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Soil Science Society of America Journal 65:613-623 (2001)
© 2001 Soil Science Society of America

DIVISION S-1 - SOIL PHYSICS

Tortuosity, Diffusivity, and Permeability in the Soil Liquid and Gaseous Phases

P. Moldrupa, T. Olesena, T. Komatsub, P. Schjønningc and D.E. Rolstond

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

Corresponding author (i5pm{at}civil.auc.dk)


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Tortuosity phenomena of pore space influence the transport of water, solutes, and gases in soil. This study presents three analyses linking tortuosity and transport in unsaturated soil. The first is a diffusion-based analysis of tortuosity in the soil water and soil air phases, related to soil surface area (SA) and pore-size distribution (PSD) (characterized by Campbell b and content of pores >30 µm). The analysis is based on recent models to predict the diffusion coefficients, Dp, of (i) a solute in soil, (ii) a gas in repacked soil, and (iii) a gas in undisturbed soil, each as a function of fluid-phase (soil water or soil air) content, {alpha}. For use in the analysis, the relation between SA and the threshold water content where solute diffusion ceases due to disconnected water films was measured for eight soils (5–46% clay). The tortuosity analysis supported by measured Dp({alpha}) data shows that SA governs and has a larger impact on liquid-phase tortuosity than PSD has on gaseous-phase tortuosity. At the same value of {alpha}, the tortuosity is typically larger in the soil water than in the soil air phase, and the difference becomes more pronounced with increasing SA and at low {alpha}. In the second analysis air permeability, ka, and gas diffusivity, DP,g, are linked in the Millington and Quirk fluid flow model to describe soil structure-forming potential and to establish a model platform to describe ka as a function of DP,g and {alpha}. Measurements on repacked, nonaggregated soil support the ka(DP,g;{alpha}) model platform, while measurements on repacked, aggregated soils and on undisturbed soils show that ka is greatly affected by soil aggregation and structure and DP,g is not. In the third analysis, a constitutive parameter model is applied to gas and solute diffusivities and air and water permeabilities in six soils along a soil texture gradient. This illustrates the different behavior of the four transport parameters with PSD and {alpha}. The liquid-phase transport parameters show a steeper decrease with {alpha} compared with the gaseous-phase parameters, in part due to the higher tortuosity in the liquid phase. Also, ka in undisturbed soil exhibited a less steep decrease with {alpha} compared with DP,g, probably due to preferential air flow in larger pores during convective transport. Any attempt to develop a unifying and PSD-dependent model for transport parameters in the soil liquid and gaseous phases will require careful distinction between repacked and undisturbed soils.

Abbreviations: BET, Brunauer–Emmett–Teller • PSD, pore-size distribution • SA, soil surface area • WLR, water-induced linear reduction


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
AN IMPORTANT GOAL of soil physics has been to understand and describe the tortuosity and connectivity of the soil fluid (water and air) phases. This is a prerequisite for describing and predicting concentration-driven (diffusive) and pressure-driven (convective–dispersive) transport of solutes and gases in variably saturated soils. Much emphasis has been put on describing the soil water permeability (soil hydraulic conductivity) as a function of soil water content. Much less emphasis has been on the soil air permeability as a function of soil air content, and on diffusive properties such as solute and gas diffusion coefficients and their variations with fluid-phase content (soil water or soil air content), probably because chemical mobility and leaching in soil historically have been assumed dominated by pressure-driven transport. However, the importance of diffusion as a controlling factor for chemical mobilization and transformations and the important interactions between diffusion-controlled and convection-controlled transport domains have been acknowledged for both liquid and gaseous-phase transport (e.g., van Genuchten and Wierenga, 1977; Brusseau, 1991). Thus, both convective (water and air permeabilities) and diffusive (gas and solute diffusion coefficients) transport parameters need to be considered to understand chemical transport in the soil fluid phases (the water and air phases).

The diffusion coefficient by definition provides basic information about the effective, tortuous pathway of the liquid or gas phase (Currie, 1960; Millington and Quirk, 1964; Epstein, 1989). Thus, new insight into solute and gas diffusivity will also probably provide valuable new insight and understanding of tortuosity in the liquid and gaseous phases and possible links to water and air permeability in variably saturated soils. Recently, a number of conceptually based, predictive models for the solute and gas diffusion coefficients in soils have been presented (Moldrup et al., 2000a, 2000b; Olesen et al., 2001). The models have been developed with careful distinction between sieved, repacked soil and undisturbed soil, and, in the case of gas diffusivity, also between dry soil and wetted soil. Parameters included in the models have been degree of phase saturation (water content or air-filled porosity), total soil porosity and PSD, the latter represented by the Campbell (1974) PSD parameter (b) and volumetric content of large pores (represented by the air-filled porosity at -100 cm H2O of soil matric head, equal to the volume of pores with an equivalent pore diameter >30 µm). The new, predictive diffusivity models together with measured diffusivity data for different soil types enable a closer look into the tortuosity of the liquid and gaseous phases of unsaturated soil.

This study presents three analyses concerning diffusive and convective transport parameters in the soil liquid and gaseous phases. The analyses are based on the classical definition of porous media tortuosity (Epstein, 1989) where the pores are assumed to be tortuous capillary tubes of uniform and similar diameter. In the first analysis, the predictive solute and gas diffusivity models together with measured data for differently textured soils are used to compare tortuosities in the soil liquid and gaseous phases. In the second analysis, gas diffusivity and air permeability (ka) are linked together in a classical fluid transport model (Millington and Quirk, 1964) to establish a conceptually based model to describe soil structure-forming potential and to predict ka. In the third analysis, the Campbell type constitutive parameter model (Campbell, 1974) is applied for all four transport parameters considered (gas and solute diffusivities and air and water permeabilities) to illustrate differences between the diffusive and convective transport parameters in the soil gaseous and liquid phases.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Most data used in this study are from our recent publications. Supplementary measurements on the same soils as used previously were carried out to allow a more comprehensive parameter analysis.

Soil specific SA was measured on eight differently textured soils (Lundgaard, Ødum, L1–L6; see Table 1) by the N2 Brunauer–Emmett–Teller (BET) method (e.g., Pennell et al., 1995) using a Shimadzu Automatic Surface Area Analyzer (Gemini 2375, Shimadzu Scientific, Columbia, MD). Standard deviations for triplicate measurements were generally below 10%. Soils L1 through L6 had been sampled at six locations along a naturally occurring texture gradient identified in an arable field near Lerbjerg, Denmark (Schjønning et al., 1999; Olesen et al., 1999), with clay content ranging from 11 to 46%. Soils L1 through L6 therefore have the same mineralogy and have received the same soil management, including the type and quality of mineral fertilizers and organic manure (Schjønning et al., 1999), allowing a more straightforward evaluation of the impact of soil texture on other soil parameters. The other two soils (Lundgaard, Ødum) are also from arable fields. For all soils, sampling included the 0- to 20-cm plough layer soil. The eight soils have similar contents of soil organic matter (Table 1), and thus the possible effects of differing organic matter contents on N2-BET SA measurements (Pennell et al., 1995) can be disregarded in this study. To obtain the volumetric soil surface area, SAvol (m2 cm-3), the measured specific SA (m2 g-1) was multiplied by the soil bulk densities used in the previous solute diffusion studies for the eight soils (Olesen et al., 1996, 1999).


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  		Table 1. Soil characteristics and specific and volumetric surface areas (SA)

 
Solute diffusivity (solute diffusion coefficient) as a function of soil water content was measured on the eight soils in Table 1 by Olesen et al. (1996)(1999). No significant difference between using sieved, repacked and undisturbed soils when measuring solute diffusivity has been observed (Olesen et al., 2001). Air permeability and gas diffusivity (gas diffusion coefficient) as a function of soil air content were measured on undisturbed L1 through L6 soil by Schjønning et al. (1999). Gas diffusivity as a function of soil air content were measured on sieved, repacked L1, L3, L5, and Lundgaard soils by Moldrup et al. (2000b). In this study, air permeability as a function of air-filled porosity was measured on L1, L3, L5, and Lundgaard soils on the same sieved, repacked soil samples as used by Moldrup et al. (2000b). Air permeability was measured at 20°C by the steady-state method of Grover (1955), where air at a constant pressure difference (here 4.5 cm H2O), displaced by a descending float chamber, flows through the soil column at a rate that is proportional to the air permeability (Moldrup et al., 1998). To evaluate the effect of soil aggregation on the gas transport parameters, two different packing procedures were used (Moldrup et al., 2000b). In the case of Lundgaard and L1, soil was thoroughly mixed with water to the desired water content and packed in 100-cm3 soil columns. Visually, the soils became more aggregated at the higher soil water contents. In the case of L3 and L5, to avoid formation of aggregates, soil was wetted to the desired water contents by adding water from the bottom of the core until the desired amount of water had infiltrated and the columns were allowed to equilibrate for 3 to 4 wk.

Solute and Gas Diffusivity Models
Solute Diffusivity
In solute diffusion studies, it is typically observed that the ratio of relative solute diffusion coefficient (DP,l/D0,l) by volumetric soil water ({theta}) increases linearly with {theta} (Porter et al., 1960; So and Nye, 1989; Olesen et al., 1996, 1999, 2000, 2001). This ratio

(1)
is called the (liquid phase) impedance factor in most solute diffusion studies. In Eq. [1] DP,l is the solute diffusion coefficient in soil (cm3 soil water cm-1 soil h-1) as defined by Fick's law of diffusion, and D0,l is the solute diffusion coefficient in water (cm2 h-1). It is noted that in most gas diffusion studies the same kind of ratio is typically called the pore continuity or the continuity index (Ball et al., 1988; Schjønning, 1989). Interestingly, the gaseous-phase impedance factor (or pore continuity) does not increase linearly in a similar way with soil air content (e.g., Schjønning, 1989). This already implies a basic difference in geometry and connectivity between the soil liquid and gaseous phases.

Figure 1 shows the liquid-phase impedance factor, Eq. [1], derived from measured solute diffusion data for soils L1, L3, and L5 (Olesen et al., 1999). As expected, fl decreases linearly with decreasing soil water content and approaches zero at a certain threshold soil water content, {theta}th > 0 (Fig. 1). This is likely because the water films surrounding the soil particles become disconnected at a certain soil water content, the value of which will depend on the soil type (soil SA). In more clayey soils (e.g., L5 in Fig. 1), the high SA will result in lower water film thicknesses (compared with a sandy soil at the same water content) and thus in a higher value of {theta}th. Also, the soil water close to the soil mineral surfaces will exhibit higher viscosity (Kemper et al., 1964; Stigter, 1980), and this will lower the solute diffusion coefficient. The viscosity effect on solute diffusion coefficient should, at the same soil water content, be most pronounced for soils with high SA. Thus, both phenomena (water film discontinuity and increased water viscosity) would suggest an increasing {theta}th with increasing SA, in agreement with Fig. 2 .



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  		Fig. 1. Liquid-phase impedance factor, fl, as a function of soil water content, {theta}, for a sandy (L1), a loamy (L3), and a clayey (L5) soil. The intercept with the {theta}-axis defines the threshold water content, {theta}th, where solute diffusion ceases. Data from Olesen et al. (1999)

 


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  		Fig. 2. Threshold soil water content for solute diffusion, {theta}th, as a function of volumetric soil surface area, SAvol. Data from Olesen et al. (1996)(1999) and present study

 
Figure 2 shows the measured relation between {theta}th and soil SA for the eight soils in Table 1. The value of {theta}th for each soil was found by linear interpolation of the fl({theta}) data as shown in Fig. 1. As the solute impedance factor and diffusivity are typically related to volumetric soil water content, it seems conceptually correct to use volumetric soil surface area, SAvol, instead of specific SA in the analysis. A highly significant (r2 = 0.98) but nonlinear relation between {theta}th and SAvol is observed (Fig. 2)

(2)

The likely reason for the nonlinear behavior is that both viscosity and water film discontinuity play a role, and that water is mostly located in water films in clayey soils, while at a relatively higher water content, water will tend to be located in water menisci in sandy soils (Campbell and Mulla, 1990; Petersen et al., 1996).

Olesen et al. (2001) found, based on data for 23 differently textured soils, that fl can be predicted by fl = 1.1({theta} - {theta}th). The term ({theta} - {theta}th) can be interpreted as the effective water content available for solute diffusion and 1.1 is a factor describing the meandering of the diffusive pathway. Inserting Eq. [2] and fl = 1.1({theta} - {theta}th) into Eq. [1] gives

(3)

Gas Diffusivity
In dry (void of water), sieved and repacked porous media Moldrup et al. (2000b) found that gas diffusivity was best described by the Marshall (1959) model

(4)
where DP,g is the gas diffusion coefficient in soil (cm3 soil air cm-1 soil h-1), D0,g is the gas diffusion coefficient in air (cm2 h-1), and {epsilon} is the air-filled porosity (volumetric soil air content). In completely dry soil, {epsilon} will equal the soil total porosity, {Phi}.

Adding a linear reduction term to Eq. [4] ({epsilon}/{Phi}, where {Phi} is soil total porosity) to account for water-induced changes in air-filled pore shape and configuration in a wet compared with a completely dry soil, the water-induced linear reduction (WLR) model

(5)
was found to accurately describe gas diffusivity in sieved and repacked soils at different soil water contents and total porosities (Moldrup et al., 2000b).

In the case of undisturbed soil, Moldrup et al. (2000a) found that both soil type and content of large pores apparently influenced gas diffusivity. At a soil water content corresponding to -100 cm H2O of soil water matric head, the following expression was found to describe well gas diffusivity for soils with different texture, from different soil horizons and representing different soil management

(6)
where DP,g100 is the gas diffusion coefficient at -100 cm H2O, and {epsilon}100 is the air-filled porosity at -100 cm H2O (corresponding to the volumetric content of soil pores with an equivalent pore diameter >30 µm). Figure 3 illustrates the performance of the DP,g100({epsilon}100) expression, Eq. [6], compared with the measured data for 144 undisturbed soils from Denmark, United Kingdom, and the Netherlands; see Moldrup et al. (2000a) for data references. The width of the 95% prediction interval is 0.09, and the coefficient of regression is r2 = 0.98. In Fig. 3, a mean value of DP,g100({epsilon}100) measurements on between three and six closely sampled (within 0.5 m2) undisturbed soil cores was used for each soil, since the local-scale variability in the gas diffusion coefficient typically was small. Figure 3 also illustrates the level of {epsilon}100 for most soils: 80% of the soils are within 0.1 < {epsilon}100 < 0.3 m3 m-3, while some very sandy soils from Denmark and Holland have {epsilon}100 values above 0.3 m3 m-3, and a few very clayey soils have {epsilon}100 values below 0.1 m3 m-3. Hence, the common range for {epsilon}100 is between 0.1 and 0.3 m3 m-3.



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  		Fig. 3. Predicted, Eq. [6], and measured gas diffusivity in undisturbed soil at -100 cm H2O of soil water matric head (DP,100/D0). Data for 144 soils. Fine broken lines represent 95% prediction interval. Coarse broken lines represent different values of content of large pores, {epsilon}100, in Eq. [6]

 
By combining Eq. [6] with the Campbell (1974)–Burdine (1953) permeability-water retention model, modified for gas diffusivity (Moldrup et al., 1996), a DP,g({epsilon}) model for undisturbed soils was derived (Moldrup et al., 2000a),

(7)
where b is the Campbell (1974) PSD index (corresponding to the slope of the soil water retention curve in a log-log coordinate system). Equation [7] accurately predicted gas diffusivity when tested against independent DP,g({epsilon}) data for 24 differently textured, undisturbed soils.

Diffusion-Based Tortuosity Analysis
Tortuosity (T) is here defined as the ratio of the average capillary tube length, Le, to the length of the porous media (soil sample), L, along the major flow (diffusion) axis, in a tortuous (sinuous) capillary tube of uniform diameter

(8)

This definition follows Carman (1937)(1956), Currie (1960), Scheidegger (1974), Ball (1981), and Epstein (1989). As discussed by Epstein (1989), several definitions of tortuosity and tortuosity factor have been used in the literature, which contributes to a general confusion concerning use of the term tortuosity. Assuming diffusion in a porous medium with pores consisting of tortuous and nonconstricted tubes with uniform and similar diameter, the relation between diffusivity and Le/L becomes (Currie, 1960; Ball, 1981; Epstein, 1989)

(9)
where {alpha} is the volumetric fluid-phase content (equals {theta} in the case of solute diffusion and {epsilon} in the case of gas diffusion). Combining Eq. [8] and [9] gives

(10)

Combining Eq. [10] with the presented Dp/D0({alpha}) models for soil liquid and gaseous phases, Eq. [3] through [7], enables a diffusion-based analysis of tortuosity. The tortuosity can also be considered to be the squareroot of (1/f), where f is the impedance factor (Eq. [1]).

Figure 4a shows T in the soil liquid phase as a function of soil water content and soil SA, based on the solute diffusivity model, Eq. [3]. As no significant difference in solute diffusion coefficients between sieved, repacked, and undisturbed soil has been observed (Olesen et al., 2000), the calculated tortuosity curves in Fig. 4a are assumed representative for both sieved and undisturbed soil. A large effect of both water content and soil type (SA) is seen for tortuosity in the soil liquid phase (Fig. 4a).



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  		Fig. 4. Influence of fluid-phase (water or air) content, soil volumetric surface area (SAvol), and content of large pores ({epsilon}100) on tortuosities in (a) the soil liquid phase and (b) the soil gaseous phase. Solid lines are calculated from Eq. [10] combined with Eq. [3] through [7], assuming {Phi} = 0.5. Open triangles are data from literature for gas diffusion in completely dry soil

 
Figure 4b shows T in the soil gaseous phase as a function of air content, based on the gas diffusivity models (Eq. [4], [5], and [7]) and with careful distinction between (i) dry, sieved and repacked soil, (ii) wet, sieved and repacked soil, and (iii) wet, undisturbed soil. The low gaseous-phase tortuosity in completely dry soil and the pronounced effects of added water (creating water menisci and thus discontinuous diffusion pathways) to increase tortuosity in wet soil, especially at low {epsilon} values (high water contents), are evident from Fig. 4b. Data from literature (listed in Moldrup et al., 2000b) for gas diffusivity in completely dry soil is marked with open triangles in Fig. 4b. Data support the model and show that the tortuosity in the gas phase in completely dry soil is close to one. For wet soil, the tortuosity in undisturbed soil is larger than in sieved soil, probably due to local-scale zones with higher-than-average bulk density and/or water content that hinders the diffusive gas flux.

Varying the value of {epsilon}100 (content of large pores) within the typical interval (0.1–0.3 cm3 cm-3; see Fig. 3) and with b = 6 (loam soil) gives minor effects on tortuosities. Varying both {epsilon}100 and the Campbell PSD index (b) gave similar effects (not shown). The analysis implies that PSD (b, {epsilon}100) has limited effects on gaseous-phase tortuosity; that is, the effects are minor compared with the water-induced effects on gaseous-phase tortuosity in a wet soil compared with a completely dry soil at the same soil air content (Fig. 4b), and also minor compared with the huge effect of soil type (SA) on liquid-phase tortuosity (Fig. 4a).

Measured solute and gas diffusion coefficients support the tortuosity analysis. Figure 5a shows tortuosity, T, as a function of fluid-phase content, {alpha}, in four differently textured sieved and repacked soils (Lundgaard sand, L1 loamy sand, L3 sandy clay loam, L5 sandy clay), obtained by inserting measured values of solute and gas diffusivities in Eq. [10]. No soil type effect on gaseous-phase tortuosity was seen, whereas there was a pronounced effect of soil type on liquid-phase tortuosity, in agreement with Fig. 4. Interestingly, the liquid- and gaseous-phase tortuosities became very similar for the more sandy soils (Lundgaard and L1). This implies that the effective diffusive pathways in coarser textured soils with low SA are quite similar in the liquid and gaseous phases. In contrast, a pronounced difference occured for more finely textured soils where the large soil SA will create a thin and tortuous liquid phase, while there still is a low tortuosity in the gaseous phase. Thus, the differences between the tortuosities in the liquid and gaseous phases of sieved, repacked soil becomes larger with increasing soil SA, especially at lower fluid phase contents (Fig. 4a, 4b, and 5).



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  		Fig. 5. Measured tortuosities as a function of fluid-phase content in the soil liquid phase (from solute diffusion data, {alpha} = {theta}) and soil gaseous phase (from gas diffusion data, {alpha} = {epsilon}) for (a) four sieved, repacked soils and (b) undisturbed compared with sieved, repacked L1 soil. Data from Olesen et al. (1996)(1999), Schjønning et al. (1999), and Moldrup et al. (2000b)

 
Figure 5b shows gaseous-phase tortuosities in sieved L1 soil and in undisturbed L1 soil, as compared with liquid-phase tortuosities in L1 soil. The gaseous-phase tortuosities are larger in undisturbed than in sieved L1 soil, in agreement with Fig. 4b. This was also observed for L3 and L5 soils (not shown). For L1 soil, the gaseous-phase tortuosities in undisturbed soil are larger than the liquid-phase tortuosities (Fig. 5b), again likely due to local zones with higher-than-average bulk density and/or water content that reduce the diffusive gas flux. For undisturbed L3 and L5 soils, the tortuosities in the gaseous phase were similar to or lower than in the liquid phase (not shown).

Another interesting question is at what soil water content the liquid- and gaseous-phase tortuosities become similar? Figure 6 shows the relative soil water content ({theta}/{Phi}) where the models predict equal values of T in the soil liquid and gaseous phases of sieved, repacked soil. In this case, the gas diffusivity model, Eq. [5], was written with respect to soil water content using {epsilon} = {Phi} - {theta}. Also shown in Fig. 6 are the actual values for soil L1 (sandy), soil L3 (loamy), and soil L5 (clayey) obtained from the measured DP({alpha}) relations for solute and gas diffusivity in repacked soil. Good agreement between data-derived and model-predicted values for the point of equal tortuosity ({theta}/{Phi} value at equal tortuosity) in the liquid and gaseous phases is seen. The point of equal tortuosity will take place around half saturation ({theta}/{Phi} = 0.5) for the sandy L1 soil, whereas the point of equal tortuosity will be around {theta}/{Phi} = 0.7 for the clayey L5 soil and thereby takes place in a much wetter soil. This is because a higher soil water content is needed in the clayey L5 soil to counterbalance the massive effect of the high SA on liquid-phase tortuosity (Fig. 4a). Hence, the higher water content will decrease the high liquid-phase tortuosity and at the same time increase the gaseous-phase tortuosity (due to a reduced soil air content), causing the tortuosities in the liquid and gaseous phases to approach each other.



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  		Fig. 6. Relative soil water content where the soil liquid- and gaseous-phase tortuosities become the same. Solute and gas diffusion data from Olesen et al. (1999) and Moldrup et al. (2000b) used to obtain the observed values (symbols) for L1, L3, and L5 soils. Model prediction (solid line) from Eq. [10] combined with Eq. [3] and [5]

 
Air Permeability and Gas Diffusivity
The Link to Soil Structure
More than 40 yr ago, Kirkham et al. (1958) suggested that gas permeability (ka) measurements on soils at natural field capacity (around -100 cm H2O of soil water matric head) can provide information about changes and differences in soil structure. Later studies including Ball et al. (1988), Schjønning (1989), Blackwell et al. (1990), Moldrup et al. (1999), Schjønning et al. (1999), Iversen et al. (2001), and Poulsen et al. (2001) have looked at soil structure based on gas permeability and/or gas diffusivity measurements. Tortuosity expresses a structural condition of soil and may therefore in itself be taken as an index of structure and may be obtained from diffusivity studies (Eq. [10]). Compared with studies of water flow, however, surprisingly little effort has been devoted to air permeability studies considering that ka is easy and rapid to measure compared with water permeability (Iversen et al., 2001; Poulsen et al., 2001). Kirkham et al. (1958) suggested ka itself, Groenevelt et al. (1984) and later Blackwell et al. (1990) used the so-called pore continuity (ka/{epsilon}, where {epsilon} is soil air content), and Ball (1981) used the equivalent pore diameter

(11)
as a soil structure characterizing parameter. Equation [11] was, to the authors knowledge, first derived by Millington and Quirk (1964), based on the same definition of the porous medium as used in the tortuosity analyses (pores being uniform, tortuous, and nonjointed tubes of similar diameter) and by combining Ficks' law for diffusive transport with Poiseuille's law for convective fluid transport. Ball (1981) independently derived the same model and extended it to consider a porous medium with pores being jointed tubes of different diameter.

Following Kirkham et al. (1958) and Moldrup et al. (1999), the equivalent pore diameter at -100 cm H2O of soil water matric head, d100, is suggested as a soil structure index. The value is obtained by using the measured air permeability and gas diffusivity at -100 cm H2O (ka100 and Dp,g100) in Eq. [11].

Figure 7 shows the values of d100 for six soils (L1–L6) sampled along a natural clay gradient, calculated from Eq. [11] using measured air permeabilities and gas diffusivities at -100 cm H2O. Three cases are considered, namely (i) sieved, repacked and nonaggregated soil (L3 and L5, measured in this study), (ii) sieved and repacked soil that have been standing in lysimeters for 17 mo exposed to freeze–thaw and dry–wet cycles as well as frequent tillage, and subsequently undisturbed soil samples were retrieved from the lysimeters (called structurally disturbed soil in Fig. 7; data from Schjønning et al., 1999), and (iii) undisturbed soil samples from the six field locations (data from Schjønning et al., 1999). The sieved, repacked and nonaggregated soil have a d100 value around 50 µm (49 for L3 and 51 for L5). The structurally disturbed soil has developed some structure during the 17 mo (new structure), probably in the form of aggregation due to biotic as well as abiotic mechanisms, and has d100 values between 100 and 250 µm. The d100 value is increasing with clay content, probably because larger clay content promotes formation of soil aggregates. For the undisturbed soil samples, d100 is even larger (250–500 µm) as more permanent and more continuous macropores will add to the overall soil structure (old structure). Only in the case of L6, d100 values were not significantly different (overlapping values of standard deviations for six closely sampled soil core, not shown) between structurally disturbed and undisturbed soil. The large difference between the three situations (repacked, structurally disturbed, and undisturbed) with respect to soil structure and air flow behavior is obvious from Fig. 7 (note logarithmic d100 axis).



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  		Fig. 7. Equivalent pore diameter at -100 cm H2O of soil water matric head (d100) for six soils (L1–L6) sampled along a soil texture gradient. Pore diameter was calculated from measured gas diffusivities and air permeabilities, Eq. [11]. The structurally disturbed soil is repacked soil that has been allowed to develop soil structure for 17 mo. Data from Schjønning et al. (1999) and present study

 
Figure 8 shows measured Dp,g/D0 and ka as functions of soil air content for the Lundgaard and L1 soils. The packing and wetting procedure visually caused increased aggregation (pellets of cohering soil particles) at higher soil water contents (lower soil air contents) for these two soils (see Materials and Methods section). Interestingly, gas diffusivity appears unaffected by soil aggregation, as also suggested by Flegg (1953), and by the fact that the measured Dp,g/D0({epsilon}) values are well predicted by the WLR gas diffusivity model for sieved, repacked soil (Eq. [5]). Air permeability acts in an opposite manner to gas diffusivity and shows an increase in ka with decreasing soil air content, with the highest values of ka obtained at the lowest soil air contents (Fig. 8). At first sight, this appears illogical; however, the explanation is that at low soil air contents, the soil water contents were high and the mixing of the soil caused soil aggregation. This creates more continuous gas-phase pathways, causing pronounced higher air permeability as the convective gas flow is more likely to take place in the larger, more continuous pores (preferential gas flow effects). In accordance with this, the equivalent pore diameters (d) are increasing with decreasing soil air content for both soils, from {approx}40 µm at the highest {epsilon} values to d > 100 µm at the lowest {epsilon} values. Figure 8 proves that soil aggregation has little effect on gas diffusivity but dramatic effects on air permeability.



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  		Fig. 8. Measured air permeabilities and gas diffusivities in two soils where the packing procedure has resulted in the formation of soil aggregates at higher soil water contents (lower air contents). Solid line is model prediction for gas diffusivity in repacked soil, Eq. [5]. Data from present study and Moldrup et al. (2000b)

 
For comparison, the similar measurements for soils L3 and L5 (sieved, but nonaggregated due to another wetting procedure) showed little or no change in d with soil air content. For gas permeability in sieved, nonaggregated soil, a conceptually based model platform for describing ka({epsilon}) may therefore be derived from the Millington and Quirk (1964) fluid flow model. Combining Eq. [11] and [5] and isolating ka yields

(12)

Figure 9c shows that Eq. [12] with d taken as a constant (50 µm = d100) describes the measured air permeabilities in the sieved, repacked and nonaggregated L3 and L5 soils reasonably well. In accordance with this, the WLR gas diffusivity model (an inherent part of Eq. [12]) describes well the measured gas diffusivity data for sieved L3 and L5 soil (Fig. 9a).



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  		Fig. 9. Measured gas diffusivities and air permeabilities in L3 sandy clay loam and L5 sandy clay. (a), (c) Sieved, repacked soil. Model predictions (solid lines) by Eq. [5] and [12]. (b), (d) Undisturbed soil (closed symbols) and structurally disturbed soil (open symbols). Note the different ka axes on (c) and (d). Data from present study and Schjønning et al. (1999)

 
Figure 9b shows gas diffusivities measured on undisturbed L3 and L5 soil samples. The gas diffusivities in undisturbed soil are generally equal to or lower than those for sieved, repacked soil (in agreement with the tortuosity analysis of Fig. 4b), and show a minor soil type effect, in agreement with the Campbell b dependent gas diffusivity model for undisturbed soil (Eq. [7]). Gas permeabilities for the undisturbed soils are in this case one order of magnitude larger than for the sieved soil (compare Fig. 9c and 9d) and show a more pronounced soil type effect. For structurally disturbed soil (measurements at only two soil air contents for each soil), both gas diffusivites and, especially, air permeabilities are lower than in undisturbed soil. This is because a natural soil structure is not yet fully developed after 17 mo, also evident from Fig. 7.

The difference between transport parameters for sieved and undisturbed soil seems related to the magnitude of transport velocity. For the slowest process, solute diffusion, Olesen et al. (2000) found no difference between sieved and undisturbed soil. Some difference is evident for gas diffusion (compare Fig. 9a and 9b) and a pronounced difference is evident for convective gas transport (Fig. 9c and 9d). Figure 9 represents, to our knowledge, the first direct comparison of gas diffusivity and permeability for both undisturbed and sieved, repacked (nonaggregated) soil.

It is not immediately feasible to combine Eq. [11] with gas diffusivity models for undisturbed soil, for example Eq. [7], to derive ka({epsilon}) models for undisturbed soil. Besides the fact that d is very sensitive to small changes in soil structure including soil aggregation (Fig. 8 and 9), the rate of decrease in ka with {epsilon} in the case of undisturbed soil is not necessarily well described by the same term as used in the gas diffusivity model, 2 + 3/b (see Eq. [7]), because of preferential flow effects during convective gas transport. This can be shown by a constitutive function analysis for each transport parameter.

Constitutive Function Analysis of Transport Parameters
This analysis assumes the general validity of the Campbell (1974) constitutive parameter model

(13)
where p is the transport parameter (solute diffusivity, gas diffusivity, air permeability, or water permeability) and {alpha} as before is the fluid-phase content (soil water content, {theta}, for solute diffusivity and water permeability, and soil air content, {epsilon}, for gas diffusivity and air permeability). In a log(p/p*)–log({alpha}/{alpha}*) coordinate system, Eq. [13] will yield a straight line with slope {eta}. The dataset (p*, {alpha}*) represents the parameter value, p*, at a chosen reference value of fluid-phase content, {alpha}*. In the present analysis {alpha}* is taken as the highest fluid-phase content where a parameter measurement was available.

Figure 10a shows p/p*({alpha}/{alpha}*) for gas diffusivity and air permeability in undisturbed L1 soil. Two things are evident. First, the Campbell constitutive function describes well the two gas transport parameters within the soil air content interval where measurements were available. This was generally the case for gas diffusivity and air permeability as well as for solute diffusivity in the Lerbjerg soils (L1–L6) with the coefficient of regression (r2) >0.97 in all cases. Second, the air permeability shows a smaller decrease with {alpha} (smaller value of {eta}) than does gas diffusivity, probably due to preferential air transport in the larger pores during convective air flow.



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  		Fig. 10. Parameter analysis based on the Campbell-type model, Eq. [13]. (a) Model fit of air permeability and gas diffusivity relations for soil L1. (b) Slope of Campbell-type model, {eta}, as a function of Campbell pore-size distribution index, b. The {eta} interval for water permeability is calculated by the Alexander and Skaggs (1986) and Campbell (1974) expressions ({eta} = b + 3 and 2b + 3). The {eta} values for the remaining parameters are based on data from Olesen et al. (1999) and Schjønning et al. (1999) for soils L1 through L6

 
Figure 10b shows the Campbell constitutive function slope, {eta}, as a function of the Campbell PSD index, b, for air permeability (undisturbed soil), gas diffusivity (undisturbed soil) and solute diffusivity for the L1 through L6 soils. The value of {eta} for air permeability increases with b, in agreement with the model proposed by Moldrup et al. (1998). Based on data for mainly sandy and loamy soils, they suggested {eta} = 1 + 0.25b, but from the data on Fig. 10b, {eta} = 1 + 0.05b would better describe air transport in the six Lerbjerg soils. Hence, the model {eta} = 1 + 0.25b would predict too steep a slope of the air permeability relations, especially for the finely textured L4 through L6 soils. In agreement with this, Moldrup et al. (1998) noted that {eta} = 1 + 0.25b could not provide a satisfying description for across soil textural groups (see Table 1 of Moldrup et al., 1998). Both studies therefore imply that the {eta}(b) description for ka needs further improvement.

The value of {eta} is decreasing with b for gas diffusivity and, for the interval two to three, is in agreement with the {eta} model by Moldrup et al. (1996), {eta} = 2 + 3/b; the latter was also used in this study (Eq. [7]). For the more finely textured soils (high b values), the effect of local zones of varying bulk density or water content will probably be more pronounced, yielding a lower {eta} value. The different behavior of gas diffusivity compared with air permeability is likely because air permeability is greatly affected by soil structure while gas diffusivity is not (Fig. 8 and 9). For all six soils, {eta} for air permeability is smaller than for gas diffusivity, and the difference is increasing with decreasing b (more coarsely textured soils). For sieved, repacked soil (L3 and L5), however, the {eta} values for air permeability and gas diffusivity were similar and close to 2.5 (not shown in Fig. 10), in agreement with the model platform for nonaggregated soil, Eq. [12]. These observations also support the hypothesis of preferential flow in the larger pores (structural pores) in undisturbed soil during convective air transport.

For solute diffusivity, {eta} is increasing with b, in agreement with the model by Olesen et al. (1996). Rearranging the model by Olesen et al. (1996) to obtain a {eta} expression yields {eta} = 1 + 0.3b. This is in reasonable agreement with the data for the six Lerbjerg soils, especially for the more finely textured L4 through L6. The {eta} values for solute diffusivity is generally higher than for the two gas transport parameters (gas diffusivity and air permeability; Fig. 10b), again implying a much larger tortuosity in the soil liquid than in the soil gaseous phase. The difference is especially pronounced for the finely textured L4 through L6 soils with high SAs.

No measurements for water permeability were available for the L1 through L6 soils. However, Poulsen et al. (1998)(1999a) showed that {eta} for air permeability is typically well above the Alexander and Skaggs (1986) model, {eta} = b + 3, and is typically well predicted by the original Campbell (1974) model, {eta} = 2b + 3. These two {eta} models have been plotted in Fig. 10b to indicate a likely range of water permeability. It is seen that {eta} for water permeability is at a completely different and higher level than the three remaining transport parameters. This is caused by the effect of water retention (water adsorption) to the soil particles, creating a much steeper decrease in water permeability with {alpha} than for the three other parameters. Comparing the {eta} functions for solute diffusivity and water permeability is interesting in that solute diffusivity {eta} reveals the basic effects of the soil liquid-phase tortuosity, while the difference between solute diffusivity {eta} and water permeability {eta} reveals the basic effect of soil water adsorption on the water permeability function. Thus, by measuring both solute diffusivity and water permeability as a function of soil water content, it should be possible to distinguish between the effects of liquid-phase tortuosity and water retention on water permeability.


   RESULTS AND DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
To put the present work into perspective, Fig. 11 shows possible links and key soil physical parameters when trying to tie the diffusive and convective transport parameters in the soil liquid and gaseous phases into a unifying model concept. A complete literature review on the transport parameters and links between them is not attempted. Instead, the new knowledge obtained in this and other recent studies is combined, pointing towards new research possibilities. Five potential links are shown in Fig. 11. A discussion of each of the links follows:

  1. Solute and gas diffusivities can be used to describe and understand liquid- and gaseous-phase tortuosities in soil, as shown in the first analysis of this study. The tortuosities in the two phases behave quite differently, and typically the tortuosity at a given degree of phase saturation is larger in the liquid than in the gaseous phase, the exception possibly being for undisturbed, coarser textured soils (Fig. 5b). The liquid-phase tortuosity strongly depends on soil type (soil SA), while only a limited soil-type dependency is seen for gas diffusivity and only in the case of undisturbed soil. Gas diffusivity is different in dry, repacked soil as compared with wet, repacked soil and with undisturbed soil, while no difference between repacked and undisturbed soil has been observed for solute diffusivity. In spite of this, the same diffusivity–tortuosity models have traditionally been used for both solute and gas diffusivity in chemical transport and fate models. No distinction has been made between undisturbed soil (field studies) and sieved, repacked soil (most laboratory studies). Also, the most commonly used and soil type independent model (Millington and Quirk (1961)) was originally derived for permeability, not for diffusivity. This study suggests that different approaches are needed for diffusivity in the liquid and gaseous phases and in disturbed vs. undisturbed soil to improve the present chemical transport and fate models with respect to chemical diffusion. Such models are now available (Eq. [4]–[7]).
  2. Gas diffusivity and air permeability in combination can be used to describe soil structure, as shown in the second analysis of this study. Air permeability is strongly affected by soil structure, including soil aggregation, while gas diffusivity is not (see Fig. 8 and 9). The reason for this is that diffusion and convection will probably not see the same pore structure as convective air transport will take place preferentially through the large-pore network, especially in well-structured undisturbed soils. The equivalent pore diameter (as defined by gas diffusivity and air permeability in Eq. [11]) is a useful parameter for describing soil structure, and may even be used as a preliminary model platform for describing air permeability as a function of soil air content (Eq. [12]). However, at present no realistic and accurate models for air permeability in undisturbed soil, taking into account the large effect of soil type and soil structure, exist. Moldrup et al. (1998) suggested that the best approach presently available is to measure air permeability for at least one (fairly high) value of soil air content and use this as a reference point (matching point) value in Eq. [13]. This in combination with using {eta} = 2 in Eq. [13] gave a good prediction for most soils (Moldrup et al., 1998). Air permeability becomes an increasingly important parameter in environmental soil studies, for example when modeling and designing soil vapor extraction–soil venting systems for cleanup of contaminated soils (Poulsen et al., 1999b). Development of predictive air permeability models for undisturbed soil is an important scope of future research.
  3. Linking air and water permeability is not immediately feasible due to the different geometries and tortuosities of the gaseous and liquid phases. Studies based on the Brooks and Corey (1966)–Burdine (1953) and the van Genuchten (1980)–Mualem (1976) permeability–water retention models in combination with measured data for sieved, repacked soils have shown somewhat promising results for simultaneous predictions of gas and water permeabilities, but the soil structure effects in undisturbed soil, obvious from Fig. 10, cannot at present be handled by such models. Pore-scale network models may therefore be a more promising approach in linking water and air permeabilities in undisturbed soil systems (Fisher and Celia, 1999). Poulsen et al. (1998)(1999a) found that different Campbell-type models were needed to describe water permeability in repacked and in undisturbed, relatively wet soils, emphasizing the importance of distinguishing between sieved and undisturbed soil. Loll et al. (1999) found a high correlation between air permeability at -100 cm H2O of soil water matric head and saturated water permeability in undisturbed soils, probably because the fluid flow in both cases is governed by transport in the larger soil pores. This may form the basis for improved models to simultaneously predict water and air permeability in undisturbed soil, based on macroporosity (here defined as {epsilon}100) and other water retention parameters, that is, from the Campbell, multiparameter Campbell, Brooks–Corey, or van Genuchten–Mualem type water retention models. Additionally, for highly structured soils with cracks, worm holes, or root channels, two- or multiregion flow models are needed to predict water and air permeability (Wilson et al., 1992; Chen et al., 1993; Durner, 1994).
  4. The link between solute diffusivity and water permeability has to our knowledge not been explored before. As discussed (Fig. 10), the difference between solute diffusivity and water permeability may be used to separate the effects of water retention (water adsorption) and liquid-phase tortuosity on water permeability. Both water retention and tortuosity will be highly dependent on SA (Petersen et al., 1996; Or and Tuller, 1999; present study), and the soil SA will govern how much water is adsorbed and how much is capillary bound (Or and Tuller, 1999), which will control the flow geometry of the mobile liquid phase. The threshold water contents for solute diffusion, {theta}th, for the eight soils in Table 1 all correspond to a soil water matric head less than -3000 cm H2O. The cause of {theta}th for solute diffusivity, discontinuous water films and increased water viscosity close to soil particle surfaces, will logically affect water permeability in an analogous way. Hence, it is likely that {theta}th for solute diffusion will be related to the residual water content, {theta}r, where water permeability ceases, as defined in the Brooks and Corey (1966) and van Genuchten (1980) models. Thus, it seems promising to develop expressions for {theta}r based on soil SA.
  5. An exciting possibility for cross-linking the parameters in Fig. 11 was explored by Jacobsen et al. (1999). By noting that gas diffusivity at -100 cm H2O and saturated water permeability in undisturbed soil were highly correlated, and both were highly correlated with {epsilon}100, new types of predictive models for unsaturated hydraulic conductivity were developed. The macroporosity, {epsilon}100, together with PSD and soil SA may therefore provide a good platform for describing gas diffusivity, air permeability, and water permeability in unsaturated, undisturbed soil.



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  		Fig. 11. Parameters, links, and differences to be considered towards a unifying concept of diffusive and convective transport parameters in the soil liquid and gaseous phases

 
New links between transport parameters have been suggested in Fig. 11. In this respect it is important to distinguish between (air or water) saturated and unsaturated soil, between sieved, repacked and undisturbed soil, and also between measurements on different soil sample sizes (e.g., Iversen et al., 2001). The scale issue is outside the scope of this study where all measurements are from soil samples of similar size, but the scale issue is essential in obtaining accurate transport models and is likely to be much more pronounced for convective than for diffusive transport.


   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
New and interesting insight into the liquid and gaseous phases of unsaturated soils has been obtained. At a given value of soil air content, the tortuosity in the gaseous phase of wet soil is larger than in a completely dry soil and, also, is typically larger in undisturbed soil compared with sieved, repacked soil. At a given value of fluid-phase (water or air) content, the liquid-phase tortuosity will typically be equal to or larger than the gaseous-phase tortuosity, the likely exception being coarser textured undisturbed soils.

Liquid-phase tortuosity is strongly soil type dependent and related to soil SA and liquid-phase geometry, Gaseous-phase tortuosity is less soil type dependent and related to gaseous-phase connectivity (connectivity of air-filled pores). This has important implications towards understanding and modeling diffusive and convective transport in the gaseous and liquid phases of unsaturated soil.

The observed differences between transport parameters in undisturbed compared with repacked soil greatly depend on the velocity of the transport process. No significant difference between solute diffusivity in repacked and undisturbed soil has been observed. Gas diffusivity in repacked and undisturbed soil only differ a little, and it is typically lower in undisturbed soil, probably because of local-scale zones with higher bulk density or water content that hinders the diffusive gas flux. Air permeability in repacked and undisturbed soil is very different and is typically much higher in undisturbed soil, probably because of preferential air flow during convective transport.

Along a natural soil texture gradient, the Campbell constitutive function parameter ({eta}) increased slightly with Campbell PSD index (b) for air permeability, decreased slightly with b for gas diffusivity, increased with b for solute diffusivity, and, in comparison, {eta} was estimated to strongly increase with b for water permeability. The analysis is in agreement with previous models developed for each of the four transport parameters and may suggest the possibility for developing a unifying and soil type dependent model concept for diffusive and convective transport parameters in the soil liquid and gaseous phases. Key soil physical parameters for this include pore size distribution, soil volumetric SA, and soil structure parameters (including {epsilon}100 and d100). The observations in this study emphasize that careful distinction between repacked and undisturbed soil systems is required for such model development.


   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", 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 research grant from the Japanese Ministry of Education, Science, Sports and Culture (Monbushu International Scientific Research Program: Joint Research no. 12555156).

Received for publication July 6, 2000.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES