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VARIABLY SATURATED THREE-DIMENSIONAL RAINFALL DRIVEN
GROUNDWATER PUMPING MODEL
By
AHMET DOGAN
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
1999
To my beloved and devoted father, Mehmet, the most honorable person in my little
world, who was struggling with lung cancer although he had never smoked at all.
Unfortunately, he passed away and walked to his beloved God on April 13, 1999, while
his only son was away from home to complete his Ph.D. study. I believe that my father is
at rest in peace now because he lived such a beautiful life to help others, and to comfort
others just for the sake of all mighty God.
ACKNOWLEDGMENTS
I wish to express my deep and sincere gratitude to Dr. L. H. Motz, my supervisor
and mentor during my long Ph.D. study, for his helpful support and wise guidance during
this study. Special appreciation is extended to Dr. K. Hatfield for his help, support, and
helpful technical discussions. I would also like to thank Dr. W. D. Graham for her
guidance and feedback. I appreciate Dr. K. L. Campbell for his valuable help and
guidance, especially about evapotranspiration. Particular appreciation is also extended to
Dr. R. J. Thieke. He is one of the most enthusiastic teachers in our department, and I
learned a lot in his class. I would like to thank Dr. R. W. Healy, the author of the model
code VS2D, for his comments and for providing me the most current version of VS2D. I
also give thanks to God for giving me the opportunity to meet and study with these great
people at the University of Florida and finish my Ph.D. study.
I would like to express my deep appreciation to my beloved late father Mehmet,
who passed away on 13th April 1999. He sacrificed a lot to raise me and to support my
education. He was a great man in my life, and I will try to follow in his footsteps to be a
good man. I thank my devoted mother Ayse from the bottom of my heart for her prayers,
unbelievable continuous support, and encouragement. I will never forget my beloved
wife Havva's help and support. She was always there to help me and support me any
time, anywhere. She sacrificed a lot to provide me comfort and a good study
environment during this Ph.D. study. My special appreciation is also extended to my
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little son Mehmet and my baby girl Ayse Hilal. I forgot my troubles and found peace of
mind when I was playing with them. Finally, special thanks are given to my sisters
Selime and Bedia for their love and prayers.
iv
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ................................................................................................. iii
LIST OF TABLES ........................................................................................................... viii
LIST OF FIGURES............................................................................................................ ix
ABSTRACT....................................................................................................................... xi
CHAPTERS
1
INTRODUCTION..........................................................................................................1
2
LITERATURE SURVEY ..............................................................................................6
Historical Development of Groundwater Hydrology and Hydraulics......................6
Research in Saturated Flow......................................................................................8
Unsaturated Flow Studies ......................................................................................16
Variably Saturated Flow Studies............................................................................23
Available Hydrologic Computer Models ...............................................................29
3
DERIVATION OF THE VARIABLY SATURATED GROUNDWATER
FLOW EQUATION.....................................................................................................43
General Three-Dimensional Saturated-Unsaturated Groundwater Flow
Equation ...........................................................................................................43
Conceptualization ............................................................................................43
Continuity Equation .........................................................................................45
Storage Term....................................................................................................47
Darcy-Buckingham Equation...........................................................................49
Governing Equation (Modified Richards’ Equation).......................................52
In the saturated zone: .................................................................................53
In the unsaturated zone: .............................................................................53
Hydraulic Conductivity..........................................................................................55
Sink/Source Term ..................................................................................................60
Determination of Evapotranspiration...............................................................63
Estimation of input parameters for PET calculations ................................66
Determination of transpiration (or root water uptake) ...............................70
Evaporation ................................................................................................78
Pumping and Recharge Wells ..........................................................................80
v
Drains, Sinkholes, and Springs ........................................................................81
Boundary Conditions .............................................................................................82
Specified Flux Boundary Condition ................................................................82
Specified Head Boundary Condition ...............................................................83
Variable Boundary Condition ..........................................................................83
River Boundary ................................................................................................85
General Head Boundary...................................................................................86
4
MATHEMATICAL MODEL DEVELOPMENT AND NUMERICAL
SOLUTION OF THE MODIFIED RICHARDS EQUATION....................................88
Conceptualization of the Model.............................................................................89
Spatial Discretization .......................................................................................91
Temporal Discretization...................................................................................92
Finite-difference Formulation of the Governing Equation ....................................93
Mixed Form of Richards Equation and Modified Picard Iteration
Scheme.............................................................................................................98
Boundary Conditions .....................................................................................103
Prescribed head boundaries......................................................................104
Prescribed flux boundaries.......................................................................105
River boundary.........................................................................................111
Overland flow and ponding......................................................................112
Rainfall and evaporation boundaries........................................................112
Dewatering of a Confined Aquifer.................................................................114
Iteration Levels...............................................................................................114
Conductance Terms (CNi+1/2,j,k) .....................................................................115
Matrix Equation Solver (Preconditioned Conjugate Gradient Method
{PCGM}) .............................................................................................................118
5
VERIFICATION OF THE MODEL..........................................................................124
Example 1 ............................................................................................................124
Example 2 ............................................................................................................128
Example 3 ............................................................................................................133
Example 4 ............................................................................................................138
Example 5 ............................................................................................................141
Example 6 ............................................................................................................143
Example 7 ............................................................................................................146
6
APPLICATION OF THE MODEL............................................................................150
Application of the Model to a Two-Dimensional Infiltration and
Evapotranspiration Problem.................................................................................150
Application of the Model to an Unconfined Sand Aquifer Pumping Test...........155
7
APPLICATION OF THE MODEL TO A FIELD CONDITION IN NORTH
CENTRAL FLORIDA ...............................................................................................162
Description of the Study Area..............................................................................162
Location .........................................................................................................162
vi
Climate...........................................................................................................164
Geology..........................................................................................................164
Groundwater Hydrology ................................................................................166
Application of the Model .....................................................................................168
Selection of the Model Area ..........................................................................168
Boundary Conditions .....................................................................................168
Meteorological Data.......................................................................................170
Evapotranspiration .........................................................................................171
Lakes ..............................................................................................................172
Three-Dimensional Discretization .................................................................174
Two-Dimensional Discretization ...................................................................176
Description of Input Parameters for the Two-dimensional Simulation
of the Model...................................................................................................178
Model Results ................................................................................................179
8
SUMMARY AND CONCLUSIONS ........................................................................185
Applicability Limitations of the Model ..............................................................189
Future Study........................................................................................................189
LIST OF REFERENCES .................................................................................................192
APPENDICES
A FORTRAN CODE OF VARIABLY SATURATED THREE-DIMENSIONAL
RAINFALL DRIVEN GROUNDWATER PUMPING MODEL..............................208
B INPUT FILES FOR THE MODEL SIMULATION IN THE UECB.........................230
BIOGRAPHICAL SKETCH ...........................................................................................256
vii
LIST OF TABLES
Table
page
2.1 Summary of selected saturated-unsaturated flow models...........................................31
5.1 Parameters used for example 1 .................................................................................127
6.1 Parameters used for the VS2D problem....................................................................153
6.2 Parameters for the unconfined aquifer pumping problem.........................................160
7.1 Geologic layers in the Upper Etonia Creek Basin (based on Motz et al., 1993).......165
7.2 Hydrogeologic units of the Upper Etonia Creek Basin (based on Motz et al.,
1993) ........................................................................................................................167
7.3 Regional and Local Rainfall Data During the Simulation Period.............................171
7.4 Lake Barco Pan Evaporation Coefficients ................................................................172
7.5 Lake stages in the model domain..............................................................................174
7.6 Parameters used for the model application in the UECB area. .................................178
8.1 Summary of new model. ...........................................................................................188
B.1 Isoil matrix for material properties of the model domain in hydrologic
simulation of UECB, where, 1: Upper Floridan Aquifer (limestone); 2, 3, 4:
Confining Unit (Hawthorn Group); 5: Surficial Aquifer (sand); and 0: no
material.....................................................................................................................230
B.2 Ibound matrix for the boundary properties of the model domain in hydrologic
simulation of UECB, where, 1: Active cell; 0: inactive cell; -2: fixed head
cell for Crystal Lake, -3:fixed head cell for Magnolia Lake, 9: general head
boundary cell, 7: rainfall and evapotranspiration boundary cell. .............................232
B.3 Meteorological data for the period September 1, 1994-August 31, 1995 ................234
B.4 Initial pressure heads (m) and geometric elevations in the model domain for
hydrologic simulation of the UECB.........................................................................242
viii
LIST OF FIGURES
Figure
page
3.1 Conceptualization of hydrologic system.....................................................................43
3.2 Representative unit volume of an aquifer. ..................................................................44
3.3 Flow chart describing the principle sink/source terms in the model...........................60
3.4 Flow chart for actual transpiration calculations in the model.....................................62
3.5 Flow chart for the evapotranspiration calculations (Fares, 1996)...............................65
3.6 Schematic of the plant water stress response function, ar(h) (Feddes et al.,
1978). Water uptake below h1(air entrainment pressure, saturation starts) and
above h4 (wilting point) is set to zero. Between h2 and h3 water uptake is
maximum. The value of h3 varies with the potential transpiration rate Tp. ..............75
3.7 Water stress function as a function of pressure head and potential
transpiration (Jensen, 1983). ......................................................................................77
4.1 Schematic representation of the physical components and the interaction
among them. .............................................................................................................90
4.2 Vertical discretization of the model............................................................................92
4.3 Flow into and out of cell i, j, k....................................................................................94
4.4 Diagram for calculation of vertical conductance in case of semi-confining
units..........................................................................................................................118
4.5 PCG methods (Schmit and Lai, 1994). .....................................................................121
5.1 Comparison of the numerical model with results of Paniconi et al. (1991)..............128
5.2 Comparison of the numerical model with the analytical solution of Srivastava
and Yeh (1991).........................................................................................................133
5.3 Comparison of the numerical model with the analytical solution of Srivastava
and Yeh (1991) for layered soils. .............................................................................136
5.4 Comparison of the numerical model with experimental results of Vauclin et
al. (1979). ...............................................................................................................139
5.5 Three-dimensional model domain description for example 5 ..................................141
5.6 Water-table elevations resulting from 3-D simulation of example 5 at y = 0
for various time values.............................................................................................142
5.7 Three-dimensional water-table recharge. Water-table elevations are shown at
the end of 8-hr rainfall of 0.148 m/hr......................................................................143
5.8 Three-dimensional water-table recharge and pumping. Water-table elevations
are shown at the end of a 4-hr rainfall of 0.148 m/hr and 6.25 m3/hr pumping.......144
5.9 Three-dimensional pumping from water-table. Water-table elevations are
shown at the end of a 4-hr pumping period at the rate of 6.25 m3/hr.......................144
5.10 Three-dimensional recharge to the water-table. Water-table elevations are
shown at the end of a 4-hr injection period at a rate of 6.25 m3/hr..........................145
ix
5.11 Three-dimensional recharge to the water table. Water-table elevations are
shown at a cross-section in the x-z plane at j = 1 for different time values.
Injection well is located at (i, j) = (15, 15) at k = 2, 3, 4, 5, 6 with the rate of
6.25 m3/hr.................................................................................................................145
5.12 Problem definition sketch for example 7. ...............................................................147
5.13 Water-table position at the steady-state condition for example 7. ..........................148
5.14 Three-dimensional view of the water-table for example 7. ....................................149
6.1 Description of the problem of Lappala et al. (1987).................................................151
6.2 Comparison of the results of VS2D and the current model. .....................................155
6.3 Cross section for the unconfined aquifer pumping problem.....................................157
6.4 Comparison of the pumping test results of Nwankwor et al.(1992) and the
current model results................................................................................................161
7.1 September 1994 water table map in the UECB and the location of the model
domain (Source: Sousa, 1997). ................................................................................163
7.2 Topographic surface of the model area.....................................................................164
7.3 The model boundaries and September 1994 water table map. .................................169
7.4 Lake levels in the model domain. .............................................................................173
7.5 Horizontal discretization of the three-dimensional model domain...........................175
7.6 Vertical discretization of the two-dimensional model domain. ................................177
7.7 The model results versus the observed data in the well C520 during the period
of Sepember,1994-September, 1995........................................................................181
7.8 The Rainfall and Evapotranspiration components in the model area. ......................182
7.9 Total head contours at time =150 days .....................................................................183
7.10 Moisture content profiles at different times of the simulation................................184
x
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
VARIABLY SATURATED THREE-DIMENSIONAL RAINFALL DRIVEN
GROUNDWATER PUMPING MODEL
By
Ahmet Dogan
December 1999
Chairman: Louis H. Motz
Major Department: Civil Engineering
Many water-resource management and environmental quality problems require a
better understanding of the complete hydrological process, which can be described only
by using a variably saturated groundwater flow model. A new saturated/unsaturated
three-dimensional rainfall-driven groundwater-pumping model has been developed to
understand the response of a variety of hydrogeologic systems to both natural and
anthropogenic impacts. This model was designed to simulate all of the important
elements of the hydrological cycle other than the runoff and seepage processes as
accurately as possible using physically based assumptions and equations.
The uniqueness of the model is its hydrological and hydrogeological completeness
such that the model runs using rainfall and climatologic data and calculates the threedimensional pressure distribution over the entire hydrogeologic domain. The model also
calculates the potential evapotranspiration for given climatological data. In the model,
xi
the greatest effort has been devoted to an improved conceptualization of the unsaturated
zone, which is a crucial part of the hydrological system in a groundwater basin. This is
because the unsaturated zone plays an important role in many hydrological processes such
as recharge to groundwater, surface-groundwater interaction, solute transport, and
evapotranspiration.
Recent advances in modeling variably saturated flow are incorporated into the
model. The model simulates the hydrogeologic system by solving the nonlinear threedimensional mixed form of the Richards equation using the modified Picard iteration
scheme and preconditioned conjugate gradient method. A new convergence criterion is
used for faster convergence in the iterations. The model treats the complete subsurface
regime as a unified whole, and flow in the unsaturated zone is integrated with saturated
flow in the underlying unconfined and confined aquifers. The model has the capability to
simulate pumping from the aquifer and artificial recharge. A transpiration and an
evaporation model are integrated into the groundwater flow equation as sink terms.
Input data for the model are the hydrogeologic and geometric properties of the
flow domain, meteorological data, vegetative cover, and soil moisture characteristics.
The output is in the form of groundwater heads, moisture contents, and actual
evapotranspiration. The model has been verified against other model results from the
literature.
xii
CHAPTER 1
INTRODUCTION
The management and control of our water resources requires the conception,
planning, and execution of designs to make use of water without causing harm to the
environment. Approximately forty percent of the water used for all purposes in the
United States is derived from groundwater sources (Heath, 1983). Groundwater is a vital
and very vulnerable source of water supply. The main source of recharge for the
groundwater is precipitation, which may move through the soil directly to the
groundwater or it may enter surface-water bodies such as rivers, streams, lakes, and
wetlands and percolate from these water bodies to the groundwater. Interception,
depression storage, evapotranspiration, and soil moisture capacity must be satisfied
before any large amount of water can percolate to the groundwater. Precipitation can
supply large quantities of water for groundwater easily in such places where sandy soils,
flat topography, and high water-tables occur, i.e., in Florida. The surface-water and the
groundwater are strongly interrelated, and the use of one source may affect the water
available from the other source. Both surface water and groundwater should be
considered together in plans for water-resources development.
The groundwater-surface-water interaction process involves infiltration,
evapotranspiration, runoff, and seepage between streams and aquifers. A surface-water
model or a groundwater model alone cannot accurately simulate this process. Instead, a
complete hydrological system model is required, which can simulate the rainfall-runoff
1
2
relation, evapotranspiration, unsaturated flow, saturated flow, seepage, and pumping from
the aquifer.
There are two main reasons to develop and rely upon hydrologic models, i.e., to
understand why a flow system is behaving in a particular observed manner and to predict
how a flow system will behave in the future. In addition, models can be used to analyze
hypothetical flow situations in order to gain generic understanding of that type of flow
system. The first step in studying a groundwater system is to develop a conceptual
model, which describes the real hydrogeologic system. After conceptualization of the
real system, a mathematical model is developed to solve some form of the basic equations
of groundwater flow. Mathematical models can be classified as analytical or numerical
models, depending on the solution technique. Analytical models can be solved rapidly,
accurately, and inexpensively. Numerical models sometimes must be used when there is
a very complex hydrogeologic system where hydrogeologic and hydraulic properties vary
within the model area. Numerical solutions to the groundwater flow equations require
that the equations be reformulated using one of the approximation techniques, e.g., finitedifference, finite element, or the method of characteristics. The requirements of water
resources planning have made numerical model simulations of the hydrologic response of
groundwater basins a very important technique. Successful resolution of many waterresources management and environmental quality problems is possible through a better
understanding of the hydrological processes that take place near the ground surface, i.e.,
in the unsaturated, or vadose, zone.
A new saturated/unsaturated three-dimensional rainfall-driven groundwaterpumping model, described in this dissertation, has been developed to understand better
3
groundwater level fluctuations and help to make reasonable groundwater policies. The
model was designed to simulate all of the important elements of the hydrological cycle as
accurately as possible in a manner that all assumptions and equations are physically
based. The uniqueness of the model is its three-dimensional hydrological and
hydrogeological completeness and better conceptualization of the unsaturated zone. The
unsaturated zone is a crucial part of the hydrological system in a basin. It plays an
important role in many modeling applications, e.g., for recharge estimation, surfacegroundwater interaction, solute transport, and evapotranspiration calculations. Therefore,
in the model, the main emphasis is given to simulation of the unsaturated zone, the
infiltration process, evapotranspiration, and the root water uptake process.
The model utilizes the finite-difference technique to solve the three-dimensional
form of the variably saturated groundwater flow equation. The finite-difference grids can
be generated as variable or constant size. The upper boundary in the model is at ground
surface, and the upper boundary conditions are determined using soil and meteorological
data. The upper boundary condition can be a positive flux boundary (i.e., before ponding
occurs) or a fixed head (i.e., after ponding occurs) during a rainfall event. It can be a
negative flux boundary or a fixed head boundary during the evaporation process. The
model treats the complete subsurface regime as a unified whole, and the flow in the
unsaturated zone is integrated with saturated flow in the underlying unconfined and
confined aquifers. This is achieved by solving the complete three-dimensional nonlinear
Richards equation (1931) throughout the whole flow domain. The model allows
modeling of heterogeneous and anisotropic geologic formations. It has the capability to
simulate anthropogenic effects such as pumping from an aquifer and artificial recharge.
4
A plant root water uptake (transpiration) model and an evaporation model are included as
sink terms in the groundwater flow equations. The model also includes a module to
calculate the potential evapotranspiration values for a given location and climatologic
data based on the Priestly and Taylor (1972) equation.
The required input data for the model are hydrogeologic and geometric properties
of the flow domain, meteorological data, vegetative cover, and soil type data for the
calculation of evapotranspiration, rainfall data, and soil-water characteristics. The output
provides groundwater heads in terms of pressure head, moisture-content profile in the
unsaturated zone, actual evapotranspiration, and exchange of water between surfacewater and groundwater systems.
A groundwater setting in north Florida was selected as an example of the model’s
application. Florida has a unique hydrogeologic character with its flat topography, heavy
subtropical rainfalls, wetlands, high water-tables, and sandy soils, which cause significant
interactions between groundwater and surface-water systems. Florida’s continuing
population growth has resulted in an increasing demand on the water supply. This
increasing demand mainly will be met using the state’s groundwater resources. However,
excess usage of groundwater for public water supply, irrigation, and industry has led to
negative impacts, including saltwater intrusion, the lowering of lake and wetland water
levels, and the reduction of spring flow and stream flow. This problem is especially true
for north Florida.
Using the deterministic, fully distributed, physically based integrated hydrological
model, the effects of human interventions and effects of naturally varying recharge in the
5
form of rainfall patterns on the hydrological cycle can be determined. Using this model, a
more informed basis for policy and decision-making can be created.
CHAPTER 2
LITERATURE SURVEY
Historical Development of Groundwater Hydrology and Hydraulics
Although groundwater has been used since early times, an understanding of the
origin of groundwater as related to the hydrologic cycle was established only in the later
part of the seventeenth century. Several incorrect hypotheses explaining the occurrence
of groundwater were given by such early Greek philosophers and historians as Homer
(about 1000 BC), Anaxagoras and Herodotus (fifth century BC), Plato (427-347 BC), and
Aristotle (384-322 BC). Plato thought that one huge underground cavern in the earth was
the source of all rivers and that water flowed back from the ocean to this cavern.
Surprisingly, however, Plato’s opinion includes an accurate description of the hydrologic
cycle (Baker and Horton, 1936). The Roman philosophers followed the Greek teachings
and contributed little to the subject. These hypotheses were unquestioned until the end of
the seventeenth century. The Roman architect Marcus Vitrivius (15 BC-58 AD) was
probably the first in the recorded history to have a correct grasp of the hydrologic cycle.
He realized that the mountains receive a large amount of water from melting snow that
seeps through the rock strata and emerges as springs at lower elevations. Al-Biruni (9731048) accurately explained the mechanics of groundwater movement as well as the
occurrence of natural springs and artesian wells "on the principle of water finding its own
level in communicating channels" (Kashef, 1986). Bernard Palissy (1509-1589) is
6
7
recognized as the first in modern history to explain the hydrologic cycle, the origin of
springs, and the relationship between wells and rivers (Cap, 1961). The first field
measurements were made by Pierre Perrault (1608-1680). He studied evaporation and
capillary rise and measured the rainfall and runoff of the upper drainage basin of the
Seine River in France (De Wiest, 1965). The findings of Perrault were verified several
years later by EdmÀ Maiotte (1620-1684), whose report appeared in 1686 after his death.
Outstanding documents on the subject of artesian wells were written in 1715 by Antonio
Vallisnieri, President of the University of Padua, Italy (De Wiest, 1965).
In the nineteenth century, quantitative measurements were initiated by Darcy
(1856) and supplemented by the analytical work of Dupuit (1863), Thiem (1906), and
Forchheimer (1898). This work stimulated groundwater research in the twentieth century
and shifted groundwater hydrology from a descriptive subject to a more rigorous
analytical science (Kashef, 1986).
There have been three revolutions in physical hydrogeology: the historic set of
experiments carried out by Darcy (1856) in Dijon, France; the transient well-hydraulics
analysis by C.V. Theis in 1935; and the introduction of large digital computers in the
early 1960s. Darcy developed an empirical law on which nearly all subsequent work has
been based, and Theis developed a methodology for both the in-situ measurement of
hydrologic properties of geologic formations and the prediction of the response of
groundwater systems to pumping. Digital computers provide the means for assessment of
groundwater resources on a regional scale within the context of the full hydrologic cycle
(Freeze and Back, 1983).
8
Research in Saturated Flow
Two- and/or three-dimensional water flow through saturated porous media has
been known in its steady-state form since the work of Forchheimer (1898) in the late
nineteenth century. His understanding was based on an analogy with the heat-flow
equation. Theis invoked the same analogy in 1935 in presenting a solution to the
transient form of the flow equation, although he did not present the fundamental
differential equation itself. Since the movement of fluids in geological materials can be
understood based on treating fluid flow as a process mathematically analogous to heat
conduction in solids, the working mathematical model for the transient groundwater flow
is the partial differential equation of heat conduction, originally proposed by Fourier.
Fourier’s theory was published in 1822 with additional works of Laplace, Lagrange,
Monge, and Lacroix. Darcy was aware of the studies of Fourier, Ohm, and Poiseuille and
made use of them in his work (Narasimhan, 1998a).
During the latter half of the nineteenth century, Boussinesq, Dupuit, Forchheimer,
Adolph Thiem, and Gunther Thiem made important contributions to the development of
the science. Dupuit (1863) developed a linear constitutive law, similar to Darcy’s, based
on hydraulic theory rather than experimental evidence. He also produced the first formal
mathematical analysis of a groundwater hydraulics problem, that of radial flow toward a
pumped well in an unconfined aquifer. The assumptions invoked in his analysis, namely,
that the hydraulic gradient is equal to the slope of the water-table and that it is invariant
with depth, have come to be known as the Dupuit assumptions, and methods based on
these assumptions are still in wide use today.
9
Chamberlin (1885) is generally recognized as initiating the science of
hydrogeology in the United States. He outlined the seven prerequisites for artesian flow
and described the hydrogeologic properties of water bearing beds in his 1885 report. If
Gauss was the “prince of mathematicians,” then surely Forchheimer was the prince of
groundwater hydraulics" (Freeze and Back, 1983). Forchheimer (1898) was the first to
note the analogy between groundwater flow and heat flow, and he was the first to use the
Laplace equation in the description of steady-state groundwater flow. He clarified the
Dupuit assumptions and recognized that steady-state flow in unconfined aquifers under
the Dupuit assumptions would obey the Laplace equation with respect to the square of the
hydraulic head rather than the hydraulic head itself.
Dupuit’s formula for the discharge from a well in an unconfined aquifer required
advanced knowledge of the radius of the zone of influence at steady-state. Adolph Thiem
carried out extensive field investigations to clarify the controls on the radius of influence
in 1870. His son Gunther Thiem (1906) was the first to recognize that Dupuit’s equations
could be applied at any two points on the profile of the cone of depression around a well.
This realization led to the first inverse application of a solution to the steady-state flow
equation and, hence, to the first use of pumping tests as a practical tool for in-situ
measurement of the hydraulic properties of geologic formations.
During the last part of the nineteenth century, nearly the same important
developments were duplicated in the United States because of the poor interchange of
information between Europe and the United States. C. S. Slichter of the U. S. Geological
Survey, working twelve years after Forchheimer and apparently unaware of his work,
utilized the same heat-flow literature to arrive at the Laplace equation and flow-net
10
construction. Another important contribution of Slichter was the investigation of the
physical significance of hydraulic conductivity, which was treated only as an empirical
constant by Darcy. He identified the geometric and viscous drag components of hydraulic
conductivity.
In the evolution of the ideas pertaining to the flow of fluids in geological media,
the period 1920-1940 must rank as truly remarkable (Narasimhan, 1998a). Oscar
Meinzer of the U.S. Geological Survey was one of the most famous hydrogeologists
during the early decades of the twentieth century in the United States. His two classic
water-supply papers (Meinzer, 1923 and 1928) are still reprinted and widely used today
(Freeze and Back, 1983). His major contribution to the understanding of the physics of
groundwater flow came in his 1928 paper on the compressibility of the artesian aquifers
wherein he invoked the effective stress principle. Meinzer recognized that the water in a
confined aquifer supports part of the overlying load and that aquifers compact when fluid
pressure is decreased. Although Terzaghi (1925) developed the basic concept of
effective stress in a laboratory soil column, Meinzer’s realization that the same concept
applied to aquifers was a major breakthrough.
Weber (1928) made a successful attempt to analyze the unsteady flow of water to
a fully penetrating gravity well in an unconfined aquifer for the first time. In the 1930s,
the results of mathematical and experimental studies in the petroleum reservoir
engineering field were utilized by researchers in the groundwater field. Muskat (1934)
presented a detailed analysis of transient flow of compressible fluids in oil and gas
reservoirs. In the field of groundwater hydrology, Theis (1935) set up and obtained a
solution to the parabolic equation of groundwater flow similar to that of Muskat (1934).
11
He verified the credibility of his model by applying it to Wenzel’s Grand Island,
Nebraska field data from an unconfined aquifer. Wenzel (1942) brought Theis’ theory
into practical use by publishing a table of the exponential function values that appeared in
Theis' solution. Theis’ work has been a milestone in groundwater hydrology and his
model is still used frequently today. Theis was careful in his paper to spell out the
assumptions on which his method was based, i.e., it applies to an idealized aquifer
configuration. The history of the subsequent development of the methodology of aquifer
hydraulics is largely a history of the systematic removal of his assumptions one by one.
Jacob (1946) extended Theis’ method to heterogeneous media when he published
a paper on radial flow to a leaky aquifer, which opened up a new area of research relating
to multiple aquifer systems in groundwater hydrology and petroleum engineering.
The auger-hole methods and piozemeter methods were pioneered by Kirkham and
coworkers (Kirkham, 1946; Luthin and Kirkham, 1949; van Bavel and Kirkham, 1948).
These methods improved the estimation of the hydraulic conductivity of the saturated soil
below the water-table and are still being used.
Boulton (1954) pioneered the analysis of unconfined aquifers. He investigated the
transient flow of water to a well in an unconfined aquifer. Instead of solving the highly
complex flow process in the unsaturated zone embodied in Richards’ equation, Boulton
simplified the effect of the unsaturated zone by introducing an empirical constant that
accounted for the delayed yield from the storage. As an approximation, he assumed that
the drainage from the unsaturated zone was an exponential function of time. The
resulting governing equation was solved for potentials within the saturated domain, while
yet approximately accounting for the contribution from the unsaturated zone by means of
12
a time dependent source term. His model still continues to be used by hydrogeologists
with minor modifications and extensions (Narasimhan, 1998a).
The effects of anisotropy and heterogeneity of the aquifers on flow were
investigated by Maasland (1957). Maasland also outlined the relationships between
stratified heterogeneous systems and homogeneous anisotropic systems in his paper.
During the early 1960s, doubts were expressed about the validity of Jacob’s
development of the groundwater flow equation. The doubts were centered around the
fact that the effective stress laws he invoked assumed that only vertical stress existed. A
full analysis should have dealt with the interaction between a three-dimensional stress
field and a three-dimensional fluid flow field. Hydrogeologists discovered that Biot
(1941, 1955), a physicist working in a petroleum research institute, had already coupled a
three-dimensional stress field with the fluid -flow field. His work was interpreted in terms
of hydrogeological notation by Verruijt (1969) and placed in the context of earlier
groundwater development. In the mean time, De Wiest (1966) improved the Jacob
equation with respect to the variation of hydraulic conductivity with effective stress but
not with respect to the storage side of the equation. Cooper (1966) clarified the
relationship between the development of the flow equation in fixed coordinates and
deforming coordinates. Cooper concluded that Jacob’s equation was correct for almost
all practical applications. Cooper and a group of hydrogeologists led by him made many
contributions to groundwater hydrology. These include interpretation of data from a slug
test (Cooper et al., 1967), analysis of transient pressure data from an anisotropic aquifer
(Papadopulos, 1965), transient flow of water to a well of large diameter (Papadopulos and
Cooper, 1967), and the response of a well to seismic waves (Cooper et al., 1965).
13
The study of leaky aquifers was pioneered by Jacob and his student Hantush.
Hantush (1960) considered the effects of aquitard storage in his leaky aquifer solution.
Hantush (1964) provided a comprehensive summary of developments related to leaky
aquifers as well as other aquifer configurations in his paper “Hydraulics of Wells”. Toth
(1963) produced a set of analytical solutions to the steady-state boundary value problem
representing regional flow in a vertical profile. Neuman and Witherspoon (1969)
presented a complete solution that considers both water released from storage in the
aquitard and drawdowns in the hydraulic head in the unpumped aquifer.
A significant research milestone of the 1960s was the development of numerical
models. The era of the digital computer had started and computer development was
advancing with incredible rapidity. The digital computers provided the possibility of
solving transient flow problems in complex geological systems, which are impossible to
solve in closed form solutions. The early numerical solutions were based on the finitedifference method and the method of relaxation, both of which were known before the
advent of computers. Stallman introduced finite-difference concepts into the
hydrogeological literature in 1956. Much later, Nelson (1968) used the finite-difference
method to study the inverse problem studies. The finite-element method (Clough, 1960),
which was initially designed for solving structural engineering problems, was soon
adapted to solve steady-state and transient problems of groundwater flow (Javandel and
Witherspoon, 1968).
Remson et al. (1965) helped popularize the computer modeling approach by
developing a steady-state computer model to predict the effects of a proposed surfacewater reservoir on the heads in an unconfined regional aquifer. Freeze and Witherspoon
14
(1966) presented numerical solutions that allowed consideration of more complex watertable configurations and geologic environments.
By the early 1970s, computer simulation of aquifers was widely used in water
resources evaluations. This advance resulted largely from the development,
documentation, and availability of two aquifer simulation programs, the first by Pinder
and Bredehoeft (1968) of the U.S. Geological Survey, and the second by Prickett and
Lonnquist (1971) of the Illinois State Water Survey. The U.S. Geological Survey model
has been continually updated over the years. The first attempt to create a complete
hydrologic response model was made by Freeze and Harlan in 1969. Freeze (1971), who
was one of the pioneer numerical modelers, developed a three-dimensional variably
saturated transient groundwater flow model. His model was in finite-difference form and
used the line successive over relaxation method to solve the governing equation. The
finite-difference models of Freeze (1971) and Cooley (1971) however are not robust
because they incur numerical instabilities and convergence difficulties (Clement et al.,
1994). In the late 1970s, research emphasis was shifted from resources development
issues to environmental issues pertaining to chemical contamination. Since the
contaminant transport path typically goes through the unsaturated zone, soil scientists and
agricultural engineers began to investigate unsaturated soil characteristics and flow
processes in the vadose zone. Most of the researchers focused on unsaturated-saturated
flow problems. Neuman (1973), Brandt et al. (1971), and Haverkamp et al. (1977) are
among those researchers.
In the 1980s, topics such as leaky aquifers and unconfined aquifers gradually
receded from researchers’ focus of attention. Interest steadily grew in characterizing flow
15
processes in the vadose zone, which is the path between wastes deposited at the land
surface and the water-table at depth. In the latter part of the 1980s, the motivation was to
develop better computer models and to search for better numerical techniques to solve
governing nonlinear partial differential equations. Advancements in computer technology
eased the researchers’ job and motivated them to attempt to solve more complex,
challenging, and time consuming groundwater problems. Parallel to these advancements,
studies on numerical solution techniques increased rapidly. New numerical methods
were developed and applied in models. The boundary integral method (Liggett and Liu,
1983) and the analytic element method (Strack, 1989) were relatively new techniques that
were applied in models. Many sophisticated groundwater models were developed in the
late 1980s. The most well known of these groundwater models, MODFLOW, was created
by McDonald and Harbaugh (1988) of U.S. Geological Survey. MODFLOW is still
widely used by hydrogeologists.
In the 1990s, more challenging problems begun to be dealt with. Attempts were
made to couple variably saturated flow models, root water uptake models, and
groundwater models to simulate the complete hydrological process. With the help of
high-speed computers, hydrogeologists started modeling the surface-water groundwater
interaction process, and surface-water flow models were coupled with the groundwater
models. MODFLOW and BRANCH models were coupled by Swain and Wexler (1992)
of U.S. Geological Survey in 1992 to simulate non-steady river flow interaction with
groundwater in a successful coupled model referred to as MODBRANCH. Yeh et al.
(1996) developed a three-dimensional finite-element saturated unsaturated flow and
transport model. During the 1990s, contaminant transport and consequently unsaturated
16
saturated flow studies became very important because of increasing environmental
awareness and multi million dollar support of government agencies such as the U.S.
Department of Energy (DOE), U.S. Nuclear Regulatory Commission (NRC), U.S.
Environmental Protection Agency (EPA), and others. Contaminant transport is beyond
the scope of this dissertation. The studies for unsaturated flow and variably saturated
flow problems are described in the subsequent sections.
Another very important development was the introduction of Geographic
Information Systems (GIS) to water resources research in the 1990s. With the help of
GIS methods and the graphical interface programs, numerical models became very userfriendly in terms of input data and post processing of the output data. This new technique
provided an interactive environment in which model grids, spatially referenced to a base
map, can be generated on the computer screen and the model results can be seen on the
screen immediately. It provides the capability for modelers to create, apply, and revise
groundwater models quickly and in a way never possible before. The first example of
this type of model is GWZOOM by Yan and Smith (1995), who created a system based
on GIS that works interactively with MODFLOW. Application of GIS to groundwater
problems is a very rapidly growing research area today, and it will be one of the primary
interests of researchers in the 21st century.
Unsaturated Flow Studies
The unsaturated zone in the hydrologic cycle transmits water falling or ponded on
the land surface to underlying groundwater or temporarily stores water near the land
surface for plant use. The first researchers who dealt with the unsaturated zone were soil
17
physicists. Later, agricultural engineers investigated the behavior of the unsaturated zone
around the plant root zone above the water-table. Starting with Terzaghi (1925), civil
engineers and geotechnical engineers became interested in the unsaturated zone to deal
with seepage and ground settlement problems.
The first research about unsaturated flow dates back to the early twentieth century.
This was conducted by Edgar Buckingham (1867-1940), who was a physicist at the
Physical Laboratory of the Bureau of Soils, U.S. Bureau of Agriculture. His theoretical
and experimental studies on the dynamic movement of soil gases and soil moisture led to
a major contribution to the foundation of soil physics. His first paper was published in
1904, but his major contribution to unsaturated flow research was his second paper,
which was published in 1907. This paper reported the results of studies on the movement
of soil moisture. Based on the works of Fourier and Ohm, Buckingham rigorously
defined the concept of capillary potential and proposed that the steady flux of moisture
through an unsaturated soil is directly proportional to the gradient of the potential, with a
coefficient of proportionality being a function of capillary potential. The mathematical
form of this statement was much the same as that of Darcy’s law, except that the
parameter of proportionality was recognized by Buckingham to be a function of capillary
potential. It is remarkable that Buckingham, who was probably unaware of Darcy’s work
(Sposito, 1987) gave a theoretical basis for Darcy’s empirical law and extended the law to
the unsaturated zone. Buckingham provided a paradigm and unified the flow process in
the unsaturated and saturated zones. Some soil physicists persuasively argue that the
phrase “Darcy-Buckingham’s law” should be used in place of Darcy’s law (Narasimhan,
1998b). Buckingham appears to be the first researcher to address the transient movement
18
of water in the subsurface, and he is also widely known for developing the dimensional
analysis “pi theorem” (Buckingham, 1914).
At about the same time, Green and Ampt (1911) proposed a simple approximation
to quantify the vertical infiltration of water into an unsaturated soil. The Green and Ampt
idealization assumes that a sharp, piston-like zone of saturation advances with time as
water infiltrates into a soil. This approximation is still widely used.
Gardner and Widtsoe (1921) attempted to quantify the unsteady moisture
movement in unsaturated soils in terms of a transient diffusion equation analogous to
Fourier’s transient heat conduction equation. They did not achieve satisfactory agreement
between experiment and theory, because they did not account for the dependency of
hydraulic conductivity on capillary potential suggested a decade earlier by Buckingham.
They tried to fit experimental data to a linear partial differential equation, when in fact a
nonlinear parabolic equation should have been used. In 1924, Terzaghi experimentally
studied the deformation of water-saturated clays and established the relationship among
external stresses, pore-water pressure, and deformation. Although his paper is classified
under the soil mechanics discipline, he proceeded to write down and solve the equation
for transient movement of water in a one-dimensional clay column by analogy with the
heat conduction equation (Narasimhan, 1988a). Tensiometers had become well
developed by the efforts of Willard Gardner and his coworkers in the late 1920s. Gardner
et al. (1922), in an abstract, published the first reference to the tensiometer (Narasimhan,
1998a), an instrument that has played a vital role in the evaluation of modern soil physics.
Because of the tensiometer, routine measurements of moisture-content and its relation to
capillary pressure had become possible (Richards, 1928). Combining Buckingham’s
19
(1907) work on the equation of water motion in unsaturated soils with the newly available
soil moisture retention curves, Richards (1931) formally wrote down, for the first time,
the nonlinear partial differential equation describing transient flow of water in unsaturated
soils. He defined the moisture capacity as the slope of the moisture-content versus
capillary pressure curve. The Richards equation remained unsolved for nearly two
decades because of its nonlinearity. Klute (1952), Philip (1955), and others began to
obtain solutions for the Richards equation under highly simplified conditions using
numerical methods in the early 1950s.
Richards et al. (1956) demonstrated a method by which the hydraulic conductivity
function could be estimated in the field by measuring the depth profile of gauge pressure
head as well as moisture-content as a function of time during the redistribution of soil
moisture immediately following an infiltration event. In this experiment, the soil
moisture distribution was measured rather laboriously by the gravimetric method.
Gardner and Kirkham (1952) used neutron scattering to estimate quantitatively the soil
moisture, and this process developed into a workable field neutron probe, which is very
useful in measuring the soil moisture profile. During the 1960s, the field method of
Richards and his coworkers was improved by other researchers by utilizing the neutron
probe.
Gardner (1957) found that there is an exponential relationship between the
hydraulic conductivity and gauge pressure head over a limited range of gauge pressure.
This relationship made it possible for him to solve the Richards equation. The works of
Gardner (1957) and Philip (1955) continue to influence present day research relating to
hydraulic characterization of unsaturated soils in the field. Veihmeyer and Hendrickson
20
(1955) presented a comprehensive literature review about the relationship between
transpiration and the soil moisture. In 1957, Philip published his famous “Theory of
Infiltration” series in which he developed an infiltration equation based on the Richards
equation and Klute’s (1952) equations for finite and infinite soil profiles. In 1957,
Gardner developed several steady-state solutions of the unsaturated moisture equation
with the application to evaporation from a water-table. Gardner (1960) did another study
about the plant root and soil moisture interaction and its dynamic behavior. The relation
of the root distribution to water uptake as a function of soil suction and water availability
was described by Gardner (1964). Brooks and Corey (1964) developed analytical
expressions to define the relationship between unsaturated hydraulic conductivity and soil
moisture-content based on the statistical predictive conductivity model of Burdine (1953).
Brooks and Corey (1964) obtained fairly accurate results with their equations.
In the 1970s, research continued to find better and more effective solutions to the
Richards equation to describe the infiltration process in the soil. Ripple et al. (1972)
investigated the relationship between bare soil evaporation and a high water-table.
Meteorological and soil equations of water transfer were combined in order to estimate
approximately the steady-state evaporation from bare soil under conditions of a high
water-table. Warrick (1975) described a one-dimensional linearized analytical solution to
the moisture flow equation for arbitrary input using simplified boundary conditions.
Mualem (1976) derived a new model for predicting the hydraulic conductivity based on
the soil-water retention curve and the conductivity at saturation. Mualem’s derivation
leads to a simple integral formula for the unsaturated hydraulic conductivity, which
makes it possible to derive closed-form analytical expressions, provided suitable
21
equations for the soil-water retention curves are available. van Genuchten (1980)
developed a closed-form equation for predicting the unsaturated hydraulic conductivity
using Mualem’s (1976) model. van Genuchten’s closed form expression is still used
widely by hydrogeologists.
Haverkamp et al. (1977) developed one-dimensional moisture-content based
numerical models to solve the Richards equation for infiltration. Six different numerical
models were developed and compared to each other in terms of numerical errors and
computer time requirements. Those models were verified using experimental values and a
comparison was made between one of the six models and a calculated analytical solution
of Philip (1957) and Parlange (1971).
During the 1980s, groundwater contamination came into sharp focus because of
leaky gasoline tanks and other industrial wastes. Several government agencies gave large
financial support to contaminant transport studies, which require unsaturated flow
analysis. This motivation increased the number of investigations concerned with vadose
zone hydrology. The use of numerical models for simulating fluid flow and mass
transport in the unsaturated zone became increasingly popular. A lot of effort was made
to develop these kinds of models. A comprehensive list of numerical codes for singlephase (water) flow in the vadose zone is given by Stephensen (1995).
Knoch et al. (1984) developed a one-dimensional physically based computer
model for predicting direct recharge to groundwater. Yeh et al. (1985) presented the
results of a stochastic analysis of unsaturated flow in heterogeneous soils. Results of their
stochastic theory for flow in heterogeneous soils were compared with experiments and
field observations. Effects of anisotropy on recharge, irrigation and surface runoff
22
generation, and waste isolation were discussed in the paper. Broadbridge and White
(1988) presented an analytical solution for a nonlinear diffusion-convection model
describing constant rate rainfall infiltration in uniform soil and the application of the
solutions (White and Broadbridge, 1988). Hills et al. (1989) developed a onedimensional model for infiltration into very dry soil. Numerical instability and
convergence problems caused by a sharp wetting front in very dry soil conditions were
dealt with. A two-step Crank-Nicholson procedure was developed, in which the first step
estimates the material properties and the second step uses the temporal averages of these
properties to calculate the unknown pressure head or moisture-content.
In the 1990s, more complex geometric situations such as layered and
heterogeneous soils, and variably saturated, multidimensional problems were
investigated. Warrick and Yeh (1990) presented a one-dimensional solution for a layered
soil profile. Ross (1990) developed efficient numerical methods for infiltration using the
Richards equation, proposing the "Advancing Front Method" as a better method for
infiltration redistribution and drainage problems. Celia et al. (1990) developed a general
mass-conservative numerical solution for the unsaturated flow equation. They reported
that the pressure based form of the Richards equation generally yields poor results and is
characterized by large mass balance errors. Conversely, mass is perfectly conserved in
numerical solutions based on the mixed (head and moisture-content) form of the Richards
equation.
Yeh and Harvey (1990) investigated various approaches for determining effective
conductivity values for heterogeneous sands and compared them to laboratory
measurements. Warrick et al. (1990, and 1991) and Broadbridge and Rogers (1990)
23
presented analytical solutions for steady and transient infiltration processes by solving the
Richards equation. An analytical solution for one-dimensional, transient infiltration in
homogeneous and layered soils was developed by Srivastava and Yeh (1991) using
exponential functions describing hydraulic conductivity, pressure head, and moisturecontent. Paniconi et al. (1991) evaluated iterative and non-iterative methods for the
solution of the Richards equations. They presented four different non-iterative solution
techniques and concluded that a second order accurate two-level "implicit factored" noniterative technique is a good alternative to iterative methods. Barry et al. (1993)
developed a class of exact analytical solutions for the Richards equations. Tracy (1995)
developed a three-dimensional analytical solution for unsaturated flow using a simplified
boundary condition. Chang and Corapcioglu (1997) presented a study on the effects that
roots have on water flow in unsaturated soils and included a root distribution model.
Variably Saturated Flow Studies
The necessity of modeling variably saturated flow was brought about by drainage
problems, which include both saturated and unsaturated flow. The first, relatively simple
variably saturated flow research was done in the field of agricultural engineering in the
late 1950s. These studies were limited to the one-dimensional drainage problems (e. g.,
Day and Luthin, 1956).
Starting in the late 1960s, rapidly developing computer technology motivated
researchers to analyze the entire hydrologic cycle. Sophisticated numerical methods and
high-speed computers made modeling of the entire geologic formation possible, from
ground surface to the impermeable bottom of a confined aquifer. The solution of the
24
highly nonlinear governing equation with very complex boundary conditions and
heterogeneous geologic formations combined with the transient nature of the problem,
required using very powerful high-speed computers.
Finite-difference approximations were widely used in several early studies. Most
of these variably saturated models were one-dimensional (e. g., Freeze and Harlan, 1969).
Transient numerical models that integrate the saturated and unsaturated zones
were pioneered by Rubin (1968), who developed the first multidimensional variably
saturated model. He developed a model using Darcy's flow equation (but it was actually
the Richards equation) for two-dimensional, transient water movement in a rectangular
unsaturated or partly unsaturated soil domain. He used alternating direction and implicit
difference methods. His paper was followed by several other two-dimensional
applications to various problems, i.e., Hornberger et al. (1969), Taylor and Luthin (1969),
Verma and Brutsaert (1970), and Cooley (1971). These studies used the Laplace equation
in the saturated zone and are thus limited to near-surface flow in homogeneous,
incompressible, and unconfined aquifers. All these studies were for small regions with
simplified boundary configurations. In late 1960s, only Jeppson (1969) considered a
variably saturated flow regime on a basin wide scale but with the restriction of steadystate condition.
In 1971, a remarkable three-dimensional transient, saturated-unsaturated flow
model was developed by Freeze (1971). The model was designed as a regional model
applicable to any groundwater basin. The complete subsurface regime was treated as a
unified whole by solving the variably saturated flow equation in the unsaturated zone and
the saturated flow equation in the underlying unconfined and confined aquifers. This
25
was the first complete three-dimensional hydrologic response model. Jacob’s (1940)
equation, as clarified by Cooper (1966), as a saturated equation and the Richards (1931)
equation as an unsaturated equation were combined and solved in terms of the pressure
head in deforming coordinates to take into account the compressibility of the formation.
In the same year, Cooley (1971) developed a finite-difference method for unsteady flow
in variably saturated porous media. He applied his model to a single pumping well
successfully. According to Wise et al. (1994), the Freeze (1971) and Cooley (1971)
models are not robust because they incur numerical instabilities and convergence
difficulties.
In most applications, the pressure-based form of the variably saturated flow
equation is used. Celia et al. (1990) and Kirkland (1991) claimed that the numerical
solution of the pressure-based Richards equation has poor mass-balance properties in the
unsaturated zone.
Brutsaert (1971) developed a two-dimensional model by solving the mixed form,
i.e., the pressure head and moisture-content based, Richards equation, using the finitedifference method. Neuman (1973) developed a numerical model (UNSAT2) similar to
Rubin (1968), but it was a finite-element model. Narasimhan et al. (1978) created
TRUST, which is pressure-based for variably saturated flow and moisture-content based
for unsaturated flow problems. An integrated finite-difference method with a mixed
explicit -implicit time stepping procedure was used in TRUST.
In the 1980s, research studies on groundwater and solute transport increased
greatly. In this period, many remarkable research papers were published and the most
well known groundwater models MODFLOW and FEMWATER were created. van
26
Genuchten (1980) presented his closed form equation describing the relationship between
the hydraulic conductivity and pressure head. Yeh and Ward (1980) developed the twodimensional finite element variably saturated code called FEMWATER, which was
updated as a three-dimensional finite element model by Yeh and Cheng (1994) as
3DFEMWATER. Cooley (1983) presented his two-dimensional variably saturated finite
element model. In that paper, a new method for locating the position of a seepage face
was presented. Huyakorn et al. (1984 and 1986) developed two- and three-dimensional
finite element variably saturated flow models, respectively. In those models, the lower
and upper (LU) decomposition method, Newton-Raphson method, Picard iteration
schemes, and strongly implicit procedures were used. Voss (1984) of the U.S. Geological
Survey developed a two-dimensional finite-element simulation model called SUTRA for
saturated-unsaturated, fluid density-dependent groundwater flow with energy transport.
van Genuchten and Nielsen (1985) modified the original van Genuchten (1980) formula
to allow a non-zero value of specific moisture capacity in the saturated zone, which
makes the formula very useful for variably saturated flow models. Allen and Murphy
(1986) presented a variably saturated model that solved the mixed form of the Richards
equation using the finite-element method and the Gauss elimination technique during
iterations.
Kuiper (1986) compared seventeen different methods for solution of the
simultaneous nonlinear finite-difference approximating equations for groundwater flow in
a water-table aquifer in three-dimensions. The best methods were found to be those using
Picard iteration implemented with the preconditioned conjugate gradient method.
27
Lappala et al. (1987) developed a computer model, VS2D, for solving problems of
variably saturated, single-phase flow in porous media in two dimensions. Nonlinear
boundary conditions treated by the model included infiltration, evaporation, seepage
faces, and water extraction by plant roots. Subsequently, Healy (1990), who was one of
the co-authors of VS2D, added a solute transport capability to the VS2D and named the
new model VS2D. In 1986, another comprehensive saturated unsaturated flow model
named SHE was developed in Europe by a multinational group consisting of the Danish
Hydraulic Institute, SOGREAH of France, and the British Institute of Hydrology (Abbott
et al., 1986 a, b). SHE was revised and developed as MIKE SHE in 1995 by Refsgaards
and Storm (1995) of the Danish Hydraulic Institute. MIKE SHE is a complete hydrologic
system model that simulates overland and channel flow, snowmelt, evapotranspiration,
and saturated-unsaturated flow.
In the 1990s, research interests were focused on developing more numerically
stable, fast converging, and more accurate numerical methods to solve complex,
nonlinear partial differential equations. Another area of interest was finding exact
analytical solutions to the nonlinear Richards equation. During this period, a large
number of papers were published dealing with mathematical solution techniques,
coupling of surface-water and groundwater techniques, and GIS application techniques in
groundwater hydrology.
Kirkland et al. (1992) presented a successful and efficient example of a finitedifference solution to two-dimensional, variably saturated flow problems. However, the
objective of Kirkland et al. was the development of competitive numerical procedures to
solve infiltration problems in very dry soils. Thus, they did not take into account the
28
effects of specific storage in their fundamental flow equation, and, consequently, their
model cannot be used to model accurately a wide variety of variably saturated flow
problems (Clement et al., 1994).
Clement et al. (1994) presented an algorithm for modeling variably saturated flow
in the two-dimensional finite-difference form. The mixed form of Richards equation was
solved in finite-differences using a modified Picard iteration scheme to determine the
temporal derivative of water content. Wise et al. (1994) presented a sensitivity analysis
of the same variably saturated flow model to soil properties. They showed that the
location of the phreatic surface and height of the seepage face are functions of the
capillary forces exerted in the vadose zone.
Yan and Smith (1994) attempted to integrate the South Florida Water
Management Model (SFWMM) (MacVicar et al., 1983) and MODFLOW to simulate the
hydrogeologic system of south Florida. They presented the algorithm of the proposed
model that is conceptualized and constructed with a reasonable level of detail regarding
the simulation of surface-water movement, groundwater movement, and interactions
between the surface-water and groundwater systems. The movement of water outside of
the aquifer is simulated using SFWMM, and the water movement within the aquifer
system is simulated using MODFLOW. The models are linked by processes that include
recharge, infiltration, changes in soil moisture in the unsaturated zone, evapotranspiration
from the unsaturated and saturated zones, and flow between surface-water bodies and the
aquifer as recharge or discharge. No further action has been taken to create the real
model, and it has remained as a proposal (personal communication with Yan, August,
1996). Their evapotranspiration and infiltration formulation was implemented in MIKE
29
SHE as an alternative to the complex ET modules of MIKE SHE to use in South Florida
Water Management District projects (Yan et al., 1998).
Clement et al. (1996) compared modeling approaches for steady-state unconfined
flow. The Dupuit-Forchheimer, the fully saturated flow, and the variably saturated flow
models were compared for problems involving steady-state unconfined flow through
porous media. The variably saturated flow model was found to be the most
comprehensive of the three.
Parallel to those studies mentioned above, a vast amount of research exists
concerning evapotranspiration calculations. Although evapotranspiration is an important
element of the saturated-unsaturated flow model, the theory of evapotranspiration and
development of the evapotranspiration models are beyond the scope of this dissertation.
In this study, one of the most widely accepted and physically based evapotranspiration
models, i.e., the Priestly-Taylor (1972) model, was selected for use in the numerical
model.
Available Hydrologic Computer Models
In this section, widely used, numerical saturated-unsaturated groundwater flow
models that provided insight and guidance in the development of the model in this study
are discussed briefly (Table 2.1). MODFLOW is included in this discussion because of
its very well organized modular structure and multi-layer aquifer simulation capability in
the saturated zone, although it is a saturated flow model only.
MODFLOW. MODFLOW is a three-dimensional finite-difference ground-water
flow model (McDonald and Harbaugh, 1988). It has a modular structure that allows it to
30
be modified easily to adapt the code for a particular application. Many new capabilities
have been added to the original model of 1988.
MODFLOW simulates steady and unsteady flow in an irregularly shaped flow
system in which aquifer layers can be confined, unconfined, or a combination of confined
and unconfined. Flows from external stresses such as flow to wells, areal recharge,
evapotranspiration, flow to drains, and flow through river beds can be simulated.
Hydraulic conductivities or transmissivities for any layer may differ spatially and be
anisotropic. However, they are restricted to having the principal direction aligned with
the grid axes. The storage coefficient may be heterogeneous, and the model requires
input of the ratio of vertical hydraulic conductivity to distance between vertically adjacent
block centers (vertical conductance). Specified head and specified flux boundaries can be
simulated.
In MODFLOW, the ground-water flow equation is solved using the finitedifference approximation. The flow region is considered to be subdivided into blocks in
which the medium properties are assumed to be uniform. In the vertical direction, zones
of varying thickness are transformed into a set of parallel "layers". The associated matrix
problem can be solved by choosing one of several solver routines that are available, i.e.,
the strongly implicit procedure, slice successive over relaxation method, and
preconditioned conjugate gradient method. Mass balances are computed for each time
step and as a cumulative volume from each source and type of discharge.
In order to use MODFLOW, initial conditions, hydraulic properties, and stresses
must be specified for every model cell in the finite-difference grid. The primary output is
31
hydraulic head. Other output includes the complete listing of all input data, drawdowns,
and water-budget data.
Table 2.1 Summary of selected saturated-unsaturated flow models
Title
MODFLOW
Developer
3-D finite-difference, distributed,
saturated groundwater flow model.
Boussinesq equation is solved.
Yeh et al., 1996
3-D finite-element, distributed,
saturated-unsaturated flow and
transport model. Pressure based
Richards equation is solved.
Skaggs, 1980
Lumped model, approximate
methods are used. Variably saturated,
designed for drainage and irrigation
problems
2-D finite-element, distributed,
variably saturated flow and transport
model. General variably saturated
density dependent flow equation is
solved.
2-D finite-difference, distributed flow
and transport model. Pressure based
form of general variably saturated
flow equation is solved.
Modified VS2DT with coupled ET
calculations and surface-water
groundwater interaction modules. 2D Richards equation is solved.
3-D finite-difference, distributed,
saturated-unsaturated model. ET
calculations, surface-water
interactions are considered. Richards
and Boussinesq equations are solved.
3-D finite-element and finitedifference variably saturated flow
and transport model for fractured
porous media. Mixed form of the 3D Richards equation is solved.
1-D finite-element variably saturated
flow and solute and heat transport
model. Mixed form of the 1-D
Richards equation is solved using a
new convergence criterion.
FEMWATER
DRAINMOD
Voss, 1984
SUTRA
VS2DT
Lappala et al.,
1987 and Healy,
1990
Bloom et al.,
1995
WETLANDS
MIKE SHE
Distinct features
McDonald and
Harbaugh, 1988
Abbott et al.,
1986 a, b, and
Refsgaard and
Storm, 1995
FRAC3DVS
Therrien, and
Sudicky, 1996.
HYDRUS
Vogel et al.,
1996.
Limitations
No unsaturated flow
modeling, requires user
supplied ET and recharge
values.
No root water uptake
feature, requires user
supplied net rainfall,
infiltration capacity, and
ET.
Not distributed, no 3-D
simulation, and no
confined aquifer or
pumping.
No 3-D simulation, lack of
ET and surface-water
groundwater interaction.
No 3-D simulation, poor
surface-water groundwater
interaction.
No 3-D simulation.
Unsaturated zone is 1-D
vertical. Difficulties exist
in coupling 3-D saturated
and 1-D unsaturated zone
modules.
No ET calculations or
surface water groundwater
interaction.
No 2-D or 3-D simulations.
32
The main limitation of the model is that it lacks the capability to simulate flow in
the unsaturated zone. Although MODFLOW can simulate evapotranspiration, recharge,
areal recharge, and river-groundwater interaction, some of the model parameters have to
be provided by the user to compensate for the lack of unsaturated zone simulation. These
parameters includes the maximum evapotranspiration rate, net recharge to the watertable, etc. Therefore, the model may give erroneous results because of the user defined
parameters.
FEMWATER. FEMWATER is a 3D flow and contaminant transport finiteelement density-driven coupled or uncoupled model used to simulate both saturated and
unsaturated conditions (Yeh et al., 1996). FEMWATER was formed by combining two
older models, 3DFEMWATER (flow) and 3DLEWASTE (transport), which were written
by Yeh and Cheng (1994), into a single coupled flow and transport model.
The governing equation for flow is the three-dimensional Richards equation
modified to include a density dependent flow term and to consider consolidation of the
aquifer. There are four types of iteration methods for solving the linearized matrix
equations of the governing equation, i.e., successive point and block iteration, polynomial
preconditioned conjugate, and incomplete Choleski preconditioned conjugate gradient
methods.
The model requires identification of material properties representing the
hydrogeologic and transport characteristics of soil contained within the model. The
moisture-content, relative conductivity, and water capacity versus pressure head curves
should be supplied to the model. Initial conditions and four kinds of boundary conditions,
namely Dirichlet, Neumann, Cauchy, and variable boundary conditions, can be assigned
33
by selecting nodes or element faces. Features such as wells, constant head, and no-flow
boundaries can be defined. Transient data (such as recharge or well pumping), which is
typically available in hydrograph form, can be input and edited graphically. These data
can then be interactively assigned to a single element or a series of elements. The model
outputs are pressure head, moisture-content, Darcy velocity, and concentration values at
each node.
FEMWATER has no capability to calculate the evapotranspiration losses, which
have to be supplied to the model separately. The model does not calculate the rainfallrunoff process. Therefore, the net precipitation has to be supplied to the model as a
boundary condition. Vegetative cover, precipitation, and evaporation interactions are not
considered in the model. The evapotranspiration losses calculated outside of the model
can be applied only at the ground surface as a boundary condition, and transpiration
losses below the ground surface around the root zone are not considered.
DRAINMOD. DRAINMOD is a lumped hydrologic simulation model developed
by Skaggs (1980). The model simulates the hydrology of poorly drained, high water-table
soils on an hour-by-hour, day-by-day basis for long periods of the climatological record
(e.g., 40 years). The model predicts the effects of drainage and associated water
management practices on water-table depths, the soil-water regime, and crop yields.
Initially, DRAINMOD was used as a research tool to investigate the performance
of a broad range of drainage and subirrigation systems and their effects on water use, crop
response, land treatment of wastewater, and pollutant movement from agricultural fields.
The specific objectives of DRAINMOD are to simulate the performance of water-table
management systems and to simulate lateral and deep seepage from the field.
34
DRAINMOD was developed for field-sized units with parallel subsurface drains. Most
of the hydrologic components considered in the water balance are formulated in the
model. DRAINMOD uses approximate methods to characterize the rates of infiltration,
drainage, evapotranspiration, and the distribution of soil water in the profile instead of
using numerical solutions to nonlinear differential equations. However, the estimates
provided by this approximate method are comparable to exact methods. A general flow
chart for DRAINMOD is given by Skaggs (1980). The following are the model
components:
1. Precipitation (hourly data is suited);
2. Infiltration: the Green-Ampt equation is used to compute infiltration;
3. Surface drainage: the average depth of depression storage, which must be
satisfied before runoff;
4. Subsurface drainage: the rate of subsurface-water movement into drain tubes or
ditches;
5. Subirrigation;
6. Evapotranspiration;
7. Soil water distribution; and
8. Rooting depth.
The input data can be summarized as follows: soil property inputs; hydraulic
conductivity; soil-water characteristics; drainage volume - water-table depth relationship;
upward flux; Green-Ampt equation parameters; trafficability parameters; crop input data;
drainage system parameters; surface drainage; and effective drain radius. Outputs are:
yearly rankings of parameters such as number of working days and relative yields; daily
35
and monthly summaries of many of the output parameters; relative yield; and daily watertable depths and drainage volumes for each year simulated.
Sensitivity analyses were conducted for different soils and water management
systems of North Carolina. The results are presented in the DRAINMOD reference report
(Skaggs, 1980). These results indicate that errors in the hydraulic conductivity (K) have
the greatest effect on predicting the water-table depth and water content of the soil
profile. DRAINMOD model was tested for use in humid and semi-arid climatic regions.
Skaggs (1980) reported the following limitations of this model:
1. DRAINMOD should not be applied to situations that are widely different from
conditions for which it was developed, without further testing; and
2. The field should have parallel subsurface drains.
In addition to those limitations, the model is not distributed so that heterogeneity
in the field can not be simulated. DRAINMOD uses approximate methods, which
requires extensive efforts to find appropriate coefficients for different geological
formations. The model was not designed to model confined aquifers, and lateral
movement of moisture in the unsaturated zone was not considered in the model.
SUTRA. SUTRA is a two-dimensional finite-element simulation model for
saturated-unsaturated, fluid-density-dependent ground-water flow with energy transport
or chemically-reactive single-species solute transport (Voss, 1984). SUTRA can be used
for areal and cross-sectional modeling of saturated ground-water flow systems and for
cross-sectional modeling of the unsaturated flow zone. SUTRA can also be used to
simulate solute transport to model natural or man-induced chemical species transport
including processes of solute sorption, production, and decay, and it may be applied to
36
analyze ground-water contaminant transport problems and aquifer restoration designs. In
addition, solute transport simulation with SUTRA may be used for cross-sectional
modeling of saltwater intrusion in aquifers in near-well or regional scales.
The model employs a two-dimensional hybrid finite-element and integrated-finitedifference method to approximate the governing equations. These equations describe the
two interdependent processes that are simulated: (1) fluid density-dependent saturated or
unsaturated ground-water flow, and (2) transport of a solute in the ground water and solid
matrix of the aquifer.
Important limitations of the program are: it is two-dimensional, which is not
convenient to regional modeling of aquifers; and it does not have any capability to
calculate evapotranspiration losses and overland flow. The model’s main emphasis is
solute transport and variable density flow, and it is not designed to simulate the complete
hydrological system such as extensive pumping from multiple aquifers that are highly
interactive with a river system and overland flow.
VS2DT. VS2DT solves problems of water and solute movement in variably
saturated porous media. The origin of the VS2DT is VS2D developed by Lappala et al.
(1987). Healy (1990) added a transport module and renamed the model as VS2DT. The
finite-difference method is used to approximate the flow equation, which is developed by
combining the law of conservation of fluid mass with the Darcy-Buckingham equation
and the advection-dispersion equation. The model can analyze problems in one and two
dimensions with planar or cylindrical geometries. There are several options for using
boundary conditions that are specific to flow under unsaturated conditions. There are
infiltration with ponding, evaporation, plant transpiration, and seepage faces. Solute
37
transport options include first-order decay, adsorption, and ion exchange. Extensive use
of subroutines and function subprograms provides a modular code that can be easily
modified for particular applications.
For the flow equation, spatial derivatives are approximated by the centraldifference method. Time derivatives are approximated by a fully implicit backwarddifference scheme. Nonlinear conductance terms, boundary conditions, and sink terms
are linearized implicitly using previous iteration step values. The relative hydraulic
conductivity is evaluated at cell boundaries by using full upstream weighting, the
arithmetic mean, or the geometric mean of values from adjacent cells. Saturated hydraulic
conductivities are evaluated at cell boundaries by using distance-weighted harmonic
means. Nonlinear conductance and storage terms can be represented by algebraic
equations or by tabular data.
The model requires initial conditions be input in terms of pressure heads or
moisture-contents for flow simulations and concentrations for transport simulations.
Hydraulic and transport properties of the porous media are also required. Flow
simulations require values for saturated hydraulic conductivity and for relative hydraulic
conductivity and moisture-content as functions of pressure head. Transport simulations
require values for dispersivity and molecular diffusion. Simulation results can be output
in terms of pressure head, total head, volumetric moisture-content, velocities, and solute
concentrations.
The main shortcomings of the model are that it can not simulate threedimensional problems. Also, it does not consider stream flow-groundwater interaction, it
38
does not consider interception losses from rainfall, and it cannot simulate multi-aquifer
systems.
WETLANDS. WETLANDS is a mathematical model for one- or twodimensional water flow and solute movement in variably- saturated multi-layered porous
media featuring optional surface-water bodies (ponds) and multiple root zones (Bloom et
al., 1995). The Richards equation and the advective-dispersive equation are solved
numerically using a finite-difference approximation, and the interaction of water levels in
the ponds with the surrounding soils are continually and dynamically adjusted.
A Priestly-Taylor model is used to simulate evapotranspiration by one to three
plant species. The controlling parameters can be specified to track seasonal variation in
sunlight, temperature, and rainfall. Solute transport can be affected by plant uptake,
passive sinks, or a variety of sorption phenomena as well as by water transport in the soil
and ponds.
WETLANDS is a finite-difference model using the strongly implicit method to
simulate water and solute transport. WETLANDS is a descendant of VS2DT, but it has
been modified to support simulations of shallow systems featuring ponds (or lakes, rivers,
etc.) with coupled evapotranspiration of multiple plant species. Output can consist of
matrices of head and solute concentration, temporal traces of stream flow and solute
concentration, mass balance monitoring, and data-sets depicting the water-table at any
given time.
The limitations of WETLANDS are that it is a two-dimensional model, and it
cannot simulate multi-aquifer systems or sink/sources such as pumping, drains, and
springs.
39
MIKE SHE. MIKE SHE is a distributed, physically based, three-dimensional,
finite-difference saturated unsaturated hydrological system model (Refsgaard and Storm,
1995). The MIKE SHE model was derived from the SHE model (Abbott et al., 1986a
and 1986 b). The model is applicable to a wide range of water resources problems related
to surface-water and ground-water management, contamination, and soil erosion. It is
designed as a modular structured model so that it can be easily modified or expanded. It
has the following components: evapotranspiration (ET), unsaturated zone flow (UZ),
saturated zone flow (SZ), overland and channel flow (OC), and irrigation module (IR)
components.
Different time scales can be used for different flow processes throughout the
simulation. For example, smaller time steps can be used in the unsaturated zone than are
used in the saturated zone. However, the UZ, ET, and OC modules use identical time
scales. This feature saves computer memory and enables faster simulations.
The UZ module plays an important role in the MIKE SHE, because all the other
components depend on boundary data from the UZ module. The flow is one-dimensional
vertically in UZ module. The governing equation for flow is the one-dimensional form of
the Richards equation. The UZ includes root extraction for the transpiration process,
which is explicitly incorporated in the equation by sink terms. The integral of the sinks
over the entire root zone depth gives the total amount of actual evapotranspiration. If the
root zone is homogeneous in certain regions, then the UZ calculations are only performed
in one representative column within those regions, and then lumped together for each
homogeneous region. The relationship between the moisture-content and pressure head
and hydraulic conductivity is a necessary input to the UZ module.
40
The ET module uses meteorological and vegetative input data to predict the total
evapotranspiration and net rainfall amounts. In the calculation of net rainfall amounts, the
processes of interception by the canopy, drainage from the canopy, evaporation from the
canopy surface, evaporation from the soil surface, and plant root water-uptake are
considered. An evapotranspiration model developed by Kristensen and Jensen (1975) is
used in the ET module.
The OC model is designed to route the excess ponded water on the ground-surface
towards the river system. The exact route and quantity is determined by the topography
and flow resistance as well as the losses due to evaporation and infiltration along the flow
path. Both the overland flow and the channel flow are modeled by approximations of the
St. Venant equations.
The SZ component of MIKE SHE calculates the saturated subsurface flow in the
catchment by solving the quasi-three-dimensional Boussinesq equation. MIKE SHE
allows for a fully three-dimensional flow in a heterogeneous aquifer with shifting
condition between unconfined and confined conditions. The spatial and temporal
variations of the hydraulic head are described by the nonlinear Boussinesq (1868)
equation and solved numerically by an iterative finite-difference technique. Successive
over relaxation and the preconditioned conjugate gradient solution techniques are
available in the model. In structure and flexibility, the SZ module is similar to
MODFLOW.
There is a difficulty in the linkage between the unsaturated zone and the saturated
zone in MIKE SHE because the UZ and SZ components run parallel, and thus they are
not solved in an integrated form. This difficulty has been solved by using an iterative
41
procedure based on mass balance calculations for the entire column including horizontal
flows in the saturated model. Because of the one-dimensional structure of the UZ
module, horizontal moisture movement in the unsaturated zone cannot be simulated in
the model. The model uses two different governing equations in the unsaturated and
saturated zones although one equation, the Richards equation, could be used for both
zones.
GMS (Groundwater Modeling System). GMS is not a groundwater model but,
instead, it is a groundwater modeling environment developed by the Department of
Defense (Yeh et al., 1996). GMS integrates and simplifies the process of groundwater
flow and transport modeling by seamlessly integrating all the tools needed for a
successful study. GMS supports the following models: MODFLOW, MODPATH,
MT3D, FEMWATER, SEEP2D, and RT3D.
FRAC3DVS. FRAC3DVS was developed by Therrien and Sudicky (1996). It is a
three-dimensional finite-element model that simulates saturated-unsaturated groundwater
flow and solute heat transport in porous or discretely fractured porous media. Galerkin
finite-element or finite-difference schemes can be selected. A conjugate-gradient like
solver is used to solve the systems of equations, and a full Newton-Raphson iteration
scheme is used to linearize the non-linear mixed form of the Richards equation.
HYDRUS. HYDRUS is a one-dimensional variably saturated groundwater flow
and solute and heat transport model developed by Vogel et al. (1996). It solves the mixed
form of the Richards equation using a new convergence criterion (Huang et al., 1996) to
speed up the iterative solution process. The model allows hysteresis to occur in both the
42
soil-water retention and hydraulic conductivity functions. It can simulate root-water
uptake, and it also includes heat transfer and heat movement simulations.
CHAPTER 3
DERIVATION OF THE VARIABLY SATURATED GROUNDWATER FLOW
EQUATION
General Three-Dimensional Saturated-Unsaturated Groundwater Flow Equation
Conceptualization
A general three-dimensional saturated-unsaturated groundwater flow equation can
be derived by considering the hydrological events and parameters that are depicted in
Figure 3.1. Using the concept of conservation of mass in a hydrological system such as
the one shown in Figure 3.1, the governing equation for groundwater flow can be derived.
Figure 3.1. Conceptualization of hydrologic system.
43
44
The conservation of mass concept considers that the sum of the inputs to the
system minus the sum of the outputs from the system is equal to change of mass in the
system per unit time. The mathematical description of the conservation of mass can be
described in terms of a unit volume taken from an interior location in the groundwater
system (Figure 3.2).
(qz)out
(qy)out
∆x
(qx)out
∆z
(qx)in
∆y
(qy)in
(qz)in
Figure 3.2 Representative unit volume of an aquifer.
45
Continuity Equation
The general groundwater flow equation is developed based on the mass continuity
(mass conservation) equation. The mass continuity equation can be written for a unit
volume of an aquifer (Figure 3.2) as
I−O±W =
dS
dt
(3.1)
where I is the mass inflow rate in the x, y, and z directions [MT-1], O is the mass outflow
rate in the x, y, and z directions [MT-1], W is a sink/source term representing the mass of
water injected into or removed from the aquifer per unit time [MT-1], and dS/dt is the
change in mass storage (S) per unit time [MT-1].
The mass inflow rates at x, y, and z in the x, y, and z-directions respectively are
I x = (ρ q x )dz dy
(3.2a)
I y = (ρ q y )dx dz
(3.2b)
I z = (ρ q z )dx dy
(3.2c)
where ρ is the fluid (water) density[ML-3], and q is the specific discharge, i.e., the Darcy
flux [LT-1].
Similarly, the mass outflow rates at x+dx, y+dy, and z+dz in the x, y, and z
directions (approximately by Taylor series expansion) respectively are
46
O x = (ρ q x )dz dy +
∂
(ρ q x )dz dy dx
∂x
(3.3a)
O y = (ρ q y )dx dz +
∂
(ρ q y )dx dz dy
∂y
(3.3b)
O z = (ρ q z )dx dy +
∂
(ρ q z )dx dy dz
∂z
(3.3c)
where dx dy dz is the volume of the unit representative element.
The right hand side of equation (3.1) can be written as
dS ∂
= [(ρ θ) dx dy dz]
dt ∂t
(3.4)
where θ is the volumetric moisture-content of the medium[L3L-3].
If equations (3.2a),(3.2b),(3.3c), and (3.4) are substituted into equation (3.1) and
rearranged, then,
 ∂ (ρ q x ) ∂ (ρ q y ) ∂ (ρ q z ) 
∂ (ρ θ)
+
+
−
+w =
∂y
∂z 
∂t
 ∂x
where w =
(3.5)
W
mass of water injected or removed from a unit volume of the aquifer
dx dy dz
per unit time [ML-3T-1].
47
Equation (3.5) is the continuity equation. There is only one equation and there are
five unknowns, i.e., the three components of the Darcy flux (qx, qy, qz), and θ and ρ.
Thus, it is necessary to formulate four more equations to solve equation (3.5) for the five
unknowns.
Storage Term
The storage term on the right-hand side of the equation (3.5) can be expanded by
defining Sw as the saturated fraction of the porous medium and substituting this parameter
into the storage term in equation (3.5). This yields
∂ (ρ η S w )
∂S
∂η
∂ρ
= ρ η w + ρ Sw
+ η Sw
∂t
∂t
∂t
∂t
where η is the porosity and S w =
(3.6)
θ
.
η
Using the chain rule for differentiation, equation (3.6) can be rewritten in terms of
the pressure head h = p/γ [L] as
∂ (ρ η S w )
∂S ∂h
∂η ∂h
∂ρ ∂h
= ρη w
+ ρ Sw
+ η Sw
∂t
∂h ∂t
∂h ∂t
∂h ∂t
(3.7)
The first term of equation (3.7) accounts for the change in fluid storage due to a
change in the volumetric water content. It actually describes the effects of draining and
filling the pores. The first term can be redefined as
48
∂ η S w ∂θ
= C( h )
=
∂h
∂h
(3.8)
where C (h) [L-1] is the slope of the moisture retention curve. This term is called the
specific moisture capacity, and it expresses the volume of water released per unit volume
of unsaturated zone for a unit decrease in pressure head h.
The second term in equation (3.7) accounts for the change in fluid storage due to
the compressibility of the solid matrix:
∂η
= αρg
∂h
(3.9)
where α is the solid matrix compressibility [LT2M-1], and g is the acceleration of gravity
[LT-2].
The third term accounts for the change in fluid storage due to fluid
compressibility:
∂ρ
= β ρ 2g
∂h
where β is fluid compressibility [LT2M-1]. Equations (3.8) through (3.10) can be
combined and substituted into equation (3.7) to give
(3.10)
49
∂ (ρθ)
∂h
= ρ [ C( h ) + S w S s ]
∂t
∂t
(3.11)
where Ss is the specific storage [L-1] defined as
Ss = ρ g(α + η β)
(3.12)
The specific storage represents the volume of water released per unit volume of
aquifer per unit decline in pressure head. Equation (3.11) is the second equation of the
five equations necessary to solve for the five unknowns in Equation (3.5). This equation
is based on the assumptions that the aquifer and water are slightly compressible in the
saturated confined zone but incompressible in the unsaturated zone and in the unconfined
saturated zone. Therefore, Ss approaches zero in the unsaturated and the unconfined
aquifer zones because of its dependency on the compressibility of the solid matrix and
fluid. On the other hand, although C(h) can have significant values in the unsaturated
zone, it has very small values approaching zero in the saturated zone. This is because
C(h) is the slope of the moisture retention curve, which is zero in the saturated zone, i.e.,
the moisture-content is constant in the saturated zone.
Darcy-Buckingham Equation
Darcy developed his well-known formula for saturated flow conditions, and
Buckingham developed nearly the same relationship for unsaturated flow conditions.
Combining those two formulas results in the Darcy-Buckingham equation, which is used
as the flux equation for both saturated and unsaturated zones (Narasimhan, 1998b).
50
In Darcy’s equation, the flux is linearly proportional to the hydraulic gradient, and
the proportionality constant is defined as the hydraulic conductivity. In Buckingham's
equation, in the unsaturated zone, the proportionality constant is not linear and the
hydraulic conductivity is a function of both the pressure head and the medium properties
of the unsaturated zone.
Defining the hydraulic head (or total head), H, as
H=h+z
(3.13)
where h is the pressure head [L] and z is the elevation head [L], the Darcy-Buckingham
equation can be written as
→
q = − K ij (h)
∂H
∂l
(3.14)
→
where q is the specific discharge [LT-1] in the x, y, and z directions, Kij (h) is the
hydraulic conductivity [LT-1] in the x, y, and z directions, and l represents the unit
distances in the x, y, and z directions.
Hydraulic conductivity is not only a function of the porous medium but also of the
fluid properties. Hubert (1956) pointed out that hydraulic conductivity is directly
proportional to the square of the mean grain size diameter (d2) and the specific weight of
the fluid (ρg) and inversely proportional to the fluid viscosity (µ) (Bear, 1972). Together
with Darcy’s original observation and dimensional analysis, the hydraulic conductivity
51
can be expressed as K = Cd2ρg/µ. The term Cd2 is a property of the soil itself, and it is
called the intrinsic permeability, k. The coefficient C in the intrinsic permeability (k)
represents the grain-size distribution, the sphericity and roundness of the grains, and the
nature of their packing. The hydraulic conductivity K is written as K = kρg/µ.
K(h) is function of the pressure head in the unsaturated zone, but it is constant and
equal to saturated hydraulic conductivity in the saturated zone, i.e., Kij (h)= Ks. Some
typical values of Ks can be found on page 29 in Freeze and Cherry (1979).
In general, in a three-dimensional flow field, the hydraulic conductivity tensor
could have nine components. However, the hydraulic conductivity tensor is symmetric
such that Kxy(h) = Kyx (h), Kxz (h) = Kzx (h), and Kyz(h) = Kzy(h), and thus it has only six
components:
K xx

K ij =  K xy
 K xz

K xy
K yy
K yz
K xz 

K yz 
K zz 
(3.15)
If the principle axis of anisotropy is aligned with the principle axis of flow, then
only three non-zero hydraulic conductivity terms remain, i.e., Kxx (h), Kyy (h), and Kzz (h).
Thus, the hydraulic conductivity tensor becomes
K xx
K ij =  0
 0
0
K yy
0
0 
0 
K zz 
(3.16)
52
Governing Equation (Modified Richards’ Equation)
Substituting equation (3.16) into the Darcy-Buckingham equation (3.14) in the x,
y, and z directions results in:
q x = −K x (h )
∂H
∂h
= − K x (h )
∂x
∂x
(3.17a)
q y = −K y (h )
∂H
∂h
= −K y (h )
∂y
∂y
(3.17b)
q z = −K z (h )
∂H
∂h
= − K z (h )( + 1)
∂z
∂z
(3.17c)
These equations (3.17 a-c) are the third, fourth and fifth equations of the five
equations required in order to solve the five unknowns of equation (3.5).
Assuming that water density does not vary spatially, such that
∂ρ
= 0 , and
∂x i
substituting the Darcy-Buckingham equation (equation (3.17)) and the general saturatedunsaturated storage term (equation (3.11)) into the continuity equation (equation (3.5))
gives the following form of the three-dimensional Richards equation:
∂h


∂h
∂h
 ∂ (K x (h )( )) ∂ (K y (h )( ∂y )) ∂ (K z (h )( + 1)) 
∂h
∂x +
∂z

 ± Q ext = [ C(h ) + S w Ss ]
+
(3.18)
∂x
∂y
∂z
∂t




53
where Qext is a volumetric source or sink term, which is obtained by dividing w by the
density of water, Q ext =
w
[L3L-3T-1].
ρ
Equation (3.18) is the general three-dimensional saturated-unsaturated flow
equation that is called the “modified Richards equation” due to the inclusion of the
saturated zone, which is achieved by modifying the general storage term on the right hand
side of equation (3.5).
The expressions for C(h) and K(h) are both highly nonlinear, which makes the
solution of the governing equation very difficult and complex.
In the saturated zone:
C(h) = 0;
K(h) = Ks = constant;
Sw = 1; and
h ≥ h air entry .
Therefore, in the saturated zone, equation (3.18) becomes
∂h


∂h
∂h
 ∂ ( K x ( )) ∂ (K y ( ∂y )) ∂ (K z ( + 1)) 
∂h
∂x +
∂z

 ± Q ext = Ss
+
∂x
∂y
∂z
∂t




In the unsaturated zone:
C(h) ≠ 0;
C(h) >> Swr Ss ;
(3.19)
54
S wr =
θ
< 1;
η
h < h air entry ; and
K(h) = function of the pressure head.
Therefore, in the unsaturated zone, equation (3.18) becomes
∂h


∂h
∂h
 ∂ (K x (h )( )) ∂ (K y (h )( ∂y )) ∂ (K z (h )( + 1)) 
∂h
∂x +
∂z

 + Q ext = C(h )
+
∂x
∂y
∂z
∂t




(3.20)
The right side of the equation (3.20) describes the effects of draining and filling
the pores in the unsaturated region. Thus, expressing this concept in terms of the
temporal change in moisture-content would be more appropriate than expressing it in
terms of pressure head. In other words, the term C(h)(∂h/∂t) = (dθ/dh)(∂h/∂t) can be
written more appropriately in its original simpler form, i.e., ∂θ/∂t.
Using the term ∂θ/∂t in equation (3.20) converts the pressure-based modified
Richards equation into a mixed form of the modified Richards equation. Celia et al.
(1990) showed that the modified Picard iterative procedure for the mixed form of the
Richards equation is fully mass conserving in the unsaturated zone. By contrast, the
conventional pressure-based, backward Euler finite-difference formulations exhibit poor
mass-balance behavior according to Clement et al. (1994) and Celia et al. (1990). The
reason for this is that the discrete analogs of ∂θ/∂t and C(h) ∂h/∂t are not equivalent even
though the time derivative of the moisture-content, ∂θ/∂t, is equal to C(h)∂h/∂t, which is a
55
mathematically valid approximation (Clement et al., 1994). This inequality is amplified
owing to the highly nonlinear nature of the specific capacity term, C(h). Using the
modified Picard iteration method eliminates this problem by approximating directly the
temporal term ∂θ/∂t with its algebraic analog (Clement et al., 1994). The algebraic
approximation of the temporal term (∂θ/∂t) and the modified Picard iteration method are
described in detail in Chapter 4 of this study.
Hydraulic Conductivity
The hydraulic conductivity, K, is constant with respect to time and equal to the
saturated hydraulic conductivity, Ks, in the saturated zone. In this study, a relative
hydraulic conductivity term, Kr, is used. Kr is the ratio of the unsaturated hydraulic
conductivity to the saturated hydraulic conductivity, K(h)/ Ks and thus K = Kr Ks.
The hydraulic conductivity in the unsaturated zone is defined as a function of the
pressure head, which can be derived from moisture-retention (or moisture characteristic)
curves, h versus θ. Several researchers have developed relationships between K(h) and
moisture retention curves. Measuring pressure head and moisture-content, h versus θ, is
easier than measuring pressure head versus K(h), so therefore h versus θ relationships are
very useful in determining unsaturated hydraulic conductivity values.
The specific moisture-content C(h) is defined as the slope of the moisture
retention curve. It can be found by taking the derivative of the moisture-content with
respect to pressure head, h, or
56
C(h ) =
dθ
dh
(3.21)
Three different moisture-content-pressure head-hydraulic conductivity algebraic
relationships are used in this model study, i.e., the Brooks and Corey (1964) equations,
the van Genuchten and Nielsen (1985) relations, and the general power formula.
Brooks and Corey method. The Brooks and Corey (1964) equations are
θ − θr  h a 
Se =
= 
θs − θ r  h 
Se =
θ − θr
=1
θs − θ r
λ
when h < ha, and
(3.22)
when h ≥ ha
(3.23)
where Se is the effective saturation, θr is the residual water content, and θs is the saturated
moisture-content, which is generally equal to the porosity of the formation (η), and λ is a
pore size distribution index that is a function of soil texture. The term ha is the bubbling
(or air entry) pressure head, equal to the pressure head required to desaturate the largest
pores in the medium, and it generally is less than zero.
The hydraulic conductivity is defined as
K (h )  h 
Kr =
= 
K sat  h a 
Kr = 1
−2−3λ
when h ≥ h a
when h < h a ; and
(3.23)
(3.24)
57
The specific moisture capacity C(h) can be calculated from
 λ
C(h ) = −(n − θ r )
 ha
C( h ) = 0
 h

 h a



− ( λ +1)
when h < h a , and
when h ≥ h a
(3.25)
(3.26)
van Genuchten and Nielsen method. van Genuchten and Nielsen (1985)
developed a closed-form equation for hydraulic conductivity as a function of the pressure
head using the moisture retention curve:
Kr =
K ( h)
= (1 + β ) −5 m / 2 (1 + β ) m − β m
Ks
[
]
Kr =
K (h )
=1
Ks
h≥0
for
2
for
h < 0 , and
(3.27)
(3.28)
n
 h 
 , ha is air entry (or bubbling) pressure head[L], and n is a fitting
where β = 

h
 a 
parameter in the moisture retention curve, or m =1-1/n.
This closed form equation can be obtained by applying the fitting curve technique
to measured, or experimental, moisture-content-pressure head data. The moisturepressure head data generally fit the following equations:
Se =
θ − θr
= (1 + β) −m
θs − θ r
if
h ≤ 0 , and
(3.29)
58
Se =
θ − θr
=1
θs − θ r
if
h>0
(3.30)
where θr is the residual water content, and θs is the saturated moisture-content, which
generally equals the porosity of the formation (η).
For the moisture-content relations, Paniconi et al. (1991) modified van Genuchten
and Nielsen’s relation (equation (3.28) and (3.29)) in the form
θ(h ) = θ r + (θs − θ r )(1 + β) − m
for
h ≤ h 0 , and
θ(h ) = θ r + (θs − θ r )(1 + β 0 ) − m + Ss (h − h 0 )
for
(3.31)
h > h0
(3.32)
where Ss is the value of specific storage for the pressure head h that is greater than the air
n
h 
entry pressure, β 0 =  0  , and h0 is a parameter determined on the basis of continuity
 hs 
requirements imposed on Ss, which implies that
Ss =
(n − 1)(θs − θ r ) h
n −1
h s (1 + β) m+1
(3.33)
n
h =h 0
For a given value of Ss , equation (3.33) can then be solved for h0.
The specific moisture capacity C(h) can be calculated from
C(h ) =
(n − 1)(θs − θ r ) h
h s (1 + β) m+1
n
n −1
when h ≤ h 0 , and
(3.34)
59
C( h ) = 0
when h > h 0 .
(3.35)
For Ss = 0 and h0= 0, equations (3.29-35) revert to their original form in van
Genuchten and Nielsen (1985).
General power formula. A general power formula also can be used if there is a
moisture-content-pressure head (h-θ) data set. The hydraulic conductivity K can be
described as a function of the effective saturation, Se:
Kr =
where Se =
K (h )
= Se n
Ks
(3.36)
θ − θr
and n is a parameter that has to be estimated by calibration. As a
θs − θ r
guideline, the exponent n is usually relatively small for sandy soils (between 2 to 5) and
larger for clayey soils (between 10 to 20). The value of n influences the percolation rate in
the soil and thereby influences the actual evaporation rate.
In this method, any kind of soil moisture retention curve (θ-h) can be used, but the
data should be supplied in a tabular form to the model. The model calculates the
intermediate values using interpolation methods.
The specific moisture capacity can be calculated using tabular values of the soil
moisture retention curve with the following formula:
C(h m+1/ 2 ) =
∂θ θ m+1 − θ m
=
∂ h h m+1 − h m
(3.37)
60
Sink/Source Term
The volumetric sink/source term (Qext ) [L3T-1L-3] in equation (3.18) represents
the volume of water removed or injected per unit time from a unit volume of soil due to
sinks such as root water uptake in the unsaturated zone and pumping from wells and flow
from drains in the saturated zone. It is a source term in case of artificial recharge or
injection. Qext can be expressed as Qext = Wr + Ww + Wd where Wr represents root water
uptake, Ww represents well recharge or discharge, and Wd represents a drain. The
components of the sink/source terms of this study are briefly described in figure 3.3.
Figure 3.3 Flow chart describing the principle sink/source terms in the model.
61
Although evaporation and rainfall can be considered as sink and source terms,
respectively, they are treated in this study as upper boundary conditions. This is because
these processes occur on the land surface. Rainfall is applied to the land surface as a flux
boundary condition, and evaporation is separated from evapotranspiration by a procedure
that is described in the following sections. Then, it is also treated as a flux boundary
condition on the land surface. It could be in the form of soil evaporation or direct
evaporation from surface-water bodies such as lakes and rivers. The transpiration, or root
water uptake, is considered a sink term and applied to the cell nodes in the unsaturated
zone where it is occupied by roots.
The main part of the sink/source term in the unsaturated zone is the transpiration
(or root water uptake) process. To calculate the actual transpiration, the following steps
are taken (see figure 3.4):
1. Potential evapotranspiration (PET) is determined using one of the two
methods:
a. Pan evaporation method; or
b. Physically based equations (i.e., Priestly and Taylor Equation).
2. Potential evaporation (Ep) and potential transpiration (Tp) are determined from
PET as follows:
a.
PET = Ep + Tp;
b. Ep = PET*exp(-0.4 LAI) where LAI = leaf area index; and
c. Tp = PET - Ep
3. The actual transpiration (TA) is determined using two different options:
a. The method of Feddes et al. (1978); or
62
b.The method of Lappala et al. (1987), i.e., VS2D.
4. The actual evaporation (EA) is determined using the method of Lappala et al.
(1987). The actual evaporation is not treated as a sink term but it is
considered in the top boundary condition as a negative flux, and the modified
Richards equation takes EA into account at the top boundary.
The procedure outlined above is briefly summarized in figure 3.4, and it is also
described in detail in the following sections.
Figure 3.4 Flow chart for actual transpiration calculations in the model.
63
Determination of Evapotranspiration
Evapotranspiration in this study is considered to be the combination of
transpiration and evaporation. Potential evapotranspiration (PET) calculations can be
carried out using two different options. The first option is the easiest and the most
empirical one, i.e., the pan evaporation technique. If there are not enough available
climatologic data to calculate PET using one of the physically based equations, then the
pan evaporation method can be used. The pan evaporation method requires daily
measured pan evaporation values and a pan coefficient. The PET is calculated as
PET = Cpan*Epan
(3.38)
where Cpan is pan coefficient, which is generally equal to approximately 0.7, and Epan is
the measured pan evaporation in [L/T].
The second method of PET calculations is physically based, i.e., based on the
energy conservation method. Although there are many equations in the literature to
calculate PET, the Priestly-Taylor equation was selected for use in this study. The
Priestly and Taylor (1972) equation is a derivative of the Penman equation (Fares, 1996).
It is advantageous among the others because it requires the least amount of input data.
Despite the empirical nature of its proportionality factor α, the Priestly-Taylor equation is
based upon physical theory, and it reduces input data requirements (Buttler and Riha,
1989). It is also a simplified form of the Penman equation and is most reliable in humid
climates where an aerodynamic component has been deleted and the energy term
64
multiplied by a constant α (Jensen et al., 1989). The Priestly-Taylor equation can be
written as
PET = α
∆
(R n − G )
λ(∆ + γ )
(3.39)
where PET is potential evapotranspiration in mm t-1 (t is time unit), Rn is net solar
radiation [W m-2], ∆ is the gradient of saturation vapor pressure-temperature curve
evaluated at the air temperature Ta, λ is latent heat of vaporization [J m-3], G is soil heat
flux [W m-2], γ is the psychrometric constant (0.067 kPa oC-1) and α is an empirical
parameter that depends on the nature of the surface, the air temperature, and time of day
and which varies from 1.05 to 1.38 (Viswanadham et al., 1991). Values of α are
generally between 0.6 and 1.1, according to Spittlehouse and Black (1981). Priestly and
Taylor (1972) obtained a mean value of α equal to 1.26 for an extensive wet surface in
the absence of advection. Jury and Tanner (1975) showed that α increases with heat
advection from surrounding areas and suggested a procedure for adapting the PriestlyTaylor equation to such conditions (Fares, 1996).
The Priestly-Taylor equation requires values for six input parameters: the PriestlyTaylor coefficient (α), slope of the saturation vapor pressure curve for water (∆), net
radiation (Rn), soil heat flux (G), psychrometric constant (γ), and latent heat of
vaporization (λ). Fares (1996) developed an analytical model to calculate the above
parameters using available meteorological data, i.e., maximum and minimum daily
temperatures for a given location, day of the year, altitude and latitude of the location, and
65
albedo coefficient of the surface. If the albedo coefficient is equal to 0.05, the calculated
value of PET will be the potential evaporation directly from a free water surface.
However, if the albedo coefficient is equal to 0.15 (i.e., for pine trees), the result will be
the PET for pine trees. The flow chart of the calculations of PET using the PriestlyTaylor equation is presented in figure 3.5.
Figure 3.5 Flow chart for the evapotranspiration calculations (Fares, 1996).
66
Estimation of input parameters for PET calculations
All the time units in PET calculations are in day. If it is desired, during the model
simulation daily values of PET can be converted into hourly values by assuming that
there is a sinusoidal distribution of the PET process based on a daily cycle in which PET
reaches its maximum value around noon time and reaches its minimum value around
midnight.
Calculation of the daily net radiation (Rn). Daily total values of Rn (MJ m-2 t-1)
can be determined from daily total incoming solar radiation, Rs (MJ m-2 t-1), and outgoing
thermal or long-wave radiation if they are not measured in the field. The following
relationship was proposed by Penman (1948) and modified by Wright(1982) to estimate
Rn :
4
 T 4 + Tmin
k
Rn = (1 − β ) Rs − σ  max k
2

(
)

 R

 a1 − 0.139 ed  a s + b 

 Rso

(3.40)
where β is albedo (or reflectivity) coefficient of the surface, Rs is daily total incoming
solar radiation, σ is the Stephan-Boltzman constant (4.903x10-9 MJ m-2 d-1 K-4), Tmaxk and
Tmink are the maximum and minimum daily air temperatures (oK), respectively, ed is
saturation vapor pressure (kPa) at the dewpoint temperature (which is taken as the
minimum daily temperature), Rso (MJ m-2 d-1) is daily total clear sky short wave radiation,
and a1, a, b are empirical coefficients.
67
Accepted values for the albedo coefficient (β) are 0.05 for water surfaces; a range
of 0.15 to 0.60 for bare soil surfaces; 0.25 for most agricultural crops; and 0.1 for forests
(Fares, 1996). Wright (1982) estimated the a1 empirical coefficient using
a1 = 0.26 + 0.1 e −(0.0154 ( J −180 ) )
2
(3.41)
where J is the day of the year (1-365). If there are few clouds (Rs/Rso > 0.7), then a =
1.126 and b = -0.07; if it is cloudy (Rs/Rso < 0.7), then a = 1.017 and b = -0.06. Clear sky
short wave radiation (Rso) can be estimated from the Jensen et al. (1989) relationship:
R so = 0.75 R A
(3.42)
where RA is extraterrestrial radiation (MJ m-2 d-1). Although solar radiation (Rs) can be
measured using sophisticated instruments, the estimation of Rs using RA is also possible
using the Doorenbos and Pruitt (1977) equation:
n

R s =  0.25 + 0.5 R A
N

(3.43)
where n is the number of actual bright sunshine hours and N is the maximum possible
sunshine hours for that location. RA can be calculated using the following set of
equations developed by Duffied and Beckman (1980):
68
R A = 37.58 d r {w s sin(φ) sin(δ) + cos(l) cos(δ) sin( w s )}
(3.44)
where φ and l are the longitude and latitude of the location in radians respectively (-E,
+W, -S, +N), dr is the relative distance of the earth from the sun, δ is declination in
radians, and ws is sunset hour angle in radians, which can be calculated as
J 

d r = 1 + 0.033 cos2π

 365 
(3.45)
 284 + J 
δ = 0.4093 sin 2π

365 

(3.46)
w s = Arc cos(− tan(φ) tan(δ) )
(3.47)
The average daily soil heat flux was approximated by Wright and Jensen (1972) as
G = (Ta − Tp ) c s
(3.48)
where Ta is average daily air temperature (0C) at the height z, and Tp is the average daily
temperature (0C) at that height for the previous three days. Parameter cs is the general
heat conductance for the soil surface (Allen et al., 1989). G can be neglected if it makes a
very small contribution to the PET.
The slope of the saturated vapor function(∆) can be calculated by taking the
derivative of the saturated vapor pressure (ed) equation with respect to temperature T
(Tetens, 1930) :
69
 16.78 T + 117 
e d = exp

 T + 237.3 
∆=
4098 e d
(T + 237.3)2
(3.49)
(3.50)
The psychometric constant(kPa 0C-1) can be calculated as follows:
γ=
cp P
λε
(3.51)
where cp is the specific heat of moist air at constant pressure (1.01x10-3 MJ kg-1 C-1), P is
atmospheric pressure (kPa), ε is the ratio of molecular weights of air to water (0.622), and
λ is the latent heat of vaporization (MJ kg-1), which can be calculated using the Harrison
(1963) relationship:
λ = 2.5 – ( 2.361x10-3 ) Ta
(3.52)
where Ta is the air temperature in 0C. The atmospheric pressure at a given altitude,
assuming a constant temperature lapse rate, can be calculated based on Burman et al.
(1987) as
70
P
P= 0
Y
 g 


 T0 − Y(z − z 0 )   ΩR 


T0


(3.53)
where P0 and T0 are known atmospheric pressure (kPa) and absolute temperature (0K) at
elevation z0 (m), and P is the desired pressure estimate at elevation z. The parameter Ω is
the assumed constant adiabatic lapse rate. R is the specific gas constant for dry air (
286.9 Jkg-1 oK-1. Allen et al. (1989) suggested a Y value of 0.0065 K m-1 for saturated air
and of 0.01 K m-1 for dry air. The gravitational acceleration g is equal to 9.806 m s-2.
Reference values for P0, T0, and z0 are set to those for the standard atmospheric pressure
definition at sea level, which are 101.3 kPa, 288 0K, and 0 m respectively.
Using all the above relationships, i.e., equations from (3.40) through (3.53), the
PET in equation(3. 39) can be calculated.
Determination of transpiration (or root water uptake)
Water is extracted from the unsaturated zone through plant roots. There are two
approaches to calculating root water uptake. The first one considers properties of a single
root (microscopic approach), and the second one considers the integrated properties of the
entire root system (macroscopic approach). In this study, the macroscopic approach is
followed.
The first step in root water uptake calculations is to determine the potential
transpiration rate. Then the next step is to find out how much of this potential
transpiration rate can actually occur under the restriction of soil and available moisturecontent conditions.
71
Potential transpiration (Tp) can be calculated as a fraction of the potential
evapotranspiration (PET) as a function of leaf area index (LAI) of the soil surface
(McCarthy and Skaggs, 1992; Fares,1996; McKenna and Nutter, 1984). Developed by
Ritchie (1972) and modified by McKenna and Nutter (1984) , the potential evaporation
from soil surface, Ep, and potential transpiration, Tp, can be calculated as follows:
TP = PET − E P
(3.67)
E P = (PET ) exp(−0.4 LAI)
(3.68)
where LAI is the leaf area index. This term is defined as the ratio of total area of leaves
to the area of ground surface, and it can vary through the year depending on the type of
vegetation.
To calculate the actual amount of water taken up by a root system, a root water
uptake sink term, Wr, which represents the volume of water taken up by the roots per unit
volume of the soil in unit time [L3 L-3 T-1], was defined by Feddes et al.(1978) as
Wr (h) = a r (h ) Wp (z)
(3.69)
In equation (3.69) Wp (z) is the potential water uptake sink term [L3 L-3 T-1],
which is a function of depth and root density. It can be defined as the maximum possible
water uptake in favorable conditions, i.e., sufficient moisture-content around the root
zone. Wp stands for the distribution of the potential transpiration throughout the entire
root zone. The water stress response function ar (h) in equation (3.69) determines the
72
degree of restriction of the potential transpiration based on the available soil moisturecontent and potential transpiration rate.
Wp can be calculated using a root density distribution function and potential
transpiration rate. In the literature, many root distribution and water uptake functions can
be found (Molz, 1981). In this study, three different distribution functions were
considered.
The first root distribution model (Feddes et al., 1978) assumes a uniform
distribution of water uptake throughout the root zone, or
Wp =
1
Tp
Zr
(3.70)
where Zr is the bottom of the root zone depth, and Tp is the potential transpiration [LT-1].
The second root distribution model (Prasad, 1988) is a linearly decreasing water
uptake model starting with a maximum value at the top and zero at the bottom of the root
zone, or
Wp (z) =
2Tp 
z 
1 − 
Zr  Zr 
(3.71)
where z is the current depth and Zr is the bottom of the root zone depth.
The third root distribution model was developed for this study by modifying the
logarithmic root distribution model of Jensen (1983). His original relationship was
73
log W p ( z ) = log Ro − Rd z
(3.72)
where R0 is a value of Wp at soil surface, and Rd is a parameter dependent on the crop and
soil. Equation (3.72) can be written as
Wp (z) =
R0
= R 0Cd z
R dz
10
(3.73)
where Cd is a crop and soil coefficient, which has a value in the range of 0.1 < Cd < 1.
This third method is very flexible, and it can be applied to various vegetation
cover scenarios by changing the Cd values. If there is a uniform root distribution, then a
Cd value close to 1.0 is chosen. If there is a linearly decreasing crop distribution, then a
Cd value between 0.5 and 0.8 is chosen. Finally, if there is denser root distribution at the
top and much less root distribution at the bottom, i.e., there is grass and some bushes and
several pine trees in a unit area of the soil, then a Cd value less than 0.5 is chosen.
Hansen et al. (1976) gave a few characteristic values for Rd to calculate Cd values. In this
study, Cd values are assumed to be constant in time although the root depth can vary in
time. Integrating Wp(z) along the root zone gives the potential transpiration:
∫ R 0Cd dz = ln C0 d (Cd
zr
z
o
R
Zr
)
− 1 = Tp
(3.74)
74
Solving equation (3.74) for R0 and substituting that into equation (3.73) gives the
following expression for the potential root water uptake function:

 ln C
Wp (z) =  Z d Tp C d z
 C r −1 

 d
(3.75)
If Zr is a function of time, i.e., the annual vegetation with varying root depth, then
it can be calculated using Borg and Grimes (1986) relationship:
Z r ( t ) = Z T (0.5 + 0.5 sin[3.03( t / t T ) − 1.46])
(3.76)
where tT, and ZT are time to plant maturity and maximum rooting depth to be achieved at
t = tT respectively.
The water stress response function ar(h) is a prescribed dimensionless function of
the soil water pressure head (0≤ ar ≤ 1). A schematic plot of the stress response function
used by Feddes et al. (1978) is shown in Figure3.6.
In this function, the wilting point is defined as the minimum moisture-content (or
corresponding pressure head) at which plant roots cannot extract any more water from the
surrounding soil.
75
Figure 3.6 Schematic of the plant water stress response function, ar(h) (Feddes et al.,
1978). Water uptake below h1(air entrainment pressure, saturation starts) and above h4
(wilting point) is set to zero. Between h2 and h3 water uptake is maximum. The value of
h3 varies with the potential transpiration rate Tp.
In this study, a water response function ar(h) was developed by modifying the
relationship suggested by Kristensen and Jensen (1975):
 h −h
T
a r (h) = a = 1 −  fc
h −h
Tp
wp
 fc
c3
 Tp



(3.77)
where Ta, and Tp are the actual and potential transpiration respectively, and hfc and hwp are
pressure heads at field capacity and at wilting point of the soil around the root zone,
respectively. Field capacity is defined as the moisture-content (or corresponding pressure
head) at which gravitational drainage ceases. h is the pressure head around the root zone,
and C3 is a parameter greater then T p. The parameter C3 defines the shape of the water
stress function. For example, if C3 is equal to Tp, then the water stress function will
change linearly between 0 at hwp and 1 at hfc. The relationship in equation (3.77) is shown
in figure 3.7.
76
If equation (3.75) and equation (3.77) are substituted in equation (3.69) and
written in terms of pressure head, then the actual root water uptake model is obtained as
C3

  h fc − h  TP
Wr (h, z, t) = 1 − 

  h fc − h wp 


 lnC d
 z
Tp C d
 Z r
 C d − 1 

(3.78)
The integration of the root water uptake function, equation (3.78), along the root
zone gives the actual transpiration rate. The actual transpiration is restricted according to
the available moisture in the root zone, soil type, root distribution type, and the potential
evapotranspiration rate.
The model has an option to calculate the actual transpiration using a method based
on VS2D (Lappala et al., 1987). This method is relatively easy to use and requires the
input of the root pressure, which is the pressure applied by the roots on the surrounding
soil to extract water, and the root activity function. The root water uptake (wr) [T-1] for
each cell having a root zone is calculated using equation 3.79:
w r = (K sat ) hrz K r (h ) r (z, t ) (h root − h )
(3.79)
where r (z, t) is the root activity function of depth and time [L-2], hroot is the pressure head
in the root zone [L] for the entire root system, and (Ksat)hrz is the average horizontal
conductivity (equal to (Kx + Ky)/2.0).
77
Figure 3.7 Water stress function as a function of pressure head and potential transpiration
(Jensen, 1983).
The total extraction by roots in a given column of cells can be calculated from
Q r = ∑ j=1 ( w r dx dy dz) j
J
(3.80)
where J is the total number cells in the column where roots are present.
If water is freely available to the plants, it is possible that a flux from the soil that
is larger than the potential transpiration rate may be computed using equations 3.79 and
3.80. Consequently, if the calculated value of Qr is larger then Tp, the value of wr is
adjusted by the ratio
Tp
Qr
, such that (wr) i j k =
Tp
Qr
(wr) i j k.
The root activity function r (z, t) is defined as the length of roots in a given
volume of soil divided by that volume. This function is assumed to vary linearly between
78
the root activity at the bottom of the root zone and the root activity at top of the root zone.
Therefore, the root activities at the bottom and top of the root zone and the root depth as a
function of time need to be provided to the model.
Evaporation
Evaporation can take place from open water bodies or from soil surface, and it can
vary according to whether there is vegetation cover or not. Potential evaporation, Ep, is a
fraction of potential evapotranspiration (PET) and can be calculated using equation 3.68.
If there is an open water body such as lakes, rivers, etc., then Ep will be equal to PET by
equating LAI to zero in equation 3.68. Actual evaporation is determined by considering
the amount of rainfall intercepted by vegetation and the available moisture in soil.
Some researchers have reported that interception and transpiration are related and
transpiration will not start before the intercepted water is dried out by direct evaporation
(Hansen et al., 1976). To the contrary, Jensen (1979) and Van der Ploeg and Benecke
(1981) claim that both transpiration and evaporation from intercepted water can occur
simultaneously.
Interception (I) is the amount of water held by vegetation leaves during rainfall.
Jensen (1979) proposed that the maximum interception storage of the crop, Im, is linearly
related to the leaf area index, LAI:
Im = Cint LAI
when P≥ Im
(3.81a)
Im = P
when P< Im
(3.81b)
79
where Im is the maximum interception capacity [L], P is precipitation [L], and Cint is an
interception coefficient that can be taken as 0.2 for forest and for regions that have high
trees (Rutter et al., 1975) and 0.05 for short vegetation and agricultural crops (Jensen,
1979) when precipitation is measured in mm.
The demand for potential evaporation is met by intercepted water if it is sufficient.
If the intercepted water does not satisfy Ep, available water in the soil is used to satisfy the
Ep demand. The Ep demand is satisfied as long as the soil medium can conduct water to
the soil surface at a rate equal to Ep rate. As the soil near the surface becomes drier, then
soil evaporation is reduced to below the potential value. A physically based relationship
of Lapalla et al. (1987) is used for the prediction of actual evaporation from soil. The
relationship is described as
E a = (K sat ) z K r SRES (h atm − h top )
for Ea < Ep
(3.82 a)
for Ea ≥ Ep
(3.82 b)
and
Ea = Ep
where Ea is the actual soil evaporation [LT-1], Ep is the potential soil evaporation [LT-1],
hatm is the pressure potential of the atmosphere [L], htop is the pressure head at first node
on the land surface [L], and SRES is the surface resistance [L-1].
The atmospheric pressure potential hatm can be calculated using the Kelvin
equation (Lappala et al., 1987):
80
h atm =
RT
ln(h r )
M wg
(3.83)
where R is the ideal gas constant [ML2T-2 oK-1Mol-1], T is absolute temperature [oK], Mw
is the mass of water per molecular weight [M Mol-1], hr is relative humidity [L0], and g is
the gravitational acceleration [LT-2].
SRES can also be calculated, assuming that atmospheric pressure is applied to the
land surface, and the pressure at the top cell is applied to the node of the first cell.
Therefore, SRES will be the part of the conductance term between the top cell node and
the land surface such that
SRES = [2.0 / DZ top ] K c /(K sat z ) top
(3.84)
where DZtop is the thickness of the first top cell, Kc is the saturated hydraulic conductivity
of the crust material, and Ksat z is the saturated hydraulic conductivity of the first cell on
the land surface.
The calculated actual evaporation is introduced as a negative flux boundary on the
land surface as a boundary condition.
Pumping and Recharge Wells
Wells are used for withdrawing water from the saturated zone of the aquifer or
adding water to the aquifer. The well discharge rate (Q) [L3T-1] should be specified for
each cell of the saturated zone. Negative values of Q are used to indicate a pumping well,
81
while positive values of Q indicate a recharge well. The specified Q value for each cell is
converted to a sink/source term Ww (Ww is part of main sink/source term Qext as
discussed above) by dividing the total Q by the number of cells. The node containing the
well itself is considered to be outside of the model, and six surrounding nodal blocks are
treated with the appropriate side as a flux boundary (Freeze, 1971). Such an approach
does not provide an exact duplication of flow conditions near the well, but it prevents the
well from becoming unsaturated immediately and unrealistically. For example, if a well
is open to five unconfined aquifer cells with uniform grid size (dx dy dz), then Ww for
each individual cell surrounding the cell containing the well itself will be calculated as
Ww = (Q/(5 dx dy.dz))/6 [T-1]. If the top cell becomes unsaturated during water-table
drawdown, Ww is adjusted for the remaining four cells.
Drains, Sinkholes, and Springs
Drains are treated as specified head sink terms in this model. Drain heads are
specified for each cell in the saturated zone. Drainage flow occurs only in the saturated
zone if the water-table is above the position of the drains. The drain head should be
specified for each cell containing a drain. When the hydraulic head in the cell drops
below the specified drain head, the drainage flow ceases. It will start again if the watertable rises above the specified drain head. Drainage flow is calculated based on the head
difference between the calculated hydraulic heads of the cell and the specified drain head.
The head difference is multiplied with the drain conductance that represents the resistance
to the flow because of material around the drain, the number of the holes in the drainpipe,
and converging streamlines in the immediate vicinity of the drain. After drain discharge
(Qd = Conductance * Head Difference) is calculated, it is converted into a drain sink term
82
Wd (which is part of total sink term W) by dividing the calculated discharge by the total
volume of the drain cell such that Wd = Qd/(dx dy dz). The drain conductance is a
lumped parameter describing all of the head losses between the drain and the region of
the cell.
Direct recharge to an aquifer through a sinkhole can be treated as a negative drain
(source term). Springs are treated the same as drains (sink terms) by assigning a proper
conductance and a specified discharge head to each spring.
Boundary Conditions
Boundary conditions on all six sides of the flow domain must be known prior to
solving the governing groundwater flow equation. Typically, three types of boundary
conditions can be described along the boundaries: specified flux (Neumann); specified
pressure head (Dirichlet); and variable (between Neumann and Dirichlet) boundary
conditions. In each boundary condition, the prescribed values either can be constant or
they can vary with time.
Specified Flux Boundary Condition
Specified flux boundary conditions can be used to describe the rainfall
(infiltration), evaporation, and seepage processes. These processes are treated as sink or
source terms at the boundary element faces. No-flow boundaries and impervious
boundaries with zero flux also can be classified in this category. A specified boundary
condition can be defined formally as
83
q1 / 2 ( h ) = q p ( x b , y b , z b , t )
(3.85)
where q1/2 (h) is the flux at the boundary, and qp is the specified flux (evaporation or
infiltration rates) at the boundary nodes.
A detailed mathematical description of the boundary conditions in terms of the
finite-difference method and iteration procedures is presented in chapter 4 of this study.
Specified Head Boundary Condition
Specified head boundary conditions can be defined if there is a constant head
water body such as a lake, river, etc. The specified head boundary condition can be
written formally as
h = h d (x b , y b , z b , t )
(3.86)
where h is the pressure head at the boundary node, and hd is the specified head at the
boundary coordinate of (xb, yb, zb).
Variable Boundary Condition
Variable boundary conditions are used to describe the evaporation from the soil
surface and infiltration due to rainfall. These two hydrologic events have two stages such
that in one stage they are described as a flux boundary and in the other stage they are
treated as a constant specified head boundary condition. Variable boundary conditions
are called “variable” because they vary between flux boundary and specified head
84
boundary conditions during the simulation depending on the potential evaporation, the
conductivity of medium, and the availability of water such as rainfall.
Infiltration is a two-stage procedure. In the first stage, all rainfall enters the
system at the applied rate as long as the conductivity and sorptive capacities of the
medium are not exceeded. If these capacities are exceeded, water ponds on the surface
and infiltration decreases asymptotically to a value equal to saturated hydraulic
conductivity of the medium (Lappala et al., 1987). Rubin (1966) and Smith (1972)
presented extensive discussions of the ponding process and reported that surface runoff
cannot occur until ponding has begun. Using this concept, a maximum ponding depth
value (hpmax) is assigned for each top surface cell in the flow domain to handle the
rainfall-runoff-infiltration procedure. At land surface, two boundary conditions are
possible:
q1 / 2 ( h ) = p p ( x b , y b , z b , t )
if t ≤ tp
(3.87a)
h = h p max ( x b , y b , z b , t )
if t > tp
(3.87b)
and
where q1/2 (h) is the vertical flux at the boundary, pp is the rainfall rate flux, and tp is time
of ponding, which is determined during the iterations.
Evaporation is also a two-stage procedure at the boundary nodes, and it is
analogous to precipitation. It is dependent on both the potential evaporative demand of
the atmosphere and the ability of the porous medium to conduct water to the surface.
During the first stage of evaporation, there is an outward flux boundary at the surface.
85
This continues as long as water is conducted to the top layer soil where the soil moisturecontent is greater than the residual moisture-content, which can also be described as (hmin)
in terms of pressure head. In the second stage, evaporation ceases because of a lack of
moisture, and boundary conditions are set to hmin minimum pressure so that no more
water can be extracted from the soil. These two stages are described mathematically as
q1 / 2 ( h ) = E a ( x b , y b , z b , h , t )
if h1/2 ≥ hmin
(3.88a)
h = h min ( x b , y b , z b , t )
if h1/2 < hmin
(3.88b)
and
where Ea is the actual evaporation, h1/2 is the pressure head at the face of the boundary
cell, and hmin is the minimum pressure at the residual water content level.
River Boundary
In the model, one-dimensional steady-state open-channel flow can interact with
the underlying porous medium. It is necessary to specify the river heads for each time
step. Actually the river boundary acts as a specified sink/source term in the top boundary.
The flow exchange between the river and the porous media can be calculated using
Darcy's equation based on the head gradient between the river and the underlying porous
media and the hydraulic conductance of the river bed material:
qr = Cr
Hr − H
DZ r
(3.87)
86
where qr is the aquifer river exchange per unit length of the river [L3T-1L-1], Cr is the
conductance term for the river bed (calculated by averaging the river bed hydraulic
conductivity and the hydraulic conductivity of the first cell beneath the river bed), Hr is
the river head, H is the hydraulic head in the first cell beneath the river bed, and DZr is
the distance between the river bottom and the first cell beneath the river bed.
At each time step, using the prescribed river heads, Hr, and heads of the porous
medium H from the previous iteration level, the flow exchange qr between the river and
underlying groundwater system is calculated using equation (3.87). In the new iteration
level, the calculated qr is used as a specified flux boundary condition for top cells having
river segments superimposed on them. This procedure is followed for each river cell for
every time step.
General Head Boundary
A general head boundary can be used to provide a flow into or out of an active
cell at a boundary from an external source far from the boundary. That flow is calculated
as a function of the head difference between the active cell and the external source and
the conductance between the external source and the boundary cell. Functionally, this
type of boundary is similar to a drain or a river boundary (McDonald and Harbaugh,
1988). To specify a general head boundary condition, the head at the external source
(Hext) and the distance (Xghb) between the source and the boundary cell (which is used to
calculate the conductance between the boundary and the external source) need to be
specified. The flow between the external source and the boundary cell is calculated using
equation 3.88.
87
q ghb = −
K
(H ext − H)
X ghb
(3.88)
where qghb is flux into or out of the boundary cell [LT-1], K is the hydraulic conductivity
between the boundary cell and the external source [LT-1], Xgbh is the distance between the
boundary cell and the external source [L], Hext is the specified head of the external source
[L], and H is the head at the boundary cell.
CHAPTER 4
MATHEMATICAL MODEL DEVELOPMENT AND NUMERICAL SOLUTION OF
THE MODIFIED RICHARDS EQUATION
As described in this chapter, a new variably saturated rainfall-driven groundwaterpumping model has been developed. The model simulates a hydrogeologic system by
solving the nonlinear, three-dimensional form of the modified Richards equation
continuously throughout the whole flow domain from ground surface to the impervious
bottom of the lowest layer. The finite-difference method with a variable finite-difference
grid is used to solve the governing equation. The upper boundary in the model is at land
surface, and the upper boundary conditions are determined using soil and meteorological
data. The model treats the complete subsurface regime as a unified whole, and the flow
in the unsaturated zone is integrated with saturated flow in the underlying unconfined and
confined aquifers. The model allows modeling of heterogeneous and anisotropic
geologic formations. A plant root water uptake (transpiration) model and an evaporation
model are included in the governing flow equation as a sink term and boundary condition,
respectively, in the model.
In the previous chapter, the partial differential form of the three-dimensional
modified Richards equation was developed. Boundary conditions and sink/source terms
were mathematically described. This new numerical model was developed based on
those partial differential equations. The model was mathematically conceptualized and
developed using finite-difference approximation methods. The resulting finite-difference
88
89
equations form matrixes that are quite large depending on how many nodes and time
steps are considered. For example, if a groundwater basin is discretized into 30 rows, 30
columns, and 30 layers, then the model generates one equation with seven unknown
heads (one for the cell itself and six for the neighboring cells) for each of 30x30x30 =
27,000 cells. The resulting matrix has to be solved for each time step, which would
require excessive computer memory and time if a direct solution method were used to
obtain the solution. Thus, iteration methods generally are required to solve matrices of
this size. In this investigation, the modified Picard scheme of Celia et al. (1990) with the
preconditioned conjugate gradient method was selected because this method has been
demonstrated by other investigators (e.g., Clement et al., 1994) to be more mass
conservative than other iteration methods.
Conceptualization of the Model
The model has a modular structure similar to MODFLOW (McDonald and
Harbaugh, 1988), which makes it possible to modify the model by adding new features
later. Each item in the source/sink terms and boundary conditions is modeled as a
module of the main body of the model.
The physical system considered in this study includes the coupled physics of the
soil, vegetation, and atmospheric events such as precipitation, daily maximum and
minimum temperatures, etc. This system, which is conceptualized to be threedimensional (but consisting of a vertically dominant flow system in the unsaturated zone
and a horizontally dominant flow system in the saturated zone), is depicted in Figure 4.1
with the interlinking flow processes. The flow processes in the variably saturated zone
90
are of primary concern in this study. The simulation of soil moisture flow under moistu
re extraction by crop roots involves consideration of several variables and factors, which
are listed below (see Figure 4.1):
Rainfall data;
Meteorological data for prediction of evapotranspiration;
Crop characteristics such as root depth, growth pattern, leaf area index, and root
water uptake pattern; and
Soil layering and soil properties for each soil layer.
With the input information indicated above, the model predicts evaporation loss,
transpiration loss, soil moisture-content, pressure head, and, consequently, the position of
the water-table for each time interval.
Atmospheric Boundary layer
Precipitation
Transpiration
(Root Water Uptake)
Evaporation
Unsaturated/Saturated Zone
Infiltration+Drains + Pumping + Leakage to/from Water Bodies
Leakage
Confining Unit
Leakage
Confined Aquifers and Confining Units
+
Pumping
Figure 4.1 Schematic representation of the physical components and the interaction
among them.
91
Spatial Discretization
A variable grid size can be used in the model to overcome the difficulties of the
nonlinearity of the unsaturated zone and to reduce computer time to solve the whole flow
domain. The nonlinearity of the hydraulic process creates large gradients in soil pressure
and soil moisture-content during the infiltration and evapotranspiration processes. It is
therefore important to select appropriately small node increments in the z-direction. As a
general guideline, a small-scale spatial resolution is recommended in the top nodes and
around the capillary fringe zone of the model domain. The unsaturated and unconfined
zones should be discretized with enough resolution not to have large mass balance errors
and numerical instability during the simulation. The confined zone can be modeled as a
large one-layer cell (in the vertical direction) if the physical properties of the confined
zone are the same over the entire zone. A confining unit can be modeled if it has some
storage capacity, or it can be incorporated in the vertical conductivities between two
aquifer layers. If there is a possibility that the upper part of a confined aquifer might
become unconfined and thus unsaturated during a simulation, then the upper part also
should be discretized into layers, and soil moisture properties should be input along with
the properties of the confining units at the beginning of a simulation. Proper vertical
discretization is very important in that a good discretization saves computer time while
correctly describing physical phenomena that change significantly with depth in the
variably saturated region (see Figure 4.2). Horizontal discretization is done spatially
based on the variation of soil, vegetation types, and aquifer properties.
92
Figure 4.2 Vertical discretization of the model.
Temporal Discretization
During long-term simulations, usage of small time steps results in excessive
amounts of computer time and memory requirements. Therefore, a time step calculation
adjustment procedure should be employed based on test runs to achieve optimum
simulation time. During a rainfall period, small time steps should be used. After rainfall
ceases, time steps can be increased by multiplying the time step interval by a user defined
factor that increases the time step until the prescribed maximum iteration number is
reached for that time step increment, or until the maximum time step increment is
reached, whichever comes first. During long-term dry periods, the maximum allowable
time step is used. Other than the dependency on the upper boundary condition, the model
adjusts the time step by comparing the maximum head changes in the flow domain with
the user defined allowed maximum head change. The model estimates the maximum
head change for the next time step by linearly extrapolating the maximum previous time
93
step head changes. If that estimate is larger than the user specified allowable head
change, then the new time step is decreased by a proportionality factor, which is the ratio
of the allowable head change to the estimated head change. A time step reduction occurs
if the maximum number of iterations is exceeded without the solution converging. In that
case, the time step is reduced by a user supplied factor.
Finite-difference Formulation of the Governing Equation
The groundwater flow equation in finite-difference form is developed by applying
the continuity equation to a unit volume of aquifer such that the sum of all flows into and
out of a cell must be equal to the rate of change in storage within the cell. Under the
assumption of constant groundwater density, the continuity equation (equation (3.5)) can
be written in a finite-volume format as
∆H
∑ Q i + Q e = SS ∆t
∆V
(4.1)
where Qi is the flow rate into (or out of) the cell [L3T-1] from (or to) adjacent cells, Qe is
the external flow due to sink/source terms, SS is a general storage term [L-1] equivalent to
equation (3.11), ∆H is the change in hydraulic head in the cell with respect to ∆t, the time
step, and ∆V is the volume of the cell [L3]. The right hand side of the equation (4.1) is
equivalent to the volume of water taken into storage over a time interval ∆t given an
increase in hydraulic head of ∆H. In equation (4.1), inflows and sources are considered
as positive, and outflows and sinks are considered as negative (Figure 4.3).
94
Figure 4.3 Flow into and out of cell i, j, k.
Figure 4.3 depicts a cell at i,j,k in the x-direction and two adjacent cells at i-1,j,k
and i+1,j,k. From the Darcy-Buckingham equation (3.17), Qi-1/2,j,k (inflow to cell i,j,k)
can be written as
Q i −1 / 2, j,k = CN i −1 / 2, j,k (dy j dz k )(H i, j,k − H i −1, j,k )
(4.2)
where Hi,j,k and Hi-1,j,k are the hydraulic heads (H = z + p / γ) at nodes i,j,k and i-1,j,k,
respectively, Qi-1/2,j,k is the volumetric fluid discharge through the face between cells i1,j,k and i,j,k [L3T-1], dyjdzk is the area of the cell face perpendicular to the flow
direction, and CNi-1/2,j,k is the conductance between cells i,j,k and i-1,j,k calculated as
CN i−1/ 2, j,k = −
(K s x K r ) i−1/ 2, j,k
(dx i + dx i−1 ) / 2
(4.3)
95
where (Ks)x is the saturated hydraulic conductivity [LT-1] in the x-direction between the
cells i,j,k and i-1,j,k, and (Kr)i-1,j,k is dimensionless relative hydraulic conductivity
( K r (h ) =
K (h )
) between the nodes i,j,k and i-1,j,k that is calculated by averaging the
Ks
relative hydraulic conductivities of cells i,j,k and i-1,j,k. Kr is a function of the pressure
head h ( h i , j,k = H i, j,k − Z i, j,k ) which takes values between 0.0 for dry, unsaturated
conditions and 1.0 for saturated conditions. (Methods of averaging hydraulic
conductivities are discussed later). Similarly, the flow between cells i,j,k and i=1,j,k, or
Qi+1/2,j,k (i.e., the outflow from cell i,j,k), can be written as
Q i +1 / 2, j,k = CN i +1 / 2, j,k (dy j dz k )(H i +1, j,k − H i, j,k )
CN i+1/ 2, j,k = −
(K s x K r ) i+1/ 2, j,k
(dx i + dx i+1 ) / 2
(4.4)
(4.5)
If the same equations are written for y and z directions by visualizing the same
orientation in figure 4.3 in the y and z-directions, the following relations are obtained for
discharges and related conductances:
Inflow and outflow of cell i,j,k in y-direction:
Q i, j−1 / 2,k = CN i , j−1 / 2,k (dx i dz k )(H i, j,k − H i , j−1,k )
CN i, j−1/ 2,k = −
(K s y K r ) i, j−1/ 2,k
(dy j + dy j−1 ) / 2
Q i, j+1 / 2,k = CN i , j+1 / 2,k (dx i dz k )(H i, j+1,k − H i, j,k )
(4.6)
(4.7)
(4.8)
96
CN i, j+1/ 2,k = −
(K s y K r ) i, j+1/ 2,k
(dy j + dy j+1 ) / 2
(4.9)
Inflow and outflow of cell i,j,k in z-direction:
Q i, j,k −1 / 2 = CN i, j,k −1 / 2 (dx i dy j )(H i , j,k − H i, j,k −1 )
CN i, j,k −1/ 2 = −
(K s z K r ) i, j,k −1/ 2
(dz k + dz k −1 ) / 2
Q i, j,k +1 / 2 = CN i, j,k +1 / 2 (dx i dy j )(H i , j,k +1 − H i, j,k )
CN i, j,k +1/ 2 = −
(K s z K r ) i, j,k +1/ 2
(dz k + dz k +1 ) / 2
(4.10)
(4.11)
(4.12)
(4.13)
Equations from (4.2) to (4.13) account for the flow into and/or out of cell i,j,k
from the six adjacent cells. External flows coming in and out of cell i,j,k and
sources/sinks are included in the Qe term of the continuity equation (4.1) as follows:
(Q e ) i , j,k ,n = CNSi , j,k ,n (H i, j,k − HS n ) + (Q se ) i, j,k ,n
(4.14)
where (Qe) i,j,k,n represents the flow from the nth external source/sink into cell i,j,k [L3T-1].
The first term of equation (4.14) represents specified head source/sink terms (i.e., drains,
sink holes, root water uptake, etc.), where CNSi,j,k,n is the conductance term between the
aquifer cell and the nth source/sink environment [L2T-1] and HSn is the specified head for
97
that source/sink. The second term represents the specified flux type source/sink [L3T-1]
(i.e., pumping or injection wells), and Qse is the specified flux for that source/sink.
In general, if there are N external sources and sinks in a single cell, the combined
flow is expressed by
Qe
i , j, k , n =
∑n=1 CNSi, j,k ,n (H i, j,k − HSn ) + ∑n =1 (Q se ) i, j,k ,n
n=N
n=N
(4.15)
Applying the continuity equation (4.1) to cell i,j,k, and taking into account the
flows from the six adjacent cells as well as the source/sink flow rates, Qe, gives
Q i−1/2, j,k − Q i+1/2, j,k + Q i, j−1/2,k − Q i, j+1/2,k + Q i, j,k −1/2 − Q i, j,k +1/2 + (Q e ) i, j,k
= SSi, j,k
ÄH i, j,k
Ät
(4.16)
ÄVi, j,k
The right hand side of the equation (4.16) can be written in backward difference
form with respect to time as follows:
SSi , j,k
∆H i, j,k
∆t
∆Vi, j,k = SSi, j,k
(H im, j+,k1 − H im, j,k )
∆t
dx i dy j dz k
(4.17)
where SSi,j,k is the general storage term defined in equations (3.11) and (3.18) and which
can be written in finite-difference form as
SSi , j,k = C(h i , j,k ) + (S w Ss ) i, j,k
(4.18)
98
When the equations from (4.2) to (4.13) together with the equation (4.17) are
substituted into equation (4.16), after rearranging there results
m +1
CN i −1 / 2, j, k (
m +1
m +1
H i , j,k − H i −1, j, k
m +1
+ CN i , j−1 / 2, k (
dx i
m +1
m +1
) − CN i +1 / 2, j, k (
m +1
H i , j,k − H i , j−1, k
dy j
m +1
m +1
H i +1, j, k − H i , j, k
m +1
) − CN i , j+1 / 2, k (
dx i
m +1
)
m +1
H i , j+1,k − H i , j, k
dy j
)
(4.19)
m +1
+ CN i , j, k −1 / 2 (
m +1
+
(Q e ) i , j, k
dx i dy j dz k
m +1
m +1
H i , j,k − H i , j,k −1
dz k
m +1
= SS i , j, k
m +1
) − CN i , j,k +1 / 2 (
m +1
m +1
H i , j,k +1 − H i , j, k
dz k
)
H im, j+, k1 − H im, j, k
t m +1 − t m
where m denotes the previous time step, and m+1 denotes the current time step.
Mixed Form of Richards Equation and Modified Picard Iteration Scheme
As discussed in chapter 3, the mixed form of the Richards equation has
advantages over the pressure-based Richards equation because the former is more mass
conservative then the latter. The mixed form can be solved in a computationally efficient
manner, and it is capable of modeling a wide variety of problems, including infiltration
into very dry soils (Kirkland et al., 1992). The modified Picard iterative procedure for the
mixed flow form of the Richards equation is fully mass conservative in the unsaturated
99
zone. A detailed analysis of the mass conservative property of the modified Picard
solution to the mixed form of Richards equation has been given by Celia et al. (1990).
The essence of the Picard iteration technique is to iterate with all the linear
occurrences of hm+1 at the current (k+1) iteration level and the nonlinear occurrences at
the previous (k) iteration level.
The mixed form of the Richards equation is obtained by writing the storage term
in its simpler form. Following that, the right hand side of the equation (3.18) becomes
(C(h ) + S w Ss ) ∂H =  ∂θ + S w Ss  ∂H = ∂θ + S w Ss ∂H
∂t
 ∂h
 ∂t
∂t
∂t
(4.20)
The time derivative of the moisture-content term in equation (4.20) is discretized
according to the Picard iteration scheme as
m +1, k +1
− θ im, j,k 
∂θ  θ i, j,k
=

∂t 
∆t


(4.21)
where m is the time step, and the superscript k represents the Picard iteration level. The
term θ im, j+,k1,k is expanded using a first-order truncated Taylor series in terms of the
pressure-head perturbation arising from Picard iteration about the expansion point
(θ
m +1, k
i , j, k ,
)
h im, j+,k1,k as (Celia et al., 1990):
100
θim, j+,k1,k +1 ≅ θim, j+,k1,k +
m+1,k
[
dθ
H im, j+,k1,k +1 − H im, j+,k1,k
dh i, j,k
]
(4.22)
Substituting equation (4.22) into equation (4.21) with the definition of C (h) =
dθ
dh
as the specific water capacity, the time derivative of the moisture-content is approximated
as
m +1, k
m
 H im, j+,k1,k +1 − H im, j+,k1,k 
∂θ  θ i , j,k − θ i, j,k 
m +1, k
≅

 + C(h ) i, j,k 
∆t
∂t 
∆t






(4.23)
The first term on the right side of equation (4.23) is an explicit estimate for the
time derivative of moisture-content, based on the kth Picard level estimate of pressure
head. In the second term of the right side of Equation (4.23), the numerator of the
bracketed fraction is an estimate of the error in the pressure head at node i,j,k between
two successive Picard iterations. Its value diminishes as the Picard iteration process
converges. As a result, as the Picard process proceeds, the contribution of the specific
water capacity, C(h), is diminished (Clement et al., 1994). After equation (4.23) is
substituted into equation (4.20) replacing the right side of the equation (4.19):
101
m +1, k +1
m +1, k
CN i −1 / 2, j, k (
dx i
m +1, k +1
m +1, k
+ CN i , j−1 / 2,k (
+ CN i , j,k −1 / 2 (
m +1, k
m +1, k +1
dy j
m +1, k
) − CN i , j+1 / 2,k (
m +1, k +1
H i , j,k − H i , j,k −1
dz k
m +1, k +1
m +1, k
m +1, k +1
H i +1, j, k − H i , j,k
) − CN i , j,k +1 / 2 (
dx i
m +1, k +1
)
m +1, k +1
H i , j+1,k − H i , j,k
dy j
m +1, k +1
)
m +1, k +1
H i , j,k +1 − H i , j,k
dz k
)
(4.24)
 θ im, j+, k1,k − θ im, j, k 
 H im, j+,k1,k +1 − H im, j+, k1,k 
m +1, k
C
(
h
)
=  m +1

+

i , j, k 
− t m 
t m +1 − t m
 t


(Q e ) i , j,k
dx i dy j dz k
+ (S w S s )
m +1, k
) − CN i +1 / 2, j,k (
H i , j,k − H i , j−1, k
m +1, k +1
m +1, k
+
m +1, k +1
H i , j,k − H i −1, j,k
H im, j+,k1,k +1 − H im, j, k
t m +1 − t m
where h is pressure head equal to (H - z). The moisture-content and the specific moisture
capacities are functions of pressure head, not functions of total (hydraulic) head.
Therefore, they are evaluated using h values at each iteration level.
The finite-difference expressions for the spatial and temporal derivatives in
equation (4.24) are rearranged by collecting all the unknowns on the left side and all the
knowns on the right side, in agreement with equation (3.18):
cH im, j+,k1,−k1+1 + bH im, j+−11,,kk+1 + aH im−1+,1j,,kk+1 + dH im, j+,k1,k +1
(4.25)
+
eH im+1+,1j,,kk+1
+
fH im, j++11,,kk+1
+
gH im, j+,k1,+k1+1
= RHS i, j,k
102
where the coefficients a, b, c, d, e, f, g, and RHS are defined as
a=−
b=−
c=−
e=−
f =−
g=−
p1 =
p2 =
CN im−1+1/ ,2k, j,k
(4.26)
dx i
CN im, j+−11,/k2,k
(4.27)
dy j
CN im, j+,k1,−k1 / 2
(4.28)
dz k
CN im+1+1/ ,2k, j,k
(4.29)
dx i
CN im, j++11,/k2,k
(4.30)
dy j
CN im, j+,k1,+k1 / 2
(4.31)
dz k
C(h im, j+,k1,k )
(4.32)
t m+1 − t m
t
S w Ss
m +1
m
(4.33)
−t
d = −(a + b + c + e + f + g + p1 + p 2 )
s=
(4.34)
θ(h im, j+,k1,k ) − θ(h im, j,k )
(4.35)
t m+1 − t m
RHSi, j,k = s − p1 * H im, j+,k1,k − p 2 * H im, j,k −
Q em+1,k
dx i dy j dz k
(4.36)
103
In equations (4.25) through (4.36), the superscript m+1,k+1 denotes that the value
is unknown (to be calculated), the superscript m+1,k means that the value is known from
the previous iteration level in the current time step, and the superscript m means that the
value is known from the previous time step in the last iteration level.
Equation (4.25) is the final form of the finite-difference equation for one cell.
This equation applies to all interior cells in the flow domain. At the boundary nodes, this
equation is modified to reflect the appropriate boundary conditions. The resulting system
of linear equations can be written in a matrix form such that:
[A] {H} = {RHS}
(4.37)
where [A] is a square matrix consisting of the coefficients(a,b,c,d,e,f,g) of the finitedifference equation(4.25), {H} is unknown total head values for current time step, and
{RHS} is the forcing vector consisting of known values from the previous time step or
previous iteration values.
Boundary Conditions
Equation (4.25) is modified at the boundary nodes to reflect the boundary
conditions. There are basically two types of boundary conditions included in the model,
i.e., prescribed head or prescribed flux boundaries. In addition to these boundary
conditions, a seepage boundary condition may occur where water leaves the model
domain from saturated surfaces such as stream banks, levees, etc. However, in this study,
the seepage boundary is not included because its effects are considered negligible in a
regional scale model.
104
To specify prescribed flux boundary conditions on the outer face of boundary
nodes, fictitious or imaginary nodes are added outside of the boundary (i.e., at node 0 and
node n+1 in the x-direction, node 0 and node m+1 in the y-direction, and node 0 and l+1
in the z-direction are imaginary nodes). Those nodes are treated as if they were real
nodes in the derivation of the finite-difference equations at the boundaries but then
subsequently the corresponding conductances and heads belonging to those imaginary
nodes are canceled mathematically in the main finite-difference equation (4.25). Thus,
these imaginary nodes do not appear in the final form of the governing equation. Then,
internally, the prescribed known flux values are added to the right side of the equation
(4.25) by the model.
Prescribed head boundaries
The prescribed head boundary, called the Dirichlet boundary condition in the
literature, is defined where total heads are known. Dirichlet boundaries are described by
H i, j,k = (H d ) i, j,k
(4.38)
where (Hd)i,j,k is the known head at Dirichlet boundary nodes i,j,k. The matrix coefficients
a, b, c, e, f, and g in the finite-difference equation (4.25) are zero, and d equals to unity.
The corresponding term RHSi,j,k in the forcing vector {RHS} is equal to the known heads
(Hd)i,j,k.
105
Prescribed flux boundaries
The prescribed flux boundary condition is called the Neumann boundary
condition in the literature. Rainfall and evaporation are described with Neumann
boundary conditions. Using the Darcy-Buckingham equation, known values of normal
fluxes, qn, are specified at Neumann (or flux) boundary nodes. If a no-flow boundary
condition occurs, then the following equation is written for a no-flow boundary to the
right as
H N +1, j,k = H N , j,k
(4.39)
where HN+1,j,k is the imaginary node outside of the boundary in x-direction.
The horizontal flux at a (right) boundary node is written in (forward) finitedifference form as
Q N = Q N +1 / 2, j,k = CN N +1 / 2, j,k (dy j dz k )(H N +1, j,k − H N, j,k )
H N +1, j,k =
QN
+ H N, j,k
CN N +1 / 2, j,k (dy j dz k )
(4.40)
(4.41)
where QN is the specified flux in the x direction (positive outward direction) at node
N+1/2,j,k where N+1/2 is the right boundary in x-direction. CNN+1/2,j,k is a conductance
term, but it will disappear inside equation (4.25) when it is multiplied by coefficient e
(equation (4.29)). Therefore, it is not necessary to calculate conductance terms at the
106
boundaries. Equation (4.41) is substituted in to equation (4.25) and rearranged to obtain
the finite-difference equation at the right boundary (node i = N):
+1, k +1
m +1, k +1
m +1,k +1
m +1, k +1
cH m
N , j, k −1 + bH N , j−1, k + aH N −1, j,k + (d + e) H N , j, k
+1, k +1
+ fH m
N , j+1, k
+1, k +1
+ gH m
N , j, k +1
= RHS N , j,k
QN
+
dx N dy j dz k
(4.42)
where it should be noticed that term e in (d+e) disappears such that (from equation 4.33):
d + e = −(a + b + c + e + f + g + p1 + p 2 ) + e
(4.43)
= −(a + b + c + f + g + p1 + p 2)
Similarly, at the left boundary in the x-direction:
Q x1 = Q1 / 2, j,k = CN 1 / 2, j,k (dy j dz k )(H1, j,k − H 0, j,k )
H 0, j,k =
Q x1
+ H1, j,k
CN 1 / 2, j,k (dy j dz k )
(4.44)
(4.45)
where QX1 is the specified flux at the left boundary in the x direction (positive inward
direction) at node 1/2,j,k. Equation (4.45) is substituted into equation (4.25) and
rearranged to obtain the finite-difference equation at the left boundary (node i=1):
107
cH1m, j+,k1,−k1+1 + bH1m, j+−11,,kk+1 + (d + a )H1m, j+,k1,k +1
+
+1, k +1
eH m
2, j, k
+
fH1m, j++11,,kk+1
+
gH 1m, j+,k1,+k1+1
= RHS1, j,k
(4.46)
Q x1
+
dx 1dy j dz k
where it should be noticed that term a in (d+a) disappears such that (from equation 4.33)
d + a = −(a + b + c + e + f + g + p1 + p 2 ) + a
(4.47)
= −(b + c + e + f + g + p1 + p 2)
Similarly, at the front boundary in the y-direction at node i, j = 1, k,
Q y1 = Q i,1 / 2,k = CN i,1 / 2,k (dx i dz k )(H i ,1,k − H i, 0,k )
H i , 0, k =
Q y1
CN i ,1 / 2,k (dx i dz k )
(4.48)
+ H i,1,k
(4.49)
where Qy1 is the specified flux at front boundary in the y-direction (positive inward
direction) at node i,1/2,k. Equation (4.49) is substituted into equation (4.25) and
rearranged to obtain a finite-difference equation at the front boundary (node j = 1):
cH im,1+,k1,−k1+1 + aH im,1+,k1,k +1 + (d + b)H im,1+, k1,k +1
+
eH im+1+,11,,kk+1
+
fH im, 2+,1k,k +1
+
gH im,1+,k1,+k1+1
= RHS i,1,k +
Q y1
dx i dy1dz k
(4.50)
108
where it should be noticed that term b in (d+b) disappears such that (from equation 4.33)
d + b = −(a + b + c + e + f + g + p1 + p 2 ) + b
(4.51)
= −(a + c + e + f + g + p1 + p 2)
Similarly, at the rear boundary in the y-direction at node i, j = M , k,
Q M = Q i,M +1 / 2,k = CN i,M +1 / 2,k (dx i dz k )(H i,M +1,k − H i,M ,k )
H i,M +1,k =
QM
+ H i,M +1,k
CN i,M +1 / 2,k (dx i dz k )
(4.52)
(4.53)
where QM is the specified flux at the back boundary in the y-direction (positive outward)
at node i,1/2,k. Equation (4.55) is substituted into equation (4.26) and rearranged to
obtain a finite-difference equation at the back boundary (node j = M):
cH im,M+1,,kk−+11 + bH im,M+1−,1k,+k1 + aH im,M+1,,kk +1 + (d + f )H im,M+1,,kk +1
+
eH im+1+,1M,k,+k1
+
gH im,M+1,,kk++11
= RHSi ,M ,k
QM
+
dx i dy M dz k
(4.54)
where it should be noticed that the term f in (d+f) disappears such that (from equation
4.34)
109
d + f = −(a + b + c + e + f + g + p1 + p 2) + f
(4.55)
= −(a + b + c + e + g + p1 + p 2)
The top boundary in z-direction is very important because major inflows and
outflows, i.e., precipitation and evaporation, take place at this boundary. The boundary
condition is subject to changing from a Dirichlet to a Neumann condition or from a
Neumann to a Dirichlet condition during a simulation and this change is tracked by the
model during a simulation to make necessary changes in the boundary conditions from
Dirichlet to Neuman or vice versa. The Neumann boundary condition at land surface
(node L+1/2, figure 4.2) can be derived from
Q T = Q i , j,L+1 / 2 = CN i , j,L +1 / 2 (dx i dy j )(H i, j,L +1 − H i, j,L )
H i, j,L +1 =
QT
+ H i, j,L
CN i , j,L+1 / 2 (dx i dy j )
(4.56)
(4.57)
where QT is the specified flux at the top boundary (ground surface) in the z-direction
(positive outward) at node i,j,L+1/2. Equation (4.59) is substituted into equation (4.25)
and rearranged to obtain the finite-difference equation at the top boundary (node k = L):
cH im, j+,L1,−k1+1 + bH im, j+−11,,kL+1 + aH im−1+,1j,,kL+1 + (d + g )H im, j+,L1,k +1
+
eH im+1+,1j,,kL+1
+
fH im+1+,1j,,kL+1
= RHSi , j,L
QT
+
dx i dy jdz L
(4.58)
110
where it should be noticed that term g in (d+g) disappears such that (from equation 4.33)
d + g = −(a + b + c + e + f + g + p1 + p 2 ) + g
(4.59)
= −(a + b + c + e + f + p1 + p 2)
Generally, the bottom boundary condition is at an impervious layer, or no flow
boundary, such that QB is equal to zero at the bottom of the impervious layer (node i, j, k
= 1/2). The flux from the bottom boundary is expressed by Darcy's flow equation:
Q B = Q i , j,1 / 2 = CN i , j,1 / 2 (dx i dy j )(H i, j,1 − H i, j,0 )
(4.60)
If equation 4.60 is solved for H at the bottom boundary for a given specified flux
boundary condition, then,
H i, j, 0 =
QB
+ H i, j,1
CN i, j,1 / 2 (dx i dy j )
(4.61)
where QB is the specified flux (generally QB is zero) at the bottom boundary (impervious
layer) in the z-direction (positive inwards) at node i,j,k=0+1/2. Equation (4.61) is
substituted into equation (4.25) and rearranged to obtain the finite-difference equation at
the bottom boundary (node k=1):
111
bH im, j+−11,,k1 +1 + aH im−1+,1j,,k1 +1 + (d + c)H im, j+,11,k +1 + eH im+1+,1j,,k1 +1
+
fH im+1+,1j,,k1 +1
+ gH im, j+, 21,k +1
QB
= RHSi , j,1 +
dx i dy j dz k
(4.62)
where it should be noticed that term c in (d+c) disappears such that (from equation 4.33)
d + c = −(a + b + c + e + f + g + p1 + p 2 ) + c
(4.63)
= −(a + b + e + f + g + p1 + p 2)
River boundary
River boundary conditions can be applied to the top cells where a river crosses a
top cell. Indeed, the river boundary condition is a special form of a prescribed head
sink/source term, in which the river stage is specified although the cell containing the
river segment is active itself. The flow exchange between the river and the underlying
porous medium is calculated in the model using the river head, the head in the underlying
porous medium, and equation 3.87. A river may occupy the face of the cells partially or
completely. If the river partially occupies the top face of a cell, the conductance term Cr
in equation (3.87) should be reduced proportionally based on the percentage of the cell
occupied by the river. If the river occupies more than one cell, the head in these cells
occupied by the river segment is set to the river head Hr as a fixed head boundary
condition. In that case, the leakage between the river and the porous medium occurs as a
function of head difference and conductance between the river cells and the underlying
porous medium cells.
112
If a time series of river stage data (i.e., a hydrograph) for the period of simulation
is available, the river head Hr is set as the elevation of the water surface of the river,
which is used to calculate the river-aquifer exchange flow using equation (3.87). River
leakage to/from the groundwater (qr) is calculated during each iteration using the head in
the first node vertically under the riverbed. The model calculates qr at the nodes that have
river segments and applies it as a source/sink term to the right hand side of the equation
4.54 implicitly.
Overland flow and ponding
The overland flow component is considered as a loss after ponding occurs relative
to the groundwater system. In this study, no overland flow calculations are included,
which restricts the model to areas where the runoff process has minimal recharge effects
on the groundwater system. In the top boundary condition, a maximum ponding depth is
assigned for each top cell. During a rainfall event, the top boundary condition is changed
from a flux boundary to a fixed head boundary if the ponding depth is reached. Then, the
runoff process is assumed to occur as long as the maximum ponding depth is maintained
on the land surface during the rest of the rainfall event. During the simulation, if the
boundary conditions change from a flux boundary to a fixed head boundary or vice versa,
the calculations during that time step are repeated using the new boundary condition.
Rainfall and evaporation boundaries
Rainfall and evaporation are nested in the model as top boundary conditions. A
time series of rainfall data is supplied to the model as input. The model calculates the net
infiltration by calculating the net influx at the top first two nodes using the modified
Richards equation, which involves the hydraulic head gradient and the saturated hydraulic
113
conductivities of the top soil material at the land surface. Rainfall reaching the ground
surface is applied to the top boundary as a prescribed flux boundary condition. The
model checks the total head at the first node on the land surface. If it is greater than the
maximum ponding depth, this means that the infiltration capacity is reached, and the
model changes the boundary condition from a flux boundary to a prescribed head
boundary condition. The initial infiltration capacity can be higher than the value for the
saturated hydraulic conductivity because the hydraulic gradient can be greater than 1.0.
This can occur if the first node can be saturated but the second node from top is dry so
that it has a negative pressure head, which creates a very large hydraulic gradient. This
large hydraulic gradient can cause the infiltration rate to be greater than the value for the
saturated hydraulic conductivity.
Evaporation is evaluated using the same procedure as infiltration. The maximum
potential evaporation is calculated based on the climatological data supplied to the model.
Then, actual evaporation values are calculated based on the pressure head at the land
surface, the atmospheric pressure head, and the soil surface resistance. Similar to the
infiltration process, the evaporation process is also treated as a prescribed flux boundary
(outward from ground surface) until the soil moisture-content is reduced to a specified
minimum (e.g., the atmospheric pressure potential). After that, the boundary condition
becomes a prescribed head boundary condition (set as the minimum pressure head). The
procedures for calculating potential evaporation and actual evaporation are described in
detail in chapter 3.
114
Dewatering of a Confined Aquifer
Dewatering of a confined aquifer does not occur in the classical sense in this
model. If dewatering of the upper part of a confined aquifer is likely, then that part of the
aquifer should be discretized, and soil pressure-moisture characteristics for that part of
the aquifer should be input at the beginning of the simulation. Thus, that part of a
confined aquifer has to be treated as a variably saturated zone. During a simulation, the
model gives a warning if dewatering occurs.
Iteration Levels
In this model, two different iteration procedures are used as nested iterations in
each time step. In the outer iteration level, the modified Picard linearization iteration
scheme is used. The inner iteration procedure is for the conjugate gradient method for the
solution of the system of finite-difference equations (equation 4.37) subject to the
boundary conditions (equations 4.38 - 4.63).
A new convergence criterion is used for the modified Picard iteration scheme in
this study based on Huang et al. (1996). The new convergence criterion was derived
using a Taylor series expansion of the water content. The maximum change in the
moisture-content at every modified Picard iteration level is calculated. If the maximum
change in the moisture-content is smaller than the user defined moisture tolerance, then
the next time step calculations are started. This new convergence criterion is very useful
especially in such cases where the moisture-content changes dramatically with small
changes in the pressure head. Huang et al. (1996) reported that using the new
convergence criteria saves considerable computer time in simulations.
115
Conductance Terms (CNi+1/2,j,k)
Since the block-centered finite-difference scheme is used in this model, it is
necessary to average the conductance terms for adjacent blocks to obtain the conductance
between adjacent cells. Several authors have evaluated methods for determining these
intercell conductance terms. Lappala et al. (1987) suggested that the distance-weighted
harmonic mean is better than other methods in the saturated zone, although the distanceweighted arithmetic mean is better for some situations in the unsaturated zone. In
particular, using the distance-weighted arithmetic mean in the unsaturated zone yields
better results for very dry soils, because that method allows the wetting front to move
forward during the infiltration process.
The distance-weighted arithmetic means (in the unsaturated zone, i.e., Kr < 1.0)
for the unsaturated hydraulic conductivity and conductance are determined using the
following equations, respectively:
(K s K r ) i+1/ 2, j,k =
dx i (K s x K r )i , j,k + dx i+1 (K s x K r )i+1, j,k
dx i + dx i+1
 dx i (K s x K r ) i, j,k + dx i+1 (K s x K r ) i+1, j,k
CN i +1/2, j,k = −
(dx i + dx i+1 ) 2 /2




(4.64)
(4.65)
The right side of a cell in the x-direction is the left side of the next cell. Similarly,
the back side of a cell is the front side of the next cell in the y-direction, and the top of a
cell is the bottom of the next cell in the z-direction. Therefore, only the left, rear, and top
sides of the conductances of the cells are calculated. Equation (4.69) is the conductance
116
of the right side of a cell in the x-direction. The conductance of the rear side of a cell in
the y-direction is determined as
 dy j+1 (K s y K r ) i, j+1,k + dy j (K s y K r ) i, j,k
CN i, j+1/2,k = −

(dy j + dy j+1 ) 2 /2





(4.66)
The conductance of the top side of a cell in the z-direction is calculated by
 dz k (K s z K r ) i, j,k + dz k +1 (K s z K r ) i, j,k +1 

CN i, j,k +1/2 = −
(dz k + dz k +1 ) 2 /2


(4.67)
In the saturated zone (Kr = 1.0), the distance-weighted harmonic means for the
saturated hydraulic conductivity and conductance are calculated using the following
equations:
(K s ) i+1/ 2, j,k =
(dx i + dx i+1 ) / 2
dx i / 2
dx i+1 / 2
+
(K s ) x i, j,k (K s ) x i, j,k
CN i+1/ 2, j,k = −(
2(K s ) x i , j,k (K s ) x i+1, j,k
dx i+1 (K s ) x i, j,k + dx i (K s ) x i+1, j,k
(4.68)
)
(4.69)
Equation (4.69) is the conductance of the right side of a cell in the x-direction in
the saturated zone. The conductance of the rear side of a cell in the y-direction is
determined from
117

2(K s ) y i, j+1,k (K s ) y i, j,k
CN i, j+1/2,k = −
 dy j+1 (K s ) y
+ dy j (K s ) y i, j+1,k
i, j, k





(4.70)
The conductance of the top side of a cell in the z-direction (vertical conductivity)
is calculated by


2(K s ) z i, j,k +1 (K s ) z i, j,k

CN i, j,k +1/2 = −
 dz k +1 (K s ) z
+ dz k (K s ) z i, j,k +1 
i, j, k


(4.71)
If two layers are separated by a semi-confining unit that does not have any storage
capacity, then it is not necessary to include the semi-confining unit as a discrete layer
(Figure 4.4). Instead, the confining unit is included implicitly by calculating the vertical
conductance between two such upper and lower layers as
CN i, j,k +1 / 2 =
1
dz
dz k / 2
dz k +1 / 2
+ c +
(K s ) z i , j,k K c (K s ) z i , j,k +1
(4.72)
118
Figure 4.4 Diagram for calculation of vertical conductance in case of semi-confining
units.
In general, Kc is much smaller than the upper and lower hydraulic conductivities.
Therefore, the terms involving (Ks)i,j,k and (Ks)i,j,k+1 are negligible in equation (4.72) so
that the expression for vertical conductance becomes
CN i , j ,k +1 / 2 =
Kc
dz c
(4.73)
Equation (4.73) is equivalent to the leakance (K'/b') of a confining unit.
Matrix Equation Solver (Preconditioned Conjugate Gradient Method {PCGM})
The conjugate gradient method was originally proposed by Hestenes and Stiefel
(1952) to solve a system of linear algebraic equations in the form of [A]x = b, where [A]
is an n x n symmetric, positive-definite matrix. If exact arithmetic is used, convergence
will occur in m (m < n) iterations, where m is the number of distinct eigenvalues of [A].
119
The rate of convergence can be improved significantly if the original system can be
replaced by an equivalent system in which the modified matrix has many unit eigenvalues
(Schmit and Lai, 1994). The central idea of preconditioning is to construct a
transformation which has this effect on [A]. The preconditioning matrix [ M] is chosen
such that [A] = [M+N] and is symmetric and positive-definite, in which [M] is a matrix
easy to invert and resembles [A] as much as possible. The matrix produced in
groundwater-flow models generally is symmetric and positive-definite.
After each iteration, a system of linearized algebraic equations (Ax = b) is first
derived from equation (4.25) then solved using the preconditioned conjugate gradient
(PCG) method after incorporating the boundary conditions. The PCG method has a
number of attractive properties when used as an iterative method. Sudicky and Huyakorn
(1991) expressed the advantages of the PCG procedure as compared to other iterative
methods. The PCG method solves matrix problems by minimizing residuals, and it is a
good choice for solving transient problems (elliptical partial differential equations) with
banded matrices in the form of Ax = b. The PCG method never requires the complete
matrix A. It needs only the vector product Apk, where pk is the directional vector at PCG
iteration level k. To compute the initial directional vector pk, the PCG method requires
an initial estimate for heads h0, containing the pressures at all nodes. These initial
estimates are iteratively updated as the process converges toward the solution. The
solution for a given time step is usually a good estimate for the next, so the solution
converges quickly within a few iterations.
Convergence depends on the type of preconditioning. The Jacobi-iteration
preconditioner was chosen in this study because of its simplicity and easy application.
120
Other methods, such as polynomial preconditioning and incomplete Cholesky
preconditioning, also can be used. The incomplete Cholesky preconditioner and Jacobi
preconditioner methods were tested in a small scale matrix solution, and they both
converged very quickly. Since the applicability of the Jacobi method is very easy, it was
chosen to be used in this study. Various preconditioners may be used in the PCG method.
Among the different preconditioners there often is a direct relationship between increased
efficiency and increased computer storage (Meijerink and van der Vorst, 1974). To avoid
this tradeoff, the only preconditioner considered is that which produces a solver that has
computer storage requirements less than the strongly implicit procedure (SIP) as
programmed for most ground-water flow problems. The SIP method requires additional
computer storage equal to four arrays with dimensions equal to the number of grid nodes
(McDonald and Harbaugh, 1988, chap. 12). The influence of the preconditioner on the
solution procedure depends on [M] in the preconditioned conjugate gradient (PCG)
algorithm. When [M]-1 = [I], the PCG algorithm becomes a pure conjugate gradient
method. The more closely [M]-1 approximates [A]-1, the faster the convergence will be.
Figure 4.6 describes the relationship between the PCG, CG, and direct methods.
121
Figure 4.5 PCG methods (Schmit and Lai, 1994).
The general PCG method and the basic iteration procedure can be developed as
[A] {x}= {b}
(4.74)
A = M+N
(4.75)
M x k +1 = M x k + b − A x k
(4.76)
where k is the iteration index. If (b-Axk) is the residual rk of the original set of equations
(4.74) at the kth iteration, and sk = xk+1-xk then
M sk = r k ⇒
s k = M −1r k
(4.77)
The new x values then may be calculated as xk+1 = xk + sk. Some functions of sk
are used to calculate new xk+1 values in PCG method. The new change in x values (pk) is
calculated using the change from prior iteration pk-1, in addition to the vector sk of
equation (4.77). The PCG method algorithm is as follows:
122
r 0 = b − Ax 0
(4.78 a)
s k = M −1r k
(4.78 b)
s k

p =
s k + β k p k −1

for k = 0
k
(4.78 c)
for
k>0
x k +1 = x k + α k p k
(4.78 d)
r k +1 = r k − α k Ap k
(4.78 e)
where
βk =
{s k }T r k
{s k −1}T r k −1
(4.78 f)
αk =
{s k }T r k
{p k }T Ap k
(4.78 g)
where the superscript T indicates the transpose of the vector. Because rk+1 can be
calculated using the last statement, b need not be saved within the solver algorithm.
Iteration parameters α and β are calculated internally such that they are composed of
successive updating vectors, which are being calculated at each iteration.
The most important property of a good preconditioning matrix is that it should be
solved easily for sk in Equation (4.78b). The Jacobi preconditioning matrix M can be
formed from the diagonal terms of the original matrix A such that
123
M = a i,i
(4.79)
[M-1] can easily be inverted from [M] for equation (4.78b), which brings very fast
convergence to the iteration procedure.
In each Picard iteration level, [A] and {b} are updated and then several numbers
of PCG iterations are accomplished for convergence. PCG iterations are called as inner
iterations in this study. The total number of iterations for one time step is calculated as
the sum of the inner iterations for all updates of [A] and {b}. For any one [A] and {b},
the inner iterations continue until final convergence criteria are met.
CHAPTER 5
VERIFICATION OF THE MODEL
The numerical model developed in this study was verified using five examples
from the literature and two examples developed for this study. The boundary conditions,
input variables, and soil hydraulic properties for all the examples are given in detail in
this chapter. The first example is a one-dimensional problem, which deals with time
dependent recharge to a uniform soil column. In the second example, the numerical
model was tested against an analytical solution for one-dimensional infiltration to a
uniform soil column. The third example is an analytical solution for one-dimensional
infiltration to a two-layered soil having different soil properties. The fourth example is
for a two dimensional recharge experiment, and the fifth example is a three-dimensional
recharge (mounding) problem generated from the fourth example. The sixth example is
also a three-dimensional problem developed for this study in which a recharge area and a
pumping well exist. The seventh example is a comparison to a three-dimensional steadystate pumping problem.
As demonstrated in this chapter, the numerical model results reproduced very
closely the results of the five examples from the literature.
Example 1
Paniconi et al. (1991) described two test case problems (test cases 2 and 3) based
on the unmodified Richards equation to compare the performance of six different time
124
125
discretization strategies for simulations. A very dense fine grid, small time increments,
and a highly accurate Newton iterative solution were used as the base case “exact”
solution for these test problems. In this example, test case 2 was chosen to verify the new
model.
Paniconi et al. (1991) simulated the problem of infiltration and redistribution in a
10-m column with a flux at the surface that increased linearly with time and a constant
pressure head at the bottom to allow drainage. The boundary conditions for this example
are
q = t / 64 m / hr
at z = 10 m
h=0
at z = 0
(5.1)
(5.2)
The system is initially in hydrostatic equilibrium, i.e., h+z is constant over the
entire flow domain.
The material properties for this problem are derived from van Genuchten and
Nielsen’s (1985) closed form equation. For moisture-content, Paniconi et al. (1991)
modified van Genuchten and Nielsen’s relation to permit a non-zero value of specific
moisture capacity in the saturated zone. The hydraulic conductivity and moisture-content
equations are
[
Kr =
K (h )
= (1 + β) − 5m / 2 (1 + β) m − β m
Ks
Kr =
K (h )
=1
Ks
if
]
2
h≥0
if
h<0
(5.3a)
(5.3b)
126
and
θ(h ) = θ r + (θs − θ r )(1 + β) − m
if
h ≤ h0
θ(h ) = θ r + (θs − θ r )(1 + β 0 ) − m + Ss (h − h 0 )
(5.4a)
if
h > h0
(5.4b)
n
 h 
 , hb is air entry (or bubbling) pressures head[L-1], n is fitting parameter
where β = 

 hb 
in the moisture-retention curve, and m =1-1/n. θr is the residual water content, and θs is
the saturated moisture-content, which generally is equal to the porosity (η) of the
n
h 
formation. β 0 =  0  , and Ss is the value of specific storage when the pressure head h is
 hs 
greater than h0, which is a parameter determined on the basis of continuity requirements
imposed on Ss, which implies that
Ss =
( N − 1)(θs − θ r ) h
h s (1 + β) m+1
n −1
(5.5)
n
h =h 0
For a given Ss, equation (5.5) can be solved for h0.
The specific moisture capacity C (h) can be calculated from
127
C(h ) =
(n − 1)(θs − θ r ) h
C(h) = S s
n −1
h s (1 + β) m+1
n
when h ≤ h 0 ; and
when h > h 0
(5.6a)
(5.6b)
For a given Ss = 0 and h0= 0 these modified equations turn back to their original
form of the 1985 Van Genuchten and Nielsen equation.
The parameters appearing in equations (5-3a) through (5-6b) are given in Table 51. The column was discretized into 100 increments for the finite-difference formulation.
Grid information and other simulation parameters are summarized in Table 5-1. The
pressure head values versus elevation after 1, 2, 4, 10, and 32 hours of infiltration are
shown in figure 5-1. The simulation (CPU) time of this problem was about 4 minutes
using a 350-MHz, 128-MB ram, Pentium II computer.
Table 5.1 Parameters used for example 1
Flow domain
Relative hydraulic conductivity Kr(h)
Moisture-content θ(h)
Saturated hydraulic conductivity, Ks
Saturated moisture-content, θs
Residual moisture-content, θr
Air entry (bubbling) pressure head, hb
h0
Van Genuchten parameter, n
M= 1-1/n
Specific storage (h > h0 ), Ss
Top boundary condition
Bottom boundary condition
Initial pressure heads
Grid characteristics
Nodal spacing, dz
Time increment, dt
Maximum simulation time
10-m soil column
Equations (5.3a) and (5.3b)
Equations (5.4a) and (5.4b)
5 m/hr
0.45
0.08
-3.0 m
-0.19105548 m
3
0.667
0.001m-1
Specified flux, q = t/64 m/hr
Constant head, h=0 m
Hydrostatic equilibrium, h+z=0
Uniform grid with 100 elements
0.1 m
Variable from 0.001 hr to 0.1 hr
32 hr
128
10.00
t = 0 hr
1
2
4
Elevation (m)
8.00
10
32
6.00
4.00
2.00
The Numerical Model Results
Paniconi et al, (1991) Results
0.00
-10.00
-8.00
-6.00
-4.00
-2.00
0.00
Pressure Head h(m)
Figure 5.1 Comparison of the numerical model with results of Paniconi et al. (1991).
Example 2
Analytical solutions for one-dimensional, transient infiltration toward the watertable in homogeneous and layered soils developed by Srivastava and Yeh (1991) were
used to test the numerical model in examples two and three. In their solution, the
Richards equation governing one-dimensional vertical flow in the unsaturated zone is
linearized using exponential hydraulic-conductivity pressure head and moisture-contentpressure head relations. Although the exponential relations may be very restricted for
129
practical applications, they do serve as a means for verifying many numerical models for
unsaturated flow, especially for infiltration in very dry, layered soils where numerical
models often suffer from convergence and mass balance problems (Srivastava and Yeh,
1991).
The following relations were used for hydraulic conductivity and moisture-content
as a function of the pressure head:
K = K seα h
(5.7)
θ = θ r + (θs − θ r ) e α h
(5.8)
and
where K, and Ks are unsaturated and saturated hydraulic conductivities, respectively[LT1
], h is pressure head [L], θr and θs are residual and saturated moisture-contents
respectively, and α is a soil pore-size distribution parameter representing the rate of
reduction in hydraulic conductivity or moisture-content as h becomes more negative.
The one dimensional Richards equation is linearized using equations (5.7) and
(5.8) such that
∂ 2K
∂K α(θs − θr ) ∂K
+α
=
2
∂z
Ks
∂t
∂z
(5.9)
In this problem, one-dimensional vertical infiltration toward the water-table
through a homogeneous soil was considered. L is the depth to the water-table, so that z =
130
0 at the water-table and z = L at the ground surface. The bottom boundary is considered a
prescribed head boundary condition with h0 = 0. At t = 0, qa is the initial flux at the soil
surface, which determines the initial pressure distribution in the soil (along with h0), and
qb is the prescribed flux at the soil surface for times greater than 0. For convenience, the
following dimensionless parameters are defined and used in the rest of the equations:
z* = α z
(5.10a)
L* = α L
(5.10b)
K
Ks
(5.10c)
K* =
qa* = qa / Ks
(5.10d)
q b* = q b / K s
(5.10e)
and
t* =
áK s t
è s −è r
(5.10f)
Using equation (5.10), the linearized Richards equation and the initial and
boundary conditions can be written as
∂ 2K*
∂z*
2
+
∂K * ∂K *
=
∂z *
∂t *
(5.11)
K * (z* ,0) = q a * − (q a * − e αh 0 )e − z* = K 0 (z* )
(5.12a)
K * (0, t * ) = e αh 0
(5.12b)
131
and
∂K *
+ K*
= q b*
∂z*
z =L
*
(5.12c)
*
After taking Laplace transformations of equation (5.11) and the corresponding
boundary conditions, the following Laplace-space particular solution is obtained:
K=
K 0 (z* )
+ (q b* − q a * )e ( L * − z * ) F(s)
s
(5.13)
and


1/ 2


sinh[ z * ( s + 0.25) ]
1
F ( s) = 

s 1
sinh[ L* ( s + 0.25)1 / 2 ] + ( s + 0.25)1 / 2 cosh[ L* ( s + 0.25)1 / 2 ] 

 2
(5.14)
The inversion of this Laplace space solution is obtained using a numerical
inversion method developed by De Hoog et al. (1982). The solution of equation (5.13) is
for K*, which is substituted into equation (5.7) to obtain the pressure head (h) values.
The values of the parameters used in equations (5.8) through (5.12) are shown in
Table 5.2. The analytical solution and the numerical model solution results show a very
close match between these two models (see figure 5.2). The pressure head curves are
drawn for time 0, 1, 3, 5, 10, 15, 20, 30, 50, 75, and 100 hrs in figure 5.2 for both models.
132
Flow domain
100 cm uniform soil column
Hydraulic conductivity
Equation (5.7)
Moisture-content
Equation(5.8)
1.0 cm/hr
0.40
0.06
qa
0.1 cm/hr
qb
0.9 cm/hr
α
0.1/cm
Equation (5.12a)
Equation (5.12b)
Equation (5.12c)
Nodal spacing, dz
1 cm
Time increment, dt
Varies from 0.001 hr to 0.1 hr
100 hr
133
100
t = 0 hr
1
90
3
5
80
10
70
15
60
50
20
40
30
30
50
20
75
10
100
Numerical Solution
Analytical Solution
0
-25
-20
-15
-10
-5
0
Figure 5.2 Comparison of the numerical model with the analytical solution of Srivastava
and Yeh (1991).
Example 3
This example is actually a continuation of example 2. Srivastava and Yeh (1991)
used the same method to find an analytical solution for layered soils by considering the
case where the soil profile consists of two distinct soil layers. The datum (z = 0) is
134
assumed to be at the interface between the two layers. In the following notation,
subscript 1 denotes the lower layer, and subscript 2 denotes the top layer. The
dimensionless parameters are defined using the same notation as in example 2:
z* = α1z for − L1 ≤ z ≤ 0.0
so that L1*-=α1L1
(5.15a)
z* = α 2 z for 0.0 ≤ z ≤ −L 2
so that L2*-=α2L2
(5.15b)
K1* = K1/Ks1
qa1=qa/Ks1
qb1=qb/Ks1
(5.15c)
K2* = K2/Ks2
qa2=qa/Ks2
qb2=qb/Ks2
(5.15d)
and
t* =
α1K s1t
θs1 − θ1r
(5.15e)
Srivastava and Yeh obtained the following Laplace-space solution for the two
layered soils:
K1 =
K *10
− 4(q * b1 − q *a1 )e ( L*2 −z* ) / 2 F1 (s)
s
(5.16a)
K2 =
K *20
− 4(q *b1 − q *a1 )e ( L*2 −z* ) / 2 F2 (s)
s
(5.16b)
and
where K*10 is the initial condition for layer 1 such that
135
K *1 (z * ,0) = q a1 − (q a1 − e
αqh 0
)e − ( L1* +z* ) = K *10
(5.17a)
and K*20 is the initial condition in layer 2 such that
K *2 (z* ,0) = q a 2 − {q a 2 − [q a1 − (q a1 − e
αqh0
)e − L1* ]α2 / α1 }e − z* = K *20 (5.17b)
and
F1 (s) =
q sinh[p(L1 + z* )]
Ds
F2 (s) =
1 K s1
{
sinh(qz* )[sinh(pL1* ) + 2p cosh(pL1* ) −
2D s K s 2
(5.17c)
(5.17d)
K s2
sinh( pL1* )] + 2q sinh( pL1* ) cosh(qz* )}
K s1
and
D s = s{−[sinh(pL1* ) + 2p cosh(pL1* )] * [sinh(qL 2* )
K
+ 2q cosh(qL 2* )] + s 2 (1 − 4q 2 ) sinh( pL1* ) sinh(qL 2* )}
K s1
where p = (s + 0.25)1 / 2 , q = (β s + 0.25)1/ 2 , and β =
(5.17e)
α1K s1 (θs 2 − θ r 2 )
.
α 2 K s 2 (θs1 − θr1 )
Equation (5.16) can be inverted using the numerical inversion method of De Hoog
et al. (1982), and the resulting K values are substituted into equation (5.7) to obtain
pressure heads the same as in example 2.
136
The parameters used in the equations from (5.15 through 5.17) are shown in table
5.3. The results of the analytical solution and the numerical solutions for times 0, 0.1,
0.5, 1, 2, 5, 10, 15, 20, 30, 50, 75, and 100 hrs are plotted in the same graph in Figure 5.3.
They showed an excellent match between the numerical and analytical models.
200.00
t = 0 hr
t = 0.1 hr
160.00
0.5
1
Elevation z (cm )
2
120.00
5
5
80.00
10
20
40.00
30
50
Analytical model
Numerical Model
75 &
100
hrs
0.00
-50.00
-40.00
-30.00
-20.00
-10.00
0.00
Pressure head h (cm)
Figure 5.3 Comparison of the numerical model with the analytical solution of Srivastava
and Yeh (1991) for layered soils.
137
Flow domain
Layer 1
Layer 2
100 cm soil column
100 cm soil
column
Hydraulic conductivity
Equation (5.7)
Moisture-content
Equation (5.8)
1.0 cm/hr
10.0 cm/hr
0.40
0.06
qa
0.1cm/hr
qb
0.9cm/hr
α
0.1/cm
0.1/cm
Equation (5.17a)
Equation (5.17b)
h0 = 0.0 at z = -L1
Nodal spacing, dz
Time increment, dt
Prescribed flux, qb
2 cm
2 cm
100 hr
138
Example 4
A transient, two-dimensional, variably saturated water-table recharge problem was
selected to verify the performance of the numerical model for a two-dimensional flow
case. The experiment has been presented in detail by Vauclin et al. (1979). The same
example was also used by Clement et al. (1994) to verify their two-dimensional variably
saturated model. The flow domain consists of a rectangular soil slab 6.00 m by 2.00 m,
with an initial horizontal water-table located at a height of 0.65 m. At the soil surface, a
constant flux of q = 0.14791 m/hr is applied over a width of 1.00 m in the center. The
remaining soil surface is covered to prevent evaporation losses. Because of the
symmetry, only the right side of the flow domain needs to be modeled. The modeled
portion of the flow domain is 3.00 m x 2.00 m, with no-flow boundaries on the bottom
and on the left side because of the symmetry. At the soil surface, the constant flux of q =
0.14791 m/hr is applied over the left 0.50 m of the top of the modeled domain. The
remaining 2.50-m soil surface at the top of the modeled area is a no-flow boundary. The
water level at the right side of the model is maintained at 0.65 m as a fixed head
boundary. A no-flow boundary is specified above the water-table at the right side of the
model domain.
The soil hydrologic properties from Vauclin et al. (1979) are given in the Table
5.4. Clement et al. (1994) fitted the soil properties of Vauclin et al. (1979) to the van
Genuchten (1980) model to estimate soil properties αv and nv of the Van Genuchten
model.
139
The specific storage was neglected in this problem by Clement et al. (1994),
because changes in storage are facilitated by the filling of pores, which overshadows the
effects of compressibility. Therefore, the specific storage coefficient was set to zero for
this example. The transient position of the water-table is plotted for time 0, 2, 3, 4, and 8
hrs in Figure 5.4. The results of the numerical model closely agree with the
experimentally observed values reported by Vauclin et al. (1979) (see Figure 5.4).
0.148 m/hr
2.00
Water Table Position (m)
1.60
t = 8 hr
1.20
t = 4 hr
t = 3 hr
t = 2 hr
0.80
t = 0 hr
0.40
Comparision of the Numerical Model with Experimental Results
Numerical Model
Vauclin et al. (1979) Experimental Results
0.00
0.00
0.50
1.00
1.50
2.00
2.50
3.00
X- Coordinates (m)
Figure 5.4 Comparison of the numerical model with experimental results of Vauclin et al.
(1979).
140
Flow domain
3.00 m x 2.00 m
Hydraulic conductivity, K(h)
Equation (5.3a)
Moisture-content, θ(h)
Equation(5.4 a)and (5.4b)
0.35 m/hr
0.30
0.01
Air entry pressure, he=1/αv
h0
1/3.3 m
0.0 cm/hr
van Genuchten parameter, nv
4.1
Specific storage, Ss
0.0
Impervious no-flow boundary
Prescribed flux(left 0.50 m) and no-flow
boundary(right 1.50 m)
Hydrostatic equilibrium with horizontal watertable at 0.65 m (i.e., h+z=.65)
30x40=1200 cells with size of dx = 0.1 m and
dz = 0.05 m
Time increment, dt
8.00 hr
141
Example 5
This problem is a continuation of example 4 in that the y-dimension is added to
the problem of example 4. In this three-dimensional example, a (6 x 6 x 2 m) soil cube
with a recharge area of (1 x 1 m) at the center of the soil surface of the cube is surrounded
by a 0.65 m fixed head water body. This problem resembles a water-table-mounding
problem in a square island. Only a quarter portion of the flow domain needs to be
modeled because of the symmetry (Figure 5.5).
Figure 5.5 Three-dimensional model domain description for example 5
The modeled portion of the flow domain is (3 x 3 x 2 m) discretized into 30 rows,
30 columns, and 40 layers. All the cells are uniform in size with dz = 0.05 m, dx = 0.1 m,
and dy = 0.1 m. All the soil parameters and hydrologic parameters are exactly the same
as in the two-dimensional case (i.e., example 4).
142
As output, water-table positions throughout time for a selected cross section (at y
= 0.0 m) and water-table elevations at a certain point in time (after 8 hrs of recharge, in
this example) are plotted in figure 5.6 and figure 5.7, respectively.
0.148 m/hr
2.0 m
Time (hr)
t=0
t=2
t=3
t=4
t=8
1.2
1.1
Water Table Elevation (m)
1
0.9
Water table elevations at y=0.0, for different times
0.8
0.7
0.6
0.5
0
1
2
3
X-axis (3.0 m)
Figure 5.6 Water-table elevations resulting from 3-D simulation of example 5 at y = 0 for
various time values.
143
Recharge Area (0.1 48 m/ hr)
(0.5x0.5 m^ 2)
2.00 m
Wat er table elevations at t he end of 8 hr rainfall.
Figure 5.7 Three-dimensional water-table recharge. Water-table elevations are shown at
the end of 8-hr rainfall of 0.148 m/hr.
Example 6
In this example, a pumping/injection well is added to the problem in example 5.
The pumping/injection well is located at the center of the model domain (i.e., at I =15,
J=15). The well is pumped from layers K = 2, 3, 4, 5, and 6 at a rate of 1.25 m3/hr from
each cell for a total of 6.25 m3/hr.
For this example, three different scenarios were tested. In the first scenario, the
pumping well is turned on along with a recharge (rainfall) rate of 0.148 m/hr to the area
of 0.5 x 0.5 m2 in the southwest corner of the model domain. The results at the end of a
4-hr simulation of the first scenario are presented in Figure 5.8. The second scenario
consists of only pumping without any recharge (rainfall). The results at the end of a 4-hr
simulation of this test are presented in Figure 5.9. In the third scenario, the wells are used
144
as injection wells instead of pumping wells, and water is injected at the same rate as the
pumping rate in scenario 1 without any recharge. The results of this last test are
presented in Figure 5.10 and Figure 5.11.
Recharge Area (0.148 m / hr)
(0.5x0.5 m^2)
2.00 m
Pumping f rom Cells i,j=(15,15) at layer s k=2,3,4,5,6.
After 4 hour s pumping with the rate of 1.25 m^ 3/ hr fr om each cell.
Figure 5.8 Three-dimensional water-table recharge and pumping. Water-table elevations
are shown at the end of a 4-hr rainfall of 0.148 m/hr and 6.25 m3/hr pumping.
Pumping with the total rate of 6.25 m^3/ hr.
Pump is located at i,j =(15,15) at layer s k= 2,3,4,5,6.
Figure 5.9. Three-dimensional pumping from water-table. Water-table elevations are
shown at the end of a 4-hr pumping period at the rate of 6.25 m3/hr.
145
Inject ion Well with the total rate of 6.25 m^3/ hr.
Pump is located at i,j =(15,15) at layers k= 2,3,4,5,6.
Figure 5.10. Three-dimensional recharge to the water-table. Water-table elevations are
shown at the end of a 4-hr injection period at a rate of 6.25 m3/hr.
Injection Well
0.72
Water Table Elevations (m)
0.7
t = 4.0 hr
t = 3.0 hr
0.68
t= 1.5 hr
t = 0.5 hr
0.66
t = 0.1 hr
Original water table before injection starts
0.64
0.00
1.00
2.00
3.00
X- axis at j=1 (3.00 m with dx = 0.1 m)
A cross-section at x-z plane at j=1.
Water Table elevation for different times during the injection of 6.25 m3/hr.
Figure 5.11 Three-dimensional recharge to the water table. Water-table elevations are
shown at a cross-section in the x-z plane at j = 1 for different time values. Injection well
is located at (i, j) = (15, 15) at k = 2, 3, 4, 5, 6 with the rate of 6.25 m3/hr.
146
Example 7
A three-dimensional pumping problem was selected from 3DFEMWATER (Yeh,
and Cheng, 1994). This example involves steady-state flow to a pumping well in a model
domain bounded on the left and right by hydraulically connected rivers, on the front,
back, and bottom by impervious rock formations, and on the top by the soil-air interface
(Figure 5.12). A pumping well is located at (x, y) = (540, 400 m). Initially, the watertable is assumed to be horizontal and 60 m above the bottom of the aquifer. The water
level at the well is then lowered to a height of 30 m. This height is held until a steadystate condition is reached. The porous medium in the region is assumed to be anisotropic
and to have saturated hydraulic conductivity components Kx = 5 m/d, Ky = 0.5 m/d, and
Kz = 2 m/d. The porosity of the medium is 0.25, and the residual moisture capacity is
0.0125. The unsaturated characteristic hydraulic properties of the medium are given as
θ = θr +
θs − θ r
1 + (α h a − h ) β
 θ − θr 
Kr = 

 θs − θ r 
(5.18)
2
(5.19)
where ha, β, and α are the parameters used to compute the water content and relative
hydraulic conductivity. The values of ha, β, and α are 0, 0.5 and 2.0, respectively.
147
Figure 5.12 Problem definition sketch for example 7.
The initial condition is set as H = 60 m, or h = 60 - z. The boundary conditions
are as follows: the pressure head is assumed hydrostatic on the two vertical planes located
at x = 0 and 0 < z < 60, and x = 1000 and 0 < z < 60, respectively; and no-flow
boundaries are imposed on all the other boundaries of the flow domain.
The model domain is discretized with 27 x 17 x 100 = 45,900 node centered cells.
The discretization increments in the x direction change from 50 to 20 m, while the
increments in the y direction are constant at 47 m. The discretization increments in the z
direction change from 20 m in the saturated zone to 0.25 m in the unsaturated zone.
The newly proposed numerical model results and the results of Yeh and Cheng
(1994) very closely match each other (figure 5.13). A cross-section in the x-z plane
passing through the center of the well shows both results. A three-dimensional view of
148
the steady-state condition of the flow domain calculated by the model in this study is
presented in Figure 5.14.
The convergence criteria in the Picard iteration scheme was 0.0001 moisturecontent difference, and it was 0.00001 in the preconditioned conjugate solution scheme.
The simulation run took 24 minutes 54 seconds using an Alpha work-station computer
and DEC Fortran compiler. The total number of Picard iterations to reach a steady-state
Elevation (m)
condition was 84.
60.00
60.00
40.00
40.00
20.00
20.00
3DFEMWATER
Model Results
0.00
0.00
0.00
200.00
400.00
600.00
800.00
Horizontal Distance (m)
Figure 5.13 Water-table position at the steady-state condition for example 7.
1000.00
149
Figure 5.14 Three-dimensional view of the water-table for example 7.
CHAPTER 6
APPLICATION OF THE MODEL
In this chapter, the results of applying the numerical model to a problem that
consists of evaporation, transpiration, and rainfall events are discussed. This problem is
based on a problem solved by Lappala et al. (1987) using the VS2D model. The
numerical model was also used to simulate an unconfined aquifer pumping test. The
results are compared to Nwankwor et. al. (1984), who reported the results of a detailed
pumping test in an unconfined sand aquifer located at the Canadian Forces Base, Borden,
Ontario, Canada.
Application of the Model to a Two-Dimensional Infiltration and Evapotranspiration
Problem
This problem based on Lappala et al. (1987) is a relatively complex twodimensional problem involving rainfall events, evaporation, and transpiration. The
simulated section consists of a 1.5 m thick clay layer, which overlies a 0.6 m thick gravel
layer (figure 6.1). A discontinuous 0.3 m thick sand layer is embedded in the clay at a
depth of 0.3 m. The width of the simulated section is 3.0 m. The sand layer extends from
the left-hand side boundary for a distance of 1.5 m. During the simulation, the sand layer
acts as a capillary barrier, affecting infiltration, evaporation, and root water uptake rates.
In the sand layer, relatively greater negative pressures occur during the simulation
because the sand releases its moisture-content quickly because of its relatively large
hydraulic conductivity and relatively small porosity. Consequently, the relatively greater
150
151
negative pressures occur because of the relatively smaller moisture-content. This
negative pressure zone creates a capillary barrier that blocks the water moving vertically.
Figure 6.1 Description of the problem of Lappala et al. (1987)
Four recharge periods totaling 77 days are simulated. For the first period, rainfall
is applied for one day at a rate of 25 mm/day. In the second period, bare-soil potential
evaporation occurs at a rate of 2.0 mm/day for 30 days. This is followed in the third
period by another one-day duration rainfall event at a rate of 25 mm/day. The fourth
period lasts for 45 days and consists of both evaporation and evapotranspiration. The
user-defined variables that control evaporation and evapotranspiration are assumed to
remain constant throughout the simulation, with the exception of the potential
evapotranspiration rate (PET), root depth, and root pressure. These parameters vary
linearly during each ET period, which is 30 days. This means that the parameter values
are given for time = 0, 30, 60, and 90 days, and the intermediate values of those
parameters are calculated by linear interpolation between the known values at time 0, 30,
152
60, and 90 days. For example, the value of root depth is 0.0 cm at day 0, and 35 cm at
day 30. Therefore, it will be 17.5 cm at day 15.
The input data for this problem are listed in table 6.1. The grid contains 572
nodes consisting of 26 layers and 22 columns that are variably spaced. The initial
conditions consist of an equilibrium head profile specified above a fixed water-table at a
depth of 2.0 m. The minimum pressure head is set at -1.00 m. The hydraulic properties
of the three different soils are represented by the Brooks-Corey functions (equations 3.223.26).
During the second and fourth periods when evaporation and transpiration occur,
the initial time steps are decreased to 0.00001 day to achieve convergence. When
evapotranspiration occurs from fine-grained materials overlying coarse-grained materials
that contain a water-table, it becomes particularly difficult to achieve convergence in a
numerical solution (Lappala et al., 1987).
The output file consists of pressure heads and moisture-contents at four locations
in the simulated region. Those four locations were selected to be the same depth of 0.33
m from the land surface but at horizontal distances of 0.11 m, 1.46 m, 1.54 m, and 2.89 m
from the left hand side boundary, respectively. The first two locations are in the sand,
and the other two locations are in the clay layers. After 60 days of simulated
evapotranspiration, the pressure head difference between two adjacent locations (one
location in the sand at 1.46m and the other in the clay layer at 1.54m) starts to increase
and reaches approximately 7 m.
153
Table 6.1 Parameters used for the VS2D problem.
Flow domain
Saturated moisture-content, θs and
residual moisture-content respectively.
Brook and Corey Parameters, hb and λ
Evapotranspiration parameters (change
linearly as time progresses)
Potential transpiration (cm/day)
Root depth (cm)
Root pressure (cm)
Root activity at the bottom
Root activity at the top
Potential evaporation (cm/day)
Surface resistance (1/cm)
Atmospheric pressure potential (cm)
Initial condition
Time increment, dt
3.00 m x 2.10 m
Brooks and Corey Relation (equation 3.23)
Brooks and Corey Relation (equation 3.22)
5.0 cm/day for clay
100.0 cm/day for sand
300.0 cm/day for gravel
0.45-0.15 for clay
0.40-0.08 for sand
0.42-0.05 for gravel
-50 cm, 0.6 for clay
-15 cm , 1.0 for sand
-8 cm, 1.2 for gravel
1.0x10-6 cm-1 for all materials
Prescribed fixed pressure head (0.55 m)
2.5 cm/hr rainfall for the first day;
2.0 cm/day potential evaporation for 30 days;
2.5 cm/hr rainfall for another 1 day; and
potential evapotranspiration for next 45 days.
0th day
30th day
60th day
90th day
0.0
0.0
0.45
0.60
0.0
35
35
35
-8,000
-8,000
-12,000
-15,000
0.2
0.2
0.2
0.2
0.9
0.9
0.9
0.9
0.2
0.2
0.2
0.2
0.6
0.6
0.6
0.6
-100,000 -100,000
-100,000 -100,000
Hydrostatic equilibrium with horizontal watertable at 10 cm from the bottom of the aquifer;
minimum pressure head is -100 cm.
22 cells in the x-direction; dx values in cm:
22.5, 22.5, 15, 15, 15, 15, 11.25, 11.25, 7.5, 7.5,
7.5, 7.5, 7.5, 7.5, 11.25, 11.25, 15, 15, 15, 15,
22.5, and 22.5
26 cells in the z-direction; dz values in cm:
3.0, 3.0, 3.0, 4.5, 4.5, 6.0, 6.0, 6.0, 6.0, 9.0, 9.0,
9.0, 9.0, 12.0, 15.0, 15.0, 12.0, 9.0, 6.0, 6.0, 6.0,
9.0, 9.0, 9.0, 9.0, and 15.0
Varies from 0.00001 day to 0.15 day.
77 days
154
The problem was solved again using the most current version of VS2D to plot its
results together with the numerical model results of this study in the same graph (figure
6.2). Both the model and VS2D results match very well for the pressure heads in the sand
layer at all simulation times. During early time (before transpiration starts), the clay layer
results are in agreement in both models, but at later times, especially after transpiration
becomes more effective, the pressure heads calculated by the numerical model of this
study are slightly less than those calculated by VS2D. This difference may be due to
hydraulic properties of the clay soil. Since the numerical model of this study takes into
account moisture-content and pressure head at the same time (i.e., the mixed form of the
modified Richards equation is used), the results of pressure heads are different from
VS2D. To increase or to decrease the moisture-content of the clay layer requires a
significant amount of pressure change compared to the sand layer. In another words, a
very slight moisture-content change in the clay layer requires substantial pressure head
change.
The numerical model of this study gives successful results. Since the new model
is written in terms of the mixed form of the modified Richards equation, it is more mass
conservative then VS2D, which solves the pressure-based Richards equation. The slight
pressure difference in the results at the clay layer may be caused because of this
difference between these models.
155
0
Pressure Head (m)
-500
-1000
-1500
Sand at 1.46 m
Sand at 0.11 m
Clay at 1.54 m
Clay at 2.89 m
VS2D Results
-2000
0
7
14
21
28
35
42
49
56
63
70
Time (Days)
Figure 6.2 Comparison of the results of VS2D and the current model.
Application of the Model to an Unconfined Sand Aquifer Pumping Test
The numerical model was compared to pumping test results described by
Nwankwor et al. (1984) to test its ability to simulate an unconfined aquifer pumping
problem. This application illustrates how improved estimate of the specific yield (or
storage coefficient) of an unconfined aquifer could be obtained. It also helps to clarify
77
156
the conflicting theories in the literature concerning the delayed yield concept in
unconfined aquifers.
Simulating an unconfined aquifer pumping test is a challenging problem in
numerical modeling. The difficulty arises as pumping proceeds and the water-table
declines. The thickness of the aquifer is not constant with respect to time or space, and
also there are significant vertical hydraulic gradients, particularly during the early
pumping period. As a consequence of these difficulties, there are still uncertainties
regarding the source of the water released from storage. Nwankwor et al. (1992), and
later Akindunni and Gillham (1992), tried to explain the physical behavior of the pumped
aquifer using their models. They developed an idea contrary to Neuman's (1972) concept
of instantaneous and complete release of water at the water-table. Their idea is that the
response of an unconfined aquifer to pumping is largely controlled by the magnitude of
vertical hydraulic gradients developed above the moving water-table and the resulting
variations in the values of apparent specific yield. They concluded that the unsaturated
zone plays a significant role in the delayed yield concept (Akindunni and Gillham, 1992).
The unconfined aquifer in the test results reported by Nwankwor et al. (1984) is
shown in figure 6.3 as a cross section. The aquifer is 9 m thick and is composed
primarily of horizontal, discontinuous lenses of medium-grained, fine-grained and silty
fine-grained sand based on Sudicky (1986). The water-table was located 2.3 m below the
land surface at the time the pumping test was conducted, although its position could have
a seasonal variation of up to 1.0 m. A thick deposit of clayey silt underlies the aquifer.
The pumping well has an inner diameter of 0.15 m with a 4 m screen located at the
bottom of the aquifer. Data were collected from piezometers installed at different radial
157
distances and terminated at different depths. The test lasted for about 24 hours at a
discharge rate of 60 l/ min. Complete details of the instrumentation and test procedures
are included in Nwankwor et al. (1984,1992).
Figure 6.3 Cross section for the unconfined aquifer pumping problem.
The domain was discretized for this study into a three-dimensional variably sized
grid. Finer discretization was used close to the well around the top of the screen. It was
also necessary to extend the fine discretization to some distance above the top of the
capillary fringe in order to ensure that a reasonable number of nodes was specified within
the zone where the magnitude of specific water capacity and hydraulic conductivity
varied significantly with changes in pressure head (Akindunni and Gillham, 1992). The
158
model simulated a pumping period of 1440 minutes (24 hours). The external boundary
was kept at 100 m as a no-flow boundary condition to ensure that all the flow originated
from the discretized flow domain. Beyond 70 m from the well axis, drawdowns were
insignificantly small, so therefore choosing the external no-flow boundary at 100 m away
from the well axis is reasonable. Parameters required by the numerical model were
chosen in a manner that was independent of the pumping test. Only the value of principal
hydraulic conductivity at saturation was adjusted to improve the fit between the field data
and the model results. The vertical conductivity at the well screen location was increased
on the order of 105 to enable water to flow freely inside the well casing, and the
horizontal conductivity at the well screen location was also increased 100 times its
original value to represent the real pumping well situation, based on Halford (1997). The
well screen was discretized into 8 cells with the dimensions of 0.15 x 0.15 x 0.50m.
Therefore, the total discharge was divided by 8 and then distributed as fluxes to the
appropriate sides of the surrounding 6 cells of each of the cells containing the well screen.
The values of the parameters used in the simulation are shown in Table 6.2. The
curve-fitting parameters required to represent the van Genuchten (1980) θ-h relationship
(equation 6.2) were obtained from the laboratory drainage experiment reported by
Nwankwor et al. (1984). The porosity of the aquifer material (0.4) was obtained from the
pressure head -moisture-content profiles of Nwankwor et al. (1992).
The saturated principal hydraulic conductivities were obtained from results of the
permeameter tests reported by Sudicky (1986). K-θ relations are given by equation 6.1:
K x (θ) = K y (θ) = a θ b
(6.1a)
159
Kz(θ) = 0.64* Kx(θ)
(6.1b)
The moisture-content-pressure head relation is shown in equation 6.2:


1
θ(h ) = (θ s − θ r ) * 

n
1 + (α h ) 
m
(6.2)
where n, and α [1/L] are van Genuchten parameters, and m = 1-1/n. θs and θr are
saturated and residual moisture-contents.
The simulated and measured time-drawdown graphs were compared at horizontal
distances of 5 and 15 m from the pumping well. The measurements were made at the
depth of 7 m from the surface (2 m from the bottom of the aquifer). There is good
agreement between the field data and the model results (figure 6.4). The results clearly
indicate that the numerical model can simulate the delayed yield effect on time-drawdown
curves as observed in the field study of Nwankwor et al.(1992).
160
Table 6.2 Parameters for the unconfined aquifer pumping problem.
Flow domain
100m x 100m x 9m
Equation (6.1)
Equation (6.2) van Genuchten (1980)
Kx =
Ky =
Kz =
6.6 x10-5 m/s
6.6 x10-5 m/s
4.2 x10-5 m/s
0.37
0.07
van Genuchten parameters, n and α :
6.095 and 1.9 1/m
3.25x10-4 1/m
Impervious no-flow boundary
No-flow boundary
Hydrostatic equilibrium with horizontal watertable at 6.7 m
49 x 49 x 49 = 117649 cells with various sizes
of dx, dy, dz changing from 0.15m to 10m.
Time increment, dt
Varies from 0.001 min to 1.0 min
1440 min (24 hr)
161
Drawdown (m) at a depth of 7 m
10
Model results at 5 m distance from the well axis
Model results at 15 m distance from the well axis
Field measurements at 15 m distance from the well axis
Field measurements at 5 m distance from the well axis
1
0.1
0.01
0.001
0.1
1
10
100
1000
Time (min)
Figure 6.4 Comparison of the pumping test results of Nwankwor et al.(1992) and the
current model results
10000
CHAPTER 7
APPLICATION OF THE MODEL TO A FIELD CONDITION IN NORTH CENTRAL
FLORIDA
The model developed in this study was applied to a field situation that includes
rainfall, evaporation, transpiration, lakes, and a stream. The area of this field application,
approximately eight square kilometers, is part of the Upper Etonia Creek Basin (UECB),
which is located in parts of Alachua, Bradford, Clay, and Putnam counties in northcentral Florida. The model domain is 2,800 m by 2,800 m and includes parts of
Magnolia, Lowry, and Crystal lakes and Alligator Creek. The model extends vertically to
the bottom of the upper Floridan aquifer.
Description of the Study Area
Location
The model area is bounded by Magnolia Lake on the south, Alligator Creek on the
east, Lowry Lake on the northeast, and Crystal Lake on the southwest. The area lies
between 29049'20" and 29051'40" north latitude and 82000'50" and 82002'42" west
longitude (see Figure 7.1). The topographic surface of the model area is relatively
smooth (see Figure 7.2). Lowry Lake and Magnolia Lake are located on the Camp
Blanding Military Reservation.
162
163
° 05′ W
82° 00′ W
3306000
BLUE POND
3304000
LOWRY LAKE
3302000
40.12
0 km
1 km
2 km
3 km
4 km
29 ° 50′ N
3300000
C RYSTAL
LAKE
37.98
LAKE BEDFORD
3298000
LOCH LOMMOND
LAKE
BROOKLYN
30.68
3296000
3294000
LI TTLE
SANTA FE
LAKE
LAKE
GEN EVA
27.65
3292000
OLDFIELD
POND
29 ° 45′ N
SANTA FE
LAKE
HALFMOON
LAKE
3290000
396000
398000
Well Location
40
27.65
Water Table Contour (m, NGVD).
Contour Interval = 2 m.
400000
402000
Blue Pond
Crystal Lake
Lake Bedford
Loch Lommond
Santa Fe Lake
404000
52.12 m, NGVD
30.87 m, NGVD
28.65 m, NGVD
25.65 m, NGVD
42.87 m, NGVD
Lake Surface Elevation (m, NGVD).
Coordinates are UTM (meters), Zone 17.
Figure 7.1 September 1994 water table map in the UECB and the location of the model
domain (Source: Sousa, 1997).
164
LOWRY LAKE
MAGNOLIA LAKE
CRYSTAL LAKE
Figure 7.2 Topographic surface of the model area.
Climate
The climate in the UECB is classified as humid subtropical (Sousa, 1997). The
average annual temperature is approximately 22 oC. The area receives more than half of
its annual rainfall between June through September. Precipitation in the winter and early
spring typically is the wide spread type associated with frontal activity. Most of the
rainfall in the summer is in the form of local showers and thunderstorms. A notable
feature is that the average rainfall for June is about double the average rainfall for May
(Clark et al., 1964).
Geology
The majority of the surficial geologic deposits in the UECB consist of
unconsolidated to semi-consolidated sand, clayey sand, marl, and shell. The thickness of
these sediments ranges from 6 to 60 m, and the sediments are associated with the
Pleistocene and Pliocene periods. These deposits are underlain by the Hawthorn Group, a
165
marine deposit of Miocene age, which consists of clay, quartz sand, carbonate, and
phosphate (Clark et al., 1964). The Ocala Limestone lies below the Hawthorn Group.
This formation ranges in thickness from 60 to 120 m and is of the Late Eocene period.
The major geologic layers in the area of the UECB are shown in Table 7.1.
Table 7.1. Geologic layers in the Upper Etonia Creek Basin (based on Motz et al., 1993)
Approximate
Geologic
Stratigraphic
Thickness
Unit
(m)
Age
Pleistocene
Post-Hawthorn
and Recent
Deposits
10-100
Lithology
Discontinuous beds of loose sand,
clayey sand, sandy clay, marl, and
shell
Pliocene
Post-Hawthorn
10-100
Deposits
Miocene
Hawthorn
and limestone
100-400
Group
Late Eocene
Ocala
Clay, clayey sand, sandy clay, shell,
Interbedded clay, quartz sand,
carbonate, and phosphate
200-400
Porous limestone
500-1200
Interbedded limestone and dolomite
300-800
Interbedded limestone and dolomite
Unknown
Interbedded dolomite and anhydrite
Limestone
Middle
Avon Park
Eocene
Formation
Early
Oldsmar
Eocene
Formation
Paleocene
Cedar Keys
Formation
Sources: Bermes et al. 1963; Clark et al. 1964; Fairchild 1972; Hoenstine and Lane 1991;
Leve 1966; Miller 1986; and Scott 1988.
166
Groundwater Hydrology
The hydrogeologic units in the UECB consist of the surficial aquifer system, the
upper confining unit, and the Floridan aquifer system (see Table 7.2). The surficial
aquifer system is the uppermost of the three units, and depths of its water table are 1-5
meters below the ground surface. The Hawthorn Group makes up the upper confining
unit for the Floridan aquifer system. This unit is bounded by upper and lower confining
units in the Hawthorn Group. An intermediate aquifer consisting of permeable lenses of
limestone occurs locally in some parts of the UECB, but it is areally discontinuous and
not well mapped or quantified. The Floridan aquifer system is the deepest of the three
units. It is under confined conditions within the study area, and it is separated from the
surficial aquifer system by the Hawthorn Group confining unit. The Floridan aquifer
system is comprised of two zones. A low permeability layer of limestone and dolomite
separates the two zones into the upper Floridan aquifer and the lower Floridan aquifer.
The bottom of the Floridan aquifer system is bounded by beds of low permeability
anhydrite in the Cedar Keys Formation, which serves as the lower confining unit of the
Floridan aquifer system (Miller, 1986).
There are numerous lakes in the region of the UECB. Lake levels and surficial
aquifer water table heads are higher than the Floridan aquifer head, and the lakes and the
surficial aquifer are sources of recharge to the Floridan aquifer system. Rainfall is the
primary source of recharge to the surficial aquifer. The lakes discharge water to the
Floridan aquifer system through the leaky confining unit. The lakes and the surficial
aquifer exchange water with each other depending on their respective water levels during
different seasons of the year.
167
Table 7.2 Hydrogeologic units of the Upper Etonia Creek Basin (based on Motz et al.,
1993)
Geologic
Age
Pleistocene
and Recent
Pliocene
Geologic
Unit
Pleistocene
and Recent
deposits
Pliocene
deposits
Miocene
Hawthorn
Group
Late
Eocene
Ocala
Limestone
Hydrologic Unit
Description
Surficial Aquifer System
Consists of sands, clayey sand,
and shell. Thickness ranges from
6 to more than 35 m
Upper
Confining
Unit of the
Floridan
Aquifer
Upper
Confining Unit
of the Hawthorn
Group
Intermediate
Aquifer System
Lower
Confining Unit
of the Hawthorn
Group
Upper
Floridan
Aquifer
Consists of clay, marl, and
discontinuous beds of sand, shell,
dolomite, and limestone.
Thickness ranges from 45 to 135
m.
Consists mainly of limestone of
high primary and secondary
porosity. Thickness ranges from
90 to 215 m.
Middle
Avon Park Floridan
Middle Semi- Consists of leaky, low
Eocene
Formation Aquifer
confining
permeability limestone and
System
Unit
dolomite. Thickness ranges from
15 to 60 m.
Early
Oldsmar
Lower
Consists primarily of interbedded
Eocene
Formation
Floridan
limestone and dolomite.
Aquifer
Thickness ranges from 335 to 455
m.
Paleocene
Cedar Keys Lower Confining Unit of the
Consists of low permeability
Formation Floridan Aquifer System
anhydrite beds.
Sources: Clark et al. 1964; Hoenstine and Lane 1991; Miller 1986; Scott 1988; and
Southeastern Geological Society 1986.
Long term records for the UECB indicate a general decline in the potentiometric
surface in the upper Floridan aquifer. Water withdrawal by pumping from wells is the
major cause of declining water levels and the potentiometric surface in the upper Floridan
168
aquifer. The primary center of pumping that affects the upper Floridan aquifer in Clay
County is metropolitan Jacksonville in Duval County (Sousa, 1997).
Application of the Model
Selection of the Model Area
A 2,800 m by 2,800 m area in the UECB was selected to apply the current model.
This area was chosen based on the September 1994 water table map of the UECB such
that there are no flow boundaries along streamlines and fixed head boundaries along
Crystal Lake, Magnolia Lake, Lowry Lake, and Alligator Creek. This area is a recharge
area for the upper Floridan aquifer, and the potentiometric surface of the upper Floridan
aquifer is almost constant at approximately 25 m, mean sea level, over all the area.
Therefore, the upper Floridan aquifer in that area was modeled using general head
boundaries. There are three observation wells in the surficial aquifer in the selected area,
two of which are located between Magnolia Lake and Lowry Lake and one of which is
located between Magnolia Lake and Crystal Lake (see Figure 7.3).
Boundary Conditions
Three types of boundary condition were used in this simulation, i.e., fixed head
boundaries, no-flow boundaries, and general head boundary conditions. The fixed head
boundaries were located around the lakes. Based on historical observations that the lake
levels in Magnolia Lake, Lowry Lake, and Crystal Lake were stable throughout the period
of simulation, it was reasonable to choose the lakes as fixed head boundaries for the
surficial aquifer. Based on the September 1994 water table map, streamlines were chosen
169
as no-flow boundary conditions for the surficial aquifer at the remaining borders of the
flow domain (see Figure 7.3).
0
200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800
2800
2800
NO FLOW BOUNDARY
2600
2600
LOWRY LAKE
DA
RY
2400
2200
BO
UN
2200
2000
FL
OW
2000
2400
1600
1600
rC
r ee
k
MODEL DOMAIN
1400
at o
(m)
1800
C0522
NO
1800
1200
Al
lig
1200
1400
1000
800
C0521
1000
800
600
600
400
CRYSTAL
200LAKE
400
C0520
NO FLOW BOUNDARY
0
0
MAGNOLIA
LAKE
200
0
200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800
Observation Wells
(m)
Figure 7.3 Model boundaries and September 1994 water table map.
A general head boundary condition was created around the upper Floridan aquifer.
The boundary condition for the upper Floridan aquifer was chosen as a no-flow boundary
170
by knowing that the potentiometric surface of the lower Floridan aquifer is equal to the
potentiometric surface of the upper Floridan aquifer, which creates a no-flow boundary.
The top boundary condition was chosen as a specified flux (evaporation and
rainfall) boundary condition. The observed daily potential evapotranspiration (pan
evaporation times pan coefficient) and rainfall data (daily values) were entered in the
model, and the model calculated the evaporation and rainfall values at intermediate time
steps and applied those values to the uppermost active grid cell as a specified flux. The
values for intermediate time steps were calculated using a linear interpolation technique.
The cells outside of the model borders were specified as inactive cells. The model
does not make any calculations for inactive cells, but the finite-difference method
requires them to be entered as part of the model in the input file. These inactive cells
complete the system of equations to create a symmetric matrix of the system of equations.
Meteorological Data
Over the last five years, rainfall data have been collected at Magnolia and Lowry
lakes as a part of a project conducted in the UECB area by University of Florida
investigators for SJRWMD. Rain gages are located at each of the two lakes in the model
domain. The gage at Magnolia Lake came on line on April 18,1995 and is monitored by
the USGS. The gage at Lowry Lake is monitored by SJRWMD. Lowry Lake appears to
receive more precipitation then Magnolia Lake. The regional rainfall data and local
rainfall data during the simulation period of the model are given in Table 7.3.
171
Table7.3 Regional and Local Rainfall Data During the Simulation Period
Gainesville
Regional
Jacksonville
Lake City
Ocala
Sep-94
Oct-94
Nov-94
Dec-94
Jan-95
Feb-95
Mar-95
Apr-95
May-95
Jun-95
Jul-95
Aug-95
Federal
Point
294500N
813200W
Putnam
105
133.27
cm
33.48
18.64
10.57
8.89
3.18
2.49
0.61
4.50
10.67
19.69
14.02
23.72
294100N
823000W
Alachua
8
NA
cm
9.83
11.28
2.03
3.40
11.73
2.87
9.65
12.57
6.27
21.67
11.28
22.30
303000N
814200W
Duval
60
130.35
cm
24.87
25.98
8.86
10.01
4.85
5.26
9.32
4.50
4.50
13.59
24.00
25.22
301100N
823600W
Columbia
113
140.94
Cm
10.77
24.16
2.06
3.51
11.13
4.50
7.54
10.19
10.90
24.08
21.64
20.96
291200N
820500W
Marion
106
131.04
cm
11.46
11.56
9.25
6.88
6.65
3.94
12.42
8.59
7.82
29.67
11.68
16.36
AVG
Year Avg.
12.54
150.44
10.41
124.89
13.41
160.96
12.62
151.41
11.36
136.27
Latitude
Longitude
County
Yrs. in Op.
30yr Avg.
Local
Lowry
Magnolia
Lake
Lake
295115N 294930N
820100W 820100W
Clay
Clay
NA
NA
NA
NA
cm
cm
3.66
3.23
14.78
13.06
5.92
5.23
4.57
4.04
7.34
6.49
4.29
3.79
10.72
9.47
13.26
11.71
11.71
9.86
31.62
27.64
25.73
15.72
17.02
23.90
12.55
150.62
11.18
134.14
Evapotranspiration
Daily pan evaporation measurements made by the Department of Agronomy at the
University of Florida in Gainesville, Florida were used to calculate evaporation and
transpiration values used in the model as a prescribed flux boundary. Monthly pan
coefficients were used from a study at nearby Lake Barco (see Table 7.4). The potential
evapotranspiration (PET) values were obtained by multiplying the pan coefficients by the
measured pan evaporation values. Then, these PET values were separated into
components internally in the model as potential evaporation (PE) and potential
transpiration (PT) based on the leaf area index of the area.
172
The actual evaporation (AE) was calculated by the model using the PE as a
function of the available moisture content at the top nodes for top boundary condition
calculations at every time step. The actual transpiration (AT) was calculated and
distributed to the grids along the root zone as a function of root depth, root distribution
function, and available moisture content along the root zone. All these calculations were
repeated at every time step and at every Picard iteration level.
Table 7.4. Lake Barco Pan Evaporation Coefficients
Month
Pan Coefficients
January
0.61
February
0.78
March
0.83
April
0.89
May
0.87
June
0.88
Source: Sousa, 1997.
Month
July
August
September
October
November
December
Pan Coefficient
0.87
0.96
0.94
0.96
0.95
0.90
Lakes
Water budget calculations for Lowry Lake and Magnolia Lake have been done by
Sousa (1997). According to his study, the major inflow components for Lowry Lake are
rainfall, surface-water inflow, and surficial aquifer inflow. The major inflow components
for the Magnolia Lake are surface-water inflow and rainfall. In both of the lakes, the
contribution of the runoff (overland flow) is very minimal, and accordingly it was
neglected in the model application. The basic outflow components for both of the lakes
are the surface-water outflow, evaporation, and vertical leakage (Sousa, 1997). The lake
levels during the simulation period were almost constant with only small fluctuations (see
Table 7.4 and Figure 7.4). Therefore, the assumption of fixed head boundary conditions
for the lakes is very reasonable.
173
42
40
L a k e S ta g e s (m , N G V D )
38
Lake Stages
Lowry Lake
Magnolia Lake
Cyrstal Lake
36
34
32
30
9/1/94
11/30/94
2/28/95
Figure 7.4 Lake levels in the model domain.
5/29/95
8/27/95
174
Table 7.5 Lake stages in the model domain
Date
Lowry Lake
09/02/94
10/03/94
11/02/94
11/28/94
12/29/94
01/30/95
02/27/95
03/31/95
04/28/95
05/31/95
07/04/95
07/28/95
09/05/95
40.121
40.054
40.076
40.066
40.03
40.039
40.002
39.987
40.033
39.996
40.115
40.191
40.201
Magnolia
Lake
37.985
37.915
37.939
37.896
37.841
37.847
37.805
37.799
37.844
37.829
37.96
38.03
38.082
Crystal Lake
30.874
30.874
31.111
31.111
31.102
31.085
31.069
31.069
31.154
31.127
31.279
31.386
31.642
Three-Dimensional Discretization
The model area was discretized into 25 rows and 25 columns in the horizontal x
and y directions in variable sizes (see Figure 7.5). The grid sizes were chosen smaller (80
m) near the lakes and larger (300 m) away from the lakes. The model domain was
discretized into 60 layers in the vertical direction, with the largest grid in vertical size at
the bottom and with the smallest grid size (0.30 m) at the ground surface and around the
capillary fringe zone. The first layer is 80 m thick and contains only the upper Floridan
aquifer. Layers two and three are 10 m and 15 m thick, respectively, and are in the upper
Floridan aquifer or the lower part of the confining unit depending on their elevation.
Layer four is 45 m thick, and it contains only the middle part of the confining unit.
Layers five and six are 10 and 3 meters thick, respectively, and contain the upper part of
the confining unit and/or the lower part of the surficial aquifer depending on their
elevations. Layers seven and eight are 2 and 1 meters thick, respectively, and contain the
175
upper part of the confining unit and the saturated part of the surficial aquifer. Above
layer eight, the layer sizes were chosen small, changing between 0.30 m and 0.80 m and
being smaller around the water table (capillary fringe zone) and ground surface and larger
in the rest of the area.
0
200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800
2800
2800
NO FLOW BOUNDARY
2600
2600
ND
2000
FL
OW
2000
1600
1800
C0522
NO
1800
2400
2200
BO
U
2200
1600
re
ek
MODEL DOMAIN
1400
at
or
C
(meters)
LOWRY LAKE
AR
Y
2400
1200
Al
lig
1200
1400
1000
800
A
C0521
1000
800
600
600
400
400
CRYSTAL
200LAKE
C0520
NO FLOW BOUNDARY
0
MAGNOLIA
LAKE
200
0
0
200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800
Observation Wells
(meters)
A-A A cross-section for 2-dimensional simulation
Figure 7.5. Horizontal discretization of the three-dimensional model domain.
The three-dimensional simulation required excessive computer time for one
complete year of hydrologic simulation. The vertical grid sizes were not small enough to
A
176
simulate the nonlinear hydrogeologic nature of the unsaturated zone. The computer
memory did not allow use of a finer grid size in the vertical direction, because increasing
the number of layers increases the size of the system of equation matrix to be solved in
each iteration in the order of the square of the layer numbers. Therefore, it was very
difficult to calibrate the model parameters. Using the initial hydrogeologic values
obtained from Motz et al. (1993), the model results have the same pattern as the real data,
but the model underestimated the water table elevation most of the time. The time
required to calibrate the model parameters (i.e., hydraulic conductivities, leaf area index,
root depth, and moisture content parameters) was excessive. For example, each
simulation for a year, with the largest time step as 0.25 day, required 18 hours using a 500
MHz Pentium computer.
Two-Dimensional Model Discretization
To decrease the time for the simulation and to increase the computer efficiency, a
cross section (A-A cross-section in Figure 7.5) was chosen between Magnolia Lake and
Crystal Lake, which also included one of the observation wells, to make a twodimensional simulation. In the two-dimensional application, the same horizontal
discretization of the three-dimensional application was used (25 columns). In the vertical
direction, a finer discretization was used, and the model domain was divided into 99
layers starting with the upper Floridan aquifer as the first layer, which is 80 m thick. The
next two layers are 10 m and 15 m thick, which are in the upper Floridan aquifer and the
lower part of the confining unit depending on their elevation. Layer four is 45 m thick
and it contains only the confining unit. Layers five and six are 10 and 3 meters thick,
respectively, and contain the upper part of the confining unit and/or the surficial aquifer,
177
depending on their elevations. The next two layers are 1 meter each, and they contain
parts of the confining unit and saturated parts of the surficial aquifer. Beginning with
layer nine, the layer sizes were chosen very small, changing between 0.15 and 0.33 m and
being very small around the water table (capillary fringe zone) and the ground surface
(see Figure 7.6).
60
Lake Crystal
Magnolia Lake
40
Elevation (m)
20
Cross-Section at Y=120 m
0
Ground Surface
Top of the Hawthorn
Top of the Upper Floridan
Water Table
Piezometric Elevation of the Upper Floridan
-20
-40
-60
0
250
500
750 1000 1250 1500 1750
Distance (m) in the X-Direction
2000
Figure 7.6 Vertical discretization of the two-dimensional model domain.
2250
2500
178
Description of Input Parameters for the Two-dimensional Simulation of the Model
The hydrogeological and geometrical parameters used in the model applications
are tabulated below in Table 7.6. The hydrogeological data were obtained from Motz et
al. (1993). The input files for the initial pressure heads, elevations of the geologic layers,
the arrays showing the boundary characteristics (Ibound), and the soil type (Isoil) are in
Appendix B.
Table 7.5 Parameters used for the model application in the UECB area.
Flow domain
2800 m x 190 m
Brooks and Corey Relation (1964)
Moisture Content, θ(h)
Brooks and Corey Relation (1964)
503 m/day for upper Floridan aquifer
0.008 m/day for the confining unit
Kx = Ky = 7.62 m/day , Kz=5.62 m/day for the
surficial aquifer
Saturated moisture content, θs
0.45 for sand (surficial aquifer only)
Residual moisture content, θr
0.08 for sand (surficial aquifer only)
Brook and Corey (1964) Parameters,
-0.85 m, 1.0 for sand (surficial aquifer only)
hb and λ
0.0003 for the surficial aquifer;
Specific storage, Ss (1/day)
0.0001 for the confining units; and
0.0001 for the upper Floridan aquifer.
No-flow boundary
Rainfall data for Magnolia Lake during the
period September 1994-September 1995; and
pan evaporation data for Gainesville during the
period September 1994-September 1995.
179
Table 7.5-continued
Evapotranspiration parameters
0th day
100th day
150th day
365th day
Root Depth (m)
0.35
0.50
0.50
0.35
Root Pressure (m)
-80
-80
-120
-80
Root activity at the bottom
0.2
0.2
0.2
0.2
Root activity at the top
0.5
0.7
0.9
0.9
Leaf Area Index
2.0
5.0
1.25
1.0
Interception Coefficient
0.2
0.2
0.2
0.2
Surface Resistance (1/m)
2.0/dz(k)
2.0/dz(k)
2.0/dz(k)
2.0/dz(k)
Atmospheric pressure potential (m)
-1,000
-1,000
-1,000
-1,000
(assuming the parameters change
linearly as time progresses)
Hydrostatic equilibrium with September 1994
water table elevations at the surficial aquifer;
Initial Conditions
minimum pressure head is -3.5 m; and initial
heads for the confining unit were interpolated
between the surficial aquifer and the upper
Floridan aquifer.
Grid Characteristics
25 cells in the x-direction. Variably sized dx
values changed from 80 m to 300 m.
99 cells in the z-directions. Variably sized dz
values changed from 0.15 m at the ground
surface to 80 m at the upper Floridan aquifer.
Time increment, dt
Varied from 0.01 to 0.125 days
365 days
Model Results
The two-dimensional simulation of the model area gave reasonable estimates of
the water table elevation at the observation well C520 under the influence of daily rainfall
180
and evapotranspiration data (see Figure 7.7). The simulation took 35 minutes of
computer time using a Pentium II 350 MHz computer
The calculated potential evaporation and the interpolated rainfall data are also
presented in Figure 7.8. Total head contours at time = 150 days are shown in Figure 7.9.
The head contours shows clearly that both lakes recharged water into the upper Floridan
aquifer and received water from the surficial aquifer. The moisture content profiles at
different times are plotted in Figure 7.10, which show the change in the moisture content
as a function of depth.
The model estimated the water table elevations very close to the observed data in
the first 8 months of the simulation period. The mean of the differences between the
observed data and the model results is 0.024 m, and of the standard deviation is 0.11 m.
During the last four months from June to September, the model slightly overestimated the
water table elevations. The explanation for this over estimation could be because of two
reasons:
1. The model does not take surface runoff into account and half of the annual
rainfall was received during the last four-month period.
2. The model does not allow rainfall to infiltrate into the ground after ponding
starts. Since daily rainfall values were used, all of the rainfall infiltrated into the
ground gradually. For example, even if a very intense rainfall fell in one hour in
one day, this intense rainfall would be treated in the model as if it rained gradually
over 24 hours. This gradual daily rainfall intensity and gradual infiltration
without ponding might have caused this overestimation during the last four
months of the simulation.
181
42
40
Elevation (m , NG VD)
38
36
Model Results
Observed Water Table Elevation in the Well C520
Magnolia Lake
Cyrstal Lake
34
32
30
9/1/94
10/31/94
12/30/94
2/28/95
4/29/95
6/28/95
8/27/95
Day
Figure 7.7 Model results versus the observed data in the well C520 during the
period of September,1994-September, 1995.
182
0.08
Evapotranspiration and Rainfall
Rainfall and Evaporation(m/day)
Rainfall
Evapotranspiration
0.06
0.04
0.02
Day
Figure 7.8 Rainfall and evapotranspiration components in the model area.
7-Aug
7-Jul
6-Jun
6-May
5-Apr
5-Mar
2-Feb
2-Jan
2-Dec
1-Nov
1-Oct
31-Aug
0
Crystal Lake
Magnolia Lake
Water Table
Surficial Aquifer
Well C520
183
0
500
1000
Figure 7.9 Total head contours at time =150 days
1500
2000
2500
184
56
Simulated Moisture Content Profile at Location of C520 Well
t = 365 day
t = 120 day
t = 20 day
Elevation(m )
52
48
44
40
0.1
0.2
0.3
0.4
0.5
Moisture Content at Well C520
Figure 7.10 Simulated moisture content profiles at different times during the simulation
CHAPTER 8
SUMMARY AND CONCLUSIONS
In this dissertation study, a complete three-dimensional variably saturated
numerical groundwater flow model was created. The model can simulate most of the
hydrologic events with the exception of surface runoff. Surface runoff is assumed to be a
loss term from rainfall after ponding starts. To describe soil hydrologic properties, which
are nonlinear functions in the model, three different options are available, i.e., the Brooks
and Corey (1964) equation, the van Genuchten and Nielsen (1985) equation, and the
power formula.
The governing equation of the model is the three-dimensional modified Richards
equation. The mixed form of the modified Richards equation, i.e., both moisture-content
and pressure based, is solved using the modified Picard iteration scheme based on Celia
et al. (1990). This mixed form of the Richards equation is more mass conserving, and it
does not require additional effort to solve compared to the pressure based form. The
resultant flow equation is written in terms of the total hydraulic head and moisture
content as the dependent variables in a fully implicit block-centered backward difference
finite-difference scheme. The resultant system of equations is solved using the
preconditioned conjugate gradient method, which is very robust and which converges to a
solution relatively quickly if a good preconditioner matrix is provided. A new nonlinear
convergence criterion derived using a Taylor series expansion of the water content is used
for the Picard iteration scheme. The new nonlinear convergence criterion, which based
185
186
on Huang et al. (1996), is computationally more efficient than the classical convergence
criterion, which is based on the maximum head difference between two consecutive
iterations. The new convergence criterion, which is based on the maximum change in the
moisture content in the unsaturated zone, reduces the total simulation time significantly
compared to the conventional method.
Pumping from a cell is simulated by distributing the total pumping discharge to
the neighboring six cells of the pumped cell as surface fluxes from their appropriate faces,
based on Freeze (1971). This method prevents the pumped cells from going dry during
the pumping unrealistically.
Conductances between block-centered nodes are calculated using two options, i.e.,
the arithmetic mean in the unsaturated region and the geometric mean in the saturated
region. Generally, the geometric mean yields more accurate results, but the arithmetic
mean is more suitable in the unsaturated region in the case of infiltration in an initially
very dry soil medium. When the soil is initially very dry, the hydraulic conductivity on
the dry side of the wetting front will be very small (nearly zero) and the resulting average
hydraulic conductivity also will be very small, if not zero, if the geometric mean is
calculated. Consequently, the wetting front will not move forward. However, calculating
the arithmetic mean will result in a larger, non-zero conductance, and the wetting front
will move forward into a very dry soil.
The model is capable of simulating sink and source terms that include pumping,
recharge, and drains. Rainfall and evaporation are simulated as upper boundary
conditions. Transpiration calculations can be done using two options: (1) the model
option (equation 3.78), and (2) the method used in VS2D by Lappala et al. (1987). The
187
model has a subprogram to calculate potential evapotranspiration from climatologic data
that are available for most of the United States.
The model was verified using five different examples chosen from the literature
and two examples created for this study. These examples involve one-dimensional, twodimensional, and three-dimensional variably saturated flow problems. Almost all aspects
of the model were verified using these examples. The results of the model and the
examples are in very close agreement, which proves that the model is suitable for
application.
As shown in chapter 6, a two-dimensional rainfall and evapotranspiration
simulation was done successfully. The model also was used to simulate an unconfined
aquifer pumping problem in three-dimensions, and these results help to support the
hypothesis that delayed yield actually happens in an unconfined aquifer during pumping
due to the effect of the unsaturated zone. In Chapter 7, a field application is described in
three- and two-dimensions. The three-dimensional simulation with 60 layers did not give
successful results because of the relatively coarse vertical discretization in the unsaturated
zone. The coarse discretization caused instability in solving the Richards equation
because of its highly nonlinear properties in the unsaturated zone. Finer discretization
resulted in extremely large matrices that could not be solved because of limitations of the
computer memory and speed. Therefore, a cross section on a streamline from the threedimensional discretization was selected and rediscretized into 99 layers in the vertical
direction. The two-dimensional simulation gave very successful results, which matched
the observed data over the period of simulation.
188
Table 8.1 summarizes the features of the new model. The only items lacking from
the model are surface runoff and the coupling of a more realistic river flow. Seepage face
effects are ignored, because these are considered negligible in regional scale problems.
These lacking items need to be studied as a future work. In the next section, as a frame
work, the theoretical steps of how to integrate a surface-flow calculation with the
groundwater system are presented.
Table 8.1 Summary of new model.
Property:
Dimensions
Saturated/unsaturated
flow
Evapotranspiration
calculations
Transpiration (root
water uptake)
Rainfall data
Governing equation
Numerical formulation
Solution technique
Iteration techniques
Pumping
Confined/unconfined
aquifer simulation
Boundary conditions
River and groundwater
interaction
Output options
Description:
Three-dimensional (or can be collapsed to one- or twodimensions)
Continuous modeling of variably saturated Darcian
groundwater flow
Pan evaporation, or
Priestly-Taylor method using meteorological data
VS2D method (Lappala et al., 1987), or
Modified Feddes et al.(1988) method
Daily or hourly rainfall data input is possible.
Mass conserving mixed (θ- and h-based) form of the modified
Richards equation
Block centered, fully implicit, backward finite difference
formulation
Preconditioned conjugate gradient method (PCGM)
Modified Picard iteration for outer iteration scheme and PCGM
iteration for inner iteration scheme.
Pumping discharge from a cell is distributed as facial fluxes to
its neighboring cells for a better conceptualization of pumping.
Confined aquifers can be modeled without any input data of
unsaturated material properties.
Specified head, or flux boundary, variable boundary
(rainfall/ponding), and general head boundary conditions can be
chosen.
Water infiltration to porous medium to/from a river, requires
specifying head in river and conductance.
Pressure head, total head, moisture content, and saturation ratio
profiles at any cross-section for a given time or at any location
throughout time.
189
Applicability Limitations of the Model
The main drawbacks of the model are the lack of the runoff and seepage face
simulations. Therefore, the model should not be applied where surface runoff is a major
component of the hydrologic cycle. Also, the model is very sensitive to vertical
discretization, and thus it requires finer discretization near the ground surface and in the
vicinity of the water table, which is especially true if the capillary rise of the soil is
higher. Therefore, this model is very useful in small-scale applications where finer
vertical discretization is mathematically feasible. In regional applications, this model can
be used to predict the unconfined aquifer properties, i.e., specific yield, rainfall recharge
relation, etc. Special care should be given to the selection of time steps because larger
time steps may cause instability in simulations during the rainfall and evaporation
processes.
Future Study
To make this model a more complete hydrological numerical model in order to
simulate any hydrological event, a comprehensive surface flow submodel needs to be
developed and integrated into the current model. The following procedure represents a
framework for creating a surface submodel and integrating it into the current model.
The governing equation for surface flow can be developed by assuming that the
kinematic wave interpretation of the equation of motion is valid for the surface flow of a
region. This requires assuming that the river-bottom slope and water-surface slopes are
equal and those acceleration effects are negligible. After these assumptions are made,
flow at any point in the river can be calculated from Manning‘s formula:
190
Qr =
A 2/3 1/2
R S0
n
(7.1)
where Qr is the river flow, A is the cross sectional area of the river, R is the hydraulic
radius of the river, and So is the river bed slope.
The continuity equation for a river segment can be written as
∂A ∂ (Q r )
+
+ qr = 0
∂t
∂x
(7.2)
where qr, the aquifer river exchange per unit length of the river [L3T-1L-1], is calculated
from the Darcy-Buckingham equation:
q r = Cr
hr − h
dz r
(7.3)
where Cr is the conductance term for the river bed, which is calculated by averaging the
river-bed hydraulic conductivity and the hydraulic conductivity of the first cell beneath
the river bed; hr is the river head; h is the hydraulic head in the first cell beneath the river
bed; and dzr the distance between the river bottom and the first cell beneath the river bed.
At each time step, using previously calculated h and hr values, new hr values
would be calculated from equations (7.1 and 7.2). Then, using the new hr values, the new
qr values would be calculated using equation (7.3). In the new time step, qr would be
191
used as a specified flux boundary condition for the top cells having river segments in
them. This procedure would be followed for each river cell for every time step.
To solve equation (7.2), an initial condition, a downstream boundary condition in
the form of hydrograph, and a Manning roughness coefficient for the river-bed would be
required.
Application to Finite-Difference Scheme
In the model, the river boundary is treated as a specified head source/sink term
located only in the top boundary cells. The fluxes passing from or to the underlying
porous medium could be calculated using the procedure described above. River stages
would be calculated at each time step using the river boundary conditions (generally a
hydrograph), Manning's equation, and the momentum equation. To calculate river stages,
iteration would be required until the river and porous medium pressure heads did not
change any more. The steps of the iteration are as follows:
Step 1: The leakage qr is calculated (equation 7.3) using the most current
riverhead Hr and porous medium pressure head Hi.
Step 2: The new Hr values are calculated substituting qr of step 1 into equation
(7.2).
Step 3: The new Hi is calculated from the groundwater flow equation (using the
modified Picard-iteration and PCG methods) using the qr value of step 1 as
source/sink term.
Step-4: If convergence criteria are not satisfied for both Hr and Hi, it is necessary
to return to step 1.
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APPENDIX A
THE FORTRAN CODE OF VARIABLY SATURATED THREE-DIMENSIONAL
RAINFALL DRIVEN GROUNDWATER PUMPING MODEL
C LAST CHANGE: KH 21 JUL 99 9:10 PM
C************************************************************************************
C THIS IS THE MAIN PROGRAM TO SOLVE THREE-DIMENSIONAL GROUNDWATER FLOW
C EQUATION UNDER VARIABLY SATURATED CONDITIONS.
*
C THE GOVERNING EQUATION IS THE MIXED FORM OF THE MODIFIED RICHARDS
C EQUATION. IT IS SOLVED BY AN IMPLICIT FINITE-DIFFERENCE EQUATION USING
C MODIFIED PICARD ITERATION SCHEME AND PRECONDITIONED CONJUGATE
C GRADIENT METHOD.
*
C
THE FOLLOWING RELATIONSHIPS ARE REQUIRED FOR SIMULATION:
*
C 1. RELATIVE HYDRAULIC CONDUCTIVITY VERSUS PRESSURE HEAD
*
C 2. MOISTURE CONTENT VERSUS PRESSURE HEAD
*
C 3. ROOT ACTIVITY FUNCTION AS A FUNCTION OF TIME AND DEPTH
*
C 4. SPECIFIC MOISTURE CAPACITY (FIRST DERIVATIVE OF MOISTURE CONTENT VS.
C
PRESSURE HEAD).*
C************************************************************************************
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
C
CHARACTER*8 BUF1,BUF2
INTEGER DXNUM,DYNUM,DZNUM,UNSTDY,IBOUND,ICONF,ISOIL
C
INTEGER*2 TMPHOUR, TMPMINUTE, TMPSECOND, TMPHUND
DOUBLE PRECISION HMAX,XKM1(300000),MAXKC,TIMWRT(6),HMINI,RCHPER(4)
2,RTBOT(4),RTTOP(4),RTDPTH(4),LAI(4),SRES(4),HROOT(4),ETPER(4),
3HA(4)
COMMON /CODE/IBOUND(300000),ISOIL(300000)
COMMON /BNDRY/HPOND(100,100),QRAIN(1,1,365),QEVAP(1,1,365)
COMMON /TETA/THETA1(300000),XDIST(100),XVEC(100),YVEC(100)
COMMON /DIMENS/ ICSTRT,IBSTRT,IGEND,IFEND,IEEND,NUMCEL,DXNUM,DYNUM
1,DZNUM,NDXDY
COMMON /VECTS/AVEC(300000),BVEC(300000),CVEC(300000),
1 DIAGNL(300000),EVEC(300000),FVEC(300000),GVEC(300000),
2RHSVEC(300000),HOLDR(300000),HOLD(300000),QEXVEC(300000),
3DX(100),DY(100),DZ(300),ZVEC(300000),THETAO(300000),THETAN(300000)
4,P1VEC(300000),P2VEC(300000),HSTRT(300000),HNEW(300000),SW(300000)
COMMON /MTRX3D/SATKZ(100,100,300),SATKX(100,100,300),
1SATKY(100,100,300),RCON(100,100,300),CAP(300000),QEX(300000),
2TETADT(300000),SS(300000),ICONF(300000)
COMMON /ET/QRTDPH,QRTTOP,QRTBOT,QHROOT,QPET,QPEV,QSRES,QHA,RAIN,
1 QTOT(100,100),EVAP,QRATIO,ZROOT(100,100)
PARAMETER(DZERO=0.0D+00)
OPEN (11,FILE='EVAPRC.DAT',STATUS='UNKNOWN')
OPEN( 17,FILE='DWDP15.DAT',STATUS='UNKNOWN')
OPEN (13,FILE='TIME.DAT',STATUS='UNKNOWN')
OPEN( 54,FILE='DWSR5.DAT',STATUS='UNKNOWN')
208
209
OPEN (57,FILE='DWSR15.DAT',STATUS='UNKNOWN')
OPEN (21,FILE='ALI21.DAT',STATUS='UNKNOWN')
OPEN (14,FILE='DWDP5.DAT',STATUS='UNKNOWN')
OPEN (47,FILE='POND.DAT',STATUS='UNKNOWN')
DATA DXNUM,DYNUM,DZNUM,DTMIN,DTMAX,TIMEND,UNSTDY
1 /25,1,99,0.01,.125,365,1/
DATA TIMWRT,RCHPER,ETPER/0.0,5,20,50,120,365.0,
11,40,128,185,0,100.,150,365/
DATA LAI,SRES,RTBOT,RTTOP/2.,5.,1.25,1.0,4*.6,
1 4*.2,.5,.7,.9,.9/
DATA RTDPTH,HROOT,HA/.35,.5,.5,.35,-80.,-80.,-120.,-80.,
14*-1000.0/
ITERMX=50
HMINI=-1000.
PRINT*,'UNSTEADY= ',UNSTDY
IF (DZNUM.LE.1) STOP 'YOU MUST HAVE MORE THAN ONE DZ'
NDXDY=DXNUM*DYNUM
NUMCEL=NDXDY*DZNUM
DT=DTMIN
ICSTRT=NDXDY+1
IBSTRT=DXNUM+1
IASTRT=2
IGEND=NUMCEL-NDXDY
IFEND=NUMCEL-DXNUM
IEEND=NUMCEL-1
ISIMST=111
C START SIMULATION WITH TIME STEP INCREMENTS
C
CALL TIME(BUF1)
C
PRINT*, 'STARTING TIME=', BUF1, ' DT = ',DT
C
PRINT*,'SIMULATION STARTS'
C
WRITE(13,*),'TIME STATRT= ', BUF1, 'DT= ',DT
TOLD=0.
NWRITE=1
TNEW=TOLD+DT
ISTRT=0
210
CALL UNSAT(UNSTDY,TOLD,TNEW,KP,ISTRT,TETMAX,HMAX,DELXHD)
ISTRT=1
WRITE(11,*) 'TIME ','PETREAD ','PETCALC ','QPEV ',
1'ACTET ','QPT
','ACTPT ','QRAIN ','RAIN'
1 CONTINUE
C
IF(TOLD.GE.1) ITERMX=5
ISTRT=ISTRT+1
TNEW=TOLD+DT
IPOND=0
C ADJUST THE TIME STEP ACCORDING TO NUMBER OF ITERATION FROM PREVIOUS
TIME STEP
IF (KP.LE.17) THEN
DT=1.1*DT
IF (DT.GT.DTMAX) DT=DTMAX
TNEW=TOLD+DT
ELSE IF(KP.GE.18) THEN
DT=DT*.2
IF (DT.LE.DTMIN) DT=DTMIN
TNEW=TOLD+DT
ELSE
TNEW=TOLD+DT
END IF
C DETERMINE THE RECHARGE PERIOD
IF (TOLD.LT.RCHPER(1)) THEN
IRAIN=1
IEVT=1
NPER=1
ELSE IF (ABS(TOLD-RCHPER(1)).LE.DT)
1 THEN
NN=1
DTMIN=0.1
DT=DTMIN
TNEW=TOLD+DT
IRAIN=1
IEVT=1
NPER=2
PRINT*,TOLD,TNEW,NPER,DT
ELSE IF (ABS(TOLD-RCHPER(2)).LE.DT) THEN
DTMIN=0.1
DT=DTMIN
TNEW=TOLD+DT
IRAIN=1
IEVT=1
NPER=3
PRINT*,TOLD,TNEW,NPER,DT
ELSE IF (ABS(TOLD-RCHPER(3)).LE.DT) THEN
DTMIN=0.1
DT=DTMIN
TNEW=TOLD+DT
IRAIN=1
IEVT=1
NPER=4
PRINT*,TOLD,NPER,DT
END IF
C DETERMINE THE ET PERIOD AND INTERPOLATE THE ET VARIABLES
211
C
C
C
C
C
C
IF (TOLD.LE.ETPER(2).AND. TOLD.GE.ETPER(1)) THEN
IET=1
TIMRAT=(TOLD-ETPER(1))/(ETPER(2)-ETPER(1))
QPEV=PEV(1)+(PEV(2)-PEV(1))*TIMRAT
QPET=PET(1)+(PET(2)-PET(1))*TIMRAT
QLAI=LAI(1)+(LAI(2)-LAI(1))*TIMRAT
QRTDPH=RTDPTH(1)+(RTDPTH(2)-RTDPTH(1))*TIMRAT
QHROOT=HROOT(1)+(HROOT(2)-HROOT(1))*TIMRAT
QSRES=SRES(1)
QHA=HA(1)
QRTBOT=RTBOT(1)
QRTTOP=RTTOP(1)
ELSE IF (TOLD.LE.ETPER(3).AND.TOLD.GT.ETPER(2)) THEN
IET=2
QSRES=SRES(2)
QHA=HA(2)
QRTBOT=RTBOT(2)
QRTTOP=RTTOP(2)
ELSE IF (TOLD.LE.ETPER(4).AND.TOLD.GT.ETPER(3)) THEN
IET=3
QSRES=SRES(3)
QHA=HA(3)
QRTBOT=RTBOT(3)
QRTTOP=RTTOP(3)
END IF
N=TOLD
TIMEDF=TNEW-N
IF (TIMEDF.GT.1) THEN
TNEW=N+1
TIMEDF=1
END IF
RAIN=-((QRAIN(1,1,N+1)-QRAIN(1,1,N))*TIMEDF+QRAIN(1,1,N))
PTEVA=(QEVAP(1,1,N+1)-QEVAP(1,1,N))*TIMEDF+QEVAP(1,1,N)
QPET=PTEVA*(1.-EXP(-0.4*QLAI))
QPEV=PTEVA-QPET
XINTP=0.2*QLAI/1000.
RAIN=RAIN+XINTP
IF(RAIN.GT.0.0) RAIN=0.0
PRINT*,'TOLD= ',TOLD, ' DT= ',DT,' TNEW= ',TNEW,' TIME=',BUF1,
1'TIMEDIF=',TIMEDF
WRITE(*,279) QRAIN(1,1,N+1),RAIN,QPET,QPEV
WRITE(13,*) 'TNEW= ',TNEW, ' DT= ',DT,' TOTAL KP= ',KP
212
WRITE(13,279) QRAIN(1,1,N+1),RAIN,QPET,QPEV
279 FORMAT(1X,'QRAIN= ',F10.4,' RAIN=',F10.4,' QPET=',F10.4,'
1 QPEV=',F10.4)
IF (TOLD.GT.TIMEND) GOTO 9001
3 IF(IPOND.EQ.1) CALL UNSAT(UNSTDY,TOLD,TNEW,KP,
1ISTRT,TETMAX,HMAX,DELXHD)
C FIRST PICARD ITERATION STARTS HERE
KP=0
KC=0
2 KP=KP+1
IF (KP.GE.ITERMX.OR.KC.GE.10000) GOTO 331
C CREATE THE SYSTEM OF EQUATIONS MATRIX FROM FINITE DIFFERENCE EQUATIONS
IMIXED=01
CALL VECCRT(UNSTDY,TOLD,TNEW, IMIXED)
C RECALCULATE SOME OF THE MATRIX ELEMENTS ACCORDING TO BOUNDARY
CONDITIONS
CALL BOUND(IEVT,IRAIN,TOLD,KP,HMINI)
IF(TOLD .EQ.0.0 .AND. KP.EQ.1) THEN
PRINT*, TOLD,TNEW
C 'WRITE INITIAL CONDITIONS TO OUT PUT FILES'
DO 108 I=1,DXNUM
DO 108 J=1,DYNUM
DO 108 K=1,DZNUM
IN=I+(J-1)*DXNUM+(K-1)*DXNUM*DYNUM
IF (HOLD(IN).GT.-999.) THEN
WRITE(NWRITE+30,79) .1*XVEC(I),ZVEC(IN),HNEW(IN),I,K
1 ,THETAN(IN)
IF(I.EQ.6) WRITE(NWRITE+20,79) ZVEC(IN),
1 HNEW(IN),THETAN(IN),I,K,HNEW(IN)-ZVEC(IN)
END IF
108 CONTINUE
CLOSE (21)
CLOSE(31)
PRINT*, 'NWRITE= ',NWRITE
NWRITE=NWRITE+1
ENDIF
DO I=1,NUMCEL
XKM1(I)=HOLD(I)
END DO
C CALL THE MATRIX SOLVER IN PRECONDITIONED CONJUGATE GRADIENT METHOD
CALL PCGM(XKM1,DIAGNL,RHSVEC,NUMCEL,ISIMST,KC,MAXKC)
C CHECK WHETHER CONVERGENCE CRITERIA REACHED OR NOT IN THIS PICARD
ITERATION LEVEL
C IN DO LOOP 15
DO II=1, NUMCEL
HNEW(II)=XKM1(II)
END DO
CALL UNSAT(UNSTDY,TOLD,TNEW,KP,ISTRT,TETMAX,HMAX,DELXHD)
IF(KP.EQ.1) TETMAX=.25
WRITE(*,122) KP,HMAX,KC,TETMAX,TOLD,DELXHD
WRITE(13,122) KP,HMAX,KC,TETMAX,TOLD,DELXHD
122 FORMAT(1X,'KP= ',I3,' KPMAX =',F8.3, ' KC= ',I4,
1 ' TETMAX= ',F7.5,'TOLD=',F7.3,'MXHD',F7.2)
213
C
IF CONVERGENCE CRITERIA IS SATISFIED,PROCEED FOR THE NEXT TIME STEP
IF (TETMAX.LE.0.005)THEN
331 IPOND=0
IREITR=0
IF(IFLUX.EQ.1)CALL POND(IREITR,IRAIN,IFLUX,NPOND1,TNEW,RAIN,QPEV)
PRINT*, IFLUX,'NUMBER OF PONDED CELL= ',NPOND1
NPOND=0
DO 15 I=1,DXNUM
DO 15 J=1,DYNUM
DO 15 K=1,DZNUM
II=I+(J-1)*DXNUM+(K-1)*DXNUM*DYNUM
IF(IBOUND(II).EQ.7) THEN
IF(HNEW(II).GT.1.01*HPOND(I,J)) THEN
NPOND=NPOND+1
PRINT*,'PONDING AT',I,J,K, 'AT TIME',TNEW,HNEW(II),HPOND(I,J)
WRITE(13,247)I,J,K,TNEW,HNEW(II),HPOND(I,J)
WRITE(47,247) I,J,K,TNEW,HNEW(II),HPOND(I,J)
247 FORMAT('PONDING ',3I3,' AT TIME= ',F6.2,'HNEW,HPOND= ',2F9.3)
HNEW(II)=HPOND(I,J)
HOLD(II)=HPOND(I,J)
HOLDR(II)=HPOND(I,J)
C
THETAO(II)=TETAS WILL BE DONE IN UNSAT SUBROUTINE
IBOUND(II)=-9
IPOND=1
IFLUX=1
END IF
END IF
15 CONTINUE
IF (IPOND.EQ.1.OR.IREITR.EQ.1) THEN
PRINT*,NPOND,NPOND1
PAUSE 'NPOND, NPOND1(POND DAN GELEN)'
DO 16 II=1, NUMCEL
HOLD(II)=HOLDR(II)
HNEW(II)=HOLDR(II)
THETAN(II)=THETAO(II)
THETA1(II)=THETAO(II)
16 CONTINUE
PRINT*, 'REITERATE, IFLUX= ',IREITR,IFLUX,TOLD,TNEW
C
PAUSE 'IRETIERATION'
GOTO 3
END IF
INDEXT=TNEW
C WRITE(101,*) 'TIME ',' PETREAD ',' PETCALC',' QPEV', ' ACTET', ' QPT'
C ,'ACTPT ',' QRAIN ','RAIN'
WRITE(11,141)TNEW,QEVAP(1,1,INDEXT),PTEVA,QPEV,EVAP,QPET,
1 QTOT(6,1)*QRATIO/(DX(6)*DY(1)),QRAIN(1,1,INDEXT),RAIN
141 FORMAT(10E11.4)
C
WRITE(101,*) 'HROOT=',QHROOT
C IF THIS IS A STEADY STATE SIMULATION( UNSTDY=0) PRIBT THE OUTPUTS AND STOP
IF(UNSTDY.EQ.0) GOTO 333
DO K=1,DZNUM
K520=7+(1-1)*DXNUM+(K-1)*NDXDY
C
C
214
K522=7+(1-1)*DYNUM+(K-1)*NDXDY
IF (ABS(HNEW(K520)-ZVEC(K520)).LT.DZ(K)) WT520=HNEW(K520)
END DO
IC520=7+(1-1)*DXNUM+(26-1)*NDXDY
IC521=6+(1-1)*DXNUM+(99-1)*NDXDY
IC522=6+(1-1)*DYNUM+(98-1)*NDXDY
IC523=8+(1-1)*DXNUM+(26-1)*NDXDY
WRITE(41,144) TNEW,HNEW(IC520),WT520,
1 HNEW(IC520)-ZVEC(IC520)
WRITE(42,144) TNEW,HNEW(IC521),THETAN(IC521),
144 FORMAT(4F12.3)
C CHECK THE TIME IF IT IS TIME FOR OUTPUT PRINTING
IF (ABS(TNEW-TIMWRT(NWRITE)).LT.DT/2.) THEN
333 PRINT*,'TNEW= ',TNEW,' TIMWRITE= ',TIMWRT(NWRITE),' NWRITE=',
1 NWRITE
PRINT*,'KP= ',KP,' KC= ',KC
C PRINT THE RESULTS FOR THIS TIME STEP
DO 107 I=1,DXNUM
DO 107 J=1,DYNUM
DO 107 K=1,DZNUM
IN=I+(J-1)*DXNUM+(K-1)*DXNUM*DYNUM
IF (IBOUND(IN).NE.0) THEN
WRITE(NWRITE+30,79) XVEC(I)*.1,ZVEC(IN),HNEW(IN),J,K,
1SW(IN)
IF(I.EQ.6) WRITE(NWRITE+20,79) ZVEC(IN),
1 HNEW(IN),THETAN(IN),I,K,HNEW(IN)-ZVEC(IN)
END IF
107 CONTINUE
CLOSE(NWRITE+20)
CLOSE(NWRITE+30)
C
CLOSE(NWRITE+40)
C IF NUMBER OF ITERATION EXCEEDS THE MAXIMUM ALLOWABLE NUMBER OF
ITERATION
C THEN WRITE THE RESULTS AND HALT THE PROGRAM
IF (KP.GT.ITERMX+1.OR.KC.GE.10000) THEN
CLOSE(14)
CLOSE(15)
CLOSE(NWRITE+20)
CLOSE(NWRITE+30)
PRINT*,'ALLOWABLE MAXIMUM NUMBER OF ITERATION IS EXCEEDED!!!'
GOTO 9001
END IF
215
79 FORMAT(3F12.4,2I3,F12.4)
78 FORMAT(2I8,F10.2)
NWRITE=NWRITE+1
ENDIF
DO I=1,NUMCEL
C
IF(IBOUND(II).GT.0) THEN
HOLDR(I)=HNEW(I)
THETAO(I)=THETAN(I)
THETA1(I)=THETAN(I)
HOLD(I)=HNEW(I)
C
END IF
END DO
TOLD=TNEW
C SOLUTION IS OBTAINED FOR THIS TIME STEP GOTO 1 FOR NEW TIME STEP
IF (UNSTDY.EQ.0) GOTO 9001
GOTO 1
ENDIF
C GOTO NEXT PICARD ITERATION
DO I=1,NUMCEL
C
IF (IBOUND(II).GT.0) THEN
THETA1(I)=THETAN(I)
HOLD(I)=HNEW(I)
C
END IF
END DO
GOTO 2
77 FORMAT(F10.4)
9001 END
C CALL TIME(BUF2)
C
PRINT*, 'TIME START=', BUF1
C
PRINT*, 'TIME END=', BUF2
C
WRITE(13,*) 'TIME START= ',BUF1,' TIME END= ',BUF2
C
END
**********************************************************************
**********************************************************************
INCLUDE 'UNSAT2.FOR'
INCLUDE 'VECRT2.FOR'
INCLUDE 'MATVEC.FOR'
INCLUDE 'PCGM.FOR'
INCLUDE 'BOUND2.FOR'
INCLUDE 'INTCON.FOR'
INCLUDE 'POND.FOR'
C
INCLUDE 'READ2D.FOR'
C
INCLUDE REALCN.FOR
**********************************************************
C****************************************************************************************
*
C THIS IS THE SUBROUTINE TO CALCULATE UNSATURATED SOIL CHARACTERISTICS
*
C (DX, DY, DZ, AND BROOK-COREY COEFICCIENTS ARE READ).
*
C****************************************************************************************
*
SUBROUTINE UNSAT(UNSTDY,TOLD,TNEW,KP,ISTRT,TETMAX,HMAX,XXMAX)
INTEGER DXNUM,DYNUM,DZNUM,IBOUND,ICONF,UNSTDY,ISOIL
216
DOUBLE PRECISION XDIST(100),THETA1(300000),BROOK(5,8),LAMBDA,
1ZELEV(300),Z1(100,100),Z2(100,100),Z3(100,100),Z4(100,100),
2Z5(100,100),ZTOP(100,100)
COMMON /TETA/THETA1,XDIST,XVEC(100),YVEC(100)
COMMON /BNDRY/HPOND(100,100)
1,QRAIN(1,1,365),QEVAP(1,1,365)
1,DZNUM,NDXDY
2TETADT(300000),SS(300000),ICONF(300000)
NUMCEL=DZNUM*NDXDY
IF (ISTRT.EQ.0 ) THEN
OPEN (156,FILE='GEOLAA.DAT',STATUS='UNKNOWN')
OPEN (157,FILE='PIEZAA2.DAT',STATUS='UNKNOWN')
OPEN (117,FILE='ZVEC.INP',STATUS='UNKNOWN')
OPEN (155,FILE='MODXY.DAT',STATUS='UNKNOWN')
OPEN(48,FILE='BROOK.INP',STATUS='UNKNOWN')
OPEN(158,FILE='EVAPRE2.INP',STATUS='UNKNOWN')
C
OPEN(117,FILE='HEDST2.INP',STATUS='UNKNOWN')
READ(158,*)
DO I=1,365
READ(158,*) QEVAP(1,1,I),QRAIN(1,1,I),NI
C
PRINT*,QEVAP(1,1,I),QRAIN(1,1,I),NI,I
C
PAUSE 1
END DO
DO I=1,5
READ(48,*) (BROOK(I,J),J=1,8)
END DO
ITERMX=7
HMINI=-1000.
IPMP=0
READ(155,*)
DO I=1,DXNUM
J=I
READ(155,*) DX(I),DY(J),XVEC(I),YVEC(J)
END DO
READ(117,*)
DO K=1,DZNUM
READ(117,*) DZ(K),ZELEV(K)
END DO
READ(157,*)
READ(156,*)
DO J=1,DYNUM
DO I=1,DXNUM
READ(157,*)II,JJ,XVEC(I),YVEC(J),Z1(I,J),Z2(I,J),
1Z3(I,J),Z4(I,J),Z5(I,J)
READ(156,*) II,JJ,XX,YY,ZTOP(I,J),ZZ,ZZZ
END DO
217
END DO
CALL INTCON(IBOUND,DXNUM,DYNUM,DZNUM,'IBOUND.INP')
CALL INTCON(ISOIL,DXNUM,DYNUM,DZNUM,'ISOIL.INP')
DO 33 K=1,DZNUM
DO 33 J=1,DYNUM
DO 33 I=1,DXNUM
II=I+(J-1)*DXNUM+(K-1)*NDXDY
IF(IBOUND(II).EQ.0) GOTO 33
IF (ISOIL(II).EQ.5) HSTRT(II)=Z1(I,J)
ZVEC(II)=ZELEV(K)
C
IF(IBOUND(II).EQ.9) IBOUND(II)=-1
IF((HSTRT(II)-ZVEC(II)).LT.-3.5) HSTRT(II)=ZVEC(II)-3.5
C
IF(ISOIL(II).GE.1.AND.ISOIL(II).LE.4) ICONF(II)=1
HPOND(I,J)=ZTOP(I,J)
HOLD(II)=HSTRT(II)
HOLDR(II)=HOLD(II)
HNEW(II)=HOLD(II)
33 CONTINUE
END IF
TETMAX=0.0
HMAX=0.0
XXMAX=0.0
C
C
C
C
C
C
C
C
C
DO 555 K=1,DZNUM
DO 555 J=1,DYNUM
DO 555 I=1,DXNUM
II=I+(J-1)*DXNUM+(K-1)*DXNUM*DYNUM
IF(IBOUND(II).EQ.0) GOTO 555
IF(HNEW(II).GE.HPOND(I,J))THEN
PRINT*,HNEW(II),I,K
HNEW(II)=HPOND(I,J)
END IF
NSOIL=ISOIL(II)
IF (NSOIL.EQ.2.OR.NSOIL.EQ.3.OR.NSOIL.EQ.4) NSOIL=2
IF (NSOIL.EQ.5) NSOIL=3
SATKX(I,J,K)=BROOK(NSOIL,1)
SATKY(I,J,K)=BROOK(NSOIL,2)
SATKZ(I,J,K)=BROOK(NSOIL,3)
SS(II)=BROOK(NSOIL,4)
QHB=BROOK(NSOIL,5)
IF (NSOIL.EQ.0)PRINT*,QHB,BROOK(NSOIL,5),NSOIL,IBOUND(II)
TETAR=BROOK(NSOIL,6)
TETAS=BROOK(NSOIL,7)
LAMBDA=BROOK(NSOIL,8)
IF(IBOUND(II).EQ.-9) THEN
THETAO(II)=TETAS
THETA1(II)=TETAS
END IF
IF(I.LE.3.AND.ISOIL(II).LE.4.AND.K.GT.1)
1SATKZ(I,J,K)=10.*SATKZ(I,J,K)
218
C
DO N=1,5
C
WRITE(*,3)(BROOK(N,IX),IX=1,8)
C
END DO
3 FORMAT(8F12.3)
XHDMAX=HNEW(II)-HOLDR(II)
DIFHED=HOLD(II)-HNEW(II)
DELTAH=ABS(DIFHED)
C DELXHD IS THE MAXIMUM HEAD CHANGE BETWEEN TWO TIME STEPS
DELXHD=ABS(XHDMAX)
IF( DELXHD.GT.XXMAX) XXMAX=DELXHD
IF (DELTAH.GT. HMAX) HMAX=DELTAH
HSMALN=HNEW(II)-ZVEC(II)
IF (HSMALN.LE.(HMINI-ZVEC(II))) THEN
C
PRINT*, HMINI,HNEW(II),I,K
C
PAUSE 'UNSAT1'
C
HNEW(II)=HMINI
TERM=ABS(QHB/HSMALN)
THETAN(II)=(TETAS-TETAR)*TERM**LAMBDA+TETAR
C
TETRAN=(THETAN(II)-TETAR)/(TETAS-TETAR)
RCON(I,J,K)=(1.0/TERM)**(-2.-3.*LAMBDA)
CAP(II)=-(TETAS-TETAR)*(LAMBDA/QHB)*(1.0/TERM)**(-LAMBDA-1.0)
ELSE
IF (HSMALN.LT.QHB) THEN
TERM=ABS(QHB/HSMALN)
THETAN(II)=(TETAS-TETAR)*TERM**LAMBDA+TETAR
C
TETRAN=(THETAN(II)-TETAR)/(TETAS-TETAR)
RCON(I,J,K)=(1.0/TERM)**(-2.0-3.0*LAMBDA)
CAP(II)=-(TETAS-TETAR)*(LAMBDA/QHB)*(1.0/TERM)**(-LAMBDA-1.0)
C
PRINT*, QHB,HSMALN,HNEW(II),I,K
C
PAUSE 'UNSAT2'
ELSE IF(HSMALN.GE.QHB) THEN
THETAN(II)=TETAS
CAP(II)=0.0
RCON(I,J,K)=1.0
END IF
END IF
IF (ISTRT.NE.0) THEN
IF (UNSTDY.EQ.1) TETADT(II)=(THETAN(II)-THETAO(II))/(TNEW-TOLD)
ELSE
END IF
DELTET=ABS(THETAN(II)-THETA1(II))
IF (DELTET.GT.TETMAX) TETMAX=DELTET
SW(II)=THETAN(II)/TETAS
555 CONTINUE
IF (ISTRT.EQ.0) THEN
DO I=1,NUMCEL
THETAO(I)=THETAN(I)
THETA1(I)=THETAN(I)
END DO
END IF
RETURN
END
***************************************************************************
219
C
LAST CHANGE: KH 9 JUN 99 1:01 PM
SUBROUTINE VECCRT(UNSTDY,TOLD,TNEW,SW)
INTEGER DXNUM,DZNUM,DYNUM,NUMCEL,UNSTDY,ICONF(300000),
1IBOUND(300000),ISOIL(300000)
DOUBLE PRECISION AVEC(300000),BVEC(300000),CVEC(300000)
1,DIAGNL(300000),EVEC(300000),FVEC(300000),GVEC(300000),
2RHSVEC(300000),HOLDR(300000),HOLD(300000),SATKZ(100,100,300)
3,RCON(100,100,300),CAP(300000),P1VEC(300000),P2VEC(300000),
4SS(300000),THETAN(300000),THETAO(300000),TETADT(300000),
5QEXVEC(300000),QEX(300000),DX(100),DY(100),DZ(300),ZVEC(300000)
6,SATKX(100,100,300),SATKY(100,100,300),HSTRT(300000),SW(300000)
COMMON /MTRX3D/SATKZ,SATKX,SATKY,RCON,CAP,QEX,TETADT,SS,
1IBOUND,ICONF,ISOIL
1,DZNUM,NDXDY
COMMON /VECTS/ AVEC,BVEC,CVEC,DIAGNL,EVEC,FVEC,GVEC,RHSVEC,
1HOLDR,HOLD,QEXVEC,DX,DY,DZ,ZVEC,THETAO,THETAN,P1VEC,P2VEC,HSTRT
C*******************************************************************
C ICONF=1 MEANS THE LAYER IS SITRICLY CONFINED
PARAMETER (DZERO=0.0D0)
C
OPEN (3,FILE='VECTOR.OUT')
C
PAUSE 'YOU ARE IN VECCRT (ISIMST,UNSTDY,TOLD,TNEW)'
C KX/KZ RATIO IS XZRAT, KY/KZ IS YZRAT
C
XZRAT=1.0
C
YZRAT=1.0
C INITIALIZE ALL VECTORS AS ZERO
DO I=1,300000
AVEC(I)=DZERO
BVEC(I)=DZERO
CVEC(I)=DZERO
DIAGNL(I)=DZERO
EVEC(I)=DZERO
FVEC(I)=DZERO
GVEC(I)=DZERO
RHSVEC(I)=DZERO
P1VEC(I)=DZERO
P2VEC(I)=DZERO
END DO
C
WHICH AQUIFER IS CONFINED IUPCNF
IUPCNF=0
DO 111 K=1,DZNUM
DO 111 J=1,DYNUM
DO 111 I=1,DXNUM
C EVEC(I)=-(CNI+1/2,J,K)/DXI
C
CNI-1/2,J,K=-2(KSKR)I+1/2,J,K/(DXI+DXI+1)(EQ 4.69)
C KSKRAVE=(EQUATION 4.68)
C----SKIP CALCULATIONS IF CELL IS INACTIVE
C IF THE CELL IS INACTIVE OR FIXED HEAD MAKE ALL THE VECTORS ZERO FOR INACTIVE
CASE
C AND DIAGNL UNITY AND RHSVEC IS HOLD FOR FIXED HEAD CASE
IF (I.EQ.1) THEN
AVEC(II)=0.0
EVEC(II-1)=AVEC(II)
ELSE
220
IF (K.GT.IUPCNF) THEN
AVEC(II)=2.0D0*(DX(I)*SATKX(I,J,K)*RCON(I,J,K)+DX(I-1)*
1SATKX(I-1,J,K)*RCON(I-1,J,K))/(DX(I)*(DX(I)+DX(I-1))**2.0)
ELSE
AVEC(II)=2.0D0*SATKX(I,J,K)*SATKX(I-1,J,K)/
1(DX(I)*(DX(I)*SATKX(I-1,J,K)+DX(I-1)*SATKX(I,J,K)))
END IF
EVEC(II-1)=AVEC(II)
END IF
IF (J.EQ.1) THEN
BVEC(II)=0.0
FVEC(II-IBSTRT+1)=BVEC(II)
ELSE
BVEC(II)=2.0D0*(DY(J)*SATKY(I,J,K)*RCON(I,J,K)+DY(J-1)
1*SATKY(I,J-1,K)*RCON(I,J-1,K))/(DY(J)*(DY(J)+DY(J-1))**2.0)
ELSE
BVEC(II)=2.0D0*SATKY(I,J,K)*SATKY(I,J-1,K)/
1(DY(J)*(DY(J)*SATKY(I,J-1,K)+DY(J-1)*SATKY(I,J,K)))
END IF
FVEC(II-IBSTRT+1)=BVEC(II)
END IF
IF (K.EQ.1) THEN
CVEC(II)=0.0
GVEC(II-ICSTRT+1)=CVEC(II)
ELSE
CVEC(II)=2.0D0*(DZ(K)*SATKZ(I,J,K)*RCON(I,J,K)+DZ(K-1)*
1SATKZ(I,J,K-1)*RCON(I,J,K-1))/(DZ(K)*(DZ(K)+DZ(K-1))**2.0)
ELSE
CVEC(II)=2.0D0*SATKZ(I,J,K)*SATKZ(I,J,K-1)/
1(DZ(K)*(DZ(K)*SATKZ(I,J,K-1)+DZ(K-1)*SATKZ(I,J,K)))
END IF
GVEC(II-ICSTRT+1)=CVEC(II)
END IF
IF (IBOUND(II).GT.0 .AND. UNSTDY.EQ.1) THEN
P1VEC(II)=CAP(II)/(TNEW-TOLD)
P2VEC(II)=SW(II)*SS(II)/(TNEW-TOLD)
ELSE
P1VEC(II)=0.0
P2VEC(II)=0.0
ENDIF
111 CONTINUE
C IF CELL IS FIXED HEAD THEN DIAGNL=UNITY
DO 19 K=1,DZNUM
DO 19 J=1,DYNUM
DO 19 I=1,DXNUM
IF (IBOUND(II).LE.0) THEN
AVEC(II)=DZERO
BVEC(II)=DZERO
CVEC(II)=DZERO
EVEC(II)=DZERO
FVEC(II)=DZERO
GVEC(II)=DZERO
221
HOLD(II)=HSTRT(II)
NEIGHBOURING CELLS WILL NOT GET ANY FLOW
IF (IBOUND(II).EQ.0) THEN
EVEC(I-1+(J-1)*DXNUM+(K-1)*NDXDY)=DZERO
AVEC(I+1+(J-1)*DXNUM+(K-1)*NDXDY)=DZERO
BVEC(I+(J)*DXNUM+(K-1)*NDXDY)=DZERO
FVEC(I+(J-2)*DXNUM+(K-1)*NDXDY)=DZERO
GVEC(I+(J-1)*DXNUM+(K-2)*NDXDY)=DZERO
CVEC(I+(J-1)*DXNUM+(K)*NDXDY)=DZERO
HOLD(II)=-9999.9
END IF
RHSVEC(II)=HOLD(II)
DIAGNL(II)=1.00
END IF
19 CONTINUE
C
C
DO 10 II=1,NUMCEL
DIAGNL(II)=-(AVEC(II)+BVEC(II)+CVEC(II)+EVEC(II)+FVEC(II)+
1
GVEC(II)+P1VEC(II)+P2VEC(II))
RHSVEC(II)=TETADT(II)-P1VEC(II)*HOLD(II)-P2VEC(II)
1
*HOLDR(II)-QEXVEC(II)
END IF
10 CONTINUE
79 FORMAT(8F8.2)
78 FORMAT(6F9.2)
77 FORMAT (4E12.4,I5)
RETURN
END
**********************************************************************
C LAST CHANGE: KH 21 JUN 99 1:46 AM
C THIS SUBROUTINE MULTIPLIES MATRIX A BY ZKM1 VECTOR FOR PCGM
SUBROUTINE MATVEC(VEC,RES,ISIMST)
IMPLICIT DOUBLE PRECISION (A-H,P-Z)
INTEGER DXNUM,DYNUM,DZNUM,NDXDY,ISIMST
DOUBLE PRECISION RES(300000),VEC(300000)
1,DZNUM,NDXDY
4,P1VEC(300000),P2VEC(300000),HSTRT(300000)
DO 100 IA=1,NUMCEL
IF (IA.LE.IGEND) THEN
RES(IA)=DIAGNL(IA)*VEC(IA)+EVEC(IA)*VEC(IA+1)
1+FVEC(IA)*VEC(DXNUM+IA)+GVEC(IA)*VEC(NDXDY+IA)
IF (IA.LT.IBSTRT.AND. IA.GT.1) THEN
RES(IA)=AVEC(IA)*VEC(IA-1)+RES(IA)
ELSE IF (IA.GE.IBSTRT.AND.IA.LT.ICSTRT) THEN
RES(IA)=BVEC(IA)*VEC(IA-DXNUM)+AVEC(IA)*VEC(IA-1)+RES(IA)
C AVEC=EVEC,BVEC=FVEC,CVEC=GVEC BECAUSE OF SYMMETRY
ELSE IF (IA.GE.ICSTRT.AND. IA.LE.IGEND) THEN
RES(IA)=CVEC(IA)*VEC(IA-NDXDY)+BVEC(IA)*VEC(IA-DXNUM)
222
1+AVEC(IA)*VEC(IA-1)+RES(IA)
END IF
ELSE
IF (IA.GT.IGEND.AND.IA.LE.IFEND) THEN
1+AVEC(IA)*VEC(IA-1)+DIAGNL(IA)*VEC(IA)+EVEC(IA)*VEC(IA+1)
2+FVEC(IA)*VEC(DXNUM+IA)
ELSE IF (IA.GT.IFEND.AND.IA.LE.IEEND) THEN
1+AVEC(IA)*VEC(IA-1)+DIAGNL(IA)*VEC(IA)+EVEC(IA)*VEC(IA+1)
ELSE IF (IA.GT.IEEND) THEN
1+AVEC(IA)*VEC(IA-1)+DIAGNL(IA)*VEC(IA)
ENDIF
END IF
100 CONTINUE
RETURN
END
***************************************************************************
C*********************************************************************
C THIS IS THE SUBROUTINE TO SOLVE THE SYSTEM OF EQUATIONS USING *
C PRECONTIONED CONJUGATE GRADIENT METHOD.
*
C*********************************************************************
SUBROUTINE PCGM(XKM1,DIAGNL,RHSVEC,NUMCEL,ISIMST,KC,MAX)
C XKM1 CORRESPONDS TO HOLD, AND XK RESPONDS TO HNEW TO BE CALCULATED
C AT THE END OF PCGM
IMPLICIT DOUBLE PRECISION (A-G,O-Z)
DOUBLE PRECISION RKM1(300000),SKM1(300000),DIAGNL(300000),
1RHSVEC(300000),XK(300000),RK(300000),XKM1(300000),RES(300000),
2MAX,ALPHAK,BETAK,PK(300000),PKM1(300000)
C
PCGM ITERATION STARTS HERE
KC=1
CALL MATVEC(XKM1,RES,ISIMST)
DO 101 I=1,NUMCEL
101 RKM1(I)=RHSVEC(I)-RES(I)
1 CONTINUE
MAX=0.0
STR2=STR1
STR1=0.0
DO 99 I=1,NUMCEL
IF (ABS(RKM1(I)).GT. MAX) MAX=ABS(RKM1(I))
99 CONTINUE
C CHECK IF THE CONVERGENCE CRITERIA FOR PCGM METHOD IS SATISFIED OR NOT?
IF (MAX.LE.0.000001) GOTO 1001
DO 104 I=1,NUMCEL
SKM1(I)=RKM1(I)/DIAGNL(I)
104 STR1=STR1+SKM1(I)*RKM1(I)
IF (KC.EQ.1) THEN
BETAK=0.0
DO I=1,NUMCEL
PK(I)=SKM1(I)
END DO
ELSE
BETAK=STR1/STR2
223
DO 105 I=1,NUMCEL
105 PK(I)=SKM1(I)+BETAK*PKM1(I)
END IF
PAP=0.0
CALL MATVEC(PK,RES,ISIMST)
DO 201 I=1,NUMCEL
201 PAP=PAP+PK(I)*RES(I)
ALPHAK=STR1/PAP
DO 110 I=1,NUMCEL
110 XK(I)=XKM1(I)+ALPHAK*PK(I)
DO I=1,NUMCEL
PKM1(I)=PK(I)
XKM1(I)=XK(I)
END DO
DO 111 I=1,NUMCEL
111 RK(I)=RKM1(I)-ALPHAK*RES(I)
DO I=1,NUMCEL
RKM1(I)=RK(I)
END DO
KC=KC+1
IF (KC.GT.100000) GOTO 1001
GOTO 1
77 FORMAT(F12.3)
78 FORMAT (2F10.3)
1001 RETURN
END
C****************************************************************************************
*
C THIS IS THE SUBROUTINE TO RECALCULATE THE COEFFICIENTS AT THE BOUNDARY
ACCORDING *
C BOUNDARY CONDITIONS. IT ALSO CALCULATE THE EVAPOTRANSPIRATION.
*
C****************************************************************************************
*
SUBROUTINE BOUND(IEVT,IRAIN,TOLD,KP,HMINI)
INTEGER DXNUM,DZNUM,DYNUM,NUMCEL,ICONF,IBOUND,ISOIL
DOUBLE PRECISION RTACT(300000)
COMMON /BNDRY/HPOND(100,100),QRAIN(1,1,365),QEVAP(1,1,365)
1,DZNUM,NDXDY
2TETADT(300000),SS(300000),ICONF(300000)
COMMON /ET/QRTDPH,QRTTOP,QRTBOT,QHROOT,QPET,QPEV,QSRES,QHA,RAIN,
1 QTOT(100,100),EVAP,QRAT1,ZROOT(100,100)
C READ THE RAINFALL EVAP AND HPOND VALUES FOR TOP BOUNDARY
224
C SIDE BOUNDARIES WILL BE TREATED AS NO FLOW OR FIXED HEAD BOUNDARIES
PARAMETER (DZERO=0.0D+00)
C GENERAL HEAD BOUNDARY CONDITION WILL BE APPLIED IF IGHB=1
XDISTR=90000.
XDISTL=90000.
YDISTF=90000.
YDISTB=90000.
IGHB=01
IF (TOLD.GE.0.0 .AND. KP.EQ.1 ) THEN
C CHECK IF THERE WILL BE RAINFALL OR ET CALCULATIONS?
IF (IRAIN.EQ.1 .OR. IEVT.EQ.1) THEN
C
CALL READ2D(QRAIN,DXNUM,DYNUM,'QRAIN')
C
CALL READ2D(HPOND,DXNUM,DYNUM,'HPOND')
C
CALL READ2D(KACTBN,DXNUM,DYNUM,'KACTBN')
***********************************************************************
C
RAIN=-0.000835
END IF
ENDIF
DO 411 K=1,DZNUM
DO 411 J=1,DYNUM
DO 411 I=1,DXNUM
ZROOT(I,J)=9999.99
IF(IBOUND(II).EQ.7) THEN
ZROOT(I,J)=ZVEC(II)+DZ(K)/2-QRTDPH
END IF
411 CONTINUE
DO 111 K=1,DZNUM
DO 111 J=1,DYNUM
DO 111 I=1,DXNUM
C IF THE BOUNDARY CELL IS INACTIVE OR FIXED HEAD THEN SKIP CALCULATIONS
IF (IGHB.EQ.1) THEN
IF (IBOUND(I+1+(J-1)*DXNUM+(K-1)*NDXDY).EQ.0
1 .AND.IBOUND(I+(J-1)*DXNUM+(K-1)*NDXDY).EQ.9) THEN
GHBR=HSTRT(II)
QEX(II)=(GHBR-HOLD(II))*SATKX(I,J,K)*RCON(I,J,K)/(XDISTR*DZ(K))
RHSVEC(II)=RHSVEC(II)-QEX(II)
END IF
IF (IBOUND(I+(J-1)*DXNUM+(K-1)*NDXDY).EQ.9
1 .AND.IBOUND(I+1+(J-1)*DXNUM+(K-1)*NDXDY).GT.0) THEN
GHBL=HSTRT(II)
QEX(II)=(GHBL-HOLD(II))*SATKX(I,J,K)*RCON(I,J,K)/(XDISTL*DZ(K))
END IF
IF (IBOUND(I+(J)*DXNUM+(K-1)*NDXDY).EQ.0
1 .AND.IBOUND(I+(J-1)*DXNUM+(K-1)*NDXDY).EQ.9) THEN
GHBB=HSTRT(II)
QEX(II)=(GHBB-HOLD(II))*SATKY(I,J,K)*RCON(I,J,K)/(YDISTB*DZ(K))
END IF
IF (IBOUND(I+(J-1)*DXNUM+(K-1)*NDXDY).EQ.9
1 .AND.IBOUND(I+(J)*DXNUM+(K-1)*NDXDY).GT.0) THEN
GHBF=HSTRT(II)
225
QEX(II)=(GHBF-HOLD(II))*SATKY(I,J,K)*RCON(I,J,K)/(YDISTF*DZ(K))
END IF
END IF
C IBOUND(II).EQ.7 MEANS THAT THE UPPERT MOST FIRST ACTIVE CELL IN WHICH ET
AND RAINFALL
C CALCULATIONS WILL TAKE PLACE
IF (IBOUND(II).EQ.7) THEN
QSRES=2.0/DZ(K)
EVAP=-SATKZ(I,J,K)*RCON(I,J,K)*QSRES*(QHA-HOLD(II))
IF (EVAP.LT.0) EVAP=0.0
IF (EVAP.GT.QPEV) EVAP=QPEV
RHSVEC(II)=RHSVEC(II)+(EVAP+RAIN)/DZ(K)
END IF
C
END IF
C
ROOT WATER UPTAKE CALCULATIONS
IF (ZVEC(II).GE.ZROOT(I,J)) THEN
RTACT(II)=(QRTTOP-QRTBOT)*(ZVEC(II)-ZROOT(I,J))/QRTDPH+QRTBOT
ENVPRS=HOLD(II)
IF(ENVPRS.GT.QHROOT) THEN
QEXVEC(II)=-SATKX(I,J,K)*RCON(I,J,K)*RTACT(II)*
1 (QHROOT-ENVPRS)
ELSE
QEXVEC(II)=0.0
END IF
ELSE
QEXVEC(II)=0.0
END IF
END IF
111 CONTINUE
C CHECK THE TOTAL PET, AND ADJUST THE QEXVEC(II)
IF (QPET.NE.0.0) THEN
DO J=1,DYNUM
DO I=1,DXNUM
PTOT=0.0
DO K=1,DZNUM
IF(ZVEC(II).GE.ZROOT(I,J).AND.IBOUND(II).GT.0) THEN
PTOT=PTOT+QEXVEC(II)*DZ(K)*DX(I)*DY(J)
END IF
END DO
IF(I.EQ.6) PTOT1=PTOT
QTOT(I,J)=PTOT
END DO
END DO
DO J=1,DYNUM
DO I=1,DXNUM
IF (QTOT(I,J).GT.QPET*DX(I)*DY(J)) THEN
QRATIO=QPET*DX(I)*DY(J)/QTOT(I,J)
IF(I.EQ.6) QRAT1=QRATIO
ELSE
QRATIO=1.0
IF(I.EQ.6) QRAT1=QRATIO
END IF
226
DO K=1,DZNUM
IF(ZVEC(II).GE.ZROOT(I,J).AND.ZROOT(I,J).GT.0) THEN
QEXVEC(II)=QEXVEC(II)*QRATIO
RHSVEC(II)=RHSVEC(II)+QEXVEC(II)
END IF
END DO
END DO
END DO
END IF
RETURN
END
C *******************************************************************
C THIS SUBROUTINE CHECKS THE UPPER BOUNDARY CONDITION IF THE
C PONDING CEASES OR NOT
SUBROUTINE POND(IREITR,IRAIN,IFLUX,NPOND,TOLD,RAIN,QPEV)
INTEGER DXNUM,DZNUM,DYNUM,NUMCEL,ICONF,IBOUND,ISOIL
C
DOUBLE PRECISION
COMMON /BNDRY/HPOND(100,100)
1,QRAIN(1,1,365),QEVAP(1,1,365)
2TETADT(300000),SS(300000),ICONF(300000)
1,DZNUM,NDXDY
IFLUX=0
IREITR=0
NPOND=0
C IF IPOND=1 THERE IS A NEW PONDING
C IF IFLUX=1 CHECK THE OLD PONDING IF IT IS ENDED
DO 111 K=1,DZNUM
DO 111 J=1,DYNUM
DO 111 I=1,DXNUM
C IF THE BOUNDARY CELL IS INACTIVE OR FIXED HEAD THEN SKIP CALCULATIONS
IF (IBOUND(II).EQ.-9) THEN
NPOND=NPOND+1
IFLUX=1
C
IF(HOLD(II).LE.-100) IDRY=1
C CHECK IF THERE IS ANY CHANGE IN BOUNDARY CONDITION FROM FIXED HEAD TO
FLUX AGAIN
DHZB=HOLD(II)-HOLD(I+(J-1)*DXNUM+(K-2)*NDXDY)
DHXL=HOLD(II)-HOLD(I-1+(J-1)*DXNUM+(K-1)*NDXDY)
DHXR=HOLD(II)-HOLD(I+1+(J-1)*DXNUM+(K-1)*NDXDY)
DHYF=HOLD(II)-HOLD(I+(J-2)*DXNUM+(K-1)*NDXDY)
DHYB=HOLD(II)-HOLD(I+(J)*DXNUM+(K-1)*NDXDY)
C
QZT=EVAP*DX(I)*DY(J)
227
QZB=GVEC(I+(J-1)*DXNUM+(K-2)*NDXDY)*DZ(K-1)*DHZB*DX(I)*DY(J)
QXL=EVEC(I-1+(J-1)*DXNUM+(K-1)*NDXDY)*DX(I-1)*DHXL*DY(J)*DZ(K)
QXR=AVEC(I+1+(J-1)*DXNUM+(K-1)*NDXDY)*DX(I+1)*DHXR*DY(J)*DZ(K)
QYF=FVEC(I+(J-2)*DXNUM+(K-1)*NDXDY)*DY(J-1)*DHYF*DX(J)*DZ(K)
QYB=BVEC(I+(J)*DXNUM+(K-1)*NDXDY)*DY(J+1)*DHYB*DX(J)*DZ(K)
IF(QZB.LE.0) QZB=0.0
IF(QXL.LE.0) QXL=0.0
IF(QXR.LE.0) QXR=0.0
IF(QYF.LE.0) QYF=0.0
IF(QYB.LE.0) QYB=0.0
QTEST=QZB+QXL+QXR+QYF+QYB
IF (QTEST.GT.1.02*ABS(RAIN*DX(I)*DY(J))) THEN
IBOUND(II)=7
IREITR=1
PRINT*,'PONDING ENDED',I,J,K, 'AT TIME',TOLD,HOLD(II),HPOND(I,J)
WRITE(13,247)I,J,K,TOLD,QTEST,RAIN*DX(I)*DY(J)
WRITE(47,247) I,J,K,TOLD,QTEST,RAIN*DX(I)*DY(J)
247 FORMAT('PONDING ENDED',3I3,' AT TIME=',F6.2,' QTEST,RAIN=',2F9.3)
END IF
END IF
111 CONTINUE
RETURN
END
***********************************************************************
C**************************************************************************
C THIS IS THE SUBROUTINE TO CREATE THE IBOUND AND ISOIL VECTORS WHICH *
C ARE USED TO INDICATE CERTAIN PROPERTIES OF EACH CELL, I.E., ACTIVE *
C OR INACTIVE, SOIL TYPE ETC.
*
C**************************************************************************
SUBROUTINE INTCON(ICONE,DXNUM,DYNUM,DZNUM,VARNAM)
C READ THE CODES OF EACH CELL IN THE FLOW DOMAIN BY ACCEPTING GLOBAL
C COORDINATE SYSTEM, I.E., BOTTOM LEFT CORNER IS THE ORIGIN, X-AXIS IN
C HORIZONTAL DIRECTION, Y-AXIS IS PERPENDICULAR TO THE SCREEN, AND
C Z-AXIS IS IN VERTICAL DIRECTION
IMPLICIT DOUBLE PRECISION (A-H,P-Z)
INTEGER DXNUM,DYNUM,DZNUM, ICODE(100,100,100),ICONE(300000)
CHARACTER VARNAM*12,YN*1
PRINT*,'DO YOU WANT TO READ ',VARNAM,' FROM FILE? PRESS 1 FOR YES'
READ(*,*) IREAD
C
IREAD=1
IF (IREAD.EQ.1) THEN
IF (VARNAM.EQ.'IBOUND.INP') NUNIT=7
IF (VARNAM.EQ.'ISOIL.INP') NUNIT=8
IF (VARNAM.EQ.'IVEGET.INP') NUNIT=9
OPEN(NUNIT,FILE=VARNAM,STATUS='UNKNOWN')
REWIND(NUNIT)
DO 1121 K=1,DZNUM
DO 1121 J=1,DYNUM
1121 READ(NUNIT,4001)(ICODE(I,J,K),I=1,DXNUM)
GOTO 20
ELSE
OPEN(NUNIT,FILE=VARNAM,STATUS='UNKNOWN')
END IF
C ICODE<0 CONSTANT HEAD
228
C
C
ICODE=0 NOFLOW
ICODE>0 ACTIVE HEAD
J=0
K=0
10 K=K+1
J=0
11 J=J+1
PRINT*,'READ THE ',VARNAM,' VARIABLE ALONG THE X-AXIS FOR',K
IF(DYNUM.EQ.1) J=1
DO 12 I=1,DXNUM
PRINT*,'I= ',I
12 READ(*,*)ICODE(I,J,K)
IF (DXNUM.EQ.1) GOTO 31
IF (DYNUM.EQ.1.OR.J.GE.DYNUM) GOTO 31
PRINT*,'HOW MANY ROWS SAME AS PREVIOUS ROW ?'
READ(*,*) NOY
IF (NOY.EQ.0) GOTO 11
JEND=J+NOY
IF (JEND.GE.DYNUM) JEND=DYNUM
DO 29 JI=J+1,JEND
DO 29 II=1,DXNUM
29 ICODE(II,JI,K)=ICODE(II,JI-1,K)
J=JEND
IF (JEND.LT.DYNUM) GOTO 11
31 PRINT*,'HOW MANY LAYERS SAME AS PREVIOUS LAYER ?'
READ(*,*) NOK
IF (NOK.EQ.0 .AND.KEND.LE.DZNUM) GOTO 10
KEND=K+NOK
IF(KEND.GE.DZNUM) KEND=DZNUM
DO 9 KK=K+1,KEND
DO 9 JK=1,DYNUM
DO 9 IK=1,DXNUM
PRINT*,'KK=',KK,ICODE(IK,JK,KK-1)
9 ICODE(IK,JK,KK)=ICODE(IK,JK,KK-1)
K=KEND
IF (KEND.LT.DZNUM) GOTO 10
20 DO 21 K=1,DZNUM
DO 21 J=1,DYNUM
DO 21 I=1,DXNUM
II=I+(J-1)*DXNUM+(K-1)*DYNUM*DXNUM
21
ICONE(II)=ICODE(I,J,K)
23 CONTINUE
C
PRINT*, 'DOU YOU WANT TO CHANGE ANY OF THE ',VARNAM,' CODE ?(Y/N)'
YN='N'
C
READ(*,*) YN
IF (YN.EQ.'Y') THEN
PRINT*, 'ENTER THE I,J,K INDICES, AND NEW CODE'
READ(*,*) I,J,K,NEWCOD
ICODE(I,J,K)=NEWCOD
ICONE(I+(J-1)*DXNUM+(K-1)*DYNUM*DXNUM)=NEWCOD
GOTO 23
END IF
IF (IREAD.NE.1 .OR.YN.EQ.'Y') THEN
DO 121 K=1,DZNUM
DO 121 J=1,DYNUM
229
121 WRITE(NUNIT,4001)(ICODE(I,J,K),I=1,DXNUM)
CLOSE(NUNIT)
END IF
4001 FORMAT(50I2)
4002 RETURN
END
**********************************************************************
APPENDIX B
INPUT FILES FOR THE MODEL SIMULATION IN THE UECB
Table B.1 Isoil matrix for material properties of the model domain in hydrologic
simulation of UECB, where, 1: Upper Floridan Aquifer (limestone); 2, 3, 4: Confining
Unit (Hawthorn Group); 5: Surficial Aquifer (sand); and 0: no material.
k/i 1
99
98
97
96
95
94
93
92
91
90
89
88
87
86
85
84
83
82
81
80
79
78
77
76
75
74
73
72
71
70
69
68
67
66
65
64
63
62
61
60
59
58
57
56
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
7
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
9
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
230
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
231
Table B.1-continued
(k/i) 1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
4
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
4
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
4
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
0
0
0
0
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
0
0
0
0
0
3
2
2
2
2
2 2
2
2
2
2
2
2 2
2
2
2
2
2
2
2
0
0
0
0
0
2
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
k/i
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
232
Table B.2 Ibound matrix for the boundary properties of the model domain in hydrologic
simulation of UECB, where, 1: Active cell; 0: inactive cell; -2: fixed head cell for Crystal
Lake, -3:fixed head cell for Magnolia Lake, 9: general head boundary cell, 7: rainfall and
evapotranspiration boundary cell.
(k/i)
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
99
98
97
96
95
94
93
92
91
90
89
88
87
86
85
84
83
82
81
80
79
78
77
76
75
74
73
72
71
70
69
68
67
66
65
64
63
62
61
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
233
Table B.2-continued
(k/i)
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
45
44
43
42
41
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 -2
0 -2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
7
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6 -2 -2 -2
1
1
1
1
1
1
1 -3 -3 -3 -3 -3 -3 -3 -3 -3 -3
0
0
0
0
0
5
9
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
9
0
0
0
0
0
4
9
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
9
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
9
0
0
0
0
0
3 9
0
0
0
0
0
0
0
0
0
0
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
-3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
9
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
9
0
0
0
0
0
1
9
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
9
0
0
0
0
0
(k/i)
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
234
Table B.3 Meteorological data for the period September 1, 1994-August 31, 1995
Date
Gaines.
Evap.
(m)
Pan
Coeff.
Lowry
Precip.
(m)
01-Sep-94
02-Sep-94
03-Sep-94
04-Sep-94
05-Sep-94
06-Sep-94
07-Sep-94
08-Sep-94
09-Sep-94
10-Sep-94
11-Sep-94
12-Sep-94
13-Sep-94
14-Sep-94
15-Sep-94
16-Sep-94
17-Sep-94
18-Sep-94
19-Sep-94
20-Sep-94
21-Sep-94
22-Sep-94
23-Sep-94
24-Sep-94
25-Sep-94
26-Sep-94
27-Sep-94
28-Sep-94
29-Sep-94
30-Sep-94
01-Oct-94
02-Oct-94
03-Oct-94
04-Oct-94
05-Oct-94
06-Oct-94
07-Oct-94
08-Oct-94
09-Oct-94
10-Oct-94
11-Oct-94
12-Oct-94
13-Oct-94
14-Oct-94
5.33E-03
4.57E-03
4.32E-03
5.84E-03
3.30E-03
4.83E-03
5.33E-03
5.08E-03
1.27E-03
4.57E-03
3.05E-03
4.57E-03
4.57E-03
5.84E-03
3.05E-03
5.08E-03
4.32E-03
3.56E-03
4.32E-03
3.81E-03
1.78E-03
4.83E-03
4.83E-03
3.30E-03
1.02E-03
5.08E-04
3.81E-03
4.06E-03
3.56E-03
4.32E-03
5.08E-03
2.03E-03
2.03E-03
4.32E-03
1.78E-03
4.32E-03
4.06E-03
4.06E-03
3.81E-03
2.79E-03
1.02E-03
5.08E-04
5.08E-04
2.29E-03
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.94
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
2.54E-04
3.05E-03
2.54E-04
1.52E-03
7.62E-04
0.00E+00
0.00E+00
2.54E-04
9.91E-03
1.27E-03
4.32E-03
0.00E+00
3.81E-03
2.54E-04
0.00E+00
9.40E-03
1.02E-03
5.08E-04
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
6.35E-03
1.65E-02
7.37E-03
2.54E-03
0.00E+00
0.00E+00
0.00E+00
7.62E-03
2.54E-04
2.29E-03
7.39E-02
2.54E-04
0.00E+00
1.24E-02
Magnolia
Precip
(m)
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
2.24E-04
2.69E-03
2.24E-04
1.35E-03
6.72E-04
0.00E+00
0.00E+00
2.24E-04
8.75E-03
1.12E-03
3.81E-03
0.00E+00
3.36E-03
2.24E-04
8.30E-03
8.97E-04
4.49E-04
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
5.61E-03
1.46E-02
6.50E-03
2.24E-03
0.00E+00
0.00E+00
0.00E+00
6.72E-03
2.24E-04
2.02E-03
6.53E-02
2.24E-04
0.00E+00
1.10E-02
Potential
Evapotrans.
(m)
5.014E-03
4.298E-03
4.059E-03
5.491E-03
3.104E-03
4.536E-03
5.014E-03
4.775E-03
1.194E-03
4.298E-03
2.865E-03
4.298E-03
4.298E-03
5.491E-03
2.865E-03
4.775E-03
4.059E-03
3.343E-03
4.059E-03
3.581E-03
1.671E-03
4.536E-03
4.536E-03
3.104E-03
9.550E-04
4.775E-04
3.581E-03
3.820E-03
3.343E-03
4.059E-03
4.877E-03
1.951E-03
1.951E-03
4.145E-03
1.707E-03
4.145E-03
3.901E-03
3.901E-03
3.658E-03
2.682E-03
9.754E-04
4.877E-04
4.877E-04
2.195E-03
235
Table B.3-continued
Date
Gaines.
Evap.
(m)
Pan
Coeff.
Lowry
Precip.
(m)
15-Oct-94
16-Oct-94
17-Oct-94
18-Oct-94
19-Oct-94
20-Oct-94
21-Oct-94
22-Oct-94
23-Oct-94
24-Oct-94
25-Oct-94
26-Oct-94
27-Oct-94
28-Oct-94
29-Oct-94
30-Oct-94
31-Oct-94
01-Nov-94
02-Nov-94
03-Nov-94
04-Nov-94
05-Nov-94
06-Nov-94
07-Nov-94
08-Nov-94
09-Nov-94
10-Nov-94
11-Nov-94
12-Nov-94
13-Nov-94
14-Nov-94
15-Nov-94
16-Nov-94
17-Nov-94
18-Nov-94
19-Nov-94
20-Nov-94
21-Nov-94
22-Nov-94
23-Nov-94
24-Nov-94
25-Nov-94
26-Nov-94
27-Nov-94
28-Nov-94
29-Nov-94
30-Nov-94
01-Dec-94
02-Dec-94
3.56E-03
2.29E-03
3.05E-03
4.32E-03
2.79E-03
3.81E-03
3.56E-03
3.30E-03
3.05E-03
4.06E-03
3.56E-03
3.56E-03
7.62E-04
4.06E-03
1.27E-03
1.02E-03
1.52E-03
2.29E-03
3.56E-03
3.56E-03
2.79E-03
3.56E-03
4.06E-03
2.29E-03
3.30E-03
2.54E-03
2.79E-03
2.03E-03
1.78E-03
5.08E-04
3.05E-03
3.81E-03
0.00E+00
1.27E-03
1.78E-03
1.78E-03
3.56E-03
2.29E-03
2.29E-03
2.54E-03
3.81E-03
1.78E-03
2.03E-03
1.52E-03
2.54E-03
2.03E-03
2.54E-03
1.27E-03
2.79E-03
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.95
0.90
0.90
2.54E-04
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
1.02E-03
0.00E+00
6.86E-03
2.29E-03
7.62E-03
2.54E-04
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
2.79E-03
2.03E-03
1.04E-02
0.00E+00
0.00E+00
3.63E-02
3.56E-03
1.02E-03
0.00E+00
0.00E+00
5.08E-04
1.02E-03
2.54E-04
2.54E-04
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
7.62E-04
2.54E-04
0.00E+00
0.00E+00
Magnolia
Precip
(m)
2.24E-04
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
8.97E-04
0.00E+00
6.06E-03
2.02E-03
6.73E-03
2.24E-04
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
2.47E-03
1.79E-03
9.20E-03
0.00E+00
0.00E+00
3.14E-03
8.97E-04
0.00E+00
0.00E+00
0.00E+00
4.49E-04
8.97E-04
2.24E-04
2.24E-04
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
6.73E-04
2.24E-04
0.00E+00
0.00E+00
Potential
Evapotrans.
(m)
3.414E-03
2.195E-03
2.926E-03
4.145E-03
2.682E-03
3.658E-03
3.414E-03
3.170E-03
2.926E-03
3.901E-03
3.414E-03
3.414E-03
7.315E-04
3.901E-03
1.219E-03
9.754E-04
1.463E-03
2.172E-03
3.378E-03
3.378E-03
2.654E-03
3.378E-03
3.861E-03
2.172E-03
3.137E-03
2.413E-03
2.654E-03
1.930E-03
1.689E-03
4.826E-04
2.896E-03
3.620E-03
0.000E+00
1.207E-03
1.689E-03
1.689E-03
3.378E-03
2.172E-03
2.172E-03
2.413E-03
3.620E-03
1.689E-03
1.930E-03
1.448E-03
2.413E-03
1.930E-03
2.413E-03
1.143E-03
2.515E-03
236
Table B.3-continued
Date
Gaines.
Evap.
(m)
Pan
Coeff.
Lowry
Precip.
(m)
03-Dec-94
04-Dec-94
05-Dec-94
06-Dec-94
07-Dec-94
08-Dec-94
09-Dec-94
10-Dec-94
11-Dec-94
12-Dec-94
13-Dec-94
14-Dec-94
15-Dec-94
16-Dec-94
17-Dec-94
18-Dec-94
19-Dec-94
20-Dec-94
21-Dec-94
22-Dec-94
23-Dec-94
24-Dec-94
25-Dec-94
26-Dec-94
27-Dec-94
28-Dec-94
29-Dec-94
30-Dec-94
31-Dec-94
01-Jan-95
02-Jan-95
03-Jan-95
04-Jan-95
05-Jan-95
06-Jan-95
07-Jan-95
08-Jan-95
09-Jan-95
10-Jan-95
11-Jan-95
12-Jan-95
13-Jan-95
14-Jan-95
15-Jan-95
16-Jan-95
17-Jan-95
18-Jan-95
19-Jan-95
20-Jan-95
1.27E-03
1.02E-03
2.29E-03
2.79E-03
2.54E-03
1.52E-03
1.78E-03
1.27E-03
2.79E-03
2.29E-03
2.29E-03
1.27E-03
1.52E-03
1.02E-03
5.08E-04
1.02E-03
2.03E-03
2.79E-03
5.08E-04
5.08E-04
0.00E+00
1.52E-03
2.03E-03
1.78E-03
2.03E-03
2.29E-03
2.03E-03
1.27E-03
1.02E-03
1.27E-03
7.62E-04
3.30E-03
1.27E-03
1.52E-03
2.03E-03
2.79E-03
3.56E-03
4.32E-03
2.29E-03
1.27E-03
2.03E-03
2.29E-03
1.27E-03
5.08E-04
1.52E-03
1.02E-03
1.52E-03
1.52E-03
1.02E-03
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.90
0.61
0.61
0.61
0.61
0.61
0.61
0.61
0.61
0.61
0.61
0.61
0.61
0.61
0.61
0.61
0.61
0.61
0.61
0.61
0.61
0.00E+00
0.00E+00
7.87E-03
0.00E+00
2.54E-04
0.00E+00
2.54E-04
0.00E+00
4.06E-03
0.00E+00
0.00E+00
0.00E+00
2.54E-04
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
9.65E-03
8.89E-03
3.81E-03
0.00E+00
0.00E+00
2.54E-04
0.00E+00
0.00E+00
7.11E-03
3.30E-03
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
3.25E-02
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
2.64E-02
1.17E-02
2.54E-04
0.00E+00
0.00E+00
5.08E-04
2.54E-04
Magnolia
Precip
(m)
0.00E+00
0.00E+00
6.96E-04
0.00E+00
2.24E-04
0.00E+00
2.24E-04
0.00E+00
3.59E-03
0.00E+00
0.00E+00
0.00E+00
2.24E-04
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
8.52E-03
7.85E-03
3.36E-03
0.00E+00
0.00E+00
2.24E-04
0.00E+00
0.00E+00
6.28E-03
2.92E-03
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
2.87E-02
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
2.33E-02
1.03E-02
2.24E-04
0.00E+00
0.00E+00
4.49E-04
2.24E-04
Potential
Evapotrans.
(m)
1.143E-03
9.144E-04
2.057E-03
2.515E-03
2.286E-03
1.372E-03
1.600E-03
1.143E-03
2.515E-03
2.057E-03
2.057E-03
1.143E-03
1.372E-03
9.144E-04
4.572E-04
9.144E-04
1.829E-03
2.515E-03
4.572E-04
4.572E-04
0.000E+00
1.372E-03
1.829E-03
1.600E-03
1.829E-03
2.057E-03
1.829E-03
1.143E-03
9.144E-04
7.747E-04
4.648E-04
2.014E-03
7.747E-04
9.296E-04
1.240E-03
1.704E-03
2.169E-03
2.634E-03
1.394E-03
7.747E-04
1.240E-03
1.394E-03
7.747E-04
3.099E-04
9.296E-04
6.198E-04
9.296E-04
9.296E-04
6.198E-04
237
Table B.3-continued
Date
Gaines.
Evap.
(m)
Pan
Coeff.
Lowry
Precip.
(m)
21-Jan-95
22-Jan-95
23-Jan-95
24-Jan-95
25-Jan-95
26-Jan-95
27-Jan-95
28-Jan-95
29-Jan-95
30-Jan-95
31-Jan-95
01-Feb-95
02-Feb-95
03-Feb-95
04-Feb-95
05-Feb-95
06-Feb-95
07-Feb-95
08-Feb-95
09-Feb-95
10-Feb-95
11-Feb-95
12-Feb-95
13-Feb-95
14-Feb-95
15-Feb-95
16-Feb-95
17-Feb-95
18-Feb-95
19-Feb-95
20-Feb-95
21-Feb-95
22-Feb-95
23-Feb-95
24-Feb-95
25-Feb-95
26-Feb-95
27-Feb-95
28-Feb-95
01-Mar-95
02-Mar-95
03-Mar-95
04-Mar-95
05-Mar-95
06-Mar-95
07-Mar-95
08-Mar-95
09-Mar-95
10-Mar-95
2.29E-03
2.03E-03
2.54E-03
2.54E-03
2.03E-03
2.29E-03
2.54E-03
2.03E-03
1.27E-03
1.52E-03
2.03E-03
2.54E-03
3.56E-03
2.79E-03
4.83E-03
3.30E-03
3.81E-03
3.05E-03
3.81E-03
3.05E-03
2.79E-03
2.79E-03
1.78E-03
1.78E-03
1.27E-03
2.54E-03
1.27E-03
2.54E-03
2.03E-03
3.05E-03
1.02E-03
1.27E-03
4.06E-03
3.05E-03
2.54E-03
2.79E-03
4.83E-03
3.81E-03
4.32E-03
1.02E-03
2.03E-03
1.27E-03
2.79E-03
4.32E-03
3.05E-03
3.30E-03
6.35E-03
3.05E-03
4.83E-03
0.61
0.61
0.61
0.61
0.61
0.61
0.61
0.61
0.61
0.61
0.61
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.78
0.83
0.83
0.83
0.83
0.83
0.83
0.83
0.83
0.83
0.83
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
1.52E-03
0.00E+00
0.00E+00
2.54E-04
5.08E-04
0.00E+00
0.00E+00
1.32E-02
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
2.16E-02
1.78E-03
2.54E-04
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
4.32E-03
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
1.27E-03
1.02E-03
7.62E-04
0.00E+00
0.00E+00
0.00E+00
7.62E-04
0.00E+00
2.01E-02
0.00E+00
0.00E+00
Magnolia
Precip
(m)
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
1.35E-03
0.00E+00
0.00E+00
2.24E-04
4.49E-04
0.00E+00
0.00E+00
1.17E-02
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
1.91E-02
1.57E-03
2.24E-04
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
3.81E-03
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
1.12E-03
8.97E-04
6.73E-04
0.00E+00
0.00E+00
0.00E+00
6.73E-04
0.00E+00
1.77E-02
0.00E+00
0.00E+00
Potential
Evapotrans.
(m)
1.394E-03
1.240E-03
1.549E-03
1.549E-03
1.240E-03
1.394E-03
1.549E-03
1.240E-03
7.747E-04
9.296E-04
1.240E-03
1.981E-03
2.774E-03
2.179E-03
3.764E-03
2.576E-03
2.972E-03
2.377E-03
2.972E-03
2.377E-03
2.179E-03
2.179E-03
1.387E-03
1.387E-03
9.906E-04
1.981E-03
9.906E-04
1.981E-03
1.585E-03
2.377E-03
7.925E-04
9.906E-04
3.170E-03
2.377E-03
1.981E-03
2.179E-03
3.764E-03
2.972E-03
3.368E-03
8.433E-04
1.687E-03
1.054E-03
2.319E-03
3.584E-03
2.530E-03
2.741E-03
5.271E-03
2.530E-03
4.006E-03
238
Table B.3-continued
Date
Gaines.
Evap.
(m)
Pan
Coeff.
Lowry
Precip.
(m)
11-Mar-95
12-Mar-95
13-Mar-95
14-Mar-95
15-Mar-95
16-Mar-95
17-Mar-95
18-Mar-95
19-Mar-95
20-Mar-95
21-Mar-95
22-Mar-95
23-Mar-95
24-Mar-95
25-Mar-95
26-Mar-95
27-Mar-95
28-Mar-95
29-Mar-95
30-Mar-95
31-Mar-95
01-Apr-95
02-Apr-95
03-Apr-95
04-Apr-95
05-Apr-95
06-Apr-95
07-Apr-95
08-Apr-95
09-Apr-95
10-Apr-95
11-Apr-95
12-Apr-95
13-Apr-95
14-Apr-95
15-Apr-95
16-Apr-95
17-Apr-95
18-Apr-95
19-Apr-95
20-Apr-95
21-Apr-95
22-Apr-95
23-Apr-95
24-Apr-95
25-Apr-95
26-Apr-95
27-Apr-95
28-Apr-95
4.32E-03
3.30E-03
6.10E-03
7.11E-03
4.83E-03
3.30E-03
1.78E-03
1.78E-03
3.81E-03
4.83E-03
4.06E-03
4.57E-03
5.59E-03
5.84E-03
4.06E-03
5.59E-03
5.08E-03
3.81E-03
4.83E-03
5.33E-03
5.33E-03
0.00E+00
3.05E-03
5.33E-03
4.83E-03
3.05E-03
0.00E+00
2.29E-03
3.56E-03
4.32E-03
5.08E-03
5.84E-03
4.06E-03
1.27E-03
5.33E-03
5.33E-03
5.33E-03
4.83E-03
6.10E-03
6.10E-03
5.08E-03
6.35E-03
4.83E-03
5.33E-03
6.60E-03
2.79E-03
4.83E-03
3.30E-03
4.57E-03
0.83
0.83
0.83
0.83
0.83
0.83
0.83
0.83
0.83
0.83
0.83
0.83
0.83
0.83
0.83
0.83
0.83
0.83
0.83
0.83
0.83
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.89
0.00E+00
0.00E+00
0.00E+00
5.08E-04
1.02E-03
0.00E+00
3.43E-02
1.78E-03
5.08E-04
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
5.08E-04
0.00E+00
4.60E-02
4.06E-03
0.00E+00
0.00E+00
0.00E+00
1.22E-02
5.03E-02
0.00E+00
2.29E-03
2.54E-04
0.00E+00
1.22E-02
2.29E-03
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
5.08E-04
0.00E+00
0.00E+00
1.52E-03
0.00E+00
0.00E+00
2.51E-02
0.00E+00
0.00E+00
2.18E-02
0.00E+00
Magnolia
Precip
(m)
0.00E+00
0.00E+00
0.00E+00
4.49E-04
8.97E-04
0.00E+00
3.03E-02
1.57E-03
4.49E-04
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
4.49E-04
0.00E+00
4.06E-02
3.59E-03
0.00E+00
0.00E+00
0.00E+00
1.08E-02
4.44E-02
0.00E+00
2.02E-03
2.24E-04
0.00E+00
1.08E-02
2.02E-03
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
1.60E-02
0.00E+00
0.00E+00
2.62E-02
0.00E+00
Potential
Evapotrans.
(m)
3.584E-03
2.741E-03
5.060E-03
5.903E-03
4.006E-03
2.741E-03
1.476E-03
1.476E-03
3.162E-03
4.006E-03
3.373E-03
3.795E-03
4.638E-03
4.849E-03
3.373E-03
4.638E-03
4.216E-03
3.162E-03
4.006E-03
4.427E-03
4.427E-03
0.000E+00
2.713E-03
4.747E-03
4.295E-03
2.713E-03
0.000E+00
2.035E-03
3.165E-03
3.843E-03
4.521E-03
5.199E-03
3.617E-03
1.130E-03
4.747E-03
4.747E-03
4.747E-03
4.295E-03
5.425E-03
5.425E-03
4.521E-03
5.652E-03
4.295E-03
4.747E-03
5.878E-03
2.487E-03
4.295E-03
2.939E-03
4.069E-03
239
Table B.3-continued
Date
Gaines.
Evap.
(m)
Pan
Coeff.
Lowry
Precip.
(m)
29-Apr-95
30-Apr-95
01-May-95
02-May-95
03-May-95
04-May-95
05-May-95
06-May-95
07-May-95
08-May-95
09-May-95
10-May-95
11-May-95
12-May-95
13-May-95
14-May-95
15-May-95
16-May-95
17-May-95
18-May-95
19-May-95
20-May-95
21-May-95
22-May-95
23-May-95
24-May-95
25-May-95
26-May-95
27-May-95
28-May-95
29-May-95
30-May-95
31-May-95
01-Jun-95
02-Jun-95
03-Jun-95
04-Jun-95
05-Jun-95
06-Jun-95
07-Jun-95
08-Jun-95
09-Jun-95
10-Jun-95
11-Jun-95
12-Jun-95
13-Jun-95
14-Jun-95
15-Jun-95
16-Jun-95
4.57E-03
6.60E-03
5.33E-03
6.60E-03
7.62E-03
6.10E-03
4.32E-03
6.35E-03
7.11E-03
6.60E-03
7.11E-03
4.83E-03
4.83E-03
3.05E-03
5.33E-03
5.59E-03
6.35E-03
6.60E-03
7.37E-03
6.86E-03
6.35E-03
4.57E-03
5.33E-03
6.10E-03
8.13E-03
7.11E-03
7.37E-03
6.60E-03
7.87E-03
6.35E-03
6.35E-03
5.84E-03
5.84E-03
8.38E-03
5.84E-03
7.62E-04
3.05E-03
0.00E+00
3.81E-03
6.86E-03
7.62E-03
5.84E-03
7.37E-03
6.35E-03
5.84E-03
5.08E-03
6.35E-03
6.86E-03
8.38E-03
0.89
0.89
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.88
0.88
0.88
0.88
0.88
0.88
0.88
0.88
0.88
0.88
0.88
0.88
0.88
0.88
0.88
0.88
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
5.08E-03
0.00E+00
0.00E+00
0.00E+00
5.08E-04
0.00E+00
4.32E-02
1.83E-02
7.62E-04
0.00E+00
5.08E-04
0.00E+00
0.00E+00
0.00E+00
6.35E-03
2.06E-02
0.00E+00
0.00E+00
1.19E-02
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
5.08E-04
9.40E-03
0.00E+00
3.05E-03
3.30E-03
4.14E-02
5.84E-02
2.54E-03
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
9.04E-02
4.06E-03
0.00E+00
0.00E+00
0.00E+00
1.50E-02
Magnolia
Precip
(m)
0.00E+00
0.00E+00
2.54E-04
0.00E+00
0.00E+00
0.00E+00
7.62E-04
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
1.47E-02
2.69E-02
1.27E-03
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
1.27E-03
3.38E-02
0.00E+00
0.00E+00
1.22E-02
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
1.02E-03
6.60E-03
0.00E+00
2.29E-03
4.06E-03
3.73E-02
5.33E-02
5.33E-02
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
5.59E-02
3.30E-03
0.00E+00
0.00E+00
0.00E+00
3.81E-03
Potential
Evapotrans.
(m)
4.069E-03
5.878E-03
4.641E-03
5.745E-03
6.629E-03
5.304E-03
3.757E-03
5.525E-03
6.187E-03
5.745E-03
6.187E-03
4.199E-03
4.199E-03
2.652E-03
4.641E-03
4.862E-03
5.525E-03
5.745E-03
6.408E-03
5.966E-03
5.525E-03
3.978E-03
4.641E-03
5.304E-03
7.071E-03
6.187E-03
6.408E-03
5.745E-03
6.850E-03
5.525E-03
5.525E-03
5.083E-03
5.083E-03
7.376E-03
5.141E-03
6.706E-04
2.682E-03
0.000E+00
3.353E-03
6.035E-03
6.706E-03
5.141E-03
6.482E-03
5.588E-03
5.141E-03
4.470E-03
5.588E-03
6.035E-03
7.376E-03
240
Table B.3-continued
Date
Gaines.
Evap.
(m)
Pan
Coeff.
Lowry
Precip.
(m)
17-Jun-95
18-Jun-95
19-Jun-95
20-Jun-95
21-Jun-95
22-Jun-95
23-Jun-95
24-Jun-95
25-Jun-95
26-Jun-95
27-Jun-95
28-Jun-95
29-Jun-95
30-Jun-95
01-Jul-95
02-Jul-95
03-Jul-95
04-Jul-95
05-Jul-95
06-Jul-95
07-Jul-95
08-Jul-95
09-Jul-95
10-Jul-95
11-Jul-95
12-Jul-95
13-Jul-95
14-Jul-95
15-Jul-95
16-Jul-95
17-Jul-95
18-Jul-95
19-Jul-95
20-Jul-95
21-Jul-95
22-Jul-95
23-Jul-95
24-Jul-95
25-Jul-95
26-Jul-95
27-Jul-95
28-Jul-95
29-Jul-95
30-Jul-95
31-Jul-95
01-Aug-95
02-Aug-95
03-Aug-95
04-Aug-95
6.35E-03
7.11E-03
6.35E-03
6.60E-03
5.59E-03
7.62E-03
6.86E-03
4.83E-03
6.10E-03
1.02E-03
3.81E-03
5.33E-03
5.33E-03
5.33E-03
6.60E-03
6.35E-03
5.08E-03
5.59E-03
5.59E-03
6.35E-03
6.10E-03
5.84E-03
7.37E-03
6.35E-03
4.57E-03
3.56E-03
7.11E-03
4.57E-03
5.33E-03
5.33E-03
4.57E-03
7.62E-04
5.59E-03
6.86E-03
6.60E-03
4.06E-03
4.83E-03
4.57E-03
6.86E-03
2.79E-03
5.08E-03
5.08E-03
6.86E-03
4.06E-03
6.35E-03
6.60E-03
6.10E-03
2.03E-03
4.83E-03
0.88
0.88
0.88
0.88
0.88
0.88
0.88
0.88
0.88
0.88
0.88
0.88
0.88
0.88
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.87
0.96
0.96
0.96
0.96
0.00E+00
5.08E-04
1.47E-02
2.54E-04
1.02E-03
4.06E-03
2.54E-04
0.00E+00
2.74E-02
9.14E-03
0.00E+00
2.03E-02
2.03E-02
0.00E+00
0.00E+00
0.00E+00
5.08E-04
0.00E+00
7.62E-04
0.00E+00
1.78E-03
0.00E+00
0.00E+00
4.80E-02
0.00E+00
0.00E+00
0.00E+00
0.00E+00
1.78E-03
1.02E-02
4.45E-02
0.00E+00
0.00E+00
6.32E-02
0.00E+00
7.37E-03
3.56E-03
3.05E-03
5.66E-02
0.00E+00
2.03E-03
1.37E-02
2.54E-04
0.00E+00
0.00E+00
0.00E+00
4.98E-02
0.00E+00
1.63E-02
Magnolia
Precip
(m)
2.54E-04
4.06E-03
3.79E-02
0.00E+00
5.00E-04
2.79E-03
2.54E-04
0.00E+00
2.57E-02
5.33E-03
0.00E+00
1.55E-02
1.88E-02
0.00E+00
1.02E-03
0.00E+00
0.00E+00
0.00E+00
0.00E+00
3.05E-03
5.08E-04
2.54E-04
0.00E+00
1.27E-02
1.52E-03
0.00E+00
5.08E-04
0.00E+00
8.89E-03
1.27E-03
4.62E-02
0.00E+00
0.00E+00
2.41E-02
1.02E-03
2.29E-03
3.30E-03
5.84E-03
2.77E-02
7.62E-04
0.00E+00
1.63E-02
0.00E+00
0.00E+00
0.00E+00
4.57E-03
7.11E-02
0.00E+00
2.21E-02
Potential
Evapotrans.
(m)
5.588E-03
6.259E-03
5.588E-03
5.812E-03
4.917E-03
6.706E-03
6.035E-03
4.247E-03
5.364E-03
8.941E-04
3.353E-03
4.694E-03
4.694E-03
4.694E-03
5.745E-03
5.525E-03
4.420E-03
4.862E-03
4.862E-03
5.525E-03
5.304E-03
5.083E-03
6.408E-03
5.525E-03
3.978E-03
3.094E-03
6.187E-03
3.978E-03
4.641E-03
4.641E-03
3.978E-03
6.629E-04
4.862E-03
5.966E-03
5.745E-03
3.536E-03
4.199E-03
3.978E-03
5.966E-03
2.431E-03
4.420E-03
4.420E-03
5.966E-03
3.536E-03
5.525E-03
6.340E-03
5.852E-03
1.951E-03
4.633E-03
241
Table B.3-continued
Date
Gaines.
Evap.
(m)
Pan
Coeff.
Lowry
Precip.
(m)
05-Aug-95
06-Aug-95
07-Aug-95
08-Aug-95
09-Aug-95
10-Aug-95
11-Aug-95
12-Aug-95
13-Aug-95
14-Aug-95
15-Aug-95
16-Aug-95
17-Aug-95
18-Aug-95
19-Aug-95
20-Aug-95
21-Aug-95
22-Aug-95
23-Aug-95
24-Aug-95
25-Aug-95
26-Aug-95
27-Aug-95
28-Aug-95
29-Aug-95
30-Aug-95
31-Aug-95
6.10E-03
6.10E-03
6.10E-03
6.35E-03
3.81E-03
6.35E-03
4.06E-03
6.60E-03
4.57E-03
5.59E-03
6.35E-03
4.06E-03
7.62E-03
5.84E-03
6.10E-03
5.08E-03
4.57E-03
3.56E-03
4.32E-03
3.56E-03
1.52E-03
2.79E-03
4.83E-03
5.84E-03
5.59E-03
4.06E-03
5.59E-03
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
0.96
2.54E-04
0.00E+00
0.00E+00
1.27E-03
0.00E+00
7.62E-03
0.00E+00
1.93E-02
2.54E-04
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
3.30E-03
0.00E+00
0.00E+00
0.00E+00
9.40E-03
6.86E-03
4.55E-02
1.52E-03
8.89E-03
0.00E+00
0.00E+00
0.00E+00
0.00E+00
Magnolia
Precip
(m)
0.00E+00
5.33E-03
7.62E-04
6.35E-03
1.78E-03
3.56E-03
2.54E-04
3.45E-02
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
0.00E+00
5.84E-03
2.54E-04
2.54E-04
0.00E+00
1.68E-02
7.62E-03
4.50E-02
1.52E-03
8.89E-03
1.77E-03
0.00E+00
7.62E-04
0.00E+00
Potential
Evapotrans.
(m)
5.852E-03
5.852E-03
5.852E-03
6.096E-03
3.658E-03
6.096E-03
3.901E-03
6.340E-03
4.389E-03
5.364E-03
6.096E-03
3.901E-03
7.315E-03
5.608E-03
5.852E-03
4.877E-03
4.389E-03
3.414E-03
4.145E-03
3.414E-03
1.463E-03
2.682E-03
4.633E-03
5.608E-03
5.364E-03
3.901E-03
5.364E-03
Maximum
3.494E+04
8.382E-03
9.600E-01
9.042E-02
7.112E-02
Minimum
0.000E+00
0.000E+00
0.000E+00
0.000E+00
0.000E+00
Average
3.476E+04
3.866E-03
8.702E-01
4.127E-03
3.698E-03
242
Table B.4 Initial pressure heads (m) and geometric elevations in the model domain for
hydrologic simulation of the UECB.
i
j
x
(m)
Initial piezometric elevations in
Geometric elevations (m)
September 2nd ,1994
Water- Upper part Middle part Lower part
Upper
Topogra- Elevations Elevations of
table
of the
of the
of the
Floridan
phic
of the top the bottom
y elevations confining confining confining aquifer
surface
of the
of the
(m)
unit
unit
heads
unit
heads
(m)
elevations
Hawthorn
Hawthorn
(m)
heads(m)
(m)
heads(m)
(m)
(m)
(m)
40
31
29.104
27.74
26.576
24.68
31
30
-48.35
1
1
40
2
1
120
40
31
29.101
27.78
26.569
24.67
31
30
-48.47
3
1
200
40
31
29.095
28.12
26.555
24.65
31
30
-48.64
4
1
280
40
32.28
29.985
28.45
26.925
24.63
36.8
30
-48.79
5
1
360
40
33.1
30.553
28.86
27.157
24.61
40.3
30
-48.95
6
1
550
40
37
33.271
29.86
28.299
24.57
53.05
30
-49.33
7
1
850
40
39.5
35
31.62
29
24.5
50.89
30
-49.91
8
1
1095
40
39
34.632
31.64
28.808
24.44
46.46
30
-50.37
9
1
1230
40
38.34
34.161
31.38
28.589
24.41
41.01
30
-50.62
10
1
1310
40
38
33.917
31.18
28.473
24.39
38
30.01
-50.76
11
1
1390
40
38
33.911
31.14
28.459
24.37
38
30.04
-50.9
12
1
1470
40
38
33.902
31.15
28.438
24.34
38
30.08
-51.04
13
1
1550
40
38
33.896
31.15
28.424
24.32
38
30.11
-51.18
14
1
1630
40
38
33.89
31.13
28.41
24.3
38
30.12
-51.31
15
1
1710
40
38
33.884
31.12
28.396
24.28
38
30.13
-51.44
16
1
1790
40
38
33.878
31.12
28.382
24.26
38
29.91
-51.57
17
1
1870
40
38
33.872
31.09
28.368
24.24
38
29.54
-51.7
18
1
1950
40
38
33.866
31.09
28.354
24.22
38
29.16
-51.82
19
1
2030
40
38
33.86
31.08
28.34
24.2
38
29
-51.95
20
1
2110
40
38
33.854
31.08
28.326
24.18
38
29
-52.07
21
1
2190
40
38.01
33.855
31.08
28.315
24.16
38.11
29
-52.18
22
1
2270
40
37.93
33.79
31.03
28.27
24.13
38.03
28.78
-52.3
23
1
2355
40
37.95
33.798
31.03
28.262
24.11
38.06
28.41
-52.41
24
1
2500
40
38.4
34.101
31.24
28.369
24.07
38.84
27.99
-52.61
25
1
2700
40
40
35.206
32.01
28.814
24.02
42.14
28
-52.86
1
2
40
120
31
29.104
27.72
26.576
24.68
31
30
-48.47
2
2
120
120
31
29.098
27.78
26.562
24.66
31
30
-48.58
3
2
200
120
31
29.095
28.08
26.555
24.65
31
30
-48.73
4
2
280
120
32.27
29.978
28.45
26.922
24.63
36.79
30
-48.89
5
2
360
120
35
31.883
28.86
27.727
24.61
40.4
30
-49.04
6
2
550
120
39.49
35.014
29.85
29.046
24.57
52.95
30
-49.42
7
2
850
120
39.49
34.993
31.99
28.997
24.5
50.56
30
-50
8
2
1095
120
39.49
34.975
31.81
28.955
24.44
49.32
30
-50.46
-50.71
9
2
1230
120
38.32
34.144
31.36
28.576
24.4
47.45
30
10
2
1310
120
37.99
33.907
31.19
28.463
24.38
45.84
30.01
-50.85
11
2
1390
120
38
33.908
31.15
28.452
24.36
38
30.05
-50.99
12
2
1470
120
38
33.902
31.15
28.438
24.34
38
30.08
-51.13
13
2
1550
120
38
33.896
31.14
28.424
24.32
38
30.12
-51.27
14
2
1630
120
38
33.89
31.1
28.41
24.3
38
30.17
-51.41
15
2
1710
120
38
33.884
31.12
28.396
24.28
38
30.13
-51.54
16
2
1790
120
38
33.878
31.11
28.382
24.26
38
29.91
-51.67
17
2
1870
120
38
33.872
31.08
28.368
24.24
38
29.54
-51.8
243
Table B.4-continued
i
j
x
(m)
Upper
table
of the
of the
of the
Floridan
phic
surface
of the
of the
(m)
unit
unit heads
unit
heads (m) elevations Hawthorn
Hawthorn
(m)
heads(m)
(m)
heads(m)
(m)
(m)
(m)
120
38
33.866
31.09
28.354
24.22
38
29.16
-51.92
18
2
1950
19
2
2030
120
38
33.857
31.09
28.333
24.19
38
29
-52.04
20
2
2110
120
38
33.851
31.09
28.319
24.17
38
29
-52.16
21
2
2190
120
37.98
33.831
31.07
28.299
24.15
38.1
29
-52.28
22
2
2270
120
37.99
33.832
31.06
28.288
24.13
38.12
28.77
-52.39
23
2
2355
120
38
33.833
31.06
28.277
24.11
38.13
28.41
-52.51
24
2
2500
120
38.48
34.157
31.28
28.393
24.07
38.7
27.98
-52.71
25
2
2700
120
40
35.206
32.01
28.814
24.02
40.88
28
-52.96
1
3
40
200
31
29.098
27.71
26.562
24.66
31
30
-48.64
2
3
120
200
31
29.095
27.81
26.555
24.65
31
30
-48.73
3
3
200
200
31
29.089
28.14
26.541
24.63
31
30
-48.9
4
3
280
200
32.28
29.982
28.45
26.918
24.62
37.06
30
-49.02
5
3
360
200
33.07
30.529
28.83
27.141
24.6
41.74
30
-49.16
6
3
550
200
35.06
31.91
29.81
27.71
24.56
54.08
30
-49.53
7
3
850
200
39.31
34.864
31.9
28.936
24.49
51.68
30
-50.1
8
3
1095
200
39.14
34.727
31.79
28.843
24.43
49.18
30
-50.56
-50.81
9
3
1230
200
38.43
34.221
31.42
28.609
24.4
49.29
30
10
3
1310
200
38.21
34.061
31.3
28.529
24.38
44.81
30.03
-50.95
11
3
1390
200
38.26
34.09
31.31
28.53
24.36
39.83
30.07
-51.09
12
3
1470
200
38
33.902
31.21
28.438
24.34
38
30.08
-51.23
13
3
1550
200
38
33.896
31.08
28.424
24.32
38
30.15
-51.37
14
3
1630
200
38
33.89
31.03
28.41
24.3
38
30.21
-51.5
15
3
1710
200
38
33.884
31.12
28.396
24.28
38
30.15
-51.64
16
3
1790
200
38
33.875
31.04
28.375
24.25
38
29.84
-51.77
17
3
1870
200
38
33.869
31.06
28.361
24.23
38
29.55
-51.9
18
3
1950
200
38
33.863
31.06
28.347
24.21
38
29.11
-52.02
19
3
2030
200
38
33.857
31.05
28.333
24.19
38
29.13
-52.14
20
3
2110
200
38.11
33.928
31.13
28.352
24.17
38.11
29.01
-52.26
21
3
2190
200
37.97
33.824
31.06
28.296
24.15
37.98
29.06
-52.38
22
3
2270
200
38.25
34.014
31.19
28.366
24.13
38.4
28.67
-52.5
23
3
2355
200
38.27
34.019
31.19
28.351
24.1
38.63
28.4
-52.61
24
3
2500
200
39.01
34.528
31.54
28.552
24.07
39.68
28.06
-52.81
25
3
2700
200
40.01
35.213
32.02
28.817
24.02
44.72
27.99
-53.06
1
4
40
280
31
29.095
27.78
26.555
24.65
31
30
-48.79
2
4
120
280
31
29.092
27.8
26.548
24.64
31
30
-48.89
3
4
200
280
31
29.089
28.05
26.541
24.63
31
30
-49.02
4
4
280
280
32.26
29.965
28.44
26.905
24.61
38.16
30
-49.15
5
4
360
280
33.09
30.543
28.84
27.147
24.6
42.54
30
-49.28
6
4
550
280
35.53
32.239
30.04
27.851
24.56
51.82
30
-49.64
7
4
850
280
39.55
35.032
32.02
29.008
24.49
51.74
30
-50.2
8
4
1095
280
39.23
34.79
31.83
28.87
24.43
50.12
30
-50.66
9
4
1230
280
38.64
34.368
31.52
28.672
24.4
47.62
30.01
-50.91
10
4
1310
280
38.27
34.103
31.32
28.547
24.38
44.9
30.02
-51.05
11
4
1390
280
38
33.908
31.18
28.452
24.36
41.64
30.06
-51.2
12
4
1470
280
38
33.902
31.25
28.438
24.34
38
30.1
-51.34
13
4
1550
280
38
33.893
31.08
28.417
24.31
38
30.13
-51.47
244
Table B.4-continued
i
j
x
(m)
Upper
table
of the
of the
of the
Floridan
phic
surface
of the
of the
(m)
unit
unit heads
unit
Hawthorn
(m)
heads(m)
(m)
heads(m)
(m)
(m)
(m)
280
38
33.887
31.02
28.403
24.29
38
30.15
-51.61
14
4
1630
15
4
1710
280
38
33.881
31.22
28.389
24.27
38
30.03
-51.74
16
4
1790
280
38
33.875
31.06
28.375
24.25
38
29.91
-51.87
17
4
1870
280
38
33.869
31.11
28.361
24.23
38
29.54
-52
18
4
1950
280
38
33.863
31.17
28.347
24.21
38
29.27
-52.13
19
4
2030
280
38
33.857
31.1
28.333
24.19
38
29.01
-52.25
20
4
2110
280
38.27
34.04
31.22
28.4
24.17
38.41
28.98
-52.37
21
4
2190
280
38.25
34.017
31.23
28.373
24.14
38.25
28.87
-52.49
22
4
2270
280
38.67
34.305
31.4
28.485
24.12
39.22
28.78
-52.6
23
4
2355
280
38.65
34.285
31.38
28.465
24.1
39.48
28.42
-52.72
24
4
2500
280
39.08
34.574
31.57
28.566
24.06
41.01
27.98
-52.91
25
4
2700
280
40.28
35.402
32.15
28.898
24.02
46.84
27.99
-53.17
1
5
40
360
31
29.092
27.75
26.548
24.64
31
30
-48.95
2
5
120
360
31
29.089
27.77
26.541
24.63
31
30
-49.04
3
5
200
360
31
29.086
28.08
26.534
24.62
31
30
-49.16
4
5
280
360
32.29
29.983
28.45
26.907
24.6
38.95
30
-49.28
5
5
360
360
33.16
30.589
28.87
27.161
24.59
43.28
30
-49.41
6
5
550
360
35.59
32.278
30.07
27.862
24.55
52.1
30
-49.76
7
5
850
360
39.46
34.966
31.97
28.974
24.48
51.93
30
-50.31
8
5
1095
360
39.35
34.874
31.89
28.906
24.43
50.27
30
-50.77
9
5
1230
360
38.8
34.477
31.6
28.713
24.39
47.28
30
-51.02
10
5
1310
360
38.58
34.317
31.48
28.633
24.37
45.31
30.01
-51.16
11
5
1390
360
38.54
34.283
31.45
28.607
24.35
43.26
30.04
-51.3
12
5
1470
360
38.25
34.074
31.29
28.506
24.33
40.19
30.08
-51.44
13
5
1550
360
38
33.893
31.3
28.417
24.31
38
30.1
-51.58
14
5
1630
360
38
33.887
31.45
28.403
24.29
38
30.12
-51.72
15
5
1710
360
38
33.881
31.27
28.389
24.27
38
30.11
-51.85
16
5
1790
360
38
33.875
31.25
28.375
24.25
38
29.88
-51.98
17
5
1870
360
38
33.869
31.25
28.361
24.23
38
29.5
-52.11
18
5
1950
360
38
33.86
31.18
28.34
24.2
38
29.14
-52.23
19
5
2030
360
38.23
34.015
31.21
28.395
24.18
38.6
28.99
-52.36
20
5
2110
360
38.49
34.191
31.32
28.459
24.16
38.8
28.96
-52.48
21
5
2190
360
38.34
34.08
31.52
28.4
24.14
38.34
28.99
-52.59
22
5
2270
360
38.93
34.487
31.53
28.563
24.12
39.67
28.74
-52.71
23
5
2355
360
38.94
34.488
31.52
28.552
24.1
40.15
28.39
-52.83
24
5
2500
360
39.38
34.784
31.72
28.656
24.06
42.56
27.99
-53.02
25
5
2700
360
40.51
35.563
32.26
28.967
24.02
48.56
28
-53.27
1
6
40
440
31
29.089
27.67
26.541
24.63
31
30
-49.11
2
6
120
440
31
29.086
27.8
26.534
24.62
31
30
-49.2
3
6
200
440
31.56
29.475
28.09
26.695
24.61
35.09
30
-49.32
4
6
280
440
32.37
30.036
28.48
26.924
24.59
39.46
30
-49.44
5
6
360
440
33.42
30.768
29
27.232
24.58
43.98
30
-49.56
6
6
550
440
35.85
32.457
30.19
27.933
24.54
53
30
-49.89
7
6
850
440
38.74
34.462
31.61
28.758
24.48
52.11
30
-50.43
8
6
1095
440
39.37
34.885
31.89
28.905
24.42
50.29
30
-50.89
9
6
1230
440
38.94
34.575
31.67
28.755
24.39
47.43
30
-51.13
245
Table B.4-continued
i
j
x
(m)
Upper
table
of the
of the
of the
Floridan
phic
surface
of the
of the
(m)
unit
unit heads
unit
Hawthorn
(m)
heads(m)
(m)
heads(m)
(m)
(m)
(m)
440
39.27
34.8
31.82
28.84
24.37
47.18
30.02
-51.27
10
6
1310
11
6
1390
440
39.1
34.675
31.72
28.775
24.35
44.57
30.08
-51.42
12
6
1470
440
38.86
34.501
31.6
28.689
24.33
41.64
30.06
-51.55
13
6
1550
440
39.84
35.181
32.08
28.969
24.31
44.68
30.02
-51.69
14
6
1630
440
38
33.887
32.3
28.403
24.29
38
30.04
-51.83
15
6
1710
440
38
33.881
31.86
28.389
24.27
38
30.07
-51.96
16
6
1790
440
38
33.872
32.06
28.368
24.24
38
29.69
-52.09
17
6
1870
440
38
33.866
31.64
28.354
24.22
38
29.42
-52.22
18
6
1950
440
39.16
34.672
31.68
28.688
24.2
42.28
29.01
-52.35
19
6
2030
440
38.81
34.421
31.5
28.569
24.18
40.79
28.99
-52.47
20
6
2110
440
38.66
34.31
31.41
28.51
24.16
39.28
28.88
-52.59
21
6
2190
440
38.43
34.143
31.68
28.427
24.14
38.43
28.97
-52.71
22
6
2270
440
38.78
34.382
31.45
28.518
24.12
39.03
28.51
-52.82
23
6
2355
440
39.12
34.611
31.61
28.599
24.09
40.25
28.33
-52.94
24
6
2500
440
39.84
35.106
31.95
28.794
24.06
43.42
28.06
-53.13
25
6
2700
440
40.58
35.609
32.3
28.981
24.01
46.76
27.99
-53.38
1
7
40
520
31
29.086
27.77
26.534
24.62
31
30
-49.27
2
7
120
520
30.95
29.048
27.78
26.512
24.61
31.77
30
-49.36
3
7
200
520
31.5
29.43
28.05
26.67
24.6
34.66
30
-49.47
4
7
280
520
32.67
30.243
28.62
27.007
24.58
40.6
30
-49.58
5
7
360
520
33.78
31.017
29.18
27.333
24.57
45.11
30
-49.7
6
7
550
520
35.89
32.482
30.21
27.938
24.53
51.49
29.99
-50.02
7
7
850
520
38.49
34.284
31.48
28.676
24.47
51.9
30
-50.56
8
7
1095
520
39.75
35.151
32.09
29.019
24.42
51
30
-51
9
7
1230
520
39.82
35.188
32.1
29.012
24.38
49.62
30
-51.25
10
7
1310
520
40.01
35.318
32.19
29.062
24.37
49.64
30
-51.39
11
7
1390
520
40.02
35.316
32.18
29.044
24.34
48.11
30.01
-51.53
12
7
1470
520
40.45
35.611
32.39
29.159
24.32
47.52
29.99
-51.67
13
7
1550
520
41.26
36.172
32.78
29.388
24.3
49
29.89
-51.81
14
7
1630
520
41.45
36.299
32.87
29.431
24.28
49.03
29.89
-51.94
15
7
1710
520
40.84
35.866
32.55
29.234
24.26
47.52
29.84
-52.08
16
7
1790
520
41.05
36.007
32.65
29.283
24.24
48.4
29.69
-52.21
17
7
1870
520
39.85
35.161
32.03
28.909
24.22
44.83
29.35
-52.33
18
7
1950
520
39.66
35.022
31.93
28.838
24.2
43.89
29.09
-52.46
19
7
2030
520
38.83
34.435
31.5
28.575
24.18
41.26
28.77
-52.58
20
7
2110
520
38.79
34.398
31.47
28.542
24.15
39.66
28.78
-52.7
21
7
2190
520
38.53
34.21
31.59
28.45
24.13
38.53
28.71
-52.82
22
7
2270
520
39.06
34.575
31.59
28.595
24.11
38.91
28.58
-52.93
23
7
2355
520
39.36
34.779
31.72
28.671
24.09
40.54
28.34
-53.05
24
7
2500
520
40.03
35.239
32.04
28.851
24.06
43.53
27.98
-53.25
25
7
2700
520
40.83
35.784
32.42
29.056
24.01
45.44
27.99
-53.5
1
8
40
600
31
29.083
27.79
26.527
24.61
31
30
-49.43
2
8
120
600
31.08
29.136
27.84
26.544
24.6
32.89
30
-49.52
3
8
200
600
32.15
29.879
28.37
26.851
24.58
38.58
30
-49.62
4
8
280
600
33.14
30.569
28.85
27.141
24.57
42.57
30
-49.73
5
8
360
600
34.13
31.259
29.35
27.431
24.56
46.21
30
-49.85
246
Table B.4-continued
i
j
x
(m)
550
Upper
table
of the
of the
of the
Floridan
phic
surface
of the
of the
(m)
unit
unit heads
unit
Hawthorn
(m)
heads(m)
(m)
heads(m)
(m)
(m)
(m)
600
35.78
32.405
30.15
27.905
24.53
51.2
30
-50.16
6
8
7
8
850
600
38.42
34.235
31.44
28.655
24.47
51.72
29.99
-50.68
8
8
1095
600
40.11
35.4
32.26
29.12
24.41
51.74
30
-51.13
9
8
1230
600
40.45
35.629
32.41
29.201
24.38
51.79
30
-51.37
10
8
1310
600
40.81
35.875
32.59
29.295
24.36
51.75
29.98
-51.51
11
8
1390
600
41.35
36.247
32.85
29.443
24.34
51.35
29.93
-51.65
12
8
1470
600
42.5
37.046
33.21
29.774
24.32
52.49
29.84
-51.79
13
8
1550
600
42.5
37.04
33.32
29.76
24.3
52.54
29.84
-51.93
14
8
1630
600
42.5
37.034
33.21
29.746
24.28
51.7
29.91
-52.06
15
8
1710
600
42.45
36.993
33.35
29.717
24.26
52.89
29.83
-52.19
16
8
1790
600
41.71
36.469
32.97
29.481
24.24
50.76
29.62
-52.32
17
8
1870
600
40.53
35.637
32.37
29.113
24.22
47.32
29.28
-52.45
18
8
1950
600
39.48
34.893
31.84
28.777
24.19
44.22
28.85
-52.58
19
8
2030
600
39.09
34.614
31.63
28.646
24.17
41.95
28.74
-52.7
20
8
2110
600
38.95
34.51
31.55
28.59
24.15
40.02
28.73
-52.82
21
8
2190
600
38.63
34.28
31.59
28.48
24.13
38.63
28.71
-52.94
22
8
2270
600
39.23
34.694
31.67
28.646
24.11
39.38
28.54
-53.05
23
8
2355
600
39.54
34.905
31.82
28.725
24.09
40.82
28.31
-53.17
24
8
2500
600
40.22
35.369
32.14
28.901
24.05
43.3
28.01
-53.36
25
8
2700
600
41.05
35.938
32.53
29.122
24.01
44.29
28.01
-53.61
1
9
40
680
31
29.077
28.03
26.513
24.59
31
30
-49.59
2
9
120
680
31.86
29.676
28.22
26.764
24.58
38.98
30
-49.67
3
9
200
680
33.14
30.569
28.86
27.141
24.57
46.69
30
-49.78
4
9
280
680
33.7
30.958
29.13
27.302
24.56
46.53
30
-49.89
5
9
360
680
34.36
31.417
29.45
27.493
24.55
47.09
29.99
-50
6
9
550
680
36.06
32.598
30.29
27.982
24.52
51.85
30
-50.31
7
9
850
680
38.77
34.477
31.61
28.753
24.46
51.17
29.98
-50.82
8
9
1095
680
40.16
35.435
32.28
29.135
24.41
51.68
30
-51.25
9
9
1230
680
40.71
35.811
32.54
29.279
24.38
52.07
30.01
-51.49
10
9
1310
680
41.45
36.323
32.9
29.487
24.36
51.9
29.98
-51.63
11
9
1390
680
42.07
36.751
33.2
29.659
24.34
51.05
29.93
-51.78
12
9
1470
680
42.85
37.291
33.58
29.879
24.32
53.73
29.81
-51.91
13
9
1550
680
42.5
37.04
33.32
29.76
24.3
52.69
29.9
-52.05
14
9
1630
680
42.5
37.034
33.12
29.746
24.28
51.26
30.02
-52.18
15
9
1710
680
43.08
37.431
33.67
29.899
24.25
55.04
29.83
-52.32
16
9
1790
680
41.51
36.326
32.87
29.414
24.23
50.22
29.56
-52.45
17
9
1870
680
40.9
35.893
32.56
29.217
24.21
48.57
29.26
-52.57
18
9
1950
680
39.75
35.082
31.97
28.858
24.19
45.4
28.76
-52.7
19
9
2030
680
39.72
35.055
31.94
28.835
24.17
43.53
28.88
-52.82
20
9
2110
680
39.22
34.699
31.68
28.671
24.15
40.83
28.72
-52.94
21
9
2190
680
38.73
34.35
31.75
28.51
24.13
38.73
28.71
-53.06
22
9
2270
680
39.44
34.838
31.77
28.702
24.1
40.69
28.5
-53.17
23
9
2355
680
39.64
34.972
31.86
28.748
24.08
41.26
28.29
-53.29
24
9
2500
680
40.21
35.362
32.13
28.898
24.05
43.62
28.11
-53.48
25
9
2700
680
41.31
36.117
32.66
29.193
24
44.66
28.04
-53.73
1
10
40
760
32.33
30.005
28.46
26.905
24.58
41.73
30
-49.74
247
Table B.4-continued
I
j
x
(m)
Upper
table
of the
of the
of the
Floridan
phic
surface
of the
of the
(m)
unit
unit heads
unit
Hawthorn
(m)
heads(m)
(m)
heads(m)
(m)
(m)
(m)
760
32.55
30.156
28.56
26.964
24.57
44.75
30
-49.83
2
10
120
3
10
200
760
33.31
30.685
28.94
27.185
24.56
48.52
30
-49.93
4
10
280
760
34.05
31.2
29.3
27.4
24.55
47.76
29.99
-50.04
5
10
360
760
34.78
31.708
29.66
27.612
24.54
48.13
29.99
-50.15
6
10
550
760
36.47
32.882
30.49
28.098
24.51
50.42
30.02
-50.44
7
10
850
760
39.03
34.656
31.74
28.824
24.45
50.76
29.99
-50.95
8
10
1095
760
40.57
35.719
32.49
29.251
24.4
51.93
30
-51.38
9
10
1230
760
41.29
36.214
32.83
29.446
24.37
52.17
30
-51.62
10
10
1310
760
41.89
36.628
33.12
29.612
24.35
52.1
29.99
-51.76
11
10
1390
760
42.42
36.993
33.38
29.757
24.33
52.29
29.96
-51.9
12
10
1470
760
42.58
37.099
33.44
29.791
24.31
53.25
29.86
-52.04
13
10
1550
760
42.5
37.037
33.4
29.753
24.29
53.08
29.84
-52.17
14
10
1630
760
42.59
37.094
33.43
29.766
24.27
53.06
29.85
-52.31
15
10
1710
760
42.34
36.913
33.3
29.677
24.25
52.88
29.65
-52.44
16
10
1790
760
41.9
36.599
33.07
29.531
24.23
51.43
29.64
-52.57
17
10
1870
760
41.1
36.033
32.65
29.277
24.21
49.12
29.21
-52.7
18
10
1950
760
40.21
35.404
32.2
28.996
24.19
45.93
28.92
-52.82
19
10
2030
760
39.65
35.003
31.91
28.807
24.16
43.89
28.71
-52.94
20
10
2110
760
39.48
34.878
31.81
28.742
24.14
41.52
28.65
-53.06
21
10
2190
760
38.83
34.417
31.83
28.533
24.12
38.83
28.55
-53.18
22
10
2270
760
39.54
34.908
31.82
28.732
24.1
40.13
28.59
-53.29
23
10
2355
760
39.74
35.042
31.91
28.778
24.08
41.43
28.27
-53.41
24
10
2500
760
40.29
35.415
32.16
28.915
24.04
44.35
28.02
-53.6
25
10
2700
760
41.74
36.418
32.87
29.322
24
45.33
28.02
-53.85
1
11
40
850
33.13
30.562
28.85
27.138
24.57
47.26
30
-49.91
2
11
120
850
33.35
30.713
28.95
27.197
24.56
49.02
30
-50
3
11
200
850
34
31.165
29.28
27.385
24.55
48.35
29.99
-50.1
4
11
280
850
34.64
31.61
29.59
27.57
24.54
48.98
29.99
-50.2
5
11
360
850
35.35
32.104
29.94
27.776
24.53
49.1
30.01
-50.31
6
11
550
850
36.96
33.222
30.73
28.238
24.5
48.98
30.02
-50.61
7
11
850
850
39.35
34.877
31.9
28.913
24.44
50.56
30.14
-51.1
8
11
1095
850
40.87
35.929
32.63
29.341
24.4
51.8
30.03
-51.53
9
11
1230
850
41.74
36.529
33.05
29.581
24.37
51.79
30.02
-51.77
10
11
1310
850
42.05
36.74
33.2
29.66
24.35
52.09
29.96
-51.91
11
11
1390
850
42.21
36.846
33.27
29.694
24.33
52.65
29.89
-52.05
12
11
1470
850
42.36
36.945
33.33
29.725
24.31
52.73
29.8
-52.18
13
11
1550
850
42.49
37.03
33.39
29.75
24.29
53.34
29.67
-52.32
14
11
1630
850
42.67
37.15
33.47
29.79
24.27
54
29.63
-52.45
15
11
1710
850
42.45
36.99
33.35
29.71
24.25
53.19
29.69
-52.58
16
11
1790
850
41.87
36.575
33.05
29.515
24.22
51.4
29.39
-52.71
17
11
1870
850
41.23
36.121
32.72
29.309
24.2
49.3
29.04
-52.84
18
11
1950
850
40.29
35.457
32.24
29.013
24.18
46.7
28.72
-52.96
19
11
2030
850
40.05
35.283
32.1
28.927
24.16
44.21
28.57
-53.09
20
11
2110
850
39.82
35.116
31.98
28.844
24.14
41.87
28.51
-53.2
21
11
2190
850
39.7
35.026
31.91
28.794
24.12
40.18
28.61
-53.32
22
11
2270
850
38.96
34.502
31.95
28.558
24.1
38.96
28.33
-53.43
248
Table B.4-continued
i
j
x
(m)
Upper
table
of the
of the
of the
Floridan
phic
surface
of the
of the
(m)
unit
unit heads
unit
Hawthorn
(m)
heads(m)
(m)
heads(m)
(m)
(m)
(m)
850
39.84
35.109
31.96
28.801
24.07
41.74
28.11
-53.55
23
11
2355
24
11
2500
850
40.5
35.562
32.27
28.978
24.04
43.82
27.82
-53.74
25
11
2700
850
41.95
36.562
32.97
29.378
23.99
45.45
27.85
-53.99
1
12
40
1050
35
31.862
29.77
27.678
24.54
50.3
30
-50.29
2
12
120
1050
35.02
31.873
29.78
27.677
24.53
50.31
30
-50.38
3
12
200
1050
35.61
32.283
30.07
27.847
24.52
50.48
29.99
-50.48
4
12
280
1050
36.31
32.77
30.41
28.05
24.51
50.45
29.99
-50.58
5
12
360
1050
37.04
33.278
30.77
28.262
24.5
50.31
30.03
-50.69
6
12
550
1050
37.85
33.836
31.16
28.484
24.47
50.31
30.68
-50.97
7
12
850
1050
40.16
35.441
32.29
29.149
24.43
50.54
30.69
-51.45
8
12
1095
1050
41.23
36.175
32.81
29.435
24.38
51.81
30.02
-51.87
9
12
1230
1050
41.82
36.579
33.09
29.591
24.35
51.81
30.03
-52.1
10
12
1310
1050
42.08
36.758
33.21
29.662
24.34
52.19
29.88
-52.24
11
12
1390
1050
42.34
36.934
33.33
29.726
24.32
53.26
29.53
-52.38
12
12
1470
1050
42.45
37.005
33.37
29.745
24.3
54.28
29.07
-52.51
13
12
1550
1050
42.47
37.013
33.37
29.737
24.28
54.58
28.84
-52.65
14
12
1630
1050
42.39
36.951
33.32
29.699
24.26
53.86
28.93
-52.78
15
12
1710
1050
42.13
36.76
33.18
29.6
24.23
52.52
28.84
-52.91
16
12
1790
1050
41.85
36.558
33.03
29.502
24.21
50.88
28.67
-53.04
17
12
1870
1050
41.53
36.328
32.86
29.392
24.19
48.95
28.4
-53.16
18
12
1950
1050
41.16
36.063
32.66
29.267
24.17
46.68
27.98
-53.29
19
12
2030
1050
40.92
35.889
32.53
29.181
24.15
44.29
27.84
-53.41
20
12
2110
1050
40.79
35.789
32.46
29.121
24.12
41.91
27.89
-53.53
21
12
2190
1050
40.8
35.79
32.45
29.11
24.1
40.2
27.82
-53.64
22
12
2270
1050
40.53
35.595
32.3
29.015
24.08
39.84
27.66
-53.76
23
12
2355
1050
39.24
34.686
32.11
28.614
24.06
39.24
27.41
-53.87
24
12
2500
1050
40.41
35.493
32.22
28.937
24.02
41.98
27.03
-54.06
25
12
2700
1050
42
36.594
32.99
29.386
23.98
44.21
27
-54.3
1
13
40
1350
37.81
33.817
31.15
28.493
24.5
52.2
29.94
-50.83
2
13
120
1350
37.93
33.898
31.21
28.522
24.49
52.29
29.95
-50.92
3
13
200
1350
38.4
34.224
31.44
28.656
24.48
52.36
29.92
-51.02
4
13
280
1350
38.5
34.291
31.49
28.679
24.47
52.28
29.96
-51.12
5
13
360
1350
38.99
34.631
31.72
28.819
24.46
52.28
29.99
-51.23
6
13
550
1350
40.49
35.672
32.46
29.248
24.43
52.27
30.62
-51.51
7
13
850
1350
41.78
36.563
33.08
29.607
24.39
52.3
30.06
-51.98
8
13
1095
1350
42.51
37.062
33.43
29.798
24.35
51.82
29.42
-52.39
9
13
1230
1350
42.92
37.343
33.63
29.907
24.33
51.72
29.54
-52.62
10
13
1310
1350
42.99
37.389
33.65
29.921
24.32
52.38
29.15
-52.76
11
13
1390
1350
43.04
37.418
33.67
29.922
24.3
53.66
28.49
-52.89
12
13
1470
1350
42.98
37.373
33.63
29.897
24.29
54.25
28.53
-53.03
13
13
1550
1350
42.84
37.266
33.55
29.834
24.26
54.19
28.29
-53.16
14
13
1630
1350
42.75
37.197
33.49
29.793
24.24
53.61
28.15
-53.29
15
13
1710
1350
42.63
37.104
33.42
29.736
24.21
53.56
28.43
-53.42
16
13
1790
1350
42.47
36.986
33.33
29.674
24.19
52.29
27.94
-53.54
17
13
1870
1350
42.31
36.865
33.24
29.605
24.16
50.81
27.7
-53.67
18
13
1950
1350
42.15
36.747
33.14
29.543
24.14
47.27
27.51
-53.79
249
Table B.4-continued
I
j
x
(m)
Upper
table
of the
of the
of the
Floridan
phic
surface
of the
of the
(m)
unit
unit heads
unit
Hawthorn
(m)
heads(m)
(m)
heads(m)
(m)
(m)
(m)
1350
41.88
36.552
33
29.448
24.12
46.4
27.24
-53.91
19
13
2030
20
13
2110
1350
41.92
36.571
33.01
29.439
24.09
44.73
27.21
-54.03
21
13
2190
1350
41.25
36.096
32.66
29.224
24.07
41.54
27.42
-54.15
22
13
2270
1350
41.65
36.37
32.85
29.33
24.05
44.31
26.91
-54.26
23
13
2355
1350
41.41
36.196
32.72
29.244
24.03
44.25
26.94
-54.38
24
13
2500
1350
39.66
34.962
32.92
28.698
24
39.66
26.81
-54.57
25
13
2700
1350
41.48
36.221
32.72
29.209
23.95
44.65
26.86
-54.8
1
14
40
1600
40.49
35.681
32.47
29.269
24.46
53.85
29.82
-51.26
2
14
120
1600
40.66
35.797
32.56
29.313
24.45
53.92
29.88
-51.36
3
14
200
1600
40.87
35.941
32.66
29.369
24.44
53.96
29.97
-51.45
4
14
280
1600
41.33
36.26
32.88
29.5
24.43
53.92
29.86
-51.56
5
14
360
1600
41.58
36.432
33
29.568
24.42
53.94
29.78
-51.67
6
14
550
1600
42.5
37.07
33.45
29.83
24.4
53.94
28.99
-51.94
7
14
850
1600
43.09
37.471
33.73
29.979
24.36
53.09
28.88
-52.4
8
14
1095
1600
43.51
37.753
33.91
30.077
24.32
51.88
28.25
-52.81
9
14
1230
1600
43.74
37.905
34.02
30.125
24.29
51.75
27.9
-53.04
10
14
1310
1600
43.58
37.79
33.93
30.07
24.28
52.15
27.88
-53.17
11
14
1390
1600
43.42
37.672
33.84
30.008
24.26
52.84
27.72
-53.31
12
14
1470
1600
43.25
37.547
33.75
29.943
24.24
53.03
27.01
-53.44
13
14
1550
1600
43.16
37.478
33.69
29.902
24.22
53.73
27.19
-53.57
14
14
1630
1600
43.16
37.472
33.68
29.888
24.2
54.69
27.47
-53.7
15
14
1710
1600
42.97
37.33
33.57
29.81
24.17
54.53
26.89
-53.83
16
14
1790
1600
42.74
37.163
33.45
29.727
24.15
52.95
26.83
-53.96
17
14
1870
1600
42.56
37.031
33.35
29.659
24.13
52.94
26.51
-54.09
18
14
1950
1600
42.33
36.864
33.22
29.576
24.11
51.83
25.92
-54.22
19
14
2030
1600
42.77
37.166
33.43
29.694
24.09
51.24
26.16
-54.34
20
14
2110
1600
43.2
37.458
33.63
29.802
24.06
51.17
25.97
-54.46
21
14
2190
1600
43.83
37.893
33.94
29.977
24.04
51.41
25.78
-54.58
22
14
2270
1600
44.3
38.216
33.88
30.104
24.02
50.52
26.22
-54.69
23
14
2355
1600
44.1
38.07
34.05
30.03
24
50.94
26.3
-54.81
24
14
2500
1600
44
37.991
33.45
29.979
23.97
47.3
26.52
-55
25
14
2700
1600
40.05
35.211
32.29
28.759
23.92
40.05
26.55
-55.24
1
15
40
1750
41.5
36.382
32.97
29.558
24.44
54.88
29.8
-51.51
2
15
120
1750
41.51
36.386
32.97
29.554
24.43
54.88
29.8
-51.6
3
15
200
1750
41.86
36.628
33.14
29.652
24.42
54.93
29.71
-51.7
4
15
280
1750
42.2
36.863
33.31
29.747
24.41
54.92
29.73
-51.81
5
15
360
1750
42.53
37.091
33.46
29.839
24.4
54.88
29.75
-51.92
6
15
550
1750
42.95
37.379
33.66
29.951
24.38
54.88
29.06
-52.19
7
15
850
1750
43.61
37.826
33.97
30.114
24.33
53.12
28.8
-52.65
8
15
1095
1750
44.04
38.115
34.17
30.215
24.29
51.78
27.92
-53.05
9
15
1230
1750
44.05
38.116
34.16
30.204
24.27
51.77
27.9
-53.28
10
15
1310
1750
43.95
38.04
34.1
30.16
24.25
52
27.84
-53.41
11
15
1390
1750
43.78
37.915
34.01
30.095
24.23
52.5
27.35
-53.55
12
15
1470
1750
43.54
37.741
33.88
30.009
24.21
53.04
27.12
-53.68
13
15
1550
1750
43.35
37.602
33.77
29.938
24.19
53.18
26.94
-53.82
14
15
1630
1750
43.22
37.505
33.7
29.885
24.17
52.78
26.79
-53.95
250
Table B.4-continued
i
j
x
(m)
Upper
table
of the
of the
of the
Floridan
phic
surface
of the
of the
(m)
unit
unit heads
unit
Hawthorn
(m)
heads(m)
(m)
heads(m)
(m)
(m)
(m)
1750
43.07
37.394
33.61
29.826
24.15
52.15
26.92
-54.08
15
15
1710
16
15
1790
1750
42.86
37.241
33.5
29.749
24.13
51.96
26.77
-54.22
17
15
1870
1750
42.59
37.046
33.35
29.654
24.11
52.2
26.32
-54.35
18
15
1950
1750
42.39
36.897
33.24
29.573
24.08
51.94
26.1
-54.47
19
15
2030
1750
42.56
37.01
33.31
29.61
24.06
51.58
25.93
-54.6
20
15
2110
1750
43.02
37.326
33.53
29.734
24.04
51.13
25.85
-54.72
21
15
2190
1750
43.61
37.733
33.81
29.897
24.02
50.51
25.96
-54.84
22
15
2270
1750
44.3
38.21
33.88
30.09
24
50.17
26.05
-54.96
23
15
2355
1750
44.3
38.204
33.86
30.076
23.98
50.07
26.19
-55.08
24
15
2500
1750
44.3
38.192
33.45
30.048
23.94
48.34
26.45
-55.27
25
15
2700
1750
40.23
35.331
32.01
28.799
23.9
40.23
26.5
-55.51
1
16
40
1850
42.05
36.764
33.24
29.716
24.43
55.52
29.03
-51.67
2
16
120
1850
42.25
36.901
33.33
29.769
24.42
55.63
29.04
-51.76
3
16
200
1850
42.66
37.185
33.53
29.885
24.41
55.71
28.87
-51.86
4
16
280
1850
42.83
37.301
33.62
29.929
24.4
55.62
29.18
-51.97
5
16
360
1850
43.13
37.508
33.76
30.012
24.39
55.56
28.9
-52.08
6
16
550
1850
43.54
37.786
33.95
30.114
24.36
55.57
28.49
-52.35
7
16
850
1850
44.5
38.446
34.41
30.374
24.32
53.1
27.78
-52.81
8
16
1095
1850
44.79
38.634
34.53
30.426
24.27
51.85
27.12
-53.21
9
16
1230
1850
44.64
38.523
34.44
30.367
24.25
51.77
27.37
-53.44
10
16
1310
1850
44.38
38.335
34.31
30.275
24.23
52.09
26.83
-53.57
11
16
1390
1850
44.28
38.259
34.25
30.231
24.21
52.79
26.16
-53.71
12
16
1470
1850
43.88
37.973
34.04
30.097
24.19
53.11
26.49
-53.84
13
16
1550
1850
43.69
37.834
33.93
30.026
24.17
52.35
26.13
-53.98
14
16
1630
1850
43.67
37.814
33.91
30.006
24.15
51.87
25.92
-54.11
15
16
1710
1850
43.3
37.549
33.72
29.881
24.13
51.19
26.38
-54.24
16
16
1790
1850
42.88
37.249
33.5
29.741
24.11
51.74
25.72
-54.38
17
16
1870
1850
42.57
37.026
33.33
29.634
24.09
51.81
25.81
-54.51
18
16
1950
1850
42.36
36.873
33.21
29.557
24.07
51.3
25.51
-54.64
19
16
2030
1850
42.77
37.154
33.41
29.666
24.05
51.84
25.5
-54.77
20
16
2110
1850
42.53
36.98
33.28
29.58
24.03
50.42
25.54
-54.89
21
16
2190
1850
43.55
37.685
33.78
29.865
24
50.35
25.73
-55.01
22
16
2270
1850
42.58
37
33.28
29.56
23.98
49.11
25.64
-55.13
23
16
2355
1850
42.59
37.001
33.28
29.549
23.96
47.68
25.81
-55.26
24
16
2500
1850
40.87
35.788
32.4
29.012
23.93
43.03
25.73
-55.45
25
16
2700
1850
40.25
35.342
32.08
28.798
23.89
40.25
25.86
-55.69
1
17
40
1950
42.86
37.325
33.64
29.945
24.41
56.24
28.27
-51.82
2
17
120
1950
42.82
37.294
33.61
29.926
24.4
56.21
28.26
-51.92
3
17
200
1950
43.04
37.445
33.71
29.985
24.39
56.21
28.18
-52.02
4
17
280
1950
43.44
37.722
33.91
30.098
24.38
56.16
28.41
-52.13
5
17
360
1950
43.8
37.971
34.09
30.199
24.37
56.19
28.2
-52.23
6
17
550
1950
44.44
38.413
34.39
30.377
24.35
55.45
27.82
-52.5
7
17
850
1950
45.29
38.993
34.8
30.597
24.3
53.11
27.12
-52.96
8
17
1095
1950
45.47
39.107
34.86
30.623
24.26
51.74
26.69
-53.36
9
17
1230
1950
45.13
38.86
34.68
30.5
24.23
51.9
26.4
-53.59
10
17
1310
1950
44.8
38.623
34.5
30.387
24.21
51.96
26.15
-53.72
251
Table B.4-continued
i
j
x
(m)
Upper
table
of the
of the
of the
Floridan
phic
surface
of the
of the
(m)
unit
unit heads
unit
Hawthorn
(m)
heads(m)
(m)
heads(m)
(m)
(m)
(m)
1950
44.63
38.498
34.41
30.322
24.19
52.31
25.83
-53.86
11
17
1390
12
17
1470
1950
44.21
38.201
34.2
30.189
24.18
52.39
25.59
-54
13
17
1550
1950
43.86
37.95
34.01
30.07
24.16
51.94
25.68
-54.13
14
17
1630
1950
43.71
37.839
33.92
30.011
24.14
51.81
25.71
-54.27
15
17
1710
1950
43.22
37.487
33.67
29.843
24.11
51.1
25.31
-54.4
16
17
1790
1950
42.89
37.25
33.49
29.73
24.09
50.77
25.36
-54.54
17
17
1870
1950
42.45
36.936
33.26
29.584
24.07
50.77
25.32
-54.67
18
17
1950
1950
42.31
36.832
33.18
29.528
24.05
50.1
25.15
-54.8
19
17
2030
1950
42.23
36.77
33.13
29.49
24.03
51.81
25.08
-54.93
20
17
2110
1950
41.8
36.463
32.91
29.347
24.01
48.76
25.22
-55.06
21
17
2190
1950
42.18
36.723
33.08
29.447
23.99
47.84
25.31
-55.18
22
17
2270
1950
41.37
36.15
32.67
29.19
23.97
46.04
25.19
-55.31
23
17
2355
1950
41.21
36.032
32.58
29.128
23.95
44.24
25.32
-55.43
24
17
2500
1950
40.25
35.348
31.94
28.812
23.91
40.25
25.18
-55.63
25
17
2700
1950
40.25
35.336
31.99
28.784
23.87
40.25
25.21
-55.87
1
18
40
2050
43.27
37.609
33.83
30.061
24.4
56.35
27.99
-51.98
2
18
120
2050
43.42
37.711
33.9
30.099
24.39
56.44
27.99
-52.07
3
18
200
2050
43.72
37.918
34.05
30.182
24.38
56.31
28.22
-52.17
4
18
280
2050
44.11
38.188
34.24
30.292
24.37
56.65
27.99
-52.28
5
18
360
2050
44.58
38.514
34.47
30.426
24.36
56.65
27.91
-52.39
6
18
550
2050
45.42
39.093
34.88
30.657
24.33
55.41
26.96
-52.66
7
18
850
2050
45.96
39.456
35.12
30.784
24.28
53.49
26.84
-53.12
8
18
1095
2050
45.95
39.437
35.09
30.753
24.24
51.93
26.09
-53.51
9
18
1230
2050
45.56
39.155
34.88
30.615
24.21
51.91
25.93
-53.74
10
18
1310
2050
45.16
38.869
34.68
30.481
24.19
52.26
25.87
-53.88
11
18
1390
2050
44.89
38.677
34.53
30.393
24.18
52.56
25.69
-54.01
12
18
1470
2050
44.43
38.349
34.29
30.241
24.16
52.25
25.52
-54.15
13
18
1550
2050
43.9
37.972
34.02
30.068
24.14
51.51
25.12
-54.29
14
18
1630
2050
43.61
37.763
33.86
29.967
24.12
51.24
24.8
-54.42
15
18
1710
2050
43.16
37.442
33.63
29.818
24.1
50.33
25.12
-54.56
16
18
1790
2050
42.55
37.009
33.31
29.621
24.08
49.32
25.04
-54.7
17
18
1870
2050
42.15
36.723
33.1
29.487
24.06
48.47
25.06
-54.83
18
18
1950
2050
41.91
36.546
32.97
29.394
24.03
50.42
24.89
-54.97
19
18
2030
2050
41.52
36.267
32.77
29.263
24.01
46.48
25.07
-55.1
20
18
2110
2050
41.15
36.002
32.57
29.138
23.99
45.76
25.03
-55.23
21
18
2190
2050
40.75
35.716
32.36
29.004
23.97
43.62
24.92
-55.36
22
18
2270
2050
40.67
35.654
32.31
28.966
23.95
42.7
25.11
-55.48
23
18
2355
2050
40.25
35.354
32.12
28.826
23.93
40.25
25.02
-55.61
24
18
2500
2050
40.25
35.345
32.03
28.805
23.9
40.25
25.1
-55.81
25
18
2700
2050
40.25
35.333
31.97
28.777
23.86
40.25
24.99
-56.06
1
19
40
2150
43.91
38.051
34.15
30.239
24.38
56.44
27.99
-52.12
2
19
120
2150
43.94
38.069
34.15
30.241
24.37
56.45
27.99
-52.22
3
19
200
2150
44.21
38.255
34.29
30.315
24.36
56.54
28.04
-52.32
4
19
280
2150
44.84
38.693
34.6
30.497
24.35
57.01
27.88
-52.43
5
19
360
2150
45.43
39.103
34.89
30.667
24.34
57.3
27.9
-52.54
6
19
550
2150
46.11
39.57
35.21
30.85
24.31
55.75
27.13
-52.81
252
Table B.4-continued
i
j
x
(m)
Upper
table
of the
of the
of the
Floridan
phic
surface
of the
of the
(m)
unit
unit heads
unit
Hawthorn
(m)
heads(m)
(m)
heads(m)
(m)
(m)
(m)
2150
46.61
39.908
35.44
30.972
24.27
54.11
26.84
-53.26
7
19
850
8
19
1095
2150
46.51
39.823
35.36
30.907
24.22
51.87
26.02
-53.66
9
19
1230
2150
45.94
39.415
35.06
30.715
24.19
52.32
25.79
-53.89
10
19
1310
2150
45.47
39.083
34.82
30.567
24.18
52.81
25.86
-54.02
11
19
1390
2150
45.09
38.811
34.62
30.439
24.16
52.8
25.67
-54.16
12
19
1470
2150
44.41
38.329
34.28
30.221
24.14
52.21
25.14
-54.3
13
19
1550
2150
43.84
37.924
33.98
30.036
24.12
51.27
25.02
-54.44
14
19
1630
2150
43.4
37.61
33.75
29.89
24.1
50.42
24.97
-54.57
15
19
1710
2150
42.72
37.128
33.4
29.672
24.08
49.01
24.85
-54.71
16
19
1790
2150
42.12
36.702
33.09
29.478
24.06
46.78
25.03
-54.85
17
19
1870
2150
41.74
36.43
32.89
29.35
24.04
45.48
24.92
-54.99
18
19
1950
2150
41.37
36.165
32.69
29.225
24.02
45.42
24.94
-55.12
19
19
2030
2150
41.05
35.935
32.53
29.115
24
42.49
24.92
-55.26
20
19
2110
2150
40.73
35.705
32.35
29.005
23.98
42.63
24.92
-55.4
21
19
2190
2150
40.41
35.475
32.18
28.895
23.96
41.24
25.02
-55.52
22
19
2270
2150
40.25
35.357
32.05
28.833
23.94
40.25
24.72
-55.65
23
19
2355
2150
40.25
35.348
32.01
28.812
23.91
40.25
24.92
-55.79
24
19
2500
2150
40.25
35.339
31.98
28.791
23.88
40.25
25.01
-55.99
25
19
2700
2150
40.25
35.327
31.97
28.763
23.84
40.25
24.99
-56.24
1
20
40
2250
44.41
38.398
34.39
30.382
24.37
56.82
27.86
-52.27
2
20
120
2250
44.53
38.479
34.44
30.411
24.36
56.88
27.86
-52.37
3
20
200
2250
45.17
38.924
34.76
30.596
24.35
57.7
27.78
-52.47
4
20
280
2250
45.72
39.306
35.03
30.754
24.34
57.86
27.86
-52.57
5
20
360
2250
46.26
39.681
35.29
30.909
24.33
57.78
27.77
-52.68
6
20
550
2250
46.71
39.987
35.5
31.023
24.3
55.37
27
-52.95
7
20
850
2250
47.34
40.413
35.79
31.177
24.25
54.34
26.62
-53.41
8
20
1095
2250
47.02
40.174
35.61
31.046
24.2
52.24
25.75
-53.8
9
20
1230
2250
46.3
39.664
35.24
30.816
24.18
53.07
25.79
-54.03
10
20
1310
2250
45.63
39.189
34.89
30.601
24.16
53.42
25.58
-54.17
11
20
1390
2250
45.04
38.77
34.59
30.41
24.14
53.09
25.06
-54.3
12
20
1470
2250
44.36
38.288
34.24
30.192
24.12
52.59
24.97
-54.44
13
20
1550
2250
43.56
37.722
33.83
29.938
24.1
51.49
24.74
-54.58
14
20
1630
2250
42.83
37.205
33.46
29.705
24.08
49.87
24.58
-54.72
15
20
1710
2250
42.17
36.737
33.12
29.493
24.06
47.26
24.8
-54.86
16
20
1790
2250
41.69
36.395
32.86
29.335
24.04
45.52
24.7
-55
17
20
1870
2250
41.29
36.109
32.66
29.201
24.02
43.45
24.61
-55.14
18
20
1950
2250
40.97
35.879
32.49
29.091
24
41.02
24.62
-55.28
19
20
2030
2250
40.66
35.656
32.32
28.984
23.98
40.63
24.57
-55.42
20
20
2110
2250
40.41
35.475
32.19
28.895
23.96
40.4
24.51
-55.56
21
20
2190
2250
40.25
35.357
32.01
28.833
23.94
40.25
24.25
-55.69
22
20
2270
2250
40.25
35.351
32.01
28.819
23.92
40.25
24.54
-55.82
23
20
2355
2250
40.25
35.345
31.99
28.805
23.9
40.25
24.7
-55.96
24
20
2500
2250
40.25
35.333
31.97
28.777
23.86
40.25
24.78
-56.17
25
20
2700
2250
40.25
35.321
31.97
28.749
23.82
40.25
24.86
-56.43
1
21
40
2350
45.25
38.98
34.8
30.62
24.35
58.13
27.44
-52.41
2
21
120
2350
45.38
39.068
34.86
30.652
24.34
58.16
27.44
-52.51
253
Table B.4-continued
i
j
x
(m)
Upper
table
of the
of the
of the
Floridan
phic
surface
of the
of the
(m)
unit
unit heads
unit
Hawthorn
(m)
heads(m)
(m)
heads(m)
(m)
(m)
(m)
2350
45.87
39.408
35.1
30.792
24.33
58.31
27.44
-52.61
3
21
200
4
21
280
2350
46.45
39.811
35.38
30.959
24.32
58.12
27.42
-52.71
5
21
360
2350
47.01
40.2
35.66
31.12
24.31
57.84
27.32
-52.82
6
21
550
2350
47.71
40.681
36
31.309
24.28
56.73
26.63
-53.09
7
21
850
2350
48.15
40.974
36.19
31.406
24.23
54.35
26.18
-53.54
8
21
1095
2350
47.53
40.528
35.86
31.192
24.19
52.45
25.28
-53.94
9
21
1230
2350
46.48
39.784
35.32
30.856
24.16
53.49
25.09
-54.17
10
21
1310
2350
45.65
39.197
34.9
30.593
24.14
54.11
24.95
-54.3
11
21
1390
2350
44.87
38.645
34.5
30.345
24.12
54.19
24.67
-54.44
12
21
1470
2350
44.01
38.04
34.06
30.08
24.11
53.55
24.42
-54.58
13
21
1550
2350
43.14
37.425
33.61
29.805
24.09
52.58
24.17
-54.72
14
21
1630
2350
42.39
36.894
33.23
29.566
24.07
50.84
24.06
-54.86
15
21
1710
2350
41.76
36.447
32.91
29.363
24.05
48.36
24.12
-55
16
21
1790
2350
41.25
36.084
32.64
29.196
24.03
46.02
24.07
-55.14
17
21
1870
2350
40.86
35.805
32.44
29.065
24.01
43.33
24
-55.29
18
21
1950
2350
40.58
35.603
32.28
28.967
23.99
41.43
23.96
-55.43
19
21
2030
2350
40.35
35.436
32.16
28.884
23.97
41.08
23.83
-55.57
20
21
2110
2350
40.25
35.357
32.08
28.833
23.94
40.25
23.85
-55.71
21
21
2190
2350
40.25
35.351
32.01
28.819
23.92
40.25
23.92
-55.85
22
21
2270
2350
40.25
35.345
31.99
28.805
23.9
40.25
23.95
-55.99
23
21
2355
2350
40.25
35.339
31.98
28.791
23.88
40.25
24.19
-56.13
24
21
2500
2350
40.25
35.33
31.97
28.77
23.85
40.25
24.41
-56.35
25
21
2700
2350
40.25
35.315
31.96
28.735
23.8
40.25
24.43
-56.62
1
22
40
2450
46
39.502
35.17
30.838
24.34
59.45
27
-52.54
2
22
120
2450
46.03
39.52
35.18
30.84
24.33
59.43
27
-52.64
3
22
200
2450
46.64
39.944
35.48
31.016
24.32
58.95
27.09
-52.74
4
22
280
2450
47.25
40.368
35.78
31.192
24.31
58.42
27.08
-52.85
5
22
360
2450
47.84
40.778
36.07
31.362
24.3
57.91
26.96
-52.96
6
22
550
2450
48.66
41.343
36.47
31.587
24.27
57.87
26.53
-53.22
7
22
850
2450
48.98
41.552
36.6
31.648
24.22
54.4
25.82
-53.68
8
22
1095
2450
48.09
40.914
36.13
31.346
24.17
52.38
24.85
-54.07
9
22
1230
2450
46.66
39.904
35.4
30.896
24.14
54.13
24.55
-54.3
10
22
1310
2450
45.66
39.201
34.9
30.589
24.13
54.95
24.41
-54.43
11
22
1390
2450
44.65
38.488
34.38
30.272
24.11
55.18
24.15
-54.57
12
22
1470
2450
43.61
37.754
33.85
29.946
24.09
55.01
23.75
-54.71
13
22
1550
2450
42.68
37.097
33.37
29.653
24.07
54.05
23.53
-54.85
14
22
1630
2450
41.9
36.545
32.98
29.405
24.05
52.08
23.59
-54.99
15
22
1710
2450
41.3
36.119
32.66
29.211
24.03
49.88
23.53
-55.14
16
22
1790
2450
40.82
35.777
32.42
29.053
24.01
47.17
23.46
-55.28
17
22
1870
2450
40.48
35.533
32.24
28.937
23.99
43.97
23.38
-55.42
18
22
1950
2450
40.23
35.352
32.1
28.848
23.97
41.34
23.11
-55.57
19
22
2030
2450
40.25
35.36
32.03
28.84
23.95
40.25
23.05
-55.71
20
22
2110
2450
40.25
35.354
32.01
28.826
23.93
40.25
23.11
-55.86
21
22
2190
2450
40.25
35.348
32.01
28.812
23.91
40.25
23.03
-56
22
22
2270
2450
40.25
35.342
32
28.798
23.89
40.25
23.29
-56.14
23
22
2355
2450
40.25
35.336
31.98
28.784
23.87
40.25
23.69
-56.29
254
Table B.4-continued
i
j
x
(m)
Upper
table
of the
of the
of the
Floridan
phic
surface
of the
of the
(m)
unit
unit heads
unit
Hawthorn
(m)
heads(m)
(m)
heads(m)
(m)
(m)
(m)
2450
40.25
35.324
31.97
28.756
23.83
40.25
24.05
-56.53
24
22
2500
25
22
2700
2450
40.25
35.312
31.95
28.728
23.79
40.25
24
-56.81
1
23
40
2550
46.78
40.045
35.55
31.065
24.33
60.24
26.99
-52.67
2
23
120
2550
47.09
40.259
35.71
31.151
24.32
60.08
26.98
-52.77
3
23
200
2550
47.92
40.837
36.11
31.393
24.31
59.72
27.12
-52.87
4
23
280
2550
48.08
40.946
36.19
31.434
24.3
58.99
26.97
-52.98
5
23
360
2550
48.41
41.171
36.35
31.519
24.28
58.13
26.85
-53.09
6
23
550
2550
49.27
41.764
36.76
31.756
24.25
58.03
26.61
-53.36
7
23
850
2550
49.94
42.218
37.07
31.922
24.2
54.2
25.63
-53.81
8
23
1095
2550
48.78
41.394
36.47
31.546
24.16
53.03
24.42
-54.2
9
23
1230
2550
47.28
40.335
35.71
31.075
24.13
54.37
24.39
-54.42
10
23
1310
2550
46.14
39.531
35.12
30.719
24.11
54.92
24.25
-54.56
11
23
1390
2550
45.43
39.028
34.76
30.492
24.09
54.95
23.99
-54.7
12
23
1470
2550
43.73
37.835
33.9
29.975
24.08
55.46
23.48
-54.84
13
23
1550
2550
42.36
36.87
33.21
29.55
24.06
54.09
23.59
-54.98
14
23
1630
2550
41.67
36.381
32.85
29.329
24.04
51.94
23.78
-55.12
15
23
1710
2550
40.78
35.752
32.4
29.048
24.02
50.38
23.35
-55.26
16
23
1790
2550
40.39
35.473
32.19
28.917
24
47.31
23.36
-55.4
17
23
1870
2550
40.21
35.341
32.1
28.849
23.98
44.29
23.18
-55.55
18
23
1950
2550
40.25
35.363
31.89
28.847
23.96
40.25
22.82
-55.69
19
23
2030
2550
40.25
35.357
32
28.833
23.94
40.25
23.05
-55.84
20
23
2110
2550
40.25
35.351
31.99
28.819
23.92
40.25
22.87
-55.99
21
23
2190
2550
40.25
35.342
31.98
28.798
23.89
40.25
22.65
-56.13
22
23
2270
2550
40.25
35.336
31.96
28.784
23.87
40.25
23.35
-56.28
23
23
2355
2550
40.25
35.33
31.98
28.77
23.85
40.25
23.52
-56.43
24
23
2500
2550
40.25
35.318
31.96
28.742
23.81
40.25
23.91
-56.68
25
23
2700
2550
40.25
35.306
31.94
28.714
23.77
40.25
23.96
-57
1
24
40
2650
47.86
40.795
36.08
31.375
24.31
60.72
26.99
-52.8
2
24
120
2650
47.8
40.75
36.05
31.35
24.3
60.74
26.99
-52.9
3
24
200
2650
48.25
41.062
36.27
31.478
24.29
60.34
27.01
-53
4
24
280
2650
48.47
41.213
36.38
31.537
24.28
59.18
26.95
-53.11
5
24
360
2650
48.9
41.511
36.59
31.659
24.27
57.94
26.94
-53.21
6
24
550
2650
49.66
42.034
36.95
31.866
24.24
56.64
26.24
-53.48
7
24
850
2650
50.72
42.761
37.46
32.149
24.19
54.45
25.81
-53.93
8
24
1095
2650
49.39
41.815
36.77
31.715
24.14
52.24
24.43
-54.32
9
24
1230
2650
47.55
40.518
35.83
31.142
24.11
54.2
24.32
-54.54
10
24
1310
2650
46.48
39.766
35.29
30.814
24.1
55.15
24.38
-54.68
11
24
1390
2650
45.41
39.011
34.75
30.479
24.08
55.93
24.2
-54.82
12
24
1470
2650
43.58
37.724
33.82
29.916
24.06
54.52
23.73
-54.95
13
24
1550
2650
41.77
36.451
32.91
29.359
24.04
54.52
23.46
-55.09
14
24
1630
2650
40.82
35.78
32.42
29.06
24.02
53.59
23.36
-55.24
15
24
1710
2650
40.5
35.55
32.25
28.95
24
48.12
23.34
-55.38
16
24
1790
2650
40.05
35.229
32.02
28.801
23.98
47.51
23.42
-55.52
17
24
1870
2650
40.25
35.366
32.01
28.854
23.97
40.25
23.19
-55.67
18
24
1950
2650
40.25
35.357
31.91
28.833
23.94
40.25
23.02
-55.81
19
24
2030
2650
40.25
35.351
32.02
28.819
23.92
40.25
22.97
-55.96
255
Table B.4-continued
i
j
x
(m)
Upper
table
of the
of the
of the
Floridan
phic
surface
of the
of the
(m)
unit
unit heads
unit
Hawthorn
(m)
heads(m)
(m)
heads(m)
(m)
(m)
(m)
2650
40.25
35.345
31.99
28.805
23.9
40.25
22.89
-56.11
20
24
2110
21
24
2190
2650
40.25
35.339
31.99
28.791
23.88
40.25
22.98
-56.26
22
24
2270
2650
40.25
35.333
31.99
28.777
23.86
40.25
23.19
-56.41
23
24
2355
2650
40.25
35.327
31.97
28.763
23.84
40.25
23.52
-56.57
24
24
2500
2650
40.25
35.315
31.96
28.735
23.8
40.25
24.05
-56.83
25
24
2700
2650
40.25
35.3
31.93
28.7
23.75
40.25
23.99
-57.17
1
25
40
2750
48
40.89
36.15
31.41
24.3
60.98
27
-52.92
2
25
120
2750
48.01
40.894
36.15
31.406
24.29
60.94
27
-53.02
3
25
200
2750
48.17
41.003
36.23
31.447
24.28
60.01
27.03
-53.12
4
25
280
2750
48.75
41.406
36.51
31.614
24.27
58.88
27
-53.23
5
25
360
2750
49.03
41.599
36.65
31.691
24.26
57.81
26.96
-53.33
6
25
550
2750
49.94
42.227
37.08
31.943
24.23
55.14
26.13
-53.6
7
25
850
2750
50.87
42.863
37.52
32.187
24.18
54.4
25.78
-54.05
8
25
1095
2750
49.45
41.854
36.79
31.726
24.13
52.48
24.47
-54.43
9
25
1230
2750
47.76
40.662
35.93
31.198
24.1
54.02
24.5
-54.66
10
25
1310
2750
46.41
39.711
35.25
30.779
24.08
55.19
24.37
-54.79
11
25
1390
2750
44.64
38.469
34.35
30.241
24.07
56.13
24
-54.93
12
25
1470
2750
43.02
37.329
33.53
29.741
24.05
54.85
23.71
-55.06
13
25
1550
2750
41.67
36.378
32.85
29.322
24.03
49.94
23.48
-55.2
14
25
1630
2750
40.89
35.826
32.45
29.074
24.01
46.46
23.42
-55.34
15
25
1710
2750
40.43
35.498
32.21
28.922
23.99
48.18
23.52
-55.49
16
25
1790
2750
40.25
35.366
32.02
28.854
23.97
40.25
23.39
-55.63
17
25
1870
2750
40.25
35.36
32.01
28.84
23.95
40.25
23.23
-55.77
18
25
1950
2750
40.25
35.354
32.04
28.826
23.93
40.25
23.1
-55.92
19
25
2030
2750
40.25
35.348
32
28.812
23.91
40.25
22.94
-56.07
20
25
2110
2750
40.25
35.342
32.01
28.798
23.89
40.25
22.93
-56.22
21
25
2190
2750
40.25
35.336
32.02
28.784
23.87
40.25
23.04
-56.37
22
25
2270
2750
40.25
35.33
31.98
28.77
23.85
40.25
23.21
-56.52
23
25
2355
2750
40.25
35.324
31.98
28.756
23.83
40.25
23.58
-56.68
24
25
2500
2750
40.25
35.312
31.95
28.728
23.79
40.25
24.01
-56.95
25
25
2700
2750
40.25
35.297
31.93
28.693
23.74
40.25
24
-57.32
BIOGRAPHICAL SKETCH
Ahmet Dogan was born in Beysehir, Turkey, in 1968. He graduated from the
Middle East Technical University in Ankara, Turkey, with a Bachelor of Science degree
in civil engineering in June 1991. He immediately started working for Tanrikulu
Construction Company as the chief engineer for the construction of a sewerage system in
the town of Kadinhani near Konya, Turkey. In September 1991, he was admitted to the
Master of Science program in hydraulics at the Middle East Technical University. He
earned his Master of Science degree in June, 1993, with a thesis entitled "Flow Around
Bridge Piers." He started working on his Ph.D. study at the same university in the water
resources research group. In 1995, he was awarded a scholarship from the Higher
Education Council (YOK) of Turkey to pursue his Ph.D. in the United States of America.
He was admitted to the Ph.D. program in hydrology/water resources in the Civil
Engineering Department at the University of Florida in the spring of 1995. Every
academic year of his Ph.D. study, he received an award for "Academic Achievement by
an International Student" from the Office of International Studies and Programs in
recognition for earning a cumulative 4.0 grade point average. He also received an award
for "Outstanding Academic Achievement by an International Student" from the College
of Engineering in April 1998. After graduation, Ahmet Dogan will start teaching in the
Civil Engineering Department at the Suleyman Demirel University, Isparta, Turkey.
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