Overland Water Flow and Solute Transport:
Model Development and Field-Data Analysis
Fariborz Abbasi1; Jirka Simunek2; M. Th. van Genuchten3; Jan Feyen4; Floyd J. Adamsen5;
Douglas J. Hunsaker6; Theodore S. Strelkoff7; and Peter Shouse8
Abstract: The application of plant nutrients with irrigation water is an efficient and cost-effective method for fertilizer application toenhance crop production and reduce or eliminate potential environmental problems related to conventional application methods. In thisstudy, a combined overland water flow and solute transport model for analysis and management of surface fertigation/chemigation ispresented. Water flow is predicted with the well-known Saint-Venant’s equations using a control volume of moving cells, while solutetransport is modeled with the advection-dispersion equation. The 1D transport equation was solved using a Crank-Nicholson finitedifference scheme. Four, large-scale, field experiments were conducted on blocked-end and free draining furrows to calibrate and verify
the proposed model. The results showed that application of solute during the entire irrigation event, or during the second half of theirrigation for blocked end conditions with appropriate inflow rates, produced higher solute uniformity than application of solute during thefirst half of the irrigation event. Measured fertilizer distribution uniformity of the low quarter ranged from 21 to 76% while fertilizer
distribution uniformity of the low half values varied between 62 to 87%. The model was subsequently applied to the experimental data;
results showed good agreement with all field data. Water balance errors for the different experiments varied from 0.004 to 1.8%, whereas
fertilizer mass balance errors ranged from 1.2 to 3.6%. A sensitivity analysis was also performed to assess the effects of longitudinal
dispersivity parameter on overland solute concentrations. A value of 10 cm for dispersivity provided a reasonable fit to the experimental
data.
DOI: 10.1061/共ASCE兲0733-9437共2003兲129:2共71兲
CE Database subject headings: Overland flow; Furrow irrigation; Dispersion; Fertilization; Data analysis.

Introduction
Fertilizers are widely applied to agricultural fields using surface
fertigation, i.e., application of fertilizers with irrigation water.
However, there are still no adequate guidelines for the proper
1
Institute for Land and Water Management, Katholieke Universiteit
Leuven, Belgium, Vital Decosterstraat 102, 3000-Leuven, Belgium.
E-mail: FRABBASI@hotmail.com
2
George E. Brown Jr. Salinity Laboratory, USDA-ARS, 450
West
Big
Springs
Dr.,
Riverside,
CA 92507.
E-mail:JSIMUNEK@ussl.ars.usda.gov
3
George E. Brown Jr. Salinity Laboratory, USDA-ARS, 450 West Big
Springs Dr., Riverside, CA 92507. E-mail: RVANG@ussl.ars.usda.gov
4
Institute for Land and Water Management, Katholieke Universiteit
Leuven, Belgium, Vital Decosterstraat 102, 3000-Leuven, Belgium.
E-mail: JAN.FEYEN@agr.kuleuven.ac.be
5
U.S. Water Conservation Laboratory, USDA-ARS, 4331 East Broadway Rd., Phoenix, AZ 85040. E-mail: FADAMSEN@uswcl.ars.ag.gov
6
U.S. Water Conservation Laboratory, USDA-ARS, 4331 East Broadway Rd., Phoenix, AZ 85040. E-mail: DHUNSAKER@uswcl.ars.ag.gov
7
U.S. Water Conservation Laboratory, USDA-ARS, 4331 East Broadway Rd., Phoenix, AZ 85040. E-mail: FSTRELKOFF@uswcl.ars.ag.gov
8
George E. Brown Jr. Salinity Laboratory, USDA-ARS, 450 West Big
Springs Dr., Riverside, CA 92507. E-mail: PSHOUSE@ussl.ars.usda.gov
Note. Discussion open until September 1, 2003. Separate discussions
must be submitted for individual papers. To extend the closing date by
one month, a written request must be filed with the ASCE Managing
Editor. The manuscript for this paper was submitted for review and possible publication on October 12, 2001; approved on July 3, 2002. This
paper is part of the Journal of Irrigation and Drainage Engineering,Vol. 129, No. 2, April 1, 2003. ©ASCE, ISSN 0733-9437/2003/271– 81/$18.00.

design and management of surface fertigation. The method has
certain advantages as compared to conventional application methods, such as reduced energy, labor, soil compaction, and machinery costs. Moreover, it allows growers to apply nutrients in small
amounts throughout the season in response to crop needs without
the possibility of crop damage or soil compaction caused bymechanized application methods. Concerns about the use of surface fertigation stem from the alleged low uniformity and efficiency of surface irrigation systems. Treadgill 共1985兲 conducted achemigation survey in the United States and found that while
61% of microirrigation and 43% of sprinkler irrigation systems
used chemigation, only 3.5% of the surface irrigation systems
utilized this technique. Likely reasons for the limited use foundfor surface fertigation were the typically low uniformity of surface irrigation systems and fertilizer losses due to runoff 共Treadgill et al. 1990兲. However, more recent findings by Hanson et al.
共1995兲 on 959 irrigation fields in California showed that the uniformity of border and furrow irrigation was generally higher thanthat for all other irrigation systems in the area. These findings
indicate that further studies on surface fertigation are needed.
Jaynes et al. 共1992兲 conducted a fertigation experiment andcompared the leaching behavior of two conservative mobile tracers. They studied the fate of Br applied in the irrigation water ando-triflouromethyl benzoic acid sprayed on the soil surface before
irrigation. Their results implied that application of fertilizers in
irrigation water applied in level basins could potentially increase
deep leaching of agricultural chemicals. Earlier studies further
indicated rapid leaching along preferential flow paths whenchemicals were applied with the irrigation water 共Bowman andRice 1986; Jaynes et al. 1988兲. Preferential flow was also ob-

JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / MARCH/APRIL 2003 / 71

Table 1. Infiltration and Roughness Parameters, Field and Solute Properties, Furrow Section and Geometry Parameters as the Input Values during

Model Performance for Different Field Experiments
ExperimentParameter
Field length, L (m)Furrow spacing, W (m)Field slope, S o (m•m⫺1 )Hydraulic section parametersa

␳ 1 (m3.33⫺2␳ 2 )
␳ 2 (-)
Furrow geometry parametersb

␴ 1 (m1/2␴ 2 )
␴ 2 (-)
Infiltration parametersa(-)

k (m3 •min⫺a•m⫺1 ) cc (m3 •m⫺1 ) c
Manning, n (m1/6)
Time of cut-off, T co (min)Inflow rate, Q o (L•s⫺1 )Input solute concentration, C 0 , 共g Br•L⫺1兲
Solute application time, T f , 共min兲
Dispersivity, D l , 共cm兲100%

First half

Second half

Free draining

115
1
0.0001115
10.0001115
10.0001115
10.0001

0.326
2.789

0.326
2.789

0.326
2.789

0.326
2.789

0.871
0.635

0.871
0.635

0.871
0.635

0.871
0.635

0.75
0.00209
0.006
0.066
140
1.29
2.36
140
10

0.75
0.00218
0.006
0.066
140
1.32
2.79
70
10

0.75
0.00288
0.001
0.081
140
1.28
5.35
70
10

0.78
0.00212
0.006
0.071
275
1.07
6.30
275
10

Furrow section parameters for A 2 R 4/3⫽␳ 1 A ␳ 2 .
Furrow geometry parameters for y⫽␴ 1 A ␴ 2 .
cExpressed in terms of k (m•min⫺a) and c (m) for 100%, FH, and SH experiments. The EVALUE uses the procedure reported by Strelkoff and Clemmens
共2000兲 to convert the values.
ab

served when tracers were applied to the soil by conventional
methods before irrigation events 共Rice et al. 1986兲. Boldt et al.
共1994兲 presented a numerical model for surface fertigation management in surged irrigation of furrows. The writers applied themodel to hypothetical conditions for various soils, infiltration, andfurrow inflow rates in order to develop optimum fertigation management strategies. The simulation results showed that to obtainproper fertigation efficiency on coarse-textured soils, fertigation
could be applied during any or all parts of a surge cycle but
should be carried out during all cycles. Moreover, fertigation applications on medium- and fine-textured 共low permeability兲 soils
should be made during the entire surge cycle and should be applied either during the entire irrigation event or only during advance surges.
Playan and Faci 共1997兲 evaluated the uniformity of surface
fertigation by conducting a series of field experiments on
blocked-end borders. They also developed a fertigation model
based on the 1D pure advection equation to simulate the experimental data. Results revealed that the fertilizer distribution uniformity of the low half (DU LH ) ranged from 3 to 52%, while thewater DU LH ranged from 64 to 97%. The best fertigation uniformity was achieved by applying a constant rate during the entire
irrigation event. By contrast, a short-duration fertilizer application
produced relatively low uniformity. Garcia-Navarro et al. 共2000兲
investigated the effects of the dispersion coefficient on overland
solute concentration by executing several experiments on an impervious border. The results exhibited a constant value of 0.075
m2/s for the indicated parameter. A comparison between numerical results based on the advection-dispersion equation 共GarciaNavarro et al. 2000兲 and a pure advection model 共Playan and Faci
1997兲 showed that the advection-dispersion model better estimated the observed solute uniformity in the border experiments.

The main objective of this paper is to present a combinedmodel of overland water flow and solute transport for assessing
and managing surface fertigation. Results of several field experiments on both blocked-end and free-draining furrows involving
surface chemigation are presented, and the data are used to calibrate and verify the model. Uniformity of the applied solute pulse
and the effect of longitudinal dispersivity on overland solute concentrations were also evaluated.

Model System Definition
The system presented in this paper combines an overland waterflow and solute transport model and applies it to furrows/borders.
The integrated model numerically solves the 1D dispersionadvection and simplified form of the Saint-Venant’s 共Chow 1959兲
equations as the governing equations in a decoupled fashion. In
spite of 2D water flow and solute transport in and below furrows,
the authors believe that a simple 1D model can provide plenty ofuseful information without requiring as many input parameters asneed for conducting a 2D analysis. Assuming 1D water flow andsolute movement in overland studies, such as furrows, is not too
problematic because of the inherent small confined flow areas infurrows. In fact, overland water and solute transport analysis in
furrows, because of indicated reason above, could be closer to 1D
rather than wide borders and rivers in particular.
The model is applied to evaluate several alternative fertigation
practices used in free-draining and blocked-end furrows. It uses
geometry of the furrow, infiltration, roughness, flow, and soluteproperties listed in Table 1 as the input parameters for each furrow and as output, the model predicts overland solute concentrations, flow rates/velocities, flow areas/depths, advance-recession
trajectories, and water and solute losses through runoff at the end

72 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / MARCH/APRIL 2003

of furrow in case of free drainage. Furthermore, the quality of theirrigation and fertigation event is evaluated using the distribution
uniformity of the low quarter (DU LQ ) and low half (DU LH ) indices 共Merriam and Keller 1978兲. As indicated earlier, the developed model is mainly an overland model and cannot provide somuch subsurface information such as distribution of infiltrated
water and solute. The model just provides water and fertilizer
infiltrated at the surface using simple empirical infiltration models
共Kostiakov, Kostiakov-Lewis, Philip, and modified Kostiakov
branch function兲 at any desired station along the monitored
furrows/borders, which in turn from those results quality of theirrigation and fertigation can be evaluated using distribution uniformity indices. The model also provides total values of water andsolute infiltrated throughout the monitored furrows/borders. Initialand boundary conditions required for the solution of the flow
equations are well described in relevant references 共Abbasi et al.
1999兲. Details about the proposed model are discussed in thefollowing sections.

Z⫽c⫹k␶ a ;␶⭐␶ BZ⫽c B ⫹b␶;

␶⬎␶ B

(6)

where ␶⫽infiltration opportunity time 共min兲; ␶ B ⫽inundation time
共min兲; and c (m3 •m⫺1 ), c B (m3 •m⫺1 ), a (-), k (m2 •min⫺a), andb (m2 •min⫺a)⫽empirical constants.
The above flow and transport models suffer from several limitations. One limitation is the inherent Fickian approximation in
Eq. 共1兲, which is only valid at large distances from the injection
point. The Fickian approximation is valid for rivers with large
flow cross-sectional areas, rather than for furrows with relatively
small flow rates and areas. A second limitation deals with the
hypothesis behind the Saint-Venant’s equations that continuity
and momentum equations are vertically averaged 共Playan andFaci 1997兲.

Solution Techniques

Governing Equations
Hydrodynamic dispersion and advection are the two most important processes governing solute transport during surface fertigation. Using the hypotheses of Holley 共1971兲 that differential convective transport and turbulent diffusion processes may be
combined in gradient diffusion terms, the 1D cross-sectional average dispersion equation for surface fertigation is as follows
共Cunge et al. 1980兲:冉⳵ 共 AC 兲 ⳵ 共 AUC 兲
⳵C
⳵
⫹
⫽
AK x
⳵t
⳵x
⳵x
⳵x

冊

(1)

where C (kg•m⫺3 ) and U (m•s⫺1 )⫽cross-sectional average concentration and velocity, respectively; A (m2 )⫽flow area; K x(m2 •s⫺1 )⫽dispersion coefficient; and t (s) and x (m)⫽time andspace, respectively. The dispersion coefficient K x incorporates
both dispersion due to differential advection and turbulent diffusion 共Cunge et al. 1980兲. The dispersion coefficient for transport
in soils can generally be described by 共Bear 1972兲
␪K x ⫽D l q x ⫹␪D d ␶ 0(2)⫺3

where ␪⫽volumetric water content 共cm •cm 兲; ␶ 0 ⫽tortuosity
factor 共-兲; q x ⫽flux density at location x (m•s⫺1 );D l ⫽longitudinal dispersivity 共m兲; and D d ⫽molecular diffusion infree water 共m2•s⫺1兲. Eq. 共2兲 for overland flow, letting ␪⫽1 and␶⫽1, simplifies to3

K x ⫽D l U x ⫹D d

(3)
⫺1

where U x ⫽overland flow velocity at location x (m•s ).
The solution of the advection-dispersion Eq. 共1兲 requires asolution for the Saint-Venant’s 共Chow 1959兲 equations governing
water flow. The Saint-Venant’s and advection-dispersion equations are solved sequentially at each time step. The simplified
Saint-Venant’s equations, without inertia terms, are as follows:⳵A ⳵Q ⳵Z
⫹
⫹
⫽0
⳵t
⳵x
⳵t
⳵y
⫽S 0 ⫺S f
⳵x⫺1

Water Flow Model
The governing equations for water flow 关Eqs. 共4兲 and 共5兲兴 weresolved in the form of a zero-inertia model, ZIMOD 共Abbasi et al.
1999兲 using a control volume of moving cells and then linearized
by means of a Newton-Raphson algorithm. The resulting algebraic equations were solved iteratively by applying the Gaussianelimination technique. The ZIMOD can simulate all phases ofboth border and furrow irrigation systems with either free draining or blocked-end conditions. The model was verified and validated against several field experiments under different irrigation
methods, soil infiltration properties, field lengths, and slopes.
More details about the ZIMOD are given by Abbasi et al. 共1999,2003兲.
Fertigation Model
Numerical solutions of the 1D advection-diffusion equation, subject to appropriate initial and boundary conditions, under conditions of overland flow require a great deal of care because of thedominance of the advection term in Eq. 共1兲. Most finite-difference
approaches for calculating the convection term of Eq. 共1兲 areplagued by problems of numerical oscillations and/or artificial or
numerical dispersion, which is sometimes larger than the actual
physical dispersion. A Crank-Nicholson finite difference schemewith a small truncation error of the order of 关 (⌬x) 2 ⫹(⌬t) 2 兴 wasused to numerically solve Eq. 共1兲. The following discretization
was used:
共 AC 兲 in⫹1 ⫺ 共 AC 兲 in⌬t
⫹ 共 1⫺ f 兲

(4)
⫽f

and

冋

⫹f

⫹ 共 1⫺ f 兲n⫹1
共 AUC 兲 in⫹1 ⫺ 共 AUC 兲 i⫺1

x i ⫺x i⫺1n
共 AUC 兲 in ⫺ 共 AUC 兲 i⫺1

x i ⫺x i⫺1n⫹1
n⫹1
Kx i⫹1/2
A i⫹1/2(5)

where Q⫽flow rate 共m •s 兲; y⫽flow depth 共m兲; S 0 ⫽field slope
共m•m⫺1兲; S f ⫽hydraulic resistance slope 共m•m⫺1兲; andZ⫽infiltrated water volume per unit length of the field 共m3•m⫺1兲.
3

The infiltrated water volume was formulated using the modified
Kostiakov branch function 共Strelkoff et al. 1999兲

冋

冏

冏

⳵C n⫹1
⳵C n⫹1n⫹1n⫹1⫺Kx i⫺1/2
A i⫺1/2
⳵x i⫹1/2
⳵x i⫺1/2
x i⫹1/2⫺x i⫺1/2nn
Kx i⫹1/2
A i⫹1/2

冏

册

冏

⳵C n
⳵C n
n
n
⫺Kx i⫺1/2
A i⫺1/2
⳵x i⫹1/2
⳵x i⫺1/2
x i⫹1/2⫺x i⫺1/2

册(7)

JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / MARCH/APRIL 2003 / 73

where subscript i and superscript n denote the space and timeintervals, respectively; and f⫽real constant within the interval 0⭐ f ⭐1. When f ⫽0 the solution system is explicit and when f⫽1 the system is implicit. For implicit schemes, which are unconditionally stable under most circumstances, the resulting set of
equations must be solved simultaneously. In implicit approaches,
if f is large enough, the value of the time increment, ⌬t, becomesless dependent on the space increments, ⌬x. These schemes generally provide a numerical method that is computationally very
efficient.
Discretization of the remaining terms on the right hand side of
Eq. 共7兲 and rearrangement leads to the following linear algebraic
equation at each time step:n⫹1n⫹1⫹ 共 d i 兲 C in⫹1 ⫹ 共 b i 兲 C i⫹1⫽e i
共 a i 兲 C i⫺1

(8)

where
a i ⫽⫺

n⫹1
f ⌬t 共 AU 兲 i⫺1

x i ⫺x i⫺1
b i ⫽⫺

d i ⫽A in⫹1 ⫹
⫹

⫺

n⫹1
n⫹1
2 f ⌬tKx i⫺1/2
A i⫺1/2

共 x i⫹1 ⫺x i⫺1 兲共 x i ⫺x i⫺1 兲n⫹1n⫹1
2 f ⌬tKx i⫹1/2
A i⫹1/2

共 x i⫹1 ⫺x i⫺1 兲共 x i⫹1 ⫺x i 兲f ⌬t 共 AU 兲 in⫹1
x i ⫺x i⫺1⫹

e i ⫽ 共 AC 兲 in ⫺⫹(10)

共 x i⫹1 ⫺x i⫺1 兲共 x i⫹1 ⫺x i 兲(11)

共 x i⫹1 ⫺x i⫺1 兲共 x i ⫺x i⫺1 兲
共 1⫺ f 兲 ⌬tn关共 AUC 兲 in ⫺ 共 AUC 兲 i⫺1兴
x i ⫺x i⫺1冋n
C in ⫺C i⫺1x i ⫺x i⫺1

n
C i⫹1
⫺C in

册

Numerical solutions of Eq. 共8兲 often exhibit oscillatory behavior
and/or artificial numerical dispersion near relatively sharp concentration fronts. These problems can be especially serious whenconvection dominates dispersion such as during overland flow.
Undesired oscillations can be limited or avoided by selecting appropriate space and time discretizations. Two dimensionless numbers, the Peclet and Courant numbers, may be used to characterize the space and time discretizations for the fertigation model.
The Peclet number is ratio of the convective and dispersive transport terms relative to coarseness of the spatial grid size as follows:
P e⫽

(12)

Eq. 共8兲 is a tridiagonal system of linear equations that can besolved efficiently using the Thomas algorithm 共Thomas 1949兲.
While the solution of Eq. 共8兲 depends on a numerical solution of
the water flow equations, the reverse is not true, because flow
equations can be solved independently of the advection-diffusion
equation. Hence, the two models can be formulated and solved ina sequential but uncoupled fashion.
The solution of Eq. 共8兲 requires that the flow area and thecross-sectional average velocities for each node be obtained firstfrom the solution of the flow equations. The surface irrigation
model itself does not provide flow velocities directly; those must
be derived by dividing the flow rates, Q i , by the flow areas, A i ,for each node, e.g., U i ⫽Q i /A i . Therefore, the procedure applied
at each time step is to first solve the flow equations and then,given the flow areas and velocities, to solve the advectiondispersion equation to obtain overland solute concentrations in
the irrigation water at different locations along the field.

Boundary and Initial Conditions
Eq. 共8兲 is a linear parabolic partial differential equation that requires one upstream and one downstream boundary condition, as
well as knowledge of the initial state. The upstream boundary
condition is the known applied solute concentration as a function

U x ⌬xKx(13)

To achieve acceptable numerical results, the spatial discretizationmust be kept relatively fine to maintain a low Peclet number.
Numerical oscillations can be virtually eliminated when the local
Peclet numbers do not exceed about 5, while relatively small but
acceptable oscillations can be obtained with local Peclet number
as high as 10 共Huyakorn and Pinder 1983兲.
A second dimensionless number, the Courant number, C r ,characterizes the relative extent of numerical oscillations. The
Courant number is closely associated with the time discretization
as follows:
C r⫽

2 共 1⫺ f 兲 ⌬tnn
Kx i⫹1/2
A i⫹1/2
x i⫹1 ⫺x i
共 x i⫹1 ⫺x i⫺1 兲nn⫺Kx i⫺1/2
A i⫺1/2

Time and Space Discretizations

(9)

n⫹1
n⫹1
A i⫹1/2
2 f ⌬tKx i⫹1/2n⫹1n⫹12 f ⌬tKx i⫺1/2
A i⫺1/2of time. The downstream boundary condition is generally a zeroconcentration gradient. A zero value was used as the initial condition along the entire furrows.

U x ⌬t⌬x(14)

In order to achieve stable oscillation-free solutions, the Courant
number generally must remain less than about one. The maximum
time and space discretizations for the fertigation model are calculated using Eqs. 共13兲 and 共14兲 and considering the Peclet and
Courant numbers equal to 10 and 1, respectively. A constant valueof U x at inlet was used in calculating time and space increments.
The solution of the water flow equations is implemented withvariable space increments, ⌬x i , to enable special conditions associated with the water flow regime. ⌬x i is determined simultaneously with the flow areas and flow rates at each time step during the advance phase of irrigation. Unlike the surface irrigation
model, the fertigation model is executed with constant ⌬x to facilitate adherence to the stability criteria. The nodal values of theflow areas, A i , and the velocities, U i , obtained from the numerical solution of the surface irrigation model, are associated withthe variable grid system involving different space intervals, ⌬x i .
Hence, they must be interpolated to constant space intervals, ⌬x
to be used in numerical solutions of Eq. 共8兲. Solutions of Eq. 共8兲
at each time step are subsequently obtained at constant ⌬x tosimulate overland solute concentrations along the field.

Infiltrated Solute Mass
The mass of solute F z (kg/m) infiltrated through the soil surface
into the soil between two consecutive time steps can be estimatedusing the overland solute concentrations and infiltrated amount ofwater as
n

F z⫽

兺

i⫽1

74 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / MARCH/APRIL 2003

共 Z it⫹⌬t ⫺Z it 兲 ⫻共 C it⫹⌬t ⫹C it 兲

2

(15)

Fig. 1. Plane view of the furrow irrigation field experiments at Maricopa Agricultural Center near Phoenix, Ariz. 共axes are not to scale兲

where n⫽number of nodes along the length of the field; andZ ti ⫽volume of infiltrated water per unit length at node i and timelevel t.
The mass of fertilizer losses due to runoff from the field downstream, F R (kg), can be estimated using outflow rates and overland water concentrations as follows:
F R⫽

共 Q Rt⫹⌬t ⫹Q Rt 兲

2⫻

共 C it⫹⌬t ⫹C it 兲

2

⫻⌬t

(16)

where Q R ⫽outflow rate 共m3•s⫺1兲 at the end of field.
Irrigation and surface fertigation uniformities were evaluatedusing the distribution uniformity of the low quarter (DU LQ ) andlow half (DU LH ) as described by Merriam and Keller 共1978兲 forthe infiltrated irrigation depth.

Field Experiments
Four, large-scale, furrow experiments, one under free draining
and three under blocked-end conditions, were conducted at the
Maricopa Agricultural Center 共MAC兲, Maricopa, Phoenix, Ariz.
in 2001 to calibrate and validate the developed model. The experiments were carried out on 115-m furrows spaced 1 m apart.
Each experiment included three irrigated furrows, one monitored,
nonwheel furrow in the middle, and two nonmonitored wheel
furrows, one at each side. A plane view of the experiments isshown in Fig. 1. The experiments are discussed in detail in thefollowing sections.

Free Draining (FD) Experiment
Experiment one was conducted under free draining conditions.
The experiment was run for two successive irrigation events with10 days apart. The first irrigation lasted 275 min and the second
140 min. The average inflow rates were 1.07 and 1.03 L•s⫺1 forthe first and second irrigations, respectively. Bromide, based on
CaBr2 component, was injected at a constant rate during the entire
first irrigation event. The average bromide concentration for the
first irrigation cycle was 6.3 g Br•L⫺1. The second irrigation was
carried out with unamended water 共without bromide兲.

Blocked-End Experiments
Experiments two, three, and four were carried out on blocked-end
furrows, using the following three solute application options 共referred to further as the 100%, FH and SH experiments, respectively兲:• Bromide applied during the entire irrigation event 共100%兲;• Bromide applied during only the first half of the irrigation
event 共FH兲; and• Bromide applied during the second half of the irrigation event
共SH兲.
Each of the three experiments included one irrigation, whichlasted 140 min. The average inflow rates for these three runs were1.29, 1.32, and 1.28 L•s⫺1, respectively. Water reached the end of
the field after 107 and 114 min for the 100% and FH experiments,
respectively, but did not reach the end for the SH experimentbecause of high roughness and high infiltration properties in that
plot. Furthermore, the averaged applied bromide concentrations
for the 100%, FH, and SH experiments were 2.36, 2.79, and 5.35
g Br•L⫺1, respectively. As indicated above, the solute application
time for the 100% experiment was 140 min and 70 min for the FH
and SH experiments. Relatively long solute applications wereused for these experiments because of the low solute uniformity
reported in the literature for borders with short application times
共Playan and Faci 1997兲. Also, short fertilizer application times
共about 10 min or so兲 are not common in practice. The following
additional measurements were made for both the free-draining
and blocked-end experiments:• The maximum nonerosive flow rate for the field site was measured to be 1.5–1.6 L•s⫺1 in a pretest before performing themain experiments.
• The inflow rate was measured using 2.5-cm 共1-in.兲 size double
water meters 共with two water meters on the same line兲. They
were calibrated at the field site before being used. The inflow
rate was delivered and measured at each furrow separately.
Water meter readings were recorded every 5–10 min during
the experiments.
• The monitored furrows were marked with stakes at 20-m intervals. Advance and recession times as well as water depth
measurements were recorded at those stations along the monitored furrows. Water depths, measured using staff gauges
placed at the bottom of the furrows, were initially recorded
every minute for the first 10 min after advance was reached,
and then every 5–10 min. They were also recorded at 2–5
min. intervals after shutting off the inflow rate until recession
was complete.
• The applied bromide was metered to each experimental furrow
separately using small solute boxes, each having a capacity of
4 L and equipped with valves and floaters. The bromide solution was prepared in advance in 220-L barrels. The soluteboxes were then calibrated for the desired concentrations before each experiment. Later, each barrel was connected to a

JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / MARCH/APRIL 2003 / 75

Fig. 2. Schematic of the inflow system used for the field experiments 共not to scale兲

•

•

•

•

solute box using 1.25-cm 共1/2-in.兲 flexible tubing. The bromide solution was applied at the head of the furrows where themetered water was introduced 共Fig. 2兲.
Overland water samples for analysis of bromide concentrations were taken at three different locations—10, 60, and 100
m from the inlet for the blocked-end experiments and 5, 60,
and 110 m for the free-draining experiment. Water samples atthe different stations were collected as soon as water reached aparticular station. Samples were initially taken at 1 min withthe sampling interval increased gradually to every 10 min andthen decreased to every 2–3 min after terminating the solute
application 共such as for the FH experiment on the blocked-end
furrows兲. The samples were kept in an air-conditioned roomand later analyzed for bromide concentrations with a Lachat
QuikChem flow injection analyzer using the standard colorimetric procedure.
Elevations of the experimental furrows were recorded using astandard surveyor’s level at the stations where the water
depths and advance times were recorded. The average longitudinal slope of the field was 0.0001 m•m⫺1 共Table 1兲.
The geometry of the monitored furrows was also measured
before and after the irrigation at three different locations along
the monitored furrows. The average furrow cross-section (␳ 1and ␳ 2 ) and geometry (␴ 1 and ␴ 2 ) parameters 共Elliot andWalker 1982兲 were determined 共Table 1兲 and later used as
input to the model.
Runoff water from the free-draining experiment was collected
at the end of the furrows and pumped out using a small pump,because the furrows were almost level. The pumped water was
weighed each 5 min during the irrigation time using an electrical scale.

Results and Discussion
Infiltration and Roughness Parameters
Several methods, such as ring infiltrometers, blocked furrow infiltrometers, a recirculating method, inflow-outflow method, procedures based on advance stream data, and volume balance areavailable for estimating infiltration parameters. The simplest

method is probably the ring infiltrometer, but this method is notprecise for cracking clay, freshly tilled soils, and furrows. While
procedures based on advance phase data are relatively easy andsimple for estimating infiltration parameters, results may not adequately represent the infiltration process during the entire irrigation cycle. Fangmeier and Ramsey 共1978兲 found that such procedures based only on the advance phase tend to underestimate
infiltration during subsequent phases of the irrigation process.
Volume balance methods are usually more appropriate for determining field-scale infiltration parameters. These methods calculate the infiltrated water as the difference between the inflow volume and the volume stored on the soil surface. In this study, weused the EVALUE program developed by Strelkoff et al. 共1999兲
to estimate the infiltration and roughness parameters for theblocked-end furrow experiments. The EVALUE estimates the infiltration parameters in a modified Kostiakov branch function, Eq.
共6兲, by matching measured growth of infiltrated volumes to those
calculated on the basis of the measured advance. The techniquealso generates an estimate of the roughness coefficient in the
Manning formula. It assumes that water velocities are small 共generally lower than 0.1 m•s⫺1兲, which is common for surface irrigation, particularly for flat fields in which case the water surface
slope, ⳵y/⳵x, can be considered equal to the friction slope, S f .
The hydraulic radius, R, is found using the measured flow depths
and the either power law or trapezoidal cross section shape. Local
discharges are estimated using an integrated form of the continuity equation, Eq. 共4兲.
Infiltration and roughness parameters for the blocked-end experiments as estimated using the EVALUE program are given in
Table 1. Both the estimated infiltration and roughness values wereconsistent between the 100% and FH experiments, whereas theinfiltration coefficients k and the Manning n were substantiallyhigher for the SH experiment. Based on field observations, we
suspected higher infiltration and roughness values for the SH experiment, because with almost the same inflow rate water failed toreach the end of the field, Fig. 3共d兲, while it reached the end in
107 and 114 min for the 100% and FH experiments, respectively.
The exponent ‘‘a’’ in Eq. 共6兲 was found to be essentially the samefor all experiments and we obtained relatively small values for thefitting of parameter ‘‘c.’’ A sensitivity analysis showed that the

76 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / MARCH/APRIL 2003

Fig. 3. Measured and simulated advance and recession times for the different field experiments: 共a兲 free draining experiment; 共b兲 100%experiment; 共c兲 first-half experiment; 共d兲 second-half experimentmodel was not sensitive to this parameter. For example, increasing ‘‘c’’ from 0.001 to 0.006 increased the total advance time at
the end of the field by only about 2–3 min 共results not shown兲.
The EVALUE program was specifically designed for blockedend fields without runoff, so the infiltration parameters for the FD
experiment were estimated using the following volume balance
equation:V i ⫽V y ⫹V z ⫺V R
3

(17)

3

where V i (m )⫽inflow volume; V y (m )⫽stored volume on thesoil surface; V z (m3 )⫽infiltrated volume; and V R (m3 )⫽runoff
volume at the end of the field.
The stored volume on the soil surface was estimated from themeasured flow depths using trapezoidal interpolation. The estimated infiltration parameters using this approach were more orless the same as those obtained for the 100% and FH experiments
using the EVALUE program 共Table 1兲. The infiltration rate did
not reach a steady state 共final infiltration rate兲 within the durationof all experiments. For this reason, we only estimated the coefficients a, k, and c for Eq. 共6兲. The average value of the Manning n
for the blocked-end experiments was taken as the roughness value
for the FD experiment

Advance and Recession Trajectories
Simulated and observed water advance and recession times for all
four experiments are shown in Fig. 3. In general, excellent agreement exists between the simulated and measured results for all
runs in both the advance and recession phases. In a few cases, themodel slightly underestimated, Fig. 3共c兲, or overestimated, Fig.
3共d兲, the advance and recession times, respectively. Likely reasons for this were spatial and temporal variability in infiltration
and roughness parameters and nonuniformity of the field slope
along the furrows. The water balance errors were 1.8, 1.3, 0.004,
and 1.6% for the 100%, FH, SH, and FD experiments, respectively. Results were obtained using the same time increments,⌬t⫽3 min for all simulations.

Overland Solute Concentration
Observed and simulated overland solute concentrations at different stations along the experimental furrows are shown in Fig. 4.

Some deviations in solute concentrations are apparent at the beginning of the experiments. This is due mainly to undesired inflow rate changes. Although the solutes were applied uniformly
during the application time, any variation in inflow rate, which iscommon especially at the beginning of an irrigation event, couldchange the applied solute concentration. Any changes at thesource point should be visible at especially the first stations close
to the input source. A minor part of the concentration variations
could possibly be attributed also to dispersivity. The dispersivity
parameter, D l , was found to be 10 cm using model calibration.
Calibration procedure and the effects of D l on the results are
discussed in detail in the following section.
The applied solute concentration for the FD experiment Fig.
4共a兲 decreased from an average of 6.75 to 6 g Br•L⫺1 after 110
min because of increasing inflow rate from an average of 0.97 to
1.14 L•s⫺1. Notice that the peak values of the relative concentration at the three monitoring stations did not show any decay intime and remained more or less the same because of a relatively
long application time and the short field length. The steep rise at
the beginning of each break through curve 共BTC兲 was due to thehigh flow velocity for the surface irrigation, as compared to flow
and transport fluxes in soils, which caused concentrations reaching the maximum rate shortly after water flow arrived a given
station.
From Fig. 4, we can conclude that the model simulated all
experiments relatively well. The model predicted slightly longer
duration of the solute peak at the third monitored stations 共100 m兲
for the FH and SH experiments, Figs. 4共c兲 and 4共d兲, as comparedto field data, most likely due to spatial variability in soil infiltration and roughness properties. The later also caused relatively
poor arrival estimate of bromide to station 100-m at FH experiment, Fig. 4共c兲.

Dispersion Coefficient
A sensitivity analysis was carried out to assess the effects of thedispersivity, D l in Eq. 共3兲, on overland solute concentrations andalso to estimate a reasonable value for this parameter. The model
was run with three different values of D l , 1, 10, and 100 cm.
Results for the FH experiment were selected and shown in Fig. 5.
Clearly, all three D l values produced more or less the same fit
with measured data at all stations. Simulated peak values of the

JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / MARCH/APRIL 2003 / 77

Fig. 4. Relative solute concentrations for the different experiments: 共a兲 free draining experiment; 共b兲 100% experiment; 共c兲 first-half experiment;
共d兲 second-half experimentFig. 5. Sensitivity of the overland solute concentrations to different values of the dispersivity parameter performed using the first-half experimental data: 共a兲 10-m station; 共b兲 60-m station; 共c兲 100-m station
78 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / MARCH/APRIL 2003

Infiltrated Solute Mass

Fig. 6. Predicted infiltrated solute mass along the field for thedifferent experiments

relative solute concentrations remained 1.0 for all monitored stations regardless of the selected D l -value because of the minor
effect of the dispersivity values on the results. A value of 10 cm
was chosen for D l in all subsequent simulations. Effect of selected D l values on solute concentrations in other experiments
共FD, 100%, and SH兲 was almost negligible 共not shown here兲. The
effect of dispersivity on the solute uniformity was also negligible.
Neither oscillation behavior nor numerical diffusion were
found even with applying D l ⫽0 共pure advection兲 likely becauseof choosing appropriate values for the Peclet and Courant numbers. Solute BTCs remained as a rectangle, with relative soluteconcentration equal to 1.0, while applying D l ⫽0 关in this case ⌬x
was assumed to be 0.5 m and ⌬t was determined using Eq. 共14兲
while in other cases ⌬x and ⌬t were directly calculated from Eqs.
共13兲 and 共14兲兴.
Assuming D l ⫽10 cm, the dispersion coefficient K x in Eq. 共3兲
varied between 4⫻10⫺8 and 0.00745 m2•s⫺1 in our study. This isbecause K x strongly depends upon the overland flow velocity,which varies with time and distance. In addition to a transient
velocity distribution, K x in overland studies also can be affectedby shear velocity, hydraulic radius, and flow area/flow depth
共Cunge et al. 1980兲. As indicated earlier, Garcia-Navarro et al.
共2000兲 obtained a much larger, constant value of 0.075 m2•s⫺1 forK x by calibrating their model to their steady-state experimental
data using an impervious border. Still larger K x values are reported in the literature in hydrological studies. For instance, Brebion et al. 共1971兲 obtained a value of 2 m2•s⫺1 for K x in the
Vienne River in France. We believe that smaller obtained values
for K x in this study are more physical than the previously reported
values for borders and rivers, one to several magnitudes of orders
larger than physical values. Another possible reason for the discrepancy could be associated to small confined flow area 共andrelatively short lengths in this study兲 in furrows, which consequently result in almost instantaneous solute mixing, compared towide and longer borders and particularly rivers. In fact, K x values
in both borders and rivers, resulted in 1D analysis, represent an
average for the longitudinal and transverse dispersion coefficientwhile in furrows essentially represent longitudinal dispersion coefficient because of the indicated reason above.
Table 2. Measured and Predicted Water and Solute Distribution

Uniformities of the Low Quarter, DU LQ , and Low Half, DU LH , forDifferent Field Experiments 共%兲
Predicted Water Predicted Solute Measured Solute
Experiment

DU LQ

DU LH

DU LQ

DU LH

DU LQ

DU LH

100%
First-half, FH
Second-half, SH
Free draining, FD

77.8
74.6
52.8
57.8

85.3
83.5
70.8
73.3

77.8
32.2
83.0
57.8

85.3
62.2
91.6
73.3

75.6
20.6
38.7
34.5

87.4
61.8
73.9
71.9

The amount of infiltrated bromide along the furrows 共Fig. 6兲 was
calculated with Eq. 共15兲 using the predicted infiltrated water
depths and overland solute concentrations. The uniformity of thechemigation process was also evaluated using DU LQ and DU LH
terms and presented in Table 2 for the different field experiments.
Bromide application decreased with distance for the FD experiment, Fig. 6. In addition, both the water and solute uniformities
for this experiment were lower than for the 100% and SH experiments 共Table 2兲. These results were due mostly to the applied
inflow rate and applied solute mass. Irrigation uniformity depends
greatly upon the inflow rate. As stated above, the average inflow
rate for the FD experiments was only 1.07 L•s⫺1, which is lower
than for the other three experiments 共about 1.3 L•s⫺1兲. Consequently, much more water 共and hence solutes兲 infiltrated at small
distances from the inlet. Furthermore, the average applied soluteconcentration during the first 110 min of irrigation 共6.75
g Br•L⫺1兲 was somewhat higher than for the rest of the irrigation
共6 g Br•L⫺1兲. An overall large amount of infiltrated bromide for
the FD experiment was due to a longer 共2– 4 times兲 solute application time as compared to the other experiments.
The 100% and FH experiments provided higher irrigation
DU LH values while the 100% and SH experiments gave higher
solute DU LH values 共Table 2兲. This means that applying the fertilizer during the entire irrigation, or during the second half of theirrigation, with an inflow rate close to the maximum nonerosive
inflow rate, can provide high irrigation and fertilization uniformity. However, applying fertilizers during the entire irrigation
event may cause losses of the fertilizers due to deep percolation.
Applying fertilizers during the second half of the irrigation event,or even later, particularly under blocked-end conditions, could
significantly reduce the solute deep percolation and increase solute uniformity. However, in the presence of runoff 共i.e., freedrainage兲, applying the solute during the second half of the irrigation will increase solute losses due to runoff from the end offield. A comparison between solute uniformities obtained in thisstudy and those reported in the literature for borders 共Playan andFaci, 1997; Garcia-Navarro et al. 2000兲 shows a higher soluteuniformity for our furrow experiments. This may be attributed toour relatively long solute application times, as well as the inherently higher irrigation uniformity of furrows.
The agreement between measured and predicted results for thesolute DU LH was satisfactory, whereas the agreement for the
DU LQ values was not as good as DU LH . The discrepancy between measured and predicted values may be attributed to thelimited number of measured points used in the calculations. The
measured DU LQ and DU LH values were calculated based on fourmeasured points located 0, 10, 60, 100 m away from the inlet forthe blocked-end experiments. The measured points for the FD
experiment were located at 0, 5, 60, and 110 m from the inlet.
More measurement points, probably, would have improved consistency. The model gave the same water and solute uniformities
for the 100% and FD experiments, because the solutes were continuously applied with irrigation water. Low water uniformity for
the SH experiment 共Table 2兲 was due to high infiltration rate andlikely also the roughness at the end of the field which resulted inwater never reaching the end of the furrow. The model accounted
for over 96% of the applied solute. Fertilizer mass balance errors
for the 100%, FH, SH, and FD experiments were 2.2, 3.6, 2.3, and
1.2%, respectively. Moreover, measured and predicted fertilizer
losses through the runoff at the end of the FD experiment were
found to be 1.46 and 1.36%, respectively.

JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / MARCH/APRIL 2003 / 79Summary and Conclusions
A combined overland water flow and solute transport model wasdeveloped and used to evaluate several alternative fertigation
practices used in furrow irrigation field experiments. Contrary to
the former solutions for 1D advection-dispersion equation reported in the literature in hydrological 共Leendertse 1970; Brebion
et al. 1971兲 and fertigation 共Playan and Faci 1997; GarciaNavarro et al. 2000兲 studies, which applied two-step schemes toavoid numerical dispersion and used Courant number as stability
criteria, in the present study a Crank-Nicholson finite-difference
scheme and two dimensionless numbers, Peclet and Courant numbers, were used in solving the advection-dispersion equation. The
applied scheme did not show oscillation behavior and numericaldispersion most likely due to the appropriate selected values for
the Peclet and Courant numbers 共10 and 1, respectively兲.
In spite of recognized 2D water flow and solute movement infurrows, presented 1D solution of the overland water flow andtransport equations adequately described and provided a reasonable picture of the intended chemigation likely due to inherent
small flow areas and relatively larger surface flow velocities infurrows, which in turn reduce lateral flow and solute displacement. Both measured and predicted water and solute uniformities
showed a high dependency to the inflow rates, as the upstream
boundary condition, soil infiltration, and roughness properties, aswell as solute application time, whereas dispersivity and concentration of the input solute had a minor influence on the fertigation
uniformities.
Although the experiments were conducted adjacent to eachother, estimated soil infiltration and roughness characteristics revealed some spatial variability among the monitored furrows that
affected water advance and recession trajectories, and consequently the irrigation and solute uniformities. Both field measured
and model predictions concluded that application of fertilizers
during either the entire irrigation event or the second half of theirrigation, resulted in higher solute uniformity under blocked-end
conditions. Also, a sensitivity analysis showed that the longitudinal dispersivity did not play an important role in our surface
chemigation experiments, nor had a significant effect on the overland solute concentrations. This may have been partly due to relatively short experimental furrows in this study and partly due tosmall confined flow areas in furrow irrigation. We believe that thepresented model is capable of assessing more complicated andrealistic fertigation alternatives. However, more research for verifying of the obtained dispersivity parameter in longer furrows andvalidation of the model particularly for fertigation processes in
irrigated borders are recommended.

Acknowledgments
The first writer is very grateful to Jack Jobes and JoAn Fargerlundof the U.S. Salinity Laboratory, Riverside, Calif., Carl Arterberry,
Don Powers, and Candy Van Meeteren of the U.S. Water Conservation Laboratory, Phoenix, Ariz., for their help during executing
the field experiments and subsequent laboratory analyses. Thanks
to Robert L. Roth, director of Maricopa Agriculture Center, Phoenix, Ariz., and his colleagues for their kind collaboration during
the field experiments and initial laboratory analysis. The writers
also gratefully acknowledge two anonymous reviewers for theirhelpful comments and suggestions.

Notation
The following symbols are used in this paper:A ⫽ cross section flow area 共m2兲;a ⫽ exponent of Kostiakov branch infiltration
equation 共-兲;b ⫽ constant in Kostiakov branch infiltration equation
共m2•min⫺a兲;C ⫽ solute concentration 共g•L⫺1兲;
C r ⫽ Courant number 共-兲;c,c B ⫽ constants in Kostiakov branch infiltration
equation 共m3•m⫺1兲;D d ⫽ molecular diffusion in free water 共m2•s⫺1兲;D l ⫽ longitudinal dispersivity 共cm兲;DU LH ⫽ distribution uniformity of low half 共%兲;DU LQ ⫽ distribution uniformity of low quarter 共%兲;F R ⫽ mass of solute losses due to runoff 共kg兲;F z ⫽ mass of infiltrated solute 共kg•m⫺1兲;K x ⫽ dispersion coefficient 共m2•s⫺1兲;k ⫽ constant in Kostiakov branch infiltration equation
共m2•min⫺a兲;n ⫽ Manning’s roughness 共m1/6兲;P e ⫽ Peclet number 共-兲;Q ⫽ flow rate 共m3•s⫺1兲;Q o ,Q R ⫽ inflow and outflow rates 共m3•s⫺1兲;
q x ⫽ flux density 共m•s⫺1兲;R ⫽ hydraulic radius 共m兲;S f ,S 0 ⫽ friction and field slopes 共m•m⫺1兲;t co ⫽ cut-off time 共min兲;U ⫽ flow velocity 共m•s⫺1兲;V i ,V R ,V y ,V z
⫽ inflow, runoff, stored, and infiltrated volumes 共m3兲;y ⫽ flow depth 共m兲;Z ⫽ infiltrated volume per unit length 共m3•m⫺1兲;⌬t ⫽ time increment 共min兲;⌬x ⫽ space increment 共cm兲;␪ ⫽ water content 共cm3•cm⫺3兲;␳ 1 ,␳ 2 ⫽ hydraulic section parameters (m3.33⫺2␳ 2 ,-);
␴ 1 ,␴ 2 ⫽ furrow geometric section parameters (m1/2␴ 2 ,-);
␶ ⫽ infiltration opportunity time 共min兲;␶ B ⫽ inundation time 共min兲; and␶ 0 ⫽ tortuosity factor共-兲.

References
Abbasi, F., Jolaini, M., Moaiyeri, M., Rezaee, H. T., and Shooshtari, M.
M. 共1999兲. ‘‘Development a mathematical model to evaluate and design of surface irrigation systems.’’ Technical Report No. 122, Iranian
Agricultural Engineering Research Institute, Karaj, Iran 共in Persian兲.
Abbasi, F., Shooshtari, M. M., and Feyen, J. 共2003兲. ‘‘Evaluation of thevarious surface irrigation numerical simulation models.’’ J. Irrig.
Drain. Eng., in press.
Bear, J. 共1972兲. Dynamics of fluids in porous media, Elsevier Science,
New York.
Boldt, A. L., Watts, D. G., Eisenhauer, D. E., and Schepers, J. S. 共1994兲.
‘‘Simulation of water applied nitrogen distribution under surge irrigation.’’ Trans. ASAE, 37共4兲, 1157–1165.
Bowman, R. S., and Rice, R. C. 共1986兲. ‘‘Transport of conservative tracers in the field under intermittent flood irrigation.’’ Water Resour. Res.,22, 1531–1536.
Brebion, S., Lebrun, B., Chevereau, G., and Preissmann, A. 共1971兲.
‘‘Modeles mathematiques de la pollution.’’ IRCHA, Centre de Recherche, 91 Vert-lepetit, France 共in French兲.
Chow, V. 共1959兲. Open channel hydraulics, McGraw-Hill, New York.
Cunge, J. A., Holly, F. M., and Verwey, A. 共1980兲. Practical aspects ofcomputational river hydraulics, Pitman, London.
Elliot, R. L., and Walker, W. R. 共1982兲. ‘‘Field evaluation of furrow
infiltration and advance functions.’’ Trans. ASAE, 25共2兲, 396 – 400.

80 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / MARCH/APRIL 2003

Fangmeier, D. D., and Ramsey, K. K. 共1978兲. ‘‘Intake characteristics ofirrigation furrows.’’ Trans. ASAE, 21, 696 –700.
Garcia-Navarro, P., Playan, E., and Zapata, N. 共2000兲. ‘‘Solute transportmodeling in overland flow applied to fertigation.’’ J. Irrig. Drain.
Eng., 126共1兲, 33– 40.
Hanson, B., Bowers, W., Davidoff, B., Kasapligil, D., Carvajal, A., andBendixen, W. 共1995兲. ‘‘Field performance of micro-irrigation systems,
Micro-irrigation for a changing word: Conserving resources/
preserving the environment.’’ Proc., 5th Int. Micro-Irrigation Congress, 769–774.
Holley, E. R. 共1971兲. ‘‘Transverse mixing in rivers.’’ Report No. S132,
Delft Hydraulic Laboratory, Delft, The Netherlands.
Huyakorn, P. S., and Pinder, G. F. 共1983兲. Computational methods insubsurface flow, Academic, London.
Jaynes, D. B., Bowman, R. S., and Rice, R. C. 共1988兲. ‘‘Transport ofconservative tracers in the field under continuous flood irrigation.’’
Soil Sci. Soc. Am. J., 52, 618 – 624.
Jaynes, D. B., Rice, R. C., and Hunsaker, D. J. 共1992兲. ‘‘Solute transport
during chemigation of a level basin.’’ Trans. ASAE, 35共6兲, 1809–
1815.
Leendertse, J. J. 共1970兲. ‘‘A water quality simulation model for wellmixed estuaries and coastal seas: Vol. I., Principles of computation.’’Rand Corporation Memorandum, RM-6230-RC.
Merriam, J. L., and Keller, J. 共1978兲. Farm irrigation system evaluation:
A guide for management, Utah State Univ., Logan, Utah.
Playan, E., and Faci, J. M. 共1997兲. ‘‘Border irrigation: Field experimentand a simple model.’’ Irrig. Sci., 17共4兲, 163–171.
Rice, R. C., Bowman, R. S., and Jaynes, D. B. 共1986兲. ‘‘Percolation ofwater below an irrigated field.’’ Soil Sci. Soc. Am. J., 50, 855– 859.
Strelkoff, T. S., and Clemmens, A. J. 共2000兲. ‘‘Approximating wetted
perimeter in a power-law cross section.’’ J. Irrig. Drain. Eng., 126共2兲,
98 –109.
Strelkoff, T. S., Clemmens, A. J., El-Ansary, M., and Awad, M. 共1999兲.
‘‘Surface irrigation evaluation models: Application to level basin in
Egypt.’’ Trans. ASAE, 42共4兲, 1027–1036.
Thomas, L. H. 共1949兲. ‘‘Elliptic problems in linear difference equations
over a network.’’ Report of Watson Science Computing Laboratory,
Columbia Univ., New York.
Threadgill, E. D. 共1985兲. ‘‘Chemigation via sprinkler irrigation: Current
status and future development.’’ Appl. Eng. Agric., 1共1兲, 16 –23.
Threadgill, E. D., Eisenhauer, D. E., Young, J. R., and Bar-Yosef, B.
共1990兲. ‘‘Chemigation.’’ G. J. Hoffman, T. A., Howell, and K. H.
Solomon 共eds.兲 Management of farm irrigation systems, ASAE, St.
Joseph, Mich., 747–780.

JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE / MARCH/APRIL 2003 / 81