EPA-600/5-74-013
MAY 1974
Socioeconomic Environmental Studies Series
A Water Quality Model for a Conjunctive
Surface-Groundwater System
Office of Research and Development
U.S. Environmental Protection Agency
Washington, D.C. 20460
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RESEARCH REPORTING SERIES
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This report has been assigned to the SOCIOECONOMIC ENVIRONMENTAL STUDIES
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environmental planning techniques at the regional, state and local levels,
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EPA-600/5-74-013
May 1974
A WATER QUALITY MODEL FOR A
CONJUNCTIVE SURFACE-GROUNDWATER SYSTEM
Armando I. Perez
Wayne C. Huber
James P. Heaney
Edwin E. Pyatt
Department of Environmental Engineering Sciences
University of Florida
Gainesville, Florida 32611
Project Number 16110 GEW
Program Element Number 1BA030
Project Officer
Mr. Edmond P. Lomasney
Region IV
U.S. Environmental Protection Agency
1421 Peachtree St., N.E.
Atlanta, Georgia 30309
Prepared for
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
WASHINGTON, D.C. 20460
For s»le by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 - Price $350
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ABSTRACT
Increasing pressures toward efficient decision-making in water pollu-
tion control are creating the need for improved pollutant routing
models. Such mathematical models help understand cause and effect re-
lationships between sources of pollutants and their ensuing concentra-
tions at various locations in a basin.
Considered in this study were both flow and water quality processes
occurring on the ground surface, in the unsaturated soil zone and
in the saturated or groundwater zone. The objective was to improve
already available formulations for the above processes and subsequently
to develop a methodology for interfacing the individual models.
Emphasis was placed on the modeling of agricultural pollution. For
this reason, nitrogen and phosphorous were the main substances con-
sidered. The selection of the Lake Apopka basin in Central Florida
as the study area was made in accordance with these project goals.
The data base for this basin was limited, but better than that for
other candidate areas in Florida. However, this data base appeared
typical of that available to engineers in the analysis of a practical
problem. The scope of this particular study precluded additional
sampling. Current data limitations precluded a complete verification
of the model. However, various general conclusions could be drawn.
In the future, the formulation could be viewed as an instrument for
structuring data gathering efforts.
This report was submitted in fulfillment of Grant Number 16110 GEW,
by the University of Florida under sponsorship of the Environmental
Protection Agency. Work was completed as of March, 1972.
11
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CONTENTS
Page
Abstract ii
List of Figures iv
List of Tables xi
Acknowledgments xii
Sections
I Conclusions 1
II Recommendations 3
III Introduction 4
IV Surface Water Models 10
V Unsaturated Soil Zone Models 31
VI Groundwater Models 51
VII Interfacing of Models 67
VIII Application to the Study Area 79
IX Results and Conclusions 131
X References 166
XI Publications 182
XII List of Symbols 183
XIII Appendices 186
A. Details of Surface Water Models 187
B. Details of Unsaturated Zone Model 201
C. Details of Groundwater Model 225
D. Details of Interfacing Methodology 232
E. Details on Application to the Lake Apopka Basin 236
F. Summary of Additional References on Agricultural
Nutrients 257
G. Listing of Computer Programs 259
iii
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FIGURES
No. Page
1 Vertical cross-section of a typical basin 6
2 Linkages involving various physical locations and
mathematical models 7
3 Location map of the Lake Apopka basin 8
4 Plan view of application site for runoff portion
of the Storm Water Management Model 19
5 Lateral view of overland flow on a catchment 23
6 Illustration of a simple case involving the receiving
water model 27
7 Typical receiving water model grid for two-dimensional
flow 29
8 Typical soil profile including-location of unsaturated
zone 32
9 Graphical representation of vertical flow in the
unsaturated zone 39
10 Illustration of the numerical solution for unsaturated
flow 44
11 Schematic representation of soil system for water
quality formulation 47
12 Vertical cross-section of a groundwater system 52
13 Grid structure superimposed on a groundwater cross-
section 57
14 Details of a groundwater grid system 64
iv
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Figures-continued
No. Page
15 Illustration of the different models applied in
a basin 68
16 Flow diagram fjor the various models, illustrating
data .interfacing 74
17 Schematic representation of the use of overlay
technique with data sets 76
18 Schematic representation of time-averaging process
for interfacing flows between overland flow and
receiving models 78
19 Illustration of the types of models in the executive
program and the use of data sets for interfacing 80
20 Map of the Lake Apopka basin showing approximate
location of drainage boundary and muck farms 82
21 Illustration of the topographic features of the
sandy soils and other hydrologic features in
the Lake Apopka basin 84
22 Typical piezometric contours, (ft), in the Floridan
Aquifer. Lake Apopka is only weakly connected to
the aquifer at Gourd Neck Springs. Hence, piezometric
values are not constant across the lake. Vertical
cross-sections conform to two-dimensional flow within
the Floridan Aquifer 86
23 Directions of horizontal flow components within
the shallow formations, inferred from topographic
data. Also indicated are main and auxilliary cross-
sections selected for analysis 88
24 Lateral view of cross-section AA1, (Figure 23) 89
25 Reproduction of project map containing locations of
various sampling points 91
26 Reproduction of soil classification overlay map 94
27 Illustration of the lumping of two successive
catchments into one 96
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Figures-continued
No. Page
28 Illustration of the topographic contours of the
sandy soils and delineation of overland flow
catchments 97
29 Reproduction of an overlay map showing the layout
of overland flow catchments in the sandy soils 98
30 Approximate location of catchments in the sandy
areas feeding into Lake Apopka 99
31 Map of Zellwood Drainage District showing location
of pumps and flap valves 102
32 Simplified lateral view of vertical section EE1
(Figure 31) in Zellwood Drainage District 103
33 Plan and lateral views of an isolated portion of the
drainage system in the muck farms 104
34 Reproduction of runoff map for the Zellwood Drainage
District in the muck farms 106
35 Runoff map for the Duda Drainage District in the
muck farms 107
36 Reproduction of map showing the different farms within
the Zellwood Drainage District 108
37 Illustration of a typical receiving water model junction
in the muck farm canals 110
38 Map of the Duda Drainage District showing simplified
connections for the receiving water model 111
39 Location of various junctions for the receiving water
model application in Lake Apopka. Also shown are
inputs for the lake simulation 114
40 Cross-sectional view of the unsaturated zone, showing
boundary conditions and other model features 115
41 Partial view of cross-section AA1 (Figure 23), illustrating
the idealization of the unsaturated zone geometry 117
vi
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Figures- continued
No. Pag
42 Estimated pressure head versus volumetric moisture
fraction curve for sandy soils 119
43 Estimated pressure head versus volumetric moisture
fraction curve for muck soils
44 Lateral view of cross-section AA' (Figure 23),
indicating flow boundary conditions 123
45 Assumed permeability (gal/day/ft2) regions in
section AA' (Figure 23) 124
46 Node classification system for groundwater flow in
section AA1 (Figure 23) 126
47 Water quality node classification system in section
AA1 (Figure 23) 128
48 Computational flow chart of program package, showing
the use of data sets 13°
49 Catchment geometry and boundary conditions used in
kinematic wave overland flow model demonstration run 132
50 Predicted outflow hydrograph from catchment number 1
in sandy soils area
51 Catchment geometry and boundary conditions used in
demonstration run of runoff portion of Stormwater
Management Model
52 Predicted outflow hydrograph for catchment 24 in
the Duda District 136
53 Map of Zellwood drainage district showing location
of various junctions 138
54 Predicted stages at junction 57 in Zellwood district 159
55 Predicted flows between junctions 56 and 57 in Zellwood
district 139
vii
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Figures-continued
No. Page
56 Predicted nitrate and phosphorous concentrations at
junction 57 in Zellwood district 140
57 Predicted stages at junction 3 in Lake Apopka 141
58 Predicted flows from junction 3 to junction 8 in
Lake Apopka 141
59 Predicted nitrate concentrations at junction 3 in
Lake Apopka 142
60 Boundary conditions used in a run of the unsaturated
zone model for the sandy soils 144
61 Predicted moisture profiles in the sandy soils at
different time steps
62 Predicted nitrate concentration profiles in the sandy
soils at different time steps i46
63 Predicted heads in cross-section AA1(Figure 23) after
iterations
64 Predicted nitrate concentration contours in cross-
section AA' (Figure 23) after one hour of input 149
65 Predicted nitrate concentration contours in cross-
section AA1 (Figure 23) after one day of input
66 Predicted nitrate concentration contours in cross-
section AA' (Figure 23) after 10 days of input 151
67 Predicted peak flows and peak phosphorous fluxes
in catchment 1 for various rainfall intensities 153
68 Predicted nitrate concentration profile for hypothe-
tical soil under assumed high velocity conditions 154
69 Predicted nitrate concentration profiles for assumed
nigh value of release rate by muck 156
70 Lateral view of a portion of cross-section AA1(Figure 23) 157
Vlll
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Figures-continued
No. Page
71 Predicted recharge velocities resulting from various
increases in water table elevation 158
72 Predicted nitrate concentration contours in cross-
section AA* (Figure 23) after 68.8 days (1656) hours
of input 160
73 Plan and leteral views of a hypothetical catchment 188
74 Hypothetical rainfall rates, infiltration rates and
ponding depths used in overland flow calculations 190
75 Graphical representation of computational steps in
subroutine AVE involving infiltration 196
76 Graphical representation of computations in sub-
routine AVE involving rainfall 197
77 Lateral view of overland flow situation, indicating
catchment divisions : 199
78 Illustration of unsaturated zone profile, showing
nodes and boundary conditions 202
79 Illustration of an isolated portion of an unsaturated
zone system, showing flow model data 212
80 Illustration of an isolated portion of an unsaturated
zone system, showing quality model data - 215
81 Isolated portion of a groundwater grid system, showing
geometry and assumed heads and permeabilities 227
82 Generalized hierarchial representation of overlay
procedure 233
83 Job Control Language (JCL) instructions for overlay
procedure valid on IBM 370/65 computer at the
University of Florida 234
84 List of major interfacing variables for the various
models 235
ix
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Figures-concluded
No. Page
85 Illustration of fertilizer application on a plot in
the sandy soils 239
86 Assumed pressure versus hydraulic conductivity
relationship for sandy soils 242
87 Assumed pressure versus hydraulic conductivity
relationship for muck soils 243
88 Illustration of layout of mole drains 244
89 Illustration of water uptake by citrus 247
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TABLES
1. List of Hydrologic Data for Lake Apopka Basin 251
2. List of Agricultural Data for Lake Apopka Basin 253
3. Water Quality Data for the Lake Apopka Basin 254
4. Miscellaneous Data Used in Computer Programs for
Lake Apopka Basin 256
XI
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ACKNOWLEDGMENTS
This report essentially constitutes the doctoral dissertation of
Dr. Armando I. Perez in the Department of Environmental Engineering
Sciences at the University of Florida. The authors wish to thank
Dr. M.D. Mifflin and Dr. J.F. Burns of the Geology and Industrial
and Systems Engineering Departments, respectively, for their valuable
advice. Within the department, Dr, P.L. Brezonik made many helpful
comments. Student assistants Ray Cribb and Jack Riggenbach performed
data reduction work and Mssrs. W. Alan Peltz and Wilfredo Albanes
conducted numerous programming tasks. Computations were performed
at the Northeast Regional Data Center at the University of Florida.
Numerous individuals from various agencies furnished invaluable assis-
tance in the gathering and interpretation of data for this study.
Among these are Mr. Darwin Knockenmus of the U.S. Geological Survey,
Mr. C.W. Sheffield of the Orange County Pollution Control Department
and Mr. Peter B. Rhoads of the East Central Florida Regional Planning
Council. Personnel of the Central and Southern Florida Flood Control
District provided helpful guidance. Dr. R. Allan Freeze,now at the
University of British Columbia, and Mr. H.N. Holtan of the Agricul-
tural Research Service (USDA) offered very useful insights on model-
ing techniques. Various university departments, especially the
Institute of Food and Agricultural Sciences (IFAS), provided helpful
assistance. The listing of all contributing individuals from these
and other agencies and institutions would make this section inter-
minable.
The support of this study by the Office of Research and Monitoring,
Environmental Protection Agency, is gratefully acknowledged. Mr.
E.P. Lomasney, Project Officer, provided useful guidance through-
out the study.
XII
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SECTION I
CONCLUSIONS
The results indicate that it is feasible to model a conjunctive
surface-groundwater system. Current data limitations in the rather
typical data base available for the study area precluded a complete
verification of the model. However, the formulation can be used as
a guide for data collection in this and other similar areas.
While it is desirable to conduct modeling efforts using a systems
approach, it appears that a close integration of the various indivi-
dual programs would be premature. In their present relatively inde-
pendent status, the programs are more accessible for the introduction
of improvements. Also, the high computation cost and programming
complexities inherent in a fully interfaced approach seem to be un-
warranted in the light of present need for additional data.
Much remains to be accomplished in the usage and interpretation of
agricultural information affecting water quality. A better quanti-
tative insight into the mechanisms of various processes (e.g. soil
erosion, nutrient reactions in the soil) would provide direction to
future sophistication in modeling efforts.
Regarding the study area (Lake Apopka basin, Florida), overland flow
nutrient contribution to the lake appears to be of little significance.
Subsurface seepage of fertilizer nutrients from the sandy soils areas
to the lake is likely to be small, due to the groundwater flow condi-
tions, which tend to transport substances downward and not laterally.
Using the available data, computations indicate that more comprehensive
simulation of the muck farms would require additional information on
flow rates and on the seemingly important muchanism of nutrient release
by the soil. The predicted movement of substances within the lake is
slow, and quality calculations would be much improved by field data
allowing proper consideration of source/sink effects involving the
bottom sediments.
Considering source-sink effects in subsurface water, some preliminary
modeling was conducted for the unsaturated zone (e.g. nutrient release
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by muck soil). However, inclusion of source/sink calculations in
groundwater would have required usage of the scarce data available in
the literature for dissimilar areas. It was felt that further effort
in this regard would not be realistic and would detract from efforts
in other subsystem models.
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SECTION II
RECOMMENDATIONS
In formulating needed modeling improvements, it is suggested that first
each model be considered individually. These modifications should be
carried out in conjunction with data gathering efforts. Work on com-
puter programming of interfacing aspects should follow the verification
and calibration processes.
For improvements in describing surface and shallow subsurface phenomena,
it would be worthwhile to further examine the watershed model developed
recently by the Agricultural Research Service (1971). This ARS model
does not include quality calculations, but its flow routines incorpor-
ate numerous agricultural features (e.g. soil medium discontinuities
and the effect of plants) which are pertinent to sediment transport
(thus, also to transport of nutrients).
In agricultural areas where groundwater flow conditions change slowly,
it seems that the vertical, two-dimensional groundwater model applied
to selected cross-sections provides satisfactory insight. Careful
consideration should be given to computation cost prior to utilizing
a three-dimensional formulation. Additional data on source/sink pro-
cesses in groundwater are an essential requirement for better pre-
dictions.
The receiving model, used in connection with the muck farms and Lake
Apopka, is quite flexible and versatile, and should be widely applica-
ble. Its predictive quality could be improved by data allowing proper
consideration of nutrient cycling phenomena, especially those involving
sediments.
It appears that quality data collection work in the Lake Apopka basin
should center on the release of nutrients by the muck soil and on
sediment-water interchanges in the lake. Better information on the
various flows (i.e. drainage pumpage and irrigation records) occurring
between the lake and the muck farms would allow more precise answers
on nutrient fluxes.
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SECTION III
INTRODUCTION
OVERVIEW
In recent years, increasing pressures toward efficient decision-making
in water pollution control have created the need for improved pollu-
tant routing models. Such mathematical models help determine cause
and effect relationships between sources of pollutants and the ensuing
concentrations at various locations in a basin. A better knowledge
and understanding of these relationships should, hopefully, promote the
expenditure of economic resources where they are most needed.
A basin-wide study of water quality must take into account the various
hydrologic processes such as rainfall, runoff, infiltration and ground-
water flow that occur in nature. These processes determine the spatial
and temporal distribution of water in the basin. As the water moves
within the system, it is subjected to a set of physical, chemical and
biological driving forces which affect its quality characteristics.
Numerous models have already been developed to simulate the above
hydrologic and water quality processes. However, many of these for-
mulations have been produced by researchers working independently
within their own area of interest, such as meteorology or agriculture.
For this reason, much work remains to be done in modifying these
individual models to render them useful for the unifying purpose of
predicting water quality changes in a basin. It also is apparent
that much remains to be accomplished in the area of interfacing these
models once they are made compatible.
The objectives of this investigation were two-fold: improving exist-
ing formulations for the various processes occurring in a basin and
developing a methodology for interfacing these formulations. Additional
information on these goals is presented following a description of
the physical system.
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DESCRIPTION OF THE PHYSICAL SYSTEM
Figure 1 constitutes a vertical cross-section of a basin, and indicated
therein are some of the physical features and hydrologic processes
which are of interest in this study. The movement of water in the
system can be predicted using various hydraulic relationships. These
relationships utilize values of initial and boundary conditions (e.g.,
rainfall) to determine the magnitude and direction of flow velocity
at different locations. Associated with,the movement of water are
physical, chemical and biological processes which affect its quality.
The processes acting upon a given particle of water are largely deter-
mined by the particle's location in the system. For instance, scour-
ing and deposition are phenomena occurring in surface runoff, while
dispersion may occur in both surface and groundwater flow.
A common approach was taken in modeling the surface, unsaturated flow
and groundwater phenomena: the flow or quantity aspects were simulated
first, followed by the water quality aspects. At each time step, the
velocities at various physical locations were determined from the hy-
draulic formulations. The calculated velocity field was then used as
input to the water quality models, which routed the movement of pollu-
tants through the system. This sequential approach was required be-
cause certain water quality phenomena (e.g., dispersion) are affected
by velocity.
From Figure 1 it can be seen that the path of any particle of water
may cross different regions which, in turn, involve different models.
This fact points out the need to formalize methods for utilizing these
models sequentially and for transmitting information from one model to
another. Some of these interfacing aspects are depicted in Figure 2.
The boxes in this figure represent physical locations in a basin, which
may be sources or sinks of pollutants. The arrows denote movement of
these pollutants among the various locations. Two major items are
considered in routing this movement: flows and pollutant concentrations.
A more detailed explanation of the interfacing methodology is presented
later.
OUTLINE OF WORK
The resulting computer models were tested in the Apopka basin, a pre-
dominantly agricultural area in Central Florida. The location of this
basin is shown in Figure 3. In this report, discussions involving the
development and discussion of mathematical techniques precede those
pertaining to the application phase of the study.
For organizational purposes, both the hydrologic and water quality
models were classified according to the major physical location where
they occur, namely: surface, unsaturated soil zone and groundwater
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f H T
PRECIPITATION
UNSATURATED INFILTRATION
ZONE
SURFACE
BODY OF
WATER
GROUND WATER
FLOW
SATURATED
ZONE
UNSATURATED
DISCHARGE ZONE
Figure 1. Vertical cross-section of a typical basin.
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GROUND SURFACE
Processes)
Fertilizer inputs
Animal waste inputs
Infiltration
Pollutant
Concentrations
UNSATURATED
ZONE
Processes'
Advection
Dispersion
Source-sink
Decay
Recharge
or
Discharge
Pollutant
Concentrations
Overland Flow
Pollutant Concentrations
SURFACE BODY
OF WATER
Processes)
Advection
Diffusion ,
Source-sink
Decay
Recharge
or
Discharge
Pollutant
Concentrations
6ROUNDWATER
(SATURATED)
ZONE
Processes
Advection
Dispersion
Source-sink
Decay
Figure 2. Linkages involving various physical locations and
mathematical models.
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00
C DRAINAGE
BOUNDARY
CITY OF
WINTER GARDEN
Figure 3. Location map of the Lake Apopka basin.
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zone. One section is devoted to each of these three classes, in the
order above. Each section contains a brief survey of the pertinent
literature (for a more comprehensive review refer to Perez's disser-
tation (1972)) and discussions on the selection of the formulations
used and the corresponding modifications introduced. In most cases,
mathematical complexities and lengthy derivations are relegated to the
appendices which also include sample calculations. In the discussion
of all major formulations, the consideration of flow or quantity pro-
cesses will precede that of water quality aspects.
Following the three sections covering the individual models, a section
is provided to discuss the interfacing methodology. Subsequently, the
study area description and other application aspects are presented.
The last section is devoted to the results and conclusions.
Although the methodology developed herein is intended to be applicable
to any basin, primary emphasis in this study is placed on agricultural
pollution and on the routing of nitrogen and phosphorous. Thus,
water quality discussions in the following chapters will be centered on
these two substances. Also, discussions on agricultural pollution cal-
culations, data usage and other matters related to agriculture are
included in various appendices. It is strongly suggested that before
readers attempt to follow the sample model calculations and computer
program summaries provided in Appendices A, B and C, they be familiar
with the respective mathematical formulations explained in Sections
IV, V and VI.
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SECTION IV
SURFACE WATER MODELS
GENERAL DISCUSSION
Various hydrologic phenomena occur on the surface of any basin, namely:
(1) precipitation, (2) evaporation, (3) evapo-transpiration, (4) over-
land flow or runoff and (5) flow in surface bodies of water (e.g., lakes
and streams). Consideration of the first three phenomena above, and
their associated water quality processes, is usually not a major problem
for two reasons. Firstly, better data generally exist for these items
than for other surface and subsurface phenomena. Secondly, their com-
putations and mathematical formulations are relatively simple.
For these reasons, the attention paid in this study to precipitation,
evaporation and evapo-transpiration is somewhat limited. In this sec-
tion, brief literature reviews and discussions on the significance of
these topics are included. In the application phase of the project, the
available data were coupled with simplifying assumptions about the be-
havior of these processes.
On the other hand, the processes of runoff and flow in surface bodies
of water are covered extensively herein. Consideration of these pro-
cesses begins with a discussion of the types of variables involved and
a review of the literature. The discussion then covers basic formula-
tions and modifications used in this study.
The following subsections cover each of the five hydrologic phenomena
listed above, in the same order. For each phenomenon, discussions on
quantity aspects generally precede those involving water quality,
PRECIPITATION
Quantity Aspects
The phenomenon of precipitation includes both rainfall and snow, A
recent publication by the U. S. Department of Agriculture and Depart-
ment of Commerce C1969) provides monthly precipitation probabilities
10
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for various climatic divisions in this country. In many states, such
as Florida, the regional and temporal variability of precipitation has
warranted the compilation of frequency data (Butson and Prine, 1968,
U. S. Weather Bureau, 1961).
Even within a specific geographic region, the measurements at various
gaging stations have been found to differ significantly, depending
upon the type of storm and the local topographic conditions. Conse-
quently, the average precipitation on a given area is usually computed
by assigning certain weighting factors, commonly known as Thiessen
weights, to the measurements at the various gages (Linsley and Franzini,
1964). A computerized method has been developed by Diskin
Another application of computer models in hydrology is in the area of
rainfall synthesis, in which the production of synthetic or artificial
records is accomplished through the coupling of actual records and
statistical techniques (Yevdjevich, 1972). A model for synthetic
streamflows has been discussed by Fiering (1971).
It might be pointed out that while the studies discussed above apply
to precipitation in general, the behavior of snow after it reaches
the ground differs markedly from that of rainfall (Eagleson, 1970).
Much work has already been performed in the analysis of the meteoro-
logical variables affecting the behavior of snowpack and the produc-
tion of snowmelt. For this reason, these relationships are not con-
sidered in this investigation. Furthermore, the location of the study
area in Florida precludes any verification of snowmelt formulations.
Quality Aspects
The concentration of any substance in rainfall depends upon a myriad
of factors. One of these is the intensity-duration relationship of
the storm. As a general rule, the concentrations measured in the
rain decrease exponentially with time. The first rain droplets remove
a large amount of gaseous and particulate matter from the atmosphere,
and carry them downward. The availability of these substances in the
atmosphere then decreases, and therefore their presence in subsequent
rainfall becomes more scarce. Gambell and Fisher have found that the
total rainfall relationships vary with type of storm.
Another major factor affecting the amounts of atmospheric constituents
in rain is the rate at which they are made available by their different
sources. These sources can be natural or man-made processes. For in-
stance, decreasing Cl concentrations in the air with increasing dis-
tance inland suggest to some investigators that the main source Of this
ion is sea water, swept upward by the winds (Weibel £t al_. , 1966) . The
presence of various forms of nitrogen in the atmosphere stems from or-
ganic decay processes in the surface of the earth, photochemical
11
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reactions in the atmosphere and several human activities such as
application of fertilizers. Weibel and co-workers have deemphasized
the significance of the formulation of NOa from lightning. Chemical
reactions within the atmosphere are illustrated by the conversion of
N0~ to NO".
Transport processes also play an important role in determining the
availability of substances in the atmosphere. Pesticides such as
DDT, for example, have been found to be carried by the wind for long
distances (Weibel ejt al., 1966). These pesticides can exist in vapor
form or they can be adsorbed on particulates, Their physical state
affects their diffusion rates.
In short, the concentrations of substances in rainfall are influenced
by man, physical and chemical reactions and meteorological conditions
in the atmosphere. Unfortunately, these factors are not well under-
stood from a quantitative standpoint. Consequently, the development
of mathematical models or even substantive regression equations to
describe these processes would appear to be premature.
Instead, the approach taken in this study is to demonstrate the use
of rainfall quality data collected in the vicinity of the study area.
The quality of rainfall should not be ignored in this or any other
basin. Weibel et_ al. (1966) for instance, have found that ammonia and
nitrate concentrations in rainfall are sometimes higher than the thres-
hold values for algal nuisances.
EVAPORATION
Quantity Aspects
The loss of water from surface bodies to the atmosphere is generally
important in a basin-wide hydrologic balance. Evaporation computations
are applied to relatively large lakes and streams which exist on a
perennial basis. Temporary or small accumulations of water on soil
fall within the realm of evapo-transpiration formulas. These formulas
are discussed in the following subsection.
It is of interest to explore first the types of variables involved in
the mechanics of the evaporation process. This discussion will be
followed by an examination of field measurement considerations, and
finally by a discussion of water quality effects.
The basic relationship governing the rate of evaporation is (Eagleson,
1970):
12
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where E = evaporation rate, in/hr
K = transfer coefficient, 1/hr
e = vapor pressure, in
z = elevation, in
The value of the vapor pressure is in turn affected by other variables
such as salinity and total atmospheric pressure. In fact, the inter-
actions between meteorological variables are so complex that it is
difficult in practice to compute the effect on vapor pressure of any
single parameter.
Commonly, evaporation estimates are made in the field using pans. This
process involves three phases. First, the actual pan evaporation losses
are measured and recorded, Secondly, the basin-wide evaporation losses
are estimated from measured values of all the other remaining variables
in the water budget. Finally, a pan coefficient (proportionality fac-
tor) is found which will best relate the two results (Thornthwaite and
Mather, 1955; Eagleson, 1970; Kohler, 1954).
Since the primary goals of this investigation were to emphasize the
regional water quality transport processes, no mathematical formulations
were attempted herein. The use of pan data in actual evaporation compu-
tations is discussed later.
Quality Aspects
Water quality effects of evaporation are basically the following:
(1) concentration changes in the water produced by changes in volume,
(2) evaporation of non-ionic substances and (3) changes in the heat
content of lakes and streams, which affect chemical and biological
reaction rates. The first item can be handled adequately by simple
mass-balance computations and therefore one may proceed to examine the
remaining two items.
Ions tend to remain in solution and do not escape with the evaporating
water. The only significant non-ionic substance subject to evaporation
is ammonia (Stratton, 1969). The relative amounts of the gaseous and
ionic forms of ammonia are controlled by the equilibrium reaction
NH3 + H20 £ NHt + OH" (2)
Thus as the pH increases, the amount of NH9 dissolved in water also
increases. Ammonia will evaporate if its partial pressure in the
water is greater than that in the atmosphere. For this case, a first-
order reaction equation has been proposed to describe the transfer.
13
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Predictive efforts attempted by Stratton were not very successful,
however.
Regarding heat losses from impoundments due to evaporation, one widely
used technique is that developed by Edinger and Geyer (1965). Huber and
Perez (1971) discuss other formulations, including Kohler's (1954)
relationship based on equation (1). In areas where evaporation measure-
ments are available, a simpler calculation may be conducted using the
amount of water lost and the latent heat of vaporization. Evaporative
heat losses will cause temperature drops in the water. As a rule,
chemical and biological reaction rates tend to decrease after such
losses.
EVAPO-TRANSPIRATION
Quantity Aspects
The transfer of water from the soil and plants to the atmosphere is
known as evapo-transpiration. Due to the great variety of soil and
plant characteristics, the direct computation of this transfer can
become burdensome. For this reason, hydrologists often compute all
other items in the water budget first and then obtain evapo-transpira-
tion by subtraction. This is generally a satisfactory method when the
land use patterns in the basin are essentially static. If this is not
the case, particularly if heavy irrigation is contemplated, a direct
approach is preferable. Some direct methods have been developed by
Penman (1948), Thornthwaite and Mather (1955, 1957) and Freeze (1967).
In 1962, Blaney and Griddle dealt with irrigation water requirements,
which are significantly affected by evapo-transpiration.
Numerous studies are available on the estimation of evapo-transpiration
rates for specific geographic regions. In the State of Florida, one
recent paper has been that by Hashemi and Gerber (1967) on evapo-
transpiration from citrus areas. Another investigation, carried out
by Stewart e^ al^. of the USDA Agricultural Research Service (1969),
deals with the effect of plant density and water table depth on evapo-
transpiration. As in the case of rainfall and evaporation, little
emphasis was placed in this study on the use of sophisticated formula-
tions for evapo-transpiration.
Quality Aspects
The water quality implications of evapo-transpiration cover basically
just one of the factors previously mentioned in connection with evapo-
ration from impoundments: the loss of ammonia to the atmosphere. Its
extent would be largely determined by the pH of the water in the soil
and in plants. Little information is available on this subject.
14
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OVERLAND FLOW (RUNOFF)
General Considerations
When rainfall arrives at the ground surface, a portion of it penetrates
the soil, another portion evaporates and the rest becomes overland
flow or runoff. This separation process is quite complex, and reason-
able success in simulating it necessitates proper interfacing of the
mathematical models for the individual phenomena involved. For
instance, data such as ponding depths and infiltration rates are
commonly used in both runoff and infiltration formulations.
Numerous overland flow models are available in the literature, ranging
from the theoretical to the semiempirical. A brief review of the major
types of models is given below. Additional discussion of these may be
found in Perez's dissertation (1972).
LITERATURE REVIEW
Background
The characteristics of the numerous available runoff models are so
varied that an effective system of classification for purposes of
discussion is rather elusive. Instead, the approach taken herein is
to present some representative models in the approximate chronological
order in which they were created.
Quantity Aspects^
Much of the early work in runoff modeling was prompted by developments
in the understanding of the infiltration process as presented by Horton
(1940), Phillips (1957) and Holtan (1961). In 1964, Overton compared
the formulation of the above investigators and summarized their simi-
larities. It recent years, other infiltration formulations (Freeze,
1969) have emphasized a numerical solution involving Darcy's Law. These
and other aspects are covered in greater depth in the chapter on un-
saturated zone models.
Infiltration relationships are now being combined with surface flow for-
mulations to produce comprehensive watershed models, e.g., Holtan and
Overton (1964), Muggins and Monke (1968) and Onstad and Brakensiek
(1968). In 1970, Sinha and Lindahl presented a paper on an operational
watershed model which applied Nash's (1957) technique for routing over-
land flow through surface reservoirs. In a related paper, Glymph,
Holtan and England (1971) discussed the watershed model developed by the
Agricultural Research Service (USDA) and particularly its application to
various study areas in the United States.
15
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Much research has been conducted recently in the areas of hydrograph
analysis and optimization of hydrologic parameters (Overton, 1968).
Also, James et al. (1970) developed a self-calibrating version of
the Stanford Watershed Model (Crawford and Linsley, 1966), and titled
it the Kentucky Watershed Model. The latter formulation is written
in FORTRAN language. One drawback of both the Stanford and Kentucky
models is that they utilize a large number of variables, many of which
cannot be measured directly.
Most of the above mentioned models were intended for application in
predominantly rural areas. However, some models have been developed
which are better suited for urban situations. One example is the
Stormwater Management Model, formulated by the University of Florida,
Metcalf and Eddy, Inc. and Water Resources Eng., Inc. (Environmental
Protection Agency, 1971). Basically, the runoff portion of this model
utilizes Manning's equation to compute overland flow and Horton's equa-
tion to determine infiltration.
Another significant development in the field of runoff modeling has
been the use of the kinematic wave approach along with the method of
characteristics (Eagleson, 1970). As its name suggests, this approach
determines the propagation of flood waves or disturbances down a catch-
ment. A characteristic is defined as the path, in the horizontal and
vertical directions, traveled by a given disturbance. Eagleson's
analytical formulation assumes constant rainfall and infiltration.
Improvements have been proposed by Eagleson and Maddaus (1969), Bravo
et_ al. (1970) and Harley, Perkins and Eagleson (1970).
Quality Aspects
Two main processes affect runoff water quality: (1) uptake of soluble
substances by the water, and (2) scouring and deposition of solids.
Very few of the overland flow models contain any water quality formu-
lations. One exception is the Stormwater Management Model. The re-
lationships utilized in this model are rather simple, but they are com-
mensurate with the accuracy of the water quality data that are typically
available.
.Selec_tipn__of_ Runoff Models
Two basic conditions were utilized to select the types of runoff models
to be used in this study. The first was that the number of variables
should be small, and just sufficient to properly represent the natural
system. The purposes of this condition were to avoid the use of -
extraneous variables with little physical significance and to minimize
computation time. This condition eliminated both the Stanford and
Kentucky models from use in this investigation. Since the emphasis
herein is on development of a methodology, it was thought that the
16
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calibration and modification of either of these models would be cum-
bersome and lie beyond the overall project objectives.
The second condition used was the availability of water quality formu-
lations in the runoff models considered. This condition was seldom
satisfied, as mentioned above, and therefore the relative ease with
which quality aspects could be added to the available quantity models
was investigated. As a result, the watershed models of the type
developed by the Agricultural Research Service were eliminated from
consideration for use. This decision was made with some reluctance
since these model types, especially the ARS model (Agricultural Research
Service, 1971), contain many desirable features. They are based on data
collected routinely by agricultural agencies, such as soil characteris-
tics, and their results have been verified in practice. However, the
present structures of these models are not well-suited for the systema-
tic routing of pollutants in time and space. It was felt that the
changes required to achieve this purpose would extend beyond the scope
of this study.
In addition to these two conditions, another factor was taken into
account: the applicability of the available models to predominantly
urban or rural situations. In view of the lack of a general model to
cover both cases, and the apparent difficulties in developing one, the
decision was made to use two runoff models in this study.
The Runoff portion of the Stormwater Management Model was selected for
application in urban areas and in agricultural regions in which the
natural hydrology has been profoundly affected by man. These man-made
effects include channelization, pumpage of water and other aspects of
artifical drainage. The structural characteristics of this model, such
as numbering systems for catchments and corresponding receiving ele-
ments, were found to be useful for this type of application. Also the
model considers water quality processes.
On the other hand, the modeling of overland flow in natural rural water-
sheds was carried out using the kinematic wave approach formalized by
Eagleson (1970) and others. To apply this method, it is first necessary
to divide the study area into catchments. The physical information on
these catchments, including length, width, slope and Manning's rough-
ness factor (see Appendix A), is then tabulated. Finally, the model
computes the ponding depths, velocities and flows at different locations
within the catchments. In the quality calculations, these velocities
are used to compute the erosion of soil particles. The uptake of dis-
solved substances by the flowing water is also considered.
17
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Runoff Portion of the Storm Water Management Model
Background —
This runoff model was developed as a part of the comprehensive Storm
Water Management Model (Environmental Protection Agency, 1971). The
four volume report on this comprehensive model, including a user's
manual, provides good documentation on formulas, flowcharts and pro-
gramming methodologies. For this reason, the discussions below are
restricted to the fundamental quantity and quality aspects of this run-
off formulation. In its original form, the runoff program simulates
the flow over catchments and then into gutters and through pipes, as
shown in Figure 4. In this study, only the runoff flow over catch-
ments is of interest and the latter aspects will not be discussed.
Quant i ty__A_sp_e_c_t s —
The quantity or flow calculations involve the rainfall hyetograph,
infiltration rates and runoff flows. A set of sequential computations
is carried out at each time step within the period of analysis.
The first operation carried out at a given time step is to add the
proper incremental amount of rainfall to each catchment in the study
area. Infiltration is then computed for each catchment using one of
two possible methods. In the original model, the infiltration rates
are calculated from Morton's (1940) exponential function, of the form
It - fo + (f£ - £0) e"at (3)
where I = infiltration rate for a given catchment at time t, in/hr
fo = final infiltration rate, in/hr
f. = initial infiltration rate, in/hr
a = a constant, 1/hr
t = time, hr
An alternative approach, used in this study, is to obtain the infil-
tration values from an unsaturated zone flow model. Specifically, the
sequential method utilized herein was to run the infiltration model
(see Section V) first, the results of which were stored for later use
in the runoff formulation.
The next operation performed by the runoff model is to compare the
current water depth (ponding depth) at each catchment to a specified
detention depth. If the former exceeds the latter, the outflow rate
from the catchment in question is computed using Manning's equation.
To satisfy continuity of mass, the Newton-Raphson iterative technique
18
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CATCHMENT BOUNDARY
RUNOFF CATCHMENT
Figure 4. Plan view of application site for runoff portion of the
Storm Water Management Model.
-------
is utilized to determine compatible values of water depth and outflow
rate. The remaining quantity computations involve flows in gutters and
pipes, which are not relevant to this study.
(Duality Aspects —
In its original form, the runoff quality model considers three pollu-
tants: suspended solids, BOD and MPN bacterial counts. It assumes
that these pollutants are conservative, due to the short transport
times involved. In this study, emphasis was given to agricultural
rather than urban runoff and therefore only suspended solids were
considered. Furthermore, the urban-oriented formulation for this
parameter was modified to improve its validity in rural areas.
One basic assumption used in the urban runoff model is that the rate
of suspension of any pollutant at any given time is proportional to
the mass of pollutant remaining on the ground at that time, thus imply-
ing a first-order reaction process. Accordingly, the governing equa-
tion becomes
P « P0e"kt (4)
where P • mass of pollutant on catchment at time t, Ib
P0 » mass of pollutant on catchment at the beginning of
the rainstorm, Ib
k » a rate parameter, assumed to vary in direct proportion
to the rate of runoff, 1/hr
t = time, hr
The mass suspension rates computed from equation (4) were then con-
verted to fluxes by dividing by the corresponding runoff flows and
applying a conversion factor. It might be pointed out that in an
urban area the value of PO» above, can be related to factors such as
land use and street cleaning practices.
An examination of any rural area, however, reveals that some assump-
tions of the formulation require revision. First, it would appear
that the availability of soil for erosion is, for practical purposes,
unlimited and not dependent upon erosion rates in the immediate past.
Thus, the assumption of a first-order reaction process may not be
justified. Also, it seems that erosion would occur only if a critical
scouring velocity is exceeded. For these reasons, equation (4) was
replaced by other equations. These revised equations are very similar
to those developed for the erosion formulation in the kinematic wave
20
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model, and to avoid duplication a detailed explanation will be presented
in the discussion of that model.
In agricultural areas, the main pollutants usually considered are
nitrogen and phosphorous, since these substances are contained in
fertilizers. It is interesting to note that recent studies (Stanfprd,
England and Taylor, 1970) indicate that while nitrogen is readily
leached through the soil, predominantly in the form of nitrate, phos-
phorus tends to become fixed to the soil particles, thus becoming
available for transport through runoff. Accordingly, only phosphorous
is considered in the erosion formulation.
Kinematic Wave Runoff Model
General discussion —
As indicated previously, this model is intended for use primarily in
rural watersheds where man's impact, through such features as drainage
canals, is of little significance. The mathematical formulation of
this model is based on Eagleson's (1970) kinematic wave relationships.
For detailed information on this topic, the reader is referred to
Eagleson's text.
The main advantage of the kinematic wave method is that it constitutes
an analytical solution for a given set of data inputs, and therefore
no numerical or iterative techniques are required. However, a price is
paid in terms of restrictions that must be placed on the input data.
The basic restriction is that the values of rainfall and infiltration
be time-invariant. Thus, for any period of study, average rates of
these two processes must be used instead of the actual values. As a
result, the runoff hydrographs at the various catchments become dis-
torted with respect to their true shape. For the purposes of this
study, this distortion was deemed acceptable in view of: (1) the damped
nature of rural runoff dynamics and (2) the lack of need to simulate
the effect of man-made responses on a short-term basis, such as arti-
fical drainage through canals.
To handle water quality aspects, several additions were made to the
original kinematic wave formulation. The most significant of these
was the provision for computing flow velocities at various points in
each catchment instead of just at the downstream outlets, thus obtain-
ing a better sample of these values. These velocities were then,used
to compute soil erosion rates.
Quantity aspects —
Preliminary comments.— When rainfall occurs in a given region, its rate
often exceeds the infiltration rate into the soil. If this is the case,
water will accumulate on the ground surface until the gravitational
forces overcome the forces resisting motion, and flow begins to take
21
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place. The nature of this flow depends upon the geometry of the
terrain. In relatively small and flat catchments, the predominant
phenomenon is the thin-sheet flow generally designated as overland
flow. Usually, the outflow from these catchments is then routed to
small stream channels, and subsequently to higher order streams. In
this particular study, the modeling efforts were directed only to these
overland flow phenomenon. An illustration of this type of flow and
some of the variables affecting it is given in Figure 5.
After consideration of the predominant forces acting on the water, the
overland flow formulation becomes that for steady, uniform flow in a
wide channel, namely
TO = Yy sin 9 (5)
where TO = bottom shear stress, lb/ft2
y = unit weight of water, lb/ft3
y = water depth, ft
6 = angle of bottom slope
In turn, T0 is a function of the Reynolds number and surface roughness
factor, in a manner depending upon whether the flow is laminar or
turbulent.
The formula used to compute the flow at a given location in a catch-
ment is, then,
m ,,x
q « ay (6)
3
where q = flow per unit catchment width, ft /sec/ft
m = a constant, found to be 5/3 for turbulent flow
(sin 6) /2 for the above value of m, ft /3/sec
n = Manning's roughness coefficient
a «
The mean velocity of flow (ft/sec) can be obtained by dividing the
flow per unit width, above, by the water depth. If this operation is
carried out, the resulting expression for velocity becomes
v = ay"1'1 (7)
22
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RAINFALL=i
K>
f I GROUND
V t
INFILTRATION =f
ADVANCING -^
WAVE FRONT!
RECEIVING
ELEMENT
Figure 5. Lateral view of overland flow on a catchment.
-------
It follows from equations (6) and (7) that it is necessary to determine
the water depth prior to computing velocity and flow. This is a signi-
ficant factor from the standpoint of structuring the computer program.
The objective of the formulation is basically to predict the velocities
at different times and locations in a given area. For this purpose, the
overland flow is assumed to have the characteristics of a kinematic
wave. The mathematical properties of this wavelike motion are obtained
by the method of characteristics. In essence, this technique estab-
lishes the path (in time and space) of wave propagation, known as the
characteristic curve, along which certain relationships hold. The
variable specifically considered in the computer program is y, the
flow depth.
Quality Aspects —
Two basic phenomena are dealt with in the model: suspension and
dissolution. For the former, two different modeling approaches were
considered.
The first approach was based on the concept of critical scouring
velocity. This critical velocity is computed as follows (Einstein,
1964 Environmental Protection Aenc 1971 :
veocty. s crca veocy s compue
1964; Environmental Protection Agency, 1971)
v = R [kcs 1)d]
^ 1 1 J
V2
where v = critical scouring velocity, ft/sec
c
n = Manning's roughness coefficient
R = hydraulic radius, assumed herein to be equal to the
depth of water, ft
k = Shield's coefficient
S = specific gravity of soil
o
d = diameter of soil particle, ft
If the flow velocity is less than vc, no scouring takes place. If the
opposite is true, scouring occurs and the uptake rate of a substance by
the overland flow is set equal to a predetermined value, P^. This
formulation was abandoned due to the difficulty in estimating an upper
limit to this uptake rate.
Instead the approach used involved the duBoys sediment load formula
(Einstein, 1964) :
24
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T ^
c; . _££
"b y J
where q = flux of sediment, Ib/sec/ft of width
s
y = water depth, ft
ijj = a constant for a given bed composition
TO = bed shear stress, Ib/sq ft = y yS,
S, = bed slope
T = critical shear stress, lb/ft2
cr
y = unit weight of water, lb/ft3
The computation of the phosphorous content of sediment loads is dis-
cussed in other portions of this work.
The formulation for the uptake rate of dissolved substances by the
overland flow consists of a first-order reaction process, in which the
rate decreases with time. It may be noted that the flux of substances
at the downstream end of a catchment at any time is a result of prior
events which occurred at upstream locations. For this reason, a time
lag adjustment factor was used in the dissolved substances calculations.
Summary of Computer Program —
The main purpose of the kinematic wave program, known as KWAVE, is to
compute runoff outflows from catchments into receiving elements. These
receiving elements can be either large bodies of water, such as lakes
and rivers, or simply surface depressions which are occasionally filled
with water. The hydrologic data inputs to the program are the values
of rainfall, evaporation and infiltration at the various time steps
within the study period considered. As mentioned previously, to have
manageable equations it is necessary to determine the time-average
values of the above variables, which are then used instead of the raw
data.
In addition to hydrologicaldata, the program requires geometric infor-
mation on each of the catchments, which are delineated in an approxi-
mately rectangular fashion. This information includes average length and
width, slope and hydraulic friction factor (Manning's n). Also, a num-
bering system that links the subcatchments to the corresponding receiving
bodies of water is provided.
25
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The program has the capability of routing dissolved and suspended sub-
stances from the catchments into the receiving elements at every time
step. For this purpose, the rates of dissolution and suspension must
be computed from some basic data. 'These data include nutrient concen-
trations in fertilizers and their application rates, and effective
diameter of soil particles.
Additional details on both the flow and quality computations in the
kinematic wave model are provided in Appendix A. These discussions
include sample calculations and descriptions of the various subroutines.
RECEIVING WATER MODEL
Quantity Aspects
The hydrodynamic response of any body of water is determined by two
fundamental relationships: the equation of motion and the equation
of continuity. The quantity program solves these two equations simul-
taneously in simulating the effects of hydrologic inputs to the receiv-
ing waters. In most practical situations, which involve time-varying
hydrologic inputs (i.e., boundary conditions) and other complexities,
an analytical solution of the above equations would become unwieldy.
The approach taken is, therefore, to utilize numerical techniques in
which the values of the variables are only considered at certain points
in space and time.
An illustration of a simple hypothetical situation involving the receiv-
ing model is given in Figure 6. In this situation, the urban runoff
from a city is discharged at a single location. Also, the river is
divided into a number of reaches. The values of the hydraulic variables
are then computed at the nodes dividing the various reaches. It may be
noted that the input hydrograph must be provided in discrete, not con-
tinuous, form.
Hydraulic computations within the river or channel involve a stepwise
iterative procedure. Given the hydraulic heads at two adjoining nodes,
the hydraulic gradient and corresponding flow along the link is computed.
As a result of this flow, the head at the downstream node is altered.
The data required for each reach consist of surface area, width, length,
depth and friction coefficient. At each node, the variables utilized
are head and bottom elevations.
In many bodies of water, such as lakes and bays, the simple channel flow
system depicted in Figure 6 would not adequately represent the actual
situation. Instead, the flow in these water bodies is basically two-
dimensional within a horizontal plane. Accordingly, the basic system
of links (channels) and nodes (junctions) will constitute a grid cover-
ing the entire water body. An example of such a grid system is shown
26
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N)
DOWNSTREAM
FLOW
NOT DISCRETIZED HYDROGRAPH A
ORIGINAL
TRANSFORMED
UPSTREA
BOUNDARY
CONDITION
Figure 6. Illustration of a simple case involving the receiving
water model.
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in Figure 7 for the receiving portion of the Storm Water Management
Model (Environmental Protection Agency, 1971) used in this study. In
this case, the hydraulic computations follow an iterative procedure simi-
lar to that described above. The main complication added is the need
to consider typically several channels in satisfying the continuity
relationship at a given node. This and other factors tend to increase
the computation time significantly.
Conceptually, both the location of the nodes and lengths of the links
are arbitrary, and therefore the general pattern of the grid system can
be irregular. However, from prior computational experience at the
University of Florida, it has been concluded that efficiency is maxi-
mized as the areas delineated by the links approach the form of equi-
lateral triangles. Selection of a fairly uniform link length through-
out the system has also been found useful.
Within these guidelines, the node locations should be selected to lie
as near as possible to the location of natural or man-made boundary
conditions. In Figure 7, for instance, the node in the upper right
hand corner of the bay was located in the vicinity of B, the runoff
outfall point. Also, a tidal flow condition, basically of sinusoidal
form, is provided at the mouth of the bay as shown. Other nodes should
be placed in the vicinity of water quality sampling stations, if any,
to facilitate calibration of the model.
In this study, several modifications were made to the original receiv-
ing model. In the original version, the outflow from a body of water
was calculated using either a ^sinusoidal formula, valid in estuaries,
or a weir equation, applicable to reservoirs. Another option was added
herein; namely, to specify an outflow hydrograph of any shape. This op-
tion is applicable in the case of regulated canals.
Another improvement in the model consists of added flexibility in the
types of inputs that can be provided to a receiving body. In the
initial program package developed for the Environmental Protection
Agency, the body of water could only receive the computed outflows from
the urban runoff model. At the present time, the receiving body can
also be provided flows computed by KWAVE, the rural runoff model. It
may be noted that the location of the catchments in KWAVE should be made
compatible with the location of the nodes near the periphery of the
receiving body. These rural flows must be considered point sources for
modeling purposes.
Finally, it was recognized that quite often small receiving bodies flow
into larger ones. One example would be drainage-canals flowing into
lakes. For this reason, an interfacing methodology was developed for
such a case.
28
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to
<£>
RUNOFF FLOW
TLET
OUTLET FLOW
BOUNDARY CONDITION,
Figure 7. Typical receiving water model grid for
two-dimensional flow.
-------
Quality Aspects
The water quality subroutines of the receiving model consider both
conservative and nonconservative pollutants. In routing these pollu-
tants, the subroutines utilize the hydraulic data generated by the
quantity program and the boundary fluxes. The quality processes formu-
lated are advection, first-order decay and sources and/or sinks. The
mechanism of diffusion was neglected. Nutrient cycling relationships
from an ecosystem standpoint (Chen, 1969; DiToro, O'Connor and Thomann,
1970) were not included.
For a single branch system such as that shown in Figure 6, the basic
finite-difference equation applied at a given junction or node is:
Ac . Ac .
where j = junction number
c = pollutant concentration, dimensionless
t = time, sec
v = velocity, ft/sec
x = distance, ft
k = decay constant, I/sec
S = combined source/sink term, I/sec
Thus the first term in the right-hand side of equation (10) represents
advection, followed by decay and source/sink, respectively. At nodes
where more than two links meet (see Figure 7) , the advection term above
is modified to include weighting factors corresponding to these links.
For additional informatin of the weighting procedure above, and on the
sequential structure of the other calculations, the reader is again
referred to the main source (Environmental Protection Agency, 1971) .
The quality program generates the concentrations of pollutants at all
nodes for every time step.
30
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SECTION V
UNSATURATED SOIL ZONE MODELS
INTRODUCTION
Following its entrance into the soil, water moves through it in a
manner controlled by soil permeability, antecedent moisture condi-
tions and other parameters. At this point, a semantic distinction
can be made. According to McGauhey and Winneberger (1965), the term
infiltration should refer exclusively to the vertical flow of water at
the ground surface. On the other hand, the term percolation should be
reserved to describe the flow at lower points within a soil profile.
The approach taken herein was to develop a formulation to predict flow
and water quality conditions for a soil zone bounded by: (1) the ground
surface at the top, and (2) the water table at the bottom. Such a zone
and its boundaries are shown in Figure 8. The soil zone enclosed by
these boundaries is denoted as the unsaturated zone because, in the
general case, the void space in the soil is not totally occupied by
water. Under some conditions, such as heavy rainfall, portions of the
soil zone defined above may become saturated. Such a situation simply
constitutes a special case of the general formulation. Another special
case occurs in regions where groundwater discharge conditions and high
evaporation rates are manifest in upward flows through the soil system.
It can be seen that the formulation for the unsaturated zone encom-
passes both infiltration and percolation as defined above. Although
these two processes differ with respect to location, their mathematical
representation is the same for practical purposes. For convenience,
only the term infiltration is used in subsequent discussion, in a manner
inclusive of both connotations above. Also, the term unsaturated flow
is often utilized interchangeably with infiltration.
The state of the art in "modeling this process varies according to whether
its quantity or quality aspects are being considered. Since the quantity
and quality processes are intimately related, the ultimate choice of
model must constitute a compromise between the techniques available for
the two aspects.
31
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TT1T4 4 4 4 RAINFALL
°^^Sw*- RUNOFF MODEL
T ;>r ;r T '"I' -'>'H »i» >i///i/
Y77T74I4
^ INFILTRATION
UNSATURATED
ZONE MODEL
WATER TABU
GROUND WATER MODEL
Figure 8. Typical soil profile including location of
unsaturated zone.
32
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Investigations conducted on the flow aspects of infiltration ate quite
varied. One type of approach consists of empirical relationships
derived from field work, such as soil percolation tests (McGauhey,
Krone and Winneberger, 1966). Several researchers (Holtan, 1961) have
developed formulations which now rely strongly on soil data collected
by agencies such as the Soil Conservation Service. Other workers have
obtained more theoretical relationships based essentially on Darcy's
Law and the equation of conservation of mass (Klute, 1952).
In the area of water quality, work has been less extensive. Some de-
tailed field studies have been conducted (Parizek et_ al_., 1967) in
certain geographic regions. Also, some water quality formulations have
been proposed to predict the concentrations of substances in the soil
system (Tanji et^ aj^., 1967). Most of these formulations, however, have
been developed for substances other than nutrients.
STUDIES ON QUANTITY ASPECTS
Field Studies
Much of the hydraulic field work performed in soils has been with
the intention of ascertaining the usefulness of a given soil as a
wastewater treatment system. Sometimes the application method consists
of intensive irrigation of crops. Thus, the maximum rate at which the
wastewater can be applied to the soil is of great importance. Studies
involving these aspects include those by McGaughey and Winneberger
(1965), Frederick (1948), McGaughey, Krone and Winneberger (1966), and
Winterer (1922).
A comprehensive study conducted at Pennsylvania State University
(Parizek e_t al., 1967) gives further insight into the importance of
various geologic formations in site selection. It also provides some
data on pollutant removals by the soil.
Quasi-Theoretical Approach
The formulations classified herein under this category are those whose
basic premise is that the infiltration rate is inversely proportional
to the degree of saturation of the soil. Various researchers have
arrived at this relationship from more complex flow equations, and others
have shown its applicability using soil tests. Gradually these quasi-
theoretical formulations have been expanded to produce comprehensive
watershed models.
In 1940, Horton defined the term infiltration capacity as the maximum
rate at which soil can absorb incoming rain. He stated that this
value decreases exponentially with time approaching a constant value
asymptotically. In 1955, Philip reviewed the governing equations for
33
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infiltration, including Darcy's law and developed modified relationships.
Later Philip (1957) conducted a study of the performance of the Horton
and Kostiakov equations. As a result, he proposed the following infil-
tration equation:
i = st1/2+ A (11)
where i = cumulative infiltration, in
s = sorptivity, in/hr1'2
t = time, hr
A = infiltration parameter, in
Sorptivity is a measure of the capacity of the soil to absorb or desorb
liquid by capillarity. The work of Holtan (1961) also related infil-
tration to the available soil porosity. The relationship proposed was
f - f = a Fn (12)
where f = infiltration rate, in/hr
f = lower limit of infiltration rate, in/hr
a = a constant
F = potential volume of infiltration (a function of time,
P based on porosity and plant root structure)
n = a constant
In 1964, Holtan modified the usage of the above formulation and mentioned
the testing of the improved method in various watersheds in the United
States. In the same year, Overton developed an integrated form of the
basic relationship given in equation (12), and compared it to the equa-
tions proposed by other researchers.
An early version of the watershed model developed by the Agricultural
Research Service was applied to an area in Florida with good results
(Sinha and Lindahl, 1970). Further efforts have resulted in a more
versatile model, which utilizes soil and crop data collected routinely
by agricultural agencies (Agricultural Research Service, 1971).
Since the Agricultural Research Service (ARS) watershed model was
structured around infiltration relationships, other hydrologic flows
34
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such as overland flow are essentially computed indirectly by means of
a budgeting process. The success of the ARS model, therefore, appears
to depend largely on the proper calibration of its infiltration formu-
lation. On the other hand, hydrologic models such as the Stanford
Watershed Model (Crawford and Linsley, 1966) and the Kentucky Watershed
Model (James, 1970) are calibrated primarily using gaged streamflows.
In many areas where soils are highly permeable and subsurface flows are
significant, as in the state of Florida, it would appear that the ARS
model offers greater advantages.
Along with the increased efforts in the mathematical formulation of
infiltration, field data have been collected to help verify and imple-
ment these models. Some examples of such studies are those by Free,
Browning and Musgrave (1940), Stewart, Powell and Hammond (1963), Holtan
et^ al., and Choate and Harrison (1968).
Theoretical Approach
Much of the foundation work in the mathematics of flow through soils
was completed in the 1950s. A 1952 paper by Klute presented the de-
rivation of the flow governing equation using two basic relationships:
the equation of conservation of mass and Darcy's law. In its original
form, the governing equation contained two unknowns: moisture content
and hydraulic potential. Klute pointed out the difficulties in solving
this expression analytically and proposed a numerical solution instead.
Other papers similar to the above appeared in subsequent years. Freeze
(1969), thoroughly delineated both the theory and the numerical methodo-
logy. Since this methodology was used in this study, it is discussed
in some depth below.
Freeze began his formulation by defining the term hydraulic head as:
$ ~ z + 4> (13)
where = hydraulic head, ft
z = elevation head, ft
ty - pressure head, ft
The latter variable is commonly known as tension when its value is nega-
tive. In any system, the water flows in the direction of decreasing
head, and the relationships controlling this flow are of interest.
The first relationship considered was Darcy's law, which for simplicity
is written in the x-direction only:
35
-------
= -K(x,y,z,\JO
(14)
where K denotes the hydraulic conductivity, ft/sec, which under saturated
conditions equals the soil permeability.
The second basic relationship written was the equation of continuity,
for the case of unsteady flow of a compressible fluid, namely
3(pv J 3(pv
9x
3y
3(pv
3z
(P6)
where
p = fluid density, slug/ft
6 = moisture content, %/lOO
v ,v , and v = velocities in the three coordinate directions,
x y
ft/sec
After some simplifying assumptions, such as vertical flow only, homo-
geneity and incoropressibility, equations (14) and (15) were combined
to yield
3z
KOJO
36.
3t
(16)
Some observations can be made about equation (16) above. Firstly, note
that either fy or may be eliminated using equation (13). Secondly,
the difficulty presented by having two variables, say $ and 6, is
apparent. The assumption is commonly made that and 6 are related by
means of a single-valued function, which can be determined experimentally
for a given soil. However, in practice the relationship varies according
to whether the soil is being wetted or dried, that is, a hysteresis
effect exists. Secondly, the hydraulic conductivity is a function of
pressure head (and also, therefore, of moisture content). Data on some
or the relationships above are available for representative soils (Holtan
et_ aJL, 1968),
The approach used in the formulation of an infiltration model is to
first eliminate either ty or 9 from equation (16). Klute's method was
to eliminate 1(1 and use a 6-based equation. On the other hand, Freeze
eliminated 6 and obtained the following equation
36
-------
9
•5T
(17)
where C(i|0 is defined as
dc{)/di|j
(18)
and thus can be evaluated from the moisture-pressure curve for a given
soil.
In the numerical solution, a value of ^ is predicted at various loca-
tions in the porous medium using equation (17) . A graphical represen-
tation of the situation for a one-dimensional, vertical flow is provided
in Figure 9. The various nodes in the soil profile at which equation
(17) is applied are shown. Also indicated are the flow boundary condi-
tions at the upper and lower boundaries. Once the values of 4>at the
nodes have been computed, the corresponding flow velocities can be
readily determined from Darcy's law.
Many other formulations have been developed based on the continuity
equation and Darcy's law, (e.g., Fok and Hansen, 1966). Other authors
have concentrated on specific portions of the general unsaturated flow
problem. For instance, Rubin and Steinhardt (1963) directed their
efforts toward the study of the upper boundary conditions in the soil
profile specifically, ponding.
The approaches discussed so far have assumed the existence of a homo-
geneous soil. In 1962, Hanks and Bowers developed a numerical solution
for flow through layered soils. Remson, Fungaroli and Hornberger
followed in 1967 with another nonhomogeneous formulation. In 1970,
Fok developed rules for modeling layered soil systems.
In 1970, Green et jil_. developed equations for two-phase flow (air and
water) and considered the effect of hysteresis on soil moisture rela-
tionships. A paper by Freeze and Banner (1970) considered these pro-
blems of air entrapment and hysteresis.
STUDIES ON QUALITY ASPECTS
Background
As in the case of flow aspects, available water quality studies range
from field and laboratory work to complex mathematical models. Some
quality data collected in connection with wastewater recharge have been
37
-------
tabulated by Parizek et_ al_., (1967) and McGauhey, Krone and Winneberger
(1966). The balance of the major water quality research in the unsatu-
rated zone may be broken down into three categories: (1) semiquantita-
tive generalizations on the fate of various substances in unsaturated
porous media, (2) developmental mathematical models and (3) applied
mathematical models. The latter two types are based on the type of
flow formulation depicted in Figure 9. The three categories listed
are discussed below,
Fate of Substances
Considering nitrogen first, its chemistry in soils is strongly affected
by biological activities. Nitrogen in organic form or in fertilizers
can be converted to ammonia and then to the nitrate form. Also, nitrate
may be reduced through nitrate to N2 and then lost to the atmosphere by
volatilization.
The degree of mobility of the various forms of N in soils varies con-
siderably. Recent lysimeter studies (Bailey, 1968) have indicated that
more than 99 percent of the soluble nitrogen in percolating water was in
the form of nitrate, less than 1 percent was present as ammonia and only
a trace of nitrite was detected. However, this breakdown can vary de-
pending on the type of soil. In peat and muck soils, the ammonium
form is present in large amounts in the percolating water. The expla-
nation lies in the reducing power of these soils.
As discussed previously (equation 2), the ammonium ion and ammonia can
coexist in water, and their relative amounts depend largely on pH. The
ammonia molecule can be adsorbed by the soil, sometimes with the forma-
tion of NH+. Also, volatilization of the ammonia occurs when its con-
centration is high and the ability of the soil to react with it is low.
Volatilization is generally high from peat and muck soils (Bailey, 1968),
Numerous studies have also been conducted on the leachability of nitro-
genous compounds. In 1939, Benson and Barnette directed their attention
to the movement of fertilizers. Bates and Tisdale (1957) investigated
the leachability of nitrate. Several experiments conducted by Bailey
(1968) revealed that the leachability of organic N compounds was
limited.
Turning now to phosphorous, the behavior of this substance differs
markedly from that of mobile nitrate-nitrogen. Bailey (1968) has
pointed out the large P-fixation capacities of some soils, and stated
that most phosphorous compounds react strongly with the soil, with
very little leaching into the subsurface water system.
Stanford, England and Taylor (1970) have compiled information on
fertilizer use in the United States and its water quality effects.
38
-------
04
-------
The figures indicate that about 90 percent of the total N used was
in the form of ammonia or ammonium salts. Just after application,
therefore, the availability of highly leachable nitrate is not very
significant. The ultimate presence of this ion in subsurface waters
largely depends upon the degree of nitrification.
On the other hand, pollution from agricultural phosphorous occurs mainly
as the erosion of soil material and its movement into surface bodies
of water, and much less as transport in the unsaturated zone. It
appeared, therefore, that modeling of phosphorous should be emphasized
in runoff models and limited in unsaturated flow models.
Developmental Models
One of the earlier papers on water quality modeling in soils was
written by Day and Forsythe (1957) who distinguished between dispersion
and diffusion. They also reviewed Scheidegger's (1954) mathematical
study on dispersion. Other papers that followed included those by
Ashcroft et^ a_l., (1952), Dagan (1971), and Nakano and Murrman (1971).
Bear (1972) presents a comprehensive review of dispersion and many other
aspects of flow in porous media.
One significant conclusion can be drawn from a review of the above
papers; namely, that in most cases, the effect of diffusion is very
small relative to that of dispersion. The only exception to this
rule would appear to be the case where the flow approaches stagnation.
Applied Models
The papers which fall under this classification are those in which
the authors, rather than proposing a new approach, have instead
attempted to adapt existing formulations to improve their predictive
capacity. Some of these are: Miller, Biggar and Nielsen (1965),
Dyer (1965), Warrick, Biggar and Nielsen (1971), Biggar, Nielsen and
Tanji (1966), Tanji et^ al_. (1967), and Oster and McNeal (1971).
DEVELOPMENT OF MATHEMATICAL MODELS
General Comments
The quantity and quality portions of any hydrologic model must be
closely tied together. For this reason, the decisions on what quantity
and quality formulations should be used could not be made independently.
An overall model structure had to be developed which would accommodate
both aspects.
Beginning with the flow or quantity models, it seemed that a choice
could be made between two basic types: (1) watershed models such as
40
-------
that developed by the Agricultural Research Service (ARS) (1971) or
(2) numerical solution of the unsaturated flow equation (Freeze, 1969).
The first choice would offer several advantages; for instance, the fact
that it has been verified at several locations in the United States,
and its strong reliance on factors such as soil classification and crop
type. However, the ARS model has one main drawback; namely, that it
does not include water quality aspects.
The numerical solution approach also would offer some advantages, such
as a more precise treatment of boundary conditions (refer to Figure 9)
and the routing of water through the soil system. On the other hand,
this method tends to overlook some problems encountered in practice.
For example, many soils are not a continuous porous medium but rather
contain appreciable discontinuities such as void spaces. For instance,
it appears that much of the water that enters a cultivated soil moves
down the walls of the plant roots. Furthermore, the soil moisture data
necessary to utilize this approach, especially those on hydraulic con-
ductivity, are often unavailable or erratic.
Turning to quality models, it seems that the most desirable method
to route substances through soils is that based on solving the convec-
tion-dispersion equation numerically. Empirical and statistical rela-
tionships do not provide sufficient accuracy or understanding of the
system. The numerical approach permits a detailed examination of pro-
cesses which occur over short distances, and is flexible enough to allow
the addition or elimination of certain terms, such as those involving
sources and sinks.
For the above reasons, the decision was made to develop the water quality
formulation around the numerical solution of the convection-dispersion
equation. The remaining question was to determine what quantity formu-
lation would be most desirable. A careful review of the ARS model in-
dicated that the task of introducing a water quality submodel of the
above type in it would be beyond the scope of this study.
Accordingly, the quantity model selected was that based on the numeri-
cal solution of the unsaturated flow equation. It is recognized that
this approach is built upon idealized conditions, such as continuity
of the porous medium. Therefore, several features were added to the
basic formulation to remedy these deficiencies. These features include
computation of water withdrawals by plants from the soil at different
depths and also withdrawals through artificial drainage devices.
In both the quantity and quality aspects, the decision was reached to
consider flow in the vertical direction only. The mathematics of a
model involving more than one dimension would have been extremely
complex. Also, in many basins the slopes of the water table are
relatively gentle, indicating the lack of appreciable moisture gradients
41
-------
in the horizontal direction. In a region where this is not the case,
the vertical flow condition can be approached by partitioning the
region into subareas.
Additional details on the flow and quality models are given below.
Quantity Aspects
The flow formulation used herein is based primarily on Freeze's (1969)
work. The principal function of this method is to predict the value of
hydraulic pressure at different locations (nodes) in the soil profile.
This is accomplished by solving the governing equation for unsaturated
flow in finite-difference form. Once these pressure values have been
determined, the corresponding velocities at every node can be computed
using Darcy's law. These velocities are then stored for subsequent use
in the quality calculations.
It is worthwhile to review the mechanics of the finite-difference
approach. The governing equation being solved is equation (17),
discussed previously, which for convenience is rewritten below
__
3z
(19)
in which, again, z = vertical coordinate, ft
t = time, hr
ty = hydraulic pressure (head), ft
K = hydraulic conductivity, ft/hr
y-i flO /+ * • •
C = -7-j- = slope of the moisture versus tension curve,
^ I/ft
S = sink term representing withdrawals of water by
plants and/or artificial drainage devices, 1/hr
The sink term, S, constitutes an addition to the basic equation. It
can be observed that derivatives are taken with respect to two variables:
space and time. The basic approach in the numerical solution is there-
fore to compute the head at a given point in space and time in terms of
head values at other locations in the space and time coordinates.
42
-------
This approach is illustrated in Figure 10. Two time steps are shown,
each having a set of boundary conditions and the same set of nodes.
At any time step, the values of the top boundary variables can be
obtained from hydrologic records. In turn, the value of the bottom
variable, the net recharge, must be obtained by interfacing with the
groundwater (saturated) flow model. For purposes of this discussion,
it is assumed that the n-th time step on the right represents the pre-
sent time. At this time, the values of head at each node must be com-
puted. The heads at the previous time step have already been calculated.
Equation (19) can be rewritten in finite difference form (Freeze, 1969)
and applied to the j-th node in the profile as follows:
,n
At
,n
__
Az
_1_
Az
^ + ^ -
-------
COORDINATE
AXES
2
CO
TIME STEP: n-i TIME STEP.n
Dn~l c*n-l Dn r-n
IV t I J ,
X*x*x"xVxVx/X /?' X X y> x x >"/Y' ^ X x y x x
TIME(t)
.__ ^TjH
j p—*- c? i
NOTATION
9 H R - RAINFALL RATE
E = EVAPORATION RATE
Q = NET RECHARGE
WATER TABLE? 2 <>
2
J v
Figure 1G. Illustration of the numerical solution for unsaturated
flow.
-------
Equation (20) relates the value of hydraulic head at an interior node,
labeled j, to the values at its two adjoining nodes. An equation of
the general form of (20) can be written for every node in the profile.
However, the specific forms of the equation applied at the top and bottom
notes are special cases. The type of relationship used in these cases
consists of Darcy's law applied between the boundary node in question
and its adjoining node, with the corresponding flow set equal to the
corresponding known boundary condition flow.
For a profile containing L nodes (see Figure 10), there exists a system
of L equations, with the same number of unknowns; namely, the head
values at each node.
The system of equations can be written as.
' ' ' ° ' ' ' ° - D 'CD
3 • • • ° ' ' ' ° = D2 (2)
0 .„ 0,
DL
(22)
The values of the coefficients A, B, C and D above can be computed
from expressions which are derived in Appendix B, These derivations
basically consist of regrouping of the terms found in equation (20).
Expressions are also included for the coefficients used for the top
and bottom nodes, which constitute special cases.
The left-hand side of system (22) above is commonly termed a tri-diagonal
matrix. Due to this special form, the system is more easily solvable
by an elimination procedure than by matrix inversion. A Gaussian
45
-------
elimination technique is used at each time step to obtain the desired
set of values of head.
Prior to discussing the computer program for unsaturated flow, it is
pertinent to mention other features of the formulation. One of these
is the handling of a ponding condition at the surface. This situation
arises when the net rainfall rate (rainfall minus evaporation) exceeds
the saturated conductivity of the soil, and can continue for several
time steps. As long as a ponding condition exists, the head at the top
node is set equal to the ponding depth. This in effect constitutes the
upper boundary condition, superseding the application of Darcy's law.
Other significant features, mentioned previously, are the computations
of water uptake by plants and withdrawals through artificial drainage
devices. These calculations, as applied to the Lake Apopka basin, are
illustrated in Appendix E.
Some general sample calculations for the unsaturated zone flow model
are provided in Appendix B. Also included is a description of the
program subroutines.
Quality Aspects
At each time step, the quality formulation utilizes output from the flow
program to route the movement of substances through the soil system.
This output is basically in the form of moisture contents and flow
velocities at each node.
In this investigation, it is assumed that the water quality processes
that occur in the soil do not affect its hydraulic characteristics.
This assumption is not always valid (McGauhey and Winneberger, 1965).
However, it is felt that the assumption is justifiable in the context
of the airms of this study, for various reasons. Firstly, the emphasis
herein is on agricultural pollution in general, in which the phenomenon
of clogging due to the action of microorganisms is much less significant
than that in wastewater recharge. Secondly, there appears to be a lack
of quantitative relationships linking these biological and hydraulic
aspects.
A typical soil profile is shown in Figure 11, This profile is broken
down into nodes. At the present time, assumed to be the n-th time step,
the values of the moisture content and the velocity at each node are
available from the unsaturated zone flow program. Also shown in this
figure are boundary conditions affecting water quality.
The above boundary conditions can be classified as natural, such as
rainfall and evaporation, or man-made, such as fertilizer input. The
water quality implications of the phenomena in the first category have
46
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RAINFALL
EVAPORATION
uuuumi
L-\9
-FERTLI2ER INPUT
KNOWN VARIABLES AT EACH NODE
Oj * MOISTURE AT JTH NODE
J T Vj * VELOCITY AT JTH NODE
ip
NET RECHARGE
Figure 11. Schematic representation of soil system for
water quality formulation.
47
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been discussed previously. As might be recalled, no mathematical for-
mulations were developed in this study for rainfall and evaporation.
The results obtained by Stanford, England and Taylor (1970) have been
used herein to formulate simplifying assumptions about the travel of
substances through the soil. For instance, it is assumed that all the
nitrogen originating from fertilizers and traveling through the soil is
in the form of nitrate. Another tentative assumption made is that all
the phosphorous applied through fertilizers is fixed to the soil.
Turning now to the processes occurring within the soil profile itself,
one consists of changes in concentrations due to changes in the moisture
content with time and space. Another is advection; that is, the trans-
port of substances along with the moving stream of water. The spreading
phenomenon of dispersion is due to the tortuosity of the paths through
which water particles travel in the porous medium. This phenomenon is
generally much more significant than diffusion, which is caused by ther-
mal motion of the molecules. In addition to the above processes, other
physical, chemical and biological phenomena can occur which either add
or withdraw substances from the flowing water. Terms designated as
source/sink are generally used to represent a combination of these
phenomena ,
Accordingly, the governing differential equation for water quality in
a soil profile can be written as:
(60) . -v + KD(V) * S0 - S. (23)
where t = time, hr
6 = moisture content, %/100
c «3 concentration of a given substance, dimensionless
v = velocity, ft
z = elevation, ft
2
KD(V) = dispersion coefficient (dependent upon velocity), ft /hr
S = source, 1/hr
o
S. = sink, 1/hr
48
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The left-hand side of equation (23) above can be expanded to yield
the following expression:
c!i+8— = -v — + K (v) — + S -S (24)
Equation (24) above can, in turn, be written in finite difference form
and applied to the j-th node in the soil profile at the n-th time step.
The resulting expression, after some algebra, is
en - e11"1
n n _n n „ ,^Nnjn
V , C •
_ j
At At 2Az 2Az (Az)
c^
-l , ., sn
* So. - Si..
As in the unsaturated flow formulation, the subscripts and superscripts
on the variables above denote the node and the time step, respectively.
The derivation of equation (25) is given in Appendix B. Also included
in this appendix are derivations for the expressions used at the top
and bottom node, which constitute special cases.
The dispersion coefficients in equation (25) are indicated to be a
function of velocity; a thorough discussion is found in Bear (1972).
The actual formula used to compute the dispersion coefficient is based
on experimental results given by Hoopes and Harleman (1965):
KD = (vo,)
I^c
v
(26)
where V = kinematic viscosity of water (a function of temperature),
cm^/sec
a3 = a constant
v = velocity, cm/sec
2
C = coefficient of intrinsic permeability, cm
3, = a constant
49
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Additional discussion on equation (26), its units and the values of
its constants is provided in Appendix B. This equation is representative
of the data from which it was derived and is unlikely to be universally
applicable.
Returning to equation (25), it may be noted that this expression
basically relates the concentration of a substance at a given node to
the concentrations at its two adjoining nodes. These three variables
are the only unknowns in the equation. The values of all other vari-
ables are either known or can be computed from available data.
It follows then, that equation (25) can be rewritten in the form:
-(COV^ + (BC)V? - (AC)V?+1 = (DC)!? (27)
The derivations of the expressions used to compute the coefficients
AC1?, BC1?, CC1? and DC1? above are given in Appendix B for the case of
an interior node and also for the cases of the top and bottom nodes.
These derivations consist basically of regrouping of terms,
One water quality equation of the general form of (27) above can be
written for every node in the soil profile. The resulting system of
equations has the same structure as that shown in system of equations
(22). Whereas the unknowns in (22) are the 4>'s, the unknowns in the
new system are the concentrations, computed at each time step in the
program by Gaussian elimination.
Appendix B contains sample quality calculations and a description of
program subroutines.
50
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SECTION VI
GROUNDWATER MODELS
INTRODUCTION
The groundwater, or saturated, zone may be defined as the region of
porous media in which the void spaces are totally occupied by water.
The delineation of this region is illustrated in Figure 12 which re-
presents a typical vertical cross-section in a basin. Shown therein
are the ground surface, water table and two different geologic forma-
tions. According to the above definition, the water table constitutes
the top boundary of the groundwater region. For purposes of analysis,
it is desirable to define lateral and bottom boundaries for the system,
shown as dotted lines. The location of these boundaries is facilitated
by the existence of certain hydrogeologic features, (e.g. aquicludes).
Somewhat arbitrary boundaries can also be used.
The movement of water within a given groundwater region is controlled
by the flows occurring across its boundaries, such as recharge or dis-
charge across the water table, and by hydraulic relationships within
the region itself. More specifically, flow through saturated porous
media is controlled by two basic relationships: Darcy's law and the
equation of conservation of mass. The former states that the velocity
at a certain point in the groundwater system is proportional to the
hydraulic potential gradient at that location. The corresponding con-
stant of proportionality, determined empirically for any given type of
material, is known as the permeability or hydraulic conductivity. Dis-
cussions of this constant, its determination and the range of validity
of Darcy's law are given by Todd (1967) and Bear (1972). These two
relationships can be combined into a single expression, which can be
used to predict flow velocities at different locations in the ground-
water system. Data required to carry out this predictive process include
flow boundary conditions and an initial hydraulic potential field.
As water moves through a groundwater system, it is subjected to physical,
chemical and biological processes which affect the concentrations of its
water quality constituents. Mathematically, some of these processes may
be represented as terms in a water quality equation. Such processes
51
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GROUND SURFACE
en
isi
WATER TABLE
(TOP BOUNDARY)
UNSATURATED ZONE
UNSATURATED ZONE
— SURFACE BODY OF
WATER
SATURATED ZONE
Figure 12. Vertical cross-section of a groundwater system.
-------
include advection, dispersion, diffusion, and sources and sinks. The
source-sink rates, thought to be generally important, can only be roughly
estimated from available data. An equation of the above type can be
used to predict the concentrations of various substances at different
locations within the groundwater system. Some terms in the equation,
such as dispersion and advection, are a function of velocity.
LITERATURE REVIEW
Summary of Purposes
As in other hydrologic processes, a stepwise procedure is used herein
in the modeling of groundwater. That is, at any time the flow or
quantity computations are conducted first, followed by those involving
water quality.
Quantity Aspects
Basic Theory-
Flow of groundwater is controlled by two relationships: Darcy's law
and the conservation of mass equation (Todd, 1967) . The former can be
written in one -dimensional form as:
Vx •
where v * average flow velocity in the.x-direction, ft/sec
^\ -
K « permeability or hydraulic conductivity, ft/sec
« hydraulic head, ft
x « space coordinate, ft
The term £$> above is known as the hydraulic gradient. Similar expres-
sions can be written for the y and z space coordinates.
The second relationship, the conservation of mass equation, is also
known as the equation of continuity. Assuming a steady-state situation
(i.e., time- invariant boundary conditions) and an incompressible fluid,
the continuity equation can be expressed as
3v 3v 3v
x _._ _ y . z
+ -r-«- + -=-*•« 0 (29)
3x 3y 3z
53
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Substitution of the expressions of Darcy's law (28), for each of the
three coordinate directions, into equation (29) yields the steady-state
form of Richard's equation (Freeze and Witherspoon, 1966), as follows:
~9x"I"*••""*7 ''•r*~~i ' a«i"^A»/ »"J^;\ • -*^[n^»J *"J-*^'\ ~ u (30)
In the special case of a homogeneous medium, the permeability is constant
and equation (30) may be simplified to
which is the Laplace equation. All the above equations, especially (28)
and (29) serve as a basis for any steady-state groundwater formulation.
The historical development of the major types of formulations is dis-
cussed below.
Hydraulics of Wells and Aquifers—
Much of the early work in ground water centered around radial flow into
a pumping well. In 1906, Thiem investigated the steady- state aspects
of well hydraulics. As a result, formulations for both confined and
unconfined water-bearing formations, or aquifers, are available (Todd,
1967) . In 1935, Theis developed the basic unsteady- state formulation
for well flow. The main value of these well formulas, and their modi-
fied versions, lies in their use in pumping tests to determine aquifer
permeabilities and other parameters.
In later years, other well formulations have been developed to allow
less restrictive and more realistic assumptions on boundary conditions,
An example of a realistic assumption is the existence of flow between
a given aquifer and an adjacent semi- confining stratum (Hantush and
Jacob, 1955).
Other improvements in well hydraulics formulations have permitted the
analysis of complex boundary conditions and geometric configurations.
According to Sternberg (1971), the handling of complicated geometric
boundaries is best carried out by analog, rather than digital, computer
techniques. Analog models have been used successfully to predict the
response of aquifers to heavy pumpage in many areas, such as Las Vegas
Valley (Domenico, Stephenson and Maxey, 1964). Details on the use of
analog techniques are discussed in recent texts by Domenico (1972) and
Bear (1972).
54
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Although analog models allow great flexibility in the analysis of
boundary conditions, the overall capability of digital models is
stronger. As a consequence, numerous digital aquifer models are pre-
sently available. Examples are those developed by Trelease and Bittinger
(1963), Engineering-Science (1971), Sechrist e_t al_. (1971), Bredehoeft
and Finder (1970) and Neuman and Witherspoon (1970).
Groundwater flow systems—
Another major type of research effort in ground water has been the
development of the flow systems approach. In contrast to the above
emphasis on radial flow occurring in a certain portion of a ground
water basin, this approach is more concerned with the basin as a whole
and with the understanding of general flow patterns. Thus, the wide-
spread use of terms such as aquifer versus aquiclude (also, confined
and unconfined), which tend to denote absolutes, is replaced in the
systems approach by consideration of relative values of permeability
at various locations. All geologic formations are assumed to have
some permeability, however small it might be.
Also, the flow systems method places great weight on the delineation of
recharge and discharge areas, and the relation of these features to the
overall movement of water in the saturated zone. It follows that for a
study of basin-wide water quality with various sources of pollutants,
the systems approach appears to offer great flexibility and promise.
Some major developments in this approach are discussed below. <
Much of the fundamental work in groundwater flow systems was carried
out by Hubbert in 1940. In 1962 and 1963, Toth applied the principles
proposed by Hubbert to the analysis of groundwater motion in small
basins. Further application of flow theory to geology was conducted
by Maxey and Hackett (1963).
In 1966, Freeze and Witherspoon utilized the groundwater flow systems
approach in developing a two-dimensional steady-state model. In a sub-
sequent paper, Freeze and Witherspoon (1967) discussed geohydrologic
features such as water table configuration and the occurrence of re-
charge and discharge, and their formulation as boundary conditions in
the mathematical model. In their steady-state formulation, these boun-
dary conditions are assumed to be time-invariant. Also contained in
this paper are useful rules of thumb which permit the deduction of
general flow patterns in a basin from geologic and geographic informa-
tion .,':-'.'
The two-dimensional model developed by Freeze and Witherspoon (1966)
can be applied to selected vertical cross-sections of the type shown
in Figure 12. The sufficient condition for the validity of the two-
dimensional approach therein is that no significant flow actually occurs
in the direction incidental to the plane of the section. Accordingly,
55
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the cross-sections to be used in the analysis of a basin should be care-
fully chosen using potentiometric data which indicate flow paths. This
process is illustrated in Section VIII.
Assuming that the two-dimensional assumption is valid for the selected
cross-section, the first step in using the model consists of super-
imposing a grid system, as shown in Figure 13. The main function of
the formulation is to predict the value of hydraulic head at each node
in the system. To achieve this objective, Freeze and Witherspoon
utilize a finite-difference form of equation (30) as the basis of an
iterative approach. The final values of head obtained at each node are
then used to compute the corresponding flow velocities by means of
Darcy's law (28).
In an attempt to increase the generality of his flow systems approach,
Freeze proposed in 1971 a three-dimensional, unsteady-state formulation.
At the present time, complete verification results of this numerical
model are not available. Many of the basic numerical techniques used
by the above and other workers in subsurface hydrology have been dis-
cussed in a text by Remson, Hornberger and Molz (1971).
Estimation of Hydrogeologic Parameters—
The successful verification of any groundwater model depends heavily
upon the estimation of hydrogeologic parameters such as permeability.
A general discussion on parameter estimation methodologies in hydro-
logy has been presented by Jackson and Aron (1971).
The available works on parameter estimation in groundwater may be classi-
fied according to whether they are primarily applicable to aquifer pump-
age problems or to general flow systems. In the former category, papers
are available by Haimes, Perrine and Wismer (1968), Yeh and Tauxe (1971)
and Kleinecke (1971).
Parameter estimation in general flow systems involves great difficulty.
This is due to the large number of different geologic formations and
the areal variability of flow boundary conditions such as recharge and
discharge. Some of the problems have been recognized by Nelson (1960),
Freeze and Witherspoon (1967), and Simpson, Kisiel and Duckstein (1971).
Selection of Flow Model—
A review of the main goal of this particular study, namely the devel-
opment of a basin-wide relationship, indicated that the most desirable
type of flow model was that based on the flow systems concept. Also,
the emphasis of this approach on recharge and discharge relationships
makes it well-suited to analyze agricultural effects. Accordingly, the
two-dimensional, steady-state formulation by Freeze and Witherspoon
(1966) was selected.
56
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• GROUND SURFACE
tn
UNSATURATED ZONE
WATER TABLE
UNSATURATED ZONE
V
- SURFACE BCOY OF
O O O O O O——(
) o O O O O -O- 0
) O O O
C3 Cl O
D O——O O
O O -O O 1
Figure 13. Grid structure superimposed on a groundwater cross-section.
-------
Some consideration was given to using the three-dimensional, unsteady-
state model developed by Freeze (1971). However, the computational
cost to be brought about by an additional dimension was thought to be
unwarranted. Also, in many basins adequate potentiometric data are
available, allowing a fairly accurate delineation of cross-sections
for analysis. Finally, in agricultural basins, boundary conditions
(such as water table elevations) do not vary rapidly but rather under-
go long-range and seasonal changes (Orsborn, 1966). Therefore, the
use of a transient formulation can be effectively substituted by that
of a steady-state approach at a series of relatively long time in-
tervals.
The two-dimensional, steady-state model used allows the computation of
the velocity field and is thus compatible with most of the available
water quality formulations.
Quality Aspects
Background—
The concentration of various substances in ground water is controlled
by physical, chemical and biological actions.
The physical category includes dispersion and diffusion. These two
phenomena involve spreading and mixing of substances, but their causal
mechanisms are different. Dispersion is due to velocity fluctuations
resulting from the tortuosity of the flow paths in the medium. Dif-
fusion, on the other hand, is associated with random thermal motion of
molecules.
The significance of chemical reactions in ground water varies with the
type of medium and other natural conditions. In clay soils, the pre-
.dominant reactions would occur in the form of ion exchange. Limestones
tend to dissolve in water, producing high hardness values. All these
reactions occur until equilibrium is reached.
Biological actions are the most difficult to quantify. However, it
may be speculated that these activities are less complex in the ground-
water region than in the unsaturated zone. The reason would lie in the
filtering of organisms and organic matter by the soil, sometimes within
the first few feet of travel. In any case, data on the impact of bio-
logical activity are very scarce.
Dispersion and Diffusion—
According to Day and Forsythe (1957), diffusion of ions is a process
independent of hydrodynamic dispersion. Work on both of these phenomena
has been carried out by Scheidegger (1954), Bear (1961), Scheidegger
(1961), Bachmat and Bear (1964), and Brutsaert (1968). A comprehensive
review is found in Bear (1972).
58
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Scheidegger's 1961 paper further developed the mathematics of dispersion.
Theoretical arguments up to that time indicated that there were two
possible expressions relating the factor of dispersion to the velocity:
D = av (32)
or,
D = av2 (33)
where D = dispersion factor
a = geometrical dispersivity
v = velocity.
After reviewing experimental results, Scheidegger concluded that (32)
was the correct expression.
Turning to the relative importance of dispersion versus diffusion,
it may be recalled that the latter has been usually neglected in
calculations. In 1967, Hoopes and Harleman tested the validity of the
above simplification by means of a tracer study. They found that the
omission of diffusion terms from the general dispersion-diffusion
expressions did not significantly increase the discrepancy between
calculated and observed concentrations. This result was obtained for
both the longitudinal and transverse directions.
As a part of the same investigations, Hoopes and Harleman (1965)
developed expressions for the dispersion coefficient. One equation
proposed was:
(34)
where Kn = dispersion coefficient, cm2/sec
v = kinematic viscosity of water, cm2/sec
C = coefficient of intrinsic permeability, cm2
v = flow velocity, cm/sec
a3 33 = constants
59
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The above expression (the range of validity of which, as mentioned
earlier, is uncertain) is also used herein for dispersion calculations
in the unsaturated zone. In that regime, the intrinsic permeability
is derived from the variable hydraulic conductivity.
Source- sink Factors—
The fact that various reactions occur between the moving water and the
geologic formations has been recognized for many years as shown by the
work of Kaufman et^ al_. (1961), Champlin and Eicholz (1968), Scalf et al.
(1969), and Stanford, England and Taylor (1970).
Consideration of the available literature on source-sink interactions
reveals the lack of a unifying mathematical formulation. However,
the approach proposed by Back and Hanshaw (1970) appears promising.
These researchers have pointed out that perhaps the hydraulic potential
energy should not be considered as a separate entity, but rather should
be integrated with other chemical and thermal energy factors. The
pursuit of this approach, while worthwhile, would extend beyond the
scope of this particular study.
Selection of Quality Model—
The selection of the water quality formulation involved decisions on
the various processes to be included in the governing equation* In
view of the available data on the relative importance of dispersion
and diffusion, the latter phenomenon was neglected. Also, source-sink
terms were not considered due to lack of data. However, the type of
advection-dispersion model chosen below is one in which source-sink
terms could be incorporated with relative ease. Nitrate is the only
substance being routed at present.
An extensive survey of two-dimensional advection-dispersion formu-
lations has been conducted by Bedient (1972) as part of an estuarine
water quality simulation study. These numerical procedures utilize
a previously computed set of velocity components at the grid nod.es,
along with initial and boundary conditions, to calculate the pollu-
tant concentrations at all the nodes. Bedient 's work was based on a
model developed by Shamir and Harleman (1966), which is effective in
reducing numerical error.
In its original form the Shamir and Harleman model was designed to
solve the basic transport equation in an orthogonal curvilinear coor-
dinate system. It was subsequently modified by Bedient for applicabil-
ity to any two-dimensional coordinate system, as follows:
60
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where c = concentration of substance, dimensionless
u, v = horizontal and vertical velocities, respectively,
ft/sec
t = time, sec
x, y = coordinate directions, ft
spersion coefficients
respectively, ftVsec.
D , D = dispersion coefficients in the x and y directions,
x
QUANTITY MODEL
Basically, the approach used herein (Freeze and Witherspoon, 1966)
consists of applying a finite-difference form of Richard's equation
(30), in only two dimensions, to every node in a grid system such as
that shown in Figure 13. The values of hydraulic head at each node
are predicted using the above finite-difference equations as part of
an iterative procedure. Subsequently, the two components of velocity
at the nodes are computed from Darcy's law.
For discussion purposes, it is worthwhile to select a small section of
a large grid system, such as that shown in Figure 13, for detailed
analysis. This isolation process is illustrated in Figure 14. Indi-
cated in the upper left hand portion of the figure is an arbitrary
section of a grid system. Also shown is information on the coordinate
axes and the horizontal and vertical spacings.
The interior node having coordinates (I, J), and its four adjoining
nodes, are shown in more detail in the right-hand side of Figure 14.
The finite difference solution procedure to predict the head at the
central node utilizes the heads at its adjoining nodes and both the
horizontal and vertical permeabilities at all five nodes. In addition
to the above, permeabilities and heads are evaluated at locations mid-
way between the nodes.
In two dimensions, the original form of the governing equation (30)
reduces to
The finite-difference form of the first term in the above equation,
applied to node (I, J) can be written as:
61
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3x
(37)
A counterpart expression can be developed for the second term in (36)
above. Following the expansion of midpoint terms and regrouping in
equation (37) and its counterpart, the following expression is ob-
tained (Freeze and Witherspoon, 1966):
+ aK (I,J-1)
-------
After a desired tolerance in the predicted values of head has been
achieved, it is possible to compute the horizontal and vertical com-
ponents of velocity at the various nodes. For this purpose, Darcy's
law is applied in finite-difference form in the s- and z-directions.
Sample calculations for the groundwater flow model and a discussion of
its subroutines are provided in Appendix C.
QUALITY MODEL
The function of the quality formulation is to predict the concentrations
of a given pollutant^ at all nodes in a grid system such as that shown
in Figure 13. For this purpose, it utilizes the velocity components
determined by the flow model along with initial and boundary conditions
for the pollutant in question. Specifically, the quality model solves
the advection-dispersion equation (35) using various numerical techniques.
It may be noted that the y-coordinate in equation (35) corresponds to
the vertical z-coordinate used in the flow model, as shown in Figure
14. Since the solution procedure for the quality model has been dis-
cussed extensively by Bedient (1972), only a brief explanation thereof
is given below.
The original Shamir and Harleman (1966) formulation, adapted by Bedient
to cartesian coordinates, combined the techniques of several investi-
gators to minimize numerical error. One of these techniques is the
application of weighting factors to the various time and space finite
difference terms in the governing equation (35). The optimal values
of these weights were determined by Stone and Brian (1963) using
Fourier analysis.
Another feature incorporated into the formulation is the alternating
direction implicit (ADI) scheme proposed by Peaceman and Rachford
(1955). At any given time, the ADI procedure first solves a form of
equation (35) in the horizontal direction, and subsequently solves a
similar form in the vertical direction. The solutions obtained in the
two above steps consist of intermediate, or temporary, concentration
values at each node in the grid system.
The ADI method can be discussed further by referring to the right-
hand portion of Figure 14. Considering the horizontal direction first,
the following generalized finite-difference equation can be written
for a given node with coordinates (i,j) after rearranging terms in
equation (35):
-C. .C* . . + B. .c* . - A. .c* . = D. . f401
1.3 i-l>3 1,3 i»3 L3 1+1.3 i>3 ^ }
63
-------
J + 20
r
COLUMN INDEX
-*- X
x = HORIZONTAL SPACING
z = VERTICAL SPACING
Figure 14, Details of a groundwater grid system.
-------
where c* = intermediate concentration of pollutant at a given
node
A, B, C = left-hand side coefficients (constants)
D = right-hand side constant
Since equation (40) can be written for each node, a system of such
equations is required to describe the entire grid structure. The
resulting system of equations is of tri-diagonal form, and the values
to£ concentration are obtained using the elimination procedure algorithm
'developed by Thomas (1949) . It may be recalled that this solution
scheme is analogous to that used in the unsaturated zone models.
In the vertical direction, an equation similar to (40) above can be
written for the same node, namely:
-cr .c**. . + BT .c**. - A: .c**. , = or
1,3 i,3-l 1.3 1,3 1,3
where c** = second intermediate pollutant concentration
A", B*, C' = second set of left-hand side coefficients
D" = second right-hand side constant
Likewise, the system of equations involving all nodes is solved by
means of the Thomas algorithm.
After the two basic steps in the ADI procedure have been completed, the
quality formulation computes the final concentrations throughout the
system using an explicit scheme. The intermediate values of concen-
tration, determined above, are used in these computations.
Finally, it is pertinent to mention briefly the types of boundary con-
ditions used in the formulation, in order of increasing complexity:
1. A value of concentration, c, is specified at a given
boundary node.
2. The concentration gradient is set equal to zero at a
given boundary. This is known as the reflection boundary
condition, and it implies that no dispersion occurs across
the boundary.
3. The concentration gradients on both sides of the boundary
are set equal.
4. The difference between the mass flux entering a control
volume in the vicinity of the boundary and the flux leaving
it is equated to the change in storage;
65
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The usage of some of these boundary conditions is illustrated in
Section VIII.
66
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SECTION VII
INTERFACING OF MODELS
INTRODUCTION
To predict flow and water quality conditions on a basin-wide basis,
it is necessary to link together the various computer models discussed
previously. Figure 15 consists of a vertical cross section of a typi-
cal basin which indicates the formulations applied to different por-
tions of the system.
The interfacing methodology consists of the exchange of flow and
quality information between the different models shown in Figure 15.
In developing this methodology, the trade-off between two factors must
be considered: (1) computational convenience and (2) conformity with
the natural system. Due to size limitations in computer storage
facilities, it is generally infeasible to carry out the computations
for all the models on an integrated basis. Thus, it becomes necessary
to establish a sequence in which the various programs should be run.
Ideally, this order would follow the paths of water particles as they
move within the system. However, since most basin-wide flow patterns
are multidirectional, a compromise sequential system must be developed.
The discussions below include further consideration of stepwise com-
putations and their corresponding data linkages. Information on the
types of data used in these transfers is provided. Also included
is a discussion of programming and usage factors,
GENERAL COMMENTS ON SEQUENTIAL APPROACH
At present, the only feasible methodology is to run the various pro-
grams in a stepwise fashion. The selection of this sequential pro-
cedure should be made in a manner that minimizes distortion of the
natural processes.
Prior to developing a stepwise approach for a given study area, it is
advisable to run the individual models under a variety of boundary con-
ditions. Since these boundary conditions represent the interactions
67
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I RAINFALL
KINEMATIC WAVE
OVERLAND FLOW
MODEI
1
IRAir
1
RU.L
f '
STORMWATER
/- RUNOFF
,
II
MANAGEMENT
MODEL -v.
00
LAKE
RECEIVING WATER
MODEL
6ROUNDWATER MODEL
Figure 15. Illustration of the different models applied in a basin.
-------
between the different models, the various runs should provide insight
into sensitivity aspects, The relative impact of any given model upon
another will depend upon the particular basic under study. However,
it is still possible to propose some generalizations on sensitivity
which should hold under most circumstances. For instance, short-term
infiltration rates in the unsaturated flow model should produce rela-
tively little impact on the overall groundwater flow patterns, which
are long-term results of basin geometry and permeability configuration
(Freeze, 1969). On the other hand, the infiltration rates following
a rainstorm should have a profound effect on the ensuing overland flow
(Agricultural Research Service, 1971).
In addition to the need for more detailed analyses of model interactions
in any given basin, it is anticipated that improvements in the indivi-
dual formulations might be desirable. To facilitate the introduction
of these improvements in the future, an effort was made to maintain
the computer models as self-sustaining entities, linked together by
a relatively flexible set of interfacing instructions.
More specifically, it appears that one of the basic problems that will
require increased attention is that of estimation of model parameters,
such as permeability. At present, several optimization techniques
have been employed to estimate permeabilities in groundwater systems,
e.g., Kleinecke (1971). It might become desirable to extend the use
of these techniques on a basin-wide basis. In that case, it is likely
that the overall calibration task might best be accomplished by using
a decomposition, rather than a unified, approach.
In summary, this discussion has pointed out the need to preserve the
identity of the various computer models, at least until an improved
state of the art permits the development of a single, global model.
Thus, the decision at hand consists of determining the order in which
the models are to be activated. It might be recalled that each hydro-
logic model has a water quality counterpart, which must be run immed-
iately following the flow computations. Therefore the sequential un-
certainty involves only the hydrologic processes.
INTERFACING STRATEGY
Spatial and Temporal Factors
At this point, it pertinent to discuss the significance of the matter
of flow paths with respect to the two basic mathematical approaches
to fluid mechanics. The first of these, the Lagrangian method, follows
the spatial coordinates of a given fluid parcel as it moves with time.
The alternate Eulerian approach describes temporal changes in fluid
motion at prespecified locations in the system (Li and Lam, 1964), The
latter approach has been widely considered to be more convenient and it
69
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underlies most available computer models, including those used in this
study. Accordingly, a detailed treatment herein of particle paths
was considered to be unwarranted. In future efforts, it might be
worthwhile to utilize the Lagrangian technique for processes such as
runoff quality, in which soil particles may undergo successive cycles
of scouring and settling. In this study, however, the use of any
time history information is restricted to the determination of boun-
dary conditions between the various hydrologic regimes in a basin.
Since the accurate estimation of boundary conditions plays a signi-
ficant role in determining overall predictive success, it is impor-
tant to consider the interplay of these conditions in establishing
the stepwise procedure for calling the various models. One immediate
problem in this regard is the determination of which model should be
run first. Intuitively, it seems that the model in question should
be one whose main boundary conditions occur early in the hydrologic
chain of events. Another desirable feature of this model would be its
relative insensitivity to short-term feedback from other models. In
this study, the initial model selected was that for the unsaturated
zone, which is thought to be the likely choice for most basins.
Illustration of the basic features of this model, including its
boundary conditions, is given in Figure 9.
Unsaturated Zone Model
The top two boundary conditions in this model, rainfall and evaporation,
are among the better known variables in any hydrologic system. Under
conditions of high rainfall intensity, a ponding situation will arise
at the ground surface, and in this case the formulation will require
the setting of a maximum ponding depth (Freeze, 1969). The determina-
tion of this value is a somewhat arbitrary process, but any reasonable
value will result in a hydraulic gradient which approaches unity in
the vicinity of the ground surface. That is, unless the node spacing,
Az, is very small, the value of the hydraulic gradient at the upper
nodes should be relatively insensitive to the maximum ponding depth
used.
The bottom boundary flow in the soil profile is equal to the net
recharge into the groundwater system. It is proposed that an initial
recharge rate be obtained prior to running the unsaturated model, in
the following stepwise fashion: (1) conduct several runs of the
groundwater flow model under a variety of water table elevations;
(2) store the computed sets of recharge velocities at the upper
groundwater system nodes; and (3) select the recharge value corres- •
ponding to the initial water table location used for the unsaturated
model. This bottom boundary condition is assumed to remain constant
over relatively short-term changes in the vertical soil profile, in-
cluding saturation of some nodes. The justification for this assumption
70
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lies in the fact that groundwater flow systems are sluggish in respond-
ing to external conditions, and are thus unlikely to produce signifi-
cant short-term feedback.
Finally, some initial conditions must be specified in the unsaturated
model. In the flow portion of the formulation, these conditions con-
sist of moisture content values at the various nodes in the soil pro-
file. It is difficult to determine these values exactly except by
direct measurement, but it should be possible to provide reasonable
estimates based on basin hydrology and land use. For instance, in
agricultural areas it is common practice to maintain the soil mois-
ture near field capacity (0.1 atmospheres) to facilitate water uptake
by crops (Choate and Harrison, 1968). This objective is generally
achieved by providing sufficient irrigation water to offset evapora-
tive losses from the soil. In uncultivated soils, evaporative losses
during dry periods typically produce a moisture profile in which the
moisture content decreases with elevation (Freeze, 1969). If a no-
flow condition were to exist within the soil, the total head at all
nodes would be the same. Thus, changes in elevation head between
nodes would be offset by opposite and equal changes in pressure head.
In the water quality portion of the unsaturated zone^model, the main
boundary condition is that which specifies the amounts of substances
made available by man's activities on the ground surface. In this
study, this type of boundary condition is set at the uppermost node
by specifying the concentrations of nitrogen and phosphorous origina-
ting from fertilizer application. A discussion of references on other
agricultural sources of nutrients, such as animal wastes; is provided
in Appendix F. Also, the nutrient contents of rain water have been
found to be significant by some investigators (Weibel et al., 1966).
If such a contribution is to be accounted for by means of another
surface boundary condition, it is advisable to utilize rainfall quality
data collected at the particular basin in question. In addition to
boundary conditions, the quality model requires the specification of
-initial conditions, in the form of concentrations at the various nodes.
These values can only be estimated roughly in the absence of field data.
It may be concluded that the unsaturated zone model plays an important
role in the overall system, in the capacity of a link, or buffer, be-
tween the surface and groundwater regimes.
Surface and Groundwater Models
The degree of connection between surface and groundwater systems
depends largely on whether a recharge or discharge condition exists.
Under a recharge situation, the two systems are generally separated by
the unsaturated zone, as can be seen in Figure 15. Consequently, it
is possible to consider the surface and groundwater regimes in a
71
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quasi-independent fashion. That is, any feedback between the two
regimes is transmitted only indirectly through the unsaturated zone
model.
On the other hand, groundwater discharge to surface bodies of water
occurs directly. This is the case with base flow in streams and up-
flow in springs, for instance. Thus, the corresponding modeling
approach is somewhat different.
Since the unsaturated zone model was selected in this study as the
first one to be run in the modeling cycle, the question remained as
to which system should be considered next, surface or ground water.
In view of the fact that changes occur much more rapidly in the sur-
face system than in ground water, the order developed consisted of
the surface followed by ground water. The overland flow and ground-
water models utilize information generated by the unsaturated zone
formulation in the vicinity of its top and bottom nodes, respectively.
The surface water processes are covered by two overland flow models
and the receiving model. The first overland flow formulation and
the receiving model are modified versions of those found in the
Stormwater Management Model (Environmental Protection Agency, 1971).
This overland flow"formulation is proposed for use in areas where
artificial drainage and channelization are widespread. The second
overland flow model, to be utilized in relatively undeveloped areas,
is based on the kinematic wave approach (Eagleson, 1970). The types
of areas in which these two approaches are utilized are illustrated
in Figure 15.
In both overland flow formulations, the infiltration rates (flows
between the two upper nodes) obtained from the unsaturated zone model
are used in conjunction with rainfall hyetographs to calculate flow
hydrographs at the various catchment outlets. Also computed at the
outlets are nutrient transport rates. This flow and quality infor-
mation is in turn used as input to the receving water model.
Turning to the groundwater model, it receives input from the unsat-
urated zone model in the form of water table elevations and concen-
trations of nutrients at the bottom node of the soil profile. Follow-
ing heavy rainstorms, it is possible that the soil profile may become
saturated throughout. In any case, the groundwater model utilizes
the computed water table elevation as a boundary condition at the end
of the unsaturated zone simulation period. At the same time, the
groundwater model takes into account flow information at the boundaries
which are not linked with other models, such as the bottom horizontal
boundary. After the velocity field has been computed, the convection-
dispersion formulation routes the movement of nutrients located at the
bottom of the unsaturated zone.
72
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PROGRAMMING AND USAGE ASPECTS
Figure 16, constitutes a flow diagram involving the various formula-
tions. Beginning at the upper portion of this figure with the unsat-
ura'ted zone model, the computations are next carried out for the over-
land flow and receiving models. The calculations for the latter two
models are branched out, depending upon whether or not man-made drain-
age is prevalent in any given area. Finally, the groundwater computa-
tions are completed.
The dotted arrows in Figure 16 denote the interfacing or boundary
condition data which are transferred between any two models. These
data consist typically of hydraulic heads, velocities and concentra-
tions. Some of the specific items are indicated in the figure.
Changes in the linkage structure might be deemed desirable depending
upon local conditions.
From a programming standpoint, it becomes apparent that the stepwise
computational procedure shown in Figure 16 should be controlled by a
main, or executive, program. Accordingly, the individual computer
models become subroutines (sub-programs) of this main program. In
turn, each model subroutine contains other smaller subroutines.
With regard to data storage and retrieval, two alternate programming
techniques were considered. The first consisted of the specification
of various data storage areas which would be shared by two or more
subroutines. The advantage of the use of these storage areas, tech-
nically known as COMMON blocks, is that computed values are trans-
ferred implicitly between the pertinent subroutines, without the
need for input/output instructions. However, the main drawback of
this method lies in the fact that its total storage requirements for
a basin of reasonable size can approach or exceed the capacity of the
computer facilities. This was the case with the University of Florida
system, an IBM 360/65 with a total capacity of approximately one
million bytes. Since this is a comparatively large system, the
storage problem is likely to be aggravated at most facilities else-
where. Accordingly, the use of COMMON blocks was restricted to-
internal linkages between the quantity and quality portions of each
model, and another method was sought for transfers between these
major models.
The alternate approach used, designed to minimize storage utilization,
is known as an overlay technique. The effect of this method is that,
at any time in a computer run, the storage being utilized involves
only the main program and the particular model subroutine being
executed. Following the execution of a given model, its results must
be stored for subsequent use by another model. For this purpose,
active computer storage is not used, but rather an external unit
73
-------
. Perform Trial -Runs
1 for all Models *
I (optional) >
X
/inflltrotlon ,
/ Rates /
Run Unsaturated
Zone Model
\
Is land channelized ?
\Inflltratlon
\ Rates
/ /
/ /
/ t
Yes
1 Run Kinematic
/ Wove Overland
/ Flow Model
1
1
\
\
\
\
\
\
\
Water table \
elevation,
Nutrient concentre
1 Flow Rat
(Nutrient 1
1
1
\
\
\
tloni — ^
es
:lux«s
No
Run Receiving
Water Model
For Drainage
Canals
1
Run Receiving
Water Model
For Lakes
And/Or Streams
1
Run Groundwater
Model
\
\
\
Run Storm water
Management
Runoff Model
^N
iFlow Ro
1 Nutrient
V\
\ Hydraulic
j Nutrient
1
i
/Flow Re
/Nutrient
^
les
Fluxes
Heads
Concentrations
Figure 16. Flow diagram for the various models,
illustrating data interfacing.
74
-------
utilized. The unit in question, known as a data set (DATA SET) is
operated in a fashion similar to that of a tape, but has the advan-
tage of more rapid access. The use of the overlay procedure and a
data set is illustrated in Figure 17.
Appropriate WRITE/READ statements must be included in the model sub-
routines to transfer data to and from data sets, respectively, A
large number of data sets may be used in connection with a given pro-
gram. A given data set may be used successively to store different
variables, and in this case it must be designated as temporary, or
scratch. On the other hand, a permanent data set may be created for
data transfer exclusively between two given model subroutines.
Additional details on the use of data sets within an executive pro-
gram are provided in Appendix D. Also contained are the job control
language (JCL) instructions used. These JCL statements will require
modifications prior to usage in other computer facilities.
TIME INCREMENTS AND SPATIAL PARTITIONING
Due to the significant expense involved in running the computer pack-
age, it is infeasible to utilize it on a continuous basis. Instead,
it is advisable to reserve its use to simulate the effect of signi-
ficant events, such as a high intensity rainfall following the appli-
cation of fertilizer.
It is important to note that, while the simulation periods for all
the models shown in Figure 16 begin at the same point in time, the
total length of these periods need not be the same. For instance, the
overland flow model might be run only for several hours, but the
receiving model might be run for several days thereafter to allow
sufficient time for significant pollutant movement.
Also, a wide difference in the significance of the time element exists
between the flow and quality portions of the groundwater model. Since
the former is relatively insensitive to short-term changes in flow
boundary conditions, it should seldom be justified to run it with a
frequency greater than, for instance, once a month. That is, the
velocity field could be considered invariant within such a period.
However, to adequately trace the impact of a given input of pollutants
to a groundwater system, the total simulation time required would be
in the order of months and years. Thus, in strict terms, the use of
several velocity fields would be required in addition to that in exis-
tence at the time of the pollutant input.
It follows that discrepancies also exist with regard to the length of
time increments used in the various models. As a rule, the models which
75
-------
!l
MAIN (EXECUTIVE) PROGRAM
OVERLAY
NO. I
BOUNDARY
I
UNSATURATED
ZONE MODEL
TIME :t.
UNSATURATED ZONE
MODEL ENDING
EXECUTION
WRITE
INFILTRATION
RATES
COMPUTER TIME
I
i
KINEMATIC WAVE
OVERLAND FLOW
MODEL
DATASET "A"
STORES INFILTRATION
RATES
READ
INFILTRATION
RATES
OVERLAY
N0.2
] BOUNDARY
TIME = t2
OVERLAND FLOW
MODEL BEGINNING
EXECUTION
Figure 17. Schematic representation of the use of overlay technique with data sets
-------
are run earlier in the package are those which utilize the shorter
time steps. Thus, the results of the overland flow model, for instance,
must be averaged over longer intervals prior to transfer into the re-
ceiving model." This averaging process is shown in Figure 18.
It is also pertinent to point out the interfacing complexities in-
troduced by partitioning the physical region to which a given class
of model applies. Schematically, this concept of partitioning was
first presented in Figure 16, in which the overland flow calculations
are branched out to indicate two different types of surface terrain:
(1) artificially drained (channelized) land and (2) nonchannelized
areas. In a typical basin, it is likely that these two types will be
interspersed, effectively producing proliferation of branches at the
level of overland flow calculations. Similarly, the areal partition-
ing of the unsaturated zone on the basis of soil classification will
produce branching at that level. Care must be taken to insure that
a proper interfacing between the various model branches is conducted.
It would be difficult to further illustrate the above comments on
temporal and spatial aspects of interfacing without reference to a
specific study area. For this reason, additional discussion of these
matters is deferred to Section VIII, involving the Lake Apopka basin.
77
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00
OVERLAND FLOW MODEL
TIME STEP = At
RECEIVING WATER MODEL
TIME STEP = AT= 3At
OUTFLOW HYDROGRAPH
AT END OF SIMULATION
PERIOD
Figure 18. Schematic representation of time-averaging process for interfacing
flows between overland flow and receiving models.
-------
SECTION VIII
APPLICATION TO THE STUDY AREA
INTRODUCTION
Prior to discussing the application of the computer model package to
the Lake Apopka basin in Central Florida, it is worthwhile to review
some general aspects of the various models and their interfacing
methodology. A schematic representation of the executive, or main,
program and the order of its individual model subroutines is pro-
vided in Figure 19. Also illustrated in this figure is the use of
data sets for input/output operations between models.
In Appendices A, B and C, sample calculations for the different models
are illustrated using hypothetical data. Additional insight into
these calculations will be provided by discussions on the usage of
data from the study area. It must be emphasized that the scope of
this study precluded a sampling program. Whenever necessary, values
were obtained from the literature, and some of these were modified
to reflect specific conditions in the Lake Apopka basin. Most of the
assumptions in agricultural calculations were formulated following
personal communications with research personnel of the University of
Florida Institute of Food and Agricultural Sciences (IFAS).
One interfacing aspect which requires much judgment is the partition-
ing of the basin into relatively homogeneous units, to improve the
predictive quality of the various models. For instance, the unsat-
urated zone can be partitioned according to soil type, among other
possibilities. For any given model, this spatial division process
could be carried out by either one of two methods: (1) reusing the
same program for all the pertinent partitions, in each case utilizing
the appropriate parameters; and (2) making available a separate pro-
gram for each partition, and imbedding the necessary parameters with-
in each program. While the latter method minimizes the transfer and
storage of data, the former is more efficient from a computational
standpoint.
79
-------
START
00
o
Unsaturated Zone Flow
Unsaturated Zone Quality
WRITE
WRITE
Runoff Flow
Runoff Quality
READ
WRITE
Receiving Water Flow
Receiving Water Quality
READ
Groundwater Flow
Groundwater Quality
READ
Data Set "A1
Data Set "B"
•I si M
Data Set "C
END
Figure 19. Illustration of the types of models in the executive program and
the use of data sets for interfacing.
-------
Both of the above approaches are applied to the study area. For ex-
ample, method (1) is used in connection with the various catchments
in the kinematic wave overland flow model. On the other hand, method
(2) is applied to the unsaturated zone.
The next item to be discussed consists of a description of the study
area, involving both geographic and water quality aspects. Nitrogen
and phosphorous are the main quality parameters considered. Each
model is then discussed individually. Subsequently, the interfacing
methodology is explained further, with emphasis on partitioning as-
pects. Finally, objectives considered in devising various simulation
runs are presented.
DESCRIPTION OF THE STUDY AREA
LjOcation^and Size
The Lake Apopka basin is located in the central portion of the State
of Florida, approximately 15 miles (24 km) west of the city of Orlando
(see Figure 3). A map of the basin, showing its drainage boundary, is
provided in Figure 20, The drainage divide lies much closer to the
southern shore of the lake than that originally proposed by the U.S.
Geological Survey (Lichtler, Anderson and Joyner, 1968). One reason
for this modification is that the southern divide used is a secondary
topographic ridge, which should nevertheless constitute the lake's
effective overland flow drainage boundary for most rainstorms. Also,
any subsurface contributions from the omitted area could be accounted
for by specifying boundary conditions beneath the selected divide.
Accordingly, the effective basin area considered was reduced from the
U.S.G.S. estimate of 180 square miles (461 km2) to approximately 120
square miles (307 km2).
As can be observed, Lake Apopka occupies a relatively large portion of
the basin. Its surface area is about 30,000 acres (120 km2) and its
mean depth is only 6 feet (1.8 m) (Sheffield and Kuhrt, 1969). The
mean stage of the lake is 67 feet (20 m) M.S.L.
General Hydrology
The mean annual temperature at Orlando, Florida is 71.5°F (22°C) and
the average yearly rainfall is 51.4 inches (131 cm) (Lichtler, Anderson
and Joyner, 1968). Studies conducted for Orange County (Anderson and
Joyner, 1966), on which most of the basin lies, indicate that about
70 percent of the rain falling on the county returns to the atmosphere
by evaporation and transpiration. The remaining 30 percent is broken
down as follows: approximately two-thirds flows out of Orange County
as surface runoff and one-third flows out under ground. These figures
are presented only with the purpose of providing some insight into
81
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-^ v^
f
-------
general conditions in the area. More specific information on rainfall
and evaporation rates is available from various gaging stations. Hydro-
logic budgeting efforts conducted within the Lake Apopka basin are
discussed briefly in subsequent sections and in Appendix E.
Natural Features and Human Activities
For modeling purposes, the hydrqlogic system was broken down into three
regimes: (1) surface waters, (2) unsaturated zone and (3) ground-
water zone.
Surface Waters—
As a whole, the topography of the basin is of gentle relief, but it
varies locally depending upon the type of soil. The areas east, west
and south of Lake Apopka, shown in Figure 21, are underlain by sandy
s,pils and for the most part are utilized for the cultivation of citrus.
Chemical fertilizers are used in these farming operations. The terrain
in these citrus areas undulates, with a maximum elevation of 250 feet
(76 m) M.S.L. Due to the large number of ridges and local depressions,
the overland flow pattern is quite complex. For rainstorms of moderate
intensity, only a small portion of the total citrus area should con-
tribute overland flow to Lake Apopka. That is, the contributing catch-
ments would be only those located in the vicinity of the lake shores.
In contrast to the undulating topography of the sandy soils, the land
located on the north shore of the lake exhibits an almost completely
flat surface. This region (see Figure 20), originally a part of the
lake's littoral zone, was reclaimed by drainage during the 1940s for
farming purposes (Sheffield and Kuhrt, 1969). It is composed of a
nutrient-rich organic soil known as muck. The total area of the muck
farms is approximately 18,000 acres (72 km2), or about one-third of
the original lake surface area.
Natural, periodic inundation of the muck lowlands has been prevented
by the construction of an earthen dike along the north shore of Lake
Apopka. A drainage system, composed of mole drains, canals and pumps
has been developed to transport excess rain water into the lake. The
nitrogen and phosphorous contents of this drainage water are generally
high (Sheffield and Kuhrt, 1969). These nutrients originate from
fertilizers and also from the muck soil itself.
In addition to farming activities, other man-made effects, shown in
Figure 21, are manifest in the basin. One of these is the Apopka-
Beauclair Canal, constructed in 1948 to withdraw water from Lake
Apopka. With the construction of a lock structure in the canal four
years later, the lake became a managed reservoir (Sheffield and Kuhrt,
1969). The small city of Winter Garden, located on the southeastern
83
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APOPKA-
BEAUCLAIR
CANAL
WINTER GARDEN
CANAL
SANDY SOILS
Figure 21, Illustration of the topographic features
of the sandy soils and other hydrologic
features in the Lake Apopka basin.
84
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shore of the lake, discharges treated municipal and citrus processing
waste water into Lake Apopka.
Another source of flow and associated nutrients to Lake Apopka is
Gourd Neck Spring,, located on the southwestern tip of the lake. The
spring water is thought to originate from a highly permeable deep
formation, known as the Floridan Aquifer (Anderson, 1971). Lake water
quality samples taken by the Orange County Pollution Control Department
in the vicinity of the spring suggest that its nitrogen contribution
is greater than its contribution of phosphorous.
Unsaturated Zone—
The movement of water through the unsaturated zone strongly depends
upon the type of soil present. Literature data from Florida on the
two predominant soil types in the basin, muck (Harrison and Weaver,
1968) and fine sand (Stewart, Powell and Hammond, 1963) indicate their
large capacity to transmit water under saturated conditions. Under
unsaturated conditions, which constitute the general case, hydraulic
conductivities decrease appreciably with decreasing water content.
Unfortunately, very little quantitative information on unsaturated
flow relationships is available for the two soils under consideration.
The thickness of the unsaturated zone (vertical distance between
ground surface and water table) can be estimated at different locations
from U.S. Geological Survey data on shallow observation wells. In
the Lake Apopka basin, the water table is relatively flat, indicating
the lack of appreciable moisture gradients in the horizontal direction
of the unsaturated soil zone. For this reason, the use of a one-
dimensional flow model (Freeze, 1969) for this zone was thought to be
justified. Additional schematic information on the unsaturated zone
will be provided in other discussions.
Groundwater Zone—
Two main geologic formations exist in the basin: (1) a shallow stratum
composed of sand and clays (elastics) with some low permeability layers;
and (2) a deep formation composed of highly permeable limestone, known
as the Floridan Aquifer (Lichtler, Anderson and Joyner, 1968). Data
collected by the U.S.G.S. in Orange County (Lichtler and Joyner, 1966)
indicate that the thickness of the clastic formation fluctuates around
100 feet (30.5 m) in the vicinity of Lake Apopka.
Flow in the Floridan Aquifer occurs predominantly in the horizontal
direction, with the exception of a significant vertical component in
the Gourd Neck Springs area. A typical piezometric map for this
aquifer is provided in Figure 22. It indicates that the general
direction of flow is from southwest to northeast.
85
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Figure 22. Typical piezometric contours, (ft), in the Floridan
Aquifer. Lake Apopka is only weakly connected to
the aquifer at Gourd Neck Springs. Hence, piezometric
values are not constant across the lake. Vertical
cross-sections conform to two-dimensional flow within
the Floridan Aquifer.
86
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The hydraulic head at a given location in the Floridan Aquifer is
not constant, but varies seasonally in response to rainfall patterns
(Lichtler and Joyner, 1966). The extent of these fluctuations depends
upon factors such as degree of proximity to recharge areas. However,
a comparison of piezometric countours at low-water, normal and high-
water conditions (Lichtler, Anderson and Joyner, 1968) indicated
relatively little change in the general direction of flow. This
conclusion is less likely to be valid in basins where seasonal pumpage
effects are significant.
For modeling purposes, it is necessary to consider the selection of
geologic cross sections satisfying the condition of two-dimensional
flow (Freeze and Witherspoon, 1966). An examination of Figure 22
reveals that a large number of roughly parallel sections could be
drawn for the Floridan Aquifer along the dip of its piezometric
surface. Some of the possible choices, valid only for this formation,
are illustrated.
For a more complete examination of the system, however, it is desirable
to locate vertical cross sections which include all geologic strata of
interest. Accordingly, consideration of conditions in the overlying
clastic formation becomes necessary. Referring to Figure 24, a rough
assumption can be made that in the shallow groundwater system, the
horizontal component of flow occurs radially into or out of Lake
Apopka. A review of ground surface contours and U.S.G.S. shallow
well data for the citrus areas indicates that this horizontal com-
ponent is oriented toward the lake. On the north shore, the fact that
the ground elevation of the muck farms is only 65 feet (20 m) M.S.L.,
or 2 feet (0.6 m) lower than the mean lake stage, points out that some
flow must occur from the lake to the farms. The directions of horizon-
tal flow components in the shallow zone, described above, are shown
in Figure 23.
Given the flow patterns in both the shallow and deep formations, it
was possible to select a vertical cross section for analysis. The
location of this section, labeled as AA1, is shown in Figure 23. A
lateral view of AA1 is provided in Figure 24. The location of the
approximate boundary between the shallow stratum and the Floridan
Aquifer is shown. It is recognized that this aquifer extends down-
ward several hundred feet but, for practical reasons, an artificial
bottom boundary was defined as shown.
It may be observed that the vertical thickness of the unsaturated
zone varies considerably along the cross section. In the southwestern
portion , the water table surface is just slightly higher than the
piezometric surface of the Floridan Aquifer, shown in Figure 22. In
the muck farm areas, the water table, controlled by artificial drain-
age methods, lies approximately 2 feet (0.6 m) below the ground.
87
-------
B
f
B'
1
SAN DYxS OILS
C*
. /'
N
Figure 23. Directions of horizontal flow components
within the shallow formations, inferred
from topographic data. Also indicated
are main and auxilliary cross-sections
selected for analysis.
88
-------
200
ISO
iso
140
120
100
SO
SO
40
20
0
-20
-40
-60
-80
-100
• 120
-140
SHALLOW FORMATION
CLASTICS
FLORIDA AQUIFER
LIMESTONE
APPROXIMATE
GEOLOGIC BOUNDARY
Figure 24. Lateral view of cross-section AA1, (Figure 23)
-------
Finally, the thickness of the unsaturated zone for the region northeast
of the muck farms is assumed to be approximately 5 feet (1.5 m).
Additional data on this cross section, including permeability estimates,
are provided in subsequent sections.
In Figure 23, sections BB' and CC1 were other original candidates for
analysis. These sections extended downward only through the clastic
formations, since their orientation would preclude their validity
within the Floridan Aquifer. Preliminary computer results indicated
the existence of relatively little flow in the vicinity of Lake Apopka
within both BB1 and CC'. Due to their low degree of interaction with
the lake, further analyses of these sections were not conducted.
WATER QUALITY STUDIES
Reports dating back to 1947 indicate a gradual decline of water quality
conditions in Lake Apopka, mainly in the form of nutrient enrichment
(Florida State Board of Health, 1964). In a recent statistical
comparison of several lakes in Florida, Putnam e_t al. (1969) found
Lake Apopka to be the most eutrophic. However, cause and effect rela-
tionships between the various external sources of nutrients and lake
water quality are somewhat elusive.
Natural tendencies of lakes toward (or away from) eutrophy have been
a subject of controversy among scientists (Brezonik, 1969). In the
1930s and 1940s the predominant view was that lakes follow a slow but
irreversible path from their initial state of oligotrophy (low nutrient
content) to eutrophy. According to this theory, man-made, or cultural,
activities merely accelerate a natural process.
In recent years, an opposing theory has been proposed which states
that lakes naturally tend to evolve from eutrophy to oligotrophy but
that this process can be reversed by external processes such as man's
activities (Mackereth, 1965; Hutchinson, 1969). Studies of lake core
samples appear to verify this theory. Additional support for it
derives from ecological principles. According to Odum (1969), the
eutrophic state has many characteristics of immature ecosystems, such
as low stability and low species diversity, while the reverse is true
of the oligotrophic state. Thus, he reasoned, a lake should evolve
toward the oligotrophic state if left alone. This latter theory offers
encouragement to efforts to control or reverse eutrophication.
Several studies have been conducted in the Lake Apopka area to in-
vestigate the relative impact of the various nutrient sources on water
quality. In 1968, Forbes reported the results of a sampling program
to compare the nutrient levels in the muck farm ,canals to those in
the lake itself. He pointed out the difficulty in establishing a cause
90
-------
• TfcW-^fc'vrtA
jB:.*V(\^-;:
*" .: t
Figure 25. Reproduction of project map containing locations of various
sampling points.
91
-------
and effect relationship between the farming operations on the muck
soil and the earlier reported deterioration of lake water quality.
A comprehensive study was completed in 1968 by Schneider and Little,
which included consideration of various nutrient sources in addition
to the muck farms. For instance, nitrogen and phosphorous samples
were collected from shallow wells in the citrus groves. Also, mea-
sured nutrient concentrations in rain water indicated that its contri-
bution to the lake was significant. Schneider and Little placed great
emphasis on their lake sediment sampling program and on the probable
importance of nutrient recycling from these bottom deposits to the lake
water, Finally, they recommended further work on the development of
a nutrient budget for Lake Apopka.
In 1969, Sheffield and Kuhrt proposed a nutrient budget, which was
further discussed by Sheffield in 1970. A major conclusion derived
from this budget was that the muck farms constituted the principal
source of nutrients to the lake, In 1972, Heaney, Perez and Fox
developed a nutrient budget for the muck farms area. This study is
briefly discussed in Appendix E,
In the early phases of this investigation, the coordinates of the
various sampling points used by previous investigators were established
and their location indicated on a master map. The sampling points
included surface water quality stations used by different agencies,
shallow wells and deep U.S.G.S. wells. A reproduction of the project
map with the sampling points is provided in Figure 25.
To.avoid confusion, a symbol system was developed for the different
types of stations and agencies. A list of the various data sources
used in this study is given in Appendix E.
USAGE OF MODELS
According to a previously developed format, the computer-models in
this study are classified as to their zone of applicability, namely;
surface, unsaturated zone and groundwater zone. The model usage as-
pects for these three categories are discussed below.
Surface Water Models
The surface water processes considered are overland flow and flow in
bodies of water. Two overland flow models have been discussed pre-
viously: (1) the kinematic wave formulation (Eagleson, 1970), and
(2) the runoff portion of the Stormwater Management Model (Environ-
mental Protection Agency, 1971). The latter model, originally developed
for urban runoff systems, is relatively well suited for adaptation to
agricultural areas in which the influences of man are greatly felt.
92
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Thus, the Stormwater Management Model was selected for use in the
highly channelized muck farms.
On the other hand, the kinematic wave model is better suited for areas
with few artificial drainage controls. Accordingly, it was chosen for
application to the sandy, undulating terrain.
It is recognized that, due to the high infiltrative capacity of both
the muck and sandy soils, the phenomenon of overland flow appears to
be of limited significance in the Lake Apopka basin. However, the
application of the two model types was deemed desirable for illustrat-
ing their data requirements and interfacing aspects with other models.
The surface bodies of water considered were the drainage canals in the
muck farms area and Lake Apopka itself. The receiving model portion
of the Stormwater Management Model (Environmental Protection Agency,
1971) was utilized for the flow and quality computations in these bodies
Further discussion on the applications of this model follows that on
the overland flow formulations.
Kinematic Wave Model
Data Sources—
The terrain east, west and south of Lake Apopka (see Figure 22) is
composed primarily of sandy soils. The predominance of this type of
soil is shown in Figure 26, which consists of a soil classification
map of the area. The map in this figure is a reproduction of a plas-
tic overlay map developed from U.S. Soil Conservation Service data
(1967) for Orange and Lake counties. The scale of the plastic over-
lay is the same as that of U.S. Geological Survey quadrangle maps
(1:24000).
Also illustrated in Figure 21 are'some of the numerous localized ridges
and depressions, which result in a complex overland flow pattern. ,
These topographic features are provided in detail in U.S.G.S. quad-
rangle maps. The undulating nature of the local topography indicated
the need for partitioning the area into numerous idealized plane
catchments,
Regarding land use, earlier reports (Schneider and Little, 1968;
Sheffield and Kuhrt, 1969) have indicated that the main activity in
this area is the cultivation of citrus. Since land use is an impor-
tant factor affecting overland flow and water quality aspects, addi-
tional information was sought on other activities. One source of
data consisted of aerial maps available from Soil Conservation Service
soil surveys (1960). Another source consisted of acreage figures for
various agricultural and nonagricultural uses, compiled also by the
Soil Conservation Service (SCS). These figures, available from local
93
-------
Figure 26. Reproduction of soil classification overlay map
94
-------
SCS offices, were tabulated for each of the various soil type regions
shown in Figure 26. In this study, land use data were utilized as a
guide for subjective estimation of lumped model parameters, such as
Manning's surface roughness.
Layout of Catchments—
It then became necessary to consider the division of the area into
catchments. For simplicity, it was assumed that the entire area was
underlain by sandy soils, and thus soil type was excluded as a criterion
for partitioning. Similarly, the scope of this particular investigation
precluded the utilization of land use data in the division process.
Therefore, the layout of catchments was carried out using only topo-
graphic criteria.
In the kinematic wave model used in this study, no provision is made
for a given plane catchment receiving discharge from another upstream
catchment. Thus, this situation, illustrated in Figure 27, part (a),
could not be allowed to exist in the catchment layout. Instead, the
alternative approach required by the present model would be to con-
sider the upstream and downstream catchments as a single entity.
This simplification is shown in part (b) of the same figure.
In general, the result of the simplification applied in (b) is
decreased model accuracy due to the lumping of catchment slopes.
However, in the sandy soils under consideration, the topography was
such that the cascading case depicted in (a) was uncommon. In basins
where cascading is widespread, it would be advisable to improve the
capabilities of the kinematic wave model to handle this situation.
In this regard, a review of recent studies at the Massachusetts In-
stitute of Technology (Maddaus and Eagleson, 1969; Harley, Perkins
and Eagleson, 1970), among other sources, should be useful.
In the sandy soils of the Lake Apopka basin, the predominant pattern
is that of radial flow into local depressions. This situation is
illustrated in Figure 28. Also shown in this figure is the'division
of the land surface into catchments. To avoid any lateral flows be-
tween catchments, their boundaries were drawn normal to contour lines.
Ideally, the catchment slopes should be uniform, and their shape
rectangular.
The catchment layout for the sandy soils area is provided in Figure 29.
This figure consists of a reproduction of a plastic overlay map,
similar to that used in delineating soil types (Figure 26). While most
catchments drain into local depressions, others provide runoff to Lake
Apopka. The latter catchments, of greater interest in this study, are
depicted in Figure 30.
95
-------
RAINFALL
vo
.CATCHMENT I
SLOPE =S,
PART A SIDE VIEW OF STREAM
DOWNSTREAM CATCHMENTS
EQUIVALENT CATCHMENT
SLOPE = s' = Si + Sg
PART B SIDE VIEW OF IDEALIZED
EQUIVALENT CATCHMENT
Figure 27. Illustration of the lumping of two successive
catchments into one.
-------
CATCHMENT
BOUNDARY
SYMBOLS
I * EFFECTIVE LENGTH
^EFFECTIVE \VfDTH
CONTOUfeS
Figure 28. Illustration of the topographic contours of the
sandy soils and delineation of overland flow
catchments.
97
-------
I
•
Figure 29. Reproduction of an overlay map showing the layout of
overland flow catchments in the sandy soils.
98
-------
LAKE
APOPKA
Figure 30. Approximate location of catchments in the sandy
areas feeding into Lake Apopka.
99
-------
Model Parameters—
Regarding model parameters, the information contained in Figure 29
allowed the determination of the approximate length, width and slope
of each catchment. The value of the Manning roughness coefficient
(n = 0.025) was estimated subjectively from land use information,
with the aid of a table (Henderson, 1966). Assuming turbulent flow
conditions, the exponent parameter, m, in the formulation was set
equal to 5/3 (Eagleson, 1970).
In the quality portion of the model, the grain size selected was that
proposed by Gottschalk (1964) for fine sands (1/8 mm). The correspond-
ing parameters for the DuBoys sediment transport formula were obtained
from Einstein's (1964) work. Estimates of fertilizer nutrient content
and loading rates in the citrus groves were made available by the East
Central Florida Regional Planning Council. The quality assumptions
and calculations involving nitrogen and phosphorous are discussed in
Appendix E. References on nutrient availability at the ground surface
originating from animal wastes (not of interest in this study area)
are provided in Appendix F.
Finally, the overland flow model receives input from the unsaturated
zone model, in the form of infiltration rates. The spatial variability
of the thickness of the unsaturated zone in the sandy soils was shown
in Figure 24. However, partitioning of the unsaturated zone on this
basis was deemed unwarranted from a model demonstration standpoint.
Instead, an average thickness value was estimated. Additional details
are given in discussions of the unsaturated zone model.
Runoff Portion of the Stormwater Management Model
The runoff portion of the Stormwater Management Model, which was orig-
inally developed for use in urban basins (Environmental Protection
Agency, 1971), was adapted for application to the organic (muck) soils
area north of Lake Apopka. The general location of this area, compris-
ing approximately 18,000 acres, was shown in Figure 20. Due to the
high permeability of the muck soils (Harrison and Weaver, 1958) and
the widespread existence of mole drains in the area, it is unlikely
that significant ponding or overland flow will occur, except under
rainstorms of very high intensitites. Therefore, the application of
the Stormwater runoff model was primarily for illustrative purposes,
especially with regard to the interfacing aspects with the receiving
model. At this point, it is pertinent to interject a brief discussion
of the general hydrology of the muck farms.
Hydrologic Inflows and Outflows—
From a hydrologic budgeting standpoint, the muck farms area is much
more complex than the sandy soils region. The various surface inflows
and outflows (aside from rainfall and evaporation) for the muck farms
100
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are illustrated in Figure 31 for the Zellwood Drainage District. The
least significant of the factors shown is considered to be the over-
land flow from the higher, sandy soils. On the other hand, the pump-
age rates into Lake Apopka are known to be high, especially following
large rainstorms. Unfortunately, these pumpage records, made avail-
able through the East Central Florida Regional Planning council
(ECFRPC), constitute rough estimates based on approximate duration
of pumpage and head differential. Furthermore, the rates are given
as district totals, and are not broken down by individual pumps.
In dry periods, irrigation water for the muck farms is obtained from
Lake Apopka through several flap valves located along the earthen
dike. The approximate number and diameter of these inlets, shown
also in Figure 31, were obtained from Zellwood Drainage District re-
cords. However, no records are kept on the operation of the valves.
Rough calculations indicate that at any time in which valves are open,
the resulting flows could be significant.
In the subsurface water category, the inflows and outflows for the
muck farms area are also not well known. Section EE1, shown in
Figure 31, can be examined to illustrate the groundwater regime. A
lateral view of this section is provided in Figure 32. Considering
first the left-hand portion of the section, it becomes apparent that
seepage could occur from Lake Apopka to the muck farms through the
earthen dike, although the amount is uncertain. Also, groundwater
flow originating in the sandy soils could enter the muck farms. The
extent of the latter contribution would depend upon the actual water
table elevation in the sandy areas and the local permeability.
Additional discussion on this point is provided in the sub-section on
groundwater.
It follows from the above comments on the nature of the surface and
subsurface flows that the system at hand is indeterminate. At best,
only rough estimates or ranges of inflows and outflows could be used
in budgeting attempts for the general area (Heaney, Perez and Fox,
1972). This hydrologic uncertainty affected the calculations for
runoff and other phenomena.
Layout of Runoff Catchments-
The purpose of the runoff formulation is to compute the flows and
nutrient fluxes into the drainage canals. The flow and quality pro-
cesses in the canals are handled by the receiving model. A diagram of
an isolated portion of the drainage system under study is provided in
Figure 33. Part (a) of this figure constitutes a plan view of the
isolated unit.
Indicated therein are the mole drains, which are spaced approximately
20 feet C6 m) apart. A lateral view of the system is shown in part (b).
101
-------
N
\ ZELLWOOD DRAINAGE
\ DISTRICT BOUNDARY
\
\
\
\
v
\
\
DRAINAGE
CANAL
LAKE APOPKA
2 VALVES OF
18 DIAMETER
Figure 31. Map of Zellwood Drainage District showing location of
pumps and flap valves.
102
-------
EARTHEN
DIKE
SANDY
SOILS
WATER
TABLE
.ZELLWOOD DISTRICT
INFLOW FRONT
LAKE APOPKA
\
INFLOW FROM
SANDY SOILS
V ^'
Figure 32. Simplified lateral view of vertical section EE1 (Figure 31) in
Zellwood Drainage District.
-------
RAINFALL
11 m 11 m
.RUNOFF
MOLE DRAIN
PART(A) PLAN VIEW
PARK BILATERAL VIEW OF FF1
Figure 33. Plan and lateral views of an isolated portion of the drainage
system in the muck farms.
-------
The receiving model applied to the canals utilizes flow and -quality
inputs from two sources: (1) overland flow (runoff model) and (2)
mole drains (tmsaturated zone model).
Considering drainage aspects from a broader standpoint, the muck farms
are effectively divided into two systems of roughly equal acreage:
the Zellwood Drainage District (shown in Figure 31) and the Duda
Drainage District. Figure 34 is a reproduction of a working map of
the former, indicating the numbers used for the various catchments and
receiving junctions. A similar map is provided for the Duda area,
in Figure 35. In the runoff simulation, these two districts were
considered as independent entities, and calculations were conducted
separately.
Model Parameters—
In the flow computations, two main parameters were required: land
slope and Manning's roughness (n). Due to the flat nature of the
muck farms surface, the former could not be estimated from topographic
maps. Instead, an arbitrary value of 10~u ft/ft was selected for all
catchments in Zellwood and Duda. The value of the roughness coeffi-
cient used was n = 0.04.
An important parameter required in the quality portion of the model
was the grain size of the muck soil. In the absence of data, the
assumption was made that, in suspension, muck is somewhat akin to silt.
Accordingly, a grain size of 10 mm was used. Other corresponding
sediment load parameters were estimated by extrapolation of the tables
given by Einstein (1964). Additional information on the suspended
solids calculations are given in Appendix A.
The nutrient contents of the suspended solids loads, computed above,
were estimated with the aid of fertilizer data provided by the East
Central Florida Regional Planning Council CECFRPC). Specifically,
data on nutrient content in fertilizer, loading rates and crop types
were obtained for each major farm. A map showing the subdivision of
the Zellwood district into farms is provided in Figure 36. Yearly
average nutrient loading rates were computed for the entire muck
farms area by Heaney, Perez and Fox (1972). The estimation of nutrient
contents in suspended solids using the above yearly rates is explained
in Appendix E.
Receiving Water Model
In this study, the receiving formulation was utilized in a stepwise
fashion. First, it was applied separately to the Zellwood and Duda
districts in the muck farms area. Subsequently, the output from these
two programs was used as input to the receiving program for Lake Apopka.
105
-------
/ -, 1, ',
'l
ZELLWOOD DRAINAGE
ANCt
WATER CONTROL DISTRICT
UNIT NO. IAN* UNIT NO. 2
OPANGE COVNTY
FLQ01OA
Figure 34. Reproduction of runoff map for the Zellwood Drainage
District in the muck farms.
106
-------
© f
V,-\ ,-ir-L_l
DUOA \U ® j
CLAP
JUNCTION NOS.
PUMP
SUBCATCHMENT
PIPE
FARM BOUNDARY
SUBCATCHMENT BOUNDARY
CANAL
[WEST SIDE MUCK FARMS
Figure 35. Runoff map for the Duda Drainage District
in the muck farms.
107
-------
Hug*,
T
i&&=.
MAP OF
ZELLWOOD DRAINAGE
ANQ
'' WATER CONTROL DISTRICT
UNIT NO. t*no UNIT NO. 2
OffAM&E COUNTY
_i J-
Figure 36. Reproduction of map showing the different farms within the
Zellwood Drainage District.
108
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Application to Drainage Canals—
To consider the different variables and parameters used in the model,
it is worthwhile to isolate a small portion of a hypothetical canal
system for closer examination. This isolation process is carried out
in Figure 37. Shown therein are the boundaries of a typical receiving
unit, known as a junction. Associated with each junction are its bot-
tom elevation and water surface area. These parameters were estimated
from canal geometry data obtained by the East Central Florida Regional
Planning Council (ECFRPC) from personnel associated with drainage opera-
tions. Most of the canal geometry data were quite rough approximations,
since no accurate records exist and the canals are frequently dredged.
However, supplementary field measurements would have been beyond the
scope of this study.
In addition to these parameters, an initial value of water surface
elevation must be provided at the beginning of a simulation run. The
Manning coefficient value used for the canals was n » 0.025.
With regard to water quality calculations, the absence of data
dictated that a value of zero be specified for source/sink and decay
rates. Likewise, the initial nitrogen and phosphorous concentrations
were set at zero for convenience. The zero values above were not
used in all computer runs. For instance, concentration data collected
by the Orange County Pollution Control Department were sometimes used
as initial values.
From the previous discussion of the runoff model (Figure 33), it may
be recalled that a channel junction receives flow and quality input
from both runoff and the mole drains. In addition to these two sources
shown in Figure 33, provision was made to include vertical ground-
water flow through the channel bottom. In the muck farms, it is
thought that this flow occurs in the upward direction, and its magni-
tude was obtained from hydrologic budgeting estimates by Heaney, Perez
and Fox (1972). Data on the quality characteristics of this upflowing
water are not available. Evaporation rates were computed from pan
evaporation rates for the U. S. Weather Bureau station at Lisbon,
Florida. Using a pan coefficient of 0.78 (Heaney and Huber, 1971),
the computed rates were approximately SO in/yr (127 cm/yr).
Turing to broader scale aspects of usage, the receving model was applied
separately to the Zellwood and Duda drainage districts (see Figures
34 and 35). In the Duda area, certain simplifying assumptions were
made on the linkages between canals and pumps. These simplifications
are illustrated in Figure 38.
The flow portion of the receving model involves iterative calculations
to determine the stage at the various junctions, in response to pumpage
and other hydrologic inputs to the channels. Since the pumping figures
109
-------
CM
_J
<
Z
<
I
| CANAL 5
I
L
SURFACE
AREA
n r
CANAL 6
Figure 37. Illustration of a typical receiving water model junction in the
muck farm canals.
-------
EUSTIS
3
/ l(?) '—•
FRANKS ' lw I ._
/4 * 2 I [4
DUO A \
CLAP
JUNCTION NOS.
PUMP
SUBCATCHMENT
PIPE
FARM BOUNDARY
SUBCATCHMENT BOUNDARY
CANAL
[WEST SIDE MUCK FARMS
Figure 38. Map of the Duda Drainage District showing
simplified connections for the receiving
water model.
Ill
-------
available for both Zellwood and Duda were not broken down by indivi-
dual pumps, arbitrary weights (percentages of total pumpage) were
assigned to the pumps. For instance, pump numbers 1 through 4 in the
Zellwood District were given weights of 0.20, 0.35, 0.25, and 0.20,
respectively.
Application to Lake Apopka—
While the surface inflows from the muck farms occur as point sources,
any overland flows from the sandy areas (see Figure 30} consist of
distributed input. Data for other inflows and outflows (see Figure 21)
are available from other sources. Preliminary measurements of Gourd
Neck Spring flow by the U.S.G.S., taken during low piezometric condi-
tions in the Floridan Aquifer, indicated an approximate value of 18 MGD
(0.79 m3/sec). It was assumed herein that the spring flow varied
linearly with the head differential between the aquifer and the lake,
and thus a yearly average flow value of 22 MGD (0.96 m3/sec) was ob-
tained. The average annual outflow from the Winter Garden canal was
estimated by the East Central Florida Regional Planning Council
(ECFRPC) to be 2.1 MGD (0.092 m3/sec). Quality data for this canal
are available from the Orange County Pollution Control Department.
Daily flows at the Apopka-Beauclair Canal were tabulated by the ECFRPC.
Also, evaporation rates were estimated at 50 inches per year (127 cm/yr)
as discussed previously.
The receiving model has the capability of incorporating subsurface
inputs in addition to the surface flows discussed above. However, in
the case of Lake Apopka the former were neglected. As pointed out
by Anderson (1971), lateral seepage from the surrounding sandy areas
to the lake probably accounts for only a small percentage of total
lake inflow. Also, a comparison of the mean lake stage, 67 ft (20 m),
M.S.L. to the piezometric map of the Floridan Aquifer (see Figure 22)
indicates that upward flow occurs in the southwestern half of the
lake, and downward flow of possibly the same order of magnitude takes
place in the northeastern portion. Although Anderson's report implies
that these two flows are not offsetting, a simplifying assumption of
no vertical flow throughout the lake was made for model demonstration
purposes.
No data are available for the flows from the lake to the farms through
the flap valves, or for seepage through the earthen dike. For this
reason, these two factors could not be included in the analysis..
The development of a link-node, or channel-junction, system for Lake
Apopka was carried out by first considering the location of the various
surface inflows and outflows. It was concluded that boundary nodes
should be located at approximately equally spaced intervals along the
periphery of the lake, and as close as possible to the inflow/outflow
locations. Next, the positioning of the interior nodes was considered
112
-------
in the light of prior computational efforts in the Florida Everglades
by Heaney and Huber (1971). Programming experience gained in that
study indicated the desirability of designing a link-node structure
in which: (1) links should form sets of approximately equilateral
triangles and (2) triangles should have about the same size, Another
criterion for locating nodes was proximity to the Orange County Pollu-
tion Control Department sampling stations. The resulting link-node
system developed using these criteria is shown in Figure 39. The flow
and quality inputs from the catchments in the sandy areas were assigned
as point sources to the nearest boundary node.
The geometrical data required for a typical junction consist of bottom
elevation and water surface area. For the purposes of this study, the
lake bottom was assumed to be flat, although contour maps of the bottom
surface (Sheffield and Kuhrt, 1969) indicate some areal variability.
The principle for delineating the surface area assigned to each junc-
tion is analogous to that used in determining Thlessen weights in
rainfall calculations (Linsley and Franzini, 1964).
Quality computations in the lake receiving model require data on source/
sink and decay rates, in addition to initial pollutant concentrations.
In general, the same guidelines that applied to the drainage canals
were used in the lake. In their 1968 study of Lake Apopka, Schneider
and Little pointed out the importance of nutrient cycling between water
and sediments, and mentioned the work of Kuznetsov (1968) in the Soviet
Union. A review of Kuznetsov's paper indicated that little attention
was devoted to phosphorous, and the applicability of his nitrogen rates
to Lake Apopka was considered uncertain. However, this and other papers
should provide insight into the methodology for obtaining these needed
source/sink rates.
Unsaturated Zone Model
General Description—
Boundary conditions- The main boundary conditions involved in this
model are: rainfall, evaporation and net recharge. These parameters
are illustrated in Figure 40. In the Lake Apopka basin, rainfall data
were available from the U. S. Weather Bureau stations at Clermont and
Lisbon. Pan evaporation measurements were available at the latter
station. Also, daily rainfall records for various stations within the
muck farms area were provided.
Regarding the usage of these data, it might be first pointed out
that, due to the high infiltrative capacity of the soils in the basin,
the utilization of average daily (or even hourly) rainfall rates
would have resulted in the prediction of no overland flow. Thus,
synthetic rainstorms were constructed somewhat arbitrarily from the
existing data, in which the intensity was expressed as an exponentially
113
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INPUTS AT JUNCTION
N-FLUX, Ib/doy P-FLUX, Ib/doy
1
2
4
5
9
25
27
500
200
300
200
200
200
200
21,600
21,600
16,200
9.TOO
—
9,700
3,230
810
860
650
433
—
433
108
Figure 39. Location of various junctions for the
receiving water model application in
Lake Apopka. Also shown are inputs for
the lake simulation.
114
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RAINFALL EVAPORATION
'.'S/S/S//S/S/SS / S^S / / ///. \-7>f/ .' X
MOLE DRAIN
PLANTS
DIRECTION OF FLOW
WHEN FULL
WATER TABLE
NET RECHARGE
Figure 40. Cross-sectional view of the unsaturated zone, showing boundary conditions
and other model features.
-------
decaying function of storm duration. This relationship is considered
to be approximately valid for thunderstorms (Eagleson, 1970). In
future studies, it would be desirable to obtain better rainfall infor-
mation, since it is an important factor in determining so-called first
flush effects in overland flow quality.
In evaporation calculations, a pan coefficient of 0.78 (Heaney and
Huber, 1971) was utilized. The assumption was made that, since soils
in the basin are irrigated (especially muck), water would always be
available to satisfy the potential evaporation. Estimates of recharge
rates were obtained from the groundwater model.
Additional features- In addition to the external flows, the unsaturated
zone model developed herein considers two features: (1) withdrawals
of water by plants and (2) outflows via mole drains. These two fea-
tures are also shown in Figure 40. Their corresponding mathematical
formulations are presented in Appendix E.
Areal partitioning criteria— A simplistic partitioning process for the
unsaturated zone was conducted to promote the homogeneity of model
parameters within each region. Accordingly, the criteria considered
for this division process were those parameters which were thought
to vary areally in the basin. Since lack of data precluded considera-
tion of areal variations in rainfall and evaporation, the remaining
factors were: (1) net recharge rate, (2) soil type, (3) crop type,
and (4) existence of mole drains. Items (3) and (4) were considered
homogeneous within each soil type, thus leaving only (1) and (2) as
effective criteria.
The next step in the partitioning process was to consider the vari-
ability of recharge conditions within the two major soil types, muck
and sand. An examination of Figure 24 indicated that, while ground-
water flow conditions might be assumed to be homogeneous within the
muck farms, wide differences could exist between conditions in the
sandy soils in the southwest portion of the basin and those in the
sandy soils north of the muck farms. Accordingly, three unsaturated
zone programs were used: two for the sandy soils and one for the
muck farms. The distinctive features of each partition, such as soil
mositure-pressure-conductivity relationships, plant root depth and
location of mole drains (if any), are imbedded in its corresponding
computer program.
Sandy Soils—
Description of areas— The two separate sandy soils regions discussed
above are shown again in Figure 41. For the purposes of this study,
the soil hydraulic characteristics, crop information and fertilizer
loading rates were assumed to be equal for both regions. Also, both
unsaturated zones were assumed to be homogeneous. However, their
geometrical characteristics and recharge rates differ.
116
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IDEALIZED SECTION I
SSS f/Sf fff
IDEAUZED SECTION 2
SS
./•yxx x^x
',—
SANDY SOILS
REGION I
t, = EFFECTIVE THICKNESS
MUCK FARMS
SANDY SOILS
REGION £
Figure 41. Partial view of cross-section AA» (Figure 23), illustrating the
idealization of the unsaturated zone geometry.
-------
To conform with geometrical assumptions inherent in the model, the
ground and water table surfaces were redrawn in Figure 41 as parallel,
horizontal lines. The resulting distortion, especially apparent in
the northeastern region, is due in a large measure to the vertical
scale exaggeration in the figure. Thus, horizontal moisture gradients
are likely to be of little actual significance. The horizontal lin-
earization process for the water table is carried out only for inter-
nal calculations in the unsaturated zone model. Prior to the transfer
of data to the groundwater model, any computed change in water table
elevation is added uniformly to the original irregular surface.
Flow parameters- Included in both programs for the sandy areas are
function subprograms containing the algebraic expressions for the
following soil hydraulic relationships: (1) moisture versus pressure
head, (2) conductivity versus pressure head, (3) slope of the moisture
versus pressure head curve, as a function of the latter and (4)
pressure head versus moisture content. Using data for Florida sandy
soils (Stewart, Powell and Hammond, 1963), it was possible to construct
a curve for item (4), which is shown in Figure 42. The value of sat-
urated hydraulic conductivity (permeability) assumed herein was 3
inches per hour (7.6 cm/hr). A plot of item (2) is provided in
Appendix E.
Regarding evapo-transpiration from citrus orchards, a yearly mean
rate of 0.12 in/day (0.3 cm/day) was estimated from studies by Hashemi
and Gerber (1967), It was assumed that transpiration from the plants
accounted for four-fifths of this total water loss, the remaining por-
tion being evaporation from the soil. The root depth of citrus was
estimated at 5 ft (1.5 m) from a table provided by Eagleson (1970).
The water uptake rate by citrus was assumed to vary with depth in the
following manner: 75 percent of the total demand is removed from the
upper one-fourth of the root zone and the remaining 25 percent from the
lower three-fourths.
Quality parameters— Fertilizer application rates and nutrient contents
for the citrus areas in the basin were made available by the East
Central Florida Regional Planning Council. Other interspersed land
uses exist in the sandy soils, but for convenience these loading
rates were prorated on the basis of citrus versus total acreage, and
then assumed to apply uniformly over the sandy soils.
Due to lack of appropriate data, the setting of source and sink terms
for both nitrate and total phosphorous was based on rough assumptions.
It would appear that sandy soils should not release significant amounts
of nutrients, and thus the source terms for both substances were set
at zero. The estimation of sink rates was more elusive. Since Schnei-
der and Little (1968) reported high N and P in their shallow sampling
wells in the citrus areas, trial values of zero for these two
118
-------
-300
£-200
H
•^
o
5
to
u
I
•100
SANDY SOILS
O.I 0.2 0.3 0.4
VOLUMETRIC MOISTURE FRACTION.fl
Figure 42. Estimated pressure head versus volumetric
moisture fraction curve for sandy soils.
O.K) 0.20 0.90
0.40 (00 C«0 O.TO
VOLUMETRIC MOSTURE FRACTION,8
HBO
Figure 43. Estimated pressure head versus volumetric
moisture fraction curve for muck soils.
119
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substances were used. Also, little information was available on the
nutrient uptake rates by citrus crops. The assumption was made that
the ratios of yearly nitrate and phosphate uptake to the corresponding
amounts applied in fertilizer were the same as those obtained for the
muck farms. The corresponding calculations are shown in Appendix E.
The dispersion coefficients at the various nodes, which depend upon
the corresponding velocities, were computed using the Hoopes and
Harleman (1965) formula discussed earlier. The original constants
used therein for sand were utilized in this study.
Muck Farms—
Description of area— The surface of the muck farms is very flat.
Likewise, the water table is maintained at approximately 2 feet (0.6 m)
below ground level with the aid of mole drains and drainage canals.
Except for these discontinuities in the soil medium, the system is
compatible with the assumption of one-dimensional, vertical flow.
Also, the horizontal cross-sectional area occupied by the drainage
canals is only a small percentage of the total muck farms area. The
behavior of the mole drains was incorporated into the model (see
Appendix E ).
Flow parameters— The unsaturated flow model contains function sub-
programs for all the soil hydraulic relationships listed previously
in the sandy soils section. The pressure head versus moisture curve
used (Weaver and Speir, 1960) is shown in Figure 43. The conductivity
versus pressure head curve is provided in Appendix E. As in the sandy
soils, this curve is hypothetical. Very little information is avail-
able on this aspect of muck soils. Based on studies by Harrison and
Weaver (1958), the saturated conductivity of muck was assumed to be
7.5 inches per hour (19 cm/hr).
Since the moisture content of the muck soils is carefully controlled,
initial moisture conditions were assumed equal to field capacity at
every node in the profile. The water requirements of the vegetable
crops grown in the area were estimated at 0.15 inches per day (0.38 cm/
day) (Harrison, Stewart and Speir, 1960). The root depth for all these
crops was assumed to be 1 foot (0.3 m). Calculations for mole drains
were based on a formula developed in the Netherlands (Weaver and .Speir,
1960) and are explained in Appendix E.
Quality parameters- The initial nitrate and total inorganic phosphor-
ous concentrations at the top node of the soil profile were estimated
from the results of a nutrient budget study conducted by Heaney, Perez
and Fox (1972). The corresponding calculations, which were based on
assumptions about nutrient movement and availability, are briefly dis-
cussed in Appendix E. Plant nutrient uptake rates were estimated from
data contained in this nutrient budget study, such as crop harvest
120
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weights, N and P contents and duration of growing seasons. The cal-
culation of these uptake rates is also discussed in Appendix E.
Although fertilizers provide nutrient input to the percolating water,
it appears that the major source of nutrients to this water is the
muck soil itself. The water quality data collected by Hortenstine
and Forbes (1971) for both fertilized and nonfertilized plots in the
muck farms area provide support to this theory. Unfortunately, the
driving forces and mechanisms affecting the release of these nutrients
are not well understood. Muck of the research work on muck soils has
dealt with consolidation aspects (Stephens, 1969), with little empha-
sis on water quality.
With regard to source/sink effects, the above quality data suggested
the existence of a large net source term for both N and P. The values
of these terms, however, were quite elusive and therefore several trial
values were used. Likewise, the initial concentrations at the various
nodes were sometimes set at zero for convenience, or assigned some
assumed values. The discussions on the individual computer runs pro-
vide additional information on these trial values.
In the computation of dispersion coefficients, the constants used
were those associated with sand and are likely to be higher than
corresponding values for muck. The reason for this approach was lack
of data for muck soils.
Groundwater Model
Description of the System—
Location of cross-section— The main cross-section considered in this
study was one originating at the southwest portion of the basin and
extending to its northeast boundary (see Figures 23 and 24). Two
main geologic formations exist in the area: the shallow clastic
aquifer and the deep and highly permeable Floridan Aquifer. The flow
pattern within this cross-section depends upon the permeabilities of
these formations and the flow boundary conditions. Water quality
conditions are influenced by the activities occurring on the surface
of the basin.
Delineation of boundaries— In the model developed by Freeze and
Witherspoon (1966), the shape of vertical cross-sections considered
was rectangular, with an irregular top boundary representing the
water table. Also, they assumed that no flow occurred across the left,
bottom and right boundaries of the system.
While many well-defined basins should conform fairly satisfactorily
with these no-flow assumptions, it becomes apparent that some viola-
tions should be expected in more complex systems. Such is the case
121
-------
in the cross-section under study, shown in Figure 44. Since flow in
the Floridan Aquifer is predominantly horizontal, the portions of the
left and right boundaries located in this aquifer required attention.
Reasonable estimates of flow across these two portions could be ob-
tained from piezometric and permeability data. Thus, the original
no-flow condition in each boundary portion was replaced by a flow
specification.
In a qualitative discussion of shallow aquifers, Freeze and Withersppon
(1967) generalized that, barring unusual permeability configurations,
the water table surface should roughly follow the topographic surface.
Thus, in many cases, topographic divides (ridges or depressions)
indicate corresponding hydrologic divides. In the shallow formation
under consideration in Figure 44, the left boundary was located in
the vicinity of a topographic ridge, and the right boundary near a
depression. An alternative location for the right boundary might
have been at the nearby ridge. In the absence of data indicating the
contrary, a no-flow condition across both boundaries selected was
assumed valid.
Finally, it is recognized that the existence of Gourd Neck Spring
implies the existence of a vertical flow component in its vicinity.
It was assumed that the bottom boundary was located at a sufficient
distance away from the spring to insure that the vertical flow
component across the boundary would, in effect, be negligible. If
additional data or calculations should indicate significant vertical
flows in the vicinity of this assumed boundary, its downward reloca-
tion would be necessary to conform with the no-flow condition.
Permeability estimates— To improve upon the simplified geologic picture
presented above, additional data were sought from well logs. The
locations of these wells, drilled by the U.S.G.S., were indicated on
the master project map reproduced in Figure 25. Since none of these
wells lay exactly on the vertical cross-section under study, the
delineation of geologic boundaries involved extrapolation and judgement.
The boundaries between the different formations, superimposed on a
grid system, are shown in Figure 45. Also indicated are the perme-
ability values assumed for these formations. In the shallow region,
these estimates are within perhaps an order of magnitude. The per-
meability value for the Floridan Aquifer was estimated from the results
of pumping tests near Orlando (Lichtler, Anderson and Joyner, 1966).
Due to the widespread existence of solution channels within the latter
formation, its permeability should be viewed from a macroscopic stand-
point. All the permeabilities indicated in Figure 45 were assumed
equal in the horizontal and vertical directions. This assumption was
dictated primarily by lack of data, since it is recognized that strati-
fication produces anisotropy.
122
-------
N)
TOPOGRAPHIC
RIDGE
HORIZONTAL
FLOW
SPECIFIED
GOURD NECK SPRING
SHALLOW FORMATION
TOPOGRAPHIC
DEPRESSION^
NO-FLOW
BOUNDARY
FLORIDAN AQUIFER
NO-FLOW _
BOUNDARY"/
HORIZONTAL
FLOW
SPECIFIED
Figure 44. Lateral view of cross-section AA' (figure 23),
indicating flow boundary conditions.
-------
tv)
K = 80
K = 200
K = 100
K = .5
K = 500
Figure 45. Assumed permeability (gal/day/ft2) regions in section AA1 (Figure 23)
Note: 1 gal/day/ft2 = 0.0017 m/day.
-------
Flow Calculations—
To aid in the selection'by the program of the proper form of equation
(38) to be used for a given node, a node classification system was
developed. Its code, illustrated in Figure 46 for the cross-section
under study, is the following: 0 - interior, 1 - water table, 2 -
outside the groundwater system, 3 - no-flow boundary and 4 - known
head. Hydraulic heads for nodes classified as (1) and (4) are read
in and kept unchanged thereafter. The heads at nodes of type (3) are
allowed to change, with the no-flow condition maintained by means of
an implicit image-node scheme explained elsewhere. Nodes classified
as (2) are avoided in all calculations.
It is pertinent to discuss the methods used for obtaining the head
values for nodes designated as (1) or (4). In the water table nodes
(1), the heads at the nodes underlying Lake Apopka and the muck farms
could be estimated with reasonable certainty. However, the water
table location in the sandy soils on both sides of the lake could only
be approximated from the limited data available from shallow observa-
tion wells.
Turning to class (4) nodes, the values specified in the row adjoining
the bottom boundary were found by interpolating between the piezometric
contours shown in Figure 22. The heads specified in the vicinity of
the left and right boundaries of the Floridan Aquifer implicitly set
the respective flow boundary conditions. Similarly, the outflow at
Gourd Neck Spring was defined by specifying the head at the node
located below its approximate outlet point. It is recognized that
the approach used for Gourd Neck Spring is overly simplistic. However,
a more realistic treatment would have required a much finer grid mesh,
and consequently an increased computational cost.
Quality Calculations-
The groundwater quality formulation, developed by Shamir and Harleman
(1966) and modified by Bedient (1972), utilizes the velocities computed
from the flow model to predict the concentrations of substances at all
nodes in the system. In this study, the quality boundary conditions
consisted of concentration values specified along the top boundary.
These concentrations were obtained from the quality portion of the
unsaturated zone model. Nitrate was the only substance routed in the
groundwater system.
Although Bedient's mathematical formulation included source/sink and
decay terms, these factors could not be utilized herein due to lack of
data. Thus, the only two driving forces taken into account were
advection and dispersion. In the computation of the horizontal and
vertical dispersion coefficients, which depend upon the respective velo-
cities, the constants used for all the geologic formations were those
proposed for sand (Hoopes and Harleman, 1965)
125
-------
t-J
OURD NECK-
I- SPRING
i I i
-2—2—2-2-2—2—2—2—2—2—
i I i i
o-o-o—o-o—o-i
I I I I I I I I I I I I
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1 I
o-o
4-4-4-4-4-4-4-4-4-4-4-4-
-4-4-4-4-4-'
4-4-4-4-4-4-'
Figure 46. Node classification system for groundwater flow in section AA'(Figure 23)
-------
In a manner analogous to that in the flow model, a node classification
system was developed for use in the quality calculations. Its code,
which is illustrated in Figure 47, is the following: 0 - exterior,
1 - left boundary, 2 - left top, 3 - left bottom, 4 - top, 5 - bottom,
6 - right top, 7 - right bottom, 8 - right boundary> 9 - interior and
10 - known concentration. Initial concentrations must be set for all
nodes not classified as (10), and the values selected depend upon the
purpose and period of a particular simulation run.
INTERFACING METHODOLOGY
General Approach
The general technique used was to develop a relatively simple execu-
tive program, which in turn calls the different models in a sequential
fashion. The order used for the various types of models was: (1)
unsaturated zone, (2) overland flow (runoff), (3) receiving model and
(4) groundwater.
The executive program utilized an overlay procedure to minimize the
active data storage requirements. As part of the overlay technique,
data sets were used to store the information utilized by the various
models above. Permanent data sets were specified, mainly for data
that were known prior to a computation run, such as catchment layouts
and canal geometry. On the other hand, temporary, or scratch, sets
were defined primarily for input/output operations between models.
Partitioning of the Study Area
To promote the homogeneity of model parameters, the Lake Apopka basin
was partitioned into subsystems within the various model categories.
A review of the previous sections indicates the following breakdown:
1. Unsaturated Zone
a. Sandy soils located around the southern half of the lake
b. Muck farms lying north of the lake
c. Sandy soils north of the muck farms
2. Overland Flow (Runoff)
a. Sandy soils described in (la)
b. Zellwood district in muck farms
c. Duda district in muck farms
d. Sandy soils indicated in (Ic)
3. Surface Bodies of Water (Receiving Model)
a. Drainage canals in Zellwood district
b. Drainage canals in Duda district
c. Lake Apopka
4. Groundwater
a. S. W. to N. E. cross-section
127
-------
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Figure 47. Water quality node classification system in section AA1 (Figure 23).
-------
An individual computer model was utilized for each of the subsystems
listed above. In the executive program, these models appeared in an
order dictated by their category, such as unsaturated zone or ground-
water. The sequencing of models within a given category was determined
arbitrarily.
jFlow Chart of Models and Data Sets
A graphical illustration of the various subsystem models run and their
order in the executive program is provided in Figure .48. The permanent
data sets are represented by the boxes on the left hand side of this
figure. One permanent set is used for each model. The temporary
data sets, and some of the variables stored therein, are depicted in
the right hand portion. The arrows denote the flow of information
between these data sets and the models, thus representing READ and
WRITE statements. Not shown in this figure are the data sets used for
interfacing the flow and quality portions of both the Stormwater
Management Model runoff and receiving models, since additional infor-
mation on these programming aspects can be obtained elsewhere (Environ-
mental Protection Agency, 1971). A list of variables associated with
each temporary data set is given in Appendix D.
OBJECTIVES OF COMPUTER RUNS
In this investigation, the main purposes of the various simulation
runs were to gain an intuitive understanding of system behavior and
to ascertain the relative importance of different model parameters
upon this overall behavior. Since many of the values of these para-
meters are not available, the model at the present stage could be
viewed as an instrument for structuring data gathering efforts.
129
-------
w
o
PERMANENT
DATA SETS
MODELS
C STARTJ
r^ 1—1—1 --
j hr S.W. SANDS UNSATURATED ZONE^I
TEMPORARY(INTERFACING)
DATA SETS
RAINFALL, EVAPORATION, RECHARGE
QUALITY INPUT
X
SANDS UNSATURATED ZONE
|~{-4^
r—H
IS.W. SANDS OVERLAND FLOW t
I N.E. SANDS OVERLAND FLOW |*"
| ZELLWOOD MUCK FARMS RUNOFF
H
«-r —
i
1DUDA MUCK FARMS RUNOFF J~* J
. — ~T- •:,'..':.-' \. IJ
"r— , ~T" i -L
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1 DUDA MUCK FARMS RECEIVING
1 , ,
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RAINFALL, INFILTRATION f*^
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FLOWS, QUALITY CONCENTRATIONS |«^
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WATER TABLE (HEADSJ QUALITY
CONCENTRATIONS AT TOP
I
( END )
Figure 48. Computational flow chart of program package, showing the use of
data sets.
-------
SECTION IX
RESULTS AND CONCLUSIONS
INTRODUCTION
This section discusses the results of the various simulation runs
conducted in this study and the conclusions derived from them. The
main items considered are sample results for the individual models,
interpretation of these results for different subsystems within the
study area and evaluation of the impact of different model parameters.
These trial runs indicated the need for additional data for proper
calibration, especially for the quality models. Thus, comprehensive
interfacing runs were felt to be unwarranted. Instead, a limited
amount of interfacing was conducted for the purpose of checking pro-
gramming instructions. Conclusions and suggestions for further re-
search are presented from the standpoint of both general modeling
efforts and the Lake Apopka basin.
SAMPLE RESULTS QF INDIVIDUAL MODELS
Representative computational results are discussed for each of the
three model types: (1) surface water, (2) unsaturated zone, and
(3) groundwater zone. Category (1) includes overland flow (runoff)
and flow in surface bodies of water.
Surface Water
Overland Flow-
Two separate overland flow models were utilized, depending on the
degree of channelization in a given area: (1) the kinematic wave
method (Eagleson, 1970) and (2) the runoff portion of the Stormwater
Management Model (Environmental Protection Agency, 1971).
Kinematic wave model— This model was applied to the sandy soils in
the Lake Apopka basin. Its results are illustrated for a catchment
located north of the muck farms area. The catchment geometry param-
eters and the assumed boundary conditions are indicated in Figure 49.
Also shown are the ponding depths obtained from the unsaturated zone
131
-------
CM
ro
CATCHMENT NUMBER I
LENGTH = 14,000 ft
WIDTH = ' 7,000 ft
SLOPE = 0.0021 ft/ft
PART A CATCHMENT BOUNDARY
10
ui
7.5 *
M
i£3
of 2
^.
I 2 3 4 5 6 7 8 9 10 II 12 13 14 IS
NOTE:At=IOmln TIME STEP
PART B ASSUMED BOUNDARY CONDITIONS
Figure 49. Catchment geometry and boundary conditions used in kinematic wave
overland flow model demonstration run.
-------
model and used as inputs to this formulation. The necessary averag-
ing calculations involving these boundary conditions are explained
more fully in Appendix A.
The predicted outflow hydrograph resulting from the conditions depicted
in Figure 49 is shown in Figure 50. Its shape and characteristics
conform to those proposed by Eagleson for these relative values of
storm duration and time of concentration, It may be observed that
the rainfall intensity used is very high. Due to the high infiltra-
tive capacity of these sandy soils, ordinary rainstorms would produce
little or no overland flow. This conclusion has been confirmed by
agricultural personnel in the area.
Also plotted in Figure 50 is a graph of the predicted phosphorous flux
versus time at the catchment outlet. As was the case with the outflow
hydrograph, only severe rainstorms should produce appreciable sediment
loads in the sandy areas. The calculation of phosphorous content in
sediments, discussed in Appendix E, was based on several simplifying
assumptions. A comparison of predicted loads in other catchments
under similar hydrologic conditions pointed out the significance of
the slope on sediment transport. That is, erosion from catchments
having gentle slopes should occur only under extremely high rainfall
conditions.
Stormwater runoff model- This formulation was utilized for the arti-
fically drained muck farms on the north shore of Lake Apopka. Figure
51 constitutes a map of a catchment in the Duda district. The hydro-
logic boundary conditions for this catchment are also indicated therein.
The numerical technique used in this model permitted the utilization
of the original boundary condition values, without the intermediate
averaging procedure required in the analytical kinematic wave method.
All the overland flow from the catchment is assumed to constitute a
point source at an inlet point. The computed flow hydrograph at this
point is plotted in Figure 52. For ordinary rainstorms, overland
flow is of relatively little significance in the muck farms.
The model computations yielded a suspended solids flux of zero for
all time intervals. Thus, the phosphorous contribution was also zero.
The same result was obtained throughout the muck farms and it appears
reasonable in view of the essentially flat ground surface in the area.
Receiving Water—
The receiving model was applied to the drainage canals in the muck farms
and subsequently to Lake Apopka. The results for these two areas are
discussed separately.
133
-------
30001-
2000
O
1000
FLOW
o
V.
.0
10
30 30 40 50 60 70 80 90
TIME, mm
100 110 120 130 140 ISO
Figure 50. Predicted outflow hydrograph from catchment number 1 in
sandy soils area.
-------
en
CATCHMENT NUMBER 24
IN DUDA DISTRICT
DUDA DISTRICT
BOUNDARY
If
* * ' ' / X^ 7' 1
T 24 PIAM VIFW »
(NLET 2*1 PLAN VIEW
NOT TO SCALE
APOPKA-BEAUCLAIR CANAL'
AREA- 147 acres
SLOPE = HP* ft/ft
PART A CATCHMENT GEOMETRY
u
<
a:
10
1 2 3
TIME, hr
PART B ASSUMED BOUNDARY CONDITIONS
Figure 51. Catchment geometry and boundary conditions used in demonstration
run of runoff portion of Stormwater Management Model.
-------
o
o>
w
16
15
14
13
12
II
|0
9
t<
* r
3 6
5
4
3
2
-r\
:i \
:| \
\
\
I X
i
{
x\
TIME, hr
Figure 52. Predicted outflow hydrograph for catchment 24 in the Duda District.
-------
Muck farms canals— The location of two adjoining junctions (numbers
56 and 57) in the Zellwood area is shown in Figure 53 and also in
Figure 34. The period under study was that during the simulated rain-
storm given in Figure 49. Indicated in Figure 53 are the pumpage rates
used in the simulation. These assumed rates were quite high because
they were based upon maximum pumping capacities.
Figure 54 contains a plot of the calculated stage at the upstream
junction for several time steps. The corresponding heads at the
downstream junction are slightly lower. As can be observed, the stage
decreased sharply after the rainfall ceased. Several flow values
between the two points are plotted in Figure 55.
Turning to the water quality results, Figure 56 contains a plot of
nitrogen and phosphorous concentrations versus time, at the upstream
junction (57). Nutrient inputs originated primarily from the mole
drains. Source/sink reactions between water and soil were not con-
sidered.
Calibration of the relationships shown in Figures 54, 55 and 56 was
precluded by the lack of short-term data.
Lake Apopka- Figure 39 constitutes a map of Lake Apopka with the approx-
imate location of the junctions defined for the, lake receiving model.
Average inputs used for the simulation are also shown. The hypothe-
tical rainfall conditions and pumpage rates used were the same as those
utilized for the muck farms simulation. Overland flow contributions
from the sandy soils catchments were neglected.
The predicted stages for junction 3 (see Figure 39) are plotted in
Figure 57. In the first few hours the stage increased rapidly, due
primarily to rainfall. Thereafter it rose more slowly, in response
only to the pumpage inflows. Most of the corresponding stages at
other junctions were within 0.01 ft of the values obtained for junc-
tion 3. This indicated the lack of appreciable hydraulic gradients
across the lake. The predicted flows between junctions 3 and 8 are
provided in Figure 58.
To investigate the movement of nutrients, their initial concentrations
at the various junctions were set at zero. Also, source/sink and decay
effects were neglected, thus treating these substances as conservative.
Predicted increases in concentrations resulting from the imposed boun-
dary loads were rather small. System response can be illustrated by
an examination of the results obtained for junction 3, located in the
vicinity of the pumping centers. Concentrations are plotted in Figure
59. Predicted concentrations for junctions farther away from the pumps
were much lower. The generally low values obtained point out the need
to consider water quality changes over longer time horizons.
137
-------
DRAINAGE CANALS
PUMP FLOWS:
JUNCTION FLOW, ft3/sec
300
200
200
200
4
9
25
27
Figure 53,
Map of Zellwood drainage district showing location
of various junctions.
138
-------
59
345
TIME, hfs
Figure 54. Predicted stages at junction 57
in Zellwood district.
'0 I
456
TIME, hrs
Figure 55. Predicted flows between junctions
56 and 57 in Zellwood district.
139
-------
II -
—n
o
D
NQf
P
567
TIME, hr
8
10 II
0.5
°'4 ^
E
*•
O
0.3
0.2
UJ
O
o
O.I
12 13 14 15
Figure 56. Predicted nitrate and phosphorous concentrations at junctibn 57
in Zellwood district.
-------
M.9
M.O
12
TlME.hr*
Figure 57. Predicted stages at junction 3 in Lake Apopka
(90
too
ISO
I
100
•0
If
TIME.hr*
ti
Figure 58. Predicted flows from junction 3 to junction 8
in Lake Apopka.
141
-------
2x10^-
10"
a:
UJ
o
z:
O
O
10
12 15
TIME, hrs
18
21
24
Figure 59. Predicted nitrate concentrations at junction 3 in Lake Apopka.
-------
Unsaturated Zone
The unsaturated zone model was applied to the muck farms areas and two
areas in the sandy soils. Results for the sandy region north of the
muck farms are presented herein.
Figure 60 consists of a graphical representation of the boundary
condition flow values used in a particular run. The three boundary
conditions are: rainfall, evaporation, and recharge. Computed soil
moisture profiles at different times are plotted in Figure 61. In
the initial profile, the moisture content decreased with elevation.
Subsequently, as a consequence of high rainfall, the moisture at the
upper nodes increased significantly. The uppermost node became sat-
urated and a temporary ponding condition resulted. Gradually the en-
tire profile approached saturation. Withdrawal of water by plants
was found to have little appreciable effect in this short simulation
run. However, it should be important on a long term basis. In the
muck farms, the predicted withdrawal flows at the mole drains were
significant, with a corresponding impact on soil moisture conditions.
In practice, the efficiency of the mole drains might be much lower.
The water quality results corresponding to the sandy soil profile
shown in Figure 60 are presented in Figure 62. Indicated therein are
the upper boundary condition, which was set at the initial time step,
and the predicted concentration profiles at successive time steps.
Source/sink reactions between the percolating water and the soil and
nutrient withdrawals by plants were not considered in this run. In-
sufficient time was provided in this run for significant downward
movement of nitrate, with buildup expected to have occurred between
the top two nodes. Thus, these results illustrate primarily the
phenomenon of dispersion.
Data were not available to verify or calibrate the quality model in
the sandy soils. In the muck farms, previous data collected during
both wet and dry periods indicated fairly uniform concentration pro-
files with depth. Also, the measured concentrations were rather high.
These data suggested the importance of nutrient release from the muck
soil, and therefore, the need to gain further insight into source/
sink rates.
.Groundwater
The groundwater model results presented herein are those for vertical
cross-section AA1, which extends from the southwest portion of the
Lake Apopka basin toward the northeast (see Figure 22). The location
of this section was illustrated previously. Also discussed earlier
were its estimated boundary conditions and permeability configuration.
143
-------
5 10
TIME STEP
NOTE-TIME STEP IS 10 min
EVAPORATION =
A z = 17cm.
RECHARGE = 0.10 ft /day CONSTANT
Figure 60. Boundary conditions used in a run of the unsaturated zone
model for the sandy soils.
-------
.GROUND SURFACE
Z
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X
t
5?
o
1
o;
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CD
5
ID
z
UJ
Q
O
z
NOTE!TIME STEP = lOmin,
0.10 0.20 0.30 0.40 0.43
VOLUMETRIC MOISTURE FRACTION^ SATURATION
Figure 61. Predicted moisture profiles in the sandy
soils at different time steps.
145
-------
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TIME STEP = At = lOmin. ._
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,
10
-6
I0"5 10" K) -
NITRATE CONCENTRATION, mg/l
10
10
Figure 62. Predicted nitrate concentration profiles in the sandy soils at different
time steps.
-------
Figure 63 is a reproduction of a computer tabulation of predicted hy-
draulic heads for all nodes within the cross-section. These were the
values obtained after 100 iterations of the steady-state flow model.
The values for water table nodes and other nodes of known head (classes
1 and 4, respectively) were provided initially and remained unchanged
throughout.
To graphically analyze the flow patterns in this section, it would be
possible to draw flow nets, that is, orthogonal families of equi-
potential and stream lines. However, prior to utilizing this approach
it would be desirable to improve the predictive accuracy of the model
by obtaining additional data on boundary conditions and permeabilities.
Specifically, data for the sandy soils area north of the muck farms
and its underlying formations are scarce. Several preliminary runs
for this cross-section indicated that the overall flow patterns are
more sensitive to changes in the permeability configuration than to
fluctuations in flow boundary conditions.
The water quality portion of the groundwater model was tested by"
specifying nitrate concentrations of unity at various nodes along
the water table boundary of the cross-section. The ensuing concen-
trations within the section were predicted and tabulated at different
times. Figures 64, 65 and 66 constitute contour maps obtained from
computer printouts of these values. The respective times within the
simulation period are indicated.
To investigate the gradual increase in concentrations at the various
nodes as a result of the boundary conditions, the initial values of
these concentrations were arbitrarily set equal to a small positive
number. This number (10"31) was used rather than zero to avoid the
prediction of very small negative concentrations resulting from num-
erical oscillations. An examination of Figures 64, 65 and 66 reveals
that the movement of nitrate in groundwater is predominantly down-
ward, with little contribution to Lake Apopka. The area of Gourd
Neck Spring merits further investigation since short-circuiting effects
might be significant.
Numerical instabilities are likely to result in this model in situa-
tions where the time step exceeds the time of water travel between
any two nodes. However, the problem can be avoided effectively by
reducing the time step. Additional insight and comments about this
matter are provided by Bedient (1972).
DISCUSSION OF INVESTIGATIVE RUNS
The main features considered in these simulation runs were trial
variations in model parameters. These changes were dictated by the
individual model results.
147
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COLUMN NUMBER
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Figure 63. Predicted heads in cross-section AA1 (Figure 23) after 100 iterations,
-------
20
23 4 5 67 8 9 10 II 12 13 14 15 16 17 18 19
COLUMN NUMBER
Figure 64. Predicted nitrate concentration contours in cross-section
AA1 (Figure 23) after one hour of input.
-------
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(Figure 23) after one day of input.
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Figure 66. Predicted nitrate concentration contours in cross-section AA'
(Figure 23) after 10 days of input.
-------
SURFACE WATER
Overland Flow
To investigate the effect of extreme rainfall events on overland
flow, several simulated high intensity storms were applied to a catch-
ment in the sandy soils area (see Figure 49). The rainfall hyeto-
graphs used were multiples of that shown in Figure 49. Thus, an
"intensity ratio" is defined as the ratio of hyetograph ordinates to
those of Figure 49.
The results of the various runs are provided in Figure 67. The plot
of peak outflows versus rainfall intensity ratio shows a steady rise
with increases in the latter. A similar trend is found for the phos-
phorous flux. In catchments having milder slopes, phosphorous flux
rates were much lower.
Receiving Waters—
In an effort to gain insight into system behavior, several trial runs
were conducted for the muck farms, using various boundary conditions
and assumptions. However, the lack of accurate records, especially on
pumpage figures and on the efficiency of the mole drains, precluded a
thorough calibration of the receiving model.
Likewise, usage of this costly program for Lake Apopka was kept to a
minimum. Data on source/sink reactions between the lake water and
the bottom sediments would be required for effective calibration.
Also, usage of longer time steps and simulation periods would be
necessary to deal with long-term eutrophication trends.
UNSATURATED ZONE
In the quality run for the sandy soil (see Figure 62), the illustration
of the phenomenon of advection would have required a much longer
simulation period. The effects of this phenomenon can be observed in
Figure 68. Plotted therein are the results of a simulation run in
which arbitrarily high values of velocity were used. At time step
n = 10, the maximum concentration point was located at the second
node from the top.
Since the predicted concentration profile in the muck soil did not
agree with available data for the area, additional attention was paid
to nutrient releases by the soil. For this purpose, the effects of
several trial release rates were investigated in the quality model.
These rates were assumed to be time-invariant and constant with depth.
These assumptions were quite simplistic because the release mechanisms
are likely to be numerous and sensitive to environmental factors.
152
-------
in
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5000
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Figure 67. Predicted peak flows and peak phosphorous fluxes in
catchment 1 for various rainfall intensities.
-------
NITRATE CONCENTRATION, mg/I
Figure 68. Predicted nitrate concentration profile for hypothetical soil
under assumed high velocity conditions.
-------
The source rate estimates used were necessarily rough. Some data on
muck consolidation in the Everglades (Stephens, 1969) and on the nu-
trient content of muck (Heaney, Perez and Fox, 1972) were used in
attempts to determine upper bounds of nutrient release (see Appendix E).
The concentration profiles obtained for a given source rate are shown
in Figure 69. The assumed hydrologic conditions are also indicated.
In the initial time steps, the predicted concentration profiles were
fairly uniform with depth. Subsequently, the downward velocities
produced profiles in which the concentrations increased with depth.
For dry periods in which soils approach an equilibrium no-flow con-
dition, the concentration profiles should be relatively uniform.
Lower values of nutrient release were used in other trial runs.
Similar shapes of concentration profiles were obtained.
GROUNDWATER ZONE
In the groundwater model runs, emphasis was placed on investigating
model sensitivity to changes in flow and quality boundary conditions.
Preliminary computer runs indicated significant model sensitivity to
permeability changes. However, an organized analysis of this latter
effect was not possible within the scope of this study.
The flow boundary conditions considered were water table elevations,
which fluctuate in response to rainfall and evaporation. Of specific
interest was the sensitivity of groundwater recharge rates to these
fluctuations. The investigation of this relationship is illustrated
in Figure 70. Isolated in this figure is a portion of the groundwater
cross-section considered in this study, namely, the region extending
from the southwestern boundary of the basin to the vicinity of Gourd
Neck Spring (see Figure 22). The analytical procedure used consisted
of the following steps: (1) apply uniform increments of water table
elevation (head) to the top nodes, (2) run the groundwater model for
each new increment and (3) store the vertical flow velocities at the row
of nodes immediately below the top row. The average velocity at these
nodes was assumed to constitute the recharge rate in each case.
The computed recharge rates resulting from various water table heads
are plotted in Figure 71. As expected, recharge increased with higher
water tables. However, the response sensitivity was considered to be
sufficiently small to allow the usage of time-average recharge rates
in subsequent interfacing activities between the unsaturated zone and
groundwater models. Somewhat similar response characteristics were
obtained for the sandy soils area north of the muck farms.
The results shown in Figures 64, 65 and 66 indicated a downward move-
ment of nitrate from the surface sources toward the Floridan Aquifer.
155
-------
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RELEASE RATE=4.4 8 x I04b
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X
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NITRATE CONCENTRATION,mg/I
-------
W END OF A A1
Cn
LAKE APOPKA
Figure 70. Lateral view of a portion of cross-section AA» (Figure
23)
-------
c/i
oo
I 2
HEAD INCREMENT, ft
Figure 71. Predicted recharge velocities resulting from various increases
in water table elevation.
-------
Additional information was sought by extending the simulation period.
The predicted contours following 68.8 days of input are given in Figure
72. In the right hand portion of the figure, the advancing high con-
centration (10"*) front has been slowed down. This is due to the low
permeability section in the vicinity (see Figure 45).
The high vertical velocities in the upper nodes of the sand formations
dictated that a time step no greater than one day be used. For this
reason, simulation over long time periods would have been very costly.
SUMMARY
The results obtained indicate that it is feasible to model a conjunc-
tive surface-groundwater system. Current data limitations precluded
an adequate calibration of the model. However, the formulation can
be used as a guide for data collection.
While it is desirable to conduct modeling from a systems viewpoint,
it appears that a close integration of the various individual pro-
grams would be premature. In their present relatively independent
status, the programs are more accessible for the introduction of im-
provements. Also, the high computation cost and programming complex-
ities inherent in a fully interfaced approach seem to be unwarranted
in the light of data deficiencies.
SUGGESTIONS FOR FURTHER WORK
Discussed below are possible improvements that might be incorporated
into the methodology and some extensions of its usage. Considered
first are improvements of the individual models. Data collection and
parameter optimization factors are discussed next. Finally, several
comments about the study area are included. It is likely that some
of these comments will be applicable to other locations.
In the formulation of improvements, a compromise must be established
between accuracy and cost. The desirable position of this balance
depends strongly on the nature of the study area and the planned use
of the model. The numerous cost-efficiency tradeoffs can perhaps be
most effectively evaluated by decision-makers, who are generally the
ultimate users of models (Kisiel, 1971).
Individual Models
Surface Water Models-
Kinematic wave model- The little apparent significance of overland
flow in the Lake Apopka basin and the lack of appropriate data precluded
adequate verification of the kinematic wave relationships. Nevertheless,
it is felt that several improvements therein should be given considera-
tion, as follows:
159
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IO'1
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Figure 72. Predicted nitrate concentration contours in cross-section AA!
(Figure 23) after 68.8 days (1656 hours) of input.
-------
1. Allowance should be made for a given catchment to provide flows
to another catchment. Recent reports from the Massachusetts institute
of Technology (Harley, Perkins and Eagleson, 1970) contain useful
information on this matter.
2. A more rigorous interfacing between overland flow and flow through
the unsaturated zone might be desirable. Smith and Woolhiser (1971)
developed a numerical formulation for the simultaneous consideration
of these two regimes. Their methodology, Which is quite general,
includes provision for cascading catchment planes. However, its
computational costs might be relatively high.
3. Attempts should be made to account for the effects of agricultural
practices in the model. As indicated by Hills (1971), the infiltra-
tion capacity of a soil depends more on treatment factors (e.g., com-
paction) than on its natural characteristics. In 1971, Seginer devel-
oped a model for surface drainage of cultivated fields, which gives
consideration to furrows and their hydraulic characteristics. The
formulation developed by the Agricultural Research Service (1971) is
also quite valuable. Testing of Manning coefficient relationships for
vegetated areas (Corps of Engineers, 1956) would be desirable.
4. Much remains to be accomplished in the matter of suspended solids
and phosphorous fluxes in overland flow. A better representation of
erosion phenomena might be obtained by utilizing a Lagrangian modeling
approach, in which the path of the particles are followed as they move
through the system. Additional agricultural data are required to
improve estimates of phosphorous content in suspended soil particles.
Stormwater Management Model- The stormwater model offers advantages
in its inherent structure for designating catchments and reading in
data in an organized fashion. Its structure is also well suited for
interfacing with the receiving model. However, several modifications
to the formulation should be contemplated:
1. At present, the model contains calculations for hydraulic elements
which are not found in agricultural areas, such as pipes and gutters.
These calculations were bypassed in this study, but for more intensive
use of the model it might be worthwhile to carry out a more thorough
elimination and rewriting process. The same argument applies to water
quality computations, s*uch as those for biochemical oxygen demand
(BOD) and coliform bacteria counts.
2. Consideration should be given to improvements regarding closer
interfacing with unsaturated zone flows, accounting for agricultural
practices and sediment load calculations. The same specific comments
provided for the kinematic wave model apply herein.
161
-------
Receiving water model— The performance of the flow portion of this
model, in both the drainage canals and Lake Apopka, was satisfactory.
Due to the flexibility of its link-node structure and its ability to
consider various boundary conditions, the model has great applica-
bility to practical situations. Its major drawback lies in its high
computational cost. Suggested improvements lie in the area of quality
calculations:
1. For a complete understanding of nutrient behavior, it would be
desirable to incorporate an ecosystem model in the quality subroutines.
Such models have been proposed by numerous investigators, including
Chen (1969), Bloom, Levin and Raines (1969) and Chen and Orlob (1972).
Since the amount of data required to calibrate an ecosystem model is
rather massive, the development of a data collection strategy would
appear worthwhile. In this regard, Park and Wilkerson (1971) provide
insight into the interaction between statistical studies and sampling
programs. In a recent paper, Moore (1972) discussed the use of filter
theory in data collection, and the tradeoffs between spatial and tem-
poral frequency in sampling.
2. Since often appreciable amounts of nutrients are contained in soil
particles, it would be pertinent to devote further attention to the
phenomenon of sediment transport. The consideration of this phenome-
non within the receiving bodies of the Lake Apopka basin (and other
agricultural basins) would appear to be of interest. In a study of
cohesive soils, Partheniades (1965) concluded that erosion rates were
rather independent of the concentration of suspended sediment, and
arrived at several other generalizations regarding erosion and deposi-
tion factors. In 1968, Etter <5t aJL discussed the importance of parti-
cle sizes and their relationship to flocculation. In addition to in-
formation contained in such studies, data are required on nutrient up-
take and realease rates from sediments.
Unsaturated Zone Model—
The most important suggested improvements for this model are on the
subject of water quality. Following is a list of suggestions:
1. It might be desirable to integrate this model more fully with an
overland flow formulation (Smith and Woolhiser, 1971), as discussed
previously.
2. There is a general scarcity of data on hydraulic conductivities
under unsaturated conditions, especially for muck soils. \ More infor-
mation on this matter would be desirable.
3. Much attention should be paid to the impact of agricultural activi-
ties on flow conditions in soils. Some of the basic relationships
developed in this study for water uptake by plants could be improved in
162
-------
their representation of seasonal patterns and the effects of different
crops. Another possible feature to be included would be the flow of
water through the voids between the plant roots and the soil. This and
other features are contained in a model developed recently by the
Agricultural Research Service (ARS) (1971)•
4. A considerable amount of work remains to be done on the determina-
tion of source/sink rates for nutrients in soils. At present, the
qualitative understanding of the mechanisms affecting these rates is
very limited. It is suggested that the model developed herein be used
as a guide for data collection and for verification purposes. It is
also thought that the ARS model could be improved by the incorporation-
of water quality routines.
Groundwater Model—
It is felt that the decision to utilize a steady-state two-dimensional
model, explained in previous sections, was justified in terms of cost
and the accuracy desired for most agricultural situations. However,
in basins where pumpage and other boundary conditions produce signifi-
cant changes in piezometric conditions over time and space, the use of
an unsteady-state three-dimensional formulation (Freeze, 1971) may be
necessary. Most of the remaining comments deal with general approaches
to model calibration:
1. In groundwater systems, it is commonplace to find that more than
one set of heads and permeabilities will satisfy certain known flow
conditions. This problem of indeterminacy poses serious difficul-
ties to the calibration of both the flow and quality portions of the1
model. The possible use of an optimization approach to the estimation
of model parameters is discussed later.
2. It appears worthwhile, on a long-range basis, to pursue the global
energy approach proposed by Back and Hanshaw (1970). The success of
this method, which was discussed previously, would depend on both a
better understanding of energy phenomena and intensive sampling. It
would seem appropriate to devote the original modeling efforts in
connection with this approach to well-defined geologic systems and
substances of simple chemical behavior.
3. Additional quantitative information should be obtained on source/
sink reactions of nutrients in groundwater. Since agricultural
phosphorous is likely to be fixed to the shallow soil zone, more
attention might be devoted to anaerobic nitrogen reactions.
Data Collection and Parameter Estimation
The problem of indeterminacy, which is quite significant in ground-
water models, is also present in other subsystem formulations. Thus
163
-------
the consideration of data collection aspects is extremely complex,
Several data collection strategies have been developed whose basic
objective is to minimize the variance between observed and predicted
values (Matalas, 1967). However, in the modeling of a conjunctive
surface-groundwater system the precise definition of an objective func-
tion is an elusive matter. This problem is due to the stepwise (flow
followed by quality) nature of the predictive process, and the large
number of subsystem models. In designing sampling programs, it would
seem reasonable to assign higher priority to the data that are more
likely to affect pollution abatement decisions. Kisiel (1971) has
provided useful insight into the use of decision theory to determine
the value of data.
In view of the imperfect nature of any data base, it is also worth-
while to enhance the predictive quality of models by improving the
estimates of various parameters. Optimization methods for flow resis-
tance in overland flow (Schreiber and Bender, 1972) and for per-
meability in groundwater (Kleinecke, 1971) are available. The latter
contains restrictive assumptions such as homogeneity of the medium
and the occurrence of flow in two dimensions only. Since available
optimization techniques involve only flow phenomena, the development
of counterparts would be required to calibrate quality processes,
The multiplicity of subsystems within any given basin suggests the use
of a decomposition, rather than integrated, approach for both data col-
lection and parameter optimization. Much remains to be accomplished
on the quantitative understanding of interactions between subsystem
models through boundary conditions, and on the effect of flow predic-
tion improvements upon water quality modeling. Regarding the latter,
perhaps the global energy approach suggested by Back and Hanshaw (1970)
could be used to eliminate the need for the classical two-step (flow,
then quality) approach.
Suggestions for the Lake Apopka Basin
Due to the current interest in the water quality of Lake Apopka and
its possible restoration, it seems pertinent to summarize the main
data needs and potential model uses:
1. Available data were inadequate to determine a reasonably accurate
hydrologic budget of the muck farms area (Heaney, Perez and Fox', 1972).
Pumpage figures are quite rough and no records are kept on the opera-
tion of the flap valves. It is suggested that better data be obtained
for these flows. Groundwater flows in the area also merit further
attention.
2. In the muck farms, the impact of fertilizer application appears to
be less significant than that of nutrient releases by the soil. Data
164
-------
leading to a better understanding of the mechanisms affecting this
release would be very desirable.
3. The nitrogen contributions from the citrus groves to the groundwater
system might be appreciable. However, due to the piezometric conditions,
any fluxes into Lake Apopka are likely to be small. The only signi-
ficant contribution to the lake could occur at Gourd Neck Spring.
4. The results of the receiving model indicate that the horizontal
movement of substances in Lake Apopka is quite slow. Thus, it might
be worthwhile to incorporate the phenomenon of diffusion into the
model. A better quantitative understanding of source/sink reactions
between the lake water and the bottom sediments would be very help-
ful. Research on sediment transport aspects, including particle size
and scouring characteristics, should be useful. Also, the mixing effect
of wind should be included. This effect might account for the fairly
uniform (in the horizontal sense) pattern of nutrient concentrations
in the lake.
5. Overland flow appears to be of very limited significance in the
Lake Apopka basin. In future modeling efforts in this basin, it
might be desirable to bypass consideration of this phenomenon, except
in the case of very high intensity storms,
6. The methodology developed herein could be used as a guide for
data collection in this and other basins. Following satisfactory
verification, the model could also be utilized as a tool for simu-
lating different pollution abatement measures.
165
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SECTION X
REFERENCES
1. Agricultural Research Service, U. S. Dept, of Agriculture, Techni-
cal Bulletin No. 1435, Beltsville, Maryland. 1971. (unpublished).
2. Anderson, W. and B. F. Joyner. Availability and Quality of Surface
Water in Orange County, Florida. Map Series No. 24, Florida Board
of Conservation, Tallahassee, Florida. 1969.
3. Anderson, W. Hydrologic Considerations in Draining Lake Apopka—
A Preliminary Analysis. Open File Report 71004, U. S. Geological
Survey, Tallahassee, Florida. 1971.
4. Ashcroft, G., et_ al_. Numerical Method for Solving the Diffusion
Equation: I. Horizontal Flow in Semi-Infinite Media. Soil Sci
Am Proc. 2(5:522-525, 1962.
5. Bachmat, Y. and J. Bear. The General Equations of Hydrodynamic
Dispersion in Homogeneous, Isotropic Porous Mediums. Journal
of Geophysical Research. £9(12):2561-2567, June 1964.
6. Back, W. and B. B. Hanshaw. Comparison of Chemical Hydrogeology
of the Carbonate Peninsulas of Florida and Yucatan. Journal of
Hydrology. j_0:330-368, 1970.
7. Bailey, G. W. Role of soils and Sediment in Water Pollution
Control. EPA Southeast Water Lab, Athens, Georgia, March 1968.
8. Bates, T. E. and S. L. Tisdale. The Movement of Nitrate-Nitrogen
through Columns of Coarse-Textured Material. Soil Sci Soc Am Proc.
Vol.21, 1957. ' " :
9. Bear, J. On the Tensor Form of Dispersion in Porous Media.
Journal of Geophysical Research. 60_(4) : 1185-1198, Apr.il 1961.
10. Bear, J. Dynamics of Fluids in Porous Media. American-Elsevier.
1972.
166
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11. Bedient, P. B. A Two-Dimensional Transient Numerical Model for
Radionucleide Transport in Tidal Waters. Master of Science Thesis,
Environmental Engineering Dept., University of Florida, Gaines-
ville. March 1972.
12. Benson, N. and R. N. Barnette. Leaching Studies with Various
Sources of Nitrogen. J Amer Soc Agron. Vol. 31, 1939
13. Bigger, J W., D. R. Nielson and K. K. Tanji. Comparison of
Computed and Experimentally Measured Ion Concentrations in Soil
Column Effluents. Amer Soc Agric Engr Trans. £: 784-787.
14. Blaney, H. F. and W. D. Griddle. Determining Water Requirements
in Irrigated Areas from Cliraatological and Irrigation Data.
U. S. Dept. of Agric. Technical Paper No. 96. 1950.
15. Bloom, S. G., A. A. Levin and G. E. Raines. Mathematical Simula-
tion of Ecosystems- A Preliminary Model Applied to a Lotic Fresh-
water Environment. Battelle Memorial Institute Report, Columbus,
Ohio. April 1969.
16. Bravo, C. A. et. al^. A Linear Distributed Model of Catchment
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165. Trelease, F. J, and M. W. Bittinger, Mechanics of a Mathmatical
Groundwater Model. J Irrig and Drainage Div, ASCE. 89(l):51-62,
March 1963.
166. U. S. Army Corps of Engineers. Everglades Gaging Program Pro-
gress Report—Everglades Area, Florida. Report No. 6. U. S.
Army Engineer Dist., Meteorology and Regulation Section, Jackson-
ville, Florida. Dec. 1956.
\
167. U. S. Dept. of Agric. and U. S. Dept. of Commerce. Monthly
Precipitation Probabilities by Climatic Divisions. Miscellaneous
Publ. No. 1160. 1969.
168. U. S. Weather Bureau. Rainfall Frequency Atlas of the U. S.
Technical Paper No. 40. May 1961.
169. Warrick, W. A. J. W. Biggar and D. R. Nielsen. Simultaneous
Solute and Water Transfer for an Unsaturated Soil. Water Resources
Research. 1(5):1216-1225, Oct. 1971.
180
-------
170. Weaver, H. A. and W. H. Speir. Applying Basic Soil Water Data
to Water Control Problems in Everglades Peaty Muck. ARS 41-40,
Agric. Res. Service, U.S.D.A. Nov. 1960.
171. Webber, L. R. and T. H. Lane. The Nitrogen Problem in the Land
Disposal of Liquid Manure. In: Animal Waste Management. Proc.
Cornell University Conf. on Agricultural Waste Management, Syra-
cuse, New York. Jan. 1969. p. 124-130.
172. Weibel. S. R., e£ al.. Pesticides and Other Contaminants in Rain-
fall and Runoff. J. AWWA. 518:1075-1084, 1966.
173. Williford, J. W., e£ al_. The Movement of Nitrogenous Fertilizers
through Soil Columns. In: Collected Papers Regarding Nitrates
in Agricultural Waste Waters. FWQA, Dec. 1969.
174, Winterer, E. V. Percolation of Water Through Soils. Calif.
Aqric. Exp. Sta. Annual Rept. from July 1, 1920 to June 20,
1922.
175. Yeh, W. and G. W. Tauxe. Quasilinearization and the Identification
of Aquifer Parameters. Water Resources Research. 2.C2) :375-381,
April 1971.
181
-------
SECTION XI
PUBLICATIONS
Perez, A. I., W. C. Huber, J. P. Heaney and E. E. Pyatt. A Water
Quality Model for a Conjunctive Surface— Groundwater System.
(Presented at the Seventh American Water Resources Conference,
Washington, D. C. October 1971.)
Perez, A. I., W. C. Huber, J. P. Heaney and E. E. Pyatt. Back-
ground of a Surface— Groundwater Quality Model. (Presented at ASCE
Water Resources Engineering Conference, Atlanta, Georgia. January
1972.)
Perez, A. I. A Water Quality Model for a Conjunctive Surface—
Groundwater System. Doctoral Dissertation, Environmental Engineering
Sciences Department, University of Florida, Gainesville. June 1972.
Perez, A. I., W. C. Huber, J. P. Heaney and E. E. Pyatt. A Water
Quality Model for a Conjunctive Surface— Groundwater System: An
Overview. Water Resources Bulletin. 8^(5): 900-908, October 1972.
182
-------
SECTION XII
LIST OF SYMBOLS
a = constant in infiltration equation; also, constant in
dispersion equation
A = infiltration parameter
A, B, C = left-hand side coefficients in tridiagonal matrix
AC, BC, CC = left-hand side coefficients in tridiagonal matrix
c • concentration; also, a subscript indicating corrected
value
c* = intermediate value of concentration
C B slope of moisture versus pressure head diagram
C B intrinsic permeability
d B soil particle diameter
d = thickness of "equivalent layer" for mole drains
o
e
D B right-hand side coefficient in tridiagonal matrix; also,
dispersion coefficient
DC = right-hand side coefficient in tridiagonal matrix
D = distance
5
D . D = dispersion coefficients in x- and y-directions
x y
e = vapor pressure
E = evaporation rate
f = infiltration rate
f. = initial infiltration rate
f = final infiltration rate
F = potential volume of infiltration
P
g = gravity
183
-------
h = distance between water table and drain
i = rainfall intensity; also, cumulative infiltration
i, I = column index
i = rainfall excess, (rainfall less infiltration)
I = total infiltration; also, infiltration rate
j, J - row index
k = Shield's parameter; also, decay coefficient
K = hydraulic conductivity or permeability
K_ = dispersion coefficient
L ~ catchment length; also, length of drains
m = exponent constant in kinematic wave formulation
(=5/3 for turbulent flow)
n = time step; also, Manning's roughness coefficient; also,
a constant; also, a subscript indicating new value
p = subscript indicating predicted value
P = mass of pollutant; also phosphorus
P = initial mass of pollutant
q = overland flow per unit width
q = flux of sediment
S
Q = flow rate; also groundwater recharge
r - nutrient release rate
R = rainfall rate; also rainfall less evaporation; also,
hydraulic radius
R = total rainfall
R = distance ratio
s
S = source/sink; also, sorptivity; also mole drain spacing
S, = bed slope
D
S. = sink rate of nutrients from water
i
S = specific gravity
S_ = source rate of nutrients to water
t = time
t = time of concentration
c
184
-------
tf = duration of infiltration
t = rainfall duration
u = nutrient uptake rate
u, v, w = velocities
v = critical scouring velocity
c
w = relaxation factor in groundwater flow calculation; also,
water withdrawal rate per unit depth
W = water withdrawal rate; also, weight
x, y, z - space coordinates
y = flow depth in overland flow
y = flow depth at catchment outlet
Li
z = elevation head
a = square of ratio of horizontal to vertical spacing in
groundwater model; also, constant in kinematic wave over-
land flow computations; also, decay constant in infiltra-
tion equation
a_, 3 = constants in dispersion equation
•5 3
Y = unit weight of water
6 = volumetric moisture content (fraction); also angle of
catchment slope
v = kinematic viscosity of water
p = density of water
= hydraulic head
ty = pressure head; also, constant in sediment load calculations
T = shear stress at the stream bed
T = critical shear stress for erosion
cr
A list of computer variable names is provided in the comments section
at the beginning of each program.
185
-------
SECTION XIII
APPENDICES
Page
A. Details of Surface Water Models 187
B, Details of Unsaturated Zone Model 201
C. Details of Groundwater Model 225
D. Details of Interfacing Methodology 232
E. Details on Application to the Lake Apopka Basin 236
F. Summary of Additional References on Agricultural Nutrients 257
G. Listing of Computer Programs 259
186
-------
APPENDIX A
DETAILS OF SURFACE WATER MODELS
INTRODUCTION
Since adequate information is available on the Stormwater Management
Model runoff and receiving portions (Environmental Protection Agency,
1971), the discussion on surface water models is confined to the kine-
matic wave overland flow formulation as described in Section IV. The
mathematics of this formulation are covered thoroughly by Eagleson
(1970) and, therefore, the emphasis herein is on illustrations and
sample calculations. Quality computations are included. Finally, a
description of the various program subroutines is provided. Consistent
English units are used throughout. Metric equivalents are not given
since the data are hypothetical.
SAMPLE CALCULATIONS
Catchment Geometry
The geometric characteristics of a hypothetical catchment are illus-
trated in Figure 73. Part (a) of this figure is a plan view of the
catchment and its boundaries and assumed dimensions are indicated.
A side view of the catchment is presented in part (b) on which the
slope is shown.
Hydraulic Parameters
As discussed in Section IV, the flow rate at the catchment outlet can
be computed from the formula:
q - °tym (42)
where q = discharge per unit width, ft3/sec/ft
y = depth of water, ft
a and m = constants
187
-------
oo
oo
TOPOGRAPHIC
CONTOUR
LINES
SURRVCE BODY
OF WATER
PART A - PLAN VIEW
f f FTRAINFALL
100 ft
PART 8 - LATERAL VIEW
Figure 73. Plan and lateral views of a hypothetical catchment.
-------
The values of a and m vary according to whether the flow is laminar or
turbulent. Eagleson has pointed out that since both flow regimes are
likely to occur on vegetated surfaces, a compromise value of m « 2
might be used (Morton, 1938) . The corresponding value of a must then
be determined from field measurements,
The lack of appropriate data in the Lake Apopka basin, and possibly many
other areas, precludes the statistical estimation of the parameter a.
For this reason, the flow was assumed to be turbulent (m = 5/3) . This
assumption permitted the direct calculation of from:
(43)
where 6 = catchment slope angle
n = Manning roughness coefficient
In this case, the units of a are (ft1/ '/sec).
The value of n can be estimated from tabulated values for different
types of land surfaces (Henderson, 1966). In vegetated areas, it appears
that n varies with flow depth (U. S. Army Corps of Engineers, 1956). For
simplicity, a fixed value of n • 0,05 is assumed for the hypothetical
catchment in question.
Hydrologic Input Conditions
The hydrologic inputs required to perform the overland flow calculations
are rainfall rates, infiltration rates and ponding depths. The latter
two are obtained from the unsaturated zone model. The assumed values
of these three variables are indicated in Figure 74. Evaporation rates
are not considered.
Preliminary Calculations
Since the analytical kinematic wave formulation requires the usage of
constant rainfall and infiltration inputs, preliminary calculations
must be conducted to obtain time-average values for these variables.
The first step is to inspect the ponding depth values to determine the
time at which maximum depth occurred. This time is then assumed to be
the duration, tr, of the simplified rainstorm, which is 15 min (Figure
74).
Next, the total rainfall, R-, is computed as follows:
189
-------
LJ
CC
Sc
e
7
i
6
5
3
2
£-
\
-
-
•
1
1
t 1 1 III
10
IS
20
25
30
< c
(E ""_
£ CK
O
4
3
1
(
-
] 1
i i i i ili
35 10 15 20 25 30
0. ?
UJ 2
1 - 1
10
1 l
15
20
25
30
TIME, min
Figure 74. Hypothetical rainfall rates, infiltration rates
and ponding depths used in overland flow
calculations.
190
-------
in-min
[8(2.5} + 7(5) * 4(5) + 2(5) + 1(5) * 0.5(5)] hr _ .
60 min/hr -- lt54 in
and the synthetic rainfall intensity becomes
= 0.102 in/rain = 0.00858 ft/min f or 0 < t < t
— — r
= 0 for t > t
Turning to the infiltration calculations, it is assumed that the time
at which ponding disappears constitutes the end of the infiltration
process. An inspection of Figure 74 reveals that the duration of this
process is tf » 27.5 min.
The total amount of infiltration for that period is
in-min
j m [1(2.5) + 1.5(5) * 1.5(5) + 2(5} + 0.5(5) + 0.5(5)] hr
T "" 60 min/hr
* 0.538 in
Accordingly, the synthetic infiltration rates becomes
= 0.0196 in/min = 0.00162 ft/min for 0 £ t 5 tf
for t > t£
Kinematic Wave Calculations
Two main cases arise in the formulation, depending on the relative
values of the storm duration, tr, and the time of concentration, tc.
The latter is computed from the formula:
191
-------
Li
1/m
(44)
where L = catchment length, ft
i^ = rainfall excess (rainfall less infiltration in ft/min)
Substitution of the appropriate parameters in (44) yields:
c
100(0.00858
1.49flO V
0.05[100 J
- 0.00162)'2/3
1/2 f60 sec)
[ minj
3/3 f ,
(2800)
[565J
0.6
=2.6 min
The correction factor in the denominator was necessary to convert a from
(ft1/3/sec) to (ft1'3/min). For the particular catchment and hydrologic
conditions in question, t exceeds t .
JL C
It is now possible to compute the depths of water, and the correspond-
ing velocities and flows, at various times. These values are computed
herein only at the catchment outlet. The calculations for the first
time step (t = 5 min) are given below.
The equations used to obtain the flow depth are
y = (i - f)t 0 £ t £ t
(45)
(46)
A more complex relationship must be used for the case t >-tr. Since in
this case t < t < t ,
£ — — f
the depth becomes
(0.00696)(2.6) = 0.0181 ft
192
-------
The corresponding velocity is
v = ay51"1 = (9.42)(0.0181)°>66 = 0.705 ft/sec
and the flow per unit width becomes
q = aym = vy = 0.01275 ft3/sec/ft.
Multiplying this unit flow by the width (ft), the resulting catchment
outflow is
Q = qw = (0.01275)(60) = 0.765 ft3/sec.
Sediment Load Calculations
The sediment transport formula used in this study is that developed by
duBoys (Einstein, 1964):
To Tcr
Y (47>
' t
where q = flux of sediment, Ib/sec/ft of width
ty = a constant for a given bed composition
T a bed shear stress, lb/ft2
o
T a critical shear stress, lb/ft
cr
2
3
Y a unit-weight of water, lb/ftj
The bed shear stress can, in turn, be expressed as:
T = yys (48)
where y = flow depth, ft
S, = slope of the bed
193
-------
Substitution of equation (48) into (47) yields
q
s
T
Y
(49)
which is the actual expression utilized herein. Values of if> and Tcr
are tabulated (Einstein, 1964) for various grain sizes.
Assuming that the soil in the hypothetical catchment is fine sand,
the sediment load at the first time step (t = 5 min) is:
n — fc o*z v i n5> /*i QI v
n *- ID.^J ^ J.U ) I J. • o J. ^
Te V** • **** *fcv ^ V ^ • w * — — ,/ V ~ "" ^1***** •* v x- « 4
s I 62.4
=1.47 Ib/sec/ft
Thus, the total catchment load becomes
Qs = wqg =88.3 Ib/sec.
The calculations on the nitrogen and phosphorous fractions in the
sediment load are shown in Appendix E.
PROGRAM DESCRIPTION
The kinematic wave overland flow program is composed of a main program
(MAIN) and several subroutines, AVE, WAVE, RECV, HIDUR, LODUR, RQUAL
and WRIT. All these subroutines are involved in quantity computations,
with the exception of RQUAL and WRIT. The former handles water quality
computations and the latter writes out the overall results.
Main Program
The first task of MAIN is to read in the necessary hydrologic, geometric
and quality data. Included among the geometric information is the
numbering system that connects the catchments with the receiving elements.
Following the data input, MAIN calls subroutines AVE, WAVE^ RECV, RQUAL
and WRIT, in that order. Subroutine AVE performs the averaging compu-
tations for rainfall, evaporation and infiltration, and thus prepares
these data for use in the kinematic wave calculations. The task of
WAVE is to carry out these calculations for all catchments and time
steps. It ultimately computes flows into the receiving elements.
Subroutine RECV then computes the total flows into each receiving element
194
-------
from all its feeder catchments. The functions of RQUAL and WRIT were
explained above.
Subroutine AVE
This subroutine determines the mean values of infiltration and rain-
fall for the time period considered. The stepwise procedure used to
obtain these values, discussed below, are also shown in Figure 75 and 76.
!
Beginning with infiltration, AVE first determines the time step after
which this process no longer occurs. The method used is to scan the
infiltration results from the unsaturated zone program and establish
when the ponding depth becomes zero. The time interval corresponding
to this event is considered to constitute the end of the infiltration
process. The scanning process described above is illustrated as Step 1
in Figure 75.
In actual practice the runoff process will probably produce a zero
ponding depth condition at an earlier time than that obtained from the
above scanning method. The reason is that the unsaturated zone model
assumes a laterally stagnant, non-dynamic, ponding situation. Never-
theless, it is thought that the approximation used is adequate for the
modeling of rural watersheds. In the future, one method to obtain
more accurate results might be to interface the infiltration and over-
land flow models at every time step.
Following determination of the duration of infiltration, AVE computes
the time-average value of the infiltration rate. This calculation is
depicted as Step 2 in Figure 75. Finally, updated values of the infil-
tration rate are stored for all the time intervals in the study period,
in the manner shown in Step 3 of the same figure.
Subroutine AVE then considers the rainfall process, and converts the
original irregular-shaped hyetograph into a step function containing
the same total amount of rainfall. This conversion process is begun
by scanning the output from the unsaturated zone program, determining
the time at which the maximum ponding depth occurs and arbitrarily
setting the duration of the synthetic storm equal to the above time.
This process is illustrated in Step 1 of Figure 76. The subroutine
then computes the total rainfall from the original record, as shown
in Step 2 of the same figure. Finally, it constructs a synthetic step
function hyetograph (Step 3) using the duration, tr, determined above.
Subroutine WAVE
To better represent the overland flow process, it is desirable to apply
the .kinematic wave relationships not only at each catchment outlet,
but also at several points along the length of the catchment. For this
195
-------
STEP I SCAN PONDING DEPTHlWATER DEPTH)
DATA FROM INFILT TO DETERMINE
END OF INFILTRATION PERIOD.
2
ASSUMED END OF
INFILTRATION PROCESS
23 V M-l M
TIME INTERVAL NUMBER
i
UJ
I
O
I
b
i
STEP 2 USE THE INFILTRATION RATES FOR M
PERIODS ABOVE FROM INFILT TO
OBTAIN AN AVERAGE VALUE.
2 3 V M-l
TIME INTERVAL NUMBER
M
STEP 3 USING THE RESULTS OF (2) ABOVE
CONSTRUCT AN EQUIVALENT STEP
FUNCTION.
2 3 ^-l M
TIME INTERVAL NUMBER
M + l M+2
Figure 75. Graphical representation of computational steps
in subroutine AVE involving infiltration.
196
-------
MAXIMUM DEPTH-?
STEP I SCAN PONDING DEPTHS FROM INFILT
TO DETERMINE TIME OF MAXIMUM
PONDING DEPTH
f
DEPTH.
1 1
23 4 5 6 78
TIME INTERVAL NUMBER
I
SHADED AREA IS TOTAL RAINFALL
TEP2 USE ORIGINAL HYETOORAPH TO
COMPUTE TOTAL RAINFALL.
2 3 4 f 5 6 789
TIME INTERVAL NUMBER
UJ
*
a:
ce
SHADED AREA SAME AS
IN (2) ABOVE
STEP 3 CONSTRUCT SYNTHETIC RAINFALL IN
STEP FUNCTION FORM.
123456 7 •
TIME INTERVAL NUMBER
Figure 76. Graphical representation of computations
in subroutine AVE involving rainfall.
197
-------
purpose each catchment is divided into a given number of intervals,
and the formulation is applied to the downstream end of each interval.
This division process is illustrated in Figure 77. In this study, only
one interval, comprising the entire catchment, was considered. This
was acceptable because of the small size of each catchment.
The type of formulas used at each catchment location largely depends
on the relative values of tr, the storm duration as defined in AVE,
and tc, the time of concentration. The latter term can be defined as
the maximum amount of time during which an increase of depth can occur
on the surface. While tr is assumed to be the same throughout the
basin, tc varies with location. At each time step and each catchment
interval, WAVE compares these values. If the storm has a relatively
high duration, namely tr exceeds tc, subroutine HIDUR is called. Con-
versely, if tc is greater than tr, LODUR is utilized. Each of these
two subroutines computes the depth of water at each catchment location.
After the flow depths have been determined, WAVE proceeds to calculate
their corresponding velocities and flows by means of equations (7)
and (6). These values are then stored for future use.
Subroutine RECV
At each time step, this subroutine computes the total flow entering
each receiving element from its feeder catchments. To accomplish this
purpose, it considers the stored flows corresponding only to the down-
stream interval in each catchment.
Subroutine HIDUR
The kinematic wave computations carried out in this subroutine corres-
pond to the case tr > tc as mentioned above. These computations branch
out according to three possible cases. To determine which particular
case holds, HIDUR compares the relative values of t, the time corres-
ponding to the time step in question, to three other parameters: tc, tr
and t^. The latter parameter is defined as the time at which the depth,
y, at a given location vanishes. At the conclusion of these calculations,
control is returned to subroutine WAVE.
Subroutine LODUR
This subroutine is utilized in a relatively low duration situation;
namely, tr < tc. The computations herein are broken down into three
classes, depending on the relative values of t^, defined above, and
tp, which represent the time at which the depth begins to decrease.
Each one of these three cases is in turn divided into three sub-cases.
As in HIDUR, control is returned to WAVE dollowing completion of the
computations.
198
-------
RAINFALL
to
to
INFILTRATION,
RECEIVING
ELEMENT
Figure 77. Lateral view of overland flow situation, indicating catchment divisions.
-------
Subroutine RQUAL
This subroutine utilizes the duBoys sediment transport equation (Einstein,
1964) to compute the suspended solids fluxes at the catchment outlets.
It also calculates the nitrogen and phosphorous fluxes at the outlets.
Subroutine WRIT
After subroutine RQUAL has completed its calculations, subroutine WRIT
is called to print out the results obtained. The most important re-
sults are the flows and concentrations at the various catchments, and
the total inputs to the receiving elements.
2<)0
-------
APPENDIX B
DETAILS OF UNSATURATED ZONE MODEL
INTRODUCTION
The first item to be discussed is the derivation of coefficients used
in the matrix solutions of both the flow and quality portions of the
unsaturated zone model as developed in Section V. Subsequently, sample
calculations are provided for one time step. Finally, a discussion of
the various subroutines is given.
DERIVATION OF COEFFICIENTS
Flow Model
The basic link-node structure used in a given soil profile is shown
in Figure 78. Also indicated therein are the boundary conditions at
the top and bottom of the profile.
The flow governing equation used in this study was that developed by
Freeze (1969), with an additional sink term:
s = J-
3t 3z
(50)
where ip = pressure head, cm
C(ip) = slope of the moisture versus pressure head curve, I/cm
t = time, min
S = sink, 1/min
z = elevation, cm
K(i|0 = hydraulic conductivity, cm/min
201
-------
TOP BOUNDARY CONDITIONS:
K>
o
to
RAINFALL
EVAPORATION
GROUND
SURFACE
S////J///S//SSSSSSS S TX / X / /^//SSSSSS ' S S
L-l
I
NODE
NUMBERS.
.WATER TABLE
BOTTOM BOUNDARY CONDITION: RECHARGE
Figure 78. Illustration of unsaturated zone profile, showing nodes
and boundary conditions.
-------
The sink terra above includes withdrawals by plants and artificial
drains. Its usage is illustrated in Appendix E.
jnterior Node—
The finite-difference expression corresponding to equation (50), written
for an interior node (1 < j < L), is:
;M
At
_L
Az
Az
n
n
(51)
in which the subscripts denote the node and the superscripts the time
step. The expressions for ^,, if)__, and ty-r-r-r are the following:
+ *?*. *
(52)
*
II
,n
(53)
(54)
These expressions are valid only for n > 2. Otherwise, the definitions
Provided by Rubin and Steinhart (1963) should be used.
203
-------
Since the flow governing equation form desired is:
V?1-
it becomes necessary to regroup terms in equation (51). Expanding
first, the following equation is obtained:
C55)
2Az
2Az
2Az
2Az
2Az
2Az
2Az
^- (56)
Rearranging terms and multiplying both sides by (-1), the expression
becomes
2Az
.. -.-
2Az 2Az
At
2Az
2Az 2Az
(57)
Note that the terms on the right-hand side are known quantities.
Comparing equation (57) to (55) it follows that:
2Az
(58)
204
-------
Bi = 2Az * 2Az* * At
-n
2Az
(59)
(60)
n
At
2Az 2Az
(61)
The expressions for the four coefficients above are valid for any in-
terior node. However, the expressions for the top and bottom nodes
constitute special cases.
Top Node-
The boundary condition at the ground surface (j « L) is (Freeze, 1969):
JfWU
- 1
(62)
where R « net rainfall infiltration (rainfall less evaporation),
cm/rain
If a sink term is included, equation (62) can be rewritten as:
R - S(Az/2) - K
(63)
Where • ijj + z
The finite-difference expression for (63), after some rearranging,
becomes
205
-------
s"(Az/2)Az
K
r i )
n-|
' '"I ^ *L-1 "
K
r i -\
n-T
Thus, the coefficient expressions become
A? - 0
L ~ L
(64)
(65)
(66)
RAz
" ZL + ZL-1
S"(Az/2)Az
Li
(67)
Bottom Node—
The boundary condition at the bottom node (j = 1) can be written as:
K
3z
= Q + s"(Az/2)
(68)
Where Q = groundwater recharge (positive downward), cm/min
After some manipulations, the following expression is obtained:
QAz . „ S1(Az/2)Az
K
, j »
n- -y
% .
• *1 '2
K
r i ^
n-f
* i
(69)
206
-------
It follows that the coefficient expressions are:
cn = o
QAz
""*
Zl + Z2
(70)
(71)
(72)
Quality Model
The quality governing equation consists of a convective-dispersion
formulation with provision for source/sink reactions:
where
KnCv)
S - S.
o i
D
(73)
time> min
moisture fraction, cm/cm
concentration, mg/1
velocity (positive upward), cm/min
elevation, cm
dispersion coefficient, cm2/min
source, mg/l/min
sink, mg/l/min
Since the flow through the soil will be downward following rainstorms,
it is convenient to re-define the velocity sign convention as positive
downward. The resulting expression becomes:
207
-------
v * VCV) t S - S. (74)
9t 9z D ' 3z2 01
The computation of source/sink terms is illustrated in Appendix E. To
solve (74) numerically, it is necessary to regroup its terms and obtain
an expression of the form:
V? - (AB)ncn+i = (DC)J (75)
The expressions for coefficients AC, BC, CC and DC are derived below.
Interior Node—
For an interior node (1 < j < L), equation 74 can be written in finite
difference form as:
(en - eH n (cn - c
-"1 3 3 J + en I J J
j At j At
n n , n n
'
After some algebra, the following expressions are obtained
vn
en _ 9n-i en 2KD
BCn -
BC
.
j « it C78)
v"
cc" ' - + C793
208
-------
rn _ _ n
UL. = o
(80)
Top Node-
At the uppermost node (j = L), the dispersion and source/sink terms are
neglected. Its concentration at the first time step is set, and its
value at subsequent steps is computed as follows:
1. Determine the distance, Ds, that the substance traveled between the
current and the previous time step.
(81)
2. Calculate the ratio, Rs, of this distance to the midpoint distance
between the two upper nodes
R =
s (Az/2)
(82)
3. Develop a mass-balance relationship involving the current and pre-
vious time steps:
L L
L L S
(83)
4. Substitute (82) into (83) and rearrange terms, to yield:
n
L n-i
en L
vl^At
i ^
f Az/2 J
(84)
Thus, coefficients AC, BC, CC and DC are not used explicitly to compute
the concentrations at the top node.
209
-------
Bottom node—
Due to the absence of two adjacent nodes, required to evaluate a second
derivative, the dispersion term is not considered at the bottom node
(j « 1). The finite-difference form of equation (74) at this location
is:
ffln ftn-i) f n n-il f n n
niei - 91 J + en K " c! J _ n [C2 " cl
1 At °1 At " vl I
C - Si
°i h
(85)
The resulting expressions for the coefficients, after regrouping of
terms,
(86)
At
At Az
(87)
C? = 0
(88)
c n-i
- s
n
(89)
Discussion of Dispersion Equation—
The expression used to compute the dispersion coefficient was derived
by Hoopes and Harleman (1965) for sands based on laboratory experiments.
It is unlikely to be generally applicable, but was useful for this
study. The expression is:
(90)
V
where v = kinematic viscosity of water, craVsec
a_ = 88 (for sand)
«J
v = velocity, cm/sec
210
-------
C = coefficient of intrinsic permeability, cm2
B = 1.2
The usage of this formula requires that the computed velocities at the
nodes, which have units of Ccm/min) be converted to (ft/sec).
Intrinsic permeabilities are computed from values of the hydraulic
conductivity. Since the kinematic viscosity of water is a function
of temperature, the value corresponding to the mean annual temperature
in the study area was used.
After KD is computed, it must be converted to units of cm2/min for
usage in the quality calculations. The various conversion factors used
in connection with equation (90) are given in the listing of the pro-
gram (see Appendix G) .
SAMPLE CALCULATIONS
In both the flow and quality models, the calculations follow a two-step
procedure: 1. compute the values of the coefficients of each node, and
2. solve the resulting system of equations using Gaussian elimination
(Hadley, 1964). Step (1) is illustrated below for a single node in
the soil profile. Space limitations preclude the illustration of step
(2) . The soil type considered herein was the sand of the Lake Apopka
basin, whose estimated hydraulic characteristics are shown in Figures
42 and 86.
Flow Model
An isolated portion of a soil profile is shown in Figure 79. The
calculations to be discussed are those for node j = 5 and time step
n « 7. Indicated in the figure are the pressure heads for this node,
and its two adjoining nodes, at the previous time step (n = 6) . Since
the computations of withdrawals (sinks) by plants and artificial drains
are explained in Appendix E, these sinks are considered to be zero in
this discussion.
The coefficients AL,Bg, C75 and D* can be obtained from the formulas
for an interior node, that is, equations (77), (78), (79) and (80).
A review of these expressions indicates that to determine certain values
of K and C, it is first necessary to compute ty*, ^TT7 and ^TTT?
5 5 5
Equations (52), (53) and (54) for these three variables contain terms
for pressure heads at the present time step, which are yet unknown.
Thus these values must be estimated by any one of several methods,
211
-------
to
»-«
tsJ
TIME STEP
B=6
TIME STEP
„= 7
J JJ S s S J J S J
PRESSURE HEADS cm
ATn=6
V4 = -60.1
Y5 = -72.4
V6 = -80.3
COORDINATE AXES:
L
>J/J*V*>
.
'
GROUND SURFACE
•rs s s f f f f
j=5
J=4
o—
Az-IOcm
1=
?
At * lOmln.
WATER TABLE
Figure 79. Illustration of an isolated portion of an unsaturated zone system, showing
flow model data.
-------
including regression. For simplicity, these pressure heads are assumed
to be equal to those in the previous time step. Substitution into
equations (52), (53) and (54) yields:
cm
, 7 f- 72.4 - 80.3 - 80.3 - 72.4) _. ,
^ = - - - . _ 76i3
5 '
, . [- ^O.l - 72.44- 72.4 - 60.1J . . w>2
5 *" '
7 (• 72.4 - 72.4)
"'ill B [ - 2 - J a - 72'4
The conductivities and slope parameters corresponding to .these inter-
mediate head values can be estimated from Figures 86 and 42. The
resulting coefficients are:.
D; - (0.0038,(- 80.3) * [(0.00378) (10). 0.076. 0.096)^ „.,,
+ CO. 0048) (- 60.1) + 0.076 - 0.096 + 0 • - 0.266
Thus, the basic finite-difference equation for j • 5 and n • 7 is:
- (0.0048)l(i7 + (0.0124W7 - (0.0038)i|»7 = (- 0.266) (91)
it 5 6
Following the computation of pressure heads throughout the soil profile
using the elimination technique, it is possible to determine the flow
velocity at each node. The finite- difference formula for Darcy's law,
applied to node j = 5, is:
213
-------
C92)
where v denotes the velocity (cm/min) , which is defined as positive in
the downward direction, and z the elevation (cm) . It may be noted that
Z - Z = 2Az = - 20.0 cm
For demonstration purposes, it may be assumed that $1 = -56.1, if;7.
- 69.0 and i^7 = - 78.1. From Figure 86, the conductivity becomes
K7 = 0.074 cm/min
Substitution into equation (92) produces
v' . - (0.074) [(- 56a> - I" "•" * <- = 0.0074 cm/min
5 ^
Quality Model
The computations of nutrient withdrawals by pjlants and drains, and of
source/sink reaction rates are shown in Appendix E. For this reason,
the terms involving these processes are set equal to zero below.
An isolated portion of a soil profile is depicted in Figure 80. Shown
therein are assumed values of moisture content, 9, for three nodes at
two time steps. Also indicated is the velocity, v, at the middle node
at the current time step n = 10, and the concentration at n = 9. In
practice, these moisture and velocity values are obtained from the flow
model. The concentrations at the various nodes at the current time
step constitute the unknowns.
To compute the values of the coefficients AC, BC, CC and DC at node
j =4, it is first necessary to determine the dispersion coefficients
at that location. As can be seen in equation (90), the computation of
this coefficient requires the values of the kinematic viscosity and
hydraulic conductivity.
Values for the kinematic viscosity of water are available for different
temperatures (King and Brater, 1963). The subsurface water temperature
214
-------
ro
t->
tn
TIME STEP TIME STEP
n = 9 n=IO
GROUND
/ / / s / /
VALUES
®!
e|
el
c!
VALUES
v,"
®4
.2
_
// /"/ / S / / S S S / /
- 0.24
= 0.20
= 0.16
= 5.2 ma/I i - R
ATn=lO= J = 4
= 0.06 cm/mi n . _ 3
= 0.001 cm/iec
= 0.25
mnornuATc
CUUKUI NA 1 1
AXIS
t
1
<:
b
<
, i
^ ^
L f
<
At = lOmin.
>
>
r
Az = 10 cm
?
/
/WATER
TABLE
Figure 80. Illustration of an isolated portion of an unsaturated zone system,
showing quality model data.
-------
in the Lake Apopka basin was estimated to be 71.5°F, which is the
annual mean value for Orlando, Florida (Lichtler, Anderson and Joyner,
1968). Accordingly,
v = 1.040 x io~5 ft2/sec = 1.12 x 10~8 cm2/sec
To determine the hydraulic conductivity, it is necessary first to
determine the pressure head corresponding to the moisture content at
j = 4. From Figure 42,
= - 59.0 cm
The corresponding hydraulic conductivity value, K, obtained from Figure
86, is converted to intrinsic permeability by the relationship
C« K v/g (93)
where g = gravitational acceleration. Thus:
c « (0.107 cm/min) l1-9^^)"7^/^] = 2.04 x io~8 cm2
Substitution of the above values into equation (90) yields
- n 12 x lo-'ifflinf1'0 x 10"3C2.Q4 x lo-y-n
- (1.12 x 10 )(88)^ i.i2 x io-« J
= 1.98 x 10"5 cmVsec = 1.19 x io"3 cmVmin
The desired coefficients AC, BC, CC and DC are computed from equations
(77), (78), (79) and (80), respectively, as follows:
ACio _ (M6- } + [1.19 x 1Q-3] _
A(\ - [2(10)J I io2 J " °*0030119
216
-------
0.25 - 0.20
I 10 I
0.25
10
cc10 =
tf
.06 ] fl.19 x lp-3<
(10)J + I 10*j
= 0.0030119
DC
.. = o * 0 * [t°-25ff'25?) . 0.13
The finite-difference equation for j = 4 and n = 10 is, therefore
- (0.0030119)c10 + (0.0750238)c10 - (0.0030119)c10 = (0.13)
The system of equations involving all nodes is solved in a manner
analogous to that in the flow model.
PROGRAM DESCRIPTION
Quantity Portion
Summary—
The purpose of the model is to solve numerically the one-dimensional,
unsteady-state flow equation for an unsaturated porous medium. Ini-
tially, the program considers a given vertical column of soil with a
certain pressure (moisture content) profile. This column is parti-
tioned into a discrete number of nodes, as shown in Figure 78. At
each time step, the program determines the values of hydraulic pressure
resulting from the values of two boundary conditions, namely: (1)
the combined effect of rainfall and evaporation at the top of the pro-
file, and (2) net recharge rate at the bottom. Since these boundary
conditions change with time, it is necessary to conduct an inter-
facing with the overland flow and groundwater programs.
In addition to the initial tension profile and boundary conditions,
some relationships involving moisture, tension pressure and hydraulic
conductivity must be provided as inputs to the model. These relation-
ships are essentially fundamental properties of each soil. Therefore,
the program computations must be conducted independently for each major
217
-------
soil type. Special formulas are also included to compute withdrawals
(sinks) of water at each node in the soil profile.
The output from the unsaturated zone flow model consists basically of
the values of moisture content and velocity, obtained from the corres-
ponding pressure heads at the various nodes. One table is printed out
for every time step. The velocities affect the movement and disper-
sion of substances dissolved in the water.
The computer model consists of a main program (designated as MAIN)
and several subroutines: COEF, EXTRA, VFLOW, WTABLE, PSIVAL, PSIMID,
SINK and MATRIX. Also, four function subprograms are utilized to per-
form moisture-tension-conductivity conversions. The names of these
functions are: CSL, CK, TENS and HUMID. A summary of the main pro-
gram, subroutines and functions follows.
Main Program—
The main program first reads in the data. A partial list of data is
given below:
1. Number of nodes and time steps.
2. Rates of rainfall, evaporation and net recharge flow at each time
step.
3. Vertical distance between node and interval between time steps.
4. Initial values of moisture content for all nodes.
Subsequently, some of the values above are converted into consistent
units. In the first time step, the moisture values given are used
to evaluate the various soil moisture functions.
In the second time step (N =2), the information from the previous step
and the boundary conditions are utilized to compute the values of
pressure at each node. In carrying out these computations, MAIN calls
subroutines PSIMID, COEF, PSIVAL (in that order) and various function
subprograms. Thereafter, it calls subroutine VFLOW to determine
the vertical velocities at each node in this time step.
For the following time steps (N > 2), the calculations are slightly more
involved than those for N = 2. Subroutines EXTRA and WTABLE are called
in addition to those above. Tables of moisture values for each node are
printed.
At any time step, the option of plotting the moisture versus depth
profile can be exercised. This can be accomplished by calling a spe-
cial subroutine package.
Subroutine COEF-
At the j-th node in the soil profile, it is possible to write an equa-
tion of the form of (55) which relates the pressure head, \p, at the
218
-------
node in question to the t|»'s at its adjoining nodes. The coefficients
associated with these three heads are designated in the program as
A?, B? and C?, in which the superscript n denotes the time step. The
coefficient designated as D? contains all known constants in the equa-
tion. In a soil profile containing L nodes, it follows that L equa-
tions of the general form described above can be written. The solu-
tion of the resulting system of equations (22) requires that L values of
the coefficients A., B., C, and D1? be calculated at every time step.
Subroutine COEF performs the calculations for these coefficients.
Two nodes, one lying on the upper boundary and another on the lower
boundary of the soil profile, have only one adjacent node instead of
the usual two. The calculation of the coefficients associated with
these two boundary equations is therefore a special case. It consists
of the application of Darcy's law between the boundary node in question
and its adjacent node. The flow between the nodes is set equal to the
known boundary condition flow.
The values of A1?, B., C. and D1} at each node obtained by subroutine
J «* J J
COEF are used elsewhere in the program to determine the tension head at
every node.
Subroutine EXTRA-
To compute the coefficients mentioned above in subroutine COEF, it is
necessary to determine first the value of some moisture properties of
the soil, such as hydraulic conductivity, at every node. Unfortunately,
these properties are a function of the respective pressure heads, which
are not yet known for the time step in question.
For this reason, these pressure heads most be predicted using their
computed values from previous time steps. Subroutine EXTRA performs
these extrapolations. It may be noted that EXTRA is not called at
219
-------
time step N « 2, since only one set of previous pressure heads is
available at that time.
Subroutine PSIMID-
To determine the coefficients An, B1?, C? and D1? in subroutine COEF,
it is first necessary to evaluate certain soil moisture functions, such
as conductivity. The numerical technique requires that some of these
functions be evaluated, not at the nodes themselves, but rather at
points midway between the nodes. Subroutine PSIMID computes the
values of pressure head, r|), at such midpoint locations, using linear
interpolation.
Subroutine SINK-
The option exists in this program to specify certain rates of with-
drawal of water (sinks) at each node in the soil profile. These with-
drawals can consist of water uptake by plants and/or artificial drain-
age. In the former category, the uptake rates must be determined from
climatic and agricultural information. In the latter case, any one of
several semi-empirical formulas to compute drainage rates may be used.
Additional information on these two types of calculations is given in
Appendix E.
Subroutine SINK adds the plant and drainage withdrawals for each node
in the soil column. The resulting values of total sink flows are stored
and then utilized in subroutine COEF to calculate the D" coefficients.
Subroutine PSIVAL-
The purpose of subroutine PSIVAL is to assemble the information needed
to compute the values of pressure head, \p, at each node, and then to
call subroutine MATRIX to carry out the computations.
220
-------
First, PSIVAL determines whether the top node is saturated or not.
Saturation at this node can be caused by two conditions: (1) net
rainfall rate (rainfall minus evaporation) greater than the saturated
conductivity of the soil, or (2) existence of a ponding situation due
to high previous rainfall. If either one of the above conditions holds,
the definition of the boundary condition at the top of the profile is
achieved by setting the pressure head at the top node equal to the pond-
ing depth. If these conditions do not hold, the top node is not satu-
rated and the usual upper boundary condition involving Darcy's law
applies.
After the proper boundary condition above has been selected, subroutine
MATRIX is called to compute the values of pressure head at each node.
Subroutine PSIVAL then predicts the depth of ponding (if any) for the
next time step.
Subroutine MATRIX-
This subroutine assembles the previously determined values of A1?, B? and
C? into a left-hand side matrix, and the values of D1? into a right-hand
side matrix, as shown in system of equations (22). It then proceeds
to solve for the unknown values of pressure head, ty» at each node.
As mentioned previously, the left-hand side matrix is of tri-diagonal
form and therefore the system is most easily solvable by an elimination
procedure. The specific technique used herein is Gaussian elimination.
It may be noted that the occurrence of a ponding condition at the top
of the profile introduces some changes in the calculations.
Subroutine WTABLE-
This subroutine determines the elevation of the water table at each time
step. To achieve this, it checks the previously computed values of
pressure head, starting at the bottom node and proceeding upward. After
221
-------
it locates the first unsaturated node (pressure head less than zero),
it computes the water table elevation.
Subroutine VFLOW-
Subroutine VFLOW first utilizes the values of pressure head, determined
by subroutine MATRIX, to find the total head (elevation plus pressure)
at each node. It then determines the flow of moisture at each node
from Darcy's law.
Plotting Subroutines Package—
This package is composed of subroutines GRAPH, PINE, CURVE, PPLOT and
function ACOS. The option to plot any tabulated results may be exer-
cised from the main program by calling subroutine GRAPH. This subrou-
tine, in turn, controls the flow of information between the other sub-
routines in the package.
Function Subprograms—
Four function subprograms are used in this computer model. They all
involve basic soil moisture relationships, and their names and purposes
are:
1. Function CSL. Computes the slope of the moisture versus pressure
head curve, as a function of the latter.
2. Function CK. Finds the hydraulic conductivity for a given value of
pressure head.
3. Function TENS. Converts values of moisture content into pressure
heads.
4. Function HUMID. Converts pressure heads into moisture contents.
The above relationships may be determined in the field for any given
soil type, or may be approximated using literature data for similar
soils. The latter approach was used herein. Segmented straight lines
were used to fit non-linear relationships.
222
-------
Quality Portion
Summary—
The main purpose of SOILQUAL is to compute, at every time step, the
concentrations of any given substance at the various nodes in a soil
profile. The required input data for this program can be broken down
into two basic types. The first category consists of water quantity
and flow data, obtained by interfacing with the unsaturated flow pro-
gram. These data include values of flow velocity and moisture content
at every node, in addition to initial and boundary conditions. The
second type is composed of water quality information, such as fertili-
zer availability and rate coefficients for source and sink reactions in
the soil.
The two types of data discussed above are used by the program to compute
the values of coefficients ACn, BC1?, CC?, and DC? in equation (75) at
•^ J J J
every node. Some of these coefficients vanish in the case of the top
and bottom nodes, which have only one adjacent node instead of the
usual two. The program then proceeds to solve a system of simultan-
eous equations to obtain the desired concentrations at every node.
The computer model consists of a main program, designated as MAIN, and
subroutines GROUP, MAT2 and OUTRIT. Following is a discussion of
both the main program and the subroutines.
Main Program—
Initially, MAIN sets up data storage blocks from which to retrieve
water quantity information previously obtained by the flow program.
Other blocks are designated to store water quality data read into the
model, and other water quality parameters generated by the model itself.
223
-------
After reading in the data, the program performs some simple calcula-
tions involving conversion factors, to obtain the desired consistency
in units. Next, it utilizes hydraulic results from the flow program
to compute the value of the dispersion coefficient at every node.
Subsequently, a loop involving all time steps carries out the follow-
ing instructions, in order: (1) calls subroutine GROUP to assemble at
every node the various terms in equation (74) into an equation of the
form of (75), and (2) calls subroutine MAT2 to perform the matrix
operations required to solve for the concentrations of substances at
each node. At the conclusion of this time loop, MAIN calls subroutine
OUTRIT to print out all the pertinent results.
Subroutine GROUP-
At every time step, GROUP determines the values of coefficients AC, BC,
CC and DC for every node in the soil system. Recall that the top and
bottom nodes constitute special cases. The structure of this subroutine
is very similar to that of subroutine COEF in program INFILT.
Subroutine MAT2-
This subroutine inserts the values of AC1?, BC1? and CC1? made available
by GROUP into a left-hand side matrix. Also, it inserts the corres-
ponding values of DC. into a right-hand side vector. Subsequently it
solves the system of equations in a stepwise fashion using Gaussian"
elimination. It may be noted that subroutine MAT2 is a modified
version of subroutine MATRIX in the unsaturated zone flow model.
Subroutine OUTRIT—
After subroutines GROUP and MAT2 have been called for all the time steps
in the period of study, MAIN calls subroutine OUTRIT to print out the
results. The most significant information provided by OUTRIT is a table
of concentration values at all nodes for all time steps.
224
-------
APPENDIX C
DETAILS OF GROUNDWATER MODEL
INTRODUCTION
Shown herein are sample calculations for the groundwater flow model
as described in Section VI. Space limitations preclude the presenta-
tion of counterparts for the quality formulation (Bedient, 1972).
Also included are descriptions of the subroutines in both the f;ow
and quality models. Consistent English units are used. Inasmuch as
the data are hypothetical, equivalent metric units are not given.
FLOW CALCULATIONS
The basic finite-difference relationship (Freeze, 1966) used in the
calculations expresses the hydraulic head at a given node in terms of
the heads at its adjoining nodes and other constants:
aKvCl,J)cf)(I,J+l)]
+ ccKv(I,J-l) + aK
where = hydraulic head, ft
K, , K = horizontal and vertical permeabilities, gal per
* v day/ft2
o
||| (dimensionless)
Ax * horizontal spacing, ft
Az = vertical spacing, ft
225
-------
Any consistent permeability units could be utilized in this formula,
since Kh and Kv act as weighting factors.
To illustrate the usage of equation (94), it is possible to isolate a
portion of a grid system, as shown in Figure 81 along with assumed
values of the permeabilities and geometric spacings. Also shown are
the heads at the various nodes. A new value of head at node (I,J) can
be determined by substitution into (94), as follows
(20) (70.0) + (20)(74.0)
+
'100'
50
V J
(40) (69.1) +
20 + 20 +
100
SO I
« f
rioo'
L50J
(40) (75. 0)1
(40) +
100
[50
^ /
(40)
= 72.2 ft
Next, a relaxation factor is applied to the computed value to promote
convergence in the calculations. The formula used for this purpose is
= wn(I,J)
(I.J)
(95)
in which w denotes the relaxation factor. The subscripts p, n and c
indicate the previous, new and corrected values, respectively. Expres-
sion (95) above is the same as equation (39). In this study, the value
selected for w was 1.85 (Freeze, 1971). Substitution of the appropriate
values into equation (95) yields
(I,J) - (1.85) (72.2) + (-0.85) (71.1) = 73.2 ft
Equations (84) and (95) are applied to the various nodes in the
groundwater system in an iterative fashion. It may be assumed that
after all iterations have been completed, the values of head (ft)
at the five nodes in question are:
226
-------
COORDINATE AXES:
IZ
Ni
fO
HORIZONTAL AND VERTICAL
PERMEABILITES, gol/doy/sq ft
FOR ALL NODES:
Kv=40
VALUES OF HEAD ft AT
n-th ITERATION'.
(I-I,J) = 70.0
4>(I,J) = 71.1
0(I + I,J) = 74.0
(I,J-l)= 69.1
0(I,J*I)= 75.0
Figure 81. Isolated portion of a groundwater grid system, showing
geometry and assumed heads and permeabilities.
-------
(I-l,J) • 71.5
(I,J) * 73.0
<|>(I+1,J) - 74.4
-------
and PREDIS. Following is a description of both the main program and
its subroutines.
Main Program—
The main program first reads in the data, some of which are listed
below:
1. A matrix (NCLASS) for node classification. The numbering system
indicates, for instance, whether the node lies on the water table or
another boundary, or whether its head value is known. This matrix
remains unchanged throughout the program,
2. Horizontal and vertical permeabilities for every node,
3. The values of head at nodes where it is known, and trial values
at other nodes.
The program then proceeds to write out the matrix PHI, which contains
the information in item (3) above. Next, the iterative procedure to
calculate the head at each node where it is unknown is begun. The head
at a given node is expressed in terms of the heads and permeabilities
of its surrounding nodes, its own permeability and the horizontal and
vertical spacings. However, the exact form of the expression varies
with the node type as designated in NCLASS. For this reason, several
conditional statements in the program check the value of NCLASS for the
node under consideration to ascertain the proper formula to be used.
Most nodes are of the interior type, that is, surrounded by four ad-
jacent nodes, and the standard formula for computing head is applica-
ble, Other nodes lie on no-flow boundaries of the grid system and are
thus surrounded by fewer than four nodes. In this case, subroutine
BOUND is called to perform the! head computation. If a certain discon-
tinuity effect occurs at the water table, the head at the affected
node is determined by subroutine TABLE. In all the above cases, a
relaxation factor is applied to the value of head following its cal-
culation.
After the iterative process to determine the heads throughout the grid
system has been completed, MAIN prints out these values in tabular
form. It may be noted that any grid nodes located above the water
table lie outside the groundwater system. For identification purposes,
their head values are printed out as zeros.
Next, subroutine EQUIP is called to determine the location of points
of equal head. Subsequently, MAIN calls subroutine VEL to determine
the velocities throughout the grid system. The last subroutine called
is PREDIS. Its function is to prepare the results obtained for use as
input to the water quality program.
229
-------
Subroutine BOUND—
The subroutine was designed to compute the head at nodes located on a
no-flow boundary. The existence of such a boundary can be brought
about by an impervious formation or by the flow patterns of a basin.
From Darcy's law, it follows that the hydraulic head on both sides of
a no-flow boundary must be equal.
Since nodes lying on boundaries are surrounded by fewer than the usual
four nodes, BOUND must first establish the location of the missing
node(s) (up or down, right or left) with respect to the boundary node
in question. Subsequently, it implicitly creates an "image" or imagin-
ary node at that location. To satisfy the no-flow condition, the head
at this image node is set equal to that of the node located directly
opposite it across the boundary. Finally, the computation for the head
at the boundary node in question is carried out,
Subroutine TABLE—
In the unlikely event that a node lying to the right or left of an
interior node is located above the water table, the head computation
becomes another special case. Subroutine TABLE first establishes which
of the adjacent nodes (left, right, or both) lie above the water table.
It then branches out to perform the appropriate calculation.
The general condition described above is brought about by large varia-
tions in the slope of the water table. By decreasing the spacings in
the grid system, the need to utilize subroutine TABLE can be minimized.
Subroutine EQUIP-
This subroutine determines the location in the grid system of points
having certain values of head. It uses linear interpolation to achieve.
this purpose. A table of results is printed.
Subroutine VEL—
Subroutine VEL uses Darcy's law to compute the horizontal and vertical
velocity components throughout the grid system. It then uses these
components to determine the resultant velocity vector at each node.
The direction of this vector is computed as an angle, in degrees,
measured counter-clockwise from the positive horizontal axis. Both
the magnitude and direction of the velocity vectors are printed out
in tabular form.
Subroutine PREDIS-
The function of PREDIS is to assemble the results obtained throughout
program PHI for subsequent transfer to the groundwater quality node.
The variables used in the interfacing are the velocity components,
horizontal and vertical spacings, time step and others. This inter-
facing can be carried out using punched cards or by means of common
storage locations.
230
-------
Quality Portion
The computer program used herein has basically the same sequence of
operations as that developed by Shamir and Harleman (1966), with some
modifications introduced by Bedient (1972). Since the latter has
provided a detailed description of the program, only a brief summary is
given below.
The computer model consists of a main program (MAIN) and the following
subroutines: CINPUT, TRIX, TRIY, EXPLIC, COEFX, COEFY, CFEXP, VCALC
and OUTPUT. The function of MAIN is to read in preliminary data,
compute the dispersion coefficients, and control the sequence of opera-
tions performed by the various subroutines.
Subroutine CINPUT reads in concentration data used as initial and
boundary conditions. The purpose of COEFX and COEFY is to provide the
values of the coefficients and right-hand side constants for all equa-
tions of the type of (40) and (41), respectively. The two systems of
equations corresponding to these types are solved by TRIX and TRIY.
After the computations involving the alternating direction implicit
procedure have been completed, subroutine EXPLIC obtains the final
concentration values using an explicit scheme. The coefficients used
in this matrix solution are provided by CFEXP.
If desirable, it is possible to read in velocity values from a tape
by calling VCALC. This subroutine also performs some necessary aver-
aging calculations. Finally, subroutine OUTPUT prints out a table of
pollutant concentrations at each grid node, at any given time step.
231
-------
APPENDIX D
DETAILS ON INTERFACING METHODOLOGY
A generalized schematic representation of the overlay procedure used
in this study is provided in Figure 82 which shows the basic branching
system containing the main program and various model subroutines. Each
of these model.ysubroutines corresponds to a major partition in the
hydrologic system and, in turn, contains other subroutines of lesser
hierarchy.
In the present formulation, any given subroutine can only be called
once during a simulation run. For instance, subroutines SUB1A and SUB2A
might in effect be identical, but their names cannot be the same. The
capability of re-using a subroutine name at different times in the
simulation would have required additional job control language (JCL)
statements and input/output instructions. These complexities were
thought to be unwarranted from a demonstration standpoint. Also, the
current relative inefficiency of the computations can be reduced by
the use of object decks, preferably within a program library.
Example JCL instructions for the overlay procedure applicable on the
IBM 370/165 at the University of Florida are given in Figure 83. The
overlay requires that the program setup be carried out through the
linkage editor, instead of the less sophisticated loader. Source or
object programs may be used, but their locations in the deck differ.
The different data sets are defined toward the end of the deck. In
the University of Florida system, data sets 5, 6 and 7 are reserved
for reading, writing and punching cards, respectively.
Lists of interfacing variables between the different types of model
subroutines are provided in Figure 84. The pertinent information for
the Stormwater Management Runoff and Receiving Models is available
elsewhere (Environmental Protection Agency, 1971).
A listing of the main program used for an overlay run may be found
toward the beginning of Appendix G. Defined therein are the names of
model subroutines and data sets used for the various subsystems of the
Lake Apopka basin. Also included is the calling sequence for the
model subroutines.
232
-------
MAIN
PROGRAM
to
SUB I
SU8IA
SUB IB
SUB 1C
SUB 2^-
SUB2A
SUB2B
SUB2C
MODEL SUBROUTINE
SECONDARY SUBROUTINE
SUB 13
SUB I3A
SUB I3B
SUB I3C
COMPUTER TIME
Figure 82. Generalized hierarchial representation of overlay procedure.
-------
ro
04
// EXEC F4GCLM , PARM.LKED = 'LIST, MAP, VLY'
// F$RT. SYSIN DD *•
(Insert Source Programs)
/*
// LKED. SYSIN DD *
(Insert Object Programs) Arbitrary Name
OVERLAY ALP
INSERT SUB I, SUB IA , SUB IB , SUB 1C , ...
OVERLAY ALPHA
INSERT SUB 2 , SUB 2A , SUB 2B , SUB 2C,
OVERLAY ALPHA
INSERT SUB I3,SUBI3A,SUBI3B,SUBI3C,
/ -K- -s-Data Set Number
/*"•*"
GO. FT01F001 DD UNIT = SYSDA, SPACE = (CYL ,{ 2,1))
GO.FT02F001 DD UNIT = SYSDA, SPACE = (CYL , (2 ,1))
//GO. SYSIN DD*
Figure 83. Job control language (JCL) instructions for
overlay procedure valid on IBM 370/65 computer
at the University of Florida,
-------
DATA SET
From Model
Unsaturoted
Zone
Time Step
No. of Time Steps
Rainfall Rates
Infiltration Rates
Ponding Depths
To Modeh
Kinematic Wave
Overland Flow
Kinematic Wave*
Overland Flow
Time Step
No. of Time Steps
Flows
Nutrient Fluxes
Receiving
(Lake Apopko)
Stormwoter Management
Runoff
Time Step
No, of Time Steps
FJows in Mole Drains
Nutrient Fluxes through
Mole Drains
Receiving
(Muck farms canals)
Unsaturated
Zone
Water Table Elevations
(heads)
Nitrate Concentrations
Groundwoter
Figure 84, List of major interfacing variables for the
various models.
235
-------
APPENDIX E
DETAILS ON APPLICATION TO THE
LAKE APOPKA BASIN
INTRODUCTION
This appendix contains brief discussions on hydrologic and nutrient
budgets conducted within the Lake Apopka basin. These are followed
by explanations of various agricultural calculations. Finally, several
lists of data used in this study are provided.
HYDROLOGIC AND NUTRIENT BUDGETS
Budgeting efforts have been attempted for two portions of the study
area: (1) Lake Apopka and (2) the muck farms. Calculations for the
former were carried out as a supplement to the more detailed modeling
computations which constituted the main thrust of this study. The
latter budget was conducted by Heaney, Perez and Fox (1972) for the
East Central Florida Regional Planning Council.
Lake Apopka Budget
Numerous sources and sinks of water and associated nutrients were
considered in the analysis. For some of the budget items, such as
Gourd Neck Spring and the Apopka-Beauclair Canal, the available data
were adequate. Most of the data deficiencies involved items related
to the muck farms, namely: (1) pumpage rates, (2) flows through the
flap values, and (3) seepage through the earthen dike.
The approach taken in the hydrologic budget was to lump the three items
above into an unaccounted-for inflow term. Monthly estimates of other
sources and sinks, along with lake stage data, were used to solve for
the unaccounted-for inflow. The results obtained for years 1961 through
1970 indicated the existence of significant average annual flows from
the muck farms to the lake. However, in some months the computed
flows occurred from the lake to the farms.
Measured nitrogen and phosphorous concentrations at the various sources
and sinks were utilized in the estimation of a nutrient budget for the
236
-------
lake. Due to lack of data, nutrient cycling between the lake sediments
and the water was not considered in the calculations. Also, the Hydro-
logic indeterminacy involving the three items at the lake-muck farm
interface posed severe difficulties in computing a net nutrient flux.
At best, only limiting values could be estimated. In any case, the
various budget results indicated a significant overall net influx of
nutrients into Lake Apopka.
Space limitations preclude the presentation of the specific assumptions
and results of both the hydrologic and nutrient budgets. However, this
information has been tabulated and filed for possible subsequent use.
An examination of the hydrologic budget developed by Anderson (1971)
should also be useful.
Muck Farms Budget
The hydrologic indeterminacy mentioned above, among other data defi-
ciencies, precluded an accurate estimate of both the hydrologic and
nutrient budgets within the muck farms. Since the overall results
may be found in a report by Heaney, Perez and Fox (1972), it is more
pertinent to discuss certain items which provided insight into modeling
approaches.
One of these items consisted of the nutrient contribution by rainfall.
It was determined that this source is significantly smaller than ferti-
lizer application. Therefore, rainfall was not considered in defining
the upper boundary conditions in the quality portion of the unsaturated
zone model. This omission was also made in the sandy soils area.
Computed yearly average fertilizer application rates were used to
estimate nutrient concentrations at the uppermost node of the soil
profile. Estimated harvest weights and nutrient contents of the var-
ious vegetable crops in the muck farms were utilized to determine nu-
trient withdrawal (sink) rates from the soil.
Finally, consideration was given to the release of nutrients by the
muck soil. If the soil losses in the muck farms were comparable to
those in the Florida Everglades of 1 in/yr (2.5 cm/yr) (Stephens,
1969), the corresponding nutrient losses could be very significant.
CALCULATIONS
Agricultural calculations conducted in the Lake Apopka basin involved
both types of overland flow models and the unsaturated zone formulation.
237
-------
Overland Flow
The usage of the duBoys formula (Einstein, 1964) to compute sediment
loads at the catchment outlets was illustrated in Appendix A. It is
also of interest to consider the routing of phosphorous and nitrogen
in overland flow.
Certain soils (e.g., muck) contain large amounts of nutrients, pre-
dominantly in organic form. Thus, these nutrients are transported
as part of suspended solids loads. Due to the short duration of over-
land flow phenomena, it was considered unlikely that significant amounts
of nutrients would be released by the soil particles. However, it is
expected that significant releases could occur in the receiving bodies
of water.
Since organic N and P fluxes can be estimated from soil chemical data
and predicted suspended soilds loads, separate attention was given to
the inorganic forms of these nutrients.
Phosphorous Routing—
The only inorganic phosphorous source considered herein was fertilizer
application. Data on fertilizer loading rates for the study area were
coupled with generalizations on phosphorous behavior in soils (Stanford,
England and Taylor, 1970).
Agricultural phosphorous is fixed rapidly to most soils, and thus tra-
vels with suspended particles in overland flow. For this reason, the
weight fraction of this substance in the soil was sought.
The calculations may be illustrated by referring to Figure 85. Part
(a) of this figure constitutes a top view of an arbitrary square plot
within the sandy soils region. The area of this plot is one acre.
Part (b) is a lateral cross-section of the plot. Illustrated is the
estimated phosphorous loading rate corresponding to fertilizer appli-
cation in citrus groves of 60 Ib/ac/yr (64.4 kg/ha/yr), obtained from
the East Central Florida Regional Planning Council. Since the citrus
groves occupy approximately one-half of the total area of sandy soils,
the areal average application rate was assumed to be 30 Ib P/ac/yr
(32.2 kg/ha/yr).
Several simplifying assumptions were necessary to estimate the distri-
bution of the applied P within the soil profile. One wa-s that 90
percent of the applied amount remained within the top six inches of the
profile. It was also assumed that the distribution was uniform with
depth within this range (see Figure 85).
Following application, phosphorous is slowly removed from the soil by
plants. The remaining amount at any time depends, therefore, on the
238
-------
K>
04
AREA = 1 acre
r
•O
'
"1
CITRUS GROVES
PART A PLAN VIEW
P APPLICATION RATE
3O
M ll?l
G1
P CONCENTRATION
-. PROFILE I
WATER TABLE '
v
PART B LATERAL VIEW
Figure 85. Illustration of fertilizer application on a plot in the
sandy soils.
-------
time elapsed since application and the growth characteristics of the
crop. A gross assumption was made that, on a time-average basis, the
P remaining in the soil was one-half of the applied amount.
The concentration of P within the top six inches (15 cm) can be com-
puted as:
(0.90) [IJ(30 Ib)
°P ~ f i f+\ ( f**}
= 6.18 x lo'" = 9.93 x
About 43 percent of the sandy soil is occupied by voids. Thus, the
concentration in the soil itself becomes
' - 100 r - 1 f)ft x in-3 Ib P _ i 77 x ln-2 Kg P
Cn TTnTrTTp-D ~ 1'U° X J-U •——= = 1.73 x 10 —s—*
°P (100-43) P ft3 sand m3 sand
That is, suspended solids fluxes must be multiplied by c to obtain
the phosphorous fluxes.
The calculations for the muck soils were based on similar assumptions.
The value of Cp obtained was 4.25 x 10"3(lb P/ft3 muck); or metric -
6.92 x 10'2(Kg P/m3 muck).
Nitrogen Routing—
According to Stanford, England and Taylor (1970), nitrogen applied in
fertilizers is generally readily converted to nitrate ion. This sub-
stance leaches rapidly through soils.
In this study, the contribution of agricultural nitrogen to overland
flow was viewed to occur, not as part of an erosion process, but rather
as diffusion upward from the soil water. It was thought that the rate
of transfer might decrease exponentially ^ith time, and such a relation-
ship was incorporated into the model. However, due to lack of data
these transfers were tentatively assumed to be zero.
240
-------
Unsaturated Zone
Calculations for the unsaturated zone involved consideration of diverse
items, ranging from soil curves to source/sink effects within the soil.
Hydraulic Conductivity Curves—
Plots of assumed pressure versus conductivity relationships for sandy
and muck soils are provided in Figure 86 and 87, respectively. Only
the saturated conductivities for these two soils could be estimated
with reasonable accuracy (Stewart, Powell and Hammond, 1963; Harrison
and Weaver, 1958). No data were.available for unsaturated conditions.
In the muck, this deficiency is less likely to be severe because the
soil is maintained close to field capacity. The curve for sandy soil
was set by visual comparison to curves reported by Freeze (1969).
Mole Drains—
The expression used to compute the flows through the mole drains in
the muck farms was the following (Harrison, 1960):
4ifJ2d* + hl
Q . 15iL| L (96)
where Q = discharge, m3/hr per unit length in m
K = hydraulic conductivity, m/hr
h = vertical distance between water table and drain, m
do - thickness of an "equivalent layer", m
e
S = spacing, m
The discharge on a per unit area basis, q, can be obtained by dividing
the above expression by the spacing, as illustrated in Figure 88,
part (a):
.9.
S C97)
Accordingly, the units of q are m3/hr/m2, or m/hr. For consistency
with equation (50) in Appendix B, the units of q are converted to cm/mini
241
-------
10
-ISO
E
u
-100
tu
o:
en
-------
K>
-30O
o
^200
tr,
^
CO
w
LU
£-100
0.01
MUCK SOILS
S»vsK
0.1 0.2 0.3 03275
HYDRAULIC CONDUCTIVITY, K, cm/min
0.4
Figure 87,
Assumed pressure versus hydraulic conductivity
relationship for muck soils.
-------
NJ
•MOLE DRAINS-
PART A PLAN VIEW OF UNIT ELEMENT
rj[
I ',!
X X X
CAN At. A
II
I
t_JUNCTION |
AN
ll
II
I
1,1
1
!!
OUTLINE OF CATCHMENT
PART B TOP VIEW OF THE MOLE DRAINS
AND CANAL SYSTEMS
Figure 88. Illustration of layout of mole drains.
-------
100 ^
6o2hT
Finally, the discharge must be expressed on a per unit depth basis:
in which Az represents the vertical spacing (cm) . The variable q .
(1/min) represents the sink term for mole drains. d
To compute the contribution of the mole drains to a given junction in
the receiving model, it is necessary to determine the total length
of drains feeding into such a junction. An approximate relation for
this length, L, is:
(99)
in which A represents the area of the runoff catchment and S denotes
the spacing. The total flow then becomes
QL (ioo)
where Q is the flow per unit length computed from equation (96) •
Appropriate unit conversions must be carried out in equation (100).
The nutrient fluxes into the receiving junctions were computed using
the flows from equation (100) and concentration data collected by
Hortenstine and Forbes (1971) .
Top Boundary Condition—
The concentration value at the top node, resulting from fertilizer
loads, depends upon the time elapsed since application, uptake by plants
and other conditions in the soil. For simplicity, the computation of
this value is illustrated for the situation immediately following
application.
245
-------
The loading rate for phosphorous in the sandy soils was estimated at
30 Ib/ac/yr (32.2 Kg/ha/yr) . It is assumed that following application,
the entire amount becomes uniformly distributed with depth, down to
a distance Az/2 from the ground surface. The vertical spacing, pre-
viously selected, was 17 cm. For convenience, a saturation condition
may be assumed to exist at the top node. Accordingly, the volumetric
moisture fraction is 0.43.
The concentration in the soil water, c, can be computed as follows:
30 Ib x 454^ x I03m&
_ weight m _ IP _ g _
volume - 0>43 x 1 ac x 43<560 £_ x 8>5 cm x 1 J£ x 2g>32 i
elC • OU Cm XL
= 9.2^
Analogous calculations may be conducted for nitrogen.
Plant Withdrawal s-
Quantity— Data on evapo-transpiration rates for a given area provide
the starting point for calculations involving water uptake by plants.
For instance, for the citrus groves in the Lake Apopka basin it was
estimated that four- fifths of the total water loss from the soil con-
stituted transpiration by the crops. Data on total evapo-transpiration
rates for citrus groves in Florida have been compiled by Hashemi and
Gerber (1967).
It is known that water uptake by crops is not uniform throughout the
root zone but, rather, decreases with depth. Several distributions of
this uptake were considered in preliminary calculations. However, it
was found that overall model results for short simulation periods were
insensitive to plant withdrawals, which are significant only on a long-
term basis. For this reason, it was decided to assume a uniform with-
drawal pattern and thus simplify programming instructions.
To illustrate the computation of the sink terms, it may be assumed that
the total uptake by citrus is 0.10 in/ day (0.25 cm/ day) . Assuming also
a root zone depth of five feet (1.5 m) and the- designation of 10 nodes
within that zone (see Figure 89) the withdrawal rate at a given node is '
W = 0'?° in/day = 10-2 in/day = 1.76 x 10-5 cm/min
ID nodes
246
-------
K)
J-UPTAKE RATE = 0.10in /day
Figure 89. Illustration of water uptake by citrus.
-------
This value must be converted to a per unit depth basis:
1.76 x io-5 -SL
w = ~ = - SH> - = 1.03 x io-6 4-
Az e t+ v in A cm v 1 mm
5 ft x 30.4 x
The units of w are consistent with those specified for a sink in equa-
tion (50).
In the computer program, the soil tension at each node is checked
prior to computing the sink term, if the tension exceeds that for the
wilting point, a withdrawal of zero is specified. It is perhaps likely
that, in practice, a no-withdrawal condition might result from even
smaller tensions.
Quality— The nutrient withdrawal pattern within the soil profile should
follow that of water uptake, since nutrients are dissolved in water.
However, for simplicity, nutrient uptake was assumed to be uniform with
depth. Also, seasonal effects and crop growth patterns were not
considered.
In the muck farms, the nutrient uptake rates were estimated from the N
and P contents of the crops and their harvest weights. Yearly with-
drawals were 96.0 Ib/ac (103 Kg/ha) for N and 8.9 Ib/ac (9.6 Kg/ha)
for P (Heaney, Perez and Fox, 1972) .
Calculation of the nitrogen sink term may be illustrated by converting
the withdrawal rate into equivalent units of nitrate
96.0 Ib N = 96.0 x = 427 Ib NOJ * 194 Kg NOJ
Assuming a root depth of one foot (0.3 m),and a node spacing of 6.7 cm,
the uptake for a given node becomes, on a per month basis:
u= 427 -x x - - - -,or
'ac-yr 12 moj 1 ft x 30.5 cm >
U-7.8
• ac-mo ha-mo
248
-------
For consistency within equation (73), the desired units for this
withdrawal rate are (rag/ 1/min):
u =
/ \
7.8 ^
1 ra°J
mg
1 *\ 99 f* JllJtll
10.35 -rr-
ID
mo .
(1 ac)(6.7 cm) |4.05 x 10" —^—16
^ ac-craj
2.97 x IQ-" mg/1
9 min
in which 6 is the volumetric moisture fraction. Assuming 6 = 0.75,
the uptake becomes
u = 3.97 x 10-* mg NOT/1/min
The calculations for phosphorous follow the same steps.
Little information was available on nutrient uptake by the citrus in
the Lake Apopka basin. One approach used to estimate this uptake was
to examine the ratios of plant nutrient withdrawals to fertilizer load-
ing rates in the muck farms and attempt a correlation. Data on this
aspect of citrus would be desirable.
Soil Source/Sink Effects-
Since these soils are unlikely to release nutrients, the soil medium
should act as a net sink for both nitrate and phosphorous, especially
the latter. However, due to lack of data, these sink terms were tenta-
tively set at zero.
On the other hand, muck soils are known to release nutrients. Little
quantitative information is available on the release mechanism, but for
illustrative purposes it may be tied to the consolidation of these soils
(Stephens, 1969).
The nutrient release resulting from a hypothetical one inch (2.54 cm)
yearly soil loss can be estimated from data on nutrient contents of
muck (Forbes, 1971). Since these data are expressed on a percent dry
weight basis, it is necessary to estimate the weight of one inch of
soil over an area of, say, one acre (0.4 ha): Assuming that the dry
muck unit weight is 100 lb/ft3 (1606 Kg/m3) the weight can be computed
as:
249
-------
W = 1 inx-L^lx 4,356 x 10" -£- x 1 ac x 100 -£r = 3.63 x lo^b
L£ 1T1 3-C
- 1.65 x 10s Kg
The nitrogen content of muck is 2.79 percent. Thus, the amount of this
substance released is:
r = x 3.63 x 105 = 10.1 x 103 = 10.9 x 103
100 ac ha
Assuming that this loss occurs over a period of one year and that all
the nitrogen is converted to the nitrate form
r - 10.1 x 10' x = 4.48 x 10" „ 4.82 x 10*
ac-yr "tw* *w ha-yr
The units for this release rate are the same as the original units used
for the harvest removal estimates in the muck farms. Therefore, con-
version into the desired sink units of mg/i/min is carried out in the
same manner as in the plant nutrient withdrawal calculations. Hope-
fully, a better understanding of release mechanisms should suggest
improvements in the corresponding calculations.
LIST OF DATA
The scope of this investigation precluded a separate sampling program.
Nevertheless, substantial amounts of available data collected by various
agencies were assembled. These data are summarized in Tables 1 through
4. Indicated are the sampling frequency, agency and other pertinent
information.
Most of these data have been punched on computer cards. The original
tabulations, which are mostly in computer coding sheets, are also on
file.
250
-------
TABLE 1
List of Hydrologlc Data for Lake Apopka Basin
fO
in
Item
Pumping rates
from muck, farms
to Lake Apopka
Rainfall
Stage-area data,
before and after
muck farms
Pan evaporation
Pan evaporation
Lake stage data
Rainfall data
Apopka— Beauclalr
Canal flows
Piezoicetrlc
levels
Location
Zellwood and
Duda muck farms
Clermont and
Lisbon stations
near Lake
Apopka
Lake Apopka
Lisbon station
near Lake Apopka
Lisbon station
Lake Apopka
Stations In
muck farms
North shore of
Lake Apopka
Orange County
and Lake County
Source
East Central
Florida Regional
Planning Council
(ECFRPC)
ESSA (Heather
Bureau)
uses
ESSA (Heather
Bureau)
ESSA (Weather
Bureau)
uses
Muck farms
uses
DSGS
Dates
1967-1970
1936-1970
1961-1970
1967-1970
1936-1970
1961-1970
1967-1970
1968-1971
Sampling Frequency
Dally, with gaps
Monthly and daily
. Monthly
Daily
Daily
Daily, with gaps
Daily
Twice per year
Comments
Hot broken down
by individual
pumps
Monthly 1936-1970
Daily 1967-1970
Areas available on
0.1 ft. stage
intervals
-------
TABLE 1—Continued
to
tn
K>
Zten
Coordinates of
veils
Rainfall data
Apopka-Beauclair
Canal flows
Location
Orange County
and Lake County
Apopka-Bcaucla.lt
Canal
North shore of
Lake Apopka
Source
uses
ECFRPC
uses
Dates
1936-1967
Sampling Frequency
Monthly
Monthly, with
numerous gaps
Consents
Longitude and
latitude
-------
TABLE 2
U*t of Agrlcnltural Data for Lake Apopka Basin
Ite»
Land use data
Acres planted
with a given crop
Tertlllzer types
for various crops
Portions of faros
lying in each
runoff subcatchsent
Amounts of ferti-
lizer applied to
•various crops
Areas of runoff
sub catchment
Farm areas
Location
Lake Apopka
has la
Muck ferns
Kuck faros
Muck, farms
Mock f aras
Mock farms
Mock farms
Source
Soil Conservation
Service (SCS)
East Central
Florida Regional
Planning Council
ECFKPC
ECFBPC
ECFEPC
ECFKPC
ECFSFC
Sates
Recent
1970-1971
1970-1971
1970-1971
1970-1971
1970-1971
1970-1971
Sampling Frequency
Weekly
COBEBditS
-------
TABLE 3
Water Quality Data for the Lake Apopks Basin
(S3
tn
Item
Water quality
parameters,
Including N
and P
Water quality
parameters, as
above
Water quality
parameters, as
above
Water quality
parameters, as
above
Typical values of
various water
quality parameters
Various water
quality parameters
Ions and solids ,
Sediment core
samples including
Location
21 stations in
Lake Apopka
Shallow wells
around Lake
Apopka
Stations in
muck farms
canals
3 stations in
Winter Garden
Canal
10 stations in
Lake Apopka
21 sampling
stations in
Lake Apopka
21 bottom
sampling
stations in
Lake Apopka
Stations as
shown in
Source
Orange County
Pollution
Control Dept.
(OCPCD)
OCPCD
OCPCD
OCPCD
Florida State
Board of Health
(FSEH)
FSBH
OCPCD
EPA
(Schneider
Dates
1967-1970
1970
1968-1970
1969-1970
October 1962
and January
1963
First half
of 1964
Summer 1967
Report dated
1968
Sampling Frequency
Once every month
or 2 months
Monthly
Monthly
Every 2 weeks
Every 2 weeks
Once
Approximately 3
tines each station
Comments
-------
TABU 3—Continued
Is)
VI
cn
Item
nutrients and
solids
Nutrient
concentrations
Effect of lake
bottom on fish
feed organisms
Biological
organisms
(mainly bottom)
Phosphorous data
Chemical and
biological
parameters
Nutrient
concentrations
Location
report
Rainwater,
shallow wells
and artesian
veils in Lake
Apopka basin
Lake Apopka
Lake Apopka
Muck farms
and Lake
Apopka
Lake Johns
(south of
Lake Apopka)
Percolating water
in muck faros
Source
and Little)
Appendix B
EPA
(Schneider
and Little)
Appendix C
FSBH report
dated 1964
OCPCD
Or. Forbes
(IFAS, Univ.
of Florida)
OCPCD
Drs. Forbes and
Hortenstlne
(IFAS, Univ. of
Florida)
Dates
Report dated
1968
1947-1963
1967-1970
1967-1970
1966-1969
1971
Sampling Frequency
Once
Every 2 years
Every 3 months
Monthly
Every 3 months
Seasonal
Comments
Qualitative
comments for
each station
Being prepared
for publication
-------
TABLE 4
Miscellaneous Data Used in Computer Programs for Lake Apopka Basin
ro
tn
Item
Permeability data
Soil charac-
teristics :
BOisture^tension-
conductivlty
Geometry of
subcatchments and
canals . Station
locations. Link-
age data.
Sub catchment
geometry and
linkage data
Fertilizer data
and ebil charac-
teristics
Location
Vertical cross-
sections in
Lake Apopka
basin
Muck soils and
sandy soils in
Lake Apopka
basin
Zelluood and
Duda muck
farms . Lake
Apopka.
Sandy soils
Muck and
sandy soils
Source
DSGS geologic
data
Various Soil
Conservation
Service data
Assorted naps
USGS quadrangle
maps
East Central
Florida Regional
Planning Council
Dates
Recent
Recent
Recent
Recent
Sampling Frequency
Comments
Used is ground-
water model
Used in unsatu-
rated zone
program
Used in runoff
and receiving
models
Used in runoff
model for undu-
lating areas
(sandy soils)
Used in quality
portion of
unsaturated
zone program
-------
APPENDIX F
SUMMARY OF ADDITIONAL REFERENCES
ON AGRICULTURAL NUTRIENTS
Although the state of the art in hydrologic modeling is at a relatively
advanced stage, much remains to be accomplished in the modeling of
agricultural water quality. Hopefully, the references mentioned herein
should be of some value in formulating needed improvements.
One source of nutrients not considered in this study was animal wastes,
due to their insignificance in the Lake Apbpka basin. However, this
source can often be important, as pointed out by Townshend, Reichert
and Nodwell (1969). These researchers provided figures on nutrient
contributions by different animals, expressed in terms of human popu-
lation equivalents. They also discussed trends in animal populations
in Ontario, Canada. Webber and Lane (1969) studied the different
nitrogen compounds contained in manure and the ensuing reactions in
the soil.
Water quality characteristics of runoff from cattle feedlots were
investigated by Norton and Hansen (1969). Their study comprised both
a mathematical development of overland flow and a quality sampling pro-
gram. Their results should provide useful insights into water quality
modeling approaches.
Other investigations on overland flow quality include those by Gilchrist
and Gillingham (1970) and by Nelson and Romkens (1970). These two
papers deal primarily with field studies on phosphorous movement re-
sulting from different fertilizer application rates, and could be used
as guides for sampling programs in a given area. Data on the concen-
tration of different ions in overland flow have been collected by Gburek
and Heald (1970). Bacterial levels in runoff have been reported by
Kunkle (1970).
The calculation of an inventory of nutrient sources and sinks in a
given area can be facilitated by the use of the tabular approach pro-
posed by Schultz (1970). This was the basic approach used for the
257
-------
budgeting calculations in the muck farms of the Lake Apopka basin.
Insight into the economics of fertilizer application is provided in
a paper by Fox and Allee (1970).
258
-------
APPENDIX G
LISTING OF COMPUTER PROGRAMS
Included herein are listings for the following models: (1) unsaturated
zone, (2) kinematic wave overland flow and C3) groundwater. Listings
for the stormwatermanagement runoff and receiving programs are avail-
able elsewhere (Environmental Protection Agency. 1971).
A listing of the executive program, used in connection with the overlay
technique, is provided first. Usage of this program is optional.
259
-------
C
C
C-
C
PAIN PROGRAM FOR INTERFACING
COKVCN BLOCKS ON?,. UNSAT1 AND NUM1 RE-NAMED ID TO SAVE CORE SPACE
CQMGN/DSETS/NGWIF,NUUIF1,NUNOV1,NUNRE1,NUNGWUNUNRE2»NUNIF2«
NU!40V2tNUN3V3iNUNOV
-------
13
I*
15
16
17
18
19
20
21
NUNOV4
NUMF3
NUISGW2
NUNGW3
NOVRE1
NOVRE2
NRERE1
NRERE2
NOVRE3
UNZON3
UNZON3
UNZON2
UNZON3
OVLAN3
OVLArK
REC1
REC2
OVLAN1
OVLAN2
UN Z ON 3
GWBLOK
GWBLOK
REC1
REC2
REC3
REC3
REC3
c
c
c
c
c
c
c
c
c
c
END
SUBROUTINE UNZONl
COKMON/DSETS/NGWIP,NUNIFl,.NUNOVl.,NUNREl,NljNGWl,NUNRE;2,NUNIF2I,
1 MUNCV2,NUNOV3,NUNIOV4,NUNIF3,NUNGW2,NUNGW3,NOVREl,NOVRE2tNREREl,
2 NRERE2,NOVR£3
RETURN
ENC
SUBROUTINE UNZON2
CONMCN/DSETS/NGWIF,NUNIFl,.NLNOVl,NUNREl,NUNGWl,NUiMRE2,NUNIF2,
1 NUNOV2,NUNOV3tNUNOV4,NUNIF3,NUNGW2,NUNGW3,NOVREl»NOVRE2»NREftEl,
2 NRERE2.NOVRE3
RETURN
END
SUBROUTINE UNZON3
COMMON/OS ETS/NGW IF, NUNIFl,.NUNOVl,NUNREl,NUNGWl,NUNRE2tNUNlF2i
I NUNOV2tNUNOV3,NUNOV*»,NUNIF3,NUNGW2,NUNGW3|NOVREl,NOVRE2tNREREl,'
2 NRERE2.NOVRE3
CALL UNFLO
CALL SCILQU
CALL IRAN
RETURN
ENO
SUBROUTINE UNFLO
C UNSATURATED FLOW (INFILTRATION) PROGRAM tNFL i
C CNE-CIMSNSIONAL (VERTICAL) UNSTEADY STATE INFL 2
C AT EACH TIME STEP, THE PROGRAM USES BOUNDARY CONDITIONS AND INFL 3
C INFORMATION FROM ThF PREVIOUS TIME STEPS TO COMPUTE THE TENSION INFL *
C HEAD, (PSD, AT EACH NODE IN THE VERTICAL PROFILE. INFL 5
C THEREAFTER, THE PROGRAM COMPUTES AND STORES MOISTURE FLUXES INFL 6
C BETWEEN NOCES^ HYSTERESIS EFFECTS ARE NOT CONSIDERED INFL 7
C .
C LIST OF VARIABLES
C NNODES'NO OF NOCES IN PROFILE
C NSTEPS»NO OF TIME STEPS
C TSTART-STARTING TIME OF SIMULATION, IN MIN.
C TENO-.ENDING TIME OF SIMULATION! IN MIN
C R.AIN-RAINFALL RATE
C EVAP-EVAPORATION RATE
C GWFLUW«NET RECHARGE AT BOTTOM BOUNDARY
C DEL/.-VERTICAL SPACING, IN CM.
C CELT-TIME STEP, IN MIN.
C THENOT-INITIAL MOISTURE CONTENT
C ZDRAIN-ELEVATION OF ARTIFICIAL DRAIN
C ZBOTOM-ELEVATION OF BOTTOM NODE
C PSJNOT-INITIAL PRESSURE HEAD VALUE
C PSI»PRESSURE HEAD
C CSLOPE-SLOPE OF MOISTURE VS TENSION CURVE
C COND*HYDRAULIC CONDUCTIVITY
C NOCSAT-NO OF UPPERMOST NODE THAT IS SATURATED
C A- A-COEFFICIENT IN TRIUIAGONAL MATRIX USED TO FIND PSI VALUES
C B- B-COEFF1CIENT IN TRICIAGONAL MATRIX USED TO FIND PSI VALUES
C C« C-COEFFICIENT IN TRIDIAGONAL MATRIX USED TO FIND PSI VALUES
C C« C-COEFFICIENT FOR RIGHT HAND SIDE OF MATRIX SCHEME USED TO FIND
C PSI VALUES
C PI,P2.P3,-SYMBOLS FOR PSI AT MIDPOINT LOCATIONS (SEE FR£EZE,1969)
C C1,C2«CONDUCTIVITIES CORRESPONDING TO P1,P2
C R-EXCESS RAINFALL (RAINFALL MINUS EVAPORATION)
C CELPSI-DIFFERENCE IN PSI VALUES
261
-------
C ^AdV»ZQeL=eL6VAT!ON AT UPPER AND LOWER NODES, RESPECTIVELY
C PHIABV,PHIB2L=PRESSURE HEADS AT UPPER AND LOWER NODES,RESP£CTIVELY
C GRAC12=PRFSSURE HEAD GRADIENT BETWEEN NODES 1 AND 2
C ZTA3LE=ELEVATION OF WATER TABLE
C NNSAT*INCEX COUNTING MW3ER FOR NUMBER OF SATURATED NODES*
C STARTING AT BUTTOM
C HPCND=PONDING DEPTH AT GROUND SURFACE
C FSIGND=PSESSURE HcAD VALUE AT TOP NODE
C SATCCN=SATURATEC CONDUCTIVITY
C CPLANT=FLOW UPTAKE BY PLANTS AT A NODE
C CORAIN=FLOW WITHDRAWAL BY DRAIN AT A NOOe
c CSIC;K=TOTAL SINMWITHDRAWAD AT A NODE
C XL=ARRAY FOR CUEFFICIENTS A,B AND C (SEE ABOVE)
C RHS=RIGHT-HAND SIDE VECTOR FOR 0-COEFFICIENTS (SEE ABOVE)
C TEKS=TENSION PRESSURE
C CK=CONDUCTIVITY
C CSL-SLOPE OF MOISTURE VS PRESSURE HEAD CURVE
C HUMID = MOISTURE CONTENT
C £TCRCP=WATER WITHDRAWAL(E-T) BY CROPS IN STUDY AREA
C WILTK=WILTING POINT MOISTURE OF A GIVEN SOIL
C RINF=INFILTRATIQN RATE (DOWNWARD VELOCITY BETWEEN TWO TOP
C NODES) IN CENTIMSTERS/M1N
C ORGOT-DEPTH OF ROOT £QN6 (FT)
C GWCLC=FLQW OUT CF A MOLE DRAIN (CFS/FT)
C JTA3=NOU£ AT WHICH WATER TABLE IS LOCATED
C HTAS=HEIGHT OF WATER TABLE ABOVE BOTTOM OF UNSATURATED PROFILEt
C IN F6ST,
COMMON/ 10 /PSINOT(50>,RAIN(20),EVAP(20) ,GV>FLOW(20) , ZTABLE120) i
1 PSI(5G,20)»M50,20)«B(50».2G),C(50»20) ,0150,20) ,E ( 50..20) ,F (50V 20)
2 CSLOPE(50,20), TH£TA( 50, 20) , THENOT1 50) , VELZ I 50, 20) ,PSI1 (50,20) ,.
3 PSI2(50*20>tPSI3(5C»20>f. NOOSAT(20),
4 CUNC(SO»20),PSIBOT(50)t"HPONO(501,QSINK(50i20)tQNET12(50>»
COMMCN/LAB/ TITLE ( 20 ) , XLABI 11 ) , YLAB ( 6) (HORI1(20) , VERT (6)
DIMENSION G1650)
DIMENSION TIT(20)
COM^CN/CROP/ETCROPtWILTM
COMMON/ROOT/ DROCT
COMKCN/MOLE/CMOLC150)
tINFL 9
INFL 10
IMFL 11
INFL 12
INFL 13
INFL I*
INFL 15
INFL 16
50)
COKMCN/PONO/DPO-NDf 50)
CQMMCN/T IKE/NT I fE,DTIME
COMKON/SOIL/KSOIL
COMMCN/LCCMOL/LfOLE
COM^ON/WTAB/ JTAB( 50 ) i HTAB( 50)
COMKCN/DSETS/NGWIF,NUNlFl,.NUNaVl,NUNREl,NUNGWlfNUNRe2,NlJNIF2.
1 NUrtOV2,NUNOV3,NUNOV4fNUNIF3fNUNGW2(NUNGW3,NOVREl»MOVR62iNRER
101
100
120
150
141
142
2 NRERE2.NOVRE3
READ (5tlOO) NNODES.NSTEPS
WRITE (.6,101) NNODES
FORMAT (' NO OF NQOes=«,I5)
FORMAT (213)
NTIME'NSTEPS
READ (5,120) TSTART.TEND
FORMAT (2F8.2)
RAINFALL IN INCHES/HOUR
READ 15,140) IRAIN(I),I»i,NSTEPS)
FORMAT (15F5.2)
EVAPORATION IN INCHES/HOUR
READ (5,150) (EVAP(I),I»1,NSTEPS)
FORMAT (15F5.2)
WRITE (6,141)
FORMAT C1',20X,'RAINFALL AND
WRITE (6,142)
lYT STEP''5X''RAIMFALL'*7X''EVAPORATION')
EVAPORATION IN INCHES/HOUR')
INFL 17
INFL 18
INFL 19
INFL 20
INFL 21
INFL 22
INFL 2»
INFL 26
INFL 27
INFL 28
INFL 29
INFL 30
INFL 31
INFL 32
*NFL H
INFL 3*
INFL 35
262
-------
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
144
143
160
162
164
163
195
161
190
130
170
180
205
210
7500
7600
200
300
WRITE (6,144) I.RAIMI I),EVAP( I)
FORMAT ( 110,5X,F10.3,7X,F10.3)
CONTINUE
GRCUNCWATER FLOW IN FT/DAY
REAC 15,1601 (GwFLOwi i), I»I,NSTEPS>
FORMAT (15F5.2)
WRITS (6,162)
FORMAT (20X, 'GRCUNDWATER FLOW IN FT/OAY')
DO 163 I*1,NSTEPS
WRITE (6,164) I.GWFLOU(I)
FORMAT ( • TIME STEP • , I 5,.'RECHARGE • ,F5. 2)
CONTINUE
REAC (5,195) OELZ.DELT
FORMAT 12F6.2)
WRITE (6,161) DELi.DELT
FORMAT (' SPACING IN CM ISSF8.3 ,5X,.'TIME STEP IN MIN IS',F3.3>
REAC PRESSURE HEADS RATHER THAN MOISTURE CONTENTS
READ (5,190) (THENOTU),J'1,NNODES)
FORMAT (10F6.2)
DTIME-DELT
REAC (5,130) (PSINOT(J),J«1,NNODES)
FORMAT (1PF6.2)
REAC (5,170) (ZTABLEl I ) , I»l .NSTEPS)
FORMAT (1CF6.2)
FOR SANDY SOILS IN STUDY AREA VALUES BELOW ARE ARBITRARY
SINCE TERRAIN IS UNDULATING. ALSQi NO DRAINS EXIST
IN THESE PARTICULAR SANDY SOILS.
READ (5,180) ZDRAIN,ZBOTOM,NOD
FORMAT (2F6.2.I5)
LMGLE-NOO
READ (5,205) ( HORIZt I ) , I»l, 20)
READ (5,205) ( VERT(I) ,.I »1,6)
FORMAT (20A4)
REAC (5,210) (TIT (I), 1*1,20)
FORMAT (20A4)
REAC IN WITHCRAWAL RATE FOR CROPS IN SANDY SOILS (HEREiCITRUS)
IN I.NCHES/DAY. FROM HASHEMI AND GERBER (1967).
REAC (5,7500) ETCROP.WILTM, DROOT
FORMAT (F5.3t2F5.2>
KEAi:(5,7600) KSCIL
FORMAT (12)
SOIL CLASSES AS FOLLOWS — l.Srt SAND..2. MUCK, 3.NE SAND
CONVERT RAW DATA ON RAINFALLrEVAPORATION,ANO GROUNOWATER FLOW
INTO PROPER AND CONSISTENT UNITS.
CO 200 I=il,NSTEPS
CONVERT INCHES/HOUR TO CM/MIN, BELOW
RAIN(I) = RAIN( I)»0. 0423333
RRAIM I) = RAIN< I )
EVAP(I) = EVAP( I )»0. 0423333
CONVERT FT/DAY TO CM/MIN,.BELOW
GWFLOW(I) * GWFLOW( I)*0. 02116666
CONTINUE
CONVERT CROP WITHDRAWAL FROM IN/DAY TO CM/MIN
ETCROP=ETCROP»0.00176
CONVERT WILTING POINT MOISTURE FROM PERCENT TO FRACTION
WILTM»WILTM/100-
• FOR THE 1ST TIME STEP,. COMPUTE VALUES OF VARIABLES-
WHICH DEPEND ON PSI
N « 1
NOOSAT(N) = 1
CO 300 J^l.NNODES
PSKN.J) • PSINOT(J)
X » PSINOT(J)
CSLOPE(N.J) = CSL(X)
COND(NiJ) * CK(X)
CONTINUE
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
36
37
38
39
40
41
42
43
44
45
46
47
50
51
52
53
54
55
56
57
58
*Q
3 y
60
61
62
63
64
65
66
67
68
69
70
71
72
73
76
77
78
79
80
263
-------
L = NNODES
CALL VFLUW(N,L,CELZ,ZBOTOM)
LOCATE WATER TABLE
CALL WTABLE(N,L,DELZ»ZBOTOM)
WRITE (6,3000)
WRITE (6,3500) N
CO 2100 J=1,L
WRITE (6,2500) J,PSI(N,J1
2100 CONTINUE
CQ6300J=1»L
X=PSI(N, J)
THETA(N, J) = HUMJCU)
6300 CONTINUE
WRITE OUT MOISTURE VALUES
KRITE(6,5000)
V»RITE(6,5500) N
D06200J=l,L
WRITE(6,6500) J , THETAIN , J >
6200 CONTINUE
PREPARE INFORMATION FOR PLOTTING ROUTINE
INFL
INFL
81
82
NPLOT=L
NSTEP2=L
NCURVE=l
CO 3800 J*1,L
GlKJsj
G(K+1)=TH£TA(N, J)
K=K+2
3800 .CONTINUE
CALL GRAPH 1/2) AND
C SIMILARLY WITH CSLQPt. (RU91N AND STEINHARDT, 1963).
C THUS, FORMULAS FOR PSI ( I ) ,-PSI ( 1 1 ) ,P SI U 1 1 I ARE SPECIAL
C CASES OF THOSE USED FOR N 2.
c ------------------------ „ ------- _ ----- . -----
N = 2
NOCSAT(N), * 1
C FIMC VALUES OF PS K I ) , PS I( I I ) , PSI ( 1 1 IJ
CALL PSIMID(NtL)
C FIND COEFFICIENTS -A, B, C»D
CALL COEF(N,L,,UELZ,iQOTOM^NOD,.OELT)
C FIND VALUES OF PSI
CALL PSIVAL(N,L,06LT,OELZ)
C FIND VERTICAL VELOCITIES
CALL VFLOW(N,L, CElZ , ZBQTQM)
C LOCATE WATER TACLf:
CALL WTABLE(N,L,DELZ,ZOOTOM)
WRITE (6,3000)
N
2200
WRITE (6,3500)
DO 2200 J=1,L
WR'ITE (6,2500)
CONTINUE
J,PSI(N,J)
X=PSI(N,J)
THETA(N,J)=HUMIC(X)
6<>00 CONTINUE
WRITE OUT MOISTURE VALUES
WRITE16.5000)
WRITQ(6,5500) N
D06600J=1»L
WRITE(6,6500> J,THETA(N»JI
6600 CONTINUE
PREPARE INFORMATION FOR PLOTTING ROUTINE
83
84
85
86
87
88
89
90
91
92
93
95
96
97
98
99
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFL
INFU
INFL
INFL
INFL
INFL
INFL100
INFL101
INFL102
INFL103
INFL104
I NFL 1 05
INFL106
INFL107
INFL108
JNFL109
INFLllO
INFLlll
INFL112
INFL113
INFL1L4
INFL115
INFL1L6
'INFL117
INFL118
INFL119
NPLOT=«L
NST£P2»L
INFL120
INFL121
INFL122
INFL123
INFL124
INFH25
INFL126
IMFC127
INFL128
INFL129
INFL130
INFL131
IMFL132
INFL1J3
264
-------
CO 3900 J = l,L
G(K)=J
G(K+l)=THETA
C FluO VERTICAL VELOCITIES
CALL VFLOWt,N,L,DELZ,ZEOTOM)
C LOCATE WATER TABLE NODE FOR NEXT TIME STEP
CALL WTABLE(N,L,CELZ,ZBOTOM)
WRITE OUT PSI VALUES
WRITE (6,3000)
FOR ALL TIME STEPS
3000 FORMAT f • I1 , 20X , 'TABLE OF PRESSURE HEADS IN CM1)
3500
WRITE (6.3500) N
FORMAT (5X, 'TIME STEP=«, 15)
CO 2 COO J*lfL.
WRITE 16,2500) J,.PS1(N,J)
FOR.VAT ( I5.E8.2)
CONTINUE
HUMICU)
VALUES (PER CENT BY VOLUME)..
2500
2000
C
X=PSI(N, J)
THiiTAIN, J)
4500 CONTINUE'
; WRITE OUT MOISTURE
WRITE (6, 5000),
5003 FORMATC I'iZGX, 'TABLE OF MOISTURE VALUES AS PER CENT BY VOLUME')
WRITE (6,5500) N
5500 FORMAT 15X,'T1ME STEP=»,I5)
U06000J=1,L
WRITE (6,6500) J , TH£TA( N,.J )
FORMAT ( I5.F6.1)
CONTINUE
PREPARE INFORMATION FOR PLOTTING ROUTINE
KKK=l
NPLOTaL
NSTEP2»L
6500
6000
DO 3700 J*l,L
G(K)=.J
G(K+1)»THETA(N,J)
3700 CONTINUE
CALL GRAPH ( NPLOT,.KKK,NSTEP2,NCURVE ,G, TI T)
1000 CONTINUE
C-J.
C WRITE DATA FOR QUALITY PORTION
' WRITE lNUNIF3)DELi»OELT,NSTEPSfNNODES,DRQQT
DO 4700 N=1,NSTEPS
WRITE (NUNIF3)(COND(N,.J)..J*l»-NNQDeS)
4700 CONTINUE
CO 4800 N*1,NSTEPS
WRITE (NUNIF3)(VELZ(N,J)rJ"ltNMODES)
INFL134
INFH35
INFL136
INFL137
INFL138
INFLL39
INFL140
INFL141
INFL142
INFLL43
INFL144
INFLU5
INFL146
INFL147
INFL148
INFL149
INFL150
INFL151
INFL152
1NFL153
INFL154
INFL155
INFL156
INFH57
INFL158
INFL159
INFL161
INFH62
INFH63
INFL164
INFH65
TMFI 166
INFH67
INFL168
INFH69
INFL170
INFL171
1NFL172
INFL173
INFL174
INFL175
INFL176
INFL177
INFL178
INFL179
INFU.UO
INFL181
INFL182
INFL183
1NFL184
INFL185
INFL186
INFL187
INFL188
INFL189
INFL190
INFL191
INFL192
265
-------
4800 CONTINUE
CO 4900 N*1,NSTEPS
WRITE (NUNIF3H THETAIN,J),J = 1,NNODES)
4900 CONTINUE
WRITE: (NUNIF3)(QMOLD(N),N=1,NSTEPS)
END FILE NUNIF3
REWIND NUNIF3
RETURN
ENC
SUBROUTINE COEF(NrL»DELZ,.ZBOTOM,NOD,OELT>
C COMPUTES VALUES OF COEFFICIENTS A,B,C,D
COMMON/10 /PSINOT(50),RAIN(20),EVAP(20),GWFLOW(20),ZTA8LE(201 ,
1 PSI<50,20),A(50,20>,B(5C,20).Cl50,.20)fDl50,20)tE(50,.20) »F150,20)
2 CSLOPE(50,20>,THETA(50,20),THENOT(50),VELZ(50,20),PSIH50,20),
3 PSI2(50,.20) ,PSI3(50,20),. NODSATI20),
4 CCNC(50,20),PSIBOT(50),HPONO(50),QSlNK(50,20),QNET12t50)i
5 QPLANT(50,20)
CALL SINK(N,L,NCD,.DPLT,DELZ J
C DEFINE A.B.C.D AT J=l (BOTTOM)
PSI12=(PSI(N,J) +PSHN,J + 11 )/2.
CONC12-CMPSI12)
Zl=ZBOTOM*100./3.29
Z2=/.l +OELZ
A(N,J)*1.
Q(N,J)*1.
C(IN, J)=0.
C CROU.NCWATER FLOW IS POSITIVE DOWNWARD
CINiJ)=Z2-Zl-(-GWFLOW(N)*DELZ/COND12)
1 - (-(QSINK(N,J)»(DELZ/2.)»OELZ)/COND12)
LL = L-l
CO 300 J=.2,LL
PI = PSIKN.J)
P2 = PSI2JN.J)
P3 = PSI3IN.J)
CK1 = CK(P1)
CK2 = CK(P2)
A.(N.J) " CK1/I2»CELZ)
C(N.J) = CK2/(2«DELZ)
Cl = A(N,J)
B2 = ClN^J)
B3 = CSL(P3)»DELZ/DELT
B(N,J) = B1+B2VB3
C VALUES OF QSINK ARE INPUTS TO 0 BELOW
C(N,J) * AIN»J)*PSI(N-lf J*l) * (B3-Bl-B2)»PSUN-l,J)
1 * C(N,J)»PSI(N-1,.J-1) * CK1 - CK2 - (0 SINK (N, J ) *DELZ)
3(?0 CONTINUE
C DEFINE A.B.C.D AT GROUND SURFACE
A(NiL)=0.
B(N,L)=L.
X=PSI2(N,L)
R=RAIN(N)-EVAP(N)
ZLl=ZBOTOM»100./3.28+(L-2)*DELZ
ZL=ZL1+DELZ
C NET RAINFALL IS POSITIVE DOWNWARD
C(N,L) = (R»DELZ/CK(X) )-ZL*:ZLl - (QSINK(N,Jl»CDELZ/2. )»OELZ/CK(X) )
RETURN
ENO
SUBROUTINE EXTRA IN.U
C USING EXTRAPOLATION FORMULAS. PREDICTS VALUES OF
C PSI FOR THIS TIME STEP.
COMMON/ID /PSINOT(50),RAIN(20),EVAP(20].GKFLOWC20)tZTABLE120)t
1 PSI(50,20),A(50i20>,G(50,20),C(50,20),0(50,20),E<50,20)»F150,20)
2 CSLOPE(50,20>f THETAl 50, 20) .THENOTl 50), VELZ( 50,20) .PSI 1 (50*20) ».
3 PSI2(50,20),PSI3(50,20)r NODSAT(20>,
4 CQNC(59,2Q)tPSIBQT(50),HPONO(5Q}fQSINK(50t20)iQNET12(50}»
266
INFL196
INFL197
INFL198
.INFL200
INFL201
INFL202
INFL203
INFU204
INFL206
INFL207
INFL208
INFL209
INFL210
INFL211
INFL212
INFL2L3
INFL214
INFL215
INFL216
INFL217
INFL218
INFL219
INFL220
IKFL221
INFL222
INFL223
INFL224
INFL225
INFL226
INFL227
INFL228
INFL229
INFL230
INFL231
INFL232
INFL233
INFL234
INFL235
INFL236
INFL237
INFL238
INFL239
INFL240
INFL241
INFL242
INFL243
INFL245
INFL246
EXTR 1
EXTR 2
EXTR 3
.EXTR 5
EXTR 6
EXTR 7
EXTR 8
-------
C
C
C
C
VALUES FROM
FUNCTIONS
PREVIOUS
500
C
C
C
C
C
C
500
5 CPLANTJ50.20)
C0500J=1,L
CELPSI = PSilN-l,J) - PSKN-2.J)
PSKN,J> = PSHN-l.J) * DELPSI
INSTEAD OF EXTRAPOLATING AS ABOVE, USE
TIME STEP TO EVALUATE THE VARIOUS SOIL
PSI(N,J)=iPSI(N-l,J>
CONTINUE
RETURN
END
SUBROUTINE VFLOWfN,L,DELZtZBOTOM)
COPKCN/IO /PSINCT(50),RAIN(20),EVAP(20) »GWFLQW«20»,ZTABtEC20)»
1 PSI(50»20J.A(50»20»,B(50;20UC(50,20),t)t50,20) ,E150,20)>F159,20)
2 CSLQPE150,20),THETA£50,20),THENOT(50),VEL2(50,2Q),PSU(50,20),
3 PSI2(50,20 ),PSI3t50,20)f. NQOSAT(20)»
^ CCJNC(50,20) t.PSIBOTI50),HPONO(50),QSINK(50,20) , ONE T12 150) ,
5 CPLANT(50»20)
APPLICATION OF CARCY'S LAW TO FIND FL£W VELOCITIES.
COMPUTE FLOW VELOCITY BETWEEN L-TH AND (L-IJ-TH NODES
THIS FLOW DEFINED AS INFILTRATION
PSITCP= tPS.I(N,LJ+PSI(N.L-l))/2«
X=PSITOP
CONTCP=CK(X)
CCNC(N,LJ=CONTOP
GRTCP=(PSI(N,L)+OELZ-PSI(N,L-1))/OELZ
VELZ(NfU)aCONTOP*GRTOP
RINFtN)=VELZtN,L)
UTILIZATION OF CONDUCTIVITY AT EACH INTERIOR NODE AND TENSION
AT TWO ADJOINING NODES.
COWNWARD VELOCITY DEFINED AS POSITIVE
C0500J=1,L
X=PSItN,J)
COND(N,J)=CK(X)
CONTINUE
NN » L - 1
CO 700 J = 2>NN
700
800
900
950
C
C
ZA6V=,(ZBCTOM/3.23J*100.+(M-ll»OELZ
PHI
ZBEL=. IZBOTUM/3.28)*100.*(MM-1)»DELZ
PHlD£L=ZBEL-*-PSI(N,J-l)
GRAC=(PHIAa'/-PHI3EL )/(2.*DELZ)
VeLZ(N,JI ^ CONCtN, J)»GftAO
CONTINUE
COMPUTE NET FLOW BETWEEN 1ST AND 2ND NODES
J=l
PSU2=(PSI(N,J)*PSHNtJ + in/2.
CGN012=CK(PSI12)
CONO(N,J)»COND12
PHIABV=( ZBOTOM/3.28)«100.-«:DELZ + PSI(N,J«-1)
PHIBEL=(ZBOTOM/3.28)»100.tPSI(N,JJ
GRACl2=CPHIABV-PHI8eLl/OELZ
CNEVl2(fJ)= COND12»GRAD12
VELZ(N,J)*QNET12tN)
WRITE <6,800) N
FORMAT (• l«, IOX ,' DOWNWARD VELOCITIES IN CM/MIN AT TIME STEP",I5)
CO 950 J=5ltL
WRITE 16^900) J,VELZ(N,.J)
FORMAT i I5.E8.2)
CONTINUE
RETURN
END ,
SUBROUTINE WTA8LE1 N,L»DELZ» ZBOTOM)
LOCATES WATER TABLE BASED ON LOCAL FLOW AND
HEAD CONDITIONS
COMMON/ID /PSINCT(50»,RAIN( 20 ), EVAP ( 20) ,GW=LOW<20> , 2TABLE 120) ,
EXTR 9
EXTR 12
EXTR 13
EXTR 14
EXTR 15
EXTR 16
EXTR 17
VFLO 1
,VFLO 3
VFLO 4
VFLO, 5
VFLO 6
VFLO 7
VFLO 8
VFLQ 9
VFLO 10
VFLO 12
VFLO 13
VFLO 14
VFLO 15
VFLO 16
VFLO 17
VFLO 18
VFLO 19
VFLO 20
VFLO 21
VFLO 22
VFLO 23
VFLO 24
VFLO 25
VFLO 26
VFLO 27
VFLO 28
VFLO 29
VFLO 30
VFLO 31
VFLO 32
VFLO 33
VFLO 34
VFLQ 35
VFLO 36
TABE 1
TA8E 2
TABE 3
267
-------
C-
C
TABE 2
TABE 21
TABE 22
PSIV 1
PSIV 2
PSIV 3
1 PSI(50,20), A(50,20),B( 50,.2C ) ,.C( 50, 20) ,0( 50,20) ,E (50,20) ,F 150,20) .TABE 5
2 CSLUPE(50,2G),THETA(50,20),THENOT(50),VELZ(50,20>,PSI1<50»20)t TABE 6
3 PSI2<50,20),PS13(50,20), NODSAT(20), TABE 7
4 CCNDt50,20),PSIBOT<50),HPOND(50),QSINK<5C,20),ONEU2(50), TABE 8
5 CPLANT(50,20) - TABE 9
COMMCN/WTAB/JTAB(50),HTAB(50)
C ASSUME THAT NOOi J»l CHOSEN LOW ENOUGH SO THAT IT WILL TABE 10
C ALWAYS BE SATURATED. FREEZE'S EQ.<12) HOLDS, TABE 11
C SCAN PSI VALUES FOR SATURATION FROM BOTTOM NODE UPWARDS. TABE 12
IF (N.EQ.l) HPOND(N)-0,
NfJSAT«l TABE 13
DUbOOJ«2,L TABE 14
IF(PSI(N,J).GE.O.) NNSATiNNSAT*! TABE 15
IF(P$l(iJrJ).LT.C.) GO TO 900 TABE^ 16
000 CONTINUE TABE 17
900 NOCSAT(N) » NNSAT
JTABINJ'NODSATtN)
ZTAQLEIN) =ZBOTOM+,
1 PSI<50,20),A150,20),B(50,20),C(50,20),D(50,20),E(50,.20),F(53,20)
2 CSLOPE(50,20),TH6TM50,20),THENOT(50),VELZ(50,20),PSI1(50,20) ,
3 PSI2(50,20),PSI3(50,20)h NODSAT(20),
4 CONC(50,20>,PSIBOT(50),HPOND(50),QSINK(50,20),QNET12(50)*
5 QPLANT(50,20)
COKMON/POND/CPOND(50)
C SET MAXIMUM PONCING DEPTH, IN CENTIMETERS
PCNMAX-5.
C TWOPAIN CASES ARISE 1) UPPER NODE UNSATURATED OR R MSAT'D)
C 2) UPPER.NODE SATURATED (WITH A FINITE HEAD)
C CAS5 2 IS STARTED WHEN R SATURATED K,. AND CONTINUES AS LONG AS
C CONTINUITY RELATIONSHIPS INDICATE EXISTENCE OF A FINITE PONDING
C DEPTH AT THE GROUND SURFACE.
IFtfc.EQ.l) HPONC(N)*0.
IF (N.EQ.2) HPGND(N)«0.
C THIS SUBROUTINE GETS CALLED ON 2ND TIME STEP
IF {N.EQ.2).GO TO 700
HPONO(N)-HPONClN-i) * ((RAIN(N-1)-EVAP(N-1>)-VEL2(N-l»L))«OELT
IF (HPONC(N).LT.O.) HPOND(N)«0.
C PONDING DEPTH MUST BE NON-NEGATIVE
700 IF(HPGND(N)*GT.3.) GO TO 300
x«o.
CQNSAT'CK(X)
R-RAININ)-EVAP(N)
C CHECK 'PREDICTED' VALUE OF PSI AT TOP NODE t IF NOT SATURATED
C SCLVE CASE 1.
IF (PSKN.L) .LT.O.) GO TO 850
IF (R.GT.CONSAT) HPOND(N)»(R-CONSAT)»DELT
IF(R.GT.CONSAT) GO TO 300
,PSIV 5
PSIV 6
PSIV 7
PSIV 8
PSIV 9
C-
C
C
SOLVE CASE 1,BELOW.
850 INOEX'l
NDUPMY»999
MJNSAT-NCUMMY
CALL MATRIX(N,L, INDEX,NUNSAT,DELZ,)
HPCMD(N)-,0.
GO TO 150
SOLUTION OF CASE 2, 8ELOW
SET CONDUCTIVITY AT GROUNQ EQUAL TO SATURATION VALUE.
300 IF
-------
PSIGND=HPDNO(N)
PS1GNC=AMAX1(0.,PSIGND)
PSHNtDsPSIGNO
INCEX«2
SEARCH FOR FIRST UNSATURATED NODE FROM TOP
DO 400 1*2, L
K»L-m
IF (PSIU.K).LT.O.) GO TO 500
400 CONTINUE
500 NUNSAT«K
IF INUNSAT.cC.l) NUNSAT»999
CALL MATRIX(N,L,INDEXiNUNSAT,DELZ)
150 DPCNDEI50,201 »F {50,20 1
2 CSLOPEtSO,20)»TM=TA(50i20>,THENOT[50)*VELZ<50,20),PSIl<50»20),
3 PSI2«50,20) »PSI3(5Q,20),. NODSAT(20)t
A CCNC(50»20),PSIBOT(50l|HPONO<50>»QSINKt50,20),QNETl2(50)i
5 CPLANT(50,20)
IF (N.EQ.2.) GO TO 510
C04lOJ»ltU
IF U.EQ.l.) GO TO 610
IF (J.EQ.U) GO TO 610
PSIlCN»J)«(PSI(N-ltJ)«iPSI(N-UJi-m-PSI(N»J+mPSI»RAINt20>,EVAP«20)tGWFLOWC20)»*TABLEUOH
1 P$H5Q,20),A<50,20),8( 50f20),C(50i20),D<50,20),E(50t20),Fl50f20)
2 CSLOPE(50,20)tTHETA(50i20),THENOTt50),VEU(50t20),P9IU50«20)f
3 PSI2(50^0)tPSI3(50f20)» NOOSAT(ZO),
4 COND(50,20)tPSlBOT(50)iHPONOt50).QSlNK(50i20itQNET12(SO)«
5 QPLANT(50»20)
COMMON/CROP/ ETCROP. W ILTM
COMMCN/RCOT/CROOT
COMMON/MOLe/CMOLO( 50 I
COMMGN/SOILVKSOU
DEPTH OF SATURATED SECTION A80VE DRAIN
PREVIOUS TIME STEP
C COMPUTE 1ST THE
C FROM RESULTS OF
PMIO
PHID
PHID
PMIO 10
PMID 11
PMID 12
PMIO 13
PMID 14
PMID 15
PflO 16
PMIO 17
PMID 18
PNIO 19
PWIO 20
PMIO 21
PMIO 22
PMIO 23
PMID 24
PMIO 25
PMIO 26
PMID 27
PMID 28
PMID 29
PMIO 30
PMID 31
PMIO 32
PMIO 33
PMID 34
PMIO 35
PMID 36
PMID 37
PMID 38
PMID 39
PMID 40
*$INK 3
SINK 4
SINK 5
SINK 6
SINK 7
SINK 8
SINK 9
269
-------
NOG IS NODE NO AT WHICH CRAIN LOCATED
C
C
C
C
C
C
C
C
IF(PSItN-ltJ3.GE.O.) KK=KK-H
JHPSKN-lrJJ.LT.O.) GO TO 600
500 CONTINUE
600 HCRAIN=KK*UELZ/100.
COMPUTE FLOW TO GRAINS (SEE HARRISON)f IN METERS/HOUR.
X=PSI(N-lfNQD)
:K(X)»60./100.
C
C
750
700
850
S = 20. 73.28
XLEN=il.
AREA=.XLEN»S
FLOW IN CFS/METER 3ELOW
CCFSPM=QCRAIN«S«XLEN«O.C098
CONVERT TO CFS/FT BELOW
CCFSPF=QCFSPI*/3.28
C
C
SINK 10
SINK 11
SINK 12
SINK 13
SINK 14
SINK 15
SINK 16
SINK 17
SINK IS
SINK 19
SINK 22
SINK 23
SINK 24
CONVERT FROM M/J-R TO CM/MIN
GDRAIN=QDRAIN»100./60.
CCNVEKT TO ( 1/MIN)
CDRAJN=QDRAIN/DELZ
NO DRAINS EXIST IN SANDY SOILS OF STUDY AREA
IF (KSQIL.NE.2) QDRAIN=0.
COMPUTE PLANT REQUIREMENTS WITH DEPTH.
TEMPORARILY ASSUMED TO BE ZERO.
COMPUTE PLANT WITHDRAWALS FOR ALL NODES
ASSUME UNIFORM WITHDRAWAL AT TOP HALF OF PROFILE.
WITHDRAWAL AT EACH NODE CANNOT EXCEED AVAILABLE MOISTURE*
AVAILABLE MOISTL'RE = MOISTUKE CONTENT AT NODE (USE VALUE FROM
LAST TIME Pfc-Riqoi MINUS CONTENT AT WILTING POINT
COMPUTE WITHCRAWALS AT EACH NODE
LRCCT=ORGCT*30.5/D£L2
LLCW=L-LRCOT
IF +QDRAIN
SINK UNITS ( L/MINI
CONTINUE
RETURN
END
SUBROUTINE MATRIXIN.L, INDEXrNUNSAT, DELZ )
COMMON/ 10 /PSINCT(50),RAIN(20),EVAP(20),GWFLOW(20),ZTABLEl20)i
PSI(50i20)»AJ50,20),B( 50, 20 ) ,C ( 50, 20 ) ,D( 50,20) ,E ^50^20) .F150.20)
CSLOP£(50,20)tTHETA(50i20),THENOT(50),VELZ(50,20),PSIH50.20) ,
PSI2(50,20) ,PSI3(50,20), NODSAT(20)t
CGNG(50,20) ,PSIBOT(50) , HPQNOl 50) ,QSINK ( 5C, 20) fQNET12(50) t'
CPLANT(50,20)
DIMENSION XLl50i20),RHS(50)
USES GAUSSIAN TECHNIQUE TO COMPUTE PSI VALUES RECURSIVELY*
INITIALIZE LEFT HAND SIDE ARRAY (XL)
CQ550I*ltL
SINK 25
SINK 26
SINK 27
SINK 28
SINK 32
SINK 33
SINK 34
SINK 35
SINK 36
SINK 37
MATR I
, HATR
MATR
MATR
MATR
MATR
HATR
MATR
MATR 10
MATR 11
MATR 12
3
4
5
6
7
8
9
270
-------
XL(1,,J) = 0.
520 CONTINUE
550 CONTINUE
C INSERT VALUES INTO RIGHT H&NO SIDE ARRAY tRHS)
D0600J*1,L
I = J
RHS( I)=0(N,JJ
600 CONTINUE
C INSERT VALUES OF A.BiC INTO LEFT HAND SIDE
C ARRAY (XL)
I»l
JNOCE-I
XL(Un«B(N,JNOC£)
XL(I,H-1)*-A(N,JNODE)
M = L-1
CQ660I=.2tM
JNCCE=I
XLI UI-1)=-C(N, JNODE)
XL(UI)»8(N,JNOC£}
XL(UI+1)=-A-XL( I«J)/XL( JtJ)
'WRITE 16,1900) JtXL(J,J)
FORMAT (I5,F5.2)
IPREVaI-1
JNODE=»I
XL
-------
C «RITE<6,2700) INDEX
2700 FORMAT(2Xi» INDEXES 13)
IF (INDEX. EQ.1J PSI (N, I )=RHS( I J
IF (INDEX. EU. 2) PSICN, I )*PSI(N,.I)
C COMPUTE VALUES OF PSI RECURSIVELY
LL=L-1
00570I=.lt.LL
K=L-t+l
C
C
570
C
C
600
610
620
C
C
700
710
720
,315
C
C
500
510
535
520
C
C
JNCUE1=JNODE-1
IF ( JNODE1.EQ.NUNSAT) GO TO 580
PSI(NtJNODE-l)*RHSU-l)-XL(K-l,K)»PSKN,JNODE)
GO TO 570
PREVIOUS STATEMENTS APPLYING OARCY'S LAW TO 1-ST UNSATURATED NODE
UIRECTLY HAVE BEEN DELETED
CONTINUE
RETURN
END
FUNCTION TEMS(Y)
TENSION VERSUS MOISTURE CONTENT RELATIONSHIPS
FORMULAS BELOW FOR SANDY SOILS
IF (Y.LT.9.5) GC TO 600
IF (Y.GT.42.) GO TO 610
TENS — EXPt (131. 456-YI/27. 4112)
GO TO 620
TENS a-EXP( ( 13.6879-Y)/0.9433)
GO TO 620
TENS --EXPM42.5178-Y1/0.1335)
IF (Y.GE.43.3) TENS=0.
RETURN
END
FUNCTION CK(X)
CONDUCTIVITY VERSUS PSI RELATIONSHIPS
FORMULAS BELOW FOR SANDY SOILS
IF (X.GE.O.) GO TO 315
C=ADS(X>
IF (G.LT.50.) GO TO 700
IF (G.GT.150.) GO TO 710
CK = 0.5391 - 0.1056»ALOG(Q)
GO TO 720
CK = 0.1264 - 0.00009242»ALOG(Q)
GO TO 720
CK = 0.01119 - O.COC238i»ALOG(Q)
RETURN
CK=0.126
RETURN
END
FUNCTION CSL(X)
C (DTHETA/UPSI) VERSUS PSI RELATIONSHIPS
FORMULAS BELOW FOR SANDY SOILS
IF U.GH.O.) GC TO 535
P=AQS(X)
IF (P.LT.25.) GO TO 500
IF (P.GT.85.) GO TO 510
CSL = (27.4112/PJ/100.
GO TC 520
IF (P.LE.l.) GO TO 535
CSL = (0.1335/P )/100,
GO TC 520
CSL = (0.9433/P )/100.
GO TO 520
CSL=0.
RETURN
END
FUNCTION HUMID(X)
COMPUTES MOISTURE CONTENT FOR A GIVEN PSI.
FORMULAS BELOW FOR SANDY SOIL.
81
MATR 82
MATR 83
MATR 84
MATR 85
MATR 86
MATR 87
MATR 88
MATR 89
MATR 90
MATR 91
MATR 92
MATR 93
MATR 94
MATR102
MATR103
MATR104
STEN
STEN 2
STEN 3
STEN 4
STEN 5
STEN 6
STEN 7
STEN 8
STEN 9
STEN 10
1
2
3
4
5
6
7
8
9
STEN 11
STEN 12
SCK
SCK
SCK
SCK
SCK
SCK
SCK
SCK
SCK
SCK 10
SCK U
SCK 12
SCK 13
SCK 14
SCK 15
SCK 16
SCSL 1
SCSL
SCSL
SCSL
SCSL
SCSL
SCSL
SCSL 9
SCSL 10
SCSL 12
SCSL 14
SCSL 15
SCSL 16
SCSL 17
SHUM 1
SHUM 2
SHUM 3
272
-------
IF (X.GC.O.) GO TO 330 SHUM 4
S*ABS(X) SHUM 5
IF (S.LT.26.) GO TO 815 SHUM 6
IF tS.GT.86.) GO TO 825 SHUM 7
HUMIC=131. 45-27. 41»ALOG
-------
AYB=Yl
CONTINUE
IXA=AXA+.5
IYB=AYB+.5
250 IFUXA^LUO.OR.IXA.GT.IOO) GO TO 260
IF< IYA.LT.O.OR.IYA.GT.50) GO TO 260
CALL PPLOT( IXA, IYA,NSYMiNCT)
260 CONTINUE
YA= (NMAYB-AYA) )/< AX8-AXA)
IYAaAYA+YA+0.5
IFHXA.LE.IX6) GO TO 250
GO TO 400
C SET PARAMETERS FOR Y DIRECTION
C
290 CONTINUE
IFIAYB.GT.AYAJ GO TO 295
AYB=Yl
AYA=Y2
AXB=X1
AXA=X2
295 CONTINUE
IXA=AXA+.5
IYA=AYA*.5
lYB-AYBi-.S
300 CONTINUE
IFtlXA.LT.O.OR.IXA.GT.ICO) GO TO 310
IFIIYA.LT.O.OR.IVA.GT.50) GO TO 310
CALL PPLOTCIXA,IYA,NSYM,NCT>
310 CONTINUE
IYA=lYA+\
XA=(N*JAXB-AXA) )/(AYB-AYA)
IXA=XA*AXA+0.5
N=N+1
IF(IYA-IYB) 300i320,AOO
320 IXA * IXB
GO TO 300
400 RETURN
END,
SUBROUTINE CURVE (X, Y,NPT»NCV,NPLOT )
DIMENSION X( 10 1,10), Yl 10 1,10),. NPT( 10 »
COCMON/LAB/ TITLEl20)tXLAB(ll),.YLAB(6),HORlZ(20),VERTI6»
XMAX=,X(li.l)
XMIN=XMAX
YMAX=,YMIN
CO 205 L*l»NCV
IF(NPTM.EG.n) GO TO 205
CO 204 N=il»NPTM
IF(X(N,L).LT.XMIN> XMIN=X(NtL)
IF(X
-------
RANGE=RANGE/(10.««N)
L=RANGE+1.001
206 CONTINUE
IF(L.EG.2) GO TO 209
IFIL.GT.4) GO TC 207
L=4
207 IF
-------
COMMON/LAS/ TITLE ( 20 ) ,XLAB( 11) , YLAB ( 6) , HORIZ ( 20) , VERT 16 J
DATA SVM / 4H**»», 4H+> + +:, 4H'1", 4HXXXX, 4H....f 4H2222i
4H , 4HIIII, 4H ---- /
IFIK-99) 209,220i23Q
200 AI51-IY,IX+1)»SYM(K)
RETURN
220 CCNTINUE
I«0
WRITE (6,103) TITLE, NOT
CO 225 11*1,6
1 = 1 + 1
WRITE (6, 101) YL£B( II ),( A< UJ),J=1,101)
IF( II.E0.6) GO TO 228
CO 224 JJ*i,9
I-I + l
IFd.NE.28) GO TO 221
WRITii(6,i06) VERT(5),VERT(6)r(A(I,J),J»i,101)
GO TO 224
221 CONTINUE
IFd.NE.24) GO TO 222
WRITE (6, 106) VEIU( 1),VERT<2),(A(UJ),J«1,101)
GO TC 224
222 IFl I.NE.26) GO TO 223
WRITE I 6, 106) VERT ( 3 ),VERT(4),{A(UJ)»J» 1,101)
GO TO 224
223 CCNTINUE
WRITE(6,100)IA( UJ),J«1,101I
224 CONTINUE
225 CONTINUE
228 CONTINUE
WR1TE(6,102) XLAD
hR!Tc(6rl05) .HORIZ
100 FORMAH18X» 101A1)
101 FORWAT(F17,3,1X,101A1)
102 FOKMAT(F20.1,10F10.1)
103 FORMAT ( 1H1 , 16X, 20A4* 10X , 'ELEMENT NUMBER1, 14)
105 FORMAT(/40X,10A4)
106 FORMAT(3X,2A4,7X,101A1)
230 CO 250 Ml, 50
CO 240 J*l,101
240 AII, j)»svym
A(I,l)=SYMt8)
250 CONTINUE
CO 260 Jil,101
260 A(51,J)-SYM(9J
DO 270 Ml, 101, id
270 A(5UI)-SYM(8)
CO 290 Mil, 41, 10
A(1,1)»SYM(9)
AU,101MSYM(9)
290 CONTINUE
RETURN
C-ND
FUNCTION ACOS(X)
ARC - (1.0 - X) / (1.0 + X)
TARG » SQRT(ARG)
ACQS - 2.0 * ATAN(TARG)
RETURN
END
SUBROUTINE SOILCU
C PROGRAM ''SOILQUAL1 SLQU 1
c ------------------ SLCU 2
C SLCU 3
C PERFORMS WATER QUALITY COMPUTATIONS AT THE SLQU 4
C UNSATURATED SOIL ZONE. PROCESSES CONSIDERED ARE SLQU 5
C ADVECTION, DISPERSION, ABSORPTION AND RELEASE SLCU 6
C CF SUBSTANCES BY THE SOIL. FINITE DIFFERENCE SLQU 7
276
-------
C EQUATIONS UTILIZE PREVIOUSLY COMPUTED VELOCITIES SLCU 8
C AT EVERY NODE IN SOIL PROFILE. AT EACH TIME SLCU 9
C STEPt PROGRAM COMPUTES CONCENTRATION OF SLQU 10
C SUBSTANG6IS) AT EACH NOCE. TRIOIAGONAL MATRIX SLCU 11
C SCHEME IS USED. INTERFACES WITH '1NFILT* PROGRAM. SLCU 12
C • SLGU 13
C PARTIAL LIST OF VARIABLES— SLCU 14
C NNQOES-NC. OF NOCES IN SOIL PROFILE SLCU 15
C NSTEPS'WO. OF TIME STEPS IN PERIOD OF STUDY SLCU 16
C CELT^TIME INTERVAL SLQU 17
C DELZ*VERTICAL INTERVAL SPACING SLCU 18
C TSTART-STARTING TIME OF PERIOD OF ANALYSIS SLCU 19
C TEKD=ENDING TIME OF PERIOD OF ANALYSIS SLCU 20
C FERT(N1«F£RTILIZER INPUT OF N AT N-TH TIME STEP SLCU 21
C FERTPIMJ-FE/UILIZER INPUT OF P AT N-TH TIME STEP SLCU 22
C RAINtNJaRAI.NFALL RATE AT N-7H TIME STEP SLCU 23
C EVAP«G^OUNCWATER FLOW AT BOTTOM NODE AT N-TH TIME STEP SLQU 25
C SQUN-AVgRAGf: (TEMPORAL AND SPATIAL) SOIL RELEASE OF NITRATE-N
C IN (LB/FT-MONTH-ACRE)
C SCUH»AVERAG£ SOIL RELEASE QF TOTAL INORGANIC PHQSPHOftOUS
C IN (LB/FT-MONTH-ACRE)
C SI\\-AV£RAGe SOIL UPTAKE OF NITRATE-N
C IN (LB/FT-MONTH-ACRE)
C SlNP-AVERAGE SOIL UPTAKE t)F TOTAL INORGANIC PHOSPHOROUS
C IN (LB/Fr-MONTH-ACRE)
C SQURMN.D-RATE OF N RELEASE BY SOIL AT N-TH TIME STEPiL-TH NODE SLCU 26
C IN LB/MONTH . $LCU 27
C SCURP(N,L)»RATE CF P RELEASE BY SOIL AT N-TH TIME STEPIL-TH NODE SLCU 28
C IN .LO/MONTH SLQU 29
C SINKN(N,L)»RATE OF N UPTAKE BY SOIL AT N-TH TIME STEP.L-TH NODE SLCU 30
C IN L3/MONTI: SLCU 31
C SlNKPtN,L)-RATE QF P UPTAKE BY SOIL AT N-TH TIME STEP.L-TH NODE SLCU 32
C IN LB/MONTH SLCU 33
C VELZ^CONC£NTRATION OF M-TH SUBSTANCE AT N-TH TIME STEP* SLQU 38
C L-TH NODE (IN MG/L) SLQU 39
C CKCIN.D-DISPERSION COEFFICIENT FOR N»TH TIME ST6Pt L-TH NODE SLQU 40
C IN (SO CM/MIN) SLCU 41
C CROPtN)*CROP TYPE GROWN AT N-TH TIME STEP SLQU 42
C THETA(NiLI-MOISTURE CONTENT AT N-TH TIME STEP,L-TH NODE SLCU 43
C VISKIN*KINEMATIC VISCOSITY OF WATER SLCU 44
c ALPHAS-COEFFICIENT USED TO COMPUTE DISPERSION COEFFICIENT SLCU 45
C (SEE REDOELLtP.16) SLQU 46
C BETA3«COEFFICIENT ALSO USED IN DISPERSION COEFFICIENT FORMULA; SLCU 47
C AS ABOVE SLQU ^8
C CCNC(N,L)»HYDRALLIC CONDUCTIVITY (LENGTH/TIME) FOR SLCU 49
C N-TH TIME STEP»L-TH NODE SLQU 50
C CCNV-FACTOR USED TO CONVERT CONO {CM/MIN) TO
C PERMEABILITY UNITS I SO CM) * 1.91E-7 SLQU 52
C CPSRIN.D-PERMEABILITY IN UNITS OF UENGTH»»2) SLCU 53
C HEREtSQ CM SLQU 54
C ACIN,.L>»COeF FOR CONCENTRATION AT UPPER NODEt SLCU 55
C N-TH TIME STEP AT L-TH NODE SLCU 56
C BClNrD-COEF FOR CONCENTRATION AT MIDDLE NODE» SLQU 57
C N-TH TIME STEP AT L*TH NODE SLQU 58
C CC(N,.L)»CO£F FOR CONCENTRATION AT LOWER NODE, SLQU 59
C N-TH TIME STEP AT L-TH NODE SLQU 60
C CCIN.U-TERM FOR KNOWN VALUESf SLQU 61
C N-TH TIME STEP AT L-TH NODE SLQU 62
C BFLUX(N)«FLUX AT BOTTOM NODE AT N-TH TIME STEP SLCU 63
C IN LB/AC/YEAR
C CFACTl=CQNV FACTOR FROM LB/MO-ACRE TO MG/MIN-ACRE-10.35
C CFACT2-CCNV FACTOR FflOM (OELZ(CM)*1ACRE) TO LITERS-4.05«I10»»4) SLQU 66
C CFL-CONV FACTOR FROM (CM/tUN)M MG/L) TO ILB/AC/MIN) SLCU 67
277
-------
c
c
c
c
c
c
c
c
c
c
c
c-
c
*8.90»(10*»-2)
CYEAR=CONV FACTOR FROM (CM/MIN)•(MG/L) TO (LB/ACRE-YR) * 4.70E03
KMUNTH=MCNTH IN WHICH SIMULATION TAKES PLACE
K=INCEX DENOTING MONTH
CINITIL)=1NITIAL CONCENTRATION AT L-TH NODE
FSAT=FRACTION OF SATURATION (PERCENT/100.)
CRCl]T*UEPTH OF ROOT ZONE (FT)
UNIT=UPTAKE RATE OF N BY PLANTS (L8/AC-YR)
UPHOS=UPTAKE RATE OF P BY PLANTS (LB/AC-YR)
CMCLC(M,N)=CONC. OF M-Th POLLUTANT AT MOLE DRAIN AT STEP N
CTA3LE=CONCENTRATION OF SUBSTANCE AT VlATER TABLE NODE
.•_.._._.._••.•..*_»—*—«*-*-p «• — -•>—•*•• — ••-••-•-'•«*•**»_••*•*• MMV«»M
SET UP COMMON STORAGE AREAS
CCMKCN/IC /NNODES,NSTEPS,DELTrOELZ,TSTART,TEND,FERTN(50),
1 FE«TP150),RAIH150),EVAP(50),GWFLOW(50.) , SOURN ( 50 ,20 ) ,SOURP(50j20)
2 ,SINKN(50, 20),SIMKP(50,20),VELZ(50,20) ,SMOIST,PSI ISO', 20) ,
3 C(2.5C,20),CROP(5G),THETA(50,.20),VISKIN,ALPHA3,BETA3,CIJND(50,20)
4 AC(50,20),BC150,20),CC(50,20),DC(50,20),BFLUX(50),CFL
5,CU-4V,K,L,M ,CPER(50,20),CKO(50,20),CFACTl,CFACT2^N,CINITt50)
6 ,CYEAR
CO^MCN/SOIL/KSOIL
COMMCN/CMOLE/CMOLDt 2,50)
COMMCN/LCCMOL/LMOLc
COMMCN/CTAB/CTAQLE(2,50)
COfCCN/OSETS/NGV«IF,NUMIFl,.NilNOVl»NUNREl,NUNGWl,.NUNRE2,NUNIF2»
1 NU^CV2,NUNOV3,NUNOV4,NUNIF3,NUNGW2,NUNGW3,NOVRE1,.NOVRE2,NRERE1,
2 NKER£2,NOVRE3
READ (NUNIF3)OELZ,DELT,NSTEPS,NNOOES,OROOT
CELZ=6.7
CELT=10
NSTEPS=15
NNGDES=10
1,NMONTH)
100
105
130
C
c
c
150
160
165
KMIJNTH=7
READ IN CATA
READ (5,100) (FERTN(K),K=
FORMAT (12F5.2)
READ (5, 100 )(F£RTP(K),.K = 1,NMONTH)
READ (5,105) SOUN»SOUP,SINN,SINP
FORMAT (4F5.2)
READ (5,130) (CROP(K),K=1,NMONTH)
FORMAT (12F5.2)
READ (5,140) VISKIN,ALPHA3,BETA3
,5)
IN SO CM
SMOIST,VMAX,.CFL,CONV,.CFACT1,CFACT2
,5,FlQ.4,F10.9,F10.5,F10.4tF10.2)
(CINIT(L),L=1,NNODES)
.2)
ZEROS, ABOVE, EXCEPT TOP NODE
FORMAT (3F10,
•CCNV« BELOW
READ (5,150)
FORMAT (2F10,
READ (5,160)
FORMAT (10F5,
INITIALIZE AT
.CYEAR
EVENTUALLY MIGHT NEED ONE INITIAL CONCENTRATION VECTOR
FOR EACH SUBSTANCE
READ (5,165) UNIT,UPHOS
FORMAT (2F5.2)
WRITE (6»700) DELZ.DELT
700 FORMAT (• SPACING IN CM' ,.F7.2, • TIME STEP IN MIN SF7.2)
C CONVERT SATURATION MOISTURE VALUE FROM PERCENT TO FRACTION
SMQIST»SMOIST/100.
K'KMCNTH
C CONVERT FROM »PER FT1 TO 'PER NODE' BASIS
C COULD MULTIPLY VALUES BELOW BY SEASONAL AND SPATIAL FACTORS
SLCU 68
SLGU 69
SLGU 70
SLCU 71
SLQU 72
SLCU 73
SL
SLCU 75
SLCU 76
,SLCU 77
SLCU 78
SLQU 79
SLCU 80
SLCU 86
SLCU 89
SLCU 92
SLQU 93
SLCU 94
SLCU 95
SLCU123
SLCU124
SLQU125
SLQU126
SLCU127
SLCU128
SLCU129
SLCU130
SLCU131
SLQU132
SLQU135
278
-------
5/30.5
SCURPCKtL)=SOUP»C£LZ«C.5/30.5
SINKMK,L) = SINN»DELZ»0.5/30.5
SINKPIK,L)*SINP«OELZ«0.5/30.5
NN1=NNODES-1
CC 850 L=i2iNNl
SCURP(K,L)=SQUP»D£LZ
SJNKN(K,L)=SINN«DELZ
SIKKP(K,L)=SINP»CELZ
850 CONTINUE
730
/30
/30
/30
185
C
C
C
180
170
C
C
C
C
C
C
C
C
C
c
C«*»
C
210
220
C««*
«»
230
C*»
C«»*
375
SCU*K(K,L)=SOUN«CELZ«0.5/30.5
SI,NKMK,L)-SINN«CELZ*0.5/30.5
SINKP(K,L)=SINP*CELZ*0.5/30.5
COMPUTE N AND P UPTAKE BY PLANTS AT NODES WITHIN ROOT ZONE
NRCCT*ORCOT*30.5/DELZ
NLCwER=NNODES-NRCOT
IF INLOW6R.LT.1) NLOWER*!
CO 105 L^NLOWER,NNOCHS
IF '( MOWER. EQ-10) G0 T0 185
SlNKMK,L»*SINKMK,Ll*CUNIT/12.)*DELZ/tDROOT«30.5l
siNKP(K.L)»SINKP(K,L)t(UPHOS/l2.)*DELZ/(nROOT*30.5)
CONTINUE
00 170 N-.liNSTEPS
CO 180 L-l.NNODES
COMPUTE SOURCE AND SINK TERMS FOR EVERY TIME STEP WITHIN
THE CONTh IN QUESTION
RATE UNITS BELOW ARE LB/MO-ACftE
SOURNIN,L)=SOURN(K,L»
SCURP(N,L)=SCURP(K,L)
SINKN(N,L)=SINKN(K,L)
SINKPIN,L)»SINKP(K,L)
CONTINUE
CONTINUE
FOR ALL TIME STEPS, COMPUTE THE VALUES OF
VARIABLES TO BE USED IN FINITE DIFFERENCE FORMULATION
NOTE—PRESENTLY,. VALUES OF ' SOURN •, • SOURP *, ' SINKN' AND'SINKP*
ARE READ IN. INSTEAD, THESE COULD BE COMPUTED USING FUNCTIONAL
RELATIONSHIP (E.G.,SINUSOIDAL) INVOLVING MIN AND MAX VALUES.
COMPUTE DISPERSION COEFFICIENTS BELOW—
VISKIN=0.00001040 (BASED ON ORLANDO,FLA. MEAN TEMP 71.5 F)
IN (SO FT/SEC.)
CONVERT VISKIN FROM
-------
CO 200 N-1,NST£PS
CO 250 L^l.NNODES
C NOTES—
C ALPHA3=38 FOR SAND
C BETA3=1.2 IHOOPES AND HARLEMAN,1965)
C BELCH. CONVERT HYDRAULIC CONDUCTIVITIES TO APPROPRIATE
C UNITS OF PERMEABILITY (FROM CM/MIN TO SQ CM)
CPER(N,L)*=CONO(N,L)*CONV
TERMl±VISKIN»ALPHA3
C CONVERT VELOCITY FROM CM/MIN TO CM/SEC
VZ=VELZ(N,L)/60.
VZ=ABS(VZ)
TERf2=VZ MCP6R(N,L)»»0.5)/VISKIN
IF (TERM2.LE.O.) GO TU 235
C 'CKi;1 BELOW SHOULD HE IN (SC CM/SEC).
CKC(N,L)=TERM1MTERM2)*«BETA3
GO TO 245
235 CKC(N.L)=iO.
C CONVtRT DISPERSION COEF FROM SQ CM/SEC TO SQ CM/MIN
245 CKC(N,L)=CKD(N,L)»6Q.
250 CONTINUE
200 CONTINUE
C PREPARE FINITE DIFFERENCE RELATIONSHIPS FOR EACH TIME STEP
C GOVckMNG EQUATION (ORIGINAL FORM) IS —
C CITHETA»C)/DT = C»D(ThcTA)/C(T) * THETA*D(C)/D(T)
C = -V*C(C)/C(Z)
C +CKD(V)*D2(C)/D2(Z)
c + SOUR
C - SINK
C NOTE- IN THIS MCCEL. VELOCITY WAS PRE-DEFINED AS (+) IN DIRECTION
C OF DECREASING NODE NUMBER (DOWNWARD). THUS IN EFFECT THE
C (-) VELOCITY TERM ABOVE 3ECOMES ( +.).
C PRQCECURE- CCLLECT TERMS ASSOCIATED WITH UNKNOWN CONCENTRATIONS
C FOR UPPER, MIDDLE AND LOWER NODES
C CU FOR ALL SUBSTANCES
MPCL-2
DO 300 M^J,MPOL
CO 450' N=U,NSTEPS
CALL GROUP
C SOLVE TRIOIAGONAL SCHEME* BELOW
CALL MAT2
450 CONTINUE
C WRITE OUT RESULTS. BELOW
CALL OUTRIT
500 CONTINUE
RETURN
END
SUBROUTINE GROUP
C GROUPS TERMS INTO COEFFICIENTS A,B,C AND D IN TRIDIAGONAL
C SCHEME. UPPER AND LOWER NODES ARE SPECIAL CASES
COMMON/10 /NNQDF,S.NSTEPS,DELT,DELZ,TSTART.TEND,FERTN(50) ,
1 Ft*TP(50).!UIN(5CM, EVAPl 501 .GWFLOW ( 50) . SOURNtSO ,20 ) ,.SOURP (50'.20)
2 ,SINKN(50,20),SINKP(50,20),VELZ(50,20),SMOIST,PSH50,20),
3 C(2,50,20)fCROP(50)«THETA(50,20),VISKIN,ALPHAS,BETA3.CONDI50,20)
4 AC(50,20),BC(50,20),CC(5G,20),DC(50,20) ,BFLUX(5,0) ,CFL
5,CONV,K,L,M ,CPER(50,20),CKD(50,20),CFACTlfCFACT2,.N,CINITt50)
6 .CYEAR
DIMENSION SOURt 50, 20 I ,-S INK ( 50,20)
SLCU15T
SLCU158
SLQU163
SLCU164
SLCU165
SLGU166
SLGU167
SLCU16S
SLCU169
SLCU170
SLQU171
SLCU172
SICU173
SLCU174
SLOU175
SLCU177
SLCU178
SLQU179
SLCU180
SLCU181
SLCU182
SLCU183
3LCU184
SLCU185
SLQU186
SLCU187
SLCU188
SLCU189
SLCU192
GRUP
GRUP
GRUP
SL
GRUP
GRUP
.GRUP
GRUP
GRUP
1
2
3
5
6
7
8
9
C-
c
c
300
150
CIRCUMVENT CALCULATIONS FOR 1-ST TIME STEP
INSTEAD, INITIALIZE CONCENTRATION VALUES
IF (N.GT.l) GO TO 150
CO 300 L^l.NNODES
C(M,N.LHUNIT(L)
CONTINUE
GO TO 250
SET SOURCES AND SINK VALUES ACCORDING TU SUBSTANCE
DO 350 LiliNNODES
GRUP 10
GRUP 11
GRUP 12
GRUP 13
GRUP 14
GRUP 15
GRUP 16
GRUP 17
280
-------
SUBSTANCE 1
IF (M.EQ.l)
IF (N.EO.l)
SUBSTANCE 2
IF (M.EQ.2)
IF (M.EQ.2)
350 CONTINUE
IS NITRATE NITROGEN
SOUR(N,L)=SOURN(N,L)
SINK(N,L)=S1NKN(N,L)
IS TOTAL INORGANIC-PHOSPHOROUS
SOUR(N,L)=SOURP(N,L)
SINK/OELT)
* (THETA(N,L)/DELT) +: IVELZ(N,L)/OELZ)
'CC(N|L>=0.
CC(N,.L)= (THETA < N, L ) *C (M,.N-1,L)/DEL T>
L +.( (SOUR ( Ni D-S INK(N,L) >*CFACT1 /1CFACT2* (DELZ/2. ) »THETA(N,L)) 1
C
C
PERFCftM FOR INTERMECIATE NODES
CU 100 L=i2iLL
AC(NiL)=(VELZ(N,L)/(2.»DELZ) )+.( CKOlN.L ) / IDELZ»»2) )
3C(NiL>=( (THETA(NtL)-THETA(N-l,L),)/DELT) + ( THETA(N»L) /DEIT)
I + (2.»CKOIN>L)/(DELZ»»2) )
CC(NtL)-(-VSLZ(N,L)/«2.*DELZ))*(CKD(N|L)/(DELZ#*2))
CC(N,L)= (THETA(N,L)»C(M,.N-1,L)/DELT)
1 t ( (SOUR (N,L)-SINK ( N, L ).) »CFACT1/ (CFACT2«OELZ»THETAIN, L))
100 CONTINUE
C-
C
C
C
c
c
c
PERFORM FOR TOP NODE ,L=NNODES
L=NSCDES
AC(NiL)=0.
BC(N,L>= «ThETA(N,L)-T»-ETA(N-l,L))/DELT) * (THETA (N,L) /CELT)
I - (VELZtN.D/OELZ)
CC(NrL)» - VELZIN.D/OELZ
CC(N,L)= (THETA(N,L^•C(M^N-1,L)/DELT)
1 *{(SOUR(NtL)-SINK(N,L))«CFACT1/(CFACT2*(DELZ/2.)»THETAtNiL)»)
nMfSINKP(50,20),VELZ(50,20) , SMOl ST, PSH 50,20) ,
3 C(2,50t20),CROP(501tTHETA<50,-20),VI SKIN,ALPHA3,BeTA3,COND(50(20)
4 AC«50t20),BCC50,20),CC(50,20>,DC(50,20).BFLUXl50)iCFL
5,CONV,K,L,M ,CPER(50,20),CKO(50,20),CFACTlhCFACT2,NtCINITt50)
6 ,CY£AR
CO^VCN/LOCMOL/LfOLE
CCMMC'J/CMCLE/CMCLO( 2i 50)
COMKCN/WTAB/JTABI50)|HTAB(50)
CO^CN/CTAB/CTA6LE( 2i50)
DIMENSION XL(5Ci20»,RHS(50)
CIRCUMVENT CALCULATIONS FOR 1-ST TIME STEP
IF (N.£(,».!> GO TO 530
USES GAUSSIAN TECHNIQUE TO COMPUTE CONCENTRATION VALUES
INITIALIZE LEFT HAND SIDE ARRAY (XL)
C0553I=1,NNODES
C0520J = li-NNODES
XL(UJ) = 0.
520 CONTINUE
550 CONTINUE
C-
C
INSERT VALUES INTO RIGHT HAND SIDE ARRAY (RHS)
C0600J=1,NNODES
I*J.
GRUP 18
GRUP 19
GRUP 20
GRUP 21
GRUP 22
GRUP 23
GRUP 24
GRUP 25
GRUP 27
GRUP 28
GRUP 29
GRUP 30
GRUP 31
GRUP 32
GRUP 33
GRUP 35
GRUP 37
GRUP 38
GRUP 39
GRUP 40
GRUP 41
GRUP 42
GRUP 43
GRUP 44
GRUP 45
GRUP 47
GRUP 48
GRUP 49
MTUO
MTWO
MTWQ
MTWO
SL
MTWO
MTWO
,MTWO
MTWQ
1
2
3
4
6
7
a
9
MTWO 10
MTWO 11
MTWO 12
MTWO 13
MTWO 14
MTWO 15
MTWO 16
MTWO 17
MTWO 18
HTWO 19
MTWO 20
MTWO 21
MTWO 22
MTWQ 23
MTWO 24
281
-------
600
RHS( I)=DC(N,J)
CONTINUE
C-
c
c
660
C
1400
C
C
200C
CISCO
INSERT VALUES OF A,B,C INTO LEFT HAND SIDE
ARRAY (XL)
1=1
JNCCE*!
XL«I,I)=BCM
JNODE=»I
XL»
-------
570
C(P,N,JNODE-l)
CONTINUE
'RHS(K-l) - XL(K-1,.K)»C(M,N,JNOOE)
C-
C
MTWO 90
MTWO 91
MTWQ 92
C
C
C-
SET CONC. AT MOLE DRAIN BELOW
LMQLE=i
CMOLD(M,N)=CIM,N,LMOLE)
SET CONC. AT WATER TABLE NOCE
JTAB(N)=1
JJ=JTAB
-------
C •*••••••«*»» •••••••^••••» • ••^ J» «• *»-W ^ffc^«**^it»^» <••**• ^«* ^••••i
C CONSIDER OUTPUT TO OVERLAND FLOW MODEL
C --------
WRITE (NUNOV4>NTIME,DTIME
WRITE (6..110) NTIME,OTIME
110 FORMAT (I5.F5.1)
C CONVERT RRAIN FROM CM/MIN TO IN/HR
DO 100 N^ltNTIME
RRAIN(N)=RRAIN(N)*60./2.5*
WRITE (NUNOV4)RRAIN(N)
WRITE (6,120) RRAIN(N)
120 FORMAT (10X.E8.2)
100 CONTINUE
C CCNVERT RINF FROM CM/MIN TO CM/HR
CO 200 NMtNTIME
RINF(N)»RINFIN)«60.
WRITS (NUNOVORINFH)
WRITE (6,120) RINF(N)
200 CONTINUE
WRITE (NUNOV,N«liNTIME)
WRITS (6»130) (DPONO(N) ,N*1,NTIME)
130 FORMAT (5X.10E8.2)
END FILE NUNOV4
REWIND NUNOV4
C CONSICER CUTPUT TO RECEIVING MODEL IN MUCK FARMS—
C ---------
C FLUW AND QUALITY IN MOLE DRAINS
IF (KSOIL.NE.2) GO TO 300
WRITE (NUNRE1) NTIME.OFIME
C WRITE FLCW IN CFS PER UNIT FOOT OF MOLE LENGTH
WRITE (NUNRE1) ( QMOLOJ N ) ,N*i,NTIME )
C WRITE N-FLUX IN L8/DAY PER FT OF MOLE LENGTH
M»l
CO 400 N*1,NTIME
C CONVERT MGD»MG/L TO LB/OAY
FLUXN(N)«.{QMQLO(N)/1.5S)*(CMOLD(H,N) )»8.34
WRITE (NUNRE1) FLUXN(N)
400 CONTINUE
C WRITE P-FLUX IN LB/DAY PER FT OF MOL6 LENGTH
K»2
DO 500 N»sl,NTIME
FLUXP(N)^(QMOLC(N)/l.SSt*(CMOLD(M,N))*8.3A
WRITE (NUNRE1I FLUXP(N)
500 CONTINUE
C TO COMPUTE FLOWS AND FLUXES FOR EACH CANAL JUNCTION,
C yUST MULTIPLY ABOVE VALUES BY (AREA OF CATCHMENT(SQ FT) /20t)
C SINCE SPACING OF MOLE DRAINS IS APPROXIMATELY 20 FT.
300 CONTINUE
C CONSIDER OUTPUT TO GROUNDWATER MODEL
C -----------
C WATER TABLE ELEVATION AND CONCENTRATIONS—
C TRANSFER VALUES AT LAST TIME STEP
NN»NTIME
CMNN)*CTABLEU|NN)
CP(NN)"CTABLE(2,NN)
C WRITE WATER TABLE HEIGHT AND NITROGEN CONC. ONLY
WRITE (NUNGW3) I-T3
WRITE (NUNGW3) CN3
END FILE NUNGM3
REWINC NUNGW3
RETURN
ENU
SUBROUTINE OVLAN1
COMPON/DSETS/riGWlF,NUNIFl,NUNOVl,NUN*El,NlJNGWlt.NUNRE2,NUNIF2V
1 NUNOV2,NUNaV3,NUNOV4,NLiNIF3,NUNGW2,NUNGW3,NOVRELf.NOVRE2,NREREl,
284
-------
c
c
c
c
c
c
c
c
c
c
c
c
c.
c.
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
2 NRERE2.NOVRE3
RETURN
END
SUBROUTINE OVLAN2
OVERLAND FLOW PROGRAM »KWAVE'
THIS MODEL USES KINEMATIC WAVE RELATIONSHIPS
TO COMPUTE FLOWS FROM PLANE CATCHMENTS TO
RECEIVING ELEMENTS. CATCHMENTS 00 NOT FEED INTO EACH OTHER.
A RECEIVING ELEMENT MAY BE CONNECTED TO MORE THAN ONE
CATCHMENT. SCANNING PROCEDURE USED TO OBTAIN
EQUIVALENT STORM DURATION, AVERAGING PROCEDURE
US£0 TO OBTAIN EQUIVALENT 'MEAN (CONSTANT) RAINFALL
ANC INFILTRATION RATES. QUALITY RELATIONSHIPS ARE
INCLUDED. PROGRAM APPLIED TO SANDY SOILS IN STUDY AREA
(APCPKA BASIN).
PARTIAL LIST OF VAR1ABLES-
NRbC»NO. OF RECEIVING ELEMENTS (TOTAL)
NDEP-.NO. UF RECEIVING ELEMENTS CONSTITUTING «SMALL»
SURFACE DEPRESSIONS (TYPED
NfiCCY- NG. OF RECEIVING ELEMENTS LOCATED ON
•LARGE' BOCY OF WATER (TYPE2)
NCATCHUJ-NO. OF CATCHMENTS FEEDING TO I-TH
RECEIVING ELEMENT
NTYPEIIJ-TYPE OF I-TH RECEIVING ELEMENT (1 OR 2 ABOVE)
XU)*X-COOROINATE OF I-TH RECEIVING ELEMENT
Y(M«Y-CGQRDINATE OF I-TH RECEIVING ELEMENT
NTC-TOTAL NO. OF CATCHMENTS
XLMIK,M)'LENGTH THROUGH K-TH INTERNAL IN K-TH CATCHMENT
•IN FEET
XLENGJKHLENGTH OF K-TH CATCHMENT I IN THE HORIZONTAL)
IN FEET
WIOTH(K)»WIDTH CF K-TH CATCHMENT
IN FEET
SLOPE(K)«SLOPE CF K-TH CATCHMENT
•IN (FT/FT)
ALPHA(K)«ALPHA FACTOR OF K-TH CATCHMENT (SEE EAGLESON)
(OlfENSIONLESS)
EMK)»MANNING 'N' ROUGHNESS COEF FOR K-TH CATCHMENT
EM(K)«M-FACTOR OF K-TH CATCHMENT (SEE EAGLESON)
(HEREi EMIKJ-2.)
NLIN'KJKJa NO. OF RECEIVING ELEMENT TO WHICH K-TH CATCHMENT
IS LINKED
MINTING. OF INTERVALS INTO WHICH EVERY CATCHMENT IS DIVIDED
NT»NO. OF TIME STEPS IN PERIOD
CT-TIME INCREMENT IN MINUTES
XRAIN(L)«RAINFALL RATE AT L-TH TIME STEP
IN (INCHES/HOUR)
X1NFIDINFILTRATION RATE AT L-TH TIME STEP (FLOW BETWEEN UPPER
TWO NODES IN UNSATURATED MODEL) IN (CM/HOUR)
XPCNCID-PONOING DEPTH AT L-TH TIME STEP (FROM UNSATURATEO
MOCEDt IN CM
LSTOP-TIME STEP AT WHICH PONDING DEPTH BECOMES ZERO
XF-AVERAGE INFILTRATION RATE FOR I L LSTOP
IN (FT/MIN)
NTR-NO. OF TIME STEP AT WHICH MAXIMUM PONDING DEPTH OCCURS
TR-TIME CORRESPONDING TO ?NTR' ABOVE*ASSUMED TO BE EQUIVALENT
STORM DURATION, IN MIN
XIxAVERAGI? RAINFALL RATE FOR 1 L LSTOP
IN (PT/MIN)
YlENGm^ELEVATION DIFFERENCE BETWEEN UPPER ANDLOWER PORTIONS
OF K-TH CATCHMENT,. IN FT
SL(K)»»TRUE« LENGTH OF K-TH CATCHMENT
IN FEET
SLM(K,M)*TRUE LENGTH,AS ABOVE.THRU M-TH INTERVAL
IN FEET
285
1
2
3
4
5
6
7
8
9
KINW
KINW
KINW
KINW
KINW
KINW
KINW
KINW
KINW
KINW 10
KINW 11
KINW 12
KINW 13
KINW 14
KINW 15
KINW 16
KINW 17
KINW 18
KINW 19
KINW 20
KINW 21
KINW 22
KINW 23
KINW 24
KINW 25
KINW 26
KINW 27
KINW 28
KINW 29
KINW 30
KINW 31
KINW 32
KINW 33
KINW 34
KINW 35
KINW 36
KINW 37
KINW 38
KINW 39
KINW 40
KINW 41
KINW 42
KINW 43
KINW 44
KINW 45
KINW 46
KINW 47
KINW 48
KINW 49
KINW 50
KINW 51
KINW 52
KINW 54
KINW 55
KINW 56
KINW 57
KINW 58
KINW 59
KINW 60
KINW 61
KINW 62
KINW 63
KINW 64
-------
C SINTH(K)-SINE QF ANGLE THETA WITH HORIZONTAL FOR K-TH CATCHMENT KINW 65
C TC(KrM)=TIME OF CONCENTRATION IN K-TH CATCHMENT KINH 66
C AT M-TH POINT, IN MIN KINW 67
C VEL(K,L.M)=OUTFLOW VELOCITY IN K-TH CATCHMENT AT OUTLET KINW 68
C AT L-TH TIME STEP AND AT M-TH POINT, IN FT/MIN KINW 69
C YL(K,.L,M)=OEPTH IN K-TH CATCHMENT AT L-TH TIME STEP,AT M-TH POINT KINW 70
C IN INCHES KINW 71
C CUNIT(K,L,M)=UNIT OUTFLOW,K-TH CATCHMENT AT L-TH TIME STEP, KINW 72
C AT M-TH POINT* IN (CUBIC FEET PER MIN/FT) KINW 73
C CTCT,(K,L,.M) = CUTFLOW AT K-TH CATCHMENT AT L-TH TIME STEP, KINW 74
C AT M-TH POINT, IN (CUBIC FT / SEC) KINW 75
C CRCV(I,L)<=INFLOW INTO I-TH RECEIVING ELEMENT AT L-TH TIME STEP KINW 76
C IN (CUBIC FE6T/SECI KINW 77
C TI(K,M)=T1M£ AT WHICH YL VANISHES IN K-TH CATCHMENT.AT M-TH POINT KINW 78
C IN MIN KINW 79
C TP(K,M)=TIME AT WHICH YL BEGINS TO DECREASE AFTER ACHIEVING KINW 80
C PEAK DEPTH, AT K-TH POINT, IN MIN KINW 81
C VBAR(K,L)=AVERAGE VELOCITY AT K-TH CATCHMENT AT L-TH TIME STEP KIKW 82
C IN FT/SEC KINW 83
C YBAR(K,L)=AVERAGE DEPTH AT K-TH CATCHMENT AT L-TH TIME STEP,IN FT KINW 8*
C PMS(K,L)=RATE OF INPUT (SUSPENSION) OF SOLIDS INTO WATER KINW 85
C IN LB/ACRE/SEC KINW 86
C PMD(K,L)=.RATE OF INPUT OF DISSOLVED SUBSTANCES INTO WATER KINW 87
C IN LB/ACRE/SEC KINW 88
C R=HYCRAULIC RADIUS, IN FT KINW 89
C VCRIT=AVERA3E VELOCITY AT WHICH PARTICLES OF DIAMETER *0« KINW 90
C WILL BE TRANSPORTED, IN FT/SEC KINW 91
C CK=SHIELC'S CONSTANTS.C56 KINW 92
C SS=SPECIFIC GRAVITY OF PARTICLE (APPROX.=2.) KINW 93
C C=CIAMETER OF PARTICLE (ASSUMED TO Be D50) ,IN FT KINW 94
c PS«AX=MAXIMUM SCOURING RATE IN LB/ACRE/SEC KINW 95
C CPOL(K,L)*FLUX OF SUSPENDED POLLUTANT (PHOSPHOROUS IN THIS CASE)
C IN OUTLET FLCW (L3/OAY)
C RO=DENSITY OF WATER KINW 98
C PZESC=INITLAL RATE OF UPTAKE OF DISSOLVED SUBSTANCE (NITRATE-N
C IN THIS CASE) IN ,T1(200,20UTPI200,1),XLM(200,I),MINT,SLMt200 ,1),
6 YL(200,20,l),VEL(200,.20,l), QTOT(2CO,20U) »
7VBAR(200,20),YBAR(200,.20),PMS(200,20),PKD(200,20),CKrSS,D^PSMAX,
8 CPOL(2()0,20).RO,PZEROtCOECAY,CDIS(200,20)
9 ,X(200),YI200),I,K,L,M
COK^CN/DSETS/NGWlF,NuNIFl,NL.NOVl,NUNREl,NUNGWlrNUNRE2,NUNIF2i
1 NUNCV2,NUNOV3,NUNOV4,NUNIF3».NUNGW2,NUNGW3,NOVREl,NOVRE2iNRERElt
2 NRERE2fNOVRE3
C,
C
C
C
C
READ INPUT DATA BELOW
REAC PHYSICAL DATA AND LINKAGE DATA BETWEEN RECEIVING ELEMENTS
AND CATCHMENTS
286
KINW116
KINW1X7
KINW118
KINWH9
KINW120
-------
C
C
C
C
C
C
100
120
125
110
135
130
140
C
C
C
150
160
350
450
210
170
180
READ (5,100) NREC,NOEP,NBODY,.MINT,NTC
FORMAT (515)
CO 125 I*1,NREC
READ (5,120) II,NTYPEm,NCATCH(I),X(I),Y(I)
FORMAT ( I3,I2,12,4X,2F5.2)
CONTINUE
READ (5,110) (NCATCHII),1=1,NREC)
FORMAT (2512)
DO 130 K*1,NTC
REAC (5,135) KK,NLlNK(K),.XLENG{K)^WIOTH{K)rSLOPE(K)
. ,EN(K)(EM(K)
FORMAT (14, 3X,I3,2F5.0,F5.4,F5.3,F5.2)
SET VALUES OF EMK) FOR TURBULENT FLOW
£M(K)=5./3.
KC(K)sKK
CONTINUE
REAC HYDROLOGIC DATA
READ (5,140) NT.CT
FORMAT ( I5,F5.2)
FOR THE PRESENT, ASSUME OATA BELOW HOLDS FOR EVERY CATCHMENT
DO 160 Lnl,MT
READ (5,150) XRAIN(L),XINF(L),.XPONO(L)
FACTOR*!.3
ADJUST RAINFALL INTENSITY FOR VARIOUS RUNS
XRAIN(LI»XRAIML»*FACTO.R
UNITS FOR 'XRAIM-IN/HR ,. FOR 'XINF '-CM/HR , FOR «XPOND'*CM
NOTE— VARIABLES ABOVE COULE BE OBTAINED THROUGH INTERFACING
INSTEAD OF READING IN
FORMAT (3F10.2)
CONTINUE
REAC INUNOV4)NT,DT
CO 350 L*1»NT
PEAO (NUNOV4) XRAIN(L)
CCUTINUE
WRITE (6»2IO) (XRAINtL),L*l,NT)
CO 450 L*1»NT
Rfc'AC (NUNOV4) XINF(L)
CONTINUE
WRITE (6,210) (XINFU),L*1,,NT)
READ (NUNOV4)(XPONO(L).L*1,NT)
WRITE (6,210) {XPONDtL),L«l,NT)
FORMAT (5X.12E8.2)
REWIND NUNOV4
REAC (5,170) CK,SS,0
FORMAT (2F5«l,F7.5l
READ 15,180) PSMAX,PZERO,CDECAY
FORMAT (3F5.1)
CALL AVE
WAVE
RECV
RQUAL
WRIT
CALL
CALL
CALL
CALL
RETURN
C DEBUG SUDCHK,TRACE,INIT,SUBTRACE
END
SU&rtOUTlNE AVE
C PERFORMS AVERAGING COMPUTATIONS INVOLVING
C RAINFALL AND INFILTRATION. PREPARES DATA FOR USE 'IN
C KINEMATIC WAVE CALCULATIONS.
COMMON/ID / NR£C».NDEP,NBODYrNCATCH(70>,KC(200J
1,NTYPE (2 ),NTC,XLENG( 200),.WIDTH! 200)» SLOPE 1200) t
2 iLPHA(200),EN(200),EM(200),NLPJK(200»,NT,DT,XRAIN(200),
3 XINF(2C»,XPONO<20>,ISTOP,XF,NTR,.TR,XI,
4 YLENG(200),SLt200),SINTh(2CO),TC(200t1).
5 QRCV(200,23)tTI(200,20),.TP(200,l),XLM(200,l)«MINT,SLMI200»l)»
6 YL<200,20,lJ,VEH200i20,.l), QTOT(200|20,ll f
7VBAR(200r20),YBAR(200,20),PHS(2COf.20),PMO{200,20)
8 CPCL<200*20),RC»PZERO,CDECAY,CDIS(200».20)
287
KINH121
KINW122
KINW123
KINW124
KINW125
KINW126
KINW127
KINM128
KINW129
KINW130
KINW131
KINW132
KINWL33
KINW134
KINW136
KINW137
KINW138
KINM139
KINM140
KINW141
KINW142
KINW1A3
KINW144
KINW147
KINW148
KUW149
KINW150
KINW151
KINH152
KINM153
KINW155
KIKW1S6
AVE 1
AVE 2
AVE 3
AVE t
-------
9 iX(200) ,.Y(200),ItK,L,M
C SCAN TIME HORIZON TO FIND TIME STEP AT WHICH AVE 15
C PONDING CEPTh BECOMES 'PERMANENTLY' ZERO. WORK AVE 16
C PACKWARD IN TIKE. AVE 17
CG 200 L=a,NT AVE 18
NDUNKYsNT+1-L AVE 19
IF fKC(200l
ltNTYPE(2)fNTC,XLENG<2CO)t.WICTH<2QO),SLOPE{200),
2 ALPHAl200).EN(200)iEM(2CO)tNL!NK(200),NT,DTiXRAIN<200)»
3 XINF120),XPONC(20),LSTOP,XF,,NTR,TR,XI,
4 YLENG(200)tSL(200),SlNTH{2CO»,TC(2C0.1)r
5 ORCV(200,20),TI(200,20)i.TP{200tl),.XLM(200tl)tMINTfSLMl200ill*
6 YL«200,20?l),VEL(200i2C».l) , QTOT (200,20 ,1) »
7VBAR(200»2C) »YBAR( 2CO,20),PMS( 2CO,20),PMD(200,20) ^C K ,-SS ,D t PSMAX ,
ti CPCL(20J,20>,RG,PZEROrCDECAY,CDIS(200t20)
9 fX(200)r.Y<200)t IiKiLtH
CO 200 KnltNTC WAVE 17
YLEhGlK)=XLcNG(K)*SLOPE(K) WAVE 18
RSC*XLENG(K)»»2 4 YLENG(K)»»2 WAVE 19
SL
-------
C TURBULENT FLOW ASSUMED ABOVE
200 CONTINUE
PERFORM THE FOLLOWING CALCULATIONS FOR EVERY INTERVAL
CO 700 M=U,MINT
COMPUTE TIME OF CONCENTRATION BELOW
CO 300 K^ltNTC
SLK(K,M)=SL(K)*(M/FLCAT(MINTJ)
XNUM=SLM(K,M)« UXI-XF)»M l.-EM(K)) )
CONVERSION BELOW— 60 SEC/MIN
XC£N=ALPHA(K J »60.
EMINV=1.0/EM(K)
TC(K,M)= (XNUM/XOEN) »*EMlNV
300 CONTINUE
WRITE (6,3501 TR
350 FORMAT CIS1 STORM DURATION*' ,F7. 2, 'MIN* )
CO 360 K=a,NTC
WRITE (6,370) K,TC(K,M)
370 FORMAT (' CATCH N0= ' , 15 , • TIME OF CONC=',F7.2,
360 CONTINUE
C FOR EACH CATCHMENT COMPUTE DEPTH AT OUTLET
C
C
C
DEPENDING ON RELATIVE VALUES OF TR AND TC(K).
CASE 1 TR TCIK,M)
CASE 2 TR TC(K,M)
CO 400 K^ltNTC
IF (TR.GE.TC(K,M))
IF (TR.LT.TCtK.M))
SUBROUTINE HIDUR
SUBROUTINE LOOUR
CALL HIOUR
CALL LOOUR
*MIN» )
400 CONTINUE
C CCNPUTE VELOCITIES AND FLOWS USING DEPTHS
CO 500 K=U,NTC
DO 6CO L=a,MT
C 'VEL1 BELOW IN FT/SEC
C (COMPATIBLE KITK MANNING'S N IN ALPHA COMPUTATION*
C PREVENT BASE ZERO IN OPERATION BELOW
If (YL(K>L,M).LE.O.) YL (K ,.L ,M) =0. 1E-15
VbL(K,L,M) = ALPHA(K.) » I YL( K ,L,.M ) »• ( EM(K )-L. > )
C CLNVERT TO (FT/MN)
VeL(K,L,M)=VEL(K,L,M)»60.
C 'CUNIT' BELOW IN (CU FT PER MIN/FT)
CUNIT =VEL(K,L,M) « YL(K,L,,M)
C 'OTOT' BELOW IN (CU FT/SEC)
C3»0.0167
QTCT(K,L,M)=CUNIT • WIDTH(K)*C3
600 CCNTINUE
500 CONTINUE
700 CONTINUE
RETURN
C DEBUG SUBCHKtTRACE,IN1T.SUBTRACE
END
SUBROUTINE RECV
C COMPUTES TOTAL FLOW INTO EACH RECEIVING ELEMENT
C FROM EACH CATCHMENT,FOR EACH TIME STEP
COMtfCN/ID / NREC,NCEP,NBODY,NCATCH{70),KC(200)
L,NTYP£<2),NTC,XLENG(2CO),WIQTH<2QO),SLOPE<200)t
2 ALPHA(200),EN(200),EM(200),NLINK(200)tNTtO-VrXRAINI200) t
3 XINF(20),XPCND{20),LSTCP,XF,NTR,TR,XI,
4 YLSNG(200),SL(200),SINTH(2CO),TC(200,l)l
5 QRCV(200,20),TI( 20(J, 20),.TP( 200,1 ),XLM(2CC,1) ,MI NT, SLM(200il) ,
6 YL(200,20,l),VEL(200,20t.l)t QTOT (200,20,1) t
7VBA^(200^20),YBAR(200,2C>,PMS(200,.20},PMO(200»20)»CK>5S,U»PSHAX»
8 CPCL(200,20),RC,PZERO,CCECAY,COIS(200,20)
9 ,X(200)^Y(200),I,K,L,M
C WORK ONLY WITH LAST INTERVAL OF EACH CATCHMENT
M*MINT
C INITIALIZE ARRAY BELOW TO ZERO
CO 200 L=»1,.NT
CO 300 I*1,NREC
289
WAVE 23
WAVE 24
HAVE 25
WAVE 26
HAVE 27
WAVE 28
WAVE 30
WAVE 30A
WAVE 31
WAVE 32
WAVE 33
WAVE 34
WAVE 35
WAVE 36
WAVE 37
WAVE 38
WAVE 39
WAVE 40
WAVE 41
WAVE 42
WAVE 43
WAVE 44
WAVE 45
WAVE 46
WAVE 47
WAVE 48
WAVE 50
WAVE 51
(HAVE 53
WAVE 54
WAVE 55
WAVE 56
HAVE 57
RECV 1
RECV 2
RECV 3
RECV 14
RECV 15
RECV 16
RECV 17
RECV 18
-------
CRCV(I,L)=0.
300 CONTINUE
200 CONTINUE
CO 400 L=1,NT
CO 500 K»itNTC
INDEX=NLINK(K)
•CJRCV BELOW IN ICU FT/SEC)
CRCVlINDEX,L)=QRCV(INDEXED
500 CONTINUE
400 CCNTINUE
RETURN
END
SUBROUTINE HICUR
UTILIZES KINEMATIC
TR TC(K,y) [HIGH
+ QTOT(KtL»M)
FOR THE CASE
C UTILIZES KINEMATIC WAVE RELATIONSHIPS
C TR TC(K,y) [HIGH CURATION)
CCMKCN/IC / NREC,NOEP,N30DY,NCATCH(70),KC(200)
1,NTYPE(2J,NTCtXLENG( 200 ) ,-W ICTH( 200) .SLOPE (200) ,
2 ALPHA(200), EN(200),EM(200),NLINK(200),NT,DT,XRAINt2001 .
3 XI-;F(20),XPCND(20),LSTOP,XF,NTR,TR,XI,
4 YLEi-iG(200),SL(2CO),SINTH(2GO)tTC(200tl) ,
5 (JRCV(2QOt20),TI<200i20),.TP(200,l),XLM<20Ci 1) .MINT , SLM (200 .1) t
6 YLC200,20,l),VEL(20Q,20,.l). OTOTl200.20,1)t
7VBA3(200,20),YBAR<200,20),.PMS(200,.20),PMD(200,20) ,CK*SS,D«PSMAX,
8 CPCL(200,20),RC,PZERO,COECAY,CDIS(200,20)
9 ,X(200) »Y(200)t I,.KTL,M
C CETrRMINE THK,M),TIME AT WHICH YUK.-L.M) VANISHES
THK,M)«TR +
-------
c-
c
c
CASE 1 OELOW
COMPUTE DEPTH AT DIFFERENT TIMES
100 CO 150 L^1,.NT
T=L*CT
110 IF IT.GT.TR) GO TO 120
120
130
140
150
GO TC 150
IF (T.GT.TP(KtM)) GO TO 130
YL{KfL,M)»XI*TR-XF«T
GO TO 150
IF (T.GT.THK.M) ) GO TO 140
TERM1=(XI-XF)«»0.5
TERM2=(XI»( *0.
CONTINUE
RETURN
• (XI-XF))*«0.5
C-
c
c
CASE 2 BELOW
COMPUTE DEPTH AT DIFFERENT TIMES
200 CO 250 L*l•NT
T=L»CT
210 IF (T.GT.TR) GC TO 220
YL(K,L,M)=(XI-XF)*T
CO TO 250
220 IFU.GT.TPCK.M) ) GO TO 230
YL(K,LiM)teXI*TR-XF«T
GO TC 250
230 YL(K,L,M)=0.
250 CONTINUE
RETURN
C-
C
C
CASE 3 BELOW
COMPUTE DEPTH AT DIFFERENT TIMES
300 CO 350 L^liNT
T=L»CT
310 IF (T.GT.TR) GC TO 320
YL(KrLtM)*tXI-XF)»T
GO TO 350
320 IF (T.GT.THKiM) ) GO TO 330
YL(K*L,M)=XI»TR-XF*T
GO TC 350
330 YLCKtL,M)=0.
350 CONTINUE
RETURN
END
SUBROUTINE RCUAL
C PERFORMS WATER QUALITY COMPUTATIONS AT THE VARIOUS
C SUBCATCHMENTS. RELATES THESE TO INFLOWS INTO
C RECEIVING ELEMENTS.
COMMON/IC / NRECiNOEPfNBODYtNCATCHt70)»KC(200)
1 ,NTYPE(2 ) ,NTCtXLENG( 2CO ) ,-WlDTH( 200)» SLOPE (200) *
2 ALPHA! 200), EN( 200), EMI 2CO),NLlNM2CO)t NT tDTtXRAlNt 2001 t
3 XINF(20),XPCNO(20),LSTOP,XF,NTR,TR,XI,
4 YLENGI200)tSL(2CO),SlNTH(2CO>,TC(200tl),
5 QRCVt200.20),TI<200,20)iTPC200,l),XLM(200,U,MINT,SLMI200»lU
6 YL(2CO,20, i),VEL(200t20|.l), QTOT<200,20t1 ) »
:7VBAR(200t20>.YBAR(2COt20)fPMS(200f20},PMD(200»20),CK,SS,D^PSMAX,
a CPULI200.20J,RO,PZ6RO,CDECAY,CDIS(200i20)
9 ,X«2GOUY<200) , ItK.LtM
COKMGN/NUM3/QPHCS(2CO,20) ,QNIT(200,20)
C-
C
C
AT EACH TIME STEP,
CATCHMENT BASED ON
DO 100 L*liNT
COMPUTE MEAN VELOCITY FOR EACH
VELOCITIES AT ITS INTERVAL POINTS.
291
LOOV 23
LQOV 24
LOOV 25
LCOV 26
LODV 27
LCOV 28
LODV 29
LCDV 30
LCDV 31
LODV 32
LOOV 33
LOOV 34
LODV 35
LCOV 36
LOOV 37
LCDV 38
LCDV 39
LODV 40
LODV 41
LCCV 42
LCDV 43
LODV 44
LOOV 45
LCCV 46
LCDV 47
LODV 48
LODV 49
LCDV 50
LCDV 51
LCOV 52
LODV 53
LOOV 54
LCCV 55
LODV 56
LODV 57
LCOV 50
LCDV 59
LODV 60
LCDV 61
LGDV 62
LODV 63
LCDV 64
LODV 65
LODV 66
LCDV 67
LCDV 68
LODV 69
LODV 70
LCDV 71
RQUA 1
RQUA 2
RCUA 3
RCUA A
RCUA 15
AQUA 16
RQUA 17
RCUA 18
-------
DO 200 K=il,NTC
M
C
c
C
c
c
c
c
c
c
c
c
c
c
c
c
SUM2»0.
CO 300 M*il, MINT
SUM2*SUM2*YL(K,L,M)
300 CONTINUE
C4-0.0167
•V8AR' BELOW IN FT/M1N
VBAR(K,L)»(SUM1/MINT)
•YOAR1 BELOW IN FT
YBArUK,L)«SUM2/MINT
20»0 CCNTINUE
100 CONTINUE
COMPUTE VALUES CF PMS(K,L) AND PMD
-------
570
TADJ^T- UXLENG(K)/2.)/VBAR/43560.
CSFLUX UNITS IN (LB/SEC)
CCIS(K,L>=DSFLUX«86«00.
IF OUTFLOW RATE IS ZERO » SET COIS»0.
IF (QTOT2^)fTI(200|20)^TP(200tl)»XLM(200|l}fMINT*SLM(200Ut*
YL(200,20»l>fVEU(20Q,2C,l), QTOT(200(20il)t
7V&AR(200»20>tYBAR(200,2G),PMS(200i20)iPMO(200t20}»CKfSStO»PSMAXi
8 CPCL(200t20)tROtPZERO,CDECAY».CDIS(200,20)
9 ,X(200>F.Y(20P),IiK,L,M
COM^ON/NUM3/CPHCS(200i20) »GNITI200*20)
M»MINT
WRITE (6,300)
FORMAT (' STEP
WRITE (6t350)
FORMAT (•
CO 100 K^l.NTC
CO 200 L*ltNT
WRITE (6^250) LtKiQTOT(Kf.L*.M)*CPOL (K,L) »CDIS(K*L)
FORMAT (2I5,3F10.2)
CCNTINUE
CONTINUE
RETURN
END
SUBROUTINE OVLAN3
COMMON/DSETS/NGKIF,NUNIFltNUNOVUNUNREl*NUNGWl»NUNRE2*NUNIF2t
1 NUMOV2tNUNOV3iNUNOV4»NUNIF3*NUNGW2,NUNGW3,NOVREl,NOVRE2*NRERElt
2 NRERE2.NOVRE3
RETURN
END.
293
RQUA 68
RCUA 69
RCUA 70
RCUA 71
RGUA 72
RQUA 74
RQUA 75
RQUA 75B
ftCUA 77
RCUA 78
RCUA 79
RQUA BO
RGUA 81
RCUA 82
RGUA 83
WRIT 1
WRIT 2
5
6
CAT FLOW
P LOAD
N03-N LOAO»J
WRIT 13
WRIT 14
(CFS) (LO/DAY) (LB/OAY) ')
WRIT 18
WRIT 19
WRIT 20
WRIT 21
WRIT 22
-------
SUBROUTINE OVLAN'.
CUKlrfON/DSETS/NGV
-------
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c..,
PHIPRV = PREVIOUS VALUE OF PHI, IN FT.
PHINEW = UPDATEC VALUE OF Phi, IN FT.
PHICCR = CORRECTED VALUE OF PHIt IN FT. , AFTER RELAXATION
DELPHI =• DIFFERENCE BETWEEN PRESENT AND PREVIOUS VALUE OF PHI
TOL = ABSOLUTE VALUE OF DELPHI, ABOVE (TQLERENCE)
FRAC=FRACTION OF NODES FOR WHICH AN ERROR LARGER
THAN TOLERANCE VALUE IS ALLOWED
NPER = NO. OF NCCES CORRESPONDING TO FRACTION
PHISYM = SPECIAL VALUE OF PHI DETERMINED FROM
ABOVE
SYMMETRY FORMULA
PHILO = LOWER BCUNO VALUE USED IN EQUIPQTENTI AL CALCULATIONS
PHIHI = UPPER UCUND VALUE OF PHI USED IN EOUl
POTENTIAL CALCULATIONS
REMUREM2 » REMAINDER TERMS USED TO DETERMINE LOCATION OF EQUI POTENTIALS
XORIG * VALUE OF X-CCOKDINATE AT ORIGIN
OHPHI = HORIZONTAL GRADIENT OF PHI
DVPHI = VERTICAL GRADIENT OF PHI
VELH = HORIZONTAL VELOCITY { FT/DAY )
VELV = VERTICAL VELOCITY < FT/DAY )
VELR = MAGNITUDE OF VELOCITY RESULTANT
VH,VV,VR - ABSOLUTE VALUES OF VELH, VELV, VELR
SINA=SIN£ OF ANGLE OF RESULTANT VELOCITY
THEFA*ANGLE OF RESULTANT WITH RESPECT TO POSI
ABOVE
TIVE
HORIZONTAL AXIS, COUNTERCLOCKWISE, IN DEGREES
ANGLE ~ MATRIX USED TO STORE VALUES OF THETA ABOVE
IDIV = NO. OF DIVISIONS ALONG HORIZONTAL
JDI7 = NC. OF DIVISIONS ALONG VERTICAL
XMI.N » MINIMUM VALUE OF ABSCISSA ,IN FT.
XtfAX * MAXIMUM VALUE OF ABSCISSA , IN FT.
YMIN = MINIMUM VALUE OF ORDINATE , IN FT.
YMAX = MAXIMUM VALUE OF ORDINATE , IN FT.
L = fATRIX USED TO STORE VALUES OF KCLASS » ABOVE
U = HORIZONTAL VELOCITY (POSITIVE TO RIGHT)
VY ~ VERTICAL VELOCITY (POSITIVE DOWNWARD)
DCL '. DISPERSION COEF ALONG DIRECTION OF FLOW
CCT * DISPERSION CUEF TRANSVERSE TO DIRECTION
OF FLOW
NUMBERING SYSTEM FOR DISPERSION NODE CLASS MATRIX KCLASS
0 = EXTERIOR
I * LEFT GOUNCARY
2 » LEFT TOP
3 =» LEFT BOTTOM
4 = TOP
5 = DCTTOM
6 = RIGHT TOP
7 = RIGHT BOTTOM
8 = RIGHT BOUNDARY
9 = INTERIOR
10 = KNOWN CONCENTRATION
PHI 7
COK»«CN/ZEROTH/C,CSTAR,CSS., I I, JJ , OT, OX, DY,-DCL,DCT,L,
IA1,A2,A3,.A
-------
c
c
c
c
c
c
c
c
c
c
c
c
c
READ (NUNGW2). HT2
HT2=C.
REAC (NUNGW3). HT3
HT3=0.
READ IN NITROGEN CONCENTRATIONS
REAC (MUNGWl) CM
CN1=0.
PEAC (NUNGW2) CN2
CN2=l.
REAC (NUMGW3) CN3
CN3=0.
REWIND NUNGWI
REWIND NUNGW2
REWINC NUNG/J3
REAC (5,100) NROW,NCOL
WRITE (6, ICO) NROW,NCOL
100 FORMAT (213)
•NCLASS' CODE IS 0 0 FOR
2 FOR AOOVE WATER TABLE,
HEAD.
INTERIOR
3 FOR NO
PHI 20
PHI 21
PHI 22
NODES,I FOR WATER TABLE* PHI 23
FLOW BOUNDARY, 4 FOR KNOWN PHI 24
PHI 25
CO 110 J=a,NROW PH4 26
C PEAC IN FROM BOTTOM TO TOP
REAC (5,120) (NCLASS(I,J> ,I»l,NCOL) PHI 27
120 FORMAT (2011) PHI 29
110 CCNTINUE PHI 30
C WRITE (6,127)
127 FORMAT (• FLOW CLASS MATRIX")
CO 125 J=1,NROW
JJ=NROW-J+1
C WRITE (6*126) ( NCLASSU , JJ ), 1 = 1 tNCOL)
126 FOkMAT (2011)
125 CONTINUE
CO 133 I=1,NCOL PHI 31
F6AD (5,140) (C-ROUNCU UWTABLEU),BOTOM(1,I),BOTOM(2,I)) PHI 32
C WRITE (6*141) (GROUND( I ),.WTABLEU),BQTOK.EQ.l) PHItI,J)=PHI{I»J)+HT1
181 CCNTINUE
296
-------
CO 182 1=^13,14
CG 182 J5.UNROW
IF (NCLASS.C I,J).EQ.1» PH II I, J >*PHI( I, J )+HT2
182 CCNTINUE
CO 183 I=a5,NCQL
CO 183 J=-1,NROW
IF INCLASS(IiJ).EQ.l) PHI(I,J>-PHI(I,J)+HT3
183 CONTINUE
REAC (5,200) OELX,D£LZ,S,ZZERO
C WRITE (6,200) DELX,OELZ,S,ZZERO
200 FORMAT (4F10.2)
REAC (5,210) ITMAX.TOLER,OMEGA
C WRITE (6,211) ITKAX,TOLER',OMEGA
210 FORMAT (15,2F5.4)
211 FORMAT (IX,I5.2F5.2)
C SPECIFY ELEVATION OF LOWER BOUNDARY,BELOW.
READ (5,220) ELECT
C WRITE (6,220) ELBOT
220 FORMAT (F6.1)
PEAC (5,230) NREG
C WRITE (6,230)NREG
230 FORMAT ( ID
C READ J COORDINATES FOR EACH REGION,STARTING WITH
C LOrtER LEFT-HAND CORNER,THEN CLOCKWISE.
CU 240 1=1,NREG
REAC (5, 250) UK I),K1( I ),L2(I),K2(I ) ,L3 < I ) ,K3( I ) ,LM I) ,K4l!) )
C WRIT£(6,250)(LUI),K1(I),L2(I),K2( I),L3(I),K3(I),L4(I),K4LI) )
250 FQRPAT (812)
240 CONTINUE
CO 270 1^1,NREG
REAC (5,260) ( PHI ( LU I ) ,K1( I) ),- PHI (L2 ( I) ,K2( I ) ).
I PhI(L3(I),K3(I>), PHI(L4(I),K4(I)))
C WRITE (6,260) ( PHI(LllI),K1(I)), PHI(L2(1),K2t PHI(L4(I),K4(I)» )
260 FORMAT (4F6.2)
270 CONTINUE '
REAC15.2BO) ( JT ABLE( I), I*1,.NCOL )
CO 290 JROW=1,NROW
REAC (5,285) (KCLASS(I,JROW),I=1,NCOL)
C READ IN FROM TOP TO BOTTOM IN THIS CASE
285 FORfAT (2012)
290 CONTINUE
280 FORMAT (1912)
C.
C
COMPUTE VALUES THAT HILL REMAIN PERMANENT
ALPHA = (CELX/DELZ)»»2
COMPUTE HEAD FOR ALL NODES IN EACH RECTANGULAR REGION
CO 300 M=1,NREG
PHIREG*tPHI(Ull*>t.KllM) )«PHI(L2(M)»K2(K) )+PHI(L3(M) ,K3lM)»
+PHI1L4
PHI(I,J)=PHIREG
CONTINUE
CCNTINUE
CONTINUE
GO TO 320
GO TU 320
GO TO 320
WRITE OUT INITIAL VALUES OF PHI
WRITE (6,325)
325 FORMAT (30X,•INITIAL PHI MATRIX")
CO 330 J»ltNROW
K=NRCH*1-J
PHI 61
PHI 62
PHC 63
Phi 64
PHI 65
PHI 66
PHI 67
PHI 68
PHI 69
PHI 70
PHI 71
PHI 72
PHI 73
PHI 74
PHI 75
PHI 76
PHI 77
PHI 78
PHI 79
PHI 80
PHI 81
PHI 82
PHI 83
PHI 84
PHI 85
PHI 86
PHI 87
PHI 88
PHI 89
PHI 90
PHI 91
PHI 92
PHI 93
PHI 94
PHI 95
PHI 106
PHI107
PHU08
PH1109
PHI110
PHI111
PHI112
PHI113
PH1114
PHI11S
PHU16
PHI117
PHI118
PHI119
PHI120
PHI121
PHI122
PHI123
PHI124
PHI125
PHI126
PH1127
PHI128
PHI129
PHI130
297
-------
c.
c
WRITE (6,340) (PHUIiK) , I=1,NCOL)
340 FORMAT (20F5.1)
330 CONTINUE
"*"sTART*ITERATIVE PROCESS TO FIND PHI VALUES
NNCDES=NCOL«NRQW
CO 400 M=5l,ITMAX
c
c
GO TO 700
GO TO 700
C
C
C
C
C
C
C
540
550
700
500
450
C
C
C
c
c
c,
c
850
800
400
CO 450
CO 500 I*1,NCOL
SCAN TO CETERMINE NOCE CLASS
PHIPRV=PHI
IF (\CLASS(I,J).EQ.4)
TYPc BELOW IS IMTERIOR
IF (NCLASS
-------
980 FORMAT (5X..20I5)
CO 990 J=1,NROW
995
990
WRITE (6,995» ( Ji ( PhK UK ) , 1 =1 ,NCOL ) )
FORMAT ( I5,20F5.11
CONTINUE
CALL EQUIP( I,J,C£LX,CELZ,ELBOT,NROW,NCOL)
C INSERT 'CLEAN-UP1 ROUTINES PRIOR TO PLOTTING
C INSERT PLOTTING ROUTINES
CALL VEL( I,J,DELX,OELZ, NROW.NCOL)
CALL PRECIS ( I, J, DELX.DELZ.NROW.NCOL.ZZERQ)
RETURN
END
SUBROUTINE BOUNC < I, J, ALPHA)
C THIS SUBROUTINE SELECTS THE PROPER NUMERICAL EXPRESSION
C TO BE USED FUR ANY 'BOUNDARY* NODE.
COKMCN/FINIT/NCL*SS<20,20),PERKH120,20) , PERKV(20t 20) ,PHI(20,20> ,
1 JTABLE(20),KCLASS( 20,20)
C POINTS ON BOUNDARY ARE THOSE SATISFYING I=1,I=NCOL
C AND/CR J=l. NOTE J=l HAS 2 CORNER POINTS.
IF. (J.NE.l) GO TO 1000
C FORMULAS BELOW FGR J-t
IF (l.EQ.l) GO TO 1200
IF (I.EQ.NCOL) GO TO 1300
GO TO 1400
12CO Xi\UMER= (2»PERKH I, J )»PHIl 1 + 1, J ) *2*ALPHA»PERKV< I , J) »PHI ( H Jf 1) )
XCcKCK= 12*PERKH I,J)+2«ALPHA»PERKV(I,J))
GO TU 1500
1300 XNUMER=(2»PERKH( I-1,J)»PHI< 1-1 , J ) t,2»ALPHA«PERKV( I » J ) *PHI (IiJ+l>.>
XDENCM= (2«PERKH I-liJ) +! 2*ALPHA*PERK VU ,J ) )
GO TO 1500
1400 XNUMER= (PERKHt I-l»J)«PHU-lfJ)+PERKHUiJ)»PHI
-------
4000
C
C
C
5500
5000
C
C
PHIUiJ)* (PHI(I-ltJ)4PHI( !4ltJ})/2.
GO TO 5000
IF IHHKI*liJ).EQ.O.J GO TO 5500
PERFORM FOR LEFT NODE PKI»0.
PHISYH=IPhIt IfJ*l) + PHU I-1,J-1) ) /2.
FORMULA BELOW GOOD FOR HOMOGENEOUS MEDIUM ONLY.
XNUMER = (8»PHISYM44»PHI( Hl.J) 4. 3» (PHI ( I *J«1) +
1 PHIt I,J-1I I 1
XDcNCJM»18.
PHK I,J)=iXNUMER/XDENOM
GO TC 5000
PERFORM FOR RIGHT NODE PHI'C.
PHISYM - IPHl(I,J4.1)4PHlU4;UJ-l)}/2.
XNUMER» (8*PHISYM+4*PHI( I-1,J) * 3* ( PHI ( I , J*i) +
1 PHI )
XCENOM»lfl.
PHId, Jl-XNUMER/XDENOM
RETURN
ENCr
SUBROUTINE EQUIP ( I , J,DELX, DELZ ,ELBOT,NROU,NCOL)
THIS SUBROUTINE COMPUTES ELEVATIONS (MSL) OF
POINTS OF KNOWN HEAD, 6Y INTERPOLATION, FOR EACH COLUMN
TABL 13
TABL 14
TABL 15
TABL 16
TABL 17
TABL 18
TABL 19
TABL 20
TABL 21
TABL 22
TABL 23
TABL 24
TABL 25
TABL 26
TABL'27
TAOL 28
TABL 29
TABL 30
TABL 31
EGUI 1
EGUI 2
EQUI 3
COMMCN/FINIT/NCLASS( 20, 20 ) , PERKH( 20 » 20 > »PERKV( 20 ,20 ) ,.PHI{20»20) , ECUI 4
C
C
C
105
C
C
400
300
200
100
450
C
1 JTABLEI20),KCLASS(20,20)
SPECIFY LCWER AND UPPER BOUNDS FOR HEAD, BELOW
PHILO»50.
NL*PHILO
PHIHI-125.
NHaPHlHI
SCAN ALL COLUMNS FOR EACH HEAD VALUE
HEKE, INCREMENT IS 5 FT.
WRITE (6,105)
FORMAT (•!',//, 5X, 'LOCATION OF EQUIPOTENTIAL POINTS IN
1 ' )
CO 100 LlNL,NH,5
DO 200 Ul.NCOL
JTAB'JTABLEI IJ-1
CO 300 J^liJTAB
REMUPHldi J)-L
IF (REM1.GE.O.) MM»l
IF (REM1.LT.O.) PM«0
REM2=PHId»J + l)-L
IF (REM2.GE.O.) NN»1
IF (REM2.LT.O.) NN*0
KC ( 1 U v lU U A. hi h4
\&Vi>it^'\^t*
'IF (NSUM.NE.l) GO TO 300
WH£N HEAD VALUE FALLS BETWEEN NODES, INTERPOLATE
TO FIND ELEVATION
RI»ABS(REMi)
R2»ABS(REM2)
ZINTn (Rl«DELZ)/(Rl+R2)
ZL» ELBOT 4 U-1)«OELZ 4 ZINT
IF (NCLASS (I.JJ.EQ.2) GO TO 300
IF(NCLASS(I, J41).£Q.2) GO TO 300
WRlTt*(6, 400 ) L,I»J,ZL
FORMAT (2Xt »PHI»S I3» 'COL"1 , 13, 'ROW*' , 1 3t>*ELEV«* ,F7.2)
CONTINUE
CONTINUE
CONTINUE
WRITE (6,450)
FORMAT ( '1', 20X, 'INTERPOLATIONS IN THE HORIZONTAL')
SCAN ALL ROWS FOR EACH HEAD VALUE
XQRIG'O.
ITMAX-NCOL-l
CO 500 L*?NL,NHf 5
CO 600 JMtNROW
CO 700 IM,ITMAX
REM.la*PHI ( It J )-L
ECUI 5
EGUI 6
EQUI 7
EGUI 8
EGUI 9
EGUI 10
EQUI 11
EQUI 12
EGUI 13
GRID SYSTEMEGUI 14
EQUI 15
EQUI 16
EGUI 17
EGUI 18
EQUI 19
EQUI 20
EGUI 21
EGUI 22
EGUI 23
EQUI 24
ECUI 25
EQUI 26
EGUI 27
EQUI 28
EGUI 29
EGUI 30
EGUI 31
EQUI 32
EGUI 33
ECUI 34
EGUI 35
EQUI 36
EQUI 37
EQUI 38
EQUI 39
EQUI 40
EGUI 41
EGUI 42
EGUI 43
EQUI 44
EGUI 45
EGUI 46
EQUI 47
EQUI 48
EQUI 49
300
-------
c
c
INTERPOLATE
800
700
600
500
C
C
50
IF (REM1.GE.O. MM»l
IF (REMl.LT.O. MM.-0
REM2-PHI I I+:i,J -L
IF (REM2.CE.O. NN«1
IF IREM2.LT.O. NN-0
NSUM-HM*NN
IF (NSUM.NE.l) GO TO 700
WHEN HEAD VALUE FALLS BETWEEN NODES,
TO FIND HORIZONTAL POSITION.
Rl»ABSIREMl>
R2-ADSIREM2)
XINTMR1«OELX)/(R1*R2)
XL-XCRIG+:i I-1)*CELX+XINT
IF INCLASSiI,J).EQ.2) GO TO 700
IF (NCLASS11+1,J).EQ.2) GO TO 700
WRITE (6,800) LiI.J.XL
FORMAT (2X,.«PHI-»,.I3,ICOL-I,I3» 'ROW*1,1 3r«X*» »F8.1)
CONTINUE
CONTINUE
CONTINUE
RETURN
END
SUBROUTINE VEL ( I,J,CELX^DELZ.NROW.NCOL)
THIS SUBROUTINE COMPUTES THE VELOCITY VECTOR AT
EACH NODE IN THE GR 1C SYSTEM (EXCEPT AT THE BOUNDARY NODESk)
COKJ"!ON/PINIT/NCLAS$(20t20}fPERKH(20,20),PERKV(20i20)»PHI<20,20)
1 JTAELE(20),KCLASS120,20)
COMMGN/VELOZ/VELH(20,20)».VELV<2C!,20UVELR(20»20),ANGLE120»'20)
COPMCN/ZERaTh/C,C$TARfCSS,Il,JJ,DT,.DX»DYf.DCL*DCT,L,
lAl,A2,A3,A4,BltB2».83,KFX,.CFX,KLX>CLX,KFY,CFY1KLY,CLY»
2U»VY,ZPiGPiGPC»
3XMIN,XMAX,IDIV,YMIN,YMAX,JOIV
DIMENSION CI20,20 ) ,CSTAR(20,20),CSS(20,20»«U(20,20),VYl20*20)
DIMENSION L(20,20)
INITIALIZE VELOCITIES TO ZERO
CO 50 I»1»NCCL
DO '50 J-l.NROW
VELH(I,JJ»0.
VELV(I,.J)»0.
CONTINUE
DO 100 t-l.NCOL
CO 200 Jil.NROW
'STAR* OUT ALL NON-INTERIOR NODES.
IF (NCLASS(I,J).EQ*2) VELVlI,J)«9$99999.
VELHU,J>«9999999.
VELRl UJ) "9999999.
ANGL£(I,J)-9959999.
GO TO 200
(NCLASSIItJ).E0.2)
(NCLASSt I»J).ECJ.2}
(NCLASSIIiJ).EQ.2>
[NCLASSd, JJ.EQ.2.)
IF
IF
IF
IFI
SET HORIZONTAL AND VERTICAL VELOCITIES AT ZERO AT NO-FLOW
IF (NCLASSI I,J).EQ.3) VEIVU,J)»0.
EGUI
EOUI
EQUI
EQUI
EGUI
EQUI
EQUI
EQUI
EQUI
EQUI
EQUI
EQUI
EQUI
EQUI
EQUI
EQUI
EQUI
EQUI
ECUI
EQUI
EQUI
EGUI
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
IF
IF
IF
IF
(NCLASSII,J).EQ.3)
(NCLASSIIiJ).EQ.3)
(NCLASSII,J).EU.3)
(NCLASSII,J).EQ,3)
C
C
150
VELH(I,J)*0.
VELR(I,J}-0.
ANGLEU, J)»0»
GO TO 200'
JRQW-NROW+1-J
IF (KCLASS(I,JRCW).NE.9> GO TO 150
USc 2-POINT FORMULA FOR NUMERICAL DIFFERENTIATION
FIND PHI GRADIENT IN HORIZ..VERTICAL
CHPhI — -PHI(IfJ-m/f2.*DELZ)
COMPUTE VELOCITIES IN FT/DAY
VELHII,J) • PERKHII,J)*0.134*DHPHI
VELV(I,J) * PERKV(I,J)*0.134*DVPHI
GO TO 180
DETERMINE HORIZONTAL GRADIENTS
IF (KCLASSII,JRCW).EQ.2) DH-0.
IF iKCLASStI,JROW).EQ.A) DH-0.
IF (KCLASSII»JROW).EQ.6) -DH-0.
80UNDARYVEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
VEL
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
1
2
3
4
5
6
8
9
10
11
12
12A
12B
12C
120
12E
12F
13
14
15
16
17
18
19
20
20 A
20B
20C
200
20E
20F
21
22
23
24
25
26
27
28
29
30
301
-------
c
c
IF
IF
IF
IF
IF
FOR
(KCLASSt I,JROW).ECU1)
(KCLASSt I» JROWJ..EQ.3)
(KCLASSII,JRCW).EQ.7)
(KCLASS1I,JRCW).EQ.8)
(KCLASS(ItJROW).6a.5)
DH=-(PHI< I+:i,J) - PHI (UJn/DELX
DH*-(PHI
KCLASSI It JRCWJ.EQ.4)
KCLASSt I,JROW).EQ.6)
FOR POLLUTANT INPUT POIN
CALCULATIONS IN 'GWQUAL1
300
400
450
700
500
600
200
100
800
850
910
VH=VcLH(I,JJ
VH= ABS(VH)
VV = VELV( UJ)
VV= AES(VV)
VR = VELR(I»J)
VR = ABS(VR)
IF (VR.LT.0.1E-12) VR=Q.1E-12
SINA = VV/VR
A = ARSIN(SINA)»180./3,1416
START SEPARATING INTO QUADRANTS
NDUNMY = MNSUM-1.
IF(NCUMMY) 300,400t500
THETA = 180. + A
3RD CiUADRANT
GO TO 600
IF(VELV< ItJI.LT.O.) GO TO 450
CISTINGUISH BETWEEN 2ND AND 4TH QUADRANTt- ABOVE
THETA = 180. - A
GO TC 700
THETA i 360. - A
GO TC 600
THETA = A
1ST GUADRANT
ANGLEtItJ) = THETA
CONTINUE
CONTINUE
WRITE OUT VELOCITY AND ANGLE
WRITE (6,800)
FORMAT (•!•,//,30X,'VELOCITY -FT/DAY - AND ANGLE - DEC - MATR1XM
WRITE (6,850) tIfI=l,NCOL)
FORMAT (2016)
CO 900 J=1,NROW
K=NRCW+1-J
WRITE (6..910) tVELR(l,K),l = l,NCOL)
FORMAT (20F6.U
WRITE (6,920) (ANGLE!I,K),I=1,NCOL)
VEL 34
VEL 35
VEL 36
VEL 37
VEL 38
VGL 39
VEL 40
VEL 41
VEL 42
VEL 43
VEL 44
VEL 45
VEL 46
VEL 47
VEL 48
VEL 49
VEL 50
VEL 51
VEL 52
VEL 53
VEL 54
VEL 56
VEL 57
VEL 58
VEL 59
VEL 60
VEL 61
VEL 62
VEL 63
VEL 64
VEL 65
VEL 66
VEL 67
VEL 68
VEL 69
VEL 70
VEL 71
VEL 72
VEL 73
VEL 74
VEL 75
VEL 76
VEL 77
VEL 78
VEL 79
VEL 80
VEL 81
VEL 82
VEL 83
VEL 84
302
-------
920 FORMAT (20F6.1,/M
900 CONTINUE
RETURN
ENU
SUBROUTINE PRECIS (I,J.DELX,DELZ,NROW,NCQL,ZZERO)
c THIS SUBROUTINE PREPARES THE INPUT INTO THE
C GROUNCWATER DISPERSION MODEL
COyMCN/FINIT/NCLASS(20,20),PERKHl20t20>,PERKV(20,20),PHI(20,20),
1 JTABLE(20),KCLASS(20,2C)
COMMCN/VELOZ/VELM20,20j,.V£LV(2Q*20),.VELRt20,20) ,ANGLE 1201'20>
COKMCN/ZEROTH/C,CSTAR,CSS,II,JJ,DT,DX,DY,OCL,OCT,L,
1AI,A2,A3,A4, 61,e2,83,KFX,.CFX,,KLXf.CLX,KFY,CFYf,KLY,CLY,.
2U,VY,ZP,GP,GPC,
3XMIN,.XMAX,IDIV,YKIN»YMAX,.JDIV
COMMON/DSETS/NGWIF,NUNIF1,NLNOV1,,NUNRE1,NUNGW1,NUNRE2,NUNIF2,
1 NUNCV2,NUNOV3,NUNOV4,NUNIF3,NUNGW2fNUNGW3^NOVREl^NOVRE2iNREREl,
2 NRERE2.NOVRE3
DIMENSION C(20,20),CSTAR(20,20),CSS(20,20),U(20,20) ,VY120«0)
DIMENSION L(20,20)
DIMENSION PERMH(20,20),PERMV(20,20)
C
C SET INDEXES FOR ROWS AND COLUMNS
750
700
C
C
C.
C
850
800
JJ=NROW
SET COUNCARIES IN BOTH DIMENSIONS
XMIN»0.
XMAXsCELXMNCOL-1)
YMIN=ZZ£RO
YMAX=ZZERO+OELZ»(NROW-1 )
TRANSFER NODE INFORMATION TO K-MATRIX
CO 700 I=*1,NCOL
CO 750 JROW=l»NROW
L( I,JROW)=»KCLASS( It JROWl
CONTINUE
CONTINUE
SET UP ARRAYS FCR VERTICAL AND HORIZONTAL
VELOCITIES
CO 300 MUNCOL '
CO 850 JROW»1,NROW
J=NRLW+l-JRQW
UUtJROW)»VELHI I»J>/864CO.
VY(I>JROW)=-VELVII,J)/86400.
PERMH( I, JRQWI*PERKH( J, J )
PERMVtlf JROW)»PERKV( I,J)
CONTINUE
CONTINUE
WRITE OUT DATA FOR INTERFACE WITH DISPERSION MODEL,
WRITE (6,900)
900 FORMAT ('1*,20X,•INTERFACE DATA LISTING1)
WRITE (6,910) IltJJ
910 FORMAT (215)
WRITE (6,920)
920 FORMAT <20X,'VERTICAL VELOCITIES IN FT/SEC1)
DO 925 JROW»1,NROW
WRITE 16 ,.930) ( VY( I, JROK ),, I»l,,NCOL)
930 FORMAT (10E8.2)
925 CONTINUE
WRITE (6,940)
940 FORMAT (20X,'HORIZONTAL VELOCITIES IN FT/SEC•)
CO 945 JRQW»ltNROW
WRITE (6,950) (U(I,JROW),I«1,NCOL)
950 FORMAT uoE8.2)
945 CONTINUE
WRITE (6,960)
960 FORMAT ('1«,20X,'DISPERSION NODE CLASS MATRIX*)
VEL 85
VEL 86
VEL 87
VEL 88
PREO 1
PRED
PREO
PRED
PREO
PREO
PREO 8
PR£0 9
PREO 10
PRED 11
PRED 12
PREO 13
PRED 14
PRED 15
PREO 16
PREO 17
PRED 18
PRED 19
PREO 20
PREO 21
PRED 22
PRED 23
PREO 24
PRED 25
PRED 26
PREO 27
PREO 28
PRED 29
PREO 30
PREO 31
PREO 32
PREO 33
PRED 34
PRED 35
PRED 36
PRED 37
PRED 38
PREO 39
PREO 40
PRED 41
PREO 42
PRED 43
PREO 44
PRED 45
PRED 46
PREO 47
PRED 48
PftED 49
PREO 50
PRED 51
PREO 52
PftEO 53
PREO 54
PREO 55
PRED 56
PR60 57
PREO 56
303
-------
c
c
c
c
c
970
965
1001
CO 965 JROW=l,NROW
WRITE (6,970) ( L( I , JROW ),.!»!, NCOL )
FORMAT (2012)
CONTINUE
WRITE INTERFACE INFORMATION ON
WRITE (7..10Q1) IDIV,XMIN,XMAX
FORMAT (I5,2F10.2)
WRITE (7,1001) JDIV.YMIN.YMAX
CO 599 J=.1,JJ
WRITE (7..651) (HI, J), 1 = 1,11)
CARDS
651
599
602
601
606
605
612
611
616
615
(U(I,J), 1=1,11)
(U(
1=1, II)
(VY( I,J), 1 = 1,11)
(VY(I,J), 1-1,1!)
WRITE (NGWIF)
FORMAT (4012)
CONTINUE
CO 601 J=.1,JJ
WRITE (7*602)
WRITE (NGWIF)
FORMAT (10ES.2)
CCNTINUE
CO 605 J=1,JJ
WRITE (7,606)
WRITE (NGWIF)
FORMAT (10E3.2)
CONTINUE
CO 611 J=1,JJ
WRITE (7,612) (PERMHlI,J),1=1,II)
WRITE (NGWIF) (PERMh(I,J).tI»if II)
FORMAT (10E8.2)
CCNTINUE
CO 615 J=1,JJ
WHITE (7..616) IPERMV( I.J),1 = 1,II)
WRITE (iJGWIF) (PERMV( UJ),1-1,11)
FORMAT (10E8.2)
CCNTINUE
ENC FILE NGWIF
REWIND NGWIF
RETURN
ENC
SUEROUTINE GWQUAL
C PRGGKAM GWQUAL.
C COMPUTES CONCENTRATIONS OF A POLLUTANT AT ALL NODES
C IN A GRIC SYSTEM.
C« DEFINITION OF SYMBOLS
C* A1-A4 WEIGHTING COEFFICIENTS ON THE CONVECTIVE TERMS
C« Bl-83 WEIGHTING COEFFICIENTS ON THE TIME TERMS
C» CFX BOUNDARY VALUE OF CONCENTRATION
C» CFY BOUNDARY VALUE OF CONCENTRATION
C* CLX BOUNDARY VALUE OF CONCENTRATION
C» CLY BOUNDARY VALUE OF CONCENTRATION
C» GGUIT REFERENCE VALUE TO ENC CALCULATIONS
C» CCL*CIFFUSION COEFFICIENT IN THE X DIRECTION
C* DCT=DIFFUSION COEFFICIENT IN THE Y DIRECTION
C« CT=VALUE OF THE TIME STEP
C« CTICE=T1ME STEP BETWEEN DIFFERENT TIDAL VELOCITIES
C« CTMAX=MAXIMUP VALUE OF CT
C» DTMIN=MINIMUM VALUE OF CT
C« CTSTEP«MULTIPLIER FOR SUCCESSIVE TIME STEPS
C« CX=GRID SPACE INTERVAL IN THE X DIRECTION
C» CY»GRID SPACE INTERVAL IN THE Y DIRECTION
C» IDIV=.NUMBER OF GRID SPACES IN THE X DIRECTION
C» JDIV=NUMBER OF GRID SPACES IN THE Y DIRECTION
C» II=NUMBER OF GRIC POINTS IN THE X DIRECTION
C» JJ=NUMBER OF GRID POINTS IN THE Y DIRECTION
C» ICOUNT*COUNTER FOR. THE NUMBER OF TIME STEPS
C» IFREC=FRECUENCY FOR READING NEW TAPE VELOCITIES
C» ITICE*REFERENCE PARAMETER BETWEEN MAIN AND VCALC
C» JCUIT,JQUIT=SUBSCRIPTS FOR THE REFERENCE POINT
PRED 59
PRED 60
PRED 61
PRED 62
PRED 63
PRED 64
PRED 65
PRED 66
PRED 67
PRED 69A
PRED 70
PRED 71
PRED 71A
PRED 72
PRED 73
PRED 74
PRED 75
PRED 76
PRED 77
PRED 78
PRED 79
PRED 80
PRED 81
304
-------
ISET»JSST=COURDINATES FOR POINTS OF INPUT ON BOUNDARY*
JMIN»JMAX=LIMITING POINTS FOR INPUT ON THE BOUNDARY
KIN*INTEGER SPECIFYING THE TYPE OF INPUT
SPECIFYING
SPECIFYING
SPECIFYING
SPECIFYING
KFX=INT6GER
KFY=INTEGER
KLX=INTEGER
KLY= INTEGER
NHR=INTEGER
THE
THE
THE
THE
THE
E.G.
B.C.
B.C.
B.C.
TIME
ON
ON
ON
ON
FOR
THE
THE
THE
THE
LEFT BOUNDARY
TOP BOUNDARY
RT. BOUNDARY
BOT. BOUNDARY
PRINTING
TRI-DIAGONAL MATRIX
MATRIX
MATRIX
ALGORITHM
SPECIFYING
?VN=NUMBER OF TIME STEPS
BETAXtbETAY*CQEFFlCIENTS IN THE
GX,GY*C06FFICIENTS IN THE TRI-DIAGONAL
ZX,ZY=COEFFICIENTS IN Tt-.E TRI-OIAGCJNAL
ZP,GP,GPC»VARIABLES USEC IN THE THOMAS
DEL1-DEL10=R.H.S. GOEFFICIENTS
R01-RC3*R.H.S. COEFFICIENTS
P1-P3,01-03 »Sl-S3*R.t-.S. COEFFICIENTS
TICE=;TIME TO CALL UP A NEW TIDAL VELOCITY FIELD
TINPUT=LENGTH OF TIME OF INPUT
(j=V:LOCITY FIELC IN THE X DIRECTION
VY=veLCCITY IN THE Y DIRECTION
V£L=NAME OF THE TIDAL VELOCITY FIELD ON TAPE
LtI,J)= SPECIFIES THE GEOMETRIC CONFIGURATION
CU,J)=RESULTING COMCENTRAT ION FIELD
CSTAR(ltJ>»!NTERMEDIATe CONCENTRATION FIELD
CSSf I,J)=INTERMECIATE CONCENTRATION FIELD
C*
C»
C»
C*
C*
C*
C*
C»
C*
C*
C*
C*
C*
C*
C*
C*
c»
C*
C*
C*
c»
c»
C*
C*
C —
C
C
C
C
C
C
C
C
C
C
C
C
C —
c «****« **««««*•******»<
DIMENSION C(23,21),CSTARt23,21),CSS(23».21l»U(46.34) ,VYU6«34>
CI^ENSION L(23,21),VELU9,34)
DIMENSION GX(23J,ZX(23J
DIMENSION GY(2l I,ZY(211
DIMENSION PERMH(20,2QJ,PERMV(20,20)
DIMENSION DCLt20,20)tDCT(20,20)
DIMENSION TITLE(IO)
COMfCN/ZEROTH/C,CSTAR,CSS»II,JJ,OT,DX,DY»DCL,.DCT,L»
lAl,A2,A3f A4,Bl,a2,B3,KFXt.CFX,KLX,CLX,KFYiCFY,KLY,CLYi.
2UtVY,2P,GP,GPC,IFRSQ,ICOUNT,VELi
3XMlNrXMAXiIOIV,YMIN,YMAX,JDlV
COMMCN/FIRSr/ZX,BET4X,GXhDELl,DEL2fDEL3iDEL^,0£L5
COMMCN/SECONC/ZY,B€TAY,.GY,DEL6,DEL7tDEL8»UEL9,OEL10tROl»R02,R03
CUMMO.N/THrRO/PltP2,P3,P<»»P5,Ql,Q2«.Q3,Sl»S2»S3
COMMCN/CN/CNltCN2tCN3
COfyON/DSETS/NGWlF,NUNlFl,NUNOVl,NUNREl,NUNGHl«NUNRE2»NUNIF2»
1 NU,NOV2f KUNaV3,N'L'NOV4,NUNIF3,NUNGW2rNUNGV«3»NOVREltNUVRe2»NREREl,
2 NRERE2»NOVRE3
NUMBERING SYSTEM FOR DISPERSION NODE CLASS MATRIX KCLASS
0 = EXTERIOR
1 = LEFT BOUNDARY
2 = LEf-T TOP
3 = LEFT BOTTOM
-------
WRITE (6,2000) XMAX.DX
2000 FORMAT (' XMAX='.F10.1,5X,' DX *'»F10«AI
C GRID SIZc IN THE Y CIRECTION
READ 15,1000) JCIV,YMIN,YMAX,DY
WRITE (6r2001) YKAX.OY
2001 FORMAT (' YMAX=',.F10.4,5X, ' DY »',F10.4)
C
C
SET BOUNCARY CONDITIONS
REAC (5,2003) KFX,CFX,KLX,CLX
READ (5,2003) KFV,CFY,KLY,CLY
2003 FORMAT (2(13,F7.2))
WRITE (6,2004) KFX,CFX,KLX,CLX
2004 FORMAT!' KFX=',I3,.' CFX=',F7.2,' KLX = ',I3,« CLX=',F7.2)
WRITE (6,2035) KFY,CFY,KLY,CLY
2005 FORMAT (• KFY=',I3,« CFY»«,F7.2t' KLY*',I3,*
CONNECTIVE TERM WEIGHTING COEFFICIENTS*
REAC(5,1001) A1,A2,A3,A4
1001 FORMAT (5F10.5)
WRITE (6,2006) A1,A2,A3,A4
C
C
2006 FORMAT!' Al=l,F10.5,' A2*.«t-F10.!
READ (5,103) IFREO.TMAX
103 FORMAT!I3tF10.5)
UMTS, OF TMAX ARE DAYS
A3=' ,F10.5 , «
TERMINATION OF CALCULATION
CCUIT» IQUIT.JQUIT
C CONCENTRATION FOR
REAC (5,1004)
C
C TIME STEP SECUENCE NO. OF STEPS, Mlivu MAXt STEP SIZE
REAC (5,1000) NN,DTMIN,.DTMAX,OTSTEP
WRITE (6,7008) NN,DTMIN,DTMAX,DTSTEP
2008 FORMAT I* ',15,' TIME STEPS',' OTMIN* »,F10.4,' OTMAX* f,F10.4t
1' CTSTEP= SF10.4) •-
C UNITS GF OTMIN AND CTMAX MUST BE CONVERTED FROM HOURS TO SEC
CTMIN=DTMIN»360Q.
LTMAX=DTMAX*3600.
C
C
INPUT CONCENTRATION (TYPE,FROM JMIN TO JMAX)
REAC (5,1203) KIN,JMIN,JMAX,ISET^JSETtTINPUT
; UNITS CF TINPUT ARE CAYS
1200 FORMAT (5I3.F10.5)
fcRITE (6,1500) ISET,JMIN,.JMAX
1500 FORMAT (' INPUT AT I«»,I3,' BETWEEN J««,I3,' TO J»»tI3)
WRITE(6,1501) TINPUT
1501 FORMATC TINPUT* ', F10.2 )
1 SPACE INCREMENTS
DIVJaJDIV
CWI=It)IV
; NUMBER CF GRID POINTS IN X DIRECTION
II=ICIV-H
; NUMBER OF GRID POINTS IN THE Y CIRECTION
JJ=JCIV+1
WRITE (6,2007) II,JJ
2007 FORMAT (« II=',I3,« JJ*',I3>
C
C
REAC AND WRITE THE L ARRAY
WRITE16.85)
85 FORMAT!/' L(I,J) ARRAY FOLLOWS')
IF (11-21) 90,90,.95
90 CO 92 J=l,JJ
91 READ (5,98)!L(I,J),1=1,II)
91 REAC (NGWIF) ( L ( I ,-J ), 1 = 1, II )
92 WRITE 16..99) (L
GO TO 89
95 CO 97 1=1,11
306
-------
96 READ 15,98) (L( IRGV , J ) ,-J*!, JJ >
97 WRITEl6t99) ( Ll IREV, J 1 ,.J = 1» J J )
98 FORNAT UOI2)
99 FORMAT « A« ',4012)
89 CONTINUE
C
C INITIALIZE ALL B.C. TO ZERO
CO 100 I=5li II
DO 100 J-=UiJJ
CH,J)=0.
CSS( I,JJ=50.
100 CONTINUE
CSTART=0.1E-30
C TEMPORARILY, USE NON-ZERO VALUES TO AVOID UNDERFLOWS
CO 150 Inltll
00 150 J=.1,JJ
IF (LI UJJ.EO.O) GO TO 150
C( I, J)=CSTART
CSTARtlf J)»CSTAftT
CSS( I,J)*CSTART
150 CONTINUE
C
C SET THE VELOCITY FIELD
DO 101 I'l.H
CO 101 J^ltJJ
U(I,J)=0.
VYI I, J)=0.
101 CONTINUE
C
C FIRST TIKE STEP
CT=DTHIN
C
C SET CONSTANT VALUES
110 T*0.
ICOUNT»0
TIUE=0.
CTICE-600*IFREQ
CT=CT/OTSTEP
CONVERT TMAX AND TINPUT FROM DAYS TO SECONDS
TMAX=TMAX»86<»00.
TINPUT=TINPUT*86400.
B2-0.1667
B3=B2
C INSTEAD OF USING VCALC,- REAO VELOCITY FIELD BELOW
C VELOCITIES ARE IN (FT/SEC)
DO 601 J=.ifJJ
C READ 15,602) 606) ( VYU , J > i 1 = 1, II)
READ (NGWIF) ( VYI I » J ) i I=l« II )
606 FORMAT MOES.2)
605 CONTINUE
C READ IN hORIZCNTAL AND VERTICAL PERMEABILITIES
CO 611 J=il»JJ
C REAO{5,612» ( PERMH( I » J ) , 1 = 1 , 1 1 }
READ (NGW1FMPERMH1 I,JJ,I = 1, JI)
612 FORMAT ( 10E8.21
611 CONTINUE
00 615 J«,1,JJ
C REAO (5,6161 ( PERMV( I, J J rI«UII )
READ(NGWIF) ( PERMV( I , J ) , 1*1, U )
307
-------
616 FORMAT (1068.2)
615 CONTINUE
REHlND NGWIF
C
C COMPUTE DISPERSION COEFFICIENTS
C BASED ON HOOPES AND HARLEMAN FORMULAS
C FORMULATION BELOW COMPUTES COEFFICIENTS IN ( SQ CM/SEC)
VISKIN-O.C0001040
C VISKIN ABOVE IN (SQ FT/SEC), FOR T6MP 71.5 F
C CONVcRT TO (SQ CM/ScC) BELOW
VISKIN»VISKIN*0.001Q75
C BELCW CONVERT PERMEABILITY IN (GPO/SQ FT) TO ISO CM)
C ALSO* CONVERT VELOCITY FROM (FT/SEC) TO (CM/SEC)
ALPHA3»88.
BETA3-1.2
DO 350 I=U,II
CO 350 J^ltJJ
C USE ABSOLUTE VALUES UF VELOCITY
UABS=;ABS(U< I, J) )*30.S
VAUS = ABS(VY{ I,J) )*30.5
Ph=PeRMH(I,J)«0.542E-9
PV=PERMV( I,J)»0.542E-9
TERM1=VISKIN*ALPHA3
TER^2=IUABS»(PH»»0.5M/VISKIN
CCLU,J)aTERMl»(TERM2«*BETA3)
TERM3«(VABS»(PV»*0.5))/VISKIN
DCTU,J)=TERM1»(TERM3»»BETA3)
350 CONTINUE
C CONVERT CCL AND OCT FROM ( SQ CM/SEC) TO I SQ FT/SEC)
CO 360 Mlt II
DO 360 J»1,JJ
DCLU,J)=DCL< I* J)»0. 001075
CCT(I,J)=»DCT(I,J)*0.001C75
360 CONTINUE
C
C BEGIN TIPE 00 LOOP
C
400 CO 530 M4l,NN
DARG^DT
IF (CARG-CTMAX) 430i4ZOi420
430 CONTINUE
C OEGIN TIKE STEPPING SEQUENCE
IF(M-3) 402i402»403
402 DT-f«3630.
GO TC 419
403 IF (M-8) 404t405»40S
404 CT»DT«DTST6P
GO TO 419
405 IF(M-ll) 406,407,408
406 CT»48.f3600.
GO TO 419
407 CT-120.«3600.
GO TO 419
408 IFIM-13) 404,409,420
409 DT-DTMAX
GO TO 419
420 CT-OTMAX
419 CONTINUE
C
C CHECK FCR SINGLE OR COUBLE INPUT
C STATEMENTS BELOW EXCLUDED TEMPORARILY
C IF (KIN. EC. 1) GO TO 433
C T1-43200.-OT
C T2-43200.+TINPUT
C IF (T.LE.T2.AND.T.GE.T1) GO TO 435
433 CONTINUE
IF .(T-TINPUT) 435,435,437
308
-------
435 CONTINUE
C CALL SUB. CINPUT TO SET INPtT VALUES
CALL CINPUTUWIN,JMAX,ISET*JSET,KINfTiTINPUT)
437 CONTINUE
ICCUNT-ICOUNT*!
T*T+DT
C
C SAVE C(N) BY STORING IN CSS
DO 440 Itl*II
00 440 J^l.JJ
440 CSSII,J)* C( ItJ)
C
/• — — M.^•••^••B—«•* »»"•*•*• ^•••» — **"™**^—™*—<—> »»^^*^»"«
C BYPASS USAGE INSTRUCTIONS FOR SUBROUTINE VCALC
C SET UP MEW VEL FIELC AT TIDE INCREMENTS
C IF UCOUNT.EC.l) GO TO 700
C IF (T.EU.TIOE) CO TO 700
C GO TO 800
C 700 CONTINUE
C CALL VCALC (ITICE)
C CONTINUE
C 800 CONTINUE
C THREE EQUATION SCHEME FOR SOLUTION
C SOLVE THE.X EQN. FOR CSTAR
CALL TRIX
445 CCNTINUE
C RESTCRE C TO C«N) FROM CSS
CO 480 lilt II
CO 480 JiltJJ
480 C(I«J)*CSS(IrJ)
C SCLVE THE Y EQN. FOR CSS
CALL TRIY
CONTINUE
C SCLVE EXPLICITLY FOR CtN+11
CALL EXPLIC
439 CCNTINUE
C PRINT THE CONCENTRATION CIST. IF T EQUALS TIDE
C IF (ICOUNT.EQ.il GO TO 250
C IF (T.EQ.TIOE) GO TO 250
C GO TO 285
C 250 CONTINUE
C IF HITICE/NHR)«NHR.NE.IT10£) GO TO 275
C PRINT OUT RESULTS AT ALL TIKE STEPS
CALL OUTPUT!T,DTMAX)
275 CONTINUE
ITICE-ITICE*l
TIDE-TIDE+OTIDE
285 CCNTINUE
IF(T-TMAX) 520r540,540
520 IF (CUQUIT,JQUIT).-CQ.um 530i540f540
530 CONTINUE
540 CONTINUE
RETURN
END
SUBROUTINE VCALCtITICE)
C SUBROUTINE VCALC
DIMENSION C(23,21),CSTAR<23i21).CSS(23»2I)tUl46«34»,VY(46rf34l
DIMENSION L(23i2DtVEL(49,34)
DIMENSION DCL<20,20)»OCT<20,20)
COI"MCN/ZEROTH/C,CSTARtCSSrIItJJtDT*OXtDY»OCLtDCT»l»
lAl,A2,A3,A4,Bl,B2iB3»KFXiCFX,KLX»CLXfKFYrCFY,KLYiCLY>
2UrVY^ZP,GP,GPC,IFREQ,ICOUNTiVEL
C REAO IN VELOCITY FIELD FROM TAPE
IF
-------
CO 18 NFREQ=1, IFREQ
18 REAC (11) VEL
IMINC = (72-IS«m/iFR£Q
GO TO 45
21 CONTINUE
IF (ITIOE.EU.IW1N01 GO TO 25
DO 23 NFREQ=1, IFREQ
23 REAC (11) V6L
GO TO 45
25 REWIND 11
CO 27 NFREQ=1, IFREQ
27 REAC (11 ). VEL
IWIND=IWIND-K72/IFREQ)
45 CONTINUE
18=45
i=ra
JR=33
JJM1=JR-1
C TRANSFER VELOCITY FIELD FROM VEL TO U AND VY
C STORE U AND VY FOR CCD-EVEN AND EVEN-ODD POINTS
5 CO 10 J=i2tJJMl»2
U( IT J)=VEL( I, J)
VYU fJ)=VEH I, J41)
10 CONTINUE
1 = 1-1
IF U) 50,50,15
15 DO 20 J=U,JR,2
UU,JJ=V£L( I, J>
VY( UJ) = VEL( I,J + ir
20 CONTINUE
1 = 1-1
GO TC 5
50 CCNTINUE
C CALC FOR OCD-ODO PUINTS
I=IB
40 CO 190 J=vl,JR,2
C K CHECK FOR U AND VY
IF (K-9) 55, 75^ 110
55 IF (K-l) 100.110,. 60
60 IF (K-3) 120,130,65
65 IF 1K-6) 145,100, 70
70 IF (K-8) 100,110, 75
75 U(M,N»= .25«(Ul I-1^J)+:U( I + l,4)fU( I, J-l )+U{ 1 1 J+l
VY(^^N) = .25»(VY( 1-1, J)4iVY(I-H,J)*:'VY(
GO TC 190
100 UIM,N)»0.
VY(M^N)=0.
GO TC 190
110 U(MfN)».5*lUU,J-U)+U
-------
GO TC 190
170 U(I*,N) =
190 CONTINUE
1=1-2
IF (1-1) 350,40,40
350 CCNTINUE
CO 700 M*2,46,2
DO TOO N=i2,34i2
J = N/2
U(l,J>=U(M,N)
VY< UJ> = VYIM,N)
700 CONTINUE
WRITE CUT VELOCITY FIELD
NHR=L8/IFREQ
IF (UTICE/NHRMNHR.NE.ITIOE) GO TO 500
WKITE (6,410) TIME
VjRITL 16,420)
CO 400 IR*l» II
WRITE (6,401) I,
-------
SM=2.
GO TO 100
70 CONTINUE
40 JM1=J-1
SIGM=1.
SM = 0.
IF (JJ-J) 55V55.A2
TOP CR BCTTOM
42 IF (K-9) 45,100,100
45 IF (K-i) 20,100,46
46 IF (K-3) 50,55,47
47 IF(K-5) 50,55,48
48 IF (K-7) 50,55,100
TOP CONDITION
50 JM1=J+1
GO TC (30,60,70),.KFY
GO TO 100
BOTTOM CONDITION
55 JP1=J-1
GO TO (110,115,120) ,KLY
110 SIGP=1.
SP=Q.
GO TC 141
115 SIGP=-1.
SP=2.
GO TO 141
120 CONTINUE
100 JPl=J+i
SP=0.
141 CONTINUE
CALL COEFX( I,J)
IF (K.EU.10) GO TO 191
IF (I-M) 425,425r375
375 IF (V(I-1» -<»00,425,400
C LEFT OR RIGHT
400 IF (K-l) 450,425,401
401 IF (K-10) 402,191,425
402 IF (K-9) 403*450,191
403 IF (K-^) 425,450.,404
404 IF (K-6) 450,475,475
C LEFT BOUNDARY CONDITION
425 GO TO (430,431,432)tKFX
430 CIMU=C( I+1,J)
CIP1J»C( I+1,J)
GO TO 485
431 CItUJa-C
CIP1J = C( I+1,J)
V(I)= BETAX*2.#ZX( I)
GO TO 485
432 CONTINUE
C R+GHT BOUNDARY CONDITION
475 GO TO (480,481, 482), KLX
480 CIP1J=C< I-l.J)
CIM1J*C< I-l.J)
V(I)= EETAX-(ZP»GP)/V(I-l)
GO TO 485
481 CIP1J=-CII-1, J)+2.*CIUJ)
CIM1J=»C( I-1,J)
V(I)= BETAX + 2.*GX(n-(ZP«GP)/V(I-U
GO TC 485
*82 CONTINUE
C INTERIOR CONDITION
450 CIPlJaC( I+l.J)
312
VJI)= BETAX-(2P»GP)/V(I-l)
485 CONTINUE
-------
CIJ=C(I,J)
CIJMl = SIGM«Ct It JK1.)+SK«C( I»J)
CIJP1=SIGP»C(I,JP1J+SP*C(I, J)
RHS =. SUMF(OELl,CIMiJ,OeL2,CU,DEL3,CIPlJ,DEL4,CIJMl,DEL5i
1CIJP1)
IB=l
IN=2
172 IF II-M1 186,186,187
187 IF (V(I-D) 188,186,188
186 MI)=RHS
GO TO 189
188 W(II = RHS-tW< I-l)*Zf>)/V< I-U
1B9 CONTINUE
C CHECK FCR TYPE 2 B.C.
GO TO 195
191 W(1I=C(UJ)
V( I )=.BETAX
195 CONTINUE
C WRITE OUT W AND V i TEMPORARILY
C WRITE <6,.2500) I W( I ), 1 = 1, 11 )
C WRITE (6,2530) ( V( IJ , I* 1 i-I I )
2500 FORMAT (10E8.2)
C BEGIN CSTAR CALCULATION
CSTARdRtJ)=W« IR)/VUR)
ISCL=IR-M
00 2^0 I=?1,.ISOL
IRtV=lR-I
CALL CQEFXIIREV.J)
IF (V(IKEV» 230,210,230
210 CSTAk
C SUB COEFX
DIMENSION C(23,21),CSTAR{23,21),CSSI23,2l),UU6,34) .VY146V34)
DIMENSION LI23.21)
CIMENSION DCLJ20,20),DCT(20,20)
DIMENSION GX(23)>IX<23)
CO^MCNVZcROTh/C.CSTARtCSS,!!, J J ,DT, OXtDY^CL.OCT.L,
lAl,A2,A3»A4,Bl,B2,a3,KFX,CFX,KLX,.CLX,KFY,.CFY,KLY«CUY^
2U,VY,ZP,GP^GPC,IFREQ.ICOUNT
CayMON/FlRST/ZX,aETAX,GXrDELl,DEL2,OEL3,OEL4,OEL5
CXSC-DX«DX
DYSQ=.OY»CY
21 K=LII,J)
IF(K-10)9,9,70
9 CONTINUE
2X(II»B3/DT-DCL(I,J)/DXSO-(A4»U«I,J))/OX
BETAX = B1/DT + 2.0*CCL( I,.J I /OXSQ + t A4-A3)*U( I , J)/OX
GX(I)=B2/DT-CCL(I,J)/OXSQ *A3*U(I«J)/OX
CEL1=B3/CT*A2*U1I,J)/DX
CEL2 = 81/DT-2.0«CCT{I,J>/DYSQ*;IA1-A2)#U(I,J}/DX
CEL3*Q2/CT-A1*U(I,J)/OX
CEL4=DCT(I,J)/DYSQ*VY(I,J)/(2.0»DY)
10 CEL5sCCT
-------
50 IF (L(.I-ltJ)-lO) 51,59,57
51 IF (U I-1,J>- 9) 55,60,59
55 IF (L(I-UJ)-l) 20,57,56
56 IF (1(1-1,J)-4) 57,60,60
57 ZP=ZX,CSS(23,21),U(46,34) ,VY146,'34)
DIMENSION L(23,21)
DIMENSION DCL(20»20),DCT(20,20)
DIMENSION GY.(21)rZYC2l)
DIMENSION V(23)iW(23)
CO^NCN/ZEROTh/C,CSTAR,CSS,IUJJ,OT,DX»DY,DCL,DCT,L,
l/Sl,A2,A3^A4,Bl,B2,B3,KFX,CFX,KLX,CLX,KFY^CFY,KLY^CLY^
2U,VY,ZP»GP,GPC,IFREQ,ICOUNT
COV^CN/SECOND/ZY,B£TAY,GY,OEL6,DEL7,OEL8,.DEL9,DELlOfR01,R02,R03
SUMF{Sl,Pl,S2,P2,S3,P3t,S4,.P4, S5, P5I »Sl
314
-------
K-l
CD 550 I=iM, 11
J"0
20 J=J + l
If (JJ-J1. 55Cu2i»Zl
21 K=L(IfJ>
IF fK-U 20,140*140
140 K=J
C FIND TH£ LOWEST BOUND
5
IF (l(I, JREVl) 10,10,15
10 J=J*i
GQ TC 5
15 JBOTsJREV
DO 465 j=N,jear
IF(J-l) 320t320.3lQ
320 IWl*l*vl
SC re (330,33l»3321tKFX
330
GO TG 370
331 SIGM=,-1,
SH=2.
GO TC 370
332 CONTINUE
310
IF UI-l> 53,.58,42
C LEFT OR RIGHT
42 IF (K-9> 45,370,370
*5 IF U-l> 20,57,
IF (K..EQ.IOJ GO TO
IF JJ-N) 475,475,575
575 IF (VOJ-1J) 600,475,600
C TOP CR BOTTOM
600 IF 1K-1) 460,460,601
60L IF
-------
V(J)=8ETAY
GO TO 455
CUM— C(I,J + 1)*2.»CII*.J)
CIJP1»C{ I,J*1J
V( J)-CETAY-»-2.»ZY{J)
GO TO 455
432 CONTINUE
C 8CTTCM CONDITION
485 GO TO (490,491(4921,KLY
490 CIJP1*C( ItJ-1)
CIJM = Ct I.J-1)
VIJ)*.SETAY -tZP»GP)/VU-l)
GO TO 455
491 CIJP1— C( I, J-l)+2.»Cl Ii-J I.
VUMBETAY t2.»GY(J)-CZP»GP)/V{ J-l)
GO TO 455
492 CONTINUE
C INTERIOR CONDITION
460 CIJP1*C( I.J+l)
CIJM1»C( I»J-i)
VUU.BETAY -(ZP«GP)/V(J-1)
455 CONTINUE
CIJ=C(I»J)
CIMlJaSIGM*CUMl,.J)+SM»CCLX,KFY,CFY^KLY,CLY,
2U,VY,ZP,GP,GPCt IFREQ.ICOUNT
316
-------
COMMCN/S6COND/ZY,QETAY,GY,DEL6,DEL7,DELG»DEL9,OEL10»ft01fR02,R03
DXSC=CX»CX
DYSG»OY*OY
21 K»LCI,J)
IF (K-10) 9,9,70
9 CONTINUE
ZY< J)»-DCT(I,J)/DYSQ-A4*VY( 1, J)/DY+B3/DT
BETAY=Bl/DTf2.0«CCT(I,J>/OYSCH:.(A/DY + B2/DT
CEL6=>-DCTU,J)/CYSQtU2-.5>»VYU,J)/OY+B3/OT
DEL7^2.0«DCT(I,JJ/DYSQ-(A2-A1)«VY(I,J)/DY
DEL8*-CCTU,Jl/CYSQ-(Al-.5)«VYt I ,-J )/DY + B2/OT
CEL9=*-B3/DT
CEL10—B2/DT
R01=01/DT
R02=B2/DT
10 R03*B3/DT
TOP CR BOTTOM
IF 55,.60,59
55 IFCL(IiJ-l)-l) 20,60,56
56 IF(LII,J-U-3) 57,60,51
51 IFU(IrJ-n-3) 57,60,58
58 IF(L(IrJ-D-7) 57..60.60
57 2P=ZY{J)
GO TO (52,53,54),,KFY
52 GP=GY(J-1)+ZY(J-1)
GPC»GY(J)
GO TO 100
53 GP»GYU-l)-ZYU-l>
GPOGYUl
GO TO 100
54 CONTINUE
59 ZP«ZY(J)
GP-0,
GPC=GY(J)
GC TC 100
INTERIOR CONDITION
60 ZP-ZY(J)
GP=GY(J)
GPC»GYtJ)
GO TO 100
20 ZP»0.
BETAY-0.
GP»0.
GPC'O.
GO TO 100
TOP CONDITION
34 ZP*0.
GP»0.
GO TO <35136i37>,,KFY
35 GPC=GY(J)*ZY(J1
GO TC 100
36 GPC-GY(J)-ZYU)
GO TO 100
37 CONTINUE
BOTTOM CONDITION
33 GP»GY(J-1)
GO TO I38,39,40),KLY
38 ZP*ZY(J)^GY(J)
GPC»0.
GO TO 100
39 ZP*ZYtJ)-GY(J)
317
-------
GPC*G.
GO TC 100
40 CONTINUE
42 ZP=0.
GP=0.
GPC=0.
EETAY=1.0
GO TO 100
70 CONTINUE
100 CONTINUE
RETURN
ENC
SUbROUTINE EXPLIC
C SUBEXPLIC
DIMENSION C(23,21),CSTARt23,21),CSS<23,21),U(46,34) ,VYt46,'34)
DIMENSION L(23,21)
DIMENSION CTEMP(23)
DIMENSION DCL(20,20),DCT(20,20)
COMVCN/ZEROTH/C,CSTAR,CSS,II,JJ,DT,DX,DYrDCL,DCT,L,
1A1,A2,A3,A4,Bl,B2,B3,KFX,CFX.KLX,CLX,KFY,CFY,KUY,CLYr
2U,VY_ZP,GP,GPC,IFREG,ICCUNT
CCMtfON/THIRD/Pl,P2,P3,P4,P5,
-------
SP=0.
GO TO 141
160 SIGP=-1.
SP=2.
GO TO 141
170 CONTINUE
LOO JPi-J+1
SIOP=i.
SP=0.
141 CONTINUE
CSIGM«SIGM«C(I,JMl)+SM*C(I,J)
CSIGP*51GP«CIItJP1)+SP»C(I,JJ
CSSIGM=SIGM«CSS(I,JM1)+SM»CSS(ItJ)
CSSIGP»SIGP*CSS(IiJPl) + SP*CSSU»J)
CALL CFEXP(IiJ)
C LEFT OR RIGHT
IF IK-10) 149,41, 33
149 IF (K-9) 11,80.41
11 IF CK-1> 241,33,12
12 IF (K-4) 33,80,13
13 IF IK-6) 80,34,34
C LEFT CONCITION
33 GC TC (35,37,39)»KFX
35 CTEMPUJ* P2*C< HI, J>*Ql«CSTAR+:
2S3«CSSIGP
18=1
IN=2
GO TO 85
C RIGHT CONDITION
75 CTEMP( I»*CTEMP( I)+P1»C( I,.J)*P2»C {I-i, J)*P4»CSIGW*:P5»
1 CSIGP +01»CSTAR( I-ltJ)+Q2*CSTAR( I»J)+S1»CSSIGM+S2»
2CSS( I,J)+S3«CSSIGP
GO TC 85
241 CTEWPI-D^O.
GO TO 85
41 CTEP-PUUCUtJ)
85 CONTINUE
CO 210 I=«.l, II
210 CUtJI»CT6MPCl)
280 CONTINUE
RETURN
END
SUBROUTINE CFEXP(I.J)
C SUB CFEXP
DIMENSION C(23,2U,CSTAR(23,21),CSS<23,21),U(46,34),VY146,I34)
CIKENS10N LI23.2U
DIMENSION DCL(20^20"),DCT(20,20)
COfMQN/ZEROTH/C,CSTAft,CSS,II,JJ,DT,OX,DY>OCL»OCT,l.,
A2,A3f.A4, B1,B2,B3,KFX,CFX,KLX,.CLX,KFY,CFY,KLY,CLY».
319
-------
2U,VY,ZP,GP,GPC, IFREQ, ICGUNT
COPKON/THiU.>/Pl,P2,P3,P4,P5,Ql,Q2,Q3,Sl»S2,S3
K=LU,J)
IF (K-10) 75,75,100
T5 CONTINUE
OYSC=DY*OY
CTX=CT/DX
CTX2=DT/(CX*DX)
CTYsCT/DY
CTY2=DT/(CY»OY)
P1=*1.0 + U1-A2)«(UU,J)»DTX*VY(I,J).*DTY)
P2*A2»U( I,J)«DTX
P3=-A1»U< I,J)»DTX
P4=A2»VY< I,J)«OTY
P5=-Al«VY(I,J)»RTY
G1=CCL( I,.J)»CTX2 + A4»U» I, J)*OTX
C2*-2.6»CCL( I, J>»DIX2 + (A3-A
-------
GO TO 50
130 CO 140
CUSET,J).*0.
CSTAR(ISET,J)=0.
CSSl ISET»,J)=0»
140 CONTINUE
GO TO 59
INSTRUCTIONS BELOW ARE FOR TOP INPUT
150 CONTINUE
IF (T-TINPUTJ 160,170,170
160 CO 165 1*1,7
JSET=8
C(1,JSET)-CN1
CSTARU,JSET>»CNl
CSS(I,JSET)=CNl
165 CCIvTINUc
CO 166 I=il3rl4
CU»JSET)=*CN2
CSTAR(I,JSET>*CN2
CSS( I,JSET)«CN2
166 CONTINUE
1=15
CU,JSET)=CN3
CSrAR(I,JSET)-CN3
CSS(I.JSET)=CN3
I»16
JSET=7
C(IiJSET)*CN3
CSTARdf JSET)*CN3
CSS(I,JSET)»CN3
1 = 17
JScT*.6
C(I,JS£T>.»CN3
CST4R(I,JSET)»CN3
CSS1I,JSET)»CN3
I»18
JSET-6
CU,JSET)»CN3
CSTARU, JSET»»CN3
CSS(ItJScT>=CN3
I»19
JSET*8
CII,JSET)3CN3
CSTARd, JSET)»CN3
CSS(I.JSET)»CN3
GO TC 50
170 CO 175 UJMIN.JMAX
Cdf JSETJ«0.
CSTARCIf JSET)*0.
C'SS(I»JSET)»0.
175 CONTINUE
50 CONTINUE
60 RETURN
END
SUBROUTINE OUTPUTtT.DTMAX).
C SUB CUTPUT
DIMENSION C(23,21),CSTAR(23,21) iCSSU3»21 ) ,Ut 46.3A J
DIMENSION L(23,21)
DIMENSION DCL(20».20)>OCT(20»20)
COKMON/ZEROTH/C.CSTAR,CSS,II,JJ,DT^OX|DY»DCLfOCT*L^
2U.VYt.ZP»GP,GPC, IFREQ.ICOUNT
C PRINT CONCENTRATIONS IN BAY
TH»T/3600.
IF (II-ll) I25il25,135
125 CONTINUE
321
-------
GC TC 620
610 WRITE(6,600) T
600 FORMAT ( 'ISSOXf 'TIME* 'fFll.At1 SECS»»«**')
GO TO 630
620 TH=T/3600.
WRITG< 6*615) T>
615 FORMAT (' l« » SOX , ' TIME- SF11.4,' HRS •»«*»•)
630 CONTINUE
WSITE<6,100)
100 FORMAT (//30X,'C CONCENTRATIONS ARE1)
CO 440 J^lfJJ
440 WRITE(6,400) J , iC(11J)»1 = 1111)
400 FORMAT I • J = ', 13, 1 IE 11.3)
GO TO 2000
135 CONTINUE
GO TO 120
110 WRITE (6,300) T
GO TO 130
120 WRITE (6i200) TH
130 CONTINUE
WRITE (6,111)
111 FORMATCO',1 1*1 1-2 1-3 1-4
1 l»6 I«7 1=8 I»9 I»10
21' )
CO 150 J=1,JJ
150 WRITE (6*250) J, (C( I,J),l = i,U•
230 CONTINUE .
WRITE (6>112)
112 FORMATCO1,1 I»12 I»13 I»14 I»15
16 . 1=17 I«18 I»19 I«20 I»21
22')
CO 160 J=.1,JJ
160 WRITE (6,250) Jt(C(IiJ>tIa12»19)
200 FORMAT <'1',40X,' C CONCENTRATIONS AT TIME- T» SF8.2,.1 HRSJ)
300 FOrtMAT Cl',40X,' C CONCENTRATIONS AT TIKE T= «,F8.2,; SECS1)
250 FORMAT C J = •, 13, HE 11. 3 )
2CTOO CONTINUE
10 CONTINUE"
RETURN
ENU
/*
//LKED.SYSIN DC *
CVERLAY ALPHA
INSERT UNZON1
OVERLAY ALPHA
INSERT UNZON2
CVERLAY ALPHA
INSERT UNZON3,UNFLO,COEF,EXTRA,VFLOW,WTA8LEtPSIVAL,PSIMID
INSERT SINK,MATRIX,TENS,CK.CSL,HUMID,GRAPH,PINE,CURVE,PPLOT
INSERT SCILQU,GROUP,MAT2,.OUTRIT,TRAN
CVERLAY ALPHA
INSERT OVLAN1
OVERLAY ALPHA
INSERT OVLAN2,AVE,WAV6tRECVrHIDURrLODUR»RQUAL,WRIT
CVtRLAY ALPHA
INSERT CVLAN3
CVERLAY ALPHA
INSERT OVLANA
CVERLAY ALPHA
INSERT RECl
CVERLAY ALPHA
INSERT REC2
CVERLAY ALPHA
INSERT REC3
CVERLAY ALPHA
INSERT GWBLCK,FHI,80UND,.TAOLE,EQUIP,VEL,PREDIS,GWQUALiVCALC
INSERT TftIX,COEFXtTRIYrCOEFY,EXPLIC,CINPUTtOtTPUT
/*
322
-------
//GQ.FT01FC01 CC
//GO.FT02FU01 CD
//GO.FT03F001 CC
//GO.FT04F001 CC
//GC.FTObFOOl CD
//GO.FT09F001 CD
//GO.FT10FC01 CD
//GO.FTHFOOl DO
X/GO.FT12F001 CD
//GO.FT13FC01 CD
/7 SPACR=(TRKf (
//GO.FT1AF001 CD
//GO.FT15F001 CD
//GO.FT16FC01 CD
//GO.FT17F001 CC
//GO.FT18F001 CD
//GC.FT19F001 CD
//GO.FT20F001 CD
//GO.FT21F001 CD
//GC.FT22F001 CC
//GO.SYS1N CD *
SYSCA,.$PACE = (CYL 1 < 2 , 1 ) )
UNIT=SYSC/»t SPACE=( CYL t < 2, 1) )
SYSCA, SPACE = (CYL,.{ 2, 1) )
SYSCA,SPACE = (CYL» ( 2, U )
SYSCA,.SPACE = (CYL ,.< 2 1 1 ) )
UNIT=SYSOAiSPACE = (CYLtJ 2, 1) )
UNIT=SYSCA,SPACE=( CYLr< 2 t L»)
UNIr = SYSCAfSPACEa(CYL^^2t D)
UNI T = SYSDA, SPACE*tCYL t < 2 . 1 ) )
DSN= ALAN. APQPKA4,UNIT = 23U,DI SP"< ,CATLG) ,
20i5),RLSE)tVQL=SER»USER01,OCB*(RECFM*VS,BLKSIZE»920)
Ui\l T = SYSCAf SPACE = (CYL t < 2, 1 ) )
UNI T=SYSCA,SPACE = (CYU< 2, 1 ))
UN IT=SYSCA,SPACE= t CYL t ( 2, 1) )
UNIT = SYSCA, SPAC£ = ( CYLr( 2, 1) )
UN.I T=SYSCA, $PACE = (CYLt-< 2, 1 ) )
UNIT=SYSDA,SPACE=tCYL»< 2 , 1) )
SYSCA,SPACE=( CYL , \ 2, L) )
SYSCA,SPACE = «CYL»< 2t I) )
DSN*ALAN. APOPKAA,DISP=OLD»,UNI T*2314» VOL»SER-USER01
OO.S. GOVERNMENT PRINTING OFFICE: »74 582-413/76 1-3
323
-------
1
5
Accession Number
2
Organization
Department of
Subject field it Group
SELECTED WATER RESOURCES ABSTRACTS
INPUT TRANSACTION FORM
Environmental Engineering Sciences
University of Florida
Gainesville. Florida 32611
Title
A Water Quality Model for a Conjunctive Surface-Groundwater System
10
Authors)
Perez, Armando I.
Huber, Wayne C.
Heaney, James P.
Pyatt, Edwin E.
ix. Protect Destination
2] Note
EPA 16110 GEW
22
Citation
Environmental Protection Agency report number, EPA-600/5-7U-013,
May
23
Descriptors (Starred First)
*Surface-groundwater relationships, *Surface runoff, *Groundwater flow,
*Unsaturated flow, *Water pollution, Path of pollutants, Agricultural watershed,
Florida, Nutrients, Nitrates, Phosphates, Hydrologic models, Mathematical models
25
Identifier* (Started Pint)
'Agricultural pollution, Lake Apopka Florida, Water quality models
27
increasing pressures toward efficient decision-making in water pollution con-
trol are creating the need for improved pollutant routing models, Such mathematical
models help understand cause and effect relationships between sources of pollutants
and their ensuing concentrations at various locations in a basin.
Considered in this study were both flow and water quality processes occurring
on the ground surface, in the unsaturated soil zone and in the saturated or ground-
water zone. The objective was to improve already available formulations for the
above processes and subsequently to develop a methodology for interfacing the
individual models.
Emphasis was placed on the modeling of agricultural pollution. For this rea-
son, nitrogen and phosphorous were the main substances considered. The selection of
the Lake Apopka basin in Central Florida as the study area wds made in accordance
with these project goals. The data base for this basin was limited, but better
than that for other candidate areas in Florida. However, this data base appeared
typical of that available to engineers in the analysis of a practical problem. The
scope of this particular study precluded additional sampling. Current data limita-
tions precluded a complete verification of the model, However, various general
conclusions could be drawn. In the future, the formulation could be viewed as an
instrument for structuring data gathering efforts.
Abstractor
Wayne C. Huber
Institution
University of Florida
WRIIC
------- |