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DIVISION S-1—SOIL PHYSICS

Using Particle-Size Distribution Models to Estimate Soil Hydraulic Properties

Dep. of Civil and Environmental Engineering, Clarkson Univ., 8 Clarkson Ave., Potsdam, NY 13699-5710

^* Corresponding author (sep{at}clarkson.edu)

	ABSTRACT

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

 
Mathematical models for soil water retention characteristic [h()] and unsaturated conductivity function [K()] from particle-size distribution (PSD) and bulk density data are indirect and empirical approaches to estimate these hydraulic functions. Often times, mathematical models are fit to sparse PSD data sets to provide the input for h() and K() functions. The objective of this study was to determine whether the choice of a particular mathematical model to represent the continuous PSD significantly affects the predicted soil hydraulic properties. We considered four PSD models with between one and four fitting parameters. For most of the soils, the one-parameter Jaky model for generating PSD input, resulted in the best estimate of soil hydraulic properties. This finding indicates that the linear relationship between the PSD and the void-size distribution (VSD) (or between particle volume and pore volume) defined by the AP model would be not appropriate for most soils. This result suggests that the nonlinear relationship between the PSD and the VSD of the Jaky model would more appropriate. The PSD generated by other models provided better input to the h() and K() functions for clay soils.

Abbreviations: AIC, Akaike's information criterion • AP, Arya-Paris • h(), water retention curve • K(), hydraulic conductivity function • ONL, offset-nonrenormalized lognormal • PSD, particle-size distribution • RMSE, root mean square error • SL, simple lognormal • VSD, void-size distribution

	INTRODUCTION

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

 
THE SOLUTION of equations describing the flow of water in unsaturated soils requires the expression of two soil hydraulic properties, the water retention curve and the hydraulic conductivity function. The water retention curve (h[]) describes the relationship between the pressure head, h, and the volumetric water content, . The hydraulic conductivity function (K[]) describes the relationship between the unsaturated hydraulic conductivity, K, and . Direct measurements of both functions are relatively time-consuming in laboratory and field conditions.

Particle-size distribution data have been widely used as a basis for estimating soil hydraulic properties. Significant contributions were made by Arya and Paris (1981) to predict h() curves using the PSD. Their physico-empirical approach is based mainly on the similarity between shapes of the cumulative PSD and h() curves. Various authors have developed similar models (Haverkamp and Parlange, 1986; Wu et al., 1990; Smettem and Gregory, 1996; Zhuang et al., 2001). The AP model was later refined by Arya et al. (1999a).

Arya et al. (1999b) also derived a model to compute K() directly from the PSD. Unlike other models, the need for both measured h() and the saturated hydraulic conductivity is eliminated. They found that the shapes of the predicted K() were similar to those of the measured data and the average of root mean square errors (RMSEs) for all textures was similar to that which had been observed with other hydraulic conductivity prediction models (e.g., Schaap and Leij, 1998).

Since most models mentioned above are based on the PSD information, details of a PSD curve may affect estimates of h() and K(). For example, Arya et al. (1999a)(b) suggested that PSDs comprised of at least twenty fractions are necessary to reasonably calculate their own models. However, experimental PSD data sets usually have a limited number of data points. For example, the Korean soil database containing 1387 soils has seven PSD data points for each soil (Hwang et al., 2002). Soils in the international Unsaturated Soil Database (UNSODA) used by Arya et al. (1999a)(b) usually have from four to eleven PSD data points. This indicates that a procedure is needed to generate a detailed PSD from a limited number of experimental PSD data points to provide accurate prediction of h() and K().

Little consistency exists among the selection of the PSD interpolation methods for this approach. For example, Haverkamp and Parlange (1986) and Smettem and Gregory (1996) adopted the van Genuchten (1980) function to fit the experimental cumulative PSD data. Wu et al. (1990) used a logistic function, whereas Zhuang et al. (2001) adopted two functional relationships, that is, the van Genuchten (1980) function for sand and loamy sand soils and a logarithm function for other textures, to generate detailed PSD. Recently, Hwang and Powers (2002) adopted a simple lognormal distribution function for this procedure. In addition, among several studies using the AP and Arya et al. (1999b) models, most researchers did not describe explicitly how detailed PSD data were generated from experimental PSD data points (e.g., Arya and Paris, 1981; Gupta and Ewing, 1992; Jonasson, 1992; Basile and D'Urso, 1997; Arya et al., 1999a,b), whereas Schuh et al. (1988) adopted a b-spline interpolation procedure (Rogers and Adams, 1976) and Mishra et al. (1989) used a simple lognormal distribution function.

Besides the PSD models mention the above, other diverse PSD models have been proposed to generate detailed PSD from experimental data. These include lognormal models with different number of parameters (Buchan et al., 1993), exponential and power-law distribution models (Rousseva, 1997), models based on the water-retention curve (Fredlund et al., 2000), the Gompertz model (Nemes et al., 1999), the fragmentation model (Bittelli et al., 1999), a Weibull distribution function (Assouline et al., 1998; Kotrechko et al., 2001), a model estimating PSD from limited soil texture data (Skaggs et al., 2001), and a model based on the relationship between the cumulative PSD and the fractal dimension of the PSD (Medina et al., 2002). Hwang et al. (2002) compared the capability of seven PSD models with different underlying assumptions to fit experimental PSD data of 1387 soils in the Korean soil database. They found that the four-parameter Fredlund et al. (2000) model performed best even when three statistical criteria that impose a penalty for additional fitting parameters were used to assess the quality of the fit. Nemes et al. (1999) evaluated four different procedures to interpolate PSDs to achieve compatibility within soil databases. They found that a log-linear interpolation procedure was the least accurate for estimating missing particle-size classes for the soil databases they studied.

There has been little attempt to quantitatively investigate the effect of the choice of a PSD model on the prediction of h() and K() curves. Thus, the objectives of this study were to (i) determine whether estimates of these functions are affected by the selection of a PSD model; and, (ii) determine if the use of a PSD model with better fitting ability also provides better predictions of h() and K() data. To achieve these objectives, we selected the AP model (Arya and Paris, 1981; Arya et al., 1999a) because it has been used extensively for h() estimation (e.g., Schuh et al., 1988; Mishra et al., 1989; Gupta and Ewing, 1992; Jonasson, 1992; Basile and D'Urso, 1997; Nimmo, 1997). For the K() estimates, we used a model developed by Arya et al. (1999b). To generate detailed PSD curves from experimental PSD data, we used four parametric models, each with a different number of parameters. Statistical analyses were then used to compare published h() and K() data to predictions based on these detailed PSD curves and the AP and Arya et al. (1999b) models.

	Background

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

 
The AP model assumes that the VSD is directly related to the PSD. The PSD curve is divided into n mass fractions, and the solid mass in each fraction (w_i) is assembled to form a hypothetical, cubic close-packed structure consisting of uniform size spherical particles. The pore volume in each mass fraction is calculated from the bulk density and particle density measured on the natural structure soil, progressively summed and considered filled with water. The volumetric water content (_i) is obtained by summing the individual pore volumes divided by the bulk volume of the sample. An equivalent pore radius (r_i) is calculated for each mass fraction and converted to soil water pressure head using the Young-Laplace equation for capillary rise in a tube. Calculated pressure heads are sequentially paired with calculated water contents to obtain h().

To formulate a relationship between r_i and particle radii (R_i), Arya and Paris (1981) assumed that r_i is determined by scaling the pore length. Pore lengths based on spherical particles were scaled to natural pore lengths using a scaling parameter, , with an average value of 1.38. The resulting relationship between r_i and R_i is described by

[1]

where e is the void ratio, and n_i is the number of spherical particles with R_i. Arya et al. (1999a) proposed three methods to estimate , which is equal to log N_i/log n_i where N_i is the scaled number of spherical particles in the corresponding natural structure soil. The first method was to relate log N_i and log n_i using a logistic growth curve. The second method was to relate log N_i and log (w_i/R_i³) linearly. The third method was to give a single value for each soil texture by fitting a linear regression with zero intercept to the plot of log N_i vs. log n_i. They found that Method 1 is the most appropriate formulation for . In our study, we adopted Method 2 because (i) Method 2 was found to be comparable with Method 1 (Arya et al., 1999a), and (ii) the Arya et al. (1999b) model used to predict K() in our study adopted Method 2 to calculate pore radii from particle radii.

The Arya et al. (1999b) model assumes that equivalent capillary tubes can represent soil pores and that the water flow rate is a function of pore size. The VSD is derived from the PSD using the AP model. The pore size is converted to the pore flow rate that then is used to calculate hydraulic conductivity using Darcy's law. Calculated hydraulic conductivities are sequentially paired with calculated water contents to obtain K().

Detailed information on the equations and parameters required to estimate h() and K() are presented by Arya and Paris (1981) and Arya et al. (1999a)(b).

	MATERIALS AND METHODS

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

 
The UNSODA Hydraulic Property Database
Experimental h() and K() data, PSD, bulk density, and particle density data were obtained from the UNSODA hydraulic property database (Nemes et al., 2001). Among the 45 soils adopted from the UNSODA database, Arya et al. (1999a) used 24 soils for calibrating their three methods to formulate , and used the remaining soils (21) for testing the three methods. In our study, 24 soils used for formulating were again adopted to calculate h() (Table 1). Through this selection, we may eliminate the variation in that would be possible when other soils are used. The 24 soil data sets represented a range of textures that include sand, sandy loam, loam, silt loam, and clay. These soil data sets had 4 to 11 experimental PSD data points and between 6 to 30 experimental h() data points.

Selection of PSD Models
To estimate h() and K() from a PSD using the AP and Arya et al. (1999b) models, we need the PSD curve divided into n fractions and w_i calculated from the difference in cumulative mass fraction corresponding with successive particle sizes. Interpolation in some fashion is required to estimate cumulative mass fractions corresponding with successive particle sizes from a limited number of experimental PSD data points. Parametric PSD models are typically used to provide a continuous mathematical representation of the complete PSD.

In general, the fitting performance of a PSD model improves as the number of fitting parameters increases (Hwang et al., 2002). Since one of the aims of this study was to evaluate if the PSD model with greater number of fitting parameters also predicts better soil hydraulic properties, we needed to select PSD models with different numbers of fitting parameters. Non-parametric models, such as cubic spline interpolation (Kastanek and Nielsen, 2001), were not considered. The PSD in soil is frequently assumed to be approximately lognormal (Shirazi and Boersma, 1984; Campbell, 1985; Buchan, 1989). Hwang et al. (2002) compared the capability of seven parametric PSD models with different underlying assumptions, including five lognormal models and the Gompertz and Fredlund models. Among the PSD models studied by Hwang et al. (2002), we selected four PSD models with different number of parameters ranged from one to four. Each PSD model may yield different w_i, providing different predicted h() or K() pairs. Three lognormal models previously studied by Buchan et al. (1993) and Hwang et al. (2002) were used: the one-parameter Jaky model (Jaky, 1944) with a sigmoid half of a Gaussian lognormal distribution, which was first introduced into the soil science literature from the geotechnics by Buchan et al. (1993); a simple lognormal (SL) model with two parameters (Buchan, 1989); and one modified lognormal model with three parameters, that is, an offset-nonrenormalized lognormal model (ONL) (Buchan et al., 1993). Buchan et al. (1993) provides details of the three lognormal models. The Fredlund et al. (2000) model with four parameters was also tested. The four models considered in this study are listed in Table 2.

Calculation of Water Retention and Hydraulic Conductivity Curves
We fitted four PSD models to the experimental PSD data for each soil to estimate a full range of PSD from the limited number of experimental data. An iterative nonlinear regression procedure was employed to find the values of the fitting parameters that give the ‘best fit’ between the PSD model and the experimental PSD data. This procedure was done using the SOLVER routine of Microsoft Excel software (Microsoft Corp, Redmond, WA) (Wraith and Or, 1998; Hwang et al., 2002). At least three different initial parameter estimates for each soil were used to ensure the iterations converged on a unique solution. Akaike's information criterion (AIC; Akaike, 1973) was used to compare the quality of model fits while taking into account for the variable number of parameters in the PSD models. The AIC was defined by

[2]

where m_i(d_i) and are observed and predicted cumulative mass fractions of the particles with diameter smaller than d_i, respectively, N is the number of PSD data, and p is the number of model parameters.

The cumulative mass fraction at each fraction boundary was estimated using the above fitting results for four PSD models. We divided a PSD curve for each soil into twenty size fractions (i.e., n = 20) with fraction boundaries at particle diameters of 1, 2, 3, 5, 10, 20, 30, 40, 50, 70, 100, 150, 200, 300, 400, 600, 800, 1000, 1500, and 2000 µm (Arya et al., 1999a,b). This yielded twenty corresponding pairs of mass fraction, w_i, and mean particle radii, R_i, for each PSD model. Values of h_i(_i) and K_i(_i) were then calculated by the procedures outlined by Arya and Paris (1981) and Arya et al. (1999a)(1999b).

Impact of Different PSD models on Water Retention and Hydraulic Conductivity Predictions
Three PSD models (Jaky, SL, ONL) used in our study have the same underlying assumption that the PSD in soil is lognormally distributed. This implies that the PSD inputs derived from these models could result in similar predictions of the h() and K() functions. Therefore, to determine whether statistically identical estimates resulted between the PSD model pairs, including the Fredlund model, we conducted paired t tests on each log-transformed predicted h_i or K_i values at selected water contents.

The r², AIC, and RMSEs were used for statistical comparison to determine whether a PSD model that better fits experimental data also results in better estimates of experimental h(_i) and K(_i) data. The prediction of h() and K() data with the AP and Arya et al. models resulted in 20 data points for each soil. To interpolate predicted h_i and K_i values at experimental water contents from these data points, we adopted the van Genuchten (van Genuchten, 1980) and the van Genuchten–Mualem (Yates et al., 1992) models. The SOLVER routine of Microsoft Excel software (Microsoft Inc., Redmond, WA) was used to fit these nonlinear functions to predicted data. The r², AIC, and RMSEs were computed using log-transformed experimental and predicted h_i and K_i values.

	RESULTS AND DISCUSSION

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

 
Fitting Ability of Particle-Size Distribution Models
Among all 24 soils and all of the models, values of AIC for the PSD models fit to experimental PSD data ranged from -66.8 to -9.8 (Fig. 1). The smaller value of AIC indicates better fitting performance. As expected, the Fredlund model with the greatest number of parameters had smaller AIC values than the Jaky model with only one parameter. There was no distinct difference in the fitting performance between the SL and the ONL models, suggesting that it may result in the similar performance for predicting h() and K().

View larger version (22K): [in this window] [in a new window]   Fig. 1. Box plot for Akaike's information criterion (AIC) percentiles as the goodness-of-fit of four PSD models for all 24 soils.

View larger version (20K): [in this window] [in a new window]   Fig. 2. Comparative fit of (a) the PSD and (b) h() curves using four PSD models for a sand (UNSODA No. 3132).

View larger version (30K): [in this window] [in a new window]   Fig. 3. An illustration of experimental and predicted h() curves using four PSD models for each textural class, (a) sand (UNSODA 4650), (b) sandy loam (UNSODA 3310), (c) silt loam (UNSODA 1341), and (d) clay soils (UNSODA 1400).

View larger version (31K): [in this window] [in a new window]   Fig. 4. An illustration of experimental and predicted K() curves using four PSD models for each textural class, (a) sand (UNSODA 4650), (b) sandy loam (UNSODA 4160), (c) loam (UNSODA 2531), and (d) clay soils (UNSODA 1400).

The Fredlund model, which had the best ability among the models to fit PSD data, did not show the best performance for estimating h() and K() (Fig. 3 and 4). This trend was confirmed by the r², AIC, and RMSE analyses on the predicted and experimental h_i or K_i data for each PSD model (Fig. 5). The Jaky model had the highest r² value and the smallest AIC (or RMSE) values in both h() and K() functions. The superiority of the Jaky model for predicting h() and K() indicates that the model with greater fitting ability to the PSD did not guarantee better prediction of soil hydraulic functions.

View larger version (44K): [in this window] [in a new window]   Fig. 5. Comparison of experimental and predicted (a) pressure heads and (b) hydraulic conductivities for four PSD models. The 24 soils are pooled for pressure heads and 12 soils for hydraulic conductivities.

As shown in Fig. 1, the Fredlund model showed the best fitting for experimental PSD data while the opposite was observed in the Jaky model. Therefore, if the AP and Arya et al. (1999b) models are correct, the Fredlund model should show the best estimates for the soil hydraulic data. However, the Jaky model showed the best estimates, showing the possibility of incorrectness of the AP and Arya et al. (1999b) models in estimating soil hydraulic properties.

Errors associated with the AP model could result from the assumptions employed. These include (i) the assumption of discrete particle-size domain, and (ii) the assumption that the pore volume in the ith particle-size fraction (V_vi) is proportional to w_i (e.g., Schuh et al., 1988). Most soils consist of mixed particles, with smaller grains occupying voids between larger grains (Schuh et al., 1988). Moreover, natural packing densities (and porosities) vary with particle size (Gupta and Larson, 1979).

The AP model defines the particle volume (V_pi) and V_vi, respectively, by

[3]

[4]

where _s is the particle density. This definition results in the linear proportionality between V_pi and V_vi. Also, the AP model defines the linear relationship between the PSD and the VSD (Eq. [1]). However, the linear relationship between the PSD and the VSD (or between particle volume and pore volume) would be not appropriate for the multicomponent particle systems such as soils. To investigate the relationship between the PSD and the VSD, we plotted the PSD and the VSD of two soils (sand and silt loam soils) as affected by the PSD models. Figure 6 shows that the VSD predicted from each PSD model was very similar to its corresponding PSD and the Jaky model predicted a slightly broader PSD and VSD than other models. If the linear relationship between the PSD and VSD assumed by the AP model were correct, the most appropriate representation for the VSD would result from the Fredlund model showing the best performance for fitting experimental PSD data. However, the Jaky model showed the best performance for predicting the h() curve. For example, its AIC values were smaller than those of other models (Table 3). This indicates that the VSD resulting from the Jaky model may be more suitable than those resulted from other PSD models. This result suggests that the nonlinear relationship between the PSD and the VSD would be more appropriate. In a study on the relationship between the PSD and the VSD in multicomponent sphere packs, Rouault and Assouline (1998) showed that the linear relationship might not be adequate. Their study proposed a probabilistic approach to compute the VSD of a sphere pack given its PSD. The approach was applied to power-function, Gaussian and lognormal PSDs. In the case of the PSD with power-function, they found that the corresponding VSD was bell-shaped. For the two other cases, no similarity was found between the distributions themselves, indicating that the relation between the PSD and the VSD was not a simple linear relationship. Hence, they concluded that (i) the linear relationship assumed by Arya and Paris (1981) and Haverkamp and Parlange (1986) might be inadequate for multicomponent particle system such as soils where the pore volume may be not proportional to the particle volume, and (ii) the nonlinear relationship would be more appropriate for reproducing the VSD.

View larger version (33K): [in this window] [in a new window]   Fig. 6. Examples of the PSD and the VSD of two soils as affected by the PSD models: (a1) the PSD and (a2) the VSD of a sand soil (UNSODA No. 4650), and (b1) the PSD and (b2) the VSD of a silt loam soil (UNSODA No. 1341).

[5]

where q_i is the volumetric flow rate for a single pore and N_pi is the number of pores in the ith pore-size fraction. Because q_i and N_pi are a function of r_i (Arya et al., 1999b), Q_i also depends on the VSD. When the Jaky model was used for fitting the PSD, the resulting VSD was a slightly broader than other PSD model (see [a2] and [b2] in Fig. 6), indicating that the Jaky model had greater probability densities in the VSD than other PSD models in smaller and larger pore sizes. As an example, we plotted r_i vs. Q_i for a sand soil (UNSODA No. 4650) (Fig. 7). The Jaky model showed greater Q_i in larger pore sizes. Other soils showed similar trends (the data and figures are not shown). So, its resulting K() curve showed higher K_i values over high saturation range (Fig. 4a). Since the prediction from the Jaky model showed the best performance for predicting K() curve for most soils (Table 3 and Fig. 5), it suggests that the VSDs represented by the Jaky model may be more appropriate for K() prediction than those resulted from other PSD models, even though the Jaky model showed the poorest PSD fitting.

View larger version (20K): [in this window] [in a new window]   Fig. 7. An example of ri vs. Qi plot for a sand soil (UNSODA No. 4650).

In addition, we used textural class average values for the parameters needed in the AP and Arya et al. (1999 a,b) models. However, there are a wide range of variations in PSD, bulk density, mineralogy, and organic matter content within a textural class because of various geographical origins of the UNSODA soil database (Nemes et al., 2001). Grouping soils together solely on the basis of textural nomenclature may introduce variations up to 25 percentage points in the mass fraction for some particle-size ranges (Arya et al., 1999a, b). Thus, textural similarities do not necessarily translate into hydrophysical similarities. The AP model is based on the assumption that the size of the particles and the bulk density are primary determinants of the pore size. The Arya et al. (1999b) model assumes that the water flow rate is a function of pore size as determined from particle sizes and bulk density. Therefore, the only input data required to calculate h() and K() curves are a PSD and bulk density. However, real soils may have aggregation of primary particles into secondary and tertiary particles, root channels, and microcracks, suggesting that these factors could not be fully represented only by the PSD and bulk density. Schuh et al. (1988) suggested that the superiority of laboratory methods over the AP model in A horizons would be expected because of the presence of organic matter and structure, which is not accounted for in the AP model. Also, several researchers have developed and investigated several methodologies to predict h() and K() curves from both textural and structural information (e.g., Nimmo, 1999; Lin et al., 1999; Zeiliguer, 1999). They found that structural features have strong effects on h() and K() and that mathematical functions that consider structural information give better prediction over the texture-based model.

Texture may affect the performance of PSD models for the prediction of h() and K(). In contrast with other soil classes, the Jaky model was not superior to other models for clay soils (Fig. 8). This trend can be confirmed by the illustrations shown in Fig. 3d and 4d and greater AIC and RMSE values. The poorer performance of the one-parameter Jaky model for high-clay soils is not surprising. The AP model is based on an assumed correspondence between PSD and pore-size distribution, and more specifically on the assumption of water retention within pores by capillary action. However, the clay fraction retains water primarily by surface sorption, rather than by capillary action, so that the AP model is strictly inadequate as a model to predict the contribution of the clay fraction to bulk hydraulic properties. Basile and D'Urso (1997) suggested that the AP model is a commonly accepted method for rigid soils with medium grain size but limitations to its application for fine textured soils occur, because of the dominant role of internal structure in such soils. Therefore, while the other PSD models with greater number of parameters (e.g., SL, ONL, Fredlund) may partly be able to compensate for this inadequacy, the one-parameter Jaky model presumably cannot accommodate the more complex water retention behavior of high-clay soils.

View larger version (72K): [in this window] [in a new window]   Fig. 8. Variation of AIC and root mean squared errors (RMSEs) associated with estimates of (a) h() curve and (b) K() curve for each textural class.

	CONCLUSIONS

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

	ACKNOWLEDGMENTS

 
We thank Dr. Walt Russell of USDA-ARS, George E. Brown Jr. Salinity Laboratory for his kind supply on the UNSODA database and Mr. Kwang-Pyo Lee of Seoul National University in South Korea for careful preparation of soil database used in our study. This research was partly supported by the U.S. Department of Energy, Environmental Management and Science Program (DE-FG07-99ER 15006).

Received for publication May 29, 2002.

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

 

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