Original Research

D. Dourado Neto
Q. de Jong van Lier*
M.Th. van Genuchten
K. Reichardt
K. Metselaar
D.R. Nielsen

Alternative Analytical Expressions
for the General van Genuchten–
Mualem and van Genuchten–
Burdine Hydraulic
Conductivity Models
The van Genuchten expressions for the unsaturated soil hydraulic properties, first published in 1980, are used frequently in various vadose zone flow and transport applications
assuming a specific relationship between the m and n soil hydraulic parameters. By comparison, probably because of the complexity of the hydraulic conductivity equations, themore general solutions with independent m and n values are rarely used. We expressed thegeneral van Genuchten–Mualem and van Genuchten–Burdine hydraulic conductivity equations in terms of hypergeometric functions, which can be approximated by infinite series
that converge rapidly for relatively large values of the van Genuchten–Mualem parametern but only very slowly when n is close to one. Alternative equations were derived thatprovide very close approximations of the analytical results. The newly proposed equations
allow the use of independent values of the parameters m and n in the soil water retention
model of van Genuchten for subsequent prediction of the van Genuchten–Mualem andvan Genuchten–Burdine hydraulic conductivity models, thus providing more flexibility infi ng experimental pressure-head-dependent water content, θ(h), and hydraulic conductivity, K(h), or K(θ) data.

The unsaturated hydraulic conductivity is a key property for modeling water
Alternatives for the van Genuchten
hydraulic conductivity equations arepresented using approximations of
infinite series of hypergeometric functions. These approximations matchvery closely to the analytical results
and allow the use of independent
parameter values in the soil water
retention model for subsequent prediction of the hydraulic conductivity.

flow in the vadose zone. This property can be measured directly (Klute and Dirksen, 1986)
or indirectly (van Genuchten and Leij, 1992) or estimated implicitly using inverse methods (Hopmans and Šimůnek, 1999; Butters and Duchateau, 2002). Based on pore size
distribution theories proposed by Childs and Collis-George (1950), Burdine (1953), andMualem (1976), among others, several approaches for predicting the unsaturated hydraulicconductivity from measured soil water retention data have been presented in the literature.
These theories produced predictive expressions for the relative hydraulic conductivity, Kr,
which is the ratio between the hydraulic conductivity, K, at a certain water content, θ, orpressure head, h, and the hydraulic conductivity at saturation, Ks:

K r ( θ, h ) =K ( θ, h )Ks[1]

A frequently used equation for describing the soil water retention curve is (van Genuchten, 1980)
D. Dourado Neto, Q. de Jong van Lier, and
K. Reichardt, Univ. of São Paulo, Piracicaba (SP), Brazil; M.Th. van Genuchten, Dep.
of Mechanical Engineering, Federal Univ. of
Rio de Janeiro, Rio de Janeiro (RJ), Brazil; K.
Metselaar, Dep. of Environmental Sciences,
Wageningen Univ., Wageningen, the Netherlands; and D.R. Nielsen, Dep. of Land, Air andWater Resources, Univ. of California, Davis,
CA 95616. *Corresponding author (qdjvlier@esalq.usp.br).
Vadose Zone J. 10:618–623
doi:10.2136/vzj2009.0191.
Received 22 Dec. 2009.
Posted online 26 Apr. 2011.
© Soil Science Society of America
5585 Guilford Rd., Madison, WI 53711 USA.

All rights reserved. No part of this periodical may
be reproduced or transmi ed in any form or by anymeans, electronic or mechanical, including photocopying, recording, or any information storage andretrieval system, without permission in writing fromthe publisher.

2011, Vol. 10

Se =θ−θ r ⎡n −m
= ⎢1 + (α h ) ⎤⎥⎦
θ s −θ r ⎣[2]

where Se is the effective saturation, θ s and θ r are the saturated and residual soil water contents, respectively, and α, m, and n are empirical parameters. As shown by van Genuchten(1980) and van Genuchten and Nielsen (1985), Eq. [2] may be combined with the poresize distribution models of Burdine (1953) or Mualem (1976) to lead to predictive expressions for the relative hydraulic conductivity of unsaturated soils. The Burdine (1953) and
Mualem (1976) models differ in the manner in which effective pore radii are estimated(Vereecken, 1995). The model of Mualem (1976) is given by
K r ( Se ) ={∫}{ ∫ ⎡⎣1 h( S )⎤⎦ dS }

S el

Se

0

⎡1 h ( S e ) ⎤ d S e
⎣
⎦

1

2

0

e

2

e

www.VadoseZoneJournal.org | 618[3]

where l is a mostly empirical pore-connectivity or tortuosity
parameter. Its value was initially estimated by Mualem (1976) tobe 0.5, although widely different values have since been reported,
generally between about −6 and 6 (e.g., Yates et al., 1992; Schaapand Leij, 2000; Wösten et al., 2001).
Following the theory of Burdine (1953), another equation for Kr
is given by
K r S e = S eλ

( )

Se

2

1

2

∫0 ⎡⎣1 h( Se )⎤⎦ d Se

[4]∫0 ⎡⎣1 h( Se )⎤⎦ d Se

1/ n
S e1/ m ⎞⎟f ( S e ) = ∫ ⎜⎜⎟⎟ d S e
0 ⎜ 1− S 1/ m ⎟
⎝
⎠
e
Se ⎛
⎜

Using the substitution y = Se1/m in Eq. [6] yields a more tractableform:
f ( Se ) = m∫

The combination of Eq. [2] with either Eq. [3] or [4] allows direct
evaluation of the integrals in the Kr(Se) expressions. Closed-form
equations for Kr were derived by van Genuchten (1980) by expressing m in Eq. [2] as a function of n. Van Genuchten and Nielsen(1985) later presented solutions for Kr(S e) without assuming adependency between the m and n parameters. These solutions
contained the incomplete beta function, which they evaluatednumerically using continued fractions. Probably because of computational demands (Ross, 1992), the general solutions have beenused only rarely in vadose zone hydrologic applications. One recent
exception was given by Regalado (2005). The predictive qualities ofthe equations with m expressed as a function of n were evaluatedby Stephens and Rehfeldt (1985), Yates et al. (1992), Khaleel andRelyea (1995), and Cornelis et al. (2005), among others.
In the following, we revisit the general solutions of van Genuchten
and Nielsen (1985), show that it is feasible to derive alternative
formulations for the closed-form Kr expressions, and provide analytical approximations that are very easy to evaluate.

6Theory

S e1/m0− 1/ ny m−1+(1/ n ) (1− y )

[7]

dy

In van Genuchten (1980), m and n were assumed to be related by

m = 1−

where λ, similar to l in Eq. [3], is a pore-connectivity or tortuosity
parameter, generally assumed to be 2.0. Modifications of Eq. [3]and [4], including the use of other values for l or λ, have been proposed by Marshall (1958), Wyllie and Gardner (1958), Millington
and Quirk (1961), Kunze et al. (1968), and Farrell and Larson(1972), among others.

[6]

1
n

[8]

so that Eq. [7] simplifies tof ( Se ) = m∫S e1/m0− 1/ n(1− y )

[9]dywhich can be integrated immediately to yieldf ( S e ) = 1−(1− S e1/ m )

m

[10]

Substitution of Eq. [10] into [5] leads then to the following expression for Kr:m ⎤2⎡
K r ( S e ) = S el ⎢1−(1− S e1/ m ) ⎥ ( n > 1)
⎣⎢
⎦⎥

[11]

General van Genuchten–Mualem Equations
with Independent m and n Values
The more general approach with independent values of m and n
in Eq. [2] was considered in detail by van Genuchten and Nielsen(1985). They were able to integrate Eq. [7] to obtain
⎛
1
1⎞ ⎛
1
1⎞
f ( S e ) = mI ζ ⎜⎜⎜ m + ,1− ⎟⎟⎟ B ⎜⎜⎜ m + ,1− ⎟⎟⎟
⎝
n
n⎠ ⎝
n
n⎠

[12]

where Iζ and B are the regularized incomplete and complete beta
functions, respectively (Abramowitz and Stegun, 1972), andζ = Se1/m. Because f(1) = mB(m + 1/n,1 − 1/n), the general van
Genuchten–Mualem equation with independent m and n becomes

The Original van Genuchten–
Mualem Equations

2
⎡ ⎛
1
1 ⎞⎤
K r ( S e ) = S el ⎢ I ζ ⎜⎜⎜ m + ,1− ⎟⎟⎟⎥
⎢⎣ ⎝
n
n ⎠⎥⎦

We first briefly review the original approach followed by van
Genuchten (1980) and van Genuchten and Nielsen (1985).
Inverting Eq. [2] and substitution into Eq. [3] leads towhich holds for values of m and n restricted by m > 0 and n > 1.

⎡⎤2l ⎢ f ( Se ) ⎥K r ( Se ) = Se ⎢⎥⎢⎣ f (1) ⎥⎦
where

[5]

[13]

Equation [13] contains the regularized incomplete beta function,Iζ, evaluated by van Genuchten and Nielsen (1985) using a continued fraction approximation. They found the approximation
to require, in most cases, five terms in the expansion to obtainaccuracy in Kr of at least four significant digits. A few more terms
were recommended for relatively large values of the parameter m.

www.VadoseZoneJournal.org | 619

We revisited Eq. [7] and found an alternative formulation for Eq.
[12], in terms of the hypergeometric function F (Abramowitz andStegun, 1972, p. 558):

f (Se ) =mnSe1+(1/mn) F ⎡⎢m +(1 n),(1 n); m +(1 n)+1; Se1/m ⎤⎥
⎣
⎦
mn +1

[14]

with m > −1/n and n > 1. Equations [12] and [14] are completely
equivalent (cf., Abramowitz and Stegun, 1972, p. 263, Eq. 6.6.8) inthat for any r = m + 1/n > 0 (and hence m > −1/n) and s = 1 − 1/n> 0 (and hence n > 1):
rI ζ ( r , s ) B ( r , s ) = ζ r F ( r ,1− s; r +1; ζ )[15]where ζ = Se1/m.

and
∞⎛
k −1
1
n + i −1 ⎞⎟⎟⎟β = ( mn +1) ∑ ⎜⎜⎜∏⎜ mn + kn +1 i=1
i
⎠⎟
k=1⎝

[22]

Equations [21] and [22] were found to converge rapidly when
the parameter n is relatively large; however, convergence of especially Eq. [22] is very slow when n becomes less than about 2 or
3. The convergence of η(Se) given by Eq. [21] can be verified byd’Alembert’s ratio test (e.g., Rudin, 1976), which makes use of theparameter L as follows
ak+1
k→∞ akL = lim[23]

where ak+1 and ak are two consecutive terms of the infinite series
given by Eq. [21]. Substituting these into Eq. [23] gives

Abramowitz and Stegun (1972, p. 556, Eq. 15.1.1) presented thefollowing definition for the hypergeometric function:

L = lim S e1/ mk→∞( a )k ( b )k z k
F ( a, b; c ; z ) = ∑( c )k k !
k =0
∞

[16]

where (a)k, (b)k, and (c)k are Pochhammer symbols (Abramowitz
and Stegun, 1972, p. 256, Eq. 6.1.22) of the formk( x )k = x ( x +1)…( x + k −1) = ∏( x + i −1)

[17]

i =1

Substitution of Eq. [17] with x = a, x = b, and x = c in Eq. [16] yields

z k k ( a + i −1)( b + i −1)∏ ( c + i −1)
k=0 k ! i =1∞

F ( a, b; c ; z ) = ∑[18]( kn + mn + 2 ) kn + mn +1( kn + mn + 2n +1) kn + n( mn + n +1) [24]According to d’Alembert’s ratio test, a series converges absolutely
if L < 1 but does not converge if L > 1. If L = 1 or the limit fails toexist, then the test is inconclusive. In our case, we have 0 ≤ L < 1
for all 0 ≤ Se < 1, n > 1, and m > 0, which shows that η(Se) given
by Eq. [21] converges.
An analogous procedure for the series in Eq. [22] shows that L =1, which means that d’Alembert’s ratio test is inconclusive. In thiscase, Raabe’s test (Rudin, 1976) may be used to prove convergence.
Raabe’s R test states that if L = 1, and if⎛ a⎞⎟
R = lim k ⎜⎜⎜ k+1 −1⎟⎟ <− 1
k→∞ ⎜
⎝ ak
⎠⎟

[25]

and hence for n > 1
⎛
1 1
1
1m⎞
F ⎜⎜⎜ m + , ; m + +1; S e ⎟⎟⎟ =
⎝
⎠
n n
n
⎡
⎤⎡
⎤
∞
S ek/ m k ⎣⎢ m + (1 n ) + i −1⎦⎥ ⎣⎢(1 n ) + i −1⎦⎥∑∏m + (1 n ) + i
k=0 k ! i =1

then the series will converge. In our case, R is given by[19]

R = lim kk→∞kn + mn +1− 2 kn2 − n( mn + n +1)( kn + mn + 2n +1) kn + n( mn + n +1)

[26]

1
= −2
n

Substituting Eq. [19] into [14] and the result into Eq. [5] yields
the general solution

which shows that β given by Eq. [22] converges for n > 1 (tomake R < −1).

⎡ S ( l /2)+(1/ mn)+1 ⎡1 +η( S )⎤ ⎤ 2ee ⎦⎥⎣K r ( S e ) = ⎢⎢
⎥
1
+β
⎣⎢
⎦⎥

We calculated values of β for each combination of n = 1.1, 1.2,…, 10.0, 11, 12, …, 40 and m = 0.05, 0.10, …, 0.95. The numberof terms required for each combination was found to dependon the particular n and m values. Relatively small values of nand values of S e close to 1 in particular required a large numberof terms. As a criterion for convergence, we used the parameterδ β given by[20]( 0 ≤ Se ≤1, n > 1 and m > 0 )in which∞⎛k
S ek/ mn− 1 + i −1 ⎞⎟⎟η( S e ) = ( mn +1) ∑ ⎜⎜⎜∏ i ⎟⎟⎝ mn + kn +1 i=1
⎠
k=1⎜

[21]

www.VadoseZoneJournal.org | 620δβ =β( p )−β( p −1)[27]β( p )

where β(p) is the value of β after calculating 10p terms, with p being
an integer within the interval 0 ≤ p ≤ 12. When, after 10p terms,
δ β became <10−4 , evaluation of the series solution would be terminated. If no convergence was reached after 1012 terms (p = 12),the calculation was aborted and flagged as inconclusive.

which, when substituted into Eq. [28], gives
K r ( S e ) = S eλ I ζ ⎡⎢ m + ( 2 n ),1−( 2 n )⎤⎥⎣⎦Equation [34] can also be formulated in terms of the hypergeometric function as follows:g (Se ) =The Original van Genuchten–
Burdine Equations
The derivations of the original (restricted) and general van
Genuchten–Burdine expressions parallel those of the van
Genuchten–Mualem expressions. Inversion of Eq. [2] and substitution into Eq. [4] yields
λ g ( Se )K r ( Se ) = Se

[28]

g (1)

where

mnSe1+(2/mn) Fm +(2/ n),(2/ n); m +(2/ n)+1; Se1/m
mn + 2(n > 2)[36]

Substituting Eq. [36] into [28] and evaluating the hypergeometricfunctions analogously as shown for the Mualem case (Eq. [16–19])
yields the general van Genuchten–Burdine solution:

K r ( Se ) =S eλ+(2/ mn)+1 ⎡⎣1 +ζ( S e )⎤⎦1 +χ
( 0 ≤ Se ≤1, n > 2, and m > 0 )[37]

in which

1/ m ⎞λ / nSe ⎛ S⎟g ( S e ) = ∫ ⎜⎜⎜ e 1/ m ⎟⎟ d S e
0 ⎜ 1− S
⎝
⎠⎟
e

[29]

Using again the substitution y = Se1/m, Eq. [29] becomesg ( Se ) = m∫S e1/m0− 2/ ny m−1+(2/ n ) (1− y )

[30]

dy

Assuming that m is related to n by

m = 1−

[35]

2
n

[31]

allows the integral in Eq. [30] to be evaluated, resulting in
g ( S e ) = 1−(1− S e1/ m )

m

[32]

∞⎛
k
S ek/ m2 n− 1 + i −1 ⎞⎟⎟ζ( S e ) = ( mn + 2 ) ∑ ⎜⎜⎜∏⎟i⎝ mn + kn + 2 i=1
⎠⎟
k=1⎜

[38]

∞⎛
k
1
2n− 1 + i −1 ⎞⎟⎟⎟χ = ( mn + 2 ) ∑ ⎜⎜⎜∏i⎝ mn + kn + 2 i=1
⎠⎟
k=1⎜

[39]

Analogous to the case of η(S e) (Eq. [21]), it can be shown thatd’Alembert’s L parameter for ζ(Se) (Eq. [38]) equals Se1/m, whichindicates that Eq. [38] converges. For χ (Eq. [39]), L = 1 andRaabe’s R parameter becomes 2/n − 2; R is less than −1 for n > 2,
which shows that Eq. [39] then converges also.
Values of χ (Eq. [39]) were calculated for each combination ofn = 2.1, 2.2, …, 10.0, 11, 12, …, 40 and m = 0.05, 0.10, …, 0.95.
Analogously to the invoked procedure for β, we used as a criterionof convergence for χ the parameter δχ given byδχ =Substituting Eq. [32] into [28] yieldsm⎤⎡
K r ( S e ) = S eλ ⎢1−(1− S e1/ m ) ⎥ ( n > 2 )
⎢⎣
⎥⎦

[33]

General van Genuchten–Burdine Equations
with Independent m and n Values
Following van Genuchten and Nielsen (1985), direct integrationof Eq. [29] for independent values of m and n (restricted to m > 0
and n > 2) leads tog (Se ) = mIζ ⎢⎡m +(2 n),1−(2 n)⎥⎤ B ⎢⎡m +(2 n),1−(2 n)⎥⎤
⎣
⎦ ⎣
⎦

[34]

χ ( p )−χ ( p −1)
χ( p )[40]

where χ(p) is the value of χ after calculating 10p terms, with p
being an integer. The remaining calculations were similar to thosefor β as described above.

6 Results
Equations [20] and [37] are equivalent to the general van
Genuchten–Mualem and van Genuchten–Burdine expressions(Eq. [11] and [33], respectively) for the unsaturated soil hydraulic conductivity assuming independent m and n values. Generally,
only a few terms were needed when n was relatively large (n > 5).

www.VadoseZoneJournal.org | 621

The variables η (Eq. [21]) and ζ (Eq. [38]) were found to converge
within 1000 terms for 0 < Se ≤ 0.99 for all combinations of n andm analyzed, using as a criterion that the relative change in η or ζ
became <10−4 compared with the 10,000-terms estimate. The β(Eq. [22]) and χ parameters, however, converged very slowly, especially for values of n close to 1 and 2, respectively. As an example,
Fig. 1 shows the convergence of β for several combinations of n andm. Notice that an inordinate number of terms are needed whenn becomes relatively small (approaching the lower limit of 1.0).
Convergence of χ was found to be even slower.
Using an efficient Fortran routine running on a state-of-the art
microcomputer, it took >12 h to evaluate 1012 terms of Eq. [22]for each combination of m and n. For n ≤ 1.5, convergence of βevaluated with Eq. [22] and invoking the described criteria would
take even more terms (i.e., at least 1013 terms or >120 h of computing time); the corresponding values were therefore not calculated.
For 1.6 ≤ n ≤ 40 and 0.05 ≤ m ≤ 0.95, we evaluated β using 2185
combinations of m and n, and fitted to the results the equation
βi = a + bL( n; n0 , ν ) + cL( m; m0 , μ ) + dL( n; n0 , ν ) L( m; m0 , μ ) [41]where βi is the fitted approximation of β, L is the cumulative
Cauchy distribution function defined by⎛ f − f 0 ⎞⎟1 1
L( f ; f 0 , φ) = + arctan⎜⎜⎟
2 π⎝⎜ φ ⎠⎟⎟[42]

Fig. 1. Estimate of parameter β as a function of the number of terms
evaluated in Eq. [22].
Table 1. Values and standard errors of fitting parameters of Eq. [41] for
estimation of β (Eq. [22]) for 1.6 ≤ n ≤ 40 and 0.05 ≤ m ≤ 0.95.
Parameter

Value

SE

a

0.09389

b

16.92

1.688

c

−0.29230.06759d−30.777.103

0.01607n0

1.325

0.0032m0−0.3778

0.32359ν

−0.3958

0.00301μ−1.7900.2700and a, b, c, d, n0, m0, ν, and μ are adjustable parameters. Of several
alternative expressions we examined, Eq. [41] gave by far the best
fit to the 2185 points (R 2 = 0.9995; F = 6.0 × 105). Values andstandard errors of a, b, c, d, n0, m0, ν, and μ are listed in Table 1.

Table 2. Values and standard errors of fitting parameters of Eq. [43]for estimation of χ (Eq. [39]) for 3.1 ≤ n ≤ 40 and 0.05 ≤ m ≤ 0.95.

q

19.43

1.481

i for χ was derived using 1900 values
A similar approximation χof χ calculated with Eq. [39] for various combinations of m and n
across the interval 3.1 ≤ n ≤ 40 and 0.05 ≤ m ≤ 0.95:

r−0.3891

0.06589s−33.87

5.718

n1

2.561

0.0052m1−0.30470.22473τ−0.7068

0.00512ρ

−1.8210.2187i p + qL( n; n , τ ) + rL( m; m , ρ ) + sL( n; n , τ ) L( m; m , ρ ) [43]
χ=
1
1
1
1

where L is defined by Eq. [42] as before. Values and standard errorsof the parameters p, q, r, s, n1, m1, ρ, and τ for the best fit (R 2 =0.9997; F = 8.9 × 105) are shown in Table 2.
Together with a computer routine to evaluate the relatively quickly
converging η (Eq. [21]) or ζ (Eq. [38]) (we suggest 1000 terms fora <0.01% error), Eq. [41] or [43] can be used to estimate the slowly
converging β or χ functions, respectively, thus allowing calculation of Kr for any combination of the n and m parameters with an
insignificant computing time. Figure 2 shows an example of Kr
calculated as a function of Se for n = 4, l = 0.5, and λ = 2, assuming
five values for m: 0.05, 0.25, 0.5, 0.75, and 0.95. For the Mualem
case, m = 0.75 corresponds to Eq. [11] with m = 1 − 1/n. For the
Burdine calculations, m = 0.5 corresponds to Eq. [33] with m =Parameterp

Value
0.1282SE
0.01706

1 − 2/n. Values calculated with these equations (thick lines in Fig.
2) showed an essentially perfect match to those obtained with theapproximate equations.

6 Concluding Remarks
The general expressions of van Genuchten and Nielsen (1985) for
predicting the unsaturated hydraulic conductivity based on thetheories of Burdine (1953) and Mualem (1976) can be formulated
in terms of hypergeometric functions. The hypergeometric functions can be approximated by infinite series, which converge very

www.VadoseZoneJournal.org | 622

Fig. 2. Relative hydraulic conductivity (Kr) as a function of the effective saturation (Se) evaluated for n = 4, l = 0.5, and λ = 2 by the
Mualem Eq. [20] and the Burdine Eq. [37] (thin lines) and for m =0.05, 0.25, 0.50, 0.75, and 0.95 by Eq. [11] and [33] (thick lines).

slowly when the parameter n is close to 1 or 2 for the Mualem
and Burdine solutions, respectively. Alternative equations may be
used to obtain very close approximations of the infinite series forselected values of the van Genuchten m and n parameters, thus
allowing the use of the general expressions without a need for timeconsuming computational processes.
The newly proposed equations permit the use of independent
values of the parameters m and n in the soil water retention modelof van Genuchten (1980) for subsequent prediction of the van
Genuchten–Mualem and van Genuchten–Burdine hydraulicconductivity models. Thus, more flexibility is allowed in fitting
experimental θ(h) and K(h) or K(θ) data.

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