Received: 19 January 2024

Accepted: 7 March 2024

DOI: 10.1002/vzj2.20332
Vadose Zone Journal

TECHNICAL NOTE
S p e c i a l S e c t i o n : Tr i b u t e t o R i e n v a n G e n u c h t e n , R e c i p i e n t o f t h e 2 0 2 3 Wo l f P r i z e f o r A g r i c u l t u r e

Comment on “Modified expression for unsaturated hydraulicconductivity according to Mualem–van Genuchten to allow
proper computations at low pressure heads” by M. Heinen
Marcia S. Batalha1
Camila R. Bezerra-Coelho2
Martinus Th van Genuchten1,3
Jian Su1

Elizabeth M. Pontedeiro1,3

1 Department of Nuclear Engineering, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil
2 Department of Civil Engineering, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil
3 Department of Earth Sciences, Utrecht University, Utrecht, The Netherlands

Correspondence
Marcia S. Batalha, Department of Nuclear
Engineering, Federal University of Rio deJaneiro, Ilha do Fundão, Rio de Janeiro,
Brazil.
Email: marciabatalha.nunes@gmail.com
Assigned to Associate Editor Jirka Simunek.
Funding information
FAPERJ (Fundação Carlos Chagas Filho de
Amparo à Pesquisa do Estado do Rio deJaneiro), Brazil, Grant/Award Number:SEI-260003/003011/2023

Abstract
This comment concerns evaluation of the Mualem–van Genuchten (MvG) unsaturated soil hydraulic functions at very low (negative) pressure heads as describedrecently in an important study by Heinen. He showed that below some critical pressure head, the unsaturated hydraulic conductivity should be approximated by a powerfunction, even when evaluated using double precision computations. We show thata more precise approximation is possible when the approximation is formulated interms of effective fluid saturation (Se ) rather than the pressure head, h. More general
constraints are also provided when the approximated hydraulic conductivity equation should be used. The alternative formulation and constraints were implemented
earlier in the SOHYP and RETC software packages.

Heinen (2023) recently showed that the Mualem–van
Genuchten (MvG) unsaturated soil hydraulic conductivity(van Genuchten, 1980) at very low (negative) pressure heads
should be evaluated by means of an approximate powerfunction of the soil water pressure head. The MvG functions for soil water retention and the unsaturated hydraulicconductivity are given by:
𝑆𝑒 (ℎ) =𝜃 (ℎ) − 𝜃𝑟1=
𝜃𝑠 − 𝜃𝑟[1 + |𝛼ℎ|𝑛 ]𝑚Abbreviations: BiMvG, bimodal van Mualem–van Genuchten; MvG,
Mualem–van Genuchten.

(1)

[
(
( )
)𝑚 ]2
𝐾 𝑆𝑒 = 𝐾𝑠 𝑆𝑒 𝜆 1 − 1 − 𝑆𝑒 1∕𝑚(2)

respectively, where θ is the volumetric water content(cm3 /cm3 ), θs and θr denote the saturated and residual watercontents (cm3 /cm3 ), respectively, Se is effective water saturation (-), h is the pressure head (cm), K is the hydraulicconductivity (cm/day) given here as a function of Se , Ks is thesaturated hydraulic conductivity (cm/day), and α (cm−1 ), n (), m (-), and λ (-) are quasi-empirical shape parameters, with
m = 1 − 1/n. Heinen showed that direct evaluation of Equation (2) produces inaccuracies at very low negative valuesof the pressure head h, even with double precision calculations. To avoid these problems, he recommended to use an

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© 2024 The Authors. Vadose Zone Journal published by Wiley Periodicals LLC on behalf of Soil Science Society of America.
Vadose Zone J. 2024;e20332.
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approximate log-linear approximation of K(h) at low pressure
heads, which he derived by reformulating Equation (2) into
the form:[𝑛 −𝑚𝜆1 − (1 −𝐾 (ℎ) = 𝐾𝑠 (1 + |𝛼ℎ| )Core Ideas∙ Accurate evaluation of the Mualem–van
Genuchten (MvG) unsaturated hydraulic conductivity in very dry soils sometimes require someapproximation for certain hydraulic parametersets.
∙ The approximation is shown more accurate when
formulated in terms of effective fluid saturation, Se ,rather than the pressure head, h, as used by Heinen(2023).
∙ More flexible constraints are also possible when
the Se –based approximation is to be used.

]2|𝛼ℎ|𝑛−1(1 + |𝛼ℎ|𝑛 )𝑚(3)

which, using his notation, can be written as[
𝐾 (ℎ) = 𝐾𝑠 𝑠

−𝑚𝜆

(

𝑠−1
1−
𝑠

)𝑚 ]2

= 𝐾𝑠 𝑠−𝑚𝜆 (1 − 𝑥𝑚 )2 (4)where𝑠 = 1 + |𝛼ℎ|𝑛

and

𝑥=1−

1
𝑠

(5)

Since x is very close to 1.0 when |ℎ| becomes large, and mis always positive (0 < m < 1), the term xm in Equation (4)can be approximated by a Taylor series expansion around x =1 to give
1 − 𝑥𝑚 = 1 − [1 − 𝑚(1 − 𝑥)] + ⋯ ≈ 𝑚(1 − 𝑥)(6)

Genuchten (1978) and later implemented also in the RETC
software package of van Genuchten et al. (1991). The earlier
approach was based directly on Equations (1) and (2). Whena soil or other porous medium dries out nearly completely,|ℎ|→∞ and Se →0. For very small Se values, one may approximate the term (1 − Se 1/m )m in Equation (2) also in terms ofa Taylor series expansion around 1.0 to give

which, with s ≈ |𝛼ℎ|𝑛 for large values of |ℎ|, simplifies
Equation (3) to𝐾 (ℎ) = 𝐾𝑠 𝑚2 |𝛼ℎ|−(2+𝑚𝜆)𝑛(7)Heinen evaluated this equation for a coarse sand (his sample 10134) whose MvG parameters are listed in Table 1. Hesuggested that quadruple precision may be needed to properly
evaluate Equations (2) or (3) for very low negative pressure
heads (i.e., large values of |ℎ|), unless Equation (7) is used or
special formulations are implemented. Results are shown in
Figure 1 of Heinen (2023).
Our concern is that Equation (7) is not as accurate as
approximations used earlier in the SOHYP code of van

1(1 − 𝑥)𝑚 = 1 − 𝑚𝑥 + 𝑚(𝑚 − 1)𝑥2 − …
2(8)

where x is now given by x = Se 1/m . Using only the first two
terms gives)𝑚(
1 − 𝑆𝑒 1∕𝑚 ≈ 1 − 𝑚𝑆𝑒 1∕𝑚(9)in which case K(Se ) becomes[]2
𝐾(𝑆𝑒 ) = 𝐾𝑠 𝑆𝑒 𝜆 𝑚𝑆𝑒 1∕𝑚 = 𝐾𝑠 𝑚2 𝑆𝑒 𝜆+2∕𝑚(10)Since for large values of |ℎ| one has also

T A B L E 1 Parameter values of the unimodal (MvG) and bimodal (BiMvG) van Genuchten–Mualem unsaturated hydraulic functions used inthis comment.

Parameter

Unit

Coarse sand 10134

Soil 2Mining slag

MvG

MvG

BiMvG

θr

3

cm /cm

0.035390.035390.031

θs

cm3 /cm3

0.36683

0.36683

0.390

α (or α1 )1/cm

0.02135

0.021350.0406n (or n1 )–7.2372

1.050

15.0

3

λ

–

Ks

cm/day0.00010.0001−0.555101.3839

101.3839

40.0

w1

–

–

–

0.666

α2

1/cm

–

–

0.0308

n2

–

–

–

3.455

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3 of 6F I G U R E 1 HYPROP observed data and fitted BiMvG hydraulic properties of a Brazilian mining slag. The hydraulic conductivity in the lower
figures is plotted both versus pF = log |ℎ| and the volumetric water content.

𝑆𝑒 = [

11
1−𝑛] ≈𝑚𝑛 = |𝛼ℎ|𝑛 𝑚|𝛼ℎ|1 + |𝛼ℎ|this would reduce Equation (10) to Equation (7) as derived
by Heinen. We will show below that Equation (10) is moreaccurate than Equation (6), mostly because it avoids theapproximation given by Equation (11). Equation (10) is alsoeasier to implement than the methodology worked out by
Heinen involving critical pressure heads below which herecommended to use Equation (7).
Heinen assumed that the critical pressure heads below
which the power function should be used are those where
deviations of 1% or 5% would occur between the Equations (2)and (7). He showed that these pressure heads very muchdepend on the value of n (or m = 1 − 1/n) in the conductivity expression, Equation (2). Based on an empiricalevaluation of a large database of hydraulic properties, including especially data from the soils catalogue of Mualem (1976),
van Genuchten (1978) proposed alternative constraints forEquation (10):𝑆𝑒 < 0.06
or

1 − (1 − 𝑆𝑒 1∕𝑚 )𝑚 < 5 × 10−5

(11)

(12a)

(12b)

Equation (12a) was found to be the best controlling condition for using Equation (10) rather than Equation (2) directly,
when n is relatively large, typical of soils having a relatively narrow pore size distribution such as the example usedby Heinen when n = 7.2372. This is shown by comparing
deviations (DEVs) between the exact hydraulic conductivity equation and the approximations. DEV is defined here
as the percentage deviation of the invoked approximations,that is, Equation (7) of Heinen (2023) or Equation (10) of
van Genuchten (1978), from the exact equation given by
Equation (2), that is:[
DEV = 100]Equation (7) or Equation (10)−1Equation (2)(13)

Hydraulic conductivities given by Equation (2) were calculated with elevated precision using Wolfram Mathematica(2021) version 12.3.1. The results in Table 2 show that for
Heinen’s soil sample 10134 the DEVs at around h = −77.3 cm
are about +5% when Equation (7) is used, while Equation (10)gives DEV values of less than −0.4%. Similarly, DEV values
at around h = −96.5 cm are about +1% using Equation (7), but

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BATALHA ET AL.

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T A B L E 2 Values of the pressure head, h, where deviations (DEVs) of +5% and +1% occur between Equations (7) and (2), and the deviations
between Equations (10) and (2) at those same pressure heads. Also listed are DEV values at pressure heads when constraints (12a) and (12b) arereached. Results are for Heinen’s Soil 10134, for Soil 2 having the same properties as Soil 10134 but a much lower n value, and for a Brazilian
mining slag having bimodal soil hydraulic functions. Only the parameters of the macropore part of the hydraulic properties of the mining slag are
given (i.e., using the first terms of Equations (14) and (15) with w1 = 1.0 and w2 = 0.0, but keeping θr , θs , and Ks at their values listed in Table 1).
n (-) or n1 ,n2 (-)h(cm)

Se (-)

𝟏 − (𝟏 − 𝑺𝒆 𝟏∕𝒎 )𝒎(Equation 12b)K
(cm/day)

DEV (%)

Material

Equation (7)Equation (10)Soil 10134

7.2372

−73.1

0.060

0.0330

0.110

+7.5

−0.534

Soil 10134

7.2372

−77.3

0.043

0.02230.0505

+5.0

−0.36

Soil 10134

7.2372−96.5

0.01100.00459

0.00214+1.0−0.074

Soil 10134

7.2372−180.30.000220.000050

2.53 × 10−7

+0.0011−0.0008

Soil 2

1.05−853.00.863

0.00221

4.94 × 10−4

+5.0

−4.41

Soil 2

1.05−3.94 × 103

0.801

0.000452

2.07 × 10−5

+1.0−0.898

Soil 2

1.05−3.22 × 104

0.721

0.000050

2.53 × 10−7

+0.01

−0.01

Soil 2

1.05≈ −1030

0.060

0.0

0.0

–

–

Mining slag

15.0

−30.010.060

0.04590.4018+9.7−0.3

Mining slag

15.0

−30.790.043

0.03170.2314+5.0−0.2

Mining slag

15.0

−34.270.00970.00650.0223

+1.0

−0.04

0.000050

1.63 × 10−50.008

−0.0004

Mining slag

15.0

−47.450.0001less than −0.1% when Equation (10) is used as the approximate equation at low pressure heads. We selected thesepressure heads (−77.3 and −96.5 cm) since they were usedby Heinen when the deviations were 1% and 5% using his
approach. Also included in Table 2 are the results for conditions (12a) and (12b). Please note that Equation (7) always
leads to K(h) values that are higher than the exact equation,
while Equation (10) leads to much smaller deviations and are
below those calculated with Equation (2). No double precision computations were used for the approximate and all otherequations.
When using condition (12b), the suggested change from
Equation (2) to Equation (10) occurred at about h =−180.3 cm (Table 2). Deviations with Equation (2) are then
only about 0.001% for both Heinen’s Equation (7) and van
Genuchten’s Equation (10). This shows that both equations
can be used when condition (12a) is followed, which implies
that Heinen’s suggestion to estimate pressure heads where thedeviations would become less than 1% and 5% is not needed
anymore.
The above results for condition (12a) appropriate for relatively high n values. Condition (12b) on the other hand isimportant when n is small and approaches its lower limitof 1.0. This we tested by considering a soil with the same
hydraulic properties as used by Heinen, except for assumingn to be 1.05 rather than 7.2372 (Soil 2 in Table 1). Valuesof n close to 1.0 are typical of soils having relatively broadpore- or particle-size distributions. For Soil 2, condition (12b)is reached when h is about −32,200 cm, even though Se isthen still 0.721. Absolute DEV values are now less than 0.01%

using both Heinen’s Equation (6) and van Genuchten’s Equation (9). This is in sharp contrast to Heinen’s soil, whichreached condition (12a) already at a pressure head of about−77.1 cm. Using Heinen’s 1% and 5% DEV values, theseoccurred for Soil 2 at pressure heads of about −853 and−3940 cm, respectively, while they were much lower (onlyabout 0.01%) when condition (12b) is used.
We note that conditions (12a) and (12b) were formulated
in the late 1970s when the computational time on a laptop
computer was often an important issue. Laptop computers atpresent are much faster, which would allow one to make theconditions more stringent, also with single precision computations. For example, using cutoff values of 0.04 and 10−8 cm
in Equations (12a) and (12b), respectively, would reduce theabsolute DEV values to far less than 0.001% using both
Equations (7) and (10).
Still other provisions or constraints are desirable, such as
those implemented in the RETC software where the upper
limit of n was fixed at 15 and the unsaturated hydraulicconductivity was set at zero when reaching values below
10−30 cm/day. Limiting the hydraulic parameter values isimportant in some cases. One such example we encounteredwhen using the dual-porosity (bimodal) descriptions of thehydraulic properties to obtain more flexibility in describing
observed data. Within the MvG context, those bimodal functions are given by (Durner, 1994; Priesack & Durner, 2006):
𝑆𝑒 (ℎ) =𝑤1
𝑤2
𝜃 (ℎ) − 𝜃𝑟=[
]𝑚1 + []𝑚𝑛
𝜃𝑠 − 𝜃𝑟
1 + |𝛼1 ℎ| 1
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[
[
(
(
(
)𝑙 {
)𝑚 ]
)𝑚 ]}2
𝑤1 𝛼1 1 − 1 − 𝑆1 1∕𝑚1 1 + 𝑤2 𝛼2 1 − 1 − 𝑆2 1∕𝑚2 2
𝑤 1 𝑆 1 + 𝑤2 𝑆 2( )𝐾 𝑆𝑒 = 𝐾𝑠()2
𝑤1 𝛼1 + 𝑤2 𝛼2where wi (i = 1, 2) are weight factors associated with themacropore and micropore regions (with w1 + w2 = 1), andαi , ni , and mi (i = 1, 2) are shape parameters for the bimodal
model, with mi = 1 − 1/ni .
We applied the bimodal formulation to HYPROP measured hydraulic property of a Brazilian mining residue, which
had been subjected to high-temperature (pyrometallurgical)
mineral extraction. Disposing of the residues required asafety assessment in Brazil for which hydraulic properties
were needed. Figure 1 shows the excellent fit we obtainedwith the dual-porosity (bimodal van Mualem–van Genuchten[BiMvG]) formulation, which was slightly better than theclassical MvG model. Table 1 lists the unsaturated parameterswith the BiMvG model. The saturated hydraulic conductivity was fixed at 40 cm/day as estimated with independent
field infiltration experiments. The BiMvG model fit using
HYPROP-Fit (Pertassek et al., 2015) produced the imposed
limiting value of 15.0 for the exponent n1 of the macropore
region, described by the first part of the second term withinbrackets of Equation (14). This suggests that evaluation ofK(Se ) or K(h) requires careful programming of the type used
for RETC. Similar concerns arise when different hydraulicmodels (e.g., Seki et al., 2023) are combined to obtain more
flexibility in describing observed unsaturated soil hydraulicdata.
Because of the large value of n, the retention curve of
the mining slag is very steep and approximate equations will
generate similar and close approximations to the exact curvealready at relatively low pressure heads. Our main concern
now is the first term of Equation (15). Assuming w1 = 1
and w2 = 0, the +1% and +5% Heinen deviations (DEV values) occurred at about −30.8 and −34.3 cm, respectively.
By comparison, the MvG deviations were then less than
−0.2% and -0.04%, respectively. These differences in the deviations are not significant when used at lower pressure heads,
such as when condition (12b) is invoked, in which case his about −47 cm and K(h) about 10−5 cm/day (Table 2).
The RETC code assumes applicability of Equation (10) until
10−30 cm/day (which in this case occurred already at a pressure head of about −310 cm), after which K is assumed to be
zero.
As discussed well by Heinen, combining Equation (7) orEquation (10) with Equation (2) will produce small discontinuities in the calculated unsaturated hydraulic conductivity at

5 of 6(15)

the critical pressure heads or fluid saturation values where theapproximate equations are first used. Heinen overcame thiscontinuity by shifting Equation (7) at the critical pressure headto the value obtained there with Equation (2), leading to Equation (21) of Heinen (2023). One could make a very similar
shift when using Equation (9). However, since the DEV values of Equation (10) are much lower (often by a factor of morethan 10), we believe that such a shift is not needed when Equation (10) is used, implying that a very minimal discontinuityin K(h) or K(Se ) is acceptable for most or all applications.
The above analyses show that Equation (10) from van
Genuchten (1978) is preferred over Equation (7) from Heinen(2024), especially for soil hydraulic properties where n is relatively close to the lower limit of 1.0, in which case condition(12b) is important. The differences with Heinen’s equation are
far less at higher n values. Application of conditions (12a,b) issufficient for all cases we looked at, while placing some limiting values on n and K(h) (e.g., 15 and 10−30 cm/day in RETC)
is also important.
We conclude that one should be aware that for some soils
at very dry conditions (either at very low pressure heads hor very low degrees of saturation Se ), floating point imperfections in computer codes can occur for the Mualem-based
K(h) and K(Se ) expressions. Proper approximations have been
proposed by Heinen (2023) for K(h) and by van Genuchten(1978) for K(Se ). Depending on the type of computations, one
can clearly use either approach. In general, the van Genuchten
K(Se ) approximation is preferred since it has negligible deviations in the dry moisture range below the critical pressure
head or fluid saturation points, as opposed to Heinen’s K(h)approximation.
AU T H O R C O N T R I B U T I O N S
Marcia Batalha: Data curation; writing—review and editing.
Camila R. Bezerra-Coelho: Data curation; writing—review
and editing. Elizabeth M. Pontedeiro: Formal analysis;
methodology; writing—review and editing. Martinus Th
van Genuchten: Writing—review and editing. Jian Su:
Supervision; writing—review and editing.
AC K N OW L E D G M E N T S
This study was financed in part by FAPERJ (Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio deJaneiro, Processo SEI-260003/003011/2023) in Brazil.

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C O N F L I C T O F I N T E R E ST STAT E M E N T
The authors declare no conflicts of interest.
ORCID
Marcia S. Batalha https://orcid.org/0000-0002-3725-6099
Martinus Th van Genuchten https://orcid.org/0000-00031654-8858
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Heinen, M. (2023). Modified expression for hydraulic conductivity
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10.1002/vzj2.20279
Mualem, Y. (1976). A catalogue of the hydraulic properties of unsaturated soils (Research Project no. 442). Technion, Israel Institute ofTechnology.
Pertassek, T., Peters, A., & Durner, W. (2015). HYPROP-FIT softwareuser’s manual, version 3.0. UMS GmbH.
Priesack, E., & Durner, W. (2006). Closed-form expression for the multimodal unsaturated conductivity function. Vadose Zone Journal, 5(1),121–124. https://doi.org/10.2136/vzj2005.0066
Seki, K., Toride, N., & van Genuchten, M. Th. (2023). Evaluation of ageneral model for multimodal unsaturated soil hydraulic properties.
Jourmal of Hydrology and Hydromechanics, 71(1), 22–34. https://doi.org/10.2478/johh-2022-0039

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van Genuchten, R. (1978). Calculating the unsaturated hydraulicconductivity with a new, closed-form analytical model (Research
Report no. 78WR8). Water Resources Program, Department of Civil
Engineering, Princeton University. https://www.pc-progress.com/
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van Genuchten, M. Th., Leij, F. J., & Yates, S. R. (1991). The RETC
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03615995004400050002x
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How to cite this article: Batalha, M. S.,
Bezerra-Coelho, C. R., Pontedeiro, E. M., vanGenuchten, M. Th., & Su, J. (2024). Comment on“Modified expression for unsaturated hydraulicconductivity according to Mualem–van Genuchten to
allow proper computations at low pressure heads” by
M. Heinen. Vadose Zone Journal, e20332.
https://doi.org/10.1002/vzj2.20332

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