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HYDRAULIC ANALYSIS OF SURFACE
IRRIGATION SYSTEMS USING THE
FINITE ELEMENT METHOD

presented by
Walid Hani Shayya

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a

 

HYDRAULIC ANALYSIS OF SURFACE
IRRIGATION SYSTEMS USING THE
FINITE ELEMENT METHOD

By

Walid Hani Shayya
A DISSERTATION

Submitted to
Michigan State University ,
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
in

Agricultural Engineering

Department of Agricultural Engineering

1991

ABSTRACT

HYDRAULIC ANALYSIS OF SURFACE IRRIGATION
SYSTEMS USING THE FINITE
ELEMENT METHOD

By

Walid Hani Shayya

A mathematical formulation for the hydraulic analysis of ﬂow conditions
in furrow and border irrigation systems is presented in this research study.
The methodology is based on applying the one-dimensional Galerkin
formulation of the ﬁnite element method to the numerical solution of the
hydrodynamic or the so-called Saint-Venant equations. Numerical
developments of the complete and simpliﬁed forms of the hydrodynamic
equations were prepared using both linear and quadratic one—dimensional
ﬁnite element forms of these equations. The studied models include the
hydrodynamic, zero-inertia, and kinematic wave models. A general
one-dimensional surface irrigation computer model (FE-SURFDSGN) was
developed based on this formulation. This computer model simulates the
various phases of ﬂow in border and furrow irrigation systems using the
hydrodynamic, zero-inertia, and kinematic wave models. Currently, only the
kinematic wave ﬁnite element analysis is fully operational for a complete

irrigation cycle in the present version of the computer model. The

Kostiakov-Lewis equation was used as the inﬁltration function in this
development even though both the mathematical development and the
developed ﬁnite element model allow for utilization of any other inﬁltration
function. Actual ﬁeld measurements were utilized to validate
FESURFDSGN. These data were taken from previous studies that were
conducted in Colorado and Idaho. Although the computer model is still in the
developmental stage, its application to the simulation of the various phases of
ﬂow in surface irrigation systems is very reasonable as demonstrated through
the various runs that were conducted. The results of this research work
indicate that the ﬁnite element method provides accurate simulation of the
ﬂow conditions in both border and furrow irrigation systems. These results
also suggest that the method developed through this research can be used as
an eﬁ'ective tool for the hydraulic analysis of ﬂow conditions in surface

irrigation systems.
Approved : D

erssor

W

Chairperson

 

Copyright by

WALID HANI SHAYYA

1991

DEDICATION

I would like to dedicate this research work to my parents, Hani and
Salwa, for their love, devotion, tremendous support, and belief in me. My
appreciation to them is beyond words. They always encouraged me to pursue
my dreams and never failed to extend their help. Even though they had no
clue about what I was truly working on, without their encouragement and
support this research work would have not been realized.

I would also like to dedicate this work to my best friend and closest
companion Cathy Cullen. Her loving support, caring, patience,
understanding, and friendship were an important source of strength. Her
continued encouragement, assistance, and personal support made the
completion of this study a reality. I deeply appreciate all that she has done
with and for me.

ACKNOWLEDGNIENTS

I would like ﬁrst to take this opportunity to thank my major professor, Dr.
Vincent Bralts, for his constant encouragement, continued moral and ﬁnancial
support, genuine help, and excellent guidance. His eﬁ‘orts in making this
learning experience worthwhile are deeply appreciated and his sincere belief
in me kept me going.

Special thanks are reserved to my guidance commim members, Dr.
Donald Edwards, Dr. Raymond Kunze, Dr. Larry Segerlind, and Dr. David
Wiggert. The valuable suggestions and advice extended by both Dr. Edwards
and Dr. Raymond Kunze brought this study to a successful completion. A
particular debt of gratitude is owed to Dr. Larry Segarlind for warmly sharing
his own experience as a teacher, researcher, and learner and for providing an
opportunity to put complex numerical theory into practice. The constructive
criticism and valuable suggestions and assistance of Dr. David Wiggert are
very much appreciated.

My appreciation is also extended to Dr. Charles Cress who originally
served on my committee. I would also like to express my sincere gratitude to
Dr. Robert von Bernuth, the chairman of our department, who took the time to
read and correct my dissertation. His helpful suggestions and advice were
critical in making this study successful.

Special thanks are extended to the faculty, staﬁ', and graduate students of
the Department of Agricultural Engineering at Michigan State University, a
very special group that I have had the privilege of being one of its members.

Finally, I would like to express my deepest appreciation to the Hariri
Foundation and particularly to its founder Mr. Raﬁq Hariri. Their ﬁnancial
support gave me the opportunity to come to this country and further my
education. It is very diﬁcult to put into words my genuine appreciation to the
Foundation. However, I would like to try and express my deep gratitude and
admiration to them, a feeling that is shared by thousands of other scholars
that were privileged by receiving the help and support of the Hariri
Foundation. Without their support, none of this would have been possible.

TABLE OF CONTENTS

 

Page

LIST OF TABLES ............................................ xi
LIST OF FIGURES ........................................... xii
LIST OF SYMBOLS .......................................... xvi
I. INTRODUCTION ........................................ 1
A. Scope and Objectives ................................. 2

II. REVIEW OF THEORY AND LITERATURE .................. 4
A. Basic Background of Surface Irrigation .................. 5

1. Flow Description ................................. 6

2. Flow Phases .................................... 6

B. Surface Irrigation Inﬁltration .......................... 7

1. Inﬁltration Equations ............................. 7

2. Parameter Estimation of Inﬁltration Equations ....... 11

3. Eﬂ'ect of Inﬁltration on System Performance .......... 14

C. Surface Flow Equations ............................... 15

1. Saint-Venant Equations ........................... 15

2. Uniform Flow Equations .......................... 17

D. Surface Irrigation Models .............................. 19

1. Hydrodynamic Model ............................. 20

2. Zero-Inertia Model ............................... 26

3. Kinematic Wave Model ............................ 31

4. Volume Balance Model ............................ 35

E. Numerical Solution of the Hydrodynamic Equations ........ 37

1. Finite Diﬁ‘erence ................................. 38

2. Finite Element .................................. 38

F. Synopsis ............................................ 39

III. METHODOLOGY ........................................ 41
A. Research Approach ................................... 42

B. Theoretical Development .............................. 45

1., Development of the Saint-Venant Equations .......... 45

Continuity Equation ..........................

Forces Acting on the Fluid Element .............

Unsteady Momentum .........................

Steady Momentum ...........................
2. Finite Element Formulation Using Linear Elements.

a. Formulation of the Continuity Equation Using

Linear Elements .............................

b. Formulation of the Unsteady Momentum Equation

Using Linear Elements ........................

c. Formulation of the Steady Momentum Equation

Using Linear Elements ........................

d. Formulation of the Zero-Inertia Equation Using

linear Elements .............................

e. Formulation of the Kinematic Wave Equation Using

Linear Elements .............................

f. General Finite Element Formulation Using Linear

Elements ...................................

3. Finite Element Formulation Using Quadratic Elements .

a. Formulation of the Continuity Equation Using

Quadratic Elements ..........................

b. Formulation of the Unsteady Momentum Equation

Using Quadratic Elements .....................

c. Formulation of the Steady Momentum Equation

Using Quadratic Elements .....................

d. Formulation of the Zero-Inertia Equation Using

Quadratic Elements ..........................

e. Formulation of the Kinematic Wave Equation Using

Quadratic Elements ..........................

f. General Finite Element Formulation Using

Quadratic Elements ..........................

4. Finite Element Development Using nonstandard

Galerkin Formulation .............................

5. Finite Element Development Using Lumped Formulation

a. Linear Element ..............................

b. Quadratic Element ...........................

Dimct Stiffness Method ...........................

Finite Diﬁ’erence Solution in Time ..................

Implementing the Solution Procedure ................

a. Relationships Among Flow Geometry Parameters . .

b. Uniform Flow Equations ......................

c. Inﬁltration Functions .........................

9-9 9'?

90:45»

45
50
56
63
65

65
73
77
78
82
.83

87
87

101
107
109
111
113

117
118
119
122
127
128
133
133
135
137

IV. RESULTS AND ANALYSIS ................................ 138

A. Finite Element Model Formulation ...................... 139

1. Assembling System of Equations .................... 139

2. Incorporating Boundary Conditions ................. 151

3. Numerical Solution ............................... 153

B. Finite Element Computer Model ........................ 166

1. Model Components ............................... 166

2. Data Input ...................................... 167

3. Model Output ................................... 174

C. Results and Comparison ............................... 175

1. Actual Field Data ................................ 176

2. Model Runs ..................................... 179

3. Sensitivity Analysis .............................. 196

D. Discussion .......................................... 212

V. CONCLUSIONS AND RECOMMENDATIONS ................ 219
A. Conclusions ......................................... 219

B. Recommendations .................................... 221

VI. LIST OF REFERENCES .................................. 222
APPENDICES ............................................... 234
Appendix A. FE-SURFDSGN Computer Model Listing .......... 235
Appendix B. FE-SURFDSGN Graphics Routine Listing ......... 264

Table 1.

Table 2.

Table 3.

Table 4.

Table 5.

Table 6.

Table 7.

Table 8.

Table 9.

Table 10.

LIST OF TABLES

Summary of the parameters It, and It, in the uniform ﬂow
equations ........................................

Deﬁnition of the parameters c, and c2 in Equation [4.20] for

equations with odd numbers in the global system of
equations ........................................

Deﬁnition of the parameters c, and c, in Equation [4.20] for

equations with even numbers in the global system of
equations ........................................

Dimensions of the matrices [Ca]. [Kc]. [Ac]. [Pa]. and [X0] in
full and handed forms ..............................

General information and furrow evaluation data for the
Colorado study sites that were used in model testing
(source: Elliott at al., 1982b) .........................

Additional data for model testing (source: Walker and
Skogerboe, 1987) ..................................

Rate of advance predictions for several time steps using the
kinematic wave model of FE-SURFDSGN ..............

Rate of advance predictions for several time weighting

coeﬁcients using the kinematic wave model of
FE-SURFDSGN ...................................

Rate of advance predictions for several a weighting

coeﬁcients using the kinematic wave model of
FE-SURFDSGN ...................................

Input data to investigate the sensitivity of the model to the
change of various physical parameters .................

138

162

163

164

183
184

203

204

205

207

Figure 1.

Figure 2.

Figure 3.

Figure 4.

Figure5.
Figure6.

Figure7.

Figure8.

Figure9.

Figure 10.

Figure 11.

LIST OF FIGURES

An enlarged ﬂuid element within a furrow with a spatially
varied unsteady ﬂow ..............................

Acting forces on an enlarged ﬂuid element within a furrow
with a spatially varied unsteady ﬂow .................

Finite element discretization. (A) Furrow ﬂow. (B)
Generic one-dimensional linear element ...............

Finite element discretization. (A) Furrow ﬂow. (B)
Generic quadratic element in the system of local
coordinates. (0) Generic quadratic element in the system
of natural coordinates .............................

Plot of em as a function of time ......................

Schematic diagram of surface ﬂow and inﬁltration during
the ﬁrst time step of the linear one-dimensional ﬁnite
element solution ..................................

Schematic progression of surface ﬂow and inﬁltration over
linear one-dimensional ﬁnite element grid with a constant
time step ........................................

Schematic diagram of ﬂow phases in surface irrigation
systems: (a) advance phase, and (b) pending phase ......

Schematic diagram of ﬂow phases in surface irrigation
systems: (a) depletion phase, and (b) recession phase . . . .

Linear plot of storage requirements of the global stiﬁ'ness
or capacitance matrices versus total number of linear
elements ........................................

Semi-log plot of storage requirements of the global stiffness
or capacitance matrices versus total number of linear
elements ........................................

46

51

68

90
131

147

148

149

150

165

166

Figure12.

Figure 13.

Figure 14.
Figure 15.
Figure 16.
Figure 17.

Figure 18.

Figure 19.

Figure 21.

Figure22.

Linear plot of storage requirements of the global stiffness
or capacitance matrices versus total number of quadratic
elements ........................................

Semi-log plot of storage requirements of the global stimiess
or capacitance matrices versus total number of quadratic
elements ........................................

First screen of input data into FE-SURFDSGN. ........
Second screen of input data into FE-SURFDSGN. .......
Third screen of input data into FE-SURFDSGN. ........

Predicted and observed advance for irrigation 1, group 1,
furrow 5 on Benson farm using the kinematic-wave ﬁnite
element model which was based on the nonstandard
Galerkin formulation (source of actual data: Elliott at al.,
1982b) ..........................................

Predicted and observed advance for irrigation 5, group 2,
furrow 1 on Benson farm using the kinematic-wave ﬁnite
element model which was based on the nonstandard
Galerkin formulation (source of actual data: Elliott at al.,
1982b) ..........................................

Predicted and observed advance for irrigation 2, group 3,
furrow 5 on Matchett farm using the kinematic-wave ﬁnite
element model which was based on the nonstandard
Galerkin formulation (source of actual data: Elliott et al.,
1982b) ..........................................

. Predicted and observed advance for irrigation 1, group 4,

furrow 5 on Matchett farm using the kinematic-wave ﬁnite
element model which was based on the nonstandard
Galerkin formulation (source of actual data: Elliott et al.,
1982b) ..........................................

Predicted and observed advance for irrigation 8, group 2,
furrow 3 on Printz farm using the kinematic-wave ﬁnite
element model which was based on the nonstandard
Galerkin formulation (source of actual data: Elliott at al.,
1982b) ..........................................

Predicted and observed advance for irrigation 1, group 1,
furrow 1 on Printz farm using the kinematic-wave ﬁnite

167

168
175
176
178

187

188

189

190

191

Figure23.

Figure 25.

Figure26.

Figure 27.

Figure28.

Figure29.

element model which was based on the nonstandard
Galerkin formulation (source of actual data: Elliott at al.,
1982b) ..........................................

Predicted and observed advance for irrigation on Flowell
continuous wheel furrow using the kinematic-wave ﬁnite
element model (source of actual data: Walker and
Skogerboe, 1987) ..................................

. Predicted and observed advance for irrigation on Flowell

continuous nonwheel furrow using the kinematic-wave
ﬁnite element model (source of actual data: Walker and
Skogerboe, 1987) ..................................

Predicted and observed advance for irrigation on Kimberly
continuous ﬂow wheel furrow using the kinematic-wave
ﬁnite element model (source of actual data: Walker and
Skogerboe, 1987) ..................................

Predicted and observed irrigation data for irrigation 1,
group 1, furrow 5 on Benson farm using the
kinematic-wave ﬁnite element model (source of actual
data: Elliott at al., 1982b, advance data; Oweis, 1983,
recession data) ...................................

Predicted and observed irrigation data for irrigation 5,
group 2, furrow 1 on Benson farm using the
kinematic-wave ﬁnite element model (source of actual
data: Elliott et al., 1982b, advance data; Oweis, 1983,
recession data) ...................................

Predicted and observed irrigation data for irrigation 2,
group 3, furrow 5 on Matchett farm using the
kinematic-wave ﬁnite element model (source of actual
data: Elliott et al., 1982b, advance data; Oweis, 1983,
mcession data) ...................................

Predicted and observed irrigation data for irrigation 1,
group 4, furrow 5 on Matchett farm using the
kinematic-wave ﬁnite element model (source of actual
data: Elliott et al., 1982b, advance data; Oweis, 1983,
recession data) ...................................

. Predicted and observed irrigation data for irrigation 8,

group 2, furrow 3 on Printz farm using the kinematic-wave
ﬁnite element model (source of actual data: Elliott at al.,
1982b, advance data; Oweis, 1983, recession data) ......

xiv

192

193

194

195

196

197

198

199

200

Figure 31.

Figure 32.

Figure33.

Figure34.

Figure35.

Figure36.

Figure 37.

Figure38.

Figure39.

Predicted and observed irrigation data for irrigation 1,
group 1, furrow 1 on Printz farm using the kinematic-wave
ﬁnite element model (source of actual data: Elliott et al.,
1982b, advance data; Oweis, 1983, recession data) ......

Sensitivity of simulated advance and recession curves of
FE-SURFDSGN kinematic wave model to inﬂow rate, Q,,

at ﬁeld inlet ......................................

Sensitivity of simulated advance and recession curves of
FE-SURFDSGN kinematic wave model to ﬁeld slope, So . .

Sensitivity of simulawd advance and recession curves of
FE-SURFDSGN kinematic wave model to Manning's
coeﬁcient, n .....................................

Sensitivity of simulated advance and recession curves of
FE-SURFDSGN kinematic wave model to the hydraulic
section parameter, p, ..............................

Sensitivity of simulated advance and recession curves of
FE-SURFDSGN kinematic wave model to the hydraulic

section parameter, p, ..............................

Sensitivity of simulated advance and recession curves of
FE-SURFDSGN kinematic wave model to the inﬁltration
function coeﬁcient, k ..............................

Sensitivity of simulated advance and recession curves of
FE-SURFDSGN kinematic wave model to the inﬁltration
function exponent, a ...............................

Sensitivity of simulated advance and recession curves of
FEoSURFDSGN kinematic wave model to the inﬁltration
function coeﬁcient, j}, ..............................

201

208

209

210

211

212

213

214

215

[Ac]

LIST OF SYMBOLS

([Cal +9Athal)

space weighting coeﬁcient of the nonstandard Galerkin
formulation

cross-sectional area of ﬂow

empirical parameter in the Kostiakov-Lewis inﬁltration equation
soil parameter in the Philip’s inﬁltration equation

furrow hydraulic section parameter

furrow hydraulic section exponent

global capacitance matrix

celerity wave

Chezy coeﬁcient

element capacitance matrix

time step

change in the stored volume within the element during time dt
length of the cell

drag coeﬁcient (AS,)

empirical parameter in the US. Soil Conservation Service
inﬁltration equation

empirical parameter in the Kostiakov-Lewis inﬁltration equation

force due to ﬁiction in a ﬂuid element

{F5}
{Fa}
{Fa}.

{F0};

'11
i

‘5‘»

{F }
71
72

hi

if

Ail“ - 9) {Fe}. + alpalb)
global force vector

global force vector at time a
global force vector at time b

pressure force

force due to the weight of a ﬂuid element in the direction of ﬂow
Darcy-Weisbach’s roughness coeﬁcient

empirical parameter in the US. Soil Conservation Service
inﬁltration equation

element force vector

furrow ﬂow geometry empirical ﬁtting constant

furrow ﬂow geometry empirical ﬁtting exponent

speciﬁc weight of water

acceleration due to gravity

empirical parameter in the US. Soil Conservation Service
inﬁltration equation

suction at the wetting front in the soil in inﬁltration

ﬁnal inﬁltration rate

inﬁltration rate per unit length

coeﬁcient in the general uniform ﬂow equation

coeﬁcient in the general uniform ﬂow equation

global stimiess matrix

element stiffness matrix

empirical parameter in the Kostiakov-Lewis inﬁltration equation
hydraulic conductivity in wetted zone

{d’a}

{<5}

{4’}
{4’}.

{4’}:

length of the element
mass of the ﬂuid element

shape function at node i
Manning’s roughness coefﬁcient
([Ca] - (1 - 9)At[Kal)

global vector of the derivatives of the unknowns, A and Q , with
respect to time

global vector of unknowns, A and Q

element vector of the derivatives of unknowns, A and Q , with
respect to time

element vector of unknowns

element vector of unknowns at time a
element vector of unknowns at time b
space-averaging coeﬁcient

wetted perimeter

inlet ﬂow rate into a furrow or inlet ﬂow rate per unit width into
a border

ﬂow rate

element residual vector

global residual vector

furrow hydraulic section parameter
furrow hydraulic section exponent
density of water

hydraulic radius which represents the ratio of the cross-sectional
area of ﬂow, A, to the wetted perimeter, P

slope of the channel bed

slope of energy line or friction slope

furrow ﬂow geometry empirical ﬁtting constant
furrow ﬂow geometry empirical ﬁtting exponent
integrand of the direction of ﬂow

sorptivity which is a soil parameter in the Philip’s equation
time at which the ﬁnal inﬁltration rate is reached
saturated soil moisture content

initial soil moisture content

time weighting coeﬁcient

time at which the advancing front reaches distance 8
inﬁltration opportunity time

time

top width of ﬂow cross section

total inﬂow during a time step, dt

total inﬁltration during a time step, (it

total outﬂow during a time step, dt

surface volume of water

inﬁltrated volume of water

ﬂow velocity

weighting function

Escoﬁer stage variable

distance along the direction of ﬂow

ﬂow depth

inﬁltration volume per unit length

I. INTRODUCTION

Surface irrigation is the most important method of irrigation in the world.
Kay (1986) reported that surface irrigation still covers more than 95% of all
irrigated land worldwide in spite of the popularity of sprinkler and trickle
irrigation systems. In the United States, around 62% of irrigated land is
currently under surface irrigation (Bajwa et al., 1987). These ﬁgures reﬂect
the importance of this type of irrigation practice and the need for additional
improvements in both the design and operation of surface irrigation systems.
As Bassett et al. (1980) described it, this process may be accomplished through
the ”skillful combination of experience and thorough understanding of the
processes involved". Recent theoretical developments in the area of surface
irrigation system design should serve as an excellent means for better
understanding the physical processes involved in the hydraulic design and
ﬁeld evaluation of surface irrigation systems. These developments involve the
application of numerical analyses to the modeling of surface irrigation systems
which will in turn increase the accuracy of the design and improve the

performance of the system with the least incurred cost.

The earliest models of surface irrigation systems dealt only with water
advance down the border or furrow. Among the many early works that were
based on this approach are the developments by Lewis and Milne (1938), Hall
(1956), Philip and Farrell (1964), and Hart et a1. (1968). These approaches
were based on the principle of conservation of mass together with assumptions
regarding average depth of surface ﬂow. These have resulted in assumptions
that water at the upper end of the border is at normal depth and that both

1

surface and subsurface water proﬁles are of a monomial power law of a ﬁxed or
assumed degree. The application of these assumptions produced acceptable
results at times and fundamental errors at other times (Strelkoﬂ' and
Katopodes, 1977a).

An alternative approach is to numerically solve the two partial
diﬂ'erential equations that govern the unsteady water ﬂow conditions in open
channels. These partial differential equations, or the so-called Saint-Venant
equations, consist of an equation of continuity and an equation of motion. The
latter equations can be developed based on applying the principles of
conservation of mass and momentum or energy to a controlled volume of ﬂow
in a channel. The hydrodynamic equations, which have been studied since the
turn of century, describe the unsteady spatially varied ﬂow of water across the
soil surface. With the recent rapid advancement in numerical techniques and
the computing potentials of computers in general and microcomputers in
particular, the possibilities and alternatives of solving surface irrigation
problems based on the numerical approximations of the hydrodynamic
equations look increasingly promising. In recent years, the hydrodynamic
equations were applied to the analysis of overland ﬂow in watershed
hydrology, open channel ﬂow, and surface irrigation. These equations were
utilized either in complete form or after implementing some simplifying
assumptions which would result in the zero-inertia and the kinematic wave
models.

A. Scope and Objectives

The general scape of this research work was to develop ﬁnite element
Galerldn formulations of the complete and simpliﬁed forms of the
hydrodynamic equations (equations that were developed based on the

application of the conservation of mass and momentum principles) as applied

to the simulation of ﬂow conditions in surface irrigation systems. The physical

ﬂow conditions in border and furrow irrigation systems were represented by a
mathematical model that could incorporate the aforementioned complete and
simpliﬁed forms of the hydrodynamic equations with a ﬁnite element
numerical solution procedure, initial and boundary conditions, and other

necessary physical parameters. The speciﬁc objectives of this research were

1.

To develop a ﬁnite element solution procedure of the Saint-Venant
equations for the hydraulic analysis of surface irrigation systems.

To create a general solution approach that will accommodate the available
mathematical models of the Saint-Venant equations in the analysis of
surface irrigation systems.

To develop an approach to easily incorporate the varying boundary
conditions of the advance, pending, depletion, and recession phases of
surface irrigation into the solution process with minimal arbitrary or
experimental parameters.

To develop a computer model that will utilize the above mathematical
concepts for the hydraulic analysis of ﬂow conditions in border and furrow

irrigation systems.
To numerically evaluate and compare the results of the ﬁnite element

model to actual ﬁeld measurements from existing surface irrigation
systems.

II. REVIEW OF THEORY AND LITERATURE

The analysis of surface irrigation systems is a process that involves many
parameters. One approach. for the design is to establish analytical
relationships among the various factors aﬁ'ecting the ﬂow conditions in surface
irrigation problems. These factors include length of the ﬁeld, inﬂow time,
inﬂow rate, surface rtmoﬂ‘, deep percolation, application depth, soil intake
characteristics, and land slope. This approach has been utilized in the design
of surface irrigation systems for many years. However, it is incapable of
deﬁning or estimating the spatially distributed ﬂow conditions in surface
irrigation systems.

The process of overland ﬂow across a soil surface is both spatially varied
and unsteady. The principles of continuity of mass and continuity of
momentum or energy can be applied to describe overland ﬂow conchtions in
irrigation systems. Applying these concepts will result in the so-called
hydrodynamic equations that are commonly known as the Saint-Venant
Equations. These equations, which have been studied since the turn of
century, describe the unsteady spatially varied ﬂow of water across soil
surface. Originally, graphical solutions were utilized for the solution of the
above equations. However, the application of these equations was limited to
restrimd, simpliﬁed cases (Strelkoﬁ', 1970). With the advent of high speed
digital computers, numerical approximations of the Saint-Venant equations
Mcame feasible.

This review expounds the basic background of surface irrigation as
related to surface ﬂow description and inﬁltration. Also, it elucidates the

4

theory in the literature on the numerical solution procedures of the
Saint-Venant equations, particularly those utilizing the ﬁnite difference and
ﬁnite element methods as well the method of characteristics to describe the
spatially distributed ﬂow conditions in surface irrigation systems.

A. Basic Background of Surface Irrigation

The basic modeling of surface irrigation systems involves many factors
that are generally common to the available types of systems. These types
include borders, furrows, and basins (Walker and Skogerboe, 1986). The
hydraulic ﬂow characteristics of basins is a special case of the border ﬂow
(Bassett et al., 1980). For this reason, only the hydraulics of borders and
furrows will be discussed in this section. The geometry of ﬂow in both furrow
and border irrigation systems are generally similar. The major difference
arises from the width of the channel which is narrow for furrows and usually
wide for borders. The width of the border strip is generally wide enough to
ignore the contribution of channel walls to both ﬂow retardate and inﬁltration,
an assumption which is not applicable to irrigation furrows (Bassett et al.,
1980).

The different types of surface irrigation systems involve numerous
physical characteristics that may be deﬁned by common terminology. This
terminology, together with the basic concepts of surface irrigation systems, is
well documented in the literature. This section will review the basic concepts
of surface irrigation systems without much detail. The reader is referred to
Walker and Skogerboe (1986), Bassett et a1. (1980), and Kay (1988) for more
in-depth description of the diﬂ‘erent types of surface irrigation systems.

1. Flow Description

The ﬂow of free water in surface irrigation systems is gradually varied
and unsteady. The inﬁltration of water into the soil seems to dominate this
hydraulic characteristic of ﬂow. Bassett et a1. (1980) reported that the state of
water ﬂow in surface irrigation systems is mostly turbulent er transitional, a
ﬂow regime that is characterized by a Reynold’s number around or above 1000
even though numbers well below 1000 are frequently encountered. The
Reynold’s number is a dimensionless ratio of the inertial te viscous forces
(Binder, 1973).

The ﬂow regime of water in surface irrigation systems is usually
sub-critical. This is characterized by Froude numbers well below unity. The
Froude number is a dimensionless ratio of the inertial to the gravitational
forces. Bassett et al. (1980) reported that critical and super-critical ﬂow
regimes might occur just behind the wetting ﬁ'ent in the advance phase of both
sloping borders and furrows, and just ahead of the drying front during the

recession phase.

2. Flow Phases

There are four phases of ﬂow in a typical surface irrigation cycle. These
include the advance, pending, depletion, and recession phases. The advance
phase represents the ﬁrst portion of irrigation time during which water
advances down the furrow or border. The pending phase starts at the end of
the advance phase when the advancing front reaches the end of the furrow or
border. This phase extends till water is shut-eﬂ' at the inlet boundary. The
duration of this phase is zero if the inlet water is turned eﬁ‘ before water
reaches the end of the furrow or border. The next phase of irrigation
represents the portion of the total irrigation time between inlet ﬂow shut-eff

and the beginning of water recession at the inlet boundary. The ﬁnal phase of
ﬂow is the recession phase which represents the portion of the total irrigation
time between the beginning of water recession at the inlet boundary and the
complete disappearance of water hour the furrow or border.

B. Surface Irrigation Inﬁltration

The simulation of ﬂow in surface irrigation systems relies on the
knowledge of the hydraulic characteristics and inﬁltration (Walker and
Humpherys, 1983, and Strelkeﬁ‘ and Seuza, 1984). However, the most critical
step in studying the hydraulics and distribution of water in surface irrigation
systems is to establish a reasonable estimate of the inﬁltration function.
Clemmens (1981) described the. determination of the inﬁltration
characteristics of the soil as "the biggest stumbling block in accurately
describing or predicting the irrigation process". Inﬁltration has always been
referred to as the impeding factor to improving both the design and
performance of surface irrigation systems (Elliott at al., 1982a, and Elliott et
al., 1983b).

1. Inﬁltration Equations

There are many inﬁltration equations that have been developed
throughout the years. Walker and Skegerbee (1987) classiﬁed these equations
into thme general categories. These include theoretically, physically, and
empirically based equations. A good example of an inﬁltration equation that
was developed based on the single-phase ﬂew solution of the one-dimensional
Darcy equation is the Richard’s equation (Walker and Skegerbee, 1987). This
equation has the form

a[K(e)(§-- 1)]

39
5'"- az [2.1]

 

where 8 is the soil moisture content on volume basis, It is the total pressure
head, K is the hydraulic conductivity, 2 is the vertical distance downward from
the soil surface, and t is the inﬁltration opportunity time.

The Green and Ampt equation is a good example of the physically based
equations. It was developed based on the assumption that soil could be
modeled as a bundle of capillary tubes (Hillel, 1980). The Green and Ampt
equation has the form (Walker and Skogerboe, 1987)

_ (er-0.)",
1-x[1+ Z ] [221

where I is the inﬁltration rate, 0, is the saturated soil moisture content, 0,- is

 

the initial soil moisture content, 11’ is the suction at the wetting front in the
soil, K is the hydraulic conductivity in the wetted zone, and Z is the cumulative
inﬁltration depth at any speciﬁc point in the space dimension.

Inﬁltration is a complex physical process that is very diﬁcult to
characterize in irrigated ﬁelds where anistropic and heterogeneous conditions
usually prevail. This leads to the conclusion that an empirical approach to
assess this process is more practical than a purely theoretical approach (Blair
and Smerdon, 1987).

There are many empirical inﬁltration equations that were developed in
the literature. These empirical inﬁltration equations have always been
expressed in either exponential or power form. However, the inﬁltration
equations in power forms have been widely adopted to estimate inﬁltration in
surface irrigation problems because of the simplicity and practicality of these
e(minions (Fok. 1967).

Empirical inﬁltration equations are the result of ﬁtting observed
inﬁltration data to explicit time-dependent equations. Many inﬁltration
equations fall under this category. A good example of these equations are the
Kostiakov and the Kostiakov-Lewis equations. The Kostiakov equation is one
of the earliest equations of inﬁltration (Hillel, 1980). This equation has the

form
I = alct("” [2.3]

where t is the inﬁltration opportunity time and a and k are two empirical

parameters obtained from inﬁltration tests in the ﬁeld. The major draw back
of this equation is that the inﬁltration rate approaches zero at long times.
Hence, it is more applicable to horizontal rather than vertical inﬁltration. The
Kostiakov-Lewis equation has the same form as the Kostiakov equation with
an added term to correct the latter problem of the Kostiakov equation (James,
1988). This results in

I = akt“"’+ﬁ, [2.4]

where 15, is an empirical parameter that represents the inﬁltration rate as the

inﬁltration opportunity time, t, becomes considerably large. Elliott and
Walker (1980 and 1982) used the Kostiakov-Lewis function which has the
additional term for the asymptotic long-time inﬁltration. Their results
suggested that the Kostiakov-Lewis inﬁltration equation is highly eﬁ'ective in
pmdicting inﬁltration if a steady state inﬁltration rate can be assessed.

Clemmens (1981) suggested using a two-branch frmction of the Kostiakov
equation where the cumulative inﬁltration depth, Z, is represented by two
equations. The ﬁrst equation applies before the steady state inﬁltration is
reached while the second applies after. Clemmens’ proposed equations for the
cumulative inﬁltration have the form

10

Z=kt‘ for 25:,

= kr,‘ + i,(t — t,) for t > t, [2.5]

where i, is the ﬁnal inﬁltration rate and t, is the time at which the ﬁnal
inﬁltration rate is reached.

Philip (1957a and 1957b) developed an equation similar to the
Kostiakov-Lewis equation but with more physical signiﬁcance. His equation
has the form

2 =s:°-“+A: [2.6]
where S is the sorptivity. The parameter A was deﬁned by Philip as

A =K, + I =K,+[§Ko-K,] [2.7]

8

where K, and K, are soil parameters. Kunze and Ker-Keri (1983) and Kunze

and Shayya (1990) deﬁned the parameter A as the hydraulic conductivity of
the soil and suggested the adjustment of the sorptivity term by a factor to
implement this denotation. Fangmeier and Ramsey (1978) made a comparison
between the Philip equation and the Kostiakov-Lewis equation. They
concluded that the Philip equation provided better estimates of inﬁltration
compared to the Kostiakov-Lewis equation. However, the ceeﬁcients in the
Philip equation are more diﬁcult to obtain.

The US. Soil Conservation Service (1974 and 1984) developed an
equation that relates the cumulative inﬁltration and the opportunity time as

z=eH+g [in

where e, f, and g are empirical parameters.

11

2. Parameter Estimation of Inﬁltration Equations

Determining inﬁltration parameters in the various empirical inﬁltration
equations previously discussed is a very critical step in the design and
operation of surface irrigation systems. One approach for the estimation of
these parameters can be accomplished through the utilization of direct ﬁeld
measurements of inﬁltration using ring inﬁltrometers. However, such
measurements do not reﬂect the actual hydraulic characteristics at the time of
irrigation (Beuwer, 1957, and Blair and Trout, 1989). They can also be very
time consuming and costly. Many scientists attempted to assess the
variability of inﬁltration parameters and inﬁltration along furrows using
inﬁltrometers and moisture measurements (Bautista and Wallender, 1985;
Izadi and Wallender, 1985; and Bali and Wallender 1986). Brakensiek et al.
(1979) discussed the application of a new inﬁltremeter system and its
utilization in the estimation of inﬁltration parameters in the Green and Ampt
function. Blair and Trout (1989) presented a ﬁeld guide for the construction
and operation of a recirculating inﬁltrometer, a device that can be used in the
measurement of inﬁltration and the estimation of inﬁltration parameters in
various inﬁltration functions.

Another approach for the parameter estimation of inﬁltration equations
is to implement a numerical solution procedure of surface ﬂow equations in
conjunction with advance, storage, and inﬂow-outﬂow ﬁeld measurements.
Christiansen et al. (1966) and Fangmeier and Ramsey (197 8) implemented the
volume balance method in the estimation of inﬁltration parameters in the
Kostiakov equation. Elliott at al. (1983a) simulated advance trajectories of
water in furrows using assumed inﬁltration frmctions and the zero-inertia
model. They then estimawd the actual ﬁeld inﬁltration using actual ﬁeld
advance trajectories and simulated advance trajectories by the zero-inertia

12

model. Katopodes et al. (1991) presented a procedure for estimating both
inﬁltration parameters and soil roughness in surface irrigation systems using
a linearized zero-inertia model.

In 1987, Kaytal et al. developed an inﬁltration equation that accounts for
the two-dimensional inﬁltration ﬂow conditions in furrows. Their approach
was an attempt to develop inﬁltration functions that are applicable throughout
the growing season instead of one irrigation only. The approach that they
followed included the numerical solution of the Richard’s inﬁltration equation
in furrows using the ﬁnite element method. The Richard’s equation was
utilized in conjunction with basic soil data and furrow shape parameters in
order to estimate the parameters of a power inﬁltration function. The latter
function had two independent variables which include the top width of ﬂow as
well as the cultivation depth.

More mcently, Walker and Busman (1990) discussed an approach to use a
kinematic wave simulation model in conjunction with the simplex method to
determine the inﬁltration parameters ﬁ'em early stages of furrow advance.
Their approach was based on minimizing the diﬁ‘erences between predicted
and measured advance rates. They concluded that this procedure will provide
estimates of the inﬁltration parameters with suﬁicient accuracy.

The Kostiakov and Kostiakov-Lewis inﬁltration equations are the most
widely used inﬁltration equations in surface irrigation problems. The
parameter-estimation of these equations was the subject of many
investigations. Norum and Gray (1970) presenwd a method for deriving the
values of the parameters in the modiﬁed Kostiakov function. Smith (1972)
used surface irrigation data and a kinematic wave model to characterize the
inﬁltration parameters in the Kostiakov-Lewis equation. Elliott at al. (1982a
and 1983b) presented a method for deriving the values of the parameters in the

13

modiﬁed Kostiakov function using dimensionless advance curves and furrow
irrigation ﬁeld data. Elliott and Walker (1980 and 1982) reported on the use .
of two-point volume balance methodology to establish the parameters of the
Kostiakov-Lewis function based on advance data measurements. Sirjani and
Wallender (1989) attempted to approximate the mean and variance of the
parameters in the Kostiakov-Lewis inﬁltration equation using the ﬁrst order
analysis.

One last method for estimating the inﬁltration parameters in various
inﬁltration functions is to analyze the inﬂow-outﬂow measurements. The
inﬁltration is calculated as the difference between measured water inﬂow and
outﬂow from a border or furrow section. However the major disadvantage of
this method is the sensitivity of the inﬁltration estimates to the accuracy of
ﬂow measurements (Trout and Mackey, 1988b). The accuracy of these
measurements depend on many factors. These factors are aﬁ'ected by the ﬂow
characteristics and the geometry of furrows and borders. Bautista and
Wallender (1985) reported that inﬁltration rate is usually greater in blocked
furrows with ﬂowing water compared to stagnant tests.

Strelkoﬁ‘ and Souza (1984) considered six diﬁ‘erent schemes for
incorporating the variable depth eﬁ'ect into the computations of inﬁltration in
furrows. They concluded that the "wetted perimeter based on local depth is the
best choice of traverse length to characterize furrow intake in mathematical
models of furrow irrigation". Izadi and Wallender (1985) statistically
examined furrow hydraulic characteristics in space and time and related these
characteristics to inﬁltration. Trout (1986) conducted a study to measure the
eﬂ‘ect of both overland ﬂow velocity and furrow hydraulic parameters on
inﬁltration. His conclusion was that furrow inﬁltration increases by
increasing the wetted perimeter of ﬂow. He also reported that ﬂow velocity is
inversely related to furrow inﬁltration rate.

14

To summarize, there are various methods for obtaining inﬁltration data
of surface irrigation systems. These include the ring inﬁltrometer, pending,
two—point, blocked furrow, inﬂow/outﬂow, and recirculating inﬁltrometer
methods. As outlined in this section, these procedures were utilized directly
and indirectly by many scientists in the parameter estimation of the
inﬁltration functions for surface irrigation systems. The reader is referred to
the texts by James (1988) and Walker and Skegerbee (1987) for a detailed
description of the various inﬁltration measurement devices and the procedures
followed in their implementation in the ﬁeld.

3. Effect of Inﬁltration on System Performance

Improving the eﬁciency of furrow irrigation systems can be attained by
accurate assessment of inﬁltration (Elliott and Walker, 1980, and Elliott and
Walker, 1982). On the other hand, accounting for the spatial variability in
inﬁltration is very essential in evaluating the performance of surface irrigation
systems (Bautista and Wallender, 1985, and Davis and Fry, 1963). Trout and
Mackey (1985 and 1988a) attempted to quantify the eﬁ'ect of both inﬂow and
inﬁltration variability on the uniformity of water application and runoff in
furrows. They concluded that the consequences of inﬂow and inﬁltration
variability are excessive deep percolation as well as runoﬂ’lesses in some areas
of the ﬁeld while other areas receive inadequate amounts of water. Sirjani and
Wallender (1989) conducted a study to assess the eﬂ‘ect of temporal and spatial
variability of inﬁltration on the performance of furrow irrigation systems.
Fonteh and Pedmore (1989) developed a kinematic wave furrow irrigation
model using a physically based inﬁltration function. Their model could
simulate spatially varied inﬁltration along furrows using geostatistics and
their results seemed to be most accurate under ﬁne textured soil conditions.

15

The spatial and temporal variability in inﬁltration seems to affect crop
yield. Kemper et al. (1982) rede that yield decreases substantially by low
uniformity of water distribution throughout the length of the run. They
presented several treatments which can increase or decrease inﬁltration in
order to improve uniformity. One approach for reducing inﬁltration rate
involves the practice of surge irrigation. Surge irrigation is deﬁned as "the
intermittent application of irrigation water to furrows or borders" (Bishop et
al., 1981). Surge ﬂow creates a series of on and oﬁ‘ inﬂow conditions at the
inlet (Izuno and Pedmore, 1986). The characterization of inﬁltration under
surge irrigation systems was studied by many scientists (Bishop et al., 1981;
Izuno et al., 1985; Kemper et al., 1988; and Samani et al., 1985). The reader is
referred to these references for more details.

C. Surface Flow Equations

The principles of mass, and momentum (or energy) can be utilized to
describe the ﬂow of water over soil surface. These principles result in two
ﬁrst-order, nonlinear partial diﬁ‘erential equations. The resultant differential
equations approximate the spatially varied and unsteady ﬂow conditions in
open channels and surface hydrology. These same equations can also be
applied to the hydraulic analysis of water ﬂow conditions in surface irrigation
systems.

1. Saint-Variant Equations

The two equations that result ﬁem applying the above principles are
known as the Saint-Venant equations. The ﬁrst equation results from

16

applying the conservation of mass principle to a control volume of ﬂow in
surface irrigation systems. It is usually referred to as the continuity equation
and has the form

8A 39 _
ar+ax+"° [2.9]

where A is the cross sectional area of ﬂow, Q is the ﬂow rate, I is the
inﬁltration rate per unit length, r is time, and x is the distance along the
direction of ﬂow. The second equation is referred to as the momentum
equation and results from applying the conservation of momentum principle to
the same ﬂuid element. This equation has the form (Strelkoﬁ', 1969)

.. .29. 33.2 -2: 91 _1- 12
S°‘Sl+[A2g)ax+[1 Angax+[Ag)at [2.10]

where y is the ﬂow depth, 3 is the acceleration due to gravity, T is the top

width of ﬂow cross section, So is the slope of the channel bed, and S, is the
friction slope. Equation [2.10] describes unsteady non-uniform ﬂow
conditions. The last term in the above equation cancels out if steady
non-uniform ﬂow conditions are presumed. Of note, with steady uniform ﬂow
assumptions, the last three terms of [2. 10] mutually cancel.

Another approach can be followed to develop an equation which can
replace [2. 10]. This development can be based on the principle of conservation
of energy. However, both the approach and the resultant equation will be
inherently diﬂ'erent from the equation developed based on the momentum
approach (Martin and Wiggert, 1975; Yen, 1973; and Strelkoﬂ', 1969).

Brutsaert (1971) veriﬁed the Saint-Venant equations experimentally for
open channel ﬂow conditions. His approach included the comparison of the
results from the numerical solution of these equations to the ﬂow

17

measurements of a physical model. The solution procedure encompassed the
implementation of a ﬁnite diﬁ‘erence computational algorithm with implicit
and explicit diﬁ‘erences at interior cells and boundaries, respectively. He
concluded that the Saint-Venant equations represented the physical ﬂow
system reasonably well. However, these positive results are only attainable
under proper mathematical conditions and appropriate description of
boundary conditions, hydraulic resistance to ﬂow, - and channel parameters.
The formulation and veriﬁcation of the Saint-Venant equations for other
related problems were the concern of many scientists. These analyses were
reported in many references including Morgali and Linsley (1965), Kruger and
Bassett (1965), Weeding (1965b). Brakensiek (1966), Chen and Hansen (1966),
Ram (1966), Liggett and Woolhiser (1967), Strelkoﬂ' (1969), Brutsaert (1971).
and Katopodes and Schamber (1983). These references concluded basically
that the application of the Saint-Venant equations to surface ﬂow problems
produwd good results when the various assumptions implemented in the
development of these equations were not violated. Many of these assumptions
were discussed in details in the journal articles by Strelkoﬂ‘ (1969) and Yen
(1973).

2. Uniform Flow Equations

There are three popular equations for establishing relationships among
ﬂow rate, slope of channel bed, and channel geometry. These equations are
essential for deﬁning the friction slope in the momentum equation, [2. 10].
Originally, these equations were deve10ped for the analysis of uniform ﬂow
conditions in open channels. Thus, these equations are frequently referred to
as uniform ﬂow equations. However, these equations may be used to
approximate the friction slope for nonuniform, unsteady, turbulent ﬂow
conditions at a given instant (Morgali and Linsley, 1965). The ﬁrst equation is

18

known as the Chezy equation after it was introduced by a French engineer of
that name in 1768 (Henderson, 1966). The development of the Chezy equation
was based on the dimensional analysis of the resistance equation with the
assumption that ﬂow conditions are uniform. The Chezy equation has the

form
Q = CAR "’1',"2 [2.11]

where C is the Chezy coemcient and R is the hydraulic radius which

represents the ratio of the area of ﬂow to the wetted perimeter.

A more practical formula was developed in 1889 by an Irish engineer by
the name of Robert Manning (Chow, 1959). The Manning equation is widely
used for steady ﬂow conditions of incompressible ﬂuids in prismatic open
channels (Streeter and Wylie, 1979). The Manning equation is an empirical
equation with the form

-4 .1. .1.
Q - n R S, [2.12]
where n is Manning’s roughness coeﬁcient. The Manning equation is very

popular in many Western countries (Henderson, 1966). Besides, it has well
documented roughness coeﬁcients that were developed over the years.

The third equation is the Darcy-Weisbach equation that was originally
developed for pipe-ﬂow. This equation can be expressed as (Brater and King,
1976)

Q = ifs-AR ”’5,“ [2.13]

where f is Darcy-Weisbach’s roughness coeﬁcient.

19

The relationships between C, n , and f can be summarized by

 

C =%R"‘ [2.14]
and
8 2
f= if}, [2.15]

These roughness coeﬁcients change with varying conditions (Walker and
Skogerboe, 1987). A complete discussion of these ﬂow equations can be found
in Chow (1959), Henderson (1966), Streeter and Wylie (1979), White (1979).
and Bassett et al. (1980).

D. Surface Irrigation Models

There are several mchniques that have been developed over the years to
model surface irrigation processes mathematically. Most of these techniques
are based on the application of the principles of conservation of mass and
momentum which are referred to as the Saint-Venant equations. The
Saint-Venant equations are applied either in complete or simpliﬁed forms.
This results in four general models, one complete form and three simpliﬁed
forms. The application of the simpliﬁed forms of the Saint-Venant equations
to the analysis of surface irrigation systems oﬂ‘er simpler and faster surface
irrigation modeling approaches. However, the accuracy of the results is
reduced either slightly or appreciably depending on the level of simpliﬁcations.

Strelkoﬁ‘ (1970) classiﬁed the numerical procedures that may be
implemented in the solution of the complete or simpliﬁed ﬁrst-order nonlinear
equations of hydrodynamic models into two categories. The ﬁrst approach is to
convert theoriginal system of partial diﬂ‘erential equations into an equivalent

20

system of ordinary differential equations. This transformation is
accomplished by changing the independent variables through the introduction
of an alternate coordinate system that is inclined at an angle to the original
space-time system (Strelkoﬁ’, 1970). The resulting ordinary system of
diﬂ‘erential equations is then solved numerically using methods such as the
ﬁnite diﬁ‘erence method. The resultant equations are algebraic instead of
being ordinary differential equations. This approach is referred to as the
method of characteristics. The theoretical basis for the method of
characteristics is reviewed in details by Stmeter and Wylie (1967 ), Strelkoﬁ‘
(1970), Liggett and Cunge (1975), and Wylie and Streeter (1983). The second
approach involves the implementation of one of many available numerical
solution schemes which directly replace the ﬁrst-order nonlinear partial
diﬁ‘erential equations by diﬁ'erence quotients. This approach results in a
system of algebraic equations instead of the original partial diﬂ'erential
equations.

1. Hydrodynamic Models

Mathematical models that result from applying the Saint-Venant
equations to the analysis of surface ﬂow problems without any simpliﬁcations
are refemd to as hydrodynamic models. These models have high potential to
be very accurate in predicting ﬂow conditions in surface ﬂow problems and
surface irrigation systems in particular. However, these models depend
heavily on the accuracy of the provided input information (Strelkoﬂ‘ and
Katopodes, 1977a). Bassett et al. (1980) described the roughness and
inﬁltration parameters as the two main sources of input error which affect the
precision of such models. Besides, the complex, delicate nature of the

21

hydrodynamic models represents their major drawback. This results in
lengthy computation times and makes such models expensive to operate when
it comes to computer cost (Bassett, 1980, and James, 1988).

Over the years, hydrodynamic models were developed by many scientists
based on the method of characteristics and applied to the analysis of ﬂow
conditions in open channels and surface irrigation systems. Bassett (1972)
developed a model of water advance in border irrigation by solving both the
continuity and momentum equations in their complete form for unsteady
spatially varied ﬂow conditions using the method of characteristics as
described by Streeter and Wylie (1967 ). In his development, Bassett utilized a
ﬁxed mctangular grid in the space-time plane. He used the Kostiakov
inﬁltration function and applied the Chezy equation to the evaluation of
ﬁiction slope. He concluded that the developed model predicts ﬂow conditions
in borders with acceptable accuracy. This conclusion was based on the
comparison of model results to actual laboratory and ﬁeld measurements.
Kincaid et al. (1972) and Sakkas and Strelkoﬂ‘ (1974) also applied the method
of characteristics to the solution of the complete hydrodynamic equations for
border irrigation advance. They reported that their developed models
predicted the ﬂow conditions in borders reasonably well. However, their
developments covemd only the advance phase of irrigation instead of the
complete phases of surface irrigation. However, they claimed that their
approaches can be extended to the simulation of the recession phase with
proper boundary conditions.

Bassett and Fitzsimmons (1976) extended the work of Bassett (1972) by
presenting a mathematical model for the analysis of the complete phases of the
irrigation process in borders. Their model was based on the same approach,
i.e, the application of the equations of continuity and momentum in their
complete forms using the method of characteristics. They reported good

22

results in general. However, they have also reported some problems of
instability and the high execution costs which were associated with the
mquired extensive computer time. Katopodes and Strelkoﬂ' (1977a) applied
also the method of characteristics to the approximation of the hydrodynamic
equations in borders. In their study, a comparison of the results from their
developed model and actual ﬁeld measurements were made and the associated
accuracy and costs of model execution were assessed. They concluded that the

solution results were correct to second order accuracy.

The major disadvantage of the method of characteristics usually lies in
the total number of unknowns. By implementing this method, the two
nonlinear partial diﬁ‘erential equations represented by equations [2.9] and
[2.10] are transformed into four ordinary diﬁ‘erential equations (Katopodes
and Strelkoﬂ‘, 1977a). These equations are

 

dour-(e); _ I(v-c)

—dt — g[.S'o S,+ Ag ] [2.16]
along

dx

3 _ v + c [2.17]
and

d(v-co)_ _ I(v+c)

_dt - g[So S,+ Ag ] [2.18]
along

dx

E - v - c [2.19]

23

where v is the ﬂow velocity, to is the Escoﬁer stage variable (Escoﬂier and

Boyd, 1962), and c is the celerity wave. The celerity and Escoﬁer stage
variables are deﬁned as

c = £14- [2.20]
and
(0: F” [221]
o C

where y is the ﬂow depth. The Escoﬁer stage variable, re, reduces to 2c when
the above method is applied to the hydraulic analysis of borders.

The ordinary diﬂ‘erential equations ([2.16] to [2.19]) that were the result
of applying the method of characteristics to the complete ﬁrst order, nonlinear
hydrodynamic equations were never applied successfully to the simulation of
ﬂow conditions in furrow irrigation systems (Walker and Skogerboe, 1987).

The ﬁnite diﬂ‘erence schemes for the direct solution of the Saint-Venant
equations are preferred by many scientists. Using these techniques, the
hydrodynamic equations are solved at a ﬁnite number of grid points in the
space-time plane (Liggett and Cunge, 1975). Two basic types of
ﬁnite-diﬁ‘erence schemes are usually used in the literature (Strelkoﬂ‘, 1970,
and Liggett and Cunge, 1975). The ﬁrst represents the explicit schemes where
the algebraic equations are arranged to be solved for one unknown at a time.
These numerical schemes, although simple, are unstable and usually require
excessive computation times due to the need for the selection of small time
steps (Liggett and Woolhiser, 1967, and Strelkoﬁ', 1970). This prompted many
investigators to avoid using these schemes, a trend which is especially true in
recent developments. On the other hand, the implicit schemes solve for a

24

group of unknowns simultaneously rather than one at a time. These
numerical schemes permit larger time steps but require the solution of a
system of nonlinear simultaneous equations at each time stop. They are
usually more desirable in the direct solution of the Saint-Venant equations
because of their stability and high accuracy. Various analyses of stability and
accuracy of both explicit and implicit ﬁnite diﬁ‘erence schemes, as applied to
the direct solution of the complete continuity and momentum equations, can be
found in Liggett and Woolhiser (1967 ), Strelkoﬁ' (1970), Price (1974), and
Liggett and Cunge (1975).

A number of scientists have reported on the successful application of
implicit ﬁnite diﬁ‘erence numerical schemes to the direct solution of the
Saint-Venant equations in open channels. The latest development was
reported by Swain and Chin (1990) for modeling unsteady ﬂow in regulated
open channels. Their development would also allow for the simulation of
hydraulic structures, an option that is not available in many current open
channel models.

Walker and Skegerbee (1987) presented an implicit ﬁnite diﬁ‘erence
scheme for the direct solution of the continuity and momentum equations in
furrows. Their scheme was based on the Eulerian integration approach, a
numerical procedure which approximates the hydrodynamic equations using
the concept of multi-cell deforming control volume. Using this approach, the
continuity and momentum equations as represented by [2.9] and [2. 10],
respectively, become

[0(QL " QR) + (1 - 9) (Q1 " Qﬂla‘ " [¢(AL 'l' Zr. "A1 " Zr)l8‘
"" [(1 "’ if) (A): + Zn - Au " 2.015! = 0 [2.22]

and

25

 

l[¢(QL-QJ)+(1’¢)(QR-QM)] +a[ (P +Q’IAg)r-(P +Q'IA8)L]
g 8: 5x

 

+(1-0)[ 5x

‘Soeer. + (1 - Win] -So(1 - 9) [M1 + (1 - (Mal
+9[¢Dt + (1 -¢)D.l + (1 - 9) [$01 + (1 -¢)Dul = 0 [2.23]

(P +Q2/A8)u-(P +Q2/Ag),]

where A is the cross sectional area of ﬂow, Q is the ﬂow rate across the

respective cell boundaries, Z is the inﬁltrated volume per unit length, ﬁt is the
time step, 5: is the length of the cell, 0 is the time-averaging coeﬁcient to
account for the nonlinear variation in the ﬂow proﬁle over time, o is the
space-averaging coeﬁcient to account for the nonlinear variation in the ﬂow
proﬁle over the cell length, D is the drag (AS,), and P is the pressure force. The
subscripts J and M in [2.23] represent the left and right cell boundaries,
respectively, at time t,- -, while the subscripts L and R represent the left and
right cell boundaries, respectively, at time t,. Walker and Skegerbee (1987)
presented a detailed description of their mathematical development from its
inception to the ﬁnal form as represented by the nonlinear algebraic equations
in [2.22] and [2.23]. They discussed the implementation and applications of
this implicit ﬁnite diﬂ'erence scheme to furrow and border irrigation systems.
Also, they discussed the basic advantages of this procedure over the method of
characteristics approach. The primary advantage relates to the number of
computational unknowns which is half as much in the former method
compared to the latter method.

As to the application of the ﬁnite element method to the numerical
solution of hydrodynamic equations, Katopodes (1984) developed a dissipative
Galerkin scheme for the solution of these equations as applied to open
channels with discontinuous ﬂow. All the energy diﬁ‘using terms were

26

neglected in the hydrodynamic equations and his development was restricted
to one-dimensional ﬂow in a prismatic channel with a rectangular cross
section. Katopodes reported excellent results using the dissipative Galerkin
scheme and was very optimistic about the utility of the ﬁnite element method
in computing surges and shocks in open channels. Akanbi and Katopodes
(1988) extended the work of Katopodes (1984) to two-dimensional overland
ﬂow problems. Their development was for the solution of ﬂood wave
propagation on initially dry land. They were very successful in solving the two
dimensional shallow-water equations with a dissipative Galerkin scheme
which involved a deforming and moving computational grid.

To conclude this section, it is clear that the one-dimensional gradually
varied hydrodynamic equations together with appropriate initial and
boundary conditions are capable of high accuracy. However, the numerical
solution of either the two partial differential motion equations or the four
ordinary diﬁ‘erential equations developed from applying the method of
characteristics is costly due to excessive requirement of computations which
can be time-consuming. This makes the solution of the simpliﬁed forms of the
Saint-Venant equations more desirable especially when such models produce
acceptable results (Miller and Cunge, 1975). The models that are based on the
complete hydrodynamic equations are not usually intended for the design or
speciﬁc applications because of the high operational costs. However, such
models can be used as standards of comparison for the simpliﬁed models which
have the potential to produce good results at minimal cost.

2. Zero-Inertia Models

The zero-inertia model is a simpliﬁed form of the hydrodynamic model.
The continuity equation ([2.9]) is kept unchanged while the acceleration and

27

inertial terms in the momentum equation ([2.10]) are ignored based on the
assumption that such terms are negligible in most ﬂow conditions of surface
irrigation systems (Strelkoﬂ‘ and Katopodes, 1977a). This assumption results
in the following simpliﬁed momentum equation

The zero-inertia model was ﬁrst proposed by Brakensiek et al. (1966) in
the context of ﬂood routing where he described the process of propagation of
ﬂood hydrographs through the watershed channel system. Brakensiek (1966)
discussed the appropriateness of the assumptions of the zero-inertia model.
His results revealed that the zero-inertia model produwd excellent results
compared to the full hydrodynamic model in the regions of slowly accelerating
ﬂow conditions in watershed channel systems. More recently, the accuracy of
the zero-inertia model in channel routing was the center of attention of many
investigations including those of Ponce et al. (197 8), Ponce and Theurer (1982),
and Ponce (1987).

Strelkoﬁ' (1972) and Katopodes (1974) were the ﬁrst to apply the
zero-inertia model to surface irrigation. Their applications were prepared for
borders. The solution technique that they followed was similar to the
nonlinear shooting technique which was utilized by Brakensiek et al. (1966).
However, the latter solution approach had some convergence problems
towards the end of the depletion phase. Strelkoﬁ' and Katopodes (1977a)
presented a new numerical approach for the solution of the zero-inertia model.
The numerical approach they developed precluded the problems that were
encountered in their previous work. Their model examined the process of
irrigation as a deforming control volume with upper and lower boundaries.
This numerical approach included the solution of a system of nonlinear

28

equations for each time step by ﬁrst linearizing and solving the system using
the double-sweep technique as described by Liggett and Cunge (197 5). The
selected time step was constant. Their linearization technique produced very
realistic results for the prediction of both advance and recession phases in
border irrigation systems.

The zero-inertia model developed by Strelkoﬂ‘ and Katopodes (1977a) was
the center of attention for many scientists. Several investigations were
conducted to reﬁne and verify the model against ﬁeld measurements. Strelkoﬁ'
and Katopodes (1977b) discussed establishing appropriate boundary
conditions for the zero-inertia model when a free over-fall downstream
boundary occurs. Clemmens and Fangmeier (1978) discussed the ways to
improve the numerical solution of the model when diked-end conditions occur
at the downstream boundary following the completion of the advance phase.
Clemmens (197 9) veriﬁed the zero-inertia model for advance and recession in
blocked-end borders with actual ﬁeld measurements. He concluded that the
agreement was generally good. However, he reiterated the notion that
successful application of the zero-inertia, or any other mathematical model of
surface irrigation, is dependent on the accuracy of inﬁltration and soil
roughness measurements. Elliott et al. (1982b) developed a mathematical
model to simulate the advance phase of ﬂow in furrow irrigation based on the
zero-inertia assumptions. The approach they followed in their development
was similar to that of Strelkoﬂ‘ and Katopodes (1977a) which included the
integration of the governing equations over ﬁnite cells in the space-time plane.
The cross sectional area of ﬂow and wetted perimeter were related to ﬂow
depth through the implementation of power curve relationships. Their results
revealed that the zero-inertia model simulated the hydraulics of advance
phase of furrow irrigation very effectively. J aynes (1986) presented a
numerical procedure for the solution of sloping and level berders based on the

29

zero-inertia model. His model utilized a ﬁnite diﬂ'erence scheme to model both
advance and recession phases of ﬂow conditions. The depth gradient term in
the simpliﬁed momentum equation was expressed explicitly and averaged over
the entire border. The author reported that this simpliﬁcation made the model
simpler to program and the required computer code less cumbersome.

The zero-inertia approximation of the full hydrodynamic model was
utilized by many investigators for different applications in surface irrigation.
Fangmeier and Strelkoﬁ‘ (1979) evaluated the U. S. Soil Conservation design
criteria for sloping borders without runoﬁ' using a mathematical procedure
which was based on the zero-inertia model. They concluded that the design
charts of the U. S. Soil Conservation Service (1974) were reasonable for graded
borders. However, they suggeswd that these charts be supplemented by a
mathematical model, similar to their developed model. This would provide
those designing irrigation systems with speciﬁc guidelines on the range of
applicability of the charts for various irrigation ﬂow parameters. Rayej and
Wallender (1985) develowd a nonlinear zero-inertia model for furrow
irrigation. Later, they developed a zero-inertia model for surge ﬂow irrigation
bawd on their previous work (W allender and Rayej, 1985). The non-linearity
of the governing equations allowed for simultaneous modeling of wet and dry
sections of furrows. Their model provided adequate simulations of the advance
and recession phases of ﬂow when compared to three sets of ﬁeld data that
utilized for comparison. Schwankl and Wallender (1988) studied the effect of
the spatially-varying inﬁltration and wetted perimeter on furrow advance and
inﬁltrated water distribution using a similar zero-inertia model developed by
Rayej and Wallender (1985). More recently, Schmitz and Seus (1990)
presenmd an analytical solution of the zero-inertia model as applied to the

30

advance phase in sloping and level borders. Their development was based on
the assumption of a "moving momentum representative cross section in the
water body".

The zero-inertia model was analyzed by many investigators to study its
general response. This was accomplished through the process of
non-dimensionalizing the solution to produce families of dimensionless
advance curves in borders, level basins, and furrows (Katopodes and Strelkoﬂ',
1977b; Clemmens and Strelkoﬂ‘, 1981; Strelkoﬂ' and Clemmens, 1981; and
Elliott at al., 1983a). These curves were restricted solely to the advance phase.
Hence, they cannot be utilized to predict the entire irrigation process.
However, these evaluations have produced very important design
methodologies for level and sloping basins and borders (Clemmens and
Strelkoﬂ‘, 1979, and Strelkoﬁ' and Clemmens, 1981).

In short, the zero-inertia model represents a simpliﬁed hydrodynamic
model that is intermediate in computational approach between the fully
hydrodynamic model represented by [2.9] and [2. 10] and the kinematic wave
model which will be discussed in the following section. The zero-inertia model
was utilized in various areas of open channels and surface hydrology as well as
ﬂood-routing and surface irrigation. The zero-inertia approximation
transforms the system of partial differential equations in [2.9] and [2. 10] from
hyperbolic to parabolic form. This reduces the computer time requirements for
the execution of this group of models. Many studies revealed that the
zero-inertia model produces excellent results in the simulation of the hydraulic
behavior of basin, border, and furrow irrigation systems as long as the
assumptions of the model are not violated. The zero-inertia model appears to
be very applicable to the above areas of irrigation since the assumption of
neglecting the inertial terms in the momentum equation is realistic given the
low values of Froude numbers prevailing under actual ﬁeld conditions

31

(Strelkoﬁ' and Katopodes, 1977b, and Clemmens, 1978). The computational
cost of the zero-inertia model is certainly cheaper than that of the complete

hydrodynamic model.

3. Kinematic Wave Models

The Kinematic wave model is the most simpliﬁed form of the
hydrodynamic model. This model is based on the assumption that the inertial
terms in the momentum equation together with the term that describes the
pressure variation in the direction of ﬂow are negligible ([2.10]). The
continuity equation ([2.9]) is kept unchanged. The simpliﬁed momentum
equation has the form

s,=s, [225]

This above equation implies that ﬂow is at normal depth throughout the
domain of solution (Bassett et al., 1980). The kinematic wave approximation
is only applicable when the slepe of the channel bed is steep. Given the
relation that is depimd in [2.25], a uniform ﬂow equation such as Chezy
([2.11]), Manning ((2.121), or Darcy-Weisbach ([2.13]) may be used to relate
ﬂow rate (Q) to ﬂow depth (y) or cross-sectional area (A).

The kinematic wave model was named after identifying the fact that the
method projects the movement of a ln'nematic shock wave. Since every
kinematic wave model utilizes a uniform equation to establish ﬂow-depth
relationship, these models are ﬁequently referred to in the literature as
"uniform depth" or "uniform ﬂow" models (Walker and Skogerboe, 1987).

The kinematic wave method as a technique was ﬁrst proposed by
Lighthill and Whitham (1955) for modeling overland ﬂow. Later, it was
utilized in . the solution of watershed problems and in predicting ﬂood

32

movements in rivers (Henderson and Weeding, 1964; Weeding, 1965a; and
Weeding, 1965b). Woolhiser and Liggett (1967) examined the errors
introduced by the application of the kinematic wave model to overland ﬂow
problems. Their work showed that the simpliﬁed hydrodynamic model based
on the kinematic wave assumptions is applicable to these problems within a
certain range of input parameters. Following their application, the kinematic
wave theory was heavily utilized in many investigations on surface runoff in
watershed hydrology (Brakensiek, 1967; Woolhiser, 1969; and Singh, 1975).
Singh (197 6) conducted a study to assess the discretization error of four
diﬂ'erent ﬁnite difference numerical schemes frequently used in solving the
kinematic wave equations. He examined the problems of convergence and
stability of these schemes. Since then, many other investigations were
conducted in diﬁ'erent areas of surface hydrology and open channel ﬂow using
kinematic wave models. The most recent is a study by Hromadka and DeVries
(1988) where they examined the use of the kinematic wave method in open
channel ﬂow routing of runoff hydrographs. Their work concentrated on the
signiﬁcance of the computational errors in the application of numerical
prewdures of the kinematic wave models. They were also interested in
assessing the eﬁ‘ect of the various assumptions implemented with the
kinematic wave model as opposed to the complete hydrodynamic model.

The utilization of the kinematic wave theory in hydrologic applications
was extended to sloping, free draining borders by Chen (1970) and Smith
(197 2). In his development, Chen based the solution of the kinematic wave
model on the method of characteristics with the help of initially prescribed
initial and boundary conditions. Chen indicated that the depth or discharge of
ﬂow at any time and distance from the inlet can "be determined from the
family of characteristic curves in the x, t-plane". In his conclusions, he stated
that "the kinematic wave method may only be valid for super-critical ﬂow. For

33

other than super-critical ﬂow, the more general hydrodynamic approach
should be adop ". Smith (1972) discussed two methods for solving the
kinematic wave model in ﬂood wave movement and attenuation in dry alluvial
channels. The ﬁrst included the application of the method of characteristics
outlined by Chen (1970). This approach reduces the partial differential
equation as represented by the continuity equation in [2.9] to
two-characteristic ordinary diﬁ'erential equations in the x, t-plane combined
with a third equation for shock movement. The second included the solution of
the same partial diﬁ’erential equation using ﬁnite difference approximation
and a rectangular grid. After comparing the results of both numerical
approaches to available ﬁeld data, Smith reported that the kinematic wave
assumption was very reasonable for the particular cases of unsteady wave ﬂew
addressed. These cases included sloping border irrigation and ephemeral ﬂood
routing. He cited both the work of Woolhiser and Liggett (1967) and Tinney
and Bassett (1961) as the proper indication for the appropriateness of the
kinematic wave assumption under the conditions of his study. Once again,
Smith reported that the results of the kinematic wave models were more
sensitive to the inﬁltration frmction.

The formulation of free boundary problems in surface irrigation using
both complete hydrodynamic and kinematic wave models was investigated by
Shaman and Singh (1978 and 1982). In their work, Sherman and Singh
presenmd explicit formulations of the free boundary problems in surface
irrigation. Their approach was also based on the method of characteristics.

As was the case with other models, the kinematic wave model was ﬁrst
developed for border irrigation, and later applied to furrow irrigation systems.
Walker and Humpherys (1983) highlighted three essential modiﬁcations in the
mathematical prewdures utilized in the analysis of border irrigation systems
before these analyses are implemented in the simulation of furrow irrigation

34

systems. These include the description of the geometry of ﬂow cross section,
implementation of an inﬁltration function that accounts for both steady and
time dependent inﬁltration rates, and assessing the eﬂ‘ect of the wetted
perimeter on the inﬁltration function. Walker and Humpherys accounted for
the ﬁrst two modiﬁcations in their study which included the development of an
implicit ﬁnite diﬁ‘erence scheme to the direct solution of the kinematic wave
furrow inigatien model. Their scheme was based on the Eulerian integration
approach which represents the numerical approximation of the continuity
equation based on the concept of multi-cell deforming control volume. Using
this approach the continuity equation became

{[993 + (1 - 9)Qul - [GQL + (1 - 9)Q:]}5t
+ {[Mz. + (1 ~¢Mul - [M1 + (1 -¢)Aul}5x
+ {(4)21 + (1 -¢)Zrl - Nil: + (1 - ¢)Zul}5x = 0 [2.26]

The subscripts J and M represent the left and right cell boundaries,

respectively, at time t, _ , while the subscripts L and R represent the leﬁ: and
right cell boundaries, respectively, at time t,. Walker and Humpherys gave a
detailed description of their mathematical development. In their conclusions,
they reported that the kinematic wave analysis is "a satisfactory tool to predict
water advance, intake, and runoﬁ' ﬁ-om sloped furrow irrigated systems".
After comparing the characteristic furrow model to the integral model, they
reported that "the integral model was superior on the basis of its adaptability
to both surged and continuous ﬂows, less sensitivity to the size of the time
step, and the numerical stability of the solution". This was supported by the
work of Walker and Lee (1981). Izuno and Pedmore (1985) developed yet
another ln'nematic wave model for surge and continuous irrigation of furrows.
The surge inﬁltration function utilized in their study was based on the

35

two-branch function of the Kostiakov equation as suggested by Clemmens
(1981). They reported acceptable model predictions of advance under surged
and continuous furrow irrigation applications.

The kinematic wave assumption simpliﬁes the analysis of surface
irrigation systems immensely. However, this same assumption limits the use
of kinematic wave models to the he draining graded borders and furrows with
relatively smep slopes (James, 1988). Hence, the model is inapplicable to
dead-level ﬁelds and diked borders or furrows. The limitations can be
attributed to the facts that the normal depth is inﬁnite in the ﬁrst case while
the kinematic wave solution would be only inﬂuenced by the upstream
boundary conditions (no downstream boundary conditions could be imposed on
the ﬂow) in the second case (Bassett et al., 1980).

4. Volume Balance Models

Volume balance models are the most simpliﬁed form of the fully dynamic
equations. These models neglect the entire momentum equation and
implement some approximations to the continuity equation. The continuity
equation ([2.9]) is applied to the entire ﬂow proﬁle at once (Bassett et al.,
1980). The continuity equation is integrated over space, which represents the
length of the advancing stream, to produce (Hart et al., 1968, and Bassett et
al., 1980)

_dV,(t) dV,(t)
Q" dt + d:

 

[2.27]

where Q, is the inlet ﬂow rate, V,(t) is the surface volume of water, and V,(t) is

the inﬁltrated volume of water. Equation [2.27] can be further integrated over
time to yield

36

Qot = V,(‘) + V,(t) [2.28]

The surface and inﬁltrated volumes of water can be determined by integrating
V,(t) and V,(r) over the advance distance and substituting the results in [2.28].
This results in (Walker and Skogerboe, 1987)

Q,,r= JA(S,t)d3 + flows [2.29]
0 0

where s is the integrand of x and Z is the inﬁltrated volume per unit length.

Since the momentum equation which describes the temporal and spatial
variation is completely ignored, the volume balance model is based on the
assumption that the average area of ﬂow is a constant, A-. If the inﬁltration is
considered to be a function of intake-opportunity time, [2.29] reduces to

Q0! =Xx + Ila - t,)ds [2,30]
0

where t - t, is the intake-opportunity time and t, is the time at which the
advancing ﬁ-ont of the stream reaches distance 3 .

The various volume balance models presented in the literature can be
classiﬁed into four diﬁ'erent categories. The ﬁrst category is based on the
recursive approach. The work by Hall (1956) and Strelkoﬂ' (1977) represent
two good examples of this category of volume balance models. While Hall
( 1956) solved the border advance problem, Strelkoﬁ‘ modeled all phases of
irrigation in borders including the depletion and recession phases. The second
category of the volume balance models is based on the Kernel function
approach. The work by Hart et al. (1968) was based on this approach. Their
work was also applied to border irrigation systems using the Kostiakov
inﬁltration equation. The third category of volume balance models utilizes the

37

Laplace transform approach. In this approach the Laplace transform of [2.30]
is established. A good example of such an approach is the research work by
Philip and Farrell (1964). The last group of volume balance models is based on
the power advance approach. Among the many researchers that followed this
approach were Fok and Bishop (1965), Wilke and Smerdon (1965), Chen
(1966), and Singh and Chauhan (1972).

Since the volume balance model has the basic approximations of the
kinematic wave model together with many other assumptions, it must at least
have the same limitations that were highlighted in the previous section. As a
matter of fact, the volume balance models have additional inaccuracies since
the shape frmctions are presumed arbitrarily (Bassett et al., 1980). However,
the volume balance models have the cheapest execution costs among the
various models covered to this point. This makes this approach good for quick,
rough initial calculations.

E. Numerical Solution of the Hydrodynamic Equations

The solution of the full hydrodynamic equations or a simpliﬁed form of
these equations requires the implementation of a numerical solution
procedure. The partial diﬁ‘erential equations that are represented by the
Saint-Venant equations could be transformed to ordinary differential
equations after utilizing the method of characteristics. Then, the resultant
ordinary diﬂ‘erential equations are usually solved numerically. Another
alternative implements the transformation of the above partial differential
equations to a system of algebraic equations amenable to solution.

There are various numerical techniques for solving diﬁ‘erential equations.
The ﬁnite diﬁ'erence and ﬁnite element methods represent the two most widely
used procedures for obtaining numerical solutions to both ordinary and partial

38

diﬁ‘erential equations. These numerical procedures are utilized in many
problems especially overland ﬂow problems which cover various areas of
hydraulics and hydrology including ﬂood routing and open channel ﬂow.

1. Finite Diﬂerence

The ﬁnite diﬁ'erence method approximates ordinary or partial diﬁ'erential
equations with diﬁ'erence equations. Using this numerical method, a
continuum is replaced by a series of discrete points between which the
diﬁ'erentials are approximated. The ﬁnite diﬁ'erence method has been the
most widely used procedure for approximating diﬁ'erential equations
numerically. It was heavily utilized repeatedly in the solution of
shallow-water equations which describe ﬂow conditions in both open channels
and overland ﬂow (Liggett and Woolhiser, 1967). Over the last several years,
many scientists utilized the ﬁnite diﬁ‘erence method in the direct or indirect
solution of the simpliﬁed or complete hydrodynamic equations as applied to
surface irrigation problems. The review of such work was brieﬂy outlined in
the previous section on surface irrigation models and therefore will not be
repeated in this section.

2. Finite Element

The ﬁnite element method utilizes an integral formulation to generate a
system of algebraic equations after approximating a continuum with a
continuous piecewise smooth functions (Segerlind, 1984). This method was
initially developed for the analysis of problems in structural mechanics. Later,
the ﬁnite element method was applied to the numerical solution of various

39

classes of problems. The application of the ﬁnite element method to many ﬂuid

mechanics problems serves as a good evidence for the wide spread use of this
numerical technique.

The application of the ﬁnite element method to overland ﬂow problems
was the subject of many investigations. Among the many early publications
are the studies by Guymon (1972), Taylor et al. (1974), Judah et al. (1975),
Desai (1979), Ross et al. (1977; 1979; and 1980), and Heatwole et al. (1982). All
of these researchers have reported some success in utilizing the ﬁnite element
method for modelling the various physical processes in overland ﬂow problems
which are governed by the shallow-water equations. The most recent
developments in this area were the studies by Katopodes (1984), Akanbi and
Katopodes ( 1988), Hu et al. (1989), Kaneko (1989), Kashiyama and Kawahara
(1989), and Vieux (1989).

On the other hand, the application of the ﬁnite element method to
irrigation problems has Men very limimd. Much of the prior work focused on
utilizing the ﬁnite element formulation in the analysis of both pressure and
ﬂow conditions in sprinkler and drip irrigation systems as well as
pipe-network analyses. The application of the ﬁnite element method in these
areas was unique in the sense that the solution processes didn’t start from the
partial or ordinary diﬁ‘erential equations but rather from utilizing the direct
stiﬁ‘ness procedure of the ﬁnite element method. The reader is referred to the
work by Bralts (1981) and Bralts and Segerlind (1985) for a detailed
description of this approach.

F. Synopsis

After reviewing the theory and literature that is pertinent to this research

work, it was clear that an extensive amount of work has been done in the area

40

of numerical analysis of surface irrigation systems based on the hydrodynamic
equations. However, what seems to be lacking is a general numerical
formulation of the complete and simpliﬁed forms of the hydrodynamic
equations in one model. Even though many comparison were made in the
literature among the various forms of the hydrodynamic equations as applied
to surface irrigation systems, these comparisons were not done on the same
basis. On the other hand, it was observed that the ﬁnite element method was
not exploited in the area of numerical analysis of surface irrigation systems.
The application of the ﬁnite element method to many ﬂuid mechanics
problems serves as a good evidence for the wide spread use of this numerical
technique. This also serves as an indication that the method could successfully
be used for the hydraulic analysis of surface irrigation systems.

Based on the above, it was felt that there is a need to develop a
mathematical formulation of the Saint-Venant equations for the analysis of
surface irrigation systems using the ﬁnite element method. The numerous
features of this numerical technique makes it attractive to the solution of
initial and boundary value problems that can be described by ﬁrst or second
order partial diﬁ'erential equations. The simplicity in handling boundary
conditions and the ability of the method to accurately handle complex solution
domains are two of the many important features of the ﬁnite element method.

III. METHODOLOGY

The ﬂow of water across the soil surface in any surface irrigation systems
is spatially varied. Moreover, the hydraulic design and analysis of surface
irrigation systems is a time dependent process. The hydraulics of ﬂow in both
sloping furrow and border irrigation systems is governed by two ﬁrst-order
partial diﬂ‘erential equations. These unsteady ﬂow equations were developed
originally by A.J.C. Barre De Saint-Venant in 1871 (Miller and Yevjevich,
1975). The development of the Saint-Venant equations was based on the
application of the conservation of mass and momentum principles to the
analysis of surface ﬂow conditions in open channels. Since their development,
the Saint-Venant equations have been used in many areas of hydrology
including the study of river ﬂoods and propagation of tides in river channels.
These equations were later used in the analysis of ﬂow conditions in surface
irrigation systems.

There are four general mathematical schemes that result from applying
the Saint-Venant equations to the hydraulic analysis of surface ﬂow problems
in general and surface irrigation problems in particular. These mathematical
approaches result in one of the following models: hydrodynamic, zero-inertia,
kinematic wave, and volume balance models. The primary difference among
the above list of models lies in the number of assumptions that are
implemented in the Saint-Venant equations.

In order to analyze various surface ﬂow problems in any of the above
models, a numerical procedure needs to be implemented. Historically, these
numerical procedures have been based on the ﬁnite diﬁ‘erence method, the

41

42

method of characteristics, or a combination of both. Such models were
developed over the years by many scientists as was discussed in the previous
chapter. More recently, the ﬁnite element numerical procedure was
implemented in the solution of surface ﬂow problems. However such
developments were limited to the areas of surface hydrology and open
channels.

The sole purpose of this research study is to develop a methodology for
implementing the Galean formulation of the ﬁnite element numerical
prowdure to the hydraulic analysis of ﬂow conditions in surface irrigation
systems. A general ﬁnite element model will be developed to perform the
hydraulic analysis of surface irrigation problems using the hydrodynamic,

zero-inertia, and ln'nematic wave models.

A. Research Approach

Five fundamental objectives are presented as the goals of this research.
The approaches utilized to achieve these objectives are delineated below.

Objective 1. To develop a ﬁnite element solution procedure of the
Saint-Venant equations for the hydraulic analysis of

surface irrigation systems.

The approach to be followed under Objective 1 will be to apply the
Galean formulation of the ﬁnite element method to the solution of the
Saint-Venant equations using both linear and quadratic one-dimensional
elements. The Galean formulation will be applied to both the continuity and
momentum equations with respect to the space coordinate for a ﬁxed instant of
time. Each will result in a system of ﬁrst-order differential equations in the
time domain. The resultant two systems of ordinary differential equations will
be combined into one general system. Then, a ﬁnite diﬁ‘erence approximation

43

in the time domain will be applied to the ﬁnal general system of equations to
generate a system of algebraic equations which will then be solved iteratively
over time. The direct stiﬁ‘ness procedure will be utilized in building global
systems of equations at various time steps.

Objective 2. To create a general solution approach that will
accommodate the available mathematical models of the
Saint-Venant equations in the analysis of surface

irrigation systems.

The approach to be followed under Objective 2 will be to establish the
coeﬁcients that will implement the various assumptions utilized in
establishing the hydrodynamic, zero-inertia, and kinematic wave
mathematical models from the Saint-Venant equations. By choosing these
coeﬁcients, the solution process would be performed based on the selected
model. The general development would apply to the selected model and the
solution process will never be altered by the choice of the mathematical model.

Objective 3. To develop an approach to easily incorporate the varying
boundary conditions of the advance, pending, depletion,
and recession phases of surface irrigation into the solution
process with minimal arbitrary or experimental

parameters.

The approach to be followed under Objective 3 will include the
modiﬁcation of the ﬁnal system of equations to incorporate given boundary
conditions under varying physical phases of an irrigation cycle. The possibility
of implementing this approach should be straight forward since including
boundary conditions at a later stage of the solution process is one of the
primary features of the ﬁnite element method. The dimensions of the total
system of equations will remain unchanged at any instance in time.

44

Objective 4. To develop a computer model that will utilize the above
mathematical concepts for the hydraulic analysis of ﬂow
conditions in border and furrow irrigation systems.

The approach to be followed under Objective 4 will be to implement the
ﬁnite element mathematical development of the motion equations in building
a computer model that will simulate the advance, pending, depletion, and
recession phases of both furrow and border irrigation systems. The computer
model will be developed to run on any IBM-compatible microcomputer with a
Random Access Memory (RAM) of 512 Kbytes or more and an MS-DOS version
2.00 or higher.

Objective 5. To numerically evaluate and compare the results of the
ﬁnite element model to actual ﬁeld measurements from

existing surface irrigation systems.

The approach to be followed under Objective 5 will be to compare the
results to be obtained from running the developed ﬁnite element computer
model to those reported from actual ﬁeld measurements for existing surface
irrigation systems. A graphics routine will be developed to display both
simulated and actual data of the various ﬂow phases of irrigation on the same
graph. The graphical display will include plots of actual and predicted
advance and recession trajectories of ﬂow and will be revealed after the
conclusion of any surface irrigation simulation run. This will be utilized to
evaluate the utility of the developed ﬁnite element model and its ability to
accurately simulate the hydraulic conditions of ﬂow in surface irrigation

systems.

45

B. Theoretical Development

There are two basic equations that can be utilized in the hydraulic
analysis of ﬂow conditions in open channels and surface irrigation systems.
The two equations, or the so-called Saint-Venant equations, were developed
based on applying the conservation of mass and momentum principles to ﬂow
conditions. This research will utilize these equations as the basis for the
hydraulic analysis of various surface irrigation systems. The ﬁnite element
method will be used then to solve these equations numerically. The resultant
numerical model will then be applied to the analysis of hydraulic conditions in
furrow and border irrigation systems. This general development may also be
applied to the analysis of ﬂow conditions in open channels with or without
inﬁltration. However, the assumption that the slope of channel bed is mild
will be utilized in the development, an assumption which is very reasonable in
surface irrigation systems but not necessarily true in many open channels.

1. Development of the Saint-Venant Equations

The development of the Saint-Venant equations will be repeated in this
section to delineate the important principles of these two equations as related
to surface irrigation.

ammonium

The volume of water stored in a ﬂuid element (Figure 1) within a spatially
varied furrow is represented by the following relationship

dvc=Vis-V~-Vl [3-1]

 

Figure 1. An enlarged ﬂuid element within a furrow with a spatially varied
unsteady flow.

47

where W, is the change in the stored volume within the element during time

dt, V... is the total inﬂow during dt, V... is the total outﬂow during tit, and V,
is the total inﬁltration during dt. The total inﬂow during a time step dt is
described as ‘

_ Qio)+Qr(r+a)
.1 . 1
«Mm—11

_13th
'Qd‘+2ard’z

~er [3.2]

 

where Q,(,, and Q10“, are the inﬂow rates at times t and t+dt, respectively.

The total outﬂow during the time step dt is

V _(Qza)+Q2(:+a)}l
“" 2

elm-2% 1+11e+sdx1+3§"71“‘

 

 

 

ao
_ 12 IHQ 4'3de 2
— th + ax dxdr + 2 at (It
~ th +g—g-dxdt [3.3]

where Q”, and Q20...) are the outﬂow rates at times t and t+dt, respectively.

The total inﬁltration volume during the time step dt is

V +V ,
V:=( «e 210 do)!”

 

48

where Vm and Vum.) are the average inﬁltration rates at times t and t+dt,

respectively. These two terms can be developed separately as

I +1 )
VIO)=[ 1(02 2t!)
I+idx
=[I+( ; )yx

181 2
=Idx+2axdx

~Idx

I :+ +1 +
Vuuﬂ= -[ 1( ”2 at: ")1:

=Idx +gdtdx 4-;Xl :dtlitdx

and

 

 

where ’10) and Inc“) are the left boundary inﬁltration rates per unit length at

times t and t+dt, respectively, and In, and In...) are the right boundary
inﬁltration rates per unit length at times t and t+dt, respectively. After
substituting the above two terms, V, becomes

Vz=%[(1dx)+{mxgt +—dtdx+; (”swat and:

=Idxdt+§gt£dt2dx+4 130+: a? d”its

~ I dxdt [3.4]

 

 

Substituting [3.2]. [3.3], and [3.4], in [3.1] results in

49

av, = [94:] -[th 4-838-dxdt] - [Idxdt] [3.5]
01‘

4V: - 99. _

75m.- 8:: dx Idx [3.6]

The storages in the element at times t and t+dt can be deﬁned by the
following two terms

_ 18A 3
“Mk-+231“

~ Adx [3.7]

= [ A1041) +A2(:+&)]dx
2

.§[(..%.).[(A.g..).iﬂf_d‘l.]]..

v00“)

an
_ 13A 2 18A 39%“)
-Adx+2axdx+zatdxdt+ 23‘ dxdt
13A
~Adx+2(-2--$dxdt)
3A

‘=Adx +-a—t-dxdt [3.8]

50

where Am, and Au...) are the left boundary areas of ﬂow at times t and t+dt,

respectively; A”, and Aw“, are the right boundary areas of ﬂow at times t and
t+dt, respectively; and V4,, and V4,...) are the storages at times t and t+dt,
respectively. Substituting [3.7] and [3.8] in [3.6], the continuity equation

would result as follows

39 dx -Idx = V.(.+a)- V4.)

 

d:
_[(Adx +§dxd:)-(Adx)]
' dt
aA
=$dx
01'
%+%+I = o [3.9]

kW

There are three forces which act upon the ﬂuid element in Figure 2.
These include the force due to the weight of the ﬂuid element, the pressure
force which represents the resultant of two pressure forces acting on the
upstream and downstream boundaries of the ﬂuid element, and the friction
force which results from the resistance to water ﬂow due to the viscous force
along the weMd perimeter of the element. These forces can be assessed as
discussed below.

i. W The weight of the ﬂuid element can be expressed as

 

F” =[Fpo)+:w(t+l)] [310]

51

T V /\
i
x ‘F.

0

I 0

dx

 

 

Figure 2. Acting forces on an enlarged ﬂuid element within a furrow with a
spatially varied unsteady ﬂow.

52

where Fm and FIKH") are the forces due to the weight of water at times t and

t+dt, respectively. The ﬁrst component of [3.10] can be written as

A +A ,
F..(:)""Y[ __n(:) 2 a {'41

___+[ (A)+(A2+§:-dx)]dx

.. 12A; 2
—7Adx+2axdx

~ yAdx [3.11]

where yis the speciﬁc weight of water and Am, and Aw, are the cross sectional

areas of ﬂow at the upstream and downstream boundaries of the ﬂuid element,
respectively, at time t. The second component of [3.10] can be deﬁned as

 

A +A
Fw+‘)=7[ I(H-t) 2(1+l)]dx

2
_1 .3_A;_ 3A 3(A +£2dx)
-2[[A + at dt)+[[A +$dIJ+Tdt dx
M
= 1.31 1% . L“ W“)
{Adz-+23, dtdx+2axdx + 23‘ dtdx
13A 13A
—7[Adx +5-3de +§$dtdx:l
=yAdx magnum [3-121

where Am“, and An“, are the cross sectional areas of ﬂow at the upstream

and downstream boundaries of the ﬂuid element, respectively, at time t+dt.
Substituting [3.11] and [3.12] in [3.10] results

53

F = [yAdx]+[-{Adx+y%dtdx]
" 2
78A

=1Adx +55“: [3.13]

 

The component of the weight of the ﬂuid element with the direction of
ﬂow, F”, is obtained from

F,,=sina-F,

where a is the angle between the lower boundary, or the bottom of the

channel, of the ﬂuid element and the horizontal plane. Ifthe slope of the lower
boundary (SQisassumedtobesmall,thesineoftheanglecanbe
approximated by the tangent. This results in

F" 2 Tan (1 - F,
= Sol”.
where So is the slope of the furrow or border. Substituting [3.13] in the above
equation results in

F,,=yAs.,dx +-;%dxdxso [3.14]

The second term in the equation above is negligible compared to the ﬁrst term
which contains the cross sectional area of ﬂow. Therefore, Equation [3.14]
reduces to

F,,=7Asodx [3.15]

ii. W: The pressure force, 1",, which represents the resultant

of the pressure forces acting on the upstream and downstream boundaries of
the ﬂuid element can be written as

54

F, :17, -F, [3-161

where F, and F, are the pressure force acting on left and right boundaries,

respectively. The pressure force acting on the left boundary of the ﬂuid
element can be determined as follows

F, =[ F1(:)+F1(¢+a)]

 

2

=-[(yhA)+y(Ah +9-glldt)]

1824—}:
=1hA+2 a: [3.17]

where Pm and F10“, are the pressure forces acting on the left boundary at

times t and t+dt, respectively; 7 is the speciﬁc weight of water; and h is the
distance ﬁ'om the water surface to the centroid of the left and right areas of
ﬂow. The pressure force acting on the right side boundary of the ﬂuid element

canbewrittenas

Fz___[pz(o+:w+a)]

=%[(7Ah +1?dx)+ [YA]! +Y—d" Hi“; +— ‘1‘) (1‘)]

-7“, +73§x_hdx+1;3(__vh)dt+xa(1 a: “)d,

 

2 a: 2 a:
~7Ah+ya§x—'ldx+;i—gth)dt [3.18]

where Fate and Fm...) represent the pressure forces which act on the right

boundary of the ﬂuid element at times t and t+dt, respectively. Substituting
[3.17] and [3.18] in [3.16] produces

55

_ 19511 _ 2L" 114.3.)
F—(yhA+ ) (yhA-t-yaxduwz at d1)

= ‘7——dx [3.19]

Potter and Wiggert (1991) developed the following relationship using the
Leibnitz rule from calculus (refer to page 463 of their text):

M-

8h .4 [320]
Substituting [3.20] in [3.19] results in
1:54.423! [3.21]

31

iii. W: The friction force, F, , which results from the

resistance to water ﬂow can be expressed as
F,= 1,194: [322]

where t, is the shear stress at and P is the wetted parameter. The slope of the

ﬁction slope can be deﬁned as (Potter and Wiggert, 1991)

55% [3.23]

Equation [3.22] can be rearranged as

1.,“de
47

M 741
R}

 

Fl:

 

[324]

56
where R =AIP. Substituting Equation [3.23] in [3.24] results in

«mm

The conservation of momentum in Figure 2 follows Newton’s second law
of motion (Walker and Skogerboe, 1987). It states that the resultant force
which acts on the ﬂuid element in motion is equal to the rate of momentum
change within the element and the momentum ﬂux across the element
boundaries. This can be represented by the following relationship ‘

£(Force3) = (PQV)... - (PQV)... +1? [326]

where m is the mass of the ﬂuid element, Q is the ﬂow rate, v is the ﬂow

velocity, p is the density of the ﬂuid, 2 (Forces) is the summation of forces
(F... +F, -F,), (va),, is the average momentum ﬂux into the element, (va),.,
is the average momentum ﬂux out of the element, and (d (mv)/d t) is the
average momentum change during the time increment, dt. The ﬁrst two right
side terms of Equation [3.26] can be evaluated individually as follows :

(va),, =§[(va)+(va +§§ﬂdtﬂ

= va +%[v%%+g 3;] [3.27]

57

m.- at M a]
+2[0(Q +de)[v +gdx)+pa[(g +34va +3610] dt]

3:

 

3v
=va+pv§dx+anxdx+p§axdx

+P%[(v+3«]*9*=2+(a+e)w

a: a:

=va+pv§dx+pdix
22: ac a’_a_ ( 9.9. )2. iv.
+ 2 [(v +axd‘][‘5.' axa ah)“ max“ ar+axaxd"
=PQV+PV%%41+Pandx+pd[vaaQ+v:—gtdxm]

+pd: £9an 23va 2 32v
2[7a;a.“* tat—a.“ mamma]
_p_d_t[393vdx BQ a’v _dx]

ax a: ax axe: [328]

Ifthe second and third order differential products are assumed negligible, then
[3.28] mduces to

39 pvdtaQ +dexav

 

av
va--va+pv§x-dx+pQ$dx-+ 2 3t+ 2 a: [3.29]
The change of momentum, d(mv), can be determm' ed as
d (mv) = ORV)“. - (mv), [3.30]

where (mv),,.,, and (mv), are the average momentums at times t and t+dt,

respectively. The ﬁrst term of the right side expression of [3.30] can be
expressed as

58

(mv), = pAuhvuhdx [3.31]
where A...' and v.,..,b are the average cross sectional area of ﬂow and average

ﬂow velocity, respectively, at time t. These two terms can be evaluated
separately as shown below

Alm+A2m
A..." — —2

wees
‘ 2

1 3A
— A +2-de [332]

 

V +V
v = no) 20)
“k . 2

=(v)-]- v+§dx)
2
18v

= V + 5;” [3.33]

 

Substituting the above two terms in [3.31] results in

13A lav
(0W), = 0(A +5-311va +§$dxjdl

.. 32v. 22.9: 21939: .
-p(Av+2axdx +2axdx +4axdxdx)

= pAvdx [3.34]

where both second and third order differential products are assumed

negligible. The second term of the right hand side expression of Equation
[3.30] can be evaluated as

59

(mv),“, = M*t+av“h+adx [3.35]

where 4...“, and v are is the average cross sectional area of ﬂow and

”ho-d:

the average ﬂow velocity, respectively, at time t + dt. These two terms can be
evaluated individually as

_ Akt+a)+A7(1+a)
“5+4: - 2

=;[(..§.).[(..%.).§i§ﬂmll

13A 13A 13A 1 32A
—A +5-37dt +§$a +5511! 4.5%“

31] 1M 132A
-A +Edt 4-53-11! +555“

V _ v1(:+dt)+A2(t+&)
“5+4: 2

-;[(..g.).[[..g.].ig%ﬂ.]]

13v 13v 13v 1 32v
—V +§§dt 4-5ng +E'a—tdt +5753?“

3v 13v 1 8’v
— v +§dt +§$dx +§ﬁdxdt

 

Substituting the above two terms in [3.35] produces

60

_ 3A 13A 1321‘]
(MV),+a-D[A +— at -—+dt +2ax —dx +5733“)

3v 18v 132v
(v+atdt+§axdx +§axatdxdtyl

av. 32». am .
=O(Avdx+Aatdtdx+2ax:+ deatdxdt)

3-..». --——w)
43333 23W $333.03]
4:33am 13.-.3322)

When the second, third, and fourth order differentials are assumed negligible,
[3.36] reduces to

Gav)”, = p(Avdx +A gaudy: + v 95¢]:de [3.37]

Substituting [3.34] and [3.37] in [3.30] results in

d(mv)= (Mvdx+pA gtv-dtdx+pv%dtdx)- -(pAvdx)

01'

d(mv)_ 91d: Q11
d‘ =pA a: dx+pv 3: dz [3.38]

 

The individual relationships that were developed to this point as
presented by [3.27], [3.29], and [3.38] are substituted in [3.26] to yield

61

 

3_Q_ pvdtaQ Qthav
2(Forces)= (va+pvaxdr+anxdx+ 2 81+ 2 at)

pvdtBQ detav
""("Q +— 2 5+— 2 —ar)

+[pAgtv dx+pv%dx)

dx+anxdx+pA— —dx+pvaAdx

3:
av av 3A
an ”a?” v.37)“

34:8

pv

9(v

$48

01'

_ _ 39 av av 3A
F,,+F, F,—p(v—- +Q-5x-+Aat+v 3 ——)dx [3.39]

Substituting the expressions for F", 1",, and F, (i.e., Equations [3.15], [3.21],

and [325]) in [3.39] results in

yAsodx— Hugh -=yAS,dx -p(v-a-Q+an+A%"- HM}: [3.40]

The speciﬁc weight, 7, may then be replaced by pg where g is the acceleration
due to gravity. The next step will be to divide [3.40] by pgAdx to produce

8y 3Q av av 8A
50- a: -=S, —1X(v —+an+Aa—t+v vat] [3.41]
or
S°_sf=8y an Qav 13v vaA [3.42]

 

 

.$+gA ax +gAdx+g a: +gA 3t

62

Since the discharge rate of ﬂow, Q, is a preferable term over the ﬂow

velocity, v, Equation [3.42] can be rewritten as a function of Q instead of v
based on the expression

 

=A —+v — [3.43]

where §is the independent variable of diﬁ'erentiation which represents either
1 or t. Equation [3.43] can be rearranged into

4,322.- e) .3...
Substituting Equation [3.44] in [3.42] results in

By +(Q/A)aQ er
3°" Sf=ax gA dx+gALA_’(A%g— anD

*;[‘=(‘ar 39 Ml] (Mai

 

Q3: gA a:
Hay EgaQ 023A 139

”a: ngax gA’ax+gA 3:

Since both the cross sectional are of ﬂow, A, and the ﬂow depth, y, are

exclusively independent variables, one of these variables can be used instead
ofboth. Ifthe channel is assumed to be prismatic, the term aA/ax in the above
equation can be replaced with Tay/ax. The resultant equation is

63

- _ﬂ 2?. 32% 3.92
So-S,—[l gA3Jax+gA23x+gAat [3.45]

Equation [3.45] can be rewritten exclusively in terms of A instead of both A and
y. This step results in

- 1-2.“. 9: £29. 1.22
So-S,—[T gA’)8x+gA’8x+gA at [3.46]

4W

If a steady momentum is assumed, the change in momentum within the
ﬂuid element with respect to the time domain is assumed negligible. This
reduces Equation [3.26] to

24F 01'6“) = WV)“ - 039V):- [3‘47]

Substituting the results that were obtained earlier as represented by
Equations [3.27] and [3.29] in Equation [3.47] leads to

_ 92 2: 1:22:29. _PQd‘.3l
2(Forces)—[va+pvaxdx+anxdx+ 2 at+ 2 at)

_ pvdtaQ detav
(”9” 2 31+ 2 at)

_ 29. 2V.
_pvaxdx+pgaxdx

_ .32. 2". ’
_p(v ax +an}! [3.48]

The expressions that correspond to the various forces that are acting on
the ﬂuid element as represented Equations [3.15], [3.21], and [3.25] are then
substituted in the equation above to produce

64

yASudx -7A 24:: -yAs,dx = p[v%%+g %]dx [3.49]

Following the same step as discussed in the previous section which include the
replacement of the speciﬁc weight, 7, by pg and the division of [3.49] by pgAdx

gives

3y -L Q .231
So-E-SI—gA (v ax +an)

01'

 

50-35%3 ”if: Q: [3.50]

The change in ﬂow velocity with respect to x, av/ax, in the above equation is
then replaced by the expression of [3.44] to produce

50-3.9.2 92.14299. .Q.[_1.[A§_Q. «MD
+-———-—— [3.51]

Since the channel is assumed to be prismatic, the term aA/ax can be
exchanged by Tay/ax. Equation [3.51] reduces to

so-s,=[ _Q’T]ay+ 293—9- [3.52]

311’ a! +gA2 ax
Equation [3.52] can be rewritten in terms of A instead of y. The resultant
equation is

2

f-gi'mg—A3 ax +g—A2 ax

65

2. Finite Element Formulation Using Linear Elements

The numerical solution of the Saint-Venant equations will be presented in
this section. This analysis is accomplished using the Galean formulation of
the ﬁnite element method. The space dimension is discretized using linear
elements. This development is prepared for the solution of furrow irrigation
systems, bearing in mind that the border is a special case of the furrow
irrigation problem.

LEW
Elements

A system of linear equations is generated by evaluating the weighted
residual integral

R<x>= fwm(% +aQ+I [3541

which is the result of integrating the product of the continuity equation ([3.9])
and a weighting function, W(x), over the length of the element. The Galean
formulation of [3.54] is based on considering the shape functions N,- and N,- as
the weighting functions at nodes i andj, respectively (Segerlind, 1984). Since
the element selected in Figure 3 is linear, there will be two linear equations
(shape functions) for each element.

The ﬁnite element formulation is applied after representing the
unknowns by linear approximations of the form

4".) =Ni¢i +Nj¢j [3.55]

01'

 

 

 

 

 

 

 

 

F <—
F 4—
(A)
Section F-F
.—p s
O + O 9 4 ﬂ ‘
_. x i j
H L +1
(B)

Figure 3. Finite element discretization. (A) Furrow ﬂow. (B) Generic
one-dimensional linear element.

67

¢
¢(0) = [NI Ni] {¢i}

I
= [N]{¢} [356]
where o is the unknown; N,- and Ni are the shape functions at nodes i and 1',

respectively; and $.- and 45- are the values of the unknowns at nodes i and j,
respectively. The convention that { } and [] represent a vector and matrix
quantities, respectively, will be followed throughout this development.

Based on [3.56], the cross sectional area of ﬂow, A, and the ﬂow rate, Q,
can be represented by

A<x.:)=N.A.(r)+N,A,(:)

= [N‘- ”fl {2‘}

l

= [N l {A (0} [3.57]
and
Q<x.t)=N.Q.(t)+N,Q,(r)
Q
=‘”‘ Moi}
=[N]{Q(t)} [353]

The shape functions for the one-dimensional linear element are expressed

 

N.- = I [3.59]

 

[3.60]

where L is the length of the element.

The shape functions are written in the local system of coordinates which
allows for easier integration over the element. The shape functions for a
coordinate system locamd at node i are obtained from [3.59] and [3.60] by
replacingx with X5+s and resulting in

”i ='—L—— ”F
= 1 -i- [3.61]
and
N1 ___ (X.- +2) -X.-
=% [3.62]
Equations [3.61] and [3.62] can be rewritten in the matrix form
[N1=[N.- Ni]
=[1_% 1%] [3.63]

The partial derivative of [3.58] with respect to x is computed by
evaluating the partial derivatives of Ni and N , since the nodal values Q30) and
91(2) are constants with respect to the x - or 3 -space dimension. The partial
derivative of [3.58] becomes

69

=[B]{Q} [3.64]

dimensionare
95:25:...1. [365]
31 as L '
and
3N, 3N, 1
ii??? [3'66]

Equations [3.66] and [3.66] then rewritten in the matrix form

_ 51".: '1”; ~
[B]-[ 8x ax]
l 1
=[-Z z] [3.67]
Substituting [3.67] in [3.64] produces
19... .. _.1. l 9‘

The partial derivative of [3.57] with respect to time is compuwd by taking
the partial derivatives of A. and A j with respect to time and multiplying the
results by the shape functions. This assumption is true since the latter are
considered to be constant with respect to time. The resultant equation is

70

+37%
'3!
[

3A,:

= l-i i <5.»
911i.

L31 .
= [N] {A} [3-69]

 

 

Substituting the shape functions in [3.54] results in the system of
equations

8’
35% fM(%+%€-+I}tx [3.70]
x. .

and

12]": IN —+%€-+I}t [3.71]

Rewriting Equations [3.70] and [3.71] in matrix form results in

‘9’}. [N]; ._.,).
a. [N _._.,).

’1

3A 29.
=IIN] [— —+ ax H}: [3.72]

where [N]’ is the transpose of matrix [N].

‘

 

 

Substituting [3.68] and [3169] in [3.72] yields

71

XI

{Rf’} = [INJ'([N1{A}+[BJ{Q}+I)dx
8‘

or

"I ’1 ’1 '

{RP} = ftm’uv] {de + I [NI’ [3] {Q }dx + fm’ldx [3.73]
‘l ‘1 ‘1

The individual terms of [3.73] are integrated separately over the length of
the element in the local system of coordinates to result in

’1 L
fin/1’ [N] {A }dx = [W (~14: {A}
0

‘1

 

 

 

 

L 1-% s s
=1 2. [l-Z 71am}
L

' s s s
41-] [1+]
=f L fLmA}
o 1_£ f. ‘—

Ll LJL L2

".1; E]

3 6
= 5 _ {A}

.5 34
4.]: :14}
=[C.]{A} [3.74]

72

‘1 l- .
ﬂm’wnmdx = [[Nl’wmo}
0

3‘

r

[h

1...

ll
0%9
'—‘7
bl '—
l‘l H
l___.l
a
”A.
D
we

 

 

(4+1) (Li)
2 L 2
L L L (“{9}

II
a
a

 

 

ll
y—s
Nlt— MI
m
D
boa

 

 

= [KJ {Q}

[3.75]

The inﬁltration from the element can be expressed in various forms.
These forms result in several equations that were discussed in the previous
chapter. Many of these equations are expressed as a function of inﬁltration

opportunity time only. The inﬁltration term is then constant with respect to

the x - or .7 -space dimension and can be moved outside the integral. This

 

 

 

 

 

 

resultsin
.
a L 1. 1-5
[[mex =1JlN]’dw =1]. as
x‘ 0 0 i
L L .
11:: r-éj
=I<Z>=-I<-:>
[2. L 2
=-I{F.}

[3.76]

73

Substituting [3.74]. [3.75]. and [3.76] in [3.73] results in

 

 

 

 

' 1 1‘ L‘
(‘ __l: 2 1 A5 -5 —§ Qi _ -5
{NHL 2]{A.}* 1 1 {2,} "-1; [3'7"
.. 2 2 . L 2]
or
{R.‘"} = [C.1{A}+IK.1{Q}-‘I{F.} [3.781

 

A system of equations is generamd by evaluating the following weighted
residual integral for the linear element

R(x)= IW(x)[(%—EQT:)%+%%+E%§§-(So-S,)]d.x [3.79]
which is the result of integrating the product of the unsteady momentum
equation, [3.46], and a weighting function, W(x), over the length of the element.
Based on the previous discussion, the Galean formulation of [3.79] utilizes
the shape functions N,- and Ni as the weighting functions at nodes i and j,
respectively.

Substituting the shape functions in [3.7 9] and rewriting the results in a
matrix form produces

74

 

 

“I N 1 Q2 314 | 2Q 8Q l 1 89 S S l

{I (a ={R(¢)}= I {[1 8 3Jax g‘va 8A 81 (o 1)]
"(a x,“ 1 Q2 aAIZQaQI IBQ S S l}
3!: {(1 gASJax gAza 3A 3' (o 1)] J

or
{R‘.:’}=f[N1[f—nguﬁm’ﬁ—ﬁdx
’1 l 3Q ’1
T__ _ T _
+1911“ 3: dx {[N] (50 s,)dx [3.81]

The partial derivative wax is determined by evaluating the derivative of
[3.57] with respect to the space dimension. This is accomplished by evaluating
the partial derivatives of N,- and N, with respect to x since the nodal values AU)
and Aja) are constant with respect to the x — or s -space dimension. This
results in

i”;

a+aa
BN

.1 {2:}

= [B] {A} [3.82]

a’la’
$4.55

._"_.
a’Iﬁ

The partial derivative BQ/ax was developed earlier as expressed by
Equation [3.68].

75

The partial derivative 89/31 is determined by evaluating the derivative of

[3.58] with respect to time. This is accomplished by evaluating the partial
derivatives of Q,- and Q,- with respect to time and multiplying the results by the
shape functions N,- and N, that are constant with respect to time. This results

in

”‘29:.”
”a:

348

g
a:

=[N‘ N]<

 

348 3L8

 

=[N]{Q} [3.83]

Using [3.68], [3.82], and [3.83], the individual terms of Equation [3.81] are
integrated individually as follows :

81
r Lg: _ 1 a2
{[NJL. 34’]:a=ﬁm[1‘373)w”“d3

{A}

II
/—\
'~ll*-‘

I

ﬁle

U N

\.=;/

l l

.— NI
N'H NI:

 

 

L5-

2
=[%-E%-,)[KJ{A} [3341

76

[thfo anx= [lb/17:? —[B]{Q}ds

L V1-5
3] H am

0 _
[ L

 

 

-l
-39. 2
1

{Q}

.Nlr— Nit-j .

 

 

-—, [KJ {Q}

]INI’E‘E? —dx= fin/ff; —[N1{Q}ds
.

-_1_ 1'

S
'34 L [1‘3 H‘MQ}

chat-

b-lh

_._1.

axll‘ ulh
bolt“ call“

1
=g—AICJ {Q}

[3.85]

[3.86]

77

‘1 L
fin/1’ (S. - sad: = (S. — S,) [IA/I’d:
x‘ 0

r

 

 

S‘
L l-Z
=(so-s,)f« s as
o L E J
L1
2
=(So—Sf)<£>

 

 

2

=(So- 3,){F. }

Substituting [3.84], [3.85], [3.86], and [3.87] in [3.81] produces
{33)}: (T'ﬁ3)[m{“+[3w )[K.]{Q}
{2%)[C.]{Q}-(So-Sy){l'.}
=[£X)[c.]{a}+[%-gQ—:,]IK.]{A}
:QH][K1{Q}—-<so Slum

or

{RS’}=IC..1{Q}+[K..1{A}+[K..l{Q}-{F.}

 

[3.87]

[3.88]

[3.89]

The residual vector of the linear element for the steady momentum
equation, [3.53], is developed in the similar way as the unsteady momentum

equation. The result is

 

 

r8, 1
1.9.“. 24 32.22- ..
(0}: RM {N‘IIT 8 3]&X 4.818231 (so Sfildt
{R.. { *
R30}: 3’ l 2 8A
l”[[f'§‘=]s'*§%%“i""]“
[3t
‘1 2
= ].~.r[[.;.-§ﬁ]%.§%§-(s.-s,.].. [3.901

01‘

{R‘.?}- [IN] (73921587573) a+lwf§f¢§¢

I!
- ftNl’(S.—S,>dx [3.91]
x.

Substituting [3.84], [3.85], and [3.87] in [3.91] yields

2
{RS’}=[-11-.-£-3)[K.]{A}+[:Q'—.)[K.]{Q}—.(S -S,){F.} [3.921
or
{£9} = [K..1{A}+[K..1 {Q} -{F..} [3.93]

 

A smcial case of the momentum equation is the zero-inertia equation
which is based on the assumption that the change in momentum is negligible.
This assumption drastically reduces the momentum equation and greatly
simpliﬁes the process of numerical analysis. Based on the above assumption,
[3.26] reduces to

79

XForces) = 0 [3.94]
Substituting the forces in [3.94] results in

F, ,,-—+F -=F, —0 [3.95]

Substituting [3.15]. [3.21], and [3.25] in [3.95] produces

yASodx -1A §dx -yAS,dx = 0 i [3.96]
01'

3.45%:- [3.97]
or

13‘ -(s -=s,) -o [3.98]

The residual vector of the linear element for the zero-inertia momentum
equation, [3.98], can be developed in a similar way as discussed in the previous
section. The result is

 

 

R0)
{R“’}= { (.,}= f »
R1 ’ 13A
IN.(———(so-s,)}zw
[a
=fl~1 [——-(S.- S,))4r

01'

80

., 8:
{RS’}= [INF-111%dx-IIN1W80-Sﬂdx [3.991
I. I.

The ﬁrst term of [3.99] is developed based on Equation [3.82]. This
results in

I" 1 as " 1
r _ r_
[IN] i371: - {IN} Tm {Am

‘1

{A}

 

 

alt-
p—s
lMIM- NI

1
= [K14] {A} [34ml

The second term of [3.99] was developed earlier in [3.87]. However, the
friction slope, 5,, can be expressed as a function of ﬂow rate, Q, and
cross-sectional area of ﬂow, A , using any of the uniform ﬂow equations that

were reviewed in Chapter II. If the Manning equation is selected, Equation
[3.87] becomes

8’
II” 11' (So "' Sﬂdx = (so "' sf)
1,

81

£1
2

E
2

_£ (50.31),-

"2 (so-s,»
. "Q; .
-2 5“”[W’I
24 an >
ssw-[Ajkan]:
' "Qi ‘
(WT
"Q1 >
AM” J

 

 

 

 

 

NH
2 5,, 2

 

 

 

[3.101]

Substituting [3.100] and [3.101] in [3.99] yields the system of equations

{RS’} r;-

 

 

 

' 1 1‘ '1; "20:”
"2' 5 {14.};(4 “mower?
-1 l A]. L029} 2 so
. 2 2. _§A}Rfﬂ

 

 

 

[3.102]

Equation [3.102] is rewritten in a form similar to that of the steady momentum
formulation which is expressed by [3.93]. This results in

01'

u¢2=%

 

 

"-1 1] "E "’9‘ o ’
2 2 {141+ 2A.’R.-"’

_.l. .1. A1 0 E "2Q!

» 2 2- . 2AM”.

 

 

 

=%[Kc]{A}+[K.g]{Q}'So{Fe}

{3:} -%{i::.}

82

{RS’}=[K..1{A}+IK..1{Q}-{F.} [3.1031

 

Surface irrigation analysis can be performed based on both the continuity
and momentum equations, or the so-called the Saint-Venant equations ([3.9]
and [3.46]). Various assumptions are implemented to simplify the momentum
equation ([3.46]) to the so-called kinematic wave approximation. This
approximation is based on the assumption that the inertial terms together
with the term that describes the pressure variation of ﬂow in the momentum

equation are negligible. The simpliﬁed momentum equation then has the form
So-S, = 0 [3.104]

while the continuity equation ([3.9]) is kept unchanged. These assumptions
imply that surface ﬂow is at normal depth throughout the domain of solution.
Based on the above assumptions, Equation [3.104] is utilized in [2.11], [2. 12],
or [2.13] to calculate the ﬂow rate, Q, at nodes i and j. Ifthe Manning equation
([2. 12]) is selected, the following equations result

2.- = ﬁAﬁf’St"
and

91‘ = i“???
01'

(lkf’sryi—Qﬁo [3.105]

83
and
1
(;R,?”sgf)a,-Q,=o [3.106]

where R, and R, are the hydraulic radii at nodes 1' and j, respectively. Since we

are primarily interested in the contribution of nodes 1' and j to the element,
only halfof the ﬂow terms in [3.105] and [3.106] will be utilized as the element
contribution to the ﬁnal system of equations. Equations [3.105] and [3.106]
can be combined together in a matrix form to yield

ii -- 0
RS' 0 A' .-

i .. .. {.‘}+ 2' lgHg} [3,107]
0 RES; 1 0 "

 

 

[King] {A } 4" [Km] {Q} "' {F2} = {0} [3-108]

where {17,} = {g}.

 

The linear one-dimensional ﬁnite element formulation of the continuity
equation, [3.78], can be rewritten as

A [-1 1‘ 'J;
(,__21 22Q._‘2
““1 6L 2]{A}* -1 .1_{Q.-} Ce ””91

 

 

 

On the other hand, [3.89], [3.93], [3.103], and [3.108] can be expressed in the
general form

84

 

 

(02‘93) +5 (Cz‘cs) ‘

(0 =9, 201 C: 01} — 2 2 2 {A3}
{R3,} 6[ C1 201]{Qj +1 -(Cz"'C3) (Oz-€94.51 A1

L 2 2 2.

 

 

 

’-c,+co c, '

2 2 Q1 fi
* c. we {OJ-{13} [3'1”]

2 2

 

 

 

where co, cl, c2, c,, c,, c,, c,, f}, and f,- are coeﬁcients that vary based on the

selmd model.

There are four diﬁ‘erent models that will result based on the previous
discussion. These include

i. W: This model is the result ofcombining both the
continuity equation and the unsteady momentum equation. The various

coeﬁcients in [3.110] may be expressed as
l l 2
., «=1.» 0].], we} 4%)

L L
c, = 0, c,- 0, f} =E-(So-Sf)‘, and j} =-2-(So-S,)j -

co

ii. W : This model is the result of combining both
the continuity equation and the steady momentum equation. The various

coeﬁcients in [3.110] may be expressed as

1 2
90:09 61‘0’ 'c’=(T} cﬁhgﬁ} c‘=[§z}

L L
c,=o, cj=o, 1:.=-2-(s,-s,),, and f,=§(s,-s,)j.

85

iii. W: Thismodelisthe resultofcombiningboththe
continuity equation and the zero-inertia assumptions that result in [3.98]. The
various coeﬁcients in [3.110] may be expressed as

L n2 1
Co=§A—2-RQ—m-, c,=0, Cz=(7) 63-40, C4=0,

c, = 0, c. = 0, f, =§Sw and fj-=%So,-

iv. WW : This model is the result of combining both
the continuity equation and the kinematic wave or the uniform ﬂow
assumptions that result in [3.108]. The various coeﬁcients in [3.110] may be

expressedas

Co =-1, C1 = O, 02 = 0, C3 = 0, C4 = 0,
c,=%-R,”S&”, afﬁrm, f..=o, and 1;=o-

Equations [3.109] and [3.110] are solved simultaneously for every
element to determine both the ﬂow rate, Q, and the cross sectional area of ﬂow,
A , at each node. These equations are combined together to produce one system
of equations for every element. The resultant system of equations has the

general form

 

 

 

 

 

 

 

 

 

 

.. 2 o 1 0' A.-
R. Lo 2c 0 c Q.
(. _‘ II: =_ l l 1
{R5 R,,*610 2 o‘A,’
.Rui. .0 cl 0 2“. .Qi.
' o -1 o 1 ' FA.‘
+1 -Cz+C3+C,- -C4+c° €2-03 C4 Q;
o -1 o 1 ‘A,’
-0244}, -C4 €2-C3+Cj 044130. Q”
r’IiL‘
2
f.
“4],; [3.111]
2
bf]. J

 

 

01'

{R”1=1€1{¢1+1K1{¢}-{F1 13.1121

87

3. Finite Element Formulation Using Quadratic Elements

The numerical solution of the Saint-Venant equations is repeated in this
section. However, the analysis this time is accomplished based on the c sub
Galerkin formulation of the ﬁnite element method and using quadratic

element.

 

A system of equations is generated by evaluating the weighted residual
integral

’2
R(x)= IWQ)[2£—+%+I)dx [3.113]
x,

which is the result of integrating the product of the continuity equation ([3.9])
and weighting function, W(x), over the length of the quadratic element. The
Galean formulation of [3.113] is based on considering the shape functions M,
"iv and N, as the weighting functions at nodes i, j, and h, respectively. Since
the elementinFigure 4 is quadratic, there will be a system ofthme equations
for every element.

The ﬁnite element formulation is applied after representing the
unknowns by linear approximations of the form

9‘.) =Ni¢i +Nj¢j +Nk¢h [3.114]

01'

“7

 

 

 

 

 

 

 

(A) F‘—
...
F‘s
_,x 1 j ﬂk a .
P—L—4
I(B)|

 

 

 

Figure 4. Finite element discretization. (A) Fur-row flow. (13) Generic
quadratic element in the system of local coordinates. (C) Generic
quadratic element in the system of natural coordinates.

89

4».
¢m=lNi N1 N2] ¢i
4%

=[N]{¢} [3.115]

whereo is the unknown; N,,N,, andN, are the shape functions at nodes 1,}, and

k, respectively; and Q, 41,, and d», are the values ofthe unknowns at nodes i, j,
andk, respectively.

Based on [3.115], the cross sectional area of ﬂow, A, and the ﬂow rate, Q,
are expressed as
AW) =N1A.(t)+N,-A,-(t)+N.'A.(t)
A10)
= [Ni Ni NJ A10)
A2“)
= [N] {Am} [3.116]

and
Qua‘)=N5Qi(t)+Nij(‘)+NtQA:(‘)
Q10)
=[N5 N} NJ 20)
Qt“)
= [N] {Cm} [1117}

The shape functions for the one-dimensional quadratic element are
expressed by the following equations (Segerlind, 1984)

N.=22;(x-X,)<x-X.> 13.1181

N,=—-E;(x-X.)(x-X.) 13.1191

N.=%(x-X.-)(x-X,) 13.1201

whereL isthelength ofthe element.

The shape functions can be written in the system of local coordinates
which allow [for easier integration over the element. The shape functions for a
system of coordinates located at node 1' are obtained from [3.118], [3.119], and
[3.120] by replacing x with X, +s (Figure 4a). This results in the following
equations:

2
N.=z;o:.+s—X,)or.+s-X.)

=22;(s-%)(s-L) 13.1211

4
Nj=“i'5(xi+3-Xi)(xi+s“xi)

=-£3(s)(s -L) [3.122]

N.=i%0t.+s-X.>a(.-+s-X,-)
2 L
=17“)(’ -5) [3.123]

The shape functions are then rewritten in the system of natural
coordinates which consist of a pair of length ratios as shown in Figure 4b.
These ratios are deﬁned as (Segerlind, 1984)

L -s s
It =(T) and [2.-.(2) [3.124]

91

where s is the distance from node i. The shape functions which are expressed

by [3.121], [3.122], and [3.123] can be rewritten in the system of natural
coordinates and the results are

1129-12—3
132.51%)

= (12 " 11) ('11)
= I: — 1112 [3.125]

~j=1%)(%:—‘—J

= 112 [3.126]

s ] {2.1 -L
~t=(z.-.—)

_ 2‘ [13.29.

" L AL L

=lz(lr11)

=13-I1l. [3.127]

 

 

The system of natural coordinates is very essential for directly evaluating
various integrals that contain the shape functions directly as will be
highlighted in the subsequent discussion.

The partial derivative of Equation [3.117] with respect to x is computed by
evaluating the partial derivatives of N,, N,, and N, with respect to 1: since the
nodal values Q,(t), Q,-(t), and Q,(t) are assumed constant with respect to the x -
or 3 -space dimension. The partial derivative of [3.117] becomes

92

= [B] [9] [3.128]

The derivatives of the shape functions with respect to the x - or .1 -space
dimension are as follows:

 

4 3L
=34 -7) [3.129]
3N a~._
ax): as: 22((3 -L)+s)
s L
=_;(, -5) [3.130]
5’.”_e_2”_.2-_2_ -2 .,
:1: as L2 2 S
4 L
74"?) [3.131]

The derivatives of the shape ﬁmctions which are expressed by [3.129],
[3.130], and [3.131] is expreswd in the system of natural coordinates as

§N_.__1_ 4s-3L
as L L

 

=i—(31,-1,-2) [3.132]

93

a’lﬁ

I
t‘l-h PIA hlh

'2: —L)

‘3 +_..s_-£.

[L L

((2-11) [3.133]

.1.)

{2.1; s-L
[L L

 

a?
1L

81:5

ll
l‘lr— ("It-

 

1
7912-1!) [3.134]

The partial derivative of Equation [3.116] with respect to time is
computed by evaluating the partial derivatives of the cross sectional are of
ﬂow, A, at nodes i,j, and k. The products of the resulting functions and the
respective shape functions which are considered constant with respect to time

produce

We at
””3:

8A
31“" "arm's:

raAi}

= [N,. N, 111,11

3L? 9"I

DA,

 

 

3|

= [N] {A} [3.135]

Substituting the shape functions in [3.135] results in the system of
equations

‘1

R,‘"= IN,(%‘3—+%+I)¢x [3.136]
x,
" 3A

11]": JN,(-$+%%+I [3.137]
‘1

Rf"=f1v —+%%+I}1x [3.138]

Equations [3.137], [3.137], and [3.138] can be rewritten in the matrix form

‘

IN%( —+—+I}h
my}: 2:: =‘i"i[w+é§*’)“
[N _._.,)..

J

’1
= f 11111135933“ ' 13.1391
8‘

V

 

 

where [N]’ is the transpose of matrix [N].

Substituting [3.128] and [3.135] in [3.139] results in

3!
{Rf’} = [[N1’11N11A1+131{Q}+ndx
x,

95

‘0 ‘1 ‘1
{RP} = [1N1’1N11A1dx + [1N1’1311Q1dx + [1N1’Idx 13.1401
81 81 *1

The individual terms of Equation [3.140] are then integrated over the
length of the quadratic element in either system of local or natural
coordinates. However, the system of natural coordinates is used in the
development of this section since such a system simpliﬁes the integration
process immensely. The ﬁrst right side term of Equation [3.140] is evaluamd

88
‘3 L
[1N1’1N11A1dx = [[N1’ 11111411111
2, 0
1
= [[1,1me
0
1 ’12 ‘11,:
=L [ 41.1. [If-1.1. 41.1. é-WMA}
0 £22 “1112
. I.‘ - 2131. +131.” 4131. - 4131’ 1:1: — 131. - 1.1: +1.21.
=L 4131-4131: 16133 4113-41313 dam
° 133-131271113 +1313 41113-41313 1; ~21312+1313
[3.141]
where s= (L1,), 312:1. and I.=(1-I.>~
The various integrals in [3.141] are reduced to the form
[1; 1(1- 19""’d1,=11:((—:—):$)) [3.142]

96

where l‘(n + l) = n! (Abramowitz and Stegun, 1964). The integrals of the matrix

in [3.141] are then evaluated separately as follows:

[a1-211.+z:z:>az.= [11131-2 [21,122+ [211.21.
0 0 0 o

 

 

_ 4101 _ 3111 + 2121
'(4+0+1)1 (3+1+1)1 (2+2+1)1
=1——1260 [3.143]
, 2, 3111 2121
[(4% 411% {(3+1+1)! (24-244)!)
=é5 [3.144]
2, , ,2 _ 2121 _ 3111 _ 1131
[“12 "Ihllhuﬁ‘u’ (2+2+1)1 (3+1+1)1 (1+3+1)1
1 2121 _—4
T(2+2+1)1" 120 [1145]
‘ 2121
22 _
[(la‘lzm‘l (2+2+1)1)
=% [3.146]
1
3 ,2 _ 1131 _ 2121
[(4A12-41112M1r“((1+3+1)1 (2+2+1)1)
=38E [3.147]

97

1
, ,, _ 0141 _ 1131 2121
[(4'2""+"l’)dl"(0+4+1)1 (1+3+1)1)+(2+2+1)1
16
=12?) [3.148]
Substituting [3.143] through [3.148] in [3.1411 results in
=- L 4 2 -1 4.-
[1N]’[N]{A}dx=3-5[ 2 16 2 J A,
‘1 -1 2 4 A;
=[C.]{A} [3.149]

The second integral of [3.140] is evaluated using the same procedure.
This results in

H L
[111/1’13] 1914: = [1111713121191
3, 0

l
= [1111]" 181Ld1.{Q1
0

1 g-hh 1 l 1
=1, [ 41,1, [3(34-11-2) Z(-4I,+4I.) Z(3lz-lt)]dlz{Q}

o g’hk
, -1,’+4z,’1,-31,z,’—21,’+ 21,1, -111,’1,+4z,1,’+41,3 Mfg-31,134,
= 121113-4li'lz-81112 ~161113+161312 121113 -411’12
° 313-4I1I§+11’12-213+21112 -4l§+8l113-411’12 313-41113H1’lz
41219} [3.150]
where s=(Lt.1. $151.. and lx=(1-la)-

The integrals of the matrix in [3.150] can be evaluated separately as

follows:

1
fH.’+4t.’4-31.I:-21?+2I.Iodl.=
0

 

-212.
’24
1
2 2 _ 1121 __ 2111
{(121,5-41,1,-81,1,)d1,_1 (1+2+1)1) (2+1+1)1)
1111
(l+1+1)!
-212
'24
_ 2 0131 1121
11312 “MIR” 29+”‘le" 3((04-3211») (1+2+1)1
J 2111 _ 0121 + 1111
T(2+1+1)1 (0+2+1)1 (1+1+1)1

-_4_
“'24

1

98

310!

211!

 

(3+0+1)!+

_ 112! _ 210!
(1+2+1)! (2+-0+1)!

z 2 3 _ 211!
{(4554-4142 +4I,)clI,—-8(———(2+ 1 + 1)! )4-
310!
(3+0-1-l)!
-1_6
'24
112!

l
I(—lél,l,’+ 1611219111, = -1
o

(1+2+l)!

J“

(2+-1+1)!

+ 111!
(1+1-1-1)!

 

1!2!

(1+2+1)!

211!
(2+1 + l)!

 

J

].-.o

J
J

J

[3.151]

[3.152]

[3.153]

[3.154]

[3.155]

99

0. 3! 112!
I941” +8“: 411W12= 4((0+3+ _—)+8((11)! +2+ 1)!)

2!!!
(2+ 1+1)!

2 2111 1121
[(4% 3% I‘W’ A((2+-1+1») 3((1+2-1-1)!]

3!0!
(3 +04» 1)!

 

Ellis

1
2 2 _ 1121 _ 2111
{(121,12-412191112-1 (”2+1”) (2+1+1)1)

1
MS:

0.31 1121
1‘31“ 4"“1‘9‘19: 3((0+3+1_—)1] 4((1+2+1)1)

2 2111
T(2+ 1+ n1

_1_2
'24

 

Substituting [3.151] through [3.159] in [3.150] results in

1 -12 16 - Q.-
[[111] [B]{Q}dx= ﬁ[—16 0 16] Q,-
4 -16 12 Q,

= [K2] {Q}

[3.156]

[3.157]

[3.158]

[3.159]

[3.160]

100

The third integral of [3.140] is evaluated using a similar procedure as
shown above. This results in

8. L
r _ r
{[N] Id: -I {[N] d:

1
_ r
-1 {[N] 1.1112

1 112'112
=1Lf 41,12 12 [3.161]
0 122‘“:

where s=(LI2), :—:=L: and 11=(1'12)'

The integrals of the vector in [3.161] are evaluated separately as follows:

1
2 _ 2101 _ 1111
[0"1‘94972-10411» (1+1+1)1

 

 

 

1
=3 [3.162]
' 1111
{mwlﬁ‘huoatm}
4
=6 [3.163]
1
2 _ 0121 _ 1111
{(B'I‘W’rmnn)! (1+1+1)1
1
-3 [3.164]

Substituting [3.162]. [3.163], and [3.164] in [3.161] produces

101

.2 l
[[Nfldx = 53+}

1

=—I{F.1 [34651

Substituting [3.149]. [3.160], and [3.165] in [3.140] results in

L 4 2 -1 A.-
{Rf’}=-3-6[ 2 16 2 HA2}

-1 2 4 2
+%[ :1126 1: 16Hg,}-%{j} [3.166]
4 —16 12 Q, -1
or
{RP}=10.1{A1+1K.1{Q}-I1F.1 13.1671

 

A system of equations is generated by evaluating the following weighted
residual integral

 

a
_ LE. 24;. 2939 139- _
Raylwmﬂr 3.43]ax+g,12ax+g A3, (s2 s,)]dx [3.168]

which is the result of integrating the product of the unsteady momentum
equation, [3.46], and a weighting function, W(x), over the length of the

102

quadratic element. As discussed in the previous section, the Galean
formulation of [3.168] is based on considering the shape functions N2, N11 and N,
as the weighting functions at nodes LL and 1:, respectively.

Substituting the shape functions in [3.169] and rewriting the results in a
matrix form yields the system of equations

1 Q2 3’1 M39139
IN‘KT gA’)ax+ +gA23x+g A 8: --(So S,)]dx
R."

.. _ 2.2”” 0’ 34 2030130
{R3}- Rj f~'{(T gT3J3x+ +gA’&+g Aat ”(So S,)]dx
Rf"

r

 

 

2.: a. 2—Aza.+- A a. --<so S1]

J

[1- Q__’_]3_A_+ZQBQ 180
T

’2
_ 1' 1 Q2 3A _2_Q_aQ __3_Q__
-1... 1.- 31m 7. .2]

Or
(a _ T :Q anx
+f1~1’g—A—anx-‘ [IN] (S.— spar [3.1691

The partial derivative 311/811 is determined by evaluating the derivative of

[3.116] with respect to the space dimension. This is accomplished through the
evaluation of the partial derivatives of the shape functions with respect to :1
since the nodal values A2“), A m, and A2“, are assumed constant with respect to
thex-ors-space dimension . This results in

103

M1

3A awA 3111,.
27 Ta? it"s-‘1
_ ﬂ 3111,. an. 2‘
’[ax 3x" 317] "
A1

=[B]{A} [3.170]

The partial derivative aQ/ax was evaluated earlier as expressed by
Equation [3.128].

The partial derivative of anat is determined by evaluating the derivative

of [3.117] with respect to time. This is accomplished through the evaluation of
the partial derivative of ﬂow rate, Q , at nodes i, j, and h, respectively, with
reswct to time and multiplying the results with the respective shape functions
which are assumed constant with respect to time. This results in

9g 391392-39.
at=11I,.-(.J,—t-‘+11I.at41112—37

= [N2 N1. N211

 

 

:32.
a:
92
a:
g
a:
Q.-
=[Ni N} ”1] Q
Q.

= [M {Q} [3.171]

The various terms of Equation [3.169] are integrated individually aﬁer
utilizing the results of Equations [3.170], [3.128], and [3.171]. The ﬁrst term is
determined as

104

1 Q2
[IN 1’[;-%—;)—dx= [IN] ’[;-7[IBMIA}
[=--—[: [IN1’ IBILdtIA} [3.172]

T A3

The right side integral of [3.172] is evaluated separately as

112- 112
[INI’IB1Ldz2=L[{ 41.12 [[ﬁ-(slz-Irz) 541341.) %(3£2"1)]d12

13-1112
2 -12’+412’12-31212’-2122 +21212 -1112’12+41212’+4123 41312—31213-12
= 121213-41312-81212 -161212’+1612’1~2 121213-41312 412
° 313-41113+I?12-213+21112 -4I§+81113-4l?lz 313-41112’+11’12
[3.173]
where s= (L1,), -—ZZ=L, and I2=(1- £2)

The integrals of the matrix in [3.173] were evaluated previously.
Substituting [3.151] through [3.159] in [3.173] yields

-12 16 -
[[N]’[B]Ld12=2—14-|:-16 0 16] [3.174]
4 -16 12

Substituting [3.174] in [3.172] yields

171. ’J’ --(% £21013}? 30; inEJ

2
=[%—EQZ-5)[Kc]{A} [3.175]

105

The second integral of [3.169] is evaluated using the same prowdure. The
result is

39d, _. .22
[[M’ (2 .12]: - [ IN1’[g A2)[Blds{Q}

20 [ ‘ 1'
= — [N] IBJLdAIQ} [3.1761
[1121’ I

The right side integral of [3.176] was evaluated previously ([3.174]).
Substituting [3.174] in [3.176] results in

-12 16 - Q;
212 22. = 22; (1) _
1m (.7sz (.A2J24[16 .1. 12M

Q1

=[f—f;)1x.1 1121 [3.177]

The third integral of [3.169] is evaluated using a similar prowdure as
discussed in the earlier discussion. The result is

[[Nl'g-l-[ )—dx= [IN1’£—[ )[Nlds{Q}

1
=(EIZJJM 111112111210} 13.1781
0

The right side integral of [3.178] was evaluated previously ([3.1491). The
results are summarized as

1 L 4 2 -l
[[Nl’INJLdI2=-33 2 16 2 [3.179]
o -1 2 4

Subsﬁtuﬁng.[3.179] in [3.178] results in

106

1t~1’(.’J%7=(.‘:J(’J[§2 1} é‘HégJ

{310.110} 13.1801

00

The last integral of [3.169] is evaluated using the same procedure as
discussed above

’3 L
[IN]’ (S. - S,)dx = (S. - S,) [We
3‘ 0

1
= (so-3,) [ [1111’th
0

l g'hh
=L(S.-S,)[ 41112 12 13.1811

0 (22",112

The integrals of the vector in [3.181] were evaluated earlier in [3.162] through
[3.164]. Substituting these equations in [3.181] produces

a l
ﬂWTtk-SMR=kaﬁ(%)[%
a 1
=(So-S,){F.} [3.182]

Substituting [3.175]. [3.177], [3.180], and [3.182] in [3.169] produces the
system of equations

107

Q.-

{11: =(2%J(%J[.’2 1:6 éngi}
{21.-2%[2‘4-[322 3233]}

k

l
{4} [3.183]
1

01'

IR“’}=[i)ICIIQ}+ l—Q: [K]{A}
a 8A 1: T 8A3 c

.22
842

= [(3.] {Q} + [KM] {A} + [K229] {Q} - (So-Sf) {F.} [1184]

)[K.] {Q}- (So-Sf) {Fe}

 

The residual vector of the thme-nodal quadratic element for the steady
momentum equation, [3.53], is developed by following the same procedure of
the unsteady momentum equation. The result is

108

 

 

"a
1 Q2 811
f"{["mJ7’%%“’°"f’J’
x2 1
RI“ *1
(c 2.. (C) -2 l-.Q_2 9.4. 31.232- -
{Rm-{:16 - {NMT sA’J3x+gA’3x (3° S’)]dx>
k
‘0
1 9’ a4
111—1—1
1" J
" r 1 0’ 64 2080
= [[N] [[i-‘g—Aj]$+m$-(so-Sl)]dx [3185]
‘1

or

7 ’1
-[IN1’IS.-S,)dx [3.186]
‘1

Substituting [3.175]. [3.177], and [3.182] in [3.186] results in

-12 16 -4 A.-
{RS’}=[l-£ [i] -16 0 16 A.
T 3‘3 24 4 -16 12 1

A1

-12, 16 — Q.-
-2%[(%)[-16 0 16 0,-
3 4 -16 12 Q,

L 1
—(s2-s,)[3){4} 13.1871
1

109

1. - 1-2.2. 29
IRA-[T 8A2)[K.]{A}+(g 2)[K.]{Q}- -(S. -S,)IF.}

= [KM] {Al-t [K229] {Q}-{F..} [3.188]

 

The residual vector of the quadratic element for the zero-inertia
momentum equation, [3.177], is developed based on the same prowdure that
was followed in the previous section. The result is

[NE‘M 3; ”(so 5,)[0
R.“

{112’} = R.” =1[N[Ta‘-—-<S.-S,)]4x

Rf"

Y

 

 

‘1 ‘1
IRS’} = [IN1’[-;—)%dx - [ INJ’ 18.—Spa 13.1891
81 :1

Substituting [3.175] and [3.182] in [3.189] results in the system of equations

«71-1-1011? if. 2H2}-.. 7121:}

110

where (NT) is the new coeﬁcient in [3.98] instead of (1/1' - Q2/3113). The friction

slope, S,, is expressed as a function of ﬂow rate, Q, and cross sectional area of
ﬂow, A , using any of the uniform ﬂow equations that were reviewed in Chapter
H. Ifthe Manning equation is selected, [3.190] becomes

1 1 ' -12 16 - Ai L (SO—SI)I
_ 4 - 16 12 At (50-5,).

1 '-12 16 —4 A13. L So.-
'24] —16 0 16[A, +% 462 7 4.32,.
_ 4 -16 12 A, s,, so,
1 -12 16 —4 A.-
=(7J( [-1. o .2],
4 -16 12

i
24
1 A.
"Q5
AR.-
L {A an 213]] L 45-
+_ __ o,-
+6291?!“ 6 so.
"Qt
.4er
The system of equations above can be rewritten in a similar form as that of the
steady momentum formulation ([3.1881). This results in

 

 

 

 

111

{RS’}=(-11.-)[;14-)[::126 :3; ['qu

 

 

 

 

 

A1
. 2 .
" Q}, 0 0
AR, Q
L "2Q; i
+-6- 0 :4ij” 0 {Qj}
2 0.
0 0 " Q‘
L 43313”.
S...
'17; 430’, [3.191]
so:
01'
IRS’} =%IK.1 {A}+[K.g] 101-3.111.}
= [K22] {A} + [Kg] {Q} - {FJ [3.192]

 

As was discussed previously, various assumptions are implemented to
simplify [3.46] to the so-called kinematic wave approximation. Under these

assumptions, the momentum equation reduces to
82-3,: 0 [3.193]

Equation [3.193] is utilized in the Manning equation ([2.12]) to calculate
the flow rate, Q , at individual nodes. This results in

1
Q: = 741an 51:?

112

1
0.- = mm
and
1
Q; = :Athm 31:2

where R2, 11,-, and R2 are the hydraulic radii at nodes 1', j, and k, respectively.

The above equations can be rewritten as

(ikf’sgz’y’gw [3.194]

(inf’so’fyrgﬁo [3.195]
and

(£12223 51:2” )3: - Qt = 0 [3-196]

Since we are primarily interested in the contribution of nodes i , j, and k to the

individual element, only half of the ﬂow terms in Equations [3.194] through
[3.196] will be utilized as the element contribution to the ﬁnal system of

equations. Thus, Equations [3.194], [3.195], and [3.196] can be rewritten in
the matrix form

1 Rims? 0 0 AI [-2 0 0 91 0
3 0 211%,}? 0 A, +‘ 0 -1 0 Q, =[0} [3.197]
0 0 11335;,” A. [0 0 '1' Q. 0

 

01’

[Kn] [4 } + [Km] {Q} - {F1} = {0}

113

 

The one-dimensional quadratic element formulation of the continuity
equation, [3.167], is rewritten as

L 4 2 -1 A.-
{Rf’}=§[2 16 2[A,-

LE .13 "4.. Pi}
24 24 24 Qi 6
16 16 -4I-
+ -24 0 54- {Qj}-<—6’£> [3.198]
1 _1_6 1.2 9* .12
_ 24 24 24 . ‘7‘

 

 

 

 

On the other hand, Equations [3.183], [3.187], [3.191], and [3.197] are
expressedin the general form

114

4C1 2‘1 ‘c1 Q
{R:)}='3£0'[ 2‘1 16‘'1 2‘1]{Q}}

 

 

 

 

 

'- C1 201 401 Q‘
"-(cz-ca)+c.- 16(c2-ca) -4(c2-ca)]
2 24 24 A.
+ ‘16“: '" 93) Ci 16(02 ' Cs) A;
24 24 A
4(02‘03) ”16(02'03) (oz-ca)“. "
L 24 24 2 .

"-c2-1-co2 16c, -4c2 '
2 24 24
Q.- I?
-l6c 16c
‘ ——‘ 0. - 4,;

24 ”W 24
404 -l&‘ C4 '7' Cu Q‘ 1;

2'47 24 2 _

 

[3.199]

 

 

 

 

 

where 02,-, co}, 02,, c2, c2, c2, c2, (:2, 1:2, (:2, ﬂ, f}, and j; are coeﬁcients that vary based
on the selected model.

There are four diﬂ'erent models that will result based on the previous

discussion. These include

i. W : This model is the result of combining both the

continuity equation and the unsteady momentum equation. The various

coeﬁcients in [3.199] may be expressed as

co,- = 0, co, = 0, co, = 0, ,- ,-

c1=e1 we} (—-—-J $12271

L L
f2=%(So-S,)i, f}=E(So-S,)j, and 125360-89,-

115

ii. W: This model is the result ofcombining both
the continuity equation and the steady momentum equation. The various

coeﬁcients in [3.199] may be expressed as

c22=0, coj=0, c22=0, 02:0, c-=0, c2=0,

2
we we) we}

L L
£=E(So-Sj)p 15=3(So-S,)j. and ft=%(so'sr)1'

iii. W : This model is the result of combining both the
continuity equation and the zero-inertia assumptions that produced [3.98].
The various coeﬁcients in [3.199] may be expressed as

Ln’Q, 4Lanj Ln’Q,
3112’an °’ 611,211,?” °‘ 311312;"

 

co, =

L L L
.0 ='6'Sw fj =3’Soj’ and ﬂ =35}: °

iv. W: This model is the result of combining both
the continuity equation and the kinematic wave or the uniform ﬂow

assumptions that produced [3.193]. The various coemcients in [3.199] may be
expressed as

116

Equations [3.198] and [3.199] should be solved simultaneously for every
element to determine both the ﬂow rate, Q, and the cross sectional area of ﬂow,
A , at each node. However, these systems of equations are combined together
to produce one system of equations for every element. The resultant system of

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

equations will have the form
V (.r
11:2) ' 4 0 2 0 —1 0 ‘ '49
R77) 0 4c, 0 2c, 0 -c2 Q,
R‘.
“225:4 "mi 2 0 16 0 2 0 [11,»
R53 30 0 2c, 0 16¢, 0 21:, Q,
Rf? -1 0 2 0 4 0 11',
[R22 - O -61 0 201 0 461‘ [an
' 12 16 4 '
° "22 0 a 0 "2: .4,
.2... 7+7» .193 .122 :13 ~46 6
2 2 24 24 24 A.[ f
16 16 Q. '
+ 0 -'2—4 0 0 0 '2: [A] '4:}L
.16c, -16c, 16c, 16c2 Q,’ ‘ 4f ’
24 24 Ci c0) 24 24 A; i
-I.L
0 i 0 -16. o 2 Qt _
24 24 24 ‘ ‘ 5
325. 1C: '1605 -16C4 C5+Ck C4+Cu t I; J
_ 24 24 24 24 2 2 .
[3200]
or
{R"’}=IC]{¢}+IK1{¢}-IF} [32011

Where Cs = C: - Cg.

117

4. Finite Element Development Using Nonstandard Galerkin
Formulation

An oscillation free solution can be generamd using the nonstandard
Galerkin formulation of the linear ﬁnite element problem. The formulation
utilizes asymmetric weighting functions for approximating the space
derivatives. These asymmetric weighting functions have the form (Allen et al.,
1988; Lapidus and Pinder, 1982)

X.—x (1' i)(x-xj)

 

 

 

= ;_
N,- L + 311 L2 [3.202]
x "X1 (1 ”X00 ‘X")

where L is the length of the element (refer to Figure 3). The shape functions

are then substituted as the weighting functions in Equations [3.72], [3.80],
[3.90], and [3.99] for the continuity, unsteady momentum, steady momumtum,
and simpliﬁed momentum (with zero-inertia assumption) equations,
respectively. Then the entire integration process is repeated. After completing
this step, the resultant element stiﬁ‘ness matrix, [Kg], will have the form

1" N 1v. N-
flu]? -—-=-dx f [N] [snows —102{N}[%x-' aaxjdsm
'-_1.+2 1-2‘
2 2 2
= __1._2 1+2 {¢}=[K.]{¢} 132041
1 2 2 2.

 

 

where d> represents either A or Q .

118

On the other hand, the time derivatives in the above equations may be
approximated using yet another nonstandard Galerkin procedure. This
procedure involves the lumping of the coeﬁcients of the time derivative. In
order to accomplish this step, the variation of 8.4/3: and anat with respect to x
are assumed constant within the midpoints of adjacent linear elements
(Segerlind, 1984). This step is discussed in the following section for both linear
and quadratic elements.

5. Finite Element Development Using Lumped Formulation

The ﬁnite element development of the variation of the 321/81 and 39/8:

terms of the Saint-Venant equations with respect to the space dimension was
perfomd in the previous two sections based on the so-called consistent or
standard ﬁnite element Galerkin formulation. This formulation was utilized
in establishing the weighting functions together with approximating the
variation of time derivatives of area of ﬂow and ﬂow rate with respect to x for
both the continuity, [3.9], and momentum, [3.46], equations. This
development was based on the assumption that the variations in the time
derivative are linear and quadratic for the linear and quadratic ﬁnite
elements, respectively (Segerlind, 1984). The linear and quadratic shape
functions were then used as a weighting coeﬁcients for the linear and the
quadratic element formulations, respectively. The resultant element
capacitance matrix for the linear element in both the continuity and
momentum equations had the form

L 2 1
[C‘]='6[1 2] [3205]

On the other hand, applying the consistent ﬁnite element formulation to the
quadratic element resulted in the capacitance matrix

119

L 4 2 -1
[c,]=§ 2 16 2 [3.206]
-1 2 4

An alternative to the above approach for deﬁning the variation of 34/3:

and age: with respect to x is to assume that these are constant within the
midpoints of adjacent linear elements (Segerlind, 1984). This concept is
referred to as the lumped or nonstandard ﬁnite element Galerkin formulation.
Under this assumption, the variation of both 3A Id: and anat with respect to x
are written explicitly using the step ﬁmction. This concept is applied to both
linear and quadratic element formulations.

8.an

As discussed by Segerlind (1984), the variation of both BA/at and 39/81
within the element can be written using the step functions

3A _ L 3A5 L aAJ'
Emil-("53:44 7)]? ”m"
and
24414-21211 21%
L '0 for s<§
where h(s-§)=+' L} [3209]
L1 for s>§

 

 

The selection of the step ﬁmction in [3.209] is based on the assumption that the
variations of am: and 39/3: with respect to x are constant within the
midpoints of adjacent elements.

120

The quantities multiplying 21,-, Aj, Q,, and Q, are the new shape functions,
N,’ and N}, where

N;=1-h( -92.) ' [3210]

. . L
N; = ’{S '3) [3211]

Substituting [3.210] and [3.211] in [3.207] and [3.208] results in

.394, .34,
at“): 4"— a: ”'1”. Ta?
=[N‘]{A} [32121
and
gob 44%”! 33-1
=[N‘1{Q} [32131

Substituting the same shape functions (Equations [3.210] and [3.211]) as the
weighting functions in the ﬁrst right side term of [3.73] and the third right side
term of [3.81] results in

‘1 L
ftN'l’IN‘l {A 14:: = [[N’I’IN‘JMA}
3‘ 0

= [Q] {A} [3214]

I, z.
fuv‘l’uv‘l {12147: = I [W 141141121

=[C.]{Q} [3215]

121

L
where [6;] = fw‘fmjds [3.216]
0

The element capacitance matrix, [Q]. can be readily evaluawd since mm = l,

NIN,‘ = 0, and N,-'N,-‘ = 1. These terms are the result of the characteristics of the
shape functions. Each shape function has a value of one at its own node and
zero at the other nodes (Segerlind, 1984). Substituting these terms in [3.216]
results in

=£[l 0] [3.217]

The resultant new capacitance matrix in [3.217] together with the new
stiﬁiess matrix in [3.204] are then utilized in establishing a general linear
element system of equations for the nonstandard ﬁnite element Galerkin
formulation. This is accomplished by following the same approach that
resulted in [3.111] which represents the general linear element system of
equations developed based on the consistent ﬁnite element formulation. The
resultant general system of equations will have the form

122

 

 

 

 

 

 

 

 

 

 

 

 

'RS’“ '1 o o 0‘ A]
R“? 0 c 0 0 Q-
(. _‘ In =_ l 4 r
{R)}— R3); 0 0 l 0 A1»
#5:}, _0 o o c,_ g,
' 0 -1+a 0 1-01 ' ME
+1 (‘l+a)cs+c£ ('1+a)c4+co (l'aks (l-G)C4 Qt>
2 o -1-a o l-a ‘4,
(—l-a)c, (-l-a)c, (l+a)c,+cj (1+ak‘+c°.(QL
3w
2
21:13 [3.218]
T
J:- .
01'
{R“’}=ICI{¢}+1K1{¢}-{F} [3219]

where co, c,, c,, c,, c,, c,, cj, ﬂ, and f; are as deﬁned previously.

b-Quadratiamemaut

In this action, the same approach is followed as shown above. The
resulting variations of both 824/31 and age: within a quadratic element are
written using the step functions

%a)=[1'h["%)]%+[h( '13]

{141-2214212134

123

%«>=[l-h(s-%J]%+[h(s-%)]

 

 

5L 89,- 51. BQ.
{1-h(s— 6 II a‘ +[h[s- 6 )] a: [3.221]
L 0 for s<l—6'
where h[s-—)=< > [3.222]
6 L
l for S)-
. 51
' W
5L 0 for 615—:-
and h[ -—)=< > [3.223]
6 5L
1 for S)?

 

 

It is important at this point to emphasize the fact that the selection of the step
functions in [3.222] and [3.223] is bawd on the assumption that the variations
ofaAlat and anat with respect to x are limited to three constant values within
the length of the quadratic element. These values are constant in the intervals
0 to L/6, LI6 to 5L/6, and 5L/6 to L.

The quantities multiplying A,, A,, A.” Q,, Q, and Q, are the new shape

functions (N,', N}, and NI) where
~;=1-h(s—%) [3224]
H1 6114-2)] 225
N; =h( -%} [3.226]

Substituting [3.224]. [3.225], and [3.226] in Equations [3.220] and [3.221]
results in .

124

.BA .aA_, .aA,
—(x)= Ni— at +jN. J-aT +N:— at

= [N‘] {A} [3227]
and

w OiaQ ean+ oan
'37“): N‘a:+";'_ a: ”‘3‘?

=[N’]{Q} [3.228]

After implementing the same shape functions (Equations [3.224], [3.225], and
[3.226]) as the weighting functions in the ﬁrst right side term of Equation
[3.140] and the third right side term of Equation [3.169], the following
equations result

‘1 L
ﬁN‘l’ [N‘] {A 14: = f [N‘I’ [M14141
0

x‘

= [Ce] {4} [3229]

‘1 1.
[INT [N‘] {214: = [[N‘l’mm}
x. 0

= [Cc] {Q} [3230]
L

where [cc] = fpv‘fuv’ws [3231]
0

The element capacitance matrix, [Cc], can be readily evaluated since MN: = 1,

MN; = 0, MN; = 0, Mn; = 1, Nj‘N; = o, and NIN: = 1. Substituting these terms in
[3.231] results in

9%.. P-
'2'
‘41.
i
ll

9%
3.
.2,

OIF
I

 

E
6
o
0

COO

a~|t~

1 0
0 4
0 0

660

125

MN; MN,
MN; NM 4:
MN; MN;

0 i-ooo
0ds+ 010
o 4000

00 0
4L
+0-6—0-t-
00 0

0
0
1

L000
ds+ 000d:

-001
C

00
00
00

0

o
.1:
6

[3.232]

The resultant capacitance matrix in [3.232] is utilized in establishing a
new general system of equations for the quadratic element. This is
accomplished by following the same approach that resulted in [3.200] which
represents the general system of equations based on the consistent ﬁnite

element formulation. The resultant system of equations has the form

{R“’}=<

01'

 

 

 

 

 

 

 

126

 

 

 

 

 

o o 0" 'A.‘
0 0 0 Q,
o 0 o A,
4
4c, 0 0 Q1.
0 1 o A',
0 0 c,‘ £ng
16
° ‘22
16c, 16c,
24 24
0 0
Ci 901'
16
° '52
-16CS -1604
24 24

{Rm}= [C]{¢}+[K]{¢}-{F}

2

 

 

 

 

6
1?
46L

6 .
4f;
411-

6

 

 

111.

[3.233]

[3.234]

where ca, coj, cu, c,, c,, c,, 0., 0,, c,-, c,, f}, and f,- are as deﬁned previously and

C5 = 02- C3.

The lumping of the coeﬁcients of the time derivative usually produces
smoother numerical solutions compared to the standard ﬁnite element
Galerkin formulation (Allen et al., 1988).
formulation has fewer constraints on the time step compared to the consistent
ﬁnite element formulation. The consistent formulation will violate physical
reality and produce numerical oscillations if the selected value of the 0

The lumped ﬁnite element

127

parameter is less than 2 I 3 in the two-level time scheme (Segerlind, 1984). The
latter scheme will result from implementing the ﬁnite diﬁ‘erence solution in
time as will be demonstrated in the next two sections.

6. DirectStiffnessMethod

Segerlind (1984) deﬁned the direct stiﬂness method as "the procedure for
incorporating the element matrices into the ﬁnal system of equations".
Equations [3.219] and [3.233] represent the general ﬁnite element system of
equations for the linear and quadratic elements, respectively. These element
matrices can be incorporated into the ﬁnal system of equations in a
straightforward manner. The element matrices will be assembled into a
banded system of equations as discussed in the text by Segerlind (1984). The
zero coeﬁcients outside the bandwidth will not to be stomd. The bandwidth is
deﬁned as one plus the greatest distance between the last non-zero coeﬁcient
and the diagonal coeﬁcient in a row (Segerlind, 1984). Since both the linear
and quadratic elements of the ﬁnite element grid of the surface irrigation
problems will be numbered successively from left to right as shown in Figures
3 and 4, the bandwidths of these problems will be four and six, respectively.
The reason the bandwidths are of values of twice as much as the classic linear
and quadratic one-dimensional grids is due to the number of unknowns per
node. As discussed earlier, there are two unknowns per node for the surface
irrigation problems which include the area of ﬂow, A , and the ﬂow rate, Q ,
compared to only one unknown per node for the classic one-dimensional ﬁnite
element grids.

By applying the direct stimress procedure, the following global system of
equations result

{R0} = [Ca] {‘1’} + [K0] {¢} - {F0} [3235]

128

where {R0} is the global residual vector, {1b} is the global vector of unknowns

A and Q, {<5} is the global vector of the unknown derivatives with respect to
time, [C0] is the global capacitance matrix, [K0] is the global stiﬁress matrix,
and {F6} is the global force vector.

The system of equations in [3.235] represents a system of nonlinear
ﬁrst-order diﬁ'erential equations in the time domain. The boundary conditions
should be incorporated into this system before it can be worked out further as
will be shown in the next section.

7. Finite Diﬂerence Solution in Time

After applying the direct stiﬂness procedure as discussed in the previous
section, the ﬁnite element solution of both the continuity and the unsteady
momentum equations for both linear and quadratic elements results in a
general system of linear-ordinary differential equations in the time domain
(Equation [3.235]). This system of equations must be solved numerically to
account for the variation in time. Several procedures can be implemented for
the numerical solution of [3.235]. The ﬁnite diﬁ'erence method is the most
commonly used method to approximate d{d>(t)}/dt and d>(t) at successive points
in the time domain (Segerlind,,1984).

The mean value theorem for diﬁ‘erentiation can be applied to the solution
of [3.235] as discussed by Segerlind (1984). Given any function 41(1) and the
interval [a,b] as shown in Figure 5, the mean value theorem states that there
isa§betweena andb suchthat

129

W)

 

 

 

a g b t

Figure 5. Plot of ¢(t) as a function of time (Segerlind, 1984).

130

6(6)-¢(a)= (b «@349 [3.236]

01‘

511 _ Nb) -¢(a)
d: (§) - A: [3237]
where At = a -b.
The value of Ma) can be approximated as
dt
Ma)=¢(§)-(§-a);;(§) [3238]
01'
dd»
¢(§)=¢(a)+(§-a)3;(§) [3239]
Substituting [3.237] in [3.239] results in
¢(§)=¢(a)+wgf“a—)(§-a) 132401
or
«5.) = «0+ [MM-Mano
= (1 -e) ¢(a)+9 ¢(b) [3241]

where 9 = (§-a)/At.

The results of Equations [3.237] and [3.241] can be generalized for column
vectors (Segerlind, 1984). The resultant two equations at t = g will be

4’ - <5
{1%)}: {¢}={ }.~{ 1. [3242]

131

{4’} = (1 - 9) {4’}. + 9 {4’};

[3.243]

where {tb}, and {d>}, are two column vectors containing the nodal values at

times a and b , respectively.

A similar approach can be followed to develop an equation for {F} at t = §.

The resultant equation is

{Fa} =0 ‘9) {Fa}.+o {Fa},

[3.244]

where {F }, and {F }, are the force vectors at times a and b, respectively.

Equations [3.242], [3.243], and [3.244] can now be replamd in [3.235].

The result is

{‘1’}. - {4?}.
At

-(1-9){Fo}, -9{Fo}, =10}

 

[Ca] ( )+ [Kc] «1 " 9) {¢}. + 0{¢}6)
01'

([Ca] + ONIKal) {4’}. = ([Ca] - (1 - GWIKal) {4’}.
+At((1 - 0) {Fa}, + 9{Fa},)

or
[110]er = [P6] {4’}. + {F5}

where

[Ac] = ([Ca] + GNIKal).

[Pa] = ([Ca] ’ (1 ' GWIKGD. and
{F5} = At((1 - 9) {Fa}. + 9{Fa},)-

[3.245]

[3246]

[3.247]

132

Equation [3.246] represents a general two-level-time scheme that gives
the nodal values, {d>},, in terms of a set of known values, {tb},, and the force
vectors at times a and b . The parameter 9 should be speciﬁed to obtain a
solution to [3.246]. The range of values of 9 is in the interval [0,1]. Selecting 9
would in turn determine the location of g at which the mean value theorem is
applied.

There are four popular methods that result from four choices of 9
(Segerlind, 1984). These methods include the following :

i. W: This method is obtained by owdﬁ'ins
§=a. The resultant 9 will be 6 =0. Equation [3.246] reduces to

[Ca] {4’}. = ([Ca] - NlKol) {<5}. + NW6}. [3248]

ii W: This method is obtained by specifying
§= At/2 +11. The resultant 6 will be 9 = 0.5. Equation [3.246] mduces to

([Ca] 4.923161) {‘5}; = [We] "% [K0]) {4’}. + [% {Fa }. '4'? {F0 :1.) [3249]

iii. W: This methodis obtained by specifying§=2Atf3+a.
The resultant 9 will be 6 = 213. Equation [3.246] reduces to

[Ical+-2§-1K.1){o}.=(tcal-%‘-1K.1){o}.+[%{Fa},+%‘-{Fa},] 132501

iv. BMW : This method is obtained by specifying
§= b. The resultant 0 will be 9.: 1. Equation [3.246] reduces to

([Ca] +NIK6D {4’}; = [Ca] {4’}. +44%}, [3251]

The global matrices [Ca] and [K0] and the global vector {Fa} which result

ﬁ'om applying the direct stiﬁ‘ness procedure should be modiﬁed before the

133

initiation of the solution process. This modiﬁcation is necessary to include the
known nodal boundary conditions without which the problem is undeﬁned.
The resultant system of equations is solved then for every time step using a
direct approach such as Gaussian elimination.

8. Implementing the Solution Procedure

In order to implement the ﬁnite element Galerkin formulation of the
motion equations as was described earlier in this section, many relationships
should be established to reduce the number of dependent variables in the
solution process. These include establishing relationships among ﬂow
geometry parameters. Also, such relationships should be implemenmd in
order to mduce the number of dependent parameters in the uniform ﬂow
equations which are utilized to establish the friction slope. Moreover, the
inﬁltration function that will be used in the solution process should be
selected.

 

In order to apply the ﬁnite element Galerkin formulation to the numerical
solution of the complete or simpliﬁed forms of the hydrodynamic equations in
furrows or borders, some mathematical relationships ought to be established
among ﬂow geometry parameters. These relationships are to reduce the
number of dependent variables in the ﬁnite element formulations that were
discussed in the Theoretical Development section of this chapter. The number
of dependent variables can be mduced by relating the depth of ﬂow, y , and the
cross-sectional area of flow, A , in furrows. One approach is to select the
following power curve (Elliott at al., 1982; Walker and Skogerboe, 1987):

y = 6,11" [3252]

134

where o, and o, are empirical ﬁtting constants. These constants reduce to

unity in borders given a unit width of ﬂow. The next step is to establish a
mathematical relationship between the top width of ﬂow, T, and the

cross-sectional area of ﬂow, A. Assuming a parabolic cross sectional area in
furrows, the top width can be represented as (Chow, 1959)

T =5; [3.253]

= 71A [3254]

where 7, and 72 are empirical ﬁtting constants with y, = 3120, and 7, =1- 0,.

Equation [3.254] is only applicable to furrows with parabolic cross sections.
These parameters are considered constants for any given furrow given the
assumption of a prismatic channel. The parameters 7, and y, in [3.254] will
reduce to 1 and 1.5, respectively, when the case of borders is considered
(T =1, given a unit width of ﬂow). Equation [3.252] is utilized in the ﬁnite
element formulations of the zero-inertia and the steady and unsteady

hydrodynamic models.

135

kW

The uniform ﬂow equations (Equations [2.11], [2.12], and [2.13]) that

were reviewed in the previous chapter can be written in this general form

 

 

 

[3.255]

where the parameters It, and k, are shown in Table l, A is the cross-sectional

area of ﬂow, R is the hydraulic radius, P is the wetted perimeter, Q is the ﬂow
rate, and S, is the friction slope. The number of dependent variables in [3.255]
can be reduced by mathematically relating the wetted perimeter, P, and the
cross-sectional area of flow, A, in furrows. Again, if a power curve is selected,
the following relationship can be established

P=B.A” 132561

where B, and B, are empirical ﬁtting constants. Substituting Equation [3.256]

in [3.255] results in

2’53“"
= k‘AZ‘rb‘

 

3!

 

= [3257]

136

Table 1. Summary of the parameters It, and k, in the uniform ﬂow equations.

 

 

 

n is Manning's roughness coeﬁcient,

C is Chezy’s roughness coeﬁcient,

f is Darcy-Weisbach’s roughness coeﬁcient, and
g is the acceleration due to gravity.

 

137

where p, and p, are empirical ﬁtting constants that are controlled by the

hydraulic section of the furrow or border. The parameters p, and p, will vary
based on the selected uniform ﬂow equation. The above parameters are held
constant for any given irrigation event. The parameters [3, and 6, reduce to 1
and 0, respectively, when the case of borders is considered given a unit width
of flow. This reduces the parameters p, and p2 to 1 and 2 +b,, respectively.

cmmtiQLEuncﬂons

Any of the empirical inﬁltration functions that were reviewed in the
previous chapter can be utilized to estimate inﬁltration in the ﬁnite element
development of the motion equations as applied to surface irrigation problems.
However, many scientists preferred to use the Kostiakov-Lewis inﬁltration
equation in their developments. Although this inﬁltration function neglects to
account for the eﬁ'ect of wetted perimeter changes on inﬁltration, it was
reported in many studies that the Kostiakov-Lewis function produces good
results especially when the deﬁnition of the parameters a, k, and 13 was based
on ﬂow rates typical of the normal irrigating conditions. The Kostiakov-Lewis
function has the form

2:21“ +f,z [3.258]
01'

I=g=akr"+yg [3259]

where a, k, and ﬂ, are ﬁtting constants, Z is the cumulative inﬁltration, I is the

inﬁltration rate, and t is the opportunity time, to be diﬁ‘erentiated from the
time t.

IV. RESULTS AND ANALYSIS

The Galerkin formulation of the ﬁnite element method was used to solve
the complete and simpliﬁed forms of the hydrodynamic model using both
linear and quadratic one-dimensional elements. The Galerkin formulation
was ﬁrst applied to both the continuity and momentum (in its complete or
simpliﬁed forms) equations with respect to the space coordinate for a ﬁxed
instant of time. This results in a system of ﬁrst-order ordinary diﬂ'erential
equations in the time domain. Then, a ﬁnite diﬂ‘erence approximation in the
time domain was applied to the ﬁnal system of equations to generate a
numerical solution. The direct stiﬂ‘ness procedure was utilized in building the
global systems of equations at various time steps. The ﬁnal system of
equations were then modiﬁed to incorporate the boundary conditions of the
advance, ponding, depletion, and mcession phases. The dimensions of the total
system of equations remain unchanged after the application of these boundary
conditions at any given time step. The ﬁnite element Galerkin formulation
was then utilized in building a general computer model. The latter model can
be used in the analysis of water ﬂow conditions in surface irrigation systems.
Currently, only the kinematic wave ﬁnite element analysis is fully operational

in the present version of the computer model.

The ﬁnite element Galerkin formulation that was applied in the
development of the computer model will be discussed in this chapter, together
with the analysis of the results from applying this model to the simulation of
ﬂow conditions in surface irrigation systems.

138

139

A. Finite Element Model Formulation

The ﬁnite element Galerkin formulation of the complete and simpliﬁed
forms of the Hydrodynamic model was developed in the Chapter 111 using both
linear and quadratic elements. This development produced a general system
of equations for each of the latter elements (refer to Equations [3.218] and
[3.233]). The generalized system of equations reduces to either the complete
hydrodynamic model, the steady hydrodynamic model (where the time
dependent term in the momentum equation is assumed negligible), the
zero-inertia model, or the kinematic wave model through the selection of the
appropriate coeﬁcients which result in the respective model. The system of
equations for the various elements are then assembled into a global system of
equations using the direct stiffness procedure. This latter represents a system
of nonlinear ordinary diﬂ'erential equations. The mean value theorem for
diﬁ'erentiation is then applied to change this ordinary system of equation into
a system of nonlinear algebraic equations. The next step includes the
modiﬁcation of this system of equations to incorporate the various boundary
conditions. Then, the resultant system of equations is solved numerically
using the Gauss elimination. These steps will be described in further detail in
the following sub-sections using the linear element. It is important at this
point to remember that the procedure that is followed in accomplishing the
various tasks that were highlighted in this paragraph is similar for both linear
and quadratic elements. The only difference between the two is reﬂected in
the original equations of each element.

1. Assembling System of Equations

The ﬁnite element solution starts with one two-nodal element during the
ﬁrst time step (Figure 6). This element has two nodes: an upper node at the

140

ﬂow inlet boundary and a lower node at the tip of the advancing ﬁont. Since
the lower node is selected at the tip of the advancing front, both the ﬂow rate
(Q) and the cross-sectional area of ﬂow (A) are zero at that node. The only
unknowns during the ﬁrst time step are the length of the element, which
represents the distance that the moving front has advanced during the time
step At,, and the cross-sectional area of ﬂow (A) at the upper boundary. This is
true for all models except the kinematic wave model where the cross-sectional
area of ﬂow (A) at the upper boundary of the ﬁrst element is determined from
the uniform ﬂow relationship depicwd in Equation [3.254]. As time
progresses, an element is added to the system of equations during each
subsequent time step (Figure 7). The advance phase (Figure 8a) is concluded
once the advancing front reaches the end of the furrow or border. After the
completion of the advance phase, the number of elements remains the same
alter subsequent time steps during the pending (Figure 8b) and depletion
(Figure 9a) phases. After the cutoﬁ' time of water inﬂow the number of
elements is reduced starting from the upstream end of the furrow or border as
the cross-sectional area of ﬂow approaches zero at various nodes. The solution
process is concluded once the receding front reaches the end of the furrow or
border at the end of the recession phase (Figure 9b).

In order to demonstrate the direct stiffness procedure which is
implemented in assembling the global system of equations, four examples will
be presented in this section using linear elements. These examples will
pertain to the full hydrodynamic, steady hydrodynamic (hydrodynamic II),
zero-inertia, and kinematic wave models, respectively. These examples are
based on the discussion that was presented in Section B-5 of Chapter III. The
number of elements that will be used in each of these examples is 3 which
results in four nodes. The global capacitance matrix ([CGD, stiﬁress matrix
(Hail). and force vector ({Fa}) in Equation [3.232] which has the form

141

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{Ra} = [C0] {4’} + [K0] {‘1’} - {Fa}

will have the form

Example kW

 

 

 

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151

2. Incorporating Boundary Conditions

Most of the boundary conditions in the one-dimensional surface irrigation
problem involve boundaries with known-values. These are referred to as
boundary conditions of the ﬁrst kind or Dirichlet boundary conditions.
Generally, there are ﬁve of these boundary conditions in sloping border and
furrow irrigation systems. The ﬁrst is the upstream boundary during the
advance phase (Figure 8a) where the inlet ﬂow to the furrow, or the unit inlet
ﬂow to the border, is known. This represents the case where the ﬂow rate at
node 1 (Q,) is known. During the advance phase, another known boundary
exists at the tip of the advancing front. At this boundary, both the ﬂow rate
and the cross-sectional area of ﬂow at that node are zero. Once the advancing
front reaches the end of the furrow or border, two boundary conditions may
occur. The ﬁrst involves the case when tail water is draining from the end of
the ﬁeld. At this instance, a ﬂow is assumed to be uniform at the downstream
node of the last element. The other boundary could involve the case where a
barrier at the downstream node of the last element stops the forward ﬂow of
water. This boundary translates to a ﬂow rate of zero at the downstream node
of the last element while the cross-sectional area of ﬂow increases as the water
ponding phase (Figure 8b) starts. The last boundary represents the upstream
boundary as the receding edge moves downstream during the recession phase
(Figure 9b) of ﬂow. Both the ﬂow rate and cross-sectional area of ﬂow at the
upper node of the upstream element of the receding front are zero.

As for the mcession phase, the upstream node that has a cross-sectional
area of ﬂow approaching zero dictates that the receding front has reached that
node which would mean that the last boundary that was described in the last
paragraph is applicable. Then, the receding front moves to the upper node of

152

the next downstream element if the same conditions occurs. This process
continues on until the receding ﬁ'ont reaches the upper node of the last
downstream element.

The various boundary conditions are implemented in the solution process
of the ﬁnite element Galerkin formulation so as to result in a well posed
problem. These known boundary conditions are implemented after the global
system of equations is assembled using the direct stiﬁiess prowdure of the
ﬁnite element method as was discussed in the previous section. The global
system of equations at each time step is modiﬁed to incorporate the various
known boundary conditions starting with the initial conditions where the
global system of equations contains the contribution of only one element. The
same procedure is then repeated at subsequent time steps until the solution

process of the various phases of ﬂow is concluded.

The general form of the global system of equations was developed in
Section B-7 of Chapter III after implementing the mean value theorem for
differentiation. This resulted in Equation [3.246] which has the form

(1%] + GAIIKal) {‘3}. = ([Ca] - (1 - 9)At[Kol) {4’}. + At((1 - 9) {Fa}, + 9{Fa},)
[4.16]

In order to implement known boundary conditions, the system of equations
above ([4.16]) should be modiﬁed at each time step. However, the modiﬁcation
should not be done before determining the matrices [Ag] and [Pa] and the vector
{F5} in the equation

[Ac] {4’}. = [Pa] {4’}. + {F5} = {So} [418}

where

153

[Ag] = ([Ca] + OAthal).

[Pa] = ([Ca] - (l - OWIKal). and
{F5} = At((1- 9) {Fa}. + 9{Fa},)-

Ifthe number of equations is n and d», is the known value (I: is 1,2,3...” or n),

the modiﬁcation of the system of equations in [4.18] is accomplished using the
following steps:

1. Subtract the product A,,,¢,, with i= l,2,....,n ﬁ-om the corresponding

coeﬁcient in the vector {F5}.
2. Replace the coeﬁcients in row I: and column I: of the matrix [Ag] by zeros.
3. Replace the coeﬁcient A... in the [Ag] matrix by one.
4. Add the product n.9,, with i = 1,2, ....,n to the corresponding coeﬁcient
in the vector {F5}.
5. Replace the coemcients in row It and column I: of the matrix [Pa] by zeros.
6. Replace the coeﬁcient in row I: of the {F5} vector by tb...

The nodal values 4’» and 45, represent <b, at times b and a , respectively. This

would allow the known boundary conditions to be speciﬁed as inﬂow or outﬂow
hydrograph.

3. Numerical Solution

After assembling the system of equations using the direct stiﬂ'ness
procedure as was discussed in Section A-1 of this chapter, the system of
equations is solved iteratively at each time step. The iterative solution
procedure is a necessity since the algebraic equations that make up the global
system of equations, which resulted ﬁnm implementing the mean value
theorem for diﬁ'erentiation, is nonlinear.

154

The general ﬁnite element Galerkin formulation of the complete and
simpliﬁed forms of the hydrodynamic model which were developed in Chapter
111 using both linear and quadratic elements has some particular
characteristics that are speciﬁc to the selected model. The diﬁ'erence is
primarily between the full hydrodynamic model on the one hand and the thme
simpliﬁed forms of the hydrodynamic model (namely the steady
hydrodynamic, zero-inertia, and kinematic wave models) on the other hand.
The complete hydrodynamic model is based on two unsteady nonlinear partial
diﬁ'erential equations. The simpliﬁed hydrodynamic models are based on one
unsteady partial diﬁ‘erential equation (continuity equation) and one steady
partial diﬁ'erential equation (simpliﬁed momentum equation). Therefore, the
mean value theorem of diﬁ‘erehtiation applies to both system of equations in
the case of full hydrodynamic model while it applies only to the continuity
equation in the case of the simpliﬁed hydrodynamic models. This would then
translate into the fact that the ordinary diﬁ‘erential equations with odd
numbers in equation [3.235], which represent the contribution of the
continuity equation to the ﬁnal global system of equations, should be
transformed to algebraic equations using the mean value theorem of
diﬁ'erentiation. However, the equations with even numbers in [3.235] would
need the same procedure only for the case of the complete hydrodynamic
model. In other words, the mean value theorem for diﬁ'erentiation is not
needed in the latter case when considering the simpliﬁed hydrodynamic
models. Instead, these equations should have the form

[K.1{¢}.={F.}, [4.191

In order to satisfy both [4.16] and [4.19] for the complete and simpliﬁed
hydrodynamic models, the following equation is developed:

(1%] + Cleal) W}. = (1%] - Cleal) {‘1’}. + €2{Fa }, + 010%}, [420]

155

where the parameters c1 and c, are deﬁned in Tables 2 and 3.

The Gauss elimination routine was used in this deve10pment for the
solution of the non-linear system of simultaneous algebraic equations which
ememd from the global system of ordinary differential equations based on the
discussion earlier in this section. The Gauss elimination routine
(GAUSSBND) implements the solution of the system of equations in two steps.
The ﬁrst involves the forward elimination which reduces the set of algebraic
equations to an upper triangular system. The next phase utilizes the
backward substitution to produce a solution to the unknowns in the system of
equations. GAUSSBND implements partial pivoting, a step that involves the
switching of rows to make the largest element the pivot element. This step is
accomplished without exchanging the rows physically, but by keeping track of
the order of equations in an array. The partial pivoting makes the routine
applicable to the solution of sparse and ill-conditioned system of equations.

The developed Gauss elimination routine was modiﬁed to solve the ﬁnal
system of equations in a banded form since the resultant system of equations
is handed with bandwidths of 4 and 6 for the linear and quadratic elements
(refer to Section B-6 of chapter III), respectively. For this reason, the system
of equations is assembled in banded form using the direct stiﬁ‘ness procedure.
This step reduces the computer storage requirements of the [AG], [Pa], [Co], and
[K9] matrices in Equations [4.17] and [4.19]. The dimensions of these matrices
are deﬁned in Table 5 when these matrices are stored in full and handed forms.
The advantages of this approach are demonstrated in Figures 10, 11, 12, and
13 for both linear and quadratic elements.

156

Table 2. Deﬁnition of the parameters c1 and c2 in Equation [4.20] for
equations with odd numbers in the global system of equations.

Complete Hydrocbrnamic Model ,
(Hydrodynamic Model I)

 

Steady Hydrodynamic Model
‘ (Hydrodynamic Model 11)

Zero-Inertia Model

 

 

Kinematic wave Model

 

 

157

Table 3. Deﬁnition of the parameters c, and c, in Equation [4.20] for
equations with even numbers in the global system of equations.

Complete Hydrotbnamic Model
(Hydrodynamic Model I)

Steatbr Hydrodynamic Model
(Hydrotbrnamic Model H)

Zero-Inertia Model

 

 

 

 

I Kinematic wave Model

 

158

Table 4. Dimensions of the matrices [Co], [KG], [Ag], [Pa]. and [Xe] in full and
handed forms. '

 

Number of Columns, c

Matrix Linear Element Quadratic Element
(n,=2 and n=e+1) (n,=3 and n=2c+l)

 

 

 

 

 

 

 

 

 

number of nodes per element

a]: the matrix that corresponds to [A0] in Equation [4.18] once the
GAUSSBND subprogram is called

[Ca]
[Kc]
| I...
| a».
[Xal
r : number ofrows = 2n
c : number of columns
e : number of elements
n: number ofnodes = (n,-1)e +1
n,:
[X

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163

In order to demonstrate the assembling of the global system of equations

in a banded format, the four examples that were presented earlier will be

presented in this section in a banded form. The global capacitance ([Co]) and

stiﬁ‘ness ([KGD matrices of Equation [3.232] will have the form

Example BMW

[co] 0;-

 

[ta-é

 

 

 

 

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[428]

166

B. Finite Element Computer Model

A general one-dimensional surface irrigation computer model
(FE-SURFDSGN) was developed based on the ﬁnite element Galerkin
formulation of the complete and simpliﬁed hydrodynamic equations. The
model was developed based on the methodology of Chapter III. It utilizes the
ﬁnite element Galerkin formulation for performing the analysis of the surface
irrigation problem using either linear or quadratic elements. The model was
develomd to run on any IBM-compatible microcomputer with a Random
Access Memory (RAM) of 512 Kbytes or more and an MS-DOS version 2.00 or
higher.

1. Model Components

FE-SURFDSGN was developed in a modular format. It has a main
program that addresses a series of routines. These routines vary in size and
complexity. Some of these routines are devoted solely to input and output
while others deal with the setting up of the global systems of equations using
the direct stimiess procedure of the ﬁnite element method and the numerical
solution of the developed system of equations at each time step. The program
allows the user to produce data ﬁles that include the data for both the advance
and recession curves of ﬂow in surface irrigation. FE-SURFDSGN has various
levels of tabular output which is also controlled by the user. It has a
companion graphics routine which, if selected, produces a graphical output of
the advance and recession curves in addition to the plot of actual ﬁeld
measurements. FE-SURFDSGN was developed and compiled in Power Basic
which is the product of Spectra Publishing Company. The size of the program
is very small compared to the number of various functions and options that it
has. The size of the listing of FE-SURFDSGN doesn’t exceed 90 Kbytes which

167

also includes the listing of the graphics routine. The listing of both the
FE-SURFDSGN program and the graphics routine are presented in
Appendices A and B, respectively.

The model can simulate the various phases of ﬂow in furrows and borders.
These include the advance, ponding, depletion, and recession phases. This
analysis is performed based on either the standard or nonstandard ﬁnite
element Galean formulation and using either linear or quadratic elements.
The analysis can be carried out based on any combination of the above options
and using the ﬁnite element formulation of the complete hydrodynamic
(hydrodynamic model I), steady hydrodynamic (hydrodynamic model 11).

zero-inertia, or kinematic wave model.

2. Data Input

There are three screens of data input to FE-SURFDSGN. The ﬁrst input
screen (Figure 14) has the following entries:

Irrigation Method: This input allows the user to select the irrigation
method which could either be furrow ((3) or border (.).

Method of Solution: This entry allows FE-SURFDSGN to perform the
analysis based on the ﬁnite element formulation of the complete
hydrodynamic (.). steady hydrodynamic (.). zero-inertia ([2), or
kinematic wave model (.).

Type of Element: The user has to select the type of element which could
either be linear (ED or quadratic (@3).

Level of Printing: The level of printing can be selected by entering an
integer in the range of 0 to 3. By selecting E), the model would only print
a general summary of the simulation process. On the other hand,

168

extensive output is produced when the level of printing is selected at I.
This output would include a printout of the intermediate steps of
computation which might amount to pages and pages of printout. This
level of printing is only needed for debugging FE-SURFDSGN.

Output Device: The last entry on the ﬁrst screen would allow the user to
choose the output device which could be the screen (.). a temporary ﬁle
SURFDSGNDUT (.), or the printer ((3).

The second screen (Figure 15) allows the user to select many speciﬁc
parameters which aﬁ'ect the speed and accuracy of the solution. These
parameters include the following:

Time Step: The time step (8:) should be entered in minutes.

FE-SURFDSGN utilizes constant time steps to solve the time-dependent
surface irrigation problem. However, the model can be modiﬁed to use
variable time steps if the user sees an advantage in implementing such a
change.

Maximum Number ofIterations: This entry represents the maximum
number of iterations that the model is allowed to perform before
convergence is reached at various time steps. If the maximum number of
time steps has Men reached and the solution did not converge to the
desimd accuracy that is speciﬁed by the user, the program proceeds to the
analysis of the subsequent time step.

Allowable error: This error represents the maximum allowable error for
the accumulated deviation between the results of the previous and

current iteration at all nodes.

Time Weighting Coefﬁcient: The time weighting coeﬁcient (0) is a
number in the range of 0 to 1 (refer to Section B-7 of Chapter III).

169

 

 

 

 

 

 

Figure 14. First screen of input data into FE-SURFDSGN.

170

 

 

 

 

 

 

Figure 15. Second screen of input data into FE-SURFDSGN.

171

Space Weighting Coefﬁcient: The space weighting coeﬁcient (a) is a
number in the range of 0 to I (refer to Section 34 of Chapter III).
Selecting a at 0 would result in the standard ﬁnite element Galerkin

formulation. However, if 0 < a 51 is selected, the nonstandard ﬁnite
element Galerkin formulation results.

Top Width Coefﬁcient: The coeﬁcient 3/ 2 in Equation [3.254] is
included sometimes in the coeﬁcient 0,. Ifthis was the case, a value of 1
should be selected. Otherwise, a value of 1.5 should be keyed in as an
input to this entry.

Consistent or Lumped: This allows the user to select either the
consistent (1) or the lumped formulation (0) (refer to Section B-5 of
Chapter III).

The last screen (Figure 16) allows the user to enter the hydraulic
parameters of the furrow or border. These parameters include the following:

Farrow Length: This entry represents the length of the border or furrow

in meters.

Time of Cutoﬂ! The cutoﬂ‘ time in minutes represents the time when
water ﬂow into the upstream boundary of the furrow or border is turned
off.

Inlet Flow Rate: The inlet ﬂow rate into the furrow or the unit inlet ﬂow
rate into the border should be entered in liters/minute. This entry is
currently considered as the average inﬂow rate at the upper boundary of
the furrow or border. However, the model can be modiﬁed to handle an
inﬂow hydrograph. This step could be accomplished through the
deﬁnition of inlet ﬂow rates at various time steps starting with time zero
until water inﬂow is turned oﬂ‘.

172

 

mam: masters;senesceazssxsmxonV
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Figure 16. Third screen of input data into FE-SURFDSGN.

173

Slope of Channel Bed: This entry represents the slope of channel bed
which should be entered as a fraction. The program currently deﬁnes the
slope at all nodes to be the same. However, this could also be changed in
FE-SURFDSGN to allow for the set up of an array that contains the slope
of channel bed at various nodes.

Manning’s Roughness Coeﬁicient: This entry pertains to the
roughness coeﬁcient in the Manning’s equation. The program currently
uses the Manning equation as the sole uniform ﬂow equation. However,
various uniform ﬂow equations may be incorporated into the model based
on the discussion in Section B-8 of Chapter III.

Flow Geometry Parameter, 0,: This hydraulic parameter represents

the coeﬁcient in Equations [3.252] and [3.254] that were presenmd in
Mon B-8 of Chapter III. These equations are power functions that
correlate the ﬂow rate and top width of ﬂow, respectively, to the
cross-sectional area of ﬂow. The parameter 0, reduces to 1.5 when
borders are considered.

Flow Geometry Parameter, 0,: This hydraulic parameter represents

the exponent in the same equations above (i.e, Equations [3.252] and
[3.254]). Again, this parameter reduces to 1 when the user chooses to
analyze borders.

Hydraulic Section Parameters, p, and p2: These parameters are

empirical ﬁtting constants which are controlled by the hydraulic section
of the furrow as shown in Equation [3.257] (refer to Section B-8 of Chapter
111). These parameters reduce to 1 and 3.333, respectively, when the case
of borders is considered.

174

Inﬁltration Function Coefﬁcient, h: This is the ﬁrst coeﬁcient in the
Kostiakov-Lewis equation (see Equation [3.258] in Section B-8 of Chapter
III). This input should be entered in m’lm/mi n‘. The Kostiakov-Lewis
function is currently the only available inﬁltration function in the
developed computer model. However, other inﬁltration functions can be
easily incorporated into FE-SURFDSGN.

Inﬁltration Function Exponent, a: This parameter represents the
exponent in the Kostiakov-Lewis equation.

Inﬁltration Function Coefﬁcient, f. This is the second coeﬁcient in
the Kostiakov-Lewis equation. This input should be entered in m’Imlmin .

3. Model Output

The developed ﬁnite element surface irrigation model displays a
summary output which represent both the ﬂow rate and the cross-sectional
area of ﬂow at all nodes at various time steps starting from initial conditions
until the conclusion of the recession phase. Also, FE-SURFDSGN produces
two data ﬁles which can subsequently be used to plot the simulated advance
and recession curves of ﬂow in surface irrigation. The program reports to the
user the execution time of the computer after the completion of the simulation
run for all phases of ﬂow.

As was discussed earlier, FE-SURFDSGN has a companion graphics
routine which produces a graphical output of the advance and recession curves
in addition to the plot of actual ﬁeld measurements for those curves. The
utility of the graphics routine of FE-SURFDSGN is demonstrated in the
following section where comparisons between simulated model data and actual

ﬁeld measurements are made.

175

FE-SURFDSGN has various levels of tabular output which is also
controlled by the user. The level of printing can be selected by entering an
integer in the range of 0 to 3. By selecting O, the model would only print a
general summary of the simulation process. However, extensive outputs are
possible by selecting numbers in the range of 1 to 3 (refer to the Data Input
section). The level of data output becomes more and more extensive when
numbers closer to the upper range are selected. The latter outputs would
include printouts of the intermediate steps of computation which amount to
pages and pages of printout.

C. Results and Comparison

A computer model (FE-SURFDSGN) was developed based on the ﬁnite
element Galean formulation of the complete and simpliﬁed forms of the
hydrodynamic model. The model was originally developed to simulate the
advance phase of water in border and furrow irrigation systems. The model
was then extended to simulate the ponding, depletion, and recession phases of
ﬂow conditions in border and furrow irrigation systems. This was
accomplished through incorporating the proper boundary conditions that
correspond to these phases of ﬂow into the ﬁnite element Galerkin formulation
as discussed in Section A-2 of this chapter.

This section includes a summary of the comparisons that were made
between the simulated model data and actual ﬁeld measurements. A
summary of the actual ﬁeld measurements will be presented ﬁrst, together
with the input parameters that were utilized in running FE-SURFDSGN as
well as the sources of these data. Then, plots of actual ﬁeld data and predicted

176

data using the developed computer model will be presented. Finally, a
sensitivity analysis will be presented to show the model’s sensitivity to various

input parameters.

1. Actual Field Data

In order to validate any surface irrigation model, actual ﬁeld
measurements are a necessity. The input data used for running
FE-SURFDSGN which was based on the Galerkin formulation of the complete
and simpliﬁed forms of the hydrodynamic model are presented in Table 5.
These data were repon by Elliott et al. (1982b). The data were originally
collected from furrow irrigation evaluations at three Colorado locations during
the Summer of 1979 by Colorado State University researchers (Elliott, 1980).
The three sites belonged to thme farms that were privately owned. Walker
and Skogerboe (1987) repomd that this study involved six furrows (three
groups of two furrows each) at each site during the 1979 irrigation season. The
reader is referred to the publications by Elliott (1980) and Elliott et al. (1982b)
for a detailed description of the Colorado study. Additional input data for
model testing were used from four different Utah and Idaho tests. These data
were collemd by the researchers in the Department of Agricultural and
Irrigation Engineering at Utah State University from two locations in Utah
and Idaho. These data are described in Table 6 and were taken from Walker
and Humpherys (1983) and Walker and Skogerboe (1987).

The observed advance data for the Colorado study were taken from the
journal article by Elliott et al. (1982b) where those of the Utah and Idaho
studies were taken from the text by Walker and Skogerboe (1987 ). The
observed mcession data for the Colorado study data were taken from the
research study by Oweis (1983).

177

Table 5. General information and furrow evaluation data for the Colorado
study sites that were used in model testing (source: Elliott et al.,

1982b).

---

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Clay loam Loam to clay loam Loamy sand

CropTwe Corn Corn Corn

Fun-row Length (m) 625 425 350

Spacing Between 1.524 0.762 1.524
Wetted FuI'I‘OWI (m)

Irrigation Event, Group 1, 1, 5 5, 2, 1 2, 3, 5 1, 4, 5 8, 2, 3 1, 1, 1
#, Furrow #

Time of Cutoﬂ‘ (min) 690 619 1364 1478.5 171 248

Inlet Flow, Q. (lps) 2.78 1.17 0.92 0.85 2.77 4.81

81090. So (m I m) 0.0044 0.0044 0.0095 0.0092 0.0025 0.0023

Manning’s Roughness, n .03 .02 .02 .03 .02 .03

Furrow Geometry 0.92 0.72 2.18 .87 1.13 1.78
Parameter, o,

Furrow Geometry 0.65 0.64 0.79 0.56 0.75 0.72
Parameter, 01

Hydraulic Section 0.46 0.34 1.35 0.30 0.73 0.92
Parameter, p1

Hydraulic Mon 2.86 2.84 3.00 2.73 2.98 2.91
Parameter, p,

Inﬁltration Function {
Coeﬁcient, k 0.0252 0.0173 0.0033 0.0011 0.0161 0.0078
(m’lmlmin‘)

Inﬁltration Function
Exponent, o 0.02 0.01 0.40 0.48 0.02 0.40

Inﬁltration Function
Coeﬁcient, f. 0.00023 0.00008 0.00003 0.00003 0.00040 0.0014
(m’lmlmin)

 

 

 

 

 

 

Table 6. Additional data for model testing (source: Walker and Skogerboe,

1987).

Parameter

   
 

Soil Type

Flowell

Nonwheel

178

Kimberly

N onwheel

 

 

   

FurrowLength (m)

 

    

Time of Cutoﬂ‘ (min)

 

   

Inlet Flow, Q, (1178)

 

 
   

Slope, s, (m I m)

 

    

Manning's Roughness, n

 

Furrow Geometry
Parameter, o1

 
    

 

Fur-row Geometry
Parameter, o2

 
    

 

  
    

Hydraulic Section
Parameter, p1

 

  
    

Hydraulic Section
Parameter, p,

 

      
 
   

Inﬁltration Function
Coeﬁcient, k
(m’lmlmtn‘)

 

    
   

Inﬁltration Function
Exponent, a

 

     
 
   

Inﬁltration Function
Coeﬁcient, f.
(m’Im/mtn)

 

 

 

179

2. Model Runs

FE-SURFDSGN was developed as a general computer model for
simulating the ﬂow conditions in borders and furrows based on the
methodology that was discussed in the previous chapter. The various models
that are available include the hydrodynamic and kinematic-wave models as
well as the zero-inertia model. Currently, only the kinematic wave ﬁnite
element analysis is fully operational in the present version of FE-SURFDSGN.
There are still some problems in predicting the rate of advance in both the
hydrodynamic and zero-inertia models. More work is being done to complete
the development of these options in the computer model so that the general
ﬁnite element deve10pment is operational for the complete and simpliﬁed
forms of the hydrodynamic equations.

In order to demonstrate the effectiveness of the ﬁnite element formulation
in the numerical solution of the hydrodynamic equations, various simulation
runs were conducted using the kinematic-wave model. Comparing model
results to actual ﬁeld measurements demonstrate the effectiveness of the
presented methodology and the utility of the nonstandard Galerkin
formulation of the complete and simpliﬁed forms of the hydrodynamic
equations as applied to the analysis of furrow and border irrigation systems.

The ﬁnite element model generated an advance curve for each column of
the input data in Tables 4 and 5. These curves are shown in Figures 17
through 25 together with the plot of the measured advance data as rede by
Elliott et al. (1982b) and Walker and Skogerboe (1987). The parameters or and
9 were selected at 0.25 and 0.5, respectively, for all these runs. The selection
of these parameters was based on the sensitivity analysis that will be
presented in the following subsection. The time step, At, was selected at 5

minutes for all these runs.

180

In general, it is obvious from these runs (Figures 17 through 25) that
model predictions are very reasonable. This is apparent where the simulated
rate of advance is very consistent with actual ﬁeld measurements in almost all
cases. The model predictions are good indicators of the effectiveness of the
presented methodology in this research study.

As to the speed of the developed model, the latter prediction runs took
approximately 50 seconds on average on a 386 IBM compatible machine.
However, the average execution time could be cut at least in half if the length
of the time step, At, is doubled. The accuracy of the simulated model results
will not be aﬂ‘ected even if At was doubled or quadrupled as will be
demonstrated again in the subsequent subsection on sensitivity analysis.

FE-SURFDSGN was also utilized in simulating the complete irrigation
cycle for the furrow tests from the Colorado study that were presenwd in Table
4. The irrigation cycle includes the advance, ponding, depletion, and recession
phases of ﬂow. The results of these simulation runs are presented in Figures
26 through 31. The plot of the measured advance and recession data were
taken from Elliott et al. (1982b) and Oweis (1983), respectively. These runs
were conducted using a At of 10 minutes. The parameters a and 9 were
selected at 0.25 and 0. 5, respectively.

Again, it is apparent from Figures 26 through 31 that the simulated rates
of advance and recession are very consistent with actual ﬁeld measurements.
This shows that FE-SURFDSGN can be used as an eﬂ'ective tool for simulating
the irrigation cycle in furrows and borders including the advance, ponding,
depletion, and recession phases of ﬂow.

181

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196

3. Sensitivity Analysis

FE-SURFDSGN was utilized to carry out simulation runs for
investigating the sensitivity of the model to various physical input parameters.
The sensitivity of the model to the selected time step (At), time weighting
coefﬁcient (6), and the parameter a (a parameter that dictates if the standard
or nonstandard Galerkin ﬁnite element formulations are utilized) were also
studied. The linear ﬁnite element formulation of the kinematic-wave model
was used for this purpose. The expected trends should be representative of the
complete and simpliﬁed forms of the hydrodynamic equations. This is
especially true when considering the sensitivity of the model to various
physical input parameters.

The ﬁrst phase was to study the eﬁ'ect of the selected time step, At, on

model predictions as related to accuracy and numerical oscillations. The
various time steps selected in this sensitivity analysis were 2.5, 5, 10, 15, 20,
30, 40, and 50 minutes. On the other hand, the selected physical input data
were those that correspond to irrigation 5, group 2, and furrow 1 of the Benson
farm (refer to Table 5). The results that reveal the sensitivity of the model to
the selected time steps are presented in Table 7. These results indicate that
selecting a A: as high as 20 or 30 minutes would produce results that are
reasonably accurate. This would certainly shorten the execution time which is

as low as 5 seconds on a 386 IBM compatible microcomputer.

The sensitivity of the model to the selection of various values of the
weighting coeﬁcient 0 and a was studied next. The results from these runs
are presented in Table 8 and 9 for 0 and a, respectively. A conservative time
step was used in all these runs. The A: was chosen at 10 minutes even thou@
values as high as 20 or 30 minutes could have been selected.

197

Table 7. Rate of advance predictions for several time steps using the
kinematic wave model of FE-SURFDSGN.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Time Step
Advance Time (min)
(min) 2.5 5 10 _15 20 30 49— 50
10 26.76 25.91 23.93
20 52.61 52.66 50.92 46.40
30 77.83 77.41 77.72 75.21 67.65
40 102.40 102.29 102.44 98.82 87.80
50 126.34 126.04 125.57 106.94
60 149.65 149.49 148.66 149. 77 149.65 144. 19
70 172.37 172.12 171.87
80 194.49 194.31 194.18 194.80 187.28
90 216.03 215.81 215.42 214.53 216.67
100 237.00 236.81 236.31 235.41 228.27
110 257.42 257.22 256.99
120 277.29 277.09 276.91 276.94 275.36 278.44 279.24
130 296.62 296.43 296.07
140 315.43 315.23 314.90 314.89
150 333.72 333.54 333.35 332.76 331.56 337.75
160 351.50 351.31 351.11 351.32 354.17
170 368.79 368.62 368.30
180 385.59 385.41 385.19 385.15 384.61 383.00
190 401.91 401.75 401.60
200 417.77 417.61 417.39 416.80 414.85 422.66
210 433.17 433.01 432.79 432.62 433.27
220 448.12 447.98 447.85 ‘ 447.94
230 462.64 462.49 462.37
240 476.72 476.60 476.41 476.27 476.41 477.15 473.66
250 490.39 490.26 490. 14 487.21
260 503.65 503.53 503.47 502.91
270 516.50 516.40 516.28 516.28 515.31
280 528.97 528.87 528.77 528.67 530.39
290 541.05 540.97 540.94
300 552.76 552.68 552.64 552.41 552.81 552.14 549.87
310 564.11 564.04 563.97
320 575.10 575.04 575.05 574.81 576.68
# of Elements 128 64 32 22 16 11 8 7
Exec. Time (sec) . 415.8 104.7 28.1 15.3 8.7 4.4 2.5 2.0

 

198

Table 8. Rate of advance predictions for several time weighting coeﬁcients
using the kinematic wave model of FE-SURFDSGN.

Time Weighting
Advance Time Coefﬁcient, 0
(min)

mammalian—_—

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10 21.91 22.99 23.93 24.76 25.50 26.16 26.75
20 51.40 51.21 50.92 49.83 48.63 47.38 46.11
30 82.82 80.11 77.72 74.94 72.54 70.35 68.47
40 110.63 105.77 102.44 99.00 96.14 93.40 90.88
50 133.90 128.34 125.57 122.00 118.78 115.43 112.25
60 155.66 151.50 148.66 144.58 140.80 136.89 133.20
70 182.69 176.42 171.87 166.82 162.43 157.99 153.77
80 212.90 199.99 194.18 188.44 183.53 178.54 173.79
90 238.92 220.77 215.42 209.39 204.02 198.53 193.31
100 260.60 241.48 236.31 229.81 223.99 218.05 212.37
110 278.37 263.78 256.99 249.77 243.50 237.09 230.98
120 309.78 284.91 276.91 269.19 262.47 255.66 249.13
130 354.04 ' 303.34 296.07 288.08 280.95 273.76 266.84
140 398.02 322.03 314.90 306.47 298.96 291.40 284.11
150 435.98 342.37 333.35 324.39 316.51 308.60 300.96
160 468.43 360.95 351.11 341.82 333.59 325.36 317.39
170 495.91 376.97 368.30 358.77 350.23 341.69 333.41
180 518.84 394.54 385.19 ' 375.26 366.42 357.60 349.03
190 537.71 413.46 401.60 391.31 382.19 373.10 364.25
200 553.03 428.93 417.39 406.91 397.53 388.19 379.09
210 599.93 442.81 432.79 422.07 412.45 402.89 393.56
220 1 674.94 461.38 447.85 436.81 426.98 417.20 407.65
230 479.59 462.37 451.13 441.10 431.13 421.38
240 491.66 476.41 465.05 454.83 444.68 434.75
250 506.62 490.14 478.57 468.19 457.88 447.77
260 531.53 503.47 491.70 481.17 470.71 460.46
270 554.35 516.28 504.44 493.79 483.20 472.81
280 568.53 528.77 516.82 506.05 495.34 484.83
290 576.67 540.94 528.83 ‘ 517.96 507.15 496.53
300 587.92 552.64 540.49 529.53 518.64 507.92
310 616.11 563. 97 551. 80 540. 77 529.80 519.01

643.11 575.05 562. 77 551.68 540.65 529.79

Tune(aec)

199

Table 9. Rate of advance predictions for several a weighting coeﬁcients
using the kinematic wave model of FE-SURFDSGN.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Advance Time WeightingaCoefﬁcient
(min) 0 0.10 0.20 0.25 0.30 0.40 0.50
10 23.93 23.93 23.93 23.93 23.93 23.93 23.93
20 47.86 49.24 50.40 50.92 51.40 52.27 53.02
30 75.21 76.24 77.24 77.72 ' 78.19 79.11 80.01
40 100.83 101.55 102.15 102.44 102.73 103.32 103.98
50 122.84 124.39 125.25 125.57 125.86 126.37 126.82
60 144.66 146.77 148.16 148.66 149.09 149.83 150.40
70 169.11 170.30 171.38 171.87 172.32 173.15 173.81
80 192.97 193.22 193.83 194.18 194.54 195.25 195.90
90 213.56 214.26 215.06 215.42 215.78 216.43 217.07
100 232.82 234.62 235.83 236.31 236.73 237.46 238.10
110 253.43 255.38 256.53 256.99 257.40 258.14 258.80
120 274.30 275.61 276.51 276.91 277.28 277.98 278.66
130 293.95 294.67 295.62 296.07 296.46 297.17 297.87
140 312.89 313.33 314.41 314.90 315.32 316.05 316.74
150 331.07 331.91 332.90 333.35 333.75 334.46 335.15
160 347.78 ' 349.69 350.68 351.11 351.50 352.21 352.92
170 364.57 366.68 367.82 368.30 368.73 369.45 370.18
180 382.95 383.60 384.70 385.19 385.63 386.33 387.05
190 400.40 400.20 401.15 ‘ 401.60 402.02 402.70 403.43
200 414.93 415.89 416.93 417.39 417.82 418.53 419.28
210 429.01 431.10 432.28 432.79 433.24 433.94 434.70
220 444.88 446.28 447.36 447.85 448.29 448.96 449.72
230 460.16 460.88 461.91 462.37 462.81 463.48 464.26
240 473.87 474.75 475.91 476.41 476.87 477.56 478.35
250 487.88 488.48 489.64 490.14 490.61 491.28 492.07
260 501.46 501.95 503.00 503.47 503.92 504.57 505.36
270 513.06 514.66 515.77 516.28 516.75 517.43 518.24
280 525.13 527.03 528.23 528.77 529.25 529.92 530.74
290 538.94 539.35 540.43 540.94 541.40 542.05 542.87
300 551.13 551.06 552.14 552.64 553.11 553.77 554.60
310 561.25 562.23 563.44 563.97 564.44 565.14 565.98
320 572.08 573.37 574.53 575.05 575.51 576.17 577.01
Execution 45.7 33.9 29.4 28.1 28.5 33.9 34.2

Time (sec)

200

The results in Table 8 show that an optimum value of 0.5 should be
chosen for the time weighting coeﬁcient, 9. This would result in the highest
accuracy and the least numerical oscillations. The numerical oscillations have
shown to be substantial for 9 < 0.5 while the accuracy is also lower. Selecting 9
at values in the range of 0.5 to 0.7 produwd results as stable as 9 = 0.5 but not
necessarily as accurate. This was observed in the runs that were presented in

Table 8 and in other simulation runs with different input parameters.

On the other hand, it was observed that selecting a at 0.25 would produce

optimum results both in terms of numerical accuracy and stability. Even
though the results in Table 9 show similar results using the various levels of a
that were selected (i.e. 0, 0.10, 0.20, 0.25, 0.30, 0.40, and 0.50). high levels of
numerical oscillations were observed in other simulation runs when a was set
at 0 which corresponds to the standard Galerln'n formulation of the ﬁnite
element method. Numerical oscillations were never observed in any of the
simulation runs when a was selected within the range of 0.1 to 0.4. These
conclusions are consistent with those established by Lapidus and Pinder
(1982) and Allen et al. (1988).

The next step was to investigate the sensitivity of FE-SURFDSGN to
physical input parameters. The data of irrigation 5, group 2, and furrow 1 of
the Benson farm were again used for this purpose (Table 5). These input
parameters were designated as the reference data. Variations of these
parameters were then established on both sides of the latter data. The
reference and formulated input data are summarized in Table 10. These data
were used to investigate the sensitivity of the model to variations in physical
parameters. The runs were conducted by selecting all of the reference input
data except for the input parameter that was varied during the respective run.
The results of these simulation runs are presented in Figures 32 through 39.

201

Table 10. Input data to investigate the sensitivity of the model to the change
of various physical parameters.

Parameter Reference Med:
High

Inlet Flow, Q, (1pc)

 

Slope, Ssh/m)

Manning’s Roughness, n
Hydraulic Section
Parameter, p,

 

 

 

Hydraulic Section
Parameter, p2

 

Inﬁltration Function
Coefﬁcient, k
(m’lmlmin‘)

Inﬁltration Function
Exponent, a

Inﬁltration Function
Coeﬂicient, fo

(rn’lm/min)
Farrow Length :
Time of Cutoﬁ' :
Time Step, At
Time Weighting Coefﬁcient, 9 :
Weighting Coeﬁcient, a :
Type of Element :

 

 

 

 

 

 

 

 

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210

The sensitivity of the model to the inﬂow rate at ﬁeld inlet, Q0, is very

clear in Figure 32. By comparing the predicted advance and recession curves
of the reference data (the curve that is represented by a solid line in Figure 32
as well as Figures 33 through 39) to the other curves, it is clear that as the
inﬂow rate increases the pmdicmd advance curve becomes more linear while
the predicted recession curve is not aﬁ'ected to any extent. It is also obvious
ﬁ'om these curves that as the inﬂow rate increases from 0.6 to 1.75 l/sec, the
advance rate becomes more sensitive to Q. When the average inﬂow rate for a
selected run is low, the rate of advance diminishes.

The sensitivity of the model to the slope of the furrow is illustrated in
Figure 33. Even though the variation in the slope between various runs was
not considerable, it is obvious that there are some variations in both the
advance and the recession trajectories. The trend seems to be comparable to
that observed in the case of the inﬂow rate shown in Figure 32. One can also
observe that as the slope, So, increases the predicted rates of advance and
recession become less sensitive to the change in slope. This trend becomes
clear when a comparison is done among the advance and recession curves of
0.0044, 0.0066, and 0.0088 ml m runs. The sensitivity of the model to the
variation in the slope should not be of any concern since accurate ﬁeld slope
measurements could be easily obtained.

Figure 33 illustrates the sensitivity of the model to the Manning's
roughness coeﬁcient, n. By examining the advance and recession curves in
this ﬁgure, it is clear that these curves are sensitive to the variations in the
Manning‘s coeﬁcient. This could be of some concern since it is not easy to
assess the Manning’s coeﬁcient very accurately in the ﬁeld. The curves in

211

Figure 33 depict that there is an inverse relationship between the rates of
advance and recession, on the one hand, and the Manning’s roughness

coeﬁcient on the other.

The sensitivity of the model to the hydraulic section parameters, p, and

p,, is shown in Figures 35 and 36. While the predicted advance and recession
curves appear not sensitive to p,, these curves are sensitive to the variations in
the hydraulic section parameter, p,. The curves in Figure 36 reveal a similar
trend as that depicted in Figure 34 where the sensitivity of the model to the
Manning's coeﬁcient is presenmd. However, it seems that the predicted
advance and recession curves become less sensitive to the change in p, when

the latter parameter increases.

The sensitivity of the model to the three inﬁltration parameters It, a , and

j}, is illustrated in Figures 37, 38, and 39, respectively. The predicted rate of
advance curve seems to be very sensitive to the inﬁltration function
coeﬁcients, k and 11,, but less sensitive to the inﬁltration function exponent, a .
The predicted recession curve, however, shows moderate sensitivity only to the
inﬁltration function coeﬁcient, )3, with no sensitivity to either the inﬁltration
coeﬁcient, k, or the exponent, a. The curves in Figures 37, 38, and 39 show the
inverse relationship between the rate of advance and the inﬁltration
parameters. These ﬁgures also depict that as the inﬁltration function
parameters decrease the predicted rate of advance becomes less sensitive to
the changes in these parameters.

212

D. Discussion

From the results presented in this chapter, it can be concluded that the
ﬁnite element formulation of the hydrodynamic equations can be successfully
used in the hydraulic analysis of ﬂow conditions in sloping furrows and
borders. It was demonstrated that the one-dimensional linear ﬁnite element
formulation produced excellent predictions of the advance and recession
trajectories for almost all the simulation runs conducted under the available
ﬁeld tests. Besides, the computer execution times of the developed computer
model (FE-SURFDSGN) were very reasonable even when very small time
steps were selected. Although the kinematic wave model applications were the
only runs that were presented in this manuscript, similar results are expected
from the other models that are not complete in FE-SURFDSGN at the present
time. These analyses should be undertaken some time in the future to conﬁrm
this claim.

As to the type of element used, it was observed that the quadratic ﬁnite
element formulations of the complete and simpliﬁed forms of the
hydrodynamic equations did not produce the expected results. Because of the
higher order of the quadratic element, it seems that whatever was gained from
the more appropriate approximation of the element to both surface and
subsurface proﬁles of ﬂow was lost due to the unstable behavior of the

problem. This was partially expected since higher order elements were known
to exhibit this kind of behavior in time-dependent problems.

The development of the nonstandard ﬁnite element Galerkin formulation
was prepared after realizing that the standard Galerkin formulation of the
complete and simpliﬁed forms of the hydrodynamic equations had many
problems of numerical instabilities. These problems were avoided in the linear

ﬁnite element developments through the application of the principle of

213

upstream weighting, a step that was accomplished using non-linear shape
functions instead of the usual linear functions. On the other hand, it was
observed that the standard ﬁnite element formulation in time, or the so-called
consistent formulation, was highly unstable. This necessitated the application
of yet another nonstandard ﬁnite element Galean formulation in time for
both linear and quadratic ﬁnite element formulations. The latter nonstandard
Galerkin formulation was referred to in the literature as the lumped

formulation.

The problems of numerical instability of the standard ﬁnite element
Galerkin formulation could be attributed to the nature of the hydrodynamic
equations. These two ﬁrst-order partial diﬁ‘erential equations which represent
the equations of continuity and momentum are both hyperbolic. Numerical
solutions of hyperbolic diﬁ’erential equations are known to exhibit more
numerical instabilities compared to parabolic and elliptic partial differential
equations (Allen et al., 1975).

The goals of this research have been realized by accomplishing the
following ﬁve fundamental objectives. The approaches utilized to achieve
these objectives are delineated below.

Objective 1. Develop a ﬁnite element solution procedure of the
Saint-Venant equations for the hydraulic analysis of

surface irrigation systems.

The approach that was followed under Objective 1 was to develop a ﬁnite
element Galean formulation of the hydrodynamic equations using linear
one-dimensional elements. The ﬁnite element development resulted in a
general system of ﬁrst-order diﬁ‘erential equations for each individual element
in the space domain. The system of ordinary diﬁ‘erential equations included
the derivatives of the unknowns (cross-sectional area of ﬂow, A , and ﬂow rate,

214

Q) with respect to time. This system incorporated the contributions of both the
continuity and momentum equations which constitute the hydrodynamic or
so-called Saint-Venant equations. Each of the hydrodynamic equations
resulted in a system of ordinary differential equations, and the resultant two
systems were combined into one general system of elemental equations. The
mean value theorem for diﬂ’erentiation was then applied to transform the
global system of ordinary diﬁ‘erential equations into a system of algebraic
equations. The global system of algebraic equations should be solved
iteratively over time. The elemental equations should be assembled into a
global system of equations at various time steps using the direct stiﬁ'ness
procedure. After deriving the standard ﬁnite element Galerkin formulation of
the hydrodynamic equations using linear elements, the same development was
repeated using quadratic one—dimensional elements. The latter development
was an attempt to investigate if such an approach would yield faster and more
accurate results. By examining the shape of both the advance and recession
fronts, it was felt that quadratic elements would model both fronts more
adequately compared to linear elements. The ﬁnal step under Objective 1 was
to repeat the previous developments using nonstandard ﬁnite element
Galerkin formulations of the hydrodynamic equations. The one-dimensional
linear and quadratic element developments were repeated under this step.
This action was deemed necessary after realizing that the standard Galerkin
formulations produced numerical oscillations. The latter problem could be
attributed to the unsteady nature of the problem and the presence of sharp
advancing fronts during the advance phase.

Objective 2. Create a general solution approach that accommodates the
available mathematical models of the Saint-Venant

equations in the analysis of surface irrigation systems.

215

The approach that was followed under Objective 2 was to repeat all the
developments of Objective 1 for the simpliﬁed forms of the hydrodynamic
equations. These models included the hydrodynamic II (the model with a
steady momentum equation), zero-inertia, and kinematic wave models. The
next step was to derive a general ﬁnite element representation that would
accommodate the complete as well as the simpliﬁed forms of the hydrodynamic
model. The general model has different coeﬁcients which vary based on the
selected model. By choosing the coeﬁcients of the desired model, the solution
process would accomplish the analysis of ﬂow conditions in surface irrigation
based on the corresponding model of the hydrodynamic equations. The general
development was repeated using quadratic elements. The general linear and
quadratic ﬁnite element developments could be implemented in developing
computer models that simulate the ﬂow conditions in surface irrigation
systems. Such models would be independent of the selected form of the
hydrodynamic equations. The general ﬁnite element formulations using linear
and quadratic elements are distinct when considering the elemental
equations. However, the numerical solution of the resultant system of
equations is independent of the selected type of element after the elemental
equations are assembled into a global system of equations using the direct
stimress procedure.

Objective 3. Develop an approach to easily incorporate the varying
boundary conditions of the advance, ponding, depletion,
and recession phases of surface irrigation into the solution
process with minimal arbitrary or experimental
parameters.

The approach that was followed under Objective 3 was to develop a
procedure for incorporating the appropriate boundary conditions under
varying physical phases of ﬂow in an irrigation cycle into the ﬁnal system of

216

equations. Prior to the application of this step, the global system of equations
would have been assembled using the direct stiﬁ'ness procedure. Also, the
mean value theorem for differentiation would have been applied to transfer
the system of ordinary diﬁ‘erential equations into a system of algebraic
equations. Since most of the boundary conditions in the one-dimensional
surface irrigation problem involve boundaries with known-values, a six-step
process was devised to incorporate the diﬁ'erent boundary conditions. The
developed procedure allows the known boundary conditions to be speciﬁed as
inﬂow and outﬂow hydrographs. The global system of equations would be
modiﬁed at each time step to incorporate the various known boundary
conditions starting with the initial conditions where the global system of
equations incorporates the contribution of only one element. The same
procedure would then be repeated at subsequent time steps until the solution
process of the various phases of ﬂow is concluded. The dimensions of the total
system of equations would be kept the same at any instance in time.

Objective 4. Develop a ﬁnite element computer model for the hydraulic
analysis of ﬂow conditions in border and furrow irrigation
systems.

The approach that was followed under Objective 4 was to implement the
ﬁnite element mathematical development of the motion equations in building
a computer model that can be utilized in simulating the advance, ponding,
depletion, and recession phases of ﬂow in both furrow and border irrigation
systems. The analysis could be performed based on either the standard or
nonstandard ﬁnite element Galerkin formulations and using either linear or
quadratic elements. Any combination of the above options could be selected
and the ﬁnite element analysis could be performed using the complete
hydrodynamic (hydrodynamic model I), steady hydrodynamic (hydrodynamic

model 11), zero-inertia, or kinematic wave models. Currently, the kinematic

217

wave model is the only model that is fully operational in the present version of
the program. The developed computer model has a companion graphics
routine which produces a graphical output of the advance and recession curves
in addition to the plot of actual ﬁeld measurements. The computer model was
prepared in a modular format and was developed to run on any
IBM-compatible microcomputer with a Random Access Memory (RAM) of 512
Kbytes or more and an MS-DOS version 2.00 or higher. The program is very
concise in size compared to the number of functions and options that it
embodies. It makes use of the banded form of the global matrices that result
from the ﬁnite element solution. It directly stores the non-symmetrical square
matrices in a banded form. The program has a routine that solves the global
system of algebraic equations based on the method of Gaussian elimination.
The latter routine was developed to solve matrices that are stored in a banded
form. This drastically mduces the execution time of the program and makes
the simulation run time highly eﬁcient.

Objective 5. Evaluate the predications of the ﬁnite element model using
actual ﬁeld measurements from some existing surface

irrigation systems.

The approach that was followed under Objective 5 was to compare the
results obtained from running the developed ﬁnite element computer model to
those reported from actual ﬁeld measurements for some existing surface
irrigation systems. Actual ﬁeld measurements were taken from furrow
irrigation evaluations at three Colorado locations and two different locations
in Utah and Idaho. The data were originally collected by Colorado State
University researchers and the researchers in the Department of Agricultural
and Irrigation Engineering at Utah State University, respectively. The actual
data were taken from Elliott et al. (1982b), Walker and Humpherys (1983),
Walker and Skogerboe (1987), and Oweis (1983). These data included the

218

physical input parameters as well as the actual measurements of both the
advance and recession phases of ﬂow. The developed graphics routine was
used to display both simulated and actual data of the various ﬂow phases of
irrigation on the same graph. The graphical display included plots of actual
and predicted advance and recession trajectories of ﬂow. The results indicated
that the developed ﬁnite element model simulates the hydraulic analysis of

ﬂow conditions in surface irrigation systems fairly well, under the investigated

conditions.

V. CONCLUSIONS AND RECOMIMENDATION S

A. Conclusions

Based on the results and discussion presented in this study, the following

conclusions can be made:

1. A general ﬁnite element formulation was successfully developed for the
numerical solution of the complete and simpliﬁed forms of the
hydrodynamic equations as applied to the hydraulic analysis of surface
irrigation systems. This formulation allows for the implementation of the
various forms of the Saint-Venant equations without the need for
modifying the solution process.

2. The developed computer model (FE-SURFDSGN) based on the ﬁnite
element formulation of the hydrodynamic equations has proven to be an
eﬁ'ective tool in the hydraulic analysis of ﬂow conditions in furrow and
border irrigation systems. Even though not all the forms of the
hydrodynamic equations are fully operational in FE-SURFDSGN at the
present time, it was shown that the model produces excellent results when
compared to actual ﬁeld measurements for existing systems in Colorado,
Idaho, and Utah. These results were obtained by using the ﬁnite element
kinematic wave model with linear elements. Other models are expected to

produce results at least as good as those presented using the kinematic
wave model, if not better.

3. FE-SURFDSGN model predictions appear to be very consistent with
actual ﬁeld measurements for all phases of ﬂow under the studied cases.
The varying boundary conditions of the advance, ponding, depletion, and

219

220

recession phases of surface irrigation were easily implemented into the
solution process by modifying the system of equations after it had been
assembled at various time steps.

. The standard ﬁnite element Galerkin formulation using linear elements

was shown to have some problems with respect to numerical instabilities.
However, the nonstandard Galerkin formulation resulted in stable

solutions for the various simulation runs conducted.

. Even though the model was developed to run on any IBM compatible

microcomputer, the average time for completing a simulation run was still
approximately 50 seconds on a 386 machine. The selected time steps
under those runs were very conservative in almost all cases. It was shown
that good accuracy could be achieved by using larger time steps (20 or 30
minutes), resulting in execution times as low as 5 seconds. Time steps as

high as 20 minutes were very feasible for all the simulation runs
conducted.

. Due to the banded form of the global square matrices, it was possible to

analyze systems with as much as 200 elements on any IBM compatible
microcomputer with 512 Kbytes or more of memory. This system of
equations would contain around 400 equations to be solved simultaneously
at various time steps starting from initial conditions. Since the system
was assembled without storing any zeros outside the bandwidth and was
solved using a modiﬁed form of the Gauss elimination method, the
execution time for solving these systems of simultaneous equations was a
ﬁ'action of the time that would have otherwise been needed.

. The input data necessary for running the model was kept to a bare

minimum. All the input parameters required by the model are consistent

with those needed with any other numerical surface irrigation model.

221

B. Recommendations

. Further work should be devoted towards completing the ﬁnite element

formulation of the remaining forms of the hydrodynamic equations using

linear elements.

. More simulation runs should be conducted using the various models of the

hydrodynamic equations to compare the accuracy of these models using
actual ﬁeld observations. These runs would make it feasible to assess the
accuracy of the various models and to estimate the trade oﬁ's in using any
of the forms of the hydrodynamic equations.

. The speed and accuracy of FE-SURFDSGN should be compared to existing

ﬁnite diﬁ‘erence surface irrigation models to establish the advantages and
disadvantages of numerical methods as applied to the solution of ﬂow
conditions in surface irrigation systems.

. FE-SURFDSGN should be modiﬁed to allow for the simulation of surge

irrigations. Such a modiﬁcation could be easily accomplished in
FE—SURFDSGN since the ﬁnite element development allows for the
incorporation of various kinds of boundary conditions including an inﬂow
hydrograph at ﬁeld inlet.

. Devote more time to investigate the ﬁnite element formulation of the

simpliﬁed and complete forms of the hydrodynamic equations using
quadratic elements. Since the amount of time that was devoted to
developing and debugging the options of FE-SURFDSGN that correspond
to quadratic elements was not extensive by any means, further work is
needed to investigate whether these elements would work and produce
results comparable to, or even better than, those observed with linear

element solutions.

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Rayej, M. and W.W. Wallender. 1985. Iﬁrrrow irrigation simulation time
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230

Ross, B.B., D.N. Contractor, and V.O. Shanholtz. 1980. A one-dimensional
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Sakkas, J .G. and T. Strelkoﬁ‘. 1974. Hydrodynamics of surface
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Samani, Z.A., W.R. Walker, and LS. Willardson. 1985. Inﬁltration under
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Schmitz, G.H. and G.J. Seus. 1991. Mathematical zero-inertia modeling of
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Schwankl, L.J. and W.W. Wallender. 1988. Zero inertia furrow modeling with
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Segerlind, L.J. 1984. Applied Finite Element Analysis. John Wiley and Sons,
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Shaman, B. and V.P. Singh. 1978. A kinematic model for surface irrigation.
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231

Smith, RE. 1972. Border irrigation advance and ephemeral ﬂood waves.
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233

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APPENDICES

APPENDIX A
FE-SURFDSGN Computer Model Listing

REM *tti*ttt*‘k*******kttti’*******************t‘ktﬁtttittt*ﬁttttttﬂ'tti'titt

 

REM* *
REM * FINITE ELEMENT SURFACE IRRIGATION *
REM * DESIGN PROGRAM *
REMA *
RBMt *
REM * Program SURFDSGN.BAS *

REM *ttitiiiittt*tttit*tittitttitﬁiitiii*itiiiﬁiiiii*ttitiitittttitﬁt‘ktt

*
REM : Developed By
g2: : Walid H. Shayya
Egg * Michigan State University
REM :
4

July 30, 1991
REM

REM *iiititiiiiitti’ﬁtiittitiﬁiititﬂiiiiﬁitit‘kitt**t**t********ﬁﬁ*****ti

COMMON NGphFilS

%DimAry = 302
kDimCol = 7
%DimC02 = 11
%DimAryh = 151

DIM C(tDimAry,1),S(%DimAry)

dim x(§DimAry,%DimCoZ),y(%DimAry,%DimCol),zttDimAry,§DimCol)

dim phi(§DimAry,1),Kmatrx(%DimAry,§DimCol),Cmatrxt‘DimAry,%DimCol)
dim temp1(%DimAry,%DimCol),Forcet‘DimAry,1),ForceP(%DimAry,1)

dim FetarttDimAry.1),philttDimAry,1l,pfx(%DimAry,1)

dim Amatrxt§DimAry,&DimC02),Pmatrx(%DimAry,%DimCol)

dim PrevPhi(%DimAry,1)
dim PFplustt§DimAry,1)
dim tempFl(tDimAry,1),tempP2(%DimAry,1)

dim Lengtht§DimAryh,1),Infil(%DimAryh,1).TofOpp(§DimAryh,1)
dim SO(%DimAryh,1),TopWidth(%DimAryh,l),InfilP(%DimAryh,1)
dim FsubitiDimAryh,1),FsubjttDimAryh,1),FsubkttDimAryh,1)
dim CoeflttDimAryh,3),Coef2ttDimAryh,3).Coef3ttDimAryh,3)
dim Coefd(iDimAryh,3),Coef5tkDimAryh,3)

dim CzitkDimAryh,1),Czj(%DimAryh,1),CzkttDimAryh,1)

dim Elthmtx(6,6),Elthmtxt6,6),TempKt6,6),TempCt6,6),NofPhi(%DimAryh,1)
dim ATmp$(11)

DIM Grpthnt(4), XX%(10), YY%(10), CCSth), AASth)
GOSUB InitialScrn

*

t

t

*

Department of Agricultural Engineering *
t

t

t

i

t

.L
T

 

4+-

DEFINITION OF VARIABLES

 

“““‘

NumElem% : Total number of elements
NumNodet : Total number of nodes
NP% : Total number of unknowns (2 * NumNode%)

NumElmNode§ : Number of nodes per element (2 for Linear Element and

SE35

““‘ \Q“‘ “‘ ‘ “‘ “ “‘ ‘\ ‘§ ‘ Q‘Q‘QQQQ“ ‘ ““ ““‘§‘§‘§‘ “‘~“““ ‘

NumBandW% :

NumofPhi%
phi(i,1)
phil(i,1)
Force(i,1)
ForcePti,1)
Fstarti,l)
Kmatrxti,i)
Cmatrxti,i)
Amatrx(i,i)
Pmatrxti,i) .
Elthmtxt6,6l
Elthmtx(6,6)
NofPhi(10,1):
Coefti,5) :
VARTheta :

O. O. O. O.

ALPHA

236

3 for Quadratic Element)
Band width of the global stiffness and capacitace
matrices
Number of known Phi values
Vector of unkowns O and A
Vector of unkowns Q and A at time t - 8T
Force vector
Force vector of the previous time step
Combination of the vectors Forceti,l) and ForceP(i,l)
Global stiffness matrix [K]
Global capacitance matrix [C]
[C] + 6.5r.[r<] = [A]
[C] - (1 - 9).5T.[K] = [P]
Element stiffness matrix
Element capacitance matrix
Number representing known Phis as boundary conditions
Coefficients in the element [C] and [K] matrices
Is the parameters that determines the model for the
finite difference solution in time where

9 - 0 for forward difference

9 = 1/2 for central difference

8 - 1 for backward difference

8 = 2/3 for Galerkin
A coefficient that determines if the formulation is
either a finite element Galerkin formulation or non-
Galerkin formulation. If Alpha - 0 then the solution
is Galerkin. Otherwise, the solution is asymetric.

 

Qin :
Rhol & RhoZ :

Sigmal and
Sigma2 :
Gravity :
IrrMethodS
MethodS
Phase% :

PhaseRec% :

Inlet flow rate
Characteristics of the hydraulic section of the furrow
or border where

2 1.33 u2
A R = ul A
Rhol - ul
RhoZ - u2

For the case of the border irrigation, the values of
Rhol and RhoZ are 1 and 3.33, respectively.

Empirical fitting constants controlled by the
characteristics of the hydraulic section of the furrow
or border where

62
y = 01 A
Sigmal = 01
Sigma2 = 02

For the case of the border irrigation, the values of
Sigmal and Sigma2 are 1.5 and 1.0, respectively.
Gravitational acceleration (9.81 m/sec“2)
Irrigation method withsolving the surface irrigation
“P" for Purrow irrigation
"B" for.Border Irrigation
Numerical method for solving the surface irrigation
problem where the variable Methods equals
"H“ for Hydrodynamic Model I (Continuity and
Unsteady momentum equations)
"5" for Hydrodynamic Model II (Continuity and
steady momentum equations)
"2" for Zero Inertia Model
"K" for Kinematic Wave Model
An integer that represents the phase of flow where
"1' - Advance
"2' - Ponding
An Integer to denote if the recession phase has
started (1) or not yet (0)

§ ‘ ‘ ~ ~ ‘ ‘ ‘ ‘ ‘ ‘ ‘ Q,‘ Q ‘ Q ‘ ‘ ‘ ‘ Q ~ ~ ‘ Q ‘ Q ‘ ‘ ‘ Q Q ‘ Q ‘ ‘ ‘ V ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘

+

FDorFE%

CONSISOILUMP‘:

TypElemS

FGNamS
LevPrt$

SelPrtOptS

DeltaT
TotalTime
TotalLength

NumStep§
Length(i,1)
Infil(i,1)

InfilP(i,1)
kofInf
fsubO

aofInf
TofOpp(i,1)

SO(i,1)
ManngN
CoefTW
Fsubi(i,1),

GOSUB READING
GOSUB INITIAL
FGNamS ' "SURFDSGN.CFG“

FilIsIn§ = FNExists%(FGNam$)
IE FilIsIn§ - True% THEN
OPEN 'I',t2,FGNam$

INPUT #2,IrrMethod$

INPUT #2,Method$

INPUT t2,TypElem$

INPUT #2,Se1Prt0pt$

INPUT #2,iprint%

5E3?

Determines if Ai & 01 to be used in matrices for
calculating coefficients c1, c2, c3, c4, and c5 (FD)
instead of A1, Aj, Ak, Qi, Qj, and Qk (FE) where
"1" selects the finite difference and
"0" selects the finite element approach
Determines if the consistent or the lumped finite
element formulations for the change of Phi with respect
to time ought to be used where
”1" selects the consistent formulation and
"0' selects the lumped formulation
Specifies the type of element where the variable
TypElemS equals

"L" for linear element

”Q" for quadratic element
Represents the name of the configuration file
Specifies the level of print out which varies from
O to 2. Specifying 0 produces no print out and 2
extensive print out.
A variable to select the output device. In order to
select the output device, you ought to enter the
following for the variable SelPrtOptS

"S" for screen

”P" for printer

"F" for the data file “FESIDP.OUT"
Time step, 5T
Total elapsed time since time 0
Accumulated length of flow for the advance phase
and constant thereafter
Number of elapsed time steps
Length of element 1
Infiltration depth at individual nodes (I). The
Kostiakov - Lewis relation will be used for
determining I (m3/sec/m) where

(a-l)
I a a k t + f
O
Infiltration depth at previous time step.
The coefficient k in the infiltration equation
The coefficient f in the infiltration equation
0

The exponent a in the infiltration function
Time of opportunity 1 in the infiltration function
at individual nodes
Slope of furrow or border at the individual node (So)
Manning's roughness coefficient (n)
Top width coefficient which could either be 1.5 or 1

subj(i,1), and Fsubk(i,1) : Coefficients f , f , and f

i j k
in the force vector at nodes i, j, and k, respectively.

A
T

INPUT
INPUT

INPUT
INPUT
INPUT
INPUT
INPUT
INPUT

238

#2,VARTheta
#2,ALPHA

#2,DeltaT
#2,numiter%
#2,AllError
#2,TotNumStep%
#2,CONSISorLUMP%
#2,CoefTW

INPUT
INPUT
INPUT
INPUT
INPUT
INPUT
INPUT
INPUT

INPUT
INPUT
INPUT
INPUT
CLOSE
END IF

IrrMethodS = UCASE$(IrrMethodS)
Methods - UCASE$(Method$)
TypElemS 8 UCASE$(TypElemS)
SelPrtOptS = UCASE$(SelPrtOptS)

LOCATE 8,8

PRINT "Irrigation Method
LOCATE 10,8

PRINT "Method of Solution . . . . . . . . .
LOCATE 12,8

PRINT "Type of Element
LOCATE 14,8

PRINT "Level of Printing . . . . .
LOCATE 16,8

PRINT "Output Device (’S',
PRINT SelPrtOptS

Reiteratel:

SelStrg$ - 'YyNn"

Strg$ = "Modify the above ('Y', yes,
RowStrg§ a 20

ColStrgt = 8

call SelStrgEntry

Ansl$ = SelOptS

Ansl$ = UCASE$(AnSIS)

IF Ansl$ = "Y" THEN
GOSUB READING
SelStrgS - "FfBb'
Strg$ = "Irrigation Method . . .
RowStrgi = 8
ColStrg% = 8
call SelStrgEntry
IrrMethodS a SelOptS

SelStrgS = 'HSZKhszk"

Strg$ - "Method of Solution
RowStrg‘ = 10

ColStrg% = 8

call SelStrgEntry

Methods = SelOptS

#2,FurLength
#2,TimCut
#2,Qin
#2,kofInf
#2,fsub0
#2,Slope
#2,aofInf
#2,ManngN

#2,Rhol
82,Rh02
#2,Sigma1
#2,Sigma2
#2

('F', furrow, '3', border)

(H, S, 2, or K)

('L', linear, 'Q', quadratic)

(select 0, 1, 2, or 3)

screen, 'F', file, 'P', printer)

’N', no) : "

(’F’, furrow, 'B',

O.

":IrrMethodS
';Method$
';TypElem$

":iprintt

"e
I

border) :

(select H, S, Z, or K) :

239

SelStrg$ = ”Lqu"

Strg$ = “Type of Element . . . ('L’, linear, 'Q', quadratic) :
RowStrg% = 12

ColStrg% = 8

call SelStrgEntry

TypElem$ = SelOpt$

SelStrg$ - ”0123"

Strg$ - ”Level of Printing . . . . . . (select 0, 1, 2, or 3) :
RowStrg§ - 14

ColStrg% - 8

call SelStrgEntry

LevPrt$ - SelOpt$

iprintt = VAL(LevPrt$)

SelStrg$ = "PpSst"

Strg$ = ”Output Device ('8’, screen, 'F', file, 'P', printer) :
RowStrg% = 16

ColStrg% = 8

call SelStrgEntry

SelPrtOptS 8 SelOpt$

GOTO Reiteratel
end if

IrrMethodS = UCASE$(IrrMethodS)
Methods 8 UCASE$(Method$)
TypElem$ = UCASE$(TypElemS)
SelPrtOptS - UCASE$(SelPrtOptS)

1P SelPrtOptS - "s" THEN
OPEN 'SCRN:' FOR OUTPUT AS #1
ELSEIP SelPrtOptS - 'P' THEN
OPEN 'LPT1:' FOR OUTPUT AS #1
ELSEIF SelPrtOptS - "F“ THEN
OPEN 'SURFDSGN.OUT' FOR OUTPUT AS #1
END IF

GOSUB READING

LOCATE 9,14

PRINT "Time Step, 8T . . . . . . . (min) = ';DeltaT
LOCATE 10,14

PRINT “Maximum Number of Iterations . . = “:numiter%
LOCATE 11,14

PRINT "Allowable Error . . . . . . . . . = ";

PRINT USING “4.444494%";A11Error
LOCATE 12,14

PRINT "Maximum Number of time Steps
LOCATE 13,14

PRINT "Time Weighting Coefficient, 9 . . = ":VARTheta
LOCATE 14,14

PRINT “a (Select a-O for Galerkin)
LOCATE 15,14

PRINT "Top Width Coefficient (1 or 1.5) = ":CoefTW
LOCATE 16,14

PRINT "Consistent (1) or Lumped (0) . . = ":CONSISorLUMP%

Reiterate2:

SelStrg$ - "YyNn'

Strg$ - "Modify the above ('Y', yes, 'N', no) : "
RowStrg% = 21

ColStrg§ - 14

call SelStrgEntry

AnsZS 8 SelOpt$

AnsZ$ = UCASE$(AnsZS)

":TotNumStep§

“:ALPHA

240

IF Ans2$ ' "Y" THEN
GOSUB HEADING
LOCATE 9,14
PRINT "Time Step, 8T . . . . . . . (min) = ";
INPUT ",DeltaT
LOCATE 10,14
PRINT "Maximum Number of Iterations . . ,
INPUT ",numiter%
LOCATE 11,14
PRINT "Allowable Error . . . . . . . . . = ”;
INPUT ",A11Error
LOCATE 12,14
PRINT "Maximum Number of Time Steps . . ";
INPUT ",TotNumStep%
LOCATE 13,14
PRINT “Time Weighting Coefficient, 9 . . = ",°
INPUT '",VARTheta
LOCATE 14,14
PRINT "a (Select a=0 for Galerkin) . . . ;
INPUT '“,ALPHA
LOCATE 15,14
PRINT "Top Width Coefficient (1 or 1.5) ,
INPUT '",CoefTW
LOCATE 16,14
PRINT "Consistent (1) or Lumped (0) . . = ";
INPUT ",CONSISorLUMP%
GOTO Reiterate2

END IF

freqprintt x 1
Gravity = 9.81

GOSUB READING

LOCATE 8,8

PRINT 'Purrow Length . . . . . . . . . . . . . (m) = ":FurLength
LOCATE 9,8

PRINT 'Time of Cutoff . . . . . . . . . . . . (min)
LOCATE 10,8

PRINT "Inlet Flow Rate . . . . . . . (liters/sec) - ';Qin
LOCATE 11,8

“:TimCut

PRINT "Slope of Channel Bed . . . . . . (fraction) 3 ":Slope
LOCATE 12,8

PRINT "Manning Roughness Coefficient, n . . . . . . = “:ManngN
LOCATE 13,8

PRINT "Flow Geometry Parameter, 01 . . . . . . . . = ';Sigma1
LOCATE 14,8

PRINT "Flow Geometry Parameter, 02 . . . . . . . . = ';Sigma2
LOCATE 15,8

PRINT "Hydraulic Section Param., Rho 1 . . . . . . = ":Rhol
LOCATE 16,8

PRINT "Hydraulic Section Param., Rho 2 . . . . . . = “:RhoZ
LOCATE 17,8

PRINT "Infiltration Function Coeff., k (m‘3/m/min‘a)= ';kofInf
LOCATE 18,8

PRINT "Infiltration Func. Exponent, a . . . . . . . = '3aof1nf
LOCATE 19,8

PRINT "Infiltration Function Coeff., f (m‘3/m/min) = ":fsubO

Reiterate3:

SelStrg$ = 'YyNn“

Strg$ - “Modify the above ('Y', yes, 'N', no) : "
RowStrgi = 21

ColStrgi = 8

call SelStrgEntry

Ans3$ - SelOpt$

Ans3$

= UCASE$(Ans3S)

IF Ans3$ = ”Y" THEN
GOSUB READING
LOCATE 8,8

PRINT 'Furrow Length

INPUT '”,FurLength
LOCATE 9,8

PRINT “Time of Cutoff .

INPUT “",TimCut
LOCATE 10,8

PRINT "Inlet Flow Rate

INPUT ",Qin
LOCATE 11,8

PRINT "Slope of Channel Bed .

INPUT '",Slope
LOCATE 12,8
PRINT “Manning Roughness Coefficient,
INPUT "",ManngN
LOCATE 13,8

PRINT "Flow Geometry Parameter, 01

INPUT "",Sigma1
LOCATE 14,8

PRINT “Flow Geometry Parameter, 02

INPUT '",SigmaZ
LOCATE 15,8

PRINT "Hydraulic Section Param., Rho 1

INPUT ",Rh01
LOCATE 16,8

PRINT "Hydraulic Section Param., Rho 2

INPUT '",Rh02
LOCATE 17,8

PRINT "Infiltration Function Coeff., k (m‘3/m/min“a)

INPUT ",kofInf
LOCATE 18,8

PRINT ”Infiltration Func. Exponent, a .

INPUT "”,aofInf
LOCATE 19,8

PRINT "Infiltration Function Coeff., f

INPUT '",fsub0
GOTO Reiterate3

END IF

OPEN "O“,02,FGNam$

WRITE
WRITE
WRITE
WRITE
WRITE

WRITE
WRITE

WRITE
WRITE
WRITE
WRITE
WRITE
WRITE

WRITE
WRITE
WRITE
WRITE
WRITE

#2,IrrMethod$
#2,Method$
#2,TypElem$
#2,SelPrtOpt$
#2,iprint%

#2,VARTheta
#2,ALPRA

#2,DeltaT
#2,numiter§
#2,AllError
#2,TotNumStep§
#2,CONSISorLUMP%
#2,CoefTW

42,FurLength
#2,TimCut
#2,Qin
42,kofInf
#2,fsub0

ikll

. . . . . (m)

. . . . (min)

(liters/sec)

. (fraction)

0 O O O O O

(m‘3/m/min)

WRITE
WRITE
WRITE

WRITE
WRITE
WRITE
WRITE
CLOSE #2

Qin = Qin / 1000
NumStep§ = 0

GOSUB READING
LOCATE 10,8

#2,Slope
42,aofInf
#2,ManngN

#2,Rhol
#2,Rh02
#2,Sigmal
#2,Sigma2

Sb42

PRINT "Enter File Name of Advance and Recession Data (No extension) : "3

INPUT "",NGphFil$
xxx = TIMER

if iprint% >= 0 AND SelPrtOptS <> "S" THEN

ATmp$(1)=STRING$ (5." ")+DATE$

ATmp$(1)-ATmp$(1)+STRING$ (41,- ")+TIME$

ATmp$(2)=STRING$ (69,"*")
ATmp$(3)="*"+STRING$ (67," ")+w*~
ATmp$(4)-"*“+STRING$ (25," v)+n
ATmp$(5)="*“+STRING$ (3," ")

Output from

"+STRINGS (25," ")+"*"

ATmp$(S)=ATmp$(5)+"FINITE ELEMENT SURFACE IRRIGATION DESIGN PROGRAM"
ATmP$(5)=ATmp$(5)+", FE-SURFDSGN“+STRING$ (3," -)+«*«

ATmp$(6)-ATmp$(3)
ATmp$(7)-ATmp$(2)
ATmp$(8)="*"+STRING$ (6," ")

ATmp$(8)=ATmp$(8)+"Developed By : Walid H. Shayya"+STRING$ (31," ")+"*"

ATmp$(9)-'*”+STRING$ (21," ~)

ATmp$(9)=ATmp$(9)+"Department of Agricultural Engineering

ATmp$(10)=“*"+STRING$ (21," T)

ATmp$(10)=ATmp$(10)+"Michigan State University

ATmp$(1l)=ATmp$(2)
for icntt = 1 to 11
print #1, ATmp$(icnt%)

next icntt

print #1,"

print #1,"'

print #1,"“

end if

if iprintt >= 0 AND SelPrtOptS <> "8"

in

THEN

PRINT #1,"Output File for Recession and Advance Data . . . . . : ":NGphFilS;
PRINT #1,".PRG"

PRINT #1,"Irrigation Method . . . ('F', furrow, '8', border) : ":IrrMethodS
PRINT #1,"Method of Solution . . . . . . . (R, S, Z, or K) : ":MethodS
PRINT #1,"Type of Element ('L', linear, 'Q', quadratic) : “:TypElemS
PRINT #1,“Level of Printing . . (select 0, 1, 2, or 3) : ":iprint%
PRINT #1,"Output Device ('8', screen, 'F', file, 'P', printer) : ";

PRINT #1, SelPrtOptS '

PRINT 91,

PRINT {1,"Time Step, 5T . . . (min) = ';DeltaT

PRINT #1,'Maximum Number of Iterations = ":numiter%

PRINT #1,"Allowable Error . . . . . . . . . = ";

PRINT #1, USING "t.#t#i###";A11Error

PRINT #1,'Maximum Number of Time Steps . . = ":TotNumStep%

PRINT #1,”Time Weighting Coefficient, 9 . . - ":VARTheta

PRINT #1,'a (Select a=0 for Galerkin) . . . = ":ALPHA

PRINT #1,'Top Width Coefficient (1 or 1.5) = ":CoefTW

PRINT #1,"Consistent (1) or Lumped (0) = ":CONSISorLUMP%

PRINT 41,

PRINT 41,'Furrow Length . . . . . . . . . (m) = ":FurLength

PRINT {1,"Time of Cutoff . . . . . . (min) = ":TimCut

243

PRINT #1,"In1et Flow Rate . . . . . . . (liters/sec)
PRINT #1,“Slope of Channel Bed . .‘. . . . (fraction)
PRINT #1,"Manning Roughness Coefficient, n . . . . . .

PRINT #1,'Hydraulic Section Param., 01 . . . . . . . .
PRINT #1,”Hydraulic Section Param., 02 . . . . . . . .
PRINT #1,"Hydraulic Section Param., Rho 1 . . . . . . -
PRINT #1,"Hydraulic Section Param. , Rho 2 . . . a
PRINT #1, "Infiltration Function Coeff., k (m‘3/m/min a)-
PRINT #1,'Infiltration Func. Exponent, a . . . . -

PRINT #1,"Infiltration Function Coeff., f (m‘3/m/min) -
PRINT #1, '"
end if

IF TypElem$ 8 "L" THEN
NumElmNode% = 2
ELSE
NumE lmNode% = 3
END IF
NumBandW% = NumElmNode% * 2
NClm% = 2*NumBandW%-1

cls

GraphFillS = NGphFil$ + ".PRG"
GraphFilZS = NGphFil$ + ".REC"

open "O",t2,GraphFill$
open 'O",#3,GraphFilZ$

TotalTime = 0
TotalLength = 0
write 42, TotalTime,TotalLength

GOSUB GetBasicElmntMtx

if iprint§ >- 2 then

call matrixprt(E1thmtx(),NumElmNode2§,NumElmNodeZt,". .

I ) .
call matrixprt(Elthmtx(),NumElmNodeZS,NumElmNode2%,". .

end if

if iprint% >= 0 then
print #1,STRING$(78,"*")
end if

IF SelPrtOptS = "F" or SelPrtOptS = "P" THEN
GOSUB READING

END IF
NumElem% 8 1
Phaset = 1

PhaseRec% a 0
STPExc% = 0 ' A flag that stops program execution

DO
IF SelPrtOptS = "F" or SelPrtOptS = "P" THEN
locate 10,14
print "Completed Time Steps . . . . . . . . . . . . :
locate 12,14
print "Completed Time of Current Simulation Run (min):
END IF

IF Phasei = 1 AND PhaseRec% - 0 THEN
IF Methods = "K" OR NumElem% = 1 THEN

”:Qin*1000
":Slope
":ManngN
';Sigma1
":Sigma2
":Rhol
“:RhoZ
":kofInf
';aofInf
';fsub0

. . [C(e)]
. . [k(e)]

":NumStep%

":TotalTime

NumofPhi% = 4

NofPhi(1,1) = 0.5

NofPhi(2,1) = 1

NofPhi(3,l) = 2 + (NumElmNode%-2) + (NumElem%-1)*(NumElmNode%-l)
NofPhi(4,1) = NofPhi(3,1)-0.5

ELSE

NumofPhi§ - 3
NofPhi(1,1) - 1
NofPhi(2,1) = 2 + (NumElmNodeﬁ-Z) + (NumElemt-l)*(NumElmNode%-1)
NofPhi(3,1) = NofPhi(2,1)-0.5
END IF

ELSEIF Phase% = 2 AND PhaseRec% = 0 THEN
IF Methods = "K” THEN
NumofPhi% a 2
NofPhi(1,1)
NofPhi(2,1)
ELSE
NumofPhii =
NofPhi(1,1)
END IF
END IF

IF PhaseRec% 8 1 THEN

PrPhi% - 0

IF Phase‘ - 1 THEN
PrPhi% 8 2
NofPhi(1,1) - 2 + (NumElmNodet-Z) + (NumElem%-1)*(NumElmNode%-l)
NofPhi(2,1) = NofPhi(1,1)-0.5

END IF

NumofPhi§ = NumNodRec8*2 + PrPhi%

FOR jjt% - 1 to NumNodRec§
NofPhi(jjt§*2-1+PrPhi§,1) = jjt%-.5
NofPhi(jjt§*2+PrPhi%,1) - jjts

NEXT jjtt

END IF

IF NumElmNode% = 3 THEN
NumNodet - NumElem%*2+1
ELSEIF NumElmNode% = 2 THEN

NumNodet = NumElem§+1
END IF

NP% = NumNode% * 2

IF Phaset s 1 THEN
FOR SPct§ - 1 TO NumNode%
SO(SPct%,1) I Slope
NEXT SPct§
END IF

IF NumElmNode§ a 3 THEN
LocL§ s NumElem%*4+1
LocQ! = NumElem%*4-2

ELSEIF NumElmNodet = 2 THEN
LocL§ - NumElem%*2+1
LocQ§ - NumElem§*2

END IF

IF Phase% 8 1 AND PhaseRec% = 0 THEN
IF NumElem% = 1 THEN
Phi(2,1) = Qin
Phi(1,1) s ( (Phi(2,1)“2*ManngN“2) / (Rhol*SO(1,1)) )“(1/Rh02)
PhiSet - 0.02*Phi(1,1)
IF NumElmNode% - 3 THEN
Phi(4,1) s Qin*VARTheta
Phi(3,1) - ( (Phi(4,1)‘2*ManngN“2) / (Rhol*SO(1,1)) )“(1/Rh02)
END IF
ELSE
IF NumElmNode‘ =
Phi(LocL%-4,1)
Phi(LocL%-3,l)
Phi(LocL%-2,1)
Phi(LocL%-1,1)
ELSE

0.5
1

"H
H

THEN

Phi(LocL%-8,1)
Phi(LocL%-7,1)
Phi(LocL%-6,l)
Phi(LocL%—5,1)

nuuuw

245

Phi(Loth-2,1) = Phi(LocL%-4,1)
Phi(LocL8-1,1) = Phi(LocL8-3,1)
END IF
END IF
ELSEIF Phaseﬁ = 1 AND PhaseRec§ = 1 THEN

IF NumElmNode§ = 3 THEN
Phi(LocL§-4,1) - Phi(LocL%-8,1)
Phi(LocL%-3,1) - Phi(LocL%-7,1)
Phi(LocL%-2,1) = Phi(LocL%-6,1)
Phi(LocL%-l,l) a Phi(LocL%-5,l)

ELSE
Phi(LocL%-2,1)
Phi(LocL%-l,1)

END IF

END IF

IF Phase%- 1 AND PhaseRec% - 0 THEN
FOR ICTt 8 1 TO NumNode% - 1
IF NumElmNode§ = 3 AND ICT% - NumNode%-1 THEN
TofOpp(ICT%,l) - DeltaT*VARTheta
ELSE
TofOpp(ICT%,1) = DeltaT + TofOPP(ICT%,1)
END IF
NEXT ICT%
ELSEIF Phase% 3 2 AND PhaseRec§ a 0 THEN
FOR ICT% = 1 TO NumNode%
TofOpp(ICT§,1) - DeltaT + TofOpp(ICT§,1)
NEXT ICT%
ELSEIF PhaseRect - 1 THEN
FOR ICT§ a NumNodRec%+l TO NumNode%
TofOpp(ICT%,1) s DeltaT + TofOPP(ICT%,1)
NEXT ICT%
END IF

if iprint% >= 2 then

Phi(LocL%-4,1)
Phi(LocL%-3,l)

PRINT #1, "Number of elements : ":NumElem%
PRINT #1, "Number of nodes/element : ":NumElmNode‘
PRINT #1, "Band Width : ":NumBandW%
PRINT #1, "Total number of nodes : ":NumNode%
end if
iloop% - 0

FOR JTmpi a 1 TO NP%
PrevPhi(JTmp%,1) = 0
NEXT JTmpt

DO
iloopl = iloopi + 1
IF SelPrtOptS 3 "F" or SelPrtOptS 8 "P" THEN
locate 14,14
print "Completed Iterations Within Current Time Step : “:iloopt-l
END IF

IF Phase% = 1 THEN
IF NumElmNode§=3 then
TmpI - kofInf*TofOpp(NumElem%*2-1,l)“aofInf

TmpI = TmpI+ fsub0*TofOpp(NumElem%*2-1,1)
TmpJ a kofInf*TofOpp(NumElem%*2,1)‘aofInf
TmpJ - TmpI+ fsub0*TofOpp(NumElem%*2,1)
TmpJ a TmpJ + Phi(LocQ%+1,1)

ELSE

TmpI = kofInf*TofOpp(NumElem%,1)“aofInf
TmpI = TmpI + fsub0*TofOpp(NumElem§,1)
TmpI = VarTheta * TmpI + Phi(LocQ%-1,1)
END IF
IF NumElmNode§=3 then
Length(NumElem%,1)=(VARTheta*Phi(LocQ%,1)*60*DeltaT)/TmpI

246

ELSE

Length(NumElem%,1)-(VARTheta*Phi(LocQ#,l)*60*DeltaT*(l-APLHA))/TmpI

END IF

IF Length(NumElem%,1)<0 THEN Length(NumElem%,l)=0

PRINT #1,"Iter #";

PRINT #1, USING "###";iloop%;

PRINT #1,SPACE$(57);

PRINT #1,USING"#####.### m";Length(NumE1em%,1)
ELSE

PRINT #1,"Iter #";

PRINT #1, USING "###";iloop#
END IF

FOR ICT% - 1 TO NPt
FOR JCT‘ - 1 TO NClm%
Kmatrx(ICT%,JCT%) = 0
Cmatrx(ICT%,JCT%) = 0
Amatrx(ICT§,JCT%) = 0
NEXT JCT%
FOR JCTi = 1 TO NClmt+NumBandW$
Amatrx(ICT%,JCT%) = 0
NEXT JCT%
Force(ICT%,1) = 0
NEXT ICT%

for ict§ = 1 to NPi

Pfx(ict%,1) = Phi(ict%,1)

if Pfx(ict%,1) < 0 THEN Pfx(ict%,1)=0
next ict%

IF NumElmNode% = 3 THEN
LocL% = NumElem%*4+l

ELSEIF NumElmNode% s 2 THEN
LocL% = NumElem%*2+1

END IF

IF Phaset = 1 THEN
Pfx(LocL%,l) = 0
Pfx(LocL§+1,l) = 0

END IF

if iprintt >= 3 then
call matrixprt(Length(),NumElem%,1,". . . . {L} . . . . ")
call matrixprt(TofOpp(),NumNode%,1,". . . . (1} . . . . ")

call matrixprt(SO(),NumNode%,1,". . . . (SO) . . . . ")

call matrixprt(Phi(),NP§,l,". . . . {Phi} . . . . ")

call matrixprt(ForceP(),NP%,l,". . . . (Fa) . . . . “)
end if

CALL GetElmntMthoef

if iprint% >= 3 then
call matrixprt(Fsubi().NumElem%,l,". . . . (Fsubi) . . . . ")

call matrixprt(Fsubj(),NumElem%,1,". . . . {Fsubj} . . . . ")

call matrixprt(Fsubk(),NumElem§,1,”. . . . (Fsubk) . . . . ")

call matrixprt(TopWidth(),NumNode%,1,". . . . (T) . . . . ")
end if

CALL BuildGlobalMtx
if iprint‘ >= 3 then

call matrixprt(Cmatrx(),NP§,NClm%,". . . . [C] . . . . ")
call matrixprt(Kmatrx(),NP%,NClm%,". . . . [K] . . . . “)
call matrixprt(Force(),NP%,1,". . . . {F} . . . . ")
call matrixprt(Phi(),NP%,1,". . . . [Phi] . . . . ")

end if

GOSUB SolveTimeStepl
CALL ModfyGlobMtx

247

if iprint% >= 3 then

call matrixprt(Amatrx(),NP8,NClm%,". . . . [A] . . . . ")
call matrixprt(Pmatrx().NP%,NClm#,". . . . [P] . . . . ")
call matrixprt(Fstar(),NP§,1,". . . . (F*) . . . . “)

end if

call matrixVectmult(Pmatrx(),NP%,NumBandW%,phil(),Temp1())
call matrixadd(temp1(),NP%,1,Fstar(),NP%,1,PFplust(),NP%,1)
call GAUSSBND(Amatrx()'NP#,NumBandW%,PFplust().Phi())

if iprint§ >2 0 then
IF iloop% = 1 then

print #1,” A1 Q1 A2 02 A3";
IF Phaset - 1 THEN
print #1,USING ' Q3 A4 Q4 Elem### Length ":NumElem%
ELSE
print #1," Q3 A4 Q4"
END IF
if NumNode%>4 then
print #1," A5 Q5 A6 Q6 A7";
print #1,“ Q7 A8 Q8 ..etc."
end if
end if
call Vectoerrt(Phi().NP%,1)
end if

DiffError a 0

FOR JTmp% = 1 TO NP§ STEP 1
DiffError = DiffError + ABS(PrevPhi(JTmp%,1)-Phi(JTmp%,1))
PrevPhi[JTmp%,1) = Phi(JTmp%,1)
NEXT JTmp§
AllowErr = AllError+numiter§*AllError/IO
LOOP UNTIL (iloop%>2 AND DiffError<=AllowErr) OR (iloop%>numiter%)

TotalTime = TotalTime + DeltaT

NumStept - NumStep§ + 1

IF Phase! = 1 THEN
TotalLength - TotalLength + Length(NumElem%,1)
write #2, TotalTime,TotalLength

END IF

for j II=1 to NPS
Phil(j,l) = Phi(3,1)
ForceP(j,1) = Force(j,1)
next 3

FOR ICT% = 1 TO NumNode%
InfilP(ICT%,1) = Inf11(ICT%,1)

NEXT ICTt

if iprint% >= 0 then
print #1," A1 01 A2 Q2 ";
print #1,”A3 Q3 A4 Q4 ==>Time Stepz“;

print #1, USING ”###";NumStep%
if NumNode§>4 then

print #1," A5 05 A6 06 ":
print #1,'A7 Q7 A8 QB "
end if

print #1,STRING$(78,"+")
PRINT #1,SPACE$(25);'Simulated Time :";Tota1Time;"min“
print #1,STRING$(78,'*')

end if

2PL8

IF Phase% - 1 THEN
IF TotalLength<FurLength THEN
NumElem‘ = NumElem% + 1
ELSE
Phase} - 2
CLOSE #2
END IF
END IF

IF PhaseReci - 0 THEN
IF TotalTime>=TimCut THEN
PhaseRec% = 1
NumNodRec% = NumNodRec§ + 1
Phi(l,l) n 0

Phi(2,1) = 0
write #3, TotalTime,TotalRec
END IF
END IF

IF PhaseRec§ s 1 THEN
TotalRec = 0
NumNodRec# = 1
1'2
WHILE j <- NumElem§
IF Phi(j'2-1,1) <-PhiSet THEN

Phi(j*2-1,l) = 0
Phi(j*2,1) = 0
NumNodRec% = NumNodRec% + 1
TotalRec = TotalRec + Length(j-1,1)
ELSE
j = NumElem%
END IF
j=j+1
WEND
IF TotalRec <> 0 THEN
write #3, TotalTime,TotalRec
END IF
IF NumNodRec§ >- NumElem§ THEN STPExc% = 1
END IF

IF NumStep§>=TotNumStep§ THEN
STPExc! - 1
END IF
LOOP UNTIL STPExc§ = 1

YYY = TIMER
IF SelPrtOpt$ = "F” or SelPrtOpt$ = "P" THEN
locate 10,14
print "Completed Time Steps . . . . . . . . . . . . : ":NumStep%
locate 12,14
print “Completed Time of Current Simulation Run (min): ":TotalTime
locate 16,14
PRINT USING ”Time of Execution for this Run :#####.## ":YYY-XXX:
PRINT "sec"
LOCATE 21,1
PRINT
END IF
PRINT #1, '"
PRINT #1, USING "Time of Execution for this Run : ######.## ":YYY-XXX;
PRINT #1, I'sec"
PRINT #1, USING ' : ######.## ":(YYY-XXX)/60;
PRINT #1, "min"
CLOSE
LOCATE 23,40
PRINT "Press any key to see next screen ..."
WHILE INKEYS =”'

249

WEND

GOSUB HEADING

LOCATE 10,8

SelStrg$ = 'YyNn"

Strg$ - "Would you like to run the graphics routine ('Y', yes, 'N’, no) ? “

RowStrg§ = 10

ColStrg% = 8

call SelStrgEntry

AnsGS = SelOpt$

AnsGS = UCASE$(AnsGS)

IF AnsGS = "Y" THEN
CHAIN "SURFGRPH.EXE"

END IF '

CLS

END

InitialScrn:
REM titttt*tti*ttttiit*ittwtittittttttit**it*tittttttittitttttttttt*ti

REM * A subroutine to display the first screen. *
REM tittti*titiitttittit*tttit*tttiittttttitﬁitt******t*******t*******

KEY OFF

XX%(1) = 5: XX%(2) = 23: XX%(3) ' 23: XX§(4) 8 22: XX§(5) ' 10
YY%(1) = 2: YY%(2) = 4: YY%(3) = 8: YY%(4) 3 10: YY§(5) 8 12
XX%(6) = 17: YY%(6) = 14

CC$(1) = CRR$(201): CC$(2) = CRR$(205): CC$(3) = CHR$(187)
CC$(4) = CHR$(186): CC$(5) I CRR$(204): CC$(6) = CRR$(185)

CC$(7) 8 CRR$(200): CC$(8) = CRR$(188)

A1$ - CCS(1) + STRING$(54, CC$(2)) + CC$(3)
A25 - CC$(5) + STRING$(54, CC$(2)) + CC$(6)
A3$ I CC$(7) + STRING$(54, CC$(2)) + CC$(8)

AA$(1) - ' FINITE ELEMENT SURFACE IRRIGATION DESIGN MODEL "
AA$(2) 8 ”VERSION 1.00"
AA$(3) ' ”Developed by”
AA$(4) a "Walid H. Shayya"
AA$(5) = "Department of Agricultural Engineering"
AA$(6) r ”Michigan State University"
CLS
. LOCATE 4, 1

PRINT SPACE$(12); A1$
FOR Iloop§ = 1 TO 5
PRINT SPACE$(12): CC$(4); SPACE$(54); CC$(4)
NEXT Iloopi
PRINT SPACE$(12): A25
FOR Iloopﬁ a 1 TO 9
PRINT SPACE$(12); CC$(4): SPACE$(54); CC$(4)
NEXT Iloopt
PRINT SPACE$(12): A3$
FOR Iloopt = 1 TO 6
LOCATE YY%(Iloop%) + 4, XX%(Iloop§) + 12
PRINT AA$(Iloop%)
NEXT Iloopt

LOCATE 23, 48

COLOR 15, 0

PRINT “Press any key to continue."
COLOR 7, 0

WHILE INKEY$ = "“: WEND
RETURN
READING:
rem tsatasasasasstastsistesttqstttsstttttettttseattestsasstststtsttsta

rem * A subroutine to print page heading. *
rem itiiitttttit*ttttttt*tttttiitttttttttt*ttttt*iiittiﬁttiitttttitttﬁ

cls
LOCATE 2,24

250

PRINT ”FINITE ELEMENT SURFACE IRRIGATION"
LOCATE 3,24

PRINT " DESIGN PROGRAM”

LOCATE 4,23

PRINT STRING$(35,196)

return

SUB SelStrgEntry
rem *****t*******t***t*********t**t**t*******t************************
rem * A subprogram for entering one charcter input to a selected *
rem * string variable. *
rem *ttt*ttt*t****t*t*t***t**ttttttttw************t*******i*t*********
SHARED SelOpt$,SelStrgS,Strg$,RowStrg%,ColStrg%
LOCAL BCount%

BCount% 0
SelOpt$ = ””
WHILE INSTR(1,SelOpt$,ANY SelStrg$) = 0
locate RowStrg%,ColStrg%
IF BCount% <> 0 THEN BEEP
PRINT Strg$;
SelOpt$ = INPUT$(1)
PRINT SelOpt$

BCount% = 1

WEND
END SUB

SolveTimeStepl:
rem ****i****************iitttttttttii*tttﬁttttt**t********t***t******
rem * A subroutine to build the ordinary differential equation in *
rem * time. It will construct the following system of equations: *
rem * *
rem * [A] {Q} ‘ [P] {O} + {F*) *
rem * b a *
rem * where, *
rem * [A] = [C] + 9.5T.[K] *
rem * [P] = [C] - (1 - 9).5T.[K] *
rem * (F*} - 5T.(1 — GHF} + 8T.9.(F} *
rem * a b *

rem *iiﬁ******************************************ﬁ*************ii*it*

Paraml = VARTheta * DeltaT * 60
Param2 - (VARTheta - 1) * DeltaT * 60
Param3 - - Param2

ParamlM = 1

Param2M - 0

Param3M - 0

IF MethodS = "R” THEN

call matrixNumult(Kmatrx(),NP%,NClm%,Param1,temp1())
else

call matrixNumultMod(Kmatrx(),NP%,NC1m%,Param1,Param1M,temp1())
end if

if iprint% >= 3 then
call matrixprt(temp1(),NP%,NC1m%,". . . . . 9.5T.[K] . . . . . ")
end if

call matrixadd(temp1(),NP%,NClm%,Cmatrx(),NP%,NClm%,AmatrX(),NP%,NClm%)
if iprint% >8 3 then

call matrixprt(Amatrx(),NP%,NClm%,". . [A] = [C] + 6.5T.[K] . .")
end if

251

IF Methods 8 "H" THEN
call matrixNumult(Kmatrx(),NP8,NClm%,Param2,temp1())
else
call matrixNumultMod(Kmatrx(),NP§,NClm#,Param2,Param2M,temp1())
end if
if iprintt >8 3 then
call matrixprt(temp1(),NP%,NClm%,". . . . - (1 - 9).5T.[K] . . . . ")
end if

call matrixadd(templ(),NP§,NClm#,Cmatrx(),NP%,NClm%,Pmatrx(),NP§,NClm%)
if iprintt >= 3 then

call matrixprt(Pmatrx(),NP%,NClm%,". . [P] 8 [C] - (1 - 8).5T.[K] . .")
end if

IF Methods 8 "H" THEN

call matrixNumult(ForceP(),NP%,1,Param3,TempF1())
else

call matrixNumultMod(ForceP(),NP%,1,Param3,Param3M,TempF1())
end if

if iprint% >8 3 then
call matrixprt(TempFl(),NP%,1,". . . . (1 - 9).5T.{F}a . . . . ")
end if

IF MethodS = "R” THEN
call matrixNumult(Force(),NP%,1,Param1,TempF2())
else '
call matrixNumultMod(Force(),NPt,1,Param1,Param1M,TempF2())
end if

if iprinti )- 3 then
call matrixprt(TempF2(),NP%,1,". . . . 9.5T.(F)b . . . . ")
end if

call matrixadd(TempF1(),NP§,1,TempF2(),NP%,l,Fstar(),NP§,1)
if iprint% >8 3 then

call matrixprt(Fstar().NP%,1,". [F*) = 5T.(1-9)(Fa) + 6T.9.{Fb} .")
end if

return

GetBasicElmntMtx:
rem *ttttttttttt*titiiit*****t*ﬁ*****tttttti*tti*itititttttittttittwtt
rem * A subroutine for getting the constant coefficients of the *
rem * element stiffness and the capacitance matrices. This *
rem * subroutine works for both linear and quadratic elements. *

rem ﬁ****it**titiiﬁt*t****t*ttt**t*i*iiiiitttiiﬁﬁitiiitﬂtitttﬁitiititi

NumElmNode2§ 8 NumElmNodei * 2

IF NumElmNode2% 8 6 THEN
IF CONSISorLUMP§ 8 1 THEN
RESTORE 100
FOR Icount‘ 8 1 TO NumElmNodeZ‘
FOR Jcountt 8 1 TO NumElmNode2%
READ Elthmtx(Icount¥,Jcountt)
NEXT Jcount!
NEXT Icount%
ELSE
RESTORE 125
FOR Icount‘ 8 1 TO NumElmNodeZt
FOR Jcountt 8 1 TO NumElmNode2%
READ Elthmtx(Icount§,Jcountt)
NEXT Jcount%
NEXT Icountt
END IF
RESTORE 150
FOR Icounti 8 1 TO NumElmNode2§
FOR Jcountt 8 1 TO NumElmNodeZt
READ Elthmtx(Icount%,Jcount§)

SMEZ

Elthmtx(Icount%,Jcount%)
NEXT Jcount%
NEXT Icount%

Elthmtx(Icount%,Jcount%) / 24

ELSEIF NumElmNodeZi 8 4 THEN

IF CONSISorLUMPt 8 1 THEN
RESTORE 200
FOR Icount% 8 1 TO NumElmNode2%
FOR Jcountﬁ 8 1 TO NumElmNodeZi
READ E1thmtx(Icount§,Jcount‘)
NEXT Jcount!
NEXT Icount%
ELSE
RESTORE 225
FOR Icounti 8 1 TO NumElmNodeZ‘
FOR Jcountt 8 1 TO NumElmNode2§
READ E1thmtx(Icount%,Jcounti)
NEXT Jcount%
NEXT Icount‘
END IF
RESTORE 250
FOR Icount§ 8 1 TO NumElmNode2§
FOR Jcount% 8 1 TO NumElmNode28
READ Elthmtx(Icount%,Jcount%)
NEXT Jcount%
NEXT Icount‘
REM ** Non-Galerkin Finite Element Formulation except if ALPHA=0

Elthmtx(1,2) = Elthmtx(1,2) + ALPHA / 2
Elthmtx(1,4) = Elthmtx(1,4) - ALPHA / 2
Elthmtx(3,2) a Elthmtx(3,2) - ALPHA / 2
Elthmtx(3,4) = Elthmtx(3,4) + ALPHA / 2
END IF
100 DATA 4, o, 2, 0,-1, 0
DATA 0, o, o, o, o, 0
DATA 2, 0,16, 0, 2, 0
DATA 0, o, o, o, o, 0
DATA -1, o, 2, o, 4, 0
DATA 0, o, o, o, o, o
125 DATA 5, o, o, o, o, 0
DATA 0, o, o, o, o, 0
DATA 0, 0,20, 0, o, 0
DATA 0, o, o, o, o, 0
DATA 0, o, o, o, 5, 0
DATA O, o, o, o, o, o
150 DATA 0,-12, o, 16, o, -4
DATA 0, o, o, o, o, 0
DATA 0,-16, o, o, o, 16
DATA 0, o, o, o, o, 0
DATA 0, 4, 0,—16, o, 12
DATA 0, 0, o, o, o, o
200 DATA 2, o, 1, 0
DATA 0, o, o, 0
DATA 1, o, 2, 0
DATA 0, o, o, o
225 DATA 3, o, o, 0
DATA 0, o, o, 0
DATA 0, o, 3, 0
DATA 0, o, o, o
250 DATA 0, .5, o, .5
DATA 0, o, o, 0
DATA 0, .5, o, .5
DATA 0, o, o, o

RETURN

253

sub GetElmntMthoef
rem *ttittttttiittittitttt*ti***itt**iﬁ*tttiit*ttiittiitﬁtiiitttttittt
rem * A subroutine for determining the coefficients C, C2, C3, and*
rem C4 of the element stiffness and the capacitance matrices. Also,*
rem the coefficients fi, fj, and fk for the force vector are *
rem determined. This subroutine works for both linear and *
rem quadratic elements. *
rem *tttwtttitititititit*itttttttttitttt*ttwtt*tttiittttittttttitititt
SHARED NumElem%,Gravity,Phi(),NumElmNode%,Method$,Sigma1,Sigma2
SHARED NumNode§,Fsubi(),Fsubj(),Fsubk(),ManngN,SO(),Rhol,Rh02,Length()
SHARED TopWidth(),Coef1().Coef2()'Coef3().Coef4().Coef5(),CONSISorLUMP%
SHARED Pfx().Czi(),Czj(),Czk(),CoefTW

LOCAL Count%,AreaC§,FlowC%,TempV1,TempV2,TempV3,Sfi,Sfj,ka,NodeC%,itr%
LOCAL ct§,Expnt

IF NumElmNode% 8 2 THEN

& i i i

Temle 8 2
TempV2 8 2
TempV3 8 2
ELSEIF NumElmNode% 8 3 THEN
Temle 8 6
TempVZ 8 2
TempV3 8 4
END IF
IF Methods 8 "Z" THEN
Expnt 8 1
ELSE
Expnt 8 2
END IF

FOR Count% 8 1 TO NumElem%
REM *ﬁttﬁﬁtt*********************t*************t****t***********tt
REM * Determine the coefficients c , c , c , c , c , f , f , a f *
REM * 1 2 3 4 5 i j k*
REM * of the individual element matrices for all the methods. *
REM ********************i***************************************tt
IF NumElmNode% 8 2 THEN
AreaC§ 8 Count§*2-1
FlowC§ 8 Count%*2
NodeC% 8 Counti
ELSEIF NumElmNode§ 8 3 THEN
AreaC§ 8 Count%*4-3
FlowC§ 8 Count%*4-2
NodeC% 8 Count§*2-1
END IF
IF Methods <> "K“ THEN
REM itﬂirtit*******wttttttttttttttttttit*ttttttttttttttttttttttt
REM * If kinematic wave, the force vector of the 2nd, 4th, 5 *
rem * 6th equations is placed in the stiffness matrix. *
rem *titttttitit*wtitt**w*i******************t*ttttttiitttttittt
FOR itr% 8 1 TO NumElmNode%
ct§ 8 (itr%-1)*2
IF Methods 8 "H" AND Pfx(AreaC§+ct%,1)<>0 THEN
Coef1(Count%,itr%) 8 1 / (Gravity * Pfx(AreaC%+ct%,l))
ELSE
Coef1(Count%,itr%) 8 0
END IF
IF (Methods 8 "H" OR Methods 8 ”8“) AND (Pfx(AreaC%+ct#,1)<>0) THEN
Coef3(Countt,itr#)8Pfx(FlowC§+ct§,1)‘2/(Gravity*Pfx(AreaC%+ctt,1)‘3)
ELSE
Coef3(Count%,itr%) 8 0
END IF
IF Pfx(AreaC%+ct%,1)<>0 THEN
TopWidth(NodeC%+itr§-1,1)8(CoefTW*Pfx(AreaC8+ct§,1)“(1-Sigma2))/Sigma1
Coef2(Count%,itr§) 8 1 / TopWidth(NodeC§+itr§-1,l)

254

ELSE
TopWidth(NodeC%+itr%-1,1) 8 0
Coef2(Count%,itr%) 8 0
END IF
IF Pfx(AreaC§+ct%,1)<>0 AND Methods <> "2" THEN
Coef4(Count%,itr%)82*Pfx(FlowC#+ct§,1)/(Gravity*Pfx(AreaC%+ct§,1)“2)
ELSE
Coef4(Count%,itr%) 8 0
END IF
Coef5(Count§,itr%)-Coef2(Count%,itr%)-Coef3(Count§,itr%)
NEXT itri
IF Pfx(AreaC§,1) <> 0 THEN
Sfi 8 (Pfx(FlowC§,1)‘Expnt*ManngN‘2)/(Rhol*(Pfx(AreaC%,l))‘Rh02)
ELSE
Sfi 8 0
END IF

IF Pfx(AreaC%+2,1) <> 0 THEN
Sfj 8 (Pfx(FlowC%+2,1)“Expnt*ManngN“2)/(Rhol*(Pfx(AreaC%+2,1))‘Rh02)
ELSE
Sfj 8 0
END IF
IF MethodS 8 '2" THEN
Czi(Count%,1) 8 Sfi * Length(Count§,1)/TempV1
Czj(Count%,1) 8 Sfj * Length(Count§,1)/TempV3
Sfi 8 0
Sfj 8 0
END IF
Fsubi(Count%,1) 8 Length(Count%,1)*(SO(NodeC%,l)-Sfi)/TempV1
Fsubj(Countt,1) 8 Length(Count%,1)*(SO(NodeC%+1,1)-Sfj)/TempV1
IF NumElmNode% 8 3 THEN
IF Pfx(AreaC%+4,1) <> 0 THEN
ka 8 (Pfx(FlowC§+4,1)‘Expnt*ManngN‘2)/(Rhol*(Pfx(AreaC%+4,1))“Rh02)
ELSE
ka 8 0
END IF
IF Methods 8 "2" THEN
Czk(Count%,1) 8 ka * Length(Count%,1)/TempV1
ka 8 0
END IF
Fsubk(Count%,1) 8 Length(Count%,1)*(SO(NodeC%+2,1)-ka)/TempV1
END IF
ELSE
Sfi 8 Rhol*(Pfx(AreaC%,1))‘(Rh02-2)*SO(NodeC%,1)
Sfj 8 Rhol*(Pfx(AreaC%+2,1))“(Rh02-2)*SO(NodeC§+1,1)
Fsubi(Count§,1) 8 (Sfi“.5)/(ManngN*TempV2)
Fsubj(Count%,1) 8 (Sfj“.5)/(ManngN*TempV3)
IF NumElmNode% 8 3 THEN
ka 8 Rhol*(Pfx(AreaC%+4,1))‘(Rh02—2)*SO(NodeC%+2,1)
Fsubk(Count%,1) 8 (ka“,5)/(ManngN*TempV2)
END IF
END IF
NEXT Count%
END SUB

sub BuildGlobalMtx

rem titttttttttitiﬁt*ﬁttitiﬁttttiiitti*iiiitiiiﬁititittitttttt*ttﬁﬁttt
rem * A subroutine for building the global stiffness and *
rem * capacitance matrices, and the global force vector. *
rem tittit*itttttitttt*tii*tﬁttttttttttittttittttt*iitﬁtttttttitttttii
SHARED NumElem§,Coef(),NumElmNode%,Kmatrx().Cmatrx().Elthmtx(),Force()
SHARED Length(),Elthmtx(),TempK(),TempC(),Method$,ManngN,Infil()

SHARED kofInf,fsub0,aofInf,Tof0PP().Fsubi(),Fsubj().Fsubk(),CONSISorLUMP%

255

SHARED Coef1(),Coef2(),Coef3(),Coef4(),Coef5(),DeltaT,FDorFE%,InfilP()
SHARED C21():Czj(),Czk(),NumBandW%
SHARED ALPHA,Phase$

LOCAL NumElmNode2%,Countt,Count2%,LocatC%,LocatN%,Ict%,Jct%,TempV1
LOCAL TempV2,TempV3,TempV4,TempV5

NumElmNode2% 8 NumElmNode% * 2

IF NumElmNodet 8 2 THEN
Temle 8 2

ELSEIF NumElmNodet 8 3 THEN
Temle 8 6

END IF

IF NumElem% 8 1 OR Phase%<>1 THEN
Umet% 8 NumElemt

ELSE
Umet% 8 NumElem§ - 1

END IF

FOR Count% 8 1 TO Umet%
REM *tttttti***********ttttitttttitttttt*tttit*iittttiﬁiﬁttttiiititt
REM * Construct the individual element matrices for all the *
REM * different models. *
REM titﬁtttttttt*tiﬁti*ﬁtttiittttiﬁiittitttiittitttttttittittttttﬁti
IF NumElmNode2% 8 6 THEN
LocatCt 8 Count%*4-4
LocatN§ 8 Count%*2-2
ELSEIF NumElmNode2% 8 4 THEN
LocatC% 8 Count%*2-2
LocatNt 8 Countﬁ-l
END IF
FOR Ictt 8 1 TO NumElmNode2%
FOR Jctt 8 1 TO NumElmNode2%
TempK(Ict4,Jct%) 8 Elthmtx(Ict%,Jct%)
TempC(Ict%,Jct%) 8 Elthmtx(Ict§,Jct%)
NEXT Jct§
NEXT Ict%
IF Methods 8 "H" THEN
IF CONSISorLUMPt 8 1 THEN
IF FDorFEk-l THEN
IF NumElmNode2% 8 6 THEN
TempC(2,2) 8 4 * Coef1(Count%,1)
TempC(2,4) 8 2 * Coef1(Count§,2)

TempC(2,6) 8 - Coef1(Count§,3)
TempC(4,2) 8 2 * Coef1(Count%,l)
TempC(4,4) 8 16 * Coef1(Count§,2)
TempC(4,6) 8 2 * Coef1(Count%,3)
TempC(6,2) 8 - Coef1(Count%,1)
TempC(6,4) 8 2 * Coef1(Count%,2)
TempC(6,6) 8 4 * Coef1(Count%,3)

ELSEIF NumElmNode2§ 8 4 THEN
TempC(2,2) 8 2 * Coef1(Count§,1)
TempC(2,4) 8 Coef1(Count%,2)
TempC(4,2) 8 Coef1(Count§,1)
TempC(4,4) 8 2 * Coef1(Count%,2)

END IF

ELSE

IF NumElmNode2% 8 6 THEN

TempC(2,2) 8 4 * Coef1(Count§,1)

TempC(2,4) 8 2 * Coef1(Count§,1)
TempC(2,6) 8 - Coef1(Count%,1)
TempC(4,2) 8 2 * Coef1(Count%,1)
TempC(4,4) 8 16 * Coef1(Count§,1)
TempC(4,6) 8 2 * Coef1(Count§,1)

TempC(6,2) - Coef1(Count%,1)

SHHS

TempC(6,4) 8 2 * Coef1(Count%,1)
TempC(6,6) 8 4 * Coef1(Count%,1)
ELSEIF NumElmNode2% 8 4 THEN

TempC(2,2) 8 2 * Coef1(Count§,1)
TempC(2,4) 8 Coef1(Count%,1)
TempC(4,2) 8 Coef1(Count§,l)
TempC(4,4) 8 2 * Coef1(Count§,1)
END IF
END IF

ELSE
IF FDorFE%81 THEN
IF NumElmNode2% 8 6 THEN
TempC(2,2) 8 5 * Coef1(Count§,1)
TempC(4,4) 8 20 * Coef1(Count%,2)
TempC(6,6) 8 5 * Coef1(Count§,3)
ELSEIF NumElmNode2% 8'4 THEN
TempC(2,2) 8 3 * Coef1(Count%,1)
TempC(4,4) 8 3 * Coef1(Count§,2)
END IF
ELSE
IF NumElmNode2% 8 6 THEN
TempC(2,2) 8 5 * Coef1(Count%,1)
TempC(4,4) 8 20 * Coef1(Count§,1)
TempC(6,6) 8 5 * Coef1(Count%,1)
ELSEIF NumElmNode28 8 4 THEN
TempC(2,2) 8 3 * Coef1(Count%,1)
TempC(4,4) 8 3 * Coef1(Count%,1)
END IF
END IF
END IF
END IF
IF Method$ 8 ”K" THEN
IF NumElmNode2% 8 6 THEN
REM itititit*******tttiﬂ*ttttittttttit*tttttttttttttttttttttti
REM * If kinematic wave, the force vector of the 2nd, 4th, s *

rem * 6th equations is placed in the stiffness matrix. *
rem ittitttttttiittttttttttttttttt*ttt*ttttttttttttttttittttit

TempK(Zp 2’ .

TGU'IPK(20 1)
TempK(4p4)
TempKMo 3)
TempK(Go 6)

.5
-Fsubi(COUNT§,1)

1
-4*Fsubj(COUNT%,1)
.5

TempK(6,5) -Fsubk(COUNT§,1)
ELSEIF NumElmNodeZt 8 4 THEN
TempK(2,2) 8 .5
TempK(2,1) 8 -Fsubi(COUNT%,1)
TemPKHHl) ' .5
TempK(4,3) 8 -Fsubj(COUNT%,1)
END IF
ELSEIF Methods 8 "2"
IF FDorFE%81 THEN
IF NumElmNode2% 8

THEN

6 THEN

TempK(2,1) 8 - Coef2(Count%,1) / 2
TempK(2,3) 8 2 * Coef2(Count%,2) / 3
TempK(2,5) 8 - Coef2(Countt,3) / 6
TempK(4,1) 8 - 2 * Coef2(Count%,1) / 3
TempK(4,5) 8 2 * Coef2(Count%,2) / 3
TempK(6,1) 8 Coef2(Count§,1) / 6
TempK(6,3) 8 - 2 * Coef2(Count%,2) / 3
TempK(6,5) 8 Coef2(Count%,3) / 2

ELSEIF NumElmNode2% 8 4 THEN
TempK(2,1) 8 Coef2(Count%,1) *
TempK(2,3) 8 Coef2(Count%,2)
TempK(4,1) 8 Coef2(Count%,1)

(-.5 + ALPHA/2)
(.5 - ALPHA/2)
(-.5 - ALPHA/2)

O I

TempK(4,3) 8 Coef2(Count%,2)

END IF
ELSE

IF NumElmNodeZ! 8 6 THEN
TempK(2,1) 8 - Coef2(Count%,1) / 2
2 * Coef2(Count%,1) / 3

TempK(2.3)
TempK(Zr 5)
TempK(4.1)
TempK(40 5)
TempK(6.1)
TempK(Gr 3)
TempK(Gr 5)

i

257

(.5 + ALPHA/2)

Coef2(Count%,l) / 6

- 2 * Coef2(Count§,1) / 3
2 * Coef2(Count%,l) / 3
Coef2(Count%,1) / 6

- 2 * Coef2(Count%,l) / 3

Coef2(Count§,1)

ELSEIF NumElmNodeZ§ 8 4 THEN

TempK(2yl)
TempK(2,3)
TempK(4,1)
TempK(4,3)

END IF

END

IF NumElmNode2%
TempK(2,2)
TempK(‘h 4)

END
ELSE

IF FDorFE§=1 THEN

IF

IF

Coef2(Count%,1)
Coef2(Count%,1)
Coef2(Count%,1)
Coef2(Count%,1)

6 THEN

Czi(COUNT%,l)
4*Czj(COUNT%,1)
TempK(6,6) 8 Czk(COUNT%,1)
ELSEIF NumElmNode2% 8 4 THEN

TempK(2,2) 8 Czi(COUNT§,1)
TempK(4,4) 8 Czj(COUNT§,1)

IF NumElmNodeZ‘ 8 6 THEN

TempK(Z, 1)
TempK(2,2)
TempK(2,3)
TemPK (20 4)
TempK(2,5)
TempK(2p6)
TempK(‘h 1)
TempK(4,2)
TempK(‘L 5)
TempK(4,6)
TempK(GI 1)
TempK(GI 2)
TempK(6( 3)
TempK(6p4)
TempK(6p5)
TempK(6,6)

/

ﬁﬁ’i

- CoefS(Count%,1) / 2

IINNI

Coef4(Count§,1) / 2
* Coef5(Count%,2) /
* Coef4(Count§,2) /
Coef5(Count%,3) / 6
Coef4(Count§,3) / 6
2 * Coef5(Count%,1)
2 * Coef4(Count%,1)
2 * Coef5(Count%,2) /
2 * Coef4(Count§,2) /

Coef5(Count%,1) / 6
Coef4(Count%,1) / 6

- 2 * Coef5(Count%,2)
- 2 * Coef4(Count§,2)

Coef5(Count§,3)
Coef4(Count%,3)

ELSEIF NumElmNode2% 8 4 THEN
TempK(2,1) 8 Coef5(Count%,1)

EN
ELSE

TempK(2,2)
TempK(2,3)
TempK(ZI 4)
TempK(4p 1)
TempK(4,2)
TernpK(4r3)
TempK(4p4)
D IF

Coef4(Count§,1)
Coef5(Count%,2)
Coef4(Count%,2)
Coef5(Count§,1)
Coef4(Count%,1)
Coef5(Count%,2)
Coef4(Count§,2)

IF NumElmNode2§ 8 6 THEN
TempK(2,1) 8 - Coef5(Count§,1) / 2
8 - Coef4(Count%,1) / 2
2 * Coef5(Count%,1) / 3
2 * Coef4(Count%,1) / 3
- Coef5(Count%,1) / 6

TempK(2,2)
TempK(Zr 3)
TempK(2p4)
TempK(2,5)
TempK(2p6)

\\

Iitiil’ﬁﬁ

2
(-.5 + ALPHA/2)
( .5 - ALPHA/2)
(-.5 - ALPHA/2)
( .5 + ALPHA/2)
3
3
/ 3
/ 3
3
3
/ 3
/ 3
2
2

(-.5 + ALPHA/2)
(-.5 + ALPHA/2)
(.5 - ALPHA/2)
(.5 - ALPHA/2)
(-.5 - ALPHA/2)
(-.5 - ALPHA/2)
(.5 + ALPHA/2)
(.5 + ALPHA/2)

Coef4(Count%,1) / 6

TempK(4,1)
TempK(4p2)
TempK(‘h 5)
TempK(4r 6)
TempK(6pl)
TempK(SpZ)
TempK(6'3)
TempK(6,4)
TempK(6,5)
TEMPK(616)

258

- 2 * Coef5(Count%,1)
- 2 * Coef4(Count%,1)
2 * Coef5(Count§,1) /
2 * Coef4(Count§,1) /
Coef5(Count%,l) / 6
Coef4(Count§,l) / 6
- 2 * Coef5(Count%,l)
- 2 * Coef4(Count%,l)
/ 2
/ 2

Coef5(Count%,l)
Coef4(Count%,1)

ELSEIF NumElmNode2% 8 4 THEN

EN

TempK(2.1)
TempK(Z: 2)
TempK(Zr 3)
TempK(ZI 4)
TempK(4,1)
TempK(4,2)
TempK(4y3)
TempK(4p 4)
D IF

END IF

END

REM
REM
REM
REM
FOR

IF

Coef5(Count%,1)
Coef4(Count%,l)
Coef5(Count%,1)
Coef4(Count§,1)
Coef5(Count%,1)
Coef4(Count%,1)
Coef5(Count§,1)
Coef4(Count§,1)

‘ ﬁ ﬁ ﬁ ’ W t i

hihi\~\.

\C\

ALPHA/2)
ALPHA/2)
ALPHA/2)
ALPHA/2)
ALPHA/2)
ALPHA/2)
ALPHA/2)
ALPHA/2)

*ititﬁitti*************ttt*t*i*******i**************tt*i***ttiit

* Construct the global stiffness and capacitance matrices, and *
* build the global force vector.
it*t*t*********t***tttttttittitttttttittiittttittttttitt****t*it
Ict% 8 1 TO NumElmNode2§
FOR Jctﬁ 8 1 TO NumElmNodeZ‘
IRow§ 8 Ictt+LocatC§

JCol% 8 Jctt+LocatC§+NumBandW%-IRow§

IF TempC(Ict%,Jct%) <> 0 THEN
IF NumElmNode2§ 8 6 THEN

TempV2 8 TempC(Ict%,Jct%) * Length(Count%,1) / 30
ELSEIF NumElmNode2# 8 4 THEN
TempV2 8 TempC(Ict%,Jct§) * Length(Count%,l) / 6

END IF

Cmatrx(IRow%,JCol%) 8 TempV2 + Cmatrx(IRow%,JCol%)
END IF

NEXT Jctt
NEXT Ictt
FOR COUNT2§ 8 1 TO NumElmNodet

TempV38kofInf*TofOpp(LocatN%+COUNT2%,1)“aofInf

TempV38(TempV3+Tof0PP(LocatN§+COUNT2§,l)*fsubO)

Infil(LocatN%+COUNT2§,1) 8 TempV3

Tempv48(TempV3-InfilP(LocatN%+COUNT2§,1))/(De1taT*60)
TempVS 8 - Length(Count%,1)*TempV4/TempV1

If COUNT2§ 8 2 AND NumElmNode% 8 3 THEN
TempVS 8 TempVS * 4
END If

Force(LocatC%+Count2%*2-1,1)8TempV5+Force(LocatC%+Count2%*2-1,1)

NEXT COUNT2%
IF MethodS <> “K" THEN

Force(LocatC§+2,1) 8 Force(LocatC%+2,1)+Fsubi(COUNT%,1)

IF NumElmNode§ 8 2 THEN

Force(LocatC§+4,1) 8 Force(LocatC%+4,1)+Fsubj(COUNT§,1)

ELSEIF NumElmNode‘ 8 3 THEN

Force(LocatC%+4,1) 8 Force(LocatC%+4,1)+4*Fsubj(COUNT%,1)
Force(LocatCt+6,1) 8 Force(LocatC§+6,1)+Fsubk(COUNT§,1)

END IF

END IF

NEXT Count%
END SUB

Kmatrx(IRow§,JCol%) 8 TempK(Ict%,Jct§) + Kmatrx(IRow%,JCol%)

*

259

sub ModfyGlobMtx

*ﬁiitttittt****iﬁitﬁtiﬁtiiiiiiti*t**t************tiﬁiititittttﬁtit

rem
rem
rem
rem
rem
rem
rem
rem
rem
rem
rem
rem
rem
rem

t

I I I I I I I I I I

t

A subroutine for modifying the global system of equations *
for known Phi boundary conditions. This subroutine modifies *
the [A] and [P] matrices by deleting rows and columns. It is *
always assumed that the unknown to be modified is the flow Q *
which represents the second unknown at each node. *
However, if the Area of flow at the node has a fixed value as *
a boundary condition, then this can still be incorporated into *
the final system of equations using this subroutine. This step*

is accomplished by entering the number of node minus 0.5 *
as the node number to be modified. In other words, the *
first value in the array NofPhi(1,1) will be *

NofPhi(1,1) 8 Node Number of 1st boundary - .5 *

titiii*tttitﬁﬁittttttﬂiiitti*tﬁittitﬁtti**ﬁtﬁ*ﬁ**t******it**ﬁ*t***

SHARED NumElemt,Amatrx(),Pmatrx(),Method$,NumNode§,Phi()'Phil()
SHARED NumofPhi%,NumBandW%,Fstar(),NP§,NofPhi()
LOCAL Ict§,Jct§,PrvN§,KPhiN%,TT1%,TT2%,TT3%,TT4%

FOR Icti 8 1 TO NumofPhii

KPhiN§ 8 NofPhi(Ict%,1) * 2

PrvN§ 8 KPhiNt - 1

FOR Jct§ 8 KPhiN§+1 TO NumBandW8+PrvNS
TT1% 8 NumBandW8 - Jct§ + KPhiN%

TT2§
TT3%

NumBandW% + Jct% - KPhiN§
NumBandWi - PrvN% + KPhiNt

TT4% 8 NumBandW% + PrvN% - KPhiN%
IF Jctt <= NPi THEN

Fstar(Jct%,1) 8 Fstar(Jct%,1)-Amatrx(Jct§,TT1%)*Phi(KPhiN§,1)
Fstar(Jct%,1) 8 Fstar(Jct%,1)+Pmatrx(Jct%,TT1%)*Phil(KPhiN%,l)
Amatrx(KPhiN%,TT2§) 8 0.

Amatrx(Jct%,TT1%) 8 0.

Pmatrx(KPhiN%,TT2§) 8 0.

Pmatrx(Jct§,TT1§) 8 0.

END IF
IF PrvN% > 0 THEN

Fstar(PrvN%,1) 8 Fstar(PrvN%,1)-Amatrx(PrvN%,TT3%)*Phi(KPhiN§,1)
Fstar(PrvN§,1) 8 Fstar(Prth,1)+Pmatrx(PrvN%,TT3%)*Phil(KPhiN%,1)
Amatrx(PrvN%,TT3§) 8 0.

Amatrx(KPhiN%,TT4%) 8 0.

Pmatrx(PrvN%,TT3%) 8 0.

Pmatrx(KPhiN%,TT4%) 8 0.

PrvN% 8 PrvN% - 1

END IF
NEXT Jct%
Amatrx(KPhiN%,NumBandW§) 8 1.
Pmatrx(KPhiN%,NumBandW%) 8 0.
Fstar(KPhiN%,1) 8 Phi(KPhiN%,1)

NEXT Ictt

end sub

sub matrixNumult(x(2),rx%,cx%,var1,z(2))
rem titttﬁttttttttttttttitttttttitttitﬁtttttittttt*tittttttttttttttttt

rem * A subroutine for matrix multiplication by a constant. *
rem *ttittittttttttttt*ttttttttitttttttttttttittttiiwtttitttittttttt*t

local 11%,jj8

for 11% 8 1 to rx‘
for jj% 8 1 to cxﬁ

z(ii%,jj%) 8 0
next jji

next ii%

260

for 11% 8 1 to rx%
for jj% 8 1 to cx%
z(ii§,jj%) 8 varl * x(ii%,jj%)
next jj%
next 11%
end sub

sub matrixNumultMod(x(2),rx%,cx%,var1,var2,z(2))
rem titttitﬁttttttitttttttti**tt*tiitttt*t***ttitittttittﬁttttittitttt
rem * A subroutine for matrix multiplication by two constants. The *
rem * for the unsteady part of the problem while the second is for *
rem * the steady part of the problem when other than full dynamic *

rem * equations are used. *
rem ii*****tti*ttt*ﬁ*****ﬁ*tﬁtii*tit*iitti*tﬁtttti**t*******ﬁ***i*****

local 11%,jj%

for 11% 8 1 to rx‘
for jj% 8 l to cx%
z(11%,jj%) 8 0
next jj%
next 11%

for 11% 8 l to rx%
for jj% 8 1 to cxt
if int(ii%/2)*2 8 118 then
z(ii%,jj%) 8 var2 * x(ii%,jj%)
else
z(11%,jj%) 8 varl * x(ii§,jj§)
end if
next jj%
next 118
end sub

sub matrixprt(x(2),rx%,cx%,var$)
rem *ttttt************itititit*iitttittt*itt*ttttttt*tttititttitiitiit

rem * A subroutine for printing matrices. *
rem *tttttittttttttttt*tttit*ittitttttﬁtttttttti*ttttttttittitttttttit

local ii%,jj%

PRINT #1,
PRINT #1,
PRINT #1, space$(20):var$
PRINT #1,
for 11% 8 1 to rx‘
for jj% 81 to cx%
PRINT #1, using "##.#### ";x(ii%,jj%):
next jj%
PRINT #1,
next 11%
PRINT #1,
PRINT #1,
end sub

sub Vectoerrt(x(2),rx%,cx%)

rem ﬁﬁtﬁ**t*i**i*t*******i*ttt*t***********i**tittﬁ***ﬁ**ﬁ********ittt

rem * A subroutine for printing matrices transposes. *
rem *ttttttwit*ttitwtttttttttt*ttﬁtﬁiti*tt***ttt*tttt*ﬁ*tt**t***t*t***

local ii%,jj%

for 11% 8 1 to cx%
for jj% 81 to rxt
PRINT #1, using "##.#### ";x(jj%,ii%);
if int(jj%/8)*8 = jj% then PRINT #1,
next jj%
PRINT #1,
next 11%
end sub

261

sub matrixadd(x(2),rx%,cx%,y(2),ry%,cy%,z(2),rz%,cz%)

rem itit*ttttttitttﬂtttiiiﬂittiiiitttﬁtitttittitﬂittitititttiittitttii

rem * A subroutine for matrix addition.

*

rem it*iti*tittttitttttittttitﬁtititit*i*i***************************t

local ii%,jj%,kk%

if cxt<>cy% or rx%<>ry% then
PRINT #1, "Matrices can't be added 1!!!"
goto quitadd

end if

r2% 8 rx%
c2% 8 cy%_

for 11% 8 1 to rx%
for jj% 8 1 to cx%
z(ii%,jj%) 8 0
next jj%
next 11%

for 11% 8 l to rx%
for jj% 8 l to cx%
z(ii%,jj%) 8 x(ii%,jj%) + y(ii%,jj%)
next jj%
next 11%

quitadd:
end sub

SUB GAUSSBND(X(2),NEQU%,Bndeth%,C(2),Z(2))

REM ﬁttiit**************ii**t**tittt**tt*t**titiﬁtﬁﬁtiitﬁﬁitttiﬁti*tit

REM * A subprogram that implements the Gaussian Elimination

REM * procedure to the solution of a system of equations.

REM * subprogram takes the bandwidth into account when solving the
REM * system of equations. However, if your system is not banded,
REM * you should use the total dimension of the matrix as the

REM * bandwidth.

I I I I I I

REM **tt***t*ﬁﬁi*itﬁti****tt***ttt*ﬁﬁ**tﬂttt**ﬁ**t**t**t*tﬁ****iﬁt*tti

LOCAL UPLMT%,MXMUM,PVT%,K§,1%,J§,DUM,DUM§,II‘,Factor,SumOfX

SHARED S‘)

for 11% 8 1 to NEQU%
2(1i%,1) 8 0
next 11%

UPLMTZG‘ 8 3 * Bndeth% - 1
FOR 1% 8 1 TO NEQU%
S(I8) 8 ABS(X(I%,1))
FOR J‘ 8 2 TO 2 * Bndeth% - 1
IF ABS(X(I%,J%)) > S(I%) THEN
S(I§) 8 ABS(X(I§,J%))
END IF
NEXT J%
NEXT 1%

FOR K% 1 TO NEQU§-1
PVT‘ K8
MXMUM 8 ABS(X(K§,Bndeth8)/S(K%))
UPLMT‘ 8 K% + Bndeth% - 1

IF UPLMT‘ > NEQU% THEN UPLMT% 8 NEQU%

FOR 118 8 K% + 1 TO UPLMT% '
IF Bndeth§+K§-II% > 0 THEN
DUM 8 ABS(X(II%,Bndeth%+K§-II%)/S(II%))
IF DUM > MXMUM THEN
MXMUM 8 DUM
PVT% 8 II%

262

END IF
END IF
NEXT 11%

IF PVT% <> K% THEN
FOR III% 8 Bndeth% TO UPLMTZG%
DUM 8 X(PVT%,III%-PVT%+K%)
X(PVT%,III%-PVT%+K%) 8 X(K%,III%)
X(K%,III%) = DUM
NEXT III%
DUM 8 C(PVT%,1)
C(PVT%,1) 8 C(K%,1)
C(K%,1) 8 DUM
DUM 8 S(PVT%)
S(PVT%) = S(K%)
S(K%) 8 DUM
END IF

UPLMT2% 8 3*Bndeth%-1

IF UPLMT2% > NEQU% THEN UPLMT2% 8 NEQU%

FOR I% 8 K% + 1 TO UPLMT%
Factor 8 X(I%,Bndeth%-I%+K%)/X(K%,Bndeth%)
FOR J% 8 Bndeth%+1 TO UPLMT2G%

X(I%,J%-I%+K%)8X(I%,J%-I%+K%) - Factor*X(K%,J%)

NEXT J%
C(I%,1) 8 C(I%,1) - Factor * C(K%,1)

NEXT I%

NEXT K%

Z(NEQU%,1) 8 C(NEQU%,1) / X(NEQU%,Bndeth%)
FOR 1% 8 NEQU%-1 TO 1 STEP -1
SumOfX 8 0
FOR J% 8 1% + 1 TO NEQU%
IF Bndeth%+J%-I% <8 UPLMT2G% THEN
SumOfX 8 SumOfX + X(I%,Bndeth%+J%—I%) * Z(J%,1)
END IF
NEXT J%
Z(I%,1) 8 (C(I%,1) - SumOfX)/ X(I%,Bndeth%)
NEXT 1%
END SUB

sub matrixVectmult(X(2),NEQU%,Bndeth%,C(2),Z(2))

rem *ittttttitiitt*tiitﬁﬁt*iittitttttttﬁttit*ttttittttittt*tiitiiiittt
rem * A subroutine for matrix multiplication. This routine would
rem * only be usable to mutiply a matrix [X] by a vector {C}. The
rem matrix should be banded and the band width should be given.
rem Moreover, the the matrix should be based in a banded form with
rem the dimensions of [X] as follows: NEQU% rows and 2*Bndeth%-1
rem columns. If the data is not passed as such, errors will occur.*

rem The subroutine doesn't have any error checks. *
rem itttttttitttitit*ttt*tttittiittﬁtt*tttttittttttttttttiittttttttttt

I I I I I

I I I I I

LOCAL I%,J%,II%

for 11% 8 1 to NEQU%
Z(1i%,1) 8 0
next 11%

FOR I% 8 1 TO NEQU%
FOR J% 8 1 TO 2*Bndeth%-1
IF -Bndeth%+J%+I% > 0 AND -Bndeth%+J%+I%<=NEQU% THEN
Z(I%,1) 8 Z(I%,1) + X(I%,J%) * C(-Bndeth%+J%+I%,1)
END IF
NEXT J%
NEXT I%
END SUB

263

INITIAL:
REM it*ttttttti**tt***ttttt*tiﬂirt*tttﬁttttittttttttttititﬂtttititﬁtit
REM * A Subroutine to initialize parameters. *
REM ititttittiﬁttt*itttt*tﬁtttttiﬁittttttﬁttititittttttitﬁﬁttttﬁttttti
FDorFE% 8 1 '<88881 is finite difference, 0 is finite element88
False%8 0
True% 8 1

IrrMethodS 8 ”F"
Methods 8 "K"
TypElem$ 8 "L“
SelPrtOpt$ 8 "F"

 

 

 

iprint% 8 0

VARTheta 8 .50 ' <

ALPHA 8 .25

DeltaT 8 5 ' <==minutes
numiter% 8 10 ' <

AllError 8 0.0005

TotNumStep%8 20
CONSISorLUMP% 8 0 '<8=881 is consistent, 0 lumped-888
CoefTW 8 1.0 '<888888 Coefficient of Top Width (possible values are 1 a 1.5)

FurLength 8 625

TimCut 8 300 'min

Qin 8 2.78 'liters/sec
kofInf 8 0.0252

fsub0 8 0.00023

Slope 8 0.0044

aofInf 8 0.02

ManngN 8 0.03

IF IrrMethodS 8 "B" THEN

Rhol 8 1
Rhoz 8 3.3333333
Sigmal 8 1.5
Sigma2 8 1
ELSE
Rhol 8 0.46
Rhoz 8 2.86
Sigmal 8 0.92
Sigma2 8 0.65
END IF
RETURN

DEF FNExists%(FilNam$)
REM *iii*tttttittttttitﬁtittittttﬁttttttttttiitt*tittttttttttttitiitti
REM * The function Exists% returns a non zero integer value if the *
REM * file specified by FilNamS is on the current disk drive. *

REM *iﬁﬁ*itiiitﬁitﬁiiﬁﬁtitt**t**************ﬁi*t****ﬁt*ttttﬂiiiiiiittt

Shared False%,True%
LOCAL ExistF%

ON ERROR GOTO FileError
ExistF% 8 True% 'initial

OPEN FilNamS FOR INPUT AS #9
IF ERR80 THEN CLOSE #9
GOTO Finish

FileError:
ExistF% 8 FALSE% ' File doesn't exist
RESUME NEXT

Finish:

ON ERROR GOTO 0

FNExists%8ExistF%
END DEF

APPENDIX B

FE-SURFDSGN Graphics Routine Listing

REM ********t*****tt**tiit*tiiittiﬁtiiit*iti***********tﬁ**************ﬁ

REM" 4:
REM * Program SURFGRPH.BAS *
REM* 8
REM itit******titt*tttttttttttiitﬁtttttittitt***ttttii**tt**t*tﬁit*ttttt
REM" 8
REM * Developed By *
REM * *
REM * Walid H. Shayya *
REM * a
REM * Michigan State University *
REM * *
REM * July 30, 1990 *
REM‘ *
REM *itttti**ti**t*ttt******itittittittt*ttt*i*t*t********t**ttttttttitt

COMMON NGphFil$
%Maprts 8 200

DIM Varbll(%Maprts), Varb12(%Maprts)

DIM XPOS%(6), YPOS%(11), XPnts(%Maprts), YPnts(%Maprts)
DIM Length(%Maprts,3), Time(%Maprts,3)

DIM Grpthnt(4), XX%(10), YY%(10), CC$(10), AA$(10)

SCREEN 0

IF NGphFil$ 8'" THEN
GOSUB InitialScrn

END
CLS

IF

'ON ERROR GOTO Trap
NumOfPrb% 8 l
GOSUB READING

LOCATE 8, 10
IF NGphFil$ 8" THEN
INPUT "Enter the name of file to plot (no extension) : “, NGphFil$

END

IF

F11Prgl$ 8 NGphFil$ + ".PRG"
FilPrg2$ - NGphFil$ + ".REC"

FilInS - NGphFil$ + ".OUT"

CheckEnt$ 8 “"
GOSUB GetFilNam

GOSUB LdScrnFl

IF CheckEnt$ 8 "R“ OR CheckEnt$ 8 "B“ THEN
NumOfPrb% 8 NumOfPrb% + 1
GOSUB LdScrnFll 'open file for data output

END

IF

IF CheckEnt$ 8 ”A” OR CheckEnt$ 8 "B” THEN
NumOfPrb% 8 NumOfPrb% + 1
GOSUB OpnFil 'open file for data output

END

IF

GOSUB Initial

SM54

265

GOSUB HEADING
GOSUB GetScren

REM ********t***************ﬁ****************ttti*t******ﬁt*******t**tit

REM * Prepare Screen for output.
REM *ﬁtttttttttttttttttttttitttttttttttitit.titttttttt*titttttﬂtitttitit

CALL PrntScrn

GOSUB PrepDGrph

CALL FindRange2(MaxLength / 10)
CALL FindRange1(MaxTime / 10)
CALL Labeleis

GOSUB GetDat

as 8 INPUT$(1)
SCREEN 0
END

InitialScrn:

REM tttiit*ittitttttti*************ﬁ*t*ittittitiititiititttttt*tt**tt*

REM * A subroutine to display the first screen.
REM tttitiittittttttttitﬁt*tttttttﬁt*t*ttittttttﬁi*tttittiiitttitttttt

KEY OFF

XX%(1) 8 5: XX%(2) 8 23: XX%(3) 8 23: XX%(4) 8 22: XX§(5) 8 10
YY%(1) 8 2: YY%(2) = 4: YY%(3) 8 8: YY%(4) 8 10: YY§(5) 8 12
XX%(6) = 17: YY%(6) 8 14

CC$(1) 8 CHR$(201): CC$(2) 8 CHR$(205): CC$(3) 8 CHR$(187)
CC$(4) 8 CHR$(186): CC$(5) 8 CHRS‘ZOQ): CC$(6) 8 CHR$(185)
CC$(7) 8 CHR$(200): CC$(8) 8 CHR$(188)

A15 8 CC$(1) + STRING$(54, CC$(2)) + CC$(3)
A25 = CC$(5) + STRING$(54, CC$(2)) + CC$(6)
A35 8 CC$(7) + STRING$(54, CC$(2)) + CC$(8)

AA$(1) 8 " FINITE ELEMENT SURFACE IRRIGATION DESIGN MODEL “
AA$(2) 8 “VERSION 1.00”

AA$(3) 8 "Developed by"

AA$(4) 8 "Walid H. Shayya"

AA$(5) 8 "Department of Agricultural Engineering"

AA$(6) 8 "Michigan State University"

CLS

LOCATE 4, 1

PRINT SPACE$(12): A1$
FOR Iloop§ 8 1 TO 5
PRINT SPACE$(12); CC$(4); SPACE$(54); CC$(4)
NEXT Iloopt
PRINT SPACE$(12): A2$
FOR Iloopﬁ 8 1 TO 9
PRINT SPACE$(12); CC$(4); SPACE$(54); CC$(4)
NEXT Iloop§
PRINT SPACE$(12); A3$
FOR Iloopi 8 1 TO 6
LOCATE YY§(Iloop%) + 4, XX%(Iloop%) + 12
PRINT AA$(Iloop%)
NEXT Iloop%

LOCATE 23, 48

COLOR 15, 0

PRINT "Press any key to continue."
COLOR 7, 0

WHILE INKEYS = "“: WEND
RETURN

266

HEADING:
REM ititit*ttttﬁtttttttittttitttttttttitit*tttitittittttttﬁ****ttt*ttt*ﬁtit
REM * Heading Number 1

REM tittit*ttttttttttttttttttttitttttitittnttintittitttttttttttittttititttt
CLS

LOCATE 2, 16

PRINT " FINITE ELEMENT SURFACE IRRIGATION DESIGN MODEL "

LOCATE 3, 16

PRINT " GRAPHICS ROUTINE”

LOCATE 4, 16

PRINT STRING$(48, 196)

RETURN

GetScren:
REM *ttttttttttttttitttittitttﬁtttttittiit*ﬁ*ttt*tttttttttttttttitttttttttt

REM * Check the screen number. *
REM *ﬁtttttiit*tiﬁ*ttttit.tttittﬁtttttititttitttittitttittttttﬂﬁitttttitttt
LOCATE 8, 10
PRINT "Please select the number of the graphics screen from the following :”
LOCATE 11, 10: PRINT "Press ": : COLOR 0, 7: PRINT ' 2 ”3 : COLOR 7, 0
PRINT ' for a high resolution,graphics screen (640x200 pixels),'
LOCATE 13, 16: COLOR 0, 7: PRINT ' 9 "; : COLOR 7, 0
PRINT ' for an enhanced resolution graphics screen (640x350 “;
LOCATE 14, 20
PRINT "pixels), or“
LOCATE 16, 16: COLOR 0, 7: PRINT ' S "; : COLOR 7, 0
PRINT ' for special screen number 10 (640x200 pixels) : ";
WHILE SanumS 8 ": SanumS 8 INKEYS: WEND
SanumS 8 UCASE$(SanumS): PRINT SanumS
IF SanumS <> '2" AND SanumS <> '9' AND SanumS <> 'S' THEN
BEEP: SanumS 8 "“: LOCATE 23, 42: PRINT "Press either ': : COLOR 0, 7
PRINT "2": : COLOR 7, O: PRINT ", '; : COLOR O, 7:
PRINT "9"; : COLOR 7, 0: PRINT ' or “; : COLOR 0, 7:
PRINT 'S'; : COLOR 7, 0
PRINT ' to proceed."
GOTO GetScren
END IF
RETURN

GetFilNam:

REM itit****ﬁﬁﬁt*it***********tt*******iﬁﬁti******t**i*i***t*****ﬁ*********

REM * A subroutine to check if to plot any additional data files.
REM *tiiﬁtttiiittiii*ttttitittttiiitﬁiittﬁitttittit!*ttittiﬁtttﬁtttﬁiﬁiitti

LOCATE 11, 10
PRINT "Would you like to plot any additional data files ?”:
LOCATE 14, 10: PRINT ”Press ": : COLOR 0, 7: PRINT " R "; : COLOR 7, 0
PRINT " for simulated recession curve,"
LOCATE 16, 16: COLOR O, 7: PRINT " A '; : COLOR 7, 0
PRINT " for a plot of actual advance data curve, or"
LOCATE 18, 16: COLOR 0, 7: PRINT ” B “: : COLOR 7, 0
PRINT " for a plot of both curves, or"
LOCATE 20, 16: COLOR O, 7: PRINT " N "; : COLOR 7, 0
PRINT ' for a plot of neither curve : “:
WHILE CheckEnt$ 8 "": CheckEnt$ 8 INKEYs: WEND
CheckEnt$ 8 UCASE$(CheckEntS): PRINT CheckEnt$
IF CheckEnt$ <> "R" AND CheckEnt$ <> "A" AND CheckEnt$ <> "B" _
AND CheckEnt$ <> "N" THEN
BEEP: CheckEnt$ 8 ”": LOCATE 23, 40: PRINT "Press either ": : COLOR 0, 7
PRINT "R"; : COLOR 7, 0: PRINT ", “; : COLOR 0, 7: PRINT "A";
COLOR 7, 0: PRINT ", ": : COLOR 0, 7: PRINT "B":
COLOR 7, 0: PRINT ”, or "; : COLOR O, 7: PRINT "N";
COLOR 7, O
PRINT " to proceed."

267

GOTO GetFilNam
END IF
RETURN

OpnFil:

REM i***********ti**********t*t**ﬁ*********iit*********t**************

REM * Open file and initialize numbers.
REM tit.*ﬁtitt*itiitﬁttttttttttttttttttitﬁtttttttttttittttittttttttktt
OPEN "I", ll, FilIn$
NumOfPts§(NumOfPrb%) 8 0
IILPS 8 1
WHILE NOT EOF(1)
INPUT #1, Time(IILP%,NumOfPrb%), Length(IILP§,NumOfPrb%)
IILP't 8 IILP% + 1
NumOfPts%(NumOfPrb%) 8 NumOfPts%(NumOfPrb§) + 1
WEND
CLOSE #1
RETURN

PrepDGrph:
REM ttti*ﬁtitt*tiittttttittttt*******ttttttttttitttﬁtiittﬁitittttttttttttit
REM * Subroutine to prepare for doing graph.
REM *******it**tttitttttitittt************it***t*****tttttﬁttttitﬁitttttttt
MaxLength 8 0
MaxTime 8 0

FOR J 8 1 TO NumOfPrb%
FOR I 8 1 TO NumOfPts%(J)
IF Length(I,J) > MaxLength THEN MaxLength 8 Length(I,J)
IF Time(I,J) > MaxTime THEN MaxTime 8 Time(I,J)
NEXT I
NEXT J
RETURN

GetDat:

REM *ttt*ﬁ***********************i****************i**ﬁtittitiﬁtittttti

REM * A subroutine initialize the array.
REM *****ttittitttti*tittititt*tttttittititttitttttt*tttttititttitittt
FOR Jlopt 8 1 TO NumOfPrb%

FOR Ilop§ 8 1 TO NumOfPts§(Jlop%)
Varb12(Ilop%) 8 Length(Ilop§,Jlop%)
Varbll(Ilop%) 8 Time(Ilop%,Jlop§)

IF Varb12(Ilop%) < 0 THEN Varb12(Ilop§) 8 0
IF Varbll(Ilop%) < 0 THEN Varbll(Ilop%) 8 0
NEXT Ilop‘
CALL PrepPoints(Varb12(), Varbll(). Jlopi)
NEXT Jlop%
RETURN

LdScrnFl:

REM *ittﬁtti********ititit*ﬁtﬁitﬁttiitti*******t*****itﬁttttﬁtttiitit*

REM * Subroutine to load file containing saved screen for file 1.
REM tittiiittttttttiiiit*titﬁtttﬁititttﬁtt********ﬁttitttiittttittittt
OPEN "I", #1, FilPrng
NumOfPts%(1) 8 0
IILPt 8 1
WHILE NOT EOF(1)
INPUT #1, Time(IILP§,1), Length(IILP%,1)
IILP‘ 8 IILP‘ + 1
NumOfPts%(1) 8 NumOfPts%(1) + 1
WEND
CLOSE #1
RETURN

268

LdScrnFll:

REM *iittiiiiiittttitttﬁ*tﬁttiii*******tﬁii*ﬁ**tiiﬁﬁitiﬁitﬁﬁitttitttti

REM * Subroutine to load file containing saved screen for file 2.
REM *ittitiiittttt*tttttttttitii*tttttttttittittittitittitttttittitttt
OPEN "I”, {1, FilPrgZS
NumOfPts%(2) 8 0
IILP% 8 1
WHILE NOT EOF(1)
INPUT #1, Time(IILP§,2), Length(IILP%,2)
IILP% 8 IILP% + 1
NumOfPts%(2) 8 NumOfPts%(2) + 1
WEND
CLOSE #1
RETURN

Initial:

REM it*tit*tttittﬁﬂit.******tiﬁittti*ttiiitﬁ*ttﬁi*****i*ﬁ******ﬁ*****t

REM * Subroutine to Initialize.
REM *ttttt*tttit*itttiittttitttﬁtttttttttttttttttiittﬁtttttttttttitttt

XPos%(l) 8 21: YPos%(1) 8 13
XPos%(2) 8 17: YPos%(2) 8 25
XPos%(3) 8 13: YPos%(3) 8 38
XPos%(4) = 10: YPos%(4) 8 50
XPos$(5) 8 7: YPos%(5) 8 63
XPos§(6) 8 3: YPos%(6) 8 75
Grpthnt(1) 8 4.0

Grpthnt(2) 8 8.0

Grpthnt(3) 8 12.0
Grpthnt(4) 8 16.0

RETURN

Trap:

REM ***t*******ttttt**t**iit*ﬁtiﬂﬁtt*t***tiﬁ**********ﬁﬂ***ﬁ*****ﬁ*t

REM * A subroutine to handle error.

REM t*tttt*i*t***t***t*tiiitititttﬁttiitﬁttt*iﬁtt******t*t****t*i**t
LOCATE 24, 1

PRINT SPC(79):

LOCATE 23, 35

PRINT "Error Number ”; ERR; " has occurred 1";

LOCATE 24, 35

PRINT " Press any key to continue.";

RESUME Qtrap

RETURN

Qtrap:
WHILE INKEYS = "": WEND
SCREEN 0
CLS
END

SUB DoGrph (X1, Y1, X2, Y2)
REM *tttttttti*tttit*********tt*tt********i**tttit*ttiititttitittttﬂtt
REM * This subprogram recieves 2 points (x1,y1) and (x2,y2) and
REM * connects the line between these two points.
REM ﬁttﬁtt*ttttttitttttt*tti*ttitttttitttttttt*tiiitttttiittttti*tt*t*
LINE (X1, Y1)-(X2, Y2)

END SUB

SUB DoGrphZ (X1, Y1, X2, Y2)
REM ittitttttiitttttititittttiit*ttitittttttttttitt**ttttt*tittttt*ttt
REM * This subprogram recieves 2 points (x1,y1) and (x2,y2) and
REM ' connects the line between these two points.
REM ittttt*ﬁtt*titt*t**t*t*itttititttﬂiﬁittt*tttttittttttttttﬁtﬁtﬁtttt
SHARED Numbrt
IF Numbrt 8 2 THEN
LINE (X1, Y1)-(X2, Y2), I . 8888
ELSEIF Numbr! 8 3 THEN

269

LINE (X1, Y1)-(X2, Y2). , , 1111
ELSE
LINE (X1, Y1)-(X2, Y2)
END IF
END SUB

SUB DoGrphl (X1, Y1)

REM *tittiiitt*iﬁtittttitttttittitiitttiﬁtﬁtit*****************tt*t***

REM * This subprogram draws a point (x1,y1).
REM titttttiﬁiﬂittttttit*ttttttﬁit*****t*t**tt***********tttttttittitt
Circle (X1, Y1),.5

END SUB

SUB FindRangeZ (Var2)

REM itﬁitttttt*tttiti**********tti*ﬁ*********t*****i****t*********ti

REM * A subroutine to find the Range 1.
REM tttt*****t*t*****t**ititttttttitttit*tti*itttﬂttitttitittttitiit
SHARED Rangel, Form1$
'LOCAL II
FOR II 8 90 TO 10 STEP -5
IF Var2 > II THEN
Rangel 8 II + 5
FormlS 8 “tit!”
GOTO QSB
END IF
NEXT II
IF Var2 > 8 THEN
Rangel 8 10
Forml$ 8 "90%!"
GOTO QSB
END IF
FOR II 8 10 TO 2 STEP -2
IF Var2 > II THEN
Rangel 8 II + 2
FormlS 8 “ti.#"
GOTO QSB
END IF
NEXT II
FOR II 8 2 TO 1 STEP -1
IF Var2 > II THEN
Rangel 8 II + 1
Form1$ = "##.9"
GOTO QSB
END IF
NEXT II
FOR II 8 1 TO 0 STEP -.5
IF Var2 > II THEN
Rangel 8 II + .5
Form1$ 8 "##.O”
GOTO QSB
END IF
NEXT II
Rangel 8 .5
Form1$ 8 "##.#"
QSB:
END SUB

SUB FindRangel (Varl)

REM ****i*t*t******t*it*ititiiitﬁt*ttittttttiitttiiﬁtttiitiiﬂttitttt

REM * A subroutine to find the Range 2.
REM *iittttttt********************************it***************t****
SHARED Range2, Form2$, CheckEnt$
'LOCAL 11
FOR II 8 190 TO 10 STEP -5
IF CheckEnt$=”A” THEN
IF Varl > II THEN

270

Range2 8 II + 5
Form2$ 8 ”####”
GOTO 0882
END IF
ELSE
IF Varl > II - 5 THEN
Range2 8 II + 5
Form2$ 8 "it?!”
GOTO 0882
END IF
END IF
NEXT II
IF Varl > 8 THEN
Range2 8 10
Form2$ 8 "##t!"
GOTO 0882
END IF
FOR 11 8 10 TO 2 STEP -2
IF Varl > II THEN
Range2 8 II + 2
Form2$ 8 "##.O'
GOTO 0882
END IF
NEXT II
FOR II 8 2 TO 1 STEP -1
IF Varl > II THEN
Range2 8 II + 1
Form2$ 8 “9%.!"
GOTO 0882
END IF
NEXT II
FOR II 8 1 TO 0 STEP -.5
IF Varl > II THEN
Range2 8 II + .5
Form2$ 8 "#9.!"
GOTO 0882
END IF
NEXT II
Range2 8 .5
Form2$ 8 "##.#"

QSBZ:
END SUB

SUB Labeleis

REM *tttttiitittti**t***ti*************tititﬁttitttttitttttttiﬁi*ttt

REM * A subroutine to label X and Y axis.

REM *ttttiiitttititttt*ttt*tttitittt*titttiﬁtttt*titiiiittﬁiittititt
SHARED Rangel, Forml$, Directs, XPOS§(), YPOS%(), BoundS, Range2, Form2$
'LOCAL VertPost, TLabe11%, TLabeth, HorizPost, f1$, £25, Varblﬁ

£25 8 Forml$
£15 8 Form2$
FOR TLabe11% 8 0 TO 5

LOCATE XPOS%(TLabell% + 1), 9

PRINT USING f1$; Range2 * (TLabe11% * 2);
NEXT TLabe11§
FOR TLabe12% 8 0 TO 5

LOCATE 22, YPos§(TLabe12% + 1)

PRINT USING f2$; Rangel * (TLabe12% * 2);
NEXT TLabelZ$

END SUB

271

SUB PrepPoints (XPnts(), YPnts(), NumOfItert)

REM **************i*****ti*****titittttitt*********t**t******t******

REM * A subroutine to draw points.
REM *tttttttﬁttttittttttitttiﬁﬁtiiittttiit*tittttttt*ttiiiittitttttt
SHARED Rangel, Range2, NumOfPts%(), NumOfPrb%, CheckEnt$
'LOCAL Pont‘, r1, r2
r1 8 Rangel
r2 8 Range2
FOR Font} 8 1 TO NumOfPts%(NumOfIter%)
XPnts(Pont%) 8 XPnts(Pont§) / r1 * 10
YPnts(Pont%) 8 YPnts(Pont%) / r2 * 10
NEXT Pont%
IF NumOfItert 8 1 THEN
FOR Pont% 8 2 TO NumOfPts%(NumOfIter§)
CALL DoGrph(XPnts(Pont% - 1), YPnts(Pont% - 1), XPnts(Pont%),
YPnts(Pont%))
NEXT Pont%
ELSEIF NumOfIter% 8 2 AND (CheckEnt$ = “R“ OR CheckEnt$ 8 "8”) THEN
FOR Pont§ 8 2 TO NumOfPts§(NumOfIter%)
CALL DoGrph(XPnts(Pont% - 1), YPnts<Pont§ - 1), XPnts(Pont§),
YPnts(Pont§))
NEXT Pontt
ELSE
FOR Pent} 8 1 TO NumOfPts§(NumOfIter§)
CALL DoGrph1(XPnts(Pont%), YPnts(Pont%))
NEXT Pontt
IF CheckEnt$ 8 “A” THEN
FOR ILP 8 2 to 6 step 2
Circle (ILP, 83),.5
NEXT ILP
LOCATE 6, 22
PRINT ”Actual Data"
ELSEIF CheckEnt$ 8 'B' THEN
FOR ILP 8 2 to 6 step 2
Circle (ILP-20, -20),.S
NEXT ILP
LOCATE 24, 9
PRINT "Actual Data";
END IF
END IF
END SUB

SUB PrntScrn

REM *itttﬂﬂitt****t*****it*****i**t****ﬁ*tiitiiiﬁtﬁﬁﬁﬁttittittttitti

REM * A subroutine to plot the X and Y axis.

REM tiiﬁitﬂtitﬁttﬁ*tit**i**i***tttﬁttittit********ﬁttiﬁﬁﬁittﬂtittiii
SHARED Grpthnt(), SanumS, CheckEnt$

'LOCAL VertPos‘, TLabe11§, TLabeth, HorizPos%, £15, £25, Varbli

IF SanumS = '2' THEN
SCREEN 2

ELSEIF SanumS = "9" THEN
SCREEN 9

ELSE
SCREEN 10

END IF

WINDOW (-24, -28)-(105, 115)

LINE (-24, -28)-(105, 115), p B
LINE (~23.75, -28)-(104.75, 115): r B

LINE (0, O)-(100, 100): r B

FOR VertPos% 8 0 TO 100 STEP 20
LINE (0, VertPos%)-(-2.5, VertPos%)
LINE (100, VertPos§)-(97.5, VertPosi)
NEXT VertPos§

272

FOR ICyclet 8 1 TO 5
FOR JCyclet 8 1 TO 4
LOCAT2 8 Grpthnt(JCyc1e%) + (ICycle% - 1) * 20
LINE (0, LOCAT2)-(-1.S, LOCAT2)
LINE (100, LOCAT2)-(98.5, LOCAT2)
NEXT JCycle§
NEXT ICyclet

FOR HorizPos§ 8 0 TO 100 STEP 20
LINE (HorizPost, 0)-(HorizPos§, -4)
LINE (HorizPos‘, lOO)-(HorizPos§, 96)
NEXT HorizPos%
FOR ICycle§ 8 1 TO 5
FOR JCycle§ 8 1 TO 4
LOCAT2 8 Grpthnt(JCycle§) + (ICycle% - 1) * 20
LINE (LOCAT2, 0)-(LOCAT2, -2)
LINE (LOCAT2, 100)-(LOCAT2, 98)
NEXT JCyc1e§
NEXT ICyclet

LOCATE 24, 41

PRINT "Distance (m)";
LOCATE 12, 3

PRINT “Time"

LOCATE 13, 3

PRINT ”(min)";

LOCATE 2, 18
PRINT “Finite Element Surface Irrigation Design model";

IF CheckEnt$-"A” OR CheckEnt$8'N" THEN
LINE (2, 90)-(8, 92)
LOCATE 5, 22
PRINT "Simulated Data"

ELSEIF CheckEnt$ 8 ”B" THEN
LINE (71, -20)-(77, -19)
LOCATE 24, 65
PRINT “Simulated Data“:

ELSEIF CheckEnt$ 8 "R" THEN
LINE (822, -20)-(-16, -19)
LOCATE 24, 7
PRINT "Simulated Data“;

END IF

END SUB