In Situ Method To Measure Effective and Sorption ... - ACS Publications

David Werner and Patrick Höhener *. Swiss Federal Institute of Technology (EPFL), ENAC-ISTE-LPE, CH-1015 Lausanne, Switzerland. Environ. Sci. Technol...
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Environ. Sci. Technol. 2003, 37, 2502-2510

In Situ Method To Measure Effective and Sorption-Affected Gas-Phase Diffusion Coefficients in Soils DAVID WERNER# AND PATRICK HO ¨ HENER* Swiss Federal Institute of Technology (EPFL), ENAC-ISTE-LPE, CH-1015 Lausanne, Switzerland

Transport modeling, risk assessment, and the evaluation of remediation strategies at contaminated sites require the knowledge of gas diffusivities in soil. A field method is presented, which determines the tortuosity factor and the mass fraction in the air phase of a volatile compound in situ. The compound is injected into the unsaturated zone together with a conservative gaseous tracer to form a point source. Concentrations are monitored at the injection point during 8 h and evaluated with an analytical equation for reactive transport. The air-filled porosity is determined independently. From these data, both the effective and the sorption-affected diffusion coefficients are obtained. Results are reported for volatile organic pollutants in both a lysimeter and a sandy soil. The measurements show good reproducibility. Batch experiments suggest that tracers were not truly conservative at subsurface temperatures. This may lead to a systematic underestimation of the effective diffusion coefficient by less than 10%, but the sorptionaffected diffusion coefficients were probably overestimated by 15-20%. Nevertheless, the in situ method can avoid considerable uncertainties associated with choosing appropriate empirical relationships for the tortuosity factor or deviations from natural soil conditions in laboratory experiments.

Introduction Gas-phase diffusion dominates the migration of natural gases and volatile pollutants in the unsaturated zone in the absence of pressure gradients. It is crucial for vapor migration of volatile organic chemicals at contaminated sites (1), for the supply of molecular oxygen to soil organisms (2), or for 222Rn migration from soil to houses (3). Modeling of gas-phase diffusion in the unsaturated zone requires the knowledge of the gas diffusivities. The molecular diffusion coefficients in free air can be measured or estimated from empirical relationships (4). However, the gas-phase diffusion in the unsaturated zone differs from the diffusion through free air. Solid and liquid obstacles reduce the cross-sectional area and increase the mean path length in soils. This effect is identical for different gases but is specific for a certain soil at a specific water content. Furthermore, the gas-phase diffusion in the unsaturated zone is affected by partitioning into the soil water, onto the air-water interface, and into or onto the solids (5). It spreads only the contaminants in the * Corresponding author phone: 0041 21 693 57 50; fax: 0041 21 693 28 59; e-mail: [email protected]. # Current address: Department of Civil and Environmental Engineering, Stanford University. 2502

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gas phase, and the contaminants with a low mass fraction in the soil gas will therefore diffuse more slowly than conservative gases, depending on the soil and chemical properties of the diffusing substance (6). Finally, methods to determine diffusion coefficients for nonconservative gases need to take into account degradation or reaction processes. For gas-phase diffusion in a porous medium, coefficients are defined with proportionality factors accounting for the effects of the physical reduction and/or partitioning (7, 8). The effective diffusion coefficients De (5) are used to calculate gas fluxes from Fick’s first law or to interpret steady-state vapor concentration profiles. They account for the reduced cross-sectional area and the increased mean path length in soils. The sorption-affected diffusion coefficients Ds describe the transient diffusion. They account for the effect of the increased mean path length and the partitioning into or onto stationary phases (8). The marked difference between values for the effective diffusion coefficients obtained from the existing empirical relationships (9), laboratory tests (10), and field experiments calls for accurate in situ measurements of diffusion parameters. Both, flux chamber and point injection methods are used to determine the effective diffusion coefficients De directly in the field. The flux chamber methods can only be used in surface soils. A cylinder is inserted into the soil and a gaseous tracer is supplied to the confined core from a wellstirred reservoir in a diffusion chamber placed on the cylinder (11). In the point injection methods soil gas probes or access needles are used to inject the gaseous tracer at the depth of interest. In the procedure reported in ref 2 a volume of gaseous tracer is injected into the soil, and the change in concentration at the injection point is measured as a function of time. Jellick and Schnabel (12) regard as unrealistic the initial conditions of the analytical solution for spherical diffusion used by (2) and propose a finite difference model as an improvement. Ball et al. (13) use a special probe with a Geiger-Mu ¨ ller tube to measure the concentration decline of a radioactive tracer gas, 85Kr, directly at the injection point. Hers et al. (14) evaluate an alternative method originated by Johnson et al. (15). It determines the effective diffusion coefficients from the mass fraction of gaseous tracer found within a certain radius from the injection point. Kreamer et al. (8) use a continuous point source of gaseous tracer rather than an instantaneous point source and measure the tracer concentration at some distance from the source. Their method allows for studying diffusion across extended distance. Sorption and partitioning into the water phase are included in the theories of studies (8) and (14). However, (14) report only data for He, which is generally considered to be a conservative tracer. Kreamer and coworkers (8) conclude that the potential of their tracer test to determine sorption characteristics in situ remains undemonstrated as uncertainties in the data greatly outweigh any probable effect of dissolution or sorption. This study furthers the development of the instantaneous point source diffusion experiments. The data obtained include compounds, for which absorption, adsorption, and/ or dissolution is relevant. The gaseous compound of interest is injected together with a conservative gaseous tracer to form an instantaneous point source in the unsaturated zone. The decline of the injected compounds in concentration is monitored at the injection point. The air-filled porosity is determined independently. The tortuosity factor is obtained from the conservative tracer data. The mass fraction in the soil gas phase of the compounds of interest is determined in situ by comparing the data of these compounds with the conservative tracer data. Both the effective and the sorption10.1021/es020101s CCC: $25.00

 2003 American Chemical Society Published on Web 04/24/2003

affected diffusion coefficients of the compounds of interest are obtained from these parameters. In this study, the compounds of interest are chlorinated ethenes or petroleum hydrocarbons, and the tracers are chlorofluorocarbons (CFCs) or sulfur hexafluoride (SF6). The tracers are recalcitrant under aerobic conditions (16). Degradation of the compounds of interest is included in the basic theoretical approach. It is found to be insignificant in the pristine soil investigated. For comparison, both the effective and sorption-affected diffusion coefficients are calculated from sorption data measured in batch experiments, using soil core data and empirical relationships for the tortuosity factor.

Theory Relevant Assumptions. The theory presented below relies on the following assumptions: The soil can be described as a homogeneous porous medium with uniform and constant properties consisting of soil air, soil water, and the solid matrix, where all solid surfaces are wetted. The partitioning of the gaseous compounds between those phases can be described by an instantaneous, reversible linear equilibrium. Gas-phase diffusion is the only relevant transport mechanism, and degradation occurs in the water phase and can be described by a single first-order rate constant. Dimensions and definitions of all parameters are given under notation. Partitioning of Gaseous Compounds in Soil. The partitioning of a gaseous compound is described using the airwater partitioning coefficient or Henry’s law constant H and the air-solid partitioning coefficient Ks. The latter is defined as the ratio between the Henry’s law constant and the solidwater partitioning coefficient Kd. The mass fraction of the compound in the soil air fa is calculated as (17)

Ca(r,t) )

(1)

θw f Hθa a

∂/∂tCa ) faτDm‚∆Ca - fwkwCa

(3)

where ∆ stands for the Laplace operator, τ the tortuosity factor, and Dm the molecular diffusion coefficient. The parameters accounting for degradation can be lumped into a single apparent first-order degradation constant kapp ) fwkw. Solution for an Instantaneous Point Source. For a point source of mass m0 released at r ) 0 in an infinite porous medium with uniform and constant properties, the concentration at distance r and time t is (20)

[

- r2 - kapp‚t 4faτDm‚t

]

(4)

1 ‚ ‚ exp[-kapp‚t] 8θa(faτDmπ)1.5 t1.5

(5)

Degradation. For a correct interpretation of the experimental data, one needs to assess the relevance of degradation. From eq 5 it follows that

[

]

Cr,1(0,t)

Cr,2(0,t)

)

[( ) ( ) ] Dm,2 Dm,1

1.5

fa,2 fa,1

0.5

- (kapp,1 - kapp,2)‚t (6)

The difference in the apparent first-order degradation rate constant kapp,1 - kapp,2 is thus equal to the slope of a linear regression of the left-hand side of eq 6 versus t. Mass Fraction in the Air Phase. In the following only the case where degradation is slow during the time scale of the experiment (kapp,1*t ≈ 0 and kapp,2*t ≈ 0) is discussed. The exponential factor in eq 5 can then be neglected. From eq 5 it follows that

[

(2)

In soils with high air-water interfacial areas, certain compounds such as long-chain alkanes can be deposited on the air-water interface (6, 19), and an additional summand accounting for this needs to be added to the denominator of eqs 1 and 2. Gas-Phase Diffusion in Soil: General Equation. If degradation can be described by a single first-order degradation rate constant kw and occurs in the water phase, the governing equation for diffusion-dominated transport in a porous medium with uniform and constant properties can be written as (20)

exp

Vinfa

Cr(0,t) )

ln

where θa, θw, and θt denote the air-filled, water-filled, and total porosity, respectively, and Fs denotes the density of the solids. The inverse value of fa can be interpreted as the retardation coefficient R, a quantity used for the interpretation of advective partitioning tracer tests in the unsaturated zone (18). The mass fraction in the water phase fw is calculated as

fw )

8θa(faτDmπ‚t)

1.5

For a nondegrading tracer (kapp ) 0) this equation describes a normal distribution with standard deviation σ ) (2faDmt)0.5, which can be used to characterize the average distance of diffusion. Basic Equation. The theory of the in situ method is developed for two compounds, termed 1 and 2. For compound 1 the apparent diffusion coefficients and the degradation rate constant will be measured. Compound 2 serves as tracer. A small volume of gas containing both compounds is injected into the soil to form a point source. It is assumed that the mass fraction in the soil gas fa,2 of the tracer and kapp,2 are known. Ideally, the tracer is conservative with fa,2 ) 1 and kapp,2 ) 0. The method requires the knowledge of experimentally determined or estimated molecular diffusion coefficients Dm for both compounds. The soil gas concentrations are standardized with respect to the concentration in the injected gaseous compound mixture Cin ) m0/Vin to obtain the relative concentrations Cr ) Ca/Cin. At distance r ) 0 and time t, the relative concentration is

ln 1 fa ) Fs(1 - θt) θw 1+ + Ks θ a Hθa

m 0 fa

][ ]

fa,1 Cr,2(0,t) ) fa,2 Cr,1(0,t)

2

‚

Dm,2 Dm,1

3

(7)

Tortuosity Factor. The tortuosity factor τ is obtained from the data of the tracer, from which one knows the mass fraction in the gas-phase fa,2. The air-filled porosity θa needs to be determined independently. From eq 5 it follows that

Vin

τ)

2/3

4πθa

2/3

1 ‚ 1/3 Cr,2(0,t)2/3fa,2 Dm,2 t

(8)

Apparent Diffusion Coefficients. The apparent diffusion coefficients De and Ds can be calculated from the mass fraction in the soil gas fa, the tortuosity factor τ, and the air-filled porosity θa. A detailed derivation is given in ref 5, where De is called the effective diffusion coefficient and Ds is the transient effective diffusion coefficient. Inserting the above expressions for fa and τ it becomes evident that De can be obtained from the data as follows:

De,1 ) θaτDm,1 )

(

Vin 1 4π Cr,2(0,t)

)( )( ) 2/3

θa fa,2

1/3

Dm,1 1 ‚ Dm,2 t

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TABLE 1. Relevant Physical-Chemical Properties of Volatile Compounds and Ks from Batch Experiments air-water partitioning constant H (Henry const.) (mol cm-3 air)/ (mol cm-3 water) molecular diffusion coeff Dm at 25 °Ca (cm2 s-1)

compound SF6 CFC-12 CFC-114 CFC-11 CFC-113 chloroethene trans-dichloroethene cis-dichloroethene benzene toluene n-octane

0.089 0.089 0.076 0.083 0.073 0.102 0.093 0.094 0.090 0.082 0.067

at 10 °C

at 15 °C

air-solid partitioning coeff Ks(batch,sterile) [(mol cm-3 air)/ (mol g-1 solid)]

105d

126d 9.1b 36c 2.7b 8.5b 0.75b 0.29b 0.11b 0.15b 0.16b 57b

15 ( 8e 22 ( 2 f 20 ( 4 f 24 ( 8 f 20 ( 3 f 13 ( 3e 2.3 ( 0.2e 1.5 ( 0.2e 4.1 ( 1.8 f 2.9 ( 0.5 f 4.1 ( 0.9 f

7.5b 31c 2.2b 7.1b 0.63b 0.23b 0.09b 0.12b 0.13b 36b

a Estimated according to Fuller et al. (4) with molar liquid volumes published in ref 28. Molecular diffusion coefficients were corrected for the appropriate temperature T using Dm(T)/Dm(298.15 K) ) (T/298.15)1.75 (4). b According to ref 29. c According to ref 30. d According to ref 31. e At 10 °C, sand from lysimeter, additional values: Ks ) 14 ( 3 for CFC-12 and 24 ( 6 for CFC-11. f At 15 °C, sand from field.

and Ds is

Ds,1 ) fa,1τDm,1 )

( )( )

4/3 1 Cr,2(0,t) Dm,2 4π C (0,t)2 Dm,1 r,1

2

Vin fa,2 θa

2/3

‚

1 (10) t

Error Analysis. Only errors in parameters with a high exponent in eqs 9 and 10 will make a relevant contribution to the overall error of the apparent diffusion coefficients. Assuming that Cr(0,t), θa, fa,2, and Dm,1/Dm,2 can be measured or estimated with an error of 10%, whereas Vin and t have negligible error, the theoretical overall error of the apparent diffusion coefficients as determined by this method is 13% for the effective diffusion coefficient De and 33% for the sorption-affected diffusion coefficient Ds. The method is therefore more robust for De. The error contribution of Cr(0,t) can be reduced by calculating the apparent diffusion coefficients as an average of measurements at different times t.

Materials and Methods Chemicals. Chlorofluorocarbons CFC-11, CFC-12, and CFC114, chloroethenes, and the petroleum hydrocarbons were obtained from Fluka (Buchs, Switzerland), CFC-113 from Merck (Dietikon, Switzerland), and SF6 from Carbagas (Lausanne, Switzerland). All chemicals had >0.98% purity. The relevant physical-chemical properties of these compounds are given in Table 1. Gaseous Compound Mixtures and Analytical Procedures. Dual or multiple mixtures were prepared by enclosing the organic compounds as liquid or gas in a flask with a large headspace. Gaseous concentrations of the compounds and the tracers were determined by gas chromatography. A detailed description of the gaseous compound mixtures and of the analytical procedures used in this study is available as Supporting Information. Lysimeter. A cylindrical field lysimeter (1.2 m diameter, 2.5 m depth), similar to the one described in ref 1, was packed with a coarse sand originating from the Rhone river delta in Lake Geneva. The sand had an organic carbon content foc of 0.2 ( 0.1% (weight percent of the dry sand) and the following grain size distribution: < 4 mm: 99.0% of weight; < 2 mm: 89.7%; < 1 mm: 78.5%; < 0.5 mm: 50.2%; < 0.25 mm: 19.1%; < 0.10 mm: 2.5%. The resulting total porosity of the porous 2504

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FIGURE 1. Soil profile, soil gas probe, and experimental setup in the field. medium was θt ) 0.42 ( 0.03. Two 1/16′′ stainless steel capillaries were installed at 1.5 m depth in the center of the lysimeter. One served as the injection capillary and the other one as the sampling capillary. Above ground, gastight sample lock syringes with removable needles (80727-series, Hamilton, Bonaduz, Switzerland) were connected to the capillaries. Time domain reflectometry (TDR) probes (SDEC, Reignac s. Indre, France) were installed at 1.5 and 1.3 m depth, and a thermometer was installed at 1.5 m depth. The water-filled porosity θw determined with the TDR probes was 0.06 ( 0.02, and the temperature was between 9° and 12° C. Field Experiment. A sandy soil near Lyngby (Denmark) was used to test the method in a natural setting. The surface vegetation was a lawn. A dark brownish topsoil was followed by a loamy, orange colored transition zone, which was then followed by the actual sand layer (Figure 1). The grain size distribution of the sand at the injection depth of 1.3 m was < 5 mm: 98%, < 2 mm: 97%, < 1 mm: 96%, < 0.5 mm: 90%, < 0.2 mm: 59%, < 0.1 mm: 16%, < 0.05 mm: 4%. The foc of the sand was 0.04 ( 0.02%. Two soil cores were taken:

One before the field experiments and the other one 30 days later, after the field experiments, to determine the water contents (3.3 and 3.5% by weight) and the total porosity θt (0.31 ( 0.03) at the depth of the injection point. Soil cores were homogeneous within the sandy layer to a depth of 2 m. The water-filled porosity θw was 0.060 for the first core and 0.063 for the second core. The temperatures in a nearby sandy soil measured between 1 and 1.5 m depth ranged from 10° to 15 °C (21), and the outside temperatures ranged from 0° to 20 °C during the experiments. Soil Gas Probes. Soil gas probes were constructed from stainless steel tubes with an inner diameter of 4 mm, into which three 1/16′′ stainless steel capillaries were inserted (Figure 1). The dead volume of the capillaries was 0.7 mL. One capillary was used to inject the gaseous compound mixture into the soil, the remaining ones to withdraw samples. At the bottom end, the tips of the capillaries were protected from clogging or contamination by soil with steel wool. Holes in the lowermost 5 cm of the outer tube allowed gases to diffuse in or out. The void volume of this section is less than 0.5 mL. In the uppermost hole, the void space between capillaries and outer tube was filled as closely as achievable by welding to avoid diffusion inside the tube. Soil gas probes were installed at the field site by predrilling a hole of 3 cm diameter. The probes were inserted into the predrilled hole and then hammered an additional 35 cm into the sand, and the predrilled hole was backfilled with sand. This procedure reduced the risk of clogging the holes in the probe with topsoil. Experimental Procedures. Before every experiment the flask with the mixture of gaseous compounds was equilibrated at the outside temperature. Then 5 mL of the headspace gas was drawn into a gastight sample lock syringe, followed by 5 mL of clean air to reduce the risk of vapor condensation. The syringe was locked, and the vapors were allowed to mix with the air by diffusion for 15 min. The concentration of the gaseous compounds in the syringe was determined by withdrawing from the syringe three samples of 20 µL, which were diluted and stocked in air-filled 64 mL flasks with Teflon Mininert valves (Supelco, Buchs, Switzerland). The gaseous compound mixture (10 mL) was then injected into the soil through the injection capillary, followed by 1 mL of clean air. Samples (100 µL) were taken both before the injection (background) and at regular time intervals for 7 to 8 h thereafter through the sampling capillary and stocked in gastight sample lock syringes until analysis. Before every sample 2.5 mL of soil gas was withdrawn to flush the sampling capillary. Batch Experiments: Determination of Apparent FirstOrder Degradation Rate Constants. Glass vials (64 mL) with Mininert valves were filled with excavated sand. Resulted total porosities θt ranged from 0.4 to 0.50. Equal amounts of a gaseous compound mixture (50 µL) were injected into a sand-filled and an empty flask serving as control, and the concentrations were monitored by sampling 100 µL of soil gas at six equal time intervals during 33 h. The constant temperatures was 10 °C (sand from lysimeter) and 15 °C (sand from field). The vials were initially shaken to facilitate a homogeneous distribution of the gaseous compounds. The apparent first-order degradation rate constant kapp was calculated from the concentration decline in the sand-filled vial. The soil was thereafter dried at 105 °C and moistened again to allow germination of microorganism spores. This procedure was repeated three times, and an identical control experiment was performed with the sterilized soil. Batch Experiments: Determination of Air-Solid Partitioning Coefficients. Vials (64 mL) with Mininert valves were filled with sterilized (see the section above), moistened sand and kept at constant temperatures. Equal amounts of a contaminant vapor mixture (50 µL) were injected into three sand-filled and three empty vials, and the concentrations

were monitored by sampling 100 µL of soil gas at six equal time intervals during 8 h. With an assumed density of 2.5 for the solids, θa, θw, and θt were calculated. The mass fraction in the gas-phase fa of each compound was determined by dividing the mass of tracer in the gas phase through the mass of tracer injected. Air-solid partitioning coefficients Ks were calculated from fa by using eq 1 and the Henry’s law constants given in Table 1 for the respective temperatures.

Results In Situ Experiments: Relative Concentrations. The measured relative concentrations in the soil gas Cr(0,t) are plotted as a function of time in Figure 2 for the experiments in the sand-filled lysimeter and in Figure 3 for the field experiments. The relative concentrations are shown on a logarithmic scale, which provides more details at small concentrations. The relative concentrations at the injection point fall by 3 orders of magnitude within the first hour and then decline more gradually with time. At the end of the experiments the concentrations are 4-5 orders of magnitude lower than the injected concentrations and are close to the detection limit of the analytical system. Differences in the plotted relative concentrations visualize those in the molecular diffusion coefficient Dm and/or the mass fraction in the gas-phase fa. For instance, the relative concentrations of benzene, toluene, and n-octane are higher than those of the respective CFCtracers, indicating a more significant partitioning and a slower gas-phase diffusion of the former. Relative concentrations calculated by eq 5 with the fitted parameters τ and fa,1 and with the assumption that degradation is negligible (kapp ≈ 0) are shown as the solid or dashed lines in the plots. Molecular diffusion coefficients Dm from Table 1 corrected for 10 °C (lysimeter) and 13 °C (field experiment) and air-filled porosities θa from Table 2 were used to evaluate the data. The tortuosity factor τ was determined from the data of the tracer, assuming fa,2 ) 1. The mass fraction in the soil gas fa,1 of compound 1 was then determined from the data of compound 1. As can be seen in Figures 2 and 3 the mathematical model fits the data well. Deviations are observed in some data points from the field at low concentrations. Deviations affect both compounds in a similar matter and therefore have less impact on parameters such as the apparent first-order degradation rate constant kapp or the ratio of the mass fraction in the gas-phase fa,1/fa,2, because these parameters are determined from concentration ratios. Degradation. For every experiment, the apparent firstorder degradation rate constants of the compound 1 were calculated from eq 6 and reported in Table 2. Errors are given as the confidence intervals of the slope of the linear regression line used for the determination of kapp,1 - kapp,2. An estimate for the overall uncertainty in measuring small kapp,1 with the in situ method can be obtained from experiments A, E, and F, where both compounds are nondegradable tracers. Experimental kapp,1 between (1 d-1 are observed, due to experimental error near the analytical detection limits. Degradation rate constants < 1 d-1 can therefore not be determined reliably. All measured kapp,1 fall within this range, and are < 1 d-1. Moreover, kapp,1 is often negative, indicating an increase in the concentration of the degradable compound (chloroethenes, benzene, toluene, or n-octane) at the injection point as a function of time relative to the nondegradable tracer rather than a decrease. Degradation was neglected in the further evaluation of the data. Mass Fraction in the Air Phase. The ratio of the mass fractions in the air phase fa,1/fa,2 was calculated according to eq 7 with the molecular diffusion coefficients Dm listed in Table 1. The ratios were calculated for every measurement and then averaged. The error of the mean value was calculated as the experimental standard deviation from the eight measurements. Results are reported in Table 2. VOL. 37, NO. 11, 2003 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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FIGURE 2. Measured standardized concentrations (symbols) and model data assuming no degradation (eq 5, parameters as described in text, lines) from the lysimeter experiment A-D. CFC-12 data is not shown for experiment A to ensure readability. Apparent Diffusion Coefficients. Apparent diffusion coefficients were calculated from the measured data and the mean measured air-filled porosity θa with the approximation fa,2 ) 1 for the tracer (compound 2). The apparent diffusion coefficients were calculated for every measurement according to eqs 9 and 10 and then averaged. The experimental error of the mean value is given as the standard deviation of the measurements. Proportionality factors for the apparent diffusion coefficients and the molecular diffusion coefficient Dm are reported in Table 2. Batch Experiments. Apparent first-order degradation rate constants kapp measured in batches with pristine and with sterilized soil were < 1.0 d-1 with no significant difference between pristine and sterilized soil (data not shown). Airsolid partitioning coefficients Ks were determined in triplicate with sterilized soil, and the error is given as the standard deviation of the three measurements (Table 1). Empty control flasks were leak tight during the experiments. The ratios of the mass fractions in the gas-phase fa,1/fa,2 were calculated according to eq 1 with the Henry’s law constant H given in Table 1 for the respective temperature, Ks as determined from the batch experiments (Table 1), and the porosity data (lysimeter: θw ) 0.06, θa ) 0.36, θt ) 0.42, field experiment: θw ) 0.06, θa ) 0.25, θt ) 0.31). The calculated ratios fa,1/fa,2 are compared with the corresponding ratios measured in situ in Figure 4a. Calculated mass fractions in the soil gas of the tracers are fa,2 ) 0.79 for SF6 in the lysimeter, fa,2 ) 0.75 for CFC-12, fa,2 ) 0.73 for CFC-11, and fa,2 ) 0.73 for CFC-113 in the field. This suggests that the tracers were not truly conservative.

Discussion Assumptions of the Mathematical Model. The evaluation of the data measured in situ relies on the mathematical model 2506

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presented in the theory section and on several assumptions. The first assumption is that the soil can be described as a homogeneous porous medium with uniform and constant properties. With eq 4 it can be calculated that CFC-12 falls to approximately 10% of the concentration Cr(0,t) within a radial distance of 80 cm after 8 h at the field site. Most of its mass remains in the soil volume at a shorter radial distance and within the homogeneous sandy soil below the B horizon shown in Figure 1. Within the 8 h duration of experiments, no significant changes of the soil properties occurred. Thus the assumption of a homogeneous medium was reasonable and held. The deviation of measured concentrations from the model prediction observed in some field experiments for small concentrations are not systematic and were attributed to the increased experimental error for concentrations close to the detection limit of the analytical system. According to eq 4 the tracer SF6 reached the wall of the lysimeter at 60 cm distance from the injection point toward the end of the experiments, but this is not expected to influence Cr(0,t) significantly. A slight tendency toward higher than predicted concentrations of SF6 can actually be observed in Figure 2 for the last data point. However, if the results in Table 2 are calculated without this last measurement, the difference is less than 5% for all reported values. The second assumption is the instantaneous linear equilibrium partitioning of the compounds between the different phases of the soil. The stable measured concentration ratios suggest that such equilibrium exists. An initial decrease in fa was observed for more hydrophilic and/or less volatile tracers, thus resulting in a negative kapp for many in situ experiments. This suggests that kinetic processes may affect the soil gas concentrations in the first hours. With the increase of time the concentrations at the injection point change more slowly and equilibrium conditions are reached

FIGURE 3. Measured standardized concentrations (symbols) and model data assuming no degradation (eq 5, parameters as described in text, lines) from the field experiment E-I. Repeated experiments (Grep and Hrep) are not shown. CFC-114 data are not shown for experiment E to ensure readability. with a higher probability. Consequently, no significant time trend was found for mass fractions in the gas phase determined from 5 h after the injection onward. The third assumption is that the gas-phase diffusion is the only relevant transport mechanism. Advective gas transport could be created due to gravity-driven density dependent vapor transport (22, 23). Gravity-driven advection may be important in coarse soils with a high permeability (23) but is not expected to be important in the sandy soils investigated here. Furthermore the injected gaseous compounds are initially diluted quickly by diffusion due to steep concentration gradients. The concentrations measured at the injection point after 1 h are far below the concentrations, where gravity-driven advection is important (23).

Finally, a point source was assumed, while the actual initial tracer distribution was extended. Hers et al. (14) plot the analytical equations for diffusion from a point source and diffusion from a small extended sphere with a constant initial concentration. The simpler equation for a point source provides an accurate description after an initial time lag much shorter than the 1 h between injection of the tracers and the first sampling. Apparent First-Order Degradation Rate Constants. For both the in situ method and the batch experiments the apparent first-order degradation rate constants kapp in the two unacclimated sandy soils were < 1 d-1 for all VOCs tested. The error in determining kapp from the results of the in situ method was too big to determine any value below this limit VOL. 37, NO. 11, 2003 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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FIGURE 4. Comparison of ratios fa,1/fa,2 (a) and sorption-affected diffusion coefficients Ds (b) as measured in situ and as calculated from batch experiments and the Millington and Quirk relationship (26) for the tortuosity factor.

TABLE 2. Results of the in Situ Method exp.

compound 1

compound 2 (tracer)

A A B C D

CFC-12 CFC-11 chloroethene trans-dichloroethene cis-dichloroethene

SF6 SF6 SF6 SF6 SF6

E E F G Grep H Hrep I

CFC-114 CFC-11 CFC-113 benzene benzene toluene toluene n-octane

CFC-12 CFC-12 CFC-12 CFC-11 CFC-11 CFC-113 CFC-113 CFC-113

a Calculated according to eq 6 with k app,2 ) 0. to eq 10 with fa,2 ) 1.

b

kapp,1a (d-1)

fa,1/fa,2b (-)

De,1/Dm,1c (-)

Ds,1/Dm,1d (-)

Lysimeter: θa ) 0.36 ( 0.03 0.1 ( 0.4 0.5 ( 0.3 -0.3 ( 0.5 -0.7 ( 0.3 -0.0 ( 0.3

0.82 ( 0.11 0.84 ( 0.11 0.70 ( 0.14 0.45 ( 0.07 0.27 ( 0.03

0.19 ( 0.01 0.19 ( 0.01 0.21 ( 0.01 0.20 ( 0.01 0.20 ( 0.01

0.44 ( 0.04 0.45 ( 0.03 0.41 ( 0.06 0.26 ( 0.04 0.15 ( 0.02

Field: θa ) 0.25 ( 0.03 0.0 ( 0.1 0.8 ( 0.1 -0.5 ( 0.4 -0.8 ( 0.3 -1.1 ( 0.4 -0.4 ( 0.4 0.1 ( 0.4 -0.2 ( 0.4

0.93 ( 0.03 0.88 ( 0.15 0.83 ( 0.16 0.22 ( 0.04 0.29 ( 0.09 0.27 ( 0.05 0.30 ( 0.03 0.66 ( 0.14

0.09 ( 0.01 0.09 ( 0.01 0.11 ( 0.01 0.10 ( 0.01 0.09 ( 0.01 0.10 ( 0.02 0.11 ( 0.01 0.09 ( 0.02

0.33 ( 0.03 0.32 ( 0.05 0.38 ( 0.09 0.08 ( 0.01 0.10 ( 0.03 0.10 ( 0.02 0.13 ( 0.02 0.23 ( 0.07

Calculated according to eq 7. c Calculated according to eq 9 with fa,2 ) 1.

accurately. Therefore, the in situ method for the determination of degradation rate constants is only potentially usable in highly active soils for easily biodegradable compounds. Disappearance rate constants observed mainly during the first 5 h in sterile batches are most likely explained by partitioning kinetics. For in situ experiments increased sorption will reduce the mass fraction in the gas-phase fa. This will, according to eq 5, result in a higher gas-phase concentration Cr(0,t) at the injection point. Therefore, a slow decrease of fa,1 of the less volatile compound (chlorinated ethenes, benzene, toluene, or n-octane) with respect to fa,2 of the tracers (CFCs, SF6) can explain negative values for kapp,1, as have been measured in many in situ experiments. Mass Fraction in the Gas Phase. The ratios of the mass fractions in the gas-phase fa,1/fa,2 determined in situ and with partitioning coefficients measured in batch experiments agree well as shown in Figure 4a. The mass fraction in the soil gas fa can also be calculated by using empirical relations to estimate the air-solid partitioning coefficient Ks in eq 1 from the organic carbon content foc and the octanol-water partitioning coefficient Kow. Using an empirical relationship given in ref 24 to estimate Ks, the agreement between the fa calculated and determined in batch experiments was found to be of a factor of 2 or better (Figure S1, Supporting Information). According to the batch experiments the tracers (CFCs, SF6) were not truly conservative with fa,2 ranging from 0.73 for CFC-11 and CFC-113 at the field site to 0.79 for SF6 in the lysimeter. Therefore one will systematically overestimate fa,1 by 20-30%, if this parameter is calculated from fa,1/fa,2 with 2508

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d

Calculated according

the approximation fa,2 ) 1. CFCs and SF6, although reported to be conservative tracers (11, 15, 25), apparently partition into the aqueous phase or sorb slightly to the solid matrix of the investigated sands or to the air-water interface at subsurface temperatures. Batch experiments with CFCs at 25 °C in lysimeter sand indicate negligible sorption (17). This indicates that the measured tracer sorption at 10 or 15 °C was not an artifact. However, the batch experiment approach has its own limitations, e.g. air-water interface adsorption (19) cannot be determined independently. For a more thorough discussion of the batch experiments, please refer to ref 20. Effective Diffusion Coefficients. The proportionality factor between the effective and the molecular diffusion coefficient De /Dm measured in the lysimeter or at the field site on different days fall for each porous medium within a relatively narrow range. This is expected as soil conditions, namely the soil water content, did not change significantly in the experimental period. The effective diffusion coefficients De determined with the in situ method showed good reproducibility (Table 2). Robustness with respect to errors in the input parameters is given by the nature of eq 9: errors due to uncertainty of θa decrease because θa is raised to the power of 1/3. Likewise, an overestimation of the mass fraction in the soil gas fa,2 of the tracer by 25% due to the assumption fa,2 ) 1 leads to an underestimation of De by 9% only. The proportionality factor between De and Dm can also be estimated from porosity data with empirical relationships for τ. The empirical relationship τ ) θa2.33/θt2 proposed by Millington and Quirk (26) gives a proportionality factor of

0.19 ( 0.02 in the lysimeter and 0.10 ( 0.03 in the field site. Estimation of τ with the empirical relationship τ ) 0.66 proposed by Penman (27) gives a proportionality factor of 0.24 ( 0.01 for the lysimeter and 0.17 ( 0.02 for the field site. According to Jin and Jury (25) those two relationships often define a lower and an upper limit for the actual τ. The effective diffusion coefficients De in the soils investigated are better described by the Millington and Quirk relationship (26) as compared to the Penman relationship (27). This good agreement between the measured De and De calculated according to Millington and Quirk may not be true in other soils (25). In conclusion, there is a significant difference between De calculated from different relationships, but there is no obvious a priori choice for the best relationship. The in situ method avoids this uncertainty. Sorption-Affected Diffusion Coefficients. The sorptionaffected diffusion coefficients Ds measured in situ by using the approximation fa,2 ) 1 are compared in Figure 4b with Ds calculated from the batch experiment data and the Millington and Quirk relationship (26) for the tortuosity factor τ according to eqs 1 and 10. The Ds determined in situ are in qualitative agreement with those calculated from the results of the batch experiments. The proportionality factor between Ds and the molecular diffusion coefficient Dm depends both on the properties and the condition of the soil and on the chemical properties of the compounds. Within a given porous medium the proportionality factor is the highest for volatile and hydrophobic compounds such as the CFCs and the lowest for less volatile and/or hydrophilic compounds such as toluene or benzene (Table 2). For the repeated experiments (G, Grep) and (H, Hrep), the Ds measured in situ agree within error limits. The sorption-affected diffusion coefficients Ds determined in situ (fa,2 ) 1) are, however, on average 18% higher than those calculated from the batch data by using the Millington and Quirk relationship (Figure 4b). The observed difference can be explained by the approximation fa,2 ) 1 in the evaluation of the field data, whereas the batch experiments suggest fa,2 ≈ 0.75 for the tracers used in this study. This results in a systematic overestimation of Ds by 15-20%. More work is needed to understand the nature and kinetics of sorption of tracers and VOCs at subsurface temperatures.

Nomenclature Ca

concentration of the compound in the soil air (g cm-3)

Cin

concentration of the compound in the injected compound mixture (g cm-3)

Cr

relative concentration of the compound, Ca/Cin (-)

C0

initial concentration in the soil air in batch experiments (g cm-3)

De

effective diffusion coefficient (cm2 s-1)

Ds

sorption-affected diffusion coefficient (cm2 s-1)

Dm

molecular diffusion coefficient (cm2 s-1)

fa

fraction of the total mass of compound found in the soil air (-)

foc

organic carbon content (% dry weight)

fw

fraction of the total mass of compound found in the soil water (-)

H

Henry’s law constant [(mol cm-3 air)/(mol cm-3 water)]

kapp

apparent first-order degradation rate constant (s-1 or d-1)

Kd

solid-water partitioning coefficient [(mol g-1 solid)/(mol cm-3 water)]

Ks

air-solid matrix partitioning coefficient [(mol cm-3 air)/(mol g-1 solid)]

kw

first-order degradation rate constant in the water phase (s-1 or d-1)

m0

total mass of the compound (g)

R

distance from the point source (cm)

t

time (s)

Vin

volume injected (cm3)

D

Laplace operator

θa

air-filled porosity (cm3 cm-3 soil)

θt

total porosity (cm3 cm-3 soil)

θw

water-filled porosity (cm3 cm-3 soil)

Fs

density of the solids (g cm-3)

τ

tortuosity factor (-)

Acknowledgments Financial support is provided by the Board of the Federal Institutes of Technology, and by the Swiss National Science Foundation (Grant 21-57115.99). The study benefited from the cooperation within the European project Groundwater risk assessment at contaminated sites GRACOS, EVK1-CT1999-00029. The authors thank Gabriele Pasteris and Nathalie Dakhel for help in operating the lysimeter and Peter Kjeldsen, Mette Christophersen, Mette Broholm, Trine Bjerre, and Lone E. Holtegaard for their support in Denmark. The comments of Jennifer Lu and three unknown reviewers helped to improve this manuscript.

Supporting Information Available Detailed information of the preparation of gaseous compound mixtures and the analytical procedures used in this study, concentrations of the gas mixtures used (Table S1), a protocol for the data evaluation, and comparision of batch data with theoretical data obtained from estimations from organic matter content and octanol-water partitioning constants (Figure S1). This material is available free of charge via the Internet at http://pubs.acs.org.

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Received for review May 24, 2002. Revised manuscript received March 12, 2003. Accepted March 14, 2003. ES020101S