EPA-600/2-78-041
March 1978 C
Environmental Protection Technology Series
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S. Environmental
Protection Agency, have been grouped into nine series. These nine broad cate-
gories were established to facilitate further development and application of en-
vironmental technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields.
The nine series are
1 Environmental Health Effects Research
2 Environmental Protection Technology
3 Ecological Research
4. Environmental Monitoring
5. Socioeconomic Environmental Studies
6. Scientific and Technical Assessment Reports (STAR)
7 Interagency Energy-Environment Research and Development
8. "Special" Reports
9. Miscellaneous Reports
This report has been assigned to the ENVIRONMENTAL PROTECTION TECH-
NOLOGY series This series describes research performed to develop and dem-
onstrate instrumentation, equipment, and methodology to repair or prevent en-
vironmental degradation from point and non-point sources of pollution. This work
provides the new or improved technology required for the control and treatment
of pollution sources to meet environmental quality standards.
This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.
-------
EPA-600/2-78-041
March 1978
ASSESSING THE SPATIAL VARIABILITY OF IRRIGATION WATER APPLICATIONS
by
David Karmeli
LeRoy J. Salazar
Wynn R. Walker
Department of Agricultural and Chemical Engineering
Colorado State University
'Fort Collins, Colorado 80523
Grant No. R804828
Project Officer
Arthur G. Hornsby
Source Management Branch
Robert S. Kerr Environmental Research Laboratory
Ada, Oklahoma 74820
ROBERT S. KERR ENVIRONMENTAL RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
ADA, OKLAHOMA 74820
-------
DISCLAIMER
This report has been reviewed by the Robert S. Kerr Environmental
Research Laboratory, U.S. Environmental Protection Agency, and approved
for publication. Approval does not signify that the contents necessarily
reflect the views and policies of the U.S. Environmental Protection Agency,
nor does mention of trade names or commercial products constitute
endorsement or recommendation for use.
11
-------
FOREWORD
The Environmental Protection Agency was established to coordinate
administration of the major Federal programs designed to protect the quality
of our environment.
An important part of the Agency's effort involves the search for
information about environmental problems, management techniques and new
technologies through which optimum use of the Nation's land and water
resources can be assured and the threat pollution poses to the welfare of
the American people can be minimized.
EPA's Office of Research and Development conducts this search through a
nationwide network of research facilities.
As one of these facilities, the Robert S. Kerr Environmental Research
Laboratory is responsible for the management of programs to: (a) investigate
the nature, transport, fate and management of pollutants in groundwater;
(b) develop and demonstrate methods for treating wastewaters with soil and
other natural systems; (c) develop and demonstrate pollution control tech-
nologies for irrigation return flows; (d) develop and demonstrate pollution
control technologies for animal production wastes; (e) develop and demon-
strate' technologies to prevent, control or abate pollution from the petroleum
refining and petrochemical industries; and (f) develop and demonstrate tech-
nologies to manage pollution resulting from combinations of industrial waste-
waters or industrial/municipal wastewaters.
This report contributes to the knowledge essential if the EPA is to meet
the requirements of environmental laws that it establish and enforce pollu-
tion control standards which are reasonable, cost effective and provide
adequate protection for the American public.
William C. Galegar (/
Director
Robert S. Kerr Enviornmental
Research Laboratory
111
-------
ABSTRACT
The current state-of-the-art regarding the spatial distributions of
irrigation water applications under surface, sprinkler, and trickle irrigation
systems has been assessed. The analyses found in the literature and several
new uniformity concepts have been integrated into models which can be used
in both field and research applications. These models simulate the spatial
distributions of applied irrigation water under specified design and
operating conditions.
The performance of an irrigation system has been described by a series
of "quality" parameters relating to: (1) uniformity in an irrigated field;
(2) adequacy of the irrigation system in meeting crop requirements; (3) volume
of applied water wasted as deep percolation; and (4) in the case of surface
irrigation, the water leaving the field as tailwater.
Verification of the models developed during this project was made
against most of the data identified in the literature as well as an intensive
collection effort as part of this project. The results illustrate both the
use of the analytical approach and the procedures for field data collection.
This report was submitted in fulfillment of Grant No. R-804828, Colorado
State University, under the sponsorship of the U.S. Environmental Protection
Agency. This report covers the period October 1, 1976 to September 30, 1977,
and work was completed as of September 30, 1977.
IV
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CONTENTS
PAGE
Foreword iii
Abstract iv
List of Figures vii
List of Tables x
Acknowledgements xii
1. Introduction 1
2. Conclusions 3
3. Recommendations 5
4. Spatial Distribution in Surface Irrigation 6
Irrigation Quality Parameters 7
Factors Affecting Irrigation Quality 12
Formulation of Model and Data Requirements 39
Model Field Studies 43
5. Spatial Distributions in Sprinkler Irrigation 94
The Uniformity Concept 95
A Model to Describe Distribution Patterns Based on
Linear Regression 100
Model Studies 115
Summary 121
6. Spatial Distribution in Trickle Irrigation 125
Factors Affecting Trickle Irrigation Quality 126
Effluency and Uniformity Concepts 130
Description of Suggested Model 133
Summary 143
References 144
Appendices
A. Techniques for Determining Soil Infiltration
Characteristics 149
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PAGE
B. Equipment, Forms, and Sample Data 158
C. Computer Program for Furrow Irrigation Model ". .167
D. I. Predicting Soil Moisture Depletion from Crop and
Climatic Data 174
II. Soil Moisture Results . .180
E. Computer Program for Sprinkler Irrigation Model 181
F. Computer Program for Trickle Irrigation Model 194
VI
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LIST OF FIGURES
FIGURE PAGE
1 Distribution of infiltrated depths along irrigated run ... 8
2 Power fit curve of dimensionless distribution for surface
irrigation 11
3 Infiltration curves (example) 15
4 Definition sketch for Hall method of determining the
advance of water down a border 30
5 Flow chart for computer model of furrow irrigation system. . 41
6 Layout of field used for model verification 44
7 Field topographic profile : 46
8 Infiltration results from cylinder infiltrometers 49
9 Infiltration results from "inflow-outflow" method 50
lOa Effects of wetted perimeter on furrow infiltration 51
lOb Example of the effect of wetted perimeter on intake rate
from Ramsey and Fangmeier, 1976 52
lla Infiltration results from blocked furrow measurement .... 56
lib Mean infiltration curve from blocked furrow measurement. . . 57
12 Advance-recession for Irrigation 5 59
13a Furrow inflow and outflow for Furrow 1, Irrigation 5 .... 60
13b Furrow inflow and outflow for Furrow 5, Irrigation 5 .... 61
14a Estimated infiltration patterns using blocked furrow
equation and intake opportunity times from advance-
recession data 62
14b Estimated infiltration patterns using blocked furrow
equation and intake opportunity times from advance-
recession data 63
vii
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FIGURE PAGE
15a Furrow cross sections: Before irrigation 1 64
15b Furrow cross sections: After irrigation 1 65
15c Furrow cross sections: After irrigation 2 66
15d Furrow cross sections: Before irrigation 5 67
16 Mean furrow cross sections 68
17 Averaged Manning's n and Sayre-Albertson X (for
experimental furrows from Ramsey and Fangmeier, 1976) ... 77
18a Surface storage estimates: Irrigation 1 79
18b Surface storage estimates: End of Irrigation 2 80
18c Surface storage estimates: Furrow 1, Irrigation 5 81
18d Surface storage estimates: Furrow 5, Irrigation 5 82
19a Comparison of forced power fit to representative blocked
furrow equation for advance stage of irrigation 90
19b Predicted advance under variations of surface storage for
Irrigation 5 91
20 Typical effects of water distribution patterns on a crop
under irrigation assuming no runoff (Hansen, 1960) 96
21 Histogram of application depth versus area irrigated. ... 97
22 Normalized, nondimensional frequency curve 97
23 Linear regression fit of nondimensional distribution curve
for sprinkler patterns 101
24a-e Fits of actual data into normal and linear regression
distributions in sprinkler irrigation 102-106
25 Y , UCL and Y . as functions of the irrigation
max mm
quality (b) 108
26 Uniformity coefficients assuming linear relationships of
precipitation depth and area irrigated 109
27 Dimensionless linear distribution curve indicating surplus
and deficient application and extent of each 110
Vlll
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FIGURE PAGE
28 Increase in total water applied to bring ctAD of field
to a minimum depth of YD .................. 112
29 Increase in excess water to bring aAp of field to a
minimum depth of YD .................... 113
30 The proposed model with YD, the dimensionless requirement . 115
K
31 The proposed model with Y <_ Y .............. 116
j\ ~~~ 111X11
32 The proposed model with Y. < Y < 1.0 .......... 117
r r nun R —
33 The proposed model with Y > Y ............. 118
K lUciX
34 Linear uniformity coefficient (UCL) as a function of wind
velocity for various spacings and pressures in sprinkler
irrigation ......................... 123
35 Dimensionless emitter discharge versus relative location
on lateral ......................... 136
36 Logarithmic curve fit for pressure versus location
(r*=0.97) and discharge versus location (r2=0.97) ..... 139
37 Logarithmic curve fit for pressure versus location
(r2=0.93) and discharge versus location (r2=0.93) ..... 140
Al Blocked furrow infiltrometer
IX
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LIST OF TABLES
TABLE PAGE
1 Philip comparison of infiltration equations
(Philip, 1957d) . . „ . 18
2 Soil profile classification. .... 45
3 Effect of flow measuring device on surface storage 53
4 Infiltration using Christiansen method . . 54
5 Comparison of infiltration rates predicted by Christiansen
technique with blocked furrow prediction and furrow inflow-
outflow difference «... . . 55
6 Comparison of infiltrated volumes as predicted by blocked
furrow infiltration equation and furrow inflow-outflow ... 55
7 Seasonal variation in surface storage for various depths
of flow 69
8 Furrow area comparisons between actual and mean fitted
parabolic representation or trapezoidal assumptions for
irrigation 5 71
9 Manning's roughness factor (M ) calculated for Irrigation 5. 73
10 Manning's roughness factor (M ) through the season ...... 74
11 Selected t-tests on roughness variations, Irrigations 1
and 2 . 76
12 Surface storage areas through the season 78
13 Surface storage factors. .... 84
14 Furrow inflow characteristics. . . 85
lii Model input parameters for irrigation 5 with different
surface storage assumptions 87
16 Actual and predicted quality parameters, dimensionless
distributions and water volumes for Irrigation 5 ...... 89
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TABLE PAGE
17 The effects of skewness and kurtosis on sprinkler
irrigation design parameters (Seniwongse et al., 1972) ... 99
18 Differences between actual, linear (CL) and normal (e^)
distributions for the various fractions of area X
(Karmeli, 1977) 119
19 Comparison between efficiencies estimated by UCL and UCH . .122
20 Case studies results (Y =1.0) 141
Al Rate of advance and required computations for determination
of K, n, and F 155
A2 Data and results from Griddle et al. (1956) using inflow-
outflow method 157
Dl Soil moisture history for first five irrigations 180
D2 Soil moisture results for irrigations 4 and 5 from soil
moisture samples 181
El Sprinkler pattern - sample 184
XI
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ACKNOWLEDGEMENTS
The authors express their gratitude to the many people who contributed to
this work. In a special way the writers express their deepest appreciation to
Tom Ley, Lee Wheeler and Michelle Salazar.
The contributions of Tom Ley and Lee Wheeler in the field data collection
and analysis, literature reviews, and computer programming were essential to
the timely completion of this report. The typing and review of the initial
drafts of the manuscript by Michelle Salazar greatly aided the authors in the
final completion of the report.
The helpful comments of Dr. William Hart in the collection and analysis of
field data are appreciated as is the help of the many people who aided in the
data collection at one time or another.
Finally, the authors wish to thank the project officer, Dr. Arthur G.
Hornsby, whose direction maintained the proper scope of the project.
XII
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SECTION 1
INTRODUCTION
Irrigation of agricultural lands where natural precipitation has been
inadequate for profitable crop production has stabilized the food and fibre
supplies in many areas. However, the return flows from the irrigated areas
generally carry at least some concentrations of salts, insecticides,
herbicides, and sediments which subsequently degrade the quality of receiving
waters. In a number of regions, these pollutant loads are limiting the
number of uses to which the available water resource can be applied.
The physical and chemical processes affecting the quantity and quality
of irrigation return flows are complex. In order to evaluate the effectiveness
of various management alternatives for controlling these processes, it is
necessary to simulate their behavior and interactions. The tools for
performing the simulations, which are commonly called mod.els, are mathematical
relations applied to appropriate boundary conditions.
Recent assessments of irrigation return flow modeling agree upon the need
for future research concerned with the problem of spatial variability. Large
hydrologic systems encompass widely varying climatic, soil, and crop condi-
tions from one locale to another. In such cases, many parameters and data
are not normally distributed. Thus, the arithmatic mean may not be a good
estimation of central tendency, yet most modeling efforts necessarily make
such an assumption.
Probably the most complex and important segment of the irrigation system
is the region between the field or crop surface and the bottom of the root
zone, a vertical increment varying up to approximately two meters. Detailed
models of this subsystem have been formulated for small scale analyses .where
comparatively large quantities of data are available and where much is known
concerning the system being modeled. As these models are applied to larger
areas, a relative lack of detailed data necessitates simplifying assumptions
leading to reduced modeling reliability. Two of the important simplifications
being made are: (1) that the soil hydraulic and chemical properties are
uniformly distributed and (2) that the irrigation water applications are
uniformly distributed.
The problem of spatial variability in soil properties must be left to
qualified researchers in the fields of soil physics, soil chemistry, etc.
This report suggests an analysis of water application uniformity as
influenced by irrigation method, design and operation of the system, and
various soil and crop conditions. The specific objectives of this effort
are:
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(1) To assess the current state-of-the-art for describing irrigation
application uniformities under surface, sprinkler, trickle, and
subsurface irrigation methods;
(2) To formulate a general submodel of application uniformity as
affected by irrigation method, system design and operation, crop
conditions including height, density, and variety, and soil
characteristics such as infiltration and redistribution, initial
soil moisture, and soil salinity. This submodel would also predict
field tailwater; and
(3) To verify the submodel using available field data.
The performance of an irrigation system is described by quality parameters
which relate to: 1) the uniformity of distribution in the irrigated field,
2) the adequacy of the irrigation in meeting crop requirements (water storage
efficiency), 3) the amount of total applied water which is made available for
plant use (water application efficiency), and 4) the fraction of applied water
resulting in deep percolation. In addition, for surface irrigation, the
quality of irrigation is also described by the percent of total applied water
that is runoff.
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SECTION 2
CONCLUSIONS
1. The presently available models for obtaining and describing the spatial
distribution of irrigation water and the ones outlined and initiated in
this research indicate that it is feasible to estimate spatial distribu-
tions and irrigation efficiencies with given design and operating
conditions.
2. In this study, the successful use of a specific model for prediction of
irrigation quality parameters in furrow irrigation has been demonstrated.
The model requires only knowledge of soil infiltration characteristics,
furrow shape and roughness, furrow inflow, and soil moisture deficit as
well as physical characteristics of the irrigated field (slope, length,
and furrow spacing).
The model indicates that an irrigation system may be modeled to assess
the impact of changes in irrigation system operation and design on deep
percolation, tailwater. and deficiently irrigated area if recession may
be neglected. The model has the potential for assessing environmental
and economic impact of varying design and operation of a furrow irriga-
tion system.
3. The models for surface irrigation require much research effort so that
they can evolve into models which are more usable and applicable to
wide ranges of conditions. Specifically, research may concentrate in:
a. The development of empirical and theoretical means of
estimating recession.
b. The improvement of techniques to estimate surface storage.
Changes in channel geometry and roughness through the season
will have to be emphasized.
c, The establishment of relationships between soil infiltration
and other soil characteristics (moisture, tillage, etc.).
4. The models for sprinkler irrigation seem to satisfy most requirements
towards understanding distribution and efficiencies. Various regional
strategy models can be developed on this basis and results will have to
be verified on a large scale.
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5. The model prepared for trickle irrigation has to be further elaborated
and also verified. Also, more soil moisture distribution inputs will
have to be analyzed and defined.
6. Integration into a comprehensive model which will optimize the design
or operation of an irrigation system within environmental, social, and
economic constraints will expand the models usability.
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SECTION 3
RECOMMENDATIONS
In order to expand the usability of the model in surface irrigation, and
to insure a good predictive capability, the following research needs are
recognized:
1. The methods for assessing infiltration characteristics must be evaluated
for applicability to specific conditions.
2. General relationships which relate soil type, soil structure, soil
moisture, surface storage, and tillage practice to infiltration
characteristics will expand the usability of the furrow irrigation model.
3. General relationships which relate soil type, irrrigation practice,
crop and crop stage, tillage practice, and channel cross section to
roughness characteristics will allow for better estimates of surface
storage.
4. The relationships for estimating of the recession curve from basic
knowledge of infiltration characteristics at the end of irrigation,
surface storage, roughness, and slope should be established.
5. Integration into the model of a submodel which will predict
evapotranspiration of the crop in different sections of the field based
on the previous history of climatic conditions and the previous distri-
bution history as calculated by the model through the season will increase
the model's accuracy in predicting deep percolation and underirrigation.
This may be accomplished by integration of suitable methods for
estimating soil moisture depletion (Appendix D).
6. The model must be tested on a wide variety of conditions for proper
verification. Model evaluation should include the sensitivity of
irrigation quality to variations in surface storage, infiltration
characteristics and inflow variability. Recession must be studied in
order to see if variation in input parameters have a significant effect
on irrigation quality.
In the area of sprinkler irrigation, the effects of changes in wind
direction, temperature, and pressure on the irrigation quality must be
assessed in order to determine effects on the distribution patterns. Likewise
for trickle irrigation, the wetting patterns in different soil profiles under
different operating conditions must be assessed in order to determine the
validity of assumptions that have been made in relation to the availability
of the water applied by the trickle system to the plant root system.
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SECTION 4
iPATIAL DISTRIBUTION IN SURFACE IRRIGATION
INTRODUCTION
As competing uses for water continue to increase demands for a share of
critically short water supplies, and as environmental degradation becomes a
major concern, the better allocation of water becomes a necessity. The opti-
mal distribution of water to agricultural, municipal, industrial, and recrea-
tional entities requires that water be used and reused whenever possible, and
if discharged into the environment, it must contain a minimum of contaminants.
Agriculture is the economic base for many of the western states. Crop
production is impossible without irrigation in some parts and in other parts
marginal or nonprofitable farming has become highly profitable through
irrigation.
The western £"cates contain a ;najor portion of the energy reserves of the
United States in the form of coal, natural gas, and oil shale. Tie scarcity
of water is ;i major impediment to the development of these energy reserves.
The procurement of necessary water supplies for development of energy
-resources will require either a retirement of agriculturally productive land
or a more efficient use of water by the agricultural community which will
allow the cooxistence of agriculture and energy resource development.
Purpose and Objectives of Study
Modifications in the design or operation of surface irrigation systems to
obtain different spatial distributions of applied irrigation waters can result
in changes in the following:
1, crop productivity
~. return flow quality and quantity
;. energy consumption
4, total use of water
3, labor input into the system
o. initial and amortized costs of an irrigation system.
This work deals primarily with determining the state of the art in
evaluating the spatial distribution of applied water in surface irrigation
with a view toward evaluating performance and potential performance of
surface irrigation systems which may either be operational or in the design
stage. It presents a specific case study of a furrow irrigation system.
-------
Specifically, the objectives of this study are:
1. Determine analytical tools and techniques for evaluating factors
which influence the performance of a surface irrigation system.
2. Develop and test a general model of a furrow irrigation system which
will predict the spatial distribution of applied irrigation water
under different specified operating or design conditions.
IRRIGATION QUALITY PARAMETERS
A schematic representation of a typical spatial distribution of
irrigation water in surface irrigation is illustrated in Figure 1. In the
discussions which follow, the water consumptively used which is in excess of
that stored in the soil root zone above field capacity is assumed negligible.
The following terms are defined with reference to Figure 1:
1. Volume of water retained in the root zone of the plant after
irrigation, VRZ (Area ABDGKA) . This is the portion of applied
water which remains available for plant use after an irrigation.
2. Volume of soil moisture deficiency remaining after an irrigation,
VDF (Area GDEG) . This is the volume of water. required by the root
zone after irrigation to reach field capacity.
3. Volume of deep percolated water, VDP (Area KGHIK).
4. Volume of tailwater, VTW (Area BCDB) .
5. Volume of water delivered to irrigated field, VAP (Area ACDIA) .
6. Volume of water required in root zone to overcome total soil
moisture depletion, VNW (Area ABEKA) .
7. Total volume of infiltrated water, VIN (Area ABDGIA) .
8. Actual spatial distribution of infiltrated water, ASD (Line.IHGD).
9. Mean infiltrated depth of water, DAP (Line JHF) .
Several concepts which describe irrigation quality have been discussed in
the literature, and are summarized as follows:
Israelsen's Efficiency Concept
Israelsen (1950) defined the following commonly used concept of water
Ea =
application efficiency, E :
3.
-------
Distance Along Irrigated Run (L)
A
B
JSZ
*—
0.
Q
c
o
o
K
J
DMA SMD DAP DI
DM I
SMD--soil moisture deficiency before irrigation (depth)
DAF--mean infiltrated depth of water
DI --depth of infiltration at any distance along irrigated
run
DMI--minimum infiltrated depth along run
DMA--maximum infiltrated depth along run
GDEG--soil moisture deficiency after irrigation per unit
field width (VDF)
KGHIK--volume of deep percolation per unit field width (TOP)
BCDB--volume of tailwater per unit field width (VTW)
ACDIA--total volume of water applied per unit field width (VAP)
Figure 1. Distribution of infiltrated depths along irrigated run.
8
-------
While this term generally describes the percentage of delivered water
which can be beneficially used by a crop, it is inadequate in describing the
overall quality of an irrigation since it does not indicate the actual
uniformity of irrigation, the amount of deep percolation, or the extent of
underirrigation.
Christiansen Uniformity Coefficient
Generally, more uniform infiltration of water over a field results in
better crop response, especially in crops without extensive root systems.
Christiansen (1942) developed a uniformity coefficient (C ) which is
given as :
cu = t1-0
where Zx = summation of deviations from mean depth infiltrated
n = number of observations.
A perfectly uniform irrigation is one in which Cu is equal to 100 percent.
The shortcomings in describing an irrigation by Cu is that several possible
distributions may have the same uniformity coefficients, but the effects on
the crop may be quite different (Howell, 1964).
Hansen 's Efficiency Concepts
Hansen (1960) used three concepts to more adequately describe the quality
of an irrigation. The first concept is water storage efficiency (Es), and is
indicative of the percentage of needed water which is supplied to the plants
by an irrigation. The second, water distribution efficiency (Ej) , is identi-
cal to the Christiansen uniformity coefficient. The third, consumptive use
efficiency (Eu) , deals with the plant's ability to use the stored water.
Water Storage Efficiency--
Water storage efficiency (Es) is defined as:
Low financial returns from irrigated crops may occur with combinations
of low Es and high Ea. Hansen indicates studies by Meyer et al. (1953) and
Ross (1953) where yields were significantly increased from better storage
efficiencies. Low Es may be a result of inadequate farmer knowledge of crop
water requirements, nonuniform distribution of infiltrated water, due to
design or operational factors, or inadequate supplies of water for fulfillment
of crop water requirements.
Consumptive Use Efficiency--
Consumptive use efficiency (Eu) is defined as:
-------
w
Eu = ^- x 100 (4)
d
where W = crop consumptive use of water (transpiration and water retained in
plant tissue)
W
', = net amount of water depleted from root zone.
The term Wj includes all the water evapotranspired, both beneficially and non-
beneficially, and the water lost to deep percolation. The plant spacing,
amount of foliage, height of ridges, and depth and uniformity of infiltration
are factors in determining the consumptive use efficiency. Losses due to
evaporation may be extremely difficult to isolate from consumptive plant use.
Karmeli's Dimensionless Distributions and Quality Parameters
The description of water distribution by concepts which relate to tail-
water, deep percolation, and adequacy in meeting the crop requirements permits
the designer or farmer to readily relate to those familiar quantities. The
operation and design of a system can be adjusted to achieve optimal results
when such quantities can be predicted. Karmeli (1977) related to these
quantities in order to describe the quality of irrigation.
Discussion of Concepts--
Karmeli (1977) described the distribution of infiltrated water over an
irrigated field by a dimensionless curve. A typical curve is illustrated in
Figure 2 for a furrow irrigation system with monotonically decreasing depth of
infiltration along the run. The infiltrated depth of water is nondimension-
alized by dividing it by the mean soil moisture deficiency (SMD). The
distance along the run is nondimensionalized by dividing it by the total
length of run (L). The X coordinate is measured from the end of the
irrigated run.
The usefulness of the dimensionless representation (Y = c + aX ) used
by Karmeli is that:
1. The uniformity of irrigation and relative amounts of deep percola-
tion or underirrigation are evident from simple visual inspection.
2. The effects of varying design and operation variables on the
overall system performance can be easily assessed by superposition
of the curves.
3. From the representative equation (Y - c + aX ), Es and Cu may be
established through graphical means or by integration and manipula-
tion of the representative equation. If a tailwater reuse system
is part of the irrigation system, Ea may also be obtained.
Recognizing that deep percolation and tailwater may have significantly
different environmental and economic impacts, Karmeli (1977) also proposed
that two efficiencies which relate to tailwater and deep percolation be used
10
-------
Fraction of Length (x)
1
/"\ ^"\
r^o.o
in O
0) "o
"c CC
.2 c
(fl O
o> o
£.£: 1 n
•*-^ 1 .U
o ^
—
0
H
\
G
F
0.5 0
1 B
r
i c
E|^X^
^^^^
^^^
_^^^
_
.0
1
(
f
Dl
v
.
I.
'/
y
]
_
^, A
7 A
max.
Y = c + aX
Y--dimensionless infiltration ratio = depth infiltrated/
mean soil moisture deficiency
a—Y - Y .
max mm
D—exponent of distribution
HAFH--dimensionless total applied volume
GEFG--dimensionless volume of deep percolation
DCED--dimensionless volume of soil moisture deficiency
remaining after irrigation
BACB--dimensionless runoff volume
Figure 2. Power fit curve of dimensionless distribution
for surface irrigation.
11
-------
to describe an irrigation system. These are the tailwater efficiency (ETW)
and the deep percolation efficiency (EDP). With reference to Figure 1, these
are:
VTW
ETW = 1 - (5)
EDP - 1 - ^ (6>
It may be noted that the percent of total applied water which is tailwater
(PTW) and the percent of total applied water which deep percolates (PDP) are
simply:
VTW
PTW = (1 - ETW) x 100 = J^£ x 100 (7)
and
PDP = (1 - EDP) x 100 = ^- x 100 (8)
also:
E = (100 - (PTW + PDP) (9)
3.
In this work, the five irrigation quality parameters used to describe an
irrigation system are Ea, ES, PTW, PDP, and the dimensionless distribution
function (Y = c + aXb).
In accordance with the selected quality parameters, an irrigation in
which all applied water remains in the root zone and no moisture deficiency
is present after irrigation would be an irrigation in which Ea and Es are
100 percent, PDP and PTW are 0 percent, and the constants of the dimensionless
distribution are: c = 1, a = 0, and b = 0.
The optimal combination of quality parameters attainable is constrained
by economic, crop, climatic, environmental, and social factors. For a given
system, a decrease in Es may result in loss of productivity due to crop
moisture stress and salt accumulation but may result in energy and labor
savings. Changing the design or operation of the system to obtain a higher
Es may result in higher PDP and PTW and may result in increased total water
cost, drainage costs, reduction in yield due to lost fertilizers, or increase
in cost of a reuse system.
FACTORS AFFECTING IRRIGATION QUALITY
Introduction
All of the irrigation quality parameters previously discussed may be
computed without resorting to actual field measurements if the following are
known:
12
-------
1. rate of advance of the irrigation front
2. rate of infiltration of water into the soil
3. inflow into border or furrow
4. total time of irrigation
5. times of recession of water from the soil surface
6. soil moisture deficiency before irrigation
The distribution of infiltrated water in the soil profile may be computed
if times of advance and recession of the water from the soil surface and rate
of infiltration of water into the soil profile are known. Inflow and duration
of irrigation are specified or measured. Soil moisture deficiency can be
established from field measurement or knowledge of crop water use and soil
moisture history.
The basic factors affecting the phases of water movement over and through
the soil surface are:
1. soil infiltration characteristics
2. slope of irrigated run
3. geometric configuration and roughness of furrow or borders
4. length of run
S. volume inflow into furrow or border
6. total time of irrigation
These factors are discussed in the following section with respect to
their influence on 1) infiltration, 2) advance and 3) recession. In addition,
a brief discussion on surface storage as affecting advance and recession is
presented.
Infiltration
Infiltration rate describes the rate at which water is absorbed by the
soil when water is applied to the soil surface.
Infiltration rate as used in border irrigation and sometimes in furrow
irrigation has the dimensions of velocity (L/T), and is the depth of water
entering the soil profile per unit time. It can also be thought of as the
volume of water absorbed by a unit area per unit time (L^/(L^T)). The metric
units commonly used to express infiltration rate are mm/hr or mm/min. In
furrow irrigation, when infiltration rate is expressed as a depth per unit
time, an equivalent depth is usually implied since movement is horizontal as
well as vertical. This depth is obtained by dividing the volume rate of
infiltration per unit of furrow length by the product of unit length and
furrow spacing. In furrow irrigation, infiltration rate is commonly expressed
as the volume absorbed by a unit length of furrow in a unit time (L3/(LT)).
When the infiltration rate is expressed as L/T, the cumulative
infiltration is expressed as a depth (L). In furrow irrigation, when
infiltration rate is expressed as (L3/(LT)), the cumulative infiltration is
expressed as (L^/L).
13
-------
Most soils exhibit an initially high infiltration rate which decreases
with time and eventually reaches a constant or nearly constant rate called the
"basic intake rate." Figure 3 demonstrates typical behavior of infiltration
rate with time as well as cumulative infiltration with time.
Factors Affecting Infiltration--
The difficulty in assessing the infiltration characteristics of a soil
under field conditions is due to the variability with time and space of those
factors which affect infiltration. These factors must be understood lest the
problem be greatly oversimplified.
Physical soil properties—The porosity of a soil is a major factor
affecting infiltration rate because hydraulic conductivity is greatly
dependent on soil particle spacing, especially when the soil approaches
saturation. The porosity of a soil is primarily dependent on soil texture
and soil structure. However, the soil structure is by no means stable in
irrigated soils either with time or space. Wetting and drying affects the
soil structure, especially near the soil surface. The breakdown of aggregates
on the soil surface and the rearrangement of soil particles by the moving
water also affects the structure. Most soils exhibit nonuniforrnity of
texture and structure with depth. Different crops and cropping patterns can
cause soil structure to either improve or deteriorate.
Soil tillage can have a profound influence on the infiltration rate of
the soil due to the physical disturbance of the soil surface by the tillage
implements as well as compaction caused by the tractor and implement wheels.
In furrow irrigation, the compaction caused by the tractor wheels may cause
the furrow bottom to exhibit much lower infiltration characteristics than the
uncompacted sides.
Particular soil characteristics such as the type of clay or amount of
clay may cause soils to crack upon drying thus providing for a high initial
infiltration rate. These soils may swell quickly upon wetting to greatly
reduce the infiltration rate.
Soil moisture characteristj.cs--The infiltration characteristics of
irrigated soils are greatly influenced by the moisture content of the soils.
Philip (1957a) demonstrated the relationships between moisture content and
infiltration rate. Very seldom is an agricultural soil uniformly wet prior
to an irrigation. In general, most crops take the highest percentage of
their soil moisture requirement from the upper part of the root zone. The
variation in rooting depth through the season as well as the distribution of
the root system cause the soil moisture profile to vary through the season.
Channel characteristics, water temperature, and air entrapment--In
border irrigation, the channel shape does not normally affect the infiltration
rate significantly as the entire border is usually covered with water. In
furrow irrigation, the wetted perimeter (and therefore effective surface area
of infiltration) is usually much smaller per unit of land area irrigated than
it is for border irrigation. Wetted perimeter varies both with time and
distance along the furrow since it is determined by the hydraulic conditions
of shape, roughness, slope, and flow rate at any point in the furrow.
14
-------
00
X
0
I
§
•H
to
0
^H
bO
•H
tu
(uuuu
15
-------
Channel characteristics which affect the depth of water on the soil
surface also affect the infiltration rate since the hydraulic head at the
infiltration surface varies in proportion to the depth. This in turn affects
the hydraulic gradient in the soil. This effect is usually negligible except
during initial infiltration stages (Philip, 1958).
Water temperature affects viscosity which in turn affects the hydraulic
conductivity of the soil. However, this effect is usually assumed negligible
in agricultural practice.
Air entrapment in the soil may be less significant in furrow irrigation
since much of the air within the soil profile may escape while the discontin-
uous water medium exists between furrows. In border irrigation where a large
portion of the land surface is covered by a sheet of water, the effect may be
more pronounced.
The Theory of Infiltration and Descriptive Models--
Philip model--Philip (1954, 1957a-d, 1958) developed and discussed
comprehensively the theory of infiltration and developed a solution with a
physical basis which describes infiltration into a homogeneous soil with
uniform moisture content. He discussed the bearing of different moisture
profiles on the behavior of infiltration, the influence of the initial
moisture content, and the effect of water depth over the soil. He also
compared the several different empirical equations with the proposed equation.
Philip (1957a) rewrote an equation which Klute (1952) had derived for
water flow in an unsaturated medium as follows:
— = V-(KV$) (10)
where 6 = moisture content expressed in volumetric terms and liquid density
assumed constant
<\ t\ r\
V = the vector differential operator = i x— + j 4— + k -r—
i, j, k are unit vectors in the x, y, z directions, respectively
K = the unsaturated hydraulic conductivity
$ = the total potential.
When the total potential consists only of gravitational and capillary
pressure components then the following holds:
* = (4, + z) (11)
where fy = pressure potential
z = gravitational potential
16
-------
Also, since V(i|; + z) = Vij; + Vz, equation (10) takes the form:
- = V-(KVijJ + KVz) = V-KV^ + V-KVz , (12)
ot
and since gravity acts only in the downward (z) direction
|i - V- [KV« , || (13)
ip and K are assumed to be single valued functions of 6. Equation (13) can be
written as:
H • "0>W)*H
where D = diffusivity = K ^-r- .
do
For movement in the vertical direction and with z as the vertical
coordinate, positive downward Equation (14) becomes:
If water covers the soil surface, the soil surface is at saturation, 6 .
Also, considering a semi-infinite column, and the soil at a constant initial
moisture content 9n, the following boundary conditions and initial conditions
hold:
6 = 6 at t = 0z>0
n
6 = 6 at t>0z = 0
o —
Philip obtained a series solution for the partial differential equation
(Equation 15) of the form:
D = St1/2 + At + Bt3/2 + Ct2... (16)
where D = cumulative depth of infiltration
S,A,B,C = functions of 6.
The series converges rapidly and normally only the first two terms of the
equation are used, thus:
1/2
D * St ' + At (17)
The values for the coefficients in the clay loam soil illustrated by
Philip illustrate the insignificance of expanding beyond the first few terms.
The values of the coefficients in this case are:
17
-------
S = 1.254 x 10"2
A = 4.654 x 10"6
B = 1.405 x 10"9
C = 8.961 x 10"14
The infiltration rate into a soil is found by differentiating Equation
(17) to obtain:
* ft'1/2. A (18)
2
From Darcy's Law the rate of movement of water through a soil approaches the
saturated hydraulic conductivity at large time when the soil is covered by a
thin layer of water. In this case:
I = KQ (19)
where K = saturated hydraulic conductivity.
In this case, equations (18) and (19) do not exactly agree at large
times.
Philip suggested that the coefficient S be called sorptivity since it is
a measure of the capillary uptake or removal of water and since the equation
applies both to the capacity of a medium to absorb or desorb liquid by
capillarity. The gravitational effects are embodied in the second term of
equation (17). For horizontal flow this second term drops out.
Philip (1957d) obtained values for S and A by obtaining experimental
values of cumulative depth infiltrated at t = 1000 and 10,000 seconds, me
equations obtained by substituting the two times with corresponding depths
in equation (17) can be solved simultaneously to obtain approximate values
of S and A.
Philip noted in an example the closeness with which his two-term equation
approximated actual infiltration and compared it with two empirical equations
(Table 1).
TABLE 1. PHILIP COMPARISON OF INFILTRATION EQUATIONS
(PHILIP, 1957d)
Method of Computation
Experimental Analysis
Horton Equation
Kostiakov Equation
Philip Equation
t =
D (cm)
cum
4.477
8.147
4.225
4.449
105 sec
% Error
0
+82.
-5.6
-.63
t =
D (cm)
cunr J
18.670
29.412
15.395
17.753
106 sec
% Error
0
+58.
-18.
-4.9
18
-------
It must be noted that the Philip equation was derived for a soil with
uniform moisture content. Philip did discuss the effects of different uniform
moisture contents on the sorptivity term of the infiltration equation. The
nonuniform water contents and nonhomogeneity of soil associated with agricul-
tural practice may be the greatest sources of error when this equation is used
in field situations.
Green-Ampt model—Green and Ampt (1911) derived an equation for vertical
infiltration into a soil. They suggested the following equation based on the
then known laws for flow through capillaries:
(P/S) t = L - (a + K)-ln[l + L/(a + K)] (20)
where P = permeability of soil to water
S = porosity
t = time
L = distance to which water has penetrated
a = height of water on the soil surface
K = capillary potential at the wetting front
For horizontal movement they obtained the equation:
(PK/S) t = L2/2 (21)
Their derivation implies that all pores are filled (saturated) out to the
wetting front.
Philip, in his series, showed that the Green-Ampt infiltration equation
is a special case of the solution which he derived. Fok (1975) demonstrated
a comparison between the Green-Ampt and two-term Philip equation, and also
examined the time limits for which the Green-Ampt model is valid.
Gardner-Widstoe model--Gardner and Widstoe (1921) developed a semi-r
empirical equation for infiltration by assuming that capillary potential (t|0
could be related to the soil-water content as:
\l> = c'/P + b' (22)
where p = volumetric water content of the soil
b',c' = constants
The infiltration equation which they derived is the following:
f. = f + (f - f ) e"Bt (23)
t oo O °°
19
-------
where f = infiltration rate at time t
f = infiltration rate at t = 0
o
f = final infiltration rate
CO
3 = a soil dependent parameter which can be determined with knowledge
of f^, f , and infiltration rate at a specific time t
Kostiakov models--In design, experiments, or in evaluation of irrigated
lands, two infiltration equations have been primarily used. These are the
Kostiakov (1932) and the modified Kostiakov equations which are empirical.
The Kostiakov equation is:
i = kt" (24)
where i = infiltration rate at time t
k,n = constants obtained through soil infiltration trials.
The equation may be integrated to obtain the cumulative infiltrate n depth
(D) at time (t):
D = At8 (25)
where
n+1
B = n+1
The modified Kostiakov equation is:
, n
i = k't .+ C (26)
where i = infiltration rate at time t
C = infiltration rate at time t = °°.
This equation may be integrated to obtain cumulative infiltration:
D = A't8' + Ct (27)
where D = cumulative intake depth at time t
_ _
n' + l
B'= n' + l
Techniques for Determining Infiltration Characteristics--
At present, theoretical approaches which relate infiltration to basic
factors affecting it cannot be used to establish the infiltration character
istics of an irrigated soil. Field trials, or in depth knowledge by
20
-------
experienced personnel, must be used to establish infiltration characteristics
under the wide assortment of field conditions.
The previously discussed factors which affect infiltration indicate that
the same methods for assessing the infiltration characteristics may not be
applicable to both borders and furrows. The methods used for assessing
infiltration characteristics must be assessed for applicability to specific
conditions.
Three techniques which have been used to determine infiltration rates in
both furrows and borders are:
1. cylinder infiltrometer
2. basin infiltrometer
3. volume balance techniques based on the rate of advance of the water
front and estimated or measured values for surface storage.
In addition, two commonly used techniques in furrow irrigation are:
4. blocked furrow infiltrometer
5. inflow-outflow measurements on segments of the furrows
Description of these methods is given in Appendix A.
Cylinder infiltrometers--A cylinder infiltrometer measures primarily the
vertical rate of movement in the soil from a ponded surface if it is buffered
by an outer ring filled with water. This downward rate of movement has been
considered by many to simulate the movement of water into the soil during
border irrigation.
In unbuffered cylinder infiltrometers which are sometimes used to
determine furrow infiltration characteristics, the movement of water is
initially vertical until the water passes beneath the cylinder walls, after
which time movement is both vertical and radial.
Advantages are:
1. Small amount of water is required for a test.
2. The cylinders are easy to transport and install.
Disadvantages are:
1. Effects of flow over land surface are not accounted for.
2. Soil disturbance during installation may significantly affect
infiltration.
3. Entrapped air below the ponded surface and between cylinder walls
may significantly reduce infiltration.
21
-------
4. Variability in soil characteristics requires large numbers of trials
in order to establish mean infiltration characteristics.
5. In furrows the lateral movement through the sides of the furrow is
not represented. This area represents an uncompacted, highly
permeable surface.
In furrow studies by Davis and Fry (1963), the infiltration rates
determined with cylinder infiltrometers were in all cases one-half to one-
fourth those determined by one of the other methods.
Basin infiltrometers--The principles are the same as for the cylinder
infiltrometers. However, since a larger area is used, a single sample takes
into account more of the actual field variability. Edge effects are also not
as important. The disadvantages over the ring infiltrometer are primarily in
the greater amount of water required for a single test and greater installa-
tion time.
Volume balance techniques--Shu11 (1964), Singh and Chauhan (1973)
Christiansen, et al. (1966), Norum and Gray (1970), and other researchers,
have presented techniques for calculating infiltration rates from volume
balance methods. These methods normally use the entire border or furrow
as an infiltrometer. One of these (Christiansen) is presented in Appendix A.
Advantages are:
1. The border or furrow is not disturbed, and conditions under normal
irrigating practice are well accounted for if a reasonable approxi-
mation of surface storage can be obtained.
2. Only advance data and cross-sectional flow area are needed for the
determination of infiltration characteristics.
Disadvantages are:
1. The infiltration equation is reliable only during the initial stages
of irrigation, i.e., data are obtained only during the time required
for the water front to reach the end of the irrigation run.
2. Accurate estimates of surface storage may be difficult to make.
Blocked furrow infiltrometer--The blocked furrow infiltrometer measures
the infiltration of water into the soil profile over a short segment of
furrow and from a ponded surface storage.
Advantages are:
1. A limited supply of water only is needed.
22
-------
2. The water level in the furrow can be set to represent actual depth
during irrigation and variability in infiltration rate due to
changes in depth of water in the channel may be assessed.
Disadvantages are:
1. The infiltrometer does not account for variability in soil
characteristics along the furrow unless a large number of samples
are studied.
2. The infiltrometer does not account for the effects of changes in
furrow cross-section during irrigation or soil particle orientation
and deposit caused by the flowing water.
Inflow-outflow method—The inflow-outflow method allows for determination
of furrow intake rates by measuring the inflow and outflow from a reach of a
furrow.
Advantages are:
1. The influence of flowing water on the infiltration characteristics
is accounted for.
2. A substantial portion of the furrow, usually 30-75 meters, provides
a large sample area of infiltration.
Disadvantages are:
1. Most outflow-measuring devices generally obstruct the flow, causing
it to back up in the furrow, thus increasing the cross-sectional
area of infiltration. This is especially true in fields with a
small slope.
2. The build up of surface storage between measuring devices is
generally neglected and it is assumed that the difference between
inflow and outflow is the intake rate for the portion of the furrow
between the two measuring devices. The error accompanying this is
especially significant during early stages of measurement as the
furrow surface storage build up is not reflected in the inflow and
outflow measurements.
3. The requirement to average the time at the inflow-measuring device
with the time at the outflow-measuring device, as suggested by
Griddle et al. (1956), does not permit accurate assessment of the
initial furrow infiltration, i.e., the furrow infiltration rate
associated with a given average time is the infiltration rate at
only one point in the furrow.
Rate of Water Front Advance
The movement of water front down the length of the irrigated field is
designated the advance phase. The advance phase is a case of unsteady
23
-------
nonuniform flow over a porous bed. A complete solution of this problem
requires consideration of mass and energy (or momentum) conservation.
The rate of advance of the water front in surface irrigation is primarily
dependent on channel (1) inflow, (2) cross section, (3) roughness characteris-
tics, (4) slope, (5) infiltration characteristics.
Inflow during the advance phase is assumed to be a constant in most
studies for advance calculations. However, under field conditions, inflow
varies from one channel to the next and also with time during the irrigation.
Varying differences in head across the inflow structures, variation in the
manufacture of inflow structures, operational conditions, differences in well
discharge due to varying pumping levels, and inability to adjust flows
precisely results in time and space variations for most inflows.
The border or furrow cross section at any time is a function primarily
of design, implements used to construct the channel, and flow history of the
channel. The channel cross section, especially for furrows, changes from
irrigation to irrigation due to scouring and sediment redeposit by the flowing
water.
The roughness characteristics of irrigation channels also vary with time
and space. Roughness characteristics are dependent on the ground surface
roughness and vegetative growth. The soil surface roughness changes primarily
because of the erosive effects of flowing water. The vegetative roughness is
a function of the density and diameter of stems, the rigidity of the stems,
the amount of other vegetative matter on the surface, and the water flow
depth and velocity. The roughness, in spite of its variability, is usually
characterized by a single roughness parameter such as Manning's n.
The channel slope determines the rate at which energy is imparted to the
flowing water. Although the land surface slope is never completely uniform
a single average slope is considered in most analyses.
Infiltration characteristics, as previously discussed, vary both in time
and space under most conditions. However, normally a single infiltration vs.
time relationship is assumed to be valid for the irrigation under
consideration.
Approaches to obtaining a solution for the rate of advance in surface
irrigation fall into two principal categories:
1. hydrodynamic
2. volume-balance
The hydrodynamic approach is based on equations of mass and energy (or
momentum) conservation which relate flow depth and velocity in a.n open
channel. Solutions of the complete flow equations, known as the St. Venant
equations, have been obtained by several authors. Among these authors are
Kruger and Bassett (1965); Wilke (1968); Strelkoff (1970); Bassett (1972);
Kincaid, et al. (1972); and Bassett and Fitzsimmons (1974). The solutions to
24
-------
these equations have been accomplished numerically through the method of
characteristics as generally stated by Streeter (1965), and as presented and
discussed for surface irrigation by Strelkoff (1970) . Simplified forms of
the St. Venant equations have been presented by some in attempts to reduce
the complexity of the numerical solution.
The volume-balance approach is based on the principle of conservation of
mass. In this approach, a constant average depth or area of surface storage
obtained through field measurement or estimated through other means supplants
the momentum equations necessary to establish the surface profile. In spite
of this seemingly gross assumption, comparisons with field data of rate, of
advance illustrated by several authors indicates that the assumptions yield
reasonable approximation of actual advance (Hall, 1956; Davis, 1961; Fok and
Bishop, 1965; Wilke and Smerdon, 1965; and Hart, et al., 1968).
Hydrodynamic Models--
General equations --The equations of motion for unsteady spatially varied
flow are the St. Venant equations :
(29)
where A = cross-sectional area of flow
B = the top width of flow
g = weight to mass ratio
I = infiltration per unit length of channel
S = slope of channel
S,. = resistance slope
t = time
V = average velocity of flow at distance x
x = distance along the channel
y = depth of flow.
Equation (28) is developed from mass conservation, and equation (29) can
be developed from either momentum or energy conservation. In general, a com-
plete irrigation may be simulated with the full dynamic models using the
methods of characteristics if input variables can be described adequately.
However, Bassett and Fitzsimmons (1974) have stated that even for short, 30
meter borders, the cost of a computer run can be $15 to $25 per computer run.
25
-------
The full dynamic models as used by Bassett and Fitzsimmons (1976) and
others may be simplified by dropping out selected terms. The results are
then approximations to the complete solution. One of these is summarized
in the following discussion.
Zero-Inertia model—The following analysis was presented by Strelkoff
(1973) and is applied to flow in an irrigation border.
If the equations of motion are applied to flow in a border with a unit
width stream, the equations become:
If the inertial terms are neglected in equation (31), the equation
becomes :
I = So - Sf
The neglect of the inertial terms leads to the conclusion that the water
surface slope is just the difference between the channel slope and the slope
of the energy grade line. The rationale for neglecting the velocity terms in
equation (31) stems from flood routing work which concludes that the first
two terms of the equation are a small part of the remaining terms and may be
safely neglected. The slope of the energy grade line can be determined if
the channel resistance is known. Assuming the resistance can be expressed by
a Chezy-type relationship, the following relationship holds:
V = C /R^ (33)
where V = velocity of flow
C = Chezy resistance coefficient
R, = channel hydraulic radius
S,. = slope of energy grade line
For border irrigation the hydraulic radius is approximately equal to the
depth of flow in the channel:
Rh = y (34)
From equation (33) :
..2 2..2
V V
y
26
-------
where q = unit flow at point under consideration.
Equation (32) can then be written as:
2
|£ = S - -§-— (36)
H "V O / S
The first two terms of equation (30) can be written as:
With the preceding considerations, equations (30) and (31) may be written
as:
where -r- = I of equation (30) .
at
The flow profiles are separated by constant time increments (6t) so that
for each successive time increment the water advances a distance increment
(6xk) . Thus, after the i"1 time increment, the distance that the water has
traveled is:
i
a. = I 6x (40)
1 k=l k
For each time step, profile computations begin at the tip with a first
estimate for the incremental advance, 6£. A surge propagation velocity (W)
is implied:
W - I! (41)
In the vicinity near the surge front it is assumed that the average flow
velocity in the cross section is equal to the advance rate, W. Near the tip,
the channel slope is negligible compared to the water surface slope. Equation
(39) can be written as:
, 2 22 2 2
* - -v
_ _
dx _2 3 _2 3 r2 r2
Cy Cy Cy Cy
The above differential equation, with variables x and y, can be solved through
separation of variables if W is a parameter. C can be evaluated directly
from Manning's equation as:
C . (43)
27
-------
Equation (42) can then be written as:
d -w2 -w2
= ~ =
yl/3~ (1.486}2 ..4/5
The above equation can be integrated:
4/3, f -w n , ,,.,.,
dy = - 5- dx (45)
J (1.486)
When the above equation is integrated from the tip Si to a point x, the depth
at any point, x, may be calculated as:
(1.486)'
With the approximation that:
(46)
fix. = w
-------
If, upon reaching the head of the border, q compares with the discharge
into the border, then the assumed values for W, and therefore, 6£ has been
found. If the comparison is not close enough, a new value for W is assumed,
and the entire sequence of steps back to the head of the border is repeated.
Volume Balance Models--
General approach--The volume balance equations provided the first
mathematical analysis of the advance of the water front in irrigation. In
essence the volume balance methods replace energy (or momentum) conservation
with constant surface storage depth or cross-sectional flow area. The
basic volume balance equation is here stated as:
Qt = W(yx + zx) (52)
where Q = inflow into border or furrow
t = total time of advance
W = width of border or spacing between furrows
y = mean depth of surface storage over advance distance x or mean
equivalent depth in furrow irrigation (mean furrow surface storage
area divided by furrow spacing)
z = mean depth of infiltrated water over distance of advance x or mean
equivalent depth in furrow irrigation
x = distance of advance at time t
Lewis and Milne (1938) wrote the volume balance equation for border
irrigation as:
x
qt = yx + I z(t-t ) ds (53)
0
where q = inflow per unit width of border
t,y,x = as in equation (50)
t = time at which water reached point s on the field
t-t = total intake opportunity at a distance x along the border
z(t-t ) = cumulative infiltration associated with a distance s along
the border
ds = incremental distance along border.
One of the solutions to this form of the volume balance equation is
discussed in a following section.
29
-------
Hall recursive model--Figure 4 represents the surface and subsurface
profiles during the advance of the water front down an irrigated border.
Hall (1956) proposed a recursive method for estimating flow in
irrigation.
With the infiltration function z(t-t ) known, the horizontal lines z1,
z™, z... representing infiltrated depths at constant intervals of time can
be drawn. The advance distances at times At, 2At, 3At... are represented by
the vertical lines x,, x?, x_, ....
The subsurface profile is calculated at constant intervals of At, and
consequently, the subsurface profiles can be constructed by joining the
opposite corners of the resulting rectangles in Figure 4. Now the rate of
advance can be determined if the shape of the surface and subsurface profiles
are known since equation (52) can be written as:
y 7ra
m
(54)
where y = a y
1 y Jm
z = a z
z m
y , z = maximum depth of water on the soil surface and maximum depth
of infiltrated water, respectively
a , a = surface and subsurface shape factors, i.e., the ratio of the
" z actual surface or subsurface profile area to the area of the
respective circumscribed rectangle.
Figure 4. Definition sketch for Hall method of determining
the advance of water down a border.
The parameter ay, depends on gravitational and resistance forces as well
as the soil infiltration characteristics. The product of Oy and ym is a
constant (a necessary consequence of the volume balanced formulation) during
the advance stage, and that their product y includes the surface storage due
30
-------
to surface irregularities (puddling). Hall suggests that ym is the normal
depth of flow at the head of the border, and it can be computed using
Manning's equation as:
M q
where M = Manning's roughness factor (n)
q = flow rate per unit border width (ft /sec)
S = slope of the border
Hall suggests that: .5 <_a <_1.
The function az is dependent on the infiltration function as well as
the hydraulic characteristics of the system. The subsurface shape factor
(a ) is assumed constant for the leading profile with values between 0.5 and 1,
L,
At time t,:
z1 = z(At) (56)
For this time period and rearranging equation (54),
AX]L = _ (57)
During the second time step, the water advances a distance Ax . The
additional surface storage is given as y Ax2- The subsurface storage in
advance increment Ax2 is Ax2OZlzj. The increase in storage under increment
Ax., is area ebdf in Figure 4. By the trapezoidal rule this increment (Av,-)
is:
z - z. + z1 - 0
Av12 - (- ^ i ) AXj (58)
Therefore, during the second time increment a volume balance produces:
Z2 ~ zl + zl ~ °
qAt = ( = ) Ax1 + (y + a z ) Ax_ (59)
l Z1 i t
In general:
Ax + - - Ax
Ax. = [ ? _, 1 ^ j (6Q)
or
31
-------
i-1 z. - z. .
v ,- i-p+1 i-p-1 , ,
^ ( - iTZ - ^ — AxJ
p=i 2(y-a z)
For very small time increments a -> 1/2. Thus, knowing the incremental
Li .
i
distances, the total distance of advance may be obtained as:
n
x = Axj + I Ax. (62)
i=2
Davis (1961) adapted a similar recursive technique to solve the advance
problem in furrow irrigation.
Wilke-Smerdon model--Philip and Farrell (1964) developed a solution to
the advance problem in surface irrigation by working with the Lewis and Milne
Volume Balance Equation (equation (53)). They derived the solution for a
border irrigation system, but stated that it could easily be extended to
furrow irrigation. The portion of equation (53) under the integral sign may
be expressed as:
z(t-ts) ds = z(t-ts) X'(ts) dts (63)
0
Equation (53) becomes:
qt = yx + ) z(t-ts) X'(tg) dts (64)
subject to y >_ 0, dz/dt >_ 0, d2z/dt2 <_ 0.
Taking the Laplace transform of 64 and using the convolution theorem of
the Laplace transform, Philip and Farrellobtained the general solution:
—^ (65)
9 9
s L(z) + ys
where s indicates the Laplacian operator. Applying the general solution to
the Kostiakov equation, Philip and Farrellobtained the specific solution:
B
[(-^-) T(l-B)]n
m L *• J ^ ' J
?(2 + ^. C66)
y n=0
where F = symbol for Gamma function.
32
-------
The above series converges rapidly for small values of time. Philip
and Parrel developed a solution for large times, but did not state the time
ranges where equation (66) converges rapidly.
This equation, as rearranged by Wilke and Smerdon (1965), is:
AtB n
[(-^-) r(i-B)]n
CX co LV. <, ; «. ;j
m i"
where c = average depth of storage in border irrigation or average area of
surface storage in furrow
q = unit width inflow in borders or total inflow into furrow
t = time of advance
x = distance wetting front has advanced
F = symbol for the Gamma function
n = an integer
A,B = constants of Kostiakov equation (equation (25)) where
D = cumulative infiltration depth in borders, or cumulative
infiltration volume per unit of furrow length in furrows
Wilke and Smerdon (1965) solved equation (67) and obtained a family of
linear, dimensionless curves relating the quantities c x/qt and At^/cm, with
the exponent of the infiltration equation (B) as a parameter.
Since the series in equation (67) is not rapidly converging at large
times, solutions were obtained at several values of At /cm and the dimension-
less curves were extrapolated to large times. Solutions which are appli-
cable to many field situations were obtained. The solutions were fitted to
straight lines for each value of B. The general solution when the lines
were forced through qt/c x = 1 at kxa/c = 0 is:
^ m m
31- - 1 + P ^ (68)
C X C J
m m
where P = a coefficient dependent (slope of dimensionless solution curves)
only on B.
Their comparisons with field data indicate good correlation between
predicted and actual advance distances.
Chen and Hansen (1966) pointed out that the curves of qt/cmx versus
At^/cm proposed by Wilke and Smerdon were not straight lines. In a closure
33
-------
to the article, Wilke and Smerdon agreed that the relationship was not linear,
but that the linear relationship yielded good comparisons with field data.
In the closure they presented dimensionless curves which hold for all times.
Because the hydraulic roughness is difficult to characterize in furrows,
Wilke and Smerdon (1965) circumvented the use of Manning's equation by
establishing an empirical relationship between depth of flow, furrow inflow,
and slope in order to establish a value for cm. Since the furrows were
approximately parabolic, the surface storage could be expressed as a function
of depth. Their empirical relationships are:
and
= 2.75do5/3 (70)
2
where c = mean surface storage area (ft )
d = depth of flow at head of furrow (feet)
q = stream size cfm
s = slope of ground surface, percent
Empirical relationships for surface storage such as those established by
the Wilke-Smerdon study may be used only for furrows of the same general shape
and roughness as those studied, but similar relationships may be established for
other conditions.
Fok-Bishop model--Fok and Bishop (1965) developed a volume balance
technique for estimating the rate of advance in borders and furrows. The
assumptions, in addition to those made for all volume-balance methods are:
1. The advance function follows a power law:
x = atb (71)
where x = distance of advance
a,b= constants of advance equation
t = time of advance for distance x.
2. The average surface storage depth is primarily a function of the
maximum depth at the head of the border (y ) and the exponent of
the advance function.
3. The exponent of the advance function can be approximated from
simple knowledge of the exponent of the Kostiakov infiltration
equation.
34
-------
The validity of a power law advance as suggested by Fok and Bishop has
been discussed by Hart, et al., (1968), who have shown that with a function
of the Kostiakov form to describe infiltration, the advance will not be a
power function of time initially, but will approach a power function as time
increases. Field data indicate, however, that the advance may in many cases
be approximated as a power function.
In their analysis, Fok and Bishop suggested that the b in equation (71)
may be approximated by an empirical relationship as :
b = e-. (72)
where n+1 = exponent of Kostiakov infiltration equation (i= kt ).
The average subsurface storage depth (z) is given by the relationship:
* KL
where F = Kiefer correction factor (Kiefer (1959)) =
TTteT + • • •] (74)
b b+1 2(b+2) J
The authors estimated the mean surface storage depth (y) in borders as:
y
^ = T7b (75)
With the stated approximations, equation (52) becomes:
v n+1
Qt = W(T^ + T^TTTT^TTT t:") x (76)
(77)
The advance distance at any time in a border is then:
Ot
Equation (77) modified for furrow irrigation is:
„ Qt _
m T
m WFktn+1
(78)
H-b
where u,m = area coefficients for cross sectional area at head of furrow
in terms of the normal depth
35
-------
Q = furrow inflow
W = furrow spacing.
Fok, et al. (1971) determined that in the case where the infiltration
equation is given by the modified Kostiakov type, equation (77) may be
expressed as :
x = - 2t - (79)
y „, n+1 ,,. v J
m Fkt Lt .
+b
Recession of Water from the Soil Surface
General Discussion--
Recession is defined as the phase of irrigation marked by the
disappearance of water from the soil surface. The times at which water
recedes from the soil surface along an irrigation run at the end of irriga-
tion are difficult to predict accurately. Gravitational forces, amount of
surface storage at the end of irrigation, resistance forces, and infiltration
rate determine the time at which recession starts and the rate at which it
proceeds down the border or furrow after water is cut off at the head of the
furrow.
The full dynamic model presented by Bassett and Fitzsimmons (1976)
defines the recession phase as well as the advance phase for border irriga-
tion, and has been used to study irrigations in very short experimental
borders. The cost of using this model in general design practice is
prohibitive.
On soils with high infiltration rates, most of the water in surface
storage at the end of irrigation may enter the soil profile. In soils with
low infiltration rates, much of the water remaining in surface storage at the
end of irrigation may run off. In irrigated furrows, the ratio of surface
storage to total infiltrated water is in many cases small, and the time of
recession is assumed negligible in its contribution to infiltrated water. As
expressed by Wilke and Smerdon (1969), "the assumption that surface storage
is negligible would not hold for soils having very slow infiltration rates,
or for unusually large furrows or stream sizes." The assumption that surface
storage is negligible is seldom true in irrigated borders.
Empirical Approaches- -
A few highly empirical approaches have been suggested for estimating
recession. Fok (1964) proposed a recession function of the form:
xr = g(t - tr)h (80)
where x = distance of water recession measured from the head of the
border
36
-------
g,h = empirical constants
t = time at which recession reaches x
t = time at which recession begins.
The empirical constants g and h are specific to a given set of conditions
and may be obtained through field trials.
Fok and Bishop (1969) estimated recession by inserting depth (y) = 0,
and to = time at which water is shut off at the head of the border in equation
(77) to obtain a new length L'. Using this L1 in equation (77), they obtained
the time t1 which would satisfy L' when reinserting the flow depth. Connec-
ting the points L = 0, and t = t with the points L1, t1 on a graph of L
versus t, they assumed that the straight line connecting the two points
represented the recession curve.
Wu (1972) presented an analysis of recession and related it to runoff
analysis from a watershed. No verification of his approach was given,
however, and the necessary constants and coefficients applicable to agricul-
tural crops would have to be developed before such an approach could be used.
Surface Storage
»
Surface storage is dependent on all those factors which affect flow in
open channels as well as the infiltration characteristics of the soil.
Changes in furrow cross section, roughness, and infiltration characteristics
throughout an irrigation or season may cause substantial variations in
surface storage.
An accurate determination of surface storage may or may not be necessary.
If the time of irrigation is long, an error in surface storage estimates
which in turn causes variations in advance or recession time estimates may be
insignificant, since the error in estimation of intake opportunity times may
be a small fraction of the total irrigation time. On the other hand, a very
accurate representation of surface storage may be critical in determining
infiltration on a tight soil by volume-balance procedures since an over-
estimation of surface storage may yield unrealistically negative or low.
infiltration values, and an underestimation will yield high values. In
borders, the volume of surface storage is generally a much larger proportion
of the total water introduced into the channel than in furrow irrigation.
Thus, surface storage estimates may be more critical in border irrigation
than in furrow irrigation.
The slope of the channel determines the rate at which energy is imparted
to the flowing water. The rate at which energy is dissipated is dependent on
the furrow roughness and the flow characteristics of the water. The cross-
sectional area of flow can usually be estimated in a section of the channel
through use of an equation such as Manning's equation if the channel cross
section, roughness, slope, and flow rate are known, and if uniform flow can
be assumed.
37
-------
Manning's equation can be expressed as:
r,n 1/6
U
(81)
M
n
where V = mean velocity of flow in channel
V* = shear velocity (XgRu S- or /p)
R, = hydraulic radius
g = gravitational constant
S,, = energy gradient
T = boundary shear
p = fluid density
C1 = constant depending only on the system of units
M = Manning's friction factor
Although Manning's friction factor (Mn) has usually been assumed constant,
Kruse. et al. (1965). Ramsey and Fangmeier (1976) and others have shown that
M has considerable variation with depth in furrows,
fjayre and Albertson (1961) sought to incorporate channel geometry with
roughness height in a single parameter. Their equation is:
— - 6.06 log ~
1 A
where x - resistance parameter, constant for given channel shape and
roughness.
Kruse, et al. (1965) related the standard deviation (a) of equally
spaced measurements of roughness height along the longitudinal profile of
the furrow bottom, to the resistance parameter x for a particular channel
shape. Heerman, et al. (1969) showed that the laboratory results of Kruse
could be directly applied to controlled field conditions for a furrow irriga-
tion system. At present the application of approaches such as that used by
Heerman are hampered by lack of knowledge of the interrelationships between
factors which affect the roughness. Thus, empirical relationships which
relate a constant roughness to surface storage are most commonly used.
The assumptions of constant channel geometry and channel relative
roughness which are generally used to estimate surface storage in both
hydrodynamic and volume balance models may be the greatest source of error
38
-------
entering into advance rate computations for agricultural practice where such
conditions are generally not found.
FORMULATION OF MODEL AND DATA REQUIREMENTS
Introduction
The quality parameters selected to describe the performance of surface
irrigation systems are restated:
1. Water application efficiency (E )--percentage of total applied
water which remains available to plant use after an irrigation.
2. Water storage efficiency (E )--percentage of total soil moisture
deficiency which is met by an irrigation.
3. Percentage of total applied water that deep percolates (POP).
4. Percent of total applied water which is tailwater (PTW).
5. Karmeli dimensionless distribution curve (Y = c + aX ).
It is possible to calculate the above parameters for an irrigation if
the following are known:
1. intake opportunity times at various locations in a field
2. a representative infiltration function for the soil
3. level of soil moisture depletion before irrigation
4. total applied water.
The "intake opportunity time," the total time water is present at a
point on the field, is determined from the time-distance relationships of
the advance and recession phases. The infiltration function is derived
through techniques of Appendix A. Soil moisture depletion is either deter-
mined through soil moisture measurements or through mathematical models which
relate soil moisture, crop, and climatic conditions to crop water use. The
total volume applied is either measured by volumetric or flow rate measuring
devices, or by simple estimation from experience.
The procedure for determining distribution of irrigation water is
essentially the same for borders and furrows. The following section illus-
trates the approach for a furrow irrigation system, but with minor modifica-
tion it can be adapted to border irrigation.
39
-------
A Specific Model for Furrow Irrigation
The Model--
In this model (Figure 5), the total inflow is determined from specified
values of the inflow rate and the duration of irrigation. The infiltration
at any point along the furrow is determined from knowledge of the intake
opportunity time at different stations of the furrow and from measured infil-
tration data which is fitted to one or more empirical infiltration equations.
The soil moisture deficiency is determined from knowledge of crop evapotrans-
piration through the season or from soil moisture samples taken the day
before irrigation.
The advance curve is estimated using the Wilke-Smerdon technique. This
technique was chosen over the others discussed in the literature because it
is simple to apply and because it has been verified by several field investi-
gations. The relationship between the variables affecting advance is given
by:
~ =
rrT
I + P
At
B
(68)
m
where q = furrow inflow
t = time of advance to distance x
c = average cross-sectional area of flow during the advance phase
of irrigation
x = distance of advance
P = a constant depending only on the exponent of the infiltration
equation B
A,B= constants of Kostiakov infiltration equation.
The use of this equation for advance calculations requires that the
infiltration be expressed by the Kostiakov equation, at least during the
advance phase. The model requires input of the soil infiltration charac-
teristics either in the form of the Kostiakov or modified Kostiakov equations
(equations (25) and (27)). The procedure for fitting the modified Kostiakov
equation to the Kostiakov equation for advance calculations is explained in a
later section.
The constant c in equation (68) is predicted from empirical relation-
ships such as those developed by Wilke and Smerdon (1965):
where
0.4
5/3
(69)
(70)
m
40
-------
Start
J
Read W
yes
ther
combination
for same field
Read remaining
model input
parameters
/Write input7
parameters /
Determine power
fit to infiltra-
tion eq. for Wilke
-Smerdon model
Calculate surface
storage if C .EQ.
0.0 m
Calculate time vs.
distance using
Wilke-Smerdon
model
Fit advance point;
to power curve
Calculate intake
opportunity time-
neglect recession
L
Call Exit J
fWrite qulity
irameters,vol-
(umesjaleo vol-j
ime distribu-
tions along
run
•aiculate dimen-
sionless distribu-
tion (y=c+ax )
Calculate deep
)ercolation & def-
.cient volumes &
'raction of requi
ement met in eacl
Oa of field
Calculate E , E ,
a s
PDF, PTW
Calculate volumes
Infiltrated, app-
lied, in root zone
Calculate cumula-
tive deey percol-
fttion & deficienc;
Calculate infil-
trated volumes a-
Long run (m /m)
Figure 5. Flow chart for computer model of furrow irrigation system.
41
-------
where d is the upstream depth in the furrow. The upstream depth, d , may
be computed using the Manning's equation and an estimate of surface roughness.
A relationship has been developed in this study which relates c to Manning's
roughness factor (M ), inflow (q), slope (s), and geometry of trie furrow.
This relationship is:
Mnq STF
c = SSF x STC x (-—) (82)
m g.b
where STC, STE = constants depending only on furrow geometry
SSF = an empirical coefficient which is the ratio of mean surface
storage area to the surface storage at the head of the furrow,
assumed constant. Wilke and Smerdon (1969), Davis and Fry
(1963), and others have used SSF factors ranging from 0.75 to
0.77. For the soil under investigation this constant was
determined to be 0.80.
Data Requirements for Model Input--
The input to the model consists of the following eight sets of data. If
a specific irrigation site and management practice are to be evaluated, then
items three to eight may be considered to be specified and determined from
site measurements. If effects of changes in management or design practice
are to be investigated, any of the data may be altered to reflect such changes.
1. Constants SSF, STC, STE of equation (82), as well as an estimate of
Manning's roughness factor. The quantity, c , can be estimated
from the equation, or read in as a value known from previous
experience.
2. Constants of Kostiakov or modified Kostiakov equation. This
requires field measurement of infiltration or previous experience
with similar conditions.
3. Furrow spacing. This allows conversion of infiltration volumes to
equivalent depths.
4. Average furrow slope
5. Furrow inflow rate
6. Duration of irrigation
7. Length of furrow
8. Soil moisture deficiency (SMD) at time of irrigation.
Data Needs for Assessing Model Usefulness in Irrigation Practice--
In determining the usefulness of a given model for general irrigation
practice, the variability and relative importance of the input variables must
be assessed. Thus, in addition to obtaining the basic model input parameters,
42
-------
the variability in the following, during an irrigation and through the season,
were assessed:
1. average cross section of surface storage during advance phase and
continuing phases of irrigation
2. roughness
3. geometry of furrows
4. inflow
Additional Data Needs for Model Verification--
The following data were needed to verify the volumes and quality
parameters predicted by the model for a specified set of conditions:
1. Furrow outflow. This a-llows calculation of runoff.
2. Waterfront advance and recession. These data, in conjunction with
the representative infiltration equation, soil moisture deficiency,
and total applied water, allow for estimation of:
a. distribution of infiltrated depths along irrigated run
b. volumes of'deep percolation and water retained in root zone
c. PDF and E
d. accuracy of model advance estimates, and relative importance
of recession.
MODEL FIELD STUDIES
Introduction
The validity of the model was tested on a specific farm over the first
five irrigations of the season. No alteration in the farmer's irrigation
practice was attempted in this study because the applicability of the model
to field irrigation practice was desired.
Site Selection and Layout
The site used for model verification was selected on the basis of
relatively uniform slope and soil. The layout of the site selected for this
study is illustrated in Figure 6. A description of the soil profile along
the irrigated run is contained in Table 2.
Measurements were taken on the same six furrows during each irrigation.
Three of these furrows (advance furrows) were used for advance, recession,
surface storage, inflow, and outflow data. The other three furrows (guard
furrows) were used to establish infiltration by inflow-outflow, cylinder
43
-------
Concrete Supply Ditch
0)
c
o
o
*-
CO
u+uu-*
1 +00
2+00
3+00
4+00
5+00
6+00
64-QO
~ J7 \J
(
-
-
-
-
3 r
)
)
y| t
F
(
\ r
i
t
i
i
i
1 C
F
i
i
i
3 r
2
'
)
si /
F
^
(
\ r
3
i
i
i
i d
F4
5 1
.
)
)
^J /
.
i
(
\ r
F5
si G
F€
.542m
0.0045 m/m
Clay Loam
Location: Benson Farm, Greeley
Slope
Soil:
G--guard furrow
N--nonirrigated furrow
A--advance-recession furrows
K--inflow-outflow flumes (cutthroat)
D--blocked furrow infiltrometers
x--trapezoidal flumes for determining infiltration
from inflow-outflow method
o--ring infiltrometers
Fl, F2, F3, F4, F5, F6—data furrows
Figure 6. Layout of field used for model verification.
N
t
44
-------
TABLE 2. SOIL PROFILE CLASSIFICATION
STATION (meters)
Layer (cm) 0+00 1+00 2+00 3+00 4+00 5+00 6+00
0-15
15-30
30-60
60-90
90-120
Soil Type:
1:
2:
3:
4:
5:
1 1
1 1
2 2
3 2
3 3
Clay Loam
Calcareous
Calcareous
Sandy Loam
Sand with
1
1
1
2
3
1
1
1
4
5
1
1
1
4
5
Clay Loam-Transition
Clay Loam
Pebbles
1
1
1
4
5
1
1
1
5
5
Between 1 § 3
infiltrometer and blocked furrow methods. In all cases, care was taken to
ensure a minimum of disturbance to the furrows under investigation. Because
of the nonuniformity in slope at the lower end of the furrow, measurements
were taken only from stations 0+00 to 6+25.
Procedure for Data Procurement
Once the field had been selected, based on length, preliminary
measurements of slope, and soil uniformity, the procedure discussed below was
used to obtain required data. Equipment and data forms used are given in
Appendix B, along with a sample set of data.
1. Field Profile. The field was staked out in 25 meter intervals, and
elevations at each 25 meter station along the irrigation run were
established. Elevations and distances were plotted, and a least squares
linear regression determined the best fit slope. The slope was assumed
constant through the period of study (Figure 7).
2. Furrow Profile (Cross Section). Furrow profile measurements in each of
the three advance furrows of Figure 6 were taken at 100-meter intervals
before and after each irrigation at the same location.
A camera, profilometer, and identification plaque were used for
each measurement. The profilometer consisted of a 5 cm x 5 cm x 44 cm
board with a series of movable rods graduated in centimeters mounted
along the board at 2 cm intervals. To obtain profile data, the profile
board was set across the furrows and the rods were lowered until they
touched the ground surface at which time the photographs were taken.
45
-------
ro
•H
<4H
O
O
•H
•H
P-,
-------
3. Soil Moisture. Moisture samples at several depth intervals (0-15,
15-30, 30-60, 60-90, 90-120 cm) were taken at 100-meter intervals
before and after each irrigation. Soil moisture depletion between
irrigations was also determined using the modified Penman equation to
calculate evapotranspiration (Appendix D). Necessary climatic data were
collected at the Northern Colorado Research Demonstration Center, a
distance of two miles from the site under investigation.
4. Infiltration. Blocked furrow and cylinder infiltrometer measurements,
when taken, were taken the day before irrigation. Cylinder measurements
were taken before only one irrigation because of the variability obtained.
Blocked furrow measurements were taken for irrigations 4 and 5 after it
became apparent that infiltration data obtained by volume balance and
inflow-outflow techniques were not yielding good values for infiltra-
tion. With the blocked furrow infiltrometer, the water was maintained
at approximately the same level as water normally flowed in that section
of channel.
Infiltration measurements by the inflow-outflow method were taken
using W.S.C. (trapezoidal) flumes to measure infiltration at the head,
middle, and end of the irrigation run as indicated in Figure 6. Flows
were monitored throughout the irrigation. The trapezoidal flumes, when
calibrated, were found to deviate substantially from the SCS discharge
relationships (SCS, 1962). They did, however, agree with the calibra-
tion of Chamberlain and Robinson (1960). Surface storage was measured
between flumes to determine the buildup of surface storage caused by the
outflow measuring flume.
Inflow, advance and surface storage measurements required for
volume balance determinations of infiltration are discussed in separate
sections.
5. Furrow Inflow-Outflow. Flow into and out of the advance furrows during
irrigation was measured using cutthroat flumes. Because throat widths
varied from .932 to 1.062 inches, separate discharge curves were needed
for each flume. The flumes with smallest, intermediate, and largest
throat widths were calibrated, and discharge coefficients and exponents
were plotted against throat width. The exponents and coefficients for
intermediate sizes were taken from these plots. The outflow flumes were
placed at station 6+25.
6. Furrow Surface Storage. Depth and top-width measurements of flow for
each of the advance furrows were taken as often as time permitted during
the advance stage. Depths were determined using a centimeter scale with
a flat 2 cm x 5 cm base. Measurements were taken at 25 meter intervals
during the time the water advanced to station 2+00, after which time
measurements were taken at 50-meter intervals. Depth and top-width
measurements were monitored two or three times between the time when the
advance phase ended and the time when irrigation ended. Depth and width
measurements in conjunction with the furrow profiles were used to
determine surface storage.
7. Furrow Advance Data. The time at which the water front reached each
station was recorded in each of the advance furrows for stations spaced
47
-------
25 meters apart starting from the time water began to discharge at
station 0+00.
8. Recession. Recession at any given station was recorded as the time at
which water movement over the soil surface ceased completely. The
inflow and outflow measurement flumes were removed before the end of the
irrigation to preclude any effects due to the backed up water on
recession.
Data Analysis
In addition to furrow spacing, length, slope (Figure 7) which were
determined at the beginning of the season, the following system character-
istics were determined through the data analysis:
1. infiltration
2. surface storage
3. furrow inflow and inflow variability
4. soil moisture
5. volumes of infiltration, deep percolation, tailwater; and
quality parameters (E , E , PDF, PTW).
a s
Soil Infiltration Characteristics
Cylinder Infiltrometer Analysis--
Cylinder infiltrometer measurements were discarded as being an
ineffective means of assessing the infiltration characteristics for the soil
under study. Figure 8 demonstrates the variability and extremely low infil-
tration rates predicted by the infiltrometers.
Inflow-Outflow--
Inflow- outflow measurements were also discarded as useless for
determining infiltration rates for the furrows under study. Results of
infiltration tests at two different locations during irrigation 3 are shown
in Figure 9. At later stages of irrigation the asymptotic infiltration rate
indicated by the inflow-outflow method is 3 to 5 times higher than that
indicated by the difference in inflow and outflow over the entire length of
the advance furrows. The reason for this extreme over prediction of actual
infiltration rate is discussed in the following paragraph.
The trapezoidal flumes used to measure the inflow and outflow for a
section of furrow in all cases back up the water to some extent. The
increase in depth above the non-obstructed depth of flow in the furrow
depends on the flow rate through the flumes. Table 3 indicates the depths of
flow, cross-sectional area of flow, and wetted perimeter between the flumes
used to obtain infiltration results of Figure 9. The effect of increasing
flow depth (and wetted perimeter) on the infiltration rate is illustrated in
Figures lOa and lOb.
48
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Blocked Furrow Infiltration I
Furrow I , Station 0 + 25
July22, 1977
2 Days after Irrigation 5
Benson Farm, Greeley
Wetted Perimeter 33.7cm
X-Sectional Flow Area 90.3cm2
Wetted Perimeter Increased
to 43.0cm „ v
Infiltration Rate =
X-Sectional Flow Area 187.1cm,* 638cm/m-min
i
i
Infiltration Rate =93 cm3/m-min
i i i i
100 200
Time ( minutes)
300
Figure 10a. Effects of wetted perimeter on furrow infiltration.
51
-------
0.16
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TABLE 3. EFFECT OF FLOW MEASURING DEVICE ON SURFACE STORAGE
Location of Flumes Station (meters)
0+25, 0+75
Q. =1.49 £/sec
5+75, 6+25
Q. =1.17 a/sec
0+27
0+50
0+73
5+77
6+00
6+23
Depth (cm) Wetted Perimeter* (cm)
2.8
3.3
10.1
2.5
2.6
7.5
27.4
29.4
47.0
26.2
26.6
41.5
* Wetted perimeter calculated assuming mean parabolic shape in advance
furrows holds for section between flumes.
Volume Balance--
Infiltration equations were obtained through the volume balance method
proposed by Christiansen, et al. (1966) (Appendix A). The method presented a
gross inability to adequately represent the infiltration rate of the soil
under study in spite of the fact that actual surface storage volumes were
used and not extrapolated as in the examples by Christiansen. No advance
data at less than two hundred meters could be used if a high correlation
between Da and t was to be obtained (Table 4). Several values of C for the
modified Kostiakov equation yielded equally good fits of Da vs. t on log-log
paper and the highest correlation coefficient (r2) was not necessarily that
fit which resulted in the best representation of infiltration. This occurred
in spite of the fact that the correlation coefficient for the fitted power
curve of time versus distance is greater than 0.99.
Table 4 shows partial data and computations for the establishment of the
infiltration equation by the Christiansen volume-balance technique for furrow
1, irrigation 5. Table 5 demonstrates five different infiltration equations
for the same furrow. The first four are derived from the Christiansen
method and the fifth is from blocked furrow data. In the third and fourth
equation, the C value for the Christiansen method is that obtained from
blocked furrow infiltrometer data. The importance of the C value for the
soils under study is also evident from the comparisons with furrow inflow-
outflow results as indicated in Table 5. Because of its inability to generate
the proper C value for the soil under study without resorting to additional
measurements and assumptions, the Christiansen volume-balance technique was
also discarded.
Blocked-Furrow Infiltrometer--
The results of blocked-furrow studies resulted in a good representation
of soil infiltration characteristics as verified by the results of inflow and
outflow measurements taken on the advance furrows (Table 6). Blocked furrow
results are presented for irrigation 5 in Figures lla and lib. Seven blocked-
furrow infiltrometers were installed, however, two sets of data were discarded
because the monotonically decreasing infiltration rates were suddenly
replaced by increased infiltration rates not considered plausible.
53
-------
TABLE 4. INFILTRATION USING CHRISTIANSEN METHOD*
L (meters)
100
200
300
400
500
600
t (min)
28.75
56.5
89.0
129.0
172.5
233.25
D (meters)
.00322
.00289
.00269
.00256
.00243
.00243
qt D i*
L s ~ 2
.00993
.01003
.01088
.01219
.01335
.01536
qt C2t
L °s 2
.00929
.00876
.00882
.00930
.00948
.01011
L = 6.0 f86 r2 = .992
Cj = 0.000000 m/min
C9 = 0.000045 m/min
3
Q = 1.16 liters/sec = .06972 m /min
W = 1.524 m
3
q = .04575 m /m-min ~
D = mean surface storage area (m ) /furrow spacing
(m)
Fk
D = qt/L - D - Ct/2 =
a M s '
Case 1: C, = 0.0 m/min, stations 1+00 ->- 6+00
Infiltration Rate
Case 2: C, = 0.0 m/min, station 1+00 neglected
Case 3: C2 = .000045 m/min, stations 1+00 -»• 6+00
D =
a
i =
a
i =
.00457 t'21 r2 = .86
.00113 t"'79 (m/min)
1738 t"'74 (cm2/m-min)
= .00293 t<3° r2 = .97
- 70
.00111 t (m/min)
70 ?
1709 t (cm /m-min)
a
i =
.00769 f°4 r2 =
.000321 t"'96
+ .000045 (m/min)
-.96
,37
i = 473 t
Case 4: C« = .000045 m/min, station 1+00 neglected D =
i =
+ 68.5 (cm /m-min)
.00575 f10 r2 =
.00631 t"'9°
+ .000045 (m/min)
-.,90
.90
i = 966 t
+ 68.5 (cm /m-min)
For detailed example, see Appendix A.
54
-------
TABLE 5. COMPARISON OF INFILTRATION RATES PREDICTED BY CHRISTIANSEN
TECHNIQUE WITH BLOCKED FURROW PREDICTION AND FURROW INFLOW-
OUTFLOW DIFFERENCES
Equation time
70
i = 1264 t~* 8+68.5 * 3(
inflow- outflow
difference ***
- 7Q
i = 1738 t **
i = 1709 t~'7° **
i = 473 t~'76+68.5 **
- QD
i = 966 t +68.5 ** \
[min) I (cm /m-min) time
)0 83.3 65
81.6
19.2
31.5
70.5
> 74.2 <
^min) I (cm /m-min)
n 76.2
73.9
9.9
17.6
69.4
^ 71.2
* Equation obtained from blocked furrow studies.
** Different possible equations obtained using Christiansen method (see
Table 4).
*** Difference between inflow at 0+00 and outflow at 6+25 divided by
distance between inflow-outflow devices (625 meters).
TABLE 6. COMPARISON OF INFILTRATED VOLUMES AS PREDICTED BY
BLOCKED-FURROW INFILTRATION EQUATION AND FURROW
INFLOW-OUTFLOW
Irrigation Furrow Location Time ( Volume-Infiltrated Q.3) Comparison
5 (meters) Elation™" ^low-Outflow Difference % Difference
1 0+00-^6+25
5 O+OCH-6+25
2S61
7432
2601
7402
17.0
41.3
16.6
41.2
15.2"
38.9
14.44
37.0
1.1
2.4
2.2
4.2
6.9
6.2
15.3
11.4
End of Advance Phase.
End of Recession Phase.
3Surface Storage 2.2 m3.
4 3
Surface Storage 2.1 m .
55
-------
X 10
7
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Irrigation 5
July 18,1977
Benson Farm,Greeley
Station!
• 0+25
O I +25
0 4 + 25
A 5 + 25
D 6 + 25
a
0 b
100
200 300
Time ( minutes)
400 500
Figure lla. Infiltration results from blocked furrow measurement.
56
-------
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•2 3
Day before Irrigation 5
July 18, 1977
Benson Farm, Greeley
Mean Cumulative Infiltration from
5 Infiltrometers
Representive Infiltration
Equation:
D = 2747 ta22+ 68.5t
D =( cm3/meter of furrow)
100
200
300 400 500
Time { minutes)
600
700
Figure lib. Mean infiltration curve from blocked furrow measurement.
57
-------
The adequacy of the equation which was fitted to the infiltration data
(Figure 8) was verified by comparing the total volume infiltrated at the end
of the advance phase and at the end of irrigations with the infiltrated
volumes predicted by the representative blocked- furrow equation (Table 6).
The total actual volume infiltrated was found by graphically integrating
the area between inflow and outflow curves (Figures 13a, b) up to the desired
time and subtracting the measured volume of surface storage. The volume
predicted by the blocked- furrow equation is obtained by: 1) using Figure 12
to establish intake opportunity times; 2) plotting the volumes infiltrated
along the run (Figures I4a, b) ; and 3) graphically integrating the volume
along the run.
Table 6 indicates that the blocked furrows over predict the furrow infil-
tration rate. However, the results are within the range in which field
values of infiltration can be obtained. When flows into and out of the
furrows are known (as was the case), the coefficients of the blocked-furrow
equation may be adjusted so that the infiltrated volumes are more closely
predicted. In this study, values were not adjusted.
Surface Storage- -
The analysis of. surface storage is presented to varying degrees for
irrigations 1, 2, and 5. Irrigations 1 and 2 were chosen since the most
rapid changes in furrow cross- section and roughness were assumed to occur for
these irrigations. Irrigation 5 represents a nearly stable late season
furrow cross section and roughness. The analysis of surface storage follows.
Observations and General Information--
The furrow cross sections are illustrated in Figures 15a-15d. The
changes in the flatness of the furrow bottom, the steepness of the furrow
sides, and width of the furrow are indicative of the scouring action of the
flowing water. The change in depth-width relationships between the end of
irrigation 1 and the end of irrigation 2 is partially due to the remaking of
the furrows after the first irrigation.
The scouring of the furrows at the upper end is especially evident in
the wide-deep trapezoidal furrow cross sections at the head of the furrow,
and the resulting sedimentation is evidenced by the wide shallow channel at
the lower end of the irrigated furrows for irrigation 5 (Figure 15d) .
Furrow Cross Sections- -
The parabolic representations of the furrows are determined from a
furrow cross section in Figures 15a-15d. The mean relationships between
depth and width are illustrated in Figure 16. If the depth (d) and width (x)
relationships of a parabola are known, then the area can be found as :
(83)
58
-------
800 r Furrow 5
700
300 -
Recession
Intake Opportunity Time
300
200
00
0+00
Recession
Irrigation 5
July 19, 1977
Benson Farm, Greeley
Total Irrigation Time : 691 min
O Data Points
2+00 4+00 6+00
Station ( meters )
Figure 12. Advance-recession for Irrigation 5.
59
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67
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Furrow X-Section before First Irrigation
.0.5 5 ,_ A _ _ j.s
Furrow X-Section after First Irrigation
°'29
L29
Furrow X-Section after Second Irrigation
X=8.8da395r A=l2.7dL39
Furrow X-Section before Fifth Irrigation
= l0.4d°'265r A=l6.5dL26
First and Fifth Irrigation Comparison
0510
Distance from Center (cm)
Figure 16. Mean furrow cross sections.
68
-------
where a, b = coefficient and exponent of the parabolic x, y relationship
where x = a y^l and x is measured from the center line of the
parabola
A = area of parabola corresponding to depth d
d = vertical distance from bottom of furrow (parabola)
The mean relationship between d and A in Figures 15a-15d was determined by:
1. Graphically establishing the area of each cross section at depths
of 1, 2, and 3 centimeters from the furrow bottom at the furrow
centerline,
2. calculating the mean area for each depth, and
3. performing a least squares regression between d and A to determine
the best fit power curve.
The mean relationships derived are given in Table 7 and depicted in
Figure 16 for Figures 15a-15d. The relationship between x and d was deter-
mined by differentiating the area equations (equation (83)). The relation-
ship between depth and wetted perimeter (WP) was determined by graphically
determining WP for several values of d on the representative parabola, and
fitting these points to a power curve also.
TABLE 7. SEASONAL VARIATION IN SURFACE STORAGE
FOR VARIOUS DEPTHS OF FLOW
Irrigation
Ib
la
2a
5b
Area Equation
7.
12.
12.
16.
7
2
7
5
1 50
1 29
1 39
a
1 26
d
1
1
1
1
A
7.
12.
12.
16.
7
2
7
5
d
2
2
2
2
21
29
33
39
A
.8
.8
.3
.4
d
3
3
3
3
A
40.0
50.3
58.5
65.7
b -- before irrigation
a — after irrigation
d — depth (cm)
A -- Area (cm )
Although many of the cross sections in irrigation 5 appear trapezoidal,
no trapezoid which would adequately describe the width, centerline depth, and
area relationships was found. This is primarily because in very few cases is
the furrow invert actually horizontal.
69
-------
The maximum deviations from the computed mean parabolic relationships
between depth and area occurred for the fifth irrigation because the furrows
were less parabolic and differences between upper and lower end were greatest.
A comparison between the areas computed under different assumptions is illus-
trated in Table 8. A flow depth of 2 centimeters was chosen for comparison
since this flow depth occurred commonly along the furrow during irrigation.
Table 7 indicates the seasonal variation in cross sectional area at
different depths. In the following sections, unless specified otherwise, the
areas referred to are derived from parabolic assumptions.
Furrow Roughness--The Manning equation can be expressed as:
a/?
•00465QS/ A R//3 (84)
2
where A = flow area (cm )
S = slope (meters/meter)
Q = flow rate (liters/sec)
R, = hydraulic radius = (A/WP) where WP is wetted perimeter
M = Manning's roughness factor.
If the water can be assumed to flow at normal depths and the cross
section can be assumed to be a parabol^. so that the area and wetted perimeter
can both be expressed as power functions of the depth of flow as described
previously, equation (84) becomes:
1d)S .00465 sl/2 V 5/3 B.-2/3 e
M = - - - = - * - C~T)d (85)
The quantities Aj, Bj, cj, and e are functions only of the parabolic furrow
cross section where:
B 2
Ad = cross sectional area of flow (cm )
Q
c, d = wetted perimeter
The constants which depend only on furrow cross section may be consolidated
so that equation (85) becomes:
M .00465 S1/2 ,B3
Mn = - Q - C2d (86)
70
-------
TABLE 8. FURROW AREA COMPARISONS BETWEEN ACTUAL AND MEAN FITTED PARABOLIC
REPRESENTATION OR TRAPEZOIDAL ASSUMPTIONS FOR IRRIGATION 5.
„ Station
Furrow , ,
(meters)
1 0+25
1+25
2+25
3+25
4+25
5+25
6+25
2 0+25
1+25
2+25
3+25
4+25
5+25
6+25
3 0+25
1+25
2+25
3+25
4+25
5+25
6+25
Fitted parabola:
Other fits
T°P W^dth „ f . Area (cm2)
at d=2 cm m Fitted Graphical
Parabolic Integration
29.5 2 39.5 47.0
25.5
25.5
24.5
25.0
27.5
32.0
26.0
21.0
27.0
21.5
19.0
22.0
27.0
25.0
27.0
25.0
25.0
24.5
21.5
45.5
34.0
41.0
35.5
39.0
60.0
44.0
26.5
43.0
38.0
28.0
33.5
44.5
40.0
45.0
41.0
41.5
37.0
31.5
30.0 $ $ 37.5
Mean 39 . 6
Standard Deviation 7.3
A = 16.5 d1'26 (cm2)
Difference
(cm2)
-7.5
-6.0
+ 5.5
-1.5
+4.0
+ .5
-20.5
-4.5
+ 13.0
- .5
+2.0
+ 11.5
+6.0
-5.0
-0.5
-5.5
-1.5
-2.0
+2.5
+8.0
+2.0
-0.1
7.3
Error
-16.0
-13.2
+ 16.2
- 3.2
+ 11.3
+ 1.3
-34.2
-10.2
+49.1
- 8.1
+ 5.3
+41.1
+ 17.9
-11.2
- 1.3
-12.2
- 3.7
- 4.8
+ 6.8
+ 25.4
+ 5.3
2
Assumption* Mean Area (cm ) Standard Deviation
a = 45° 46.6
a = 60° 43.6
a = 68.5° 40.6
3.9
3.9
3.9
* Mean area is estimated from measured top width of flow and
measured centerline depth assuming flat bottom.
71
-------
where C2 = A15/3/c12/3
From equation (86) , Mn can be computed for any section of the furrow if
the normal depth, furrow slope, and flow rate in that section are known. In
the following analysis, uniform flow, with the rate of head loss equal to the
channel slope, was assumed and, consequently, the measured depth of flow in
the furrows was assumed to be the normal depth.. The values for C and B in
equation (86) were assumed to be those for the" mean parabolic representations
for each irrigation. Flow rates in any section, i, of the furrow were
estimated as follows:
1. The time required for the soil to approximately reach its basic intake
rate is the time at which difference between the inflow and outflow
flumes approached within 10 percent of the minimum difference in flow
rate between the two flumes minus the time at which the water reached
the outflow flumes (Figures 13a and 13b).
2. Using the stations which had approximately reached the basic intake
rate, the flow was approximated by the following equation:
Q. = Q. - AQ L. (87)
xi xin x i ^ J
where Q. = flow at a distance L. from the head of the furrow
1 (liters/sec)
Q. = flow rate at head of furrow (liters/sec)
AQ = decrease in flow rate per meter of furrow length estimated as:
(Q- - Q t)/625 meters for times approaching the end of
irrigation, Q = outflow at station 6+25
L. = distance at which Q. is estimated
Using the above procedure, a maximum overestimation of 10 percent in
flow rate may be assumed at any station if the infiltration characteristics
of the soil are approximately homogeneous.
The roughness along the furrow may be estimated by equation (86) . An
incomplete set of calculations for furrows 1 and 5 of irrigation 5 are
demonstrated in Table 9. The table indicates a good correlation between
values calculated by parabolic assumptions and values calculated by graph-
ically determining flow area and wetted perimeter for use in Manning's
equation.
Table 10 represents a summary of M through the irrigation season as
well as variations in roughness between upper and lower portions of the
furrows. The highest roughness values during any irrigation occur at the
lower end of the furrow in irrigation 1, and coincide with the lowest flow
-------
TABLE 9.
MANNING'S ROUGHNESS FACTOR (M ) CALCULATED FOR IRRIGATION 5
„ Time Station
Furrow (min) (meters)
1 134 0+00
0+75
285 0+00
0+75
1+25
1+75
2+25
2+75
3+25
468 0+00
0+75
1+25
1+75
2+25
2+75
3+25
3+75
4+25
4+75
5+25
5 158 0+00
0+75
1+25
252 0+00
0+75
1+25
1+75
2+25
2+75
3+25
495 0+00
0+75
1+25
1+75
2+25
2+75
3+25
3+75
4+25
4+75
5+25
Parabolic fit assumed
Depth
(cm)
— _
1.7
_ _
2.0
2.0
1.6
1.7
1.7
1.9
—
1.8
2.0
1.6
1.5
1.5
2.0
1.7
1.6
1.8
1.7
2.0
1.6
__
2.0
1.5
1.7
2.0
1.5
1.5
__
2.4
1.8
2.0
1.5
1.5
1.5
1.6
1.7
1.5
2.5
A = 16.5
2
Values for A and WP determined
Width AQ/AL
(cm) (£/sec-m)
.00125
27 .00125
.00125
30
26
26
26
23
23
.00125
30
25
26
25
24
24
28
25
25
28
.00123
25
26
.00123
25
27
25
24
23
23
.00123
26
27
25
24
24
24
26
22
29
23
d1'26 WP = 20.
Q
(£/sec)
1.16
1.07
1.16
1.07
1.0
.94
.88
.82
.75
1.18
1.09
1.02
.96
.90
.84
.77
.71
.65
.59
.52
1.06
.97
.91
1.03
.94
.88
.81
.75
.69
.63
1.06
.97
.91
.84
.78
.72
.66
.60
.54
.48
.41
9 d'31.
graphically from Figure
M1
n
—
.011
—
.015
.016
.011
.014
.015
.020
—
.012
.016
.011
.011
.011
.021
,017
.016
.023
.023
—
.017
.012
—
.017
.011
.015
.022
.014
.015
__
.024
.015
.019
.012
.013
.014
.018
.022
.020
.061
15d.
M2
n
—
--
—
--
.019
--
.012
--
.019
—
--
.018
--
.008
--
.021
--
.014
--
.022
—
--
.013
—
--
.013
--
.021
--
.016
—
—
.017
—
.013
—
.016
—
.018
__
.048
Difference
—
--
__
--
.003
--
.002
--
.001
--
--
.002
--
.003
--
.000
--
.002
--
.001
—
--
.001
—
--
.002
--
.001
--
-.001
__
_-
.002
—
.001
-_
.002
—
.004
—
.013
73
-------
TABLE 10. MANNING'S ROUGHNESS FACTOR (M ) THROUGH THE SEASON
Irrigation r Furrow
., Furrow T ,,,
No. Inflow
i
JU
2
5
n -- Number of
M -- Manning's
S -- Standard
1
j.
1
1
1
1
3
5
1
3
5
1
3
5
1
5
1
c
•j
1.45
1.50
1.17
1.04
1.17
1.04
1.17
1.04
1.57
1.42
1.64
1.57
1.42
1.64
1.57
1.42
1.62
1.12
1.16
1.16
1.09/1.09
1.06/1.03
1.16
1.16
1.09/1.06
/1. 03
observations
n
deviation from
Times
t=73
t=152
t=453
t=610
t=453
t=610
t=453
t=610
t=540
t=555
t=578
t=540
t=555
t=578
t=540
t=555
t=578
t=65
t=134
t=285
47/96
158/252
t=134
t=285
96/158
252
mean
Location
50
-------
rate of any irrigation. The lowest roughness values occur during the second
irrigation which also had the highest flow rate of the season. The results
of several t tests performed on the roughness factors are given in Table 11.
The t-tests indicate a difference in roughness between the upper and lower
ends of the furrows with significance levels higher than 90 percent. Even at
the end of irrigation 2 when the difference between the upper and lower
portions of the furrow was least, the roughness factors were different at the
99 percent level of significance. No significant difference was found
between the roughnesses at the beginning and end of irrigation 1, the reason
being that the decrease in absolute roughness caused by scouring was offset
by increases in relative roughness associated with lower flow rates at the
end of irrigation. The general conclusions of the roughness study are in
agreement with the studies of Ramsey and Fangmeier (1976), Figure 17.
Estimating Furrow Cross-Sectional Area of Flow--If the furrow cross
section can be assumed to be a parabola, Manning's equation can be used to
estimate the furrow cross-sectional flow area in terms of only M , furrow
inflow (Q), and the slope of the furrow (S).
With the area and wetted perimeter expressed as power functions of
depth, an equation for surface storage as a functionRof known slope (S), flow
rate (Q), roughness (N^), and cross section (A = Ad 1 and WP = C-d6) was
established: Manning's equation can be written as discussed previously.
D
n .00465 A5/V/2 .00465 (^d V/V/2
^ ~ 2/3 = e~173 ( -1
MRWP ' Mn(C1d ) /A
Rearranging the above equation:
1
Q M C'" 5 2
Bl ' 3 e (89)
.00465 S1/2
Substituting the above depth in the area equation established for a
given parabolic cross section, the area can be expressed as:
Bl
Q M C 2/3 4- B - \ e Q M n
A - A r n . J 1 3 3 - n r niR rani
1 L 1 /o r /rJ ~ U1 >- 1 I^J V.yuJ
.00465 S ' A
where V., R = functions only of channel shape, i.e.,
B
r 2/3 r
rr
Di = V l 3 Bi -
.00465 A
75
-------
TABLE 11. SELECTED t-TESTS ON ROUGHNESS VARIATIONS, IRRIGATIONS 1 AND 2
Irriea- Time 0
, . , ,. , , Assumptions
tion (mm) (S,/sec) r
1 t=610 1.04 1, 2, 3
1 t:=73 Q1=1.45 1, 2, 3
t 2=610 Q2=1.04 1, 2, 4
1 tl 152 Q1=1.45 1, 2, 3
t2=610 Q2=1.04 1, 2, 4
2 t=540 1.58
500 1.42 1, 2, 4
577 1.63
M ,
nl
0+500
o nl nz—
M ,=.0258
n2
S =.0078 H : m ,-m ,<0
2 a nl n2
0+500
0+50
-------
to
q
o
ro
o
d
OJ
o
d
(ft
»»—
o
ro
C\J
O
O
V)
•«—
o
ro
to
O
O
o
or
O CM
d o
d
in
o
o"
o
c>
10
o
o
c>
o
o
OJ
o
GO
o
CO
V)
(fl
o
t/)
o
.3
rt
4->
D
6
•H
fH
(D
p^
X
CJ
X
§
fH
0)
X
rt
cn
VO
13
0)
rt
rt
< u,
S9n|DA IHO
bo
•H
77
-------
R =
The flow areas in terms of M, Q, and S are listed in Table 12 for the
irrigations considered.
TABLE 12. SURFACE STORAGE AREAS THROUGH THE SEASON
__
Time Period Cross Sectional Flow Area (cm )
Before Irrigation 1 96.6[M Q/S°'5]0t7°
After Irrigation 1 95.7[Mn Q/S°'5]0'67
After Irrigation 2 102.2[MR Q/S°'5]°'68
Before Irrigation 5 121.6[M Q/S°'5]°'73
Q -- furrow inflow (liters/sec)
M -- Manning's n
S -- slope (meters/meter)
Surface Storage Factors--The surface storage factors as used here are
analogous to the shape factor used in border irrigation, i.e., the ratio of
mean depth of flow to the normal depth of flow at the head of the border is
called the shape factor. The surface storage factor (SSF) is defined here
as:
SSF = A/Ah (91)
where A = mean cross sectional surface storage area
Ah = area associated with normal depth of flow at the head of the
furrow.
The cross sectional area of surface storage was obtained from the depth
measurements taken during the irrigation, using the area equations derived
from parabolic assumptions. The surface storage estimates obtained for a
selected number of times during irrigations 1, 2, and 5 are illustrated in
Figures 18a-18d. The mean surface storage values were obtained by graphically
integrating the area under the surface storage curves (Figures 18a-18d), and
dividing by the distance of advance (L), or by the distance between the
inflow and outflow flumes during continuing phases of irrigation. Surface
storage values computed from depth and width measurements at the locations of
known profile are also illustrated in Figure 18c and 18d for irrigation 5.
78
-------
60
50
40
30
20
10
n
Time = 610 min June 4, 1977
Q= 1.04 2 /sec Benson Farm, Greeley
OMean Parabolic Cross-
Section before
Assumed Valid
D
ODD
a DMean after
Irrigation
a
Irrigation
Parabolic Cross -Section
Assumed Valid
for Area
Estimates from Depth
Measurements
-
i i i i
i i
u
7 50
a>
<40
a>
o>
230
o
O
60
50
40
30
20
10
0
Time= 453mm
Q= I.l7e/sec
a
a
D
Time = 73min
Q=I.I7 e /sec
o
D
3
CO
n
i i i i i i
100 200 300 400 500
Distance Along Furrow (meters)
600
Figure 18a. Surface storage estimates: Irrigation 1.
79
-------
60
50
40
30
20
10
0
N- 60
E
u
- 50
0
0)
£ 40
0)
k-
o
co 20
u
o 10
3
co o
60
50
40
30
20
10
0
Furrow 5 June 15, 1977
0 - 1 fi4 0/*er Benson Farm> Greeley
W i.DHK/sec 0 Surface Storage Estimates
from Parabolic Mean Furrow Cross-
Sections after Irrigation 2
°ooooo o°o
O
-
-
l l l i l i
Furrow 3
Q = 1.42 2/sec
o o
o o
o
-
—
1 1 1 1 1 1
Furrow 1
Q=l.57e/sec
»
0 0
°o0oo°0oo
-
-
1 1 1 1 1 1
0 100 200 300 400 500 600
Distance Along Furrow (meters)
Figure 18b. Surface storage estimates: End of Irrigation 2.
80
-------
60
N
£ 50
S 40
IB
<
o> 30
o>
o
o 20
w
10
0)
u
o
60
50
40
30
20
10
0
I98min
1.062 /sec
Time = I57min
Q = 1.072/sec
a
Time = 96 min
Q = 1.09 B /sec
July 19,1977
Benson Farm , Greeley
• • A Estimates from Graphical
Integration at Known Cross-Section
O a A Estimates from Mean Parabolic
Furrow Cross -Sections and Measured
Depths
100 200 300 400 500
Distance Along Furrow (meters)
600
Figure 18c. Surface storage estimates: Furrow 1, Irrigation 5.
81
-------
60
50
40
30
20
10
0
60
50
E
o
o>
o 30
20
Time = 220min
Q= 1.16 e/ sec
Time =
Q =
77min
.l6e/sec
0>
o
s, 10
3
v> o
60
50
40
30
20
10
n
\
o Time = 65min
0 • G
° A
o o
-
—
i
= 1.172 /sec 1Q7_
July 19,1977
Benson Farm,Greeley
• • A Estimates from Graphical
Integration at Known Cross-Sections
O DA Estimates from
Mean Parabolic
Furrow Cross -Sections and Measured
Depths
\ i i
i i
100 200 300 400 500 600
Distance Along Furrow (meters)
Figure 18d. Surface storage estimates: Furrow 5, Irrigation 5.
82
-------
The values of surface storage (Ah) associated with a normal depth at
the head of the furrow were not taken from direct measurements as the slope
at distances less than 50 meters varied from 0.1 to 0.45 percent. Values for
Ah were estimated from the derived equations in Table 12, using the flow into
the furrow, the mean furrow slope, and the mean roughness calculated for the
upper reaches of the furrow (Table 10).
The surface storage factors as calculated under different conditions and
assumptions are given in Table 13 for irrigations 1, 2, and 5. Table 13
demonstrates a noticeable decrease in shape factor as the distance of advance
increases for irrigation 5, During the latter stages of irrigation 1, the
surface storage factor increases due to decreased inflow rate and the accom-
panying difference in M between the upper and lower sections of the furrows.
Irrigation 2 indicates surface storage factors during the continuing phases
of irrigation stabilizes at about 0.8.
Due to variability in surface storage factors during advance indicated
by Table 13, a value obtained by taking the mean of the surface storage
factors of advance distances of 200, 400, and 600 meters was selected as the
surface storage value for use in the suggested model.
Furrow Inflows and Inflow Variability--
Table 14 indicates several furrow inflow characteristics for irrigations
2, 3, 4 and 5. Irrigation 1 is not included in the flow analysis because of
a gradually increasing river water supply which caused an abnormal variability
in furrow inflow of nearly fifty percent. Also in this first irrigation, the
flow was cut back by the farmer by almost fifty percent halfway through the
irrigation because of the large amount of tailwater. During irrigations 2,
3, 4 and 5, the farmer established the siphon flow rate from experience at
rates which would allow the water to reach the end of the furrow within a
"reasonable" time, which, in all cases, was less than five hours or 40
percent of his normal twelve-hour irrigation sets.
Table 14 indicates a difference between the lowest and highest mean
irrigation inflows of 0.44 liters per second. The variation of mean single
furrow inflows about the mean inflow for each irrigation considered is
characterized by a standard deviation of only 0.08 liters per second. This
value is indicative of the degree of accuracy with which the farmer was able
to control his flow rates by visual inspection of the siphon discharges.
The standard deviation of 0.06 liters per second indicated from the
differences between specific observations of single furrow discharges and
mean single furrow discharges, is indicative of the variation in observed
flows caused by:
1. the time required for the head to stabilize in the head ditch at
the beginning of an irrigation,
2. obstructions to the siphon discharges such as weeds,
3. fluctuations in well discharge or river water supply,
83
-------
TABLE 13. SURFACE STORAGE FACTORS
Irrigation Furrow Time. _. Advance. Q M Ah A SSF Comment
6 (mm) Distance (m) Q/sec) n
1 3 73
152
610
2 1 540
3 550
5 578
5 1 27
65
134
177
220
5 47
96
157
198
252
600
C
C
C
C
C
100
200
425
515
595
150
265
425
500
600
1.45
1.50
1.04
1.57
1.42
1.64
1.17
1.18
1.15
1.17
1.14
1.09
1.00
1.03
1.05
1.03
.017
.017
.014
.011
.011
.011
.0.17
.0.17
.017
.017
.017
.017
.017
.017
.017
.017
48
49
29
39
36
40
50
50
49
50
49
47
44
45
46
45
42
38
36
31
31
30
49
44
40
36
36
36
40
38
34
33
.88
.78
1.24
.80
.86
.75
.98
.88
82
.72
.74
.77
.91
.84
.74
.73
1
1
2
3
3
3
4
4
4
4
4
4
4
4
4
4
SSF -- Surface storage factor = mean area of surface storage in furrow (A)/
surface storage area at head of furrow (Ah)
C -- Water has reached end of furrow
Ah -- Estimated by assuming normal depth at head of furrow, mean furrow
slope and with furrow profile as indicated below.
Comment: 1) Mean parabolic profile at beginning of irrigation and M for
entire furrow during advance stages
2) Mean parabolic profile at end of irrigation 1 assumed and M
for 50_
-------
TABLE 14. FURROW INFLOW CHARACTERISTICS
Irrigation
2
3
4
5
Number of
Observations
3
3
2
3
Qi
(A/sec)
1.53
1.48
1.12
1.09
Seasonal
Population
Qs = Z Q../n
i
Z Z ((} . --(}. )
i f
T y T fo -C
1 1 1 «lft Q
Characteristics
n
4
11
i£) 126
Mean
1.31
0.00
0.00
Standard
Deviation
.23
.08
.06
0 = seasonal mean furrow inflow
Q. =
Q..p =
Q..C
population consisting of mean furrow inflow rates for irrigation i
where i = 2, 3, 4, 5
population consisting of mean single furrow flow rates for
irrigation i and furrow f through whole season: f = 1, 3, 5
except for irrigation 4 where f = 1, 5
population consisting of instantaneous furrow inflow for t
observation for furrow f and irrigation i: t is variable
depending on the number of times at which inflow is observed
during an irrigation but in all cases 10
-------
was assumed, to be at field capacity after the first irrigation., Assuming the
clay loam soil has a moisture holding capacity of 17 cm/meter (Jackson,
1973), a maximum depletion of 8 cm indicates that the soil is at less than 50
percent depletion at any time assuming a rooting depth of 120 cm.
Volumes and Quality Parameters Required for Model Verification--
The volumes and irrigation quality parameters required for model
verification are presented only for irrigation 5. The volumes are derived in
accordance with the following assumptions:
1. The entire soil profile was at field capacity after the first
irrigation, and the rate of depletion of soil moisture was uniform
throughout the field.
2. Deep percolation is negligible (see Appendix D).
3. The soil moisture depletion before irrigation 5 can be estimated by
subtracting the mean cumulative depths infiltrated during irriga-
tions 2, 3, and 4 from the cumulative evapotranspiration between
irrigation 1 and 5 as computed from the Penman equation.
With the assumption of no deep percolation, the volume infiltrated is
synonymous with the volume applied to the root zone. Thus, the following
equations hold:
1. VDP = 0
2. VAP = V. , where V. = total inflow volume (area under inflow
in ln curves (Figures 13a, 13b))
3. VTW = V . , where V = total volume discharged from outflow
out out ,... , . . , , u • i • *
flumes obtained by graphical integration
in Figures 13a, b
4. VRZ = VIN = V. = V .
in out
5. VNW = SMD x L x W
where SMD = total soil moisture deficiency before irrigation
as indicated by evapotranspiration studies
L = distance between inflow and outflow flumes
W = spacing of furrows
The irrigation quality parameters are calculated as established in the
introduction. The summary is presented in the following section.
Summary and Discussion of Input Parameters--
A summary of input parameters for irrigation 5 is given in Table 15.
Because it was desired to assess the effects of surface storage miscalcula-
tions and assumptions on the model's ability to predict rate of advance and
efficiencies, several assumptions on surface storage were used in the model.
These included variations in Manning's roughness factor (M^ from the minimum
86
-------
TABLE 15. MODEL INPUT PARAMETERS FOR IRRIGATION 5 WITH DIFFERENT SURFACE
STORAGE ASSUMPTIONS
Comment
Input Parameters
W(m) L(m) S(m/m) 1?TA, STC STE SSF M ™SET A1 B1
*• J *• J *• } (mm) n (mm)
I 1A 1.542 625 .0045 128 121.6 .73 .80 .017 691 5747, .22
II 1A
I 2A
II 2A
I 3A
II 3A
I 4B
II 4B
I 5C
II 5C V
Comments :
I --
II --
1 —
2 —
3 —
4 —
5 --
A --
B --
C --
Definitions
W --
L --
S --
DTA --
STC, STE, SSF
121.6 .73 .017
95.7 .67 .025
95.7 .67 .025
121.6 .73 .011
121.6 .73 .011
121.6 .73 .015
121.6 .73 .015
96.6 .70 .017
^ ^ ^ 96.6 .70 .017 ^ ^
Furrow 1
Furrow 5
Mean roughness for stations 0-350; Irrigation 5
Maximum roughness at any time during season
C' Q
(4/sec)
,68.5 1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
16
06
16
06
16
06
16
06
16
06
Minimum roughness during season for stations 0+00 ->• 3+00
Mean seasonal roughness from stations 0+00 •> 3+00
Mean furrow roughness at start of first irrigation
Mean profile for irrigation 5 used to estimate surface
Mean seasonal profile considered to be profile at end
second irrigation
Furrow profile at start of irrigation season
:
Furrow spacing (meters)
Furrow length (meters)
Furrow slope (meters/meter)
Mean depth of soil moisture deficiency (mm)
-- Coefficients for mean area of surface storage, cnu
Q M, CTC
storage
of
, where
cm2 = SSF x STC x
(v i
-r) where
M --
n
TIMSET --
A',B',C? --
Furrow inflow (&/sec)
Manning's n
Total time of irrigation
Coefficients of cumulative modified-Kostiakov equation
where cumulative infiltration D (cm /meter of furrow
length) is D = A tB + CT
87
-------
to the maximum values encountered during the season. These variations also
included the extremes of seasonal parabolic cross-sectional area.
Because the Wilke-Smerdon advance model was developed for soils whose
infiltration characteristics can be described by the Kostiakov equation
rather than the modified Kostiakov equation, the following procedure is used
in the model to convert the modified Kostiakov equation to suitable form:
1. The total time of irrigation (TIMSET) divided by four to obtain a
time (TIM 4).
2. TIM 4 is divided into tenths.
3. The cumulative infiltration is established for each of the time
increments from time equal zero to time equal to TIM 4 by the
modified Kostiakov equation.
4. These cumulative infiltrations are fitted to a power curve as a
function of time by a least squares procedure.
5. This new curve is assumed to represent the intake characteristics
during advance.
For all purposes, other than advance predictions, the modified Kostiakov
equation is used. The modified equation was fitted only for the initial
fourth of the total irrigation time because the empirical criteria for furrow
irrigation is that the water reach the end of the run in one-fourth of the
total irrigation time.
A comparison of the power fit to the modified Kostiakov equation is
illustrated in Figure 19a for the duration of advance phase.
Results and Discussion
Figure 19b illustrates the advance curves developed from field
observation as well as the curves generated by the model under the extreme
variations in surface storage predicted by roughness and furrow cross section
previously discussed. The figure indicates the extremes in predicted rates
of advance. These extremes, when compared with actual advance data are
indicative of the maximum error which could have occurred if furrow cross-
section and roughness had been measured only once during the five-irrigation
period and assumed constant throughout the period. The actual and estimated
quality parameters are given in Table 16. Also illustrated is the fraction,
Y, of soil moisture deficiency which was satisfied in a given tenth of the
field.
Table 16 indicates a maximum deviation of 5 percent in predicted and
actual quality parameters. The close agreement under the various surface
storage assumptions indicates a relative insensitivity of the quality param-
eters to surface storage variations which affect the rate of advance of the
water front.
88
-------
TABLE 16. ACTUAL AND PREDICTED QUALITY PARAMETERS, DIMENSIONLESS
DISTRIBUTIONS AND WATER VOLUMES FOR IRRIGATION 5
Furrow Combination E
a
1 Actual
1 A
2 A
3 A
4 B
5 C
80.
82.
82.
83.
83.
83.
2
5
7
2
1
0
E
31.
32.
31.
32.
32.
32.
9
2
8
5
4
4
POP
0.0
0.0
0.0
0.0
0.0
0.0
PTW
19.
17.
18.
16.
16.
17.
8
S
3
8
9
0
DAP
48.5
48.1
48.1
48.1
48.1
48.1
VRZ
38.
39.
39.
40.
40.
39.
.9
,7
,3
,0
.0
,9
VDP
0.0
0.0
0.0
0.0
0.0
0.0
VTW
9.6
8.4
8.8
8.1
8.1
8,2
Y=c+ax
.
.26+.
.26+.
.27+.
.27+.
.27+,
2 .
.
llx-89
12x-9°
, , .88
llx
,. .89
llx
.89
r minimum
Furrow Actua
5 1 A
2 A
3 A
4 B
84.
88.
87.
89.
89.
89.
9
8
9
6
5
3
33.
31.
31.
31.
31.
31.
9
6
3
9
9
8
0.0
0.0
0.0
0.0
0.0
0.0
15.
11.
12.
10.
10.
10.
3
2
1
4
5
7
43.7
43.9
43.9
43.9
43.9
43.9
37.
39.
38.
39.
39.
39.
,0
,0
,6
,4
.3
.3
0.0
0.0
0.0
0.0
0.0
0.0
6.7
4.9
5.3
4.6
4.6
4.7
= .
-
.25+.
.24+.
.25+.
.25+.
.25+.
995
-
,, .89
13x
.. .89
14x
13x-88
13x'88
13x'88
r^ minimum
=
995
Distribution of Infiltrated Water Along Furrow as
Fraction of Requirement Met for Combination 1A Furrows 1,
Furrow 1
Furrow 5
O-.l
.1-.2
.2-. 3
.3-. 4
.4-. 5
.5-. 6
.6-. 7
.7-. 8
.8-. 9
.9-1.0
Actual*
.36
.35
.35
.35
.34
.34
.33
.32
.31
.30
Predicted
- .36
.36
.35
.34
.33
.32
.31
.30
.28
.27
Actual*
.56
.36
.35
.34
.33
.32
.31
.31
.30
.29
Predicted
.36
.36
.35
.34
.33
.32
.30
.29
.27
.26
x. = fraction of furrow
1 length
y. = fraction of
requirement met
*Actual distribution
calculated from intake
opportunity times (from
advance-recession curves)
and the representative
infiltration equation.
1. Average roughness for stations 0+00 •* 3+50, irrigation 5 = .017.
2. Maximum roughness at any time in season = roughness = .025.
5. Minimum seasonal roughness for stations 0+00 -+ 3+00 = .011.
4. Mean seasonal roughness at head of furrow = .015.
5. Mean furrow roughness at start of first irrigation = .017.
A. Mean profile for irrigation 5 used to estimate surface storage.
B. Mean seasonal profile, considered to be the profile at end of second
irrigation, used to estimate surface storage.
C. Mean furrow profile at start of irrigation season used to estimate
surface storage.
89
-------
IO
u
o
o
e
u
o
0
D = 5747t°-22
D = 36681°-40
D = cm /meter of furrow
100 200
Time ( minutes )
300
Figure 19a. Comparison of forced power fit to representative blocked
furrow equation for advance stage of irrigation.
90
-------
300r-
250 -
200 -
150
100
0)
C
i 50
u
C
o
•o
0
250
200
I 50
00
50
0
Furrow
Q=l.06
'/^July 19, 1977
Benson Farm,Greeley
- Extreme Variations in Predicted
Advance with Min.and Max.
Surface Storage for Season
Assumed
Storage Computed with Rough-
ness and Parabolic Cross -
section from Irrigation 5
o Advance Data
Furrow 5
0=1.16
100 200 300 400 500 600
Distance of Advance (meters)
Figure 19b. Predicted advance under variations of surface
storage for irrigation 5.
91
-------
Summary
The state of the art of spatial distribution in surface irrigation is
presented in this study. Specific results of an extensive literature review
have been integrated into a specific model for prediction of overall system
performance in furrow irrigation when recession may be neglected. The model
has successfully been used to determine the quality of an irrigation.
Specifically, this study has indicated the following:
1. Several methods for establishing the infiltration rates of a soil
exist; however, these must be assessed for applicability to specific
situations. Infiltration may be significantly affected by changes in
wetted perimeter, and it should be established by methods which permit
water to infiltrate through the approximate wetted perimeter which it
has during irrigation.
2. Several methods (hydrodynamic and volume balance) for estimating the
rate of advance in surface irrigation have been proposed which yield
good comparisons with field data. The selection of method is primarily
dependent on the simplicity of the mathematical model desired and on the
relative importance of accurate surface storage estimates.
3. The inability to predict recession of water from soil surface is the
main obstacle to simulation of entire irrigations through mathematical
modeling except in cases where recession can be neglected, determined
from field experience, or (in border irrigation) where economic con-
straints permit the use of full dynamic models (Bassett and Fitzsimmons,
1976). Full hydrodynamic models have been tested only under very
limited experimental conditions and applicability to field situations
(cost permitting) must be tested before the model can be used in design
or evaluation practice.
4. An inability to predict roughness characteristics of small channels with
shallow flow depth, changing cross section, and crop characteristics
prohibits the accurate estimation of surface storage values both in
volume balance and hydrodynamic models. However, errors in predicted
advance times due to surface storage error which lead to errors in
intake opportunity times may be insignificant compared to the duration
of irrigation.
5. Few attempts have been made to integrate advance recession and
infiltration models into general models which will predict the
performance of an irrigation system.
This study has indicated, for a specific case of a currently operational
furrow irrigation system, the following:
1. The most difficult factor to assess which affected the quality of
irrigation was the establishment of a representative infiltration
function. Estimates of infiltration characteristics varied signifi-
cantly depending on the method used to establish them. Infiltration was
92
-------
significantly affected by changes in wetted perimeter and infiltration
had to be established by methods which permit water to infiltrate
through the approximate wetted perimeter which it has during irrigation.
2. The cross-sectional characteristics of the furrow changed significantly
through the irrigation season. However, in all cases the mean cross-
sectional area and wetted perimeter could be derived and expressed as
simple power functions of depth by assuming a parabolic shape for the
furrows.
3. The roughness of the furrows as characterized by Manning's roughness
factor could not be considered a constant along the length of the
furrows, through the season, or with differing flow rates.
4. With the parabolic shape'of the furrow known, the surface storage at any
location could be approximated as a function only of furrow roughness,
flow rate, and slope.
5. In spite of the variations in the surface storage during the advance
stage, the assumption of constant mean area of surface storage does not
lead to gross over or underestimation of rate of advance.
6. In spite of cross-sectional and roughness variations through the season,
the assumptions of a constant cross section and roughness do not signifi-
cantly affect irrigation quality parameters as predicted by the suggested
model.
7. The suggested model closely predicts the irrigation quality parameters
as verified by comparison to field data results.
93
-------
SECTION 5
SPATIAL DISTRIBUTIONS IN SPRINKLER IRRIGATION
INTRODUCTION
The distribution of water in a field under sprinkler irrigation is
primarily a function of design, operational, and climatic factors. Effects
of soil characteristics on the distribution are considered negligible.
The purpose of this work is to review the literature dealing with various
means of describing the quality of irrigation as well as to suggest a compre-
hensive and useful technique for assessing the performance of a sprinkler
irrigation system.
Various factors affect the uniformity of application in sprinkler
irrigation. Some of these are:
1. Climatic (wind speed and direction)
2. Design (sprinkler and lateral spacing, lateral diameters, nozzle type
and size, user height, and design operating pressure)
3. Operational (operating pressures and maintenance condition)
Insight into the performance of an irrigation system can be obtained if the
pattern (distribution) of water application can be established for a specific
set of conditions.
If the water application pattern of a single sprinkler is known for a
given set of climatic, design, and operational conditions, then the overlapped
patterns of several sprinklers can be established analytically for different
lateral and sprinkler spacings. The distribution of water in the overlapped
patterns may usually be described by fitting the distribution to one of
several functional relationships, e.g., normal, gamma, exponential. The per-
formance of an irrigation system (quality of an irrigation) can be assessed
if the distribution is known.
A measure of the quality of a sprinkler irrigation is its uniformity of
water application. This uniformity has commonly been expressed in terms of a
uniformity coefficient. This uniformity coefficient has often been the basis
for comparisons of sprinkler performance.
In addition to the uniformity coefficient, two other irrigation quality
parameters have been used to indicate the performance of an irrigation system.
94
-------
These are the water storage efficiency (Eg) and the water application
efficiency (Ea). These are defined as:
amount of water stored in root zone
a ~ total amount of water applied
amount of water stored in root zone
s total amount of soil moisture deficit in the root zone
E
The water storage efficiency is also known as the availability factor (Hart
and Reynolds, 1965).
THE UNIFORMITY CONCEPT
The first uniformity concept was developed by Christiansen (1942), and is
commonly called the Christiansen Uniformity Coefficient (UCC):
n
- xl
I |x.
• , 1
UCC = 100 1 - — I (92)
\ nx ' J
where x. = individual observation of applied water
x = mean depth for all observations
n = total number of observations
Christiansen's uniformity coefficient is widely used, and a UCC of 70 or
greater is considered acceptable by designers.
The significance of the uniformity coefficient and that of the two
efficiency parameters (E and E ) is illustrated in Figure 20.
3. S
Wilcox and Swailes Q_947) suggested another uniformity coefficient, UCW.
n |xi - x|
They replaced £ , the absolute value of the mean deviation about
i=l
the mean from the UCC, with S, the standard deviation (the sum of the squares
of the deviations from the mean).
UCW = 100[1 - ^] (93)
x
Hart (1961) and Hart and Reynolds (1965) developed a uniformity
coefficient for the Hawaiian Sugar Planter's Association (HSPA) by assuming
the precipitation from commonly used sprinklers under standard spacings is
normally distributed and, therefore, its pattern (Figure 21) can be described
by a normal distribution.
95
-------
Ea = 100%
Es=50%
(a)
UCC=75%
Ea = 90%
UCC = 85%
Ea =60%
ES=IOO% UCC =95%
(O
Figure 20. Typical effects of water distribution patterns on a crop
under irrigation assuming no runoff (Hansen, 1960).
96
-------
Area Receiving Depth x or More , acres
36 9 12 15
o>
tt)
.2
0) X
^•* ^
o —
£ 0
o.
0)
o
Figure 21. Histogram o£ application depth versus area irrigated
(Hart and Heerman, 1976).
Dimensionless Area , a
0.2 0.4 0.6
0.4
o.
9)
O
Q>
c
o
"v>
0.8
1.2
1.6
0.8
1.0
C Dimensionless Requirement2
Needed Depth of
Water to Replenish
Root Zone
Figure 22. Normalized, nondimensional frequency curve.
97
-------
A dimensionless frequency curve is established (Figure 22) by dividing
the application depth by the mean application depth. If the water distribu-
tion is normal, then the absolute value of the mean of the deviation about the
n |xi - x|
mean, [ £ ], equals 0.798 S, where S is the standard deviation. The
i=l
HSPA uniformity coefficient or the UCH is defined as:
7QQ C
UCH = 100[1 - _ ] (94)
x
The UCC is equal to the UCH for a normal distribution.
Hart and Reynolds {1965) stated that the UCH is easier to calculate than
the UCC because the data does not need to be arrayed for the UCH as it does
for the UCC.
A physical interpretation of the UCH i_s that 79 percent of the pattern
area will receive an application of (UCH)(x) or more, when the water distri-
bution's normal (Hart and Heerman, 1976) as the area under a normal curve
from_(x - 0.798 S) to °° is about 79 percent of the total area under the curve.
UCH x is the lower limit of x. in this fractional area.
Seniwongse, et al. (1972) investigated the effect of kurtosis and
skewness on the uniformity coefficients (skew and kurtosis are two statistical
parameters which describe the departure of a distribution pattern from a
normal distribution and aid in characterizing the frequency curve of a
distribution pattern).
Seniwongse, et al. (1972) and Chaudry (1976) studied the gamma, poisson,
and exponential distributions as possible ways to describe the distribution
from overlapped sprinklers. They both found that of the three, the gamma
distribution was the best fit to the actual data.
2
Seniwongse, et al. (1972) used the X test (95 percent confidence level)
to compare the normal and gamma fits of actual data. He found that the normal
fit is better over the entire range of UCC values.
Chaudry (1976) concluded that for the gamma distribution, skewness
affected the relationships between the uniformity coefficients; UCC, UCW and
UCH, and also that skewness must be taken into account.
The analysis of Seniwongse and Chaudry showed that when UCC, UCW, and UCH
are greater than 70 percent, the skewness or kurtosis has no significant
effect on the values of E or E (Table 17).
3. S
Relationships were established between the various uniformity
coefficients:
UCC = 0.030 + 0.958 UCH; R2 = 0.888 (Hart and Heerman, 1976) (95)
98
-------
2
0
i— i
E-
^
U
HH
C£
a;
rH
PJ
X
2
rH , — i
pi /— ,
CL, CM
co t~-
cn
2 •-<
O ^
co •>
1™H rH
co rt
O
H P
»— )
S co
w
a M
2 q
CD
C CO
CD
CO
^
X
p +•
•H p
S a
rH C.
O f-
Mn CL
•rl >_
3
X
u
j 3
^
)
)
i
)
r -i
^ <^~J
U
D
5 g
r*v$
CM
•
rH O
^.
LO vO
• •
CM vO
r^ vo
^J" ^O
a •
tO vO
t^~ vO
vO CO
vO CO
rH O^
, •
tO CN
00 CM
00 ^
\Q \,O
. .
O 0
00 vO
vO vO
t^^ vO
tO CM
r>! t-^
t^ vO
03 Ri
O CT>
00 to
O 0
to to
vO
•
CM
^.
0
LO
O"!
LO
•
t-^.
O^
•^
•
0
4.
LO
•
i— 1
f~-w
CO
a
rH
r^
[ —
CM
LO
«
LO
t-^
CM
LO
•
0
vO
•^J-
t\
f--
vO
^
Rj
00
vO
o
to
vO
to
1
o
to
CO
o
•
0
00
CM
•
^«
1
vO
rH
vO
o
•
0»
LO
00
LO
t^.
.
rH
1— 1
cn
o
o
1— 1
.
vO
f-
rH
'*
LO
LO
o
to
vO
•
0
1
LO
t-^
00
o
•
t--
00
00
•
o
1
LO
•
to
vO
o
•
to
vO
rH
vO
rH
,
CM
CM
LO
CM
,
o
rH
t
o
vO
1— 1
00
LO
R3
CM
LO
0
to
CM
•
Cn
1
o
vO
t^.
o
•
CTl
vO
to
•
00
1
0
•
CM
LO
o
•
00
•*t
^3-
^-
0
CM
1— 1
*^-
vO
o
o^
*
to
CM
vO
i--!
1— 1
,0
CM
LO
o
to
99
-------
UCH = 10.943 + 0.756 (UCC) + 0.013 (UCC)2; R2 = 0.998 (96)
UCW = -11.287 + 0.92 (UCC) + 0.0020 (UCC)2; R2 = 0.997 (97)
(Seniwongse, et al., 1972)
Two other parameters for evaluating overlapped sprinkler patterns are
the PEU and the PEH.
PEU = i^- (98)
Nq x
where £ x. = the sum of the 25 percent of the observation with the lowest
'" values of application depth
N = the number of observations in the lowest 25 percent
q
x = the average of all observations.
The HSPA pattern efficiency (PEH) assumes normal distribution:
PEH = 1 - 1.27 - (99)
A MODEL TO DESCRIBE DISTRIBUTION PATTERNS BASED ON LINEAR REGRESSION
It is recognized that there exists a need for a single, easily usable
and accurate method that will enable the establishment of the sprinkler system
distribution pattern and also supply other irrigation quality parameters.
A model is suggested (Karmeli, 1977) whose properties enable the
characterization of precipitation patterns of sprinklers, mainly in reference
to efficiency and other irrigation parameters.
The model is based upon the dimensionless cumulative frequency curve of
the infiltration depth (Y) and the fraction of area (X), which is represented
by the linear regression function (Figure 23).
Y = a + bX (100)
where Y = dimensionless precipitation depth
X = fraction of area
a,b= linear regression coefficients (constants).
The least squares method (minimization of sum of squares of deviations of
estimated values from the observed ones) is used to fit a straight line
(linear regression) to the frequency curve. Examples of actual cumulative
frequency curve and its normal and linear fits are given in Figure 24a-e.
100
-------
0.5
Fraction of Area (X)
.0
Figure 23. Linear regression fit of a nondimensional
distribution curve for sprinkler patterns.
The dimensionless sprinkler frequency curve usually takes the "S" shape,
as_the distribution pattern usually tends towards a normal distribution. The
s/~y has a relatively small value when the pattern is highly uniform and most
of the distribution is about_the mean (Figure 24d). However, when the pattern
tends to be less uniform, s/y would increase as the deviation from the mean is
larger, and the "S" shape of the distribution curve would stretch out to
behave more like a straight line (Figure 24a).
It may be hypothesized that the normal fit would be most suitab1' for
distributions where s/y tends to be small. However, the linear fic may be
just as good, as most of the distribution curve would tend to concentrate
around the mean, and errors at both extremes of the_frequency curve would be
relatively limited. For distributions where the s/y is larger (less fitted to
normal), the linear fit would better predict the overall distribution pattern,
as errors on both extremes of the frequency curve would be of smaller
magnitude.
The suggested model, based on the linear regression fit (equation (97))
has some basic properties where for Y = 1.0 (average precipitation depth
entering the soil profile = depth designed to replenish soil moisture defi-
ciency), X = 0.5 (half of the area irrigated). The regression coefficient,
101
-------
2.0 r
Linear Regression Fit
Normal Distribution Fit
Actual Data
Y=-Q.289 + 2.579X
r2= 0.974
UCC = 32.7
UCH = 39.7
UCL= 35.5
16
8
-
—
y
i i i i i i , i , i i i
0 0.2 0.4 0.6 0.8 1.0 1.2
-rrrrfl
1.4 1.6 1.8
Histogram for Dimensionless Precipitation
Depth of Irrigated Areas
0.2 0.4 0.6 0.8
Fraction of Area (X)
.0
Figure 24a. Fits of actual data into normal and linear regression
distributions in sprinkler irrigation.
102
-------
2.0,-
1.5
H"
o.
0>
o
c
o
1.0
in
w
o
'w
c
0)
E
0.5
Linear Regression Fit
Normal Distribution Fit
——. Actual Data
= 0.3085
r2=0.973
UCC =63.6
UCH = 64.I
•=:=0.45
y
I.383X
rr
ffTTT
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Histogram for Dimensionless Precipitation
Depth of Irrigated Areas
0.2 0.4 0.6 0.8
Fraction of Area IX)
1.0
1.2
Figure 24b. Fits of actual data into normal and linear regression
distributions in sprinkler irrigation.
103
-------
2.0 r
Linear Regression Fit
NormaL,Distribution Fit
Actual Data
Y=O.OI22+I.9755X
r2 =0.970
0.4 0.6 0.8
Fraction of Area (X)
Figure 24c. Fits of actual data into normal and linear regression
distributions in sprinkler irrigation.
104
-------
1.5
Q.
O
c
o
"o
1.0
U)
1 °'5
C
0>
E
Linear Regression Fit
Normal Distribution Fit
Actual Data
Y=0,75I + 0.470X
r2 = 0.932
32
24
16
8
n
—
-
-
-rT
-~
UCC = 87.5
UCH = 87.2
UCL =88.3
r=r = O.I6
y
1 h— i v
0.6 0.8 1.0 1.2 1.4
Histogram for Dimensionless Precipitation
Depth of Irrigated Areas
0.2 0.4 0.6 0.8
Fraction of Area (X )
1.0
Figure 24d. Fits of actual data into normal and linear regression
distributions in sprinkler irrigation.
105
-------
Linear Regression Fit
Normal Distribution Fit
Actual Data
UCC=74.8
UCH=73.6
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Histogram for Dimensionless Precipitation
Depth of Irrigated Areas
Y= 0.514 + 0.972 X
0.4 0.6 0.8
Fraction of Area (X)
Figure 24e. Fits of actual data into normal and linear regression
distributions in sprinkler irrigation.
106
-------
b (slope of the line), represents Ymax - Ymin (difference between minimal and
maximal wetting zones of the field). The regression coefficient, a,
(equation (97)) is the estimated minimal precipitation depth (Ymin) and also
Ymin = 1 - °«5b as [a + (a+b)]/2 = 1.0. The estimated maximal precipitation
depth (Ymax) equals a + b and also Yfflax = 1 + 0.5b.
A DISTRIBUTION COEFFICIENT (UCL) BASED ON LINEAR REGRESSION
A distribution coefficient (UCL) based on the linear regression model is
derived to describe the uniformity of distribution. A scaled, nondimensional
mean deviation, 2[0.5 • 0.5(Ymax - Y)], from Figure 25 is substituted into
equation (92).
UCL = 1 - 2[0.5 • 0.5(Y - Y)]
L max
and _
UCL = 1 - 0.51
Since (Y - Y) = 0.5 • b:
v max }
UCL = 1 - 0.25 • b (101)
and b = 4.0 - 4.0 • UCL (102)
Relationships between Y , Y . , b and UCL are given in Figure 25.
max mm
Linear regression frequency curves are given in Figure 26 for different
uniformity coefficients (UCL), as derived from the linear regressions:
50 percent, 60 percent, 70 percent, 80 percent, 90 percent and 95 percent,
and the fraction of irrigated area, X, to given dimensionless precipitation
depth, Y.
The use of the model allows reaching additional information regarding
deficient and surplus volumes, area fractions, as well as changes required
to adjust to requirements. Irrigated areas and precipitation depths are
usually classified into three main groups: deficient (d), surplus (S), and
average (A). The average area fraction and depth relate to the design values
with some allowable deviations (Figure 27).
Y and Y are arbitrary tolerances of depth of water, dependent on the
s a
crop.
The following are some related formulations, based on linear regression
fits:
YD - a
area deficiently irrigated, A = r— = X (103)
where YD = maximal depth in the area irrigated deficiently.
107
-------
a.
c
o
V
E
-1.0
0.5
1.0 1.5 2.0 2.5
Irrigation Quality (b)
3.0
Figure 25.
Y , UCL and Y . as functions of the irrigation
max mm &
quality (b).
108
-------
2.0
o
c
-------
.0
Fraction of Area (X)
Figure 27. Dimensionless linear distribution curve indicating
surplus and deficient application and extent of each.
110
-------
b • ai + b> where ai - Ymin
a = Y .
rain
YD = a' + b(l - a) XD, where XD = AQ =
Y - a
a' = YD - b(l - a) (~-) = YD - (1 - a) (YQ - a)
Fractional increase in total water applied, ITWA, is defined as,
Y'. + Y'
mm max y, +
TTWA - New Volume 2 min max a' + (a* + b)
Old Volume ~ Y~ TY = 2 = 2
mm max
ITWA = a' + = YD - (1 - a) (YD - a) + (104)
Ys - a
area irrigated in surplus, A = 1 r— = 1 - X (105)
j L) J
where Y = minimal depth in the area irrigated in surplus.
o
Area irrigated which falls within allowable deviations from the mean is:
Aa = 1 - (AD + As). (106)
- YD + 1 - \
average depth in the deficient area, YD = ^ (107)
- Ys + l + I
average depth in the surplus area, Y^ = ^ (108)
Volume estimates may be obtained if total area (Aj) and average depth
are given. The excess volume applied (V£) would be:
Y - Y
V = max S . n -A • A_
VE 2 UA AS ^T
where D. = average depth applied
A™ = total area irrigated.
The increase in total water applied which is necessary to bring the field
to a minimum application of Y may be established from equation (100) and
equation (104) as shown in Figure 28 and where:
111
-------
Fraction of Area (X)
Figure 28. Increase in total water applied to bring
depth of YD>
of field to a minimum
a Ap = fraction of field deficiently irrigated
which must be brought up to Yn
The increase in excess volume of water when the minimum eipplication is
increased to Yp is illustrated and calculated as shown in Figure 29 and
equation (110).
The previously developed performance parameters were not related to the
water requirements of the crop. The efficiencies as defined by Israelsen
(1950) and Hansen (1960) indicated the adequacy with which water requirements
are met. Hart and Reynolds (1965) introduced this consideration in their
analysis of water distribution.
Accordingly, a dimensionless ratio, YD, is defined:
R
RJ
depth required in the root zone to overcome total deficiency
mean application, x
XR = fraction of the field that did not receive the required depth, Y,
112
-------
Fraction of Area (X)
Figure 29. Increase in excess water to bring
aA of field to a minimum depth of Y.,.
Y - Y
, , ., max s. ,
Area 1 = ( ^ 5 As>
Y' - Y
. ™ r max s.. , ,
Area 2 = ( ) A,
Y - a
Y - a'
A; - i - x; = i - C-V-)
The fractional increase in excess water as the result of bringing aA of
the field to a minimum depth of Y is then defined as IEW.
IEW =
Area 2
Area 1
- Y
max s
Y - Y
Y - a'
Y - a
(HO)
113
-------
YR - a
XR = g— from YR = a + b XR
See Figure 30.
The water storage efficiency, E , is:
Ec = l ~ v n from Figure 30 (111)
S I yj * J, • U
YR * E
(-S—> XR
= 1
substituting for XR and a = 1 - b/2 = Y .
(YR - (1 -
Th« water application efficiency, E., is:
Y - Y
E. = 1 - (-^~ - -) (1-XD) = 1 - Area 2 on Figure 30 (114)
A Z K
Substituting for Xn and Y =a + b=l+-=-,
6 R max 2 '
- YR YR - l +
- ^) (1 - ~ - - -•)] (115)
The fraction of deep percolation, Dp, represents the fraction of the
total water applied that percolated past the root zone.
"* ' Y Y
"* Area 2 max "
n
D
p - 1.0 " 2 - R'
1 + I - YR YR - 1 + \
Dp = ( - 2 - ^ (1 - -^b - 5
If Yr = 1.0, the case where the desired application is equal to that amount
to completely overcome deficit in the root zone, then equations (113), (115)
and (116) reduce to ,
E = 1 - (118)
r\ O
Dp - | (119)
114
-------
i Ymax
ex
4)
O
o
o
Q.
u
in
jtt
c.
o
in
c
0)
E
YB--,
0.5
Fraction of Area (X)
Figure 30. The proposed model with Y , the dimensionless requirement.
K
The development of EA, Eg, and Dp (equations (114), (116) and (117))
reveal that the only variables they are dependent on are YR and b. Equations
(114) through (119) represent the case where 1.0 < YR <^ ^max- For other
practical values of YR, the ranges and necessary equations for computation of
Eg, Ea and Dp are illustrated in Figures 31, 32 and 33. The model, as
developed, allows for accurate determination of irrigation quality when the
system's performance, b, and level soil moisture depletion, Y^, are given.
Various efficiencies can be calculated and accordingly, optimize the return
flows. The use of the model can be helpful in establishing strategies of
total water used on a farm, area, and watershed in relation to its uses and
losses.
MODEL STUDIES
The purposes of the model verification were to establish the accuracy of
the estimates in relation to the actual performances of the sprinklers and
also to compare the model to the previous available ones.
A sample of 36 sprinkler distributions with s/y values varying from 0.16
to 0.88 (Table 18) had their actual field data arranged as precipitation-area
115
-------
HI-
Q.
a>
Q
c
o
o
a>
«/>
V)
c
-------
0.5
Fraction of Area (X)
.0
Figure 32. The proposed model with Ymin < YR <_ 1.0
Y - Area A
K
TTo
Y . = a = 1 - TT
mm 2
Y -fl - -I
R >• 9'
V /" K f. ."\ V
' YR - ( 2 } V
- Y
R min
R
YR-
2b
Area A
YR
2-b«Y
R
D = 1 - E,
p A
117
-------
E
b
0.5
Fraction of Area (X)
.0
Figure 33. The proposed model with YR > Y
max
E = 1.0
a
D = 0
P
118
-------
TABLE 18. DIFFERENCES BETWEEN ACTUAL, LINEAR (eL) and
NORMAL (eN) DISTRIBUTIONS FOR THE VARIOUS
FRACTIONS OF AREA X (Karmeli, 1977)
Group
I
II
III
UCL
88.3
84.2
80.7
79.4
78.0
77.0
73.8-
75.7
74.6
75.7
72.6
75.2
67.6
65.4
63.5
63.1
58.8
54.9
58.7
54.2
54.4
51.2
50.6
49.6
43.9
45.5
44.1
40.7
35.4
37.5
35.5
38.0
30.2
31.8
26.4
23.0
S/Y
.16
.21
.25
.27
.29
.29
.32
.33
.33
.34
.36
.36
e. -
LI
S
.39
.39
.48
.51
.53
.53
.54
.56
.57
.58
.59
.60
e. =
SL =
.64
.65
.66
.69
.74
.75
.76
.79
.81
.82
.88
.88
e, =
S.
eL
.033
.023
.052
.049
.057
.057
.047
.072
.031
.053
.045
.072
.050 ?,. =
NI
.015 Se =
.035
.062
.037
.057
.059
.039
.044
.098
.061
.034
.080
.057
.055 e^ =
.029 SN =
.073
.026
.042
.069
.122
.090
.105
.074
.072
.073
.142
.098
.082 BJJ
.032 SM =
eN
.030
.025
.042
.038
.054
.066
.054
.066
.043
.045
.030
.061
.047
.014
.056
.065
.036
.056
.042
.075
.039
.111
.074
.091
.105
.072
.069
.025
.124
.081
.093
.124
.180
.132
.170
.106
.166
.147
.209
.195
.144
.041
119
-------
fraction cumulative curves and approximated by both normal and linear
regression fits. Nine points of different area fractions along the normal
and linear regression fits were chosen (X = .1, .2, ..., .9) to test the
approximation of each fit to the actual field data, which is taken off the
cumulative frequency curve (representing the actual distribution). The
differences between linear (AL) and normal (AN) vs. the actual distribution
are represented by £(AL)2 and 2(AN)2 (the summations of the squared differ-
ences between the approximation fits and the actual field values) or by e,
and eN:
•L
and
where N - number of observations.
The 36 sprinkler distributions were_classified into three groups_on the
basis of the s/y values. Group I has s/y < 0.38; Group II, 0.38 < s/y < 0.62;
and Group III, s/y > 0.62 and student "t" tests were performed.
The results were such that the e^ values of the three different groups
did not differ from one another. However, it was established that the e^
values for Group III differed significantly from either of the e^ values of
Group I at 95 percent confidence level or Group II at 90 percent confidence
level.
It was also established that there was no significant difference between
CL and ejvj values for Groups I and III. However, e^ and e^ values for Group
III vary significantly (at a 90 percent confidence level).
The differences between the e^ values for Group III vs. Groups I and II
also led to a significant relationship between s/y and e^, given by
s/y = 0.21 + 3.75 CN, r2 = .73.
The results (Table 18) tend to show a significant difference between
linear and normal estimates where the distribution pattern is of low quality:
s/y larger than 0.62. Where the sprinler distribution is of medium or good
quality, s/y < 0.62, there is no significant difference between estimates of
either linear or normal approximations. Also, it may be concluded that while
the linear fit supplies good estimates for the whole range, the normal fit
estimates are good for the medium and good patterns only.
Another study was performed to establish the efficiencies Ea and ES
from the suggested linear model and in comparison to values reached from the
UCH model.
Nineteen single sprinkler patterns were selected. The Ea and ES values
for the UCH model were interpolated from Hart and Reynolds results (1965),
and that from the linear model by using a computer program (Appendix E) . The
120
-------
results are given in Table 19a, b, c, where the percent difference was
calculated as follows:
_ ,._,- UCL value - UCH value inri
percent difference = UCH value X 10°
From the results (Table 19), it is apparent that there is an insignificant
difference in the calculation of Ea and Es using either the UCL or UCH fit.
The reason that very little difference occurred is that only medium to high
uniformities were looked at. This is the range where the UCH and UCL fits
are very close in their accuracy. The UCH table does not go lower than 60,
hence the reason for no comparison at lower values. However, from the study
by Karmeli (1977), the UCL would be more accurate at lower uniformities.
To evaluate the suggested model and coefficient of uniformity, 19 sets
of single sprinkler patterns were done. The sprinklers operated under
varying pressures (40, 50, 60 psi) and varying wind velocities, and using a
computer program, it was overlapped into various spacings (20', 30', ...,
70') x (20', 30', ..., 70') as well as various geometrical arrangements
(rectangular and triangular). The total number of combinations studied was
798.
The actual data for each combination was transformed into a dimensionless
frequency curve and fitted into the linear regression model. The linear
regression was found to fit the frequency curve very well. The determination
coefficient, r , was < 0.800 for only 5.12 percent of the 798 patterns;
0.800 - 0.899 for 14.15 percent; 0.900 - 0.949 for 28.90 percent; and > 0.950
for 51.83 percent of the patterns.
The results were arranged so that the linear uniformity coefficient,
UCL, is related to wind velocities for the various pressures and spacings
(Figure 34). As expected, higher pressures and closer spacings improved the
irrigation quality (higher UCL) in an exponential rather than linear form.
Also, the relationship between UCC and UCL showed great similarity, as
expected:
UCL = .011 + 0.985 UCC (r2 = 0.998) (122)
SUMMARY
The state-of-the-art of spatial distribution in sprinkler irrigation is
presented in this study. A prediction model enabling the study of spatial
distribution in sprinkler irrigation and irrigation efficiencies is
suggested. This model is based on the use of the linear regression curve
fit.
The linear regression model, Y = a + bX, is an accurate and relatively
easy method for describing sprinkler distribution patterns. It was found
to approximate very well the large sample tested. This approximation
proved to produce good estimates for both high and low quality distributions.
121
-------
TABLE 19. COMPARISON BETWEEN EFFICIENCIES
ESTIMATED BY UCL AND UCH
(a) H
UCH
.948
.879
.831
.880
.606
.665
.754
.803
(b) Y
UCH
.821
.781
.770
.757
.833
.921
.950
.880
= HR = 0.
b(UCL)
.181
.466
.691
.482
1.591
1.371
.998
.774
R=0.9
b(UCL)
.727
.907
.958
1.023
.644
.316
.202
.489
(c) YR = H = 1.
UCH
.930
.896
.734
.881
.715
.695
.665
.606
b(UCL)
.282
.417
1.094
.482
1.184
1.259
1.372
1.591
8
E (UCH)
a
.799
.794
.780
.794
.683
.713
.752
.770
Ea(UCH)
.852
.833
.825
.822
.856
.891
.894
.877
.0
Ea(UCH)
.965
.948
.866
.940
.856
.847
.832
.802
E (UCL)
a.
.800
.799
.785
.798
.688
.714
.755
.777
Ea(UCL)
.852
.831
.825
.817
.862
.895
.900
.879
Ea(UCL)
.965
.948
.863
.940
.852
.843
.829
.801
% diff.
.13
.60
.59
.53
.81
.13
.43
.96
% diff.
.02
-.23
.01
-.57
.66
.42
.67
.19
% diff.
-.02
-.01
-.32
-.03
-.47
-.51
-.42
-.11
Es(UCH)
.999
.992
.975
.992
.858
.892
.943
.962
Es (UCH)
.946
.925
.920
.913
.952
.990
.999
.975
E=(UCH)
.965
.948
.866
.940
.856
.847
.832
.802
Es(UCL)
1.000
.999
.981
.998
.861
.892
.944
.972
Es (UCL)
.947
.923
.917
.908
.957
.994
1.000
.976
Es(UCL)
.965
.948
.863
.938
.852
.843
.829
.801
% diff.
.100
.665
.595
.585
.303
.045
.106
1 . 008
% dief.
.09
.17
-.35
-.54
.58
.41
.10
.13
% diff.
-.02
-.01
-.32
-.03
-.47
-.51
-.42
-.11
122
-------
4.0
c
0)
u
o
o
c
o
Regression:
Spacing
I. (30'x 30')
IA.(30'x 30')
2. (50'x 50' )
2A.(50'x 50' )
3. (70'x 70' )
3A.(70'x 70- )
= ox
b.
Y-b, X
Pressure (psi)
60
50
60
50
60
50
Wind
.2 3.0
M
in
o>
0>
o
«
c
I
jQ
2.0
O
3
O
c
o
1.0
4 6 8 10
(W)- Wind Velocity, m.p.h.
12
14
Figure 34. Linear uniformity coefficient (UCL) as a function of wind
velocity for various spacings and pressures in sprinkler
irrigation.
123
-------
In the higher quality distributions, the linear regression model
estimated-the actual field data as well as the normal model. However, in
lower quality distributions (UCC < 55.0) the linear regression model proved
significantly better than the normal model in its estimates.
A distribution coefficient (UCL) based on the linear regression model
was derived to describe the uniformity of distribution. This distribution
coefficient has a high correlation to Christiansen's Coefficient of
Uniformity(UCC).
Various irrigation quality parameters related to volume and area
quantities may be estimated when the distribution pattern is described by
the linear regression coefficients a and b.
The UCL model is easier to use than previously developed models or
uniformity coefficients used in the design of sprinkler irrigation systems.
Given a value for the irrigation quality b, from the UCL model, and a
chosen value for YR, the dimensionless precipitation depth required to over-
come total deficiency in the root zone, important parameters and efficiencies
can be calculated.
124
-------
SECTION 6
SPATIAL DISTRIBUTION IN TRICKLE IRRIGATION
INTRODUCTION
Trickle Irrigation is a method for applying water directly to the root
zone of a plant through an extensive above ground pipe network. Outlet
devices called tricklers discharge the water to the surface and are usually
at a designed spacing to allow for one or more tricklers per plant. A
typical trickle irrigation design will normally include several subunits
incorporating a manifold and several emitter lateral lines; a main line from
the water source; and a central head at the source, which includes water
measuring devices, valves, injectors, controllers and a filtering system.
Trickle irrigation has developed into a means for more localized and
efficient application of irrigation water. Tricklers along the lateral lines
act as pressure dissipation devices and point source applicators, such that
high frequency, small volume applications may result. This allows for an
irrigation method where 90 to 100 percent, depending on design considerations,
of the applied irrigation water is beneficially transpired by the irrigated
crop and little or none is lost to deep percolation or evaporation. Upon
discharge from the trickier, water is distributed through the soil profile
limited by constraints on horizontal flow in the soil. Present day design
techniques take this into consideration and result in designs which will
adequately wet the desired soil volume and at the same time reduce the losses
already mentioned.
An ideal trickle irrigation system would be one where there is a uniform
discharge from each and every emitter, resulting in uniform water application
throughout the crop. Design of such a system, however, is hampered by
several factors which reduce the possibilities of achieving uniform discharge
from each emitter. This will be discussed later. Presently, design tech-
niques also take these factors into consideration and may result in applica-
tion uniformities upward of 90 percent. High values of overall application
efficiency, Ea, (Karmeli and Keller, 1975) are also being realized. Ea is
defined as the product of the uniformity of application and the ratio of the
amount of water transpired to the amount of water applied to the least
watered areas. Thus, the main problems in achieving uniform water applica-
tion to the crop stems from the problem of achieving uniform emission from
the point sources along the laterals.
125
-------
FACTORS AFFECTING TRICKLE IRRIGATION QUALITY
Several factors affect emitter discharge rates and can cause considerable
variation. Studies to date have been primarily concerned with two, which
seem to have the greatest effect: 1) emitter characteristics and manufac-
turing variability of emitters, and 2) pressure distribution throughout the
pipe networks due to frictional head losses. Losses (or gains) due to
elevational differences have been studied and results from a level system
analysis can be extended to include these differences with relative ease, as
shown by Solomon and Keller (1975) and Karmeli and Keller (1975). Other
factors such as clogging of emitters and variation in water temperature in
the system affect the emitter discharge rates, but have not been as
thoroughly analyzed.
Emitters operating at a reference pressure head will not discharge the
same due to design emitter characteristics and manufacturing variations.
Karmeli and Keller (1975) and Howell and Hiler (1974) have proposed that
emitter flow is characterized by:
q = KdHX (123)
where q = emitter discharge rate, Jlph (gph)
K, = discharge coefficient characterizing each emitter
H = pressure head at the emitter, m(ft)
x = discharge exponent which characterizes the emitter flow regime.
For orifice and nozzle type emitters, x = 0.5, for long path emitters,
0.5 <_ x <_ 1.0 and for pressure compensating emitters, 0.0 <^ x <^ 0.5. For a
complete discussion of emitter flow regimes, see Karmeli and Keller (1975).
Manufacturing variations will disallow any set of emitters in having the
same K, value. These variations tend to be normally distributed about a mean
value. Karmeli and Keller (1975) have developed a parameter called the
manufacturer's coefficient of emitter variation which characterizes emitter
flow rates at a given pressure head as a normal distribution:
vm = a/y (124)
where v = manufacturer's coefficient of emitter variation
m
a = standard deviation of emitter discharges at a reference head
u = mean discharge of emitters at a reference head.
When there is more than one emitter per plant, a system coefficient of
variation, discussed by Karmeli and Keller (1975) and developed more exten-
sively by Solomon and Keller (1975) and Solomon (1977), results:
126
-------
v = e gm = v e-1/2 (125)
e u m
a
m
V = - —
e u
where v = system coefficient of variation
e = number of emitters per plant (e = 1 if 1 emitter is shared by
2 or more plants) .
Keeping in mind the normal distribution concept, it can be seen for a sample
set of emitters operating at a reference head, 68 percent will have a dis-
charge within +_ 1 standard deviation of the mean, about 95 percent will have
a discharge within +. 2 standard deviations of the mean and 98 percent will
have a discharge within ± 3 standard deviations of the mean. Solomon (1977)
states that values of v range from a low of 0.02 to a high of 0.40.
Emitter design characteristics, i.e. discharge exponent, x, and unit-to-
unit variations between emitters, have been shown to have definite effects on
trickle irrigation system efficiencies as related to application uniformities,
Karmeli and Keller (1975), Solomon and Keller (1975) and Solomon (1977).
However, a review of the analyses for characterization of pressure distribu-
tions within the pipe distribution network is in order before the magnitude
of these effects can be appreciated.
Several studies have been undertaken to characterize the pressure
distributions along a lateral line. On flat terrain, variations in pressure
are due to pipeline friction losses. Curves of pressure versus position
along a lateral have been developed and have shown that the general shape and
characteristics are the same and are practically independent of the emitter
characteristics and amount of head loss (Wu and Gitlin (1973) and Karmeli and
Keller (1975)). A general rule of thumb from sprinkler irrigation design
practices has been carried over to trickle irrigation lateral design, and
that is to keep the difference in discharge between emitters operating
simultaneously at a maximum of 10 percent. Depending on emitter character-
istics, this will allow for a 10 to 20 percent variation in pressure head
along the lateral.
Studies by Howell and Hiler (1974), Karmeli and Keller (1975), Solomon
and Keller (1975) and Wu and Gitlin (1973) have shown methods for estimating
to a reasonable degree of accuracy the pressure distribution along an emitter
lateral line. These studies have also elucidated several methods for
designing trickle systems once the pressure variation has been determined.
Wu and Gitlin (1973) showed dimensionless pressure curves for emitter lines
developed through iterative computations. From there, they developed design
techniques involving varying emitter orifice diameters, varying lengths of
microtube at the discharge points and varying emitter spacings to achieve
uniform applications. Unfortunately, use of different size emitters is
impractical due to installation and maintenance constraints. Also, uniform
spacing of emitters may normally be desired.
127
-------
Karmeli and Keller [1975), in their development of pressure head versus
position along a lateral line relationship, have shown interesting results
for all cases studied. They have shown that the emitter discharging the
average flow rate qa, operating at the average head, Hg, for the entire lateral
is located, in all cases, approximately 40 percent of the total lateral
length from the inlet end. Thus, most of the head loss occurs near the inlet
end of the lateral, where the flow is the greatest. This is for a uniform
size lateral. They have also shown that approximately 77 percent of the
total head is lost by this point. These results are based upon iterative
use of the Hazen-Williams equation for finding the lateral head loss:
j = K()- D-- (126)
where J = the head loss gradient, m/100 m (ft/100 ft)
12
K = constant, 1.21 x 10 for metric units and 1050 for English units
0 = flow rate in the lateral, &ps (gpm)
C = friction coefficient for pipe used
D = inside diameter of the pipe, mm (in).
From their results, design techniques were developed for design of laterals
to achieve the appropriate pressure distributions and thus good application
uniformity. This will be discussed in more detail later.
Solomon and Keller (1975) further developed these concepts and derived
two-dimensional equations which will characterize the pressure head anywhere
within a subunit consisting of a manifold and emitter lateral lines. Con-
sidering three types of manifolds: uniform size, tapered and one with
pressure regulators at the entrance to each lateral, they were able to fit
equations to the general head loss curve. They treated the manifold as a
type of lateral and characterized the flow from the manifold into any lateral
by:
Q = KD HX (127)
where Q = flow rate into the lateral, £ps (gpm)
Kn = discharge coefficient of the lateral
H = lateral inlet pressure, m (ft)
X = lateral discharge exponent.
For constant diameter pipe manifolds, the pressure head distribution anywhere
within a subunit can be characterized by:
128
-------
H(M, L) = [1.3E(IO - 0.3][E(M) H(0, 0) + (l-E(M)
(128)
1) + 0.3H(1, 0)][E(M) + (1-E(M, 0)]X/X
where H(M, L) = head at the relative manifold and lateral positions M and L
E(L) = fraction of the total head loss that remains to be lost from
L to the end of the lateral
E(M) = fraction of the total head loss that remains to be lost from
M to end of the manifold
E(L) or E(M) = exp[((L or M)/0.39)r ((1-0.39)/!- (L or M)))S ln(0.23)]
r = 1.19086 determined for best fit to
s = 0.0555 general head loss curve
X = lateral discharge exponent
x = emitter discharge exponent.
For completely tapered manifolds, the head loss per unit length is constant
and the pressure head distribution is :
H(M, L) = [1-.3E(L) - 0.3][(1-M) H(0, 0) + M*H(1, 0)] +
(129)
X/X
0.3 H(l, 0)][(1-M) + M]
For manifolds with pressure regulators at the entrance to each lateral,
H(M, L) = H(l, 1) + E(L) [R-H(1, 1)] (130)
where R = constant output head of the regulators.
In order to use equations (131) and (132), the ratio of lateral discharge
exponent X to emitter discharge exponent x, must be known. Solomon and
Keller (1975) analyzed the data of a graph showing X as a function of the
head loss ratio of a lateral for values of x presented by Karmeli and Keller
(1975). Their results found that:
X =
A ,,, 0.852 n 0.9
x + 0.62 x G
1.148
where G, = lateral head loss ratio expressed as lateral head loss divided by
the average pressure head in the lateral. G.. is very nearly
constant for all laterals.
Solomon and Keller (1975) continued by presenting histograms of subunit
pressure distributions for the three types of manifolds and uniformly spaced
129
-------
reference points. They also varied the lateral head loss ratio G and emitter
discharge exponent, x. Their findings indicated that the histograms were
nearly the same shape and were skewed towards the low pressure head end of
the subunit.
It has been fairly well documented that the pressure head distribution
within a trickle lateral is not uniform and doesn't follow a normal distribu-
tion. Thus, it is obvious to see that discharge variations will occur from
uniformly spaced emitters along the lateral.
Both Karmeli and Keller (1975) and Solomon and Keller (1975) have
presented extensive research results characterizing lateral pressure and flow
rate distributions. The combined effects of pressure distribution, emitter
characteristics and manufacturing variation were shown. Karmeli and Keller
(1975) presented graphs of maximum and minimum discharge ratios versus head
loss ratios for various emitter discharge exponents. Also presented were
approximate equations for these graphs so that estimates of the maximum and
minimum discharge ratios could be obtained. Solomon and Keller (1975)
presented several flow rate distribution histograms which delineated the
effects of varying the manufacturer's coefficient of variation; varying
emitter discharge exponents; and the combined effect of the two for a fairly
large head loss ratio. These histograms were derived with 1:1 ratios of
manifold loss to lateral loss. They show that flow rates from emitters are
normally distributed about a nominal discharge. An important conclusion from
their results is the fact that the manufacturer's coefficient of variation
can have an even greater effect on flow differences than pressure variations
within the pipe network.
EFFICIENCY AND UNIFORMITY CONCEPTS
Efficiency of applying water by trickle irrigation as proposed by
Karmeli and Keller (1975) is dependent upon two concepts. These are: 1) the
transpiration ratio, TR, the ratio of water transpired to the ratio of water
applied to the least watered areas, and 2) the uniformity of application or
emission uniformity, EU. From Karmeli and Keller (1975), characteristics of
the emitters which affect efficiency are:
1. variations in discharge rate due tp manufacturing variations
2. closeness of discharge-pressure relations to design specifications
3. emitter discharge exponent, x
4. possible range of suitable operating pressures
5. pressure loss in lateral lines due to emitter connections
6. susceptibility to clogging or fouling
7. stability of pressure-discharge relationship over a long time period.
130
-------
Important design criteria which affect efficiency are:
1. efficiency of filtration
2. permitted variations in pressure head allowed
3. base operating pressure used
4. degree of flow (or pressure) control used
5. relationship between discharge and pressure at the control head
6. allowance for temperature correction in long path emitters
7. chemical treatment to dissolve mineral deposits
8. use of secondary safety screening
9. incorporation of flow monitoring
10. allowance for reserve system capacity or pressure to compensate for
reduced flow due to clogging.
Good management is also a must to achieve high efficiencies. They have
defined application efficiency as:
E = (TR) (EU) (132)
d
The ratio of transpiration to application actually depends highly on
good management. Some excess water will be required for leaching and to
allow for a small margin of safety. Karmeli and Keller (1975) have suggested
a reasonable design value of TR = 0.90. Reliable scheduling procedures and
use of volumetric monitoring should be used to determine the proper
application volume.
EU is a concept proposed by Karmeli and Keller (1975) which takes into
account the uniformity of emitter discharge throughout a system. They have
noted that it is of primary concern since sufficient flow capacity to insure
adequate irrigation of the least watered areas is the most important objective
of irrigation system design. Excess watering can also cause problems, so
they also developed a concept called absolute emission uniformity, EU . EU
and EU are defined as follows: a
3.
EU = 100 x — (133)
1 qn qa
EU = 100 x 4-C— + ~ ) (134)
a 2^q q ' ^ '
Ma nx
where EU = emission uniformity, percent
131
-------
EU = absolute emission uniformity, percent
3.
a = average of the lowest -r of the emitter flow rates, £ph (gph)
q = average of all emitter flow rates, £ph (gph)
cl
q = average of the highest -5- of the emitter flow rates, &ph (gph).
1\. Q
Values of TR and EU are used in the design procedures as efficiency
concepts for computing the gross depth of application, the irrigation interval,
and the required system capacity. Recommended values of EU of 94 percent or
above are desirable and design EU's below 90 percent should not be considered.
An important design consideration which directly affects the quality and
safety of trickle irrigation systems is the percentage of wetted area, P.
Soil type and movement of water through the soil, plus lateral and emitter
spacings, affect the percentage of wetted area, P. Karmeli and Keller (1975)
give a complete discussion of this parameter and suggest for safety and
productivity reasons that P should not fall below a recommended design
minimum of 33 percent. However, in wide- spaced crops P should not be too
large, since many advantages of trickle irrigation depend on keeping strips
between rows dry.
Karmeli and Keller (1975) have developed design equations for EU and EUa
from nominal emitter discharge versus pressure head curves, which are usually
supplied by the manufacturer. These equations take into consideration
manufacturing variability and minimum and maximum discharge ratios. They are
as follows:
qn
EU = 100(1 - 1.27 v) — (135)
EU = 100(1 _ 1.27 v) j fcp + jp) (136)
Ha 4x
where EU = design emission uniformity, percent
EU = design absolute emission uniformity, percent
a _!
v = system coefficient of variation = e 2 v
a = minimum emitter discharge computed with the minimum pressure
using the nominal emitter discharge versus pressure head
relationship, &ph (gph)
q = average discharge of all emitters, &ph (gph)
3.
q = maximum emitter discharge computed with the maximum pressure using
the nominal emitter discharge versus pressure head relationships,
£ph (gph)
132
-------
With the use of these design estimates for emission uniformity and
relatively straightforward design techniques for choosing the proper emitter,
design of the laterals and manifold, and layout of the system, overall
designs which will result in the desired uniformities of application are the
end product. In trickle systems the uniformity of application depends
completely on the uniformity of emitter discharge throughout the system. The
design techniques outlined in Karmeli and Keller (1975) are based on pressure
variation and emitter discharge variation concepts such that a high degree of
uniform water application can be achieved.
Solomon and Keller (1975) applied the concepts of EU and EUa to their
research results of flow distributions in various subunit situations. They
generated tables of EU and EUa from their results and further substantiated
their findings concerning flow rate distributions. For instance, with low
values of lateral head loss ratio, G, and an emitter discharge exponent of
x = 0.0, which largely eliminates the head loss from consideration, subunit
EU values were found to be unacceptable if the system coefficient of emitter
variation, v, becomes too large.
DESCRIPTION OF SUGGESTED MODEL
A computer model was developed for the purpose of characterizing the
pressure head and emitter discharge distribution relationships along the
lateral for various conditions. Once these distributions are calculated,
least squares curve-fitting techniques are utilized to fit the distributions
to exponential or logarithmic type functions. The parameters of these
functions are then used in efficiency calculations in order to describe the
quality of an irrigation which would utilize the given lateral.
The model computes the head loss along the lateral starting from the
distal end with a known pressure head and emitter discharge assuming uniform
pipe size laterals on level ground. The discharge distribution follows
directly from the pressure distribution with the use of the emitter discharge
equation as described. The Hazen-Williams equation is used to determine the
head loss along the lateral.
Upon calculation of the head loss gradient,
D-4-87 (137)
\_i
where J = the head loss gradient, m/100 m (ft/100 ft)
12
K = constant, 1.21 x 10 for metric units; 1050 for English units
Q = inlet flow rate to lateral, £ps (gpm)
C = friction coefficient, usually 150 for plastic pipe
D = diameter of lateral, mm (in),
133
-------
The head loss along the lateral can be calculated at points on the lateral by
an iterative procedure. Two variations may be used involving different
required reduction coefficients due to outlets along the lateral.
The first variation, presented by Karmeli and Keller (1975), is used when
JLF ,C ,1.852
a C value is known:
e
AU
AH =
100 T
e
where AH = head loss for length of lateral, L, m (ft)
J = head loss gradient, m/100 m (ft/100 ft)
L = length of lateral considered, m (ft)
F = coefficient to compensate for discharge along pipe and is
dependent on the number of outlets
C = friction coefficient for lateral with emitters attached.
C
The second variation utilizes a reduction coefficient presented by Geohring
(1976):
AH =
0
where F(N ) = -0.63867(N -1-8916) + 0.35929
N = number of outlets.
o
With either of these variations it is possible to determine the pressure head
and thus the discharge distributions along the lateral at predetermined points,
i.e. the location of each emitter.
The model utilizes the second variation with the reduction factor
proposal by Geohring (1976), since Ce values are difficult to obtain.
However, the model can easily be adapted for use with the first variation if
a Ce value is known. A reference pressure head, i.e. 10 m, is assumed at the
distal end of the lateral, for instance, at the last emitter. The pressure
head and discharge distributions are then calculated working backwards along
the lateral from emission point to emission point. The sum of the emitter
discharges is thus the inlet lateral flow rate.
Once the pressure head versus location and discharge versus location
distributions are established, the data is fitted to both exponential and
logarithmic type functions. The exponential distribution is given by:
y = aebx (140)
where a, b = coefficients of the curve fit.
134
-------
The logarithmic distribution is given by:
y = a + b In x (141)
where a, b = coefficients of the curve fit.
The mean discharge of the emitters along the lateral is calculated
utilizing the a and b coefficients from the curve fit, depending on which
curve fit results in a higher correlation coefficient. The mean discharge
is then utilized to create a nondimensional discharge distribution. The
relative position along the lateral is formed using the emitter position and
lateral length. The model then fits the dimensionless discharge versus
relative location distribution to both functions. The mean discharge is also
used in design EU and EUa calculations as presented by Karmeli and Keller
(1975).
The dimensionless discharge versus relative location distribution is
used for efficiency calculations similar to those prepared by Karmeli (1977),
where a linear distribution is presented as a characterization of sprinkler
irrigation, and a dimensionless requirement is used in the efficiency calcu-
lations. Assuming the curve fits are good (case study results will be
presented showing the logarithmic curve fit to be best) the a and b
coefficients are used to calculate irrigation efficiencies. Figure 35 is a
general illustration of the fitted curve and will aid in describing how the
efficiencies are calculated.
A dimensionless ratio for an irrigation, YR, is defined as:
Y _ the volume required in the root zone to overcome the total deficiency
R ~ mean application
The efficiencies calculated are:
_ . ,. ,.,,. . volume of water put into root zone
E = water application efficiency = . . , ^- =-.—•=
a FF } total volume applied
r- ^ ,-,-. . volume of water put into root zone
E = water storage efficiency = —^ -= *- , , ,
s o j volume of water root zone can hold
The percent deep percolation, D , is also calculated:
D _ volume of water which percolates past root zone ,nn
p total volume applied
It can be seen that the area under the curve in Figure 35, ABCDEF is the
total volume applied and is equal to unity. When y . < yR < y , the volume
of water entering the root zone is ABDEF. Therefore^11 max
1.0
_ ABDEF ABDEF v v f ,-,-,, ,.,„..
Ea = ABCDEF = TTO~ = Vr + J fW dx C142)
X
r
135
-------
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136
-------
E » =
(yR)(D yR
n - ( i - r i - ri-F i fi44"i
p " (-ABCDEFJ100 ~ l 1 J100 " U VlOO U44J
When YR < Y • , i.e., when the volume of water required in the root zone
is' less than the volume applied to the least watered area, then Ea = YR, and
Es = 1.0, because it is implied that the root zone is completely filled. The
percent deep percolation is still Dp = (l-Ea)}oO> or Dp = (l-YR)ioO- When
YR > Ymax, i.e. the volume of water required in the root zone is greater than
the volume applied to the most watered areas, Ea = 1.0, because the total
volume applied enters the root zone. However, Es = Ea/YR or Ea = l.O/Y^,
because the volume of water entering the root zone is less than the require-
ment. The percent deep percolation now equals zero.
Thus, it is obvious that the efficiencies are dependent on the curve fit
correlation coefficient and the coefficients a and b.
A computer model was prepared to handle several trickle lateral
conditions, i.e. various diameters for a given length and a given emitter
spacing. Also, several values of the dimensionless requirement, yR, can be
input, such that a comparison of the resulting efficiencies for a given
lateral in a given situation is accommodated.
The model outputs are: the total lateral head loss; the head loss
gradient; the mean emitter discharge; the results of the curve fits for
pressure vs. location, discharge vs. location and dimensionless discharge vs.
relative location, plus these distributions; the design emission uniformity;
the absolute design emission uniformity; and E , E and D for various values
of YD (Appendix F). asp
K
Model Studies
Several trial conditions involving five different lateral diameters,
four different lateral lengths and two different emitter spacings were
evaluated by the suggested model.
1. Lateral diameters, mm: 9.4, 12.8, 16.4, 20.8, 26.8
2. Lateral lengths, m: 33.53, 67.10, 100.58, 201.15
3. Emitter spacings, m: 1.0, 2.0
Each lateral diameter was tried while the lateral length and emitter
spacing was held constant. A single-exit emitter with the following
characteristics was chosen:
q = 0.5471 H°'606
137
-------
v = 0.05
where q = emitter discharge, £ph
H = pressure head at the emitter, m
v = manufacturer's coefficient of variation.
The design of each lateral proceeded from the distal end of the lateral
where a reference pressure head of 10 m and the corresponding emitter dis-
charge was assumed. The appropriate distributions resulted, with approxi-
mately 77 percent of the total head loss occurring within the first 35
percent to 45 percent of the lateral. This doesn't hold true for the short
length, larger diameter laterals; however, these are extreme situations.
For all cases the results of the logarithmic curve fit were better than
those of the exponential curve fit, i.e. higher correlation coefficients
resulted. For 1.0 meter emitter spacings, the logarithmic curve fit gave
correlation coefficients ranging from 0.95 to 0.97, while the exponential
curve fit gave correlation coefficients from 0.85 to 0.89. For 2.0 meter
emitter spacings the logarithmic curve fit gave correlation coefficients from
0.93 to 0.95 while the exponential curve fit gave correlation coefficients
from 0.85 to 0.87. Figure 36 is an illustration of pressure head and dis-
charge distribution for a 12.8 mm diameter, 33.5 m long lateral with emitters
spaced at 1.0 m. The actual fitted curve is also shown. Figure 37 is an
illustration of the pressure head and discharge distribution for a 9.4 mm
diameter, 100.6 m long lateral with emitters spaced at 2.0 m. The fitted
curve is also shown. These cases represent the best and worst cases in terms
of the correlation coefficient for the logarithmic curve fit. In general, it
is possible to characterize discharge distributions on trickle lateral with a
logarithmic model. However, it may also be possible to achieve exact curve
fits for these distributions with a function of the form:
Y = c + aXb.
This, however, requires the addition of another parameter to describe the
distribution, plus a trial and error solution for the best fit function.
A summary of emission uniformity and efficiency calculation results with
YR = 1.0 is given in Table 20. The emission uniformity and absolute emission
uniformity were calculated using the minimum emitter discharge and the
maximum and minimum emitter discharges, respectively, in design equations
presented by Karmeli and Keller (1975). The water application efficiency,
water storage efficiency and percent deep percolation were calculated using
parameters from logarithmic curve fits for the'discharge distributions.
The EU and EUa values for each situation, except the longer length-
smaller diameter laterals, were relatively high and within acceptable limits.
This is due to the fact that each lateral was designed using the second
method: starting from the distal and working towards the inlet end. An
upper limit of EU = 93.65 percent and EU = 93.65 percent was observed in all
3.
138
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cases. This value is a maximum and is attributed to the manufacturer's
coefficient of variation for the emitter used. In this case, v = 0.05.
Relatively high values resulted for Ea and Es when Yr = 1.0. Other
values of YR also resulted in relatively high Ea and Es values. Also, the
percent deep percolation when YR - 1.0 was minimal. From Table 20 it can
be seen that although high values of Ea and Es may be the case, a good
irrigation may not be the result when the corresponding EU and EUa values
are studied, i.e. D = 9.4 mm, L = 201.15 m, SE = 1.0 m.
From Table 20, values of Ea from 0.99 to 1.00 indicate good water
application when the corresponding EU and EUa values are also included.
Values of Ea below 0.99 have corresponding EU and EUa values yc. or below 90
percent. It should be remembered that Table 1 is a summary of the results
when the irrigation just achieves completely refilling '.lie root zone or when
YR = 1.0. The results must be interpreted individualIv for different values
of YR. Thus, further study and development of this n.^\ -od for characterizing
the efficiencies of a trickle irrigation system is indicated.
142
-------
SUMMARY
A review of available literature has been presented which summarizes and
assesses the current state of the art of trickle irrigation for achieving
uniform application of irrigation water. Howell and Hiler (1974) and Wu and
Gitlin (1973) have developed methods for characterizing the pressure head
distribution along an emitter lateral line. Using results of their studies,
design techniques were developed. Howell and Hiler (1974) proposed varying
the length of lateral line to achieve uniform application. Wu and Gitlin
(1973) have proposed varying the emitter orifice diameter, varying a micro-
tube diameter or length at each point source, or varying the emission point
spacing in order to achieve uniformity of application.
Studies by Karmeli and Keller (1975), Solomon and Keller (1975) and
Solomon (1977) have presented "typical" head loss relations for emitter
lateral lines. Solomon and Keller (1975) and Solomon (1977) have shown the
effects of pressure variations and emitter variations on flow rate distribu-
tions within a system subunit. Karmeli and Keller (1975) have used efficiency
and emission uniformity concepts to develop design techniques which will
result in acceptable application uniformity. These design techniques take
into account pressure variations and emitter discharge variations in the
design of emitter lateral lines and manifolds.
The concept of emission uniformity, EU, which is used extensively in
design techniques developed by Karmeli and Keller (1975), gives an insight as
to the attainable efficiency and uniformity of application for a trickle
irrigation system. They have also presented field procedures for the evalua-
tion of EU for in situ trickle systems.
It is possible to achieve high application uniformities and efficient
use of irrigation water in trickle irrigation if proper design techniques are
utilized.
The proposed model suggests that the distribution of water in trickle
irrigation may be characterized by a logarithmic function of the form
(Y = a + b In X). In turn, the coefficients of this function are utilized
in establishing the quality of irrigation parameters: water application
efficiency, water storage efficiency and percent deep percolation. Caution
must be used with this model, since little account of soil wetting patterns
is taken into consideration.
143
-------
REFERENCES
Bassett, D. L., 1972. A Mathematical Model of Water Advance in Border
Irrigation. Transactions of the ASAE, 15(5):992-995.
Bassett, D. L. and D. W. Fitzsimmons, 1974. A Dynamic Model of Overland Flow
in Border Irrigation. Am. Soc. Agri. Eng. , Paper No. 74-2529.
St. Joseph, Michigan.
Bassett, D. L. and D. W. Fitzsimmons, 1976. Simulating Overland Flow in
Border Irrigation. Transactions of the ASAE, 19(4):666-671.
Chamberlain, A. R. and A. R. Robinson, 1960. Trapezoidal Flumes for
Open-Channel Flow Measurement. Transactions of the ASAE, 3(2):120-128.
Chaudry, Fazal H., 1976. Sprinkler Uniformity Measures and Skevmess. Journal
of Irrigation and Drainage Division, ASCE, 102(IR4), December.
Chen, Cheng-lung and V. E. Hansen, 1966. Theory and Characteristics of
Overland Flow. Transactions of the ASAE, 9(1):20-26.
Christiansen, J. E., 1942. Irrigation by Sprinkling. California Agricultural
Experiment Station Bulletin, No. 570.
Christiansen, J. E., A. A. Bishop, F. W. Kiefer, Jr., and Yu-Si Fok, 1966.
Evaluation of Intake Rate Constants as Related to Advance of Water in
Surface Irrigation. Transactions of the ASAE, 9 (5):671-674.
Griddle, W. D., S. Davis, C. H. Pair and D. G. Shockley, 1956. Methods for
Evaluating Irrigation Systems. Agricultural Handbook No. 82, Soil
Conservation Service, USDA.
Davis, John R., 1961. Estimating Rate of Advance for Irrigation Furrows.
Transactions of the ASAE, 4(1):52-57.
Davis, John R. and A. W. Fry, 1963. Measurement of Infiltration Rates in
Irrigated Furrows. Transactions of the ASAE, 6(4):318-319.
Fok, Yu-Si, 1964. Analysis of Overland Flow on a Porous Bed with Application
to the Design of Surface Irrigation Systems. Unpublished Doctoral
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144
-------
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Howell, Terry A. and Edward A. Hiler, 1974. Designing Trickle Irrigation
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146
-------
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147
-------
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Technical Report No. 13. Water Resources Institute. Texas ASM
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Problem. Journal of the Irrigation and Drainage Division, ASCE,
91(IR3):23-34, Proc. Paper 4471.
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Cutback Stream Sizes for Irrigation Furrows. Transactions of the ASAE,
12(5):634-637.
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Irrigation and Drainage Division, ASCE, 98(IR1):223-240, Proc. Paper
8764.
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Irrigation. Journal of the Irrigation and Drainage Division,
99(IR2):157-167, June.
148
-------
APPENDIX A
TECHNIQUES FOR DETERMINING
SOIL INFILTRATION CHARACTERISTICS
149
-------
APPENDIX A. TECHNIQUES FOR DETERMINING SOIL INFILTRATION CHARACTERISTICS
I. CYLINDER (RING) INFILTROMETER
A. Equipment Suggested
1. Cylinder(s) (a single cylinder for unbuffered tests; a cylinder
and buffer cylinder for buffered tests)
2. Means for measuring level of water in cylinder
3. Impermeable material to prevent infiltration while water is
introduced
4. Means of installing cylinder(s) in ground
5. Water
6. Forms (see Appendix B)
B. Procedure
1. Install infiltrometer with minimum disturbance of soil.
2. Cover bottom of cylinder with impermeable material to prevent
soil disturbance when water is introduced.
3. Place water in cylinder and record initial reading.
4. Read water level at succeeding times and record.
5. Determine cumulative infiltration depths with time from data.
C. Data Analysis
1. Plot cumulative depth infiltrated versus time
2. Determine best fit infiltration equation or use the plotted
data to determine cumulative infiltration at any time. If a
functional relationship is desired between cumulative infiltra-
tion and depth, the plotted data may be fitted to a Kostiakov,
modified Kostiakov, Philip, or other suitable equation. The
Kostiakov, modified Kostiakov or Philip equations may be
fitted as follows:
Kostiakov
a) Plot depth-time relationship on log-log paper. Use either
a least squares analysis or "eye-ball" the best fit line
to determine constants of Kostiakov equation.
150
-------
Modified Kostiakov
b) If the curve on log-log paper curves upward an equation of
the modified Kostiakov type may be better. This may be
accomplished by assuming values for the long time infil-
tration constant C. The values of (D-Ct) versus t are
plotted until a best fit line is obtained. From this best
fit line the values of A' and B' for the modified
Kostiakov equation are obtained.
Philip Equation
c) Values for the Philip 2 term equation may be obtained by
inserting the time and cumulative depth at two different
times into the Philip equation and solving the two
resulting equations simultaneously.
The infiltration rates at any time may be obtained by
differentiating the cumulative infiltration equations.
II. BASIN INFILTROMETER
A. Equipment
1) Metal sheets or wood planks which can be driven into ground or
dug into ground with minimum of soil disturbance to form a basin
2) Impermeable sheet to prevent infiltration and soil disturbance
during time water is introduced into basin
3) Means for measuring depth of water at succeeding times
4) Means for installing the infiltrometer
5) Water supply
6) Watch
7) Forms
B. Procedure--Same as Cylinder Infiltrometer
C. Analysis—Same as Cylinder Infiltrometer
III. VOLUME BALANCE METHODS
A. Christiansen, et al. (1966)
1) Assumptions
a) Water front follows a power relationship: L = a t
where L = distance of advance at time t
t = time of advance
b = exponent of advance function, i.e. arithmetic
slope of L vs. t on log-log plot
a = coefficient of advance function, i.e. intercept
of L vs. t relationship on log-log plot at t=l
151
-------
b) The infiltration rate of a soil can be expressed either
by the Kostiakov or modified Kostiakov equation
c) The average depth of water on the soil is a constant
2) Data requirements
a) Rate of advance
b) Mean depth of surface storage in borders or equivalent
depth in furrows (mean flow area/furrow spacing)
c) Inflow per unit width of border or inflow divided by
furrow spacing for furrows
3) Discussion and procedure
Writing the continuity equation as :
Qt = D WL + D WL (A-l)
or equivalently
qt = DsL + DaL (A-2)
where L = distance of advance in time
Q = rate of flow entering border or furrow, assumed
constant
W = spacing of furrows or width of border strip
q = rate of flow per unit width (Q/W)
D = average depth of water that has entered soil in
time t
D = average depth of water on surface. For furrows this
is the mean cross sectional area of flow divided by
the furrow spacing.
Assuming an infiltration equation of the Kostiakov type,
Kiefer (1959) determined that the average depth infiltrated at
time t during the advance phase could be given ats :
_ bKt_(n+1)rl_ 1
~ L (
__
a ~ n+1 b b b+1 2(b+2) •••
where K, n = constants of Kostiakov infiltration rate equation
(I = Kt")
If a factor F is defined as:
then D can be expressed as
3.
n _
a ~ (n+1) (n+2)
F can be approximated as :
152
-------
F . i (A_6,
Using the modified Kostiakov equation to express infiltration:
C t F K t"+1 Ct K tn+1 ( ,
Da - b+T + (n+1) (n+2) ' 1>02 F 2~ + (n+1) (n+2) (A'7)
Christiansen, et al. (1966) stated that the above equation could
be used for most values of b and n and t <_ 1000 minutes . Ds
may be obtained through actual measurements of surface storage
or estimated through empirical means such as those which relate
channel cross section, slope, inflow, and furrow roughness,
e.g., Manning's equation.
Fok and Bishop (1965) suggested that Ds can be
approximated as (Do)/(l+b) where Do is the depth at the head
of the border. Equation A-2 may then be written as:
and
D + D = D + , ., (A-8)
s a s (n+1) (n+2) *• J
D = F K t = 31 . D (A-9)
a (n+1) (n+2) L s l J
Thus
, if the plot of (7 -- D ) versus t results in a
Li S
straight line on log- log paper for the period of water front
advance, the slope of the line is n+1 and the intercept at t=l
F K
is ~? — TTT — oT • From the slope and intercept as well as
(n+1) (n+z)
equation A-3 or A-6, the constants of the Kostiakov
equation may be determined.
If a straight line does not result the authors suggest
that the modified Kostiakov equation may be established as
follows :
n F K tn+1 qt n Ct ,, ...
Da = (n+1) (n+2) = L - Ds - T ^~^
The best value of C is that value which yields the best
straight fit line of Da vs. t on log- log paper. The K and n
values are determined as for the Kostiakov equation.
Example of Procedure for Determining Constants of Kostiakov
Equations
The following example is taken from Christiansen, et al .
(1966) with all data converted to metric units.
153
-------
The advance data and associated Da values are given in Table
Al. If D (column 3) is plotted versus t to obtain n+1 and the
intercept at t=l, the following results:
TABLE Al. RATE OF ADVANCE AND REQUIRED COMPUTATIONS
FOR DETERMINATION OF K, n, AND F
L
(meters)
0
30.5
61.0
91.4
121.9
152.4
182.9
213.4
t
(rain)
0
13
42
90
158
239
333
457
3 D - •=— (m) Additional Required
L S Z
c = 0 Information
-0.00305
0.00603
,0.0116
0.0179
0.0246
0.0304
0.0358
0.0426
Q = .0227 m /min
W = 1.067 m
q = .0213 m /m-min
DS= .003048 m
L = 8.32 t'53
F K
= .00146 m
+ 1
b
F
K
i
.54
.53
1.18
.001029 m/min = 1.029 mm/min, therefore
1.029 t~-46 mm/min
It may be noted that at small times Da may be a negative
number. This is due to the assumption of constant surface storage
per unit length. In cases where the surface storage is over-
estimated the method may even produce negative values for
infiltration.
IV. "INFLOW-OUTFLOW" TECHNIQUE
The inflow-outflow method for estimating furrow infiltration
requires only inflow-outflow measurements with time over a section
of furrow. The following procedure is used:
A. Set two measuring devices in the furrow at a selected distance
apart. This distance is usually 30-75 meters, but in all cases
should allow for an accurately measurable difference in discharge.
154
-------
B. Measure and record inflow and outflow with corresponding times from
the selected furrow section. The difference in inflow and outflow
is the infiltration rate for the section under consideration.
C. Analyze the results as in Table A2.
D. Determine the best fit equation (Kostiakov, modified Kostiakov,
Philip, etc.) which will represent infiltration rate versus time
over the time of a normal irrigation, or use the actual data plots
to characterize infiltration rates with time. The cumulative
infiltration may either be obtained through integration or the
infiltration rate equation or through graphical integration of the
area under the infiltration rate versus time curve.
V. BLOCKED FURROW INFILTROMETER
A. Equipment—see Figure Al for example
B. Procedure Used for Blocked Furrow Measurement of Infiltration
1. Install infiltrometer (Figure Al) with minimum disturbance
of soil surface.
2. Cover furrow bottom with impermeable material; fill to a level
at which water normally flows in the channel (in Figure Al
tip of hook gauge is set at initial water level).
3. Remove impermeable material and maintain level of water at
initial depth (tip of hook gauge in Figure Al) by regulating
the inflow.
4. Measure depth of water in supply reservoir with time and record.
C. Analysis of Data
1. Obtain cumulative depth vs. time from data and convert
cumulative depth to volume.
2. Plot cumulative volume vs. time to obtain cumulative infiltra-
tion per meter of furrow length.
3. Fit Kostiakov, modified Kostiakov, Philip or other suitable
equation to the plot, or use the line connecting the points
to obtain cumulative infiltration at any time. Infiltration
rate if needed may be obtained either by differentiating the
equations or determining the slope of the cumulative
infiltration equation.
155
-------
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os
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9 9 LO
LO J O O
00-00 v— ' O M
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157
-------
APPENDIX B
EQUIPMENT, FORMS, AND SAMPLE DATA
158
-------
APPENDIX B. EQUIPMENT, FORMS, AND SAMPLE DATA
EQUIPMENT USED
1. Cutthroat flumes (one-inch
nominal throat width)
2. Trapezoidal flumes (small 60°V)
3. Level-Rod-Engineer Chain
4. Stakes
5. Flags
6. Profilometer
7. Soil auger
8. Soil cans
9. Hand level
10. Ring - 30 cm (I.D.) cylinders
with hook gauge and scale
11. Block furrow infiltrometer
(Figure Al) with accessories
including impermeable lining
for furrow bottom
12. Shovel
13. Sledge hammer and block of wood
14. Depth-width gauge with flat
2 cm x 5 cm bottom
15. Camera and picture
identification marker
PURPOSE
Measure furrow inflow-outflow
Measure infiltration by inflow-outflow
method
Measure elevations and distances
Station markers
Mark exact location where furrow
pictures are taken
Measure furrow cross section
Obtain moisture samples
Store moisture samples
Level flumes
Measure ring infiltration
Measure furrow ponded infiltration
Install flow measuring devices and
build up furrows
Install blocked furrow infiltrometer
and rings
Measure depth of flowing water and
top width of flow
Take profile pictures as presented by
profilometer. (Location, date, and
irrigation is indicated by a marker
next to profilometer)
159
-------
DATA FORMS AND SAMPLE DATA
The following data forms are presented:
1) Water advance/recession
2) Cylinder infiltrometer
3) Blocked furrow infiltrometer
4) Inflow-outflow data for determination of infiltration
5) Furrow inflow-outflow-surface storage
6) Soil moisture content data
Each set of data is identified in the upper left hand of the data form
as follows:
Location (or identification) F.., F2, I, F_, S
where F, = Farm (B = Benson, _G = Northern Colorado Research Demonstration
Center, H = Horticulture Farm)
F2 = Field number
I = Irrigation number
F_ = Furrow Number
S = Station
160
-------
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FUK3CW IKFLOW-CU7FLOW-SUSFACE STORAGE DATA
Location(F.F.I,7.3) B/1/5/5/X Length 625m Observer LJS
Soil Clay Loaro
Remarksl Start 5:54
Crop Corn
Furrow Spacing 1.342m
Date 7/19/77
Inflow = Flume #1 Outflow = Flume #4
fime
«in4
•*
£
'•out Q
time
station
d/W
;ime
^in
''out
;ime
station
d/W
13
0.25
1.04
start
26
0+00
3.0/33
0+25
1.7/28
0+50
2.3/29
0+75
2.5/28
1+00
2.2/27
1+10
0/0
270
.265
1.16
start
210
o+oc
3.0/30
o+25
1.7/30
6+75
2.0/30
1+25
2.0/26
1+75
1.5/26
2+25
1.6/26
2+75
1.5/23
27
0.267
1.17
end
28
292
.120
.28
end
230
3+25
2.0/25
3+75
2.6/27
4+2$
L.9/23
*+75
2.4/24
5+25
L.8/26
5+50
1.5/28
5+75
2.0/24
70
0.268
1.18
306
.125
.30
5+95
0/0
102
0.259
1.10
start
60
2+00
0/0
1+75
2.3/27
1+50
5.0/27
1+25
2.7/25
1+00
2.0/26
0+75
2.5/27
D+50
2.2/30
0+25
1.5/30
330
.125
.30
start
270
0+00
3.0/30
0+25
1.9/30
0+75
2.0/30
1+25
2.0/26
1+75
1.6/26
2+25
1.7/26
2+75
1.7/23
110
0.26^
1.16
end
70
0+00
5.0/3C
365
.258
1.10
end
300
3+25
[L.9/23
3+75
M/2?
4+25
2.5/24
4+75
L.9/24
5+25
L.7/29
5+75
W23
125
0.264^
1.15
^55
.265
1.16
169
C.267
1.17
start
125
0+00
5.0/30
6+25
L.6/30
C+75
L.7/27
1+25
^2/24
1+75
2.0/26
2+25
1.7/25
2+75
1.5/24
3+25
2.0/25
480
.138
.36
start
455
0+00
3.0/30
0+25
2.0/28
0+75
1.8/30
1+25
2.0/25
1+75
1.6/26
2+25
1.5/25
2+75
1.5/24
210
0.263
1.14
end
1^3
3+75
L.8/25
4+25
0/0
510
.269
1.19
end
480
3+25
2.0/24
3+73
1.7/28
4+25
Le/25
4+65
L.8/2?
5+25
L.7/28
5+75
2.0/25
230
618
.268
1.18
.147
.40
256
«•.
c.oo
0.00
start
169
0+00
5.0/30
6+25
L.7/30
0+75
L.7/27
l+i!5
2..0/25
1+75
L.8/26
2+25
1.8/25
2+75
2.0/2}
3+^3
1.9/25
264
«.—
0.10
0.20
end
185
3+75
•:.0/26
4+25
L.8/24
4+75
?.l/23
5+15
0/0
162
-------
Cylinder Infiltrometer
Location S/lA/2/1 t-PO-'i+4Q, cbscrvpr Ignacio Garcia
Date. 7/6/77
Crop
Kemarks: ^xte-nely low intake indicated by all ir.filtror.eters excej.t #2.
Cylinder # 2
time(min)
watch
diff
CUK
0
2
6
10
15
25
30
60
90
120
180
240
200
335
infiltratior.(cm)
depth
13.20
-13.09
12.91
12.80
12.65^
12.^5
12.35
11.80
11.50
11.10
1C.V>
10.00
9.55
9.25
diff
.
cum
0
.11
.29
.40
.55
.75
.85
1.4C
1.70
2.10
2.80
3.20
• sp
3.95
Cylinder # 3
time(nin)
watch
diff
CUE
0
1
2
5
10
20
30
oO
90
120
180
240
298
inf iltration( en)
depth
7.13
7.07
7.05
7.02 j
7.02
6.95
6.90
5. So
5.70
6.65
6.60
5.55
5.50
diff
— r- '
CU.TI
0
.06
.08
.11
.11
.18
.23
.33
' .-+3
.48
.53
.58
,63 ,
163
-------
Location
Soil type Clay Loam
Blocked Furrow Infiltrometer
B/l/5/2/0+25^ Observer Hart
Crop Corn
Pate 7/18/77
D (cm /m)=Depth(cumulative)xA
Remarks: 1 day before irrigation 5. Blocked furrow plates 1 meter apart.
A—Cross-sectional area of cylindrical supply reservoir
Infiltrometer # 1
time(min)
watch
8:30
—
«
8:37
8:40
8:45
8:50
8:55
—
—
8:60
9:05
9:10
9:15
9:20
9:25
9:30
9:40
—
10:05
10:30
diff
Refi^
cum
—
' —
7
10
15
20
25
~
—
30
35
40
45
50
55
60
65
--
95
120
infiltration(cm)
depth
11.3
25.4
10.1
11.5
13.1
15.5
17.1
18.5
18.9
12.4
12.9
14.3
15.2
16.0
17.1
18.0
18.7
20.0
20.0
12.0
15.9
19.4
diff
14.1
0.0
1.4
1.6
2.4
1.6
1.4
0.4
.0
0.5
1.4
.9
0.8
1.1
0.9
0.7
1.3
0.0
0.0
3.9
3.9
cum
0.0
_-
__
15.5
17.1
19.5
21.1
22.5
—
23.4
24.8
25.7
26.5
27.6
28.5
29.2
30.5
—
—
34.4
Infiltroraeter # 1
time(min)
watch
11:00
11:30
---
-_
12:00
___
—
1:03
—
—
2:10
3:10
4:10
tiff
ejim
150
180
— «.
__
210
__
—
273
—
—
340
400
460
infiltration(cm)
depth
23.0
26.7
26.7
21.3
24.3
' 27.0
10.2
14.9
20.0
10.7
14,0
19.5
27.2
t
diff
JL-'L.
0.0
0.0
_!.£_
2.7
0,0
*tZJ
5.1
' 0.0
3.3
_S;L5_
7»7
cura
41. 9
45.6
•«P
*.«
48.6
__
--
56.0
—
—
64.4
69.9
77.6
I :r_tj , .
164
-------
y.
§*
O A
-P
S o
O S
ILTRAT]
utflow
SI
3
O
OJ
rA
CO
»
M
•>
F*
f*T
s^
g
•H
"S
o
•H
03
fl)
\
•H
H
ti a>
o B
•H ^
JJ
it! R
rl 0
•P -H
rH -P
•H eO
a X) OO
(N O^
H rA
OJ IA
C^ ON
H rA
OJ IT
tN O--
HtA
ON OJ
i£> ON
rnrA
to ON
to OO
rH IA
rH IA
toco
rH tA
$8
rH tA
rH IA
^J- to
rH rA
S3
rH tA
88
o o
I
1
t
-p
J
O"
x-
H
1
II
S
OJ
H
tA
0
to
OJ
OO
oj
0
rA
OJ
OJ
rA
IA
0
J-
a
•o
tN
OJ
CN
OJ
CN
OJ
IN
OJ
OJ
UA
OJ
10
OJ
UN
tD
OJ
tN
OJ
IA
OJ
cs:
EH
-
uj
O
1
IA
OJ
ON
OJ
CO
OJ
O
*
IA
O
tA
fA
^
•
KN
0
*
8
o
1
JS
4->
p<
0)
13
«
OJ
OJ
[N
Oj
IN
OJ
OJ
f\l
t\l
S?
6
!
H
i!
?
O
II
O
H
00
ON
OJ
s
H
S
O
S
H
•
O
H
OJ
S
ON
OJ
1
A
•P
a.
-------
0>
+>
rt
Q
fcO
•rl
01
4J
S-,
p<
o
PH
c
o
•H
4i
05
O
•rl
& a
O
to
Moisture
depth
(mm)
%
Moisture
(vol)
x •
H a
3 V
m -a
%
Moisture
(wt)
_ . . I
Moisture
(g)
H S-->
0*3^-^
«
« -rj *-»
C d >, to
nj a b ^^
M -3
Tare +
soil
Moisture
(g)
« ^
C bo
S^
J.
g
J3^
U8
4) ««^
«
ttlfe
5
<0
"~" -"
166
-------
APPENDIX C
COMPUTER PROGRAM FOR FURROW IRRIGATION MODEL
167
-------
APPENDIX C. COMPUTER PROGRAM FOR FURROW IRRIGATION MODEL
EXPLANATION OF PROGRAM*
The program is constructed in accordance with the flow chart of Figure 5.
The primary symbols for the various parameters are defined at the beginning of
the program in alphabetical order. Comment cards are inserted into the program
to facilitate the users understanding.
In addition to the selected quality parameters discussed in Chapter 2,
the program output indicates advance times and distances along the run. It
also indic?tes the distribution of infiltration in each tenth of the field
along the run.
Recession is assumed to be instantaneous at the end of irrigation.
*Note: If an estimate of surface storage (CM) is to be calculated within
the program, CM must still be fed in as zero. If a nonzero value
for CM is read into the program then SSF, STC, STE, MN, and S are
not read in. If CM is to be calculated within the program then the
surface storage is calculated as CM = SSF x STC x (PM)STE. The
S'*
user is referred to the section on data analysis if SSF, STC, and
STE are to be used in the program.
168
-------
PROGRAM IRFU?R(INPUT,OUTPUT, TAPE5=INPUT, TAPE6«OUTPUTI
C...A,B,C VALUES FOR CUMULATIVE INFILTRATION EQUATION—Z.»A»T»»8»C»T«
C... Z..CM«3 PER METER OF FURROW LENGTH
C....ACOF.BCOF —ADVANCE POKER FUNCTION CONSTANTS-X*ACOF»T»»8COF—OBT AIMED
5 C... BY FITTING VALUES OBTAINED BY HILKE-SNEROON MODEL TO A
C... POWER CU?VE
C. .ACOF1.RC3F1.CC3F1-VAI.UES FOR OIMENSIONLESS DISTRIBUTION CURVE
C... Y*;COFl*ACOFl»XXo«1COFl
10 r°IlAP,8P—COEFFICIENT AND EXPO-'NT OF ITILTRATION EQUATION OCTAINED BY
i,... FITTING POINTS FROM L-t, ,ATIVE INFILTRATION EQUATION TO A
C.. POWER CURVE rOR USE IN HILKE-SMEROON ADVANCE MODEL — IE POINTS
C.. FROM 7= A»T»»3 » CM ARE FITTED TO OBTAIN 2 = AP»T»»BP
C...APRAT(I)*VOLUMEOF HATER INFILTRATED IN SECTION I OF FIELD/VOLUME
15 C... REQUIRED TO FILL *OOT ZONE IN SECTION I— (1*1,10)
C...CM 'AVERAGE CROSS-SE3TIONAL FLOW AREA—C£NTIMETERS*»2
C...DEFVOLT--TOTAL DEFICIENT VOLUME IN ON£ FURROW LENGTH
C...DEFVOL(I» —DEFICIENT VOLUME OF INFILTRATION BETWEEN POINTS X(II,X
C...DEPVL —VOLUME INFILTRATED PER METER OF FURROH LENGTH AT POINT I
20 C...DTA — MEA* SOIL MOISTURE DEFICIENCY BEFORE IRRIGATION(MILLIMETE*S>
C...DTAV—VOLUME REQUIRED BY ROOT ZONE TO REACH FIELD CAPACITY-M»»3 PER 1ETER
C... OF FURROH LENGTH PER FURROH SPACING
^..EA'-APPLICATION EFFi:iENCY=VOLRZT/QTOT X 100
,...ES—HATE* STORAGE EFFICIENCY*VOLRZT/VR£0 • 100
25 C...LENGTH LENGTH OF FJRROH—METERS
-•-...LNGlDsSEGMENT OF FIELD UNDER CONSIDERATION—WHEREI-1,10
C...MN—MANNINGS N AT HEAD OF FURROM UNDER CONSIDERATION
^...NCI=NO. OF INTAKE EQJATIONS UNDER CONSIDERATION
C...NOE*NO. OF DIFFERENT VALUES OF DTA CONSIDERED
33 C...NLE»NO. OF FURROH LENGTHS UNDER CONSIDERATION
C...NQU=N3. OF FURROW STREAM FLOHS UNDER CONSIDERATION
C'. ..NSL*NUMq-R OF SLOPES UNDER CONSIDERATION
C...3 —INFLOW INTO FURRO* — LITERS PER SECOND
C...QN3M=FUR*OW INFLOW--1ETERS«3/MIN
35 C...QTOT—TOTAL VOLUME APPLIED TO FURROW — METERS" 3
C...POP—PERCENT OF TOTA. APPLIED HATER THAT DEEP PERCOLATES
C...PTH--PER:ENT OF TOTA. APPLIED W»TE° THAT is TAIL HATER
C...S=SLOPE
C...SMD—MEAN SCIL MOISTJRE DEFICIENCY REFOPE IRRIGATION—MILLIMETERS
i»0 C...SSF SURFACE STORAG- FACTOR IE. HEAN X-SECTIONAL AREA OF SURFACE STORAGE
C... OUUNG AOVANCE/SURFACE STORAGE AT HEAD OF FURROW
C...STC.STE —COEFFICIENT AND EXPONENT OF EQUATION HHICH RELATES SURFACE STORAGE
C... TO MANNINGS N, DISCHARGE,ANO SLOPE OF FURROH—FURROW PROFILES
C... MUST BE EXPRESSIBLE AS PARAqQLAS-STE.STC ARE FUNCTIONS ONLY OF THE
itS C ,. PARABOLIC DIMENSIONS OF THE FURROH SO THAT AREA ANO WETTED
c... PERIMETER A*E KNOWN IF DEPTH is KNOWN
C...T.X—TIME(MIN) V XCOISTANCE ALONG FURROW IN METERS)
S...TIMSET—TOTAL TIME WATER FLOWS INTO HEAD OF FURROW
C...TINF—TOTAL TIME WAT-R INFILTRATES AT POINT I
50 C...TRE—TIME REQUIRED FOR WATER TO REACH END OF FURROW
C...VDEF10«II=VOLU1E OF OEFICIENCY IN SECTION I OF THE FIELO<«oo FTM vs.o-P365 OPT-I wdr/T? is.oi.13.
60 C. ..H —SPACING OF FURROWS—METERS
REAL L£NCTH,LNG ,HN
DIMENSION S«10I,AP?AT(10), nTA(10»,Q(10),LENGTH(50).
IVOB(350 I. TINF(2SO),X(250),T(250> ,OEPVL(250),OEFVOL(Z50),
2 A(10 I,3(10),VDP10(10),VOF10(10),LNG(10»,VRZ(10). C(IO)
65 it.xxtZQ) .TPK10) ,OPI (101 , YFI100I , XN (100 I , TT (100)
1 ^EAD(5,19) W
19 FORMAT(F10.3I
IF(W.EQ.O.O) CALL EXIT
REAO (5,2) CM
70 2 FORMAT(F10.6)
IF(CM.JT. .01) GO TO "tl
READ (5,26) MN
26 FORMAT(FIO.'J)
REAO(5,27)STC,STE
75 27 FORMAT<2F10.
-------
60 FORMATI5I5)
READ (5,6".) (A( IC),B< Id ,C(IC),IC»1,HCI)
61. FORMAT<6F10.«.)
READ (5, 62)
70 FORMAT(5F10.2>
90 REAt)(5,6S) (IKiai ,I3 = 1,NQU)
68 FOR*AT<6F10.3)
C t • t
HRITE(6,<»2> X
42 FORMATUX,»H=»,F1C.3)
95 HRITE<&,<»<.> SSF,ST:,STE
1.". FOf9.C(IC»,IC*l,NCII
81. FORMAT! 1X,*A( IC),8(IC),C(IC)*,F10.D,Fia.3,F10.1)
MRITE(6,»2> (SCSI .IS = 1,S'SLI
135 t? FO?HAT(lX,»SUOPE*,5Fl3.2)
1113 KRITEI6.88I ((JCIQ),IQ = 1,NQUI
88 FO»MAT(1X,«I3( IQ)»,i,X,6F10.3l
C...
00 59 IC=1,NCI
00 59 IS-l.NSL
115 TFIT=TI1SET/!»0.
DO 18 1:1,10
TPim = TFIT»FLOAT(I)
OPKIOAI ICI«TPI(II»*8(IC)»C( lO'TPKIl
OR03CAM IR.FUR9 COC b<.CO FTN V3.0-P365 OPT*i 09/OV/7 18.ul.li.
38 CONTINJE
120 CALL FIT ( TPI, DPI.10 , C. 0 .AP,BP,9P2 »
19 FORMAT(1X,»MOCIFIE3 A ( 1C) 3 ( 1C ) , , R»*2 AR£»,F10, i) , F10, Z ,F1 0.3)
COFs.6ir«6-.li.215*(ALOG(DP) I
APrAP/lOCOOOO.
125 00 59 10*1, tOE
00 59 IQ=1,N3U
IF(CH.LE..005I CM=SSF»STC*(HN»Q(I(;)/S(ISI'».5|»»ST£
WRITE(6,10) CH
10 FORMAT(iX,»SURFACE STORAGE', F10 .<» I
110 DO 59 IL'l.NLE
OELL=LENr,TH(IL»/10.
0£LX'L£NGTH(ILI/<.D.
QM3H=Q(IQ)». 001*60.
US CM2*C«/1COOO.
TRI=TI1SET
TRI2*2.»TRI
12
DELN»T?I/30.
OELfsOELN
6 CONTINJE
11.5
C... CALCULATE TIME VS DISTANCE USING WItKE-SHERDON EaUATION—OIVinE TIMSET
C 3r 30 AND CALCULATE AOVANCt DISTANCES UNTIL F.NO OF FIELD 15 REACHED—
C THIS IS INITIAL TRE — OIVIOE TRE flv 1.0 AND CALCULATE ADVANCE DISTANCES
150 z AGAIN TO OBTAIN MORE POINTS ON ADVANCE CURVE
7 J-J»1
KOUNT'KOUNT*!
T(JI«TU-1)*3ELT
OIOP»AP»T(J»»»BP/C12
155 X(J)s(JM3M»T( J» )/ C
IF(X(J).GE.LENGTH(ILI) GO TO 8
IF G3 TO 56
IF(T(J) .GE.2880.) SO TO 56
170
-------
SO TO 7
1*1 8 T*r»T(J>
DEL1-T?E/t.O.
IFnELT.fQ.OtL1) 63 TO
IFIBELr.E1.atLN) 0€LT«
IFnELT.EQ.OttH) G3 Tfl
1M ^ CONTINJE
IF«OUNT.GT.S C) KOJNT
«,?'n.i)
3D CONTIIUE
ccor=o.o
IRFU»<» COC »i»cn FT* V1.I-PM*
C...FIT «T»»NCE POINTS T3 »o»e^ cum
CALL FITITT.XM.KNT, ccoF,*coF,f!COF,»ii
21
1 TI>(F(1( 'TIlSlT
C...CALCUL«T£ UIST^ISUTIDNS Of I .LT.LENGTH(IL) I GO TO S
C. ..CALCUL»T£ TOTAL AMOU^TSOF V0° T, IFF V CL T . QTO T
I in>«TIMSET«. ttl'bl.
.CALCULATE DESHET t Fc ICI ENCI ?S ANT »E9Ct.N T4GES
FA«VOL*?T/1TOT»10C.
ES«( (V<£0-OE- VOLT I/VRE1I •100.
t. CALCULATE VOLJ^E OF 3EFP PgoCOL AT IDH, OEFJ £1 E»C», AS MIL A$
C IN tACH TENTH OF FIfLO
»D^1CI1I= j.O
VOF10I1) «C.3
LNG(l) >OELL
f«L>«VOPia (Nil t-VCP(t)
GO TO 3fc
3S CONTINJE
IF(X(LI.GF. LNG(NLI)60 TO 13
50 TO I*.
Hi 13 NL«*L*1
IFCX(L) .GE.LENSTMdLI ) GO TO
LNG»3ELL
GO TO I*
H COHTINjr
CDC
171
-------
00 16 NN=1.10
APRATINNIMlHZtNNHVOPlQCNNI-VOFlOlNNM/VRZlO
16 CONTINJE
HRITEt&,52>
52 FORMATC1X, • I S 0 L Q VREQ OEFVOLT QTQT V3LRZT
1QRUNOF VOPT ES PTW POP EA »>
WRITE (6,51) IC.IS,IO,IL,IQ,VREQ,OEFVOLT,O.TOT,VOLRZT,QRUNOF,VOPT,ES
1,PTM,PDP,EA
51 FORMAT(lX,5I3tF9.3, 5F10 .2, ,I'l ,10 I
22 FORMAT( 1X,*APRAT(I)»,10F10.2)
56 CONTINJE
59 CONTINJE
GO TO I
270 END
SUBROUTINE FIT I X ,Y ,NPR,CCOF, ACOF ,BCOF,R2I
C...THIS SUBROUTINE FITS POINTS TO A »OWER CURVE-r=ACOF»X»»8COF- AND CALCULATES
C... CORRELATION COEFF ICI- NT-R*"Z
DIMENSION X(200I , Y< 200 ) ,XLN(200 ) , VLN(20l) )
5 SLNX'0.0
SLNY=C.O
SSLNX^O.O
SSLN*=J.O
SLNXY=S.fl
10 C...
00 "»2 1 = 1, NP?
XLN(I)=«LOG(X (III
X = SLNX»XLN(I)
15 SLNYsSLNY»YLN(I)
SSLNX=SSLNX»XLN(I)'»2
SSLNY=SSLNY»YLN/
-------
Ml 1.51.2
SSF.STC.STE .77 131.63
MN* .01 I
NCI* 1 NSL= 1 NDE= 1 NL£ =
A(ICI ,H(ICI,C( 1C) 571,7.
.73
NQU= 1
SLOPE(IS)
OTC(ID)
LENGTHULI
UdQI
MODIFIED
SURFACE STORAGE
.CQi.5
121. tO
635.00
1.160
) ,,R2 ARt
68.6
38.1671
ADVANCE TIME
6.91
13.83
20.73
27.6",
3C .992
AOVANC£ DISTANCE
1.9.31
82.59
110.90
136.03
159.21
180.16
201.38
220.71
339.30
257.18
27<».i»3
291.12
307.33
323.09
338.1,5
353.!. I.
368.10
382.1.*,
396.50
1,10.?8
".23.82
1,37.13
<,50.20
1.63.08
1,75.75
1,88.31.
500.56
512.70
521,.i9
536.52
51.8.31
559.76
571.17
A,COF,BCOF,R2 13.699
I S 0 L Q VREQ
11111 123.360
CCOFl,ACO-l,BCOFl
2M.85 593.62
21.8.76 601..66
255.67 615.59
.684 1.000
OEFVOLT QTQT VOLRZT
83.67 1,8.09 39.69
.Ib2 .IH. .892 .996
VRZfl)
VOFIO(I)
APPAT(I)
OFPC13S6.
12,336
3.000
.J6
12.316
0.000
7.9<,3
.36
13.336
0.000
8.038
.35
30000S PAGES P
-------
APPENDIX D
I. PREDICTING SOIL MOISTURE DEPLETION
FROM CROP AND CLIMATIC DATA
II. SOIL MOISTURE RESULTS
174
-------
APPENDIX D. I. PREDICTING SOIL MOISTURE DEPLETION
FROM CROP AND CLIMATIC DATA
INTRODUCTION
The approach to estimating soil moisture depletion as used by Heerman
and Kincaid (1974) requires knowledge of climatic conditions, crop conditions,
and antecedent soil moisture conditions. The soil moisture depletion is
established by:
1. calculating the potential evapotranspiration (E ) of a well
watered reference crop using the Penman equation^ and knowledge of
climatic conditions,
2. multiplying Etp by coefficients which depend on crop, crop stage,
and antecedent moisture,
3. doing a moisture balance to determine the soil moisture depletion
based on crop water use, evaporation, rainfall and irrigation
amount to determine the soil moisture depletion on day i.
The procedure is explained in the following section.
USE OF PENMAN EQUATION TO ESTIMATE POTENTIAL EVAPOTRANSPIRATION
The Penman equation as used by Heerman and Kincaid (1974) is:
E.n - 0.000673 {C,(R + G) + 15.36 C_(A, + b,U9) (e o - e,) (D-l)
up J.T1 Z o O Z Z CL
The terms are defined and expanded in the following:
E = Potential evapotranspiration from a well watered
™ reference crop in inches/day
Cl "
C.. - is a coefficient dependent only on temperature
A - slope of saturation vapor pressure- temperature line
Y - psychrometric constant
C2 = 0.959 - 0.0125 T + 0.00004534 T2 \l 3)
175
-------
T - the mean daily temperature for the day or period for which the
equation is used.
Rn = (1 - a) Rs - Rb (D-4)
R - net radiation in cropped area
n rjr
a - shortwave reflectance or albedo. Although this coefficient varies
from 0.20-0.25 depending on the crop, it is usually taken to be
0.23 for calculation purposes
R - mean solar radiation in langleys/day -- this is a measured quantity
usually — if R data are not available, R may be estimated as
follows: (Jensln, 1973) S
R = (0.23 + 48S) R (D-5)
S A.
S - ratio of actual to possible sunshine
R. - extra-terrestrial radiation, or
r\
R = (0.35 + 0.61S) R (D-6)
^ J \J
R - cloudless day solar radiation received at earth's surface
Rb - (*1 (RS RSO + b -
R, - net outgoing thermal radiation
a1 , b.. - constants depending on region (arid, semiarid, etc.)
(Jensen, 1973)
Rbo = £oT4 = (-0.02 + 2.61 exp[(-7.77 x 10~4) (273. -T) 2] 11. 71 x 10~8T4 (D-8)
Rbo - net outgoing radiation on a clear day (longwave)
t, - emissivity constant
-8
a - Stefan-Boltzman constant = 11.71 x 10
T - mean temperature in "Kelvin
(T. - T ) .100
G = — — - i~ - (Jensen, 1973) (D-9)
G - average daily soil heat flux (langleys/day)
T. - mean air temperature in °C for period i-1
176
-------
T. - mean air temperature in °C for period i+1
At - time in days between midpoints of two periods
a?, b_ - constants for wind term for a given location (Jensen, 1973)
U_ - windspeed in miles/day at 2 meters in height; if windspeed at
2 meters is not known, use the following formula:
U2 = Uz(2/z)°'2 (D-10)
where z = elevation of known windspeed
e (t ) + e ft . )
o z*- max' zVmiir n^<.* 1077^ r^ 11^
e = * (Jensen, 19/3) (0-11)
Z 2
e - mean saturation vapor pressure
e(t ) - saturation vapor pressure at maximum temperature
e(t . ) - saturation vapor pressure at minimum temperature
e at any temperature is given by:
ez(t) = -0.6959 + 0.2946 T - 0.005195 T2 + 89. x 10"6T3
(Heerman and Kincaid, 1974) (°-12)
e (t) - saturation vapor pressure at T °F in millibars
e, = e x RH or e (t min) x RH at minimum temperature (D-13)
e, - mean dewpoint temperature
RH - mean daily relative humidity
OBTAINING SOIL MOISTURE DEPLETION WITH CROP, SOIL, AND CLIMATIC CONDITIONS
The determination of daily water use by a specific crop requires the use
of a crop coefficient (Kco) which is defined as the ratio of the crop ET to
the potential ET for a given day under well watered conditions. This
empirical coefficient varies with time depending on type of crop, planting
date and effective cover date or expected date of peak use. K is given as:
K = Ar3 + Br2 + Cr + D (D-14)
where A, B, C, D = constants which change once effective cover has been
reached (Jensen, 1969; Jensen, et al., 1970, 1971).
177
-------
r = before the effective cover date, r is the fraction of time from
planting to effective cover. After effective cover date, r is the
number of days beyond the effective cover data.
The daily crop evapotranspiration depends on the soil water availability,
and decreases as soil moisture deficiency increases. The crop coefficient
K must be multiplied by a stress factor (Ks) in order to account for the
decreased evapotranspiration under stress. K is given as:
log[(l + 100) (1 - D /D )]
K
s log(lOl)
where D - soil water depletion
P
D - total available water within the root zone at field capacity.
The effects of Kco and Ks are combined by multiplying them to obtain a new
coefficient (K ) .
Immediately following an irrigation or rainfall, additional water use
because of evaporation from the soil surface occurs. This additional water
use, because of evaporation on a given day, is determined by the equation:
Etr = V0'9 - y Etp (D-16)
where K = a coefficient equal to
0.8 the first day after a rain or irrigation
0.5 the second day after a rain or irrigation
0.3 the third day following a rain or irrigation.
If Kc = 0.9, or if no rainfall has occurred within three days, then
E is equal to zero.
The water balance equation, for a specific crop, on a specific day, is:
D = D + K E. + E. - R. (D-17)
p. p(._i;) c tp tr i
where D - the depletion on day i and
pi
R. - sum of irrigation and rainfall for day i.
Using the above equations with crop, climatic, and soil factors known,
an irrigation budget may be worked out for the irrigation season.
178
-------
II. SOIL MOISTURE RESULTS
TABLE Dl. SOIL MOISTURE HISTORY FOR FIRST FIVE IRRIGATIONS
Irrigation
1
2
3
4
5
Date
June 4
June 15
June 28
July 7
July 19
D (mm)
"i
40
44
66
67
82
D (mm)
*cum
40
84
150
217
299
Dap.
44
43
47
41
41
Dap
rcum
44
87
134
175
216
SMD*
b
40
44
67
87
128
SMD
a
0
1
20
46
87
D = mean soil moisture depletion from crop and climatic
"i data to irrigation i (using Penman equation)
D = mean cumulative soil moisture depletion
P
cum
Dap. = mean depth applied in irrigation i
Dap = mean cumulative depth applied
SMD, = soil moisture deficit before irrigation
SMD = soil moisture deficit after irrigation
cl
*Assume deep percolation only if SMD, < Dap. and assume water use
uniform throughout field.
179
-------
TABLE D2. SOIL MOISTURE RESULTS FOR IRRIGATIONS
4 AND 5 FROM SOIL MOISTURE SAMPLES
Station
(meters)
0+00
1+00
2+00
3+00
4+00
5+00
6+00
*Moisture
Symbols:
Level
(cm)
0-15
15-30
30-60
60-90
90-120
0-15
15-30
30-60
60-90
90-120
0-15
15-30
30-60
60-90
90-120
0-15
15-30
30-60
60-90
90-120
0-15
15-30
30-60
60-90
90-120
0-15
15-30
30-60
60-90
90-120
0-15
15-30
30-60
60-90
90-120
samples tafcen
Percent Moisture* (wt) for Irrigations 4, 5
4B
25.2
25.7
28.3
29.3
29.5
25.4
27.7
34.0
24.2
26.9
23.4
21.7
27.1
27.2
26.3
21.5
20.4
20.8
26.6
22.1
18.7
19.5
20.4
23.8
18.7
18.2
18.8
22.5
26.9
19.2
21.5
19.7
22.9
26.3
20.3
one day before
4, 5 = irrigations 4 and 5,
B = before
A = after
irrigation
irrigation
4A
30.2
29.6
33.0
31.4
32.2
29.8
32.6
27.9
25.0
24.6
30.5
29.8
30.6
25.3
24.8
26.2
26.4
23.6
26.2
21.7
26.1
26.5
19.5
23.7
15.9
26.4
26.2
25.0
23.8
15.6
33.6
26.9
28.3
26.5
16.8
irrigation
5B
23.4
25.5
50.0
28.6
28.7
23.5
26.8
24.6
21.2
25.3
22.3
22.9
26.9
22.9
24.0
17.8
17.9
19.4
24.8
17.0
15.8
18.4
20.2
21.2
10.5
13.8
16.0
22.5
19.8
10.8
17.1
15.6
20.7
17.6
12.9
and two
5A
30.0
31.0
33.8
30.6
29.1
31.4
30.8
23.9
19.3
23.9
29.3
29.4
28.6
k J. 6
23.0
28.0
26.5
26.2
24.4
18.1
26.3
25.9
22.9
19.3
12.9
27.3
27.5
24.8
20.3
10.1
28.4
25.7
23.4
20.8
14.2
days after
respectively
180
-------
APPENDIX E
COMPUTER PROGRAM FOR SPRINKLER IRRIGATION MODEL
181
-------
APPENDIX E. COMPUTER PROGRAM FOR SPRINKLER IRRIGATION MODEL
EXPLANATION OF PROGRAM
The input is as follows:
1. Read in all the conditions for the single sprinkler pattern (SSP)
test.
2. Read in the size of the two dimensional sprinkler pattern array,
m(rows) x n(columns) and the location of the sprinkler head during
the test. Note how the location of the sprinkler head is found in
Figure El.
3. Read in the Row SSP by rows, 1 row per data card. All values
should be integers and I/loo*-*15 of an inch as can be seen in the
listing of the SSP in Figure El.
4. Repeat steps 1-3 for each new sprinkler test.
5. The last sprinkler test must be followed by a data card with a
negative integer value for the test number.
6. The overlapped sprinkler pattern can be chosen rectangular or
triangular depending upon the value given within the program for
PATTN.
PATTN = 4 This pattern will have rectangular spacing, QQ x PP. QQ is the
spacing perpendicular to the wind. PP is the spacing parallel to
the wind. The spacing is in multiples of the can spacing for the
particular test.
PATTN = +3 This pattern will have a triangular spacing, QQ x PP x QQ/2
where the wind is perpendicular to the laterals.
jWind
-QQ
•Overlapped Pattern
-QQ/2
PP
i
182
-------
4J
CU
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ex
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E-1 CU
< fn
O- -H
c£
PJ
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HH *rt
05 C
CU -H
co s
t— i
tu
w
PJ C/)
i-J Qi
CO fn
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CD
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t>0
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CD
f^
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CU
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LO
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rt
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ooooooooooooooooo
OOOOOOOi— ii— ir-Hi— lOOOOOO
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I~H r*^ oo ^0 t*** ^o ^^ ^^" 0*1 ^^ ^o t**^ ^~ ^^ ^^ r^ to
i-H LT) vO r^ t*^ 00 C^l t^O LO ^J" Tj" \D CM O^ O^ t*1^ CM
r—l ,_) r^ ,—j f~4 ,__( ^_|
o^-t^oovooocN^-cMLovotoa>cni^LOO
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(/) i— 1 rH rH r— 1 I— 1
rt
PJ
00
1
oo^-r^o^ooooooaioooit^h-r^^j-r-io
^D ^2 ^3 ^f t**^ 00 OO 00 00 ^* t^» \O t*^ LO r-H ^D ^D
C5 C5 ^^ ^^ i~t LO ^D t^*1 \O ^0 t^ ^D ^*O f™H ^3 CD CD
'OOOOOOrHlOtOMr-lr-lOOOOO
183
-------
PATTN = -3 This pattern will have triangular spacing. QQ x PP, PP/2.
The wind is paralle to the laterals.
Wind
-Overlapped Pattern
PP
-QR
PP/2
7. A value for YR or (YYR) is chosen within the program. YR is the
dimensionless requirement.
Y _ depth required in the root zone to overcome total deficiency
*n
x (the mean application)
THE OVERLAPPING PROCESS
1. Rectangular spacing. First the SSP array is sent down to Subroutine
OSSP. The information sent down is M, N (the size of the SSP), PP,
QQ (the sprinkler spacing), A (the SSP array), Pattern = 4. Index = 1
if listing of OSSP is wanted, 0 if not. The subroutine returns with
the OSSP ar£ay. The OSSP array is then sent to Subroutine ALPHA which
calculated x, Coefficient of Variation, S (standard deviation), UCH and
UCC.
Subroutine Beta then sorts the data into a(X, Y) arraty, with which
REGRES fits to a linear regression by the least squares method. ALO is
the a and AL1 is b in Y = a+bX from the UCL. RRL is the R2 (coefficient
of determination) for the fit.
2. The triangular spacings are dealt with in much the same way except the
overlapping concept is different. From the figure below it can be seen
that the triangular spacing overlap is the result of overlapping two
rectangular spacing overlaps.
Rectangular
Overlap
Triangular
Overlap
For this reason OSSP is called twice.
184
-------
EFFICIENCIES
The efficiencies can be calculated once the b(Y = a+bX) and YR values are
known. (There are four different cases for calculating the efficiencies
depending upon the value of YR.)
OUTPUT
The output is tabulated for each SSP. Each of the spacing combinations
are printed along with the values of many parameters calculated for each case.
The test conditions are first printed. Then the spacing is listed under
OSSP. The remaining parameters are defined below.
XBAR = mean application (1/100 inches)
SD = standard deviation
SD/XBAR = S/X"
UCH = coefficient of uniformity (HSPA)
UCC = Christiansen's coefficient of uniformity
AL = linear regression coefficient a ,y _ , .
BL = linear regression coefficient b '
RRL = r^ for linear fit of data
EA = water application efficiency
ES = water storage efficiency
DP = fraction of total water deep percolated.
TABLES OF EFFICIENCIES
Since EA, £5, and Dp depend only on YR and b, tables can be printed if
desired by inserting CALL EFFIC (z) just before the END card of the main
program. These tables can then be utilized as a tool in design once b, YR
are determined from a given sprinkler, spacing, wind and operating conditions.
185
-------
PHOGRAH NAME (INPUT, OUTPUT, TAPE5*INPUT,TAPE6«OOTPUT)
DIMENSION U(*g,*0>. AC.0,1.0). BUCftitl. CUl.tl), mOOt, V(«M>t
1NTES(900), PRESSU(900>, NIN(900), CANSPCC9JO), INS (900), INC 1900),
ZSTDOV(900I. COV(900I< HOC(900), CUCC900), AALO(900), AALK9II).
3ARRL(900>, E»(900), ESI900), OOP(900), XMEAN(9«»),
«XHIN(900>
REAL NOZ1, NOZZ, NOZZ1I900), NOZZ2I900)
INTEGER P. Q, PP. OOt PQt P«N, PM»X, OP, PATTN, SPNS(900),
1SPWE(900)
1C * 0
Zfl
Z5
50
C»
c»
c»
c»
c»
c*
c»
c»
C
c»
c»
c»
c»
•PROCEDURE FOR READING IK DATA
•DREAD IN VALUES FOR RAH SPRINKLER PATTERN TEST CONOITIONS»»NTEST-
•TEST NUMBER.NOZi'OIA OF LARGE NOZZLE,NOZZ*DIA OF SMALLER NOZZLE.
•ANG'ANGLE OF NOZZLE,PRESS=OPERATING PRESSURE AT SPRINKLER.P.S.I.)
•HIND=NINO SPEEO(M.P.H.),CSPCG=SPACING OF PHECIPATION COLLECTION
• OANSIFT.),TIME*OURATION OF TESTIMIN.)
2) READ IN SIZE CF THE SINGLE SPRINKLER PATTERN MATRIZAND THE
LOCATION OF THE SPRINKLER HEAD IN MULTIPLES OF CAN SPACING
•3) READ IN RAM SPRINKLER PATTERN 3Y ROMS ,IRON PER DATA CARD. VALUES
•ARE IN INTEGER FORH AND IN ONE-HUNORETHSOF AN INCH!.15IN.»15I
•X, At, FA.O, Fl (lad.J), J = 1, M
IZ FORMATI25I3I
HRITE(6,19)
19 FORMAT(i40X, 1ZH LIST OF 5S°, //I
00 20 I = 1, M
20 HRITE(b, 21) (IA(I,J), J = l.N)
Zl F09MATI5X.20I'.)
DO «.0 I = 1, M
DO 50 J = l.N
50 A(I,J) = IA(I,J)
i»0 CONTINUE
SPRINKLER SPACING IS IN MULTIPLES OF THE CAN SPACING
PP IS THE SPRINKLER SPACING IN 1IRECTION OF THE KINO
QQ IS SPRINKLER SPACING IN DIRECTION PERPENDICULAR TO THE
PATTN = -3 POR QQXP»,PP/2 L4TEP4L PARALLEL TO NINO
PATTN T 1 FOR QCXPPXQQ/2 L4TERAL PERPENDICULAR TO WIND
PATTN s i, FOR RECTANGULAR SPACING
CHOOSE THE DESIRED SFUINKLF* PATTERN HFR£
NAME
COC SltOO FTN V3.0-P36S OPT =
13.39.ZH.
PATTN = l.
DO 70 I =
00 71 J =
Pt> s I
QO = J
1C = IC*1
NTES(IC) =
N077KIC)
N07Z2(IC>
PRESSU < Id
MIN(IC) -
C4NSPC (IT)
INSIIC) =
iwri 1C) =
SPNSIIC;) =
SPWEdCI =
IF(P«TTN .
<*, lit, 2
I. li», ?
NTtST
- NOZ1
= NOZ2
= PRfSS
KIND
- CSPCG
M
M
PP
on
NE. 1.1 GO TO 23
186
-------
C«« RECTANGULAR SPACING
HO C»»" OSSP OVERLAPS SSP FOR PPCROHS BY QQ(COLUMNS) SPRINKLER HE*0 SPACING
CALL OSSPIM, N, PP. 00, A, B, it, 01
C"»»LPHA CALCULATES THE UCC AND UCH
K C»« BETA SORTS THE DATA FROM SMALLEST TO LARGEST
c. .,,..„.„.„.. ................ ..................................................
CALL ALPHAIPP, 00, B, XBAR, SO, CV, UCH, UCC)
CALL BETA(PP,QQ,6, X, Y)
PQ = PP»OQ
90 c»»»»»»» •••••*•••••••*•»••••»••••••»•••«•»••••••»*•••»•••••••»•••••••»•••»••••••
C»*» REOES USES THE LEAST SQUARES CURVE FITTING PROCEDURE TO FIND THE
C»»" LINEAR REGRESSION COEFFICIENTS A AM) B FROM Y=A»8X HHERE Y IS THE
C»« DINFNSIONLESS INFILTRATED DEPTH < Y=OEPTH/AVERAGE DEPTH) X IS THE
C"»» FRACTION OF THE FIELD RECEIVING THE DEPTH Y OR LESS
,5 c»...». .................... .................... .................I...............
CALL REGRES(X, Y. It PQ, It 110, AL1, RRL)
GO TO 29
23 IFIPATTN + 3) 30, 30, 31
30 P s PP/2
100 0 - 2*QQ
CALL OSSPIM, N, PP, Q, A, 8, U, 1)
CALL OSSP(PP, Q, PP, QQ. B, C, -3, 1)
CALL ALPHACPP, OQ, C, XBAR, Sn, CV, UCH, UCC)
CALL BETA(PP,OQ,C, X, Y>
105 PQ = PP»QQ
CALL "EGRF.StX, Y, 1, PQ, 1, «LO, ALL RRL)
3
-------
DOPIICI'OOPP
155 GO TO 71
703 EAIICM1.
OOP(IC)=0.
escici=i./YYR
71 CONTINUE
160 70 CONTINUE
GO TO 7
333 CONTINUE
00 100 I = 1. 1C
IFd .NE. i .AND. NTESCII .ED. -msd-m GO TO 88
165 WRITE (6,151» YYR.PATTN
151 FORMAT< 1X,///,5X,»THE VALUE F09 YK IS =»,F5.Z,«X,»THE PATTERN IS
1 • ,121
HRITEC6,150)
150 FORMAT (
-------
SUBROUTINE EFFICIZ)
DIMENSION YR(ll), E(20), XB<20)
00 39911*1.3
IF(II.GT.l) GO TO 405
5 MRITEC6.500)
500 FORMAT!1H1,* TABLE OF HATER APPLICATION EFFICIENCIES',////.TSO,
1'VALUES OF YR (OIMENSIONLCSS SOIL HATER DEFICIT IN ROOT ZONE)*)
GO TO 398
405 CONTINUE
10 IFdI.GT.2) GO TO 406
MRITE(6,501)
501 FORMATUHl,'TABLE OF MATES STORAGE EFFICIENCIES',////,TSO ."VALUES
10F YR (OIMENSIOkLESS SOIL WATER DEFICIT IN ROOT ZONE)*)
GO TO 398
15 406 HRITE(6,502)
502 FORMATdHl,' TABLE OF THE FRACTION OF TOTAL HATER DEEP PERCOLATED*
I,////,TSO,'VALUES OF YR IOIMENSIONLESS SOIL HATER DEFICIT IN ROOT
2ZONE)')
398 CONTINUE
20 00 400 1-1,20
KK«0
X801
C CALCULATE THE HATER STORAGE EFFICIENCY, £S
15 388 EIKK)*1.
GO TO 401
C CALCULATE THE FRACTION OF DEEP PERCOLATION, OOP
389 E(KK)*1.-YR(KK)
GO TO 1,01
1,0 701 CONTINUE
IF(YR(KK).GT.l. » GO TO 702
IFIII-2) 353, m, 386
3«3 EIKKI «YR (KK)-(YR(KKl -1.^X9(1 1 /2.)**2/(2.»XB( II)
GO TO <«01
«,5 C CALCULATE THE WATER STORAGE EFFICIENCY, ES
3»lt E(KK)=l.-UYR(K*)-l.tXB(I)/2.l»»2/(2.»XBU>*YR(KKI)>
GO TO 401
C CALCULATE THE FRACTION OF DEEP PERCOLATION, OOP
386 EtKKI*l.-YR(KK)+(YR
GO TO 401
C CALCULATE THE FRACTION OF DEEP PERCOLATION, OOP
75 3«2 E(KK)«0.
401 CONTINUE
IF(I.NE.l) GO TO 385
HRITE<6,503)
503 FORMAT(1X,T50,1HF5.2,2X)//I
80 399 MRITEI6.504I Xfl (I ( , IE ( JJ) , JJ=1, 111
504 FORHAT(1HO,T40,F5.2,T50, 11(F5.3,2X)I
IFU.NE.8) GO TO 400
HRITE(6,506I
SOS FORHAT(lHt,T5, 'VALUES OF IRRIG. QUAL. 3»(
85 400 CONTINUE
399 CONTINUE
RETURN
END
189
-------
SUBROUTINE OSSPItt. N. P, Q, », 8, P»T, INDEX)
C»»» IF INOEX*0 THE OSSPIOVERLAPPEO SPRINKLER PATTERN) H^LL NOT BE
C***PRINTED. IF INDEX «1 THE OSSL KILL BE PRINTED
OINENSION AUO.tOI. BUO.fcOI. I9(4lttO)
5 INTEGER P. Qt R. St PAT
IF(PAT .HE. *) GO TO 80
00 25 K» It P
00 SO L * It Q
IF(L .GT. N .OR. K .GT. H) GO TO <»1
IB SUM * 0.0
00 i»0 R - K, Ht P
00 50 S * L. N, Q
50 SUN = SUH * A0 MRITE(6, SO) P. Q. PAT
60 FOP,HAT( /////, 5X, 13H LIST OF OSSP. 5X, 12. JH X. 12, 10X,
18H PATTEPN, IH, ///)
00 70 K » 1» P
70 HRITEI6. 751 (IB(K.L) . L - 1, Ql
<»5 75 FORHATI5X, 2515,/I
76 RETURN
END
SUBROUTINE ALPHA (H, N. A, X3AP, SO. CV, UCH, UCC)
OIHENSICK SCtO, 401
SUM4 =3.0
SUH«A ' 0.0
5 HN = M»N
A*IN - IN
00 100 I = 1. 1
00 11C J - 1. N
SUMA = SU1A » A(I,J)
10 110 SUMAA = 3UMAA * A(I.JI**2
100 CONTINUE
X««P = SUM«/«MN
SO = S(KT( (<>Uf AA-SUM«»SUMA/AMN»/ (
cw - sn/xna^
15 UCH = 1.3 - 0.7e7C»
-------
SUBROUTINE 9ETA(M,N.A,X,YI
DIMENSION Ad.C.'.OI, ASRAY(300>. XMOOI. Y<30fl>
MN = M*N
AMN = MN
5 HNN = MN-1
DO 10 J=1,M
JJ * (J-l)*Ntl
DO 2V L=1.N
ARRAY) JJtL-ll = A(J,L)
1(J 20 CONTINUE
ID CONTINUE
DO 10 I - 1. MNN
11=1*1
no <>o J=II.MN
.5 IF0 II1,MN
AI = I
60 Yd) = A»RAY( II/X8AR
30 RETURN
END
SUBROUTINE
-------
TEST HUM
131.1
:OO«OINATE
-0 -C
-0 -0
-0 -3
-3 -C
-0 1
-0 5
1 6
3 7
3 6
3 6
1 7
1 6
-0 3
-0 1
-3 -0
-a -o
-0 -0
NOZZLE ANGLE PRFSS WIND OIKcCT CAN SPCG
172 .091, 7. 1,0. C 1.6 VAU 5.
OF SPPINKLE" HEAD, 8 EAST 9 SOUTH OF N-M CORNER
LIST OF SSP
• 0 ~0
-0 -0
7 9
8 8
8 8
8 8
8 9
7 8
7 9
6 7
7 7
5 7
1 1.
-0 1
-0 -9
THE VALUE FOR
TEST N9Z1 N07.2
-g
1
f,
7
9
10
12
13
12
1C
8
8
3
-0
ft IS
•C 1
6 7
8 8
12 12
14 13
12 15
15 H,
17 16
9 1?
0 9
7 9
5 7
-C 2
= l.OC
PRESS HINH
131,1 .17? .091. 1,0.
CSSP
* *11C.
i. e> 73.
i, 8 55.
1.10 1.1..
*12 36.
*1* 31.
6 6 1.8.
6 8 36.
flC 29.
f.12 2i».
611. 20.
8 8 27.
81o 22.
812 18.
811. 15.
1010 17.
1012 11..
1C1* 12.
1212 12.
1211. 10.
1H1. 8.
XflAR
125 7.
1.17 11.
G62 *.
050 5.
70f 7.
',61, 10.
9** 7.
70( *.
367 l,.
<>72 5.
976 7.
531 l».
025 I,.
351. <>.
712 5.
620 3.
68^ 3.
586 l».
236 3.
1,88 It.
990 *.
SO
151.
096
819
729
781.
1,91
608
1.58
>82
760
291
125
219
*8*
6?6
*51
635
806
777
398
1,36
0 1.5;
sn/x. 11
16 13
12 13
9 a
8 9
a 8
6 5
1 2
PATTERN
1 -0
6 3
8 6
7 7
5 6
6 5
6 5
7 6
7 7
7 8
8 6
8 6
6 5
3 1
-0 -0
IS i.
-0
-0
2
7
8
7
8
8
8
8
8
1.
1
-0
-0
MIN TEST
30.0
-0 -0
-0 -0
-0 -0
2 -0
6 -0
6 -0
6 1
7 1
5 1
* -0
1 -0
-0 -0
-0 -0
-C -0
-0 -0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
53P CUT
17 1
IICH
.95*6
.8883
.93C7
.9032
.8296
.7130
.8781,
.9015
.8928
.8181,
.7095
.8861
.8535
.8133
.7J87
.81,13
_.8236
.6969
.7565
.6629
.6311
*
UCC AL BL
.9095
.7672
.8593
.7915
.651.1
.1,510
.71,75
.8013
.7700
.6221,
.1.311.
.7588
.6876
.6066
.1.081
.6813
.6127
.3708
.5009
.311.1
.20*3
.1811
.1.656
.2815
.tl71
.6917
1.091.1
.5051
.3971,
...601
.7552
1.1372
.1.821.
.62*8
.7868
1.1839
.6.375
.771,6
1.2585
.9983
1.3719
1.5911,
RRL
.7827
.8968
.947I,
.92*3
.9 1.* 2
.9*67
.9567
.9501
.8723
.891,*
.92*5
.905*
.9207
.8919
.9380
.9063
.8369
.92*9
.8900
.9081
.(803
.977*
.9*18
.96*8
.9*74
.9135
.8632
.9369
.950 3
.9*25
.9056
.8579
.9397
.9219
.9017
.8520
.9203
.9032
.8*27
.8752
.8285
.8011
EA tS
.977*
.9*18
.96*8
.9*79
.9135
.8632
.9369
.9503
.9*25
.9056
.8579
. 9397
.9219
.9017
.H52C
.9203
.9032
.8*27
.8752
.8285
.8011
DP
.0226
,0582
.0352
. 0521
.0865
.136S
.0631
. 0*97
.0575
. 09**
. 1*21
.0603
.0781
.0983
.1*80
.0797
.0968
.1573
.12*8
.1715
.1989
192
-------
1 TA9LE OF WATER APPLICATION EFFICIENCIES
VALUES OF YR (DIMENSIONLESS SOIL HATER OEFICIT IN ROCT ZONE)
.50 .63 .70 .10 .90 1.00 1.10 1.20 1.1
1.1(0 1.50
.1C
.2C
.It
.tO
.5r
.fcC
.7;
.HC
.90
l.CC
I. 1C
l.?C
1.3C
l.*C
l.'ii
1.6C
1.70
l.«C
1.>>7
.1.J8
.600
.603
.600
.600
.600
.600
.600
.600
.591
.595
.!>93
.501
.576
.568
.559
.5^0
.5*0
.531
.523
.510
.730
.70C
.700
.700
.730
.700
.69A
.691.
.687
.680
.672
.062
.653
.61,?
.632-
.6*2
.611
.630
.5*9
.577
.800
.800
.800
.800
.797
.792
.781,
.775
.765
. 755
.71.1.
.733
.722
.711
.619
.517
.676
. 66<>
.652
.61.0
• 900
.900
.896
.887
.877
.867
.855
.81,1,
.832
.820
.808
.796
.781.
.771
.759
.71.7
.735
.722
.71C
.697
.987
.975
.962
.950
.938
.925
.912
.900
.887
.875
.862
.850
.837
.825
.813
.800
.788
.775
.762
.750
1.000
1.000
.996
.987
.977
.967
.955
.11. 1.
.932
.920
.9C8
.896
. as*
.871
.859
.81.7
.835
.822
.81C
.797
1.000
l.OCO
1.003
1.000
.997
.992
.98<*
.975
.965
.955
.91.1.
.933
.922
.911
.899
.887
.876
.8(1.
.852
.81,0
1.000
1.000
1.1)30
1.000
1.000
1.0 00
.998
.991.
.987
.980
.972
.962
.953
.91.3
.932
.922
.911
.903
.889
.877
l.CCO
l.OCO
1.000
1.000
1.000
1. 000
l.OCG
1.000
.999
.995
.990
.983
.976
.968
.959
.950
. T.G
.931
.920
.910
1.000
1.1)00
1.000
1.000
1.000
1.050
1.003
1.000
l.OOC
1.003
.999
.996
.991
.986
.979
.972
.96c
.6C
. i'
.1C
.90
i.o:
1.1C
1.2C
1..U
1.1.1
1.5:
1.6C
1.7C
l.BL
1.9'.
2.00
1.0G3
l.OOC
1.000
1.000
1 . C o 0
l.t^C
1.330
i.OOQ
1.000
1.053
.<>S8
.912
.183
.971
.958
.9<>l»
.921
.911
.893
.«75
1.033
1.000
1.000
1.300
l.OCO
l.OCO
l.GCJ
l.OCO
.998
.992
.983
.972
.9*0
.900
.500
.50Q
.500
. r>00
.500
.500
.501
.501.
.509
.511.
.521
.528
.536
.51*1.
.553
.5b3
.".00
.1.00
.1.00
. ".CO
.1.011
.1.00
.1.00
.1.00
.1.01
.1.05
. 1. 10
.«.17
.1.21.
.1.J2
.1.1.1
.i»60
.1.63
.1.69
.U80
.1.90
.300
.330
.300
.330
.3CC
.3JO
.332
.316
.312
. 32G
.328
.337
.^1,7
.357
.367
.378
.389
.1.SO
.1.11
.1.22
.230
.230
.200
.200
.202
.238
.216
.225
.235
.21.5
.256
.267
.278
.289
.301
.312
.321.
.336
.31.8
.360
.133
.103
.lOi.'
.113
.123
.133
.11.5
.156
.168
.180
.192
.201.
.216
.229
.21.1
.253
.265
.278
.290
.303
.312
.025
.037
.050
.063
.075
.087
.130
.112
.125
.137
.150
.162
.176
.188
.230
.212
.225
.237
.250
U.OOC
.000
.001.
.013
.023
.033
.01,5
.056
.0*8
.080
.092
.101,
.116
.129
.11.1
.153
.165
.178
.19C
.203
3. COO
0.200
O.uCO
.CCO
.003
.008
.016
.025
.035
.01.5
.056
.667
.078
.089
.101
.113
.121,
.136
.11.8
.160
0.000
0.030
0.030
0. 300
0.000
.000
.032
.336
.013
.023
.328
.338
.01.7
.057
.068
.078
.019
.103
.111
.12)
0.300
0.000
0.300
3.000
0. 01.0
b. OCO
0.003
.000
.oci
.005
. 010
.317
.1*21.
.032
.041
.053
.CbC
.Cb9
.080
. 090
0.000
0. 000
0.000
0. JOO
j.OOD
0.330
0.300
O.C30
0. 303
C.303
.001
.031.
.339
.311.
.021
.328
.336
. 01.1.
.353
. 363
193
-------
APPENDIX F
COMPUTER PROGRAM FOR TRICKLE IRRIGATION MODEL
194
-------
APPENDIX F. COMPUTER PROGRAM FOR TRICKLE IRRIGATION MODEL
EXPLANATION OF PROGRAM
Required input data for this program are: the lateral length and
diameter, the emitter spacing, characteristics of the emitter, and friction
characteristics of the lateral pipe.
The computation of the pressure head and discharge distributions along
the lateral is through an iterative procedure utilizing the Hazen-Williams
equation for head loss.
A least squares curve-fitting technique is employed to fit the generated
data to a logarithmic type function: y = a + b In x. Coefficients of the
curve fit, a and b are used in calculating the quality parameters, and the
emission uniformity.
The output of the model includes all input data, the distributions
generated, the mean emitter discharge, the lateral inlet flow rate and
pressure head, the design EU and design EU , and the results of the curve
fits. a
195
-------
»'0<;*AM T=TCKLE (INPUT, CCTPUT. TAPE5=I NPUT , TAPF.6 = OUTPUT )
C...THIS "DOG-JAM CALCULATES THE PRESSURE HCAO ANO 1ISCHA1GE ALONG A LATERAL AT
C...POFOFTF-MINEO POINTS. THE PRESSU'E HEAO AND DISCHARGE VERSUS LOCATION ARE
C...THFN BITTED TO & \_r GSR I THPIO TYPE FUNCTION.
5 C...THE nTSCHlPnr VFRCliS LOCATION O-ISTRIBUTI ON IS THEN NCNOIMENSIONALIZED «NO
c...FiTm TO n LOGASIIHMIC TYPF FUNCTION IN O^FCE" TO CALCULATE EFFICIENCIES.
C...OQ=OFSIGN DISCHARGE OF 1 LATE* AL . L PS. INITIAL ESTICATtS OF 00 CAN BE MADE
C...USINO. TH£ PPfinUCT OF THF NU«3EP nf EM!TTFRS AND THE DISCHARGE OF A GIVEN
C...FMITT1-0 AT fl KF.FEPENCF PRESSURE HEAD.
10 C...HI=PPFS3U'»r HfAO AT T C[ EKOAhCr TO THE LATERAL, 1.
C...CF. = "ICTJCN COI-FFICIFNT FCP PIPF KITH EMITTERS ATTACHED.
C...IF 4 VAU£ CF.CE IS NOT KNCHN, THrN fiE = C.j. WHEN THIS OCCURS, THE HEAD LOSS
C... AL >Hr- TH-! L4TFP1L I? C«LCULATEO *ITH ANOTHER RE3UCTION COEFFIC IENT.FN.
C...C»H47£N-^ILLIAMc FtTCTIfN CTEFFICIENT FO^ PIFE.
15 C... TIVF POSTICN ALCNL LATERAL.
c...PH=pot"su',E HFAD AT 'ELATI.'E POSITION ,n.
C...EO=r"ITTEJ iTSrHAfcGF AT PFLATJVE FOSI TTnN.LPH.
c...oELH=T3TAu ^ran HLCSS ALONG LATFRAL.I.
I5...0EQ= TI1L NSIONLCSS f«ITTF' ^ISC^A'^,F FOR CUCUE FIT.
'0 C. ..OXL = CIMtl{SIONLFSS POSITION ALONG LATERAL FCR CUFVE FIT.
C. ..EQMAVaMAVIKLI cMITTfJ, l~ISCHA^GE. LPH.
C...EOMIN=MINIPUM-:MITTtf DISC" AS''. I, LPH.
C...EU-1FSinK E'^ISSIO^ UNIFC'CITY, PERCENT.
C. ..FU« = AiJ«GLUTF TFSIGN tMISSICH UN IFOR fl TY, PE'CFNT.
V* C... YRi'lf PTH --FHIIPfO IN "CCT 7PNF. TO OVERCOME TOTAL HFF 1C I'NCV /AVERAGE OEPTH
C... APPLien.
C...NYPrNO. Of YO FSTIMTES.
IHlMU1< l/ALUt CF NO KOI » I NS I O'l AL 5ISCHARRF R4TIC.
AXI"UM VALUE CF NCNTIMENSIONAL OI^CH ARGF RATIO.
M C. ..E A = l«6T-~ APPLICATION EFFICIENCY. OtFINEH AS TH? VOLUME OF WATER STORED IN
C...THE COOT /ONf/TTHf TOTAL VOLHME ACPLIEO.
C. ,.rs=W(!Tr ' ^TQOAGF tf FIFCIENCY. OEFINFT AS THE VCLUHF OF MATCR APPLIED TO THE
C.^'OOT *ONF/TH|. VCLUCE CF WATEC THAT TH£ POQf 7QNg CAN HOLO.
C. ..DP=PF°CrNT n€F" PECCOL»TICN.
.(,»
IF (OF. £3.1. 0) GO To ICO
PROGRAM TOICKLF 0110 (SI.CO CT N V3.0-P365 OPT = 1 C9/15/77
60 23C Rt AT(S,11) O.L ,S6
11 TOPMAT (tFlO.3 )
IF(0.-'Q.3.0> GO TO KD
I"--(Cr.E:). C.OIGO TO =0
C...CALCULATlQ\r)f: imH^GF.PBESSURF.HEAO AN1 TOTAL LATERAL HEAD LiJSSWHEN A CE
65 C...VALUF IS KNOWN FOh' TH? DFCUCTION COEFFICIENT IN THF HA7EN-WILLI AMS E7N.
CALL OfSTGM(PH,D,C,L,SF,
GO TO til
C... CALCULATION Oc TISCHARGf, P»ESSU»E HEAT ANO TOTAL LATF°AL LOSS WHEN Ct
C...IS UNKNOWtl. REOllfTION FACTOR FN=:T . 6 3167*N ••-! .«91S * G . ^592^ , WHERE N=NO. OF
7C C...FMITTF»<:, IS USEO I" THH HA^N-WILL I4MS EQN.
•50 CALL 1EJIO,H(PH,XL,£ 1 , OQ ,HI , NE , J . DELH , 0 ,C ,L ,SE, - AN=( Af OFL2*XL(NFI «(aCOFL?*XL(NE>»ALOG(XL(NF)H-eCOFL?*XL(NE))/X
IL(Nr)
FUA=UO.»(1.G-1.27*VI'C.5»(EOMIN/'J1EAN+QN£AN/EQ-1AX)
SO 03 f N=1,NF
C!*L (Nt=XL (") /L
OEC(NI=EO(N»/01f tN
3 CONTINUE
196
-------
<}5
CALL LOGFIT(nXL.OEa,KF,ACOFL3,BCOFL3,RZL5>
NRiTF.<6,i.ii ca,Hi,cE,c,Kn,x,v
1,1 FORMAT! lHl,*nC3*.3,3*,*x = »,Ff ,3,3X,»V*»,F*.3»
HRITEfh, 3110, ?FiL
,F&. G . 3X .
3X,*SE=«,F6. 1 , 3X ,«L*' ,F3 ,Z)
K, J.NF
t -OPMATdX, 'TOTAL LATFUAL HtAO LOS «»» ,F9. I..2X ,»METEB»/1X, » MEAN EMTT
1TF° OISCHAPGf = »,FU. At ?X, 'LITER PER HOUR»/1X ,»HEAC LOSS GRAQIEKT = »
»KUWBER OF
33 FO°MAT(1X,*OFSIGN fMISSICM UNIFORfl TY= », Fid . 2, ?X, 'PERCENT «/ IX, »OES
1ION OBS.1LUTE EKISSJOK '-', T ORMIT' - *. Fl ". . 2 , 2X . »PE"CE NT* I
W-dTlfc. 9» «CDFH,:JCCFL1. .1
S FOK1AT CX.'LOGACITHMIC CURVE FIT FrtR ORgsSURC HE A0» ,5X, 'A = ', F10. 5
110
115
10 FOFMATI Ix.'LOGHCITHs,.,. CURVE FIT FQ" EMIITE» CISCH A»GE» ,5X, »A = », 5X
l.F10.'i,->X,»1=«fF10.S,l5X,5M»»'? = ,F10.5)
MRITtCh. ^0001 «CCFL3,erOFL3,R?L'
3000
Ni.ES'; LOGORITHMtC CU^VE FIT F Ok LMI7ti-< "ISCHAR
,5,%X,»3=»,F10.5,SX,';HP»»a=,F10.5)
5 FOCMATdX.'PPFStUi-t HFAn*,5X, 'LOCATION', SX,»EMITTFR OISCH ARGE •, 5X,
1'RFLATIVE LnO«TinN«,5x,*CIMENSIONLESS EMITTER OISC.HSRGE'1
W^ITr(6,f, I (PHI I) ,XL(I» ,EQ(II,OXLlI),DFfJ(I),I = l,NE)
S FOP-1AT iaX,F8. <,, CX,Ffl.l.,10X,F»l^,l JX,F8.i),20X,FH,l.|
C...CAtCULATION OF fFFiriEKCIEStEA,ES,OP.
73 03 15 JJ = 1,NVP
IF(*0(JJ) .LE. V^INI GO TO 70
IF (YRt J II ,GE. YMOX) GC TO 71
TPICKLf
CDC
FTM VJ.O-
np = lC3.*( l.C-EAl
Gn Tn 7?
70
135
QP:!!?).* (l.C-FAl
GO TO ^
133
135
I)P=1C3.* H.C-EAI
7? IFItS.GT. t.CI £3
l lX,»YRs»,F10.
1*«ATFR -;
35 CONTINUE
GO TO ZOO
300 CALL EXIT
END
APPLICATION f FFI C IENC »=». Fli) . <»/ IX ,
£tP °E ^C JL ATI ON=»,F
197
-------
CALL LOr,FIT(nXL,DrO,KF,«COFL3,BCOFL3,RaL!)
WRITF<6»/nCCFL3)
115 Y»1IN=CCOFL''
IF (VO(JJ) .LE. V1IN) GO TO 70
IF (V91 J)l .GE. YMSX) GC TO 71
P=OGOAM TPICE''CENT OitP OE^C JL ATI ON=» , F
UC.".)
135 35 CONTINUE
GO TO 200
300 CALL EXIT
ENtl
198
-------
aFSIGN(PH,XL,FC,DO,.HI,NE,J,UtLH,0,C,L,5E,KO,X,C£>
C...THIS SUBROUTINE CALCULATE? THfc PPESSU3E HEAO Ann 1ISCHA9GE nIST"IBUTI ONS
C...ALCNr, AM EMTTFC LATERAL, STARTING F«!0*t THE DISTAL ESD OF THE LATERAL WITH A
C...KNOWM(PFFCBtNrEI PRFSSU&E HEAD ANI5 OICHA3GE.
DIMFNSTCK PH(3QC).XL('OC)tEQf'00),FNC'COI,PPH(300I.EEQ(300),FI38u)
1.FTOPI330)
RtAl L.KC.J.JAI :CCI
XL(1)=1.0
PPHdl =SUhH
IS S=C.O
DO IOCS I=?,NF
,)A (I) = ^l?f>9c!.»
11=1-1
IF ?/XN)
20 ACOPL=(SY-nCOFL'SLNX)/X>4
»TCP=(SYLNX-(SLNX«SYI/XNI««2
199
-------
100* .0203 HI' 10.06 CE' 0. C» ISO. KO* .5*7 *« .606 V> .OSO
0* 9.1.0 SE* 1.0 L» 33.S3
TOTAL LATERAL HEAD LOSS' .0632 METER
MEAN EMITTER DISCHARGE' 2.2128 LITER PER HCUR
HEAD LOSS GRADIENT' 1.5077 M/100M
NUMBER OF EMITTERS' 33
DESIGN EMISSION UNIFORMITY* 93.55 PERCENT
DESIGN ABSOLUTE EMISSION UNIFORMITY* 93. -.061.27 «"2« .972*1.
CIMENSIONLESS EMITTER DISCHARGE
1.8026
1.0022
1.0020
1.1017
1 . 00IV
1.0012
1.0009
1. 0307
1»0005
1.. 00 01.
1., 0002
1.0000
.,9999
.. 9998
..9996
..9995
. 999%
.9991.
.9993
.9992
.9991
.9991
.9991
.9990
.9990
.9990
.9989
.9989
.9989
.9989
.9989
.9989
.9989
200
-------
TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA-600/2-78-041
3. RECIPIENT'S ACCESSION-NO.
4. TITLE AND SUBTITLE
ASSESSING THE SPATIAL VARIABILITY OF IRRIGATION
WATER APPLICATIONS
5. REPORT DATE
March 1978 issuing date
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
David Karmeli, LeRoy J. Salazar, and Wyrin R. Walker
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Department of Agricultural and Chemical Engineering
Colorado State University
Fort Collins, Colorado 80523
10. PROGRAM ELEMENT NO.
1BB039
11. CONTRACT/GRANT NO.
R-804828
12. SPONSORING AGENCY NAME AND ADDRESS
13. TYPE OF REPORT AND PERIOD COVERED
Robert S. Kerr Environmental Research Laboratory-Ada,0
Office of Research and Development
U.S. Environmental Protection Agency
Ada. Oklahoma 74820
Final
U. SPONSORING AGENCY CODE
EPA/600/15
15. SUPPLEMENTARY NOTES
16. ABSTRACT
The current state of the art regarding the spatial distributions of irrigation
water applications under surface, sprinkler, and trickle irrigation systems has been
assessed. The analyses found in the literature and several new uniformity concepts
have been integrated into models which can be used in both field and research
applications. These models simulate the spatial distributions of applied irrigation
water under specified design and operating conditions.
The performance of an irrigation system has been described by a series of
"quality" parameters relating to: (1) uniformity in an irrigated field; (2) adequacy
of the irrigation system in meeting crop requirements; (3) volume of applied water
wasted as deep percolation; and (4) in the case of surface irrigation, the water
leaving the field as tailwater.
Verification of the models developed during this project was made against most
of the data identified in the literature as well as an intensive collection effort as
part of this project. The results illustrate both the use of the analytical approach
and the procedures for field data collection.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.IDENTIFIERS/OPEN ENDED TERMS
c. COSATI Field/Group
Irrigation
Efficiency
Sprinkler irrigaiton
Border irrigation
Furrow irrigation
Irrigation efficiency
Irrigation uniformity
Trickle irrigation
98 C
8. DISTRIBUTION STATEMENT
Release Unlimited
19. SECURITY CLASS (ThisReportj
Unclassified
21. NO. OF PAGES
213
20. SECURITY CLASS (Thispage)
Unclassified
22. PRICE
EPA Form 2220-1 (9-73)
201
•U.S. GOVERNMENT PRINTING OFFICE : 1978 0-728-335/6081
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