DIVISION S-l—SOIL PHYSICS
Relationship between the Hydraulic Conductivity Function
and the Particle-Size Distribution
Lalit M. Arya,* Feike J. Leij, Peter J. Shouse, and Martinus Th. van Genuchten
ABSTRACT
We present a model to compute the hydraulic conductivity, K, as afunction of water content, 6, directly from the particle-size distribution(PSD) of a soil. The model is based on the assumption that soil pores
can be represented by equivalent capillary tubes and that the waterflow rate is a function of pore size. The pore-size distribution is derived
from the PSD using the Arya-Paris model. Particle-size distribution
and A'(0) data for 16 soils, representing several textural classes, wereused to relate the pore flow rate and the pore radius according to
qr, = erf, where q, is the pore flow rate (cm3 s ') and r, is the poreradius (cm). Log c varied from about -2.43 to about 2.78, and x
varied from =2.66 to -4.71. However, these parameters did not exhibita systematic trend with textural class. The model was used to independently compute the A (0) function, from the PSD data for 16 additionalsoils. The model predicted A'(O) values from near saturation to very
low water contents. The agreement between the predicted and experimental A'(0) for individual samples ranged from excellent to poor,
with the root mean square residuals (RMSR) of the log-transformed
K(Q) ranging from 0.616 to 1.603 for sand, from 0.592 to 1.719 forloam, and from 0.487 to 1.065 for clay. The average RMSR for all
textures was 0.878.

R

4OWLEDGE OF THE HYDRAULIC CONDUCTIVITY aS afunction of water content, K(Q), or pressure head,
K(h), is essential for many problems involving waterflow and mass transport in unsaturated soils. A variety
of field and laboratory techniques have been developed
to directly measure the unsaturated hydraulic conductivity (e.g., Klute and Dirksen, 1986; Green et al., 1986;
Dirksen, 1991). While research continues to improvephysical measurements of flow parameters and analytical techniques (e.g., Smettem and Clothier, 1989; Eching
et al., 1994; Wildenschild et al., 1997), it is unlikely thatone standard methodology will become available thatis satisfactory for all applications. Hence, considerable
efforts have been devoted to the indirect estimation of
the hydraulic conductivity function (e.g., van Genuchten
and Leij, 1992; Mualem, 1992). These efforts are justified, since direct measurements of the hydraulic properties are costly and time consuming and the results arevariable, error-prone, and applicable to only a narrow
range of saturation. Additionally, the number of measurements required to adequately characterize K in thefield is prohibitive given the natural soil variability.
Estimation of the hydraulic conductivity function istypically based on models that consider the pore-size
US Salinity Lab., USDA-ARS, 450 W. Big Springs Rd., Riverside,
CA 92507. Received 2 Oct. 1998. *Corresponding author (larya@ussl.
ars.usda.gov).

distribution of a soil (e.g., Childs and Collis-George,
1950; Marshall, 1958; Mualem, 1976; van Genuchten,1980). Input data for these types of models generally
include a measured or estimated soil water retentionfunction, /z(0), and the saturated conductivity, Ky The
hydraulic conductivity is derived by integration of elementary pore domains, represented by a specific poreradius. The range of the predicted K(ft) function is generally limited to the saturation range for which /i(0) data
are available. A more generalized formulation includes
pore tortuosity, pore connectivity, or pore interaction
terms (e.g., Mualem and Dagan, 1978). However, sincegeometric descriptions of these parameters are notavailable, they are evaluated empirically from more easily measured soils data such as soil texture, bulk density,and organic matter content (e.g., Schuh and Bauder,1986; Wosten and van Genuchten, 1988; Vereecken etal., 1990; Vereecken, 1995). Model parameters are often
expressed as averages for the different textural classes,with considerable uncertainty in predicted K(Q) functions. The problem is compounded as the number ofparameters increases, especially if h(Q) input data arealso based on a model (e.g., Wosten and van Genuchten,
1988; Vereecken, 1995). Thus, there is a need for models
that accurately reflect phenomenological properties ofobserved hydraulic conductivity functions, but that contain as few unknown parameters as possible (e.g., vanGenuchten and Leij, 1992; Poulsen et al., 1998).
The utility of pore-size distribution models to predictthe hydraulic conductivity function, K(Q), from the water retention characteristic, /z(6), and the saturated hydraulic conductivity, Ks, implies that K(Q) can be related
to the same basic soil properties that are commonlyused to characterize /z(9) and Ks. This may be accomplished with so-called pedotransfer functions that empirically predict hydraulic properties from basic soil
properties (e.g., Bouma and van Lanen, 1987; van Genuchten and Leij, 1992). However, soil properties seldom,
if ever, directly constitute formative elements of themodels. Commonly available models are mathematicaldescriptions of the shapes of h(Q) and K(Q) curves withunknown parameters. Much of the effort appears tohave been focused on relating parameters of mathematical functions to basic soil properties (e.g., Schuh and
Bauder, 1986; Wosten and van Genuchten, 1988; Schuhand Cline, 1990; Vereecken et al., 1990; Vereecken,
1995).
Basic soil properties such as PSD and bulk density
Abbreviations: PSD, particle-size distribution; RMSR, root meansquare residuals.

Published in Soil Sci. Soc. Am. J. 63:1063-1070 (1999).

1063

1064

SOIL SCI. SOC. AM. J., VOL. 63, SEPTEMBER-OCTOBER 1999

are widely available for many soil types and can be
accurately and routinely determined in laboratories
around the world. Hence, formulation of hydraulicproperties entirely in terms of basic soil properties
should be of considerable benefit. Translation of the
PSD into a corresponding water retention function has
been accomplished using the concept that the pore-size
distribution is directly related to PSD (e.g., Arya andParis, 1981; Tyler and Wheatcraft, 1988; Rieu and Sposito, 1991; Arya et al., 1999). In view of the above, it
is possible to also quantitatively relate the hydraulicconductivity of a soil to its PSD. The objective of thisresearch was to formulate a relationship between thehydraulic conductivity function and the PSD, and toevaluate predictions of the conductivity obtained bythis relationship.

MODEL DEVELOPMENT
Following Hagen-Poiseuille's equation and following
other studies in soil hydraulics, we base this model on
the premise that flow in soil pores is a function of thepore radius, with pore radii determined by the PSD.
Additionally, we assume that the hydraulic conductivityof a soil, at a given saturation, is made up of contributions from flow in pores that remain completely filledwith water at that saturation; that is, the contributionof partially drained pores to overall flow is insignificant.
Although these are simplified assumptions, they help
to formulate a model of flow which can be calibrated
for more complex media.
Based on the above simplifications, the hydraulic conductivity of a soil with its pore volume divided into npore-size fractions, and with fractions 1 through i filledwith water, can be expressed according to Darcy's law
asTable 1. Variables used in the equations.
Variable

A
a
e
H
K
L

n,
N,

Ne,

e,

9,
r,

R,
S

s.
w.

X

a
11
8,
Pb
P.
*

Description
cross-sectional area, bulk sample
pore cross-section, single porevoid ratio (p, - pb)/pb
hydraulic head
hydraulic conductivity
flow path length
number of spherical particles, ith fraction
scaled number of spherical particles, ith fraction
number of pores, ith fraction
volumetric flow rate, ith pore fraction
volumetric flow rate for a single pore,ith pore fraction
mean pore radius, /th fraction
mean particle radius, ith fraction
shape factor
ratio of maximum water content to porosity
mass fraction, solid particles, ith fraction
exponent on pore radius
scaling parameterviscosity of water
volumetric water content, ith fraction
bulk density
particle density
porosity

Dimension!

u
L

\

IK3
L»
L.T-'
L.
-

UTUT-'
LP
L,

wr'

-

-1

M. L' T 'U U5
M, U33
Ms Lr
I^Lf 3t For dimensional analysis, L = length, M = mass, and T = time, with
subscripts b for bulk, e for effective, in for measured, p for pore, s forsolid and saturated, and w for water.

Li = 1, 2, ..., n[1]where K(Qt) is the hydraulic conductivity of the sample(cm s~') corresponding with water content 0, (cm3 cm"3),
L is the sample length (cm), AH is the hydraulic headdifference (cm) across the sample length in the directionof flow, A is the cross-sectional area of the sample (cm2),
and Qj is the volumetric outflow rate (cm3 s"1) contributed by they'th pore fraction. A complete list of variables
used in the equations is provided in Table 1.
The water content, 6,, can be calculated from PSD,
porosity, and maximum measured water content information, according to
i = 1, 2, ..., nwhere $ is the total porosity (cm3 cm"3), 5W is the ratioof measured saturated water content to total porosity,and Wj is the mass fraction of particles (g) in the y'th
particle-size fraction. To compute water content, the
PSD is divided into n fractions, and the difference incumulative percentages corresponding with successive
particle sizes is divided by 100 to obtain values of Wj
such that the sum of all wf is unity. The pore volume
associated with the solid mass in each fraction is calculated on the basis of the assumption that the bulk andparticle densities of the bulk sample apply to each fraction. The volumetric water content is given by a summation of the individual pore volumes (from fraction j =I to fraction jf = i) divided by the bulk volume of thesample. Further information on scaling water content
from the PSD can be found in Arya and Paris (1981)
and Arya et al. (1999). The value of Sw normally varies
from 0.85 to 0.95 for nonswelling soils. For soils withhigh clay content, 5W may be progressively modified withchanges in sample volume that accompany wetting anddrying (e.g., Gamier et al., 1997).
The volumetric flow rate, Qh can be written as thesum of the flow rates of individual saturated pores within
the /th pore fraction of a particular soil sample. Thus,
Qi = qF*[3]
where qt is the volumetric flow rate for a single pore(cm3 s"1) and Npi is the number of pores in the ith poresize fraction.

Flow in a Single Pore
Flow at the microscopic pore scale cannot be precisely
defined. We conceptualize an individual pore as anequivalent capillary tube in which flow occurs according
to the Hagen-Poiseuille law for capillary flow (e.g., Batchelor, 1967; Hillel, 1971). The flow rate, <?„ for an individual pore can be calculated from
irrf
[4]
~^~
Sr\
LT
where r, is the mean pore radius (cm) for the ith pore
fraction, pw is the density of water (g cm'3), g is theacceleration due to gravity (cm s~2), AH is the pressure
head difference across the flow path (cm), 5 is a shape

ARYA ET AL.: HYDRAULIC CONDUCTIVITY FUNCTION FROM PARTICLE-SIZE DISTRIBUTION

factor, T) is the viscosity of water (g cm"1 s"1), and L is
the length of the flow path (cm). The term (AH/L) in
Eq. [4] can be eliminated if flow occurs under a unit
gradient. The exponent x and the shape factor S are
dimensionless parameters. For cylindrical tubes of uniform diameter, x = 4 and 5 = 8.
Soil pores, however are neither circular nor uniform.
Nor are they straight. Therefore, an adjustment of parameters S and x is necessary to adapt Eq. [4] for natural
materials. Flow of water in soil pores is affected by anumber of factors, including pore size, the spatial andrelative distribution of pores of various sizes, poreshape, tortuosity, pore connectivity, fluid properties,
and as the degree of saturation (e.g., Mualem, 1992;
Corey, 1992). However, it is intuitive from the HagenPoiseuille equation and other studies in soil hydraulics(e.g., Childs, 1969) that pore size should have an overwhelming effect on flow rate. While factors such as
pore shape and tortuosity are important, a satisfactorygeometric description of these factors is as yet unavailable. Therefore, we follow a pragmatic approach, assume a unit gradient, and combine all factors except forpore size in a single empirical variable. As a result, Eq.
[4] is modified toq, = erf[5]in which c and x can be evaluated empirically using
experimental data.
The Pore Radius, r,
The pore radius, r,, can be calculated from the PSD
according to Arya and Paris (1981), usingr, = 0.816JR,[6]
where Rt is the mean particle radius for the ith particlesize fraction, e is the void ratio of the natural-structured
soil sample, n, is the number of equivalent sphericalparticles in the fraction, and a, is the scaling parameter
defined by Arya et al. (1999) asa, = logty/logn,[7]
where Nt represents the scaled number of hypotheticalspherical particles of radius R, required to trace thetortuous pore length contributed by nt natural particles
in the actual sample. Note that the terms e and nj~ ai )in Eq. [6] are dimensionless. The equivalent number ofspherical particles, «, in the ith particle-size fraction canbe calculated from
^

[8]

1065

Table 2. Parameters of Eq. [10] relating log N, to log(w/R?), andthe goodness of fit, r2, for four soil textural classes.!
Textural class

Soils

Data pairs

a

b

r2

Sand
Sandy loam
Loam
Clay

6
6
4
5

62

-2.478-3.398-1.681-2.600

1.490
1.773
1.395

0.882
0.952
0.936
0.954

75
SO
88

1.305

t cf. Arya et al., 1999.

Table 2 summarizes parameters a and b and the goodness of fit, r2, for four soil textures included in the studyof Arya et al. (1999). Particle-size distribution, bulk
density, and soil water characteristic data to establish
Eq. [10] for the soils in Table 2 were obtained from the
UNSODA soil hydraulic properties database (Leij etal., 1996). Once Eq. [10] is specified, there is no needto know the water retention characteristic, h(Q).

Number of Pores, Nfi
We assume that ratio of the total pore area exposedat the cross-sectional area of a sample is the same aseffective porosity of the sample. Thus, for a naturalstructure sample of unit massAfl, = cMb[11]where Ape is the total effective pore area exposed at the
sample cross section, 4>e is the effective porosity givenby 4>e = Sw [1 - (Pb/Ps)], and Ab is the cross-sectional
area of the sample given by Ab = (l/pb)2'3. Following
Arya and Paris (1981), we distribute the effective porearea among the various pore-size fractions according tosolid mass, wh in the various particle-size fractions, that
is,
[12]
Api =
where Afi is the effective pore area attributed to the ith
pore-size fraction. The number of pores in the ith pore
fraction, Nph exposed at the cross-sectional area, can beobtained by dividing Api by the cross-sectional area ofa single pore, api. Thus,
Nfi = Api/afi[13]2
where ap, = nr, and rt can be determined from PSD
data according to Eq. [6].

The Hydraulic Conductivity Function,
For flow under a unit gradient, Eq. [I], [3], and [5]can be combined to formulate the A"(6,) function as= ~ 2 (erf) Np-

i = 1, 2, ..., n[14]•"& ;=1

The void ratio, e is given bye = (PS - Pb)/Pb[9]3
where ps is the particle density (g cm~ ) and pb is thebulk density (g cm"3) of the natural-structured (undisturbed) sample.
The scaled number of spherical particles, TV, can beobtained from the PSD using the empirical relationship(Arya et al., 1999):
3

logty = a + 61og(w,/jf? )[10]

Substitution for r, (Eq. [6]) and Nfj (Eq. [11], [12], [13])results in Eq. [15], which describes /C(0,) expressly interms of parameters of the PSD and packing characteristics of the sample.

i = 1, 2, ..., n
[15]

1066

SOIL SCI. SOC. AM. J., VOL. 63, SEPTEMBER-OCTOBER 1999

The summation in Eq. [14] and [15] for i = n leads to/ = i, 2,..., n

the value of the saturated conductivity, Ks.

[16]

and

MATERIALS AND METHODS
Soils data used in this study were taken from the UNSODA
hydraulic properties database (Leij et al., 1996). The parameters c and x were calibrated by fitting experimental conductivity data to the flow model given by Eq. [5]. The parameters
were determined for sand, sandy loam, loam, and clay textures.
Within each textural class, three or four data sets were used
to evaluate c and x. In the second part of this study, theempirical values of c and x were used to independently predict
K(Q) functions from PSD data for different data sets. Gross
textural characteristics and bulk density data for soils used inthis study are given in Table 3.

where (Q,)m is an estimate of the experimental volumetricflow rate for the z'th pore fraction. An approximation of themeasured flow rate for a single pore, (qi)m, in the /th pore
fraction can be obtained from(*-)m = (G/yw,,[i[18]

Replacing qt in Eq. [5] by (q,)m and rewriting the equation inlogarithmic form, we havelog(g,)m = logc + xlogr/[19]

The parameters c and x were evaluated from plots of log(<?<)m vs. log rt data. The UNSODA data sets used in this study
and the values of log c and x along with goodness of fit, r2,
are presented in Table 3.

Evaluation of Parameters c and x
First, the cumulative PSD curve for the soil was converted
into a corresponding pore-size vs. water content curve on thebasis of procedures initially proposed by Arya and Paris (1981)
and later refined by Arya et al. (1999). Measured values of thehydraulic conductivity, K(9,i)m, corresponding with calculated
water contents, were interpolated from plots of the experimental K(Q) curves. Since each calculated water content corresponded with a pore radius, it was possible to generate K(Qj)m
vs. r, pairs across a range of water contents. Backward calculations were then performed to obtain estimates of the measured
values of the pore flow rates, (<7,)ra, corresponding with thepore sizes with which K(Bi)m values were paired. It follows
from Eq. [1] that multiplying K(6i)m by the cross-sectional
area of the sample yields the total flow for the sample perunit time and per unit gradient. This flow volume represents
the sum of flow volumes being contributed by all pore fractions
that remain filled at the water content, (9/)m. Thus,

Calculation of Hydraulic Conductivity Function, K(Qt)
The PSD is divided into n particle-size fractions (usually
20 or more uniformly spaced increments of logarithm of theparticle size) to yield n corresponding pairs of mass fraction,
wit and mean particle radii, /?,-. Each H>, is converted to an
equivalent number of spherical particles, «,-, using Eq. [8]. The
scaled number of spherical particles, Nh is calculated for eachfraction using Eq. [10] and Table 2. Particle numbers obtained
from Eq. [8] and [10] are used in Eq. [7] to calculate thescaling parameter, a, for each particle-size fraction. Mean /?,-,«,, and a, for each fraction and the void ratio, e, for the naturalsoil sample (Eq. [9]) are used in Eq. [6] to compute meanpore radii, /•/. Values of water content, 6,, are obtained bysumming the pore volumes from fraction / = 1 to /, adjusting

Table 3. UNSODA soils used for model calibration and testing, model parameters log c and x in Eq. [19], goodness of fit, r2, shapefactor log S, and root mean square residuals (RMSR) of log-transformed A'(0)tak and A'(H)nK.al, data.
Model test

Model calibration
Texture

Codet

Clay

Sand
o/
/o ————

Sand

Average
Sandy loam

Average
Loam

AverageClayPbg cm~3
1.600
1.622
1.490
1.67
1.590

1050
4650
4661
1464
1061

3.2
1.0
3.0
2.4
4.0

92.2
94.0.
92.9
93.0
91.0

1381
1130
4160
4162

11.0
7.0
11.2
19.4

61.2
71.0
66.3
65.0

1.730
1.560
1.700

3192
4102
4610
4101

16.3
18.3
24.0
16.3

44.2
40.0
39.0
41.6

1.530
1.110

4681
4121
2360

Average
Average for all textures
t UNSODA code.
1 Bulk density.

11.6
7.2
20.5

54.4
53.2
45.0

1.520
1.440
1.500

1.080
1.110
1.420

log c
2.590
2.507
2.226
0.171
1.642
1.849
-0.952-0.808-2.4260.231-0.871
2.392
2.647-0.766
2.780

x
4.714
4.471
4.095
2.808
3.718
3.999
3.000
2.882
2.664

3.351
3.063
4.139
4.072
3.413
4.344

r

2

0.978
0.970
0.883
0.963
0.916
0.913
0.977
0.981
0.972
0.987
0.964
0.996
0.965
0.999
0.976

logS
2.949
3.032
3.313
5.368
3.897
3.690
6.491
6.347
7.965
5.308
6.410
3.147
2.891
6.305
2.759

2.647
-0.226-1.203-0.262

4.258
3.625
3.227
3.531

0.972
0.969
0.997
0.986

2.892
5.765
6.742
5.801

-0.488
0.482

3.506
3.602

0.976
0.938

6.027
5.507

Codet
4000
1063
1043
4171
1054

2531
1370
4600
2530
4620
3190
2361
1400
4680
4120
2362

Clay

1.5
4.0

3.0
11.7
2.5

Sand

pb

93.8
93.0
93.0
65.1
97.2

gcnr 3
1.460
1.550
1.593
1.480
1.540

15.7
25.9
24.0
22.6
24.0
16.5

41.0
46.0
39.0
45.0
47.9

1.461
0.950
1.040
1.361
1.210
1.410

3.5

57.0
58.0
47.3
50.8
63.0

1.387
1.450
1.100
1.240
1.250

4.7

13.8
11.8
3.8

39.0

RMSR
0.669
0.638
1.603
0.616
1.563
1.000

0.592
1.719
1.155
0.638
1.397
0.657
0.996
0.516
0.487
1.065
0.549
0.542
0.670
0.878

ARYA ET AL.: HYDRAULIC CONDUCTIVITY FUNCTION FROM PARTICLE-SIZE DISTRIBUTION

for effective saturation, and then dividing the sum by the bulk
volume of the sample, given by (1/pb)- Pore volume associatedwith each particle-size fraction equals (Wi/pf)e. Combining
these terms and simplifying yields Eq. [2], which gives 0i in
terms of mass fraction, w,, effective saturation, Sw, and porosity, 4>.

Equation [5] is used to compute flow rate for a single pore,<?,, in each pore-size fraction with c and x given in Table 3.
As discussed above, estimates of c and x can be obtained using
Eq. [16] through Eq. [19]. The number of pores, in each poresize fraction, Npi, is calculated using Eq. [11], [12], and [13].
Pore flow for a single pore is multiplied by Npi to obtain theflow rate, <2,, for the rth pore-size fraction (Eq. [3]). A summation of Q, from fraction j = 1 to i yields the flow rate for the
sample corresponding with 6;. Since all flow rates correspondwith a unit gradient, dividing SpiQy by the bulk cross-sectional
area of the sample, Ab, gives the hydraulic conductivity,(Eq. [15]).

RESULTS AND DISCUSSION
Pore Flow in Idealized Capillary Tubes
and Soil Pores
Figure 1 shows an example of the relationship between the logarithm of the pore flow rate, q, and log r,
for 16 soils, representing sand, sandy loam, loam, andclay textures. The Hagen-Poiseuille pore flow rates werebased on Eq. [4], assuming that pores are circular andstraight capillary tubes. The experimental pore flow
rates were calculated from measured /C(0) data using
Eq. [16], [17], and [18]. Similar plots for individual textural classes showed a strong linear relationship between
logg, and logr, (see parameters and r2 values in Table3). We conclude that a linear relationship between logg,
and logr, holds not only for the straight capillary tubes,
but also for tortuous porous materials. The difference
between the two is indicated by the slopes and interceptsof the regression lines (see Table 3). In the case of the- - -Hagen-Poiseuille equation (S=8, x=4)

o
CO

"Eo

-5

1
I

Experimental data (Eq [20])-15

Log q, = 0.482 * 3.602 log r,; r2 = 0.938'

S.

3 Clays (4681, 4121, 2360)

O)

o

-20

4 Loams (3192. 4102, 4610, 4101)
4 Sandy loams (1381, 1130, 4160, 4162).
5 Sands (1050, 4650, 4661, 1464, 1061)

-25

-5

-3

-2

-1

0

Log pore radius, r, (cm)Fig. 1. Relationship between pore flow rate and pore radii for 16
soils, representing sand, sandy loam, loam, and clay textures.

1067

Hagen-Poiseuille equation, x = 4, S = 8, T\ = 0.008904
poise (25°C), and g = 980 cm s~2. For the soil samples,the value of x is equal to the coefficient of regression
and S can be calculated from logc = log(-rrpw glS-r\). For
our calculations, we assumed the viscosity of water at
25°C to be applicable to all samples, since all experimental data were obtained at room temperature. The calculated values of log c, Jt, and S for four textural classes
are summarized in Table 3. Results show that x varies
from =2.66 to =4.71. Values of S, on the other hand,are quite large and vary within and between texturalclasses by orders of magnitude.
The deviations in x and S of the soils from those fora bundle of straight and circular capillary tubes reflect
the combined effects of pore size, pore-size distribution,pore shape, tortuosity, interaction between pores, viscosity of water, and possibly other factors. The valuesof x and 5 (Table 3) in our analysis represent only thegross properties of soil materials. It is quite possiblethat they both vary with pore size in a given sample.
The lack of a systematic trend in x and S with texturalclass is not surprising. Soils within a textural class represent wide variations in PSD, bulk density, and organic
matter content. Our grouping of x and S by texturalclass is only for convenience. Further investigation of
the behavior of x and S is beyond the scope of this initialmodel of £(0) based on the PSD. However, results in
Fig. 1 demonstrate that the model for flow in straightcapillary tubes can be modified to adequately describe
the macroscopic flow behavior in porous media.

Predicted Hydraulic Conductivity Functions
The empirically estimated values of c and x wereused in Eq. [5] to independently calculate hydraulicconductivity functions, K(Q), of 16 soils. These soils are
identified in Table 3. Examples of predicted and experimental data are presented in Fig. 2a (sand), 2b (loam),and 2c (clay). Overall, the shapes of the predicted /C(9)functions were similar to those of the measured data in
the range of wetness for which measured data wereavailable. However, quantitative agreements between
predicted and experimental values varied, partly because the majority of the experimental data sets werelimited to a relatively narrow range of water contents(usually in the wet range). The model, on the other
hand, predicted hydraulic conductivity values across awide range of water contents — from near saturation
to extreme dryness. However, because of a lack of measured data in the dry range, model predictions cannot
be validated.
Figure 3 shows a comparison of the logarithms ofexperimental vs. predicted hydraulic conductivity values
for all soils on a 1:1 scale. The regression line between
the experimental and predicted K(Q) values closely
matches the 1:1 line with an r2 of 0.807. The RMSRs of
the log-transformed predicted and experimental K(Q)data ranged from 0.616 to 1.603 for sand (average 1.000),from 0.592 to 1.719 for loam (average 0.996), and from
0.487 to 1.065 for clay (average 0.670). The average
RMSR for all textures was 0.878. Similar RMSR values

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SOIL SCI. SOC. AM. J., VOL. 63, SEPTEMBER-OCTOBER 1999

ALL TEXTURES ( Evaluation)ooooo

UNSODA soils
Clay: 2361, 14OO. 468O, 4120, 2362
O
Loam: 2531, 1370, 46OO, 253O, 4620, 319O
Sand: 4OOO. 1063, 1043, 4171. 1O54

CO

1

o

o

«"

0 - RMSR = 0.878

•o

«
E

,
-1

8

I

-2

o

"5
ra

-

-

ro -3
o
-4

Sand (UNSODA 1054), predicted

O

Sand (UNSODA 1054), experimentsO

Sand(UNSODA4171), experimenta

Sand (UNSODA 4171), predicted

-5
-6
0.00

0.10

0.20

0.30

0.40

0.50

Water content, cm3/ cm3IO

to- - - Loam (UNSODA 3190), predicted
O

O

0.00

Loam (UNSODA 3190), experimental
Loam (UNSODA 1370), predicted
Loam (UNSODA 1370), experimental

0.60

0.40

0.20

0.80

Water content, cm3/ cm3

10'F- - - Clay (UNSODA 4120), predicted

10°

O

Clay {UNSODA 4120), experimental——— Clay (UNSODA 2362), predicted
O

Clay (UNSODA 2362), experimental

10-

t>

"5

CO

•6
>,

,oco

io~4

oooo

10'6
0.20

0.30

0.40

0.50
3

3

Water content, cm / cm

-5

-4

-3

-2

-1

1

Log K predicted, cm/day
Fig, 3. Comparison of predicted and experimental hydraulic conductivity. Test results for 16 soils are pooled.

T3

————

-6

0.60,

have been observed with other hydraulic conductivity
prediction models (e.g., Schaap and Leij, 1998). The
scatter in the data is not surprising in view of the manysources of variation in the experimental as well as input
data. Measurement of hydraulic conductivity is usuallydifficult, and large differences in experimentally measured hydraulic conductivity between replicated samples of the same soil are not uncommon (e.g., Dirksen,1991). There are many examples in the literature showing such variations, as well as large variations between
measurements and prediction models (e.g., El-Kadi,
1985; Mishra and Parker, 1992; Yates et al., 1992; Tamari
et al., 1996; Poulsen et al., 1998). These variations may
be attributed to differences in PSD, bulk density, mineralogy, microaggregation, and organic matter content
within a textural class. We used textural class averagevalues of the model parameters, c and x. However, it
should be noted that textural similarities do not necessarily translate into hydrophysical similarities. Groupingsoils together solely on the basis of textural nomenclature may introduce variations of up to 25 percentage
points in the mass fraction of some particle-size ranges(L.M. Arya, 1983, unpublished data, based on USDASCS [1974]). Significant variations in bulk density andorganic matter content also exist within a textural class.
Variations in experimental procedures and associatederrors may introduce additional noise in the experimental data. Hydraulic properties of soils in the UNSODA
database (Leij et al., 1996) are heterogenous and exhibit
large within-texture variations (see Fig. 2, for example).
These data were contributed by researchers in the USA,
Netherlands, United Kingdom, Germany, Belgium,
Denmark, Russia, Italy, Japan, and Australia, who useddifferent experimental procedures. In view of the above
Fig. 2. Comparison of predicted and experimental hydraulic conductivity functions: (a) sand, (b) loam, and (c) clay.

ARYA ET AL.: HYDRAULIC CONDUCTIVITY FUNCTION FROM PARTICLE-SIZE DISTRIBUTION

uncertainties, we believe that the predictions of the proposed PSD-based model of the hydraulic conductivity
are quite reasonable. Predictions of the model may befurther improved if measurable physical characteristics
of soils that influence flow processes can be identified.

SUMMARY AND CONCLUSIONS
Research reported here describes a method to compute the hydraulic conductivity function from the PSD
and bulk density data. The pore-size distribution isbased on the Arya-Paris model (1981) as modified by
Arya et al. (1999). The pore flow rate is related to poreradii in a manner similar to that for idealized capillarytubes, described by the Hagen-Poiseuille equation.
However, the model parameters for flow in natural soil
are determined empirically. They deviate significantly
from those for the idealized model, and reflect the effects of pore size, shape, and other unknown factors.
The effects of packing density and pore tortuosity areincorporated by the scaling parameter, a, of the AryaParis model. The model predicts hydraulic conductivity
function across a wide range of water contents, fromnear saturation to extreme dryness. Unlike other models, the need for a measured soil water retention function
is eliminated because the pore-size distribution is derived directly from the PSD. Likewise, the need for ameasured saturated hydraulic conductivity as a matching point is also eliminated. Our flow model requires
only two parameters of concern: c and x. The behaviorof these parameters may be further elucidated as themodel is subjected to further tests and scrutiny.
Predictions of the hydraulic conductivity for a numberof soils, representing a range in texture, bulk density,and organic matter content were reasonable. The
RMSEs of log-transformed predicted and experimental
data ranged from a low of 0.487 to a high of 1.562, theaveraged being 0.878. Heterogeneity in experimental
data and nonconformity between gross textural classattributes and hydrophysical behavior of individual soils
are believed to be responsible for uncertainties in themodel's predictions.

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SOIL SCI. SOC. AM. J., VOL. 63, SEPTEMBER-OCTOBER 1999

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the Hydraulic Properties of Unsaturated Soils. 11-13 Oct. 1989.
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Vereecken, H. 1995. Estimating the unsaturated hydraulic conductivity from theoretical models using simple soil properties. Geoderma 65:81-92.
Vereecken, H., J. Maes, and J. Feyen. 1990. Estimating unsaturated
hydraulic conductivity from easily measured soil properties. Soil
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Wildenschild, D., K.H. Jensen, KJ. Hollenbeck, T.H. Illangasekare,
D. Znidarcic, T. Sonnenborg, and M.B. Butts. 1997. A two-stage

procedure for determining unsaturated hydraulic characteristics
using syringe pump and outflow observations. Soil Sci. Soc. Am.
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Yates, S.R., M.Th. van Genuchten, and F.J. Leij. 1992. Analysis ofpredicted hydraulic conductivities using RETC. p. 273-283. In
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