Soil Science Society of America Journal 65:1577-1584 (2001)
© 2001 Soil Science Society of America
DIVISION S-1 - SOIL PHYSICS
Tortuosity of an Unsaturated Sandy Soil Estimated using Gas Diffusion and Bulk Soil Electrical Conductivity
Comparing Analogy-based Models and Lattice–Boltzmann Simulations
A. H. Weerts*,a,b,
D. Kandhaic,
W. Boutenb and
P. M. A. Slootd
a Unilever Research Colworth, Colworth House, Sharnbrook, Bedford MK 44 1LQ, UK
b Institute for Biodiversity and Ecosystem Dynamics, Physical Geography, Faculty of Science, Universiteit van Amsterdam, Nieuwe Achtergracht 166 1018 WV Amsterdam, The Netherlands
c Kramers Laboratorium voor Fysische Technologie, Faculty of Applied Sciences, Technical Univ. Delft, Prins Bernhardlaan 2628 BW, Delft, The Netherlands
d Instituut voor Informatica, Faculty of Science, Universiteit van Amsterdam, Kruislaan 403 1098 SJ Amsterdam, The Netherlands
* Corresponding author (albrecht.weerts{at}unilever.com)
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ABSTRACT
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Prediction of gas and ion diffusion in unsaturated soils has been obstructed by the inability to independently measure tortuosity parameters. Analogies between transport processes are often suggested to assess these nonmeasurable parameters. However, it is unclear whether these analogies are valid in unsaturated soils. Therefore, to obtain a more fundamental insight into the differences between ion and gas diffusion in an unsaturated sandy soil a different approach than current empirical models is required. In this study, fitted tortuosity parameter values of analogy-based models describing the gas diffusion coefficient and bulk soil electrical conductivity (ion diffusion) of a sandy soil are compared. Gas diffusion in unsaturated soil is most affected by connectivity in contrast to electrical conductivity that is among other factors primarily determined by tortuosity. Measured gas diffusion and electrical conductivity are also compared with lattice–Boltzmann simulated gas and ion diffusion as a function of water content in a thin section (two dimensional) of a sandy soil. Results of the lattice-Boltzmann simulations are in qualitative agreement with measured diffusional ratios and support the findings with the analogy-based models. Lattice-Boltzmann simulations may be a valuable tool to overcome empirism of current models and increase our insight in transport properties of unsaturated soils in the coming years.
Abbreviations: 2D, two dimensional • 3D, three dimensional • BGK, Bhatnagar Gross Krook Method • Dp,w, effective ion diffusion coefficient of the bulk soil • Dw,diffusion coefficient of an ion in free water • Fg, hydraulic conductivity of bundled straight capillaries (Ksoil/Kcap) • GIS, geographical information system • l.u., lattice untis • LBM, lattice-Boltzman method • na, gas phase tortuosity parameter • nw, water phase toturosity/connectivity and correlation factor • sw, relative saturation • 
a, soil surface conductivity • w, soil water conductivity • 
s, saturatedwater content • w, water content • 
, pressure head • d, finite value of pressure head • 0, air entry value
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INTRODUCTION
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UNDERSTANDING the mechanisms of water and diffusive transport in unsaturated porous media is a challenge for scientists. Despite the amount of theoretical and experimental work, many aspects of transport in unsaturated soils are still poorly understood. The dependency of diffusive transport coefficients on water content is one of the topics in need of further investigation.
A lot of research has been devoted to transport coefficients of soils as a function of water content for both gas and ion diffusion and electrical conductivity (Collin and Rasmuson, 1988; Freijer, 1994; Klute and Letey, 1958; Moldrup et al., 1997; Mualem and Friedman, 1991; So and Nye, 1989). Note that ion diffusion in a porous medium can be very slow and is often estimated using electrical conductivity measurements. Electrical conductivity and ion diffusion in noncharged porous media are related through Dp,w = D
(Schwartz et al., 1993), where Dw is the diffusion coefficient of an ion in free water, Dp,w is the effective ion diffusion coefficient of the bulk soil, w is the soil water conductivity, and a is the electrical conductivity of the bulk soil. The measured diffusion data are often described with empirical models containing one or several nonmeasurable parameters that have to be obtained by fitting the data with the model. The value that must be assigned to the fitted parameters is often limited to the specific study involved. However, in an attempt to obtain tortuosity parameters from gas diffusion measurements to describe the unsaturated hydraulic conductivity and implicitly bulk soil electrical conductivity, Weerts et al. (2000) suggested that there may be a difference between gas and ion diffusion pathways in soils.
Massively parallel and interactive simulations of transport in porous media with lattice-gas and lattice-Boltzmann methods (LBMs) have arisen hope of gaining more insight in transport processes in porous media (Di Pietro et al., 1994; Gunstensen and Rothman, 1993; Martys and Chen, 1996). Both methods are cellular automata and can therefore be represented as an array of discrete variables following specific rules (Chopard and Droz, 1998). The underlying discreteness of cellular automata makes it well suited for simulation on parallel computers. Moreover, both lattice-gas and -Boltzmann methods enable direct simulation of gas and ion diffusion in soils. For example, Alvarez-Ramirez et al. (1996) and Martys (1999) showed the possibilities of diffusion simulation with the lattice-Boltzmann method in saturated and unsaturated model porous media.
In this contribution, we explore the differences between gas and ion diffusion coefficients as a function of water content in an unsaturated sandy soil (noncharged porous medium). First, we will compare connectivity and tortuosity parameters obtained by fitting measured gas diffusion coefficients and bulk soil electrical conductivity of unsaturated sandy soil to analogy-based models (Mualem and Friedman, 1991; Weerts et al., 2000) to investigate whether there exists difference and similarities in tortuosity and connectivity affecting gas diffusion and electrical conductivity. Second, we simulate gas and ion diffusion in a digitized two dimensional thin section of a similar sandy soil with the lattice-Boltzmann method to obtain diffusion coefficients as a function of water content. Finally, we compare the lattice-Boltzmann simulations with the measured diffusion coefficients.
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MATERIALS AND METHODS
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Soil Samples
Undisturbed soil cores were taken from the field in 0.05-m-diameter stainless steel rings of 0.05 and 0.10 m long. The samples are from an eolian deposited fine sand horizon (cover sand, Appelscha) with a very low organic mattter content (<0.15%). A thin section of a similar horizon of eolian deposited fine sand was obtained from the study conducted by Koster (1978)(p. 195). This thin section was photographed (Leica MPS 52 mounted on a Leica M420, Leica, Northvale, NJ) and then digitized. Figure 1a shows the digitized sample of 1444 by 2388 pixels each side of 4.4 µm. The porosity of this sample is 0.43.
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Fig. 1. (a) Overview of digitized sample (gray/dark represents soil matrix and white represents pore space). (bcde) Part (200 by 200 pixels) of digitized sample (indicated in a with black lines) at different water contents (black represents soil matrix, gray represents water phase, and white represents gas phase). The size of the digitized sample is 1444 by 2388 pixels and each side of a pixel is 4.4 mm.
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Measurement of Electrical Conductivity
The undisturbed 0.05-m soil cores were first saturated with water then flushed with four times the pore volume of a NaCl solution of 0.518 Sm-1. The saturated flushed samples were placed on a hanging water column (filled with the same NaCl solutions as the soil samples). With the hanging water column several drainage steps were made. The pressure head and the volumetric water content were determined from the outflow of the samples and used to fit a water retention model (see subsection Analogy-based Gas Diffusion Coefficient and Bulk Electrical Conductivity Model). Time domain reflectometry (TDR) probes (L = 0.05 m, cable length 
3.0 m) were inserted vertically middle in the soil sample. For the calculation of a, the cell constant of each probe and the resistance of the TDR system (cable tester, cables, and connectors) were calibrated with a range of solutions of known electrical conductivity (Heimovaara et al., 1995). At the beginning of the experiment and after a new equilibrium was reached following a pressure step, five TDR measurements were made to obtain a good estimate of a. The surface conductance was determined by performing a measurement after drying the samples in the oven at 105°C and was found to be near zero (s 0). The values of a were corrected for the temperature differences using the method of Heimovaara et al. (1995). The whole procedure is analogous to the procedure followed by Weerts et al. (1999) except for the way the samples were saturated.
Measurement of Gas Diffusion
The experimental procedure for measuring effective diffusion coefficients was similar to the procedure of Freijer (1994) and we used a measurement apparatus similar to the one used by Reible and Shair (1982). This apparatus consists of two air chambers separated by a 0.10-m stainless steel sample ring containing soil. The air in each chamber was well mixed during the experiment by fans installed inside the chambers. Each chamber was kept in pressure equilibrium with the ambient air by vent. At the start of the experiment a known amount of gas, in our case CO2 with a gas diffusion coefficient in air of 1.27 m2d-1 at a temperature of 20°C and pressure at 1.0.105 Pa, was injected into one of the chambers, which induced diffusion from this chamber to the other. The concentration of the gas in both chambers was followed in time by regular measurements. Gas samples are withdrawn from the chambers with a syringe and were analyzed by gas chromatography. During the experiment ambient pressure and temperature were also measured. Reible and Shair (1982) gave a simplified analytical solution for diffusion in such an experimental setup. This solution is valid if both chambers have the same volume and the initial concentration of the injected gas in sample and the exit chamber are zero. For this analytical solution and other details, we refer to Reible and Shair (1982) or Freijer (1994).
Effective gas diffusion coefficients were measured at different water contents. Before saturating the sample, one measurement was performed at the original water content. The sample was supported with a fine mesh and then saturated by placing it on a saturated sand box. After saturation the sample was placed in a horizontal position with caps on both sides to allow redistribution of water. Starting from saturation, repeated measurements of the effective diffusion coefficients were performed at series of decreasing water contents. Lowering of the water content was established by placing the samples on sand boxes at a certain pressure head. After a certain water content, lowering of the water content was established by letting the samples evaporate from both open sides, again redistribution was allowed by placing the caps on both sides. No gaps were observed between the soil and the sample ring after evaporation. Finally, we dried the samples in an oven at a temperature of 105°C to measure the effective gas diffusion coefficient at zero water content. The water content at each diffusion measurement and the dry bulk density of the samples was derived from measured dry and wet weights of the sample. Total porosity was calculated from dry bulk density and measured density of the solid phase.
Analogy-based Gas Diffusion Coefficient and Bulk Electrical Conductivity Model
Water Retention Model
Water retention is described using the two-parameter junction model of Rossi and Nimmo (1994).
 | [1a] |
with
 | [1b] |
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 (natural) saturation s, Rossi and Nimmo added the parabolic equation, w,I, proposed by Hutson and Cass (1987) to the Brooks and Corey model, as described by w,II. The equation for w,II is a power law for pressure head , smaller than the air entry value, 0. The simple power law overestimates the water content at very low pressure heads. Therefore, a third part, w,III, as proposed by Ross et al. (1991) was added which makes the water content, w, equal to zero at a finite value of pressure head, d.
If s is measured and the value of d is set to -105 m of water that corresponds to pressure head at oven dryness (Ross et al., 1991), there are six unknown parameters (c, i, j, 
, 
, and 0). But four of these parameters (c, i, j, and ) are determined as analytical functions of the remaining two parameters and 0 through Eqs. [1c] and [1d]:
 | [1c] |
 | [1d] |
which insure continuity of the global function and its first derivative at the two junction points (i and 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).
Bulk Electrical Conductivity Model
Bulk soil electrical conductivity is described by
 | [2] |
where w is the water content (m3 m-3), w is the soil water conductivity (Sm-1), a is the soil bulk electrical conductivity (Sm-1), and s is the surface conductivity (Sm-1). Mualem and Friedman (1991) propose the concept that the geometry factor, Fg is equal to the ratio of the hydraulic conductivity of the soil, Ksoil(w), to the hydraulic conductivity of a bundle of straight capillaries, Kcap(w), with an identical water retention characteristic as the soil. This leads to the following expression for Fg(w):
 | [3] |
in which nw is the water phase tortuosity/connectivity and correlation factor, Sw is the relative saturation (w/s), s being the water content at (natural) saturation (m3 m-3), and is the pressure head (m). For more details, we refer to Mualem and Friedman (1991) and Weerts et al. (1999).
Gas Diffusion Coefficient Model
Likewise, the gas diffusion coefficient is described by
 | [4] |
Where Dp,g is the effective gas diffusion coefficient, D0 is the diffusion coefficient of the gas in free air, a,c is the continuous air content (m3 m-3) defined as a,c = 
- de - w, is the porosity, and de is the dead-end pore volume defined as de = -s. In analogy with the bulk soil electrical conductivity, the geometry factor for the gas phase, Fg(a,c), can be defined as the ratio of the gas permeability of the soil, Ksoil(a,c), and the gas permeability of a bundle of straight capillaries, Kcap(a,c) (Brooks and Corey, 1966; Friedman, 1993; Parker et al., 1987):
 | [5] |
where na is the gas phase tortuosity parameter. For more details concerning the gas diffusion coefficient model we refer to Weerts et al. (2000).
Fitting of Water Retention and Tortuosity Parameters
The parameters, and 0, are obtained by fitting Eq. [1a] through the measured water retention data by Simplex optimization (Nelder and Mead, 1965). If we now apply Eq. [3] and Eq. [5] in combination with Eq. [1a], this would result in unrealistic behavior near s. 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(w), which is known to increase much faster then Ksoil(w) near s (Mualem, 1986). To avoid this unrealistic behavior, we used the extrapolation as suggested by Weerts et al. (1999). The values of nw and na are obtained from fitting both models to gas diffusion measurements and bulk soil electrical conductivity measurements. Note that the averaged water retention parameters obtained from the electrical conductivity experiments were used in fitting the gas diffusion coefficient model.
Lattice-Boltzmann Simulation of Gas and Ion Diffusion
Lattice-Boltzmann Method
Lattice–Boltzmann methods originated from the lattice gas automata that are discrete models for the simulation of transport phenomena (Frisch et al., 1986). In these models, the computational grid consists of a number of lattice points that are connected with some of their neighbors (depending on the model) by a bond or link. Basically, particles move synchronously along the bonds of the lattice and interact locally according to a given set of rules in the following two stages:
1. . Propagation
In this phase particles move along lattice bonds from one lattice node to one of its neighbors.
2. . Collision
Particles on the same lattice node shuffle their momenta locally, subject to mass, momentum, or energy conservation depending on the application.
In Fig. 2
, an example of a lattice gas automaton for modeling fluid flow is shown. Note that in the collision phase, only those configurations that satisfy conservation of mass and momentum are allowed. For instance, when two particles enter the same site with opposite velocities, both of them are deflected by 60° such that the net momentum in the post collision state remains zero and when multiple post collision states are possible a random selection is made. It has been argued that lattice gas automata are similar to random walk methods when after collision a particle follows a random direction (Chopard and Droz, 1998).
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Fig. 2. Propagation and collision phases for some initial configuration. Arrows denote a particle and its moving direction. When multiple collision states are possible a random selection is made. The collision rules satisfy mass and momentum conservation.
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The major drawbacks of early lattice gas models were statistical noise, exponential complexity of the collision operator, and lack of Galilean invariance (Benzi et al., 1992). Later, many modifications have helped to resolve most of these problems and lattice gas methods have proven to be correct models for several transport phenomena (Rothman and Zaleski, 1997). Nevertheless, the apparent deficiencies of the early lattice gas methods inspired formulation of LBMs. In LBM, a population of particles is being tracked instead of a single particle and there is relatively more freedom in the definition of the collision operator (Chopard and Droz, 1998; Rothman and Zaleski, 1997). The latest major modification to date is the lattice-Bhatnagar-Gross-Krook (BGK) model, where the collision operator is based on a single-time relaxation to the local equilibrium distribution (Qian et al., 1992). This model is the simplest one in the hierarchy of LBMs and is currently widely used in practical simulations. The time-evolution of the lattice-BGK model is given by (Qian et al., 1992):
 | [6] |
where fi(r, t) is the probability of finding a particle with velocity, vi, at position, r, and time, t; 
t is the time step; vi is the velocity along link i, t is the BGK relaxation parameter, and f0i
is the equilibrium distribution function towards which the particle populations are relaxed. The first part of the right-handside of Eq. [6] represents the propagation part and the second term is the collision term. The exact definition of the equilibrium distribution and the lattice connectivity determine the transport phenomenon that is actually being modeled (Chopard and Droz, 1998). Although these models are commonly used for simulating hydrodynamics, it has been shown that diffusion phenomena can be mimicked by taking the following form for the equilibrium distribution (Chen et al., 1995)
 | [7] |
where ti is a weight-factor depending on the link i, and 
is the density that is defined as
 | [8] |
N is the number of links per lattice point. In this paper, the so-called D2Q9 model is used where each lattice point is connected to its nearest and diagonal neighbors. Rest particles are also included here. In this model ti is 4/9 for the rest particle, 1/9 for links pointing to the nearest neighbors, and 1/36 for links pointing to the next-nearest neighbors on a hexagonal lattice. The diffusion constant D in lattice units is given by (Alvarez-Ramirez et al., 1996; Chen et al., 1995):
 | [9] |
Phase Separation
The modeling of phase separating binary mixtures with LBM is still a great research challenge (Martys, 1999; Martys and Chen, 1996) and was not the objective of this study. Therefore, we did not employ numerical phase separation by the LBM. Instead, we started with a water-saturated sample, and decreased the water content stepwise by removing water starting with water from pores that had the greatest distance from pore center to the solid walls using a geographical informational system (GIS) based modeling environment (Burrough and McDonnell, 1998). We always kept one layer of water (4.4 µm) surrounding the solid walls, except for the completely dry simulations. Figures 1b through 1e show selections of 200 by 200 pixels at four different water contents.
Simulated Experiment and Data Analysis
To gain insight in the discretization error of the LBM, a resolution study was performed prior to the real simulations. For this, we computed the effective diffusion coefficient of a sample with very thin diffusive path lines. We used three magnifications of the grid cells, namely 1 by 1, 2 by 2, and 4 by 4 and thus the grid sizes were 200 by 200, 400 by 400, and 1600 by 1600 lattice points, respectively. We found that the result obtained on the 200 by 200 grid deviated <5% from that of the large 1600 by 1600 grid. For the final simulations we used grid sizes of 200 by 200 lattice points.
For all water contents, diffusion simulations were performed on 16 randomly chosen grids of 200 by 200 pixels (Fig. 1). In the lattice–Boltzmann simulations the soil (solid phase) is considered uncharged. The relaxation parameter was equal to 1.0 and solid boundaries were modeled using the bounce-back boundary condition. Diffusion was simulated in a sample with one open boundary that was held at a certain constant concentration, 1.1 lattice units (l.u.). The other boundaries were closed. The initial concentration in the sample was set to 1 l.u. The concentration as a function of time was measured just before the closed boundary at the opposite end of the open boundary. Bird (Bird, 1956) gives the analytical solution of diffusion in such a system. The simulated diffusion experiments were fitted with this analytical solution to obtain the diffusion coefficients. The computation times for the simulations varied from 72 to 160 h per grid (depending on water content of the sample) on a Pentium II machine with 512 MB main memory (Hewlett Packard, Palo Alto, CA). Note that the large computing times are caused by the fact that the diffusion takes place in thin layers.
For all water contents, the distribution of the 16 diffusion coefficients obtained was used to randomly fill a 4 by 4 matrix. On this matrix a steady-state numerical diffusion experiment using a GIS-based modeling environment (Burrough and McDonnell, 1998) was performed. This procedure was repeated 250 times (increasing this amount does not change the results) to calculate the mean effective diffusion coefficient.
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RESULTS AND DISCUSSION
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Analogy-based Gas Diffusion Coefficient and Bulk Electrical Conductivity Model
Figure 3
shows an example of the water retention fit of sample B6. Figure 4
shows an example of the fitted (a) bulk electrical conductivity model and (b) gas diffusion coefficient model to the measured data. The s and the fitted water retention parameters and nw tortuosity parameters are given in Table 1. Porosity, dead-end pore volume, and the fitted na tortuosity parameters are given in Table 2. Note that the measurement scale for both the gas diffusion coefficient (0.10 m) and the bulk electrical conductivity (0.01 m as the distance between and length of wires, 0.05 m) are slightly different.
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Fig. 4. Measured (sample B1) and fitted relative electrical conductivity (above) and measured (sample A261) and fitted relative gas diffusion coefficient (below) as a function of water content of a cover sand.
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Both the gas diffusion coefficient model and the bulk soil electrical conductivity are able to describe the measured data. Note that the bulk electrical conductivity model underestimates the electrical conductivity at lower water contents. The averaged fitted gas and water phase tortuosity parameters are not significantly different (P = 0.19), in contrast to the findings of Weerts et al. (2000), who suggested that there is a difference between gas and water phase tortuosity parameters based on predictions of hydraulic conductivity using the gas phase tortuosity parameter. Figure 5a
shows all measured gas diffusion coefficient and electrical conductivity data together with the curves obtained with the averaged values of Tables 1 and 2. The curve that describes the gas diffusion data is not as good in describing the data as the curve that fits the electrical conductivity data. The suggestion that the tortuosity parameter for the water phase is larger than for the gas phase implies that electrical current follows a more tortuous pathway than gas particles. Based on this suggestion, one would expect that the curve for gas diffusion coefficient in Fig. 5a should lie above the electrical conductivity curve. If we take into account the dead-end pores, the curve for gas diffusion coefficient will shift to the left. Even then the gas diffusion coefficient curve is lower than the electrical conductivity curve. Apparently gas diffusion is more affected by connectivity than electrical conductivity as the water phase forms a (almost) connected, but more tortuous, pathway. In other words, gas diffusion pathways are severly blocked if water is present. Therefore, it is interesting to see that in the complete dry case, the gas diffusion coefficient tend to the same ratio as obtained by electrical conductivity measurements at saturation as one also expects on theoretical grounds. This may also explain why the gas diffusion coeffient model is not as good as the electrical conductivity model, during the fitting a weighting between the complete dry case (tortuosity) and partially saturated case (connectivity) has to be made. The empirical gas diffusion model cannot account for both cases at the same time and as a results ends up somewhere inbetween.
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Fig. 5. (a) Measured and (b) lattice–Boltzman simulated diffusional ratios. The averaged curves of the analogy-based models are also indicated in Fig. 5a.
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Measurements compared with Lattice-Boltzmann Simulations of Gas and Ion Diffusion
Figure 5 shows (a) measured gas diffusional ratios and bulk soil electrical conductivity ratios and (b) the simulated lattice–Boltzmann mean diffusional ratios. Note that the porosity in the digitized thin section (0.43) is higher than the porosity of the soil samples (0.34). Qualitatively, the trend in the simulations is also present in the measurements and supports the interpretation of connectivity and tortuosity with the analogy-based models. From Figure 1b, c, d, and e, it is evident that water phase pathways are more tortuous than gas phase pathways, and that the gas phase is much more affected by the connectivity than electrical conductivity. However, the lattice–Boltzmann simulations are done in two dimensions (2D) while the diffusive processes take place in three dimensions (3D). Therefore the simulations exaggerate some of these effects. For instance, the rise of the simulated gas diffusion coefficient with increasing air content occurs at relatively high air contents in comparison with the measurements and can be attributed to an underestimation of the connectivity of the gas phase. Also, lower gas and ion diffusion coefficients compared with the measured values must be atrributed to this 2D-3D effect, as we expect an increase in connectivity in case of 3D simulations. Besides the 2D-3D effects, it is likely that the 16 samples represent the total digitized thin section insufficient. Therefore, we did not plot the standard errors that are small and only tells us how good the mediation procedure to obtain the mean D was. Because of the fact that we used a 200 by 200 grid for the lattice–Boltzmann simulations, we sometimes obtain zero values for the diffusion coefficient. This artificially causes a relatively high standard deviation that does not say too much about the real variation of the diffusion coefficient of the sample. The fact that the two curves are not smooth also shows that 16 simulations per water content is on the low side.
The procedure followed in this paper such as the mediation to obtain the mean D from the 16 simulations may seem laborious, but is given current computer memory a compromise between the spatial resolution necessary to model the desaturation proces (Fig. 1b, c, d, and e) and the representative pore structure that needs to be present in the calculation. Figure 1a shows that this last condition is not met, but simulations on larger, possibly 3D, grids would take too much computing time given current computer resources.
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CONCLUSIONS
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Gas diffusion coefficients and bulk soil electrical conductivity can be described with an empirical analogy-based model. Comparison of the fitted gas and water phase tortuosity parameter of an unsaturated sandy soil (noncharged porous medium) showed that the values were similar. However, comparison of all measured data and the averaged analogy-based models make clear that the fitted parameters have a different physical meaning. The tortuosity parameter for the water phase is mainly linked with the tortuosity of the pathways. The tortuosity parameter of the gas phase is mainly linked with the connectivity of the pathways. However, if the sample is completely dry, gas diffusion is not hindered by water any more and results in almost the same measured ratios as measured with electrical conductivity of the water saturated samples. This double effect on gas diffusion also explains why the empirical model is worse in describing the measured gas diffusion data in comparison with the electrical conductivity measurements, as it has to weight both effects during the fitting process. Because electrical conductivity, and ion and gas diffusion are affected by either tortuosity or connectivity, it is not a good idea to interchange the fitted values when one wants to predict diffusion in the other phase.
Alternatively, we also compared measured gas diffusion coefficients and electrical conductivity with lattice–Boltzmann simulations. Although the porosities of the soils used for the measurements and for the lattice–Boltzmann simulations were different, this comparison showed qualitative agreement between measurements and simulations and supports the idea that the fitted parameters of the analogy-based models are affected by either tortuosity or connectivity.
Unfortunately, we could only perform a limited amount of lattice–Boltzmann simulations because of limitations in available computing resources. With increasing computer power, these amounts of simulations will soon be possible and can even be extended to 3D. Therefore, we expect that lattice–Boltzmann simulations in combination with new imaging techniques of porous media (Torquato et al., 1999) can provide an enormous progress in understanding transport processes in unsaturated soils in the coming years.
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ACKNOWLEDGMENTS
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We are grateful to Joke Westerveld and Leen de Lange for assisting with the experiments. Koos Verstraten is thanked for reading an earlier draft.
Received for publication August 24, 2000.
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ABSTRACT
INTRODUCTION
and fitted (na) gas diffusion coefficient model parameters. Note that the gas tortuosity parameters were obtained by fitting. For more explanation see text.