Soil Physics Note

Analysis of the Role of Tortuosity and Infiltration
Constants in the Beerkan Method
Paolo Nasta*

Dep. of Land, Air and Water
Resources
Univ. of California
Davis, CA 95618

Laurent Lassabatere

Université de Lyon
Laboratoire d’Ecologie desHydrosystèmes Naturels et
Anthropisés
UMR5023 CNRS-ENTPEUniversité Lyon 1
3, rue Maurice Audin
69518 Vaulx-en-Velin, France

Maziar M. Kandelous

Dep. of Land, Air and Water
Resources
Univ. of California
Davis, CA 95618

Jirka Šimůnek

Dep. of Environmental Sciences
Univ. of California
Riverside, CA 92521

Rafael Angulo-Jaramillo

Université de Lyon
Laboratoire d’Ecologie desHydrosystèmes Naturels et
Anthropisés
UMR 5023 CNRS-ENTPEUniversité Lyon 1
3, rue Maurice Audin
69518 Vaulx-en-Velin, France

It has recently been proposed to couple the Beerkan method with the Beerkan
Estimation of Soil Transfer parameters (BEST) algorithm to facilitate the estimation of soil hydraulic parameters from an infiltration experiment. Although thissimplified field procedure is relatively rapid and inexpensive, it has been doubted if the Beerkan method can represent a valid and reliable alternative to otherconventional methods. This study explored the impact of the tortuosity parameter (p) and two infiltration constants (b and g) included in the BEST algorithm
using a sensitivity analysis applied to three experimental soils. The analysis thatwas validated using the numerical model HYDRUS 2D/3D indicates that the
tortuosity is relatively insignificant compared to parameters b and g that have alarge impact on the estimation procedure.
Abbreviations: BC, Brooks and Corey; HCF, hydraulic conductivity function; PSD, particle size
distribution; PTF, pedotransfer function; VG, van Genuchten; WRF, water retention function.

Modeling soil hydrologic processes at hillslope or catchment scales requires knowledge of the soil hydraulic parameters for land surface layers that manifest a significant spatial variability (Ritter et al., 2003).
Assessing these parameters from laboratory experiments on soil cores is often costly
and time consuming (Oliver and Smettem, 2005). Moreover, because of the usually small size of soil cores, laboratory experiments can also have the drawback ofidentifying properties not fully representative of the effective soil hydraulic properties that control hydrologic processes at much larger spatial scales (Minasny and
McBratney, 2002). For these reasons, the use of field methods seems more attractive in determining the appropriate soil hydraulic parameters (Reynolds et al.,
2002; Angulo-Jaramillo et al., 2000). In general, field methods are analyzed using either analytical or numerical solutions. In the former case, however, in which
only the steady-state flow rates are used, the time when steady-state conditions arereached is uncertain and this may lead to unreliable estimates of the soil hydraulic properties. In the latter case, the numerical inverse solution requires additional
information, such as supplementary measurements of pressure heads or soil watercontents, to ensure the parameter uniqueness (e.g., Hopmans et al., 2002).
To overcome these problems, the Beerkan method (Braud et al., 2005) was introduced to offer an alternative for estimating soil hydraulic properties while relying
on different information obtainable by simple, fast, and cheap measurements (Lassabatere et al., 2006; Mubarak et al., 2009a, 2009b; Lassabatere et al., 2010; Yilmaz etal., 2010; Xu et al., 2012). This technique is based on an infiltration experiment andadditional measurements of the dry bulk density, initial and saturated volumetricwater contents, and the particle size distribution (PSD). The soil hydraulic properties are then estimated using the BEST algorithm (described below). Infiltration
Soil Sci. Soc. Am. J. 76:1999–2005
doi:10.2136/sssaj2012.0117n
Received 2 Apr. 2012.
*Corresponding author (paolo.nasta@unina.it).
© Soil Science Society of America, 5585 Guilford Rd., Madison WI 53711 USA
All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by
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and retrieval system, without permission in writing from the publisher. Permission for printing and forreprinting the material contained herein has been obtained by the publisher.

Soil Science Society of America Journal

into the soil surface horizon is assumed not to be considerably affected by preferential flow and underlying layers of the soil profile.
The BEST algorithm is based on a series of analytical expressions to calculate the soil hydraulic parameters from theaforementioned measurements. First, the shape parameters of
the water retention function are estimated using a pedotransfer
function (PTF) from the PSD and the dry bulk density, whereas
the shape parameter of the hydraulic conductivity function depends on an unknown tortuosity parameter, p. Second, the remaining scale parameters of the water retention and hydraulicconductivity functions are determined from the analysis of measured infiltration data. Infiltration data are used in the analyticalexpression that describes the three-dimensional cumulative infiltration proposed by Haverkamp et al. (1994); however, the two
infiltration constants (b and g) in the analytical model are notknown a priori. The objective of this study was to analyze the impact of p, b, and g on the results obtained by the Beerkan method
for three experimental soils previously investigated by Lassabatere et al. (2006). Moreover, the reliability and accuracy of theBEST algorithm, which is based on the combination of a PTF
and a series of analytical solutions, may be questioned. Hence,
the BEST-derived parameters were numerically validated using
the HYDRUS 2D/3D model (Šimůnek et al., 2008).

TheoryThe BEST Algorithm

In the BEST algorithm, the soil water retention function,q(h), referred to as the van Genuchten water retention function(VG-WRF), is described by van Genuchten’s equation (van Genuchten, 1980):
q ( h =) q r +qs - q r
 1+ ( a h )n 



m

[1a]

with Burdine’s condition (Burdine, 1953):
m = 1-

2n[1b]where a [L−1] is a scale parameter for the pressure head, h [L],which is equal to the reciprocal of the inflection point (hg ) of
the water retention curve (hg = 1/a), qr [L3 L−3] and qs [L3 L−3]
are the residual and saturated water contents, respectively, and n
and m are dimensionless retention curve shape parameters. The
unsaturated hydraulic conductivity function, K(q), referred toas the Brooks and Corey hydraulic conductivity function (BCHCF), is described by the relationship proposed by Brooks andCorey (1964):

 q -qr K ( q )= Ks  qs - q r h[2]

where Ks [L T−1] is the saturated hydraulic conductivity and his a dimensionless conductivity shape parameter that is related

2000

to the retention curve shape parameters through the following
equation, expressing the capillary model:

h=

2
+2+ p
mn

[3]

where p (dimensionless) is a tortuosity parameter that is assumed
to be equal to 0 (Childs and Collis-George, 1950), 0.5 (Mualem,1976), 1.0 (Burdine, 1953), or 1.33 (Millington and Quirk,
1961). In the original procedure of Lassabatere et al. (2006), p
was set according to Burdine’s condition (p = 1.0). In the BESTalgorithm, it is also assumed that the residual water content (qr)is equal to zero (Haverkamp et al., 2005; Leij et al., 2005).
The Beerkan method requires that an undisturbed soil core
is collected to measure the dry bulk density, rb [M L−3], theinitial volumetric water content, q0 [L3 L−3], the saturated volumetric water content, qs [L3 L−3], and the PSD. Subsequently,
the experimental cumulative infiltration is measured using an annular ring of radius rd [L], with a constant zero pressure head at
the soil surface. A series of known volumes of water is poured
into the cylinder and the time is recorded when the water has infiltrated into the soil. The BEST algorithm uses the experimental
cumulative infiltration, IEXP [L], expressed per unit area of theannular ring for each time, tk [T].
The BEST algorithm starts with a PTF, which is used toestimate the shape parameters (n, m, and h) from the PSD andrb measurements (Braud et al., 2005; Lassabatere et al., 2006).
Subsequently, the analytical solution for three-dimensional infiltration (IBEST ), valid for the transient state, is used (Haverkamp
et al., 1994):

I BEST ( t ) =S t +  A ( 1- B ) S 2 + Bq +∞  t

[4]

with

g

 A = r ( q -q )
d
s
0

h
h



 B =2 - b 1-  q 0   +  q 0 
 
 

3   qs    qs 




[5]

where S [L T−1/2] is the sorptivity, q+¥ [L T−1] is the steadystate infiltration rate, assumed to be the asymptotic slope ofthe last few experimental data points, b (dimensionless) adjusts
the relation between the water diffusivity and the unsaturated
hydraulic conductivity, and g (dimensionless) corrects the analytical three-dimensional infiltration equation (Eq. [4]) for theeffects of radial expansion (Fuentes et al., 1992; Haverkamp etal., 1994). The constant g is feasible in the range of 0.6 < g < 0.8(Haverkamp et al., 1994), while b was originally constrained in
the interval of 0 < b < 1 (Fuentes et al., 1992; Haverkamp et al.,
1994; Braud et al., 2005). Subsequently, Lassabatere et al. (2009)
considered a larger range of b (0 < b < 2) to evaluate its effecton the accuracy of the three-dimensional analytical solution withrespect to numerically generated three-dimensional cumulative
Soil Science Society of America Journalinfiltration for four synthetic soils. Lassabatere et al. (2006) used
the BEST algorithm, assuming that g = 0.75 and b = 0.60 for
three experimental soils. It is still necessary, however, to better
understand the effect of these infiltration constants, along withthe tortuosity parameter, p, on the estimates of the soil hydraulicparameters.
The sorptivity (Sk) is obtained by incrementally minimizing the sum of squared deviations between the experimental data
(cumulative infiltration, IEXP,k, at time tk) and the analytical solution given by Eq. [4] (cumulative infiltration, IBEST,k, at time tk)(Lassabatere et al., 2006). The selection of the final value of thesorptivity (S) from the series of Sk values depends on the maximum transient time, tmax(k) [T], which corresponds to the upper
limit of the validity interval of Eq. [4] (Lassabatere et al., 2006):
t max ( k ) =



Sk
2 
2 
4 ( 1- B )  q +∞ - AS k 
1

2

[6]

The final value of S is chosen for the largest experimental
time, tk, that fulfills the condition tmax(k) ³ tk. This condition
offers an empirical, but relatively robust, selection of data during
the transient phase of the cumulative infiltration curve. The saturated hydraulic conductivity, Ks, is then given by the analyticalsolution for the steady-state condition:K=s q +∞ - AS 2

[7]

Finally, the retention curve scale parameter, hg [L], is calculated usinghg = -S2hc p K s ( q s - q 0 )  1- ( q 0 q s ) 



[8]

where cp (dimensionless) is a texture-dependent parameter retrieved from the shape parameters of the soil hydraulic functions(Braud et al., 2005; Lassabatere et al., 2006).

Numerical Modelh ( r , z , t=) 0 t > 0, 0 < r < rd , =
z 0

[10b]

q ( r , z , t ) = 0 t > 0, r > rd , z = 0[10c]where z = 0 is the surface of the soil surface layer, and h0 is theuniform initial matric head in the soil layer derived from the measured initial water content, q0, by inverting Eq. [1a]. Infiltration
was modeled on the part of the soil surface delimited by the ring(with radius rd) using the zero pressure head boundary condition (Eq. [10b]), while the remaining area outside of the ring had
the zero water flux (q = 0) boundary condition (Eq. [10c]). Freedrainage was set at the bottom of the modeled transport domain.
Because HYDRUS 2D/3D does not support the combination of soil hydraulic properties given by Eq. [1] (VG-WRF)
and Eq. [2] (BC-HCF) as required by the BEST algorithm, its
Look-up Table option was adapted to provide the definition of
the soil hydraulic properties (Table 1) using an external text input file, Mater.in.
We emphasize that both the BEST algorithm and the HYDRUS 2D/3D model were set up assuming a homogenous soil
profile with negligible effects of heterogeneity. Moreover, the unimodal parametric relationships (Eq. [1] and [2]) adopted in theanalytical BEST algorithm are not able to capture the presence ofmacropores in the investigated porous medium (e.g., Kodešová etal., 2006; Akay et al., 2008). Bimodal forms of the soil hydraulicproperties would certainly be more suitable to describe possible
effects of macroporosity (e.g., Durner, 1994). Although HYDRUS 2D/3D is capable of considering the bimodal soil hydraulic model of Durner (1994), the analysis was performed using theunimodal parametric relations (Eq. [1] and [2]) to be consistent
with the constraints of the BEST algorithm.

Results and Discussion
The Beerkan infiltration experiments reported in Lassabatere
et al. (2006) were performed on three soils: Roujan (loam), Chernobyl (sand), and Django Reinhardt (sand). Table 1 provides the

Table 1. Original properties and parameters for the three experimental soils

The infiltration experiments were consequently (Lassabatere et al., 2006).
modeled using HYDRUS 2D/3D, which numerically solves the Richards equation describing water
Parameter
flow in a variably saturated, isotropic, rigid, porous Sand, %Silt, %
medium (Šimůnek et al., 2008):∂q 1 ∂  ∂h  ∂ ∂h  ∂K= rK
+  K ( h )  +∂t r ∂r  ∂r  ∂z ∂z  ∂z[9]

where r is the radial coordinate [L] and z is the verticalcoordinate [L], positive upward. The surface soil layer
was represented by a three-dimensional, axisymmetric,
finite-element mesh with both depth and radius of 300
mm. The initial and boundary conditions were:h ( r , z , t ) =h 0 t =0, r > 0, z < 0
www.soils.org/publications/sssaj[10a]Clay, %
Annular ring radius (rd), mm

Initial water content (q0), m3 m−3Saturated water content (qs), m3 m−3Retention curve shape parameter nRetention curve shape parameter m

RoujanChernobyl

DjangoReinhardt

40.57
43.73
15.70

91.17
8.81
0.02

98.28
1.71
0.01

75
0.030
0.330
2.20
0.089
13.20

75
0.063
0.314
2.97
0.327
5.06

100
0.050
0.400
2.65
0.246
6.07

−99.5

−61.8

−9.6

Conductivity shape parameter hSteady-state infiltration rate (q+¥), mm s−1 1.60 ´ 10−2 2.15 ´ 10−2 6.91 ´ 10−2Maximum transient time (tmax), s
12,7701768
93Sorptivity (S), mm s−0.5
0.579
0.507
0.630
Saturated hydraulic conductivity (Ks), mm s−1 4.79 ´ 10−3 1.03 ´ 10−2 6.33 ´ 10−2Retention curve scale parameter (hg), mm

2001

Fig. 1. The effect of infiltration constants b and g on the estimates of the maximum transient time, tmax, sorptivity, S, saturated hydraulic conductivity, Ks, and retention curve scale parameter, hg for Roujan, Chernobyl, and Django Reinhardt soils. Note that the tortuosity parameter is set toits original value (p = 1).

as a function of p is rather low, however, as evidenced by the low
coefficient of variation (Table 2).
For experimental conditions with initially dry soils, when q0<< qs and thus (q0/qs)h ? 0, the low impact of p can be explainedby the following approximation of the three-dimensional infiltration Eq. [4] and [5]:textural and experimental characteristics and originally estimatedhydraulic parameters that describe the surface soil layer (measured
PSDs were depicted in Lassabatere et al., 2006, Fig. 1).

Impact of Tortuosity and Infiltration Constantsin the BEST Algorithm
After estimation of the water retention shape parameters (nand m), the tortuosity parameter (p) was used to calculate thehydraulic conductivity shape parameter (h) in Eq. [3]. Table 2reports the basic statistical properties for the shape parameter h(mean, standard deviation, and coefficient of variation) that were
derived from the four different values of p (0, 0.5, 1.0, and 1.33).
Because p is additive in Eq. [3], its variability has more weight
for lower h values, hence for the sandy soils. The variability of hg
 2-b  2 2-b

I BEST ( t ) =+S t 
q +∞  t [11]
 1S +3 
3 rd ( q s - q 0 ) Equation [11] does not depend on h (and on the tortuosity, p),which thus has a negligible impact on the optimization of S andconsequently on the estimation of Ks (Eq. [7]) and hg (Eq. [8]).
For the scale parameter hg , however, we emphasize that the varia-

Table 2. Mean, standard deviation (SD), and coefficient of variation (CV) of the analyzed soil hydraulic parameters for the threeexperimental soils.
Parameter

Mean

Conductivity shape parameter hMaximum transient time (tmax), s

12.92

Retention curve scale parameter (hg), mm−119.2

Roujan
SD

CV

0.58

%
4.5

11,1072248
20.2
Sorptivity (S), mm s−0.5
0.605
0.031
5.1
Saturated hydraulic conductivity (Ks), mm s−1 4.62 ´ 10−3 9.69 ´ 10−4 20.9

2002

30.2

25.4

Chernobyl
SD

CV

Django Reinhardt
Mean
SD

CV

4.77

0.58

%
12.21

5.78

0.58

%
10.08

1586
0.538

317
0.040

77.6
0.629

33.4
0.036

Mean

19.9
7.4
1.08 ´ 10−2 1.60 ´ 10−3 14.8
−65.9
17.3
26.3

43.0
5.7
6.17 ´ 10−2 1.20 ´ 10−3 2.0
−10.1
1.3
12.7

Soil Science Society of America Journal

tion of the texture-dependent parameter cp as a function of p inits feasible range between 0 and 1.33 is not significant in Eq. [8].
For the reasons stated above, the BEST algorithm was run
for the three soils with only the original value of h (Table 1),
resulting from the assumption of p = 1 (Burdine’s condition), as
in Lassabatere et al. (2006), while varying the constant b in thefeasible range of 0 < b < 2 and the proportionality constant g in
the feasible range of 0.6 < g < 0.8. The results of these calculations for the parameters tmax, S, Ks, and hg are presented in Fig.
1. Table 2 reports the comprehensive descriptive statistics of thesensitivity analysis. The maximum transient time, tmax (Eq. [6]),was rather variable for all three soils and tended to increase fordecreasing values of b and increasing values of g. On the other
hand, the sorptivity showed the opposite trend as a function ofb and g compared with tmax and is characterized by the lowest
variability (see Table 2). A closer inspection of Eq. [7] reveals
the role of the term A (which includes the infiltration constantg in Eq. [5]), which affects the sorptivity under steady-state conditions and subsequently Ks.
The water retention scale parameter, hg , is influenced mainly by the parameter b in so far as its values show a clear horizontal
pattern that increases with decreasing b values. This observation
confirms the theoretical statements of Fuentes et al. (1992, Fig.
1). The Django Reinhardt soil, however, compared with the other two soils, exhibited only minor variations of tmax, S, Ks, andhg as a function of g because the infiltration process in this soil isdriven mainly by gravity and the radial correction of the infiltration had only very minor effects on the results. The parameterg corrects the simplified theoretical description of the parabolic
two-dimensional wetting front and the one-dimensional wetting

front along the soil profile (see Smettem et al., 1994, Fig. 1). On
the other hand, b influences the shape of the soil water diffusivity and, consequently, of the soil hydraulic curves (Fuentes et al.,1992). Therefore, we can attribute a geometric meaning to g anda structural meaning to b.

Numerical Validation
The original parameters reported in Table 1 were used in
HYDRUS 2D/3D to obtain the simulated cumulative infiltration curves (IHYDRUS) that are compared graphically in Fig.
2a with the corresponding experimental data (IEXP for N data
points at times tk) and numerically using the root mean squared
errors (RMSEs):
RMSE =∑N
k =1

 I HYDRUS ( t k ) - I EXP ( t k ) 
N

2

[12]

The RMSEs that are reported for each soil in Fig. 2a indicate good agreement between measured and simulated cumulative infiltration curves. Small deviations (reflected by low RMSE
values) imply that the BEST-derived soil hydraulic parameters
were able to numerically generate infiltration curves that are approximately similar to the experimental data.
Figure 2b shows the water content profiles for the three soils
at the initial time (t0), maximum transient time (tmax), and final
experimental time (tend). We note that tmax > tend for the Roujan
soil, which clearly indicates that the experiment was terminatedprematurely. This was also demonstrated by a significant radial
expansion of the moisture front outside of the ring, which im-

Fig. 2. (a) Infiltration curves simulated using HYDRUS 2D/3D (IHYDRUS) (lines) with parameters reported in Table 1 compared with the experimental data (IEXP) (symbols); (b) soil water content profiles simulated using HYDRUS 2D/3D at time 0 (t0), maximum transient time (tmax), and final
time of the experiment (tend) for Roujan, Chernobyl, and Django Reinhardt soils.
www.soils.org/publications/sssaj

2003

plies that capillary forces dominated over gravity during the infiltration experiment in this particular soil. On the contrary, we
can observe that water tended to mainly infiltrate in the verticaldirection after a short time in the coarser soils (i.e., Django Reinhardt and Chernobyl).
Finally, we numerically evaluated with HYDRUS 2D/3D
the results generated using the BEST algorithm for different
combinations of the parameters p, b, and g that determined thevariability of the following soil hydraulic parameters: h, hg , andKs. We note that the other BEST parameters (qs, n, and m) were
kept constant throughout the entire analysis. One numerical
HYDRUS 2D/3D simulation was performed for each set of soil
hydraulic parameters (h, hg , and Ks) estimated using the BESTalgorithm for the corresponding set of parameters (p, b, and g).
The RMSEs between modeled (IHYDRUS) and experimental (IEXP) cumulative infiltration data were calculated using Eq.
[12]. Figure 3 shows the relative deviations from the minimum
RMSE, RD (%), given by RMSE - min ( RMSE ) 
RD = 
min ( RMSE )[13]

considered values of the infiltration constants, i.e., g = 0.75 and b= 0.60, were not optimal for the three experimental soils; however, the inaccuracy caused by p, b, and g did not significantly affectthe final estimates of the soil hydraulic parameters (h, hg , andKs) with regard to numerical modeling because all the RD values
are <10%. The sensitivity analysis also confirmed that the impactof the tortuosity parameter p (varying in the feasible range from
0–1.33) on the final results was negligible.
In Fig. 3, the minimum RMSE values are located at different
positions in the subplots for each experimental soil. We therefore
believe that to further improve the final estimates by the BESTalgorithm and to reduce the uncertainty in the estimated parameters, a specific calibration of b and g for each soil type would
be required. We also note that the first part of the BEST algorithm, in which the shape parameters of the VG-WRF (n andm) are estimated from the PSD and the soil bulk density using a
PTF, was skipped in this study. The performance of this PTF has
been already successfully verified on large experimental databases(Minasny and McBratney, 2007).

Conclusion

The Beerkan method is an alternative technique to convenwhere min(RMSE) is the minimum RMSE across all the sets oftional laboratory or field measurements for rapid and low-cost
parameters (p, g, b). The stars in Fig. 3 indicate that the originallyestimation of soil hydraulic properties that is based on trustworthy and robust analytical solutions. In
this technical note, we have assessed thesensitivity analysis of the BEST algorithm
regarding the choice of tortuosity (p) andinfiltration constants (g and b) in theirfeasible range. The results demonstratethat tortuosity (p) plays only a secondaryrole compared with constants b and g,which are more important for the estimation of the scale parameters.
Numerical simulations performed
using HYDRUS 2D/3D with the soil
hydraulic parameters estimated by the
Beerkan method (Lassabatere et al.,2006) provided a good description ofexperimental cumulative infiltration
curves, indicating reliability of this technique. We offered an interpretation of
the role of the two infiltration constants,
highlighting the pros and cons that characterize the Beerkan method. The proper
calibration of these two constants, as afunction of the soil type, could significantly further improve the estimates ofthe soil hydraulic parameters.
We therefore conclude that, for soil
surface horizons not significantly affected
Fig. 3. Relative differences (in %, with respect to the minimum RMSE) as a function of infiltration by macroporosity or preferential flow, theconstants b and g and tortuosity parameter p (results are shown only for two limiting values of p soil hydraulic parameters derived using theand not for 0.5 and 1) for the Roujan, Chernobyl, and Django Reinhardt soils. The stars indicate thepositions corresponding to g = 0.75 and b = 0.60.
2004

Soil Science Society of America Journal

BEST algorithm can be utilized in numerical models to accurately
describe water infiltration.

Acknowledgments

Paolo Nasta would like to thank Dr. Massimo Nicolazzo and Dr.
William Piras for their technical support.

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