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Published online 20 November 2006
Published in Vadose Zone J 5:1194-1204 (2006)
DOI: 10.2136/vzj2006.0014
© 2006 Soil Science Society of America
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ORIGINAL RESEARCH

Gas Transport Parameters in the Vadose Zone

Gas Diffusivity in Field and Lysimeter Soil Profiles

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

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)

Received 20 January 2006.



   ABSTRACT
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 SOIL-GAS DIFFUSIVITY MODELS
 RESULTS AND MODEL TESTS
 CONCLUSIONS
 REFERENCES
 
The main soil-gas transport parameters, gas diffusivity and air permeability, and their variations with soil type and air-filled porosity play a key role in soil-gas emission problems including volatilization of toxic chemicals at polluted sites and the production and emission of greenhouse gases. Only limited information on soil-gas transport parameters across the vadose zone is available, especially for soil layers below the root zone. In a series of studies, we developed new data for the soil-gas transport parameters in different soil profiles and tested existing and new predictive models. In this first study, we measured gas diffusivity at different soil-water matric potentials on undisturbed soil samples for three lysimeter soil profiles down to 1.4-m depth and for two field soil profiles down to 5.6-m depth, representing a total of 22 different soil layers with soil texture ranging from sand to sandy clay loam. Five commonly used predictive gas diffusivity models were tested. The three-porosity model (TPM) performed best for both shallow and deep soil layers. The tortuosity–connectivity parameter X in the TPM varied with soil texture and pore size distribution, and the TPM predicted well the depth distributions of measured soil-gas diffusivities. The TPM also requires less detailed information on the soil-water characteristic curve than other well-performing predictive models, and is therefore recommended for predicting variations in soil-gas diffusivity within the vadose zone.

Abbreviations: AIC, Akaike's information criterion • MQ, Millington and Quirk • BBC, Buckingham–Burdine–Campbell • MPD, macroporosity-dependent model • TPM, three-porosity model


   INTRODUCTION
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 SOIL-GAS DIFFUSIVITY MODELS
 RESULTS AND MODEL TESTS
 CONCLUSIONS
 REFERENCES
 
ACCURATE PREDICTIVE MODELS for the soil-gas diffusion coefficient and its variations with air-filled porosity and soil type are essential to understand and solve gas transport and emission problems from the local to the global scale, including volatilization of toxic chemicals at polluted soil sites (Jury et al., 1990; Petersen et al., 1996; Poulsen et al., 1999), production and emission of greenhouse gases (Osozawa and Hasegawa, 1995; Smith et al., 2003), and poor soil aeration and reduced aerobic microbial activity in cultivated soils (Schjønning et al., 2003). In spite of this, only limited measurements and knowledge of the soil-gas diffusivity in undisturbed soil is available, especially for deeper vadose zone profiles (>1-m depth).

The soil-gas diffusivity in natural, undisturbed soil is dependent on both soil texture and structure, therefore pore size distribution and pore connectivity and tortuosity (Schjønning et al., 1999; Moldrup et al., 2001). To evaluate in situ gas diffusion phenomena, it is often necessary to investigate the depth distribution of soil-gas diffusivity for soil profiles including several different soil layers or soil types. Washington et al. (1994) successfully simulated a steady-state 222Rn (as a tracer gas) concentration profile using field-measured soil-gas diffusivities for different soil layers from the A horizon down to the C horizon at 2-m depth. Using predicted soil-gas diffusivities based on the Millington and Shearer (1971) model, Davidson and Trumbore (1995) simulated a steady-state 222Rn profile down to 5-m depth and estimated surface fluxes of CO2 for forest and pasture lands of the western Amazon. Also, accurate description of the soil-gas diffusivity with depth is important for evaluating the removal efficiency of volatile organic contaminants from low-permeability (gas diffusion limited) regions when applying gas-transport-based in situ remediation methods such as soil-vapor extraction (Poulsen et al., 1998).

Predictive models for the soil-gas diffusion coefficient (Dp) are typically developed as a function of the soil air-filled porosity ({varepsilon}). Numerous predictive Dp({varepsilon}) models have been developed empirically, or theoretically, considering soil pore parameters such as air-filled and total porosities, pore-size distribution indexes, multiple pore regions, and a threshold air-filled porosity (where gas diffusion ceases because remaining air-filled pore spaces are isolated or blocked by water) (Moldrup et al., 2004). For simulation of field-scale gas transport and fate processes, however, simple and accurate low-parameter Dp({varepsilon}) models are desired (Werner et al., 2004).

This study on gas transport parameters in vadose zone soil profiles had the following objectives: (i) to measure soil-gas diffusivity on undisturbed soil samples in five differently textured soil profiles (three lysimeter and two field profiles) containing several morphologic horizons; (iii) to test five commonly used predictive models for soil-gas diffusivity against the measured data; and (iii) to evaluate depth variations of soil-gas diffusivity. A companion study (Kawamoto et al., 2006) investigated soil-air permeability data and predictive model tests for the same soil profiles.


   MATERIALS AND METHODS
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
SOIL-GAS DIFFUSIVITY MODELS
 RESULTS AND MODEL TESTS
 CONCLUSIONS
 REFERENCES
 
Soil Sites and Characteristics
Undisturbed soil samples were collected at five soil profiles in Denmark: three large soil bins (lysimeters) at the Danish Institute of Agricultural Sciences at Research Centre Foulum with soil from Rønhave, Foulum, and Jyndevad, and two agricultural field soils, Gjorslev and Mammen (Fig. 1 ). For convenience, the Rønhave, Foulum, and Jyndevad soils are hereafter rederred to as the lysimeter soils, and the Gjorslev and Mammen soils as the field soils.


Figure 1
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  		Fig. 1. Locations of soil sampling: three lysimeter soils came from Rønhave, Foulum, and Jyndevad; two agricultural field soils were sampled at Gjorslev and Mammen. Note that soils from Rønhave and Jyndevad were transported to lysimeters at Research Centre Foulum.

 
The soil bins included four lanes 40 m long, 2.7 m wide, and 1.5 m deep. Each lane was divided into 25 lysimeter units by concrete walls to 30 cm below the soil surface, which enabled "natural" tillage operations with all traffic on the concrete walls. The soil bins were filled with soil in 1993. Large quantities of soil were excavated from three locations with different soil textures, respecting its vertical pedological differentiation. The soil was air dried and crumbled to aggregates <20 mm. The soil bin units were filled with soil in 10-cm increments and packed to the dry bulk density found in the field. The soil was exposed to freeze–thaw and wet–dry cycles as well as frequent tillage, allowing soil structure to develop. Winters are normally mild in Denmark and during the 6-yr period from filling the lysimeters to sampling, frost had probably not reached depths below 20 cm. The annual excess precipitation amounted to 200 to 400 mm percolating into the soil, primarily during the winter. Tillage was performed to 0.2-m depth with a one-furrow moldboard plow, pulled by a tractor driving on the top of the concrete walls containing the soil. Secondary tillage was performed with a rotary harrow and drilling of seeds with a traditional drill. The soil was cropped with primarily small grain cereals. in the start of the period, however, alfalfa (Medicago sativa L.) was frequently grown due to its high potential for deep rooting in virgin soil. The soil represents an intermediate step between a remolded soil column in the laboratory and field conditions where the soil may be heavily structured and may present a large textural gradient and high heterogeneity. The field soils have been under cultivation for centuries.

For all locations, soil layers were carefully distinguished and soil samples were collected from each layer from the surface Ap horizon down to the C horizon at 1.4-m depth for the lysimeter soils, and down to the C horizon at 4.8- or 5.6-m depth for the field soils. Soil texture and physical characteristics for each layer are given in Table 1.


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  		Table 1. Soil physical characteristics for lysimeter soils (Rønhave, Foulum, and Jyndevad) and field soils (Gjorslev and Mammen).

 
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 soil cores were retrieved in metal cylinders forced into the soil by means of a hammer. The cylinders were held in the position by a flange, ensuring the vertical downward movement into the soil. After careful removal of the soil-filled cylinder, end surfaces were trimmed with a knife. For all locations, sampling included the top surface layer of the Ap horizon, representing a plow layer at 0- to 0.3-m depth, which has relatively higher soil organic matter contents (Table 1).

Measurement Methods
Soil-water retention was measured by the method of Klute (1986). The intact 100-cm3 soil cores were saturated in sand boxes and subsequently drained to the desired soil-water matric potentials ({psi} = –20, –50, –100, and –160 [or –200] cm H2O). This was achieved by the use of tension tables (fine sand or ceramics) for all potentials, except –200 cm H2O, where we used a suction plate apparatus. The Campbell (1974) soil-water retention parameter b was determined as the slope of the soil-water retention curve in a log–log coordinate system. The average value and standard deviation of b for each soil horizon are given in Table 1. The b values varied in a wide interval from 1.53 to 35.2 depending on soil texture.

Soil-gas diffusivity (Dp) was measured at each {psi} by the method described in Rolston and Moldrup (2002). An experimental setup first suggested by Taylor (1949) and further developed by Schjønning (1985a) was used. A schematic of a similar diffusion chamber setup is shown in Fujikawa and Miyazaki (2005) using different soil sample dimensions. Soil-gas diffusion was measured with O2 as the experimental gas at 25°C. The O2 diffusion coefficient in free air (D0) is 0.205 x 10–4 m2 air s–1 at 25°C (Sahores and Witherspoon, 1970). The diffusion chamber was first flushed with 100% N2, after which the upper end of the soil core was exposed to the atmosphere. Oxygen was measured in the diffusion chamber with an O2 electrode. Oxygen consumption in the soil core was considered negligible during the short periods (minutes to a few hours depending on matric potential) needed to measure the diffusion coefficient at each matric potential (Schjønning, 1985a). Oxygen concentration in the diffusion chamber was analyzed as a function of time, and the gas diffusion coefficient in soil, Dp, was calculated (Currie, 1960; Rolston and Moldrup, 2002). A total of 136 Dp measurements were obtained for the lysimeter soils and 235 for the field soils.

Statistical Analyses
Three statistical indexes were used to evaluate and compare predictive soil-gas diffusivity models. To evaluate best overall fit compared with measured data, the RMSE was used:

Formula 1[1]
where di [= (Dp/D0)predicted – (Dp/D0)measured] is the difference between the predicted and the measured values of soil-gas diffusivity (Dp/D0) 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):

Formula 2[2]
To account for the number of model parameters when comparing model performance for a given measured data set, Akaike's information criterion was used (Akaike, 1973):

Formula 3[3]
where k is the number of model parameters. A smaller (or more negative) AIC indicates better model performance (Minasny et al., 1999).

In the case of the statistical comparison based on log-transformed values of Dp/D0, the di in Eq. [1], [2], and [3] is replaced by log(Dp/D0)predicted – (Dp/D0)measured.


   SOIL-GAS DIFFUSIVITY MODELS
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 SOIL-GAS DIFFUSIVITY MODELS
RESULTS AND MODEL TESTS
 CONCLUSIONS
 REFERENCES
 
The Millington and Quirk (MQ, 1960), Eq. [4], and (1961), Eq. [5], models are the most widely used two-parameter gas diffusivity models:

Formula 4[4]

Formula 5[5]
where {varepsilon} is the air-filled porosity (m3 m–3), {phi} is the total porosity (m3 m–3), Dp is the gas diffusion coefficient in soil (m3 soil air m–1 soil s–1), and D0 is the gas diffusion coefficient in free air (m2 air s–1).

The Buckingham–Burdine–Campbell (BBC) model (Moldrup et al., 1999) is

Formula 6[6]
where the term {phi}2 corresponds to soil-gas diffusivity in completely dry soil, as suggested by Buckingham (1904), and the term 2 + 3/b is an analog to the Burdine (1953)–Campbell (1974) tortuosity model for describing unsaturated hydraulic conductivity (Moldrup et al., 1996). The BBC model, Eq. [6], was developed based on Dp/D0 and soil-water retention data for 20 undisturbed soils, and successfully tested against data for six undisturbed surface soils along a soil textural gradient with b values ranging from 4.6 to 14.

The BBC model was modified by Moldrup et al. (2000). They replaced the term {phi}2 in Eq. [6] by a highly significant correlation between soil-gas diffusivity and air-filled porosity at {psi} = –100 cm H2O based on data for 126 undisturbed soils:

Formula 7[7]
and obtained the so-called macroporosity-dependent model (MPD):

Formula 8[8]
where {varepsilon}100 is the air-filled porosity at {psi} = –100 cm H2O, and Dp,100/D0 is the soil-gas diffusivity at {psi} = –100 cm H2O. The {varepsilon}100 is the volumetric content of pores >30 µm (Schjønning, 1985b), and is related to large pores that will quickly become air filled by gravity drainage after infiltration. The MPD model was successfully tested against 21 undisturbed soils with relatively small b values (typically <10). The MPD model was further tested against data for deep subsurface soil samples (4–7-m depth) at a sandy soil profile below a coal-tar-polluted soil site, and performed better than the MQ (1961) model, Eq. [5] (Moldrup et al., 2000).

To avoid the use of the Campbell (1974) b parameter (which requires knowledge of the entire soil-water retention curve within the {psi} interval of interest), Moldrup et al. (2004) developed the three-porosity model (TPM). The TPM is derived by combining a general power-law Dp({varepsilon}) model, the Buckingham (1904) model, and the relationship between {varepsilon}100 and Dp,100/D0 in the MPD model, Eq. [7], yielding

Formula 9[9]
where the tortuosity-connectivity parameter X is found from

Formula 10[10]

The TPM predicts Dp,100/D0 as a function of three porosities:{varepsilon}, {varepsilon}100, and {phi}. Moldrup et al. (2004) tested the TPM against data for 60 undisturbed soils and found that it performed as well as the b-dependent BBC and MPD models, Eq. [6] and [8].


   RESULTS AND MODEL TESTS
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 SOIL-GAS DIFFUSIVITY MODELS
 RESULTS AND MODEL TESTS
CONCLUSIONS
 REFERENCES
 
Pore Characteristics
The volume of larger pores (e.g.,{varepsilon}100) markedly influences Dp/D0 as suggested by the MPD model, Eq. [7] and [8], and the TPM, Eq. [9] and [10]. Figure 2 shows the calculated pore volumes occupied by pores of equivalent diameters d ≥ 150, 30 < d < 150, and d ≤ 30 µm. The pore volume with d ≥ 150 µm corresponds to the difference between total porosity {phi} and volumetric water content at {psi} = –20 cm H2O. The pore volume with d ≤ 30 µm corresponds to the volumetric water content at {psi} = –100 cm H2O.


Figure 2
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  		Fig. 2. Pore volume distribution for (a) lysimeter soils (Rønhave, Foulum, and Jyndevad) and field soils at (b) Gjorslev and (c) Mammen. Average data for each soil layer are given; d is equivalent pore diameter calculated from d = 3000(–{psi})–1 where {psi} is soil-water matric potential (cm H2O) (Schjønning, 1985b).

 
The {varepsilon}100 values, corresponding to the volume of pores with d ≥ 30 µm, depend on soil texture. This is shown by the large {varepsilon}100 values for the loamy sand layers at Jyndevad (Fig. 2a) and the sand layer at Mammen (C3 horizon, Fig. 2c), and the small {varepsilon}100 values for the sandy clay loam layers at Gjorslev (Fig. 2b) and Mammen (Bt and C1 horizons, Fig. 2c). A greater amount of large air-filled pores (d ≥ 150 µm) for the lysimeter soils may be attributed to a structural disturbance due to the refilling, indicating that natural soil structure is not fully developed in contrast to the field soils. The same was seen when comparing lysimeter and field soil data for six surface soils along a natural soil texture gradient (Schjønning et al., 1999).

Soil Type Effects on Gas Diffusivity
All Dp/D0 measurements for the lysimeter soils are shown as a function of {varepsilon} in Fig. 3a . All measurements for the field soils are shown in Fig. 3b. To illustrate magnitude and distribution of Dp/D0 values, data are shown in a log–log coordinate system, and three simple power-law functions, Dp/D0 = {varepsilon}4/3 (Millington, 1959), {varepsilon}2 (Buckingham, 1904), and {varepsilon}10/3 (Millington and Quirk, 1961), are shown for comparison. For the lysimeter soils, the measured Dp/D0 narrowly ranges between Dp/D0 = {varepsilon}2 and Dp/D0 = {varepsilon}10/3. For the field soils, the measured Dp/D0 range is wider, and values of Dp/D0 < 0.0001 were observed across a wide range of {varepsilon}. There was no consistent difference in Dp/D0 measurements between shallow (0.15–1.20 m) and deeper soil layers (2.10–5.40 m) for the field soils.


Figure 3
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  		Fig. 3. Soil-gas diffusivities Dp/D0 as a function of air-filled porosity {varepsilon}: (a) all matric-potential ({psi}) measurements for lysimeter soils (Rønhave, Foulum, and Jyndevad); (b) all measurements for field soils (Gjorslev and Mammen); (c) measurements from C horizon samples with different soil textures at Mammen; and (d) measurements from sandy clay loam layers from Foulum, Gjorslev, and Mammen. Note different axes and log scale.

 
Differences in Dp/D0 within a given {psi} range greatly depend on the difference in soil texture, because at a given {psi} value, a coarse-textured soil will have higher Dp/D0 than a fine-textured soil. This is illustrated in Fig. 3c, showing measurements of Dp/D0 for C horizon samples with different soil textures (sandy clay loam, sandy loam, and sand). Values of Dp/D0 corresponding to the given {psi} interval (from –20 to –200 cm H2O) increase from finer (sandy clay loam) to coarser (sand) textured soil. There is a tendency, however, for Dp/D0 values from sandy loam to be smaller than those from both finer (sandy clay loam) and coarser soils around {varepsilon} = 0.1 m3 m–3. This supports our contention that Dp/D0 is controlled not only by soil texture, but also by soil structure and pore geometry (Schjønning et al., 1999; Moldrup et al., 2001; Tuli and Hopmans, 2004; Fujikawa and Miyazaki, 2005). This is also evident from Fig. 3d, in which the Dp/D0 values for B2 and B3 horizons at Foulum (lysimeter soil) are smaller than those for Bv horizons at Gjorslev and Mammen (field soils) at {varepsilon} < 0.1 m3 m–3, even though all the soil horizons have similar textures and porosities (Table 1).

Reference-Potential Gas Diffusivity
Measurements of Dp/D0 at {psi} = –100 cm H2O (=Dp,100/D0) are shown as a function of {varepsilon}100 for the lysimeter soils in Fig. 4a (linear scale), and for the field soils in Fig. 4c (both linear and log scales). For comparison, field-measured Dp,100/D0 values from Schjønning (1985b), obtained at the same sampling locations as the lysimeter soils, are shown in Fig. 4b (linear scale). In the figures, the Dp,100/D0 function by Moldrup et al. (2000), Eq. [7], is also shown.


Figure 4
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  		Fig. 4. Soil-gas diffusivities Dp,100/D0 as a function of air-filled porosity {varepsilon}100 for (a) lysimeter soils (Rønhave, Foulum, and Jyndevad), (b) field soils (Rønhave, Foulum, and Jyndevad) (Schjønning, 1985b), and (c) field soils (Gjorslev and Mammen). For (c), matric potential ({psi}) = –100 cm H2O data are plotted in both normal (left axis) and log scale (right axis). {varepsilon}100 = air-filled porosity at {psi} = –100 cm H2O; Dp,100/D0 = soil-gas diffusivity at {psi} = –100 cm H2O.

 
For the lysimeter soils (Fig. 4a), Eq. [7] overestimated the data for Foulum (sandy loam) and slightly underestimated the measured data for Jyndevad (loamy sand). In contrast, Eq. [7] predicted accurately the field-measured Dp,100/D0 values obtained at the same sampling locations as the lysimeter soils across the entire {varepsilon}100 range (Fig. 4b). For the field soils in this study (Fig. 4c), the measured Dp,100/D0 values are smaller than 0.002 (corresponding {varepsilon}100 values are smaller than 0.16) except for the sand layer at 5.4-m depth at Mammen (surrounded by the dotted ellipse in Fig. 4c), and Eq. [7] predicts well the measured Dp,100/D0 values across the entire {varepsilon}100 range.

The results in Fig. 4 probably reflect differences in soil structure. The field soils (Fig. 4b and 4c) possess a natural soil structure, and the measured Dp,100/D0 values agree well with Eq. [7] developed from data from undisturbed field soils (Moldrup et al., 2000). In contrast, the lysimeter soils had been standing in lysimeters for 6 yr after refill but the soil structure was not yet as fully developed as it was in the original field soils, and therefore the measured Dp,100/D0 values are less accurately described by Eq. [7].

Test of Gas Diffusivity Models
We tested five experimental models against measured Dp/D0 data: MQ (1960) and (1961), Eq. [4] and [5]; the BBC (Moldrup et al., 1999), Eq. [6]; the MPD model (Moldrup et al., 2000), Eq. [7] and [8]; and the TPM (Moldrup et al., 2004), Eq. [9] and [10]. Separate tests were done for specific data subsets: (i) all lysimeter soils (0.10–1.05 m), (ii) all field soils (0.15–5.40 m), (iii) shallow field soils (0.15–1.20 m), and (iv) deep field soils (2.10–5.40 m). The RMSE, Eq. [1], bias, Eq. [2], and AIC, Eq. [3], were calculated using both linear and log-transformed Dp/D0 data (Table 2). When calculating AIC (Eq. [3]), the number of model parameters k = 2 ({varepsilon} and {phi}) for the MQ (1960, 1961) models, and k = 3 for the BBC model ({varepsilon}, {phi}, and b), the MPD model ({varepsilon}, {varepsilon}100, b), and the TPM ({varepsilon}, {varepsilon}100, {phi}). Scatterplot comparisons of predicted and measured Dp/D0 for the MQ (1961) model and the TPM are also depicted in Fig. 5 in a log–log coordinate system.


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  		Table 2. Test of predictive soil-gas diffusivity models against data for lysimeter soils (Rønhave, Foulum, and Jyndevad) and field soils (Gjorslev and Mammen) at three soil depth intervals. Calculated RMSE, bias, and Akaike's information criterion (AIC), Eq. [1], [2], and [3], for each predictive model are given. Calculated RMSE, bias, and AIC using log-transformed soil-gas diffusivity (Dp/D0) data are also given (values in brackets).

 

Figure 5
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  		Fig. 5. Scatterplot comparison of predicted and measured soil-gas diffusivities (Dp/D0) for lysimeter soils (Rønhave, Foulum, and Jyndevad) and field soils (Gjorslev and Mammen). All measured data are given. Model predictions are: (a, b) the Millington and Quirk (1961) model (MQ) and (c, d) the three-porosity model (TPM). Calculated RMSE values using Dp/D0 data and log-transformed Dp/D0 data (values in brackets), Eq. [1], are given. Note different axes and log scale.

 
The MQ (1960) model generally overestimated the measured Dp/D0 (positive bias), and yielded RMSE values between 0.011 and 0.065 for all soil layers (Table 2), clearly providing the worst model performance among the five models tested. The MQ (1961) model performed better than the MQ (1960) model but the model predictions had a large negative bias (underestimation) for the field soils in the range Dp/D0 < 0.02 (Fig. 5b).

The BBC, MPD, and TPM models all gave good predictions. The TPM performed the best among all models, as indicated by all three statistical indexes: RMSE, bias, and AIC (Table 2). The one exception is that the MQ (1961) model performed slightly better than the TPM on the log-transformed data for the lysimeter soils—the reason being that Eq. [7] overestimated measured Dp,100/D0 for the Foulum soil (Fig. 4a), causing the TPM to generally overestimate the measured Dp/D0 for the Foulum soil (Fig. 5c).

The TPM tends to overestimate measured values in the low Dp/D0 range (<0.01 for the lysimeter soils in Fig. 5c and <0.001 for the field soils in Fig. 5d) at low {varepsilon} (typically <0.1 m3 m–3), while the TPM predicts accurately the measured values in the high Dp/D0 range. This may indicate the presence of isolated air-filled pore spaces that do not contribute to gas diffusion, affecting the TPM prediction accuracy for gas diffusivity at low {varepsilon}.

The TPM tortuosity–connectivity parameter X, Eq. [10], and its relation to the Campbell (1974) pore-size distribution parameter, b, gives insight into the effect of soil type and pore-size distribution on the model predictions. Moldrup et al. (2004) reported X values within a relatively narrow range from 2 to 3 for 60 undisturbed soils; the X values decreased with increasing b (i.e., the X values were smaller for finer textured soils). Figure 6 shows the relation between X and b for all soil samples in our study. The function X = 2 + 3/b (corresponding to the exponent term in the BBC model, Eq. [6]) is also shown in the figure. Contrary to the study of Moldrup et al. (2004), X values were located in a relatively wide range from 1.7 to 3.6. The X values decreased with increasing b values, as expected.


Figure 6
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  		Fig. 6. Relation between the Campbell (1974) pore-size distribution parameter b and the tortuosity–connectivity parameter X for 96 soils. The X = 2 + 3/b curve is also drawn.

 
For the lysimeter soils, the X values ranged narrowly from 2.4 to 3.1 irrespective of soil texture (Table 1). For the field soils (mostly sandy clay loam), the X values ranged from 1.7 to 2.7, except for the sand layer (3.0 < X < 3.6) at 5.4- to 5.6-m depth at Mammen. This adds to the present understanding of gas diffusivity in soils with different textures and pore connectivities, since the X value characterizes the Dp/D0–{varepsilon} function. A large X value corresponds to a small slope in Dp/D0 at smaller {varepsilon}/{phi} values and a steep slope in Dp/D0 close to {varepsilon}/{phi} = 1. In contrast, a small X value corresponds to a gradual increase in Dp/D0 across the whole {varepsilon}/{phi} range. Taking a coarser textured soil with a large content of macropores (high {varepsilon}100), for example the sand layer at Mammen with 3.0 < X < 3.6, air-filled pores that largely contribute to gas diffusion have already been created when the soil is drained to the air-entry matric potential, leading to a steep increase in Dp/D0. On the other hand, for a finer textured soil, a high degree of air-filled pore connectivity does not occur immediately during the early stage of drainage. Air-filled pore connectivity will be established more gradually between isolated or remote air-filled pores that did not previously contribute to gas diffusion at higher water contents, leading to a more gradual increase in Dp/D0 with increasing {varepsilon}.

The TPM, Eq. [9] and [10], is mechanistically an analog to the BBC model, Eq. [6], with the exponents X and (2 + 3/b) representing pore connectivity or tortuosity. In Fig. 6, the (2 + 3/b) curve tends to overestimate X at both low (<5) and high (>20) b values. This may be one reason why the BBC performed worse than the TPM for certain data subsets (for example, all lysimeter soils and deep field soil layers), which include soil layers with low b values (loamy sand layers at Jyndevad for lysimeter soils, and sand layer at 5.4–5.6 depth at Mammen) (Table 2).

Depth Distribution of Gas Diffusivity
The TPM, Eq. [9] and [10], performed best for the soils in this study. Comparisons of TPM model prediction with measured Dp/D0 at three depths are shown in Fig. 7 for Gjorslev (field soil). The average Dp/D0 values at each {psi} (–20, –50, –100, and –200 cm H2O), and the standard deviations of both {varepsilon} and Dp/D0 are shown in the figure. The average of {varepsilon}100 and {phi} (for five samples at each depth) were used for model prediction by the TPM. For comparison, predictions by the MQ (1961) model, Eq. [5], are also shown in the figure. In agreement with previous model tests, the TPM predicted the measured data well, and the MQ (1961) model greatly underestimated the measured data.


Figure 7
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  		Fig. 7. Comparison of the three-porosity model (TPM) and the Millington and Quirk (1961) model (MQ) with measured data for field soils at Gjorslev at the (a) 0.43-m, (b) 2.15-m, and (c) 4.75-m depth. The average of soil-gas diffusivities (Dp/D0) at each matric-potential ({psi}) stage are shown in the plots. The standard deviations of both air-filled porosity {varepsilon} and Dp/D0 are shown in error bars. Average values of the Campbell pore-size distribution parameter b are also given.

 
The air-filled porosity at {psi} = –100 cm H2O ({varepsilon}100) is closely related to the field capacity soil water content {theta}fc (Madsen, 1976). Thus, the Dp,100/D0 appears significant not only as a reference-potential soil-gas diffusivity in the TPM but also as the soil-gas diffusivity close to natural field capacity water content. Figure 8 shows the depth distribution of Dp/D0 (average and standard deviation) for Gjorslev and Mammen (field soils) at {psi} = –100 cm H2O. The TPM predictions for Dp,100/D0 using average {varepsilon}100 and {phi} values are also shown in the figure. The TPM predicted well Dp,100/D0 values across the entire depth from shallow to deep layers, and described well the variation in Dp,100/D0 through the soil profiles with several morphologic horizons.


Figure 8
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  		Fig. 8. Depth distribution of soil-gas diffusivities Dp,100/D0 for Gjorslev and Mammen (field soils) at matric potential {psi} = –100 cm H2O. Average and standard deviation of Dp,100/D0 at each depth are shown. Predicted Dp/D0 by the three-porosity model (TPM) are given by dotted and solid lines.

 
Because the TPM requires only one measurement on the soil-water retention curve (to obtain {varepsilon}100), the model is easy to apply compared with the BBC and MPD that include the Campbell (1974) pore-size distribution parameter b. The tortuosity–connectivity parameter X in the TPM, as discussed, seems logically related to soil texture and pore connectivity. In this study, we tested predictive Dp({varepsilon}) models against soils with relatively high sand contents (Table 1). Thus, the applicability of predictive Dp({varepsilon}) models to finer textured (more clayey) soils and growing media like peat is an important.


   CONCLUSIONS
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 SOIL-GAS DIFFUSIVITY MODELS
 RESULTS AND MODEL TESTS
 CONCLUSIONS
REFERENCES
 
Soil-gas diffusivities for five soil profiles with several morphologic horizons were measured at different soil-water matric potentials, and five commonly used predictive gas diffusivity models were tested against the measured data. The TPM performed the best for both shallow and deep soil layers, and predicted the depth distributions of measured soil-gas diffusivities well. The TPM also requires less detailed information on the soil-water characteristic curve than other well-performing predictive models, and is therefore recommended for predicting soil-gas diffusivities in the vadose zone.


   ACKNOWLEDGMENTS
 
This work was 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. 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. 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
 SOIL-GAS DIFFUSIVITY MODELS
 RESULTS AND MODEL TESTS
 CONCLUSIONS
 REFERENCES
 




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A. Thorbjorn, P. Moldrup, H. Blendstrup, T. Komatsu, and D. E. Rolston
A Gas Diffusivity Model Based on Air-, Solid-, and Water-Phase Resistance in Variably Saturated Soil
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