Irrig Sci (2010) 28:435–444
DOI 10.1007/s00271-009-0205-9

ORIGINAL PAPER

Comparison of numerical, analytical, and empirical models
to estimate wetting patterns for surface and subsurface
drip irrigation
Maziar M. Kandelous • Jiřı́ Šimůnek

Received: 30 June 2009 / Accepted: 17 December 2009 / Published online: 7 January 2010
Ó The Author(s) 2010. This article is published with open access at Springerlink.com

Abstract Due to increasing competition for water
resources by urban, industrial, and agricultural users, theproportion of agricultural water use is gradually decreasing. To maintain or increase agricultural production, newirrigation systems, such as surface or subsurface drip irrigation systems, will need to provide higher water use
efficiency than those traditionally used. Several models
have been developed to predict the dimensions of wetting
patterns, which are important to design optimal drip irrigation system, using variables such as the emitter discharge, the volume of applied water, and the soil hydraulicproperties. In this work, we evaluated the accuracy ofseveral approaches used to estimate wetting zone dimensions by comparing their predictions with field and laboratory data, including the numerical HYDRUS-2D model,
the analytical WetUp software, and selected empiricalmodels. The soil hydraulic parameters for the HYDRUS2D simulations were estimated using either Rosetta for thelaboratory experiments and inverse analysis for the field
experiments. The mean absolute error (MAE) was used tocompare the model predictions and observations of wettingzone dimensions. MAE for different experiments anddirections varied from 0.87 to 10.43 cm for HYDRUS-2D,

Communicated by J. Ayars.
M. M. Kandelous  J. Šimůnek (&)
Department of Environmental Sciences,
University of California Riverside,
Riverside, CA 92521, USAe-mail: jiri.simunek@ucr.edu
M. M. Kandelous
e-mail: mkandelous@gmail.com

from 1 to 58.1 cm for WetUp, and from 1.34 to 12.24 cm
for other empirical models.

Introduction
Surface and subsurface drip irrigation systems are one of themost efficient systems for irrigating crops, vegetables, andfruit trees. Optimally designed drip irrigation systems should
produce water distribution between the emitters and laterals
as uniformly as possible. Water distribution is strongly
affected by the soil hydraulic properties, spacing and depthof emitters and laterals, and the emitters’ discharge.
There are a number of models that describe infiltrationfrom a point/line source that can be used to design, install,and manage drip irrigation systems (e.g., Camp 1998;
Singh et al. 2006). Some of these analytical, numerical, andempirical models have been developed to estimate wettingzone dimensions for surface and subsurface drip irrigationfrom a point source. While empirical models have typically
been developed using a regression analysis of field observations, analytical and numerical models usually solve
governing flow equations for particular initial and boundary conditions. A different approach was suggested by
Lazarovitch et al. (2007) who introduced a moment analysis approach to describe spatial and temporal subsurface
wetting patterns for irrigation from surface/subsurface drip
irrigation systems.
On the one hand, Cook et al. (2003) developed a userfriendly software tool that uses analytical solutions toestimate the wetting pattern for different soil hydraulicproperties, emitter positions, and volumes of water applied
to homogenous soils from a surface or subsurface point
source. On the other hand, HYDRUS-2D (Šimůnek et al.
1999) is a well-known Windows-based computer software

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package that uses numerical techniques to simulate water,
heat, and/or solute movement in two-dimensional, variably
saturated porous media. This software has been evaluatedby several researchers who assessed its ability to simulate
water movement from the subsurface drip irrigation system(e.g., Šimůnek et al. 2008, and references given there). For
example, Liga and Slack (2004) used HYDRUS-2D toestimate the wetting pattern for subsurface drip irrigation,
but did not compare the results of HYDRUS-2D simulations with actual experimental data. Skaggs et al. (2004)
carried out an extensive analysis of multiple subsurface
drip irrigation field experiments, and successfully compared observed wetting patterns with HYDRUS-2D model
predictions. Lazarovitch et al. (2005) implemented into
HYDRUS-2D a new, system-dependent boundary condition that considers source properties, inlet pressure, andeffects of the soil hydraulic properties on calculated subsurface source discharge and validated resulting code
against transient experimental data. Cook et al. (2006)
compared wetting patterns estimated for both surface andsubsurface drip irrigation systems using WetUp and HYDRUS-2D. This study was purely theoretical and did not
involve any observation data. Cook et al. (2006) found thatsimilar results were provided for fine-textured soils by bothmodels. Note that models based on analytical and numerical solutions of the governing flow equation do not necessarily provide the same results, since analytical models
often involve many simplifying assumptions that may not
fully represent the observed reality.
Several empirical models have been presented in theliterature to estimate the distance of the wetting front froma surface or subsurface dripper. For example, Schwartzman
and Zur (1986) developed an empirical model to estimate
the vertical and horizontal distances of a wetting front froma surface point source. Their empirical model was developed based on experiments carried out in the field on two
soils (Gilat loam and Sinai sand), with emitter dischargesof 1.1 9 10-6 and 5.6 9 10-6 m3 s-1. Amin and Ekhmaj(2006) verified this model using several experimental
datasets and modified it by including the saturated soil
water content as one of the model parameters. Finally,
Kandelous et al. (2008) developed an empirical model for
estimating upward, downward, and horizontal distancesfrom the wetting front to a subsurface point source. Thisempirical model was developed based on data collected in
the laboratory on the clay loam soil using a subsurface
dripper installed at a depth of 30 cm and having an emitter
discharge of 2.78 9 10-7 m3 s-1.
In this paper, we compare the accuracy of these different
empirical, analytical, and numerical models for estimating
dimensions of the wetting zone, compare their predictions
against laboratory and field data, and discuss their particular advantages and disadvantages.

123

Irrig Sci (2010) 28:435–444

Description of various models
Numerical model HYDRUS-2D
Since all experiments used only one emitter as the point
source of water, water movement during infiltration could
be considered an axisymmetrical process. The following
Richards equation is the governing equation for water flow
in a homogenous and isotropic soil:
oh 1 oohooh¼
rK ðhÞ
K ð hÞ þ K ð hÞ
ð1Þ
þot r ororoz
oz
where h is the volumetric water content (L3L-3), h is thesoil water pressure head (L), t is time (T), r is the radial
space coordinate (L), z is the vertical space coordinate (L),and K is the hydraulic conductivity (LT-1). The soil
hydraulic properties were modeled using the van
Genuchten–Mualem constitutive relationships (vanGenuchten 1980) as follows:8>< hr þ h s  hrm
ð1 þ jahjn Þ
h\0ð2Þhð hÞ ¼>:h h0shm i2
K ðhÞ ¼ Ks Sle 1  1  S1=mewhereh  hrSe ¼;
hs  hrð3Þ
m ¼ 1  1=nwhere hs is the saturated water content (L3L-3), hr is theresidual water content (L3L-3), Ks is the saturated
hydraulic conductivity (LT-1), and a (L-1), n (-) and l (-)
are shape parameters.
HYDRUS-2D uses the Galerkin finite-element method
to solve the governing water flow equation. For subsurface
drip simulations, the transport domain, for which thenumerical solution was obtained, was rectangular, except
for the semicircle on the left side of the domain representing the dripper. The location of the semicircle depended on the location of the dripper. During water
application, a constant flux boundary condition was used at
the emitter. The constant water flux was calculated by
dividing the water discharge by the surface area of theemitter. At the end of the irrigation event, the emitterboundary became a zero flux boundary (Skaggs et al.
2004). The remaining part of the left boundary was a zero
flux boundary both during and after the irrigation event.
Zero flux boundary conditions were used at the right andbottom boundaries, since the computational flow domain
was large enough that these boundaries did not affect waterflow in the domain. A zero flux boundary condition was
also used at the soil surface because evaporation could be
neglected due to the surface plastic mulch used during andIrrig Sci (2010) 28:435–444

437

after irrigation. For surface drip simulations, the transportdomain was rectangular. All boundary conditions were
zero flux, except a small part at the top of the domain close
to the axis of symmetry that represented a surface dripper.
The length of this part was such to prevent surface ponding
during simulations. For all calculations, the standard version of HYDRUS-2D was used. No specialized dynamic
boundary conditions suitable for various micro irrigation
schemes (e.g., variable ponding for surface drip irrigation(Gärdenäs et al. 2005) or the effect of the back pressure forsubsurface drip irrigation (Lazarovitch et al. 2005))
recently implemented into HYDRUS-2D were used.
WetUp

evaluate them in real time. WetUp contains a database of
precalculated values for the predefined irrigation rates,
application times, antecedent water contents, and emitter
locations, and it only interpolates within this database once
input parameters are specified by the user. The database in
WetUp currently contains information for Australian soils
reported by Clapp and Hornberger (1978) and Verburg
et al. (2001), for emitter discharges varying from1.40 9 10-7 to 7.53 9 10-7 m3 s-1, and for three different initial soil matric potentials of -10, -6, and -3 m; i.e.,
for dry, moist, and wet soils. The value of aw given in the
database for clay loam was used in calculations presented
below.
For a surface source, the wetting perimeter is given byanother equation as follows:

Cook et al. (2003) developed easy-to-use software that uses
Philip’s (1984) analytical solutions for flow from a surface
or subsurface point to estimate a wetting perimeter forsurface or subsurface drip irrigation. The distance between
the emitter and the edge of the wetting front for a buriedsource is given implicitly by the following equation:
exp 2Rsin2 12 /T ðR;/Þ ¼
2cos2 12 /
 

9
8
1
1
1
2
2
>
>
>
>
R
/

Rsin
/
þ

R
þ
ln
cos
>
>
>
>
2
2
2
>
>
>
>
>
>
>
>
!
>
>
=
<
21
21
exp 2Rsin 2/  cos 2/
 ln>>sin2 12/
>
>
>
>
>
>
>
>






>
>
>
>
>
>
1
1
1
1
>
;
:  L sec2 / exp 2Rsin2 /  L sec2 / >2222T ðR; /Þ ¼ exp½Rð1  cos /Þ
9
8 

1
>
>
>
>
R2 1  cos /  Rþ
>
>
2
>
>
>
>
>
>
>
>
2
3
>
>
>
>
Rð1  cos /Þ lnð1  cos /Þ
>
>
=
<
6
7


6
7
1 6
1  cos / exp½Rðcos /  1Þ
>
7>
>
>
>
6 þ ln
7>
>
>
>
6
7>
cos
/
1

cos
/
>
>
>
4
5>
>
>
>
>
>
>
:
 Lð1  cos / exp½Rðcos /  1ÞÞ þ Lð1  cos /Þ ;10\/\ pð7Þ
2

ð4Þ

Schwartzman and Zur (1986) developed an empiricalmodel to estimate the wetting pattern from a surface point
source. The empirical model was developed using experimental results for two soils (Gilat loam and Sinai sand) andfor
twoemitterdischarges(1.1 9 10-6
and
-6
3 -1
5.6 9 10 m s ). The saturated hydraulic conductivities, Ks, were 2.4 9 10-6 m s-1 for the Gilat loam and2.4 9 10-5 m s-1 for the Sinai sand. The Schwartzman
and Zur (1986) empirical model is as follows:0:017
0:22 KSW ¼ 1:82ðVw Þ
ð8Þ
Q

0:45
0:63 KS
Z ¼ 2:54ðVw Þ
ð9Þ
Q

0\/\p

where r and / are spherical polar coordinates (L), R = awr/
2 is the dimensionless length, aw is the reciprocal of themacroscopic capillary length (L-1), and L(x) is the
dilogarithm defined as:Lð xÞ ¼ 

Zx

ln x
dx
x1

ð5Þ

1

Dimensionless time T in (4) is given by:
T¼

a3w Qt16pDh

ð6Þ

where Q is the discharge of the emitter (L3T-1) located at(s, z) = (0,0), s is the radial distance and z is the depth (L)(s = r sin/, z = r cos/), t is the actual time (T), and Dh isthe average volumetric water content change behind thewetting front (L3L-3). Although WetUp is based on theanalytical solutions described previously, it does not

Since the wetting pattern calculated by WetUp is always
elliptical, the diameters of the estimated ellipse wereselected to represent dimensions of the wetting zone.
Schwartzman and Zur (1986)

where W and Z are horizontal and vertical dimensions of
the wetting profile in meters, respectively, Vw is the total
volume of applied water (m3), Q is the emitter discharge(m3 s-1), and Ks is the soil saturated hydraulic conductivity(ms-1).

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Irrig Sci (2010) 28:435–444

Amin and Ekhmaj (2006)

Laboratory lysimeter experiments

Amin and Ekhmaj (2006) developed the following equations for estimating horizontal (R) and vertical downward(Z) distances of the wetting front from the surface drip
emitter using several different experimental datasets(Taghavi et al. 1984; Angelakis et al. 1993; Moncef et al.
2002; and Li et al. 2003) using nonlinear regression:Experiments were conducted at the central laboratory of
the College of Agricultural and Natural Resources of the
University of Tehran, Iran. Laboratory experiments wereconducted on a 2 m 9 1 m 9 1.2 m lysimeter that had
transparent walls and was filled with a clay loam soil
(33.5% clay, 39.7% silt, 26.8% sand) (Table 1). To prevent
preferential flow along the walls, lysimeter walls were firsttreated with glue and sprayed with sand to create a course
surface, before the soil was filled into the lysimeter.
A thin pipe connected a 2-cm-diameter sphere installed
30 cm below the soil surface with an emitter placed on thesoil surface. Since the subsurface sphere, made of foamed
plastic polymers to prevent clogging of the irrigation
device, was directly hydraulically connected with the surface emitter, it represented in our physical model a subsurface dripper. The subsurface sphere was installed in theclose vicinity and in the middle of one of the lysimeter
walls. The lysimeter represented a half space of a subsurface drip irrigation problem and could thus be treatedmathematically as an axisymmetrical problem. Water was
delivered to the emitter using a 20-mm nominal diameter
polyethylene pipe from a reservoir located on a scale 3 m
above the emitter. The average water discharge used in
calculations was 3.6 9 10-7 and 5 9 10-7 m3 s-1 for the
first and second irrigation experiment, respectively.
The shape of the wetting front was drawn on thetransparent wall about every 2 h until the end of irrigation.
ROSETTA (Schaap et al. 2001), a pedotransfer function
software package that uses neural network models to predict soil hydraulic parameters from soil texture and relateddata, was used to estimate the soil hydraulic parameters hr,
hs, a, n, Ks, and l for the van Genuchten–Mualem model.
Parameters required for the most complex ROSETTA
model are bulk density (1.35 g cm-3), percentages of sand(26.8%), silt (39.7%) and clay (33.5%), and water contents
for pressures of -33 and -1,500 k Pa (0.25 and 0.133,
respectively). For these input variables, ROSETTA predicted the following soil hydraulic parameters:
hr = 0.06 m3 m-3, hs = 0.41 m3 m-3, a = 2.3 m-1,
n = 1.34, Ks = 2.5 9 10-6 m s-1, and l = 0.5 (Table 2).
The average initial volumetric water content obtained frommeasurements with soil sensors was hi = 0.07 cm3 cm-3
for the first irrigation experiment and 0.11 cm3 cm-3 forthe second.

R ¼ 0:2476 Dh0:5626 Vw0:2686 Q0:0028 KS0:0344

ð10Þ

Z ¼ 2:0336 Dh0:383 Vw0:365 Q0:101 KS0:195

ð11Þ

where R and Z are horizontal and vertical dimensions of thewetting pattern (m), respectively, Dh is the average volumetric water content change behind the wetting front, Vw isthe total volume of applied water (m3), Q is the emitter
discharge (m3 s-1), and Ks is the soil saturated hydraulicconductivity (ms-1). It is often assumed, when no otherguidance is available, that Dh is equal to about half of thesaturated soil water content (i.e., Dh = hs/2).
Kandelous et al. (2008)
Kandelous et al. (2008) developed the following equations
for estimating horizontal (W), vertical upward (Z?), andvertical downward (Z-) distances (m) of the wetting front
from the subsurface emitter using dimensional analysis in amethod similar to Singh et al. (2006):
KS ð0:026Þ
W ¼ 4:244 Vw0:526
ð12Þ
QZ

ð0:156Þ
0:344 KSZþ ¼ 0:72 Vw
ð13Þ
QZ

KS ð0:167Þ
Z ¼ 0:66 Vw0:333
ð14Þ
QZ
where Vw is the volume of applied infiltrating water (m3), Q
is the emitter discharge (m3 s-1), K is the saturated
hydraulic conductivity (ms-1), and Z is the emitter installation depth (m). These equations were developed using
experimental data from subsurface drip irrigation experiment with an initially air-dried clay loam and an emitter
discharge of 2.78 9 10-7 m3 s-1, during which the
movement of the moisture front was monitored over time.

Materials and methods
The following two sections describe laboratory and field
experiments that were carried out in collecting information
about the wetting pattern for subsurface and surface drip
irrigation.

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Table 1 Soil physical properties
Clay (%)

Silt (%)

Sand (%)

Bulk density (g cm-3)

Lysimeter

33.5

39.7

26.8

1.35

Field

32.5

36.5

31

1.55

Irrig Sci (2010) 28:435–444
Table 2 van Genuchten–
Mualem’s (van Genuchten1980) soil hydraulic parameters439
hr (cm3 cm-3)

hs (cm3 cm-3)

a (m-1)n

Ks (m s-1)lLysimeter

0.06

0.41

2.3

1.34

2.5 9 10-6

0.5

Field

0.07

0.38

1

1.89

5 9 10-6

0.5

Field experiments
Field experiments were carried out at the Research Field of
the College of Agricultural and Natural Resources of the
University of Tehran, Iran, on a clay loam soil (32.5% clay,
36.5% silt, 31% sand) (Table 1). Twenty-five experiments
included 15 surface drip irrigation experiments, 5 subsurface drip irrigation experiments with an emitter depth of
15 cm, and 5 subsurface drip irrigation experiments with
an emitter depth of 30 cm. Each experiment was conductedin a different location of the same field and involved
infiltration from a single emitter. Different irrigation volumes were used for different irrigation experiments. The
emitter discharge in these experiments varied from6.94 9 10-7 to 7.14 9 10-7, 1.14 9 10-6 to 1.3 9 10-6,
and 1.71 9 10-6 m3 s-1 for the surface drip irrigation(denoted as experiments A, B, and C, respectively), andfrom 5.28 9 10-7 to 9.69 9 10-7 m3 s-1 for subsurface
drip irrigation. For each experiment, the same emitter
discharge (the average emitter discharge during theexperiment) was used in all models. Water from the reservoir was delivered with millimeter precision to theemitter using a 160-mm nominal diameter PVC pipe and asmall pump (Fig. 1). At the end of each irrigation experiment, the soil around the emitter was dug out, and the
distance of the wetting front from the emitter was measured
in the horizontal, vertical downward, and vertical upward(in SDI experiments) directions. Maximum distances
between the wetting front and the emitter in particular
directions (upward, downward, and horizontal) were usedin comparison with various models.

Rosetta was used first to provide initial estimates of thesoil hydraulic parameters from textural soil properties.
However, HYDRUS simulations with the Rosetta-estimatedparameters could not provide good descriptions of measured data. For example, the estimated saturated hydraulicconductivity (5.0 9 10-7 ms-1) was too low to give good
descriptions of moisture front movement. Note that Rosetta’s estimates are based on soil textural properties and could
thus describe reasonably well a repacked soil used in thelaboratory, but did not work well for an undisturbed soil in
the field where soil structure likely affected soil hydraulicproperties. The soil hydraulic parameters for the field
experiment were, therefore, further estimated using theinverse solution option of HYDRUS-2D and one additional
subsurface drip irrigation experiment carried out specifically for this purpose. The emitter was installed 30 cm
below the soil surface, and the average emitter discharge
was 7.5 9 10-7 m3 s-1 for 5 h. At the end of the irrigation
experiment, the soil surrounding the emitter was excavated
to expose a vertical soil profile with the emitter in thecenter. Water contents measured at soil samples taken inthree locations (0, 12.5, and 25 cm) away from the emitterand in four depths (10, 20, 30, and 40 cm) were then used in
the inverse analysis. Parameters n and Ks were optimized
using HYDRUS-2D to get better description of the measured wetting pattern for this control experiment. The final
estimated soil hydraulic parameters for a clay loam soil are
hr = 0.07 cm3 cm-3, hr = 0.38 cm3 cm-3 (measured in
the laboratory on soil samples), a = 1 m-1, n = 1.89,
Ks = 5 9 10-6 m s-1, and l = 0.5 (Table 2). The average
dry bulk density was 1.55 g cm-3. The average initial watercontent for the field experiments was 0.1 cm3 cm-3.
Statistical analysis
Mean absolute errors (MAE) between the simulated andmeasured dimensions of the wetting zone in different
directions were calculated to provide a quantitative comparison between measured and simulated data.
n
1X
MAE ¼
ð15Þ
jfi  yi j
n i¼1

Fig. 1 A schematic of a pump and a water reservoir

where fi and yi are observed and simulated values,
respectively, and n is the number of observations. All
simulated and measured data were converted into the sameunits (cm) before the MAE value was calculated for eachmodel.

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Result and discussions
Lysimeter experiments
First, various model predictions are compared against two
irrigation events carried out on the laboratory lysimeter.
The numerical model, HYDRUS-2D, the analytical model,
WetUp, and Kandelous’ empirical model were used. Notethat the other two empirical models discussed earlier are
not suitable for subsurface irrigation, since they weredeveloped for surface drip irrigation.
HYDRUS-2D and Kandelous’ empirical model performed better than the WetUp for the first irrigation event(Fig. 2). As shown in Table 3, the value of MAE for different distances varied from 0.87 to 2.14 cm for HYDRUS2D, from 1.34 to 1.81 cm for the Kandelous’ empiricalmodel, and from 7.64 to 12.72 cm for WetUp. However,for the second irrigation event, WetUp performed better
than the other models. Table 3 shows that values of MAE
varied from 3.66 to 10.43 cm for HYDRUS-2D, from 1.45
to 11.9 for the Kandelous’ empirical model, and from 3.79
to 5.25 cm for WetUp.
The soil profile was uniformly dry before the first irrigation event. In HYDRUS-2D simulations, the initial condition (given in terms of soil water content) was assumed to
be uniform and equal to the measured soil water content.

Irrig Sci (2010) 28:435–444

Consequently, the model provided good predictions of theobserved data when correct initial conditions were available. The Kandelous’ empirical model, which was developed for a soil that is similar in dryness to the soil before thefirst irrigation, also provided good estimates. However,
WetUp, which offers only three general initial conditions,
i.e., dry (used here), moist, and wet, did not predict thedimensions of the wetting pattern well.
The second irrigation experiment (Fig. 2) was conducted
two weeks after the first one when the initial soil watercontent was not uniform, but was still variable. However,
although not required by the model, we still assumed thatthe initial condition was uniform and equal to the averagesoil water content for the entire soil profile in HYDRUS-2D
calculations, in order to have the same conditions for allmodels. Because of this assumption, which did not fully
correspond with reality, the accuracy of the HYDRUS-2D
predictions decreased. The value of the MAE varied from3.66 to 10.43 cm for HYDRUS-2D, from 3.79 to 5.25 cm
for WetUp, and from 1.45 to 11.9 cm for the Kandelous’empirical model. The MAE values in Table 3 show thatwhile WetUp performed better for the second irrigation
event than the other models, it did not do so in predicting the
upward movement. The empirical model of Kandelous,
which was developed for different initial conditions (dry),
could not provide good description of the wetting pattern.
At the beginning of irrigation, when water was infiltrating into the soil close to the emitter where the soil watercontent was higher than elsewhere, observed data andpredictions by WetUp were closer to each other than atlater times when the wetting front reached the drier soil.
From this, it can be concluded that the soil water content
around the emitter was similar to the soil water content
used as an initial condition in WetUp (used with moist
initial conditions here). Thus, once the wetting front
reached the drier soil, differences between observed andestimated data started increasing. When about 10 liters ofwater was applied, WetUp estimated that the wetting front
reached the soil surface and, therefore, the line in Fig. 2
stops. Again, it was assumed that the initial water content
was equal to the average soil water content in the soil
profile in the HYDRUS-2D simulation, which was less than
the soil water content close to the emitter. Therefore, Fig. 2
shows that when the wetting front was close to the emitter,the observed front moved faster than the one predicted by
HYDRUS-2D. The difference between observations andpredictions decreased with the increasing distance of thewetting front from the emitter.
Field experiments

Fig. 2 A comparison of observed and estimated wetting dimensions
for the lysimeter experiments: the first irrigation experiment is on theleft and the second irrigation experiment is on the right123

In this section, we will compare observed data and model
predictions for both surface drip (DI) and subsurface drip

Irrig Sci (2010) 28:435–444

441

Table 3 Statistical analysis for the comparison of observed and estimated data
Experiment

Experimental conditions

Models

Wetting front dimensions
R2

MAE (cm)

Horizontal
Lysimeter

SDI-30* First irrigation

SDI-30 Second irrigation

Field

SDI-30

SDI-15

DI-A?

DI-B

DI-C

UpwardDownward

Horizontal

UpwardDownward

HYDRUS

0.99

0.99

0.99

2.04

0.87

2.14

WetUp

0.98

0.98

0.98

10.89

7.64

12.72

Kandelous
HYDRUS

0.98
0.99

0.98
0.98

0.98
0.96

1.34
8.2

1.74
3.66

1.81
10.43

WetUp

0.98

0.98

0.96

4.29

5.25

3.79

Kandelous#

0.98

0.98

0.98

8.32

1.45

11.9

HYDRUS

0.71

0.98

0.98

3. 5

1.9

3.14

WetUp

0.71

0.98

0.98

10.25

1

10.7

Kandelous

0.72

0.96

0.98

2.2

8.95

HYDRUS

0.94

0.99

2.11

WetUp

0.86

0.98

9.65

5.6

Kandelous

0.94

0.98

1.83

12.24

HYDRUS

0.96

0.89

3.05

3.59

5.93
4.34

WetUp

0.97

0.90

55.43

12.42

Schwartzman#

0.96

0.84

5.3

4.47

Amin#

0.96

0.86

1.96

3.91

HYDRUS

0.99

0.93

2.43

4.25

WetUp
Schwartzman

0.99
0.97

0.92
0.95

58.1
4.59

12.2
10.18

Amin

0.98

0.93

2.08

4.01

HYDRUS

0.98

0.97

6.36

9.76

WetUp

0.96

0.98

54.54

21.5

Schwartzman

0.98

0.97

11.55

4.42

Amin

0.98

0.97

8.14

8.47

* SDI-30 Subsurface drip irrigation with the emitter at a depth of 30 cm?

DI-A Surface drip irrigation. A with discharge varying from 6.94 9 10-7 to 7.14 9 10-7 m3 s-1; B with discharge varying from 1.15 9 10-6
to 1.3 9 10-6 m3 s-1, and C with the emitter discharge of 1.71 9 10-6 m3 s-1#
Kandelous, Schwartzman, and Amin refer to solutions given in Kandelous et al. (2008), Schwartzman and Zur (1986), and Amin and Ekhmaj(2006), respectively(SDI) irrigations. While the numerical model HYDRUS2D and the analytical model WetUp are used for both
surface and subsurface irrigation experiments, Kandelous’empirical model is used only for subsurface drip irrigation,and the empirical models of Schwartzman and Zur (1986)
and Amin and Ekhmaj (2006) are used only for surface drip
irrigation that is for conditions for which these particularempirical models were developed.
Subsurface drip irrigation
Subsurface drip irrigation experiments were conductedwith emitters installed at a depth of 15 and 30 cm below
the soil surface. A statistical comparison of observed data
and those predicted by various models is presented inTable 3. No comparison was made for the upward direction

for experiments with the emitter installed at a depth of
15 cm, since all models predicted, as was also observed,that the wetting front would reach the soil surface during
these experiments. The value of MAE varied from 2.11 to
4.34 cm for HYDRUS-2D, from 5.6 to 9.65 cm for WetUp,
and from 1.83 to 12.24 cm for Kandelous’ empiricalmodel. In general, HYDRUS-2D provided better estimates
than the other models. Although the initial soil profile wasdry for all experiments, and thus was similar to the conditions, for which the Kandelous’ empirical model wasdeveloped, the model did not provide good predictions forthe downward direction.
Under natural (field) conditions, once the wetting front
reaches the soil surface, downward water movement
increases. Since empirical models, including the one by
Kandelous, do not consider this phenomenon and estimate

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each direction (i.e., horizontal, upward, and downward)
separately, observed downward water movement was much
faster than predicted by the Kandelous’ empirical model(Fig. 3).
Similar results were obtained for the infiltration experiments with the emitter installed at a depth of 30 cm.
HYDRUS-2D provided better estimates of wetting front
movement than the other models. The MAE varied from
1.9 to 3.5 cm for HYDRUS-2D, from 1 to 10.7 cm for
WetUp, and from 2.2 to 8.95 cm for the Kandelous’empirical model (Table 3).
Surface drip irrigation
Four models, i.e., HYDRUS-2D, WetUp, Schwartzman
and Zur (1986), and Amin and Ekhmaj (2006) were used toanalyze surface irrigation experiments (Fig. 4). Statisticalcomparisons between each model’s predictions andobserved data are presented again in Table 3. The value of
MAE varied from 2.43 to 9.76 cm for HYDRUS-2D, from
12.11 to 58.1 cm for WetUp, from 4.42 to 11.55 cm for
Schwartzman and Zur’s (1986) empirical model, and from1.96 to 8.47 cm for Amin and Ekhmaj’s (2006) empiricalmodel. All models except WetUp provided good estimations of the wetting front. As shown in Fig. 4, WetUp
overestimated the movement of the wetting front, while

Fig. 3 A comparison of observed and estimated wetting dimensions
for the field subsurface drip irrigation experiments with emittersinstalled at a depth of 15 cm (left) and 30 cm (right)

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Irrig Sci (2010) 28:435–444

Schwartzman & Zur’s empirical model underestimated it.
It can be concluded that HYDRUS-2D and the Amin &Ekhmaj’ (2006) empirical model provided better predictions than the other models because HYDRUS-2D uses soil
hydraulic parameters that characterize the soil, and theAmin & Ekhmaj’ (2006) empirical model uses the Dhvariable.
It can be seen in Fig. 4 that for the experimental set C,
which had the highest emitter discharge, HYDRUS-2D did
not provide as good estimates as for the experimental sets
A and B. Ponding of water on the soil surface due to highemitter discharge could have caused the underestimation of
the horizontal distance and the overestimation of the vertical distance. Water ponding at the soil surface would
cause larger horizontal and smaller vertical movement ofwater when compared to conditions without ponding andfor the same volume of irrigation water.

Conclusions
Wetting patterns for surface and subsurface drip irrigation
were evaluated in this paper. While both laboratory andfield experiments were conducted for the subsurface drip
irrigation, only field experiments were carried out for thesurface drip irrigation. Three models were used to estimate
the wetting shape for the subsurface drip irrigation, but fourmodels were used for the surface drip irrigation. A comparison of the model predictions against observed data wascarried out and statistically analyzed.
HYDRUS-2D was used to represent numerical models
simulating water flow and solute transport in soils and to
estimate the wetting patterns for both surface and subsurface drip irrigation. The soil hydraulic parameters for the
HYDRUS-2D simulations were estimated using either
Rosetta for the laboratory experiments and inverse analysisof an additional control experiment for field conditions.
Except for the second subsurface irrigation experiment in
the laboratory and the experimental set C for the surface
drip irrigation in the field, HYDRUS-2D estimations of thewetting shape were satisfactory, with values of MAE andR-square varying between 0.87 and 4.34 cm and 0.71 and
0.99, respectively (Table 3). Although HYDRUS-2D is arather complicated model that requires a lot of input data inorder to simulate water movement in soils, this is balancedby more precise estimates of water movement from drip
irrigation systems and the ability to use it for both surface
and subsurface drip irrigation. It can be concluded thatHYDRUS-2D provides good predictions and should beselected over other models evaluated in this paper when it
is important to obtain accurate results.
Although the WetUp software is based on the analyticalmodels for estimating the wetting pattern for surface andIrrig Sci (2010) 28:435–444

443

Fig. 4 A comparison ofobserved and estimated wetting
dimensions for the field surface
drip irrigation experiments.
Experiments A (left) have adischarge varying from6.94 9 10-7 to
7.14 9 10-7 m3 s-1,
experiments B (middle) from
1.14 9 10-6 to
1.3 9 10-6 m3 s-1, andexperiments C (right) have theemitter discharge of
1.71 9 10-6 m3 s-1

subsurface drip irrigation systems, its database is currently
populated by values for selected Australian soils (thatlikely have a different type and structure than our soils) andconditions (Clapp and Hornberger 1978; Verburg et al.
2001). For soil types used in our experimental work and forconsidered initial conditions, WetUp was not a very usefultool. Its MAE and R-square values varied between 1 and58.1 cm and 0.71 and 0.99, respectively. While WetUp
requires much fewer input parameters and shorter computational time to simulate wetting shapes than HYDRUS2D, its predictions are significantly less precise. WetUp
would be a more useful tool if it evaluated analyticalsolutions directly for any entered parameters, in which case
it could be used for a much wider range of soils andconditions.
The empirical model of Kandelous et al. (2008) for estimating the wetting shape for subsurface drip irrigation provided good results only for conditions that were similar to
those for which it was developed. As shown in Table 1,values of MAE and R-square for the Kandelous’ empiricalmodel varied between 1.34 and 12.24 cm and from 0.72 to
0.98, respectively. It can be concluded that this model is not
general enough to be used for different conditions. The
model needs to be further developed by considering different
soil types, different initial soil water conditions, and perhaps
other factors. An important defect of this model, similarly to
WetUp, is that it cannot consider interactions between thewater front and the soil surface once it is reached.
The empirical model of Amin and Ekhmaj (2006),
which was developed to estimate the wetting pattern forsurface drip irrigation, provided better results than theempirical model of Schwartzman and Zur (1986), and insome case even better than HYDRUS-2D. While MAE andR-square varied between 1.96 and 8.47 cm and from 0.86
to 0.98, respectively, for Amin and Ekhmaj’s empiricalmodel of (2006), they were between 4.42 and 11.55 cm andfrom 0.84 to 0.98, respectively, for the empirical model of
Schwartzman and Zur (1986). The better predictive capabilities of the Amin and Ekhmaj (2006) model can likely be
explained by its use of Dh, i.e., the average change of thesoil water content behind the wetting front. It can thus beconcluded that consideration of a parameter related to thesoil water content is an important prerequisite for anempirical model to estimate wetting patterns for drip
irrigation.
Open Access This article is distributed under the terms of theCreative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any
medium, provided the original author(s) and source are credited.

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