Citation
Measurement and simulation of soil water status under field conditions

Material Information

Title:
Measurement and simulation of soil water status under field conditions
Creator:
Stone, Kenneth Coy, 1959-
Publication Date:
Language:
English
Physical Description:
xi, 167 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Irrigation ( jstor )
Lysimeters ( jstor )
Modeling ( jstor )
Moisture content ( jstor )
Soil infiltration ( jstor )
Soil water ( jstor )
Soil water content ( jstor )
Soils ( jstor )
Tensiometers ( jstor )
Two dimensional modeling ( jstor )
Soil moisture -- Florida ( lcsh )
Soil moisture -- Mathematical models ( lcsh )
Soil moisture -- Measurement ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1987.
Bibliography:
Includes bibliographical references (leaves 163-166).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Kenneth Coy Stone.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
030338492 ( ALEPH )
AEQ4824 ( NOTIS )
16767007 ( OCLC )
AA00004850_00001 ( sobekcm )

Downloads

This item has the following downloads:


Full Text









MEASUREMENT AND SIMULATION OF SOIL WATER STATUS
UNDER FIELD CONDITIONS







By

KENNETH COY STONE


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1987


i




MEASUREMENT AND SIMULATION OF SOIL WATER STATUS
UNDER FIELD CONDITIONS
By
KENNETH COY STONE
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
V
1937


ACKNOWLEDGEMENTS
The author would like to express his appreciation to Dr. Allen G.
Smajstrla for his guidance, assistance and encouragement throughout th
research. The author would like to thank his other committee members
for their support and guidance throughout this research and in
coursework taken under their supervision.
The author also expresses his appreciation to the Agricultural
Engineering Department for the use of its research and computing
facilities. Appreciation is also expressed to other faculty and staff
members in the department for their assistance.
Special gratitude is expressed to the authors family for their
continuous support and encouragement throughout his studies.
Finally, the author would like to express his very special
graditude to his wife, Carol, for her drafting expertise, patience and
continuous support and encouragement.


TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS
LIST OF TABLES v
LIST OF FIGURES vi
ABSTRACT x
CHAPTERS
I INTRODUCTION 1
II REVIEW OF LITERATURE 4
Measurement of Soil Water Status 4
Soil Water Content Measurement 4
Soil Wafer Potential Measurement 8
Automation of
Soil Water Potential Measurements 11
Water Movement in Soils 15
Soil Water Extraction 20
III METHODS AND MATERIALS 26
Equipment 26
Instrumentation Development 27
Field Data Collection 30
Model Development 33
One-Dimensional Model 36
Two-Dimensional Model 41
Soil Water Extraction 50
IV RESULTS AND DISCUSSION 54
Instrumentation Performance 54
Field Data Collection 60
Model Verification 70
Model Operation with Field Data 94
One-Dimensional Model 102
Two-Dimensional Model 109
Model Application 116
V SUMMARY AND CONCLUSIONS 124
Microcomputer-based Data Acquisition System 124
Field Data 124


Modeling Soil Water Movement and Extraction 125
Model Applications 125
APPENDICES
A. Alternating Direction Implicit Finite Differencing. 126
B. Listing of Soil Water Movement and Extraction Model. .137
REFERENCES 163
BIOGRAPHICAL SKETCH 167


LIST OF TABLES
Table 3-1. Microcomputer-based data acquisition system
components and approximate costs 28
Table 4-1. One-dimensional distribution of water extraction
for a 15 day drying cycle for young citrus trees
with and without grass cover 68
Table 4-2. Two-dimensional distribution of water extraction
for a 15 day drying cycle for young citrus trees
with grass cover 69
Table 4-3. Two-dimensional distribution of water extraction
for a 4 day drying cycle for young citrus trees
with and without grass cover 73
Table 4-4. One-dimensional distribution of water extraction
for a 4 day drying cycle for a grass cover crop
at water depletion levels of 20 kPa and 40 kPa ... 76
v


LIST OF FIGURES
Figure 2-1. A typical soil water characteristic curve 5
Figure 3-1. Layout of lysimeter system 31
Figure 3-2. Details of individual lysimeter soil water status
monitoring system 32
Figure 3-3. Location of tensiometers in 1985 field experiment
for young citrus trees in grassed and bare soil
lysimeters 34
Figure 3-4. Location of tensiometers in 1986 field experiment
for young citrus trees in grassed and bare soil
lysimeters 35
Figure 3-5. Schematic diagram of the finite-difference grid
system for the one-dimensional model of water
movement and extraction 38
Figure 3-6. Schematic diagram of the finite-difference grid
system for the two-dimensional model of water
movement and extraction 44
Figure 3-7. Effect of the relative available soil water
and potential soil water extraction rate on the
soil water extraction rate 53
Figure 4-1. Calibration curve of output voltage versus
pressure applied for pressure transducer No. 1. . 55
Figure 4-2. Calibration curve of digital units versus
voltage applied for the analog-to-digital
circuit used 57
Figure 4-3. Comparisons of mercury manometer manually-read
and pressure transducer automatically-read
tensiometer water potentials during drying cycles
for 2 tensiometers in the laboratory 58
Figure 4-4. Comparisons of mercury manometer manually-read and
pressure transducer automatically-read tensiometer
water potentials for all laboratory data 59
Figure 4-5. Evaluation of pressure transducer-tensiometer No. 1
by comparison of mercury manometer manually-read
and pressure transducer automatically-read water
potentials in the field 61
vi


Figure 4-6. Evaluation of pressure transducer-tensiometer No. 2
by comparison of mercury manometer manually-read
and pressure transducer automatically-read water
potentials in the field 62
Figure 4-7. Evaluation of pressure transducer-tensiometer No. 3
by comparison of mercury manometer manually-read
and pressure transducer automatically-read water
potentials in the field 63
Figure 4-8. Evapotranspiration rate from the 1985 field
experiment with a 20 kPa soil water potential
treatment young citrus tree with grass cover. ... 66
Figure 4-9. Evapotranspiration rate from the 1985 field
experiment with a 20 kPa soil water potential
treatment young citrus tree with bare soil 67
Figure 4-10. Evapotranspiration rate from the 1986 field
experiment with a 20 kPa soil water potential
treatment young citrus tree with grass cover. ... 71
Figure 4-11. Evapotranspiration rate from the 1986 field
experiment with a 20 kPa soil water potential
treatment young citrus tree with bare soil 72
Figure 4-12. Evapotranspiration rate from the 1986 field
experiment with a 20 kPa soil water potential
treatment with a grass cover 74
Figure 4-13. Evapotranspiration rate from the 1986 field
experiment with a 40 kPa soil water potential
treatment with a grass cover 75
Figure 4-14. Soil water potential-soil water content
relationship for Rehovot sand 79
Figure 4-15. Hydraulic conductivity-soil water content
relationship for Rehovot sand 80
Figure 4-16. Simulated results of soil water content profiles
for infiltration into a Rehovot sand under
constant rain intensity of 12.7 mm/hr 81
Figure 4-17. Simulated results of soil water content profiles
for infiltration into a Rehovot sand under
constant rain intensity of 47 mm/hr 82
Figure 4-18. Soil water potential-soil water content
relationship for Yolo light clay 83
Figure 4-19. Hydraulic conductivity-soil water content
relationship for Yolo light clay 84


Figure 4-20. Soil water potential-soil water content
relationship for Adelanto loam 85
Figure 4-21. Hydraulic conductivity-soil water content
relationship for Adelanto loam 86
Figure 4-22. Simulated results of soil water content profiles
for infiltration into a Yolo light clay with
initial pressure potential at -66 kPa 87
Figure 4-23. Simulated results of soil water content profiles
for infiltration into a Yolo light clay with
initial pressure potential at -200 kPa 88
Figure 4-24. Simulated results of soil water content profiles
for infiltration into a Adelanto loam with
initial pressure potential at -66 kPa 89
Figure 4-25. Soil water potential-soil water content
relationship for Nahal Sinai sand 90
Figure 4-26. Hydraulic conductivity-soil water content
relationship for Nahal Sinai sand 91
Figure 4-27. Simulated results of dimensionless soil water
content distribution for infiltration into a
Nahal Sinai sand at 57 min of infiltration time. 92
Figure 4-28. Simulated results of dimensionless soil water
content distribution for infiltration into a
Nahal Sinai sand at 297 min of infiltration time. 93
Figure 4-29. Two-dimensional model simulated results of soil
water content profiles for infiltration into a
Rehovot sand under constant rain intensity
of 12.7 mm/hr 95
Figure 4-30. Two-dimensional model simulated results of soil
water content profiles for infiltration into a
Rehovot sand under constant rain intensity
of 47 mm/hr 96
Figure 4-31. Evapotranspiration rate from the second drying
cycle of the 1985 field experiment with a 20 kPa
soil water potential treatment young citrus tree
with grass cover 98
Figure 4-32. Distribution of evapotranspiration through the
daylight hours 99
Figure 4-33. Soil water potential-soil water content
relationship for Arredondo fine sand. .
vi ii
100


Figure 4-34. Hydraulic conductivity-soil water potential
relationship for Arredondo fine sand 101
Figure 4-35. Simulation results for the one-dimensional model
with field data for the 150 mm depth 104
Figure 4-36. Simulation results for the one-dimensional model
with field data for the 300 mm depth 105
Figure 4-37. Simulation results for the one-dimensional model
with field data for the 450 mm depth 106
Figure 4-38. Simulation results for the one-dimensional model
with field data for the 600 mm depth 107
Figure 4-39. Simulation results for the one-dimensional model
with field data for the 900 mm depth 108
Figure 4-40. Simulation results for the two-dimensional model
with field data for the 150 mm depth Ill
Figure 4-41. Simulation results for the two-dimensional model
with field data for the 300 mm depth 112
Figure 4-42. Simulation results for the two-dimensional model
with field data for the 450 mm depth 113
Figure 4-43. Simulation results for the two-dimensional model
with field data for the 600 mm depth 114
Figure 4-44. Simulation results for the two-dimensional model
with field data for the 900 mm depth 115
Figure 4-45. Soil water potentials for the three irrigation
treatments at the 300 mm depth 118
Figure 4-46. Soil water potentials for the three irrigation
treatments at the 900 mm depth 119
Figure 4-47. Soil water storage for the three irrigation
treatments 120
Figure 4-48. Cumulative irrigation for the three irrigation
treatments 121
Figure 4-49. Cumulative drainage from the soil profile for
the three irrigation treatments 122
IX


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
MEASUREMENT AND SIMULATION OF SOIL WATER STATUS
UNDER FIELD CONDITIONS
By
KENNETH COY STONE
May 1987
Chairman: Allen G. Smajstrla
Major Department: Agricultural Engineering
Irrigation of agricultural crops is one of the major uses of fresh
water in Florida. Supplemental irrigation is required in Florida
because the predominant sandy soils have very low water holding
capacities, up to 70 of the annual rainfall occurs during months when
many crops are not grown, and rainfall is not uniformly distributed.
Under these conditions the scheduling of irrigation becomes extremely
important. The timely application of irrigation water can result in
increased yields and greater profits, while untimely applications could
result in decreased yields and profits, and leaching of nutrients.
A low cost microcomputer-based data acquisition system for
continuous soil water potential measurements was developed. The system
consisted of tensiometer-mounted pressure transducers, an analog-to-
digital converter and a portable microcomputer. The data acquisition
system was evaluated under laboratory and field conditions. Excellent
x


agreement was obtained between soil water potentials read with the data
acquisition system and those read manually using mercury manometers.
Experiments were conducted utilizing the data acquisition system to
monitor soil water potentials of young citrus trees in field lysimeters
to determine model parameters and to evaluate model performance.
Evapotranspiration, soil water extraction rates and relative soil water
extractions were calculated from the soil water potential measurements.
Two numerical models were developed to study the movement and
extraction of soil water. A one-dimensional model was developed to
describe a soil profile which is uniformly irrigated such as one which
is sprinkler irrigated. The model was used to simulate soil water
movement and extraction from a young citrus tree with grass cover in a
field lysimeter. Simulated results were in excellent agreement with
field observations.
A two-dimensional model was developed to describe a nonuniformly
irrigated soil profile as in trickle irrigation. The model was used to
simulate the two-dimensional movement and extraction of soil water from
a young citrus tree with grass cover in a field lysimeter. The model
results were in excellent agreement with field observations.
xi


CHAPTER I
INTRODUCTION
Irrigation of agricultural crops is one of the major uses of fresh
water in Florida. It was reported (Harrison et al., 1983) that in 1981,
41 percent of the fresh water use in Florida was for irrigation of more
than 800,000 hectares of agricultural crops. Three reasons cited for
supplemental irrigation in Florida were 1) sandy soils have very low
water holding capacities, 2) up to 70 percent of the annual rainfall
occurs during months when many crops are not grown, and 3) rainfall is
not uniformly distributed even during months of high rainfall.
Because no control can be placed on when and where rainfall occurs,
researchers must focus their attention on managing the soil water
content in the plant root zone. One method of managing the water
content in soils is to apply small amounts of irrigation water at
frequent intervals. In the absence of rainfall, this provides
agricultural plants with adequate water for growth and also minimizes
losses from percolation below the root zone. With rainfall
interactions, irrigation scheduling is more difficult; the objective is
to minimize irrigation inputs (maximize effective rainfall) while
optimizing production returns. Because of the complexity of these
interactions, numerical models are useful tools to study them. Many
researchers have developed models to use in the study of soil water
management (Smajstrla, 1982; Zur and Jones, 1981). These models allow
many different irrigation strategies to be investigated without the cost
normally associated with field experiments. These models involve
1


2
analyses of infiltration, redistribution, evaporation and other factors
which affect the movement and uptake of soil water by plants.
The objective of this research was to develop a data collection
system and a numerical model which together will provide researchers
with information needed for developing and validating models of soil
water movement, crop water use and evapotranspiration. The
instrumentation system developed will record soil water potentials on a
real time basis in order to provide input data for model verification
and validation. A microcomputer based instrumentation system will allow
the computer to monitor inputs and make decisions based on the input
data. The microcomputer will have the ability to monitor and control
events in the field.
Two numerical models will be developed to study the movement and
extraction of soil water. A one-dimensional model will be developed to
describe the movement and extraction of soil water in a soil profile
which is uniformly irrigated. A two-dimensional model will be developed
to describe the movement and extraction of soil water from a soil
profile which is not uniformly irrigated such as trickle irrigation.
The specific objectives of this research were:
1. To develop and test instrumentation for the real-time
monitoring of soil water potential under agricultural crops.
2. To record the dynamics of soil water movement and water
extraction under irrigated agricultural crops.
3. To develop a numerical model to simulate the soil water
extraction patterns observed under agricultural crops as a
function of irrigation management practices and climatic
demands.


3
4. To demonstrate the use of the numerical model in evaluating
and recommending irrigation strategies.


CHAPTER II
REVIEW OF LITERATURE
Measurement of Soil Water Status
The measurement of soil water can be classified into two
catagories: (1) the amount of water held in a given amount of soil (soil
water content), and (2) the potential, or tension with which the water
is held by the soil (soil water potential). These properties are
related to each other (Figure 2-1) and describe the ability of a soil to
hold water available for plant growth.
Soil Water Content Measurement
Several methods are available to measure soil water content. The
gravimetric method is the standard method of determining the soil water
content. This method consist of physically collecting a soil sample
from the field, weighing it, and oven drying the sample to constant
weight at 105 C. The difference in weights before and after drying is
the amount of water removed from the sample. Water contents can be
calculated on a weight basis, or on a volume basis if the soil volume or
bulk density was measured when the sample was taken. An advantage of
the gravimetric method is that it requires no specialized equipment.
Disadvantages are that it is laborious and time consuming. Destructive
sampling is required and sampling may disturb a location sufficiently to
distort results. This method does not lend itself to automation.
Neutron scattering allows the nondestructive measurement of soil
water content. A neutron moisture meter may be used in the field to
4


SOIL WATER POTENTIAL (mm)
5
SOIL WATER CONTENT (mm/mm)
Figure 2-1. A typical soil water potential-soil water content curve.


6
rapidly and repeatably measure water contents in the same location and
depth of soil. An access tube must be installed in the soil to allow a
probe to be lowered to the desired soil depth for measurements.
The neutron scattering method operates on the principle of nuclear
thermalization. Fast neutrons are emitted from a radiation source
located on a probe which is lowered into the soil. These neutrons lose
energy as they collide with hydrogen atoms in the soil and are slowed or
thermalized. Thermalized neutrons are counted by a detector which is
also located on the probe. The number of slow neutrons is an indirect
measure of the quantity of water in the soil because of the hydrogen
atoms present in water. Because the neutron method counts only hydrogen
atoms, the method must be calibrated for each specific soil type and
location. This is especially true for soils with variable quantities of
organic (natter because the neutron meter will also count hydrogen atoms
in the organic matter and cause errors in the water content
measurements.
With the neutron method, a spherical volume of soil is sampled as
neutrons are emitted from the source. The radius of the sphere varies
with the soil water content from a small radius for wet soils to 20 or
30 cm for very dry soils. This measurement over a relatively large
volume is an advantage for homogenous soils with no discontinuities.
However when discontinuities such as water tables exist, their exact
locations cannot be detected accurately. Aribi et al. (1985)
investigated the accuracies of neutron meters when measuring water
contents near boundaries. They found that significant errors occurred
when readings were taken within the top 0.4 meters in an unsaturated
soil profile.


7
Another disadvantage of the neutron method is that the equipment
cost is high. The radioactive material contained in the neutron meter
is also a disadvantage because of the health hazards associated with
radiation. The automation of a neutron meter would be very expensive
and extremely complicated.
Gamma ray attenuation is a method which can be used to determine
the soil water content. This method determines soil water content by
measuring the amount of gamma radiation energy lost as a radiation beam
is directed through a soil. The method depends on the fact that gamma
rays lose part of their energy upon striking another substance, in this
case soil water. As the water content changes, the amount of
attenuation will also change.
Various types of gamma radiation instruments are available. One
type is intended for laboratory or stationary use. This instrument has
the source and detector parallel in line at a fixed distance apart, and
a soil column is placed between them. The source and detector are then
moved along the length of the soil column to determine the water content
distribution in the column. This would not be practical for field
application.
A second type of gamma radiation instrument has been developed for
field application. This instrument requires that two parallel access
tubes be installed into the soil. The source is placed in one access
tube and the detector in the other. Radiation is focused into a narrow
beam between the source and detector, and soil water content can be
measured in very thin layers of soil.
Another type of gamma radiation instrument consists of a stationary
detector located at the soil surface and a source located on a probe


8
which is lowered into the soil. The relationship between the source and
detector is known, and thus changes in density with depth can be
measured accurately with this instrument. Because the detector is
located at the soil surface, the instrument is limited to the top 30 cm
of the profile.
Gamma radiation instruments are expensive, and they can be
hazardous due to the radioactive source. This type of instrument would
be very difficult and expensive to automate.
Additional research has been conducted to utilize other soil
properties which would lend themselves to rapid methods of soil water
content determination. Fletcher (1939) conducted work on a dielectric
method of estimating soil water content. He used a resonance method to
determine the dielectric constant of the soil. This type of instrument
consists of an ocillator circuit and a tuned receiving circuit. The
instrument is placed in the soil and allowed to equilibrate. A variable
capacitor in the receiving circuit is then tuned to resonance, and the
capacitor reading is correlated with the soil water content. This
instrument must be calibrated for each soil in which it is used. Such a
device is not known to be commercially available.
Most of the methods discussed produce indirect measurements of soil
water content. Soil water content is calibrated to other factors such
as neutron thermalization, gamma ray attenuation or dielectric
properties. Only the gravametric method yields a direct measure of soil
water content.
Soil Water Potential Measurement
An alternative to the measurement of soil water content is the
measurement of soil water potential. A tensiometer measures the


9
potential or tension of water in the soil. The tensiometer consists of
a closed tube with a ceramic cup on the end which is inserted into the
soil and a vacuum gage or manometer to measure the water potential in
the tensiometer tube. The tube is filled with water, closed and allowed
to equilibrate with the soil water potential. As the soil dries, water
in the tensiometer is pulled through the ceramic cup. The soil water
potential which pulls water through the ceramic cup is registered on the
vacuum gage or manometer. This force is also the hydraulic potential
that a plant would need to exert to extract water from the soil.
Therefore, a tensiometer measures the energy status of water in the
soil. Tensiometers left in the soil for a long period of time tend to
follow the changes in the soil water potential.
The hydraulic resistance of the ceramic cup, the surrounding soil,
and the contact between the cup and soil cause tensiometer readings to
lag behind the actual tension changes in the soil. Lags are also caused
by the volume of water needed to be moved through the cup to register on
the measuring device. The useful range of tensiometers is from 0 to -80
kPa. Below -80 kPa air enters through the ceramic cup or the water
column in the tensiometer breaks, causing the tensiometer to fail. This
measurement limitation is not serious for irrigated crops on sandy soils
because most of the available water for plant use occurs between 0 and
-80 kPa.
Another method of measuring soil water potential is the thermal
conductance method. The rate of heat dissipation in a porous material
of low heat conductivity is sensitive to the water content in the porous
material. When in contact with a soil, the water potential in the
porous material tends to equilibrate with the soil water potential.


10
Phene et al. (1971) developed an instrument to measure soil water
potential by sensing heat dissipaton within a porous ceramic. The water
potential of the porous ceramic was measured by applying heat at a point
centered within the ceramic and measuring the temperature rise at that
point. Soil water potential measurements were obtained by taking two
temperature readings. The first one was taken before the heating cycle
and the second after the heating cycle. The difference between the two
temperature measurements was the change in temperature at the center of
the sensor due to the heat applied. The magnitude of the difference
varies depending on the water content of the porous block. Phene stated
that the sensor should measure the soil water potential regardless of
the soil in which it is embedded.
In experimental applications of the heat pulse device, the accuracy
of the sensor was 20 kPa over the range 0 to -200 kPa. In Florida
sandy soils such a wide range of variability would not be acceptable
because most of the available soil water is contained in the soil
between field capacity and -20 kPa. Calibration for individual soils
couTd improve the accuracy of the instrument. This instrument has the
capability of allowing automated data collection and is nondestructive.
The cost of the individual sensors and the associated data recording
devices is relatively high.
Soil psychrometers are instruments which can be used to measure
soil water potentials in the range of -1 to -15 bars. They operate by
cooling a thermocouple junction which is in equilibrium with the soil to
the point of water condensation on the junction, and then measuring the
junction temperature as water is allowed to evaporate. Thus the
temperature depression due to evaporation can be related to soil water


11
tension. Only small responses are obtained at potentials above -1 bar,
making this instrument unsuitable for irrigation scheduling on sandy
soils where irrigations would typically be scheduled at much greater
potentials.
Electrical resistance blocks can be correlated with soil water
potential. The blocks are placed in the soil and allowed to
equilibrate with the soil water. Resistance blocks usually contain a
pair of electrodes embedded in gypsum, nylon or fiberglass.
Measurements with resistance blocks are sensitive to the electrolytic
solutes in the fluid between the electrodes. Thus resistance blocks are
sensitive to variations in salinity of soil water and to temperature
changes. Temperatures must be measured and resistance readings
appropriately corrected.
Resistance blocks are not uniformly sensitive over the entire range
of soil water content. They are more accurate at low water contents
than at water contents near field capacity. Because of this limitation,
resistance blocks can be used to complement tensiometers to measure
soil water potentials below the -80 kPa range. They are relatively
inexpensive, and an automated data collection system may be built around
these instruments for measuring water potentials in drier soils.
Automation of Soil Water Potential Measurements
The instruments discussed may all be used to measure the status of
soil water. Most of the techniques relate a soil property to another
property which is measured by the instrument. Two of the methods allow
a direct measure of a soil property. The gravimeteric method gives a
direct measure of soil water content and tensiometers give direct
measurement of soil water potential. The gravimetric measurement method


12
requires destructive sampling and does not lend itself to automation.
Tensiometers may be automated by recording the changes in soil water
potential, and they do not require destructive sampling. Therefore, to
measure the status of soil water under a growing crop, an automated data
collection system using tensiometers to measure soil water potential was
chosen.
To determine the soil water potential, the water potential within
the tensiometer is measured. The tensiometer water potential is assumed
to be in equilibrium with the soil water potential. Early methods of
measuring tensiometer water potentials used mercury manometers. Later,
mechanical vacuum gauges were used. Both methods functioned well for
manual applications. Neither, however, was readily automated.
Recent interest in better understanding the dynamics of soil water
movement, and in the development of numerical models to simulate this
process, has resulted in the need for an automated system of recording
tensiometer readings on a continuous real time basis.
Van Bavel et al. (1968) used a camera to take periodic photographs
of a tensiometer manometer board. By analyzing the photographs they
were able to record changes in potentials. This procedure was, however,
laborious and did not provide continuous soil water potential records.
Enfield and Gillaspy (1980) developed a transducer which measured
the level of mercury in a mercury manometer tensiometer. The principle
of operation of their transducer was the same as a concentric capacitor.
The level of mercury in the manometer corresponds directly to the length
of a capacitor plate. A steel tube was used as the outer capacitor
plate around a column of mercury. The nylon tube which contained the
mercury column acted as the dielectric material. The capacitance of the


13
transducer was measured and converted to length of the column of
mercury. This instrument was found to be very sensitive to temperature
fluctuations. Further research is needed on this transducer to make it
suitable for applications in automated data collection systems.
Fitzsimmons and Young (1972) used a tensiometer-pressure transducer
system to study infiltration. Their system consisted of a pressure
transducer which was connected to many tensiometers by a system of fluid
switches. This system required substantial time lags after switching a
tensiometer to the transducer before an accurate reading could be made.
This was primarily due to changes in volume due to the elasticity of the
system. This resulted in a large scanning time in order to read all
tensiometers. Long and Hulk (1980) used a similar system.
Bottcher and Miller (1982) developed an automatic manometer
scanning device which read and recorded mercury levels in manometer-type
tensiometers. The system consisted of a computer-control 1ed chain-drive
mechanism which moved photocells and light sources up and down the
manometer tubes. When the scanner passed a mercury-water interface, a
change in voltage was detected by the computer. This system is
expensive, uses specially constructed rather than commercially-available
components, and is not readily expandable. It also requires the use of
a central manometer board with connecting tubing from the various
tensiometers. It has the advantage of easy verification of computer
readings by manual reading of the manometers.
Marthaler et al. (1983) used a pressure transducer to read
individual tensiometers. The upper end of the tensiometers was closed
off with septum stoppers to provide air-tight seals through which a
needle connected to a pressure transducer was inserted. The needle was


14
inserted into a pocket of entrapped air in the tensiometer, and the
pressure transducer output was obtained immediately. This method
introduced an error because the air pocket was compressible and was
affected by the addition of air at atmospheric pressure in the needle,
transducer, and connectors. This system was designed to minimize errors
by minimizing the volume of air introduced into the tensiometer. The
tensiometer air pockets were, however, temperature sensitive and
introduced diurnal time lags because of expansion and contraction due to
diurnal temperature changes. Also, to read several instruments, the
pressure transducer was required to be moved manually. Thus, this type
of system does not lend itself to automation.
Thomson et al. (1982) used individual pressure transducers on
tensiometers with all air purged from the system to monitor soil water
potentials. Thus, they were able to avoid lag times associated with air
pockets in the tensiometers. They were able to read pressure
transducers very rapidly by electronic rather than hydraulic switching.
A soil water potential monitoring system that used pressure
transducers as used by Thomson et al. (1982) would be able to record
data from a number of sensors very rapidly. A continuous data
acquisition system based on tensiometer-mounted pressure transducers
would provide data necessary for models of movement of water in soils,
crop water use, and evapotranspiration. The use of a microcomputer to
monitor the pressure transducers would allow the system to be programmed
to make decisions based on the input data (Zazueta et al., 1984).
Also, the cost of a dedicated data acquisiton system would be greater
than that of a microcomputer-based system because of current
microcomputer costs and availability.


15
Water Movement in Soils
The general equations governing unsaturated flow in porous media
are the continuity equation and Darcy's Law. Hi11 el (1980) presented a
combined flow equation which incorporates the continuity equation
36
_= v-q S (1)
3t
where q = flux density of water,
0 = volumetric water content,
v = differential operator,
t = time, and
S = a sink or source term,
with Darcy's equation for unsaturated flow
q = K(h) vH (2)
where K = hydraulic conductivity,
H = the total hydraulic head and defined as
H = h z, and
h = capillary pressure head.
The resulting combined flow equation for both steady and transient
flows is also known as Richards equation
39
= v*(K(h>v H ) S (3)
3t
For one-dimensional vertical flow, equation (3) becomes
36 3 3h 3K(h)
= ( K(h) ) - S(z,t) (4)
3t 3Z 3Z 3z
where z = the vertical dimension.


16
The sink term is used to represent the loss or gain of water from
the soil by root extraction or by application of irrigation water from a
point source.
For implicit solutions, equation (4) must be written in terms of
only one variable, soil water content or potential. By introducing the
specific water capacity, C, defined as
d e
C = (5)
d h
and using the chain rule of calculus, equation (4) may be written in
terms of the soil water potential as
9h 3 3 h 3 K(h)
C = (K(h) ) - S(z,t) (6)
3 t 3 Z 3 Z 3 Z
For two-dimensional flow, equation (3) becomes
3 h 3 ah 3 3 h 3K(h)
C = (K(h)) + (K(h)) S(x,z,t) (7)
at 3 X 3 X 3 Z 3 Z 3 Z
Th with radial symmetry may be written as
3h 13 3 h a ah 3 K(h)
C (r K(h)) + (K(h)) - S(x,z,t) (8)
atrar 3r 3z 3Z sz
The two-dimensional flow equation in radial coordinates may be used to
describe water movement and extraction of soil water from nonuniform
water applications such as trickle irrigation.
Due to the variable nature of K(h), equations (6), (7) and (8) are
highly nonlinear, and analytical solutions are extremely complex or


17
impossible to obtain. The nonlinearity of equations (6) and (7), and
the typical variable boundary conditions, have led to the use of
numerical methods to solve practical problems of soil-plant-water
relationships, such as irrigation management for agricultural crops.
For one-dimensional flow, equation (6) has been successfully solved
using explicit finite difference methods by many researchers. Hanks and
Bowers (1962) developed a numerical model for infiltration into layered
soils. They solved the Richards equation for the hydraulic potential
using implicit finite difference equations with a Crank-Nicholson
technique which averages the finite differences over two successive time
steps. Rubin and Steinhardt (1963) developed a numerical model to study
the soil water relationships during rainfall infiltration. They used a
Crank-Nicholson technique to solve Richards equation for the water
content. Rubin (1967) developed a numerical model which analyzed the
hysteresis effects on post-infiltration redistribution of soil water.
Haverkamp et al. (1977) reviewed six numerical models of one-
dimen s torra. 1 infiltration. Each model employed different discretization
techniques for the nonlinear infiltration equation. The models reviewed
were solved using both the water content based equation and the water
potential based equation. They found that implicit models which solved
the potential based infiltration equation had the widest range of
applicability for predicting water movement in soil, either saturated or
nonsaturated.
Clark and Smajstrla (1983) developed an implicit model of soil
water flow to study the distribution of water in soils as influenced by
various irrigation depths and intensities. The model simulated water
application rates from center-pivot irrigation systems with intensities


18
of application typical of low and high pressure irrigation systems.
Their model also simulated post-infiltration redistribution.
Rubin (1968) developed a two-dimensional numerical model of
transient water flow in unsaturated and partly unsaturated soils. He
utilized alternating-direction implicit (ADI) finite difference methods.
He studied horizontal infiltration and ditch drainage with the numerical
model. Hornberger et al. (1969) developed a two-dimensional model
to study water movement in a composite soil moisture groundwater system.
They modeled the two-dimensional response of falling water tables. They
considered both saturated and unsaturated zones in their model. The
solution method used was a Gauss-Sidel iterative technique.
A two-dimensional model to simulate the drawdown in a pumped
unconfined aquifer was developed by Taylor and Luthin (1969). The model
gave simultaneous solutions in both the saturated and unsaturated zones.
They used a Gauss-Sidel iterative method to solve the flow equations.
Amerman (1969) developed two-dimensional numerical models to
simulate steady state saturated flow, drainage and furrow irrigation.
He used ADI methods to solve both the steady state saturated flow model
and the furrow irrigation model. He also used an explicit method to
solve the drainage model.
A study of the sensivity of the grid spacing for finite difference
models was reported on by Amerman and Monke (1977). Two finite
difference models of two-dimensional infiltration were analyzed. They
solved the two-dimensional flow equations with successive overrelaxation
(SOR) and alternating direction implicit (ADI) methods. They found that
smaller grid sizes were needed in regions where the hydraulic gradients


19
changed rapidly. Considerable computational savings without appreciable
loss of accuracy was achieved using irregular grid sizes.
Perrens and Watson (1977) developed a two-dimensional numerical
model of water movement to analyze infiltration and redistribution.
They used an iterative alternating direction implicit technique to solve
the flow equation. Two soil types, a sand and a sandy loam, were
simulated. Nonuniform surface fluxes were applied along the horizontal
soil surface in a step type distribution pattern. They also
incorporated hysteresis of soil hydraulic characteristics into the model
to be used in the redistribution phase of the simulations.
Researchers have also utilized two-dimensional models to study soil
water movement from trickle irrigation systems. Brandt et al. (1971)
solved the flow equation in two dimensions to analyze infiltration
from a trickle source. They developed a plane flow model in cartesian
coordinates to analyze infiltration from a line source of closely spaced
emitters with overlapping wetting patterns. They also developed a
cylinderical flow model to analyze infiltration from a single emitter
when its wetting pattern is not affected by other emitters. Both models
were solved using noniterative ADI finite difference procedures with
Newton's iterative method. The results were compared to an analytical
solution of steady infiltration and a one-dimensional solution with good
results.
Armstrong and Wilson (1983) developed a model for moisture
distribution under a trickle source. They utilized the Continuous
System Modeling Program (CMSP) to simulate the soil moisture movement.
The model calculated the net flow rates into each grid. It then
calculated the change in water content by dividing the net flow rate


20
by the volume of soil in each grid and then multiplying by the time
step. Finally, the model calculated the new water contents from the
previous water contents plus the calculated water content changes.
Model results compared favorably with field measurements.
Zazueta et al. (1985) developed a simple explicit numerical model
for the prediction of soil water movement from trickle sources. The
model was based on the mass balance equation and an integrated form of
Darcy's law. The model produced good agreement with other results
obtained with more complicated numerical methods and analytical
solutions.
Soil Water Extraction
Water uptake by plant roots has been investigated by many
researchers. Among the first researchers to attempt to describe plant
water relations were Gradmann (1928) and van den Honert (1948). Two
approaches to modeling water extraction have been utilized to describe
the water extraction by plant roots. The first, called the microscopic
approach, describes water movement to individual roots. The second
approach, called the macroscopic water extraction model, describes water
uptake by the whole root zone, and the flow to individual roots is
ignored.
Gardner (1960) developed a microscopic water uptake model. He
described the root as an infinitely long cylinder of uniform radius and
water-absorbing properties, assuming that water moves in the radial
direction only. The flow equation for such a system is
8 e
8 t
1 3 80
( r D )
r 8 r 8 r
r 8 r
(9)


21
where a = the volumetric water content,
D = the diffusivity,
t = the time and
r = the radial distance from the axis of the root.
He then obtained a solution at the boundary between the plant root
and the soil in order to maintain a constant rate of water movement to
the plant, subject to the following boundary conditions:
6 = e0, h = hQ, at t = tQ, and
dh de
2 ir r^ k() = 2 tt r1 D () = q at r = rj
(10)
dr dr
where k = the hydraulic conductivity of the soil,
r^= the radius of the root, and
q = the rate of water uptake by the root.
The solution of (9) subject to the initial and boundary conditions
in (10) is
q 4Dt
h h = dh = ( In g ) (11)
4 k r
where g = 0.57722 is Euler's constant. The diffusivity and conductivity
were assumed to be constant, this assumption was justified because D, r
and t are all in the logarithmic term, and dh is thus not very sensitive
to these variables.
Gardner (1960) also noted that the solution will behave as though
infinite for only very short times or for very low values of hydraulic
conductivity. Exact solutions of the problem for finite systems would
require taking the dependence of D and k upon the soil water potential
into account. Gardner also compared equation (11) with the steady-state
solution for flow in a hollow cylinder:


22
q tr
h hQ = dr = ( In ) (12)
4 irk a
where hQ = the potential at the outer radius of the hollow cylinder,
b = one half the distance between neighboring roots, and
h = the potential at the inner radius a.
If b = 2/Dt, then equation (12) is identical to equation (11)
except for the g term which is small compared to the logarithmic term
and may be ignored.
Gardner and Ehlig (1962) presented a macroscopic water uptake
equation in which the rate of water uptake is proportional to the
potential energy gradient and inversely proportional to the impedance to
water movement within the soil and the plant. The potential energy was
expressed as the difference between the diffusion pressure deficit and
the soil suction. The impedance was expressed as the sum of plant
impedance and soil impedance. They analyzed greenhouse experiments and
obtained results consistent with the uptake equation.
Molz and Remson (1970) developed a mathematical model describing
moisture removal from soil by plant roots. The model described one
dimensional water movement and extraction. The model used a macroscopic
water extraction term to describe moisture removal by plants. Their
formula for the extraction term was given as
R(z) D(e)
S(z,t) = T ( ) (13)
i R(z) D(e) dz
where D(e) = diffusivity,
R(z) = the effective root density,
T = the transpiration rate per unit area and
v = the root depth.


23
The model gave results that compared reasonably with experimental
results. The numerical solution of the Richards equation with the
extraction function was obttained using the Douglas-Jones predictor-
corrector method.
Nimah and Hanks (1973) developed a numerical model to predict water
content profiles, evapotranspiration, water flow in the soil, root
extraction, and root water potential under field conditions. Their
extraction term had the form
(Hroot + (1.05 z) -h(z,t)-s(z,t)) RDF(z) K(O)
A (z, t) = (14)
AX A Z
where A(z,t) = the soil water extraction rate,
Hroot = the effective water potential in the root at the soil
surface,
h(z,t) = the soil matric potential,
s(z,t) = the osmotic potential,
RDF(z) = proportion of total active roots in depth increment DZ,
K(e) = the hydraulic conductivity and
Ax = the distance between the plant roots at the point
where h(z,t) and s(z,t) are measured.
Tollner and Molz (1983) developed a macrocopic water uptake model.
The extraction function assumed that water uptake rate per unit volume
of soil is proportional to the product of contact length per unit soil
volume, root permeability per unit length and water potential difference
between soil and root xylem potential. A factor which accounts for
reduced root-soil-water contact as water is removed was included in the


. 24
extraction function. Their model predictions were comparable with
results of a greenhouse experiment.
Slack et al. (1977} developed a mathematical model of water
extraction by plant roots as a function of leaf and soil water
potentials. The model used a microscopic model to calculate water
A.
uptake which was input into a macroscopic model of water movement and
water uptake. Water movement was described by a two-dimensional radial
flow model. The model was used to estimate transpiration from corn
grown in a controlled environment under soil drying conditions. The
model predicted daily transpiration adequately for the period modeled.
Feddes et al. (1978) developed an implicit finite difference model
to describe water flow and extraction in a non-homogeneous soil root
system under the influence of groundwater. Their extraction term
assumed the extraction rate to be maximum when the soil water potential
was above a set limit. When the soil water potential fell below this
limit, the water uptake, S(h), was decreased linearly with the soil
water potential
( h h3 )
S(h) = Smax (15)
( h2 h3 )
where h2 is the set limit below which the water uptake decreases
linearly to h3. Below h3 it was assumed that no water is extracted,
then h3 may be assumed to be the wilting point. The maximum possible
transpiration rate divided by the effective rooting depth is Smax. The
model yielded satisfactory results in predicting both cumulative
transpiration and distribution of soil moisture content with depth.


25
Zur and Jones (1981) developed a model for studying the integrated
effects of soil, crop and climatic conditions on the expansive growth,
photosynthesis and water use of agricultural crops. The model utilized
Penman's equation modified by Monteith to calculate the water vapor
transport from the plant to the atmosphere. It utilized a
macroscopic soil water extraction term to calculate water uptake from
the soil. The model assumed the water uptake from the soil was the
difference between the total water potential at the root surface and the
soil water potential divided by the sum of the resistance to radial
water flow inside the roots and resistance of water flow in the soil.
The model treated the water relations in plants in considerable detail.
It was successfully tested on soybeans. However the model did not take
into consideration infiltration, drainage, evaporation from the soil
surface or upward movement of water in the soil.


CHAPTER III
MATERIALS AND METHODS
Equipment
The data acquisition system developed in this research was based on
tensiometer-mounted pressure transducers. Pressure transducers used
were Micro Switch model 141PC15D. These transducers measured pressures
from 0 to -100 kPa. They produced an analog electrical output from 1 to
6 volts which was proportional to the vacuum exerted on a membrane in
the transducers. The transducers operated much like strain gages. One
side of the membrane was open to the atmosphere and the other side was
in contact with water in the tensiometer. As pressure on the water
changed, the membrane was deformed and the voltage across the membrane
changed.
The pressure transducers required an 8-volt direct current (VDC)
power supply at 8 mA each. Because a muItiple-transducer system was
developed, a separate regulated power supply was used rather than the
microcomputer power supply. This assured adequate power for the
transducers.
The pressure transducers were temperature-compensated. Outputs
varied less than 1% of full scale output in the range of 5 C to 45 C.
The output voltages from the pressure transducers were interfaced
to a microcomputer through an analog-to-digital (A/D) board. The A/D
board was a Mico World MW-311 general purpose Input/Output Board. This
device had 8-bit resolution and measured input voltages from 0 to 5
volts. It had 16-channel multiplexing capability.
26


27
The microcomputer used was a Commodore 64. It was selected for its
programmability, its availability at local electronics supply companies,
and its low cost. An overall goal in component selection was to be able
to replace any component individually with as little down-time as
possible, and thus to assure the reliability of the system.
The microcomputer output was to a video screen, a printer, and a
cassette recorder. The cassette recorder provided storage of data for
later computations and easy transfer to other computer systems. The
printer provided a hard copy backup of data. The video screen provided
immediate interaction with the computer for programming and monitoring.
The microcomputer was programmed in the BASIC computer language.
It had approximately 32 kilobytes of programmable memory (RAM) when
using the BASIC computer language. This was sufficient to store the
calibration equations and calculated outputs from the transducer inputs.
The microcomputer also had an internal 24-hour clock which was required
for real-time data acquisition. Data acquisition system components and
approximate costs are given in Table 3-1.
Instrumentation Test Procedure
Pressure transducers were individually calibrated using a regulated
vacuum source and a mercury manometer. Measurements of transducer
voltages and mercury column heights were made at decreasing pressure
increments of approximately -10 kPa from 0 to approximately -80 kPa. To
determine whether the transducers displayed hysteretic behavior,
calibrations were continued at increasing pressure increments from -80
kPa to 0 kPa.


23
Table 3-1. Microcomputer-based data acquisition system components
and approximate costs.
Component Cost($)
Microcomputer 100
Printer 100
Cassette Recorder 60
Pressure Transducer 68
Tensiometer 40
Analog-to-Digital and
Multiplexing Circuit 200
Transducer Power Supply 50


29
A/D boards were calibrated while interfaced with the microcomputer.
Known voltages were applied to the boards and the digital outputs read
by the computer were recorded.
Calibrated pressure transducers were interfaced to the tensiometers
by drilling small holes into the walls of the tensiometers. Access
ports on the transducers were inserted into the tensiometers and the
connections were sealed with silicone sealant. The pressure transducer-
equipped tensiometers were connected to the computer using 3-lead 22
gauge electrical hookup wire.
Two tests of the functioning of the microcomputer-based tensiometer
pressure transducer system were conducted. The first was conducted in
the laboratory. Three mercury manometer tensiometers were equipped with
pressure transducers and interfaced with the microcomputer. The
tensiometers were serviced to remove air from the system. The ceramic
cups were then exposed to the atmosphere and allowed to dry for several
hours. Microcomputer readings and mercury manometer readings were taken
for comparison at random intervals during drying. Data collection
continued throughout the tensiometer range. Three replications were
taken.
The second test of the tensiometer-pressure transducer system was
conducted under field conditions. Three instrumented mercury manometer
tensiometers were installed in a field lysimeter system with grass
cover. The microcomputer was located in the field laboratory office 25
m from the sensors. Tensiometers were serviced and allowed to
equilibrate with the soil water potential. The microcomputer was
programmed to read the pressure transducers hourly. Mercury manometers
were read manually at random intervals, typically 2 or 3 times per day


30
during the field evaluation. Manual mercury manometer readings were
compared with the microcomputer readings to assess the accuracy of the
instrumentation under field conditions. Instrumentation was evaluated
for 3 1-week periods. Tensiometers were serviced to remove entrapped
air between 1-week analysis periods.
Field Data Collection
A field lysimeter system was used to monitor soil water status and
water uptake from young citrus trees. The field lysimeter system was
located at the Irrigation Research and Education Park in Gainesville,
FI. Layout of the lysimeters is shown in Figure 3-1. The lysimeters
were cylindrical steel tanks with one end open. Tanks were installed
with the open end upward, exposing a circular surface production area of
2 m2. Soil profiles were 1.8 m deep. The soil in the lysimeters was
Arredondo Fine Sand, (hyperthermic, coated, Typic Quartzipsamment),
which was hand packed to approximate the physical characteristics of an
undisturbed soil profile. Drainage of excess water from the bottom of
each lysimeter was accomplished using either a porous stone or porous
ceramic cup located in the bottom of the tank as shown in Figure 3-2. A
vacuum pump removed the water from the lysimeters and the water was
trapped in PVC cylinders. The PVC cylinders were used to measure the
drainage from each lysimeter. Automated rainfall shelters were used to
cover the lysimeters in the event of rainfall. Additional details of
the construction and operation of the lysimeters were given by Smajstrla
(1985).
Data collection was conducted in two phases. In 1985 two
lysimeters were instrumented, and data were collected as described
earlier. Each lysimeter contained a young citrus tree approximately 3


31
Figure 3-1. Layout of lysimeter system.


32
TO VACUUM
SOURCE
Figure 3-2. Details of individual lysimeter soil water status
monitoring system.


33
years old. Both lysimeters were irrigated when the water potential in
the lysimeter dropped below -20 kPa. The -20 kPa water potential
corresponded to approximately a 44% water depletion from field capacity.
One lysimeter had a grass cover crop and the second had bare soil.
These two cover practices were used to simulate different cultural
practice effects upon the young citrus in a related experiment.
Tensiometers were installed at radii of 100, 300, 500 and 700 mm
from the center of the lysimeter, and at depths of 150, 300, 450, 600
and 900 mm as shown in Figure 3-3. Data were collected with the
microcomputer based data acquisition system at 10 minute intervals and
averaged over hourly periods.
In 1986 two lysimeters with young citrus trees were instrumented
with the same water and cover treatments as in 1985. Tensiometers were
installed in the lysimeters at radii of 130, 380 and 630 mm and at
depths of 150, 300, 450, 600, 900 and 1020 mm as shown in Figure 3-4.
In addition to the two lysimeters with young citrus trees, two
lysimeters with only grass cover were instrumented. The water potential
treatments for the two grassed lysimeters were -20 kPa and -40 kPa. The
-40 kPa water potential corresponded to approximately a 50% water
depletion from field capacity.
Model Development
A computer program describing the infiltration, redistribution and
extraction of soil water was written in the FORTRAN computer language.
A Prime 550 computer was used for all computations.
Several assumptions were employed in the development and use of the
numerical models to ease the complexities associated with mathematically
describing water movement in unsaturated porous media. These


Depth (mm)
Radius (mm)
0 100 300 600 700 800
150
300
450
600
900
1800
location of tensiometers
Figure 3-3. Location of tensiometers in 1985 field experiment
for young citrus trees in grassed and bare soil
lysimeters.


Depth (mm)
Radius (mm)
0 130 380 630 800
150
300
450
600
900
1020
1800
location of tensiometers
Figure 3-4. Location of tensiometers in 1986 field experiment
for young citrus trees in grassed and bare soil
lysimeters.


36
assumptions were: (1) The physical properties of the soil are
homogeneous and isotropic; (2) The physical and chemical properties of
the soil are constant in time; and (3) The hydraulic conductivity and
soil pressure potential are single-valued functions of water content.
One-Dimensional Model
Equation (6) was used to describe the one-dimensional movement and
extraction of water. Finite difference methods were used to solve the
water potential based flow equation. The model used an implicit finite
difference technique with explicit linearization of the soil parameters.
The finite difference form of equation (3) was written as
k
k+1 k
(h h )
k i i
k+1 k+1 k k+1 k+1
K (h h ) K (h-h)
i+1/2 i+1 i i-1/2 i i-1
At
Az
where
i-1/2
AZ
K
i-1/2
K
i+1/2
+ K
i-1
(16)
(17)
and
i+1/2
i+1
(18)
The subscript i refers to distance and the superscript k refers to time.
To apply this equation to soil profiles, the soil was divided into a


37
finite number of layers or grids as shown in Figure 3-5. Equation (16)
was rearranged so that unknowns were on the left side of the equation
and the knowns were on the right side such that
At K At
i-1/2 k+1
) h + ( 1 +
2k i-1 2
AZ C A Z
At
K
i-1/2
i+1/2
1
2 k
AZ C
i
) h
k+1
k k k
At K At K K At
i+1/2 k+1 k i-1/2 i+1/2
- ( ) h = h + ( ) S
2 k i+1 i k k i
az C az C C
i i i
(19)
Equation (19) was written for each of the interior grids in the soil
profile. The top and bottom grids of the system required equation (19)
to be modified so that the equation would describe the boundary
conditions. The system of equations which are formed by applying
equation (19) to each grid produces a tridiagonal matrix which is
implicit in terms of h, water potential. The tridiagonal matrix was
solved using Gauss elimination. From explicit linearization, the
coefficients of the unknown h terms were known at each time step. Thus
equation (19) was reduced to
k+1 k+1 k+1
Ah + B h + C h
i i-1 i i i i+1
= D
(20)
where
k
At K
i-1/2
A = ( )
i 2 k
A z C
(21)
1


GRIDS
1
i-1
i
i+1
Soil surface
K
1
h. 6. C. K. .
i-l l-l i-l l-l
h. 9. C. K.
i ill
h.., 9... C. .
l+l l+l l+l i+l
9
n
Figure 3-5. Schematic diagram of the finite-difference grid
system for the one-dimensional model of water
movement and extraction.


39
At K
1+1/2
C =
i
- (-
2 k
AZ C
i
-)
(22)
B = 1 A C
i i i
k k
At K K
k i-1/2 i+1/2
D = h + (
i i k
AZ C
(23)
At
) S
k i
C
i
(24)
Boundary Conditions. The boundary condition for the surface grid
was variable. A flux boundary condition was used to simulate
infiltration during irrigation, and a no flux boundary condition was
used for redistribution. Evaporation from the soil surface was included
in the water extraction term. Equation (20) was modified for the
surface boundary condition such that
k+1
k+1
B h
1 1
+ C h
1 2
1
where
and
B = 1
(25)
(26)
At
k
D = h + (
1 1
Az
Qs
1+1/2
At
- s
k 1
*1
(27)


40
The lower boundary was also a flux boundary and was written as
k+1 k+1
Ah + B h = D (28)
n n-1 n n n
B = 1 A (29)
n n
k
At K Qb At
k n-1/2
h + ( ) S (30)
n k k n
Az C C
n n
The variables Qs and Qb were, respectively, the surface and bottom
fluxes imposed on the grid system. The bottom flux was set to either a
no flow or a gravity flow boundary condition.
Surface infiltration. The infiltration rate during irrigation was
set equal to the water application rate until the irrigation event was
completed. The total depth of water infiltrated at any time was
calculated as the summation of the incremental infiltration volumes at
all previous time steps.
Time steps. The time steps for implicit numerical techniques can
generally be much larger than those for explicit numerical techniques,
and still maintain stability. Haverkamp et al. (1977) stated that
stability conditions must be determined by trial and error because they
depend on the degree of nonlinearity of the equations.
The time step was estimated by a method given by Feddes et al.
(1978) and used by Clark (1982):
where
and
D =
n


41
X AZ
At < (31)
| Qmax |
where Qmax was the maximum net flux occuring across any grid boundary
and x is a factor where 0.015 accommodate the rapid movement of water during infiltration and early
stages of redistribution. The value of x may be assumed to be the
maximum permissible change in water content for any grid in the soil
profile.
Updating of soil parameters. The updating of the soil parameters
was achieved by use of a tabulated search to determine the corresponding
values of hydraulic conductivity, specific water capacity and water
content for the new values of soil water potential at each time step. A
logarithmic interpolation method was used to reduce computer time. The
table was set up with linearly increasing multiples of the logarithmic
values of soil water potentials.
Mass balance. A mass balance was calculated at each time step to
check the stability of the simulation model. The method used the
initial volume of water in the profile plus the infiltration and minus
the extraction by the plant. Evaporation from the soil surface was
included in the water extraction term as extraction from the surface
grids.
Because the grid system approximates a continuous system with dis
crete points, some error will result from the numerical calculations of
the simulation model. The error decreases as the grid size decreased.
Two-Dimensional Model
Water movement and extraction under young citrus trees was
simulated using a two-dimensional soil water extraction model. A radial


42
flow model may be used to describe water movement and extraction in a
soil profile which is nonuniformly irrigated as in water flow from
trickle emitters and spray jets. The radial flow model may also
describe the water extraction under widely spaced crops such as citrus.
With the assumption of radial symmetry, equation (8) was solved in
the vertical and radial directions. This equation is a second order,
nonlinear parabolic partial differential equation which expresses the
water potential distribution as a function of time and space
coordinates. No known analytical solution of equation (8) exist. Thus
it was solved numerically using finite difference equations. The model
developed in this research used an implicit finite difference technique
with explicit linearization of the soil parameters. Two solution
methods were used to solve the finite difference equations for the two-
dimensional model. It was solved implicitly using a Gauss elimination
method. Because of the time required to solve a large system of
equations by Gauss elimination, an alternation direction implicit (ADI)
method was also utilized to solve equation (8).
The finite difference equation for the solution of equation (8)
utilizing the Gauss elimination method was
k+1 k k k+1 k+1 k k+1 k+1
(h-h) K (h-h)-K (h-h)
k i,j i,j 1+1/2J i+l,j i,j -1/2J i,j i-1, j
C =
i,j 2
At AZ
k k+1 k+1 k k+1 k+1
r K (h -h)-r K (h-h)
i,j+l/2 i,j+1/2 i,j+l i,j i.j-1/2 ij-1/2 i,j i,j-1
+
2
Ar
r


43
k k
K K
1-1/2, j 1+1/2,j k
(32)
To apply this equation to the soil profiles, the soil was divided
into a finite number grids as shown in Figure 3-6. Equation (32) was
arranged such that the unknowns were on the left side of the equation
and the knowns were on the right side as follows
k
At K
1-1/2,j
- ( )
2 k
Az C '
i,j
At r
k+1
h (
1-1J 2
Ar
k
K
i.j-1/2 1-1/2, j
)
k
r C
1J 1J
k+1
h
i.j-1
k k
At ( K + K )
1-1/2,j i+1/2, j
+ ( 1 + +
2 k
AZ C
ij
k k
At ( r K + r K )
iJ+1/2 i.j+1/2 i.j-1/2 i.j-1/2
2 k
Ar r C
i,j i,j
) h
i.J
- (-
k k
At K At r K
i+1/2,j k+1 i.j+1/2 i+1/2,j
) h ( )
2k 1+1,j 2 k
az C Ar r C
i,J
i.J
i.J
k+1
h
1.J+1


44
Figure 3-
. Schematic diagram of the finite-difference grid system
for the two-dimensional model of water movement and
extraction.


k
h +
i.j
At
(
At
(33)
AZ
k k
K k
i-1/2 ,j i+1/2, j
)
k
C
i.j
k
S
C
i.j
Equation (33) was reduced to
' k+1 k+1 k+1 k+1
Ah + B h + C h + D h
i,j 1-1j i,j 1,j-l i,j 1,j i.j i+l.j
where
k+1
+ E h = F
I.j 1,j+l 1,j
k
At K
i-1/2,j
A = ( )
I.j 2 k
AZ C
i.j
k
At r K
i,j-1/2 i-1/2,j
B = ( )
i.j 2 k
Ar r C
i.j i.j
k
At K
i+1/2,j
D = ( )
i.j 2 k
AZ C
i.j
k
A t r K
i,j+1/2 i+1/2, j
E - ( )
i.j 2 k
Ar r C
i.j i.j
(34)
(35)
(36)
(37)
(38)


46
C = 1 A B -D -E (39)
i,j i,j 1,j i,j i,j
k k
At K K At
k i-1/2, j i+1/2, j k
F = h + ( ) S (40)
i,j i,j k k i,j
Az C C
1.j i.J
Boundary Conditions. The boundary condition for the surface grids
was a flux boundary condition which accounted for irrigation and
rainfall at the soil surface. At times when no irrigation occured the
surface boundary was a no flux boundary. Evaporation from the soil
surface was included in the water extraction term. The modification of
equation (34) to describe the surface boundary condition was
k+1 k+1 k+1 k+1
B h +C h + D h + E h =F (41)
l.j 1J-1 l,j l.j l.j 2,j l,j 1J+1 1J
where
C = 1 B D E (42)
l.j l.j l.J l.j
A t
k
F = h +
l.j l.j
A Z
Qs
j
k
- K
1+1/2,j
k
C
) -
A t
k
C
(43)
l.j l.j
The lower boundary condition was also represented as a flux
boundary. The lower boundary in this research was an impermeable
boundary which represented the bottom of the lysimeter. The
modification of equation (34) to describe the lower no flux boundary was


47
k+1
k+1
k+1
k+1
A h
+ B h
+
C h + E
h = F
n,j n-lj
n,j n,j-
1
n,j n,j n,j
n,j+l n,j
where
C = 1
-
A B E
n,j
n,j n,j n
k
At
K
- Qb
At
k
n-
1/2, j j
k
F = h
+ (
) -
S
n,j n
J
k
k n,j
AZ
C
C
n,j
n,j
The boundary conditions
for the radial
boundaries were
both
boundaries.
The no flux boundary condition
for the outer radius
represented the outer wall of the lysimeter
and was written
as
k+1
k+1
k+1
k+1
A h
+ B h
+
C h + D
h = F
i,m i-l,m
i,m i,m-
1
i ,m i ,m i ,m
i+l,m i,m
where
C
1
- A B -
D
i ,m
i,m i ,m
i ,m
k
k
At
K
- K
A t
k
i-
1/2,m i+l/2,m
k
F = h
+ f
1
8
(44)
(45)
(46)
(47)
(48)
i,m i,m k k i,m
Az C C
(49)
n,m n,m
The boundary condition for the inner radiirs was also represented as
a no flux boundary. For the inner radius r(i,j) = 0, and its inclusion
into equation (8) would result in a division by zero. Therefore the
second term in equation (8) was rewritten as
13 3h K3h 3 a h
(r K(h) _) = + ( K(h) _)
r3r 3 r r a r 9r a r
r 9 r
(50)


48
dh
a no flux boundary at r = 0 implies that =0. Then if
dr
dh 1 dh
= 0, the lim = 0. We rewrite equation (8) as
dr r dr
ah 3 3 h 3 a h 3 K(h)
C = ( K(h) ) + ( K(h) ) S(z,r,t) (51)
at 3 z 3 Z 3 r 3 r 3 z
The finite difference equation for equation (51) at the inner
radius may be written as
k k k
At K / At ( K + K )
i-1/2,1 k+1 i-1/2,1 i+1/2,1
- ( )h + ( 1 +
2 k 1-1,1 2 k
AZ C
1,1
A Z C
1,1
k k
At K A t K
i, 1+1/2 i+1/2,1 k+1
+ ) h ( ) h
2 k i ,1 2 k 1+1,1
A r C
A z C
1,1
1,1
At K
.i,1+1/2 k+1
- ( ) h
2 k i ,2
A r C
1,1
= h
1,1
k k
At K K
i-1/2,1 i+1/2,1
+ ( ) -
At
AZ
1,1
k i ,1
u
or as
k+1 k+1 k+1 k+1
Ah + C h + D h + E h = F
i,l i-1,1 i ,1 i,l i,l 1+1,1 i ,1 i ,2 i, 1
(52)
(53)


49
where
k
At K
i-1/2,1
A = ( ) (54)
i,l 2 k
Az C
1.1
k
At K
i+1/2,1
D = ( ) (55)
i ,1 2 k
A z C
U
k
At K
i,1+1/2
E = ( ) (56)
i,l 2 k
A r C
1,1
C = 1 A D E (57)
1,1 1.1 1,1 1,1
A t
k
F = h + _
1,1 1,1
A Z
k k
K k
i-1/2,1 i+1/2,1
)
k
C
1,1
At
k
S
k i ,1
C
1,1
(53)
Updating of the soil parameters, surface infiltration, mass balance
and time steps were implimented as described for the one-dimensional
soil water flow model.
The system of equations for the two dimensional model produces a
banded matrix. The matrix is a five banded matrix with the form of


50
Reddel and Simada (1970) utilized a Gauss elimination method for
the solution of a two-dimensional groundwater model. They used an
algorithm developed by Thurnau (1963) which operates only on the banded
part of the solution matrix. Computer storage is not required for the
matrix elements above or below the band. A minimum band width is
desirable and an appropriate choice of the grid numbering pattern can
reduce the total width of the band.
In this model of two-dimensional soil water movement and
extraction, a subroutine was written which used the BANDSOLVE algorithm
developed by Thurnau (1963) to solve the system of equations. A
subroutine was also written which solves equation (8) using an
alternating direction implicit method. The finite difference equations
for the ADI solution are presented in Appendix A.
Soil Water Extraction
A macroscopic soil water extraction term was used in this research.
The model used was based on actual field measurements reported in the
literature. Denmead and Shaw (1962) conducted experiments which
compared actual ET to potential ET of corn as a function of the
available soil water. Their study showed that under high potential ET
demands, the actual ET was considerably less than the potential rate
even though the available soil water was considered adequate. They also
observed that under low potential ET demands, the actual ET was equal to


51
the potential ET down to very low soil water contents. Ritchie (1973)
observed similar results in separate experiments.
Saxton et al. (1974) used these observed relationships in modeling
soil water movement and extraction under a corn and under a grass crop.
Smajstrla (1982) used these relationships to model the soil water status
under a grass cover crop in Florida.
The soil water extraction function used in this research was
modeled after that used by Smajstrla (1982). The soil water extraction
rate was calculated as a function of the actual evapotranspiration rate
and the current soil water status. The water extraction term was
defined as
S^ = ET RDFt- R,
(ET / R.j )
(59)
where S^ = the soil water extraction rate per soil zone,
ET = the actual evapotranspiration rate,
RDF.¡ = the relative water extraction per soil zone at field
capacity, and
R,- = the relative available soil water per soil zone defined as
( 9 9wp )
( fc % )
(60)
where 6
e
= the soil water content of the soil zone,
Wp = the soil water content at wilting point, and
0fc = the soil water content at field capacity.
The relative water extraction per soil zone (RDF) was defined as
the percentage of water extraction for the ith soil zone when soil water
is not limiting. It may also be considered to be a rooting activity
term which indicates the percentage of active roots in the ith soil
zone.


.52
Figure 3-7 shows the form of equation (59). Equation (59) permits
a rapid rate of soil water extraction when soil water is readily
available. As soil water is depleted, the water extraction function
produces a logarithmic rate of decline of soil water extraction. It
allows recovery of near potential rates of extraction during periods of
low potential ET and it rapidly limits extraction during periods of high
potential ET. It prevents the soil permanent wilting point from being
reached by limiting ET to very low rates as the permanent wilting point
is approached.
In this research equation (59) was used to limit water extraction
from a given soil zone as the water in that zone became less available.
Equation (59) was not used to limit the ET from the profile, only to
repartition the extraction within the soil profile.
Equation (59) was applied to calculate a water extraction rate from
each grid for each time step during model operation. The soil water
extraction rate per soil grid was then multiplied by the grid volume to
calculate the total water extraction at that time step. If the
calculated total water extraction rate was not equal to the actual ET,
the soil water extraction rates were linearly adjusted so that the
calculated and actual ET rates were equal.
The adjusted soil water extraction rates (S.¡) were computed as
ET
where Av is the volume of the ith soil grid.
(61)


RELATIVE SOIL WATER EXTRACTION RATE
53
Figure 3-7. Relationship of the relative available soil water and
potential soil water extraction rate on the soil
water extraction rate.


CHAPTER IV
RESULTS AND DISCUSSION
Instrumentation Performance
Three pressure transducers were individually calibrated to analyze
their response characteristics. A typical calibration curve (for
pressure transducer No. 1) is shown in Figure 4-1. The three
calibration equations are
Volts = 0.048 1 2
(kPa)
+ 0.9861
R2 = 0.999
(62)
Volts = 0.04781
(kPa)
+ 0.9867
R2 = 0.999
(63)
Volts = 0.04804
(kPa)
+ 0.9819
R2 = 0.999
(64)
The coefficients for these equations were not significantly
different ( a = 0.05), so that for many applications one equation could
be used for all 3 pressure transducers without significant error. For
the above 3 transducers, combined equations resulted in a maximum
expected error of 0.06 kPa at 1 volt (-0.3 kPa potential) and 0.27 kPa
at 5 volts (-83.67 kPa potential). Equations could therefore be
combined for many field applications. For this research, individual
transducer calibration curves were used in order to obtain the maximum
accuracies possible with the instrumentation, and because the
microcomputer memory was adequate to permit storage of the individual
calibration curves.
In the calibration procedure, hysteresis effects were studied by
measuring transducer outputs at decreasing water potentials from 0 to
-80 kPa, followed by increasing potentials from -80 to 0 kPa. For
these transducers there were no measurable hysteresis effects.
54


55
Figure 4-1. Calibration curve of output voltage versus pressure
applied for pressure transducer no. 1.


56
The A/D board was found to have a linear relationship between
voltage input and digital output as shown in Figure 4-2. The equation
which related the voltage to digital output was
Digital Units = 49.98 (Volts) R2 = 0.999 (65)
Equation (65) was combined with the individual transducer
calibration equations (62-64) to obtain relationships between
transducer-measured water potentials and the digital inputs to the
microcomputer. Equation (65) is only valid in the range of 0-5 volts
and 0-255 units for the 8-bit A/D board used. The 8-bit board,
therefore, allowed a resolution of only the nearest 0.02 volts. Greater
resolution could be achieved by the use of a 12-bit, 16-bit or other
higher resolution A/D board. A resolution of 0.02 volts is a resolution
of approximately 0.4 kPa (from equations 62-64). For this work this
degree of resolution was judged to be acceptable.
Tests of the assembled tensiometer-pressure transducer systems were
conducted in the laboratory by manually and automatically recording
tensiometer water potentials as water was allowed to evaporate from the
tensiometer ceramic cups. Figure 4-3 shows the changes in water
potential with time for 2 tensiometers. The continuous automatic
readings by the microcomputer are shown as solid lines. Open circles
show mercury manometer data that were manually read. Excellent
agreement between the automatic and manual readings was obtained. In
all cases, agreement was within 1.0 kPa.
Figure 4-4 shows a comparison of the manual and automatic readings
for the 3 tensiometers and pressure transducers tested. Agreement was
excellent, with all data points located within 1 kPa of the 100%
accuracy line.


57


58
Figure 4-3. Comparisons of mercury manometer manually-read and
pressure transducer automatically-read tensiometer
water potentials during drying cycles for two
tensiometers in the laboratory.


PRESSURE TRANSDUCER WATER POTENTIAL, kPa
59
Figure 4-4. Comparisons of mercury manometer manually-read and
pressure transducer automatically-read tensiometer
water potentials for all laboratory data.


60
To evaluate the performance of the microcomputer-based soil water
potential monitoring system under field environmental conditions, 3 1-
week studies were conducted at the IFAS Irrigation Research and
Education Park. Typical results for 1 week are shown in Figure 4-5 to
4-7 for 3 pressure transducers and tensiometers. Microcomputer data
were recorded hourly and are shown as the solid line in each figure.
These lines show diurnal cycles in soil water potential and a gradually
decreasing average daily water potential as the soil dried. Water
potential changes were slow because of the low evaporative demand and
relatively inactive grass cover during the February 25 March 1 period
during which these data were collected.
The open circles in Figure 4-5 to 4-7 show manually read mercury
manometer data. Agreement between these and the automatically read data
were excellent under field conditions. The average variation between
manually and automatically read data was 0.47 kPa. This was
approximately the 0.4 kPa resolution of the instrumentation. The
maximum deviation observed was 1.76 kPa. These data demonstrated that
an accurate, effective microcomputer-based data acquisition system was
developed for automatically recording soil water potential measurements.
Because it was microcomputer-based, the system was inexpensive and
consisted of components that were readily available from local
electronics companies.
Field Data Collection
Field data were collected at the Irrigation Research and Education
Park as previously described. The data were collected for input and
verification of the numerical models developed in this research.
- j-


TENSIOMETER WATER POTENTIAL, kPa
61
Figure 4-5. Evaluation of pressure transducer-tensiometer no. 1
by comparison of mercury manometer manually-read and
pressure transducer automatically-read water potentials
in the field.


TENSIOMETER WATER POTENTIAL, kPa
62
I 1 I 1 I
0 20 40 60 80 100 120 140
TIME, HOURS
Figure 4-6. Evaluation of pressure transducer-tensiometer no. 2 by
comparison of mercury manometer manually-read and
pressure transducer automatically-read water potentials
in the field.


TENSIOMETER WATER POTENTIAL, kPa
63
Figure 4-7. Evaluation of pressure transducer-tensiometer no. 3 by
comparison of mercury manometer manually-read and
pressure transducer automatically-read water potentials
in the field.


64
The tensiometer data were used to calculate the soil water
extraction from the lysimeters. A method described by van Bavel et al.
(1968) was used to calculate the soil water extraction. Van Bavel's
method used an integrated form of equation (1) to calculate the soil
water extraction for a one-dimensional soil profile which was described
as z
(66)
where Rz = the total soil water extraction rate for the soil profile to
a depth of z. The total soil water extraction rate calculated for the
entire profile depth z is then the ET rate.
A direct measurement of the surface evaporation was not made.
Evaporation was included in the soil water extraction from the surface
grid. The tensiometer nearest the surface was located at 150 mm. Water
extraction from this surface grid included surface evaporation and ET
for that zone. Gravity flow drainage from the lower grid was assumed.
The hydraulic conductivity for the lower grid was assigned as the flux
out of the soil profile. This was justified because observed changes in
soil water potentials for the lower tensiometer were small.
1985 Field Data. The ET rates for selected drying cycles from the
1985 field experiments are shown in Figures 4-8 and 4-9. The observed
drying cycles occured from August 20 to September 6. Figure 4-8 is the
ET rate from the 20 kPa young citrus tree with grass cover. Figure 4-9
is the ET rate for the 20 kPa young citrus tree on bare soil. Several
data points for the tree with no grass were missed throughout the
experiment due to equipment failure. Due to these problems the tree
with no grass cover was treated as a one-dimensional profile for
analysis.


65
Differences between the ET rates in Figures 4-8 and 4-9 show that
the ET rates for the tree with grass cover were greater than those of
the tree with no grass cover. The higher ET rates for the tree with
grass cover were expected and were also observed by Smajstrla et al.
(1986).
The distribution of water extraction for the two citrus tree
treatments are shown in Table 4-1. The water extraction rates are shown
as percentages of the total soil water extraction and soil water
extraction per soil zone. These values were computed for input into the
water extraction models. Table 4-1 shows the differences between the
soil water extractions for the two treatments. Approximately 97% of the
water from the tree with no grass cover was extracted from the top 750
mm of the soil profile. The water extraction for the tree with grass
cover shows that approximately 73% of the water was extracted from the
top 750 mm of the soil profile. The water extraction rates from Table
4-1 also suggest that, for the tree with grass cover, water was
extracted below the 1050 mm depth observed with the data collection
system.
A two-dimensional distribution of soil water extraction is shown in
Table 4-2 for the tree with grass cover. Table 4-2 shows that water was
extracted almost uniformly throughout the profile. Greater percentages
of water were extracted from the inner radii of the lysimeter.
1986 Field Data. The ET rates for a selected drying cycle from
1986 field experiments are shown in Figures 4-10 and 4-11. Figure 4-10
is the ET rate from the 20 kPa young citrus tree with grass cover.
Figure 4-11 shows the ET rate for the 20 kPa young citrus tree with no
grass cover. The observed drying cycles occurred from June 16 to June


10
8 -
IT3
o
c
ce
ce
>4
o_
tn
z:
*C
ce
o
a.
c
6 -
235
~1
240
245
250
JULIAN DAY
Figure 4-8. Evapotranspiration rate from the 1985 field experiments with a
20 kPa soil water potential treatment young citrus tree with
grass cover.
CT>
cn


Figure 4-9. Evapotranspiration rate from the 1985 field experiment with
a 20 kPa soil water potential treatment young citrus tree
with bare soil.
cr>
^4


68
Table 4-1. One-dimensional distribution of water extraction for a 15 day
drying cycle for Young Citrus trees with and without grass
cover.
Depth
Depth
Water
Water
Relative
Increment
Extraction
Extraction
Water
Extraction
(mn)
(mm)
(mm)
(rnn/mm)
(%)
Tree
with Grass Cover
0-375
375
9.8
0.026
32.3
375-750
375
7.3
0.019
23.8
750-1050
300
8.7
0.029
28.6
below 1050
4.7
0.016
15.2
Tree with
No Grass Cover
0-375
375
5.9
0.016
27.9
375-750
375
7.4
0.020
35.3
750-1050
300
7.0
0.024
33.5
below 1050
0.7
0.0023
3.3


69
Table 4-2. Two-dimensional distribution of water extraction for a 15
day drying cycle for young Citrus with grass cover.
Depth Depth
Increment
(mm) (mm)
Radi us
(rrm)
100
300
500
700
Relative Water Extraction

( % )
0-375
375
8.59
10.46
6.92
7.66
375-750
375
5.65
5.79
5.46
5.65
750-1050
300
8.03
8.26
3.94
4.07
below 1050
4.21
3.45
6.33
6.04
Water Extraction
(mm)
0-375
375
2.6
3.2
2.1
2.3
375-750
375
1.7
1.8
1.7
1.7
750-1050
300
2.4
2.5
1.2
1.2
below 1050
1.3
1.1
1.9
1.8
Water Extraction
(mm/mm)
0-375
375
0.0070
0.0085
0.0056
0.0062
375-750
375
0.0046
0.0047
0.0044
0.0046
750-1050
300
0.0082
0.0084
0.0040
0.0041
below 1050
0.0043
0.0035
0.0064
0.0061


70
24. Differences between the two treatments showed that ET of the
tree with grass cover was greater than that of the tree with no cover,
especially early in the drying cycle. The 1220 mm depth tensiometer for
the tree with no grass did not function properly so these data were not
included in the data anaylysis, but this did not introduce a large
error, from Table 4-1.
The water extraction distributions are shown in Table 4-3. The
percentage of water extracted from the top layer was greater for the
tree with grass cover.
Evapotranspiration data for the two grass cover treatments are
shown in Figures 4-12 and 4-13. Figure 4-12 shows ET rates of the 20
kPa grass covered lysimeter and Figure 4-13 shows ET rates of the 40 kPa
grass covered lysimeter. Comparison of Figure 4-12 and 4-13 shows that
the ET of the 20 kPa treatment was higher than that of the 40 kPa
treatment. The grass had more water available for ET in the 20 kPa
treatment.
Table 4-4 shows the water extraction distributions for the two
grass treatments. Both distributions were similar with both having
approximately the same soil water extraction percentages for each layer.
Model Verification
The accuracy of the numerical models in simulating the infiltration
and redistribution of soil water was determined by comparison with other
computer simulations from previous works in the literature.
The one-dimensional computer model was compared to the simulations
of Rubin and Steinhardt (1963) and Hiler and Bhuiyan (1971). Their work
provided data from soils with widly different hydraulic properties and
also provided simulation results which were used for model verification.


JULIAN DAY
Figure 4-10. Evapotranspiration rate from the 1986 field experiment with a
20 kPa soil water potential treatment young citrus tree with
grass cover.



Figure 4-11. Evapotranspiration rate from the 1986 field experiment with
a 20 kPa soil water potential treatment young citrus tree with
bare soil.
ro


73
Table 4-3. Two-dimensional distribution of water extraction for a 4
day drying cycle for young citrus trees with and without
grass cover.
Depth Depth Radius
Increment (mm)
(mm) (rrm)
200 600 200 600
Tree with Tree with
Grass Cover No Grass Cover
Relative Water Extraction
(%)
0-225
225
28.2
31.2
19.9
16.95
225-375
150
12.2
15.2
12.3
17.9
375-525
150
4.45
1.2
6.85
3.75
525-750
225
4.4
1.2
8.4
8.7
750-1050
300
0.75
1.2
2.6
2.65
Water Extraction
(nm)
0-225
225
1.6
1.8
0.9
0.8
225-375
150
0.7
0.9
0.6
0.8
375-525
150
0.3
0.07
0.3
0.2
525-750
225
0.3
0.07
0.4
0.4
750-1050
300
0.04
0.07
0.1
0.1
Water Extraction
(mm/mm)
0-225
225
0.0073
0.0081
0.0041
0.0035
225-375
150
0.0048
0.0059
0.0038
0.0057
375-525
150
0.0017
0.00046
0.0021
0.0011
525-750
225
0.0011
0.00031
0.0017
0.0018
750-1050
300
0.0001
0.00023
0.0004
0.0004


4
u 1 1 1 1 1 1 '
170 171 172 173 174
JULIAN DAY
Figure 4-12. Evapotranspiration rate from the 1986 field experiment with
a 20 kPa soil water potential treatment with a grass cover.
-p


4
3-|
L
170 171 172 173
JULIAN DAY
174
Figure 4-13. Evapotranspiration rate from the 1986 field experment with
a 40 kPa soil water potential treatment with a grass cover.
^1
cn


76
Table 4-4. One-dimensional distribution of water extraction for a 4 dry
drying cycle for a grass cover crop at water depletion
levels of 20 kPa and 40 kPa.
Depth
Depth
Water
Water
Relative
Increment
Extraction
Extraction
Water
Extraction
(mm)
(mm)
(rim)
(mm/mm)
(*)
20 kPa Treatment Tree with
Grass Cover
0-225
225
6.3
0.028
57.9
225-375
150
2.0
0.014
18.6
375-525
150
0.9
0.006
8.3
525-750
225
1.1
0.005
9.9
750-1050
300
0.6
0.002
5.3
40 kPa Treatment Tree with
Grass Cover
0-225
225
3.9
0.0170
63.1
225-375
150
0.8
0.0055
13.5
375-525
150
0.5
0.0036
8.7
525-750
225
0.4
0.0019
6.9
750-1050
300
0.5
0.0016
7.8
Mi


77
Rubin and Steinhardt (1963) used an implicit solution of the
Richards equation in terms of water content. Their model was used to
study constant intensity rainfall infiltration on a Rehovot sand. The
soil data were presented as analytic functions which were tabulated for
use in this work. The soil hydraulic characteristics are shown in
Figures 4-14 and 4-15.
The simulated soil profile had a uniform initial soil water content
of 0.005. A uniform grid size of 10 mm was used. Figures 4-16 and 4-17
contain plots of soil water content profiles for infiltration into the
Rehovot sand. The figures show both the results from Rubin and
Steinhardt (1963) and this work. Figure 4-16 contains the soil water
content profiles for a constant infiltration rate of 12.7 mm/hr. Figure
4-17 contains the soil water content profiles for a constant
infiltration rate of 47 mm/hr. Results of this work are in excellent
agreement with those of Rubin and Steinhardt (1963).
The work of Hiler and Bhuiyan (1971) was also used to verify the
accuracy of the model developed in this work. They used a computer
model written in CMSP to solve the Richards equation. Surface
infiltration was simulated for two soils, Yolo light clay, and Adelanto
loam. The hydraulic characteristics of these soils are presented in
Figures 4-18 through 4-21.
The simulation results for these soils are presented in Figures 4-
22 through 4-24. Figures 4-22 and 4-23 show water content profiles with
time for the Yolo light clay at different initial conditions. Figure 4-
24 shows the soil water content profiles with time for the Adelanto
loam.


78
Comparison of results for the two soils showed excellent agreement
between Hiler and Bhuiyan's CMSP model and this work. The minor
differences between the results of the models could have resulted from
the interpolation of the soil properties.
The two-dimensional model was tested by simulating infiltration for
a Nahal Sinai sand (Bresler et al. 1971). Results were compared with a
steady-state analytical solution developed by Wooding (1968) and with an
ADI-Newton numerical simulation method presented by Brandt et al.
(1971).
These models simulated the infiltration of water into the soil from
a point source. The boundary condition used for the simulations was a
constant flux for the innermost surface grid (r = 0). The flux was set
equal to the saturated hydraulic conductivity of the soil. No flow
boundaries were used for the lower and radial boundaries. The simulated
soil profile had a uniform initial soil water content of 0.037. Figures
4-25 and 4-26 show the soil characteristics for the Nahal Sinai sand.
Figures 4-27 and 4-28 show a comparison between this work and that
of Wooding (1968) and Brandt et al. (1971). The data are presented in
dimensionless form with the relative water content defined as
S(0)/S(6sat) where S(0) was defined by Philip (1968) as
e
S(e) = \ D de (67)
0n
where D = the hydraulic diffusivity and 9n = the residual soil water
content. The total infiltration time was 57 min in Figure 4-27 and 297
min in Figure 4-28.
Figures 4-27 and 4-28 demonstrate that this model accurately
simulates the results of Brandt et al. (1971). Differences between the


SOIL WATER POTENTIAL (mm)
79
Figure 4-14. Soil water potential-soil water content relationship
for Rehovot sand.


HYDRAULIC CONDUCTIVITY (mm/hr)
80
Figure 4-15. Hydraulic conductivity-soil water content relationship
for Rehovot sand.


DEPTH (mm)
SOIL WATER CONTENT (mm/mm)
Figure 4-16. Simulated results of soil water content profiles for infiltration into a
Rehovot sand under constant rain intensity of 12.7 mm/hr.


DEPTH (mm)
Figure
SOIL WATER CONTENT (mm/mm)
17. Simulated results of soil water content profiles for infiltration into a
Rehovot sand under constant rain intensity of 47 mm/hr.


SOIL WATER POTENTIAL (mm)
83
SOIL WATER CONTENT (mm/mm)
Figure 4-18. Soil water potential-soil water content relationship
for Yolo light clay.


HYDRAULIC CONDUCTIVITY {m/hr)
Figure 4-19. Hydraulic conductivity-soil water content relationship
for Yolo light clay.


SOIL WATER POTENTIAL (mm)
85
Figure 4-20. Soil water potential-soil water content relationship
for Adelanto loam.


HYDRAULIC CONDUCTIVITY (mm/hr)
86
Figure 4-21. Hydraulic conductivity-soil water content relationship
for Adelanto loam.




Pages
Missing
or
Unavailable




DEPTH (mm)
0.0 0.1 0.2 0.3 0.4 0.5
SOIL WATER CONTENT (mm/mm)
Figure 4-22. Simulated results of soil water content profiles for infiltration into
a Yolo light clay with initial pressure potential at -66 kPa.
co
^4


Full Text
xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID EV6GP5OUT_FV517H INGEST_TIME 2011-11-01T15:59:22Z PACKAGE AA00004850_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES




This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate School and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
May 1987
I'LsJLuJ' Q.
Dean, College of Engineering
Dean, Graduate School





PAGE 1

0($685(0(17 $1' 6,08/$7,21 2) 62,/ :$7(5 67$786 81'(5 ),(/' &21',7,216 %\ .(11(7+ &2< 6721( $ ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( 6&+22/ 2) 7+( 81,9(56,7< 2) )/25,'$ ,1 3$57,$/ )8/),//0(17 2) 7+( 5(48,5(0(176 )25 7+( '(*5(( 2) '2&725 2) 3+,/2623+< 81,9(56,7< 2) )/25,'$ 9

PAGE 2

$&.12:/('*(0(176 f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

PAGE 3

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

PAGE 4

0RGHOLQJ 6RLO :DWHU 0RYHPHQW DQG ([WUDFWLRQ 0RGHO $SSOLFDWLRQV $33(1',&(6 $ $OWHUQDWLQJ 'LUHFWLRQ ,PSOLFLW )LQLWH 'LIIHUHQFLQJ % /LVWLQJ RI 6RLO :DWHU 0RYHPHQW DQG ([WUDFWLRQ 0RGHO 5()(5(1&(6 %,2*5$3+,&$/ 6.(7&+

PAGE 5

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

PAGE 6

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

PAGE 7

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
PAGE 8

)LJXUH 6RLO ZDWHU SRWHQWLDOVRLO ZDWHU FRQWHQW UHODWLRQVKLS IRU $GHODQWR ORDP )LJXUH +\GUDXOLF FRQGXFWLYLW\VRLO ZDWHU FRQWHQW UHODWLRQVKLS IRU $GHODQWR ORDP )LJXUH 6LPXODWHG UHVXOWV RI VRLO ZDWHU FRQWHQW SURILOHV IRU LQILOWUDWLRQ LQWR D
PAGE 9

)LJXUH +\GUDXOLF FRQGXFWLYLW\VRLO ZDWHU SRWHQWLDO UHODWLRQVKLS IRU $UUHGRQGR ILQH VDQG )LJXUH 6LPXODWLRQ UHVXOWV IRU WKH RQHGLPHQVLRQDO PRGHO ZLWK ILHOG GDWD IRU WKH PP GHSWK )LJXUH 6LPXODWLRQ UHVXOWV IRU WKH RQHGLPHQVLRQDO PRGHO ZLWK ILHOG GDWD IRU WKH PP GHSWK )LJXUH 6LPXODWLRQ UHVXOWV IRU WKH RQHGLPHQVLRQDO PRGHO ZLWK ILHOG GDWD IRU WKH PP GHSWK )LJXUH 6LPXODWLRQ UHVXOWV IRU WKH RQHGLPHQVLRQDO PRGHO ZLWK ILHOG GDWD IRU WKH PP GHSWK )LJXUH 6LPXODWLRQ UHVXOWV IRU WKH RQHGLPHQVLRQDO PRGHO ZLWK ILHOG GDWD IRU WKH PP GHSWK )LJXUH 6LPXODWLRQ UHVXOWV IRU WKH WZRGLPHQVLRQDO PRGHO ZLWK ILHOG GDWD IRU WKH PP GHSWK ,OO )LJXUH 6LPXODWLRQ UHVXOWV IRU WKH WZRGLPHQVLRQDO PRGHO ZLWK ILHOG GDWD IRU WKH PP GHSWK )LJXUH 6LPXODWLRQ UHVXOWV IRU WKH WZRGLPHQVLRQDO PRGHO ZLWK ILHOG GDWD IRU WKH PP GHSWK )LJXUH 6LPXODWLRQ UHVXOWV IRU WKH WZRGLPHQVLRQDO PRGHO ZLWK ILHOG GDWD IRU WKH PP GHSWK )LJXUH 6LPXODWLRQ UHVXOWV IRU WKH WZRGLPHQVLRQDO PRGHO ZLWK ILHOG GDWD IRU WKH PP GHSWK )LJXUH 6RLO ZDWHU SRWHQWLDOV IRU WKH WKUHH LUULJDWLRQ WUHDWPHQWV DW WKH PP GHSWK )LJXUH 6RLO ZDWHU SRWHQWLDOV IRU WKH WKUHH LUULJDWLRQ WUHDWPHQWV DW WKH PP GHSWK )LJXUH 6RLO ZDWHU VWRUDJH IRU WKH WKUHH LUULJDWLRQ WUHDWPHQWV )LJXUH &XPXODWLYH LUULJDWLRQ IRU WKH WKUHH LUULJDWLRQ WUHDWPHQWV )LJXUH &XPXODWLYH GUDLQDJH IURP WKH VRLO SURILOH IRU WKH WKUHH LUULJDWLRQ WUHDWPHQWV ,;

PAGE 10

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} RI WKH DQQXDO UDLQIDOO RFFXUV GXULQJ PRQWKV ZKHQ PDQ\ FURSV DUH QRW JURZQ DQG UDLQIDOO LV QRW XQLIRUPO\ GLVWULEXWHG 8QGHU WKHVH FRQGLWLRQV WKH VFKHGXOLQJ RI LUULJDWLRQ EHFRPHV H[WUHPHO\ LPSRUWDQW 7KH WLPHO\ DSSOLFDWLRQ RI LUULJDWLRQ ZDWHU FDQ UHVXOW LQ LQFUHDVHG \LHOGV DQG JUHDWHU SURILWV ZKLOH XQWLPHO\ DSSOLFDWLRQV FRXOG UHVXOW LQ GHFUHDVHG \LHOGV DQG SURILWV DQG OHDFKLQJ RI QXWULHQWV $ ORZ FRVW PLFURFRPSXWHUEDVHG GDWD DFTXLVLWLRQ V\VWHP IRU FRQWLQXRXV VRLO ZDWHU SRWHQWLDO PHDVXUHPHQWV ZDV GHYHORSHG 7KH V\VWHP FRQVLVWHG RI WHQVLRPHWHUPRXQWHG SUHVVXUH WUDQVGXFHUV DQ DQDORJWR GLJLWDO FRQYHUWHU DQG D SRUWDEOH PLFURFRPSXWHU 7KH GDWD DFTXLVLWLRQ V\VWHP ZDV HYDOXDWHG XQGHU ODERUDWRU\ DQG ILHOG FRQGLWLRQV ([FHOOHQW [

PAGE 11

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

PAGE 12

&+$37(5 ,1752'8&7,21 ,UULJDWLRQ RI DJULFXOWXUDO FURSV LV RQH RI WKH PDMRU XVHV RI IUHVK ZDWHU LQ )ORULGD ,W ZDV UHSRUWHG +DUULVRQ HW DO f WKDW LQ SHUFHQW RI WKH IUHVK ZDWHU XVH LQ )ORULGD ZDV IRU LUULJDWLRQ RI PRUH WKDQ KHFWDUHV RI DJULFXOWXUDO FURSV 7KUHH UHDVRQV FLWHG IRU VXSSOHPHQWDO LUULJDWLRQ LQ )ORULGD ZHUH f VDQG\ VRLOV KDYH YHU\ ORZ ZDWHU KROGLQJ FDSDFLWLHV f XS WR SHUFHQW RI WKH DQQXDO UDLQIDOO RFFXUV GXULQJ PRQWKV ZKHQ PDQ\ FURSV DUH QRW JURZQ DQG f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f ZKLOH RSWLPL]LQJ SURGXFWLRQ UHWXUQV %HFDXVH RI WKH FRPSOH[LW\ RI WKHVH LQWHUDFWLRQV QXPHULFDO PRGHOV DUH XVHIXO WRROV WR VWXG\ WKHP 0DQ\ UHVHDUFKHUV KDYH GHYHORSHG PRGHOV WR XVH LQ WKH VWXG\ RI VRLO ZDWHU PDQDJHPHQW 6PDMVWUOD =XU DQG -RQHV f 7KHVH PRGHOV DOORZ PDQ\ GLIIHUHQW LUULJDWLRQ VWUDWHJLHV WR EH LQYHVWLJDWHG ZLWKRXW WKH FRVW QRUPDOO\ DVVRFLDWHG ZLWK ILHOG H[SHULPHQWV 7KHVH PRGHOV LQYROYH

PAGE 13

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

PAGE 14

7R GHPRQVWUDWH WKH XVH RI WKH QXPHULFDO PRGHO LQ HYDOXDWLQJ DQG UHFRPPHQGLQJ LUULJDWLRQ VWUDWHJLHV

PAGE 15

&+$37(5 ,, 5(9,(: 2) /,7(5$785( 0HDVXUHPHQW RI 6RLO :DWHU 6WDWXV 7KH PHDVXUHPHQW RI VRLO ZDWHU FDQ EH FODVVLILHG LQWR WZR FDWDJRULHV f WKH DPRXQW RI ZDWHU KHOG LQ D JLYHQ DPRXQW RI VRLO VRLO ZDWHU FRQWHQWf DQG f WKH SRWHQWLDO RU WHQVLRQ ZLWK ZKLFK WKH ZDWHU LV KHOG E\ WKH VRLO VRLO ZDWHU SRWHQWLDOf 7KHVH SURSHUWLHV DUH UHODWHG WR HDFK RWKHU )LJXUH f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

PAGE 16

62,/ :$7(5 327(17,$/ PPf 62,/ :$7(5 &217(17 PPPPf )LJXUH $ W\SLFDO VRLO ZDWHU SRWHQWLDOVRLO ZDWHU FRQWHQW FXUYH

PAGE 17

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f LQYHVWLJDWHG WKH DFFXUDFLHV RI QHXWURQ PHWHUV ZKHQ PHDVXULQJ ZDWHU FRQWHQWV QHDU ERXQGDULHV 7KH\ IRXQG WKDW VLJQLILFDQW HUURUV RFFXUUHG ZKHQ UHDGLQJV ZHUH WDNHQ ZLWKLQ WKH WRS PHWHUV LQ DQ XQVDWXUDWHG VRLO SURILOH

PAGE 18

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

PAGE 19

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f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

PAGE 20

SRWHQWLDO RU WHQVLRQ RI ZDWHU LQ WKH VRLO 7KH WHQVLRPHWHU FRQVLVWV RI D FORVHG WXEH ZLWK D FHUDPLF FXS RQ WKH HQG ZKLFK LV LQVHUWHG LQWR WKH VRLO DQG D YDFXXP JDJH RU PDQRPHWHU WR PHDVXUH WKH ZDWHU SRWHQWLDO LQ WKH WHQVLRPHWHU WXEH 7KH WXEH LV ILOOHG ZLWK ZDWHU FORVHG DQG DOORZHG WR HTXLOLEUDWH ZLWK WKH VRLO ZDWHU SRWHQWLDO $V WKH VRLO GULHV ZDWHU LQ WKH WHQVLRPHWHU LV SXOOHG WKURXJK WKH FHUDPLF FXS 7KH VRLO ZDWHU SRWHQWLDO ZKLFK SXOOV ZDWHU WKURXJK WKH FHUDPLF FXS LV UHJLVWHUHG RQ WKH YDFXXP JDJH RU PDQRPHWHU 7KLV IRUFH LV DOVR WKH K\GUDXOLF SRWHQWLDO WKDW D SODQW ZRXOG QHHG WR H[HUW WR H[WUDFW ZDWHU IURP WKH VRLO 7KHUHIRUH D WHQVLRPHWHU PHDVXUHV WKH HQHUJ\ VWDWXV RI ZDWHU LQ WKH VRLO 7HQVLRPHWHUV OHIW LQ WKH VRLO IRU D ORQJ SHULRG RI WLPH WHQG WR IROORZ WKH FKDQJHV LQ WKH VRLO ZDWHU SRWHQWLDO 7KH K\GUDXOLF UHVLVWDQFH RI WKH FHUDPLF FXS WKH VXUURXQGLQJ VRLO DQG WKH FRQWDFW EHWZHHQ WKH FXS DQG VRLO FDXVH WHQVLRPHWHU UHDGLQJV WR ODJ EHKLQG WKH DFWXDO WHQVLRQ FKDQJHV LQ WKH VRLO /DJV DUH DOVR FDXVHG E\ WKH YROXPH RI ZDWHU QHHGHG WR EH PRYHG WKURXJK WKH FXS WR UHJLVWHU RQ WKH PHDVXULQJ GHYLFH 7KH XVHIXO UDQJH RI WHQVLRPHWHUV LV IURP WR N3D %HORZ N3D DLU HQWHUV WKURXJK WKH FHUDPLF FXS RU WKH ZDWHU FROXPQ LQ WKH WHQVLRPHWHU EUHDNV FDXVLQJ WKH WHQVLRPHWHU WR IDLO 7KLV PHDVXUHPHQW OLPLWDWLRQ LV QRW VHULRXV IRU LUULJDWHG FURSV RQ VDQG\ VRLOV EHFDXVH PRVW RI WKH DYDLODEOH ZDWHU IRU SODQW XVH RFFXUV EHWZHHQ DQG N3D $QRWKHU PHWKRG RI PHDVXULQJ VRLO ZDWHU SRWHQWLDO LV WKH WKHUPDO FRQGXFWDQFH PHWKRG 7KH UDWH RI KHDW GLVVLSDWLRQ LQ D SRURXV PDWHULDO RI ORZ KHDW FRQGXFWLYLW\ LV VHQVLWLYH WR WKH ZDWHU FRQWHQW LQ WKH SRURXV PDWHULDO :KHQ LQ FRQWDFW ZLWK D VRLO WKH ZDWHU SRWHQWLDO LQ WKH SRURXV PDWHULDO WHQGV WR HTXLOLEUDWH ZLWK WKH VRLO ZDWHU SRWHQWLDO

PAGE 21

3KHQH HW DO f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

PAGE 22

WHQVLRQ 2QO\ VPDOO UHVSRQVHV DUH REWDLQHG DW SRWHQWLDOV DERYH EDU PDNLQJ WKLV LQVWUXPHQW XQVXLWDEOH IRU LUULJDWLRQ VFKHGXOLQJ RQ VDQG\ VRLOV ZKHUH LUULJDWLRQV ZRXOG W\SLFDOO\ EH VFKHGXOHG DW PXFK JUHDWHU SRWHQWLDOV (OHFWULFDO UHVLVWDQFH EORFNV FDQ EH FRUUHODWHG ZLWK VRLO ZDWHU SRWHQWLDO 7KH EORFNV DUH SODFHG LQ WKH VRLO DQG DOORZHG WR HTXLOLEUDWH ZLWK WKH VRLO ZDWHU 5HVLVWDQFH EORFNV XVXDOO\ FRQWDLQ D SDLU RI HOHFWURGHV HPEHGGHG LQ J\SVXP Q\ORQ RU ILEHUJODVV 0HDVXUHPHQWV ZLWK UHVLVWDQFH EORFNV DUH VHQVLWLYH WR WKH HOHFWURO\WLF VROXWHV LQ WKH IOXLG EHWZHHQ WKH HOHFWURGHV 7KXV UHVLVWDQFH EORFNV DUH VHQVLWLYH WR YDULDWLRQV LQ VDOLQLW\ RI VRLO ZDWHU DQG WR WHPSHUDWXUH FKDQJHV 7HPSHUDWXUHV PXVW EH PHDVXUHG DQG UHVLVWDQFH UHDGLQJV DSSURSULDWHO\ FRUUHFWHG 5HVLVWDQFH EORFNV DUH QRW XQLIRUPO\ VHQVLWLYH RYHU WKH HQWLUH UDQJH RI VRLO ZDWHU FRQWHQW 7KH\ DUH PRUH DFFXUDWH DW ORZ ZDWHU FRQWHQWV WKDQ DW ZDWHU FRQWHQWV QHDU ILHOG FDSDFLW\ %HFDXVH RI WKLV OLPLWDWLRQ UHVLVWDQFH EORFNV FDQ EH XVHG WR FRPSOHPHQW WHQVLRPHWHUV WR PHDVXUH VRLO ZDWHU SRWHQWLDOV EHORZ WKH N3D UDQJH 7KH\ DUH UHODWLYHO\ LQH[SHQVLYH DQG DQ DXWRPDWHG GDWD FROOHFWLRQ V\VWHP PD\ EH EXLOW DURXQG WKHVH LQVWUXPHQWV IRU PHDVXULQJ ZDWHU SRWHQWLDOV LQ GULHU VRLOV $XWRPDWLRQ RI 6RLO :DWHU 3RWHQWLDO 0HDVXUHPHQWV 7KH LQVWUXPHQWV GLVFXVVHG PD\ DOO EH XVHG WR PHDVXUH WKH VWDWXV RI VRLO ZDWHU 0RVW RI WKH WHFKQLTXHV UHODWH D VRLO SURSHUW\ WR DQRWKHU SURSHUW\ ZKLFK LV PHDVXUHG E\ WKH LQVWUXPHQW 7ZR RI WKH PHWKRGV DOORZ D GLUHFW PHDVXUH RI D VRLO SURSHUW\ 7KH JUDYLPHWHULF PHWKRG JLYHV D GLUHFW PHDVXUH RI VRLO ZDWHU FRQWHQW DQG WHQVLRPHWHUV JLYH GLUHFW PHDVXUHPHQW RI VRLO ZDWHU SRWHQWLDO 7KH JUDYLPHWULF PHDVXUHPHQW PHWKRG

PAGE 23

UHTXLUHV GHVWUXFWLYH VDPSOLQJ DQG GRHV QRW OHQG LWVHOI WR DXWRPDWLRQ 7HQVLRPHWHUV PD\ EH DXWRPDWHG E\ UHFRUGLQJ WKH FKDQJHV LQ VRLO ZDWHU SRWHQWLDO DQG WKH\ GR QRW UHTXLUH GHVWUXFWLYH VDPSOLQJ 7KHUHIRUH WR PHDVXUH WKH VWDWXV RI VRLO ZDWHU XQGHU D JURZLQJ FURS DQ DXWRPDWHG GDWD FROOHFWLRQ V\VWHP XVLQJ WHQVLRPHWHUV WR PHDVXUH VRLO ZDWHU SRWHQWLDO ZDV FKRVHQ 7R GHWHUPLQH WKH VRLO ZDWHU SRWHQWLDO WKH ZDWHU SRWHQWLDO ZLWKLQ WKH WHQVLRPHWHU LV PHDVXUHG 7KH WHQVLRPHWHU ZDWHU SRWHQWLDO LV DVVXPHG WR EH LQ HTXLOLEULXP ZLWK WKH VRLO ZDWHU SRWHQWLDO (DUO\ PHWKRGV RI PHDVXULQJ WHQVLRPHWHU ZDWHU SRWHQWLDOV XVHG PHUFXU\ PDQRPHWHUV /DWHU PHFKDQLFDO YDFXXP JDXJHV ZHUH XVHG %RWK PHWKRGV IXQFWLRQHG ZHOO IRU PDQXDO DSSOLFDWLRQV 1HLWKHU KRZHYHU ZDV UHDGLO\ DXWRPDWHG 5HFHQW LQWHUHVW LQ EHWWHU XQGHUVWDQGLQJ WKH G\QDPLFV RI VRLO ZDWHU PRYHPHQW DQG LQ WKH GHYHORSPHQW RI QXPHULFDO PRGHOV WR VLPXODWH WKLV SURFHVV KDV UHVXOWHG LQ WKH QHHG IRU DQ DXWRPDWHG V\VWHP RI UHFRUGLQJ WHQVLRPHWHU UHDGLQJV RQ D FRQWLQXRXV UHDO WLPH EDVLV 9DQ %DYHO HW DO f XVHG D FDPHUD WR WDNH SHULRGLF SKRWRJUDSKV RI D WHQVLRPHWHU PDQRPHWHU ERDUG %\ DQDO\]LQJ WKH SKRWRJUDSKV WKH\ ZHUH DEOH WR UHFRUG FKDQJHV LQ SRWHQWLDOV 7KLV SURFHGXUH ZDV KRZHYHU ODERULRXV DQG GLG QRW SURYLGH FRQWLQXRXV VRLO ZDWHU SRWHQWLDO UHFRUGV (QILHOG DQG *LOODVS\ f GHYHORSHG D WUDQVGXFHU ZKLFK PHDVXUHG WKH OHYHO RI PHUFXU\ LQ D PHUFXU\ PDQRPHWHU WHQVLRPHWHU 7KH SULQFLSOH RI RSHUDWLRQ RI WKHLU WUDQVGXFHU ZDV WKH VDPH DV D FRQFHQWULF FDSDFLWRU 7KH OHYHO RI PHUFXU\ LQ WKH PDQRPHWHU FRUUHVSRQGV GLUHFWO\ WR WKH OHQJWK RI D FDSDFLWRU SODWH $ VWHHO WXEH ZDV XVHG DV WKH RXWHU FDSDFLWRU SODWH DURXQG D FROXPQ RI PHUFXU\ 7KH Q\ORQ WXEH ZKLFK FRQWDLQHG WKH PHUFXU\ FROXPQ DFWHG DV WKH GLHOHFWULF PDWHULDO 7KH FDSDFLWDQFH RI WKH

PAGE 24

WUDQVGXFHU ZDV PHDVXUHG DQG FRQYHUWHG WR OHQJWK RI WKH FROXPQ RI PHUFXU\ 7KLV LQVWUXPHQW ZDV IRXQG WR EH YHU\ VHQVLWLYH WR WHPSHUDWXUH IOXFWXDWLRQV )XUWKHU UHVHDUFK LV QHHGHG RQ WKLV WUDQVGXFHU WR PDNH LW VXLWDEOH IRU DSSOLFDWLRQV LQ DXWRPDWHG GDWD FROOHFWLRQ V\VWHPV )LW]VLPPRQV DQG
PAGE 25

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f XVHG LQGLYLGXDO SUHVVXUH WUDQVGXFHUV RQ WHQVLRPHWHUV ZLWK DOO DLU SXUJHG IURP WKH V\VWHP WR PRQLWRU VRLO ZDWHU SRWHQWLDOV 7KXV WKH\ ZHUH DEOH WR DYRLG ODJ WLPHV DVVRFLDWHG ZLWK DLU SRFNHWV LQ WKH WHQVLRPHWHUV 7KH\ ZHUH DEOH WR UHDG SUHVVXUH WUDQVGXFHUV YHU\ UDSLGO\ E\ HOHFWURQLF UDWKHU WKDQ K\GUDXOLF VZLWFKLQJ $ VRLO ZDWHU SRWHQWLDO PRQLWRULQJ V\VWHP WKDW XVHG SUHVVXUH WUDQVGXFHUV DV XVHG E\ 7KRPVRQ HW DO f ZRXOG EH DEOH WR UHFRUG GDWD IURP D QXPEHU RI VHQVRUV YHU\ UDSLGO\ $ FRQWLQXRXV GDWD DFTXLVLWLRQ V\VWHP EDVHG RQ WHQVLRPHWHUPRXQWHG SUHVVXUH WUDQVGXFHUV ZRXOG SURYLGH GDWD QHFHVVDU\ IRU PRGHOV RI PRYHPHQW RI ZDWHU LQ VRLOV FURS ZDWHU XVH DQG HYDSRWUDQVSLUDWLRQ 7KH XVH RI D PLFURFRPSXWHU WR PRQLWRU WKH SUHVVXUH WUDQVGXFHUV ZRXOG DOORZ WKH V\VWHP WR EH SURJUDPPHG WR PDNH GHFLVLRQV EDVHG RQ WKH LQSXW GDWD =D]XHWD HW DO f $OVR WKH FRVW RI D GHGLFDWHG GDWD DFTXLVLWRQ V\VWHP ZRXOG EH JUHDWHU WKDQ WKDW RI D PLFURFRPSXWHUEDVHG V\VWHP EHFDXVH RI FXUUHQW PLFURFRPSXWHU FRVWV DQG DYDLODELOLW\

PAGE 26

:DWHU 0RYHPHQW LQ 6RLOV 7KH JHQHUDO HTXDWLRQV JRYHUQLQJ XQVDWXUDWHG IORZ LQ SRURXV PHGLD DUH WKH FRQWLQXLW\ HTXDWLRQ DQG 'DUF\nV /DZ +L HO f SUHVHQWHG D FRPELQHG IORZ HTXDWLRQ ZKLFK LQFRUSRUDWHV WKH FRQWLQXLW\ HTXDWLRQ B YT 6 f W ZKHUH T IOX[ GHQVLW\ RI ZDWHU YROXPHWULF ZDWHU FRQWHQW Y GLIIHUHQWLDO RSHUDWRU W WLPH DQG 6 D VLQN RU VRXUFH WHUP ZLWK 'DUF\nV HTXDWLRQ IRU XQVDWXUDWHG IORZ T .Kf Y+ f ZKHUH K\GUDXOLF FRQGXFWLYLW\ + WKH WRWDO K\GUDXOLF KHDG DQG GHILQHG DV + K ] DQG K FDSLOODU\ SUHVVXUH KHDG 7KH UHVXOWLQJ FRPELQHG IORZ HTXDWLRQ IRU ERWK VWHDG\ DQG WUDQVLHQW IORZV LV DOVR NQRZQ DV 5LFKDUGV HTXDWLRQ f§ Yr.K!Y + f 6 f W )RU RQHGLPHQVLRQDO YHUWLFDO IORZ HTXDWLRQ f EHFRPHV K .Kf f§ f§ .Kf f§ f 6]Wf f W = = ] ZKHUH ] WKH YHUWLFDO GLPHQVLRQ

PAGE 27

7KH VLQN WHUP LV XVHG WR UHSUHVHQW WKH ORVV RU JDLQ RI ZDWHU IURP WKH VRLO E\ URRW H[WUDFWLRQ RU E\ DSSOLFDWLRQ RI LUULJDWLRQ ZDWHU IURP D SRLQW VRXUFH )RU LPSOLFLW VROXWLRQV HTXDWLRQ f PXVW EH ZULWWHQ LQ WHUPV RI RQO\ RQH YDULDEOH VRLO ZDWHU FRQWHQW RU SRWHQWLDO %\ LQWURGXFLQJ WKH VSHFLILF ZDWHU FDSDFLW\ & GHILQHG DV G H & f G K DQG XVLQJ WKH FKDLQ UXOH RI FDOFXOXV HTXDWLRQ f PD\ EH ZULWWHQ LQ WHUPV RI WKH VRLO ZDWHU SRWHQWLDO DV K K .Kf & .Kf f 6]Wf f W = = = )RU WZRGLPHQVLRQDO IORZ HTXDWLRQ f EHFRPHV K DK K .Kf & f§ f§.Kff§f f§.Kff§f 6[]Wf f DW ; ; = = = 7KH WZRGLPHQVLRQDO IORZ HTXDWLRQ ZULWWHQ LQ UDGLDO FRRUGLQDWHV ZLWK UDGLDO V\PPHWU\ PD\ EH ZULWWHQ DV K K D DK .Kf & f§ U .Kff§f f§.Kff§f 6[]Wf f DWUDU U ] = V] 7KH WZRGLPHQVLRQDO IORZ HTXDWLRQ LQ UDGLDO FRRUGLQDWHV PD\ EH XVHG WR GHVFULEH ZDWHU PRYHPHQW DQG H[WUDFWLRQ RI VRLO ZDWHU IURP QRQXQLIRUP ZDWHU DSSOLFDWLRQV VXFK DV WULFNOH LUULJDWLRQ 'XH WR WKH YDULDEOH QDWXUH RI .Kf HTXDWLRQV f f DQG f DUH KLJKO\ QRQOLQHDU DQG DQDO\WLFDO VROXWLRQV DUH H[WUHPHO\ FRPSOH[ RU

PAGE 28

LPSRVVLEOH WR REWDLQ 7KH QRQOLQHDULW\ RI HTXDWLRQV f DQG f DQG WKH W\SLFDO YDULDEOH ERXQGDU\ FRQGLWLRQV KDYH OHG WR WKH XVH RI QXPHULFDO PHWKRGV WR VROYH SUDFWLFDO SUREOHPV RI VRLOSODQWZDWHU UHODWLRQVKLSV VXFK DV LUULJDWLRQ PDQDJHPHQW IRU DJULFXOWXUDO FURSV )RU RQHGLPHQVLRQDO IORZ HTXDWLRQ f KDV EHHQ VXFFHVVIXOO\ VROYHG XVLQJ H[SOLFLW ILQLWH GLIIHUHQFH PHWKRGV E\ PDQ\ UHVHDUFKHUV +DQNV DQG %RZHUV f GHYHORSHG D QXPHULFDO PRGHO IRU LQILOWUDWLRQ LQWR OD\HUHG VRLOV 7KH\ VROYHG WKH 5LFKDUGV HTXDWLRQ IRU WKH K\GUDXOLF SRWHQWLDO XVLQJ LPSOLFLW ILQLWH GLIIHUHQFH HTXDWLRQV ZLWK D &UDQN1LFKROVRQ WHFKQLTXH ZKLFK DYHUDJHV WKH ILQLWH GLIIHUHQFHV RYHU WZR VXFFHVVLYH WLPH VWHSV 5XELQ DQG 6WHLQKDUGW f GHYHORSHG D QXPHULFDO PRGHO WR VWXG\ WKH VRLO ZDWHU UHODWLRQVKLSV GXULQJ UDLQIDOO LQILOWUDWLRQ 7KH\ XVHG D &UDQN1LFKROVRQ WHFKQLTXH WR VROYH 5LFKDUGV HTXDWLRQ IRU WKH ZDWHU FRQWHQW 5XELQ f GHYHORSHG D QXPHULFDO PRGHO ZKLFK DQDO\]HG WKH K\VWHUHVLV HIIHFWV RQ SRVWLQILOWUDWLRQ UHGLVWULEXWLRQ RI VRLO ZDWHU +DYHUNDPS HW DO f UHYLHZHG VL[ QXPHULFDO PRGHOV RI RQH GLPHQ V WRUUD LQILOWUDWLRQ (DFK PRGHO HPSOR\HG GLIIHUHQW GLVFUHWL]DWLRQ WHFKQLTXHV IRU WKH QRQOLQHDU LQILOWUDWLRQ HTXDWLRQ 7KH PRGHOV UHYLHZHG ZHUH VROYHG XVLQJ ERWK WKH ZDWHU FRQWHQW EDVHG HTXDWLRQ DQG WKH ZDWHU SRWHQWLDO EDVHG HTXDWLRQ 7KH\ IRXQG WKDW LPSOLFLW PRGHOV ZKLFK VROYHG WKH SRWHQWLDO EDVHG LQILOWUDWLRQ HTXDWLRQ KDG WKH ZLGHVW UDQJH RI DSSOLFDELOLW\ IRU SUHGLFWLQJ ZDWHU PRYHPHQW LQ VRLO HLWKHU VDWXUDWHG RU QRQVDWXUDWHG &ODUN DQG 6PDMVWUOD f GHYHORSHG DQ LPSOLFLW PRGHO RI VRLO ZDWHU IORZ WR VWXG\ WKH GLVWULEXWLRQ RI ZDWHU LQ VRLOV DV LQIOXHQFHG E\ YDULRXV LUULJDWLRQ GHSWKV DQG LQWHQVLWLHV 7KH PRGHO VLPXODWHG ZDWHU DSSOLFDWLRQ UDWHV IURP FHQWHUSLYRW LUULJDWLRQ V\VWHPV ZLWK LQWHQVLWLHV

PAGE 29

RI DSSOLFDWLRQ W\SLFDO RI ORZ DQG KLJK SUHVVXUH LUULJDWLRQ V\VWHPV 7KHLU PRGHO DOVR VLPXODWHG SRVWLQILOWUDWLRQ UHGLVWULEXWLRQ 5XELQ f GHYHORSHG D WZRGLPHQVLRQDO QXPHULFDO PRGHO RI WUDQVLHQW ZDWHU IORZ LQ XQVDWXUDWHG DQG SDUWO\ XQVDWXUDWHG VRLOV +H XWLOL]HG DOWHUQDWLQJGLUHFWLRQ LPSOLFLW $',f ILQLWH GLIIHUHQFH PHWKRGV +H VWXGLHG KRUL]RQWDO LQILOWUDWLRQ DQG GLWFK GUDLQDJH ZLWK WKH QXPHULFDO PRGHO +RUQEHUJHU HW DO f GHYHORSHG D WZRGLPHQVLRQDO PRGHO WR VWXG\ ZDWHU PRYHPHQW LQ D FRPSRVLWH VRLO PRLVWXUH JURXQGZDWHU V\VWHP 7KH\ PRGHOHG WKH WZRGLPHQVLRQDO UHVSRQVH RI IDOOLQJ ZDWHU WDEOHV 7KH\ FRQVLGHUHG ERWK VDWXUDWHG DQG XQVDWXUDWHG ]RQHV LQ WKHLU PRGHO 7KH VROXWLRQ PHWKRG XVHG ZDV D *DXVV6LGHO LWHUDWLYH WHFKQLTXH $ WZRGLPHQVLRQDO PRGHO WR VLPXODWH WKH GUDZGRZQ LQ D SXPSHG XQFRQILQHG DTXLIHU ZDV GHYHORSHG E\ 7D\ORU DQG /XWKLQ f 7KH PRGHO JDYH VLPXOWDQHRXV VROXWLRQV LQ ERWK WKH VDWXUDWHG DQG XQVDWXUDWHG ]RQHV 7KH\ XVHG D *DXVV6LGHO LWHUDWLYH PHWKRG WR VROYH WKH IORZ HTXDWLRQV $PHUPDQ f GHYHORSHG WZRGLPHQVLRQDO QXPHULFDO PRGHOV WR VLPXODWH VWHDG\ VWDWH VDWXUDWHG IORZ GUDLQDJH DQG IXUURZ LUULJDWLRQ +H XVHG $', PHWKRGV WR VROYH ERWK WKH VWHDG\ VWDWH VDWXUDWHG IORZ PRGHO DQG WKH IXUURZ LUULJDWLRQ PRGHO +H DOVR XVHG DQ H[SOLFLW PHWKRG WR VROYH WKH GUDLQDJH PRGHO $ VWXG\ RI WKH VHQVLYLW\ RI WKH JULG VSDFLQJ IRU ILQLWH GLIIHUHQFH PRGHOV ZDV UHSRUWHG RQ E\ $PHUPDQ DQG 0RQNH f 7ZR ILQLWH GLIIHUHQFH PRGHOV RI WZRGLPHQVLRQDO LQILOWUDWLRQ ZHUH DQDO\]HG 7KH\ VROYHG WKH WZRGLPHQVLRQDO IORZ HTXDWLRQV ZLWK VXFFHVVLYH RYHUUHOD[DWLRQ 625f DQG DOWHUQDWLQJ GLUHFWLRQ LPSOLFLW $',f PHWKRGV 7KH\ IRXQG WKDW VPDOOHU JULG VL]HV ZHUH QHHGHG LQ UHJLRQV ZKHUH WKH K\GUDXOLF JUDGLHQWV

PAGE 30

FKDQJHG UDSLGO\ &RQVLGHUDEOH FRPSXWDWLRQDO VDYLQJV ZLWKRXW DSSUHFLDEOH ORVV RI DFFXUDF\ ZDV DFKLHYHG XVLQJ LUUHJXODU JULG VL]HV 3HUUHQV DQG :DWVRQ f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f VROYHG WKH IORZ HTXDWLRQ LQ WZR GLPHQVLRQV WR DQDO\]H LQILOWUDWLRQ IURP D WULFNOH VRXUFH 7KH\ GHYHORSHG D SODQH IORZ PRGHO LQ FDUWHVLDQ FRRUGLQDWHV WR DQDO\]H LQILOWUDWLRQ IURP D OLQH VRXUFH RI FORVHO\ VSDFHG HPLWWHUV ZLWK RYHUODSSLQJ ZHWWLQJ SDWWHUQV 7KH\ DOVR GHYHORSHG D F\OLQGHULFDO IORZ PRGHO WR DQDO\]H LQILOWUDWLRQ IURP D VLQJOH HPLWWHU ZKHQ LWV ZHWWLQJ SDWWHUQ LV QRW DIIHFWHG E\ RWKHU HPLWWHUV %RWK PRGHOV ZHUH VROYHG XVLQJ QRQLWHUDWLYH $', ILQLWH GLIIHUHQFH SURFHGXUHV ZLWK 1HZWRQnV LWHUDWLYH PHWKRG 7KH UHVXOWV ZHUH FRPSDUHG WR DQ DQDO\WLFDO VROXWLRQ RI VWHDG\ LQILOWUDWLRQ DQG D RQHGLPHQVLRQDO VROXWLRQ ZLWK JRRG UHVXOWV $UPVWURQJ DQG :LOVRQ f GHYHORSHG D PRGHO IRU PRLVWXUH GLVWULEXWLRQ XQGHU D WULFNOH VRXUFH 7KH\ XWLOL]HG WKH &RQWLQXRXV 6\VWHP 0RGHOLQJ 3URJUDP &063f WR VLPXODWH WKH VRLO PRLVWXUH PRYHPHQW 7KH PRGHO FDOFXODWHG WKH QHW IORZ UDWHV LQWR HDFK JULG ,W WKHQ FDOFXODWHG WKH FKDQJH LQ ZDWHU FRQWHQW E\ GLYLGLQJ WKH QHW IORZ UDWH

PAGE 31

E\ WKH YROXPH RI VRLO LQ HDFK JULG DQG WKHQ PXOWLSO\LQJ E\ WKH WLPH VWHS )LQDOO\ WKH PRGHO FDOFXODWHG WKH QHZ ZDWHU FRQWHQWV IURP WKH SUHYLRXV ZDWHU FRQWHQWV SOXV WKH FDOFXODWHG ZDWHU FRQWHQW FKDQJHV 0RGHO UHVXOWV FRPSDUHG IDYRUDEO\ ZLWK ILHOG PHDVXUHPHQWV =D]XHWD HW DO f GHYHORSHG D VLPSOH H[SOLFLW QXPHULFDO PRGHO IRU WKH SUHGLFWLRQ RI VRLO ZDWHU PRYHPHQW IURP WULFNOH VRXUFHV 7KH PRGHO ZDV EDVHG RQ WKH PDVV EDODQFH HTXDWLRQ DQG DQ LQWHJUDWHG IRUP RI 'DUF\nV ODZ 7KH PRGHO SURGXFHG JRRG DJUHHPHQW ZLWK RWKHU UHVXOWV REWDLQHG ZLWK PRUH FRPSOLFDWHG QXPHULFDO PHWKRGV DQG DQDO\WLFDO VROXWLRQV 6RLO :DWHU ([WUDFWLRQ :DWHU XSWDNH E\ SODQW URRWV KDV EHHQ LQYHVWLJDWHG E\ PDQ\ UHVHDUFKHUV $PRQJ WKH ILUVW UHVHDUFKHUV WR DWWHPSW WR GHVFULEH SODQW ZDWHU UHODWLRQV ZHUH *UDGPDQQ f DQG YDQ GHQ +RQHUW f 7ZR DSSURDFKHV WR PRGHOLQJ ZDWHU H[WUDFWLRQ KDYH EHHQ XWLOL]HG WR GHVFULEH WKH ZDWHU H[WUDFWLRQ E\ SODQW URRWV 7KH ILUVW FDOOHG WKH PLFURVFRSLF DSSURDFK GHVFULEHV ZDWHU PRYHPHQW WR LQGLYLGXDO URRWV 7KH VHFRQG DSSURDFK FDOOHG WKH PDFURVFRSLF ZDWHU H[WUDFWLRQ PRGHO GHVFULEHV ZDWHU XSWDNH E\ WKH ZKROH URRW ]RQH DQG WKH IORZ WR LQGLYLGXDO URRWV LV LJQRUHG *DUGQHU f GHYHORSHG D PLFURVFRSLF ZDWHU XSWDNH PRGHO +H GHVFULEHG WKH URRW DV DQ LQILQLWHO\ ORQJ F\OLQGHU RI XQLIRUP UDGLXV DQG ZDWHUDEVRUELQJ SURSHUWLHV DVVXPLQJ WKDW ZDWHU PRYHV LQ WKH UDGLDO GLUHFWLRQ RQO\ 7KH IORZ HTXDWLRQ IRU VXFK D V\VWHP LV H W U f§ f U U U U U f

PAGE 32

ZKHUH D WKH YROXPHWULF ZDWHU FRQWHQW WKH GLIIXVLYLW\ W WKH WLPH DQG U WKH UDGLDO GLVWDQFH IURP WKH D[LV RI WKH URRW +H WKHQ REWDLQHG D VROXWLRQ DW WKH ERXQGDU\ EHWZHHQ WKH SODQW URRW DQG WKH VRLO LQ RUGHU WR PDLQWDLQ D FRQVWDQW UDWH RI ZDWHU PRYHPHQW WR WKH SODQW VXEMHFW WR WKH IROORZLQJ ERXQGDU\ FRQGLWLRQV H K K4 DW W W4 DQG GK GH LU UA Nf§f WW U f§f T DW U UM f GU GU ZKHUH N WKH K\GUDXOLF FRQGXFWLYLW\ RI WKH VRLO UA WKH UDGLXV RI WKH URRW DQG T WKH UDWH RI ZDWHU XSWDNH E\ WKH URRW 7KH VROXWLRQ RI f VXEMHFW WR WKH LQLWLDO DQG ERXQGDU\ FRQGLWLRQV LQ f LV T 'W K K GK ,Q J f f r N U ZKHUH J LV (XOHUnV FRQVWDQW 7KH GLIIXVLYLW\ DQG FRQGXFWLYLW\ ZHUH DVVXPHG WR EH FRQVWDQW WKLV DVVXPSWLRQ ZDV MXVWLILHG EHFDXVH U DQG W DUH DOO LQ WKH ORJDULWKPLF WHUP DQG GK LV WKXV QRW YHU\ VHQVLWLYH WR WKHVH YDULDEOHV *DUGQHU f DOVR QRWHG WKDW WKH VROXWLRQ ZLOO EHKDYH DV WKRXJK LQILQLWH IRU RQO\ YHU\ VKRUW WLPHV RU IRU YHU\ ORZ YDOXHV RI K\GUDXOLF FRQGXFWLYLW\ ([DFW VROXWLRQV RI WKH SUREOHP IRU ILQLWH V\VWHPV ZRXOG UHTXLUH WDNLQJ WKH GHSHQGHQFH RI DQG N XSRQ WKH VRLO ZDWHU SRWHQWLDO LQWR DFFRXQW *DUGQHU DOVR FRPSDUHG HTXDWLRQ f ZLWK WKH VWHDG\VWDWH VROXWLRQ IRU IORZ LQ D KROORZ F\OLQGHU

PAGE 33

T WU K K4 GU ,Q B f f LUN D ZKHUH K4 WKH SRWHQWLDO DW WKH RXWHU UDGLXV RI WKH KROORZ F\OLQGHU E RQH KDOI WKH GLVWDQFH EHWZHHQ QHLJKERULQJ URRWV DQG K WKH SRWHQWLDO DW WKH LQQHU UDGLXV D ,I E 'W WKHQ HTXDWLRQ f LV LGHQWLFDO WR HTXDWLRQ f H[FHSW IRU WKH J WHUP ZKLFK LV VPDOO FRPSDUHG WR WKH ORJDULWKPLF WHUP DQG PD\ EH LJQRUHG *DUGQHU DQG (KOLJ f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f GHYHORSHG D PDWKHPDWLFDO PRGHO GHVFULELQJ PRLVWXUH UHPRYDO IURP VRLO E\ SODQW URRWV 7KH PRGHO GHVFULEHG RQHn GLPHQVLRQDO ZDWHU PRYHPHQW DQG H[WUDFWLRQ 7KH PRGHO XVHG D PDFURVFRSLF ZDWHU H[WUDFWLRQ WHUP WR GHVFULEH PRLVWXUH UHPRYDO E\ SODQWV 7KHLU IRUPXOD IRU WKH H[WUDFWLRQ WHUP ZDV JLYHQ DV 5]f 'Hf 6]Wf 7 f f L 5]f 'Hf G] ZKHUH 'Hf GLIIXVLYLW\ 5]f WKH HIIHFWLYH URRW GHQVLW\ 7 WKH WUDQVSLUDWLRQ UDWH SHU XQLW DUHD DQG Y WKH URRW GHSWK

PAGE 34

7KH PRGHO JDYH UHVXOWV WKDW FRPSDUHG UHDVRQDEO\ ZLWK H[SHULPHQWDO UHVXOWV 7KH QXPHULFDO VROXWLRQ RI WKH 5LFKDUGV HTXDWLRQ ZLWK WKH H[WUDFWLRQ IXQFWLRQ ZDV REWWDLQHG XVLQJ WKH 'RXJODV-RQHV SUHGLFWRU FRUUHFWRU PHWKRG 1LPDK DQG +DQNV f GHYHORSHG D QXPHULFDO PRGHO WR SUHGLFW ZDWHU FRQWHQW SURILOHV HYDSRWUDQVSLUDWLRQ ZDWHU IORZ LQ WKH VRLO URRW H[WUDFWLRQ DQG URRW ZDWHU SRWHQWLDO XQGHU ILHOG FRQGLWLRQV 7KHLU H[WUDFWLRQ WHUP KDG WKH IRUP +URRW ]f K]WfV]Wff 5')]f .2f $ ] Wf f $; $ = ZKHUH $]Wf WKH VRLO ZDWHU H[WUDFWLRQ UDWH +URRW WKH HIIHFWLYH ZDWHU SRWHQWLDO LQ WKH URRW DW WKH VRLO VXUIDFH K]Wf WKH VRLO PDWULF SRWHQWLDO V]Wf WKH RVPRWLF SRWHQWLDO 5')]f SURSRUWLRQ RI WRWDO DFWLYH URRWV LQ GHSWK LQFUHPHQW '= .Hf WKH K\GUDXOLF FRQGXFWLYLW\ DQG $[ WKH GLVWDQFH EHWZHHQ WKH SODQW URRWV DW WKH SRLQW ZKHUH K]Wf DQG V]Wf DUH PHDVXUHG 7ROOQHU DQG 0RO] f GHYHORSHG D PDFURFRSLF ZDWHU XSWDNH PRGHO 7KH H[WUDFWLRQ IXQFWLRQ DVVXPHG WKDW ZDWHU XSWDNH UDWH SHU XQLW YROXPH RI VRLO LV SURSRUWLRQDO WR WKH SURGXFW RI FRQWDFW OHQJWK SHU XQLW VRLO YROXPH URRW SHUPHDELOLW\ SHU XQLW OHQJWK DQG ZDWHU SRWHQWLDO GLIIHUHQFH EHWZHHQ VRLO DQG URRW [\OHP SRWHQWLDO $ IDFWRU ZKLFK DFFRXQWV IRU UHGXFHG URRWVRLOZDWHU FRQWDFW DV ZDWHU LV UHPRYHG ZDV LQFOXGHG LQ WKH

PAGE 35

n f H[WUDFWLRQ IXQFWLRQ 7KHLU PRGHO SUHGLFWLRQV ZHUH FRPSDUDEOH ZLWK UHVXOWV RI D JUHHQKRXVH H[SHULPHQW 6ODFN HW DO `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f GHYHORSHG DQ LPSOLFLW ILQLWH GLIIHUHQFH PRGHO WR GHVFULEH ZDWHU IORZ DQG H[WUDFWLRQ LQ D QRQKRPRJHQHRXV VRLO URRW V\VWHP XQGHU WKH LQIOXHQFH RI JURXQGZDWHU 7KHLU H[WUDFWLRQ WHUP DVVXPHG WKH H[WUDFWLRQ UDWH WR EH PD[LPXP ZKHQ WKH VRLO ZDWHU SRWHQWLDO ZDV DERYH D VHW OLPLW :KHQ WKH VRLO ZDWHU SRWHQWLDO IHOO EHORZ WKLV OLPLW WKH ZDWHU XSWDNH 6Kf ZDV GHFUHDVHG OLQHDUO\ ZLWK WKH VRLO ZDWHU SRWHQWLDO K K f 6Kf 6PD[ f K K f ZKHUH K LV WKH VHW OLPLW EHORZ ZKLFK WKH ZDWHU XSWDNH GHFUHDVHV OLQHDUO\ WR K %HORZ K LW ZDV DVVXPHG WKDW QR ZDWHU LV H[WUDFWHG WKHQ K PD\ EH DVVXPHG WR EH WKH ZLOWLQJ SRLQW 7KH PD[LPXP SRVVLEOH WUDQVSLUDWLRQ UDWH GLYLGHG E\ WKH HIIHFWLYH URRWLQJ GHSWK LV 6PD[ 7KH PRGHO \LHOGHG VDWLVIDFWRU\ UHVXOWV LQ SUHGLFWLQJ ERWK FXPXODWLYH WUDQVSLUDWLRQ DQG GLVWULEXWLRQ RI VRLO PRLVWXUH FRQWHQW ZLWK GHSWK

PAGE 36

=XU DQG -RQHV f GHYHORSHG D PRGHO IRU VWXG\LQJ WKH LQWHJUDWHG HIIHFWV RI VRLO FURS DQG FOLPDWLF FRQGLWLRQV RQ WKH H[SDQVLYH JURZWK SKRWRV\QWKHVLV DQG ZDWHU XVH RI DJULFXOWXUDO FURSV 7KH PRGHO XWLOL]HG 3HQPDQnV HTXDWLRQ PRGLILHG E\ 0RQWHLWK WR FDOFXODWH WKH ZDWHU YDSRU WUDQVSRUW IURP WKH SODQW WR WKH DWPRVSKHUH ,W XWLOL]HG D PDFURVFRSLF VRLO ZDWHU H[WUDFWLRQ WHUP WR FDOFXODWH ZDWHU XSWDNH IURP WKH VRLO 7KH PRGHO DVVXPHG WKH ZDWHU XSWDNH IURP WKH VRLO ZDV WKH GLIIHUHQFH EHWZHHQ WKH WRWDO ZDWHU SRWHQWLDO DW WKH URRW VXUIDFH DQG WKH VRLO ZDWHU SRWHQWLDO GLYLGHG E\ WKH VXP RI WKH UHVLVWDQFH WR UDGLDO ZDWHU IORZ LQVLGH WKH URRWV DQG UHVLVWDQFH RI ZDWHU IORZ LQ WKH VRLO 7KH PRGHO WUHDWHG WKH ZDWHU UHODWLRQV LQ SODQWV LQ FRQVLGHUDEOH GHWDLO ,W ZDV VXFFHVVIXOO\ WHVWHG RQ VR\EHDQV +RZHYHU WKH PRGHO GLG QRW WDNH LQWR FRQVLGHUDWLRQ LQILOWUDWLRQ GUDLQDJH HYDSRUDWLRQ IURP WKH VRLO VXUIDFH RU XSZDUG PRYHPHQW RI ZDWHU LQ WKH VRLO

PAGE 37

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f SRZHU VXSSO\ DW P$ HDFK %HFDXVH D PX,WLSOHWUDQVGXFHU V\VWHP ZDV GHYHORSHG D VHSDUDWH UHJXODWHG SRZHU VXSSO\ ZDV XVHG UDWKHU WKDQ WKH PLFURFRPSXWHU SRZHU VXSSO\ 7KLV DVVXUHG DGHTXDWH SRZHU IRU WKH WUDQVGXFHUV 7KH SUHVVXUH WUDQVGXFHUV ZHUH WHPSHUDWXUHFRPSHQVDWHG 2XWSXWV YDULHG OHVV WKDQ b RI IXOO VFDOH RXWSXW LQ WKH UDQJH RI & WR & 7KH RXWSXW YROWDJHV IURP WKH SUHVVXUH WUDQVGXFHUV ZHUH LQWHUIDFHG WR D PLFURFRPSXWHU WKURXJK DQ DQDORJWRGLJLWDO $'f ERDUG 7KH $' ERDUG ZDV D 0LFR :RUOG 0: JHQHUDO SXUSRVH ,QSXW2XWSXW %RDUG 7KLV GHYLFH KDG ELW UHVROXWLRQ DQG PHDVXUHG LQSXW YROWDJHV IURP WR YROWV ,W KDG FKDQQHO PXOWLSOH[LQJ FDSDELOLW\

PAGE 38

7KH PLFURFRPSXWHU XVHG ZDV D &RPPRGRUH ,W ZDV VHOHFWHG IRU LWV SURJUDPPDELOLW\ LWV DYDLODELOLW\ DW ORFDO HOHFWURQLFV VXSSO\ FRPSDQLHV DQG LWV ORZ FRVW $Q RYHUDOO JRDO LQ FRPSRQHQW VHOHFWLRQ ZDV WR EH DEOH WR UHSODFH DQ\ FRPSRQHQW LQGLYLGXDOO\ ZLWK DV OLWWOH GRZQWLPH DV SRVVLEOH DQG WKXV WR DVVXUH WKH UHOLDELOLW\ RI WKH V\VWHP 7KH PLFURFRPSXWHU RXWSXW ZDV WR D YLGHR VFUHHQ D SULQWHU DQG D FDVVHWWH UHFRUGHU 7KH FDVVHWWH UHFRUGHU SURYLGHG VWRUDJH RI GDWD IRU ODWHU FRPSXWDWLRQV DQG HDV\ WUDQVIHU WR RWKHU FRPSXWHU V\VWHPV 7KH SULQWHU SURYLGHG D KDUG FRS\ EDFNXS RI GDWD 7KH YLGHR VFUHHQ SURYLGHG LPPHGLDWH LQWHUDFWLRQ ZLWK WKH FRPSXWHU IRU SURJUDPPLQJ DQG PRQLWRULQJ 7KH PLFURFRPSXWHU ZDV SURJUDPPHG LQ WKH %$6,& FRPSXWHU ODQJXDJH ,W KDG DSSUR[LPDWHO\ NLORE\WHV RI SURJUDPPDEOH PHPRU\ 5$0f ZKHQ XVLQJ WKH %$6,& FRPSXWHU ODQJXDJH 7KLV ZDV VXIILFLHQW WR VWRUH WKH FDOLEUDWLRQ HTXDWLRQV DQG FDOFXODWHG RXWSXWV IURP WKH WUDQVGXFHU LQSXWV 7KH PLFURFRPSXWHU DOVR KDG DQ LQWHUQDO KRXU FORFN ZKLFK ZDV UHTXLUHG IRU UHDOWLPH GDWD DFTXLVLWLRQ 'DWD DFTXLVLWLRQ V\VWHP FRPSRQHQWV DQG DSSUR[LPDWH FRVWV DUH JLYHQ LQ 7DEOH ,QVWUXPHQWDWLRQ 7HVW 3URFHGXUH 3UHVVXUH WUDQVGXFHUV ZHUH LQGLYLGXDOO\ FDOLEUDWHG XVLQJ D UHJXODWHG YDFXXP VRXUFH DQG D PHUFXU\ PDQRPHWHU 0HDVXUHPHQWV RI WUDQVGXFHU YROWDJHV DQG PHUFXU\ FROXPQ KHLJKWV ZHUH PDGH DW GHFUHDVLQJ SUHVVXUH LQFUHPHQWV RI DSSUR[LPDWHO\ N3D IURP WR DSSUR[LPDWHO\ N3D 7R GHWHUPLQH ZKHWKHU WKH WUDQVGXFHUV GLVSOD\HG K\VWHUHWLF EHKDYLRU FDOLEUDWLRQV ZHUH FRQWLQXHG DW LQFUHDVLQJ SUHVVXUH LQFUHPHQWV IURP N3D WR N3D

PAGE 39

7DEOH 0LFURFRPSXWHUEDVHG GDWD DFTXLVLWLRQ V\VWHP FRPSRQHQWV DQG DSSUR[LPDWH FRVWV &RPSRQHQW &RVWf 0LFURFRPSXWHU 3ULQWHU &DVVHWWH 5HFRUGHU 3UHVVXUH 7UDQVGXFHU 7HQVLRPHWHU $QDORJWR'LJLWDO DQG 0XOWLSOH[LQJ &LUFXLW 7UDQVGXFHU 3RZHU 6XSSO\

PAGE 40

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

PAGE 41

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f ZKLFK ZDV KDQG SDFNHG WR DSSUR[LPDWH WKH SK\VLFDO FKDUDFWHULVWLFV RI DQ XQGLVWXUEHG VRLO SURILOH 'UDLQDJH RI H[FHVV ZDWHU IURP WKH ERWWRP RI HDFK O\VLPHWHU ZDV DFFRPSOLVKHG XVLQJ HLWKHU D SRURXV VWRQH RU SRURXV FHUDPLF FXS ORFDWHG LQ WKH ERWWRP RI WKH WDQN DV VKRZQ LQ )LJXUH $ YDFXXP SXPS UHPRYHG WKH ZDWHU IURP WKH O\VLPHWHUV DQG WKH ZDWHU ZDV WUDSSHG LQ 39& F\OLQGHUV 7KH 39& F\OLQGHUV ZHUH XVHG WR PHDVXUH WKH GUDLQDJH IURP HDFK O\VLPHWHU $XWRPDWHG UDLQIDOO VKHOWHUV ZHUH XVHG WR FRYHU WKH O\VLPHWHUV LQ WKH HYHQW RI UDLQIDOO $GGLWLRQDO GHWDLOV RI WKH FRQVWUXFWLRQ DQG RSHUDWLRQ RI WKH O\VLPHWHUV ZHUH JLYHQ E\ 6PDMVWUOD f 'DWD FROOHFWLRQ ZDV FRQGXFWHG LQ WZR SKDVHV ,Q WZR O\VLPHWHUV ZHUH LQVWUXPHQWHG DQG GDWD ZHUH FROOHFWHG DV GHVFULEHG HDUOLHU (DFK O\VLPHWHU FRQWDLQHG D \RXQJ FLWUXV WUHH DSSUR[LPDWHO\

PAGE 42

)LJXUH /D\RXW RI O\VLPHWHU V\VWHP

PAGE 43

72 9$&880 6285&( )LJXUH 'HWDLOV RI LQGLYLGXDO O\VLPHWHU VRLO ZDWHU VWDWXV PRQLWRULQJ V\VWHP

PAGE 44

\HDUV ROG %RWK O\VLPHWHUV ZHUH LUULJDWHG ZKHQ WKH ZDWHU SRWHQWLDO LQ WKH O\VLPHWHU GURSSHG EHORZ N3D 7KH N3D ZDWHU SRWHQWLDO FRUUHVSRQGHG WR DSSUR[LPDWHO\ D b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b ZDWHU GHSOHWLRQ IURP ILHOG FDSDFLW\ 0RGHO 'HYHORSPHQW $ FRPSXWHU SURJUDP GHVFULELQJ WKH LQILOWUDWLRQ UHGLVWULEXWLRQ DQG H[WUDFWLRQ RI VRLO ZDWHU ZDV ZULWWHQ LQ WKH )2575$1 FRPSXWHU ODQJXDJH $ 3ULPH FRPSXWHU ZDV XVHG IRU DOO FRPSXWDWLRQV 6HYHUDO DVVXPSWLRQV ZHUH HPSOR\HG LQ WKH GHYHORSPHQW DQG XVH RI WKH QXPHULFDO PRGHOV WR HDVH WKH FRPSOH[LWLHV DVVRFLDWHG ZLWK PDWKHPDWLFDOO\ GHVFULELQJ ZDWHU PRYHPHQW LQ XQVDWXUDWHG SRURXV PHGLD 7KHVH

PAGE 45

'HSWK PPf 5DGLXV PPf f ORFDWLRQ RI WHQVLRPHWHUV )LJXUH /RFDWLRQ RI WHQVLRPHWHUV LQ ILHOG H[SHULPHQW IRU \RXQJ FLWUXV WUHHV LQ JUDVVHG DQG EDUH VRLO O\VLPHWHUV

PAGE 46

'HSWK PPf 5DGLXV PPf f ORFDWLRQ RI WHQVLRPHWHUV )LJXUH /RFDWLRQ RI WHQVLRPHWHUV LQ ILHOG H[SHULPHQW IRU \RXQJ FLWUXV WUHHV LQ JUDVVHG DQG EDUH VRLO O\VLPHWHUV

PAGE 47

DVVXPSWLRQV ZHUH f 7KH SK\VLFDO SURSHUWLHV RI WKH VRLO DUH KRPRJHQHRXV DQG LVRWURSLF f 7KH SK\VLFDO DQG FKHPLFDO SURSHUWLHV RI WKH VRLO DUH FRQVWDQW LQ WLPH DQG f 7KH K\GUDXOLF FRQGXFWLYLW\ DQG VRLO SUHVVXUH SRWHQWLDO DUH VLQJOHYDOXHG IXQFWLRQV RI ZDWHU FRQWHQW 2QH'LPHQVLRQDO 0RGHO (TXDWLRQ f ZDV XVHG WR GHVFULEH WKH RQHGLPHQVLRQDO PRYHPHQW DQG H[WUDFWLRQ RI ZDWHU )LQLWH GLIIHUHQFH PHWKRGV ZHUH XVHG WR VROYH WKH ZDWHU SRWHQWLDO EDVHG IORZ HTXDWLRQ 7KH PRGHO XVHG DQ LPSOLFLW ILQLWH GLIIHUHQFH WHFKQLTXH ZLWK H[SOLFLW OLQHDUL]DWLRQ RI WKH VRLO SDUDPHWHUV 7KH ILQLWH GLIIHUHQFH IRUP RI HTXDWLRQ f ZDV ZULWWHQ DV N N N K K f N L L N N N N N K K f KKf L L L L L L $W $] ZKHUH L $= L L L f f DQG L L f 7KH VXEVFULSW L UHIHUV WR GLVWDQFH DQG WKH VXSHUVFULSW N UHIHUV WR WLPH 7R DSSO\ WKLV HTXDWLRQ WR VRLO SURILOHV WKH VRLO ZDV GLYLGHG LQWR D

PAGE 48

ILQLWH QXPEHU RI OD\HUV RU JULGV DV VKRZQ LQ )LJXUH (TXDWLRQ f ZDV UHDUUDQJHG VR WKDW XQNQRZQV ZHUH RQ WKH OHIW VLGH RI WKH HTXDWLRQ DQG WKH NQRZQV ZHUH RQ WKH ULJKW VLGH VXFK WKDW $W $W L N f K N L $= & $ = $W L L N $= & L f K N N N N $W $W . $W L N N L L f K K B f 6 N L L N N L D] & D] & & L L L f (TXDWLRQ f ZDV ZULWWHQ IRU HDFK RI WKH LQWHULRU JULGV LQ WKH VRLO SURILOH 7KH WRS DQG ERWWRP JULGV RI WKH V\VWHP UHTXLUHG HTXDWLRQ f WR EH PRGLILHG VR WKDW WKH HTXDWLRQ ZRXOG GHVFULEH WKH ERXQGDU\ FRQGLWLRQV 7KH V\VWHP RI HTXDWLRQV ZKLFK DUH IRUPHG E\ DSSO\LQJ HTXDWLRQ f WR HDFK JULG SURGXFHV D WULGLDJRQDO PDWUL[ ZKLFK LV LPSOLFLW LQ WHUPV RI K ZDWHU SRWHQWLDO 7KH WULGLDJRQDO PDWUL[ ZDV VROYHG XVLQJ *DXVV HOLPLQDWLRQ )URP H[SOLFLW OLQHDUL]DWLRQ WKH FRHIILFLHQWV RI WKH XQNQRZQ K WHUPV ZHUH NQRZQ DW HDFK WLPH VWHS 7KXV HTXDWLRQ f ZDV UHGXFHG WR N N N $K % K & K L L L L L L f ZKHUH N $W L $ f L N $ ] & f

PAGE 49

*5,'6 L L L 6RLO VXUIDFH K & LO OO LO OO K & L LOO K & OO OO OO LO Q )LJXUH 6FKHPDWLF GLDJUDP RI WKH ILQLWHGLIIHUHQFH JULG V\VWHP IRU WKH RQHGLPHQVLRQDO PRGHO RI ZDWHU PRYHPHQW DQG H[WUDFWLRQ

PAGE 50

$W & L N $= & L f f % $ & L L L N N $W . N L L K f§ L L N $= & f $W f 6 N L & L f %RXQGDU\ &RQGLWLRQV 7KH ERXQGDU\ FRQGLWLRQ IRU WKH VXUIDFH JULG ZDV YDULDEOH $ IOX[ ERXQGDU\ FRQGLWLRQ ZDV XVHG WR VLPXODWH LQILOWUDWLRQ GXULQJ LUULJDWLRQ DQG D QR IOX[ ERXQGDU\ FRQGLWLRQ ZDV XVHG IRU UHGLVWULEXWLRQ (YDSRUDWLRQ IURP WKH VRLO VXUIDFH ZDV LQFOXGHG LQ WKH ZDWHU H[WUDFWLRQ WHUP (TXDWLRQ f ZDV PRGLILHG IRU WKH VXUIDFH ERXQGDU\ FRQGLWLRQ VXFK WKDW N N % K & K ZKHUH DQG % f f $W N K B $] 4V $W V N r f

PAGE 51

7KH ORZHU ERXQGDU\ ZDV DOVR D IOX[ ERXQGDU\ DQG ZDV ZULWWHQ DV N N $K % K f Q Q Q Q Q % $ f Q Q N $W 4E $W N Q K f§ f 6 f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f VWDWHG WKDW VWDELOLW\ FRQGLWLRQV PXVW EH GHWHUPLQHG E\ WULDO DQG HUURU EHFDXVH WKH\ GHSHQG RQ WKH GHJUHH RI QRQOLQHDULW\ RI WKH HTXDWLRQV 7KH WLPH VWHS ZDV HVWLPDWHG E\ D PHWKRG JLYHQ E\ )HGGHV HW DO f DQG XVHG E\ &ODUN f ZKHUH DQG Q

PAGE 52

; $= $W f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n FUHWH SRLQWV VRPH HUURU ZLOO UHVXOW IURP WKH QXPHULFDO FDOFXODWLRQV RI WKH VLPXODWLRQ PRGHO 7KH HUURU GHFUHDVHV DV WKH JULG VL]H GHFUHDVHG 7ZR'LPHQVLRQDO 0RGHO :DWHU PRYHPHQW DQG H[WUDFWLRQ XQGHU \RXQJ FLWUXV WUHHV ZDV VLPXODWHG XVLQJ D WZRGLPHQVLRQDO VRLO ZDWHU H[WUDFWLRQ PRGHO $ UDGLDO

PAGE 53

IORZ PRGHO PD\ EH XVHG WR GHVFULEH ZDWHU PRYHPHQW DQG H[WUDFWLRQ LQ D VRLO SURILOH ZKLFK LV QRQXQLIRUPO\ LUULJDWHG DV LQ ZDWHU IORZ IURP WULFNOH HPLWWHUV DQG VSUD\ MHWV 7KH UDGLDO IORZ PRGHO PD\ DOVR GHVFULEH WKH ZDWHU H[WUDFWLRQ XQGHU ZLGHO\ VSDFHG FURSV VXFK DV FLWUXV :LWK WKH DVVXPSWLRQ RI UDGLDO V\PPHWU\ HTXDWLRQ f ZDV VROYHG LQ WKH YHUWLFDO DQG UDGLDO GLUHFWLRQV 7KLV HTXDWLRQ LV D VHFRQG RUGHU QRQOLQHDU SDUDEROLF SDUWLDO GLIIHUHQWLDO HTXDWLRQ ZKLFK H[SUHVVHV WKH ZDWHU SRWHQWLDO GLVWULEXWLRQ DV D IXQFWLRQ RI WLPH DQG VSDFH FRRUGLQDWHV 1R NQRZQ DQDO\WLFDO VROXWLRQ RI HTXDWLRQ f H[LVW 7KXV LW ZDV VROYHG QXPHULFDOO\ XVLQJ ILQLWH GLIIHUHQFH HTXDWLRQV 7KH PRGHO GHYHORSHG LQ WKLV UHVHDUFK XVHG DQ LPSOLFLW ILQLWH GLIIHUHQFH WHFKQLTXH ZLWK H[SOLFLW OLQHDUL]DWLRQ RI WKH VRLO SDUDPHWHUV 7ZR VROXWLRQ PHWKRGV ZHUH XVHG WR VROYH WKH ILQLWH GLIIHUHQFH HTXDWLRQV IRU WKH WZR GLPHQVLRQDO PRGHO ,W ZDV VROYHG LPSOLFLWO\ XVLQJ D *DXVV HOLPLQDWLRQ PHWKRG %HFDXVH RI WKH WLPH UHTXLUHG WR VROYH D ODUJH V\VWHP RI HTXDWLRQV E\ *DXVV HOLPLQDWLRQ DQ DOWHUQDWLRQ GLUHFWLRQ LPSOLFLW $',f PHWKRG ZDV DOVR XWLOL]HG WR VROYH HTXDWLRQ f 7KH ILQLWH GLIIHUHQFH HTXDWLRQ IRU WKH VROXWLRQ RI HTXDWLRQ f XWLOL]LQJ WKH *DXVV HOLPLQDWLRQ PHWKRG ZDV N N N N N N N N KKf KKf. KKf N LM LM LOM LM LM L M & LM $W $= N N N N N N U K KfU KKf LMO LM LMO LM LM LfM LM LM $U U

PAGE 54

N N r M M N f 7R DSSO\ WKLV HTXDWLRQ WR WKH VRLO SURILOHV WKH VRLO ZDV GLYLGHG LQWR D ILQLWH QXPEHU JULGV DV VKRZQ LQ )LJXUH (TXDWLRQ f ZDV DUUDQJHG VXFK WKDW WKH XQNQRZQV ZHUH RQ WKH OHIW VLGH RI WKH HTXDWLRQ DQG WKH NQRZQV ZHUH RQ WKH ULJKW VLGH DV IROORZV N $W M f N $] & n LM $W U N K $U N LM M f N U & N K LM N N $W . f M L M N $= & LM N N $W U U f LLM LM LM N $U U & LM LM f K LN N $W $W U LM N LM LM f K f N M N D] & $U U & LLLN K -

PAGE 55

)LJXUH 6FKHPDWLF GLDJUDP RI WKH ILQLWHGLIIHUHQFH JULG V\VWHP IRU WKH WZRGLPHQVLRQDO PRGHO RI ZDWHU PRYHPHQW DQG H[WUDFWLRQ

PAGE 56

N K LM $W $W f $= N N N L M L M f N & LM N 6 & LM (TXDWLRQ f ZDV UHGXFHG WR n N N N N $K % K & K K LM }M LM MO LM M LM LOM ZKHUH N ( K ) ,M MO M N $W LM $ f ,M N $= & LM N $W U LM LM % f LM N $U U & LM LM N $W LM f LM N $= & LM N $ W U LM L M ( ‘ f LM N $U U & LM LM f f f f f

PAGE 57

& $ % ( f LM LM M LM LM N N $W . $W N L M L M N ) K f§ f f§ 6 f LM LM N N LM $] & & M L%RXQGDU\ &RQGLWLRQV 7KH ERXQGDU\ FRQGLWLRQ IRU WKH VXUIDFH JULGV ZDV D IOX[ ERXQGDU\ FRQGLWLRQ ZKLFK DFFRXQWHG IRU LUULJDWLRQ DQG UDLQIDOO DW WKH VRLO VXUIDFH $W WLPHV ZKHQ QR LUULJDWLRQ RFFXUHG WKH VXUIDFH ERXQGDU\ ZDV D QR IOX[ ERXQGDU\ (YDSRUDWLRQ IURP WKH VRLO VXUIDFH ZDV LQFOXGHG LQ WKH ZDWHU H[WUDFWLRQ WHUP 7KH PRGLILFDWLRQ RI HTXDWLRQ f WR GHVFULEH WKH VXUIDFH ERXQGDU\ FRQGLWLRQ ZDV N N N N % K & K K ( K ) f OM OM OM OM M OM ZKHUH & % ( f OM OM OOM $ W N ) K f§ OM OM $ = 4V M N M N & f $ W N & f OM OM 7KH ORZHU ERXQGDU\ FRQGLWLRQ ZDV DOVR UHSUHVHQWHG DV D IOX[ ERXQGDU\ 7KH ORZHU ERXQGDU\ LQ WKLV UHVHDUFK ZDV DQ LPSHUPHDEOH ERXQGDU\ ZKLFK UHSUHVHQWHG WKH ERWWRP RI WKH O\VLPHWHU 7KH PRGLILFDWLRQ RI HTXDWLRQ f WR GHVFULEH WKH ORZHU QR IOX[ ERXQGDU\ ZDV

PAGE 58

N N N N $ K % K & K ( K ) QM QOM QM QM QM QM QM QMO QM ZKHUH & $ % ( QM QM QM Q N $W 4E $W N Q M M N ) K f§ f 6 QM Q N N QM $= & & QM QM 7KH ERXQGDU\ FRQGLWLRQV IRU WKH UDGLDO ERXQGDULHV ZHUH ERWK ERXQGDULHV 7KH QR IOX[ ERXQGDU\ FRQGLWLRQ IRU WKH RXWHU UDGLXV UHSUHVHQWHG WKH RXWHU ZDOO RI WKH O\VLPHWHU DQG ZDV ZULWWHQ DV N N N N $ K % K & K K ) LP LOP LP LP L P L P L P LOP LP ZKHUH & $ % L P LP L P L P N N $W . $ W N L P LOP N ) K I f f f f f LP LP N N LP $] & & f QP QP 7KH ERXQGDU\ FRQGLWLRQ IRU WKH LQQHU UDGLLUV ZDV DOVR UHSUHVHQWHG DV D QR IOX[ ERXQGDU\ )RU WKH LQQHU UDGLXV ULMf DQG LWV LQFOXVLRQ LQWR HTXDWLRQ f ZRXOG UHVXOW LQ D GLYLVLRQ E\ ]HUR 7KHUHIRUH WKH VHFRQG WHUP LQ HTXDWLRQ f ZDV UHZULWWHQ DV K .K D K U .Kf Bf f§ f§ .Kf Bf UU U U D U U D U U U f

PAGE 59

GK D QR IOX[ ERXQGDU\ DW U LPSOLHV WKDW f§ 7KHQ LI GU GK GK f§ WKH OLP :H UHZULWH HTXDWLRQ f DV GU U GU DK K D K .Kf & f§ f§ .Kf f§ f f§ .Kf f§ f 6]UWf f DW ] = U U ] 7KH ILQLWH GLIIHUHQFH HTXDWLRQ IRU HTXDWLRQ f DW WKH LQQHU UDGLXV PD\ EH ZULWWHQ DV N N N $W $W . f L N L L fK N N $= & $ = & N N $W $ W L L N f K f K N L N $ U & $ ] & $W L N f K N L $ U & K N N $W . L L B f $W $= N L X RU DV N N N N $K & K K ( K ) LO L L LO LO L L L f f

PAGE 60

ZKHUH N $W L $ f f LO N $] & N $W L f f L N $ ] & 8 N $W L ( f f LO N $ U & & $ ( f $ W N ) K B $ = N N N L L f N & $W N 6 N L & f 8SGDWLQJ RI WKH VRLO SDUDPHWHUV VXUIDFH LQILOWUDWLRQ PDVV EDODQFH DQG WLPH VWHSV ZHUH LPSOLPHQWHG DV GHVFULEHG IRU WKH RQHGLPHQVLRQDO VRLO ZDWHU IORZ PRGHO 7KH V\VWHP RI HTXDWLRQV IRU WKH WZR GLPHQVLRQDO PRGHO SURGXFHV D EDQGHG PDWUL[ 7KH PDWUL[ LV D ILYH EDQGHG PDWUL[ ZLWK WKH IRUP RI

PAGE 61

5HGGHO DQG 6LPDGD f XWLOL]HG D *DXVV HOLPLQDWLRQ PHWKRG IRU WKH VROXWLRQ RI D WZRGLPHQVLRQDO JURXQGZDWHU PRGHO 7KH\ XVHG DQ DOJRULWKP GHYHORSHG E\ 7KXUQDX f ZKLFK RSHUDWHV RQO\ RQ WKH EDQGHG SDUW RI WKH VROXWLRQ PDWUL[ &RPSXWHU VWRUDJH LV QRW UHTXLUHG IRU WKH PDWUL[ HOHPHQWV DERYH RU EHORZ WKH EDQG $ PLQLPXP EDQG ZLGWK LV GHVLUDEOH DQG DQ DSSURSULDWH FKRLFH RI WKH JULG QXPEHULQJ SDWWHUQ FDQ UHGXFH WKH WRWDO ZLGWK RI WKH EDQG ,Q WKLV PRGHO RI WZRGLPHQVLRQDO VRLO ZDWHU PRYHPHQW DQG H[WUDFWLRQ D VXEURXWLQH ZDV ZULWWHQ ZKLFK XVHG WKH %$1'62/9( DOJRULWKP GHYHORSHG E\ 7KXUQDX f WR VROYH WKH V\VWHP RI HTXDWLRQV $ VXEURXWLQH ZDV DOVR ZULWWHQ ZKLFK VROYHV HTXDWLRQ f XVLQJ DQ DOWHUQDWLQJ GLUHFWLRQ LPSOLFLW PHWKRG 7KH ILQLWH GLIIHUHQFH HTXDWLRQV IRU WKH $', VROXWLRQ DUH SUHVHQWHG LQ $SSHQGL[ $ 6RLO :DWHU ([WUDFWLRQ $ PDFURVFRSLF VRLO ZDWHU H[WUDFWLRQ WHUP ZDV XVHG LQ WKLV UHVHDUFK 7KH PRGHO XVHG ZDV EDVHG RQ DFWXDO ILHOG PHDVXUHPHQWV UHSRUWHG LQ WKH OLWHUDWXUH 'HQPHDG DQG 6KDZ f FRQGXFWHG H[SHULPHQWV ZKLFK FRPSDUHG DFWXDO (7 WR SRWHQWLDO (7 RI FRUQ DV D IXQFWLRQ RI WKH DYDLODEOH VRLO ZDWHU 7KHLU VWXG\ VKRZHG WKDW XQGHU KLJK SRWHQWLDO (7 GHPDQGV WKH DFWXDO (7 ZDV FRQVLGHUDEO\ OHVV WKDQ WKH SRWHQWLDO UDWH HYHQ WKRXJK WKH DYDLODEOH VRLO ZDWHU ZDV FRQVLGHUHG DGHTXDWH 7KH\ DOVR REVHUYHG WKDW XQGHU ORZ SRWHQWLDO (7 GHPDQGV WKH DFWXDO (7 ZDV HTXDO WR

PAGE 62

WKH SRWHQWLDO (7 GRZQ WR YHU\ ORZ VRLO ZDWHU FRQWHQWV 5LWFKLH f REVHUYHG VLPLODU UHVXOWV LQ VHSDUDWH H[SHULPHQWV 6D[WRQ HW DO f XVHG WKHVH REVHUYHG UHODWLRQVKLSV LQ PRGHOLQJ VRLO ZDWHU PRYHPHQW DQG H[WUDFWLRQ XQGHU D FRUQ DQG XQGHU D JUDVV FURS 6PDMVWUOD f XVHG WKHVH UHODWLRQVKLSV WR PRGHO WKH VRLO ZDWHU VWDWXV XQGHU D JUDVV FRYHU FURS LQ )ORULGD 7KH VRLO ZDWHU H[WUDFWLRQ IXQFWLRQ XVHG LQ WKLV UHVHDUFK ZDV PRGHOHG DIWHU WKDW XVHG E\ 6PDMVWUOD f 7KH VRLO ZDWHU H[WUDFWLRQ UDWH ZDV FDOFXODWHG DV D IXQFWLRQ RI WKH DFWXDO HYDSRWUDQVSLUDWLRQ UDWH DQG WKH FXUUHQW VRLO ZDWHU VWDWXV 7KH ZDWHU H[WUDFWLRQ WHUP ZDV GHILQHG DV 6A (7 5')W 5 (7 5M f f ZKHUH 6A WKH VRLO ZDWHU H[WUDFWLRQ UDWH SHU VRLO ]RQH (7 WKH DFWXDO HYDSRWUDQVSLUDWLRQ UDWH 5')c WKH UHODWLYH ZDWHU H[WUDFWLRQ SHU VRLO ]RQH DW ILHOG FDSDFLW\ DQG 5 WKH UHODWLYH DYDLODEOH VRLO ZDWHU SHU VRLO ]RQH GHILQHG DV ZS f pIF b f f ZKHUH H WKH VRLO ZDWHU FRQWHQW RI WKH VRLO ]RQH :S WKH VRLO ZDWHU FRQWHQW DW ZLOWLQJ SRLQW DQG IF WKH VRLO ZDWHU FRQWHQW DW ILHOG FDSDFLW\ 7KH UHODWLYH ZDWHU H[WUDFWLRQ SHU VRLO ]RQH 5')f ZDV GHILQHG DV WKH SHUFHQWDJH RI ZDWHU H[WUDFWLRQ IRU WKH LWK VRLO ]RQH ZKHQ VRLO ZDWHU LV QRW OLPLWLQJ ,W PD\ DOVR EH FRQVLGHUHG WR EH D URRWLQJ DFWLYLW\ WHUP ZKLFK LQGLFDWHV WKH SHUFHQWDJH RI DFWLYH URRWV LQ WKH LWK VRLO ]RQH

PAGE 63

)LJXUH VKRZV WKH IRUP RI HTXDWLRQ f (TXDWLRQ f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f ZDV XVHG WR OLPLW ZDWHU H[WUDFWLRQ IURP D JLYHQ VRLO ]RQH DV WKH ZDWHU LQ WKDW ]RQH EHFDPH OHVV DYDLODEOH (TXDWLRQ f ZDV QRW XVHG WR OLPLW WKH (7 IURP WKH SURILOH RQO\ WR UHSDUWLWLRQ WKH H[WUDFWLRQ ZLWKLQ WKH VRLO SURILOH (TXDWLRQ f ZDV DSSOLHG WR FDOFXODWH D ZDWHU H[WUDFWLRQ UDWH IURP HDFK JULG IRU HDFK WLPH VWHS GXULQJ PRGHO RSHUDWLRQ 7KH VRLO ZDWHU H[WUDFWLRQ UDWH SHU VRLO JULG ZDV WKHQ PXOWLSOLHG E\ WKH JULG YROXPH WR FDOFXODWH WKH WRWDO ZDWHU H[WUDFWLRQ DW WKDW WLPH VWHS ,I WKH FDOFXODWHG WRWDO ZDWHU H[WUDFWLRQ UDWH ZDV QRW HTXDO WR WKH DFWXDO (7 WKH VRLO ZDWHU H[WUDFWLRQ UDWHV ZHUH OLQHDUO\ DGMXVWHG VR WKDW WKH FDOFXODWHG DQG DFWXDO (7 UDWHV ZHUH HTXDO 7KH DGMXVWHG VRLO ZDWHU H[WUDFWLRQ UDWHV 6cf ZHUH FRPSXWHG DV (7 ZKHUH $Y LV WKH YROXPH RI WKH LWK VRLO JULG f

PAGE 64

5(/$7,9( 62,/ :$7(5 (;75$&7,21 5$7( )LJXUH 5HODWLRQVKLS RI WKH UHODWLYH DYDLODEOH VRLO ZDWHU DQG SRWHQWLDO VRLO ZDWHU H[WUDFWLRQ UDWH RQ WKH VRLO ZDWHU H[WUDFWLRQ UDWH

PAGE 65

&+$37(5 ,9 5(68/76 $1' ',6&866,21 ,QVWUXPHQWDWLRQ 3HUIRUPDQFH 7KUHH SUHVVXUH WUDQVGXFHUV ZHUH LQGLYLGXDOO\ FDOLEUDWHG WR DQDO\]H WKHLU UHVSRQVH FKDUDFWHULVWLFV $ W\SLFDO FDOLEUDWLRQ FXUYH IRU SUHVVXUH WUDQVGXFHU 1R f LV VKRZQ LQ )LJXUH 7KH WKUHH FDOLEUDWLRQ HTXDWLRQV DUH 9ROWV N3Df 5 f 9ROWV N3Df 5 f 9ROWV N3Df 5 f 7KH FRHIILFLHQWV IRU WKHVH HTXDWLRQV ZHUH QRW VLJQLILFDQWO\ GLIIHUHQW D f VR WKDW IRU PDQ\ DSSOLFDWLRQV RQH HTXDWLRQ FRXOG EH XVHG IRU DOO SUHVVXUH WUDQVGXFHUV ZLWKRXW VLJQLILFDQW HUURU )RU WKH DERYH WUDQVGXFHUV FRPELQHG HTXDWLRQV UHVXOWHG LQ D PD[LPXP H[SHFWHG HUURU RI N3D DW YROW N3D SRWHQWLDOf DQG N3D DW YROWV N3D SRWHQWLDOf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

PAGE 66

)LJXUH &DOLEUDWLRQ FXUYH RI RXWSXW YROWDJH YHUVXV SUHVVXUH DSSOLHG IRU SUHVVXUH WUDQVGXFHU QR

PAGE 67

7KH $' ERDUG ZDV IRXQG WR KDYH D OLQHDU UHODWLRQVKLS EHWZHHQ YROWDJH LQSXW DQG GLJLWDO RXWSXW DV VKRZQ LQ )LJXUH 7KH HTXDWLRQ ZKLFK UHODWHG WKH YROWDJH WR GLJLWDO RXWSXW ZDV 'LJLWDO 8QLWV 9ROWVf 5 f (TXDWLRQ f ZDV FRPELQHG ZLWK WKH LQGLYLGXDO WUDQVGXFHU FDOLEUDWLRQ HTXDWLRQV f WR REWDLQ UHODWLRQVKLSV EHWZHHQ WUDQVGXFHUPHDVXUHG ZDWHU SRWHQWLDOV DQG WKH GLJLWDO LQSXWV WR WKH PLFURFRPSXWHU (TXDWLRQ f LV RQO\ YDOLG LQ WKH UDQJH RI YROWV DQG XQLWV IRU WKH ELW $' ERDUG XVHG 7KH ELW ERDUG WKHUHIRUH DOORZHG D UHVROXWLRQ RI RQO\ WKH QHDUHVW YROWV *UHDWHU UHVROXWLRQ FRXOG EH DFKLHYHG E\ WKH XVH RI D ELW ELW RU RWKHU KLJKHU UHVROXWLRQ $' ERDUG $ UHVROXWLRQ RI YROWV LV D UHVROXWLRQ RI DSSUR[LPDWHO\ N3D IURP HTXDWLRQV f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b DFFXUDF\ OLQH

PAGE 69

)LJXUH &RPSDULVRQV RI PHUFXU\ PDQRPHWHU PDQXDOO\UHDG DQG SUHVVXUH WUDQVGXFHU DXWRPDWLFDOO\UHDG WHQVLRPHWHU ZDWHU SRWHQWLDOV GXULQJ GU\LQJ F\FOHV IRU WZR WHQVLRPHWHUV LQ WKH ODERUDWRU\

PAGE 70

35(6685( 75$16'8&(5 :$7(5 327(17,$/ N3D )LJXUH &RPSDULVRQV RI PHUFXU\ PDQRPHWHU PDQXDOO\UHDG DQG SUHVVXUH WUDQVGXFHU DXWRPDWLFDOO\UHDG WHQVLRPHWHU ZDWHU SRWHQWLDOV IRU DOO ODERUDWRU\ GDWD

PAGE 71

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

PAGE 72

7(16,20(7(5 :$7(5 327(17,$/ N3D )LJXUH (YDOXDWLRQ RI SUHVVXUH WUDQVGXFHUWHQVLRPHWHU QR E\ FRPSDULVRQ RI PHUFXU\ PDQRPHWHU PDQXDOO\UHDG DQG SUHVVXUH WUDQVGXFHU DXWRPDWLFDOO\UHDG ZDWHU SRWHQWLDOV LQ WKH ILHOG

PAGE 73

7(16,20(7(5 :$7(5 327(17,$/ N3D ‘ , 7,0( +2856 )LJXUH (YDOXDWLRQ RI SUHVVXUH WUDQVGXFHUWHQVLRPHWHU QR E\ FRPSDULVRQ RI PHUFXU\ PDQRPHWHU PDQXDOO\UHDG DQG SUHVVXUH WUDQVGXFHU DXWRPDWLFDOO\UHDG ZDWHU SRWHQWLDOV LQ WKH ILHOG

PAGE 74

7(16,20(7(5 :$7(5 327(17,$/ N3D )LJXUH (YDOXDWLRQ RI SUHVVXUH WUDQVGXFHUWHQVLRPHWHU QR E\ FRPSDULVRQ RI PHUFXU\ PDQRPHWHU PDQXDOO\UHDG DQG SUHVVXUH WUDQVGXFHU DXWRPDWLFDOO\UHDG ZDWHU SRWHQWLDOV LQ WKH ILHOG

PAGE 75

7KH WHQVLRPHWHU GDWD ZHUH XVHG WR FDOFXODWH WKH VRLO ZDWHU H[WUDFWLRQ IURP WKH O\VLPHWHUV $ PHWKRG GHVFULEHG E\ YDQ %DYHO HW DO f ZDV XVHG WR FDOFXODWH WKH VRLO ZDWHU H[WUDFWLRQ 9DQ %DYHOnV PHWKRG XVHG DQ LQWHJUDWHG IRUP RI HTXDWLRQ f WR FDOFXODWH WKH VRLO ZDWHU H[WUDFWLRQ IRU D RQHGLPHQVLRQDO VRLO SURILOH ZKLFK ZDV GHVFULEHG DV ] f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

PAGE 76

'LIIHUHQFHV EHWZHHQ WKH (7 UDWHV LQ )LJXUHV DQG VKRZ WKDW WKH (7 UDWHV IRU WKH WUHH ZLWK JUDVV FRYHU ZHUH JUHDWHU WKDQ WKRVH RI WKH WUHH ZLWK QR JUDVV FRYHU 7KH KLJKHU (7 UDWHV IRU WKH WUHH ZLWK JUDVV FRYHU ZHUH H[SHFWHG DQG ZHUH DOVR REVHUYHG E\ 6PDMVWUOD HW DO f 7KH GLVWULEXWLRQ RI ZDWHU H[WUDFWLRQ IRU WKH WZR FLWUXV WUHH WUHDWPHQWV DUH VKRZQ LQ 7DEOH 7KH ZDWHU H[WUDFWLRQ UDWHV DUH VKRZQ DV SHUFHQWDJHV RI WKH WRWDO VRLO ZDWHU H[WUDFWLRQ DQG VRLO ZDWHU H[WUDFWLRQ SHU VRLO ]RQH 7KHVH YDOXHV ZHUH FRPSXWHG IRU LQSXW LQWR WKH ZDWHU H[WUDFWLRQ PRGHOV 7DEOH VKRZV WKH GLIIHUHQFHV EHWZHHQ WKH VRLO ZDWHU H[WUDFWLRQV IRU WKH WZR WUHDWPHQWV $SSUR[LPDWHO\ b RI WKH ZDWHU IURP WKH WUHH ZLWK QR JUDVV FRYHU ZDV H[WUDFWHG IURP WKH WRS PP RI WKH VRLO SURILOH 7KH ZDWHU H[WUDFWLRQ IRU WKH WUHH ZLWK JUDVV FRYHU VKRZV WKDW DSSUR[LPDWHO\ b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

PAGE 77

,7 fR F FH ] FH !f§ RB WQ ] r& FH R D mF af§ -8/,$1 '$< )LJXUH (YDSRWUDQVSLUDWLRQ UDWH IURP WKH ILHOG H[SHULPHQWV ZLWK D N3D VRLO ZDWHU SRWHQWLDO WUHDWPHQW \RXQJ FLWUXV WUHH ZLWK JUDVV FRYHU &7! FQ

PAGE 78

)LJXUH (YDSRWUDQVSLUDWLRQ UDWH IURP WKH ILHOG H[SHULPHQW ZLWK D N3D VRLO ZDWHU SRWHQWLDO WUHDWPHQW \RXQJ FLWUXV WUHH ZLWK EDUH VRLO FU! A

PAGE 79

7DEOH 2QHGLPHQVLRQDO GLVWULEXWLRQ RI ZDWHU H[WUDFWLRQ IRU D GD\ GU\LQJ F\FOH IRU
PAGE 80

7DEOH 7ZRGLPHQVLRQDO GLVWULEXWLRQ RI ZDWHU H[WUDFWLRQ IRU D GD\ GU\LQJ F\FOH IRU \RXQJ &LWUXV ZLWK JUDVV FRYHU 'HSWK 'HSWK ,QFUHPHQW PPf PPf 5DGL XV UUPf 5HODWLYH :DWHU ([WUDFWLRQ f b f EHORZ :DWHU ([WUDFWLRQ PPf EHORZ :DWHU ([WUDFWLRQ PPPPf EHORZ

PAGE 81

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f DQG +LOHU DQG %KXL\DQ f 7KHLU ZRUN SURYLGHG GDWD IURP VRLOV ZLWK ZLGO\ GLIIHUHQW K\GUDXOLF SURSHUWLHV DQG DOVR SURYLGHG VLPXODWLRQ UHVXOWV ZKLFK ZHUH XVHG IRU PRGHO YHULILFDWLRQ

PAGE 82

-8/,$1 '$< )LJXUH (YDSRWUDQVSLUDWLRQ UDWH IURP WKH ILHOG H[SHULPHQW ZLWK D N3D VRLO ZDWHU SRWHQWLDO WUHDWPHQW \RXQJ FLWUXV WUHH ZLWK JUDVV FRYHU ‘

PAGE 83

)LJXUH (YDSRWUDQVSLUDWLRQ UDWH IURP WKH ILHOG H[SHULPHQW ZLWK D N3D VRLO ZDWHU SRWHQWLDO WUHDWPHQW \RXQJ FLWUXV WUHH ZLWK EDUH VRLO UR

PAGE 84

7DEOH 7ZRGLPHQVLRQDO GLVWULEXWLRQ RI ZDWHU H[WUDFWLRQ IRU D GD\ GU\LQJ F\FOH IRU \RXQJ FLWUXV WUHHV ZLWK DQG ZLWKRXW JUDVV FRYHU 'HSWK 'HSWK 5DGLXV ,QFUHPHQW PPf PPf UUPf 7UHH ZLWK 7UHH ZLWK *UDVV &RYHU 1R *UDVV &RYHU 5HODWLYH :DWHU ([WUDFWLRQ bf :DWHU ([WUDFWLRQ QPf :DWHU ([WUDFWLRQ PPPPf

PAGE 85

Xf n -8/,$1 '$< )LJXUH (YDSRWUDQVSLUDWLRQ UDWH IURP WKH ILHOG H[SHULPHQW ZLWK D N3D VRLO ZDWHU SRWHQWLDO WUHDWPHQW ZLWK D JUDVV FRYHU S}

PAGE 86

_ / -8/,$1 '$< )LJXUH (YDSRWUDQVSLUDWLRQ UDWH IURP WKH ILHOG H[SHUPHQW ZLWK D N3D VRLO ZDWHU SRWHQWLDO WUHDWPHQW ZLWK D JUDVV FRYHU A FQ

PAGE 87

7DEOH 2QHGLPHQVLRQDO GLVWULEXWLRQ RI ZDWHU H[WUDFWLRQ IRU D GU\ GU\LQJ F\FOH IRU D JUDVV FRYHU FURS DW ZDWHU GHSOHWLRQ OHYHOV RI N3D DQG N3D 'HSWK 'HSWK :DWHU :DWHU 5HODWLYH ,QFUHPHQW ([WUDFWLRQ ([WUDFWLRQ :DWHU ([WUDFWLRQ PPf PPf ULPf PPPPf rf N3D 7UHDWPHQW 7UHH ZLWK *UDVV &RYHU N3D 7UHDWPHQW 7UHH ZLWK *UDVV &RYHU ‘‘0L

PAGE 88

5XELQ DQG 6WHLQKDUGW f XVHG DQ LPSOLFLW VROXWLRQ RI WKH 5LFKDUGV HTXDWLRQ LQ WHUPV RI ZDWHU FRQWHQW 7KHLU PRGHO ZDV XVHG WR VWXG\ FRQVWDQW LQWHQVLW\ UDLQIDOO LQILOWUDWLRQ RQ D 5HKRYRW VDQG 7KH VRLO GDWD ZHUH SUHVHQWHG DV DQDO\WLF IXQFWLRQV ZKLFK ZHUH WDEXODWHG IRU XVH LQ WKLV ZRUN 7KH VRLO K\GUDXOLF FKDUDFWHULVWLFV DUH VKRZQ LQ )LJXUHV DQG 7KH VLPXODWHG VRLO SURILOH KDG D XQLIRUP LQLWLDO VRLO ZDWHU FRQWHQW RI $ XQLIRUP JULG VL]H RI PP ZDV XVHG )LJXUHV DQG FRQWDLQ SORWV RI VRLO ZDWHU FRQWHQW SURILOHV IRU LQILOWUDWLRQ LQWR WKH 5HKRYRW VDQG 7KH ILJXUHV VKRZ ERWK WKH UHVXOWV IURP 5XELQ DQG 6WHLQKDUGW f DQG WKLV ZRUN )LJXUH FRQWDLQV WKH VRLO ZDWHU FRQWHQW SURILOHV IRU D FRQVWDQW LQILOWUDWLRQ UDWH RI PPKU )LJXUH FRQWDLQV WKH VRLO ZDWHU FRQWHQW SURILOHV IRU D FRQVWDQW LQILOWUDWLRQ UDWH RI PPKU 5HVXOWV RI WKLV ZRUN DUH LQ H[FHOOHQW DJUHHPHQW ZLWK WKRVH RI 5XELQ DQG 6WHLQKDUGW f 7KH ZRUN RI +LOHU DQG %KXL\DQ f ZDV DOVR XVHG WR YHULI\ WKH DFFXUDF\ RI WKH PRGHO GHYHORSHG LQ WKLV ZRUN 7KH\ XVHG D FRPSXWHU PRGHO ZULWWHQ LQ &063 WR VROYH WKH 5LFKDUGV HTXDWLRQ 6XUIDFH LQILOWUDWLRQ ZDV VLPXODWHG IRU WZR VRLOV
PAGE 89

&RPSDULVRQ RI UHVXOWV IRU WKH WZR VRLOV VKRZHG H[FHOOHQW DJUHHPHQW EHWZHHQ +LOHU DQG %KXL\DQnV &063 PRGHO DQG WKLV ZRUN 7KH PLQRU GLIIHUHQFHV EHWZHHQ WKH UHVXOWV RI WKH PRGHOV FRXOG KDYH UHVXOWHG IURP WKH LQWHUSRODWLRQ RI WKH VRLO SURSHUWLHV 7KH WZRGLPHQVLRQDO PRGHO ZDV WHVWHG E\ VLPXODWLQJ LQILOWUDWLRQ IRU D 1DKDO 6LQDL VDQG %UHVOHU HW DO f 5HVXOWV ZHUH FRPSDUHG ZLWK D VWHDG\VWDWH DQDO\WLFDO VROXWLRQ GHYHORSHG E\ :RRGLQJ f DQG ZLWK DQ $',1HZWRQ QXPHULFDO VLPXODWLRQ PHWKRG SUHVHQWHG E\ %UDQGW HW DO f 7KHVH PRGHOV VLPXODWHG WKH LQILOWUDWLRQ RI ZDWHU LQWR WKH VRLO IURP D SRLQW VRXUFH 7KH ERXQGDU\ FRQGLWLRQ XVHG IRU WKH VLPXODWLRQV ZDV D FRQVWDQW IOX[ IRU WKH LQQHUPRVW VXUIDFH JULG U f 7KH IOX[ ZDV VHW HTXDO WR WKH VDWXUDWHG K\GUDXOLF FRQGXFWLYLW\ RI WKH VRLO 1R IORZ ERXQGDULHV ZHUH XVHG IRU WKH ORZHU DQG UDGLDO ERXQGDULHV 7KH VLPXODWHG VRLO SURILOH KDG D XQLIRUP LQLWLDO VRLO ZDWHU FRQWHQW RI )LJXUHV DQG VKRZ WKH VRLO FKDUDFWHULVWLFV IRU WKH 1DKDO 6LQDL VDQG )LJXUHV DQG VKRZ D FRPSDULVRQ EHWZHHQ WKLV ZRUN DQG WKDW RI :RRGLQJ f DQG %UDQGW HW DO f 7KH GDWD DUH SUHVHQWHG LQ GLPHQVLRQOHVV IRUP ZLWK WKH UHODWLYH ZDWHU FRQWHQW GHILQHG DV 6f6VDWf ZKHUH 6f ZDV GHILQHG E\ 3KLOLS f DV H 6Hf ? GH f Q ZKHUH WKH K\GUDXOLF GLIIXVLYLW\ DQG Q WKH UHVLGXDO VRLO ZDWHU FRQWHQW 7KH WRWDO LQILOWUDWLRQ WLPH ZDV PLQ LQ )LJXUH DQG PLQ LQ )LJXUH )LJXUHV DQG GHPRQVWUDWH WKDW WKLV PRGHO DFFXUDWHO\ VLPXODWHV WKH UHVXOWV RI %UDQGW HW DO f 'LIIHUHQFHV EHWZHHQ WKH

PAGE 90

62,/ :$7(5 327(17,$/ PPf )LJXUH 6RLO ZDWHU SRWHQWLDOVRLO ZDWHU FRQWHQW UHODWLRQVKLS IRU 5HKRYRW VDQG

PAGE 91

+<'5$8/,& &21'8&7,9,7< PPKUf )LJXUH +\GUDXOLF FRQGXFWLYLW\VRLO ZDWHU FRQWHQW UHODWLRQVKLS IRU 5HKRYRW VDQG

PAGE 92

'(37+ PPf 62,/ :$7(5 &217(17 PPPPf )LJXUH 6LPXODWHG UHVXOWV RI VRLO ZDWHU FRQWHQW SURILOHV IRU LQILOWUDWLRQ LQWR D 5HKRYRW VDQG XQGHU FRQVWDQW UDLQ LQWHQVLW\ RI PPKU

PAGE 93

'(37+ PPf )LJXUH 62,/ :$7(5 &217(17 PPPPf 6LPXODWHG UHVXOWV RI VRLO ZDWHU FRQWHQW SURILOHV IRU LQILOWUDWLRQ LQWR D 5HKRYRW VDQG XQGHU FRQVWDQW UDLQ LQWHQVLW\ RI PPKU

PAGE 94

62,/ :$7(5 327(17,$/ PPf 62,/ :$7(5 &217(17 PPPPf )LJXUH 6RLO ZDWHU SRWHQWLDOVRLO ZDWHU FRQWHQW UHODWLRQVKLS IRU
PAGE 95

+<'5$8/,& &21'8&7,9,7< ^PKUf )LJXUH +\GUDXOLF FRQGXFWLYLW\VRLO ZDWHU FRQWHQW UHODWLRQVKLS IRU
PAGE 96

62,/ :$7(5 327(17,$/ PPf )LJXUH 6RLO ZDWHU SRWHQWLDOVRLO ZDWHU FRQWHQW UHODWLRQVKLS IRU $GHODQWR ORDP

PAGE 97

+<'5$8/,& &21'8&7,9,7< PPKUf )LJXUH +\GUDXOLF FRQGXFWLYLW\VRLO ZDWHU FRQWHQW UHODWLRQVKLS IRU $GHODQWR ORDP

PAGE 98

'(37+ PPf 62,/ :$7(5 &217(17 PPPPf )LJXUH 6LPXODWHG UHVXOWV RI VRLO ZDWHU FRQWHQW SURILOHV IRU LQILOWUDWLRQ LQWR D
PAGE 99

3DJHV 0LVVLQJ RU 8QDYDLODEOH

PAGE 100

5(/$7,9( (9$3275$163,5$7,21 5$7( 7,0( 2) '$< 7 )LJXUH 'LVWULEXWLRQ RI HYDSRWUDQVSLUDWLRQ WKURXJK WKH GD\OLJKW KRXUV /2

PAGE 101

62,/ +$7(5 327(17,$/ PPf )LJXUH 6RLO ZDWHU SRWHQWLDOVRL ZDWHU FRQWHQW UHODWLRQVKLS IRU $UUHGRQGR ILQH VDQG

PAGE 102

+<'5$8/,& &21'8&7,9,7< PPKUf )LJXUH +\GUDXOLF FRQGXFWLYLW\VRLO ZDWHU SRWHQWLDO UHODWLRQVKLS IRU $UUHGRQGR ILQH VDQG

PAGE 103

FRQGXFWLYLW\ IXQFWLRQ SURGXFHG UHVXOWV ZLWK WKH QXPHULFDO PRGHO ZKLFK ZHUH LQ DJUHHPHQW ZLWK WKRVH IURP WKH REVHUYHG ILHOG GDWD 2QHGLPHQVLRQDO 0RGHO 7LPH 6WHSV $ VHQVLWLYLW\ DQDO\VLV ZDV FRQGXFWHG WR HYDOXDWH RSWLPXP JULG VL]HV DQG WLPHVWHSV 7KH WLPH VWHS XVHG ZDV YDULDEOH DQG WKH FULWHULD IRU GHWHUPLQLQJ WKH YDULDEOH WLPH VWHS ZHUH SUHVHQWHG LQ HTXDWLRQ f 2SHUDWLRQ RI WKH PRGHO SURYLGHG GDWD IURP ZKLFK OLPLWV IRU PD[LPXP DQG PLQLPXP WLPH VWHSV ZHUH GHWHUPLQHG 7KH PD[LPXP WLPH VWHS IRU PRGHO RSHUDWLRQ ZDV GHWHUPLQHG WR EH KRXU 2SHUDWLRQ RI WKH PRGHO ZLWK JUHDWHU PD[LPXP WLPH VWHS SURGXFHG YDULQJ UHVXOWV ZKLOH RSHUDWLRQ ZLWK VPDOOHU PD[LPXP WLPH VWHSV JDYH UHVXOWV LQ DJUHHPHQW ZLWK WKH KRXU WLPHVWHS 7KH PD[LPXP WLPH VWHS ZDV D XSSHU OLPLW IRU WKH FDOFXODWHG WLPH VWHS 7KH PD[LPXP WLPH VWHS ZDV XVHG ZKHQ ZDWHU PRYHPHQW LQ WKH VRLO SURILOH ZDV VPDOO :KHQ ZDWHU PRYHPHQWV LQ WKH VRLO SURILOH ZHUH ODUJH DV GXULQJ LQILOWUDWLRQ D PLQLPXP WLPH VWHS ZDV GHWHUPLQHG 7KH PLQLPXP WLPH VWHS IRU WKH UDQJH RI LQILOWUDWLRQ XVHG LQ WKH PRGHO ZDV GHWHUPLQHG WR EH KRXUV 7KH XVH RI D VPDOOHU WLPH VWHS VORZHG WKH PRGHO H[HFXWLRQ ZKLOH QRW LPSURYLQJ WKH PDVV EDODQFH /DUJHU WLPH VWHSV FDXVHG WKH PDVV EDODQFH WR LQFUHDVH WKXV LQGLFDWLQJ D UHGXFWLRQ LQ PRGHO DFFXUDF\ *ULG 6L]H 7KH RSWLPXP JULG VL]HV ZHUH DOVR GHWHUPLQHG E\ VHQVLWLYLW\ DQDO\VLV $ JULG VL]H RI PP ZDV GHWHUPLQHG WR EH DGHTXDWH WR GHVFULEH ZDWHU PRYHPHQW LQ WKH VRLO SURILOH IRU WKH RQHn GLPHQVLRQDO PRGHO /DUJHU JULG VL]HV GHFUHDVHG PRGHO H[HFXWLRQ WLPH EXW WKH VRLO ZDWHU GLVWULEXWLRQ ZDV QRW DGHTXDWHO\ GHVFULEHG 6PDOOHU JULG VL]HV LQFUHDVHG WKH H[HFXWLRQ WLPH ZKLOH QRW LPSURYLQJ WKH PRGHO DFFXUDF\

PAGE 104

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

PAGE 105

62,/ :$7(5 &217(17 PPPPf -8/,$1 '$< )LJXUH 6LPXODWLRQ UHVXOWV IRU WKH RQHGLPHQVLRQDO PRGHO ZLWK ILHOG GDWD IR WKH PP GHSWK f§ )LHOG GDWD ‘ 0RGHO UHVXOWV O f U

PAGE 106

62,/ :$7(5 &217(17 PPPPf -8/,$1 '$< )LJXUH 6LPXODWLRQ UHVXOWV IRU WKH RQHGLPHQVLRQDO PRGHO ZLWK ILHOG GDWD IRU WKH PP GHSWK

PAGE 107

62,/ :$7(5 &217(17 PPPPf IW )LHOG GDWD 0RGHO UHVXOWV f§ L ‘ U -8/,$1 '$< )LJXUH 6LPXODWLRQ UHVXOWV IRU WKH RQHGLPHQVLRQDO PRGHO ZLWK ILHOG GDWD IRU WKH PP GHSWK R 2n}

PAGE 108

62,/ :$7(5 &217(17 PPPPf )LJXUH 6LPXODWLRQ UHVXOWV IRU WKH RQHGLPHQVLRQDO PRGHO ZLWK ILHOG GDWD IRU WKH PP GHSWK

PAGE 109

62,/ :$7(5 &217(17 PPPPf f§ )LHOG GDWD ‘ 0RGHO UHVXOWV M ‘ -8/,$1 '$< )LJXUH 6LPXODWLRQ UHVXOWV IRU WKH RQHGLPHQVLRQDO PRGHO ZLWK ILHOG GDWD IRU WKH PP GHSWK

PAGE 110

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f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

PAGE 111

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

PAGE 112

62,/ :$7(5 &217(17 PPPPf )LHOG GDWD 0RGHO UHVXOWV B -8/,$1 '$< )LJXUH 6LPXODWLRQ UHVXOWV IRU WKH WZRGLPHQVLRQDO PRGHO ZLWK ILHOG GDWD IRU WKH PP GHSWK

PAGE 113

62,/ :$7(5 &217(17 PPPPf D f§ / RRRf§ )LJXUH )LHOG GDWD 0RGHO UHVXOWV -8/,$1 '$< 6LPXODWLRQ UHVXOWV IRU WKH WZRGLPHQVLRQDO PRGHO ZLWK ILHOG GDWD IRU WKH PP GHSWK

PAGE 114

62,/ :$7(5 &217(17 PPPPf -8/,$1 '$< )LJXUH 6LPXODWLRQ UHVXOWV IRU WKH WZRGLPHQVLRQDO PRGHO ZLWK ILHOG GDWD IRU WKH PP GHSWK

PAGE 115

62,/ :$7(5 &217(17 PPPPf )LHOG GDWD 0RGHO UHVXOWV PP 5DGLXV PP 5DGLXV -8/,$1 '$< )LJXUH 6LPXODWLRQ UHVXOWV IRU WKH WZRGLPHQVLRQDO PRGHO ZLWK ILHOG GDWD IRU WKH PP GHSWK

PAGE 116

62,/ :$7(5 &217(17 PPPPf ‘ ‘ f§ ‘f§ f§f§ )LHOG GDWD 0RGHO UHVXOWV PP 5DGLXV PP 5DGLXV ‘ L L Af§X L L O PP 5DGLXV PP 5DGLXV L L -8/,$1 '$< )LJXUH 6LPXODWLRQ UHVXOWV IRU WKH WZRGLPHQVLRQDO PRGHO ZLWK ILHOG GDWD IRU WKH PP GHSWK

PAGE 117

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f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n GLPHQVLRQDO PRGHO ZDV D QR IOX[ ZKHQ LUULJDWLRQ ZDV QRW RFFXUULQJ

PAGE 118

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

PAGE 119

62,/ :$7(5 &217(17 PPPPf N3D N3D N3D 7 f§U L '$<6 )LJXUH 6RLO ZDWHU FRQWHQWV IRU WKH WKUHH LUULJDWLRQ WUHDWPHQWV DW WKH PP GHSWK B

PAGE 120

62,/ :$7(5 &217(17 PPPPf )LJXUH 6RLO ZDWHU FRQWHQWV IRU WKH WKUHH LUULJDWLRQ WUHDWPHQWV IRU WKH PP GHSWK f

PAGE 121

6725$*( PPf )LJXUH 6RLO ZDWHU VWRUDJH IRU WKH WKUHH LUULJDWLRQ WUHDWPHQWV UR R

PAGE 122

&808/$7,9( ,55,*$7,21 WQPf N3D N3D N3D '$<6 )LJXUH &XPXODWLYH LUULJDWLRQ IRU WKH WKUHH LUULJDWLRQ WUHDWPHQWV

PAGE 123

&808/$7,9( '5$,1$*( PPf '$<6 )LJXUH &XPXODWLYH GUDLQDJH IURP WKH VRLO SURILOH IRU WKH WKUHH LUULJDWLRQ WUHDWPHQWV

PAGE 124

WUHDWPHQWV 7KH GUDLQDJH RXW RI WKH DOO RI WKH WUHDWPHQWV ZDV ODUJH DW ILUVW EHFDXVH RI WKH LQLWLDO FRQGLWLRQV 7KH GUDLQDJH IRU WKH N3D WHDWPHQW ZDV VPDOO IRU WKH GXUDWLRQ RI WKH VLPXODWLRQ SHULRG 'UDLQDJH IURP WKH RWKHU WUHDWPHQWV ZHUH KLJKHU EHFDXVH RI WKH ODUJHU DPRXQWV RI ZDWHU DSSOLHG DW HDFK LUULJDWLRQ

PAGE 125

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

PAGE 126

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

PAGE 127

$33(1',; $ $OWHUQDWLQJ 'LUHFWLRQ ,PSOLFLW )LQLWH 'LIIHUHQFLQJ $Q DOWHUQDWLQJ GLUHFWLRQ LPSOLFLW VROXWLRQ PHWKRG ZDV GHYHORSHG WR VROYH WKH WZRGLPHQVLRQDO UDGLDO IORZ PRGHO 7KH VROXWLRQ PHWKRG GLYLGHV WKH VROXWLRQ LQWR WZR KDOI F\FOHV WKXV WZR F\FOHV DUH UHTXLUHG WR VROYH IRU RQH WLPH VWHS (TXDWLRQ f ZDV VROYHG LQ WKH UDGLDO GLUHFWLRQ GXULQJ WKH ILUVW KDOI F\FOH DQG WKHQ VROYHG LQ WKH YHUWLFDO GLUHFWLRQ GXULQJ WKH VHFRQG KDOI F\FOH 7KH $', VROXWLRQ IRU HTXDWLRQ f PD\ EH ZULWWHQ DV N N N N K N K N K & f§ U B f U f§f f W U U U U U N N N K N K f§f Bf f = = = = N N N . f§ f 6]UWf ] = = f 7KH ILQLWH GLIIHUHQFH HTXDWLRQV XVHG IRU WKH VROXWLRQ RI HTXDWLRQ f ZHUH H[SUHVVHG DV N N K K f LM L& LM $W f

PAGE 128

K f§. f = = K K f K K f LOM LM LM LM LOM f $] K BBU. Bff U K K f U U U LMO LMO LMO LM U $U LM U K K f f f LLM LM LM . L M M f (TXDWLRQ f ZDV GLYLGHG LQWR WZR SDUWV DQG DUUDQJHG VXFK WKDW WKH XQNQRZQV DUH RQ WKH OHIW VLGH RI WKH HTXDWLRQV DQG WKH NQRZQV DUH RQ WKH ULJKW VLGH VXFK WKDW $W $U ULMO U LM N LM N f K LM & LM N $W U LM LM N U f LM OM f N K LM $U U & LM LM

PAGE 129

N $W U LM L M D U U & LM LM N K N N $W . N L M LM K f§ f LM $ ] & $ ] N N $W . LM LM Lf $ W LM $ ] & N N f K K f M LL$ W $= & L M N N f K K f M LL$W & LLDQG $W N L M N f K $ ] & LM N N $W . f L M LM $= & LM N f K LM f

PAGE 130

N $W LM N f K LOM $] & L}M $W BB $] N N N W M L M f & LM $ W & LM N 6 LM N $W U LM LM $ U U & LM LM $W U N LM LM $U U & LM L N N K K f LM LN N K K f f LM LM ZKHUH L UHSUHVHQWV WKH YHUWLFDO GLVWDQFH M UHSUHVHQWV WKH UDGLDO GLVWDQFH DQG N UHSUHVHQWV WKH WLPH LQFUHPHQW 7KH V\VWHP RI HTXDWLRQV ZKLFK DUH IRUPHG E\ DSSO\LQJ HTXDWLRQV f DQG f WR HDFK JULG SRLQW SURGXFHV D WULGLDJRQDO PDWUL[ DW HDFK KDOI WLPH VWHS 7KH WULGLDJRQDO PDWULFHV ZHUH VROYHG XVLQJ D *DXVV HOLPLQDWLRQ PHWKRG IRU WULGLDJRQDO PDWULFHV (TXDWLRQV f DQG f PD\ EH VLPSOLILHG VXFK WKDW HTXDWLRQ f PD\ EH ZULWWHQ DV N N N $5 K %5 K &5 K '5 LM LM LMO LM f

PAGE 131

ZKHUH N $ W U OM LM $5 f D U U LLf N $W U LM &5 f DW U LLf DQG %5 $5 &5 f $W . '5 K LM $= M LM N N $W . M LM f $= Of $W M $= & N N ‘f K K f OM $W L M $= & N N f K K f M LL$W & LLf DQG HTXDWLRQ f PD\ EH ZULWWHQ DV

PAGE 132

$9 K ZKHUH N N N %9 K &9 K M OM N $W $9 M $ = & f LM '9 Lf f N $W M & 9 f f D] & L %9 $9 & 9 DQG '9 $W N K f§ OM $= N N . f & M $W N 6 f $W U ,M $ U U LM N LM N N f K K f OM F OM N $W U LM LM f $U U & LM M N N K K f LMO M f

PAGE 133

%RXQGDU\ &RQGLWLRQV 7KH ERXQGDU\ FRQGLWLRQ IRU WKH VXUIDFH OD\HUV ZDV D IOX[ ERXQGDU\ FRQGLWLRQ ZKLFK DFFRXQWHG IRU LUULJDWLRQ DQG UDLQIDOO DW WKH VRLO VXUIDFH $W WLPHV ZKHQ QR LUULJDWLRQ RFFXUHG WKH VXUIDFH ERXQGDU\ ZDV D QR IOX[ ERXQGDU\ (YDSRUDWLRQ IURP WKH VRLO VXUIDFH ZDV LQFOXGHG LQ WKH ZDWHU H[WUDFWLRQ WHUP 7KH PRGLILFDWLRQ RI HTXDWLRQV f DQG f WR GHVFULEH WKH VXUIDFH ERXQGDU\ FRQGLWLRQ ZDV N N N $5 K %5 K &5 K OM OM '5 OM f ZKHUH N $W 4V N M '5 K OM $= $W 4V M M f B M f $ ] M OM N $W f§ M N D] & OM $W f K KN f 6N M OM O& OM f DQG %9 K ZKHUH N N &9 K '9 OM M OM %9 &9 f f DQG

PAGE 134

'9 N K OM $W BB $] N 4V M M f & OM $W & OM N 6 OM N $W U OM OM $ U U & OM OM $W U N OM OM $ U U & OM M N N K K f OMO OM N N K K f f OM OM 7KH ORZHU ERXQGDU\ FRQGLWLRQ ZDV DOVR UHSUHVHQWHG DV D IOX[ ERXQGDU\ 7KH ORZHU ERXQGDU\ LQ WKLV UHVHDUFK ZDV DQ LPSHUPHDEOH ERXQGDU\ ZKLFK UHSUHVHQWHG WKH ERWWRP RI WKH O\VLPHWHU 7KH PRGLILFDWLRQV RI HTXDWLRQV f DQG f WR GHVFULEH WKH ORZHU ERXQGDU\ FRQGLWLRQ ZHUH N N N $5 K %5 K &5 K '5 QM QM Q M QM ZKHUH $5 %5 DQG '5 DUH DV SUHYLRXVO\ GHVFULEHG DQG '5 $W N K QM D] N QM 4E $W M f f§ N 4E Q M M f & $] & QM Q M f

PAGE 135

N $W QOM N N N f K K f 6 Q M QM QM D] & & Q}M QM $W f DQG N N $9 K %9 K Q M QM ZKHUH $9 ZDV SUHYLRXVO\ GHVFULEHG DQG %9 $9 DQG N $W 4E N QM M '9 K QM D] & QM '9 Q$W f 6 & Qf f QN $W U QM QM DW U N N f K K f QM QM QQN $W U QM QM D U U & N N f K K f QMO QM QQM f 7KH ERXQGDU\ FRQGLWLRQV IRU WKH UDGLDO ERXQGDULHV ZHUH ERWK QR IOX[ ERXQGDULHV 7KH QR IOX[ ERXQGDU\ FRQGLWLRQ IRU WKH RXWHU UDGLXV M Pf

PAGE 136

UHSUHVHQWHG WKH RXWHU ZDOO RI WKH O\VLPHWHU DQG HTXDWLRQV f DQG f ZHUH ZULWWHQ DV N N $5 K %5 K '5 f LPO LP LP ZKHUH DQG $ 9 K %5 $5 f N N N %9 K &9 K '9 f L P LP LOP LP ZKHUH N N $W . $W N LOP LOP N '9 K f 6 LP LP $ ] & & LP LP N $W U LP LPO N N f K K f LPO LP D U U L P L P f 7KH ERXQGDU\ FRQGLWLRQ IRU WKH LQQHU UDGLXV ZDV DOVR UHSUHVHQWHG DV D QR IOX[ ERXQGDU\ )RU WKH LQQHU UDGLXV ULMf WKHUHIRUH LWV LQFOXVLRQ LQWR HTXDWLRQ f ZRXOG UHVXOW LQ D GLYLVLRQ E\ ]HUR 7KHUHIRUH WKH VHFRQG WHUP LQ HTXDWLRQ f ZDV UHZULWWHQ DV GHVFULEHG LQ HTXDWLRQV f DQG f VXFK WKDW WKH ILQLWH GLIIHUHQFH HTXDWLRQV IRU WKH $', VROXWLRQ RI HTXDWLRQ f ZHUH H[SUHVVHG DV

PAGE 137

N N %5 K &5 K '5 L L L ZKHUH N $W L N &5 f K $U & %5 &5 DQG N N N $9 K %9 K &9 K '9 L ZKHUH N N $W . $W N L '9 K f f $= & & N $W L N N f K K f $U & 8SGDWLQJ RI WKH VRLO SDUDPHWHUV VXUIDFH LQILOWUDWLRQ PDVV f f f f f EDODQFH DQG WLPH VWHSV ZHUH LPSOLPHQWHG DV GHVFULEHG IRU WKH RQHGLPHQVLRQDO VRLO ZDWHU IORZ PRGHO

PAGE 138

$33(1',; % /LVWLQJ RI 6RLO :DWHU 0RYHPHQW DQG ([WUDFWLRQ 0RGHO

PAGE 139

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r0$; ,=r,5f ,03/,&,7 5($/ $+4=f ,17(*(5 080 287,6723, &2817 &+$5$&7(5r 0%),/( &20021 3$5$:&,=,5f&1',=,5f6:&,=,5f+($',=,5f 1/'=,=f'(37+,=f46,5f6,=,5f, &216715'5,5f 5,5f,/: &200217,0'77,0(7287f(1'7,0('70$;'70,17',7,0(40'7,55 &20021 7$%/(6:&17f&17f6:&7f+('7f

PAGE 140

R R R 1'$7$0%),/( ,62 ,, &2 0021 0$6%',))36725)/2:,1(;75$&76725$*()/2:287 &20021 :&&:.((3)/2:O,=,5f &20021 &219',),7(50$;,7(5 &20021 465'),=,5f:&:3:&)&,&) &20021 (77615f7616f(73f5$,1f'85-6723 &20021 '80<,),(/f -),(/'f ,)657,0( 237 7(1 ,7)f-7)f7+5(655$7(,6&+f &$// ,1387 &$// 0$6%$/2f :5,7(rrff 7+( ,1,7,$/ 0$66 %$/$1&( ,6 n :5,7( rf 7,0( '7',))36725 )/2:287 :5,7(rfn 7+( ,1,7,$/ 0$66 %$/$1&( ,6 n :5,7( f 7,0( '7',))36725 )/2:287 :5,7(rf6725$*()/2:,1(;75$&7 :5,7(f6725$*()/2:,1(; 75$& 7 71(: 57,0( ,),237(4f:5,7(f71(::&,),(/',f-),(/',ff ,)6f 67$57 2) 6,08/$7,21 /223 ,7 ,0( 0,7(5 ,'$< 7,0( &217,18( &$// )/8;(73)/; (73,'$
PAGE 141

& :5,7(rf6725$*()/2:,1(;75$&7 :5,7(f6725$*()/2:,1(;75$&7 )250$7 6725$*( ?(n &80B,1),/ n( n &80B(;75$&7 n (f & :5,7(f7,1( )250$7& 7,0( n )f '2 1/ :5,7(rf '(37+,f :&,-f 15f ,),237(4Of:5,7(f:&,f15Of:&,f 15f :5,7( f '(37+,f :&,-f 15f )50$7(f )250$7( ;(;(f & 71(: 57,1( 7,1( ,),237(4f:5,7(f71(::&,),(/',f-),(/',ff ,)6f ,),37(4f:5,7(f71(::&,),(/',f-),(/',ff, ,)6f 6725$*( )/2:,1 (;75$&7 )/2:287 )250$7;)(f (1' ,) & F 0,7(5 0,7(5 ,)0,7(5r(40,7(5f 7+(1 23(1),/( n7,0(87n,267$7 ,&f ,),&1(2f *2 72 :5,7(f7,0(0, 7(5 )250$7& 7,0( n )n ,7(5 nf &/26(f (1' ,) & ,)(1'7,0(/(7,1(f *2 72 *2 72 &217,18( 23 (1),/( n),1$/327nf '2 1/ :5, 7( f +($',-f15f &/46(f &/26(OOf (1' & & 68%5287,1( 72 5($' ,1 7+( ,1387 9$/8(6 )25 7+( 6,08/$7,21 & $1' ,1,7,$/,=( 7+( 9$5,$%/(6 & 68%5287,1( ,1387 3$5$0(7(5 ,= ,5 ,% ,5r0$; ,=r,5f ,03/,&,7 5($/ $+2=f &+$5$&7(5r ,13),/(0%),/(287),/(:'),/('$7$),/( &+$5$&7(5r = 5($/ ,:&217,6725 +(f(;f ,17(*(5 f--f &20021 7$%/(6:&17f&17f6:&7f+('7f 1'$7$0%),/(,62 ,/

PAGE 142

&20021 3$5$:&,=,5f&21',=,5 f6:&,=,5f+($',=,5f 1/'=,=f'(37+ ,=f46 ,5f6,=,5f,&216715'5,5f5,5f,/2: &20020LU0'77,0(787f(1'7,0('70$;'70,172',7,0(40'7,55 &20021 0$6% ,:&217,=,5f$5($,5f7$5($,6725 &20021 '80<,),(/'f -),(/'f ,)657,0(,237 ,7(1,7)f-7)f7+5(655$7(,6&+f &20021 &219',),7(50$; ,7(5 &2 :21 465' ),=,5f :&:3:&)&,&) &20021 (77615f7616f(73f5$,1f'8576723 &01 :&&:.((3 )/2:O ,=,5f 3,( :5,7(rr! (17(5 7+( ,1387 '$7$ ),/( 5($'rf '$7$),/( 3(1),/( '$7$),/(f 5($' f= 5($'f ,62,/,13),/( )50$7$f )250$7 ,, ; $f 23(1 ), /( ,13 ),/(f & & 5($' ,1 $1' :5,7( 7+( 62,/6 '$7$ & 5($'f 1'$7$ )250$7f 5($'f+('7,f:&17,f6:&7,f&217,f 1'$7$f )50$7(f )250$7n nf &/26(f F 5($'f = 5($'f0%),/( 5($'f:'),/( 3(1),/( 0%),/(f 5($'f = 5($'n ,, ; $fnf ,237287),/( 3(1),/( 287),/(f 5($'rf ,)6 5($' rf 57,0( ), (/' ,f -),(/' ,f ,)6f 5($'f = 5($'rf7,1( (1'7,0(72' 5($'f = 5($'rf:.((3:&:3:&)& 5($'f = 5($'rf7287f-87f787f787f ,)7287f1(f *2 72 '2 ,)787fr)/$7,f*7(1'7,0(f *2 72 7287,f 7287fr)/2$7,f &217,18( &217,18( 5($'f = 5($'rf'70$;'70,10$;,7(5'7,55'85 :5,7(rf0%),/(7,0(7287f(1'7,0('70$;'70,10$; ,7(5 :5,7(f0%),/(7,0(787f(1'7,0('70$;'70,10$; ,7(5

PAGE 143

)250$7n 7+( I$66 %$/$1&( '$7$ ,6 ,1 ),/( n $ n 7+( 67$57,1* 6,08/$7,21 7,0( ,6 n ( n 7+( 2873876 :,// %( $7 ,17(59$/6 ( n 7+( (1',1* 7,0( :,// %( n( n 7+( 0$;,080 7,0( 67(3 ,6 n( n 7+( 0,1,080 7,1( 67(3 ,6 n ( n 7+( 180%(5 2) ,7(5$7,216 nf & & 5($' ,1 7+( 180%(5 2) *5,'6 $1' 7+( *5,' 6,=(6 & 5($' f = 5($'rf 1/ '=f 15 '5f :5,7(rf1/'=f15'5f )250$7& 7+( 180%(5 2) 9(57,&$/ ,1&5(0(176 n n 7+( 9(5,&$/ 63$&,1* ,6 f ( n 7+( 180%(5 2) 5$',$/ ,1&5(0(176 n n 7+( 5$',$/ 63$&,1* ,6 n (f & '(37+Of '=f '2 1/ '=,f '=,Of '(37+,f '(37+f '=,f '=,Of f &217,18( & ,)15(4 f 15 $5($Of 7$5($ ,)15*(f 7+(1 5f '5f $5($Of 3,( r 5Of'5Offrr '2 15 '5,f '5,Of 5 ,f 5,Of '5 ,f '5,Of f $5($ ,f 3,(r 5,f'5,ffrr 5 ,f'5 ,ffrrf &217,18( 7$5($ 3,( r '5f r 15 frr :5,7( r f )250$7 n 7+( 5$',86 $1' $5($6 nf :5 ,7( r f 5 ,f $5($ ,f 15f )250$7& 5$',86 nf ( ;n $5($ f (f (1' ,) & & ,1387 7+( ,1,7,$/ 327(17,$/6 )25 7+( 6<67(0 & 5($'f= & 5($'f +($',-f,5f, ,=f & 5($'rf 1/1151,19 5($'rf ,f 1/1 f 5($'rf ---f151 f '2 1/1 5($'rf +(, -f151f 5($' f= &

PAGE 144

& 5($' ,1 7+( ,1),/75$7,21 5$7( & & 5($'f 46-f,5f 5($'f = 5($'f= 5($'rf ,&2167 5($'f= & & 5($' ,1 7+( &21',7,21 2) 7+( /2:(5 %281'5< &21',7,21 & $1' 7+( 67$786 2) 7+( &255(&7,21 )$&725 & 5($'rf,/2: ,&) 5($'f = & 5($' ,1 7+( 5227 ',675,%87,21 )81&7,21 & & 5( $' f 5') -f ,5f ,=f '2 1/1 5($'rf (;,-f151f & '2 '2 ,= '2 5 +($',-.f 5'),-f 6,-f &217,18( & 1 '2 1/1 '2 / ,f 1 1 0 '2 151 '2 ---f 0 0 +($'10f +(, -f +($'10 f +(,-f +($'10f +(,-f 0 ,),19(4Of 15 0 $$$ )/$7.rr .Ofrrf )/2$7 15rrf ,)15(4Of $$$ 5')10f (;, -f ,f r $$$ &217,18( & )50$7)f )250$7$Of )250$7 (f & &$// 83'$7( f ,6725 '2 1/ 15 ,:&217, -f :& f

PAGE 145

,6725 ,6725 :&,fr'=,fr$5($-f &217,18( & )250$7 ;(O ;f f 5($'rf ,7(1 7+5(6 55$7( ,6& 5($'rf ,7),f-7),f ,7(1 f '2 ,'$< ,6&+,f ,6& &/6(f & & 5($' ,1 :($7+(5 '$7$ & 3(1 ),/( :'),/(f ,'$< (1'7,0( 5($' (1' f (73,f 7615,f 7616,f 5$,1,f ,'$
PAGE 146

,03/,&,7 5($/ $+2=f 5($/ 75$165$6:,=,5f$,=,5f.,6725,:&217 &20021 3$5$:&,=,5f &21',=,5f6:&,=,5f+($',=,5f 1/'=,=f'(37+,=f46,5f6,=,5f,&216715'5,5f5,5f, /2: &20021 0$6%,:&217,=,5f$5($,5f7$5($,6725 &20021 465'),=,5f:&:3:&)&,&) & ,)75$16(422f 7+(1 '2 1/ '2 15 6,-f 5(7851 (1' ,) 680 '2 1/ '2 15 5$6: -f :&,-f :&:3 f :&)& :&:3 f ,)5$6:,-f*7Of 5$6:, -f ,)5$6:,-f/(2f 7+(1 5$6:,-f $,-f *2 72 (1' ,) $,-f 75$16 r 7$5($ r 5$6:,-f rr 75$16 5$6:, -fr.ff 6,-f $ -f r 5') -f 680 680 6,-f &217,18( & ,) 680(4 25 ,&)(4 f 7+(1 &) (/6( &) 75$16 r 7$5($ 680 (1' ,) 680 1/ '2 15 6,-f 6,-f '=,f $5($-f r &) 680 680 6,-fr'=,fr$5($-f &217,18( 5(7851 (1' & 68%5287,1( 75,'1*,)$%&'9f & & 7+,6 68%5287,1( 62/9(6 7+( 75,',$*21$/ &2()),&,(17 0$75,; & :+,&+ 5(68/76 )520 $ 6(7 2) 6,08/7$1(286 (48$7,216 :+,&+ & &$1 %( 387 ,172 7+( )2//2:,1* )250 & $r9 %r9 &r9 & 3$5$0(7(5 ,= f ,03/,&,7 5($/ $+2=f ',0(16,21 $,=f%,=f&,=f',=f9,=f%(7$,=f*$00$,=f ,)3 ,)

PAGE 147

2 R R R R R R R R R R R R 1*0 1* &20387( ,17(50(',$7( $55$<6 %(7$ $1' *$00$ %(7$,)f %,)f *$00$,)f ',)f%(7$,)f '2 ,)3O1* %(7$,f %,f$,fr&,f%(7$,f *$00$,f ',f$,fr*$00$,ff%(7$,f &217,18( &20387( ),1$/ 62/87,21 9(&725 9 91*f *$00$1*f /$67 1*,) '2 /$67 1*. 9,f *$00$,f&,fr9f%(7$,f &217,18( 5(7851 (1' 68%5287,1( 72 &$/&8/$7( $ 0$66 %$/$1&( 2) 7+( 6<67(0 6,08/$7(' 68%5287,1( 0$6%$/,5(6(7f 3$5$0(7(5 ,= ,5 ,% ,5r0$; ,=r,5f ,03/,&,7 5($/ $+2=f 5($/ ,:&217c6725$*( &20021 3$5$:&,=,5f&1',=,5f6:&,=,5f+($',=,5f 1/'=,=f'(37+,=f46,5f6,=,5f, &216715'5,5f5,5f,/: &200217,0'77,0(787f(1'7,0('70$;'70,172',7,0( 40'7,55 &20021 0$6% ,:&217,=,5f$5($,5f7$5($ 6725$*( &20021 0$6%',))3675$*()/:,1(;75$&76725$*()/2:287 ,1,7,$/,=( 7+( &808/$7,9( ,1),/75$7,21 $1' (;75$&7,21 ,),5(6(7(4f 7+(1 )/2:,1 )/2:287 (;75$&7 ',)) 6725$*(6 '2 15 '2 1/ 6725$*( 6725$*( :&, fr'=,fr$5($-f ',)) 6725$*( c6725$*( 36725$*( ',)) ,6725$*( r 5(7851 (1' ,) ',)) 6725$*(6 & '2 15 )/2:,1 )/2:,1 '7r46-fr$5($-f

PAGE 148

,),/2:(4f )/2:287 )/2:287 '7r&2121/f f '2 1/ 6725$*( 6725$*( :&, -fr'=,fr$5($-f (;75$&7 (;75$&7 6, -fr'=,fr$5($-fr'7 &217,18( ',)) 6725$*( ,6725$*( )/2:,1 (;75$&7 )/2:287 f 36725$*( ',)) ,6725$*( )/2:,1 (;75$&7 )/2:287f r 5(7851 (1' & 68%5287,1( ,55,*$7( ,'$< ,5&2'( f & & 68%5287,1( 72 &$/&8/$7( 7+( $02817 2) ,55,*$7,21 5(48,5(' & )25 7+( 02'(/ %$6(' 21 7+( 0$;,080 62,/ :$7(5 327(17,$/ $//2:(' & 3$5$0(7(5 ,= ,5 ,% ,5r 0$; ,=r,5f ,03/,&,7 5($/ $+2=f &20021 3$5$:&,=,5f&21',=,5f6:&,=,5f+($',=,5f 1/'=,=f'(37+,=f46,5f6,=,5f, &216715'5,5f5,5f /2: &200217,0'7-,0( 787f (1'7,0( '70$; '70,1 72',7,0( 40'7,55 &20021 (77615f 7616f (73f 5$,1f '8576723 &20021 465'),=,5f:&:3:&)&,&) &20021 '80<),(/'f-),(/'f,)6 57,0(,237 7(1 ,7)f-7)f7+5(655$7(,6&+f &20021 0$6% ,:&217,=,5f$5($,5f7$5($,6725$*( & $02817 '(),&,7 ,),6&+,'$
PAGE 149

QRQ QRQQQ 76723 767$57 '85 & ,5&22( 15 46 ,f 5$,1 ,'$< f &217,18( (1' ,) & ,) 7,1( *( 76723 f 7+(1 ,5&'( '2 15 46 ,f &217,18( 5$,1,'$
PAGE 150

r &212f &21'f f r+($'f+($' f f'=,ff ,) /2: (4 f )/2:O,Of )/2:O,Of &1' f *2 72 (1' ,) & )/: f r &21' f &21' f f r+($'f+($'&f f'= ,ff &21' f &21' f f r +($' f +($' f f'= ,ff & ,)$%6)/2: ff *740f *2 72 *2 72 40 $%6)/: ff &217,18( & ,)40/(22f *2 72 '7 :.((3 r '=f 40 *2 72 '7 '70$; ,)'7*7'70$;f '7 '70$; ,)'7/7'70,1f '7 '70,1 ,),5&2'((4f '7 $0,1 '7,55 '7 f ,)'77,0(*7787,7, 0(f$1'7287,7, 0(f*72 f '7 7287,7,0(f 7,1( ,)'77,1(*7(1'7,0(f '7 (1'7,1( 7,1( ,)'77,1(*776723$1',5&2'((4f '7 76723 7,0( ,)'7/722f '7 '70,1 & :5,7(rf '740 )250$7rrrrrr '7 n) f 40 (f 5(7851 (1' & & 68%5287,1( 72 &$/&8/$7( 7+( 1(: :$7(5 327(17,$/6 & )25 7+( 21(',1( 16 1$/ 02'(/ & 68%5287,1( 5('67' 3$5$0(7(5 ,= ,5 ,% ,5r 0$; ,5r,=f ,03/,&,7 5($/ $+2=f 5($/ ..',=f$^,=f%,=f&,=f9,=f 5($/ &217$%f:&&7$%f 63&7$%f357$%f &+$5$&7(5r 0%),/( &20021 7$%/(6:&&7$%&217$%63&7$%357$%1'$7$0%),/(,62,/ &200213$5$:&,=,5f&21',=,5f6:&,=,5f+($',=,5f 1/'=,=f'(37+,=f46,5f6,=,5f,&216715'5,5f5,5f,/2: &200217,0'77,0(787f(1'7,0('70$;'70,172',7,0(40'7,55 &21021 &219',), 7(50$; ,7(5 & & 6(7 83 7+( ,17(5,25 *5,' 32,176 & '2 ./ 1/ .O &21'./f &1'./f f &21' ./f &1'./f f

PAGE 151

&O 6:&./f &167 '7 '=./frr&O & $./f &167 r &./! &167 r %./f $./f &./f './f +($'./f '7'=./f&r . f '7 r 6./f&O &217,18( & & 6(7 83 7+( %281'5< &21',7,216 & ,),&23L67(4f *2 72 & & 12 )/8; %281'5< &21',7,21 $7 7+( 685)$&( & ./ &' 1'./f &21'./ f f &O 6:&./ f &167 '7 '=./frr& & $./f &./f &167 r % ./f & ./f './f +($'./f '7'=./f&Or 46f f '7 r 6./ f& *2 72 & & &2167$17 327(17,$/ %281'5< &21',7,21 $7 7+( 685)$&( & ./ &21'f &21' f f &21' f &O 6:&f &167 '7 '=frr& & $Of &167 r &Of &167 r % f &Of f +($'f '7'=f&r . f '7 r 6f& & & 12 )/8; %281'5< &21',7,21 $7 /2:(5 %281'5< & ./ 1/ .O &21'./ f &21' ./f f ,) ,/2:(4 f &21' ./ f &O 6:&./ f &167 '7 '=./frr&O & $./f &167 r %./f $./f

PAGE 152

'./f +($'./f '7'=./f&Or . f '7 r 6./f& & & &$/&8/$7( 7+( 1(: 9$/8(6 )25 7+( +($'6 & ,) & ,),&2167(4f ,) &$// 75,'1/,)$%&' 9f '2 ./ ,&21671/ +($'./ f 9./f &217,18( & ,)+($'f*7357$%Of$1',&2167(4 f ,&2167 ,) ,&2167(4Of *2 72 4 4 $0$; 46f 40 f & &217,18( &$// 83'$7( f '2 1/ +($' f +($' f +($' f +($' f :&,f :& f :&f :&f &21' f &1' f &21'f &21'f 6:& f 6:& f 6:&f 6:&f &217,18( &$// 0$6%$/Of & 5(7851 (1' & & 68%5287,1( 72 &$/&8/$7( 7+( 7,0( 67(3 %$6(' 21 7+( 0$;,080 & 1(7 )/2: 5$7( ,172 ($&+ *5,' & )25 7+( 7:2',0(16,21$/ 02'(/ & 68%5287,1( 767(3' ,5&2'( f 3$5$0(7(5 ,= ,5 ,% ,5r 0$; ,=r,5f ,03/,&,7 5($/ $+2=f 5($/ ....1)5 &20021 :&&:.((3 )/: ,= ,5f &20021 (77615f7616f(73f5$,1f'8576723 &20021 3$5$:&,=,5f&1',=,5f6:&,=,5f+($',=,5f 1/ '=,=f'(37+,=f46,5f6,=,5f,&216715'5,5f5,5f,/2: &200217,0'77,0(787f(1'7,0('70$;'70,172',7,0(40'7,55 1)5 '2 1/ '2 15 ,),(4Of 7+(1 4 46-f (/6( &

PAGE 153

4 r &21', f &1' f f f r+($',-f+($'-ff'=,ff (1' ,) ,),(41/f 7+(1 4 ,),/2:(4Of 4 &1',-f (/6( 4 r &21' f &21' f f f r +($' f +($' f f'= ,ff (1' ,) ,)-(4Of 7+(1 4 (/6( 4 r &21', f &21' -f f f r+($', f +($' -ff5 -f 5 -Off f (1' ,) ,)(415f 7+(1 4 (/6( 4 r &2 1' f &21' -ff f r +($' -f+($' ff 5 -f 5 -ff f (1' ,) )/: -f 4 4f'= ,f 4 4f'5-f 6 -f ,)$%6)/2: -ff *71)5f *2 72 *2 72 1)5 $%6)/:, -ff &217,18( & ,)1)5/(22f *2 72 '7 :.((3 1)5 *2 72 '7 '70$; ,)'7*7'70$;f '7 '70$; ,)'7/7'70,1f '7 '70,1 ,),5&2'((4f '7 $0,1 '7,55 '7 f ,) '77,0(*(7287,7,0(f$1'7287,7,0(f*72f '7 7287,7,0(f 7,0( ,)'77,0(*776723$1',5&2'((4f '7 76723 7,1( ,) '77,0(*((1'7,0(f '7 (1'7,0( 7,0( ,) '7/(2 f '7 '70,1 5(7851 (1' &rrr & & $', 62/87,21 0(7+2' 2) 5,&+$5'6 (48$7,21 & 68%5287,1( 5('67' 3$5$0(7(5 ,= 5 % ,5r 0$; ,=r,5f ,03/,&,7 5($/ $+2=f 5($/ . .9,=f$,=f%,=f&,=f',=f+67$5,=,5f &200213$5$:&,=,5f&21',=,5 f6:&,=,5f+($',=,5f 1/ '=,=f'(37+,=f46,5f6,=,5f,&216715'5,5f5,5f,/2: &200217,0'77,0(787f(1'7,0('70$;'70,172',7,0(40'7,55 &20021 &219',), 7(50$; 7(5

PAGE 154

& & 6(7 83 7+( ,17(5,25 *5,' 32,176 )25 7+( ),567 +$/) &<&/( & 62/9( ,1 7+( 5$',$/ ',5(&7,21 & ,7(5 '2 ./ 1/ ,)./(4Of *2 72 ,)./(41/f*2 72 '2 15 .O &21'./-f &1'./-f f n &1'./ f &1'./-f f &21'./ -f &1'./-f f &1'./-f &21'./-O f f &O 6:&./-f 6:&./ f f 5O 5-Of 5-f f 5 5-f 5-Of f &167 r '7 '=./frr &O &167 r '7 '5-frr &O 5-f $,f r &167 r 5 r &,f r &167 r 5 r % ,f $,f &,f ,f +($'./ f &167 r .Of r +($'./f &167 r .O r +($'./f r +($'./ f f '7'=./f&r .O f '7'=./f&Or .O f '7& r 6./ -f &217,18( & & 12 )/8; %281'5< &21',7,21 $7 ,11(5 5$',86 & .O &21'./f&1'./ ff &1'./ f &1'./-f f &21'./ -f &1'./-f f . &O 6:&./-Of 6:&./ f f &167 r '7 '=./frr &O &167 r '7 '5-frr &O $ ,f &,f r &167 r % ,f &,f ,f +($'./ f &167r..f r +($'./f &167Or.Or+($'./O-Of &167r.r+($'./Of '7'=./f&r .O f '7'=./f&Or .O f '7&O r 6./-f & & & 12 )/8; %281'5< &21',7,21 $7 287(5 5$',86 & 15 15 .O &21'./ -f &21'./ f f &21 ./ f &1'./-f f n

PAGE 155

. &21'./ f &41'./ -f f &O 6:&./ f 6:&./ f f &167 r '7 '=./frr &O &167 r '7 '5-frr &O & $,f r &167 r &,f % ,f $ ,f ,f +($'./ f &167r..f r +($'./ f &167r.r+($./-f &167r.r+($'./Of '7'=./f&Or . f '7'=./f&Or . f '7& r 6./ -f *2 72 & & 12 )/8; %281'5< &21',7,21 $7 7+( 685)$&( & 15 &1'./ f &1'./-f f 46-f ,)-1(15f &2 1' ./ -f &1'./-f f ,)-(415f ,)-1(15f 5 5-f 5-f f ,)-1(Of &1'./ f &1'./ -f f ,)-(4Of ,)-1(Of 5 5-f 5-Of f &O 6:&./-f 6:&./-f f &167 r '7 '=./frr &O ,)-1(O$1'-1(15f &167 r '7 '5-frr &O 5-f ,)-(4O25-(415f &167 r '7 '5-frr &O $,f r &167 r 5 r ,)-(4Of $,f r &167 r &,f r &167 r 5 r ,)-(415f &,f r &167 r %,f $,f &,f ',f +($'./-f &167 r r +($'./ f &167 r r +($'./-f '7'=./f&r. .f '7'=./f&Or. .f '7&O r 6./ -f &217,18( *2 72 & & 12 )/8; %281'5< &21',7,21 $7 /2:(5 %281'5< & '2 15 .O &1'./-f &1'./ f f ,)-1(15f &1'./ -f &1'./-f f ,)-(415f ,)-1(15f 5 5-Of 5-f f ,)-1(Of &21'./ f &21'./ -f f ,)-(4Of

PAGE 156

,)-1(Of 5 5-f 5-Of f &O 6:&./ f 6:&./-f f &167 r '7 '=./frr &O ,)-1(O$1'-1(15f &167 r '7 '5-frr &O 5-f ,)-(4O25-(415f &167 r '7 '5-frr &O $,f r &167 r 5 r ,)-(4Of $,f r &167 r &,f r &167 r 5 r ,)-(415f &,f r &167 r % ,f $ ,f & ,f ',f +($'./ -f &167 r r +($'./ f &167 r r +($'./ f '7'=./f&r. .f '7'=./f&Or. .f '7& r 6./-f &217,18( rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr ,) &$// 75,' 15,)$ % & 9 f '2 15 +($'./ f 9 -f &217,18( &217,18( & & 6(7 83 7+( ,17(5,25 *5,' 32,176 )25 7+( 6(&21' +$/) & ,1 7+( 9(57,&$/ ',5(&7,21 & '2 15 ,)-(4Of *2 72 ,)-(415f*2 72 '2 ./ 1/ ./ .O &21'./ f &21'./f f &1'./ f &1'./-f f &21'./-f &1'./-f f &21'./f &21'./-f f 5O 5-Of 5-f f 5 5-f 5-Of f &O 6:&./ f 6:&./ f f &167 r '7 '=./frr &O &167 r '7 '5-frr &O 5-f $,f r &167 r &,f r &167 r %,f $,f &,f ',f '7'=./f&Or . f &167r5r.r+($'./-f &167r5r.r+($'./-f +($'./-f &167 r 5r. 5r. f r +($'./ f '7&O r 6 ./ -f &217,18( & & 6(7 83 7+( %281'5< &21',7,216 & & 12 )/8; %281'5< &21',7,21 $7 7+( 685)$&( & ./

PAGE 157

R R R R R R ./ &1'./ -f &1'./-f f 46-f &21' ./ -f &1'./-f f 5O 5-Of 5-f f &21'./ f &21'./ -f f 5 5-f 5-Of f & 6:&./f 6:&./ f f &2167 r '7 '=./frr & &167 r '7 '5-frr & 5-f $ ,f &,f r &167 r %,f &,f ',f '7'=./f&r . f &167r5r.r+($'./-f &167r5r.r+($'./-f +($'./-f &167 r 5r. 5r. f r +($'./ -f '7& r 6./-f 12 )/8; %281'5< &21',7,21 $7 /2:(5 %281'5< ./ 1/ ./ .O &21'./ f &21'./ f f . &21'./ -f &21' ./ f f 5 5-Of 5-f f &21'./ f &21'./ -f f 5 5-f 5-Of f &O 6:&./-f 6:&./ -f f &167 r '7 '=./frr &O &167 r '7 '5-frr &O 5-f $ ,f r &167 r & ,f %,f $,f ',f '7'=./f&Or . f &167r5r.r+($'./-f &167r5r.r+($'./-O f +($'./ f &167 r 5r. 5r. f r +($'./f '7& r 6./ -f *2 72 12 )/8; %281'5< &21',7,21 $7 ,11(5 5$',86 '2 ./ 1/ ./ ,)./1(Of &21'./-f&1'./O ff ,)./(4Of 46-f ,)./1(1/f. &1'./ f &1'./-f f ,)./(41/f. &21'./-f &1'./-f f . &O 6:&./-f 6:&./ f f &167 r '7 '=./frr &O &167 r '7 '5-frr &O $,f r &167 r

PAGE 158

,)./(4Of $,f r &,f r &167 r ,)./(41/f &,f r % ,f $ ,f & ,f ',f '7'=./f&Or . f &167r.r+($'./-f +($'./-f &167 r r +($'./f '7& r 6./-f &217,18( *2 72 & & 12 )/8; %281'5< &21',7,21 $7 287(5 5$',86 & '2 ./ 1/ ./ ,)./1(Of .O &21'./-f &21'./ f ` ,)./(4Of .O 46-f ,)./1(1/f &1'./ f &1'./-f f ,)./(41/f. &21'./ f &21' ./ -f f &O 6:&./ f 6:&./ f f &167 r '7 '=./frr &O &167 r '7 '5-frr &O $,f r &167 r ,)./(4Of $,f r &,f r &167 r ,)./(41/f &,f r %,f $,f &,f '& ,f '7'=./f&Or. .f &167r .r +($'./-f +($'./ f &167 r r +($'./ f '7& r 6./-f &217,18( & ,) &$// 75,'1/,) $ % & 9f '2 ./ 1/ +67$5./-f 9./f &217,18( &217,18( & ',) '2 1/ '2 15 ',) ',) $%6 +67$5 -f +($' f f+($' f f +($' f +67$5 -f &217,18( & ,)',)/7O (f *2 72 &$// 83'$7( f ,7(5 ,7(5 ,),7(5/70$; ,7(5f *2 72 & &217,18( &$// 83'$7(f

PAGE 159

2&-222222 R R R '2 1/ '2 15 +($' f +($' f +($' ` +($' f :&,-f :&, f :&, f :&,f &1' f &1' f &1', f &1', f 6:&,-f 6:& f 6:&, f 6:&, f &217,18( &$// 0$6%$/Of 5(7851 (1' 5287,1( 72 83'$7( 7+( 62,/ 3523(57,(6 68%5287,1( 83'$7( ,/3 f 3$5$0(7(5 ,= ,5 ,% ,5r0$; ,=r,5f ,03/,&,7 5($/ $+2=f &+$5$&7(5r 0%),/( 5($/ :&17f&17f 6:&7f +('7 f &20021 7$ %/(6:&17&176:&7+('71',00%),/(, 62,/ ,),62,/(4Of &$// 7%/$55 ,/223 f ,),62,/(4 f &$// 7%/5(+ ,/223 f ,),62,/(4f &$// 7%/2$0 ,/223 f ,),62,/ (4 f &$// 7%/2$0 ,/223 f ,),62 ,/(4f &$// 7%/<2/ ,/223 f ,),62,/(4 f &$// /,1($5 ,/223 f 5(7851 (1' 68%5287,1( 7%/$55 ,/223 f 7+,6 68%5287,1( 86(6 $1 ,1'(; 5287,1* 7(&+1,48( :,7+ 7$%8/$7(' '$7$ 72 ),1' 7+( &255(6321',1* 9$/8(6 2) :$7(5 &217(17 63(&,),& :$7(5 &$3$&,7< $1' +<'5$8/,& &21'8&7,9,7< )25 $ *,9(1 9$/8( 2) 62,/ 0$75,& 327(17,$/ )25 $55('21'2 ),1( 6$1' 3$5$0(7(5 ,= ,5 ,% ,5r0$; ,=r,5f ,03/,&,7 5($/ $+=f &+$5$&7(5r 0%),/( 5($/ :&17f&17f6:&7f+('7f &200213$5$:&1,=,5f&21,=,5 f6:&,=,5f+(',=,5 f 1/'=,=f'(37+,=f46,5f6,=,5f,&216715'5,5f5,5f,/2: &200217$%/(6:&17&2176:&7+('7 1',00%),/( ,62,/ +('60/ +('7f +('/*( +('7Of '2 /223 '2 15 '2 1/ ,) +(' -f /( +('60/f +(',.-f +('60/

PAGE 160

R R R R R R R ,) +(',.-f *( +('/*(f +(',.-f +('/*( 5,1'(; $/ *r+(',. -ffr((2O ,1'(; 5,1'(; ,),1'(;/7Of ,1'(; ,) +(',.-f /7 +('7 ,1'(;ff ,1'(; ,1'(; ,) +(',.-f *7 +('7,1'(;ff ,1'(; ,1'(; ,),1'(;/7Of ,1'(; ,) ,1'(; *( f ,1'(; 50 5,1'(;,1'(; 5$7,2 $/*+(',.-f+('7,1'(;ff$/2*2+('7,1'(;f +('7,1'(;ff :&1 -f :&17 ,1'(;f5$7,2r :&17 ,1'(;f:&17 ,1'(;ff 6:&, .-f 6:&7,1'(;fr(rr5$7,r$/*6:&7,1'(;f 6:&7,1'(;fff &21, .-f &217,1'(;fr(rr5$7,r$/*&17,1'(;f &217,1'(;fff &217,18( 5(7851 (1' & 68%5287,1( 7%/2$0 ,/223 f 7+,6 68%5287,1( 86(6 $1 ,12(; 5287,1* 7(&+1,48( :,7+ 7$%8/$7(' '$7$ 72 ),1' 7+( &255(6321',1* 9$/8(6 2) :$7(5 &217(17 63(&,),& :$7(5 &$3$&,7< $1' +<'5$8/,& &21'8&7,9,7< )25 $ *,9(1 9$/8( 2) 62,/ 0$75,& 327(17,$/ )25 (,7+(5 2) 7+( /2$06 3$5$0(7(5 ,= ,5 % ,5r 0$; ,=r,5f ,03/,&,7 5($/ $+2=f &+$5$&7(5r 0%),/( 5($/ :&17f&17f6:&7f+('7f &200213$5$:&1,=,5f&21,=,5f6:&,=,5f+(',=,5f 1/ '=,=f'(37+,=f46,5f6 ,=,5f, &216715'5,5f5,5f,/2: &200217$ %/(6:&17&2176:&7+('71',00%),/(,62,/ +('60/ +('7f +('/*( +('7Of '2 /223 '2 15 '2 1/ ,) +(',.-f /( +('60/f +(',.-f +('60/ ,) +(',.-f *( +('/*(f +(',.-f +('/*( 5,1'(; $/* r+(' .-f fr (( ,1'(; 5,1'(; ,) +(',.-f /7 +('7,1'(;ff ,1'(; ,1'(; ,) +(',.-f *7 +('7,1'(;ff ,1'(; ,1'(; ,) ,1'(; *( f ,1'(; ,) ,1'(; /( f ,1'(; 50 5,1'(;,1'(; 5$7,2 $/*+(' .-f+('7 ,1'(;f f$/* +('7 ,1'(;f +('7,1'(;ff :&1,.-f :&17 ,1'(;f 5$7,r :&17 ,1'(;f:&17 ,1'(;f f 6:&, -f 6:&7,1'(;fr(2Orr5$7,r$/*6:&7,1'(;f

PAGE 161

2 2 2 f R R R 6:&7,1'(;fff &0,-&-f &17 ,1'(;fr (rr 5$7,2r$/*&17 ,1'(;f &217,1'(;fff &217,18( 5(7851 (1' 68%5287,1( 7%/5(+ ,/223 f 7+,6 68%5287,1( 86(6 $1 ,1'(; 5287,1* 7(&+1,48( :,7+ 7$%8/$7(' '$7$ 72 ),1' 7+( &255(6321',1* 9$/8(6 2) :$7(5 &217(17 63(&,),& :$7(5 &$3$&,7< $1' +<'5$8/,& &21'8&7,9,7< )25 $ *,9(1 9$/8( 2) 62,/ 0$75,& 327(17,$/ )25 5(+2927 6$1' 3$5$0(7(5 ,= ,5 % ,5r 0$; ,=r,5f ,03/,&,7 5($/ $+2=f &+$5$&7(56 0%),/( 5($/ :&17f&217 f6:&7f+('7f &200213$5$:&1,=,5f&21,=,5f6:&,=,5f+(',=,5f 1/'=,=f'(37+ ,=f46 ,5f6 ,=,5f,&216715'5,5f5,5f,/2: &200217$%/(6:&17 &217 6:&7 +('7 1',0 0%),/( ,62,/ & +('60/ +('7 f +('/*( +('7Of '2 ,/223 '2 15 '2 1/ ,) +(' -f /( +('60/f +(',.-f +('60/ ,) +(',.-f *( +('/*(f +(',.-f +('/*( 5,1'(; $/* r+(' .-f fr ((2 ,1'(; 5,1'(; ,) +(' .-f /7 +('7 ,1'(;ff ,1'(; ,1'(; ,) +(',.-f *7 +('7,1'(;ff ,1'(; ,1'(; ,) ,1'(; *( f ,1'(; ,) ,1'(; /( f ,1'(; 50 5,1'(;,1'(; 5$7,2 $/*+(', .-f+('7,1'(;ff$/*+('7,1'(;f +('7,1'(;ff :&1,.-f :&17 ,1'(;f 5$7,2r:&17 ,1'(;f:&17 ,1'(;ff 6:&, .-f 6:&7,1'(;fr(rr5$7,2r$/2*6:&7,1'(;f 6:&7,1'(;fff &21 -f &217 ,1'(;fr(rr5$7,r$/*&17 ,1'(;f &217 ,1'(;fff &217,18( 5(7851 (1' & 68%5287,1( 7%/<2/ ,/223 f & 7+,6 68%5287,1( 86(6 $1 ,1'(; 5287,1* 7(&+1,48( :,7+ & 7$%8/$7(' '$7$ 72 ),1' 7+( &255(6321',1* 9$/8(6 2) & :$7(5 &217(17 63(&,),& :$7(5 &$3$&,7< $1' +<'5$8/,& & &21'8&7,9,7< )25 $ *,9(1 9$/8( 2) 62,/ 0$75,& 327(17,$/

PAGE 162

& )25 <2/2 /,*+7 &/$< & 3$5$0(7(5 ,= ,5 ,% ,5r0$; ,=r,5f ,03/,&,7 5($/ $+2=f &+$5$&7(5r 0%),/( 5($/ :&17f&217f6:&7f+('7f &0013$5$:&1,=,5f&21,=,5f 6:&,=,5f+(',= ,5f 1/'=,=f'(37+,=f46,5f6,=,5f,&216715'5,5f5,5f,/: &200217$%/(6:&17&176:&7+('71',00%),/( ,62,/ +('60/ +('7f +('/*( +('7Of '2 ,/223 '2 15 '2 1/ ,) +(',.-f /( +('60/f +(', .-f +('60/ ,) +(', -f *( +('/*(f +(',.-f +('/*( 5,1'(; $/*r+(', -ffr(( ,1'(; 5,1'(; ,) +(',.-f /7 +('7,1'(;ff ,1'(; ,1'(; ,) +(', .-f *7 +('7,1'(;ff ,1'(; ,1'(; ,) ,1'(; *( f ,1'(; ,) ,1'(; /( f ,1'(; 50 5,1'(;,1'(; 5$7,2 $/*+(',.-f+('7,1'(;ff$/ *+('7,1'(;f +('7,1'(;ff :&1,.-f :&17 ,1'(;f 5$7,r :&17 ,1'(;f:&17 ,1'(;ff 6:&,.-f 6:&7,1'(;fr(rr5$7,2r$/2*6:&7,1'(;f 6:&7,1'(;fff &21,.-f &17 ,1'(;f r (rr 5$7,r$/*&17 ,1'(;f &217,1'(;fff &217,18( 5(7851 (1' & 68%5287,1( /,1($5 ,/223 f 3$5$0(7(5 ,= ,5 ,% ,5r 0$; ,=r,5f ,03/,&,7 5($/ $+2=f &+$5$&7(5r 0%),/( 5($/ :&17f&217f6:&7f+('7f &20021 7$%/(6:&17&2176:&7+('71',00%),/(,62,/ &200213$5$:&1,=,5f&21,=,5f6:&,=,5f+(',= ,5f 1/'=,=f'(37+,=f46,5f 6,=,5f, &216715'5,5f5,5f,/: & '2 ,/223 '2 15 '2 1/ ,)+(',-.f *722f +(',-.f &$// ,173/+(',-.f&21, -.f 6:&, .f :&1, .ff &217,18( 5(7851 (1' FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF F 68%5287,1( ,173/ :;<=f

PAGE 163

2 &af &f! ,03/,&,7 5($/ $+=f 7+,6 68%5287,1( ,17(532/$7(6 /,1($5/< %(7:((1 7:2 6(76 2) '$7$ 32,176 '$7$ ,6 ,1387 $6 7$%8/$5 )81&7,216 &+$5$&7(5r 0%),/( 5($/ :7$%f;7$%f<7$%f=7$%f &20021 7$%/(6=7$%;7$%<7$%:7$%1',00%),/(,62,/ ; ;7$%f & < <7$%f = =7$%f 1 ,)$%6:f/($%6:7$%f ff *2 72 ; ;7$%1',0f & < <7$%1',0f = =7$%1',0f 1 1',0 ,)$%6:f*($%6:7$%1',0f ff *2 72 & '2 1',0 1 ,)$%6:f/($%6:7$%,fff *2 72 &217,18( ; ;7$%1f$%6::7$%1Off:7$%1f:7$%1ffr ;7$%1f;7$%1fff < <7$%1f $%6::7$%1Off :7$%1f :7$%1ff r <7$%1f<7$%1Offf = =7$%1f$%6::7$%1Off:7$%1f:7$%1Offr =7$%1f=7$%1Offf 5(7851 (1'

PAGE 164

5()(5(1&(6 $PHUPDQ &5 )LQLWH GLIIHUHQFH VROXWLRQV RI XQVWHDG\ WZR GLPHQVLRQDO SDUWLDOO\ VDWXUDWHG SRURXV PHGLD IORZ 9ROXPHV DQG ,, 3K' 'LVVHUWDWLRQ 3XUGXH 8QLYHUVLW\ : /DID\HWWH ,QG $PHUPDQ &5 DQG (0RQNH 6RLO ZDWHU PRGHOLQJ ,, 2Q VHQVLWLYLW\ WR ILQLWH GLIIHUHQFH JULG VSDFLQJ 7UDQV $PHU 6RF $JU (QJ f $ULEL $* 6PDMVWUOD )6 =D]XHWD DQG 6) 6KLK $FFXUDFLHV RI QHXWURQ SUREH ZDWHU FRQWHQW PHDVXUHPHQWV IRU VDQG\ VRLOV 6RLO DQG &URS 6FL 6RF )OD 3URF $UPVWURQJ &) DQG 79 :LOVRQ &RPSXWHU PRGHO IRU PRLVWXUH GLVWULEXWLRQ LQ VWUDWLILHG VRLOV XQGHU D WULFNOH VRXUFH 7UDQV $PHU 6RF $JU (QJ f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nV 7KHVLV 8QLYHUVLW\ RI )ORULGD *DLQHVYLOOH )/ &ODUN DQG $* 6PDMVWUOD :DWHU GLVWULEXWLRQV LQ VRLOV DV LQIOXHQFHG E\ LUULJDWLRQ GHSWKV DQG LQWHQVLWLHV 6RLO DQG &URS 6FL 6RF )OD 3URF 'HQPHDG 27 DQG 5+ 6KDZ $YDLODELOLW\ RI VRLO ZDWHU ZDWHU WR SODQWV DV DIIHFWHG E\ VRLO PRLVWXUH FRQWHQW DQG PHWHRURORJLFDO FRQGLWLRQV $JURQ (QILHOG &* DQG &9 *LOODVS\ 3UHVVXUH WUDQVGXFHU IRU UHPRWH GDWD DFTXLVLWLRQ 7UDQV $PHU 6RF $JU (QJ f )HGGHV 5$ 3.RZDOLN DQG + =DUDGQ\ 6LPXODWLRQ RI ILHOG ZDWHU XVH DQG FURS \LHOG +DOVWHG 3UHVV 1HZ
PAGE 165

)LW]VLPPRQV ': DQG 1&
PAGE 166

1LPDK 01 DQG 5+DQNV E 0RGHO IRU HVWLPDWLQJ VRLO ZDWHU SODQW DQG DWPRVSKHULF LQWHUUHODWLRQV ,, ILHOG WHVW RI PRGHO 6RLO 6FL 6RF $PHU 3URF 3HUUHQV 6DQG .. :DWVRQ 1XPHULFDO DQDO\VLV RI WZR GLPHQVLRQDO LQILOWUDWLRQ DQG UHGLVWULEXWLRQ :DWHU 5HVRXU 5HV f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nV 7KHVLV 7H[DV $ t 0 8QLYHUVLW\

PAGE 167

6PDMVWUOD $* ,UULJDWLRQ PDQDJHPHQW IRU WKH FRQVHUYDWLRQ RI OLPLWHG ZDWHU UHVRXUFHV 2IILFH RI :DWHU 5HVHDUFK DQG 7HFKQRORJ\ RI WKH GHSDUWPHQW RI ,QWHULRU :DWHU &RQVHUYDWLRQ 5HVHDUFK 3URJUDP :5&f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f 7ROOQHU (: DQG )0RO] 6LPXODWLQJ SODQW ZDWHU XSWDNH LQ PRLVW OLJKWHU WH[WXUHG VRLOV 7UDQV $PHU 6RF $JU (QJ f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

PAGE 168

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f GHJUHH LQ 0DUFK +H HQUROOHG LQ WKH 8QLYHUVLW\ RI )ORULGD LQ 0D\ WR SXUVXH WKH 3K' GHJUHH LQ DJULFXOWXUDO HQJLQHHULQJ

PAGE 169

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f "V nDPHV : -RQHV rR IHV VRU RI $JULFXOWXUDO (QJLQHHULQJ FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ )DGUR 5 =D] $VVLVWDQW 3U $JULFXOWXUDO UHWDL ,IHV£SU RI LQJLQHHULQJ FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 7 *UD\ &XUWLV $VVLVWDQW 3URIHVVRU RI &LYLO (QJLQHHULQJ

PAGE 170

7KLV GLVVHUWDWLRQ ZDV VXEPLWWHG WR WKH *UDGXDWH )DFXOW\ RI WKH &ROOHJH RI (QJLQHHULQJ DQG WR WKH *UDGXDWH 6FKRRO DQG ZDV DFFHSWHG DV SDUWLDO IXOILOOPHQW RI WKH UHTXLUHPHQWV IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 0D\ ,n/V-/X-n 4 ‘ 'HDQ &ROOHJH RI (QJLQHHULQJ 'HDQ *UDGXDWH 6FKRRO


MEASUREMENT AND SIMULATION OF SOIL WATER STATUS
UNDER FIELD CONDITIONS
By
KENNETH COY STONE
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
V
1937

ACKNOWLEDGEMENTS
•The author would like to express his appreciation to Dr. Allen G.
Smajstrla for his guidance, assistance and encouragement throughout th
research. The author would like to thank his other committee members
for their support and guidance throughout this research and in
coursework taken under their supervision.
The author also expresses his appreciation to the Agricultural
Engineering Department for the use of its research and computing
facilities. Appreciation is also expressed to other faculty and staff
members in the department for their assistance.
Special gratitude is expressed to the authors family for their
continuous support and encouragement throughout his studies.
Finally, the author would like to express his very special
graditude to his wife, Carol, for her drafting expertise, patience and
continuous support and encouragement.

TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS Ü
LIST OF TABLES v
LIST OF FIGURES vi
ABSTRACT x
CHAPTERS
I INTRODUCTION 1
II REVIEW OF LITERATURE 4
Measurement of Soil Water Status 4
Soil Water Content Measurement 4
Soil Wafer Potential Measurement 8
Automation of
Soil Water Potential Measurements 11
Water Movement in Soils 15
Soil Water Extraction 20
III METHODS AND MATERIALS 26
Equipment 26
Instrumentation Development 27
Field Data Collection 30
Model Development 33
One-Dimensional Model 36
Two-Dimensional Model 41
Soil Water Extraction 50
IV RESULTS AND DISCUSSION 54
Instrumentation Performance 54
Field Data Collection 60
Model Verification 70
Model Operation with Field Data 94
One-Dimensional Model 102
Two-Dimensional Model 109
Model Application 116
V SUMMARY AND CONCLUSIONS 124
Microcomputer-based Data Acquisition System 124
Field Data 124

Modeling Soil Water Movement and Extraction 125
Model Applications 125
APPENDICES
A. Alternating Direction Implicit Finite Differencing. . 126
B. Listing of Soil Water Movement and Extraction Model. .137
REFERENCES 163
BIOGRAPHICAL SKETCH 167

LIST OF TABLES
Table 3-1. Microcomputer-based data acquisition system
components and approximate costs 28
Table 4-1. One-dimensional distribution of water extraction
for a 15 day drying cycle for young citrus trees
with and without grass cover 68
Table 4-2. Two-dimensional distribution of water extraction
for a 15 day drying cycle for young citrus trees
with grass cover 69
Table 4-3. Two-dimensional distribution of water extraction
for a 4 day drying cycle for young citrus trees
with and without grass cover 73
Table 4-4. One-dimensional distribution of water extraction
for a 4 day drying cycle for a grass cover crop
at water depletion levels of 20 kPa and 40 kPa . ... 76
v

LIST OF FIGURES
Figure 2-1. A typical soil water characteristic curve 5
Figure 3-1. Layout of lysimeter system 31
Figure 3-2. Details of individual lysimeter soil water status
monitoring system 32
Figure 3-3. Location of tensiometers in 1985 field experiment
for young citrus trees in grassed and bare soil
lysimeters 34
Figure 3-4. Location of tensiometers in 1986 field experiment
for young citrus trees in grassed and bare soil
lysimeters 35
Figure 3-5. Schematic diagram of the finite-difference grid
system for the one-dimensional model of water
movement and extraction 38
Figure 3-6. Schematic diagram of the finite-difference grid
system for the two-dimensional model of water
movement and extraction 44
Figure 3-7. Effect of the relative available soil water
and potential soil water extraction rate on the
soil water extraction rate 53
Figure 4-1. Calibration curve of output voltage versus
pressure applied for pressure transducer No. 1. . . 55
Figure 4-2. Calibration curve of digital units versus
voltage applied for the analog-to-digital
circuit used 57
Figure 4-3. Comparisons of mercury manometer manually-read
and pressure transducer automatically-read
tensiometer water potentials during drying cycles
for 2 tensiometers in the laboratory 58
Figure 4-4. Comparisons of mercury manometer manually-read and
pressure transducer automatically-read tensiometer
water potentials for all laboratory data 59
Figure 4-5. Evaluation of pressure transducer-tensiometer No. 1
by comparison of mercury manometer manually-read
and pressure transducer automatically-read water
potentials in the field 61
vi

Figure 4-6. Evaluation of pressure transducer-tensiometer No. 2
by comparison of mercury manometer manually-read
and pressure transducer automatically-read water
potentials in the field 62
Figure 4-7. Evaluation of pressure transducer-tensiometer No. 3
by comparison of mercury manometer manually-read
and pressure transducer automatically-read water
potentials in the field 63
Figure 4-8. Evapotranspiration rate from the 1985 field
experiment with a 20 kPa soil water potential
treatment young citrus tree with grass cover. ... 66
Figure 4-9. Evapotranspiration rate from the 1985 field
experiment with a 20 kPa soil water potential
treatment young citrus tree with bare soil 67
Figure 4-10. Evapotranspiration rate from the 1986 field
experiment with a 20 kPa soil water potential
treatment young citrus tree with grass cover. ... 71
Figure 4-11. Evapotranspiration rate from the 1986 field
experiment with a 20 kPa soil water potential
treatment young citrus tree with bare soil 72
Figure 4-12. Evapotranspiration rate from the 1986 field
experiment with a 20 kPa soil water potential
treatment with a grass cover 74
Figure 4-13. Evapotranspiration rate from the 1986 field
experiment with a 40 kPa soil water potential
treatment with a grass cover 75
Figure 4-14. Soil water potential-soil water content
relationship for Rehovot sand 79
Figure 4-15. Hydraulic conductivity-soil water content
relationship for Rehovot sand 80
Figure 4-16. Simulated results of soil water content profiles
for infiltration into a Rehovot sand under
constant rain intensity of 12.7 mm/hr 81
Figure 4-17. Simulated results of soil water content profiles
for infiltration into a Rehovot sand under
constant rain intensity of 47 mm/hr 82
Figure 4-18. Soil water potential-soil water content
relationship for Yolo light clay 83
Figure 4-19. Hydraulic conductivity-soil water content
relationship for Yolo light clay 84

Figure 4-20. Soil water potential-soil water content
relationship for Adelanto loam 85
Figure 4-21. Hydraulic conductivity-soil water content
relationship for Adelanto loam 86
Figure 4-22. Simulated results of soil water content profiles
for infiltration into a Yolo light clay with
initial pressure potential at -66 kPa 87
Figure 4-23. Simulated results of soil water content profiles
for infiltration into a Yolo light clay with
initial pressure potential at -200 kPa 88
Figure 4-24. Simulated results of soil water content profiles
for infiltration into a Adelanto loam with
initial pressure potential at -66 kPa 89
Figure 4-25. Soil water potential-soil water content
relationship for Nahal Sinai sand 90
Figure 4-26. Hydraulic conductivity-soil water content
relationship for Nahal Sinai sand 91
Figure 4-27. Simulated results of dimensionless soil water
content distribution for infiltration into a
Nahal Sinai sand at 57 min of infiltration time. . 92
Figure 4-28. Simulated results of dimensionless soil water
content distribution for infiltration into a
Nahal Sinai sand at 297 min of infiltration time. . 93
Figure 4-29. Two-dimensional model simulated results of soil
water content profiles for infiltration into a
Rehovot sand under constant rain intensity
of 12.7 mm/hr 95
Figure 4-30. Two-dimensional model simulated results of soil
water content profiles for infiltration into a
Rehovot sand under constant rain intensity
of 47 mm/hr 96
Figure 4-31. Evapotranspiration rate from the second drying
cycle of the 1985 field experiment with a 20 kPa
soil water potential treatment young citrus tree
with grass cover 98
Figure 4-32. Distribution of evapotranspiration through the
daylight hours 99
Figure 4-33. Soil water potential-soil water content
relationship for Arredondo fine sand. .
vi ii
100

Figure 4-34. Hydraulic conductivity-soil water potential
relationship for Arredondo fine sand 101
Figure 4-35. Simulation results for the one-dimensional model
with field data for the 150 mm depth 104
Figure 4-36. Simulation results for the one-dimensional model
with field data for the 300 mm depth 105
Figure 4-37. Simulation results for the one-dimensional model
with field data for the 450 mm depth 106
Figure 4-38. Simulation results for the one-dimensional model
with field data for the 600 mm depth 107
Figure 4-39. Simulation results for the one-dimensional model
with field data for the 900 mm depth 108
Figure 4-40. Simulation results for the two-dimensional model
with field data for the 150 mm depth Ill
Figure 4-41. Simulation results for the two-dimensional model
with field data for the 300 mm depth 112
Figure 4-42. Simulation results for the two-dimensional model
with field data for the 450 mm depth 113
Figure 4-43. Simulation results for the two-dimensional model
with field data for the 600 mm depth 114
Figure 4-44. Simulation results for the two-dimensional model
with field data for the 900 mm depth 115
Figure 4-45. Soil water potentials for the three irrigation
treatments at the 300 mm depth 118
Figure 4-46. Soil water potentials for the three irrigation
treatments at the 900 mm depth 119
Figure 4-47. Soil water storage for the three irrigation
treatments 120
Figure 4-48. Cumulative irrigation for the three irrigation
treatments 121
Figure 4-49. Cumulative drainage from the soil profile for
the three irrigation treatments 122
IX

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
MEASUREMENT AND SIMULATION OF SOIL WATER STATUS
UNDER FIELD CONDITIONS
By
KENNETH COY STONE
May 1987
Chairman: Allen G. Smajstrla
Major Department: Agricultural Engineering
Irrigation of agricultural crops is one of the major uses of fresh
water in Florida. Supplemental irrigation is required in Florida
because the predominant sandy soils have very low water holding
capacities, up to 70» of the annual rainfall occurs during months when
many crops are not grown, and rainfall is not uniformly distributed.
Under these conditions the scheduling of irrigation becomes extremely
important. The timely application of irrigation water can result in
increased yields and greater profits, while untimely applications could
result in decreased yields and profits, and leaching of nutrients.
A low cost microcomputer-based data acquisition system for
continuous soil water potential measurements was developed. The system
consisted of tensiometer-mounted pressure transducers, an analog-to-
digital converter and a portable microcomputer. The data acquisition
system was evaluated under laboratory and field conditions. Excellent
x

agreement was obtained between soil water potentials read with the data
acquisition system and those read manually using mercury manometers.
Experiments were conducted utilizing the data acquisition system to
monitor soil water potentials of young citrus trees in field lysimeters
to determine model parameters and to evaluate model performance.
Evapotranspiration, soil water extraction rates and relative soil water
extractions were calculated from the soil water potential measurements.
Two numerical models were developed to study the movement and
extraction of soil water. A one-dimensional model was developed to
describe a soil profile which is uniformly irrigated such as one which
is sprinkler irrigated. The model was used to simulate soil water
movement and extraction from a young citrus tree with grass cover in a
field lysimeter. Simulated results were in excellent agreement with
field observations.
A two-dimensional model was developed to describe a nonuniformly
irrigated soil profile as in trickle irrigation. The model was used to
simulate the two-dimensional movement and extraction of soil water from
a young citrus tree with grass cover in a field lysimeter. The model
results were in excellent agreement with field observations.
xi

CHAPTER I
INTRODUCTION
Irrigation of agricultural crops is one of the major uses of fresh
water in Florida. It was reported (Harrison et al., 1983) that in 1981,
41 percent of the fresh water use in Florida was for irrigation of more
than 800,000 hectares of agricultural crops. Three reasons cited for
supplemental irrigation in Florida were 1) sandy soils have very low
water holding capacities, 2) up to 70 percent of the annual rainfall
occurs during months when many crops are not grown, and 3) rainfall is
not uniformly distributed even during months of high rainfall.
Because no control can be placed on when and where rainfall occurs,
researchers must focus their attention on managing the soil water
content in the plant root zone. One method of managing the water
content in soils is to apply small amounts of irrigation water at
frequent intervals. In the absence of rainfall, this provides
agricultural plants with adequate water for growth and also minimizes
losses from percolation below the root zone. With rainfall
interactions, irrigation scheduling is more difficult; the objective is
to minimize irrigation inputs (maximize effective rainfall) while
optimizing production returns. Because of the complexity of these
interactions, numerical models are useful tools to study them. Many
researchers have developed models to use in the study of soil water
management (Smajstrla, 1982; Zur and Jones, 1981). These models allow
many different irrigation strategies to be investigated without the cost
normally associated with field experiments. These models involve
1

2
analyses of infiltration, redistribution, evaporation and other factors
which affect the movement and uptake of soil water by plants.
The objective of this research was to develop a data collection
system and a numerical model which together will provide researchers
with information needed for developing and validating models of soil
water movement, crop water use and evapotranspiration. The
instrumentation system developed will record soil water potentials on a
real time basis in order to provide input data for model verification
and validation. A microcomputer based instrumentation system will allow
the computer to monitor inputs and make decisions based on the input
data. The microcomputer will have the ability to monitor and control
events in the field.
Two numerical models will be developed to study the movement and
extraction of soil water. A one-dimensional model will be developed to
describe the movement and extraction of soil water in a soil profile
which is uniformly irrigated. A two-dimensional model will be developed
to describe the movement and extraction of soil water from a soil
profile which is not uniformly irrigated such as trickle irrigation.
The specific objectives of this research were:
1. To develop and test instrumentation for the real-time
monitoring of soil water potential under agricultural crops.
2. To record the dynamics of soil water movement and water
extraction under irrigated agricultural crops.
3. To develop a numerical model to simulate the soil water
extraction patterns observed under agricultural crops as a
function of irrigation management practices and climatic
demands.

3
4. To demonstrate the use of the numerical model in evaluating
and recommending irrigation strategies.

CHAPTER II
REVIEW OF LITERATURE
Measurement of Soil Water Status
The measurement of soil water can be classified into two
catagories: (1) the amount of water held in a given amount of soil (soil
water content), and (2) the potential, or tension with which the water
is held by the soil (soil water potential). These properties are
related to each other (Figure 2-1) and describe the ability of a soil to
hold water available for plant growth.
Soil Water Content Measurement
Several methods are available to measure soil water content. The
gravimetric method is the standard method of determining the soil water
content. This method consist of physically collecting a soil sample
from the field, weighing it, and oven drying the sample to constant
weight at 105 C. The difference in weights before and after drying is
the amount of water removed from the sample. Water contents can be
calculated on a weight basis, or on a volume basis if the soil volume or
bulk density was measured when the sample was taken. An advantage of
the gravimetric method is that it requires no specialized equipment.
Disadvantages are that it is laborious and time consuming. Destructive
sampling is required and sampling may disturb a location sufficiently to
distort results. This method does not lend itself to automation.
Neutron scattering allows the nondestructive measurement of soil
water content. A neutron moisture meter may be used in the field to
4

SOIL WATER POTENTIAL (mm)
Figure 2-1.
SOIL WATER CONTENT (mm/mm)
A typical soil water potential-soil water content curve.

6
rapidly and repeatably measure water contents in the same location and
depth of soil. An access tube must be installed in the soil to allow a
probe to be lowered to the desired soil depth for measurements.
The neutron scattering method operates on the principle of nuclear
thermalization. Fast neutrons are emitted from a radiation source
located on a probe which is lowered into the soil. These neutrons lose
energy as they collide with hydrogen atoms in the soil and are slowed or
thermalized. Thermalized neutrons are counted by a detector which is
also located on the probe. The number of slow neutrons is an indirect
measure of the quantity of water in the soil because of the hydrogen
atoms present in water. Because the neutron method counts only hydrogen
atoms, the method must be calibrated for each specific soil type and
location. This is especially true for soils with variable quantities of
organic (natter because the neutron meter will also count hydrogen atoms
in the organic matter and cause errors in the water content
measurements.
With the neutron method, a spherical volume of soil is sampled as
neutrons are emitted from the source. The radius of the sphere varies
with the soil water content from a small radius for wet soils to 20 or
30 cm for very dry soils. This measurement over a relatively large
volume is an advantage for homogenous soils with no discontinuities.
However when discontinuities such as water tables exist, their exact
locations cannot be detected accurately. Aribi et al. (1985)
investigated the accuracies of neutron meters when measuring water
contents near boundaries. They found that significant errors occurred
when readings were taken within the top 0.4 meters in an unsaturated
soil profile.

7
Another disadvantage of the neutron method is that the equipment
cost is high. The radioactive material contained in the neutron meter
is also a disadvantage because of the health hazards associated with
radiation. The automation of a neutron meter would be very expensive
and extremely complicated.
Gamma ray attenuation is a method which can be used to determine
the soil water content. This method determines soil water content by
measuring the amount of gamma radiation energy lost as a radiation beam
is directed through a soil. The method depends on the fact that gamma
rays lose part of their energy upon striking another substance, in this
case soil water. As the water content changes, the amount of
attenuation will also change.
Various types of gamma radiation instruments are available. One
type is intended for laboratory or stationary use. This instrument has
the source and detector parallel in line at a fixed distance apart, and
a soil column is placed between them. The source and detector are then
moved along the length of the soil column to determine the water content
distribution in the column. This would not be practical for field
application.
A second type of gamma radiation instrument has been developed for
field application. This instrument requires that two parallel access
tubes be installed into the soil. The source is placed in one access
tube and the detector in the other. Radiation is focused into a narrow
beam between the source and detector, and soil water content can be
measured in very thin layers of soil.
Another type of gamma radiation instrument consists of a stationary
detector located at the soil surface and a source located on a probe

8
which is lowered into the soil. The relationship between the source and
detector is known, and thus changes in density with depth can be
measured accurately with this instrument. Because the detector is
located at the soil surface, the instrument is limited to the top 30 cm
of the profile.
Gamma radiation instruments are expensive, and they can be
hazardous due to the radioactive source. This type of instrument would
be very difficult and expensive to automate.
Additional research has been conducted to utilize other soil
properties which would lend themselves to rapid methods of soil water
content determination. Fletcher (1939) conducted work on a dielectric
method of estimating soil water content. He used a resonance method to
determine the dielectric constant of the soil. This type of instrument
consists of an ocillator circuit and a tuned receiving circuit. The
instrument is placed in the soil and allowed to equilibrate. A variable
capacitor in the receiving circuit is then tuned to resonance, and the
capacitor reading is correlated with the soil water content. This
instrument must be calibrated for each soil in which it is used. Such a
device is not known to be commercially available.
Most of the methods discussed produce indirect measurements of soil
water content. Soil water content is calibrated to other factors such
as neutron thermalization, gamma ray attenuation or dielectric
properties. Only the gravametric method yields a direct measure of soil
water content.
Soil Water Potential Measurement
An alternative to the measurement of soil water content is the
measurement of soil water potential. A tensiometer measures the

9
potential or tension of water in the soil. The tensiometer consists of
a closed tube with a ceramic cup on the end which is inserted into the
soil and a vacuum gage or manometer to measure the water potential in
the tensiometer tube. The tube is filled with water, closed and allowed
to equilibrate with the soil water potential. As the soil dries, water
in the tensiometer is pulled through the ceramic cup. The soil water
potential which pulls water through the ceramic cup is registered on the
vacuum gage or manometer. This force is also the hydraulic potential
that a plant would need to exert to extract water from the soil.
Therefore, a tensiometer measures the energy status of water in the
soil. Tensiometers left in the soil for a long period of time tend to
follow the changes in the soil water potential.
The hydraulic resistance of the ceramic cup, the surrounding soil,
and the contact between the cup and soil cause tensiometer readings to
lag behind the actual tension changes in the soil. Lags are also caused
by the volume of water needed to be moved through the cup to register on
the measuring device. The useful range of tensiometers is from 0 to -80
kPa. Below -80 kPa air enters through the ceramic cup or the water
column in the tensiometer breaks, causing the tensiometer to fail. This
measurement limitation is not serious for irrigated crops on sandy soils
because most of the available water for plant use occurs between 0 and
-80 kPa.
Another method of measuring soil water potential is the thermal
conductance method. The rate of heat dissipation in a porous material
of low heat conductivity is sensitive to the water content in the porous
material. When in contact with a soil, the water potential in the
porous material tends to equilibrate with the soil water potential.

10
Phene et al. (1971) developed an instrument to measure soil water
potential by sensing heat dissipaton within a porous ceramic. The water
potential of the porous ceramic was measured by applying heat at a point
centered within the ceramic and measuring the temperature rise at that
point. Soil water potential measurements were obtained by taking two
temperature readings. The first one was taken before the heating cycle
and the second after the heating cycle. The difference between the two
temperature measurements was the change in temperature at the center of
the sensor due to the heat applied. The magnitude of the difference
varies depending on the water content of the porous block. Phene stated
that the sensor should measure the soil water potential regardless of
the soil in which it is embedded.
In experimental applications of the heat pulse device, the accuracy
of the sensor was - 20 kPa over the range 0 to -200 kPa. In Florida
sandy soils such a wide range of variability would not be acceptable
because most of the available soil water is contained in the soil
between field capacity and -20 kPa. Calibration for individual soils
couTd improve the accuracy of the instrument. This instrument has the
capability of allowing automated data collection and is nondestructive.
The cost of the individual sensors and the associated data recording
devices is relatively high.
Soil psychrometers are instruments which can be used to measure
soil water potentials in the range of -1 to -15 bars. They operate by
cooling a thermocouple junction which is in equilibrium with the soil to
the point of water condensation on the junction, and then measuring the
junction temperature as water is allowed to evaporate. Thus the
temperature depression due to evaporation can be related to soil water

11
tension. Only small responses are obtained at potentials above -1 bar,
making this instrument unsuitable for irrigation scheduling on sandy
soils where irrigations would typically be scheduled at much greater
potentials.
Electrical resistance blocks can be correlated with soil water
potential. The blocks are placed in the soil and allowed to
equilibrate with the soil water. Resistance blocks usually contain a
pair of electrodes embedded in gypsum, nylon or fiberglass.
Measurements with resistance blocks are sensitive to the electrolytic
solutes in the fluid between the electrodes. Thus resistance blocks are
sensitive to variations in salinity of soil water and to temperature
changes. Temperatures must be measured and resistance readings
appropriately corrected.
Resistance blocks are not uniformly sensitive over the entire range
of soil water content. They are more accurate at low water contents
than at water contents near field capacity. Because of this limitation,
resistance blocks can be used to complement tensiometers to measure
soil water potentials below the -80 kPa range. They are relatively
inexpensive, and an automated data collection system may be built around
these instruments for measuring water potentials in drier soils.
Automation of Soil Water Potential Measurements
The instruments discussed may all be used to measure the status of
soil water. Most of the techniques relate a soil property to another
property which is measured by the instrument. Two of the methods allow
a direct measure of a soil property. The gravimeteric method gives a
direct measure of soil water content and tensiometers give direct
measurement of soil water potential. The gravimetric measurement method

12
requires destructive sampling and does not lend itself to automation.
Tensiometers may be automated by recording the changes in soil water
potential, and they do not require destructive sampling. Therefore, to
measure the status of soil water under a growing crop, an automated data
collection system using tensiometers to measure soil water potential was
chosen.
To determine the soil water potential, the water potential within
the tensiometer is measured. The tensiometer water potential is assumed
to be in equilibrium with the soil water potential. Early methods of
measuring tensiometer water potentials used mercury manometers. Later,
mechanical vacuum gauges were used. Both methods functioned well for
manual applications. Neither, however, was readily automated.
Recent interest in better understanding the dynamics of soil water
movement, and in the development of numerical models to simulate this
process, has resulted in the need for an automated system of recording
tensiometer readings on a continuous real time basis.
Van Bavel et al. (1968) used a camera to take periodic photographs
of a tensiometer manometer board. By analyzing the photographs they
were able to record changes in potentials. This procedure was, however,
laborious and did not provide continuous soil water potential records.
Enfield and Gillaspy (1980) developed a transducer which measured
the level of mercury in a mercury manometer tensiometer. The principle
of operation of their transducer was the same as a concentric capacitor.
The level of mercury in the manometer corresponds directly to the length
of a capacitor plate. A steel tube was used as the outer capacitor
plate around a column of mercury. The nylon tube which contained the
mercury column acted as the dielectric material. The capacitance of the

13
transducer was measured and converted to length of the column of
mercury. This instrument was found to be very sensitive to temperature
fluctuations. Further research is needed on this transducer to make it
suitable for applications in automated data collection systems.
Fitzsimmons and Young (1972) used a tensiometer-pressure transducer
system to study infiltration. Their system consisted of a pressure
transducer which was connected to many tensiometers by a system of fluid
switches. This system required substantial time lags after switching a
tensiometer to the transducer before an accurate reading could be made.
This was primarily due to changes in volume due to the elasticity of the
system. This resulted in a large scanning time in order to read all
tensiometers. Long and Hulk (1980) used a similar system.
Bottcher and Miller (1982) developed an automatic manometer
scanning device which read and recorded mercury levels in manometer-type
tensiometers. The system consisted of a computer-control 1ed chain-drive
mechanism which moved photocells and light sources up and down the
manometer tubes. When the scanner passed a mercury-water interface, a
change in voltage was detected by the computer. This system is
expensive, uses specially constructed rather than commercially-available
components, and is not readily expandable. It also requires the use of
a central manometer board with connecting tubing from the various
tensiometers. It has the advantage of easy verification of computer
readings by manual reading of the manometers.
Marthaler et al. (1983) used a pressure transducer to read
individual tensiometers. The upper end of the tensiometers was closed
off with septum stoppers to provide air-tight seals through which a
needle connected to a pressure transducer was inserted. The needle was

14
inserted into a pocket of entrapped air in the tensiometer, and the
pressure transducer output was obtained immediately. This method
introduced an error because the air pocket was compressible and was
affected by the addition of air at atmospheric pressure in the needle,
transducer, and connectors. This system was designed to minimize errors
by minimizing the volume of air introduced into the tensiometer. The
tensiometer air pockets were, however, temperature sensitive and
introduced diurnal time lags because of expansion and contraction due to
diurnal temperature changes. Also, to read several instruments, the
pressure transducer was required to be moved manually. Thus, this type
of system does not lend itself to automation.
Thomson et al. (1982) used individual pressure transducers on
tensiometers with all air purged from the system to monitor soil water
potentials. Thus, they were able to avoid lag times associated with air
pockets in the tensiometers. They were able to read pressure
transducers very rapidly by electronic rather than hydraulic switching.
A soil water potential monitoring system that used pressure
transducers as used by Thomson et al. (1982) would be able to record
data from a number of sensors very rapidly. A continuous data
acquisition system based on tensiometer-mounted pressure transducers
would provide data necessary for models of movement of water in soils,
crop water use, and evapotranspiration. The use of a microcomputer to
monitor the pressure transducers would allow the system to be programmed
to make decisions based on the input data (Zazueta et al., 1984).
Also, the cost of a dedicated data acquisiton system would be greater
than that of a microcomputer-based system because of current
microcomputer costs and availability.

15
Water Movement in Soils
The general equations governing unsaturated flow in porous media
are the continuity equation and Darcy's Law. Hi11 el (1980) presented a
combined flow equation which incorporates the continuity equation
36
_= - v-q - S (1)
3t
where q = flux density of water,
0 = volumetric water content,
v = differential operator,
t = time, and
S = a sink or source term,
with Darcy's equation for unsaturated flow
q = - K(h) vH (2)
where K = hydraulic conductivity,
H = the total hydraulic head and defined as
H = h - z, and
h = capillary pressure head.
The resulting combined flow equation for both steady and transient
flows is also known as Richards equation
39
— = v*(K(h>v H ) - S (3)
3t
For one-dimensional vertical flow, equation (3) becomes
36 3 3h 3K(h)
— = — c K(h) — ) - - S(z,t) (4)
3t 3Z 3Z 3z
where z = the vertical dimension.

16
The sink term is used to represent the loss or gain of water from
the soil by root extraction or by application of irrigation water from a
point source.
For implicit solutions, equation (4) must be written in terms of
only one variable, soil water content or potential. By introducing the
specific water capacity, C, defined as
d e
C = (5)
d h
and using the chain rule of calculus, equation (4) may be written in
terms of the soil water potential as
9h 3 3 h 3 K(h)
C = (K(h) ) - - S(z,t) (6)
3 t 3 Z 3 Z 3 Z
For two-dimensional flow, equation (3) becomes
3 h 3 ah 3 3 h 3 K(h)
C — = —(K(h)—) + —(K(h)—) - S(x,z,t) (7)
at 3 X 3 X 3 Z 3 Z 3 Z
Th with radial symmetry may be written as
3h 13 3 h a ah 3 K(h)
C — (r K(h)—) + —(K(h)—) - - S(x,z,t) (8)
atrar 3r 3z 3Z sz
The two-dimensional flow equation in radial coordinates may be used to
describe water movement and extraction of soil water from nonuniform
water applications such as trickle irrigation.
Due to the variable nature of K(h), equations (6), (7) and (8) are
highly nonlinear, and analytical solutions are extremely complex or

17
impossible to obtain. The nonlinearity of equations (6) and (7), and
the typical variable boundary conditions, have led to the use of
numerical methods to solve practical problems of soil-plant-water
relationships, such as irrigation management for agricultural crops.
For one-dimensional flow, equation (6) has been successfully solved
using explicit finite difference methods by many researchers. Hanks and
Bowers (1962) developed a numerical model for infiltration into layered
soils. They solved the Richards equation for the hydraulic potential
using implicit finite difference equations with a Crank-Nicholson
technique which averages the finite differences over two successive time
steps. Rubin and Steinhardt (1963) developed a numerical model to study
the soil water relationships during rainfall infiltration. They used a
Crank-Nicholson technique to solve Richards equation for the water
content. Rubin (1967) developed a numerical model which analyzed the
hysteresis effects on post-infiltration redistribution of soil water.
Haverkamp et al. (1977) reviewed six numerical models of one-
dimens torra.! infiltration. Each model employed different discretization
techniques for the nonlinear infiltration equation. The models reviewed
were solved using both the water content based equation and the water
potential based equation. They found that implicit models which solved
the potential based infiltration equation had the widest range of
applicability for predicting water movement in soil, either saturated or
nonsaturated.
Clark and Smajstrla (1983) developed an implicit model of soil
water flow to study the distribution of water in soils as influenced by
various irrigation depths and intensities. The model simulated water
application rates from center-pivot irrigation systems with intensities

18
of application typical of low and high pressure irrigation systems.
Their model also simulated post-infiltration redistribution.
Rubin (1968) developed a two-dimensional numerical model of
transient water flow in unsaturated and partly unsaturated soils. He
utilized alternating-direction implicit (ADI) finite difference methods.
He studied horizontal infiltration and ditch drainage with the numerical
model. Hornberger et al. (1969) developed a two-dimensional model
to study water movement in a composite soil moisture groundwater system.
They modeled the two-dimensional response of falling water tables. They
considered both saturated and unsaturated zones in their model. The
solution method used was a Gauss-Sidel iterative technique.
A two-dimensional model to simulate the drawdown in a pumped
unconfined aquifer was developed by Taylor and Luthin (1969). The model
gave simultaneous solutions in both the saturated and unsaturated zones.
They used a Gauss-Sidel iterative method to solve the flow equations.
Amerman (1969) developed two-dimensional numerical models to
simulate steady state saturated flow, drainage and furrow irrigation.
He used ADI methods to solve both the steady state saturated flow model
and the furrow irrigation model. He also used an explicit method to
solve the drainage model.
A study of the sensivity of the grid spacing for finite difference
models was reported on by Amerman and Monke (1977). Two finite
difference models of two-dimensional infiltration were analyzed. They
solved the two-dimensional flow equations with successive overrelaxation
(SOR) and alternating direction implicit (ADI) methods. They found that
smaller grid sizes were needed in regions where the hydraulic gradients

19
changed rapidly. Considerable computational savings without appreciable
loss of accuracy was achieved using irregular grid sizes.
Perrens and Watson (1977) developed a two-dimensional numerical
model of water movement to analyze infiltration and redistribution.
They used an iterative alternating direction implicit technique to solve
the flow equation. Two soil types, a sand and a sandy loam, were
simulated. Nonuniform surface fluxes were applied along the horizontal
soil surface in a step type distribution pattern. They also
incorporated hysteresis of soil hydraulic characteristics into the model
to be used in the redistribution phase of the simulations.
Researchers have also utilized two-dimensional models to study soil
water movement from trickle irrigation systems. Brandt et al. (1971)
solved the flow equation in two dimensions to analyze infiltration
from a trickle source. They developed a plane flow model in cartesian
coordinates to analyze infiltration from a line source of closely spaced
emitters with overlapping wetting patterns. They also developed a
cylinderical flow model to analyze infiltration from a single emitter
when its wetting pattern is not affected by other emitters. Both models
were solved using noniterative ADI finite difference procedures with
Newton's iterative method. The results were compared to an analytical
solution of steady infiltration and a one-dimensional solution with good
results.
Armstrong and Wilson (1983) developed a model for moisture
distribution under a trickle source. They utilized the Continuous
System Modeling Program (CMSP) to simulate the soil moisture movement.
The model calculated the net flow rates into each grid. It then
calculated the change in water content by dividing the net flow rate

20
by the volume of soil in each grid and then multiplying by the time
step. Finally, the model calculated the new water contents from the
previous water contents plus the calculated water content changes.
Model results compared favorably with field measurements.
Zazueta et al. (1985) developed a simple explicit numerical model
for the prediction of soil water movement from trickle sources. The
model was based on the mass balance equation and an integrated form of
Darcy's law. The model produced good agreement with other results
obtained with more complicated numerical methods and analytical
solutions.
Soil Water Extraction
Water uptake by plant roots has been investigated by many
researchers. Among the first researchers to attempt to describe plant
water relations were Gradmann (1928) and van den Honert (1948). Two
approaches to modeling water extraction have been utilized to describe
the water extraction by plant roots. The first, called the microscopic
approach, describes water movement to individual roots. The second
approach, called the macroscopic water extraction model, describes water
uptake by the whole root zone, and the flow to individual roots is
ignored.
Gardner (1960) developed a microscopic water uptake model. He
described the root as an infinitely long cylinder of uniform radius and
water-absorbing properties, assuming that water moves in the radial
direction only. The flow equation for such a system is
8 e
8 t
1 3 80
( r D — )
r 8 r 8 r
r 8 r
(9)

21
where a = the volumetric water content,
D = the diffusivity,
t = the time and
r = the radial distance from the axis of the root.
He then obtained a solution at the boundary between the plant root
and the soil in order to maintain a constant rate of water movement to
the plant, subject to the following boundary conditions:
6 = e0, h = hQ, at t = tQ, and
dh de
2 tt r^ k(—) = 2 tt r1 D (—) = q at r = r^
(10)
dr dr
where k = the hydraulic conductivity of the soil,
r^= the radius of the root, and
q = the rate of water uptake by the root.
The solution of (9) subject to the initial and boundary conditions
in (10) is
q 4Dt
h - h = dh = ( In - g ) (11)
4 * k r¿
where g = 0.57722 is Euler's constant. The diffusivity and conductivity
were assumed to be constant, this assumption was justified because D, r
and t are all in the logarithmic term, and dh is thus not very sensitive
to these variables.
Gardner (1960) also noted that the solution will behave as though
infinite for only very short times or for very low values of hydraulic
conductivity. Exact solutions of the problem for finite systems would
require taking the dependence of D and k upon the soil water potential
into account. Gardner also compared equation (11) with the steady-state
solution for flow in a hollow cylinder:

22
q tr
h - hQ = dr = ( In _ ) (12)
4 irk a¿
where hQ = the potential at the outer radius of the hollow cylinder,
b = one half the distance between neighboring roots, and
h = the potential at the inner radius a.
If b = 2/Dt, then equation (12) is identical to equation (11)
except for the g term which is small compared to the logarithmic term
and may be ignored.
Gardner and Ehlig (1962) presented a macroscopic water uptake
equation in which the rate of water uptake is proportional to the
potential energy gradient and inversely proportional to the impedance to
water movement within the soil and the plant. The potential energy was
expressed as the difference between the diffusion pressure deficit and
the soil suction. The impedance was expressed as the sum of plant
impedance and soil impedance. They analyzed greenhouse experiments and
obtained results consistent with the uptake equation.
Molz and Remson (1970) developed a mathematical model describing
moisture removal from soil by plant roots. The model described one¬
dimensional water movement and extraction. The model used a macroscopic
water extraction term to describe moisture removal by plants. Their
formula for the extraction term was given as
R(z) D(8)
S(z,t) * T ( ) (13)
i R(z) D(e) dz
where D(e) = diffusivity,
R(z) = the effective root density,
T = the transpiration rate per unit area and
v = the root depth.

23
The model gave results that compared reasonably with experimental
results. The numerical solution of the Richards equation with the
extraction function was obttained using the Douglas-Jones predictor-
corrector method.
Nimah and Hanks (1973) developed a numerical model to predict water
content profiles, evapotranspiration, water flow in the soil, root
extraction, and root water potential under field conditions. Their
extraction term had the form
(Hroot + (1.05 z) -h(z,t)-s(z,t)) RDF(z) K(O)
A(z,t) = (14)
AX A Z
where A(z,t) = the soil water extraction rate,
Hroot = the effective water potential in the root at the soil
surface,
h(z,t) = the soil matric potential,
s(z,t) = the osmotic potential,
RDF(z) = proportion of total active roots in depth increment DZ,
K(e) = the hydraulic conductivity and
Ax = the distance between the plant roots at the point
where h(z,t) and s(z,t) are measured.
Tollner and Molz (1983) developed a macrocopic water uptake model.
The extraction function assumed that water uptake rate per unit volume
of soil is proportional to the product of contact length per unit soil
volume, root permeability per unit length and water potential difference
between soil and root xylem potential. A factor which accounts for
reduced root-soil-water contact as water is removed was included in the

. ' • 24
extraction function. Their model predictions were comparable with
results of a greenhouse experiment.
Slack et al. (1977} developed a mathematical model of water
extraction by plant roots as a function of leaf and soil water
potentials. The model used a microscopic model to calculate water
A.
uptake which was input into a macroscopic model of water movement and
water uptake. Water movement was described by a two-dimensional radial
flow model. The model was used to estimate transpiration from corn
grown in a controlled environment under soil drying conditions. The
model predicted daily transpiration adequately for the period modeled.
Feddes et al. (1978) developed an implicit finite difference model
to describe water flow and extraction in a non-homogeneous soil root
system under the influence of groundwater. Their extraction term
assumed the extraction rate to be maximum when the soil water potential
was above a set limit. When the soil water potential fell below this
limit, the water uptake, S(h), was decreased linearly with the soil
water potential
( h - h3 )
S(h) = Smax (15)
( h2 - h3 )
where h2 is the set limit below which the water uptake decreases
linearly to h3. Below h3 it was assumed that no water is extracted,
then h3 may be assumed to be the wilting point. The maximum possible
transpiration rate divided by the effective rooting depth is Smax. The
model yielded satisfactory results in predicting both cumulative
transpiration and distribution of soil moisture content with depth.

25
Zur and Jones (1981) developed a model for studying the integrated
effects of soil, crop and climatic conditions on the expansive growth,
photosynthesis and water use of agricultural crops. The model utilized
Penman's equation modified by Monteith to calculate the water vapor
transport from the plant to the atmosphere. It utilized a
macroscopic soil water extraction term to calculate water uptake from
the soil. The model assumed the water uptake from the soil was the
difference between the total water potential at the root surface and the
soil water potential divided by the sum of the resistance to radial
water flow inside the roots and resistance of water flow in the soil.
The model treated the water relations in plants in considerable detail.
It was successfully tested on soybeans. However the model did not take
into consideration infiltration, drainage, evaporation from the soil
surface or upward movement of water in the soil.

CHAPTER III
MATERIALS AND METHODS
Equipment
The data acquisition system developed in this research was based on
tensiometer-mounted pressure transducers. Pressure transducers used
were Micro Switch model 141PC15D. These transducers measured pressures
from 0 to -100 kPa. They produced an analog electrical output from 1 to
6 volts which was proportional to the vacuum exerted on a membrane in
the transducers. The transducers operated much like strain gages. One
side of the membrane was open to the atmosphere and the other side was
in contact with water in the tensiometer. As pressure on the water
changed, the membrane was deformed and the voltage across the membrane
changed.
The pressure transducers required an 8-volt direct current (VDC)
power supply at 8 mA each. Because a muItiple-transducer system was
developed, a separate regulated power supply was used rather than the
microcomputer power supply. This assured adequate power for the
transducers.
The pressure transducers were temperature-compensated. Outputs
varied less than 1% of full scale output in the range of 5 C to 45 C.
The output voltages from the pressure transducers were interfaced
to a microcomputer through an analog-to-digital (A/D) board. The A/D
board was a Mico World MW-311 general purpose Input/Output Board. This
device had 8-bit resolution and measured input voltages from 0 to 5
volts. It had 16-channel multiplexing capability.

27
The microcomputer used was a Commodore 64. It was selected for its
programmability, its availability at local electronics supply companies,
and its low cost. An overall goal in component selection was to be able
to replace any component individually with as little down-time as
possible, and thus to assure the reliability of the system.
The microcomputer output was to a video screen, a printer, and a
cassette recorder. The cassette recorder provided storage of data for
later computations and easy transfer to other computer systems. The
printer provided a hard copy backup of data. The video screen provided
immediate interaction with the computer for programming and monitoring.
The microcomputer was programmed in the BASIC computer language.
It had approximately 32 kilobytes of programmable memory (RAM) when
using the BASIC computer language. This was sufficient to store the
calibration equations and calculated outputs from the transducer inputs.
The microcomputer also had an internal 24-hour clock which was required
for real-time data acquisition. Data acquisition system components and
approximate costs are given in Table 3-1.
Instrumentation Test Procedure
Pressure transducers were individually calibrated using a regulated
vacuum source and a mercury manometer. Measurements of transducer
voltages and mercury column heights were made at decreasing pressure
increments of approximately -10 kPa from 0 to approximately -80 kPa. To
determine whether the transducers displayed hysteretic behavior,
calibrations were continued at increasing pressure increments from -80
kPa to 0 kPa.

23
Table 3-1. Microcomputer-based data acquisition system components
and approximate costs.
Component
Cost($)
Microcomputer
100
Printer
100
Cassette Recorder
60
Pressure Transducer
68
Tensiometer
40
Analog-to-Digital and
Multiplexing Circuit
200
Transducer Power Supply
50

29
A/D boards were calibrated while interfaced with the microcomputer.
Known voltages were applied to the boards and the digital outputs read
by the computer were recorded.
Calibrated pressure transducers were interfaced to the tensiometers
by drilling small holes into the walls of the tensiometers. Access
ports on the transducers were inserted into the tensiometers and the
connections were sealed with silicone sealant. The pressure transducer-
equipped tensiometers were connected to the computer using 3-lead 22
gauge electrical hookup wire.
Two tests of the functioning of the microcomputer-based tensiometer
pressure transducer system were conducted. The first was conducted in
the laboratory. Three mercury manometer tensiometers were equipped with
pressure transducers and interfaced with the microcomputer. The
tensiometers were serviced to remove air from the system. The ceramic
cups were then exposed to the atmosphere and allowed to dry for several
hours. Microcomputer readings and mercury manometer readings were taken
for comparison at random intervals during drying. Data collection
continued throughout the tensiometer range. Three replications were
taken.
The second test of the tensiometer-pressure transducer system was
conducted under field conditions. Three instrumented mercury manometer
tensiometers were installed in a field lysimeter system with grass
cover. The microcomputer was located in the field laboratory office 25
m from the sensors. Tensiometers were serviced and allowed to
equilibrate with the soil water potential. The microcomputer was
programmed to read the pressure transducers hourly. Mercury manometers
were read manually at random intervals, typically 2 or 3 times per day

30
during the field evaluation. Manual mercury manometer readings were
compared with the microcomputer readings to assess the accuracy of the
instrumentation under field conditions. Instrumentation was evaluated
for 3 1-week periods. Tensiometers were serviced to remove entrapped
air between 1-week analysis periods.
Field Data Collection
A field lysimeter system was used to monitor soil water status and
water uptake from young citrus trees. The field lysimeter system was
located at the Irrigation Research and Education Park in Gainesville,
FI. Layout of the lysimeters is shown in Figure 3-1. The lysimeters
were cylindrical steel tanks with one end open. Tanks were installed
with the open end upward, exposing a circular surface production area of
2 m2. Soil profiles were 1.8 m deep. The soil in the lysimeters was
Arredondo Fine Sand, (hyperthermic, coated, Typic Quartzipsamment),
which was hand packed to approximate the physical characteristics of an
undisturbed soil profile. Drainage of excess water from the bottom of
each lysimeter was accomplished using either a porous stone or porous
ceramic cup located in the bottom of the tank as shown in Figure 3-2. A
vacuum pump removed the water from the lysimeters and the water was
trapped in PVC cylinders. The PVC cylinders were used to measure the
drainage from each lysimeter. Automated rainfall shelters were used to
cover the lysimeters in the event of rainfall. Additional details of
the construction and operation of the lysimeters were given by Smajstrla
(1985).
Data collection was conducted in two phases. In 1985 two
lysimeters were instrumented, and data were collected as described
earlier. Each lysimeter contained a young citrus tree approximately 3
â– I

31
Figure 3-1. Layout of lysimeter system.

32
TO VACUUM
SOURCE
Figure 3-2. Details of individual lysimeter soil water status
monitoring system.

33
years old. Both lysimeters were irrigated when the water potential in
the lysimeter dropped below -20 kPa. The -20 kPa water potential
corresponded to approximately a 44% water depletion from field capacity.
One lysimeter had a grass cover crop and the second had bare soil.
These two cover practices were used to simulate different cultural
practice effects upon the young citrus in a related experiment.
Tensiometers were installed at radii of 100, 300, 500 and 700 mm
from the center of the lysimeter, and at depths of 150, 300, 450, 600
and 900 mm as shown in Figure 3-3. Data were collected with the
microcomputer based data acquisition system at 10 minute intervals and
averaged over hourly periods.
In 1986 two lysimeters with young citrus trees were instrumented
with the same water and cover treatments as in 1985. Tensiometers were
installed in the lysimeters at radii of 130, 380 and 630 mm and at
depths of 150, 300, 450, 600, 900 and 1020 mm as shown in Figure 3-4.
In addition to the two lysimeters with young citrus trees, two
lysimeters with only grass cover were instrumented. The water potential
treatments for the two grassed lysimeters were -20 kPa and -40 kPa. The
-40 kPa water potential corresponded to approximately a 50% water
depletion from field capacity.
Model Development
A computer program describing the infiltration, redistribution and
extraction of soil water was written in the FORTRAN computer language.
A Prime 550 computer was used for all computations.
Several assumptions were employed in the development and use of the
numerical models to ease the complexities associated with mathematically
describing water movement in unsaturated porous media. These

Depth (mm)
Radius (mm)
0 100 300 600 700 800
150
300
450
600
900
1800
• - location of tensiometers
Figure 3-3. Location of tensiometers in 1985 field experiment
for young citrus trees in grassed and bare soil
lysimeters.

Depth (mm)
Radius (mm)
0 130 380 630 800
150
300
450
600
900
1020
1800
• - location of tensiometers
Figure 3-4. Location of tensiometers in 1986 field experiment
for young citrus trees in grassed and bare soil
lysimeters.

36
assumptions were: (1) The physical properties of the soil are
homogeneous and isotropic; (2) The physical and chemical properties of
the soil are constant in time; and (3) The hydraulic conductivity and
soil pressure potential are single-valued functions of water content.
One-Dimensional Model
Equation (6) was used to describe the one-dimensional movement and
extraction of water. Finite difference methods were used to solve the
water potential based flow equation. The model used an implicit finite
difference technique with explicit linearization of the soil parameters.
The finite difference form of equation (3) was written as
k
k+1 k
(h - h )
k i i
k+1 k+1 k k+1 k+1
K (h - h ) - K (h-h)
i+1/2 i+1 i i-1/2 i i-1
At
Az
where
i-1/2
AZ
K
i-1/2
K
i+1/2
+ K
i-1
(16)
(17)
and
i+1/2
i+1
(18)
The subscript i refers to distance and the superscript k refers to time.
To apply this equation to soil profiles, the soil was divided into a

37
finite number of layers or grids as shown in Figure 3-5. Equation (16)
was rearranged so that unknowns were on the left side of the equation
and the knowns were on the right side such that
At K At
i-1/2 k+1
) h + ( 1 +
2k i-1 2
AZ C A Z
K At K
i-1/2 i+1/2
+
2 k
AZ C
i
) h
k+1
k k k
At K At K - K At
i+1/2 k+1 k i-1/2 i+1/2
- ( ) h = h + _ ( ) - S
2 k i+1 i k k i
az C az C C
i i i
(19)
Equation (19) was written for each of the interior grids in the soil
profile. The top and bottom grids of the system required equation (19)
to be modified so that the equation would describe the boundary
conditions. The system of equations which are formed by applying
equation (19) to each grid produces a tridiagonal matrix which is
implicit in terms of h, water potential. The tridiagonal matrix was
solved using Gauss elimination. From explicit linearization, the
coefficients of the unknown h terms were known at each time step. Thus
equation (19) was reduced to
k+1 k+1 k+1
Ah + B h + C h
i i-1 i i i i+1
= D
(20)
where
k
At K
i-1/2
A = - ( )
i 2 k
A z C
(21)
l

GRIDS
1
i
•
I
i
•
i
i
•
â– 
I
i
•
I
I
I
i-1
i
i+1
I
I
I
•
I
I
I
•
I
I
I
n
Figure
Soil surface
K
1
h. . 6. . C. . K. .
i-l l-l i-l l-l
h. 9. C. K.
i ill
h.., 9... C. .
l+l l+l l+l i+l
9
n
-5. Schematic diagram of the finite-difference grid
system for the one-dimensional model of water
movement and extraction.

39
At K
1+1/2
C =
i
- (-
2 k
AZ C
i
-)
(22)
B = 1 - A - C
i i i
k k
At K - K
k i-1/2 i+1/2
D = h + — (
i i k
AZ C
(23)
At
) - s
k i
C
i
(24)
Boundary Conditions. The boundary condition for the surface grid
was variable. A flux boundary condition was used to simulate
infiltration during irrigation, and a no flux boundary condition was
used for redistribution. Evaporation from the soil surface was included
in the water extraction term. Equation (20) was modified for the
surface boundary condition such that
k+1
k+1
B h
1 1
+ C h
1 2
1
where
and
B = 1
(25)
(26)
D =
1
k
h +
1
At
— (
Az
Qs
1+1/2
At
- s
k 1
l
(27)

40
The lower boundary was also a flux boundary and was written as
k+1 k+1
Ah + B h = D (28)
n n-1 n n n
B = 1 - A (29)
n n
k
At K - Qb At
k n-1/2
h + — ( ) - S (30)
n k k n
Az C C
n n
The variables Qs and Qb were, respectively, the surface and bottom
fluxes imposed on the grid system. The bottom flux was set to either a
no flow or a gravity flow boundary condition.
Surface infiltration. The infiltration rate during irrigation was
set equal to the water application rate until the irrigation event was
completed. The total depth of water infiltrated at any time was
calculated as the summation of the incremental infiltration volumes at
all previous time steps.
Time steps. The time steps for implicit numerical techniques can
generally be much larger than those for explicit numerical techniques,
and still maintain stability. Haverkamp et al. (1977) stated that
stability conditions must be determined by trial and error because they
depend on the degree of nonlinearity of the equations.
The time step was estimated by a method given by Feddes et al.
(1978) and used by Clark (1982):
where
and
D =
n

41
X AZ
At < (31)
| Qmax |
where Qmax was the maximum net flux occuring across any grid boundary
and x is a factor where 0.015 accommodate the rapid movement of water during infiltration and early
stages of redistribution. The value of x may be assumed to be the
maximum permissible change in water content for any grid in the soil
profile.
Updating of soil parameters. The updating of the soil parameters
was achieved by use of a tabulated search to determine the corresponding
values of hydraulic conductivity, specific water capacity and water
content for the new values of soil water potential at each time step. A
logarithmic interpolation method was used to reduce computer time. The
table was set up with linearly increasing multiples of the logarithmic
values of soil water potentials.
Mass balance. A mass balance was calculated at each time step to
check the stability of the simulation model. The method used the
initial volume of water in the profile plus the infiltration and minus
the extraction by the plant. Evaporation from the soil surface was
included in the water extraction term as extraction from the surface
grids.
Because the grid system approximates a continuous system with dis¬
crete points, some error will result from the numerical calculations of
the simulation model. The error decreases as the grid size decreased.
Two-Dimensional Model
Water movement and extraction under young citrus trees was
simulated using a two-dimensional soil water extraction model. A radial

42
flow model may be used to describe water movement and extraction in a
soil profile which is nonuniformly irrigated as in water flow from
trickle emitters and spray jets. The radial flow model may also
describe the water extraction under widely spaced crops such as citrus.
With the assumption of radial symmetry, equation (8) was solved in
the vertical and radial directions. This equation is a second order,
nonlinear parabolic partial differential equation which expresses the
water potential distribution as a function of time and space
coordinates. No known analytical solution of equation (8) exist. Thus
it was solved numerically using finite difference equations. The model
developed in this research used an implicit finite difference technique
with explicit linearization of the soil parameters. Two solution
methods were used to solve the finite difference equations for the two-
dimensional model. It was solved implicitly using a Gauss elimination
method. Because of the time required to solve a large system of
equations by Gauss elimination, an alternation direction implicit (ADI)
method was also utilized to solve equation (8).
The finite difference equation for the solution of equation (8)
utilizing the Gauss elimination method was
k+1 k k k+1 k+1 k k+1 k+1
(h-h) K (h-h)-K (h-h)
k i,j i,j 1+1/2J i+l,j i,j Í-1/2J i.j i-l,j
C =
i,j 2
At AZ
k k+1 k+1 k k+1 k+1
r K (h -h)-r K (h-h)
i»j+l/2 i,j+l/2 i,j+l i,j i.j-1/2 i•j-1/2 i,j i,j-1
+
2
Ar
r

43
k k
K * K
i-l/2, j 1+1/2»j k
(32)
To apply this equation to the soil profiles, the soil was divided
into a finite number grids as shown in Figure 3-6. Equation (32) was
arranged such that the unknowns were on the left side of the equation
and the knowns were on the right side as follows
k
At K
i-l/2,j
- ( )
2 k
Az C '
i,j
At r
k+1
h - (
i-lj 2
Ar
k
K
i.J-1/2 i-l/2, j
)
k
r C
i,j iJ
k+1
h
i,j-l
k k
At ( K + K )
i-l/2, j 1+1/2,j
+ ( 1 + +
2 k
AZ C
i,j
k k
At ( r K + r K )
i.j+1/2 i.j+1/2 i,j-1/2 i.j-1/2
2 k
Ar r C
i,j i,j
) h
i.J
- (-
k k
At K At r K
i+1/2,j k+1 i, j+1/2 1+1/2,j
) h - ( )
2k 1+1,j 2 k
az C Ar r C
t,J
i.J
i,J
k+1
h
i,j+l

44
Figure 3-6. Schematic diagram of the finite-difference grid system
for the two-dimensional model of water movement and
extraction.

At
At
45
k
h +
i,j
AZ
k
k
K - k
i-l/2,j i+1/2, j
( ) -
C
i,j
k
S
k i,j
C
Equation (33) was reduced to
' k+1 k+1 k+1 k+1
Ah + B h + C h + D h
i, j 1-1»j i,j 1,j-l i,j 1,j 1,j i+l.j
where
k+1
+ E h = F
1,j i.j+1 i.j
k
At K
i—1/2,j
A = - ( )
i,j 2 k
AZ C
i.j
k
At r K
l.j-1/2 i-1/2, j
B = - ( )
I.J 2 k
Ar r C
I.J i.J
k
At K
i+1/2,j
D = - ( )
i.J 2 k
AZ C
i.j
k
A t r K
i.j+1/2 i+1/2, j
E - - (â–  )
i, J 2 k
Ar r C
i.j i.j
(33)
(34)
(35)
(36)
(37)
(38)

46
C = 1 - A - B -D -E (39)
iJ i,j 1,j i,j i,j
k k
At K - K At
k i-1/2, j i+1/2, j k
F = h + — ( ) - — S (40)
i,j i,j k k i,j
Az C C
1.j i.J
Boundary Conditions. The boundary condition for the surface grids
was a flux boundary condition which accounted for irrigation and
rainfall at the soil surface. At times when no irrigation occured the
surface boundary was a no flux boundary. Evaporation from the soil
surface was included in the water extraction term. The modification of
equation (34) to describe the surface boundary condition was
k+1 k+1 k+1 k+1
B h +C h + D h + E h =F (41)
l.j l.j-1 l,j l.j l.j 2J 1J 1J+1 1J
where
C = 1 - B - D - E (42)
l.j l.j l.J l.j
A t
k
F = h + —
l.j l.j
A Z
Qs
j
k
- K
1+1/2,j
k
C
) -
A t
k
C
(43)
l.j l.j
The lower boundary condition was also represented as a flux
boundary. The lower boundary in this research was an impermeable
boundary which represented the bottom of the lysimeter. The
modification of equation (34) to describe the lower no flux boundary was

47
k+1
k+1
k+1
k+1
A h
+ B h
+
C h + E
h = F
n,j n-lj
n,j n,j-
1
n,j n,j n, j
n.j+1 n,j
where
C = 1
-
A - B - E
n,j
n,j n, j n
,j
k
At
K
- Qb
At
k
n-
1/2, j j
k
F = h
+ — (
) -
S
n,j n
J
k
k n, j
AZ
C
C
n,j
n,j
The boundary conditions
for the radial
boundaries were
both
boundaries.
The no flux boundary condition
for the outer radius
represented the outer wall of the lysimeter
and was written
as
k+1
k+1
k+1
k+1
A h
+ B h
+
C h + D
h = F
i,m i-l,m
i,m i,m-
1
i ,m i ,m i ,m
i+l,m i,m
where
C
1
- A - B -
D
i ,m
i,m i ,m
i ,m
k
k
At
K
- K
A t
k
i-
1/2,m i+l/2,m
k
F = h
+ r
1
8
(44)
(45)
(46)
(47)
(48)
i,m i,m k k i,m
Az C C
(49)
n,m n,m
The boundary condition for the inner radiirs was also represented as
a no flux boundary. For the inner radius r(i,j) = 0, and its inclusion
into equation (8) would result in a division by zero. Therefore the
second term in equation (8) was rewritten as
13 3h K3h 3 a h
(r K(h) —) = - — + — ( K(h) —)
r3r 3 r r a r 9r a r
r 9 r
(50)

48
dh
a no flux boundary at r = 0 implies that — =0. Then if
dr
dh 1 dh
— = 0, the lim = 0. We rewrite equation (8) as
dr r dr
ah 3 3 h 3 a h 3 K(h)
C — = — ( K(h) — ) + — ( K(h) — ) - S(z,r,t) (51)
at 3 z 3 Z 3 r 3 r 3 z
The finite difference equation for equation (51) at the inner
radius may be written as
k k k
At K / At ( K + K )
i-1/2,1 k+1 i-1/2,1 i+1/2,1
- ( )h + ( 1 +
2 k 1-1,1 2 k
AZ C
1,1
A Z C
1,1
k k
At K A t K
i, 1+1/2 i+1/2,1 k+1
) h - ( ) h
2 k i ,1 2 k 1+1,1
A r C
A z C
1,1
1,1
At K
.i,1+1/2 k+1
- ( ) h
2 k i ,2
Ar C
1,1
= h
1,1
k k
At K - K At
i-1/2,1 i+1/2,1
+ _ ( ) -
AZ
1,1
k i ,1
u
(52)
or as
k+1 k+1 k+1 k+1
Ah + C h + D h + E h = F
1,1 i-1,1 i ,1 i,l i ,1 1+1,1 i,1 i,2 i, 1
(53)

49
where
k
At K
i-1/2,1
A = - ( ) (54)
i,l 2 k
Az C
1.1
k
At K
i+1/2,1
D = - ( )
i ,1 2 k
A z C
1,1
k
At K
i,1+1/2
E = - ( )
i,l 2 k
A r C
1,1
C = 1 - A - D - E
1,1 1.1 1,1 1,1
(55)
(56)
(57)
A t
k
F = h + —
1,1 1,1
A Z
k k
K - k
i-1/2,1 i+1/2,1
)
k
C
1,1
At
k
S
k i, 1
C
1,1
(53)
Updating of the soil parameters, surface infiltration, mass balance
and time steps were implimented as described for the one-dimensional
soil water flow model.
The system of equations for the two dimensional model produces a
banded matrix. The matrix is a five banded matrix with the form of

50
Reddel and Simada (1970) utilized a Gauss elimination method for
the solution of a two-dimensional groundwater model. They used an
algorithm developed by Thurnau (1963) which operates only on the banded
part of the solution matrix. Computer storage is not required for the
matrix elements above or below the band. A minimum band width is
desirable and an appropriate choice of the grid numbering pattern can
reduce the total width of the band.
In this model of two-dimensional soil water movement and
extraction, a subroutine was written which used the BANDSOLVE algorithm
developed by Thurnau (1963) to solve the system of equations. A
subroutine was also written which solves equation (8) using an
alternating direction implicit method. The finite difference equations
for the ADI solution are presented in Appendix A.
Soil Water Extraction
A macroscopic soil water extraction term was used in this research.
The model used was based on actual field measurements reported in the
literature. Denmead and Shaw (1962) conducted experiments which
compared actual ET to potential ET of corn as a function of the
available soil water. Their study showed that under high potential ET
demands, the actual ET was considerably less than the potential rate
even though the available soil water was considered adequate. They also
observed that under low potential ET demands, the actual ET was equal to

51
the potential ET down to very low soil water contents. Ritchie (1973)
observed similar results in separate experiments.
Saxton et al. (1974) used these observed relationships in modeling
soil water movement and extraction under a corn and under a grass crop.
Smajstrla (1982) used these relationships to model the soil water status
under a grass cover crop in Florida.
The soil water extraction function used in this research was
modeled after that used by Smajstrla (1982). The soil water extraction
rate was calculated as a function of the actual evapotranspiration rate
and the current soil water status. The water extraction term was
defined as
S^ = ET RDFt- R,
(ET / R.j )
(59)
where S^ = the soil water extraction rate per soil zone,
ET = the actual evapotranspiration rate,
RDF.¡ = the relative water extraction per soil zone at field
capacity, and
R,- = the relative available soil water per soil zone defined as
( 9 " 9wp )
( ®fc - % )
(60)
where 6
e
= the soil water content of the soil zone,
Wp = the soil water content at wilting point, and
0fc = the soil water content at field capacity.
The relative water extraction per soil zone (RDF) was defined as
the percentage of water extraction for the ith soil zone when soil water
is not limiting. It may also be considered to be a rooting activity
term which indicates the percentage of active roots in the ith soil
zone.

.52
Figure 3-7 shows the form of equation (59). Equation (59) permits
a rapid rate of soil water extraction when soil water is readily
available. As soil water is depleted, the water extraction function
produces a logarithmic rate of decline of soil water extraction. It
allows recovery of near potential rates of extraction during periods of
low potential ET and it rapidly limits extraction during periods of high
potential ET. It prevents the soil permanent wilting point from being
reached by limiting ET to very low rates as the permanent wilting point
is approached.
In this research equation (59) was used to limit water extraction
from a given soil zone as the water in that zone became less available.
Equation (59) was not used to limit the ET from the profile, only to
repartition the extraction within the soil profile.
Equation (59) was applied to calculate a water extraction rate from
each grid for each time step during model operation. The soil water
extraction rate per soil grid was then multiplied by the grid volume to
calculate the total water extraction at that time step. If the
calculated total water extraction rate was not equal to the actual ET,
the soil water extraction rates were linearly adjusted so that the
calculated and actual ET rates were equal.
The adjusted soil water extraction rates (S^) were computed as
ET
where Av is the volume of the ith soil grid.
(61)

RELATIVE SOIL WATER EXTRACTION RATE
53
Figure 3-7. Relationship of the relative available soil water and
potential soil water extraction rate on the soil
water extraction rate.

CHAPTER IV
RESULTS AND DISCUSSION
Instrumentation Performance
Three pressure transducers were individually calibrated to analyze
their response characteristics. A typical calibration curve (for
pressure transducer No. 1) is shown in Figure 4-1. The three
calibration equations are
Volts = 0.048 1 2
(kPa)
+ 0.9861
R2 = 0.999
(62)
Volts = 0.04781
(kPa)
+ 0.9867
R2 = 0.999
(63)
Volts = 0.04804
(kPa)
+ 0.9819
R2 = 0.999
(64)
The coefficients for these equations were not significantly
different ( a = 0.05), so that for many applications one equation could
be used for all 3 pressure transducers without significant error. For
the above 3 transducers, combined equations resulted in a maximum
expected error of 0.06 kPa at 1 volt (-0.3 kPa potential) and 0.27 kPa
at 5 volts (-83.67 kPa potential). Equations could therefore be
combined for many field applications. For this research, individual
transducer calibration curves were used in order to obtain the maximum
accuracies possible with the instrumentation, and because the
microcomputer memory was adequate to permit storage of the individual
calibration curves.
In the calibration procedure, hysteresis effects were studied by
measuring transducer outputs at decreasing water potentials from 0 to
-80 kPa, followed by increasing potentials from -80 to 0 kPa. For
these transducers there were no measurable hysteresis effects.
54

55
Figure 4-1. Calibration curve of output voltage versus pressure
applied for pressure transducer no. 1.

56
The A/D board was found to have a linear relationship between
voltage input and digital output as shown in Figure 4-2. The equation
which related the voltage to digital output was
Digital Units = 49.98 (Volts) R2 = 0.999 (65)
Equation (65) was combined with the individual transducer
calibration equations (62-64) to obtain relationships between
transducer-measured water potentials and the digital inputs to the
microcomputer. Equation (65) is only valid in the range of 0-5 volts
and 0-255 units for the 8-bit A/D board used. The 8-bit board,
therefore, allowed a resolution of only the nearest 0.02 volts. Greater
resolution could be achieved by the use of a 12-bit, 16-bit or other
higher resolution A/D board. A resolution of 0.02 volts is a resolution
of approximately 0.4 kPa (from equations 62-64). For this work this
degree of resolution was judged to be acceptable.
Tests of the assembled tensiometer-pressure transducer systems were
conducted in the laboratory by manually and automatically recording
tensiometer water potentials as water was allowed to evaporate from the
tensiometer ceramic cups. Figure 4-3 shows the changes in water
potential with time for 2 tensiometers. The continuous automatic
readings by the microcomputer are shown as solid lines. Open circles
show mercury manometer data that were manually read. Excellent
agreement between the automatic and manual readings was obtained. In
all cases, agreement was within 1.0 kPa.
Figure 4-4 shows a comparison of the manual and automatic readings
for the 3 tensiometers and pressure transducers tested. Agreement was
excellent, with all data points located within 1 kPa of the 100%
accuracy line.

57

58
Figure 4-3. Comparisons of mercury manometer manually-read and
pressure transducer automatically-read tensiometer
water potentials during drying cycles for two
tensiometers in the laboratory.

PRESSURE TRANSDUCER WATER POTENTIAL, kPa
59
Figure 4-4. Comparisons of mercury manometer manually-read and
pressure transducer automatically-read tensiometer
water potentials for all laboratory data.

60
To evaluate the performance of the microcomputer-based soil water
potential monitoring system under field environmental conditions, 3 1-
week studies were conducted at the IFAS Irrigation Research and
Education Park. Typical results for 1 week are shown in Figure 4-5 to
4-7 for 3 pressure transducers and tensiometers. Microcomputer data
were recorded hourly and are shown as the solid line in each figure.
These lines show diurnal cycles in soil water potential and a gradually
decreasing average daily water potential as the soil dried. Water
potential changes were slow because of the low evaporative demand and
relatively inactive grass cover during the February 25 - March 1 period
during which these data were collected.
The open circles in Figure 4-5 to 4-7 show manually read mercury
manometer data. Agreement between these and the automatically read data
were excellent under field conditions. The average variation between
manually and automatically read data was 0.47 kPa. This was
approximately the 0.4 kPa resolution of the instrumentation. The
maximum deviation observed was 1.76 kPa. These data demonstrated that
an accurate, effective microcomputer-based data acquisition system was
developed for automatically recording soil water potential measurements.
Because it was microcomputer-based, the system was inexpensive and
consisted of components that were readily available from local
electronics companies.
Field Data Collection
Field data were collected at the Irrigation Research and Education
Park as previously described. The data were collected for input and
verification of the numerical models developed in this research.
- j-

TENSIOMETER WATER POTENTIAL, kPa
61
Figure 4-5. Evaluation of pressure transducer-tensiometer no. 1
by comparison of mercury manometer manually-read and
pressure transducer automatically-read water potentials
in the field.

TENSIOMETER WATER POTENTIAL, kPa
62
Figure 4-6. Evaluation of pressure transducer-tensiometer no. 2 by
comparison of mercury manometer manually-read and
pressure transducer automatically-read water potentials
in the field.

TENSIOMETER WATER POTENTIAL, kPa
63
TIME, HOURS
Figure 4-7. Evaluation of pressure transducer-tensiometer no. 3 by
comparison of mercury manometer manually-read and
pressure transducer automatically-read water potentials
in the field.

64
The tensiometer data were used to calculate the soil water
extraction from the lysimeters. A method described by van Bavel et al.
(1968) was used to calculate the soil water extraction. Van Bavel's
method used an integrated form of equation (1) to calculate the soil
water extraction for a one-dimensional soil profile which was described
as z
(66)
where Rz = the total soil water extraction rate for the soil profile to
a depth of z. The total soil water extraction rate calculated for the
entire profile depth z is then the ET rate.
A direct measurement of the surface evaporation was not made.
Evaporation was included in the soil water extraction from the surface
grid. The tensiometer nearest the surface was located at 150 mm. Water
extraction from this surface grid included surface evaporation and ET
for that zone. Gravity flow drainage from the lower grid was assumed.
The hydraulic conductivity for the lower grid was assigned as the flux
out of the soil profile. This was justified because observed changes in
soil water potentials for the lower tensiometer were small.
1985 Field Data. The ET rates for selected drying cycles from the
1985 field experiments are shown in Figures 4-8 and 4-9. The observed
drying cycles occured from August 20 to September 6. Figure 4-8 is the
ET rate from the 20 kPa young citrus tree with grass cover. Figure 4-9
is the ET rate for the 20 kPa young citrus tree on bare soil. Several
data points for the tree with no grass were missed throughout the
experiment due to equipment failure. Due to these problems the tree
with no grass cover was treated as a one-dimensional profile for
analysis.

65
Differences between the ET rates in Figures 4-8 and 4-9 show that
the ET rates for the tree with grass cover were greater than those of
the tree with no grass cover. The higher ET rates for the tree with
grass cover were expected and were also observed by Smajstrla et al.
(1986).
The distribution of water extraction for the two citrus tree
treatments are shown in Table 4-1. The water extraction rates are shown
as percentages of the total soil water extraction and soil water
extraction per soil zone. These values were computed for input into the
water extraction models. Table 4-1 shows the differences between the
soil water extractions for the two treatments. Approximately 97% of the
water from the tree with no grass cover was extracted from the top 750
mm of the soil profile. The water extraction for the tree with grass
cover shows that approximately 73% of the water was extracted from the
top 750 mm of the soil profile. The water extraction rates from Table
4-1 also suggest that, for the tree with grass cover, water was
extracted below the 1050 mm depth observed with the data collection
system.
A two-dimensional distribution of soil water extraction is shown in
Table 4-2 for the tree with grass cover. Table 4-2 shows that water was
extracted almost uniformly throughout the profile. Greater percentages
of water were extracted from the inner radii of the lysimeter.
1986 Field Data. The ET rates for a selected drying cycle from
1986 field experiments are shown in Figures 4-10 and 4-11. Figure 4-10
is the ET rate from the 20 kPa young citrus tree with grass cover.
Figure 4-11 shows the ET rate for the 20 kPa young citrus tree with no
grass cover. The observed drying cycles occurred from June 16 to June

10
8 -
IT3
•o
c
ce
ce
>—4
o_
LO
ZZ
c
ce
i—
o
a.
c
6 -
235
~1—
240
245
250
JULIAN DAY
Figure 4-8. Evapotranspiration rate from the 1985 field experiments with a
20 kPa soil water potential treatment young citrus tree with
grass cover.
cn
cn

Figure 4-9. Evapotranspiration rate from the 1985 field experiment with
a 20 kPa soil water potential treatment young citrus tree
with bare soil.

68
Table 4-1. One-dimensional distribution of water extraction for a 15 day
drying cycle for Young Citrus trees with and without grass
cover.
Depth
Depth
Water
Water
Relative
Increment
Extraction
Extraction
Water
Extraction
(ITTT1)
(mm)
(mm)
(rnn/mm)
(%)
Tree
with Grass Cover
0-375
375
9.8
0.026
32.3
375-750
375
7.3
0.019
23.8
750-1050
300
8.7
0.029
28.6
below 1050
4.7
0.016
15.2
Tree with
No Grass Cover
0-375
375
5.9
0.016
27.9
375-750
375
7.4
0.020
35.3
750-1050
300
7.0
0.024
33.5
below 1050
0.7
0.0023
3.3

69
Table 4-2. Two-dimensional distribution of water extraction for a 15
day drying cycle for young Citrus with grass cover.
Depth Depth
Increment
(mm) (mm)
Radi us
(rrm)
100
300
500
700
Relative Water Extraction
•
( % )
0-375
375
8.59
10.46
6.92
7.66
375-750
375
5.65
5.79
5.46
5.65
750-1050
300
8.03
8.26
3.94
4.07
below 1050
4.21
3.45
6.33
6.04
Water Extraction
(mm)
0-375
375
2.6
3.2
2.1
2.3
375-750
375
1.7
1.8
1.7
1.7
750-1050
300
2.4
2.5
1.2
1.2
below 1050
1.3
1.1
1.9
1.8
Water Extraction
(mm/mm)
0-375
375
0.0070
0.0085
0.0056
0.0062
375-750
375
0.0046
0.0047
0.0044
0.0046
750-1050
300
0.0082
0.0084
0.0040
0.0041
below 1050
0.0043
0.0035
0.0064
0.0061

24. Differences between the two treatments showed that ET of the
tree with grass cover was greater than that of the tree with no cover,
especially early in the drying cycle. The 1220 mm depth tensiometer for
the tree with no grass did not function properly so these data were not
included in the data anaylysis, but this did not introduce a large
error, from Table 4-1.
The water extraction distributions are shown in Table 4-3. The
percentage of water extracted from the top layer was greater for the
tree with grass cover.
Evapotranspiration data for the two grass cover treatments are
shown in Figures 4-12 and 4-13. Figure 4-12 shows ET rates of the 20
kPa grass covered lysimeter and Figure 4-13 shows ET rates of the 40 kPa
grass covered lysimeter. Comparison of Figure 4-12 and 4-13 shows that
the ET of the 20 kPa treatment was higher than that of the 40 kPa
treatment. The grass had more water available for ET in the 20 kPa
treatment.
Table 4-4 shows the water extraction distributions for the two
grass treatments. Both distributions were similar with both having
approximately the same soil water extraction percentages for each layer.
Model Verification
The accuracy of the numerical models in simulating the infiltration
and redistribution of soil water was determined by comparison with other
computer simulations from previous works in the literature.
The one-dimensional computer model was compared to the simulations
of Rubin and Steinhardt (1963) and Hiler and Bhuiyan (1971). Their work
provided data from soils with widly different hydraulic properties and
also provided simulation results which were used for model verification.

I
I
I
I
JULIAN DAY
Figure 4-10. Evapotranspiration rate from the 1986 field experiment with a
20 kPa soil water potential treatment young citrus tree with
grass cover.

Figure 4-11. Evapotranspiration rate from the 1986 field experiment with
a 20 kPa soil water potential treatment young citrus tree with
bare soil.
ro

73
Table 4-3. Two-dimensional distribution of water extraction for a 4
day drying cycle for young citrus trees with and without
grass cover.
Depth Depth Radius
Increment (mm)
(mm) (rrm)
200 600 200 600
Tree with Tree with
Grass Cover No Grass Cover
Relative Water Extraction
(%)
0-225
225
28.2
31.2
19.9
16.95
225-375
150
12.2
15.2
12.3
17.9
375-525
150
4.45
1.2
6.85
3.75
525-750
225
4.4
1.2
8.4
8.7
750-1050
300
0.75
1.2
2.6
2.65
Water Extraction
(nm)
0-225
225
1.6
1.8
0.9
0.8
225-375
150
0.7
0.9
0.6
0.8
375-525
150
0.3
0.07
0.3
0.2
525-750
225
0.3
0.07
0.4
0.4
750-1050
300
0.04
0.07
0.1
0.1
Water Extraction
(mm/mm)
0-225
225
0.0073
0.0081
0.0041
0.0035
225-375
150
0.0048
0.0059
0.0038
0.0057
375-525
150
0.0017
0.00046
0.0021
0.0011
525-750
225
0.0011
0.00031
0.0017
0.0018
750-1050
300
0.0001
0.00023
0.0004
0.0004

4
Figure 4-12.
JULIAN DAY
Evapotranspiration rate from the 1986 field experiment with
a 20 kPa soil water potential treatment with a grass cover.
-P»

4
3-|
Lü
170 171 172 173
JULIAN DAY
174
Figure 4-13. Evapotranspiration rate from the 1986 field experment with
a 40 kPa soil water potential treatment with a grass cover.
^1
CT»

76
Table 4-4. One-dimensional distribution of water extraction for a 4 dry
drying cycle for a grass cover crop at water depletion
levels of 20 kPa and 40 kPa.
Depth
Depth
Water
Water
Relative
Increment
Extraction
Extraction
Water
Extraction
(mm)
(mm)
(rim)
(mm/mm)
(*)
20 kPa Treatment Tree with
Grass Cover
0-225
225
6.3
0.028
57.9
225-375
150
2.0
0.014
18.6
375-525
150
0.9
0.006
8.3
525-750
225
1.1
0.005
9.9
750-1050
300
0.6
0.002
5.3
40 kPa Treatment Tree with
Grass Cover
0-225
225
3.9
0.0170
63.1
225-375
150
0.8
0.0055
13.5
375-525
150
0.5
0.0036
8.7
525-750
225
0.4
0.0019
6.9
750-1050
300
0.5
0.0016
7.8

77
Rubin and Steinhardt (1963) used an implicit solution of the
Richards equation in terms of water content. Their model was used to
study constant intensity rainfall infiltration on a Rehovot sand. The
soil data were presented as analytic functions which were tabulated for
use in this work. The soil hydraulic characteristics are shown in
Figures 4-14 and 4-15.
The simulated soil profile had a uniform initial soil water content
of 0.005. A uniform grid size of 10 mm was used. Figures 4-16 and 4-17
contain plots of soil water content profiles for infiltration into the
Rehovot sand. The figures show both the results from Rubin and
Steinhardt (1963) and this work. Figure 4-16 contains the soil water
content profiles for a constant infiltration rate of 12.7 mm/hr. Figure
4-17 contains the soil water content profiles for a constant
infiltration rate of 47 mm/hr. Results of this work are in excellent
agreement with those of Rubin and Steinhardt (1963).
The work of Hiler and Bhuiyan (1971) was also used to verify the
accuracy of the model developed in this work. They used a computer
model written in CMSP to solve the Richards equation. Surface
infiltration was simulated for two soils, Yolo light clay, and Adelanto
loam. The hydraulic characteristics of these soils are presented in
Figures 4-18 through 4-21.
The simulation results for these soils are presented in Figures 4-
22 through 4-24. Figures 4-22 and 4-23 show water content profiles with
time for the Yolo light clay at different initial conditions. Figure 4-
24 shows the soil water content profiles with time for the Adelanto
loam.

78
Comparison of results for the two soils showed excellent agreement
between Hiler and Bhuiyan's CMSP model and this work. The minor
differences between the results of the models could have resulted from
the interpolation of the soil properties.
The two-dimensional model was tested by simulating infiltration for
a Nahal Sinai sand (Bresler et al. 1971). Results were compared with a
steady-state analytical solution developed by Wooding (1968) and with an
ADI-Newton numerical simulation method presented by Brandt et al.
(1971).
These models simulated the infiltration of water into the soil from
a point source. The boundary condition used for the simulations was a
constant flux for the innermost surface grid (r = 0). The flux was set
equal to the saturated hydraulic conductivity of the soil. No flow
boundaries were used for the lower and radial boundaries. The simulated
soil profile had a uniform initial soil water content of 0.037. Figures
4-25 and 4-26 show the soil characteristics for the Nahal Sinai sand.
Figures 4-27 and 4-28 show a comparison between this work and that
of Wooding (1968) and Brandt et al. (1971). The data are presented in
dimensionless form with the relative water content defined as
S(0)/S(6sat) where S(0) was defined by Philip (1968) as
0
S(e) = \ D de (67)
0n
where D = the hydraulic diffusivity and 9n = the residual soil water
content. The total infiltration time was 57 min in Figure 4-27 and 297
min in Figure 4-28.
Figures 4-27 and 4-28 demonstrate that this model accurately
simulates the results of Brandt et al. (1971). Differences between the

SOIL WATER POTENTIAL (mm)
79
Figure 4-14. Soil water potential-soil water content relationship
for Rehovot sand.

HYDRAULIC CONDUCTIVITY (mm/hr)
80
Figure 4-15. Hydraulic conductivity-soil water content relationship
for Rehovot sand.

DEPTH (mm)
SOIL WATER CONTENT (mm/mm)
Figure 4-16. Simulated results of soil water content profiles for infiltration into a
Rehovot sand under constant rain intensity of 12.7 mm/hr.

DEPTH (mm)
Figure
SOIL WATER CONTENT (mm/mm)
17. Simulated results of soil water content profiles for infiltration into
Rehovot sand under constant rain intensity of 47 mm/hr.

SOIL WATER POTENTIAL (mm)
83
SOIL WATER CONTENT (mm/mm)
Figure 4-18. Soil water potential-soil water content relationship
for Yolo light clay.

HYDRAULIC CONDUCTIVITY (mm/hr)
Figure 4-19. Hydraulic conductivity-soil water content relationship
for Yolo light clay.

SOIL WATER POTENTIAL (mm)
85
Figure 4-20. Soil water potential-soil water content relationship
for Adelanto loam.

HYDRAULIC CONDUCTIVITY (mm/hr)
86
Figure 4-21. Hydraulic conductivity-soil water content relationship
for Adelanto loam.

DEPTH (mm)
0.0 0.1 0.2 0.3 0.4 0.5
SOIL WATER CONTENT (mm/mm)
Figure 4-22. Simulated results of soil water content profiles for infiltration into
a Yolo light clay with initial pressure potential at -66 kPa.
co
^4

Pages
Missing
or
Unavailable

RELATIVE EVAPOTRANSPIRATION RATE
TIME OF DAY
T I
25
Figure 4-32. Distribution of evapotranspiration through the daylight hours.
LO

SOIL HATER POTENTIAL (mm)
100
Figure 4-33. Soil water potential-soi1 water content relationship
for Arredondo fine sand.

HYDRAULIC CONDUCTIVITY (mm/hr)
.101
Figure 4-34. Hydraulic conductivity-soil water potential relationship
for Arredondo fine sand.

102
conductivity function produced results with the numerical model which
were in agreement with those from the observed field data.
One-dimensional Model
Time Steps. A sensitivity analysis was conducted to evaluate
optimum grid sizes and timesteps. The time step used was variable, and
the criteria for determining the variable time step were presented in
equation (31). Operation of the model provided data from which limits
for maximum and minimum time steps were determined. The maximum time
step for model operation was determined to be 1 hour. Operation of the
model with greater maximum time step produced varing results while
operation with smaller maximum time steps gave results in agreement with
the 1 hour timestep. The maximum time step was a upper limit for the
calculated time step. The maximum time step was used when water
movement in the soil profile was small. When water movements in the
soil profile were large, as during infiltration, a minimum time step was
determined. The minimum time step for the range of infiltration used in
the model was determined to be 0.0001 hours. The use of a smaller time
step slowed the model execution while not improving the mass balance.
Larger time steps caused the mass balance to increase, thus indicating a
reduction in model accuracy.
Grid Size. The optimum grid sizes were also determined by
sensitivity analysis. A grid size of 30 mm was determined to be
adequate to describe water movement in the soil profile for the one¬
dimensional model. Larger grid sizes decreased model execution time,
but the soil water distribution was not adequately described. Smaller
grid sizes increased the execution time while not improving the model
accuracy.

103
Model Operation. The accuracy of the one-dimensional model was
verified by simulating 35 days of observed water extraction,
infiltration and redistribution under a young citrus tree with grass
cover in a field lysimeter system. Initial conditions for the model
were obtained directly from the observed soil water potentials and
calculated root activity distributions. Observed average daily ET
values were input to the numerical model. A no flux boundary condition
was specified for the lower boundary. The surface boundary was set to a
constant flux during irrigation and no flux at the end of the
irrigation. Evapotranspiration from the top grid in the model was
accounted for in the soil water extraction term.
Simulations of the observed field data were in good agreement with
the experimental data. The simulated water contents were compared to
field data in Figures 4-35 to 4-39. These data represent the five soil
depths monitored with the data acquisition system. The simulated water
contents for the first drying cycle were in excellent agreement with the
field data.
An irrigation of 28 mm of water was applied to the lysimeter. The
model simulated this irrigation, redistribution and extraction for a
second drying cycle. The simulation results were in excellent agreement
with the field data for the second drying cycle also. The water
contents for the 450, 600 and 900 mm depths were in excellent agreement
with the observed field water contents throughout the drying cycle.
The water contents for the 150 mm depth was in good agreement throughout
most of the second drying cycle. The simulated water contents for the
300 mm depth were greater than the observed field data.

SOIL WATER CONTENT (mm/mm)
0.20
0.15
0.10
0.05
0.00
230 240 250 260 270
JULIAN DAY
Figure 4-35. Simulation results for the one-dimensional model with field data fo
the 150 mm depth.
— Field data
â–  Model results
i • 1 • r

SOIL WATER CONTENT (mm/mm)
Figure 4-36. Simulation results for the one-dimensional model with field data for
the 300 mm depth.

SOIL WATER CONTENT (mm/mm)
0.2ft
— Field data
â–  Model results
0.15-
0.10-
0.05.
230 240 250 260 270
JULIAN DAY
Figure 4-37. Simulation results for the one-dimensional model with field data
for the 450 mm depth.
O
O'»

SOIL WATER CONTENT (mm/mm)
Figure 4-38. Simulation results for the one-dimensional model with field data
for the 600 mm depth.

SOIL WATER CONTENT (mm/mm)
Figure 4-39. Simulation results for the one-dimensional model with field data
for the 900 mm depth.

109
The differences between the simulated and observed water contents
may have been due to the variations in the soil properties under field
conditions. The model assumed homogeneous and isotropic soil
conditions. This cause is supported by the random variations between
observed and simulated water contents. Overall the model simulated
changes in soil water movement and extraction with excellent agreement.
Two-dimensional Model
Time Steps. A sensitivity analysis was conducted to evaluate
optimum grid sizes and time steps for the two-dimensional model. The
time step used was variable, and the criteria for determining the
variable time step were presented in equation (31). Operation of the
model provided data from which limits for maximum and minimum time steps
were determined. These maximum and minimum time steps were determined
to be the same as used with the one-dimensional model. The maximum
allowable time step for model operation was determined to be 1 hour.
The minimum allowable time step for the range of infiltration used in
the model was determined to be 0.0001 hours.
Grid Size. The grid sizes for the two-dimensional model were also
determined by sensitivity analysis. A grid size of 30 mm was determined
to be adequate to accurately describe water movement in the vertical
direction. The radial grid size of 50 mm was determined to be adequate
to accurately describe water movement in the radial direction.
Model Operation. The two-dimensional model was used to simulate 35
days of water extraction, infiltration and redistribution under a young
citrus tree with grass cover in a field lysimeter. The initial
conditions for the model were determined directly from the observed soil
water potentials and root distributions. Observed daily ET amounts were

110
input to the model. Boundary conditions for the two-dimensional model
were all no flow boundaries when there was no infiltration. During
infiltration, a surface flux equal to the irrigation application rate
was used. When irrigation was completed, the surface boundary condition
was reset to a no flow condition.
Simulations of the observed field data were in excellent agreement
with the experimental data for the first drying cycle. The simulated
water contents were compared to the field data in Figures 4-40 to 4-44.
These data represent the 15 sites in the soil profile which were
monitored with the data acquisiton system. The observed and simulated
water contents for the drying cycle are presented for each of the four
radii measured. Simulated water contents were in excellent agreement
with the observed field water contents for almost all of the observed
sites during the first drying cycle. The 450 mm depth water contents
did not agree as well at the 100 mm radius for the later part of the
drying cycle.
Simulated soil water movement and extraction for the second drying
cycle were in good agreement with the observed field data. Simulated
water contents for the 100 and 300 mm radii were in excellent agreement
with field water contents for all five observed depths. The simulated
water contents for the outer radii were different than observed water
contents for several locations. These differences may be attributed to
variations in the soil properties or nonuniform application of
irrigation water at the surface. The soil properties were assumed to be
homogenous and isotropic. The model assumed that the irrigation
application was uniform. Overall the model results were in good
agreement with the observed field data.

SOIL WATER CONTENT (mm/mm)
0.20-
Field data
Model results
0.15-
0.15-
0.10-
0.05-
0.00.-
230
JULIAN DAY
Figure 4-40. Simulation results for the two-dimensional model with field data for
the 150 mm depth.

SOIL WATER CONTENT (mm/mm)
0.2a
0.15-
0.10-
0.05-
0.00- —
0.15-
0.10L
0.05-
o.oo4—
230
Figure 4-41.
Field data
Model results
JULIAN DAY
Simulation results for the two-dimensional model with field data for
the 300 mm depth.

SOIL WATER CONTENT (mm/mm)
JULIAN DAY
Figure 4-42. Simulation results for the two-dimensional model with field data for the
450 mm depth.

SOIL WATER CONTENT (mm/mm)
Figure 4-43. Simulation results for the two-dimensional model with field data for
the 600 mm depth.

SOIL WATER CONTENT (mm/mm)
Figure
0.20-
- - ■ — ' ■■—— —
Field data
Model results
—4,
¿ irr. i .■.■.■.■•4»
100 mm Radius
300 mm Radius
â–  i i
^ U
i i l
500 mm Radius
700 mm Radius
230 240 250 260 22
< i |
tO 240 250 260 270
0.00-
0.00
JULIAN DAY
4-44. Simulation results for the two-dimensional model with field data for
the 900 mm depth.

116
Model Applications
The numerical models developed in this research can provide
researchers with numerical models of soil water movement to study
infiltration, water extraction and solute movement. The models may be
used to determine optimum irrigation scheduling for a wide variety of
soils and atmospheric conditions. The numerical models allow the
input of historic rainfall and ET data.
To demonstrate potential applications of the numerical models
developed, the one-dimensional numerical model was used to compare three
irrigation strategies. The first strategy studied was to schedule
irrigations to occur when the water potential in the soil profile
dropped below -10 kPa at a specified depth in the soil profile. The
second was to schedule irrigations to occur when the water potential in
the soil profile dropped below -20 kPa and the third when the water
potential dropped below -30 kPa. The numerical model was modified to
monitor specific depths in the soil profile to represent tensiometer (or
other soil water status monitoring equipment) locations as in field
operations. Depths of 300 and 900 mm were monitored in this study. An
irrigation depth of 25 mm was applied when the soil water potential at
any monitoring depth dropped below the set water potential. Rainfall
interactions were not considered in these simulations but could be taken
into account with the numerical model by inputting historical data.
The inputs to the model were the initial soil water potentials
which were assumed to be at field capacity. Evapotranspiration was
input on a daily interval, and for these simulations was assumed to be a
constant 4 mm/day. The surface boundary condition for the one¬
dimensional model was a no flux when irrigation was not occurring.

117
During irrigation the surface boundary condition was a flux boundary
with the flux equal to the irrigation application rate. The lower
boundary condition for the model was a gravity flow condition to
simulate deep percolation from the soil profile.
Simulation results for the three irrigation strategies were shown
in Figures 4-45 to 4-49. Figure 4-45 and 4-46 show the soil water
contents monitored at 300 and 900 mm. From the soil water contents, it
appears that the irrigations for the -10 kPa treatment were controlled
by the potentials at the 300 mm depth because of the frequent and small
irrigations applications. The irrigations for the -20 and -30 kPa
treatments appear to be controlled by the 900 mm depth because of the
less frequent and larger irrigation applications. These demonstrate
another use of the numerical model to determine the optimum depths to
monitor for different soils and different allowable soil water
depletions.
Figure 4-47 shows the soil water storage in the profile during the
simulation period. The soil water storage for the -10 kPa treatment was
consistently more uniform, while the soil water storage for the -20 and
-30 kPa treatments varied widely.
Figures 4-48 and 4-49 shows the irrigation applied and the drainage
out of the profile. The same amount of irrigation was applied for all
three treatments. Irrigation was applied more frequently for the -10
kPa treatment. Irrigations were less frequent for the -20 and -30 kPa
treatments but they required several irrigations to bring the soil water
potentials in the soil profile above the set limits. Figure 4-49 showed
the drainage loss out of the bottom of the profile for the different

SOIL WATER CONTENT (mm/mm)
0.20-
10 kPa
20 kPa
0 10 20 30 40 50 60
DAYS
Figure 4-45. Soil water contents for the three irrigation treatments at
the 300 mm depth.
00

SOIL WATER CONTENT (mm/mm)
Figure 4-46. Soil water contents for the three irrigation treatments for
the 900 mm depth.
‘119

STORAGE (mm)
Figure 4-47. Soil water storage for the three irrigation treatments.
ro
o

250
10 kPa
20 kPa
30 kPa
I 1 1 1 1 >
30 40 50 60
DAYS
Figure 4-48. Cumulative irrigation for the three irrigation treatments.

CUMULATIVE DRAINAGE (mm)
150
125
10 kPa
20 kPa
30 kPa
100.
75 -
/
Figure 4-49.
Cumulative drainage from the soil profile for the three
irrigation treatments.

123
treatments. The drainage out of the all of the treatments was large at
first because of the initial conditions. The drainage for the -10 kPa
teatment was small for the duration of the simulation period. Drainage
from the other treatments were higher because of the larger amounts of
water applied at each irrigation.

CHAPTER V
SUMMARY AND CONCLUSIONS
A data collection system was developed which provides
researchers with information needed for developing and validating models
of soil water movement, crop water use and evapotranspiration. Two
numerical models were developed to simulate soil water movement and
extraction. These numerical models may be used to compare irrigation
scheduling strategies.
Microcomputer-based Data Acquisition System
A low-cost microcomputer-based data acquisition system to
continuously record soil water potentials was developed and tested.
Excellent agreement was obtained between soil water potentials read with
the data acquisition system and those read manually using mercury
manometers. Typical agreement was within 0.47 kPa. The maximum,
deviation observed for all evaluations conducted was 1.76 kPa. The
microcomputer based data acquisition system has the ability to monitor
inputs and make decisions based upon the input data to control events in
the field.
Field Data
The data acquisition system developed was used to measure soil
water extraction patterns under young citrus trees and under a grass
cover crop grown in field lysimeters. Observations were monitored
during two drying cycles and one irrigation. The data collected were
evaluated and parameters determined for input to the numerical
simulation models. Evapotranspiration, soil water extraction rates and
124

125
relative soil water extractions were calculated from the soil water
potential measurements.
Modeling Soil Water Movement and Extraction
A one-dimensional numerical model was developed to simulate soil
water movement and extraction under a uniformly irrigated soil profile.
The model was verified using data from the literature. It was then used
to simulate the soil water movement and extraction from a young citrus
tree in a field lysimeter. Two drying cycles and one irrigation were
simulated. Simulated results were in excellent agreement with the data
collected using the microcomputer based data acquisition system.
A two-dimensional numerical model was developed to simulate soil
water movement and extraction under a nonuniformly irrigated soil
profile. The model was compared to both one- and two-dimensional models
of soil water movement from the literature. It was then used to
simulate the two-dimensional movement and extraction of soil water in a
field lysimeter with a young citrus tree. Two drying cycles and one
irrigation were simulated. The model results were in good agreement
with the data collected using the microcomputer based data acquisition
system.
Model Applications
The numerical models may be used to study irrigation scheduling
problems. A comparison of three different irrigation scheduling
strategies was conducted to demonstrate one use of the models. These
numerical models will also be useful in other applications such as the
modeling of solute movement in soils, modeling of fertilizer movement
under a trickle emitter, or as inputs to a larger crop modeling system.

APPENDIX A
Alternating Direction Implicit Finite Differencing
An alternating direction implicit solution method was developed to
solve the two-dimensional radial flow model. The solution method
divides the solution into two half cycles, thus two cycles are required
to solve for one time step. Equation (8) was solved in the radial
direction during the first half cycle and then solved in the vertical
direction during the second half cycle. The ADI solution for equation
(8) may be written as
k+1/2
k+1/2
k+1/2
k+1/2 8 h
1
1 3 k+1/2 3 h
1 3 k+1/2 3 h
C — =
- (
(r K _
) + (r K
—) )
3 t
r
2 3r 3 r
2 3 r
3 r
k+1
k
13 k+1 3 h
13 k 3 h
+
(
(K —)
+ (K _)
)
2 3 Z 3 z
2 3 Z 3 Z
k+1/2
k+1 k
1 3 K 1
3 K 3 K
+
(
+ — (
+ )
- S(z,r,t)
2 3 z 4
3 Z 3 Z
(68)
The finite difference equations used for the solution of equation (8)
were expressed as
k+1 k
(h - h )
i,j iJ
C
i,j
At
(69)
126

127
3 3h
—(K )
3 Z 3 Z
K ( h - h ) - K ( h - h )
Í+1/2J i+l,j i,j i-1/2,j i,j i-l,j
(70)
2
Az
13 3 h 1
_(_(rK _)) = (r K (h -h )
r 3r 3r 2 i,j+l/2 i,j+l/2 i,j+l i,j
r Ar
i.j
- r K ( h - h ) ) (71)
iJ-1/2 i,j-1/2 i.j i,j-1
K - K
3 K i-1/2, j 1+1/2,j
(72)
Equation (68) was divided into two parts and arranged such that the
unknowns are on the left side of the equations and the knowns are on the
right side such that
1 At
( 2
2 Ar
ri,j-l/2
r
i.j
k+1/2
K
i-1/2,j k+1/2
) h
i. J-1
C
i.j
k+1/2
1 At ( r K
i, j+1/2 iJ+1/2
+ (
k+1/2
+ r K )
i,j-1/2 1,j-1/2
+ 1 )
k+1/2
h
i.j
2
2
Ar r C
i.j i.j

128
k+1/2
1 At r K
i,j+1/2 i, j+1/2
2
2 a r r C
i,j i,j
k+1/2
h
k k
1 At K - K
k i-1/2, j i+1/2,j
= h +-— ( )+ (
i.j
2 A z C 4 A z
k+1/2 k+1/2
1 At K - K
i-1/2,j i+1/2,j
-)
1 A t K
+ ( -
i-1/2,j
2 A z C
k k
.) ( h - h )
1-1. j i.j
i,J
1 A t K
+ (
2 AZ C
i+1/2, j k k
) ( h - h )
1+1. J i.J
i.J
1 At
2 C
i.J
i.J
and
1 At K
( -
k+1
i-1/2, j k+1
) h
2 A z C
i.J
1-1, j
k+1 k+1
1 At ( K + K )
i-1/2,j i+1/2,j
+ (
2
AZ C
i,j
k+1
+ 1 ) h
i.j
(73)
2

129
1
- ( -
2
k
At K
i+1/2,j k+1
) h
2 1+1,j
Az C
i.j
1 At
+ _— (
4 Az
k+1/2 k+1/2
K - k
t-1/2, j i+1/2, j
)
C
i,j
1 A t
2 C
i.j
k
S
i.j
k+1/2
1
At r
i.j-1/2
K
I.j-1/2
2
2
A r r
C
i.j
i.j
1
At r
k+1/2
K
i.j+1/2
i.j+1/2
2
2 Ar r C
i.j 1
k+1/2 k+1/2
h - h )
i.j-1 i.j
k+1/2 k+1/2
h - h ) (74)
i.j+1 i.j
where i represents the vertical distance, j represents the radial
distance and k represents the time increment.
The system of equations which are formed by applying equations (73)
and (74) to each grid point produces a tridiagonal matrix at each half
time step. The tridiagonal matrices were solved using a Gauss
elimination method for tridiagonal matrices. Equations (73) and (74)
may be simplified such that equation (73) may be written as
k+1/2 k+1/2 k+1/2
- AR h + BR h - CR h = DR
i.j-1 i.j i.j+l i.j
(75)

130
where
k+1/2
1 A t r K
i.j-1/2 i.J-1/2
AR = ( )
2
2 a r r
i,J
i.J
(76)
k+1/2
1 At r K
l.j+1/2 i,j+1/2
CR = ( )
2
2 a r r
i.J
i.J
(77)
and
BR = 1 + AR + CR
(78)
1 At K - K
DR = h + (
i,j
2 AZ
i-1/2 ,j i+1/2,j
k+1/2 k+1/2
1 At K - K
i-1/2, j i+1/2,j
-) + (
4 AZ
1,J
-)
1 At K
+ (-
i-1/2,j
2 az C
k k
â– ) ( h - h )
1-1J l.j
i.J
1 At K
+ ( -
i+1/2, j
2 AZ C
i.J
1 At
k k
) ( h - h )
i+l»j 1.j
2 C
i.J
i.J
(79)
and equation (74) may be written as

131
- AV h
where
k+1
k+1
k+1
+ BV h
- CV h
1-1. j
I.j
1+1 .j
k+1
1
At K
AV =
(
1-1/2,j
2
2
A Z C
)
i,j
DV
i.J
(80)
(81)
k
1 At K
i+1/2J
C V = ( ) (82)
2
2 az C
i J
and
BV = 1 + AV + C V
(83)
DV
1 At
k
h + — (
i.j
4 AZ
k+1/2 k+1/2
K - K
1-1/2,j i+1/2, j
)
C ..
i,j
1 At
k
S
i.j
1 At r
i.j-1/2
2
2 A r r
i,j
k+1/2
K
i.j-1/2 k+1/2 k+1/2
) ( h - h )
i.j-1 1. J
C
I.j
k+1/2
1 At r K
i.j+1/2 i.j+1/2
+ ( )
2
2 Ar r C
i,j i.j
k+1/2 k+1/2
( h - h )
i,j+l 1,j
(84)

132
Boundary Conditions. The boundary condition for the surface layers
was a flux boundary condition which accounted for irrigation and
rainfall at the soil surface. At times when no irrigation occured the
surface boundary was a no flux boundary. Evaporation from the soil
surface was included in the water extraction term. The modification of
equations (75) and (80) to describe the surface boundary condition was
k+1/2 k+1/2 k+1/2
- AR h + BR h - CR h
l.j-1 l.j l.j+1
DR (85)
l.j
where
k+1/2
lit Qs - K 1 At Qs
k j 1+1/2,j j
DR = h + ( ) + (
l.j
2 az C 4 az
1J
- K
1+1/2,j
-)
l.j
1 At K
+ ( -
1+1/2,j k
2 az C
l.j
1 At
) ( h - hk ) Sk
1+1, j l.j l.j
2 C
l.j
(86)
and
k+1
BV h
l.j
where
k+1
CV h = DV
2,j l.j
BV = 1 + CV
(87)
(88)
and

133
DV =
k
h
l.j
1 At
+ __ (
4 Az
k+1/2
Qs - K
j 1+1/2,j
)
C
l.j
1 At
2 C
1J
k
S
l,j
k+1/2
1
At r
l.j-1/2
K
l.j-1/2
2
2
A r r
C
l,j
1, j
1
At r
k+1/2
K
1J+1/2
1, j+1/2
2
2
A r r
C
l.j
l.j
k+1/2 k+1/2
h - h )
l.j-1 l.j
k+1/2 k+1/2
h - h ) (89)
l.j+l l.j
The lower boundary condition was also represented as a flux
boundary. The lower boundary in this research was an impermeable
boundary which represented the bottom of the lysimeter. The
modifications of equations (75) and (80) to describe the lower boundary
condition were
k+1/2 k+1/2 k+1/2
- AR h + BR h - CR h = DR
n,j-l n,j n,j+1 n,j
where AR, BR and DR are as previously described and
DR
1 At
k
h +
n,j
2 az
k
K
n-1/2J
Qb 1 At
j
) + - — (
k+1/2
K - Qb
n-1/2, j j
)
C 4 Az C
n,j n,j
(90)

134
k
1 At
n-l/2,j k k k
) ( h - h ) S
2 n-1, j n,j n,j
2 az C 2 C
n»j n,j
1 At K
+ (
(91)
and
k+1 k+1
- AV h + BV h
n-1, j n,j
where AV was previously described and
BV = 1 + AV
and
k+1/2
1 At K - Qb
k n-1/2,j j
DV = h + (
n.j
4 az C
n,j
DV
n,J
1 At
-) S
2 C
n»J
(92)
(93)
n,J
k+1/2
1 At r K
n,j-1/2 n,j-1/2
2
2 at r
k+1/2 k+1/2
) ( h - h )
n,j-l n,j
n,J
n,J
k+1/2
1 At r K
n.j+1/2 n,j+1/2
2
2 a r r C
k+1/2 k+1/2
) ( h - h )
n,j+l n, j
n,J
n,J
(94)
The boundary conditions for the radial boundaries were both no flux
boundaries. The no flux boundary condition for the outer radius (j=m)

135
represented the outer wall of the lysimeter and equations (75) and (80)
were written as
k+1/2 k+1/2
- AR h + BR h = DR (95)
i,m-l i,m i,m
where
and
A V h
BR = 1 +
AR
(96)
k+1
k+1
k+1
+
BV h
- CV h = DV
(97)
i-1 ,m
i,m
i+l,m i,m
where
k+1/2 k+1/2
1 At K - K 1 At
k i-l/2,m i+l/2,m k
DV = h + ( ) S
i,m i ,m
4 A z C 2 C
i,m i,m
k+1/2
1 At r K
i.m-1/2 i,m-l/2 k+1/2 k+1/2
+ ( ) ( h - h )
2 i,m-l i,m
2 a r r
i ,m
i ,m
(98)
The boundary condition for the inner radius was also represented as
a no flux boundary. For the inner radius r(i,j) = 0, therefore its
inclusion into equation (8) would result in a division by zero.
Therefore, the second term in equation (8) was rewritten as described in
equations (50) and (51) such that the finite difference equations for
the ADI solution of equation (51) were expressed as

136
where
and
where
DV =
k+1/2 k+1/2
BR h - CR h = DR
i ,1 i ,2 i ,1
k+1/2
1 At K
i+1/2,1 k+1/2
CR = ( ) h
2 1,2
2 Ar C
1,1
BR = 1 + CR
k+1 k+1 k+1
AV h + BV h - CV h = DV
1-1,1 i ,1 1+1,1 1,1
k+1/2 k+1/2
1 At K - K
k i -1/2,1 i+1/2,1
h + - — ( )
M
4 az C
1,1
1 At
2 C
1,1
1,1
(99)
(100)
(101)
(102)
k+1/2
1
At K
i,1+1/2
k+1/2 k+1/2
+ (-
)
( h - h
2
1,2 1,1
2
Ar C
i,l
(103)
Updating of the soil parameters, surface infiltration, mass balance
and time steps were implimented as described for the one-dimensional
soil water flow model

APPENDIX B
Listing of Soil Water Movement and Extraction Model
137

138
C
c
C This program models the water movement and extraction in both a one-
C and a two-dimensional soil profiles. The model was designed to
C study water movement and extraction and how they affect irrigation
C scheduling. The program used an input file to input the parameters
C for the model. The input file is interactively asked for during
C program execution.
C
C The soils used in this model are assumed to homogenous and isotropic
C The surface boundry is a flux boundary which may be specified in the
C input file. The boundary is assumed to be a no flux unless an
C irrigation is occuring. The lower boundary may be either a no flux
C boundary or a gravity flow. The radial boundaries for the two-
C dimensional model are both no flow boundaries.
C
C
C CONT
C COND
C DTIRR
C DZ
C DEPTH
C DR
C DT
C DTMAX
C DTMIN
C ENDTIME
C HEAD
C HEDT
C ILOW
C I CONST
C ISOIL
C MBFILE
C NL
C NR
C R
C QS
C S
C SWC
C SWCT
C TIME
C TOUT
C TOD
C WC
C WCNT
C
PARAMETER (IZ=50,1R=16,1B=IR*2+1,MAX=IZ*IR)
IMPLICIT REAL (A-H.O-Z)
INTEGER I, NUM, I OUT,ISTOP,I COUNT
CHARACTER*64 MBFILE
COMMON /PARA/WC(IZ,IR,3),C0ND(IZ,IR,3),SWC(IZ,IR,3),HEAD(IZ,IR,3),
$NL,DZ(IZ),DEPTH(IZ),QS(IR),S(IZ,IR),I CONST,NR,DR(IR) ,R(IR),IL0W
COMMON/TIM/DT,TIME .TOUT(90) .ENDTIME ,DTMAX ,DTMIN,TOD,ITIME ,QM,DTIRR
COMMON /TABLES/WCNT(120),C0NT(120),SWCT(120) ,HEDT(120)
= table of conductivities
= the hydraulic conductivity
= the duration of the irrigation event
= the vertical grid increment
= the depth of the grids
= the radial grid increment
= the time step
= the maximum time step
= the minimum time step
= the ending simulation time
= the matric water potential
= table of soil matric potentials
= the status of the lower boundary condition
= the status of the surface boundary condition
= the soil type
= the output masbalance file
= the number of vertical grids
= the number of radial increments
= the radius of the grids
= the surface flux
= the soil water extraction
= the specific water capacity
= table of specific water capacities
= the current simulation time
= the times as which the outputs should occur
= the time of day
= The water content
= table of water contents

o o o
139
$ ,NDATA,MBFILE, ISO II-
CO MMON /MASB2/DIFF,PSTOR.FLOWIN,EXTRACT,STORAGE.FLOWOUT
COMMON /WCC/WKEEP,FLOWl(IZ,IR)
COMMON /CONV/DIF,ITER,MAXITER
COMMON /QS1/RDF(IZ,IR),WCWP,WCFC,ICF
COMMON /ET1/TSNR(90),TSNS(90),ETP(90),RAIN(90),DURJSTOP
COMMON /DUMY/IFIEL0(25) ,JFIELD(25) ,IFS,RTIME,1 OPT
$ ,ITEN,ITF(25),JTF(25),THRES,RRATE,ISCH(90)
CALL INPUT
CALL MASBAL(O)
WRITE(*,*)' THE INITIAL MASS BALANCE IS '
WRITE (*,90) TIME , DT,DIFF,PSTOR .FLOWOUT
WRITE(11,*)' THE INITIAL MASS BALANCE IS '
WRITE (11,90) TIME ,DT,DIFF,PSTOR, FLOWOUT
WRITE(*,91)STORAGE,FLOWIN,EXTRACT
WRITE(11,91)STORAGE,FLOWIN,EX TRAC T
TNEW = RTIME
IF(IOPT.EQ.0)WRITE(15,924)TNEW,(WC(IFIELD(I),JFIELD(I),1),1=1,IFS)
START OF SIMULATION LOOP
IT IME=1
MITER = 0
IDAY = TIME / 24.0 + 1.0
88 CONTINUE
CALL FLUX(ETPFLX, ETP(IDAY),TSNR(IDAY),TSNS(IDAY),TOD)
CALL QSS( ETPFLX)
CALL IRRIGATE( IDAY, IRCODE )
C ONE-DIMENSIONAL MODEL
IF(NR.EQ.l) THEN
CALL TSTEP1D( IRCODE )
CALL REDST1D
END IF
C TWO-DINENSIONAL MODEL
IF(NR.GE.3) THEN
CALL TSTEP2D( IRCODE )
CALL REDST2D
END IF
C
TIME = TINE + DT
IDAY = TIME / 24.0 +1.0
TOD = TOD + DT
IF(T0D.GE.24.0) TOD = TOD - 24.0
C
IF(TIME.EQ.TOUT(ITIME).OR.TIME.EQ.ENDTIME) THEN
C OUTPUT LOOP
ITIME =ITIME+ 1
WRITE(*,90)TINE ,DT,DIFF,PSTOR,FLOWOUT
WRITE(11,90)TIME,DT,DIFF,PSTOR,FLOWOUT
90 FO RMAT (/,
$' TINE1 ,T25, '= 1 , F12.5 ,' (HRS) DT',9X,'= \F12.5
(HRS)',/,
$’ MASS BALANCE --> ERROR = \E12.5,' (mm3) TERROR
$ E12.5,/,' DRAINAGE = ' ,E12.5)

140
C
WRITE(*,91)STORAGE,FLOWIN.EXTRACT
WRITE(11,91)STORAGE.FLOWIN.EXTRACT
91 FORMAT(1 STORAGE = \E12.5,' CUM_INFIL = '.E12.5,
$' CUM_EXTRACT = ' ,E12.5,/)
C
WRITE(11,9000)TINE
9000 FORMATC TIME = ' ,F12.5)
DO 12 1=1, NL
WRITE(*,11) DEPTH(I) ,(WC(I,J,1) ,J=1,NR)
IF(IOPT.EQ.l)WRITE(15,11)(WC(I,J ,1),J=NR,1,-l),(WC(I,J ,1) ,J=1,NR)
12 WRITE( 11,11) DEPTH(I) ,(WC(I,J,1) ,J=1,NR)
11 F0RMAT(10E12.5)
95 FORMAT(E10.4 ,10X,E10.4,10X,E10.4)
C
TNEW = RTINE + TINE / 24.0
IF(IOPT.EQ.0)WRITE(15,924)TNEW,(WC(IFIELD(I),JFIELD(I),1),I=1,IFS)
IF(I0PT.EQ.3)WRITE(15,924)TNEW,(WC(IFIELD(I),JFIELD(I),1),I=1,IFS)
$ .STORAGE, FLOWIN, EXTRACT, FLOWOÃœT
924 FORMAT(18X,F9.4,20E12.5)
END IF
C
c
MITER = MITER + 1
IF(MITER/5*5.EQ.MITER) THEN
OPEN(12,FILE='TIME.0UT1',IOSTAT=IC)
IF(IC.NE.O) GO TO 9008
WRITE(12,9005)TIME,MI TER
9005 FORMATC TIME = ' ,F12.5,' ITER = ',110)
CLOSE(12)
9008 END IF
C
IF(ENDTIME.LE.TINE) GO TO 99
GO TO 88
99 CONTINUE
OP EN(17,FILE='FINAL.POT')
DO 66 1=1 ,NL
66 WRI TE (17 ,11) ( HEAD(I,J,1),J=1,NR)
CLQSE(17)
CLOSE(ll)
END
C -
C SUBROUTINE TO READ IN THE INPUT VALUES FOR THE SIMULATION
C AND INITIALIZE THE VARIABLES
C
SUBROUTINE INPUT
PARAMETER (IZ=50,IR=16,IB=IR*2+1,MAX=IZ*IR)
IMPLICIT REAL (A-H.O-Z)
CHARACTER*64 INPFILE,MBFILE,OUTFILE,WDFILE,DATAFILE
CHARACTER*! Z
REAL IWCONT.ISTOR, HE(6,5),EX(6,5)
INTEGER 11(6),JJ(5)
COMMON /TABLES/WCNT(120),C0NT(120),SWCT(120),HEDT(120)
$ ,NDATA,MBFILE,ISO IL

141
COMMON /PARA/WC(IZ,IR,3),COND(IZ,IR ,3),SWC(IZ,IR,3),HEAD(IZ,IR,3),
$NL,DZ(IZ),DEPTH( IZ),QS( IR) ,S(IZ,IR),ICONST,NR,DR(IR),R(IR),ILOW
COMMOM/irM/DT.TIME,T0UT(90).ENDTIME,DTMAX,DTMIN,T0D,ITIME,QM,DTIRR
COMMON /MASB/ IWCONT( IZ, IR) ,AREA( IR) .TAREA,ISTOR
COMMON /DUMY/IFIELD(25) ,JFIELD(25) ,IFS,RTIME,IOPT
$ ,ITEN,ITF(25),JTF(25),THRES,RRATE,ISCH(90)
COMMON /CONV/DIF,ITER,MAX ITER
COWON /QS1/RDF(IZ,IR) ,WCWP,WCFC,ICF
COMMON /ET1/TSNR(90),TSNS(90),ETP(90),RAIN(90),DUR,TSTOP
COMMON /WCC/WKEEP,FL0W1(IZ,IR)
PIE = 3.1415926
WRITE(*,*)1 ENTER THE INPUT DATA FILE1
READ(*,1Q00) DATAFILE
0PEN(8,FILE=DATAFILE)
READ(8 ,901) Z
READ(8,1005) ISOIL.INPFILE
1000 F0RMAT(A64)
1005 FORMAT( II ,1X ,A64)
OPEN( 9, FI LE= INP FILE)
C
C READ IN AND WRITE THE SOILS DATA
C
READ(9,11) NDATA
11 FORMAT(15)
READ(9,09)(HEDT(I),WCNT(I),SWCT(I),CONT(I),1=1,NDATA)
09 F0RMAT(4E12.5)
900 FORMAT(' ')
CLOSE!9)
c
READ(8,901) Z
READ(8,1000)MBFILE
READ!8,1000)WDFILE
0PEN(11,FILE=MBFILE)
READ!8,901) Z
READ(8,' (II ,1X ,A64)') IOPT.OUTFILE
0PEN(15,FILE=OUTFILE)
READ(8,*) IFS
READ( 8,*) RTIME , (I FI ELD( I) ,JFIELD( I) , 1=1,IFS)
READ(8,901) Z
READ(8,*)TII€ .ENDTIME ,TOD
READ(8,901)Z
READ(8,*)WKEEP,WCWP,WCFC
READ(8,901) Z
READ(8,*)TOUT(1)J0UT(2),T0UT(3),T0UT(4)
IF(TOUT(2).NE.-99.0) GO TO 9000
DO 7000 I = 1,90
IF(T0UT(1)*FL0AT(I).GT.ENDTIME) GO TO 7000
TOUT(I) = TOUT(1)*FLOAT(I)
7000 CONTINUE
9000 CONTINUE
READ(8,901) Z
READ(8,*)DTMAX,DTMIN,MAXITER,DTIRR,DUR
WRITE(*,710)MBFILE.TIME,TOUT(1),ENOTIME,DTMAX,DTMIN,MAX ITER
WRITE(11,710)MBFILE,TIME,TOUT(1).ENDTIME,DTMAX,DTMIN,MAX ITER

142
710 FORMAT(/,* THE Í4ASS BALANCE DATA IS IN FILE ',A40,/,
$' THE STARTING SIMULATION TIME IS ' ,E12.5,/,
$' THE OUTPUTS WILL BE AT INTERVALS 1 ,E12 .5,/,
$' THE ENDING TIME WILL BE ',E12.5,/,
$' THE MAXIMUM TIME STEP IS ' ,E12.5,/,
$' THE MINIMUM TINE STEP IS ' ,E12.5,/,
$' THE NUMBER OF ITERATIONS ',112,/)
C
C READ IN THE NUMBER OF GRIDS AND THE GRID SIZES
C
READ(8 ,901) Z
READ(8,*) NL ,DZ(1), NR, DR(1)
WRITE(*,720)NL,DZ(1),NR,DR(1)
720 FORMAT(1 THE NUMBER OF VERTICAL INCREMENTS ',110,/,
$ ' THE VERICAL SPACING IS ’,E12.5,/,
$ ' THE NUMBER OF RADIAL INCREMENTS ',110,/,
$ ' THE RADIAL SPACING IS ' ,E12.5,/)
C
DEPTH(l) = - DZ(1) / 2
DO 200 I = 2,NL
DZ(I) = DZ(I-l)
DEPTH(I) = DEPTH(1-1) - ( DZ(I) + DZ(I-l) ) / 2
200 CONTINUE
C
IF(NR.EQ.2) NR = 1
AREA(l) = 1.0
TAREA = 1.0
IF(NR.GE.3) THEN
R(1) = DR(1) / 2
AREA(l) = PIE * (R(l)+DR(l)/2)**2
DO 210 I = 2, NR
DR(I) = DR(I-l)
R(I) = R(I-l) + ( DR(I) + DR(I-l) ) / 2
AREA( I)= PIE* ( (R(I)+DR(I)/2)**2 -(R( I)-DR( I)/2)**2)
210 CONTINUE
TAREA = PIE * ( DR(1) * NR )**2
WRITE (* ,735)
735 FORMAT(/,1 THE RADIUS AND AREAS ',/)
WR ITE (* ,740) (I ,R( I) ,AREA( I) ,1=1 ,NR)
740 FORMATC RADIUS(1 ,13,') = 1 .E12.5 ,10X,' AREA = ‘ ,E12.5)
END IF
C
C INPUT THE INITIAL POTENTIALS FOR THE SYSTEM
C
READ(8,901)Z
C READ(8,910) (( HEAD(I,J,1),J=1,IR),I=1,IZ)
C
READ(8,*) NLN.NRN.INV
READ(8,*) ( 11( I), 1=1 ,NLN )
READ(8,*) ( JJ(J),J=1,NRN )
DO 1010 1=1,NLN
1010 READ(8,*) ( HE(I,J),J=1,NRN)
READ(8 ,901)Z
C

C READ IN THE INFILTRATION RATE
C
C READ(8,910) (QS(J),J=1,IR)
READ(8,901) Z
READ(8,901)Z
READ(8,*) ICONST
READ(8,901)Z
C
C READ IN THE CONDITION OF THE LOWER BOUNDRY CONDITION
C AND THE STATUS OF THE CORRECTION FACTOR
C
READ(8,*)ILOW, ICF
READ(8,901) Z
C READ IN THE ROOT DISTRIBUTION FUNCTION
C
C RE AD (8,910) ((RDF( I ,J) ,J=1 ,IR) , 1=1 ,IZ)
DO 1011 1=1,NLN
1011 READ(8,*) ( EX(I,J),J=1,NRN)
C
DO 811 K=1,3
DO 811 1=1,IZ
DO 811 J=1,1R
HEAD(I ,J ,K)=0.0
RDF(I,J) = 0.0
S(I,J) = 0.0
811 CONTINUE
C
N = 0
DO 400 1=1,NLN
DO 400 L=1,11(I)
N = N + 1
M = 0
DO 400 J=1 ,NRN
DO 400 K=1 ,JJ(J)
M = M + 1
HEAD(N,M,1) = HE(I ,J)
HEAD(N,M,2) = HE(I,J)
HEAD(N,M,3) = HE(I,J)
K1 = M
IF(INV.EQ.l) K1 = NR + 1 - M
AAA = FL0AT(K1**2 - (Kl-1)**2) / FLOAT( NR**2)
IF(NR.EQ.l) AAA = 1.0
RDF(N,M) = EX(I ,J) / 11( I) * AAA
400 CONTINUE
C
910 F0RMAT(5F10.0)
901 FORMAT(Al)
775 FORMAT( 10E12.5)
C
CALL UPDATE (1)
ISTOR =0.0
DO 310 1=1,NL
DO 310 J=1 ,NR
IWCONT(I ,J) =WC( I ,J ,1)

144
ISTOR = ISTOR + WC(I,J ,1)*DZ(I)*AREA(J)
310 CONTINUE
C
110 FORMAT( 17 ,3X,4(El2.5 ,3X) )
READ(8,*) ITEN, THRES, RRATE, ISC
RE AD (8,*) ( ITF(I) ,JTF( I) ,1=1, ITEN )
DO 1015 1=1,IDAY
1015 ISCH(I) = ISC
CL0SE(8)
C
C READ IN WEATHER DATA
C
0PEN(10, FILE=WDFILE)
IDAY = ENDTIME / 24.0 + 1.0
REA0( 10 ,1001 ,END=1002) ( ETP(I) ,TSNR(I) ,TSNS(I) ,RAIN(I) ,1=1,IDAY)
1002 WRITE(11,1003)
WRITE(11,1001)( ETP(I),TSNR(I),TSNS(I),RAIN(I),1=1,IDAY)
1001 F0RMAT(4F10.2)
CLOSE(IO)
1003 FORMAT(/,1 THE WEATHER DATA ',/)
C
RETURN
END
C
SUBROUTINE FLUX (ETPFLX , ETP, TSNR ,TSNS ,TOD)
C
C PROGRAM TO CALCULATE THE DISTRIBUTION OF POTENTIAL
C ET DURING THE DAY. THE DISTRIBUTION IS ASSUMED TO
C BE A SINE FUNCTION.
C -
C DEFINITIONS
C TSNR = TIME OF SUNRISE
C TSNS = TIME OF SUNSET
C ETP = POTENTIAL ET MM
C ETPRMX = MAXIMUM ET RATE MM/HR
C DAYL = DAY LENGTH HOURS
C ETPFLX = CURRENT ET FLUX MM/HR
C
IMPLICIT REAL (A-H.O-Z)
PI = 3.141592654
DAYL = TSNS - TSNR
ETPRMX = 2 * ETP / DAYL
OMEGA = 2 * PI / DAYL
TD = TOD - TSNR
TI = TOD
IF(TI .LT.TSNR.OR.TI.GT.TSNS) TD = 0.0
ETPFLX = (ETPRMX/2)*SIN((0MEGA*TD) - PI/2.) + ETPRMX/2
END
C
C SUBROUTINE TO CALCULATE THE WATER UPTAKE RATE FOR THE
C ONEDIMENSIONAL SOIL PROFILE
C
SUBROUTINE QSS(TRANS)
PAR/4 METER (IZ=50,IR=16,IB=IR*2+1 ,MAX=IZ*IR)

145
IMPLICIT REAL (A-H.O-Z)
REAL TRANS,RASW(IZ,IR),A(IZ.IR),K1,ISTOR,IWCONT
COMMON /PARA/WC(IZ,IR,3) ,COND(IZ,IR,3),SWC(IZ,IR,3),HEAD(IZ,IR,3),
$NL,DZ(IZ),DEPTH(IZ),QS(IR),S(IZ,IR),ICONST,NR,DR(IR),R(IR),I LOW
COMMON /MASB/IWCONT(IZ,IR),AREA(IR),TAREA,ISTOR
COMMON /QS1/RDF(IZ,IR),WCWP,WCFC,ICF
C
IF(TRANS.EQ.O.O) THEN
DO 30 1= 1,NL
DO 30 J= 1,NR
30 S(I,J) = 0.0
RETURN
END IF
K1 = 10.0
SUM = 0.0
DO 10 1 = 1,NL
DO 10 J=1 ,NR
RASW( I ,J) = ( WC(I,J,1) - WCWP ) / ( WCFC - WCWP )
IF(RASW(I,J).GT.l.0) RASW(I ,J) = 1.0
IF(RASW(I,J).LE.O.0) THEN
RASW(I,J) =0.0
A(I,J) = 0.0
GO TO 100
END IF
A(I, J) = TRANS * TAREA * RASW(I,J) ** ( TRANS /(RASW(I ,J)*K1))
100 S(I,J) = A( I ,J) * RDF (I ,J)
SUM = SUM + S(I,J)
10 CONTINUE
C
IF( SUM.EQ .0.0 .OR. ICF.EQ .1) THEN
CF = 1.0
ELSE
CF = TRANS * TAREA / SUM
END IF
SUM2 = 0.0
DO 20 1=1 ,NL
DO 20 J=1 ,NR
S(I,J) = S(I,J) / DZ(I) / AREA(J) * CF
SUM2 = SUM2 + S(I,J)*DZ(I)*AREA(J)
20 CONTINUE
RETURN
END
C
SUBROUTINE TRID(NG,IF,A,B,C,D,V)
C
C THIS SUBROUTINE SOLVES THE TRIDIAGONAL COEFFICIENT MATRIX
C WHICH RESULTS FROM A SET OF SIMULTANEOUS EQUATIONS WHICH
C CAN BE PUT INTO THE FOLLOWING FORM
C - A*V1 + B*V2 - C*V3 = D
C
PARAMETER (IZ=50)
IMPLICIT REAL (A-H.O-Z)
DIMENSION A(IZ),B(IZ),C(IZ),D(IZ),V(IZ),BETA(IZ),GAMMA(IZ)
IFP1 = IF+1

146
C NGM1 = NG-1
C -
C COMPUTE INTERMEDIATE ARRAYS BETA AND GAMMA
C
BETA(IF) = B(IF)
GAMMA(IF) = D(IF)/BETA(IF)
DO 100 I = IFPl.NG
BETA(I) = B(I)-A(I)*C(I-1)/BETA(I-1)
GAMMA(I) = (D(I)+A(I)*GAMMA(I-1))/BETA(I)
100 CONTINUE
C
C COMPUTE FINAL SOLUTION VECTOR V
C
V(NG) = GAMMA (NG)
LAST = NG-IF
DO 200 K = 1 ,LAST
I = NG-K
V(I) = GAMMA(I)+C(I)*V(1+1)/BETA(I)
200 CONTINUE
RETURN
END
C
C SUBROUTINE TO CALCULATE A MASS BALANCE OF THE SYSTEM SIMULATED
C
SUBROUTINE MASBAL(IRESET)
PARAMETER (IZ=50,IR=16,IB=IR*2+1,MAX=IZ*IR)
IMPLICIT REAL (A-H.O-Z)
REAL IWCONT,ISTORAGE
COMMON /PARA/WC(IZ,IR,3),COND(IZ,IR,3),SWC(IZ,IR,3),HEAD(IZ,IR,3),
$NL,DZ(IZ),DEPTH(IZ),QS(IR),S(IZ,IR),I CONST,NR,DR(IR),R(IR),I LOW
COMMON/TIM/DT,TIME,T0UT(90).ENDTIME,DTMAX,DTMIN,TOD,ITIME ,QM,DTIRR
COMMON /MASB/ IWCONT(IZ.IR),AREA(IR)JAREA,ISTORAGE
COMMON /MASB2/DIFF,PST0RAGE,FL0WIN,EXTRACT,STORAGE,FLOWOUT
C
C INITIALIZE THE CUMULATIVE INFILTRATION AND EXTRACTION
C
IF(IRESET.EQ.0) THEN
FLOWIN = 0.0
FLOWOUT =0.0
EXTRACT =0.0
DIFF=0 .0
STORAGES.0
DO 20 J=1 ,NR
DO 20 1=1,NL
20 STORAGE = STORAGE + WC(I ,J ,1)*DZ(I)*AREA(J)
DIFF = STORAGE - ISTORAGE
PSTORAGE = DIFF / ISTORAGE * 100.0
RETURN
END IF
DIFF= 0.0
STORAGES.0
C
DO 10 J=1,NR
FLOWIN = FLOWIN + DT*QS(J)*AREA(J)

147
IF(ILOW.EQ.1) FLOWOUT = FLOWOUT + DT*(COND(NL,J,1) )
DO 10 1=1,NL
STORAGE = STORAGE + WC(I,J,1)*DZ(I)*AREA(J)
EXTRACT = EXTRACT + S(I,J)*DZ(I)*AREA(J)*DT
10 CONTINUE
DIFF = STORAGE - (ISTORAGE + FLOWIN - EXTRACT -FLOWOUT )
PSTORAGE = DIFF / (ISTORAGE + FLOWIN - EXTRACT - FLOWOUT) * 100.0
RETURN
END
C
SUBROUTINE IRRIGATE( IDAY, IRCODE )
C
C SUBROUTINE TO CALCULATE THE AMOUNT OF IRRIGATION REQUIRED
C FOR THE MODEL BASED ON THE MAXIMUM SOIL WATER POTENTIAL ALLOWED
C
PARAMETER (IZ=50,IR=16,IB=IR*2+1 ,MAX=IZ*IR)
IMPLICIT REAL (A-H.O-Z)
COMMON /PARA/WC(IZ,IR,3),COND(IZ,IR,3),SWC(IZ,IR,3),HEAD(IZ,IR,3),
$NL,DZ(IZ),DEPTH(IZ),QS(IR),S(IZ.IR),ICONST,NR,DR(IR),R(IR),1 LOW
COMMON/TIM/DTJIME ,T0UT(90) ,ENDTIME .DTMAX ,DTMIN ,TOD ,ITIME ,QM,DTIRR
COMMON /ET1/TSNR(90) ,TSNS(90) ,ETP(90) ,RAIN(90) ,DUR,TSTOP
COMMON /QS1/RDF(IZ,IR),WCWP,WCFC,ICF
COMMON /DUMY/1FIELD(25),JFIELD(25),IFS ,RTIME,IOPT
$ ,ITEN,ITF(25),JTF(25),THRES,RRATE,ISCH(90)
COMMON /MASB/ IWCONT(IZ,IR),AREA(IR),TAREA,ISTORAGE
C
AMOUNT =0.0
DEFICIT = 0.0
IF(ISCH(IDAY).EQ.O) THEN
DO 10 J=1 ,NR
DO 10 1=1,NL
DEFICIT = DEFICIT + ( WCFC - WC(I,J,1) ) * DZ(I) * AREA(J)
10 CONTINUE
AMOUNT = DEFICIT / TAREA
DO 20 1= 1 ,ITEN
IF (ABS(HEAD(ITF(I),JTF(I),1)).GE.ABS(THRES)) THEN
RAIN(IDAY)=AMOUNT
ISCH(IDAY)= 1
END IF
20 CONTINUE
END IF
C
IF( RAIN(IDAY).LE.0.0) THEN
DO 100 1=1,NR
QS( I) = 0.0
100 CONTINUE
IRCODE = 0
RETURN
END IF
C
IF( IRCODE.EQ.O) THEN
TSTART = TIME
C DUR = RAIN(IDAY) / RRATE
RAIN(IDAY) = RRATE

non nonnn
148
TSTOP = TSTART + DUR
C
IRCOOE = 1
00 200 1=1 ,NR
QS( I) = RAIN( IDAY )
200 CONTINUE
END IF
C
IF( TINE .GE. TSTOP ) THEN
IRC0DE = 0
DO 300 1=1,NR
QS( I) = 0.0
300 CONTINUE
RAIN(IDAY) =0.0
END IF
RETURN
END
SUBROUTINE TO CALCULATE THE TIME STEP BASED ON THE
MAXIMUM ALLOWABLE CHANGE IN WATER CONTENT
FOR THE ONE-DIMENSIONAL MODEL
SUBROUTINE TSTEP1D( IRCODE )
PARAMETER (IZ=50 ,IR=16,IB=IR*2+1 ,MAX=IR*IZ)
REAL K1,K2
COMMON/PARA/WC(IZ,IR,3) ,C0ND( IZ,IR,3),SWC(IZ,IR,3) ,HEAD( IZ,IR,3),
$NL,DZ(IZ) ,DEPTH(IZ),QS(IR),S(IZ,IR),I CONST,NR,DR(IR),R(IR),1 LOW
COMMON/TIM/DT.TIME,T0UT(90),ENDTINE ,DTMAX,OTMIN ,TOD,ITIME ,QM,DTIRR
COMMON /WCC/WKEEP, FL0W1( IZ ,IR)
COMMON /ET1/TSNR(90),TSNS(90),ETP(90),RAIN(90),DUR,TSTOP
CALCULATE THE MAXIMUM FLUX FROM THE INFILTRATION AND EXTRACTION
RATE = ABS(QS(1))
EXT = ABS(S(1,1)*DZ(1))
DO 10 1=1,NL
IF(EXT.LT.ABS( S( 1,1 )*DZ( I))) EXT = ABS( S( 1,1) *DZ( I) )
10 CONTINUE
C
IF(RATE.LT.EXT) RATE = EXT
QM = RATE
C
DO 100 1=1,NL
FLOW1 (1,1)=0.0
C
IF(I.EQ.l) THEN
FLOWl(I.l) = QS( 1)
$ + ( COND( 1,1,1) + COND( 1+1,1,1) ) / 2
$ *((HEAD(1+1,1,1)-HEAD(1,1,1))/DZ(I)-1.0)
GO TO 900
END IF
C
IF(I.EQ.NL) THEN
FLOWl(I.l) =

149
$ -1.0*( CONO(1,1.1) + COND(1-1,1,1) ) / 2
$ *((HEAD(1,1 ,1)-HEAD(I-1,1,1))/DZ(I)-1.0)
IF (I LOW. EQ. 1) FLOWl(I.l) = FLOWl(I.l) - COND( 1,1,1)
GO TO 900
END IF
C
FL0W1 (1,1) =
$ -1.0*( COND( 1,1,1) + COND( 1-1,1,1) ) / 2
$ *((HEAD(1,1,1)-HEADC1-1,1,1) )/DZ( I)-1.0)
$ + ( COND( 1,1,1) + COND( 1+1,1,1) ) / 2
$ *( (HEAD( 1+1,1,1) -HEAD( 1,1,1) )/DZ( I)-1.0)
C
900 IF(ABS(FLOW1 (1,1)) .GT.QM) GO TO 200
GO TO 100
200 QM = ABS(FL0W1 (1,1))
100 CONTINUE
C
IF(QM.LE.O.O) GO TO 400
DT = WKEEP * DZ(1) / QM
GO TO 500
400 DT = DTMAX
500 IF(DT.GT.DTMAX) DT = DTMAX
IF(DT.LT.DTMIN) DT = DTMIN
IF(IRCODE.EQ.1) DT = AMIN1( DTIRR, DT )
IF(DT+TIME.GT.T0UT(ITIME).AND.TOUT(ITIME).GT.O .0)
$ DT = TOUT(ITIME) - TINE
IF(DT+TINE.GT.ENDTIME) DT = ENDTINE - TINE
IF(DT+TINE.GT.TSTOP.AND.IRCODE.EQ.1) DT = TSTOP - TIME
IF(DT.LT.O.O) DT = DTMIN
C WRITE(*,191) DT,QM
191 FORMAT(/,****** DT = *,F10.8 , ’ QM = ',E12.5)
RETURN
END
C
C SUBROUTINE TO CALCULATE THE NEW WATER POTENTIALS
C FOR THE ONE-DINENSIONAL MODEL
C
SUBROUTINE REDST1D
PARAMETER (IZ=50,IR=16 ,IB=IR*2+1 ,MAX=IR*IZ)
IMPLICIT REAL (A-H.O-Z)
REAL K1,K2,D(IZ),A(IZ),B(IZ),C(IZ),V(IZ)
REAL CONTAB(120),WCCTAB(120) ,SPCTAB(120),PRTAB(120)
CHARACTER*64 MBFILE
COMMON /TABLES/WCCTAB,CONTAB,SPCTAB,PRTAB,NDATA,MBFILE,ISOIL
COMMON/PARA/WC(IZ,IR,3),COND( IZ,IR,3),SWC(IZ,IR,3),HEAD(IZ,IR,3),
$NL,DZ(IZ),DEPTH(IZ),QS(IR),S(IZ,IR),ICONST,NR,DR(IR),R(IR),ILOW
COMMON/TIM/DT,TIME,T0UT(90),ENDTIME,DTMAX,DTMIN,TOD,ITIME,QM,DTIRR
CONMON /CONV/DIF,I TER,MAX ITER
C
C SET UP THE INTERIOR GRID POINTS
C
DO 10 KL=2 ,NL-1
Kl=( COND(KL,1,1) + C0ND(KL-1,1,1) ) / 2
K2=( COND( KL+1,1,1) + COND( KL ,1,1) ) / 2

150
Cl = SWC(KL,1,1)
C0NST1 = DT / DZ(KL)**2/Cl
C
A(KL) = C0NST1 * K1
C(KL> = C0NST1 * K2
B(KL) = 1.0 + A(KL) + C(KL)
D(KL) = HEAD(KL,1,1) + DT/DZ(KL)/C1*( K1 - K2 )
$ - DT * S(KL,1)/Cl
10 CONTINUE
C
C SET UP THE BOUNDRY CONDITIONS
C -
IF(ICOPiST.EQ.2) GO TO 800
C
C NO FLUX BOUNDRY CONDITION AT THE SURFACE
C
KL = 1
K2=( CO ND(KL+1,1,1) + COND(KL ,1,1) ) / 2
K1 = 0.0
Cl = SWC (KL ,1,1)
C0NST1 = DT / DZ(KL)**2/C1
C
A(KL) = 0.0
C(KL) = C0NST1 * K2
B( KL) = 1.0+ C (KL)
D(KL) = HEAD(KL,1,1) + DT/DZ(KL)/Cl*( QS(1) - K2 )
$ - DT * S(KL ,1)/C1
GO TO 900
C
C CONSTANT POTENTIAL BOUNDRY CONDITION AT THE SURFACE
C -
800 KL = 1
K2=( CO'ND( 2,1,1) + COND( 1,1,1) ) / 2
K1 = C0ND( 1,1,1)
Cl = SWC(1,1,1)
C0NST1 = DT / DZ(1)**2/C1
C
A(l) = C0NST1 * K1
C(l) = C0NST1 * K2
B (1) = 1.0 + C(l)
D( 1) = HEAD(1,1,1) + DT/DZ(1)/C1*( K1 - K2 )
$ - DT * S(1,1)/C1
C
C NO FLUX BOUNDRY CONDITION AT LOWER BOUNDRY
C -
900 KL = NL
Kl=( COND(KL ,1,1) + COND( KL-1,1,1) ) / 2
K2= 0.0
IF( ILOW.EQ .1) K2 = COND( KL ,1 ,1)
Cl = SWC(KL ,1,1)
C0NST1 = DT / DZ(KL)**2/Cl
C
A(KL) = C0NST1 * K1
B(KL) = 1.0 + A(KL)

151
D(KL) = HEAD(KL,1,1) + DT/DZ(KL)/Cl*( K1 - K2 )
$ - DT * S(KL,1)/C1
C
C CALCULATE THE NEW VALUES FOR THE HEADS
C -
IF = 1
C IF(ICONST.EQ.2) IF = 2
CALL TRID(NL,IF,A,B,C,D ,V)
DO 700 KL = ICONST.NL
HEAD(KL ,1,3)=V(KL)
700 CONTINUE
C
IF(HEAD(1,1,3).GT.PRTAB(l).AND.ICONST.EQ .1) ICONST = 2
IF (ICONST.EQ.l) GO TO 2000
Q =0.0
Q = AMAX1( QS(1), QM )
C
2000 CONTINUE
CALL UPDATE (3)
DO 5000 1=1,NL
HEAD (1,1,1) =HEAD (1,1,3)
HEAD( 1,1,2)=HEAD( 1,1,3)
WC(I,1,1) =WC( 1,1,3)
WC(1,1,2) =WC(1,1,3)
COND( 1,1,1)=CQND( 1,1,3)
COND(1,1,2) = C0ND(1,1,3)
SWC( 1,1,1) = SWC (1,1,3)
SWC(1,1,2) =SWC(1,1,3)
5000 CONTINUE
CALL MASBAL(l)
C
RETURN
END
C
C SUBROUTINE TO CALCULATE THE TIME STEP BASED ON THE MAXIMUM
C NET FLOW RATE INTO EACH GRID
C FOR THE TWO-DIMENSIONAL MODEL
C
SUBROUTINE TSTEP2D( IRCODE )
PARAMETER (IZ=50,IR=16,IB=IR*2+1 ,MAX=IZ*IR)
IMPLICIT REAL (A-H.O-Z)
REAL K1,K2,K4,K5,NFR
COMMON /WCC/WKEEP, FL0W1( IZ ,IR)
COMMON /ET1/TSNR(90),TSNS(90),ETP(90),RAIN(90),DUR,TSTOP
COMMON /PARA/WC(IZ,IR,3),C0ND(IZ,IR,3),SWC(IZ.IR,3),HEAD(IZ,IR,3),
$NL ,DZ(IZ),DEPTH(IZ),QS(IR),S(IZ,IR),ICONST,NR,DR(IR),R(IR),ILOW
COMMON/TIM/DT,TIME,T0UT(90),ENDTIME,DTMAX,DTMIN,TOD,ITIME,QM,DTIRR
NFR =0.0
DO 100 1=1,NL
DO 100 J =1,NR
IF(I.EQ.l) THEN
Q1 = QS(J)
ELSE
C

152
Q1 = -1.0 * (( COND(I,J,1) + C0ND(1-1,J ,1) ) / 2)
$ *((HEAD(I,J,1)-HEAD(1-1,J,1))/DZ(I)-1.0)
END IF
IF(I.EQ.NL) THEN
Q2 = 0.0
IF(ILOW.EQ.l) Q2 = COND( I,J,1)
ELSE
Q2= -1.0 * (( COND( 1+1 ,J ,1) + COND( I ,J ,1) ) / 2)
$ *( (HEAD( 1+1 ,J ,1) -HEAD( I ,J ,1) )/DZ( I)-1.0)
END IF
IF(J.EQ.l) THEN
Q3 = 0.0
ELSE
Q3= -1.0 * (( COND(I ,J ,1) + COND( I ,J-1,1) ) / 2)
$ *((HEAD(I ,J ,1) - HEAD(I ,J-1,1))/(R( J) -R( J-l)) )
END IF
IF(J .EQ.NR) THEN
Q4 = 0.0
ELSE
Q4 = -1.0 * (( CO ND( I, J ,1) + COND( I ,J+1,1))/ 2)
$ *( (HEAD( I ,J+1,1)-HEAD( I ,J ,1))/ (R( J+1) -R( J)) )
END IF
FL0W1 (I ,J) = (Q1 - Q2)/DZ( I) + (Q3 - Q4)/DR(J) - S( I ,J)
IF(ABS(FLOW1 (I ,J)) .GT.NFR) GO TO 200
GO TO 100
200 NFR = ABS(FL0W1(I ,J))
100 CONTINUE
C
IF(NFR.LE.O.O) GO TO 400
DT = WKEEP / NFR
GO TO 500
400 DT = DTMAX
500 IF(DT.GT.DTMAX) DT = DTMAX
IF(DT.LT.DTMIN) DT = DTMIN
IF(IRCODE.EQ.1) DT = AMIN1( DTIRR, DT )
IF( DT+TIME.GE.TOUT(ITIME).AND.TOUT(ITIME).GT.O.0)
$ DT= TOUT(ITIME) - TIME
IF(DT+TIME.GT.TSTOP.AND.IRCODE.EQ.1) DT = TSTOP - TINE
IF( DT+TIME.GE.ENDTIME) DT = ENDTIME - TIME
IF( DT.LE.O.0 ) DT = DTMIN
RETURN
END
C***
C
C ADI SOLUTION METHOD OF RICHARDS EQUATION
C
SUBROUTINE REDST2D
PARAMETER (IZ=50,1R=16,1B=IR*2+1 ,MAX=IZ*IR)
IMPLICIT REAL (A-H.O-Z)
REAL K1,K2,K3,K4,V(IZ),A(IZ),B(IZ),C(IZ),D(IZ),HSTAR(IZ ,IR)
COMMON/PARA/WC(IZ,IR,3),COND(IZ ,IR,3),SWC(IZ,IR,3),HEAD(IZ,IR,3),
$NL ,DZ(IZ),DEPTH(IZ),QS(IR),S(IZ,IR),ICONST,NR,DR(IR),R( IR),I LOW
COMMON/TIM/DT,TIME,T0UT(90),ENDTIME,DTMAX,DTMIN,TOD,ITIME,QM,DTIRR
COMMON /CONV/DIF,I TER,MAX I TER

153
C
C SET UP THE INTERIOR GRID POINTS FOR THE FIRST HALF CYCLE
C SOLVE IN THE RADIAL DIRECTION
C
ITER=0
700 DO 20 KL = 1,NL
IF(KL.EQ.l) GO TO 25
IF(KL.EQ.ML)GO TO 35
DO 10 J=2,NR-1
I = J
Kl=( COND(KL,J,1) + C0ND(KL-1,J,1) ) / 2
' K2=( C0ND(KL+1 ,J ,1) + C0ND(KL,J,1) ) / 2
K4=( COND(KL ,J+1,2) + C0ND(KL,J,2) ) / 2
K3=( C0ND(KL,J,2) + COND(KL,J-l ,2) ) / 2
Cl=( SWC(KL,J,1) + SWC(KL ,J ,3) ) / 2
Rl=( R(J+l) + R(J) ) / 2
R2=( R(J) + R(J-l) ) / 2
C0NST1 = -1.0 * DT / DZ(KL)**2 / Cl
C0NST2 = -1.0 * DT / DR(J)**2 / Cl / R(J)
A(I) = -1.0* C0NST2 * R2 * K3 / 2
C(I) = -1.0* C0NST2 * R1 * K4 / 2
B( I) = 1.0 + A(I) + C(I)
D( I) = HEAD(KL ,J ,1) + C0NST1 * (K2 + Kl) * HEAD(KL,J ,1) / 2
$ - C0NST1 * ( Kl * HEAD(KL-1,J ,1) + K2 * HEAD(KL+1 ,J ,1) ) / 2
$ + DT/DZ(KL)/C1*( Kl - K2 )/2 + DT/DZ(KL)/Cl*( Kl - K2 ) / 4
$ - DT/C1 * S(KL ,J) / 2
10 CONTINUE
C
C NO FLUX BOUNDRY CONDITION AT INNER RADIUS
C
J = 1
I = J
Kl = (COND(KL,J ,1)+C0ND(KL-1 ,J ,1))/ 2
K2=( C0ND(KL+1 ,J ,1) + C0ND(KL,J,1) ) / 2
K4=( COND(KL ,J+1,2) + C0ND(KL,J,2) ) / 2
K3= K4
Cl =( SWC(KL.J.l) + SWC(KL ,J ,3) ) / 2
C0NST1 = -1.0 * DT / DZ(KL)**2 / Cl
C0NST2 = -1.0 * DT / DR(J)**2 / Cl
A( I) = 0.0
C(I) = -1.0* C0NST2 * K4 / 2
B( I) = 1.0 + C(I)
D( I) = HEAD(KL ,J ,1) + C0NST1*(K2+K1) * HEAD(KL,J ,1) / 2
$ - C0NSTl*Kl*HEAD(KL-l,J,l)/2 - C0NST1*K2*HEAD(KL+1,J ,l)/2
$ + DT/DZ(KL)/C1*( Kl - K2 )/2 + DT/DZ(KL)/Cl*( Kl - K2 ) / 4
$ - DT/Cl * S(KL,J) / 2
C
C
C NO FLUX BOUNDRY CONDITION AT OUTER RADIUS
C
J=NR
I = NR
Kl=( COND(KL ,J,1) + COND(KL-1 ,J ,1) ) / 2
K2= (CON D( KL+1 ,J ,1) + C0ND(KL,J,1) ) ' 2

154
K3=( COND(KL ,J ,2) + CQND(KL ,J-1,2) ) / 2
K4 = 0.0
Cl =( SWC(KL ,J ,1) + SWC(KL ,J ,3) ) / 2
C0NST1 = -1.0 * DT / DZ(KL)**2 / Cl
C0NST2 = -1.0 * DT / DR(J)**2 / Cl
C
A(I) = -1.0* C0NST2 * K3 / 2
C(I) = 0.0
B( I) = 1.0 + A( I)
D( I) = HEAD(KL ,J ,1) + C0NST1*(K2+K1) * HEAD(KL ,J ,1) / 2
$ - C0NST1*K1*HEAD(KL-1,J,1)/2 - C0NST1*K2*HEAD(KL+1,J ,l)/2
$ + DT/DZ(KL)/Cl*( K1 - K2 )/2 + DT/DZ(KL)/Cl*( K1 - K2 )/4
$ - DT/C1 * S(KL ,J) / 2
GO TO 50
C
C NO FLUX BOUNDRY CONDITION AT THE SURFACE
C
25 00 30 J=1,NR
I = J
K2=( C0ND(KL+1 ,J ,1) + C0ND(KL,J,1) ) / 2
K1 = QS(J)
IF(J.NE.NR) K4=( CO ND (KL ,J+1,2) + C0ND(KL,J,2) ) / 2
IF(J.EQ.NR) K4 = 0.0
IF(J.NE.NR) R1 = ( R(J+1) + R(J) ) / 2
IF(J.NE.l) K3 = ( C0ND(KL ,J ,2) + C0ND(KL ,J-1,2) ) / 2
IF(J.EQ.l) K3 = 0.0
IF(J.NE.l) R2 = ( R(J) + R(J-l) ) / 2
Cl=( SWC(KL,J,1) + SWC(KL,J,3) ) / 2
C0NST1 = -1.0 * DT / DZ(KL)**2 / Cl
IF(J.NE.l.AND.J.NE.NR) C0NST2 = -1.0 * DT / DR(J)**2 / Cl / R(J)
IF(J.EQ.l.OR.J.EQ.NR) C0NST2 = -1.0 * DT / DR(J)**2 / Cl
A(I) = -1.0* C0NST2 * R2 * K3 / 2
IF(J.EQ.l) A(I) = -1.0* C0NST2 * K3 / 2
C(I) = -1.0* C0NST2 * R1 * K4 / 2
IF(J.EQ.NR) C(I) = -1.0* C0NST2 * K4 / 2
B(I) = 1.0 + A(I) + C(I)
D(I) = HEAD(KL,J,1) + C0NST1 * K2 * HEAD(KL ,J ,1) / 2
$ - C0NST1 * K2 * HEAD(KL+1,J,1) / 2
$ + DT/DZ(KL)/C1*(K1 - K2)/2 + DT/DZ(KL)/Cl*(K1 - K2)/4
$ - DT/Cl * S(KL ,J) / 2
30 CONTINUE
GO TO 50
C
C NO FLUX BOUNDRY CONDITION AT LOWER BOUNDRY
C
35 DO 40 J=1 ,NR
I = J
Kl=( C0ND(KL,J,1) + C0ND(KL-1 ,J ,1) ) / 2
K2= 0.0
IF(J.NE.NR) K4=( C0ND(KL ,J+1,2) + C0ND(KL,J,2) ) / 2
IF(J.EQ.NR) K4=0 .0
IF(J.NE.NR) R1 = (R(J+l) + R(J) ) / 2
IF(J.NE.l) K3=( COND(KL ,J ,2) + COND(KL ,J-1,2) ) / 2
IF(J.EQ.l) K3 = 0.0

155
IF(J.NE.l) R2=( R(J) + R(J-l) ) / 2
Cl = ( SWC(KL,J,1) + SWC(KL,J,3) ) / 2
C0NST1 = -1.0 * DT / DZ(KL)**2 / Cl
IF(J.NE.l.AND.J.NE.NR) C0NST2 = -1.0 * DT / DR(J)**2 / Cl / R(J)
IF(J.EQ.l.OR.J.EQ.NR) C0NST2 = -1.0 * DT / DR(J)**2 / Cl
A(I) = -1.0* C0NST2 * R2 * K3 / 2
IF(J.EQ.l) A(I) = -1.0* C0NST2 * K3 / 2
C(I) = -1.0* C0NST2 * R1 * K4 / 2
IF(J.EQ.NR) C(I) = -1.0* C0NST2 * K4 / 2
B( I) = 1.0 + A (I) + C( I)
D(I) = HEAD(KL ,J,1) + C0NST1 * K1 * HEAD(KL ,J ,1) / 2
$ - C0NST1 * K1 * HEAD(KL-1 ,J ,1) / 2
$ + DT/DZ(KL)/C1*(K1 - K2)/2 + DT/DZ(KL)/Cl*(K1 - K2)/4
$ - DT/C1 * S(KL,J) / 2
40 CONTINUE
****************************************
50 IF = 1
CALL TRID( NR,IF,A, B, C, D, V )
DO 55 J=1 ,NR
HEAD(KL ,J ,2) = V( J)
55 CONTINUE
20 CONTINUE
C
C SET UP THE INTERIOR GRID POINTS FOR THE SECOND HALF
C IN THE VERTICAL DIRECTION
C
DO 60 J=1 ,NR
IF(J.EQ.l) GO TO 12
IF(J.EQ.NR)GO TO 14
DO 70 KL=2,NL-1
I = KL
Kl=( COND(KL ,J ,3) + C0ND(KL-1,J ,3) ) / 2
K2=( C0ND(KL+1 ,J ,3) + C0ND(KL,J,3) ) / 2
K4=( C0ND(KL,J+1,2) + C0ND(KL,J,2) ) / 2
K3=( COND(KL,J ,2) + COND(KL,J-1,2) ) / 2
Rl=( R(J+l) + R(J) ) / 2
R2=( R(J) + R(J-l) ) / 2
Cl = ( SWC(KL ,J ,1) + SWC(KL ,J ,3) ) / 2
C0NST1 = -1.0 * DT / DZ(KL)**2 / Cl
C0NST2 = -1.0 * DT / DR(J)**2 / Cl / R(J)
A(I) = -1.0* C0NST1 * K1 / 2
C(I) = -1.0* C0NST1 * K2 / 2
B(I) = 1.0 + A(I) + C(I)
D(I) = DT/DZ(KL)/Cl*( K1 - K2 ) / 4.0
$ - C0NST2*R2*K3*HEAD(KL,J-1,2)/2 - C0NST2*R1*K4*HEAD(KL,J+1,2)/2
$ + HEAD(KL,J,2) + C0NST2 * ( R2*K3 + R1*K4 ) * HEAD(KL ,J ,2) / 2
$ - DT/Cl * S(KL,J) / 2
70 CONTINUE
C
C SET UP THE BOUNDRY CONDITIONS
C
C NO FLUX BOUNDRY CONDITION AT THE SURFACE
C
KL = 1

o o o o o o
156
I = KL
K2=( C0ND(KL+1 ,J,3) + C0ND(KL,J,3) ) / 2
K1 = QS(J)
K4=( COND( KL ,J+1,2) + C0ND(KL,J,2) ) / 2
Rl=( R(J+l) + R(J) ) / 2
K3=( COND(KL ,J ,2) + COND(KL ,J-1,2) ) / 2
R2=( R(J) + R(J-l) ) / 2
C1 = ( SWC(KL,J,1) + SWC(KL ,J ,3) ) / 2
CONST1 = -1.0 * OT / DZ(KL)**2 / C1
C0NST2 = -1.0 * DT / DR(J)**2 / C1 / R(J)
A( I) = 0.0
C(I) = -1.0* C0NST1 * K2 / 2
B(I) = 1.0 + C(I)
D(I) = DT/DZ(KL)/C1*( K1 - K2 ) / 4
$ - C0NST2*R2*K3*HEAD(KL,J-1,2)/2 - C0NST2*R1*K4*HEAD(KL,J+1,2)/2
$ + HEAD(KL,J,2) + C0NST2 * ( R2*K3 + R1*K4 ) * HEAD(KL ,J,2) / 2
$ - DT/C1 * S(KL,J) / 2
NO FLUX BOUNDRY CONDITION AT LOWER BOUNDRY
KL = NL
I = KL
Kl=( COND(KL,J,3) + COND(KL-1 ,J ,3) ) / 2
K2= 0.0
K4=( COND(KL ,J+1,2) + COND( KL ,J ,2) ) / 2
R1 = (R(J+l) + R(J) ) / 2
K3=( COND(KL ,J ,2) + COND(KL ,J-1,2) ) / 2
R2=( R(J) + R(J-l) ) / 2
Cl = ( SWC(KL,J,1) + SWC(KL ,J,3) ) / 2
C0NST1 = -1.0 * DT / DZ(KL)**2 / Cl
C0NST2 = -1.0 * DT / DR(J)**2 / Cl / R(J)
A( I) = -1.0* C0NST1 * K1 / 2
C( I) = 0.0
B(I) = 1.0 + A(I)
D(I) = DT/DZ(KL)/Cl*( K1 - K2 ) / 4
$ - C0NST2*R2*K3*HEAD(KL,J-1,2)/2 - C0NST2*R1*K4*HEAD(KL,J+l ,2)/2
$ + HEAD(KL ,J ,2) + C0NST2 * ( R2*K3 + R1*K4 ) * HEAD(KL,J ,2) / 2
$ - DT/C1 * S(KL ,J) / 2
GO TO 65
NO FLUX BOUNDRY CONDITION AT INNER RADIUS
12 DO 41 KL=1 ,NL
I = KL
IF(KL.NE.l) K1 = (COND(KL,J,3)+C0ND(KL-l ,J ,3))/ 2
IF(KL.EQ.l) K1 = QS(J)
IF(KL.NE . NL) K2= ( C0ND(KL+1 ,J ,3) + C0ND(KL,J,3) ) / 2
IF(KL.EQ.NL)K2 = 0.0
K4=( COND(KL,J+1,2) + C0ND(KL,J,2) ) / 2
K3= K4
Cl = ( SWC(KL,J,1) + SWC(KL ,J ,3) ) / 2
C0NST1 = -1.0 * DT / DZ(KL)**2 / Cl
C0NST2 = -1.0 * DT / DR(J)**2 / Cl
A(I) = -1.0* C0NST1 * K1 / 2

157
IF(KL.EQ.l) A(I) = -1.0* 0.0
C(I) = -1.0* C0NST1 * K2 / 2
IF(KL.EQ.NL) C(I) = -1.0* 0.0
B (I) = 1.0 + A (I) + C( I)
D(I) = DT/DZ(KL)/Cl*( K1 - K2 ) / 4 - C0NST2*K4*HEAD(KL,J+1,2)/2
$ + HEAD(KL,J,2) + C0NST2 * K4 * HEAD(KL,J ,2) / 2
$ - DT/C1 * S(KL,J) / 2
41 CONTINUE
GO TO 65
C
C NO FLUX BOUNDRY CONDITION AT OUTER RADIUS
C
14 DO 61 KL=1,NL
I = KL
IF(KL.NE.l) Kl=( COND(KL ,J ,3) + COND(KL-1 ,J ,3) } / 2
IF(KL.EQ.l) Kl= QS(J)
IF(KL.NE.NL) K2=(C0ND(KL+1 ,J ,3) + C0ND(KL,J,3) ) / 2
IF(KL.EQ.NL)K2 = 0.0
K3=( COND(KL ,J ,2) + COND( KL ,J-1,2) ) / 2
K4 = 0.0
Cl = ( SWC(KL ,J ,1) + SWC(KL ,J ,3) ) / 2
C0NST1 = -1.0 * DT / DZ(KL)**2 / Cl
C0NST2 = -1.0 * DT / DR(J)**2 / Cl
A(I) = -1.0* C0NST1 * K1 / 2
IF(KL.EQ.l) A(I) = -1.0* 0.0
C(I) = -1.0* C0NST1 * K2 / 2
IF(KL.EQ.NL) C(I) = -1.0* 0.0
B(I) = 1.0 + A(I) + C(I)
DC I) = DT/DZ(KL)/Cl*(K1 - K2) /4 - C0NST2* K3* HEAD(KL,J-1,2)/2
$ + HEAD(KL ,J ,2) + C0NST2 * K3 * HEAD(KL ,J ,2) / 2
$ - DT/C1 * S(KL,J) / 2
61 CONTINUE
C
65 IF = 1
CALL TRID(NL,IF, A, B, C, D, V)
DO 71 KL=1,NL
HSTAR(KL,J)= V(KL)
71 CONTINUE
60 CONTINUE
C -
DIF =0.0
DO 506 1=1,NL
DO 506 J=1,NR
DIF = DIF + ABS( (HSTAR( I ,J) -HEAD( I ,J ,3) )/HEAD( I ,J ,3) )
HEAD( I ,J ,3)=HSTAR( I ,J)
506 CONTINUE
C
IF(DIF.LT.l .0E-6) GO TO 701
CALL UPDATE (2)
ITER= ITER + 1
IF(ITER.LT.MAX ITER) GO TO 7 00
C
701 CONTINUE
CALL UPDATE(3)

OC-JOOOOOO o o o
158
DO 80 1=1,NL
DO 80 J=1 ,NR
HEAD( I ,J ,1)=HEAD( I ,J ,3)
HEAD( I ,J ,2}=HEAD( I ,J ,3)
WC(I,J,1) =WC(I ,J ,3)
WC(I ,J ,2) =WC(I,J ,3)
C0ND( I ,J ,1)=C0ND( I ,J ,3)
C0ND(I ,J ,2) = C0ND(I ,J ,3)
SWC(I,J,1) =SWC( I ,J ,3)
SWC(I ,J ,2) =SWC(I ,J ,3)
80 CONTINUE
CALL MASBAL(l)
RETURN
END
ROUTINE TO UPDATE THE SOIL PROPERTIES
SUBROUTINE UPDATE ( IL00P )
PARAMETER (IZ=50,IR=16,IB=IR*2+1,MAX=IZ*IR)
IMPLICIT REAL (A-H.O-Z)
CHARACTER*64 MBFILE
REAL WCNT(120),C0NT(120) ,SWCT(120) ,HEDT( 120)
COMMON /TA BLES/WCNT,C0NT,SWCT,HEDT,NDIM,MBFILE,I SOIL
IF(ISOIL.EQ.l) CALL TBLARR ( ILOOP )
IF(ISOIL.EQ .2) CALL TBLREH ( ILOOP )
IF(ISOIL.EQ.3) CALL TBLOAM ( ILOOP )
IF(ISOIL.EQ .4) CALL TBLOAM ( ILOOP )
IF(ISO IL.EQ.5) CALL TBLYOL ( ILOOP )
IF(ISOIL.EQ .6) CALL LINEAR ( ILOOP )
RETURN
END
SUBROUTINE TBLARR ( ILOOP )
THIS SUBROUTINE USES AN INDEX ROUTING TECHNIQUE WITH
TABULATED DATA TO FIND THE CORRESPONDING VALUES OF
WATER CONTENT, SPECIFIC WATER CAPACITY AND HYDRAULIC
CONDUCTIVITY FOR A GIVEN VALUE OF SOIL MATRIC POTENTIAL
FOR ARREDONDO FINE SAND.
PARAMETER (IZ=50,IR=16,IB=IR*2+1,MAX=IZ*IR)
IMPLICIT REAL (A-H,0-Z)
CHARACTER*64 MBFILE
REAL WCNT(120),CONT(120),SWCT(120),HEDT(120)
COMMON/PARA/WCN(IZ,IR,3),CON(IZ,IR,3),SWC(IZ,IR,3),HED(IZ,IR,3),
$ NL,DZ(IZ),DEPTH(IZ),QS(IR),S(IZ,IR),ICONST,NR,DR(IR),R(IR),ILOW
COMMON/TABLES/WCNT,CONT ,SWCT ,HEDT ,NDIM,MBFILE ,ISOIL
HEDSML = HEDT(118)
HEDLGE = HEDT(l)
DO 100 J=I LOOP ,3
DO 100 K=1 ,NR
DO 100 I = 1 ,NL
IF (HED (I,K ,J) .LE. HEDSML) HED(I,K,J) = HEDSML

o o o o o o o
159
IF (HED(I,K,J) .GE. HEDLGE) HED(I,K,J) = HEDLGE
RINDEX = AL 0G10(-1.*HED(I,K ,J))*4.E01-8.1EOl
INDEX = RINDEX
IF(INDEX.LT.l) INDEX = 1
IF (HED(I,K,J) .LT. HEDT(INDEX+1)) INDEX = INDEX+1
IF (HED(I,K,J) .GT. HEDT(INDEX)) INDEX = INDEX-1
IF(INDEX.LT.l) INDEX = 1
IF (INDEX .GE. 118) INDEX = 118
RM = RINDEX-INDEX
RATIO = AL0G10(HED(I,K,J)/HEDT(INDEX))/ALOG1O(HEDT(INDEX+1)/
+ HEDT(INDEX))
WCN( I ,K ,J) = WCNT( INDEX)+RATIO* (WCNT( INDEX+1)-WCNT( INDEX))
SWC(I ,K,J) = SWCT(INDEX)*1.E01**(RATI0*AL0G10(SWCT(INDEX+1)/
+ SWCT(INDEX)))
CON(I ,K,J) = CONT(INDEX)*1.E01**(RATI0*AL0G10(C0NT(INDEX+1)/
+ CONT(INDEX)))
100 CONTINUE
RETURN
END
C
SUBROUTINE TBLOAM ( ILOOP )
THIS SUBROUTINE USES AN INOEX ROUTING TECHNIQUE WITH
TABULATED DATA TO FIND THE CORRESPONDING VALUES OF
WATER CONTENT, SPECIFIC WATER CAPACITY AND HYDRAULIC
CONDUCTIVITY FOR A GIVEN VALUE OF SOIL MATRIC POTENTIAL
FOR EITHER OF THE LOAMS.
PARAMETER (IZ=50 ,IR=16 ,1 B=IR*2+1 ,MAX=IZ*IR)
IMPLICIT REAL (A-H.O-Z)
CHARACTER*64 MBFILE
REAL WCNT(120),C0NT(120),SWCT(120),HEDT(120)
COMMON/PARA/WCN(IZ,IR,3),CON(IZ,IR,3),SWC(IZ,IR,3),HED(IZ,IR,3),
$ NL,DZ(IZ),DEPTH(IZ),QS(IR),S(IZ,IR),ICONST,NR,DR(IR),R(IR),ILOW
COMMON/TA BLES/WCNT,CONT,SWCT,HEDT.NDIM.MBFILE,ISOIL
HEDSML = HEDT(49)
HEDLGE = HEDT(l)
DO 100 J = I LOOP ,3
DO 100 K = 1,NR
DO 100 I = 1,NL
IF (HED(I,K,J) .LE. HEDSML) HED(I,K,J) = HEDSML
IF (HED( I ,K ,J) .GE. HEDLGE) HED(I,K,J) = HEDLGE
RINDEX = AL0G10(-1 .*HED( I ,K,J) )*1 .E01-2.0E01
INDEX = RINDEX
IF (HED(I,K,J) .LT. HEDT(INDEX+1)) INDEX = INDEX+1
IF (HED(I,K,J) .GT. HEDT(INDEX)) INDEX = INDEX-1
IF (INDEX .GE. 49) INDEX = 49
IF (INDEX .LE. 1) INDEX = 1
RM = RINDEX-INDEX
RATIO = AL0G10(HED( I, K,J)/HEDT( INDEX) )/AL0G10( HEDT( INDEX+1)/
+ HEDT(INDEX))
WCN(I.K.J) = WCNT( INDEX) +RATI0* (WCNT( INDEX+1)-WCNT( INDEX) )
SWC(I ,K ,J) = SWCT(INDEX)*!.EOl**(RATI0*AL0G10(SWCT(INDEX+1)/

160
+ SWCT(INDEX)))
CON(I,K,J) = C0NT(INDEX)*1,E01**(RAT10*AL 0G10(CO NT(INDEX+1)/
+ CONT{INDEX)})
100 CONTINUE
RETURN
END
C
SUBROUTINE TBLREH ( ILOOP )
C THIS SUBROUTINE USES AN INDEX ROUTING TECHNIQUE WITH
C TABULATED DATA TO FIND THE CORRESPONDING VALUES OF
C WATER CONTENT, SPECIFIC WATER CAPACITY AND HYDRAULIC
C CONDUCTIVITY FOR A GIVEN VALUE OF SOIL MATRIC POTENTIAL
C FOR REHOVOT SAND
C
PARAMETER (IZ=50,IR=16,IB=IR*2+1,MAX=IZ*IR)
IMPLICIT REAL (A-H.O-Z)
CHARACTERS MBFILE
REAL WCNT(120),CONT{120),SWCT(120),HEDT(120)
COMMON/PARA/WCN(IZ,IR,3),C0N(IZ,IR,3),SWC(IZ,IR,3),HED(IZ ,IR,3),
$ NL ,DZ( IZ) ,DEPTH( IZ) ,QS( IR) ,S(IZ,IR) , ICONST,NR,DR( IR) ,R(IR) ,ILOW
COMMON/TABLES/WCNT, CONT,SWCT,HEDT,NDIM,MBFILE,ISOIL
C
HEDSML = HEDT(59)
HEDLGE = HEDT(l)
DO 100 J = ILOOP ,3
DO 100 K = 1,NR
DO 100 I = 1,NL
IF (HED (I ,K,J) .LE. HEDSML) HED(I,K,J) = HEDSML
IF (HED(I,K,J) .GE. HEDLGE) HED(I,K,J) = HEDLGE
RINDEX = AL0G10(-1.*HED(I,K,J))*1.E01-1.1E01
INDEX = RINDEX
IF (HED( I ,K ,J) .LT. HEDT( INDEX+1)) INDEX = INDEX+1
IF (HED(I,K,J) .GT. HEDT(INDEX)) INDEX = INDEX-1
IF (INDEX .GE. 59) INDEX = 59
IF (INDEX .LE. 1) INDEX = 1
RM = RINDEX-INDEX
RATIO = AL0G10(HED( I, K,J)/HEDT( INDEX)) / AL 0G10 (HEDT( INDEX+1)/
+ HEDT(INDEX))
WCN(I,K,J) = WCNT( INDEX)+RATIO*(WCNT( INDEX+1)-WCNT( INDEX))
SWC(I ,K,J) = SWCT(INDEX)*1 ,E01**(RATIO*ALOG10(SWCT(INDEX+1)/
+ SWCT(INDEX)))
CON( I ,K ,J) = CONT( INDEX) *1 .E01**( RATI0*AL0G10(CONT( INDEX+1) /
+ CONT( INDEX)))
100 CONTINUE
RETURN
END
C
SUBROUTINE TBLYOL ( ILOOP )
C THIS SUBROUTINE USES AN INDEX ROUTING TECHNIQUE WITH
C TABULATED DATA TO FIND THE CORRESPONDING VALUES OF
C WATER CONTENT, SPECIFIC WATER CAPACITY AND HYDRAULIC
C CONDUCTIVITY FOR A GIVEN VALUE OF SOIL MATRIC POTENTIAL

161
C FOR YOLO LIGHT CLAY.
C - -
PARAMETER (IZ=50,IR=16,IB=IR*2+1,MAX=IZ*IR)
IMPLICIT REAL (A-H.O-Z)
CHARACTER*64 MBFILE
REAL WCNT(120),CONT(120),SWCT(120),HEDT(120)
C0MM0N/PARA/WCN(IZ,IR,3),CON(IZ,IR,3) ,SWC(IZ,IR,3),HED(IZ,IR,3),
$ NL,DZ(IZ),DEPTH(IZ),QS(IR),S(IZ,IR),ICONST,NR,DR(IR),R(IR),IL0W
COMMON/TABLES/WCNT,CONT,SWCT,HEDT,NDIM,MBFILE ,ISOIL
HEDSML = HEDT(49)
HEDLGE = HEDT(l)
DO 100 J = ILOOP ,3
DO 100 K = 1,NR
DO 100 I = 1,NL
IF (HED(I,K,J) .LE. HEDSML) HED(I ,K,J) = HEDSML
IF (HED(I ,K ,J) .GE. HEDLGE) HED(I,K,J) = HEDLGE
RINDEX = AL0G10(-1.*HED(I,K,J))*1.E01-1.1E01
INDEX = RINDEX
IF (HED(I,K,J) .LT. HEDT(INDEX+1)) INDEX = INDEX+1
IF (HED(I ,K,J) .GT. HEDT(INDEX)) INDEX = INDEX-1
IF (INDEX .GE. 49) INDEX = 49
IF (INDEX .LE. 1) INDEX = 1
RM = RINDEX-INDEX
RATIO = AL 0G10(HED(I,K,J)/HEDT(INDEX))/AL 0G10(HEDT(INDEX+1)/
+ HEDT(INDEX))
WCN( I ,K,J) = WCNT( INDEX) +RATI0* (WCNT( INDEX+1)-WCNT( INDEX))
SWC(I ,K,J) = SWCT(INDEX)*1,E01**(RATIO*ALOG10(SWCT(INDEX+1)/
+ SWCT(INDEX)))
CON(I,K,J) = C0NT( INDEX) *1 .E01**( RATI0*AL0G10(C0NT( INDEX+1) /
+ CONT(INDEX)))
100 CONTINUE
RETURN
END
C
SUBROUTINE LINEAR ( ILOOP )
PARAMETER (IZ=50,IR=16,IB=IR*2+1 ,MAX=IZ*IR)
IMPLICIT REAL (A-H.O-Z)
CHARACTER*64 MBFILE
REAL WCNT(120),CONT(120),SWCT(120),HEDT(120)
COMMON /TABLES/WCNT,CONT,SWCT,HEDT,NDIM,MBFILE,ISOIL
COMMON/PARA/WCN(IZ,IR,3),CON(IZ ,IR,3),SWC(IZ,IR,3),HED(IZ ,IR,3),
$ NL,DZ(IZ),DEPTH(IZ),QS(IR),S(IZ,IR),I CONST,NR,DR(IR),R(IR),ILOW
C
DO 100 K= ILOOP, 3
DO 100 J=1,NR
DO 100 I = 1,NL
IF(HED(I,J,K) .GT.O.O) HED(I,J,K) = 0.0
CALL INTPL( HED( I ,J ,K), CON( I ,J ,K) ,SWC(I ,J ,K) ,WCN(I ,J ,K))
100 CONTINUE
RETURN
END
ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c
SUBROUTINE INTPL (W,X,Y,Z)

O C~) C“>
162
IMPLICIT REAL (A-H.O-Z)
THIS SUBROUTINE INTERPOLATES LINEARLY BETWEEN TWO SETS OF DATA
POINTS. DATA IS INPUT AS TABULAR FUNCTIONS
CHARACTER*64 MBFILE
REAL WTAB(120),XTAB(120),YTAB(120),ZTAB(120)
COMMON /TABLES/ZTAB,XTAB,YTAB,WTAB,NDIM,MBFILE,ISOIL
X= XTAB(1)
C Y=YTAB(1)
Z=ZTAB(1)
N=2
IF(ABS(W).LE.ABS(WTAB(1) )) GO TO 50
X=XTAB(NDIM)
C Y=YTAB(NDIM)
Z=ZTAB(NDIM)
N=NDIM
IF(ABS(W).GE.ABS(WTAB(NDIM) )) GO TO 50
C
DO 10 I=2,NDIM
N=I
IF(ABS(W).LE.ABS(WTAB(I))) GO TO 50
10 CONTINUE
50 X=XTAB(N-1)-ABS((W-WTAB(N-l))/(WTAB(N)-WTAB(N-1))*
+(XTAB(N)-XTAB(N-1)))
Y= YTAB(N— 1) -ABS((W-WTAB(N-l)) / (WTAB(N) -WTAB(N-1)) *
+(YTAB(N)-YTAB(N-l)))
Z=ZTAB(N-1)-ABS((W-WTAB(N-l))/(WTAB(N)-WTAB(N-l))*
+(ZTAB(N)-ZTAB(N-l)))
100 RETURN
END

REFERENCES
Amerman, C.R. 1969. Finite difference solutions of unsteady, two-
dimensional, partially saturated porous media flow. Volumes I and
II. Ph.D. Dissertation, Purdue University, W. Lafayette, Ind.
Amerman, C.R., and E.J. Monke. 1977. Soil water modeling II: On
sensitivity to finite difference grid spacing. Trans. Amer. Soc.
Agr. Eng. 20(3) :478-484,488.
Aribi, K., A.G. Smajstrla, F.S. Zazueta, and S.F. Shih. 1985.
Accuracies of neutron probe water content measurements for sandy
soils. Soil and Crop Sci. Soc. Fla. Proc. 44:44-49.
Armstrong, C.F., and T.V. Wilson. 1983. Computer model for moisture
distribution in stratified soils under a trickle source. Trans.
Amer. Soc. Agr. Eng. 26(6):1704-1709.
Bottcher, A.B., and L.W. Miller. 1982. Automatic tensiometer scanner
for rapid measurements. Trans. Amer. Soc. Agr. Eng. 15:1338-1342.
Brandt, A., E. Bresler, N. Diner, I. Ben-Asher, J. Heller, and D.
Goldberg. 1971. Infiltration from a trickle source: I.
mathematical models. Soil Sci. Soc. Amer. Proc. 35:675-682.
Bresler, E., J. Heller, N. Diner, I. Ben-Asher,A. Brandt, and D.
Goldberg. 1971. Experimental data and theoretical predictions.
Soil Sci. Soc. Amer. Proc. 35:683-689.
Clark, G.A. 1982. Simulation of the sensivity of water movement and
storage in sandy soils to irrigation application rate and depth.
Unpublished Master's Thesis. University of Florida, Gainesville,
FL.
Clark, G., and A.G. Smajstrla. 1983. Water distributions in soils as
influenced by irrigation depths and intensities. Soil and Crop Sci.
Soc. Fla. Proc. 42:157-165.
Denmead, O.T., and R.H. Shaw. 1962. Availability of soil water water
to plants as affected by soil moisture content and meteorological
conditions. Agron. J. 45:385-390.
Enfield, C.G., and C.V. Gillaspy. 1980. Pressure transducer for
remote data acquisition. Trans. Amer. Soc. Agr. Eng. 23(5): 1195-
1200.
Feddes, R.A., P.J. Kowalik, and H. Zaradny. 1978. Simulation of field
water use and crop yield. Halsted Press. New York.
163

164
Fitzsimmons, D.W., and N.C. Young. 1972. Tensiometer-pressure
transducer system for studying unsteady flow through soils.
Trans. Amer. Soc. Agr. Eng. 15:272-275.
Fletcher, J.E. 1939. A dielectric method of measuring soil moisture.
Soil Sci. Soc. Amer. Proc. 4:84-88.
Gardner, W.R. 1960. Dynamic aspects of water availability to plants.
Soil Sci. 89(2):63-73.
Gardner, W.R., and C.F. Ehling. 1962. Some observations on the
movement of water to plant roots. Agron. J. 54:453-456.
Gradmann, H. 1928. Untersuchungen uber die wasserverhaltnisse des
bodens ais grundlage des pflanzenwachstums. Jahrb. Wiss. Bot.,
69:1-100.
Hanks, R.J., and S.A. Bowers. 1962. Numerical solution of the moisture
flow equation for infiltration into layered soils. Soil Sci. Soc.
Amer. Proc. 26:530-534.
Harrison, D.S., A.G. Smajstrla, R.E. Choate, and G.W. Isaacs. 1983.
Irrigation in Florida agriculture in the '80s. Bulletin 196, Fla.
Coop. Ext. Serv., Univ. of Fla.
Haverkamp, R., M. Vauclin, J. Touma, P.J. Wierenga, and G. Vachaud.
1977. A compairison of numerical simulation models for one¬
dimensional infiltration. Soil Sci. Soc. Amer. Proc. 41:285-294.
Hiler, E.A., and S.I. Bhuiyan. 1971. Dynamic simulation of unsteady
flow of water in unsaturated soils and its application to
subirrigation system design. Technical Report TR-40. Texas Water
Resources Institute. College Station, TX.
Hi 11 el, D. 1980. Fundamentals of soil physics. Academic Press, New
York.
Hornberger, G.M., I. Remson, and A.A. Fungaroli. 1969. Numeric studies
of a composite soil moisture ground-water system. Water Resour.
Res. 5:797-802.
Long, F.L., and M.G. Hulk. 1980. An automated system for measuring
soil water potential gradients in a rhizotron soil profile. Soil
Sci. 129:305-310.
Marthaler, H.P., W. Vogelsanger, F. Richard, and P.J. Wierenga. 1983.
A pressure transducer for field tensiometers. Soil Sci Soc. Amer.
Proc. 47:624-627.
Molz, F.J., and J. Remson. 1970. Extraction term models of soil
moisture use of transpiring plants. Water Resour. Res. 6:1346-
1356.
Nimah, M.N., and R.J. Hanks. 1973a. Model for estimating soil water,
plant and atmospheric interrelations: I. description and
sensitivity. Soil Sci. Soc. Amer. Proc. 37:522-527.

165
Nimah, M.N., and R.J. Hanks. 1973b. Model for estimating soil water,
plant and atmospheric interrelations: II field test of model.
Soil Sci. Soc. Amer. Proc. 37:528-532.
Perrens, S.J., and K.K. Watson. 1977. Numerical analysis of two-
dimensional infiltration and redistribution. Water Resour. Res.
13(4):781-790.
Phene, C.J., 6.J. Hoffman, and S.L. Rawlins. 1971. Measuring soil
matric potential in situ by sensing heat dissipation within a
porous body: I. Theory and sensor construction. Soil Sci. Soc.
Amer. Proc. 35:27-33.
Phene, C.J., S.L. Rawlins, and G.J. Hoffman. 1971. Measuring soil
matric potential in situ by sensing heat dissipation within a
porous body: II. Experimental Results. Soil Sci. Soc. Amer.
Proc. 35:225-229.
Phillip, J.R. 1968. Steady infiltration from buried point sources and
spherical cavities. Water Resour. Res. 4:1039-1047.
Ritchie, J.T. 1973. Influence of soil water status and meteorological
conditions on evaporation from a corn canopy. Agron. J.
65:893-897.
Reddell, D.L., and D.K. Sunada. 1970. Numerical simulation of
dispersion in groundwater aquifers. Hydrology paper 41, Colorado
State University, Fort Collins, Colorado.
Rubin, J., and R. Steinhardt. 1963. Soil water relations during rain
infiltration: 1. Theory. Soil Sci. Soc. Amer. Proc. 27:246-251.
Rubin, J. 1967. Numerical method for analyzing hysteresis-affected,
post-infiltration redistribution of soil moisture. Soil Sci. Soc.
Amer. Proc. 31:13-20.
Rubin, J. 1968. Theoretical analysis of two-dimensional, transient
flow of water in unsaturated and partly unsaturated soils. Soil
Sci. Soc. Amer. Proc. 32:607-615.
Saxton, K.E., H.P. Johnson, and R.H. Shaw. 1974. Modeling
evapotranspiration and soil moisture. Trans. Amer. Soc. Agr. Eng.
17:673-677.
Slack, D.C., C.T. Haan, and L.G. Wells. 1977. Modeling soil water
movement into plant roots. Trans. Amer. Soc. Agr. Eng. 20:919-
927,933.
Smajstrla, A.G. 1973. Simulation of Miscible displacement in soils and
sensitivity to the dispersion coefficient. Master's Thesis. Texas
A & M University.

166
Smajstrla, A.G. 1982. Irrigation management for the conservation of
limited water resources. Office of Water Research and Technology
of the department of Interior Water Conservation Research Program
(WRC-80).
Smajstrla, A.G., 1985. A field lysimeter system for crop water use and
water stress studies in humid regions. Soil and Crop Sci. Soc. Fla.
Proc. 44:53-59.
Smajstrla, A.G., L.R. Parsons, F.S. Zazueta, G. Vellidis, and K. Aribi.
1986. Water use and growth of young citrus trees. ASAE Paper No.
86-2069, Amer. Soc. of Agr. Eng. St. Joseph, MI.
Taylor, G.S., and J.N. Luthin. 1969. Computer methods for transient
analysis of water-table aquifers. Water Resour. Res. 5:144-152.
Thomson, S.J., E.D. Threadgill, and J.R. Stansell. 1982.
Field test of a microprocessor based center pivot irrigation
controller. ASAE Paper No. 82-2535, Amer. Soc. of Agr. Eng.
St. Joseph, MI.
Thurnau, D.H. 1963. Algorithm 195, bandsolve. Communications of the
Association for Computing Machinery, 6(8):441.
Tollner, E.W., and F.J. Molz. 1983. Simulating plant water uptake in
moist, lighter textured soils. Trans. Amer. Soc. Agr. Eng.
26(1):87-91.
Van Bavel, C.H.M., G.B. Strik, and K.J. Burst. 1968. Hydraulic
properties of a clay loam soil and the field measurement of water
uptake by roots. I. Interpretation of water content and pressure
profiles. Soil Sci. Soc. Amer. Proc. 32:310-317.
Van den Honert, T.H. 1948. Water transport in plants as a catenary
process. Discuss. Faraday Soc. 3:146-153.
Wooding, R.A. 1968. Steady infiltration with a shallow circular pond.
Water Resour. Res. 4:1259-1273.
Zazueta, F.S., A.G. Smajstrla, and D.S. Harrison. 1984.
Microcomputer control of irrigation systems I: Hardware and
software considerations. Soil and Crop Sci. Soc. Fla. Proc.
43:123-129.
Zazueta, F.S., A.G. Smajstrla, and D.S. Harrison. 1985. A simple
numerical model for the prediction of soil-water movement from
trickle sources. Soil and Crop Sci. Soc. Fla. Proc. 44:72-76.
Zur, B., and J.W. Jones. 1981. A model for the water relations,
photosynthesis , and expansive growth of crops. Water Resour.
Res. 17:311-320.

BIOGRAPHICAL SKETCH
Kenneth Coy Stone was born June 25, 1959, in Tifton, Georgia. He
graduated from Tiftarea Academy, Chula, Georgia, in June 1977. He then
attended Abraham Baldwin Agricultural College in Tifton, Georgia, from
which he received an Associate in Science degree in mathematics in June
of 1979. In September of 1979 he began his undergraduate studies in
agricultural engineering at the University of Georgia. He graduated
from the University of Georgia with a Bachelor of Science in
Agricultural Engineering degree in June 1981. He then began his
graduate studies at the University of Georgia and received his Master
of Science (agricultural engineering) degree in March 1985. He enrolled
in the University of Florida in May 1983 to pursue the Ph.D. degree in
agricultural engineering..
167

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
ML i
D -
(l tIL
Allen G. Smajstrlaj
i Chairman
Associate Professor of
Agricultural Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
{il -
'ames W. Jones
fes sor of
Agricultural Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Fadro R. Zaz
Assistant Pr
Agricultural
retai
Ifesápr of
ingineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Assistant Professor of
Civil Engineering

This dissertation was submitted to the Graduate Faculty of the College
of Engineering and to the Graduate School and was accepted as partial
fulfillment of the requirements for the degree of Doctor of Philosophy.
May 1987
I'LsJLuJ' Q. â– 
Dean, College of Engineering
Dean, Graduate School

UNIVERSITY OF FLORIDA
llllllllllllllllllllllllllllllllllllllllllllll
3 1262 08554 0135