Evaluation of solute diffusion tortuosity factor models for variously saturated soils

Authors


Corresponding author: L. Wu, University of California, Riverside, CA 92521, USA. (laosheng.wu@ucr.edu)

Abstract

[1] Solute diffusion flux in soil is described by Fick's law along with a tortuosity factor to account for the tortuous and reduced diffusive pathway blocked by soil particles. Predictive models based on empirical or conceptual relationships with other more commonly measured soil attributes have been proposed to replace the time-consuming and multifarious laboratory measurements. However, these models have not been systematically tested and evaluated with soils of different textures under comparable conditions. This study determined solute diffusion coefficients and calculated tortuosity factors of a sand, a sandy clay loam, and a clay at various degrees of water saturation, and used the experimental data to test the predictive capabilities of these models. All the test models can fit the experimental data reasonably well as evidenced by low root mean square errors (RMSEs). When the proposed (fixed) parameter values were used, the widely accepted Millington and Quirk tortuosity model resulted in highest RMSEs for all three test soils. In terms of model efficiency as described by Akaike weight, however, the tortuosity factors of the sand and sandy clay loam soils are best represented by a quadratic function of volumetric soil water content (with the largest Akaike weights), while the combined parallel-series conceptual model assuming different configurations of film and pore water is the best for the clay soil. The Olesen power function tortuosity model has the second largest Akaike weights for the sand and sandy clay loam soils, while the So and Nye linear model has the second largest Akaike weight for the clay soil. The two-region linear model of log (tortuosity factor) versus soil water content uses a similar framework to the conceptual model, and it can satisfactorily fit to the experimental data well (low RMSEs), but with low Akaike weights due to the large number of parameters in the model. Adaption of the findings from this study may substantially improve solute diffusion modeling in unsaturated porous media.

1. Introduction

[2] Solute diffusion in soil refers to the transport of a dissolved constituent in aqueous phase from a higher concentration point toward a lower one. This concentration gradient driven process in soil determines the rate of solute transport in the vadose zone, which can influence the solute concentration meters away from the source [Helmke et al., 2004].

[3] An accurate estimation of solute diffusion flux in soil (J) is imperative. This flux (J) is described by Fick's law of diffusion in water along with a tortuosity factor (ξ) to account for the reduced cross-sectional area and longer pathway [Jury et al., 1991]:

display math

where D0 is the solute diffusion coefficient in water, Ds is the solute diffusion coefficient in soil, C denotes solute concentration, and ∂C/∂z denotes solute concentration gradient along the flux direction z. J, Ds, and ξ are defined in terms of the total cross-sectional area.

[4] The tortuosity factor (ξ) can be calculated from the ratio of solute diffusion coefficient in soil (Ds) to that in water (D0), and both Ds and D0 are experimentally determined [Millington and Quirk, 1961]. However, measurement of solute diffusion coefficient (Ds) in soil is a time-consuming and labor-intensive procedure [Dane and Topp, 2002]. Besides, experimental measurement of Ds is not an ordinary undertaking for those who frequently use this parameter [Shackelford and Daniel, 1991]. Consequently, a number of empirical tortuosity factor models have been proposed to predict solute diffusion coefficients in soils to replace the laboratory work [Moldrup et al., 1996, 2001, 2007; Olesen et al., 2001a]. These models are based on soil attributes such as soil water content, matric potential, texture, bulk density, and particle-size distribution that are known to affect the tortuosity factors [Lim et al., 1998; Phillips and Brown, 1965; So and Nye, 1989].

[5] Solute diffusion in soil may also be viewed on a microscopic scale. Under ordinary conditions, the water molecules in the aqueous phase of soil are distributed in two forms: continuous capillary tubes (pore water) and pendular rings around soil particles (film water) [Conca and Wright, 1992; Phillips and Brown, 1965; So and Nye, 1989]. The concept of pore and film water is the theoretical basis of the conceptual models to predict the solute diffusion coefficients in soils [Lim et al., 1998]. Hu and Wang [2003] used laser ablation-inductively coupled plasma-mass spectrometry to study diffusion in porous media. They found that at different soil water contents, the liquid phase is structured in different forms and geometric arrangements. At higher soil water contents (0.05 to 0.5 cm3 cm−3), the liquid phase occupies the voids that formed continuously connected pores, while at lower soil water contents (0.005 to 0.05 cm3 cm−3), the liquid phase is held in pendular form that results in a less continuous diffusive pathway. They indicated that the transition of liquid phase distribution from the continuously connected pores form to the pendular form can cause a sudden decline in solute diffusion coefficient.

[6] The tortuosity factor predictive models if validated may greatly simplify the estimation of solute diffusion coefficients in soil. Many studies have been conducted to test the applicability of the models, but none of them used comparable experimental conditions to systematically assess the effectiveness of the prediction models for soils with different textures and at various degrees of water saturation. The main objective of this study is to measure solute diffusion coefficients of three soils ranging from sand, sandy clay loam, and clay at various soil water contents to evaluate the strength and weakness of each model in terms of predictive capability, range of applicability, and requirement of input parameters.

2. Background

2.1. Empirical Tortuosity Factor Models

[7] The empirical tortuosity factor models link the tortuosity factor (ξ) to soil attributes in different mathematical forms. Among them, exponential, power, quadratic, and linear functions are the most common forms.

2.1.1. Exponential Function Models

[8] Olsen and Kemper [1968] proposed an exponential function tortuosity factor model using soil water content as the independent variable:

display math

where Ω and λ are empirical constants and math formula is the volumetric soil water content (cm3 cm−3). This model is applicable to soils at matric potential from −33 to −1500 kPa.

[9] Hamamoto et al. [2009] reported that log ( math formula) increased linearly with math formula. The linear relationship of log ( math formula) versus math formula implies a 10 based exponential function, which is mathematically in agreement with equation (2) of an e based exponential function. Hamamoto et al. [2009] further concluded that log ( math formula) versus math formula exhibited two distinctive linear regions (two-region linear model): a steeper slope (s1) when the soil water content is low and a more moderate slope (s2) when soil water content rises to higher level. This finding is in agreement with Hu and Wang [2003] who concluded that the liquid phase distribution at a relatively lower soil water content is different from that at a higher soil water content. The two-region linear model can be expressed as follows:

display math

where math formulac is the critical water content that divides the two regions, and math formulac is the tortuosity at math formulac.

[10] It is worth noting that the exponential model (equation (2)) proposed by Olsen and Kemper [1968] can be considered as a special case of the two-region model (equation (3)).

2.1.2. Power Function Models

[11] Papendick and Campbell [1980] proposed a power function tortuosity factor model in the form of

display math

where c and math formula are empirical constants.

[12] Based on the success in predicting gas diffusion in soil, Millington and Quirk [1961] developed a solute diffusion tortuosity factor model by replacing the volumetric air fraction in their gas diffusion tortuosity factor model with volumetric soil water content:

display math

where math formula, n, and m are empirical constants and φ is the total porosity (cm3 cm−3). The empirical constants were adopted from the gas diffusion model with math formula = 1, n = 10/3, and m = 2, regardless of soil type. The model has been widely applied for its simplicity, yet the outcomes were often challenged for the lack of representation. Sadeghi et al. [1989] proposed that math formula = 0.18 and n = m = 2.98 in place of the previously used constants. Since the porosity is generally considered a constant for a given soil, equation (5) is equivalent to equation (4) and consequently, our regression analysis did not include equation (5).

[13] Olesen et al. [1996] modified the Millington and Quirk [1961] solute diffusion tortuosity factor model (equation (5)) by defining the empirical constants in terms of measurable parameters through introducing the soil water retention parameter b (the slope of the Campbell [1974] soil water retention model) into this model:

display math

where math formula represents the tortuosity impedance factor ( math formula) at 100% soil water saturation, i.e., math formula, math formula, and k1, k2 are empirical constants. Olesen et al. [1996] found math formula equals 0.45 for all soils. According to Olesen et al. [1996], the term (k1+ k2b) accounts for the effect of water film thickness, water film disconnection, ion exclusion, and dead-end pores. The parameter b can be obtained via direct measurement or empirical formulas [Williams et al., 1992]. But for a given soil, k1 and k2 cannot be independently obtained since b is not a function of water content. In this paper we consider c as an empirical constant which can be obtained via the nonlinear regression. Therefore, parameter b is not required in this model.

2.1.3. Quadratic Function Models

[14] Mullins and Sommer [1986] proposed a quadratic function model by using a conjunctive component to constrain the increase of the tortuosity factor with the increase of soil water content:

display math

where δ and μ are empirical constants, with μ being a negative number to account for the constrained increasing trend.

[15] Olesen et al. [1996] further defined the tortuosity factor in equation (7) using the impedance factor at soil water saturation math formula and the threshold soil water content ( math formulath), below which the solute diffusion rate approaches zero due to poor connection of pore water and anion exclusion at extremely low soil water content:

display math

Equation (8) is mathematically similar to equation (7), the model proposed by Mullins and Sommer [1986]. The math formulath in equation (8) can be predicted from soil texture and bulk density [Olesen et al., 2000; Olesen et al., 2001b]:

display math

or by including soil water retention parameter (b) in the model:

display math

where CL is the mass fraction (g g−1) of clay (<0.002 mm), SF is the mass fraction (g g−1) of silt (0.002–0.02 mm), math formula is the soil bulk density (g cm−3), and b is the soil water retention parameter, as defined above.

[16] Olesen et al. [2001b] observed a linear relationship of impedance factor f with math formula, which implies that the tortuosity factor has the same mathematical form as the quadratic function model of Mullins and Sommer [1986]. Moldrup et al. [2007] combined the finding of Olesen et al. [2001b] and the concept of threshold soil water content ( math formulath) [Olesen et al., 1996] to develop a new tortuosity factor model, which is similar to the Mullins and Sommer [1986] model:

display math

where H is an empirical constant and math formulath is estimated by

display math

where OMF represents the organic matter fraction (g g−1), dcl is the density of clay particle (2.7 g cm−3), dom is the density of organic matter (1.0 g cm−3), and ϖ is a fraction constant that uses 1.0 for compacted or aggregated soils and 0.8 for nonaggregated and noncompacted soils with clay fraction between 5% and 40% (g g−1).

[17] Moldrup et al. [2007] further linked the Olesen et al. [2001b] model with the Campbell [1974] soil water retention model and developed a LIFE-Campbell model for the tortuosity factor. Different from many other models, this model uses the matric potential instead of soil water content as the independent variable, which incorporates the soil water energy status directly into the model:

display math

where math formulath is the intercept on the x axis of math formula/ math formula versus math formula, math formulas represents the saturated soil water content (cm3 cm−3), which is assumed equal to the soil total porosity math formula, ψ represents the soil water matric potential (cm H2O), ψe is the matric potential at air entry, and ψth is the threshold soil water matric potential below which the solute diffusion approaches zero.

2.1.4. Linear Function Model

[18] So and Nye [1989] proposed that the tortuosity factor is a linear function of bulk density and soil water content. But for a given soil, bulk density is generally considered as a constant independent of water content. In this paper we did not consider the effect of bulk density on ξ, hence the linear model can be written as

display math

where math formula and math formula are empirical constants.

2.2. Conceptual Model

[19] Lim et al. [1998] proposed that the soil liquid phase is distributed in two forms, namely, the pore water in the continuous capillary tubes and the film water along the particle-air boundaries. Based on this concept, a set of tortuosity models was proposed: (1) pore water and film water are in parallel arrangement (equation (15)); (2) pore water and film water are in series arrangement (equation (17)); and (3) a portion of the pore water and film water are in parallel and a portion in series arrangement (equation (18)).

[20] 1. Parallel Model

display math
display math

[21] 2. Series Model

display math

[22] 3. Combined Parallel and Series Model

display math

where D(S) is the solute diffusion coefficient at saturation S (%, equal to the volumetric water content divided by total porosity); D* is the solute diffusion coefficient at S = 100%; υ is the film water fraction of the residual water content (θr), which can be calculated based on the total surface area of the soil particles, an assumed water film thickness, and θr; Se is the effective degree of saturation that is equal to (S − Sr)/(1 − Sr) [Brooks and Corey, 1966]; R (equation (16)) is a factor that includes both the air-entrance effect and the effect of increasing viscosity and ionic interaction along small pores; a/r is the ratio of the radius of the hydrated ion to the radius of the pore that was set equal to 0.5 for all soil textures [Lim et al., 1998]; η is the proportion of diffusion along the paralleled arrangement; and 1 − η is the proportion of diffusion along the series arrangement. They concluded that equation (18) with a 0.5 in parallel and 0.5 in series best fitted their experimental data.

3. Materials and Methods

3.1. Sample Preparation

[23] Three soils, a sand, a sandy clay loam (SC loam), and a clay, were used in this study (Table 1). The sand was obtained from Weist Rentals and Sales® Masonry Material Supplies in Riverside, California. The sandy clay loam and clay soils were obtained from the Imperial Valley of California. Bromide concentration in the samples was undetectable. Collected samples were air dried and passed through a 2 mm opening sieve. Aliquots of the soil samples were packed into cylindrical acrylic Tempe® diffusion cells (Figure 1) to obtain predetermined bulk densities (Table 1). The diffusion cell consists of a high-flow porous ceramic plate at the bottom and connects to a tubing from which soil water can flow out.

Figure 1.

Schematic illustration of Tempe® diffusion cell.

Table 1. Physical Properties of the Three Test Soils
TextureParticle Size Distribution (g g−1, %)Bulk Density (g cm−3)Porosity (cm3 cm−3)Specific Area (m2 g−1)
Coarse SandFine SandSiltClay
Sand16.171.210.62.11.600.400.61
SC Loam0.665.312.621.51.450.4519.73
Clay0.25.536.457.91.600.4037.19

[24] Prior to the solute diffusion experiment, the soils in the prepared diffusion cells were first saturated with 0.01 M potassium bromide solution for 24 h, and then were placed into a pressure chamber for 3–10 days (until no solution came out from the pressure plates at their respective matric potentials) to equilibrate to the predetermined matric potentials. The corresponding equilibrium volumetric soil water contents at the predetermined matric potentials were recorded. The cells were then transferred to a 25°C, vapor saturated humidity chamber to equilibrate for another 7 days before a diffusion experiment. The diffusion experiments employed bromide as the tracer because it is a nonreactive element and it has negligible quantities in natural soils.

3.2. Measurement of Solute Diffusion Coefficients in Soils

[25] The diffusion experiments were conducted in the same 25°C vapor saturated humidity chamber. At time 0, a Cl saturated polythersulfone/copolymer anion-exchange membrane (PALL, SB-6407) was placed in contact with the exposed soil surface to act as the sink for Br diffusion inside the soil column [Tinker, 1969]. To minimize the convection flow between the membrane and the soil, the anion-exchange membrane was pre-equilibrated to the same matric potential along with the soil samples. A flat glass lid was laid on the top of the membrane to eliminate trapped air pockets and to ensure a good contact between the membrane and the soil surface (Figure 1). Based on the membrane's exchange capacity, at higher water contents, double or triple layers of membrane were laid to maintain a near-zero Br concentration at the soil-membrane boundary.

[26] At time t (t = 81 min in this research), the membranes were removed, rinsed with deionized water, and then dried. The bromide in the membranes was extracted by 50 mL of 0.5 M HNO3. The bromide concentrations in the extracts were determined by the spectrophotometric method [Chiu and Eubanks, 1989]. The solute diffusion coefficient in a soil at a given water content (θ) was then calculated using the measured Br quantity and time [Tinker, 1969]:

display math

where M is the amount of bromide per unit surface area that diffused into the anion-exchange membrane (g cm−2) in time t (s), C is the initial concentration of Br in the soil solution (g cm−3), and π = 3.14.

3.3. Model Evaluation

[27] Regression analysis was used to obtain the best-fit constants of each model, and the predictive capabilities of the models with empirical constants obtained from regression analysis were evaluated by comparing the model outputs with the measured data. The root mean square error (RMSE) was used to evaluate the model's goodness of fit to the experimental data [Olesen et al., 1996]:

display math

where di is the difference between the model estimated and the measured values and N is the number of measured values. A RMSE = 0 indicates a perfect model prediction of the experimental data.

[28] Another model evaluation criterion is Akaike information criterion (AIC) [Burnham and Anderson, 2002]. It is based on information theory and is widely used in model selection. Since our regressions are based on the least-squares principle, the AIC value can be calculated from the RMSE:

display math

where K is the number of parameters. Since our sample size N is small compared to the number of parameters ( math formula), the corrected Akaike information criterion (AICc) is used:

display math

[29] By itself, the AICc value calculated from a specific model has no meaning, but among the AICc values from different models, the model with the lowest AICc value is the best candidate model. The Akaike weight ( math formula) for the ith model can be calculated as follows:

display math

where math formula is the difference between AICc values from the ith model and the best model:

display math

[30] The Akaike weight ( math formula) can be interpreted as the probability that the ith model is the best among the candidate models. That is, the best model has the largest math formula.

4. Results

4.1. Exponential Function Models

[31] The exponential function tortuosity factor model (equation (2)) fits the experimental data of the sand, sandy clay loam, and clay soils equally well, as indicated by their respective low RSME values (Table 2). The best-fit parameter values are Ω = 0.001, 0.0009, and 0.0026, and λ = 7.7024, 12.1872, and 4.8717, respectively, for the sand, sandy clay loam, and clay soils. This mathematical relationship is totally empirical and the parameter values do not exhibit any pattern in terms of soil texture.

Table 2. Model Parameters and the RMSE Values for the Exponential Function Tortuosity Model [ξ = Ω exp(λθ)] and the Two-Region Linear Model [log (ξ) = s(θ − θc) + log (ξc); θθc, s = s1 and θθc, s = s2]
SoilExponential Model Equation (2)Two-Region Linear Model Equation (3)
ΩλRMSEs1s2θcξcRMSE
Sandy0.00107.70240.00245.12401.56100.34340.01650.0018
SC Loam0.000912.18720.00939.12214.63660.34760.07150.0071
Clay0.00264.87170.00166.89591.32380.23670.00970.0010

[32] The log ( math formula) versus math formula curve for the test soils were divided into two linear regions of distinctive slopes at the critical volumetric water content ( math formulac) [Hamamoto et al., 2009]. The slopes change abruptly at the critical soil water contents of 0.3434, 0.3476, and 0.2367 cm3 cm−3, respectively, for the sand, sandy clay loam, and clay soils (Figure 2 and Table 2). Again, the math formulac values do not exhibit any pattern associated with soil texture. This two-region linear relationship of tortuosity factor and volumetric water content indicates that two sets of constants, each for a different soil water regime can better represent the tortuosity factor as a function of soil water content.

Figure 2.

Experimental data (symbols) and regression lines of the two-region linear model of log (ξ) versus math formula [Hamamoto et al., 2009] for the three test soils.

4.2. Power Function Models

[33] The predictive capability of the power function model (equation (4)) was tested by comparing with constants derived from Millington and Quirk [1961], Papendick and Campbell [1980], Mehta et al. [1995], and from the nonlinear regression of the experimental data (Figure 3).

Figure 3.

Experimental data (symbols) and model predictions (lines) based on power function model (equation (4)) using the model constants derived from Millington and Quirk [1961], Papendick and Campbell [1980], Mehta et al. [1995], and nonlinear regression of experimental data.

[34] When their proposed empirical constants are used, Millington and Quirk [1961] tortuosity factor model (equation (5)) overestimates the experimental data for all the three test soils, as shown in Figure 3. The RMSE values are 0.1232, 0.0617, and 0.1193, respectively, for the sand, sandy clay loam, and clay soils (Table 3). Based on their experimental data, Papendick and Campbell [1980] proposed to use constants of c = 2.8 and math formula = 3.0 in equation (4) for all the soils. But these values do not represent the three test soils well, as shown in Figure 3 and indicated by the large RMSE values of 0.073, 0.035, and 0.113, respectively, for the sand, sandy clay loam, and clay soils.

Table 3. Model Parameters and RMSE Values by Fitting the Power Function Models of Papendick and Campbell [1980] (Equation (4)) and Olesen et al. [1996] (Equation (6)) to the Experimental Data
SoilPapendick and Campbell [1980] (Equation (4))Olesen et al. [1996] (Equation (6))Millington and Quirk [1961] (Equation (5))a
cεRMSEcRMSERMSE
  • a

    Same values of γ = 1, n = 10/3, and m = 2 were used in equation (5) for the three test soils.

Sand0.22642.51230.00201.30490.00210.1232
SC Loam7.37024.44080.00693.62570.00720.0617
Clay0.06241.40210.00150.61520.00160.1193

[35] Mehta et al. [1995] re-examined the power function model using a dune sand and a loamy soil. They found that there is a significant difference in the values of the model parameters at different soil water contents. Accordingly, they modified the empirical constants for their loamy soil as c = 2.1, math formula = 4.1 for soil water content below residual level; c = 3.8, math formula = 4.0 for soil water content near saturation level; and c = 1.3, math formula = 1.0 for soil water content between these two levels. They used c = 0.7 and math formula = 0.8 for the dune sand for all water contents because it did not show any change with respect to the soil water status. Although the textures of our test soils are not the same as those of Mehta et al. [1995], it is still meaningful to compare our experimental data with the predictions of their soils that closely match our test soils. We find that the predicted tortuosity factors for their dune sand are much higher than those for our sand soil in almost the entire range of soil water contents, but the predicted values for their loamy soil are in agreement with our measured values for the test sandy clay loam (Figure 3).

[36] The respective RMSE values given by the Olesen et al. [1996] power function model (equation (6)) are 0.0021, 0.0072 and 0.0016 for the sand, sandy clay loam, and clay soils, respectively. The best-fit parameter values are c = 1.3049, 3.6257, and 0.6152, respectively (Table 3). It is noted that, compared to the Papendick and Campbell [1980] model, although the Olesen et al. [1996] model gives slightly larger RMSE values, it has lower AICc values (Table 6). Therefore, if both the average mismatch (RMSE) and number of parameters are considered, the Olesen et al. [1996] model is better than the Papendick and Campbell [1980] model.

4.3. Quadratic Function Models

[37] There are several mathematical forms for the quadratic tortuosity factor models. Mullins and Sommer [1986] indicated that the best fitted δ and μ for equation (7) were 1.6 and −0.172 for their sandy clay loam. But these constants overestimate the measured tortuosity factors of our sandy clay loam over almost the entire range of soil water contents (Figure 4).

Figure 4.

Experimental data (symbols) and the nonlinear regression fitted lines of the quadratic function model (equation (7)) for the three test soils, and the predicted tortuosity factors for the sandy clay loam using the δ and μ values from Mullins and Sommers [1986].

[38] According to equation (7), the quadratic function model has a threshold water content at which the tortuosity factor is zero. The corresponding threshold soil water contents are 0.1197, 0.2505, and 0.1517 cm3 cm−3, respectively, for the test sand, sandy clay loam, and clay soils. Thus the model can only be applied when the soil water content is greater than the threshold water content such that the tortuosity factors are greater than 0.

[39] Equation (8) [Olesen et al., 1996] does not require any regression parameters. However, it needs an unmeasurable parameter of the threshold water content math formulath. This value can be estimated from soil particle-size distribution and bulk density according to equation (9), or together with the measured soil water retention parameter b (see equation (10)). Figure 5 shows the experimental data and the predicted values from the threshold soil water content tortuosity model (equation (8)) for the three test soils. Our data show that the incorporation of soil water retention parameter b via equation (10) does not reduce the RMSE for the three test soils. In terms of RMSE, the volumetric water content based power function model [equation (6), Olesen et al., 1996] outperforms the threshold soil water content tortuosity model (equation (8)) in all cases.

Figure 5.

Tortuosity factors from measured (symbols) and predicted (dash lines) by the threshold water content model [Olesen et al., 1996, equation (8)].

[40] Figure 6 shows the slopes (H) of the linear relationship of impedance factor ( math formula) versus water content ( math formula). The linear impedance factor model of equation (11) by Moldrup et al. [2007] generally fits the experimental data well using the θth estimated by equation (12). Nevertheless, although equation (11) is a rectified form of the quadratic function model of equation (7), the resulting RMSEs show that the model is incapable of improving the predictive capability of the tortuosity factor of the three test soils (Table 4). The H values are highly correlated with the uniformity of particle size distribution [Hamamoto et al., 2009]. Based on the H values, the sand (H = 0.11) and clay (H = 0.08) soils are considered to be rather uniform, while a greater value of H of 1.08 for the sandy clay loam soil (H = 1.08) indicates a considerably less uniform particle-size distribution. Increase in bulk density causes the decrease of H, and with a smaller H value, the tortuosity factor increases slower with the soil water content (Figure 6). Altogether, a more uniform particle size distribution and a higher bulk density can constrain the increase of tortuosity factor with soil water content.

Figure 6.

Impedance factor (f) versus volumetric soil water content ( math formula) for the three test soils.

Table 4. Parameters and RMSEs of the Mullins and Sommer [1986] Model, the Olesen et al. [1996] Model, and the Moldrup et al. [2007] Model
SoilMullins and Sommer [1986] (Equation (7))Olesen et al. [1996] (Equation (8))Moldrup et al. [2007] (Equation (11))
δμRMSEθth From Equation (9)RMSEθth From Equation (10)RMSEHθthRMSE
Sand0.2038−0.02440.00180.030.002100.00230.110.010.0031
SC Loam2.1679−0.54310.00630.140.03030.010.04441.080.120.0132
Clay0.08370.01270.00160.170.00410.010.00200.080.340.0031

[41] The LIFE-Campbell model of equation (13) is a modification of the linear impedance factor model of equation (11) [Moldrup et al., 2007] by using matric potential instead of soil water content as the independent variable. However, we encounter two problems to acquire math formulath from the intercepts on the x axis of the math formula/ math formula versus math formula linear regression (Figure 6). First, the r2 indicates that only the linear math formula/ math formula versus math formula relationship of sandy clay loam soil was significant. Second, the x axis intercepts of the sand and clay soils indicate negative soil water contents, which are unreasonable. Therefore, in order to evaluate the LIFE-Campbell model (equation (13)), we set math formulath of the sand and clay soils as 0. The x axis intercept thus occurs at math formula = 0.15 cm3 cm−3 for the sandy clay loam soil. The saturated soil water content math formulas is assumed to equal the total porosity as listed in Table 1. The air-entry matric potentials (ψe) are −40, −600, and −2040 cm H2O, respectively; and the soil water retention parameters (b) are 1.9, 1.6, and 4.9, respectively, for the sand, sandy clay loam, and clay soils. Although more parameters are used in this model, the relatively high RMSEs (data not reported) demonstrate the poor predictive capability of the model for the tortuosity factors. Use of the soil water retention parameter (b) does not improve the model predictive capability either. The poor predictive capability of the model (Figure 7) is attributed to the nonsignificant linear relation of f- math formula and the uncertainty in math formulath.

Figure 7.

Tortuosity factors of the test soils predicted by the LIFE-Campbell model of equation (13) versus experimentally determined values of the same soils.

4.4. Linear Function

[42] The parameters of the So and Nye [1989] model were obtained by fitting equation (14) to our experimental data. Judging by the RMSE, the linear model can well describe the relationship between the tortuosity factor and soil water content for the clay soil. However, RMSE values for the sand and sandy clay loam soils are large compared to other empirical models (Table 5).

Table 5. Model Parameters and RMSE Values Derived by Fitting the Linear Tortuosity Factor Model of Equation (14) [So and Nye, 1989] to the Experimental Data
SoilModel Constants and RMSEs
αβRMSE
Sand−0.00630.06180.0032
Sandy Clay Loam−0.06680.43400.0265
Clay−0.00440.05370.0014

4.5. Conceptual Model

[43] The measured solute diffusion coefficients at 100% degree of saturation (D*) are 4.20 × 10−7, 44.15 × 10−7, and 3.75 ×10−7 cm2 s−1 for the sand, sandy clay loam, and clay, respectively. Lim et al. [1998] assumed the fraction of the film water content to the total residual water content (υ) is related to the total surface area of soil particles. A υ = 0.1 was used for their coarse sandy soil. Based on the total surface areas of the test soils, an assumed water film thickness of three molecular layers of water (1 nm), and the measured residual water contents [Lim et al., 1998], the υ values are estimated to be 0.0195, 0.1907, and 0.17 for the test sand, sandy clay loam, and clay, respectively. For the combined parallel-series model of equation (18), the proportion of diffusion in parallel arrangement η and in series arrangement 1 − η was obtained by nonlinear regression. The best-fitted values are η = 0.625, 0.148, and 0.78, and thus 1 − η = 0.375, 0.852, and 0.22, respectively, for the test sand, sandy clay loam, and clay soils. The corresponding RMSE values are 0.0038, 0.0261, 0.0015, respectively.

[44] The parallel model of equation (15) poorly describes the tortuosity factors for the three soils (Figure 8). Because the parallel model has a linear relationship between the water content and tortuosity factor, it does not fit the seemingly nonlinear trend of experimental data (Figure 8). On the other hand, the series model of equation (17) and the combined model of equation (18) exhibit nonlinear relationships between the water content and tortuosity factor that are more or less in line with the experimental data (Figure 8). Overall, the series model better predicts the tortuosity factors at the lower water contents while the combined model better predicts the tortuosity factors at the higher water contents. Compared to all the previous models, the AICc value from the combined model for the clay soil is the lowest, which indicates the combined model is the best for the clay soil (Table 6).

Figure 8.

The experimental data (symbols) and the predicted lines of the Lim et al. [1998] conceptual models: parallel (equation (15)), series (equation (17)), and combined (equation (18)).

Table 6. AICc Weights for the Test Modelsa
ModelsParametersAIC Weights
SandSCLClay
  • a

    The optimal models are with the largest AICc weights (underlined), i.e., the largest probability to be the best.

Olsen and Kemper [1968] (equation (2))Ω, λ0.00180.00020.0219
Hamamoto [2009] (equation (3)) math formula0.03440.00250.1754
Papendick and Campbell [1980] (equation (4))c, ε0.12900.09940.0960
Olesen et al. [1996] (equation (6))c0.18150.19990.1826
Mullins and Sommer [1986] (equation (7))δ, μ0.65330.69810.0346
Moldrup et al. [2007] (equation (11))H0.00010.00000.0000
So and Nye [1989] (equation (14))α, β0.00000.00000.1902
Lim et al. [1998] (equation (18))η0.00000.00000.2994

[45] Despite evidence that the conceptual models [Lim et al., 1998] fail to predict the tortuosity factors over the entire range of water content for the sand and sandy clay loam, it depicts the transition of the diffusive pathway as water content changes. For example, the series model of equation (17) predicts the tortuosity factors well at lower soil water contents, indicating that the liquid phase at lower water content is composed of film water connected to pore water. At higher soil water contents, the tortuosity factors are best described by the combined parallel-series model of equation (18) when η > 0.5, showing that the solute diffusive pathway at higher soil water contents is composed of parallel continuous film water as well as pore water.

5. Summary and Conclusion

[46] This research measured solute diffusion coefficients of three soils with different textures. Using the tortuosity factors calculated from the experimental data, we evaluated the existing models that are used to describe the tortuosity factor as a function of soil water content. All the test models can represent the relationship between tortuosity factor soil volumetric water content reasonably well, as evidenced by low root mean square errors (RMSEs). However, when the proposed (fixed) parameter values were used, the widely accepted Millington and Quirk [1961] tortuosity model resulted in highest RMSEs for all the three test soils.

[47] By evaluating the predictive capabilities of tortuosity factor models in terms of their respective Akaike weights as summarized in Table 6, we conclude that the solute diffusion tortuosity factors, and consequently the solute diffusion coefficients of the sand and sandy clay loam soils, are best described by a quadratic function of soil volumetric soil water content, i.e., the Mullins and Sommer [1986] model. The Olesen power function tortuosity model has the second largest Akaike weights for the sand and sandy clay loam soils, while the So and Nye linear model has the second largest Akaike weight for the clay soil. For the clay soil, the combined parallel-series conceptual model (equation (18)) provides the best description (largest Akaike weight) for the change of tortuosity factor with soil water content. Although the conceptual models do not predict tortuosity factors of the sand and sandy clay loam soils as well as the empirically models, they nevertheless demonstrate the transition of solute diffusive pathway between film water and pore water. There is a potential to employ the series model at lower water content and the combined parallel-series model at higher water content, if model parameters of a/r, Sr, and υ articulated in Lim et al. [1998] can be further refined.

[48] The two-region linear behavior of log ( math formula) versus math formula with the model constants proposed by Mehta et al. [1995] accommodates the fact that the solute diffusive pathway may shift from meniscus form (film water) at lower water content to continuous capillary tubes (pore water) at higher water content, and consequently, the model employs two different linear functions to account for the tortuosity factors associated with the two forms of soil water. It can satisfactorily fit to the experimental data well (low RMSEs), but with low Akaike weights due to the large number of parameters in the model. Moreover, the two-region behavior of tortuosity as a function of water content is consistent with the Lim et al. [1998] conceptual model.

[49] Currently, most simulation models use the Millington and Quirk [1961] tortuosity factor model (equation (5)). Our evaluation shows that this model overestimates the experimental data for all three test soils when their suggested model parameters are used. Adaption of the findings from this study may substantially improve modeling of solute diffusion in the vadose zone.

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