Modeling of Border Irrigation in Soils with the Presence of a Shallow Water Table. I: The Advance Phase - MDPI
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agriculture
Article
Modeling of Border Irrigation in Soils with the Presence of a
Shallow Water Table. I: The Advance Phase
Sebastián Fuentes and Carlos Chávez *
Water Research Center, Department of Irrigation and Drainage Engineering, Autonomous University
of Queretaro, Cerro de Las Campanas SN, Col. Las Campanas, Queretaro 76010, Mexico;
sebastian.fuentes@uaq.mx
* Correspondence: chagcarlos@uaq.mx; Tel.: +52-442-192-1200 (ext. 6036)
Abstract: The overelevation of the water table in surface irrigation plots is one of the main factors
affecting salinization in agricultural soils. Therefore, it is necessary to develop simulation models
that consider the effect of a shallow water table in the process of advance-infiltration of the water in
an irrigation event. This paper, the first in a series of three, develops a simple mathematical model
for the advance phase of border irrigation in soils with the presence of a shallow water table. In
this study, the hydrodynamic model of the Barré de Saint-Venant equations is used for the water
surface flow, and the equations are solved using a Lagrangian finite-differences scheme, while in the
subsurface flow, an analytical solution for infiltration in soils with a shallow water table is found
using the bisection method to search for roots. In addition, a hydraulic resistance law is used that
eliminates the numerical instabilities presented by the Manning–Strickler law. The model results
for difference irrigation tests show adjustments with an R2 > 0.98 for the cases presented. It is also
revealed that, when increasing the time step, the precision is maintained, and it is possible to reduce
the computation time by up to 99.45%. Finally, the model proposed here is recommended for studying
the advance process during surface irrigation in soils with shallow water tables.
Citation: Fuentes, S.; Chávez, C.
Keywords: Barré de Saint-Venant equations; hydrodynamic model; analytical solution; flow profiles;
Modeling of Border Irrigation in Soils
irrigation tests; inverse modeling; water use efficiency; agricultural water management
with the Presence of a Shallow Water
Table. I: The Advance Phase.
Agriculture 2022, 12, 426. https://
doi.org/10.3390/agriculture12030426
1. Introduction
Academic Editors: Yanqun Zhang
Surface irrigation is one of the main factors in the fluctuation of shallow water tables in
and Jiandong Wang
agricultural plots. Around the world, surface irrigation is the most frequently used method
Received: 5 March 2022 for water application in agricultural plots, where the water is distributed over the field and
Accepted: 17 March 2022 through the soil [1].
Published: 18 March 2022 The presence of a shallow water table changes the water content of the soil profile
(Figure 1), due principally to the capillary process, which is present in porous media [2]. It
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
has been found that in shallow water conditions, irrigation can be reduced by up to 80%
published maps and institutional affil-
without affecting yield and without increasing soil salinity [3].
iations. Poor design of surface irrigation, for example losses from tailwater due to the selection
of an inappropriate irrigation flow or due to poor operation in the water distribution
network as a result of poor leveling in the ground by percolation, can cause overelevation
of the water table and the progressive salinization of the soil [4].
Copyright: © 2022 by the authors. It is estimated that 20% of the arable land and 33% of irrigated agricultural plots
Licensee MDPI, Basel, Switzerland. in the world are affected by high salinity [5]. Water-table levels play an essential role in
This article is an open access article the salt distribution in the soil profile and could be controlled by subsurface drainage [6].
distributed under the terms and However, drainage leads to an increase in irrigation costs. Therefore, it is necessary to
conditions of the Creative Commons model surface irrigation both for the surface water movement and infiltration into the soil,
Attribution (CC BY) license (https:// using an equation that considers soil with a shallow water table for the latter.
creativecommons.org/licenses/by/
4.0/).
Agriculture 2022, 12, 426. https://doi.org/10.3390/agriculture12030426 https://www.mdpi.com/journal/agriculture
Agriculture 2022,
Agriculture 12, 12,
2022, x FOR
426 PEER REVIEW 2 of 212
of 12
Figure
Figure 1. Irrigation
1. Irrigation scheme
scheme in soils
in soils with
with shallow
shallow water
water table.
table.
The border irrigation water flow can be characterized using the hydrodynamic model
The border irrigation water flow can be characterized using the hydrodynamic model
of the Barré de Saint-Venant equations, which accurately describes surface irrigation [7].
of the Barré de Saint-Venant equations, which accurately describes surface irrigation [7].
The relationship between the width and the water depth in a border makes it possible
The relationship between the width and the water depth in a border makes it possible to
to consider the equations corresponding to runoff for an infinite surface width [8]. The
consider the equations corresponding to runoff for an infinite surface width [8]. The con-
continuity equation in the hydrodynamic model is written as follows:
tinuity equation in the hydrodynamic model is written as follows:
∂h ∂q ∂I
h ∂q+ ∂I = 0
∂+ (1)
∂t +∂x +∂t = 0 (1)
∂t ∂x ∂t
and the momentum equation is as follows [9]:
and the momentum equation is as follows [9]:
2 ∂q ∂q+ gh33 − q22 ∂∂h ∂I ∂I = 0
2 ∂q ∂q
( )
h + gh 3
hh ++2hq
∂t∂t 2hq∂x + gh − q ∂x+ gh 3 ( J(−J −
J o J)o+) β+qhβqh =0 (2)(2)
∂x ∂x ∂t ∂t
where
where q(x,t)
q(x,t) = U(x,t)h(x,t)
= U(x,t)h(x,t) is theis discharge
the discharge per width
per unit unit width
of theof the border
border or the unitary
or the unitary dis-
discharge, x is the spatial coordinate in the main direction of
charge, x is the spatial coordinate in the main direction of the water movement in the the water movement in the
border, t is the time, U is the mean velocity, h is the water depth,
border, t is the time, U is the mean velocity, h is the water depth, VI =I ∂I(x,t)/∂t is the V = ∂I(x,t)/∂t is the
infiltration flow, that is, the water volume infiltrated per unit width per unit length of the
infiltration flow, that is, the water volume infiltrated per unit width per unit length of the
border, I is the infiltrated depth, g is gravitational acceleration, β = U /U is a dimensionless
border, I is the infiltrated depth, g is gravitational acceleration, β = UIX IX/U is a dimension-
parameter where UIX is the projection in the direction of the output velocity of the water
less parameter where UIX is the projection in the direction of the output velocity of the
mass due to the infiltration, Jo is the topographic slope, and J is the friction slope that can
water mass due to the infiltration, Jo is the topographic slope, and J is the friction slope
be determined by the fractal law of hydraulic resistance [10]:
that can be determined by the fractal law of hydraulic resistance [10]:
!d
3 3 Jg
d
h
h Jg
q=
q = kkν
ν ν 2 2
(3) (3)
ν
where
where is the
ν isν the kinematic
kinematic viscosity
viscosity coefficient,
coefficient, k isk aisdimensionless
a dimensionless factor
factor thatthat includes
includes thethe
effects of soil roughness, and the exponent d has a fractal interpretation. From this law,the
effects of soil roughness, and the exponent d has a fractal interpretation. From this law,
theChezy
Chezyformula
formulaisisdeduced
deducedwith
withdd==1/21/2 and
and the
the Poiseuille
Poiseuille law
law with
with dd == 1.
1.
TheThe initial
initial andand boundary
boundary conditions
conditions forfor a closed
a closed border,
border, avoiding
avoiding loss
loss byby tailwater
tailwater
outside the irrigation domain, are as
outside the irrigation domain, are as follows: follows:
q(qx,( 0x,0
) =) =0 0; ;hh(x,
( x,0
0) )== 00 (4)(4)
q (t0,t
q(0, ) =) =
q0q; q; (qx(fx, t),t=
0 f ) =00; ; hh((xxf ,f ,tt))== 00
(5)(5)
where xf (t) is the position of the wave front at time t and qo is the unitary discharge at the
where xf (t) is
entrance ofthe
theposition
border. of the wave front at time t and qo is the unitary discharge at the
entrance of the border.Agriculture 2022, 12, 426 3 of 12
Agriculture 2022, 12, x FOR PEER REVIEW 3 of 12
Numerous practical situations require a numerical solution of the Richards equation,
with dimensions depending on the complexity of the studied problem. In surface irrigation,
a one-dimensional
Numerous practical equation is sufficient
situations requiretoa numerical
represent the infiltration
solution phenomenon
of the Richards [11].
equation,
However, the equation lacks general analytical solutions,
with dimensions depending on the complexity of the studied problem. In surface irriga- and therefore complex numerical
methods
tion, are often required
a one-dimensional in order
equation to solve it
is sufficient to [12–14].
representThere are other physics-based
the infiltration phenomenon
[11]. However, the equation lacks general analytical solutions, andin
models resulting from the simplification of the initial conditions, particular
therefore the Green
complex nu-
and Ampt
merical equation
methods are[15].
oftenThis equation
required has been
in order useditin[12–14].
to solve surfaceThere
irrigation [16–18].
are other How-
physics-
ever, with
based models thisresulting
equation,from onlythe a homogeneous
simplificationmoisture of the initialprofile can be represented
conditions, in particular in the
the
soil profile.
Green and Ampt equation [15]. This equation has been used in surface irrigation [16–18].
However, Fuentes with et al.
this[19] developed
equation, onlyan analytical solution
a homogeneous of theprofile
moisture Richardscanequation using the
be represented in
Green and
the soil profile. Ampt hypotheses to describe the infiltration of water into the soil with a shallow
water table
Fuentes et al.(Pf ).[19]
Thedeveloped
solution considers
an analytical a hydrostatic
solutioninitialof the moisture
Richards distribution (Figurethe
equation using 2),
where the initial moisture is calculated with
Green and Ampt hypotheses to describe the infiltration of wateri into the the expression θ (z) = θ + (θ − θ )(z/P
o soils witho a shal- f ).
Thus,
low the moisture
water table (Pf).content at the soil
The solution surfacea is
considers θi (0) = θo initial
hydrostatic and atmoisture
the water-table surface is
distribution
(Figure 2), where the initial moisture is calculated with the expression θi(z) = θo + (θs at
θ i (P f ) = θ s . The suction in the wetting front is a linear function of the moisture content −
the
θ front, hf (θi ,θ
o)(z/Pf). Thus,
s) =
the hf (θs −content
moisture θi )/(θs − o ), soil
at θthe thatsurface
is: is θi(0) = θo and at the water-table
surface is θi(Pf) = θs. The suction in the wetting front is a linear
zf
function of the moisture
content at the front, hf (θi,θsh) f=[θhi f(z(θf )s,−θsθ]i)/(θ = hsf −(θθoo,),θsthat −
) 1 is: (6)
Pf
zf
such that hf [θi (0),θs ] = hf and hfh [θf i(P ( z ) , θ =The
θi ),θ
f fs ] = s0.
h f (infiltrated
θo , θs ) 1 − depth
Pf is defined by:
(6)
such that hf[θi(0),θs] = hf and hf[θi(Pf),θs]zZf=(t)0. The infiltrated depth is defined by:
z (t)
s − θ (z)]dz
f
I(t) = [θ (7)
I ( t ) = θs −i θi ( z ) dz (7)
0 0
that
that is:
is:
1 1
( t )=I1M −= 1[1−−1z−f (zt)f /P
( t )f ]P2 ,f IM, =IM2 =∆θP
2
I(t)/IIM ΔθPf
2 f
(8)
where
where IIM is the maximum infiltrated depth.
M is the maximum infiltrated depth.
Figure 2. Soil
Soil moisture
moisture profile with a shallow water table.
Introducing Equation (6) into the Green and Ampt equation yields the differential
equation for the analytical solution that considers a shallow water table [19]:
h + h f ( 1 − z f Pf )
dI
= K s 1 + , z f = Pf 1 − 1 − I I M (9) ( )
dt zf Agriculture 2022, 12, 426 4 of 12
Introducing Equation (6) into the Green and Ampt equation yields the differential
equation for the analytical solution that considers a shallow water table [19]:
h + hf (1 − zf /Pf )
dI p
= Ks 1 + , zf = Pf 1 − 1 − I/IM (9)
dt zf
When the water depth is independent of time, h = h, and integration of Equation (9)
with the condition I = 0 at t = 0 leads to the following infiltration equation, where hf 6= Pf :
Pf 2 Pf (h+Pf )(h+hf )IM
hf I −
Pf −
hf ) 3
(Pf −q
h i
P f − hf I
Ks t = ln 1 + 1 − 1 − IM (10)
h+hf
2 Pf (h+hf )IM
q
I
+
2 1 − 1 − IM
(Pf −hf )
and when hf = Pf : " #
I 3/2 2
Pf IM I 2
Ks t = + 1− − (11)
h + Pf IM 3 IM 3
where h ∼ = [4d/(5d + 1)]ho is the mean water depth and ho = (ν2 /gJo )1/3 (qo /kν)1/3d is the
normal depth [20].
Equation (10) reduces to the Green and Ampt infiltration equation when Pf → ∞,
considering that IM = 1/2∆θPf . In this limit, 1 – (1 – I/IM )1/2 ∼ = I/2IM = I/∆θPf holds. The
third term is of the order of 1/Pf and tends to zero. In the second term, the argument of the
logarithm tends to 1 + I/∆θ(h + hf ) and its coefficient to ∆θ(h + hf ). Finally, in the first term,
the coefficient of I → 1. Using the definition λ = ∆θ (h + hf ), the Green and Ampt equation
is deduced.
In recent years, many different software packages have been developed to model
surface irrigation (e.g., [21,22]); however, they have some limitations due to the infiltration
equations used. For example, they only represent constant initial moisture conditions along
a homogeneous soil column, and they are empirical equations for a specific irrigation event.
Furthermore, they are not representative for the soils with shallow water tables (where the
moisture profile is not constant) existing in some irrigated agricultural areas.
The objectives of this study, the first in a series of three, are: (a) to model the advance
phase of surface irrigation in a soil with a shallow water table by coupling an analytical
solution to the Barré de Saint-Venant equations and (b) to validate the model obtained with
data from an irrigation test reported in the literature. In the second paper, the three phases
of irrigation will be included: advance, storage, and recession. Finally, in the third article,
this solution will be compared with the Barré de Saint-Venant equations coupled internally
with the Richards equation.
2. Materials and Methods
Numerical Solution
The numerical solution of the Barré de Saint-Venant equations uses a Lagrangian
scheme [9,23]. The discrete form of the continuity equation is as follows:
h i
ωqL + (1 − ω)qJ δt − [ωqR + (1 − ω)qM ]δt
−[ϕhL + (1 − ϕ)hR + ϕIL + (1 − ϕ)IR ](xR − xL )
+[ω(hR + IR ) + (1 − ω)(hM + IM )](xR − xM ) (12)
−[ω(hL + IL ) + (1 − ω)(hJ + IJ )](xL − xJ )
+[ϕhJ + (1 − ϕ)hM + ϕIJ + (1 − ϕ)IM ](xM − xJ ) = Rc
Due to the numerical instability presented in the calculation cells at the beginning,
a discrete form of the momentum equation was developed, considering the followingAgriculture 2022, 12, 426 5 of 12
Agriculture 2022, 12, x FOR PEER REVIEW 5 of 12
assumptions: (a) the derivatives in time were calculated as a rectangular cell (Eulerian)
and (b) average flow and water depth coefficients were considered for the calculation of an
average friction slope. The discrete form of the momentum equation is as follows:
( )
2hq ω ( q R − q L ) + ( 1 − ω) q M − q J δt
i
( ) ( )
h
1 (−h ω−) hqM
L)
2hq ω(+qRgh −3q−L )q +2
(ω + ( 1 −J ω)δth M − h J δt
− q
3 R
+ gh −
2
{
q2 2 [ω(hR − hL ) + (1 − ω)(hM − hJ )]δt
(
n + h ϕq L + ( 1 − ϕ ) q R ( xR − xLh) − ϕq J + ( 1 − ϕ ) qiM xM − x Jo )} (13)
(13)
(1 − ϕ)q ](x − x ) − ϕq + (1 − ϕ)q (x − xJ )
+h [ϕqL +
3
( )
+gh 3( J − J o ) ωR( x R R− x L ) L+ ( 1 − ω) JxM − x J δt M M
+gh J − Jo [ω(xR − xL ) + (1 − ω)(xM − xJ )]δt
{ ( )}
+λδtqh ϕI L + ( 1 − ϕ ) I R ( xR − xL ) − ϕI J + ( 1 − ϕ ) I M xM − x J = Rm
+λδtqh{[ϕIL + (1 − ϕ)IR ](xR − xL ) − [ϕIJ + (1 − ϕ)IM ](xM − xJ )} = Rm
where Rc and Rm represent the residuals of the continuity equation and momentum, re-
spectively,
where Rc and theRm representq the
coefficient = ω[(1 − φ)qRof+ the
residuals φqLcontinuity
] + (1 − ω)[(1 − φ)qM and
equation + φqmomentum,
J], and the coef- re-
spectively, the coefficient q = ω[(1 − ϕ)q + ϕq ] + (1 − ω)[(1 −
ficient h = ω[(1 − φ)hR + φhL] + (1 − ω)[(1 − φ)hM + φhJ], taking into account the extreme
R L ϕ)q M + ϕq J ], and the
coefficient h = ω[(1 − ϕ)hR + ϕhL ] + (1 − ω)[(1 − ϕ)hM + ϕhJ ], taking 2into account the ex-
values of each calculation cell, and consequently the coefficient J = ν ( q /kν)1/d/g h 3 . The 3
treme 2 1/d
weightvalues
factors offor
each timecalculation
and space cell,
areand consequently
denoted ω and φ,the coefficient J = ν (q/kν) /gh .
respectively.
The weight factors for
In the discrete timeaand
forms, spacefactor
weight are denoted
in spaceωofandφ =ϕ, respectively.
1/2 was considered for interior
In the discrete forms, a weight factor in space of ϕ = 1/2 was considered for interior
cells [23–26]. For the last cell and the first time step, φ = π/4 was used, deduced from the
cells [23–26]. For the last cell and the first time step, ϕ = π/4 was used, deduced from the
analysis of the short-time coupling of the Saint-Venant and Richards equations [27]. The
analysis of the short-time coupling of the Saint-Venant and Richards equations [27]. The
weight factor for time was taken as ω = 0.60 [24–26,28].
weight factor for time was taken as ω = 0.60 [24–26,28].
Figure 3 shows the Lagrangian cells during the advance phase. The subscripts L and
Figure 3 shows the Lagrangian cells during the advance phase. The subscripts L and
R denote the values of each variable to the left and right of the next time ti+1, and J and M
R denote the values of each variable to the left and right of the next time ti+1 , and J and M
represent the values at the current time ti.
represent the values at the current time ti .
Figure 3. Computational grid for the advance phase.
phase.
The
The solution
solution developed
developed is is implicit, where Equations
implicit, where Equations (12)
(12) and
and (13)
(13) are
are solved
solved simul-
simul-
taneously
taneously for all cells at each time step. By linearizing the discrete equations, a system of
for all cells at each time step. By linearizing the discrete equations, a system of
2N
2N ++ 22 nonlinear
nonlinear algebraic
algebraicequation
equationwith
with2N 2N+ +4 4unknown
unknownvariables is is
variables produced,
produced,where
whereN
is
N the number
is the number ofof
computational
computational cells.
cells.The
Thesolution
solutionofofthe
thesystem
systemisisobtained
obtained byby applying
applying
the
the double sweep algorithm, which assumes a linear relation between the variation of
double sweep algorithm, which assumes a linear relation between the variation of the
the
water depth δh and the discharge δq in a given time step
water depth δh and the discharge δq in a given time step [29]: [29]:
δqi = δEqi+i 1=δh
E ii++δh + F
1 Fii+1 i +1 (14)
and for the last cell the equation is expressed in terms of the advance distance, considering
that δhN and δqN are equal to zero:
δδ = FN +1 (15)Agriculture 2022, 12, 426 6 of 12
and for the last cell the equation is expressed in terms of the advance distance, considering
that δhN and δqN are equal to zero:
δδ = FN+1 (15)
where the values of E and F are calculated by the partial derivatives of the residual equations
of continuity and momentum for each cell with respect to hL , qL , hR , and qR .
With the coefficient values E and F, the improved solution for the new values of flow,
water depth, and position of the wave front is determined by:
hni +1 = hni + δhi , 0 ≤ i ≤ N (16)
qni +1 = qni + δqi , 1 ≤ i ≤ N (17)
δxnN+1 = δxnN + δδ (18)
The convergence criterion selected for advancing in time is that the residual values
of the continuity equation (Rc) and the momentum equation (Rm) in the current iteration
must be less than 1 × 10−5 .
The analytical solution for infiltration in a soil with a shallow water table is obtained
using the bisection method, which is widely used to find the roots of a function.
3. Results
3.1. Soil Characterization
Simulations were performed with the information reported by Pacheco [30] for a
clay soil from La Chontalpa, Tabasco, Mexico, using the coupling of the Barré de Saint-
Venant equations and the analytical solution to model infiltration in a soil with a shallow
water table. The measured data from the irrigation tests were: border width B = 10.5 m,
border length L = 100 m, border slope Jo = 0.00085 m/m, moisture content at saturation
θs = 0.5245 cm3 /cm3 , and the dimensionless parameter β = 0. The values of the hydraulic
conductivity at saturation (Ks ) and the suction in the wetting front (hf ) were optimized using
the Levenberg–Marquardt algorithm [31]. In the hydraulic resistance law, Equation (3),
d = 1 was used. The values of qo , Pf , IM , initial moisture (θo ), h, Ks , and hf are reported in
Table 1 for each irrigation test.
Table 1. Values of the initial conditions in three irrigation tests.
Test qo (m3 /s/m) Pf (cm) IM (cm) θo (cm3 /cm3 ) h (cm) Ks (cm/h) hf (cm) R2
1 0.001428 152 14.54 0.3331 2.73 1.1800 23.84 0.9983
2 0.001428 50 2.15 0.4386 2.73 1.5325 44.00 0.9814
3 0.001238 52 2.32 0.4353 2.60 0.0500 10.00 0.9967
This table shows the correlation coefficient R2 between the measured data and those
calculated with the optimized parameters of Ks and hf . Good adjustments were observed
for the three irrigation tests.
3.2. Applications
Simulations corresponding to each irrigation test were performed using the values
reported in Table 1 and the optimized values of Ks and hf .
Figure 4 shows the good fit of the measured data for the advance phase and those
obtained by the proposed model when optimizing the values of Ks and hf , which is
corroborated by the high R2 values obtained.Agriculture 2022, 12, x FOR PEER REVIEW 7 of 12
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100
100
90
90
80
80
70
70
(min)
60
(min)
60
Time 50
50
40
Time
40
30
30 Pf = 152 cm
20
Pf = 152 cm
50 cm
20
10 Pf = 50
52 cm
10 Pf = 52 cm
0
0 0 20 40 60 80 100 120
0 20 40 60
Distance (m) 80 100 120
Distance (m)
Figure 4. Evolution of the advance front in the border irrigation with shallow water table.
Evolutionof
Figure4.4.Evolution
Figure ofthe
theadvance
advancefront
frontin
inthe
theborder
borderirrigation
irrigation with
with shallow
shallow water
water table.
table.
3.3.
3.3. Comparison
Comparisonofofthe
the Theorical
Theorical Advance
Advance Curve
Curve with
with the
the Measured
Measured Data
Data
3.3. Comparison
The of the
information Theorical
from theAdvance
second Curve with
irrigation the
test Measured
(P Data
f = 50 cm) was used for a more ex-
The information from the second irrigation test (Pf = 50 cm) was used for a more
The information
haustive analysis. from
Figure the second
5 shows irrigation
the advance test obtained
curve (Pf = 50 cm)
by was used for
the by
model, a more
which ex-
shows
exhaustive analysis. Figure 5 shows the advance curve obtained the model, which
haustive
good afitanalysis.
ashows between Figure
the 5 shows
measured the advance
data obtainedcurve
in obtained
the field by
and the model,
those which
obtained
good fit between the measured data obtained in the field and those obtained with shows
with the
amodel.
good fit
the model. between the measured data obtained in the field and those obtained with the
model.
70
70
60
60
50
50
(min)
40
(min)
40
Time
30
Time
30
20
20
10 Measured data
10 Model development
Measured data
0 Model development
0 0 20 40 60 80 100
0 20 40Distance (m)
60 80 100
Distance (m)
Figure 5.
Figure Evolutionof
5. Evolution ofthe
theadvance
advance front
front in
in the
the second
second irrigation
irrigation test.
test.
Figure 5. Evolution of the advance front in the second irrigation test.
Figure 6 shows that when the infiltrated depth reaches the I calculated with Equation (8),
Figure 6 shows that when the infiltrated depth reaches M the IM calculated with Equa-
the soil
Figureis noshows
longerthat
capable ofthe
storing more water.reaches
Therefore,
the IMthe soil was with
considered
tion (8), the 6soil when
is no longer capable infiltrated
of storing depth
more water. Therefore, calculated Equa-
the soil was consid-
completely
tion (8), the saturated
soil is no 36 min
longer after
capable irrigation
of storinghad started.
more water. Therefore, the soil was consid-
ered completely saturated 36 min after irrigation had started.
The surfacesaturated
ered completely and subsurface flowirrigation
36 min after profiles are
hadshown
started.in Figure 7, where it can be
observed that the infiltration over all points of the border is limited by the maximum
infiltration shown in Figure 6. It is also evident from the third curve (39 min), that a
determined region of the soil was already at saturation, because the soil could no longer
store more water. Due to the complexity of obtaining the roots in the analytical solution,
the time used for this process was 55 min; however, the time discretization can be increased
to obtain reliable results in a shorter time.Cumulative infiltration (cm
2.00
1.50
Agriculture
Agriculture2022, 12,12,
2022, x FOR
426 PEER REVIEW 8 of8 of
1212
1.00
2.50
0.50
Cumulative infiltration (cm)
2.00
0.00
0 10 20 30 40
1.50 Time (min)
Figure 6. Evolution of the infiltrated depth in a border irrigation with a shallow water table.
1.00
The surface and subsurface flow profiles are shown in Figure 7, where it can be ob-
served
0.50that the infiltration over all points of the border is limited by the maximum infil-
tration shown in Figure 6. It is also evident from the third curve (39 min), that a deter-
mined region of the soil was already at saturation, because the soil could no longer store
0.00
more water. Due to the complexity of obtaining the roots in the analytical solution, the
0 10 20 30 40
time used for this process was 55 min; however, the time discretization can be increased
Time (min)
to obtain reliable results in a shorter time.
Figure
Figure 6. 6. Evolution
Evolution ofof the
the infiltrated
infiltrated depth
depth inin a border
a border irrigationwith
irrigation witha ashallow
shallowwater
watertable.
table.
4 surface and subsurface flow profiles are shown in Figure 7, where it can be ob-
The
served that the infiltration over all points of the border is limited by the maximum infil-
Water depth (cm)
tration shown in Figure 6. It is also evident from the third curve (39 min), that a deter-
3
mined region of the soil was already at saturation, because the soil could no longer store
more water. Due to the complexity of obtaining the roots in the analytical solution, the
time used
2 for this process was 55 min; however, the time discretization can be increased
to obtain reliable results in a shorter time.
1
4
(cm) (cm)
0 20 40 60 80 100
1
depth
3
depth
13.0 min
Infiltrated
2 26.0 min
2
39.0 min
Water
51.9 min
3
1 Distance (m)
Infiltrated depth (cm)
Figure 7. Flow profiles every 13 min of a border irrigation with a shallow water table.
Figure 7. Flow profiles every 13 min of a border irrigation with a shallow water table.
0 20 40 60 80 100
3.4.1Time Step Analysis
3.4. Time
An Step Analysis
analysis of the computation time was performed at different values of δt in the
An analysis of the computation time was performed atand13.0 minvalues of δt in the
different
numerical coupling of the Barré de Saint-Venant equations the analytical solution of
2 26.0
numerical
infiltrationcoupling of the Barré
with a shallow detable.
water Saint-Venant
Table 2 equations and
shows that the min
there analytical
was solution for
no difference of
infiltration
the differentwith a shallow
values of δt water table.
studied Table
here. The2 computer there39.0
wasmin
shows thatequipment no difference
used for the
to perform the
different values ® CoreTM i7-4710 51.9tomin
simulations
3 hasof
anδtIntel
studied here. The computer
CPU @ equipment
2.50 GHz andused
32 Gbperform
of RAM.the simula-
tions has an Intel Core i7-4710 Distance
® TM CPU @ 2.50(m)GHz and 32 Gb of RAM.
Table 2. Computation time for different time steps (δt).
Figure 7. Flow profiles every 13 min of a border irrigation with a shallow water table.2
δt Computation Time (min) R
1.0
3.4. Time Step Analysis 55.0
1.5 14.5 1.0
An analysis2.0
of the computation time was 8.2
performed at different values
1.0
of δt in the
numerical coupling
5.0 of the Barré de Saint-Venant
1.3 equations and the analytical
1.0 solution of
infiltration with a shallow water table. Table 0.3
10.0 2 shows that there was no difference
1.0 for the
different values of δt studied here. The computer equipment used to perform the simula-
tions has an Intel® CoreTM i7-4710 CPU @ 2.50 GHz and 32 Gb of RAM.δt δt Computation Time Time
Computation (min) (min) R R2
1.0 1.0 55.0 55.0
1.5 1.5 14.5 14.5 1.0 1.0
2.0 2.0 8.2 8.2 1.0 1.0
5.0 5.0 1.3 1.3 1.0 1.0
Agriculture 2022, 12, 426 9 of 12
10.0 10.0 0.3 0.3 1.0 1.0
According to Figure
According 8, for 8,
to Figure design purposes
for design it is recommended
purposes it is recommendedto use to
δt use
= 10δt
s to in-s to in-
= 10
crease the According speed,
processing to Figureand 8, forgive
this design
a purposes
fast response it isthe
for recommended to use δt
advance/infiltration = 10 s to
pro-
crease the processing speed, and this give a fast response for the advance/infiltration pro-
cess inincrease
each the
plot processing
where speed,
irrigation was and this give
required, with a presence
fast response
of thefor the table.
water advance/infiltration
cess in each plot where irrigation was required, with presence of the water table.
process in each plot where irrigation was required, with presence of the water table.
100
Computation time (min) 100
Computation time (min)
10
10
1
1
0.1
0.1
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0
Time step δt (s)
Time step δt (s)
Figure 8. Evolution of computation time for different values of δt used.
Figure Evolutionof
Figure8.8.Evolution ofcomputation
computationtime
timefor
fordifferent
differentvalues
valuesofofδt used.
δtused.
For the purpose of observing whether, with a temporal increment, the accuracy of
Forthe
For thepurpose
purposeof ofobserving
observingwhether,
whether, with
withaa temporal
temporal increment,the theaccuracy
accuracyofof
the solution was lost, simulations were performed for different valuesincrement,
of δt. However, the
the solution
the solution was lost, simulations were performed for different values of δt. However, the
coupling method was usedlost, simulations
in this were performed
work yielded for different
excellent results values
and when of δt. However,
performing the the
couplingmethod
coupling methodused usedin inthis
thiswork
work yielded
yielded excellent
excellentresults
resultsand
and when
when performing
performingthe the
temporal increments the errors were minimal (Figure 9). Nevertheless, it is recommended
temporal increments the errors were minimal (Figure 9). Nevertheless, it is recommended
to usetemporal
a small δtincrements
for researchthepurposes,
errors were minimal
since (Figureto
it is necessary 9).observe
Nevertheless, it is recommended
the phenomenon in
touse
to useaasmall
smallδtδtfor
forresearch
researchpurposes,
purposes,since
sinceititisisnecessary
necessaryto toobserve
observethethephenomenon
phenomenonin in
detail in order to provide solutions for particular cases that may occur in irrigation areas
detailin
inorder
orderto toprovide
providesolutions
solutionsfor
forparticular
particularcases
casesthat
thatmay
mayoccur
occurin inirrigation
irrigationareas
areas
with detail
these characteristics.
with these characteristics.
with these characteristics.
100 100
100 100
Measured data (m)
R² = 1 R² = 1
80 80
Measured data (m)
R² = 1 R² = 1
80 80
60 60
60 60
40 40
40 40
20 20
20 20
Agriculture 2022, 12, x FOR PEER REVIEW 0 0 10 of 12
0 0 20 40 60 80 100 0 020 40 60 80 100
0 (a20
) 40 60 80 100 0 (b20
) 40 60 80 100
100 (a) 100 (b)
Measured data (m)
R² = 1 R² = 1
80 80
60 60
40 40
20 20
0 0
0 20 40 60 80 100 0 20 40 60 80 100
Simulated data (m) Simulated data (m)
(c) (d)
FigureFigure
9. Relationship between
9. Relationship measured
between data and
measured datadata andsimulated by the model
data simulated by thewithmodel optimized
with optimized
parameters at different
parameters time steps:
at different time (a) δt =(a)
steps: 1.5δt
s; =(b)
1.5δts;= (b)
2.0 δt
s; (c) δt =s; 5.0
= 2.0 (c) s;
δt(d) δt =s; 10.0
= 5.0 s. = 10.0 s.
(d) δt
4. Discussion
The numerical coupling of the Barré de Saint-Venant equations with the analytical
solution for a soil with a shallow water table allows the effect of this on the infiltrationAgriculture 2022, 12, 426 10 of 12
4. Discussion
The numerical coupling of the Barré de Saint-Venant equations with the analytical
solution for a soil with a shallow water table allows the effect of this on the infiltration
process to be analyzed and related to the maximum infiltrated depth of the soil and the
most suitable crop for soils with this characteristic, as well as showing how the advance
wave process is affected at the surface.
By optimizing the parameters Ks and hf , it was possible to represent surface irriga-
tion in soils with shallow water tables, obtaining excellent results for the coefficient of
determination for each irrigation test, which confirms that the proposed model adequately
reproduces the data measured in the field. In addition, with the values obtained it is
possible to calculate, for the next irrigation, the optimal irrigation flow that should be
applied in each border, by means of an analytical representation that takes into account
the border length, the net irrigation depth, and the characteristic parameters of infiltration
obtained through the inverse process shown here [32].
Solving the numerical coupling of the model developed by the explicit method has a
high computational cost. However, by having δt > 1 s, the computational time decreases by
up to 99.45% compared to δt = 1 s, and the results provided by the model still fit correctly.
This method makes it possible to increase the processing speed and to ideally reproduce
the advance/infiltration process in the border irrigation in soils with shallow water tables.
It is important to note that for research purposes it is advisable to use a small δt to observe
in detail what happens in the advance phase of irrigation.
By using an implicit scheme for the discretization of the equations, a higher degree
of accuracy is provided, because the solution is iterative and considers what is occurring
in the current state of the flow profiles, as well as in the previous state. In contrast, the
explicit scheme only solves the system once, without considering the current state of the
flow profiles, which can generate accuracy problems when using different values of δt.
The developed model has the advantage of not requiring as many physical parameters
associated with the plots and irrigation, without losing the representativeness of the soils,
while the use of the Richards equation requires a hydrodynamic characterization of the
soil including the characteristic moisture curve and the hydraulic conductivity curve.
This represents more exhaustive characterization work and more time to obtain solutions
in the process of border irrigation, because the equation requires a complex numerical
solution [11].
Our method is therefore a robust and efficient model for the advance phase in soils
with shallow water table, compared to others reported in the literature that use infiltration
equations without a physical basis, such as the Kostiakov (e.g., [21,23]) and Kostiakov–
Lewis equations (e.g., [22,33]).
5. Conclusions
A numerical solution of the Barré de Saint-Venant equations coupled to the analyt-
ical solution of the infiltration for a soil with a shallow water table was implemented to
describe the advance phase of border irrigation. The proposed solution of the model used
a Lagrangian finite-difference scheme for the Barré de Saint-Venant equations, while the
infiltration equation was solved by the bisection method. The model evaluation in the
simulation of the advance phase revealed that the model fits a set of experimental data.
Finally, the solution shown here can be used with excellent results to design and model
surface irrigation, since the infiltration equation used for a soil with a shallow water table
requires only a few characteristic soil parameters (Ks and hf ) that are easy to obtain.
Author Contributions: S.F. and C.C. contributed equally to this work. S.F. and C.C. All authors have
read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.Agriculture 2022, 12, 426 11 of 12
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Acknowledgments: The first author is grateful to CONACyT for the scholarship grant, scholarship
number 957179. In addition, we would like to thank the editor and the expert reviewers for their
detailed comments and suggestions for the manuscript. These were very useful in improving the
quality of the manuscript.
Conflicts of Interest: The authors declare no conflict of interest.
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