Influence of mass transfer resistance on detection of nonaqueous phase liquids with partitioning tracer tests

Advances in Water Resources 27 (2004) 429–444 www.elsevier.com/locate/advwatres

Influence of mass transfer resistance on detection of nonaqueous phase liquids with partitioning tracer tests Paul T. Imhoff *, Katayoun Pirestani Department of Civil and Environmental Engineering, 301 DuPont Hall, University of Delaware, Newark, DE 19716-3120, USA Received 18 December 2003; accepted 4 February 2004

Abstract Partitioning interwell tracer tests (PITTs) are a relatively new technique for measuring the amount of nonaqueous phase liquid (NAPL) within saturated porous media. In this work we examined the influence of mass transfer limitations on the accuracy of measured NAPL from PITTs. Two mathematical models were used along with laboratory column experiments to explore the influence of tracer partition coefficient, tracer detection limit, and injected tracer mass on NAPL measurements. When dimensionless mass transfer coefficients were small, NAPL measurement errors decreased with decreasing tracer partition coefficient, decreasing tracer detection limit, and increasing injected tracer mass. Extrapolating breakthrough curves exponentially reduced but did not eliminate systematic errors in NAPL measurement. Although transport in a single stream tube was used in the mathematical models and laboratory experiments, the results from this simplified domain were supported by data taken from a three-dimensional computational experiment, where the NAPL resided as large pool. Based on these results, we suggest guidelines for interpreting tracer breakthrough data to ascertain the importance of mass transfer limitations on NAPL measurements.
2004 Elsevier Ltd. All rights reserved. Keywords: Nonaqueous phase liquids (NAPLs); Tracers; Moment analysis

1. Introduction A particularly difficult task facing engineers and managers concerned with subsurface spills of nonaqueous phase liquids (NAPLs) is determining where the NAPL is and how much is there. Borrowing from past work in petroleum reservoir engineering [5,34] tracer techniques were developed for characterizing the NAPL source zone and assessing the performance of remediation technologies [2,21,26,29]. Partitioning tracers have been used to determine domain-average NAPL saturations [2,21] as well as the spatial distribution of the NAPL [8,19], while interfacial tracers have been used to determine the contact area between the NAPL and the aqueous phase [23]. Here, the focus is on NAPLs that are immobile, or are moving at velocities much slower than the surrounding groundwater.

* Corresponding author. Tel.: +1-302-831-0541; fax: +1-302-8313640. E-mail addresses: imhoff@ce.udel.edu, imhoff@copland.udel.edu (P.T. Imhoff), [email protected] (K. Pirestani).

0309-1708/$ - see front matter
2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2004.02.010

The theory behind partitioning interwell tracer tests (PITT) and their application to NAPL spills was first discussed by Jin et al. [21]. Simply stated, in this technique several conservative and partitioning tracers are injected into the aqueous phase and move through a porous medium contaminated with NAPL. Chromatographic separation occurs because some tracers preferentially partition into the NAPL, which is quantified through tracer concentration measurements from an extraction well. This separation is used to determine the volume of NAPL and the average NAPL saturation in the region swept by the tracers between the injection and extraction wells. The reliability of partitioning tracer measurements was examined in a recent review [29]. A number of factors were found to affect the reliability of PITTs: nonuniform NAPL distribution, possibly associated with porous medium heterogeneity; zones with high NAPL saturations, i.e., NAPL pools; unintended sorption of tracers onto the porous medium, e.g., organic matter, mineral components, or perhaps a sorptive layer resulting from surfactant flushing; changes in tracer partitioning due to changes in NAPL composition after

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remediation efforts; and biotic or abiotic degradation of tracers in situ. Each of these factors may lead to errors in measurements of NAPL volume or saturation during a PITT. Dwarakanath et al. [10] analyzed the random and systematic errors associated with PITT for NAPL measurement. They concluded that random errors for measurement of NAPL saturation would be approximately 10% or less if the retardation factors associated with the partitioning tracers were greater than 1.2. They also concluded that systematic errors due to error in the measurement of volume of produced fluid or tracer water concentrations would have a very minor effect on NAPL volume measurements. While these authors acknowledged the existence of other sources of systematic error, e.g., nonuniform NAPL distribution, they concluded that many were site specific and might be eliminated or minimized with appropriate design of the PITT. When NAPL is located nonuniformly in a domain, either as millimeter-scale ganglia or pools that are centimeter-scale and larger, the flow paths of the injected tracer solution may bypass NAPL-contaminated zones. In this case, the transfer of tracer mass from the main flow paths to the NAPL may be slow, resulting in extensive tailing of tracer breakthrough curves. The effects of nonuniform NAPL distribution and local-scale mass transfer resistance are referred to as ‘‘mass transfer limitations’’ in this paper. The effect of mass transfer limitations on NAPL measurement has been observed in laboratory experiments [7,27], in numerical simulations [39], and in a field study [3]. For example, in a recent field test where known volumes of tetrachloroethylene were spilled in a test cell, extensive tailing of tracer breakthrough curves occurred and was attributed in part to hydrodynamic inaccessibility of the NAPL [3]. Using tracer data from tracers with large partition coefficients, measured NAPL volumes were often smaller than spilled volumes, which was attributed to the influence of mass transfer limitations. While mass transfer limitations during PITTs have been observed before, in this investigation we systematically studied the influence of these limitations on measurement of NAPL saturation. Our objectives were to (1) elucidate the conditions where mass transfer resistance may be important; (2) determine which design parameters, e.g. injected tracer mass or tracer partition coefficient, may be altered to minimize the effect of these limitations; and (3) ascertain how analysis of tracer breakthrough data may be used to estimate the significance of mass transfer limitations on estimated NAPL saturation. Groundwater velocity will influence mass transfer limitations, but was not examined in this study. To achieve the study objectives, we examined tracer transport along a single stream tube in one dimension using mathematical models and laboratory experiments.

The mathematical models were selected to examine two situations where mass transfer limitations are important: model 1 where mass transfer limitations affect only the partitioning tracer, and model 2 where mass transfer limitations affect both conservative and partitioning tracers. The laboratory experiments were used to confirm predictions from the mathematical modeling. Similar one-dimensional analyses have been used to elucidate the effects of mass transfer limitations on groundwater tracer tests in laboratory [41] and field [24] experiments. While these one-dimensional analyses do not capture all processes affecting tracer transport in the field, e.g., spatially variable velocity fields, they do allow us to examine the influence of mass transfer limitations on tracer transport and NAPL measurement for simplified conditions. A similar approach using a onedimensional model was used to assess random errors in PITT measurements from measured retardation factors [10]. Finally, we examined the results from a threedimensional numerical simulation of tracer transport during a PITT in a nonuniform flow field. The results from this simulation support the general conclusions drawn from our one-dimensional analyses.

2. Mathematical models When NAPLs are spilled or leaked below the water table, they move in response to gravitational, viscous, and capillary forces. NAPLs eventually stop moving when gravitational and viscous forces are not sufficient to overcome the capillary forces that inhibit NAPL migration. The NAPLs then reside as isolated ganglia that typically occupy 1–20% of the pore space, referred to as residual saturation, or in pools that may fill >50% of the pore openings and range from centimeters to meters in scale. Centimeter-scale pools may be associated with capillary-heterogeneity trapping [11,12], while meter-scale pools may be associated with layers of low permeability material of significant spatial extent [32], e.g., a clay aquiclude. The distribution of the NAPL ganglia and NAPL pools within the subsurface will be a function of the volume and duration of the NAPL spill; NAPL density and viscosity; porous medium structure and wettability; and groundwater flow. In the analysis below the NAPL was considered immobilized in the domain as either ganglia or small centimeter-scale pools. Two mathematical models were used to evaluate the influence of mass transfer limitations on NAPL saturation measurements with partitioning tracers: a single-region model with a uniform aqueous-phase velocity field and residual NAPL saturation distributed uniformly throughout the domain, and a two-region model with a mobile and immobile domain, with all NAPL residing in the immobile domain, e.g. in NAPL pools. These two models and

P.T. Imhoff, K. Pirestani / Advances in Water Resources 27 (2004) 429–444

431

retardation coefficient. A similar approach was followed in this work, where the one-dimensional models were intended to elucidate the effects of mass transfer limitations on tracer breakthrough curves. While these models do not exactly reproduce conditions in field-scale tracer tests, they do illustrate the influence of mass transfer limitations on tracer breakthrough and thus provide insight into the effect of mass transfer limitations on NAPL saturation measurements. 2.1. Model 1: single-region Transport of a tracer that partitions into and out of an immobile NAPL distributed uniformly along the stream tube may be described by the following equations [9,18], which are dimensionless with the exception of concentration

Fig. 1. ‘‘Real systems’’ and the equivalent models for tracer transport through these stream tubes. (A) NAPL distributed in many low NAPL saturation regions, and (B) the equivalent system, a single-region model, where the NAPL is distributed uniformly at an even smaller NAPL saturation. (C) NAPL distributed in many regions of high NAPL saturation where the aqueous and nonaqueous phases are immobile, and (D) the equivalent system, a two-region model, where the NAPL is distributed in a single high saturation NAPL pool, which is in the immobile domain.

cartoons of the ‘‘real systems’’ they are intended to emulate are depicted in Fig. 1. In the single-region model mass transfer resistance between the aqueous and nonaqueous phases only affects the partitioning tracer, while in the two-region model mass transfer resistance between the mobile domain that is NAPL-free and the immobile domain containing NAPL and water affects both conservative and partitioning traces. Aqueous phase flow within the immobile domain is assumed to be negligible. Both models describe transport and retardation of tracers along a single stream tube. These one-dimensional models assume homogeneous conditions along the stream tube, in particular firstorder mass transfer rate coefficients that are invariant in space. A similar approach was used by Dwarkanath et al. [10] to evaluate random errors associated with measurement of tracer retardation coefficients. In that work, a one-dimensional transport model that included advection, dispersion, and equilibrium partitioning between tracers and NAPL were fit to both laboratory and field data. While the fits of the one-dimensional model demonstrated the reasonableness of this approach for describing tracer transport in the laboratory and the field, their model was not intended to provide an exact representation of tracer transport. Instead, it was used to estimate random errors in measurement of the tracer

oca ðRf  1Þ ocn 1 o2 ca oca þ ¼  Pna Pe oX 2 oX oT oT

ð1Þ


 ocn cn ¼ DaPna ca  oT Pna

ð2Þ

with T ¼

va t L

ð3Þ

X ¼

x L

ð4Þ

Pe ¼

va L D

ð5Þ

Rf ¼ 1 þ

Da ¼

Sn Pna ð1  Sn Þ

Kna L va nSn Pna

ð6Þ

ð7Þ

In these equations ca is the concentration of the tracer in the a phase; Rf is the retardation factor; Pna is the dimensionless nonaqueous–aqueous phase partition coefficient; Pe is the Peclet number; Da is a type of Damkohler number, and is the ratio of the rate of mass transfer to the rate of advection; va is the mean interstitial velocity of the a phase; L is a characteristic length, taken as the distance between tracer injection and the point of tracer measurement; D is the hydrodynamic dispersion coefficient; Sa is the saturation of phase a; n is porosity; and subscripts a and n refer to aqueous and nonaqueous phases, respectively. Kna is an overall aqueous-side mass transfer rate coefficient and accounts for resistances to mass transfer within the aqueous and nonaqueous phases. Eq. (1) presumes that the NAPL is distributed uniformly in a homogeneous porous medium, and the rate of change in NAPL saturation due to

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P.T. Imhoff, K. Pirestani / Advances in Water Resources 27 (2004) 429–444

dissolution is small with respect to the time scale of tracer transport. The two-film model [6] was used in defining Kna , which we assume is dominated by resistance in the aqueous phase. This is a reasonable assumption for many NAPLs of low viscosity distributed as small ganglia [14]. If, however, the NAPL is particularly viscous or distributed as large ganglia, mass transfer resistance may be dominated by the nonaqueous phase, in which case Kna is an overall NAPL-side mass transfer rate coefficient and the Damkohler number is defined as Da ¼

Kna L va nSn

ð8Þ

In the analysis below, we assume that mass transfer resistance is dominated by the aqueous phase, although we comment on how the results would be altered if nonaqueous-phase resistance was most important. Pulse inputs of tracers were used in the simulations, since this is the most common method of introducing tracers in laboratory and field experiments. In this case the influent boundary condition was 1 oca ð0; T Þ ¼ cinf a ðT Þ Pe oX

mass transfer between the mobile and immobile domain, which affected both conservative and partitioning tracers. We show below that this model provided a reasonable fit to data from laboratory experiments where the NAPL resided in a centimeter-scale pool. Transport of a tracer that partitions into and out of an immobile NAPL that resides entirely in an immobile flow domain can be described by the following equations, which are dimensionless with the exception of concentration
 m /m ocim 1 o2 c m ocm m oca a a a /a þ 1 ð11Þ  Rf a ¼ Pe oX 2 oT Rf oT oX


 /m ocim im a 1 Rf a ¼ xðcm a  ca Þ Rf oT

ð12Þ

with hm a ha

ð13Þ

cL ha v a

ð14Þ

/m a ¼ x¼

Tracers were assumed to be absent from the domain initially, i.e., ca ðX ; 0Þ ¼ cn ðX ; 0Þ ¼ 0. In applying this model to tracer transport in a single stream tube, we assumed Pe ¼ 100, which is representative of conditions observed in our laboratory experiments. Other parameters required for the solution to these equations (Da, Sn , and Pna ) were varied to examine their effect on tracer breakthrough and NAPL measurement.

In these equations /m a is the fraction of the aqueous phase that resides in the mobile domain, cm a is the tracer concentration of the aqueous phase in the mobile domain, cim a is the tracer concentration of the aqueous phase in the immobile domain, x is a dimensionless mass transfer coefficient, ha is the volumetric fraction of the entire stream tube occupied by the aqueous phase, hm a is the volume of aqueous phase in the mobile domain normalized by the total volume of the stream tube, c is a first-order mass transfer coefficient for transport between the mobile and immobile domains, and all other parameters are defined as in model 1. The initial and boundary conditions assumed for model 2 were the same as those in model 1. We also assumed Pe ¼ 100 and /m a ¼ 0:85, while other parameters required for the solution to these equations (x, Sn , and Pna ) were varied to examine their effect on tracer breakthrough and NAPL measurement.

2.2. Model 2: two-region, mobile/immobile

2.3. Time moment analysis

In the second model NAPL was assumed to reside entirely in an immobile domain, where the aqueousphase velocity was zero. The fraction of the stream tube occupied by the immobile domain and the NAPL saturation within this domain were assumed to be constant along the stream tube (see Fig. 1). Transport of the partitioning tracer from the flowing aqueous phase in the mobile domain to the NAPL in the immobile domain may be limited by mass transfer resistance between the two domains or mass transfer resistance between the aqueous and nonaqueous phases within the immobile domain. Here, we assumed that the slowest process was

The influence of physical and chemical nonequilibrium, i.e. rate-limited diffusive mass transfer and rate-limited sorption, on breakthrough of sorbing compounds has been described through time moments [13,33,36]. Time moment analysis was also used here to illustrate the influence of mass transfer limitations on transport of partitioning tracers. Because of the similarity between transport of partitioning tracers and sorbing solutes, we followed the work of Valocchi [36] and assumed a semi-infinite domain and a Dirac input of tracer mass at the column inlet. The initial and boundary conditions for this situation are

ca ð0; T Þ 

ð9Þ

0 where cinf a ¼ ca is the solute concentration of the square wave input for 0 < T < Tp , and cinf a ¼ 0 at all other times. Flux-averaged effluent concentrations at the end of the stream tube (X ¼ 1) were determined from

ceff a ð1; T Þ ¼ ca ð1; T Þ 

1 oca ð1; T Þ Pe oX

ð10Þ

P.T. Imhoff, K. Pirestani / Advances in Water Resources 27 (2004) 429–444

ca ðX ; 0Þ ¼ cn ðX ; 0Þ ¼ 0

ð15Þ

ca ðX ! 1; T Þ ¼ cn ðX ! 1; T Þ ¼ 0

ð16Þ


 1 oca MdðT Þ ca  ¼ Pe oX X ¼0 nSa L

ð17Þ

where M is the mass of tracer injected per unit column cross-sectional area, and dðT Þ is the Dirac Delta function. The equations describing model 1 (Eqs. (1) and (2)), and (15)–(17) and model 2 (Eqs. (11) and (12)), and (15)–(17) are identical in form to Eqs. (11), (17), and (28)–(30) of Valocchi [36] that were used to describe solute transport with physical nonequilibrium using a first-order rate model. Thus, by paying careful attention to differences in parameters between these sets of equations, the time moment formulas can be taken from this earlier work. The resulting time moment formulas for the first moment, and the second and third central moments are shown in Table 1. The first moments are independent of the dimensionless mass transfer coefficients (Da and x), demonstrating that mass transfer limitations should have no influence on the mean arrival time of partitioning tracers. Mass transfer resistance does influence the second and third central moments, which are measures of the degree of spreading and asymmetry of the concentration response curves. Spreading and asymmetry increase as Da or x decrease. Where mass transfer resistance is important, increases in Pna results in increased Rf (models 1 and 2) and decreased Da (model 1), both of which result in more spreading and asymmetry of the concentration response curve. Although the modeling and experimental study used pulse inputs rather than a Dirac input, the influence of the various parameters on the moments for Dirac inputs should be similar to that for pulse inputs. Thus, the observations made above will be important for interpreting results from the modeling and experimental study. 2.4. Analytical methods Simulations of each model were obtained using analytical solutions. Both models were cast in forms that were solved with CXTFIT [35]. Breakthrough curves

433

(X ¼ 1) were obtained for both models using squarewave or pulse inputs, with the number of data points varied between simulations such that a smooth transition in effluent concentrations was obtained. The density of data points was varied in selected simulations to verify that the number of data points was sufficient to characterize the breakthrough curve. Moment analysis was used to determine the mean travel time of the conservative and partitioning tracers R tF i tc dt tp 0i ð18Þ l1 ¼ R0tF a  cia dt 2 0 where l0i1 is the mean travel time of the conservative (i ¼ c) or partitioning (i ¼ p) tracer, tF is the time that the tracer measurements were terminated, and tp is the duration of the input pulse. The concentration of the tracer at t ¼ tF was designated as the tracer detection limit, cda . The retardation factor for the partitioning tracer was then computed from Rf ¼

l0p 1 l0c1

ð19Þ

and the NAPL saturation determined from Sn ¼

1  Rf 1  Rf  Pna

ð20Þ

In model 1 spreading of the conservative tracer was influenced only by the Peclet number, since Rf ¼ 1 (see Table 1). Thus, mass transfer limitations did not affect transport of the conservative tracer, and for this reason we assumed that the mean travel time of the conservative tracer was determined exactly. In model 2, though, spreading was significant for both the conservative and partitioning tracers, and both tracer breakthrough curves were integrated as shown above to determine the mean travel times.

3. Experimental study The experimental study was used to evaluate the effects of mass transfer limitations on NAPL saturation measurements under controlled conditions. Eighteen partitioning tracer tests were conducted in a 7.6-cm inside diameter, 30-cm long glass column fitted with aluminum end fittings and packed with two quartz sands: a

Table 1 Time moment formulas for Dirac input in a homogeneous stream tube Momenta

Model 1

l01

XRf

l2

2XR2f Pe

l3

12XR3f Pe2

a

Model 2 XRf

þ

2X ðRf 1Þ Da

ðRf 1ÞRf f 1Þ þ 12X Pe þ 6X ðR Da Da2

l01 is the first moment, while ln is the nth central moment.

2XR2f Pe

m 2

þ 2X ðRfx/a Þ

12XR3f Pe

m 2

/a Þ þ 12X ðRfPex

Rf

m 3

a Þ þ 6X ðRfx/ 2

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P.T. Imhoff, K. Pirestani / Advances in Water Resources 27 (2004) 429–444

Fig. 2. Schematic of cell used in column experiments. Dimensions of the NAPL lens (l  w  h) differ between packings and are given in the text.

rectangular coarse lens of 12/20 Accusand (d50 ¼ 1:105 0:014 SD mm, d60 =d10 ¼ 1:231 0:043 SD, SD ¼ one standard deviation [31]) that spanned the column diameter surrounded by a fine medium of 40/50 Accusand (d50 ¼ 0:359 0:01 SD mm, d60 =d10 ¼ 1:200 0:018 SD [31]). The column and sand lens are shown in Fig. 2. Both sands were obtained from Unimin Corportation (Le Sueur, MN). The column was saturated with water, and then HPLC grade trichloroethylene (TCE) dyed with Oil Red O (Aldrich Chemical Company) at a dye concentration of 120 mg/l was injected into the coarse sand lens with a glass syringe to form a NAPL pool. The NAPL saturation within the pool was Sn 65–85%, based on the volume of NAPL injected and the measured porosity and dimensions of the coarse sand lens. The columnaverage NAPL saturation was Sn ¼ 0:073 0:003 in all experiments. The configurations of the NAPL pools varied between experiments: experiments 7, 9, and 10 had lenses 7.5 · 7.6 · 2.15 cm (packing 1), length · width · height; experiments 1, 3, 5, 6, and 11 had lenses 14.5 · 7.6 · 1.2 cm (packing 2); and experiments 2, 4, 8, and 12–18, 7.5 · 7.6 · 1.9 cm (packing 3). Partitioning tracer tests were conducted by injecting a pulse of tracer solution into the column followed by TCE-saturated water to inhibit NAPL dissolution. Isopropyl alcohol (IPA, Fisher Scientific Company, Fair Lawn, NJ), 2,3-dimethyl-2-butanol (DMB, Aldrich Chemical Company, Milwaukee, WI), and 1-hexanol (HEX, ICN Biomedicals Inc., Aurora, OH) were selected as tracers and mixed with distilled, deionized water before injection. IPA is a conservative tracer and was injected in all experiments, while DMB and HEX are partitioning tracers and were injected together or alone depending on the experiment. When injected together at high concentrations, DMB and HEX sometimes interacted resulting in nonlinear partitioning behavior [18]. To eliminate the possible influence of these data on the analysis of mass transfer effects, experiments where both DMB and HEX were injected are not included in the results presented below. In this case, experiments from packing 1 were eliminated.

The Darcy flux was maintained at 0.16 m/day in the experiments reported below. Effluent samples from the column were collected in precapped gas chromatography (GC) autosampler vials using a fraction collector and analyzed by GC flame ionization detection. Detection limits for all tracers were approximately 1–2 mg/l. In the experiments reported below, except for experiment 18, IPA and DMB were the tracer pair injected. Partition coefficients for these two tracer were determined by batch partitioning experiments conducted at 15, 22, and 25 C to span the range of experimental temperatures (18–24 C), and are reported elsewhere [18]. IPA partitioning into TCE was negligible, while DMB partitioning was significant and a function of experimental temperatures, with partitioning described by a van’t Hoff-type expression log10 Pna ¼ 6:74 þ

1741 K A

ð21Þ

where A is the temperature in degrees Kelvin [18]. In experiment 18, HEX partitioning was measured at the experimental temperature (22.0 ± 1.1 C) following the same procedures used for IPA and DMB. The partition coefficient for HEX was Pna ¼ 19:3, which is 2.8 times greater than that for DMB at 22 C. The experiments differed in the configuration of the NAPL lens, short and thick for some experiments and long and thin in others; the tracer pulse size, between 0.09 pore volumes (PV) to 1.14 PV; concentrations of the injected tracers, from 25 to 1760 mg/l; and partitioning tracer, DMB or HEX. This allowed us to examine the role of injected tracer mass and tracer partition coefficient on NAPL measurement.

4. Results 4.1. Mathematical models 4.1.1. Model 1: single-region Example tracer breakthrough curves (X ¼ 1) are shown for conservative and partitioning tracers using model 1 in Fig. 3. Data are plotted versus pore volumes, defined here as the volume of the pore space occupied by the aqueous phase alone. In these simulations Pe ¼ 100, Pna ¼ 8, and Sn ¼ 0:03. Asymmetric spreading of the partitioning tracers systematically increased as Da decreased from Da ¼ 10 to Da ¼ 0:5. While time moment analysis indicates that the first central moment is unaffected by mass transfer limitations, these plots suggest that if Da is too small and the tracer detection limit too high, insufficient data may be collected to characterize the tracer tail and the mean tracer arrival time. This in turn should lead to underestimation of NAPL in the system, if extrapolation of the data is not performed or

P.T. Imhoff, K. Pirestani / Advances in Water Resources 27 (2004) 429–444

o

ca /c a [-]

Da = 0.5

0.01

0.001 0

1

2

3

4

Pore Volumes [-] Fig. 3. Breakthrough curves for model 1 for a conservative tracer and partitioning tracers with three different Damkohler numbers. For these simulations Pe ¼ 100, Pna ¼ 8, Sn ¼ 0:03, Tp ¼ 0:064 PV, and X ¼ 1.

is insufficient to predict trends in the data below the detection limit. The Damkohler number will vary between systems and is a function of the parameters shown in Eq. (7). Using empirical correlations for Kna from the literature [17,25,28] that show Kna to be a function of Sn and va , Da were estimated for L ¼ 5 m, nSn ¼ 0:01, Pna ¼ 40, and va 6 1:0 m/day for typical unconsolidated porous media. For these empirical correlations and this set of conditions, Da P 10. Thus, for unconsolidated media where NAPL is distributed uniformly as ganglia throughout the domain, mass transfer limitations are not expected to significantly affect tracer transport. However, if NAPL ganglia are distributed nonuniformly, flow bypassing occurs and Kna averaged over the system is significantly smaller than that estimated from the empirical correlations that assume uniform NAPL distributions. In such cases, system-average Da may be less than 10. This situation has occurred in laboratory experiments where dissolution of nonuniformly distributed NAPL was studied [4,30]. For example, in one experiment in a two-dimensional flow cell with nonuniform NAPL distribution, the average Kna was 3 orders of magnitude lower than that obtained from experiments with a uniform distribution of NAPL ganglia [4]. Thus, mass transfer limitations between partitioning tracers and NAPL may be important under some field settings. Partitioning tracer mass was incorporated as a model parameter by normalizing the tracer detection limit, cda , with a concentration that represents the mass of injected tracer divided by an effective stream tube volume [41]. The effective stream tube volume represents the total capacity of the stream tube for the tracer, and is the aqueous phase pore volume times the retardation factor. With this definition the normalizing concentration is expressed as M=ðV ha Rf Þ, where M is the mass of tracer injected in the system, V is the volume of the stream

1

) [-]

Da = 1

0.1

0.1

o a /(Tpc a /R f

Da = 10

tube, and ha is the aqueous phase volume fraction. A similar normalizing concentration was used in a study of solute transport in NAPL-free porous media [41]. The mass of tracer injected can be expressed as M ¼ Tp V ha c0a , where Tp is the time of the injected tracer pulse in aqueous phase pore volumes, and c0a is the tracer concentration in this pulse. In this case M=ðV ha Rf Þ ¼ Tp c0a =Rf . To illustrate the influence of mass transfer limitations on NAPL detection, the error in measurement of NAPL saturation is shown in Fig. 4 as a function of Sn and cda =ðTp c0a =Rf Þ, the normalized tracer detection limit. All results are shown for Da ¼ 1, which was selected to represent conditions were mass transfer limitations are important. Effluent tracer concentration data were not extrapolated. The three curves shown represent errors in computed Sn of 20%, 40%, and 80%. For fixed Sn and a fixed mass of injected tracer M ¼ Tp c0a V ha , the error in NAPL measurement increases as cda increases. To maintain an error in computed Sn of 20% as Sn decreases from Sn ¼ 0:05 to Sn ¼ 0:002, either cda must decrease or Tp c0a =Rf must increase. Curves of similar shape were found for Da ¼ 0:1. To illustrate the reasonableness of the scaling and the effect of tracer slug size, data are shown for three different pulse sizes in Fig. 4. Pulses were varied from Tp =Rf ¼ 0:01 to Tp =Rf ¼ 0:25; the mass of tracer injected in the system M was kept the same for all pulses by adjusting the influent tracer concentration as required. The results indicate that there is a negligible difference in required tracer detection limit for the different slug sizes. Instead, what is of primary importance is the mass of tracer injected into the system, relative to the tracer detection limit. The curves in Fig. 4 are not intended to be used as design parameters for the field, since Da is unknown before conducting any field test. However, these curves illustrate the importance of both tracer

0.01

80 40 20

T p /R f = 0.25 T p /R f = 0.05 T p /R f = 0.01

d

Conservative

c

1

435

0.001

0.0001 0.00

0.01

0.02

0.03

0.04

0.05

0.06

S n [-] Fig. 4. Normalized tracer detection limits required for determination of Sn that is accurate to within 20%, 40%, and 80% of its actual value when model 1 applies. For these simulations Pe ¼ 100, Pna ¼ 8, Da ¼ 1, and X ¼ 1.

P.T. Imhoff, K. Pirestani / Advances in Water Resources 27 (2004) 429–444

1

0.1

0.01

c

d o a /(Tpc a

/R f ) [-]

10

Da = 10 Da = 1 Da = 0.5

0.001

0.0001 0.00

0.01

0.02

0.03

0.04

0.05

0.06

Sn [-] Fig. 6. Effect of Da on normalized tracer detection limits required for determination of Sn to within 20% of its actual value when model 1 applies. For these simulations Pe ¼ 100, Pna ¼ 8, Sn ¼ 0:03, and X ¼ 1.

Da 10 5

1

0.5

1000

Sn = 0.05

Sn = 0.03

100

Sn = 0.01 Sn = 0.002

10

d

detection limit and mass of tracer injected in measuring NAPL with PITTs. Tracer breakthrough curves are typically extrapolated, usually with an exponential function, to minimize data truncation errors [2,3]. If an exponential function is fitted to the tailing concentrations and extrapolated analytically to zero for the simulation results shown in Fig. 4, the required tracer detection limit (cda ) for an equivalent NAPL measurement error increases. This is illustrated in Fig. 5, where curves are shown representing results for 20% error in Sn measurement when data are extrapolated and when they are not. Parameters common to both sets of simulations are given in the figure caption. While exponential extrapolation reduces the measurement error, the error still increases as the NAPL detection limit increases. The influence of Da on NAPL measurement error is shown in Fig. 6. As Da increases, the required tracer detection limit for a 20% error in Sn also increases. Or, stated in a slightly different way, as mass transfer resistance increases, Da decreases and a smaller cda is needed to achieve a prescribed error in Sn measurement. In recent laboratory [7,18] and field [3] experiments to estimate NAPL saturation from partitioning tracer tests, investigators observed that measured Sn or NAPL volume decreased with increasing Pna . This observation was unexpected, since based on an analysis of random errors the imprecision in Sn should decrease with increasing Pna [10]. These investigators hypothesized that the influence of mass transfer limitations might be more significant for larger Pna . To further explore these observations, simulations were run with a fixed tracer pulse size (Tp ¼ 0:20 when Sn ¼ 0) and injected concentration c0a ¼ 1000 mg/l, but with varying Pna . Because Da is inversely related to Pna (see Eq. (7)), Da decreases as Pna increases. In Fig. 7(A) the required detection limit for a 20% error in NAPL

ca [mg/l]

436

1

0.1 0

20

40

60

80

100

120

140

160

P na [-]

(A)

Da 10 5

1

0.5

100 1

Sn = 0.05 Sn = 0.01

0.1

Sn = 0.002

TF [-]

c da /(T pc oa /R f ) [-]

Sn = 0.03

10

0.01 Extrapolated 0.001

Not Extrapolated

1 0.0001 0.00

0 0.01

0.02

0.03

0.04

0.05

0.06

Sn [-] Fig. 5. Effect of extrapolation on normalized tracer detection limits required for determination of Sn to within 20% of its actual value when model 1 applies. For these simulations Pe ¼ 100, Pna ¼ 8, Sn ¼ 0:03, Da ¼ 1, and X ¼ 1.

(B)

20

40

60

80

100

120

140

160

Pna [-]

Fig. 7. Tracer detection limits (A) or throughputs (B) required for determination of Sn to within 20% of its actual value when model 1 applies. For these simulations Pe ¼ 100, Pna ¼ 8 when Da ¼ 10, Sn ¼ 0:03, Tp c0a ¼ 200 mg/l, and X ¼ 1. Da changes as Pna increases according to Eq. (7).

P.T. Imhoff, K. Pirestani / Advances in Water Resources 27 (2004) 429–444

4.1.2. Model 2: two-region, mobile/immobile Example tracer breakthrough curves (X ¼ 1) are shown in Fig. 8 for conservative and partitioning tracers using model 2 for two different values of the dimensionless mass transfer coefficient. In these simulations Pe ¼ 100, Pna ¼ 8, Sn ¼ 0:03, and /m a ¼ 0:85. With model 2 mass transfer limitations between mobile and immobile domains affect transport of both the conservative and partitioning tracers. Asymmetric spreading of these tracers increases as x decreases from x ¼ 1 to x ¼ 0:1. If the mass transfer coefficient is too small and 0c the tracer detection limit too high, both l0p 1 and l1 would be underestimated, although underestimation 0c of l0p 1 would be more than l1 as can be inferred from

1 ω = 1, Conservative ω = 1, Partitioning

ca /coa [-]

saturation is shown as a function of Pna . As Pna increases from Pna ¼ 8 to Pna ¼ 160, Da decreases from Da ¼ 10 to Da ¼ 0:5 and the required detection limit for a 20% error in Sn decreases approximately 1 order of magnitude for all values of Sn tested. Thus, while the random error associated with the measurement of Sn decreases with increasing Pna [10], the systematic error due to mass transfer limitations increases with increasing Pna . While laboratory tests are often conducted until effluent tracer concentrations fall below detection limits, in the field tracer tests may be terminated before this occurs, because of costs associated with pumping, sampling, and treatment and disposal of extracted groundwater. In Fig. 7(B) the time in aqueous phase pore volumes, TF , required to achieve the tracer detection limits in Fig. 7(A) are shown. TF increases by a factor of 4 to 7 as Pna increases from Pna ¼ 8 to Pna ¼ 160. Thus, in addition to requiring a lower detection limit for similar measurement error, larger values of Pna also require significantly more throughput. While extrapolation of tracer breakthrough curves will mitigate this situation somewhat, larger throughputs will be required to achieve similar detection errors in Sn as Pna increases. Finally, it is interesting to note how the results might change if mass transfer resistance is dominated by resistance in the nonaqueous phase instead of the aqueous phase. In this case Da is not a function of Pna (see Eq. (8)). While the influence of Dna on the required cda for a prescribed error in Sn should not be as dramatic as that seen in Fig. 7(A), increasing Pna is expected to result in greater error in Sn measurements. This can be seen in the time moment formulas shown in Table 1 for model 1. As Pna increases Rf increases (see Eq. (6)) and the second and third central moments also increase, indicating greater spreading and asymmetric tailing of the partitioning tracer breakthrough curve. Thus, even when mass transfer resistance is dominated by resistance in the nonaqueous phase, errors in measured Sn should increase with increasing Pna and constant cda =ðTp c0a =Rf Þ, when mass transfer limitations are significant.

437

ω = 0.1, Conservative ω = 0.1, Partitioning

0.1

0.01

0.001 0

1

2

3

4

Pore Volumes [-] Fig. 8. Breakthrough curves using model 2 for conservative and partitioning tracers for two different dimensionless mass transfer coefficients. For these simulations Pe ¼ 100, Pna ¼ 8, Sn ¼ 0:03, /m a ¼ 0:85, Tp ¼ 0:2 PV, and X ¼ 1.

Fig. 8. In this case Rf would be underestimated in Eq. (19), which would result in underestimation of Sn in Eq. (20). Thus, just as in model 1 underestimation of NAPL in the system will occur if the mass transfer coefficient is too small and the tracer detection limit too high. The retardation factor for the partitioning tracer was computed from the mean travel times of both the conservative and partitioning tracers (see Eq. (19)). In model 1 transport of the conservative tracer was assumed to be measured exactly, since mass transfer limitations did not affect transport of this tracer. In model 2, though, mass transfer limitations are significant for both tracers and significant measurement error may occur in determining the mean arrival times of both. In most field applications, equal volumes of conservative and partitioning tracers are injected during a PITT. For this reason, in illustrating the influence of mass transfer limitations on NAPL measurement with model 2, we assumed that equal volumes of both tracers were injected into the stream tube. Fig. 9 illustrates the error in Sn measurement for simulations where mass transfer limitations were important. In this case x was selected as x ¼ 0:1 to represent mass transfer limited conditions, which is within a factor of 2 of x fitted to the laboratory data reported below. The injected volume of both tracers was Tp ¼ 0:2, a typical pore volume injected in field tests [1,22]. The other parameters common to the simulation results shown in Fig. 9 are Pe ¼ 100, Pna ¼ 8, and /m a ¼ 0:85. The injected concentrations of both tracers was c0a ¼ 1000 mg/l. On the vertical axis cda was not normalized as in Fig. 4 for model 1, since the capacity of the stream tube for the partitioning and conservative tracers was different because of differing retardation coefficients. Assuming the same tracer detection limit cda for both tracers, for a given NAPL saturation as cda increases the error in NAPL saturation measurement also

P.T. Imhoff, K. Pirestani / Advances in Water Resources 27 (2004) 429–444 10

1

d

c a [mg/l]

80% Error 40% Error 20% Error

0.1 0.00

0.01

0.02

0.03

0.04

0.05

0.06

S n [-] Fig. 9. Tracer detection limits required for determination of Sn that is accurate to within 20%, 40%, and 80% of its actual value when model 2 applies. For these simulations Pe ¼ 100, Pna ¼ 8, /m a ¼ 0:85, x ¼ 0:1, Tp c0a ¼ 200 mg/l, and X ¼ 1.

increases. The required cda for a specified error in NAPL saturation decreases as Sn increases, which is opposite to the trend for model 1 shown in Fig. 4. This difference occurs because the Damkohler number was kept constant at Da ¼ 1 in Fig. 4, which implies that Kna must increase linearly with increasing Sn (see Eq. (7)), while the mass transfer coefficient c between the mobile and immobile domains was kept constant in Fig. 9. Empirical correlations for Kna associated with the dissolution of NAPL ganglia indicate that Kna / Snb , where b has ranged between b ¼ 0:6 [25] to b ¼ 1:1 [15] depending on the porous medium and the empirical correlation. A similar dependency of Kna on Sn has been used in upscaled mass transfer rate coefficients developed for systems where flow bypasses around NAPL-contaminated regions as a result of NAPL dissolution fingering [16]. The influence of injected tracer mass on NAPL saturation measurements is shown in Fig. 10. The same Tp Tp Tp Tp Tp

= 0.2, cao = 1000 mg/l = 0.1, cao = 1000 mg/l = 0.05, cao = 1000 mg/l = 0.2, cao = 500 mg/l = 0.2, cao = 250 mg/l

1

Extrapolated Not Extrapolated

c

0.1

10

d

c a [mg/l]

1

parameters were used for the simulations in Figs. 9 and 10, except that Tp was varied between Tp ¼ 0:05 to Tp ¼ 0:2 PV and the injected tracer concentrations varied between c0a ¼ 250 mg/l and c0a ¼ 1000 mg/l. Results are only shown for a NAPL measurement error of 20%. As the injected tracer pulse increases from Tp ¼ 0:05 to Tp ¼ 0:2 PV for c0a ¼ 1000 mg/l, the corresponding detection limit required for a 20% error in Sn also increases. As c0a increases from c0a ¼ 250 mg/l to c0a ¼ 1000 mg/l, for fixed Tp ¼ 0:2 PV, a similar increase in tracer detection limit is observed for a 20% error in Sn . While it is more intuitive to think of the tracer residence time, or Tp , controlling accuracy in NAPL saturation measurements under mass transfer-limited transport [37], these plots show that it is the injected tracer mass, Tp V ha c0a , that controls measurement accuracy. The injected tracer mass may be altered by changing the size of the tracer pulse or the concentrations of the injected tracers. In either case if the mass of tracers injected into the system is increased, the accuracy of Sn measurements is also increased. There is an upper limit to the improvement in measurement of Sn that may be achieved by increasing the injected tracer mass, although this depends on the significance of mass transfer limitations for the system. Just as in model 1, extrapolating the breakthrough curves analytically to zero with exponential functions fitted to the tailing concentrations results in an improvement in Sn measurements. In Fig. 11 results are shown for the 20% error simulations in Fig. 9, where data were either extrapolated or not extrapolated. Extrapolating the data for the conservative and partitioning tracers increases the required tracer detection limit by about 1 order of magnitude. The effect of the dimensionless mass transfer coefficient, x, on NAPL saturation measurement was similar to that of Da for model 1 in Fig. 6 (data not shown).

d a [mg/l]

438

0.01 0.00

0.1 0.00 0.01

0.02

0.03

0.04

0.05

0.06

S n [-] Fig. 10. Tracer detection limits required for determination of Sn that is accurate to within 20% of its actual value when model 2 applies. For these simulations Pe ¼ 100, Pna ¼ 8, /m a ¼ 0:85, x ¼ 0:1, and X ¼ 1.

0.01

0.02

0.03

0.04

0.05

0.06

Sn [-] Fig. 11. Effect of extrapolation on normalized tracer detection limits required for determination of Sn to within 20% of its actual value when model 2 applies. For these simulations Pe ¼ 100, Pna ¼ 8, Sn ¼ 0:03, 0 /m a ¼ 0:85, x ¼ 0:1, Tp ca ¼ 200 mg/l, and X ¼ 1.

P.T. Imhoff, K. Pirestani / Advances in Water Resources 27 (2004) 429–444 1 IPA Data IPA Model 1 Fit IPA Model 2 Fit DMB Data DMB Model 2 Prediction

0.1

o

ca /c a [-]

The effect of Pna on NAPL saturation measurement was also investigated for model 2, and the results are shown in Fig. 12 where the same set of parameters were used as in Fig. 9. As Pna increases the required detection limit for a 20% error in NAPL saturation decreases. In addition, the throughput required to achieve these detection limits increases with increasing Pna , just as in model 1. These results demonstrate that as Pna increases longer sampling time is required to achieve similar errors in NAPL saturation measurement when mass transfer limitations are important.

0.01

0.001

0.0001 0

4.2. Experimental study

1

Sn = 0.05

Sn = 0.03

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d

ca [mg/l]

Sn = 0.01 Sn = 0.002

0.01 20

40

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100

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160

100

120

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Pna [-]

(A) 1000 Sn = 0.05

Sn = 0.03 Sn = 0.01 Sn = 0.002

TF [-]

100

10

1 0 (B)

20

40

2

4

6

8

10

PoreVolumes [-]

A typical tracer breakthrough curve for the column experiments is shown in Fig. 13 for experiment 3, where a pulse size of Tp ¼ 0:064 PV was injected. Experiment 3 corresponds to the test with the largest injected tracer mass (both conservative and partitioning) for packing 2. The peak concentrations of both the conservative and partitioning tracers (IPA and DMB) occurs at approximately the same time, and both exhibit tailing. How-

0

439

60

80

Pna [-]

Fig. 12. Tracer detection limits (A) or throughputs (B) required for determination of Sn to within 20% of its actual value when model 2 0 applies. For these simulations Pe ¼ 100, /m a ¼ 0:85, x ¼ 0:1, Tp ca ¼ 200 mg/l, and X ¼ 1.

Fig. 13. IPA and DMB breakthrough curves for experiment 3. Fits to the IPA data using model 1 and 2 are shown, along with predicted DMB breakthrough using model 2.

ever, DMB tailing is much more significant than that for IPA. The tailing of both tracers indicate that mass transfer limitations were important in this experiment. Both model 1 and model 2 were fitted to the IPA breakthrough data using CXTFIT [35], and the fitted models are shown in Fig. 13. While model 2 was capable of capturing most of the tailing observed in the IPA data, this was impossible with model 1. Best-fit parameters for model 2 are shown in Table 2. The fraction of the aqueous phase in the mobile domain was /m a ¼ 0:919 0:004 CI (CI ¼ 95% confidence interval), while the dimensionless mass transfer coefficient between the mobile and immobile domains was x ¼ 0:285 0:028 CI. Using the best-fit model parameters determined from the IPA data, transport of DMB through the system was predicted using model 2 and the known NAPL saturation within the column. This model prediction is shown in Fig. 13. While the curvature of the data at two pore volumes is not captured very well, the exponential tailing of the DMB data is matched reasonably well. This is further support for the utility of model 2 in describing tracer transport in this experiment. In this case, mass transfer resistance between the aqueous phase in the mobile domain and the aqueous phase in the immobile domain is more significant than mass transfer resistance between the aqueous and nonaqueous phases in the immobile domain, i.e., within the region of the NAPL pool. This result may not occur in all situations where physical heterogeneities are present. As demonstrated in the modeling results reported earlier, the most accurate measurement of parameters (e.g., Sn ) occurs when the largest tracer masses are injected, if tracer breakthrough curves are measured until data decrease below tracer detection limits. For packing 2, experiment 3 corresponds to the test with the largest tracer mass, and thus we expect that D, x, and /m a are determined most accurately by fitting IPA data from

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Table 2 Best-fit parameters for model 2 to IPA breakthrough data Parameter

Packing 2 (Exp. 3) a

(3.42 ± 0.01 ) · 10 cm/min (2.96 ± 0.23) · 103 cm2 /min (2.85 ± 0.28) · 101 (9.19 ± 0.04) · 101

va D x /m a a

2

Packing 3 (Exp. 2) (3.41 ± 0.02) · 102 cm/min (3.10 ± 0.41) · 103 cm2 /min (1.91 ± 0.77) · 101 (9.63 ± 0.06) · 101

± values represent 95% confidence intervals.

this experiment. Similarly, experiment 2 was selected for fitting model parameters for experiments in packing 3, and best-fit parameters are shown in Table 2. The dimensionless mass transfer coefficient for experiment 2 was approximately 70% of fitted x for experiment 3, demonstrating the greater resistance to mass transfer between mobile and immobile water in the packing that was 1.9 cm thick versus 1.2 cm thick. In this case, the characteristic length for mass transfer between mobile water and the immobile water within the NAPL pool might be one half the thickness of the pool. The fraction of mobile water was similar in both systems, approximately 5% larger in experiment 2 than experiment 3. The influence of injected tracer mass on NAPL measurement in the experiments is illustrated in Fig. 14, where measured column-average NAPL saturations are plotted versus the dimensionless tracer detection limit. Experiments differed depending on the packing used (packing 2 or 3) and the mass of tracers injected, which was varied by changing the injected pulse size and the injected tracer concentration. In any given experiment, concentrations of conservative and partitioning tracers injected into the porous medium were nearly the same.

DMB/IPA Data: Packing #2 DMB/IPA Data: Packing #3 HEX/IPA Data: Packing #3 Model 2: Packing #2 Prediction Model 2: Packing #3 Prediction

0.08

Sn Detected [-]

0.06

0.04

0.02

0.00 0.001

0.01

c

d

o

a /(T pc a

0.1

/R f ) [-]

Fig. 14. Detected NAPL saturation versus normalized tracer detection limits for experimental data. Error bars represent ±95% confidence intervals, based on uncertainty in Pna due to temperature variations during the experiment and uncertainty in the fitted slope to the partitioning data. Predicted curves using model 2 and best-fit parameters for IPA transport are also shown.

Tracer detection limits were similar (cda 1–2 mg/l) for all experiments. Breakthrough data were not extrapolated for the results shown in this figure. For large injected tracer masses (small cda =ðTp c0a =Rf Þ) measured NAPL saturations were in close agreement with independently determined values (Sn ¼ 0:073 0:003), to within experimental error. As smaller amounts of tracer were injected into the column, partitioning tracer tests increasingly underestimated the NAPL saturation in the system. In most cases, extrapolating the tracer breakthrough data with an exponential function had a minor effect on the results. Exponentially extrapolating the data analytically to zero increased measured NAPL saturations by less than 0.015 for all experiments, except for a single experiment where after extrapolation the measured NAPL saturation increased from Sn ¼ 0:021 to Sn ¼ 0:051. Thus, just as in the simulations extrapolation of the tailing breakthrough curves reduced the measurement error but did not eliminate it. Also shown in Fig. 14 are predicted Sn using model 2 and parameters determined from experiment 3 (packing 2) or experiment 2 (packing 3) (see Table 2). Although there is some discrepancy between the experimental data and model predictions, the trends are similar with the model predicting somewhat larger detected Sn than experimental data when 0:005 < cda =ðTp c0a =Rf Þ < 0:04. These results support the use of model 2 for predicting errors in measured NAPL saturation in systems similar to those used in the experiments. Results from experiment 15 (DMB/IPA) and experiment 18 (HEX/IPA) clearly illustrate the effect of tracer partition coefficient on NAPL measurement. These experiments had nearly identical tracer pulses, Tp ¼ 0:47 PV with injected tracer concentrations of c0a 80 mg/l, but different partitioning tracers. The tracer partition coefficient for HEX was approximately 3 times larger than that for DMB. The NAPL saturation measured in experiment 15 (DMB/IPA) was Sn ¼ 0:021 0:001 CI, while in experiment 18 (HEX/IPA) Sn ¼ 0:000 0:004 CI. When the NAPL partition coefficient was Pna ¼ 7:1 (DMB, experiment 15), Rf ¼ 1:5 and the partitioning tracer test measured approximately 30% of the NAPL in the system, but when Pna ¼ 19:3 (HEX, experiment 18) Rf ¼ 2:5 and none of the NAPL was detected. The larger tracer retardation coefficient for experiment 18 resulted in greater tailing of the partitioning tracer breakthrough

P.T. Imhoff, K. Pirestani / Advances in Water Resources 27 (2004) 429–444

curve, and a more significant underestimation of the NAPL mass in the system. This result is in agreement with the observed effect of Pna on NAPL measurement in recent field experiments [3] and supports the modeling results reported earlier. Extrapolating the data exponentially did not alter this conclusion: measured NAPL saturations were Sn ¼ 0:051 0:002 CI with DMB/IPA and Sn ¼ 0:005 0:001 CI with HEX/IPA.

5. Discussion The results presented from the modeling and experimental study clearly illustrate the importance of injected tracer mass and tracer partition coefficient on NAPL measurement under mass transfer-limited conditions. Although these analyses were conducted for a single stream tube, they are illustrative of effects that may be observed in more complicated settings. This conclusion is supported by data from a recent computational experiment designed to evaluate the ability of a PITT to detect a large NAPL pool in the field [20]. Local equilibrium between the aqueous and nonaqueous phases was assumed in this modeling exercise, but spatial heterogeneity in NAPL distribution and hydraulic conductivity resulted in significant tailing in breakthrough curves. Three tracers with different partition coefficients were selected for this three-dimensional simulation, and the injected mass of each tracer was the same. The amount of NAPL detected was found to be a function of both the tracer and the tracer detection limit. The results are shown in Fig. 15 and indicate that the error in NAPL measurement increased as the tracer detection limit increased. When the detection limit was the same for all tracers, the NAPL

Relative Error in NAPL Detected [%]

80

Pna = 28.1 Pna = 10.2 Pna = 4.5

60

40

20

0 0

2

4

6

8

10

12

d

c a [mg/l] Fig. 15. Relative error in NAPL detected as a function of the tracer detection limit, cda , and the tracer partition coefficient, Pna , for a computational experiment in a three-dimensional domain. The raw data used for this analysis were taken from Jin et al. [20].

441

measurement error increased with increasing partition coefficient. Both observations are consistent with our modeling and laboratory results. While local equilibrium was assumed between the nonaqueous and aqueous phases in this computational experiment, the tracer breakthrough curves might have been modeled with rate-limited mass transfer in a simplified model. For a relatively simple heterogeneous NAPL distribution, Brusseau et al. [4] demonstrated that nonaqueous-aqueous phase mass transfer could be modeled in two ways: with a complex two-dimensional model that captured the details of the NAPL distribution and aqueous phase flow, but where local equilibrium was assumed; or with a simplified one-dimensional model with a homogeneous NAPL distribution, but where mass transfer was rate-limited. Both modeling approaches were capable of matching effluent data. The agreement between the results from the computational experiment by Jin et al. [20] and our analyses of tracer transport with simplified stream tube models is consistent with this result: the influence of the partition coefficient on NAPL detection in systems where a complex model was employed but local equilibrium was assumed was similar to that in a system where a simplified modeling approach was taken but mass transfer limitations were significant. For this reason, if the NAPL is distributed heterogeneously in the domain and the injected tracer mass and the tracer detection limit are the same, we expect that tracers with larger partition coefficients may result in underestimation of NAPL mass, if insufficient tracer mass is injected to characterize latetime tracer tails [20]. In this study we examined the influence of mass transfer limitations on NAPL measurement with PITTs using models and experiments that represented transport in a single stream tube. This allowed us to examine the influence of a number of parameters on NAPL measurement in a controlled fashion. Because the flow domains were considerably simpler than those in typical field situations, the dimensionless mass transfer rates in this study may not be representative of many field conditions. Nevertheless, the influence of mass transfer resistance on NAPL measurement observed here is illustrative of what may be observed in many field settings. The results presented above from a three-dimensional simulation of a PITT when a NAPL pool was present support this conclusion. In addition, results from a recent field experiment where a PITT was conducted in an aquifer test cell with a known volume of tetrachloroethylene [3] are consistent with observations from this study. Extensive tailing of tracer breakthrough curves indicated heterogeneity and flow bypassing, which resulted in significant mass transfer limitations between nonaqueous and aqueous phases. PITT measurements showed a systematic decrease in measured NAPL mass with increasing tracer partitioning, just as

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observed in the models and experiments in a single stream tube.

6. Conclusions In this study two mathematical models were used to describe transport of conservative and partitioning tracers in a homogeneous stream tube. In model 1 a single domain was assumed, with mass transfer limitations between the NAPL and aqueous phase. Two aqueous-phase flow domains were assumed in model 2, a mobile and an immobile domain, with all NAPL residing in the immobile domain. Mass transfer limitations occurred between these two domains and affected both the conservative and partitioning tracers. These two models were used to explore the influence of mass transfer limitations on tracer transport and NAPL measurement during a PITT. From these analyses, we observed that: (1) effects of mass transfer limitations on NAPL measurement were more significant as the dimensionless mass transfer coefficients (Da, model 1; or x, model 2) decreased; (2) NAPL measurement error decreased with decreasing tracer detection limits and increasing injected tracer mass; (3) the injected tracer mass could be increased by either increasing the tracer slug size or increasing the concentration of injected tracers, both of which resulted in similar measurement errors; (4) for fixed tracer detection limits, NAPL saturation, aqueous phase velocity field, and injected tracer mass, NAPL measurement error decreased with decreasing tracer partition coefficient; (5) the pore volumes required to achieve a specified NAPL measurement error increased as the tracer partition coefficient increased; and (6) exponentially extrapolating the tracer breakthrough curves analytically to zero decreased the measurement error, but did not eliminate it. These observations were supported by laboratory column experiments, where NAPL measurement error was also shown to decrease as the mass of injected tracers increased in the system. Model 2 described transport of the conservative and partitioning tracers reasonably well in these experiments, with model-predicted NAPL measurement errors following the experimental trends (see Fig. 14). Finally, while the mathematical models and laboratory experiments modeled transport in one-dimensional stream tubes, analysis of data from a three-dimensional computational experiment supported the conclusions from the one-dimensional tests. The effects of tracer partition

coefficient and tracer detection limit on NAPL measurement error in the three-dimensional experiment were similar to that observed in the one-dimensional stream tubes. In any particular field situation, the importance of mass transfer limitations on a PITT will likely be unknown before conducting the tracer test. Our results indicate that when mass transfer limitations are important results will be more accurate when the tracer detection limit is small or the injected tracer mass is large. Careful attention to sampling techniques and analytical methods may be helpful in lowering tracer detection limits. On the other hand, highly concentrated tracer solutions or large tracer pulses might be injected to reduce systematic errors. Unfortunately, high tracer concentrations may result in flow instabilities [38] or nonlinear tracer partitioning [18,40]. Both high tracer concentrations and large tracer solution volumes will also require extensive sampling at downgradient wells to capture the complete tracer breakthrough curve, and will result in large volumes of extracted groundwater that may require treatment before disposal. Thus, injecting large tracer masses will result in increased sampling and treatment costs that will impose practical limitations on the mass of tracer that may be injected. Perhaps the most useful result from our work is that when mass transfer limitations are important, estimates of NAPL saturation in a PITT will systematically increase with decreasing tracer partition coefficient. Because multiple partitioning tracers with different Pna are typically injected in PITTs, the influence of mass transfer limitations on the systematic error in NAPL measurements can be inferred from these tests. Thus, if measured NAPL volume or NAPL saturation increase systematically with decreasing Pna , then mass transfer limitations may have significantly affected tracer transport. In this case, estimated Sn from partitioning tracers with small Pna may be more accurate than estimates derived from tracers with large Pna . If mass transfer limitations are not important, the random error associated with measurement of tracer retardation should increase with decreasing Pna , making the estimated Sn from tracers with large Pna more precise, if sufficient data are collected to characterize the breakthrough curves for compounds with large Pna [20]. Thus, with careful attention to data trends associated with tracers with different Pna , the importance of mass transfer limitations on tracer breakthrough data and NAPL measurements may be assessed.

Acknowledgements The authors thank Minquan Jin of Duke Engineering Services, Inc. for providing raw data that were used to prepare Fig. 15. The authors thank Yousef Jafarpour

P.T. Imhoff, K. Pirestani / Advances in Water Resources 27 (2004) 429–444

and Kelly M. Spivey for assistance with the laboratory experiments. The helpful comments from three anonymous reviewers are also gratefully acknowledged. This research was supported in part by the National Science Foundation Grant No. 9984715.

[20]

[21]

References [22] [1] Annable MD. Interdisciplinary program in hydrologic sciences. Personal Communication, University of Florida; 2002. [2] Annable MD, Rao PSC, Hatfield K, Graham WD, Wood AL, Enfield CG. Partitioning tracers for measuring residual NAPL: results from a field-scale test. J Environ Eng 1998;124(6):498–503. [3] Brooks MC, Annable MD, Rao PSC, Hatfield K, Jawitz JW, Wise WR, et al. Controlled release, blind tests of DNAPL characterization using partitioning tracers. J Contam Hydrol 2002;59: 187–210. [4] Brusseau ML, Zhang Z, Nelson NT, Cain RB, Tick GR, Oostrom M. Dissolution of nonuniformly distributed immiscible liquid: intermediate-scale experiments and mathematical modeling. Environ Sci Technol 2002;36(5):1033–41. [5] Cooke CE. Method of determining fluid saturations in reservoirs. US Patent No. 3 590 923; 1971. [6] Cussler EL. Diffusion: mass transfer in fluid systems. Cambridge, MA: Cambridge University Press; 1988. [7] Dai D, Barranco Jr FT, Illangasekare TH. Partitioning and interfacial tracers for differentiating NAPL entrapment configuration: column-scale investigation. Environ Sci Technol 2001;35: 4894–9. [8] Datta-Gupta A, Yoon S, Vasco DW, Pope GA. Inverse modeling of partitioning interwell tracer tests: a streamline approach. Water Resour Res 2002;38(6), 10.1029/2001/WR000597. [9] Dwarakanath V. Characterization and remediation of aquifers contaminated with nonaqueous phase liquids using partitioning tracers and surfactants. PhD thesis, University of Texas, Austin; 1997. [10] Dwarakanath V, Deeds N, Pope GA. Analysis of partitioning interwell tracer tests. Environ Sci Technol 1999;33(21):3829–36. [11] Fu X, Imhoff PT. Mobilization of small DNAPL pools formed by capillary entrapment. J Contam Hydrol 2002;56:137–58. [12] Glass RJ, Conrad SH, Peplinski W. Gravity-destabilized nonwetting phase invasion in macroheterogeneous porous media: experimental observations of invasion dynamics and scale analysis. Water Resour Res 2000;36(11):3121–37. [13] Harvey CF, Gorelick SM. Temporal moment-generating equations: modeling transport and mass transfer in heterogeneous aquifers. Water Resour Res 1995;31(8):1895–911. [14] Hatfield K, Stauffer TB. Transport in porous media containing residual hydrocarbon. 1. Model. J Environ Eng 1993;119(3):540– 58. [15] Imhoff PT, Arthur MH, Miller CT. Complete dissolution of trichloroethylene in saturated porous media. Environ Sci Technol 1998;32(16):2417–24. [16] Imhoff PT, Farthing MW, Miller CT. Modeling NAPL dissolution fingering with upscaled mass transfer rate coefficients. Adv Water Resour 2003;26(10):1097–111. [17] Imhoff PT, Jaffe PR, Pinder GF. An experimental study of complete dissolution of a nonaqueous phase liquid in saturated porous media. Water Resour Res 1994;30(2):307–20. [18] Imhoff PT, Pirestani K, Jafarpour Y, Spivey KM. Tracer interaction effects during partitioning tracer tests for NAPL detection. Environ Sci Technol 2003;37(7):1441–7. [19] James AI, Graham WD, Hatfield K, Rao PSC, Annable MD. Optimal estimation of residual non-aqueous phase liquid satura-

[23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]

[31]

[32] [33]

[34]

[35]

[36]

[37]

[38]

443

tions using partitioning tracer concentration data. Water Resour Res 1997;33(12):2621–36. Jin M, Butler GW, Jackson RE, Mariner PE, Pickens JF, Pope GA, et al. Sensitivity models and design protocol for partitioning tracer tests in alluvial aquifers. Groundwater 1997;35(6):964–72. Jin M, Delshad M, Dwarakanath V, McKinney DC, Pope GA, Sepehrnoori K, et al. Partitioning tracer test for detection, estimation, and remediation performance assessment of subsurface nonaqueous phase liquids. Water Resour Res 1995;31(5): 1201–11. Jin M, Jackson RE. INTERA, Inc. Personal Communication, 9111A Research Blvd., Austin, TX; 2002. Kim H, Rao PSC, Annable MD. Gaseous tracer technique for estimating air-water interfacial areas and interface mobility. Soil Sci Soc Am J 1999;63:1554–60. McKenna SA, Meigs LC, Haggerty R. Tracer tests in a fractured dolomite 3. Double-porosity, multiple-rate mass transfer processes in convergent flow tracer tests. Water Resour Res 2001; 37(5):1143–54. Miller CT, Poirier-McNeill MM, Mayer AS. Dissolution of trapped nonaqueous phase liquids: mass transfer characteristics. Water Resour Res 1990;26(11):2783–96. Nelson NT, Brusseau ML. Field study of the partitioning tracer method for detection of dense nonaqueous phase liquid in a trichloroethene-contaminated aquifer. Environ Sci Technol 1996; 30(9):2859–63. Nelson NT, Oostrom M, Wietsma TW, Brusseau ML. Partitioning tracer method for the in situ measurement of DNAPL saturation: influence of heterogeneity and sampling method. Environ Sci Technol 1999;33(22):4046–53. Powers SE, Abriola LM, Weber Jr WJ. An experimental investigation of nonaqueous phase liquid dissolution in saturated subsurface systems: transient mass transfer rates. Water Resour Res 1994;30(2):321–32. Rao PSC, Annable MD, Kim H. NAPL source zone characterization and remediation technology performance assessment: recent developments and applications of tracer techniques. J Contam Hydrol 2000;45:63–78. Saba T, Illangasekare TH. Effect of groundwater flow dimensionality on mass transfer from entrapped nonaqueous phase liquid contaminants. Water Resour Res 2000;36(4):971–9. Schroth MH, Ahearn SJ, Selker JS, Istok JD. Characterization of Miller-similar silica sands for laboratory hydrologic studies. Soil Sci Soc Am 1996;60:1331–9. Schwille F. Dense chlorinated solvents in porous and fractured media. Chelsea, MI: Lewis; 1988. Srivastava R, Brusseau ML. Nonideal transport of reactive solutes in heterogeneous porous media. 1. Numerical model development and moments analysis. J Contam Hydrol 1996;24(2): 117–43. Tomich JF, Dalton RL, Deans HA, Shallenberger LK. Single-well tracer method to measure residual oil saturation. J Petrol Technol 1973;(February):211–8. Toride, N, Leij, FJ, van Genuchten, MTh. The CXTFIT code for estimating transport parameters from laboratory or field tracer experiments. US Salinity Laboratory, Agriculatural Research Service, US Department of Agriculture, Research Report 137, Riverside, CA; 1995. Valocchi AJ. Validity of the local equilibrium assumption for modeling sorbing solute transport through homogeneous soils. Water Resour Res 1985;21(6):808–20. Whitley Jr GA, McKinney DC, Pope GA, Rouse BA, Deeds NE. Contaminated vadose zone characterization using partitioning gas tracers. J Environ Eng 1999;125(6):574–82. Willson CS, Pau O, Pedit JA, Miller CT. Mass transfer rate limitation effects on partitioning tracer tests. J Contam Hydrol 2000;45:79–97.

444

P.T. Imhoff, K. Pirestani / Advances in Water Resources 27 (2004) 429–444

[39] Wilson DJ, Burt RA, Hodge DS. Mathematical modeling of column and field dense nonaqueous phase liquid tracer tests. Environ Monit Assess 2000;60:181–216. [40] Wise WR, Dai D, Fitzpatrick EA, Evans LW, Rao PSC, Annable MD. Non-aqueous phase liquid characterization via partitioning

tracer tests: a modified Langmuir relation to describe partitioning nonlinearities. J Contam Hydrol 1999;36:153–65. [41] Young DF, Ball WP. Column experimental design requirements for estimating model parameters from temporal moments under nonequilibrium conditions. Adv Water Resour 2000;23:449–60.