Published online February 27, 2007

Upscaling Schemes and Relationships for the Gardner and van Genuchten Hydraulic
Functions for Heterogeneous Soils
Jianting Zhu,* Michael H. Young, and Martinus Th. van Genuchten
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 thelarge (field or watershed) scale. Effective parameters are
essential for upscaling point-level measurements andprocesses 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 (andhence upscaling) schemes. Sharma and Luxmoore (1979)
pointed out that the soil–plant–atmospheric interactions
are very complex in terms of evaluating the influence ofsoil heterogeneity on water budget. They found that results very much depended on the coefficient of variationand 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 boththe 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. Theirresults showed that heterogeneity has a stronger effecton 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 theunsaturated zone using an analytical framework (Kimet al., 1996). Zhu et al. (2004) established optimal averaging schemes and probability distribution functions forparameters of several hydraulic functions, using correspondence among those hydraulic models by preserving the macroscopic capillary length and predicting thesame vertical moisture flux. In more recent studies, Zhu
and Mohanty (2006) investigated the effective hydraulicparameters for transient infiltration in terms of the
optimal averaging schemes for the input hydraulic andenvironmental parameter fields. Specifically, they explored the effects of microtopography (as reflected insurface 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 (thirdorder 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 sameor similar hydrologic responses. For example, Warrick(1995) discussed some of the features that need to be

Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

ABSTRACT
Upscaled soil hydraulic properties are needed for many large-scalehydrologic applications such as regional and global climate studies andinvestigations 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 aheterogeneous soil formation are then derived by conceptualizing theformation as an equivalent homogeneous medium that satisfies the
upscaling criteria. Upscaling relationships between the Gardner andvan Genuchten models can then also be established for steady-statevertical flow using the statistical structures of the hydraulic parametersof 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 theeffective 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, butcorrelation between the hydraulic parameters within the hydraulicproperty models can significantly affect p-norm relationships.

U

NSATURATED flow in the vadose zone is often simulated using closed-form expressions for the soil hydraulic properties involving the soil water retention andunsaturated hydraulic conductivity functions. Some of
the more commonly used hydraulic functions include theGardner–Russo (Gardner, 1958; Russo, 1988), Brooks–
Corey (Brooks and Corey, 1964), and van Genuchten(van Genuchten, 1980) models. Soil hydraulic property

J. Zhu and M.H. Young, Desert Research Institute, Division of Hydrologic Sciences, 755 East Flamingo Rd., Las Vegas, NV 89119; M.Th.
van Genuchten, U.S. Salinity Lab., 450 West Big Springs Rd., Riverside, CA 92507-4617. Received 14 Mar. 2006. *Corresponding author(Jianting.Zhu@dri.edu).
Published in Vadose Zone Journal 6:186–195 (2007).
Original Researchdoi:10.2136/vzj2006.0041ª Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA

186

187

Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

www.vadosezonejournal.org

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 (us) and residual (ur) water contents, and minimizing the difference between the water retention curves.
Morel-Seytoux et al. (1996) converted Brooks–Coreyparameters 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 establishedparameter equivalence among the hydraulic conductivity
functions by preserving the macroscopic capillary lengthand predicting the same vertical moisture flux across thesoil surface. Their results showed that the hydraulicparameters 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 hydraulicproperties for steady-state vertically dominated flow.
We propose two new criteria to upscale the Gardner andvan 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 thesoil surface moisture content. The criteria require thatthe 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 forland–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 headat the soil surface are assumed homogenous for typicalremote-sensing pixel or large-scale hydro-climate model
grid scales as considered in this study, since the scaleof 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 hydraulicparameters to account for uncertainties in the spatiallyvariable 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., typicalremote-sensing pixel or a larger scale model grid) basedon 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 het-

erogeneous soil formation as an equivalent homogeneous medium such that the equivalent homogeneous
soil will produce the same ensemble-mean flux acrossthe land–atmosphere boundary and the same ensemblemean surface moisture content as the heterogeneous
medium. Mathematically, we express the effective parameters using the p-norm, or the p-order power average ofthe 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 pnorm 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, thep-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 interms of the p-norm values) are less sensitive to the variability of the hydraulic parameters and other relatedsoil-water conditions such as water table depth and thepressure head at the soil surface. In other words, if the
upscaling schemes of the hydraulic parameters for largescale applications are less sensitive to the soil heterogeneity and other conditions, conceptualization of theheterogeneous medium as an equivalent homogeneous
medium is more feasible and the upscaling schemes areeasier to implement.
One complication with upscaling schemes for larger
scale hydrologic applications is that they differ whendifferent soil hydraulic property functions are used.
While significant efforts have been made to establish theequivalence 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 andstill largely unresolved issue. Therefore, our first specificobjective 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 tovariations in random hydraulic parameter fields compared with the effective parameters.
MATERIALS AND METHODS
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 thewater 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 morecomprehensive overview of alternative functional expres-

188

VADOSE ZONE J., VOL. 6, FEBRUARY 2007

sions for the soil hydraulic properties, including those used inthis study.
Gardner Model

Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

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

Se 5 [exp(20:5aG h)(1 1 0:5aG h)]0:8[1]

K 5 KsG exp(2aG h)[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:

Se 5

K5

1
[1 1 (avG h)n ]m

[3]


2
KsvG 1 2 (avG h)mn [1 1 (avG h)n ]2m[1 1 (avG h)n ]0:5m[4]In Eq. [1] to [4], Se = (u 2 ur)/(us 2 ur) is the effective degree ofsaturation, u is the volumetric water content, ur is the residual
volumetric water content, us is the saturated volumetric watercontent, 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;a, m, and n are empirical hydraulic shape parameters, andm 5 1 2 1/n, while the subscripts G and vG refer to Gardnerand van Genuchten model parameters, respectively. Being thehydraulically 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)

z0 5

1
F(q/Ks )a[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 isdefined as
ah0

F(q/Ks ) 5

Kr (s)ds
0 Kr (s) 1 q/Ks#

[6]

in which s is a dummy integration variable denoting the dimensionless quantity ah in the hydraulic conductivity functions. For the Gardner model, Kr(s) 5 exp(2s), in which case
Eq. [6] can be integrated analytically to giveFG (q/Ks ) 5 ln

1 1 q/Ks
exp(2ah0 ) 1 q/Ks

[7]

which, with Eq. [5], leads toq 5 Ks

1 2 exp[a(z0 2 h0 )]exp(az0 ) 2 1[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 agiven soil water pressure head, h0, and water table depth, z0, is
obtained by assuming some initial value for the flux and theniteratively adjusting its value until Eq. [5] is satisfied within aprescribed tolerance.

Field Measurements
In this study, we used measured hydraulic properties fromthe 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 werecollected 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 theintake rate was at or near steady state. At that time, the tension infiltrometer was reset to a lower tension level. Four tofive tension steps were used for each test, typically at 12, 9, 6, 3,
and |0 (saturation) cm.
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) andnumerical inversion of the data using parameter estimation.
Assuming applicability of Gardner’s exponential K(h) function (Eq. [2]), three-dimensional, steady-state infiltration froma circular source is given by (Wooding, 1968)
i(h) 5 KsG exp(2aG h) 1 1

4
praG



[9]

where i(h) is the steady-state infiltration rate at supply tensionh, and r is the radius of the tension infiltrometer. The first termof Eq. [9] within the second set of parentheses accounts forvertical gravitational flow, and the second term for lateral flow
due to capillarity. Equation [9] contains two unknowns: KsG
and aG. 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 aG in Eq. [9] (e.g., Logsdon and Jaynes, 1993).
In addition to Wooding’s solution to obtain KsG and aG, acomplete set of van Genuchten soil hydraulic parameters(i.e., KsvG, avG, n, and us) 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 fordeveloping 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 thesaturated hydraulic conductivity, Ks, and the a parameterfor both models, obtaining CC(KsG, aG) 5 0.24, and CC(KsvG,
avG) 5 0.78. The parameter sets in Table 1 involve the Gardnerand van Genuchten saturated hydraulic conductivities (KsG and
KsvG), the a parameter for the two models (aG and avG), and189

Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

www.vadosezonejournal.org

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

the van Genuchten n parameter. Since we were interested in
the effective degree of saturation at the soil surface, the us andur parameters of the hydraulic property function are not relevant in this study.

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 theother 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 avG together with a constant parameter n, or a variable n with a constant avG, 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 fieldmeasured hydraulic parameters for establishing the upscaling
relationships, we also used regenerated fields of 10 000 ranTable 1. Basic statistics of 84 field-measured soil hydraulic parameter† values.
KsG
21

Mean
Variance
SD
CV

cm d
212.433
25579.630
159.936
0.753

aG

KsvG

avG

21

21

21

cm
0.227
0.010
0.098
0.433

cm d
176.73121883.190
147.930
.837

cm
0.092
6.96E-4
0.026
0.287

n

us
3

2.565
0.387
0.622
0.242

23

cm cm
0.212
2.33E-3
0.048
0.228

† G, Gardner model; vG, van Genuchten model; Ks, saturated hydraulic conductivity; a and n, shape parameters; us, saturated volumetricwater content.

dom values for the hydraulic parameters that may be crosscorrelated. The cross-correlated random fields of Ks and awere generated using the spectral method proposed by Robinet 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 byR(f) 5f12 (f)[f11 (f)f22 (f)]1/2[10]

where f11(f) and f22(f) are the power spectra of random Fields 1
and 2, respectively, and f12(f) is the cross spectrum between
Fields 1 and 2. Having |R|2 5 1 indicates perfect linear correlation between the random fields. In this study, 10 000 realizations of hydraulic parameter values were regenerated to match
the field-measured parameter target statistics to various degrees. The parameters Ks and a can be satisfactorily fittedusing lognormal distributions (Smith and Diekkruger, 1996;
Nielsen et al., 1973). In this study, both Ks and a were assumed
to obey lognormal distributions.
A total of six hydraulic parameter sets of 10 000 data points
was generated for both the Gardner and van Genuchten
models. The first four sets all assumed nearly full correlation
between Ks and a, with correlation coefficients being in therange between 0.95 and 0.99. The differences among the fourparameter 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 a as the field-measured data (about 0.24 and0.74 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 theactual field-measured variance values.

Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

190

VADOSE ZONE J., VOL. 6, FEBRUARY 2007

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 andEq. [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 calculatedusing the two upscaling criteria we proposed (i.e., preservation of the surface flux and the surface effective degree ofsaturation). Specifically, for the Gardner model the effective
hydraulic parameters aGeff and KsGeff for a and Ks are calculated using[exp(20:5aGeff h0 )(1 1 0:5aGeff h0 )]0:8
5 [exp(20:5aG h0 )(1 1 0:5aG h0 )]0:8[11]

1 2 exp[aGeff (z0 2 h0 )]5 qG
exp(aGeff ) 2 1[12]

KsGefffields than the effective parameters since the larger randomparameter 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 mainidea is that the optimal p-norm for a particular hydraulicparameter will guide how best to upscale that parameter tolarger scale applications. Here we express the effective hydraulic parameters, as defined by Eq. [11] through [14], as
p-norm power averages [i.e., jeff 5 ĵ( p)] where j can be any of
the hydraulic parameters in this study (aG, avG, KsG, or KsvG).
We now restate one of the objectives of this study: Can wefind relationships between the p-norms (the optimal upscaling
schemes) for the parameters of the two different hydraulicfunctions 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), andobtain from this effective a and Ks values for both the Gardnerand van Genuchten models using Eq. [11] through [14]. Next,
the corresponding p-norms for the hydraulic parameters arecalculated iteratively using the following equations:"

KsGeff 5 (1/N) oi51

respectively. For the van Genuchten model, the effective hydraulic parameters avGeff and KsvGeff are obtained with

"

[13]

aGeff 5 (1/N) oqvG (KsGeff , aGeff , n, m, h0 ) 5 qvG

[14]

"

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 ĵ( p) for any set of N
random parameter values ji is given by (Korvin 1982; Greenet al., 1996),

"

#1/p

N

ĵ(p) 5 (1/N) o

i51

jip

[15]

As used in this study, the p-norm is a generalization of thearithmetic ( p 5 1), geometric ( p Y 0) and harmonic ( p 5 21)
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, thep-norm should be less sensitive to variations in the random j

i51

i51

#1/paG[17]#1/pKs vG
pKs vG
KsvGi

i51

[18]

#1/pav G

N

avGeff 5 (1/N) o[16]

apGiaGN

KsvGeff 5 (1/N) o
"

pKs G
KsGi

N

[1 1 (avGeff h0 )n ]2m 5 [1 1 (avG h0 )n ]2m

respectively, where q is the ensemble steady-state flux (again
either evaporation or infiltration) across the land–atmosphericboundary. The right-hand sides of these equations are themean (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 ona 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 assumingthat the equivalent homogeneous soil will produce the sameensemble-mean flux across the land–atmosphere boundary andthe same ensemble-mean surface effective degree of saturation.
We used the effective hydraulic parameters to predict the largescale mean flux exchange across the land–atmospheric boundary and to preserve the mean effective degree of saturation at
the land surface.

#1/pKs G

N

pav G
avGi

[19]

where pKsG is the p-norm for the Gardner Ks, paG for the
Gardner a, pKsvG for the van Genuchten Ks, and pavG for thevan Genuchten a.
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 watertable 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 * 5 avG h0 andz0 * 5 avG z0 , respectively, where avG is the mean of the fieldmeasured van Genuchten avG values. In cases where aG h0 andaG z0 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 andthe water table depth, i.e., h0 * 5 avG h0 and z0 * 5 avG z0 .
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 bycalculating 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:

rb 5_b4X 2 bX __b4X _ 1 _bX _

[20]

191

where b denotes any quantity of interest (i.e., any of pKsG, paG,
pKsvG, pavG, KsGeff , aGeff , KsvGeff , and avGeff). 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 fieldmeasurements (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.

ability, which should make them less sensitive to theparameter variances, and hence to soil heterogeneity.
The lesser variability of p-norm values relative to thevariabilities (heterogeneities) of the input hydraulic parameters is important when trying to define effective
media. A motivation for using effective parameters isthat 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* 510.0), while the maximum relative difference for the
p-norm values was about 0.25 (for the Gardner a whenz0* 5 10). Figure 2 shows that the p-norms for the aparameter are generally more variable than the effective
parameters. In fact, the p-norm variability in a 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 a has more variability relative to the van Genuchten a. We believe that
this is related to the rapid decrease in the Gardner water
retention curve when soils are relatively dry (i.e., atlarge 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 thevan 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 a for both the
Gardner and van Genuchten models as functions of the

RESULTS AND DISCUSSION
Figure 2 shows a scatter plot of relative differencesin effective parameter values vs. corresponding p-norms
for the two assumed variance levels (i.e., one and fourtimes 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 combinationof model (Gardner or van Genuchten) and parameter
type (a or Ks). Depending on the value of h0*, flow caneither represent evaporation (when h0* . z0*) or infiltration (when h0* , z0*). In general, relative differencesin 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 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 muchsmaller variation in the p-norms than in the effective
parameters is encouraging when trying to estimate theeffective hydraulic parameters of relatively large land
areas. The results imply that upscaling schemes in termsof p-norms will absorb some hydraulic parameter vari-

0.50
Ks (G,10)
Ks (G,5)Relative Difference (p-norm)

Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

www.vadosezonejournal.org

0.40

Ks (G,2.5)
Ks (vG,10)
Ks (vG,5)

0.30

Ks (vG,2.5)
alpha (G,10)alpha (G,5)

0.20alpha (G,2.5)
alpha (vG,10)0.10alpha (vG,5)alpha (vG,2.5)

0.00
0.00

1:1 line

0.10

0.20

0.30

0.40

0.50

Relative Difference (Effective Parameters)
Fig. 2. Scatter plot of relative differences in effective values of the saturated hydraulic conductivity, Ks, and shape parameter a 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* 5 avG z0 , where avG is the mean of the field-measured van Genuchten
avG values and z0 is the water table depth.

192

VADOSE ZONE J., VOL. 6, FEBRUARY 2007

2.5

(a)

0.5
-0.5
-1.5

2.0

4.0

6.0

8.0

10.0

12.0

h0 *
0.0

(b)
-0.5
-1.0
-1.5

-2.5
-3.0
0.0

Gardner (1)Gardner (2)Gardner (3)Gardner (4)Gardner (5)van Genuchten (1)van Genuchten (2)van Genuchten (3)van Genuchten (4)van Genuchten (5)

5.0

(a)

4.0
2.0

4.0

6.0
h0*

8.0

10.0

12.0

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

dimensionless pressure head at the soil surface h0*, forconditions when the dimensionless water table depth
z0* 5 5. The figure legends (1) through (5) indicate various statistical structures of the heterogeneous local-scale
hydraulic parameters. Since z0* 5 5, the left half ofthe figures holds mostly for infiltration (h0* , z0*) andthe right half for evaporation (h0* . z0*). As seen by thelower slopes of the red (van Genuchten) traces vs. theblack (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 forthe 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 hydraulicparameters change (i.e., the black traces vary much morein both plots). These results suggest that the upscaling
schemes for the van Genuchten model are better definedand less variable when both the environmental condi-

3.0

p-norm (Ks) vG

-2.0

2.0
1.0
0.0
-1.0
-2.0
-3.0
-4.0
-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

p-norm (Ks) G
0.0

(b)
-1.0

p-norm (alpha) vG

-2.5
0.0

p-alphaReproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

p-Ks

1.5

tions (e.g., the pressure head values at the soil surface)
and the hydraulic parameter variabilities change.
Figures 4, 5, and 6 show p-norm relationships between
the Gardner and van Genuchten parameters for threeprogressively smaller dimensionless depths of the watertable (z0*): 10, 5, and 2.5, respectively. In each figure,we plotted a total of seven curves. Of these, six curves
resulted from the 10 000 randomly regenerated values of
the input hydraulic parameters, while the seventh curveshows results based on the measured field data. Notethat all of the curves in Fig. 4 to 6 have a mean input parameter value equal to the field-measured data. Amongthe six curves calculated from the 10 000 regenerated
data points, those with solid symbols depict situations
where a and Ks are correlated, with the coefficients (CC)in the range between 0.95 and 0.99.
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, weused four parameter autocorrelation lengths with a relatively large range (from 2 to 50 times the grid length values)-2.0(1) 10000 pts, CL=5, CC(G)>0.9,CC(vG)>0.9(2) 10000 pts, CL=2, CC(G)>0.9, CC(vG)>0.9-3.0(3) 10000 pts, CL=10, CC(G)>0.9, CC(vG)>0.9(4) 10000 pts, CL=50, CC(G)>0.9, CC(vG)>0.9-4.0(5) 84 measured pts, CC(G)=0.2401, CC(vG)=0.7408(6) 10000 pts, CL=5, CC(G)=0.2382, CC(vG)=0.7498(7) 10000 pts, CL=5, CC(G)>0.9, CC(vG)>0.9, Var 4X larger-5.0-5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5

0.0

p-norm (alpha) G
Fig. 4. The p-norm relationships between the Gardner (G) and van
Genuchten (vG) models for (a) saturated hydraulic conductivity,
Ks, and (b) the a shape parameter when the dimensionless watertable depth z0* 5 10. CL is correlation length divided by grid lengthand CC is the correlation coefficient between a and Ks.

193

www.vadosezonejournal.org

1.0

1.0

(a)

(a)
0.5

p-norm (Ks) vG

p-norm (Ks) vG

-0.5
-1.0

0.0

-0.5
-1.5
-2.0
-2.0

-1.0

0.0

1.0

-1.0
-1.0

2.0

-0.5

p-norm (Ks) G
0.0(1) 10000 pts, CL=5, CC(G)>0.9,CC(vG)>0.9

2.5

(b)

p-norm (alpha) vG(6) 10000 pts, CL=5, CC(G)=0.2382, CC(vG)=0.7498(7) 10000 pts, CL=5, CC(G)>0.9, CC(vG)>0.9, Var 4X larger

-1.5

-2.0
-2.0

-1.5

-1.0

1.0

1.5

(b)

(1) 10000 pts, CL=5, CC(G)>0.9, CC(vG)>0.9(3) 10000 pts, CL=10, CC(G)>0.9, CC(vG)>0.9(4) 10000 pts, CL=50, CC(G)>0.9, CC(vG)>0.9(5) 84 measured pts, CC(G)=0.2401, CC(vG)=0.7408

-1.0

0.5

(2) 10000 pts, CL=2, CC(G)>0.9, CC(vG)>0.9(3) 10000 pts, CL=10, CC(G)>0.9, CC(vG)>0.9

-0.5

0.0

p-norm (Ks) G(2) 10000 pts, CL=2, CC(G)>0.9, CC(vG)>0.9

p-norm (alpha) vG

Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

0.5
0.0

-0.5

0.0

p-norm (alpha) G

1.5(4) 10000 pts, CL=50, CC(G)>0.9, CC(vG)>0.9(5) 84 measured pts, CC(G)=0.2401, CC(vG)=0.7408(6) 10000 pts, CL=5, CC(G)=0.2382, CC(vG)=0.7498

0.5(7) 10000 pts, CL=5, CC(G)>0.9, CC(vG)>0.9, Var 4X

-0.5

-1.5
-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

p-norm (alpha) G

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

Fig. 6. The p-norm relationships between the Gardner (G) and van
Genuchten (vG) models for (a) saturated hydraulic conductivity,
Ks, and (b) the a shape parameter when the dimensionless watertable depth z0* 5 2.5. CL is correlation length divided by grid
length and CC is the correlation coefficient between a and Ks

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 thesituation where 10 000 points were regenerated using acorrelation coefficient that was the same as for the fieldmeasured parameters. Curve (7) shows results when thevariance was four times the field-measured variance.
Figure 4 shows p-norm relationships between the
Gardner and van Genuchten models for Ks and a whenz0* 5 10 (note that z0* 5 10 is approximately equivalent
to z0 [dimensional] 5 10 m). Our calculations showed thatthe p-norm relationships are very similar in cases ofdeeper water tables (i.e., z0* . 10, results not shown)compared with cases for which z0* 5 10 (shown in Fig. 4).
However, the p-norms span a wider range when z0* . 10
than when z0* 5 10, especially for the Gardner model.
When the water table was relatively deep (z0* $ 10),
p-norms for the van Genuchten model were well definedand relatively constant for a wide range of h0* values,
covering both evaporation and infiltration conditions. In
comparison, p-norms for the Gardner model were notwell 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 thehydrologic processes considered in this study, especiallyfor relatively high pressure heads at the soil surface (drysurfaces) 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 theGardner relative to the van Genuchten function.
Figure 5 shows p-norm relationships between the
Gardner and van Genuchten models for both a and Ks
when z0* 5 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 variancesin the input hydraulic parameters. Conversely, p-norms
for the Gardner model spanned a much smaller range ofvalues 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 a and Ks

Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

194

VADOSE ZONE J., VOL. 6, FEBRUARY 2007

when z0* 5 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, andwhen the pressure head at the surface soil is lower, pnorms for either the Gardner or van Genuchten models
would be comparable. We believe that the behavior of theGardner model p-norm for large z0* can be also partly
attributed to the fact that the Gardner a is typically about
twice as large as the van Genuchten a. The Gardner a has
first-order effects on the hydraulic conductivity functions,which makes the flux very sensitive to the value of theGardner a. Therefore, direct use of a in the Gardner hydraulic conductivity model makes p-norm schemes very
sensitive to changes in z0*.
The results from Fig. 4 to 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
10 000 randomly regenerated parameter realizations. In
other words, the two vastly different data sets (in termsof 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 hydraulicparameters within the same hydraulic property modelsignificantly impacts the p-norm relationships as well as
the p-norm values. For all scenarios considered in thisstudy, smaller correlation levels that are strictly based onfield-measured data cause the p-norm relationships toshift significantly compared with hydraulic parametersthat have correlation coefficients between 0.95 and 0.99.
Moreover, correlation between the hydraulic parameters within the same hydraulic property model has amore significant impact on the p-norm values (especiallyfor the Gardner model) and p-norm relationships than
do the variances of the hydraulic parameters.
From Fig. 4 through 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 thathydraulic 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 a and Ks change, the corresponding upscaling p-norm relationships shift significantly, thus
illustrating the importance of parameter correlationin terms of dictating scaling relationships for different
hydraulic property models. The results also show that
correlation length (CL) of hydraulic parameters has arelatively 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.

CONCLUSIONSIn 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) todemonstrate how limited field data can be used in a
relatively large-scale upscaling context, and (ii) to determine if limited field characterization is sufficientfor 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 inthis study, we showed that p-norms and their relationships were similar using 84 field-measured
hydraulic parameter values and 10 000 randomly
regenerated hydraulic parameter realizations when
upscaling the flux across the land–atmosphericboundary 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 a 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 inthe input hydraulic parameters and other hydrologic conditions.
4. In general, p-norms for the Gardner model wereless well defined than for the van Genuchten
model, and may in fact be more difficult to use thanthe van Genuchten model in the upscaling context when the water table is relatively deep (suchas 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 asoften 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 forthe van Genuchten model became also less welldefined and more sensitive to changes in the pressure head at the soil surface.

Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

www.vadosezonejournal.org

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 andvan 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 tothe 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
Ankeny, M.D., T.C. Kaspar, and R. Horton. 1988. Design for an automated tension infiltrometer. Soil Sci. Soc. Am. J. 52:893–896.
Brooks, R.H., and A.T. Corey. 1964. Hydraulic properties of porous
media. Hydrol. Pap. 3. Colorado State Univ., Fort Collins.
Casey, F.X.M., and N.E. Derby. 2002. Improved design for automatedtension infiltrometer. Soil Sci. Soc. Am. J. 66:64–67.
Gardner, W.R. 1958. Some steady state solutions of unsaturated moisture flow equations with applications to evaporation from a watertable. Soil Sci. 85:228–232.
Green, T.R., J.E. Contantz, and D.L. Freyberg. 1996. Upscaled soilwater retention using van Genuchten’s function. J. Hydrol. Eng.
1(3):123–130.
Hills, R.G., D.B. Hudson, and P.J. Wierenga. 1992. Spatial variability at
the Las Cruces trench site. p. 529–538. In M.Th. van Genuchten
et al. (ed.) Indirect methods for estimating the hydraulic propertiesof unsaturated soils. Univ. of California, Riverside.
Kim, C.P., and J.N.M. Stricker. 1996. Influence of spatially variablesoil hydraulic properties and rainfall intensity on the water budget.
Water Resour. Res. 32:1699–1712.
Kim, C.P., J.N.M. Stricker, and R.A. Feddes. 1997. Impact of soil heterogeneity on the water budget of the unsaturated zone. Water
Resour. Res. 33:991–999.
Kim, C.P., J.N.M. Stricker, and P.J.J.F. Torfs. 1996. An analytical framework for the water budget of the unsaturated zone. Water Resour.
Res. 32:3475–3484.
Korvin, G. 1982. Axiomatic characterization of the general mixture
rule. Geoexploration 19:267–276.
Leij, F.J., W.B. Russell, and S.M. Lesch. 1997. Closed-form expressions for water retention and conductivity data. Ground Water 35:
848–858.
Lenhard, R.J., J.C. Parker, and S. Mishra. 1989. On the correspondence between Brooks–Corey and van Genuchten models. J. Irrig.
Drain. Eng. 115:744–751.
Logsdon, S.D., and D.B. Jaynes. 1993. Methodology for determining
hydraulic conductivity with tension infiltrometers. Soil Sci. Soc.
Am. J. 57:1426–1431.
Morel-Seytoux, H.J., P.D. Meyer, M. Nachabe, J. Touma, M.Th. van
Genuchten, and R.J. Lenhard. 1996. Parameter equivalence for the
Brooks–Corey and van Genuchten soil characteristics: Preserving
the effective capillary drive. Water Resour. Res. 32:1251–1258.
Mualem, Y. 1976. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res. 12:513–522.

195

Nielsen, D.R., J.W. Biggar, and K.T. Erh. 1973. Spatial variability offield-measured soil-water properties. Hilgardia 42(7):215–260.
Reynolds, W.D., and D.E. Elrick. 1991. Determination of hydraulicconductivity using a tension infiltrometer. Soil Sci. Soc. Am. J. 55:
633–639.
Robin, M.J.L., A.L. Gutjahr, E.A. Sudicky, and J.L. Wilson. 1993.
Cross-correlated random field generation with the direct Fouriertransform method. Water Resour. Res. 29:2385–2397.
Russo, D. 1988. Determining soil hydraulic properties by parameter
estimation: On the selection of a model for the hydraulic properties.
Water Resour. Res. 24:453–459.
Schaap, M.G., and F.J. Leij. 1998. Database-related accuracy and uncertainty of pedotransfer functions. Soil Sci. 163:765–779.
Scheid, F. 1968. Theory and problems of numerical analysis. Schaum’s
Outline Ser. McGraw Hill Book Co., New York.
Sharma, M.L., and R.J. Luxmoore. 1979. Soil spatial variability and its
consequences on simulated water balance. Water Resour. Res. 15:
1567–1573.
Simunek, J., K. Huang, M. Sejna, and M.Th. van Genuchten. 1999. The
HYDRUS-2D software package for simulating two-dimensional
movement of water, heat and multiple solutes in variably-saturated
media. Version 2.0. IGWMC-TPS-53. Int. Ground Water Model.
Ctr., Colo. School of Mines, Golden.
Simunek, J., D. Wang, P.J. Shouse, and M.Th. van Genuchten. 1998.
Analysis of field tension disc infiltrometer data by parameter estimation. Int. Agrophys. 12:167–180.
Smith, R.E., and B. Diekkruger. 1996. Effective soil water characteristics and ensemble soil water profiles in heterogeneous soils. Water
Resour. Res. 32:1993–2002.
Stoer, J., and R. Bulirsch. 1980. Introduction to numerical analysis.
Springer-Verlag, New York.
van Genuchten, M.Th. 1980. A closed-form equation for predicting
the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J.
44:892–898.
Warrick, A.W. 1995. Correspondence of hydraulic functions for unsaturated soils. Soil Sci. Soc. Am. J. 59:292–299.
Warrick, A.W. 2003. Soil water dynamics. Oxford Univ. Press,
New York.
Warrick, A.W., and T.-C.J. Yeh. 1990. One-dimensional, steady verticalflow in a layered soil profile. Adv. Water Resour. 13:207–210.
Wooding, R.A. 1968. Steady infiltration from a large shallow circular
pond. Water Resour. Res. 4:1259–1273.
Young, M.H., K. Adams, S. Mizell, L. Chen, T. Bullard, and S. Grant.
2005. Assessment and improvement of hydraulic characterizationof alluvial fans to support estimates of recharge and initial abstraction. DRI Tech. Rep. DHS Publ. 41218. Desert Res. Inst., Las
Vegas, NV.
Young, M.H., E.V. McDonald, T.G. Caldwell, S.G. Benner, and D.G.
Meadows. 2004. Hydraulic properties of a desert soil chronosequence in the Mojave Desert, USA. Vadose Zone J. 3:956–963.
Zaslavsky, D. 1964. Theory of unsaturated flow into a non-uniform
soil profile. Soil Sci. 97:400–410.
Zhu, J., and B.P. Mohanty. 2002. Spatial averaging of van Genuchten
hydraulic parameters for steady state flow in heterogeneous soils:
A numerical study. Vadose Zone J. 1:261–272.
Zhu, J., and B.P. Mohanty. 2003. Effective hydraulic parameters forsteady state flows in heterogeneous soils. Water Resour. Res. 39(8):1178, doi:10.1029/2002WR001831.
Zhu, J., and B.P. Mohanty. 2006. Effective scaling factor for transient
infiltration in heterogeneous soils. J. Hydrol. 319:96–108.
Zhu, J., B.P. Mohanty, and N.N. Das. 2006. On the effective averaging
schemes of hydraulic properties at the landscape scale. Vadose
Zone J. 5:308–316.
Zhu, J., B.P. Mohanty, A.W. Warrick, and M.Th. van Genuchten. 2004.
Correspondence and upscaling of hydraulic functions for heterogeneous soils. Vadose Zone J. 3:527–533.