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Published online 27 October 2005
Published in Soil Sci Soc Am J 69:1902-1911 (2005)
DOI: 10.2136/sssaj2004.0238
© 2005 Soil Science Society of America
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Soil Physics

Comparison of Unimodal Analytical Expressions for the Soil-Water Retention Curve

Wim M. Cornelisa,*, Muhammed Khlosia, Roger Hartmanna, Marc Van Meirvennea and Bruno De Vosb

a Dep. Soil Management and Soil Care, Ghent Univ., Coupure links 653, B-9000 Ghent, Belgium
b Institute of Forestry and Game Management, Ministry of the Flemish Community, Gaverstraat 4, B-9500 Geraardsbergen, Belgium

* Corresponding author (wim.cornelis{at}UGent.be)


   ABSTRACT
TOP
 NOTES
 ABSTRACT
INTRODUCTION
 REVIEW OF SOME SOIL-WATER...
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
This study was conducted to evaluate ten closed-form unimodal analytical expressions to describe the soil-water retention curve, in terms of their accuracy, linearity, Akaike Information Criterion (AIC), and prediction potential. The latter was evaluated by correlating the model parameters to basic soil properties. Soil samples were taken in duplicate from 48 horizons of 24 soil series in Flanders, Belgium. All sample locations were under forest and hence the samples had, besides their difference in texture, a high variety in bulk density ({rho}b) and organic matter content (OM). The van Genuchten model with m as a free parameter showed the highest overall performance in terms of goodness-of-fit. It had the highest accuracy, the highest degree of linearity, and the lowest AIC value. However, it had a low prediction potential. Imposing the constraint m = 1 – 1/n and hence reducing the number of model parameters by one, increased the prediction potential of the model significantly, without loosing much of the model's accuracy and linearity. A high degree of accuracy and linearity was also observed for the two Kosugi models tested. Restricting the bubbling pressure to be equal to zero resulted in a rather high prediction potential, which was not the case when keeping the bubbling pressure as a free parameter. A major drawback of van Genuchten and Kosugi type models is that they do not define the soil-water retention curve beyond the residual water content. We further demonstrated that the performance of all but one model in terms of their match to the data increased with increasing clay content and decreasing sand content, which is contradictory to the deterministic character of these models. Bulk density and OM did not have a significant effect on the accuracy of most models.

Abbreviations: A1, Assouline et al. (1998) with five free parameters • A2, Assouline et al. (1998) with four free parameters • AIC, Akaike Information Criterion • BC, Brooks and Corey (1964) • K1, Kosugi (1994) • K2, Kosugi (1996, 1997) • OM, organic matter content • PTF, pedotransfer function • R, Russo (1988) • RMSE, root of mean squared error • RN, Rossi and Nimmo (1994) • SWRC, soil-water retention curve • T, Tani (1982) • VG1, van Genuchten (1980) with five free parameters • VG2, van Genuchten (1980) with four free parameters


   INTRODUCTION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
REVIEW OF SOME SOIL-WATER...
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
WATER RELATIONS are among the most important physical phenomena that affect the use of soils for agricultural, ecological, environmental, and engineering purposes. To formulate soil-water relationships, soil hydraulic properties are required as essential inputs. The most important hydraulic properties are the soil-water retention curve (SWRC) and the hydraulic conductivity. The SWRC describes the relationship between the soil's matric potential {psi} and its water content {theta} and is a unique soil characteristic.

To be useful in modeling processes depending on soil-water relationships, a continuous representation of the SWRC is required and needs to be incorporated in predictive models. One of the most manifest examples is the use of the SWRC to indirectly determine the unsaturated hydraulic conductivity, using statistical pore-size distribution models (see e.g., Mualem, 1986, for a review), which are well represented by the SWRC. Such SWRCs can be obtained by fitting closed-form analytical expressions containing several parameters to discrete ({theta}, {psi}) data sets, which can be obtained through laboratory experiments or from pedotransfer functions (PTFs) that estimate distinct SWRC data pairs. The most widely adopted and best-performing PTFs enable, however, to directly predict the parameters of some closed-form analytical expressions (Cornelis et al., 2001). Applications of closed-form analytical expressions can also be attractive for other reasons, apart from their incorporation in predictive models. Van Genuchten et al. (1991) mention their applicability in more efficiently representing and comparing hydraulic properties of different soils and soil horizons, in scaling procedures for characterizing the spatial variability of soil hydraulic properties across the landscape and in interpolating and extrapolating to parts of the soil-water retention or hydraulic conductivity curves for which little or no data are available.

The objective of our study was to evaluate ten closed-form unimodal analytical expressions, including those reported by Brooks and Corey (1964) (BC), van Genuchten (1980) (VG1 and VG2), Tani (1982) (T), Russo (1988) (R), Rossi and Nimmo (1994) (RN), Kosugi (1994)([K1], 1996, 1997 [K2]) and Assouline et al. (1998) (A1 and A2), in terms of their accuracy, linearity, AIC, and prediction potential. These models were retained in this study because they are widely adopted and cited, and because of their relative simplicity, which is needed to be easily incorporated into predictive pore-size distribution models for the hydraulic conductivity.


   REVIEW OF SOME SOIL-WATER RETENTION CURVE APPROACHES
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 REVIEW OF SOME SOIL-WATER...
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Many functions to represent the SWRC have been proposed for modeling purposes. In this section, we will try to give an overview of the different expressions for the SWRC that have been reported in literature, with special attention to the expressions that are evaluated in this study. However, this review does not pretend to be complete and focuses only on unimodal expressions. One of the first expressions for the SWRC was the still widely used four-parameter power function presented by Brooks and Corey (1964) (BC model):

[1]
where {theta}s and {theta}r are the soil-water content at saturation and the residual soil-water content respectively, {psi}b is the bubbling pressure or air-entry value, and {lambda} is a pore-size distribution factor affecting the slope of the curve. The residual water content has been generally defined as the water content at which water movement ceases (Nitao and Bear, 1996), as the air-dry water content (Shao, 2000), as the water content close to the permanent wilting point of most plants, that is, at {psi} = –1.5 MPa (van Genuchten, 1980), or simply as a fitting parameter equal to the water content where the differential soil-water capacity d{theta}/d{psi} becomes zero (van Genuchten and Nielsen, 1985). The parameter {psi}b is assumed to be related to the maximum size of the pores forming a continuous network of flow channels within a soil. The discontinuous character of Eq. [1] is generally considered as a disadvantage, particularly in describing the SWRC near saturation (van Genuchten and Nielsen, 1985). Nevertheless, Eq. [1] is historically one of the most-widely used functions by soil scientists, hydrologists, and engineers.

Brutsaert (1966) evaluated several distribution functions to describe the soil's pore-size distribution, which can then be converted to a SWRC using the Young-Laplace equation. Ahuja and Swartzendruber (1972) inserted the power form of the hydraulic conductivity-soil-water content function suggested by Brooks and Corey (1964) and Brutsaert (1967)(1968) into the basic form of the diffusivity function (Bruce and Klute, 1956) to obtain their SWRC. Campbell (1974) presented a SWRC similar to Eq. [1], but with {theta}r = 0. Clapp and Hornberger (1978) and Hutson and Cass (1987) suggested replacing the sharp corner of Eq. [1] with a parabolic curve, leading to a smoothly joined two-part SWRC. Other expressions that are often cited are those presented by Visser (1966), Laliberte (1969), Gardner et al. (1970), White et al. (1970), and Su and Brooks (1975).

The most-widely adopted alternative for the BC model is the expression introduced by van Genuchten (1980). Originally, the model contained five parameters:

[2]
where {alpha}, and n and m are parameters respectively related to {psi}–1 and the curve's slope at its inflection point. These parameters all depend on the pore-size distribution, as was the case with the parameters {psi}b and {lambda} in the BC model. Although van Genuchten and Nielsen (1985) found the five-parameter form of Eq. [2] superior to the four-parameter form with m = 1 – 1/n, the latter form might be recommended when only a limited range of retention data (usually in the wet range) is available, since keeping both n and m independent may lead to uniqueness problems in the parameter estimation process and consequently a less accurate description of the SWRC in the dry range (van Genuchten et al., 1991). In our study, both the five-parameter form and the four-parameter form, with m = 1 – 1/n, of Eq. [2] will be evaluated and are denoted as VG1 and VG2, respectively. Compared with the BC model, the van Genuchten (1980) model has a continuous character due to its inflection point. Note that Eq. [2] with m = 1 was earlier used by Ahuja and Swartzendruber (1972), Endelman et al. (1974), and Varallyay and Mironenko (1979).

Tani (1982) (T model) introduced a three-parameter expression:

[3]
where {psi}o is the soil-water potential at the inflection point. Because of its simplicity, this model has been widely used for modeling water movement in soils (e.g., Suzuki, 1984).

Russo (1988) proposed a four-parameter model (R model), which produces Gardner's (1958) exponential model for the water conductivity-capillary potential relationship when incorporated into Mualem's (1976) model for the relative hydraulic conductivity:

[4]
where m' is a parameter which accounts for the dependence of the tortuosity and the correlation factors on the water content, and {alpha}' is related to the width of the pore-size distribution. Note that m' corresponds to the shape factor in Mualem's expression for the relative hydraulic conductivity and {alpha}' to the slope of Gardner's (1958) exponential equation. The reciprocal of {alpha}' can be interpreted as the air-entry value. Equation [4] is further similar to Tani's (1982) expression where 2/(m' + 2) = 1 and 0.5 {alpha}' = 1/{psi}o.

Ross et al. (1991) modified Campbell's equation (1974) to force the SWRC to predict zero soil-water content at oven dryness. Combining the Ross et al. (1991) correction, which includes the Campbell model (1974) with the parabolic correction near saturation proposed by Hutson and Cass (1987), Rossi and Nimmo (1994) developed a four-parameter sum model and a three-parameter junction model that covers the entire range from saturation to oven-dryness. Their sum model (RN model), which originally included seven parameters and showed a higher accuracy in their study compared to their junction model, was written as:

[5]
where {psi}i is the soil-matric potential at the junction point where the two curves join, {psi}d is the soil-matric potential at oven dryness, and ß and {gamma} are shape parameters. The term {theta}I represents the Hutson and Cass (1987) parabolic curve that joints the Campbell function (1974) at the junction point {psi}i. The Ross et al. (1991) correction is included in the expression for {theta}II. Further, using data sets from Schofield (1935) and Campbell and Shiozawa (1992), Rossi and Nimmo (1994) showed that at very low soil-water content, the latter becomes proportional to the logarithm of the soil-matric potential, as can be recognized as well in {theta}II. Equation [5] contains seven parameters. However, two of them can be determined by the conditions that ensure the continuity of both Eq. [5] and its first derivative to {psi}i. Here we have chosen to explicitly determine ß and {gamma} as analytical functions of {psi}b, {psi}i, {psi}d, and {lambda}:

[6]

[7]

When setting {psi}d arbitrarily at –106 kPa (Ross et al., 1991; Rossi and Nimmo, 1994), and with Eq. [6] and [7], the number of model parameters can be reduced to four. Other functions that describe the SWRC between saturation and oven dryness include inter alii those by Campbell and Shiozawa (1992), Fayer and Simmons (1995), Morel-Seytoux and Nimmo (1999), and Webb (2000).

Kosugi (1994) proposed a five-parameter expression (K1 model) that resulted from applying three-parameter lognormal distribution laws to the pore-size distribution function and to the pore capillary pressure potential distribution function. The resulting expression for the SWRC is:

[8]
where "erfc" denotes the complementary error function, {psi}o represents the mode of the pore capillary pressure potential distribution, that is, it's peak, and {sigma} is the variance of the distribution of ln[r/(rmax – r)], in which r is the pore radius and rmax the maximum pore radius. Later, Kosugi (1996)(1997) modified Eq. [8] to have a relatively simpler functional form introducing the restriction that {psi}b = 0 (K2 model). The new expression hence becomes:

[9]

Assouline et al. (1998) proposed a conceptual model, which is based on the assumption that the soil structure results from an uniform random fragmentation process where the probability of fragmentation of an aggregate is proportional to its size, and that a power function relates the volume of the aggregates to the corresponding pore volume. The fragmentation process determines the particle-size distribution of the soil. The transformation of the particle volumes into pore volumes via a power function and the adoption of the capillarity equation leads to the following expression:

[10]
where {psi}L is the soil-matric potential limit of the domain of interest of the SWRC under study corresponding to {theta}L, and {xi} and {eta} are parameters depending on the packing and shape of the particles and hence on the pore-size distribution. The parameter {xi} further depends on the bubbling pressure potential {psi}b. Equation [10] contains five parameters and will be referred to as the A1 model. However, Assouline et al. (1998) suggested reducing the number of parameters to three by choosing the value of ({theta}L, {psi}L) according to the specific soil type under consideration since "there is no need to consider the water retention curve beyond a capillary head, {psi}L, that corresponds to a very low water content, {theta}L, at which the hydraulic conductivity is negligible." It should be noted that strictly speaking in case of the Assouline et al. (1998) model, {psi} is the capillary potential and hence does not include adsorption forces. This means that Eq. [10] is not defined in the very low matric potential range where adsorption forces become dominant, although mathematically speaking, {psi}L can tend to –106 kPa (or even –{infty}) at zero water content. Since choosing an exact value of ({theta}L, {psi}L) is not evident, Assouline et al. (1998) proposed to truncate {psi}L at –1.5 MPa, as was suggested by van Genuchten (1980). The latter alternative was also evaluated here with {theta}L as a free parameter. The number of parameters in Eq. [10] hence reduces to four (A2 model). It should further be noted that when applying the A1 or A2 model as described above, Eq. [10] is not defined for soil-water contents below {theta}L.

The expressions described above were, although they are in fact simple curve-fitting equations, mainly based on pore-size distribution functions in combination with the bundle-of-capillaries concept, in which the pores are represented by cylindrical capillary tubes obeying the Young-Laplace equation. Recently, new theories have been developed, including a pore-scale network theory (Reeves and Celia, 1996; Fischer and Celia, 1999; Held and Celia, 2001a, 2001b), and a theory first presented by Tuller et al. (1999) in which (1) a pore is represented as being composed of an angular pore cross-section connected to slit-shaped spaces, and (2) the soil-matric potential is related not only to capillary forces, but also to adsorptive forces (see e.g., also Or and Tuller, 1999, 2002; Tuller and Or, 2001).


   MATERIALS AND METHODS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 REVIEW OF SOME SOIL-WATER...
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Evaluation Data Set and Soil Sample Analysis
The study was based on soil samples taken in duplicate from 48 horizons of 24 soil series in Flanders, Belgium. They were collected in the context of assessing the predictive quality and usefulness of the Belgian soil map and historical forest soil profile data for mapping purposes. All sample locations were under forest and at each location the samples were taken from Ah and E horizons down to a depth of 30 cm. The soils used in this study cover a wide range of textures within Flanders (Fig. 1) . They were classified according to soil taxonomy (Soil Survey Staff, 2003) as Spodosols, Entisols (suborders Psamments, Fluvents, and Aquents), Alfisols, and Inceptisols.



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  		Fig. 1. Variation of clay, silt, and sand content in the dataset.

 
Undisturbed soil samples were taken using the core method. A Riverside auger was used to prepare a flat sampling platform at a predetermined depth within a specific horizon after which standard sharpened steel 100-cm3 Kopecky rings (height = 5 cm, diameter = 5.3 cm) were driven into the soil using a dedicated ring holder (Eijkelkamp Agrisearch Equipment, Giesbeek, the Netherlands). In hard layers, a percussion-free hammer was applied for hammering the ring holder with a minimum of vibration into the soil. The soil-filled cylinder was carefully removed from the ring holder and the oversized sample was trimmed flush using a sharp knife. Cylinders with stones, charcoal, or roots larger than 2 mm in diameter were rejected and resampled in the same horizon. The samples were then covered with plastic lids, which prevented them from drying out and transported in special carrying cases to the laboratory to minimize disturbances (De Vos et al., 2005).

The particle-size distributions were determined on disturbed samples using a Coulter LS200 laser diffractometer (Beckman Coulter, Fullerton, CA). Organic matter content ranged from 2.3 to 130.0 g kg–1 and was determined by means of the Walkley and Black (1934) method. Bulk densities {rho}b varied from 0.76 to 1.78 Mg m–3. They were measured by weighing the 100-cm3 sized undisturbed soil samples at –10 kPa and substracting the corresponding mass of water measured on a 25-cm3 sized subsample. The spread of both OM and {rho}b is illustrated in Fig. 2 .



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  		Fig. 2. Variation of bulk density and organic matter content in the dataset.

 
The samples' SWRC was constructed by measuring soil-water content at nine soil-matric potentials using the undisturbed soil samples. For the pressure potentials ranging from –1 to –10 kPa, the sand box apparatus (Eijkelkamp Agrisearch Equipment, Giesbeek, the Netherlands) was used. Each sample, which was covered with a nylon cloth at its cutting edge, was placed on the sandbox in 1 mm of water and gently pressed downward to create a good contact between the sample and the sand. To saturate the samples by capillary rise, the water level on top of the sand was raised until 2.5 cm (halfway the sample height). Once the samples were saturated, a suction was applied by adjusting the suction regulator of the sandbox apparatus. After having reached equilibrium between the applied pressure and the quantity of water in the sample, the samples were removed from the sandbox, weighed, placed back on the sandbox, and the suction applied on the sample was increased. After having determined the sample weight at –10 kPa, a subsample was taken, it was weighed, placed in the oven at 105°C for 24 h, and weighed again to determine the water contents at pressures between –1 and –10 kPa. This also allowed calculating {rho}b. The sandbox was thus used to determine five ({theta}, {psi}) data pairs on one single soil sample. This sample was further divided into two undisturbed subsamples using sharpened steel 20-cm3 cylinders and into two disturbed subsamples. The undisturbed subsamples were used to determine water content at –20 and –33 kPa and the disturbed subsamples for water content determination at –100 and –1500 kPa using pressure chambers (Soilmoisture Equipment, Santa Barbara, CA). After having obtained equilibrium between the applied pressure and the quantity of water in the sample, the samples were weighed and placed in the oven at 105°C for 24 h. Then they were weighed again and water content was calculated.

Parameter Estimation
Seven different closed-form unimodal analytical equations containing three to five model parameters were compared in this study. Slightly modified versions of three of the seven models were evaluated as well, which brings the total number of expressions evaluated in this study to ten. The parameters of these models were obtained by fitting the models to the observed SWRCs. The nonlinear least-squares analysis was conducted using a quasi-Newton algorithm (Press et al., 1992). It is an iterative method implying an initial estimate of the parameters. The approach is based on the partitioning of the total sum of squares of the observed values into a part described by the fitted model and a residual part of observed values around those predicted with the model. The objective of the curve fitting process is to find an equation that maximizes the sum of squares associated with the model, while minimizing the residual sum of squares or sum of squared errors, SSE. The latter reflects the degree of bias and the contribution of random errors, and was computed as:

[11]
where b is a parameter vector containing the p parameters that need to be estimated, j = 1, 2 ... N with N the number of soil-water retention data for each soil sample and equal to nine in our study, {theta}j is the soil-water content corresponding to the jth data pair for each soil, and obs and fit denote observed and fitted values, respectively. The quasi-Newton routine was performed employing the mathematical software program MathCad (Mathsoft, Cambridge, MA). It resulted in slightly better fits, that is, lower SSE values, compared with the conjugant-gradient method (Press et al., 1992) and Levenberg-Marquardt's maximum neighborhood method modified by More et al. (1980). In selecting values for the initial estimates of the model parameters in the iterative procedure, data of fitted parameter values for different soils reported in literature were, if available, considered. When not available, routinely rerunning the program with different initial parameter estimates was performed. This should have prevented convergence of SSE in local minima in the objective function. To avoid negative {theta}r or {theta}L values, we introduced the constraint {theta}r or {theta}L ≥ 0, except for the RN model. The constraint {theta}s = {theta}–1kPa was used to keep the {theta}s parameter close to the near saturation value at –1 kPa, which reduces the number of parameters of each expression with one. We did not use the porosity calculated from {rho}b and particle density for this purpose, since the latter was not determined in our study. Finally, we introduced the constraint {psi}L < –1500 kPa in case of the A1 model, which was the lower limit of our database. Otherwise unrealistic fits were produced for those data sets where {psi}L was calculated to be larger than –1500 kPa. Furthermore, it reduced the dependency of the model to initial estimates of its parameters considerably.

Evaluation Methods
Several statistical indices can be applied to assess the ‘goodness-of-fit’ of a given model. In this study, the fitting accuracy of the different models was determined by using the root of the mean of squared errors, that is, RMSE, the coefficient of determination R2, and the AIC, which were calculated for each soil sample.

The RMSE (m3 m–3) is an indication for the overall error of the evaluated function and should approach zero for best model performance. The R2 is a measure for the linearity between observed and fitted data. An R2 value that approaches unity, means that the measured and fitted data pairs are linearly located around the line of perfect agreement (or 1:1 line) or that the fitted curve is of comparable shape as the measured discrete curve. The AIC index, which is often used for model-discrimination tests (Akaike, 1974), was computed as

[12]
where p is the number of model parameters. The "best" model is the one that minimizes AIC, or in other words, which combines the lowest SSE value with the lowest number of model parameters. Although computers can nowadays easily handle models with many parameters, overparameterization should be avoided as it results in a non-identifiable model, that is, a model leading to sample configuration probabilities identical to those of a simpler model with fewer parameters, in large variances of the estimated model parameters for similar soils, and in a high degree of correlation between the parameters (or low parameter uniqueness) if the number of observations is limited as is often the case with laboratory-determined SWRCs. Further, it is advantageous to minimize the number of model parameters when attempting to predict the SWRC in terms of parameters of closed-form analytical expressions from readily available data using PTFs. To facilitate the comparison between the different expressions, the mean of RMSE, of R2, and of AIC was calculated for each expression.

We further computed the Pearson coefficient rsp of correlation between model parameters and soil properties, including {rho}b, OM, and sand, silt, and clay content. This index was used as a measure for the prediction potential of the model, in that the higher the correlation, the higher becomes the prediction potential of the parameters in the model.


   RESULTS AND DISCUSSION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 REVIEW OF SOME SOIL-WATER...
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
Evaluation of the Models
Table 1 shows the values of the statistical indices, which were computed to evaluate the ten closed-form analytical expressions. When considering the mean of RMSE, the VG1 model showed the lowest values, meaning that the fitted curve produced the highest match with the measured SWRC. The VG1 model led to the best fit in 67% of the soil samples. Second best was the K1 model, followed by the K2 model, the VG2 model, the A1 model, and the A2 model. The worst models were the RN model and the T model. Intermediate results were obtained with the R model and the BC model. As regards the mean of R2, a similar trend could be observed, with VG1 as the best model in terms of linearity, closely followed by K1, K2, VG2, A1, and A2. The mean of AIC again showed a similar trend. VG1 resulted in the lowest AIC value, but now followed by K2 and then K1, VG2, A2, and A1. In the case of for example, the VG1 model, the positive effect of a reduced SSE was higher than the negative effect associated with an increased number of model parameters. The fact that it has one parameter more compared with most other models did not counterbalance its high performance. Overall, the VG1 with five model parameters scored best, followed by the K1, K2, VG2, A1, and A2 model. Intermediate results were observed for the R and BC model. The least performing were the T and RN model.


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  		Table 1. Statistical indices of the models.{dagger}

 
Table 2 summarizes the Pearson correlation coefficients rsp computed between all parameters on the one hand and the two most significant soil properties on the other hand. As could be expected, {theta}s, which was constrained at {theta}–1 kPa, was highly correlated to {rho}b and to a lesser extent to clay content. This was also concluded by Vereecken et al. (1989) based on their principal component analysis. The variation in {theta}r (or {theta}L) was for all models to a relatively high extent explained by {rho}b and clay content as well. All models showed comparable prediction potential for {theta}r (or {theta}L), except the BC model, which showed significant lower rsp values. The other parameters, which mainly determine the specific shape of the SWRC, showed lower correlations. Highest rsp values were observed for the T model, which only has one additional parameter. Unfortunately, this model performed rather poorly in terms of goodness-of-fit to SWRC data. Relatively high values were also observed for the two additional parameters of the K2 and VG2 models. The A2 model, which also has two additional parameters, showed a relatively low correlation for its {xi} parameter. The lowest values were computed for the R and RN models. Further, the models with three additional parameters, such as VG1, K1 and A1, also showed an overall low correlation. In the case of the VG1 model, this was merely due to a low correlation with the parameter m. The K1 and A1 models showed a relatively high correlation with only one of the additional parameters.


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  		Table 2. Pearson correlation coefficient between model parameters and basic soil properties.

 
It should be noted that the above conclusions were drawn on a limited data set representing 48 horizons of 24 soil series (from soils which were under forest) and covering eight soil textural classes. As an illustration for the variability of the data set, all original SWRC points are depicted in Fig. 3 .



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  		Fig. 3. Observed soil-water retention curves grouped per soil textural class and VG1 model fitted to the pooled data sets.

 
Behavior of the Models
To illustrate the behavior of the ten closed-form analytical expressions compared in this study when fitted to soil-water retention data of a relative coarse-textured soil (sand, {rho}b = 1.668 Mg m–3, OM = 37.98 g kg–1) and fine-textured soil (silt loam, {rho}b = 1.682 Mg m–3, OM = 2.71 g kg–1), observed and fitted data are compared in Fig. 4 . When considering the sand, all models gave relatively good and realistic fits, at least when the soil-matric potential remains higher than the lower limit of the data sets, that is, higher than –1500 kPa. The only model that showed a reliable behavior beyond the driest measured point is the RN model, which is not surprisingly as it was developed for that purpose. All other models resulted in a SWRC that is undefined for soil-water contents below {theta}r or {theta}L. This is a serious drawback since many water related processes such as deflation of soil particles by wind (Cornelis et al., 2004), microbial activity, and N mineralization in soils (De Neve and Hofman, 2002), methane oxidation in soils (De Visscher and Van Cleemput, 2003), and applications in for example, colloid science (Blunt, 2001) and food technology (Weerts et al., 2003) are affected by soil-water contents well below residual. On the other hand, the discontinuous character of the BC model and the K1 model did not seem to be problematic for sand, at least in comparison with our limited number of observations near saturation.



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  		Fig. 4. Observed and fitted soil-water retention curves for sand and silt loam. The subscripts ‘S’ and ‘SiL’ denote sand and silt loam, respectively.

 
With respect to the silt loam example, the performance of the BC, T, R, and RN model seemed to be reduced compared with the sand example. The BC model produced a relatively poor match near saturation, due to its discontinuous character and its unrealistic high estimated bubbling pressure value. The T model and the R model seemed to be unrealistic over the whole data range. This is in the case of the T model due to the exponential term, which when indexed to the inflection point {psi}o, shows a typical sigmoid shape. When multiplying the exponential term with 1 + {psi}/{psi}o, the sigmoid shape becomes even more pronounced, for the effect of this term is that it increases {theta} when {psi} decreases. The R model even showed a discontinuity at the inflection point 0.5 {alpha}' or 1/{psi}o, which occurs at relatively high m' values. The higher the inflection point (which is the case as the soil texture becomes finer), the lower the {alpha}' value, and hence the higher the m' value should be to keep the curve straight near saturation. Compared with the T model, the 1 + {psi}/{psi}o term is here augmented with a power 2/(2 + m'), and hence its effect becomes more pronounced as m' decreases. The RN model showed a poor fit near saturation. This is because the inflection point {psi}i should be high enough to ensure an acceptable fit in the logarithmic part of the SWRC (including the point at oven dryness). It further performed rather poor in the dry range, due to its low flexibility in the shape of the curve. This is associated with a lower degree of freedom in the dry range compared with the other models where {theta}r or {theta}L are free parameters. Finally, both forms of the van Genuchten (1980) model, VG1 and VG2, and the Kosugi (1994)(1996, 1997) models, K1 and K2, showed very good fits to the silt loam data, but have still the drawback of an undefined SWRC for soil-water contents below {theta}r. Also the two Assouline et al. (1998) functions, A1 and A2, described the SWRC rather well for the silty loam. As both are mathematically not defined at soil-water contents lower than {theta}L, the curve was not drawn beyond that point (which is only apparent for A2 in Fig. 4).

In Table 3, parameter values are given for the different models and for different soil textural classes. They were obtained by curve fitting the models to the whole data set for each soil textural class. These data can be useful to the reader as initial estimates when attempting to use one of the evaluated expressions. In the case of the BC model and the van Genuchten (1980) model, existing PTFs that are widely reported can also be used for that purpose. Table 3 further illustrates that the parameter values of n of the VG2 model follows a more pronounced trend compared with n calculated for the VG1 model, in that for example, the curves become steeper (lower n) as the soils become finer in texture.


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  		Table 3. Average model parameters for different soil textural classes.{dagger}

 
Effect of Soil Properties on the Model Performance
To assess the dependency of the model performance on soil properties, the SSE computed per soil sample for each model was correlated to {rho}b, OM, and sand, silt and clay content (see Table 4). The Pearson correlation coefficient between SSE and {rho}b was not significant at the 0.05 level for all models except the RN model. When correlating SSE and OM no significance was found at the 0.05 level for all models. The performance of the models, except the RN model, was thus not affected by {rho}b and OM, which can vary substantially in forest soils. When relating the model's SSE values to soil texture, the performance of the models appeared to increase as the soils became finer in texture, that is, higher in clay and silt content (negative correlation), except for the RN model. The opposite was true when considering sand content. This demonstrates once more that it is simply the specific mathematical form of the models that determines their performance, rather than the physical meaning of their parameters or their conceptual background. Such deterministic models were derived by applying distribution laws to pore-size distribution functions, in combination with capillarity laws. The lower the dominance of the capillary forces over the adhesive and osmotic forces in retaining water to the soil matrix, as is the case when soils become higher in clay and OM, the lower the performance of such deterministic models is expected to be, whereas in our study, the opposite was observed.


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  		Table 4. Pearson correlation coefficient between SSE and basic soil properties for each model.

 
The poor overall performance of the RN model can mainly be attributed to its poor fits when textures became relatively fine. So, the RN model is, although it has a more realistic shape than all the other models evaluated here and describes the SWRC over the complete range of soil-water contents (from saturation to oven dryness), not a reliable alternative for the superiorly performing VG1, VG2, K1, or K2 model, at least when using data sets with a limited number of data pairs, as is most often the case in practice. If more data are available, then the RN model could perhaps perform better as was demonstrated by Rossi and Nimmo (1994) for seven soils.


   CONCLUSIONS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 REVIEW OF SOME SOIL-WATER...
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
Using a limited data set taken from 48 horizons of forest soils in Flanders, Belgium and representing eight soil-textural classes, we have evaluated ten closed-form unimodal analytical expressions for the SWRC. It was shown that the van Genuchten (1980) model with five model parameters had the highest performance in terms of the RMSE, R2 and AIC. However, its prediction potential was rather poor, due to the low correlation between the m parameter and basic soil properties. Reducing the number of parameters to four, increased the prediction potential of the model significantly, without losing much of its performance. A high performance was also observed for the five-parameter and four-parameter Kosugi models (1994, 1996, 1997) and for the five-parameter and four-parameter Assouline et al. (1998) models. Yet, these models had, except for the four-parameter Kosugi (1996)(1997) model, a low prediction potential.

A major drawback of these models is that they do not define the soil-water content vs. soil-matric potential relationship beyond the residual water content. The only model we evaluated that is able in doing so is the Rossi and Nimmo (1994) model. However, it showed the lowest performance in terms of goodness-of-fit, at least when using a limited number of nine data pairs as was the case in our study. It further showed a low prediction potential. Therefore, more recently developed expressions for the SWRC between saturation and oven dryness need to be evaluated or new expressions should be developed.

Finally, it was shown that the performance of all models in terms of their match to the data, increased with increasing clay content and decreasing sand content, except for the Rossi and Nimmo (1994) model, which is contradictory to the deterministic character of these models. Furthermore, it was shown that {rho}b and OM, at least within the range of our data set, did not have a significant effect on the accuracy of all models, except the least performing ones.


   ACKNOWLEDGMENTS
 
The database used in this work was setup in the framework of project 00/05 of the VLINA research program (1995-2001), which was funded by the Ministry of the Flemish Community and conducted by Ghent University and the Institute for Forestry and Game Management. The authors thank Maarten Vanoverbeke, Koen Willems, Ann Capieau and Jan Restiaen for their assistance in sampling, analysis and data-reporting. The helpful suggestions of two anonymous reviewers, and of Wolfgang Durner and Horst Gerke are also greatly acknowledged.


   NOTES
TOP
 NOTES
ABSTRACT
 INTRODUCTION
 REVIEW OF SOME SOIL-WATER...
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Mention of the company name is for the convenience of the reader and does not constitute any endorsement in whatever sense from the authors.

Received for publication July 12, 2004.


   REFERENCES
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 REVIEW OF SOME SOIL-WATER...
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 




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M. Khlosi, W. M. Cornelis, A. Douaik, M. Th. van Genuchten, and D. Gabriels
Performance Evaluation of Models That Describe the Soil Water Retention Curve between Saturation and Oven Dryness
Vadose Zone J., February 1, 2008; 7(1): 87 - 96.
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