did not exceed 0.1, indicatinginsignificant correlation with that parameter. When individual compounds were consid-ered, r 2 ranged between approximately 0.75 for toluene and naphthalene to about 1.0for TCE and DCB. For a given sorbent, r 2 ranged between approximately 0.74 for

Ž . Ž .Zimmerman 1998 to 0.99 for Brusseau et al. 1990 . Therefore, further improvementŽ .of Eq. 10 is possibly tenuous given the uncertainty in the reported values due to

experimental and curve-fitting procedures.The average particle size employed in all the studies cited under sorption nonequilib-

rium is slightly higher than that of the particles associated with the studies in Group 1Ž .Table 1 but smaller than the size of the aggregates of Group 2. It is interesting toobserve that the equation derived for mass-transfer coefficient between mobile and

Ž . Ž .Fig. 2. Effects of a pore-water velocity, and b residence time on mass-transfer coefficient of organiccompounds.


Ž Ž ..immobile water for fine, nonaggregated particles Eq. 8a is very close to the best fitŽ .relation derived for sorbing tracer see Fig. 2b .

Ž .While it is not clear why the mass-transfer coefficient a or k fit to breakthroughdata depends on residence time, several speculations may be offered. First, it is possiblethat the mass-transfer coefficient is not constant, but varies with residence time due to

Žvariations in either the molecular diffusion coefficient or the diffusion path length Eqs.Ž . Ž .. Ž .5 and 6 . Nkedi-Kizza et al. 1983 observed that the mass-transfer coefficient issensitive to solution concentration. If this is the case, then the dependence of mass-trans-

Ž .fer coefficient on length or time scale could be due to changes in the local aqueoussolution concentration driving diffusive mass transfer. Second, the mass-transfer coeffi-cient is constant but the apparent variations with residence time probably reflects a morecomplex mass transfer rate process than what could be accounted for by a first-ordermass transfer formulations. Theoretical analysis and experimental data presented by Rao

Ž .et al. 1980a,b showed that, for non-sorbing tracers, a mass-transfer model that onlyconsiders average concentration within mobile and immobile water regions yields amass-transfer coefficient that decreases with increasing time of diffusion. Third, theapparent variations of the mass-transfer coefficient as observed here may be a result ofthe heterogeneous advection processes that have been obscured by the use of a Fickian

Ž Ž ..diffusion model Eq. 1 .

4.3. Comparison with field mass-transfer Õalues

Ž Ž . Ž . Ž .The usefulness of the laboratory-developed relationships Eqs. 8a , 8b , 8c andŽ ..10 lies in the ability of these relationships to predict field values. Under fieldconditions, the length of the system and pore-water velocities could be orders ofmagnitude higher than those usually employed in the laboratory. Furthermore, thelaboratory data from which these relationships were developed were obtained usinghomogenized, packed soil samples, and the influence of field heterogeneity on thevalues of the nonequilibrium parameters is unknown. Four published field data sets wereutilized to evaluate the validity of the developed relationships. Two of these data sets are

Ž .related to physical-nonequilibrium Pang and Close, 1999; Harmon et al., 1992 , whileŽthe other two are related to sorption nonequilibrium Ptacek and Gillham, 1992; Goltz

.and Roberts, 1986 .

4.3.1. Data set 1Ž .The first data set is that of Pang and Close 1999 . The authors conducted a field

study at Burnham in Canterbury, New Zealand. The highly heterogeneous aquifermaterial on site consisted of medium to coarse alluvial gravel with interlayers of clay

Ž . Žand silt. Rhodamine RWT solution, an ideal tracer, was introduced as a pulse 10–17.min into the injection wells below the water table. BTCs were generated by collecting

samples from observation wells located at approximately 20, 40, 65 and 90 m downgra-dient of the injection well. The pore-water velocity ranged between 0.37 and 4.32 mrh.A 3-D model, which assumes a AconstantB mass-transfer between mobile and immobilewater regions, was used in Pang and Close study. The values of a were obtained fromthe optimized values of b and v.


Ž .The values of a reported by Pang and Close 1999 were plotted against residencetime as shown in Fig. 3. This figure also shows the best fit lines determined for the three

Ž Ž .categorized laboratory studies that are related to physical nonequilibrium Eqs. 8a –Ž ..8c . As evident from Fig. 3, the mass-transfer coefficient in Pang and Close study is

Ž .inversely proportional to residence time, confirming the laboratory findings Fig. 1b .Furthermore, all the values of a in Pang and Close study fall between the lines

Ž . Ž .corresponding to Eqs. 8b and 8c . This, in fact, is what would be expected based onŽ .the average particle size reported at the Burnham Site i.e., 1.3–2.0 cm , which falls

Ž . Žbetween the range of average aggregate size ca. 0.4 cm and that of the particles )2. Ž . Ž . Ž . Ž .cm associated with Eqs. 8b and 8c , respectively. The ability of Eqs. 8a – 8c to

reasonably estimate Pang and Close field mass-transfer data, given that the averagelength-scale and average velocity in their study are about two orders of magnitude and20 times higher than the ones associated with the laboratory studies, is an indication ofthe applicability of the derived laboratory relationships.

4.3.2. Data set 2Ž .The second group of data set is that of Harmon et al. 1992 . They conducted a field

experiment at the Moffett Field in Mountain View, California. The aquifer consisted ofsand and gravel particles, interspersed with layers of silts and clays. Approximately,45% of the mass fraction on site consisted of particles with a nominal diameter of 0.31cm. Field breakthrough data for trichloroethylene, carbon tetrachloride, and vinylchloride were collected from an observation well 1 m away from the injection well. Theauthors were able to successfully simulate breakthrough data using sorption parametersthat were derived from laboratory-scale batch experiments, while pore-water velocity

Fig. 3. Laboratory- and field-scale physical nonequilibrium mass-transfer coefficients. Pang and Close fieldŽ .data were estimated from their Fig. 3c .


Ž . Ž 2 .0.11 mrh and dispersion coefficient 0.032 m rh were derived from field-scalebromide tests. The retardation coefficients of the three compounds were 11.0, 6.0 and

Ž . Ž .5.5, respectively. Harmon et al. 1992 reported similar pore diffusivity coefficients DpŽ y9 2 .for the three compounds 2.8=10 cm rs . Using the values of D , as0.15 cm andp

Ž . Ž Ž ..u s0.04 reported by Harmon et al. 1992 , the values of a were determined Eq. 5im

and plotted in Fig. 3. The vertical error bars reflect the uncertainty associated with a

due to the uncertainty in D .pŽ .Based on particle size, Eq. 8b would be the closest to use for predicting physical

nonequilibrium mass-transfer coefficients at the Moffett site. Comparison betweenŽ .values of mass-transfer coefficient determined using Eq. 8b and those reported by

Ž .Harmon et al. 1992 shows that the mass transfer-coefficients were overestimated. It isŽ .possible that the mass transfer as dealt with in Harmon et al. 1992 may have

contribution from sorption nonequilibrium. This results in a lower mass-transfer coeffi-Ž .cient than what would be expected based on Eq. 8b . Nevertheless, given the uncer-

Ž .tainty associated with Eq. 8b and that with the determined mass-transfer values, thepredictive relationship may still serve as a useful tool from a practical perspective.

4.3.3. Data set 3Ž . Ž .Eq. 10 was compared to a field data set collected by Ptacek and Gillham 1992 at

the Borden Site in Southern Ontario, Canada. The authors determined k for the sameŽ .organic compounds utilized in their laboratory experiments Table 2 at pore-water

velocities of 11.25 and 15.8 cmrh. Their values were derived from BTCs generated byŽ .collecting samples over a short length scale 10 cm . Residence times for their

Ž .experiments 1.00–2.44 h fell within the range of values observed in the laboratory thatŽ . Ž .were used to obtain Eq. 10 . Using Eq. 10 , predicted k values based on the field

values of Õ , L and R of Ptacek and Gillham were 2.0–5.7 times lower than theo

reported ones, but fell within or close to the upper 95% confidence limit of theŽ .predictive relationship Fig. 4 . Deviations between predicted values of k and those

obtained by Ptacek and Gillham in the field could be due to the fact that the soil in thefield was not homogenized.

4.3.4. Data set 4Ž .Goltz and Roberts 1986 also conducted a field experiment at the Borden Site. They

Ž y4 y1.obtained identical values of k 1.7=10 h for carbon tetrachloride, and tetra-chloroethylene by analyzing aqueous samples collected 5 m from a line of injectionwells. The average-pore water velocity was 0.33 cmrh, and the retardation coefficientswere 1.8 and 3.0, respectively. Based on this, the residence time would be 2.7=103 and4.5=103 h, approximately 18 times higher than the largest value associated with thelaboratory studies listed in Table 2. The mass-transfer coefficient was about two ordersof magnitude lower than the lowest laboratory value cited in this study. Brusseau et al.Ž . Ž .1991a suggested that the low k value obtained by Goltz and Roberts 1986 could bedue to heterogeneity in hydraulic conductivity and sorption capacity. Fig. 4 shows,

Ž .however, that the low k values of Goltz and Roberts 1986 are due to the largeresidence time of the field experiment.


Fig. 4. Laboratory-and field-scale sorption nonequilibrium mass-transfer coefficients.

Ž .Based on Goltz and Roberts large-scale field experiment, it appears that Eq. 10 maybe extended to large values of residence time. However, since the developed predictiverelationships were obtained based on statistical procedures, demonstrated applicability toavailable data sets may not necessarily translate to their applicability to other investiga-tions having parameters that are beyond the range examined herein. Therefore, addi-tional field data are needed to support the use of these relationships for transportscenarios involving residence times beyond the values explored in this study.

4.4. Simulation of field data

Ž .Eq. 10 was used to simulate field transport data for 1,4-dioxane, tetrahydrofuranŽ .and diethyl ether reported by Bahr 1989 . The field study was conducted to assess the

effectiveness of well-purging for aquifer decontamination at the Gloucester Landfill nearOttawa, Ontario. Significant concentrations of tetrahydrofuran and diethyl ether hadbeen detected in the test area at depths 12–23 m. Field desorption data of the three

Ž .organic compounds Fig. 11 of Bahr were obtained by analyzing samples collectedfrom a sampling well located at a distance 2.5 m from the injection well. Reportedretardation coefficients were 1.6 for 1,4-dioxane, 2.2 for tetrahydrofuran, and 3.3 fordiethyl ether.

During an approximately 6-day test period, clean water was injected through aninjection well. Injection flow rate was 140 mlrmin during the first 24.7 h and was thenreduced to 94.6 mlrmin for approximately 94 h. The mean transit time of an ideal tracerduring the phase of injecting the higher flow rate was found to be 12.1 h. Thiscorresponds to an average pore-water velocity of 0.207 mrh. For the lower flow rate,


the velocity was determined from the ratio of the two flow rates and the velocityŽ .corresponding to the higher flow rate i.e. 0.207=94.6r140s0.14 mrh . The disper-

Ž .sion coefficient was determined as the product of the reported dispersivity 2 cm andpore-water velocity. The mass-transfer coefficient for each compound under the two

Ž .flow conditions was estimated using Eq. 10 .

Fig. 5. Simulation of field transport using laboratory-derived mass transfer coefficients. Symbols are Bahr’sfield data, solid lines are for Fs0, and dotted lines are for Fs0.38.


ŽTo generate model simulation for the three compounds, the value of F the fraction. Ž .of instantaneous sorption sites needs to be determined. Karickhoff 1980 and Karick-

Ž .hoff and Morris 1985 evaluated sorption kinetics for several hydrophobic organiccompounds using a two-site nonequilibrium model. These authors found that approxi-mately 50% or less of the sorption sites were in equilibrium. This is in agreement with

Žthe average value of F found from the collected laboratory studies listed in Table 2 i.e.. Ž .0.38 . But the standard deviation associated with this value was relatively high 0.23 ,

indicating a significant scattering between the reported values. High uncertainty in thevalue of F can be found even for a certain solute within a single investigator. Brusseau

Ž .et al. 1991a reported a relationship between F and K . At approximately similarow

residence times, no apparent trend has been observed between F and log K , SOC, orow

k for the data cited in this study. Two different values of F have been used to simulateBahr’s data: one representing the average value for the studies listed in Table 2 and the

Ž .other is 0 the case for one-site nonequilibrium model . The latter value represents thelower limit for this parameter and implies that all the sorption sites are under nonequilib-rium conditions.

Simulation of the field data was accomplished using the CXTFIT code version 2.0Ž .Toride et al., 1995 . In order to determine the initial point for simulating lower velocityconditions, the maximum field concentration after reducing the flow rate was matched

Ž .with the same simulated concentration assuming lower velocity at all times . The timefor the simulated concentration was then adjusted for this and all the points afterward.

As shown in Fig. 5, better simulation was obtained with Fs0, especially for diethylether and for tetrahydrofuran at the lower velocity. More importantly, the majority of thedata points for the three compounds fall between the simulation lines corresponding tothe two F values. This illustrates the ability to simulate the transport of organiccompounds under field conditions using mass-transfer values that are estimated from Eq.Ž .10 and further demonstrates the practicality of this equation.

5. Conclusions

Mass-transfer coefficient in transport models employing first-order rate limitationsdepends on flow conditions, particle size, and the extent of interaction between thesolute and the porous media. The ability to reasonably estimate field mass transfercoefficients from relationships developed using laboratory data provides evidence thatsupports the apparent time dependence mass-transfer coefficient. Although it is unclearwhy the coefficient does not retain a constant value, the developed relationships alongwith the existing model formulations may prove useful for practical applications.

Acknowledgements

This research was funded in part by the Scientific Research Council at the UnitedArab Emirates University. I am grateful to A.M.O. Mohamed and E. Abdul Hafiz fortheir comments on earlier versions of this manuscript. I also appreciate the usefulcomments from the anonymous reviewers.


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