WATER RESOURCES RESEARCH, VOL. 32;NO. 5, PAGES 1251-1258, MAY 1996

Parameter equivalence for the Brooks-Coreyand van Genuchten soil characteristics:
Preserving the effective capillary drive
Hubert J. Morel-Seytoux,’ Philip D. Meyer,* Mahniood Nachabe,” Jaoudat
Touma,4 M. T. van Genuchten,s and Robert J. Lenhard
Abstract. This paper provides a simple way to convert Brooks-Corey (BC) parameters to
van Genuchten (vG) parameters and vice versa, for use primarily in situations where
saturated conditions are likely to be encountered. Essential in this conversion is thepreservation of the maximum value of a physical characteristic, the “effective capillary
drive” H,, [Morel-Seytoux and Khanji, 1974], defined with a good approximation for a soil
water and air system as H,., = Jg k,, dh,, where k,, is relative permeability (orconductivity) to water and II, is capillary pressure (head), a positive quantity. With thisconversion, infiltration calculations are essentially insensitive to the model used torepresent the soil hydraulic properties. It is strictly a matter of convenience for the userwhich expression is used. On the other hand, the paper shows that other equivalences maylead to great variations in predictions of infiltration capacity. Consequently, the choice ofthe proper equivalence to use in calculations for rainfall-runoff modeling or for low-level
radioactive waste disposal design is a serious matter.

Introduction
Depending on circumstances, it is convenient for analyticalderivations or for numerical work to favor one type of expression for the hydraulic conductivity and water retention properties over another. However, the question can be legitimately
raised as to the influence of that choice on the results. To
answer this question conclusively, one would need to carry out
the computations twice, once with each expression. However, it
is sometimes not possible to perform the analytical derivations
for both. For analytical derivations the Brooks and Corey [ 1964](BC) formulae tend to be easier to use. On the other hand,some codes for numerical solution of the unsaturated flow
equation use the v a n Genuchten [1980] (vG) expressions exclusively. If codes provide both options, it still remains todefine the values of the parameters for both sets of expressions. Even if the data, on which one set of expressions wascalibrated, and the criterion used for calibration were readily
available, the criterion for goodness of tit of the other expression is, as always in calibration, somewhat arbitrary and furthermore need not be the same. As in statistical analysis, whendifferent probability density functions with a small number ofparameters (say two) are fitted to the same data, one must
decide which density characteristic is to be preserved. That
‘Department of Civil Engineering, Colorado State University, Fort
Collins.
‘Pacific Northwest Laboratories. Richland, Washington.
‘Department of Civil, Environmental, and Architectural Engineering, University of Colorado, Boulder.
‘Dcpartcmcnt dcs Eaux Continentales, lnstitut Francais de Recherche Scientifique pour le Developpement en Cooperation (ORSTOM),
Montpellier, France.
‘U.S. Salinity Laboratory, USDA Agricultural Research Service,
Riverside, California.
Copyright 1996 by the American Geophysical Union.
Paper number 9 6 W R 0 0 0 6 9 .
0043- 1397/96/96WR-00069$09.00

same decision must be made to define “equivalent” parameters for the Brooks-Corey and van Genuchten expressions. In
what sense are they to be equivalent? Curve fitting different
analytical expressions to the same set of data, by the classicalunconstrained least squares technique, will not inherently preserve an “important” characteristic contained but somewhat
hidden in the data. This naturally begs the question: What is an
important characteristic to be preserved?
Surprisingly, few formal attempts [e.g., Lenhard et al., 1989;
Russo et al., 1991] at defining an equivalence have been reported in the literature, because, clearly, it is an issue that musthave confronted many code users and modelers.

Objective
The main objective of this paper is to provide a simple way
to convert Brooks-Corey (BC) parameters to van Genuchten(vG) parameters and vice versa for use primarily in situations
where saturated conditions are likely to be encountered. There
are applications where the proper description of the soil properties at high water contents is crucial. Watershed rainfallrunoff modeling is one, and capillary barrier design is another.
Essential in this conversion is the preservation of the maximum
value of a well-defined physical characteristic, the effective
capillary drive fI,.,,, [Morel-Seytoux and Khanji, 1974, 1975],
given with a good approximation for a soil water and air system
aswhere k,,,, is relative permeability (or conductivity) to waterand 11, is capillary pressure (head), or suction, a positive quantity, the negative of the matric head. H,.,,, has received othernames in the literature, such as “macroscopic capillary length”[Philip, 1985; White and Sully, 1987].

1252

MOREL-SEYTOUX ET AL.: SOIL PARAMETER EQUIVALENCE

Brooks-Corey and van Genuchten Expressions

Physical Significance of HcM
There are an infinite number of ways to define an equivalence between the two sets of parameters. Thus there must bea physical justification for the one proposed here. It is known[e.g., Morel-Seytoux and Khanji, 1974] that the effective capillary drive plays a significant role in determining the infiltration
rate into a vertical soil column under a ponded condition at thesurface. Under such conditions one can show (Morel-Seytoux
and Khanji, 1974] that the “infiltration capacity” (which is theinfiltration rate when infiltration takes place at capacity), thus
more precisely defined as the capacity infiltration rate, is givenby the expressioni = I;‘[ (5 - 0,) HcM + I]//31

Soil Hydrologic Characteristics
Normalized water content is defined as

0* = (0 - I),)/(0 - 0,)

where 0 is (volumetric) water content and 0, is residual watercontent. The BC expressions arc
11, = /1,,0* -h’(2)wherei; is hydraulic conductivity at natural saturation (or fieldsaturation, that is, porosity minus trapped air content), 6 is(volumetric) water content at natural saturation, 8, is uniform
initial water content, I is cumulative infiltration depth up to
time t, and /3, the viscous correction factor, is a dimensionlessparameter which theory [Morel-Seytoux, 1969] and numericalexperiments [Morel-Seytoux and Billica, 1985] indicate is essentially a constant for a given soil and varies little from soil tosoil. From sand to clay, Morel-Seytoux and Khanji [ 1974] founda range in p of 1.1 to 1.7 and estimated an average value of 1.3.
In the case /3 = 1, (2) has precisely the functional form of theGreen-Ampt formula [e.g., Morel-Seytoux and Khanji, 1974].
The parameter p deviates from 1 for two reasons. First, thereal wetting front profile does not have a saturated rectangularshape. Second, in the wetted region, water and air flow simultaneously, and the overall viscous resistance of the complex
mixture exceeds that of water alone.
From (2) an expression for the two-phase sorptivity [MorelSeytoux and Khanji, 1974, 1975] can be defined as the constant
Sz in the expression for infiltration rate at early times, namely,(5)11, 2 II,,(6)k, = O*P

11, 2 II,,

0)

k,= 1

11, 5 ltcp

(7b)

No physical significance is hereby attached to the parameterh,, (also denoted h,,) even though it is called herein entry (orbubble) pressure. In the application for infiltration problems
these must be the curves corresponding to wetting conditions.
To reduce the number of parameters to two, a relation developed by Corey [1977] is used:

p=3+2M(8a)

M = (p - 3)/2

(8b)

or

The vG expressions are,lc = $ [e* -(l/n11 _ 1 ]I”’

(9)

where a-’ .IS a parameter with dimension of length. In the casewhen the following relation between 111 and rz holds:
nt = 1 - (l/n)(10a)

n = l/(1 - 171)

( 1Ob)orEquation (3) defines S, in terms of the primary parameters
H,, and I?. It has also been shown [e.g., Warrick and Broadridge, 1992], but from the single-phase flow point of view, that

The only difference between S, and Sz is in the estimation of/3 and 2b. White and Sully [1987] recommended for b a valueof 0.55, regardless of soil, thus 2b = 1.10. Given the fact that
/3 = 1.30 includes the viscous resistance of the air phase (neglected in Richards equation) in addition to the deviation of
the water content profile from a rectangular shape, one would
expect p to be somewhat larger than 26. The numbers are
remarkably close. For a value of 2b = 1.1 the early infiltration
rate is 95% that given by the Green-Ampt formula, and with p= 1.3 the early infiltration rate is 88% of it. Given the relative
constancy of /3 or 2b, the value of N,,,, can be obtained withgood accuracy by an infiltration test. The steady state value
reached by the infiltration test yields k;. The early part of theinfiltration test yields HcM. (Naturally, the entire observed
curve can be used to perform the calibration, for example, byusing a least squares fit.) The value of f3 is measured at the endof the experiment; that of ei is measured at the beginning of
the experiment.

thenk, = t)*‘l/2’{1_ [I _ t)+llhl~]~t}?

(11)

To reduce the vG expressions to two parameters, (10) is always
implied throughout this article.
Applicability of the BC Expressions Near or at Saturation
It is sometime stated that the B C function does not apply tothe saturated portion of fl(/l,). Whereas it is correct that (6)does not apply for It, s h,,, k,, is nevertheless a continuousfunction of /I,, keeping the value 1 in the capillary fringe (0 5h, 5 tt,,). At h, = II,, there is a disconlinuity in slope ofk,,(h,), but the integral given by (1) is perfectly regular andintegrable at that point. Note that the variable of integration ish, not 0 or f3*. Specifically, consider the integral defined by (1)and carry out the integration step by step as indicated presently:

”

dir,

k, zT do* = II,,

MOREL-SEYTOUX ET AL.: SOIL PARAMETER EQUIVALENCE

1254

20

-vG -0-C!
,i

Soil 2- Brooks-Corey, TOlJMA---- Brooks-Corey, CHEMFLO-.-.-. van Genuchten, TOUMA

0

0.5

1

1.5

Dimensionless Capillary Pressure, ah,Figure 3. Relative permeability as a function of dimensionless capillary pressure head, a/r,, for the case I?I = 0.2 thus
p = 11.

The reader should note that a-r is quite different from /I,,, in
fact, about twice as large. Figure 2 shows plots of k,, versush,,
expressed in units of h,, or in other words versus the dimensionless variable /r,/lr_. The curves are quite different, but
because H,, was preserved, the areas under the two curves areequal as can be assessed visually.
Numerical illustration-VG parameters known. Say m =0.2 (thus p = 11) and a-’ = 1. Equation (17) gives h,, =0.061 X 1 or II,, = O.Obla-‘. Now II,, is only a very small
fraction of a-‘. Vice versa, if II,, is taken as unity, a-’ hasvalue 16.35 X 1 (or a -’ = 16.35/r,,), the value that one
would read from Figure 1 for p = 11. Quite clearly, one
cannot assume that a-’ and II,, have the same value. Figure 3
displays the curves of relative permeability as a function ofcapillary pressure, expressed in units of a-’ or in other words
as a function of the dimensionless capillary pressure variable
0111,. Again, the curves are quite different, but because H , ,was preserved, the areas under the two curves are equal.
Numerical Verification
We stated earlier that H,,,,, the maximum effective capillary
drive, was the important characteristic to preserve and that, if
preserved, prediction of infiltration would hardly be affected by
the choice of representation of the soil hydrologic properties.
Several numerical codes were used to predict the infiltration
capacity of a soil column for p = 4, 11, and 20. The value of
H,,, was taken in the three cases to have value 40 cm. Hydraulic conductivity at natural saturation had value 1 mm perhour. The value .of 6 was taken as 0.3, that of 8, was taken as
0.05. For the uniform initial water content 6; a value of 0.1 wasselected. Table 1 shows the corresponding values of the p a rameters. The boundary condition at the soil surface was oneof zero capillary pressure. The soil column is assumed semiinfinite.
Figures 4 and 5 display the comparison of infiltration rates

0

2

0

4
6
Time, hour

a

-I
10

Figure 4. Infiltration rate as a function of time for soil number 2 in Table 1 using the Brooks-Corey (BC) or van Genuchten (vG) expressions according to the proposed equivalence
and as calculated with the two numerical codes, CHEMFLO
and Touma’s [ 1984].

and cumulative infiltration depth, respectively, versus time forsoil 2 using the BC and vG expressions and two numericalcodes [Nofziger et al., 1989; Touma, 1984]. It is clear that theresults are essentially unaffected by the choice of the expressions or the choice of the numerical code utilized. However, it
is interesting to note that the differences between codes, especially with the vG expressions, arc almost of the same order ofmagnitude as that resulting from the choice of expressions forthe soil properties using a given code. On the other hand,
Figure 6 displays the same comparison for soil 2 for the samem = 0.2 and a- ’ = 416 cm but using Lenhard et al.'s [1989]
equivalence, which in this case yields /> = 11.26 and II,, =345 cm thus If,, = 545 cm. This latter value of 545 cm isclearly vastly different from 40 cm given in Table 1. From thedependence of infiltration rate at early time on the square rootof If,, in (3) one would expect the early infiltration rates to be- Brooks-Corey, TOUMA

Table 1. Table of Correspondence for the BC and vG
Parameters and Relation to Effective Capillary Drive forThree Hypothetical Soils

H C-n,?
Soil
1
2
3

4
11
20

p M

cm

0.5
4
8.5

40
40
40

L

a-‘,

a-‘/

cm

111

n

cm

H <n,

35
25.4
23

0.667
0.200
0.105

3
1.25
1.12

67
416
1323

1.67
10.4
33.1

---- Brooks-Corey, CHEMFLO-.-.-’ van Genuchten, TOUMA

If

0'
0

‘.“...... van Genuchten, CHEMFLO
2

4
6
Time, houra10

Cumulative infiltration depth as a function of lime
for soil number 2 in Table 1, using the BC or vG expressions
according to the proposed equivalence and as calculated withthe two numerical codes, CHEMFLO and T o u m a ' s [1984].
Icigure 5.

MOREL-SEYTOLJX ET AL.: SOIL PARAMETER EQUIVALENCE

:

:
\

.

Soil 2‘,cr- Brooks-Corey, M.-S.,

\

\
‘.

-\ ‘\

1255

TOUMA
---- Brooks-Corey, L. et al., TOUMAl

Soil-I-VG.

Soil-2.VG
Soil-3-VG
Soil-2-K
SOiKl-BC

.
A
Ll

.
h
u

‘..\
‘*..

%_

2

--._

---_ ---_---_ ---_-----_____

\

0
0

2

4

2

4

8

Figure 6. Infiltration rate as a function of time for soil number 2 in Table 1, using the BC expressions according to the
proposed equivalence, M-S (this study), and that of Lenhard etal. [ 1989], and calculated with Touma's [ 1984] numerical code.

roughly in the ratio m = 3.7. Indeed, the curves lookquite different, and the values are roughly in that ratio. On theother hand, with the proposed equivalence the normalized
infiltration rate curves with either the BC or the vG expressions
are very similar between soils 1, 2, and 3.
In Figure 7, using a van Genuchten code for the same
boundary conditions and parameters and soils 1, 2, and 3 ofTable 1, with the same values of N,,,, k, 6, 8,, and Bi, theonly difference in results is due to the difference in the p or tn
values, or in other words in the shape of the water content
profiles. Clearly, from Figure 7 such a shape has little influenceon the infiltration rates, which confirms the fact that the parameters /3 and h are practically constant regardless of soil
texture or structure. This follows from the fact that the difference in sorptivity as given in (4) or (3), and consequently, thedifference in infiltration rate is due to the parameter h or p,
and since the infiltration rates are essentially the same, regardless of the value of parameter p, this implies that h or fl arepractically constant for all soils.
To further test the importance of the equivalence procedureon the results of infiltration calculations, three of the soils usedby Lenhard et al. [ 1989] to illustrate their procedure were

Figure 7. Infiltration rate as a function of time for soil numbers 1, 2, and 3 in Table 1, using the BC and vG expressions
according to the proposed equivalence and calculated with van
Genuchten's [ l980] numerical code.

studied. For given known vG parameters one determines thecorresponding BC curves by the two procedures, labeled M-S(this study) and L. et al. (Lenhard et al. [1989]), respectively,
and one compares them with the vG procedure. Figures 8a, 8b,
and 8c show saturation versus tension for the three soils using
the two equivalences. It is clear that the Lenhard et al. procedure does a better job at matching the vG curve for each soil.
Intuitively, one might think that because the /I, curves look
more alike with the Lenhard et al. procedure that it would lead
to better infiltration predictions. This is a case where unsupported intuition fails. Figures 9a, 9b, and 9 c show relative
permeability versus tension for the three soils. The M-S procedure gives a closer comparison, and it is very clear that the
Lenhard et al. procedure does not preserve the area under the
curve. For both soils I and 4 the areas are quite different. As
expected, Figures 10a, 10b, and 1Oc show greatly different
curves of infiltration rates and cumulative infiltration with the
Lenhard et al. procedure. The better performance of the proposed procedure is understandable because P,~~H,,,, wherep,,, is density of water and g is acceleration of gravity, is theenergy per unit volume expanded by the capillary forces during
the flow of water from a region of a high given saturation (orlow capillary pressure) to a region of a different (low) satura-(a) Soil.8 0 . 6
E
2
m 0.4
cn

- - - - BC: M.-S.
- - - - BC: L. el al.
?
0200
Tension [cm]

400

0

J
10

Time, hour

108Time, hour’

6200
Tension [cm]

400

0

5

10

15

20

Tension [cm]Figure 8. Saturation as a function of tension (centimeters) for soil numbers (a) I, ( b ) 4, and (c) 5 used in
Lenhard et al.'s [ 1989] paper, using the vG expressions and the BC expressions according to the proposed M-Sequivalence and the Lcnhard et al. equivalence.

1256

MOREL-SEYTOUX ET AL.: SOIL PARAMETER EQUIVALENCE(b) Soil 4

!

A 0.8
.z
s
7 0.6
s
ar 0.4
.z
m
$ 0.2
C

- - - - BC: M.-S.

,i?Tq=mq+

‘._

0.0 :

1 I, I

0200
Tension [cm]

0

400

200
Tension [cm]

400

0

5
10 15
Tension [cm]20

Figure 9. Relative permeability as a function of tension (centimeters) for soil numbers (a) 1, (b) 4, and (c)5 used in Lenhard et al.‘s [1989] paper, using the vG expressions and the BC expressions according to the
proposed M-S equivalence and the Lenhard et a]. equivalence.
tion. H,,, is a dynamic capillary concept, whereas the 0* (II,)
curve is a static one. H , , is a potential in the traditional fluid
mechanics sense as shown by Morel-Seytoux a n d Khanji [l974].
The value of Hen, between the maximum and the minimum

2a(a) Soil 1- van Genuchten- - - - BC: M.-S.
-.-.-.- BC: L. et al.

20

values of saturation is independent of the shape of the saturation profile between these two values. Figures lla, llb, andllc show profiles of water contents at time 1 day for the threesoils. The profiles match well with the M-S procedure and arevery different with the Lenhard et al. [1989] procedure. The
numerical calculations for these three examples were carriedout with the code STOMP [White et al., 1992].

Influence of Initial Conditions

Y

I

I

I

I

I

I

I

0.0

0.5

1.0

1.5

2.0

2.5

3.0

It may be worthwhile to mention that using H,, in (2)implies that the initial value of water content is low to moderate. Otherwise, one would have to use the more generaldefinition of the effective capillary drive H,( 0,):A<,

Time (days)

H,(R) =

(1Ya)or
H,(e,) =0.0

0.5

1.0 1.5 2.0
Time (days)$^x 30-

2.5

3.0

-.s

cl

(c) Soil 5- 16 5E_ ,2 %.
5

v
J 20-

0.0

0.5

1 .o

1.5 2.0
Time (days)

2.5

3.0 *

Figure 10. Infiltration rate and cumulative infiltration depth
as a function of time for soil numbers (a) 1, (b) 4, and (c) 5
used in Lenhard e t al.'s [ 1989] paper, using the vG expressions
and the BC expressions according to the proposed M-S equivalence and the Lenhard et al. equivalence and calculated with
the numerical code STOMP.

Hcb,[ 1 - ; (2) -“““‘+‘I(1Yb)

where hci is h, for 0 = O,, p and M being related by (8). One
can verify that the deviation from H,.,, is significant only for
relatively large values of 0:. For p = 5 thus M = 1 thenormalized initial water content would have to be 0.84 for it tohave a 10% influence o n the value of the effective capillary
drive, For p = 11 thus M = 4 the value of Of would have tobe 0.83. For f37 = 0.5 the percentage of decrease in effective
capillary drive would be 1% for 11 = 5 and 0.3% for p = 11.
Thus it is clear that in the low to moderate range of watercontents, the initial value of the water content has very little
impact on the effective capillary drive. In that range of initialvalues the preservation of the maximum value of the effective
capillary drive will guarantee the insensitivity of the model
used to represent the soil hydraulic properties on the calculations of infiltration capacity. Of course, the value of the infiltration rate will change but practically entirely from the c o n tribution of the square root of (U - ei) which appears in theexpression for the infiltration rate in (2) and (3). If, on theother hand, the initial water contents are very high and giventhat at the boundary under ponded conditions saturation
holds, then capillary drive is not a force; sorptivity is essentiallyzero and equivalence is not important. Gravity dominates theflow. The equivalence is important when in some regions water

,r

MOREL-SEYTOUX ET AL.: SOIL PARAMETER E Q U I V A L E N C E- van Genuchten

$

0
5

$

10

2
-0
2

15

s
p
CI

1257

(c) Soil 5

20
25

1

30
0.2

0.4 0.6 0.6
Saturation

1.0

0.4

0.6

0.6

1 .o

Saturation

_.

0.2
*

0.4 0.6 0.6
Saturation

1.0

Figure 11. a, b and c. Water content profiles at time 1 day for soil numbers 1, 4, and 5 used in Lenhard etal.'s [I989] paper, using the vG expressions and the BC expressions according to the pioposed M-S equiva-

lence and the Lenhard et al. equivalence and calculated with the numerical code STOMP.

contents are low to moderate and in others they are at or nearsaturation. This is the case that was investigated which is crucial for watershed rainfall-runoff modeling, where saturation
under rainfall occurs at the soil surface, whereas below the soil
is initially dry, and in capillary barrier design, where saturation
occurs at the interface between the coarse material and thefine material above it.

Conclusions
An equivalence was defined which provides a simple way toconvert BC parameters into vG parameters and vice versa,
when one set is known and preserves the value of the effective
capillary drive thus making infiltration capacity calculations
insensitive to the model used to represent the soil hydraulicproperties, It is strictly then a matter of convenience for theuser which expression is used. Naturally, this conclusion applies only for situations which lead to high water contents insome part of the domain of concern. Because the second criterion for the equivalence preserves the asymptotic behavior of
capillary pressure at low water contents and thus the hydraulicgradient, there is at least plausible ground that the equivalence
will be reasonably satisfactory in that range also. However, thisshould be tested for situations involving drainage and evapotranspiration. It is recommended that it be done before acceptance for use in these situations. Finally, other equivalences
should be developed between other types of expressions, particularly three-parameter ones.

Appendix
The numerical evaluation of the effective capillary driveusing the van Genuchten functions for the soil capillary andhydraulic properties follows. The dimensionless effective capillary drive, defined as
SUM=0

I

I

k,W,) dh,(AI)0

was evaluated numerically by Gaussian quadrature using 10,
20, 60, 120, and 256 Gauss points. Results for 120 and 256
Gauss points were essentially identical. A rational function was
fitted by least squares to the results of the numerical integration using 256 Gauss points and the optimized expression isSUM =0.046m + 2.07m2 + 19Sm’
1 + 4.7m + 16,n2(A2)

Relative deviation percentages are aiways less than 0.5% form values between tn = 0.7 (where SUM = 0.64) and m = 0.2(where SUM = 0.1) and 1.6% for m values between 0.2 and0.05 (where SUM = 0.0078). The practical range of m isroughly between 0.67 and 0.05, corresponding to a range ofp
between 4 and 40. Thus, within that range the accuracy of therational expression is quite sufficient. The differences in the
results for soils I, 2, and 3 of Table 1 using the vG or BC
expressions are sufficiently stnall that part of the difference may
have come from these small errors, but from a practical pointof view that has essentially no impact on the merit of the
proposed equivalence.
Acknowledgments. The investigation was carried out while the firstauthor was on an Associated Western Universities Fellowship at Pacific Northwest Laboratory in spring 1994. Financial assistance fromthe U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research is appreciated. Thanks are extended to Mark Rockholdfor many useful discussions and suggestions in the preparation of themanuscript. The first version of this paper was submitted on November
17,1994 and acknowledged as received by the WRR office on November 28, 1994 as WR94-733.

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Corey, A., Mechanics of heterogeneous fluids in porous media, 150 pp.,
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1258

MOREL-SEYTOUX ET AL.: SOIL PARAMETER EQUIVALENCE

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R. J. Lenhard and P. Meyer, Pacific Northwest Laboratories. Richland, WA 99352. (e-mail: pd_meyer@ccmail.pnl.gov)
H. J. Morel-Seytoux, Hydrology Days Publications, 57 Sclhy Lane,
Athcrton, CA 94027-3926. (e-mail: morelsey@leland.stanford.edu)
M. Nachabe, Department of Civil, Environmental, and Architectural Engineering, University of Colorado, Boulder, CO 80309.
(e-mail: nachabe@gpsr.colostate.edu)
J. Touma, Departement des Eaux Continentales, Institut Francais
de Recherche Scientifique pour le Developpement en Cooperation(ORSTOM), 34032 Montpellicr. France. (e-mail: touma@rio.net)
M. T. van Genuchten, U.S. Salinity Laboratory, USDA Agricultural
Research Service, 4500 Glenwood Drive, Riverside, CA 92507.
(e-mail: vangenu@ito.umnw.ethz.ch)(Rcceivcd May 22, 1995; revised January 2, 1996;
accepted January 4, 1996.)