Click
Here

WATER RESOURCES RESEARCH, VOL. 45, W12415, doi:10.1029/2009WR007941, 2009

for

Full
ArticleNumerical evaluation of a set of analytical infiltration equations
L. Lassabatere,1 R. Angulo-Jaramillo,2,3 J. M. Soria-Ugalde,4 J. Šimůnek,5and R. Haverkamp3
Received 4 March 2009; revised 19 June 2009; accepted 19 August 2009; published 19 December 2009.

[1] In this paper, a set of analytical infiltration equations that are commonly used toevaluate one- and three-dimensional water infiltration from a surface disc source isstudied. Both the quasi-exact analytical formulation and the related approximations for
short and long times are assessed. The analytical properties of the quasi-exact formulation
are evaluated using a proposed scaling procedure in order to define the validity domains ofrelated approximations. Both quasi-exact and approximate analytical equations are thenstudied with respect to their ability to reproduce numerically generated cumulative
infiltrations from a 10 cm radius disc source for four soils (sand, loam, silt, and silty clay)
at several initial saturations. The quasi-exact formulation is suitable for sand, loam, andsilt when their soil-dependent and saturation-independent shape parameters, g and b, areproperly chosen (between 0.75 and 1 and 0.3 and 1.7, respectively). Approximations
derived for the same shape parameters can also be used, provided that their use is restricted
to proposed validity intervals. However, none of these equations applies for silty clay,
since its hydraulic properties do not fulfill the conditions required for the use of the quasiexact formulation.
Citation: Lassabatere, L., R. Angulo-Jaramillo, J. M. Soria-Ugalde, J. Šimůnek, and R. Haverkamp (2009), Numerical evaluation of
a set of analytical infiltration equations, Water Resour. Res., 45, W12415, doi:10.1029/2009WR007941.

1. Introduction[2] Modeling water fluxes in the vadose zone linking
surface waters with groundwater is important for understanding the hydrological cycle and the transfer of watertransported contaminants. Such modeling is usually basedon the Richards’ equation describing variably saturated
water flow, unless preferential flow is involved [e.g.,
Feddes et al., 1988]. In any case, such modeling requires
knowledge of soil hydraulic properties, such as the water
retention curve h(q) and the hydraulic conductivity function
K(q).
[3] Analysis of water infiltration experimental data has
become a widely used practice for obtaining soil hydraulicproperties [e.g., Perroux and White, 1988; Jacques et al.,2002]. Several experimental devices, often based on thetension disc infiltrometer or in situ lysimeters, have beendeveloped for this purpose. Collected infiltration data arethen inversely analyzed using either analytical or numerical models [e.g., Šimůnek and van Genuchten, 1996;
Mallants et al., 1997; Šimůnek et al., 1998; AnguloJaramillo et al., 2000]. Among many water infiltration
1

Division Eau et Environnement, LCPC, Bouguenais, France.
Laboratoire des Sciences de l’Environnement, Ecole Nationale desTravaux Publics de l’Etat, Vaulx-en-Velin, France.
3
Laboratoire d’Étude des Transferts en Hydrologie et Environnement,
CNRS, Grenoble, France.
4
Departamento de Ingenierı́a Geomática e Hidráulica, Universidad deGuanajuato, Guanajuato, Mexico.
5
Department of Environmental Sciences, University of California,
Riverside, California, USA.
2

Copyright 2009 by the American Geophysical Union.
0043-1397/09/2009WR007941$09.00

experiments, the Beerkan method, consisting of water
infiltrating from a single infiltration ring at a zero pressure
head [Braud et al., 2005], proved to be a low cost androbust method for characterizing soil hydraulic properties,
even at the watershed scale [Haverkamp et al., 1996;
Galle et al., 2001; Braud et al., 2003]. Lassabatere et al.
[2006] proposed the ‘‘BEST’’ (Beerkan Estimation of Soil
Transfer parameters) algorithm for estimating the entire
set of unsaturated soil hydraulic properties using theinverse analysis of particle size distribution and the water
cumulative infiltration curve obtained by the Beerkan
method.
[4] Independent of the water infiltration device, there isgreat need for an accurate description of one- (1-D) andthree-dimensional (3-D) water infiltration from a surface
disc source. Several analytical solutions have been proposed
to provide either approximate [Braester, 1973; Salvucci
and Entekhabi, 1994] or exact [Green and Ampt, 1911;
Parlange et al., 1985; Basha, 1999, 2002; Zhu and Mohanty,
2002] solutions for 1-D water infiltration. However, most ofthese solutions are based on simplifying assumptions related
to soil hydraulic properties, such as constant soil water
diffusivity, or use inappropriate boundary conditions, suchas a constant infiltration flux. Based on the analytical modelfor 1-D ponded cumulative infiltration [Parlange et al.,
1985; Haverkamp et al., 1990], and its extension to three
dimensions for a surface disc source [Smetten et al., 1994],
Haverkamp et al. [1994] proposed a set of analyticalequations that were adapted for a constant water pressure
head at the soil surface (hsurf) and a uniform initial waterpressure head profile h0 (z) = h0. Such a model proved to beconsistent with infiltration data [Clausnitzer et al., 1998]. A
specific set was derived specifically for the Beerkan exper-

W12415

1 of 20

LASSABATÈRE ET AL.: EVALUATION INFILTRATION

W12415

iment, i.e., for hsurf = 0 and qsurf = qs, but can be generalized
for all other pressure heads, either negative or positive [Rosset al., 1996].
[5] The 3-D cumulative infiltration, I3D (t), is related to
the 1-D cumulative infiltration, I1D (t), by an additional term
linear with time [Smetten et al., 1994], usually written using
equation (4) of Haverkamp et al. [1994]:
g S2
t
I3D ðtÞ ¼ I1D ðtÞ þrd Dq

W12415analytical method is based on the hypothesis that functions
K(q) and D(q) are linked with the shape parameter b as
follows:
01

Zqs

ðq  q0 ÞDðqÞ C
B
dq C
B
f ðqÞ
B
C
K ðqÞ  K0
BCq
¼ f ðqÞB1  b qs
C
BCZ
Ks  K0
B
C
ð
q

q
Þ
D
ð
q
Þ
0
@
dqA
f ðqÞ

ð1Þ

ð4Þ

q0

where Dq ( = qs  q0) stands for the difference between thesaturated water content, qs, and the initial water content,
q0, t is time, rd is the radius of the disc source, g is a shapeparameter, and S stands for the sorptivity. The shapeparameter g is defined using the ratio between two
estimators of the sorptivity [Haverkamp et al., 1994],^ the first of which is more precise [Parlange, 1975]:
S and S,

for all water contents q 2 [q0, q1].
[7] To simplify the solution, Haverkamp et al. [1994]
derived the following direct time expansions, which can be
regarded as good approximations for very short times, shorttimes, and long times, respectively:
pﬃ
t

ð5aÞ



pﬃ
ð2  b Þ
DK þ K0 t
tþ
3

ð5bÞ

  2
1
1 S
ln
2ð1  b Þ
b DK

ð5cÞ

Oð1Þ

g¼

pﬃﬃﬃﬃﬃﬃﬃ ^2
S
0:3 2
S

I1D ðtÞ ¼ S
ð2aÞ
Oð2Þ

S2 ¼ 2Zqs

I1D ðt Þ ¼ S
ðq  qo ÞDðqÞ dq
f ðqÞ

ð2bÞ

q0
þ1
ðt Þ ¼ K s t þ
I1DS^2 ¼ 2 ðqs  qo Þ

ZqsDðqÞ dqð2cÞ

q0

where D stands for the soil diffusivity, which is definedusing the water retention curve h(q) and the hydraulicconductivity function K(q) as follows:DðqÞ ¼ K ðqÞ

dh
dq

ð2dÞ

f in (2) stands for the flux concentration function [Philip,
1973], which can be precisely estimated using [Parlange,
1975]:
f ðqÞ ¼

2ðq  q0 Þ
qs þ q  2 q0

Time derivatives of the quasi-exact formulation and of its
approximations provide corresponding models for infiltration rates.
[8] The use of this set of analytical formulations requires
adequate values for the shape parameters g and b. While intheir earlier work, Haverkamp et al. [1994] proposed anaveraged value of 0.75 for g and 0.6 for b, in their later
studies [Haverkamp et al., 1999, 2005] they considered theshape parameters as dependent on both the soil type and theinitial water content. On the basis of equations (2a) and (4),
Fuentes et al. [1992] proposed a direct formulation for the
shape parameters as a function of the initial water content,
q0, and hydraulic functions K(q) and D(q):
Zqs

ð2eÞ

K ðqÞ  K ðq0 Þ qs  q0DðqÞ dq
Ks  K ðq0 Þ q  q0

q

b ðq0 Þ ¼ 2  2 0[6] The 1-D cumulative infiltration equation can be derived
using the analytical method for solving the 1-D Richards
equation described by Haverkamp et al. [1990]. This leads
to a quasi-exact implicit formulation for the 1-D cumulative
infiltration [Haverkamp et al., 1994, equation (5)]:ð6aÞDðqÞ dq

q0


0 
13
DK
Bexp 2b S 2 ðI1D ðt Þ  K0 tÞ þ b  1C7
2 DK 2
1 6
2
DK
6
C7
t¼
ðI1D ðt Þ  K0 tÞ  lnB@A5b
1  b 4 S2
S2
22ðqs  q0 Þg ðq0 Þ ¼

pﬃﬃﬃﬃﬃﬃﬃ
0:3 Z qs

Z qsDðqÞ dqq0

ð6bÞðqs þ q  2q0 ÞDðqÞ dq

q0

ð3Þ

where DK (= Ks  K0) stands for the difference between
hydraulic conductivities at the soil surface, Ks, and at initialsaturations, K0 [ = K(q0)], and where b is defined as an‘‘integral’’ shape parameter [Haverkamp et al., 1994]. The

Zqs[9] The use of the approximate equations requires knowledge of their validity domains. Since no study has yet
focused on the proper definition of these time intervals,
there is a risk that these equations may be used for times forwhich they are no longer valid.

2 of 20

LASSABATÈRE ET AL.: EVALUATION INFILTRATION

W12415[10] The objective of this paper is thus to study this set of
equations, including both the quasi-exact formulation(equations (1) and (3)) and the related approximations(equation (5)) with respect to the optimal choice of the
shape parameters b and g, and their dependence on the soil
type and initial water content conditions, and the definition of
appropriate validity domains for the related approximations.
[11] First, we will define the scaled (dimensionless) onedimensional cumulative infiltration, I*(t*), and the scaled
difference between three- and one-dimensional (3-D –1-D)
cumulative infiltrations, DI*(t*). We will then show thatDI*(t*) is a linear function of the scaled time t*, using theshape parameter g as the proportionality constant. We will
also show that I*(t*) depends only on the shape parameterb. We will evaluate I*(t*) with respect to its approximate
formulations and their validity domains as a function of the
shape parameters. The scaling procedure will allow us totransform these dimensionless intervals into real time intervals for both 1-D and 3-D cumulative infiltrations. A similar
approach will be used for infiltration rates.
[12] In the second step, the quasi-exact formulation andits approximations will be tested against numerically generated 1-D and 3-D cumulative infiltration data. The 3-D
case will be for infiltration from a 10 cm radius disc source
for four different soil types (sand, loam, silt, and silty clay)
and several initial saturation conditions. Appropriate values
for the shape parameters b and g, and their dependence on
the soil type and initial water content conditions, will thenbe estimated by fitting previously scaled numerical data toscaled functions I*(t*) and DI*(t*). The fitted shape parameters will then be used in dimensional equations (1) and (3)and the adequacy of this analytical model will be discussed
for each soil type and each initial saturation. Finally, thevalidity domains of the approximations will be determined
as a function of the shape parameters for each soil type andinitial saturation conditions, and the accuracy of both scaled
and dimensional approximations will be discussed.

2. Theory: Analytical Analysis

I1D ¼




1
expðb I*Þ þ b  1
I*  ln1bb

DI* ¼ gt*

ð7aÞ

S2
t*
2 DK 2S4DI*
2 DK 2 rd Dq

ð7dÞ

ð7eÞ

These formulations define the scaling factors related to time(g t), 1-D cumulative infiltration (g I), and the difference
between 3-D and 1-D cumulative infiltrations (g DI), as
previously proposed [Parlange et al., 1982; Rasoulzadeh
and Sepaskhah, 2003; Haverkamp et al., 2005; Varado etal., 2006].
[14] Notice that the scaled difference DI*(t*) is proportional to the scaled time t*, and that the shape parameter gserves as the proportionality coefficient. The scaled 1-D
cumulative infiltration I*(t*) depends only on the shapeparameter b. Approximations of (7a) and their validity
domains, as well as its derivative with time t*, i.e., thescaled infiltration rate, q*(t*), can therefore be studied onlywith respect to b. Notice also that when the shape parameterb is close to zero, equation (7a) leads to the Green and
Ampt model [Green and Ampt, 1911] (see Appendix A):
t* ¼ I*  lnð1 þ I*Þð8Þ

2.2. Analysis of I*(t*) and Related Approximations[15] Since for long times, I*(t*) tends to infinity, the rightterm of equation (7a) can be simplified using the asymptotic
behavior of exponential and logarithmic functions, leading
to (see Appendix A):
 
1
1
ln
Iþ1 *ðt*Þ ¼ t* þ
ð1  b Þb

ð9Þ[16] At very short times, I*(t*) can be developed using asquare-root-of-time series [Philip, 1969]. Implementation ofsuch serial expansion into equation (7a) leads to the following
definitions for the first- and second-order approximations:pﬃﬃﬃﬃﬃﬃﬃ
2t*

ð10aÞ

pﬃﬃﬃﬃﬃﬃﬃ 2  b
2t* þ
t*
3

ð10bÞ

IO*ð1Þðt*Þ ¼
IO*ð2Þðt*Þ ¼[17] Validity domains of these approximations are directly
linked to their accuracy, and thus to corresponding relative
errors:ð7bÞ

where 1-D and 3-D cumulative infiltrations are related to thescaled 1-D cumulative infiltration, I*(t*), and the scaled
difference between 3-D and 1-D cumulative infiltrations,DI*(t*), throught¼

S2
I* þ K0 t
2 DK

I3D ¼ I1D þ

2.1. Scaling Procedure and Scaled Variables I*(t*)
and DI*(t*)[13] The 3-D and 1-D cumulative infiltration equations(equations (1) and (3)) can be modified to result in simpler
equations using the following procedure:
t* ¼

W12415x*Ioð1Þðt*Þ ¼

x*Ioð2Þðt*Þ ¼

ð7cÞ
* ðt*Þ ¼x Iþ1

3 of 20I*ðt*Þ  I O*ð1Þ ðt*Þ
I*ðt*Þ
I*ðt*Þ  I O*ð2Þ ðt*Þ
I*ðt*Þ* ðt*Þj
jI*ðt*Þ  I þ1
I*ðt*Þ

ð11aÞ

ð11bÞ

ð11cÞ

LASSABATÈRE ET AL.: EVALUATION INFILTRATION

W12415

W12415

Table 1. Analytical Expressions and Approximations for Scaled and 3-D Dimensional Cumulative Infiltration and Infiltration Rate
Scaled Cumulative Infiltrationh

I*(t*, b)1t* = 1b
I*  ln

All timesexpðbI  Þþb1bi

pﬃﬃﬃﬃﬃﬃﬃ
2 t
pﬃﬃﬃﬃﬃﬃﬃ 2b2 t + 3 t*
1
t* + 1bln b1Very short times
Short times
Long times

Scaled Infiltration Rate

Time Restriction

q*(t*, b)

Time RestrictionNoneb
1 + expðb I  Þ1

None

a

t*Io(1) t(x*, b)t*Io(2)(x*, b)at*I+1(x*, b)a

Dimensional Cumulative Infiltration
I3D(t)
g S2
K0 + r Dq t
d
pﬃ
S t

Very short times
Short times
Long timesSpﬃ ð2bÞgS 2
t+
3 DK + K0 + rd Dq t

gS 2
1
1 S2
Ks + r Dq t + 2 ð1bÞ ln b DKd t*qo(1)(x*, b)a1

 t*+1(x*, b)b

 t*qo(2)(x*, b)a

Dimensional Infiltration Rate

Time Restriction

q3D(t)None2
DK q* + K0 + g r SDq
dS2
2 DK I* +All times

1
pﬃﬃﬃﬃﬃ
2 t
2b
1
pﬃﬃﬃﬃﬃ+ 3
2 t

S2
 2 DK
Io(1)(x*, b)2 t*
S2 2 DK 2 t*Io(2)(x*, b)2S 2 DK
I+1(x*, b)
2 t*Time Restriction
None

pﬃ
S t

pSﬃﬃﬃ +
2t



ð2bÞ
gS 2
DK + K0 + r Dq
d
3
gS 2
Ks + r Dq
dS2
 2 DK
qo(1)(x*, b)2 t*
S2 2 DK 2 t*qo(2)(x*, b)2S 2 DKq+1(x*, b)
2 t*aCalculated numerically from the study of the error functions.
Calculated numerically from the study of the error functions or estimated through equation (13).

bSince it can be shown numerically that the function x*I+1 (t*)
is monotonously decreasing in the interval R*+, defining abijection from R*+ to (0,1), the inverse function t*I+1 (x*)
can also be defined. Therefore, for any time t*  t*I+1 (x*),
the relative error x*(t*) will be smaller than the given* (t*) is thus definedtolerance x*. The validity domain for I+1
as the time interval [t*I+1 (x*), + 1). A similar approachdemonstrates that validity domains for I*
O(1)(t*) and I*
O(2)(t*)correspond to time intervals [0,t*Io(1)(x*)] and [0,t*Io(2)(x*)],respectively, where t*
Io(1)(x*) and t*Io(2)(x*) are the inverse
functions of x*
Io(1)(t*) and x*
Io(2)(t*), respectively. Dimensionless times t*I+1 (x*), t*Io(1)(x*), and t*Io(2)(x*) are referred
to as long-time, first-order and second-order approximation
validity times. Since there are no analytical expression
available for evaluating these validity times, they will becalculated numerically in the Results section.
2.3. Analysis of q*(t*) and Related Approximations[18] An actual scaled infiltration rate q*(t*) can beobtained by evaluating the derivative of I*(t*) (7a) withrespect to time t*, leading to (see Appendix B):
q*ðI*Þ ¼ 1 þbexpð b I*Þ  1

t*qþ1 ðx*Þ ¼

B
1
B
lnB
ð1  b Þ @

1 1
b b
1þ
x* C
C
1 C
A
1þ
x*

ð13Þ

1
I*

ð14aÞ

1
b
þ1
I*
2

ð14bÞ

q*
Oð1Þ ðI*Þ ¼q*Oð2Þ ðI*Þ ¼

These approximations can be combined with the shorttime expansions of I*(t*) to give the analytical expressions for the first- and second-order approximations forq*(t*):

ð12Þ

[19] When time t* tends to infinity, I*(t*) also tends toinfinity, and thus q*(I*) tends to unity, which defines thesteady state approximation q*+1(t*). Given a tolerance x*,
the flow rate is lower than unity plus the tolerance x*,
provided time is larger than the steady state validity timet*
q+1 (x*) defined as (see Appendix B):
0

The validity domain for the steady state expansion q*+1(t*)
is then defined as [t*q+1 (x*), + 1).
[20] As time approaches zero, I*(t*) tends to zero, andthus q*(t*) approaches infinity (see (12)). Expanding
equation (12) for small I* leads to (see Appendix B):

1 12
q*
Oð1Þ ðt*Þ ¼ pﬃﬃﬃ t*
2

ð15aÞ

1 1 2  b
2 þq*Oð2Þ ðt*Þ ¼ pﬃﬃﬃ t*
3
2

ð15bÞ

Note that the steady state and short-time approximationsof q*(t*) correspond to time derivatives of approximations of I*(t*). Note also that, from a mathematical pointof view, deriving the expressions for q*(t*) using serial
expansions, rather than taking directly the derivative of
the serial functions for I*(t*), is the proper approach (seeAppendix B). It has been previously demonstrated that
directly derivating term by term serial expansions canlead to inappropriate functions; the direct derivation ofserial functions requires certain conditions.
[21] As was the case for cumulative infiltration, thevalidity domains for the first- and second-order approxima-

4 of 20

LASSABATÈRE ET AL.: EVALUATION INFILTRATION

W12415

tions of infiltration rates can be evaluated from relative
errors:x*qoð1Þ ¼

x*qoð2Þ ¼

q*ðt*Þ  q*Oð1Þ ðt*Þ
q*ðt*Þ
q*ðt*Þ  q*Oð2Þ ðt*Þ
q*ðt*Þ

ð16aÞ

derivation produces the same expressions as those developed by Haverkamp et al. [1994], i.e., equation (5).
[24] The discrepancy between the dimensional approximations and the dimensional quasi-exact formulation can belinked to the accuracy of the scaled functions. Let the*(t*) and y*(t*),
relative errors be defined for scaled, yapprox
and dimensional, yapprox(t) and y(t), variables as follows:

ð16bÞ
x approx ðt; yÞ ¼

Like those for scaled cumulative infiltration, functions
x*qo(1) and x*qo(2) are monotonously increasing (see the Result
section). Therefore, the validity domains for approximations q*O(1), q*
O(2) are defined as time intervals (0,t*qo(1)(x*)]
and (0,t*qo(2)(x*)], where t*
qo(1)(x*) and t*
qo(2)(x*) are inverse
functions of x*qo(1)(t*) and x*qo(2)(t*), respectively. The firstand second-order validity times, t*qo(1)(x*) and t*
qo(2)(x*),
respectively, can, again, be calculated numerically (see theResult section). The steady state validity time is definedexactly using equation (13).
[22] The scaled variables and their analytical properties, i.
e., their related approximations and validity times, are given
in Table 1. Since in this section, both the cumulative
infiltration and the infiltration rate have been shown to betime and b dependent, they have to been referred to as I*(t*,b) and q*(t*, b). Similarly the scaled difference between
3-D and 1-D, DI*(t*, g), has to be regarded as both timeand g dependent.

x*approx ðt*; y*Þ ¼

I3D ðtÞ ¼


S2
2DK 2 tI*;b
þ K0 t
S2
2 DK



S2
2 DK 2 t
S4I*;bþ K0 t þ
2
2 DK
2 rd Dq DK 2S22 DK t DI*;gS2

x approx ðt; I1D Þ ¼

x approx ðt; q1D Þ ¼

x approx ðt; I3D Þ ¼

x approx ðt; q3D Þ ¼

ð17aÞ



y*ðt Þ  y*approx ðt Þy*ðt Þ

ð18aÞ

ð18bÞ

1
x*
ðt*; I*Þ
K0 t* approx
1þ
DK I*

1
x*
ðt*; q*Þ
K0 1 approx
1þ
DK q*

1
 x*
ðt*; I*Þ
K0
g S 2 t* approx
1þþ
DK rd Dq I*


ð19aÞ

ð19bÞ

ð19cÞ

1
 x approx*
ðt*; q*Þ ð19dÞ
K0
g S2
1
1þ
þ
DK rd Dq q*


1

ð17bÞ

Infiltration rates can be derived as time derivatives of theabove equations:
2DK 2 tS2;b
þ K0 þ g
q3D ðt Þ ¼ DK q*
2
rd Dq
S

yðt Þ

Insofar as q*(t*) is between infinity and unity, 1/q* andt*/I* are always between zero and unity, implying that:1þ
2DK 2 tq1D ðt Þ ¼ DK q*
;
b
þ K0
S2



yðt Þ  yapprox ðt Þ

where y stands for either dimensional q or I, and y* forscaled I* or q*, in both one dimension and three
dimensions. The use of scaling equation (17) leads to thefollowing relationships:

2.4. Derivation of Dimensional I1D(t), I3D(t), q1D(t),and q3D(t) and Their Validity Domains[23] Dimensional 1-D and 3-D quasi-exact cumulative
infiltrations can then be defined using the scaled cumulative
infiltration, the scaled difference between 3-D and 1-D
cumulative infiltrations, and the definition of scaling factors(g t, g I, and g DI) using equation (7):I1D ðt Þ ¼

W12415

ð17cÞ

ð17dÞ

Similarly, the dimensional approximations can be derived
from equation (17) and the scaled approximations. Such

ðt*; y*Þ  x approx ðt; y1D Þ  x approx*
ðt*; y*Þ ð20aÞ
x*
K0 approx
DK1x*
ðt*; y*Þ  x approx ðt; y3D Þ  x approx*
ðt*; y*Þ
K0
g S 2 approx
þ1þ
DK rd Dq
ð20bÞ

Equation (19) shows that relative errors for dimensional
approximations are smaller than the errors for scaled
approximations for both one dimension and three dimensions. Consequently, relative errors are smaller than thetolerance x in intervals with boundaries defined as the
product of the time scaling factor and the scaled validity
times.

5 of 20

LASSABATÈRE ET AL.: EVALUATION INFILTRATION

W12415

Table 2. Values for Hydrodynamic Parameters for the FourStudied Soils
Sand

Loam

Silt

Silty Clay

n
2.68
1.56
1.37
1.09
I
0.5
0.5
0.5
0.5
0.045
0.078
0.034
0.07
qr
0.43
0.43
0.46
0.36
qs
Ks (cm min1) 4.95  101 1.73  102 4.17  103 3.33  104hg (cm)
6.90
27.8
62.5
200

several uniform initial saturation conditions (i.e., h0, q0),
which correspond to conditions, for which equation (3) was
developed [Haverkamp et al., 1994]. Soil hydraulicparameters for four studied soil types, i.e., sand, loam, silt,
and silty clay, were obtained from the ROSETTA database[Schaap et al., 2001] implemented into HYDRUS codes[Šimůnek and van Genuchten, 2008; Šimůnek et al., 2008].
Their water retention curves and hydraulic conductivity
functions were described using the van Genuchten-Mualem
model [van Genuchten, 1980; Mualem, 1976]:8 n m>< qðhÞ ¼ qr þ ðqs  qr Þ 1 þ jhj
hg>:qðhÞ ¼ qs[25] Such an approach leads to the following definitionof the validity time for any dimensional approximation
yapprox(t), revealing its dependency on the initial saturation
degree Se0:
tyxDapprox ðSe0 ; b Þ ¼

tyxDapprox ðSe0 ; b Þ ¼

S2
ty* approx ðb Þ
2 DK 2S 2 ðqðSe0 Þ; qs Þ
2 ðKs  K ðqðSe0 ÞÞÞ

ð21aÞ

t*
ðb Þ
2 y approx

tyxDapprox ðSe0 ; b Þ ¼ RðSe0 Þ tyxDapprox ð0; b Þ

ð22aÞ

S 2 ðqðSe0 Þ; qs Þ
Ks2
2
S ðqr ; qs Þ ðKs  K ðSe0 ÞÞ2

ð22bÞ

RðSe0 Þ ¼

m¼1Reference validity times and ratios are discussed in the
Results section. The analytical properties of the dimensionalvariables (approximations and validity times) are presented
in Table 1.

3. Material and Methods: Numerical Modeling[26] Beerkan infiltration experiments (for ponding conditions, i.e., qf = qs) were modeled for four soil types and for

h<0

ð23aÞ

h0

1
n

ð23bÞ

h

1
K ðhÞ ¼ Ks  Kr ðhÞ ¼ Ks  Sel 1  1  Se=m

ð21bÞ

where x is equal to either 1 or 3 and ty_approx*(b) stands forthe corresponding scaled validity time. When the parameterb varies little with the initial saturation degree, the scaled*(b), remains constant. Dimensional
validity time, ty_approx
times for nonzero initial saturation degrees txD
y_approx(Se, b)
can then be calculated using the dimensional time for a zero
initial saturation degree txD
y_approx(0, b) using:

W12415

m i2

ð23cÞ

where qr and qs denote the residual and saturated watercontents, respectively; Kr and Ks are the relative andsaturated hydraulic conductivities, respectively, hg is thescale parameter for water pressure, n is a pore size
distribution index, and l is a pore connectivity parameter,
assumed to be 0.5 by Mualem [1976]. Soil hydraulicparameters for four soil textural classes are given in Table 2.
[27] One dimensional water flow was modeled using
HYDRUS-1D [Šimůnek et al., 2008]. The transport domain
was assumed to be 2 m deep for sand, 1 m for loam and silt,
and 40 cm for silty clay. For the initial conditions, values ofh0 and q0 were chosen to represent saturations common forfield conditions, as well as to explore the entire saturation
range from dry soils to soils close to full saturation, i.e.,from about 5% to 95% (Table 3). Free drainage (or unit
hydraulic gradient) was selected as the lower boundary
condition, which is often used to represent conditions with adeep water table. Finite elements (FE) of 2 mm were used todiscretize the transport domain for all soils except silty
loam, for which 0.2 mm elements were used for runs with
low initial pressure heads (5  104 and 2  1010 cm). This
spatial discretization was chosen using a sensitivity
analysis. Several FE meshes were tested, and numerical

Table 3. Initial Conditions in Terms of Pressure Heads h0 and Corresponding Saturation Degrees Se0 Used in Numerical Simulations for
Four Studied Soils
Sand

Loam

Silt

Silty Clayh0 (cm)Se0 (%)

h0 (cm)Se0 (%)

h0 (cm)Se0 (%)

h0 (cm)Se0 (%)

5
10
20
40
60
80
100
1000

80.2,
44.0
16.1
5.2
2.6
1.6
1.1
0.02

20
40
60
80
100
300
500
103
5  103
104

84.5
69.4
59.1
51.9
46.6
26.2
19.7
13.4
5.5
3.7

40
60
80
100
500
5  1035  1042  105

89.0
83.5
78.9
75.0
45.6
19.8
8.4
5.0

40
60
80
100
103
5  1045  1055  106108
2  1010

98.7
98.1
97.4
96.9
85.4
60.8
49.5
40.2
30.7
19.1

6 of 20

W12415LASSABATÈRE ET AL.: EVALUATION INFILTRATION

results were plotted against mesh density. Maximum time
was chosen to be higher than time needed to reach a steadystate (a constant flow rate at the top). Minimum time steps
were fixed at 103 min for sand, loam and silt, and 104 minfor silty clay. Precision tolerances for pressure heads andwater contents were fixed for all soils at HYDRUS default
values of 1 cm and 0.001, respectively.
[28] Three-dimensional water flow was simulated using
the axisymmetrical option of HYDRUS [Šimůnek et al.,2006]. Transport domains were assumed to be 40 cm in bothradial and vertical directions. Quadrilateral FE elements of
2 mm were used for all soils. The same initial conditions
were used as in the 1-D analysis (Table 3). No flux boundary
conditions were used at all boundaries except for the bottomof the transport domain where free drainage was used, andthe center part (a radius of 10 cm) of the soil surface
representing the infiltration disc where zero pressure head
was applied. Parameters of the time discretization were thesame as for 1-D numerical simulations.
[29] Simulation results included pressure head and watercontent maps, as well as dimensional cumulative infiltrations and infiltration rates. Other calculations and minimum
least squares error optimizations were carried out using thescilab free code [Campbell et al., 2006].

4. Results[30] In what follows, we will first discuss the propertiesof the scaled quasi-exact functions I*(t*, b) and q*(t*, b)
with respect to the shape parameter b and their approximations and associated validity domain. The scaled difference
between 3-D and 1-D cumulative infiltrations will not be
discussed any further because of its simplicity (a simple
linear law). Second, the accuracy of analytical models will
be evaluated by comparing them with numerically generated cumulative infiltrations for selected scenarios involving
3-D and 1-D water infiltrations into four soil types (sand,loam, silt, silty clay) and several initial saturation conditions. This comparison will allow us to discuss the shapeparameters and validity domains of approximations.
4.1. Analytical Analysis of I*(t*, b) and q*(t*, b)4.1.1. Case of the Usual Value b0 = 0.6[31] Figure 1 shows scaled cumulative infiltration I*(t*,b 0) and infiltration rate q*(t*, b 0), as well as their long time/steady state, and first- and second-order approximations. All
functions were evaluated for the shape parameter b 0 = 0.6.
Related validity times are calculated assuming a tolerance
x*0 of 2.5%. The scaled cumulative infiltration I*(t*, b0)
and infiltration rate q*(t*, b 0) curves have required
properties, i.e., I*(t*, b0) increases with time t* and isconcave, and q*(t*, b 0) decreases toward unity and isconvex.
[32] The proposed analytical expressions (10) and (9) aregood approximations of the scaled quasi-exact formulation(7a) for I*(t*, b0) in their validity intervals. Similarly, theproposed analytical expressions (15) are accurate approximation in their validity intervals for q*(t*, b0). Thisconfirms the suitability of the proposed analytical expressions. As illustrated in Figures 1c– 1d, the related relative
errors x*I+1 (t*, b 0) and x*q+1 (t*, b0) are monotonously* (t*, b0), x*qo(1)(t*, b0),
decreasing and x*Io(1) (t*, b0), x Io(2)* (t*, b 0) are monotonously increasing functions.
and x qo(2)W12415Consequently, the long-time approximation validity time
t*I+1(x*, b0) and the steady state validity time t*q+1(x*, b 0)
are decreasing functions and the first- and second-ordervalidity times t*
Io(1)(x*, b 0), t*
Io(2)(x*, b 0), t*
qo(1)(x*, b 0),
and t*qo(2)(x*, b 0) are increasing functions (Figures 1e – 1f).
For small tolerances x*, the validity times increase in thefollowing order: first order < second order < long time/steady state. For instance, for x* = x 0* = 2.5%, these
correspond to 0.006 < 0.74 < 2.73 and 0.0015 < 0.26 < 4.13
for I*(t*, b 0) and q*(t*, b 0), respectively. When the tolerance
x* increases, this order shifts to first order < long time/steadystate < second order (Figure 1e – 1f), and when x* > 0.15 tolong time < first order < second order. Intersections between
the long time/steady state and second-order validity times
correspond to a relative error of 5.5% and 12% for I*(t*, b 0)
and q*(t*, b 0), respectively. This means that the quasi-exact
formulations I*(t*, b0) and q*(t*, b 0) can be replaced firstwith their second-order and then long-time/steady state
approximations while having relative errors lower than 5.5%
and 12%, respectively.
[33] It may be noted that relative errors are higher for
q*(t*, b 0) than for I*(t*, b 0) (Figures 1c – 1d). This meansthat the use of the approximation is less accurate for the
infiltration rate than for the cumulative infiltration. Therefore, the validity domains for q*(t*, b0) need to be reduced
for the first- and second-order approximations, andincreased for the steady state approximation to ensure thesame precision as for I*(t*, b 0).
4.1.2. Sensitivity Analysis of I*(t*, b) and q*(t*, b)
as a Function of b[34] The scaled functions I*(t*, b) and q*(t*, b) were alsostudied with respect to the value of their shape parameter b.
The value range of this parameter is not known a priori,even though it was originally restricted to between zero andunity for mathematical reasons related to the integration
method developed by Haverkamp et al. [1990] andgeneralized by Ross et al. [1996]. In the latter study, theparameter b is regarded as an integral shape parameter, assuggested by Haverkamp et al. [1994], and thus allowed tohave larger values. In agreement with its definition by (6a),(as suggested by Fuentes et al. [1992]) b is restricted here tovalues smaller than two (a positive value of the integrals 6a).
The same constraint appears from the analysis of the firstand second-order approximations of I*(t*, b). Since
I*O(2)(t*, b) is defined as I*O(1)(t*, b) plus a second term
involving b (see equation (10)), this term must be positive toensure that I*
O(1)(t*, b)  I*
O(2)(t*, b)  I*(t*, b). If such an
inequality was not ensured, the second-order approximation
would be less precise than the first-order approximation,
which would be meaningless. Such a condition implies that:2b
 0; i:e:; b  2
3

ð24Þ

The scaled functions are therefore studied for b 2 [0,2]. As
explained above, the case with b = 0 corresponds to theGreen and Ampt formulation.
[35] A change of the parameter b affects the scaled
functions I*(t*, b) and q*(t*, b) without changing theirmonotonic behavior and analytical properties. In particular,
relative errors of approximations keep the same monotonic
properties with regard to time, and likewise similar

7 of 20

W12415LASSABATÈRE ET AL.: EVALUATION INFILTRATION

W12415Figure 1. One-dimensional scaled analytical models for b0 = 0.6. (a) Scaled cumulative infiltration
I* (t*, b 0) and (b) scaled infiltration rate q*(t*, b 0) (black lines) along with related approximations (graylines) and validity times (diamonds, very short time; gray circles, short time; gray squares, long time)
calculated for the tolerance x*0 = 2.5%. Relative errors (x*I or x*)
q of approximations as a function of time
t* for (c) scaled cumulative infiltration and (d) scaled infiltration rate. Validity times (t*I or tq*) as afunction of the tolerance x* (an inverse function) for (e) scaled cumulative infiltration and (f) scaled
infiltration rate.

definitions of validity domains and validity times t*Io(1)(x*,b), t*Io(2)(x*, b), t*
I+1 (x*, b), t*
qo(1)(x*, b), t*
qo(2)(x*, b), andt*q+1 (x*, b).
[36] Cumulative infiltrations I*(t*, b) and infiltration
rates q*(t*, b) are decreasing functions with regard to b
(Figures 2a and 2b). This decrease can be demonstrated forI*(t*, b) by studying the right term of equation (7a) withregard to b. The effect of b on the validity times t*Io(1)(x*,b), t*Io(2)(x*, b), and t*I+1 (x*, b) is illustrated in Figures 2c –2f. For the tolerance x*0, the time needed to apply thelong-time approximation, t*I+1 (x*0, b), decreases with b(Figure 2c). This results from the fact that q*(t*, b) decreases
with b, i.e., a steady state is reached earlier when b increases(Figure 2b). The second-order approximation validity time,t*Io(2)(x*0, b), roughly decreases with b (Figure 2c). The timeinterval that requires the application of the quasi-exact
formulation I*(t*, b) instead of its approximations, i.e., thetime between the second-order and long-time approximations
validity domains, reduces with b (Figure 2c). On the contrary,
the first-order approximation validity time t*
Io(1)(x*0, b) is anincreasing function until it reaches the value of the secondorder validity time t*Io(2)(x*0, b) for b = 2. This equality resultsfrom the fact that the first- and second-order approximations
correspond when b = 2.
[37] Similar conclusions can be drawn for higher tolerances x*. The value of x* only affects the order between thevalidity times. For tolerances x*0 = 2.5% and x*1 = 5%, thevalidity times are in the following order: t*Io(1)(x*0, b) t*Io(2)(x*0, b)  t*
I+1 (x*0, b) and t*Io(1)(x*1, b)  t*
Io(2)(x*1, b) t*I+1 (x*1, b) (Figures 2c and 2d). For a tolerance x*
2 = 7.5%

8 of 20

W12415LASSABATÈRE ET AL.: EVALUATION INFILTRATION

W12415Figure 2. One-dimensional scaled analytical model as a function of b (from zero Green and Ampt
(GA) model to two). (a) Scaled cumulative infiltrations I*(t*, b), (b) infiltration rates q* (t*, b), andvalidity times for four tolerances: (c) x*0 = 2.5%, (d) x*1 = 5%, (e) x*2 = 7.5%, and (f) x*3 = 10%.

and values of b between 0.2 and 1.8, the order becomes:t*Io(1)(x*2, b)  t*
I+1(x*
2, b)  t*
Io(2)(x*2, b). The same orderapplies for a tolerance x*
3 = 10% and for any value of b
(Figure 2f). This means that the second-order and long-time
approximations can be successfully used to model infiltration for all values of b provided that a tolerance of 7.5% is
acceptable for I*(t*, b). Similar results are obtained for thevalidity times related to the scaled infiltration rates, exceptthat in the latter cases, the related relative errors are higher.
4.2. Reproduction of Numerically Generated I1D(t),I3D(t), q1D(t) and q3D(t)4.2.1. Numerical Cumulative Infiltrations[38] Numerical results are presented in terms of the dependence of 1-D and 3-D cumulative infiltrations upon initialsaturations (Figures 3a– 3d) and corresponding water content

profiles for a specific initial saturation (Figure 3e – 3h). For
sand, both 1-D and 3-D cumulative infiltrations depend very
little upon initial saturations, and are quasi linear (Figure 3a).
Dimensionality mainly affects the slope, with an additional
contribution of the 3-D lateral flux, due to capillary forces, tothe 1-D vertical infiltration dominated by gravity. On thecontrary, cumulative infiltrations are strongly saturation
dependent for the other soils (Figures 3b– 3d). The finer thesoil, the more initial saturation dependent cumulative infiltrations. Moreover, as the soil becomes finer, the concavity of
cumulative infiltrations becomes more pronounced, and thedifferences between 1-D and 3-D infiltrations become
smaller. This means that water infiltration is less gravity
and more capillarity driven. These numerical results are incomplete agreement with experimental data [e.g., Smith etal., 2002].

9 of 20

LASSABATÈRE ET AL.: EVALUATION INFILTRATION

W12415

W12415Figure 3. Numerically simulated data. Cumulative infiltrations (1-D and 3-D) as a function of the initialsaturation Se0 for a 10-cm radius disc source for (a) sand, (b) loam, (c) silt, and (d) silty loam (see Table 3
for Se0 values). Water contents versus time (min) for (e) sand (h0 = 40 cm, Se0 = 10%), (f) loam (h0 =100 cm, Se0 = 45%), (g) silt (h0 = 500 cm, Se0 = 45%), and (h) silty clay (h0 = 5 104 cm, Se0 = 50%); theisoclines corresponds to h0 to zero by h0=10 increments.
4.2.2. Modeling Scaled Data Using the Quasi-Exact
Implicit Formulation[39] In all cases, the correspondence between numericaland analytical models is evaluated using model relative
errors defined as follows:Erðx; yÞ ¼

^yð xÞ  yð xÞ
yð xÞ

ð25Þ

where y(x) is the numerically generated value at point x and^y(x) is the corresponding analytical model value. Note that

positive values indicate model overestimations, whereas
negative values indicate model underestimations.
[40] Scaled differences between 3-D and 1-D cumulative infiltrations DI*num(t*) follow the same linear law for
all initial saturations for loam and silt (Figure 4). A linear
trend is also followed for sand, but the proportionality
coefficient differs for the higher initial saturation Se0 of
80%. The proportionality coefficient g depends upon thesoil type as detailed in Table 4. Optimized values of g areclose to the usual value of 0.75, as suggested by Haverkamp
et al. [2005], for loam and silt, but substantially higher for

10 of 20

W12415LASSABATÈRE ET AL.: EVALUATION INFILTRATION

W12415Figure 4. Scaled differences between 3-D and 1-D cumulative infiltrations DI*num(t*) (numericallygenerated) and DI*(t*, g) (obtained from quasi-exact formulations) as a function of (a, c, e, g) soil, initialsaturation and parameter g and (b, d, f, h) model errors. Initial saturations are shown on the right.
sand (1.03). Although all these values are significantly
higher than those calculated using the relationship (6b) ofFuentes et al. [1992] (Table 4), they are in agreement withthe previous work of Warrick and Lazarovitch [2007]. In
particular, the latter authors also found lower values for
medium textured soils (loam, silty loam) and larger values

for coarser soils (sand) and finer soils (clay). Relative errors
show that the linearity is well followed for large times t*,
but deviations from it may not be negligible close to zero
(Figures 4b, 4d, and 4f), as previously highlighted by
Smetten et al. [1994] and Turner and Parlange [1974]. In
conclusion, the linear extension of 1-D cumulative infiltra-

11 of 20

LASSABATÈRE ET AL.: EVALUATION INFILTRATION

W12415

Table 4. Proposed Values for the Shape Parameters b and g
Related to the Use of Quasi-Exact Formulation as a Function of theSoil Typea
g

Sand
Loam
Silt
Silty clayb

Optimized

Fuentes

Optimized

Fuentes

1.03
0.756
0.748
0.977

0.587
0.593
0.59
0.575

0.334
1.25
1.56
1.65

0.535
1.22
1.46
1.90

a
Quasi-exact formulations use equation (7). Also given are Fuentes
values estimated [Fuentes et al., 1992] using equations (6).

tion to 3-D using equation (7b) is accurate provided that thevalue of the shape parameter g is chosen based on the soil
type, and that the time is long enough.
[41] As with the scaled difference DI*num(t*), the scaled
cumulative infiltration I*
num(t*) does not depend significantly on the initial saturation (Figure 5) and a unique valueof b can be optimized independently of the initial saturation
for sand, loam, and silt (Table 4). The resulting quasi-exact
formulation for I*(t*, b) is in good agreement withnumerical data I*num(t*) (Figures 5a, 5c, and 5e). Modelerrors show a little autocorrelation (non random distributionof errors as a function of time), but are always lower than
10%, and usually below 5% (Figures 5b, 5d, and 5f).
Although the model errors are slightly higher for silt, they
stay in an acceptable range. It should be noted that for loam
and silt, the model errors can be higher for very small times(Figures 5d and 5f). The optimized values of the shapeparameter b vary significantly between different soil types,
with a minimum value of 0.344 for sand and a maximum
value of 1.56 for silt. These values differ significantly fromthe common value of 0.6 [Haverkamp et al., 1994], but
resemble values calculated using the Fuentes et al. [1992]
relationship (6a). These results prove that the quasi-exact
formulation (7a) is accurate for modeling the scaled
cumulative infiltration, provided that the shape coefficientb reflects the soil type and is taken as proposed in Table 4.
[42] However, the textural class of silty clay remains an
exception. Scaled numerically generated curves I*num(t*) do
not fall on one line (Figure 5g), and model errors for several
initial saturations are on the order of 10% and show a verystrong autocorrelation (Figure 5h). The sign of the model
errors indicates that numerically generated infiltration data
are highly underestimated at low times t* and overestimatedat larger times t*. Thus, the analytical model underestimatesthe concavity of numerically generated infiltration curves.
Moreover, this discrepancy depends strongly upon theinitial saturation (Figure 5h), indicating that an averagevalue of b cannot be considered for all initial saturations.
The linear law for the scaled 3-D – 1-D difference is also nolonger valid for silty clay (Figure 4g). Numericallygenerated differences DI*num(t*) show a significant concavity that depends strongly on the initial saturation (Se0).
Relative errors are no longer acceptable for any consideredtimes t*(Figure 4h). This indicates that equation (7) is nolonger valid for modeling scaled cumulative infiltrations insilty clay. This discrepancy results from the nonfulfillmentof the constraints related to equation (4), which is
documented in Figure 6 (a plot of relative errors between

W12415the right and left terms of equation (4) for the four soils).
Note that these relative errors are significantly larger forsilty clay than for the other three soils (Figure 6).
4.2.3. Modeling Dimensional Data Using the
Quasi-Exact Formulation[43] Dimensional modeled cumulative infiltrations arecalculated using scaled cumulative infiltrations for optimized values of g and b, and using scaling equation (17).
This procedure is equivalent to the use of equations (1) and(3) along with optimized values of g and b. Model errors
are calculated using one- and three-dimensional numericallygenerated data and equation (25). Resulting model errors arein most cases lower than 10%, and quite frequently below
5% (Figures 7a – 7f). Therefore, equations (3) and (1) are
validated for such applications. Again, silty clay is an
exception, with relative errors higher than 10% and evenclose to 50% in certain cases (Figure 7g– 7h), resulting fromthe inadequacy of the scaled quasi-exact formulation as
discussed above.
4.2.4. Modeling Scaled Data With Approximations
and Related Validity Domains[44] Approximate solutions are first studied with regard
to scaled numerically generated data. The long-time andclose-to-zero approximations are illustrated in Figures 8a,8c, 8e, and 8g, together with scaled numerically generated
data. The accuracy of approximations is evaluated using
equation (25) and displayed in Figures 8b, 8d, 8f, and 8h.
Maximum errors are then calculated in the validity intervals,
determined for the tolerance x*0 = 2.5% and the optimized
values of the shape parameter b.
[45] As with the quasi-exact formulation, the accuracy ofapproximate equations is not acceptable for silty clay
(Figures 8g– 8h). The model relative errors reach absolutevalues that are, in most cases, higher than 10 –20%, which
renders these approximations unsuitable to properly model
water infiltration into silty clay. This was expected, as
approximate equations were derived from the quasi-exact
formulation, which was also not suitable for silty clay, as
discussed above.
[46] The long-time and close-to-zero approximations
seem accurate for sand, loam and silt (Figures 8a, 8c, and8e) in their corresponding validity domains. The first-orderapproximation is characterized by a very sharp increase ofmodel errors for all three soils (Figures 8b, 8d, and 8f). Very
low values of the first-order validity time t*Io(1)(x*0, b) for
sand and loam, along with too few numerically generated
data points in the related validity interval [0, t*
Io(1)(x*0, b)],
did not allow for precise estimation of the model errors. For
silt, the first-order approximation has a maximum model
error close to 10% in the validity domain [0, t*Io(1)(x*0, b)].
For the second-order approximation, the model errors
increase with time for sand (Figure 8b). For loam and silt,
the model errors are quite high for very small times,
especially for low initial saturations (Figures 8d and 8f).
The errors then sharply decrease before reaching aminimum, and then begin increasing again (Figures 8d
and 8f; arrows indicate the error minima). For these twosoils, the error minima are located within the validity
domains [0, t*
Io(2)(x*0, b)]. Thus, after disregarding higherrors close to zero (local phenomenon), the maximum error
in absolute value is evaluated for the time interval [t*Io(2)(x*0,b)/10, t*Io(2)(x*0, b)]. In these intervals, the model relative

12 of 20

W12415LASSABATÈRE ET AL.: EVALUATION INFILTRATION

W12415Figure 5. Scaled 1-D cumulative infiltrations I*num(t*), the quasi-exact formulations I*(t*, b) and (a, c,e, g) estimated global values of b and (b, d, f, h) their model errors accuracy. Initial saturations are shownon the right.

errors are never higher than 5%. For the long-time
approximations, the model errors are decreasing functions(Figures 8b, 8d, and 8f). When restricted to their validity
intervals, [t*I+1(x*), +1), the model errors are at most 5%
for sand and loam, and a little higher for silt (between 5 and7.5%). In conclusion, the approximations can be considered
to be good estimations of numerically generated data intheir corresponding validity domains.

4.2.5. Modeling Dimensional Data With
Approximations and Related Validity Domains[47] The relative error functions evaluating the accuracyof the dimensional approximate expansions in reproducing
dimensional numerically generated data are similar to thoseshown in Figure 8 (data not shown). As with the scaled data,
the maximum errors of dimensional expansions were determined for their validity domains, defined using scaled
validity times and time scaling factors (equation (22)).

13 of 20

W12415LASSABATÈRE ET AL.: EVALUATION INFILTRATION

W12415Figure 6. Relative error between the right and left terms of equation (4) for the four soils in function to
the saturation degree during water infiltration between an initial saturation degree Se0 = 25% and water
saturation.
Again, to avoid high error values very close to zero time,the maximum errors were evaluated over the intervals
xD
xD(Se0, b)/10, tI_approx(Se0, b)], as with the scaled[tI_approx
data.
[48] The maximum errors for different soils and dimensions and zero initial saturation are given in Table 5. Errors
for the first-order approximations for loam and sand could not
be evaluated for the same reasons as with scaled data. Again,
most errors were smaller than 5%, which indicates that thevalidity times were well chosen, and that the approximations
were sufficiently accurate for their corresponding domains.
As with the quasi-exact formulation, the approximations are
no longer valid for silty clay, for which they produce errors
higher than 20– 25%. This indicates that the approximations,as with the quasi-exact solution, are not suitable for fine soils.
The difficulty of applying common analytical models to
describe infiltration into fine soils has been reported manytimes [e.g., Lin and McInnes., 1995; Messing and Jarvis,
1993, Vandervaere et al., 2000a, 2000b].
[49] The results discussed above indicate that the proposed validity intervals ensure an acceptable error of model
predictions for sand, silt and loam. Values of the dimensional
validity times are calculated using equation (22a) from thereference validity times tIo(1)(0, b), tIo(2)(0, b) and tI+1(0, b).
Due to great differences in values of scaling factors fordifferent soil types (sand < loam < silt < silty clay),corresponding reference times are of a different order ofmagnitude (Table 5 and Figure 9a). Their values provide
practical information with respect to infiltration experiments, which usually have duration on the order of 1 to 2 h,and the cumulative infiltration is described at approximately
one minute time intervals. For sands and loams, the validity
times of the first-order approximations are too short toallow determination of sorptivity using the usual method[Vandervaere et al., 2000a, 2000b]:
dI1D
dI3D
lim pﬃ  lim pﬃ  S
t!0 d tt!0 d tð26ÞThe validity time for the second-order approximation,tIo(2)(0, b), is on the order of two minutes (Table 5). Thissuggests that, except for specific experimental protocols
and setups (precise measurements of the infiltrated height
and time), commonly collected information using the
Beerkan protocol may require the use of the quasi-exact
formulation or the long-time approximation (Figure 9a).
For loam, the second-order approximation can be used
only when short time steps are used in the experimental
protocol. However, steady state conditions are reachedonly after about 3 h, which may be at the very end of
infiltration experiments. For silt, the validity times for the
first- and second-order approximations are on the order offew hours (Table 5), suggesting that the use of theseapproximations may be sufficient. On the contrary, steadystate conditions are only reached after about 12 h, which isimpractical for most experimental conditions, and thus thelong-time approximations may not be used.
[50] When soils are initially wet, these restrictions may berelaxed, since dimensional times depend on the initialsaturation (Figure 9b). In particular, it has been shown thatvalues of b do not depend upon the initial saturation degree.
Thus, as described in the theory section, the validity times
are obtained by multiplying the reference validity time withthe ratio R (equation (22)). These ratios can be reduced byseveral orders. Note that, independent of the soil type(except for sand close to saturation), there is a decreasing
linear relation between the ratio R and the saturation degree
Se0 (Figure 9b). As a result, the validity times could be
estimated using values given in Table 5 and assuming asimple decreasing linear relationship between R and theinitial saturation (Se0).

5. Conclusions[51] Understanding and modeling the physical process ofwater infiltration into the soil are necessary for proper water
management. The quasi-exact formulation and its related
direct approximations developed by Haverkamp et al.

14 of 20

W12415LASSABATÈRE ET AL.: EVALUATION INFILTRATION

W12415Figure 7. Accuracy of (a, c, e, g) I3D(t) and (b, d, f, h) I1D(t) defined using equations (1) and (3) withregard to numerically generated data. Initial saturations are shown on the right.

[1994] for describing 1-D and 3-D water cumulative
infiltrations are studied in this paper. The 3-D case involves
a specific case of water infiltrating from a surface discsource.
[52] A scaling procedure was proposed in the first part ofthe paper to derive equations describing one- and threedimensional cumulative infiltrations and infiltration rates

from the scaled cumulative infiltration and the scaled
difference between 3-D and 1-D cumulative infiltrations.
The analysis of analytical properties of scaled functions
allows (1) complete demonstration of the analytical expressions of both scaled and dimensional direct approximationsand (2) determination of related validity domains. In thesecond part of the paper, the focus was on the accuracy of

15 of 20

W12415LASSABATÈRE ET AL.: EVALUATION INFILTRATION

W12415Figure 8. Scaled 1-D cumulative infiltrations I*num(t*) and first-order, second-order, and long-time
approximations. (a, c, e, g) Numerical and analytical data and (b, d, f, h) relative errors between analyticaland numerical curves (points) and between approximations and the quasi-exact formulation (lines). The
arrows indicate the minimum second-order approximation model error.
the analytical models with respect to numerically generated
one- and three-dimensional cumulative infiltrations for four
soils (sand, loam, silt, and silty clay) and several initialsaturations. The quasi-exact formulation was validated forsand, loam, and silt, provided that shape parameters g and b,

regarded as integral shape constants, are chosen as afunction of the soil type. The proposed approximations
derived for the same shape parameters and the proposed
validity intervals were also validated for the same soils.
Optimized values of the shape parameters b and g were

16 of 20

LASSABATÈRE ET AL.: EVALUATION INFILTRATION

W12415quite different from the averaged values suggested by
Haverkamp et al. [1994] (i.e., 0.75 and 0.6, respectively),
but were close to those predicted by the Fuentes formulation[Fuentes et al., 1992], especially for b. None of theanalytical functions worked well for silty clay (a finetextured soil), probably due to its unique hydraulic properties, which do not fulfill the conditions required for thevalidity of the quasi-exact formulation.
[53] This study proves that Haverkamp’s formulations
can be used to describe both 1-D and 3-D cumulative
infiltrations, provided that great care is taken to determine
the values of the corresponding shape parameters. Theirvalues for three soil types from the ROSETTA database,i.e., sand, loam, and silt, are recommended. When the use of
the quasi-exact formulation appears to be too complicated(due to its implicit formulation), direct approximations canbe used, provided that these are restricted to the proposed
validity intervals. Proposed values for validity times for
sand, loam, and silt can be considered as good indicators forthe choice between long-time, first-, or second-orderapproximations to analyze infiltration data.
[54] Further research is needed in several directions. First,
the difference between 3-D and 1-D cumulative infiltrationsin this study was only evaluated for one radius of the discsource, while Warrick and Lazarovitch [2007] and Warrick
et al. [2007] have shown the dependency of the parameter gupon the geometry of the infiltration source. The validityof the linear relationship for the difference between 3-D and
1-D cumulative infiltrations versus time, and the existenceof the proportionality coefficient g, needs to be confirmed
for other radii of the disc source. Second, the dependence ofthe shape parameters b and g on soil textural parameters
should be established. Estimation of b using the approach
used here for all textural classes of the ROSETTA database
could confirm the hypothesis of an increase in b values forfiner soils observed in this study. Third, there is a need to
study constraints on soil hydraulic properties with regard to
the conditions required by the quasi-exact formulation(equation (3)). This could help in defining the soils forwhich the quasi-exact formulation can be used, and those forwhich its use is not suitable. In particular, one should define
the shape parameters of the analytical expressions of water
retention and hydraulic conductivity functions, which fulfill
conditions of the quasi-exact formulation. Additionally,
Table 5. Dimensional Validity Times for Zero Initial Saturationand Errors of Approximations in Reproducing Dimensional 1-D
and 3-D Numerical Cumulative Infiltrations
tIo(1)
Sand
Loam
Silt
Silty clay
Sand
Loam
Silt
Silty clay
Sand
Loam
Silt
Silty claytIo(2)

tI+1<100
2030002032004704005000
3h05800
3h0
12h080
1-D Maximum Model Errors
NA
1.3%
NA
4.8%
8.7%
5.2%
25.6%
18.4%
3-D Maximum Model Errors
NA
7.6%
NA
3.7%
9.9%
4.9%
17.3%
17.8%

1202700
3h150
19h220
2d14h
3.4%
3.6%
5.7%
17.9%

W12415Figure 9. (a) Dimensional validity times for four types ofsoil and a zero initial saturation degree and time limits withregard to experimental constraints (horizontal lines) and(b) decrease of dimensional validity times versus the initialsaturation Se0.

correct values of b and g need to be implemented into theBEST algorithm to avoid erroneous estimations of soil
hydraulic characteristics.

Appendix A: Analysis of Scaled Cumulative
Infiltration I*(t*)[55] This appendix presents analytical properties of I*(t*)
and, in particular, its asymptotic behavior close to infinity or
zero, when values of b are between zero and two. Let thefunction I*(t*) be implicitly defined as follows:1expðb I*Þ þ b  1
t* ¼
I*  ln1bb|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

ðA1Þ

RHS

[56] When t* tends to infinity, I* also tends to infinity.
Using the properties of exponential and logarithmic
functions, the following simplifications can be performed:

5.6%
2.40%
5.1%
21.8%

17 of 20





 
expðbI*Þ þ b  1expðbI*Þ
1ln
 ln¼ bI* þ ln
b
b
b
ðA2Þ

LASSABATÈRE ET AL.: EVALUATION INFILTRATION

W12415

Implementing (A2) into (A1) and resolving I* as afunction of t* leads to the formulation of the long-time
approximation:
Iþ1* ðtÞ  t* þ

 
1
1
ln
ð1  b Þ
bW12415[59] From the property of the exponential function, it isclear that q*(I*) tends toward unity for long times. Given atolerance x, q*(I*) for time t* is as follows:
q*ðI*ðt*ÞÞ  1 þ x

ðA3Þ

ðB2Þ

When using equation (B1b) and evaluating for I* we get[57] Close to zero, I*(t*) can be analyzed using a squareroot-of-time series:
I*ðt*Þ ¼

Xþ1
k¼1

k

Ak t* =2

pﬃﬃﬃﬃ
t* þ A2 t* þ ðt*Þ

0

ðA5Þ

where (t*n) represents neglected terms:
þ1Xk

Ak t =2

ðA6Þ

knþ1

The right term of equation (A1), RHS, can be developed as aserial function when implementing the serial expansions of
the exponential and logarithmic functions; then gathering
exponent terms, we get
t* ¼




3
1 2
A2 2  b
3
A1 t* þ A1 A2  1
t =2 þ t =2
2
3
2

pﬃﬃﬃﬃﬃﬃﬃ 2  b
2t* þ
I*
t*
Oð2Þ ðt*Þ ¼3[58] The scaled infiltration rate q*(t*) can be obtainedby evaluating the time derivative of both terms of theequation (A1), leading to
1¼




1
d
expðb I*Þ þ b  1
q*  q*
ln
1b
dI*
b

ðB1aÞ

which leads toq*ðI*Þ ¼ 1 þbexpð b I*Þ  1

q*Oð2Þ ðI*Þ ¼

ðB5aÞ

1
b
þ1
I*
2

ðB5bÞ

These approximations can be combined with the short-time
expansions of I*(t*) to give

ðA8bÞ

Appendix B: Analysis of Scaled Infiltration
Rate q*(t*)

ðB4Þ

1
I*

q*
Oð1Þ ðI*Þ ¼

ðA7Þ

ðA8aÞ

B
1
B
lnB
ð1  b Þ @

1 1
b b
1þ
x* C
C
1 C
A
1þ
x*

This corresponds to equation (13) for the steady statevalidity time in the theory section.
[60] Close to zero time, the scaled cumulative infiltration
I*(t*) tends to zero. Hence, the scaled infiltration flux q*(t*)
tends to infinity (equation (B1b)). Short time approximations of q*(I*) can be defined using the series expansions ofexp(x) and (1 + x)1 leading topﬃﬃﬃ
From equation (A7), we can obtain A1 = 2 and A2 = 2b
3 .
The expressions for the first- and second-order approximations can be written aspﬃﬃﬃﬃﬃﬃﬃ
I*2t*
Oð1Þ ðt*Þ ¼

ðB3Þ

where t*q+1 is the time that corresponds to I*q+1 . This can bederived from the equation (A1), leading tot*qþ1 ðx*Þ ¼

ðt *n Þ ¼1bln 1 þ
bx

t*  t*qþ1

ðA4Þ

considering only the first two terms of (A4), thiscorresponds toI*ðt*Þ ¼ A1

I*  Iqþ1
* ¼

1 12
q*
Oð1Þ ðt*Þ ¼ pﬃﬃﬃ t*
2

ðB6aÞ

1 1 2  b
2 þq*Oð2Þ ðt*Þ ¼ pﬃﬃﬃ t*
3
2

ðB6bÞ

[61] These expressions correspond to equations (14) and(15) in the theory section. Note that these approximations,such as q*(t*), are not defined at time zero (they converge
to infinity when time approaches zero). Also note that the
proposed approach to determine the expressions forq*O(1)(t*) and q*
O(2)(t*) using the serial expansions is a more
proper way than directly calculating derivatives from theanalytical expressions for I*O(1)(t*) and I*O(2)(t*) (equations(A8a) and (A8b)). Actually, derivating term by term a serial
function does not always lead to a proper definition of its
derivate. To do so, the series require specific analyticalproperties [Ayres and Mendelson, 1999].

ðB1bÞ

References
Angulo-Jaramillo, R., J. P. Vandervaere, S. Roulier, J. L. Thony, J. P.
Gaudet, and M. Vauclin (2000), Field measurement of soil surface

18 of 20

W12415LASSABATÈRE ET AL.: EVALUATION INFILTRATION

hydraulic properties by disc and ring infiltrometers: A review andrecent developments, Soil Tillage Res., 55, 1 – 29, doi:10.1016/S01671987(00)00098-2.
Ayres, F., Jr., and E. Mendelson (1999), Schaum’s Outline of Calculus,
McGraw-Hill, New York.
Basha, H. A. (1999), Multidimensional linearized nonsteady infiltration
with prescribed boundary conditions at the soil surface, Water Resour.
Res., 35, 75 – 93, doi:10.1029/1998WR900015.
Basha, H. A. (2002), Burger’s equation: A general nonlinear solution of
infiltration and redistribution, Water Resour. Res., 38(11), 1247,
doi:10.1029/2001WR000954.
Braester, C. (1973), Moisture variation at the soil surface and the advanceof the wetting front during infiltration at constant flux, Water Resour.
Res., 9, 687 – 694, doi:10.1029/WR009i003p00687.
Braud, I., R. Haverkamp, J. L. Arrúe, and M. V. López (2003), Spatial
variability of soil surface properties and consequences for the annual
and monthly water balance of a semiarid environment (EFEDA experiment), J. Hydrometeorol., 4, 121 – 137, doi:10.1175/1525-7541(2003)
004<0121:SVOSSP>2.0.CO;2.
Braud, I., D. De Condappa, J. M. Soria, R. Haverkamp, R. AnguloJaramillo, S. Galle, and M. Vauclin (2005), Use of scaled forms of theinfiltration equation for the estimation of unsaturated soil hydraulic properties (Beerkan method), Eur. J. Soil Sci., 56, 361 – 374, doi:10.1111/
j.1365-2389.2004.00660.x.
Campbell, S., J. P. Chancelier, and R. Nikoukhah (2006), Modeling and
Simulation in Scilab/Scicos, Springer, New York.
Clausnitzer, V., J. W. Hopmans, and J. L. Starr (1998), Parameter uncertainty analysis of common infiltration models, Soil Sci. Soc. Am. J., 62,1477 – 1487.
Feddes, R. A., P. Kabat, P. J. T. Vanbakel, J. J. B. Bronswijk, and
J. Halbertsma (1988), Modelling soil water dynamics in the saturated
zone—State of the art, J. Hydrol., 100, 69 – 111, doi:10.1016/00221694(88)90182-5.
Fuentes, C., R. Haverkamp, and J. Y. Parlange (1992), Parameter constraints on closed-form soil water relationships, J. Hydrol., 134, 117 –
142, doi:10.1016/0022-1694(92)90032-Q.
Galle, S., et al. (2001), Estimation of soil hydrodynamic properties of theDonga watershed (CATCH-Bénin), paper presented at GEWEX 4thInternational Conference, Global Energy and Water Cycle Exp., Paris,
10 – 14 Sept.
Green, W. H., and G. Ampt (1911), Studies on soil physics, 1. The flow of
air and water through soils, J. Agric. Sci., 4(1), 1 – 24, doi:10.1017/
S0021859600001441.
Haverkamp, R., J. Y. Parlange, J. L. Starr, G. Schmitz, and C. Fuentes(1990), Infiltration under ponded conditions: 3. A predictive equation
based on physical parameters, Soil Sci., 149, 292 – 300, doi:10.1097/
00010694-199005000-00006.
Haverkamp, R., P. J. Ross, K. R. J. Smetten, and J. Y. Parlange (1994),
Three-dimensional analysis of infiltration from the disc infiltrometer: 2.
Physically based infiltration equation, Water Resour. Res., 30, 2931 –
2935, doi:10.1029/94WR01788.
Haverkamp, R., J. L. Arrúe, J. P. Vandervaere, I. Braud, G. Boulet, J. P.
Laurent, A. Taha, P. J. Ross, and R. Angulo-Jaramillo (1996), Hydrological and thermal behaviour of the vadose zone in the area of Barrax
and Tomelloso (Spain): Experimental study, analysis and modeling,
Proj. UE EV5C-CT 92 00 90, Eur. Union, Brussels.
Haverkamp, R., F. Bouraoui, C. Zammit, R. Angulo-Jaramillo, and J. W.
Delleur (1999), Soil properties and moisture movement in the unsaturated zone. 2931 – 2935, in The Handbook of Groundwater Engineering,edited by J. W. Delleur, pp. 5.1 – 5.50, CRC Press, Boca Raton, Fla.
Haverkamp, R., D. Debionne, P. Viallet, R. Angulo-Jaramillo, and D. de
Condapa (2005), Soil properties and moisture movement in the unsaturated zone, in The Handbook of Groundwater Engineering, edited byJ. W. Delleur, pp. 1 – 59, CRC Press, Boca Raton, Fla.
Jacques, D., B. P. Mohanty, and J. Feyen (2002), Comparison of alternative
methods for deriving hydraulic properties and scaling factors from single-disc tension infiltrometer measurements, Water Resour. Res., 38(7),1120, doi:10.1029/2001WR000595.
Lassabatere, L., R. Angulo-Jaramillo, J. M. Soria Ugalde, R. Cuenca,
I. Braud, and R. Haverkamp (2006), Beerkan estimation of soil transfer
parameters through infiltration experiments—BEST, Soil Sci. Soc. Am.
J., 70, 521 – 532, doi:10.2136/sssaj2005.0026.
Lin, H. S., and K. J. McInnes (1995), Water flow in clay soil beneath atension infiltrometer, Soil Sci., 159, 375 – 382, doi:10.1097/00010694199506000-00002.

W12415

Mallants, D., D. Jacques, P. H. Tseng, M. T. van Genuchten, and J. Feyen(1997), Comparison of three hydraulic property measurement methods,
J. Hydrol., 199, 295 – 318, doi:10.1016/S0022-1694(96)03331-8.
Messing, I., and N. J. Jarvis (1993), Temporal variation in the hydraulicconductivity of a tilled clay soil as measured by tension infiltrometers,
J. Soil Sci., 44, 11 – 24, doi:10.1111/j.1365-2389.1993.tb00430.x.
Mualem, Y. (1976), A new model for predicting the hydraulic conductivityof unsaturated porous media, Water Resour. Res., 12, 513 – 522,
doi:10.1029/WR012i003p00513.
Parlange, J.-Y. (1975), On solving the flow equation in unsaturated soil by
optimization: Horizontal infiltration, Soil Sci. Soc. Am. J., 39, 415 – 418.
Parlange, J.-Y., I. Lisle, R. D. Braddock, and R. E. Smith (1982), The threeparameter infiltration equation, Soil Sci., 133, 337 – 341, doi:10.1097/
00010694-198206000-00001.
Parlange, J.-Y., R. Haverkamp, and J. Touma (1985), Infiltration under
ponded conditions: 1. Optimal analytical solution and comparison withexperimental observations, Soil Sci., 139, 305 – 311.
Perroux, K. M., and I. White (1988), Designs for disc permeameters, Soil
Sci. Soc. Am. J., 52, 1205 – 1215.
Philip, J. R. (1969), Theory of infiltration, Adv. Hydrosci., 5, 215 – 296.
Philip, J. R. (1973), On solving the unsaturated flow equation: 1. The fluxconcentration relation, Soil Sci., 116, 328 – 335, doi:10.1097/00010694197311000-00002.
Rasoulzadeh, A., and A. R. Sepaskhah (2003), Scaled infiltration equations for furrow irrigation, Biosyst.Eng., 86, 375 – 383, doi:10.1016/
j.biosystemseng.2003.07.004.
Ross, P. J., R. Haverkamp, and J.-Y. Parlange (1996), Calculating parameters for infiltration equation from soil hydraulic functions, Transp.
Porous Media, 24, 315 – 339, doi:10.1007/BF00154096.
Salvucci, G., and D. Entekhabi (1994), Explicit expressions for GreenAmpt (delta function diffusivity) infiltration rate and cumulative storage,
Water Resour. Res., 30, 2661 – 2663, doi:10.1029/94WR01494.
Schaap, M. G., F. J. Leij, and M. T. van Genuchten (2001), ROSETTA: A
computer program for estimating soil hydraulic parameters with hierarchical pedotransfer functions, J. Hydrol., 251, 163 – 176, doi:10.1016/
S0022-1694(01)00466-8.
Šimůnek, J., and M. T. van Genuchten (1996), Estimating unsaturated soil
hydraulic properties from tension disc infiltrometer data by numerical inversion, Water Resour. Res., 32, 2683 – 2696, doi:10.1029/96WR01525.
Šimůnek, J., and M. T. van Genuchten (2008), Modeling nonequilibrium
flow and transport processes using HYDRUS, Vadose Zone J., 7, 782 –
797, doi:10.2136/vzj2007.0074.
Šimůnek, J., R. Angulo-Jaramillo, M. G. Schaap, J. P. Vandervaere, and
M. T. van Genuchten (1998), Using an inverse method to estimate thehydraulic properties of crusted soils from tension-disc infiltrometer
data, Geoderma, 86, 61 – 81, doi:10.1016/S0016-7061(98)00035-4.
Šimůnek, J., M. T. van Genuchten, and M. Šejna (2006), The HYDRUS
software package for simulating two- and three-dimensional movementof water, heat, and multiple solutes in variably saturated media, version
1.0, technical manual, PC Progress, Prague.
Šimůnek, J., M. Šejna, H. Saito, M. Sakai, and M. T. van Genuchten(2008), The HYDRUS-1D software package for simulating the onedimensional movement of water, heat, and multiple solutes in variably
saturated media, version 4.0, HYDRUS Software Ser. 3, Dep. of Environ.
Sci., Univ. of Calif., Riverside.
Smetten, K. R. J., J. Y. Parlange, P. J. Ross, and R. Haverkamp (1994),
Three-dimensional analysis of infiltration from the disc infiltrometer:
1. A capillary-base theory, Water Resour. Res., 30, 2925 – 2929.
Smith, R. E., K. R. J. Smetten, P. Broadbridge, and D. A. Woolhiser (2002),
Infiltration Theory for Hydrology Applications, Geophys. Monogr. Ser.,vol. 129, AGU, Washington.
Turner, N. C., and J. Y. Parlange (1974), Lateral movement at the peripheryof a one-dimensional flow of water, Soil Sci., 118(2), 70 – 77,
doi:10.1097/00010694-197408000-00002.
Vandervaere, J. P., M. Vauclin, and D. E. Elrick (2000a), Transient flowfrom tension infiltrometers: II. Four methods to determine sorptivity andconductivity, Soil Sci. Soc. Am. J., 64, 1272 – 1284.
Vandervaere, J. P., M. Vauclin, and D. E. Elrick (2000b), Transient flowfrom tension infiltrometers: I. The two-parameter equation, Soil Sci. Soc.
Am. J., 64, 1263 – 1272.
van Genuchten, M. T. (1980), A closed form equation for predicting thehydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J., 44,892 – 898.
Varado, N., I. Braud, P. J. Ross, and R. Haverkamp (2006), Assessment ofan efficient numerical solution of the 1D Richards’ equation on bare soil,
J. Hydrol., 323, 244 – 257, doi:10.1016/j.jhydrol.2005.07.052.

19 of 20

W12415LASSABATÈRE ET AL.: EVALUATION INFILTRATION

Warrick, A. W., and N. Lazarovitch (2007), Infiltration from a strip source,
Water Resour. Res., 43, W03420, doi:10.1029/2006WR004975.
Warrick, A. W., N. Lazarovitch, A. Furman, and D. Zerihun (2007),
Explicit infiltration function for furrows, J. Irrig. Drain., 133(4),307 – 313, doi:10.1061/(ASCE)0733-9437(2007)133:4(307).
Zhu, J., and B. P. Mohanty (2002), Analytical solutions for steady-statevertical infiltration, Water Resour. Res., 38(8), 1145, doi:10.1029/
2001WR000398.

W12415

R. Haverkamp, Laboratoire d’Etude des Transferts en Hydrologie etEnvironnement, CNRS, F-38041 Grenoble, France.
L. Lassabatere, Division Eau et Environnement, LCPC, Route de
Bouaye, F-44341 Bouguenais, France. (laurent.lassabatere@lcpc.fr)
J. Šimůnek, Department of Environmental Sciences, University ofCalifornia, Riverside, CA 92521, USA.
J. M. Soria-Ugalde, Departamento de Ingenierı́a Geomática e Hidráulica,
Universidad de Guanajuato, Avenida Juárez 77, 36000 Guanajuato, Mexico.




R. Angulo-Jaramillo, Laboratoire des Sciences de l’Environnement,
Ecole Nationale des Travaux Publics de l’Etat, Rue Maurice Audin,
F-69518 Vaulx-en-Velin, France.

20 of 20