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Soil Physics

Soil Water Retention

I. Introduction of a Shape Index

^a Laboratoire d'Etude des Transferts en Hydrologie et Environnement, LTHE (UMR 5564, CNRS, INPG, UJF, IRD), BP 53X, 38041, Grenoble, Cedex 9, France
^b Instituto Mexicano de Tecnología del Agua (IMTA), Paseo Cuauhnáhuac 8532, Col. Progresso 62550 Jiutepec, Morelos, Mexico
^c Dep. of Civil Engineering, California State Univ., Long Beach, CA 90840, USA
^d CSIRO Land and Water, Indooroopilly, Qld. 4067, Australia

^* Corresponding author (haverkamp{at}hydrowide.com)

	ABSTRACT

TOP ABSTRACT INTRODUCTION SOIL DATABASES PARAMETERIZATION OF RETENTION... SHAPE INDEX CONCLUSIONS REFERENCES

 
Knowledge of soil water retention is fundamental to quantify the flow of water and dissolved substances in the subsurface. Water retention is often quantified with models fitted to observed retention points. Interpretation and conversion of parameters from different models is subjective and prone to error. We examined 461 retention curves from the UNSODA database and 660 from the GRIZZLY database. Parameters of the Brooks-Corey (BC) and van Genuchten (vG) equations were fitted to the retention data. The shape parameters in these functions (, m, and n) are closely correlated to soil texture and may be predicted with so-called pedotransfer functions (PTFs). Among the scale parameters, the saturated water content _s proved to be a robust fitting parameter regardless of parameterization. Reliable optimization of the residual water content _r is more difficult; without any constraint it was negative for 54.4% of the GRIZZLY samples, and its value was strongly correlated to the shape parameters. The BC- and vG-shape parameters are often converted assuming = mn, which is incorrect when or mn is large (e.g., > 0.8). To facilitate the interpretation, conversion, and optimization of retention parameters, we introduce a water retention shape index P. This index constitutes an integral measure of the slope of the retention curve and characterizes the retention behavior of a particular soil with a single number. A value for the index can be estimated directly from retention data. For the majority of the samples P ranged between 0 and 0.4; rarely did P exceed 3, which is the maximum expected for fractal behavior. The value for P was related to soil texture: fine-textured soils tend to have smaller values than coarse-textured soils. The shape index provides a benchmark for conversion and comparison of parameters.

Abbreviations: BC, Brooks-Corey Model • PTF, pedotransfer function • RMSE, root mean square error • vG, van Genuchten Model

	INTRODUCTION

TOP ABSTRACT INTRODUCTION SOIL DATABASES PARAMETERIZATION OF RETENTION... SHAPE INDEX CONCLUSIONS REFERENCES

 
ELUCIDATING AND QUANTIFYING flow and transport in the vadose zone requires knowledge of the retention function, which relates soil water pressure head with soil water content, and the conductivity function, which relates hydraulic conductivity with soil water content or pressure head. Here we focus on the description of the water retention function.

Many different equations have been proposed to parameterize water retention data (e.g., Gardner, 1958; Brooks and Corey, 1964; Brutsaert, 1966; Haverkamp et al., 1977; van Genuchten, 1980; Kosugi, 1994; Assouline et al., 1998). We will consider the two most popular among them, the Brooks and Corey (BC) equation:

[1]

and the van Genuchten (vG) equation:

[2]

In these two equations, is the volumetric water content (L³ L^–3), and _r and _s are water content scaling parameters that denote the (minimum) residual and (maximum) saturated volumetric water content, respectively. The pressure head h (L), which is here expressed in centimeters of water, is taken as the absolute value of the more commonly used matric head for notational convenience. The pressure head is scaled by h_BC or h_vG. Brooks and Corey (1964) refer to as a pore-size distribution index whereas Haverkamp et al. (2002a) denote it as the shape parameter for the BC equation. The vG equation contains two shape parameters, m and n; their product mn is denoted as the shape factor. Frequently m is predicted according to:

[3]

where we will refer to k as the user index. We will utilize m_k and n_k to indicate that the shape parameters are related by Eq. [3]. Integer values are often chosen for k in accordance with the closed-form analytical expressions selected by the user for the hydraulic conductivity. The user index is set to 1 for the popular conductivity model by Mualem (1976) and k = 2 for the conductivity model by Burdine (1953). Constraints are frequently imposed on _r and m (van Genuchten et al., 1991). We will use specific terminology for the constraining modes: the nonresidual mode if there is no residual water (_r = 0) and the user mode if m is predicted from n with the user index k.

The BC equation represents a limiting case of the vG equation: Eq. [2] reduces to Eq.[1] for pressure heads very large compared with the scaling pressure, with shape parameters related according to mn. The conversion between the BC and vG equations is therefore often based on the equality = mn, even close to saturation (e.g., Lenhard et al., 1989; Rawls et al., 1992; Russo et al., 1991; Morel-Seytoux et al., 1996). The scaling parameters for the BC and vG equations tend to be very similar and they are often assumed equal.

The prediction of hydraulic parameters from more easily accessible field data, such as textural information and dry bulk density, has been in vogue for several decades as a substitute for time-consuming and error-prone experimental procedures to measure hydraulic properties in the laboratory or field. Such predictive approaches are often referred to as PTFs. Empirical approaches use regression or neural network analysis to explore relationships between hydraulic parameters and textural, structural or other soil attributes (e.g., Gupta and Larson, 1979; Cosby et al., 1984; Rawls and Brakensiek, 1985; Schaap et al., 1998). The empiricism has the well-known drawback that predictions should only be made under conditions "similar" to those for which the model was calibrated. In the physically based approaches, which are far less numerous (e.g., Arya and Paris, 1981; Haverkamp and Parlange, 1986; Arya et al., 1999), the water retention curve is predicted from the cumulative particle-size distribution function by way of the pore-size distribution.

Optimizing retention parameters of different functions or for different constraints obviously leads to nonunique parameter sets. If the parameters are to be used in other functions and/or for different constraints, some type of conversion needs to be employed to obtain consistent parameters. For example, BC parameters will have to be converted to vG parameters for use in a computer model that simulates unsaturated flow model based on the vG equation. No matter how the original retention data were optimized, the conversion should yield a unique parameter set for a particular retention function.

Finally, it may be necessary to impose constraints during the optimization of hydraulic parameters to ensure that optimized retention data converge to the actual data or to minimize the number of parameters. Guidance is needed to constrain or to eliminate parameters.

The objectives of this paper are to illustrate potential pitfalls with the estimation and use of retention parameters, and to propose an integral shape parameter, the water retention shape index, to facilitate the interpretation, conversion, and optimization of parameters. The water retention shape index should conveniently quantify the retention behavior of a soil. This will help with the classification of soils and with the development and application of PTFs. The shape index should enable the conversion of parameters obtained for different parametric models, and may guide the optimization of retention parameters. Two established databases of soil and hydraulic properties are used to meet these objectives.

	SOIL DATABASES

TOP ABSTRACT INTRODUCTION SOIL DATABASES PARAMETERIZATION OF RETENTION... SHAPE INDEX CONCLUSIONS REFERENCES

 
Textural and retention data are taken from the databases UNSODA (Leij et al., 1996) and GRIZZLY (Haverkamp et al., 1998). These databases have completely different sample populations, so each was optimized and analyzed separately.

UNSODA
We selected 461 samples from UNSODA for further analysis. Each sample has at least seven water retention data points (drying branch) that were determined in the laboratory, often on "undisturbed" core samples using pressure outflow procedures. Textural classification was available for 406 samples (Fig. 1a) . Most samples have a coarse texture, there are samples with both low and high silt percentages but virtually none with a clay percentage above 60. Further details on UNSODA can be found in Leij et al. (1996).

View larger version (22K): [in this window] [in a new window]   Fig. 1. Textural distribution of databases: (a) 406 samples from UNSODA and (b) 660 samples from GRIZZLY.

[4]

where F(D) is the cumulative particle-size distribution curve, D is the effective diameter of a soil particle [L], D_g is a scale parameter [L], and M and N are shape parameters for the particle-size distribution. Parameter sets {M, N, D_g} were obtained by fitting Eq. [4] to a subset of 245 samples of UNSODA with at least five particle-size data, also using the Gauss–Marquardt method.

Table 1 contains the linear correlations between textural and retention parameters for UNSODA as well as with a parameter P that will be defined later in this paper. The retention parameters were obtained for the vG equation without any constraints (i.e., _r, _s, h_vG, m, and n are all fitted). Because M and N, and also m and n, are optimized independently, we consider both individual parameters and their products in the regression analysis. Especially for M and N, the higher correlation in case of the product MN indicates that M and N explain different types of variability and should therefore be optimized independently. Table 1 suggests that the shape factor mn may be predicted from textural parameters. The scale parameters _s and h_vG exhibit far less correlation with texture since they depend more on soil structure (Haverkamp et al., 2002a). The scale parameter _r has some correlation with MN.

View larger version (46K): [in this window] [in a new window]   Fig. 2. Histogram of dry bulk density, b, and organic matter content, OM, for 660 samples from GRIZZLY.

To ensure that a consistent set of drying data was used, we discarded five samples for which _s/ < 0.70 (Haverkamp et al., 2002b). The soil porosity (L³ L^–3) of the GRIZZLY samples was either available as an experimental value, or was computed from bulk density and an assumed particle density of 2.65 g cm^–3. The saturated water content _s was obtained from optimizing the vG equation with k = 2 and _r as fitting parameter.

	PARAMETERIZATION OF RETENTION DATA

TOP ABSTRACT INTRODUCTION SOIL DATABASES PARAMETERIZATION OF RETENTION... SHAPE INDEX CONCLUSIONS REFERENCES

 
Constraints
The BC equation (Eq. [1]) has potentially four retention or hydraulic parameters that can be fitted (i.e., three scale parameters _s, _r, and h_BC, and one shape parameter ) while the vG Eq. [2] has a maximum of five for optimization (i.e., three scale parameters _s, _r, and h_vG, and two shape parameters m and n). Limiting the number of optimization parameters is desirable to simplify (physico-) empirical predictions with PTFs and to avoid physically unrealistic values, nonuniqueness and high correlations among parameters. As mentioned, popular constraining modes are setting _r = 0 (nonresidual mode) and predicting m according to Eq. [3] (user mode).

We examined the optimization of parameters to retention data from UNSODA in its entirety. When _r was a fitting parameter, the median r² value for optimization of retention data was 0.973 for the BC equation and 0.947 if k = 1 and 0.943 if k = 2 for the vG equation. The relatively poor fit for the vG equation is the result of convergence problems for some samples when _r is a fitting parameter. If we set _r = 0, the median r² for the BC equation and the vG equation with user index k = 1 and k = 2 were 0.886, 0.978, and 0.981. Setting _r = 0 thus actually yielded better optimizations for the vG equation. The data were best described with Eq. [2] by using all five parameters in the optimization (r² = 0.983). Although the user index imposes the constraint that n > k, the optimization was only slightly poorer for larger k (r² = 0.978 for _r = 0 and k = 5).

The histogram of mn_0,2, that is, for _r = 0 and k = 2, is given in Fig. 3 . Note that mn_0,2 is less than unity for more than 96% of the samples. The relatively low values are the result of the constraint _r = 0. The average shape factor mn increases from 0.30 to 0.51 if _r is optimized rather than fixed to zero, this interdependency between _r and mn will be further discussed later on.

View larger version (32K): [in this window] [in a new window]   Fig. 3. Histogram of the vG-shape factor mn0,2 optimized for 660 samples from GRIZZLY.

Saturated Water Content
The independence of _s was examined by considering the influence of parametric model, user and residual mode on _s for samples from GRIZZLY. Figure 4a shows _s for the BC equation as a function of _s for the vG equation (k = 2) for _r = 0. The results point to a more or less linear relationship between the two with a correlation coefficient r² = 0.966. We obtained a standard error of water content of 0.0266 cm³ cm^–3, which is of the same order of magnitude as the measurement error of 0.02 cm³ cm^–3 reported for field experiments (e.g., Sinclair and Williams, 1979; Haverkamp et al., 1984). The value for _s tends to be lower for the BC equation, presumably because of the singularity at h = h_BC in Eq. [1] and a median for h_BC of 25.5 cm versus a median h_vG of 31 cm. Figure 4b shows a similar comparison between _s for k = 1 and k = 2 with again _r = 0. There is a very strong linear relationship and _s seems to be barely affected by the user index. Figure 4c presents a scattergram of _s for optimized _r as a function of _s for _r = 0 for the Burdine mode. We used only the 253 samples where optimization yielded a nonzero _r. There is a strong linear relationship, which suggests that _s is mostly independent of the other scaling parameter, _r.

View larger version (48K): [in this window] [in a new window]   Fig. 4. Saturated water content obtained from different parameterizations for 660 samples from GRIZZLY as a function of s for the van Genuchten (vG) equation with the Burdine condition (k = 2) and r = 0: (a) s for the Brooks-Corey (BC) equation (k) with r = 0, (b) s for the vG equation with the Mualem condition (k = 1) with r = 0, and (c) s for the vG equation with the Burdine condition (k = 2) with r > 0.

Residual Water Content
The physical interpretation of _r has not been completely resolved and this parameter is poorly correlated with other soil properties (Luckner et al., 1989; Tietje and Tapkenhinrichs, 1993). We consider it merely an empirical fitting parameter restricted to the range of data points used and with an optimization error that is considerably larger than for _s, partly because _r is an extrapolated water content at "infinite" h (Schaap and Leij, 1998). The vG and BC equations provide a relatively poor description of retention data for dry conditions (h >> 1000 cm) and several parametric models have been proposed where this description does not merely rely on _r (Ross et al., 1991; Rossi and Nimmo, 1994; Fayer and Simmons, 1995). Haverkamp et al. (2002b) pointed out that for soils with a unimodal pore-size distribution, the value for _r is affected by hysteresis and is determined by antecedent wetting and drying cycles and the time allowed for equilibration.

As mentioned, fitting with the vG equation and no constraint on _r yielded a negative _r for 54.4% of the samples from GRIZZLY. For optimization purposes this is quite acceptable since _r is merely an empirical parameter. However, such optimized _r values lack a conceptual basis and provide little confidence for independent predictions and comparisons. We will therefore impose the usual constraint that _r 0 (Leij et al., 1996) with the understanding that _r is not a unique soil characteristic but somewhat of a fudge factor. Especially for coarse soils and pressure heads below 1000 cm, setting _r equal to zero may often provide an acceptable description of the retention data while simplifying the optimization and prediction of hydraulic parameter sets.

Shape Factor
Optimization of GRIZZLY retention data revealed the aforementioned increase in average shape factor mn from 0.30 to 0.51 if _r was included in the optimization. We will further examine the effect of _r for the vG equation with k = 2 for 660 soils from GRIZZLY. We used _r = 0.005 cm³ cm^–3 as a limiting value below which _r was fixed at zero (Leij et al., 1996). Only for 253 samples was _r > 0. Figure 5 shows the shape factor mn₂ as a function of mn_0,2 for those soils. The value of mn₂ increases considerably compared with mn_0,2, showing that the magnitude of shape parameters is affected by constraints on _r. Differentiating Eq. [2] with respect to yields the soil water capacity, which is proportional to both _s–_r and mn. Setting _r = 0 increases the value for _s–_r as compared with the case where _r is unconstrained. To maintain a similar capacity for a particular soil mn_0,2 hence needs to be smaller than mn₂.

View larger version (14K): [in this window] [in a new window]   Fig. 5. The shape factor mn with r optimized as a function of mn0,2 with r = 0 for the van Genuchten equation for 253 samples from GRIZZLY subject to the Burdine condition (k = 2).

View larger version (15K): [in this window] [in a new window]   Fig. 6. The van Genuchten shape factor mn0,k for three different user indices, k, as a function of the Brooks-Corey shape parameter 0 for 660 samples from GRIZZLY.

The customary optimization practice has been to use those parameters and user indices that yield the best fit (lowest RMSE). This may lead to inconsistent optimizations and a mismatch in values for a particular retention parameter. Figure 6 already demonstrated that the value for the optimized shape factor mn_k depends on the choice of k. Hence, we examined the effect of k on the optimization of retention parameters for 660 samples in GRIZZLY. Figure 7a shows the normalized vG parameters _s, h_vG, and mn_0,k as a function of k for _r = 0. Normalized BC parameters _s, h_BC, and are also included with corresponding dependent variable k. The normalization was done with respect to the highest upper limit of the 95% confidence interval for a particular parameter obtained from eight parameterizations with _r = 0 (k = 0.5, 1, 2, 3, 5, 7, 10, ) and also, although not included in Fig. 7a, two with _r 0 (k = 1, 2). The error bars denote the normalized 95% confidence limits. All parameters decrease with k, especially the decrease of h_vG and mn_k for k 2 is noteworthy. The estimates for these parameters are less precise than for _s judging by the wider confidence intervals. We also investigated the behavior of the root mean square for error (RMSE), which was adjusted to account for the degrees of freedom:

[5]

where K is the number of data points for the retention curve, mp is the number of model parameters, and (h_j) and _j denote the observed and optimized water content. Figure 7b displays the RMSE_adj values averaged for the 660 samples as a function of k for the same conditions as used for Fig. 7a and also for k = 1 and 2 when _r is a potential optimization parameter (_r 0). The optimization appears most accurate if _r is flexible and k is small and least accurate for the BC equation. However, the differences in RMSE_adj are fairly small.

View larger version (19K): [in this window] [in a new window]   Fig. 7. The behavior of optimized hydraulic parameters for GRIZZLY as a function of the user index for the van Genuchten equation with k representing the Brooks-Corey equation: (a) values for s, hvG or hBC and mn0,k or 0 divided by their maximum obtained for either r = 0 or r > 0 as a function of k (error bars denote 95% confidence limits) and (b) adjusted RMSE as a function of k with r fixed (=0) or optimized (0).

	SHAPE INDEX

TOP ABSTRACT INTRODUCTION SOIL DATABASES PARAMETERIZATION OF RETENTION... SHAPE INDEX CONCLUSIONS REFERENCES

Operational Definition
In the description of the retention curve, the shape parameters m and n determine the inflection point of the water retention curve and the slope at that point (van Genuchten, 1980). The BC parameter plays a similar role. A shape index might therefore be defined using an average slope around the inflection point rather than mn. Bloemen (1980) characterized the particle-size distribution in this manner using a discrete approach. Fuentes (1985) was the first to advocate the use of some integral or composite characteristic to describe the average slope of the retention curve for different parametric models. For this purpose, we introduce the following water retention shape index:

[6]

where P is an integral measure of the slope of the log-transformed retention curve. Note that the slope of the water retention curve has traditionally been expressed by the soil water capacity, C = d/dh. An expression for P can be obtained by integrating closed-form expressions for the retention curve or by direct computation from experimental water retention data.

The above definition of P is related to that used in fractal analysis of retention data where a plot of ln versus ln h may be used to determine fractal lengths (Tyler and Wheatcraft, 1988). The mass-fractal dimension of the solid matrix, D_m, and the volumetric water content are related under simplifying conditions by (Crawford, 1994):

[7]

where d_e is the embedding dimension, which is equal to 3 for soil cores and aggregates. Substitution of Eq. [7] into Eq. [6] leads to the simple relationship:

[8]

The upper limit for P should hence be 3 provided that the water retention curve is described according to Eq. [7] (i.e., exhibits fractal behavior). Note that a similar relationship is used to determine D_m from the slope of a logarithmic plot of bulk density versus particle radius (Anderson and McBratney, 2002).

Experimental Shape Index
The shape index defined by Eq. [6] may be determined from experimental data. We used a simple first-order difference method for differentiation and integration to calculate P for samples from the UNSODA and GRIZZLY databases with monotonically increasing h and decreasing , according:

[9]

where K denotes the number of observed retention points for a sample in the database. Alternatively, an equation may be fitted to the data and substituting this equation in Eq. [6] would yield the parametric shape index further discussed in Leij et al. (2005).

Calculated P values for 406 samples from UNSODA and 612 samples from GRIZZLY were distributed approximately lognormally (Fig. 8) . Forty-eight samples of GRIZZLY could not be used for this computation because of the monotonicity condition on h and . The P-values tend to be somewhat smaller for GRIZZLY than for UNSODA, which is likely due to the larger fraction of fine-textured soils in the GRIZZLY database (Fig. 1). Anderson and McBratney (2002) reported that the mass-fractal dimension D_m is typically in the range 2.5 to 3. This suggests values for P that are mostly between 0 and 0.5 (Eq. [8]), which is in accordance with Fig. 8. Retention data will not necessarily follow the idealized fractal behavior, but very rarely does the shape index exceed 3 (the upper limit or Euclidean dimension imposed by fractal theory).

View larger version (27K): [in this window] [in a new window]   Fig. 8. Distribution of shape index, P, calculated for 461 soil samples from UNSODA and 612 samples from GRIZZLY.

View larger version (15K): [in this window] [in a new window]   Fig. 9. Shape index computed from retention data as a function of clay and sand percentage plotted for 51 data points for: (a) UNSODA, and (b) GRIZZLY.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION SOIL DATABASES PARAMETERIZATION OF RETENTION... SHAPE INDEX CONCLUSIONS REFERENCES

 
We have used retention data for 461 samples of the UNSODA and 660 of the GRIZZLY databases (Fig. 1). The distribution of basic soil properties of the GRIZZLY samples was shown in Fig. 2. Retention parameters of the BC and vG equations were fitted to the retention data. The shape parameters (, m, and n) were closely related to soil texture, while the scale parameters (_s, _r, and h_BC or h_vG) exhibited far less correlation with soil texture (Table 1).

The residual water content _r is generally a poor fitting parameter with a vague conceptual basis and poor predictability. During optimization of water retention data, _r is sometimes set to 0. This may not matter for the goodness of fit, but _r tends to affect the optimized values for other parameters. The distribution of the shape factor mn_0,2, that is, for _r = 0 and user index k = 2, was shown in Fig. 3. The average of mn_0,2 is 0.30, but in the residual mode when _r is optimized the corresponding average mn₂ increased to 0.51. On the other hand, the concept of a saturated water content _s is not ambiguous. The optimized value for _s remains pretty much the same for each soil regardless of parametric model, user or residual mode (Fig. 4). Likewise, shape parameters are largely independent of _s.

The dependency of the shape factor on _r was further illustrated with Fig. 5. The value of mn₂ increases considerably compared with mn_0,2: an increase in _r is presumably compensated by an increase in mn to maintain a similar soil water capacity for a particular soil. The BC- and vG-shape parameters have commonly been converted assuming = mn. This hypothesis ignores the dependency of mn_k on the user index and that of both and mn_k on _r. Only if m_k0 will the vG equation resemble a power function for the major part of the retention curve with mn_k regardless of user index, _s or _r. Figure 6 displays mn_0,k for k = 0.5, 2, and 10 as a function of ₀; the greatest discrepancy occurred for large shape parameters (say ₀ > 0.8) and smallest k.

A user index k is often used to constrain the shape parameters in the vG equation. Popular values for k are 1 or 2 corresponding to the conductivity models by Mualem (1976) and Burdine (1953). The shape parameters m_k and n_k as well as the scale parameter h_vG depend on the choice of k (Fig. 7a). Setting aside the conductivity model, the choice of k is arbitrary. Lower values for k result in a slightly better fit (Fig. 7b).

We introduced a water retention shape index P to characterize the retention behavior of a particular soil with a single number and to mitigate problems with converting parameters of different retention functions and parameter dependency. The index is defined by Eq. [6] and may be related to the mass-fractal dimension according to Eq. [8]. A value for the shape index was estimated directly from retention data of the two databases, its distributions are shown in Fig. 8. For the majority of the samples P was below 0.4. The shape index was strongly correlated with texture for both databases (Fig. 9). Finer textured soils tend to have lower P. It should be noted that textural class has been popular to provide convenient, albeit rough, estimates of hydraulic parameters (Rawls and Brakensiek, 1985; Leij et al., 1996). We are particularly interested in the use of P as a benchmark for conversions. In Leij et al. (2005), we will derive parametric expression for P based on the BC and vG equations and formulate equations for parameter conversions.

	ACKNOWLEDGMENTS

 
This study was partially supported by the European Union DGXII-Environment Program through funding of the project AgriBMPWater (EVK1-1999-00117).

	REFERENCES

TOP ABSTRACT INTRODUCTION SOIL DATABASES PARAMETERIZATION OF RETENTION... SHAPE INDEX CONCLUSIONS REFERENCES

 

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