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Soil Science Society of America Journal 64:1588-1594 (2000)
© 2000 Soil Science Society of America

DIVISION S-1-SOIL PHYSICS

Predicting the Gas Diffusion Coefficient in Repacked Soil

Water-Induced Linear Reduction Model

P. Moldrupa, T. Olesena, J. Gamsta, P. Schjønningb, T. Yamaguchic and D.E. Rolstond

a Environmental Engineering Lab., Dep. of Civil Engineering, Aalborg Univ., Sohngaardsholmsvej 57, DK-9000 Aalborg, Denmark
b Dep. of Crop Physiology and Soil Science, Danish Institute of Agricultural Sciences, Research Centre Foulum, P.O. Box 50, DK-8830 Tjele, Denmark
c Dep. of Civil and Environmental Engineering, Faculty of Engineering, Hiroshima Univ., 1-4-1 Kagamiyama, Higashi-Hiroshima, 739, Japan
d Soils and Biogeochemistry, Dep. of Land, Air and Water Resources, Univ. of California, Davis, CA 95616 USA

i5pm{at}civil.auc.dk


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 Materials and methods
 Conclusions
 REFERENCES
 
Investigations of gas transport and fate processes in packed soil systems require knowledge of the gas diffusion coefficient, DP, as a function of air-filled porosity, {epsilon}. On the basis of the literature, data from six studies over the porosity range of 0.1 to nearly 1.0, it is reconfirmed that the Marshall (1959) model better predicts DP({epsilon}) in completely dry, repacked porous media than do the Penman (1940) and Millington (1959) models. The smaller DP value in wet soil, as compared with dry soil at the same air-filled porosity, is accounted for by introducing a water-induced linear reduction (WLR) term, equal to the ratio of air-filled porosity to total porosity, in the DP({epsilon}) model. By adding the WLR term in each of the three DP({epsilon}) models for dry porous media, the so-called WLR(Marshall), WLR(Penman), and WLR(Millington) DP({epsilon}) models for wet soil are developed. To test the three WLR models, DP was measured at different soil-water contents in six differently textured (6–38% clay) repacked soils. The WLR (Marshall) model accurately and best described DP({epsilon}) for all six soils and additional soils from the literature. All three WLR models performed better than previous DP({epsilon}) models. This study implies that the smaller DP in a wet soil, which is due to water-induced changes in air-filled pore shape and pore connectivity, can be described by a simple, linear function of relative air-filled porosity. The WLR(Marshall) model represents a conceptual and accurate model to predict DP({epsilon}) in sieved, repacked soil.

Abbreviations: WLR, water-induced linear reduction • MQ, Millington and Quirk • PMQ, Penman-Millington-Quirk


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
Materials and methods
 Conclusions
 REFERENCES
 
THE GAS DIFFUSION COEFFICIENT in soil, and its dependency on air-filled porosity, markedly influences transport, retardation, and degradation of greenhouse gases and volatile organic chemicals in soils (Kruse et al., 1996; Petersen et al., 1996). Gas transport and fate experiments using intact soil cores and sieved, repacked soils cores are expected to differ, among other reasons because the diffusion properties are different for undisturbed and repacked soils. The gas diffusion coefficient in natural, undisturbed soil is dependent on both soil type and structure (Schjønning et al., 1999). In agreement with this, we recently showed that that relative gas diffusivity (ratio of gas diffusion coefficients in soil and in air, DP/D0) in undisturbed soil can be accurately predicted from soil-water characteristics and macroporosity (defined as air-filled porosity at a soil matric potential of -100 cm H20) (Moldrup et al., 2000). Previously, we observed that gas diffusion in sieved, repacked soil appears to be much less soil-type dependent than gas diffusion in undisturbed soil, and we fitted available gas diffusivity data for repacked soil to the socalled Penman-Millington-Quirk (PMQ) model (Moldrup et al., 1997):

(1)

where DP is the gas diffusion coefficient in soil (cm3 soil air cm-1 soil sec-1), D0 is the gas diffusion coefficient in free air (cm2 air sec-1), {epsilon} is the soil air-filled porosity (cm3 soil air cm-3 soil), {Phi} is the soil total porosity (cm3 cm-3), and m is a fitting constant. Equation [1] with m = 6 best described gas diffusivity in sieved, repacked soil, on the basis of data for pure sand and five different loamy soils (Moldrup et al., 1997). However, the PMQ model is hitherto considered empirical (calibration model) and only tested against limited data.

The objective of this study is to derive a conceptually based and accurate model for gas diffusivity in sieved, repacked soil. Model development and tests will be based on gas diffusivity data from literature plus additional gas diffusivity measurements on six differently textured soils.


   Materials and methods
TOP
 ABSTRACT
 INTRODUCTION
 Materials and methods
Conclusions
 REFERENCES
 
Six soils were used in the diffusion experiments. The soils differed with respect to soil depth, texture, and amount of organic matter. Three soils, Lerbjerg 1, Lerbjerg 3, and Lerbjerg 5, were retrieved from the 0- to 20-cm depth at three locations along a naturally occurring texture gradient, identified in an arable field near Lerbjerg, Denmark (Schjønning et al., 1999). The clay content ranged from 11 to 38% clay. One soil was retrieved from the 0- to 20-cm depth at Lundgaard, Denmark. Two soils were retrieved at a former manufactured gas plant site at Hjørring, Denmark. In situ remediation for coal tar compounds have been carried out at the site since 1993. The soil samples were collected at a non-contaminated location. The soil type is mainly loamy sand, but more finely textured horizons were identified. The two soils were collected in the unsaturated zone from a loamy layer in the 3.5- to 4-m depth and a sandy layer in the 7- to 8-m depth, respectively (the ground water level is approximately at 10-m depth). The soils were air-dried and sieved to <2 mm. The physical properties of the soils are shown in Table 1 .


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  		Table 1 Soil physical properties of the six soils used for oxygen diffusivity measurements

 
The soils were packed in 100-cm3 cores (0.034-m length and 0.061-m inner diameter). Lerbjerg 1, Lundgaard and Hjørring 7–8 m were thoroughly mixed with water to the desired water content and then packed in 1.7-cm layers. Lerbjerg 3 and Lerbjerg 5 were packed with air-dried soil to avoid formation of aggregates when mixing with water. The cores containg Lerbjerg 3 and Lerbjerg 5 were wetted to the desired water contents by adding water from the bottom of the core. The soil cores were then weighed until the desired amounts of water was sucked up by the soil and the cores were then allowed to equilibrate for three to four weeks. When adding water to air-dry Hjørring 3.5- to 4-m soil, the soil has a tendency to shrink in the soil cores. To avoid this the soil was first wetted to 0.06 g g-1 by spraying water on the air-dried soil and then equilibrated for 3 d. After equilibration, the soil cores were packed and wetted to the chosen water contents by the same procedure as for Lerbjerg 3 and Lerbjerg 5.

Measurements were carried out at three soil-water contents for the Hjørring 7- to 8-m soil, four soil-water contents for the Lundgaard and Hjørring 3.5- to 4-m soils and five soil-water contents for the Lerbjerg 1, Lerbjerg 3, and Lerbjerg 5 soils (three replicates). Soil cores were analyzed for air diffusivity by the method suggested by Taylor (1949) with the equipment described by Schjønning (1985). Soil gas diffusion was measured with oxygen as the experimental gas at 20°C. Test of the equipment without soil in the cores showed that the measured O2 diffusion coefficient in free air varied by less than 2% from the theoretical value (0.205 cm2 s-1). Measurements of the oxygen consumption rate for each soil measured as an average consumption rate over 48 h at approximately -100 cm soil-water potential showed that the error in the measured Dp value, because of oxygen consumption, did not exceed 1.5%.

Model Development
Gas Diffusivity in Dry Soil
To understand fully the gas diffusion process in wet soil, a prerequisite is to understand the gas diffusion process in completely dry soil (void of water). Figure 1 shows data for gas diffusivity in completely dry porous media (sand, soil, glass beads, combinations thereof, glass wool, steel wool) collected from six different studies (Penman, 1940; Taylor, 1949; van Bavel, 1952; Currie, 1960; Papendick and Runkles, 1965; Reible and Shair, 1982). Following Millington and Quirk (1960), we tested three different models for gas diffusivity in dry porous media, the first by Penman (1940):

(2)
the second by Marshall (1959):

(3)
and the third by Millington (1959):

(4)



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  		Fig. 1 Gas diffusivity in completely dry porous media (sands, soils, glass beads, combinations hereof, glass wool, steel wool). Measured data (Penman, 1940; Taylor, 1949; van Bavel, 1952; Currie, 1960; Papendick and Runkles, 1965; and Reible and Shair, 1982) and predictions by three classical gas diffusivity models, Eq. [2] through [4]

 
Overall (for all 36 measured values in Fig. 1; ), the Marshall model, based on randomly inter-connected pores with two or more exits, performed best. Following Moldrup et al. (2000), we based model comparisons on root mean square error of prediction, RMSE. The Marshall (1959) model performed best with a followed by the Millington model and the Penman model . This is in agreement with the conclusions drawn by Millington and Quirk (1960) and Reible and Shair (1982), based on sub-sets of the data shown in Fig. 1. However, looking at the porosity range of most interest for soil studies, the conclusion becomes different. For example, within the range , the Marshall model again performs best but the Penman model is almost as accurate followed by the Millington model . Thus, all three models perform well in describing the measured data in the range 0.2 < {epsilon} < 0.5 (Fig. 1). We note that Shimamura (1992) found that the above diffusivity models for dry porous media were not valid for strongly–repeatedly compacted sands probably because of the development of local zones non-accesible for gas diffusion.

Gas Diffusivity in Wet Soil
Papendick and Runkles (1965) observed that gas diffusivity in wet media was lower than gas diffusivity in dry media at the same air-filled porosity. This is probable due to a change of the pore shape and configuration of air-filled pores when the porous media become wet, which causes increased tortuosity for gas transport (i.e., reduced diffusive gas flux).

We suggest that the increased tortuosity in wet soil, as compared with a dry soil at the same air-filled porosity, can be accounted for by introducing an additional, water-induced linear reduction (WLR) of diffusivity with air-filled porosity. To observe the criterion that the diffusivity model modified for wet soil conditions should equal the diffusivity model for dry soil in the case of , the WLR term must therefore equal {epsilon}/{Phi}. Introducing the WLR term in Eq. [2] through [4], respectively, to account for the effects of water-induced changes in pore shape and configuration, gives the WLR (Penman) model:

(5)
the WLR(Marshall) model:

(6)
and the WLR(Millington) model:

(7)

Figure 2 illustrates the WLR concept for one of the six soils from the present study. The figure shows the same data for dry porous media as in Fig. 1 (but only for {epsilon} < 0.5), the data for (wet) Lerbjerg 3 sandy clay loam, the Marshall (1959) model, Eq. [3], for dry porous media and the WLR(Marshall) model, Eq. [6], for wet soil. The reduction in gas diffusivity in the wet soil, as compared with dry soils at the same air-filled porosity, is obvious (Fig. 2), in agreement with observations by Papendick and Runkles (1965).



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  		Fig. 2 Comparison of gas diffusivity in completely dry porous media (open circles, same data as in Fig. 1) and in wet soil. Predictions by the Marshall (1959) model, Eq. [3], for dry soil, and the new WLR (Marshall) model, Eq. [6], for wet soil

 
Interestingly, inserting in Eq. [1] and re-arranging gives Eq. [5]. Thus the PMQ model with m = 6 can now be explained conceptually as a WLR model based on the Penman (1940) model for gas diffusivity in completely dry soil.

Test of Water-Induced Linear Reduction Models
Figure 3 shows the performance of the WLR (Marshall) model tested against the gas diffusivity data for the six soils measured in this study plus three additional, differently-textured soils from the literature. For Miles loamy sand and Sharpsburg silty clay loam (Xu et al., 1992), the measurements were carried out at four different bulk densities and thus four different soil total porosities. Although we did not have the value of soil total porosity for each measurement, it is obvious from Fig. 3g and 3h that the WLR model describes the data well within the range of soil total porosities considered by Xu et al. (1992). This was also the case for two additional soils (also a loamy sand and a silty clay loam) from Xu et al. (1992). As the soils in Fig. 3 represent different soil texture (6–54% clay) and bulk densities, the WLR (Marshall) model seems generally valid. For comparison, the Marshall (1959) model for gas diffusivity in dry soil is also shown in Fig. 3. The point where the WLR(Marshall) and the Marshall (1959) models meet corresponds to the soil total porosity, i.e. at {epsilon} = {Phi}. The effect of added water [difference between Marshall and WLR (Marshall) models] in increasing the tortuosity and, thus, decreasing the diffusivity is evident.



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  		Fig. 3 Performance of the WLR(Marshall) model, Eq. [6], when tested against gas diffusivity data for nine differently textured, sieved, and repacked soils. For comparison, the Marshall (1959) model, Eq. [3], for gas diffusivity in dry soil is shown. (a) through (f) data from this study, (g) through (h) data from Xu et al. (1992) at four different soil total porosities, the range of total porosity for each soil is shown and used in Eq. [6], and (i) data from Karimi et al. (1987)

 
The worst performance of the WLR (Marshall) model was observed when tested against data for Yolo loam (Petersen et al., 1994) as shown by Fig. 4 . In this case, the WLR (Millington) model best described the measured diffusivities, especially when near to {epsilon} = {Phi}. As shown in Fig. 1, the Millington (1959) model, Eq. [4], in a few cases described gas diffusivity in dry soils equally well or better than the Marshall (1959) model, Eq. [3]. Thus, the same can likely be expected for the WLR (Millington), Eq. [7], compared with the WLR (Marshall), Eq. [6], models in the case of wet soils. However, all three new WLR models, Eq. [5] through [7], performed with adequate prediction accuracy for most practical purposes, Fig. 4.



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  		Fig. 4 Performance of the three new WLR models, Eq. [5] through [7], when tested against data for Yolo loam (data from Petersen et al., 1994). The dotted, vertical line represents soil total porosity,

 
To evaluate this further, an overall model test was carried out, on the basis of data for 11 sieved, repacked soil where DP({epsilon}) and {Phi} values were available. The 11 soils include the six from the present study and five from literature. The five are Yolo loam (Petersen et al., 1994), Pachappa sandy loam (Jin and Jury, 1996), Gila silt loam (Shearer et al., 1973), Brookings silty clay loam (Papendick and Runkles, 1965), and West Covina clay (Karimi et al., 1987). The results of the model tests are shown in Fig. 5 , depicting scatter-plots of predicted versus measured DP/D0({epsilon}) for all 11 soils. Besides the WLR models, the widely used Penman (1940), Eq. [2], Millington and Quirk (1960):

(8)
and Millington and Quirk (1961)

(9)
models were tested. While the Penman (1940) and Millington and Quirk (1961) models are well known and have been used in numerous transport simulation studies, the Millington and Quirk (1960) model is less used and in some cases may be superior to the Penman (1940) and Millington and Quirk (1961) models, as discussed by Washington et al. (1994) and Jin and Jury (1996).



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  		Fig. 5 Scatter-plot comparison of model performance when tested against gas diffusivity data for 11 differently-textured, sieved and repacked soils. (a) through (f) Test of six different models against all data , and (g) through (i) test of the three new WLR models, Eq. [5] through [7], against low relative gas diffusivities,

 
On the basis of both visual comparisons and the RMSE values (Fig. 5), it can be concluded that the Penman (1940) and the Millington and Quirk (1960) models generally over estimate the measured gas diffusivities, while Eq. [9] performs best among the three classical models but generally under estimates the measured gas diffusivities (Fig. 5c). The new WLR models perform better than the three classical models. The WLR (Penman) and WLR (Marshall) models perform significantly better than the remaining models; in agreement with the finding that the Penman and Marshall models performed best for dry porous media within the porosity range of most relevance for soil studies ({epsilon} <= 0.5 m3 m-3; see Fig. 1). A separate model test for the six soils from the present study showed the same model performance ranking as the test for all 11 soils, with the WLR (Marshall) model performing best , and the Penman (1940) model worst . It is noteworthy that the WLR (Marshall) model is superior to all other models at low gas diffusivities (DP/D0 < 0.1, Fig. 5g–i) where gas diffusion often becomes limiting for reaction processes, especially aerobic biodegradation due to oxygen limitations.

We recognize that Schaefer et al. (1997) measured argon diffusivity for four differently textured repacked soils, applying a different measurement method (column method with gas sampling within the soil column) than used for the 11 soils in Fig. 5. Schaefer et al. (1997) obtained, as noted by themselves, much higher gas diffusivities than measured for similar soils (with respect to texture and soil total porosity) in other studies. We found this is the case compared with the studies by Shearer et al. (1973), Petersen et al. (1994), Jin and Jury (1996) and the present study, all using classical one- or two-chamber measurement methods. We will not speculate on the reasons for the large deviations between the measurement methods but relying on the thoroughly validated chamber methods (Rolston, 1986), we have chosen not to include the data by Schaefer et al. (1997).

We also recognize the model concepts by Schaefer et al. (1997) and several previous studies that at very low air-filled porosities, gas diffusion is limited by inter-connected water film and non-accessible intra-particle pores and, therefore, approaches zero. However, this will likely happen only at very low {epsilon} values (e.g., {epsilon} < 0.05 cm3 cm-3 based on the data by Schaefer et al. (1997)). None of the data considered here (Fig. 5i) or in the study by Sallam et al. (1984) (gas diffusivity in Yolo loam at very low air-filled porosities) suggested that DP/D0 effectively approached zero. In summary, modified measurement methods for gas diffusivity likely need to be compared and validated against the classical one- and two chamber methods, and more measurements of gas diffusivity at very low air-filled porosities are needed to understand the influence of inter-connected water films and intra-particle pore space and to further test possible limitations of the WLR models in the case of near-water-saturated soils.

A test of the WLR (Marshall) model with focus on the influence of soil total porosity is shown in Fig. 6 . The data are for Macmerry-Winton clay loam (Arah and Ball, 1994) representing one DP/D0 value at each of five different soil total porosities (five different bulk densities). It appears that the WLR (Marshall) model adequately takes into account the effect of soil total porosity on the measured gas diffusivities. The packed Macmerry-Winton clay loam (Fig. 6) contained some soil aggregates (<5 mm). The six soils in the present study also included soil samples that showed some degree of aggregation (i.e., the soil samples that had been thoroughly mixed with water before packing; Lerbjerg 1, Lundgaard and Hjørring 7–8 m). However, this did not significantly affect the measured gas diffusivities or the WLR model prediction accuracy (Fig. 5). This is in agreement with the study by Flegg (1953) that found negligible effect of aggregation on gas diffusivity in repacked soil. Millington and Shearer (1971), on the basis of the data of Currie (1961), found an effect of aggregation but several of the soils considered had been heated to 800°C to stabilize soil crumbs which may have altered the pore network and structure markedly. On the basis of the present data, we conclude that the effects of soil aggregation seems minor and that the WLR (Marshall) models seem valid for repacked soil with some degree of aggregation as well as for non-aggregated soils.



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  		Fig. 6 Test of the WLR(Marshall) model, Eq. [6], against gas diffusivity data for a clay loam at five different soil total porosities (data from Arah and Ball, 1994). For comparison, the Marshall (1959) model for gas diffusivity in dry soil is shown

 
An important part of evaluating a predictive model is to define when the model may not be valid. The WLR models are likely not valid for very organic soils, because the organic matter will exhibit different tortuosity compared with mineral soils (Moldrup et al., 2000). We tested the WLR (Marshall) model against the limited data for gas diffusivity in wet steel wool by Reible and Shair (1982) and found that the WLR (Marshall) model adequately predicted the gas diffusivity in this high-porosity, rigid material. Thus, the reason why the WLR models may fail for high-organic soils is not the typically high total porosity of these soils but probably the change in structure of the non-rigid organic matter when wetted. The WLR models should therefore be tested against gas diffusivity data for repacked, high-organic soils when such data become available.

The WLR models are not valid for gas diffusivity in strongly compacted or repeatedly compacted soils. Shimamura (1992) showed for a number of sandy materials that in this case a part of the total pore space will be isolated and not available for gas diffusion. This threshold air-filled porosity where gas diffusivity ceased was between 0.1 and 0.2 m3 m-3. Tested against the data by Shimamura (1992), the WLR Marshall model performed well at higher air-filled porosities where the sands had been compacted only one or two times to reach the desired soil-water content, but over-predicted gas diffusivities at lower air-filled porosities where the samples had been repeatedly compacted (results not shown).

Finally, we emphasize that the WLR models should not be used to predict gas diffusivity in natural undisturbed soils or in intact soil samples. In the case of undisturbed soils, the model by Moldrup et al. (2000), taking into account soil type and macroporosity, is recommended.


   Conclusions
TOP
 ABSTRACT
 INTRODUCTION
 Materials and methods
 Conclusions
REFERENCES
 
A so-called water-induced linear reduction (WLR) class of models for gas diffusivity in wet, repacked soil is introduced by modifying classical models for completely dry porous media with an additional linear reduction in gas diffusivity with relative air-filled porosity. The WLR term describes the effects of changing pore shape and configuration in a wet soil compared with a dry soil at the same air-filled porosity. The hitherto empirical PMQ gas diffusivity model by Moldrup et al. (1997), Eq. [1] with , is identified as a WLR model based on the Penman (1940) model for dry media.

The Marshall (1959) model was found to best describe gas diffusivity in dry porous media. The WLR model based on the Marshall (1959) model for dry media yields (Eq. [6]):

(10)

Equation [10] gave an accurate description of gas diffusivity in 11 differently textured soils (6–54% clay) both at low and high gas diffusivities and in a clay loam soil at 5 different soil total porosities. Equation [10] is recommended for use in gas transport and fate studies where sieved, repacked soil is used.

We emphasize that the WLR (Marshall) model, Eq. [10], is derived and validated only for sieved, repacked soils. The WLR model should not at present be used for high-organic soils before sufficient model tests can be performed. The WLR model is not valid for strongly–repeatedly compacted soils and undisturbed soils. For undisturbed soils, we refer to the predictive DP/D0 model by Moldrup et al. (2000) that takes into account soil type and macroporosity.Millington Schearer 1971


   ACKNOWLEDGMENTS
 
This work was supported by the Danish Technical Research Council, Research Talent Project entitled: "New methods for measuring and predicting liquid and gaseous phase transport properties in undisturbed soils," the project "Continuation and evaluation of in-situ remediation at Hjørring former manufactured gas plant site" funded by the Danish EPA in cooperation with NIRAS Consulting Engineers A/S, the county government of Northern Jutland and the municipal government of Hjørring, grant 5P42ESO4699 from the National Institute of Environmental Health Sciences, NIH, and the U.S. EPA (R819658) Center for Ecological Health Research at U.C. Davis. The contents of this publication are solely the responsibility of the authors and do not necessarily represent the official view of the NIEHS, NIH, or EPA. The authors gratefully acknowledge a travel grant from the Japanese Ministry of Education, Science, Sports and Culture (Monbushu International Scientific Research Program: Joint Research No. 12555156).

Received for publication November 8, 1999.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 Materials and methods
 Conclusions
 REFERENCES
 




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T. Olesen, J. Gamst, P. Moldrup, T. Komatsu, and D. E. Rolston
Diffusion of Sorbing Organic Chemicals in the Liquid and Gaseous Phases of Repacked Soil
Soil Sci. Soc. Am. J., November 1, 2001; 65(6): 1585 - 1593.
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J. Gamst, T. Olesen, H. De Jonge, P. Moldrup, and D. E. Rolston
Nonsingularity of Naphthalene Sorption in Soil: Observations and the Two-Compartment Model
Soil Sci. Soc. Am. J., November 1, 2001; 65(6): 1622 - 1633.
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P. Moldrup, T. Olesen, T. Komatsu, P. Schjonning, and D.E. Rolston
Tortuosity, Diffusivity, and Permeability in the Soil Liquid and Gaseous Phases
Soil Sci. Soc. Am. J., May 1, 2001; 65(3): 613 - 623.
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