Scaling Parameter to Predict the Soil Water Characteristicfrom Particle-Size Distribution Data
Lalit M. Arya,* Feike J. Leij, Martinus Th. van Genuchten, and Peter J. Shouse
ABSTRACTal., 1996) or neural network techniques (Schaap andBouten, 1996). Because the soil water retention curveis essentially a pore-size distribution curve, it is requiredas the primary input in models of the hydraulic conductivity based on pore-size distribution (e.g., Marshall,
1958; Millington and Quirk, 1961; Mualem, 1976). Thus,
accurate water retention curves are of great importance.
The first attempt to directly translate the particle-size
distribution into a soil water characteristic was made by
Arya and Paris (1981). The basis for their model is a
close similarity between the shapes of the particle-size
distribution and the water retention curve. In the model,
the pore size that is associated with a pore volume isdetermined by scaling the pore length. Since particle
size is normally expressed in terms of equivalent
spheres, Arya and Paris (1981) estimated pore lengths
for the various fractions of the particle-size distribution
curve by summing the diameters of spherical particles
in the fraction. Pore lengths based on spherical particles
were scaled to natural pore lengths using a scaling parameter, a, with an average value of 1.38.
A similar model was later proposed by Haverkamp
and Parlange (1986) and tested on a coarse-textured
sand. Tyler and Wheatcraft (1989) interpreted a. as being
the fractal dimension of a tortuous pore. Since then,there has been a growing interest in the use of fractals
to predict hydraulic properties from particle-size distributions (e.g., Rieu and Sposito, 1991; Tyler andWheatcraft, 1992; Shepard, 1993). However, it should
be noted that fractal scaling, inasmuch as it is concernedwith the nature of fragmentation, accounts only for theeffects of tortuosity of pore lengths, but not for otherfactors that influence water retention, such as packing
density, chemical characteristics of solid surfaces, organic matter content, fluid properties, and air entrapment.
Later investigations by Arya et al. (1982) showed thatthe average a varied among textural classes and ranged
in value from 1.1 for finer textures to 2.5 for coarsetextured materials. A similar range of values was reported by Tyler and Wheatcraft (1989) for the fractal
dimension. Yoshida et al. (1985) also reported higher
values of a for coarse-textured materials. Several researchers (e.g., Schuh et al., 1988; Mishra et al., 1989;
Gupta and Ewing 1992; Jonasson, 1992; Basile and
D'Urso, 1997; Nimmo, 1997) have suggested that predictions of water retention curves would improve if a were
formulated such that it varies over the range of particle sizes.
The single values of a in the original study of Aryaand Paris (1981) were obtained by minimizing the sumof squares of deviations between the measured and calculated pressures. The objective of the present study is
to investigate of relationships between a and the particle-size distribution.

The Arya-Paris model is an indirect method to estimate the soil
water characteristic from particle-size data. The scaling parameter,a, in the original model was assumed constant for all soil textures.
In this study, a is defined as a, = (logJV//log«,-), where n, is the numberof spherical particles in the ith particle-size fraction (determined by
the fraction solid mass, wh and mean particle radius, R,) and /V/ is thenumber of spherical particles of radius R, required to trace the pore
length generated by the same solid mass in a natural structure soil
matrix. An estimate for log A',- was obtained by either relating log A',
to log n, using a logistic growth equation or by relating log A', linearly
to log (w,IR}) based on the similarity principle. For any given texture,both approaches showed that a was not constant but decreased withincreasing particle size, especially for the coarse fractions. In addition,a was also calculated as a single-value average for a given texturalclass. The three formulations of a were evaluated on 23 soils that
represented a range in particle-size distribution, bulk density, andorganic matter content. The average a consistently predicted higher
pressure heads in the wet range and lower pressure heads in the dry
range. The formulation based on the similarity principle resulted inbias similar to that of the constant a approach, whereas no bias wasobserved for the logistic growth equation. The logistic growth equation
implicitly accounted for bias in experimental procedures, because it
was fitted to log N, values computed from experimental soil water
characteristic data. The formulation based on the similarity principle
is independent of bias that might be inherent in experimental data.

T

HERE is AMPLE JUSTIFICATION for indirect methods of
estimating soil hydraulic properties from routinely

available taxonomic data (e.g., Bouma and van Lanen,

1987; van Genuchten and Leij, 1992). The effects oftexture, bulk density, and organic matter on soil water
retention and hydraulic conductivity have long beenrecognized. However, an explicit formulation of the relationship between texture and hydraulic properties ofthe soil remains a challenge because of the very complex
pore-particle geometry. Hence, empirical approaches topredicting hydraulic properties at specific points of thewater content-pressure-hydraulic conductivity curves
from texture, bulk density, mineralogy, and organic matter content by using multiple regression techniques orneural network analyses remain popular (e.g., Rajkai
and Varallyay, 1992; Tietje and Hennings, 1995). Mathematical representations of the water content-pressurehydraulic conductivity curves as a continuous function(Brooks and Corey, 1964; Mualem, 1976; and van Genuchten, 1980) require one or more fitting parameters,
which are normally evaluated from the basic soil properties through regression (Kool et al., 1987; Rajkai etLalit M. Arya, Feike J. Leij, Martinus Th. van Genuchten, and PeterJ. Shouse, USDA-ARS, U.S. Salinity Lab., 450 W. Big Springs Rd.,
Riverside, CA 92507. Received 1 April 1998. *Corresponding author(larya@ussl.ars.usda.gov).
Published in Soil Sci. Soc. Am. J. 63:510-519 (1999).

510

ARYA ET AL.: SOIL WATER CHARACTERISTICS FROM PARTICLE-SIZE DISTRIBUTION

THEORYThe Arya-Paris Modelr, = 0.816

In the following we will review pertinent aspects of themodel. The particle-size distribution curve is divided into nsize fractions, and the solid mass in each fraction is assembled
to form a hypothetical, cubic close-packed structure consistingof uniform size spherical particles. The pore volume in each
assemblage is calculated from the bulk density and particle
density measured on the natural structure soil. Arya and Paris(1981) found n = 20 as a reasonable number of fractions, withfraction boundaries at particle diameters of 1, 2, 3, 5, 10, 20,
30, 40, 50, 70, 100, 150, 200, 300, 400, 600, 800, 1000, 1500,
and 2000 u,m. Starting with the first fraction, calculated porevolumes are progressively summed and considered filled withwater. Summations of filled pore volumes are divided by thebulk volume to obtain volumetric water content at the upperbounds of successive mass fractions. An equivalent pore radius
is calculated for each fraction and converted to soil waterpressure head using the capillary equation. Calculated pressure heads are sequentially paired with calculated water contents to obtain a soil water characteristic curve. The matrixof the medium formed in this manner is termed the ideal
matrix to indicate that the geometry of this matrix is welldefined. Properties of the ideal matrix can be used as a basis
to characterize deviations in the desorption behavior of more
complex structures.

Basic Relationships
The pore volume, Vfi (cm3 g"1), associated with the solid
mass in the rth particle-size fraction, is represented as a single
cylindrical capillary tube, and is given by
VH = Ple =

[1]

\P5

where w, is the fraction solid mass (g g"1), ps is the particle
density (g cm"3), e is the void ratio, rt is the pore radius (cm),and /, is the pore length (cm g"1)- The void ratio, e, is givenbye = (ps - Pb)/Pb[2]
where pb is the bulk density of the natural soil (g cm"3). The
water content, 9, (cm3 cm"3), is obtained from successive summations of water-filled pore volumes according toe, =i = l,2...n[3]

where <$> is the total porosity (cm3 cm"3), and 5W is the ratioof measured saturated water content to theoretical porosity.
The number of spherical particles, n/ (g"1), for each fractionof the particle-size distribution is calculated fromnt

= 3w,/(4Tr Ps #3)

511

[4]

where Rt is the mean particle radius (cm) for the rth particlesize fraction. For an ideal soil consisting of uniform size spherical particles in a cubic close-packed assemblage, /, can be
estimated by /, = 2«, R,, and the pore radius is related to theparticle radius by

rt = 0.816[5]For a natural soil, made up of the same solid mass but with
nonspherical particles that are arranged randomly, /, can be
estimated by /, = 2n™7?,, and the corresponding pore radius isrelated to the particle radius by[6]

where a is the scaling parameter. Note that n,(1 a) is dimensionless. Calculated pore radii, /•„ are converted to equivalentpressure heads, \h\ (cm water), using the capillary equation:_ 2-y cos Q[7]

Pw g r,where y is the surface tension at the air-water interface (gs~2), © is the contact angle, pw is the density of water (g cm"3),
and g is the acceleration due to gravity (cm s~2). The model
assumes perfect wettability and, hence, © = 0°. A complete
list of variables used is given in Table 1.

Scaling of the Soil Water Characteristic
Scaling of the soil water characteristic from the ideal to thenatural soil is schematically shown in Fig. 1. The curve for theideal soil was calculated from the particle-size distribution
data using Eq. [1] through [5] and Eq. [7]. The curve for thenatural soil was drawn through the experimental data. Data
used in Fig. 1 are from the loam soil 4101 in the UNSODA
database (Leij et al., 1996). Scaling involves moving the soil
water characteristic curve for the ideal soil along either the 6or h axis such that it closely matches the experimental curvefor the natural soil. It requires mention that this scaling is not
to be confused with the concept of geometric similitude ofMiller and Miller (1956).
In this study, we scaled the pressure head, h. For any watercontent, 6,, the pressure heads for the ideal and natural soils
are different. They are denoted as ht (ideal) and /z, (natural)in Fig. 1. In order to match ht (ideal) to ht (natural), the poreradii calculated from the capillary equation (Eq. [7]) must be
scaled to equivalent pressure heads for the natural soil. Since
volumetric water contents are obtained by progressive summation of fractional pore volumes, scaling can be accomplishedby pairing each calculated fractional pore volume with theequivalent pore radius on the experimental soil water characteristic curve. In this study, the calculated effective pore volume was matched to experimental saturated water content,
and the experimental saturated water content was assumed
to correspond to \h\ = 10~2 cm water.
Table 1. Variables used in the equations.
Variable

Descriptionevoid ratio, defined as (p, — pb)/|>i,

g
h,
/,
Hi

acceleration due to gravitypressure head, ith fraction
pore length, ith fraction
number of spherical particles, ith fraction,
obtained from the particle-size distribution
data
scaled number of spherical particles, ith fraction,
to trace the pore length in the corresponding
natural soil
pore radius, ith fraction
particle radius, ith fraction
ratio of experimental saturated water content toporosity
pore volume, ith fraction
solid mass, ith fraction
scaling parametersurface tension (air-water)
volumetric water content, ith fraction
contact angle, degrees
particle density
bulk density
density of waterporosity

N,
r,
R,
S,
Vfl
w,
OL
•y
H,
0
p,
pb
p»
c|>

Dimension!
14 L,"3
LT' 2
L.
1

LP Mr
Mr1

14 M.-11

Ms Mr

MT" 2
U U"3
M, Lr3
Ms Lb"3M,L" 314 U->t For dimensional analysis, L = length, M = mass, and T = time, with
subscripts b for bulk, p for pore, s for solid, and w for water.

512

SOIL SCI. SOC. AM. J., VOL. 63, MAY-JUNE 1999

The assumption that a pore volume can be represented bya single cylindrical capillary tube makes the scaling convenient.
One needs to know the pore volume and pore length to calculate the pore radius. Pore volume can be easily calculated
from the particle-size distribution data using Eq. [1]. However,pore lengths can only be approximated. Arya and Paris (1981)
used the diameter of the spherical particle as a measuring unitof length. Equations [5] and [6] result from the argument thata pore length can be estimated by the number of particles
that lie along the pore path times the length contributed byeach particle; i.e., the particle diameter. Thus, for an ideal
soil, the pore length is estimated by 2n, Rt. However, for thecorresponding natural structure soil, we assume that nf spherical particles of radius /?, are needed to trace the entire pore
length; i.e., the pore length is approximated by 2nfRt.
The parameter a needs to be estimated in order to scalepore length in an ideal soil to that of a corresponding naturalsoil. In an ideal soil, the pore length is equal to the sum ofphysical lengths that are contributed by particle diametersarranged in straight columns. In contrast, particles in a naturalsoil may contribute to pore lengths in more than one dimension. In addition, the pressure head depends not only on thepore size but is also affected by organic matter, solutes, andelectrochemical properties of the solid surfaces. Therefore, aneeds to be estimated from experimental soil water characteristic data; it should be considered more of a fitting parameter
than a geometrical constant.

Table 2. Textural classes and UNSODA codes for soils used forformulating and testing the scaling parameter a.

MATERIALS AND METHODS

Estimating Pore Length in Natural Structure Soil

Experimental soil water characteristic, particle-size distribution, bulk density, and particle density data were obtained
from the UNSODA hydraulic property database (Leij et al.,The relationship between the number of spherical particles
in the ideal soil, n, (g~')> and the number of spherical particles
required to trace the pore length in the corresponding natural
structure soil, A/, (g~')> is given by

Texturalclass
Sand
Sandy loam
Loam
Silt loam
Clay

UNSODA codes

Use

1050, 1460, 2100, 3132, 3340, 4650
1464, 4000, 4521, 4661
1130, 1131, 1381, 3310, 4160
OOOOf, 1112, 1121, 1380, 1450, 4162
1370, 2531, 4610
2530, 3303, 4101, 4600
1341, 4081, 4510, 4531, 4670
2000, 3260, 4510, 4531, 4673
1400, 2361, 3282, 4121, 4681
2360, 3281, 4120, 4680

Formulating a
Testing
Formulating a
Testing
Formulating ot
Testing
Formulating ct
Testing
Formulating a
Testingt This soil is not in UNSODA.

1996). Forty-seven data sets, representing a range of texturesthat include sand, sandy loam, loam, silt loam, and clay, wereselected for this study. For each textural class, four to fivesoils were used to develop a formulation for a and anotherfour to five soils were used for independent testing of theformulations. All soils are identified in Table 2. The particlesize distribution curves for the selected soils were divided inton fractions according to the method used by Arya and Paris(1981), and all parameters required in Eq. [1] through Eq. [6]were calculated.

nr =—•— Natural soil
- - - Ideal soil
(UNSODA No. 4101)

ora, = log NJlog n([8]

UNSODA Loams (No. 1370,2531,4610)
O Log Ni data——— Fitted logistic growth curve

1
I-2
Water content, Q (cm3cm~3)
Fig. 1. Schematic representation of the relationship between water
characteristics of an ideal soil and the corresponding natural structure soil.

-1

9

10

Log n, (ideal matrix)
Fig. 2. Relationship between the number of spherical particles required to trace the pore length in a natural structure soil (log N,)
and the number of spherical particles forming the corresponding
ideal soil (log n,) for loam soils (from Table 2).

ARYA ET AL.: SOIL WATER CHARACTERISTICS FROM PARTICLE-SIZE DISTRIBUTION

Table 3. Fitted values for (log ty.),, (log TV,-),, JJL, A log N,, andA log n, and goodness of fit, r2, for five textural classes.
Textural class (log JV,)( (log N,),r!
A log JV, A log HI
H
0.00032 0.880
16.602 0.609 1.734
0.996
Sand
Sandy loam
16.983 0.553 2.492 1.849
0.559
0.966
16.614 0.510
Loam
0.980
2.242 1.977
0.628
0.940
0.684
19.686 0.457 1.902
0.719
Silt loam
Clay
0.960
21.685 0.289 4.766 2.648
1.993
Values of n, were calculated from the particle-size distribution
data using Eq. [4]. To evaluate Nf, calculated water contents
for the ideal soil were paired with pressure heads on theexperimental soil water characteristic curve (see Fig. 1). These
pressure heads were converted to pore radii using Eq. [7], andthe corresponding pore lengths were calculated using Eq. [1].
The calculated pore lengths were divided by 2/?, to obtain Nt.
These steps can be captured in the following expression for
N,.

N, = 7.371 w, e A^/p, Rs[9]where hmi is the measured pressure head. The number 7.371
represents a composite of the constants in Eq. [1] and Eq. [7]and has the unit of cm~4.
Normally no retention data are available, and a or Nj mustbe estimated from the particle-size distribution data. We plotted log Nj vs. log n, for different textural classes to investigate
the behavior of a according to Eq. [8]. Figure 2 shows asomewhat nonlinear relationship between the two variables
for loam soils (see Table 2 for UNSODA codes). Similar
relationships were obtained for other textures that were examined. In the following we investigate the relationship of log
NI to parameters of the particle-size distribution.

12<oMethod 1: Logistic Growth Curve
The relationship between log Nj and log nh as depicted inFig. 2, closely follows the logistic growth equation (Thornley,
1990)

Y=

+ (Y, - Yi) exp(-|x*)8

where Y represents the dependent variable log Nh (A is therate coefficient, and x is the independent variable log n,. The
subscripts i and f signify the initial and final values of log Nhwhich correspond to particle-size fractions with the smallest
and largest particle numbers, respectively. Equation [10] wasfitted to data depicted in Fig. 2, using a least squares method(Marquardt, 1963). For a large number of soils, the initialvalues of log A/,- and log n, were negative and, therefore, we
had to shift the X and Y axes, transforming Eq. [10] to(Y + AY) =

Ax)]

(Yf -

[11]

The least squares procedure provided the fitted initial andfinal values of log Nj and jx, as well as those of A (log TV,) andA (log «,). These values are summarized in Table 3 along withthe coefficient of determination, r 2 , between log Nj and log
n,-, determined according to Eq. [9] and Eq. [4]. Smooth curves
that show the relationship between log Nt and log n, for all
textural classes are presented in Fig. 3. Variables x and Y in
Eq. [11] were replaced by log n, and log Nh respectively. An
explicit expression for a, was obtained by substituting a, logn, for Y (see Eq. [8]) on the left hand side of Eq. [11].
Method 2: Linear Fit between Log N, and Log (w,IRftlog N,, = a + b log (w,/R?)
-• -

Sand

———
— - -

Sandy loam
Loam

———
— —

Silt loam
Clay

10

12

[12]

Combining Eq. [12] with Eq. [4] and [8], a can be expressedas

UNSODA soils (see Table 2)

-2

[10]

According to Eq. [4] log n; is linearly dependent on log (wjR]). Since Nt represents only a scaled-up number for each «,,log NI must exhibit a similar linear relationship with log (wj
RJ). An example of the plot of log A/i against log (wJRf) forclay soils is presented in Fig. 4. Similar relationships were
found for sand, sandy loam, and silt loam textures. A linear
relationship between log N, and log (wJRT) implies the existence of two particle-size distributions of similar shape, one
represented by nt and the other by A/,. Shape similarities between particle-size distributions and soil water characteristics
support this contention. We, therefore, explored formulations
for a based on a linear relationship between log N, and log(wJRf). The generalized form of this relationship, as illustrated
in Fig. 4, is

16

3

513

la + b log

a, = '——————log «,[13]

Equation [13] represents an explicit formulation for a in termsof particle-size distribution parameters. Parameters of regression for Eq. [12] for five soil textures are summarized in Table 4.
14

16

Log n, (ideal matrix)
Fig. 3. Smooth curves (fitted logistic growth equation) showing therelationship between the number of spherical particles required tomeasure the pore length in a natural structure soil (log Nt) andthe number of spherical particles forming the corresponding ideal
soil (log n,) for sand, sandy loam, loam, silt loam, and clay soils.

Method 3: Constant a
For each soil texture, an average value of a was estimatedby fitting a linear regression with zero intercept to the plotsof log NI vs. log n, data. The slope of the line should represent
an average a. This approach is essentially the same as thatused by Arya and Paris (1981). Average a values estimated514

SOIL SCI. SOC. AM. J., VOL. 63, MAY-JUNE 1999

16
14

CLAY (5 SOILS)

12

Log N,: Y = -2.600 + 1.305 X ; r2 = 0.954

10

O
O>
o

O

0 -

-2

10

12

14

Log (w/Rj3)
Fig. 4. Relationship between the number of spherical particles accounting for pore lengths in natural structure soils (log A',-) and the fraction
solid mass (»',) and particle size (R,) for clay soils.
in this manner were 1.285, 1.459, 1.375, 1.150, and 1.160 for
the sand, sandy loam, loam, silt loam, and clay textures, respectively.

Model Tests
The three formulations for a, described above, were tested
independently on four to five soils for each textural class.
These soils are also identified in Table 2. Calculated valuesof a were substituted in Eq. [6] to calculate pore radius, rt,for each particle-size fraction. Combining Eq. [6] with Eq. [7]yieldshi = 0.18/7?,[14]Equation [14] expresses h, explicitly in terms of a,, n,, Rit ande. The number 0.18 represents a composite of constants in Eq.
[6] and Eq. [7], and has the unit of cm2.
The three formulations of a were used to independently
predict pressure heads for 23 soils, using particle-size distribution, bulk density, and particle density data. The calculatedpressure heads were paired with corresponding water contents
Table 4. Parameters of Eq. 2[12] that relate log N, to log (»f,//??)and the goodness of fit, r , for five soil textures._______
Textural class No. of soils Data pairs
a
b
r1
62
Sand
6
-2.478 1.490 0.882
75-3.398 1.773
0.952
Sandy loam
6
Loam
4
50
-1.681 1.395 0.936
77
Silt loam
5-2.480 1.353
0.965
88-2.600 1.305
0.954
5
Clay

obtained from Eq. [3] to yield soil water characteristics. Predicted pressure heads were compared with the corresponding
experimental pressure heads for each a formulation.

RESULTS AND DISCUSSION
Nature of the Scaling Parameter a
Values of a, based on Methods 1 and 2 (Eq. [11] andEq. [13]), were calculated as a function of log n{ forseveral soil textures. UNSODA soils used for this purpose are identified in Table 2 as those used for formulating a. Experimental values of ot for the same soils werealso calculated using Eq. [8], in which log TV/ values wereobtained directly from measured soil water characteristic curves. Examples of variations in ot with log «, areshown in Fig. 5a and 5b for sandy loam and clay textures,
respectively. Note that large values of log «, coincidewith small particle sizes (Eq. [4]). Thus, one should
expect to see a relationship between ct and Rt, as well.
We prefer the particle number n, because it includes
the effects of both the solid mass and the particle size.
Similar results were obtained for the sand, loam, andsilt loam textures. The trends in a values calculated
from Methods 1 and 2 and those from experimental
soil water characteristics are essentially identical. The
experimental data show some scatter, probably because
experimental a values were computed from different

ARYA ET AL.: SOIL WATER CHARACTERISTICS FROM PARTICLE-SIZE DISTRIBUTION
8.5(a) Sandy loam (5 soils)

7.5

-

6.5
5.5

D

4.5 - D

S

£

3.5

2.5

e

1.5
0.5
-0.5
-1.5

-2.5
-3.5

O

Logistic growth equation (method 1)
Log linear fit (method 2)

-4.5

.

O

B

|

-5.5
-6.5
-7.5

R .

Experimental

,
2

3

4

Log r>i (ideal matrix)(b) Clay (6 soils)

I

0.

515

range of 1500 to 2000 (Jim is also the range of the particlesize distribution that coincides with the soil water characteristic near saturation. Because of experimental uncertainties in soil water characteristic data, estimating
NI accurately near saturation is relatively difficult. We,
therefore, suggest that no significance be attached tovalues of a when log n, < 0.
The degree of variation in a from clay to sand size
particles is relatively small, and an average value of a
may well represent the entire particle-size distribution,
except near saturation. The constant value of 1.38 suggested by Arya and Paris (1981) worked well enoughfor many but not all particle-size distributions. The results in Fig. 5a and 5b and those for sand, loam, andsilt loam textures, however, indicate that overall a values vary from one textural group to another, and thata single value may not be appropriate for a particular
textural class. Arya et al. (1982) analyzed 181 New Jersey soils and found average a values that varied from1.26 for silt loams to 2.10 for sandy clay loams. Schuh(1992) evaluated variation in a for a large number ofsoils in North Dakota. He concluded that a could varyfrom as low as 0.95 for silt loams to 1.3 for loams. In
addition, he found that a was not constant over the fullrange of the soil water characteristic curve for several
textural classes.

Effect of a on Predictions of Soil
Water Characteristic

8
CO

O

Logistic growth equation (method 1)

Q

Log linear fit (method 2)

|

4

5

6

Experimental

7

8

9

10

11

12

13

14

Log DJ (ideal matrix)

Fig. 5. Scaling parameter a as a function of the number of sphericalparticles in the ideal soil (log n,-): (a) sandy loam soils; and (b)clay soils.

soil water characteristics within each textural class. It isnoteworthy that a values based on Methods 1 and 2
differ from each other only slightly, and both represent
an average for the scatter of the experimental a values.
Linear regression between Methods 1 and 2 gave r2 >0.96 for all textures. Those between Methods 1 or 2 andexperimental data yielded r2 from 0.78 to 0.99.
It is clear from Fig. 5 that a is not a constant; rather,a decreases as the number of particles decreases. Thechange in a with particle number is relatively small for
most of the range of particle-size distribution. A sharp
decrease in a, however, occurs at low particle numbers
when log «, is in the range of 0.5 to 2.0. The transition
point is smallest for the sand texture and largest for theclay texture. Large negative and large positive valuesof a occur in an erratic manner below log «, < 0. We
attribute this behavior to errors in the estimation ofvery small numbers of particles. It is not unusual tocompute particle numbers of <1 in the diameter rangeof 1500 to 2000 jjurn for many particle-size distributions.
However, we believe the error is large. The diameterIn the following we discuss the effects on predictionsof the soil water characteristic of the use of a single
value of a vs. considering a as a continuous function.
Soil water characteristics were predicted for four to fivetest soils within each textural group using a values thatwere based on Methods 1, 2, and 3. Typical examples
of predicted and experimental soil water characteristics
for clay, loam, sandy loam, and sand textures are presented in Fig. 6. Results show that all three a functions
adequately predicted the shape of the experimental soil
water characteristic. However, the differences in predicted and experimental pressures varied in magnitude
from soil to soil. Differences between predicted andexperimental pressures occurred more frequently at ornear the point where soil water characteristic curves
exhibit a sharp change. However, no particular saturation could be identified where a given formulation fora produced consistently poor or good predictions, particularly for Methods 1 and 2. A constant a, on theother hand, yielded in a majority of cases lower absolutepressures in the dry range and higher absolute pressuresin the wet range.
The overall predictive ability of a for the three formulations was evaluated by comparing experimental withcalculated pressure heads on a 1:1 plot. Pairs of predicted and experimental pressures for all soils and all
textural groups were pooled, and the logarithm of experimental pressure head was plotted as a function of thelogarithm of calculated pressure head. The results arepresented in Fig. 7. The large scatter in the data is a
result of the fact that, while a wide range of soil water

516

SOIL SCI. SOC. AM. J., VOL. 63, MAY-JUNE 1999
10' F(a) Test soil: Clay - UNSODA 3281

(b) Test soil: Loam - UNSODA 2530

:

10"

10B

10s
10' r

10*

10' r
£

2

Predicted SWC (a: method 1)

10'r

Predicted SWC (a: method 2)

Predicted SWC (a: method 1)

Predicted SWC (method 3. a = 1.16)

Predicted SWC (a: method 2)

Experimental SWC

Predicted SWC (method 3, a = 1.375)
Experimental SWC
10"

10"
0.00

0.10

0.20

0.30

0.40

0.00

0.10

(c) Test soil: Sandy loam - UNSODA 1380

-

- - -

0.30

0.40

o.so

Water content, 9 (cm3 cm-3)

Water content, 0 (cm3cm-3)

—

0.20

(d) Test soil: Sand UNSODA 4661

Predicted SWC (a: method 1 )Predicted SWC (a: method 1 )Predicted SWC (a: method 2 )Predicted SWC ( a: method 2 )Predicted SWC (method 3, a = 1.459 )Predicted SWC (method 3, a = 1.285)

Experimental SWC

Experimental SWC

10'

104

-

10'

10'

10'

r

r

10"

0.00

10°

0.10

0.20

0.30

0.40

0.50

Water content, 0 (cm3 cm-3)

0.10

0.20

0.30

0.40

0.50

Water content, 0 (cm3 cm-3)

Fig. 6. Comparison of measured and predicted soil water characteristics (SWC): (a) clay soil, (b) loam soil, (c) sandy loam soil, and (d) sand.

characteristics and particle-size distributions occur
within the same textural class, only textural class averagevalues of a were used in calculating pressure heads.
Despite the scatter, Method 1 (Fig. 7a) showed an overall good agreement between experimental and predictedpressures. The linear regression had an r2 of 0.925 andthe regression line differed only slightly from the 1:1
line. In contrast, Method 2 (Fig. 7b) appears to produce
bias. Although the regression appears to be good with anr2 of 0.892, the method over-predicted absolute pressure
heads in the wet range and under-predicted in the dry
range. The regression line deviated from the 1:1 linein the dry range. Predictions with constant a (Fig. 7c)
produced results similar to those in Method 2. Although
the value of r2 in this case was 0.912, the deviation of
the regression line from the 1:1 line was much more
pronounced than for Method 2, as is reflected by theintercept of -0.795 and slope of 1.32. Additionally,

ARYA ET AL.: SOIL WATER CHARACTERISTICS FROM PARTICLE-SIZE DISTRIBUTION

over- and under-predictions of pressures with constanta were also more pronounced than with Method 2. Thus,
based on these results one would conclude that Method
1 is the most appropriate formulation for a and Method
3 the least, even when a different average a is used for
different textural classes.
While comparing Methods 1 and 2, it must be notedthat Method 1 is based on values of log Nt derived fromthe experimental soil water characteristic. Use of fiveparameters (Table 3) also assured a better fit of thelogistic growth equation to experimental log Nt data.
Naturally, then, a values based on this fitted relationshipshould produce the best overall agreement with theexperimental data. Method 2, on the other hand, is notbased on fitting an equation to a trend in the data, but
on similarity in particle-size distributions between theideal soil and its counterpart, natural soil. We postulated
for this purpose a linear relationship between log Nt
and \og(Wi/R*) and did not consider inherent nonlinearity. Therefore, results from Method 2 will conform to
the experimental data only if the similarity principle isvalid and experimental data have been obtained withoutbias. The behavior of the constant a function is partiallyclarified by the results in Fig. 5 and 6, which show that
constant a. is likely to introduce large errors in the wet
range and some errors in the dry range.
At this point we may not be able to critically differentiate between Methods 1 and 2, while yet other methods
may even be better suited to determine a. Although
Method 1 appears more attractive, this formulation is
not independent of the nature of experimental soil water
characteristic data. Our data sets were heterogeneous,and they were contributed by researchers in the UnitedStates, Netherlands, United Kingdom, Germany, Belgium, Denmark, Russia, Italy, and Australia, who eachused different experimental procedures. In addition,systematic and random errors are likely in experimental
determination of the soil water characteristic. For measurements obtained with a pressure plate apparatus(Klute, 1986), loss of contact between the sample andthe ceramic plate as the sample is progressively desaturated can result in serious overestimation of watercontent in the dry range. In the wet range, structural
disturbances (e.g., puddling and sealing, incomplete saturation) may lead to under-estimation of equilibrium
water contents. If these errors are present to some extent
in all measured soil water characteristic data, then thebias apparent with Method 2 is understandable (Fig.
7b). Of course, other sources of variability exist that
may mask any systematic trends. For example, groupingsoils together solely on the basis of textural nomenclature may introduce variations <25 percentage points inthe mass fraction for some particle-size ranges (Arya,unpublished data, 1983 based on USDA-SCS [1974]).
Significant variations in bulk density and organic matter
content also exist within a textural class. Thus, widelydiffering soil water characteristic curves may be obtained for soils of the same textural class.

SUMMARY AND CONCLUSIONSThis study proposes two new formulations for thescaling parameter a in the Arya and Paris (1981) model

517

CD

Y = 0.1313 + 0.9530 X; r2 = 0.925

1

2

———

Regression line

- - - -

1:1 Line3

4

Log calculated pressure head, |h| (cm water)(b) Variable a (method 2 )£iTEST SOILS

No.

Clay

4

Silt loam
Loam
Sandy loam
Sand

5
4
6
4

£
&

3
M

I
0)

I

Y = -0.084 + 1.094X; r2 = 0.892

1

2

——•

Regression line

- - - -

1:1 Line3

4

Log calculated pressure head, |h| (cm water)

4 (1.160)
5 (1.150)
4 (1.375)
6 (1.459)
4 (1.285)

Y = - 0.795 + 1.320 X; ft = 0.913___

Regression line

- - - -

1:1 Line

Log calculated pressure head, |h| (cm water)
Fig. 7. Comparison of calculated and experimental pressures. Test
results for 23 soils are pooled: (a) Method 1, (b) Method 2, and(c) Method 3.

518

SOIL SCI. SOC. AM. J., VOL. 63, MAY-JUNE 1999

of the soil water characteristic, which translates a particle-size distribution curve into a corresponding soil water characteristic curve. The scaling parameter is used
to estimate pore lengths in natural structure samples.
If the solid mass, w,, in a given particle-size fraction withmean particle radius #, is equivalent to nt uniform size
spherical particles, the pore length generated by theseparticles, when arranged in a cubic, close-packed assemblage, can be approximated by 2n, Rt. However, tracing
the pore length generated by the same amount of solid
mass in a natural structure assemblage would require
NJ spherical particles of radius Rh where Nt = nf.
The new formulations express Nt as a continuous function of particle-size distribution parameters. In one formulation a is predicted according to Eq. [11], and log
NI is related nonlinearly to log n, using a logistic growthequation, while in another formulation, a is defined by
Eq. [13], and log Nt is related linearly to log (wJR})
based on the principle of similarity.
The predictive ability of the new formulations andthat of the original constant a approach were compared.
Predictions of soil water characteristics for a range ofsoil textures with varying bulk densities and organic
matter contents showed reasonable to excellent agreement with experimental data. Use of a constant a usually
led to under-predictions in the dry range and over-predictions in the wet range. The predictions appeared unbiased if the formulation for a was based on the logistic
growth equation. On the other hand, the formulation
based strictly on the principle of similarity usually produced bias similar to that of the constant a. approach.
Numerous problems exist with experimental data, including random and systematic errors, and it may not
be possible to critically differentiate between the formulations based on tests with a heterogeneous data set.
Uniform data sets, free of bias and errors, are needed tomore critically evaluate indirect methods for estimating
soil hydraulic properties.

OLESEN ET AL.: SOLUTE DIFFUSION AND ADSORPTION ALONG A SOIL TEXTURE GRADIENT

519