Correspondence and Upscaling of Hydraulic Functions for Steady-State Flowin Heterogeneous Soils
Jianting Zhu,* Binayak P. Mohanty, Arthur W. Warrick, and Martinus Th. van Genuchten

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

ABSTRACT

different types of hydraulic functions to correspond(i.e., to give similar or identical results for a given flow
scenario). Lenhard et al. (1989) developed equivalentvan Genuchten (1980) and Brooks and Corey (1964)
parameters based on the shapes of retention curves.
Morel-Seytoux et al. (1996) provided a way to convert
Brooks–Corey parameters to van Genuchten parameters based on preserving the macroscopic capillary
length and the asymptotic behavior of the soil water
retention curve.
A related issue of concern for heterogeneous fieldsoils is the upscaling of hydraulic parameters. Soil hydraulic functions are generally valid only at the point
or local scale. When they are used in larger (plot, field,
watershed or regional)–scale models, major questions
remain about how best to average the spatially variable
hydraulic properties over a heterogeneous soil volume.
The main objective of this study is to investigate how
the hydraulic parameters of commonly used hydraulicfunctions correspond and what the correspondence implies in terms of averaging schemes of the hydraulicproperties for steady-state vertical flow in heterogeneous soils at the larger scale. More specifically, we aim toestablish relationships of probability distribution functions and averaging schemes (in terms of p norms, as
will be discussed below) among the parameters of those
hydraulic functions. Our study focuses especially on
correspondence between the Brooks–Corey and van
Genuchten models.

Soil hydraulic parameters at relatively large scales (e.g., remotesensing footprints) are important for land–atmosphere interaction andgeneral circulation models as well as other applications. Within this context, we investigated two major issues involving soil hydraulic properties: (i) hydraulic parameter correspondence among some of the morecommonly used soil hydraulic conductivity functions (i.e., the Gardner[G], Brooks–Corey [BC], and van Genuchten [VG] equations) and(ii) their application to upscaling of hydraulic properties for steadystate flow in heterogeneous soils. We first establish parameter equivalence among the conductivity functions based on preserving macroscopic capillary lengths and predicting the same vertical water flux.
Next we investigate the significance of parameter equivalence on averaging schemes for the hydraulic parameters to allow predictions of
the ensemble characteristics for steady-state flow. Results show thatthe hydraulic parameters correspond very well and that the same rules
can be used for averaging the parameters of different hydraulic conductivity functions when predicting ensemble evaporation rates fromheterogeneous soils having a relatively large suction at the soil surface(e.g., a dry surface condition and/or a shallow groundwater table).
On the other hand, when the surface suction is finite (especially when
the suction is relatively small and/or the groundwater table is deep),
it is more difficult to obtain correspondence between the parameters ofthe different conductivity models. The hydraulic functions correspondespecially poorly when infiltration is considered. Parameter equivalence between the hydraulic functions is always satisfied for the caseof evaporation from a shallow water table, as long as the macroscopic
capillary length is preserved.

S

imulations of unsaturated flow in the vadose zone
typically use closed-form functional relationships torepresent the water-retention and unsaturated hydraulicconductivity functions. The Gardner exponential model(Gardner, 1958), the Brooks and Corey piecewise continuous model (Brooks and Corey, 1964), and the van
Genuchten model (van Genuchten, 1980) are some of themore widely used hydraulic conductivity functions. Conditions for which alternative forms of the hydraulic functions give the same or similar responses are importantin many applications. Warrick (1995) investigated the correspondence of hydraulic functions and discussed someof the features that need to be preserved in order for

SOIL HYDRAULIC PROPERTY MODELSAND MACROSCOPIC CAPILLARY LENGTH
Soil hydraulic behavior is characterized by the soil
water retention curve, which defines the water content
as a function of suction, and the hydraulic conductivity
function, which establishes the relationship between thehydraulic conductivity and the water content or suction.
A brief review of hydraulic conductivity models usedin this study is given below. Interested readers are referred to Leij et al. (1997) and Warrick (2003) for morecomprehensive reviews and discussions of various closedform expressions for the soil hydraulic properties, including the hydraulic conductivity models used here.
The unsaturated hydraulic conductivity, K, is typically
expressed by an equation of the form

J. Zhu, Department of Biological and Agricultural Engineering, 301B
Scoates Hall, Texas A&M University, College Station, TX; B.P. Mohanty, Department of Biological and Agricultural Engineering, 301C
Scoates Hall, Texas A&M University, College Station, TX 778432117; A.W. Warrick, Department of Soil, Water and Environmental
Science, 429 Shantz Building, #38, University of Arizona, Tucson,
AZ 85721; and M.Th. van Genuchten, George E. Brown, Jr. Salinity
Laboratory, 450 West Big Springs Road, Riverside, CA 92507-4617.
Received 4 Feb. 2004. Special Section: Uncertainty in Vadose Zone
Flow and Transport Properties. *Corresponding author (jzhu@cora.
tamu.edu).

K ⫽ KsKr(␣h, ␩)[1]where Ks is the saturated hydraulic conductivity, Kr is
the relative hydraulic conductivity, ␣ is known as thepore-size distribution parameter, h is the suction, and␩ is an empirical parameter characterizing the shape

Published in Vadose Zone Journal 3:527–533 (2004).
 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA

Abbreviations: BC, Brooks–Corey; G, Gardner; pdf, probability distribution function; VG, van Genuchten.

527

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528

VADOSE ZONE J., VOL. 3, MAY 2004

(nonlinearity) of the hydraulic conductivity function.
For the three models considered here, ␣ is denoted as
␣G for the Gardner model, ␣BC for the Brooks–Coreymodel, and ␣vG for the van Genuchten model. By analogy, ␩ is nonexistent for the Gardner model, and denoted ␭ for the BC model and m for the VG modelusing the conventional nomenclature. Please note that
␣h is a dimensionless quantity.
Gardner Model∞1
Hc ⫽ K ⫺s冮Kdh

Considering the case of evaporation from a very dry
soil surface (hdry → ∞) of a soil profile that contains awater table (hwet ⫽ 0) and substituting Eq. [2] and [3]into Eq. [5] then leads to the following expressions for themacroscopic capillary length for the G and BC models:HcG ⫽ 1/␣G

[7]⫹ 1) ⫹ 2
冢 ␣1 冣 ␭(ᐉ␭(ᐉ ⫹) ⫹ 1

[8]

and

The relative hydraulic conductivity as used by Gardner (1958) is given by
Kr(␣Gh, ␩) ⫽ exp(⫺␣Gh)

HcBC ⫽[2]

Kr(␣BCh, ␭) ⫽ (␣BCh)⫺␤

when

␣BCh ⬎ 1[3a]

Kr(␣BCh, ␭) ⫽ 1

when

␣BCh ⱕ 1[3b]where ␤ ⫽ ␭(ᐉB ⫹ 1) ⫹ 2, and ᐉB is the pore-connectivity
parameter that accounts for the presence of a tortuous
flow path, generally assumed to be 2.0 (Burdine, 1953).
This model has been successfully used to describe conductivity data for relatively homogeneous, mostly coarsetextured soils. The model may not describe data wellat or near the air-entry (or bubbling) suction (h ⫽ 1/␣BC)
where the curve is only zero-order continuous (i.e., theslope is discontinuous).
van Genuchten Modelvan Genuchten (1980) combined his proposed S-shaped
soil water retention function with the theoretical poresize distribution model of Mualem (1976) to obtain thefollowing hydraulic conductivity function:[4]

HcvG ⫽⫹ 2.07m ⫹ 19.5m
冢 ␣1 冣冢 0.046m
冣1 ⫹ 4.7m ⫹ 16m
2

3

2

vG

Macroscopic Capillary Length
The macroscopic capillary length (Hc) as frequently
used in analyses of unsaturated flow from a source at
suction hwet into a soil at suction hdry is given by (e.g.,
Philip, 1985)
hdry

[5]

hwet

where Kwet ⫽ K(hwet) and Kdry ⫽ K(hwet). For systems
that span across saturated (hwet ⫽ 0) and very dry conditions (hdry → ∞), Eq. [5] reduces to the effective capillary
drive as used by Morel-Seytoux and Khanji (1974, 1975)
and Morel-Seytoux et al. (1996), among others:[9]

HYDRAULIC PARAMETER
CORRESPONDENCE
Synthesis of Earlier Results
Morel-Seytoux et al. (1996) proposed two criteria forequivalence of the BC and VG parameters. Their primary
criterion was to preserve the effective capillary drive
given by Eq. [6]. Their secondary criterion was to preserve the asymptotic behavior of the retention curve at
low water contents. These two criteria lead to冢冣冢

0.046m ⫹ 2.07m2 ⫹ 19.5m3 3␭ ⫹ 1
␣vG
⫽
␣BC
1 ⫹ 4.7m ⫹ 16m2
3␭ ⫹ 2

冣 [10a]

and
␭⫽n⫺1

[10b]

Other approximate relationships between the BC and VG
parameters were suggested by van Genuchten (1980):
␣BC ⫽ ␣vG

[11]

␭⫽n⫺1

where ᐉM is a tortuosity or pore-connectivity parameter
estimated by Mualem (1976) to be about 0.5 as an average for many soils, n is a shape factor, and m ⫽ 1 ⫺ 1/n.

Hc ⫽ (Kwet ⫺ Kdry)⫺1 冮 Kdh

B

For the VG model, the macroscopic capillary length canbe approximated fairly accurately using the expression(Morel-Seytoux et al., 1996)

Brooks and Corey (1964) used the following empirical
relationship between K and h:{1 ⫺ (␣vGh)mn[1 ⫹ (␣vGh)n]⫺m}2[1 ⫹ (␣vGh)n]mᐉM

B

BC

Brooks–Corey Model

Kr(␣vGh,m) ⫽

[6]

0

[12]

Lenhard et al. (1989) obtained yet other relationships
by equating the specific moisture capacity halfway between the saturated (␪s) and residual (␪r) water contents
and minimizing the difference between the two BC and
VG water retention curves:
␣vG⫽ [0.72 ⫺ 0.35exp(⫺n4)]1/␭
␣BC

兵[0.72 ⫺ 0.35exp(⫺n4)]n/(1⫺n) ⫺ 1其1/n␭ ⫽ (n ⫺ 1)(1 ⫺ 0.5

n/(n⫺1)

)

[13]
[14]

While these relationships are potentially useful for establishing parameter correspondence among the different hydraulic property models, all are in essence mathematical manipulations based on matching of hydraulicproperty curves, and as such may lack physical meaning
in hydrologic applications. In this study we propose analternative primary equivalence criterion based on hy-529

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www.vadosezonejournal.org

draulic behavior equivalence. Our approach forces thepredicted flux across the soil surface to be the samefor the different hydraulic conductivity functions, rather
than matching the hydraulic property functions themselves. Warrick (1995) pointed out that hydraulic behavior equivalence depends on the hydrologic scenario being considered. We selected the flux across the soil surface for this purpose since this is an important quantity
in upscaling of hydrologic process from local scale tofootprint scale.
Our second equivalence criterion remains the sameas that used by Morel-Seytoux et al. (1996), that is, preserving the macroscopic capillary length. This criterion
is important since, as we will show later, for the asymptotic case when the flux is large (such as for evaporationfrom a soil with a shallow water table), flux equivalenceof various hydraulic functions will always be satisfied
as long as the macroscopic capillary length is preserved.

Correspondence with the Gardner Model
Since the Gardner model contains only one shape (nonlinearity) parameter, while the other two models containtwo parameters, we first determine how Gardner’s ␣Gshould correspond with the ␣’s of other two models
using the primary criterion of flux equivalence. We thendiscuss conditions for which both criteria will be satisfied.
When the suction at the soil surface is relatively large(h0 → ∞), application of Darcy’s Law leads to (Zhu andMohanty, 2002)
z0 ⫽1
F(q/Ks,␩)␣[15]

where z0 is the depth from the soil surface to the watertable, q is the steady-state flux rate, and function F isdefined by∞

Kr(s,␩)ds
K(s,␩)⫹ q/Ks0 r

F(q/Ks,␩) ⫽ 冮[16]

where s is a dummy integration variable. For the G model,
Kr(s, ␩) ⫽ exp(⫺s), in which case Eq. [16] can be integrated analytically to giveFG(q/Ks,␩) ⫽ ln[1 ⫹ (q/Ks)⫺1][17]

in which case Eq. [15] reduces toz0 ⫽ln[1 ⫹ (q/Ks)⫺1]
␣G

[18]

By equating Eq. [15] and [18], any arbitrary hydraulicconductivity function will produce equivalence to the
Gardner function in terms of predicting the same dimensionless flux q/Ks if␣F(q/Ks,␩)⫽␣G ln[1 ⫹ (q/Ks)⫺1]

Fig. 1. Plots showing ␣ parameter equivalence when h0 → ∞ between(a) the Gardner and van Genuchten models and (b) the Gardnerand Brooks–Corey models.

[19]

Equation [19] was applied to both the BC and VG
hydraulic conductivity equations. F(q/Ks, ␩) in Eq. [19]required numerical integration for this purpose, which
was achieved using Romberg integration (e.g., Scheid,
1968; Stoer and Bulirsch, 1980).

Figure 1 shows ␣ parameter equivalence based on
Eq. [19] for large surface suctions (h0 → ∞) between
the Gardner and van Genuchten models (Fig. 1a), and
the Gardner and Brooks–Corey models (Fig. 1b). Notethat parameter equivalence does not depend on thedepth to the water table, z0. The figure shows that in orderfor the Gardner and van Genuchten models to correspond, ␣vG must be smaller than ␣G, while for Brooks–
Corey model ␣BC can be either larger or smaller than
Gardner ␣G. Also, the ratio ␣BC/␣G decreases as ␭ increases. Other major findings are that, for the VG model,
when m ⫽ 0.5 (i.e., n ⫽ 2) the ratio ␣vG/␣G varies overa relatively small range, from about 0.4 to 0.5, whenq/Ks varies three orders of magnitude (Fig. 1a), and for
the BC model, a ␭ value between 0.42 and 0.83 leads
to only a small range in ␣BC/␣G values as shown in Fig. 1b.
Equating Eq. [7] and [9] to preserve the macroscopic
capillary length for the G and VG models leads to
␣vG 0.046m ⫹ 2.07m2 ⫹ 19.5m3⫽
␣G
1 ⫹ 4.7m ⫹ 16m2

[20]Equation [20] shows that ␣vG/␣G ⫽ 0.4052 for m ⫽ 0.5.
In other words, when the m is about 0.5 for the VG
model, a ␣vG/␣G value of about 0.4 will produce the same
evaporative flux and also preserve the macroscopic
length when using the G and VG models.
Equating Eq. [7] and [8] similarly preserves themacroscopic capillary length for the G and BC models:530

VADOSE ZONE J., VOL. 3, MAY 2004

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

␣BC ␭(ᐉB ⫹ 1) ⫹ 2 3␭ ⫹ 2⫽⫽
␣G
␭(ᐉB ⫹ 1) ⫹ 1 3␭ ⫹ 1[21]This equation shows that ␣BC/␣G ⫽ 1.444 for ␭ ⫽ 0.42,
and 1.286 for ␭ ⫽ 0.83. These values are close to thoseobtained from Fig. 1b for ␭ ⫽ 0.42 and 0.83, respectively.
This means that for ␭ values between approximately
0.42 and 0.83, the ␣ parameter ratio required to preserve
the macroscopic capillary length will also predict approximately the same flux.
For a special scenario where q/Ks is large (such as forevaporation from soil having a shallow water table),
Eq. [16] can be approximated (with Kr neglected in the
denominator) to∞∞Kr(s,␩)ds≈ (q/Ks)⫺1 冮Kr(s,␩)ds
K (s,␩) ⫹ q/Ks
0 r
0

F(q/Ks,␩) ⫽ 冮⫽ (q/Ks)⫺1␩Hc[22]
Fig. 2. Plots showing ␣ parameter equivalence between the Brooks–
Corey and van Genuchten models when h0 → ∞.

Substituting Eq. [22] back into Eq. [15] leads to
q
H
⫽ c
Ks
z0

[23]

An important conclusion of Eq. [23] is that flux equivalence of various hydraulic functions will always be satisfied as long as the macroscopic capillary length is preserved.

Correspondence between the Brooks–Coreyand van Genuchten Models
If the suction at the soil surface is finite (h0), application of Darcy’s Law in a form similar to Eq. [15], andusing the BC and VG functions leads toz0 ⫽1
F1BC(q/Ks,␭)
␣BC

[24]

z0 ⫽

1
F1vG(q/Ks,m)
␣vG

[25]

and

respectively, where␣BCh0

F1BC(q/Ks,␭) ⫽Kr(s,␭)ds

冮 Kr(s,␭) ⫹ q/Ks

[26]

0

and
␣vGh0

F1vG(q/Ks,m) ⫽Kr(s,m)ds

冮 Kr(s,m) ⫹ q/Ks

[27]

0

By equating Eq. [24] and [25], predicting the same
q/Ks requires
F1vG(q/Ks,m)/␣vG ⫽ F1BC(q/Ks,␭)/␣BC[28]

and preserving the macroscopic capillary length impliesthat␤ ⫺ (␣BCh0)1⫺␤␣BC(␤ ⫺ 1)[1 ⫺ (␣BCh0)⫺␤]⫽1
␣vG[1 ⫺ Kr(␣vGh0,m)]␣vGh0冮 Kr(s,m)ds

[29]

0

The left-hand side of Eq. [29] for the BC model was
obtained by substituting Eq. [3] into Eq. [5] and notingthat hdry ⫽ h0 and hwet ⫽ 0. The right-hand side of Eq.
[29] was similarly obtained for the VG model by substituting Eq. [4] into Eq. [5]. Given q/Ks and the BC ␭and ␣BC, the equivalent VG m and ␣vG must be solved
simultaneously from Eq. [28] and [29]. The roots for mand ␣vG were found by a combination of successive approximations and the van Wijngaarden–Dekker–Brent
method (Brent, 1973; Press et al., 1992).
Please note that when ␣BCh0 ⱕ 1, it is impossible topreserve the macroscopic capillary length between the
BC and VG models since the BC model uses a functionthat is only piecewise continuous. To circumvent thisproblem, we used an alternate correspondence when
␣BCh0 is less than some threshold ␣hcrit (the value of ␣hcrit
is usually somewhat larger than 1). We simply assumed
that ␣vG/␣BC remains constant when ␣BCh0 ⬍ ␣hcrit. The
correspondence between the VG n and BC ␭ is thenestablished based on the requirement of having the samevertical flux.
Figure 2 shows ␣ parameter equivalence (␣vG/␣BC) between the BC and VG models as a function of ␭ when
h0 → ∞ for different q/Ks values. Figure 3 similarly demonstrates parameter equivalence when h0 → ∞ between
the BC ␭ and VG n parameters. For comparison purposes, these two figures also show the three parametric
relationships that were established previously (i.e., Eq.
[10]–[14]). Notice that the VG n and BC ␭ parameters
are linearly related. As the flux increases, both n and␣vG/␣BC must also increase to predict the same flux withthe two models. When the flux is very small, such as
for evaporation from a deep water table, the parametric
relationships of Morel-Seytoux et al. (1996) are closest
to our results (see Fig. 2). Also, please notice the rela-531

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Fig. 3. Parameter equivalence when h0 → ∞ between the Brooks–
Corey ␭ and van Genuchten n.

tively large deviations with our results in Fig. 2 when␣vG and ␣BC are simply equated (Eq. [12]).
Figure 4 shows parameter equivalence between the
Brooks–Corey and van Genuchten models when h0 is
finite and ␭ ⫽ 0.83. Results show that when ␣BCh0 is
small, the equivalent VG n needed to predict the same
flux varies considerably as a function of ␣BCh0. This ispartly due to the fact that the BC and VG models correspond poorly for relatively small values of ␣BCh0. In mostpractical applications, little or no evaporation probably
will take place for small ␣BCh0 values. When ␣BCh0 is
small enough, flow will shift from evaporation to infiltration. In other words, this shows that parameter correspondence between the BC and VG models is moredifficult to achieve for infiltration than for evaporation.

IMPLICATIONS FOR HYDRAULICPARAMETER UPSCALING
In this study we interpret hydraulic property upscaling
as deriving “equivalent” homogeneous soil hydraulicproperties to account for uncertainties in the spatiallyvariable hydraulic parameters (e.g., Zhu and Mohanty,

2002; Zhu and Mohanty, 2003). The equivalent homogeneous soil hydraulic properties should produce the sameensemble flux as that corresponding to random fieldsof the soil hydraulic parameters. In practical applications, two special types of heterogeneity need to be distinguished: (i) vertical layering (heterogeneity) where
variations in soil properties are in the vertical directionand (ii) vertically homogeneous soil columns with variations in soil properties in the horizontal plane. Our study
focuses on the latter case, where the variability is in thehorizontal plane. For meso- or regional-scale Soil–Vegetation–Atmosphere Transfer (SVAT) schemes in hydroclimatic models, pixel dimensions may range from several
hundred square meters to several square kilometers,
while the vertical scale of relevant subsurface flow processes near the land–atmosphere boundary (top few
meters) will be relatively small. For such a large horizontal scale, areal heterogeneity in soil hydraulic properties is likely much more important than vertical
heterogeneity. Our assumption of having predominanthorizontal heterogeneity will not apply to scenarios
where the vadose zone is deep and significantly layered,
nor to regions where the topography varies considerably
such that mutual lateral interactions occur between theparallel soil columns.
In previous studies (Zhu and Mohanty, 2002, 2003) wetried to answer the question: For a typical soil textural
combination in a real field condition, what are the effective (or average) homogeneous hydraulic properties forthe entire field (pixel) if the soil hydraulic properties
can be estimated accurately for each individual texture?
In this study the questions we try to address are:1. If the optimal p-norm (best averaging scheme, as
described and explained below) value is known forthe parameters of one hydraulic function that willproduce a certain ensemble flux into or from aheterogeneous field, what should be the corresponding p-norm of other hydraulic functions inorder to produce the same ensemble flux?
2. If the probability distribution function (pdf) of hydraulic parameters from one function is known,
what are the pdfs of the parameters of the otherhydraulic functions in order to produce the same
statistics of the flux field?
Specifically, we try to establish the probability distribution function and the p-norm relationships between the
BC and VG hydraulic parameters based on the correspondence between these two models developed above.
The p-norm or p-order power average ␣ˆ (p) for a setof N random parameter values ␣i is given by (Korvin
1982; Green et al., 1996)

冤N␣ˆ (p) ⫽ (1/N) 兺 ␣ip

Fig. 4. Parameter equivalence between the Brooks–Corey and van
Genuchten models when h0 is finite and ␭ ⫽ 0.83.

i⫽1

冥

1/p

[30]

The power average is a generalization of the arithmetic,
geometric and harmonic averages. The arithmetic ( p ⫽1), geometric (p → 0), and harmonic (p ⫽ ⫺1) means
are all particular cases of the power average.
For our example calculations we adopt typical valuesof ␣BC ⫽ 0.0478 (1/cm) and CV(␣BC) ⫽ 0.85, where over-Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

532

VADOSE ZONE J., VOL. 3, MAY 2004

Fig. 5. Comparison of probability density distribution of the Brooks–
Corey ␣BC parameter and the van Genuchten ␣vG parameter whenq/Ks ⫽ 0.01 and ␭ ⫽ 0.83.

bars denote average values. Based on the input statistics, a random field of 10 000 values for ␣BC was generated using the spectral method proposed by Robin et al.
(1993). After generating the random field, the corresponding fields of the VG parameters that will produce
the same flux can be calculated using the previously established relationships for parameter equivalence. After
this the pdfs of the VG parameters can be calculated.
The p-norm relationship between the BC and VG models that will produce the same ensemble flux can also
be established.
Figure 5 is a comparison of pdfs of the ␣BC and the␣vG parameters when q/Ks ⫽ 0.01 and ␭ ⫽ 0.83. Results
show that for the two extremes of ␣BCh0 (very large andvery small values; see Fig. 5) the corresponding ␣vG
distribution is the same as the input ␣BC. The only difference is in their average values, with ␣vG being smaller
than ␣BC. The entire VG ␣ pdf scales back to smaller
values, which reflects the fact that for the two extremesof large and small surface suctions, the corresponding␣vG and ␣BC parameters are proportional, while the ratioof ␣vG over ␣BC is always ⬍1 (Fig. 4). In the vicinity ofh0 ⫽ 1/␣BC, the corresponding VG ␣ changes from its
highest limit at h0 → 0 to its lowest limit at h0 → ∞. Thistransition in the VG ␣ explains the double-hump pdf
shape for the van Genuchten ␣ when ␣BCh0 ⫽ 1.0 (seecurve denoted by diamonds in Fig. 5).
Figure 6 shows probability density distributions of the
VG n when q/Ks ⫽ 0.01 and ␭ ⫽ 0.83. The structureof the VG n pdf can be explained by the local-scalecorrespondence of the models shown in Fig. 4. Whenh0 → ∞, the VG n will approach an asymptotic value ofabout 3.7, which is not related to ␣BC. This fact is reflected in the delta function pdf distribution for the VG
n seen in Fig. 6. When h0 is small or when h0 ⫽ 1/␣BC,
the VG n pdf distribution is similar to the input lognormal distribution of ␣BC. However, at and near the transition point (bubbling suction) of the Brooks–Coreymodel (h0 ⫽ 1/␣BC ), the value of n is large and its distribution more stretched, leading to a high mean value
and large standard deviation for the corresponding
VG n.
Figure 7 plots p-norm values for the ␣vG parameter

Fig. 6. Probability density distribution of the van Genuchten n parameter when q/Ks ⫽ 0.01 and ␭ ⫽ 0.83.

in comparison with those for the ␣BC parameter whenq/Ks ⫽ 0.01 and ␭ ⫽ 0.83. The comparison is not meant
to suggest optimal p-norm values (or the best averaging
scheme) that would produce the ensemble flux for heterogeneous soils. Rather the optimal p-norm values forthe ␣BC and ␣vG parameters that would produce the sameensemble flux are compared for different surface suction
conditions. Zhu and Mohanty (2002) established general
guidelines for spatial averaging of VG parameters thatwill produce the same ensemble flux for steady-stateevaporation and infiltration. It is interesting to observethat at both ends of the surface suction values, the optimal p-norms for the VG model and those for the BC
model are the same. In other words, the same averaging
rules can be used for both models. However, in the
middle range of the surface suction values, the p-norms
for the ␣vG are generally higher than those for the ␣vG
to predict the same ensemble flux.

CONCLUSION
For a large surface suction (such as for a very dry
surface), when m is approximately 0.5 (n ≈ 2), the Gardner and van Genuchten models gave similar fluxes.
When ␭ was between 0.42 and 0.83, the Gardner and
Brooks–Corey models were found to correspond the best.
In general, the ␣BC and ␣vG parameters corresponded
very well and were found to be proportional. Furthermore, the VG n and BC ␭ parameters had a linear
relationship. This means that the p-norm and the corresponding probability density functions of the BC andVG parameters should be the same. Since they have alinear relationship, we can use the same rules for upscaling of the van Genuchten and Brooks–Corey hydraulicparameters when predicting maximum potential evaporation rates (i.e., evaporation from soils having verylarge soil surface suctions).
For the more general case where the surface suction
is finite, it is more difficult for the hydraulic parametersof different conductivity models to correspond in termsof generating the same fluxes. The correspondence depends on the value of h0. The smaller h0, (i.e., scenarios
closer to infiltration), the more difficult it is to achievecorrespondence. For infiltration, the most difficult as-Reproduced from Vadose Zone Journal. Published by Soil Science Society of America. All copyrights reserved.

www.vadosezonejournal.org

533

Fig. 7. Plots comparing values of the p-norm for the van Genuchten ␣ parameter with those for the Brooks–Corey ␣ parameter when q/Ks ⫽0.01 and ␭ ⫽ 0.83.

pect is the first-order discontinuous shape of the Brooks–
Corey model. This feature makes its correspondence tothe van Genuchten model difficult when the suction in
the domain drops close to or below the threshold point(1/␣BC) of the function, which is more relevant to infiltration scenario. In case of evaporation from a shallow
water table, parameter equivalence of the different hydraulic functions will always be satisfied as long as themacroscopic capillary length is preserved.
ACKNOWLEDGMENTS
This project is supported by NASA (Grants NAG5-8682
and NAG5-11702) and also in part by SAHRA (Sustainabilityof Semi-Arid Hydrology and Riparian Areas) under the STC
Program of the National Science Foundation, Agreement no.
EAR-9876800.

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