oEPA
United States
Environmental Protection
Agency
Environmental Research
Laboratory
Dululli MN bb804
EPA 600 3 80 079
August 1980
Research and Development
Impacts of
Coal-Fired Power
Plants on Local
Ground-Water
Systems
Wisconsin Power
Plant Impact Study
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RESEARCH REPORTING SERIES
Research reports of the Office of Research and Development, U.S. Environmental
Protection Agency, have been grouped into nine series. These nine broad cate-
gories were established to facilitate further development and application of en-
vironmental technology. Elimination of traditional grouping was consciously
planned to foster technology transfer and a maximum interface in related fields.
The nine series are:
1. Environmental Health Effects Research
2. Environmental Protection Technology
3. Ecological Research
4. Environmental Monitoring
5. Socioeconomic Environmental Studies
6. Scientific and Technical Assessment Reports (STAR)
7. Interagency Energy-Environment Research and Development
8. "Special" Reports
9. Miscellaneous Reports
This report has been assigned to the ECOLOGICAL RESEARCH series. This series
describes research on the effects of pollution on humans, plant and animal spe-
cies, and materials. Problems are assessed for their long- and short-term influ-
ences. Investigations include formation, transport, and pathway studies to deter-
mine the fate of pollutants and their effects. This work provides the technical basis
for setting standards to minimize undesirable changes in living organisms in the
aquatic, terrestrial, and atmospheric environments.
This document is available to the public through the National Technical Informa-
tion Service, Springfield, Virginia 22161.
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EPA-600/3-80-079
August 1980
IMPACTS OF COAL-FIRED POWER PLANTS
ON LOCAL GROUND-WATER SYSTEMS
Wisconsin Power Plant Impact Study
by
Charles B. Andrews
Mary P. Anderson
Institute for Environmental Studies
University of Wisconsin-Madison
Madison, Wisconsin 53706
Grant No. R803971
Project Officer
Gary E. Glass
Environmental Research Laboratory-Duluth
Duluth, Minnesota
This study was conducted in cooperation with
Wisconsin Power and Light Company,
Madison Gas and Electric Company,
Wisconsin Public Service Corporation,
Wisconsin Public Service Commission,
and Wisconsin Department of Natural Resources
ENVIRONMENTAL RESEARCH LABORATORY-DULUTH
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
DULUTH, MINNESOTA 55804
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DISCLAIMER
This report has been reviewed by the Environmental Research Laboratory
Duluth, U.S. Environmental Protection Agency, and approved for publication.
Approval does not signify that the contents necessarily reflect the views
and policies of the U.S. Environmental Protection Agency, nor does mention
of trade names or commercial products constitute endorsement or recommenda-
tion of use.
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FOREWORD
The U.S. Environmental Protection Agency was established to coordinate
our country's efforts toward protecting and improving the environment.
Research projects in a multitude of scientific and technical areas are
necessary to monitor changes in the environment, to discover relationships
within that environment, to determine health standards, and to eliminate
potential hazards.
One such project, which the EPA is supporting through its Environmental
Research Laboratory in Duluth, Minnesota, is the study "The Impacts of Coal-
Fired Power Plants on the Environment." This interdisciplinary study, based
at the Columbia Generating Station, near Portage, Wis., and involving
investigators and experiments from many academic departments at the
University of Wisconsin, is being carried out by the Environmental
Monitoring and Data Acquisition Group of the Institute for Environmental
Studies at the University of Wisconsin-Madison. Several utilities and state
agencies are cooperating in the study: Wisconsin Power and Light Company,
Madison Gas and Electric Company, Wisconsin Public Service Corporation,
Wisconsin Public Service Commission, and Wisconsin Department of Natural
Resources.
Reports from this study will be published as a series within the EPA
Ecological Research Series. These reports will include topics related to
chemical constituents, chemical transport mechanisms, biological effects,
social and economic effects, and integration and synthesis.
This report describes the research undertaken by the Hydrogeology
Subproject of the Columbia project. This research includes monitoring of
ground-water flows and temperatures at the Columbia site, development of
mathematical models for predicting such flows, and applications of the
models to several problems.
Norbert A. Jaworski
Director
Environmental Research Laboratory
Duluth, Minnesota
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ABSTRACT
Quantitative techniques for simulating the impacts of a coal-fired power
plant on the ground-water system of river flood-plain wetland in central
Wisconsin were developed and tested by using field data collected at the
site of the 500-MW Columbia Generating Station. The most important effects
were those related to the construction and operation of the 200-ha cooling
lake and the 28-ha ashpit.
Several two-dimensional vertically oriented steady-state models of the
ground-water flow system were used to simulate ground-water flows before and
after the filling of the cooling lake and ashpit. The simulations and
supporting field evidence indicated that the creation of the cooling lake
greatly altered the configuration of the local flow systems and Increased
the discharge of ground water to the wetland west of the site by a factor
of 6.
Chemical changes in the ground-water system were minor. The plume of
contaminated ground water originating from the ashpit was confined to a
relatively small area near the ashpit. Thermal changes in the ground-water
system are a major impact of the operation of the cooling lake inasmuch as
the lake loses water to the ground-water system at a rate of 2 x 10^ m^ per
day. The wetland, which is a major ground-water discharge area, has
undergone a rapid and dramatic change in vegetation as a result of changes
in both water temperature and water levels.
Ground-water temperatures in the vicinity of the cooling lake were
monitored in detail for 1 1/2 yr. The response of subsurface temperatures
temperature was simulated by means of a mathematical model, and predictions
were made of the long-term changes expected in substrate temperatures in the
wetland adjacent to the power plant. In addition, the use of ground-water
estimate ground-water flow rates away from the cooling lake was
investigated.
The model, which couples equations describing ground-water flow with
those describing heat transport in the subsurface, was used to simulate the
seasonal temperature fluctuation within seven cross sections oriented
parallel to the direction of ground-water flow and downgradient from the
cooling lake. Simulated temperature patterns agreed well with field data,
but were very sensitive to the distribution of subsurface lithologies. The
predictive simulations suggest that by 1987 temperatures at a depth of 0.6 ra
will not fall below 8°C within 200 m of the dike, and that peak temperatures
near the dike will be 10-15°C above normal and will occur in October and
November rather than in August. The increase in ground-water temperatures
by 1987 will also result in a 24% increase in ground-water flow. The
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subsurface stratigraphy of the site is such that major changes in
near-surface temperatures will only occur within 340 m of the dikes, but, if
the different, impacts could extend to much greater distances from the
cooling-lake dikes.
This report was prepared with the cooperation of faculty and graduate
students in the Department of Geology at the University of Wisconsin-
Madison.
Most of the funding for the research reported here was provided by the
U.S. Environmental Protection Agency, but funds were also granted by the
University of Wisconsin-Madison, Wisconsin Power and Light Company, Madison
Gas and Electric Company, Wisconsin Public Service Corporation, and
Wisconsin Public Service Commission. This report was submitted in
fulfillment of Grant No. R803971 by the Environmental Monitoring and Data
Acquisition Group, Institute for Environmental Studies, University of
Wisconsin-Madison under the partial sponsorship of the U.S. Environmental
Protection Agency.
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CONTENTS
Foreword
Abstract iv
Figures viii
Tables xii
1. Introduction 1
2. Conclusions and Recommendations 4
Recommendations for siting future power plants 6
Recommendations for future research 6
3. The Site of the Columbia Generating Station 8
4. The Impact of a Power Plant on the Ground-Water System
of a Wetland 19
Configuration of the ground-water flow system 19
Wetland water levels • 23
Ground-water temperature 24
Relationship between ground-water temperatures .and
flow rates . . . . • 26
Changes in water chemistry 26
5. Thermal Alteration of Ground Water Caused by Seepage
from the Cooling Lake • 30
Mathematical model 30
Results 35
6. Long-Term Temperature Changes in the Ground Water
of the Wetland 41
Simulation studies 41
Results and discussion 43
References 51
Appendices
A. Data specifications for the water-flow model • . * 57
B. Techniques for determining boundary conditions and
parameters in the heat-flow model 61
C. A finite element program to simulate single-phase heat
flow and conservative mass transport in a aquifer 73
D. Estimation of aquifer parameters by using subsurface
temperature data • • 195
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FIGURES
Number Page
1 The site of the Columbia Generating Station during
construction of the first unit (24 May 1971) 9
2 The site of the Columbia Generating Station after
construction of both units (12 August 1977) ........... 10
3 Main features of the site of the Columbia Generating Station ... 11
4 Water flows and energy flows at the site of the Columbia
Generating Station ............. 12
5 Potentiometric surface contours in the area of the
Columbia Generating Station 13
6 Representative stratigraphic cross sections of the subsurface
at the site of the Columbia Generating Station 14
7 Topographic map of the site of the Columbia Generating Station . . 16
8 Peat thicknesses in the subsurface at the site of the Columbia
Generating Station • 18
9 Simulated head distribution in cross-secion A-A1 of Figure 6 before
filling of the Columbia cooling lake 20
10 Simulated head distribution in cross-section A-A1 of Figure 6 after
filling of the Columbia cooling lake 22
11 Water levels in the wetland west of the Columbia cooling lake
before and after filling of the lake 24
12 Ground-water temperature variations in wells west of the
Columbia cooling lake 25
13 Stiff diagrams of water in the cooling lake and ground water ... 27
14 Stiff diagrams of water in the ashpit and in a well cased to 1.5 m
below the surface and located 2 m west of the ashpit ...... 28
15 Location of seven cross sections of the Columbia site for which
ground-water temperature distributions were simulated ..*••* 31
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16 Observed temperatures in the subsurface west of the Columbia
cooling lake and temperatures simulated by the
mathematical model ........tit. 36
17 Observed (solid line) and simulated (dashed line) ground-water
temperature distributions in cross-section A-A1 of Figure 15 .. 37
18 Observed (solid line) and simulated (dashed line) ground-water
temperature distributions in cross-section B-B1 of Figure 15 .. 38
19 Seasonal fluctuations of ground-water temperature in cross-section
A-A1 of Figure 15 at a depth of 4.5 ra at distances of 2 m, 15 m,
50 m, and 84 m west of the cooling-lake dike ..... 39
20 Simulated temperatures (solid lines) and observed temperatures
(open circles) in cross-section A-A1 of Figure 15 at a depth
of 3*28 m at various distances west of the cooling lake .... 40
21 Cooling lake inlet temperatures for 1975-87 used in the
simulations of long-term temperature change in the
ground-water system 42
22 Predicted ground-water temperatures from 1975 to 1987 at a distance
of 2 m west of the cooling-lake dike in cross-section A-A'
of Figure 15 at a depth of 0.6 m 44
23 Pre-lake temperatures and simulated temperatures 2 m west of the
cooling-lake dike in cross-section A-A1 of Figure 15 at a
depth of 0.6 m 45
24 Temperatures 150 m west of the cooling-lake dike in cross-section
A-A' of Figure 15 at depths of 0.6 m and 3 m for the simulated
period 1975-87 46
25 Temperatures 150 m west of the cooling-lake dike in cross-section
A-A' of Figure 15 at a depth of 0.6 m 47
26 Temperatures in cross-section B-B' of Figure 15 at a depth of
0.6 m and at (a) 2 m west of the dike, (b) 50 m
west of the dike, and (c) 200 m west of the cooling-lake
dike 48-50
A-l Location of monitoring wells at the Columbia site 58
B-l Location of temperature sampling points at the Columbia
Generating Station site 62
B-2 A simple circuit for measuring temperature ... 65
B-3 Temperature-resistance characteristics of Yellow Springs Series
400 thermistors and the change in resistance with
a 0.1° change in temperature . * 67
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B-4 Potted thermistor assembly showing the details of the
thermistor mounts 67
B-5 Schematic of thermistor placement in subsurface wells at the
Columbia Generating Station site 69
B-6 Schematic of digital bridge circuit used as a portable field
meter for measuring ground-water temperatures 71
B-7 Details of (a) the needle probe and (b) the experimental
arrangement to measure thermal conductivities of
unconsolidated materials 72
B-8 Thermal conductivities of three ground-water samples from the
Columbia Generating Station site which were analyzed with
the needle probe ' 72
C-l The basic procedure for linking the ground-water flow and the
transport equations of the model 73
C-2 Examples of refinement of a finite element grid 81
C-3 Examples of numbering of nodes in a structure ...*. 93
C-4 Cartesian orientation of finite element grid 94
C-5 Typical elements of finite element grid and correct method of
numbering the elements 95
C-6 Numbering of nodes and elements in rectangular grid . . 95
C-7 Two variants of the rectangular grid that can be generated .... 96
C-8 Numbering of sides of an element 97
C-9 Finite element grids used to discretize a linear heat
transport problem •• 104
C-10 Input data used to model one-dimensional heat transport
with all linear elements . • 105
C-ll Program output for one-dimensional heat transport problem with
linear elements • 107
C-12 Data deck used to model one-dimensional heat transport
with mixed elements Ill
C-l3 Program output for the one-dimensional heat transport problem
with mixed elements 113
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C-14 Analytical solutions (solid lines) and numerical solutions
(dots) for the linear heat transport problem for t = 25, 50,
and 75 days 118
C-12 Areal view of the Mohawk River Valley showing location of the
Schenectady and Rotterdam well fields 121
C-16 Cross-sectional view of the Mohawk River alluvial aquifer
along section A-A1 of Figure C-15 121
C-17 Grid used to discretlze the Mohawk River problem 122
C-18 Data deck used to model heat flow in the Mohawk River
alluvial aquifer 123
C-19 Program output for Mohawk River problem 125
C-20 Schematic cross section of the Columbia Generating Station site
along an east-west line 131
C-21 The grid used to dlscretize the cross section simulated at the
Columbia Generating Station site 131
C-22 Data deck used to model heat flow at the Columbia Generating
Station site 132
C-23 Program output for Columbia Generating Station problem 134
C-24 Grid used to discretize the aquifer simulated in the
heat pump problem . 138
C-25 Data deck used to model the heat pump simulation with no
regional ground-water flow 139
C-26 Program output for the heat pump problem 141
C-27 Program flow chart 147
D-l Temperatures (°C) in a section of the marsh adjacent to the
cooling lake at the Columbia Generating Station site on
7 October 1977 200
D-2 Temperatures in a 60-m portion of cross-section B-B1 of Figure 6
adjacent to the drainage ditch east of the cooling lake on
9 June 1977 and 24 October 1977 201
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TABLES
Number Page
1 Horizontal Hydraulic Conductivities for the Lithologies in
the Subsurface of the Columbia Generating Station 15
2 Selected Chemical Concentrations in Ground Water Near the
Ashpit of the Columbia Generating Station . . * 29
3 Parameters Used in the Simulations for the 1 to del Describing
Thermal Alteration of Ground-water 35
B-l Temperature Sampling Points: Locations, Depths, liethods, and
Frequency of Readings 63
B-2 Heat Capacities of the Common Components of Unconsolidated
Glacial Materials 72
C-l Parameters for Both Mass and Heat Transport Problems 76
C-2 Parameters, Initial Conditions, and Boundary Conditions Used for
the Heat-Flow Equation in the Linear Heat Transport Problem . . 106
C-3 Analytical Solution at t - 50 Days for the Problem Posed in
Figure C-9 and Finite Element Numerical Solutions
at t « 50 Days for Two Grid Configuations and Two Types of
Boundary Conditions 117
C-4 Parameters, Initial Conditions, and Boundary Conditions Used for
the Simulation of Temperatures in the Mohawk River Alluvial
Aquifer 120
C-5 Parameters, Initial Conditions, and Boundary Conditions Used in
the Columbia Generating Station Problem . » i « » 130
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SECTION 1
INTRODUCTION
Large industrial facilities are common features of the American
landscape, but knowledge of the mechanisms by which they change natural
systems is inadequate. The Hydrogeology Subprojeot of the interdisciplinary
research project "The Impacts of Coal-Fired Power Plants on the Environment"
quantified the effects of construction and operation of the Columbia
Generating Station on the hydrogeologic system of an adjacent wetland. The
investigators developed techniques for simulating the observed changes and for
predicting potential impacts at other sites. Impacts on the ground-water
system of the wetland resulted from the construction of a 200-ha cooling lake
and a 28-ha ashpit. Specifically, this study assesses the alteration of three
characteristics of the ground-water system: (1) the quantity of ground-water
flow from the cooling lake and the ashpit into the wetland and surface-water
levels in the wetland, (2) the quality of ground water and surface water as
expressed by concentrations of the common cations and anions, and (3) the
temperatures of ground water and surface water at the site.
Data on ground-water temperatures were collected from 1971, 4 yr before
the operation of the Columbia Generating Station, through 1977. The basic
monitoring network consisted of 100 small-diameter wells and a subsurface
network of 64 temperature monitoring points.
During the initial phase of the study the ground-water flow systems in the
vicinity of the generating station were delineated and the way in which the
flow systems were altered by the construction and operation of a cooling lake
and ashpit were documented. A finite difference model was used to simulate
the alterations in the configuration of the ground-water flow system.
Analyses of changes in the chemical quality and temperature of ground water
were descriptive during this phase of the study.
The second phase of the study dealt with the monitoring and simulation of
the transfer of one byproduct of the generating process, specifically heat,
away from the site via the ground-water system. The study of the transfer of
heat was chosen rather than the transfer of chemical byproducts because: (1)
temperatures can be monitored inexpensively and rapidly at a large number of
points, (2) the processes of heat transfer are well understood, and (3) the
initial phase of the study had shown that alterations in the temperature of
ground water were large, whereas alterations in the chemical characteristics
of ground water were small. The development of the capability to simulate the
transfer of heat is a prerequisite to studying the flow of a chemical away
from the site. The processes determining the chemical composition of ground
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water are poorly understood, and it is difficult to obtain representative
ground-water samples.
Data on ground-water temperatures were collected for an 18-mo period
during 1976-77. in the vicinity of the cooling lake of the Columbia Generating
Station. The finite element method, using isoparametric quadrilateral
elements, was used to solve the partial differential equations describing
ground-water flow and heat transport in the subsurface. The model was tested
by comparing the observed ground-water temperatures in the vicinity of the
cooling lake with predicted temperatures. The simulated temperatures were in
close agreement with the observed temperatures.
The model was then used to simulate the long-term (12-yr) effects of
seepage from the cooling lake on ground-water temperatures near the lake. The
temperature data collected at the Columbia Generating Station site and the
model were also used to refine estimates of ground-water' flow from the cooling
lake calculated by using potentiometric data. A third application of the
model was the simulation of the impacts on ground-water temperatures on the
use of heat pumps for residential heating and cooling (Andrews 1978).
Although ground-water systems have been explored for possible energy
storage or as a source of energy, the published literature contains no
previous work documenting the effects of a power plant on a ground-water
system. The possibility of temporary storage of energy in the form of heated
water in aquifers has been explored by Meyer and Todd (1973), Hausz and Meyer
(1975), Kley and Nieskens (1975), Molz et al. (1976), Tsang et al. (1976), and
Werner and Kley (1977). Gass and Lehr (1977) advocated the use of ground
water itself as an energy source. Gringarten and Sauty (1975), Intercom?
Resource Development and Engineering (1976), and Tsang et al. (1976) developed
models of the transport of heat in ground-water systems to study the
feasibility of temporary storage of heated water in aquifers. Mercer et al.
(1975) modeled the movement of heat and water in a hydrothermal system.
Alterations in the ground-water system induced by activities at a
power-plant site can bring about significant changes in nearby flora, fauna,
and surface waters. For example, in a wetland environment, here defined as an
area where the water table is at or near the surface at all times, alterations
in ground-water quality and ground-water discharge rates have been shown to
alter wetland ecosystems (Bay 1967, Dix and Smeins 1967, Walker and Coupland
1968, Millar 1973). In addition, Vadas et al. (1976) and Gibbons (1976) found
that changes in wetland water temperatures led to changes in wetland
ecosystems.
Boulter (1972) studied the extent of water-table drawdown after a wetland
is ditched. But no one has investigated the changes that occur in water
levels, water quality, or water and substrate temperatures in a wetland when a
ground-wafer system discharging into a wetland is altered. In the wetland
adjacent to the Columbia Generating Station, wetland vegetation changed
quickly and markedly following changes in water temperatures, water levels,
and water flows caused by the presence of the cooling lake (Bedford 1977).
During the first 2 yr of operation of the Columbia Generating Station, a
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community previously dominated by sedge meadow species was replaced by
emergent aquatic species and annuals (Bedford 1978).
This report deals with the hydrogeologic environment of the site of the
Columbia Generating Station and the major effects of power-plant construction
and operation; the impacts of the generating station on the ground-water
system; the simulation of heat flow in the subsurface near the site; the
expected long-term changes in ground water at the site; and the major
conclusions and recommendations of the study. The appendices contain a
description of the field methods used in the study (appendix A and appendix
8), the computer program for solving the model of heat and water flow in
shallow ground-water systems (appendix C), and a discussion of the use of the
heat- and water-flow model to refine estimates of flow using temperature data
(appendix D). An additional application of the model is reported by Andrews
(1978).
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SECTION 2
CONCLUSIONS AND RECOMMENDATIONS
The cooling lake at the Columbia Generating Station has dramatically
altered the water supply to the marsh west of the lake. Before the filling of
the lake, ground water, the major source of water to this area, discharged at
a rate of less than 0.03 m3/s annually. The discharge rate increased by a
factor of 6 after the filling of the cooling lake. The increased water supply
to the marsh has raised surface-water levels in the sedge meadow approximately
10 cm above pre-lake levels, and the water levels now have much less seasonal
variability.
The changes in ground-water patterns have been confined to the area
between the station's cooling lake and ashpit and the Wisconsin River. To the
east of the cooling lake, a drainage ditch limits the effects of the lake on
the ground-water system to a narrow band east of the lake, except in the
northeast corner of the lake where there is no ditch. In this area the water
table has risen several feet, but flow in this area is small.
Thermal patterns in the marsh have been significantly altered by the
cooling lake. Temperatures of the ground water that discharges in the marsh
now average several degrees above prior average ground-water temperatures, and
temperatures in the marsh are out of phase with seasonal air-temperature
patterns. Much of the sedge meadow no longer freezes over in winter, and
temperatures Just below the surface are as high as 25° C prolong the growing
season. Winter temperatures as high as 21° C and summer temperatures as low
as 8° C were observed. In some areas the peak ground-water temperature
occurred during the winter months. Changes in vegetation within the thermally
altered zone were documented by Willard et al. (1976) and are thought to occur
in response to changes in ground-water temperatures.
As with changes in ground-water flows, the thermal alteration of ground
water was confined to the area west of the cooling lake. Moreover,
significant changes in temperatures extended only to about 100 m west from the
dike. The seasonal maximum ground-water temperature at any depth is a
function of distance from the cooling lake and the distribution of subsurface
lithologies.
The movement of heat in the subsurface was simulated by using the finite
element method to approximate the differential equations for water and heat
flow. The simulated temperature patterns agreed well with field data, but
were very sensitive to the distribution of subsurface lithologies. Detailed
stratigraphic information was necessary to obtain reliable results. The
problem of simulating heat flow in the subsurface is further complicated by a
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poor understanding of how to estimate dispersivity. Results presented in this
study show that the amplitude of the seasonal temperature wave is best
simulated when small dispersivities are used.
Long-term simulations of thermal alterations in the ground-water system of
the marsh indicate that the cooling lake will significantly alter substrate
temperatures but only within 350 m of the cooling lake dike. By 1987 peak
temperatures near the dike were elevated 10-15° C above normal levels and
lagged behind seasonal temperature changes by 1-2 months. At 150 m from the
dike, peak temperatures at a depth of 0.6 m were only elevated 2,3° C above
normal levels in the summer, but winter temperatures were elevated 6-10° C
above normal levels. Temperatures within the wetland varied spatially, but
trends were expected to be similar to those simulated in the two cross
sections modeled. Seepage rates will increase 20$ as the result of the
increase in temperature of seepage waters.
The simulations indicate that winter temperatures will be warm enough to
prevent formation of an ice cover in almost the entire marsh between the
cooling lake and the Wisconsin River. Near the dikes a relatively warm
microclimate will exist during most of the winter, since the temperature of
the discharging water will be above 20° C during the early winter.
The potential for significant thermal alteration of surface-water bodies
located in ground-water discharge zones is great. The temperature of the
ground water may rise considerably near a cooling lake, even though the total
heat flux is small. At the site of the Columbia Generating Station H0% of the
water pumped into the cooling lake seeps into the ground-water system, yet
less than 2% of the waste heat load is discharged to the ground-water
reservoir. The discharge rate of water from the generating station to the
cooling lake is 1 x 10^ m3 per day, and the temperature of the surface water
is increased 10-15° C. However, because the seepage rate is only 2 x 101* m3
Per day, much of the heat load is dissipated through evaporation. The
percentage of the heat load that leaves a cooling lake via seepage to the
ground-water system is unlikely to be greater at other power-plant sites.
Seepage from the ashpit averages 0.3-0.6 m3/s. Surface water in the
wetland within 50 m of the ashpit has been contaminated by seepage through the
ashpit dikes, and total dissolved solids in the surface waters have increased
over fivefold above background levels in this area. However, significant
chemical degradation of ground water, judged by the concentration of the
common cations and anions, has not occurred in the vicinity of the ashpit.
A plume of contaminated ground water is slowly moving eastward from the
coal pile. Three years after coal was piled in the area, a plume of water
with a sulfate concentration greater than 1,000 mg/liter extended to a depth
°/ greater than 30 m and outward from the limits of the coal pile more than
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RECOMMENDATIONS FOR SITING FUTURE POWER PLANTS
All new power plants should be required to gauge accurately (within 150
all water flows into and out of the plant and the plant site. In addition,
precipitation into and evapotranspiration from all bodies of open water, such
as ashpits and and cooling lakes, should be monitored. Seepage from ashpits
and cooling lakes cannot be determined accurately by direct methods; instead,
seepage should be calculated indirectly by a mass balance approach. Data on
water flows, precipitation, and evapotranspiration are required for
calculating a mass balance. Control rooms of power plants are now designed to
measure accurately mass flows within the generating station, and therefore
water flows between the environment and the plant can also be carefully
monitored.
Coal-storage areas should be designed so that water infiltrating the coal
pile cannot reach the water table. Since crushed coal is highly permeable and
leachable, an open coal pile allows most of the annual precipitation to pass
through the pile, which creates water similiar to acid mine drainage. Unless
infiltration is controlled or the infiltrating water is captured and treated,
a volume of contaminated water equal to the product of the annual
precipitation and the area of the coal pile will flow into the subsurface.
Estimations of impacts of generating stations should recognize that
significant thermal alterations of nearby surface waters can occur even when a
closed-cycle cooling lake is used.
Dikes of all ashpits and cooling lakes should be lined with several inches
of a relatively impermeable clay material. Seepage can be controlled, and
sloppy designs should not be tolerated. More seepage can usually be allowed
from a cooling lake than from an ashpit, since cooling-lake water is usually
less threatening to the environment.
Simulation techniques should be used to evaluate potential seepage from
ashpits, cooling lakes, and coal-storage areas in the design stage of these
structures. Designs should minimize ground water seepage and contamination.
Competent geologic consulting firms should be retained to conduct these
studies.
Although seepage from a cooling lake may contribute to alterations in an
adjacent wetland ecosystem, the high leakage rates are not undesirable from a
lake-management standpoint. Seepage to the ground-water system prevents an
increase in total dissolved solids in the lake. Such an increase would
necessitate periodic flushing of the lake and the consequent release of saline
water.
RECOMMENDATIONS FOR FUTURE RESEARCH
More field data are needed on the distribution of temperature or
conservative chemical characteristics, or both, in small ground-water systems
subjected to stresses that are altering these characteristics from background
levels. The flow of mass and energy in ground-water systems is much more
difficult to quantify than the flow of water, mainly because the former is
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sensitive to the hydraulic conductivity distribution. The development of
quantitative techniques to date has been hampered by the lack of field data to
verify the models. As demonstrated in this study, it may be more fruitful to
monitor temperature changes than chemical changes in a ground-water system if
the goal is to determine the system's capability to transport mass and energy.
Temperature can be inexpensively and rapidly monitored at a large number of
points, which is not the case for chemical characteristics.
Existing ground-water models will be adequate in most cases for
quantifying impacts of power plants on hydrogeologic systems. The best
existing models for these purposes are those devised by the U.S. Geological
Survey. Future research should not be directed towards developing new models
for these purposes, unless the investigator can show clearly that existing
models are deficient.
More study is needed of the factors controlling dispersivity, and the
appropriateness of the concept should be evaluated.
An evaluation is needed of the prediction ability of models used when
historical data are not available; The magnitude of error associated with
predictions should be quantified. It may be that transport models should not
be used for predictions if historical data are not available for calibration.
A theoretical evaluation of the validity of using either chemical or
energy distribution in the subsurface to determine hydraulic conductivity
distributions is needed. The question to be asked is whether or not the
inverse approach can produce useful approximations in a problem that is more
complex than those investigated by Bredehoeft and Papadopulos (1965).
-7-
-------
SECTION 3
THE SITE OF THE COLUMBIA GENERATING STATION
The Columbia Generating Station is a 1,000-MW coal-fired electric-power-
generating complex located on the flood plain of the Wisconsin River 5 miles
south of Portage, Wis. Construction began in 1971, and the two 500-MW units
began operation in May 1975 and May 1978. The 1,620-ha site on which the
facility was constructed consisted of an extensive marsh, mainly sedge
meadows, and wetland forests with sandy upland knolls (Figure 1).
Construction activities have dramatically altered the site. A 200-ha cooling
lake, a 28-ha ashpit, a 16-ha coal pile, and the generating station itself are
now the dominate features of the site (Figure 2, Figure 3).
The power plant dynamically interacts with the environment. On an
average day the first unit alone burns 2,000 tons of coal, producing 385 MW of
electricity, 1,000 MW of waste heat, and 552 tons of coal residue. This rate
of coal burning is approximately 38 kg/s, which represents over 1,000 MW of
power of which only about 360 MW are converted to electricity. The cooling
lake and the ashpit are the direct recipients of most of the byproducts. The
thermal and chemical characteristics of the byproducts differ markedly from
the thermal and chemical characteristics of water in the adjacent wetland
environments (Figure 4).
The site of the Columbia Generating Station is located in a regional
discharge area (Figure 5). The discharge area includes the wetlands on the
site of the power plant, the wetlands to the east of the site, which are used
for mint farming, and extensive wetlands along the lower reaches of Rocky Run
Creek. The marsh on which the site is located occupies a former river
channel.
The geology beneath the site of the Columbia Generating Station consists
of Irregularly eroded Upper Cambrian quartz sandstone overlain by sediments
from glacial lake bottoms and drift deposits of the Green Bay lobe of the
Wisconsin ice sheet. The bedrock is composed of Upper Cambrian sandstones and
Precambrian granites which occur 125 m below the surface. The surface is
covered by a layer of peat 1.5 m thick, which overlies a thin layer of organic
clay and silt. The peat and clay are in turn underlain by alluvial sands with
clay lenses. Representative stratigraphic cross sections are shown in Figure
6. The hydraulic conductivity of the clay is an order of magnitude less than
the peat and thus controls the rate of ground-water discharge (Table 1).
Vertical hydraulic conductivities were estimated to be 5 to 20 times less than
horizontal hydraulic conductivities.
-8-
-------
I
\o
I
Figure 1. The site of the Columbia Generating Station during construction of the first unit
(24 May 1971).
-------
i
(—•
r
Figure 2. The site of the Columbia Generating STation after construction of both units
(12 August 1977).
-------
Figure 3. Main features of the site of the Columbia Generating
Station.
-11-
-------
Figure 4. Water flows - annual averages 1CP m3 day lt and energy flows - approximate values 1MW =
239 kcal/sec, at the site of the Columbia Generating Station.
-------
Figure 5. Potentiometric surface contours in the Cambrian sandstone
(meters above mean sea level) in the area of the Columbia
Generating Station. Location is shown by diamond on insert
map of the State of Wisconsin.
-13-
-------
COOLING LAKE
UJ
UJ
A'
LEGEND
Med. -coarse sand with grovel
Fine to very fine sand
Fine sand with 10% silt
Weathered sandstone
Sandstone
Sandy silt to sandy clayey silt
Grey sandy silt with organic matter
Varved clay with sand seams
Figure 6. Representative stratigraphic cross sections of the subsurface at
the site of the Columbia Generating Station. Insert shows
locations of sections at the site.
-14-
-------
TABLE 1. HORIZONTAL HYDRAULIC CONDUCTIVITIES FOR THE LITHOLOGIES
IN THE SUBSURFACE OF THE COLUMBIA GENERATING STATION
Horizontal hydraulic
Lithology conductivity (m/day)a
Medium-coarse sand with gravel 30
Fine to very fine sand 10
Fine sand with 10$ silt 6
Weathered sandstone 8
Sandstone 3—4
Peat 0.4—3
Sandy silt to sandy clayey silt 0.4
Gray sandy silt with organic matter 0.04
Varved clay with sand seams 0.02—0.04
Conductivities were determined from a pump test, slug tests, laboratory
permeameter tests, and estimates based on lithology and grain-size analyses.
The 1,900-ha site ranges in elevation from 237 to 248 m above mean sea
level. The higher areas are well drained since the underlying soil is fine to
medium sand. The elevation of most of the site, however, is less than 239 m,
and these low areas are usually wet throughout the year because of
ground-water discharge (Figure 7).
A well-developed ground-water flow system exists in the Cambrian
sandstone. The discharge areas of the aquifer include the site, the wetlands
west of the cooling lake, the wetlands east of the lake, and extensive
wetlands along the lower reaches of Rocky Run Creek (Figure 3). The sandstone
aquifer has an areal extent of approximately 200 km^ extending from the site
to the southeast (Figure 5). Discharge from this aquifer is nearly constant
all year and averages about 0.013 m^/s per kilometer perpendicular to the
direction of flow.
The low areas also receive ground-water recharge from the upland areas
located on the site and adjacent to the site. These upland areas act as
recharge areas during precipitation events, when the infiltrating water flows
toward the low areas. Travel time for water from the upland areas to the low
-15-
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Figure 7. Topographic map of the site of the Columbia Generating Station.
-16-
-------
areas is usually less than 3 months, but flow rates are seasonally variable.
Before construction of the cooling lake, approximately 505t of the ground water
discharging in the low areas in the spring originated from the upland areas on
and near the site; during the summer the percentage dropped to less than 5%.
The ground-water inflow rates to the low areas are greatest where the
peat deposits are thinnest (Figure 8), namely in areas recently reworked by
fluvial processes. The peat deposits are generally less than 2 m thick west
of the railroad track; east of the railroad track, kettles in the outwash and
ice-contact drift have been filled in by organic deposits as thick as 10 m.
North of the ashpit alluvial deposits are more complex than in other parts of
the site; sediments range from coarse to very fine, indicating that
depositional processes characteristic of both the Wisconsin River and Duck
Creek have formed these deposits.
The cooling lake was designed to dissipate the waste heat, which is
discharged from the plant after steam has been used to generate the
electricity. The 200-ha lake has an effective length of 5,500 m. The lake
and the ashpit were formed by dikes 5 m high constructed entirely of local
silty sand. The western dikes around the cooling lake were lined with
bentonite. Water is pumped into the cooling lake from the Wisconsin River at
an average rate of 50,000 m3/day. Of this water 2Q% discharges into the
ashpit, 40$ is lost by ground-water seepage, and 40 J is lost by evaporation.
Hot water is discharged at a rate of 1 x TO** n>3 into the north end of the
lake east of the central divide (Figure 3). Water circulates in a clockwise
direction and is withdrawn from the north end of the lake west of the central
divide. Average residence time of a water particle in the lake is 5 days.
Water temperaturess at the discharge point average 10-15° C higher than
temperatures at the intake and decrease exponentially with distance from the
discharge point. The average annual range of lake temperature is 0-45° C.
Temperatures in the lake vary only slightly with depth.
Cooling towers have been built to dissipate some of the additional waste
heat that is being generated by the second unit. However, the cooling lake is
projected to receive 80% of the annual heat load; The discharge of the
additional waste heat into the cooling lake is estimated to increase lake
temperature by approximately 8° C when the cooling towers are not operating
(Wisconsin Public Service Commission 1974).
-17-
-------
INTERVALS
Less than 2.'. J I
Between 2' and 5'. FTT3
Between 5' and 10'. E2D
Between 10' and 20:
Over 20*.... ....ESi
Figure 8. Peat thicknesses in the subsurface at the site of the Columbia
Generating Station.
-18-
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SECTION 4
THE IMPACT OF-A POWER PLANT ON THE
GROUND-WATER SYSTEM OF A WETLAND
CONFIGURATION OF THE GROUND-WATfifl FLOW SYSTEM
The first goal of the hydrogeologic investigation at the site of the
Columbia Generating Station was to determine the impact of the station on
ground-water flows, temperature, and chemistry in the adjacent wetland
(Anderson and Andrews 1977, Andrews and Anderson 1978). To determine the
physical setting in the ground-water system of the wetland before operation of
the plant, data were collected beginning in the summer of 1971. The position
of the water table was monitored in 80 small-diameter observation wells, and
vertical head gradients were measured in 19 nested observation wells.
Two-dimensional steady-state models of several representative vertical
cross sections oriented perpendicular to the western dike of the cooling lake
were generated. (Figure 6 illustrates two of these cross sections.) These
models were then combined to construct a quasi-three-dimensional model of the
flow system. This model was used to simulate the head distribution before and
after filling of the lake and to compute the total ground-water discharge to
the wetland in the area affected by the lake. The vertically oriented cross
section models were patterned after the regional ground-water model of Freeze
and Witherspoon (1966). The finite difference equations were solved by using
the method of successive over-relaxation. A 25-by-50 node grid was used for
all simulations. Horizontal nodal spacing ranged from 15m near the dike to
120 m farther from the dike, and vertical spacing ranged from 0.6 m near the
surface to 30 m at depth. Details of the modeling procedure can be found in
Andrews (1976).
Field data were used to check the validity of the model, and the model
then computed ground-water discharge to the wetland. The simulated head
distribution in cross-section A-A' of Figure 6 before filling of the cooling
lake is shown in Figure 9. The ground-water flow system was modeled to a
depth of 150 m, but for clarity only the upper 30 m are shown. Horizontal
flows ranged from 0 to 4.1 cm/day. Vertical flows were mostly toward the
surface at rates of 0-0.16 cm/day. Approxmiately 300 m west of what is now
the western dike, vertical flows were as high as 0.53 cm/day. Total discharge
to the wetland in the cross section shown in Figure 9 was 1.00 m3/day per
meter dike length, of which 0.43 m3/day per meter discharged in the area of
high vertical flows shown in the figure.
A steady-state model was used to simulate pre-lake conditions, although
the water table fluctuated about 0.5 m/yr in the lowlands and as much as
-19-
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ELEVATION (METERS)
.30-.76m/yr
.!5-.30m/yr
.76-l.52m/yr
.30-.76m/yr
0.0-.30 m/yr
.30-76 m/yr
0.0-.30m/yr
Figure 9. Simulated head distribution in cross-section A-A * of Figure 6
before filling of the Columbia cooling lake. Equipotential
lines are labeled in meters. Vertical discharge rates are
given across the top of the figure.
-20-
-------
2 m/yr in the uplands. However, the water-table gradient remained nearly
constant throughout the year, except in the upland areas where the gradient
increased markedly during periods of recharge. The head distribution
presented in Figure 9 approximates the average annual head distribution.
The initial filling of the cooling lake began 13 June 1974 with water
pumped from the Wisconsin River. Pumping stopped on 17 July, when the level
was 0.5 m below the fill line. By 1 September the lake was almost empty, and
on 4 November 1974 pumping resumed. The water level reached the fill line on
2 January 1975. Water levels in observation wells monitored during the
filling of the lake showed increases in vertical gradients in the area west of
the cooling lake (Andrews 1976).
The filling of the lake greatly altered the head distribution in the
ground-water system (Figure 10). What had been one ground-water system became
three systems. Ground water that formerly discharged to the wetland began to
discharge near the drainage ditch east of the cooling lake. The lake became
the only source of water for the discharge area west of the lake. For the
cross section in Figure 10, the simulation predicted that the ground-water
discharge to the wetland west of the dike was 3.5 m^/day per meter and that
the flow into the drainage ditch was 3.7 m^/day per meter. As in the pre-lake
simulation, vertical discharge rates were highest approximately 240-300 m west
of the dike. In this main discharge area vertical flow rates varied from 1.02
to 1.34 cm/day. Elsewhere, vertical flows ranged from 0.24 to 0.94 cm/day.
The position of the main discharge area varied with the cross section modeled.
A comparison of total discharge rates to the wetland for Figure 9 and Figure
10 shows that discharge to the wetland in that cross section after filling of
the lake was 3 1/2 times greater than discharge before filling.
The effect of the cooling lake on the head distribution in several other
cross sections was also simulated. In all simulations the horizontal
component of flow was assumed to be perpendicular to the dike. Changes in
configuration of the flow system in other cross sections modeled were similar
to the changes shown in Figure 9 and Figure 10. In some cross sections,
however, the discharge to the wetland was 15 times greater after the filling
of the cooling lake. Total discharge to the area of the wetland affected by
the lake was estimated to be six times greater after filling.
The assumption of steady-state conditions after filling of the cooling
lake is acceptable in view of the stabilization of ground-water levels within
a month of the filling of the lake. However, since the water in the cooling
lake is warmer than the ground water in the system before the lake was filled,
the ground-water system is not yet in equilibrium with the altered thermal
regime. The model allows for variation in hydraulic conductivity as a result
of differences in ground-water temperatures. For the simulation presented in
Figure 10, the temperatures in the ground-water system were asssumed to vary
from 26° C directly beneath the cooling lake to 10° C at a depth of 20 m below
the lake. The relationship between ground-water temperature and ground-water
flows is discussed in section 5.
From the simulations and the supporting field evidence, the following
observations can be made. On the east and south sides of the cooling lake,
-21-
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ELEVATION (METERS)
.30-.76m/yr
.l5-.30m/yr
.30-.76m/yr
.76-l.52m/yr
l.52-3.04m/yr
.76-l.52m/yr
.30-l.52m/yr
INFLOW
7.6-25.4m/yr
Figure 10. Simulated head distribution in cross section A-A ' of Figure 6
after filling of the Columbia cooling lake. Equipotential
lines are labeled in meters. Vertical discharge rates are
given across the top of the figure.
-22-
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the drainage ditch effectively limited the impact of the lake on the
ground-water system to a relatively narrow zone. On the north and northeast
sides of the lake where there is no drainage ditch, the filling of the cooling
lake resulted in a rise of ground-water levels of almost 1 m. This rise
reversed flow in this area such that water now flows from the lake eastward
beneath the coal pile and discharges in the wetland area east of the site.
The average annual seepage from the cooling lake is estimated to be 0.18 to
0.27 m^/s. Flow rates fluctuate about 20$ during the year as a result of
changing ground-water temperatures.
The average annual seepage from the cooling lake will increase by
approximately 20% with both generating units in operation. The discharge
rates will also increase because of the formation of springs in the wetland
west of the cooling lake. Springs have formed since the filling of the
cooling lake, because vertical hydraulic gradients in many areas near the dike
exceeded 1 m/m, which approximates the critical gradient for the onset of
heaving in unconsolidated materials. In addition, increased temperatures in
the marsh have speeded up the process of peat decomposition, and deeper water
levels in the marsh have resulted in the floating of parts of the peat mat.
The result of both these processes is a decrease in the total load on the
confining silt-clay layer, a decrease that enhances the onset of heaving.
Heaving is not predictable, and therefore accurate estimates of its rapidity
and of its impact on flow rates are impossible. Potentiometric data from
observation wells in the marsh indicate that the springs that have formed
during the first 2 yr since the cooling lake was filled have not significantly
altered flow rates. Increases in flow rates caused by these processes will
probably not exceed 8% during the next 10 yr.
WETLAND WATER LEVELS
The wetland west of the cooling lake lies in two distinct drainage basins,
each of which is connected by distinct channels to tributaries of the
Wisconsin River. The wetland area adjacent to the cooling lake and north of a
line 350 m south of the intake channel drains to Duck Creek, and the area
south of this line drains to Rocky Run Creek via a small channel (Figure 3).
Ground-water inflow to the northern basin averages approximately 0.09+0.03
mVs, and inflow to the southern basin averages approximately 0.10+0.03 m3/s.
The water level in the wetland is a function of the ground-water inflow
rate, the rate of water leaving the wetland via surface outflow, the rate of
precipitation and evapotranspiration, and the wetland basin shape. The rate
at which water leaves the wetland is a function of the water level and the
shape of the channels draining the wetlands. On a daily basis
evapotranspiration from each wetland basin seldom exceeds 0.04 m3/s. Because
of the shape of the channels that drain the basins, a change in outflow from
0.04 to 0.20 m3/s raises water levels less than 5 cm.
Water levels in the wetland have been almost constant since the cooling
lake was filled. The level averages about 10 cm higher than previous levels
(Figure 11). Wetland water levels do fluctuate during the winter when ice
clogs the drainage channels and during floods on the Wisconsin River.
-23-
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0)
1C
IU
ui 238.05
UI
UI
_l
E
UI
237.74
237.44
cooling lake filled
JAN
1972
JAN
1973
JAN
1974
JAN
1975
JAN
1976
JAN
1977
Figure 11. Water levels in the wetland west of the Columbia cooling lake
before and after filling of the lake.
Before the cooling lake was filled, ground-water inflow rates in the
summer to each of the wetland basins were only 0.2 to 0.3 m3/s.
Evapotranspiration often exceeded inflow, and there was no outflow from the
wetlands via the channels. During the late summer water levels were lowered
as much as 0.6 m.
The peat substrate in the wetland has been altered by the increased flow
rates within the wetland, the increased water temperatures, and the increased
water levels. Some peat has been eroded, and new channels have been cut in
the wetland as the internal drainage system adjusts to the increased flow.
Other parts of the peat mat have floated to the surface and been broken down
by various physical and biological processes. Peat decomposition rates have
Increased because of the increased water temperatures and increased oxygen
content of discharging ground water. The plant community is rapidly
responding to the new environment as evidenced by the replacement of formerly
dominant sedges, Carex laoustris and Carex stricta, by plants more tolerant of
deeper water, primarily Sagittaria latifolia and flumex orbiculata (Willard et
al. 1976).
GROUND-WATER TEMPERATURE
Before the cooling lake was filled, the temperature of ground water
discharging to the wetland was approxmiately equal to the average annual air
temperature (10° C). The cooling lake, now the source of water discharging to
the wetland, has an average annual temperature between 10° C and 17° C,
depending on location in the lake.
In areas with a layer of relatively high permeability near the dike,
discharge rates are high and seepage from the lake has a travel time of less
than 1 yr before discharging to the wetland. Consequently, the range of
ground-water temperatures is similar to that of the lake water, although the
extremes are somewhat attenuated. The temperature of water in the western
-------
25 r
20
15
u
OT
111
Ul
10
•I I I I I I I I I I I I I
I I I I I I I I I
I I I I I
JAN MAR MAY JULY SEP MOV
1975
JAN MAR MAY JULY SEP
1976
NOV JAN MAR MAY
1977
Figure 12. Ground-water temperature variations in wells west of the Columbia
cooling lake. Curve A: data from a well cased to 4 m below the
surface and located 3 m west of the dike. Curve B: data from a
well cased to 2.5 m below the surface and located 60 m west of
the- dike.
portion of tne lake from May 1975 to May 1976 ranged from 1° C to over 30° C.
Ground-water temperature in a well 3 m west of the dike varied from 2° C to
25° C (Figure 12, curve A), and in areas west of tnis well, attenuation of
extremes was even greater (Figure 12, curve B).
To measure the heat discharge into the ground-water system, 47 thermistors
were installed in the summer of 1976 to depths of 10 m below the surface along
cross-section A-A' of Figure 6. Temperature data were collected twice a week
to document seasonal trends in the variation of ground-water temperature. The
temperature changes observed in the ground-water system are described in
detail in section 5. Preliminary calculations based on these thermistors
suggest that approximately 16 MM, less than 5% of the heat released to the
cooling lake, is discharged to the ground-water system. Ground-water
temperatures increased as much as 20° C, however, in some areas west of the
cooling lake.
Normally, the species of sedges present in the marsh emerge early and die
back late in the growing season. During the winter of 1975, however, sedges
in some areas of the marsh turned green in December because of the warm water.
In the spring these sedges were dead, possibly because they had used up their
stored food reserves. In 1976 these vegetation changes occurred over a
considerably larger area (Willard et al. 1977).
-25-
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RELATIONSHIP BETWEEN GROUND-WATER TEMPERATURES AND FLOW RATES
The groaid-water system was not yet in equilibrium with the altered
thermal regime. Because ground-water flow rates depend on temperature of the
ground water, leakage is greatest in late summer when the water is the
warmest. Furthermore, as ground-water temperatures rise in response to heat
input from the cooling lake, the flow rates will increase. The ground-water
flow model predicted that flows in the cross section shown in Figure 10 will
increase by 24J when the system reaches equilibrium. A model which couples
ground-water flow with heat flow is discussed in section 5.
CHANGES IN WATER CHEMISTRY
During the preoperational period (August 1972 to September 1974) 144 water
samples were taken from observation wells on the site. During the first 7
months of operation 134 samples were collected from 57 wells. Some of the
types of water on the site are illustrated by Stiff diagrams in Figure 13 and
Figure 14. '
Significant changes in the ionic composition of ground water and surface
waters in the wetland since the generating station began operation were
observed only near the ashpit. Because of the similarity in the chemical
composition of cooling-lake water and ground water, no significant changes are
likely to be observed in ground-water quality near the cooling lake. If
changes do occur, they are likely to be in sodium concentrations caused by
changes in ion-exchange processes associated with higher flow rates.
Major changes are likely to occur in ground-water quality near the ashpit
because ground water and water in the ashpit differ in quality. Prediction of
these changes is not yet possible because water levels in the ashpit have
fluctuated widely during the first 2 yr of operation and because the chemical
composition of ashpit waters has varied widely. Ground-water flow rates from
the ashpit have consistently ranged from 0.3 to 0.6 rn^/s. The filling of the
pit with ash has reduced seepage through the bottom, but this reduction has
been offset by increased flow through the dikes caused by elevated water
levels. Most ground-water flow from the ashpit now occurs through the dikes,
and most of the seepage water discharges near the dikes.
Some change has been observed in ground-water quality near the ashpit.
Calcium and sulfate concentrations have increased significantly in two wells
close to the ashpit dike (Table 2). Changes in water quality have been noted
in other wells farther from the dike, but these changes have not been as great
(Table 2). The plume of contaminated ground water appears to be confined to a
relatively small area near the dikes. However, pronounced increases in
specific conductivity in surface waters in the marsh have been measured up to
50 m from the dike, which suggests that water is leaking from the ashpit
through the ground-water system and is discharging to the wetland where the
water then spreads outward. Head gradients in observation wells support this
hypothesis.
The most significant degradation of ground-water quality has occurred in
the vicinity of the 17-ha coal pile. Although background sulfate
-26-
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- cr
S0=
(a)
COOLING LAKE
— cr
HCO-
Mg —
- so;
(b) SHALLOW GROUNDWATER
_ -n-
Co —
»
Mg -
— cr
- so;
(c) DEEP GROUNDWATER
1
CATIONS
1 2
ANIONS
EQUIVALENTS PER MILLION
Figure 13. Stiff diagrams of water in the cooling lake (a) and ground water
(b and c), Data were collected in fall 1975. Ground water (b) is
from a well cased to 2 m below the surface and located 50 m west
of the north end of the cooling lake; ground water (c) is from a
well cased to 6 m below the surface and located 60 m west of the
south end of the cooling lake.
-27-
-------
concentrations are less than 20 mg/liter, a plume of contaminated groind water
has been created with a sulfate concentration of greater than 1,000 mg/liter.
This concentration exists to a depth of more than 30 m near the coal pile and
extends more than 100 m to the east of the pile. The plume is gradually
spreading east, toward the wetland discharge area (Andrews 1976).
Ca
**-
Ca**-
Mg"-
— cr
- so;
(FALL 1975)
— cr
— HCOj * COj
-so;
(b) GROUNDWATER WEST OF ASH PIT (1972-1973)
Mg" -
- Cl"
— so
(C) GROUNDWATER WEST OF ASH PIT (FALL 1975)
J_
3
J_
J_
_L
2
CATIONS
1 2 3
ANIONS
EQUIVALENTS PER MILLION
Figure 14. Stiff diagrams of water in the ashpit (a) and in a well cased to
1.5 m below the surface and located 2 m west of the ashpit
(b and c). Comparison of (b) and (c) shows the change in water
quality after the partial filling of the ashpit.
-28-
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TABLE 2. SELECTED CHEMICAL CONCENTRATIONS IN GROUND WATER NEAR
THE ASHPIT OF THE COLUMBIA GENERATING STATION
(a/1)
K+
Na+
Ca++
Mg++
304
ci-
(m/1)
K+
Na+
Ca++
Mg++
son
ci-
Ashpit
water
2.3
10.2
77.8
10.6
44.6
10.6
Ashpit
water
2.4
10.3
54.7
4.3
36.0
10.5
1972-73
1.2
2.2
13.9
11.1
5.4
3.4
1972-73
1.7
4.3
18.8
17.3
7.7
2.8
A. West
1 m west
1975-76
9.2
8.5
16.9
9.8
18.5
4.5
B. North
1 m north
1 975-76
0.75
2.6
23.3
22.5
9.6
4.2
of the
1977
6.9
13-2
21.6
14.7
10.5
6.5
of the
1977
1.5
5.1
37.0
33.8
22.0
8.5
dike
1972-73
1.2
3.6
31.6
20.3
0.8
4.49
dike
1972-73
2.0
5.2
22.0
18.0
9.7
7.2
75 m west
1975-76
0.75
3.5
28.8
22.0
0.2
2.64
25 m north
1975-76
0.75
3.1
22.6
18.8
9.3
3.2
1977
0.5
4.0
24.3
16.9
0.1
5.0
1977
0.5
3.7
21.5
25.8
14.0
3.5
-29-
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SECTION 5
THERMAL ALTERATION OF GROUND WATER CAUSED BY
SEEPAGE FROM THE COOLING LAKE
In addition to monitoring and modeling groind-water flow rates near the
Columbia Generating Station, we investigated the effect of the station's
cooling lake on groind-water temperatures (Andrews and Anderson 1979).
Ground-water temperatures in the vicinity of the cooling lake were monitored
in detail in the field for 1.5 yr. The presence of the cooling lake, which
loses water to the ground-water system at a rate of 2 x 10^ m3 per day, has
created a zone of thermally altered gromd water, but the zone is confined to
a relatively small area hydraulically downgradient from the cooling lake.
A mathematical model was developed to simulate the response of subsurface
temperatures to seasonal changes in lake and air temperatures. To create the
model, equations describing groind-water flow were coupled with equations for
heat flow in the subsurface. An equation describing the rate of heat loss
from the marsh surface was used as one of the boundary conditions for the
heat-flow model. The model was solved numerically by using the finite element
technique. The model assumed that the flow of heat and water from the cooling
lake occurred in planes normal to the plane of the dike. The flux of water
and heat was modeled in seven of these planes (Figure 15), which represent
two-dimensional vertical cross sections of the ground-water system. Results
from each of the cross sections were combined to obtain the total flow of heat
and water outward from the west dike of the cooling lake.
Simulated temperature patterns agreed well with field data, but were very
sensitive to the distribution of subsurface lithologies. Results from a
predictive simulation suggest that operation of the second 500-MW unit will
increase ground-water temperatures less than 5° C at distances greater than
15 m from the cooling lake. The results of this study suggest that the
potential for significant thermal alteration of surface water bodies located
in ground-water discharge areas is slight.
MATHEMATICAL MODEL
Governing Equations
The equation describing the two-dimensional flow of water through a
nonhomogeneous aquifer may be witten
-<>•'• J'1-2 • '"
-30-
-------
CD
GENERATING
STATION
MINT
FARM
DRAIN
D1
Figure 15. Location of seven cross sections of the Columbia site for which
ground-water temperature distributions were simulated.
-31-
-------
wnere KJJ = hydraulic conductivity, L/t; = head, L; and x-|,X2 = cartesian
coordinates, L. Tne ground-water velocity or specific discharge can then be
determined from
qi * ~Kij Ih ' (2)
The movement of heat in a ground-water system can be described
mathematically assuming that: (1) Thermal equilibrium between the liquid and
the soil particles is achieved instantaneously, (2) the densitiy of the soil
particles is constant, (3) the heat capacity is constant, and (4) the chemical
system is inert. Under these assumptions the heat transport equation is
-P-°.l.'-1.* '
where T = temperature, T; Djj = coefficient of dispersion, H/TtL; pCw = heat
capacity of the saturated media, H/L^T; pCw = heat capacity of water, h/L^T;
and Qi s specific discharge in direction x^, L/t.
If the medium is isotropic, the coefficient of dispersion is a second-rank
tensor (Scheidegger 1961) composed of two parts: (1) The thermal conductivity
of the medium and (2) the coefficient of mechanical dispersion, which
represents the mixing caused by the heterogeneity of the velocity field. In
this analysis the coefficient of dispersion was assumed to have the following
form (Heddel and Sunada 1970):
D
q2q2
'22 " *22 + "L ~~q~ + *r
D21
2 2 1/2
where <1 " (ij + ^ » K11 = K22 = thermal conductivity, H/TtL (thermal
conductivity is assumed to be independent of direction); QL = longitudinal
(horizontal) dispersivity, L; and <*T = transverse (vertical) dispersivity, L.
Boundary Conditions
for the ground-water flow model, heads were specified at the water table
and along the vertical boundary east of the cooling lake. Tne model can be
modified to consider the time-dependent case of a moving water table, but for
the problem considered here fluctuations of the water table were negligible.
Mo-flow boundaries were specified along the lower boundary and at the
Wisconsin River.
for the heat-flow model, a heat flux was specified along the upper
boundary and along the vertical boundary east of the cooling lake. The fluxes
of heat across these boundaries were specified to be proportional to the water
-32-
-------
flow across the boundaries. Along that part of the upper boundary
representing the marsh, a heat flux resulting from atmospheric exchanges was
specified. The other two boundaries were specified as no-flow boundaries.
The process of heat exchange between the marsh surface and the atmosphere
is complex, and mathematical representations are necessarily rough
approximations. For this study the boundary between the marsh and the
atmosphere was assumed to occur in the vegetation at the height viiere transfer
of heat and water vapor by turbulent exchange becomes effective. This level,
known as the zero-displacement plane, was assumed equal to 0.63 of the
vegetation height. Heat flow from the ground surface to the zero-displacement
plane was assumed to occur by conduction and free convection. The heat flow
at this boundary can be expressed as
IB = \ (Tm * V + hm (em ~ e
where Tg^y = effective sky temperature, T; Tm = temperature at the marsh
surface, T; Ta = air temperature above the plane of zero displacement, T; qs =
heat flux from the marsh boundary, H/tL2; h™ = mass transfer coefficient,
H/L2Tt; hy = heat transfer coefficient, H/L^Tt; Y = psychrometer constant,
assumed equal to 0.66 mbar/T, M/t2LT; em = saturation vapor pressure at the
marsh boundary temperature, M/t2L; ea = partial vapor pressure of water in
air, M/t2L; ? = marsh boundary long-wave emissivity, dimension! ess; a =
Stefan-Blotzmann constant, H/T^tL2; otg = marsh boundary solar absorptivity
dimensionless; qsun = solar flux incident on the marsh boundary, H/tL2; and a
= percentage of open water in the marsh, dimensionless.
In addition to areas of dense vegetation, the marsh also contains some
areas of open water. For the latter it was assumed that radiative exchange
and solar absorption occur at the surface.
The heat and mass transfer coefficients (hy and h^) are equal if the
roughness height is the same for both. According to Mitchell et al. (1975)
these coefficients are given by an equation of the form
hm = hv = ^ PCpu/ln(Z
where k = Karman coefficient, dimensionless; PCp = heat capacity of air,
H/L^T; u = air velocity, L/t; Za = reference height, L; Zo = surface roughness
height, L; and d = height of the zero- displacement plane, L. The surface
roughness height can be estimated from vegetation height by using the
relationship given by Tanner and Pelton (1960), log Zo - 0.997-log 0.833 ,
where ^ is the average height of the vegetation.
Following the method of Jobson (1972), we used daily average values for
air temperature (Ta) and vapor pressure (ea) in Eq. (6). Moreover, Eq. (6)
can be simplified when temperature data at the marsh surface are available,
and the terms involving higher order dependence on Tm are approximated by
linear relations. Then, the heat transfer from the marsh surface is
proportional to the temperature difference between the marsh surface and the
-33-
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air temperature at 2 m above the surface. The constants of proportionality
were determined for a cross section for which data had been collected, and
these values were used when simulating heat flow in the other cross sections.
The heat-flow rates calculated by this method agreed well with values
calculated by using Eq. (6).
Solution Procedure
The finite element method (Zienkiewicz 1977, Finder and Gray 1977) was
used to solve fiq. (1) and Eq. (3) subject to the boundary conditions described
in the previous section. Each cross section modeled was divided into 100-150
quadrilateral elements. The procedure used to link the equations was: (a)
Eq. (1) was solved for head at each node and Eq. (2) was solved for velocity;
(b) fiq. (3) was solved for temperature at each node; (c) the solution to Eq.
(3) was stepped forward in time by the Crank -Nicolson approximation for the
time derivative for a specified number of time steps; (d) the hydraulic
conductivity distribution, which is a function of temperature, was adjusted
for the new temperature distribution; and (e) steps (a)-(d) were repeated.
Required
The physical properties required as input parameters are thermal
conductivity, specific heat capacity, hydraulic conductivity, and porosity for
each material type in the subsurface and dispersivity.
The values of the parameters for each of the subsurface materials used in
the simulations are listed in Table 3- Thermal conductivities were determined
by the needle probe technique (Von Herzen and Maxwell 1959). Heat capacities
were determined by standard additive techniques (Van Wijk and deVries 1963).
Horizontal hydraulic conductivities were estimated from aquifer test data,
slug tests, and grain-size analyses. Vertical hydraulic conductivity in each
element vets assumed to be one-tenth of the horizontal hydraulic conductivity.
Porosities, which were needed to estimate pCs, were approximated from
grain-size analyses.
Longitudinal dispersivity was assumed constant for all elements and was
estimated by a trial-and-error adjustment procedure. Lateral dispersivity is
usually assumed to be one-quarter to one-tenth of longitudinal dispersivity
(Cherry et'al. 1975). For the simulations reported here, the ratio of
longitudinal to transverse dispersivity was assumed to be 4. The field data
were best simulated by using a longitudinal dispersivity of 10 cm. When
higher values were used the amplitude of the annual temperature wave observed
in the field could not be reproduced. In Figure 16 the observed temperatures
2 m west of the dike at a depth of 3 m are plotted along with the simulated
temperatures obtained by using various values for longitudinal dispersivity.
Values for longitudinal dispersivity reported in the literature range from
0.1 cm for laboratory studies with homogeneous sands (Hoopes and Harleman
1967) to over 100 m for field studies (Bredehoeft et al. 1976). The value of
dispersivity is known to be related to media inhomogeneities and the scale of
the problem, but the exact relationship is unknown. For the problem described
-34-
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TABLE 3- PARAMETERS USED IN THE SIMULATIONS FOR THE MODEL
DESCRIBING THERMAL ALTERATION OF GROUND-WATER
' Lithology
Horizontal
hydraulic Thermal Heat
conductivity conductivity capacity
(m/day) (1(H cal/m sec°C) (cal/cm3<>c) Porosity
Medium coarse sand 30
with gravel
Fine to very fine 10
sand
Weathered sandstone 8
Sandstone 3.5
Peat 3.0
Sandy silt to sandy 0.4
clayey silt
Gray sandy silt with 0.04
organic matter
Varbed clay with 0.03
sand seams
0.45
0.51
0.51
0.64
0.12
0.42
0.42
0.40
0.64
0.67
0.67
0.60
0.91
0.72
0.72
0.74
0.33
0.39
0.39
0.26
0.75
0.48
0.48
0.52
RESULTS
Field Data
Temperatires were monitored twice a week at 48 points in cross-section
A-A1 in Figure 15 for 13 months and at 18 points in cross-section B-B1 in
Figure 15 for 6 months. Temperatures were recorded at the surface and at
depths to 10 m below the surface. In situ thermistors, hardwired to a central
control box, were used to obtain ground-water temperatures in cross-section
A-A1. Temperatures in cross-section B-B1 were measured by lowering a
thermistor probe into three 3.175-cm wells. The thermistors were calibrated
to ±0.1° C. A high-resolution digital ohmmeter was used to measure the
resistance of the thermistors. Surface-water temperatures at two sites along
cross-section A-A' were monitored continuously with liquid expansion
thermographs.
-35-
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3cm
— — o90 cm
o30cm
o|0cm
1978
Figure 16. Observed temperatures in the subsurface west of the Columbia
cooling lake and temperatures simulated by the mathematical
model. The ratio of longitudinal dispersivity (CXL) to
transverse dispersivity (al) is set at 4. ( ) is observed
temperatures with o£ = 3 cm. ( ) is simulated temperatures
with OL = 90 cm. (o) indicate maximum and minimum simulated
temperatures for cxL/ctT = 4 with L equal to 3, 10, 30, and 90 cm.
Tfte temperature distributions recorded in cross-sections A-A' and B-B| at
3-mon?h intervals are shorn in Figure 17 and Figure 18. Dissimilarities in
the temperature distributions recorded in the two cross sections are
attributed to differences in the subsurface materials, which result in
different distributions of ground-water velocity Average sround-water
velocity is slower in cross-section B-B' by a factor of 2. The fluctuations
In average lake temperature and the seasonal fluctuations of 8™^-^er
temperature at several distances from the dike along cross-section A-A at a
depth of 4.5 m are illustrated in Figure 19. The lag time »>et«en tto
occurrence of-the maximum temperature in the ground water and the maximum
cooling-lake temperature gradually increases, and the amplitude of tne
fluctuation decreases with distance from the dike.
-36-
-------
(b) JANUARY 27, 1977
0 METERS
Figure 17. Observed ( ) and simulated ( ) ground-water temperature
distributions in cross-section A-A' of Figure 15.
-37-
-------
Simulations
Good agreement between simulated and actual temperatures was obtained when
the model was used to simulate temperatures in cross-sections A-Af and B-B'
(Figure 17, Figure 18, Figure 20), but only because detailed data on the
subsurface distribution of materials had been obtained. The temperature
patterns observed in the subsurface were very sensitive to the distribution of
layers with low hydraulic conductivity. The model was most sensitive to the
extent and depth of the clay layer under the peat (Figure 8) and to a clay
layer that is present in some areas at a depth of 6-8 m. Over 70 borings were
made in the marsh to determine the subsurface lithology.
(a)
5
10
15
APRIL 26, 1977
-15e-
100
0 METERS
Figure 18. Observed ( ) and simulated ( ) ground-water temperature
distributions in cross-section B-B1 of Figure 15.
-38-
-------
30
O
ui
ff
D
ff
UJ
20
10
UJ
0
1976
N
M A
1977
M
Figure 19. Seasonal fluctuations of ground-water temperature in cross-
section A-A' of Figure 15 at a depth of 4.5 m at distances of
2, 15, 50, and 84 m west of the cooling lake dike. The
fluctuations of average lake temperature are also shown ( )
-39-
-------
30 II l| I l[ I I I H I I I I I I I I I I I I I I I I I I I I I I M I I I I I l| I I I I I I I I I ' I I M I "IT
2 METERS WEST OF DIKE
07 \
25 P
30 M 111111111111111; 1111111111 " I " I' M 1111111111111' 11 • 11 •
5O METERS WEST OF DIKE
25-
i.. I.. 11. i.. 111111 h , 1111 1
JJASONDJFMAMJJASONDJFMA
1976 1977 1978
JJflSONDJFMAMJJASONDJFMA
1976 1977 1978
n j i ij 11 j 1111111' 11«l^n ITl »1TTT
15 METERS WEST OF DIKE
1 | i 11 I 11 I 11 I I | M | I 111 I | I 11 11 11 I 11 I I I I | " I'
84 METERS WEST OF DIKE
JJASONDJFMAMJJASONDJFMA
1976 1977 1978
Q h I I 1 1 I 1 • I I I I M I I I I I I I I I I I 1 I I I I I . I I I I I I I I I I I I I I I I I 1 I II I I I I I ! I I I I 1
JJASONDJFMAMJJASONDJFMA
1976 1977 1978
!
25
1111| • 11111111111" | M I' • I " I " I " I''
32 METERS WEST OF DIKE
„
JJASONDJFMAMJJASONDJFMA
1976 1977 1978
Figure 20. Simulated temperatures (— —) and observed temperatures (o) in
cross-section A-A1 of Figure 15 at a depth of 3.28 m at 2, 15,
32, 50, and 84 m west of the cooling lake dike.
-40-
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SECTION 6
LONG-TERM TEMPERATURE CHANGES IN THE GROUND WATER OF THE WETLAND
SIMULATION STUDIES
Simulations were used to predict the long-term changes in substrate
temperature that may occur in the wetland adjacent to the Columbia Generating
Station; Wetland water levels and ground-water flow rates into the wetland
had stabilized during the first 2 yr of operation of the generating station,
but neither ground-water temperatures nor wetland vegetation (Bedford 1977)
had yet reached equilibrium. Prediction of the nature and magnitude of change
in vegetation was not possible without an understanding of the probable
changes in substrate temperatures.
The model that was described in section 5 to simulate the response of
subsurface temperatures to changes in the temperatures of the cooling lake and
the air was used to simulate seasonal temperature patterns for the 12-yr
period from November 1974 to January 1987. The governing equations, boundary
conditions, microclimatic model, and parameters remained the same in the
long-term simulation. Changes in temperature for the 12 yr were modeled in
the same two planes, which represent two-dimensional vertical cross sections
of the ground-water system (A-A1 and B-B1 in Figure 15).
Cooling-lake temperatures and air temperatures for the 10-yr period
1978-87 were synthesized by repeating five times the actual temperatures
recorded from January 1976 to January 1978. For the period after April 1978
when two generating units were assumed to be operating, the temperatures were
adjusted to account for the added heat load. When the temperature at the
generating station outlet rose above 40° C, it was assumed that the entire
heat load from the second generating unit was dissipated in the cooling
towers. The cooling-lake temperatures at the generating station inlet for the
period 1975-87 are shown in Figure 21.
The simulations predict that temperatures at a depth of 0.6 m will not
fall below 8° C within 200 m of the dike by 1987 and that peak temperatures
near the dike will be 10-15° C above normal and will occur in October and
November rather than in August. The subsurface stratigraphy of the site is
such that major changes in near-surface temperatures will only occur within
350 m of the dikes, but if the stratigraphy were different the effects could
extend to much greater distances from the cooling-lake dikes.
The temperature changes simulated in the ground-water system for the
period 1975-87 are only an approximation of the actual temperatures that may
exist in the ground-water system during this period. Lake and air
-41-
-------
I 1 1 1 1 1 1
III
78 79 80 82
YEAR
83 84 85 86 87
Figure 21. Cooling-lake inlet temperatures for 1975-87 used in the
simulations of long-term temperature change in the ground-water
system. Actual temperatures are shown from May 1975 to March
1978. Temperatures from March 1978 to January 1987 were
synthesized from the 1976-78 temperature record.
-42-
-------
temperatures may deviate widely from those assumed in this report, and
processes not accounted for in the model, such as substrate decomposition and
flood-induced erosion, may alter ground-water flow rates. These simulations
are, however, a reasonable approximation of the changes to be expected in
ground-water and substrate temperatures in the vicinity of the cooling-lake.
RESUUS AND DISCUSSION
The simulation of temperatures in the ground-water system near the
cooling-lake shows that temperatures in the vegetation rooting zone in the
wetland will reach a new steady-state condition that is much different from
the prevailing condition before the filling of the cooling-lake. The annual
temperature patterns near the dike will stabilize by 1980, but at 150 m from
the dike temperatures will not reach equilibrium by 1987.
Simulated temperatures in cross-section A-A1 (Figure 15) 2 m from the dike
from 1975 to 1987 at a depth of 3 m deviated from those at 0.6 m by less than
1.0° C (Figure 22). When pre-lake temperatures are compared with simulated
temperatures at 0.6 m depth and 2 m west of the dike in cross-section A-A1 for
the periods 1978-86 and 1984-86, the change in annual temperature patterns is
pronounced. In the period 1984-86 the temperature does not fall below 14° C,
whereas before the cooling lake was filled, substrate temperatures at this
depth approached freezing and in the 1976-78 period the temperature fell below
5° C (Figure 23). The peak temperatures occur approximately 45 days later in
the period 1976-78 than in the pre-lake period and approximately 30 days later
in the 1984-86 period than in the pre-lake period. The lag decreases as the
ground-water temperature rises because the hydraulic conductivity of the
subsurface material increases with temperature,vwhich increases the
ground-water flow velocity.
The simulated temperatures in cross section A-A1 150 m from the dike for
the period 1975-87 are shown in Figure 24. Pre-lake temperatures at a depth
of 0.6 m and temperatures simulated for the periods 1976-78 and 1985-87 are
shown in Figure 25. The attenuation in the annual temperature patterns
between the pre-lake period and the 1976-78 period is a result of the
increased ground-water flow rate in the latter period. At a distance of 150 m
from the dikes, several years are required before the impact of the
temperature variations in the lake become pronounced at a depth of 0.6 m. By
1986 the change in temperatures is pronounced with maximum and minimum annual
temperatures elevated several degrees centigrade above pre-lake temperatures.
In cross-section A-A1 (Figure 15) most of the ground-water discharge
occurs within 200 m of the dike. Consequently, the atmospheric heat flux
beyond this distance is much larger than the ground-water heat flux, and thus
substrate temperatures in 1987 at a depth of 0.6 m are only slightly altered
from pre-lake conditions. Ground-water temperatures at greater depths do show
increases, but the rate of increase at a depth of several meters is much
slower than shown in Figure 24 for a point at a depth of 0.6 m and a distance
of 150 m from the dike. (Text continues on p. 48.)
-43-
-------
i 1 1 1 1—in 1
5 76 77 78 79 80 81 82 83 84 85 86 87
Figure 22. Predicted ground-water temperatures from 1975-87 at a distance
of 2 m west of the cooling-lake dike in cross-section A-A1 of
Figure 15 at a depth of 0.6 m.
-44-
-------
JAN
MAY
SEPT
JAN
MAY
SEPT
JAN
Figure 23. Pre-lake temperatures and simulated temperatures 2 m west of the
cooling-lake dike in cross-section A-A' of Figure 15 at a depth
of 0.6 m. ( ) = simulated temperatures from 1984 to 1986;
( ) = actual temperatures from 1976 to 1978; (•••) = simulated
temperatures from 1976 to 1978 assuming that the cooling lake
had not been present.
-45-
-------
O
o
UJ
oc
30
25
2O
UJ
o.
2
UJ 10
I I I
i i i r
75 76 77
78 79 80 81 82
YEAR
83 84 85 86 87
Figure 24. Temperatures 150 m west of the cooling-lake dike in cross-
section A-A1 of Figure 15 at depths of 0.6 m and 3 m for the
simulated period 1975-87. (—), 0.6 m level; ( ), 3 m level.
-46-
-------
MAY
SEPT
JAN
MAY
SEPT
JAN
Figure 25. Temperatures 150 m west of the cooling-lake dike in cross-section
A-A' of Figure 15 at a depth of 0.6 m. ( ) = simulated
temperatures for 1984-86; ( > = actual temperatures for 1976-78;
(•••) = simulated temperatures for 1976-78.
-47-
-------
The temperature changes simulated for cross-section A-A' are believed to
be representative of the changes in temperature that will occur in the
vegetation rooting zone in the wetland along the dike. The ground-water flow
patterns, which are determined by the subsurface stratigraphy, determine that
the obvious temperature changes will occur within 350 m of the dike. The
increase in temperatures will be somewhat greater in the southern part of the
wetland since the average annual lake temperature increases from approximately
21° C at the intake to 2^° C at the south end of the lake. This difference is
shovm in Figure 26 (a-c), which presents simulated temperatures in
cross-section B-B' of Figure 15. This cross section is 200 m south of the
cooling lake intake and 1,500 m south of cross-section A-A1. Temperatures
were simulated for 1985-8? at a depth of 0.6 m and at 2, 50, and 200 m west of
the dike. Temperatures 2 m west of the dike (Figure 26a) averaged about 3° C
higher than temperatures at a similar point on cross-section A-Ar.
MAY
SEPT
JAN
MAY
SEPT
JAN
Figure 26a.
Temperatures in cross-section B-B1 of Figure 15 at a depth of
0.6 m and at 2 m west of the cooling-lake dike. ( ) =
simulated temperatures for 1985-87; (•") = simulated tempera-
tures for 1976-78 assuming no cooling lake was present.
-48-
-------
UJ
QC
oc
UJ
Q.
30
25
20
15
nt>
10
JAN
MAY
SEPT
JAN
MAY
SEPT
JAN
Figure 26b.
Temperatures in cross-section B-B* of Figure 15 at a depth of
0.6 m and at 50 m west of the cooling-lake dike. ( ) =
simulated temperatures for 1985-87; (•••) = simulated
temperatures for 1976-78 assuming no cooling lake was present.
-49-
-------
JAN
MAY
SEPT
JAN
MAY
SEPT
JAN
Figure 26c.
Temperatures in cross-section B-B' of Figure 15 at a depth of
0.6 m and at 200 m west of the cooling-lake dike. ( ) =
simulated temperatures for 1985-87; (•••) = simulated
temperatures for 1976-78 assuming no cooling lake was present.
-50-
-------
R INFERENCES
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Wisconsin-Madison, Madison, Wis.
Andrews, C.B. 1976. An analysis of the impact of a coal-fired power plant on
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Andrews, C.B., and M.P. Anderson. 1978. Impact of a power plant on the
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APPENDIX A
DATA SPECIFICATIONS FOA THE HATER-FLOW MODEL
This appendix details procedures used at the site of the Columbia
Generating Station to gather field data on the hydrogeologic system. The data
were collected from 1971 to 1975. They were used to document changes in the
system caused by construction and operation of the generating station and to
establish the boundary conditions and parameters for the model of the
ground-water flow system.
DATA FOR SPECIFYING BOUNDARIES
Side Boundaries
The position of the site in the regional ground-water flow system was
needed to determine the size of the area to be modeled. Water-level data and
well logs were collected from more than 100 domestic wells in the vicinity of
the Columbia Generating station. Most of the data were obtained from the
U.S. Geological Survey in Madison, Mis. The data were used to estimate
aquifer transmissivity, flow rates, and the extent of the regional
ground-water flow system. Transmissivity was estimated from pump tests and
specific capacity data. Flow rates were computed by using the estimated
transmissivity value and the water-table gradients existing in the field. The
extent of the flow system was estimated by contouring water-level data from
the wells open to the sandstone aquifer in the vicinity of the site.
The information on the regional ground-water system was used to justify
the assumption that the left and bottom boundaries of the cross section
modeled were no-flow boundaries and to specify the flow rates for the right
boundary of the cross sections modeled.
Upper Boundary
Over 80 well point piezometers 3.2 cm in diameter were installed on the
site to monitor the position of the water table. The piezometers had 46-cm,
80-gauge screens. They were driven in place in the lowlands and augered into
place in the uplands. They were positioned so that they were open to the
aquifer Just below the water table. Monitoring was done with a steel
measuring tape at monthly intervals beginning in summer 1971. The water
levels in two wells were recorded continuously. Beginning in fall 1974 water
levels were recorded at 14 points in the wetlands on a monthly basis. All
observation points were surveyed to establish relative elevations. The
locations of the wells are shown in Figure A-1.
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• U.W. MONITORING WELLS
© D.N.R. WELLS
* WELLS THAT HAVE BEEN REMOVED
61
• 58
MILES
Vt
77-79
•
KILOMETERS
Figure A-l. Location of monitoring wells at the Columbia site.
-58-
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Water-level data from all monitoring wells were punched onto IBM Cards. A
computer program was written in PL/1 (Programming Language, University of
Maryland) that graphs the water level in each well by water year, computes the
rate of change in water levels, and computes the water-table gradient between
the observation points and the rate of change in the gradients. These
water-level data were used to specify ground-water potential along the upper
boundary of the cross sections modeled and to justify the use of a
steady-state model.
DATA FOR SPECIFYING STATE VARIABLES
Permeabilities
The range of permeabilities for the various materials on the site was
determined by combining data from a pump test, slug tests, and laboratory
permeability tests with lithologic and grain-size information. Permeability
distribution.in the system was then modeled by using lithologic and grain-size
information from over 100 borings on the site. (The locations of the borings
are shown in appendix B.) A pump test was run in the alluvial materials on
the site in winter 1972 by the Lane Engineering Firm of Chicago, 111. The
data were analyzed by using the Dupuit-Theim equation for steady radial flow
without vertical movement. Slug tests were run on 23 of the observation wells
on the site. Water was withdrawn from the wells, and the recovery time was
recorded. The data from these tests were analyzed with the techniques
developed by Cooper, Bredehoeft, and Papadopulos (1965).
Laboratory permeabilities were run on 23 samples with soil test
permeameters under both falling and constant head conditions and under various
density conditions. Grain-size analyses (percentages of medium sand, fine
sand, silt, and clay) were run on 43 diverse samples from the site. The
information was then correlated with the permeability tests, and a working
model was formulated between lithology and permeability. This correlation of
permeabilities with lithologies was used with the logs of the borings to
develop seven cross-sectional permeability models.
Temperature
Temperatures were monitored bimonthly beginning in 1973 by lowering a
Yellow Springs Instrument thermistor into observation wells. Before the
temperature was recorded, the observation well was pumped with a hand-operated
vacuum pump until the temperature of the water became nearly constant.
CALIBRATION DATA
In addition to the data collected to define the boundary and state
variables in the system, extensive field data were collected to provide a
check on model simulations.
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Piezometer Nests
Nineteen observation wells were installed in the lowlands at the site.
They were open between 4.5 and 7.6m below the water table. Seven wells were
installed in the uplands so that they were open between 9.1 and 18.3 m below
the water table. These wells were placed adjacent to an observation well open
just below the water table. Water levels in these wells were monitored at
monthly intervals. The vertical gradients and the rate of change of vertical
gradients have been computed for each of these wells for each monitoring date.
The data from these wells were used to judge the fit of the models'
simulations of the ground-water system to the actual situation.
Direct Measurements of Flow
Surface water in the wetland west of the cooling lake drains through
well-defined channels during low river stages. The amount of discharge
through these channels was monitored several times during fall 1975 on
overcast days with a rod-suspended pygmy current meter. Measurements were
taken at 0.6 of the total depth at 1-ft intervals across the streams. By
similar methods the flow in the drainage ditch on the east side of the cooling
lake was measured several times at both the northeast corner and the southwest
corner of the cooling lake.
Discharge into the drainage ditch on the east side of the lake from the
ground-water system was monitored with seepage collectors. Seepage collectors
are 55-gal barrels cut in half whose open end is positioned into the substrate
at the bottom of the ditch (Lee 1977). On the closed end, which is also
submerged, the spout is covered by a plastic bag. The flow of water into the
plastic bag is recorded, which provides a direct measurement of the
ground-water inflow into the ditch in the area covered by the barrel. By
means of the seepage collectors, rates of ground-water flow into the drainage
ditch on the east side of the lake were determined at various locations on
several different dates.
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APPENDIX B
TECHNIQUES FOR DETERMINING BOUNDARY CONDITIONS AND
PARAMETERS IN THE HEAT-FLOW MODEL
The model developed to simulate the transport of heat away from the
cooling lake at the Columbia Generating Station requires for its solution the
specification of initial and boundary conditions for the partial differential
equations describing heat flow and the equations describing water flow. Five
sets of parameters are also required: Hydraulic conductivity, storage
coefficient, thermal conductivity, heat capacity, and dispersivity. The field
and laboratory techniques used to define the boundary conditions and the
parameters needed for the water-flow equation were described in appendix A.
The field techniques used to measure temperature and the laboratory techniques
used to measure thermal conductivity and heat capacity are described in this
appendix. The technique used for specifying the dispersivity parameters is
described in section 6.
TEMPERATURE MEASUREMENTS
Temperatures were recorded weekly at 40-110 points in the subsurface in
the vicinity of the Columbia Generating Station site from August 1976 to
January 1978 (Figure B-1). The depths at which temperatures were taken, the
dates when readings were taken, and the manner in which temperatures were
measured are listed in Table B-1.
Most temperatures in the field were measured with an electronic device
with a thermistor as the temperature sensor. A thermistor sensor system was
used because of its accuracy and its versatility. A relative accuracy of
0.1°C is easy to maintain in the field, and probes could be buried, lowered
down wells, or located far from an accessible point. Liquid expansion
thermographs with an accuracy of + 0.5°C were used for continuous temperature
recording because these mechanical devices are much easier to maintain and
less expensive than a continuous recording device with a thermistor sensor.
An electronic temperature measuring device is extremely simple in concept
(Figure B-2). It consists of a thermistor whose electrical resistance is
almost entirely a function of its temperature as the sensing device and an
electronic measuring device to measure either directly or indirectly the
resistance of the sensor, generally an ohmmeter or a wheatstone bridge. Two
types of thermistors and three measuring devices were used at the Columbia
Generating Station site.
-61-
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Figure B-l.
Location of temperature sampling points at the Columbia
Generating Station site.
-62-
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TABLE B-l. TEMPERATURE SAMPLING POINTS: LOCATIONS, DEPTHS,
PROBES, AND FREQUENCIES OF READINGS
Depth below
Sampling surface at which
point Distance temperature
number from dike (m) was measured (m)
130
131
132
133
134
135
35
2 0.
3.
15 0.
1.
6.
32.5 0.
1.
6.
51
86
131.5
15,
05,
15,
51,
10,
15,
52,
10,
3 2,3,4
0.
6.
0.
3.
9.
0.
3.
9.
91, 1.52,
10, 8.70
46, 0.91
05, 4.57
76
46, 0.91
05, 4.57
76
,5,6,7,8,9
Probe used
to take
temperature
in situ
in situ
in situ
in situ
in situ
in situ
probe
probe
probe
probe
probe
probe
probe lowered
Approximate
frequency
of readings
8/8/76-8/77 weekly
9/77-1/78 monthly
8/8/76-8/77 weekly
9/77-1/78 monthly
8/8/76-8/77 weekly
9/77-1/78 monthly
8/8/76-8/77 weekly
9/77-1/78 monthly
8/8/76-8/77 weekly
9/77-1/78 monthly
8/8/76-8/77 weekly
9/77-1/78 monthly
8/8/76-1/17/78
36
37
68
24
28
140
31
61
194
2,4,6,7,8
2,3,4,5
4,6,8,10,12
3,4,5,6,7,8,9,10
2,4,6,8,10,12,13
2,4,6,8
down 3.175 cm;
PVC pipe
probe lowered
down 3.175 cm;
PVC pipe
probe lowered
down 3.175 cm;
PVC pipe
probe lowered
down 3.175 cm;
galvanized pipe
probe lowered
down 3.175 cm;
PVC pipe
probe lowered
down 3.175 cm;
PVC pipe
probe lowered
down 3.175 cm;
galvanized pipe
weekly
8/10/77-1/18/78
weekly
8/10/77-1/18/78
weekly
8/10/77-1/18/78
weekly
5/31/77-1/18/78
weekly
8/10/77-1/18/78
weekly
4/22/77-1/18/78
weekly
-63-
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TABLE B-l (continued).
Depth below
Sampling surface at which Probe used
point Distance temperature to take
number from dike. (m) was measured (m) temperature
Approximate
frequency
of readings
143
43
75
57
L1/L2
37
126
73 west side of
ditch
74 east side of
ditch
14 m east of
ditch
Lake inlet
and outlet
Areal coverage
of marsh (not
shown in
Figure B-l)
Transect 1 0.2 m
to transect
42 (not shown
on Figure B-l)
2,4,6,8
2,4,6,8,10
probe lowered
down 3.175 cm;
galvanized pipe
probe lowered
down 3.175 cm;
galvanized pipe
1,2,3,4,5,6,7,8,9 probe lowered
down 2.54 cm;
PVC pipe
1,2,3,4 probe lowered
down 3.175 cm;
PVC pipe
4,6,8,10,12,16 probe lowered
18,20.21 down 3.175 cm;
galvanized pipe
10,12,14,16,18,20 probe lowered
down 3.175 cm;
galvanized pipe
8,10,12,14
probe lowered
down 3.175 cm;
galvanized pipe
thermistor
0.1, 0.9 m at 45 in situ
locations in marsh
1.0, 3.0 m
shallow
temperature
probe
4/22/77-1/18/78
weekly
4/22/77-1/18/78
weekly
5/31/77-1/18/78
weekly
5/31/77-1/18/78
weekly
5/31/77-1/18/78
weekly
10/1/76-12/12/78
monthly
10/1/76-12/12/78
monthly
daily average
value recorded
every day plant
is operating
10/28/76, 11/6/76
11/13/76, 12/4/76
9/30/77, 10/23/77,
12/03/77, 1/10/78,
3/30/78
-64-
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THERMISTOR
POWER
SUPPLY
VARIABLE
RESISTOR
Figure B-2. A simple circuit for measuring temperature.
-65-
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Thermistors
Most of the temperature data at the Columbia Generating Station site were
taken with Yellow Springs Instrument Company Series 400 thermistors. These
thermistors are calibrated by Yellow Springs, and all thermistors of this type
have the same calibration curve with a maximum error of 0.1°C. The probes are
guaranteed to remain calibrated for 1 yr. The temperature-resistance
characteristics of these probes, as well as the resistance change for a 0.1° C
temperature change, are shown in Figure B-3. The thermistors were used in
three modes: as sensors that were lowered down a wall, as sensors mounted on
2-m brass rods, and as sensors placed semipermanently in the subsurface.
The easiest and least expensive means for measuring temperature in the
subsurface is to lowsr a probe down a fluid-filled well and take readings at
various depths on the way down. The only problem with this technique is that
the temperature profile in the well will not correspond to the temperature
profile in the suftsurface materials if the temperature gradients are above a
critical value. For a 3.175-cm well the critical gradient is approximately
1°C/m at 10°C and 0.2°C at 25°C (Sammel 1968). The critical gradient was
exceeded in many cases at the Columbia site. The error induced in the
temperature readings is probably less than +_ 0.5°C if the critical gradient is
exceeded by less than 1 order of magnitude. Near the surface, where the
gradient can be large, the induced error was greater. Nevertheless, because
of tne advantages of simply lowering a probe down a well, this technique was
used to record temperatures at many locations in the subsurface.
The thermistors that were lowered down wells were purchased as a unit
designed for this purpose. These thermistors were sealed by the manufacturer
in an epoxy resin and attached to a shielded cable. The maximum diameter of
the assembly was 0.4 cm. The time constant of these thermistors is 7 s. A
thermistor assembly as described above was mounted inside a 0.95-cm brass rod,
2.1 m long, for use as a shallow temperature probe for accurately measuring
temperatures near the surface. The probe could easily be pushed into the soft
sediments on the Columbia site to a depth of 2 m. The thermistors purchased
as a unit were calibrated with respect to each other so that temperatures
between probes could be compared with an accuracy of +• 0.05°C.
In one plane perpendicular to the dike (Figure fi-1), 42 thermistors were
mounted in six wells with six to eight thermistors in each well. These
thermistors were purchased as unmounted thermistors. Each thermistor was
soldered to three wires on a cable, two wires were soldered to one lead, and
the assembly was potted with silicon sealer inside a piece of tygon tubing
0.64 cm by 5 cm (Figure B-4). The wires soldered to each thermistor were of
sufficient length to reach the surface. All the thermistors to be placed in a
well were bound together and wired to a connector that was placed at the
surface when the thermistors were inserted in the well. A cable was wired
from the thermistors at each of the six wells to a control box on the dike.
The control box on the dike was equipped with a 48-position switch to allow
access to each thermistor. Three wires were connected to each thermistor so
that the lead-wire resistance could be measured and subtracted from the
resistance measured for the thermistor plus lead wire.
-66-
-------
40
o 30
o
U.
O
IE 20
UJ
Q.
CO
S
X
o
10
8000
7000
6000
CO
2
5000
UJ
4000 z
CO
CO
10 20
TEMPERATURE °C
30
3000
2000
1000
Figure B-3. Temperature-resistance characteristics of Yellow Springs Series
400 thermistors ( — = right axis) and the change in resistance
with a 0.1° C change in temperature ( = left axis).
.64cm
SILICON SEALER
TYGON THERMISTOR 32GAUGE SOLDER 22 GAUGE
TUBING LEADS JOINT COPPER LEADS
Figure B-4.
Potted thermistor assembly showing the details of the
thermistor mounts.
-67-
-------
The wells into which the thermistors were placed were 10-m deep,
water-filled wells constructed with 3.175-cm schedule-40 PVC pipe jetted into
place to minimize disturbance of subsurface materials. After the cables were
placed in the wells, Fiberglas insulation was stuffed into the remaining space
to minimize convection. Thermistors at depths of 0.1, 0.4, and 0.8 m were
installed near the well rather than in the well. The field arrangement is
shown schematically in Figure B-5. The thermistors on each cable were not
calibrated relative to each other, but the cables were pulled twice during the
18 months of temperature-data collection to check for deviation from the
calibrated range.
In addition to the in situ thermistors described above, 200 additional
thermistors (Fenwall Disc Type JB3U1) were placed at 60 locations in 12 lines
perpendicular to the dike in the marsh at depths of 0.1 and 0.9 m. Each line
of thermistors was wired to a control box on the dike . Four sets of readings
were taken in October and November 1976, but in December the multicolored
22-gauge copper wire, which was used to connect the thermistors to the control
boxes on the dikes, became incorporated in the winter dwellings of the marsh
muskrats .
Meters Used for Temperature Monitoring
Because of delays in obtaining equipment, three meters were used during
the 17 months of temperature monitoring. All meters were frequently
calibrated against a known resistance.
From August through December 1976 all thermistor resistances were measured
on a Shall era ft four-dial wheatstone bridge with a 5 A full scale needle
galvanometer. Resistances could be read accurately to ± 3ft at 4,000 n with
this meter, and it probably had an operating accuracy of +_ 10 tt.
A Digitec digital recording ohmmeter mounted in a vehicle was used for
measuring resistances from January 1977 through January 1978. Most of the
resistance measurements were made with this instrument. This instrument had
an operating accuracy at 4,000 ft of approximately +_ 4 fl and was operated at
all times in the field with a 12 VDC to 120 VAC square wave converter and at a
temperature of at least 20°C.
A four-dial wheatstone bridge with a digital null meter was constructed as
a backup for the digital ohmmeter and as a portable field meter (Figure B-6).
This meter had a working accuracy at 4,000 ft of approximately ±5 & and was
used primarily with the shallow temperature probe for measuring temperatures
near the surface.
Teature
Two resistances for each location were recorded in the field, a line
resistance and a total resistance. The resistances were then coded, punched,
cross checked for errors, and then converted to temperature by the following
equation (Steinhart and Hart 1968):
T"1 - A + B log R + C(log R) ,
-68-
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CONTROL BOX
MARSH SURFACE
\
Ill
o
4
u
10
WELL CASING,
PVC
WELL DIAMETER'
1 3.175cm
131.5
86
51
METERS
32.5
FROM WEST EDGE
OF
15
DIKE
2 O
-30
Figure B-5.
Schematic of thermistor placement in subsurface wells at the
Columbia Generating Station site. The wells shown correspond to
locations 130-135 of Figure B-l. Horizontal dimensions are
distorted, but vertical dimensions are approximately correct.
1.1 V-r-
DECADE
RESISTANCE
BOX
ANALOGIC AN 2545
DIGITAL PANEL METER
Display=
xlOOO
:"••
•...'
•'"":
•.."
2
?
•
7
Figure B-6.
Schematic of digital bridge circuit used as a portable field
meter for measuring ground-water temperatures.
-69-
-------
where the constants A, 8. and C were determined from the thermistor
calibration data, and T~^ is the inverse Kelvin temperature.
The relative accuracy of all temperature measurements taken in the study
is within + 0.5°C of the actual temperature of the thermistor probe. The main
sources of error were thermistor calibration, meter drift, and incorrect meter
calibration. The correspondence of probe temperatures to actual subsurface
temperatures is unknown. The deviation between these temperatures is a
function of the mode of measurement used and was assumed to be small in all
cases.
THERMAL CONDUCTIVITY DETERMINATION
The thermal conductivities of unconsolidated materials at the Columbia
Generating Station site were measured by the needle probe method (Von Herzen
and Maxwell 1959). In the needle probe method, which was developed for
determining the thermal conductivity of deep sea sediments, a thin needle is
inserted into a sample and is heated along its length. The sample can be as
small as 3 cm in radius and 7 cm long. From a record of temperature increase
in the needle over time, the thermal conductivity can be easily calculated.
The probe and the experimental arrangement for measuring thermal conductivity
for this study are shown in Figure B-7 . The thermistor in the probe was
calibrated to 0.05°C. The needle probe was not checked for accuracy against a
known standard.
The theory for the needle probe has been worked out in detail by Jaeger
(1958). His analysis shows that temperature increase of the probe is
approximated by
), t>a2/a,
1.7811 *
where a = thermal diffusivity of the sediment sample, L2/t; a = probe radius,
L; t = time, t; and q = heat input per unit length per unit time, H/tL.
A plot of T versus In t will give a straight line, the slope of which
determines K for known values of q. The data collected from measurements of
the thermal conductivity of three samples from the Columbia Generating Station
site are shown in Figure 8-8. All samples were checked for reproducibility,
the standard error was less than 3>, and in all measurements thermal
conductivity was assumed to be independent of direction.
DETERMINATION OF HEAT CAPACITY
Since heat capacity of a composite material can be determined from a
weighted average of the heat capacities of the individual parts in the
composite, it was assumed that the heat capacity of the unconsolidated
materials at the Columbia Generating Station site could be determined from the
relation
pCs " pCwXw + pCorgXorg + pCqtZXqtz + pCclayXclay '
-70-
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NEEDLE CROSS SECTION
2O GAUGE
'HYPODERMIC
EPOXY
THERMISTOR
.STAINLESS
STEEL
HEATING
WIRES
LOW RESISTANCE WIRES
WHICH CONNECT TO HIGH
'RESISTANCE HEATING WIRE
AT BASE OF NEEDLE
THERMISTOR LEADS
4-PIN
CONNECTOR
Figure B-7. Details of (a) the needle probe and (b) the experimental arrange-
ment to measure thermal conductivities of unconsolidated
materials. The power supply was used to supply a constant
voltage to the needle probe. The voltage was then monitored on
the volt meter, and the ohmmeter was used to record the resis-
tance changes in the thermistor embedded in the thermistor.
where Xw, Xorg, Xqtz, and Xciay denote fractional parts of the material made
up of water, organic material, quartz and feldspar, and clay minerals,
respectively. The heat capacities of these four components are listed in
Table B-2. In a given sample all volatiles were assumed to be organic, all
materials greater than 0.001 mm in diameter were assumed to be quartz and
feldspars, and all materials less than 0.001 mm in diameter were assumed to be
clay minerals. The water content of the samples was determined in the
Quarternary Laboratory at the University of Wisconsin-Madison, and the
organic, sand and silt, and clay fractions were determined by the Soil and
Plant Laboratory of the University of Wisconsin-Madison.
-71-
-------
(a)
34
u
w 32
K
3
<
tc
HI
£30
Z
u
1-
28
. . . .....J .... -v
-
0
_
• .,
^^^^^
A ^^^^^
•^^^
l^r
^JQ
- ^^**
^* +
•*^
<* , -, . 1 ....1 . . . I . ...
(b)
u
• 36
Ul
3
£T
^
a
u 34
a.
Z
u
K
32
. ...... ..| , - • , .^,.
•X
^p
— 1^
Jr
^^
,j^^
^^^
— ^^
./
Jr
^
«'
.^
, , 1 , , ..1 , ,,!,,,,
10 100 1000 10 100 I00(
SECONDS SECONDS
(C) 44
42
u
w 4°
K
2 38
|36
IX
«- 34
32
. ..,...., .
X
/
/*
\ / , 1
X . , 1 , . ..1 , ..!..,,
10 100 IOOO
Figure B-8.
SECONDS
Thermal conductivities of three ground-water samples from the
Columbia Generating Station site which were analyzed with the
needlfe probe. Thermal conductivities were calculated to be
4.9 x 10~3 cal/°C cm sec for medium to fine sand (a), 5.0 K 10~3
cal/°C cm sec for fine sand (b), and 1.5 x 10~3 cal/°C cm sec
for hemic peat (c).
TABLE B-2. HEAT CAPACITIES OF THE CCMMON COMPONENTS OF
UNCONSOLIDATED GLACIAL MAT£fiiALSa
Component
Heat capacity
(cal/m3°C)
Quartz and feldspar
Clay minerals
Organic matter
Water e 25°C
0.51 x 106
0.51 x 106
0.60 x 106
1.00 x 106
aAdapted from Van Hijk and de Vries (1963).
-72-
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APPENDIX C
A FINITE ELEMENT PROGRAM TO SIMULATE SINGLE-PHASE HEAT
FLOrt OH CONSERVATIVE MASS TRANSPORT IN AN AQUIFER
BRIEF DESCRIPTION OF THE MODEL
The following computer code is designed to solve the two-dimensional
equations of ground-water flow and heat or mass transport for an aquifer by
using the finite element method. The program was written to facilitate the
analysis of ground-water contamination in shallow glacial aquifers. The
program solves for aquifer potentials and temperatures or concentrations.
However, simultaneous simulation of the transfer of heat and mass in an
aquifer is not possible. The aquifer to be simulated may be artesian, a water
table, or a combination of both. It may be heterogeneous and anisotropic and
may have irregular boundaries. Tne program was designed to simulate
cross-sectional problems, but it has also been used successfully to simulate
areal problems.
The basic procedure is to solve a ground-water flow problem for potentials
in an aquifer and tnen to use the potential information to compute a velocity
distribution in the aquifer. The heat or mass transport equation is then
solved for temperatures or concentrations (Figure C-1).
Solve for ground-
water potentials
in the aquifer.
If a transient problem
repeat for specified number
of time steps using the
Crank-Nicolson method to
step the solutions forward
in time.
Compute ground-
water velocities.
Solve for nodal
temperatures or
concentrations.
Figure C-1.
The basic procedure for linking the ground-water flow and the
transport equations of the model. In general, smaller time steps
will be used for the heat/mass transport equation than for the
water-flow equation.
-73-
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This flexible program includes these features:
1) Meuman, Dirichlet, and mixed boundary conditions can be handled.
2) The basic element shape is a quadrilateral. The sides of the elements can
be linear, quadratic, or cubic, and thus element sides can have two, three,
or four nodes. Any given element can have four linear sides or any mix of
linear, quadratic, and cubic sides. This feature allows considerable
flexibility in designing a finite element grid.
3) The program handles hydrodynamic dispersion.
4) Point, line, or areal sources of water, heat, or mass can be represented.
5) The problem need not be a linked problem; if desired, the model can be used
to solve only for aquifer potentials.
6) hydraulic conductivity can be represented as a function of temperature.
The program is relatively simple to use, but those unfamiliar with the
development of the equation describing the convective-dispersive transport of
neat or mass in an aquifer, or those unfamiliar with the limitation of the
numerical approximation technique, may experience difficulties in using the
program. Oscillations and instabilities are frequently encountered in solving
the convective-dispersive equation because (1) the equation behaves like a
hyperbolic partial differential equation when convective transport dominates
over diffusive transport; (2) in the numerical technique used in the program
velocities are not continuous everywhere; and (3) the numerical scheme cannot
transmit sharp contaminant fronts. Oscillations and instabilities can
generally be dampened by reducing element size or by reducing the time step.
THE EQUATIONS SOLVED BY Trie] PROGRAM
The program was written to solve the differential equation describing mass
or energy transport in a ground-water aquifer. Since the rate of transport is
a function of ground-water velocity, the first step in solving the transport
equation is to solve the ground-water flow equation so that ground-water
velocities can be determined.
Hater-Flow Equation
The following partial differential equation, which can be used to describe
ground-water flow in a confined or water-table aquifer, is solved by the
program
aJT^il-f^ + W-S|*=0 , i.J-1.2 ,
3xi ij 3xJ 8t (c-i)
where %j are the components of the hydraulic conductivity tensor, which may
be a function of temperature, Lf1; b = aquifer thickness, L; = ground-water
potential, L; w = flux of recharge per unit area, L/T; and S = storage
coefficient for a confined aquifer, or the specific yield in a water-table
-74-
-------
aquifer, dimension! ess. Boundary coalitions may be some combination of the
following:
- - - R - p = ° '
where ^i^l
D' . . =04 - + o
11 L q
qlq2
D ' „ . — V, n ~ \. ""W ~ «m / I >
21 12 VT, T7 q
q2q2
q '
aL = longitudinal dispersivity, L; Ts = transverse dispersivity, L;
velocity or specific discharge, which is calculated by
q = mean velocity, (q-| +
The remainder of the parameters are defined differently for a mass
transport problem and for a heat transport problem (Table C-1).
Boundary conditions may be some combination of the following:
C = constant, (C-6)
= constant, (C-7)
-
9xi
3C
^ a ^ * ' (C-8)
-75-
-------
TABLE C-1. PARAMETiSRS FOR BOTH MASS AND HEAT TRANSPORT PROBLEMS
Mass transport problem
Heat transport problem
Components of molecular
diffusion tensor, L2/t;
C rfass concentration, M;
g Unity, dimensionless;
Effective porosity, dimen-
sionless;
A Mass recharge rate, M/t.
Components of the thermal
conductivity tensor for the
saturated medium, H/t LT;
Temperature, T;
Heat capacity of water,
Heat capacity of the saturated
medium, H/L^T;
Heat generation or recharge
rate, H/L3t.
ac
(C-9)
3C
a C - C«
(C-10)
where C = concentration beyond an exterior boundary.
A physical significance can be ascribed to each of the terms in £q. (C-4),
since the equation can be thought of as a mass balance equation which states
that flow into and out of a differential volume sums to zero. The processes
described by each of the terms are: transfer by molecular diffusion,
3 3C
•jjc~ ^D£4^~^ » transfer by velocity fluctuations, called dispersive
-76-
-------
0 ; source
transfer, •- (g D'^ -) . convective transfer, g q± -
sr
or sink, R: storage changes, p ~ .
at
The use of Eq. (C-4) to describe mass or heat transport in an aquifer
implies the following assumptions: (1) the system is chemically inert, (2)
the aquifer is incompressible, (3) density is constant within the system, (4)
all transport occurs by molecular diffusion, mechanical dispersion, and
convective transport, (5) flow can be represented in two dimensions, and (6)
the flow of water is laminar. Bear (1972) and Konikow and Grove (1977)
present derivations of t,q. (C-4) to describe solute transport in ground water.
The development of Eq. (C-4) to descrioe energy transport in ground water will
be discussed.
THE APPROXIMATION TECHNIQUE
The finite element method is used in the program to approximate Eq. (C-1)
and Eq, (C-4) to solve for the unknown potentials and concentrations in an
aquifer. The finite element method, like the finite difference method, is a
discretization technique. Tne continuous aquifer is represented by a number
of regions, called elements, each bounded by a set of nodes. Integral
equations describing water flow and heat or mass flow are developed for each
region, giving two sets of simultaneous algebraic equations which are solved
to determine the unknown potentials and concentrations. The finite element
method generates the integral equation from the governing differential
equation and evaluates the integral over an element by a numerical integration
scheme. Thus, the finite element method operates on a region, whereas the
finite difference technique only operates on a point.
The finite element method has several advantages over the finite
difference method for solving Eq. (C-1) and Eq. (C-4), and, in fact, the
finite difference method may be unacceptable for solving fiq. (C-4) (binder
1973). The advantages are: (1) The finite element method is characterized by
less overshoot and less numerical smearing of a concentration front (Finder
and Gray 1977); (2) velocities can have a functional representation, which
reduces oscillations in the solution to Eq. (C-4); (3) boundary conditions are
much easier to treat in the finite element method, and (4) irregular
boundaries are easier to handle with the finite element method.
Two techniques can be used in formulating the approximating integral
equations that are the foundation of the finite element method. The procedure
used in this study to develop the required finite element formulation is the
fiayleigh-flitz procedure, or the minimum potential energy procedure, which is
based on the calculus of variation. The other procedure, the Galerkin method,
is more general in application and has been in favor in the literature. It
was used by Pinder and Gray (1977) to develop the approximating integral
-77-
-------
equations for equations similar to Eq. (C-1) and Eq. (C-4). Both methods
yield identical finite element formulations for Eq. (C-1) and Eq. (C-4).
finite £lemejit Formulation
The basis of the variational principle is to develop an integral equation
describing the total energy within a region and then to minimize the integral.
This principle implies that potentials and concentrations in an aquifer will
always be at levels that minimize total energy for a given set of boundary
conditions.
Integral equations can be developed from first principles by quantifying
potential energy in the system, but the Euler-Lagrange equation (Myers 1971)
facilitates the process by transforming the governing partial differential
equations to integral equations that describe total energy within a region.
The variational statements for Eq. (C-1) and Eq. (C-4) for an element
within the aquifer are
2
H - 1/2 // KO--) dxdx - 1/2 // [2W+ + s-] dx ; (C-11)
ar o o
- p -fT-lbdx.dx- + 1/2 § H(C - 2CC»)bds + 1/2 § gq . C bdS
o t i L TL.
f C—
+ 2 § gq± C«CbdS ;
where nw = total potential for water flow, n T = total potential for heat or
mass transport, h = convection coefficient (units depend on type of problem),
and b = aquifer thickness.
The next step is to minimize Eq. (C-11) and Eq. (C-1 2), which is
accomplished by setting the first derivatives of Hw and n^ equal to zero.
Before this is done, it is convenient to introduce the following approximating
functions to describe the system at any interior point as a function of nodal
values:
* - [N] {N> , (C-13)
3<>/3x - [n,x] {*N} , , and
-78-
-------
3C/3x = [N'.x] {CN> , (C-16)
where $ , C = potential and concentration respectively at any interior point;
= ground-water potentials at nodal points in an element, a column matrix;
= concentrations at nodal points in an element, a column matrix; and [N],
[N ' ] = shape functions which relate nodal values of potential or concentration
to values at any point in the element, a row matrix. This program allows each
element to have a minimum of 4 and a maximum of 12 nodes. For example, if an
element has six nodes the shape function has the following form:
where i = . . . $5 are nodal values of potential.
Derivation of the coefficients of the shape functions is not
straightforward, but an understanding of how they are derived is not necessary
for a general understanding of the finite element method. It is sufficient to
know that if an element side has two nodes the shape function is linear, if
the element side has three nodes the shape function is quadratic, and if the
element side has four nodes the shape function is cubic. Therefore, the more
nodes in an element, the higher the order of the polynominal used to
approximate an interior point. This program uses isoparametric elements of
the serendipity family. (See Pinder and Gray 1977 for more details.)
To solve the flow problem Eq. (C-11) is then minimized as follows:
an.,
'ij[N,xi]i [N,Xj.
T
- S - , [N]1 [N] 3X3x = 0
The integrations are performed using Gaussian quadrature (Zienkiewicz
1971). After the equation is integrated, it is convenient to combine terms
and put the equation in the following form:
8{V
[S] UN> - [C]—§r- [R] , (C-19)
where [s] = bKij[M,Xi]T [w,xj] 8x18x2, which is often called the element
structure matrix; [c] = S [NJT [N] 3XT3x2, which is called the element
capacitance matrix; and [r] = W [N] 3x-|3x2, which is called the element
recharge matrix.
An equation in the form of Eq. (C-19) is developed for each element in an
aquifer. These equations are then added to give the following equation:
[s] UN} = [c] -^l^1- [R] , (c-20)
-79-
-------
where [S] = global structure matrix, [C] = global capacitance matrix, and [fi]
= global recharge matrix.
The column matrix will have one column for each node, and the square
matrices will have a column and a row for each node. The square matrices will
be symmetric and banded, and only half the matrix will be stored in the
program.
The time derivative is evaluated by the Crank-Nicolson approximation:
([C] H-f^- [S]) {*N}1+1 - ([C] - [S] |^) {^ + [R] At . ((,_21)
The final matrix equation is solved by Gaussian elimination (Cook 1974).
To solve the mass or heat transport problem, Eq. (C-12) is minimized as
follows:
[Dtj [N,x±]T [N.Xj] +g D'^ [N.x^^N.
T 3{CKT} T
- g 1 [N,x]T[N]b3x3x{C}- p " [N]
H[N]T [N]bdS{CN) - HfNjC^dS + q±g[N]T [N]bdS
^C^ [N]bdS -
0 . (C-22)
This equation can be reduced to a form analogous to Eq. (C-19). As
before, one equation is developed for each element, and these equations are
summed to give an equation analogous in form to Eq. (C-20). The time
derivative is again evaluated by the Crank-Nicolson approximation. The final
matrix equation, which is assymmetric due to the presence of the terms
[N,XiJT[N] in Eq. (C-22), is solved by the Gauss-Doolittle method (Desai and
Abel 1972).
CONSIDERATIONS IN DESIGNING A FINITE ELEMENT GRID
The basic step in using a finite element scheme is to subdivide the region
of interest into an assemblage of smaller regions called elements. The
process of discretizing is an exercise of engineering judgment in which one
must choose the number, shape, size, and configuration of elements in such a
way that the original continuum is simulated as closely as possible. A grid
configuration that provides a greater number of nodes will generate a more
accurate solution, but at the same time will lead to more computational
effort. In general, the mesh should be refined in the region of steep
gradients. The element shapes are quadrilaterals having straight boundaries,
although curved boundaries are also possible.
-80-
-------
All boundary conditions described for Eq. (C-1) and (C-4), except for
constant flux boundaries, are handled explicitly by the program and are
explained in the data-input section of the appendix. A constant flux boundary
of zero flux is assumed by the finite element method if no other boundary
condition is specified. A finite flux boundary is treated by assigning a line
source or sink of the appropriate magnitude. Four sample problems will be
presented to illustrate the discretization procedure.
In designing the finite element grid, 10 considerations should be
observed.
1) All elements should be made nearly rectangular, since distorted element
shapes decrease the solution accuracy.
2) Elements with more than four nodes should be used sparingly. Although an
element with more than four nodes uses a higher order approximation function
for inte'rior points, replacing an element with six nodes by two elements with
four nodes each usually provides as good an approximation with less
computational effort. (This occurs because a lower order Gaussian quadrature
scheme can be used for elements with only four nodes.) Elements with more than
four nodes are best utilized for refining the finite element grid in critical
areas (Figure C-2).
Figure C-2.
Examples of refinement of a finite element grid. Left: bad
use of multi-node sides; center: the preferred way of
discretizing a region; right: good use of multi-node sides.
3) Nodes should be placed close together in areas where system parameters
exhibit spatial changes, in areas where velocity is relatively rapid or the
velocity distribution is complex, and in areas where a sharp temperature or
concentration front is expected.
4) Exterior element boundaries across which a flux of mass or energy occurs
must have only two nodes.
5) The velocity field near an exterior element boundary across which a flux of
mass or energy occurs must not be complex. Preferably, all flow is normal to
the boundary. This can be insured by setting up dummy elements along a
boundary with a high hydraulic conductivity in a direction normal to the
boundary. Likewise, the boundary should be oriented normal to one of the
principle directions of the hydraulic conductivity tensor.
6) Boundaries within the area to be modeled should be located accurately.
Distant boundaries can be located approximately with fewer nodes by expanding
-81-
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the grid. Elements with more than four nodes are very useful for expanding
the grid away from critical areas.
7) Constant concentration boundaries are generally unrealistic for the
transport equation. Transport into the system can best be represented by
multiplying the water flux across the boundary by the concentration exterior
to the system.
8) The grid should be oriented to coincide with the principle directions of
the hydraulic conductivity tensor.
9) The core requirements and computation time are proportional to the number
of nodes representing the aquifer and to the number of times system parameters
or time steps are changed.
10) If no boundary condition is specified along an exterior boundary, a
no-flow boundary is assumed.
DATA DECK INSTRUCTIONS
The Data Groups
All data are read into the model with a free format, except for
alphanumeric data. In free format, the data requested by one read statement
may be put on one input card or as many input cards as the reader desires.
Bach bit of data, however, must be followed by either a comma or a blank. The
data for each read statement must begin on a new card. Two read statements
cannot read from the same card. The data and program output for four sample
problems will be presented to illustrate the data deck preparation procedures.
Several aspects of the data preparation that may be confusing are explained in
more detail in the data notes following this section. The read statements
that must be included on every program execution are underlined.
Group 1—
This group of data, which is read by both the main program and the data
input subroutine, contains data required to dimension the model and data
required to determine the type of problem.
Read Statement Variable Definition
J. and £ TITLE Any title that the user wishes to print on
two lines at the start of output. The input
must contain two cards.
3 LM Number of elements
LM Number of nodes
MBAND Bandwidth (notes 1 and 2)
-82-
-------
KPRlNT
LOW
MQ
NLA
CFACT
and
2 and 8
VV(I)
An integer that can have values in the range
of one to five.
1—problem to be solved is a steady-state
ground-water problem.
2—problem to be solved is a transient
ground-water problem.
3—problem to be solved is a steady-state
ground-water problem linked to a transient
heat or mass transfer problem.
4—problem to be solved is a transient
ground-water problem linked to a transient
heat or mass transfer problem.
5—problem to be solved is a transient mass
or heat transfer problem.
An integer that can have values of 0 or 1.
Set to 0 to suppress printing of input data;
otherwise set to 1.
Set to 99 for calculating coefficient
matrix stability for a transient problem
(see note 3); otherwise set to zero.
Set to 2 if one card is to be read in for
each element giving the element node
numbers. Set to 0 if the spatial
structure for the problem is rectangular
and all elements have only four nodes
(data group II).
An integer specifying the method to be used
for calculating velocities in a linked
problem. Set to 1 if velocities are to be
calculated at each of the Gauss points; this
is generally the better method. Set to 2 if
velocities are to be calculated only at the
center of each element. Set to 0 if the
problem is not linked problem.
Set as 1fi13; this value is used to maintain
the specified boundary conditions. Set
higher if specified boundaries are not
reproduced by model.
Format statement for printing out nodal
values. Two cards are required "(note 4).
Format statement for printing out element
values. Two cards are required.
-83-
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Group II—
The data contained in this group of cards are required to specify the
spatial structure of the nodal array. The type of data to be input is
determined by the value previously specified for variable MQ. If the
structure is rectangular with four nodes per element, the program
automatically generates a grid and numbers the nodes and elements, and the
data input is simple. If the structure is not rectangular, the location of
each node must be input as must cards specifying the nodes defining each
element (notes 5, 6, 7, and 8).
Read Statement
1
Variable
Q
If MQ equals 2 only:
one card for each J
node XLOC
iLOC
one card for each
element
J
NOD(K,1)
NODU.2)
NOD(K,3)
NODOC.5)
NOD(K,1)
IF MQ equals 0 only:
KSPACX
KSPAC1
BU)
5
6
Group III--
MLS
Definition
Input the alphanumeric characters—up II.
node number
X location of the node
\ location of the node
element number
The element node numbers. The node numbers
for each element must be listed counter-
clockwise, starting at the corner nearest
the origin. List the nodes in sequence;
if a node is not a corner node, place an
asterisk after it. Repeat the first node as
the last value on each card.
Number of nodes in the X direction.
Number of nodes in the X direction.
KSPACX values, listing the X coordinates in
order; smallest value should be listed
first.
KSPAC1 values, listing the X coordinates in
order; .largest value should be listed
first.
Optional (note 8).
Optional (note 8).
This group of data contains information on the initial conditions and on
the specified boundary conditions. The data to be read in depends on the
value specified for KTXPE.
-84-
-------
Read Statement Variable
1 W
If KTlPfi equals 1:
2 HfiA(I)
IF KTlrti equals 2:
ALPH
ALPhM
KBOUN
IF KTlPt; equals 5:
2
HI (I)
HfcA(l)
ALPHA
ALPHAM
KBOUND
PC*
Definition
Input the alphanumeric characters—
111.
Array used to specify constant potential
boundaries. A zero is entered if a
potential is not specified at the node;
otherwise the specified potential value is
entered. See note 9 for the format for
entering these values.
The length of each time step.
Multiplication factor for each time step.
Length of the next time step is equal to
the length of the old time step multiplied
by ALPHM. The program works most
efficiently if ALPHM-equals 1.
Set to 1 if subroutine BOUND is to be called
at each time step during execution of water-
flow equation. Set to 2 if subroutine BVAL
BVAL is to be called at each time step during
execution of water-flow equation (note 13).
Set to 99 when determining structure matrix
stability for water-flow equation (note 3).
Otherwise set to 0.
Array used to specify the initial potential
at each node (use format explained in note
9).
See above.
The length of time step for heat/mass
equation.
Multiplication factor for the time step.
An integer then can havve values 0, 1, 2,
or 99. See KBOUN above, except KBOUND
applies to the heat or mass flow equation.
heat capacity of water, or 1 in a mass
transport problem with units of meters,
grams, and ppm.
An array used to specify the initial
temperature or concentration at each
node (note 9).
-85-
-------
HEA(I)
HEAD(I)
IF KTlPfi equals 4 or 5:
ALPH
ALPHM
BOUN
ALPHA
ALPttAM
BOUND
PCW
See above. Specified potential values.
Array used to identify specified
concentration boundaries. A zero is entered
if temperature or concentration is not
specified at the node; otherwise the
specified value is entered (note 9).
See description above.
See description above.
See description above.
5 A(I) See description above.
6 HfiA(I) See description above.
7 h£AD(I) See description above.
Group IV—
This group of data contains information on the model parameters.
Head Statement Variable Definition
1
MMA1A
FACTh
FACTV
FACTwS
FACTH1
FACTV 1
FACTHS
Input the alphanumeric characters—
Group IV.
An integer, less than 51, specifying the
number of different material types in the
problem. A homogeneous area has only one
material type.
This card contains multiplication factors
for the parameters to be input later.
Factor for X direction hydraulic
conductivity.
Factor for 1 direction hydraulic
conductivity.
Factor for storage coefficient.
Factor for X direction thermal conductivity.
Factor for 1 direction thermal conductivity.
Factor for specific heat capacity of
saturated medium.
-86-
-------
FACTDL
FACTDT
MA1(J)
MMATA cards,
one card for each
material type.
KX
Mi
STO
PX
PI
PCX
DIFF(J,1)
DIFF(J,2)
Factor for longitudinal dispersivity.
Factor for transverse dispersivity.
Type of material array. Integer values
in the range of 1-50 are entered for each
element, indicating the type of material in
that element (format explained in note 9).
Integer number in the range of 1-15
identifying the material type.
X-direction hydraulic conductivity
1-direction hydraulic conductivity
Storage coefficient
X direction thermal conductivity, or
molecular diffusion coefficient.
Y direction thermal conductivity or
i direction molecular diffusion coefficient.
Heat capacity of the saturated medium, or in
a mass transfer problem the porosity.
Dispersivity coefficient, lateral.
Dispersivity coefficient, transverse.
NOTE: Only values for J, WX, HI, and STO need be entered if KT1PE equals 1 or
2. If KTIPE equals 5, values for WX, Wl, and STO must not be entered.
Group V—
This group of data contains information on sources and sinks. Line,
areal, or point or sinks can be handled directly. Sources are positive, sinks
are negative. If there are no or sinks, three cards must be entered, each
with two zeros.
Bead Statement Variable
1 Q
2 KGEfl
KGENH
3 INFLOW(J)
HEAT(J)
Definition
Input the alphanumeric characters—Group V.
Set to 1 if there is a uniform source or
sink of water over at least one element in
the problem; otherwise set to zero.
Set to 1 if there is a uniform source of
energy or mass over one element in the
problem; otherwise set to 0.
Include this only if KGErt is not equal to 0.
One value for each element specifying the
rate of water generation per unit area per
unit time in the element (note 9).
Include this only if KGENH is greater than
0. One value for each element specifying
-87-
-------
10
11
12
Group VI—
LWATEfl
LhEAT
NWAT£R(J)
AW ATM (J)
NHEAT(J)
AHEAT(J)
LI NEW
LINEH
NLINEW(i,1)
NLINEH(I,2)
ALINEU(I)
NLINEH(I,1)
NLINEH(I,2)
ALlNEh(I)
the rate of mass or heat generation per unit
area per unit time in the element (note 9).
Integer values specifying the number of
point or sinks of water.
The number of point sources or sinks of
energy or mass.
Node number of a point source or sink of
water.
Rate of water pumpage per unit time. Enter
sufficient cards to contain a node number
and a rate for each water point source.
Mode number of point source of energy or
mass.
flate of energy or mass generation per unit
time at the specified source. Enter
sufficient cards to contain a node number
and a rate for each point source.
An integer specifying the number of line
sources or sinks of water.
An integer specifying the number of line
sources or sinks of heat.
Element number of line source of water and
boundary code (note 10).
Enter sufficient cards to contain an element
number and a code for each line source of
water.
Enter one value for each line source giving
the rate of water recharge or discharge per
unit length per unit time.
Element number of line source of heat or
mass and boundary code (note 8). Enter
two values for each line source of heat or
mass.
Enter one value for each line source giving
rate of heat input or output per unit
length per unit time.
Data specifying if the problem is a cross-section problem or an areal
problem.
-88-
-------
Read Statement Variable
1 Q
2 KAREAL
3 EH
ITER
BOT(I)
Definition
Input the alphanumeric characters—Group VI.
Set to 0 for cross-section problems; set to
1 for areal problems.
Error criteria on ground-water potential
for a steady-state problem. Solution is
reached by an iterative procedure in which
transmissivities are adjusted until the
potential changes by less than the error
criteria.
Number of iterations permitted for a
steady-state solution to be reached.
Include only if KAREAL equals 1. Enter the
Dottom elevation of the aquifer at each
node (note 9).
Initial potential values in the aquifer
(use format explained in note 9). Do not
input values if the initial values were
input in data Group III.
Group VII—
This group of data contains information on the location of flow and
convective flux boundaries. These types of boundaries are only used in the
mass or heat transport equations. Skip cards 2-7 of this group if the problem
is not a linked one. Flow and convective boundaries are discussed in data
note 11.
Read Statement
1
Variable
Q
NCONV
NCOMI,2)
CONV(I)
Definition
Input the alphanumeric characters—Group
VII.
An integer specifying the number of
convective boundaries.
Element number of the element containing
this type of boundary.
This boundary code identifying the side
across which the flux occurs.
The value of the convection (transfer)
coefficient for a convective boundary.
Include sufficient cards to contain three
values for each boundary of this type.
-89-
-------
TlNF(I)
LEL
NtiLU.2)
AEL(I,2)
The temperature at a distance from the
boundary (temperature at infinity).
List one value for each boundary identified
by card 3 of this group, and list them in
the same order as on card 3.
An integer specifying the number of element
sides across,which a flux of water occurs.
Element number and the boundary code.
Include two values for each element boundary
of this type.
One value for each element boundary of this
type specifying the temperature or
concentration of the incoming fluid.
Group VIII--
This group of data contains information needed for calculating a mass
balance. The data identifies the exterior boundaries on which potential,
temperature, or concentration are specified (refer to data note 12).
.Head Statement Variable
LFLUXw
LFLUXH
NFLUXto(I,1)
NFLUXW(I,2)
Definition
Input the alphanumeric characters—
Group VIII.
An integer specifying the number of
exterior element sides on which potential
is specified.
An integer specifying the number of exterior
element sides on which temperature or
concentration is specified.
Include only if LFLUXW is greater than 0.
Use sufficient cards to list two values
for each element side.
element number of element with a specified
potential on an exterior side.
Side index number, identifying which of the
four element sides the flux is across. An
integer in the range of 1 to 4. See
explanation notes.
NFLUXH(I,1) Include only if LFLUXh is greater than
NFLUXH (1,2) zero. Format is the same as above, except
that here boundaries must be identified
on which temperature or concentration is
specified.
-90-
-------
Group IX—
This group of data contains information that controls the flow of the
program and allows access to routines for changing boundary conditions at each
time step, and it includes a routine for a moving boundary in a cross-section
problem (see data note 14).
Read Statement
1
2
Variabig
Q
LA
LD
LE
LF
LG
LH
N TIME (I)
5
6
If KTYPE equals 1:
7 Ml*
Definition
Input the alphanumeric characters—Group IX.
Set to 0 unless ENTfil LAKE is to be called;
if so set to 1.
Set to 0 unless ENTfli CHANG or ENTR1 CHAN
is to be called; if so set to 1.
Set to 0 unless water flows are to
recompute at uneven intervals; if so, L£
should equal the number of intervals at
which water flows are to be recomputed
(must be less than 100).
Set to 0 unless the problem is to be posed
in radial coordinates; if so, set to 1.
Set to 0 unless the problem is to have a
moving boundary; if so, set LG equal to the
number of nodes which are to move.
Set to 0.
Include only if Lfi is greater than 0.
LE values listing the time steps at which
flows are to be recomputed; values must be
listed in increasing order.
Include only if LF is equal to 1. Input
1,1,1,0,0,0 if problem is to be quasi-radial
with aquifer thickness being equal to the
x coordinate. Input -1,1,1,0,0,0 if problem
is to be quasi-radial with aquifer thickness
being equal to the y coordinate.
Input values needed by EflTRi LAKE.
Input values needed by subroutine ADJUST
(note 14).
Set to 1 if potentials are to be written
on file 14; otherwise set to 2.
-91-
-------
If KTYPE equals 2:
7
MA
MB
MW
MC
MD
If KTlflS equals 3, 4, or 5:
7 MO
MP
MV
MQ
MR
MS
MT
MU
ME
Total number of time units for the
simulation.
Number of time steps between printing of
head values.
Number of time steps between writing heads
on file 14.
Number of time steps between printing of
mass balance.
Number of time steps between printing of
flows.
(MA, MB, MW, MC, and MD must not equal
zero.)
Total number 'of time units for the
simulation.
Number of time steps between printing nodal
values.
Number of time steps between printing nodal
values on file 14.
Number of time steps between printing of
mass balance; if set to a negative number,
mass balances will not be computed which
results in a considerable savings in CPU
time.
Number of time steps between recomputing
water flows.
Set to 1 if initial hydraulic conductivity
values are to be adjusted for changing
temperature in the aquifer; otherwise set to
0 (note 15).
Set to 1 if using dispersivities greater
than zero; otherwise set to 0.
Set to 1 if water flows are not to be
printed when flows are recomputed;
otherwise set to 0.
(MO, MP, MV, MQ, and MR must not
equal zero.)
Set to 1 if the program is to stop after
water flows are computed; otherwise set
to 0.
Data Explanation Motes
1) Program size—
The storage required for the program is approximately
2000 + LN(11 + MBAND x 4) + LM x 25,
-92-
-------
where LN is the number of nodes, LM is the number of elements, and MBAND is
the bandwidth.
The second statement in the main program is used to change the amount of
storage allocated for the program. The parameter variable N specifies the
maximum number of nodes, the parameter variable L specifies the maximum number
of elements, and the parameter variable M specifies the maximum bandwidth. On
a UNIVAC 1110, if the program is compiled in FORTRAN V, the combination of 297
nodes, 260 elements, and a bandwidth of 13 is the maximum permissible. This
arrangement uses 64K of memory. If the program is compiled in ASCII FORTRAN
on a UNIVAC 1110, the program size could be quadrupled.
If the program is being used only to solve potential problems, the value
of the parameter variable M in statement of 1 of the main program need only be
equal to one-half the bandwidth plus one. For a linked problem M must be
equal to or less than the bandwidth.
Also, if LWATtifl, LrifcAT, NWATiiR, NhiiAT, LINEW, LlNEH (data Group V), NCONV,
LiiL (data Group VII), LFLUXfc, or LFLUXH (data Group VIII) exceed 50, statement
1 of the main program must be changed. The parameter variable Z must be
changed from Z = 50 to 2 = the maximum value of the above-listed variables.
2) Bandwidth--
The bandwidth depends upon the largest difference between any two nodes in
a single element and is equal to the maximum difference between any two nodes
in an element in a structure plus one (Figure C-3).
10
11
12
10
11
12
(a) bandwidth equals five
(b) bandwidth equals six
Figure C-3. Examples of numbering of nodes in a structure.
The nodes in a structure should be numbered so as to make the bandwidth as
small as possible.
3) Calculation of Stability for a Transient Problem—
The EIGtitt subroutine computes the maximum (critical) time step that can be
used in a transient simulation to insure that the solution computed by the
Crank-Nicolson method will be nonoscillatory. The routine is based on a
modified Eigen value extraction technique developed by ^yers (1978). The
scheme slightly underestimates the maximum (critical) time step. Tne equation
used to estimate the critical time step is
-93-
-------
At
1
-- min
'1.1
LN
1,1
where Atc = safe estimate of the critical time step, S±tj = the jth entry in
the ith row of the structure matrix, and C^ i = the diagonal term in the ith
row of the capacitance matrix.
The critical time step is a function of system parameters, the x and y
spacing, and boundary conditions. The routine prints out the maximum time
step for each node in the structure and prints out the critical time step.
4) Standard FORTRAN formats—
Standard FORTRAN, formats are used for printing node and elements values.
Values are printed consecutively starting with the value for node or element
1. The format input statements allow the user to specify how the values are
to be printed out. This option can be very helpful if an irregular grid is
used.
Typical formats for printing out node and element values for a grid 5
nodes by 5 nodes would be: (1X,5(5G12.6/)) for the node values; and
(1X,4(4G12.6/)) for the element values. (The word FORMAT is not used; the '('
goes in column one of the data card.) The node values would then be printed
out on five lines, five values on each line.
5) Orientation of the Finite Element Grid—
The finite element grid must be orientated correctly, since flows and
boundary conditions will be treated incorrectly if the grid is orientated
incorrectly. The program requires a standard cartesian orientation (Figure
C-4).
Figure C-4. Cartesian orientation of finite element grid.
Values in the x direction must increase from left to the right and values in
the y direction must increase from the bottom to the top.
-94-
-------
6) Element Node Numbering—
Typical elements are shown in Figure C-5 to illustrate the manner in which
element node numbers must be input to the program.
The input data:
element # the node numbers
1 2 6 5* 4* 3 1
2 587 3 4* 5
3 69856
Figure C-5. Typical elements of a finite element grid and the correct method
of numbering the elements.
7) The Rectangular Grid Generator—
The rectangular grid generator generates a grid in which the nodes and
elements are numbered in the form shown in Figure C-6.
1 5 9
1
2
2
3
3
4
4
6
S
7
6
8
7
10
8
11
9
12
Figure C-6. Numbering of nodes and elements in a rectangular grid.
The program assigns to node 1 the first x and y coordinates read in. For
proper orientation the list of x-coordinate values read in must be arranged so
the smallest value is read in first, and the list of y coordinates must be
arranged so that the largest value is read in first.
8) Selection of Variants of Rectangular Grid—
The program can create the two variants of a rectangular grid shown in
Figure C-7.
To select the options shown in Figure C-7:
1) set KSPACX = -KSPACX.
2) set MLS = 0 for option (a); set MLS = 1 for option (b).
-95-
-------
Figure C-7.
Two variants of a rectangular grid that can be generated.
Left: grid with sloping upper surface; right: grid with all
rows sloping.
3) Input KSPACX values for the D array. Input one value for each set of nodes
in the x direction. These values specify how much the y coordinates of the
nodes are to be adjusted from the values specified by C(I). For the examples
illustrated above the values in the D array would be: 0, 0.5, 1.0, 1.5, 2.0,
2.5. These values need not increase or decrease monotonically.
9) Format for Reading in Array Data—
Several options are provided to the user to simplify the often arduous
task of initializing array value. The user must initially specify one value
to be assigned to every position in the array. The user then has the options
of (1) leaving the array as is, (2) reinitializing the entire array by
inputting one value for each position in the array, or (3) selectively
changing the initialization. The last option is accomplished by specifying
the array position that is to be reinitialized and the new value. An array
position will correspond to a node or element number. This format may be used
when initializing fl, R1, HfiA, HEAD, MAT, and EOT.
The format to be used is:
Bead Statement Variable
1 QI
TL
FACT
Definition
Value to be assigned initially to each
position in the array.
Three options:
a. set to -1 if one value is to be input
for each node or element.
b. set to 0 if all values in the array
equal QI.
c. set to a positive integer specifying
the number of array values to be
different from QI.
Factor by which each of the array values
is to be multiplied.
If IZ = -1:
-96-
-------
(sufficient cards to
contain one value for
each node or element)
If IZ - 0:
If IZ > 0:
(sufficient cards to
contain IZ x 2 values)
10) Boundary Codes—
Enter the values to be assigned to each node
or element. Values are to be entered
consecutively beginning with node or
element 1. After one value is entered for
each node or element, enter 999 which is a
sentinel character.
No cards are needed.
Enter a node or element number and the value
to be assigned to this node or element.
Enter two values for each array position
that is to be reinitialized.
When specifying line fluxes and boundary fluxes, the side of an element
across which the flux is occurring must be designated. The sides of an
element are coded as shown in Figure C-8.
Figure C-8. Numbering of sides of an element.
11) Flow Boundaries—
Data Group VII allows the user to specify several types of boundary
fluxes. The type of boundary fluxes permitted across an element side are:
convective, where transfer is equal to AH(Ti - T°°);
flow out, where transfer is equal to V^A;
flow in, where transfer is equal to V^T»A;
where A = the length of the element side multiplied by the aquifer thickness;
H = a transfer coefficient; for heat transfer the units are, H/L2tT; T^ =
temperature or concentration on the boundary; T°° = temperature or
concentration beyond the boundary; and V^ = water velocity normal to the
boundary.
If the boundary is to have more than one type of flux, for example, a flow
and a convective flux, it should be treated for purposes of data input as
though there were two separate boundaries.
-97-
-------
12) Mass Balance—
The mass balance routine computes an approximate mass balance. Several
types of boundary fluxes are computed only approximately by the mass balance
routine, and often the mass balance errors calculated by this routine will be
much higher than they really are. The program generally has mass balance
errors of less than 1% for both the water-flow and the transport equations.
The user must Identify the specified head and 'temperature or concentration
boundary conditions. The program identifies the other boundary conditions and
sources and sinks.
13) Boundary Conditions That Vary With Time—
Because the program could handle a variety of boundary conditions that
might change at each time step, a program to cover all possibilities could not
be written. The user must write additional routines to change boundary values
at each time st^p.
Several entry points in subroutine BOUNDA are provided for this. Starting
addresses for all arrays in which a user is likely to change values at each
time step are passed to this subroutine as well as the time step counter KS.
The entry points available in this subroutine are:
(a) ENTRY LAKE
The routine is called if LA = 1 (data Group IX). This routine is called
immediately after the data input routine and is called only once for a given
program execution. The routine can be used to read in information to be used
in altering boundary conditions during execution of the program.
(b) ENTRY BOUND
Called during solution of the water-flow equation by subroutine LOAD if
KBOUN (data Group III) equals 1, or called during solution of the transport
equation by subroutine LOAD if KBOUND equals 1.
(c) ENTRY BVAL
Called during solution of the water-flow equation by subroutine LOAD if
KBOUN equals 2, or called during solution of the transport equation by
subroutine LOAD if KBOUND equals 2.
(d) ENTRY CHANG
Called during solution of the transport equation by subroutine LOAD if LD
= 1 (data Group IX).
-98-
-------
(e) ENTR1 CHAN
Called during solution of the water-flow equation by subroutine LOAD if LD
= 1. The program logic is shown on the program flow chart, which shows the
position of the calls to the entry points in BOMDA by subroutine LOAD.
14) Documentation for Moving Boundary Routine—
For cross-section problems the surface boundary can be programmed to move
in response to changing recharge rates. Only boundaries with boundary code 1
may move (note 10). The moving boundary routine is called by setting the 5th
value (LG) on card 1, Group IX, equal to the number of nodes that will move.
The following cards are then added in the appropriate position in data group
IX.
Read Statement
1
Variable
NSTEP
NPRINT
NPR
ERROR
FACTOR
WOV (J, 1)
NMOV(J,2)
BMOV(J,3)
BMOV(J,6)
Definition
Number of time periods to be stimulated.
Number of time periods between printing of
potentials.
Number of time periods between printing
flows and mass balance.
Change allowed in node location before
structure matrix is recomputed.
Factor to be multiplied by recharge rates.
Four values for each moving boundary:
Element number on left of node*,
Element number on right of node*,
Storage coefficient,
Initial recharge rate.
•Note: If several nodes are to be constrained to rise and fall at the same
rate, the left and right element number for each node of this type
must be preceded by a minus sign (-).
After the last card of Group IX one card is added for each pumping period.
Read Statement Variable Definition
BKS Identifier for this time period.
TIME Length of this time period.
FACT Factor that relates initial recharge rates
to recharge rates at this time step. The
current recharge rates are calculated by
multiplying FACT by initial recharge rate.
15) Hydraulic Conductivity as a function of Temperature—
The program in subroutine F£ adjusts hydraulic conductivities for changing
temperature distributions. The following relations are used:
-99-
-------
(1.917 - 0.05635T + 0.007 IT2) x 10~3, and
where p = the kinnematic viscosity in centime ter-gram-second units, and T =
the temperature in degrees centigrade.
The relationship is only valid for temperatures between 0° and 50°C. The
relation also assumes that the initial hydraulic conductivities are specified
at a temperature of 15°C. The relationship could easily be reprogrammed to
cover a different temperature range or to relate hydraulic conductivities to
concentration.
DERIVATION OF A HEAT TRANSPORT EQUATION
The processes that control the transport of 'heat in an aquifer are in many
ways analogous to the processes that control the transport of mass in an
aquifer. The partial differential equation describing the transport of mass
is well known, and its derivation is clearly explained in Konikow and Grove
(1977) and is rigorously derived in Bear (1972). Rather than derive the heat
transport equation from first principles, which would in many respects repeat
Konikow and Grove (1977), we show how the mass transport equation can be
modified to apply to the transport of heat.
These assumptions are made in this derivation:
1) The aquifer is incompressible and chemically inert with respect to the
fluid;
2) Fluid density is constant;
3) Hydraulic head is the only driving force; coupled processes, Onsanger
relationships, and density driven convections are not considered;
4) Fluid flow is laminar; and
5) Divergence of velocity equals zero.
In addition to these five assumptions the generalized mass transport
equation
3 / r, 3C . _,_ 3 ,_. 3C , 3C _ 3C _
^'ijT^+T^'uT^-'i^- •-»--•• (c_23)
as derived by Bear (1972) and Konikow and Grove (1977), assumes the following
about the processes that transport mass in an aquifer: (1) Velocity driven
convection, molecular diffusion, and mechanical dispersion are the only
transfer processes; and (2) no transfer of mass occurs within the solid phase.
In an aquifer the basic modes of heat transfer are (Lagarde 1965, quoted
in Bear 1972, p. 640) heat transfer through the solid phase by conduction,
-100-
-------
heat transfer through the fluid phase by conduction, heat, transfer through the
fluid phase by convection, and heat transfer by dispersion.
Each of the terms in Eq. (C-23) is discussed sequentially to demonstrate
how a similar form for each term can be used to describe the transport of heat
in an aquifer.
Molecular Diffusion
Heat, unlike mass, is readily transmitted through the solid phase of most
porous media if a temperature gradient exists. In the derivation of the first
term of Eq. (C-23), it was assumed that transport does not occur in the solid
phase. The coefficient of molecular diffusion in the mass transport equation
is defined on the basis of tortuosity (Bear 1972), a concept that is
applicable if transfer only takes place through the fluid phase.
The transfer of heat by molecular diffusion (conduction) through the solid
phase and the fluid phase can be treated by assuming a simple parallel
conduction model in which conduction through the fluid phase occurs separately
but simultaneously with no interchange of heat between the two media. The
real situation is more complex since heat is interchanged continuously between
the two phases. Experimental data (Houpert 1965, quoted in Bear 1972, p. 646)
suggest that the simple model is sufficient, and the assumption of
simultaneousness is valid if the time period is greater than a few minutes.
Therefore, heat conduction can be described by
no », - 3C v . /, x 9 f, oC
" (kf ")
x± a^ (c_24)
where kfjj = coefficient of thermal molecular diffusion for the fluid phase,
ksjj = coefficient of thermal molecular diffusion for the solid phase, and n =
the porosity.
Dispersive Transfer of heat
The form of the second term in Eq. (C-23) was derived from experimental
data for mass transport in porous media. Green (1963) concludes on the basis
of experimental data that in the range of Darcian flow the influence of the
passage of heat through the solid phase will be insignificant and that the
dispersive transfer of mass and heat will be identical. (Dispersive transfer
is a term used to describe convective transfer that occurs due to velocity
fluctuations in the fluid.)
Internal Generation of Heat
neat will be generated by a moving fluid because of energy dissipation by
viscous stresses. The rate of dissipation, E, can be estimated from
E = pqg
-101-
-------
where g = the gravitational constant and p = fluid density.
A fluid flowing at a rate of 1 m/day, with a gradient of 1 m/m, will
dissipate energy at the rate of 10"'' 3 cal/m3 per day. Therefore, heat
dissipation is considered to be negligible.
The Absorption of Heat by the Solid Phase
The absorption of heat by the solid phase can be treated analogously to
the adsorption of mass by the solid phase when the process is characterized by
a linear adsorption isotherm. The last term of E. (C-23) can be modified
(Bear 1972, Pickens and Lennox 1977) to
3C . ,. . . 3C
n"3t + (1~n) Alt (C-25)
to treat the adsorption of mass by the solid phase , where A = the adsorption
distribution coefficient that describes the ratio of solute on the solid phase
to solute in the fluid phase .
If A is defined to be the ratio of heat in the solid phase to heat in the
fluid phase, Eq. (C-25) can be used to describe the absorption of heat by the
solid phase in a porous medium.
The heat transport equation can then be obtained by substituting the terms
developed above into Eq. (C-23):
By convention the concentration of heat is generally expressed as a
temperature, where concentration equals temperature multiplied by the heat
capacity. The heat capacities of a fluid and solid phase are generally not
the same. Each term in Eq. (C-26) refers to heat transfer in either the solid
or the fluid phase; no term refers to a combined heat transfer. Making the
appropriate substitutions, Eq. (C-26) can be written as:
- R - « - >c (1-) * - ° • (c-27)
where pCy, = heat capacity of the fluid, H/L3°C; and pCsoi = heat capacity of
the solid phase, H/L^OC. The equation can be simplified by introducing the
following conventions.
1) %j = (1-n) pCgoikSij + npC^kfij,
-102-
-------
where K^j are the components of the thermal conductivity tensor for the
saturated media. It is much simpler to determine experimentally the combined
coefficient than the individual ones, which are a function not only of phase
composition but also of phase geometry.
2} The absorption distribution coefficient A = pCgoi/pC^. This relationship
holds by definition.
3) The heat capacity of a whole is equal to the sum of the heat capacities of
its parts: pCs = npCw + (1-n) pCsol,
Equation (C-27) then reduces to the following form when the substitutions
are made which is in the same form as Eq. (C-23):
SAMPLE PROBLEMS
Linear Heat Transport
This example problem demonstrates the use of all linear elements and the
use of mixed higher order elements to solve a one-dimensional convective heat
transport problem. The problem analyzed and the finite element grids used to
discretize the problem are show in Figure C-9. The parameters used in the
problem are listed in Table C-2. The input data and the program-generated
output for both sample simulations are listed in Figures C-10, C-11, C-12, and
C-13. The simulated temperatures are nearly identical for both the grid with
mixed elements and the one with all linear elements (Table C-3). Simulated
temperatures differ by less than 0.5°C.
In analogy to Ogata and Banks (1961), an analytical solution to the linear
convective heat transport problem is:
x-q't XqpCw x+q't
T = T /Z [erfc( r^) + exp(-
Z(K't/pCg) Z(K't/pCs)
where q1 = qpCw/pCs, and K.' = Kfc + qaj pCw, in which the following boundary
conditions are used:
T(x,0) = 0, x > 0;
T(0,t) = T0, t >_ 0; and
T(»,t) =0, t >_ 0.
The analytical solutions at t = 25, 50, 75 days are shown with the
simulated temperatures in Figure C-14. The analytical solution at t = 50 days
is compared to the simulated temperatures at t = 50 days in Table C-3. In
addition, the simulated temperatures for the all linear element grid with
nodes 21 and 22 specified at 15°C at t = 50 days are also presented in Table
-103-
-------
1 3 57 91
1
2
3
4
5
(3 15 17 19 2
6
7
8
9
10
68 1012
14 16 18
20
22
(a)
36 8 10 14 16 18
22 24 26
29
31
1 '
<
>4 2
1
II 1
3
12
19
4
'20
5 '
'27 6
7
, (b)
2 57 9 13 15 17 21 23 25 28 30 32
t
*
1
Vio-c
V = . 1786m /day
T0=I5"C
(c)
Figure C-9.
Finite element grids used to discretize a linear heat
transport problem. The problem is depicted in (c). The
grid with all linear elements is shown in (a), and the grid
with mixed higher order elements is shown in (b). Both
node numbers and element numbers are shown in (a) and (b).
-104-
-------
Convective test problem rectangular matrix
Units used are meters, days, celsius, calories
10 22 4
30001 1E13
(lx,2G12.6)
(lx,7G12.6)
GROUP II
11 2
0 4 7 8 11 12 16 18 20 24 28
4 0
GROUP III
.510 1E6
10 0 1
0 4 1
1 100 2 100 21 105 22 105
0 0 1
GROUP IV
1
11111111
1 0 1
1110 55000 55000 700000 .1 .1
GROUP V
0 0
0 0
0 0
GROUP VI
0
GROUP VII
1
1 3 0
10
1
10 4 0
15
GROUP VIII
2 0
1 3 10 4
GROUP IX
00000000
75 50 150 50 150 0 1 0
Figure C-10. The input data used to model one-dimensional heat
transport with all linear elements.
-105-
-------
TABLE C-2. PARAMETERS, INITIAL CONDITIONS, AND BOUNDAR1 CONDITIONS
USED FOh THE HEAT-FLOW EQUATION IN ThE LINEAR HEAT TRANSPORT PROBLEM3
V
Kt
PCy
PCS
time step
Parameters
0. 1786 m/day
0.55fx 105 cal/m day °C
0.1 x 10? cal/m2 °C
0.7 x 106 cal/m2 °C
0.5 day
Initial
conditions
TA = 10°C
at all
nodes
Boundary
conditions
Dirichlet (convective flow)
boundaries at x = 0 and
x = 28
Temperature of incoming
fluid: 15°C
No flow at all other
boundaries
aThe water-flow equation was solved for head gradient of 0.1786 m/m with
hydraisic conductivity and porosity set to unity to obtain the specified
velocity.
-106-
-------
CONVECTIVE TEST PROBLEM RECTANGULAR MATRIX
UNITS USED ARE METERS, DAYS, CELSIUS, CALORIES
04//07/78
17:01:09
#OF NODES 22 # OF ELEMENTS 10 BANDWIDTH 4
VELOCITIES ARE BEING CALCULATED AT CURRENT TIME STEP 1
THE X-SPACING IS
0.0 4.0000 7.0000 8.0000 11.0000 12.0000 16.0000 18.0000
20.0000 24.0000 28.0000
THE Y-SPACING IS
4.0000 0.0
EQUATION 2 TIME STEP .500000 HEAT OR MASS CAPACITY COEFFICIENT .100000+07
PARAMETER FACTORS--IN ORDER 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000
§ MATERIAL HOR PERM VER PERM STORAGE HOR THERM COND VERT THERM COND SPECIFIC DISPERSION COEFS
I H E AT
1 1.000 1.000 .0000 .5500+05 .5500+05 .7000+06.1000+00.1000+00
THE VALUES IN THE TYPE OF MATERIAL MATRIX ARE
1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000
1.00000 1.00000 1.00000
INFORMATION FOR MASS OR HEAT TRANSFER ACROSS A SPECIFIED HEAD BOUNDARY
ELEMENT NUMBER--BOUNDARY CODE—TEMPERATURE OR CONCENTRATION OF INCOMING FLUID
CONVECTIVE OUT AND CONDUCTIVE BOUNDARY INFORMATION ELEM //--BOUNDARY CODE —TRANSFER COEFFICIENT
1 3 .000000
Figure C-ll. Program output for one-dimensional heat transport problem with linear elements.(continued)
-------
WATER FLOW BOUNDARIES—ELEM « AND BOUNDARY CODE
1 3 10 4
POTENTIAL DISTRIBUTION AT TIME STEP
.000000
100.0000
100.714
101.250
101.429
101.964
102.143
102.857
103.214
103.571
104.286
105.000
100.0000
100.714
101.250
101.429
101.964
102.143
102.857
103.214
103.571
104.286
105.000
B. FLUX RECHARGE
B. FLUX DISCHARGE
CHANGE IN STORAGE
QUANTITY PUMPED
DIFFERENCE
CUMULATIVE MASS BLANCE
.714288
.714281
.000000
.000000
.715256-05
RATES FOR THIS
.714288
.714281
.000000
.000000
.715256-05
STEP
O
oo
TIME STEP .000000
FLOWS IN THE X DIRECTION
.714281 .714284 .714285 .714286 .714287 .714286 .714287
.714288 .714288 .714288
FLOWS IN THE Y DIRECTION
.000000 .715256-06 .119209-06 .357628-06 .000000 .476837-06 .000000
.000000 -.476837-06 .000000
THE DISPERSION ROUTINE IS BEING USED
Figure C-ll. (continued)
-------
TEMPERATURE DISTRIBUTION AT TIME STEP
25.0000
10.0001
9.99935
10.0014
10.0074
9.99971
9.96143
9.90413
10.3286
11.4248
14.1568
15.3485
10.0002
9.99949
10.0016
10.0074
9.99992
9.96156
9.90422
10. 3286
11.4248
14.1569
15. 3486
l
»-•
s
CONDUCTIVE TRANSFER
CONVECTIVE TRANSFER--OUT
CONVECTIVE TRANSFER—IN
B. FLUX RECHARGE
B. FLUX DISCHARGE
CHANGE IN STORAGE
QUANTITY PUMPED
DIFFERENCE
CUMULATIVE MASS BLANCE
.000000
.178571+09
.267858+09
.000000
.000000
.892868+08
.000000
429.500
TEMPERATURE DISTRIBUTION AT TIME STEP
10.0037
9.97586
9.94439
10.0029
10.4781
10.9284
12.8394
14.0114
14.5750
15.1468
14.9093
10.0039
9.97601
9.94460
10.0029
10.4783
10.9285
12.8395
14.0115
14.5750
15.1468
14.9093
50.0000
RATES FOR THIS TIME STEP
.000000
.357147+07
.535716+07
.000000
.000000
.178568+07
.000000
8.37500
Figure C-ll. (continued)
-------
CONDUCTIVE TRANSFER
CONVECTIVE TRANSFER-OUT
CONVECTIVE TRANSFER-IN
B. FLUX RECHARGE
B. FLUX DISCHARGE
CHANGE IN STORAGE
QUANTITY PUMPED
DIFFERENCE
CUMULATIVE MASS BLANCE
.000000
.357165+09
.535716+09
.000000
.000000
.178550+09
.000000
896.781
RATES FOR
THIS TIME STEP
.000000
.357283+07
.535716+07
.000000
.000000
.178431+07
.000000
14.2500
o
TEMPERATURE DISTRIBUTION AT TIME STEP
10.0302 10.0304
10.5080 10.5082
11.6597 11.6599
12.1306 12.1306
13.4834 13.4837
13.8607 13.8608
14.9332 14.9333
14.9278 14.9278
15.0309 15.0309
15.0045 15.0045
14.9912 14.9913
75.0000
CONDUCTIVE TRANSFER
CONVECTIVE TRANSFER-OUT
CONVECTIVE TRANSFER-IN
B. FLUX RECHARGE
B. FLUX DISCHARGE
CHANGE IN STORAGE
QUANTITY PUMPED
DIFFERENCE
CUMULATIVE MASS BLANCE
.000000
.535367+09
.803574+09
.000000
.000000
.268205+09
.000000
1372.56
RATES FOR THIS TIME STEP
.000000
.358047+07
.535716+07
.000000
.000000
.177668+07
.000000
7.93750
Figure C-ll. Program output for one-dimensional heat transport problem with linear elements,
-------
CONVECTIVE TEST PROGRAM
UNITS USED ARE METERS, DAYS, CELCIUS, CALORIES
7 32 12
30021 1E13
(lx,2G12.6/3G12.6/2G12.6/2G12.6/4G12.6/2G12.6/2G12.6/4G12.6/2G12.6/2G12.6/
3G12.6/2G12.6/2G12.6)
(lx,7G12.6)
GROUP II
104
200
344
442
540
654
750
8 7. 4
970
10 8 4
11 8 3
12 8 1
13 8 0
14 9 4
15 9 0
16 11 4
17 11 0
18 12 4
19 12 3
20 12 1
21 12 0
22 16 4
23 16 0
24 18 4
25 18 0
26 20 4
27 20 2
28 20 0
29 24 4
30 24 0
31 28 4
32 28 0
1 2 5 4* 3 1 2
2 5 7* 9* 13 12* 11* 10 8* 6* 3 4* 5
3 13 15* 17* 21 20* 19* 18 16* 14* 10 11* 12* 13
4 21 23 22 18 19* 20* 21
5 23 25* 28 27* 26 24* 22 23
6 28 30 29 26 27* 28
7 30 32 31 29 30
(continued)
Figure C-12. The data deck used to model one-dimensional heat transport
with mixed elements.
-Ill-
-------
GROUP III
.5 1 0 1E6
10 0 1
041
1 100 2 100 31 105 32 105
001
GROUP IV
1
11111111
101
1110 55000 55000 700000 .1 .1
GROUP V
0 0
0 0
0 0
GROUP VI
0
GROUP VII
1
130
10
1
7 A 0
15
GROUP VIII
2 0
1374
GROUP IX
00000000
75 50 150 50 150 0 1 0
Figure C-12. The data deck used to model one-dimensional heat transport
with mixed elements.
-112-
-------
CONVECTIVE TEST PROGRAM
UNITS USED ARE METERS, DAYS, CELCIUS, CALORIES
04/07/76
1,7:00:46
t OR NODES 32 // OF ELEMENTS 7 BANDWIDTH 12
VELOCITIES ARE BEING CALCULATED AT CURRENT TIME STEP 1
EQUATION 2 TIME STEP .500000 HEAT OR MASS CAPACITY COEFFICIENT .100000+07
PARAMETER FACTORS—IN ORDER 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000
MATERIAL HOR PERM VER PERM STORAGE HOR THERN COND
VERT THERM COND SPECIFIC
HEAT
1.000
1.000
.0000
.5500+05
.5500+05
.7000+06
THE VALUES IN THE TYPE OF MATERIAL MATRIX ARE
1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000
INFORMATION FOR MASS OR HEAT TRANSFER ACROSS A SPECIFIED HEAD BOUNDARY
ELEMENT UNSER—BOUNDARY CODE—TEMPERATURE OR CONCENTRATION OF INCOMING FLUID
7 4 15.000
CONVECTIVE OUT AND CONDUCTIVE BOUNDARY INFORMATION ELEM //—BOUNDARY CODE—TRANSFER COEFFICIENT
1 3 .000000
WATER FLOW BOUNDARIES—ELEM # AND BOUNDARY CODE
13 74
POTENTIAL DISTRIBUTION AT TIME STEP
100.0000 100.0000
100.714 100.714
100.893 100.893
101.253 101.253
101.429 101.429
101.657 101.657
101.964 101.964
102.143 102.143
102.857 102.857
.000000
100.714
101.429 101.429
102.143 102.143
DISPERSION
COEFS
.1000+00 .1000+00
(continued)
Figure C-13. Program output for the one-dimensional heat transport problem with mixed elements.
-------
103.214
103.571
104.236
103.000
103.214
103.571
104.236
105.000
103.571
B. FLUX RECHARGE
B. FLUX DISCHARGE
CHANGE IN STORAGE
QUANTITY PUMPED
DIFFERENCE
TIME STEP .000000
FLOWS IN THE X DIRECTION
.714262 1.42996
FLOWS IN THE Y DIRECTION
-.476837-56 .558137-05
CUMULATIVE MASS BLANCE
.714289
.714202
.000000
.000000
.267029-04
RATES FOR THIS TIME STEP
.714289
.714262
.000000
.000000
.267029-04
.1,74388
.915181
.714290
.714292
.540157-05 -.548363-05
.00-000 -.953674-06
.714289
.000000
THE DISPERSION ROUTINE IS BEING USED
TEMPERATURE DISTRIBUTION AT TIME STEP 25.0000
10.0002
10.0058 10.0076
9.97589 9.96879
11.4717
10.0005
10.0001
9.99925
10.0025
10.007?
10.0073
9.99686
9.90961
9.7985?
10.3231
11.4719
14.1362
13.3683
10.0006
9.99989
9.99977
10.0026
10.0058
10.0079
9.9551
9.97127
9.8972?
10.373?
11.4715
14.1365
15.3502
CONDUCTIVE TRANSFER
CONVECTIVE TRANSFER-OUT
CONVECTIVE TRANSFER-IN
B. FLUX RECHARGE
B. FLUX DISCHARGE
CHANGE IN STORAGE
QUANTITY PUMPED
DIFFERENCE
CUMULATIVE MASS BLANCE
.000000
.178569-K)9
.267858+09
.000000
.000000
.589193+08
.000000
370063.
RATES FOR THIS TIME STEP
.000000
.357150+07
.535716+07
.000000
.000000
.180833+07
.000000
-22663.4
Figure C-13. (continued)
-------
TEMPERATURE DISTRIBUTION AT TIME STEP
9.98017
9.99911 10.0018
10.8995 10.8954
14.5313
50.0000
10.0049
9.96511
9.96235
9.96261
10.0013
17.1391
17.547?
17.773
12.7773
13.9537
14.5316
15.1661
14.8985
10.0050
9.93723
9.96254
9.95263
9.95-12
17.1396
17.5474
17.3997
12.9877
13.9556
14.5311
15.1663
14.5984
CONDUCTIVE TRANSFER
CONVECTIVE TRANSFER-OUT
CONVECTIVE TRANSFER-IN
B. FLUX RECHARGE
B. FLUX DISCHARGE
CHANGE IN STORAGE
QUANTITY PUMPED
Ul
1
DIFFERENCE
TEMPERATURE
10.8259
10.5035
10.6779
11.6724
12.1127
12.5753
13.4747
13.7692
14.9299
14.9297
15.0453
15.771?
14.9917
DISTRIBUTION
10.0261
17.5926
17.9895
11.6724
12.1154
12.5766
13.4742
13.5693
14.9374
14.9296
15.044?
15.7722
14.9917
.331
AT TIME STEP
10.5035
12.1154
13.8603
15.0451
CUMULATIVE MASS BLANCE
.000000
.357160-H)9
.535716+09
.000000
.000000
.17257+09
.000000
75.0000
12.1135
13.8675
RATES FOR THIS TIME STEP
.000000
.357313+07
.535716+07
.000000
.000000
.179040+07
.000000
-6372.50
Figure C-13. (continued)
-------
CONDUCTIVE TRANSFER
COSVECTIVE TRANSFER-OUT
COHVECTIVE TRANSFER-IN
B. FLUX RECHARGE
B. FLUX DISCHARGE
CHANGE IN STORAGE
QUANNTITY PUMPED
DIFFERENCE
EXGT
CUMULATIVE MASS BLANCE
.000000
.635375409
.000000
.000000
.000000
.567842409
.000000
356718.
RATES FOR THIS TIME STEP
.000000
.357859+07
.000000
.000000
.000000
.183896+07
.000000
-60682.3
Figure C-13. Program output for the one-dimensional heat transport problem with mixed elements.
-------
TABLE C-3. ANALYTICAL SOLUTION AT t= 50 DAYS FOR THE PROBLEM POSED IN
FIGURE C-9 AND FINITE ELEMENT NUMERICAL SOLUTIONS AT t = 50 DAIS FOR
TtoO GRID CONFIGURATIONS AND TWO T1PES OF BOUNDAfll CONDITIONS
-
x = 0
x = 4
x = 8
x = 10
x = 12
x = 16
x = 20
x = 24
Analytical
15. Oa
14.982
14.649
14.017
12.962
10.787
10.062
10.002
Linear
elements (a)
14.901
15.147
14.575
14.011
12.840
10.929
10.003
9.976
Mixed
elements
14.899
15.166
14.531
13.945
12.867
10.900
9.999
9.976
Linear elements
with nodes 21 & 22
specified at 15°C
15.0
15.061
14.484
13.981
13.036
10.724
10.095
9.763
.
aAll values listed are temperatures in degrees centigrade, x = 0 is at the
right in Figure C-10.
-117-
-------
0
8 12 16 20
DISTANCE (METERS)
24
28
Figure C-14.
Analytical solutions (solid lines) and numerical solutions
(dots) for the linear heat transport problem for t = 25, 50
and 75 days.
-118-
-------
C-3. Specification of the boundary temperature, rather than use of a
convective boundary, constrains the solution from oscillating as much near the
origin and is the exact boundary condition used by Ogata and Banks (1961).
The Mohawk River
This problem demonstrates the use of the program to solve an areal problem
to investigate the effect of stream infiltration on ground-water temperatures
in an alluvial aquifer along the Mohawk River near Schenectady, N.Y. (Figure
C-15).
The flood plain of the Mohawk River in the vicinity of Lock and Dam 8,
about 2 miles west of the city of Schenectady, is underlain by more than 100
ft of unconsolidated deposits. From the surface downward the deposits consist
of 30 ft of flood-plain alluvium, 20-100 ft of sandy gravel and sand, and,
immediately above the bedrock, a layer of glacial till 25-50 ft thick (Figure
C-16). The sandy gravel and sand deposit is tapped by the well fields of the
city of Schenectady and torn of Rotterdam, kvinslow (1962) monitored
temperatures in this aquifer and concluded that most of the water pumped from
the well fields originated as infiltration from the Mohawk River. This model
was programmed to determine if the temperature patterns observed by Winslow
could be simulated.
The grid used to model the aquifer is shown in Figure C-17. The boundary
conditons and parameters used in the simulation are listed in Table C-4. All
water flow was assumed to occur within the principal aquifer, and the
principal aquifer was assumed to be thermally insulated from the overlying
flood-plain deposits and the underlying glacial tills. (This assumption
greatly simplified the data input, but it could be relaxed if a better
approximation is desired.)
The temperature of the river water was changed at each time step. River
temperatures for a 1-yr period were read in at the beginning of program
execution by a call to EwTRl LAKE and were then altered at each time step by a
call to fiNTRi BOUND. ENTRi CHANG was called at each time step to compute heat
flow out of the system with the pumped water. Heat flow at the wells for each
time step was set equal to the temperature at the wells multiplied by the heat
capacity and the rate of pumpage.
The data and the program output for a 10-day simulation, with pumpage
rates only 10* of actual rates, are listed in Figure C-18 and Figure C-19. If
actual pumpage rates are used, a time step of 0.1 day must be used to insure
solution stability, unless a corrector is applied near the wells to force
stability.
-119-
-------
TABLE C-4. PARAMETERS, INITIAL CONDITIONS, AND BOUNDARY CONDITIONS USED
FOR THE SIMULATION OF TEMPERATURES IN TI£ MOHAWK RIVER ALLUVIAL AQUIFER
Hater flow
Boundary conditions
Mohawk River
All other boundaries
Parameters (principle aquifer)
K11 - K22
Wells—discharge rates
Node 70
Nodes 130, 131, 145, 146
Specified head of 100 fta
No flow
100,000 gal/day ft2
2,000,000 gal/day
4,500,000 gal/day at each node
Heat flow
Initial conditions
Boundary conditions
River
All other boundaries
Parameters
vt rt
*11 = K22
Time step
I± = 10°C
Connective flux in
No flow
1.3 x 101* cal/day ft°C
2.1 x 10* cal/ft3 oc
10 ft
1 ft
1 day
aUnits of feet, gallons, calories, and days were used in this simulation.
-120-
-------
•ROTTERDAM
WELL FIELD
SCHENECTADY
WELL FIELD
A'
0 500 1000 FEET
Figure C-15. Areal view of the Mohawk River Valley showing location of the
Schenectady and Rotterdam well fields (Winslow 1962).
MOHAWK
RIVER
0
g '5
50
u
u
LL
150
0
A1
MOHAWK
RIVER
FLOOD PLAIN DEPOSITS
SANDY GRAVEL AND SAND
GLACIAL TILL
SHALES AND SILTSTONE
1 L_
IOOO
2000
FEET
3000
4000
Figure C-16.
Cross-sectional view of the Mohawk River alluvial aquifer along
section A-A1 of Figure C-15.
-121-
-------
400
600
MOHAWK RIVER
SI400
2200
2
197
3
/
198
4
V
;
199
5
J^»
X >
200
6
t
i
7
S A
8
^
9
1
10
VEL
I
WE
_ Fl
206
12
_L
ELD
207
13
208
14
209
15
210
0
1600 3200
FEET
4800
Figure C-17. Grid used to discretize the Mohawk River problem. (The
node numbers in the first and last rows are listed.)
-122-
-------
MOHAWK RIVER PROBLEM
UNITS ARE GALLONS DAYS FEET
182 210 17
30001 1E16
(lx,15F6.2)
GROUP II
14 15
400 450 500 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600
5600 5200 4800 4480 4000 3600 3200 2800 2400 2000 1600 1200 800 400
GROUP III
111 3785.68
10 0 1
0-11
100 100 100 100 100 100 100 100 100 100 100 100 100 100 100
000000000000000
000000000
oooo-ooooo
000000000
000000000
000000000
000000000
999
001
GROUP IV
4
111111111
0-11
111111111
222222222
222222222
422222222
444222222
444422222
444444222
444444222
444444422
444444422
444444443
444444443
444444443
1 100000 1000000
2 100000 1000000
0
0
0
0
0
0
1
2
2
2
2
2
2
2
2
2
3
3
3
0
0
000
000
000
000
000
000
111
222
222
222
222
222
222
222
222
222
333
333
333
13000
13000
0000000000000
0000000000000
0000000000000
0000000000000
0000000000000
0000000000000
1
2
2
2
2
2
2
2
2
2
3
3
3 999
13000 21000 10 1
13000 21000 10 1
0000
0000
0000
0000
0000
0000
3 500 500 0 13000 13000 21000 10 1
4 0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001
GROUP VI
0 0
5 5
130 -4.5E5 131 -4.5E5 145 -4.5E5 146 -4.5E5 70 -2E5
130 1 131 1 145 1 146 1 70 1
(continued)
Figure C-18. The data deck used to model heat flow in the Mohawk
River alluvial aquifer.
-123-
-------
0 0
GROUP VI 1
1
1 3
0-11
1 1 1 1 1 10 20 30 40 50 50 50 50 50 50
1 1 1 1 1 10 20 30 40 50 50 50 50 50 50
1 1 1 1 1 10 20 30 40 50 50 50 50 50 50
1 1 1 1 1 10 20 30 40 50 50 50 50 50 50
1 1 1 1 1 10 20 30 40 50 50 50 50 50 50
99 99 99 20 20 25 35 45 50 55 55 55 55 55 55
99 99 99 99 40 45 50 55 60 60 60 60 60 60 60
99 99 99 99 99 99 65 65 65 60 60 60 60 60 60
99 99 99 99 99 99 70 70 70 70 70 70 70 70 70
75 75 75 75 75 75 75 75 75 75 75 75 75 75 75
808080808080808080808080808080
80 80 80 80 80 80 80 80 80 80 80 80 80 80' 80
808080808080808080808080808080
80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 999
100 0 1
GROUP VII
14
1 3
8 3
3000 2 3
3000 9 3
3000 3 3 3000 4 3 3000 5 3 3000 6 3 3000 7 3 3000
3000 10 3 3000 11 3 3000 12 3 3000 13 3 3000 14 3 3000
000000000000000
14
132333435363738393 10 3 11 3123 13 3 14 3
000000000000000
GROUP VIII
14 0
3 9 3 10 3 11 3 12 3 13 3 14 3
132333
GROUP IX
110000
0 220 220 1
277 860
1.0000
2.0000
3.0000
4.0000
5.0000
6.0000
7.0000
8.0000
9.0000
10.0000
10 10 10 10
435363738
220 220 1 220 220
29.5000
28.5000
27.5000
27.4600
27.4100
27.3700
27.3300
27.2900
27.2400
27.2000
10 1 1 1
111
41.0000
35.9000
30.8000
30.5000
30.2000
29.9000
29.6000
29.3000
29.0000
28.7000
Figure C-18.
The data deck used to model heat flow in the Mohawk
River alluvial aquifier.
-124-
-------
S3
Ui
I
MOHAWK RIVER PROBLEM
UNITS ARE GALLONS DAYS FEET
04/07/78
17:01:49
# OF NODES 210 # OF ELEMENTS 182 BANDWIDTH 17
VELOCITIES ARE BEING CALCULATED AT CURRENT TIME STEP 1
THE X-SPACING IS
400.0000 450.0000 500.0000 600.0000 800.0000 1000.0000 1200.0000 1400.0000
1600.0000 1800.0000 2000.0000 2200.0000 2400.0000 2600.0000
THE Y-SPACING IS:
5600.0000 5200.0000 4800.0000 4480.0000 4000.0000 3600.0000 . 3200.0000 2800.0000
2400.0000 2000.0000 1600.0000 1200.0000 800.0000 400.0000 0.0
EQUATION 2 TIME STEP 1.00000 HEAT OR MASS CAPACITY COEFFICIENT 3785.68
PARAMETER FACTORS- IN ORDER 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000
MATERIAL HOR PERM VER PERM STORAGE HOR THERM COND VERT THERM COND
1 .1000+06 .1000+06 .0000 .1300+05 .1300+05
SPECIFIC DISPERSION COEFS
HEAT
.2100+05 10.00 1.000
THE VALUES IN
1.00
2.00
2.00
4.00
4.00
4.00
4.00
4.00
4.00
4.00
4.00
4.00
4.00
1.00
2.00
2.00
2.00
4.00
4.00
4.00
4.00
4.00
4.00
4.00
4.00
4.00
2
3
4
. 1000+06
500.0
.1000-02
. 1000+06
500.0
. 1000-02
0000
0000
1000-02
.1300+05
.1300+05
. 1000-02
. 1300+05
. 1300+05
. 1000-02
THE TYPE OF MATERIAL ARE
1.00
2.00
2.00
2.00
4.00
4.00
4.00
4.00
4.00
4.00
4.00
4.00
4.00
1.00 1.00
2.00 2.00
2.00 2.00
2.00 2.00
2.00 2.00
4.00 2.00
4.00 4.00
4.00 4.00
4.00 4.00
4.00 4.00
4.00 4.00
4.00 4.00
4.00 4.00
1.00
2.00
2.00
2.00
2.00
2.00
4.00
4.00
4.00
4.00
4.00
4.00
4.00
1.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
4.00
4.00
4.00
4.00
4.00
1.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
4.00
4.00
4.00
1.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.0C
2.00
3.00
3.00
3.00
Figure C-19. Program output for Mohawk
1.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
3.00
3.00
3.00
River
1.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
3.00
3.00
3.00
1.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
3.00
3.00
3.00
1.00 1.00
2.00 2.00
2.00 2.00
2.00 2.00
2.00 2.00
2.00 2.00
2.00 2.00
2.00 2.00
2.00 2.00
2.00 2.00
3.00 3.00
3.00 3.- 00
3.00 3.00
problem. (<
.2100+05 10.00 1.000
.2100+05 10.00 1.000
.1000-02 .1000-02.1000-02
(continued)
-------
Ni
POINT SOURCES OF WATER—NODE # AND AMOUNT
130 -450000. 131 -450000.
POINT SOURCES OF HEAT
130 1.00000
131 1.00000
145 -450000. 146 -450000. 70 -200000.
145 1.00000 146 1.00000 70 1.00000
ELEVATION OF THE AQUIFER BOTTOM
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00
99.00 99.00
99.00 99.00
99.00 99.00
99.00 99.00
75.00 75.00
80.00 80.00
80.00 80.00
80.00 80.00
80.00 80.00
1.00
1.00
1.00
1.00
1.00
99.00
99.00
99.00
99.00
75.00
80.00
80.00
80.00
80.00
1.00
1.00
1.00
1.00
1.00
20.00
99.00
99.00
99.00
75.00
80.00
80.00
80.00
80.00
1.00
. 1.00
1.00
1.00
1.00
20.00
40.00
99.00
99.00
75.00
80.00
80.00
80.00
80.00
10.00
10.00
10.00
10.00
10.00
25.00
45.00
99.00
99.00
75.00
80.00
80.00
80.00
80.00
20.00
20.00
20.00
20.00
20.00
35.00
50.00
65.00
70.00
75.00
80.00
80.00
80.00
80.00
30.00
30.00
30.00
30.00
30.00
45.00
55.00
65.00
70.00
75.00
80.00
80.00
80.00
80.00
40.00
40.00
40.00
40.00
40.00
50.00
60.00
65.00
70.00
75.00
80.00
80.00
80.00
80.00
50.00
50.00
50.00
50.00
50.00
55.00
60.00
65.00
70.00
75.00
80.00
80.00
80.00
80.00
50.00
50.00
50.00
50.00
50.00
55.00
60.00
60.00
70.00
75.00
80.00
80.00
80.00
80.00
50.00
50.00
50.00
50.00
50.00
55.00
60.00
60.00
70.00
75.00
80.00
80.00
80.00
80.00
50.00
50.00
50.00
50.00
50.00
55.00
60.00
60.00
70.00
75.00
80.00
80.00
80.00
80.00
50.00
50.00
50.00
50.00
50.00
55.00
60.00
60.00
70.00
75.00
80.00
80.00
80.00
80.00
50.00
50.00
50.00
50.00
50.00
55.00
60.00
60.00
70.00
75.00
80.00
80.00
80.00
80.00
INFORMATION FOR MASS OR HEAT TRANSFER ACROSS A SPECIFIED HEAD BOUNDARY
ELEMENT NUMBER—BOUNDARY CODE—TEMPERATURE OR CONCENTRATION OF INCOMING FLUID
1 3 .00000 2 3 .00000 3 3 .00000 4 3 .00000
6 3 .00000 7 3 .00000 8 3 .00000 9 3 .00000
11 3 .00000 12 3 .00000 13 3 .00000 14 3 .00000
5 3 .00000
10 3 .00000
CONVENTIVE OUT AND CONDUCTIVE BOUNDARY INFORMATION
1 3 3000.00 2 3 3000.00 3 3
6 3 3000.00 7 3 3000.00 8 3
11 3 3000.00 12 3 3000.00 13 3
WATER FLOW BOUNDARIES--ELEM 0 AND BOUNDARY CODE
13 23 3343 5363
10 3 11 3 12 3 13 3 14 3
ELEM //--BOUNDARY CODE—TRANSFER COEFFICIENT
3000.00 4 3 3000.00 5 3 3000.00
3000.00 9 3 3000.00 10 3 3000.00
3000.00 14 3 3000.00
7 3
8 3
9 3
10 3
Figure C-19. (continued)
-------
ITERATIONS NEEDED FOR CONVERGENCE
No
vj
I
POTENTIAL DISTRIBUTION AT TIME STEP .000000
100.00 100.00 100.00
100.00 100.00 100.00
100.00 100.00 100.00
100.00 100.00 100.00
100.00 100.00 100.00
100.00 100.00 100.00
99.97 99.97 99.98
99.94 99.94 99.94
99.91 99.91 99.91
99.90 99.90 99.90
99.90 99.90 99.90
99.90 99.90 99.90
99.90 99.90 99.90
99.90 99.90 99.90
100.00
100.00
100.00
100.00
100.00
100.00
99.99
99.96
99.91
99.90
99.90
99.90
99.89
99.89
100.00 100.
100.00 100.
100.00 100.
100.00 100.
100.00 99.
99.99 99.
99.99 99.
99.98 99.
99.91 99.
99.89 99.
99.89 99.
99.89 99.
99.89 99.
99.89 99.
00
00
00
00
99
99
99
97
91
89
88
86
87
87
100.00
100.00
100.00
99.99
99.98
99.97
99.96
99.94
99.93
99.89
99.87
99.85
99.84
99.84
100.00 100.00 100.00
100.00
99.99
99.98
99.97
99.95
99.93
99.91
99.88
99.84
99.83
99.81
99.79
99.79
CUMULATIVE MASS BLANCE
B. FLUX RECHARGE
B. FLUX DISCHARGE
CHANGE IN STORAGE
QUANTITY PUMPED
DIFFERENCE
-
.200989+07
.000000
.000000
.200000+07
9886.42
99.99 99.99
99.99 99.98
99.97 99.96
99.95 99.92
99.92 99.89
99.89 99.84
99.85 99.78
99.82 99.68
99.79 99.63
99.77 99.65
99.72 99.69
99.70 99.70
99.70 99.70
RATES FOR THIS
.200989+07
.000000
.000000
-.200000+07
9886.42
100.00 100.00
99.99
99.98
99.97
99.93
99.89
99.84
99.78
99.68
99.63
99.65
99.70
99.72
99.72
TIME STEP
99.99
99.99
99.97
99.94
99.91
99.88
99.84
99.82
99.80
99.78
99.77
99.77
99.77
100.00 100.00
99.99 100.00
99.99
99.98
99.96
99.93
99.91
99.89
99.88
99.87
99.86
99.84
99.83
99.83
99.99
99.98
99.97
99.95
99.93
99.92
99.91
99.90
99.90
99.88
99.87
99.87
100.00
100.00
99.99
99.98
99.97
99.95
99.94
99.93
99.92
99.91
99.91
99.90
99.89
99.88
THE DISPERSION ROUTINE IS BEING USED
NODE # AND TEMP AT INF.
9 15.04 10 15.04
NODE # AND TEMP AT INF.
9 15.52 10 15.52
NODE // AND TEMP AT INF.
9 16.00 10 16.00
NODE // AND TEMP AT INF.
9 16.48 10 16.48
NODE // AND TEMP AT INF.
9 16.96 10 16.96
NODE # AND TEMP AT INF.
9 17.44 10 17.44
NODE # AND TEMP AT INF.
9 17.92 10 17.92
1 15.04
11 15.04
1 15.52
11 15.52
1 16.00
11 16.00
1 16.48
11 16.48
1 16.96
11 16.96
1 17.44
11 17.44
1 17.92
11 17.92
2 15.04
12 15.04
2 15.52
12 15.52
2 16.00
12 16.00
2 16.48
12 16.48
2 16.96
12 16.96
2 17.44
12 17.44
2 17.92
12 17.92
3
13
3
13
3
13
3
13
3
13
3
13
3
13
15.04
15.04
15.52
15.52
16.00
16.00
16.48
16.48
16.96
16.96
17.44
17.44
17.92
17.92
4 15.04
14 15.04
4 15.52
14 15.52
4 16.00
14 16.00
4 16.48
14 16.48
4 16.96
14 16.96
4 17.44
14 17.44
4 17.92
14 17.92
5 15.04 6
5 15.52 6
5 16.00 6
5 16.48 6
5 16.96 6
5 17.44 6
5 17.92 6
15.04 7
15.52 7
16.00 7
16.48 7
16.96 7
17.44 7
17.92 7
15.04
15.52
16.00
16.48
16.96
17.44
17.92
6 15.04
8 15.52
8 16.00
8 16.48
8 16.96
8 17.44
8 17.92
Figure C-19. (continued)
-------
NODE I AND TEMP Af INF. 1 18.40 2 18.40 3 18.40 4 18.40 5 18.40 6 18.40 7 18.40 8 18.40
9
NODE t
9
NODE 1
9
18.40 10 18.40 11
AND TEMP AT INF. 1
18.88 10 18.88 11
AND TEMP AT INF. 1
19.36 10 19.36 11
TEMPERATURE DISTRIBUTION AT
10.69
9.82
10.04
9.98
10.01
10.00
10.01
10.01
10.00
10.00
10.00
10.00
10.00
^ 10.00
N>
oo
10.70
9.82
10.04
9.99
10.00
10.00
10.01
10.01
10.00
10.00
10.00
10.00
10.00
10.00
10.76
9.81
10.04
9.99
10.00
10.00
10.01
10.02
10.00
10.00
10.00
10.00
10.00
10.00
10.91
9.77
10.04
9.99
10.00
10.00
10.00
10.02
10.00
10.00
10.00
10.00
10.00
10.00
18.40
18.88
18.88
19.36
19.36
12 18.40 13 18.40
2 18.88 3 18.88
12 18.88 13 18.88
2 19.36 3 19.36
12 19.36 13 19.36
TIME STEP
11.38
9.67
10.06
9.99
10.00
10.00
10.00
10.01
10.00
10.00
10.00
10.00
10.00
10.00
12.20
9.52
10.07
9.99
10.00
10.00
10.00
10.01
10.00
9.99
10.00
10.00
10.00
10.00
10.000
13.50
9.39
10.07
9.99
10.00
10,00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
14.99
9.46
10.02
10.00
10.00
10.00
10.00
10.00
10.00
9.99
10.00
10.00
10.00
10.00
14 18.40
4 18.88 5
14 18.88
4 19.36 5
14 19.36
16.49
9.93
9.91
10.02
10.00
10.00
10.00
10.00
10.02
10.04
10.00
10.00
10.00
10.00
17.48
10.67
9.84
10.03
9.99
10.00
10.00
10.02
9.98
9.92
10.03
9.99
10.00
10.00
18.88 6 18
19.36 6 19
17.25
10.49
9.86
10.02
10.00
10.00
10.00
10.02
9.98
9.92
10.03
9.99
10.00
10.00
16.63
10.03
9.90
10.02
10.00
10.00
10.00
10.00
10.01
10.03
10.00
10.00
10.00
10.00
.88 7
.36 7
15.86
9.69
9.96
10.01
10.00
10.00
10.00
10.00
10.00
9.99
10.00
10.00
10.00
10.00
18.88
19.36
15.15
9.52
10.01
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
8 18.88
8 19.36
14.87
9.48
10.02
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
10.00
B. FLUX DISCHARGE
CHANGE IN STORAGE
QUANTITY PUMPED
DIFFERENCE
CONDUCTIVE TRANSFER
CONVECTIVE TRANSFER—OUT
CONVECTIVE TRANSFER—IN
B. FLUX RECHARGE
.000000
.620931+12
-.755490+12
-.471091+07
CUMULATIVE MASS BLANCE
.676988+11
.000000
.130872+13
.000000
.000000
.794988+11
-.754061+11
-502208.
RATES FOR THIS TIME STEP
.759752+10
.000000
.147307+12
.000000
Figure C-19. Program output for Mohawk River problem.
-------
The Columbia Generating Station Site
This example is representative of the application described in section 5
of this report in which temperatures and water flows in the alluvial aquifer
along the Wisconsin River were simulated after construction of a cooling lake
on the river flood plain. Temperatures and water flows were simulated in an
aquifer cross section showi in the schematic diagram in figure C-20, and a
finite element grid was used to discretize the cross section (figure C-21).
The parameters, boundary conditions, and initial conditions used in the
simulation are listed in Table C-5.
The input data and the program output for a 9-day simulation are listed in
Figure C-22 and figure C-25. ENTfli BOUND was called at each time step to
change the temperature of the water in the cooling lake and the air
temperature.
-129-
-------
TABLE C-5. PARAMETERS, INITIAL CONDITIONS, AND BOUNDAR1 CONDITIONS
USED IN THE COLUMBIA GENERATING STATION PROBLEM
hater flow
Boundary conditions
River
Marsh
Lake
All other boundaries
Parameters
K11» K22
Heat flow
Initial conditions
Boundary conditions
Lake
Marsh, River
All other boundaries
Parameters
K11» K22> Pcs
Time step
Specified head at 778 fta
Specified head at 780 ft
Specified head at 789.2 ft
No flow
Ten sets of values specified
(refer to program output)
= 10°C
Convective flux in
Convective flux out and
conductive flux
No flow
Ten sets of values specified
(refer to program output)
0.33 ft
0.08 ft
3 days
afhe units of feet, gallons, calories, and days were used in this simulation.
-130-
-------
790
780
770
COOLING
LAKE
750
O
m
3801
-565
MARSH
PEATS AND CLAYS
WISCONSIN RIVER
AND DUCK CREEK
(water, marsh and swamp)
-\n , ,^—-^— 1778
.SILTS_AND^FINE_SANDS
MEDIUM TO COARSE SANDS
FINE SANDS WITH SILT AND CLAY LENSES
SANDSTONE
J
2400
FEET
Figure. C-20. Schematic cross section of the Columbia Generating Station site
along an east-west line.
UJ °
0
2 2
cr
ID 5
CO D
£ 10
l-
5 "5
O
di 20
CD
t 4°
UJ
u- 4.on
COOLING
• LAKE
-DIKE-
MARSH SURFACE
»
RIVER
^LOWLANDS
-565 -180 -60 -30 0 20 40 60 105 160 420 2396
-308 -110 282 1100
FEET FROM THE DIKE
Figure C-21.
The grid used to discretize the cross section simulated at the
Columbia Generating Station site.
-131-
-------
CGS 130'S
UNITS DSHD ARE GALLONS FOR FLOW, FT FOR DISTANCE, CAL, CENTIGRADE
105 128 10
30001 1E14
(1X.8G12.6)
(1X.7G12.6)
GROUP 22
16 8
-565 -308 -180 -110 -60 -30 0 20 40 60 105 160 260 420 1100 2396
0 -2 -5 -10 -15 -20 -40 -400
GROUP III
311 3785.68
10 0 1
0-11
89.30000000 89.30000000 89.30000000
89.30000000 89.30000000 84.50000000
80 0000000800000000 80 0000000 80 0000000
800000000 80 0000000 80 0000000 79 0000000
78.5 0000000 78.5 0 0 0 0 0 0 0 999
001
GROUP IV CGS 130'S CARDS USED FOR SAVANNAH PAPER
10
1.6 1.6 1 .7 .7 1 .2 .2
1-11
1234465
1234465
1234465
1234465
7234465
7 2 10 8 8 6 5
1 9 10 8 8 6 5
1 9 10 8 8 6 5
1 9 10 8 8 6 5
1 9 10 8 8 6 5
1 9 10 8 8 6 5
1 9 10 8 8 6 5
1 9 10 8 8 6 5
1 1 10 8 8 6 5
1 1 10 8 8 6 5
1
999
100 20 0 20000 20000 26000 .33
2
110 8000 8000 26000
3
.33 .08
.08
(continued)
Figure C-22. The data deck used to model heat flow at the Columbia
Generating Station site.
-132-
-------
100 20 0 17000 17000 21000 .33 .08
4
500 50 0 17000 17000 21000 .33 .08
5
10 10 0 8000 8000 26000 .33 .08
6
63 1 72 1 81 1 144 1 153 1 162 1
63 1 72 1 81 1 144 1 153 1 162 1
0 0
GROUP VI
0
GROUP VII
000000000000000000000000000000 00 0000 00 0
0 0
GROUP VIII
0 9
132333435363738393 10 3 11 3
GROUP IX
1 1 40 1 0 0
2 9 16 24 26 33 40 48 50 57 64 72
74 81 88 96 98 105 112 120 122 129 136 144 146
153 160 168 170 177 184 192 194 201 208 216 218 225 232 240
-1.0001 11100
11111111
1
11 1 15 5
860
857 10.2222 24.6111
858 12.5000 26.0555
859 12.7222 25.1666
860 11.8333 24.6111
11111110
-81 1
-142 -284 -142 142 284 142
000 3.15E6 6.36E6 3.15E6
12000 12000 12000 12000 12000 12000 12000 12000
12000 12000 12000 12000 12000 12000 12000 12000
0 0
000000
0000000
12000 12000 12000 12000 12000 12000 12000 12000
12000 12000 12000 12000 12000 12000 12000 12000
81 1
561.5 1123 561.5 -561.5 -1123 -561.5
-9.08E6 -1.82E7 -9.08E6 000
12000 12000 12000 12000 12000 12000 12000 12000
12000 12000 12000 12000 12000 12000 12000 12000
0 0
0000000
0000000
12000 12000 12000 12000 12000 12000 12000 12000
12000 12000 12000 12000 12000 12000 12000 12000
Figure C-22.
The data deck used to model heat flow at the Columbia
Generating Station site.
-133-
-------
CGS 130'S UNITS USED ARE GALLONS FOR FLOW,FT FOR DISTANCE,CAL,CENTIGRADE
ti OF NODES 128 9 OF ELEMENTS 105 PANDWIDTH 10
VELOCITIES ARE BEING CALCULATED AT CURRENT TIME STEP 1
THE X-SPACING IS
-565.0000 -508.0000
40.0000 60.0000
THE Y-SPACING IS
0.0 -4.0000
EDUCATION 2 TIME STEP 3
-160.0000
105.0000
-5.0000
. 00000
PARAMETER FACTORS— IN ORDER 1.60000
1
u» MATERIAL HOR PERM
1 160.0
2 1,800
3 160.0
4 800.0
5 16.00
6 200.0
7 1.800
8 1600.
9 3.200
10 280.0
VER PERM
32.00
1.600
32.00
80.00
16.00
19.20
1.600
160.0
3.200
40.00
-110.0000
-160.0000
-10.0000
HEAT OR MASS
1.60000
-60.0000 -30
260.0000 420
-15.0000 -20
CAPACITY COEFFICIENT
1.00000 .700000
.0000 0
.0000 1100
.0000 -40
3785.68
. 700000 1
STORAGE HOR THERM COND VERT THERM COND
.0000 .1400+05 .1400+. 05
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
5600.
.1190+05
.1190+05
5600.
.1190+05
5600.
.1190+05
5600.
.1190+05
5600.
.1190+05
.1190+05
5600.
.1190+05
5600.
.1190+05
5600.
.1190+05
.0 20.0000
.0000 2396.0000
.0000 -400.0000
.00000 .200000 .200000
SPECIFIC DISPERSION
HEAT
.2600+05 .6600-01
.2600+05 .6600-01
.2100+05 .6600-01
.2100+05 .6600-01
.2600+05 .6600-01
.2100+05 .6600-01
.2600+05 .6600-01
.2100+05 .6600-01
.2600+05 .6600-01
.2100+05 .6600-01
COEFS
. 1600-01
. 1600-01
. 1600-01
.1600-01
. 1600-01
.1600-01
. 1600-01
.1600-01
.1600r.01
.1600-01
Figure C-23. Program output for Columbia Generating Station problem, (continued)
-------
IN THE TYPE OF MATERIAL MATRIX ARE
LU T*»*JW»p»fc* *
1.00000
1.00000
1.00000
1.00000
7.00000
7 . 00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
1.00000
2.00000
2.00000
2.00000
2.00000
2.00000
2.00000
9.00000
2.00000
9.00000
9.00000
9.00000
9.00000
9.00000
1.00000
1.00000
3.00000
3.00000
3.00000
3.00000
3.00000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
4.00000
4.00000
4.00000
4.00000
4.00000
8.00000
8.00000
8.00000
8.00000
8.00000
8.00000
8.00000
8.00000
8.00000
8.00000
4.00000
4 . 00000
4.00000
4.00000
4.00000
8.00000
8.00000
8.00000
8.00000
8.00000
8.00000
8.00000
8.00000
8.00000
8.00000
6.00000
6.00000
6.00000
6.00000
6.00000
6.00000
6.00000
6.00000
6.00000
6.00000
6.00000
6.00000
6.00000
6.00000
6.00000
5.00000
5.00000
5.00000
5.00000
5.00000
5.00000
5.00000
5.00000
5.00000
5.00000
5.00000
5 . 00000
5.00000
5.00000
5.00000
I
I—"
Ul
INFORMATION FOR MASS OR HEAT TRANSFER ACROSS A SPECIFIED HEAD BOUNDARY
ELEMENT NUMBER—BOUNDARY CODE—TEMPERATURE OR CONCENTRATION OF INCOMING FLUID
1 i .00000 8 1 .00000 15 1 .00000 22 1 .00000
36 1
CONVECTIVE OUT AND CONDUCTIVE BOUNDARY INFORMATION
29 1
.00000
1 1 12000.0
50 1 .000000
85 1 .000000
64 1 8000.00
99 1 8000.00
8 1 12000.00
57 1 .000000
92 1 .000000
71 1 8000.00
15 1 12000.0
64 1 .000000
99 1 .000000
75 1 8000.00
ELEM //—BOUNDARY CODE—TRANSFER COEFFICIENT
22 1 12000.0 43 1 .000000
71 1 .000000 78 1 .000000
50 1 8000.00 57 1 8000.00
85 1 8000.00 92 1 8000.00
WATER FLOW BOUNDARIES--ELEM # AND BOUNDARY CODE
11 81 15 1 22 1 29 1
71 1 78 1 85 1 92 1 99 1
POTENTIAL DISTRIBUTION AT TIME STEP
89.3000
89.3000
89.3000
89.3000
89.3000
84.5000
80.0000
89.3805
89.2615
89.2224
89.2390
89.0895
89.8503
80.2692
88.70000
88.3858
86.9984
85.9284
84.7128
83.3893
82.5679
.00000
88.6525
87.9897
85.8184
85.6591
84.3900
84.3395
82,7641
36 1
88.6355
87.9564
86.7581
85.5716
84.3048
83.3358
82.8003
Figure C-23. (continued)
43 1
88.6218
87.9318
86.7171
85.5204
84.2692
83.3626
82.8165
50 1
88.4519
87.6714
86.3573
85.2024
84.2345
83.6783
83.1117
57 1
64 1
86.8035
85.9030
84.9464
84.4145
84.0487
83.8094
83.5782
-------
-51.5399 -39.3041 -31.2270 -22.3388
-31.6491 -34.3912 -33.3403 -24.6573
-30.5509 -29.3358 -28.9887 -22.8616
-53.6430 -54.0320 -52.3883 -41.7264
-48.1144 -45.1488 -44.7073 -35.9204
-34.1526 -38.6777 -33.9313 -20.7011
-35.987 -15.4392 -35.8504 -31.1358
-134.856 -156.379 -129.908 -105.243
12.9762 18.4182
THE DISPERSION ROUTINE IS
MODE t AND TEMP AT INF.
71 9.00 78 9.00
71 9.00 78 9.00
TEMPERATURE DISTRIBUTION
23.0526 10.6176
24.0723 10.8383
23.2023 11.3082
28.3750 12.0244
JL 28.4317 11.7903
w 9.36313 9.22293
I 10.6118 10.1791
9.68002 9.95359
9.38139 10.0437
9.48225 10.0148
9.42817 10.0091
9.44609 10.3090
9.35584 9.97909
9.43319 9.99364
9.38966 9.98055
9.40101 9.98236
12.3757 9.10431
BEING USED
1 25.76 8 25.76
85 9.00 92 9.00
35 9.00 92 9.00
AT TIME STEP .00000
9.98973 10.0160
9.88251 10.0196
9.35211 10.0222
9.32307 10.0226
9.80178 10.0184
10.8940 10.0030
9.97268 10.9030
10-0947 9.99947
9.99198 10.0016
9.99649 10.0011
9.99830 10.0004
10.6900 10.0791
10.8035 9.99838
10.0908 9.99995
10.0016 9.99975
10.0016 9.99369
-10.7826 -4.78335
-12.7486 -4.72700
-12.2211 -4.62473
-23.2787 -9.53972
-20.6327 -9.53262
-19.2395 -11.5253
-22.1957 -14.5765
-64.2157 -34.1659
3.91385
15 25.76
99 9.00
99 9.00
9.99720
9.99675
9.99657
9.99680
9.99758
9.98476
9.99928
10.0001
9.99978
9.99965
9.99997
10.0001
10.0001
10.00000
10.0001
10.0001
CUMULATIVE MASS BLANCE RATES
CONDUCTIVE TRANSFER
CONVECTIVE TRANSFER—OUT
CONVECTIVE TRANSFER—IN
B. FLUX RECHARGE
B. FLUX DISCHARGE
CHANGE IN STORAGE
QUANTITY PUMPED
DIFFERECE
QXQT
Figure
.114876409
.446673+08
.118618409
.000000
.000000
.188595409
.000000
32501.0
.408686
22 25.76
50 9.00
10.0002
10.0002
10.0002
10.0002
10.0001
10.0030
10.0001
10.0000
10.0000
10.0001
9.99996
9.99995
9.99998
9.99996
9.99997
9.99998
-.416066
-.530258
-.619473
-1.59924
-2.14461
-3.85953
-5.71147
-12.2147
-.678204
43 25.76
57 9.00
9.99999
9.99999
9.99999
9.99999
10.00000
10.0000
10.00000
10.00000
10.0000
10.00000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
50 25.76
64 9.00
10.0000
10.0000
10.0000
10.0000
10.0000
10.00000
10.0000
10.00000
10.0000
10.0000
10.00000
10.00000
10.00000
10.00000
10.00000
10.00000
FOR THIS TIME STEP
.114876409
.448673408
.112618409
.000000
.000000
. 188595409
.000000
32501.0
C-23. Program output for Columbia Generating Station probl
57 9.00 64 9.00
-------
80.0000 80.1100
80.0000 80.1228
80.0000 80.1019
80.0000 80.0735
80.0000 80.0499
80.0000 80.0004
79.0000 79.0329
78.5000 78.4972
78.5000 78.5814
B. FLUX RECHARGE
B. FLUX DISCHARGE
CHANGE IN STORAGE
QUANTITY PUMPED
DIFFERECE
82.1493 82.4126
81.6793 82.1055
81.6384 81.8778
81.1874 81.3707
80.7483 80.8439
80.1561 80.1600
79.0889 79.1506
79.4910 78.4856
79.0047 78.5070
CUMULATIVE MASS BLANCE
417.916
417.716
.000000
.000000
.200280
82.4586
82.1513
81.8771
81.3596
80.5631
80.1619
79.1630
78.4846
78.5075
82.4821
82.1748
81.3987
81.3761
80.8744
80.1647
79.1703
78.4842
78.5077
RATES FOR THIS
417.916
417.916
.000000
.000000
.200280
82.7735
82.4630
82.1775
81.6172
81.0584
80.2634
79.2952
78.4326
78.5085
TIME STEP
83.4091
83.2320
83.0495
82.8280
82.1134
81.2531
80.1966
78.5328
78.4860
TIME STEP .000000
FLOWS IN THE X DIPRECTION
1
U)
T4
-.
_,
— .
-.
-.
rm ^
— 1
.
™ •
-.
-.
-.
-1
-.
,
996091-02
414756-01
281315-01
378952
453898
366091
.08199
372479-01
141957
359611-01
532992-01
673201-01
.67195
208530
443338-03
-.502178-02
-.179398-01
-.315544-01
-.552483-01
-.444803
-.499161
-.117809
-.524130-01
-.533644-01
-.434412-01
-.343201-01
-.251925-01
-.515699-01
-.339984
.275349-02
-1.
-5.
-10
-16
-26
-27
-22
-17
66834
99535
.8207
.8848
.8867
.6845
.8950
.1250
-1511524
-12
-10
-7.
-7.
-1.
.6539
.0149
56603
71698
10422
.159093-01
-8.87056
-31.4491
-56.9305
-86.1615
-114.767
-105.815
-117.353
-104.200
-92.2431
-77.3605
-60.7450
-47.0623
-42.0485
-6.71201
.115948
-9.05049
-32.
-57.
-85.
-106
-122
-114
-104
-93.
-78.
-61.
-47.
-42.
0244
8404
5518
.215
.489
.864
.412
4833
5292
6645
9377
3296
-6.81793
.121541
-9.
-33
-57
-75
-82
-63
-57
-52
-47
-40
-32
-25
-20
-3.
72060
.5617
.0730
.3980
.8756
.0097
.1372
.4734
.7045
.8782
.7559
.5635
.8387
74431
.647594-01
-16.0007
-43.3976
-58.9506
-65.2530
-64.8656
-65.0647
-62.0496
-59.6368
-57.2477
-53.3715
-47.7426
-40.4953
-30.9582
-8.90867
-.394883-01
FLOWS IN THE Y DIRECTION
101.654
100.988
80.2117
109.236
8.97147
3.87355
102.223
98.5690
87.2790
86.4973
32.7361
-12.4882
100.305
96.0592
85.4840
80.4588
30.5889
-14.9164
87.9404
81.4697
70.3572
58.7120
18.1331
-13.2480
66.8651
57.0099
43.8252
29.4592
1.79816
-17.5263
45.0807
32.3655
19.3443
7.18937
-3.43804
-7.47321
18.5753
7.68152
2.90531
.918946
.399501-01
-.338393
Figure C-23. (continued)
-------
The Heat Pump Problem
This problem was coded to study the impact of the injection of cooled
waters from a heat pump into a shallow ground-water aquifer. The problem is
discussed in more detail by Andrews (1978).
The finite element grid used to discretize the problem is shown in Figure
C-24. Since the problem is symmetric, only half the system was modeled. A
novel feature of the finite element grid is that the thickness of each element
is proportional to distance along the horizontal axis from the wells—a
quasi-radial formulation. The program input data and a sample program output
are listed in Figure C-25 and Figure C-26.
A few unusual features are incorporated into this problem. The heat pump
problem was programmed to recompute flows at odd intervals, corresponding to
the end of each of the seasons. The intervals at which flows were recomputed
are listed on cards 3 and 4 of data group IX. ENTRY CHAN was called each time
flows were recomputed to read in the pumping schedule for the current
interval. Air temperatures for the entire simulation period were read in by a
call to ENTRY LAKE at the beginning of program execution.
The major subroutines and the tasks performed by each are listed in a
program flow chart (Figure C-27).
o
SCREEN FOR
SHALLOW WELL
100
1-
LJ
LJ
U.
200-
SCREEN FOR
DEEP WELL
300-
>
^^
=
i i
— •
=
«
z
1 1
oc.
•61.0 H
LJ
-91.5
60 120 240
(18.3) (73.2)
FEET
(METERS)
800 2000
(609.8)
Figure C-24. Grid used to discretize the aquifer simulated in the heat
pump problem.
-138-
-------
HOME HEATING AND COOLING—SYMMETRICAL SET-UP
UNITS FEET GALLONS DAYS CALORIES
168 198 11
30001 1E17
(lx,9(G12.6) )
(lx,8(G12.6) )
GROUP II
22 9
0 2 10 25 40 48 50 75 100 102 110 135 165 180 195 200 225 250 255 270 285 350
2000 800 250 160 120 80 60 30 0 -30 -300
GROUP III
15 1 0 3785.68
10 0 1
0-11
10 10 10 10 10 10 10 10 10
00000000000000000000000000000000000000
0000000000000000000000000000000000000000
0000000000000000000000000000000000000000
0000000000000000000000000000000000000000
0000000000000000000000000000000
999
0-11
111111111 10 10000000 10 00000000
10 00000000 10 00000000 10 00000000
10 00000000 10 00000000 10 00000000
10 00000000 10 00000000 10 00000000
10 00000000 10 00000000 10 00000000
10 00000000 10 00000000 10 00000000
10 00000000 10 00000000 10 00000000
10 0 0 0 0 0 0 0 0 999
GROUP IV
4
111.7.7111
0-11
11111111222222222222222233333333
4444444444444444444444444444444444444444
4444444444444444444444444444444444444444
4444444444444444444444444444444444444444
4444444444444444 999
1
.1 .1 0 8000 8000 17000 10 1
2
10 100 0 17000 17000 21000 10 1
3
25 250 0 17000 17000 21000 10 1
4
10 100 0 17000 17000 21000 10 1
GROUP V
0 0
6 6
125 12 0 17000 17000 21000 .33 .08
7
110 8000 8000 26000 .33 .08
8 (continued)
Figure C-25. The data deck used to model the heat pump simulation
with no regional groundwater flow.
-139-
-------
1000 100 0 17000 21000 .33 .08
9220 8000 8000 26000 .33 .08
10
175 25 0 17000 17000 21000 .33 .08
GROUP V
0 0
0 0
0 0
GROUP VI
0
GROUP VII
21
1 1 12000 8 1 12000 15 1 12000 22 1 12000
43 1 0 50 1 0 57 1 0 64 1 0 71 1 0 78 1 0 85 1 0 92 1 0 99 1 0
50 1 8000 57 1 8000 64 1 8000 71 1 8000 78 1 8000 85 1 8000 92 1 8000 99 1 8000
000000000000000000000
6
1 1 8 1 15 1 22 1 29 1 36 1
0000000
GROUP VIII
15 0
1 1 8 1 15 1 22 1 29 1 36 1 43 1 50 1 57 1 64 1 71 1 78 1 85 1 92 1 99 1
GROUP IX
100008
53 69 93 101 52 68 92 100
1 140 140 1 140 140 1 140 140 1 3
5
10 860
1.0000 29.5000 41.0000
2.0000 28.5000 35.9000
3.0000 27.5000 30.8000
11111110
Figure C-25. The data deck used to model the heat pump simulation
with no regional groundwater flow.
-140-
-------
HOME HEATING AND COOLING—SYMMETRICAL SET-UP
UNITS FEET GALLONS DAYS CALORIES
04/07/78
17:02:33
// OF NODES 198 # OF ELEMENTS 168 BANDWIDTH 11
VELOCITIES ARE BEING CALCULATED AT CURRENT TIME STEP
THE X-SPACING IS
0,0 2.0000 10.0000
100.0000 102.0000 110.0000
225.0000 250.0000 255.0000
THE Y-SPACING IS
2000.000 800.0000 250.0000
0.0
EQUATION 2 TIME STEP 15.0000
25.0000
135.0000
270.0000
160.0000
40.0000
165.0000
285.0000
120.0000
48.0000
180.0000
350.0000
80.0000
50.0000
195.0000
60.0000
75.0000
200.0000
30.0000
HEAT OR MASS CAPACITY COEFFICIENT 3785.68
PARAMETER FACTORS—IN ORDER 1.00000 1.00000 1.00000 .700000 .700000 1.00000 1.00000 1.00000
MATERIAL HOR PERM VER PERM STORAGE HOR THERM COND VERT THERM COND SPECIFIC DISPERSION COEFS
HEAT
1 . 10004-00
2 10.00
3 25.00
4 10.00
.1000+00 .0000
100.0 .0000
250.00 .0000
100.0 .0000
5600. 5600.
.1190+05 .1190+05
. 1190+05 . 1190+05
.1190+05 .1190+05
.1700+05 10.00 1.000
.2100+05 10.00 1.000
.2100+05 10.00 1.000
.2100+05 10.00 1.000
THE VALUES IN THE TYPE OF MATERIAL MATRIX ARE
1.00000
2.00000
2.00000
3.00000
4.00000
4.00000
4.00000
4.00000
4.00000
1.00000
2.00000
2.00000
3.00000
4. 00000
4.00000
4.00000
4.00000
4.00000
1.00000
2.00000
2.00000
3.00000
4. 00000
4.00000
4.00000
4.00000
4.00000
1.00000
2.00000
2.00000
3.00000
4.00000
4.00000
4.00000
4.00000
4.00000
1.00000
2.00000
2.00000
3.00000
4.00000
4.00000
4.00000
4.00000
4.00000
1.00000
2.00000
2.00000
3.00000
4.00000
4.00000
4.00000
4.00000
4.00000
1.00000
2.00000
2.00000
3.00000
4.00000
4.00000
4.00000
4.00000
4.00000
1.00000
2.00000
2.00000
3.00000
4.00000
4.00000
4.00000
4.00000
4.00000
Figure C-26. Program output for the heat pump problem.
(continued)
-------
4.00000
A. 00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4,00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
4.00000
POINT SOURCES OF WATER—NODE // AND AMOUNT
63 1.00000 72 1.00000
162 1.00000
81
1.00000
144
1.00000
153 1.00000
POINT SOURCES OF HEAT
63 1.00000 72 1.00000 81
162 1.00000
CONVECTIVE OUT AND CONDUCTIVE BOUNDARY INFORMATION
0 0 .00000
1.00000 144 1.00000 153 1.00000
ELEM //--BOUNDARY CODE—TRANSFER COEFFICIENT
HEAT FLOW BOUNDARIES—NODE // AND BOUNDARY CODE
13 23 33 43 53
6 3
7 3
8 3
9 3
TEMPERATURE DISTRIBUTION AT TIME STEP
10.0000
10. 0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10,0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
.250000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
Figure C-26. (continued)
-------
10.0000
10.0000
10.0000
10.0000
10.0000
POTENTIAL
10.00000
10.0013
10.0014
10.0015
10.0015
10.0015
10.0015
10.0016
10.0017
10.0017
10.0017
10.0018
10.0019
10.0019
10.0020
10.0020
10.0020
10.0019
10.0019
10.0019
10.0019
10.0018
10.0000
10.0000
10.0000
10.0000
10.0000
DISTRIBUTION
10.00000
10.0011
10.0012
10.0013
10.0014
10.0015
10.0015
10.0018
10.0022
10.0022
10.0023
10.0026
10.0029
10.0031
10.0033
10.0034
10.0037
10.0041
10.0042
10.0044
10.0046
10.0051
10.0000
10.0000
10.0000
10.0000
10.0000
AT TIME
10.00000
9.98874
9.98832
9.98773
9.98772
9.98799
9.98807
9.98989
9.99331
9.99365
9.99511
10.0005
10.0075
10.0109
10.0139
10.0148
10.0183
10.0201
10.0203
10.0206
10.0206
10.0198
10.0000
10. 0000
10.0000
10.0000
10.0000
STEP
10.00000
9.97939
9.97842
9.97589
9.97457
9.97279
9.97238
9.97052
9.97712
9.97797
9.98172
9.99548
10.0129
10.0215
10.0295
10.0319
10.0393
10.0383
10.0373
10.0336
10.0298
10.0222
10.0000
10.0000
10.0000
10.0000
10. 0000
000000
10.00000
9.97459
9.97322
9.96883
9.96588
9.96126
9.96016
9.95306
9.96360
9.96504
9.97125
9.99222
10.0164
10.0291
10.0419
10.0458
10.0557
10.0527
10.0502
10.0421
10.0346
10.0227
10.0000
10.0000
10.0000
10.0000
10.0000
10.00000
9.96998
9.96813
9.96082
9.95443
9.94321
9.94052
9.92129
9.94163
9.94436
9.95580
9.98861
10.0203
10.0391
10.0611
10.0683
10.0903
10.0755
10.0698
10.0527
10.0396
10.0229
10.0000
10.0000
10.0000
10.0000
10.0000
10. 00000
9.96806
9.96599
9.95687
9.94715
9.92931
9.92456
9.89284
9.92412
9.92863
9.94514
9.98716
10.0221
10.0449
10.0753
10.0860
10.1192
10.0933
10.0842
10.0588
10.0417
10.0228
10.0000
10.0000
10.0000
10.0000
10.0000
10.00000
9.96583
9.96346
9.95193
9.93387
9.89177
9.88408
9.81196
9.88125
9.88812
9.92506
9.98618
10.0242
10.0528
10.1086
10.1293
10.2003
10.1367
10.1176
10.0669
10.0442
10.0227
10.0000
10.0000
10.0000
10.0000
10.0000
10.00000
9.96433
9.96173
9.94858
9.91940
9.78885
9.72716
9.52379
9.72221
9.78247
9.90286
9.98653
10.0256
10.0565
10.1655
10.2876
10.4890
10.2951
10.1746
10.0708
10.0460
10.0225
B. FLUX RECHARGE
B. FLUX DISCHARGE
CHANGE IN STORAGE
QUANTITY PUMPED
DIFFERENCE
CUMULATIVE MASS BLANCE
.000000
.000000
.000000
.000000
.000000
RATES FOR THIS TIME STEP
.000000
.000000
.000000
.000000
.000000
Figure C-26. Program output for the heat pump problem.
-------
I. BRIEF DESCRIPTION OF THE PROGRAM ROUTINES
A. Main Routine
This routine controls the flow of the program and the program
dimensions. The parameter variables L, N, M control the maximum
number of elements, nodes, and bandwidth respectively.
1. Internal Subroutine ITERAT
This subroutine controls the solution of a steady-state areal
problem by an iterative method.
2. Internal Subroutine FLOADD
This subroutine adds in a constant specified flow rate to the
values calculated by the water-flow equation.
B. Subroutine DATAIN
This subroutine reads in the program data.
1. ENTRY DI
This entry point passes arguments from the main routine to
subroutine DATAIN.
2. Internal Subroutine ELREAD
This subroutine reads in the element node numbers.
3. Internal Subroutine P
This subroutine creates a rectangular grid and numbers the nodes
and elements in the grid.
C. Subroutine STRUCT
This subroutine assembles the global structure matrix and the global
capacitance matrix.
D. Subroutine GAUSS
This subroutine forms the element structure and capacitance matrices
by Gaussian Quadrature.
1. ENTRY DERIVE
The main entry point for subroutine GAUSS.
2. Internal Subroutine CONVEC
This routine handles the convective boundaries.
E. Subroutine SHAPE1
This subroutine computes the shape functions for a linear element.
F. Subroutine SHAPE
This subroutine computes the shape functions for elements with one or
more nonlinear sides.
-144-
-------
G. Subroutine LOAD
This subroutine computes the global recharge matrix by adding in the
various sources and sinks.
1. ENTRY HEATE
The main entry point for subroutine LOAD.
2. Internal Subroutine CB
This subroutine computes mass or heat flow across a specified
potential boundary.
H. Subroutine SOLVE
This subroutine is a symmetric banded matrix equation solver.
1. ENTRY BACK
Entry point for the back substitution of the recharge matrix.
2. ENTRY MULTI
Entry point for the multiplication in a transient problem.
I. Subroutine ASOLVE
This subroutine is an assymmetric banded matrix equation solver.
1. ENTRY ABACK
Entry point for the back substitution.
2. ENTRY AMULTI
Entry point for matrix multiplication in a transient problem.
J. Subroutine BALAN
This subroutine computes a mass balance.
1. ENTRY MASBAL
The main entry point for subroutine BALAN.
2. ENTRY BPRINT
The entry point for the printing of the mass balance.
3. ENTRY WATER
Entry point for computing water flows in each element.
4. ENTRY VELO
Computes water velocities at the Gauss points.
5. ENTRY VCENT
Computes water velocities at the center of each element.
6. Internal Subroutine FLOWW
Subroutine for computing element flows.
K. Subroutine FLOWS
This subroutine prints out element flows and nodal values.
1. ENTRY FFLOW
Entry point for the printing of nodal potentials on file 14.
-145-
-------
2. ENTRY FFFLOW
Entry point for the printing of nodal temperatures or
concentrations on file 14.
3. ENTRY WFLOW
Entry point for printing water flows for each element.
4. ENTRY HPRINT
Entry point for printing nodal temperature or concentration
values.
5. ENTRY WPRINT
Entry point for printing nodal potential values.
L. Subroutine PARAM
This subroutine computes element parameters that are a function of
velocity or temperature and localized coordinates.
1. ENTflY MECD
Entry point for computing dispersivity.
2. ENTRY CORD
Entry point for the computation of localized coordinates.
3. ENTRY PE
Entry point for computing hydraulic conductivity as a function of
temperature.
4. ENTRY PEE
Entry point for the computation of aquifer thickness in an areal
problem.
M. Subroutine BOUNDA
This routine is used to change parameter values or boundary conditions
at each time step. Five entry points, ENTRY LAKE, ENTRY BVAL, ENTRY
BOUND, ENTRY CHANG, ENTRY CHAN, are provided for user programming.
N. Subroutine EIGEN
This subroutine is used to compute the maximum stable time step for a
transient problem.
0. Subroutine ADJUST
This subroutine moves the upper boundary in a cross-section problem.
1. ENTRY ADJUST
The main entry point for subroutine ADJUST.
-146-
-------
MAIN PROGRAM
Q)—
DATAIN
Reads the data in and
prints the data
If transient
problem go to
3
PE
Adjusts hydraulic
conductivity for
temperature
STRUCT
Assembles the
structure matrix
LOAD
Assembles the load
vector—the right hand
side of tha equation
ITERATE
Called in areal
steady state problems
MASBAL
Computes
balance
a mass
ADJUST
Called in a moving
boundary problem—
adjusts the boundary
and directs program
flow to 2 or 3 depending
upon how much the boundary
is moved
WPRINT, WTLOW, BPRINT
Prints potentials' flows
and mass balance
GAUSS
Performs integration
for an element using
Gaussian Quadrature
SOLVE
Performs forward
reduction of the
structure matrix
BOUNDA
Reads in new
boundary
conditions for
the current
time step
MULTI
In a transient
problem multiplies
load vector by
potential values
at last time step
BACK
Solves for the
unknown potentials
Figure C-27. Program flow chart.
-147-
-------
If not a linked
problea STOP
STRUCTURE
Assemble the structure
natrix for equation number
two
LOAD
Assemble the load vector
for equation number two
IttSBAL
Computes a m»»» balance
for equation number two
HRPICT. BPRI8T
Prints out «ass or
energy concentrations at
nodes and the BBSS
balance
GAUSS
Performs integration
for an element using
Gaussian Quadrature
ASOLVE
Performs forward
reduction of the
structure maxtrix
(matrix is not
symmetrical)
BOUNDA
Reads in new
boundary
conditions for
the current tine
step
AMULTI
In a transient
problem multiplies
load vector by heat,
nass values at last
tine step
ABACK
Solves for the
unknown potentials
PEE
Computes aquifer
thickness
CORD
Computes
localized
coordinates
CONTROL TRAHSFER
Three possible types of
transfers—one chosen
depends upon type of
problea
SHAPE
Computes the
shape
functions
VELCO
Computes
velocity
at the Gauss
points
MECD
Computes the
dispersion
coefficients
at the Gauss
points
CONVEC
Computes
coefficients for
convective and
flow
boundaries
Figure C-27 (continued)
-148-
-------
THE FINITE ELEMENT PROGRAM LISTING
C
C
C
C
C
C1
C
C
C
C
C
C
C
THE MAIN ROUTINE FOR THE FINITE ELEMENT PROGRAM
BY CHARLES ANDREWS
MAI
MAI
MAI
MAI
MAI
MAI
MAI
MAI
MAI
THE PARAMETER STATEMENTS
L—NUMBER OF ELEMENTS, N —NUMBER OF NODES.
M— BANDWIDTH, Z—SIZE OF ARRAYS CONTAINING SOURCE
SINK, AND BOUNDARY INFORMATION
PARAMETER H=17,N=160,L=130,Z=50
**t*****************************************************
PARAMETER P=M»2-1,BBB=N
INTEGER AY(4),AZ(4)
DATA AY/4,1,4,3/
DATA AZ/3,2,1,2/
DOUBLE PRECISION S,T
REAL MAT INFLOW
COMMON/ACC/KK(12)-K1 ,K2,K3,K4
COMMON/AM/NLA,PCT/CON/MC1,MC2,HCONV
«/CONT/LA,LB,LC,LD.LE,LF,LG LH
COMMON/HI/TITLE(25),V(26),VV(26)
COMMON/HM/PXX PYY ,PXY,KAD/HH/LFLOW,LON/ME/MEQ
COMMON/ATHICK/ASIZE.NTHICK.THICK ERROR
MODEL PARAMETERS--'.
DIMENSION WXARIES
DIMENSION NWATER(Z) .AWATER(Z) ,NHEAT(Z) ,AHEAT(Z)
C— ELEVEATION AT THE BOTTOM OF THE AQUIFER
DIMENSION BOT(BBB)
C— INFORMATION ON CONVECTIVE BOUNDARIES CBALCZ 2)
• ,NCON(Z,3) .TINF(Z),CONV(Z),ALOC(Z),NEL(Z,2).AEL(Z,2),CBALU,^
C— STRUCTURE MATRICIES
DIMENSION GG(N), S( N,P) ,T( N ,P)
100
200
300
100
500
600
700
800
900
MAI 1000
MAI 1100
MAI 1200
MAI 1300
MAI 1400
MAI 1500
MAI 1600
MAI 1700
MAI 1800
MAI 1900
MAI 2000
MAI 2100
MAI 2200
MAI 2300
MAI 2400
MAI 2500
MAI 2600
MAI 2700
MAI 2800
MAI 2900
MAI 3000
MAI 3100
MAI 3200
MAI 3300
MAI 3400
MAI 3500
MAI 3600
MAI 3700
MAI 3800
MAI 3900
MAI 4000
MAI 4100
MAI 4200
MAI 4300
MAI 4400
MAI 4500
MAI 4600
MAI 4700
MAI 4800
MAI 4900
MAI 5000
MAI 5100
MAI 5200
MAI 5300
MAI 5400
MAI 5500
MAI 5600
MAI 5700
-149-
-------
CALL URDATE(IDATE.IYEAR)
CALL URTIMD(ITIME,ISEC)
FORMAKIX,'MAXIMUM NUMBER OF NODES PERMITTED EXCEEDED')
3 FORMATC1X,'MAXIMUM NUMBER OF ELEMENTS EXCEEDED')
4 FORMAK IX, 1H1)
READ 7,(TITLE(J),J=1,24)
PRINT 300 (TITLE(J),J=1,24),IDATE,IYEAR,ITIME,ISEC
7 FORMAT(12A6)
READ,LM,LN,MBAND
PRINT 8,LN,LM,MBAND
8 FORMAT(1X,'*OF NODES',14 ' * OF ELEMENTS', 1*4 ' BANDWIDTH1
IF(LM.GT.L) PRINT 3
IF(LN.GT.N) PRINT 1
MPP=MBAND»2-1
NAA=0
C*»****«**»*«*«*»***«*«***»«*****«****««*»*»«*«*««»*«*»**»*
c
c
INPUT THE DATA
CALL DATAINUM.LN MBAND,MAT,R1 ,R .HEA,HEAD,WX,WY,STO,INFLOW,PX,PY,
•PCX,HEAT,XLOC.YLOC,NOD,TPX TPY,KRANA,CFACT,CFACT1,
•ALPHA.KSTEP.KBOUND,KFT,PCW,KRAN.ALPH,KSTE,KBOUN,KF,PCH,DIFF,
•Z,NCON,TINF,CONV,ALOC,LWATER HWATER,AWATER,LHEAT,NHEAT,AHEAT.
•LFLUXW,NFLUXW,AFLUXW,LFLUXH,NFLUXH,AFLUXH,BBB,BOT,KTYPE
«,NODE,NEL,AEL)
CALL DI(KAREAL,ALPHM,ALPHAM,ER,ITER.LEL
•-LINEW,LINEH,NLINEW,NLINEH,ALINEW,ALINEH)
C
CC««**fti**«***»**«***«t****ft»»«»»«****«ttft*»t»*«ft***«*ftff**ft
C INITIALIZE THE STARTING ADDRESSES AND STARTING VALUES
C
CALL FLOWS(LM,LN,R.R1,FLOWX,FLOWY)
C
CALL BOUNDA(LM,LH.Z,R,R1.HEAD,HEA,FLOWX,FLOWY,HEAT,INFLOW
• NWATER AWATER,NHEAT.AHEAT.NCON,TINF
• CONV,ALOC,NEL,AEL,CBAL,LEL,LHEAT,LWATER,PCW,AY,AZ
•,LINEW,LINEH,NLINEW,NLINEH,ALINEW,ALINEH)
CALL PARAMUM.LN.BBB.XLOC,YLOC.NOD.PCX.DIFF,R,R1 ,WX,WY,TPX
«,TPY,BOT,KAREAL>
CALL VELOC(LN,LM,R1,WX,WY)
C
MU=0
C PROGRAM CONTROL IS ESTABLISHED
READ,LA,LD,LE,LF,LG,LH
IF(LE.GT.O) READ,(NTIME(I),1=1.LE)
IF(LF.EQ.I) READ,ASIZE,NTHICK,MTHICK ERROR,XADD,YADD
IFCLH GT.O) READ,(HSPACE(I),1=1,LH)
IF(LA.EQ.I) CALL LAKE
IF(LG.GT.O) CALL ADJUSTUG LM.LN Z.HEA.XLOC,YLOC.NOD
• ,FLOWX,FLOWY.R1)
GO T0(10.20,30,30,30), KTYPE
10 CONTINUE
MAr-1
MB=1
MC=1
MD=1
ME=1
READ.MW
GO TO 120
20 READ,MA,MB,MW,MC,MD
IF(MD.LE.O) MU=1
ME=1
GO TO 120
MAI 5800
MAI 5900
MAI 6000
MAI 6100
MAI 6200
MAI 6300
MAI 6400
MAI 6500
MAI 6600
MAI 6700
MAI 6800
MAI 6900
MAI 7000
MAI 7100
MAI 7200
MAI 7300
MAI 7400
MAI 7500
MAI 7600
MAI 7700
MAI 7800
MAI 7900
MAI 8000
MAI 8100
MAI 8200
MAI 8300
MAI 8400
MAI 8500
MAI 8600
MAI 8700
MAI 8800
MAI 8900
MAI 9000
MAI 9100
MAI 9200
MAI 9300
MAI 9*400
MAI 9500
MAI 9600
MAI 9700
MAI 9800
MAI 9900
MAI10000
MAI10100
MAI10200
MAI10300
MAI10400
HAI10500
MAI10600
MAI10700
MAI10800
MAI10900
MAI11000
MAI11100
MAI11200
MAI11300
MAI11400
MAI11500
MA111600
MAI11700
MAI11800
MAI11900
MAI12000
-150-
-------
30 READ,MO,MP.MV,MQ,MR.MS,MT MU.ME
MA = -1
MB=1
MC=1
MD=1
ME = 0
MW=9999999999
ALPH=0.0
120 CONTINUE
C NA IS A COUNTER
NA=0
C THESE LINES ESTABLISH NO CONVECTIVE BOUNDARIES IN EQN 1
NCC=NCONV
NCONV=0
C NO DISPERSION IN EQN 1
KAD = 0
C CONTROL FOR SYM OR ASSYM SOLUTION TECHNIQUE
NLB=NLA
NLA=LC
IFUTYPE.EQ.5) GO TO 160
CONTINUE
CONTINUE
124
125
C
C
C
C
EQUATION NUMBER ONE
C ADJUST PERMEABILITIES FOR CHANGING TEMPERATURES
IF(MS.EQ.I) CALL PE(MS)
C MULTIPLICATION FACTOR FOR TIME STEP
IF(KS.GT.O.OR.NA.GT.O) ALPH=ALPH*ALPHM
C INITALIZE STARTING ADDRESSES IN HEATER GAS AND BALANCE FOR EQUATION
MEQ=1
CALL LOAD(LM,LN,MBAND,NOD,NODE,R1,RI,G
».GG,INFLOW,HEA.ALPH,KBOUN,KF,PCH,KRANA,CFACT1 KS,
*Z NCON,TINF,CONV,ALOC,LWATER,NWATER,AWATER,CBAL,AY,AZ
» ,AEL,NEL,MEQ,LEL,CBALA,LINEW,NLINEW,ALINEW)
CALL BALAN(LM,LN,R1,WX,WY,STO HEA,INFLOW,Z,LWATER.AWATER
* LFLUXW,NFLUXW,WA,WB,WC,WD,WE,MPRINT:NTYPE,RI,NODE,NOD
*FLOWX FLOWY.PCW,MEQ,CBAL,CBALA,AY,AZ,NCON,KRANA)
C
C
C
130
C
C
CALL GAUSSULL-LM,LN,XLOC,YLOC,NOD,WY.WX,INFLOW,STO,ALPH,KBOUN,
* KRANA,Z,NCON,CONV,ALOC,NODE,CBAL,AY,AZ)
CALL STRUCT(LN,LM,MBAND,MPP,R1.S T,G .GG ,RI ,HEA,XLOC ,
*YLOC,NOD,NODE,CFACT1,KRANA,ALPH,KBOUN)
NAA=NAA+1
NA=NA+1
BKS=BKS-«-ALPH
IFULPH.LT.0.0001) BKS=AKS
IFCKAREAL.EQ.1) CALL ITERATCER,ITER,KSA,$12M)
KSA~0
IF(MC.GT.O) CALL MASBAL(ALPH)
M1=NA/MD
MM' — Ml^M 0
IF(MMI.EQ.NA.AND.MU.EQ.O) CALL WATER
IF(LG.GT.O) CALL ADJUS($125,$130)
MAI12100
MAI12200
MAI12300
MAI12MOO
MAI12500
MAI12600
MAI12700
MAI12800
MAI12900
MAI13000
MAI13100
MAI13200
MAI13300
MAI13400
MAI13500
MAI13600
MAI13700
MAI13800
MAI13900
MAI14000
MAI14100
MAI14200
MAI11300
MAI14400
MAI14500
MAI14600
MAI14700
MAI14800
MAI14900
MAI15000
MAI15100
MAI15200
MAI15300
MAI15400
1 MA115500
MAI15600
MAI15700
MAI15800
MAI15900
MAI16000
MAI16100
MAI16200
MAI16300
MAI16400
MAI16500
MAI16600
MAI16700
MAI16800
MAI16900
MAI17000
MAI17100
MAI17200
MAI17300
MAI17400
MAI17500
MAI17600
MAI17700
MAI17800
MAI17900
MAI18000
MAI18100
MAI18200
MAI18300
MAI18400
-151-
-------
c
c
c
c
c
MUNA/MB
M1=M1«MB
IF(MI.EQ.NA) CALL WPRINT(BKS)
M1=HA/MC
M1=M1»MC
IF(Ml.EQ.NA) CALL BPRINT
ADDS IK A CONSTANT BACKGROUND FLOW RATE
IF(XADD.EQ.O.AND.YADD.EQ.O) GO TO 145
C
145
C
C
CALL FLOADD
CONTINUE
IF(MMI.EQ.NA.AND.MU.EQ.O) CALL WFLOW(BKS)
M1=NA/MW
H1=M1»MW
IF(MI.EQ.NA) CALL FFLOW
IF(MA.LE.BKS) GO TO 155
IF(ALPHM.GT.I) GO TO 125
GO TO 130
155 CONTINUE
IF(ME.EQ.I) STOP
160 CONTINUE
£*••«»••••*»•*•*••••«•**•*••«••«•••*••*••**••*••••••ft*«««t»*
C
C EQUATION NUMBER TWO
C
£••***•••••••••••••••••t«»ll*«*«««**«»««»«»*«*t««•••*••*•**«
NTHICK=MTHICK
NCONVrNCC
KAD=MT
NUUNLB
(;•*«•*•••••••*«•«*•••*•••*•**«*t•*«*••«•»•»«•••••»••««••••••!•••
C MULTIPLICATION FACTOR FOR THE TIME STEP
IF(KS.GT.O) ALPHA=ALPHA«ALPHAM
C
C
C
INITIALIZE STARTING ADDRESSES
MEQ=2
IN HEATER AND BALANCE FOR EQUATION 2
CALL LOADUM.LN.HBAND.NOD,NODE,R,RI,G
«GG,HEAT ,HEAD ,ALPHA,KBOUND,KFT,PCW,KRAN.CFACT,KS,
»Z,NCON.TINF,CONV,ALOC,LHEAT,NHEAT AHEAT.CBAL,AY,AZ
* ,AEL,NEL,HEQ,LEL,CBALA,LINEH,NLINEH,ALINEH)
CALL BALANCLM,LN,R,PX,PY,PCX,HEAD,HEAT,Z,LHEAT,AHEAT.LFLUXH,
MAI18500
MAI18600
MAI18700
MAI18800
MAI18900
MAI19000
MAI19100
MAI19200
MAI19300
MAI19400
MAI19500
MAI19600
MAI19700
MAI19800
MAI19900
MAI20000
MAI20100
MAI20200
MAI20300
MAI20400
MAI20500
MAI20600
MAI20700
MAI20800
MAI20900
MAI21000
MAI21100
MAI21200
MAI21300
MAI21400
MAI21500
MAI21600
MAI21700
MAI21800
MAI21900
MAI22000
MAI22100
MAI22200
MAI22300
MAI22400
MAI22500
MAI22600
MAI22700
MAI22800
MAI22900
MAI23000
MAI23100
MAI23200
MAI23300
MAI23400
C
C
c
c
c
• NFLUXHtHA,HB,HCfHD,HE,MPRINT.NTYPE,RI,NODEfNODsFLOWX,FLOWY,PCWf MAI23500
* MEQ,CBAL,CBALA,AY.AZ,NCON,KRAN) MAI23600
MAI23700
MAI23800
MAI23900
CALL GAUSS(LLL,LM,LN.XLOC,YLOC,NOD,PY.PX.HEAT.PCX,ALPHA,KBOUND,KRMAI24000
•AN,Z,NCON,CONVfALOC,NODE,CBAL,AY,AZ) MAI24100
MAI24200
CALL STRUCT(LN,LM,MBAND,MPP R,S T-G,GG,RI,HEAD,XLOC,YLOC, MAI24300
« NOD.NODE,CFACT,KRAN,ALPHA.KBOUND) MAI24400
MAI24500
MAI24600
-152-
-------
CONTINUE
KS=KS+1
AKS=AKS+ ALPHA
CALL HEATE(KS)
IF(LH.EQ.O) GO TO 209
DO 208 1=1.LH
J=NSPACE(I)
YSPACEU) = R(J)
WRITEC15.V) (YSPACE(I),I=1,LH)
CONTINUE
IF(MQ.GT.O) CALL MASBAL(ALPHA)
IFUSIZE.NE.-1 .0) GO TO 212
DO 211 1=2,LN 2
200
C
C
C
C
208
209
011 RfT — l^ — RTT^
212 CONTINUE
M1=KS/MP
M1=M1«MP
IF(M1.EQ.KS) CALL HPRINT(AKS)
M1=KS/MV
M1=M1«MV
IF(M1.EQ.KS) CALL FFFLOW
M1=KS/MQ
M1=M1«MQ
IF(M1.EQ.KS) CALL BPRINT
IF(MO.LE.AKS) STOP
C NTIME CONTAINS INFORMATION ON WHEN FLOWS ARE TO BE RECOMPUTED
C NAA IS A COUNTER INDICATING THE NUMBER OF TIMES FLOWS HAVE BEEN
IF(LE.EQ.1.AND.NTIME(NAA).EQ.KS) GO TO 120
M1=KS/MR
M1=MR*M1
IF(MLNE.KS) GO TO 200
GO TO 120
C
300
FORMAT(1H1/1X,130( t*1)/1X,3('
* 2('*' ,35X,12A6,T130,'*'/1X), '
,T130, '*
,T130,'*'/1X,
,T100,A6,A2,T130.'*'/1X) ,130('*M)
C»«»»«(«»»«*««*»*««««»«*»«»«»««*»«********»*******************
C
SUBROUTINE ITERATCER.ITER.KS.*)
IF(KRANA.NE.O) RETURN
A=0
DO 1J=1,LN
B=RI(J)-RHJ)
B=ABS(B)
1 A=AMAX1(B,A)
KS=KS+1
IF(A.LT.ER.OR.KS.GT.ITER) PRINT 5.KS
IF(A.LT.ER.OR.KS.GT.ITER) RETURN
RFTII RN ll
5 FORMAT(1X//1X,'ITERATIONS NEEDED FOR CONVERGENCE ,15)
C
MAI24700
MAI2U800
MAI24900
MAI25000
MAI25100
MAI25200
MAI25300
MAI25UOO
MAI25500
MAI25600
MAI25700
MAI25800
MAI25900
MAI26000
MAI26100
MAI26200
MAI26300
MAI26UOO
MAI26500
MAI26600
MAI26700
MAI26800
MAI26900
MAI27000
MAI27100
MAI27200
MAI27300
MAI27400
MAI27500
MAI27600
MAI27700
COMPUMAI27800
MAI27900
MAI28000
MAI28100
MAI28200
MAI28300
MAI28MOO
MAI28500
MAI28600
MAI28700
MAI28800
MAI28900
MAI29000
MAI29100
MAI29200
MAI29300
MAI29MOO
MAI29500
MAI29600
MAI29700
MAI29800
MAI29900
MAI30000
MAI30100
MAI30200
MAI30300
-153-
-------
c««»*«»«*«««*•»««•»«««*»•««««««»*«««**»«»»»««»»«**««««««*»»* MAI 30500
C MAI30600
SUBROUTINE FLOADD MAI30700
C MAI30800
DO TJO KSS=1.LM MAI30900
CALL CORD(KSS) MAI31000
Y=YLOC(Kfl)-YLOC(K1)+YLOC(K3)-YLOC(K2) MAI31100
Y=ABS(Y)/2 MAI31200
X=XLOC(K1)+XLOC(K1)-XLOC(K3)-XLOC(K2) MAI 31300
X=ABS(X)/2 MAI3140G
XX=1.0 MAI31500
YY=1.0 MAI31600
IF(NTHICK.EQ.O) GO TO 135 MAI31700
YY=(YLOC(K1)+YLOC(K4))/2*ASIZE MAI31800
XX=(XLOC(K3)+XLOC(KU))/2*ASIZE MAI31900
XX=ABS(XX) MAI32000
YY=ABS(YY) MAI32100
135 CONTINUE MAI32200
FLOWY(KSS)=FLOWY(KSS)+YADD*X/YY MAI32300
FLOWXCKS3)=FLOWX(KSS)-»-XADD«Y/XX MAI32400
140 CONTINUE MAI32500
END MAI32600
-154-
-------
C*««««**«««««***»***«**»»«*«»»»****««*t«*«***««*«««»**««*«»««»«
c
C THIS ROUTINE IS USED TO READ IN THE DATA
C
SUBROUTINE DATAINCLM,LN,MBAND,MAT,R1,RtHEA,HEAD,WX,WY,STO
«,INFLOW,PX.PY.PCX,HEAT,XLOC.YLOC,NOD,TPX,TPY,KRANA,CFACT.CFACT1,
•ALPHA,KSTEP,BOUND,KFT,PCW.KRAN,ALPH,KSTE,BOUN,KF,PCH,DIFF,
*Z,NCON,TINF,CONV,ALOC,LWATERfNWATER,AWATER,LHEAT,NHEAT,AHEAT,
,LFLUXW,NFLUXW,AFLUXW,LFLUXH,NFLUXH,AFLUXH,BBB,EOT.KTYPE
*,NODE,NEL,AEL)
INTEGER BOUND,BOUN,BBB,Z
REAL INFLOW,MAT
COMMOM/HI/TITLE(25),V(26),VV(26)/HH/LFLOW,LON
COMMON/A1/M,MM,NUMNP,NSIZE/AM/NLA,PCT
COMMON/CON/MC1,MC2,NCONV
DIMENSION NCON(Z,3),TINF(Z),CONV(Z),ALOC(Z),NWATER(Z),
* AWATER(Z),NHEAT(Z),AHEAT(Z),NFLUXW(Z,2),AFLUXW(Z),NFLUXH(Z, 2) ,
*AFLUXH(Z),BOT(BBB),AEL(Z,2),NEL(Z,2)
DIMENSION MAT(LM),R1(LN),R(LN),HEA(LN),HEAD(LN),WX(LM),WY(LM)
DIMENSION STO(LM).INFLOW(LM),PX(LM),PY(LM),PCX(LM),HEAT(LM)
DIMENSION XLOC(LN),YLOC(LN),NOD(12,LM), TPX(LM),TPY(LM)
DIMENSION FF(8),Q(12), A(8 ,50) , DIFFUM ,2) , NODEC LM)
RETURN
C
C
ENTRY DI(KAREAL,ALPHM,ALPHAM,ER,ITER,LEL,
«LINEW,LINEH,NLINEW,NLINEH,ALINEW,ALINEH)
DIMENSION NLINEW(Z,2),NLINEH(Z,2),ALINEW(Z),ALINEH(Z)
£f ••*«»»««*«««««««»««*«««**«*»«*»«*«*»•*«**«««*«»««**»»«»*»«
C
DO 2 J=1,LM
HEAD(J)=0.0
2 HEA(J)=0.0
C
C
£«*«**ft**ft«*«»ft**««***ftX««»***tt«*«X«»»»****«*»****«««*««*»*»
C
7 FORMATC12A6)
READ,KTYPE,KPRINT,LON,MQ,NLA,CFACT
CFACT1=CFACT
NUMNP=LN
NSIZE=NUMNP
M=LM
MM=M
READ UO,(V(I),1=1,26)
READ 40,(VV(I),1=1,26)
IF(NLA.GT.O) PRINT 155,NLA
C
READ 7,(QU),J=1,12)
CXX»XXXXXXXftXXXX*XX*XXXXX«XXXX»*XXX*«XXX***X«*«**X**«XXX*»«XXX
C
c
c
c
DATA GROUP II
READ IN DATA ON THE SPATIAL STRUCTURE
IF(MQ.LT.0.0001) GO TO 12
DO 8 1=1,NUMNP
READ,J,XLOC(J),YLOC(J)
8 CONTINUE
DO 9 11=1,LM
CALL ELREAD
9 CONTINUE
GO TO 15
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
DAT
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
moo
1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
2500
2600
2700
2800
2900
3000
3100
3200
3300
3400
3500
3600
3700
3800
3900
4000
4100
4200
4300
4400
4500
4600
4700
4800
4900
5000
5100
5200
5300
5400
5500
5600
5700
5800
5900
6000
6100
6200
-155-
-------
12 CONTINUE
READ.KSPACX.KSPACY
CALL P(KSPACX.KSPACY)
MQ=ABS(MQ)
DO 14 J=1,LM
14 NODE
-------
31
32
33
35
C
DO 32 JJ=1,NMATA
IFCKTYPE.LT.3) READ.J ,(A(I,J>,1=1,3)
IF(KTYPE.EQ.5) READ,J,(A(I,J),1=4,8)
IFCKTYPE.GT.2) READ,J , (A(I,J),1=1,8)
DO 31 1=1 ,8
A(I,J)=A(I,J)»FF(I)
PRINT 104,J,(A(I,J),I=1 ,8)
CONTINUE
DO 34 K=1,LM
DO 33 J=1 ,50
KZX=MAT(K)
IF(KZX.NE.J) GO TO 33
WX(K)=A(1,J)
WY(K)=A(2,J)
STO(K)=A(3,J)
PX(K)=A(4,J)
PY(K)=A(5,J)
PCX(K)=A(6,J)
DIFF(K,1)=A(7,J)
DIFF(K,2)=A(8,J)
GO TO 34
CONTINUE
CONTINUE
DO 35 J=1,LM
TPX(J)=WX(J)
TPY(J)=WY(J)
PRINT 120
PRINT VV,(MAT(J),J=1,LM)
READ 7,(Q(J),J=1,12)
£•»«««*»»«»*»»««*««*«««»»«»*»««»*»««««»»«««»»«««»»««»««*«««««•««*
C
C
C
INPUT DISTRIBUTED SOURCES
READ.KGEN.KGENH
IF(KGEN.EQ.I) CALL READER(LM.INFLOW,I)
IFU.NE.999) STOP INFL
IF(KGENH.EQ.I) CALL READERUM ,HEAT ,1)
IFCI.NE.999) STOP HEAT
C«»««**»«*•**»*«**•**»»*«**««*««»«*******«**ftftx«*ft*«««ft*«*ff*»««
C
C INPUT POINT SOURCES OF WATER AND HEAT
READ.LWATER.LHEAT
IF(LWATER.EQ.O) GO TO 61
READ,(NWATER(I),AWATER(I),1=1,LWATER)
PRINT 131,(NWATER(I),AWATER(I),1=1.LWATER)
61 CONTINUE
IF(LHEAT.EQ.O) GO TO 63
READ,(NHEAT(I),AHEAT(I),1=1,LHEAT)
PRINT 132,(NHEAT(I),AHEAT(I),1=1,LHEAT)
63 CONTINUE
C
C
C INPUT LINE SOURCES OF WATER AND HEAT
READ.LINEW.LINEH
IF(LINEW.EQ.O) GO TO 66
READ,(NLINEW(I,1),NLINEW(I,2),I=1,LINEW)
READ,(ALINEW(I) ,1=1 .LINEW)
PRINT 278,(NLINEW(I,1),NLINEW(I,2),ALINEW(I),I=1,LINEW)
66 IF(LINEH.EQ.O) GO TO 68
READ,(NLINEH(I,1),NLINEH(I,2),I=1,LINEH)
READ,(ALINEH(I),1=1.LINEH)
PRINT 279,(NLINEH(I,1),NLINEH(I,2),ALINEH(I),I=1,LINEH)
DAT12600
DAT12700
DAT12800
DAT12900
DAT13000
DAT13100
DAT13200
DAT13300
DAT13400
DAT13500
DAT13600
DAT13700
DAT13800
DAT13900
DAT14000
DAT14100
DAT14200
DAT14300
DAT14400
DAT14500
DAT14600
DAT14700
DAT14800
DAT14900
DAT15000
DAT15100
DAT15200
DAT15300
DAT15400
DAT15500
DAT15600
DAT15700
DAT15800
DAT15900
DAT16000
DAT16100
DAT16200
DAT16300
DAT16400
DAT16500
DAT16600
DAT16700
DAT16800
DAT16900
DAT17000
DAT17100
DAT17200
DAT 17300
DAT17400
DAT17500
DAT17600
DAT17700
DAT17800
DAT17900
DAT18000
DAT18100
DAT18200
DAT18300
DAT18400
DAT18500
DAT18600
DAT18700
DAT18800
DAT18900
-157-
-------
68 CONTINUE
C
C
READ 7,(QU),J = 1,12)
etc***!****************************************************
C
C DATA GROUP VI
C
C INPUT INFORMATION NEEDED FOR A 2-D AREAL VIEW OF THE SYSTEM
READ.KAREAL
IF(KAREAL.EQ.O) GO TO 41
READ.ER.ITER
CALL READER(LN,BOT,I)
IFU.NE.999) STOP BOT
CALL READER(LN,R1,I)
IF(I.NE.999) STOP R1 2
10 FORMAT(2(13A6,A2/))
PRINT 276
PRINT V,(BOT(I),I=1,LN)
C
41 CONTINUE
READ 7,(Q(J),J=1,12)
£«*««>**«•*«»«***•«•*•««**««««»«»*»»*»«««»«•***««**«»**«»*»•
C INPUT CONVECTIVE BOUNDARY INFORMATION
C
C DATA GROUP VII
IFUTYPE.LT.3) GO TO 59
READ, NCONV
IF(NCONV.EQ.O) GO TO 55
READ,(NCON(I,1),NCON(I,2),CONV(I),I=1,NCONV)
55
READ,(TINF(I),1=1,NCONV)
CONTINUE
C
C
C INFORMATION NEEDED FOR HEAT OR MASS TRANSFER ACROSS A BOUNDARY
C WITH A SPECIFIED HEAD
C
READ.LEL
IF(LEL.EQ.O) GO TO 56
READ,(NEL(J,1),NEL(J,2),J=1,LEL)
READ,(AEL(J,2),J=1,LEL)
PRINT 157,(NEL(J,1),NEL(J,2),AEL(J,2),J=1,LEL)
CONTINUE
PRINT 110,{NCON(I,1),NCON(I,2),CONV(I),1=1,NCONV)
CONTINUE
READ 7,(Q(I),1=1,12)
56
59
C
C«
C
C
C
C
71
73
C
DATA GROUP VIII
INPUT LOCATION OF FLOW BOUNDARIES
INPUT NODE I AND THEN THE BOUNDARY CODE
READ,LFLUXW,LFLUXH
IF(LFLUXW.EQ.O) GO TO 71
READ,(NFLUXW(I,1),NFLUXW(I,2),I=1,LFLUXW)
PRINT 133,(NFLUXW(I,1),NFLUXW(I,2),I=1.LFLUXW)
CONTINUE
IF(LFLUXH.EQ.O) GO TO 73
READ, (NFLUXH(I,1),NFLUXH(I,2),I=1,LFLUXH)
PRINT 13* i (NFLUXH(I,1),NFLUXH(I,2),I=1,LFLUXH)
CONTINUE
READ 7,(Q(I),J=1,12)
DAT19000
DAT19100
DAT19200
DAT19300
DAT19400
DAT19500
DAT19600
DAT19700
DAT19800
DAT19900
DAT20000
DAT20100
DAT20200
DAT20300
DAT20400
DAT20500
DAT20600
DAT20700
DAT20800
DAT20900
DAT21000
DAT21100
DAT21200
DAT21300
DAT2T400
DAT21500
DAT21600
DAT21700
DAT21800
DAT21900
DAT22000
DAT22100
DAT22200
DAT22300
DAT22400
DAT22500
DAT22600,
DAT22700
DAT22800
DAT22900
DAT23000
DAT23100
DAT23200
DAT23300
DAT23400
DAT23500
DAT23600
DAT23700
DAT23800
DAT23900
DAT24000
DAT24100
DAT24200
DAT24300
DAT2H400
DAT21500
DAT24600
DAT24700
DAT2H800
DAT24900
DAT25000
DAT25100
DAT25200
DAT25300
-158-
-------
c
C FORMAT INFORMATION—INFORMATION PRINTED ON EVERY RUN
102 FORMATC1X//1X.T10,'MATERIAL',T20,'HOR PERM',T32,' VER PERM'.TSO,
"'STORAGE',T65,'HOR THERM COND1,T85,'VERT THERM COND',T103,
"'SPECIFIC DISPERSION COEFS'/1X,T103,'HEAT')
104 FORMATC1X,T12,G9.4,T22,G9.4,T34,G9.4,T50,G9.4,T68,G9.4,T88,
«G9.4,T105,G9.4,T115,2G9.4/)
105 FORMATC1X//1X,'PARAMETER FACTORS—IN ORDER ',8CG10.6,1X))
110 FORMATC1X,'CONVECTIVE OUT AND CONDUCTIVE BOUNDARY INFORMATION'
*,' ELEM # —BOUNDARY CODE—TRANSFER COEFFICIENT'
* /IX,20(IX,5(14,IX,II,1X,G11.6,2X)/))
120 FORMATC1X//1X,'THE VALUES IN THE TYPE OF MATERIAL MATRIX ARE')
131 FORMATC1X//1X,'POINT SOURCES OF WATER—NODE # AND AMOUNT'/IX,
«5(I4,1X,G12.6,5X))
132 FORMATC1X//1X,'POINT SOURCES OF HEAT1/1X,5CI4,1X,G12.6,5X))
133 FORMATC1X//1X,'WATER FLOW BOUNDARIES—ELEM # AND BOUNDARY CODE'
«10CI4,2X,I1,5X))
134 FORMATC1X//1X,'HEAT FLOW BOUNDARIES—NODE # AND BOUNDARY CODE'/
«10CI4,2X,I1,5X))
150 FORMATC1X//1X,1EQUATION 1 TIME STEP',G12.6)
151 FORMATC1X//1X,1EQUATION 2 TIME STEP',G12.6,10X,
*ITY COEFFICIENT1,G12.6)
155 FORMATC1X//1X,'VELOCITIES ARE BEING CALCULATED
*EP ',11)
156 FORMATC1X//1X,'VELOCITIES ARE BEING CALCULATED
•ME STEP')
157 FORMATC1X//1X,'INFORMATION FOR MASS OR HEAT TRANSFER ACROSS' DAT28000
•,1X,'A SPECIFIED HEAD BOUNDARYV1X,'ELEMENT NUMBER--BOUNDARY CODE'DAT28100
«,IX,'--TEMPERATURE OR CONCENTRATION OF INCOMING FLUID' DAT28200
«/1X,5(2X,I4,1X,I2,1X,2X,G10.5)/) DAT28300
C»»»««»»»»*»»»»«»««»«»«»«*»«»»»*»««*«*»«»»*»**«***»«*«*««»»»«««»«»«*»»*»DAT28400
DAT25400
DAT25500
DAT25600
DAT25700
DAT25800
DAT25900
DAT26000
DAT26100
.1XDAT26200
DAT26300
DAT26400
DAT26500
DAT26600
DAT26700
DAT26800
' DAT26900
DAT27000
DAT27100
DAT27200
DAT27300
CAPACDAT27400
DAT27500
TIME STDAT27600
DAT27700
AT THE PREVIOUS TIDAT27800
DAT27900
'HEAT OR MASS
AT CURRENT
C
C
C
THE ROUTINE THAT PRINTS OUT THE INITIAL
SPECIFIED BY THE INPUT DATA
CONDITIONS
IF(KPRINT.EQ.O) GO TO 300
PRINT 199.CTITLECJ),J=1,24)
199 FORMATC1H1,1X//,20('«',12A6,20('*')/1X,1 OX,'THE UNITS USED ARE',
*12A6))
PRINT 210,NUMNP,M
210 FORMATC1X,'NUMBER OF NODES ',13,' NUMBER OF ELEMENTS ',I3//)
PRINT 211,ALPHA,PCW
211 FORMATC1X,10X,'TIME STEP',F5.2,' UNITS',10X,'PC=',G10.5)
PRINT 220
220 FORMATCIX,'ELEMENT DATA'/1X,'NODE NUMBER',T20,'X-LOCATION',T40,
•'Y-LOCATION',T60,'SPEC. HEAD',T75,'SPEC. T',T90,'INITIAL T',
« T110,'INITIAL HEAD')
DO 240 J=1,NUMNP
PRINT 225,J,XLOCCJ),YLOCCJ),HEA(J),HEAD(J),R(J),R1(J)
225 FORMATC1X,T5,I3,T20,G10.5,T40,G10.5,T60,G10.5,T75,G10.5,T90,G10.5
« T110.G10.5)
240 CONTINUE
PRINT 250
250 FORMATC1X.2X///1X,'ELEMENT PROPERTIES'/1X,
«T22,'HCY',T32,'STO',T42,'TCX',T52,'TCY',T63,'PC',T72,
• ' THE ELEMENT NODES ',T102,'HEATIN',T112,'WATER IN')
DO 270 J=1,M
PRINT 275,J,WXCJ),WYCJ),STO(J),PX(J),PY(J),PCXCJ),CNOD(LZ,J),LZ=1,DAT31000
* 4),HEAT(J),INFLOWCJ) DAT31100
'ELEMENT1,T12,'HCX'
DAT28500
DAT28600
DAT28700
DAT28800
DAT28900
DAT29000
DAT29100
DAT29200
DAT29300
DAT29400
DAT29500
DAT29600
DAT29700
DAT29800
DAT29900
DAT30000
DAT30100
.DAT30200
DAT30300
DAT30400
DAT30500
DAT30600
DAT30700
DAT30800
DAT30900
275
276
278
FORMATC1X,T4,I3,T10,G8.3,T20,G8.3,T30,G8.3,T40,G8.3,,T50IG8.3, DAT31200
« T60,G8.3,T70,4I3,T100,G8.3,T110,G8.3) DAT31300
FORMATC1X//1X,'ELEVATION OF THE AQUIFER BOTTOM',1X/) DAT31400
FORMATCIX,'LINE SOURCES OF WATER--ELEMENT NUMBER.BOUNDARY CODE.RADAT31500
«TE'/1X,6(I4,1X,I2,1X,G10.5,2X)) HAT?,A™
-159-
-------
279 FORMAT(1X,'LINE SOURCES OF HEAT—ELEMENT NUMBER .BOUNDARY CODE, RADAT31700
*TE'/1X,6(I4,1X,I2,1X,G10.5,2X)) DAT31800
270 CONTINUE DATS 1900
300 CONTINUE DAT32000
C ROUTINE TO ORIENTATE THE MATRIX—INSURES THAT ALL PROBLEMS ARE ORIENTADAT32100
C IN THE SAME MANNER DAT32200
282
284
288
290
294
C
IF(MQ.EQ.O) GO TO 294
IF(MQ.EQ.2) GO TO 294
AC=0
AB=0
DO 282 J=1,LN
AB=AMAX1(XLOC(J),AB)
AC=AMAX1(YLOC(J),AC)
IF(MQ.EQ.4) GO TO 288
DO 284 J=1,LN
YLOC(J)=AC-YLOC(J)
IF(MQ.EQ.I) GO TO 294
CONTINUE
DO 290 J=1,LN
XLOC(J)=AB-XLOC(J)
CONTINUE
DAT32300
DAT32400
DAT32500
DAT32600
DAT32700
DAT32800
DAT32900
DAT33000
DAT33100
DAT33200
DAT33300
DAT33400
DAT33500
DAT33600
DAT33700
DAT33800
DAT33900
C*« *««««§»« t«»»«f*ci**>«««*»t*«««i*«t««*cft«*««*«t•«•••!•«t«ff«*« ««***«*** DAT34000
54
55
1
8
9
10
11
12
13
RETURN
SUBROUTINE ELREAD
DIMENSION DIGIT(IO), CHAFU80)
FORMAT(1X,'PROBLEMS—BAD CHARACTER',IX JA1,3X, •ELEMENT',16)
FORMATdX,1 PROBLEM—NOT ENOUGH NODAL DATA FOR ELEMENT ',16)
READ 1,(CHAR(J),J=1,80)
FORMAT(80A1)
KC=0
KEr4
KS=2
J=0
KL=0
NUH=0
J=J+1
IF(CHAR(J).EQ.' ') GO TO 9
1=0
1=1+1
IF(I.LE.IO) GO TO 12
PRINT 54,CHAR(J),L
STOP
IF(CHAR(J).NE.DIGIT(D) GO TO 11
NUM=10*NUM+I-1
J=J+1
IF(CHAR(J).EQ.' ') GO TO
IF(CHAR(J).EQ.'«») GO TO
GO TO 10
KL=KL+1
IFUL.EQ
13
14
DAT34100
DAT34200
DAT34300
DAT34400
DAT34500
DAT34600
DAT34700
DAT34800
DAT34900
DAT35000
DAT35100
DAT35200
DAT35300
DAT354CO
DAT35500
DAT35600
DAT35700
DAT35800
DAT35900
DAT36000
DAT36100
DAT36200
DAT36300
DAT36UOO
DAT36500
DAT36600
DAT36700
IF(KL.EQ
KL=KL+1
1)
I)
L=NUM
GO TO 8
14
IF(KC.EQ.S) GO TO 15
NOD(KC,L)=NUM
IF(KS.EQ.O) KErKE-t-2
IF(KS.EQ.I) KE=KE+1
KS=0
GO TO 8
K£=KE+1
KS=KS+1
DAT36900
DAT37000
DAT37100
DAT37200
DAT37300
DAT37400
DAT37500
DAT37600
DAT37700
DAT37800
DAT37900
DAT38000
-160-
-------
15
16
C
C'
C
1
2
3
4
20
10
11
NOD(KE,L)=NUM
IFU.LT.65) GO TO 8
PRIMT 55,L
STOP
MTM=M
DO 16 1=5,12
IF(NOD(I,L).EQ.O) GO TO 16
MTM=MTM+1
CONTINUE
NODE(L)=MTM
RETURN
SUBROUTINE P(KX,KY)
DIMENSION D(50), B(50),C(50)
J = 0
MN = 0
IF(KX.LT.O) MN=1
KX=ABS(KX)
KXA=KX-1
KYA=KY-1
DO 2 L=1 ,KXA
DO 1 K=1 ,KYA
NOD(3,J>=L+J+KYA
NODC1 ,J)=L+J
NOD(2,J)=L+J+KY
CONTINUE
CONTINUE
READ,(B(J),J=1,KX)
PRINT 10
PRINT ,(B(J),J=1,KX)
READ,(C(J),J=1,KY)
PRINT 11
PRINT, (C(J),J=1TKY)
J=0
DO 4 L=1 ,KX
DO 3 K=1 ,KY
YLOC(J)=C(K)
XLOC(J)=B(L)
CONTINUE
CONTINUE
IF(MN.EQ.O) RETURN
READ.MLD
MLB=1
IF(MLD.EO.O) MLB=KY
READ, (D(J),J=1,KX)
KXX=LN-KY+1
DO 20 K=MLB,KY
L=0
DO 20 JJ=1,KXX,KY
J=JJ+K-1
L=L+1
YLOC(J)=YLOC(J)+D(L)
RETURN
FORMAT(1X/1X,'THE X-SPACING
FORMAT(1X/1X,'THE Y-SPACING
RETURN
SUBROUTINE READERUZ ,Q ,1)
IS')
IS')
DAT38100
DAT38200
DAT38300
DAT38MOO
DAT38500
DAT38600
DAT38700
DAT38800
DAT38900
DAT39000
DAT39100
DAT39200
DAT39300
DAT39400
DAT39500
DAT39600
DAT39700
DAT39800
DAT39900
DAT40000
DAT40100
DAT40200
DAT40300
DAT40400
DAT40500
DAT40600
DAT40700
DAT40800
DAT40900
DAT41000
DAT41100
DAT41200
DAT41300
DAT41400
DAT41500
DAT41600
DAT41700
DAT41800
DAT41900
DAT42000
DAT42100
DAT42200
DAT42300
DAT42400
DAT42500
DAT42600
DAT42700
DAT42800
DAT42900
DAT43000
DAT43100
DAT43200
DAT43300
DAT43400
DAT43500
DAT43600
DAT43700
DAT43800
DAT43900
DAT44000
DAT44100
DAT44200
-161-
-------
DIMENSION Q(LZ) DAT1U300
READ,QI.IZ,FACT DAT44400
DO 1 IPs'! ,LZ DAT4M500
Q(IP)=QI DATJ44600
IF(IZ.GE.O) GO TO 2 DAT44700
READ,(Q(IP),IP=1,LZ),I DAT44800
GO TO 3 DAT4U900
1=999 DATM5000
IF(IZ.EQ.O) GO TO 3 DATU5100
READ,(J,Q(J),IP=1,IZ) DATU5200
CONTINUE DAT45300
DO 5 IP=1,LZ DAT45400
Q(IP)=Q(IP)«FACT DATM5500
RETURN DAT45600
END DAT45700
-162-
-------
*«*»»«»«»»«*»«»««
C
C
C
STR
50
STR
THIS ROUTINE ASSEMBLES THE STRUCTURE MATRICIES AND PREFORMS STR
A FORWARD REDUCTION OF THE STIFNESS MATRIX BY GAUSS ELIMINATIONSTR
STR
SUBROUTINE STRUCK LN ,LM ,MBAND ,MP ,R , S,T ,G ,GG ,RI .HEAD ,XLOC, STR
*YLOC, NOD, NODE, CFACT.TT, ALPHA, KBOUND) STR
INTEGER TT STR
REAL JAC STR
DOUBLE PRECISION AA,SE,TE,S,T STR
COMMON/AAA/XL(4),YL(4) , DETJAC, JAC( 4 ,4 )/A1/M ,MM,NUMNP .NSIZE STR
COMMON/AB/AAC6) , W( 4)/ABB/SE( 12 , 1 2) ,TE(12,12) STR
COMMON/AC/RE(12) , GE( 12)/ACC/KK( 12) ,K1 ,K2,K3,K4 STR
COMMON/HH/LFLOW .LON/HM/PXX , PYY , PXY ,KAD/AM/KSYM , PCT STR
DIMENSION XLOC(LN) ,YLOC(LN) , NOD( 1 2 , LM) , R(LN) .S(LN,MP) STR
DIMENSION T(LN,MP) ,G(LN) ,GG(LN) ,RI(LN) ,NODE(LM) ,HEAD(LN) STR
LFLOW=1 STR
AA(1)=-(1/3)«*-5 STR
STR
STR
STR
STR
STR
STR
STR
AA(2)=-AA(1)
DO 50 J=1,LN
GG( J)=0.0
DO 50 1=1 ,MP
S(J,I)=0.0
T(J,I)=0.0
C PRINT MESSAGE IF DISPERSION ROUTINE IS USED STR
IF(KAD.EQ.I) PRINT 10 STR
C STR
Q*ff««««»«»*«>«»*«*«»«*»««*»«»*«**«
-------
1«»0 CONTINUE SIR 6500
150 CONTINUE STR 6600
C STR 6700
C STR 6800
C*»t»*««»f»t«»»««•*»»•*»«»•»»»«»«»«»»»««««««»•t•«»»»«»**«»«»«»»*«««««»»#STR 5900
C STR 7000
C—ROUTINE FOR SPECIFIED HEAD BOUNDARY CONDITIONS STR 7100
DO 200 N=1,NUHNP STR 7200
IF(HEADCN).EQ.O) GO TO 200 STR 7300
IF(KSYM.EQ.O) S(N,1)=S(N,1)+CFACT STR 7400
IF(KSYM.GT.O) S(N,MBAND)=S(N,MBAND)+CFACT STR 7500
200 CONTINUE STR 7600
C STR 7700
£*••«»•»*•**••*****•«•**«»•*•**•**•*•••»«•*«***««*«»**«•»**»***«**«*«**«sjR 7800
C STR 7900
C CALL THE ROUTINE THAT CHECKS FOR THE STABILITY OF THE CHOSEN TIME STSTR 8000
KSMrKSYM STR 8100
IFUON.EQ.99.AND.TT.EQ.1) CALL EIGEN(LN ,MBAND,MP,KSM ,CFACT ,S,T ,RI ,STR 8200
•HEAD)
C GAUSS ELIMINATION—FORWARD REDUCTION OF STIFNESS MATRIX STR 8400
C ' STR 8500
IF(KSYM.EQ.O) CALL SOLVEUN MBAND ,S,T,R,RI) STR 8600
IF(KSYH.NE.O) CALL ASOLVE(S,R,RI,LN,MBAND-1,LN,MBAND»2-1,T) STR 8700
C STR 8800
C STR 8900
10 FORHAT(1X/1X,'THE DISPERSION ROUTINE IS BEING USED ) STR 9000
20 FORHAT(1X/1X,'MAX. BANDWIDTH EXCEEDED IN ELEMENT',II, STR 9100
» 'WHERE BANDWIDTH IS •,!«) STR 9200
RETURN STR 9300
C STR 9400
C STR 9500
END STR 9600
-164-
-------
£»**«*»»«***«««««**«****«««*« ft* ****•««*»«««»««* *»««!« ««»**«««•««»«««*«
C* GAU
C THIS ROUTINE FORMS ELEMENT STIFFNESS MATRIX BY GAUSS QUADRATURE 'GAU
C GAU
SU BROUTINE GAUSSC LLL , LM , LN ,XLOC , YLOC , NOD , PY , PX , HEAT , PCX , ALPHA , KBOGAU
*UND ,TT, Z , NCON , CONV ,ALOC ,NODE ,CBAL ,AY , A2) GAU
REAL JAC
INTEGER VE, TT ,Z , AY( 4) ,AZ(4)
DOUBLE PRECISION AA ,NOT,NXI ,NET,SE ,TE ,BB ,DUM1 ,DUM2 , DUM3
COMMON/AA/NXI(12),NET(12),NOT(12)
COMMON/AAA/XLC4) ,YL(4) , DETJAC , JAC( 4 ,4)
COMMON/AB/AA(6) , W( 4) /ABB/SE( 12 , 12) , TE( 12, 12)
COMMON/AC/RE(12),GE(12)/ACC/KK(12),K1,K2,K3,K4
COMMON/HH/LFLOW,LON/HM/PXX,PYY,PXY.KAD/ME/MEQ
COMMON/CON/MC1,MC2,NCONV/AS/BBC2,12)
DIMENSION NCON(Z,3),CONV(Z),ALOC(Z)
COMMON/CON1/HE(4),HEE(4,4)/AM/NLA,PCH/AT/XIH4) ,ETI(U)
COMMON/ATHICK/ASIZE.NTHICK,THICK,ERROR
DIMENSION VE(4),AVY(4),AVX<4),VEL(4)
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
DIMENSION XLOC(LN),YLOC(LN),PY(LM),PX(LM),HEAT(LM),PCXCLGAU
*M),CBAL(Z,2),AG(6),NODE(LM),NOD(12.LM) GAU
DATA XII/-1.,1.,1.,-1./ GAU
DATA VE/1 ,4,2,3/ GAU
DATA ETI/-1.,-1.,1 .,1 ./ GAU
DATA W/.3478549,.6521451,.6521451,-347S549/ GAU
DATA AG/-.5773503,.5773503,-.8611363,-.3399810,.3399810,.8611363/GAU
RETURN GAU
£«»««»«»•«»»*»«»»««*»««««««»**»««»*«»««**«**«««««««»»»«»«««»«««»*»*«««««GA[j
C GAU
ENTRY DERIVE(LLL,M4) GAU
C GAU
C GAU
£*«>*«««*«»«*«»*»«**««»*«ft»«*ft««»*«***«*««**««»*««*ft«ft*****ft**»ft* GAU
C GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
GAU
20
30
CALL CORD(LLL)
T- 1 0
CALL PEE(T,LLL)
DO 10 K=1 ,12
GE(K)=0.0
DO 10 L=1 ,12
TE(K,L)=0.0
10 SE(K,L)=0
START THE QUADRATURE LOOP
NP=2
IFCM4.GT.M) NP=H
KVE=0
DO 200 II=1,NP
DO 200 JJ=1,NP
KVE=KVE+1
IFCNP-EQ.**) GO TO 20
XI=AG(JJ)
YI=AG(II)
GO TO 30
XI=AG(JJ+2)
YI=AG(II+2)
IF(LFLOW.EQ.O) XI=0.0
IF(LFLOW.EQ.O) YI=0.0
CONTINUE
IF(M14.GT.M) CALL SHAPECLLL ,MU ,XI ,YI ,XLOC, YLOC.LN ,LM,NOD)
IFCM4.EQ.4) CALL SHAPE1(II,JJ)
c»»«»»«*»«»»«»««»«*»««»**««»««»»»»«•»«»»»«**»««
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
2500
2600
2700
2800
2900
3000
3100
3200
3300
3400
3500
3600
3700
3800
3900
4000
4100
4200
4300
4400
4500
4600
4700
4800
4900
5000
5100
5200
5300
5400
5500
5600
5700
5800
5900
6000
6100
6200
-165-
-------
175 CONTINUE
THICK=1.0
IF(NTHICK,EQ.O) GO TO 176
XX=(XLOC(K2)+XLOC(K1))/2
XX=ABS(XX)
THICK=XX*ASIZE
YY=(YLOC(K3)+YLOC(K1))/2
YY=ABS(YY)
IF(ASIZE.LT.O) THICK=-YY«ASIZE
176 DUM1=W(II)«W(JJ)«DETJAC»THICK«T
IF(M4.EQ.4) DUM1=DETJAC«THICK«T
C»«*»»»«»«*»»»»«»«««««»*»»»»»«*»»»»«»«»*»«»««««
C
IF(LFLOW.EQ.O) DETJAC=DETJAC«T«THICK
IF(LFLOW.EQ.O) RETURN
C VELEOCITY AND DISPERSION CALCULATIONS
C
VX=0.0
VY=0.0
IF(NLA.EQ.O) GO TO 178
IFCNLA.EQ.1) CALL VELO(LLL,VX,VY)
IFCNLA.EQ.2) CALL VCENTCLLL,VX,VY)
IF(KAD.EQ.I) CALL HECD(LLL,VX,VY)
C
IFCKVE.GT.4) GO TO 178
KEE=VE(KVE)
AVY(KEE)=VY
AVXCKEE)=VX
178 CONTINUE
C
£ft««f«t«»lt«*«tf*l«lf»«*f*««»f***l««*««>l*t««»ft
C
c
180
DO 200 NROW=1,M4
GE(NROW)=GE(NROW)+NOT(NROW)*HEAT(LLL)«DUM1
DO 200 NCOL=1,M4
IF(NLA.EQ.O) GO TO 180
DUM2=BB(1,NROW)»NOT(NCOL)»PCH»VX+BB(2,NROW)*NOT(NCOL)«PCH«VY
SE{NROW,NCOL)=SE(NROW,NCOL)+DUN2*DUM1
CONTINUE
DUM2=BB(1,NROW)*BB{1,NCOL)«(PX(LLL)+PXX)
S£(NROW,NCOL)=SE(NROW,NCOL)+DUM1*DUM2
DUM2=BB(2,NROW)«BB(2,NCOL)*(PYULL)
SE(NROH,NCOL)=SE(NROW,NCOL)+DUH1«DUM2
GAU
GAU
8100
8200
8300
8400
GAU 6300
GAU 6400
GAU 6500
GAU 6600
GAU 6700
GAU 6800
GAU 6900
GAU 7000
7100
7200
GAU 7300
GAU 7400
GAU 7500
GAU 7600
GAU 7700
GAU 7800
GAU 7900
GAU 8000
GAU
GAU
GAU
GAU
GAU 8500
GAU 8600
GAU 8700
GAU 8800
GAU 8900
GAU 9000
GAU 9100
GAU 9200
GAU 9300
GAU 9400
GAU 9500
GAU 9600
GAU 9700
GAU 9800
GAU 9900
GAU10000
GAU10100
GAU10200
GAU10300
GAU10400
GAU10500
GAU10600
GAU10700
GAU10800
THESE NEST TWO LINES ADD IN TRANSVERSE TERMS OF CONDUCTIVITY TENSOGAU10900
DUM2=BB(1,NROW)«BB(2,NCOL)»PXY+BB(2,NROW)*BB(1,NCOL)«PXY
SE(NROW,NCOL)=SE(NRO¥,NCOL)+DUM1*DUM2
IF(TT.EQ.O) GO TO 200
DUM3=NOT(NROW)«NOT(NCOL)«PCXULL)
IF(NLA.EQ.O) DUM3=DUM3/T
TE(NROW,NCOL) = DUM3»DUMUTE(NROW,NCOL)
200 CONTINUE
C
£•**««•*»*•»Ift***»*ft««*«»ftfft*fftt«•••*••«*•*•*««*»***»»«****
C
c
THE CONVECTIVE BOUNDARIES ARE HANDLED
IF(NCONV.EQ.O) GO TO 400
IF(MEQ.EQ.I) GO TO 400
DO 301 Isl.NCONV
11=1
IF(LLL.EQ.NCON(I,1)) CALL CONVEC
300 CONTINUE
GAU11000
GAU11100
GAU11200
GAU11300
GAU11400
GAU11500
GAU11600
GAU11700
GAU11800
GAU11900
GAU12000
GAU12100
GAU12200
GAU12300
GAU12400
GAU12500
GAU12600
-166-
-------
301
400
C
C
C
450
800
C
C
505
C
C
550
•«
:«
c
600
601
602
603
604
605
C
CONTINUE
CONTINUE
VYY=(AVY(1) + AVY(2)-t-AVY(
IF(KBOUND.EQ.99) RETURN
IF(TT.EQ.O) RETURN
t>«*«*««»1
CRANK-NICOLSON METHOD FOR TREATING TRANSIENT CONDITIONS
DO 450 NROW=1,M4
GE(NROW)=GE(NROW)*ALPHA
DO 450 NCOL=1,M4
DUM4=SE(NROW,NCOL>
SECNROW,NCOL)=ALPHA/2*SE(NROW,NCOU+TECNROW,NCOL)
TECNROW,NCOL)=TE(NROW,NCOL)-ALPHA/2*DUM4
CONTINUE
CONTINUE
**ft«*«******«***»*********t*«»*«*«*«»*****«**«****
RETURN
ROUTINE THAT ADDS IN CONVECTIVE BOUNDARY TERMS
SUBROUTINE CONVEC
LA=NCON(I,2)
MC1=AY(LA)
MC2=AZ(LA)
A=XL(MC1)-XL(MC2)
MC3=MC1+MC2
IFCMC3.EQ.5) A=YL(MC1)-YL(MC2)
A=ABS(A)
DO 505 J=1,4
HE(J)=0.0
HE(MC1)=.5
HE(MC2)=.5
ALOC(II)=A«THICK*T
CO=CONV(II)
IFCCO.LT.0.00001) GO TO 600
CBAL(II,1)=ALPHA«CO«THICK*T*A
DO 550 K=1,4
DO 550 J=1,4
SE(K,J)=HE(J)*HE(K)»CO«THICK*T«A+SE(K,J)
RETURN
CONTINUE
GO TO (603,601,601,603),LA
DO 602 J=1,4
IF(MC3.EQ.5) VEL(J)=AVX(J)
IFCMC3-NE.5) VEL(J)=AVY(J)
GO TO 605
DO 604 J=1,4
IF(MC3-EQ.5) VEL(J)=-AVX(J)
IF(MC3.NE.5) VEL(J)=-AVY(J)
CONTINUE
C0=-C0
NCO=CO
VT=VEL(MC1)/2+VEL(MC2)/2
IF(NCO.LT.995.0R.NCO.GT.1000) GO TO 700
IF(NCO.EQ.999) CO=VT«PCH*ERROR
IF(NCO.EQ.996) C0=(-VT+VYY)«PCH*ERROR
GAU12700
GAU12800
GAU12900
GAU13000
GAU13100
GAU13200
GAU13300
GAU13400
GAU13500
GAU13600
GAU13700
GAU13800
GAU13900
GAU14000
GAU14100
GAU14200
GAU14300
GAU14400
GAU14500
GAU14600
GAU14700
GAU14800
GAU14900
GAU15000
GAU15100
GAU15200
GAU15300
GAU15400
GAU15500
GAU15600
GAU15700
GAU15800
GAU15900
GAU16000
GAU16100
GAU16200
GAU16300
GAU16400
GAU16500
GAU16600
GAU16700
GAU16800
GAU16900
GAU17000
GAU17100
GAU17200
GAU17300
GAU17400
GAU17500
GAU17600
GAU17700
GAU17800
GAU17900
GAU18000
GAU18100
GAU18200
GAU18300
GAU18400
GAU18500
GAU18600
GAU18700
GAU18800
GAU18900
-167-
-------
650
IF(NCO.EQ.997) CO=ERROR
CBAL(II,1)=ALPHA«CO«THICK»T«A
DO 650 !(=•!,«
DO 650 J=1,4
SE(K,J)=HE(J)«HE(K)»CO»THICK»T«A+SE(K,J)
RETURN
£»•***»»»*•**•*•«*••««*»*««»****«««•**•«««•«««*«*•*•«»*•
c
700 CONTINUE
C CALCULATE HEAT TRANSFER OUT WITH FLOWING WATER
C
CBALCII,1)=ALPHA«VT«PCH«THICK*T«A
C
DUM1=1-AA(1)
DUM2=1-AA(2)
HE(MC1) = DUM1M
HE(MC2)=DUM2/4
VELL=VEL(MC1)
DO 720 K=1,4
DO 720 J=1,4
720 SE
-------
c«*«««*«*«««*»*«t*«««»»«**«««»*******««»****»*«**»
SUBROUTINE SHAPE1(II,JJ)
REAL JAC
DOUBLE PRECISION AA.NOT ,NXI,NET,SE,TE,BB ,DUM1,DUM2
COMMON/AA/NXI(12),NET(12),NOT(12)
COMMON/AAA/XL(4),YL(4),DETJAC,JAC(4,4)
COMMON/AB/AA(6),W(4)/ABB/SE(12,12),TE(12,12)
COMMON/AC/RE(12),GE(12)/ACC/KK(12),K1,K2,K3,K4
COMMON/HH/LFLOW,LON/HM/PXX,PYY,PXY,KAD
COMMON/AT/XII(4),ETI(4)/CON/MC1,MC2,NCONV/AS/BB(2,12)
COMPUTE THE SHAPE FUNCTIONS AND THEIR DERIVATIVES
DO 100 1=1 ,4
DUM2=(l!+ETI(I)»AA(JJ))*.25
NXI(I)=XII(I)«DUM2
NET(I)=ETI(I)*DUM1
NOT(I)=4.«DUM1«DUM2
100 CONTINUE
C COMPUTE JACOBIAN, AND ITS DETERMINANT
DO 150 1=1 ,2
DO 150 J=1 ,2
150 JAC(I,J)=0.0
DO 160 1=1,4
JAC(1,1) = JAC(1,1)-»-NXI(I)*XL(I)
JAC(1,2)=JAC(1,2)+NXI(I)*YL(I)
JAC(2,1)=JAC(2,1)*NET(I)*XL(I)
JAC(2,2)=JAC(2,2)+NET(I)*YL(I)
160 CONTINUE
DETJAC=JAC(1,1)*JAC(2,2)-JAC(2,1)*JAC(1,2)
DUM1=JAC(1,1)/DETJAC
JAC(1,1)=JAC(2,2)/DETJAC
JAC(1,2)=-JAC(1,2)/DETJAC
JAC(2,1)=-JAC(2,1)/DETJAC
JAC(2,2)=DUM1
DO 170 L=1,4
BB(1,L)=JAC(1,1)«NXI(L)+JAC(1,2)»NET(L)
BB(2,L)=JAC(2,1) »NXI(L)+JAC(2,2)*NET(L)
170 CONTINUE
RETURN
END
SHI
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SHI
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SH1
SHI
SHI
SH1
SH1
SHI
SHI
SHI
SHI
SHI
100
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3100
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3300
3400
3500
3600
3700
3800
3900
4000
4100
4200
4300
4400
-169-
-------
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
SUBROUTINE SHAPEU ,M,XI ,YI ,X ,Y ,NN ,NE , IN)
«*««»»«*»«***»*«*••««««««*****«««««*«««<»*»«
XI/YI THE GAUSS POINTS
X/Y THE X AND Y LOCATIONS OF THE ELEMENT NODES
NN NUMBER OF NODES
NE NUMBER OF ELEMENTS
IN ARRAY REFERED TO IN MAIN PROGRAM AS NOD
M REFERED TO IN MAIN PROGRAM AS Ml , NUMBER OF. NODES IN THE ELEMENT[
DET IS THE DETERMINENT, ONE QUARTER OF THE AREA OF THE ELEMENT
CALLED FROM MNPGM
PURPOSE: TO COMPUTE BASIS FUNCTIONS
ORIGINALLY PROGRAMED BY EMIL 0. FRIND
DOUBLE PRECISION DFX.DFY ,FF, BB
DIMENSION X(NN), Y(NN) , IN(12,NE)
DIMENSION ALF(4), DAX(4), DAY(4), BTX(4), BTYU) , DBX(4),
1 DBYCO
f
COMMON/AA/DFX( 12) , DFY( 12) , FF( 12)
«/AAA/ZZZCO,ZZZZZ(4),DET,DJAC(4,4)
COMMON/AS/BB(2,12)
XI/YI - LOCAL COORDINATES OF THE INTEGRATION POINTS
XIU1.-XI
XI2=1.+XI
YIIsl.-YI
YI2=1.+YI
CORNER NODE SHAPE FUNCTIONS, BASIC PART
ALF - ALPHA PART OF SHAPE FUNCTION
ALF(1)=.25«XI1«YI1
ALF(2)=.25*XI2«YI1
ALF(3)=.25*XI2«YI2
ALF(1)=.25«XI1«YI2
DAX/DAY - X- AND Y-DERIVATIVE OF ALPHA PART
DAX(1)=-.25«YI1
DAX(2)=.25»YI1
DAX(3)=.25«YI2
DAX(i|) = -.25«YI2
DAY(1)=-.25«XI1
DAY(2)=-.25«XI2
DAY(3)=.25*XI2
DAY(M)=.25«XI1
CORNER NODE SHAPE FUNCTIONS, SIDE-DEPENDENT PART
XQ1=XI-.5
XQ2=-XI-.5
YQ1=YI-.5
YQ2=-YI-.5
XC1=1.125*XI»XI-.625
XC2=2.25«XI
YC1=1.125«YI»YI-.625
YC2=2.25»YI
J1 = 1
J2=2
J3=5
FOR BETA X PART (BTX) OF SHAPE FUNCTION
DO 50 J=1,2
IF (IN(J3,D.EQ.O) GO TO 10
SHA 100
SHA 200
SHA 300
SHA 100
SHA 500
SHA 600
SHA 700
SHA 800
SHA 900
SHA 1000
SHA 1 100
SHA 1200
SHA 1300
SHA 1400
SHA 1500
SHA 1700
SHA 1800
SHA 1900
SHA 2000
SHA 2100
SHA 2200
SHA 2300
SHA 2400
SHA 2600
SHA 2700
SHA 2800
SHA 2900
SHA 3000
SHA 3100
SHA 3200
SHA 3300
SHA 3400
SHA 3500
SHA 3600
SHA 3700
SHA 3800
SHA 3900
SHA 4000
SHA 4100
SHA 4200
SHA 4300
SHA 4400
SHA 4500
SHA 4600
SHA 4700
SHA 4800
SHA 4900
SHA 5000
SHA 5100
SHA 5200
SHA 5300
SHA 5400
SHA 5500
SHA 5600
SHA 5700
SHA 5800
SHA 5900
SHA 6000
SHA 6100
SHA 6200
SHA 6^00
-170-
-------
IF (INCJ3+1,L).EQ.O) GO TO 20
GO TO 30
10 CONTINUE
LINEAR . . .
BTX(J1)=.5
BTX(J2)=.5
DBX -.BETA X DERIVATIVE
DBX(J1)=0.
DBX(J2)=0.
GO TO 40
20 CONTINUE
QUADRATIC . . .
BTX(J1)=XQ2
BTX(J2)=XQ1
DBX(J1)=-1.
DBX(J2)=1.
GO TO 40
30 CONTINUE
CUBIC . . .
BTX(J1)=XC1
BTX(J2)=XC1
DBX(J1)=XC2
DBX(J2)=XC2
40 CONTINUE
J1 = 4
J2=3
J3=9
50 CONTINUE
J1 = 2
J2=3
J3=7
FOR BETA Y PART (BTY)
DO 100 J=1,2
IF (IN(J3,L).EQ.O) GO TO 60
IF (IN(J3+1 ,D .EQ.O) GO TO 70
GO TO 80
60 CONTINUE
LINEAR . . .
BTY(J1)=.5
BTY(J2)=.5
DBY - BETA Y DERIVATIVE
DBY(J1)=0.
DBY(J2)=0.
GO TO 90
70 CONTINUE
QUADRATIC . . .
BTY(J1)=YQ2
BTY(J2)=YQ1
DBY(J1) = -1 .
DBY(J2) = 1 .
GO TO 90
80 CONTINUE
CUBIC . . .
BTY(J1)=YC1
BTY(J2)=YC1
DBY(J1)=YC2
DBY(J2)=YC2
90 CONTINUE
J1=1
J2=4
J3=11
100 CONTINUE
OF SHAPE FUNCTION
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
-SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
SHA
6400
6500
6600
6700
6800
6900
7000
7100
7200
7300
7400
7500
7600
7700
7800
7900
8000
8100
8200
8300
8400
8500
8600
8700
8800
8900
9000
9100
9200
9300
9400
9500
9600
9700
9800
9900
SHA10000
SHA10100
SHA10200
SHA10300
SHA10400
SHA10500
SHA10600
SHA10700
SHA10800
SHA10900
SHA11000
SHA11100
SHA11200
SHA11300
SHA11400
SHA11500
SHA11600
SHA11700
SHA11800
SHA11900
SHA12000
SHA12100
SHA12200
SHA12300
SHA12400
SHA12500
SHA12600
-171-
-------
c
c
c
SHAPE FUNCTION DERIVATIVE MATRIX - CORNER NODES
DO 110 J = 1,4
DFX(J) = DAX(J)«(BTX(J) + BTY(J»+DBX(J)»ALF(J)
DFY(J)=DAY(J)«(BTX(J)+BTY
-------
200
210
220
230
2UO
c
c
FF( J)= .28125*XEQ*XE1*YI2
DFY(J)=.28125»XEQ*XE1
IF (IN(11 ,L) .EQ.O) GO TO 230
IF (IN(12,L).EQ.O) GO TO 210
GO TO 220
J = J+1
DFX( J)=-.5*YEQ
DFY(J)=-YI*XI1
FF(J)=.5*XI1*YEQ
GO TO 230
J = J+1
DFX(J)=-.28125*YEQ»YE2
DFY( J)= .28125*XI1*(3.*YEQ-2.*YI*YE2)
FF(J)=.28125*XI1*YEQ*YE2
J = J + 1
DFX( J)=-.28125*YEQ*YE1
DFY(J)=-.28125*XI1*(3.*YEQ+2.«YI«YE1)
FF( J)= .28125*YEQ*YE1*XI1
CONTINUE
CONTINUE
JACOBIAN
SUM1=0.
SUM2=0.
SUM3=0.
SUMM=0.
DO 260 1=1 ,M
250 K=K+1
IF (IN(K.L).EQ.O) GO TO 250
KI=IN(K,L)
SUM1 = SUMUDFX(I)«X(KI)
SUM2=SUM2-t-DFX(I)*Y(KI)
SUM3=SUM3>DFY(I)«X(KI)
SUM1=SUM4+DFY(I)»Y(KI)
260 CONTINUE
DET=SUM1*SUM4-SUM2*SUM3
DET1=1./DET
C11=DET1*SUM4
C12=-DET1*SUM2
C21=-DET1»SUM3
C22=DET1*SUM1
DJAC(1,1)=C11
DJAC(1,2)=C12
DJAC(2,1)=C21
DJAC(2,2)=C22
SHAPE FUNCTION DERIVATIVES - GLOBAL
DO 270 J=1,M
BB(1,J) = C11»DFX(J) + C12«DFYU)
BB(2,J)=C21*DFX(J)+C22«DFY(J)
270 CONTINUE
RETURN
END
SHA19000
SHA19100
.SHA19200
SHA19300
SHA19100
SHA19500
SHA19600
SHA19700
SHA19800
SHA19900
SHA20000
SHA20100
SHA20200
SHA20300
SHA20100
SHA20500
SHA20600
SHA20700
SHA20800
SHA20900
SHA21000
SHA21100
SHA21200
SHA21300
SHA21400
SHA21500
SHA21600
SHA21700
SHA21800
SHA21900
SHA22000
SHA22100
SHA22200
SHA22300
SHA22UOO
SHA22500
SHA22600
SHA22700
SHA22800
SHA22900
SHA23000
SHA23100
SHA23200
SHA23300
SHA23'»00
SHA23500
SHA23600
SHA23700
SHA23800
SHA23900
SHA24100
SHA24200
SHA2H300
-173-
-------
c
c
c
c
c
c
c
c
c
c
c
c
c
LOA
THIS ROUTINE COMPUTES THE COLUMN MATRIX R BY ADDING IN THE VARIOUS LOA
SOURCES AND SINKS AND THEN SOLVES FOR THE UNKNOWN COLUMN MATRIX LOA
LOA
LOA
SUBROUTINE LOAD(LM,LN,MBAND,NOD,NODE,R,RI,G LOA
»,GG,HEAT,HEAD,ALPHA,KBOUND,KFT,PCW,LT,CFACT,KS, LOA
*Z,NCON,TINF,CONV,ALOC.LSOU RC,NSOU RC,ASOU RC,CBAL,AY,A Z LOA
« ,AEL,NEL,MEQ,LEL,CBALA,LINE,NLINE,ALINE) LOA
REAL JAC LOA
INTEGER Z,AY(4),AZ(4) LOA
DOUBLE PRECISION AA,NET,NXI,NOT LOA
COMMON/AAA/XL(4),YL(4),DETJAC,JAC(4,4)/AB/AA(6),W(4) LOA
COMMON/A1/M,MM,NUMNP,NSIZE/AA/NXI(12),NET(12),NOT(12> LOA
COMMON/ACC/KK(12),K1,K2,K3,K4/AM/KSYM,PCT/HH/LFLOW,LON LOA
COMMON/BAL/BLINE/CON/MC1,MC2,NCONV/CONT/LA,LB,LC,LD,LE,LF,LG,LH LOA
COMHON/ATHICK/ASIZE,NTHICK.THICK,ERROR,XADD,YADD LOA
DIMENSION NCON(Z,3),TINF(Z),CONV(Z),ALOC(Z),NSOURC(Z) LOA
DIMENSION AEL(Z,2), CBAL(Z,2), ASOURC(Z),NODE(LM),NOD(12,LM) LOA
DIMENSION R(LN),G(LN),GG(LN),RI(LN),HEAD(LN),HEAT(LM) LOA
*,NLINE(Z,2),ALINE(Z),NEL(Z,2) LOA
RETURN LOA
LOA
ENTRY HEATE(KS)
DO 1 J=1,LN
RI(J)=R(J)
G(J)=0.0
CONTINUE
BLINEsO.O
IFCLD.EQ.1.AND.MEQ.EQ.1) CALL CHAN
IF(LT.EQ.O) GO TO 300
LOA
LOA
LOA
LOA
LOA
LOA
LOA
LOA
LOA
LOA
LOA
LOA
LOA
LOA
LOA
>LOA
LOA
TRANSIENT LOOP GENERATION AND BOUNDARY CONDITIONS ARE CHANGED AT EACLOA
IF(KBOUND.EQ.I) CALL BOUND(KS) LOA
C THREE ENTRY POINTS ARE PROVIDED FOR CHANGING RECHARGE RATES LOA
C AND BOUNDARY CONDITIONS AT EACH TIME STEP—ALL ENTRIES IN BOUNDA LOA
C LOA
IF(KBOUND.EQ.2) CALL BVAL(KS.ALPHA) LOA
C LOA
IF(LD.EQ.1.AND.MEQ.EQ.2) CALL CHANG LOA
LOA
C
C
C
CONVECTIVE BOUNDARY ROUTINE
IF(HEQ.EQ.I) GO TO 50
IF(NCONV.EQ.O) GO TO 40
DO 30 I=1,NCONV
LLL=NCON(I,1)
LA=NCON(I,2)
LOA
LOA
LOA
LOA
LOA
LOA
LOA
LOA
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
2500
2600
2700
2800
2900
3000
3100
3200
3300
3400
3500
3600
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3800
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4000
4100
4200
4300
4400
4500
4600
4700
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5000
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5200
5300
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5500
5600
5700
5800
5900
6000
6100
6200
-174-
-------
MCUAYUA)
MC2=AZ(LA)
MA=NOD(MC1,LLL)
NA=NOD(MC2,LLL)
K = NA
ONV=ABS(CONV(I))
CBAL(I,2)=ONV«ALOC(I)*TINF(I)*ALPHA
C H*LENGTH PF BOUNDARY'TEMP AT INFINITY*TIME STEP
G(K)=ONV«ALOC(I)«.5*TINF(I)«ALPHA+G(K)
K = MA
G(K)=ONV*ALOC(I)*.5*TINF(I)«ALPHA+G(K)
30 CONTINUE
C
40 CONTINUE
C
C ROUTINE THAT CALCUALTES CONVECTIVE INPUTS IS CALLED
IF(LEL.GT.O) CALL CB(ALPHA)
C
C
50 CONTINUE
LOA 6300
LOA 6400
LOA 6500
LOA 6600
LOA 6700
LOA 6800
LOA 6900
LOA 7000
LOA 7100
LOA 7200
LOA 7300
LOA 7400
LOA 7500
LOA 7600
LOA 7700
LOA 7800
LOA 7900
LOA 8000
LOA 8100
LOA 8200
£«**«******«*«*»«*»«»»»****«**»»****f«*«***»*»»*f»***»****»***«*«*»«*»»»LOA 8300
C—MATRIX MULTIPLICATION LOOP LOA 8400
C LOA 8500
IF(KSYM.EQ.O) CALL MULTI LOA 8600
IF(KSYM.NE.O) CALL AMULTI LOA 8700
C LOA 8800
C»t««»*»*»*««*«•*»*»««»»«**«*«**»*»****»**»»*»**»**«*»««*»«*»*»«**«**«**LOA 8900
GENERATION TERMS ARE ADDED INTO THE RIGHT SIDE OF THE EQUATION
C
C
C
C
C
C GENERATION TERMS FROM POINT SOURCES
C
300
350
360
C
C
C
IF(LSOURC.EQ.O) GO TO 360
DO 350 LLL=1,LSOURC
K=NSOURC(LLL)
IF(ALPHA.LE.O) AS=ASOURC(LLL)
IF(ALPHA.GT.O) AS=ASOURC(LLL)»ALPHA
IF(LT.EQ.O) GG(K)=GG(K)+AS
IF(LT.EQ.I) G(K)=G(K)+AS
CONTINUE
CONTINUE
GENERATION TERMS FROM LINE SOURCES ARE COMPUTED AND ADDED IN
IF(LINE.EQ.O) GO TO 378
DO 375 L=1,LINE
LLL=NLINE(L,1)
CALL CORD(LLL)
LA=NLINE(L,2)
LY=AY(LA)
LZ=AZ(LA)
MC3=LY+LZ
IFCMC3.EQ.5) A=ABS(YL(LY)-YL(LZ))
IF(MC3-NE.5) A=ABS(XL(LY)-XL(LZ))
AS=ALINE(L)«A
IF(ALPHA.GT.O) AS=AS*ALPHA
BLINE=BLINE+AS
LY=NOD(LY,LLL)
LZ=NOD(LZ,LLL)
IF(LT.EQ.O) GO TO 372
LOA 9000
LOA 9100
LOA 9200
LOA 9300
LOA 9400
LOA 9500
LOA 9600
LOA 9700
LOA 9800
LOA 9900
LOA10000
LOA10100
LOA10200
LOA10300
LOA10400
LOA10500
LOA10600
LOA10700
LOA10800
LOA10900
LOA11000
LOA11100
LOA11200
LOA11300
LOA11400
LOA11500
LOA11600
LOA11700
LOA11800
LOA11900
LOA12000
LOA12100
LOA12200
LOA12300
LOA12400
-175-
-------
372
375
C
378
C
380
C
G(LY)=G(LY)+AS72
G(LZ)=G(LZ)+AS/2
GO TO 375
GG(LY)=GG(LY)+AS/2
GG(LZ)=GG(LZ)+AS/2
CONTINUE
CONTINUE
DO 380 N=1,NSIZE
R(N)=R(N)+G(N)
LOA12500
LOA12600
LOA12700
LOA12800
LOA12900
LOA13000
LOA13100
LOA13200
LOA13300
LOA13400
LOA13500
LOA13600
£*««««*»»•«««««««**»««»*«»»«««»««••«*••**•*•*«*•»•••«*•««*•••««««««*»»«*^QA-i 3700
C
C
C
C
GENERATION TERMS FROH SOURCES ARE ADDED IN
AND SPECIFIED HEADS ARE TREATED
LOA13800
LOA13900
LOA1HOOO
LOA1H100
LOA14200
LOA14300
LOA14400
C LOA14500
£•*••*»••••»•*****•»•**•»*»»»••••••*••*»••»«»•**»••*»»•»•« ••»««»••«»»*»«LOA 1^600
DO mo N=1,NSIZE
IF(LT.EQ.O) R(N)=CFACT»HEAD(N)+GG(N)
IF(LT.EQ.I) R(N)=R(N)+CFACT«HEAD(N)+GG(N)
C
C FORWARD REDUCTION OF THE LOAD MATRIX IS PREFORMED AND BACK
C SUBSTITUTION IS PREFORMED TO SOLVE FOR THE UNKNOWN MATRIX
C
IF(KSYM.EQ.O) CALL BACK
IF(KSYM.NE.O) CALL ABACK
550 CONTINUE
C
^•••••••••••••••••••••**tl«ftl»t**«••••••§•••§••••*•**••••*t»*••»»t*
RETURN
C
C
C
SUBROUTINE CB(ALPHA)
C
(;•**••••*»«••**«*»••••*»****••»«•*•••*•»•*•••*••«•••*•*»••»«•*«*««*
DIMENSION JA( A=XL(LY)-XL(LZ)
A=ABS(A)»T*THICK
LYsNODUY.LLL)
LZ=NOD(LZ,LLL)
COMPUTE VELOCITIES AT THE GAUSS POINTS
LOA14700
LOA14800
LOA15000
LOA15100
LOA15200
LOA15300
LOA15UOO
LOA15500
LOA15600
LOA15700
LOA15800
LOA15900
LOA16000
LOA16100
LOA16200
LOA16300
LOA16400
LOA16500
LOA16600
LOA16700
LOA16800
LOA16900
LOA17000
LOA17100
LOA17200
LOA17300
LOA17500
LOA17600
LOA17700
LOA17800
LOA17900
LOA18000
LOA18100
LOA18200
LOA18300
LOA18UOO
LOA18500
LOA18600
LOA18700
LOA18800
-176-
-------
c
c
10
20
VX IS ASSOCIATED WITH ARRAY AY
JJ=JA(LA)
II=JB(LA)
CALL SHAPEKJJ.II)
CALL VELO(LLL,VX,VY)
JJ=JC(LA)
II=JD(LA)
CALL SHAPEKJJ,!!)
CALL VELO(LLL,VXX,VYY)
IFUA.GT.2) GO TO 6
V1=(VY«A/2)
V2=(VYY«A/2)
GO TO 7
V1=(VX*A/2)
V2=(VXX»A/2)
CONTINUE
GO TO (9,8,8,9),LA
V1=-V1
V2=-V2
CONTINUE
IF(VI.GE.O) Q=V1*PCW«AEL(J,2)
IFCV1.LT.O) Q=V1*PCW«R(LY)
IF(V2.GE.O) QQ=V2»PCW«AEL(J,2)
IFCV2.LT.O) QQ=V2«PCW*R(LZ)
i = G(LY)+Q»ALPHA*FACTUQQ»ALPHA«FACT2
l = G(LZ)-t-QQ*ALPHA*FACT1+Q*ALPHA«FACT2
CBALA=CBALA+Q+QQ
CONTINUE
FORMATdX,'BOUND TEMPS • ,7( 14 ,G8 -3 ,3X))
RETURN
END
VXX IS ASSOCIATED WITH ARRAY
LOA18900
AZLOA19000
LOA19100
LOA19200
LOA19300
LOA19400
LOA19500
LOA19600
LOA19700
LOA19800
LOA19900
LOA20000
LOA20100
LOA20200
LOA20300
LOA20400
LOA20500
LOA20600
LOA20700
LOA20800
LOA20900
LOA21000
LOA21100
LOA21200
LOA21300
LOA21100
LOA21500
LOA21600
LOA21700
LOA21800
LOA21900
LOA22000
LOA22100
-177-
-------
c
c
c
FORWARD REDUCTION OF THE S MATRIX BY GAUSS ELIMINATION
SUBROUTINE SOLVECLN,MBAND,S,T,R,RI)
DOUBLE PRECISION S,T,C
COMMON/AT/ M, MM, NUMNP, NSIZE
DIMENSION S(LN,MBAND),T(LN,MBAND) ,R(LN) ,RI(LN)
C GAUSS ELIMINATION—FORWARD REDUCTION OF STIFNESS MATRIX
DO 550 N=1,NSIZE
DO 530 L=2,MBAND
IF(S(N,L).EQ.O) GO TO 530
I=N+L-1
C=S(N,L)/S(N,1)
J=0
DO 510 K=L,MBAND
510
530
550
S(I,J)=S(I,J)-C«S(N,K)
S(N,L)=C t
CONTINUE
CONTINUE
RETURN
C
C
C
C
C
C
620
630
650
660
C
BACK SOLUTION OF THE R VECTOR BY GAUSS ELIMINATION
ENTRY BACK
FORWARD REDUCTION OF THE LOAD MATRIX IS PREFORMED AND BACK
SUBSTITUTION IS PREFORMED TO SOLVE FOR THE UNKNOWN MATRIX
DO 630 N=1.NSIZE
DO 620 L=2,MBAND
IF(S(N,L).EQ.O) GO TO 620
I=N+L-1
R(I)=R(I)-S(N,L)*R(N)
CONTINUE
R(N)=R(N)/S(N,1)
DO 660 M=2,NSIZE
N=NSIZE+1-M
DO 650 L=2,MBAND
IF(S(N,L).EQ.O) GO TO 650
K=N+L-1
R(N)=R(N)-S(N,L)»R(K)
CONTINUE
CONTINUE
RETURN
C MULTIPLICATION OF THE T MATRIX BY THE NODAL VALUES AT THE LAST TIME
C
C
ENTRY MULTI
C
C MATRIX MULTIPLICATION LOOP—CRANK-NICOLSON METHOD
\*
DO 150 Hsl,NSIZE
R(N)=0.0
DO 110 M:1,MBAND
NO=H+N-1
IF(NO.GT.NSIZE) GO TO 120
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
(SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
•SOL
SSOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
SOL
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
2500
2600
2700
2800
2900
3000
3100
3200
3300
3400
3500
3600
3700
3800
3900
4000
4100
4200
4300
4400
4500
4600
4700
4800
4900
5000
5100
5200
5300
5400
5500
5600
5700
5800
5900
6000
6100
6200
6300
-178-
-------
110 R(N>=RI(NO)*TCN,M)+R\N> SOL 6400
120 CONTINUE SOL 6500
MBA=MBAND-1 SOL 6600
DO 130 M=1,MBA SOL 6700
NO=N-M SOL 6800
IF(NO.LE.O) GO TO 140 SOL 6900
130 R(N)=RI(NO)*T(NO,M+1)+R(N) SOL 7000
140 CONTINUE SOL 7100
150 CONTINUE SOL 7200
C SOL 7300
RETURN SOL 7400
END SOL 7500
-179-
-------
£*««*«*****•*•*«*•»«*»••«••«•«*«•«*»«••«»»*•»*«»»«*«*«»*»***»•« »«««»» »««ASQ -JQQ
C ASO 200
SUBROUTINE ASOLVE(B,R,RI,NEQ,IHALFB,NDIM,MDIM,T) ASO 300
C ASO 400
C ASSYMMETRIC BAND MATRIX EQUATION SOLVER ASO 500
C ORIGINALLY PROGRAMED BY J.O. DUGUID ASO 600
C ASO 700
DOUBLE PRECISION B,T,PIVOT,C ASO 800
DIMENSION B(NDIM,MDIM),R(NDIM),T(HDIM,MDIM) ASO 900
DIMENSION RI(NDIM) . ASO TOGO
NRS=NEQ-1 ASO 1100
IHBP=IHALFB+1 ASO 1200
C , ASO 1300
C TRIANGULARIZE MATRIX B USING DOOLITTLE METHOD ASO 1400
C ASO 1500
DO 20 K=1,NRS ASO 1600
PIVOT=B(K,IHBP) ASO 1700
KK=K+1 ASO 1800
KC=IHBP ASO 1900
DO 10 I=KK,NEQ ASO 2000
KC=KC-1 ASO 2100
IF(KC.LE.O) GO TO 20 ASO 2200
C=-B(I,KC)/PIVOT ASO 2300
B(I,KC)=C ASO 2400
KI=KC-f1 ASO 2500
LIMsKC+IHALFB ASO 2600
DO 10 J=KI,LIM ASO 2700
JC=IHBP+J-KC ASO 2800
10 B(I,J) = BU,J) + C»B(K,JC) ASO 2900
20 CONTINUE ASO 3000
RETURN ASO 3100
C ASO 3200
^•••••••••••••••••••••••••••••••«t»*»»«*****>«*»«»»*»«•«»«••«**««**•«**»ASO 3300
C ASO 3400
ENTRY ABACK ASO 3500
C ASO 3600
NN=NEQ+1 ASO 3700
IBANDs2«IHALFB+1 ASO 3800
DO 70 1=2,NEQ ASO 3900
JC=IHBP-I+1 ASO 4000
JI=1 ASO 4100
IF(JC.LE.O) GO TO 40 ASO 4200
GO TO 50 ASO 4300
40 JC=1 ASO 4400
JI=I-IHBP+1 ASO 4500
50 SUM=0.0 ASO 4600
DO 60 J=JC,IHALFB ASO 4700
SUM=SUM+B(I,J)«R(JI) ASO 4800
60 JI=JI+1 ASO 4900
70 R(I)=R(I)+SUH ASO 5000
C ASO 5100
C BACK SUBSTITUTION ASO 5200
C ASO 5300
R(NEQ)=R
-------
80 SUM=SUM+B(I,J)*R(JP) ASO 6300
90 R(I)=(R(I)-SUM)/B(I,IHBP) ASO 6400
100 RETURN ASO 6500
C ASO 6600
c»««*»«»»*»**********»»»****»***»»*»*»»*******»*»*****»****«»*»»»»»*«**«ASO 6700
C ASO 6800
ENTRY AMULTI ASO 6900
C ASO 7000
LN=NDIM ASO 7100
MBAND=(MDIM+1)/2 ASO 7200
DO 150 N=l,LN ASO 7300
R(N)=0.0 ASO 7400
DO 110 M=1,MDIM ASO 7500
NO=M+N-MBAND ASO 7600
IF(NO.GT.LN) GO TO 120 ASO 7700
IF(NO.LT.I) GO TO 110 ASO 7800
R(N)=RI(NO)*T(N,M)+R(N) ASO 7900
110 CONTINUE ASO 8000
120 CONTINUE ASO 8100
150 CONTINUE ASO 8200
RETURN ASO 8300
END ASO 8400
-181-
-------
£«»«»•*«»«•»«•*««t*«*««««»««*»•«•»««»«««««» *»«*»»»»«**••«««ft«»»ft»»»««ft«BAL
C THIS SUBROUTINE COMPUTES A MASS BALANCE BAL
C BAL
SUBROUTINE BALAN(LM,LN,R,PX.PY.PCX.HEAD,HEAT,Z,LSOURC,ASOURC, BAL
• LFLUX,NFLUX,A,B,C,D,E,MPRINT,NTYPE,RI,NODE,NOD,FLOWX,FLOWY,PCW BAL
» ,MEQ,CBAL,CBALA,AY.AZ,NCON,KRAN) BAL
REAL JAC BAL
INTEGER Z.AY.AZ BAL
DOUBLE PRECISION AA.NXI,NET,NOT BAL
COMMON/AAA/XL(4),YL(4),DETJAC,JAC(4.4)/AB/AA(6)-W(4) BAL
COMMON/CON/M1,M2,NCONV/AA/NXI(12),NET(12),NOT(12) BAL
COHMON/ACC/KK(12)>K1,K2,K3,K4/HH/LFLOW,LON/BAL/BLINE BAL
DIMENSION RI(LN),R(LN),PX(LM),PY(LM),NODE(LM) BAL
DIMENSION FLOHX(LM),FLOWY(LM),AY(4),AZ(4) BAL
DIMENSION NCON(Z,2),CBAL(Z,2) , NOD(12,LM) BAL
DIMENSION PCX(LM),HEAD(LN),HEAT(LM),ASOURC(Z),NFLUX(Z,2) BAL
DIMENSION LOA(4),LOB{4),LOC(4),LOD(4) BAL
DATA LOA/1,2,3,4/LOB/2,3,4,1/LOC/5,7.9,11/LOD/6 8.10,12/ BAL
RETURN BAL
£«•»**•»*«»»«•««•»*•«*»»**••«**»*»***«*•**«
ENTRY MASBAL(ALPHA)
C
C COMPUTE BOUNDARY FLUXES
C
ST=0.0
PUMP=0.0
F=0.0
LFLOW=0
RECHsO.O
DISCH=0.0
IF(LFLUX.EO.O).GO TO 15
AA(1)=0,0
AA(2)=0.0
C
DO 10 LLA=1,LFLUX
LLL =NFLUX(LLA,1)
K=NFLUX(LLA,2)
C
CALL FLOWW(LLL)
C
LT=NFLUX(LLA,2)
FLUXX=0.0
FLUXY=0.0
IFCLT.GT.2) FLUXX=DHX»ARX«PX(LLL)
IF(LT.LT-3) FLUXY=DHY*ARY»PY(LLL)
IFCMEQ.EQ.1} GO TO 8
LY=AY(LT)
LZ=AZ(LT)
LY=NOD(LY,LLL)
LZ=NOD(LZ,LLL)
RR=(R(LY)+R(LZ))/2
IFCLT.GT.2) FLUXX=FLUXX«-FLOWX(LLL)»PCW»RR
IF(LT.LT.3) FLUXY=FLUXY+FLOWY(LLL)«PCW»RR
8 CONTINUE
IFCLT.EQ.1.AND.FLUXY.LT.O)
IF(LT.EQ.1.AND.FLUXY.GT.O)
IFCLT.EQ.2.AND.FLUXY.LT.O)
IFUT.EQ.2.AND.FLUXY GT.O)
IFCLT.EQ.3-AND.FLUXX.LT.O)
IF(LT.EQ.3-AND.FLUXX.GT.O)
IF(LT.EQ.4.AND.FLUXX.LT.O)
IFCLT.EQ.4.AND.FLUXX.GT.O)
10 CONTINUE
***«»»*«**««*»««*«*««»««t»»*BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
DISCH=DISCH-FLUXY
RECH=RECH+FLUXY
RECH=RECH-FLUXY
DISCH=DISCH+FLUXY
RECH=RECH-FLUXX
DISCH=DISCH+FLUXX
DISCH=DISCH-FLUXX
RECH=RECH+FLUXX
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
BAL
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
2500
2600
2700
2800
2900
3000
3100
3200
3300
3400
3500
3600
3700
3800
3900
4000
4100
4200
4300
4400
4500
4600
4700
4800
4900
5000
5100
5200
5300
5400
5500
5600
5700
5800
5900
6000
6100
6200
6300
-182-
-------
J5 CONTINUE BAL
C*»******»*»***»*»*. **««»««*,,»«,,«««,„,», **««««„ ««,»««««««««*, »«»,,»BAL 6600
C COMPUTE RATE OF CHANGE IN STORAGE BAL 6700
n 6900
=0 .0
DO 20 LLL=1,LM
M4=H
CALL DERIVECLLL.M4) 0AL 7300
IF(MEQ.EQ.I) CALL PEECT.LLL) BAL 7400
IF(MEQ-EQ.I) DETJAC=DETJAC/T BAL 7500
j|B=0 BAL 7600
£A=0 BAL 7700
S. 9 1.1.4
L3=LOC(I) BAL 3200
W=LOD(I) BAL 8300
L5=MOD(L1,LLL) BAL 8400
L6=NOD(L2,LLL) BAL 8500
L7=NOD(L3,LLL) BAL 8600
L8=NOD(L4,LLL) BAL 8700
IF(LT.GT.O) GO TO 23 BAL 8800
RA=RA+.5«R(L5)+.5*R(L6) BAL 8900
RB=RB+.5*RI(L5)+.5*RI(L6) BAL 9000
GO TO 19 BAL 9100
23 IF(LS.GT.O) GO TO 24 BAL 9200
RA=RA+1/3»R(L5)+2/3*R(L7) + 1/3»RU6) BAL 9300
RB=RB+1/3*RI(L5) + 1/3*RIU6)+2/3»RI(L7) BAL 9400
GO TO 19 BAL 9500
24 RA=RAt-1/8«R(L5) + 1/8*R(L6)+3/8»R(L7)+3/8*R(L8) BAL 9600
RBsRB+1/8»RI(L5)+1/8»RI(L6)+3/8»RI(L7)+3/8»RI(L8) BAL 9700
19 CONTINUE BAL 9800
RA=RA/4 BAL 9900
RB=RB/4 BAL 10000
ST=ST+(RA-RB)»PCX(LLL)«U«DETJAC BAL10100
20 CONTINUE BAL10200
21 CONTINUE BAL10300
C BAL 10 400
£*««**«««*****»***«*«****««**«*»*»*»»«***»* *»*»*»»»»*»*»*»*•»****«*«* »*BAL 10500
C ADD IN PUMPING SOURCES BAL10600
C BAL10700
PUMP=0.0 BAL10800
PUMP=PUMP+BLINE BAL10900
IF(LSOURC.EQ.O) GO TO 31 BAL11000
DO 30 LLL=1,LSOURC BAL11100
PUMP=PUMP+ASOURC(LLL) BAL11200
30 CONTINUE BAL11300
31 CONTINUE BAL11400
C DISTRIBUTED SOURCES BAL11500
DO 40 LLL=1,LM BAL11600
IF(HEATULL).EQ.O) GO TO 40 BAL11700
M4=NODE(LLL) BAL1 1800
CALL DERIVE(LLL,H4) BAL11900
PUMP=PUMP+HEAT(LLL)«4«DETJAC BAL12000
40 CONTINUE BAL12100
C BAL12200
C *t ««««»»««•»**««««»*««»«»*«*•»»»*««»*»»««»««••«««»««»*«*«»»«* BALI 2300
C BAL12400
-183-
-------
c
c
43
C
C
15
48
ROUTINES TO COMPUTE FLUXES ACROSS CONVECTIVE BOUNDARIES
IF(MEQ.EQ.I) GO TO 48
IF(NCONV.EQ.O) GO TO 45
C0=0.0
COA=0.0
DO 43 I=1,NCONV
LLL=NCON(I,1)
LA=NCON(I,2)
MC1=AY(LA)
MC2=AZ(LA)
MA=NOD(HC1,LLL)
NA=NOD(MC2,LLL)
NC=CBAL(I,2)«100
IF(NC.EQ.O) COA=COA+(R(MA)/4+R(NA)/4+RI(MA)/4+RI(NA)/4)«CBAL(I,
IF(NC.EQ.O) GO TO 43
CO=CO+CBAL(I,2)-(R(MA)/4+R(NA)/4+RI(MA)/4+RI(NA)/4)*CBAL(I,l)
CONTINUE
CONTINUE
CONTINUE
C»»««»»»«»»*»»«*»»»»»*»»»»«»»*»•«»««»»»«*««««*»»«
C
c
AL = ALPHA
IFULPHA.LT.0.0001) AL=1
RECH=RECH«AL
DISCH=DISCH»AL
PUMP=PUMP»AL
CBALA=CBALA*AL
£•»»*•««•t»tt*f*•»•»**•*•*t*If«f»«*«»*•tt*»t»«t*fft«**»!«»*»* t
A=A+RECH
B=B+DISCH
C=C+ST
D=D+PUMP
0=0* CO
00=00+COA
P=P+CBALA
IF(MEQ .EQ . 1) F=RECH-DISCH-«-PUMP-ST
IF(MEQ.EQ.2) F=RECH-DISCH+PUMP-ST+CBALA+CO-COA
E=E+F
RETURN
ENTRY BPRINT
PRINT 50
IFCMEQ.EQ.2) PRINT 51,0,CO,00,COA,P.CBALA
PRINT 52,A,RECH,B,DISCH,C,ST,D,PUMP,E,F
50 FORMATCIX,T30,'CUMULATIVE MASS BLANCE1.
•T60,'RATES FOR THIS TIME STEP')
51 FORMATdX,'CONDUCTIVE TRANSFER',
« ,T30,G12.6,T60,G12.6,
• 1X/1X,'CONVECTIVE TRANSFER—OUT '.T30.G12.6,T60,G12.6
«,1X/1X,'CONVECTIVE TRANSFER—IN',T30,G12.6,T60,G12.6)
52 FORMAT(
»1Xf'B. FLUX RECHARGE',T30.G12.6,T60, G12.6.
«/1X,'B. FLUX DISCHARGE',T30,G12.6,T60.G12.6,
•/«, 'CHANGE IN STORAGE',T30,G12.6,T60,G12.6,
»/U,'QUANTITY PUMPED'.T30.G12 .6 .T60 ,G12.6 ,
•/U.'DIFFERECE • ,T30,G12.6,T60,G12.6)
RETURN
BALI 2500
BALI 2600
BAL12700
BAL12800
BAL12900
BALI 3000
BALI 3100
BALI 3200
BALI 3300
BAL13400
BALI 3500
BAL13600
BALI 3700
BALI 3800
DBAL13900
BAL14000
BAL14100
BAL14200
BAL14300
BAL14400
BAL14500
BALI 4600
BAL14700
BAL14800
BAL14900
BAL15000
BAL15100
BAL15200
BALI 5300
BAL15400
BAL15500
BAL15600
BAL15700
BAL15800
BAL15900
BALI6000
BAL16100
BALI6200
BAL16300
BAL16400
BAL16500
BALI6600
BAL16700
BAL16800
BAL16900
BAL17000
BAL17100
BAL17200
BAL17300
BAL17400
BAL17500
BAL17600
BAL17700
BAL17800
BAL17900
BAL18000
BAL18100
BAL18200
BAL18300
BAL18400
BAL18500
-184-
-------
C«f*««*»««*****»»**«»«*««***«*«»«»*«»*«»»
c
ENTRY WATER
AA(1)=0.0
AA(2)=0.0
LFLOW=0
DO 100 LLL=1,LM
CALL FLOWW(LLL)
**««*«*«*»*****»»«*»«*
C
C
C
FLOW EQUALS THE SLOPE * AREA * PERMEABILITY
FLOWX(LLL)=DHX*ARX*PX(LLL)
FLOWY(LLL)=DHY*ARY*PY(LLL)
C
100 CONTINUE
C «*««»*****»*********«**«*****«****»»***«»*««»*«»»»**»«****»****»
RETURN
C
C
C
c
ENTRY VELOC(LN,LM,RW,WX WY)
DIMENSION WX(LM),WY(LM),RW(LN)
RETURN
100
500
C
C
DO 500 1=1, Ml
K=K+1
IF(NOD(K,LLLQ) .EQ.O) GO TO 400
KI=NOD(K,LLLQ)
DHE=DHE+NXI(I)«RW(KI)
DHN=DHN+NET(I)«RW(KI)
CONTINUE
DHX=JAC( 1 , 1)*DHE+JAC( 1 ,2)*DHN
DHY=JAC(2,1)«DHE+JAC(2,2)«DHN
VX=DHX»WX(LLLQ)
VY=DHY«WY(LLLQ)
RETURN
ENTRY VCENTCLLLQ.VXQ,VYA)
THE AREA TO BE USED IN THE FORMULA Q=KIA IS COMPUTED
ARX=(YL(3) + YLC4)-YL(1)-YL(2))/2
ARX=ABS(ARX)
ARY=(XL(2)+XL(3)-XL(1)-XL(4))
ARY=ABS(ARY)/2
VXQ=FLOWX(LLLQ)/ARX
VYA=FLOWY(LLLO)/ARY
RETURN
BAL18600
BAL18700
BAL18800
BAL18900
BAL19000
BAL19100
BAL19200
BAL19300
BAL19400
BAL19500
BAL19600
BAL19700
BALI 9800
BAL19900
BAL20000
BAL20100
BAL20200
BAL20300
BAL20400
BAL20500
BAL20600
BAL20700
BAL20800
C
C
ENTRY VELO(LLLQ,VX,VY) BAL21100
BAL21200
BAL21300
BAL21400
BAL21500
BAL21600
BAL21700
BAL21800
BAL21900
BAL22000
BAL22100
BAL22200
BAL22300
BAL22400
BAL22500
BAL22600
BAL22700
BAL22800
«**«**«**«***»»»*»**«*««***«««**«««*««»***«*****»*»*****«*******»*****BAL22900
BAL23000
BAL23100
BAL23200
BAL23300
BAL23400
BAL23500
BAL23600
BAL23700
BAL23800
BAL23900
BAL2MOOO
BAL24100
BAL21200
BAL24300
BAL24400
BAL24500
BAL21600
BAL21700
BAL24800
SUBROUTINE FLOWW(LLL)
DHE=0.0
DHN=0.0
M4=NODE(LLL)
-185-
-------
CALL DERIVEULL.MH)
K=0
DO 7 1=1 ,MH
7
C
c
c
IF(NOD(K,LLL).EQ.O) GO TO H
KI=NOD(K,LLL)
DHE=DHE+NXI(I)«R(KI)
DHN=DHN+NET(I)«R(KI)
CONTINUE
DHX= JAC( 1,1) »DHE+ J AC( 1,2) »DHN
DHY=JAC(2,1)«DHE+JAC(2,2)«DHN
THE AREA TO BE USED IN THE FORMULA Q=KIA IS COMPUTED
ARX=(YL(3)+YL(4)-YL(1)-YL(2))/2
ARX=ABS(ARX)
ARY=(XL( 2)+XL{ 3)-XL( 1 )-XL( «) )
ARY=ABS(ARY)/2
ARXX=ARX
ARX=DETJAC»i»/ARY
ARY=DETJAC«J»/ARXX
t
RETURN
BAL24900
BAL2500C
BAL2510C
BAL2520C
BAL2530C
BAL25'40C
BAL2550C
BAL2560C
BAL2570C
BAL2580C
BAL2590C
BAL2600d
BAL26100
BAL26200
BAL26300
END
BAL26500
BAL26600
BAL26700
BAL26800
BAL26900
BAL27000
BAL27100
BAL27200
BAL27300
-186-
-------
C THIS ROUTINE IS USED TO PRINT FLOWS AND FLUXES FLO 100
f*
SUBROUTINE FLOWS(LM,LN ,R,R1,FLOWX.FLOWY) FLO 300
DIMENSION R(LN),RULN),FLOWX(LM),FLOWY(LM) FLO 400
COMMON/HI/TITLE(25),V(26),VV(26) FLO 500
C*«»*««**««ft*«»««**««ft**»««»««»»»t««««ft««*»ft«* pLQ ^QQ
RETURN FLO 800
ENTRY FFLOW FLO 900
VRITEdU.V) (R1(I),I=1,LN) FLO 1000
RETURN FLO HQO
ENTRY FFFLOW FLO 1200
HHITEdl.V) (R(I),I=1,LN) FLO 1300
RETURN FLO 1400
C FLO 1500
ENTRY WFLOW(S) FLO 1600
PRINT 1,S FLO 1700
1 FORMAT(1X//1X,'TIME STEP ',G12.6/1X,'FLOWS IN THE X DIRECTION1) FLO 1800
PRINT VV,(FLOWX(I),I=1,LM) FLO 1900
PRINT 2 FLO 2000
2 FORMATCJX/1X,'FLOWS IN THE Y DIRECTION') FLO 2100
PRINT VV,(FLOWY(I),1=1,LM) FLO 2200
RETURN FLO 2300
C FLO 2400
C***«*«»****«««**f**x*ft««***«x»**»*ftft«««»***x> FLO 2500
C FLO 2600
ENTRY HPRINT(S) FLO 2700
PRINT 10,S FLO 2800
10 FORMATdX//1X, 'TEMPERATURE DISTRIBUTION AT TIME STEP ',G12.6) FLO 2900
PRINT V,(R(I) ,I=1,LN) FLO 3000
RETURN FLO 3100
C FLO 3200
£t»ft«*«ft«»»*««>«««*»*>»««*«c***«*««»»»ft«««»««««*« FLO 3300
C FLO 3400
ENTRY WPRINT(S) FLO 3500
PRINT 100,S FLO 3600
100 FORMAT(1X//1X,'POTENTIAL DISTRIBUTION AT TIME STEP '.G12.6) FLO 3700
PRINT V,(R1(I),1=1,LN) FLO 3800
RETURN FLO 3900
END
-187-
-------
£**•**•••******•*•«•«***•*****«*«***»••«*•*»•***••**•*•••»***•••*«»**•»• PAP
SUBROUTINE PARAM(LM,LN,BBB,XLOC, YLOC.NOD.PCX ,DIFF ,R ,R1 ,WX,WY, PAR
C PAR
C THIS SUBROUTINE IS USED TO COMPUTE THE DISPERSION COEFFICEIENTS, PAR
C THE LOCALIZED COORDINATES, AND CHANGES IN HYDRAULIC PAR
C CONDUCTIVITY WITH TEMPERATURE PAR
C PAR
^•••••••••••••••••«»»>*«»«««»i •>»>•* «•»**>«»«»»»*««*« i ««*««***»»«**«««• PAR
• A, B, EOT, K AHEAD ' PAR
INTEGER BBB PAR
DIMENSION XLOC(LN),YLOC(LN),NOD(12,LM>,PCX(LM),DIFF(LM,2),R(LN) PAR
*, RHLN),HX(LM),WY(LM),A(LM)tB(LM),BOT(BBB) PAR
REAL JAC PAR
COMMON/AAA/XL(4),YL(4),DETJAC,JAC(4,4)/ACC/KK(12) ,K1 ,K2,K3,K4 PAR
»/HM/PXX,PYY,PXY,KAD/ AM/NLA, PCH PAR
C PAR
RETURN - PAR
C PAR
£««*••«••*«**«»»»*•«««««»*»««»*«•«*««*«*»•*«•*******»»«**
C THE ROUTINE THAT CALCULATES THE DISPERSION COEFFICIENTS
ENTRY MECD(LLL,VX,VY)
V=VX»VX+VY«VY
IF(V.LE.O) RETURN
V=SQRT(V)
DL=DIFF(LLL,1)»V
DT=DIFF(LLL,2)«V
PXX=(DL«VX*VX/V/V+DT«VY«VY/V/V)«PCW
PYY=(DT»VX«VX/V/V+DL»VY*VY/V/V)»PCW
PXY=«DL-DT)»VX»VY/V/V)*PCW
PXY=ABS(PXY)
RETURN
C
£••*»•••••**•*••••*••**»*•••*»*«•••»•*••»•«•**»****»*»*»*
C
ENTRY CORD(LLLQ)
COMMON/AF/II,JJ,L1(4)
C THIS ROUTINE COMPUTES LOCALIZED COORDINATES
K1=NOD(1,LLLQ)
K2=NOD(2,LLLQ)
K3=NOD(3,LLLQ)
K4=NOD(4,LLLQ)
AQrXLOC(KI)
BB=XLOC(K2)
C=XLOC(K3)
D=XLOC(K4)
E=AMIN1(AQ,BB,C,D)
XL(1)=AQ-E
XL(2)=BB-E
XL(3)=C-E
XL(4)=D-E
AQ=YLOC(K1)
BB=YLOC(K2)
C=YLOC(K3>
D=YLOC(K4)
E=AHIN1(AO,BB,C,D)
YL(1)=AQ-E
YL(2)=BB-E
YL(3)=C-E
YL(H)=D-E
C
C
RETURN
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
PAR
-JQQ
200
300
400
500
600
700
goo
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
2500
2600
2700
2800
2900
3000
3100
3200
3300
3400
3500
3600
3700
3800
3900
4000
4100
4200
4300
4400
4500
4600
4700
4800
4900
5000
5100
5200
5300
5400
5500
5600
5700
5800
5900
6000
6100
6200
6300
-188-
-------
C
C«»t****t*«**t*ft«**«*«***«****«*«««***xft***x«*«**««*«c«***ft«x»«*
C
ENTRY PE(MS)
C
IF(MS.EQ.O) RETURN
DO 1 J=1 ,LM
N1=NOD(1,J)
N2=NOD(2,J)
N3=NOD(3,J)
N4=NOD(4,J)
T=(R(N1) + R
c
C THE VISCOSITY RELATIONSHIP
c
U=1.917-.05635*Ti-.00071*T*T
WX(J)=1.21/U«A(J)
WY(J)=1.2VU*B(J)
1 CONTINUE
RETURN
ENTRY PEE(BBQQ.JQQ)
IF(KAREAL.EQ.O) RETURN
N1=NOD(1,JQQ)
N2=NOD(2,JQQ)
N3=NOD(3,JQQ
N4-NOD(U JOO)
C GEOMETRIC MEANS ARE CALCULATED FOR AQUIFER DEPTH
BBQQ=(R1(Sl)-BOT(N1))«(R1(N2)-BOT(N2))»(RHN3)-BOT(N3))«(R1(NM)
»OT(NU))
BBQQ=ABS(BBQQ>
BBQQ=BBQQ«*0.25
RETURN
tnu
PAR 6400
PAR 6500
PAR 6600
PAR 6700
PAR 6800
PAR 6900
PAR 7000
PAR 7100
PAR 7200
PAR 7300
PAR 7400
PAR 7500
PAR 7600
PAR 7700
PAR 7800
PAR 7900
PAR 8000
PAR 8100
PAR 8200
PAR 8300
pAR 8400
PAR 8500
PAR 8600
PAR 9100
BPAR 9200
pAR
-189-
-------
c«««»*»««««»«*««»««»»«»*«t•«»§«»«**«§«»»««»»«»««*«*«»»f«»»»»»,»,«»»,
C THIS ROUTINE IS USED TO CHANGE BOUNDARY CONDITIONS OR PARAMETERS
C AT EACH TIME STEP FIVE ENTRY POINTS ARE PROVIDED
C
SUBROUTINE BOUNDA(LM,LN,Z,R,R1.HEAD.HEA.FLOWX,FLOWY,HEAT.INFLOW
»,NWATER,AWATER,NHEAT,AHEAT,NCON,TINF,CONV,ALOC,NEL,AEL,CBAL
»,LEL,LHEAT,LWATER,PCW,AY,AZ,LINEW,LINEH,NLINEW,NLINEH,
« ALINEW.ALINEH)
INTEGER Z,AY(4),AZ(4)
REAL INFLOW,N11
COMMON/ATHICK/ASIZE.NTHICK,THICK,ERROR/BM/RYC4800,3)
*/BOUND/DIST,N1,N2,N3,N4,N5,N6,N7,N8,N9,N10
*/CONT/LA,LB,LC,LD,LE,LF,LG,LH/CON/MC1,MC2,NCONV/ME/MEQ
C—LINKING INFORMATION
DIMENSION FLOWX(LM),FLOWY(LM)
C--INFORMATION FOR POINT SOURCES
DIMENSION NWATER(Z),AWATER(Z),NHEAT(Z),AHEAT(Z)
C RECHARGE RATE INFORMATION
«, HEAT(LM),INFLOW(LM),NLINEW(Z>2),NLINEH(Z,2),ALINEW(Z),ALINEH(Z)
C—INFORMATION ON CONVECTIVE BOUNDARIES
• ,NCON(Z,3),TINF(Z),CONV(Z),ALOC(Z),NEL(Z,2),AEL(Z,2),CBAL(Z,2)
C—BOUNDARY CONDITIONS ^L,*I
DIMENSION HEAD(LN),HEA(LN)
C—INITIAL CONDITIONS AND ANSWERS
DIMENSION R(LN).RKLN)
110
111
RETURN
ENTRY LAKE
READ, DIST,N1,N2,N3,N4,N5,N6,N7,N8,N9,N10,N11
READ,M,NUMBER
DO 110 K=1,M
READ.A.B.C
HM=NUMBER-M
DO 111 K=1,MM
READ,RY(K,3),RY(K,1),RY(K,2)
RETURN
*»»
100
C
C'
C
C
59
ENTRY BVALUSQ,ALPHA)
K=KSQ*ALPHA
T1=RY(K,1)-5.0
DO 100 J=1,LEL
AEL(J,2)=T1
RETURN
EMTRY BOUND(KS)
K=KS«N10+8
TlsRY(K.I)
T2=RY(K,2)
AA=ALOG(T1)
AB=(ALOG(T2)-AA)/54
T3=AA+DIST«AB
T3=EXP(T3)
DO 59 J=N1,N2,N3
HEAD(J)=T3
DO 60 J=N1,N5,N6
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
BOU
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
2500
2600
2700
2800
2900
3000
3100
3200
3300
3400
3500
3600
3700
3800
3900
4000
4100
4200
4300
4400
4500
4600
4700
4800
4900
5000
5100
5200
5300
5400
5500
5600
5700
5800
5900
6000
6100
6200
6300
6400
-190-
-------
60 HEAD(J)=T3-2.5 BOU 6500
DO 200 J=1,LEL BOU 6600
TINF(J)=T3 BOU 6700
200 AEL(J,2)=T3 BOU 6800
IF(NCONV.LT.I) GO TO 300 BOU 6900
LEA=LEL+1 BOU 7000
DO 500 J=LEA,NCONV BOU 7100
TINF(J)=RY(K-2,1)-N11 BOU 7200
IF(RY(K-2,3).LT.50) TINF(J)=RY(K-2,3)-N11 BOU 7300
500 CONTINUE BOU 7400
PRINT 555, (NCON(J,1),TINF(J),J=1,NCONV) BOU 7500
555 FORMATCIX.'NODE # AND TEMP AT INF.',8(2X,13,2X,F5.2)) BOU 7600
GO TO 400 BOU 7700
300 CONTINUE BOU 7800
DO 700 J=N7,N8,N9 BOU 7900
700 HEAD(J)=T3-N11 BOU 8000
400 CONTINUE BOU 8100
RETURN BOU 8200
C BOU 8300
£**«***«****»****«**«*«»****««»**«*********»*««*«*«* BOU 8400
C BOU 8500
C BOU 8600
ENTRY CHANG BOU 8700
C BOU 8800
RR=R(130) + R(13D + R( 145)+R(146) BOU 8900
RR=RR/4 BOU 9000
DO 10 J=1,LHEAT BOU 9100
N=NHEAT(J) BOU 9200
10 AHEAT(J)=R(N)»PCW*AWATER(J) BOU 9300
RETURN BOU 9400
c BOU 9500
ENTRY CHAN BOU 9600
RETURN BOU 9700
END BOU 9800
-191-
-------
SUBROUTINE EIGEN(LN,MBAND.MP,KSYM,CFACT,S,T-RI,HEAD) EIG 100
DOUBLE PRECISION S,T EIG 200
DIMENSION HEAD(LN), RI(LN), SUN MBAND) ,T(LN,MBAND) EIG 300
COMMON/HI/TITLE(25),V(26),VV(26) EIG 400
PRINT 10 EIG 500
COMMON/HH/LFLOW.LON EIG 600
C EIG 700
C=1000 EIG 800
DO 1 J=1,LN EIG 900
A=0 EIG 1000
IF(KSYM.GT.O) GO TO 6 EIG 1100
IF(HEAD(J).EQ 0) A=S(J,1) EIG 1200
DO 2 K=2,MBAND 'EIG 1300
IF(HEADU).NE.O) GO TO 1 EIG 1400
E=S(J,K) EIG 1500
2 A=ABS(E)+A EIG 1600
MBA=MBAND-1 EIG 1700
DO 4 K=1,MBA EIG 1800
NO=J-K EIG 1900
IF(NO.LE.O) GO TO 5 EIG 2000
E=T(NO K+1) EIG 2100
4 A=ABS(E)+A EIG 2200
5 CONTINUE EIG 2300
B=T(J,1)».5/A EIG 2100
GO TO 8 EIG 2500
6 DO 7 K=1,MP EIG 2600
IF(S(J,K).GT.CFACT) GO TO 1 EIG 2700
E=S(J,K) EIG 2800
E=ABS(E) EIG 2900
IFU.EQ.MBAND) E=S(J,K) EIG 3000
7 A=E+A EIG 3100
B=T(J,MBAND)»0.5/A EIG 3200
8 CONTINUE EIG 3300
RI(J) = B EIG 3*»00
IF(B.LT.C) LQ=J EIG 3500
C=AMIN1(B,C) EIG 3600
1 CONTINUE EIG 3700
PRINT V,(RI(J),J=1,LN) EIG 3800
PRINT 11.C.LQ EIG 3900
STOP EIG 4000
10 FORMAT(1X/1X,'STABILITY OF SOLUTION USING CRANK NICOLSON METHOD' EIG 4100
«,4X/3X,'MAXIMUM TIME STEP FOR EACH NODE')/ EIG 4200
11 FORMAT(1X/1X,'THE CRITICAL TIME STEP IS '.G12.6,' EIG 4300
•CONSTRAINED AT NODE',15) EIG 4400
END EIG 4500
-192-
-------
c
c
c
c
SUBROUTINE THAT ALLOWS FOR A MOVING BOUNDARY IN A
SUBROUTINE ADJUST(NJUST.LM.LN,Z,HEAD,XLOC,YLOC,
I
DIMENSION NMOV(40,2), FLOWX(LM),FLOWY(LM),A(2),
DIMENSION BMOV(40,6),HEAD(LN),XLOC(LN),YLOC(LN)
READ,NSTEP.NPRINT,NPR,ERROR,FACTOR
READ , (NMOV ( J , 1) , NMOV ( J , 2), BMOV ( J , 3) , BMOV ( J , 6 ) ,
KLT=0
KSS=1
RETURN
ENTRY ADJUS(«,«)
IFCKSS.EQ.2) GO TO 5
DO 4 J=1 .NJUST
LL=NMOV(J,1)
LL=ABS(LL)
IF(LL.NE.O) GO TO 2
LL=NMOV(J,2)
LL=ABS(LL)
L=NODE(4,LL)
GO TO 3
2 CONTINUE
L=NODE(3,LL)
3 CONTINUE
BMOV(J,5)=YLOC(L)
4 CONTINUE
5 CONTINUE
KLT=KLT+1
READ,BKS,TIME,FACT
DO 6 J=1,NJUST
6 BMOV(J,4)=BMOV(J,6)*FACT
DO 8 J=1,NJUST
8 BMOV(J,4)=BMOV(J,4)/FACTOR
SLENG=0.00001
FLOW=0
DO 40 J=1,NJUST
DO 10 K=1,2
NS=1
LLL=NMOV(J,K)
IF(LLL.LT.O) NS=2
LLL=ABS(LLL)
IF(LLL.EQ.O) GO TO 10
CALL CORD(LLL)
CALL VCENTULL.VX.VY)
LA=NODE(3,LLL)
LB=NODE(4,LLL)
AA=(XLOC(LA)-XLOC(LB))
A(K)=ABS(AA)
B(K)=VY
10 CONTINUE
LC=NMOV(J,1)
LC=ABS(LC)
IF(LC.EQ.O) A(1)=A(2)
IFUC.EQ.O) B(1) = B(2)
IF(LLL.EQ.O) A(2)=A(1)
IF(LLL.EQ.O) B(2)=B(1)
IFULL.EQ.O) LB=NODE(3,LC)
VERTICAL CROSS SECADJ 100
ADJ 200
ADJ 300
ADJ 400
NODE,FLOWX.FLOWY.RADJ 500
ADJ 600
B(2),R(LN) ADJ 700
,NODE(12,LM) ADJ 800
ADJ 900
J=1,NJUST) ADJ 1000
ADJ 1100
ADJ 1200
ADJ 1300
ADJ 1400
ADJ 1500
ADJ 1600
ADJ 1700
ADJ 1800
ADJ 1900
ADJ 2000
ADJ 2100
ADJ 2200
ADJ 2300
ADJ 2400
ADJ 2500
ADJ 2600
ADJ 2700
ADJ 2800
ADJ 2900
ADJ 3000
ADJ 3100
ADJ 3200
ADJ 3300
ADJ 3400
ADJ 3500
ADJ 3600
ADJ 3700
ADJ 3800
ADJ 3900
ADJ 4000
ADJ 4100
ADJ 4200
ADJ 4300
ADJ 4400
ADJ 4500
ADJ 4600
ADJ 4700
ADJ 4800
ADJ 4900
ADJ 5000
ADJ 5100
ADJ 5200
ADJ 5300
ADJ 5400
ADJ 5500
ADJ 5600
ADJ 5700
ADJ 5800
ADJ 5900
ADJ 6000
ADJ 6100
ADJ 6200
-193-
-------
IF(NS.EQ.2) GO TO 20
C=(B(1)*A(1)+B(2)«A(2))/(A(1)/2+A(2)/2)
GO TO 10 ADJ 660°
20 CONTINUE ADJ 67°°
FLOW=B(1)»A(1)/2+B(2)«A(2)/2+FLOW ^ %*™
SLENG=SLENG+A(1)/2+A(2)/2 A°, °?°9
40 CONTINUE ™ 7000
c ADJ 7200
C LAKE LEVEL LEVELLER
HE=FLOW/SLENG
DO 50 Jrl.NJUST
IF(NMOV(J, 2). GT. -0.0001) GO TO 50
LLL=NHOV(J,2)
LLL=ABS(LLL)
IF(LLL.EQ.O) GO TO 15 1
MD=NODEC«,LLL)
GO TO 1?
15 LLL=NMOV(J,1)
LLL=ABS(LLL)
HD=NODE(3,LLL)
17 CONTINUE
50 lME ADJ 890°
5U CONTINUE ADJ
C PRINTING ROUTINE
LP=KLT/NPRINT 9300
LQ=KLT/NPR
LP=LP»NPRINT
IF(KLT.EQ.LP) CALL WPRINT(BKS) AD,
IF(KLT.EQ.LQ) CALL WFLOW(BKS) Jn] qqOO
IF(KLT.EQ.LQ) CALL BPRINT ADJIOObo
IF(KLT.GT.NSTEP) STOP
J|P n ADJ 10300
S'TOJ.I.MJUST
IF(LL.HE.O) GO TO 60 ADJ '0800
- ) .UBUO
ADJ 10900
LL) ADJ11000
rn n c ADJ 11 100
GO TO 65 Ani
60 CONTINUE ADJ
L=NODE(3,LL) ADJ
65 CONTINUE ADJlisOO
AD=BMOV(J,5)-YLOC(L) ADJ^SOO
AD=ABS(AD) ADJ 11700
AC=AHAX1(AD,AC) ADI^Rnn
70 CONTINUE JSJii'SSS
IFCAC.GT. ERROR) KSS=1 ADJ12000
IF(KSS.EQ.2) RETURN 2 ADJ12'00
PRINT nO.BKS.KLT ADJ12200
110 FORHAT(1X/1X,'STUCTURE MATRIX RECOMPUTED AT TIME STEP '.G8.3, ADJ12300
" "SR*TION ''I5) ADJ 12400
^IURN 1 ADJ 12500
EMD ADJ 12600
-194-
-------
APPENDIX D
ESTIMATION OF AQUIFER PARAMETERS BY USING
SUBSURFACE TEMPERATURE DATA
Flowing ground water* distorts the normal distribution of subsurface
temperatures. Researchers interested in determining the geothermal heat flux
have long been aware of the distorting influence moving ground water can have
on the measured geothermal gradient (Kirge 1939). Only a few ground-water
researchers, however, have attempted to use subsurface temperature information
to calculate the rate and direction of ground-water flow.
Stallman (1960) aroused interest in the use of subsurface temperature as
an indirect manifestation of ground-water velocity with a presentation of a
differential equation describing heat transport in the subsurface and with the
suggestion that temperature measurements might be a useful means for
indirectly determining aquifer characteristics. An analytical solution was
developed by Stallman (1965) for determining ground-water velocities in a
homogeneous medium when the boundary conditions of heat and water movement
are, respectively, (1) a sinusoidal temperature fluctuation of constant
amplitude at the land surface, and (2) a constant and uniform percolation rate
normal to the land surface. Bredehoeft and Papadopulos (1965) developed an
analytical solution for determining the ground-water velocity from subsurface
temperature data in an isotropic, homogeneous, and fully saturated
semiconfining layer in which all flow is vertical.
The analytical solution developed by Stallraan (1965) has not been used
extensively. Its main drawbacks are the assumptions that the water table is
at the surface and that all flow is vertical. Taking a different approach,
Nightingale (1975) estimated vertical recharge rates from an infiltration pond
with a sinusoidal temperature distribution, but he assumed that conductive
transfer processes were negligible and that velocity could be calculated by
using only the lag between surface temperatures and temperatures at a point in
the subsurface.
Several researchers have used Bredehoeft and Papadopulos's analytical
solution. Cartwright (1970) successfully used temperature anomalies to
estimate the amount of vertical movement of ground water in the Illinois
Basin. He matched the average temperature profile to the 3=-1 curve of
Bredehoeft and Papadopulos (1965) to obtain a discharge rate of 1.52 cm/yr
from the deep aquifer, a value that agrees with estimates of ground-water
discharge into streams in the basin. Sorey (1971) used the Bredehoeft and
Papadopulos technique to estimate the rate of upward movement through
semiconfining beds in the San Luis valley of Colorado and the Roswell basin of
New Mexico. His conclusions suggest that pumping tests and water-budget
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methods are often preferable because of limitations imposed by instabilities
in the borehole fluids and the measurement detail required. Boyle and Saleem
(1978) used the technique to estimate vertical flow rates through a clay-rich
glacial drift semiconfining layer in the Chicago area. They obtained good
agreement with values calculated by using the water-budget method.
The available analytical techniques only provide solutions for a very
small group of problems in which flow is in the vertical direction and
boundary conditions are specialized. In most aquifers the dominant direction
of water movement is horizontal. Noticeable temperature variations have been
observed in the horizontal direction in ground-water systems (Winslow 1962,
Schneider 1962 and 1964, Mink 1964, Parsons 1970, Suptow 1971, Cartwright
1973). The researchers all implicitly assumed that the variations in
temperature were caused by ground-water flow, but none were able to quantify
flows by using this information.
In the course of the research reported here an attempt was made to use the
measured horizontal and vertical distribution of subsurface temperatures to
estimate the rate of flow from a cooling lake situated on an alluvial aquifer.
A numerical technique was used by which ground-water velocities and hydraulic
conductivities in a two-dimensional system with nonuniform boundary conditions
can be determined by a trial-and-errror procedure from subsurface temperature
information. The procedure is not recommended for use in routine ground-water
flow system analysis since traditional methods for defining flow systems are
simpler to use and more reliable.
MATHEMATICAL MODtL
Given the following assumptions: (1) thermal equilibrium between the
liquid and soil particles is achieved simultaneously, (2) the density of the
soil particles is constant, (3) the heat capacity is constant, and (4) the
system is chemically inert, then the general differential equation for
simultaneous heat and fluid flow in a two-dimensional aquifer is
8(q,T>
where T = temperature, T; DJJ = coefficient of dispersion, H/tTL; pCw = heat
capacity of water, ti/L^T; pCs = heat capacity of the saturated media, H/L^T;
Qi = specific discharge or ground-water velocity, L/t; R = rate of heat
injection or discharge, H/L^t; and xj,S2 = cartesian coordinates L.
The goal is to solve Eq. (D-1) for the velocity distribution; then
hydraulic conductivities can be determined from
where KJJ = hydraulic conductivity tensor, L/t; and 4 = head, L.
Velocities in the aquifer can be determined from Eq. (D-1) either directly
or indirectly according to whether velocities are obtained directly by using
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the temperature distribution as a known in the differential equation, or
whether velocities are obtained as a solution to a nonlinear optimization
problem in which a set of calculated temperatures is matched to an observed
set. Direct methods require that the temperature distribution be known
completely and that it closely represent the true solution to the differential
equations (Neuman 1973). These types of solutions to the inverse problem are
currently a subject of active research and were not used in the present study.
The equation describing the two-dimensional flow of water through a
nonhomogeneous aquifer in steady state may be written as
i xj (0-3)
where h = water recharge rate per unit area, L/t; and b = thickness, L.
The ground-water velocity or specific discharge can then be determined
from
Acquisition of some information on subsurface permeability is desirable
for initial estimates of velocities by using the general differential equation
describing water flow in an aquifer. Hydraulic conductivities can then be
adjusted until the computed velocity distribution, when input into Eq. (D-1),
produces a temperature distribution that is reasonably close to the observed
temperature data.
The parameter estimation problem could be viewed as a classical nonlinear
regression problem in which a solution to the linked differential equations
[Eq. (D-1), Eq. (D-3), and Eq. (D-4)] forms the regression equation and in
which all unknown quantities are parameters (Cooley 1977). The problem was
not viewed in this manner because (1) sufficient temperature data will
generally not be available to insure a well-conditioned solution, and (2) Eq.
(D-1 ) behaves as a hyperbolic paraboloid when convective transport dominates,
which means that small errors in the input parameters can cause large errors
in the output .
Instead, a trial-and-error procedure was used to estimate hydraulic
conductivities. This technique is conceptually simple, but has several
drawbacks: (1) It can be expensive, (2) it is very time consuming, (3) an
answer cannot always be obtained, and (4) if an answer is obtinaed, it is
probably not the best estimate, and its relation to the best estimate is
unknown .
Several difficulties are encountered when Eq. (D-1) and Eq. (D-3) are
linked and solved in a trial-and-error procedure to estimate hydraulic
conductivities. The best documented difficulty is that, when hydraulic
conductivities are known, researchers have not been very successful in solving
for a known conservative contaminant distribution, even though the
conservative mass transfer problem is mathematically simpler than the heat
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transport problem. The differential equations are of the same form, but the
conservative mass transport problem has two fewer parameters because of the
common assumption that molecular diffusion, which is equivalent to thermal
conductivity, is negligible.
Attempts to simulate a known conservative contaminant distribution in an
aquifer with a deterministic model are not numerous. Finder (1973) simulated
the observed chromium contamination on Long Island; Bredehoeft and Finder
(1973) simulated the known chloride distribution in the Brunswick aquifer;
Konikow (1976) simulated the chloride distribution in the vicinity of the
Rocky Mountain Arsenal; Robertson (1974) simulated chlorides and other
contaminants in the vicinity of the Idaho Test Site; and fiobson (1978)
simulated total dissolved solids in a shallow alluvial aquifer near Barstow,
Calif. In all these simulations velocities within the aquifer were determined
by using equations similar to Eq. (D-3) and Eq. (D-4). In these equations
hydraulic conductivities were adjusted by a trial-and-error procedure until
predicted aquifer heads closely matched observed aquifer heads. An equation
similar to Eq. (0-1) was used to simulate transport of contaminants, and the
porosity and dispersivity parameters were then adjusted until the observed
chemical concentration patterns were matched. Even when hydraulic
conductivities are known, the chemical concentration distribution could not be
simulated without a trial-and-error adjustment procedure. Also, even after
the researchers had obtained what they considered to be a best fit between the
simulated and the observed data, the fit in all cases was less than perfect.
Anderson (1979) discusses some of the problems involved in applying
contaminant transport models.
Since the temperature data will generally be sparse, the trial-and-error
procedure of parameter estimation for the linked Eq. (D1) and (0-3) is
generally non-tractable unless several of the parameters are constrained. In
this analysis, in which hydraulic conductivities in a steady-state water-flow
problem were estimated, thermal conductivities, heat capacities, and
dispersivities were fixed. The rationale fbr fixing these parameters was that
the thermal conductivity and the heat capacity of most saturated glacial
materials fall within a small range, and they can be measured accurately in
the laboratory. Dispersivities were assumed to be small because element sizes
were chosen so that intra-element inhomogeneities were minimized.
These procedures left only hydraulic conductivities to be adjusted.
Generally, even with hydraulic conductivities as the only unknowns, problems
were ill conditioned .unless the hydraulic conductivities were constrained to
be within a small range. Therefore, this technique is only useful for
refining estimates of hydraulic conductivity that are determined with other
procedures.
The following procedure was used to refine estimates of the hydraulic
conductivity distribution at the Columbia Generating Station site by using
subsurface temperature data:
1) Hydraulic conductivity distributions were proposed on the basis of
stratigraphic information and field tests, and these distributions were
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then tested using Eq. (D-3) to determine if the known potential
distribution could be predicted within a set error criterion.
2) Once the potential distribution could be predicted, the relative magnitude
of the hydraulic conductivities was adjusted in a model linking Eq. (D-1)
and Eq. (D-3) to determine which factor gave the best fit to the observed
temperature data.
3) If the best fit of the temperature data obtained in step 2 was not good,
steps 1 and 2 were repeated. If the fit was judged to be acceptable,
thermal conductivities and dispersivities were changed to determine the
sensitivity of the estimate to changes in these parameters.
SOLUTION PROCEDURE
The finite element method was used to solve Eq. (D-1) and Eq. (D-3) for
temperatures in an aquifer subject to the boundary conditions described in
section 5 and appendix B. Each cross section modeled was divided into 100-150
quadrilateral elements. The procedure used to link the equations was: (1)
Eq. (D-3) was solved for head at each node and Eq. (D-4) was solved for
velocity; (2) £q. (D-1) was solved for temperature at each node; (3) the
solution to Eq. (D-1) was stepped forward in time by using the Crank-Nicolson
approximation for the tine derivative for a specified number of time steps;
(4) the hydraulic conductivities, which are a function of temperature, were
adjusted for the new temperature distribution; and (5) steps (1) to (4) were
repeated.
FIELD DATA
Temperatures were recorded weekly at 40-110 points in the subsurface in
the vicinity of the Columbia Generating Station (figure A-1) from August 1976
to January 1978. The monitoring techniques and the locations of the data
points are described in appendix A. The data collected on 7 October 1977 are
presented in a fence diagram in Figure (D-1). The data collected in
cross-section A-A' and B-B' of Figure 6 are presented in section 5.
Temperatures recorded in an array of wells on the east side of the cooling
lake are shown in Figure (D-2).
RESULTS
The temperature data from cross-sections A-A' and B-B1 were used to refine
estimates of hydraulic conductivity and to refine previous estimates of flow
in these cross sections from the cooling lake to the wetland west of the lake.
Andrews (1976) modeled flow in these cross sections and obtained average
annual flow rates of 4.5 n^/day (m^/day per meter width of the dike and 3.5
rn^/day, respectively. By using the trial-and-error procedure described in
this appendix, values of 5.2 m^/day and 4.3 n^/day were obtained for now in
these two cross sections, respectively. An error criterion of 0.05 m was used
in step 1. Thermal conductivities were set equal to values obtained in the
laboratory. Longitudinal dispersivity was set to 20 cm, and transverse
dispersivity was set to 5 cm on the basis of intro-element homogeneity
considerations.
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TwnperaturM in the Marsh on October 7, 1977
Figure D-l. Temperatures (°C) in a section of the marsh adjacent to the
cooling lake at the Columbia Generating Station site on
7 October 1977. The three-dimensional diagram depicts
temperatures in a 2,500-m section along the dike which
extends outward from the dike for a distance of 110 m and
to a depth of 10 m.
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LAKE
2
4
6
JUNE9
7.0"
10°
12.4°
8.8°
10
30
50 METERS
OCTOBER 24
30
50 METERS
Figure D-2. Temperatures in a 60-m portion of cross-section B-B' of
Figure 6 adjacent to the drainage ditch east of the cooling
lake on 9 June 1977 and 24 October 1977.
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The hydraulic conductivity distribution that best reproduced the observed
temperature data was not sensitive to changes in the thermal conductivities or
dispersivities within reasonable limits. Adjusting the thermal conductivities
and the dispersivities did, however, reduce the residuals between the observed
and the simulated temperatures. The changes in simulated temperature that
resulted when dispersivities were changed are described in section 5.
This trial-and-error procedure does not lend itself to estimation of the
standard error of the parameters in the best fit mode. It is only an
intuitive observation that the standard error of the estimated hydraulic
conductivities are reduced by using this tehcnique rather than stopping at
step 1 in the parameter estimation.
The temperature data were also used to attempt to determine flow rates in
five other cross sections (Figure 15) at the Columbia Generating Station site.
In all these cross sections only limited potential data were available, and
several hydraulic conductivity distributions could be found that would
reproduce the known potential distribution equally well. Since the
temperature data were also limited, several hydraulic conductivity
distributions which represented widely varying flow rates could reproduce the
observed data equally well. Only in the simulated cross section located near
the intake of the cooling lake were the results satisfactory. (Of the five
cross sections, the best temperature and potential data were available for
this cross section.) In this cross section the flow rates into the wetland
were estimated to average 7.4 nrVday.
CONCLUSIONS
Unfortunately, the information in Figure (0-1) was not sufficient to
estimate flow rates from the cooling lake into the wetland along the dike.
Figure (0-1) does illustrate vividly the complexity of ground-water flow
patterns and hydraulic conductivity distributions near the dike. Although
flow rates from the cooling lake to the wetland in the area shown in Figure
(D-1) are most likely all within the range of 3.5-7.5 rn^/day, the temperature
contrasts are dramatic. Small changes in peat thickness or the presence or
absence of a clay lens can cause very different temperature patterns even
though flows are approximately the same. Keys and Brown (1978) reached a
similar conclusion. The temperature data, although indicating that flows are
much greater in some areas than others, are not sufficient for determining
hydraulic conductivity distributions.
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
REPORT NO.
2.
3. RECIPIENT'S ACCESSION NO.
TITLE AND SUBTITLE
Impacts of Coal-Fired Power Plants on Local
Ground-Water Systems
Wisconsin Power Plant Impact Study
5. REPORT DATP
August 1980 Issuing Date.
i. PERFORMING ORGANIZATION CODE
AUTHOR(S)
Charles B. Andrews
Mary P. Anderson
8. PERFORMING ORGANIZATION REPORT NO.
PERFORMING ORGANIZATION NAME AND ADDRESS
Institute for Environmental Studies
Environmental Monitoring and Data Acquisition Group
University of Wisconsin-Madison
Madison, Wisconsin 53706
10. PROGRAM ELEMENT NO.
1BA820
11. CONTRACT/GRANT NO.
R803971
2. SPONSORING AGENCY NAME ANOADDRESS
Environmental Research Laboratory
Office of Research and Development
U.S. Environmental Protection Agency
Duluth, Minnesota 55804
13. TYPE OF REPORT AND PERIOD COVERED
Final 7/75 - 6/78
14. SPONSORING AGENCY CODE
EPA/600/03
5. SUPPLEMENTARY NOTES
6. ABSTRACT Quantitative techniques for simulating the impacts of a coal-fired power
plant on the ground-water system of a river flood-plain wetland were developed and
tested. Effects related to the construction and operation of the cooling lake and ash-
pit had the greatest impact. Ground-water flow system models were used to simulate
ground-water flows before and after the cooling lake and ashpit were filled. The sim-
ulations and field data indicate that the cooling lake and ashpit altered local flow
systems and increased ground-water discharge. Chemical changes in the ground-water
system were minor. Contaminated ground water was confined to a small area near the
ashpit. Thermal changes in the ground water are a major impact of the cooling lake.
Changes in water temperature and levels have altered the vegetation of the wetland, a
major ground-water discharge area. Ground-water temperatures near the cooling lake
were monitored. A model was used to simulate the response of subsurface temperatures
to seasonal changes in a lake and air temperatures. Long-term substrate temperature
changes expected in the wetland were predicted. Using ground-water temperatures to
estimate flow rates was investigated. Simulated temperature patterns agreed with
field data, but were sensitive to the distribution of subsurface lithologies. It is
predicted that by 1987 ground-water temperatures will be increased, resulting in an
increase in ground-water flow.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS C. COSATI Field/Group
Ground, water models, cooling lakes, tempera
ture monitoring, river flood-plain wetlands
coal fired power plants, ash pit, flow rate:
vegetation
Wisconsin Power Plant
Study, Impacts on Biota
Flow measurement, terraii
models, artesian water,
cooling water
06-F, T
08- D, F, H, M
10-C
13-B, M
18. DISTRIBUTION STATEMENT
Release to the public
19. SECURITY CLASS (ThisR
unclassified
215
20. SECURITY CLASS (Thispage)
unclassified
22. PRICE
EPA Form 2220-1 (Rev. 4-77) PREVIOUS EDITION is OBSOLETE
PRINTING "FFICF: 1 980--657-1 5?/OOB4
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