Agricultural Water Management 221 (2019) 334–347

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Agricultural Water Management
journal homepage: www.elsevier.com/locate/agwat

HYDRUS-2D simulations of water and potassium movement in drip irrigatedtropical soil container cultivated with sugarcane

T⁎

Katarina L. Greccoa, , Jarbas H. de Mirandaa, Laís K. Silveiraa, Martinus Th. van Genuchtenb,caDepartment of Biosystems Engineering, Luiz de Queiroz College of Agriculture, ESALQ, University of São Paulo, USP, Piracicaba, SP, Brazil
Center for Environmental Studies, CEA, São Paulo State University, UNESP, Rio Claro, SP, BrazilcDepartment of Nuclear Engineering, Federal University of Rio de Janeiro, UFRJ, Rio de Janeiro, RJ, Brazilb

A R T I C LE I N FO

A B S T R A C T

Keywords:
Subsurface drip
Fertigation
Sugarcane
Tropical soil
HYDRUS-2D

Subsurface drip irrigation is increasingly being used for sugarcane production in Southeast Brazil in attempts toconserve water, especially after extreme droughts in 2014 and 2015. Correct design of a subsurface drip systemis essential to provide information about water and solutes dynamics in the subsurface, leading to possible watersavings. While ﬁeld investigations demand much time and ﬁnancial resources, numerical models can also
provide much information about expected water and solute distributions in the soil proﬁle. The objective of thisstudy was to use HYDRUS-2D for simulations of water and potassium movement in containers packed with asubsurface drip-irrigated tropical soil cultivated with sugarcane and comparing simulated and observed data
through statistical parameters. Investigations were carried out in a greenhouse during 240 days after the tillering
stage. Spatial and temporal distributions of water content and potassium concentrations were collected using
Time Domain Reﬂectometry (TDR) to calibrate and validate HYDRUS-2D. Simulations were carried out using
hydraulic properties ﬁtted directly to measured retention date, as well as estimated using pedotransfer functions(Rosetta Lite and the class pedotransfer of Carsel and Parrish). RMSE and MAE indicated a favorable capacity ofthe model to simulate soil water content using with measured properties and a better performance than estimated properties by pedotransfer functions. For the potassium concentrations, a less than optimal performance
was obtained with HYDRUS-2D using both the measured hydraulic properties and the pedotransfer functions. A
better understanding and accurate determination of solute concentrations in soil, uptake parameters, the crop
root growth and distribution should be necessary for improve the simulations and observed data. The inability of
HYDRUS-2D to accurately estimate potassium concentrations was not necessarily a limitation of model, butrather a limitation of knowledge about the soil and plant parameters involved.

1. Introduction
First introduced in 1532, sugarcane has always been important for
Brazil’s economy. Brazil at present is not only the largest sugarcane
producer in the world, followed by India and China, but also the largest
producer of sugar and ethanol derived from sugarcane. Responsible formore than 50% of the world’s sugar market, sugarcane is cultivated innearly all parts of the country. Severe droughts during 2014 and 2015
had a direct impact on sugarcane productivity. Productivity decreased
especially dramatically in the Brazilian Southeast where rainfall remained far below the long-term average, along with high temperatures(CONAB, 2015).
Subsurface drip has become a viable alternative to traditional irrigation methods in terms of optimizing water and nutrient applications(e.g., Keller and Bliesner, 1990; Ramos et al., 2012). Investigations⁎

conducted in São Paulo State, Brazil, about the viability of subsurface
drip in sugarcane cultivation demonstrated an increase of 24% in stem
yield and 23% in sugar yield relative to no irrigation (Gava et al.,2011). Dalri and Cruz (2002) studied the eﬀect of subsurface drip frequency on the initial development of sugarcane and found an averageincrease of 45% in fresh stem and leaf mass in relation to the control
without irrigation.
Júnior et al. (2012) further showed a signiﬁcant increase in sugar(25.3 Mg ha−1) and ethanol (20 m3 ha−1) productivity with fertigationof 90 kg N ha−1 and 60 kg K ha−1 by subsurface drip. Dalri and Cruz(2008) observed increases in sugarcane production of 43 and 62% for
the second and third harvests, respectively, thus demonstrating good
adaptation with this type of irrigation.
Because of the high potassium demand of crops cultivated in tropical soils, many studies have focused on the potassium dynamics inCorresponding author.
E-mail address: klgrecco@usp.br (K.L. Grecco).

https://doi.org/10.1016/j.agwat.2019.05.010
Received 11 November 2018; Received in revised form 25 April 2019; Accepted 4 May 20190378-3774/ © 2019 Elsevier B.V. All rights reserved.

Agricultural Water Management 221 (2019) 334–347

K.L. Grecco, et al.

such soils (Miranda et al., 2005; Melo et al., 2006; Rivera et al., 2006;
Pinho and Miranda, 2014). Kolahchi and Jalali (2007) found that potassium concentrations can increase substantially in groundwater byleaching. Groundwater near agricultural areas often show potassium
concentrations above the limit for potable water (12 mg L−1) according
to World Health Organization (2011) and Griﬃoen (2001).
The rational use of natural resources, and avoiding or limiting environmental impacts because of poor management, requires a thoroughunderstanding of water and solute transport processes in irrigated soil
proﬁles. Time Domain Reﬂectometry (TDR) has many advantages formonitoring water and solute transport processes such as its precision,
the use of multiple readings, repetition without destruction of the soil,
and security (Bizari et al., 2014; Coelho and Or, 1996; Elaiuy et al.,2015; Souza and Folegatti, 2010). However, ﬁeld investigations with
TDR are often time-consuming and costly, plus may have some environmental impact because of the use of trenches during their installation.
Numerical models can provide additional insight into prevailingwater and solute transport processes in the subsurface. The HYDRUS-2D
software (Šimůnek et al., 1999a, 1999b; Šimůnek et al., 2016), in
particular, has been used in a large number of drip irrigation and/or
fertigation studies (Cote et al., 2003; Doltra and Muñoz, 2010;
Kandelous and Šimůnek, 2010; Li et al., 2015; Mguidiche et al., 2015;
Provenzano, 2007; Ramos et al., 2012; Rodríguez-Sinobas et al., 2012;
Skaggs et al., 2004).
Of these, Mguidiche et al. (2015) simulated volumetric soil watercontent in the potato crop under subsurface drip irrigation, with theresults showing relatively close agreement between simulated andmeasured data in the root zone. Provenzano (2007) showed the suitability of HYDRUS-2D to simulate water contents around a buried
emitter during irrigation. For Roberts et al. (2009) and Mguidiche et al.
(2015), a better understanding of the processes that occur in a subsurface drip-irrigated soil proﬁle, such as root growth and distribution,
as well as plant water uptake, are essential for reliable modeling of thewater and solute transport processes involved.
Other studies used HYDRUS-2D to obtain also general information
about solute transport processes in soils. For example, Ajdary et al.
(2007) reported that nitrogen leaching in ﬁelds with shallow root systems can be minimized by appropriate selection of the ﬂow rate, andthe time duration and intermittency of irrigation intervals. Doltra andMuñoz (2010) obtained acceptable accuracy of the estimated soil NNO3− contents of the upper 90 cm surface soil at four levels of fertigation during crop rotation, while Phogat et al. (2014) used HYDRUS2D to optimize the drip timescale for perennial horticultural crops toimprove fertigation eﬃciency and limit groundwater contamination.
Karandish and Šimůnek (2017) concluded that HYDRUS-2D could be
used to determine optimal strategies for deﬁcit irrigation and partial
root-zone drying, as a complement or even substitute of time-consuming and expensive ﬁeld studies.
Most or all the above studies concerned non-tropical soils. By
comparison, few if any HYDRUS-2D investigations have focused on
subsurface drip/fertigation applications of tropical soils. The objectiveof this study was to use HYDRUS-2D for simulations of water and potassium dynamics in greenhouse containers packed with a subsurface
drip-irrigated tropical soil cultivated with sugarcane and comparing
simulated and observed data using statistical parameters.

Table 1
Soil physical and chemical properties.
Properties

OxisolTexture class
Sand (%)
Silt (%)
Clay (%)
Field capacity (cm3 cm−3)
Wilting point (cm3 cm−3)
Bulk density (g cm−3)
pH (-)
Electrical conductivity (saturated paste – EC 1:5 – dS m−1)
Sodium (mg cm−3)
Potassium (mg cm−3)
Nitrate (mg cm−3)

Sandy loam
81.0
2.7
16.3
0.236
0.114
1.48
6.3
2.8
342.0
191.5
2386

According to the meteorological station installed inside the greenhouse,average temperature, global radiation and relative humidity were
24.8°C; 19.2 MJ m−2 day-1 and 78.7% during the experimental period.
Three 500-L polyethylene containers (120 cm diameter and 60 cmhigh) were uniformly packed with a sandy loam soil (Oxisol - Ustox)
collected in 0–30 cm depth with the initial soil water content of
0.15 cm3 cm−3. Packing was done from the bottom of containers using
ﬁve layers of 10 cm, which the ﬁrst (bottom) layer consisting of gravel
covered by a geotextile blanket to avoid soil losses to the drain outlet.
Subsequently four layers were packed in each container. One emitter
was installed per container at a depth of 25 cm; button drippers had a
ﬂow rate equal to 4.0 L h-1 at a nominal pressure of 100 kPa.
The RB92579 sugarcane variety Saccharum oﬃcinarum L. wasplanted using 30-day old seedlings. Soil samples were collected to determine initial soil conditions such as selected physical and chemical
properties (Table 1). A total of sixteen TDR probes (5 cm width and
10 cm length) with three stainless steel rods (3 mm thickness) were
installed in four 10-cm layers and in one side of the containers (Fig.1),
because of symmetry of the wetted soil volume measures, according toKandelous et al. (2011). Soil water contents were monitored using a
TDR calibration curve developed by Ponciano et al. (2015):

θ = 2.6953.10−5K a3 − 0.0016K a2 + 0.0395K a − 0.014(1)where θ is soil water content (L3L−3) and Ka the apparent dielectric
constant (-) obtained with TDR. TDR calibration for the potassium
concentration was performed with solution extractors to obtain a correlation between indirect and direct measurements of potassium in soil
solution. We used three treatments with three replicates: C1 for deionized water, and C2 and C3 for 500 and 2000 mg L-1 potassium concentrations, respectively. The calibration was carried out using 8 L
buckets with a drain at the bottom. Buckets were packed (using soil
layers of 5 cm) to the same bulk density as the containers and the initialsoil water content of 0.15 cm3 cm-3. TDR probes and solution extractors
for the calibration were installed in the center of each bucket. The soil
saturation was obtained by capillary ascension for 24 h.
We used the methodology of Rhoades et al. (1976) to determine thesoil solution concentration as a function of soil water content, whichenables a relationship between the electrical conductivity estimatedusing TDR (ECTDR) and the solution electrical conductivity (ECS) measured with an electrical conductivity meter. The soil solution was collected 24 h after the application of 80 kPa vacuum inside the extractor.
For the correlations were determined the potassium (CK+) and nitrate(CNO3-) concentrations since the fertilizer (KNO3) contained these nutrient sources and therefore both aﬀected ECS. CK+ and CNO3- were
measured in laboratory with a ﬂame photometer and a spectrophotometer, respectively. ECTDR and ECS readings were accomplishedweekly until the limit of 0.15 cm3 cm−3.
The experimental design was completely randomized using three
replicates. For irrigation management of sugarcane, we used the pressure head of −40 kPa (Steduto et al., 2012), as the limit without crop

2. Materials and methods
2.1. Experimental setup
The experiments was conducted during the tillering stage (2ndphenological stage) between June/2015 to January/2016 in a greenhouse at the experimental area of the Department of Biosystems
Engineering, University of São Paulo (LEB/ESALQ/USP) in Piracicaba,São Paulo, Brazil (22° 43′ 33″S, 47° 38′ 00″W, elevation of 547 m).
335

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K.L. Grecco, et al.

Fig. 1. Distribution of TDR probes in the container: (a) front vision, and (b) top vision.
Table 2Agricultural activities during the development of sugarcane.
Date

Agricultural activities

Remarks

June 9 in 2015 (day 1)
August 31 in 2015 (day 84)
September 25 in 2015 (day 109)
October 13 in 2015 (day 127)
November 11 in 2015 (day 156)
November 30 in 2015 (day 175)
December 14 in 2015 (day 189)
December 28 in 2015 (day 203)
January 11 in 2016 (day 217)
January 25 in 2016 (day 231)

Planting and irrigation
Fertigation
Fertigation
Fertigation
Fertigation
Fertigation
Fertigation
Irrigation
Irrigation
Irrigation

19.48 mm (23.70 L)
15.71 mm (17.75 L) +0.998 mg cm−3 K+
15.41 mm (17.42 L) +0.414 mg cm−3 K+
15.64 mm (17.68 L) +0.437 mg cm−3 K+
14.07 mm (16.84 L) +0.362 mg cm−3 K+
16.42 mm (18.55 L) +0.427 mg cm−3 K+
15.78 mm (17.83 L) +0.466 mg cm−3 K+
16.73 mm (18.90 L)
16.81 mm (19.00 L)
16.74 mm (18.91 L)

water stress, corresponding to 0.17 cm3 cm−3. When reaching thisvalue, irrigation started until the water content in the ﬁeld capacity(0.24 cm3 cm−3), considering the soil area of 11,309 cm2 per container.
Drainage during and after the irrigations was estimated from the volume collected through the drains. Fertigation was accomplished usinga schedule as documented by Malavolta (1994). Agricultural activities
during the experiments are summarized in Table 2.

(L3L−3), respectively, α (L-1), n and m (-) are dimensionless shapeparameters of the soil water retention curve, with m = 1-1/n, and Se iseﬀective saturation given by:

Se =The radially symmetric subsurface drip irrigation process wasmodeled using HYDRUS-2D (Šimůnek et al., 1999a) based on the following form of the Richards equation for an isotropic soil proﬁle (Coteet al., 2003; Ramos et al., 2012):(2)

where h is pressure head (L), t is time (T), r is the radial coordinate (L), z
is the vertical coordinate (L), K is the hydraulic conductivity (LT−1) andS denotes root water uptake (T−1) as described with the uptake modelby Feddes et al. (1978):S (h) = γ (h) β (r , z ) W Tp

(6)

In this study we used soil hydraulic parameters that were either
ﬁtted directly to observed retention curve (Fig. 2) using Eq. (4), or
obtained using pedotransfer functions. For the measured hydraulicdata, we used undisturbed samples taken from the containers. A total of
9 points were obtained: gravimetric water contents at pressure heads of-10, -20, -40 and −100 cm using a tension table system, and at -300,-500,-1000, -5000 and -15000 cm using the Richards pressure plateapproach (Klute, 1986).
For the pedotransfer functions we used both the Rosetta Lite function of Schaap et al. (2001) and the class pedotransfer function of Carsel
and Parrish (1988) as implemented in HYDRUS-2D. For the Rosetta
estimates we used the measured granulometric data as well as themeasured values of bulk density and water contents at -330 and
15000 cm. Table 3 lists the estimated hydraulic parameters.

2.2. HYDRUS-2D simulations∂θ (h)∂h∂ ⎡∂h∂K (h) ⎤1 ∂ ⎡=+rK (h) ⎤ +K (h)− S (h)
∂z ⎣
∂z
∂z ⎦
∂t
∂r ⎦
r ∂r ⎣

θ − θrθs − θr(3)

where γ(h) is the soil water stress response function (-), β is the normalized root water uptake distribution (L−2), Tp is the potential transpiration rate (LT−1) and W is the area of the soil surface (L2) associatedwith the transpiration process.
2.2.1. Soil hydraulic properties
The soil hydraulic properties were described using the standard
equations of van Genuchten (1980) given by:θ (h) = θr +θs − θr(1 + |αh|n )m

K (h) = Ks Sel [1 − (1 − Se1/ m)n]2(4)(5)
Fig. 2. Observed data ﬁtted soil water retention curve of the sandy loam soil.

where θr and θs are the residual and saturated soil water contents
336

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K.L. Grecco, et al.

the sink term due to root water uptake (ML−3), R is the retardation
factor (-), qr and qz are the radial and vertical volumetric ﬂuid ﬂuxes(LT-1), respectively, and Drr, Drz and Dzz are components of the dispersion tensor (L2T-1). For solute uptake we assumed a passive potassium
uptake process until some maximum value, Cs,max (Šimůnek andHopmans, 2009).
Assuming linear sorption of potassium by the solid phase, the retardation factor R is given by:

Table 3
Soil hydraulic parameters used in the HYDRUS-2D simulations.
Methods

θs(cm3 cm−3)

θr(cm3 cm−3)α(cm−1)

nK0(cm dia−1)

Direct measurement
Rosetta Lite v.1.1
Carsel and Parrish(1988)

0.544
0.418
0.387

0.104
0.066
0.039

0.0966
0.05440.0267

1.53
1.45
1.45

292.3
158.3
38.2

R=1+
2.2.2. Root water uptake
The spatial root distribution was described using the model of Vrugt
et al. (2001):⎟⎜

where ρ is soil bulk density (ML ) and Kd is the potassium distribution
coeﬃcient (L3M-1) (van Genuchten and Wierenga, 1986). Componentsof the dispersion tensor are given by the standard equations (Bear,
1988):⎟

θDrr = θDo τ + λL

(7)where β(r,z) denotes the dimensionless two-dimensional spatial distribution of root water uptake, rm is the maximum rooting length in theradial direction (L), r is the radial distance from the origin of the plant(L), pr (-) and r* (L) are empirical parameters (pr equals unity for r > r*,and some constant for r ≤ r* to be ﬁtted to the root data), zm is the
maximum rooting depth (L), z (L) is the depth in the soil proﬁle (z ≥ 0),
pz (-) and z* (L) are empirical parameters (similarly as for pr and r*, pz
equals unity for z > z*).
Only a single sugarcane plant and a simple root distribution model
were considered, in which the roots expanded vertically into all available space in the containers (zm =50 cm), but concentrated mainly
above the dripper (r* =2 cm, z* =25 cm) where water and potassium
were applied, and then extended horizontally to 30 cm (rm =30 cm).
Reductions of root water uptake due to the water stress, α1(h), were
described using the formulation of Feddes et al. (1978):⎧ 0, h > h1 or h ≤ h4
⎪ h − h1
⎪ h − h , h2 < h ≤ h12
1
α1 (h) =1,h3 < h ≤ h2
⎨
⎪h−h
4
⎪
, h4 < h ≤ h3
⎩ h3 − h4θDzz = θDo τ + λL

qd =

|q|

+ λT

|q|
qr2
|q|

qr qz
|q|

(11)

volume of solution applied/day
ASW

(12)
3

where the volume of solution applied (L ) varied for diﬀerent eventsand ASW is the surface wetted area (L2) given by

ASW = 4π (R 0 )2(13)

where R0 is the radius of semicircle (L). During irrigation, the dripper
boundary was held at a constant water ﬂux (qd). An atmosphericboundary condition was assumed for the entire soil surface. A no-ﬂow
boundary condition was established along the left and right edges of thesoil proﬁle, and along part of the bottom of the container. A drain was
assumed at the bottom of container, with the actual drainage ﬂux being
calculated similarly as Eqs. (12) and (13). The initial soil water content
was 0.15 cm3 cm−3, while the initial concentration of potassium wasestimated from measured soil chemical properties. For solute (K+)
transport, a third-type boundary condition was imposed along the

∂θRC
∂C
∂C
1 ∂ ⎡ ⎛=+ θDrz− qr C ⎞ ⎤r θDrr∂t∂r∂zr ∂r ⎢⎠⎥⎣ ⎝⎦where C is the solution concentration (ML

qz2

qz2

2.2.4. Initial and boundary conditions
A time-variable ﬂux boundary condition was applied to a semicircle
with radium (R0) of 2 cm at 25 cm depth (Fig. 3). The boundary condition with a ﬂux qd was deﬁned as:

2.2.3. Potassium transport
Simulations of potassium transport were carried out using the threedimensional radially axisymmetric advection-dispersion equation given
by:

−3

|q|

+ λT

where Do is the potassium ionic diﬀusion coeﬃcient in free water(L2T−1), τ is the tortuosity factor (-), described here using theMillington and Quirk (1961) formulation, and λ L and λ T are thelongitudinal and transverse dispersivities (L), respectively, and |q| represents the absolute value of the water ﬂux (LT−1).
The diﬀusion coeﬃcient for potassium was set at 1.7 cm2 day−1(Miranda et al., 2005). Values of λ L and Kd were derived from a simple
one-dimensional solute breakthrough experiments, while λ T was taken
to be one-tenth of the value of λ T. The 1-D transport experiments werecarried out using 20-cm PVC columns (having inside diameters of 5 cm)
ﬁlled with the same loamy sand soil as used for the containers. The
upper end of the column was ﬁxed at a positive pressure head of 1 cm,
while a geotextile membrane and metal grille were used for the bottom
boundary. The columns were packed uniformly to the same bulk density as the containers, and then saturated with deionized water for 24 h.
After saturation, the columns were washed with deionized water for
24 h to remove most or all ions from the soil exchange complex, after
which a solution of 2000 mg L−1 of potassium was applied (Pinho andMiranda, 2014). Breakthrough data were collected and subsequently
analyzed using the CFITIM code (van Genuchten, 1981) within the
STANMOD software (Šimůnek et al., 1999b).

(8)

∂ ⎡
∂C
∂C
+ θDrz− qz C⎤ − S (h) Cs
θDzz
∂z ⎣
∂z
∂z
⎦

qr2

θDrz = (λL − λL )where h1, h2, h3, and h4 are the threshold parameters. Eq. (8) assumes
that water uptake is at its potential rate when the pressure head is
between h2 and h3, decreases linearly when h > h2 or h < h3, and becomes zero when h < h4 or h > h1. The following parameters of theFeddes et al. (1978) model were used: h1 = -10, h2 = -25, h3 = -150 to-500, h4 = -16000 cm, which were taken from Taylor and Ashcroft(1972) for sugarcane.
Potassium in a soil can generally be subdivided into four diﬀerent
fractions: structural, non-exchangeable, exchangeable and in soil solution, whereby the latter can be taken up directly by plants with mass
ﬂow of water uptake (Barber, 1995; Sparks, 1987). Consequently, we
assumed that potassium was passively taken up by the root system ofsugarcane (Šimůnek and Hopmans, 2009) by moving with the soil solution.

+

(10)
−3

*
*
r ⎞⎛
z ⎞
⎧ p |r − r| ⎞ ⎛ pz |z − z| ⎞ ⎫β (r , z ) = ⎛1 −1−
exp −⎜⎛ r⎟ − ⎜⎟⎨rm ⎠ ⎝
zm ⎠
rm
zm
⎝
⎠ ⎝
⎠⎬
⎩ ⎝
⎭
⎜

ρK d
θ

(9)

), Cs is the concentration of337

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K.L. Grecco, et al.

Fig. 4. Observation nodes with coordinates (x; z) used in the HYDRUS-2D
model.

Fig. 3. A schematic view of the ﬁnite element mesh with boundary conditions
based on the experimental layout, plant, subsurface drip and drain locations.

dripper, and zero-ﬂux boundary along all sides except for a zero-concentration gradient along the drain at the bottom of the container.
2.3. Flow domain and ﬁnite element mesh
The dripper can be considered as a semicircle source (Fig. 3), because the radially symmetric of subsurface drip irrigation process(Kandelous et al., 2011; Kandelous and Šimůnek, 2010). The transportdomain was set as a rectangle (in a radial direction) with a width of
60 cm (half of the container diameter) and a depth of 50 cm, discretized
into 1959 two-dimensional elements involving 1035 nodes (Fig. 3).
Several observation nodes were included (Fig. 4) at selected locations in
the ﬂow domain to follow simulated soil water contents and potassium
concentrations versus time. The observation nodes had very similar
coordinates as the TDR probes installed in the containers to allow
comparisons of observed and simulated data.

Fig. 5. Partitioning of evapotranspiration into evaporation and transpiration
during the crop growing period (Allen et al., 1998).

LAI = −1ln(1 − SCF )
αi

(14)

where αi is a constant for radiation extinction by the canopy (-) and SCF
is the surface cover fraction (-). The values of daily potential transpiration (Tp) and soil evaporation (Es) obtained (Fig. 6) were used as
time-variable boundary conditions in the model during the experimental period. A maximum Tp of 7.9 mm occurred on day 237 (31thJanuary 2016), when the average temperature, global radiation andrelative humidity were the highest during the experiments.

2.4. Estimation of potential evapotranspiration
2.5. Statistical analysis
Potential evapotranspiration was calculated using the PenmanMonteith equation (Allen et al., 1998) as implemented in the HYDRUS1D software (Šimůnek et al., 1998). The simulations require daily estimates of potential evaporation and transpiration. Fig. 5 shows thepartitioning of potential evapotranspiration (ETp) into potential evaporation (Es) and potential transpiration (Tp) in correspondence to theleaf area index (LAI) per unit surface of soil below the canopy. At
sowing nearly 100% of ET comes from evaporation, while at full crop
cover more than 90% of ET comes from transpiration (Allen et al.,1998). Speciﬁc crop data we used were as follow: crop height (120 and
230 cm for 2nd and 3rd phenological stages, respectively); albedo (0.23
and 0.31 for 2nd and 3rd phenological stages, respectively) and LAI (2.5
and 7.5 for 2nd and 3rd phenological stages, respectively), the latter
calculated using:

Model performance for predicting soil water contents and potassium
concentrations were evaluated in terms of the Root Mean Square Error(RMSE) and the Mean Absolute Error (MAE):
n

RMSE =∑i =1 (ai − bi)2
n

(15)

n

MAE =

∑i =1 |ai − bi |n(16)where ai and bi represent the observed and simulated water contents or
potassium concentrations. The Surfer software was used for comparisons and mapping of the observed and simulated soil water content and338

Agricultural Water Management 221 (2019) 334–347

K.L. Grecco, et al.

Fig. 6. Daily potential transpiration (Tp) and potential soil evaporation (Es) estimated using the Penman-Monteith equation (Allen et al., 1998).

the electrical conductivity (ECs) by preparing diﬀerent potassium concentrations (0.02–2.00 mg cm−3) in solution with deionized water. The
following correlation was obtained (Fig. 8):

potassium concentrations at selected dates. We used kriging as thegridding method, while the statistical analyses were carried out atpreselected dates.

CKNO3 = 0.8699ECs − 0.0634

3. Results and discussion

where CKNO3 is the potassium nitrate concentration in solution (mgcm−3). Since our study focused on potassium transport, we needed toseparate the potassium and nitrate concentrations as shown Fig. 9. This
produced the following correlation (having an R2 of 0.987):

3.1. TDR calibration for potassium concentration
In this study we used an indirect TDR-based method for determining
potassium concentrations. The approach requires careful correlation of
the indirect electrical conductivity measurements (ECTDR) with directly
observed solution values (ECS) in the soil system being used. Fig. 7
shows a relatively high correlation between ECTDR and ECS as reﬂectedby an R2 value of 0.923, albeit with relatively high RMSE and MAE
values (1.447 and 1.063 dS m−1, respectively) indicating a less thatsatisfactory ﬁt. The calibration could be improved by carrying out
multiple linear regression between electrical conductivity and TDR
measured soil water contents:

C K + = 0.3760CKNO3 + 0.0281(19)

where CK+ is the potassium concentration in solution (mg cm−3).
Fig. 10 compares indirectly measured potassium concentrations using
TDR and direct measurements, as reﬂected by an R2 value of 0.995,
RMSE and MAE were 0.038 and 0.030 dS m-1, respectively.
3.2. Potassium breakthrough curve(17)

ECs = 1.9790ECTDR − 0.0630θ + 0.0391(18)

Fig. 11 shows the measured potassium breakthrough curve, plottedin terms of dimensionless concentration (from 0 for the initial concentration to unity for the inlet concentration) versus the number ofpore volumes (PV) of solution leached through the column. The use ofdimensionless time is useful since it eliminates the eﬀects of travel
distance and ﬂow velocity, which are speciﬁc to any one experiment(Skaggs and Leij, 2002):−1

where ECS is the solution electrical conductivity (dS m ), ECTDR is theelectrical conductivity estimated using TDR (dS m−1) and θ is the soil
water content (cm3 cm-3). The R2 value (0.960) was better, but theerrors remained high (0.536 and 0.400 dS m−1 for RMSE and MAE,
respectively), reﬂecting a relatively poor adjustment.
Potassium nitrate solution concentrations were correlated next with

Fig. 7. Correlation between the electrical conductivity of soil solution and the electrical conductivity measured by TDR.
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Fig. 8. Correlation between the potassium nitrate concentration and the electrical conductivity of soil solution.

PV =

ϕVs
Vc

the assumption in HYDRUS-2D of having a constant crop rooting system
in time, because the model has no options to change the root systemeach day or during a certain period. This may well be an important
limitation. For example, Assouline (2002) noted temporal changes inroot growth and distribution of drip-irrigated corn, which also depended on the drip irrigation rate within 28-day periods, while Roberts
et al. (2009) illustrated these eﬀects for successive crops of cantaloupe
and broccoli and Mguidiche et al. (2015) for potato. Fig. 12 shows thatafter day 210 simulated water contents became very similar to theobserved data before irrigation, varying from 0.16 to 0.17 cm3 cm−3.
Simulations of the second layer mostly agreed better with the data,
especially during the middle and later parts of the experiments whensimulated water contents varied between about 0.18 to 0.30 cm3 cm−3and observed values between 0.15 and 0.28 cm3 cm−3. Notable differences between simulated and observed data remained between day
60 and the ﬁrst irrigation on day 80. These changes can be explained inpart by small diﬀerences in the coordinates of the TDR probes and theobservation nodes in the model, in addition to the diﬀerences in root
growth, distribution and water uptake observed experimentally andsimulated with the model.
RMSE values of observed versus simulated data were 0.027 and
0.030 cm3 cm−3 and MAE values 0.023 and 0.024 cm3 cm−3 for theﬁrst and second layer, respectively. These values compare closely withresults of other investigations (Crevoisier et al., 2008; Ghazouani et al.,(20)

where ϕ is the soil porosity (decimal), Vs is the soil column volume(cm3) and Vc is the volume collected (cm3) from each point of breakthrough curve. From the ﬁtted dimensionless transport data, a column
Peclet number of 4.90, a retardation factor (R) of 1.79, the values of
4.08 cm for the longitudinal dispersivity (λ L) and 0.289 cm3 g−1 for the
distribution coeﬃcient (Kd), which we used subsequently in the HYDRUS-2D simulations.
3.3. Simulations of soil water content
The average soil water content of the ﬁrst (0–25 cm) layer of thecontainers as measured with eight TDR probes (observed data) and of
the eight observation nodes (simulated data) as calculated with
HYDRUS-2D, are shown in Fig. 12. Similar data are provided for thesecond (25–50 cm) layer. HYDRUS-2D overestimated volumetric watercontents of the ﬁrst layer before and at the time of the irrigations until
about day 101, with simulated values varying between 0.16 and 0.27
cm3 cm−3, compared to the observed data between 0.16 and 0.23 cm3
cm−3. The maximum simulated value was above of ﬁeld capacity,which did not occur with the observed data.
However, after this period, the model simulated smaller watercontents before irrigation until day 210. One plausible explanation isFig. 9. Correlation between the potassium concentration (CK+) and the potassium nitrate (CKNO3).
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Fig. 10. Correlation between direct and indirect (TDR) measurements of the potassium concentration.

overestimated all water contents in the soil proﬁle on this day, withnumerical results being the least accurate as reﬂected also by RMSE andMAE values of 0.067 and 0.062 cm3 cm−3, respectively. Days 86 and233, two days after irrigation, showed similar behavior as day 84 with
RMSE values of 0.051 and 0.050 cm3 cm−3 and MAE of 0.046 and0.043 cm3 cm−3, respectively.
The observed isolines were relatively horizontal and the simulated
isolines vertical on day 226, ﬁve days before irrigation. The model
underestimated the lower water contents (0.12 cm3 cm−3) and overestimated the higher values (0.30 cm3 cm-3). RMSE and MAE were
0.062 and 0.056 cm3 cm-3, respectively. The same behavior of day 226
was present in days 230 (one day before irrigation) and 240 (last observed day) with RMSE of 0.060 and 0.059 cm3 cm-3, respectively, andMAE of 0.052 cm3 cm-3 for both.
The best HYDRUS-2D simulations were obtained using the measured soil hydraulic properties. We also ran the model using hydraulicproperties based on Rosetta (Schaap et al., 2001) and the Carsel andParrish (1988) pedotransfer functions. Fig. 14 compares results using
these pedotransfer functions with the measured properties. For the ﬁrst
layer, simulations with the Rosetta function produced similar RMSE andMAE values (0.029 and 0.022 cm3 cm−3, respectively) compared tousing the measured hydraulic properties. However, RMSE and MAE for
the second layer were 0.041 and 0.036 cm3 cm−3, these values werehigher than those for the measured properties. Simulations with Rosetta
underestimated the water contents near days 125–200, especiallyduring the drying events before the mid-season irrigations.
Results with the class pedotransfer functions by Carsel and Parrish(1988) were even less accurate, giving RMSE and MAE of 0.038 and
0.030 cm3 cm−3 for the ﬁrst layer, and 0.050 and 0.044 cm3 cm−3 forthe second layer, respectively, compared to the simulations using themeasured properties. One reason may be that the pedotransfer functions by Carsel and Parrish (1988), and to a lesser extent also those by
Schaap et al. (2001), were derived nearly exclusively using temperateregion soils data, and hence may not be overly applicable to the tropicalsoil used in our study.
Most notable, again, are the very low water content values (especially in the second layer) near the middle and end of the simulations.
Simulated water contents were then only 0.09 to 0.16 cm3 cm−3 during
pre-irrigation drying, thus causing apparent water stress problems forthe sugarcane crop. One reason of the lower water contents with thepedotransfer functions is the very low value of the wilting point (and of
the residual water content), which contributed to a rapid reduction in
the water content of the lower layers, also in view of the adopted
bottom boundary condition.

Fig. 11. Observed relative potassium concentrations versus number of porevolumes leached through the column, and the ﬁtted curve using CFITIM.

2016; Kandelous and Šimůnek, 2010; Mguidiche et al., 2015; Phogat
et al., 2014; Ramos et al., 2012; Skaggs et al., 2004) and show thatHYDRUS-2D provides relatively accurate results also for subsurface drip
irrigation with sugarcane.
Fig. 13 shows the maps of observed and simulated water content at
diﬀerent dates during the experiments. The isolines had a very similar
structure on day 2, one day after 1st irrigation. However, the model
overestimated soil water contents by producing a maximum value ofabout 0.40 cm3 cm−3 compared to 0.36 cm3 cm−3 observed experimentally. Only the isolines with 0.18 cm3 cm−3 were close. RMSE andMAE were 0.048 and 0.043 cm3 cm−3, respectively. These values werehigher than those of the average water content per day in each layer.
On day 83, one day before 2nd irrigation, the observed isolines wererelatively horizontal with the water content showing lower value in a
homogeneous fashion from the container bottom to the soil surface.
HYDRUS-2D produced more concentrated isolines on the left side near
the soil surface, with simulated water contents being higher than theobserved values along the entire proﬁle. The concentrated isolines near
the dripper are indicative of the locally higher root distribution andhence water uptake. RMSE and MAE were 0.051 and 0.050 cm3 cm−3,
respectively.
Higher observed and simulated values water contents occurred on
the lower left side at day 84 (irrigation day). Nevertheless, the model
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Fig. 12. Observed and simulated soil water content during sugarcane development.

experimental period. Accurate results were obtained at the beginning,but the model slightly underestimated the concentrations in someperiods, such as between days 94–106 and isolated days as 109, 127,
219 and 233.
Most notable is that simulated potassium concentrations generally

3.4. Simulations of potassium concentrations in soil solution
Fig.15 compares observed and simulated average potassium concentrations of the 0–25 and 25–50 cm layers. HYDRUS-2D overestimated potassium concentrations of the ﬁrst layer during most of theFig. 13. Spatial distribution of observed and simulated water contents (cm3 cm−3) in the soil proﬁle at selected dates.
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Fig. 14. Soil water contents measured by TDR and simulated by HYDRUS-2D using the measured hydraulic properties, the pedotransfer function ROSETTA and soil
catalog (Carsel and Parrish, 1988).

(Doltra and Muñoz, 2010; Ramos et al., 2011; Phogat et al., 2014; Wanget al., 2014; Karandish and Šimůnek, 2017). However, all those studies
were carried out for nitrate leaching in temperate soils, and not ofpotassium. This demonstrates the importance of our study on potassium
dynamics in a tropical soil, including potential contamination of underlying groundwater.
Fig. 16 shows the distributions of observed and simulated potassium
concentration at diﬀerent dates. The isolines showed diﬀerence behavior in day 83, one day before 1st fertigation, in that the observed
isolines were mostly horizontal with the potassium concentrations declining steadily between the container bottom and the soil surface.
HYDRUS-2D, on the other hand, showed far more vertical isolines on
the left side, with lower concentrations. RMSE and MAE were 0.037 and
0.030 mg cm−3, respectively. On day 86, the model overestimated theobserved data on the left side close to dripper and underestimated themeasurements in the lower left side. The statistical parameters (RMSE
and MAE) were 0.180 and 0.135 mg cm−3, respectively.
The distributions for days 127 (the day of fertigation) and 130(three days after fertigation) were very similar to those on day 86, butwith the model now underestimating the potassium concentrations innearly the entire soil proﬁle. Day 127 showed the worst performance ofall plots, having RMSE and MAE values of 0.284 and 0.207 mg cm−3,
respectively. RMSE and MAE for day 130 were 0.117 and 0.088 mg
cm−3, respectively.
Subsequent plots in Fig. 16 were for irrigations without fertigation(days 226, 230, 233 and 240). The higher observed concentrations(0.32 to 0.26 mg cm−3) were now in the 25–40 cm layer in day 226 andthe higher simulated concentrations (0.24 to 0.21 mg cm−3) in the
20–50 cm layer with RMSE and MAE of 0.078 and 0.063 mg cm−3,
respectively. The highest potassium concentrations obtained with HYDRUS-2D remained in the same layer at day 230, but withremained constant between the fertigation periods as if the model did
not consider any leaching from the container, or other sinks like potassium uptake by the sugarcane roots. These results point to the needto understand and describe relevant root solute uptake processes andparameters to be used by HYDRUS-2D, including root growth and dynamic root distributions as discussed by Assouline (2002), Mguidiche
et al. (2015) and Roberts et al. (2009). Simulated data in the ﬁrst layer
ranged from 0.08 to 0.25 mg cm−3 and observed data from 0.09 to
0.24 mg cm−3. Results reﬂect a less than optimal performance of themodel relative to the observed data.
Simulated potassium results for the second layer intercalated between ﬂuctuated between underestimation and overestimation relative
to the observed data. Simulated concentrations during the fertigation
applications were far below the observed data, while remaining practically constant in between the applications, and then higher than theobserved data. This suggest that potassium uptake by roots was passive
and constant in each interval after the fertigation applications, as seen
for the 0–25 layer, leading to concentrations that remained constant intime.
In experimental initial conditions, it is important to notice that sodium concentration presents in chemical analyze was higher than potassium concentration. One could presume that sodium inﬂuenced potassium dispersion because its lower atomic radius (280 pm) and higher
ionization energy (496 kJ mol−1) as compared to potassium (227 pm
and 419 kJ mol−1, respectively), which would have caused more
spreading of potassium through the ﬂow domain (Andrade, 2009). Thissodium inﬂuence and potassium dispersion were not considered in thesimulations by the model.
The RMSE and MAE values for the ﬁrst layer was 0.065 and0.053 mg cm−3, respectively, and for second layer of 0.135 and
0.105 mg cm−3, respectively. Similar values were found by others343

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Fig. 15. Observed and simulated potassium concentration in soil solution during sugarcane development.

some leaching with the now potassium-free irrigation water. RMSE andMAE were 0.109 and 0.086 mg cm−3, respectively, and the higher observed concentrations (0.26 to 0.22 mg cm−3) started in 20 cm depth. A
similar behavior of days 230 and 233 was present in day 240 (lastobserved day) with RMSE and MAE of 0.072 and 0.060 mg cm−3,

overestimated values. On this day, the higher observed concentrations
started at the 25 cm depth, showing RMSE and MAE of 0.080 and
0.070 mg cm−3, respectively. The model underestimated the higher
concentrations on day 233, with the concentrations slowly decreasingfrom a depth of 35 cm until the bottom of the container, suggestingFig. 16. Spatial distributions of observed and simulated potassium concentrations (mg cm−3) in the soil proﬁle at selected dates.
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Fig. 17. Potassium concentrations in soil measured by TDR and simulated by HYDRUS-2D using the measured hydraulic properties, the pedotransfer function
ROSETTA and soil catalog (Carsel and Parrish, 1988).

respectively.
We further show in Fig. 17 how the calculated potassium concentrations in the soil proﬁle were aﬀected by using the measured
hydraulic properties and estimated (Table 3) using pedotransfer functions (Rosetta Lite by Schaap et al. (2001) and the Carsel and Parrish(1988) class pedotransfer functions). The pedotransfer functions overestimated potassium concentrations in the 0–25 cm layer almost theentire experimental period. Results of measured and estimated hydraulic properties were close in the interval of days 92–108. However,the concentrations were underestimated by the functions from day 219.
The ﬁrst layer showed RMSE and MAE values were 0.066 and0.055 mg cm−3, when we used the Rosetta (actually very similar to
those of the measured properties: 0.065 and 0.053 mg cm−3, respectively). RMSE and MAE were 0.082 and 0.068 mg cm-3 for the pedotransfer functions of Carsel and Parrish (1988).
Calculations for the 25–50 cm layer using the pedotransfer functions
were nearly identical to the results using measured hydraulic properties. RMSE and MAE were the same for measured properties and theRosetta function (0.135 and 0.105 mg cm−3, respectively), while thosefor the Carsel and Parrish functions were only minimally larger (0.137
and 0.106 mg cm−3, respectively). These results suggest that the relatively poor results for potassium fate and transport were not caused by
the hydraulic properties but rather by incomplete or inaccurate
knowledge of root growth and root distribution, and related water andsolute uptake processes and its parameters.

evaluate HYDRUS-2D performance in water and potassium dynamics.
Soil water contents monitored by TDR were predicted reasonably wellthrough the technique calibration, but potassium concentrations had a
non-consistent adjustment between variables. Parameters statisticalindicated a favorable capacity of HYDRUS-2D model to simulate soil
water content. The measured hydraulic properties were found to generally lead to better water content predictions that the estimatedfunctions. The Rosetta function actually produced very similar results
as the measured properties for the ﬁrst (0–25 cm) layer, but performed
less well for the second (25–50 cm) layer. The Carsel and Parrish (1988)
function, by comparison, performed relatively poorly for both layers.
For the potassium concentrations, a less than optimal performance wasobtained with HYDRUS-2D using both the measured hydraulic properties as well as the pedotransfer functions, thus suggesting that overallmodel performance for potassium was not related to proper estimationof the hydraulic properties. A better understanding and accurate determination of potassium concentration in soil proﬁle, uptake parameters, schematization of the spatial and temporal crop root systemshould be necessary for improve simulations and observed data. The
inability of HYDRUS-2D to accurately estimate potassium concentrations hence should not be viewed as a limitation of the model as such,
but rather a limitation of our understanding of selected crop growth
processes and root functioning.

4. Conclusions

The authors gratefully acknowledge the Coordination for the
Improvement of Higher Education Personnel (CAPES) - Finance Code001 for the scholarship granted to K.L. Grecco and to the São Paulo
Research Foundation (FAPESP) for their ﬁnancial support (Proposal:

Acknowledgements

The present research concerned a combination of observed and simulated data involving subsurface drip irrigation under sugarcane to345

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#2013/24138-1). The authors would like to thank the team of theLaboratory of Soil Physics and Water Quality of the Department of
Biosystems Engineering (USP/ESALQ). We are also grateful for important comments from the anonymous reviewers, which contributed toa signiﬁcant upgrade of the manuscript.

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