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Published in Vadose Zone Journal 3:1240-1248 (2004)
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA

REVIEWS AND ANALYSES

Review of Field Methods for the Determination of the Tortuosity and Effective Gas-Phase Diffusivity in the Vadose Zone

David Wernera,c, Peter Grathwohlb and Patrick Höhenera,*

a Swiss Federal Institute of Technology (EPFL), ENAC-ISTE-LPE, CH-1015 Lausanne, Switzerland
b University of Tuebingen, Center for Applied Geosciences, Sigwartstr. 11, Tuebingen
c Currently, School of Civil Engineering and Geosciences, Cassie Bldg., Rm. B3.14, University of Newcastle upon Tyne, NE1 7RU, UK
* Corresponding author (patrick.hoehener{at}epfl.ch)

Received 23 February 2004.



   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 THEORY
 FIELD METHODS
 FIELD DATA
 APPENDIX
 REFERENCES
 
Modeling the gas exchange flux between soil and the atmosphere, risk assessment, and the evaluation of remediation strategies at contaminated sites require the knowledge of gas-phase diffusivities in the subsurface. We review methods to measure the tortuosity factor or the effective gas-phase diffusion coefficient in situ. The strong dependency of these parameters on the structure and volume of the air-filled pore space in the subsurface calls for an accurate and robust in situ measurement. A variety of approaches have been proposed during the last decades, each based on the observation and interpretation of gaseous tracer diffusion in near-surface soils or the deeper vadose zone under various initial and boundary conditions. We briefly describe the conceptual basis and experimental setup of each method and give insight into error propagation. We then discuss 115 effective diffusion coefficients De compiled from the original method papers and applications. In situ methods and laboratory measurements on undisturbed soil cores yield comparable results. The Penman relationship, De/Dm = 0.66{theta}a, sets an upper limit for the field-determined effective diffusion coefficient in the case of uniform porosity. The Moldrup relationship, De/Dm = {theta}2.5a/{theta}t, originally proposed for sieved and repacked soils, gave the best predictions of several porosity-based relationships, but the relative deviation between observed and predicted De can be substantial. Therefore, the application of such a relationship for the site-specific modeling of gas-phase diffusion should be justified with in situ measurements, especially in heterogeneous environments.


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
THEORY
 FIELD METHODS
 FIELD DATA
 APPENDIX
 REFERENCES
 
GAS-PHASE DIFFUSION dominates the migration of natural gases and volatile pollutants in the unsaturated zone in the absence of pressure gradients. The diffusive flux of gaseous compounds through the vadose zone and their exchange at the soil–atmosphere interface is of broad interest in various fields of environmental research. Measuring diffusive O2 and CO2 fluxes for the quantification of biological activity in soils has received early interest (Raney, 1949; Lai et al., 1976). Such fluxes play a critical role in the assessment of biodegradation at contaminated sites (Suchomel et al., 1990; Hers et al., 2000a; Pasteris et al., 2002; Dakhel et al., 2003; Lee et al., 2003). The soil–atmosphere exchange of the radiatively active trace gases CH4 (Whalen and Reeburgh, 1990; Steudler et al., 1989; Mosier et al., 1991) and N2O (Jellick and Schnabel, 1986; Mosier et al., 1991; van Bochove et al., 1998; Well and Myrold, 2002) has been studied by soil scientists and climatologists. Hydrologists are interested in the diffusive flux of atmospheric tracer gases like chlorofluorocarbons (Weeks et al., 1982) and SF6 (Santella et al., 2003) from the atmosphere to the groundwater table. Radon-222 diffusion from soil to houses (Rogers and Nielson, 1991; Nazaroff, 1992; Washington et al., 1994) affects public health. Last, but not least, understanding diffusive vapor fluxes is essential for risk assessment at contaminated sites (Smith et al., 1996; Nicot and Bennett, 1998; Choi et al., 2002; Jellali et al., 2003; Christophersen et al., unpublished data).

Gas-phase diffusion in the vadose zone differs from the diffusion through free air. Solid and liquid obstacles reduce the cross-sectional area and increase the mean path length for compounds diffusing in soils. Under transient conditions, the gas-phase diffusion in the unsaturated zone is affected by partitioning of a compound into the soil water, onto the air–water interface, and into or onto the solids (Grathwohl, 1998). Compounds with a low mass fraction in the soil gas will therefore migrate more slowly than conservative gases, depending on the soil and chemical properties of the diffusing substance (Kim et al., 2001).

Effective diffusion coefficients, De, often also called intrinsic diffusion coefficients, are used to calculate diffusive gas fluxes in the vadose zone from concentration gradients according to Fick's first law or to interpret steady-state vapor concentration profiles. They account for the reduction in the diffusion effective cross-sectional area and the increased path length in soils. Their dependency on the structure and volume of the gas-filled pore space and on the temperature of the soil calls for accurate in situ measurements. We review a variety of different approaches for a determination directly in the field. Alternatives to field approaches are laboratory measurements on intact or restituted soil, or the use of empirical relationships. Compared with field methods, laboratory investigations are less susceptible to measurement errors, but can be more cumbersome. Laboratory temperatures are generally different from the field, only a small soil volume can be investigated, and a large number of measurements are required at heterogeneous sites. Empirical and semiempirical relationships (Penman, 1940; Millington and Quirk, 1961; Currie, 1970; Sallam et al., 1984; Moldrup et al., 2000) estimate diffusion parameters from soil properties such as porosity and water content. These relationships are widely used in numerical models, but their use in the calculation of diffusive fluxes from concentration profiles has its drawbacks; some of the empirical relationships are sensitive to errors in the required input parameters. For instance, the popular relationship of Millington & Quirk (1961), De = Dm, is highly sensitive because two of the three required parameters are raised to a high power. Furthermore, the predictions of the different empirical relationships differ considerably, and a priori there is no obvious choice for the best relationship. Such difficulties can be avoided by measuring the effective diffusion coefficient in the field.


   THEORY
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
FIELD METHODS
 FIELD DATA
 APPENDIX
 REFERENCES
 
Diffusive fluxes F of gaseous compounds per unit area in an unsaturated porous medium can be calculated from the measured spatial concentration gradient {partial}Ca/{partial}x of the compound of interest (denoted by subscript 1; i.e., benzene, CO2, ...) in the air-filled pore space according to a modified version of Fick's first law (Grathwohl, 1998):

[1]
where {theta}a denotes the air-filled porosity, {tau} the tortuosity factor, and Dm the molecular diffusion coefficient. Definitions of the symbols used in this article are summarized under notations. The effective diffusion coefficient De is the proportionality factor between the spatial concentration gradient and the flux per unit area:

[2]

Experimentally determined De values in partially water-saturated systems also account for diffusive fluxes in the water phase, which are generally negligible compared with the diffusive flux in the gas phase. The molecular diffusion coefficient Dm,1 in Eq. [2] can be estimated with sufficient accuracy, and corrections for the temperature dependency are available (Fuller et al., 1966). Field methods for the determination of De,1 rely on the observation of tracer diffusion for a quantification of {tau}, and additionally they require an independent determination of {theta}a, as described below (see Eq. [6]–[8]). The measurement of {theta}a can be challenging, especially at greater depths, and robustness with respect to an error in {theta}a is an important criterion when choosing a suitable field method.

The determination of the tortuosity factor {tau} is based on the observation of diffusing tracers under well-defined initial and boundary conditions. Here and henceforth we will denote tracers with the subscript 2. For the interpretation of the tracer data, the subsurface is normally 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 water wet. The partitioning of the gaseous compounds between those phases is described by an instantaneous, reversible linear equilibrium, degradation of the tracer is neglected, and gas-phase diffusion is assumed to be the only relevant transport mechanism. Under these conditions the governing equation describing the gas-phase diffusion of the tracer 2 under transient conditions is

[3]
where fa,2 stands for the mass fraction of the tracer in the soil air and {Delta} for the Laplace operator. From Eq. [3] it is obvious that a determination of {tau} from tracer data requires knowledge of fa,2. This problem is circumvented by using a tracer gas, which is very volatile and hydrophobic and does therefore neither sorb to soil solids nor partition into soil water. For such a "conservative" tracer, all molecules are assumed to be present in the soil air (fa,2 = 1). The assumption fa,2 = 1 is violated if a fraction of the molecules temporarily dissolve in the soil water or sorb to the solids or the air–water interface. For instance, even for chlorofluorocarbon tracers, fa,2 was found to be somewhat smaller than 1 at subsurface temperatures (Weeks et al., 1982; Werner and Höhener, 2003a). Therefore, the robustness of each method with respect to an overestimation of fa,2 is another important criteria, when choosing a suitable method.

By using an analytical solution to Eq. [3] for the interpretation of the tracer data, an explicit expression can be given for the calculation of the tortuosity {tau} or the effective diffusion coefficient De,1 of the compound of interest:

[4]
where A, B, ... are constants or experimentally quantified parameters. To discuss the robustness of each method with respect to error propagation, we estimate the overall relative uncertainty {Delta}De/De according to

[5]

This error propagation is based on the assumption that the parameters are uncorrelated and only first-order terms of the Taylor-series expansion are relevant. We assume that the required experimental parameters, tracer concentration ratios, and the ratio of the molecular diffusion coefficients Dm,1/Dm,2 can all be quantified with a relative error of 10%, whereas the time t of a measurement can be quantified with negligible error. Sometimes, the equivalent to a parameter A or B in Eq. [5] will be a functional value of a measured concentration ratio with its own time-dependent error propagation. For the sake of simplicity we assume that such functional values can also be quantified with 10% relative error.


   FIELD METHODS
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 FIELD METHODS
FIELD DATA
 APPENDIX
 REFERENCES
 
Flux Chamber Method
The flux chamber method has been developed to measure effective diffusion coefficients De in surface soils. Rolston and Moldrup (2002) presented a thorough review of the flux chamber method. This approach was first described by McIntyre and Philip (1964). A cylinder is inserted into the soil, and a gaseous tracer is supplied to the confined core from a well-stirred reservoir in a diffusion chamber placed on the cylinder. The experimental setup for this and other field methods is shown schematically in Fig. 1 . An analytical solution to Eq. [3] for the nonsteady diffusion of an inert gas (fa = 1) in a uniform, semi-infinite porous medium is used to interpret the change of the tracer concentration in the reservoir with time. The effective diffusion coefficient De,1 of a compound of interest (denoted by subscript 1) is calculated from

[6]
where a is the height of the reservoir, t the time of the measurement, and T is a function of the ratio between the measured and initial tracer concentration in the reservoir. An eventual constant background concentration of the tracer in the soil needs to be subtracted from both the measured and initial background concentration before building the ratio. McIntyre and Philip (1964) provide tabulated values and graphs for T. We calculate the overall relative uncertainty {Delta}De/De from Eq. [5] with the assumptions outlined in the theory and obtain a value of 26%. Rolston et al. (1991) performed a total error analysis based on their experimental design for the chamber method. They estimate the overall relative uncertainty for De to range from 10 to 40%, depending on the absolute value of the air-filled porosity {theta}a. These authors also discuss the semi-infinite soil assumption and propose use of a 15- to 20-cm soil cylinder depth to extend the time before a sufficiently large systematic error occurs. Because the analytical solution to Eq. [3] used for the data analysis does not account for tracer sorption, conservative tracer behavior is a requirement (i.e., no partitioning into the aqueous phase and no sorption to soil solids).



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  		Fig. 1. Schematic diagram of the experimental setup for the various methods.

 
The flux chamber method was used by McIntyre and Philip (1964) with O2 as the tracer and an oxygen cathode for the quantification of the tracer, by Rolston et al. (1991) with CClF3 quantified by gas chromatography, by Elberling and Nicholson (1996) with CO quantified with gas sensors, and by Ball et al. (1994) with 85Kr quantified in a Geiger–Müller tube. The design of McIntyre and Philip (1964), Elberling and Nicholson (1996), and Ball et al. (1994) allows an in situ measurement of the tracer concentration. Other design for the in situ measurement of De in an isolated core of surface soil have been described by van Bochove et al. (1998) and Rolston and Brown (1977). These designs require soil gas probes in addition to a diffusion chamber and appear to be much more complicated. They yield a diffusion coefficient, which is more representative for the whole isolated soil volume.

Instantaneous Point Source, Single-Well Methods
Instantaneous point source methods use a short pulse of tracer gas, which is introduced into the soil at the beginning of the experiment. A single-well approach was first described by Lai et al. (1976), who used syringe needles to introduce a small volume of tracer gas into near-surface soils. The concentration decline at the injection point is monitored subsequently by withdrawing small volumes of gas through the needle. The concentration decline at the injection point is described with an analytical solution to Eq. [3] for a spherical initial tracer gas distribution of constant concentration around the injection point. We believe that the expression for the tracer concentration at the injection point was wrongly derived by Lai et al. (1976), as detailed in (Werner, 2002). Consequently, these authors observed a deviation of the data from predictions with increasing time, which they attributed to a small dislocation of the injected tracer gas plume. Jellick and Schnabel (1986) regarded the initial condition of a spherical tracer gas plume with a constant concentration as unrealistic and proposed the use of a finite difference model with a cosine-like initial concentration distribution as an improvement. Werner and Höhener (2003a) circumvented the problem of the unknown initial distribution by allowing more time (>40 min) for tracer diffusion before sampling. An analytical solution to Eq. [3] for an instantaneous point source then provides a sufficiently accurate description of the tracer data. A longer delay between injection and sampling also permits bigger sample volumes because the tracer concentration gradients near the injection point decline with time. This is relevant for the use of the method at greater depth. According to Werner and Höhener (2003a) the effective diffusion coefficient De,1 of a compound of interest is obtained from

[7]
where Cr,2(0,t) is the ratio between the measured tracer concentration at the injection point and the initial tracer concentration in a small volume Vin of gas injected into the soil. With the previous assumptions we estimated the theoretical overall relative uncertainty {Delta}De/De to be 15%. This approach is remarkably robust with respect to errors in {theta}a or an eventual underestimation of fa, which is generally assumed equal to 1 for the tracers.

Lai et al. (1976) used O2 as a tracer and syringe needles fitted with a wooden collar and protected against clogging by piano wire to inject and sample the tracer at shallow depths (7.6 cm). They quantified their tracer with a portable gas chromatograph in the field. Jellick and Schnabel (1986) used N2O as tracer and a similar experimental setup. Their finite-difference approach for the evaluation of the data was used by Washington et al. (1994) to measure the diffusivity of N2O at various depths in 2-m-deep pit walls and by Ball et al. (1994), who used 85Kr as tracer and a special probe with a Geiger–Müller tube inserted into auger holes. The design of Ball et al. (1994) allows a true in situ measurement directly at the injection point and avoids any disturbance of the tracer diffusion caused by the withdrawal of samples. Werner and Höhener (2003a) used sulfurhexafluoride (SF6) or chlorofluorocarbons (CFCs) as tracers, gas-tight syringes for sample storage, and off-site gas chromatography for tracer analysis. They inserted slender probes with stainless-steel capillary tubing into auger holes and push them further into the undisturbed soil, backfilling the auger holes after installation. The soil cores from auger holes are used to determine the air-filled porosity {theta}a. Werner et al. (2004) used the same approach for the simultaneous determination of VOC-concentration gradients and effective diffusion coefficients De at a contaminated field site.

A variation of the single-well instantaneous point source approach has first been published by Johnson et al. (1998). Instead of measuring the tracer concentration at the injection point, the volume-averaged tracer concentration within a globular sphere around the injection point is determined by pumping a known amount of soil gas from the injection point after a sufficient time lag. Their interpretation of the volume-averaged tracer concentration is based on the analytical solution to Eq. 3 for an instantaneous point source of a conservative tracer (fa = 1). The effective diffusion coefficient De,1 of a compound of interest is obtained from:

[8]
where ß is a function of the tracer mass fraction recovered when sampling the spherical volume of gas around the injection point (Johnson et al., 1998). With the previous assumptions we calculated the overall relative uncertainty {Delta}De/De to be 16%. This approach is also remarkably robust with respect to errors in {theta}a. Hers et al. (2000b) provided a numerical sensitivity analysis of this method. They also compared the analytical solutions for a point and a finite source and demonstrated how they converge quickly. They considered tracer sorption and partitioning into the water phase in deriving an analog to Eq. [8]. However, tracer sorption or water-phase partitioning would certainly be a problem for the use of this method because it is not obvious whether only the tracer mass in the soil air or the total tracer mass within the corresponding spherical soil volume is recovered when pumping a large soil air sample.

Johnson et al. (1998) used SF6 as tracer, existing multilevel monitoring installations for measurements to 2 m depth, and on-site tracer analysis with an electron capture detector. Hers et al. (2000b) used He as a tracer, soil gas probes installed by direct push, and a portable He detector for measurements to 1 m depth. They later used the measured De to assess biodegradation processes in situ (Hers et al., 2000a).

Instantaneous Point Source, Inter-Well Methods
As demonstrated by Nicot and Bennett (1998), effective diffusion coefficients De can also be determined by monitoring the breakthrough of the injected tracer in some distance r from the instantaneous point source. With such an approach a larger and better-defined soil volume can be investigated. Nicot and Bennett (1998) inject 0.55 mol of ethene and propane as tracers and determined vertical and horizontal tortuosity factors {tau} in a Playa wetland by interpreting the tracer concentrations measured at up to 3 m from the injection point with a numerical model for gas-phase diffusion. Werner (2002) applied the same approach on a much smaller scale (r = 30 cm). By determining the time of the maximum tracer concentration tmax at some distance r from the injection point, he obtained De,1 of a compound of interest from

[9]

With the usual assumptions the overall relative uncertainty {Delta}De/De is 28%. Unfortunately this approach is sensitive to errors in r, which is difficult to determine accurately in the subsurface.

Continuous Point Source, Inter-Well Method
Kreamer et al. (1988) developed a special permeation device that releases compounds at a constant rate. This device can be used as a continuous point source in tracer experiments. With this approach a large amount of tracer can be released during an extended period of time without inducing a pressure gradient. The tracers migration can then be observed for greater distances, for instance up to 3 m within a week (Kreamer et al., 1988). These authors used an analytical solution to Eq. [3] for a continuous point source in a medium of infinite extension and type curve analysis to extract De from the tracer data. Another elegant approach would be to calculate De from the steady-state tracer concentration Css at distance r from the continuous point source

[10]
where q is the rate of gas diffusion from the continuous source. We mention this possibility because the tracer concentrations measured by Kreamer et al. (1988) actually reach such a steady state in proximity to the source (<1 m) and because this steady-state approach allows one to determine De without knowledge of the air-filled porosity {theta}a. With the previous assumptions we calculated the overall relative uncertainty {Delta}De/De to be 20%.

Atmospheric Tracer Method
An alternative approach for determining effective diffusion coefficients De on a large spatial scale was described by Weeks et al. (1982). These authors measured vertical concentration profiles of CFC-12 and CFC-11 in a 50-m-thick unsaturated zone and used the historic atmospheric concentration of these gases as calculated from their annual worldwide release statistics as the upper boundary condition and no flux at the groundwater table as the lower boundary condition in their numerical model. From a fit of the data with the model predictions, De can be extracted. Their analysis was complicated by the effects of solid–water–gas partitioning. The authors concluded that CFC-12 and CFC-11 were not truly conservative in the investigated vadose zone. This atmospheric tracer method could be based on other compounds like SF6 or 85Kr, but is not applicable where atmospheric concentrations vary locally (Höhener et al., 2003; Santella et al., 2003).

Comparison of Various Approaches
Our compilation of field methods demonstrates the variety of approaches for the measurement of the effective diffusion coefficient. The various methods are compared in Table 1. The chamber method and the single-well instantaneous point source methods are the best investigated. The chamber method is compatible with the measurement of the total gaseous exchange flux at the soil–atmosphere interface in flux chambers (Smith et al., 1996; Mosier et al., 1991; Steudler et al., 1989; Whalen and Reeburgh, 1990). The single-well instantaneous point source methods are compatible with soil gas monitoring in the subsurface, as demonstrated by Werner et al. (2004). Of the various single-well and instantaneous point source methods, the approach proposed by Lai et al. (1976) and refined by Jellick and Schnabel (1986) allows the fastest determination of the effective diffusion coefficient. However, it requires an accurate description of the initial condition. The approach by Werner and Höhener (2003a) is more robust and of appealing simplicity, but depends on specially designed soil gas probes with a low dead volume. The approach by Johnson et al. (1998) is robust and uses normal soil gas probes, but sampling requires pumping of a large gas volume, which might produce artifacts.


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  		Table 1. Compilation of effective diffusion coefficients measured directly in the subsurface.

 
The soil volume investigated with the in situ method is best-defined for the surface chamber method. For single-well instantaneous point source methods the investigated soil volume is ill-defined and likely to be very small (Werner and Höhener, 2002; Werner, 2002), with the notable exception of the approach presented by Johnson et al. (1998). Inter-well methods investigate a larger and better defined soil volume, but this advantage comes at the cost of a high number of measurements performed during an extended period of time (Nicot and Bennett, 1998; Kreamer et al., 1988; Weeks et al., 1982). High installation costs for measurements in the profound vadose zone will typically favor inter-well methods, whereas an investigation of the shallow vadose zone appears to be less time-consuming and more cost-effective with a number of single-well measurements at various locations. Single-well and inter-well methods can be used simultaneously or subsequently (Werner and Höhener, 2003b).

Tracers used by the various authors include noble gases such as He and 85Kr, several chlorofluorocarbons, SF6, CO, and short chain hydrocarbons like ethene and propane. The choice will typically depend on the available analytical equipment. Some authors used directly the gas of interest (O2, N2O) as their tracer because effective diffusion coefficients De can then be determined without knowledge of the molecular diffusion coefficient Dm. This approach is, however, only applicable if the compound of interest is truly conservative (i.e., no partitioning into the water phase, no sorption, no production, no degradation) and for natural gases an eventual background concentration has to be quantified. Note that the importance of partitioning or first-order degradation can be assessed in situ (Werner and Höhener, 2003a). Most authors measured the tracer directly in the field. McIntyre and Philip (1964), Elberling and Nicholson (1996), and Ball et al. (1994) found ways to avoid disturbance of the diffusion process by sampling. They quantified the tracer directly at the point of interest.


   FIELD DATA
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 FIELD METHODS
 FIELD DATA
APPENDIX
 REFERENCES
 
We compiled 115 field-determined effective diffusion coefficients De from the reviewed literature (Table 2). The ratios De/Dm are shown as a function of the air-filled porosity {theta}a of the soils in Fig. 2 . Soil types range from loam to sand. Apparently, a model solely based on {theta}a cannot explain the measured De/Dm. One such model, the relationship De/Dm = 0.66{theta}a proposed by Penman (1940), is shown in Fig. 2. Jin and Jury (1996) observed that the Penman relationship often defines an upper limit for De measured in the laboratory. In Fig. 2, only five of the measured De are significantly larger than predicted by Penman. McIntyre and Philip (1964) explained three of these ({theta}a,De/Dm) values (0.23,0.66; 0.24,0.66; 0.28,0.28) with nonuniform porosities in cracked clay and in a loamy sand. A fourth value (0.09,0.17) reported by Ball et al. (1994) is probably erroneous, since laboratory measurements on soil cores from the same location were two orders of magnitude lower. We concluded that in soils with a fairly uniform porosity, the diffusive gas-phase transport is generally reduced by at least a factor 0.66{theta}a as compared with diffusion through free air. An upper limit for the expected diffusive flux will be valuable for many practical applications.


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  		Table 2. Compilation of effective diffusion coefficients measured directly in the subsurface.

 


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  		Fig. 2. Ratios of effective and molecular diffusion coefficients determined in the field with one of the described methods as a function of the air-filled porosity of the soils. Data were obtained with the surface chamber method (circles), single-well methods (diamonds), and an inter-well method (triangles). Values were determined in various soil types ranging from loam to sand, near the surface and in the shallow and deep vadose zone.

 
There is no obvious systematic difference between diffusivities determined with different methods (Fig. 2). The data obtained with the single-well method apparently has the most pronounced scatter, which could be explained with the relatively small volume of soil investigated. This data set also represents the largest number of authors and different soil types. To date no direct comparison of different field methods at one location is available.

Three of the reviewed studies (Jellick and Schnabel, 1986; Rolston et al., 1991; Ball et al., 1994) compare in situ measurements with laboratory measurements on soil cores taken from the same or a nearby location. We combine their data in Fig. 3a . The in situ measurements and laboratory measurements on undisturbed soil cores yield comparable results, and some of the observed scatter can be attributed to actual differences in similar but not identical soil samples. Rolston et al. (1991) measured De with the chamber method and with a laboratory method on a few identical soil cores and found better agreement for this subset of data. As discussed above, one of the values (0.17,0.002) is probably erroneous. We conclude that laboratory measurements on undisturbed soil cores and in situ measurements are equivalent approaches for the determination of De.



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  		Fig. 3. (A) Ratios of effective and molecular diffusion coefficients as determined in the field and in the laboratory on undisturbed soil cores from the same location. Data from Rolston et al. (1991), Jellick and Schnabel (1986), and Ball et al. (1994). (B) Ratios of effective and molecular diffusion coefficients as determined in the field and as estimated from soil porosity data according to Moldrup et al. (2000). Data were obtained with the surface chamber method (circles), single-well methods (diamonds), and an inter-well method (triangles).

 
We assembled a subset of 81 ratios De/Dm for which both the air-filled and the total porosity of the investigated soils were reported by the authors. These data are compiled in Table 2. We used this compilation to test the empirical, porosity-based relationships proposed by Millington and Quirk (1961), Currie (1970), Sallam et al. (1984), and Moldrup et al. (2000). We calculated the RMSE (Moldrup et al., 2000) between the predictions and the values measured in situ. By this criterion, the relationship De/Dm = {theta}2.5a/{theta}t, proposed by Moldrup et al. (2000) for sieved and repacked soils, gave the best description. The applicability of this relationship for both natural and repacked soils has recently been confirmed for a sandy soil with both field (Table 2, Werner and Höhener, 2003a, 2003b) and laboratory measurements (Wang et al., 2003). The Moldrup relationship is also the least susceptible to errors in the input parameters, even though {theta}a is still raised to a high exponent of 2.5. Figure 3b compares in situ measurements with the predictions. Again, one data point (0.17,0.005) is probably erroneous. The data suggest that the air-filled and the total porosity of a soil are not always sufficient descriptors for the prediction of De. Field methods as described in this paper allow verification and should be used to justify the choice of an empirical relationship for site-specific numerical modeling.

This review describes a variety of methods for a quantification of the effective gas-phase diffusion coefficient in the vadose zone. However, only a few methods are validated at multiple sites, and a study comparing various approaches at the same site is lacking. Another research need is the evaluation of methods in heterogeneous and structured media, where some of the underlying assumptions may be violated. Field methods for the quantification of aqueous phase and solid phase partitioning, such as the one presented by Werner and Höhener (2003a), are also relevant for the understanding of diffusive vapor spreading. They should be further developed and validated.


   APPENDIX
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 FIELD METHODS
 FIELD DATA
 APPENDIX
REFERENCES
 
ACross-sectional area (cm2)aHeight of flux chamber reservoir (cm)CaConcentration of the compound in the soil air (g cm–3)CatmConcentration of the compound in the atmosphere (g cm–3)CinConcentration of the compound in the injected compound mixture (g cm–3)CrRelative concentration of the compound, Ca/CinCssSteady-state concentration (g cm–3)DeEffective diffusion coefficient (cm2 s–1)DmMolecular diffusion coefficient (cm2 s–1)FDiffusive flux (g cm–2 s–1)faFraction of the total mass of compound found in the soil airqRate of gas diffusion from source (g s–1)rDistance from the point source (cm)tTime (s)zVertical distance from the surface (cm)VinVolume injected (cm3){Delta}Laplace operatorTFunction of CrßFunction of the tracer mass fraction recovered{theta}aAir-filled porosity (cm3 cm–3 soil){theta}tTotal porosity (cm3 cm–3 soil){rho}sDensity of the solid phase (g cm–3){tau}Tortuosity factor


   ACKNOWLEDGMENTS
 
Financial support was provided by the Swiss National Science Foundation, Grant 81EL68477. We thank our partners on the European project Groundwater Risk Assessment at Contaminated Sites GRACOS for interesting discussions of the subject.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 FIELD METHODS
 FIELD DATA
 APPENDIX
 REFERENCES
 




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K. Kawamoto, P. Moldrup, P. Schjonning, B. V. Iversen, D. E. Rolston, and T. Komatsu
Gas Transport Parameters in the Vadose Zone: Gas Diffusivity in Field and Lysimeter Soil Profiles
Vadose Zone J., November 20, 2006; 5(4): 1194 - 1204.
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M. Bohy, L. Dridi, G. Schafer, and O. Razakarisoa
Transport of a Mixture of Chlorinated Solvent Vapors in the Vadose Zone of a Sandy Aquifer: Experimental Study and Numerical Modeling
Vadose Zone J., April 27, 2006; 5(2): 539 - 553.
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V. E. Turcu, S. B. Jones, and D. Or
Continuous Soil Carbon Dioxide and Oxygen Measurements and Estimation of Gradient-Based Gaseous Flux
Vadose Zone J., November 11, 2005; 4(4): 1161 - 1169.
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