VZJ sign up for etocs
HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS
QUICK SEARCH:   [advanced]


     

Published online 20 November 2006
Published in Vadose Zone J 5:1205-1215 (2006)
DOI: 10.2136/vzj2006.0030
© 2006 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
This Article
Right arrow Abstract Freely available
Right arrow Figures Only
Right arrow Full Text (PDF) Free
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Similar articles in this journal
Right arrow Similar articles in Web of Science
Right arrow Alert me to new issues of the journal
Right arrow Download to citation manager
Right arrow reprints & permissions
Citing Articles
Right arrow Citing Articles via HighWire
Right arrow Citing Articles via Web of Science (4) Citing Articles via Web of Science (4)
Right arrow Citing Articles via Google Scholar
Google Scholar
Right arrow Articles by Kawamoto, K.
Right arrow Articles by Rolston, D. E.
Right arrow Search for Related Content
PubMed
Right arrow Articles by Kawamoto, K.
Right arrow Articles by Rolston, D. E.
GeoRef
Right arrow GeoRef Citation
Agricola
Right arrow Articles by Kawamoto, K.
Right arrow Articles by Rolston, D. E.
Related Collections
Right arrow Air Permeability
Right arrow Diffusion
Right arrow Soil Physics

ORIGINAL RESEARCH

Gas Transport Parameters in the Vadose Zone: Development and Tests of Power-Law Models for Air Permeability

Ken Kawamotoa,*, Per Moldrupb, Per Schjønningc, Bo V. Iversenc, Toshiko Komatsua and Dennis E. Rolstond

a Graduate School of Science and Engineering, Saitama Univ., 225 Shimo-okubo, Sakura-ku, Saitama, 338-8570, Japan
b Environmental Engineering Section, Dep. of Biotechnology, Chemistry, and Environmental Engineering, Aalborg Univ., Sohngaardsholmsvej 57, DK-9000 Aalborg, Denmark
c Dep. of Agroecology, Danish Inst. of Agricultural Sciences, Research Centre Foulum, P.O. Box 50, DK-8830 Tjele, Denmark
d Dep. of Land, Air, and Water Resources, Univ. of California, Davis, CA 95616
* Corresponding author (kawamoto{at}post.saitama-u.ac.jp)



   ABSTRACT
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 MODEL DEVELOPMENT
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
The soil-air permeability (ka) and its dependency on air-filled porosity ({varepsilon}) govern convective air and gas transport in soil. For example, accurate prediction of ka({varepsilon}) is a prerequisite for optimizing soil vapor extraction systems for cleanup of soils polluted with volatile organic chemicals. In this study, we measured ka at different soil-water matric potentials down to 5.6-m depth, totaling 25 differently textured soil layers. Comparing ka and soil-gas diffusivity (Dp/D0) measurements on the same soil samples suggested an analogy between how the two soil-gas transport parameters depend on {varepsilon}. The exponent in a power-law model for ka({varepsilon}) was typically smaller than for Dp({varepsilon})/D0, however, probably due to the influence of soil structure and large-pore network being more pronounced for ka than for Dp/D0. In analogy to recent gas diffusivity models and in line with capillary tube models for unsaturated hydraulic conductivity, two power-law ka({varepsilon}) models were suggested. One ka({varepsilon}) model is based on the Campbell pore-size distribution parameter b and the other on the content of larger pores ({varepsilon}100, corresponding to the air-filled porosity at –100 cm H2O of soil-water matric potential). Both new models require measured ka at –100 cm H2O (ka,100) as a reference point to obtain reasonably accurate predictions. If ka,100 is not known, two expressions for predicting ka,100 from {varepsilon}100 were proposed but will cause at least one order of magnitude uncertainty in predicted ka. The ka({varepsilon}) model based on only {varepsilon}100 performed well in the model tests and is recommended together with a similar model for gas diffusivity for predicting variations in soil-gas transport parameters in the vadose zone.


   INTRODUCTION
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 MODEL DEVELOPMENT
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
KNOWLEDGE OF ka is necessary for understanding convective air and gas transport in soil in relation to analyzing and optimizing soil vapor extraction systems for cleanup of soils polluted with volatile organic chemicals (Gamliel and Abdul, 1993; Mohr and Merz, 1995; Poulsen et al., 1998, 1999). Accurate prediction of ka is also important to simulate soil-atmosphere gas exchange (Ball et al., 1997a, 1997b; Hutchinson and Livingston, 2002), as well as the transport of gaseous contaminants from soil to indoor air (Johnson and Ettinger, 1991).

Soil-air permeability is an easily measured transport parameter both in situ and in the laboratory using undisturbed or repacked soil samples at several measurement scales (Kirkham, 1947; Grover, 1955; Poulsen et al., 2001; Iversen et al., 2001a, 2001b, 2003; Jalbert and Dane, 2003). The ka value provides useful information about soil structure, and is used for characterizing soil pore geometry (Ball, 1981; Roseberg and McCoy, 1990; Granovsky and McCoy, 1997; Moldrup et al., 2001, 2003; Schjønning et al., 2002; Unsal et al., 2005). The ka value is also related to saturated and unsaturated hydraulic properties (Aljibury and Evans, 1965; Blackwell et al., 1990; Rasmussen et al., 1993), and measured ka at a given soil-water matric potential has been used to predict saturated hydraulic conductivity (Loll et al., 1999; Iversen et al., 2001b, 2003). Furthermore, the ka value is related to soil-gas diffusivity, and these two gas transport parameters have been linked together to express air-filled pore connectivity and equivalent pore diameter (Millington and Quirk, 1964; Ball, 1981; Moldrup et al., 2001).

A number of predictive models for ka as a function of {varepsilon} have been developed, based mostly on either the Millington and Quirk (1960, 1961) tortuosity model (Gomez-Lahoz et al., 1991; Moldrup et al., 1998), or on the soil-water characteristic (SWC) curve using the Burdine (1953) or Mualem (1976) capillary bundle models (Brooks and Corey, 1966; Parker et al., 1987; Lenhard and Parker, 1987; Tuli et al., 2005; Unsal et al., 2005). These models are generally expressed as a relative gas permeability, ka({varepsilon})/k*, as a function of a relative air-filled porosity, {varepsilon}/{varepsilon}*, using reference-point values of air permeability, k*, and air-filled porosity, {varepsilon}*. Analogous to predictive models for unsaturated hydraulic conductivity based on the SWC curve (Brooks and Corey, 1966; Campbell, 1974; van Genuchten, 1980), k* is often assumed to be air permeability at air saturation (completely dry condition) or intrinsic permeability estimated from saturated hydraulic conductivity (water permeability) and the corresponding {varepsilon}* to equal the total soil porosity, {phi}. In undisturbed soils, however, it has been shown that the air permeability is generally higher than water permeability by one or several orders of magnitude since air interacts with the soil matrix to a much lesser degree while water, as a polar fluid, tends to interact with the soil matrix (Aljibury and Evans, 1965; Blackwell et al., 1990; Loll et al., 1999; Iversen et al., 2001b). Furthermore, experimental difficulties occur in measuring air permeability at air saturation (Tuli and Hopmans, 2004).

Generalizing the Millington and Quirk (1961) and Brooks and Corey (1966) type models, Moldrup et al. (1998) suggested a simple power-law model for predicting ka (µm2) as a function of {varepsilon} (m3 m–3) in undisturbed soils. The power-law model can be described using air permeability and air-filled porosity at soil-water matric potential of –100 cm H2O, ka,100 (µm2) and {varepsilon}100 (m3 m–3), as reference-point values:

Formula 1[1]
where {eta} represents the combined effects of tortuosity and connectivity of air-filled pores. They found that the tortuosity–connectivity parameter {eta} can be taken as a function of the Campbell (1974) pore-size distribution parameter b (the slope of the soil-water characteristic curve in a log–log coordinate system), and suggested the expression, {eta} = 1 + 0.25b to be inserted in Eq. [1], based on 13 sandy and loamy soils with b values ranging from 4.3 to 7.0. It is noted that typical b values are in the range of 1 to 20 for most undisturbed soils (Clapp and Hornberger, 1978).

Moldrup et al. (2001) found that the expression {eta} = 1 + 0.25b (Moldrup et al., 1998) in the power-law model failed to accurately describe newly measured ka({varepsilon}) data, especially for finer textured soils, and modified the expression to {eta} = 1 + 0.05b based on six undisturbed soils along a soil-texture gradient with between 0.11 and 0.44 kg kg–1 clay content with b values ranging from 4.6 to 14.1. Moldrup et al. (2003) successfully tested Eq. [1] with {eta} = 1 + 0.05b against data for 16 undisturbed Andisols with b values ranging from 8.3 to 40.8.

The power-law ka({varepsilon}) model, Eq. [1], predicted relatively well a limited amount of measured ka data (Moldrup et al., 1998, 2001, 2003); however, the expressions for {eta}(b) varied depending on the measured data sets used for model development, implying that further improvement of the {eta}(b) expression is needed for accurate predictions of ka across soil types. Besides, only limited measurements and knowledge of ka in undisturbed soil are available, especially for deeper vadose zone soil profiles at several meters depth.

This study, one of a series on gas transport parameters in vadose zone soil profiles, focused on air permeability and had the following objectives: (i) to measure soil-air permeability on undisturbed soil samples in each of five soil profiles (three lysimeter and two field soils) containing several morphologic horizons, and compare ka({varepsilon}) to soil-gas diffusivity measurements on the same soil samples (data from Kawamoto et al., 2006); (ii) to develop predictive ka({varepsilon}) models in analogy to recent gas diffusivity models considering capillary tube models for unsaturated hydraulic conductivity, and test the modified models together with the existing models against measurements; and (iii) to validate the new ka({varepsilon}) models against independent data for undisturbed field soils.


   MATERIALS AND METHODS
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
MODEL DEVELOPMENT
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Soils and Measurement Methods
Simultaneous measurements of soil-water retention, air permeability, and gas diffusivity were performed on intact soil cores from three lysimeters at the Danish Institute of Agricultural Sciences at Research Centre Foulum with soil from Rønhave, Foulum, and Jyndevad, and two agricultural fields, Gjorslev and Mammen. For convenience, we refer to the Rønhave, Foulum, and Jyndevad soils as lysimeter soils, and the Gjorslev and Mammen soils as field soils. Soil site locations in Denmark, basic soil characteristics including soil texture, and measurements of soil-water retention and gas diffusivity are presented in Kawamoto et al. (2006).

For the lysimeter soils, between two and four intact soil cores (0.034-m length, 0.061-m i.d., 100-cm3 volume) were collected from each soil layer (a total of 36 samples). For the field soils, five 100-cm3 intact soil cores were collected from each soil layer (a total of 60 samples). The ka was measured at 25°C on all samples at four soil-water potentials ({psi}) of –20, –50, –100, and –160 (or –200) cm H2O by the steady-state method of Ball and Schjønning (2002). A constant, small air pressure difference was applied to the soil sample, and the air flow rate that is proportional to the air permeability was measured. The experimental setup and procedure used are further described by Moldrup et al. (1998), Schjønning et al. (1999), and Iversen et al. (2001a).

For an independent test of new ka({varepsilon}) models, ka was measured on 24 undisturbed field soil samples from subsurface soil horizons in the 4- to 5- and 6- to 7-m depths at Hjørring in Northern Jutland, Denmark. The sampling site was used for a manufactured-gas plant, and in situ remediation for coal tar compounds has been performed at the site since 1993. The soil types of the Hjørring subsurface soils are sandy clay loam at the 4- to 5-m depth with b values around 10, and sandy loam or loamy sand at the 6- to 7-m depth with b values from 3 to 6. The ka values were measured at four {psi} of –30, –50, –100, and –500 cm H2O. The basic soil characteristics, measurements of soil-water retention, and gas diffusivity are presented in Moldrup et al. (2000).

Statistical Analyses
Three statistical indexes were used to evaluate and compare the predictive ka({varepsilon}) models. To evaluate the best overall fit compared with measured data, the RMSE was used:

Formula 2[2]
where di [= log(ka)predicted – log(ka)measured] is the difference between the log-transformed predicted and measured values of ka at a given air-filled porosity, and n is the number of measurements. The bias was used to evaluate model overestimaton (positive bias) or underestimation (negative bias) of measured ka data:

Formula 3[3]

To account for the number of model parameters when comparing model performance for a given measured data set, Akaike's Information Criterion (AIC) was used (Akaike, 1973; Carrera and Neuman, 1986; Hwang et al., 2002):

Formula 4[4]
where k is the number of model parameters. A smaller (or more negative) AIC indicates better model performance (Minasny et al., 1999).


   MODEL DEVELOPMENT
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 MODEL DEVELOPMENT
RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Similarity between Air Permeability and Gas Diffusivity
Measurements of ka for the field soils are shown as a function of air-filled porosity ({varepsilon}) in Fig. 1a . Measurements of ka for the lysimeter soils are shown in Fig. 1b. To illustrate the magnitude and distribution of ka values, the figures show log(ka) as a function of {varepsilon} (log scale). Three simple and empirical power-law functions (representing upper and lower boundaries, and middle) are arbitrarily given for comparison. We note that measurements of ka at {psi} = –50 cm H2O for Jyndevad (lysimeter soil) are missing from Fig. 1b due to an experimental error.


Figure 1
View larger version (33K):
[in this window]
[in a new window]
  		Fig. 1. Log-transformed air permeability, log (ka), and gas diffusivity, log (Dp/D0), as a function of air-filled porosity, {varepsilon}, for (a and c) field soils (Gjorslev and Mammen) and for (b and d) lysimeter soils (Rønhave, Foulum, and Jyndevad). The Dp/D0 data are from Kawamoto et al. (2006).

 
For the field soils, the measured log(ka) (in µm2) range is wider, and very low values of log(ka) (less than –1.0) were observed within a wide range of {varepsilon}. Having a closer look, for the Gjorslev soils, most of the measured log(ka) values were positive (>0.0) at shallow soil layers (0.15–1.20 m) while they were negative (<0.0) at deep soil layers (2.10–5.40 m). For the Mammen soils, the measured log(ka) values were mostly positive at both shallow and deep soil layers, including very high log(ka) values at the 5.4-m depth (sand) at {varepsilon} around 0.3 m3 m–3. For the lysimeter soils, the measured log(ka) narrowly ranged from –0.2 to 2.1 (corresponding {varepsilon} range of 0.06–0.39 m3 m–3). The lysimeter soils had been standing in lysimeters for 6 yr after refill (Kawamoto et al., 2006) but the soil structure was not yet as fully developed as it was in the original field soils, and therefore the measured log(ka) range was different than the field soils.

Measurements of soil-gas diffusivity, Dp/D0 (Dp is the gas diffusion coefficient in soil [m2 s–1] and D0 is the gas diffusion coefficient in free air [m2 s–1]) on the same soil samples are shown in Fig. 1c for the field soils and in Fig. 1d for the lysimeter soils (data from Kawamoto et al., 2006). Comparing the measurements of log(ka) and log(Dp/D0), there appears to be similarity between the two soil-gas transport parameters. For example, the measurements of ka for the lysimeter soils (Fig. 1b) are mostly placed between the middle and lower boundary in a log–log coordinate system, this being quite similar to the measurements of Dp/D0 in Fig. 1d. Figure 1 suggests that the two soil-gas transport parameters may be linked, and that both parameters may be expressed as a function of {varepsilon} in a constitutive power-law relationship, as suggested by Moldrup et al. (2001).

Measurements of ka for three soils with different soil textures (sand, sandy loam, and sandy clay loam) are shown as a function of {varepsilon} in a log–log coordinate system (Fig. 2 ). The values of the power-law exponent (corresponding to the tortuosity–connectivity parameter) {eta} in Eq. [1] were determined as the slope of the log(ka) and log({varepsilon}) plots, and are given in the figure. Measurements of Dp/D0 corresponding to the three field soils are also shown in the figure, and {eta} values determined as the slope of log(Dp/D0) and log({varepsilon}) are also given.


Figure 2
View larger version (14K):
[in this window]
[in a new window]
  		Fig. 2. Comparison of log-transformed air permeability, log(ka), and gas diffusivity, log(Dp/D0), as a function of log-transformed air-filled porosity, log ({varepsilon}), for differently textured soils. The tortuosity–connectivity parameter ({eta}) values for air permeability and gas diffusivity are given; b = Campbell (1974) pore-size distribution parameter.

 
In general, the {eta} values for ka are smaller than those for Dp/D0, and the difference between them tends to be more significant for finer textured soils. This result coincides with past research (Moldrup et al., 2001; Tuli and Hopmans, 2004). For example, Moldrup et al. (2001) determined {eta} values for six undisturbed soils with b values ranging from 4.6 to 14.1 and showed that the {eta} values for ka are smaller than those for Dp/D0. The difference in {eta} values between ka and Dp/D0 is probably due to the influence of soil structure, including soil aggregation. During air convection, preferential air flow will occur in the large-pore networks, especially in well-structured undisturbed soils, while preferential gas diffusion will not occur. This may cause smaller values of {eta} for ka than for Dp/D0.

Power-Law Exponent (Tortuosity–Connectivity Parameter) for Air Permeability as a Function of Air-Filled Porosity
The values of {eta} for 36 soil samples from the lysimeter soils (Rønhave, Foulum, and Jyndevad) are shown as a function of the Campbell (1974) pore-size distribution parameter b in Fig. 3 . The lysimeter soils with different soil textures (sandy clay loam at Rønhave, sandy loam or sandy clay loam at Foulum, and loamy sand at Jyndevad) give a wide range of b values from 1.3 to 14.6. A shown in Fig. 3, the {eta} values decrease with increasing b values (a large {eta} corresponds to a coarser textured soil, and a small {eta} corresponds to a finer textured soil).


Figure 3
View larger version (14K):
[in this window]
[in a new window]
  		Fig. 3. The tortuosity–connectivity parameter ({eta}) values as a function of the Campbell (1974) b for lysimeter soils (Rønhave, Foulum, and Jyndevad). Three expressions for {eta}(b), {eta} = 2 + 3/b (Burdine, 1953), {eta} = 2 + 5/2b (Mualem, 1976), and {eta} = 1 + 3/b (Alexander and Skaggs, 1986) are given by solid curves. The {eta} = 1 + 0.25b (Moldrup et al., 1998) and {eta} = 1 + 0.05b (Moldrup et al., 2001) are given by broken lines.

 
Existing model expressions for {eta}(b), {eta} = 1 + 0.25b (Moldrup et al., 1998) and {eta} = 1 + 0.05b (Moldrup et al., 2001), are given in the figure by broken lines. Surprisingly, the existing expressions of {eta} greatly differed from the measured {eta}(b) in this study, while those predictive models performed well against past data (Moldrup et al., 2001, 2003). This discrepancy may be caused by the fact that the existing expressions of {eta}(b) were tested against a relatively small amount of data: 13 topsoils for {eta} = 1 + 0.25b (Moldrup et al., 1998) and six soils with similar mineralogy for {eta} = 1 + 0.05b (Moldrup et al., 2001). Also, the measured {eta} in the previous studies would be included within the scatter of {eta}(b) values in Fig. 3 (see Fig. 10b in Moldrup et al., 2001).

Moldrup et al. (1996) adopted three expressions for {eta}(b) based on capillary tube models for unsaturated hydraulic conductivity to model gas diffusivity, Dp/D0({varepsilon}). The expressions were based on Burdine (1953):

Formula 5[5]
Mualem (1976):

Formula 6[6]
and Alexander and Skaggs (1986):

Formula 7[7]

Moldrup et al. (1998) found that predictive Dp/D0 models with the three {eta}(b) expressions gave overall better predictions than other classical Dp/D0 models such as the Millington and Quirk (1960) and the Brooks and Corey (1966) based models. In addition, Moldrup et al. (1999) adopted the Burdine expression, Eq. [5], for developing a predictive Dp/D0 model, and suggested the so-called Buckingham–Burdine–Campbell (BBC) model [Dp/D0 = {phi}2({varepsilon}/{phi})2+3b, where {phi} is total porosity (m3 m–3)]. The BBC model gave good predictions against measured Dp/D0 for the five soil profiles (lysimeter and field soils) in this study (Kawamoto et al., 2006).

Analogous to the development of the predictive Dp/D0 models by Moldrup et al. (1996, 1999), the three expressions were tested against measured {eta}(b) values for ka (Fig. 3). The Burdine and Mualem expressions, Eq. [5] and [6], in general overestimated the measured {eta}(b) except for high {eta} values between 3 and 4. The Alexander and Skaggs expression, Eq. [7], gave an acceptable description of {eta}(b) within measured b values. It is important to notice that {eta}(b) for ka and Dp/D0 are described by different expressions: {eta}= 1 + 3/b for ka and {eta} = 2 + 3/b for Dp/D0 (BBC model). This coincides with the evidence that the {eta} values for ka are smaller than those for Dp/D0, as shown in Fig. 2.

Moldrup et al. (2004) derived the so-called three-porosity model (TPM), Dp/D0 = {phi}2({varepsilon}/{phi})X, including an exponent X. The value of X is found from

Formula 8[8]
where {phi} is the total porosity (m3 m–3). The TPM is mechanistically an analog to the BBC model with the exponents X and (2 + 3/b). The TPM gave the best Dp/D0 predictions for the five soil profiles (lysimeter and field soils) in this study (Kawamoto et al., 2006). If we assume that the BBC and TPM models are both valid, we can assume the exponents to be equal (i.e., X = 2 + 3/b). Substituting X into the Alexander and Skaggs expression of {eta} = 1 + 3/b, Eq. [7], we can obtain a new expression of {eta} for ka:

Formula 9[9]

Equation [9] does not require the Campbell b parameter (which requires knowledge of the entire soil-water retention curve within the {psi} interval of interest), and {eta} can be predicted by two pore indexes: the content of larger pores ({varepsilon}100) and the total porosity ({phi}). In this study, therefore, we used the Alexander and Skaggs expression, {eta} = 1 + 3/b, and the analog X-based expression, {eta} = X – 1, for further development of predictive ka models.

Reference-Potential Air Permeability
Millington and Quirk (1964) derived a link between ka and Dp/D0 by combining Fick's law for diffusive transport with Poiseuille's law for convective fluid transport, and assuming soil pores to be uniform, tortuous, and nonjointed tubes of similar diameter:

Formula 10[10]
where dg is the equivalent pore diameter for gas flow. Ball (1981) independently derived the same model and extended it to consider a porous medium with pores being jointed tubes of different diameters.

The equivalent pore diameter at the reference potential of {psi} = –100 cm H2O, dg,100, was calculated from Eq. [10] using measured ka,100 and Dp,100, and is shown as a function of {varepsilon}100 in Fig. 4 . For the lysimeter soils, the dg,100 values were generally around 150 µm. For the field soils, the dg,100 values varied widely between 10 and 320 µm for {varepsilon}100 < 0.1 m3 m–3. A majority of the values ranged between dg,100 = 100 and 200 µm, with an average of 150 µm. The wide range of dg,100 values for {varepsilon}100 < 0.1 m3 m–3 is probably due to the difference in the continuity of air-filled pores in soils. A small dg,100 value (typically for soils with very low ka,100 values) corresponds to less continuous and remote (inactive) air-filled pores where the gas convection ceases due to interconnected water films. In contrast, a large dg,100 value (from soils with high ka,100 and low Dp,100 values) corresponds to more continuous gas-phase pathways that contribute greatly to gas convection.


Figure 4
View larger version (19K):
[in this window]
[in a new window]
  		Fig. 4. Equivalent pore diameter for gas flow (estimated by Eq. [10]) at matric potential of –100 cm H2O (dg,100) as a function of air-filled porosity at matric potential of –100 cm H2O ({varepsilon}100) for lysimeter soils and field soils (shallow and deep soil layers).

 
Moldrup et al. (2000) suggested that the soil-gas diffusivity at {psi} = –100 cm H2O, Dp,100/D0 , is highly correlated to {varepsilon}100:

Formula 11[11]
Combining Eq. [11] with dg, Eq. [10], at {psi} = –100 cm H2O, we can obtain a predictive expression for ka,100 (µm2):

Formula 12[12]
If we use the average dg,100 of 150 µm for the lysimeter and field soils (Fig. 4), the term dg,1002/32 on the right-hand side is equal to 700 µm2.

Measurements of ka,100 for the lysimeter and field soils are shown as a function of {varepsilon}100 in Fig. 5 . In the figure, Eq. [12] with dg,100 = 150 µm is given by a broken curve and a simple power-law function obtained from a regression, Eq. [13], is shown by a solid curve:

Formula 13[13]
Equation [12] with dg,100 = 150 µm describes well the median values of ka,100 in the range of {varepsilon}100 > 0.1 m3 m–3, but overestimates the measurements in the range of {varepsilon}100 < 0.05 m3 m–3. The simple power-law ka,100({varepsilon}100) expression, Eq. [13], describes relatively well the measured ka,100 values in the whole range of {varepsilon}100 except for the measurements around {varepsilon}100 = 0.1 m3 m–3.


Figure 5
View larger version (21K):
[in this window]
[in a new window]
  		Fig. 5. Log-transformed air permeability at matric potential of –100 cm H2O [log(ka,100)] as a function of air-filled porosity at matric potential of –100 cm H2O ({varepsilon}100) for lysimeter soils and field soils (shallow and deep soil layers). Estimated and regression curves for ka,100 (Eq. [12] with equivalent pore diameter for gas flow estimated by Eq. [10] = 150 µm and Eq. [13]) are given.

 
New Predictive Models for Air Permeability as a Function of Air-Filled Porosity
Inserting the Alexander and Skaggs (1986) expression of {eta} = 1 + 3/b (Eq. [7]), and its analog X-based form, {eta} = X – 1 (Eq. [9]), to the power-law ka({varepsilon}) model (Eq. [1]), we can obtain two new functions:

Formula 14[14]

Formula 15[15]
The new ka({varepsilon}) functions need four model parameters, ka,100, {varepsilon}, {varepsilon}100, and b (for Eq. [14]), or {phi} (for Eq. [15]), for predictions as well as the existing power-law models with {eta} = 1 + 0.25b (Moldrup et al., 1998) and {eta} = 1 + 0.05b (Moldrup et al., 2001). Furthermore, two ka,100({varepsilon}100) functions, Eq. [12] and [13], are available to avoid the use of measured ka,100 in Eq. [14] and [15].


   RESULTS AND DISCUSSION
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 MODEL DEVELOPMENT
 RESULTS AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
Model Tests against Data for Five Soil Profiles
In this study, a total of eight predictive ka({varepsilon}) models were tested against measured data. The eight predictive models are classified into two groups (see Table 1): The first group of ka({varepsilon}) models include measured ka,100 as a reference point and four different {eta}(b) expressions: 1 + 0.25b in Moldrup et al. (1998), 1 + 0.05b in Moldrup et al. (2001), 1 + 3/b in Eq. [7], and X – 1 in Eq. [9]. In other models, the estimated ka,100 by either Eq. [12] or Eq. [13], was used together with one of the two new {eta}(b) expressions, 1 + 3/b and X – 1.


View this table:
[in this window]
[in a new window]
  		Table 1. Test of predictive air permeability as a function of air-filled porosity, ka({varepsilon}), models against data for lysimeter soils (Rønhave, Foulum, and Jyndevad) and field soils (Gjorslev and Mammen). Parameter expressions for ka at matric potential of –100 cm H2O (ka,100) and tortuosity–connectivity parameter {eta} in ka = ka,100({varepsilon}/{varepsilon}100){eta}, Eq. [1], are tabulated. Calculated RMSE, bias, and Akaike's Information Criterion (AIC) using log-transformed relative soil-air permeabilities, Eq. [2], [3], and [4], are given.

 
Separate tests were done for data subsets including: (i) all measurements for lysimeter soils (three soil profiles from 0.10- to 1.05-m depth), and (ii) all measurements for field soils (two soil profiles from 0.15- to 5.40-m depth). The RMSE (Eq. [2]), bias (Eq. [3]), and AIC (Eq. [4]) are tabulated in Table 1. Scatterplot comparisons of predicted and measured log(ka) for the two ka({varepsilon}) models with the measured ka,100, Eq. [14] and [15], and for the ka({varepsilon}) model (Eq. [15]) with the estimated ka,100 by Eq. [12], ka = 700(2{varepsilon}1003 + 0.04{varepsilon}100)({varepsilon}/{varepsilon}100)X–1, are also depicted in Fig. 6 .


Figure 6
View larger version (23K):
[in this window]
[in a new window]
  		Fig. 6. Scatterplot comparison of predicted and measured log-transformed air permeability, log(ka), for lysimeter soils and field soils (shallow and deep soil layers). Predictive models for ka as a function of air-filled porosity [ka({varepsilon})]: (a) ka = ka,100({varepsilon}/{varepsilon}100)1+3/b (Eq. [14]); (b) ka = ka,100({varepsilon}/{varepsilon}100)X–1 (Eq. [15]); and (c) ka = 700(2{varepsilon}1003 + 0.04{varepsilon}100)({varepsilon}/{varepsilon}100)X–1 (Eq. [15] with Eq. [12]); where ka,100 and {varepsilon}100 are the values at matric potential of –100 cm H2O. Calculated RMSE values by Eq. [2] using log(ka) data are given.

 
The new ka({varepsilon}) models with {eta} = 1 + 3/b and {eta} = X – 1, and the existing ka({varepsilon}) model with {eta} = 1 + 0.05b, which require measured ka,100 values as a reference point, gave good predictions (Table 1, Fig. 6a and 6b). Especially, the new ka({varepsilon}) model with {eta} = 1 + 3/b performed the best among all models, and all three statistical indexes (RMSE, bias, and AIC) indicated that the new model provided the best fit for both lysimeter and field soils (Table 1).

The ka({varepsilon}) models with estimated ka,100 gave worse predictions than those with measured ka,100, and provided at least one order of magnitude uncertainty in the prediction (Fig. 6c). This suggests that the description of ka,100({varepsilon}100) needs further improvement.

Figure 7 shows the depth distribution of log(ka) (average and standard deviation) for Gjorslev and Mammen (field soils) at three different potentials of {psi} = –20, –50, and –160 cm H2O. Three model predictions using averages of measured ka,100, {varepsilon}100, b, and X are also shown in the figure.


Figure 7
View larger version (25K):
[in this window]
[in a new window]
  		Fig. 7. Depth distribution of log-transformed air permeability, log(ka), for (a, b, and c) Gjorslev and (d, e, and f) Mammen at three different soil-water matric potentials of –20, –50, and –160 cm H2O. Averaged measured data and standard deviation of log(ka) are shown. Predicted log(ka) by three different models for ka as a function of air-filled porosity [ka({varepsilon})], ka = ka,100({varepsilon}/{varepsilon}100)1+3/b, ka = (890{varepsilon}1002.5)({varepsilon}/{varepsilon}100)X–1, and ka = 700(2{varepsilon}1003 + 0.04{varepsilon}100)({varepsilon}/{varepsilon}100)X–1, are given, where ka,100 and {varepsilon}100 are the values at matric potential of –100 cm H2O.

 
The ka({varepsilon}) model with measured ka,100 and {eta} = 1 + 3/b (Eq. [14]) (Fig. 7, solid lines) predicted well the log(ka) values throughout the entire depth from shallow to deep layers, and described well the variation in log(ka) through the soil profiles at each {psi} stage. It is noted that the ka({varepsilon}) model with measured ka,100 and {eta} = X – 1 (Eq. [8]) also predicted well the depth distributions of measured air permeabilities (not shown in Fig. 7), and the predictions overlapped the solid lines in Fig. 7 from the ka({varepsilon}) model with {eta} = 1 + 3/b (Eq. [14]).

The two ka({varepsilon}) models with estimated ka,100 (Eq. [12] and [13]) and {eta} = X – 1 gave a relatively good prediction at the Mammen site (Fig. 7c, 7d, and 7e); however, the models underestimated measured data in the shallow layers and greatly overestimated measured values in the deep and highly compacted layers at Gjorslev (Fig. 7a, 7b, and 7c).

Model Test against Independent Data
To further validate the new ka({varepsilon}) models, the models were tested against independent data for 24 undisturbed soil samples from deep subsurface soil layers at Hjørring. Figure 8a shows the relationship between measured ka,100 and {varepsilon}100 values. The estimated and regression curves, Eq. [12] with dg,100 = 150 µm and Eq. [13], are also given in the figure. Contrary to the five soil profiles (lysimeter and field soils) in Fig. 5, the estimated curve, Eq. [12], gave better agreement with measured ka,100({varepsilon}100) values than the regression curve, Eq. [13].


Figure 8
View larger version (15K):
[in this window]
[in a new window]
  		Fig. 8. Independent test of new models of air permeability as a function of air-filled porosity, ka({varepsilon}), for Hjørring subsurface soils (12 samples from 4–5-m depth and 12 samples from 6–7-m depth): (a) log-transformed air permeability at matric potential of –100 cm H2O, log(ka,100), as a function of air-filled porosity, {varepsilon}100 [estimated and regression curves for ka,100 (Eq. [12] with equivalent pore diameter for gas flow = 150 µm and Eq. [13]) are given]; (b) scatterplot comparison of predicted and measured log(ka). Calculated RMSE values by Eq. [2] using log(ka) data are given.

 
Eight predictive ka({varepsilon}) models were tested against measured data, and RMSE (Eq. [2]), bias (Eq. [3]), and AIC (Eq. [4]) are tabulated in Table 2. Scatterplot comparisons of predicted and measured log(ka) for two models, the ka({varepsilon}) model with measured ka,100, ka = ka,100({varepsilon}/{varepsilon}100)X–1 (Eq. [15]), and the ka({varepsilon}) model (Eq. [15]) with estimated ka,100 by Eq. [12] with dg,100 = 150 µm, ka = 700(2{varepsilon}1003 + 0.04{varepsilon}100)({varepsilon}/{varepsilon}100)X–1, are also depicted in Fig. 8b. In agreement with previous model tests, the three ka({varepsilon}) models with {eta} = 1 + 3/b, {eta} = X – 1, and {eta} = 1 + 0.05b, which require measured ka,100 values as a reference point, gave good predictions. The ka({varepsilon}) models with estimated ka,100 gave worse predictions similar to the previous model tests, implying again that further improvement is needed to estimate the reference-point value of ka,100.


View this table:
[in this window]
[in a new window]
  		Table 2. Independent test of predictive models of air permeability as a function of air-filled porosity, ka({varepsilon}), against data for Hjørring subsurface soil horizons (a total of 24 samples; 12 samples from 4–5-m depth and 12 samples from 6–7-m depth).

 
In model tests against measurements for the lysimeter and field soil profiles (Table 1) and for the independent data (Table 2), the new ka({varepsilon}) model with {eta} = X – 1, Eq. [15], performed well and gave accurate predictions. Compared with the models that include the Campbell (1974) pore-size distribution parameter b (which requires knowledge of the entire soil-water retention curve within the {psi} interval of interest), the ka({varepsilon}) model with {eta} = X – 1 (Eq.[15]), is easy to apply for predictions because only one measurement on the soil-water retention curve ({varepsilon}100) is needed to determine X (Eq. [8]) in the power-law exponent {eta}. Furthermore, the X in the power-law exponent {eta} originally represented the tortuosity–connectivity parameter (power-law exponent) for the gas diffusivity predictive model (three-porosity model), and X is logically related to soil texture and pore connectivity (Kawamoto et al., 2006). This suggests that X is a key parameter to obtain accurate predictions for main gas transport parameters (air permeability and gas diffusivity) in the soil vadose zone.


   CONCLUSIONS
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 MODEL DEVELOPMENT
 RESULTS AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
Soil-air permeabilities were measured at different soil-water matric potentials on undisturbed soil samples representing differently textured soil profiles. Comparing measurements of ka and Dp/D0 (data from Kawamoto et al., 2006) on the same samples suggested an analogy between how the two soil-gas transport parameters depend on {varepsilon}. The power-law exponent for ka({varepsilon}), however, was typically smaller than for Dp({varepsilon})/D0, due to more pronounced effects of soil structure on ka.

In analogy to recently developed gas diffusivity models and in line with capillary tube models for unsaturated hydraulic conductivity, two power-law ka({varepsilon}) models were developed based on the Campbell (1974) pore-size distribution parameter b or the content of larger soil pores ({varepsilon}100). Both new models require measured ka,100 as a reference point to give acceptable prediction accuracy.

Both ka({varepsilon}) models predicted well the depth distributions of measured air permeabilities at different soil sites at distances down to 6-m depth. Especially, the ka({varepsilon}) model with the tortuosity–connectivity parameter {eta} = X – 1 (where X is a function of {varepsilon}100) does not require detailed information on the soil-water retention curve and seems highly useful for predicting soil-air permeability in the vadose zone.


   ACKNOWLEDGMENTS
 
This publication was made possible by Grant no. P42 ES04699 from the National Institute of Environmental Health Sciences (NIEHS), National Institutes of Health. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of the NIEHS. This work was also partially supported by the Saitama University 21st Century Project Grant for promotion of international joint research. Part of this work was also supported by the Danish project "Concept for identifying areas where shallow aquifers are vulnerable to pesticide contamination" (KUPA) financed by the Danish Parliament. We especially acknowledge the careful and dedicated laboratory work by former M.S. students Ann Frederiksen and Marianna Irene Madsen, Aalborg University. We gratefully acknowledge a research and travel grant from the Japanese Ministry of Education, Science, Sports, and Culture (Monbukagakusyo: Research no. 18360224).


   REFERENCES
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 MODEL DEVELOPMENT
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 




This article has been cited by other articles:


Home page
 Soil Sci.Home page
S. Hamamoto, P. Moldrup, K. Kawamoto, T. Komatsu, and D. E. Rolston
Unified Measurement System for the Gas Dispersion Coefficient, Air Permeability, and Gas Diffusion Coefficient in Variably Saturated Soil
Soil Sci. Soc. Am. J., October 21, 2009; 73(6): 1921 - 1930.
[Abstract] [Full Text] [PDF]

This Article
Right arrow Abstract Freely available
Right arrow Figures Only
Right arrow Full Text (PDF) Free
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Similar articles in this journal
Right arrow Similar articles in Web of Science
Right arrow Alert me to new issues of the journal
Right arrow Download to citation manager
Right arrow reprints & permissions
Citing Articles
Right arrow Citing Articles via HighWire
Right arrow Citing Articles via Web of Science (4) Citing Articles via Web of Science (4)
Right arrow Citing Articles via Google Scholar
Google Scholar
Right arrow Articles by Kawamoto, K.
Right arrow Articles by Rolston, D. E.
Right arrow Search for Related Content
PubMed
Right arrow Articles by Kawamoto, K.
Right arrow Articles by Rolston, D. E.
GeoRef
Right arrow GeoRef Citation
Agricola
Right arrow Articles by Kawamoto, K.
Right arrow Articles by Rolston, D. E.
Related Collections
Right arrow Air Permeability
Right arrow Diffusion
Right arrow Soil Physics

HOME HELP FEEDBACK SUBSCRIPTIONS ARCHIVE SEARCH TABLE OF CONTENTS
The SCI Journals Agronomy Journal Crop Science
Journal of Natural Resources
and Life Sciences Education		 Soil Science Society of America Journal
Journal of Plant Registrations Journal of
Environmental Quality		 The Plant Genome