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Published online 16 November 2005
Published in Vadose Zone J 4:1107-1118 (2005)
DOI: 10.2136/vzj2005.0020
© 2005 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
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SPECIAL SECTION: SOIL WATER SENSING

Water Saturation Measurements by Gas Tracers in Unsaturated Porous Media—Effect of Mass Transfer Limitations

Liqing Li* and Paul T. Imhoff

Dep. of Civil and Environmental Engineering, Univ. of Delaware, Newark, DE 19716
* Corresponding author (lqli{at}ce.udel.edu)

Received 7 February 2005.



   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATHEMATICAL MODELING
 LABORATORY EXPERIMENTS
 RESULTS
 DISCUSSION
 REFERENCES
 
Water saturation is a key parameter in studies of pollutant transport in the vadose zone and an important factor controlling the rate of waste degradation in municipal solid waste landfills. The partitioning interwell tracer test (PITT) has been suggested as a useful tool for measuring water saturation over large measurement volumes in both systems. However, heterogeneous water distributions may result in mass transfer limitations, which can affect the accuracy of the measurements. In this study, we examined the influence of water saturation, Henry's Law constant, injected tracer mass, tracer quantification limit, and local rates of mass transfer between air and water on PITT measurements. A single-region model was used to describe transport of gas-phase tracers in a one-dimensional system and to investigate the effect of these factors on measurement error. To verify the conclusions drawn from this modeling exercise, laboratory-scale partitioning tracer tests were conducted in four different sand packings. Finally, the results from the mathematical modeling and laboratory experiments were used to suggest guidelines for minimizing measurement error in field-scale PITTs for water saturation measurement.

Abbreviations: DFM, difluoromethane • NAPL, nonaqueous phase liquid • PITT, partitioning interwell tracer test


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATHEMATICAL MODELING
 LABORATORY EXPERIMENTS
 RESULTS
 DISCUSSION
 REFERENCES
 
WATER SATURATION is a key parameter affecting pollutant transport in the vadose zone and it is often a limiting factor for biodegradation of wastes in solid waste landfills. In both soils and landfills core sampling and gravimetric measurement is a conventional method to quantify the subsurface water saturation or moisture content, while other methods, such as time domain reflectometry and neutron moderation, have also been used (Hiller, 1998; Gawande et al., 2003). Although these methods are widely applied, they have the same disadvantage: because of soil and solid waste heterogeneity, many measurements may be required to determine values representative of large regions. In addition, with the exception of gravimetric measurements, other techniques borrowed from soil scientists and used to measure water in landfills are thought to be influenced by the solid waste and water properties (Imhoff et al., 2003), which may vary in space and time.

The PITT is a relatively new technique for measuring water saturation in the vadose zone and has the advantage of measuring water over regions significantly larger than standard methods. The first vadose zone PITT was conducted in 1995 at Sandia National Laboratories in New Mexico to measure nonaqueous phase liquid (NAPL) and water saturations in the subsurface (Mariner et al., 1999). More tests were performed using PITT to measure NAPL saturation in the vadose zone (Whitley et al., 1999; Dwarakanath et al., 1999; Nelson et al., 1999) and to measure water saturation in lab-scale experimental systems (Deeds et al., 1999a; Dwarakanath et al., 1999). Recently, this same technology has been used to measure water in solid waste (Imhoff et al., 2003).

During a field-scale PITT, injection and extraction wells are installed such that chemical tracers flow in the zone of interest. This region typically contains mobile (e.g., air or water) and immobile (e.g., water and/or NAPL) fluid phases. Partitioning tracers and a conservative tracer are injected simultaneously in the mobile phase into the injection well and later removed from the extraction well. Partitioning tracers partition into the immobile phases, retarding the transport of these tracers with respect to a conservative tracer. A comparison of partitioning and conservative tracer breakthrough curves can be used to determine the fraction of the pore space occupied by the immobile fluid phases in the swept zone (Whitley et al., 1999; Dwarakanath et al., 1999). Thus, a PITT may be used to characterize the mean saturation of each immobile phase in the swept volume. In theory, the PITT technology may be applied to both homogeneous and heterogeneous porous media.

More than 30 PITTs have been performed across the USA to measure water or NAPLs in the subsurface, with the majority conducted in the saturated zone to detect NAPLs below the water table (Mariner et al., 1999). The effects of mass transfer limitations on NAPL measurement have been observed in some studies (Nelson et al., 1999; Dai et al., 2001; Brooks et al., 2002; Imhoff and Pirestani, 2004). In a laboratory investigation Nelson et al. (Nelson et al., 1999) found that NAPL saturations were significantly underpredicted by PITTs when physical nonequilibrium due to flow bypassing was significant. Similarly, Dai et al. (2001) found that NAPL saturation was underestimated by 35% when the NAPL was located in a high-saturation lens, a situation where mass transfer limitations were significant. In a controlled field experiment, Brooks et al. (2002) observed that tracers with large partitioning coefficients generally predicted a smaller volume of tetrachloroethylene than the actual release volume, an observation that they suggested might be due to mass transfer limitations. Imhoff and Pirestani (2004) used mathematical models and laboratory experiments to investigate the effects of mass transfer limitations on the detection of cm-scale NAPL pools. They found that NAPL detection errors increased as dimensionless mass transfer coefficients decreased, as the injected tracer mass decreased, and as the tracer detection limit increased. They also determined that exponentially extrapolating the tracer breakthrough curves when mass transfer limitations were important decreased the magnitude of the measurement error but did not eliminate it.

While considerable insight has been gained into the importance of mass transfer limitations for the detection of NAPLs in the saturated zone with PITTs, a similar understanding is lacking for measurement of water in the vadose zone. Deeds et al. (Deeds et al., 1999a) examined the accuracy of PITTs for measuring water saturations at the laboratory scale in media with homogeneous water distributions. While mass transfer limitations were not observed in their experiments when interstitial gas velocities were maintained between 0.88 to 1.40 m d–1, mass transfer limitations are expected to be more significant when PITTs are conducted in heterogeneous porous media, since diffusion pathways will be larger when local regions of high water saturation exist. A clear understanding of the factors controlling measurement error with PITTs in heterogeneous unsaturated systems does not exist.

In this work we examined the influence of mass transfer limitations on PITT measurement of water saturation in unsaturated porous media. A single-region model was used to describe transport of gas-phase tracers in a one-dimensional system and to investigate the effect of various factors on measurement error when mass transfer limitations were significant. Dimensionless groups obtained from nondimensionalizing the mathematical model and from assessing the influence of tracer quantification limits were used to characterize the measurement accuracy. To verify the conclusions drawn from this modeling exercise, laboratory-scale partitioning tracer tests were conducted in four different sand packings, two with relatively uniform water distributions and two with more nonuniform distributions. Finally, the results from the mathematical modeling and laboratory experiments were used to provide guidelines for minimizing measurement error in field-scale PITTs for water saturation measurement in soils and solid waste.


   MATHEMATICAL MODELING
TOP
 ABSTRACT
 INTRODUCTION
 MATHEMATICAL MODELING
LABORATORY EXPERIMENTS
 RESULTS
 DISCUSSION
 REFERENCES
 
Single-Region Model
Transport of a partitioning tracer into and out of uniformly distributed immobile water along a stream tube can be described by a single-region model. Single-region models have been widely used to describe mass transport processes in porous media (e.g., Valocchi, 1985; van Genuchten and Wagenet, 1989). In this model, the entire gas phase is assumed to be mobile and the water immobile. Partitioning of tracers between the gas and immobile water phase is assumed to be linear and described by a first-order rate equation, while tracer interaction with the solid phase is assumed to be negligible. The resulting dimensionless equations describing transport of a partitioning tracer are

[1]

[2]
with

[3]

[4]

[5]

[6]

[7]

[8]

[9]
where C{alpha} is the normalized tracer concentration in phase {alpha}, with subscripts "g" and "w" representing gas and water phases, respectively; c0g is a reference concentration taken here as the influent gas concentration for a pulse or step input (mol cm–3); R is the tracer retardation factor; n is porosity of the porous media; Sw is the fraction of voids filled with water; KH is the Henry's Law constant for the tracer; T is the normalized time; vg is the mean gas velocity (cm min–1); X is the normalized length; L is the length of the stream tube (cm); Pe is the Peclet number; Dg is the dispersion coefficient in the gas phase (cm2 min–1); Da is the Damkohler number; and Kwg is the first-order mass transfer rate coefficient between the gas and water phases (cm–1). The two-film model (Cussler, 1997) was used in defining Kwg, which we assume is dominated by resistance in the water phase.

For the pulse and step inputs considered below, the influent boundary condition is described by

[10]
where for pulse inputs A = 1 when 0 < T < vgt0/L (t0 is the pulse size, min); A = 0 at all other time. For step inputs A = 1 when T > 0.

Flux-averaged concentrations at the end of the stream tube (X = 1) were determined from

[11]
where Ceffg is dimensionless effluent gas concentration.

Time Moment Analysis
Time moment analysis can be used to evaluate the influence of mass transfer limitations on the breakthrough of a partitioning tracer in the system described above. Because the transport of partitioning tracers and sorbing solutes are similar, we followed the approach of Valocchi (1985) and assumed a semi-infinite domain and a Dirac input of tracer mass at the inlet to the streamtube. The resulting time moment formulas for the first moment, and the second and the third central moments are (Imhoff and Pirestani, 2004)

[12]

[13]

[14]
where µ'1 is the first moment, while µn is the nth central moment.

Equation [12] indicates that the first moment is independent of Da; that is, mass transfer limitations have no influence on the mean travel time of partitioning tracers. Mass transfer limitations do affect the second and third central moments, which are indices of the degree of spreading and asymmetry of the breakthrough curves. As the rate of mass transfer decreases (decreasing Kwg), Da decreases and spreading and asymmetry increase. Spreading and asymmetry also increase as R increases or Pe decreases. Although the modeling and experimental study reported below used pulse or step inputs rather than a Dirac input, the influence of the various parameters on the moments for Dirac inputs are similar (Valocchi, 1985). These observations about the influence of Da, R, and Pe on the time moments are discussed below when interpreting model and experimental results.

Analytical Methods
The single-region model describing tracer transport in a streamtube was solved using analytical solutions with CXTFIT (Toride et al., 1995) or MATLAB (The MathWorks, Inc., Natick, MA). Solutions were obtained for square-wave or pulse inputs and step inputs. Breakthrough curves (X = 1) were determined with the number of data points varied between simulations to verify that sufficient data points were used to characterize each breakthrough curve.

The mean travel times of the conservative and partitioning tracers were determined from moment analysis of the breakthrough curves. For pulse inputs (Valocchi, 1985)

[15]
where µ'1 is the mean travel time of the conservative (i = c) or partitioning (i = p) tracer, tF is the time the tracer measurements were terminated, and tp is the duration of the input pulse. For pulse inputs, the concentration of the tracer at t = tF was designated as the tracer detection limit, cdg. For step inputs (Buffham and Mason, 1993),

[16]
The retardation factor and the water saturation were then calculated by

[17]

[18]
Spreading of the conservative tracer was influenced only by the Peclet number, since R = 1 and the second and third moments are only affected by Pe (Eq. [13] and [14]). Because of this and our focus on the influence of mass transfer limitations on measurement error, in the computational experiments discussed below we assumed the mean travel time of the conservative tracer was determined exactly, while the mean travel time of the partitioning tracer was computed using Eq. [15] or [16].

Quantification Limit
A complicating factor influencing the analysis of tracer breakthrough curves from both laboratory and field studies is the truncation of tracer concentration data, that is, the measurement of data only up to t = tF in Eq. [15] and [16]. Because of cost and time limitations, which restrict the sampling period, or quantification limits, which restrict the range of tracer concentration that may be determined, the complete tracer breakthrough curve is never measured. In cases of mass transfer limitations, the tails of the breakthrough curves are important for computing accurate moments, and the truncation of tracer breakthrough data may result in underestimation of water saturation. In this case estimates of partitioning tracer retardation coefficients are systematically lower than actual values, since repeated application of PITT under similar operational conditions (e.g., quantification limits, injected tracer mass) results in truncated data that yield underestimates of partitioning tracer moments and R. A similar effect has been observed when moment analysis of tracer breakthrough curves were used to determine model parameters for dissolved solute transport (Young and Ball, 2000) and to determine NAPL saturations in porous media using PITT (Imhoff and Pirestani, 2004). The errors associated with data truncation are thus systematic, as opposed to random errors that cause fluctuations of measured data about mean values (Coleman and Steele, 1995).

To incorporate the effect of data truncation associated with quantification limits on the determination of water saturation, the tracer detection limit cdg was varied in the computational experiments described below and was normalized by a concentration representing the mass of injected tracer divided by an effective stream tube volume. The effective stream tube volume represents the total capacity of the stream tube for the tracer and is the gas phase pore volume multiplied by the tracer retardation factor. The concentration used to normalize the tracer detection limit is then M/(V{theta}gR), where M is the mass of tracer injected in the system, V is the volume of the stream tube and {theta}g is the volume fraction of gas phase. The mass of injected tracer can be expressed as M = TpV{theta}gc0g, where Tp is the time of the injected tracer injection in gas phase pore volumes. With this definition for M, M/ = Tpc0g/R and the normalized tracer detection limit is expressed as cdgR/. A similar definition for the normalized tracer detection limit was used in studies of dissolved solute transport (Young and Ball, 2000) and PITT's for determination of NAPL in saturated porous media (Imhoff and Pirestani, 2004).

When step inputs were used, tracer tests were terminated when the difference between the influent and effluent tracer concentration was less than a prescribed measurement error. In this analysis the prescribed measurement error was selected as three times the standard deviation of c0g measurements. This parameter was used in place of cdg for simulations where step inputs were considered.

Computational Experiments
The single-region model was used to explore the influence of Pe, Da, tracer quantification limit, injected tracer mass, data extrapolation, water saturation, Henry's Law constant, and tracer input method on water measurement accuracy. In the computational experiments these parameters were varied systematically to explore the importance of each on measurement of water in soil and solid waste with PITT.


   LABORATORY EXPERIMENTS
TOP
 ABSTRACT
 INTRODUCTION
 MATHEMATICAL MODELING
 LABORATORY EXPERIMENTS
RESULTS
 DISCUSSION
 REFERENCES
 
Experiments were conducted in four columns: two packed to create relatively uniform water distributions and two packed to create nonuniform water distributions. Partitioning tracer tests were conducted in these columns and used to compute Sw for a range of experimental conditions. The results were used to test the validity of the single-region model and to substantiate the conclusions drawn from the computational experiments.

Gas Tracers
Partitioning tracer tests require two tracers: a conservative tracer that negligibly partitions into water, onto solids, or onto the gas–water interface, and a partitioning tracer that partitions appreciably into the bulk water phase, but has minor affinity for the gas–water interface and solid surfaces. In this study, helium (He, 10000 ppm) was used as the conservative tracer, while carbon dioxide (CO2, 500000 ppm) or difluoromethane (DFM, 100 ppm) were used as the partitioning tracer. In experiments where DFM and He were used, both tracers were contained within the same gas cylinder (Scott Specialty Gas, Inc., Plumsteadville, PA) and injected simultaneously into the column. For experiments with CO2 and He, the tracers were contained in separate cylinders (Keen Compressed Gas Co., Inc, Newark, DE) and were injected sequentially: the breakthrough curve was obtained for CO2 followed by a separate test to obtain the He breakthrough curve. All gas tracers were balanced with nitrogen (N).

The pH of the distilled water used in our experiments was between 5.6 and 5.9. According to Valsaraj (1995), the effect of pH on the Henry's Law constant of CO2 is negligible in this pH range. Therefore, KH for CO2 was only related to temperature and was computed from (CRC Press, 2002)

[19]
where T is temperature (°C). The Henry's Law constant for DFM is also a function of temperature and this relationship was determined experimentally in our laboratory (Han et al., 2004)

[20]

A Shimadzu GC-14A gas chromatograph (Shimadzu Scientific Instruments, Columbia, MD) was used to measure tracer gas fluxes in our column experiments. The column effluent was directed to the detectors of the gas chromatograph and signals were recorded continually. A schematic of the experimental setup is shown in Fig. 1 . Constant influent flow rates were maintained by a mass flow controller. Before starting each experiment, the influent tracer gas bypassed the sand columns and was directly analyzed by GC to obtain the influent concentrations. Helium and CO2 fluxes were measured with a thermal conductivity detector, and DFM flux was measured with a flame ionized detector. Detection limits were 658 pg min–1 for He, 102 mg min–1 for CO2, and 15.9 pg min–1 for DFM.



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  		Fig. 1. Schematic of experimental apparatus.

 
Columns and Porous Media
Four 60-cm long columns were used in the experiments. Columns 1 and 2 were 10.0-cm-diameter cylindrical columns made of Plexiglas. Column end fittings were made of Plexiglas and fitted with stainless steel mesh, which evenly distributed incoming gas over the entire column cross-section. Columns 3 and 4 were 5.0-cm-diameter cylindrical columns made of glass (ACE Glass, Inc., Vineland, NJ). Column end fittings were made of Teflon and embedded with perforated stainless steel plates, which had the same function as the stainless steel mesh in Columns 1 and 2. All column materials were inert to the gas tracers used in this study, which was verified by testing for sorption by conducting tracer tests through dry empty columns.

A heterogeneous porous medium was created in Column 1 by embedding a fine sand layer (50/70 silica sand, d50 = 0.26 mm, U.S. Silica, IL) within a coarse sand porous medium (Accusand 12/20, d50 = 1.105 mm, Unimin Corp., MN). The fine sand lens was 20 cm long (see Fig. 2) and spanned the column diameter. The sands were washed with distilled water and then packed under water with the column mounted vertically. Aluminum dividers were used to separate the fine sand zone from the coarse sand zone when adding sands into the column. After the fine sand lens was in place, the dividers were slowly pulled out of the sands so that the edges of the lens only experienced minor disturbance. After the column was packed, suction was applied at the base of the column to drain water from the system. Water appeared to saturate the fine sand lens as seen from the column wall, while the coarse sand had significantly smaller water saturation. Gravimetric measurements were used to determine an average porosity (n = 0.345) and water saturation (Sw = 0.255) for the entire column. The column was then positioned horizontally for the gas tracer experiments.



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  		Fig. 2. Packing patterns of sand columns.

 
A homogeneous packing was created in Column 2 with C109 silica sand (d50 = 0.35-mm, Unimin Corp., MN). This column was packed vertically under water, and then suction applied at the base of the column to drain water from the system. No water pockets were observed along the column wall. When the sand column was slowly and carefully unpacked, the media appeared to be uniformly wetted. Thus, the water distribution in Column 2 appeared much more uniform than that in Column 1. The column-average porosity and water saturation were n = 0.340 and Sw = 0.253.

Column 3 was created similarly to Column 1, except that the fine sand lens formed a semi-cylinder in the middle of the column (see Fig. 2). The column-average porosity and water saturation were n = 0.358 and Sw = 0.31. Column 4 was created in a similar manner as Column 2, with the column-average porosity and water saturation n = 0.352 and Sw = 0.27.

Experimental Procedures
Partitioning tracer tests were conducted in the four experimental columns at room temperature, which varied between 20 to 24°C. Columns were weighed before and after each experiment to ensure that volatilization losses during tracer tests were negligible (<0.1%). A constant volumetric gas flow rate was maintained in each experiment with a mass flow controller and was the same in all experiments: Qg = 35 mL min–1. The average interstitial gas velocity varied between experiments and is listed along with other experimental conditions in Table 1.


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  		Table 1. Experimental conditions for tests conducted in sand packings.

 
Each experiment was started by initiating a steady stream of humidified N gas through the column. The gas stream with tracers was injected next, followed by humidified N. Column effluent was directed to the GC detectors without column separation, and detector response signals were recorded every 5 s.

With the exception of Column 1 where only DFM was used, experiments using DFM and CO2 were conducted in all columns. Two types of tracer inputs were used in the experiments: pulse and step inputs. Pulse sizes varied from 0.065 pore volume (PV) to 1.3 PV for experiments in Column 1, from 0.066 to 0.66 PV for Column 2, from 0.014 to 3.3 PV for Column 3, while only 3.4 PV pulse inputs were used in Column 4. Step input tests were only conducted in Column 4. The duration of step inputs varied from 40 to 61 min for CO2, and 57 to 111 min for DFM and He. For experiments with pulse inputs each experiment was terminated when the response signal of the detectors dropped below the detection limits. For experiments with step inputs, tests were terminated when |c0g – ceffg| dropped below three times the standard deviation of c0g measurements. The calibration curves of the detectors were measured periodically to check the linearity of tracer response. The baseline responses were checked periodically to ensure a stable background signal for each detector.

Model Fitting
To compare the mathematical modeling results with the experimental data, CXTFIT was used to fit the single-region model described above to experimental data. For each column the He breakthrough curve was used to fit vg. This value was then used as input to fit Kwg and Dg for transport of DFM through the column.

For fitting CO2 transport a slightly different procedure was followed. Because He and CO2 were contained in separate gas cylinders, CO2 and He were not measured simultaneously in the experiments. In this case vg were fitted to each He breakthrough curve and an average vg selected for fitting CO2 data and determining best-fit Kwg and Dg.


   RESULTS
TOP
 ABSTRACT
 INTRODUCTION
 MATHEMATICAL MODELING
 LABORATORY EXPERIMENTS
 RESULTS
DISCUSSION
 REFERENCES
 
Mathematical Modeling
Tracer Breakthrough Curves
Typical tracer breakthrough curves (X = 1) for pulse inputs of conservative and partitioning tracers are shown in Fig. 3 . In these simulations Pe = 60, KH = 0.49, Tp = 0.25 PV, and Sw = 0.35. Breakthrough curves differ depending on the Damkohler number, which is a measure of significance of local mass transfer limitations (Eq. [9]). As Da decreases from Da = 10 to Da = 0.1 asymmetric spreading of the partitioning tracers systematically increases. These plots suggest that if the tracer detection limit is too high or Da too small insufficient data may be collected to characterize the tracer tail and the mean tracer arrival time. This will occur even though time moment analysis indicates that the first central moment is unaffected by mass transfer limitations. Thus, for asymmetric breakthrough curves, PITT may result in a systematic underestimation of water in the system, if extrapolation of data is not performed or is insufficient to predict trends in the data below tracer detection limits.



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  		Fig. 3. Effect of Da on the asymmetry of breakthrough curves. Tp = 0.25 PV, Pe = 60, and R = 2.

 
Damkohler Number, Tracer Quantification Limit, and Injected Tracer Mass
The influence of several important parameters on the accuracy of PITTs for measuring water in porous media is illustrated in Fig. 4 . The normalized tracer detection limit is plotted vs. the Damkohler number for R = 1.1, 2.0, and 4.0 in these plots. Curves represent the required normalized tracer detection limit to achieve a specified error in water saturation, in this case an error of 5%, with the measured water saturation 95% of actual Sw. When Da is small mass transfer limitations are important and a low tracer detection limit or a large injected tracer mass are needed to achieve a 5% error. As the Damkohler number increases, mass transfer limitations are less important and a larger tracer detection limit or a smaller injected tracer mass are required to achieve the specified error. These observations are in agreement with the discussion of the time moment formulas (Eq. [12]–[14]): as Da decreases, spreading and asymmetry of the partitioning tracer breakthrough curve increases making it more difficult to measure the water.



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  		Fig. 4. Required cdgR/ to maintain a 5% error in Sw for different Damkohler and Peclet numbers. (A) R = 1.1; (B) R = 2.0; and (C) R = 4.0.

 
While it would be convenient to prescribe gas velocities that would ensure mass transfer limitations would not influence measurement error, the results of Fig. 4 illustrate the interaction between Kwg, vg, {theta}w (= nSw), and streamtube length L on measurement error [Da = KgwL/(vg{theta}w)]. If Kwg is large because of uniform water distributions, then gas velocities may be large while keeping measurement errors acceptably small. If, however, Kwg is small and/or {theta}w is large, then gas velocities may need to be much smaller to achieve acceptable measurement error. The gas velocity is a key parameter in conducting field tests, and a high gas velocity is desired for good tracer mass recovery and for completing the field test as quickly as possible.

The Damkohler number will vary between systems, depending on local rates of air–water mass transfer, the scale of the problem, water saturation, and gas velocity. The local air–water mass transfer rate coefficient has been measured in many laboratory studies in soil and granular media (Cho and Jaffe, 1990; Cho et al., 1993; Gierke et al., 1990, 1992; Berndtson and Bunge, 1991; Imhoff and Jaffe, 1994; Szatkowski et al., 1995). In these investigations, Kwg varied several orders of magnitude, with slow rates of mass transfer attributed to regions of immobile water (Cho and Jaffe, 1990; Cho et al., 1993; Brusseau et al., 1991; Gierke et al., 1992; Imhoff and Jaffe, 1994) or preferential water flow (Imhoff and Jaffe, 1994; Szatkowski et al., 1995). Based on a literature survey, Rathfelder et al. (2000) estimated that Kwg should generally range on the order 0.1 to 10.0 d–1 in vadose zone soils. Similar to Kwg, the distance between tracer injection and extraction points, volumetric water content, and gas velocities will also vary between PITT. The influence of each of these parameters on Da, for parameter values typical for laboratory and field experiments, is shown in Fig. 5 . Clearly, for some parameter combinations Da is small resulting in asymmetric spreading of tracer breakthrough curves (Fig. 3), and small cdgR/ would be required for unbiased estimates of Sw (see Fig. 4).



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  		Fig. 5. Effects of components in the Damkohler number on the value of Da. {theta}w = nSw.

 
Peclet Number
For each plot in Fig. 4 results are shown for different Peclet numbers, which are a measure of the ratio of the advective to dispersive characteristics of the system. For conditions where mass transfer limitations are important, nominally when Da < 1, the Peclet number does not have a significant influence on the critical value of cdgR/(Tpc0g) necessary to achieve a 5% error in Sw. This conclusion is independent of the retardation factor, as the Peclet number has a minor effect on the results when the retardation factor varied between R = 1.1 to R = 4.0. For R > 4.0 or Da > 1.0, though, the Peclet number may have a more significant impact on critical cdgR/ to achieve a specified error.

Data Extrapolation
The results shown in Fig. 4 and elsewhere in this work assume that no extrapolation of the data is performed. Typically, though, because of the recognized influence of data truncation, breakthrough curves are extrapolated usually with an exponential function (Keller and Brusseau, 2003). If an exponential function is fitted to the tailing concentrations and extrapolated analytically to zero, the required tracer detection limit for an equivalent measurement error in Sw increases. This is illustrated in Fig. 6 , where curves are shown representing results for a 20% error in Sw 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 is not eliminated. Thus, while the results shown in Fig. 4 and elsewhere in this work represent conditions without data extrapolation, extrapolating the data exponentially will not eliminate the systematic underestimation of Sw when mass transfer limitations are important.



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  		Fig. 6. Effect of extrapolation on normalized tracer detection limit required for a measurement error of 20% in Sw. Pe = 60, KH = 0.49, Da = 0.4, and Tp = 0.25 PV.

 
Water Saturation
The water saturation directly influences the retardation factor of a partitioning tracer (Eq. [5]) and the Damkohler number (Eq. [9]). In addition, previous research has shown that as Sw increases, Kwg decreases (Cho et al., 1994; Szatkowski et al., 1995). In this case as the water saturation of the system increases the air–water interfacial area decreases (Costanza and Brusseau, 2000; Kim et al., 1997, 2001) and the characteristic diffusion length within water films increases, which together reduce the overall rate of air–water mass transfer.

To explore the influence of Sw on PITT measurements, an empirical relationship developed by Costanza and Brusseau (2000) was used to predict the decrease of the specific air–water interfacial area (AIA) with increasing Sw.


[21]
The air–water mass transfer rate coefficient was predicted from

[22]
where kwg (cm d–1) is the air–water mass transfer coefficient. In this analysis we neglected the expected influence of Sw on kwg (decreasing kwg with increasing Sw), since we are not aware of an empirical relationship between these parameters. Instead, kwg was fixed at kwg = 0.1 cm d–1. Using the relationships between Sw and R, Da, and Kwg as indicated in Eq. [5], [9], and [21] and [22], the influence of the water saturation on cdg/c0g required to achieve a 20% measurement error of Sw is shown in Fig. 7 . Holding the injected tracer mass constant, as the water saturation increases a smaller tracer detection limit is required to limit the measurement error to 20%. Thus, measurements of Sw are expected to be more accurate for fixed cdg/c0g at smaller Sw, which is demonstrated in recent PITT tests in sands in an intermediate-scale lysimeter (Carlson et al., 2003).



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  		Fig. 7. Effect of Sw on required cdg/c0g to achieve a 20% measurement error in Sw. Pe = 85, KH = 0.49, n = 0.344, and Tp = 0.25 PV.

 
Henry's Law Constant
The influence of the Henry's Law constant on required cdg/c0g to limit the measurement error of Sw to 50% is shown in Fig. 8 , where parameters from Column 2 were used for the model simulation. Here, KH influences tracer transport through the retardation factor (Eq. [5]). As KH increases the systematic error associated with mass transfer limitations also increases, requiring smaller cdg/c0g to limit the measurement error. Experimental results using CO2 and DFM are also shown, where the tracer detection limit was adjusted to achieve a 50% measurement error of water saturation. Although the data are sparse and exhibit significant variability in required cdg/c0g, data trends are consistent with model predictions. The influence of KH on measurement error is relatively small, with required cdg/c0g decreasing by a factor of 2 to 3 as KH increases from KH = 0.4 to KH = 1.6. The influence of other parameters on measurement error, for example, Sw in Fig. 7, is more dramatic.



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  		Fig. 8. Effect of KH on required cdg/c0g to achieve a 50% measurement error of Sw in Column 2. Tp = (A) 0.25 PV or (B) 0.75 PV. The tracer detection limit was adjusted for experimental data to achieve a 50% measurement error. Model predictions are from the single region model.

 
According to the moment formulas (Eq. [14] and [15]), as KH increases R decreases and spreading and asymmetry of the partitioning tracer breakthrough curve will decrease. This implies that more accurate measurements of Sw may be made for fixed cdg for larger KH, or that larger cdg may be allowed to maintain a fixed error in Sw. However, the opposite trend is observed in Fig. 8. This can be explained by an error propagation analysis (Ku, 1966). The error propagation formula derived from Eq. [18] is shown in Eq. [23]. As seen from this equation, although the systematic error of measuring R ({Delta}R) may decrease with increasing KH, when propagated the error in Sw may increase, depending on the value of KH (1 – Sw)2. For example, for the case where Sw = 0.25, Da = 1, and Tp = 0.25 PV, when KH increases from 0.5 to 1.0, |{Delta}R| decreases from 0.213 to 0.174, but |{Delta}Sw| increases from 0.060 to 0.098. This is opposite to an earlier analysis of partitioning tracer tests for measurement of NAPL saturation (Imhoff and Pirestani, 2004). In that analysis systematic errors increased as the affinity of the partitioning tracer for the immobile phase (NAPL) increased. The reason the two results appear contradictory is that for the measurement of water in unsaturated media mass transfer limitations occurred in the immobile phase (water). For the NAPL measurements, though, mass transfer limitations occurred in the mobile phase (water). The significance of this observation is discussed further below.


[23]

Pulse vs. Step Input
The influence of tracer input method on the required cdgR/ to achieve a specified measurement error in Sw is shown in Fig. 9 . If mass transfer limitations are severe (Da < {approx}0.01), then step inputs provide an advantage over pulse inputs, as larger cdgR/ are required to achieve the same measurement error. If mass transfer limitations are less severe (Da > {approx}0.01), then tracers input as pulses result in smaller measurement errors. For most field applications, pulse inputs will be required to reduce the cost of tracer injections.



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  		Fig. 9. Comparison of step inputs and pulse inputs on the required cdgR/ to maintain a 5 or 50% error in Sw. R = 1.84, Pe = 95.

 
Experimental Results
Sixty-three experiments were conducted using two partitioning tracers to evaluate the reasonableness of the model predictions illustrated above. These experiments were conducted in sand packings where the water distribut ion was either relatively uniform or nonuniform. For the heterogeneous packings where the water was more nonuniformly distributed, tracer breakthrough curves indicated significant mass transfer limitations, while mass transfer limitations were much less important in experiments where water was more uniformly distributed. Typical breakthrough curves and model fits are shown in Fig. 10 for He and DFM transport in Column 1. Because water was distributed nonuniformly in this column, there is significant mass transfer resistance between air and water and extensive tailing of the DFM breakthrough curve. The single-region model fits the breakthrough data for both tracers reasonably well.



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  		Fig. 10. Experimental data and model fitting in Column 1. Tp = 0.43 PV, Pe = 23.7, and Da = 0.014.

 
Determination of the Air-Water Mass Transfer Rate Coefficient
CXTFIT was used to estimate Da and Pe for each column by fitting the single-region model to partitioning tracer breakthrough data from the experiments with large injected tracer masses. The air–water mass transfer rate coefficient was then determined from Eq. [9], and best-fit Kwg are reported for all experiments in Table 2. Most values fall within the range estimated by Rathfelder et al. (2000) (0.1–10.0 d–1), with the exception of experiments conducted using a step input of DFM in experimental Column 1, a pulse input of DFM in experimental Column 2, and a step input of DFM in experimental Column 4. In column nos. 2 and 4, both with relatively uniform water distributions, air–water mass transfer was fast and Kwg = 16 ± 2.6 SD d–1 (SD = standard deviation, Column 2) and Kwg = 68 ± 2.0 SD d–1 (Column 4). Air–water mass transfer coefficients were large in homogeneously packed homogeneous columns with relatively uniform water distributions, since diffusional resistance in the water phase was small and air–water interfacial areas large. In the heterogeneous packings (Columns 1 and 3), though, water saturations were high and gas permeabilities low in the fine-grained media, which resulted in more significant mass transfer limitations. In these columns Kwg ranged between Kwg = 0.12 ± 0.01 SD d–1 (Column 1) and Kwg = 2.4 ± 0.007 SD d–1 (Column 3) in case of pulse inputs of DFM. For step inputs in Column 1, Kwg = 0.038 ± 0.012 SD d–1.


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  		Table 2. Fitted Damkohler numbers and air–water mass transfer rate coefficients for experiments in sand packings.

 
Verification of Model Results
To verify the predictions from the single-region model, selected data from the 63 experiments conducted are shown in Fig. 11 and 12 . In Fig. 11 the measurement error of the known water saturation is plotted vs. cdgR/ for experiments in homogeneous porous media (relatively uniform water distribution) (Column 2) and heterogeneous media (relatively nonuniform water distributions) (Columns 1 and 3). In these experiments, cdgR/ was constant and only the size of the injected trace pulse (Tp) was varied. As cdgR/ increases measurement error also increases for Columns 2 and 3, with no significant change in measurement error over the range of cdgR/ selected in Column 1. In all cases measurement errors resulted in underestimation of the water in the sand columns.



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  		Fig. 11. Measurement error of Sw as a function of cdgR/ for three experimental columns. Error bars on experimental data represent ± one standard deviation of duplicate or triplicate measurements of water saturation. Data points without error bars are single measurements. Model simulations are predictions from the single region model for the conditions of each experimental column.

 


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  		Fig. 12. Measurement error of Sw as a function of Da. Experimental data are from different columns, with tracer detection limit adjusted to make cdgR/ = 0.02. Error bars correspond to the standard deviation of duplicate or triplicate measurements. Model predictions are from the single region model.

 
In addition to the data, predictions from the single region model are also shown in Fig. 11 for the conditions appropriate for each sand packing. Using estimated Da and Pe determined from fits to breakthrough data, a single simulation with fixed R/ was used to predict tracer breakthrough for each column, with cdg varied to obtain the curves shown. While the measurement errors are in the range of model predictions for Column 1, the experimental data do not reproduce the trends predicted from the model and the coefficient of determination (R2) is negative. This may indicate that the single region model is inappropriate for Column 1 or that variations in cdgR/ were too small and random errors in the data too large to verify model-predicted trends. Experimental data match model simulations much better for Columns 2 and 3, with R2 = 0.67 for Column 2 and R2 = 0.79 for Column 3. These results support the use of the single region model for modeling transport of partitioning tracers for column nos. 2 and 3. Thus, for some systems the single-region model may be a useful model for investigating the influence of various parameters on the accuracy of PITTs when mass transfer limitations are important. These results also support the scaling parameter cdgR/ for characterizing the influence of tracer detection limit and injected tracer mass on measurement of Sw.

Figure 12 shows the influence of Da on the measurement of Sw. Selected data are shown from Column 1 (Da = 0.044), Column 2 (Da = 6.0), and Column 3 (Da = 0.18), where the tracer detection limit was adjusted so that cdgR/ = 0.02. As the Damkohler number increases from Da {approx} 0.05 to Da {approx} 5, the systematic error in measuring Sw decreases approximately 40% for fixed cdgR/, with model predictions generally matching trends in experimental data.


   DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATHEMATICAL MODELING
 LABORATORY EXPERIMENTS
 RESULTS
 DISCUSSION
REFERENCES
 
In this work model simulations and laboratory experiments were used to evaluate the influence of various factors on water saturation measurement with PITTs under mass-transfer limiting conditions. Although tracer transport was examined in a single stream tube, observations from these tests should reflect the influence of parameters in three-dimensional systems, which was demonstrated earlier for measurement of NAPLs (Imhoff and Pirestani, 2004).

When air–water mass transfer rates are small, model simulations and laboratory experiments showed that PITTs often result in underestimation of water saturation, even if data are extrapolated exponentially. The influence of mass transfer limitations on measurement error increased with decreasing injected tracer mass, increasing Henry's Law constant, increasing gas velocity (decreasing Damkohler number), increasing tracer quantification limit, and increasing water saturation.

For partitioning tracer tests conducted to measure NAPL saturation, the influence of mass transfer limitations on measurement error can be assessed by using multiple partitioning tracers with different affinity for the NAPL in a single PITT (Imhoff and Pirestani, 2004). If tracer tests indicate decreasing amounts of NAPL as the NAPL/water partition coefficient increases (affinity for the immobile fluid increases), then it is likely that mass transfer limitations influence the results. When mass transfer limitations are important, systematic errors in NAPL saturation measurement increase as NAPL/water partition coefficients increase (Imhoff and Pirestani, 2004), while random errors for these measurements decrease with increasing partition coefficients (Jin et al., 1997).

For measurement of water in the vadose zone the situation is different. Here, because tracer diffusivities in gas are approximately 10000 times larger that those in water, mass transfer limitations are associated with diffusional resistance in the immobile phase (aqueous), rather than mass transfer resistance in the mobile phase, which is often limiting in NAPL/water systems (Imhoff and Pirestani, 2004). As the Henry's Law constant increases (affinity for the immobile phase decreases) systematic errors in water saturation increase. Unfortunately, the random error in water saturation measurements also increases with increasing Henry's Law constant. Retardation of the partitioning tracer is reduced as the Henry's Law constant increases, and the random error in the measured retardation factor and water saturation increases (Jin et al., 1997). Because both the random and systematic error in water saturation measurement increase with Henry's Law constant, water saturation measurements made using multiple tracers with different Henry's Law constants but similar injected mass and detection limit cannot be easily used to infer the significance of mass transfer limitations.

Despite this situation, it is clear that systematic errors in water saturation measurement are reduced by conducting PITTs with small gas velocities, large injected tracer masses, small tracer quantification limits, and small Henry's Law constants. To assess the significance of mass transfer limitations on water saturation measurement may require conducting multiple PITTs at different gas velocities, or conducting a single PITT with multiple tracers and different injected tracer masses or quantification limits to vary cdgR/ over a wide range. While such an approach might be used to assess the influence of mass transfer limitations on water saturation measurements from PITTs, the duration of sampling to collect the requisite breakthrough data could be excessive.

Finally, it would be ideal if test parameters could be selected to ensure that mass transfer limitations have an insignificant influence on water saturation measurement with PITTs. While in general this may not be possible, this work highlights those factors that have an important influence on water saturation measurements under mass-transfer limiting conditions and how variations in these parameters might be used to assess measurement error. It also suggests that improvements in field techniques (e.g., automatic sampling of tracer gases) that permit sampling of breakthrough curves for extensive periods may significantly improve the accuracy of PITT measurements.


   ACKNOWLEDGMENTS
 
This research was supported by the National Science Foundation Grant No. 9984715, and a grant from the Urban Waste Management and Research Center at the University of New Orleans.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 MATHEMATICAL MODELING
 LABORATORY EXPERIMENTS
 RESULTS
 DISCUSSION
 REFERENCES