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Published online 27 February 2007
Published in Vadose Zone J 6:186-195 (2007)
DOI: 10.2136/vzj2006.0041
© 2007 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
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ORIGINAL RESEARCH

Upscaling Schemes and Relationships for the Gardner and van Genuchten Hydraulic Functions for Heterogeneous Soils

Jianting Zhua,*, Michael H. Younga and Martinus Th. van Genuchtenb

a Desert Research Institute, Division of Hydrologic Sciences, 755 East Flamingo Rd., Las Vegas, NV 89119
b U.S. Salinity Lab., 450 West Big Springs Rd., Riverside, CA 92507-4617
* Corresponding author (Jianting.Zhu{at}dri.edu)

Received 14 March 2006.



   ABSTRACT
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Upscaled soil hydraulic properties are needed for many large-scale hydrologic applications such as regional and global climate studies and investigations of land–atmosphere interactions. Many larger scale subsurface flow and contaminant transport studies also require upscaled hydraulic property estimates. The objectives of this study were to develop a methodology for upscaling hydraulic property functions using a p-norm approach, to examine how p-norm values differ for two commonly used soil hydraulic property models (the Gardner and van Genuchten functions), and to investigate the relative sensitivities of p-norms and the effective hydraulic parameters to the degree of soil heterogeneity (expressed in terms of variances and auto-correlation lengths of the hydraulic parameters) and other environmental conditions. The p-norm approach expresses upscaling schemes such that it reduces their sensitivity to uncertainties (heterogeneities) in site conditions. The upscaling schemes are obtained as the result of two new criteria proposed to upscale soil hydraulic properties in this study—one preserving the ensemble vertical moisture flux across the land–atmosphere boundary, and a second preserving the ensemble soil surface moisture content. The effective soil hydraulic parameters of a heterogeneous soil formation are then derived by conceptualizing the formation as an equivalent homogeneous medium that satisfies the upscaling criteria. Upscaling relationships between the Gardner and van Genuchten models can then also be established for steady-state vertical flow using the statistical structures of the hydraulic parameters of these two models as estimated from field measurements. The upscaling scheme is demonstrated using hydraulic property data collected at 84 locations across a site in the Mojave Desert. Our results show that the p-norms generally vary less in magnitude than the effective parameters when the variances of the hydraulic parameters increase. We also show that, in general, p-norm values are better defined for the van Genuchten model than the Gardner model. Hydraulic parameter auto correlations, as defined by correlation lengths, were found to have little impact in relating the upscaling schemes (p-norm values) for the two hydraulic property models, but correlation between the hydraulic parameters within the hydraulic property models can significantly affect p-norm relationships.


   INTRODUCTION
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
UNSATURATED flow in the vadose zone is often simulated using closed-form expressions for the soil hydraulic properties involving the soil water retention and unsaturated hydraulic conductivity functions. Some of the more commonly used hydraulic functions include the Gardner–Russo (Gardner, 1958; Russo, 1988), Brooks–Corey (Brooks and Corey, 1964), and van Genuchten (van Genuchten, 1980) models. Soil hydraulic property functions are generally measured at relatively small (local) scales. Major questions remain about how to best aggregate the spatially variable hydraulic properties within a heterogeneous soil volume to obtain effective hydraulic properties (or hydraulic parameters) at the large (field or watershed) scale. Effective parameters are essential for upscaling point-level measurements and processes to larger scales, and for reducing the complexity of hydrologic systems.

During the last several decades, much work has been done to develop and examine different averaging (and hence upscaling) schemes. Sharma and Luxmoore (1979) pointed out that the soil–plant–atmospheric interactions are very complex in terms of evaluating the influence of soil heterogeneity on water budget. They found that results very much depended on the coefficient of variation and the frequency distribution function of the Miller–Miller scaling factor, as well as on the specific soil–plant–atmospheric conditions involved. Kim and Stricker (1996) used Monte Carlo simulations to investigate both the separate and simultaneous effects of horizontal heterogeneity in the soil hydraulic properties and rainfall intensity on the statistical properties for various components of the one-dimensional water balance. Their results showed that heterogeneity has a stronger effect on the annual water balance for loam than for sand. Kim et al. (1997) further investigated the impact of heterogeneity on the spatially averaged water budget of the unsaturated zone using an analytical framework (Kim et al., 1996). Zhu et al. (2004) established optimal averaging schemes and probability distribution functions for parameters of several hydraulic functions, using correspondence among those hydraulic models by preserving the macroscopic capillary length and predicting the same vertical moisture flux. In more recent studies, Zhu and Mohanty (2006) investigated the effective hydraulic parameters for transient infiltration in terms of the optimal averaging schemes for the input hydraulic and environmental parameter fields. Specifically, they explored the effects of microtopography (as reflected in surface ponding depth) and hydraulic parameter correlation on the ensemble-mean behavior, as well as on the optimal effective schemes. Zhu et al. (2006) additionally examined the impact of the skewness (third-order moment) of hydraulic parameter distributions on "effective" soil hydraulic parameter averaging schemes for heterogeneous soils in a flat landscape. Numerical or field experimental results showed that the distribution skewness is also important for determining the upscaled effective parameters, in addition to the mean and variance.

Several studies have investigated conditions for which different hydraulic property functions produce the same or similar hydrologic responses. For example, Warrick (1995) discussed some of the features that need to be preserved in order for different hydraulic property functions to give similar or identical results for a given flow scenario. Lenhard et al. (1989) related the van Genuchten (1980) and Brooks and Corey (1964) parameters by assuming similar shapes of the soil water retention curves. Lenhard et al. (1989) obtained equivalence by equating the specific moisture capacity halfway between the saturated ({theta}s) and residual ({theta}r) water contents, and minimizing the difference between the water retention curves. Morel-Seytoux et al. (1996) converted Brooks–Corey parameters to van Genuchten parameters by preserving the macroscopic capillary length and keeping the same asymptotic behavior of the soil water retention curve at low water contents. Zhu et al. (2004) similarly established parameter equivalence among the hydraulic conductivity functions by preserving the macroscopic capillary length and predicting the same vertical moisture flux across the soil surface. Their results showed that the hydraulic parameters correspond very well when predicting evaporation rates from heterogeneous soils having a relatively large soil water pressure head at the soil surface (e.g., a dry surface or a shallow groundwater table).

A main objective of this study was to express effective hydraulic parameters in terms of certain types of averages (using the averaging or upscaling schemes) of individual parameter values obtained for a heterogeneous soil formation. We examined, for this purpose, the upscaling schemes for both the Gardner and van Genuchten functions using a data set of field-measured hydraulic properties for steady-state vertically dominated flow. We propose two new criteria to upscale the Gardner and van Genuchten models, and also establish the upscaling relationships between the two models. Our criteria are based on two important hydrologic processes: the moisture flux across the land–atmosphere boundary and the soil surface moisture content. The criteria require that the upscaled hydraulic properties for both models predict the ensemble vertical water flux across the land–atmosphere boundary, and produce the ensemble surface reduced soil moisture content. These two hydrologic quantities are important in that they are key inputs for land–atmosphere feedback schemes in soil–vegetation–atmosphere transfer models in terms of partitioning upward and downward water fluxes at the land surface. Water table depth (finite) and soil water pressure head at the soil surface are assumed homogenous for typical remote-sensing pixel or large-scale hydro-climate model grid scales as considered in this study, since the scale of variations for these two quantities are usually larger than that of hydraulic properties.

In this study, we interpret hydraulic property upscaling as deriving effective homogeneous soil hydraulic parameters to account for uncertainties in the spatially variable hydraulic parameters (e.g., Zhu and Mohanty, 2002, 2003). Effective or average hydraulic properties are those that can be used for an entire field (e.g., typical remote-sensing pixel or a larger scale model grid) based on measurements at point scales, or using estimates obtained in some other way such as from soil texture. Effective soil hydraulic parameters of a heterogeneous soil formation are derived by conceptualizing the heterogeneous soil formation as an equivalent homogeneous medium such that the equivalent homogeneous soil will produce the same ensemble-mean flux across the land–atmosphere boundary and the same ensemble-mean surface moisture content as the heterogeneous medium. Mathematically, we express the effective parameters using the p-norm, or the p-order power average of the local-scale hydraulic parameter sets. The p-norm is a generalization of commonly used averaging schemes, which includes the arithmetic, geometric, and harmonic averages as particular cases. We hypothesized that the p-norm should be less sensitive to the variation in random hydraulic parameter fields than the effective parameters since the larger random parameter variance will probably offset the variability of the p-norm. Therefore, the p-norm should be better defined than the effective parameter values when describing hydraulic parameter averaging schemes. Quantitatively, the upscaling schemes are considered "better defined" if they (expressed in terms of the p-norm values) are less sensitive to the variability of the hydraulic parameters and other related soil-water conditions such as water table depth and the pressure head at the soil surface. In other words, if the upscaling schemes of the hydraulic parameters for large-scale applications are less sensitive to the soil heterogeneity and other conditions, conceptualization of the heterogeneous medium as an equivalent homogeneous medium is more feasible and the upscaling schemes are easier to implement.

One complication with upscaling schemes for larger scale hydrologic applications is that they differ when different soil hydraulic property functions are used. While significant efforts have been made to establish the equivalence among many hydraulic property models at the local scale (e.g., Warrick, 1995; Lenhard et al., 1989; Morel-Seytoux et al., 1996), this study deals with equivalence in an upscaling context, which is a significant and still largely unresolved issue. Therefore, our first specific objective was to develop a methodology for calculating the p-norm values (and therefore the upscaling schemes) based on the two new upscaling criteria. A second objective was to determine whether the optimal averaging schemes differ for different forms of the hydraulic properties, and the relationships of the averaging schemes if they indeed differ. Finally, a third objective was to examine whether the p-norms are less sensitive to variations in random hydraulic parameter fields compared with the effective parameters.


   MATERIALS AND METHODS
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Soil Hydraulic Property Models and Steady-State Local-Scale Processes
The soil hydraulic functions consist of the soil water retention function, which defines the water content as a function of the soil water pressure head, and the hydraulic conductivity function, which relates the hydraulic conductivity with the water content or soil water pressure head. A brief discussion of the soil hydraulic models used in this study is presented below. We refer to Leij et al. (1997) and Warrick (2003) for a more comprehensive overview of alternative functional expressions for the soil hydraulic properties, including those used in this study.

Gardner Model
The hydraulic functions used by Gardner (1958) and Russo (1988) are given by

Formula 1[1]

Formula 2[2]

Van Genuchten Model
Van Genuchten (1980) combined his S-shaped soil water retention function with the statistical pore-size distribution model of Mualem (1976) to obtain the following functions:

Formula 3[3]

Formula 4[4]
In Eq. [1]GoGo to [4], Se = ({theta} – {theta}r)/({theta}s – {theta}r) is the effective degree of saturation, {theta} is the volumetric water content, {theta}r is the residual volumetric water content, {theta}s is the saturated volumetric water content, h is the soil water pressure head (in this study assumed to be positive for unsaturated conditions), K is the hydraulic conductivity, Ks is the saturated hydraulic conductivity; {alpha}, m, and n are empirical hydraulic shape parameters, and m = 1 – 1/n, while the subscripts G and vG refer to Gardner and van Genuchten model parameters, respectively. Being the hydraulically active portion of the soil moisture regime, Se is an important parameter in many water balance studies and land–atmosphere parameterizations. For this reason, we use Se here to more effectively express soil moisture conditions in this study.

When the soil water pressure head at the land surface, h0, is given, application of Darcy's law leads to (e.g., Zaslavsky, 1964; Warrick and Yeh, 1990; Zhu and Mohanty, 2002)

Formula 5[5]
where z0 is the distance from the soil surface to the water table, q is the steady-state flux rate for either evaporation (positive value) or infiltration (negative value), and the function F is defined as

Formula 6[6]
in which s is a dummy integration variable denoting the dimensionless quantity {alpha}h in the hydraulic conductivity functions. For the Gardner model, Kr(s) = exp(–s), in which case Eq. [6] can be integrated analytically to give

Formula 7[7]
which, with Eq. [5], leads to

Formula 8[8]
For the van Genuchten model, F(q/Ks) in Eq. [6] requires numerical integration, which is achieved in this study using Romberg integration (e.g., Scheid, 1968; Stoer and Bulirsch, 1980). The steady-state vertical flux for either evaporation or infiltration for the van Genuchten function is calculated iteratively using Eq. [5] and [6]. The dimensionless flux q/Ks for a given soil water pressure head, h0, and water table depth, z0, is obtained by assuming some initial value for the flux and then iteratively adjusting its value until Eq. [5] is satisfied within a prescribed tolerance.

Field Measurements
In this study, we used measured hydraulic properties from the Corn Creek Fan Complex located in the Desert National Wildlife Refuge, approximately 50 km north of Las Vegas, NV (see Fig. 1 ). The soil hydraulic properties of the surface soil at each field site were determined using tension infiltrometry (Ankeny et al., 1988; Reynolds and Elrick, 1991). Data were collected using differential pressure transducers (Casey and Derby, 2002) at 15-s intervals. Manual readings were taken periodically to verify operation and to better identify when the intake rate was at or near steady state. At that time, the tension infiltrometer was reset to a lower tension level. Four to five tension steps were used for each test, typically at 12, 9, 6, 3, and ~0 (saturation) cm.


Figure 1
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  		Fig. 1. Location map of the Corn Creek Fan where the field hydraulic properties were measured.

 
Two methods of analysis were used to obtain the soil hydraulic properties from the tension infiltrometer data. This included the semiempirical approach of Wooding (1968) and numerical inversion of the data using parameter estimation. Assuming applicability of Gardner's exponential K(h) function (Eq. [2]), three-dimensional, steady-state infiltration from a circular source is given by (Wooding, 1968)

Formula 9[9]
where i(h) is the steady-state infiltration rate at supply tension h, and r is the radius of the tension infiltrometer. The first term of Eq. [9] within the second set of parentheses accounts for vertical gravitational flow, and the second term for lateral flow due to capillarity. Equation [9] contains two unknowns: KsG and {alpha}G. Using the infiltrometer for a range of tensions, paired values of the flux [i(h)] and the tension step (h) can be obtained. These paired values are subsequently used in an iterative nonlinear least-squares regression approach to estimate KsG and {alpha}G in Eq. [9] (e.g., Logsdon and Jaynes, 1993).

In addition to Wooding's solution to obtain KsG and {alpha}G, a complete set of van Genuchten soil hydraulic parameters (i.e., KsvG, {alpha}vG, n, and {theta}s) were also identified through numerical inversion of the cumulative infiltration data, given known changes in the boundary conditions (Simunek et al., 1998; Young et al., 2004). We used, for this purpose, the axisymmetric option of the HYDRUS-2D software package of Simunek et al. (1999). Additional details of the field test methodology and procedures are given by Young et al. (2005).

Results of the data analysis produced a series of fitting parameters and soil hydraulic properties that were used for developing the upscaling rules and relationships. A total of 84 hydraulic parameter sets for both the Gardner and van Genuchten models were derived from the field measurements. Basic statistical data of the 84 field-measured hydraulic parameters are listed in Table 1, including mean values, variances, standard deviations, and coefficients of variation. We additionally calculated coefficients of correlation (CC) between the saturated hydraulic conductivity, Ks, and the {alpha} parameter for both models, obtaining CC(KsG, {alpha}G) = 0.24, and CC(KsvG, {alpha}vG) = 0.78. The parameter sets in Table 1 involve the Gardner and van Genuchten saturated hydraulic conductivities (KsG and KsvG), the {alpha} parameter for the two models ({alpha}G and {alpha}vG), and the van Genuchten n parameter. Since we were interested in the effective degree of saturation at the soil surface, the {theta}s and {theta}r parameters of the hydraulic property function are not relevant in this study.


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  		Table 1. Basic statistics of 84 field-measured soil hydraulic parameter{dagger} values.

 
Effective Hydraulic Parameters
Because the van Genuchten model contains one more parameter than the Gardner model and our proposed new criteria will invoke only two equations, we assumed that the van Genuchten n parameter is a deterministic parameter for which we used the average value from our field-measured data. In practice, n can be determined with greater certainty than the other van Genuchten parameters (e.g., Schaap and Leij, 1998). In a study of van Genuchten's hydraulic model, Hills et al. (1992) also demonstrated that random variability in the water retention characteristics could be adequately modeled using either a variable parameter {alpha}vG together with a constant parameter n, or a variable n with a constant {alpha}vG, but achieved better results using the first option. Following these findings, we treated n as a deterministic parameter in this study.

To examine the appropriateness of using only the field-measured hydraulic parameters for establishing the upscaling relationships, we also used regenerated fields of 10000 random values for the hydraulic parameters that may be cross-correlated. The cross-correlated random fields of Ks and {alpha} were generated using the spectral method proposed by Robin et al. (1993). Random fields were obtained with the power spectral density function using exponentially decaying covariance functions. As an indicator of correlation between the two random parameter fields, we used the coherency spectrum given by

Formula 10[10]
where {phi}11(f) and {phi}22(f) are the power spectra of random Fields 1 and 2, respectively, and {phi}12(f) is the cross spectrum between Fields 1 and 2. Having |R|2 = 1 indicates perfect linear correlation between the random fields. In this study, 10000 realizations of hydraulic parameter values were regenerated to match the field-measured parameter target statistics to various degrees. The parameters Ks and {alpha} can be satisfactorily fitted using lognormal distributions (Smith and Diekkruger, 1996; Nielsen et al., 1973). In this study, both Ks and {alpha} were assumed to obey lognormal distributions.

A total of six hydraulic parameter sets of 10000 data points was generated for both the Gardner and van Genuchten models. The first four sets all assumed nearly full correlation between Ks and {alpha}, with correlation coefficients being in the range between 0.95 and 0.99. The differences among the four parameter sets are the normalized autocorrelation lengths, CL (i.e., the autocorrelation length divided by the grid length), which were set at 2, 5, 10, and 50 for the four sets. The fifth set had a CL of 5 and the same values of correlation coefficients between Ks and {alpha} as the field-measured data (about 0.24 and 0.74 for the Gardner and van Genuchten models, respectively). The sixth set had the same correlation characteristics as the first four data sets, but variances that were four times the actual field-measured variance values.

Given the values of the water table depth, z0, and the pressure head at the soil surface, h0, the effective degree of saturation can be calculated with Eq. [1] for the Gardner model and Eq. [3] for the van Genuchten model. The flux across the land–atmospheric boundary can be computed analytically with Eq. [8] for the Gardner model. For the van Genuchten model, the flux needs to be computed iteratively using Eq. [5] and [6] as discussed above. For either the field-measured or the regenerated hydraulic parameter data, we can now run Monte Carlo simulations to calculate both the effective degree of saturation and the flux across the land–atmospheric boundary. The effective hydraulic parameters are subsequently calculated using the two upscaling criteria we proposed (i.e., preservation of the surface flux and the surface effective degree of saturation). Specifically, for the Gardner model the effective hydraulic parameters {alpha}Geff and KsGeff for {alpha} and Ks are calculated using

Formula 11[11]

Formula 12[12]
respectively. For the van Genuchten model, the effective hydraulic parameters {alpha}vGeff and KsvGeff are obtained with

Formula 13[13]

Formula 14[14]
respectively, where Formula 14 is the ensemble steady-state flux (again either evaporation or infiltration) across the land–atmospheric boundary. The right-hand sides of these equations are the mean (ensemble) quantities for Se at the land surface (Eq. [11] and [13]) and the flux across the land–atmospheric boundary (Eq. [12] and [14]), while the left-hand sides are those based on a single set of effective parameter values. As mentioned above, the effective soil hydraulic parameters of the heterogeneous soil formation are derived by conceptualizing the soil formation as an equivalent homogeneous medium, thus assuming that the equivalent homogeneous soil will produce the same ensemble-mean flux across the land–atmosphere boundary and the same ensemble-mean surface effective degree of saturation. We used the effective hydraulic parameters to predict the large-scale mean flux exchange across the land–atmospheric boundary and to preserve the mean effective degree of saturation at the land surface.

Upscaling Schemes and Their Relationships for the Gardner and van Genuchten Models in Terms of p-Norm Values
The p-norm or p-order power average Formula 14(p) for any set of N random parameter values {xi}i is given by (Korvin 1982; Green et al., 1996),

Formula 15[15]
As used in this study, the p-norm is a generalization of the arithmetic (p = 1), geometric (p -> 0) and harmonic (p = –1) averages, which hence are all particular cases of the power average. This shows that the potential spectrum of averaging schemes is captured by the value of the p-norm. Moreover, the p-norm should be less sensitive to variations in the random {xi} fields than the effective parameters since the larger random parameter variance will offset the variability in the p-norm. The p-norm hence should be more useful than the effective parameters in describing hydraulic parameter upscaling schemes (to be discussed in more detail below). The main idea is that the optimal p-norm for a particular hydraulic parameter will guide how best to upscale that parameter to larger scale applications. Here we express the effective hydraulic parameters, as defined by Eq. [11]GoGo through [14], as p-norm power averages [i.e., {xi}eff = Formula 15(p)] where {xi} can be any of the hydraulic parameters in this study ({alpha}G, {alpha}vG, KsG, or KsvG).

We now restate one of the objectives of this study: Can we find relationships between the p-norms (the optimal upscaling schemes) for the parameters of the two different hydraulic functions that satisfy both of the upscaling criteria? To answer this question, we assume knowledge of the pressure head at the soil surface (h0) and the depth to the water table (z0), and obtain from this effective {alpha} and Ks values for both the Gardner and van Genuchten models using Eq. [11]GoGo through [14]. Next, the corresponding p-norms for the hydraulic parameters are calculated iteratively using the following equations:

Formula 16[16]

Formula 17[17]

Formula 18[18]

Formula 19[19]
where pKsG is the p-norm for the Gardner Ks, p{alpha}G for the Gardner {alpha}, pKsvG for the van Genuchten Ks, and p{alpha}vG for the van Genuchten {alpha}.

For each scenario of water table depth and pressure head at the soil surface, we can calculate one set of p-norm values. In our analysis, we used dimensionless quantities for the water table depth and the soil surface pressure head. The dimensionless pressure head at the land surface h0* and the dimensionless water table depth z0* are defined as h0* = Formula 19h0 and z0* = Formula 19z0, respectively, where Formula 19 is the mean of the field-measured van Genuchten {alpha}vG values. In cases where Formula 19h0 and Formula 19z0 are required for calculations of the p-norms for the Gardner model, we can obtain them by inverting h0 and z0 from the dimensionless pressure head at the soil surface and the water table depth, i.e., h0* = Formula 19h0 and z0* = Formula 19z0.

To examine the appropriateness and advantage of using p-norms rather than effective parameters, we further compared the sensitivity of the effective parameters and the corresponding p-norm values to the variances of the input hydraulic parameters to determine whether and to what extent the p-norms are less sensitive to parameter variances. This was done by calculating the relative difference between the effective parameters and the corresponding p-norms for two different variance levels of the input hydraulic parameters. The relative difference was calculated as follows:

Formula 20[20]
where ß denotes any quantity of interest (i.e., any of pKsG, p{alpha}G, pKsvG, p{alpha}vG, KsGeff, {alpha}Geff, KsvGeff, and {alpha}vGeff). The subscript X in Eq. [20] indicates that the quantity was obtained using variances of the field data, while the subscript 4X indicates input parameter variances that were four times those of the field measurements (i.e., the coefficient of variation was two times larger). A fourfold difference in the variances is relatively large and hence should provide a good indication of whether p-norms are indeed less sensitive to parameter variances.


   RESULTS AND DISCUSSION
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
Figure 2 shows a scatter plot of relative differences in effective parameter values vs. corresponding p-norms for the two assumed variance levels (i.e., one and four times those of the field data). Results are plotted for the Gardner and van Genuchten models assuming different dimensionless water table depths, z0* (10, 5, and 2.5). We did run simulations for a range of h0* values (i.e., pressure heads at the land surface) for each combination of model (Gardner or van Genuchten) and parameter type ({alpha} or Ks). Depending on the value of h0*, flow can either represent evaporation (when h0* > z0*) or infiltration (when h0* < z0*). In general, relative differences in the p-norms between the two variance levels are smaller than relative differences in the effective hydraulic parameter values, especially for the saturated hydraulic conductivity. The figure shows that p-norms vary far less than the effective parameter values.


Figure 2
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  		Fig. 2 Scatter plot of relative differences in effective values of the saturated hydraulic conductivity, Ks, and shape parameter {alpha} vs. corresponding p-norms for two variance levels (one and four times those observed for the field data). G: Gardner model, vG: van Genuchten model. The numbers in parentheses indicate the dimensionless water table depth, z0* = Formula 20z0, where Formula 20 is the mean of the field-measured van Genuchten {alpha}vG values and z0 is the water table depth.

 
Figure 2 indicates that results are scattered mostly below the 1:1 line, which means that p-norm relative differences are generally smaller than those for the effective parameters. In view of the large difference in the input parameter variances (a fourfold difference), a much smaller variation in the p-norms than in the effective parameters is encouraging when trying to estimate the effective hydraulic parameters of relatively large land areas. The results imply that upscaling schemes in terms of p-norms will absorb some hydraulic parameter variability, which should make them less sensitive to the parameter variances, and hence to soil heterogeneity.

The lesser variability of p-norm values relative to the variabilities (heterogeneities) of the input hydraulic parameters is important when trying to define effective media. A motivation for using effective parameters is that the larger scale process of interest in heterogeneous soils can be captured by a process that assumes only one set of soil parameters. Thus, the heterogeneous system is replaced by an equivalent homogeneous system with one constant set of hydraulic parameters. The largest relative difference in effective parameters for the two scenarios was about 0.5 (for the Gardner Ks when z0* = 10.0), while the maximum relative difference for the p-norm values was about 0.25 (for the Gardner {alpha} when z0* = 10). Figure 2 shows that the p-norms for the {alpha} parameter are generally more variable than the effective parameters. In fact, the p-norm variability in {alpha} is larger than the effective parameter variability mainly only for relatively high pressure heads at the soil surface (i.e., when the surface soil is relatively dry). Figure 2 also shows that the p-norm of the Gardner {alpha} has more variability relative to the van Genuchten {alpha}. We believe that this is related to the rapid decrease in the Gardner water retention curve when soils are relatively dry (i.e., at large pressure heads).

Two additional points from Fig. 2 are worth mentioning. First, p-norms (or the optimal upscaling schemes) are less well defined for the Gardner model than for the van Genuchten model, and may in fact be more difficult to use than the van Genuchten model in the upscaling context, especially for dry scenarios such as for many arid and semiarid conditions. Also, dry conditions are generally more difficult to deal with both in the upscaling process and when trying to establish upscaling relationship between the two soil hydraulic models.

Figure 3 shows p-norm values of Ks and {alpha} for both the Gardner and van Genuchten models as functions of the dimensionless pressure head at the soil surface h0*, for conditions when the dimensionless water table depth z0* = 5. The figure legends (1) through (5) indicate various statistical structures of the heterogeneous local-scale hydraulic parameters. Since z0* = 5, the left half of the figures holds mostly for infiltration (h0* < z0*) and the right half for evaporation (h0* > z0*). As seen by the lower slopes of the red (van Genuchten) traces vs. the black (Gardner) traces, the results indicate that upscaling schemes for the van Genuchten model are less affected by pressure head conditions at the soil surface than the Gardner model. Considering different statistical structures of hydraulic parameters, the p-norms for the Gardner model are also more variable than those for the van Genuchten model, especially for the saturated hydraulic conductivity, Ks. In general, the Gardner model p-norms vary more than the van Genuchten model p-norms when the statistical features of the hydraulic parameters change (i.e., the black traces vary much more in both plots). These results suggest that the upscaling schemes for the van Genuchten model are better defined and less variable when both the environmental conditions (e.g., the pressure head values at the soil surface) and the hydraulic parameter variabilities change.


Figure 3
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  		Fig. 3 Plots showing p-norms of (a) the saturated hydraulic conductivity, Ks, and (b) the {alpha} shape parameter for the Gardner (G) and van Genuchten (vG) models when the dimensionless water table depth z0* = 5: (1) 10 000 regenerated points, correlation length divided by grid length (CL) = 5, correlation coefficient between {alpha} and Ks (CC)(G) >0.9, CC(vG) >0.9; (2) 10 000 regenerated points, CL = 50, CC(G) >0.9, CC(vG) >0.9; (3) 84 measured points, CC(G) = 0.24, CC(vG) = 0.74; (4) 10 000 regenerated points, CL = 5, CC(G) = 0.24, CC(vG) = 0.75; (5) 10 000 regenerated points, CL = 5, CC(G) >0.9, CC(vG) >0.9, variance four times measured value.

 
Figures 4 , 5 , and 6 show p-norm relationships between the Gardner and van Genuchten parameters for three progressively smaller dimensionless depths of the water table (z0*): 10, 5, and 2.5, respectively. In each figure, we plotted a total of seven curves. Of these, six curves resulted from the 10000 randomly regenerated values of the input hydraulic parameters, while the seventh curve shows results based on the measured field data. Note that all of the curves in Fig. 4 to 6 have a mean input parameter value equal to the field-measured data. Among the six curves calculated from the 10000 regenerated data points, those with solid symbols depict situations where {alpha} and Ks are correlated, with the coefficients (CC) in the range between 0.95 and 0.99.


Figure 4
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  		Fig. 4 The p-norm relationships between the Gardner (G) and van Genuchten (vG) models for (a) saturated hydraulic conductivity, Ks, and (b) the {alpha} shape parameter when the dimensionless water table depth z0* = 10. CL is correlation length divided by grid length and CC is the correlation coefficient between {alpha} and Ks.

 

Figure 5
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  		Fig. 5 The p-norm relationships between the Gardner (G) and van Genuchten (vG) models for (a) saturated hydraulic conductivity, Ks, and (b) the {alpha} shape parameter when the dimensionless water table depth z0* = 5. CL is correlation length divided by grid length and CC is the correlation coefficient between {alpha} and Ks.

 

Figure 6
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  		Fig. 6 The p-norm relationships between the Gardner (G) and van Genuchten (vG) models for (a) saturated hydraulic conductivity, Ks, and (b) the {alpha} shape parameter when the dimensionless water table depth z0* = 2.5. CL is correlation length divided by grid length and CC is the correlation coefficient between {alpha} and Ks

 
Because the measured field data are spatially too sparse for a meaningful characterization of parameter spatial correlation structures, we used a wide range of parameter autocorrelation lengths to investigate the importance of autocorrelation length. For Fig. 4, 5, and 6, we used four parameter autocorrelation lengths with a relatively large range (from 2 to 50 times the grid length values) so as to capture possible practical field correlation conditions. The results are shown as Curves (1) through (4). Curve (5) in all figures shows results based on only 84 measured parameter points. Curve (6) represents the situation where 10000 points were regenerated using a correlation coefficient that was the same as for the field-measured parameters. Curve (7) shows results when the variance was four times the field-measured variance.

Figure 4 shows p-norm relationships between the Gardner and van Genuchten models for Ks and {alpha} when z0* = 10 (note that z0* = 10 is approximately equivalent to z0 [dimensional] = 10 m). Our calculations showed that the p-norm relationships are very similar in cases of deeper water tables (i.e., z0* > 10, results not shown) compared with cases for which z0* = 10 (shown in Fig. 4). However, the p-norms span a wider range when z0* > 10 than when z0* = 10, especially for the Gardner model. When the water table was relatively deep (z0* ≥ 10), p-norms for the van Genuchten model were well defined and relatively constant for a wide range of h0* values, covering both evaporation and infiltration conditions. In comparison, p-norms for the Gardner model were not well defined in that they tended to vary quite significantly as scenarios changed from evaporation to infiltration. These results suggest that the van Genuchten functions generally work better than the Gardner functions for the hydrologic processes considered in this study, especially for relatively high pressure heads at the soil surface (dry surfaces) such as those typical in arid and semiarid environments. Again, we believe that the steeper decline in the exponential hydraulic conductivity function in the dry range is responsible for the poorer performance of the Gardner relative to the van Genuchten function.

Figure 5 shows p-norm relationships between the Gardner and van Genuchten models for both {alpha} and Ks when z0* = 5 (i.e., a shallower water table in comparison with Fig. 4). A noticeable difference compared with Fig. 4 is that the p-norm for the van Genuchten parameters varied across a slightly wider range of values when the flow scenarios ranged from evaporation to infiltration, especially for situations with larger variances in the input hydraulic parameters. Conversely, p-norms for the Gardner model spanned a much smaller range of values than the larger z0* (i.e., deeper water table) case.

Figure 6 presents p-norm relationships between the Gardner and van Genuchten models for both {alpha} and Ks when z0* = 2.5. As z0* becomes smaller, the p-norm patterns continue to shift to a narrower range for the Gardner p-norm and a broader range for the van Genuchten p-norm. Thus, for shallower water table conditions, and when the pressure head at the surface soil is lower, p-norms for either the Gardner or van Genuchten models would be comparable. We believe that the behavior of the Gardner model p-norm for large z0* can be also partly attributed to the fact that the Gardner {alpha} is typically about twice as large as the van Genuchten {alpha}. The Gardner {alpha} has first-order effects on the hydraulic conductivity functions, which makes the flux very sensitive to the value of the Gardner {alpha}. Therefore, direct use of {alpha} in the Gardner hydraulic conductivity model makes p-norm schemes very sensitive to changes in z0*.

The results from Fig. 4 Goto 6 show that the p-norms, and their relationships, for the two hydraulic property functions are similar when comparing Curves (5) and (6). Recall that those two curves have same the means, variances, and correlation coefficients (CC), the only difference being that Curve (5) is based on only 84 measured parameter values while Curve (6) is based on 10000 randomly regenerated parameter realizations. In other words, the two vastly different data sets (in terms of data quantity) produce quite comparable results when upscaling the flux across the land–atmospheric boundary and the surface effective degree of saturation. Another finding is that correlation between the hydraulic parameters within the same hydraulic property model significantly impacts the p-norm relationships as well as the p-norm values. For all scenarios considered in this study, smaller correlation levels that are strictly based on field-measured data cause the p-norm relationships to shift significantly compared with hydraulic parameters that have correlation coefficients between 0.95 and 0.99. Moreover, correlation between the hydraulic parameters within the same hydraulic property model has a more significant impact on the p-norm values (especially for the Gardner model) and p-norm relationships than do the variances of the hydraulic parameters.

From Fig. 4 Gothrough 6, hence, an important pattern emerges. Results indicate that even when variances in the input hydraulic parameter are four times the original measured data, the p-norm relationships are still clustered in the same neighborhood as long as the correlation coefficients remain unchanged. This suggests that hydraulic parameter variances have little impact on the p-norms, as well as on the corresponding p-norm relationships between the Gardner and van Genuchten models. On the other hand, when the parameter correlations between {alpha} and Ks change, the corresponding upscaling p-norm relationships shift significantly, thus illustrating the importance of parameter correlation in terms of dictating scaling relationships for different hydraulic property models. The results also show that correlation length (CL) of hydraulic parameters has a relatively minor impact in determining the p-norm relationships of the Gardner and van Genuchten functions. This fact can be seen from the four solid curves in Fig. 4, 5, and 6. When CL varies from 2 to 50 (a difference of 25 times!), all p-norm relationships studied still cluster together and can be described by very similar relationships.


   CONCLUSIONS
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
In this study we investigated hydraulic property upscaling schemes and possible relationships between different hydraulic functions, based on a data set of field-measured hydraulic properties, by applying two new upscaling criteria. While the established framework is quite generic, we used the field-measured data for two purposes: (i) to demonstrate how limited field data can be used in a relatively large-scale upscaling context, and (ii) to determine if limited field characterization is sufficient for upscaling considerations in a wide range of hydrologic processes ranging from infiltration to evaporation. Some of the main findings of this study are summarized as follows:

  1. For the steady-state flow problem considered in this study, we showed that p-norms and their relationships were similar using 84 field-measured hydraulic parameter values and 10000 randomly regenerated hydraulic parameter realizations when upscaling the flux across the land–atmospheric boundary and the surface effective degree of saturation. For more heterogeneous sites, however, more field measurements may be required to characterize the hydraulic parameter fields in the upscaling context.
  2. The upscaling schemes were in general better defined, and had less variability, in terms of p-norms than when effective parameter values were used (see Fig. 2).
  3. For the {alpha} parameters, the p-norm relationships between the Gardner and van Genuchten models were typically similar for a broad range of scenarios considered in this study. The p-norms may diverge, however, for different levels of variability in the input hydraulic parameters and other hydrologic conditions.
  4. In general, p-norms for the Gardner model were less well defined than for the van Genuchten model, and may in fact be more difficult to use than the van Genuchten model in the upscaling context when the water table is relatively deep (such as under many arid and semiarid conditions). The exponential decline of Gardner's hydraulic conductivity with the soil water pressure head is a property that could not be handled well in the upscaling schemes for deep water table conditions (such as often occurs in arid and semiarid regions).
  5. For deep water tables (at least equivalent to 10 m), p-norms for van Genuchten parameters were relatively constant, while p-norms for Gardner parameters varied significantly, especially as flow scenarios shifted from evaporation to infiltration. As the water table became shallower, p-norms for the van Genuchten model became also less well defined and more sensitive to changes in the pressure head at the soil surface.
  6. Correlations between the hydraulic parameters within a given hydraulic property model was important for determining p-norm relationships between the Gardner and van Genuchten models.
  7. Hydraulic parameter correlation lengths appeared to have a relatively minor impact in determining the p-norm relationships between the Gardner and van Genuchten models.


   ACKNOWLEDGMENTS
 
Funding support from the Water Resources Research Act, Section 104, research grant program of the U.S. Geological Survey, DRI's Applied Research Initiative, start-up funds to the first author, and funds from the Urban Flood Demonstration Project, U.S. Army Corp of Engineers, contract no. DACW 42-03-2-0003, are greatly appreciated. The support from NSF EPSCoR Grant no. EPS-0447416 is also acknowledged.


   REFERENCES
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 




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J. Zhu
Equivalent Parallel and Perpendicular Unsaturated Hydraulic Conductivities: Arithmetic Mean or Harmonic Mean?
Soil Sci. Soc. Am. J., September 1, 2008; 72(5): 1226 - 1233.
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