estimation of advance and infiltration equations in furrow irrigation for untested discharges

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)


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


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).


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


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


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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estimation of advance and infiltration equations in furrow irrigation for untested discharges

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