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Soil Science Society of America Journal 64:527-532 (2000)
© 2000 Soil Science Society of America

DIVISION S-1-SOIL PHYSICS

Modeling the Gas Diffusion Coefficient in Analogy to Electrical Conductivity Using a Capillary Model

Albrecht H. Weertsa, Jan I. Freijerb and Willem Boutena

a Dep. Physical Geography and Soil Sci., Univ. of Amsterdam, Nieuwe Prinsengracht 130, 1018 VZ, Amsterdam, The Netherlands
b National Inst. of Public Health and the Environment, P.O. Box 1, 3720 BA, Bilthoven, The Netherlands

a.h.weerts{at}frw.uva.nl


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 Theory
 Materials and methods
 Results and discussion
 Conclusions
 Appendix
 REFERENCES
 
A conceptual model that accounts for the influence of pore geometry is presented to model gas diffusion coefficients in soils as a function of water content. To model the diffusion coefficient of the gas phase, we rewrote the Mualem and Friedman bulk electrical conductivity model. The model was calibrated using measured soil water retention curves and gas diffusion coefficients of seven mineral soil horizons. The model with only one free parameter fitted the data of five horizons. The model failed to describe the data of two clayey surface horizons. Estimated gas tortuosity parameters were tested on measured hydraulic conductivity data of three of the soils studied. The use of the gas tortuosity parameter led to overestimation of the unsaturated hydraulic conductivity at low water contents. This systematic deviation suggests that the gas (non-wetting) tortuosity parameter is not equal to the hydraulic (wetting) tortuosity parameter, although a relationship is likely.


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
Theory
 Materials and methods
 Results and discussion
 Conclusions
 Appendix
 REFERENCES
 
TRANSPORT OF GASES AND VOLATILE LIQUIDS in soils is an important topic in environmental studies. In natural soils systems, gas transport is thought to be dominated by diffusion through the gas phase (Xu et al., 1992), although convection can be of potential importance (Celia and Binning, 1992). Gas diffusion modeling in unsaturated soils requires knowledge of the effective gas diffusion coefficient as a function of water content. The soil acts as a barrier to free gas diffusion caused by the presence of liquid and solid obstacles. Therefore, the diffusion coefficient, Dp, of a gas through soil is less than that through free air, D0. The ratio Dp/D0 is called the relative diffusion coefficient. Symbol definitions and units are listed in the appendix.

Much effort has gone into finding a simple and unique relationship among Dp/D0, the volumetric air content, {theta}a, soil porosity {phi}, and other variables (Collin and Rasmuson, 1988; Jin and Jury, 1996; Moldrup et al., 1996; Troeh et al., 1982). Another approach is to directly take into account the complex geometry of the pore space using a conceptual model (Freijer, 1994; Steele and Nieber, 1994a; Steele and Nieber, 1994b). Mualem and Friedman (1991) proposed a pore space model to predict electrical conductivity in unsaturated soils, based on the idea that the geometry factor that affects hydraulic conductivity and the geometry factor that affects electrical conductivity are equal. Likewise, a model can be proposed to predict the gas diffusion coefficient in unsaturated soil, based on the idea that the geometry factor that affects gas permeability and the geometry factor that affects gas diffusion are equal.

The objective of this study is to test the Mualem and Friedman analogy to model the relative gas diffusion coefficients of seven mineral soil horizons. In addition, we compare the geometry factor for the gas and water phase for three of the soils studied.


   Theory
TOP
 ABSTRACT
 INTRODUCTION
 Theory
Materials and methods
 Results and discussion
 Conclusions
 Appendix
 REFERENCES
 
Gas diffusion is affected by three major factors, the first being the molecular diffusion coefficient of the gas in air, D0. The second factor is the continuous volumetric air content, {theta}a,c, which is equivalent to {theta}a corrected for a dead-end pore volume, {theta}de, which does not contribute to diffusion. This dead-end volume is defined as the porosity minus the natural saturation, {theta}s:

(1)

The continuous air content can now be defined as

(2)
where {theta}w is the volumetric water content. The third factor accounts for the fact that the mobility of gas molecules is influenced by the geometry of the air-filled pathways, Fg({theta}a,c), which is a function of the continuous air content. Now, Dp can be estimated with

(3)

The geometry factor Fg({theta}a,c) cannot be determined directly because there is no way to measure the geometry of the air-filled pathways. Mualem and Friedman (1991) proposed a similar model for electrical conductivity in soils, based on the hypothesis that the geometry factor affecting the bulk soil electrical conductivity is identical to the geometry factor defined for predicting the soil hydraulic conductivity. Their main assumption is that the ratio between the unsaturated hydraulic conductivity of the soil, Ksoil({theta}w), and the hydraulic conductivity of a bundle of straight capillaries, Kcap({theta}w), represents the geometry factor, Fg({theta}w), in hydraulic conductivity due to the complex geometry of the water-filled pathways:

(4)
in which {theta}w is the volumetric water content, {psi} is the pressure head, Sw is the relative saturation ({theta}w/{theta}s), and nw is the water-phase tortuosity parameter. Likewise, the geometry factor for the gas phase, Fg({theta}a,c), can now be defined as the ratio of the gas permeability of the soil, Ksoil({theta}a,c), and the gas permeability of a bundle of straight capillaries, Kcap({theta}a,c) (Brooks and Corey, 1966; Parker et al., 1987):

(5)

Inserting EQ. [5] into EQ. [3] gives

(6)
in which na is the gas-phase tortuosity parameter. Predicting the relative gas diffusion coefficient for unsaturated soils with EQ. [6] requires knowledge of the soil water retention curve and gas tortuosity parameter na.

Water retention is described using the two-parameter junction model of Rossi and Nimmo (1994):

(7a)
with

(7b)

This model is based on the Brooks and Corey (1966) water retention model, which is equivalent to the equation used by Campbell (1974), with the residual water content taken as zero. To describe the behavior of the water retention curve near {theta}s, Rossi and Nimmo added the parabolic equation, {theta}w,I, proposed by Hutson and Cass (1987), to the Brooks and Corey model, as described by {theta}w,II. The equation for {theta}w,II is a power law for pressure head {psi} smaller than the air entry value {psi}0. The simple power law overestimates the water content at very low pressure heads. Therefore, a third part, {theta}w,III, as proposed by Ross et al. (1991), was added, which makes the water content {theta}w = 0 at a finite value of pressure head {psi}d.

If {theta}s is measured and the value of {psi}d is set to -105 m of water, which corresponds to the pressure head at oven-dryness (Ross et al., 1991), there are six unknown parameters (c, {psi}i, {psi}j, {alpha}, {lambda}, and {psi}0). Four parameters (c, {psi}i, {psi}j, and {alpha}) are determined as analytical functions of the remaining two parameters {lambda} and {psi}0 through EQ. [7c] and EQ. [7d]:

(7c)

(7d)
which ensure continuity of the global function and its first derivative at the two junction points {psi}i and {psi}j (Rossi and Nimmo, 1994). Thus, the two-parameter junction water retention model has the same number of parameters as the Brooks and Corey model (Brooks and Corey, 1966). Inserting EQ. [2] and [7] into EQ. [6] and piecewise integrating (over water contents) yields:

(8)

Subscript M stands for the Mualem-type numerator in EQ. [4] and [5] (Mualem, 1976) and subscript C stands for the capillary-type denominator in Eq. [4] and [5].

It is often assumed (Fischer et al., 1997; Lenhard and Parker, 1987; Parker et al., 1987) that the tortuosity parameters of the gas (na) and the water (nw) phases are equal. If this assumption is valid then the experimentally determined value of na from gas diffusion measurements and water retention measurements can be used in the Mualem (1976) relative hydraulic conductivity function:

(9)
in which Ks is the saturated hydraulic conductivity.

The integrals, IM (Rossi and Nimmo, 1994) and IC (Weerts et al., 1999), obtained by piecewise integration over water contents are given by

(10)
and

(11)


and

(12)


In the following sections we calibrate the gas diffusion coefficient model using measured water retention curves and gas diffusion coefficients. To test, if indeed the assumption concerning the equality of the gas and water phase tortuosity parameter is valid, we compare predicted hydraulic conductivity, using gas tortuosity parameters, with measured hydraulic conductivity.


   Materials and methods
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Materials and methods
Results and discussion
 Conclusions
 Appendix
 REFERENCES
 
Experimental Data
In this study we have considered the data set of Freijer (1994) who measured gas diffusion coefficients of seven mineral horizons. Water retention curves of the Harderbos silty clay loam (HbAh), Belvedere silt loam (BeC), Eijsden silt loam (EsC), and Oss sand (OsC) soils were also obtained from Freijer (1994). Water retention data of the Kootwijk sand (KwAE, KwB, and KwC) soils were obtained from Schaap and Bouten (1996). Properties of the seven horizons are given in Table 1 . Freijer (1994) used the method of Reible and Shair (1982) to measure the gas diffusion coefficient of CO2 as a function of {theta}a. A diffusion coefficient of CO2 in free air of 1.12 m2 d-1 was used. Hydraulic conductivity data of the Kootwijk sand soils were available from previous work and were determined with the crust method (Bouma et al., 1971), the drip infiltrometer method (Dirksen, 1991), and the evaporation method (Boels et al., 1978). Hydraulic conductivity data of the Kootwijk sand C horizon (KwC) are also reported in Stolte et al. (1994)(Eolian sand).


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  		Table 1 Properties of the seven mineral horizons (Freijer, 1994)

 
Parameter Identification
Equation [7a] was fitted through the measured water retention data by Simplex optimization (Nelder and Mead, 1965). Using EQ. [8] combined with Eq. [10] results in unrealistic behavior near {theta}s. This behavior was also found when using EQ. [7a] combined with Eq. [4] for soil electrical conductivity (Weerts et al., 1999). This behavior is not a result of the choice of the water retention curve, but is caused by the use of the equation for the hydraulic conductivity of straight capillaries, Kcap({theta}w), which is known to increase much faster than Ksoil({theta}w) near {theta}s (Mualem, 1986). To avoid this unrealistic behavior, we changed the water content range for IM,II and IC,II to {theta}w,j <= {theta}w <= {theta}s, extending the power law part and ignoring the parabolic part of Eq. [10], as was suggested by Weerts et al. (1999). The value of na is obtained from fitting Eq. [8] (using, as explained above, the modified Eq. [10]) to gas diffusion measurements by minimizing the sum of squared residuals, also using the Simplex algorithm. The saturated hydraulic conductivity of the Kootwijk soils was set equal to the highest measured unsaturated hydraulic conductivity near {theta}s ({psi}->0).


   Results and discussion
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Materials and methods
 Results and discussion
Conclusions
 Appendix
 REFERENCES
 
The first set of samples contains the Kootwijk soils (Table 1, Fig. 1) , which concerned the potential impact of organic matter in sand on the gas diffusion coefficient. Optimized parameters ({lambda} and {psi}0) of all water retention curves are given in Table 2 . The data of KwAE, KwB, and KwC fit well with Eq. [8]. The second set of samples consists of soils with varying textures and structure (Table 1), ranging from silty clay loam to sand (soils HbAh, EsC, BeC, and OsC). As can be seen from Fig. 2 , the data of soils BeC and OsC are a good fit with Eq. [8]. The data of HbAh and EsC are a poor fit with Eq. [8].



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  		Fig. 1 Measured and fitted relative gas diffusion coefficient (Dp/D0, y-axis) as a function of the volumetric water content ({theta}w, x-axis) for the Kootwijk soils. Parameter values of the relative gas diffusion coefficient model are given in Table 2. Soil type is indicated on top of each graph

 

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  		Table 2 Measured model and fitted water retention parameters (for explanation see text) of the soil samples, together with the fitted gas tortuosity parameter na and the root mean squared error (RMSE) of this fit

 


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  		Fig. 2 Measured and fitted relative gas diffusion coefficient (Dp/D0, y-axis) as a function of the volumetric water content ({theta}w, x-axis) for the Harderbos, Eijsden, Belvedere, and Oss soils. Parameter values of the relative gas diffusion coeffient model are given in Table 2. Soil type is indicated on top of each graph

 
The water retention parameter {lambda} has the largest influence on the form of the fitted curves, besides the fitted gas tortuosity parameter na. With this in mind, it is interesting to look at the values of the root of the mean squared error (RMSE) of the Kootwijk horizons in Table 2. The RMSE is highest for the soil with the lowest value of {lambda}, soil KwAE, and lowest for the soil with the highest value of {lambda}, soil KwC. Qualitatively, a similar trend is observed (Table 2) for the soils in Fig. 2. A low value of {lambda} implies that the logarithmic part of EQ. [7a] describes most of the water content range of the water retention curve. The logarithmic part of EQ. [7a] is associated with adsorption as the main water retention mechanism (Rossi and Nimmo, 1994), while the power law part of EQ. [7a] is associated with capillarity as the main water retention mechanism (Rossi and Nimmo, 1994). The gas diffusion coefficient model, based on the idea that the soil can be modeled with a capillary-based geometry factor, is likely to fail for the soils where the power law part is not the most important part of the water retention curve and this is exactly what we see in Fig. 2 for soils HbAh and EsC. Note that the model fits well with the gas diffusion data of BeC soil, which has a similar texture as the EsC soil. But from Table 2 it is clear that the water retention mechanism in the BeC (capillarity) and EsC (adsorption) soils are very different. This difference in water retention behavior can be explained by the fact that the samples of the EsC were obtained from a filled lysimeter (Freijer et al., 1996) and as the lysimeter filled, the original structure (with capillary pores) of the EsC soil was destroyed.

The effect of the parameter {theta}de on the fitting results was investigated but the specific results are not shown here. From these investigations it was found that a small change of ±0.02 m3 m-3 (deviation in measured {theta}s) in the value of {theta}de, keeping the porosity constant, only leads to small changes in fitted water retention parameters {lambda} and {psi}0 and therefore has almost no effect on the fitted value of the gas tortuosity parameter na.

As mentioned before, it is often assumed that the tortuosity parameter of the gas and water phases are equal. If so, we can compare the fitted gas tortuosity parameters with hydraulic and electrical tortuosity parameter values described elsewhere (Mualem, 1976; Weerts et al., 1999). However, it is more interesting to see if this assumption is justified. Therefore, we plotted the measured and predicted unsaturated hydraulic conductivity of the three Kootwijk soils in Fig. 3 , using the fitted gas tortuosity parameters. The hydraulic conductivity near {theta}s ({psi}->0) was chosen as a reference point and therefore the model fits the data near {theta}s. The model overestimates the relative hydraulic conductivity for all three horizons at low water contents. This indicates that there may be a systematic difference between tortuosity parameters for the gas and water phase, the latter being larger than the former.



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  		Fig. 3 Measured and predicted relative unsaturated hydraulic conductivity (Kr, y-axis) as a function of the volumetric water content ({theta}w, x-axis) for the Kootwijk soils. Soil type is indicated on top of each graph

 

   Conclusions
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Materials and methods
 Results and discussion
 Conclusions
Appendix
 REFERENCES
 
The bulk electrical conductivity model presented by Mualem and Friedman (1991) was rewritten for the gas (non-wetting) phase and combined with the water retention model presented by Rossi and Nimmo (1994). To obtain a realistic solution of the integrals in Eq. [6] near {theta}s, the extrapolation suggested by Weerts et al. (1999) is used. The resulting model is capable of describing measured gas diffusion data on five different soil horizons. The model could not describe gas diffusion data of two clayey top horizons. This is probably caused by the fact that the main water retention mechanism in these two soils is adsorption instead of capillarity.

Use of the same parameter set, including the fitted gas tortuosity parameter na, for describing the relative hydraulic conductivity leads to overestimation of the relative hydraulic conductivity at low water content. Therefore, the assumption that the water-phase (nw) and gas-phase (na) tortuosity parameters are equal cannot be justified. The systematic overestimation of the relative hydraulic conductivity when using the gas-phase tortuosity parameter instead of the water-phase tortuosity parameter suggests a relationship that needs future attention.

Received for publication December 21, 1998.
   Appendix
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Materials and methods
 Results and discussion
 Conclusions
 Appendix
REFERENCES
 
Symbols
Dp, effective diffusion coefficient of a gas in a porous medium (m2 d-1)

D0, diffusion coefficient of a gas in the absence of a porous medium (m2 d-1)

Dp/D0, relative gas diffusion coefficient

Fg({theta}a,c), geometry factor for the air-filled pathways

Fg({theta}w), geometry factor for the water-filled pathways

IM,I/II/II, piecewise-integrated Mualem-type numerator over water content range of the (I) parabolic part, (II) power law part, and (III) logarithmic part of the water retention curve

IC,I/II/II, piecewise-integrated straight capillary-type denominator over water content range of the (I) parabolic part, (II) power law part, and (III) logarithmic part of the water retention curve

Ksoil({theta}w), unsaturated hydraulic conductivity (m d-1)

Kcap({theta}w), unsaturated hydraulic conductivity of a bundle of straight capillaries (m d-1)

Ksoil({theta}a,c), gas permeability (m d-1)

Kcap({theta}a,c), gas permeability of a bundle of straight capillaries (m d-1)

Kr({theta}w), relative hydraulic conductivity

Ks, saturated hydraulic conductivity (m d-1)

Sw, relative water saturation

c, constant

i subscript indicating junction point i

j subscript indicating junction point j

nw, water-phase tortuosity parameter

na, gas-phase tortuosity parameter

{psi}, pressure head (m)

{psi}0, air entry value (m)

{psi}i, pressure head at junction point i (m)

{psi}j, pressure head at junction point j (m)

{psi}d, pressure head at oven-dryness (m)

{alpha}, constant

{phi}, porosity (m3 m-3)

{lambda}, constant

{theta}a, volumetric air content (m3 m-3)

{theta}a,c, continuous air content (m3 m-3)

{theta}de, dead-end pore volume (m3 m-3)

{theta}s, natural saturation (m3 m-3)

{theta}w, volumetric water content (m3 m-3)

{theta}w,I/II/III, volumetric water content in (I) parabolic part, (II) power law part, and (III) logarithmic part of water retention curve (m3 m-3)

{theta}0, volumetric water content at air entry pressure (m3 m-3)

{theta}w,i, volumetric water content at junction point i (m3 m-3)

{theta}w,j, volumetric water content at junction point j (m3 m-3)


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Materials and methods
 Results and discussion
 Conclusions
 Appendix
 REFERENCES
 




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