estimation of advance and infiltration equations in furrow irrigation for untested discharges
DESCRIPTION
The objectives of this work are: (1) to develop an analytic procedure that permitsestimation of the advance and infiltration equations in furrow irrigation for untesteddischarges; and (2) to obtain a predictive equation that expresses the infiltration parameters as a function of the furrow wetted perimeterTRANSCRIPT
Estimation of advance and infiltration equationsin furrow irrigation for untested discharges
Jose Antonio Rodrıguez Alvarez*
Irrigation and Drainage Research Institute, Ave. Camilo Cienfuegos y Calle 27, Arroyo Naranjo,
Ciudad de la Habana Apdo, 6090 Habana 6, Cuba
Accepted 4 November 2002
Abstract
The exponents of the advance and infiltration power laws have been shown to remain practically
constant for different furrow irrigation discharges. Under this hypothesis, a procedure to estimate the
advance and infiltration equations corresponding to untested discharges was developed. The proposed
procedure was validated with different field experiments, obtaining satisfactory results for non-
erosive discharges. However, significant deviations were obtained when erosive discharges were
used. This behavior corroborates the hypothesis presented by some authors that the erosion and
sedimentation processes occurring in furrow irrigation as a consequence of high surface velocities
can reduce—and even suppress—the effect of the wetted perimeter on the infiltration rate. Finally, an
equation was derived to predict the effect of the wetted perimeter on the infiltration parameters.
# 2002 Elsevier Science B.V. All rights reserved.
Keywords: Furrow irrigation; Advance; Infiltration; Wetted perimeter
1. Introduction
The evaluation methodologies proposed for furrow irrigation systems often suggest the
performance of field trials with a range of discharges. The minimum discharge would be
the lowest discharge guaranteeing the advance front reaches the downstream end of the
furrow; the maximum discharge is the highest discharge that does not produce soil erosion
(Merriam and Keller, 1978; Walker and Skogerboer, 1987; Walker, 1989). However, in
many occasions it is impractical to evaluate such an extensive range of discharges.
Given the usual data scarcity, design methodologies for furrow irrigation systems
consider that the parameters of the infiltration equations remain constant with regard to
Agricultural Water Management 60 (2003) 227–239
* Tel.: þ53-7-911038; fax: þ53-7-911038.
E-mail address: [email protected] (J.A.R. Alvarez).
0378-3774/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 8 - 3 7 7 4 ( 0 2 ) 0 0 1 6 3 - 4
the irrigation discharge (Walker and Skogerboer, 1987; Walker, 1989). Nevertheless,
numerous research works have shown that furrow irrigation infiltration is a two-dimen-
sional process. Therefore, the infiltration rate increases with the wetted perimeter, and
consequently with the irrigation discharge (Fangmeier and Ramsey, 1978; Freyberg, 1983;
Izadi and Wallender, 1985; Samani et al., 1985; Trout, 1992; Schmitz, 1993). Therefore,
this simplification can introduce significant errors.
The influence of the wetted perimeter on furrow infiltration is an aspect that should be
considered in the design and the evaluation of these systems. Low furrow distribution
uniformities can be due not only to the spatial variability of the opportunity times and soil
properties within each furrow, but also to the variation of the irrigation discharge among
furrows (Samani et al., 1985). It is estimated that approximately one-third of the infiltration
variability can be explained by variations in the furrow wetted perimeter (Izadi and
Wallender, 1985; Oyonarte and Mateos, 1995). Research work performed with mathe-
matical models has shown that better simulation results are obtained when the effect of the
wetted perimeter on the infiltration parameters is taken into account (Strelkoff and Souza,
1984; Bautista and Wallender, 1985; Schwanki and Wallender, 1988; Bautista and
Wallender, 1993; Camacho et al., 1997).
The objectives of this work are: (1) to develop an analytic procedure that permits
estimation of the advance and infiltration equations in furrow irrigation for untested
discharges; and (2) to obtain a predictive equation that expresses the infiltration parameters
as a function of the furrow wetted perimeter.
2. Materials and methods
The simplest and most common equation representing the advance trajectory in furrow
irrigation uses the power law (Elliot and Walker, 1982; Walker and Skogerboer, 1987;
Scaloppi et al., 1995)
t ¼ pxr (1)
where t is the time of advance (min); x the distance of advance (m); while p and r are
empirical coefficients. The equivalent model in furrow infiltration is the Kostiakov
equation (Kostiakov, 1932; Smerdon et al., 1988; DeTar, 1989)
Z ¼ Ktao (2)
where Z is the cumulative infiltration in units of volume per unit length of furrow (m3/m); tothe infiltration opportunity time (min); and K and a are empirical coefficients.
The basic hypothesis adopted in this research is that the r and a exponents of the advance
and infiltration equations (Eqs. (1) and (2)) remain almost constant with the irrigation
discharge, if the inflow is not very small relative to the infiltration. This hypothesis has been
validated by numerous field trials (Fok and Bishop, 1965; Walker and Skogerboer, 1987;
Rodrıguez, 1996). Also, Garcıa (1991) and Hanson et al. (1993) showed that there is an
inverse relationship between these exponents.
The idea behind this research is to express the cumulative infiltration proportional to
flow rate raised to some power; so that the traditional measurements of volume infiltrated
per unit length can be adjusted for a discharge different from the one at which the
228 J.A.R. Alvarez / Agricultural Water Management 60 (2003) 227–239
measurements were taken. If r and a are invariant, it is the coefficients p and K of Eqs. (1)
and (2) that vary in response to different irrigation discharges. The volume balance
equation (Walker, 1989) can be evaluated at two points of the advance trajectory (half of
the furrow length, 0.5L, and the total furrow length, L) in order to determine these
coefficients
QtL ¼ AoLry þ KtaLLrz (3)
Qt0:5L ¼ Ao0:5Lry þ Kta0:5L0:5Lrz (4)
where Q is the irrigation inflow (m3/min); t0.5L the advance time to half of the furrow length
(min); tL the advance time to the total furrow length (min); Ao the area of the surface flow at
the upstream end of the furrow (m2); and ry and rz the surface and subsurface shape factors,
respectively. t0.5L and tL can be obtained from Eq. (1) as
t0:5L ¼ pð0:5LÞr(5)
tL ¼ pLr (6)
Substituting Eq. (6) in (3), we have
pQLr ¼ AoLry þ KpaLðraþ1Þrz (7)
Assigning to
VQL ¼ pQLr (8)
VyL ¼ AoLry (9)
VZL ¼ KpaLðraþ1Þrz (10)
where VQL is the volume of inflow at tL, VyL the surface flow volume at tL and VZL the
infiltrated volume at tL. Substituting the Eqs. (8)–(10) in (7)
VQL ¼ VyL þ VZL (11)
Similarly, Eq. (5) can be substituted in (4) and expressed in terms of Eqs (8)–(10), resulting
0:5rVQL ¼ 0:5VyL þ 0:5ðraþ1ÞVZL (12)
From Eq. (11) VZL is found as
VZL ¼ VQL � VyL (13)
Substituting Eq. (13) in (12), we have
VQL
VyL
¼ s (14)
where
s ¼ 0:5 � 0:5raþ1
0:5r � 0:5raþ1
Similarly, if Eq. (11) is substituted in (12)
VZL
VyL
¼ s� 1 (15)
J.A.R. Alvarez / Agricultural Water Management 60 (2003) 227–239 229
Substituting Eqs. (8) and (9) in (14), p is obtained as
p ¼ AoL1�rry
Q� s (16)
Likewise, if Eqs. (9), (10) and (16) are substituted in (15); K is found as
K ¼ QaðAoryÞ1�a
Larz
s� 1
sa
� �(17)
After performing a field trial with a certain known discharge, Qe, the pe and Ke
coefficients of the advance and infiltration equations are experimentally determined,
as well as r and a. Then for any other untested discharge, Qne, these coefficients, pne and
Kne, can be estimated considering that the r and a exponents and the furrow geometry do
not vary with the applied discharges (ry, rz, and s are constants). Applying Eq. (16) to the
evaluated discharge, Qe
pe ¼AoeL1�rry
Qe
� s (18)
For the untested discharge, Qne
pne ¼AoneL1�rry
Qne
� s (19)
Dividing (19) by (18) and simplifying, results
pne
pe
¼ AoneQe
AoeQne
(20)
Following a similar procedure on Eq. (17)
Kne
Ke
¼ Aone
Ae
� �1�aQne
Qe
� �a
(21)
The cross-sectional area of the surface flow can be estimated through the Manning
equation (Walker, 1989) as follows:
Ao ¼ Qn
60p1ffiffiffiffiffiSo
p� �1=p2
(22)
where n is the Manning resistance coefficient; So is the furrow longitudinal slope (m/m);
while p1 and p2 the coefficients depending on the furrow geometry. Substituting Eq. (22) in
(20) and (21), respectively and simplifying
pne
pe
¼ Qne
Qe
� �ð1=p2Þ�1
(23)
Kne
Ke
¼ Qne
Qe
� �ðð1�aÞ=p2Þþa
(24)
230 J.A.R. Alvarez / Agricultural Water Management 60 (2003) 227–239
Using Eqs. (23) and (24) it is possible to estimate the advance and infiltration parameters
for the untested discharges in the field trials.
2.1. Influence of the furrow wetted perimeter on the infiltration parameters
As an approximation, with constant inflow, the relationship between infiltration and
furrow wetted perimeter can be modeled as a power law (Blair and Smerdon, 1985; Trout,
1992; Utah State University, 1993; Oyonarte and Mateos, 1995)
Zne ¼ ZeWPne
WPe
� �j
(25)
where Zne and Ze are the volumes of water infiltrated in one unit of furrow length
corresponding to the WPne and WPe wetted perimeters respectively; while j represents
the empirical exponent of the power law.
Utilizing the Kostiakov infiltration model ((Eq. (2)), Eq. (25) becomes
Zne ¼ Zeta WPne
WPe
� �j
(26)
From Eq. (26) it is deduced that
Kne ¼ KeWPne
WPe
� �j
(27)
Substituting Eq. (24) in (27) results
Qne
Qe
� �ðð1�aÞ=p2Þþa
¼ WPne
WPe
� �j
(28)
If the Manning equation is used, the relationship between the discharges and the furrow
wetted perimeters results
WPne
WPe
¼ Qne
Qe
� �ð2:5=p2Þ�1:5
(29)
Substituting Eq. (29) in (28), j is finally obtained as
j ¼ 1 � a
p2þ a
� �p2
2:5 � 1:5p2
� �(30)
2.2. Field experiments
Twelve furrow irrigation events were evaluated in order to provide field data to
support the theoretical developments. The experiments were performed in four different
locations differing in soil types, longitudinal slopes and furrow geometries. Three
discharges were tested at each location. Table 1 presents the details of the four sites
and the advance parameters. In each of the 12 experiments, sets consisting of three
furrows were evaluated. The discharge applied to the central furrow of each set was
J.A.R. Alvarez / Agricultural Water Management 60 (2003) 227–239 231
measured with a Parshall flume installed at the upstream end. The longitudinal field slope
was determined by lineal regression of the soil surface elevations measured with a
conventional topographical level. The advance times were measured in the central furrow
of each set at stations located on 20 m spacing. The cross-sections of the evaluated
furrows were measured with a profilometer (Walker, 1989) at three stations, located at the
beginning, center and end of the furrows. The three resulting geometric data sets were
averaged. Manning’s resistance coefficient was estimated according to the values
proposed by Walker and Skogerboer (1987) for freshly tilled or previously irrigated,
smooth soils.
2.3. Verification of the proposed estimation procedure
The advance equation parameters (Eq. (1)) were determined for each experiment using
nonlinear regression analyses. Subsequently, the Kostiakov infiltration parameters (Eq. (2))
were obtained through the volume balance method with the two-point approach (Walker,
1989). To verify the accuracy of the obtained infiltration equations, the field measured
advance curves were compared with advance curves obtained with the SIRMOD model
(Utah State University, 1993).
The measured advance and infiltration curves derived from the field data were compared
with those estimated using the proposed procedure. The field data corresponding to the
smallest discharge at each location were used in Eqs. (23) and (24) to estimate advance and
infiltration for the other two discharges. The estimated values were compared to those
measured in the field trials. Likewise, the measured advance curves were compared with
the simulated ones, using the Kostiakov parameters estimated with different criteria of
furrow wetted perimeter influence on the infiltration parameters. The criteria used were: (1)
no influence (j ¼ 0 in Eq. (26)); (2), lineal function (j ¼ 1 in Eq. (26)); and (3) power law
(j calculated by Eq. (30)).
The infiltration parameters used in the simulation model corresponded, in all cases,
to the ones derived from the field evaluations performed with the smallest discharges.
Table 1
Main characteristics of the field evaluations
Site Rice Institute
Habana (IIA)
‘‘Urbano Noris’’
IC Holguın (UN)
‘‘Jose Marti’’ ICa
Pinar del Rıo (JM)
‘‘Cuba Libre’’ IC
Matanzas (CL)
Soilb Vertic Gleysol Eutric Vertisol Haplic Acrisol Rodic Ferrasol
Slope (m/m) 0.0020 0.00092 0.0030 0.0012
Furrow spacing (m) 1.60 1.60 1.60 1.60
Furrow length (m) 240.0 380.0 333.0 333.0
p1 0.529 0.4632 0.508 0.522
p2 1.337 1.333 1.327 1.333
Manning’s n 0.02 0.04 0.03 0.02
Discharges (L/s) 2.0; 3.0; 4.0 3.2; 6.6; 7.5 2.0; 3.0; 4.0 3.0; 4.0; 5.0
a IC: industrial complex of sugar cane.b According to FAO–UNESCO Soil Classification (FAO–UNESCO, 1988).
232 J.A.R. Alvarez / Agricultural Water Management 60 (2003) 227–239
According to the previous criterion, the SIRMOD INFILT_N parameter (which
is equivalent to j) was used to simulate infiltration in the other experimental
discharges.
3. Results
The advance and infiltration equation parameters (Eqs. (1) and (2)) obtained from the
field trials are shown in Table 2. The average surface flow velocity at the upstream end of
the furrow, Vo, calculated as the ratio between the irrigation discharge and the correspond-
ing flow area (Eq. (22)) is also presented in the table for each case. Fig. 1 presents the
advance curves simulated with SIRMOD using the Kostiakov parameters derived from the
field evaluations (Table 2). The simulated advance curves are compared with the observed
advance values.
3.1. Estimation of the advance and infiltration curves
In Fig. 2a the measured advance times are compared with the estimated ones through the
Eq. (23). Two trends can be clearly appreciated. Some experiments are located on the 1:1
line, reflecting an adequate prediction, while other experiments deviate considerably,
suggesting an overestimation of the advance times calculated with Eq. (23). A detailed
analysis of Fig. 2a disclosed that the overestimated values correspond to those field trials
that were performed with erosive discharges. It can be observed in Table 2 that the surface
velocities calculated for those trials are in excess of the 13–15 m/min recommended as a
maximum admissible range for soils with a clay texture (Walker, 1989). If the trials
performed under erosive discharges are eliminated, an excellent agreement between
estimated and measured advance times is obtained (Fig. 2b). The same can be said for
Table 2
Parameters of the advance and infiltration equations obtained from the field trials, and computed values of the
surface velocity
Site Discharge (L/s) p r K (m3/m/min2) a Vo (m/min)
IIA 2 0.03577 1.457 0.01065 0.367 14.18
3 0.03053 1.463 0.01445 0.372 15.72
4 0.02238 1.466 0.01436 0.391 16.89
UN 3.2 0.03129 1.381 0.00455 0.499 9.51
6.6 0.02625 1.383 0.00897 0.494 7.80
7.5 0.02487 1.387 0.00985 0.498 8.06
JM 2 0.03828 1.527 0.01369 0.381 12.27
3 0.03898 1.507 0.02033 0.371 13.56
4 0.02229 1.531 0.01756 0.400 14.55
CL 3 0.00704 1.731 0.00795 0.503 13.01
4 0.00613 1.729 0.00942 0.509 13.97
5 0.00488 1.729 0.00975 0.526 14.78
J.A.R. Alvarez / Agricultural Water Management 60 (2003) 227–239 233
Fig. 1. Measured and simulated advance curves. Simulations were performed using the infiltration parameters derived from the field experiments using the volume
balance approach.
23
4J.A
.R.
Alva
rez/A
gricu
ltura
lW
ater
Ma
na
gem
ent
60
(20
03
)2
27
–2
39
Fig. 2. Comparison of measured and estimated parameters: (a) advance times for all field experiments; (b) advance times for experiments with non-erosive discharges;
(c) volume infiltrated per unit length for experiments with non-erosive discharges.
J.A.R
.A
lvarez
/Ag
ricultu
ral
Wa
terM
an
ag
emen
t6
0(2
00
3)
22
7–
23
92
35
Fig. 3. Measured and simulated advance curves. Simulations are presented using different criteria for the influence of the furrow wetted perimeter on the infiltration
parameters, represented by values of phi (j) of 0, 1 and as determined using Eq. (30).
23
6J.A
.R.
Alva
rez/A
gricu
ltura
lW
ater
Ma
na
gem
ent
60
(20
03
)2
27
–2
39
the volume infiltrated per unit length estimated through Eq. (24) and those derived from the
field evaluations (Fig. 2c).
In Fig. 3 measured and simulated advance curves are presented for different criteria
regarding the furrow wetted perimeter influence on the Kostiakov infiltration parameters.
Considering the results shown in Fig. 2a, only the data corresponding to the field
evaluations performed with non-erosive discharges are presented.
4. Discussion
Table 2 serves to confirm, one more time, the basic hypothesis adopted in the
development of this work. The advance and infiltration exponents (r and a) show minor
variations with the irrigation discharge in each experimental location. Only in those trials
performed with erosive discharges, small deviations of the exponents are observed with
respect to the values obtained in the rest of the experiments. This can be explained taking
into account the effects that the erosion and deposition processes produce in the soil
physical properties, as will be discussed latter.
In spite of the simplicity and the recognized limitations of the method adopted to derive
the Kostiakov parameters from field trials (Smerdon et al., 1988; Scaloppi et al., 1995;
Rodrıguez, 1996; Valiantzas, 1997a,b); the results were sufficiently satisfactory to fulfill to
the objectives of this work. This is confirmed by the excellent agreement between the
simulated and measured advance curves for each of the field evaluations (Fig. 1).
The application of the proposed procedure for estimating the advance and infiltration
equation parameters corresponding to untested discharges led to interesting results. In the
first place, the reported reduction and even suppression of the effect of the wetted perimeter
on the infiltration rate as a response to the erosion and sedimentation processes (Trout,
1992) was confirmed. The theory behind this hypothesis states that in the evolution of the
erosive processes produced under high discharges, the susceptible material is entrained
until the supply is exhausted, producing a crusting layer that reduces the infiltration
capacity of the soil (Fernandez et al., 1995). Therefore, the dependence of the infiltrated
depth with respect to the furrow wetted perimeter is notably affected.
As observed in Fig. 2a, the field observed advance times under erosive discharges are
lower than the ones estimated with the proposed procedure (which does not take erosion/
deposition into account). Smaller than observed estimated advance times are evidence that
the soil infiltration capacity has been reduced. Once the trials performed with erosive
discharges were removed from the data set, adequate prediction of the advance times and
volume infiltrated per unit length was achieved. This is confirmed by the high determina-
tion coefficient found between the estimated and measured values, as well as the intercepts
and the slopes of the regression lines obtained, whose values are very close to zero and one,
respectively (Fig. 2b and c).
The measured and simulated advance curves (Fig. 3) showed acceptable agreement only
when the effect of furrow wetted perimeter on the Kostiakov parameters was modeled as a
power law with exponent given by Eq. (30). The poorest agreement corresponds to the
‘‘UN’’ evaluations, in which the simulated discharge practically doubles the experimental
value. However, the maximum difference between measured and simulated advance time
J.A.R. Alvarez / Agricultural Water Management 60 (2003) 227–239 237
does not exceed 5 min. If, on the other hand, it is considered that the influence of the furrow
wetted perimeter on the Kostiakov infiltration parameters can be modeled with a lineal
function (j ¼ 1), as Strelkoff and Souza (1984) proposed, or simply does not exist
(j ¼ 0), as several methodologies for designing furrow irrigation systems assume (Walker
and Skogerboer, 1987; Walker, 1989); then, significant deviations among the measured and
simulated advance curves are observed (Fig. 3).
Finally, it must be understood that the concept of cumulative infiltration dependent on
inflow raised to some power makes sense only if the inflow stays constant during the entire
event. For example, in cutback irrigation, when the inflow is cut in half once the stream
reaches the end of the field, the wetted perimeter drops substantially, and so does the
volume infiltrated per unit length per unit time, but certainly the cumulative infiltration
does not decrease, only its rate of accumulation. Therefore, the results of the approach
developed in this paper, do not seem to apply directly to a model in which the wetted
perimeter is calculated at all points in the stream at all times, and its effect upon rate of
accumulation of infiltrated volume per unit length calculated, as in the SRFR model
(Strelkoff et al., 1998).
5. Conclusions
1. The developed analytic procedure adequately estimates the advance and infiltration
equations of furrow irrigation for untested discharge in the field trials.
2. The erosion and sedimentation processes that occur in furrow irrigation as a
consequence of high surface velocities can reduce or even suppress the effect of the
wetted perimeter on the infiltration rate.
3. Under conditions of constant inflow, the influence of the furrow wetted perimeter on
the Kostiakov infiltration parameters can be satisfactorily modeled as a power law.
Acknowledgements
My thanks to Dr. Enrique Playan of the Estacion Experimental de Aula Dei, CSIC,
Saragosse, Spain, for reviewing this manuscript.
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