c/EPA
United States
Environmental Protection
Agency
Office of Research and
Development
Washington, DC 20460
EPA/600/R-92/223
November 1992
3DFEMWATER/
3DLEWASTE: Numerical
Codes for Delineating
Wellhead Protection
Areas in
Agricultural Regions
Based on the Assimilative
Capacity Criterion
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EPA/600/R-92/223
November 1992
3DFEMWATER/3DLEWASTE:
NUMERICAL CODES FOR DELINEATING
WELLHEAD PROTECTION AREAS IN
AGRICULTURAL REGIONS BASED ON THE
ASSIMILATIVE CAPACITY CRITERION
by
G.T. (George) Yeh,1 Susan Sharp-Hansen,2
Barry Lester,'Robert Strobl,'and Jeffrey Scarbrough4
The Pennsylvania State University1
University Park, PA 16802
AQUA TERRA Consultants2
Mountain View, CA 94043
GeoTrans, Inc.3
Sterling, VA 22170
AScI Corporation4
Athens, GA 30613
Project Officer
Robert Carsel
Environmental Research Laboratory
U.S. Environmental Protection Agency
Athens, GA 30613
ENVIRONMENTAL RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
ATHENS, GEORGIA 30613
Printed on Recycled Paper
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DISCLAIMER
The work presented in this document has been funded by the United States
Environmental Protection Agency. It has been subject to the Agency's peer and
administrative review, and has been approved as an EPA document. Mention of trade
names or commercial products does not constitute endorsement or recommendation for
use by the U.S. Environmental Protection Agency.
11
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FOREWORD
As environmental controls become more costly to implement and the penalties of
judgment errors become more severe, environmental quality management requires
more efficient analytical tools based on greater knowledge of the environmental
phenomena to be managed. As part of this Laboratory's research on the occurrence,
movement, transformation, impact, and control of environmental contaminants, the
Assessment Branch is developing management or engineering tools that can be used by
States to protect public drinking water wells from possible contamination.
The 1986 Amendments to the Safe Drinking Water Act require each State to develop
and submit to the U.S. EPA a wellhead protection program. As part of the program,
States must establish procedures for delineating wellhead protection areas around each
water well or well field which supplies a public water system. In order to delineate
wellhead protection areas in agricultural regions using the assimilative capacity
criterion, the 3DFEMWATER/3DLEWASTE model has been developed. These finite
element numerical codes simulate 1) flow and transport in three-dimensional variably-
saturated porous media under transient conditions, 2) multiple distributed and point
sources/sinks, and 3) processes which retard the transport of contaminants.
Rosemarie C. Russo, Ph.D.
Director
Environmental Research Laboratory
Athens, Georgia
in
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ABSTRACT
The 1986 Amendments to the Safe Drinking Water Act require each State to develop
and submit to the U.S. EPA a wellhead protection program. As part of the program,
States must establish procedures for delineating wellhead protection areas around each
water well or well field which supplies a public water system. Of the five criteria that
have been suggested by the U.S. EPA for delineating wellhead protection areas, the
assimilative capacity criterion is potentially the most accurate. It takes into account
the reduction in concentration of contaminants being transported toward a well caused
by chemical and environmental processes at the land surface and in the vadose and
saturated zones.
Nationwide, agricultural areas are located in many diverse hydrogeologic environ-
ments. Recharge and pumping rates can vary widely within an area because of
irrigation practices and/or climate. In addition, contamination scenarios must consider
multiple point and nonpoint source loadings of pesticides which vary both spatially and
temporally. In order to delineate wellhead protection areas in agricultural regions
using the assimilative capacity criterion, the use of a numerical model is needed that
accounts for 1) flow and transport in three-dimensional variably-saturated porous
media under transient conditions, 2) multiple distributed and point sources/sinks, and
3) processes which retard the transport of contaminants.
This document describes two related numerical codes, 3DFEMWATER and
3DLEWASTE, which can be used to delineate wellhead protection areas in agricultural
regions using the assimilative capacity criterion. 3DFEMWATER (Three-dimensional
Finite Element Model of Water Flow Through Saturated-Unsaturated Media)
simulates subsurface flows, whereas 3DLEWASTE (Hybrid Three-Dimensional
Lagrangian-Eulerian Finite Element Model of Waste Transport Through Saturated-
Unsaturated Media) Models contaminant transport. Both codes treat heterogeneous
and anisotropic media consisting of as many geologic formations as desired, consider
both distributed and point sources/sinks that are spatially and temporally dependent,
and accept four types of boundary conditions (i.e., Dirichlet (fixed-head or
concentration), specified-flux, Neumann (specified-pressure-head gradient or specified-
dispersive flux), and variable). The variable boundary condition in 3DFEMWATER
simulates evaporation/infiltration/seepage at the soil-air interface and, in
3DLEWASTE, simulates mass infiltration into or advection out of the system.
3DLEWASTE contains options to model adsorption using a linear, Freundlich, or
Langmuir isotherm, plus dispersion, and first-order decay.
This report was submitted in partial fulfillment of Work Assignment Number 1,
Contract Number 68-CO-0019 by AQUA TERRA Consultants, under the sponsorship of
the U.S. Environmental Protection Agency. This report covers the period May 1991 to
July 1992, and work was completed as of August 1992.
IV
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TABLE OF CONTENTS
Page
Disclaimer ii
Foreword iii
Abstract iv
Figures viii
Tables x
Acknowledgments xi
1.0 INTRODUCTION 1
1.1 Wellhead Protection Area Delineation 1
1.1.1 Issues Related to Agricultural Regions 3
1.2 The 3DFEMWATER/3DLEWASTE WHPA Model 4
1.2.1 Experience Required to Apply 3DFEMWATER/3DLEWASTE . . 6
1.2.2 Implementing a 3DFEMWATER/3DLEWASTE Modeling
Study 6
1.3 Organization of the Document 8
2.0 MODEL DEVELOPMENT, DISTRIBUTION, AND SUPPORT 9
2.1 Development and Testing 9
2.2 Distribution 10
2.3 Obtaining a Copy of the 3DFEMWATER/3DLEWASTE Model 11
2.3.1 Diskette 11
2.3.2 Electronic Bulletin Board System (BBS) 11
2.4 General/Minimum Hardware and Software Installation and Run
Time Requirements 12
2.4.1 Installation Requirements 12
2.4.2 Run Time Requirements 12
2.5 Installation 12
2.6 Installation Verification and Routine Execution 13
2.7 Code Modification 13
2.8 Technical Help 14
2.8.1 Electronic Bulletin Board System (BBS) 14
2.9 Disclaimer 15
2.10 Trademarks 16
3.0 BACKGROUND INFORMATION 17
3.1 3DFEMWATER 17
3.1.1 Governing Equations 17
3.1.2 Boundary Conditions and Transient Source/Sink Terms 21
3.1.3 Initial Conditions 28
3.1.4 Steady-State 29
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3.2 Numerical Approximation in 3DFEMWATER 30
3.2.1 Galerkin Formulation 30
3.2.2 Solution Techniques 32
3.3 3DLEWASTE 34
3.3.1 Governing Equations 37
3.3.2 Boundary Conditions and Transient Source/Sink Terms 42
3.3.3 Initial Conditions 45
3.3.4 Steady-State 45
3.4 Numerical Approximation in 3DLEWASTE 45
3.4.1 Galerkin Formulation 46
3.4.2 Solution Techniques 47
4.0 DATA INPUT REQUIREMENTS 49
4.1 3DFEMWATER Input Sequence 49
4.1.1 Data Set 1: Title of the Simulation Run 49
4.1.2 Data Set 2: Basic Integer Parameters 50
4.1.3 Data Set 3: Basic Real Parameters 51
4.1.4 Data Set 4: Printer and Disk Storage Control and Times for
Step Size Resetting 52
4.1.5 Data Set 5: Material Properties 53
4.1.6 Data Set 6: Soil Property Parameters 53
4.1.7 Data Set 7: Nodal Point Coordinates 56
4.1.8 Data Set 8: Subregional Data 56
4.1.9 Data Set 9: Element Incidences 57
4.1.10 Data Set 10: Material Type Correction 59
4.1.11 Data Set 11: Card Input for Initial or Pre-Initial Conditions . . 59
4.1.12 Data Set 12: Integer Parameters for Source and Boundary
Conditions 60
4.1.13 Data Set 13: Distributed and Point Sources/Sinks 61
4.1.14 Data Set 14: Variable Composite (Rainfall/Evaporation-
Seepage) Boundary Condition 64
4.1.15 Data Set 15: Fixed-Head (Dirichlet) Boundary Condition 68
4.1.16 Data Set 16: Specified-Flux (Cauchy) Boundary Condition .... 69
4.1.17 Data Set 17: Specified-Pressure-Head Gradient (Neumann)
Boundary Condition 71
4.1.18 Data Set 18: End of Job 73
4.2 3DLEWASTE Input Sequence 74
4.2.1 Data Set 1: Title of the Simulation Run 74
4.2.2 Data Set 2: Basic Integer Parameters 75
4.2.3 Data Set 3: Basic Real Parameters 76
4.2.4 Data Set 4: Printer and Disk Storage Control and Times for
Step Size Resetting 77
4.2.5 Data Set 5: Material Properties 78
4.2.6 Data Set 6: Nodal Point Coordinates 78
4.2.7 Data Set 7: Element Incidences 79
4.2.8 Data Set 8: Subregional Data 80
4.2.9 Data Set 9: Material Type Correction 81
4.2.10 Data Set 10: Card Input for Initial or Pre-Initial Conditions . . 82
4.2.11 Data Set 11: Integer Parameters for Sources and Boundary
Conditions 82
vi
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4.2.12 Data Set 12: Distributed and Point Sources/Sinks 84
4.2.13 Data Set 13: Variable Composite Boundary Condition 87
4.2.14 Data Set 14: Prescribed-Concentration (Dirichlet) Boundary
Condition 89
4.2.15 Data Set 15: Specified-Flux (Cauchy) Boundary Condition .... 90
4.2.16 Data Set 16: Specified-Dispersive-Flux (Neumann) Boundary
Condition 93
4.2.17 Data Set 17: Hydrological Variables 95
4.2.18 Data SetlS: End of Job 96
5.0 PARAMETER SELECTION 97
5.1 3DFEMWATER 97
5.1.1 Data Set 1: Title of the Simulation Run 97
5.1.2 Data Set 2: Basic Integer Parameters 98
5.1.3 Data Set 3: Basic Real Parameters 99
5.1.4 Data Set 5: Material Properties 102
5.1.5 Data Set 6: Soil Property Parameters 106
5.2 3DLEWASTE 110
5.2.1 Data Set 1: Title 110
5.2.2 Data Set 2: Basic Integer Parameters Ill
5.2.3 Data Set 3: Basic Real Parameters 113
5.2.4 Data Set 5: Material Properties 114
5.2.5 Data Set 17: Hydrological Variables 119
6.0 EXAMPLE PROBLEMS 121
6.1 3DFEMWATER 121
6.1.1 One-Dimensional Column 121
6.1.2 Two-dimensional Drainage Problem 122
6.1.3 Three-Dimensional Pumping Problem 125
6.2 3DLEWASTE 144
6.2.1 One-Dimensional Transport Problem 144
6.2.2 Two-Dimensional Transport in a Rectangular Region .... 147
7.0 REFERENCES 150
APPENDIX A 153
APPENDIX B 180
APPENDIX C 182
APPENDIX D 187
VII
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FIGURES
Page
3.1. Uncofined aquifer to be approximated as a variably-saturated
porous medium 18
3.2. Variable pore spacing in soil under saturated flow conditions 20
3.3. Logarithmic plot of constitutive relations for sand, clay, silty
loam, and sandy loam: (a) moisture content vs. pressure head
and (b) relative permeability vs. moisture content 22
3.4. Conceptual model and mathematical approximation for variably-
saturated flow system. Within the modeled system, transient
source/sink terms may be applied as point sources/sinks or as
distributed sources/sinks 23
3.5. Use of a pressure-head gradient boundary condition to simulate a
portion of the unsaturated zone 25
3.6. Using a series of nodes to represent a screened well interval 27
3.7. Pressure head versus time at a nodal point on the finite element
grid 28
3.8. Pressure head versus time at nodal point where steady-state
solution is being approached 29
3.9. Finite element grid for production from a single well in a
variably-saturated porous medium 30
3.10. Solution scheme for unsaturated flow analysis 33
3.11. Use of vertical or horizontal nodal slices in the block
iterative method 35
3.12. Migration of dissolved contaminants through the unsaturated zone
into unconfined aquifer system 36
3.13. Diagram showing the effect of scale on hydrodynamic dispersion
processes 38
Vlll
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3.14. Backward particle tracking to determine the starting point of an
advected particle 42
4.1. Node numbering convention for the elements 58
4.2 Node numbering convention for the elements 80
5.1 Longitudinal dispersivity versus scale with data classified
by reliability 117
5.2. Nomograph for determining Darcy velocity 120
6.1. One-dimensional transient flow through a soil column 122
6.2. Two-dimensional steady-state flow to parallel drains 124
6.3. Three-dimensional steady-state flow to a pumping well 128
6.4. One-dimensional transient transport through a horizontal column 144
6.5. Two-dimensional transient transport in a rectangular region 147
IX
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TABLES
Page
5-1. Water Density as a Function of Temperature 101
5-2. Dynamic Viscosity of Water as a Function of Temperature 102
5-3. Range of Hydraulic Conductivity Values for Various Geologic
Materials 104
5-4. Variability in Horizontal and Vertical Hydraulic Conductivities 105
5-5. Permeability of Porous Materials 105
5-6. Descriptive Statistics for Saturation Water Content (0S)
and Residual Water Content (9r) 107
5-7. Descriptive Statistics for van Genuchten Water Retention Model
Parameters, a, p, and y 108
5-8. Mean Bulk Density (g/cm3) for Five Soil Textural
Classifications 115
5-9. Range and Mean Values of Dry Bulk Density for Various Geologic
Materials 116
6-1. Input Data Set for the One-Dimensional 3DFEMWATER Problem 123
6-2. Input Data Set for the Two-Dimensional 3DFEMWATER Problem 126
6-3. Input Data Set for the Three-Dimensional 3DFEMWATER Problem 130
6-4. Input Data Set for the One-Dimensional 3DLEWASTE Problem 145
6-5. Input Data Set for the Two-Dimensional 3DLEWASTE Problem 148
x
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ACKNOWLEDGEMENTS
This document was prepared under Work Assignment No. 1 of Contract No. 68-CO-0019
by AQUA TERRA Consultants for the U.S. Environmental Protection Agency Office of
Research and Development. Robert Carsel of the Environmental Research Laboratory
in Athens, Georgia was the Technical Project Monitor and the Project Officer. We thank
him for his continuous technical and management support throughout the course of this
project.
The 3DFEMWATER/3DLEWASTE code was developed by G.T. (George) Yeh of The
Pennsylvania State University. Robert Strobl, also at The Pennsylvania State
University, upgraded the code to meet U.S. Environmental Protection Agency coding
conventions. John Kittle at AQUA TERRA Consultants reviewed the code and
suggested modifications.
The original documentation of the 3DFEMWATER/3DLEWASTE code was prepared by
G.T. Yeh. That documentation was substantially expanded and rewritten during the
course of this project. At AQUA TERRA Consultants, Susan Sharp-Hansen was
responsible for rewriting the documentation. She was assisted by Barry Lester of
GeoTrans, Inc., who wrote Section 2 and part of the introduction; by Robert Strobl, who
prepared some of the tables in the appendices; and by Jeff Scarborough of AScI
Corporation, who applied the code to the example problems. John Imhoff, the Project
Manager, supplied administrative guidance and he, Anthony Donigian, and John Kittle
reviewed the document. Technical reviewers also included David Ward and Jeff
Benegar of GeoTrans, Inc. Word processing was performed by Dorothy Inahara.
XI
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SECTION 1
INTRODUCTION
This document describes two related numerical codes, 3DFEMWATER and
3DLEWASTE. Together these codes can model flow and transport in three-dimensional
variably-saturated porous media under transient conditions, with multiple distributed
and point sources/sinks, and considering processes which retard the transport of
contaminants (i.e., dispersion, decay and adsorption). Thus, they can be used to apply
the assimilative capacity criterion to the development of wellhead protection areas.
Background information about wellhead protection area delineation criteria and
methods is provided in Section 1.1. The features and implementation of the
3DFEMWATER/3DLEWASTE codes are discussed in Section 1.2 and the contents of this
document are summarized in Section 1.3.
It is important to note that the version of 3DFEMWATER/3DLEWASTE documented in
this user's manual has substantial CPU time requirements. A faster version of the
model is currently being developed.
1.1 WELLHEAD PROTECTION AREA DELINATION
The 1986 Amendments to the Safe Drinking Water Act require each State to develop
and submit to the U.S. EPA a wellhead protection program. As part of the program,
States must establish procedures for delineating wellhead protection areas around each
water well or well field which supplies a public water system. A wellhead protection
area (WHPA) is defined as the surface and subsurface area surrounding a water well or
well field through which contaminants are likely to be transported and reach the well or
wellfield. Within the WHPA, contaminant sources need to be assessed and managed to
prevent pollution of public drinking water supplies. Existing WHP programs are
generally aimed at one of the following overall protection goals:
Provide a remedial action zone to protect wells from unexpected contaminant
releases.
Provide an attenuation zone to bring concentrations of specific contaminants
to desired levels at the time they reach the wellhead.
Provide a well-field management zone in all or part of a well's present or
future recharge area.
Five criteria have been suggested by the U.S. EPA (U.S. EPA, 1987) for delineating
wellhead protection areas that will adequately protect public water supplies. The
criteria are:
Distance, which considers a radial distance from the pumping well.
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Drawdown, which considers an area within which an aquifer's potentiometric
surface has been lowered by pumping.
Time of travel, which considers the time required for a contaminant to move
through the subsurface to a well (often only considering advection).
* Flow system boundaries, which consider the geographic or hydrologic fea-
tures that control groundwater flow.
Assimilative capacity, which considers environmental factors which reduce
the concentration of contaminants transported to a well.
One or more of the criteria may be used. The most technically demanding, but also
potentially the most accurate, is the assimilative capacity criterion. The assimilative
capacity criterion takes into account the reduction in concentration of contaminants
being transported toward a well caused by chemical and environmental processes at the
land surface and in the vadose and saturated zones.
The U.S. EPA has described six methods for applying the criteria to the delineation of
WHPAs. Listed in order of difficulty, the methods are:
Arbitrary fixed radius, which involves drawing a circle around a well. The
radius of the circle can be based on professional judgment or an established
distance criterion.
Calculated fixed radius, in which the radius of a circle around the well is
determined from an equation that considers the volume of water pumped from
a well over a specified time.
Simplified variable shapes, which makes use of "standardized forms" repre-
senting various hydrogeologic and pumping conditions. The set of
standardized forms are initially prepared using an analytical model.
Subsequent application involves selecting the most appropriate shape for a
given well.
Analytical methods, which involve the application of analytical groundwater
flow and transport models.
Hydrogeologic mapping, which makes use of geologic, geophysical, and dye
tracing techniques to map a WHPA.
Numerical models, which involve the application of numerical models of flow
and solute transport in the subsurface.
Application of the first three methods is suitable for only a very limited number of sites,
such as extensive, homogeneous, single aquifers with a relatively flat potentiometric
surface. While analytical methods are usually more technically accurate than the first
three methods, their application is still restricted to relatively simple hydrogeologic
environments. Hydrogeologic mapping may be the only reasonable method under some
hydrogeologic conditions, such as karst or fractured aquifers. However, it will also be
2
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necessary for some hydrogeologic mapping to be performed for application of either
analytical or numerical models.
Numerical models provide the greatest flexibility and accuracy in representing complex
environments and can be applied to nearly all types of hydrogeologic settings. The
models can also be used to predict the dynamic aspects of the WHPA, such as changes
in the size of the WHPA resulting from natural or man-made effects. Disadvantages for
this method include costs that are high relative to other methods and the need for
considerable technical expertise in hydrogeology and modeling. The cost may be
warranted in areas where a high degree of accuracy is desired, however. Also, due to
limitations on model grid spacing and density, numerical models are sometimes less
suitable than analytical methods for assessing drawdowns close to pumping wells.
The more rigorous the method used for WHPA delineation, the smaller the WHPA can
be without risking underprotection and the associated potential for water quality
degradation. When a smaller WHPA can be defined without generating unacceptable
risk, land use restrictions can be kept to a minimum along with the potential economic
hardships associated with such land use restrictions. The choice of WHPA delineation
methodology becomes a decision based on generating an acceptable margin of safety,
while balancing the economic hardships to affected parties with the technical and
economic feasibility of minimizing the WHPA.
1.1.1 Issues Related to Agricultural Regions
Nationwide, agricultural areas are located in many diverse hydrogeologic environments
(e.g., multiple aquifer systems, fractured and/or karst systems, and systems with wide
variations in depth to the water table). In addition, recharge can vary widely because of
irrigation practices and/or climate. Also, domestic and irrigation wells, which pump at
different and varying rates, are commonly located throughout agricultural regions.
Therefore, the ability to model transient flow conditions (i.e., transient recharge, a
fluctuating water table, and transient pumping from a variety of points in x,y,z space)
for a wide variety of hydrogeologic conditions is important.
Contamination scenarios in agricultural regions must consider multiple point and
nonpoint source loadings that vary both spatially and temporally. For example, spills,
leaks, or the direct introduction of chemicals into well casings can result in point sources
of contamination, whereas chemical application to fields can result in nonpoint sources
of contamination.
Pesticide loadings to the subsurface are affected by both surface processes and
agricultural management practices. Examples include runoff, erosion, chemical
volatilization, evapotranspiration, tillage practices, and the method, amount, and timing
of pesticide application. Most of these processes require detailed modeling of the surface
environment or zone and are not addressed in models of subsurface flow and transport.
Therefore, it is suggested that text or matrix ranking or the separate application of an
existing model be used to estimate recharge and solute loading from the surface zone to
the subsurface (e.g., PRZM-2, see Mullins et al., 1992).
The contaminants of concern in agricultural regions are predominantly organic
pesticides and nitrates. Pesticides are typically present in the subsurface in dilute
3
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concentrations. Because interest in agricultural areas is likely to focus on dilute organic
pesticides, issues such as the transport of metals, the interactions of complex mixtures,
or immiscible flow are not addressed by this model. Also, because of the complexity of
the processes associated with the transport of nitrates, nitrate contamination can not be
adequately modeled using this version of 3DFEMWATER/3DLEWASTE.
1.2 THE 3DFEMWATER/3DLEWASTE WHPA MODEL
3DFEMWATER (Three-dimensional Finite Element Model of Water Flow Through
Saturated-Unsaturated Media) can be used to investigate subsurface flows as a stand-
alone model, or it can be used to provide the hydrologic flow variables required by
3DLEWASTE. The special features of 3DFEMWATER are its flexibility and versatility
in modeling a wide range of real-world problems. The model is designed to:
Treat heterogeneous and anisotropic media consisting of as many geologic
formations as desired.
Consider both distributed and point sources/sinks that are spatially and
temporally dependent.
Accept prescribed initial conditions or obtain them by simulating a steady-
state version of the system under consideration.
Deal with a transient head variation over a freed-head (Dirichlet) boundary.
Handle time-dependent fluxes due to a varying pressure gradient along a
specified-pressure-head gradient (Neumann) boundary.
Treat time-dependent total fluxes distributed over a specified-flux (Cauchy)
boundary.
Automatically determine variable boundary conditions of evaporation, infiltra-
tion, or seepage at the soil-air interface.
Include the off-diagonal hydraulic conductivity components in the modified
Richard's equation in order to deal with cases when the coordinate system does
not coincide with the principal directions of the hydraulic conductivity tensor.
Provide three options (exact, under-, and over-relaxation) for estimating the
nonlinear matrix.
Include two options (successive subregion block iterations and successive point
iterations) for solving the linearized matrix equations.
Automatically reset the time-step size when boundary conditions or
sources/sinks change abruptly.
3DLEWASTE (Hybrid Three-Dimensional Lagrangian-Eulerian Finite Element Model of
Waste Transport Through Saturated-Unsaturated Media) uses a hybrid Lagrangian-
Eulerian approach. In comparison to conventional finite element (including both
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Galerkin and upstream finite element) or finite difference (including both central and
upstream finite difference) models, 3DLEWASTE offers several advantages. First, it
completely eliminates numerical oscillation due to advection terms. Second, it can be
applied to mesh Peclet numbers ranging from zero to infinity. (Conventional finite
element or finite difference models typically impose severe restrictions on the mesh
Peclet number.) Third, it can use very large time-step sizes to greatly reduce numerical
dispersion. In fact, the larger the time step, the better is the solution with respect to
advective transport. The time-step size is only limited by the accuracy requirement with
respect to diffusive/dispersive transport, which is normally not a very severe restriction.
Finally, the hybrid Lagrangian-Eulerian finite element approach is always superior to,
and will never be worse than, its corresponding upstream finite element method.
The 3DLEWASTE model is designed to:
Treat heterogeneous and anisotropic media.
Consider spatial and temporal distribution, as well as point sources/sinks.
Accept prescribed initial conditions or obtain them by simulating a steady-
state version of the system under consideration.
Deal with transient concentrations distributed over prescribed concentration
(Dirichlet) boundaries.
Handle time-dependent fluxes over variable boundaries.
Deal with time-dependent total fluxes over specified-flux (Cauchy) boundaries.
Handle time-dependent fluxes over specified dispersive-flux (Neumann) bound-
aries.
Include the off-diagonal dispersion coefficient tensor components in the govern-
ing equation for dealing with cases when the coordinate system does not
coincide with the principal directions of the dispersion coefficient tensor.
Provide two options of treating the mass matrixconsistent and lumping.
Provide three options (exact, under- and over-relaxation) for estimating the
nonlinear matrix.
Include a block iteration method to solve the linearized matrix equations to
eliminate the excessive storage demands of a direct band matrix solution.
Automatically reset the time-step size when boundary conditions or
sources/sinks change abruptly.
Simulate first-order contaminant decay.
Include three adsorption modelsa linear isotherm and a nonlinear Freundlich
or Langmuir isotherm.
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1.2.1 Experience Required to Apply 3DFEMWATER/3DLEWASTE
The complexity and sophistication of the 3DFEMWATER/3DLEWASTE numerical codes
limits the number of people who can successfully use the codes to apply the assimilative
capacity criterion in wellhead protection area delineation. The user community is
expected to be State personnel, as well as personnel at the U.S. EPA headquarters and
regional offices, who are experienced numerical modelers with a strong background in
hydrogeology.
1.2.2 Implementing a 3DFEMWATER/3DLEWASTE Modeling Study
Implementation of a 3DFEMWATER/3DLEWASTE modeling study represents a highly
rigorous evaluation of a wellhead site. The study is generally aimed at delineating the
WHPA with a high degree of certainty. The project team can take into consideration the
specific nature of present and future wellfields, the physical and chemical nature of
potential contaminant sources, the effect of human activities, as well as the complexity
of the groundwater flow system through which the contaminants travel.
Although 3DFEMWATER/3DLEWASTE studies can provide flexibility in defining the
hydrogeologic environment and contaminant sources, they are limited by the quantity
and quality of physical and chemical data available to define the system. When seeking
to define the zone of contribution in a WHPA using a 3DFEMWATER/3DLEWASTE
analysis, there is a law of diminishing returns. The economic benefits gained from being
able to minimize land use restrictions must be weighed against the costs of generating
the necessary data and applying the model.
Wellfield geometry and the spatial distribution of wells within a field can strongly affect
subsurface flow at regional and local scales. Using the 3DFEMWATER/3DLEWASTE
model, an investigator can consider the influence of a wellfield on the regional flow
system. On the local scale, the effects of partial penetration associated with well
screening intervals can also be considered. Localized flow patterns, which result from
perturbations to the flow field and the heterogeneous nature of the geologic medium,
influence the movement of dissolved contaminants and determine 1) the amount of time
required for a dissolved species to reach the wellfield and 2) the degree of attenuation of
the species as it approaches the field.
The 3DFEMWATER/3DLEWASTE model also allows the user to examine the influence
of temporal changes in well production on contaminant mobility. The influence of
seasonal variations in well production and other periodic variations (i.e., drought
conditions, unseasonably warm summers, etc.), can strongly affect the potential for a
contaminant to reach a wellfield at unacceptable levels or in an unacceptable amount of
time. The temporal variations in well production can be considered in conjunction with
associated temporal changes in recharge and evapotranspiration rates.
As implied above, the 3DFEMWATER/3DLEWASTE model is not limited to
discretization of the flow field into regularly shaped prismatic blocks (i.e. triangular and
rectangular prisms). Therefore, consideration of the heterogeneous nature of a modeled
system is mainly limited by either the availability of data or the computational power of
the computer utilized. There is a practical limitation on the degree of heterogeneity
which can be simulated, based on the conflict between the grid block-size restrictions
6
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needed to circumvent convergence problems and the number of blocks that a computer
can handle in a time-efficient manner. Within these restrictions, it is the model user's
goal to maximize the extent to which the influence of soil and rock-type heterogeneities
affect the flow system.
In nature, heterogeneities generate a strong control on the local pathways that the
dissolved chemicals will follow. The tendency for water to flow through low resistance
(high conductivity) pathways provides a short-circuiting effect that can accelerate the
movement of chemicals to a wellfield. In contrast, occurrences of high resistance (low
conductivity) media between the source and the screened intervals of wells can inhibit
the contaminant from reaching the water supply or attenuate the contaminant to safe
concentration levels before it reaches the water supply. The uncertainty associated with
a WHPA analysis is directly related to the presence of heterogeneity in the aquifer
properties. As the degree of heterogeneity decreases, the possibility of underestimating
or overestimating the chemical migration is reduced. On the other hand, the potential
for contamination is most uncertain when using bulk properties or using ad hoc
variances in the values of effective porosity, dispersivity and hydraulic conductivity.
Since the flow portion of 3DFEMWATER/3DLEWASTE simulates variably-saturated
conditions, a more accurate model of water storage in unconfined or partially confined
systems can be generated. The user can consider draining (and filling) of pore spaces
above the water table, which can damp the effect of time-variant changes in well produc-
tion, recharge and evapotranspiration on the flow system. Rigorous representation of
the unsaturated zone also permits examination of the influence of variable saturation on
the mobility of contaminants. Vertical infiltration through the unsaturated zone and
the associated lateral spreading of contaminants, due to the occurrence of sediment
lenses of various grain sizes, can be considered. Explicit simulation of the unsaturated
zone also allows for direct consideration of the contaminant storage capacity of the
unsaturated zone. This more accurately depicts the role of the unsaturated zone as a
source of contaminant infiltration into the saturated zone. The availability of different
adsorption models (linear, Freundlich and Langmuir) allows the user to choose a
contaminant storage capacity appropriate for the waste being modeled.
The 3DFEMWATER/3DLEWASTE model includes a relatively rigorous representation of
contaminant sources by using a variety of time-dependent boundary conditions.
Contaminant sources may be represented not only as point sources or sources of simple
geometry, as assumed in analytical solutions, but also as sources of variable geometry.
Where applicable, contaminants already present in the subsurface water and solid
matrix at the start of a modeling study can also be simulated. The use of infiltration or
recharge options available in the 3DFEMWATER/3DLEWASTE model provides a good
method of simulating contaminant sources such as spatially- and temporally-variant
pesticide or fertilizer applications to agricultural areas.
The interaction of the regional flow field and local wellfield perturbations can be
handled in two ways using 3DFEMWATER/3DLEWASTE. The localized flow field may
be implemented as a freely discretized portion of the larger system where the boundary
conditions are generally associated with the regional flow field. The problem can also be
broken up into two problems of different scales, where the regional flow system is
modeled for flow only and the local system is modeled for flow and transport with the
boundary conditions generated from the regional flow model. The degree of interaction
7
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between the two models is dictated by the degree of accuracy desired, and the placement
of local system boundaries.
1.3 ORGANIZATION OF THE DOCUMENT
This documentation contains the information needed to understand and apply the
3DFEMWATER/3DLEWASTE codes to wellhead protection area delineation problems.
Section 2 contains information on model distribution and support. In Section 3, back-
ground related to the model equations, features, and numerical approximation
techniques is presented. Section 4 is a guide to the construction of input data sets for
the code. Assistance in explaining and estimating some of the input parameters is
provided in Section 5. Five simple example problems, including the corresponding input
data files, are given in Section 6. The appendices contain more detailed information
about the numerical codes, including descriptions of the subroutines, and listings of the
maximum control parameters and program variables.
-------
SECTION 2
MODEL DEVELOPMENT, DISTRIBUTION, AND SUPPORT
NOTE: Refer to the READ.ME file for the latest supplemental information, changes,
and/or additions to the 3DFEMWATER/3DLEWASTE model documentation. A copy of
the READ.ME file is included on each distribution diskette set or it can be downloaded
from the Center for Exposure Assessment Modeling (CEAM) electronic bulletin board
system (BBS). It can be installed on a hard disk using the INSTALL (diskette) or
INSTALPZ (BBS) program. It is an ASCII (non-binary) text file that can be displayed
on the monitor screen by using the DOS TYPE command (e.g., TYPE READ. ME) or
printed using the DOS PRINT command (e.g., PRINT READ.ME).
The READ.ME file contains a section entitled File Name and Content that provides a
brief functional description of each 3 DFEM WATER/3 OLE WASTE file by name or file
name extension type. Other sections in this document contain further information
about:
System development tools used to build the microcomputer release of the
3DFEMWATER/3DLEWASTE model.
Recommended hardware and software configuration for execution of the model
and all support programs.
Program execution.
Minimum file configuration.
Sample run times.
Program modification.
Technical support.
2.1 DEVELOPMENT AND TESTING
The 3DFEMWATER/3DLEWASTE model was developed and tested on a Digital Equip-
ment Corporation (DEC) VAX6310 running under version 5.4-2 of the VMS operating
system (OS) and version 5.5-98 of VAX VMS FORTRAN- 77, and an Advanced Logic
Research (ALR) 486/25 microcomputer running under version 4.00 of IBM PC DOS and
version 2.51 of Salford FORTRAN (FTN77/486). The following FORTRAN tools were
also used to perform static evaluations of the 3DFEMWATER/3DLEWASTE FORTRAN
code on an IBM PS/2 Model 80-071 running under version 3.3 of IBM PC! DOS, MICRO
EXPRESS (ME) 486/25 and 486/33 systems running under version 5.00 of Microsoft
(MS) DOS, and a Sun SPARCstation 1+GX running version 4.1.1 of UNIX/SunOS:
-------
Ryan-McFarland FORTRAN versions 2.45, 3.10.01 (RMFORT).
Microsoft FORTRAN version 5.00 (MSFORT).
Lahey FORTRAN versions 5.01, 4.02 (F77L, F77L-EM/32).
Waterloo FORTRAN version 8.5E (WATCOM FORTRAN-77/386).
Sun FORTRAN version 1.4.
Silicon Valley FORTRAN version 2.81 (SVS FORTRAN-77/386).
In addition to the VAX and ALR systems, 3DFEMWATER/3DLEWASTE has also been
successfully executed on a PRIME 50 Series minicomputer running under PRIMOS, the
Sun SPARCstation, and the IBM PS/2 Model 80-071.
2.2 DISTRIBUTION
The 3DFEMWATER/3DLEWASTE model and all support files and programs are
available on diskette from the Center for Exposure Assessment Modeling, located at the
U.S. EPA Athens Environmental Research Laboratory, Athens, Georgia, at no charge.
The CEAM has an exchange diskette policy. It is preferred that diskettes be received
before sending a copy of the model system (refer to Section 2.3, Obtaining a Copy of the
3DFEMWATER/3DLEWASTE Model).
Included in a distribution diskette set are:
3DFEMWATER/3DLEWASTE general execution and user support guide
(READ. ME) file.
Interactive installation program (refer to Section 2.5).
Test input and output files for installation verification.
Executable task image file for the 3DFEMWATER/3DLEWASTE model.
FORTRAN source code files.
Command and/or "make" files to compile, link, and run the task image file
(*. EXE).
A FORTRAN compiler and link editor are NOT required to execute any portion of the
model. If the user wishes to modify the model, it will be up to the user to supply and/or
obtain:
An appropriate text editor that saves files in ASCII (non-binary) text format.
FORTRAN development tools to recompile and link edit any portion of the
model.
10
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CEAM cannot support, maintain, and/or be responsible for modifications that change the
function of the executable task image, MAKE, or DOS command files supplied with this
model package.
The microcomputer release of the 3DFEMWATER/3DLEWASTE model is a full
implementation of the VAX/VMS version. The microcomputer implementation of this
model performs the same function as the U.S. EPA mainframe/minicomputer version.
2.3 OBTAINING A COPY OF THE 3DFEMWATER/3DLEWASTE MODEL
NOTE: The following abbreviations are used below to represent different
quantities of computer memory:
1 k = 1 kilobyte = 1,024 bytes
1 m = 1 megabyte = 1 "meg" = 1,048,576 bytes
lb=lbyte
2.3.1 Diskette
To obtain a copy of the 3DFEMWATER/3DLEWASTE distribution model package on
diskette, send:
The appropriate number of double-sided, double-density (DS/DD 360kb) 5.25
inch, or double-sided, high-density (DS/HD 1.44mb) 3.5 inch error-free dis-
kettes.
NOTE: To obtain the correct number of diskettes, contact CEAM at
706/546-3549.
A cover letter, with a complete return address requesting the
3DFEMWATER/3DLEWASTE model to:
Model Distribution Coordinator
(ATTN: Catherine E. Green, CSC)
Center for Exposure Assessment Modeling
Environmental Research Laboratory
U.S. Environmental Protection Agency
Athens, GA 30613-0801
Program and/or user documentation, or instructions on how to order documentation, will
accompany each response.
2.3.2 Electronic Bulletin Board System (BBS)
To download a copy of the 3DFEMWATER/3DLEWASTE model, or to check the status of
the latest release of this model or any other CEAM software product, call the CEAM
BBS 24 hours a day, 7 days a week. To access the BBS, a computer with a modem and
communication software are needed. The phone number for the BBS is 706/546-3402.
Communication parameters for the BBS are:
11
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300/1200/2400/9600 baud rate.
8 data bits.
No parity.
1 stop bit.
In order to access the BBS at 9600 baud, a USRobotics Courier HST modem must be
used.
2.4 GENERAL/MINIMUM HARDWARE AND SOFTWARE INSTALLATION AND RUN
TIME REQUIREMENTS
NOTE: Refer to the READ.ME file for the latest supplemental and more complete
information, changes, and/or additions concerning specific hardware and software
installation and run time requirements.
2.4.1 Installation Requirements
3.5 inch, 1.44mb diskette drive, or 5.25 inch, 360kb diskette drive.
Hard disk drive.
Approximately 8mb free hard disk storage,
2.4.2 Run Time Requirements
386 or 486 compatible microcomputer.
MS or PC DOS version 3.30 or higher.
640k base memory.
Extended memory and hard disk storage requirements will vary with the size of the
problem being simulated. Requirements for problems similar to those found in Section 6
are:
2mb of extended (XMS) memory.
4mb free hard disk storage.
Refer to READ.ME file for suggested modification of the CONFIG.SYS and/or
AUTOEXEC.BAT DOS system configuration and start-up files.
2.5 INSTALLATION
To install the 3DFEMWATER/3DLEWASTE model and/or related support files on a
hard disk, insert the first distribution diskette in a compatible diskette drive (refer to
Section 2.4). Then type:
12
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A:\INSTALL or B:\INSTALL
at the DOS system prompt and press the key. Then follow instructions and
respond to prompts presented on the monitor screen by the interactive installation
program. Complete installation instructions are also printed on each external diskette
label. The 3DFEMWATER/3DLEWASTE distribution diskette sets implement software
product installation standards to insure the most error-free, maintainable, and user-
acceptable distribution of CEAM products. It has a "unique menu option, command, full-
screen (interactive), diagnostic, error-recovery, help, and selective installation
capabilities using state-of-the-art human-factors engineering practices and principles.
NOTE: The contents of the distribution diskettes can be copied to another set of
"backup" diskettes using the DOS DISKCOPY command. Refer to the DOS Reference
Manual for command application and use. The "backup" diskettes must be the same
size and storage density as the original, source diskettes.
2.6 INSTALLATION VERIFICATION AND ROUTINE EXECUTION
Refer to the following sections in the READ.ME file for complete instructions concerning
installation verification and routine execution of the 3DFEMWATER/3DLEWASTE
model:
File name and content.
Routine execution.
Run time and performance.
Minimum file configuration.
2.7 CODE MODIFICATION
Included in the diskette set are:
An executable task image file for the 3DFEMWATER/3DLEWASTE model.
FORTRAN source code files.
Command and/or "make" files to compile, link, and run the task image file
(*.EXE).
If the user wishes to modify the model or any other program, it will be up to the user to
supply and/or obtain:
An appropriate text editor that saves files in ASCII (non-binary) text format.
FORTRAN development tools to recompile and link edit any portion of the
model.
13
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CEAM cannot support, maintain, and/or be responsible for modifications that change the
function of any executable task image (*. EXE), DOS batch command (*. BAT), and/or
"make" utility file(s) supplied with this model package.
2.8 TECHNICAL HELP
For questions and/or information concerning:
Installation and/or testing of the 3DFEMWATER/3DLEWASTE model and/or
support programs or files, call 706/546-3590, 3548 for assistance.
3DFEMWATER/3DLEWASTE model and/or program content, application,
and/or theory, call 706/546-3 171 for assistance.
Use of the CEAM electronic bulletin board system (BBS), contact the BBS
system operator (SYSOP) at 706/546-3548.
* CEAM software and distribution Quality Assurance and Control, call 706/546-
3634.
Other environmental software and documentation distributed through CEAM,
contact the Model Distribution Coordinator at 706/546-3549.
Other support available through CEAM, contact Mr. Robert B. Ambrose, Jr.,
CEAM Manager:
By mail at the following address:
Center for Exposure Assessment Modeling (CEAM)
Environmental Research Laboratory
U.S. Environmental Protection Agency
Athens, Georgia 30613-0801
- By telephone at 706/546-3130.
By fax at 706/546-2018.
Through the CEAM BBS message menu and commands. The CEAM BBS
communication parameters and telephone number are listed above.
2.8.1 Electronic Bulletin Board System (BBS)
To help technical staff provide better assistance, write down a response to the following
topics before calling or writing. If calling, be at the computer, with the computer on,
and in the proper sub-directory when the call is placed.
14
-------
Program information:
Describe the problem, including the exact wording of any error and/or
warning message (s).
List the exact steps, command (s), and/or keyboard key sequence that will
reproduce the problem.
Machine information:
List computer brand and model.
List available RAM (as reported by DOS CHKDSK command).
List available extended memory (XMS).
List name and version of extended memory (XMS) manager (i.e., HIMEM,
VDISK, RAMDRIVE, etc.).
List available hard disk space (as reported by DOS CHKDSK command).
List the brand and version of DOS (as reported by DOS VER command).
List the name of any memory resident program(s) installed.
Printer brand and model.
Monitor brand and model.
NOTE: If contacting CEAM by mail, fax, or BBS, include responses to the above
information in your correspondence.
2.9 DISCLAIMER
Mention of trade names or use of commercial products does not constitute endorsement
or recommendation for use by the United States Environmental Protection Agency.
Execution of the 3DFEMWATER/3DLEWASTE model, and modifications to the DOS
system configuration files (i.e., /CONFIG.SYS and /AUTOEXEC.BAT) must be made at
the user's own risk. Neither the U.S. EPA nor the program authors can assume
responsibility for model and/or program modification, content, output, interpretation, or
usage.
CEAM software products are built using FORTRAN-77, assembler, and operating
system interface command languages. The code structure and logic of these products is
designed for single-user, single-tasking, non-LAN environment and operating platform
for microcomputer installations (i.e., single user on a dedicated system).
Users will be on their own if they attempt to install a CEAM product on a multi-user,
multi-tasking, and/or LAN based system (i.e., Windows, DESQview, any LAN). CEAM
15
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cannot provide installation, operation, and/or general user support under any
combination of these configurations. Instructions and conditions for proper installation
and testing are provided with the product in a READ.ME file. While
multiuser/multitasking/LAN installations could work, none of the CEAM products have
been thoroughly tested under all possible conditions. CEAM can provide scientific
and/or application support for selected products if the user proves that a given product
is installed and working correctly.
2.10 TRADEMARKS
IBM, Personal Computer/XT (PC/XT), Personal Computer/AT (PC/AT), PC
DOS, VDISK, and Personal System/2 (PS/2) are registered trademarks of
International Business Machines Corporation.
DESQview is a trademark of Quarterdeck Office Systems, Inc.
Sun and SunOS are registered trademarks of Sun Microsystems, Inc.
SPARC is a registered trademark of SPARC International, Inc.
UNIX is a registered trademark of American Telephone and Telegraph.
SVS FORTRAN-77 is a trademark of Silicon Valley Software.
PRIME and PRIMOS are trademarks of Prime Computers, Inc.
Microsoft, RAMDRIVE, HIMEM, MS, and MS-DOS are registered trademarks
of Microsoft Corporation.
Windows is a trademark of Microsoft Corporation.
RM/FORTRAN is a trademark of Language Processors, Inc.
DEC, VAX, VMS, and DCL are trademarks of Digital Equipment Corporation.
386 and 486 are trademarks of Intel Corporation.
U.S. Robotics is a registered trademark and Courier HST is a trademark of
U.S. Robotics, Inc.
16
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SECTION 3
BACKGROUND INFORMATION
3.1 3DFEMWATER
3DFEMWATER is designed to simulate the movement of moisture through variably-
saturated porous media. Typical applications include: 1) studying the influence of
transient stresses, such as well production schemes or the onset of drought conditions,
on water table elevations, and 2) generating flow fields for use in examining the
influence of physical processes such as rainfall and evapotranspiration on the movement
of dissolved contaminants through the vadose zone and into aquifers (Figure 3.1). The
complementary 3DLEWASTE model is designed to utilize the flow data generated by
3DFEMWATER simulations in order to evaluate the associated movement of dissolved
contaminants through the modeled system. The model 3DLEWASTE is described in
Section 3.3.
3.1.1 Governing Equations
The governing equation for flow of water through a variably-saturated porous medium,
as derived from mass and momentum conservation constraints, can be written:
F(h) ^L = V |K(h)
-------
Figure 3.1 UnconfLned aquifer to be approximated as a variably-saturated porous medium.
-------
K(h) = krKs (3-2)
where kris the relative permeability, ranging in value from 0.0 to 1.0, and Ksis the
saturated hydraulic conductivity (L/T). The saturated hydraulic conductivity is a flow
property of both the porous medium and fluid, and is determined by tests performed
under saturated conditions. It represents a maximum possible value of effective
hydraulic conductivity. The relative permeability term describes the influence of water
content on the magnitude of the effective hydraulic conductivity. Values of relative
permeability range from a minimum value reflecting the reduction of effective
conductivity at residual water content to a maximum of 1.0 reflecting saturated
conditions.
The change in relative permeability is caused by changes in moisture content, resulting
in the preferential movement of water through certain pathways, due to the influence of
capillary forces. As the soil becomes less saturated, water drains more readily from the
large radius pore structures, the water flow becomes restricted to pore sequences of
smaller radii (Figure 3.2) as well as those held in layers close to the soil particles. The
result of water becoming increasingly restricted to the smaller radius pathways is a
reduction in the spatially-averaged effective hydraulic conductivity.
The decrease in effective hydraulic conductivity, as reflected in the relative permeability
term, is described by pairs of empirical soil-moisture curves, These curves detail the
relationships between water content and pressure head, and between hydraulic
conductivity and water content. Soil-moisture curves are often described as coefficients
and exponents of standard analytical functions (Brooks and Corey, 1966; Mualem, 1976;
van Genuchten, 1980). The 3DFEMWATER code allows the user to define the curves
using the van Genuchten functions (1980) or as sets of paired values of relative
permeability versus moisture content and moisture content versus pressure head given
in lookup table format. The van Genuchten relationships found in 3DFEMWATER are:
kr = er U - (i
and
[1 + (h.
19
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Figure 3.2. Variable pore spacing in soil under saturated flow conditions.
where
(b - 9
T "
(3-3c)
Y = 1 - 1/p
(3-3d)
and
9W = moisture content (dimensionless)
<}> = porosity (dimensionless)
9wr = residual moisture content (dimensionless)
P, y = soil-specific exponents (dimensionless)
Ot = soil-specific coefficient (1/L)
ha = air entry pressure head (L)
9e = effective moisture content (dimensionless)
Note that the soil-moisture content is defined as the porosity multiplied by the degree of
saturation. Typical soil-moisture curves generated from Equations 3-3a and 3-3b are
presented in Figures 3.3a and 3.3b.
20
-------
NEGATIVE PRESSURE HEAD (cm)
1.0
10'
10"
i
10
10'
10
10'
10
I I
101
10-'
MOISTURE CONTENT
1.0
Figure 3.3. Logarithmic plot of constitutive relations for sand, clay, silty
loam, and sandy loam: (a) moisture content vs. pressure head, and
(b) relative permeability vs. moisture content (based on data
presented in Carsel and Parrish, 1988).
21
-------
The water capacity term or storage term used in 3DFEMWATER can be written in the
form:
F(h) = deydh (3-4)
It should be noted that, due to the relatively small influence of compressibility on water
capacity in the unsaturated zone (with respect to the drainage potential), soil and water
compressibility have been ignored in the storage term. When analytical functions are
used to describe the nonlinearity of the relative conductivity, the derivative with respect
to pressure head of the water content versus pressure head function must also be
defined analytically.
The equation governing saturated flow represents a limiting case of Richard's equation
where the relative permeability is a constant of 1.0 and the water capacity is a constant
equal to the specific yield for an unconfined aquifer or specific storage for a confined
aquifer.
3.1.2 Boundary Conditions and Transient Source/Sink Terms
Unique solutions to variably-saturated flow problems are generated by solving Richard's
equation in conjunction with 1) a set of boundary conditions defined at the physical
edges of the modeled system and 2) where appropriate, source/sink terms applied within
the system (Figure 3.4). Boundary conditions available in the 3DFEMWATER model
include fixed-head (Dirichlet) boundaries, specified-flux (Cauchy) boundaries, specified
pressure-head gradient (Neumann) boundaries and variable (head-dependent flow)
boundaries.
Fixed-head or Dirichlet boundaries are boundaries defined by prescribing pressure heads
at specified boundary nodes so that:
h = hd(xb)yb,zb)t) on Bd (3-5)
where
hd = specified pressure head (L)
Bd = portion of the system boundary subject to a Dirichlet
boundary condition
XbiYb^b = spatial coordinates on the boundary (L)
Dirichlet boundaries are typically used to define the perimeters of bodies of water, the
water table location, and leaking surface impoundments or other waste disposal
facilities containing specified levels of water. Specified pressure heads may be constant
or allowed to vary with time reflecting physical processes such as water level
fluctuations associated with seasonal changes in rainfall and evapotranspiration rates.
The specified-flux (Cauchy) boundary represents the portions of the system boundary
where infiltration or evapotranspiration rates can be quantified. The specified-flux
boundary condition can be written:
22
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CONCEPTUAL
AGRICULTURAL
AREA
WELL
MATHEMATICAL
Figure 3.4. Conceptual model and mathematical approximation for a
variably-saturated flow system. Within the modeled
system, transient source/sink terms may be applied as
point sources/sinks or as distributed sources/sinks.
23
-------
-n-krK8
-------
1
-,
ARID
J)
2
1
I
rlHl
V,
s»
1
i
J)
i
J)
^ Groun<
JUUJ
5urtac<
r
4c s
Water Table y
Figure 3.5 Use of a pressure-head gradient boundary condition to simulate a
portion of the unsaturated zone.
25
-------
h = hp(xb,yb,zb,t) on Bv (3-8a)
or
-n-krKB
-------
WELL
f. /-"^N f A
r
1 4
) I
ft »
. 4
P ~
»(
*
ft |
. t
9 '
»^
i
»f
»4
»4
1 1
t
l
^
(
>
^
i
i
i
i
k
T
t
r
t
f
f
t
1
1
I
I
Is
/
t
4
4
4
4
<
4
j
1
f
L- ELEMENT
THICKNESS
t;
i
>
r '
> '
> '
^ "
^ ^
> '
h 1
1
LEGEND
FINITE ELEMENT
NODE
() POINT SOURCE/
SINK NODE
1
'^^Screened
r-^Interval
Figure 3.6. Using a series of nodes to represent a screened well interval.
27
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vertically adjacent nodes are used to represent the screened interval of a well, the
volumetric flux must be distributed among the nodes. The most appropriate
distribution of the total flux is in proportion to the effective conductance, Ce, of the
individual nodes where the effective conductance of each node is defined as:
Ce =
(Ka)nLn
where n-1 and n are indices referring to the element below the node and the element
above the node, respectively, and 0.5L is half the thickness of an element.
Time-variant boundary conditions and source/sink flux or flux intensity rates are defined
by a series of paired time and value points. This paired data is used to assemble a look-
up table from which appropriate values are obtained using linear interpolation at
specified times of analysis. Constant values can be specified by assigning the same
value to a set of two time/data point pairs, making sure that the simulation time is fully
spanned.
3.1.3 Initial Conditions
The solution of Richard's equation also requires the initialization of pressure head
values such that:
h = hj(x,y,z,t=0) in R
(3-10)
where hj is the initial pressure head distribution (L), and R is the region of interest
(Figure 3.7). Besides providing a frame of reference for transient analyses, the initial
conditions are used to set the nonlinear parameters at the beginning of a simulation.
For transient problems, an appropriate set of initial pressure head values may either be
input directly or derived from a steady-state simulation. For more information on these
options see Section 4.1.11.
CONTINUOUS
INITIAL CONDITION
P1ECEWISB
TIME
Figure 3.7. Pressure head versus time at a nodal point on the finite
element grid.
28
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3.1.4 Steady-State
When analyzing the influence of transient stresses, such as well production schemes and
drought conditions, on the flow system, a starting point must be assumed. The user
defines boundary conditions and flow parameters as best he/she can, then does an initial
simulation to allow the system to reach an equilibrium or steady-state (Figure 3.8). The
steady-state simulation then defines the pressure head at all points in the system and it
is from this initial condition that a transient simulation is started. Although the actual
system is never really in steady-state, by using averaged conditions (i.e., rainfall, etc.) a
reasonable starting point is generated. If the steady-state simulation fails to converge
or the results poorly match field data, flow parameters and/or boundary conditions
should be adjusted to improve the starting conditions. The steady-state or equilibrium
condition is generated by removing the temporal term from Equation 3-1. The system is
then defined as the equilibrium reached under the average conditions.
Besides being used for initial conditions for a transient simulation, the steady-state flow
option can also be used in conjunction with a transient transport simulation. Since the
flow system will generally reach equilibrium under non-changing stresses faster than an
associated solute transport problem, using a steady-state flow field and average
conditions to define the advective portion of solute transport will often give a good
approximation of the change of solute distribution over time. The savings in
computational effort can be considerable and, given the uncertainty of parameters in the
system, an acceptable approximation may be reached.
STEADY-STATE
SOLUTION
INITIAL CONDITION
0
TIME
Figure 3.8 Pressure head versus time at a nodal point where a steady-state
solution is being approached.
29
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3.2 NUMERICAL APPROXIMATION IN 3DFEMWATER
The 3DFEMWATER model was developed to solve the variably-saturated flow equation
described in Section 3.1. In the model, Richard's equation (Equation 3-1) is
approximated using the Galerkin finite element technique. The time integral term in
Equation 3-1 is approximated using backwards or central (Crank-Nicholson) difference
in time. The nonlinearity of the system is treated using Picard iteration and the
generated set of linearized equations is solved using a block iterative method.
3.2.1 Galerkin Formulation
In 3DFEMWATER, Richard's equation is approximated using the Galerkin finite
element method (Finder and Gray, 1977) where the dependent variable, pressure head,
is approximated by a trial function of the form:
h = N/x^h/t) j = l,2,-.,n
(3-11)
where Nj(Xj,t) are the three-dimensional shape functions and hj(t) are nodal values of
pressure head at time t for the n nodes of which the finite element grid is comprised
(Figure 3.9).
70
66
60
35
50
45
40
35
30
15
£
20
3
300
502
400
'"y4! S / If / / f S 7 .
f. f Iff f f f f f .
\f / f f / f t f f f f /
«A^ f f ' / f f J
S f 7 / / 7 /
Y
/
t
f
1
/'
J J J J J
>
4
1
/
t
-*
1
t
*-
5
L
j^ TT *xll
c~
' S 7 / S S / S S
/ f f f / S S
f f f f f f
S /
Y
/ ,
A\,
t
y
7f
t
^
t '
/
/
' '/
0 * TO 120 100 200 275 350 400 450 500540 570«00«50 750 800150 900 950 1000
Figure 3.9. Finite element grid for production from a single well in a variably-
saturated porous medium.
30
-------
Substituting the trial functions into Equation 3-1 and applying the Galerkin criterion,
we generate a set of weighted residual minimization equations:
Jw,
:F(h)_f± - V -[krKB
-------
Bu = £ JvNi'krK,-VNjadR (3-15C)
and
C, = £ - fkrK8-VNie-VzdR + fN;eqdR + f^'n^K/h^N/ + VzflB (3-15d)
t. i * ~ *^ I
L R- R- B- J
where m is the number of elements into which the system is discretized and Ne denotes
elemental shape functions.
3.2.2 Solution Techniques
To solve the series of linearized ordinary differential equations presented in Equation 3-
15a, the time differential is replaced by a finite difference formulation, resulting in
working equations for 3DFEMWATER of the form:
Atk
- h,k + wB^V*1 + (l-w)B^V = C;kiw (3-16)
where k+1 represents the current time level, k the previous time level, At the length of
the current time step and w the time weighting function (1.0=backwards in time;
0.5= Crank Nicholson or centered in time). Note that the associated transport code,
3DLEWASTE, utilizes a backwards-in-time scheme: Therefore, when using
3DFEMWATER to generate a flow field for a 3DLEWASTE simulation, the backwards-
in-time option should be used. This prevents the possibility of a mismatch in the
interpolation of time-variant boundary condition and source/sink flux values.
For each time step, the solution method involves an outer and inner iterative scheme
(Figure 3.10) where the outer iterations control convergence of the nonlinear terms in
the equations and the inner iterative scheme controls the block-iterative method of
solving the linearized set of equations. For each nonlinear iteration, the linearized set
of equations is solved using relative permeability and storage terms updated using
pressure head values generated during the previous nonlinear (outer) iteration. Relative
permeability and storage terms for the first iteration in a time step are based on
pressure head values from the previous time step, or for the first time step, from the
initial conditions.
Because of the strong nonlinear nature of the soil moisture curves, the outer iterative
scheme may become unstable. To help circumvent this problem it is often helpful to
damp the iterative changes in the pressure head. One method of damping the iterative
changes is through the use of an under-relaxation factor. Implementation of the under-
relaxation factor for the outer iterations in 3DFEMWATER is as follows:
where u is the outer under-relaxation factor and r is the iteration number. If damping
is needed, values below one should be used. Acceleration or over-relaxation ( 1.0
-------
I
Time Step Loop
I
Assemble stiffness matrix
and load vector
Solve linearized set of
equations one block
at a time
Update load vector
using new pressure
heads
Update nonlinear
parameters
Check for convergence of
block iterative scheme
NO
t
YES
Check for convergence of
nonlinear terms if needed
I
NO
YES
Is this the last timestep?
NO
YES
End analysis
Figure 3.10. Solution scheme for unsaturated flow analysis.
33
-------
+ uhr1 (3-17)
is generally not recommended for the nonlinear iterations as it may make the solution
become unstable. For transient simulations, reduction of the time-step size can also
help increase the stability of the solution scheme. Note that sometimes steady-state
problems will be difficult to solve. In this case, it is often worth trying a transient
solution approach, using expanding time steps to approach the steady-state solution.
For each nonlinear iteration, a set of linearized simultaneous equations is solved using a
block iterative scheme. The user defines a set of subregions (or blocks) by prescribing
the nodes contained in each subregion (Figure 3.11). The code then generates a series of
connectivity arrays indicating: 1) the nodes contained in each subregion, 2) for each
node, all other nodes found in elements it is part of, and 3) which of these adjacent
nodes are located in the same subregion. The nodal equations for each subregion are
solved directly using a Gaussian solver. For each nodal equation defined in Equation 3-
15a, contributions from adjacent nodes falling outside the subregion being solved for are
generated by multiplying the matrix terms with the appropriate nodal pressure heads.
These pressure heads are generated during the last direct solution for the subregion
containing the adjacent nodes.
Subregions are generally defined as nodal planes (Figure 3.11) allowing the user to work
with a minimal half-bandwidth when the direct solver is invoked. The half-bandwidth
is defined as one plus the largest difference between the node number associated with
the nodal equation and the other nodes found in elements the node is part of and which
are in the same block as the node. As a general rule, subregions comprised of vertical or
sub-vertical nodal slices provide the smallest half-bandwidth and will perform well in
the block iterative method, although this may not always be the case. For some
problems, horizontal slicing may be advantageous. The block iterative logic contains a
relaxation factor which can be used to over-relax the solution and help accelerate the
rate of convergence. Implementation of the inner over-relaxation scheme is as follows:
+ ohr1 (3-18)
where s denotes the inner iteration number and o is the over-relaxation factor. The
optimal value of the over-relaxation factor usually falls between 1.5 and 1.9. A good
starting point is o = 1.72.
3.3 3DLEWASTE
3DLEWASTE is designed to simulate the movement of dissolved species through a
variably-saturated porous medium. Typical applications for 3DLEWASTE include the
examination of: 1) leachate migration from landfills and surface impoundments, 2) the
influence on water quality of pesticide and fertilizer applications, and 3) the
environmental impact of leaky containment structures such as underground and above
ground storage tanks (Figure 3.12). Velocity fields needed to define the advective
pathways of water bearing the chemicals are provided by associated 3DFEMWATER
simulations.
34
-------
HORIZONTAL SLICE
CO
01
' ,,,,,*
70 120 160 200 275 350 400 450 500540 570600650 750 800850 900 950 1000
Figure 3.11. Use of vertical or horizontal nodal slices in the block interative method.
-------
CO
CD
Figure 3.12. Migration of dissolved contaminants through the unsaturated zone into an unconfined aquifer system.
-------
3.3.1 Governing Equations
The governing equation for advective-dispersive solute transport through variably-
saturated porous media, based on the laws of conservation of mass and flux, can be
written in the form:
9.1 + pb. =V
-------
MEGASCOPIC SCALE SOLUTE MIGRATION
PATHWAY
Figure 3.13. Diagram showing the effect of scale on hydrodynamic dispersion
processes.
38
-------
molecular diffusion coefficient in Equation 3-20 quantifies the spreading due to
molecular diffusion.
In order to solve Equation 3-19 for a single dependent variable, the constitutive relation-
ship between the species concentrations in the dissolved and adsorbed phases must be
defined. The 3DLEWASTE code allows the user to choose from three relationships: 1)
alinear isotherm, 2) Freundlich isotherm, or 3) Langmuir isotherm. The isotherms, as
determined in laboratory partitioning experiments, can be plotted in log-log form to
derive:
log S = n log C + log K (3-21)
or
S = KG n (3-22)
where n is slope of the plot of log S versus log C and K is the S-axis intercept (Freeze
and Cherry, 1979). Equation 3-22 defines the Freundlich isotherm, which is often used
to describe the partitioning between the dissolved and absorbed phases. When the
isotherm has a slope n= 1, the isotherm is linear and the relationship can be defined as:
- = Kd (3-23)
dC
where K,, is called the distribution coefficient (LVM). Linear isotherms are often used to
describe the adsorption of hydrophobic organic compounds to organic matter in soils.
The distribution coefficient is described as a function of the organic carbon content of
the soil as:
Kd = US* (3-24)
where f^ is the fractional organic carbon content and K,,,, is the normalized distribution
coefficient. There are many published lists of values for K^, (e.g., Lyman et al., 1982;
U.S. EPA, 1986; Verschueren, 1983). Data are available primarily for pesticides and, to
a lesser degree, aromatic and polycyclic aromatic compounds. If data on K^ are not
available for a particular chemical, a value can be estimated from empirical
relationships between K^. and some other property of the chemical such as the water
volubility, S, the octanol-water partition coefficient, KDW, or the bioconcentration factor
for aquatic life, BCF. Lyman et al. (1982) tabulate 12 such regression equations
obtained from data sets of different classes of chemicals. One commonly-used
relationship (Karickhoff et al., 1979) takes the form:
Koc = 0.41KOW (3-25)
The Langmuir isotherm takes the form:
39
-------
S = Sm"KC (3-26)
1 + KG
where S_ is the maximum concentration allowed in the medium.
'max
The effective decay constant, A,, is a degradation constant that can be used to quantify
the effects of radioactive decay, or the composite effects of hydrolysis and
biodegradation. When used to quantify the effects of hydrolysis and biodegradation, the
effective decay coefficient (for a linear isotherm) takes the form:
A = l 2"b + JL (3-27)
9 + KdPb
where ^ is the first-order hydrolysis rate constant for the dissolved species, Aj is the
first-order hydrolysis rate constant for the sorbed species, and \ is the first-order
biodegradation rate constant. The dissolved species first-order hydrolysis rate can be
written in terms of the acid-catalyzed (Kg), base-catalyzed (K,,), and neutral (K,,)
hydrolysis rate constants as:
A! = KJH = KJH+] + Kn+ KJOH] (3-28)
where [H+] is the hydrogen ion concentration and [OH-] is the hydroxyl ion
concentration. The sorbed phase first-order hydrolysis rate is considered to be a
function of the acid and neutral hydrolysis rates and is usually written in the form:
A2 = ccKa[H + ] + K,, (3-29)
where a is the acid-catalyst hydrolysis rate enhancement factor for the sorbed phase
with a typical value of 10.0. Note that for a nonlinear isotherm the formulation in
3DLEWASTE is valid only if \ = A,.
The governing equation for advective-dispersive solute transport in a porous medium, as
presented in Equation 3-19, describes the transport from an Eulerian or fixed
framework. The numerical algorithm may begin to oscillate and fail to converge to a
solution of this equation when the advective term starts to dominate over the dispersive
term and the equation takes on a hyperbolic nature. Dominance of the advective term
over the dispersive term is reflected in the non-dimensional Peclet number, which is
defined as the ratio of the product of the velocity magnitude and distance advected to
the dispersion coefficient. In finite element analysis the critical Peclet number is the
local Peclet number of an element, where the local Peclet number is defined as:
P = L/O (3-30)
where L is the element length.
One method of circumventing the numerical problems (i.e., oscillation and failure to
converge) associated with Peclet numbers greater than 2 is to address the system
through a moving (i.e., Lagrangian) coordinate system. In the Lagrangian formulation
40
-------
for solute transport in a porous medium, the temporal term is defined as a material
derivative of the form:
fe + Pb *i VDC = e 9C + 9s + v.vc
b dcDF at "at
where D denotes the material derivative.
The advective term, V-VC, written in index notation becomes:
V-VC - . (3-32)
dt 9x.
where the repeated indices indicate summation. Substituting Equation 3-31 into
Equation 3-19, the governing equation for a Lagrangian framework becomes:
for a linear isotherm. The average linear velocity, V*, for a linear isotherm becomes:
V = e + pb (3-33b)
For a non-linear isotherm, the Lagrangian equation becomes:
pb-- = V < 0D -VC) - W6C + PbS) + QCin - QC (3-34a)
Dt dC at
where
V* = V/0 (3-34b)
Full implementation of the Lagrangian approach implies the solution of Equation 3-33
using a moving coordinate system. Another method of circumventing the instability
problem is to utilize a hybrid Eulerian-Lagrangian approach. Such an approach is
implemented in 3DLEWASTE. In the hybrid approach, the advective term of the
material derivative is evaluated in a Lagrangian manner by a backwards particle
tracking scheme (Figure 3.14). The particle tracking scheme generates a particle
starting location and an associated concentration, C*. This concentration, C*, is the
starting concentration of each particle which reaches a nodal point at the end of that
particular time step. The material term of Equation 3-33 is then approximated by:
41
-------
o
nodal point
initial coordinate of
particle
flowline
computed path of
particle
Figure 3.14. Backward particle tracking to determine the starting
point of an advected particle.
DC
Dt
C - C*
At
(3-35)
The diffusion-type equation is then solved using a fixed coordinate system. Note that
for a steady-state simulation, where At>°°, the logic is implemented by multiplying the
transient storage terms by zero and evaluating the advection term in a fixed coordinate
system.
3.3.2 Boundary Conditions and Transient Source/Sink Terms
Unique solutions to advective-dispersive solute transport problems are generated by
solving the governing equation (Equation 3-19) in conjunction with 1) a set of boundary
conditions, defined at the physical edges of the modeled system, and where appropriate,
2) source/sink terms applied within the system (see Figure 3.4).
Boundary conditions and source/sink terms available in the 3DLEWASTE model include:
42
-------
Prescribed-concentration (Dirichlet) boundaries
Specified-flux (Cauchy) boundaries
Specified-dispersive-flux (Neumann) boundaries
Variable boundaries
Point sources
Distributed sources
Prescribed-concentration or Dirichlet boundaries are defined by prescribing dissolved
species concentrations at specified boundary nodes as:
C = Cd (xb,yb)zb,t) on Bd (3-36)
where Cd is the specified solute concentration, Bd is the portion of the system boundary
subject to a Dirichlet boundary condition, and (xb,yb,zb) is the spatial coordinate on the
boundary. Dirichlet boundaries are typically used to test computer programs by
allowing comparisons with analytical solutions. Unlike the analogous constant-head
boundaries of flow models, constant-concentration boundaries are generally poor
approximations of contaminant source terms. Bodies of fresh water located upgradient
from contaminant sources can be approximated using constant concentration nodes.
When used to define sources, specified concentrations may be constant or allowed to
vary with time, reflecting physical processes such as degradation of the source due to
radioactive decay, hydrolysis, biodegradation, or physical removal. Concentration versus
time profiles can be defined to account for seasonal or other time-variant changes in
dissolved species levels.
The specified-flux (Cauchy) boundary represents the portions of the system boundary
where infiltration can be quantified. The specified-flux boundary has many representa-
tions including: 1) infiltration due to leachate migration from a landfill or surface
impoundment, 2) application of pesticides or fertilizer to fields, and 3) the dilution
effects of rainfall or irrigation on previously applied constituents. The specified-flux
boundary condition can be written:
n
-------
of fresh water is simulated by applying the specified-flux boundary condition and setting
the mass flux rate to zero. The automatically generated term accounting for water flow
normal to the boundary will simulate the dilution due to infiltration.
Also available in 3DLEWASTE is a specified-dispersive-flux or Neumann boundary
condition of the form:
n< -9D -VC) = qn (xb>yb>zb, t) on Bn (3-38)
where q,, (M/T/L2) is the portion of the boundary flux attributable to the concentration
and Bn is the portion of the system boundary subject to a specified-dispersive-flux
boundary condition. Note that exit boundaries can be declared using this boundary
condition and letting q^O. This physically simulates mass being advected out of the
system.
For solute transport, the variable composite boundary condition represents a combined
specified-flux/dispersive-flux boundary which allows for time-variant itilltratiorjwater-
loss rates. The boundary condition during infiltration is:
n 0 (3-39b)
and mass is advected out of the system. Like the specified-flux boundary condition, the
variable boundary can represent: 1) infiltration due to leachate migration from a landfill
or surface impoundment, 2) application of pesticides or fertilizer to fields, and 3) the
dilution effects of rainfall or irrigation on previously applied constituents. When the
boundary being modeled may be either an exit or an infiltration boundary, such as a
precipitation/ evapotranspiration boundary or a seepage face, the variable boundary
condition is the proper choice. The variable boundary condition can also be used in a
manner similar to the dispersive-flux condition to simulate strictly exit nodes.
Internal source/sink terms, as represented by the term QCin in Equation 3-19 are also
accounted for in 3DLEWASTE. As with the boundary conditions, the source/sink terms
can be constant or allowed to vary with time. Both the fluid flux rate, Q, and the
injected fluid species concentration, C^, are allowed to vary with time. Two source/sink
options are available in the code. The first is a point source/sink option and the second,
a distributed source/sink option. The first option is generally used to represent
production or injection wells. The fluid fluxes in wells are represented as volumetric
water fluxes, q^ (LVT), applied at a nodal point or to better represent a screened
interval, a column of nodal points (see Figure 3.6). If vertically adjacent nodes are used
to represent the screened interval of a well, the volumetric flux must be distributed
44
-------
among the nodes. The most appropriate method of doing this is discussed in Section
3.1.2. Note that the applied fluid fluxes must match those used in the associated flow
simulation.
The distributed source option is a source intensity that is integrated over the volume of
an element. For a distributed source element, the user defines a fluid source intensity,
02 (LVT/L3), or fluid flux rate per unit volume for each distributed source element. This
option allows a user modelling a large area to approximate the influence of a well field
within an element.
Time-variant boundary conditions and source/sink flux or flux intensity rates are defined
by a series of paired time and value points. The paired data are used to assemble a
look-up table from which appropriate values are obtained using linear interpolation at
specified times of analysis. Constant values can be specified by assigning the same
value to a set of two time/data point pairs, making sure that the simulation time is fully
spanned.
3.3.3 Initial Conditions
The solution of the governing equation for solute transport in a porous medium also
requires the initialization of concentration values such that:
C = C1(x,y,z,t= 0) in R (3-40)
where C; is the initial concentration distribution and R is the region of interest. The
initial conditions are used to define the starting water quality and soil concentration
levels for determining the fate of the dissolved constituents. Besides providing a frame
of reference for transient analyses, the initial conditions are used to set the storage
parameters for Freundlich and Langmuir isotherms at the beginning of nonlinear
simulations. For transient problems, an appropriate set of initial concentration values
may either be input directly or derived from a steady-state simulation. For more
information on these options see Section 4.2.10.
3.3.4 Steady-State
When looking for a bounding solution to determine the maximum possible concentration
levels that may be reached in a solute transport problem, a steady-state option may be
employed. In the steady-state case, the time derivatives in Equation 3-19 are discarded
and the equation, including the advective term, is solved in an Eulerian or fixed-
coordinate framework. Note that any solute source prescribed as a boundary condition
or source term becomes modeled as an infinite source. For many systems this upper
bound may be highly conservative. The steady-state option is of no use if the source is
solely defined by initial conditions.
3.4 NUMERICAL APPROXIMATION IN 3DLEWASTE
The 3DLEWASTE model was developed to simulate advective-dispersive solute
transport in variably-saturated porous media. In the model, the hybrid Eulerian-
Lagrangian governing equation (Equation 3-33) is approximated using the Galerkin
45
-------
finite element technique. The time integral term in Equation 3-33 is approximated
using backwards differencing in time. The nonlinearity of the system is treated using
Picard iteration and the generated set of linearized equations is solved using a block
iterative method.
3.4.1 Galerkin Formulation
In 3DLEWASTE, the diffusion equation is approximated using the Galerkin finite
element method where the dependent variable, concentration, is approximated by a trial
function of the form:
C = NtJCt) = 1,2,. ..,n (3-41)
where Nj(Xj,t) are the three-dimensional shape functions and Cj(t) are nodal values of
concentration at time t for the n nodes of which the finite element grid is comprised (see
Figure 3.9). Substituting the trial functions into Equation 3-33 and applying the
Galerkin criterion, we generate a set of weighted residual minimization equations of the
form:
Dt
-VOD-VC) -X(6 +PbKd)C +QCin - QR = 0 (3-42)
for the linear isotherm case, where W,are the weighting functions. For the Galerkin
method, the weighting functions are the same as the shape functions and, therefore,
Equation 3-42 can be written in the form:
PbKd):B£- VJ /)t J J 3 J
R. R. Ot R. R.
where BB is the entire region boundary. The integrals given in Equation 3-44, which are
taken over the entire region being modeled, can be replaced by the summation of
integrals taken over the volumes and surfaces of individual elements of the finite
element grid. This finite element approximation generates a set of n nodal equations of
the form:
46
-------
j = l,2,...n
where
= E f(
(3-45a)
(3-45b)
B1J = f) JvN.'-eD-VN,"
dR
(3-45c)
(3-45d)
and
in N(edR + J
(3-45e)
where m is the number of elements into which the system is discretized and Ne are the
elemental shape functions. Note that for a steady-state simulation, the full Eulerian
approach is used. The Lagrangian term DC/Dt is replaced by 3C/<5t and the Eulerian
term:
E/N.T-
-VNedR
(3-46)
is added to By.
3.4.2 Solution Techniques
To solve the series of linearized ordinary differential equations represented by Equation
3-45a, the time differential is replaced by a finite difference formulation, resulting in
working equations for 3DLEWASTE of the form:
47
-------
At
where k+1 represents the current time step, k the previous time step, and At the length
of the current time step. Note that since the transient solution scheme allows only for a
backwards difference approximation the associated flow runs should also be solved using
backwards-in-time approximation.
For each time step, the solution method involves an inner iterative scheme (see Figure
3.10) which controls the block iterative method of solving the linear equations. For
simulations where the nonlinear Freundlich or Langmuir isotherms are used, the
solution method also involves an outer iterative scheme where the iterations control
convergence of the nonlinear terms in the linearized set of equations. For each
nonlinear iteration, the linearized set of equations is solved using storage terms updated
using concentration values from the previous nonlinear (outer) iteration. Storage terms
for the first iteration in a time step are based on concentration values from the previous
time step, or for the first time step, from the initial conditions. If the outer iterative
scheme becomes unstable, it may be helpful to damp the iterative changes in the
concentration. One method of damping the iterative changes is through the use of an
under-relaxation factor. Implementation of the under-relaxation factor for the outer
iterations in 3DLEWASTE is:
c;+1= (i - u0)c;+ u0cr (3-48)
where u0 is the outer under-relaxation factor and r is the iteration number. If damping
is needed, values between 0.5 and 0.9 should suffice. Acceleration or over-relaxation
(1.0
-------
SECTION 4
DATA INPUT REQUIREMENTS
4.1 3DFEMWATER INPUT SEQUENCE
This section describes how to construct a data input file for 3DFEMWATER, the
variably-saturated flow code. Background information about the code that will aid in
building an input file, such as construction of a grid or selection of boundary condition
types, is provided in Section 3.1. In addition, help in selecting values for some of the
input parameters is given in Section 5.1.
Note that maximum control parameters are associated with a number of the input
variables. These control parameters are used in the code to specify array dimensions.
For some problems, the default values set for these parameters may be too small. If so,
they can be easily changed. The maximum control parameters and their default values
are listed in Appendix C. Note also that the logical units used by 3DFEMWATER are
defined in Appendix B.
A complete input file consists of information supplied in 18 data sets. The contents and
format of each data set are listed below. When constructing an input sequence, it is
important to note that data sets 2 through 17 must be preceded by a record which
contains a description of the data set. This can be seen in the example input sequences
provided in Section 6.1. Most of the input is entered in free-format, which means that
the spacing of the input data in a record does not need to follow a set pattern. Note
that a record can consist of multiple lines, with a line defined as up to 80 columns.
The user may choose to run the model using any set of units as long as they are consis-
tently maintained in all the input. Units of mass (M), length (L), and time (T) are
indicated in the input descriptions.
4.1.1 Data Set 1: Title of the Simulation Run
One record with FORMAT(I5,A70,2X,I2,2I1) per problem. This record contains
the following variables:
1. NPROB = Problem number (columns 1-5).
2. TITLE = Array for the title of the problem. It may contain up to 70 charac-
ters (columns 6- 75).
3. IGEOM = Integer indicating if (1) the geometry, boundary and pointer
arrays are to be printed and if (2) the boundary and pointer arrays are to
be computed or read via logical units (column 78). If IGEOM is an even
number, geometry, boundary and pointer arrays will not be printed. If
49
-------
IGEOM is an odd number, they will be printed. If IGEOM is less than or
equal to 1, boundary arrays will be computed and written on logical unit
LUBAR, but if IGEOM is greater than 1, boundary arrays will be read via
logical unit LUBAR. If IGEOM is less than or equal to 3, pointer arrays
will be computed and written on logical unit LUPAR, but if IGEOM is
greater than 3, pointer arrays will be read via logical unit LUPAR. In
summary:
IGEOM = Even No. Print the geometry, boundary, and pointer arrays.
IGEOM = Odd No. Do not print the arrays.
IGEOM <. 1 Compute and write boundary and pointer arrays.
1 < IGEOM <. 3 Read boundary arrays, compute and write pointer
arrays (not used under normal conditions).
IGEOM > 3 Read boundary and pointer arrays.
4. IBUG = Integer indicating if diagnostic output is desired to help determine
problems encountered while executing the code (column 79);
0 = no,
1 = yes.
5. ICHNG = Integer control number indicating if the cyclic change of rain-
fall-seepage nodes is to be printed (column 80);
0 = no,
1 = yes.
4.1.2 Data Set 2: Basic Integer Parameters
One record with FREE-FORMAT per problem. It contains:
1. NNP = Number of nodal points. .
2. NEL = Number of elements.
3. NMAT = Number of material types.
4. NCM = Number of elements with material property correction.
5. NTI = Number of time steps or time increments (see notes at the end of
Data Set 2).
6. KSS = Steady-state control;
0 = steady-state solution,
1 = transient-state solution (see note at the end of Data Set 2).
7. NMPPM = Number of material properties per material; this parameter
should be set equal to 6 in the present version of the code (see Data Set 5).
8. KGRAV = Gravity term control;
0 = no gravity term,
1 = gravity term included.
50
-------
9. ILUMP = Mass lumping control;
0 = no,
1 = yes.
10. IMID = Mid-difference control;
0 = no,
1 = yes.
11. NITER = Number of iterations allowed for solving the non-linear equation.
12. NCYL = Number of cycles permitted for iterating rainfall-seepage boundary
conditions per time step.
13. NDTCHG = Number of times the time-step size will be reset to the initial
time-step size; NDTCHG should be >. 1 (see Section 5.1.2.10).
14. NPITER = Number of iterations for a pointwise solution.
**** NOTE: NTI can be computed by NTI =11 + 1 + 12 + 1, where 11 is the
largest integer not exceeding Log(DELMAX/DELT)/Log(l+CHNG),
12 is the largest integer not exceeding (RTIME-DELT*((1+CHNG)
**(I1 + 1)-1)/CHNG)/DELMAX, RTIME is the real simulation time,
and DELMAX, DELT.and CHNG are defined in data set 3.
**** NOTE: A steady-state option may be used to provide either the final state
of a system under study or the initial condition for a transient-state
calculation. In the former case, KSS = 0 and NTI = 0 in this data
set. In the latter case, KSS = 0 and NTI > 0. If KSS > 0, there
will be no steady-state calculation.
4.1.3 Data Set 3: Basic Real Parameters
One record with FREE-FORMAT per problem. It contains:
1. BELT = Initial time step size, (T).
2. CHNG = Fractional change in the time-step size in each subsequent time
increment, (dimensionless decimal-point value).
3. DELMAX = Maximum value of BELT, (T).
4. TMAX = Maximum simulation time, (T).
5. TOLA = Steady-state convergence criterion, (L).
6. TOLB = Transient-state convergence criterion, (L).
7. RHO = Density of water, (M/L3).
8. GRAY = Acceleration of gravity, (L/T2); (e.g., 32.17 ft/s2or 9.81 m/s2).
51
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9. VISC = Dynamic viscosity of water, (M/UT).
10. W = Time derivative weighting factor;
0.5 = Crank-Nicolson central and/or mid-difference,
1.0 = backward difference.
11. OME = Iteration parameter for solving the nonlinear matrix equation,
0.0 < OME < 1.0 = under-relaxation,
1.0 = exact relaxation,
1.0 < OME < 2.0 = over-relaxation.
12. OMI = Relaxation parameter for solving the linearized matrix equation
pointwise;
0.0 < OMI < 1.0 = under relaxation,
1.0 = exact relaxation,
1.0 < OMI < 2.0 = over relaxation.
4.1.4 Data Set 4: Printer and Disk Storage Control and
Times for Step Size Resetting
Three records are needed per problem. The first two records are formatted input
with FORMAT (211). The third record is a FREE-FORMAT input. The number of
lines for the first two records depends on the value of NTI, the number of time
increments. The number of lines for the third record depends on the value of
NDTCHG, the number of times to reset the time-step size.
Record 1 - FORMAT(211): This record contain the following variables:
1. KPRO = Printer control for steady-state and initial conditions;
0 = print nothing,
1 = print the values for the variables FLOW, FRATE, and TFLOW,
2 = print values above plus pressure head H,
3 = print values above plus total head,
4 = print values above plus moisture content,
5 = print values above plus Darcy velocity.
2. KPR(I) = Printer control for the I-th (1=1,2,.... NTI) time step;
0 = print nothing,
1 = print the values for the variables FLOW, FRATE, and TFLOW,
2 = print values above plus pressure head H,
3 = print values above plus total head,
4 = print values above plus moisture content,
5 = print values above plus Darcy velocity.
Record 2 - FORMAT(211): This record can be used to store 3DFEMWATER
output in a binary file for use in plotting or as input to 3DLEWASTE. It
contains the following variables:
52
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1. KDSKO = Auxiliary storage control for the steady-state or initial condition;
0 = no storage,
1 = store on logical unit LUSTO.
2. KDSK(I) = Auxiliary storage control for the I-th (1=1,2,.... NTI) time step;
0 = no storage,
1 = store on logical unit LUSTO.
Record 3- FREE-FORMAT: This record contains the following variables:
1. TDTCH(I) = Time when the I-th (1 = 1,2, NDTCHG) time-step-size
resetting is needed.
4.1.5 Data Set 5: Material Properties
Either hydraulic conductivity or permeability can be input in this data set. The
flag KCP in data set 6A is used to indicate which of the two is being used. A
total of NMAT records are needed per problem, one for each material.
Record I (1 = 1, 2, .... NMAT) - FREE-FORMAT: Each record contains following
variables:
1. PROP (1,1) = Saturated xx-hydraulic conductivity or permeability of the
medium I, (L/T or L2).
2. PROP(2,1) = Saturated yy-hydraulic conductivity or permeability of the
medium I, (L/T or L2).
3. PROP(3,1) = Saturated zz-hydraulic conductivity or permeability of the
medium I, (L/T or L2).
4. PROP(4,1) = Saturated xy-hydraulic conductivity or permeability of the
medium I, (L/T or L2).
5. PROP(5,1) = Saturated xz-hydraulic conductivity or permeability of the
medium I, (L/T or L2).
6. PROP(6,I) = Saturated yz-hydraulic conductivity or permeability of the
medium I, (L/T or L2).
4.1.6 Data Set 6: Soil Property Parameters
6A. Soil Property Control Integers
One record per problem. This record is FREE-FORMATTED and contains the
following variables:
53
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1. KSP = Soil property input control;
0 = analytical input,
1 = tabular data input.
2. NSPPM = Number of points in the tabular soil property functions
when KSP = 1. The number of parameters needed to specify the
analytical soil functions per material when KSP = 0. (For analytical
soil functions, NSPPM = 5 in the current version of the code.)
3. KCP = Permeability input control;
0 = input saturated hydraulic conductivity,
1 = input saturated permeability.
6B. Analytical Soil Parameters
This subdata set is needed if and only if KSP = 0. NMAT records are needed,
one for each material type.
Record I (I = 1, 2, NMAT) - FREE-FORMAT: Each record contains the
following variables:
1. THPROP(1,I) = Residual moisture (water) content for material I, ().
2. THPROP(2,I) = Saturated moisture (water) content for material I, ().
3. THPROP(3,I) = Air entry pressure head for material I, (L).
4. THPROP(4,I) = Van Genuchten empirical coefficient alpha for material
I, (1/L).
5. THPROP(5,I) = Van Genuchten empirical coefficient beta for material
I, (")-
6C. Soil Properties in Tabular Form
This subdata set is needed if and only if KSP = 1. Four sets of records are
needed one each for pressure, water-content, relative conductivity (or relative
permeability), and water capacity, respectively. Each set contains NMAT records,
one for each material type. Thus the total number of records for this subdata set
is 4*NMAT. The number of lines in each record is determined by the input
parameter NSPPM, defined in data set 6A.
Record I (I = 1, 2, NMAT) - FREE-FORMAT: Each record contains the
following variables:
1. HPROP(1,I) = Tabular value of pressure head for the first data point of
material I, (L).
54
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2. HPROP(2,I) = Tabular value of pressure head for the second data point
of material I, (L).
NSPPM. HPROP(NSPPMJ) = Tabular value of pressure head for the NSPPM-th
data point of material I, (L).
Record (NMAT + I) (I = 1, 2, NMAT) - FREE-FORMAT: Each record contains
the following variables:
1. THPROP(1,I) = Tabular value of moisture-content for the first data
point in material I, ().
2. THPROP(2,I) = Tabular value of moisture-content for the second data
point in material 1, ().
NSPPM. THPROP(NSPPMJ) = Tabular value of moisture-content for the
NSPPM-th data point in material 1, (-).
Record (2*NMAT + I) (1 = 1, 2, NMAT) - FREE-FORMAT: Each record
contains the following variables:
1. AKPROP(IJ) = Tabular value of relative conductivity for the first data
point in material I, (--).
2. AKPROP(2,I) = Tabular value of relative conductivity for second data
point in material I, ().
NSPPM. AKPROP(NSPPMJ) = Tabular value of relative conductivity for the
NSPPM-th data point in material I, (-).
Record (3*NMAT + I) (1 = 1, 2, NMAT) - FREE-FORMAT: Each record
contains the following variables:
1. CAPROP(IJ) = Tabular value of moisture-content capacity for the first
data point in material I, (1/L).
2. CAPROP(2,I) = Tabular value of moisture-content capacity for the
second data point in material I, (1/L).
55
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NSPPM. CAPROP(NSPPMJ) = Tabular value of moisture content capacity for
the NSPPM-th data point in material I, (1/L).
4.1.7 Data Set 7: Nodal Point Coordinates
Coordinates for NNP nodes, specified in data set 2, are needed. Usually a total of
NNP records are required. However, if a group of subsequent nodes appears in
regular pattern, an automatic generation input option can be used.
Each record is FREE-FORMATTED and contains the following variables:
1. NI = Node number of the first node in the sequence.
2. NSEQ = NSEQ subsequent nodes will be automatically generated.
3. NAD = Increment of node number for each of the NSEQ subsequent nodes.
4. XNI = X-coordinate of node NI, (L).
5. YNI = Y-coordinate of node NI, (L).
6. ZNI = Z-coordinate of node NI, (L).
7. XAD = Increment of x-coordinate for each of the NSEQ subsequent nodes,
(L).
8. YAD = Increment of y-coordinate for each of the NSEQ subsequent nodes,
(L).
9. ZAD = Increment of z-coordinate for each of the NSEQ subsequent nodes,
(L).
**** NOTE: A record with nine zeroes must be used to signal the end of this
data set.
4.1.8 Data Set 8: Subregional Data
8A. Subregion Control Integer
One FREE-FORMATTED record is needed for this subdata set. It contains the
following variable:
1. NREGN = Number of subregions.
8B. Number of Nodes in Each Subregion
Normally, NREGN records are required. However, if the sequence of node num-
bers follows a regular pattern between sequential subregions, the automatic
generation input option can be used.
56
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Each record is FREE-FORMATTED and contains the following five variables:
1. NK = Subregion number of the first subregion in a sequence.
2. NSEQ = Number of subsequent subregions which will be automatically
generated.
3. NKAD = Increment of NK in each of the NSEQ subsequent subregions.
4. NODES = Number of nodes in the subregion NK.
5. NOAD = Increment of NODES in each of the NSEQ subsequent subre-
gions.
NO IE: A record with five zeroes must be used to end the input of
this subdata set.
8C. Mapping between Global Nodes and Subregion Nodes
This subdata set should be repeated NREGN times, once for each subregion. For
each subregion, normally, the number of records equals the number of nodal
points in the subregion. Automatic generation can be used, however, if the
sequence of subregional node numbers follows a regular pattern.
Each record is FREE-FORMATTED and contains the following five variables:
1. LI = Local node number of the first node in a sequence.
2. NSEQ = Number of subsequent local nodes which will be generated
automatically.
3. LIAD = Increment of LI for each of the NSEQ subsequent nodes.
4. NI = Global node number of local node LI.
5. NIAD = Increment of NI for each of the NSEQ subsequent nodes.
NO IE. A record with five zeroes must be used to signal the end of
this subdata set.
NO IE: Local node numbers have values between one and the total
number of nodes in a subregion (i.e., 1,2,.. ., NODES). Global
node numbers are associated with the entire grid and are
entered using data set 7.
4.1.9 Data Set 9: Element Incidence
Element incidence for NEL elements, specified in data set 2, are needed.
Usually, a total of NEL records are needed. However, if a sequence of element
numbers follows a regular pattern, the automatic generation input option can be used.
57
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Each record is FREE-FORMATTED and contains the following variables:
1. MI = Global element number of the first element in a sequence.
2. NSEQ = Number of subsequent elements which will be automatically
generated.
3. MIAD = Increment of MI for each of the NSEQ subsequent elements.
4. IE(MI,1) = Global node number of the first node of element MI.
5. IE(MI,2) = Global node number of the second node of element MI.
6. IE (MI, 3) = Global node number of the third node of element MI.
7. IE(MI,4) = Global node number of the fourth node of element MI.
8. IE(MI,5) = Global node number of the fifth node of element MI.
9. IE (MI, 6) = Global node number of the sixth node of element MI.
10. IE (MI, 7) = Global node number of the seventh node of element MI.
11. IE(MI,8) = Global node number of the eighth node of element MI.
12. IEMAD = Increment of IE(MI,1) through IE(MI,8) for each of the NSEQ
elements.
**** NOTE: IE(MI,1) - IE(MI,8) are numbered according to the convention
shown in Figure 4.1. The first four nodes start from the front,
lower, left corner and progress around the bottom element surface
in a counterclockwise direction. The other four nodes begin from
the front, upper, left corner and progress around the top element
surface in a counterclockwise direction.
z
i,
Figure 4.1. Node numbering convention for the elements.
58
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4.1.10 Data Set 10: Material Type Correction
This data set is required only if NCM, defined in data set 2, is greater than zero.
Normally, NCM records are required. However, if a group of element numbers
follow a regular pattern, the automatic generation input option may be used.
Each record is FREE-FORMATTED and contains the following variables:
1. MI = Global element number of the first element in the sequence.
2. NSEQ = Number of subsequent elements which will be generated automati-
cally.
3. MAD = Increment of element number for each of the NSEQ subsequent
elements.
4. MITYP = Type of material for element MI.
5. MTYPAD = Increment of MITYP for each of the NSEQ subsequent
elements.
**** NOTE: A record with five zeroes must be used to signal the end of this
data set.
4.1.11 Data Set 11: Card Input for Initial or Pre-Initial Conditions
NNP records (i.e., one record for each node) are normally needed. However, if a
sequence of node numbers follows a regular pattern, automatic generation can be
used.
Each record is FREE-FORMATTED and contains the following variables:
1. NI = Global node number of the first node in the sequence.
2. NSEQ = Number of subsequent nodes which will be generated
automatically.
3. NAD = Increment of node number for each of the NSEQ nodes.
4. HNI = Initial or pre-initial pressure head of node NI, (L).
5. HAD = Increment of initial or pre-initial head for each of the NSEQ nodes,
(L).
6. HRD = Geometrical increment of HNI for each of the NSEQ subsequent
nodes; (i.e, HNI**HRD).
**** NOTE: A record with six zeroes must be used to signal the end of this data
set.
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**** NOTE: The initial condition for a transient calculation may be obtained in
two different ways: 1) it can be read directly from data set 11, or
2) the code can perform a steady-state simulation using
time-invariant boundary conditions before automatically beginning
the transient computations. For the first case, both KSS and NTI
in data set 2 should be greater than zero. In the latter case, KSS =
0 and NTI > 0 and data set 1 1 is used to input the pre-initial
condition, which is required as the starting condition for the
steady-state iteration. In order to obtain a steady-state solution,
both KSS and NTI are set equal to zero and data set 1 1 supplies
the starting condition for the steady-state solution.
4.1.12 Data Set 12: Integer Parameters for Source and Boundary Conditions
One record per problem is needed. This record is FREE-FORMATTED and
contains the following variables:
1. NSEL = Number of distributed source/sink elements.
2. NSPR = Number of distributed source/sink profiles (i.e., time histories).
3. NSDP = Number of data points in each of the NSPR source/sink profiles.
4. KSAI = Option for the distributed source/sink profiles to be input
analytically. This variable should be set equal to zero in the current
version of the code.
5. NWNP = Number of well or point source/sink nodes.
6. NWPR = Number of well or point source/sink profiles (i.e., time histories).
7. NWDP = Number of data points in each of the NWPR profiles.
8. KWAI = Option for the well source/sink profiles to be input analytically.
This variable should be set equal to zero in the current version of the code.
9. NDNP = Number of fixed-head (Dirichlet) nodes (NDNP should be > 1).
10. NDPR = Number of fixed-head profiles (i.e., time histories) (NDPR should
be > 1).
11. NDDP = Number of data points in each freed-head profile (NDDP should be
12. KDAI = Option for the freed-head boundary value profiles to be input
analytically. This variable should be set equal to zero in the current
version of the code.
13. NVES = Number of variable composite (rainfall/evaporation-seepage)
boundary element sides.
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14. NVNP = Number of variable composite boundary nodal points.
15. NRPR = Number of variable composite profiles (i.e., time histories).
16. NRDP = Number of data points in each of the NRPR profiles.
17. KRAI = Option for the variable composite profiles to be input analytically.
This variable should be set equal to zero in the current version of the code.
18. NCES = Number of specified-flux (Cauchy) boundary element sides.
19. NCNP = Number of specified-flux nodal points.
20. NCPR = Number of specfled-flux profiles (i.e., time histories).
21. NCDP = Number of data points in each of the NCPR profiles.
22. KCAI = Option for the specified-flux profiles to be input analytically. This
variable should be set equal to zero in the current version of the code.
23. NNES = Number of specified-pressure-head gradient (Neumann) boundary
element sides.
24. NNNP = Number of specified-pressure-head gradient nodal points.
25. NNPR = Number of specified-pressure-head gradient flux profiles (i.e., time
histories).
26. NNDP = Number of data points in each of the NNPR profiles.
27. KNAI = Option for the specified-pressure-head gradient profiles to be input
analytically. This variable should be set equal to zero in the current
version of the code.
4.1.13 Data Set 13: Distributed and Point Sources/Sinks
This data set is used to supply data for both distributed sources/sinks and well
(point) sources/sinks.
13A. Distributed Sources/Sinks
The following three subdata sets are needed if and only if NSEL in data set 12 is
greater than zero. The first subdata set is used to specify the distributed
source/sink profiles. The second subdata set is used to read the global element
numbers of the distributed source/sink elements. The third subdata set is used
to assign a source/sink profile to each distributed source/sink element.
(a) Source/Sink Profiles
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There will be NSPR (see data set 12) records in this subdata set. The number of
lines in each record depends on the value of NSDP, defined in data set 12.
Record I (1 = 1, 2, NSPR) - FREE-FORMAT: Each record contains the follow-
ing variables:
1. TSOSF(U) = Time of the first data point in the I-th profile, (T).
2. SOSF(1,I) = Source/sink value (as flux rate per unit volume of
element) of the first data point in the I-th profile, (LVT/LVL); positive
for a source and negative for a sink.
3. TSOSF(2,I) = Time of the second data point in the I-th profile, (T).
4. SOSF(2,I) = Source/sink value of the second data point in the I-th
profile, (LVT/LVL); positive for a source and negative for a sink.
Up to NSDP data points.
(b) Global Element Number of All Distributed Source/Sink Elements
One record is needed for this subdata set. The number of lines in this record
depends on NSEL, defined in data set 12. The record is FREE-FORMATTED
and contains the following variables:
1. MSEL(l) = Global element number of the first distributed source/sink
element.
2. MSEL(2) = Global element number of the second distributed
source/sink element.
Up to NSEL numbers.
(c) Source/Sink Profile Type Assigned to Each Element
Usually NSEL records are needed. However, automatic generation can be used.
Each record is FREE-FORMATTED and contains the following variables:
1. MI = Compressed element number of the first element in the
sequence.
2. NSEQ = Number of elements which will be generated automatically.
3. MAD = Increment of element number for each of the NSEQ subse-
quent elements.
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4. MITYP = Source/sink profile type associated with element ML
5. MTYPAD = Increment of MITYP for each of the NSEQ subsequent
elements.
**** NOTE: A record with five zeroes must be used to signal the end of
this subdata set.
**** NOTE: Compressed element numbers have values between one and
the total number of distributed source/sink elements. Com-
pressed element one corresponds to the first element listed
in 13A(b), compressed element two corresponds to the second
global element, etc.
13B. Point (Well) Source/Sink
The following three subdata sets are needed if and only if NWNP in data set 12
is greater than zero. The first subdata set is used to specify the point source/sink
profiles. The second subdata set is used to read the global node numbers of the
point source/sink nodes. The third subdata set is used to assign a source/sink
profile to each point source/sink node.
(a) Source/Sink Profiles
There will be NWPR (see data set 12) records in this subdata set. The number of
lines in each record depends on NWDP, defined in data set 12.
Record I (I = 1, 2, NWPR) - FREE-FORMAT: Each record contains the
following variables:
1. TWSSF(U) = Time of the first data point in the I-th profile, (T).
2. WSSF(1,I) = Source/sink flow rate of the first data point in the I-th
profile, (LVT); positive for a source and negative for a sink.
3. TWSSF(2,I) = Time of the second data point in the I-th profile, (T).
4. WSSF(2,I) = Source/sink flow rate of the second data point in the I-th
profile, (LVT); positive for a source and negative for a sink.
Up to NWDP data points.
(b) Global Node Number of All Point (Well) Source/Sink Nodes
One record is needed for this subdata set. The number of lines in this record
depends on NWNP, defined in data set 12. The record is FREE-FORMATTED
and contains the following variables:
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1. NPW(l) = Global node number of the first point source/sink node.
2. NPW(2) = Global node number of the second point source/sink node.
Up to NWNP numbers.
(c) Source/Sink Profile Type for Each Node
Usually NWNP records are needed. However, automatic generation can be used.
Each record is FREE-FORMATTED and contains the following variables:
1. NI = Compressed point source/sink node number of the first node in
the sequence.
2. NSEQ = Number of subsequent nodes which will be generated auto-
matically.
3. NAD = Increment of NI for each of the NSEQ subsequent nodes.
4. NITYP = Source/sink profile type associated with node NI.
5. NTYPAD = Increment of NITYP for each of the NSEQ subsequent
nodes.
NOTE: A record with five zeroes must be used to signal the end of
this subdata set.
4.1.14 Data Set 14: Variable Composite (Rainfall/Evaporation-Seepage) Boundary
Condition
The following six subdata sets are required if and only if NVES in data set 12 is
greater than zero. The first subdata set is used to specify the rainfall/evap-
oration profiles. The second subdata set is used to assign the type of rain-
fall/evaporation profile to each of the variable composite boundary element sides.
The third subdata set is used to specify the variable composite boundary
element sides. The fourth subdata set is used to read the global nodal numbers
of all the variable composite boundary nodes. The fifth subdata set is used to
read the pending depth for each of the nodes. The sixth subdata set is used to
read the allowed minimum pressure for each of the nodes.
14A. Rainfall/Evaporation-Seepage Profiles
There will be NRPR records (see data set 12) in this subdata set. The number
of lines in each record depends on NRDP, defined in data set 12.
Record I (I = 1, 2, NRPR) - FREE-FORMAT: Each record contains the follow-
ing variables:
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1. TRF(l.I) = Time of the first data point in the I-th profile, (T).
2. RF(1,I) = Rainfall/evaporation rate of the first data point in the I-th
profile, (L/T).
3. TRF(2,I) = Time of the second data point in the I-th profile, (T).
4. RF(2,I) = Rainfall/evaporation rate of the second data point in the I-th
profile, (L/T).
Up to NRDP data points.
14B. Rainfall/Evaporation-Seepage Profile Type Assigned to Each Boundary
Element Side
At most, NVES (see data set 12) records are needed. However, automatic
generation can be used.
Record I (I = 1, 2, ) - FREE-FORMAT: Each record contains the following
variables:
1. MI = Compressed variable boundary element side number of the first
element side in a sequence.
2. NSEQ = Number of subsequent variable boundary element sides which
will be generated automatically.
3. MIAD = Increment of MI for each of the NSEQ subsequent variable
boundary element sides.
4. MITYP = Type of rainfall/evaporation-seepage profile assigned to side
MI.
5. MTYPAD = Increment of MITYP for each of the NSEQ subsequent
sides.
NO IE: A record with five zeroes must be used to signal the end of
this subdata set.
14C. Specification of Variable Composite Boundary Element Sides
Normally, NVES records are required, one each for a variable boundary element
side. However, if a sequence of variable composite boundary element side
numbers follows a regular pattern, automatic generation may be used.
Record I (I = 1, 2, ....)- FREE-FORMAT: Each record contains the following
variables:
65
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1. MI = Compressed variable boundary element side number of the first
element side in a sequence.
2. NSEQ = Number of subsequent variable boundary element sides which
will be generated automatically.
3. MIAD = Increment of MI for each of the NSEQ subsequent variable
boundary element sides.
4. II = Global node number of the first node of element side ML
5. 12 = Global node number of the second node of element side MI.
6. 13 = Global node number of the third node of element side MI.
7. 14 = Global node number of the fourth node of element side MI.
8. HAD = Increment of II for each of the NSEQ subsequent variable
boundary element sides.
9. I2AD = Increment of 12 for each of the NSEQ subsequent variable
boundary element sides.
10. 13AD = Increment of 13 for each of the NSEQ subsequent variable
boundary element sides.
11. WAD = Increment of 14 for each of the NSEQ subsequent variable
boundary element sides.
NO IE. A record with 11 zeroes must be used to signal the end of
this subdata set.
14D. Global Node Number of All Variable Composite Boundary Nodes
At most, NVNP records (see data set 12) are needed for this subdata set, one for
each variable boundary node.
Record I (I = 1, 2, . . . . ) - FREE-FORMAT: Each record contains five variables:
1. NI = Compressed variable boundary node number of the first node in
the sequence.
2. NSEQ = Number of subsequent nodes which will be generated
automatically.
3. NIAD = Increment of NI for each of the NSEQ nodes.
4. NODE = Global node number of node NI.
5. NODEAD = Increment of NODE for each of the NSEQ nodes.
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NO IE: A record with five zeroes must be used to signal the end of
this subdata set.
14E. Pending Depth Allowed for Each Variable Composite Boundary Node
Normally, NVNP records (see data set 12) are needed. However, if a sequence of
node numbers follows a regular pattern of pending depth, automatic generation
is used.
Record I (I = 1, 2, ....)- FREE-FORMAT: Each record contains the following
variables:
1. NI = Compressed variable boundary node number of the first node in a
sequence.
2. NSEQ = Number of subsequent nodes which will be generated
automatically.
3. NIAD = Increment of NI for each of the NSEQ subsequent nodes.
4. HCONNI = Ponding depth of node NI, (L).
5. HCONAD = Increment of HCONNI for each of the NSEQ nodes, (L).
**** NOTE: A record with five zeroes must be used to signal the end of this
subdata set.
14F. Minimum Pressure Head Allowed for Each Variable Composite Boundary
Node
Normally, NVNP records are needed. However, if a sequence of node numbers
follows a regular pattern of minimum pressure head, automatic generation is
used.
Record I (I = 1, 2, ....)- FREE-FORMAT: Each record contains the following
variables:
1. NI = Compressed variable boundary node number of the first node in a
sequence.
2. NSEQ = Number of subsequent nodes which will be generated
automatically.
3. NIAD = Increment of NI for each of the NSEQ subsequent nodes.
4. HMINNI = Minimum pressure head allowed for node NI, (L).
67
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5. HMINAD = Increment of HMINNI for each of the NSEQ nodes, (L).
**** NOTE: A record with five zeroes must be used to signal the end of
this subdata set.
4.1.15 Data Set 15: Fixed-Head (Dirichlet) Boundary Condition
This data set is required only if NDNP in data set 12 is greater than zero. It
consists of three subdata sets. The first subdata set is used to specify the fixed-
head profiles. The second subdata set is used to read the global node numbers of
the freed-head boundary nodes. The third subdata set is used to assign a head
profile to each Dirichlet boundary node.
ISA. Fixed-Head Profiles
There will be NDPR (see data set 12) records in this subdata set. The number of
lines in each record depends on NDDP, the number of data points in each profile.
Record I (I = 1, 2, NDPR) - FREE-FORMAT: Each record contains the
following variables:
1. THDBF(IJ) = Time of the first data point in the I-th profile, (T),
2. HDBF(1,I) = Total head of the first data point in the I-th profile, (L).
3. THDBF(2,I) = Time of the second data point in the I-th profile, (T).
4. HDBF(2,I) = Total head of the second data point in the I-th profile, (L).
Up to NDDP data points.
15B. Global Node Number of All the Dirichlet Nodes
One FREE-FORMATTED record is needed for this subdata set. The number of
lines in this record depends on NDNP, defined in data set 12.
1. NPDB(l) = Global node number of the first compressed Dirichlet node.
2. NPDB(2) = Global node number of the second compressed Dirichlet
node.
Up to NDNP numbers.
15C. Type of Head Profile Assigned to Each Fixed-Head Node
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Normally one record per Dirichlet node (i.e., a total of NDNP records) is needed.
However, if the Dirichlet node numbers follow a regular pattern, automatic
generation may be used.
Record I (1 = 1, 2, ) - FREE-FORMAT: Each record contains the following
variables:
1. NI = Compressed Dirichlet node number of the first node in the se-
quence.
2. NSEQ = Number of subsequent Dirichlet nodes which will be
generated automatically.
3. NIAD = Increment of NI for each of the NSEQ nodes.
4. NITYP = Type of total head profile assigned to node NI and NSEQ
subsequent nodes.
5. NTYPAD = Increment of NITYP for each of the NSEQ subsequent
nodes.
**** NOTE: A record with five zeroes must be used to signal the end of
this subdata set.
4.1.16 Data Set 16: Specified-Flux (Cauchy) Boundary Condition
This data set is required only if NCES in data set 12 is greater than zero. Four
subdata sets are required. The first subdata set is used to read the specified-flux
profiles. The second subdata set is used to read the type of specified-flux profile
assigned to each of the specified-flux boundary element sides. The third subdata
set is used to read the specified-flux boundary element sides. The fourth subdata
set is used to read the global nodes associated with the specified-flux boundaries.
16A. Specified-Flux Profiles
There will be NCPR records (see data set 12) in this subdata set. The number of
lines in each record depends on NCDP, defined in data set 12.
Record I (I = 1, 2, .... NCPR) - FREE-FORMAT: Each record contains the follow-
ing variables:
1. TQCBF(1,I) = Time of the first data point in the I-th profile, (T).
2. QCBF(1,I) = Normal specified-flux of the first data point in the I-th
profile, (LVT/L2); positive out from the region, negative into the region.
3. TQCBF(2,I) = Time of the second data point in the I-th profile, (T).
69
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4. QCBF(2,I) = Normal specified-flux of the second data point in the I-th
profile, (LVT/L2); positive out from the region, negative into the region.
Up to NCDP data points.
16B. Type of Specified-Flux Profile Assigned to Each Boundary Element Side
At most, NCES records (see data set 12) are needed. However, automatic genera-
tion can be used.
Record I (I = 1, 2, . . . . ) - FREE-FORMAT: Each record contains the following
variables:
1. MI = Compressed specified-flux boundary element side number of the
first side in the sequence.
2. NSEQ = Number of sides which will be generated automatically.
3. MIAD = Increment of MI for each of the NSEQ sides.
4. MITYP = Type of specified-flux profile assigned to side MI.
5. MTYPAD = Increment of MITYP for each of the NSEQ sides.
**** NOTE: A record with five zeroes must be used to signal the end of
this subdata set.
16C. Specified-Flux Boundary Element Sides
Normally, NCES records are required, one for each specified-flux boundary
element side. However, if a group of specified-flux boundary element side
numbers follows a regular pattern, automatic generation can be used.
Record I (I = 1, 2, ....)- FREE-FORMAT: Each record contains the following
variables:
1. MI = Compressed specified-flux boundary element side number of the
first element side in a sequence.
2. NSEQ = Number of subsequent specified-flux element sides which will
be generated automatically.
3. MIAD = Increment of MI for each of the NSEQ subsequent sides.
4. II = Global node number of the first node of element side MI.
5. 12 = Global node number of the second node of element side MI.
70
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6. I3=Global node number of the third node of element side MI.
7. 14 = Global node number of the fourth node of element side MI.
8. HAD = Increment of II for each of the NSEQ subsequent element
sides.
9. I2AD = Increment of 12 for each of the NSEQ subsequent element
sides.
10. I3AD = Increment of 13 for each of the NSEQ subsequent element sides.
11. I4AD = Increment of 14 for each of the NSEQ subsequent element sides.
**** NOTE: A record with 11 zeroes must be used to end this subdata set.
16D. Global Node Number of All Compressed Specified-Flux Nodes
One FREE-FORMATTED record is needed for this subdata set. The number of
lines in this record depends on NCNP, defined in data set 12.
1. NPCB(l) = Global node number of the first compressed specified-flux
node.
2. NPCB(2) = Global node number of the second compressed specified-flux
node.
Up to NCNP numbers.
4.1.17 Data Set 17: Specified-Pressure-Head Gradient (Neumann)
Boundary Condition
This data set is required if and only if NNNP in data set 12 is greater than zero.
It consists of four subdata sets. The first subdata set is used to specify the speci-
fied-pressure-head gradient flux profiles. The second subdata set is used to
assign a profile to each boundary element side. The third subdata set is used to
read the global element sides of the specified-pressure-head gradient boundary
elements. The fourth subdata set is used to read the global node numbers
associated with the specified-pressure-head gradient boundaries.
17A. Prescribed Pressure-Head Gradient Flux Profiles
There will be NNPR records (see data set 12) in this subdata set. The number of
lines in each record depends on NNDP, defined in data set 12.
Record I (I = 1, 2, NNPR) - FREE-FORMAT: Each record contains the
following variables:
71
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1. TQNBF(IJ) = Time of the first data point in the I-th profile, (T).
2. QNBF(1,I) = Normal specified-pressure-head gradient flux of the first
data point in the I-th profile, (LVT/L2); positive out from the region,
negative into the region.
3. TQNBF(2,I) = Time of the second data point in the I-th profile, (T).
4. QNBF(2,I) = Normal specified-pressure-head gradient flux of the
second data point in the I-th profile, (LVT/L2); positive out from the
region, negative into the region.
Up to NNDP data points.
17B. Type of Specified-Pressure-Head Gradient Flux Profile Assigned to Each
Boundary Element Side
At most, NNES records are needed (see data set 12). However, automatic
generation can be used.
Record I (I = 1, 2, . . . . ) - FREE-FORMAT: Each record contains the following
variables:
1. MI = Compressed specified-pressure-head gradient element side
number of the first side in the sequence.
2. NSEQ = Number of subsequent sides which will be generated
automatically.
3. MIAD = Increment of MI for each of the NSEQ sides.
4. MITYP = Type of specified-pressure-head gradient flux profile assigned
to side MI.
5. MTYPAD = Increment of MITYP for each of the NSEQ sides.
**** NOTE: A record with five zeroes must be used to signal the end of
this subdata set.
17C. Specified-Pressure-Head Gradient Boundary Element Sides
Normally, NNES records are required, one for each specified-pressure-head
gradient boundary element side. However, if a group of specified-pressure-head
gradient boundary element side numbers follow a regular pattern, automatic
generation may be used.
Record I (I = 1, 2, ....)- FREE-FORMAT: Each record contains the following
variables:
72
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1. MI = Compressed specified-pressure-head gradient boundary element
side number of the frost side in sequence.
2. NSEQ = Number of subsequent sides which will be generated
automatically.
3. MIAD = Increment of MI for each of the NSEQ subsequent sides.
4. II = Global node number of the first node of element side MI.
5,0 12 = Global node number of the second node of element side MI.
6. 13 = Global node number of the third node of element side MI.
7. 14 = Global node number of the fourth node of element side MI.
8. HAD = Increment of II for each of the NSEQ subsequent element
sides.
9. 12AD = Increment of 12 for each of the NSEQ subsequent element
sides.
10. I3AD = Increment of 13 for each of the NSEQ subsequent element sides.
11. WAD = Increment of 14 for each of the NSEQ subsequent element sides.
**** NOTE: A record with 11 zeroes must be used to end this subdata set.
17D. Global Node Number of All Compressed Specified-Pressure-Head Gradient
Nodes
One FREE-FORMATTED record is needed for this subdata set. The number of
lines in this record depends on NNNP, defined in data set 12.
1. NPNB(l) = Global node number of the first compressed specified-
pressure-head gradient node.
2. NPNB(2) = Global node number of the second compressed specified-
pressure-head gradient node.
Up to NNNP numbers.
4.1.18 Data Set 18: End of Job
If another problem is to be run, then input begins again with input data set 1. If
termination of the job is desired, a blank line must be inserted at the end of the
data set.
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4.2 3DLEWASTE INPUT SEQUENCE
This section describes how to construct a data input file for 3DLEWASTE, the transport
code. Background information about the code that will aid in building an input file,
such as the types of adsorption isotherms allowed, is provided in Section 3.3. In
addition, help in selecting values for some of the input parameters is given in Section
5.2.
Note that maximum control parameters are associated with a number of the input
variables. These control parameters are used in the code to specify array dimensions.
For some problems, the default values set for these parameters may be too small. If so,
they can be easily changed. The maximum control parameters and their default values
are listed in Appendix C. Note also that the logical units used by 3DLEWASTE are
defined in Appendix B.
A complete input file consists of information supplied in 18 data sets. The contents and
format of each data set are listed below. When constructing an input sequence, it is
important to note that data sets 2 through 17 must be preceded by a record which
contains a description of the data set. This can be seen in the example input sequences
provided in Section 6.2. Most of the input is entered in free-format, which means that
the spacing of the input data in a record does not need to follow a set pattern. Note
that a record can consist of multiple lines, with a line defined as up to 80 columns.
The user may choose to run the model using any set of units as long as they are consis-
tently maintained in all the input. Units of mass (M), length (L), and time (T) are
indicated in the input descriptions.
4.2.1 Data Set 1: Title of the Simulation Run
One record with FORNLAT(I5,A70,3X,2I1) per problem. This record contains the
following variables:
1. NPROB = Problem number (columns 1-5).
2. TITLE = Array for the title of the problem. It may contain up to 70 charac-
ters (columns 6- 75).
3. IGEOM = Integer indicating if (1) the geometry, boundary and pointer
arrays are to be printed and if (2) the boundary and pointer arrays are to
be computed or read via logical units (column 79). If IGEOM is an even
number, geometry, boundary and pointer arrays will not be printed. If
IGEOM is an odd number, they will be printed. If IGEOM is less than or
equal to 1, boundary arrays will be computed and written on logical unit
LUBAR, but if IGEOM is greater than 1, boundary arrays will be read via
logical unit LUBAR. If IGEOM is less than or equal to 3, pointer arrays
will be computed and written on logical unit LUPAR, but if IGEOM is
greater than 3, pointer arrays will be read via logical unit LUPAR. In
summary:
74
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IGEOM = Even No. Print the geometry, boundary, and pointer arrays.
IGEOM = Odd No. Do not print the arrays.
IGEOM <. 1 Compute and write boundary and pointer arrays.
1 < IGEOM <. 3 Read boundary arrays, compute and write pointer
arrays.
IGEOM > 3 Read boundary and pointer arrays.
4. IBUG = Integer indicating if diagnostic output is desired (column 80);
0 = no,
1 = yes.
4.2.2 Data Set 2: Basic Integer Parameters
One record with FREE-FORMAT per problem. It contains the following
variables:
1. NNP = Number of nodal points.
2. NEL = Number of elements.
3. NMAT = Number of material types.
4. NCM = Number of elements with material property correction.
5. NTI = Number of time steps or time increments (see notes at the end of
Data Set 2).
6. KSS = Steady-state control;
0 = steady-state solution,
1 = transient-state solution (see note at the end of Data Set 2).
7. NMPPM = Number of material properties per material; this parameter
should be set equal to 8 in the present version of the code (see Data Set 5).
8. KVI = Velocity input control;
-1 = velocity and moisture content read from data set 17,
1 = steady-state velocity and moisture content input read from
FEMWATER binary file,
2 = transient velocity and moisture content input read from FEMWATER
binary file.
9. ILUMP = Mass lumping control;
0 = no,
1 = yes.
10. IWET = Weighting function control;
0 = Galerkin weighting,
1 = upstream weighting.
75
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11. IOPTIM = Optimization control;
1 = upstream weighting optimization factor is to be computed,
0 = factor is to be set equal to 1.0.
12. NITER = Number of iterations allowed for solving the non-linear equation.
13. NDTCHG = Number of times the time-step size will be reset to the initial
time-step size; NDTCHG should be >. 1 (see Section 5.2.2.9).
14. NPITER = Number of iterations for a block or pointwise solution.
15. KSORP = Sorption model control;
1 = linear isotherm,
2 = Freundlich isotherm,
3 = Langmuir isotherm.
**** NOTE: NTI can be computed by NTI = 11 + 1+12 + 1, where II is the
largest integer not exceeding Log(DELMAX/DELT)/Log(l+CHNG),
12 is the largest integer not exceeding (RTIME-BELT*((1 + CHUG)**
(I1 + 1)-1)/CHNG)/DELMAX, RTIME is the real simulation time, and
DELMAX, BELT, and CHNG are defined in data set 3.
**** NOTE: A steady-state option may be used to provide either the final state
of a system under study or the initial condition for a transient-state
calculation. In the former case, KSS = 0 and NTI = 0 in this data
set. In the latter case, KSS = 0 and NTI > 0. If KSS > 0, there
will be no steady-state calculation.
4.2.3 Data Set 3: Basic Real Parameters
One record with FREE-FORMAT per problem. It contains the following
variables:
1. BELT = Initial time step size, (T).
2. CHNG = Fractional change in the time-step size in each subsequent time
increment, (dimensionless decimal-point value).
3. BELMAX = Maximum value of BELT, (T).
4. TMAX = Maximum simulation time, (T).
5. OME = Iteration parameter for solving the nonlinear matrix equation;
0.0 < OME < 1.0 = under-relaxation,
1.0 = exact relaxation,
1.0 < OME < 2.0 = over-relaxation.
6. OMI = Relaxation parameter for solving the linearized matrix equation
pointwise;
0.0 < OMI < 1.0 = under relaxation,
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1.0 = exact relaxation,
1.0 < OMI < 2.0 = over relaxation.
7. TOLB = Transient-state convergence criterion, (L).
8. TOLA = Steady-state convergence criterion, (L).
4.2.4 Data Set 4: Printer and Disk Storage Control and Times for
Step Size Resetting
Three records are needed per problem. The first two records are formatted input
with FORMAT(2I1). The third record is a FREE-FORMAT input. The number of
lines for the first two records depends on the value of NTI, the number of time
increments. The number of lines for the third record depends on the value of
NDTCHG, the number of times to reset the time-step size.
Record 1 - FORMAT(2I1): This record contain the following variables:
1. KPRO = Printer control for steady-state and initial conditions;
0 = print nothing,
1 = print values for the variables FLOW, FRATE, and TFLOW,
2 = print values above plus concentration,
3 = print values above plus material fluxes.
2. KFR(I) = Printer control for the I-th (1=1,2, NTI) time step; O = print
nothing,
1 = print values for the variables FLOW, FRATE, and TFLOW,
2 = print values above plus concentration,
3 = print values above plus material fluxes.
Record 2 - FORMAT(2I1): This record can be used to store 3DLEWASTE output
in a binary file for use in plotting results. It contains the following variables:
1. KDSKO = Auxiliary storage control for the steady-state or initial condition;
0 = no storage,
1 = store on logical unit LUSTO.
2. KDSK(I) = Auxiliary storage control for the I-th (1=1,2,.... NTI) time step;
0 = no storage,
1 = store on logical unit LUSTO.
Record 3- FREE-FORMAT: This record contains the following variables:
1. TDTCH(I) = Time when the I-th (1=1,2,.... NDTCHG) time-step-size
resetting is needed.
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4.2.5 Data Set 5: Material Properties
A total of NMAT records are required for this data set, one for each material.
Record I (I = 1, 2, NMAT) - FREE-FORMAT: Each record contains the
following variables:
1. PROP(1,1) = Distribution coefficient (L3/M) or Freundlich K or Langmuir K
for medium I, depending on the value of KSORP in data set 2.
2. PROP(2,1) = Bulk density for medium I, (M/L3).
3. PROP(3,1) = Longitudinal dispersivity for medium I, (L).
4. PROP(4,I) = Transverse dispersivity for medium I, (L).
5. PROP(5,I) = Molecular diffusion coefficient for medium I, (LVT).
6. PROP(6,I) = Tortuosity for medium I, (Dimensionless).
7. PROP(7,1) = Decay constant in medium I, (1/L).
8. PROP(8,1) = Freundlich N or Langmuir SMAX for medium I.
4.2.6 Data Set 6: Nodal Point Coordinates
Coordinates for NNP nodes are needed only if KVI <. 0, where NNP and KVI are
defined in data set 2. Usually a total of NNP records are required. However, if a
group of subsequent node numbers follows a regular pattern, an automatic
generation input option can be used.
Each record contains the following variables and is FREE-FORMATTED.
1. NI = Node number of the first node in the sequence.
2. NSEQ = Number of subsequent nodes which will be automatically
generated.
3. NAD = Increment of node number for each of the NSEQ subsequent nodes.
4. XNI = X-coordinate of node NI, (L).
5. YNI = Y-coordinate of node NI, (L).
6. ZNI = Z-coordinate of node NI, (L).
7. XAD = Increment of x-coordinate for each of the NSEQ subsequent nodes,
(L).
8. YAD = Increment of y-coordinate for each of the NSEQ subsequent nodes,
(L).
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9. ZAD = Increment of z-coordinate for each of the NSEQ subsequent nodes,
(L).
**** NOTE: A record with nine zeroes must be used to signal the end of this
data set.
4.2.7 Data Set 7: Element Incidence
Element incidence for NEL elements, specified in data set 2, are needed if <
O. Usually, a total of NEL records are needed. However, if a group of element
numbers follows a regular pattern, the automatic generation input option can be
used.
Each record is FREE-FORMATTED and contains the following variables:
1. MI = Global element number of the first element in a sequence.
2. NSEQ = Number of subsequent elements which will be automatically
generated.
3. MIAD = Increment of MI for each of the NSEQ subsequent elements.
4. IE(MI,1) = Global node number of the first node of element MI.
5. IE(MI,2) = Global node number of the second node of element MI.
6. IE(MI,3) = Global node number of the third node of element MI.
7. IE(MI,4) = Global node number of the fourth node of element MI.
8. IE(MI,5) = Global node number of the fifth node of element MI.
9. IE(MI,6) = Global node number of the sixth node of element MI.
10. IE(MI,7) = Global node number of the seventh node of element MI.
11. IE(MI,8) = Global node number of the eighth node of element MI.
12. IEMAD = Increment of lE(MI.l) through IE(MI,8) for each of the NSEQ
elements.
**** NOTE: IE(MI,1) - IE(MI,8) are numbered according to the convention
shown in Figure 4.2. The first four nodes start from the front,
lower, left corner and progress around the bottom element surface
in a counterclockwise direction. The other four nodes begin from
the front, upper, left corner and progress around the top element
surface in a counterclockwise direction.
79
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z
i,
Figure 4.2. Node numbering convention for the elements.
4.2.8 Data Set 8: Subregional Data
This data set is needed only if KVI <. 0, where KVI is defined in data set 2.
8A. Subregion Control Integer
One FREE-FORMATTED record is needed for this subdata set. It contains the
following variable:
1. NREGN = Number of subregions.
8B. Number of Nodes in Each Subregion
Normally, NREGN records are required. However, if the sequence of node num-
bers follows a regular pattern between sequential subregions, the automatic
generation input option can be used.
Each record is FREE-FORMATTED and contains the following five variables:
1. NK = Subregion number of the first subregion in a sequence.
2. NSEQ = Number of subsequent subregions which will be automatically
generated.
3. NKAD = Increment of NK in each of the NSEQ subsequent subregions.
4. NODES = Number of nodes in the subregion NK.
5. NOAD = Increment of NODES in each of the NSEQ subsequent subre-
gions.
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NOTE: A record with five zeroes must be used to end the input of this
subdata set.
8C. Mapping between Global Nodes and Subregion Nodes
This subdata set should be repeated NREGN times, once for each subregion. For
each subregion, normally, the number of records equals the number of nodal
points in the subregion. Automatic generation can be used, however, if the
subregional node numbers follow a regular pattern.
Each record contains five variables and is FREE-FORMATTED.
1. LI = Local node number of the first node in a sequence.
2. NSEQ = Number of subsequent local nodes which will be generated
automatically.
3. LIAD = Increment of LI for each of the NSEQ subsequent nodes.
4. NI = Global node number of local node LI.
5. NIAD = Increment of NI for each of the NSEQ subsequent nodes.
NO IE: A record with five zeroes must be used to signal the end of
this subdata set.
NO IE: Local node numbers have values between one and the total
number of nodes in a subregion (i.e., 1,2,. ... NODES). Global
node numbers are associated with the entire grid and are en-
tered using data set 6.
4.2.9 Data Set 9: Material Type Correction
This data set is required only if NCM > 0 and KVI <. 0, where NCM and KVI are
defined in data set 2. Normally, NCM records are required. However, if a group
of element numbers follows a regular pattern, automatic generation may be used.
Each record is FREE-FORMATTED and contains the following variables:
1. MI = Global element number of the first element in the sequence.
2. NSEQ = Number of subsequent elements which will be generated
automatically.
3. MAD = Increment of element number for each of the NSEQ subsequent
elements.
4. MITYP = Type of material for element MI.
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5. MTYPAD = Increment of MITYP for each of the NSEQ subsequent elements.
**** NOTE: A record with five zeroes must be used to signal the end of this data
set.
4.2.10 Data Set 10: Card Input for Initial or Pre-Initial Conditions
NNP records (i.e., one record for each node) are normally needed. However, if a
group of node numbers follow a regular pattern, automatic generation can be
used.
Each record is FREE-FORMATTED and contains the following variables:
1. NI = Global node number of the first node in the sequence.
2. NSEQ = Number of subsequent nodes which will be generated automatically.
3. NAD = Increment of node number for each of the NSEQ nodes.
4. CNI = Initial or pre-initial concentration of node NI, (M/L3).
5. CAD = Increment of CNI for each of the NSEQ nodes, (M/L3).
6. CRD = Geometrical increment of CNI for each of the NSEQ subsequent nodes
(i.e, CNI**CRD).
**** NOTE: A record with six zeroes must be used to signal the end of this data
set.
**** NOTE: The initial condition for a transient calculation may be obtained in
two different ways: 1) it can be read directly from data set 10, or
2) the code can perform a steady-state simulation using
time-invariant boundary conditions before beginning the transient
computations. For the first case, both KSS and NTI in data set 2
should be greater than zero. In the latter case, KSS = 0 and NTI >
0 and data set 10 is used to input the pre-initial condition, which is
required as the starting condition for the steady-state iteration. In
order to obtain a steady-state solution, both KSS and NTI are set
equal to zero and data set 11 supplies the starting condition for the
steady-state solution.
4.2.11 Data Set 11: Integer Parameters for Sources and Boundary Conditions
One record per problem is needed. This record is FREE-FORMATTED and
contains the following variables:
1. NSEL = Number of distributed source/sink elements.
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2. NSPR = Number of distributed source/sink profiles (i.e., time histories)
(NSPR should be > 1).
3. NSDP = Number of data points in each of the NSPR source/sink profiles
(NSDP should be > 2).
4. KSAI = Option for the distributed source/sink profiles to be input
analytically. This variable should be set equal to zero in the current version
of the code.
5. NWNP = Number of well or point source/sink nodes.
6. NWPR = Number of well or point source/sink profiles (i.e., time histories).
7. NWDP = Number of data points in each of the NWPR profiles.
8. KWAI = Option for the well source/sink profiles to be input analytically.
This variable should be set equal to zero in the current version of the code.
9. NDNP = Number of prescribed-concentration (Dirichlet) nodes (NDNP should
be > 1).
10. NDPR = Number of prescribed-concentration profiles (i.e., time histories)
(NDPR should be > 1).
11. NDDP = Number of data points in each prescribed-concentration profile
(NDDP should be > 2).
12. KDAI = Option for the prescribed-concentration boundary profiles to be input
analytically. This variable should be set equal to zero in the current version
of the code.
13. NVES = Number of variable composite boundary element sides.
14. NVNP = Number of variable composite boundary nodal points.
15. NRPR = Number of variable composite profiles (i.e., time histories).
16. NRDP = Number of data points in each of the NRPR profiles.
17. KRAI = Option for the variable composite profiles to be input analytically.
This variable should be set equal to zero in the current version of the code.
18. NCES = Number of specified-flux (Cauchy) boundary element sides.
19. NCNP = Number of specified-flux boundary nodal points.
20. NCPR = Number of specified-flux profiles (i.e., time histories).
21. NCDP = Number of data points in each of the NCPR profiles.
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22. KCAI = Option for the specified-flux profiles to be input analytically. This
variable should be set equal to zero in the current version of the code.
23. NNES = Number of specified-dispersive-flux (Neumann) boundary element
sides.
24. NNNP = Number of specified-dispersive-flux boundary nodal points.
25. NNPR = Number of specified-dispersive-flux profiles (i.e., time histories).
26. NNDP = Number of data points in each of the NNPR profiles.
27. KNAI = Option for the specified-dispersive-flux profiles to be input
analytically. This variable should be set equal to zero in the current version
of the code.
4.2.12 Data Set 12: Distributed and Point Sources/Sinks
This data set is used to supply data for both distributed sources/sinks, and point
(well) sources/sinks.
12A. Distributed Sources/Sinks
The following three subdata sets are needed if and only if NSEL in data set 11 is
greater than zero. The first subdata set is used to specify the distributed
source/sink profiles. The second subdata set is used to read the global element
numbers of the distributed source/sink elements. The third subdata set is used
to assign a source/sink profile to each distributed source/sink element.
(a) Sources/Sink Profiles
NSPR records (see data set 11) are needed. Each record contains NSDP data
points, defined in data set 11. Three numbers, representing the time, source flow
rate, and source concentration, respectively, are associated with each data point.
Record I (I = 1, 2, NSPR) - FREE FORMAT: Each record contains the follow-
ing variables:
1. TSOSF(J,I) = Time of J-th data point in I-th profile, (T).
2. SOSF(J,I,1) = Source/sink flow rate of the J-th data point in the I-th
profile, (LVWL3); positive for source and negative for sink.
3. SOSF(J,I,2) = Source/sink concentration of the J-th data point in the I-th
profile, (M/L3).
Up to NSDP data points.
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(b) Global Element Number of All Distributed Source/Sink Elements
One record is needed for this subdata set. The number of lines in the record
depends on NSEL, defined in data set 11. The record is FREE-FORMATTED
and contains the following variables:
1. LES(l) = Global element number of the first distributed source/sink
element.
2. LES(2) = Global element number of the second distributed source/sink
element.
Up to NSEL numbers.
(c) Source/Sink Profile Type Assigned to Each Element
Usually NSEL records are needed. However, automatic generation can be used.
Each record is FREE-FORMATTED and contains the following variables:
1. MI = Compressed element number of the first element in the sequence.
2. NSEQ = Number of subsequent elements which will be automatically
generated.
3. MAD = Increment of element number for each of the NSEQ elements.
4. MITYP = Source/sink profile associated with element MI.
5. MTYPAD = Increment of MITYP for each of the NSEQ subsequent
elements.
A record with five zeroes must be used to signal the end of
this subdata set.
Compressed element numbers have values between one and
the total number of distributed source/sink elements.
Compressed element one corresponds to the first element
listed in 12A(b), compressed element two corresponds to the
second global element, etc.
12B. Point (Well) Source/Sink
The following three subdata sets are needed only if NWNP in data set 11 is
greater than zero. The first subdata set is used to specify the point source/sink
profiles. The second subdata set reads the source/sink global node numbers, and
the third assigns a source/sink profile type to each node.
(a) Source/Sink Profiles
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NWPR records (see data set 11) are needed. Each record contains NWDP data
points, defined in data set 11. Three numbers, representing the time, source flow
rate, and source concentration, respectively, are associated with each data point.
Record I (I = 1, 2, .... NWPR) - FREE FORMAT: Each record contains the
following variables:
1. TWSSF(J,I) = Time of J-th data point in I-th profile, (T).
2. WSSF(J,I,1) = Source/sink flow rate of the J-th data point in the I-th
profile, (L3/T); positive for source and negative for sink.
3. WSSF(J,I,2) = Source/sink concentration of the J-th data point in the I-th
profile, (M/L3).
Up to NWDP numbers.
(b) Global Node Number of All Point (Well) Source/Sink Nodes
One record is needed for this subdata set. The number of lines in this record
depends on NWNP, defined in data set 11. The record is FREE-FORMATTED
and contains the following variables:
1. NPW(l) = Global node number of the first point source/sink node.
2. NPW(2) = Global node number of the second point source/sink node.
Up to NWNP numbers.
(c) Source/Sink Profile Type for Each Node
Usually one record per node (i.e., NWNP records) are needed. However,
automatic generation can be used. Each record is FREE-FORMATTED and
contains the following variables:
1. NI = Compressed point source/sink node number of the first node in a
sequence.
2. NSEQ = Number of subsequent nodes which will be automatically
generated.
3. NIAD = Increment of NI for each of the NSEQ nodes.
4. NITYP = Source/sink profile associated with node NI.
5. NTYPAD = Increment of NITYP for each of the NSEQ subsequent nodes.
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-J, -J, -J, -J, TV T t-^i r-p -ii
IN u it: A record with five zeroes must be used to signal the end of
this subdata set.
4.2.13 Data Set 13: Variable Composite Boundary Condition
The following four subdata sets are required only if NVES in data set 11 is
greater than zero. The first subdata set is used to specify the concentration
profiles. The second subdata set is used to assign a concentration profile type to
each of the variable composite boundary element sides. The third subdata set is
used to specify the variable composite boundary element sides. The fourth
subdata set is used to read the global nodal number of all the variable composite
boundary nodes.
ISA. Concentration Profiles
There will be NRPR records (see data set 11) in this subdata set. The number of
lines in each record depends on NRDP, defined in data set 11.
Record I (I = 1, 2, NRPR) - FREE-FORMAT: Each record contains the follow-
ing variables:
1. TCRSF(IJ) = Time of the first data point in the I-th profile, (T).
2. CRSF(1,I) = Concentration of the first data point in the I-th profile,
(M/L3).
3. TCRSF(2,I) = Time of the second data point in the I-th profile, (T).
4. CRSF(2,I) = Concentration of the second data point in the I-th profile,
(M/L3).
Up to NRDP data points.
13B. Concentration Profile Type Assigned to Each Boundary Element Side
Usually one record per variable composite boundary element side is needed.
However, automatic generation can be used.
Record I (I = 1, 2, ....)- FREE-FORMAT: Each record contains the following
variables:
1. MI = Compressed variable boundary element side of the first side in a
sequence.
2. NSEQ = Number of subsequent sides which will be generated
automatically.
3. MJAD = Increment of MI for each of the NSEQ subsequent sides.
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4. MITYP = Type of concentration profile assigned to side MI.
5. MTYPAD = Increment of MITYP for each of the NSEQ subsequent sides.
-J, -J, -J, -J, TV T /^'"PT"?
INuit: A record with five zeroes must be used to signal the end of
this subdata set.
13C. Specification of Variable Composite Boundary Element Sides
Normally, NVES records are required, one each for a variable boundary element
side. However, if a group of variable composite boundary element sides appears
in a regular pattern, automatic generation may be used.
Record I (I = 1, 2, ....)- FREE-FORMAT: Each record contains the following
variables:
1. MI = Compressed variable composite boundary element side number of
the first side in a sequence.
2. NSEQ = Number of subsequent element sides which will be generated
automatically.
3. MIAD = Increment of MI for each of the NSEQ element sides.
4. 11 = Global node number of the first node of element side MI.
5. 12 = Global node number of the second node of element side MI.
6. 13 = Global node number of the third node of element side MI.
7. 14 = Global node number of the fourth node of element side MI.
8. HAD = Increment of 11 for each of the NSEQ subsequent element sides.
9. I2AD = Increment of 12 for each of the NSEQ subsequent element sides.
10. 13AD = Increment of 13 for each of the NSEQ subsequent element sides.
11. WAD = Increment of 14 for each of the NSEQ subsequent element sides.
-j, -j, -j, -j, TV T /^'"pT"?
IN u it: A record with 11 zeroes must be used to signal the end of this
subdata set.
13D. Global Nodal Number of All Variable Composite Boundary Nodes
At most, NVNP records (see data set 11) are needed for this subdata set.
Record I (1 = 1, 2, ....)- FREE-FORMAT: Each record contains the following
variables:
88
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1. NI = Compressed variable boundary node number of the first node in the
sequence.
2. NSEQ = Number of subsequent nodes which will be generated automati-
cally.
3. NIAD = Increment for NI for each of the NSEQ nodes.
4. NODE = Global nodal number of the node NI.
5. NODEAD = Increment of NODE for each of the NSEQ subsequent
nodes.
NO IE: A record with five zeroes must be used to signal end of this
subdata set.
4.2.14 Data Set 14: Prescribed-Concentration (Dirichlet) Boundary Condition
This data set is required if and only if NDNP in data set 11 is greater than zero.
It consists of three subdata sets. The first subdata set is used to specify the
prescribed-concentration profiles, the second is used to read the prescribed-
concentration boundary nodes, and the third is used to assign a concentration
profile type to each of the Dirichlet nodes.
14A. Prescribed-Concentration Profiles
There will be NDPR records (see data set 11) in this subdata set. The number of
lines in each record depends on NDDP, defined in data set 11.
Record I (I = 1, 2, NDPR) - FREE-FORMAT: Each record contains the
following variables:
1. TCDBF(1,I) = Time of first data point in I-th profile, (T).
2. CDBF(1,I) = Concentration of first data point in I-th profile, (M/L3).
3. TCDBF(2,I) = Time of second data point in I-th profile, (T).
4. CDBF(2,I) = Concentration of second data point in I-th profile, (M/L3).
Up to NDDP data points.
14B. Global Node Number of All the Prescribed-Concentration Nodes
One FREE-FORMATTED record is needed for this subdata set. The number of
lines in this record depends on NDNP, defined in data set 11.
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1. NPDB(l) = Global node number of the first compressed prescribed-
concentration node.
2. NPDB(2) = Global node number of the second compressed prescribed-
concentration node.
Up to NDNP numbers.
14C. Type of Concentration Profile Assigned to Each Dirichlet Node
Normally one record per Dirichlet node (i.e., a total of NDNP records) is needed.
However, if the Dirichlet node numbers follow a regular pattern, automatic
generation may be used.
Record I (I = 1, 2, . . . . ) - FREE-FORMAT: Each record contains the following
variables:
1. NI = Compressed Dirichlet node number of the first node in the
sequence.
2. NSEQ = Number of subsequent Dirichlet nodes which will be automati-
cally generated.
3. NIAD = Increment of NI for each of the NSEQ nodes.
4. NITYP = Dirichlet concentration profile type assigned to node NI and
NSEQ subsequent nodes.
5. NTYPAD = Increment of NITYP for each of the NSEQ subsequent
nodes.
NO IE: A record with five zeroes must be used to signal the end of
this subdata set.
4.2.15 Data Set 15: Specified-Flux (Cauchy) Boundary Condition
Four subdata sets are required only if NCES in data set 11 is greater than zero.
The first subdata set is used to read the specified-flux profiles. The second
subdata set is used to assign the type of specified-flux profile to each of the
specified-flux boundary element sides. The third subdata set is used to read the
specified-flux boundary element sides. The fourth subdata set is used to read the
global nodal numbers associated with the specified-flux boundaries.
ISA. Specified-Flux Profiles
There will be NCPR records (see data set 11) in this subdata set. The number of
lines in each record depends on NCDP, defined in data set 11.
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Record I (I = 1, 2, NCPR) - FREE-FORMAT: Each record contains the follow-
ing variables:
1. TQCBF(U) = Time of the first data point in the I-th profile, (T).
2. QCBF(1,I) = Normal specified-flux of the first data point in the I-th
profile, (M/T/L2); positive out of the region, negative into the region.
3. TQCBF(2,I) = Time of the second data point in the I-th profile, (T).
4. QCBF(2,I) = Normal specified-flux of the second data point in the I-th
profile, (M/T/L2); positive out of the region, negative into the region.
Up to NCDP data points.
15B. Type of Specified-Flux Profile Assigned to Each Boundary Element Side
At most, NCES records (see data set 11) are needed. However, automatic genera-
tion can be used.
Record I (I = 1, 2, ) - FREE-FORMAT: Each record contains the following
variables:
1. MI = Compressed specified-flux boundary element side number of the
first side in the sequence.
2. NSEQ = Number of subsequent sides which will be generated automati-
cally.
3. MIAD = Increment of MI for each of NSEQ subsequent sides.
4. MITYP = Type of specified-flux profile assigned to side MI.
5. MTYPAD = Increment of MITYP for each of the NSEQ subsequent sides.
NO IE: A record with five zeroes must be used to signal the end of
this subdata set.
15C. Specified-Flux Boundary Element Sides
Normally, NCES records are required, one for each specified-flux boundary
element side. However, if a group of specified-flux boundary element side
numbers follows a regular pattern, automatic generation may be used.
Record I (I = 1, 2, . . . . ) - FREE-FORMAT: Each record contains the following
variables:
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1. MI = Compressed specified-flux boundary element side number of the
first element side in a sequence.
2. NSEQ = Number of subsequent element sides which will be generated
automatically.
3. MIAD = Increment of MI for each of the NSEQ subsequent sides.
4.11 = Global node number of the first node of element side MI.
5. 12 = Global node number of the second node of element side MI.
6. 13 = Global node number of the third node of element side MI.
7. 14 = Global node number of the fourth node of element side MI.
8. HAD = Increment of 11 for each of the NSEQ subsequent element sides.
9. I2AD = Increment of 12 for each of the NSEQ subsequent element sides.
10. I3AD = Increment of 13 for each of the NSEQ subsequent element sides.
11. WAD = Increment of 14 for each of the NSEQ subsequent element sides.
NO IE. A record with 11 zeroes must be used to signal the end of this
subdata set.
15D. Global Node Number of All Compressed Specified-Flux Boundary Nodes
Usually NCNP records (see data set 11) are needed for this subdata set.
However, automatic generation can be used.
Record I (I = 1, 2, . . . . ) - FREE-FORMAT: Each record contains the following
variables:
1. NI = Compressed specified-flux boundary node number of the first node
in a sequence.
2. NSEQ = Number of subsequent nodes which will be generated automatic-
ally.
3. NIAD = Increment for NI for each of the NSEQ nodes.
4. NODE = Global nodal number of the node NI.
5. NODEAD = Increment of the global nodal number for each of the NSEQ
subsequent nodes.
NO IE. A record with five zeroes must be used to signal end of this
subdata set.
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4.2.16 Data Set 16: Specified-Dispersive-Flux (Neumann) Boundary Condition
The following four subdata sets are required only if NNES in data set 11 is
greater than zero. The first subdata set is used to read the specified-dispersive-
flux profiles. The second subdata set is used to assign a specified-dispersive-flux
profile type to each boundary element sides. The third subdata set is used to
read the specified-dispersive-flux boundary element side. The fourth subdata set
is used to read the global nodal numbers associated with the specfied-dispersive-
flux boundaries.
16A. Prescribed Specified-Dispersive-Flux Profiles
There will be NNPR records (see data set 11) in this subdata set. The number of
lines in each record depends on NNDP, defined in data set 11.
Record I (I = 1, 2, .... NNPR) - FREE-FORMAT: Each record contains the
following variables:
1. TQNBF(IJ) = Time of the first data point in the I-th profile, (T).
2. QNBF(1,I) = Normal specified-dispersive flux of the first data point in
the I-th profile, (M/T/L2); positive out of the region, negative into the
region.
3. TQNBF(2,I) = Time of the second data point in the I-th profile, (T).
4. QNBF(2,I) = Normal specified-dispersive flux of the second data point in
the I-th profile, (M/T/L2); positive out of the region, negative into the
region.
Up to NNDP data points.
16B. Type of Specified-Dispersive-Flux Profile Assigned to Each Boundary Ele-
ment Side
At most, NNES records (see data set 11) are needed. However, automatic
generation can be used.
Record I (I = 1, 2, . . . . ) - FREE-FORMAT: Each record contains the following
variables:
1. MI = Compressed specified-dispersive-flux boundary element side of the
first side in a sequence.
2. NSEQ = Number of subsequent sides which will be generated automati-
cally.
3. MIAD = Increment of MI for each of NSEQ subsequent sides.
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4. MITYP = Type of specified-dispersive-flux profile assigned to side MI.
5. MTYPAD = Increment of MITYP for each of the NSEQ subsequent sides.
NO IE: A record with five zeroes must be used to signal the end of
this subdata set.
16C. Specified-Dispersive-Flux Boundary Element Sides
Normally, NNES records are required, one for each specified-dispersive-flux
boundary element side. However, if a group of specified-dispersive-flux element
side numbers follows a regular pattern, automatic generation may be used.
Record I (I = 1, 2, . . . . ) - FREE-FORMAT: Each record contains the following
variables:
1. MI = Compressed specified-dispersive-flux boundary element side
number of the first element side in a sequence.
2. NSEQ = Number of subsequent sides which will be generated automati-
cally.
3. MIAD = Increment of MI for each of the NSEQ subsequent sides.
4. II = Global node number of the first node of element side MI.
5. 12 = Global node number of the second node of element side MI.
6. 13 = Global node number of the third node of element side ML
7. 14 = Global node number of the fourth node of element side MI.
8. HAD = Increment of 11 for each of the NSEQ subsequent element sides.
9. I2AD = Increment of 12 for each of the NSEQ subsequent element sides.
10. I3AD = Increment of 13 for each of the NSEQ subsequent element sides.
11. I4AD = Increment of 14 for each of the NSEQ subsequent element sides.
NOTE: A record with 11 zeroes must be used to signal the end of this
subdata set.
16D. Global Node Number of All Compressed Specified-Dispersive-Flux
Boundary Nodes
Usually NNNP records (see data set 11) are needed for this subdata set.
However, automatic generation can be used.
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Record I (I = 1, 2, ....)- FREE-FORMAT: Each record contains the following
variables:
1. NI = Compressed specified-dispersive-flux boundary node number of the
first node in a sequence.
2. NSEQ = Number of subsequent nodes which will be generated automati-
cally.
3. NIAD = Increment of NI for each of the NSEQ nodes.
4. NODE = Global nodal number of the node NI.
5. NODEAD = Increment of the global nodal number for each of the NSEQ
subsequent nodes.
A record with five zeroes must be used to signal end of this
subdata set.
4.2.17 Data Set 17: Hydrological Variables
This data set is needed only if KVI in data set 2 is less than or equal to zero.
When KVI <; 0, two subdata sets are needed: one for the velocity field and the
other for the moisture content.
17A. Velocity Field
Usually NNP records (see data set 2) are needed. However, if the velocity values
follow a regular pattern, automatic generation can be used.
Record I (I = 1, 2, ....)- FREE-FORMAT: Each record contains the following
variables:
1. NI = Node number of the first node in a sequence.
2. NSEQ = Number of subsequent nodes which will be automatically gener-
ated.
3. NIAD = Increment of NI for each of the NSEQ subsequent nodes.
4. VXNI = X-velocity component at node NI, (L/T).
5. VYNI = Y-velocity component at node NI, (L/T).
6. VZNI = Z-velocity component at node NI, (L/T).
7. VXAD = Increment of VXNI for each of the NSEQ subsequent nodes,
(L/T).
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8. VYAD = Increment of VYNI for each of the NSEQ subsequent nodes,
(m).
9. VZAD = Increment of VZNI for each of the NSEQ subsequent nodes,
(m).
NOTE: A record with nine zeroes must be used to signal the end of
this subdata set.
17B. Moisture Content Field
Usually, NEL records (see data set 2) are needed. However, if the moisture
content values follow a regular pattern, automatic generation can be used.
Record I (I = 1, 2, . . . . ) - FREE-FORMAT: Each record contains the following
variables:
1. MI = Element number of the first element in a sequence.
2. NSEQ = Number of subsequent elements which will be automatically
generated.
3. MIAD = Increment of MI for each of NSEQ subsequent elements.
4. THNI = Moisture content of element MI, (decimal point).
5. THNIAD = Increment of THNI for the NSEQ subsequent elements,
(decimal point).
NO IE: A record with five zeroes must be used to signal the end of
this subdata set.
4.2.18 Data Set 18: End of Job
If another problem is to be run, then input begins again with input data set 1. If
termination of the job is desired, a blank line must be inserted at the end of the
data set.
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SECTION 5
PARAMETER SELECTION
This section provides guidance in selecting values for some of the parameters required
as input to the 3DFEMWATER/3DLEWASTE codes. This guidance is not intended in
any way to be used as a substitute for data collection. The most accurate model results
are obtained from simulations which are based on site-specific information. In some
cases, however, it is not feasible to measure certain parameters, and satisfactory results
may be obtained using estimated values taken from the reported ranges presented here.
For easy reference, the parameters are grouped according to the data group in which
they appear in the input data sets (see Section 4). Concepts, such as initial and
boundary conditions, isotherms, distributed and point sources and sinks, and
subregional data, were introduced in Section 3 and guidance is not provided in this
section for related parameters.
5.1 3DFEMWATER
5.1.1 Data Set 1: Title of the Simulation Run
5.1.1.1 Geometry, Boundary, and Pointer Array Control, IGEOM []
The integer IGEOM has two functions. It is used to specify if geometry, boundary, and
pointer arrays should be printed so that the user can examine them. It also controls
whether the boundary and pointer arrays are written to or read from binary files.
Boundary arrays store data related to the boundary conditions. Pointer arrays store the
global matrix in compressed form and are used to construct the subregional block
matrices. For large problems, it takes too much time to generate these arrays for each
computer execution of a particular scenario. Usually, they should be generated only
once and stored in binary files using logical units LUBAR and LUPAR (see Table B-l).
In order to compute and store the boundary and pointer arrays, the user should choose a
value for IGEOM less than or equal to one. In subsequent runs, the boundary and
pointer arrays can be read from the binary files by changing the value of IGEOM to a
number greater than three. Whenever changes are made to the model which involve the
geometry of the problem, the boundary conditions, and the configuration of the
subregions, the arrays must be generated and stored again. Note that the option
presented in the input to read boundary arrays and compute and write pointer arrays is
not used in 3DFEMWATER under normal conditions.
For the options explained above, if the number chosen by the user is even, the arrays
will be printed as output. If the number is odd, the arrays will not be printed.
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5.1.2 Data Set 2: Basic Integer Parameters
5.1.2.1 Number of Material Types, NMAT [--]
This parameter is the total number of different porous media being modeled. For
example, if the region of interest is predominantly sand with clay lenses, then the value
of NMAT should be set equal to two. When material properties are assigned to each
material type, using data set 5 (see Section 4.1.5), the first material type should be the
predominant porous medium (e.g., for the example here, the sand).
5.1.2.2 Number of Elements with Material Property Correction, NCM []
In the code, all the grid elements automatically are initialized as having a material type
of one. If the region being modelled is homogeneous, the parameter NCM is set equal to
zero. To model a heterogeneous porous medium, NCM and the parameters in data set
10 of the input (see Section 4.1.10) are used to specify which elements have different
material types associated with them. The parameter NCM is the total number of
elements which have a material type different than the first material type.
5.1.2.3 Number of Time-Steps, NTI [--]
For a constant time-step size, this number is obtained by dividing the simulation time
by the time-step size, BELT. If the time-step size is variable, this number is computed
using the formula given in the note at the end of data set 2 in Section 4.1.2. If a steady-
state solution is desired, NTI should be set equal to zero.
5.1.2.4 Steady-State Control, KSS [-]
As noted in Section 4.1.2, a steady-state option may be used to provide either the final
state of a system under study or the initial condition for a transient-state calculation.
In the former case, both KSS and the number of time steps, NTI, should be set to zero.
In the latter case (i.e., when KSS = 0 and NTI > 0), the code performs a steady-state
calculation before beginning the transient computations. If KSS = 1, no steady-state
calculation is performed. Rather, the code begins transient calculations using initial
conditions supplied in data set 11 of the input.
5.1.2.5 Gravity Term Control, KGRAV [-]
This parameter indicates if the gravity term should be included. For most cases,
KGRAV should be equal to 1. For cases when flow due to the pressure gradient is much
greater than that due to gravity, KGRAV is set to 0.
5.1.2.6 Mass Lumping Flag, ILUMP [-]
This parameter indicates if the mass matrix is to be lumped or not. Normally, one
should set this parameter to O. Without lumping, the solution is more accurate.
However, for occasions when negative concentrations or oscillating solutions occur, this
parameter should be set to 1. It has been suggested that for saturated-unsaturated flow
computations, the parameter ILUMP should always be set equal to 1.
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5.1.2.7 Mid-differencing Flag, IMID [--]
This parameter indicates if the more accurate mid-difference method should be used in
flow computations. For practical purposes, IMID = 0 should be sufficient. IMID = 1 is
used only for research purposes.
5.1.2.8 Number of Iterations for the Nonlinear Equation, NITER []
This parameter is the number of iterations allowed for solving the nonlinear equation.
Normally, NITER = 50 should be sufficient. If this number is exceeded and the solution
does not converge, the program will issue a warning message. When this occurs, the
user should first recheck the input values. If the input is correct, the program can be
re-executed using a larger value for NITER.
5.1.2.9 Number of Cycles, NCYL [-]
This parameter indicates how many cycles are used for iterating the boundary
conditions. A value of 20 should be adequate for most problems.
5.1.2.10 Number of Times to Reset the Time Step, NDTCHG [-]
This parameter indicates how many times the time-step size should be reset to the
initially small time-step size. When we start a simulation, we normally use a small
time-step size. However, for every consecutive time step, we may gradually increase the
time-step size by some amount specified by the variable CHNG in Data Set 3 in Section
4.1.3. When a steep change in boundary conditions or source/sink conditions occurs,
however, the time-step size should be reset to the initially small value. (See the
example problem in Section 6.1.1.) NDTCHG tells us how many times we want to reset
the time-step size during a simulation.
The value of NDTCHG must be at least one. If the user does not want to reset the time
step, a value of one should be entered here and a very large number, larger than the
total simulation time, should be entered for TDTCH(l) in data set 4 (see Section 4.1.4).
5.1.2.11 Number of Iterations for Pointwise Solution, NPITER [-]
This parameter is used to input the number of iterations allowed for solving the matrix
equations with the block iteration method. A value of 300 should be sufficient for most
problems. If this number is exceeded and the solution does not converge, the program
will issue a warning message. When this occurs, the users should re-execute the
program using a larger value for NPITER.
5.1.3 Data Set 3: Basic Real Parameters
5.1.3.1 Initial Time-Step Size, DELT [T]
This is the time-step size used for the first time-step computation if the variable CHNG
is not equal to 0.0. It is the time-step size used for every time step if the variable
CHNG is set equal to 0.0. It is advisable to choose the value of DELT such that:
99
-------
(F*DELX*DELX)/(DELT*K) < 1
where
DELX = the element size (L)
K = hydraulic conductivity (L/T)
F = specific storage (1/L)
For example, if F = 0.001 1/m, K = 0.00001 m/sec, and an element size of 10 m is used,
then BELT should be less than 10,000 seconds.
5.1.3.2 Fractional Change in Time-Step Size, CHNG []
This parameter specifies how much of an increase one would like to make to the time-
step size for each subsequent time step. Normally, a value from 0.0 to 0.5 can be used.
5.1.3.3 Maximum Allowable Time Step, DELMAX [T]
The maximum time-step size allowed depends on how fast the system responds to
change. Use of a value one to ten times the size of the initial time step is advised.
5.1.3.4 Maximum Simulation Time, TMAX [T]
This is the actual length of time to be simulated. If this time is exceeded before you
have made NTI step computations, the simulation will be terminated.
5.1.3.5 Steady-State Convergence Criterion, TOLA [L]
This is the absolute error allowed for assessing if a steady-state solution for hydraulic
head has converged. The value used for TOLA depends on how much the system is
disturbed. Normally, setting TOLA equal to one-ten-thousandth (0.0001) of the
maximum disturbance should be sufficient. For example, if one is conducting a
simulation of drawdown due to pumping and one expects the maximum drawdown at
steady-state will be 1 m, then a value of TOLA equal to 0.0001 m should be sufficient.
5.1.3.6 Transient Convergence Criterion, TOLB [L]
This is the absolute error allowed for assessing if the solution for hydraulic heads has
converged for each transient time step. A value equal to one-hundred-thousandth
(0.00001) of the maximum disturbance should be sufficient for most problems.
5.1.3.7 Density of Water, RHO [M/L3]
The density of water, pw, is the ratio of its mass to its volume and has SI units of kg/m3
Density varies with temperature (Table 5-1) and can be computed using regression
equations presented in CRC (1981). Density also varies with the concentration of
dissolved chemical species. Water density appears in the definition of specific storage
100
-------
TABLE 5-1. WATER DENSITY AS A FUNCTION OF TEMPERATURE
Temperature
(°C)
0
10
20
30
40
50
60
70
80
90
100
Density
(kg/m3)
999.87
999.73
998.23
995.67
992.24
988.07
983.24
977.81
971.83
965.34
958.38
Source: Mercer et al., 1982; Original Reference: CRC, 1965
and in the relationship between hydraulic conductivity and intrinsic permeability
(Section 5.1.4.2).
5.1.3.8 Dynamic Viscosity of Water, VISC [M/L/T]
The viscosity of a fluid is a measure of the forces that work against flow when a
shearing stress is applied (Lyman et al., 1982). The more viscous a fluid is, the greater
the shear stress needed to maintain a given velocity gradient. Dynamic viscosity is
often expressed in terms of poise (gram per centimeter per second) or centipoise (0.01
poise). Water has a viscosity of approximately 1 centipoise at 20°C.
Viscosity varies with temperature, as indicated in Table 5-2, and with concentration of
dissolved chemicals. The effect of pressure on fluid viscosity is generally unimportant
(Mercer et al., 1982). Note that dynamic viscosity is a term in the relationship between
hydraulic conductivity and intrinsic permeability (Section 5.1.4.2).
5.1.3.9 Time Integration Weighting Factor, W []
A value of W equal to 1.0 should be used for most practical problems (see Equation 3-
16). Setting W equal to 0.5 is normally done for research purposes to assess the
accuracy of the Crank-Nicolson scheme.
5.1.3.10 Relaxation Parameter for Solving the Nonlinear Equation, OME []
Normally this parameter should be set to 1.0 (see Equation 3-17). If the convergence
history shows signs of oscillation, then a value of 0.5 should be used. If the convergence
101
-------
TABLE 5-2. DYNAMIC VISCOSITY OF WATER AS A FUNCTION OF
TEMPERATURE
Temperature
("c)
o
10
20
30
40
50
100
Dynamic Viscosity
(centipoise)
1.7921
1.3077
1.0050
0.8007
0.6560
0.5494
0.2838
Source: CRC, 1965
history shows monotonic decreases but at a very slow rate, then OME should be set to
somewhere between 1.7 to 1.9.
5.1.3.11 Iteration Parameter to Solve the Linearized Matrix Equation, OMI [ ]
Normally this parameter should be set to 1.0 (see Equation 3-18). If the convergence
history shows signs of oscillation, then set OMI to 0.5. If the solution converges
monotonically but at a very slow rate, then set OMI to between 1.7 and 1.9.
5.1.4 Data Set 5: Material Properties
In the material properties data set, the user must input values for either hydraulic
conductivity or permeability for each aquifer/soil material type. The flag that tells the
code which of these two properties is being input is the permeability input control, KCP,
located in Data Group 6.
5.1.4.1 The Saturated Hydraulic Conductivity Tensor [L/T]
Hydraulic conductivity is the coefficient of proportionality which appears in Darcy's Law.
It expresses the ease with which a fluid can be transported through a porous medium
and is a function of properties of both the porous medium and the fluid (Mills et al.,
1985b). It is defined as the volume of water that will move in unit time under a unit
hydraulic gradient through a unit area measured at right angles to the direction of flow.
For three-dimensional flow in an anisotropic medium, hydraulic conductivity varies with
direction at any point in space and is expressed as a symmetric second-rank tensor:
Kxz
V"
yx; yy
102
-------
where Ky is the hydraulic conductivity tensor and x, y, and z are the coordinate axes of
the model grid. Because of symmetry, only 6 of the 9 terms are needed (K,^, Kyy, K^,
and Kxy = Ky,,Kxz = K,,, and K,, = K,y).
If the coordinate axes coincide with the principal directions of anisotropy, then the nine
components of the tensor reduce to K^, K^, and KZI, with the other components equal to
zero. For isotropic media, hydraulic conductivity is independent of the direction of
measurement (i.e., K^ = K^ = K^ ).
Hydraulic conductivity estimates should be based on site-specific data collection (e.g.,
pumping tests or piezometer tests). Some typical horizontal hydraulic conductivity
values for various materials are shown in Table 5-3. Note that hydraulic conductivity
varies over a very wide range. As a result, values are rarely known with more than an
order-of-magnitude accuracy. Hydraulic conductivity values for fractured rock can be
found in Mercer et al. (1982).
For many materials, the vertical hydraulic conductivity is substantially smaller than the
horizontal hydraulic conductivity (assuming horizontal bedding and measurements made
along the principal axes) (Mercer et al., 1982). Mills et al. (1985b) state that the ratio of
horizontal to vertical conductivity, known as the anisotropy ratio, is from 2 to 10 for
alluvium and glacial outwash and from 1.5 to 3 for sandstone. The variability in
horizontal and vertical conductivities for a few aquifer materials is shown in Table 5-4.
5.1.4.2 The Permeability Tensor [Lz]
Intrinsic permeability is a property of the porous medium only. It is a measure of the
resistance to fluid flow through the medium. The greater the permeability, the less the
resistance. Like hydraulic conductivity, permeability is a symmetrical second-rank
tensor. Permeability is equal to hydraulic conductivity multiplied by a scalar value, as
is seen in the following equation:
^ = Ky p/(pg) (5-2)
where
ky = permeability (L')
Ky = hydraulic conductivity (L/T)
p. = dynamic viscosity (M/L/T)
p = density (M/L3)
g = acceleration of gravity (L/T2)
As was true for hydraulic conductivity, permeability estimates should be based on site-
specific data collection. Ranges of values for permeability are shown in Table 5-3 and in
Table 5-5. Permeability is sometimes expressed in units of darcies. Conversion from
darcies to other units can be done by using the conversion factors provided at the bottom
of Table 5-3.
103
-------
TABLE 5-3. RANGE OF HYDRAULIC CONDUCTIVITY VALUES FOR VARIOUS
GEOLOGIC MATERIALS (Freeze and Cherry, 1979)
]
3
1
1
Q
i.
8
1
£
ROCKS
|
!
! 8
I-
a
!
i
J
Uncomolldaud
dspeslB
j» 0 i
J '
ll
J
'I
ll
5
'I-
J
5
:
r [
\
i
,
?
i
j
1
I
J
i
f
T
\
1
c
f
k k K K K
(
oarcyj
r-10>
_10<
-io3
-io2
- 10
"'
_io-'
-ID-*
-10°
-10^
-io-J
-io-«
-1C'7
-io-«
(cm^;
-10-3
-10^
-10-'
-in"6
-ID'7
-m*
-iO-9
-io-10
-io-11
-io-12
-io-"
-ID'"
_10"
-10 -»
(OBIS)
r-102
- 10
- 1
- 10-1
- 10-2
- 10-3
-10-*
- 10-'
-m-»
- 10-7
_ 10-*
_io-»
_ 10-K
_ 10-1:
(m/s; tgw/oay/ti-;
l_ i
-io-1
-ID'2
-10-3
-10"4
-10-'
-10-*
-10-7
-10 *
-10-9
- 10 -«
- 1C'11
_ 10 -i:
- 10 -i:
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_ 10'
-104
- 10'
- io2
- 10
- 1
_ 10-1
- 10-2
- 10 -'
-10^
_ 10-'
- 10 -«
- 10-7
Conversion Factors for Permeability
and Hydraulic Conductivity Units
cm2
ft2
darcy
m's
ft/8
U.S. gal/
day/ft2
cm2
1
9.29x10"
9.87X10'9
1.02xlO-3
S.llxlO-4
5.42xlO-10
Permeability, k*
ft2
l.OSxlO'3
1
1.06X10-11
1.1 0x10-"
3.35x10-'
5.83X10-13
Hydraulic conductivity, K
darcy
1. Olxl O3
9.42xl010
1
1. 04x10"
3.15xl04
5.49xlO'2
m/s
9.80X102
9.11X105
9.66X10'6
1
3.05xlO-1
4.72xlO'7
ft/s
3.22X103
2.99xlOe
3.17X10-"
3.28
1
1.55x1
-------
TABLE 5-4. VARIABILITY IN HORIZONTAL AND VERTICAL HYDRAULIC
CONDUCTIVITIES
Rock Types KH, m/s KV) m/s KV/KH
Shale
Siltstone-shale
Siltstone-shale
Sandstone
2.0 x ID'8
2.1 x ID'6
2.8 x 10-7
3.4 x 10-7
1.0 x 10-8
2.1 x lO'7
3.0 x 10-8
3.4 x lO'7
0.5
0.1
0.107
1.0
Source: Mercer et al., 1982; Original Reference: Colder Associates, 1977
TABLE 5-5. PERMEABILITY OF POROUS MATERIALS
Material k (m )
Argillaceous limestone 2% porosity 9.87 x 10"17
Limestone 16% porosity 1.38 x 10"13
Sandstone, silty 12% porosity 2.57 x 10'15
Sandstone, coarse 12% porosity 1.09 x 10"12
Sandstone 29% porosity 2.37 x 10'12
Very fine sand well sorted 9.77 x 10'12
Medium sand very well sorted 2.57 x 10'
Coarse sand very well sorted 3.06 x 10"
Gravel very well sorted 4.24 x 10
Montmorillonite clayb 10"17
Kaolinite clayb 10'1B
-10
,-8
a These are provided as estimates; actual values will vary.
b For the clays, only the order of magnitude is indicated.
Source: Mercer et al., 1982; Adapted from: Davis and DeWiest, 1966
105
-------
5.1.5 Data Set 6: Soil Property Parameters
As was explained in Section 3.1.1, relationships between relative permeability and water
content and between pressure head and water content must be specified in order to
solve the governing equation for unsaturated flow. The 3DFEMWATER code provides
two options for specifying these relationships. The user can 1) input parameters for
analytical expressions of these relationships, or 2) input the coordinates of
characteristic curves in tabular format. The analytical parameters are discussed first,
followed by the tabular data requirements.
5.1.5.1 Analytical Parameters
Analytical equations developed by van Genuchten (1980) are used in the code to describe
the relationship between pressure head and moisture content and the relationship
between relative hydraulic conductivity and moisture content (see Equations 3-3a
through 3-3d). In order to solve these equations, five parameters must be specified in
the input sequence for each material type: residual moisture content, saturated moisture
content, air entry pressure head, and two soil-specific empirical parameters, alpha and
beta.
5.1.5.1.1 Residual and Saturated Moisture (Water) Content [1
The volumetric moisture content, 9, is defined as:
9 = VW/VT (5-3)
where
VT = the total unit volume of a rock or soil (L3)
VW = the volume of a rock or soil occupied by water (L3)
The saturated moisture content is equal to the porosity of the medium since all of the
void space is filled with fluid. Under unsaturated conditions, however, some of the void
space is filled with air and thus, the moisture content is less than the medium's
porosity. The residual moisture content is that amount which can not be removed from
a soil by gravity drainage, even under large suction pressure, because it adheres to the
grains of the soil.
Table 5-6 lists descriptive statistics for both saturated and residual moisture content for
a variety of soil types. In addition, saturated and residual moisture content values for a
large number of soils can be obtained using the interactive computer program DBAPE
(Imhoff et al., 1990). DBAPE, which is a soils data base analyzer and parameter
estimator, is available from the U.S. EPA Center for Exposure Assessment Modeling
(CEAM) at the Environmental Research Laboratory in Athens, Georgia.
5.1.5.1.2 Air Entry Pressure Head [L]
The air entry pressure head is the threshold at which air starts to penetrate saturated
soil. It is typically a very small negative value for fine-grained materials and zero for
coarser materials. Its value can be estimated from the water retention curves of specific
106
-------
TABLE 5-6. DESCRIPTIVE STATISTICS FOR SATURATION WATER CONTENT (9.)
AND RESIDUAL WATER CONTENT (9r)
Saturation Water Content (9,) Residual Water Content (9r)
Statistic*
Soil Type x s CV n x s CV n
Clay**
Clay Loam
Loam
Loamy Sand
Silt
Silt Loam
Silty Clay
Silty Clay Loam
Sand
Sandy Clay
Sandy Clay Loam
Sandy Loam
0.38
0.41
0.43
0.41
0.46
0.45
0.36
0.43
0.43
0.38
0.39
0.41
0.09
0.09
0.10
0.09
0.11
0.08
0.07
0.07
0.06
0.05
0.07
0.09
24.1
22.4
22.1
21.6
17.4
18.7
19.6
17.2
15.1
13.7
17.5
21.0
400
364
735
315
82
1093
374
641
246
46
214
1183
0.068
0.095
0.078
0.057
0.034
0.067
0.070
0.089
0.045
0.100
0.100
0.065
0.034
0.010
0.013
0.015
0.010
0.015
0.023
0.009
0.010
0.013
0.006
0.017
49.9
10.1
16.5
25.7
29.8
21.6
33.5
10.6
22.3
12.9
6.0
26.6
353
363
735
315
82
1093
371
641
246
46
214
1183
* n = Sample size, x = Mean, s = standard deviation, CV = coefficient of
variation (percent)
** Agricultural soil, less than 60 percent clay.
Source: Sharp-Hansen et al. (1990)
Original source Carsel and Parrish (1988)
soils (Freeze and Cherry, 1979; Sharp-Hansen et al., 1990). In practice, it is regularly
assumed to be zero.
5.1.5.1.3 Van Genuchten Parameters, a [1/L]; p [--]
These are empirical parameters needed to solve the van Genuchten analytical equations
which are used to model unsaturated flow (see Equations 3-3a through 3-3d).
Descriptive statistics for these parameters have been reported by Carsel and Parrish
(1988) for a variety of soils and are shown in Table 5-7.
107
-------
o
00
5-7. DESCRIPTIVE STATISTICS FOR VAN GENUCHTEN WATER RETENTION MODEL PARAMETERS, a, B, and y
(Carsel and Parrish 1988)
Soil Type
Clay'
Clay Loam
Loam
Loamy Sand
Silt
Silt Loam
Silty Clay
Silty Clay Loam
Sand
Sandy Clay
Sandy Clay Loam
Sandy Loam
Parameter
X SD
0.008 0.012
0.019 0.015
0.036 0.021
0.124 0.043
0.106 0.007
0.020 0.012
0.005 0.005
0.010 0.006
0.145 0.029
0.027 0.017
0.059 0.038
0.075 0.037
a, cm"1
CV
160.3
77.9
57.1
35.2
45.0
64.7
113.6
61.5
20.3
61.7
64.6
49.4
Parameter 6
N
400
363
735
315
88
1093
126
641
246
46
214
1183
X
1.09
1.31
1.56
2.28
1.37
1.41
1.09
1.23
2.68
1.23
1.48
1.89
SD
0.09
0.09
0.11
0.27
0.05
0.12
0.06
0.06
0.29
0.10
0.13
0.17
CV
7.9
7.2
7.3
12.0
3.3
8.5
5.0
5.0
20.3
7.9
8.7
92
N
400
364
735
315
88
1093
374
641
246
46
214
1183
X
0.08
0.24
0.36
0.56
0.27
0.29
0.09
0.19
0.62
0.18
0.32
0.47
Parameter y
SD
0.07
0.06
0.05
0.04
0.02
0.06
0.05
0.04
0.04
0.06
0.06
0.05
CV
82.7
23.5
13.5
7.7
8.6
19.9
57.1
21.5
6.3
34.7
53.0
10.1
N
400
364
735
315
88
1093
374
641
246
46
214
1183
X = Mean, SD = Standard Deviation, CV = Coefficient of Variation (percent), N = Sample Size
* = Agricultural Soil, Clay 60 percent
-------
5.1.5.2 Tabular Data Parameters
If the user chooses to supply the soil moisture relationships in tabular form, four
parameters must be specified for each functional data point: moisture content, the
corresponding pressure head and relative permeability, and water capacity. Sets of
these parameters must be input for each type of material being simulated. The
necessary tabular data for a large number of soils can be obtained from the interactive
computer code, DBAPE, described in Section 5.1.5.1.1, with the exception of water
capacity.
Moisture content was described in Section 5.1.5.1.1. Pressure head, relative
permeability, and water capacity are briefly introduced below.
5.1.5.2.1 Relative Permeability (or Hydraulic Conductivity) []
In an unsaturated porous medium, the permeability of the water phase in the medium
is a function of the degree of saturation. The larger the degree of saturation, the larger
the permeability associated with the water phase. This unsaturated permeability is also
known as the effective permeability.
Relative permeability is defined as the ratio of the effective permeability to the
permeability at saturation. Because it is a ratio, relative permeability ranges in value
between 0.0 and 1.0. It is generally assumed that relative permeability is a scalar,
dimensionless non-linear function, even for anisotropic soils. Because of the relation of
equivalence, relative permeability is equal to relative hydraulic conductivity (Mercer et
al., 1982).
Curves showing the relationship between relative permeability and moisture content are
determined experimentally for individual soils. The tabulated data available in the
literature or in DBAPE (Imhoff et al., 1990) are extracted from these experimental
results.
5.1.5.2.2 Pressure head [L]
In groundwater hydrology, the total hydraulic head, H, is usually considered to be the
sum of two components: elevation head, z, and pressure head, h. The contribution of
velocity to the total head is neglected because velocities are usually extremely low.
Pressure head is measured in gage pressure. In the saturated zone, pressures are
greater than atmospheric and are thus recorded as positive pressures. The water table
is defined as the location at which pressure is equal to atmospheric. This implies that
pressure head is zero and the total head is equal to the elevation head. Above the water
table, pressure head is less than atmospheric and water is held in the pore spaces under
tension or suction. Thus, pressure head values in the unsaturated zone are negative.
Pressure head in the unsaturated zone is a function of moisture contentthe lower the
moisture content, the more negative the pressure head. As moisture content increases,
the surface tension forces holding the water in place between the grains of soil are
lowered, resulting in less negative pressure heads (Freeze and Cherry, 1979). The
characteristic curve showing the relationship between pressure head and moisture
content is determined experimentally for each porous medium.
109
-------
5.1.5.2.3 Moisture Content Capacity [1/L]
In an unsaturated soil, changes in moisture content, 9, are accompanied by changes in
pressure head, h. As discussed above, the 6(h) relationship results in a characteristic
curve for each soil. Example characteristic curves are shown in Figure 3.3a. The
inverse of the slope of this curve is called the water capacity, C(0), or the moisture
content capacity (Mercer et al., 1982). It is defined as:
C(9) = dO/dh (5-4)
The water capacity has no one unique value for a porous medium. Thus, the range of
values of moisture content capacity is related to the nature of the water characteristic
curve. (Mercer et al., 1982).
5.2 3DLEWASTE
5.2.1 Data Set 1: Title of the Simulation Run
5.2.1.1 Geometry, Boundary, and Pointer Array Control, IGEOM []
The integer IGEOM has two functions. It is used to specify if geometry, boundary, and
pointer arrays should be printed so that the user can examine them. It also controls
whether the boundary and pointer arrays are written to or read from binary files.
Boundary arrays store data related to the boundary conditions. Pointer arrays store the
global matrix in compressed form and are used to construct the subregional block
matrices. For large problems, it takes too much time to generate these arrays for each
computer execution of a particular scenario.
If 3DLEWASTE is being executed alone (i.e., without using 3DFEMWATER results),
these arrays should be generated only once and stored in binary files using logical units
LUBAR and LUPAR (see Table B-2). In order to compute and store the boundary and
pointer arrays, the user should choose a value for IGEOM less than or equal to one. In
subsequent runs, the boundary and pointer arrays can be read from the binary files by
changing the value of IGEOM to a number greater than three. Whenever changes are
made to the model which involve the geometry of the problem, the boundary conditions,
and the configuration of the subregions, the arrays must be generated and stored again.
If 3DLEWASTE is run after executing 3DFEMWATER for the same scenario, the
boundary array need not be recalculated (i.e., the boundary array calculated and stored
by 3DFEMWATER can be used). The pointer array should be recalculated, however.
This is done by setting IGEOM to a value greater than one and less than or equal to
three.
For each of the options explained above, if the number chosen by the user is even, the
arrays will be printed as output. If the number is odd, the arrays will not be printed.
110
-------
5.2.2 Data Set 2: Basic Integer Parameters
5.2.2.1 Number of Material Types, NMAT [--]
This parameter is the total number of different material types being modeled. When
material properties are assigned to each material type, using data set 5 (see Section
4.2.5), the first material type should be the predominant type. The number of material
types used in 3DLEWASTE need not be identical to the number specified in
3DFEMWATER.
5.2.2.2 Number of Elements with Material Property Correction, NCM []
In the code, all the grid elements automatically are initialized as having a material type
of one. To model more than one material type, the parameter NCM and the parameters
in data set 9 of the input (see Section 4.2.9) are used to specify which elements have a
material type other than material type one. The parameter NCM is the total number of
elements which have a material type different than the first material type.
5.2.2.3 Number of Time-Steps, NTI [--]
For a constant time-step size, this number is obtained by dividing the simulation time
by the time-step size, BELT. If the time-step size is variable, this number is computed
using the formula given in the note at the end of data set 2 in Section 4.2.2.
5.2.2.4 Steady-State Control, KSS [-]
As noted in Section 4.2.2, a steady-state option may be used to provide either the final
state of a system under study or the initial condition for a transient-state calculation.
In the former case, both KSS and the number of time steps, NTI, should be set to zero.
In the latter case (i.e., when KSS = 0 and NTI > 0), the code performs a steady-state
calculation before beginning the transient computations. If KSS = 1, no steady-state
calculation is performed. Rather, the code begins transient calculations using initial
conditions supplied in data set 10 of the input.
5.2.2.5 Mass Lumping Flag, ILUMP [-]
This parameter indicates if the mass matrix is to be lumped or not. Normally, one
should set this parameter to 0. Without lumping, the solution is more accurate.
However, for occasions when negative concentrations or oscillating solutions occur, this
parameter should be set to 1.
5.2.2.6 Weighting Function Control, IWET [--]
This parameter indicates if the upstream weighting function is to be used. For the
present version of code, this parameter does not affect the solution when a transient
solution is sought. If a steady-state solution is desired, one should set this parameter to
1. Thus, it is advisable to always set this parameter to 1 for the present version of the
computer code.
Ill
-------
5.2.2.7 Optimization Flag, IOPIYM [--]
This parameter specifies whether the upstream weighting factor is to be optimized. This
parameter does not affect the solution if a transient solution is sought. For a steady-
state solution, it is advisable to set IOPTIM to 1. When IOPTIM is set to O, an
upstream weighting factor of 1.0 is assumed.
5.2.2.8 Number of Iterations for the Nonlinear Equation, NITER []
This parameter is the number of iterations allowed for solving the nonlinear equation.
Normally, a value of NITER equal to 40 should be sufficient. If this number is exceeded
and the solution does not converge, the program will issue a warning message. When
this occurs, the users should re-execute the program using a larger value of NITER.
5.2.2.9 Number of Times to Reset the Time Step, NDTCHG [-]
This parameter indicates how many times one should reset the time step size to the
initially small time-step size. When we start a computation, we normally use a small
time-step size. However, for every consecutive time step, we may gradually increase the
time-step size by some amount specified by CHNG in Data Set 3 in Section 4.2.3. When
we have a steep change in boundary conditions or in source/sink conditions, we will need
to reset the time-step size to the initially small value. NDTCHG tells us how many
times we want to reset the time-step size. The value of NDTCHG must be at least one.
If the user does not want to reset the time step, a value of one should be entered here
and a very large number, larger than the total simulation time, should be entered for
TDTCH(l) in data set 4 (see Section 4.2.4).
5.2.2.10 Number of Iterations for Pointwise Solution, NPITER [-]
This parameter is used to input the number of iterations allowed for solving the matrix
equations with the block iteration method. NPITER = 300 should be sufficient for most
problems. If this number is exceeded and the solution does not converge, the program
will issue a warning message. When this occurs, the user should first recheck the input
values. If the input is correct, the program can be re-executed using a larger value for
NPITER.
5.2.2.11 Sorption Model Control, KSORP [--]
Although the Freundlich isotherm option can be used to simulate a linear isotherm by
setting the value of the exponent, n, equal to one, it is recommended that linear
isotherms be simulated using only the linear isotherm option. This is because the linear
isotherm option makes use of retarded seepage velocities, which result in a more
accurate solution for the particle tracking scheme used in 3DLEWASTE than the pore
velocities used in conjunction with the nonlinear adsorption models.
112
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5.2.3 Data Set 3: Basic Real Parameters
5.2.3.1 Initial Time-Step Size, BELT [T]
This is the time-step size used for the first time-step computation if the variable CHNG
is not equal 0.0. It is the time-step size used for every time step if the variable CHNG
is set equal to 0.0. For a steady-state computation, BELT should be chosen such that no
particle travels more than one element in one time step. For example, if an element has
a size of 10 m and the averaged velocity over this element is 0.00001 m/see, then BELT
should be less then 1,000,000 seconds. For transient computations, one should choose a
time-step size as large as possible with BELT less than BELX*BELWB, where BELX is
the size of the element and B is the dispersion coefficient. For example, if the element
size is 10 m and the dispersion coefficient is 0.00001 m2/sec, then BELT should be less
than 10,000,000 seconds.
5.2.3.2 Fractional Change in Time-Step Size, CHNG []
This parameter specifies how much of an increase one would like to make to the time-
step size for each subsequent time step. Normally, a value from 0.0 to 0.5 can be used.
5.2.3.3 Maximum Allowable Time Step, BELMAX [T]
The maximum time-step size allowed depends on how fast the system responds to
change. Use of a value one to ten times the size of the initial time step is advised.
5.2.3.4 Maximum Simulation Time, TMAX [T]
This is the actual length of time to be simulated. If this time is exceeded before you
have made NTI step computations, the simulation will be terminated.
5.2.3.5 Relaxation Parameter for Solving the Nonlinear Equation, OME []
Normally this parameter should be set to 1.0 (see Equation 3-48). If the convergence
history shows sign of oscillation, then a value of 0.5 should be used. If the convergence
history shows monotonic decreases but at a very slow rate, then OME should be set to
somewhere between 1.7 to 1.9.
5.2.3.6 Iteration Parameter to Solve the Linearized Matrix Equation, OMI []
Normally this parameter should be set to 1.0 (see Equation 3-49). If the convergence
history shows signs of oscillation, then set OMI to 0.5. If the solution converges
monotonically but at a very slow rate, then set OMI to between 1.7 and 1.9.
5.2.3.7 Transient Convergence Criterion, TOLB []
This is the relative error allowed for assessing if a solution has converged for each time
step. Setting TOLB equal to 0.000001 should be sufficient for most problems.
113
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5.2.3.8 Steady-State Convergence Criterion, TOLA []
This is the relative error allowed for assessing if a steady-state solution has converged.
TOLA = 0.00001 should be sufficient for most problems.
5.2.4 Data Set 5: Material Properties
5.2.4.1 Distribution Coefficient [L3/M]
Freeze and Cherry (1979) state that adsorption/desorption reactions for contaminants in
groundwater are normally viewed as being very rapid relative to the flow velocity and
that the amount of contaminant adsorbed is commonly a function of concentration in the
solution. At constant temperature and low-to-moderate concentrations, the functional
relationship between the adsorbed concentration, S (M/L3), and the dissolved concentra-
tion, C (M/L3), is often approximated by the Freundlich equilibrium isotherm (Helfferich,
1962):
S = KCn (5-5)
where the coefficients K and n depend on several factors, including the solute species
and the nature of the porous medium. If the isotherm is linear, n = 1, K is known as
the distribution coefficient, Kj. The derivation of the distribution coefficient, which is
different for each constituent, is discussed briefly in Section 3.3.1.
5.2.4.2 Bulk Density [M/L3]
Bulk density can be defined as the mass of a unit volume of dry soil. The soil bulk
density directly influences the retardation of solutes and is related to the structure and
texture of a soil (Mercer et al., 1982).
The bulk density of soils typically range between 1.3 and 2.0 g/cm3, but Mercer et al.
(1982) state that the bulk density can be as low as 0.3 g/cm3 for soils high in organics or
aluminum and iron hydroxides. Representative values for five different types of soils
are shown in Table 5-8. In addition, values of bulk density for a large number of soils
can be obtained from the interactive computer program DBAPE, which was discussed in
Section 5.1.5.1.1.
The bulk density of aquifer materials may differ significantly from that of soils. There-
fore, data on the ranges of bulk density for various geologic material are presented in
Table 5-9. If site-specific data are not available, the bulk density of the saturated zone
can be derived using an exact relationship between porosity, particle density and the
bulk density (Freeze and Cherry, 1979). Assuming the particle density to be 2.65 g/cm3,
we can express this relationship as:
pb = 2.65(1 - 0) (5-6)
where
pb = bulk density of the soil (g/cm3)
0 = saturated moisture content (porosity) ()
114
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TABLE 5-8. MEAN BULK DENSITY (g/cm3) FOR FIVE SOIL TEXTURAL
CLASSIFICATIONS'^
Soil Texture Mean Value Range Reported
Silt Loams 1.32 0.86-1.67
Clay and Clay Loams 1.30 0.94-1.54
Sandy Loams 1.49 1.25-1.76
Gravelly Silt Loams 1.22 1.02-1.58
Loams 1.42 1.16-1.58
All Soils 1.35 0.86-1.76
a Baes, C. F., Ill and R.D. Sharp. 1983. A Proposal for Estimation of
Soil Leaching Constants for Use in Assessment Models. J. Environ.
Qual. 12(1): 17-28 (Original reference).
"From Dean et al. (1989)
5.2.4.3 Longitudinal and Transverse Dispersivity [L]
Hydrodynamic dispersion is a non-steady, irreversible mixing process by which a
contaminant spreads as it is transported through the subsurface. It results from the
effects of two components: molecular diffusion and mechanical dispersion. The larger
the hydrodynamic dispersion term is, the larger the spreading of an initially localized
contaminant. Molecular diffusion is discussed in Section 5.2.4.4. Mechanical dispersion,
D, is caused by variations in pore velocities in a soil or aquifer material. In addition,
variations in the rate of advection caused by aquifer inhomogeneity and spatially-
variable hydraulic conductivities results in plume spreading, which is often confused
with dispersion (Keely, 1989).
Although mechanical dispersion is a second rank tensor, by assuming that a material is
isotropic with respect to dispersion, the dispersion tensor can be expressed in terms of
the average groundwater velocity and two constants: the longitudinal and transverse
dispersivity (see Equation 3-20). Longitudinal dispersivity, OCL, is defined as the
characteristic mixing length in the direction of groundwater flow and lateral
dispersivityjOj., is the mixing length in the directions perpendicular to flow.
Values for dispersivity are difficult to determine. Research has shown that the values
are dependent on the scale of the problem being studied (EPRI, 1985). This can be seen
in Figure 5.1. Usually, dispersion is determined by adjusting the dispersivity values
until modeling results match historical data (Mercer et al., 1982). Transverse
dispersivity values are commonly thought to be lower than longitudinal dispersivity
values by a factor of 3 to 20. However, recent studies suggest that transverse
dispersivity values should be at least an order-of-magnitude smaller than longitudinal
dispersivity values (Gelhar et al., 1992) and may even be close to zero (U.S. EPA, 1989).
115
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TABLE 5-9. RANGE AND MEAN VALUES OF DRY BULK DENSITY FOR VARIOUS
GEOLOGIC MATERIALS
Material
Clay
Silt
Sand, fine
Sand, medium
Sand, coarse
Gravel, fine
Gravel, medium
Gravel, coarse
Loess
Eolian sand
Till, predominantly silt
Till, predominantly sand
Till, predominantly gravel
Glacial drift, predominantly silt
Glacial drift, predominantly sand
Glacial drift, predominantly gravel
Sandstone, fine grained
Sandstone, medium grained
Siltstone
Claystone
Shale
Limestone
Dolomite
Granite, weathered
Gabbro, weathered
Basalt
Schist
Range (g/cm3)
1.18-1.72
1.01-1.79
1.13-1.99
1.27-1.93
1.42-1.94
1.60-1.99
1.47-2.09
1.69-2.08
1.25-1.62
1.33-1.70
1.61-1.91
1.69-2.12
1.72-2.12
1.11-1.66
1.36-1.83
1.47-1.78
1.34-2.32
1.50-1.86
1.35-2.12
1.37-1.60
2.20-2.72
1.21-2.69
1.83-2.20
1.21-1.78
1.67-1.77
1.99-2.89
1.42-2.69
Mean (g/cm3)
1.49
1.38
1.55
1.69
1.73
1.76
1.85
1.93
1.45
1.58
1.78
1.88
1.91
1.38
1.55
1.60
1.76
1.68
1.61
1.51
2.53
1.94
2.02
1.50
1.73
2.53
1.76
Reference: Morris and Johnson (1967); Mills et al. (1985b)
116
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10'1 10" 10
10' 10* 103 10c
Figure 5.1. Longitudinal dispersivity versus scale with data classified by reliability
(from Gelhar et al., 1992).
As initial estimates for longitudinal and transverse dispersivity, Dean et al. (19$9)
suggest the following relationships, based on values presented in the Federal Register
(1986):
OL = 0.1 x, (5-7 a)
Op = OL/S.O (5-7b)
where x, is the distance from the source to a downgradient point of interest.
5.2.4.4 Molecular Diffusion Coefficient in Water [L2/T]
As stated above, molecular diffusion and mechanical dispersion are both responsible for
the dispersion of solutes in groundwater systems. Molecular diffusion, which is a non-
reversible process, is typically small compared to mechanical dispersion and is often
neglected in groundwater studies. However, when groundwater velocities are very low,
molecular diffusion can become significant.
The flux of a solute in a fluid due to molecular diffusion is described by Pick's Law,
which states that the flux is proportional to the concentration gradient. The coefficient
of proportionality is called the molecular diffusion coefficient, am. Values for the
molecular diffusion coefficient in a fluid continuum are generally well known and are
117
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typically in the range of 10~9 m2/s or less at 20°C. If necessary, am, which varies with
temperature, can be estimated from methods described in Lyman et al. (1982).
5.2.4.5 Tortuosity []
The molecular diffusion coefficient for a solute in a porous medium is smaller than the
coefficient of diffusion in a body of water because diffusion in solids is negligible. The
amount by which the molecular diffusion coefficient is reduced is expressed by a
coefficient called tortuosity. Tortuosity is a second-rank tensor which for isotropic
conditions reduces to a scalar. It expresses the effect of the configuration of the water
occupying a porous medium (Bear and Verruijt, 1987).
De Marsily (1986) states that a medium's tortuosity, T, can be defined as:
T = 1/F<(> (5-8)
where
F = formation factor (the ratio of a rock's electric resistivity over the resistivity of
its contained water) ()
(j> = total porosity ()
The author states that tortuosity varies in practice from 0.1 for clays to 0.7 for sands.
Freeze and Cherry (1979) state that the coefficient, which is always less than one,
usually has a value between 0.01 and 0.5.
Bresler (1973), as found in Dean et al. (1989), provides the following equation to
estimate diffusion coefficient in a porous medium:
Dm = Dwaebe (5-9)
where
Dm = coefficient of diffusion in a porous medium (cm2/day)
Dw = coefficient of diffusion in water (cm2/day)
a = soil constant having a range of 0.001 to 0.005
b = soil constant having an approximate value of 10
9 = volumetric water content (cm3/cm3)
In the above equation, the term aebe represents an estimate of the soil's tortuosity.
5.2.4.6 Decay Constant [1/T]
A number of processes, such as hydrolysis and biodegradation, contribute to the
disappearance of chemicals in the subsurface. The extent to which these processes are
important depends on both environmental conditions and the chemical's properties. In
this model, the effects of individual processes on the degradation of a chemical in the
subsurface are not considered. Instead, lumped first-order decay with respect to the
concentration of the solute is assumed to occur, with a single first-order decay constant
controlling the modelled rate of disappearance in each porous material.
118
-------
When estimating a value for the first-order decay constant, one should determine which
processes are likely to be important at the study area. Hydrolysis is a potentially
significant elimination pathway for many organic chemicals. However, for chemicals
that readily biodegrade, hydrolysis may be insignificant relative to biodegradation.
Methods of estimating a first-order rate constant resulting from hydrolysis are presented
in Lyman et al. (1982). Values for hydrolysis rate constants can be found in a large
number of references, including Lyman et al. (1982), Mabey et al. (1982), and Mills et al.
(1985a).
Although biodegradation is the most significant means of removal for many organics in
the subsurface, it is a very complex and poorly understood process. Biodegradation in
the subsurface depends on a number of variable and/or unknown processes, such as the
number of microorganisms present, the availability of oxygen and other nutrients, and
the Ph and temperature of the subsurface environment (Sharp-Hansen et al., 1990).
Therefore it is very difficult to estimate the first-order decay coefficient resulting from
biodegradation. Laboratory-derived biodegradation rate constants have been compiled
by Lyman et al. (1982), Mabey et al. (1982), and Mills et al. (1985a), among others.
However, these laboratory-based values may be inappropriate for field conditions.
Therefore, considerable care should be exercised if these data are used.
5.2.5 Data Set 17: Hydrological Variables
For most wellhead protection applications of this code, the velocity field and moisture
content field will not need to be specified in the input. Instead, these variables should
be calculated and stored by the variably-saturated flow code, 3DFEMWATER. The
stored arrays of data are then accessed by 3DLEWASTE. Only when 3DLEWASTE is
executed without first running 3DFEMWATER does the user need to supply values for
these variables. Moisture content was introduced in Section 5.1.5.1.1 and will not be
discussed here.
5.2.5.1 Velocity Field [L/T]
The velocity distribution is needed to quantify transport by advection. Groundwater
velocities are routinely determined indirectly using measurements of hydraulic head,
hydraulic conductivity, and Darcy's equation. For the case when the x, y, and z axes
coincide with the principal directions of anisotropy, Darcy's Law, in terms of the Darcy
velocity, is written as:
v^-K^dh/dx (5-10a)
vy = -Kydh/dy (5-1 Ob)
v^-K.dh/dz (5-10c)
where K,, Ky, and Kj are the hydraulic conductivity values in the x, y, and z directions,
and dh/dx, dh/dy, and dh/dz are the hydraulic gradients in the x, y, and z directions. A
more generalized form can be written as:
119
-------
dh/dx - K^ dh/dy - K,z dh/dz
dh/dX - Kyy dh/dy > Ky, dh/dZ
vz = -K,, dh/dx - K,y dh/dy - K,z dh/dz
Vy = -
(5-ha)
(5-llb)
(5-llc)
Since velocity depends on the gradient as well as the hydraulic conductivity, its range is
somewhat arbitrary. A range of velocities is given in Figure 5.2.
5J
c
5
3
U5
a!
< CO
c
^
5
C/J
1
g
ifl
yr
EU
H
1
O
K q ah/it
m/s
1-
10-L
10 i
,o-l
»i
10-t-
io-7-
10-*-
10-?-
10-!°-
10-"-
10-12-
ft/d mis
I ~
- 10'
104
10 4-
- 103
10*-
_ 102
10
- 10 '
10 -ie-
_ 1
10 "U
-10-'
10-"-
_io-2
m/s
"I T <
_M4
ft/d " l0"1
_102
_ 1C'2
-- io-2
-- 10-3
= I2
ft/y
_10^
_ 10-3
-»-
_io-5
^
- 10-7
Figure 5.2. Nomograph for determining Darcy velocity (from Mercer et al.
1982)
120
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SECTION 6
EXAMPLE PROBLEMS
6.1 3DFEMWATER
To demonstrate the application of 3DFEMWATER, three simple example problems are
presented. These three problems represent one-, two-, and three-dimensional
applications, respectively. For each problem, a brief description and a correctly-
constructed input data set are given. The corresponding output is not included in this
documentation. Rather, it is distributed along with the code by the EPA Center for
Exposure Assessment Modeling (CEAM) at the Environmental Research Laboratory in
Athens, Georgia. See Section 2 for information about obtaining the code.
6.1.1 One-Dimensional Column
One-dimensional transient flow through a column is simulated in this example. The
column is 200 cm long and is 50 cm by 50 cm in cross-section (Figure 6.1). The soil in
the column is assumed to be a sandy clay loam which has a saturated hydraulic
conductivity of 31.4 cm/d, a porosity of 0.39 and a residual moisture content of 0.10. The
unsaturated characteristic hydraulic properties of the soil in the column are represented
by the van Genuchten analytical functions with the empirical coefficient alpha equal to
0.059 and the empirical coefficient beta equal to 1.48.
The initial conditions assumed are a pressure head of -90.0 cm imposed on the top
surface of the column, 0.0 cm on the bottom surface of the column, and -97.0 cm
elsewhere. The boundary conditions are as follows. No flux is imposed on the left,
front, right, and back surfaces of the column (this is done automatically by the code).
Pressure head is held at 0.0 cm on the bottom surface using a Dirichlet boundary
condition. A variable boundary condition is used on the top surface of the column with a
pending depth of zero, minimum pressure of -90.0 cm, a rainfall of 5.0 cm/d for the first
10 days, and a potential evaporation of 5.0 cm/d for the second 10 days.
The region of interest, that is, the whole column, is discretized with 1 x 1 x 40 = 40
elements with the element size equal to 50 x 50 x 5 cm. This results in 2x2x41 = 164
node points. For this simulation, each of the four vertical lines is considered a
subregion. Thus, a total of four subregions, each with 41 node points, is used for the
subregional block iteration simulation.
A variable time step size is used. The initial time step size is 0.05 days, and each
subsequent time step size is increased by 0.2 times with a maximum time step
121
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200cm
Figure 6.1. One-dimensional transient flow through a soil column.
size not greater than 1.0 d. Because there is an abrupt change in the flux value
from 5 cm/d (infiltration) to -5 cm/d (evaporation) imposed on the top surface at day 10,
the time step size is automatically reset to 0.05 d on the tenth day. Because a 20-day
simulation is to be made, 44 time steps are needed.
A pressure head tolerance of 0.02 cm is selected for the nonlinear iteration and a
tolerance of 0.01 cm is used for the block iteration. The relaxation factors for both the
nonlinear iteration and block iteration are set equal to 0.5.
The input data set for this problem, prepared according to the instructions in Sections
4.1 and 5.1, is shown in Table 6-1.
6.1.2 Two-dimensional Drainage Problem
Two-dimensional steady-state flow is simulated in this problem. The region of interest
is bounded on the left and right by parallel drains which fully penetrate the medium.
The bottom is an impervious layer and the top is an air-soil interface (Figure 6.2). The
distance between the two drains is 20 m. The medium is assumed to have a saturated
horizontal hydraulic conductivity of 0.31 m/d and vertical hydraulic conductivity of 0.12
m/d, a porosity of 0.39, and a field capacity of 0.10. The unsaturated characteristic
122
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TABLE 6-1. INPUT DATA SET FOR THE ONE-DIMENSIONAL 3DFEMWATER
PROBLEM
1 SIMULATION OF ONE-D COLUMN INFILTRATION-EVAPORATION; L=CM, T=DAY, M=G Oil
C ******* DATA SET 2: BASIC INTEGERS
164 40 1 0 44 1 61 0 0 50 20 3 100
C ******* DATA SET 3: BASIC REAL PARAMETERS
0.05DO 0.2DO l.ODO 20.0DO 2.0D-2 2.0D-2 l.ODO 7.316D12
1.1232D2 l.ODO 0.5DO 0.5DO
C ******* DATA SET 4: PRINTER, STORAGE CONTROL AND TIME STEP SIZE RESETTING
333030300030003003000033303030003000300300003
111010100010001001000011101010001000100100001
1.0D01 2.0000D1 1.0D38
C ******* DATA SET 5: MATERIAL PROPERTIES
O.ODO O.ODO 31.40DO O.ODO O.ODO O.ODO
C ******* DATA SET 6: SOIL PROPERTY PARAMETERS
050
0.100DO 0.390DO O.OODO 0.059DO 1.48DO THPROP
C ******* DATA SET 7: NODE COORDINATES
1
42
83
124
0
40
40
40
40
0
1
1
1
1
0
O.ODO
O.ODO
50.0DO
50.0DO
0.0
50.0DO
O.ODO
O.ODO
50.0DO
0.0
O.ODO
O.ODO
O.ODO
O.ODO
0.0
O.ODO
O.ODO
O.ODO
O.ODO
0.0
O.ODO
O.ODO
O.ODO
O.ODO
0.0
5.0DO
5.0DO
5.0DO
5.0DO
0.0
C ******* DATA SET 8: SUBREGIONAL DATA
4
1
0
1
0
1
0
1
0
1
0
3
0
40
0
40
0
40
0
40
0
1
0
1
0
1
0
1
0
1
0
41
0
1
0
42
0
83
0
124
0
0
0
1
0
1
0
1
0
1
0
END OF NNPLR(K)
END OF GNLR(I,1)
END OF GNLR(I,2)
END OF GNLR(I,3)
END OF GNLR(I,4)
C ******* DATA SET 9: ELEMENT INCIDENCES
1 39 1 42 83 124 1 43
00000 000
C ******* DATA SET 11: INITIAL CONDITIONS
84
0
125
0
1
0
END OF IE
1
2
43
84
125
41
0
3
38
38
38
38
3
0
41
1
1
1
1
41
0
O.ODO
-9.70D1
-9.70D1
-9.70D1
-9.70D1
-9.00D1
0.0
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
0.0
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
0.0
END OF 1C
C ******* DATA SET 12: SOURCE/SINK AND B. C. CONTROL INTEGERS
00000000412
14140000000
0
0
123
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TABLE 6-1. INPUT DATA SET FOR THE ONE-DIMENSIONAL 3DFEMWATER PROBLEM
(concluded)
C ******* DATA SET 14: VARIABLE BOUNDARY CONDITIONS
O.ODO
1
0
1
0
1
0
1
0
1
0
*******
O.ODO
1
1
0
0
0
0
0
0
3
0
3
0
3
0
DATA
42
3
0
5.0DO
0
0
0
0
1
0
1
0
1
0
SET 15:
O.ODO
83
1
0
10.
1
0
82
0
41
0
0.0
ODO 5.0DO 10.001DO -5.0DO 1.0D38 -5.(
0
0
123 164
0 0
41
0
O.ODO
0.0
-90.0DO
0.0
END OF IRTYP
41 0
0 0
END
O.ODO
0.0
O.ODO
0.0
0 0
0 0
OFNPVB
0.0
0.0
0.0
0
0 END OF ISV(J,I) ,
END OF HCON
END OF HMIN
DIRICHLET BOUNDARY CONDITIONS
124
1
0
T5TSTT)
1.0D38
0
0
OF .TOR
O.ODO
END OF
IDTYP
000
-10m
10m
Figure 6.2. Two-dimensional steady-state flow to parallel drains.
124
-------
hydraulic properties of the medium are given by the van Genuchten analytical
functions with the empirical coefficient alpha equal to 0.059 and the empirical
coefficient beta equal to 1.48.
Because of symmetry, the region to be simulated is 0.0< x < 10.0 m and 0.0< z < 10.0
m, with a width of 10 m assumed in the y-direction. A no flux boundary is imposed on
the left (x = 0.0), front (y = 0.0), back (y = 10.0), and bottom (z = 0.0) sides of the
region. Pressure head is assumed to vary from zero at the water surface (z = 2.0) to
2.0 m at the bottom (z = 0.0) on the right side (x = 10.0). Variable conditions are used
elsewhere. Pending depth is assumed to be zero meters on the whole variable
boundary. Fluxes on the top of the variable boundary are assumed equal to 0.006 m/d
and on the right side, above the water surface, are equal to zero. A pre-initial
condition for the steady-state solution is set as h = 10- z.
The region of interest is discretized with lOx 1 x 10 = 100 elements with the element
size equal to 1 x 10 x 1 cm. This results in 11x2x11= 242 nodal points. Each of the
two vertical planes is considered a subregion. Thus, a total of two subregions, each
with 121 node points, is used for the subregional block iteration simulation.
A pressure head tolerance of 0.002 m is set for the for nonlinear iteration and a value
of 0.001 m is used for the block iteration. The relaxation factors for both the nonlinear
iteration and block iteration are set equal to 0.5.
The input data set for this problem, prepared according to the instructions in Sections
4.1 and 5.1, is shown in Table 6-2.
6.1.3 Three-Dimensional Pumping Problem
Three-dimensional steady-state flow to a pumping well is simulated in this problem.
The region of interest is bounded on the left and right by hydraulically connected
rivers; on the front, back, and bottom by impervious confining beds; and on the top by
an air-soil interface (Figure 6.3). A pumping well is located at (x,y) = (540,400) in
Figure 6.3. Initially, the water table is assumed to be horizontal and is 60 m above the
bottom of the aquifer. The water level at the well is then lowered to a height of 30 m.
This height is held until a steady state condition is reached. The medium in the region
is assumed to be anisotropic and has saturated hydraulic conductivity components K^
= 0.31 m/d, Kyy = 0.03 m/d, and K^z = 0.12 m/d. The porosity of the medium is 0.10 and
the field capacity is 0.39. The unsaturated characteristic hydraulic properties of the
medium are given by the van Genuchten analytical functions with the empirical
coefficient alpha equal to 0.059 and the empirical coefficient beta equal to 1.48.
Because of symmetry, the region to be simulated is taken as 0 < x < 1000 m, 0 < y <
400 m, and 0 < z < 72 m. Two types of boundary conditions are used. Pressure head
is assumed hydrostatic on two vertical planes. The first is located at x = 0 and 0 < z
< 60 and the second, at x = 1000 and 0 < z < 60. A no flux boundary is imposed on all
other boundaries of the flow regime. The pre-initial condition for the steady-state
solution is set so that the pressure head, h = 60- z.
125
-------
TABLE 6-2. INPUT DATA SET FOR THE TWO-DIMENSIONAL 3DFEMWATER
PROBLEM
2 SIMULATION OF TWO-D STEADY DRAINAGE; L=M, T=DAY, M=KG 111
C ******* DATA SET 2: BASIC INTEGERS
242 100 1 0 00 6 1 0 0 50 20 1 100
C ******* DATA SET 3: BASIC REAL PARAMETERS
0.05DO 0.2DO l.ODO 20.0DO 2.0D-3 2.0D-3 l.ODO 7.316D10
1.1232D4 l.ODO 0.5DO 0.5DO 0.0
C ******* DATA SET 4: PRINTER, STORAGE CONTROL AND TIME STEP SIZE RESETTING
33
11
1.0D38
C ******* DATA SET 5: MATERIAL PROPERTIES
0.31DO O.ODO 0.12DO O.ODO O.ODO O.ODO
C ******* DATA SET 6: SOIL PROPERTY PARAMETERS
050
0.100DO 0.390DO O.OODO 0.059DO 1.48DO THPROP
C ******* DATA SET 7: NODE COORDINATES
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
O.ODO O.ODO
0.0 0.0
1
2
3
4
5
6
7
8
9
10
11
122
123
124
125
126
127
128
129
130
131
132
0
Q *******
2
1
0
1
0
1
0
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
0
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
0
DATA SET 8:
1
0
120
0
120
0
1
0
1
0
1
0
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
0.0
SUBREGIONAL
121 0
0 0
1 1
0 0
122 1
0 0
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
10.0DO
10.0DO
10.0DO
10.0DO
10.0DO
10.0DO
10.0DO
10.0DO
10.0DO
10.0DO
10.0DO
0.0
DATA
END OF
END OF
END OF
O.ODO
l.ODO
2.0DO
3.0DO
4.0DO
5.0DO
6.0DO
7.0DO
8.0DO
9.0DO
10.0DO
O.ODO
l.ODO
2.0DO
3.0DO
4.0DO
5.0DO
6.0DO
7.0DO
8.0DO
9.0DO
10.0DO
0.0
NNPLR(K)
GNLR(I,1)
GNLR(I,2)
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
l.ODO
0.0
126
-------
TABLE 6-2. INPUT DATA SET FOR THE TWO-DIMENSIONAL 3DFEMWATER
PROBLEM (continued)
C ******* DATA SET
1
11
21
31
41
51
61
71
81
91
0
IE
9
9
9
9
9
9
9
9
9
9
0
1
1
1
1
1
1
1
1
1
1
0
C ******* DATA SET
1
2
3
4
5
6
7
8
9
10
11
122
123
124
125
126
127
128
129
130
131
132
0
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
0
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
0
C ******* DATA SET
0
18
0
38
0
2
9: ELEMENT INCIDENCES
1
12
23
34
45
56
67
78
89
100
0
12
23
34
45
56
67
78
89
100
111
0
133
144
155
166
177
188
199
210
221
232
0
122
133
144
155
166
177
188
199
210
221
0
2
13
24
35
46
57
68
79
90
101
0
13
24
35
46
57
68
79
90
101
112
0
134 123 1
145 134 1
156 145 1
167 156 1
178 167 1
189 178 1
200 189 1
211 200 1
222 211 1
233 222 1
000 END OF
11: INITITAL CONDITIONS
10.0DO
9.0DO
8.0DO
7.0DO
6.0DO
5.0DO
4.0DO
3.0DO
2.0DO
l.ODO
O.ODO
10.0DO
9.0DO
8.0DO
7.0DO
6.0DO
5.0DO
4.0DO
3.0DO
2.0DO
l.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
12: SOURCE/SINK AND
0 0
2 0
0
0
0
0
B.C.
0
0
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
CONTROL
END OF 1C
INTEGERS
6 1
0 0
2 0
00000
127
-------
TABLE 6-2. INPUT DATA SET FOR THE TWO-DIMENSIONAL 3DFEMWATER
PROBLEM (concluded)
C ******* DATA SET 14: VARIABLE BOUNDARY CONDITIONS
O.ODO
O.ODO
1
11
0
1
11
0
1
12
20
31
0
1
0
1
0
*******
9
7
0
9
7
0
10
7
10
7
0
37
0
37
0
6.0D-3
O.ODOO
1 1
1 2
0 0
1 11
1 120
0 0
1 11
1 120
1 132
1 241
0 0
1
0
1
0
DATA SET 15:
O.ODO
111
1
0
n
112
5
0
2.0DO
113 232
1 1
0 0
1.0D38
1.0D38
0
0
0
22 143
241 242
0 0
11
-1
11
-1
0
O.ODO
O.ODO
-90.0D2
O.ODO
6.0D-3
O.ODOO
132
121
0
END
O.ODO
O.ODO
O.ODO
O.ODO
END OF IRTYP
11 11 11 11
-1 -1 -1 -1
0000 END OF ISV(J,I) J=l,4
OFNPVB
O.ODO
O.ODO END OF HCON
O.ODO
O.ODO END OF HMIN
DIRICHLET BOUNDARY CONDITIONS
1.0D38
233 234
0
0 END OF
2.0DO
IDTYP
F.Nn OF .TOR
nnn
*- 460mH
Figure 6.3. Three-dimensional steady-state flow to a pumping well.
128
-------
The region of interest is discretized with 20 x 8 x 10 = 1600 elements, resulting in 21 x
9x11= 2079 nodal points. The nodes are located at x = 0, 70, 120, 160, 200, 275, 350,
400, 450, 500, 540, 570, 600, 650, 750, 800, 850, 900, 950, and 1000 in the x-direction,
and at z = 0, 15, 30, 35, 40, 45, 50, 55, 60, 66, and 72 in the z-direction. In the y-
direction, nodes are spaced evenly at Az = 50 m. For the simulation, each of the nine
vertical planes perpendicular to the y-axis is considered a subregion. Thus, a total of 9
subregions, each with 231 node points, is used for the subregional block iteration
simulation.
The pressure head tolerance is set at .01 m for the nonlinear iteration and is .005 m
for the block iteration. The relaxation factors for the nonlinear iteration and block
iteration are set equal to 1.0 and 1.5, respectively.
The input data set for this problem, prepared according to the instructions in Sections
4.1 and 5.1, is shown in Table 6-3.
129
-------
TABLE 6-3. INPUT DATA SET FOR THE THREE-DIMENSIONAL 3DFEMWATER
PROBLEM
3 SIMULATION OF THREE-D PUMPING WELL; L = M, T = DAY, M = KG
C ******* DATA SET 2: BASIC INTEGER PARAMETERS
2079 1600 1 0 0 0 6 1 0 0 50 20 1 100
C ******* DATA SET 3: BASIC REAL PARAMETERS
0.05DO O.ODO l.ODO 20.0DO l.OD-2 l.OD-2 l.ODO 7.316D10
1. 1232D4 l.ODO l.ODO 1.5DO
C ******* DATA SET 4: PRINTER, STORAGE CONTROL AND TIME STEP SIZE RESETTING
55
11
1.0D38
C ******* DATA SET 5: MATERIAL PROPERTIES
0.31DO 0.03DO 0.12DO O.ODO O.ODO O.ODO
C ******* DATA SET 6: SOIL PROPERTY PARAMETERS
050
0.1000DO 0.390DO O.OODO 0.059DO 1.48DO THPROP
C ******* DATA SET 7: NODE COORDINATES
Oil
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0.70D+02
0.70D+02
0.70D+02
0.70D+02
0.70D+02
0.70D+02
0.70D+02
0.70D+02
0.70D+02
0.70D+02
0.70D+02
0.12D+03
0.12D+03
0.12D+03
0.12D+03
0.12D+03
0.12D+03
0.12D+03
0.12D+03
0.12D+03
0.12D+03
0.12D+03
0.16D+03
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0. OOD+00
O.OOD+00
0. OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0. OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0. OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0. 15D+02
0.30D+02
0.35D+02
0.40D+02
0.45D+02
0.50D+02
0.55D+02
0.60D+02
0.66D+02
0.72D+02
O.OOD+00
0.15D+02
0.30D+02
0.35D+02
0.40D+02
0.45D+02
0.50D+02
0.55D+02
0.60D+02
0.66D+02
0.72D+02
O.OOD+00
0. 15D+00
0.30D+00
0.35D+02
0.40D+02
0.45D+02
0.50D+02
0.55D+02
0.60D+02
0.66D+02
0.72D+02
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
O.OOD+00
0. OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0. OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0. OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0. OOD+00
130
-------
TABLE 6-3. INPUT DATA SET FOR THE THREE-DIMENSIONAL 3DFEMWATER
PROBLEM (continued)
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
0.16D+03
0.16D+03
0.16D+03
0.16D+03
0.16D+03
0. 16D+03
0. 16D+03
0.16D+03
0. 16D+03
0. 16D+03
0.20D+03
0.20D+03
0.20D+03
0.20D+03
0.20D+03
0.20D+03
0.20D+03
0.20D+03
0.20D+03
0.20D+03
0.20D+03
0.28D+03
0.28D+03
0.28D+03
0.28D+03
0.28D+03
0.28D+03
0.28D+03
0.28D+03
0.28D+03
0.28D+03
0.28D+03
0.35D+03
0.35D+03
0.35D+03
0.35D+03
0.35D+03
0.35D+03
0.35D+03
0.35D+03
0.35D+03
0.35D+03
0.35D+03
0.40D+03
0.40D+03
0.40D+03
0.40D+03
0.40D+03
0.40D+03
0.40D+03
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0.15D+02
0.30D+02
0.35D+02
0.40D+02
0.45D+02
0.50D+02
0.55D+02
0.60D+02
0.66D+02
0.72D+02
O.OOD+00
0.15D+02
0.30D+02
0.35D+02
0.40D+02
0.45D+02
0.50D+02
0.55D+02
0.60D+02
0.66D+02
0.72D+02
O.OOD+00
0.15D+02
0.30D+02
0.35D+02
0.40D+02
0.45D+02
0.50D+02
0.55D+02
0.60D+02
0.66D+02
0.72D+02
O.OOD+00
0.15D+02
0.30D+02
0.35D+02
0.40D+02
0.45D+02
0.50D+02
0.55D+02
0.60D+02
0.66D+02
0.72D+02
O.OOD+00
0.15D+02
0.30D+02
0.35D+02
0.40D+02
0.45D+02
0.50D+02
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
131
-------
TABLE 6-3. INPUT DATA SET FOR THE THREE-DIMENSIONAL 3DFEMWATER
PROBLEM (continued)
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
0.40D+03
0.40D+03
0.40D+03
0.40D+03
0.45D+03
0.45D+03
0.45D+03
0.45D+03
0.45D+03
0.45D+03
0.45D+03
0.45D+03
0.45D+03
0.45D+03
0.45D+03
0.50D+03
0.50D+03
0.50D+03
0.50D+03
0.50D+03
0.50D+03
0.50D+03
0.50D+03
0.50D+03
0.50D+03
0.50D+03
0.54D+03
0.54D+03
0.54D+03
0.54D+03
0.54D+03
0.54D+03
0.54D+03
0.54D+03
0.54D+03
0.54D+03
0.54D+03
0.57D+03
0.57D+03
0.57D+03
0.57D+03
0.57D+03
0.57D+03
0.57D+03
0.57D+03
0.57D+03
0.57D+03
0.57D+03
0.60D+03
0.60D+03
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0.55D+02
0.60D+02
0.66D+02
0.72D+02
O.OOD+00
0.15D+02
0.30D+02
0.35D+02
0.40D+02
0.45D+02
0.50D+02
0.55D+02
0.60D+02
0.66D+02
0.72D+02
O.OOD+00
0.15D+02
0.30D+02
0.35D+02
0.40D+02
0.45D+02
0.50D+02
0.55D+02
0.60D+02
0.66D+02
0.72D+02
O.OOD+00
0.15D+02
0.30D+02
0.35D+02
0.40D+02
0.45D+02
0.50D+02
0.55D+02
0.60D+02
0.66D+02
0.72D+02
O.OOD+00
0.15D+02
0.30D+02
0.35D+02
0.40D+02
0.45D+02
0.50D+02
0.55D+02
0.60D+02
0.66D+02
0.72D+02
O.OOD+00
0.15D+02
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
132
-------
TABLE 6-3. INPUT DATA SET FOR THE THREE-DIMENSIONAL 3DFEMWATER
PROBLEM (continued)
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
0.60D+03
0.60D+03
0.60D+03
0.60D+03
0.60D+03
0.60D+03
0.60D+03
0.60D+03
0.60D+03
0.65D+03
0.65D+03
0.65D+03
0.65D+03
0.65D+03
0.65D+03
0.65D+03
0.65D+03
0.65D+03
0.65D+03
0.65D+03
0.70D+03
0.70D+03
0.70D+03
0.70D+03
0.70D+03
0.70D+03
0.70D+03
0.70D+03
0.70D+03
0.70D+03
0.70D+03
0.75D+03
0.75D+03
0.75D+03
0.75D+03
0.75D+03
0.75D+03
0.75D+03
0.75D+03
0.75D+03
0.75D+03
0.75D+03
0.80D+03
0.80D+03
0.80D+03
0.80D+03
0.80D+03
0.80D+03
0.80D+03
0.80D+03
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0.30D+02
0.35D+02
0.40D+02
0.45D+02
0.50D+02
0.55D+02
0.60D+02
0.66D+02
0.72D+02
O.OOD+00
0.15D+02
0.30D+02
0.35D+02
0.40D+02
0.45D+02
0.50D+02
0.55D+02
0.60D+02
0.66D+02
0.72D+02
O.OOD+00
0.15D+02
0.30D+02
0.35D+02
0.40D+02
0.45D+02
0.50D+02
0.55D+02
0.60D+02
0.66D+02
0.72D+02
O.OOD+00
0.15D+02
0.30D+02
0.35D+02
0.40D+02
0.45D+02
0.50D+02
0.55D+02
0.60D+02
0.66D+02
0.72D+02
O.OOD+00
0.15D+02
0.30D+02
0.35D+02
0.40D+02
0.45D+02
0.50D+02
0.55D+02
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
133
-------
TABLE 6-3. INPUT DATA SET FOR THE THREE-DIMENSIONAL 3DFEMWATER
PROBLEM (continued)
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
0
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
0
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
0
0.80D+03
0.80D+03
0.80D+03
0.85D+03
0.85D+03
0.85D+03
0.85D+03
0.85D+03
0.85D+03
0.85D+03
0.85D+03
0.85D+03
0.85D+03
0.85D+03
0.90D+03
0.90D+03
0.90D+03
0.90D+03
0.90D+03
0.90D+03
0.90D+03
0.90D+03
0.90D+03
0.90D+03
0.90D+03
0.95D+03
0.95D+03
0.95D+03
0.95D+03
0.95D+03
0.95D+03
0.95D+03
0.95D+03
0.95D+03
0.95D+03
0.95D+03
0.10D+04
0.10D+04
0.10D+04
0.10D+04
0.10D+04
0.10D+04
0.10D+04
0.10D+04
0.10D+04
0.10D+04
0.10D+04
0.0
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0.60D+02
0.66D+02
0.72D+02
O.OOD+00
0.15D+02
0.30D+02
0.35D+02
0.40D+02
0.45D+02
0.50D+02
0.55D+02
0.60D+02
0.66D+02
0.72D+02
O.OOD+00
0.15D+02
0.30D+02
0.35D+02
0.40D+02
0.45D+02
0.50D+02
0.55D+02
0.60D+02
0.66D+02
0.72D+02
O.OOD+00
0.15D+02
0.30D+02
0.35D+02
0.40D+02
0.45D+02
0.50D+02
0.55D+02
0.60D+02
0.66D+02
0.72D+02
O.OOD+00
0.15D+02
0.30D+02
0.35D+02
0.40D+02
0.45D+02
0.50D+02
0.55D+02
0.60D+02
0.66D+02
0.72D+02
0.0
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0.0
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.50D+02
0.0
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0.00.0
134
-------
TABLE 6-3. INPUT DATA SET FOR THE THREE-DIMENSIONAL 3DFEMWATER
PROBLEM (continued)
C ******* DATA SET 8: SUBREGIONAL DATA
9
1 8 1 231 0
0000 0
1 230 1 1 1
0000 0
1 230 1 232 1
0000 0
1 230 1 463 1
0000 0
1 230 1 694 1
0000 0
1 230 1 925 1
0000 0
1 230 1 1156 1
0000 0
1 230 1 1387 1
0000 0
1 230 1 1618 1
0000 0
1 230 1 1849 1
0000 0
C ******** DATA SET
1
11
21
31
41
51
61
71
81
91
101
111
121
131
141
151
161
171
181
191
201
211
221
231
241
251
261
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
END OF NNPLR(9)
END OF GNLRd.l)
END OF GNLR(I,2)
END OF GNLR(I,3)
END OF GNLR(I,4)
END OF GNLR(I,5)
END OF GNLR(I,6)
END OF GNLR(I,7)
END OF GNLR(I,8)
END OF GNLR(I,9)
10: ELEMENT INCIDENCES
1
12
23
34
45
56
67
78
89
100
111
122
133
144
155
166
177
188
199
210
232
243
254
265
276
287
298
12
23
34
45
56
67
78
89
100
111
122
133
144
155
166
177
188
199
210
221
243
254
265
276
287
298
309
243
254
265
276
287
298
309
320
331
342
353
364
375
386
397
408
419
430
441
452
474
485
496
507
518
529
540
232
243
254
265
276
287
298
309
320
331
342
353
364
375
386
397
408
419
430
441
463
474
485
496
507
518
529
2
13
24
35
46
57
68
79
90
101
112
123
134
145
156
167
178
189
200
211
233
244
255
266
277
288
299
13
24
35
46
57
68
79
90
101
112
123
134
145
156
167
178
189
200
211
222
244
255
266
277
288
299
310
244
255
266
277
288
299
310
321
332
343
354
365
376
387
398
409
420
431
442
453
475
486
497
508
519
530
541
233
244
255
266
277
288
299
310
321
332
343
354
365
376
387
398
409
420
431
442
464
475
486
497
508
519
530
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
135
-------
TABLE 6-3. INPUT DATA SET FOR THE THREE-DIMENSIONAL 3DFEMWATER
PROBLEM (continued)
271
281
291
301
311
321
331
341
351
361
371
381
391
401
411
421
431
441
451
461
471
481
491
501
511
521
531
541
551
561
571
581
591
601
611
621
631
641
651
661
671
681
691
701
711
721
731
741
751
761
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
309
320
331
342
353
364
375
386
397
408
419
430
441
463
474
485
496
507
518
529
540
551
562
573
584
595
606
617
628
639
650
661
672
694
705
716
727
738
749
760
771
782
793
804
815
826
837
848
859
870
320
331
342
353
364
375
386
397
408
419
430
441
452
474
485
496
507
518
529
540
551
562
573
584
595
606
617
628
639
650
661
672
683
705
716
727
738
749
760
771
782
793
804
815
826
837
848
859
870
881
551
562
573
584
595
606
617
628
639
650
661
672
683
705
716
727
738
749
760
771
782
793
804
815
826
837
848
859
870
881
892
903
914
936
947
958
969
980
991
1002
1013
1024
1035
1046
1057
1068
1079
1090
1101
1112
540
551
562
573
584
595
606
617
628
639
650
661
672
694
705
716
727
738
749
760
771
782
793
804
815
826
837
848
859
870
881
892
903
925
936
947
958
969
980
991
1002
1013
1024
1035
1046
1057
1068
1079
1090
1101
310
321
332
343
354
365
376
387
398
409
420
431
442
464
475
486
497
508
519
530
541
552
563
574
585
596
607
618
629
640
651
662
673
695
706
717
728
739
750
761
772
783
794
805
816
827
838
849
860
871
321
332
343
354
365
376
387
398
409
420
431
442
453
475
486
497
508
519
530
541
552
563
574
585
596
607
618
629
640
651
662
673
684
706
717
728
739
750
761
772
783
794
805
816
827
838
849
860
871
882
552
563
574
585
596
607
618
629
640
651
662
673
684
706
717
728
739
750
761
772
783
794
805
816
827
838
849
860
871
882
893
904
915
937
948
959
970
981
992
1003
1014
1025
1036
1047
1058
1069
1080
1091
1102
1113
541
552
563
574
585
596
607
618
629
640
651
662
673
695
706
717
728
739
750
761
772
783
794
805
816
827
838
849
860
871
882
893
904
926
937
948
959
970
981
992
1003
1014
1025
1036
1047
1058
1069
1080
1091
1102
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
136
-------
TABLE 6-3. INPUT DATA SET FOR THE THREE-DIMENSIONAL 3DFEMWATER
PROBLEM (continued)
771
781
791
801
811
821
831
841
851
861
871
881
891
901
911
921
931
941
951
961
971
981
991
1001
1011
1021
1031
1041
1051
1061
1071
1081
1091
1101
1111
1121
1131
1141
1151
1161
1171
1181
1191
1201
1211
1221
1231
1241
1251
1261
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
881
892
903
925
936
947
958
969
980
991
1002
1013
1024
1035
1046
1057
1068
1079
1090
1101
1112
1123
1134
1156
1167
1178
1189
1200
1211
1222
1233
1244
1255
1266
1277
1288
1299
1310
1321
1332
1343
1354
1365
1387
1398
1409
1420
1431
1442
1453
892
903
914
936
947
958
969
980
991
1002
1013
1024
1035
1046
1057
1068
1079
1090
1101
1112
1123
1134
1145
1167
1178
1189
1200
1211
1222
1233
1244
1255
1266
1277
1288
1299
1310
1321
1332
1343
1354
1365
1376
1398
1409
1420
1431
1442
1453
1464
1123
1134
1145
1167
1178
1189
1200
1211
1222
1233
1244
1255
1266
1277
1288
1299
1310
1321
1332
1343
1354
1365
1376
1398
1409
1420
1431
1442
1453
1464
1475
1486
1497
1508
1519
1530
1541
1552
1563
1574
1585
1596
1607
1629
1640
1651
1662
1673
1684
1695
1112
1123
1134
1156
1167
1178
1189
1200
1211
1222
1233
1244
1255
1266
1277
1288
1299
1310
1321
1332
1343
1354
1365
1387
1398
1409
1420
1431
1442
1453
1464
1475
1486
1497
1508
1519
1530
1541
1552
1563
1574
1585
1596
1618
1629
1640
1651
1662
1673
1684
882
893
904
926
937
948
959
970
981
992
1003
1014
1025
1036
1047
1058
1069
1080
1091
1102
1113
1124
1135
1157
1168
1179
1190
1201
1212
1223
1234
1245
1256
1267
1278
1289
1300
1311
1322
1333
1344
1355
1366
1388
1399
1410
1421
1432
1443
1454
893
904
915
937
948
959
970
981
992
1003
1014
1025
1036
1047
1058
1069
1080
1091
1102
1113
1124
1135
1146
1168
1179
1190
1201
1212
1223
1234
1245
1256
1267
1278
1289
1300
1311
1322
1333
1344
1355
1366
1377
1399
1410
1421
1432
1443
1454
1465
1124
1135
1146
1168
1179
1190
1201
1212
1223
1234
1245
1256
1267
1278
1289
1300
1311
1322
1333
1344
1355
1366
1377
1399
1410
1421
1432
1443
1454
1465
1476
1487
1498
1509
1520
1531
1542
1553
1564
1575
1586
1597
1608
1630
1641
1652
1663
1674
1685
1696
1113
1124
1135
1157
1168
1179
1190
1201
1212
1223
1234
1245
1256
1267
1278
1289
1300
1311
1322
1333
1344
1355
1366
1388
1399
1410
1421
1432
1443
1454
1465
1476
1487
1498
1509
1520
1531
1542
1553
1564
1575
1586
1597
1619
1630
1641
1652
1663
1674
1685
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
137
-------
TABLE 6-3. INPUT DATA SET FOR THE THREE-DIMENSIONAL 3DFEMWATER
PROBLEM (continued)
1271
1281
1291
1301
1311
1321
1331
1341
1351
1361
1371
1381
1391
1401
1411
1421
1431
1441
1451
1461
1471
1481
1491
1501
1511
1521
1531
1541
1551
1561
1571
1581
1591
0
Q *******
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
0
DATA
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
SET 11
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
1464
1475
1486
1497
1508
1519
1530
1541
1552
1563
1574
1585
1596
1618
1629
1640
1651
1662
1673
1684
1695
1706
1717
1728
1739
1750
1761
1772
1783
1794
1805
1816
1827
0
1475
1486
1497
1508
1519
1530
1541
1552
1563
1574
1585
1596
1607
1629
1640
1651
1662
1673
1684
1695
1706
1717
1728
1739
1750
1761
1772
1783
1794
1805
1816
1827
1838
0
1706
1717
1728
1739
1750
1761
1772
1783
1794
1805
1816
1827
1838
1860
1871
1882
1893
1904
1915
1926
1937
1948
1959
1970
1981
1992
2003
2014
2025
2036
2047
2058
2069
0
1695
1706
1717
1728
1739
1750
1761
1772
1783
1794
1805
1816
1827
1849
1860
1871
1882
1893
1904
1915
1926
1937
1948
1959
1970
1981
1992
2003
2014
2025
2036
2047
2058
0
1465
1476
1487
1498
1509
1520
1531
1542
1553
1564
1575
1586
1597
1619
1630
1641
1652
1663
1674
1685
1696
1707
1718
1729
1740
1751
1762
1773
1784
1795
1806
1817
1828
0
1476
1487
1498
1509
1520
1531
1542
1553
1564
1575
1586
1597
1608
1630
1641
1652
1663
1674
1685
1696
1707
1718
1729
1740
1751
1762
1773
1784
1795
1806
1817
1828
1839
0
1707
1718
1729
1740
1751
1762
1773
1784
1795
1806
1817
1828
1839
1861
1872
1883
1894
1905
1916
1927
1938
1949
1960
1971
1982
1993
2004
2015
2026
2037
2048
2059
2070
0
1696
1707
1718
1729
1740
1751
1762
1773
1784
1795
1806
1817
1828
1850
1861
1872
1883
1894
1905
1916
1927
1938
1949
1960
1971
1982
1993
2004
2015
2026
2037
2048
2059
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0 END OF IE
: INITIAL CONDITIONS
0.60D+02
0.45D+02
0.30D+02
0.25D+02
0.20D+02
0.15D+02
0.10D+02
0.50D+01
O.OOD+00
-0.60D+01
-0.12D+02
0.60D+02
0.45D+02
0.30D+02
0.25D+02
0.
0.
0.
OOD+00
OOD+00
OOD+00
O.OOD+00
0.
OOD+00
O.OOD+00
0.
0.
0.
0.
0.
0.
OOD+00
OOD+00
OOD+00
OOD+00
OOD+00
OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
138
-------
TABLE 6-3. INPUT DATA SET FOR THE THREE-DIMENSIONAL 3DFEMWATER
PROBLEM (continued)
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
231
231
231
231
231
231 '
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
0.20D+02
0.15D+02
0.10D+02
0.50D+01
O.OOD+00
-0.60D+01
-0.12D+02
0.60D+02
0.45D+02
0.30D+02
0.25D+02
0.20D+02
0.15D+02
0.10D+02
0.50D+01
O.OOD+00
-0.60D+01
-0.12D+02
0.60D+02
0.45D+02
0.30D+02
0.25D+02
0.20D+02
0.15D+02
0.10D+02
0.50D+01
O.OOD+00
-0.60D+01
-0.12D+02
0.60D+02
0.45D+02
0.30D+02
0.25D+02
0.20D+02
0.15D+02
0.10D+02
0.50D+01
O.OOD+00
-0.60D+01
-0.12D+02
0.60D+02
0.45D+02
0.30D+02
0.25D+02
0.20D+02
0.15D+02
0.10D+02
0.50D+01
O.OOD+00
-0.60D+01
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
139
-------
TABLE 6-3. INPUT DATA SET FOR THE THREE-DIMENSIONAL 3DFEMWATER
PROBLEM (continued)
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
-0.12D+02
0.60D+02
0.45D+02
0.30D+02
0.25D+02
0.20D+02
0.15D+02
0.10D+02
0.50D+01
O.OOD+00
-0.60D+01
-0.12D+02
0.60D+02
0.45D+02
0.30D+02
0.25D+02
0.20D+02
0.15D+02
0.10D+02
0.50D+01
O.OOD+00
-0.60D+01
-0.12D+02
0.60D+02
0.45D+02
0.30D+02
0.25D+02
0.20D+02
0.15D+02
0.10D+02
0.50D+01
O.OOD+00
-0.60D+01
-0.12D+02
0.60D+02
0.45D+02
0.30D+02
0.25D+02
0.20D+02
0.15D+02
0.10D+02
0.50D+01
O.OOD+00
-0.60D+01
-0.12D+02
0.60D+02
0.45D+02
0.30D+02
0.25D+02
0.20D+02
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
140
-------
TABLE 6-3. INPUT DATA SET FOR THE THREE-DIMENSIONAL 3DFEMWATER
PROBLEM (continued)
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
0.15D+02
0.10D+02
0.50D+01
O.OOD+00
-0.60D+01
-0.12D+02
0.60D+02
0.45D+02
0.30D+02
0.25D+02
0.20D+02
0.15D+02
0.10D+02
0.50D+01
O.OOD+00
-0.60D+01
-0.12D+02
0.60D+02
0.45D+02
0.30D+02
0.25D+02
0.20D+02
0.15D+02
0.10D+02
0.50D+01
O.OOD+00
-0.60D+01
-0.12D+02
0.60D+02
0.45D+02
0.30D+02
0.25D+02
0.20D+02
0.15D+02
0.10D+02
0.50D+01
O.OOD+00
-0.60D+01
-0.12D+02
0.60D+02
0.45D+02
0.30D+02
0.25D+02
0.20D+02
0.15D+02
0.10D+02
0.50D+01
O.OOD+00
-0.60D+01
-0.12D+02
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
141
-------
TABLE 6-3. INPUT DATA SET FOR THE THREE-DIMENSIONAL 3DFEMWATER
PROBLEM (continued)
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
231
0.60D+02
0.45D+02
0.30D+02
0.25D+02
0.20D+02
0.15D+02
0.10D+02
0.50D+01
O.OOD+00
-0.60D+01
-0.12D+02
0.60D+02
0.45D+02
0.30D+02
0.25D+02
0.20D+02
0.15D+02
0.10D+02
0.50D+01
O.OOD+00
-0.60D+01
-0.12D+02
0.60D+02
0.45D+02
0.30D+02
0.25D+02
0.20D+02
0.15D+02
0.10D+02
0.50D+01
O.OOD+00
-0.60D+01
-0.12D+02
0.60D+02
0.45D+02
0.30D+02
0.25D+02
0.20D+02
0.15D+02
0.10D+02
0.50D+01
O.OOD+00
-0.60D+01
-0.12D+02
0.60D+02
0.45D+02
0.30D+02
0.25D+02
0.20D+02
0.15D+02
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
O.OOD+00
0.0
0.0
0.0
0.0
0.0
0.0-
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
142
-------
TABLE 6-3. INPUT DATA SET FOR THE THREE-DIMENSIONAL 3DFEMWATER
PROBLEM (continued)
216 8 231 0.10D+02 O.OOD+00 0.0
217 8 231 0.50D+01 O.OOD+00 0.0
218 8 231 O.OOD+00 O.OOD+00 0.0
219 8 231 -0.60D+01 O.OOD+00 0.0
220 8 231 -0.12D+02 O.OOD+00 0.0
221 8 231 0.60D+02 O.OOD+00 0.0
222 8 231 0.45D+02 O.OOD+00 0.0
223 8 231 0.30D+02 O.OOD+00 0.0
224 8 231 0.25D+02 O.OOD+00 0.0
225 8 231 0.20D+02 O.OOD+00 0.0
226 8 231 0.15D+02 O.OOD+00 0.0
227 8 231 0.10D+02 O.OOD+00 0.0
228 8 231 0.50D+01 O.OOD+00 0.0
229 8 231 O.OOD+00 O.OOD+00 0.0
230 8 231 -0.60D+01 O.OOD+00 0.0
231 8 231 -0.12D+02 O.OOD+00 0.0
0 0 0 0.0 0.0 0.0
C ******* DATA SET 12: SOURCE/SINK AND B. C. CONTROL
000
000
C ******* DATA SET
O.ODO 60.0DO
O.ODO 30.0DO
123
239 240 463
699 700 701
1159 1160 1161
1619 1620 1621
1857 221 222
458 459 460
918 919 920
1378 1379 1380
1838 1839 1840
2076 2077 111
1 161 1
163 2 1
000
0
0
0
0
0 0
0 0
0
0
15: DIRICHLET BOUNDARY
1.
1.
4
464
702
OD38
OD38
5
465
925
1162 1163
1622
223
683
921
1381
1841
112
1
2
0
1623
224
684
922
1382
1842
113
0
0
0
60.0DO
30.0DO
6 7
466 467
926 927
1164 1387
1624 1625
225 226
685 686
1145 1146
1383 1384
1843 1844
8
468
928
1388
1626
227
687
1147
1607
1845
165
0
END OF 1C
INTEGERS
2
0
2
0
0
0
0
0 0
CONDITIONS
9
469
929
1389
1849
228
688
1148
1608
1846
232
470
930
1390
1850
229
689
1149
1609
2069
233
471
931
1391
1851
452
690
1150
1610
2070
234
694
932
1392
1852
453
691
1151
1611
2071
235
695
933
1393
1853
454
914
1152
1612
2072
236
696
1156
1394
1854
455
915
1153
1613
2073
237 238
697 698
1157 1158
1395 1618
1855 1856
456 457
916 917
1376 1377
1614 1615
2074 2075
END OF IDTYP
o pvwn OF .TOR
nno
143
-------
6.2 3DLEWASTE
To demonstrate the application of 3DLEWASTE, two simple example problems are
presented. For each problem, a brief description and a correctly-constructed input data
set are given. The corresponding output is not included in this documentation. Rather, it
is distributed along with the code by the EPA Center for Exposure Assessment Modeling
(CEAM) at the Environmental Research Laboratory in Athens, Georgia. See Section 2 for
information about obtaining the code.
6.2.1 One-Dimensional Transport Problem
Transient one-dimensional transport through a horizontal column is simulated in this
example (Figure 6.4). The dimensions of the column are identical to those of the column
in Figure 6.1. It has a length of 200 cm and a 50 cm x 50 cm cross-section. Initially, the
concentration is 0.0 g/cm3 throughout the region of interest. The concentration at x = 0.0
cm is maintained at C = CO = 1.0 g/cm3 (a Dirichlet boundary). A variable boundary
condition is used to specify the natural condition of zero gradient flux at x = 200.0 cm. A
bulk density of 1.2 g/cm3 and a longitudinal dispersivity of 5.0 cm are assumed. No
adsorption or decay is allowed. A specific discharge (Darcy velocity) of 2.0 cm/d is
assumed and a moisture content of 0.4 is used.
The region of interest, that is, the whole column, is discretized with 1 x 1 x 40 = 40
elements with the element size equal to 50 x 50 x 5 cm. This results in 2x2x41 = 164
node points. For this simulation, each of the four vertical lines is considered a subregion.
Thus, a total of four subregions, each with 41 node points, is used for the subregional
block iteration simulation. A constant time-step size of 0.5 is used and a 40 time-step
simulation is run. For this discretization, the mesh Peclet number is Pe = 1 and the
Courant number is Cr = 0.5.
The input data set for this problem, prepared according to the instructions in Sections 4.2
and 5.2, is shown in Table 6-4.
c-i
Figure 6.4. One-dimensional transient transport through a horizontal column.
144
-------
TABLE 6-4. INPUT DATA SET FOR THE ONE-DIMENSIONAL 3DLEWASTE PROBLEM
1 ONE-D FIRST TYPE BOUNDARY VALUE PROBLEM WITH 3DLEWASTE L=CM,T=DAY,M=G 00
C ******* DATA SET 2: BASIC INTEGER PARAMETERS ***** ***************************
164 40 1 0 40 1 8 -1 1 0 1 1 1 100 1
C ******* DATA SET 3: BASIC REAL PARAMETERS ***** ******************************
0.50DO O.ODO l.ODO 20.0DO l.ODOO l.ODO l.OD-3 l.OD-4
C ******* DATA SET 4: PRINTER, STORAGE AND TIME RESETTING CONTROL *************
55000000005000000000500000000050000000005
11000000001000000000100000000010000000001
1.0D38
C ******* DATA SET 5: MATERIAL PROPERTIES *************************************
O.ODO 1.2DO 5.0DO O.ODO O.ODO l.ODO O.ODO O.ODO
C **** *** DATA SET 6: NODE COORDINATES ****************************************
1 40 1 O.ODO 50.0DO O.ODO O.ODO O.ODO 5.0DO
42 40 1 O.ODO O.ODO O.ODO O.ODO O.ODO 5.0DO
83 40 1 50.0DO O.ODO O.ODO O.ODO O.ODO 5.0DO
124 40 1 50.0DO 50.0DO O.ODO O.ODO O.ODO 5.0DO
000 0.0 0.0 0.0 0.0 0.0 0.0 END OF COORDINATES
C ******* DATA SET 7: ELEMENT CONNECTIVITY ***** *******************************
1 39 1 42 83 124 1 43 84 125 2 1
000000000000 END OF IE
Q ******* DATA SET 8" SUBREGIONAL DATA ***** ***********************************
4
1 3 1 41 0
00000 END OF NNPLR(K)
140 1 1 1
00000 END OF GNLR(I,1)
140 142 1
00000 END OF GNLR(I,2)
1 40 1 83 1
00000 END OF GNLR(I,3)
1 40 1 124 1
00000 END OF GNLR(I,4)
£ ******* DATA SET 10: INITIAL CONDITIONS ***** ********************************
1 3 41 l.ODO O.ODO 0.0
2 38 1 O.ODO O.ODO 0.0
43 38 1 .ODO O.ODO 0.0
84 38 1 O.ODO O.ODO 0.0
125 38 1 O.ODO O.ODO 0.0
41 3 41 O.ODO O.ODO 0.0
0 0 0 0.0 0.0 0.0 END OF 1C
C ******* DATA SET 11: SOURCE/SINK AND B. C. CONTROL INTEGERS ***** ************
0000 0000 4120
14120 00000 00000
145
-------
TABLE 6-4. INPUT DATA SET FOR THE ONE-DIMENSIONAL 3DLEWASTE PROBLEM
(concluded)
C ******* DATA SET 13: VARIABLE BOUNDARY CONDITIONS ***** **********************
O.ODO O.ODO 1.0D38 O.ODO
10010
00000 ENDOFIRTYP
1 0 0 82 123 164 41 0 0 0 0
000 0 0 000000 END OF ISV(J,I) J=l,4
1 3 1 41 41
00000 ENDOFNPVB
C ******* DATA SET 14: DIRICHLET BOUNDARY CONDITIONS **************************
O.ODO l.ODO 1.0D38 l.ODO
1 42 83 124
13110
00000 ENDOFIDTYP
C ******* DATA SET 16: HYDROLOGICAL BOUNDARY CONDITIONS ***********************
1 163 1 O.ODO O.ODO 2.0DO O.ODO O.ODO O.ODO
0 00 0.0 0.0 0.0 0.0 0.0 0.0 END OF VELOCITY
1 39 1 0.4DO 0.0
0 0 0 0.0 0.0 ENDOFTH
Q **************** END OF JOB ***** ***************************************oo
146
-------
6.2.2 Two-Dimensional Transport in a Rectangular Region
This is a two-dimensional transport problem in a rectangular region 600.0 cm long, 270.0
cm high, and 1.0 cm thick (Figure 6.5). Initially, the concentration is zero g/cm3 through-
out the region of interest. A concentration of 1.0 g/cm3 is maintained at x = 0.0 cm and
180.0 cm < y <270.0 cm by applying a Dirichlet boundary condition. A concentration of
0.0 g/cm3 is maintained at x = 0.0 cm and 0.0 cm < y < 90.0 cm and 180.0 cm < y <270.0
cm. A natural condition is imposed at x = 600 cm using a variable boundary condition
(Equation 3-39b). A single material with a bulk density of 1.2 g/cm3, a longitudinal
dispersivity of 10.0 cm, and a lateral dispersivity of 1.0 cm modeled. No adsorption or
decay is allowed. A specific discharge (Darcy velocity) of 2.0 cm/d is used and a moisture
content of 0.2 is assumed.
The region is divided into 9x9x1=81 elements, resulting inl0x!0x2 = 200 nodes.
The element size is 60.0 cm x 30.0 cm x 1.0 cm. Each of the two vertical planes is
considered a subregion. Thus, a total of two subregions, each with 100 nodal points, is
used for the subregional block iteration simulation. A time-step size of 4.5 is used and 40
time steps are simulated.
The input data set for this problem, prepared according to the instructions in Sections 4.2
and 5.2, is shown in Table 6-5.
V=2.0
c=o.o
C=1.0
00.0
270
600
Figure 6.5. Two-dimensional transient transport in a rectangular region.
147
-------
TABLE 6-5. INPUT DATA SET FOR THE TWO-DIMENSIONAL 3DLEWASTE PROBLEM
1 TWO-D FIRST TYPE BOUNDARY VALUE PROBLEM WITH 3DLEWASTE L=CM,T=DAY,M=G 00
C ******* DATA SET 2: BASIC INTEGER PARAMETERS ***** ***************************
200 81 1 0 40 1 8 -1 1 0 1 1 1 200 1
C ******* DATA SET 3: BASIC REAL PARAMETERS ***** ******************************
0.45D1 O.ODO 9.0DO 3.60D2 l.ODOO 1.78DO l.OD-3 l.OD-4
C ******* DATA SET 4: PRINTER, STORAGE AND TIME RESETTING CONTROL *************
55000000005000000000500000000050000000005
11000000001000000000100000000010000000001
1.0D38
C **** *** DATA SET 5: MATERIAL PROPERTIES *************************************
O.ODO 1.2DO 10.0DO l.ODO O.ODO l.ODO O.ODO O.ODO
C **** *** DATA SET 6: NODE COORDINATES ****************************************
1
2
3
4
5
6
7
8
9
10
101
102
103
104
105
106
107
108
109
110
0 0
9 10 O.ODO O.ODO O.ODO 6.0D1 O.ODO
9 10 O.ODO 3.0D1 O.ODO 6.0D1 O.ODO
9 10 O.ODO 6.0D1 O.ODO 6.0D1 O.ODO
9 10 O.ODO 9.0D1 O.ODO 6.0D1 O.ODO
9 10 O.ODO 12.0D1 O.ODO 6.0D1 O.ODO
9 10 O.ODO 15.0D1 O.ODO 6.0D1 O.ODO
9 10 O.ODO 18.0D1 O.ODO 6.0D1 O.ODO
9 10 O.ODO 21.0D1 O.ODO 6.0D1 O.ODO
9 10 O.ODO 24.0D1 O.ODO 6.0D1 O.ODO
9 10 O.ODO 27.0D1 O.ODO 6.0D1 O.ODO
9 10 O.ODO O.ODO l.ODO 6.0D1 O.ODO
9 10 O.ODO 3.0D1 l.ODO 6.0D1 O.ODO
9 10 O.ODO 6.0D1 l.ODO 6.0D1 O.ODO
9 10 O.ODO 9.0D1 l.ODO 6.0D1 O.ODO
9 10 O.ODO 12.0D1 l.ODO 6.0D1 O.ODO
9 10 O.ODO 15.0D1 l.ODO 6.0D1 O.ODO
9 10 O.ODO 18.0D1 l.ODO 6.0D1 O.ODO
9 10 O.ODO 21.0D1 l.ODO 6.0D1 O.ODO
9 10 O.ODO 24.0D1 l.ODO 6.0D1 O.ODO
9 10 O.ODO 27.0D1 l.ODO 6.0D1 O.ODO
0 0.0 0.0 0.0 0.0 0.0
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
O.ODO
0.0 END OF COORDINATES
C ******* DATA SET 7: ELEMENT CONNECTIVITY ***** *******************************
1
10
19
28
37
46
55
64
73
0 0
81 1 11 12 2 101 111 112 102 1
8 1 11 21 22 12 111 121 122 112 1
8 1 21 31 32 22 121 131 132 122 1
8 1 31 41 42 32 131 141 142 132 1
8 1 41 51 52 42 141 151 152 142 1
8 1 51 61 62 52 151 161 162 152 1
8 1 61 71 72 62 161 171 172 162 1
8 1 71 81 82 72 171 181 182 172 1
8 1 81 91 92 82 181 191 192 182 1
0000000000
END OF IE
C ******* DATA SET 8: SUBREGIONAL DATA ***** ***********************************
2
1 1 1 100 0
00000 END OF NNPLR(K)
1 99 1 1 1
00000 END OF GNLR(I,1)
1 99 1 101 1
00000 END OF GNLR(I,2)
148
-------
TABLE 6-5. INPUT DATA SET FOR THE TWO-DIMENSIONAL 3DLEWASTE PROBLEM
(concluded)
C ******* DATA SET 10: INITIAL CONDITIONS *************************************
1 199 1 O.ODO O.ODO 0.0
0 0 0 0.0 0.0 0.0 END OF 1C
C ******* DATA SET 11: SOURCE/SINK AND B. C. CONTROL INTEGERS *****************
0 00000008120
9 10 1200000000000
C *** **** DATA SET 13: VARIABLE BOUNDARY CONDITIONS ***************************
0. ODO O.ODO 1.0D38 O.ODO
18110
00000 ENDOFIRTYP
1 8 1 91 92 192 191 1 1 1 1
00000000 0 00 END OF ISV(J,I) J=l,4
1 9 1 91 1
00000 ENDOFNFVB
C ******* DATA SET 14: DIRICHLET BOUNDARY CONDITIONS **************************
O.ODO l.ODO 1.0D38 l.ODO
4 5 6 7 104 105 106 107
17110
00000 ENDOFIDTYP
C ******* DATA SET 16: HYDROLOGICAL BOUNDARY CONDITIONS ***********************
1 199 1 2.0DO O.ODO O.ODO O.ODO O.ODO O.ODO
000 0.0 0.0 0.0 0.0 0.0 0.0 END OF VELOCITY
1 80 1 0.2DO 0.0
0 0 0 0.0 0.0 ENDOFTH
Q **************** gj^D OF JOB ***** ****** ***** ***** ***** ***** *****#*******QQ
149
-------
SECTION 7
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Bresler, E. 1973. Simultaneous Transport of Solutes and Water Under Transient
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Brooks, R.H. and A.T. Corey. 1966. Properties of Porous Media Affecting Fluid Flow.
ASCE J. Irrig. Drain Div. 92 (IR2):61-68.
Carsel, R.F. and R.S. Parrish. 1988. Developing Joint Probability Distributions of Soil-
Water Retention Characteristics. Water Resour. Res. 24(5):755-769.
CRC. 1965. Handbook of Chemistry and Physics. 46th Edition. CRC Press, Boca
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Davis, S.N., and J.M. DeWiest. 1966. Hydrogeology. John Wiley & Sons, Inc.
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EPN600/3-89-048b.
de Marsily, G. 1986. Quantitative Hydrogeology: Groundwater Hydrology for
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Electric Power Research Institute. 1985. A Review of Field Scale Physical Solute
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Federal Register. 1986. Hazardous Waste Management System: Land Disposal
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Freeze, R.A. and J.A. Cherry. 1979. Groundwater. Prentice-Hall, New Jersey. 604
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Gelhar, L.W., C. Welty, and K.R. Rehfeldt. 1992. A Critical Review of Data on Field-
Scale Dispersion in Aquifers. Water Resour. Res. 28(7):1956-1974.
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Colder Associates. 1977. Development of Site Suitability Criteria for a High Level
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Helfferich, F. 1962. Ion Exchange. McGraw-Hill, New York, New York.
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and Parameter Estimator, DBAPE (User's Manual). EPM600/3-89/083.
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Athens, Georgia.
Karickhoff, S. W., D.S. Brown, and T.A. Scott. 1979. Sorption of Hydrophobic
Pollutants on Natural Sediments. Water Research 13:241-248.
Keely, J.F. 1989. Performance Evaluations of Pump-and-Treat Remediations.
EPA/540/4-89/005, U.S. Environmental Protection Agency, Ada, Oklahoma. 19 pp.
Lyman, W.J., W.F. Reehl, and D.H. Rosenblatt. 1982. Handbook of Chemical Property
Estimation Methods: Environmental Behavior of Organic Compounds. McGraw-
Hill, New York. 960 pp.
Mabey, W. R., J.H. Smith, R.T. Podoll, H.L. Johnson, T. Mill, T.W. Chou, J. Gates, I.
Waight Partridge, J. Jaber, and D. Vandenberg. 1982. Aquatic Fate Process
Data for Organic Priority Pollutants. EPA/440/4-81-014. Office of Water
Regulations and Standards, U.S. Environmental Protection Agency, Washington,
DC.
Meyer, C.A., R.B. McClintock, G.J. Silvestri, and R.C. Spencer. 1968. 1967 ASME
Steam Tables. 2nd Edition. The American Society of Mechanical Engineers, New
York, New York.
Mercer, J. W., S.D. Thomas, and B. Ross. 1982. Parameters and Variables Appearing
in Repository Siting Models. NUREG/CR-3066. U.S. Nuclear Regulatory
Commission, Washington, D.C.
Mills, W. B., D.B. Porcella, M.J. Ungs, S.A. Gherini, K.V. Summers, L. Mok, G.L. Rupp,
G.L. Bowie, and D.A. Haith. 1985a. Water Quality Assessment A Screening
Procedure for Toxic and Conventional Pollutants in Surface and Ground Water:
Part I. EPA/600/6-85/002a. U.S. Environmental Protection Agency, Athens, GA.
Mills, W. B., D.B. Porcella, M.J. Ungs, S.A. Gherini, K.V. Summers, L. Mok, G.L. Rupp,
G.L. Bowie, and D.A. Haith. 1985b. Water Quality Assessment: A Screening
Procedure for Toxic and Conventional Pollutants in Surface and Ground Water:
Part II. EPA/600/6-85/002b. U.S. Environmental Protection Agency, Athens, GA.
Morris, D.A., and A.I. Johnson. 1967. Summary of Hydrologic and Physical Properties
of Rock and Soil Materials as Analyzed by the Hydrologic Laboratory of the U.S.
Geological Survey. U.S. Geological Survey Water Supply Paper 1839-D, 1967.
Mualem, Y. 1976. A New Model for Predicting the Hydraulic Conductivity of
Unsaturated Porous Media. Water Resour. Res. 12(3):513-522.
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Mullins, J.A., R.F. Carsel, J.E. Scarborough, and A.M. Ivery. 1992. PRZM-2 User's
Manual, Version 1.0. Office of Research and Development, U.S. Environmental
Protection Agency, Athens, GA.
Finder, G. F., and W. G. Gray. 1977. Finite Element Simulation in Surface and
Subsurface Hydrology. Academic Press, New York.
Sharp-Hansen, S., C. Travers, P. Hummel, and T. Allison. 1990. A Subtitle D Landfill
Application Manual for the Multimedia Exposure Assessment Model
(MULTIMED). Office of Research and Development, Environmental Protection
Agency, Athens, GA.
U.S. Environmental Protection Agency. 1987. Guidelines for Delineating Wellhead
Protection Areas. EPA No. 440/6-87-010. Office of Groundwater Protection, U.S.
Environmental Protection Agency, Washington, DC. NTIS No. PB88-111430-AS.
U.S. Environmental Protection Agency. 1989. Transport and Fate of Contaminants in
the Subsurface: Seminar Publication. EPA/625/4-89/019. Center for
Environmental Research Education, U.S. Environmental Protection Agency,
Cincinnati, OH.
van Genuchten, M.T. 1980. A Closed-form Equation for Predicting the Hydraulic
Conductivity of Unsaturated Soils. Soil Sci. Sot. J. 44:892-898.
152
-------
APPENDIX A
PROGRAM STRUCTURE AND SUBROUTINE DESCRIPTIONS
A.I 3DFEMWATER
3DFEMWATER consists of a main program, FEMWAT3D, and 22 subroutines. Figure
A. 1 shows the structure of the program. The subroutines are listed in Table A-l and
the functions of these subroutines are briefly described below.
A. 1.1 Subroutine ALLFCT
This subroutine is called by subroutine GW3D to compute values for all the source/sink
and boundary nodes and elements. It uses linear interpolation of tabular data to
simulate variations in time for these conditions.
A. 1.2 Subroutine ASEMBL
This subroutine is called by subroutine GW3D. After calling subroutine Q8 to evaluate
the element matrices, it sums over all element matrices to form a global matrix
equation governing the pressure head at all nodes.
A. 1.3 Subroutine BASE
This subroutine is called by subroutines Q8DV and Q8 to evaluate the value of the
base function at a Gaussian point.
A. 1.4 Subroutine BC
This subroutine, which is called by subroutine GW3D, incorporates Dirichlet, variable
composite, specified-flux (Cauchy), and specified-pressure-head gradient (Neumann)
boundary conditions. For a Dirichlet boundary condition, an identity algebraic
equation is generated for each Dirichlet nodal point. Any other equation having this
nodal variable is modified accordingly to simplify the computation. For a specified-flux
surface, the integration of the surface source is obtained by calling subroutine Q4S and
the result is added to the load vector. For a specified-pressure-head gradient surface,
the integrations of both the gradient and gravity fluxes are obtained by calling the
subroutine Q4S. These fluxes are added to the load vector. The subroutine BC also
implements the variable composite boundary condition. First, it checks all infiltration-
evapotranspiration-seepage points, identifying any of them that are Dirichlet
153
-------
Figure A. 1. Program structure of 3DFEMWATER
154
-------
TABLE A-l. SUBROUTINES INCLUDED IN 3DFEMWATER
Subroutine
ALLFCT
ASEMBL
Called By
GW3D
GW3D
BASE
BC
Q8DV, Q8
GW3D
BCPREP
GW3D
BLKITR
DATAIN
GW3D
GW3D
GW3D
FEMWAT3D
PAGEN
DATAIN
PRINTT
GW3D
Description
Interpolates functional values for source/sink
and boundary conditions.
Evaluates the element matrices and then
sums over all element matrices to form a
global matrix equation governing the pres-
sure head at all nodes.
Evaluates the value of the base function at a
Gaussian point.
Incorporates Dirichlet, specified-flux
(Cauchy), specified-pressure-head gradient
(Neumann), and variable composite boundary
conditions.
Prepares the infiltration/seepage boundary
conditions during a rainfall period or the
seepage/evapotranspiration boundary con-
ditions during non-rainfall periods.
Solves the matrix equation with block itera-
tion methods.
Reads and prints all input information.
Controls the entire sequence of operations.
Performs either the steady-state computation
alone, or a transient-state computation using
the steady-state solution as the initial condi-
tion, or a transient computation using user-
supplied initial conditions.
Preprocesses pointer arrays that are needed
to store the global matrix in compressed form
and to construct the subregional block matri-
ces.
Prints the flow variables, which include the
fluxes through variable boundary surfaces,
pressure head, total head, moisture content,
and Darcy velocity components.
155
-------
TABLE A-l. SUBROUTINES INCLUDED IN 3DFEMWATER (continued)
Subroutine
Q4S
Q8
Q8DV
Q8TH
READN
READR
SFLOW
SOLVE
SPROP
STORE
Called By
BCPREP, BC,
ASEMBL
VELT
SFLOW
DATAIN
DATAIN
GW3D
BLKITR
GW3D
GW3D
SURF
DATAIN
Description
Evaluates the boundary surface load vector
SFLOW over a SFLOW boundary segment.
Computes the element matrices and element
load vector.
Computes the integration of N(I)*N(J) and -
N(I)* K42RAD(HT) over an element.
Evaluates the integration of moisture content
and sources/sinks over an element.
Automatically generates integer input if
required.
Automatically generates real number input if
required.
Computes the fluxes through boundaries and
the rate at which water content increases in
the region of interest.
Solves a matrix equation with the direct
band method.
Calculates the values of moisture content,
relative hydraulic conductivity, and water
capacity using van Genuchten analytical
functions.
Stores the solution in binary on logical unit
LUSTO. Information stored includes regional
geometry, subregion data, and hydrological
variables such as pressure head, total head,
moisture content, and the Darcy velocity
components.
Identifies the boundary sides, sequences the
boundary nodes, and computes the direction-
al cosine of the surface sides.
156
-------
TABLE A-l. SUBROUTINES INCLUDED IN 3DFEMWATER (continued)
Subroutine
VELT
Called Bv
GW3D
Description
Evaluates the element matrices and the
derivatives of the total head, and then sums
over all element matrices to form a matrix
equation governing the velocity components
at all nodal points.
157
-------
points. If there are Dirichlet points, they are incorporated using the method described
above. If a given point is not a Dirichlet point, the point is bypassed. Second, it
checks all rainfall-evaporation-seepage points again to see if any of them is a specified-
flux point. If it is, then the computed flux by infiltration or potential evapotranspira-
tion is added to the load vector. If a given point is not a specified-flux point, it is
bypassed. Because the infiltration-evaporation-seepage points are either Dirichlet or
specified-flux points, all points are taken care of in this manner.
A. 1.5 Subroutine BCPREP
This subroutine is called by GW3D to prepare the infiltration-seepage boundary
conditions during a rainfall period or the seepage-evapotranspiration boundary
conditions during non-rainfall periods. It decides the number of nodal points on the
variable boundary to be considered as Dirichlet or specified-flux (Cauchy) points. It
computes the number of points that change boundary conditions from 1) ponding depth
(Dirichlet types) to infiltration (specified-flux types), or 2) infiltration to pending depth,
or 3) minimum pressure (Dirichlet types) to infiltration during rainfall periods. It also
computes the number of points that change boundary conditions from potential
evapotranspiration (specified-flux types) to minimum pressure, or from ponding depth
to potential evapotranspiration, or from minimum pressure to potential evapotrans-
piration during non-rainfall periods. Upon completion, this subroutine returns the
Darcy flux (DCYFLX), infiltration/potential evapotranspiration rate (FLX), the ponding
depth nodal index (NPCON), the flux-type nodal index (NPFI.X), the minimum
pressure nodal index (NPMIN), and the number of nodal points (NCHG) that have
changed boundary conditions.
A.I.6 Subroutine BLKITR
This subroutine is called by subroutine GW3D to solve the matrix equation with block
iteration methods. For each subregion, a block matrix equation is constructed based
on the global matrix equation and two pointer arrays, GNPLR and LNOJCN (see
subroutine PAGEN), and the resulting block matrix equation is solved with the direct
band matrix solver by calling subroutine SOLVE. This is done for all subregions for
each iteration until a convergent solution is obtained.
A. 1.7 Subroutine DATAIN
Subroutine DATAIN is called by subroutine GW3D. It reads all data input described
in Section 4.1 except data set 1. It also calls subroutine SURF to identify the surface
elements and boundary nodes, and subroutines READR and READN, respectively, to
automatically generate real and integer numbers.
A. 1.8 Subroutine GW3D
Subroutine GW3D controls the entire sequence of operations. It performs either a
steady-state computation alone (KSS = 0 and NTI = 0), or a transient- state computa-
tion using the steady-state solution as the initial condition (KSS = 0, NTI > 0), or a
transient computation using user-supplied initial conditions (KSS = 1, NTI > 0).
158
-------
GW3D calls subroutine DATAIN to read and print input data; subroutine PAGEN to
generate pointer arrays; subroutine ALLFCT to obtain source/sink and boundary
values; subroutine SPROP to obtain the relative hydraulic conductivity, water capacity,
and moisture content from the pressure head; subroutine VELT to compute Darcy
velocity, subroutine BCPREP to determine if a change of boundary conditions is
required, subroutine ASEMBL to assemble the element matrices over all elements;
subroutine BC to implement the boundary conditions; subroutine BLKITR to form and
solve the subregional block matrix equations; subroutine SFLOW to calculate flux
through all types of boundaries and water accumulated in the media; subroutine
PRINTT to print out the results; and subroutine STORE to store the flow variables for
input to 3DLEWASTE or for plotting.
A. 1.9 Subroutine PAGEN
This subroutine is called by subroutine DATIUN to preprocess pointer arrays that are
needed to store the global matrix in compressed form and to construct the subregional
block matrices. The pointer arrays automatically generated in this subroutine include
the global node connectivity (stencil), GNOJCN(J,N), regional node connectivity,
LNOJCN(J,I,K), total node number for each subregion, NTNPLR(K), the bandwidth
indicator for each subregion, LMAXDF(K), and a partial fall-up for the mapping array
between global node number and local subregion node number, GNPLR(I,K), with I =
NNPLR(K) + 1 to NTNPLR(K). Here GNOJCN(J,N) is the global node number of the
J-th node connected to the global node N; LNOJCN(J,I,K) is the local node number of
the J-th node connected to the local node I in the K-th subregion; NTNPLR(K) is the
total number of nodes in the K-th subregion, including the interior nodes, the global
boundary nodes, and intra-boundary nodes; LMAXDF(K) is the maximum difference
between any two nodes of any element in the K-th subregion; and GNPLR(I,K) is the
global node number of the I-th local-region node in the K-th subregion. These pointer
arrays are generated based on the element connectivity, IE(M,J), the number of nodes
for each subregion, NNPLR(K), and the mapping between global node and local-region
node, GNLR(I,K), with 1=1, NNPLR(K). Here IE(M,J) is the global node number of J-
th node of element M; NNPLR(K) is the number of nodes in the K-th subregion,
including the interior nodes and the global boundary nodes, but not the intraboundary
nodes.
A. 1.10 Subroutine PRINTT
This subroutine, which is called by GW3D, is used to line-print the flow variables.
These include the fluxes through variable boundary surfaces, the pressure head, total
head, moisture content, and Darcy velocity components.
A. 1.11 Subroutine O4S
This subroutine is called by subroutines BC, BCPREP, and SFLOW to compute the
surface node flux of the type:
159
-------
RQ(I) = jN/qdB
(A-l)
where q is either the specified-flux, specified-pressure-head gradient flux, or gravity
flux; B is the global boundary of the region of interest; N;e is the basis functions for
nodal point i of element e; and RQ(I) is a 3DFEMWATER code parameter.
A.I.12 Subroutine O8
This subroutine is called by the subroutine ASEMBL to compute the element matrix
given by
QA(I,J) =
N,ed R
(A-2a)
QB(I,J) = J(VNie)-K.kr
-------
QRd.J) =jNieNjedR
where QR(IJ) is a 3DFEMWATER program variable. Subroutine Q8DV also evaluates
the element load vector:
QRX(I) = - fNiei-K.kr
-------
A.I.15 Subroutine READN
This subroutine is called by subroutine DATAIN to generate integer numbers for input
data sets 8, 9, 12(c), 12(f), 14(b) through 14(d), 15(c), 16(b), 16(c), 17(b), and 17(c),
which are described in Section 4.1.
A. 1.16 Subroutine READR
This subroutine is called by subroutine DATAIN to generate real numbers input for
data sets 7, 14(e), and 14(f) (see Section 4.1). Automatic generation of regularly
patterned data is built into this subroutine.
A. 1.17 Subroutine SFLOW
This subroutine is called by subroutine GW3D. It is used to compute the fluxes
through various types of boundaries and the rate at which water content increases in
the region of interest. In this subroutine, the function of variable FRATE(7) is to store
the flux through the whole boundary enclosing the region of interest. It is given by:
FRATE(7) = f(VA + Vyny + Vznz)dB (A-5)
where Vx, Vy, and Vz are Darcy velocity components, and n^, riy, and nz are the direc-
tional cosines of the outward unit vector normal to the boundary B. FRATE(l) through
FRATE(5) store the flux through the Dirichlet boundary, BD, specified-flux boundary,
Bc, specified-pressure-head boundary, BN, the seepage/evapotranspiration boundary,
Bg, and infiltration boundary, BR, respectively, and are given by:
FRATE(l) = f(VA + Vyny + Vziiz)dB (A-6a)
FRATE(2) = f(VA + Vyny + Vznz)dB (A-6b)
B.
FRATE(3) = J(Vxnx + V,n, + Vznz)dB (A-6C)
FRATE(6), which is related to the numerical loss, is given by:
162
-------
FRATE(4) = (Vxnx + Vyny + VzigdB (A-6d)
B.
FRATE(5) = (Vxnx + Vyny + Vznz)dB (A-6e)
B,
5
FRATE(6) = FRATE(7) - £ FRATE(I) (A-7)
1=1
FRATE(8) and FRATE(9) are used to store the source/sink and increased rate of water
accumulation within the media, respectively:
FRATE(8) = - fqdR (A-8)
and
FRATE(9) = (F^idR (A-9)
R "v
If there is no numerical error in the computation, the following equation should be
satisfied:
FRATE(9) = -[FRATE(7) + FRATE(8)] (A-IQ)
and FRATE(6) should be equal to zero.
A.I.18 Subroutine SOLVE
This subroutine is called by the subroutine BLKITR to solve a matrix equation of the
type:
[c]{x| = {y}
-------
A. 1.20 Subroutine STORE
This subroutine, which is called by GW3D, is used to store the flow variables in a
binary file. The stored data are intended for use in 3DLEWASTE or for plotting. The
information stored includes region geometry, subregion data, and hydrological vari-
ables such as pressure head, total head, moisture content, and Darcy velocity compo-
nents.
A. 1.21 Subroutine SURF
Subroutine SURF is called by subroutine DATAIN. It identifies the boundary sides,
sequences the boundary nodes, and computes the directional cosine of the surface
sides. The mappings from boundary nodes to global nodes are stored in NPBB(I)
(where NPBB(I) is the global node number of the I-th boundary node). The boundary
node numbers of the four nodes for each boundary side are stored in ISB(I,J) (where
ISB(I,J) is the boundary node number of I-th node of J-th side, I = 1 to 4). There are
six sides for each element. Which of these six sides is the boundary side is determined
automatically in the subroutine SURF and is stored in ISB(5,J). The global element
number, to which the J-th boundary side belongs, is also preprocessed in the subrou-
tine SURF and is stored in ISB(6,J). The directional cosines of the J-th boundary side
are computed and stored in DCOSB(I.J) (where DCOSB(I.J) is the directional cosine of
the J-th surface with I-th coordinate, I = 1 to 3). The information contained in NPBB,
ISB, and DOSB, along with the number of boundary nodes and the number of bound-
ary sides, is returned to subroutine DATAIN for other uses.
A. 1.22 Subroutine VELT
This subroutine is called by subroutine GW3D. It calls subroutine Q8DV to evaluate
the element matrices and the derivatives of the total head. It then sums over all
element matrices to form a matrix equation governing the velocity components at all
nodal points. To save computational time, the matrix is diagonalized by lumping. The
velocity components can thus be solved point by point. The computed velocity field is
then returned to GW3D through the argument. This velocity field is also passed to
subroutine BCPREP to evaluate the Darcy flux across the seepage-infiltration-evapo-
transpiration surfaces.
A.2 3DLEWASTE
LEWASTE consists of a main program, LEWAST3D, 30 subroutines, and a function.
Figure A.2 shows the structure of the program. The subroutines' and function are
listed in Tables A-2 and A-3, respectively, and the purposes of the subroutines are
briefly described below.
A.2.1 Subroutine ADVBC
This subroutine is called by GM3D to implement the boundary conditions. For a
Dirichlet boundary, the Lagrangian concentration is specified. For variable bound-
aries, if the flow is directed out of the region, the fictitious particle associated with the
boundary node must come from the interior nodes. Hence the Lagrangian concentra-
tion for the boundary node has already been computed from subroutine ADVTRN and
164
-------
Figure A. 2. Program structure of 3DLEWASTE
165
-------
TABLE A-2. SUBROUTINES INCLUDED IN 3DLEWASTE
Subroutine
Called Bv
Description
ADVBC
ADVTRN
AFABTA
ALLFCT
ASEMBL
BC
BLKITR
DATMN
FLUX
GM3D
GM3D
GM3D
GM3D
GM3D
GM3D
GM3D
GM3D
GM3D
GM3D
LEWAST3D
Applies specified-flux (Cauchy), variable, and
Dirichlet boundary conditions.
Computes the Lagrangian concentrations at
all nodes and finds in which element the
fictitious particle is located.
Calculates the values of the upstream
weighting factors along the 12 sides of all
elements.
Interpolates functional values for source/sink
and boundary conditions.
Evaluates the element matrices and then
sums over all element matrices to form a
global matrix equation governing the concen-
tration distribution at all nodes.
Incorporates Dirichlet, variable composite,
specified-flux (Cauchy), and specified-dispers-
ive-flux (Neumann) boundary conditions.
Solves the matrix equations with block itera-
tion methods.
Reads and prints system parameters, geome-
try, boundary and initial conditions, and
properties of the solute and media.
Evaluates the element matrices and the
derivatives of concentrations and then sums
over all element matrices to form a matrix
equation governing the flux components at
all nodal points.
Controls the entire sequence of operations.
Performs either the steady-state computation
alone, or a transient- state computation us-
ing the steady-state solution as the initial
condition, or a transient computation using
user-supplied initial conditions.
166
-------
TABLE A-2. SUBROUTINES INCLUDED IN 3DLEWASTE (continued)
Subroutine
LELGEN
MPLOC
NDTAU
PAGEN
PRINTT
Q4ADB
Q4BB
Q4CNB
Q8
Q8DV
Q8R
READN
Called By
GM3D
NDTAU,
ADVTRN
GM3D
DATAIN
GM3D
ADVBC
SFLOW
BC
ASEMBL
FLUX
SFLOW
DATAIN
Description
Finds the elements connecting to each node.
Locates the fictitious particle associated
with a particular node. Computes the prod-
uct of the outward unit vector with the vector
from a node on the surface to the fictitious
particle.
Determines the number of subtime steps and
the subtime step size for Lagrangian integr-
ation.
Preprocesses pointer arrays that are needed
to store the global matrix in compressed form
and to construct the subregional block matri-
ces.
Prints material flow, concentration, and
material flux output as specified by the pa-
rameter KPR.
Implements Dirichlet, specified-flux
(Cauchy), and variable boundary conditions
in the Lagrangian step computation.
Computes normal flow rates (M/T) by inte-
grating the normal fluxes (M/L2/T) over a
boundary surface.
Computes the boundary-surface matrix and
the boundary-surface load vector over a boun-
dary surface.
Computes element matrices and element load
vectors.
Computes the integration of N(I)*N(J) and -
N(I)*D>GRAD(C) over an element.
Computes the material integration and ele-
ment source integration over an element.
Automatically generates integer input if
required.
167
-------
TABLE A-2. SUBROUTINES INCLUDED IN 3DLEWASTE (concluded)
Subroutine
Called Bv
Description
READR
SFLOW
SHAPE
SOLVE
STORE
SURF
THNODE
XSI3D
DATAIN
GM3D
Q8DV, Q8
BLKITR
GM3D
DATAIN
GM3D
ADVTRN
Automatically generates real number input if
required.
Computes the flux rates through various
types of boundaries and the rate at which
material increases in the region of interest.
Computes the base and weighting functions,
their derivatives with respect to X, Y, Z, and
the Jacobian at a Gaussian point.
Solves a matrix equation with a band matrix
solver.
Stores pertinent quantities on a auxiliary
device for future uses (e.g., for plotting).
Identifies the boundary sides and sequences
of the boundary nodes, and computes the
directional cosine of the surface sides.
Computes moisture content at a node.
Computes the local coordinate of an element
given the global coordinate within that ele-
ment.
168
-------
TABLE A-3. FUNCTIONS INCLUDED IN 3DLEWASTE
Function Called By Description
FCOS MPLOC Computes the inner product of an outward
normal of the surface with a vector connect-
ing a point on the surface and the fictitious
particle to determine if the fictitious particle
lies inside the surface.
169
-------
the implementation for such a boundary segment is bypassed. For variable bound-
aries, if the flow is directed into the region, the concentration of incoming fluid is speci-
fied. The Lagrangian concentration is then calculated as:
V= /N.VnCindB/jN.VndB (A-12)
B;
where
C;* = Lagrangian concentration at the boundary node i
V n = normal vertically integrated Darcy velocity
C;,, = concentration of incoming fluid
B = global boundary of the region of interest
NI = basis function for nodal point i of element e
Specified-flux (Cauchy) boundary conditions are normally applied to the boundary
where flow is directed into the region, where the material flux of incoming fluid is
specified. The Lagrangian concentration is thus calculate as:
C;=jN,qcdB/jNiVndB (A-13)
where C," is the Lagrangian concentration at the boundary node i and qc is the Cauchy
flux of the incoming fluid.
A.2.2 Subroution ADVTRN
This subroutine is called by GM3D to compute the Lagrangian concentrations at all
nodes. It calls subroutine MPLOC to find which element a fictitious particle is located
in. It also calls subroutine XS13D to compute the local coordinate, given the global
coordinate, of the fictitious particle. If the fictitious particle associated with a particu-
lar node is located in the interior of the region, the Lagrangian concentration is
is obtained by finite element interpolation of the concentration at the previous time
step. If the fictitious particle associated with a particular node is outside the region of
interest, the Lagrangian concentration is set equal to the previous time-step concentra-
tion of the boundary node that is closest to the fictitious particle.
A.2.3 Subroutine AFABTA
This subroutine, which is called by subroutine GM3D, calculates the values of up-
stream weighting factors along 12 sides of all elements.
A.2.4 Subroutine ALLFCT
This subroutine is called by subroutine GM3D to compute values for all the source/sink
and boundary nodes and elements. It uses linear interpolation of tabular data to
simulate variations in time for these conditions.
170
-------
A.2.5 Subroutine ASEMBL
This subroutine is called by subroutine GM3D. After calling subroutine Q8 to evaluate
the element matrices, it sums over all element matrices to form a global matrix
equation governing the concentration distribution at all nodes.
A.2.6 Subroutine BC
This subroutine, which is called by subroutine GM3D, incorporates Dirichlet, variable
composite, specified-flux, and specified-dispersive flux boundary conditions. For a
Dirichlet boundary condition, an identity algebraic equation is generated for each
Dirichlet nodal point. Any other equation having this nodal variable is modified
accordingly to simplify the computation. For a variable composite surface, the
integration of the normal velocity times the incoming concentration is added to the
load vector and the integration of normal velocity is added to the matrix. For the
specified-flux boundaries, the integration of flux is added to the load vector and the
integration of normal velocity is added to the matrix. For a specified-dispersive-flux
boundary, the integration of gradient flux is added to the load vector.
A.2.7 Subroutine BLKITR
This subroutine is called by subroutine GM3D to solve the matrix equation with block
iteration methods. For each subregion, a block matrix equation is constructed based
on the global matrix equation and two pointer arrays, GNPLR and LNOJCN (see
subroutine PAGEN). The resulting block matrix equation is solved with the direct
band matrix solver by calling subroutine SOLVE. This is done for all subregions for
each iteration until a convergent solution is obtained.
A.2.8 Subroutine DATAIN
Subroutine DATAIN is called by subroutine GM3D. It reads and prints all data input
described in the Section 4.2 except data set 1. It also calls subroutine SURF to
identify the boundary segments and boundary nodes and subroutines READR and
READN, respectively, to automatically generate real and integer numbers.
A.2.9 Subroutine FLUX
This subroutine is called by subroutine GM3D. It calls subroutine Q8DV to evaluate
the element matrices and the derivatives of concentrations. It then sums over all
element matrices to form a matrix equation governing the flux components at all nodal
points. To save computational time, the matrix is diagonalized by lumping. The flux
components due to dispersion can thus be solved point by point. The flux due to
velocity is then added to the computed flux due to dispersion. The computed total flux
field is then returned to GM3D through the argument.
A.2.1O Subroutine GM3D
The subroutine GM3D controls the entire sequence of operations. It performs either a
steady-state computation alone (KSS = 0 and NTI = 0), or a transient- state computa-
171
-------
tion using the steady-state solution as the initial condition (KSS = 0, NTI > 0), or a
transient computation using user-supplied initial conditions (KSS = 1, NTI > 0).
GM3D calls subroutine DATAIN to read and print input data; subroutine LELGEN to
generate the pointer array element stencil that describes all elements connected to any
node; subroutine ALLFCT to obtain sources/sinks and boundary values; subroutine
AFABTA to obtain the upstream weighting factor based on velocity and dispersivity
(the upstream weighting factor is needed for solving the steady-state option of
3DLEWASTE); subroutine FLUX to compute material flux; subroutine ASEMBL to
assemble the element matrices over all elements; subroutine BC to implement the
boundary conditions; subroutine BLKITR to solve the resulting matrix equations with
block iteration methods; subroutine SFLOW to calculate flux through all types of
boundaries and the water accumulated in the media; subroutine PRINTT to print out
the results; subroutine STORE to store the results for plotting; subroutine THNODE to
compute the value of moisture content plus bulk density times distribution coefficient
in the case of a linear isotherm, or the moisture content in the case of a nonlinear
isotherm at all nodes; subroutine NDTAU to compute the number of subtime steps nd
the subtime step sizes used for integration in the Lagrangian step; ADVTRN to
compute the Lagrangian concentrations at all nodes; and subroutine ADVBC to
implement boundary conditions in the Lagrangian step.
A.2.11 Subroutine LELGEN
This subroutine is called by subroutine GM3D to preprocess the pointer array (the
global elements stencil), LRL(K,N), where LRL(K,N) is the global element number of
the K-th element connected to the global node N. This pointer array is generated
based on the element connectivity IE(M,J). Here IE(M,J) is the global node number of
the J-th node of element M. This pointer array is needed to facilitate the location of
fictitious particles.
A.2. 12 Subroutine MPLOC
This subroutine is called by NDTAU and ADVTRN to locate the fictitious particle
associated with a particular node. It uses the function FCOS to compute the product
of the outward unit vector with the vector from a node on the surface to the fictitious
particle.
A.2.13 Subroutine NDTAU
This subroutine is called by GM3D to compute the subtime-step size and the number of
subtime steps such that no fictitious particle travels over an element within a subtime
step. The subtime-step size and the number of subtime steps are used in subroutine
ADVTRN.
A.2.14 Subroutine PAGEN
This subroutine is called by subroutine DATAIN to preprocess pointer arrays that are
needed to store the global matrix in compressed form and to construct the subregional
block matrices. The pointer arrays automatically generated in this subroutine include
the global node connectivity (stencil), GNOJCN(J.N), regional node connectivity,
172
-------
LNOJCN(J,I,K), total node number for each subregion, NTNPLR(K), the bandwidth
indicator for each subregion, LM.AXDF(K), and a partial fill-up for the mapping array
between global node number and local subregion node number, GNPLR(I,K), with I =
NNPLR(K) + 1 to NTNPLR(K). Here GNOJCN(J,N) is the global node number of the
J-th node connected to the global node N; LNOJN(J,I,K) is the local node number of
the J-th node connected to the local node I in K-th subregion; NTNPLR(K) is the total
number of nodes in the the K-th subregion, including the interior nodes, the global
boundary nodes, and intra-boundary nodes; LMAXDF(K) is the maximum difference
between any two nodes of any element in the K-th subregion; and GNPLR(I,K) is the
global node number of the I-th local-region node in the K-th subregion. These pointer
arrays are generated based on the element connectivity, IE(M,J), the number of nodes
for each subregion, NNPLR(K), and the mapping between global node and local-region
node, GNLR(I,K), with 1=1, NNPLR(K). Here IE(M,J) is the global node number of
the J-th node of element M; NNPLR(K) is the number of nodes in the K-th subregion,
including the interior nodes and the global boundary nodes, but not the intraboundary
nodes.
A.2. 15 Subroutine PRINTT
This subroutine, which is called by GM3D, is used to line-print the simulation results.
These include the fluxes through variable boundary surfaces, the concentration, and
vertically integrated material flux components.
A.2.16 Subroutine O4ADB
This subroutine is called by subroutine ADVBC and implements Dirichlet, specified-
flux, and variable boudary conditions in a Lagrangian step computation.
A.2.17 Subroutine O4BB
This subroutine is called by subroutine SFLOW to perform surface integration of the
following type:
RRQ(I)=jN;eFdB (A-14)
where F is the normal flux and RRQ(I) is a 3DLEWASTE program variable.
A.2.18 Subroutine O4CNB
This subroutine is called by the subroutine BC to compute the surface node flux of the
type:
173
-------
RQ(I)=jNieqdB (A-15a)
where q is either the specified- (or Cauchy) flux, specified-dispersive- (or Neumann)
flux, or n'VCv; and RQ(I) is a 3DLEWASTE program variable. It also computes the
boundary element matrices:
BQ(I,J)=jNieVNjedR (A-15b)
where A^e is the basis function for nodal point j of element e, R is the region of
interest, V is the Darcy velocity, and BQ(I,J) is a 3DLEWASTE program variable.
A.2. 19 Subroutine 08
This subroutine is called by the subroutine ASEMBL to compute the element matrix
given by
QAdlJ)=jN1e9NjedR (A-16a)
QAA(I,J)= N/p^N/dR (A-16b)
R dC
QB(I,J)=J(VN,6)-eD
-------
where
Cw = dissolved concentration at the previous iteration
D = dispersion coefficient tensor
0 = moisture content
S = species concentration in the adsorbed phase
Q = source rate of water
pb = bulk density of the porous medium
X. = material decay constant
V = del operator indicating gradient
V- = del operator indicating divergence
and where QA(IJ), QAA(IJ), QB(IJ), QV(IJ), and QC(IJ) are 3DLEWASTE program
variables. Note that dS/dC should be evaluated at Cw. Subroutines Q8 also calculates
the element load vector given by:
L-[-Apb(Sw-^CJ +QCin]dR (A-16f)
i L r o^- w 1 s~* W7 ^ inj
where Cw and SW are the dissolved and adsorbed concentrations at the previous
iteration, respectively, and QR(I) is a program variable.
A.2.20 Subroutine Q8DV
Subroutine Q8DV is called by subroutine FLUX to compute the element matrices given
by:
QB(I,J)=fNieNjedR
(A-17a)
Subroutine Q8DV also evaluates the element load vector:
QRX(I)= - jNiei-9D
-------
where
Cj = concentration at nodal point j
i = unit vector along the x-direction
j = unit vector along the y-coordinate
k = unit vector along the z-coordinate
and where QRX(I), QRY(I), AND QRZ(I) are program variables.
A.2.21 Subroutine Q8R
This subroutine, which is called by subroutine SFLOW, is used to compute contribu-
tions to FRATE(8), FRATE(9), FRATE(l), and FRATE(14), discussed in Section A.2.24,
by performing material integration and element source integration over an element.
QRM=J(
ecdR (A-18a)
QDM-JsdR (A-18b)
SOSM=J[QCin(l+sign( Q))+QC(1 -sign(Q))]/2dR
where QRM, QDM, and SOSM are 3DLEWASTE program variables.
A.2.22 Subroutine READN
This subroutine is called by subroutine DATAIN to generate integer numbers for the
input data sets if required.
A.2.23 Subroutine READR
This subroutine is called by subroutine DATAIN to automatically generate real
numbers for the input data sets if required. Automatic generation-of regularly
patterned data is built into this subroutine.
A.2.24 Subroutine SFLOW
This subroutine is called by subroutine GM3D. It is used to compute flux rates
through various types of boundaries and the rate at which material increases in the
region of interest. In this subroutine, the variable FRATE(7) stores the flux through
the whole boundary. It is given by
176
-------
FRATE(7)=(FA +Fyny)dB (A-19)
where B is the global boundary of the region of interest; Fx, and Fy are the vertically
integrated flux components and n,, and iiy are the directional cosines of the outward
unit vector normal to the boundary B. FRATE( 1) stores the flux rates through a
Dirichlet boundary Bd. FRATE(2) and FRATE(3) store the flux rate through specified-
flux (Cauchy) and specified-dispersive-flux (Neumann) boundaries, respectively.
FRATE(1)= (F^+FHdB (A-20a)
FRATE(2)= lF^n, +Fyny)dB (A-20b)
yy
B,
FRATE(3)=J(Fxii, +Fyny)dB (A-20c)
FRATE(4) and FRATE(5) store incoming flux and outgoing flux rates, respectively,
through the variable boundaries B, and Bv+, as given by:
FRATE(4)= J(F)tn]t+Fyny)dB
B,-
FRATE(5)= J(FInx+Fyny)dB (A-20e)
where Bv- and Bv. are that part of variable boundary where the fluxes are directed
into the region and out from the region, respectively. The integration of Equations A-
20a through A-20e is carried out by the subroutine Q4BB.
FRATE(6), which is related to the numerical loss, is given by:
177
-------
5
FRATE(6)=FRATE( 7)-T* FRATE(I) (A-21)
FRATE(8) and FRATE(9) store the accumulate rate in the dissolved and adsorbed
phases, respectively, as given by:
FRATE(8)=
R
FRATE(9)=JpbSdR
FRATE(IO) stores the rate loss due to decay and FRATE(ll) through FRATE(13) are
set to zero as given by:
FRATE( 10)=J?t(eC +pbS)dR
R
FRATE(11)=FRATE( 12)= FRATE(13)=0
FRATE(14) is used to store the source/sink rate as:
FRATE(14)=J[QCin(l +sign(Q))+QC(l -sign(Q) )]/2dR
If there is no numerical error in the computation, the following equation should be
satisfied:
14
£ FRATE(I)=0 (A
1=7
and FRATE(6) should be equal to zero.
A.2.25 Subroutine SHAPE
This subroutine is called by subroutines Q8DV and Q8 to evaluate the value of the
base and weighting functions and their derivatives at a Gaussian point.
178
-------
A.2.26 Subroutine SOLVE
This subroutine is called by the subroutine BLKITR to solve a matrix equation of the
type:
[C]{x}={y| (A-28)
where [C] is the coefficient matrix and {x) and {y) are two vectors, {x) is the unknown
to be solved, and {y) is the known load vector. The computer returns the solution {y)
and stores it in {y). The computation is a standard banded Gaussian direct elimination
procedure.
A.2.27 Subroutine STORE
This subroutine, which is called by subroutine GM3D, stores the simulation results in
a binery file for use in plotting. The information stored includes regional geometry,
concentrations, and vertically integrated material flux components at all nodes for any
desired time step.
A.2.28 Subroutine SURF
Subroutine SURF is called by subroutine DATAIN. It identifies the boundary sides,
sequences the boundary nodes, and computes the directional cosine of the surface
sides. The mappings from boundary nodes to global nodes are stored in NPBB(I)
(where NPBB(I) is the global node number of the I-th boundary node). The boundary
node numbers of the four nodes for each boundary side are stored in ISB(I,J) (where
ISB(I,J) is the boundary node number of the I-th node of the J-th side, I = 1 to 4).
There are six sides for each element. Which of these six sides is the boundary side is
determined automatically in the subroutine SURF and is stored in ISB(5,J). The
global element number, to which the J-th boundary side belongs, is also preprocessed
in the subroutine SURF and is stored in ISB(6,J). The directional cosines of the J-th
boundary side are computed and stored in DCOSB(I.J) (where DCOSB(I.J) is the
directional cosine of the J-th surface with I-th coordinate, I = 1 to 3). The information
contained in NPBB, ISB, and DOSB, along with the number of boundary nodes and the
number of boundary sides is returned to subroutine DATAIN for other uses.
A.2.29 Subroutine THNODE
This subroutine is called by GM3D to compute (9 +pbdS/dC) for the linear isotherm
model or 9 for the Freundlich and Langmuir nonlinear isotherm models.
A.2.30 Subroutine XSI3D
This subroutine is called by ADVTRN to compute the local coordinate of an element
given the global coordinate within that element. With the local coordinate, the
Lagrangian concentration can then easily be interpolated from those on the nodes of
the element.
179
-------
APPENDIX B
INPUT AND OUTPUT DEVICES
180
-------
TABLE B-l. LOGICAL UNITS USED IN 3DFEMWATER
Logical Unit
Number
Purpose
LUSTO
LUBAR
LUPAR
LUINP
LUOUT
11
13
14
15
16
Logical unit for storing binary output for
use in 3DLEWASTE or for plotting purposes.
Logical unit for storing binary boundary
arrays, if they are generated in the present
job, for use in subsequent executions of
the same scenario.
Logical unit for storing binary pointer
arrays, if they are generated in the present
job, for use in subsequent executions of
the same scenario.
Logical unit for reading input data.
Logical unit for writing output data.
TABLE B-2. LOGICAL UNITS USED IN 3DLEWASTE
Logical Unit
Number
Purpose
LUFLW
LUSTO
LUBAR
LUPAR
LUINP
LUOUT
11
12
13
14
15
16
Logical unit for reading flow data from the
3DFEMWATER simulation.
Logical unit for storing binary output for
use in 3DLEWASTE or for plotting purposes.
Logical unit for storing binary boundary
arrays, if they are generated in the present
job, for use in subsequent executions of
the same scenario.
Logical unit for storing binary pointer
arrays, if they are generated in the present
job, for use in subsequent executions of
the same scenario.
Logical unit for reading input data.
Logical unit for writing output data.
181
-------
APPENDIX C
DEFAULT VALUES FOR THE MAXIMUM CONTROL PARAMETERS
182
-------
TABLE C-l. MAXIMUM CONTROL PARAMETERS USED IN 3DFEMWATER
Parameter
Definition
Default Value
Location
Maximum Control-Integers for the Spatial Domain
MAXNPK Maximum Number of Nodes
MAXELK Maximum Number of Elements
MXBESK Maximum Number of Boundary-Element Surfaces
MXBNPK Maximum Number of Boundary Nodal Points
MXJBDK Maximum Number of Nonzero Elements in
Any Row
MXKBDK Maximum Number of Elements Connecting
to Any Node
25578
22080
7138
7140
27
PMXSD.INC
PMXSD.INC
PMXSD.INC
PMXSD.INC
PMXSD.INC
PMXSD.INC
Maximum Control-Integers for the Time Domain
MXNTIK Maximum Number of Time Steps
MXDTCK Maximum Number of DELT Changes
100
10
PMXTD.INC
PMXTD.INC
Maximum Control-Integers for Subregions
LTMXNK Maximum Number of Total Nodal Points in any 3654
Subregion, Including Interior Nodes, Global
Boundary Nodes, and Intraboundary Nodes
LMXNPK Maximum Number of Nodal Points in any 1218
Subregion Including Interior Nodes and Global
Boundary Nodes
LMXBWK Maximum Number of the Bandwidth in any 59
Subregion
MXRGNK Maximum Number of Subregions 21
PMXSR.INC
PMXSR.INC
PMXSR.INC
PMXSRJNC
Maximum Control-Integers for Source/Sinks
MXSELK Maximum Number of Source Elements
MXSPRK Maximum Number of Source Profiles
MXSDPK Maximum Number of Data Points on Each Element
Source/Sink Profile
MXWNPK Maximum Number of Point (Well) Nodal Points
MXWPRK Maximum Number of Point (Well) Source/Sink
Profiles
MXWDPK Maximum Number of Data Points on Each Point
(Well) Source/Sink Profile
1
1
1
2
2
PMXSS.INC
PMXSS.INC
PMXSS.INC
PMXSS.INC
PMXSS.INC
PMXSS.INC
(continued)
183
-------
TABLE C-l. MAXIMUM CONTROL PARAMETERS USED IN 3DFEMWATER
(concluded)
Parameter Definition Default Value Location
Maximum Control-Integers for Specified-Flux (Cauchy) Boundary Conditions
MXCNPK Maximum Number of Specified-Flux Nodal Points 147 PMXCB.INC
MXCESK Maximum Number of Specified-Flux Element 120 PMXCB.INC
Surfaces
MXCPRK Maximum Number of Specified-Flux Profiles 1 PMXCB.INC
MXCDPK Maximum Number of Data Points on Each 2 PMXCB.INC
Specified-Flux Profile
Maximum Control-Integers for Specified-Pressure-Head Gradient Boundary Conditions
MXNNPK Maximum Number of Specified-Pressure-Head 1 PMXNB.INC
Gradient Nodal Points
MXNESK Maximum Number of Specified-Pressure-Head 1 PMXNB.INC
Gradient Element Surfaces
MXNPRK Maximum Number of Specified-Pressure-Head 1 PMXNB.INC
Gradient Flux Profiles
MXNDPK Maximum Number of Data Points on Each 2 PMXNB.INC
Specified-Pressure-Head Gradient Flux Refile
Maximum Control-Integers for Variable (Rainfall/Evaporation-Seepage) Boundary
Conditions
MXVNPK Maximum Number of Variable Nodal Points 2079 PMXRSB.INC
MXVESK Maximum Number of Variable Element Surfaces 1960 PMXRSB.INC
MXVPRK Maximum Number of Rainfall Refiles 2 PMXRSB.INC
MXVDPK Maximum Number of Data Point on Each Rainfall 2 PMXRSB.INC
Profile
Maximum Control-Integers for Dirichlet Boundary Conditions
MXDNPK Maximum Number of Dirichlet Nodal Points 210 PMXDB.INC
MXDPRK Maximum Number of Dirichlet Total Head Profiles 2 PMXDB.INC
MXDDPK Maximum Number of Data Points on Each Dirichlet 2 PMXDB.INC
Profile
Maximum Control-Integers for Material and Soil Properties
MXMATK Maximum Number of Material Types 6 PMXMS.INC
MXSPMK Maximum Number of Soil Parameters Per Material 5 PMXMS.INC
to Describe Soil Characteristic Curves
MXMPMK Maximum Number of Material Properties 6 PMXMS.INC
per Material
184
-------
TABLE C-2. MAXIMUM CONTROL PARAMETERS USED IN 3DLEWWASTE
Parameter Definition Default Value
Maximum Control-Integers for the Spatial Domain
MAXNPK Maximum Number of Nodes
MAXELK Maximum Number of Elements
MXBESK Maximum Number of Boundary-Element Surfaces
MXBNPK Maximum Number of Boundary Nodal Points
MXJBDK Maximum Number of Nonzero Elements in
Any Row
MXKBDK Maximum Number of Elements Connecting
to Any Node
Maximum Control-Integers for the Time Domain
MXNTIK Maximum Number of Time Steps
MXDTCK Maximum Number of DELT Changes
Maximum Control-Integers for Subregions
LTMXNK Maximum Number of Total Nodal Points in any
Subregion, Including Interior Nodes, Global
Boundary Nodes, and Intraboundary Nodes
LMXNPK Maximum Number of Nodal Points in any
Subregion, Including Interior Nodes and
Global Boundary Nodes
LMXBWK Maximum Number of the Bandwidth in any
Subregion
MXRGNK Maximum Number of Subregions
Maximum Control-Integers for Source/Sinks
MXSELK Maximum Number of Source Elements
MXSPRK Maximum Number of Source Profiles
MXSDPK Maximum Number of Data Points on Each Element
Source/Sink Profile
MXWNPK Maximum Number of Point (Well) Nodal Points
MXWPRK Maximum Number of Point (Well) Source/Sink
25578
22080
7138
7140
27
8
500
20
3654
1218
59
21
180
1
8
1
1
Location
PMXSD.INC
PMXSD.INC
PMXSD.INC
PMXSD.INC
PMXSD.INC
PMXSD.INC
PMXTD.INC
PMXTD.INC
PMXSR.INC
PMXSR.INC
PMXSR.INC
PMXSR.INC
PMXSS.INC
PMXSS.INC
PMXSS.INC
PMXSS.INC
PMXSS.INC
Profiles
MXWDPK Maximum Number of Data Points on Each Point
(Well) Source/Sink Profile
PMXSS.INC
(continued)
185
-------
TABLE C-2. MAXIMUM CONTROL PARAMETERS USED IN3DLEWWASTE
(concluded)
Parameter Definition Default Value Location
Maximum Control-Integers for Specified-Flux (Cauchy) Boundary Conditions
MXCNPK Maximum Number of Specified-Flux Nodal Points 8 PMXCB.INC
MXCESK Maximum Number of Specified-Flux Element 2 PMXCB.INC
Surfaces
MXCPRK Maximum Number of Specified-Flux Profiles 2 PMXCB.INC
MXCDPK Maximum Number of Data Points on Each 4 PMXCB.INC
Specified-Flux Profile
Maximum Control-Integers for Specified-Dispersive-Flux Boundary Conditions
MXNNPK Maximum Number of Specified-Dispersive-Flux 8 PMXNB.INC
Nodal Points
MXNESK Maximum Number of Specified-Dispersive-Flux 2 PMXNB.INC
Element Surfaces
MXNPRK Maximum Number of Specified-Dispersive-Flux 2 PMXNB.INC
Profiles
MXNDPK Maximum Number of Data Points on Each Specified- 4 PMXNB.INC
Dispersive-Flux Profile
Maximum Control-Integers for Variable (Run-In/Flow-Out) Boundary Conditions
MXVNPK Maximum Number of Variable Nodal Points 38 PMXRSB.INC
MXVESK Maximum Number of Variable Element Surfaces 18 PMXRSB.INC
MXVPRK Maximum Number of Rainfall Profiles 2 PMXRSB.INC
MXVDPK Maximum Number of Data Point on Each Rainfall 4 PMXRSB.INC
Profile
Maximum Control-Integers for Dirichlet Boundary Conditions
MXDNPK Maximum Number of Dirichlet Nodal Points 81 PMXDB.INC
MXDPRK Maximum Number of Dirichlet Total Head Profiles 81 PMXDB.INC
MXDDPK Maximum Number of Data Points on Each Dirichlet 2 PMXDB.INC
Profile
Maximum Control-Integers for Material
MXMATK Maximum Number of Material Types 6 PMXMS.INC
MXMPMK Maximum Number of Material Properties per 8 PMXMS.INC
Material
186
-------
APPENDIX D
PROGRAM VARIABLE DESCRIPTIONS
Information about the program variables is given in two tables in this appendix.
3DFEMWATER program variables are listed in Table D-l and 3DLEWASTE program
variables are shown in Table D-2. In the tables, the definition, type, and units of each
variable are provided. In addition, the tables indicate 1) the subroutines associated
with each variable and 2) whether a variable is an input (I), output (0), or modified
(M) variable in the subroutines. Also, if a variable is included in a COMMON block,
the COMMON block name is given.
COMMON blocks are used in 3DFEMWATER/3DLEWASTE to minimize the use of
subroutine arguments. Each COMMON block, which contains related variables, is
stored as a file separate from the 3DFEMWATER/3DLEWASTE code and is accessed
by the use of INCLUDE statements at the beginning of the main program and each
subroutine. Only those COMMON blocks needed for the execution of a subroutine are
included in the subroutine.
187
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION
Variable
AGRAV
AKPROP
(U)
AKR(LM)
AKXG(8)
AKXYG(8)
AKXZG(8)
AKYG(8)
AKYZG(8)
AKZG(8)
BFLX(I)
BFLXP(I)
C(MAXNP)
CAPROP
(U)
Units
-
L/T
--
L/T
L/T
L/T
L/T
L/T
L/T
L3/T
L3/T
L
1/L
Type
Scalar
Array
Array
Array
Array
Array
Array
Array
Array
Array
Array
Array
Array
Description
Gravity Term Included?
0.0 = no, 1.0 = yes
I-th Parameter to Describe
the Relative Conductivity
as a Function of Pressure
Head for the J-th Material
or the I-th Data Point of
Relative Conductivity for
the J-th Material
Relative Conductivity at
the I-th Node of the M-th
Element
XX-Hydraulic Conductivity
at Eight Gaussian Points
XY-Hydraulic Conductivity
at Eight Gaussian Points
XZ-Hydraulic Conductivity
at Eight Gaussian Points
W-Hydraulic Conductivity
at Eight Gaussian Points
YZ-Hydraulic Conductivity
at Eight Gaussian Points
ZZ-Hydraulic Conductivity
at Eight Gaussian Points
Present Time Flux at the
I-th Boundary Node
Previous Time Flux at the
I-th Boundary Node
Final Solution
I-th Data Point of Water
Capacity for the J-th
Material
Sub- Common
routine Block
Q8
GW3D
DATAIN
SPROP
GW3D
VELT
SPROP
ASEMBL
BC
Q8DV
Q8
Q8DV
Q8
Q8DV
Q8
Q8DV
Q8
Q8DV
Q8
Q8DV
Q8
GW3D
SFLOW
PRINTT
GW3D
SFLOW
BLKITR
SOLVE
GW3D
DATAIN
SPROP
I.M.O
I
M
0
I
M
I
0
I
I
I
I
I
I
I
I
I
I
I
I
I
I
M
M
I
M
M
0
M
M
0
I
188
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable
Units fVD
,e Description
Sub-
routine
Common
Block
I.M.O
CHNG
CMATRX
CMTRXG
(MAXNP,
JBAND)
CMTRXL
CW(MAXNP) L
DCOSB(IJ) -
Scalar
Array
Array
Array
Array
Array
DCOSB(2,1) -
Array
DCOSB(3,1) -
Array
DCYFLX(NP) L3/T Array
DELMAX T Scalar
DELT T Sealer
Multiplier for Increasing
DELT
An Array to Store the
assembled Global Matrix
Global Matrix
Assembled Matrix for a
subregion
X-Directional Cosine of
the I-th Boundary Side
Y-Directional Cosine of
the I-th Boundary Side
Z-Directional Cosine of
the I-th Boundary Side
Darcy Flux Through the
NP-th Variable Boundary
Node
Maximum Value of DELT
Time Increment
GW3D
VELT
ASEMBL
BC
BLKITR
GW3D
BLKITR
BLKITR
GW3D
DATAIN
SURF
BCPREP
BC
SFLOW
STORE
GW3D
DATAIN
SURF
BCPREP
BC
SFLOW
STORE
GW3D
DATAIN
SURF
BCPREP
BC
SFLOW
STORE
GW3D
BCPREP
PRINTT
CREAL
CREAL
ASEMBL CREAL
SFLOW
PRINTT
M
0
0
M
M
M
M
M
0
0
I
I
I
I
M
0
0
I
I
I
I
M
0
0
I
I
I
I
M
0
I
189
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARLABLE DESIGNATION (continued)
Variable
DELTO
DHQ(8)
Units
T
L
Tvoe
Scalar
Array
Description
Time Increment
Pressure Difference
Sub-
routine
Q8TH
Common
Block I.M.O
CREAL
I
DJAC L3 Scalar
DNX(8) IL Array
DNY(8) 1/L Array
DNZ(8) 1/L Array
DTH(I,M) 1/L Array
DTHG(8) 1/L Array
F(MAXNOD) - Array
FLOW(IO) L3 Array
FLX(NP) L3iT Array
FRATE(IO) I/VT Array
F1Q(4) L3/11/L2 Array
F2Q(4)
L8/T/L2 Array
Between the Present Time
Step and the Previous
Time Step at Eight Nodes
of the Element
Determinant of the
Jacobian
Partial Derivative of the
Base Function with Respect
to x
Partial Derivative of the
Base Function with Respect
to y
Partial Derivative of the
Base Function with Respect
to z
Water Capacity at the I-th
Node of the M-th Element
Water Capacity at Eight
Gaussian Points of the
Element
Array of Real Numbers
that are to be Read and
Generated Automatically
Increment of Flow
Rainfall Flux Through the
NP-th VB Node
Flow Rate
Specified Normal Flux at
Four Nodes of the Surface
Gravity Flux at Four Nodes
of a Specified-Pressure-
Head Gradient (Neumann)
Surface
BASE
BASE
BASE
BASE
Q8
READR
GW3D
BCPREP
BC
Q4S
Q4S
CFLOW
CFLOW
0
0
0
0
GW3D
SPROP
ASEMBL
SFLOW
M
0
I
I
M
0
I
190
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
GNLR(LK)
H(N) L
HCON(NP) L
HDB(J) L
HDBF(IJ) L
HMIN(NP) L
HP(N) L
HPROP(LJ) L
HT(N) L
Type
Array
Array
Array
Array
Arr
Array
Arr
Array
Array
Description
Global Node Number of the
I-th Node in the K-th
Subregion
Pressure Head at the
Present Time
Pending Depth of the NP-th
Variable Boundary Node
Total Head of the J-th
Profile at the Present
Time
Total Head of the I-th
Data Point in the J-th
Profile
Minimum Pressure Allowed
for the NP-th VB Node
Previous-Time Pressure
Head at the N-th Node
I-th Data Point of
Pressure for the J-th
Material
Total Head the N-th Node
Sub- Common
routine Block
GW3D
DATAIN
PAGEN
BLKITR
STORE
GW3D
DATAIN
VELT
GW3D
DATAIN
BCPREP
BC
GW3D
BC
GW3D
DATAIN
GW3D
DATAIN
BCPREP
BC
GW3D
ASEMBL
GW3D
DATAIN
SPROP
GW3D
PRIN'IT
STORE
VELT
I.M.O
M
0
M
I
I
0
0
I
M
0
I
I
M
I
M
0
M
0
I
I
M
I
M
0
I
M
I
I
0
HTQ(8) L Arr
HW(N) L Array
IBUG " Scalar
Total Head at Eight Nodes Q8DV
of the Element
Nonlinear Pressure Head GW3D
Iterate at the N-th Node
Diagnostic Print-Out BLKITR
Indicator
M
191
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
ICTYP(MP) -
IDTYP(NP) -
IE(M,I)
IGEOM
IHALFB
ILUMP
IMID
INDTYP(
MXTYP)
Type
Array
Array
Array
Scalar
Scalar
Scalar
Scalar
Array
Description
Type of Specified-Flux
(Cauchy) Profile Assigned
to the MP-th Side
Total Head Profile Type
of NP-th Dirichlet Node
Global Node Number of the
I-th Node of the M-th
Element if I is Between 1
and 8, Material Type of
the M-th Element if I = 9
Geometry Description
Output Control
Half Band with Plus 1
Lumping Indicator
Mid-Difference Indicator
Array of Integers that
are to be Read or
Sub- Common
routine Block
GW3D
DATAIN
BC
GW3D
DATAIN
BC
GW3D
DATAIN
SURF
PAGEN
VELT
SPROP
BCPREP
ASEMBL
BC
SFLOW
STORE
CINTE
SOLVE
Q8 OPTN
OPTN
READN
I.M.O
M
0
I
M
0
I
M
0
I
I
I
I
I
I
I
I
I
I
I
0
INTYP(MP)
IRTYP(MP)
ISB(IJ)
Generated Automatically
Array Type of Specified-
Pressure-Head Gradient
(Neumann) Flux Profile
Assigned to the
MP-th Neumam Side
Array Type of Rainfall Profile
Assigned to the MP-th
Variable Boundary Side
Array Boundary Node Number of
the First Node of the
I-th Boundary Side
GW3D
DATAIN
BC
GW3D
DATAIN
BCPREP
GW3D
DATAIN
SURF
BCPREP
BC
SFLOW
STORE
M
0
I
M
0
I
M
0
0
I
I
I
I
192
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Unite
ISB(2,I)
ISB(3,I)
ISB(4,I)
ISB(5,I)
ISB(6,I)
ISC(1,MP) -
ISC(2,MP) -
ISC(3,MP) -
Type
Array
Array
Array
Array
Array
Array
Array
Array
Description
Boundary Node Number of
the Second Node of the
I-th Boundary Side
Boundary Node Number of
the Third Node of the
I-th Boundary Side
Boundary Node Number of
the Fourth Node of the
I-th Boundary Side
Element Side Index of the
I-th Boundary Side:
l=left side, 2=front side,
3=right side, 4=back side,
5=bottom side, 6=top side
Element Number to which
the I-th Boundary Side
Belongs
Global Node Number of the
First Node of the MP-th
Cauchy Side
Global Node Number of the
Second Node of the MP-th
Cauchy Side
Global Node Number of the
Third Node of the MP-th
Cauchy Side
Sub- Common
routine Block
GW3D
DATAIN
SURF
BCPREP
BC
SFLOW
STORE
GW3D
DATAIN
SURF
BCPREP
BC
SFLOW
STORE
GW3D
DATAIN
SURF
BCPREP
BC
SFLOW
STORE
GW3D
DATAIN
SURF
BCPREP
BC
SFLOW
STORE
GW3D
DATAIN
SURF
BCPREP
BC
SFLOW
STORE
GW3D
DATAIN
BC
GW3D
DATAIN
BC
GW3D
DATAIN
BC
I.M.O
M
0
0
I
I
I
I
M
0
0
I
I
I
I
M
0
0
I
I
I
I
M
0
0
I
I
I
I
M
0
0
I
I
I
I
M
0
I
M
0
I
M
0
I
193
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
ISC(4,MP)
ISC(5,MP)
ISN(1,MP)
ISN(2,MP)
ISN(3,MP)
ISN(4,MP)
ISN(5,MPF) --
ISTYP(MP)
ISV(1,MP)
ISV(2,MP)
ISV(3,MP)
ISV(4,MP)
Type
Array
Array
Array
Array
Array
Array
Array
Array
Array
Array
Array
Array
Description
Global Node Number at the
Fourth Node of the MP-th
Cauchy Side
Boundary Side Number of
the MP-th Cauchy Side
Global Node Number of the
First Node of the MP-th
Neumann Side
Global Node Number of the
Second Node of the MP-th
Neumann Side
Global Node Number of the
Third Node of the MP-th
Neumann Side
Global Node Number of the
Fourth Node of the MP-th
Neurnam Side
Boundary Side Number of
the MP-th Neumann Side
Source/Sink Type Assigned
to the MP-th S/S Element
Global Node Number of the
First Node of the MP-th
Variable Side
Global Node Number of the
Second Node of the MP-th
Variable Side
Global Node Number of the
Third Node of the MP-th
Variable Side
Global Node Number of the
Fourth Node of the MP-th
Variable Side
Sub- Common
routine Block
GW3D
DATAIN
BC
GW3D
DATAIN
BC
GW3D
DATAIN
BC
GW3D
DATAIN
BC
GW3D
DATAIN
BC
GW3D
DATAIN
BC
GW3D
DATAIN
BC
GW3D
DATAIN
ASEMBL
SFLOW
GW3D
DATAIN
BCPREP
GW3D
DATAIN
BCPREP
GW3D
DATA.IN
BCPREP
GW3D
DATAIN
BCPREP
I.M.O
M
0
I
M
0
I
M
0
I
M
0
I
M
0
I
M
0
I
M
0
I
M
0
I
I
M
0
I
M
0
I
M
0
I
M
0
I
194
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
ISV(5,MP) -
ITIM
IWTYP(NP) -
KANALY
KCAI
KDAI
KDLAG
KDSK(I)
KDSKO
KFLOW
KGRAV
KKK
KNAI
KOUT
KPR(I)
Type
Array
Scalar
Array
Scalar
Scalar
Scalar
Scalar
Array
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Array
Description
Boundary Node Number of
the MP-th VB Side
Time Step Number
Source/Sink Type Assigned
to the NP-th Well Node
Analytical Input Control
Analytical Specified-Flux
(Cauchy) Input Control
Analytical Dirichlet
Input Control
Diagnostic Print-Out
Table Number
Auxiliary Output Control
for the I-th Time Step;
0 = no auxiliary output
1 = output stored
Disk Output Control
System Flow Counter
Index of Gravity Control
Decomposition or Back
Substitution Indicator
1 = decomposition,
2 = back substitution
Analytical Neumann Flux
Input Control
Prin-Out Table Number
Line-printer Control for
Sub- Common
routine Block
GW3D
DATAIN
BCPREP
PRINTT
GW3D
DATAIN
ASEMBL
SFLOW
ALLFCT
CCBC
CDBC
PRINTT
GW3D
DATAIN
CINTE
SFLOW
CGEOM
SOLVE
CNBC
PRINTT
GW3D
I.M.O
M
0
I
I
M
0
I
I
I
M
M
0
I
M
M
I-th Time Step:
0 = print nothing
1 = print system mass
balance plus above
2 = print pressure head
plus above
DATAIN
BLKITR
PRIN'M'
0
I
I
195
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
LES(MP)
LMAXDF(K)
LMXBW
LMXBWK
LMXNP
LMXNPK
Type
KPRO
KRAI
KSAI
KSP
KSS
KWAI
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Array
Array
Scalar
Scalar
Scalar
Scalar
Description
3 = print total head plus
above
4 = print moisture content
plus above
5 = print Darcy velocity
plus above
Output Control
Analytical Rainfall Input
Control
Analytical Distributed
Source/Sink Input Control
Soil Property Tabular
Input Control
Steady-State I.C. Control
Analytical Well Source/
Sink Input Control
Global Element Number of
the MP-th S/S Element
Maximum Difference Between
Eight Nodes of Any Element
Maximum No. of the
Bandwidth in any Subregion
Maximum No. of the Bandwidth
in Any Subregion
Maximum No. of Nodal
Points in any Subregion,
Including Interior Nodes
and Global Boundary Nodes
Maximum No. of Nodal Points
in any Subregion, Including
Interior Nodes and Global
Boundary Nodes
Sub-
routine
Common
Block
I.M.O
CINTE
CVBC
CS
SPROP CINTE
ASEMBL CINTE
CW
GW3D
DATAIN
ASEMBL
SFLOW
GW3D
PAGEN
BLKITR
DATAIN
M
0
I
I
M
0
I
0
BLKITR LGEOM
BLKITR LGEOM
196
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable
Units
Tvoe
Description
Sub-
routine
Common
Block
I,M,0
LNOJCN
(J.I.K)
LRL(I,N)
LRN(I,N)
Array
Array
Array
LTMXNK
LTMXNP
LUBAR
LUINP
LUOUT
Scalar
Scalar
Scalar
Scalar
Scalar
LUPAR
LUSTO
Scalar
Scalar
Local Node No. of the J-th
Compressed Number Connect-
ing to the I-th Local
Node in the K-th Subregion
Global Element Number of
the I-th Element Connect!
ing to the N-th Global
Node
Global Node Number of the
I-th Node Connecting to
the N-th Global Node
Maximum No. of Total Nodal
Points in any Subregion,
Including Interior Nodes,
and Global Boundary Nodes
Maximum No. of Total Nodal
Points in any Subregion,
Including Interior Nodes,
and Global Boundary Nodes
Logical Unit for Storing
Binary Boundary Arrays
Logical Unit for Input
Data
Logical Unit for Output
Data
Logical Unit for Storing
Binary Pointer Arrays
Logical Unit for Storing
Binary Output
GW3D
PAGEN
BLKITR
DATAIN
GW3D
DATAIN
SURF
PAGEN
GW3D
PAGEN
ASEMBL
BC
DATAIN
BLKITR LGEOM
STORE
GW3D
DATAIN
GW3D
DATAIN
READR
READN
GW3D
DATAIN
SURF
PAGEN
ASEMBL
BLKITR
PRINTT
READR
READN
GW3D
GW3D
STORE
M
0
1
0
M
0
1
0
M
0
1
1
0
197
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
MAXBES
MAXBNP
MAXBw
MAXEL
MAXELK
MAXM.AT
MAXNOD
MAXNP
MAXNPK
MAXNTI
MXBESK
MXBNPK
MXCDP
MXCDPK
MXCES
MXCESK
MXCNP
Type
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Description
Maximum No. of Boundary
Element Surfaces
Maximum No. of Boundary
Nodal Points
Maximum No. of Bandwidth
Maximum No. of Elements
Maximum No. of Elements
Maximum No. of Materials
Maximum No. of Data Points
to be Read
Maximum no. of Nodal
Points
Maximum No. of Nodes
Maximum No. of Time Steps
Maximum No. of Boundary
Element Surfaces
Maximum No. of Boundary
Nodal Points
Maximum No. of Data Points
on Each Cauchy-Flux Profile
Maximum No. of Data Points
on Each Cauchy-Flux Profile
Maximum No. of Specified-
Flux (Cauchy) Element Surfaces
Maximum No. of Specified-
Flux (Cauchy) Element Surfaces
Maximum No. of Specified-
Sub-
routine
STORE
PRINTT
STORE
SOLVE
SPROP
PRINTT
STORE
SPROP
READR
SPROP
BLKITR
SOLVE
PRINTT
STORE
Common
Block I.M.O
SGEOM I
SGEOM I
I
I
SGEOM I
I
I
SMTL I
I
SGEOM I
I
I
I
I
SGEOM
CCBC
CCBC
CCBC
Flux (Cauchy) Nodal Points
198
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
MXCNPK
MXCPR
MXCPRK
MXDDP
MXDDPK
MXDNP
MXDNPK
MXDP
MXDPR
MXDPRK
MXDTCK
MXJBD
MXJBDK
MXMATK
MXMPMK
MXMPPM
MXNDP
Type
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Sub- Common
Description routine Block I,M,0
Maximum No. of Specified-
Flux (Cauchy) Nodal Points
Maximum No. of Specified- CCBC
Flux (Cauchy) Profiles
Maximum No. of Specified-
Flux (Cauchy) Profiles
Maximum No. of Data Points CDBC
on Each Dirichlet Profile
Maximum No. of Data Points
on Each Dirichlet Profile
Maximum No. of Dirichlet CDBC
Nodal Points
Maximum No. of Dirichlet
Nodal Points
Maximum No. of Data ALLFCT I
Points in any Profile
Maximum No. of Dirichlet CDBC
Total Head Profiles
Maximum No. Dirichlet Total
Head Profiles
Maximum No. of DELT Changes
Maximum No. of Nonzero BLKITR I
Elements in any Row PRINTT I
Maximum No. of Nonzero
Elements in any Row
Maximum No. of Material
Types
Maximum No. of Material
Properties per Material
Maximum No. of Material SMTL
Properties per Material
Maximum No. of Data Points CNBC
on Each Neumann-Flux
Refile
199
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable
MXNDPK
MXNDTC
MXNES
MXNESK
MXNNP
MXNNPK
MXNPR
MXNPRK
MXNTIK
MXPR
MXRDP
MXREGN
MXRGNK
MXRPR
Units Type
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Description
Maximum No. of Data Points
on Each Neumann-Flux
Profile
Maximum No. of DELT
Changes
Maximum No. of Neumann
Element Surfaces
Maximum No. of Neumann
Element Surfaces
Maximum No. of Neumann
Nodal Points
Maximum No. of Neumann
Nodal Points
Maximum No. of Neumann-
Flux Profiles
Maximum No. of Neumann-
Flux Profiles
Maximum No. of Time Steps
Maximum No. of Profiles
Maximum No. of Data Points
on Each Rainfall Profile
Maximum No. of Subregions
Maximum No. of Subregions
Maximum No. of Rainfall
Sub- Common
routine Block I.M.O
SGEOM
CNBC
CNBC
CNBC
ALLFCT I
CVBC
BLKITR LGEOM I
STORE I
CVBC
MXSDP
MXSDPK
MXSEL
Profiles
Scalar Maximum No. of Data Points
on Each Element Soured
Sink Profile
Scalar Maximum No. of Data Points
on Each Element Source/
Sink Profile
Scalar Maximum No. of Source
Elements
CS
CS
200
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARLABLE DESIGNATION (continued)
Variable Units
MXSELK
MXSPMK
MXSPPM
MXWDPK
MXWNP
Scalar
Scalar
Scalar
MXSPR
MXSPRK
MXTYP
MXVDPK
MXVES
MXVESK
MXVNP
MXVNPK
MXVPRK
MXWDP
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Description
Maximum No. of Source
Elements
Maximum No. of Soil
Parameters per Material
to Describe Soil Charac-
teristic Curves
Maximum No. of Soil
Parameter Per Material
to Describe Soil
Characteristic Curves
Maximum No. of Source
Profiles
Maximum No. of Source
Profiles
Maximum No. of Integers
Allowed to be Read
Maximum No. of Data Points
on Each Rainfall Profile
Maximum No. of Variable
Element Surfaces
Maximum No. of Variable
Element Surfaces
Maximum No. of Variable
Nodal Points
Maximum No. of Variable
Nodal Points
Maximum No. of Rainfall
Profiles
Maximum No. of Data Points
on Each Well Source/Sink
Profile
Maximum No. of Data Points
on Each Well Source/Sink
Profile
Maximum No. of Well Nodal
Points
Sub- Common
routine Block I.M.O
SPROP
SMTL
CS
READN
CVBC
PRINT! CVBC
CW
CW
201
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
MXWNPK
MXWPR
MXWPRK
N
NBES
NBNP
NCDP
NCES
NCHG
NCNP
NCPR
NCYL
NDDP
NDNP
NDP
Type
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Sub- Common
Description routine Block I,M,0
Maximum No. of Well
Nodal Points
Maximum No. of Well CW
Source/Sink Profiles
Maximum No. of Well
Source/Sink Profiles
Base Functions Associated BASE 0
with 8 Nodes of the Element
Number of Boundary STORE CGEOM I
Element Surfaces
Number of Boundary Nodal STORE CGEOM I
Points
Number of Data Points on CCBC
Specified-Flux (Cauchy)
Profiles
Number of Specified-Flux CCBC
(Cauchy) Boundary Element
Sides
Number of Variable BCPREP 0
Boundary Nodes that has
Changed Boundary Conditions
Number of Specified-Flux CW
(Cauchy) Boundary Nodal
Points
Number of Specified-Flux CCBC
(Cauchy) Profiles
Number of Cycles per CINTE
Time Step
Number of Data Points on CDBC
Dirichlet Profiles
Number of Dirichlet Nodal CDBC
Points
Number of Data Points in ALLFCT I
Any Profile
202
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
NDPR
NDTCHG -
NEL
NITER
NMAT
NMPPM
NNDP
NNES
NNNP
NNP
NNPLR(K) -
NNPR
NPBB(I)
Type
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Array
Scalar
Array
Description
Number of Dirichlet
Profiles
Number of Times to Reset
Time Step Size
Number of Elements
Number of Iterations per
Cycle
Number of Materials
Number of Material
Properties per Material
Number of Data Points on
Neumann-Flux Profiles
Number of Neumann Boundary
Element Sides
Number of Neumann Boundary
Nodal Points
Number of Nodal Points
Number of Node Points in
the K-th Subregion
Number of Neumann-Flux
Profiles
Global Node Number of the
I-th Boundary Node
Sub-
r o u t
SPROP
PRINTT
STORE
BLKXTR
BLKITR
SOLVE
PRINTT
STORE
READR
GW3D
DATAIN
PAGEN
BLKITR
STORE
GW3D
DATAIN
SURF
SFLOW
STORE
Common
i Htocfe
CDBC
CGEOM
CGEOM
CINTE
CMTL
CMTL
CNBC
CNBC
CNBC
CGEOM
CNBC
I.M.O
I
I
I
I
I
M
0
I
I
I
M
0
0
I
I
203
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable
Units
Type
Description
Sub-
routine
Block
I.M.O
NPCB(MP) -
NPCNV(I)
NPCON(NP) -
Array Global Node Number of the GW3D
MP-th Cauchy Node on DATAIN
Input, then is Changed to SFLOW
Contain the Boundary Node
Number
Array Global Node Number of the GW3D
I-th Nonconvergent Node
Array Pending Condition GW3D
Indicator of the NP-th VB BCPREP
Node: 0 = this is not a BC
Pondidng-Condition Node PRINT!
for the Present Time Step,
Global Node Number = this
is a Pending-Condition Node
for the Present Time
NPDB(
MXDNP)
NPFLX(NP) -
NPITER
NPMIN(NP)
NPNB(MP)
NPR
NPROB
GW3D
DATAIN
BC
SFLOW
GW3D
BCPREP
BC
PRINTT
Array Global Node Number of the
NP-th Dirichlet Node
Array Flux Boundary Condition
Indicator of the NP-th VB
Node; 0 = this is not a
Flux-Condition Node for
the Present Time Step,
Global Node Number = This
is a Flux-Condition Node
for the Present Time
Scalar Number of Blockwise
Iterations Allowed
Array Minimum-Pressure Condition GW3D
Indicator of the NP-th VB BCPREP
Node; 0 = this is not a BC
Minimum-Pressure-Condition PRINTT
Array Global Node Number of the GW3D
MP-th Neumann Node on DATA-IN
Input, then is Changed to SFLOW
Contain the Boundary Node
Number
Scalar Number of Profiles ALLFCT
Scalar Problem Number STORE
CINTE
M
0
I
M
M
0
I
I
M
0
I
I
M
0
I
I
M
0
I
I
M
0
I
204
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
NPVB(NP)
NPW(NP)
NRDP
NREGN
NRPR
NSDP
NSEL
NSPPM
NSPR
NTI
NTNPLR(K) --
Type
Array
Array
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Array
Description
Global Node Number of the
NP-th VB Node on Input,
then is Changed to Contain
the Boundary Node Number
Global Node Number of the
NP-th S/S Well Node
Number of Data Points on
Rainfall Profiles
Number of Subregions
Number of Rainfall
Profiles
Number of Data Points on
Element-Source/Sink
Profile
Number of Element-Source/
Sink and B.C. Control
Integer
Number of Soil Parameters
per Material to Describe
Soil Characteristic Curves
Number of Element-Source/
Sink Profiles
Number of Time Increments
Total Number of Nodes for
Sub- Common
routine Block
GW3D
DATAIN
SFLOW
PRINTT
GW3D
DATAIN
ASEMBL
CVBC
BLKITR LGEOM
STORE
CVBC
CS
CS
SPROP CMTL
CS
STORE CGEOM
GW3D
I.M.O
M
0
I
I
M
0
I
I
I
I
M
NTYPE
NVES
NVNP
the K-th Subregion
Including Interior, Global
Boundary, and Intra-
boundary Nodes
Scalar Number of Integers to be
Read
Scalar Number of Variable Bounda-
Element Sides
Scalar Number of Variable
Boundary Nodal Points
PAGEN
DATAIN
READN
CVBC
PRINTT CVBC
0
0
205
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
NWDP
NWNP
NWPR
OME
OMI
PR(MXPR) L or
LVL2
PRF(MXDP, L or
MXPR) LVL2
PROP(U) L/T or
L'
Type
Scalar
Scalar
Scalar
Scalar
Scalar
Array
Array
Array
Description
Number of Data Points on
Each Well Source/Sink
Profile
Number of Well Source/Sink
Nodal Points
Number of Well Source/Sink
Profiles
Iteration Parameter for a
Non-Linear Equation
Relaxation Parameter for
Pointwise Solution
Profile Values at T
Profile Value of the Data
Point on the Profile
I-th Material Property of
the J-th Material;
1 = 1= saturated xx-
hydraulic conduc-
tivity
Sub- Common
routine Block
CW
CW
CW
BLKITR CREAL
CREAL
ALLFCT
ALLFCT
GW3D
DATAIN
VELT
ASEMBL
BC
I.M.O
I
0
I
M
0
I
I
I
QA(8,8)
QB(8,8)
Array
Array
1 = 2 = saturated yy-
hydraulic conduc-
tivity
1 = 3 = saturated zz-
hydraulic conduc-
tivity
1=4 = saturated xy-
hydraulic conduc-
tivity
1 = 5 = saturated xz-
hydraulic conduc-
tivity
1 = 6 = saturated yz-
hydraulic conduc-
tivity
Integration of N(I)
*DTH/DH*N(J)
8x8 Element Matrix
Q8
Q8DV
Q8
0
0
206
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable
QCB(J)
QCBF(IJ)
QNB(J)
QNBF(IJ)
QRX(8)
QRY(8)
QRZ(8)
QSOSM
QTHM
R(MAXNP)
RF(I,J)
RFALL(JJ)
RI(N)
RL
RLD(N)
RLDG(
MAXNP)
Units Tvoe
3 r
L3/T/L2 Array
LVT/L2 Array
(L3/T) Array
/L2
LVT/L' Array
Array
Array
Array
L3 Scalar
L3 Scalar
Array
L/T Array
L/T Array
L Array
Scalar
Array
Array
Description
Cauchy Flux of the J-th
Profile at the Present
Time
Cauchy Flux of the I-th
Data Point in the J-th
Profile
Neumann Flux of the J-th
Profile at the Present
Time
Neumann Flux of the I-th
Data Point in the J-th
Profile
X-Velocity Element Vector
Y-Velocity Element Vector
Z-Velocity Element Vector
Integration of SOURCE
Integration of DHQ*THG
Load Vector
Rainfall Rate of I-th
Data Point in J-th
Profile
Rainfall Rate of J-th
Profile at the Present
Time
Pressure Head Iterate in
BLKITR
A Working Array to Contain
the Final Solution of the
Pressure Head in BLKITR
An Array to Store the
Assembled Global Load
Vector
Global Load Vector
Sub- Common
routine Block
GW3D
BC
GW3D
DATAIN
GW3D
BC
GW3D
DATAIN
Q8DV
Q8DV
Q8DV
Q8TH
Q8TH
SOLVE
GW3D
DATAIN
GW3D
BCPREP
GW3D
GW3D
GW3D
ASEMBL
BC
BLKITR
I.M.O
M
I
M
0
M
I
M
0
0
0
0
0
0
M
M
0
M
I
M
M
M
0
M
I
207
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
RLDL(N)
RQ(8)
R1Q(4)
R2Q(4)
SOS(J) (L3/T)
/L3
SOSF(IJ) urn
L2/L
SOSM LYT
SOURCE L3/T
SS
SUBHD
T T
TDTCH(I) T
TFLOW(IO) L3
TH(I,M)
THDBF(I,J) T
THG(8)
Jjpe
Array
Array
Array
Array
Array
Array
Scalar
Scalar
Scalar
Char.
Scalar
Array
Array
Array
Array
Array
Description
Assembled Load Vector for
a Subregion
Integration of N(I).K.
(Unit Vector in Z)
Integration of N(I)*F1Q
Over the Boundary Segment
Integration of N(I)*F2Q
Over the Boundary Segment
Value of J-th Element
Source/Sink at Present
Time
S/S Rate of the I-th Data
Point in the J-th Profile
Source/Sink Strength of
the Element
Element Source/Sink
Strength
Xsi-Coordinate of the
Gaussian Point
Subheading
Time
Time of the I-th Time to
Reset the Time Step Size
to Initial Time Step Size
Total Flow
Moisture Content, at the
I-th Node of the M-th
Element
Time of the I-th Data
Point in J-th Head Profile
Moisture Content at Eight
Sub- Common
routine Block
GW3D
BLKITR
Q8
Q4S
Q4S
GW3D
ASEMBL
SFLOW
GW3D
DATAIN
Q8
Q8TH
BASE
PRINTT
ALLFCT
GW3D
DATAIN
CFLOW
GW3D
SPROP
SFLOW
PRINTT
STORE
GW3D
DATAIN
Q8TH
I.M.O
M
M
0
0
0
M
I
I
M
0
I
I
I
I
I
M
0
M
0
I
I
I
M
0
I
Gaussian Points of the
Element
208
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable
THPROP
(U)
TIME
TITLE
TMAX
TOLA
TOLB
TPRF(
MXDP,
MXPR)
TQCBF(IJ)
TQNBF(LJ)
TRF(IJ)
TSOSF(IJ)
TT
TWSSF(IJ)
UU
VX(N)
Units
T
--
T
L
L
T
T
T
T
T
--
T
--
L/T
Type
Array
Scalar
Char.
Scalar
Scalar
Scalar
Array
Array
Array
Array
Array
Scalar
Array
Scalar
Array
Description
I-th Parameter to Describe
the Moisture Content as a
Function of Pressure Head
for the J-th Material or
I-th Data Point of Moisture
Content for the J-th
Material
Real Simulation Time
Title of the Problem
Maximum Value of Time
Steady-State Tolerance
Transient State Tolerance
Time of the Data Point
on the Profile
Time of the I-th Data
Point in the J-th
Specified-Flux (Cauchy)
Profile
Time of the I-th Data
Point in the J-th Neumann
Profile
Time of the I-th Date
Point in J-th Rainfall
Profile
Time of the I-th Data
Point in the J-th Profile
Eta-Coordinate of the
Gaussian Point
Time of the I-th Data
Point in the J-th Profile
Zeta-Coordinate of the
Gaussian Point
X-Component Velocity at
the N-th Node
Sub- Common
routine Block
GW3D
DATAIN
SPROP
PRINTT
STORE
STORE
CREAL
CREAL
BLKITR CREAL
ALLFCT
GW3D
DATAIN
GW3D
DATAIN
GW3D
DATAIN
GW3D
DATAIN
BASE
GW3D
DATAIN
BASE
GW3D
VELT
BCPREP
I.M.O
M
0
I
M
0
M
0
M
0
M
0
I
M
0
I
0
0
I
209
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units Tvnp Description
VY(N) L/T Array Y-Component Velocity at
the N-th Node
VZ(N) UT Array Z-Component Velocity at
the N-th Node
Sub- Common
routine Block
SFLOW
PRINT!
STORE
GW3D
VELT
BCPREP
SFLOW
PRINTT
STORE
GW3D
VELT
BCPREP
SFLOW
PRINTT
STORE
I.M.O
I
I
I
0
0
I
I
I
I
0
0
I
I
I
I
W(8) - Array
WSS(J) L3/T Array
WSSF(LJ) I/YT Array
X(N)
Array
Weighting Function at
Eight Points of the
Element
Value of the J-th Well
Source/Sink at Present
Time
S/S Rate of the I-th Data
Point in the J-th Profile
X-Coordinate of the N-th
Node
XQ(8)
Array
X-Coordinate at Eight
Nodes of the Element
ASEMBL CREAL
GW3D M
SFLOW I
ASEMBL I
GW3D M
DATAIN 0
GW3D M
DATAIN 0
SURF I
VELT I
BCPREP I
ASEMBL I
BC I
SFLOW I
STORE I
Q8DV I
Q8 I
BASE I
Q4S I
Q8TH I
210
-------
TABLE D-l. 3DFEMWATER PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units TVHP Description
Y(N) L Array Y-Coordinate of the N-th
Node
YQ(8) L Array Y-Coordinate at Eight
Nodes of the Element
Z(N) L Array Z-Coordinate of the N-th
Node
ZQ(8) L Array Z-Coordinate at Eight
Nodes of the Element
Sub- Common
routine Block
GW3D
DATAIN
SURF
VELT
BCPREP
ASEMBL
BC
SFLOW
STORE
Q8DV
Q8
BASE
Q4S
Q8TH
GW3D
DATAIN
SURF
VELT
BCPREP
ASEMBL
BC
SFLOW
STORE
Q8DV
Q8
BASE
Q4S
Q8TH
I.M.O
M
0
I
I
I
I
I
I
I
I
I
I
I
I
M
0
I
I
I
I
I
I
I
I
I
I
I
I
211
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION
Variable Unite Type
L Scalar
AL
AM
APHA1
APHA2
APHA3
APHA4
AT
BETA1
BETA2
BETAS
BETA4
L'/T Scalar
Scalar
Scalar
Scalar
Scalar
BFLX(I) M/T Array
BFLXP(I) WT Array
Description
Longitudinal Dispersivity
Modecular Diffusion
Coefficient
Weighting Factor for
Side 1-2 Parallel to
the X-direction
Weighting Factor for
Side 4-3 Parallel to
the X-direction
Weighting Factor for
Side 5-6 Parallel to
the X-direction
Weighting Factor for
Side 8-7 Parallel to
the X-direction
Scalar Lateral Dispersivity
Scalar Weighting Factor for
Side 1-4 Parallel to
the Y-direction
Scalar Weighting Factor for
Side 2-3 Parallel to
the Y-direction
Scalar Weighting Factor for
Side 5-8 Parallel to
the Y-direction
Scalar Weighting Factor for
Side 6-7 Parallel to
the Y-direction
Boundary Flux at the I-th
Boundary Node
Value of BFI.X(I) at the
Previous Time
Sub-
routing
Q8DV
Q8
Q8
Q8DV
Q8
Common
Block
WETX
WETX
WETX
WETX
WETY
WETY
WETY
WETY
I.M.O
I
I
GM3D
SFLOW
GM3D
SFLOW
M
M
M
M
BQ(4,4)
Array A 2 by 2 Boundary
Surface Matrix
Q4CNVB
212
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units Tyne
C(N) M/L3 Array
CDB(I)
M/L3 Array
CDBF(IJ) M/L3 Array
CHNG
CMATRX
(NJ)
CMTRXG
(NJ)
CMTRXL
(NJ)
CP(N)
CQ(8)
CSQ(8)
M/L3
Scalar
Array
Array
Array
Array
M/L3 Array
M/M Array
CSTAR(N) M/L3 Array
Description
Concentration of the N-th
Node at the Present Time
Dirichlet Concentration
of the I-th Profile at
Present Time
Concentration of the I-th
Data Point in the J-th
Dirichlet Concentration
vs. Time Profile
Multiplier for Increasing
DELT
An Array to Store the
I-th Non-Zero Entry of
the N-th Equation of the
Assembled Global Matrix
Global Matrix
Assembled Matrix for a
Subregion
Concentration of the N-th
Node at the Previous Time
Dissolved Concentration
at Eight Points of an
Element
Adsorbed Concentration
at Eight Points of an
Element
Lagrangian Concentration
at the N-th Node
Sub-
routine,
GM3D
FLUX
BLKITR
SOLVE
SFLOW
PRINTT
STORE
GM3D
BC
ADVBC
GM3D
DATAIN
GM3D
FLUX
ASEMBL
BC
BLKITR
GM3D
BLKITR
GM3D
DATAIN
ASEMBL
ADVTRN
Q8DV
Q8R
Q8R
GM3D
ASEMBL
ADVTRN
ADVBC
Common
Block
CREAL
I.M.O
0
I
0
M
I
I
I
M
I
I
M
M
M
0
0
0
M
I
0
M
I
I
I
I
0
I
0
0
213
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
CVB(I) M/L3
CVBF(LJ) M/L9
CW(N) M/L3
CWQ(8) M/L3
DCOSB(IJ) -
Type
Array
Array
Array
Array
Array
DCOSB(2,1) -
Array
DCOSB(3,1) -
Array
DD L'PI!
DELMAX T
Description
Variable Concentration of
the I-th Profile at the
Present Time
Concentration of the I-th
Data Point in the J-th
Variable Concentration
vs. Time Profile
Nonlinear Iterate of the
Concentration at the N-th
Node
Iterate of the Dissolved
Concentration at Eight
Gaussian Points of the
Element
X-Directional Cosine of
the I-th Boundary Side
Y-Directional Cosine of
the I-th Boundary Side
Z-Directional Cosine of
the I-th Boundary Side
Scalar
Scalar Maximum Value of DELT
Effective Molecular
Diffusion Coefficient
Sub-
routine.
GM3D
BC
ADVBC
GM3D
DATAIN
GM3D
ASEMBL
Q8
GM3D
DATAIN
SURF
BC
Q4CNVB
SFLOW
ADVBC
Q4ADB
GM3D
DATAIN
SURF
BC
Q4CNVB
SFLOW
ADVBC
Q4ADB
GM3D
DATAIN
SURF
BC
Q4CNVB
SFLOW
ADVBC
Q4ADB
Q8DV
Common
Block I.M.O
M
I
I
M
M
0
I
M
M
0
I
I
I
I
I
M
M
0
I
I
I
I
I
M
M
0
I
I
I
I
I
CREAL
214
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
DELT T
DELTO T
DJAC L3
DNX(8) 1/L
DNY(8) 1/L
DNZ(8) 1/L
DSDCQ(8) L31M
DTAU T
DTH(LM) 1/L
DTHG(8) 1/T
ETA
F(MAXNOD) -
FLOW M/L
FQ(d) M/L'/T
Type
Scalar
Scalar
Scalar
Array
Array
Array
Array
Scalar
Array
Array
Scalar
Array
Scalar
Array
Description
Time Increment
Time Increment
Determinant of the
Jacobian
Partial Derivative of the
Base Function with Respect
to x
Partial Derivative of the
Base Function with Respect
to Y
Partial Dervative of the
Base Function wtih Respect
to z
The Derivative of Adsorbed
Concentration with Respect
to Dissolved Concentration
at Eight Points of the
Element
Sub-Time Step Size
(TH(I,M)-THP(LM)/DELT
dTH/dt at Eight Gaussian
Points of the Element
Local Coordinate of the
Particle
Array of Real Numbers
that are to be Read and
Generated Automatically
Increment of Flow
Normal Flux at Four Points
Sub- Common
routine Block
ASEMBL CREAL
SFLOW
PRINTT
NDTAU
CREAL
SHAPE
SHAPE
SHAPE
SHAPE
Q8
NDTAU
ADVTRN
GM3D
ASEMBL
Q8
XSI3D
READR
CFLOW
Q4BB
I.M.O
I
I
I
I
0
0
0
0
0
I
M
I
I
0
0
I
FRATE
M/T
Scalar
of the Element Surface
Flow Rate
CFLOW
215
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable
FX(N)
N(N)
FZ(N)
Units Tvne Description
J r '
(M/L2)/T Array X-Direction Material Flux
at the N-th Node
M/LVT Array Y-Direction Material Flux
at the N-th Node
M/LVT Array Z-Direction Material Flux
at the N-th Node
Sub-
routine
GM3D
FLUX
SFLOW
PRINT!
STORE
GM3D
FLUX
SFLOW
PRINTT
STORE
GM3D
FLUX
SFLOW
PRINTT
STORE
Common
Block I.M.O
0
0
I
I
I
0
0
I
I
I
0
0
I
I
I
GAMA1
GAMA2
GAMA3
GAMA4
GNLR(I,K) -
IBC
Scalar Weighting Factor for
Side 1-5 Parallel to
the Z-direction
Scalar Weighting Factor for
Side 2-6 Parallel to
the Z-direction
Scalar Weighting Factor for
Side 4-8 Parallel to
the Z-direction
Scalar Weighting Factor for
Side 3-7 Parallel to
the Z-direction
Array Global Nodal Number of
the I-th Local Nodal
Number in the K-th Sub-
region. This Array is an
Input for 1 = 1,2, ..,
NNPLR(K). For I =
NNPLR(K)+1, . . . NTNPLR(K),
this Array is Generated
Based on IE(NEL,8) and
Inputted GNLR.
Scalar Index of Boundary Condi-
tion Type
WETZ
WETZ
WETZ
WETZ
GM3D
DATAIN
PAGEN
BLKITR
M
M
M
I
Q4CNVB
Q4ADB
216
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
IBUG
IICTYP(MP) --
IDTYP(NP)
IE(M,I)
Tvne Description
J r *
Scalar Debugging Indicator
Array Type of Specified-Flux
(Cauchy) Profile Assigned
to the MP-th Cauchy Side
Array Type of Dirichlet Con-
centration Profile
Assigned to the NP-th
Dirichlet Node
Array Global Node Number of the
I-th Node of the M-th
Element if I is Between
1 and 8. When I = 9,
This is an Integer to
Indicate the Material
Type of the M-th
Element.
Sub-
routine, Block
BLKITR
GM3D
DATAIN
BC
ADVBC
GM3D
DATAIN
BC
ADVBC
GM3D
DATAIN
SURF
PAGEN
LELGEN
AFABTA
ASEMBL
BC
FLUX
SFLOW
STORE
THNODE
NDTAU
ADVTRN
MPLOC
ADVBC
I.M.O
M
M
I
I
M
M
I
I
M
M
I
I
I
I
I
I
I
I
I
I
I
I
I
I
IGEOM
IHALFB
ILUMP
INDTYP(
MXTYP)
INTYP(MP)
IOPTIM
Scalar Geometry Description
Output Control
Scalar Half Band Width Plus 1
Scalar Lumping Indicator
Array Army of Integers that
are to be Read or
Generated Automatically
Array Type of Specified-
Dispersive-Flux (Neumann)
Profile Assigned to the
MP-th Neumam Side
Scalar Optimizing Weighting
Factor Indicator
LELGEN CINTE
OPTN
SOLVE
READN
GM3D
DATAIN
BC
AFABTA OPTN
0
M
M
I
217
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
ISB(1,I)
ISB(2,I)
ISB(3,I)
ISB(4,I)
ISB(5,I)
ISB(6,I)
ISC(1,MP) -
ISC(2,MP) -
Type
Array
Array
Array
Array
Array
Array
Array
Array
Description
Boundary Node Number of
the First Node of the
I-th Boundary Side
Boundary Node Number of
the Second Node of the
I-th Boundary Side
Boundary Node Number of
the Third Node of the
I-th Boundary Side
Boundary Node Number of
the Fourth Node of the
I-th Boundary Side
Element Side Index of the
I-th Boundary Side:
l=left side, 2=front side,
3=right side, 4=back side,
5=bottom side, 6=top side
Element Number to which
the I-th Boundary Side
Belongs
Global Node Number of the
First Node of the MP-th
Specified-Flux (Cauchy)
Side
Global Node Number of the
Second Node of the MP-th
Specified-Flux (Cauchy)
Side
Sub- Common
routine Block
GM3D
DATAIN
SURF
BC
SFLOW
ADVBC
GM3D
DATAIN
SURF
BC
SFLOW
ADVBC
GM3D
DATAIN
SURF
BC
SFLOW
ADVBC
GM3D
DATAIN
SURF
BC
SFLOW
ADVBC
GM3D
DATAIN
SURF
BC
SFLOW
ADVBC
GM3D
DATAIN
SURF
BC
SFLOW
ADVBC
GM3D
DATAIN
BC
ADVBC
GM3D
DATAIN
BC
ADVBC
I.M.O
M
M
0
I
I
I
M
M
0
I
I
I
M
M
0
I
I
I
M
M
0
I
I
I
M
M
0
I
I
I
M
M
0
I
I
I
M
M
I
I
M
M
I
I
218
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
ISC(3,MP) -
ISC(4,MP) -
ISC(5,MP) -
ISN(1,MP) -
ISN(2,MP) --
ISN(3,MP) -
ISN(4,MP) -
ISN(5,MP) -
ISTYP(M) -
ISV(l.MP) --
Type
Array
Array
Array
Array
Array
Array
Array
Array
Array
Array
Description
Global Node Number of the
Third Node of the MP-th
Specified-Flux (Cauchy)
Side
Global Node Number of the
Fourth Node of the MP-th
Specified-Flux (Cauchy)
Side
Boundary Side Number of
the MP-th Specified-Flux
(Cauchy) Side
Global Node Number of the
First Node of the MP-th
Specified-Dispersive-Flux
(Neumann) Side
Global Node Number of the
Second Node of the MP-th
Specified-Dispersive-Flux
(Neumann) Side
Global Node Number of the
Third Node of the MP-th
Specified-Dispersive-Flux
(Neumann) Side
Global Node Number of the
Fourth Node of the MP-th
Specified-Dispersive-Flux
(Neumann) Side
Boundary Side Number of
the MP-th Neumann Side
Type of Source Profile
Assigned to the M-th
Element
Global Node Number of the
First Node of the MP-th
Variable Side
Sub- Common
routine Block
GM3D
DATAIN
BC
ADVBC
GM3D
DATAIN
BC
ADVBC
GM3D
DATAIN
BC
ADVBC
GM3D
DATAIN
BC
GM3D
DATAIN
BC
GM3D
DATAIN
BC
GM3D
DATAIN
BC
GM3D
DATAIN
BC
(IM3D
DATAIN
ASEMBL
SFLOW
GM3D
DATAIN
BC
ADVBC
I.M.O
M
M
I
I
M
M
I
I
M
M
I
I
M
M
I
M
M
I
M
M
I
M
M
I
M
M
I
M
M
I
I
M
M
I
I
219
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
ISV(2,MP) --
ISV(3,MP) -
ISV(4,MP) -
ISV(5,MP) -
ITIM
IVTYP(MP) -
IWET
IWTYP(I) -
KANALY -
KCAI
KDA1
KDIAG
KDSK(I)
Type
Array
Array
Array
Array
Scalar
Array
Scalar
Array
Scalar
Scalar
Scalar
Scalar
Array
Description
Global Node Number of the
Second Node of the MP-th
Variable Side
Global Node Number of the
Third Node of the MP-th
Variable Side
Global Node Number of the
Fourth Node of the MP-th
Variable Side
Boundary Side Number of
the MP-th Variable Side
Time-Step Index
Type of Variable Concen-
tration Profile Assigned
to the MP-th Variable Side
Upstream Weighting
Indicator
Type of Source Profile
Assigned to the I-th Node
Analytical Input Control
Analytical Cauchy-Flux
Input Control
Analytical Dirichlet Input
Control
Diagnostic Output Table
Index
Store Results on Logical
Unit 12 for the I-th
Sub- Common
routine Block
GM3D
DATAIN
BC
ADVBC
GM3D
DATAIN
BC
ADVBC
GM3D
DATAIN
BC
ADVBC
GM3D
DATAIN
BC
ADVBC
PRINTT
GM3D
DATAIN
BC
ADVBC
OPTN
GM3D
DATAIN
ASEMBL
SFLOW
ALLFCT
CCBC
CDBC
PRINTT
GM3D
DATAIN
I.M.O
M
M
I
I
M
M
I
I
M
M
I
I
M
M
I
I
I
M
M
I
I
M
M
I
I
I
0
M
M
KDSKO
Time Step? 0=no, l=yes
Scalar Disk Output Control
CINTE
220
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
3KFLOW
Type
Scalar
Description
Flow Indicator
Sub-
routine
SFLOW
Common
Block
I.M.O
I
KKK
KNAI
KOUT
KPR(I)
KPRO
KRAI
KSAI
KSORP
KSS
KVI
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
KWAI
LAMBDA
LES(I)
1/T
-1 = Initial or Pre-
initial Condition
0 = Steady-state
1. Transient
Scalar Decomposition or Back
Substitution Indicator
1 = Decomposition
2 = Back Substitution
Scalar Analytical Neumann Flux
Input Control
Scalar Output-Table Number
Index
Array Line Printing Indicator
for the I-th Time Step:
0 = print nothing
1 = print fluxes through
all types of boundaries
2 = print concentration also
3 = print material flux also
Output Control
Analytical Rainfall Input
Control
Elemen-source Input
Control
Sorption Model Indicator
Steady-State I.C. Control
Flow Variable Input
Control
Scalar Well Source Input Control
0 = Tabular Input
1 = Analytical Input
Scalar Decay Constant
Array Global Element Number of
the I-th Element-Source
SOLVE
CNBC
PRINTT
0
GM3D
DATAIN
BLKITR
PRINT!
CINTE
CVBC
CELS
THNODE OPTN
ASEMBL CINTE
CINTE
CNPS
Q8
GM3D
DATAIN
ASEMBL
SFLOW
M
M
I
I
I
I
I
M
M
I
I
221
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable
Units TV
JTP Description
Sub-
routine
Common
Block
I.M.O
LMAXDF(K)
Array
LMXBW
LMXBWK
LMXNP
LMXNPK
LNOJCN(J,
Scalar
Scalar
Scalar
Scalar
Array
LOCP
LRM(I.N)
Scalar
Array
LRN(I.N)
Array
Maximum No. of Difference
Between Nodes for Any
Element in the K-th Local
Region. This Array is
Generated from the Array
LNOJCN.
Maximum No. of the Band-
width in Any Subregion
Maximum No. of the Band-
width in Any Subregion
Maximum No. of Nodal
Points in Any Subregion,
Including Interior Nodes
and Global Boundary Nodes
Maximum No. of Nodal
Points in Any Subregion,
Including Interior Nodes
and Global Boundary Nodes
Local Node Number of the
J-th Node Connecting to
I-th Local Node for the
K-th Subregion. This Array
is Generated from GNLR and
1 =1,2,3 NNPLR(K).
Indicator of the Location
of the Fictitious Particle
Global Element Number of
the I-th Element Connect
ing to the N-th Global
Node
Global Node Number of the
I-th Node Connecting to
the N-th Global Node
GM3D
PAGEN
BLKITR
DATAIN
GM3D
PAGEN
BLKITR
DATAIN
MPLOC
GM3D
LELGEN
NDTAU
ADVTRN
MPLOC
DATAIN
PAGEN
SURF
GM3D
PAGEN
ASEMBL
BC
NDTAU
DATAIN
M
0
I
M
LGEOM
LGEOM
M
0
I
M
M
0
I
I
I
M
0
I
M
0
I
I
I
M
222
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
LTMXNK
LTMXNP
LUBAR
LUFLW
LUINP
LUOUT
LUPAR
LUSTO
Type
J r
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Description
Maximum No. of Total Nodal
Points in Any Subregion,
Including Interior Nodes,
and Global Boundary Nodes
Maximum No. of Total Nodal
Points in Any Subregion,
Including Interior Nodes,
and Global Boundary Nodes
Logical Unit for Storing
Binary Boundary Arrays
Logical Unit for Flow Data
Logical Unit for Input
Data
Logical Unit for Output
Data
Logical Unit for Storing
Binary Pointer Arrays
Logical Unit for Storing
Binary Output
Sub- Common
routine Block
LGEOM
GM3D
DATAIN
GM3D
DATAIN
GM3D
DATAIN
READR
READN
GM3D
DATAIN
SURF
PAGEN
LELGEN
ASEMBL
BLKITR
PRINTT
READR
READN
NDTAU
ADVTRN
XS13D
BC
GM3D
GM3D
STORE
I.M.O
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
M
MAXBES
MAXBNP
Scalar Element Number where
the Fictitious Particle
is Located
Scalar Maximum No. of Boundary
Element Surfaces
Scalar Maximum No. of Boundary
Nodal Points
XS13D
SGEOM
SGEOM
223
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
MAXBW
MAXEL
MAXELK
MAXMAT
MAXNOD
MAXNP
MAXNPK
MP
MXBESK
MXBNPK
MXCDP
Type
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Sub- Common
Description routine Block I,M,0
Maximum No. of Band SOLVE I
Width
Maximum No. of Elements SGEOM
Maximum No. of Elements
Maximum No. of Materials MAIL
Maximum No. of Data READR I
Points to be Read
Maximum No. of Nodal SOLVE SGEOM I
Points FCOS I
Maximum No. of Nodes
Element where the MPLOC 0
Fictitious Particle
is Located
Maximum No. of Boundary-
Element Surfaces
Maximum No. of Boundary
Nodal Points
Maximum No. of Data Points CCBC
MXCDPK
MXCES
MXCESK
MXCNP
MXCNPK
MXCPR
on Each Specified-Flux
(Cauchy) Profile
Scalar Maximum No. of Data Points
on Each Specified-Flux
(Cauchy) Profile
Scalar Maximum No. of Cauchy
Element Surfaces
Scalar Maximum No. of Cauchy
Element Surfaces
Scalar Maximum No. of Cauchy
Nodal Points
Scalar Maximum No. of Cauchy
Nodal Points
Scalar Maximum No. of Cauchy-
Flux Profiles
CCBC
CCBC
CCBC
224
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
MXCPRK
MXDDP
MXDDPK
MXDNP
MXDNPK
MXDP
MXDPR
MXDPRK
MXDTC
MXDTCK
MXJBD
MXJBDK
MXKBD
MXKBDK
MXMATK
MXMPMK
MXMPPM
Type
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Sub- Common
Description routine Block I,M,0
Maximum No. of Cauchy-
Flux Profiles
Maximum No. of Data Points CDBC
on Each Dirichlet Profile
Maximum No. of Data Points
on Each Dirichlet Profile
Maximum No. of Dirichlet CDBC
Nodal Points
Maximum No. of Dirichlet
Nodal Points
Maximum No. of Data ALLFCT I
Points in Any Profile
Maximum No. of Dirichlet CDBC
Total Head Profiles
Maximum No. of Dirichlet
Total Head Profiles
Maximum No. of DELT SGEOM
Changes
Maximum No. of DELT
Changes
Maximum No. of Nonzero SGEOM
Elements in Any Row
Maximum No. of Nonzero
Elements in Any Row
Maximum No. of Elements SGEOM
Surrounding a Global Node
Maximum No. of Elements
Surrounding a Global Node
Maximum No. of Material
Types
Maximum No. of Material
Properties per Material
Maximum No. of Material MATE
Properties per Material
225
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable
MXNDP
MXNDPK
Units
MXSDP
MXSDPK
Type
Scalar
Scalar
MXNES
MXNESK
MXNNP
MXNNPK
MXNPR
MXNPRK
MXNTI
MXNTIK
MXPR
MXRDP
MXREGN
MXRGNK
MXRPR
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Description
Maximum No. of Data Points
on Each Specified
Dispersive-Flux (Neumann)
Profile
Maximum No. of Data Points
on Each Neumann-Flux
Profile
Maximum No. of Neumann
Element Surfaces
Maximum No. of Neumann
Element Surfaces
Maximum No. of Neumann
Nodal Points
Maximum No. of Neumann
Nodal Points
Maximum No. of Neumann-
Flux Profiles
Maximum No. of Neumann-
Flux Profiles
Maximum No. of Time Steps
Maximum No. of Time Steps
Maximum No. of Profiles
Maximum No. of Data Points
on Each Rainfall Profile
Maximum No. of Subregions
Maximum No. of Subregions
Maximum No. of Rainfall
Profiles
Maximum No. of Data Points
in Any Element Source/Sink
Profile
Maximum No. of Data Points
in Any Element Source/
Sink Profile
Sub-
routine
Common
Block I.M.O
CNBC
CNBC
CNBC
CNBC
SGEOM
ALLFCT
CVBC
LGEOM
CVBC
CELS
226
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
MXSEL
MXSELK
MXSPR
MXSPRK
MXTYP
MXVDPK
MXVES
MXVESK
MXVNP
MXVNPK
MXVPRK
MXWDP
Type
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Sub- Common
Description routine Block I,M,0
Maximum No. of Source CELS
Elements
Maximum No. of Source
Elements
Maximum No. of Element CELS
Source Profiles
Maximum No. of Element
Source Profiles
Maximum No. of Integers READN I
Allowed to be Read
Maximum No. of Data Points
on Each Rainfall Profile
Maximum No. of Variable CVBC
Element Surfaces
Maximum No. of Variable
Element Surfaces
Maximum No. of Variable CVBC
Nodal Points
Maximum No. of Variable
Nodal Points
Maximum No. of Rainfall
Profiles
Maximum No. of Data Points CNPS
MXWDPK
MXWNP
MXWNPK
MXWPR
on Each Well Source/Sink
Profile
Scalar Maximum No. of Data Points
on Each Well Source/Sink
Profile
Scalar Maximum No. of Well Nodal
Points
Scalar Maximum No. of Well Nodal
Points
Scalar Maximum No. of Well
Source/Sink Profile
CNPS
CNPS
227
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
MXWPRK
N(8)
NBES
NBNP
NCDP
NCES
NCM
NCNP
NCPR
NDDP
NDNP
NDP
NDPR
NDTCHG
NEL
NITER
Tj/pe
Scalar
Array
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Description
Maximum No. of Well
Source/Sink Profiles
Base Function of Eight
Points of the Element
Number of Boundary Element
Surfaces
Number of Boundary Nodal
Points
Number of Data Points on
Specified-Flux (Cauchy)
Profiles
Number of Cauchy Boundary
Element Sides
Number of. Cycles per Time
step
Number of Cauchy Boundary
Nodal Points
Number of Specified-Flux
(Cauchy) Profiles
Number of Data Points on
Dirichlet Profiles
Number of Dirichlet Nodal
Points
Number of Data Points in
Any Profile
Number of Dirichlet
Profiles
Number of Times to Reset
Time Step Size
Number of Elements
Number of Iterations per
Sub- Common
routine Block
SHAPE
CGEOM
CGEOM
CCBC
CCBC
CINTE
CCBC
CCBC
CDBC
CDBC
ALLFCT
CDBC
CGEOM
CGEOM
BLKITR CINTE
I.M.O
0
I
I
NMAT
Scalar
Cycle
Number of Materials
MATL
228
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
NMPPM
NNDP
NNES
NNNP
NNP
NNPLR(K) -
NNPR
NODENP -
NP1
NP2
NP3
NPBB(I)
NPCB(NP) -
Type
Scalar
Scalar
Scalar
Scalar
Scalar
Array
Scalar
Scalar
Scalar
Scalar
Scalar
Array
Array
Description
Number of Material
Properties per Material
Number of Data Points on
Specified-Dispersive-Flux
(Neumann) Profiles
Number of Neumann Boundary
Element Sides
Number of Neumann Boundary
Nodal Points
Number of Nodal Points
Number of Nodes for the
K-th Subregion Including
Interior and Global
Boundary Nodes
Number of Specified-
Dispersive-Flux (Neumann)
Profiles
Nodal Point of Interest
First Node on the Surface
Second Node on the Surface
Third Node on the Surface
Global Node Number on the
I-th Boundary Node
Global Nodal Number of
the NP-Cauchy Node on
Input. Then it is Changed
to Contain the Boundary
Sub- Common
routine Block
MATL
CNBC
CNBC
CNBC
SOLVE CGEOM
READR
GM3D
DATAIN
PAGEN
BLKITR
CNBC
MPLOC
FCOS
FCOS
FCOS
GM3D
DATAIN
SURF
SFLOW
NDTAU
ADVTRN
ADVBC
GM3D
DATAIN
SFLOW
ADVBC
I.M.O
I
I
M
M
I
I
I
I
I
I
M
M
0
I
I
I
I
M
M
I
I
229
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
NPDB(NP)
NPITER
NPNB(NP)
NPR
NPROB
NPVB(NP)
NPW(I)
NRDP
NREGN
NRPR
Type
Array
Scalar
Array
Scalar
Scalar
Array
Array
Scalar
Scalar
Scalar
Description
Global Node Number of the
NP-Dirichlet Node on
Input. Then it is Changed
to Contain the Boundary
Node Number
Number of Blockwise
Iterations Allowed
Global Nodal Number of
the NP-Neumann Node on
Input. Then it is Changed
to Contain the Boundary
Node Number.
Number of Profiles
Problem Number
Global Nodal Number of
the NP-Variable Node on
Input. Then it is Changed
to Contain the Boundary
Node Number
Global Node Number of the
I-th Well Node
Number of Data Points on
Rainfall Profiles
Number of Subregions
Number of Rainfall
Sub- Common
routine Block
GM3D
DATAIN
BC
ADVBC
SFLOW
CINTE
GM3D
DATAIN
SFLOW
ALLFCT
STORE
GM3D
DATAIN
SFLOW
ADVBC
GM3D
DATAIN
ASEMBL
SFLOW
CVBC
LGEOM
CVBC
I.M.O
M
M
I
I
I
M
M
I
I
I
M
M
I
I
M
M
I
I
NSDP
Scalar
NSEL
NSPR
NTAU
Scalar
Scalar
Scalar
Profiles
Number of Data Points
in Any Element Source/
Sink Profile
Number of Source/Sink
Elemente
Number of Source/Sink
Profiles
Number of Subtime Steps
CELS
CELS
CELS
NDTAU
ADVTRN
0
I
230
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
NTI
NTNPLR(K) -
NTYPE
NVES
NVNP
NWDP
NWNP
NWPR
OME
OMI
PR(MXPR) L/T,L,
M/L3
PRF(MXDP, L/T,L,
MXPER) M/L8
PROP (1,1) L3/M
Type
Scalar
Array
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Scalar
Array
Array
Array
Description
Number of Time Increments
Total Number of Nodes for
the K-th Subregion Includ-
ing Interior, Global
Boundary, and Intraboundary
Nodes
Number of Integers to be
Read
Number of Variable Boundary
Element Sides
Number of Variable Boundary
Nodal Points
Number of Data Points in
Any Point-Source Profile
Number of Wells
Number of Well Source
Profiles
Iteration Parameter for a
Non-Linear Equation
Relaxation Parameter for
Pointwise Solution
Profile Value at Time t
Profile Value of the Data
Point on the Profile
Distribution Coefficient
or Freudlich K or Langmuir
K
Sub- Common
routine Block
CGEOM
GM3D
DATAIN
PAGEN
READN
CVBC
CVBC
CNPS
CNPS
CNPS
BLKITR CREAL
CREAL
ALLFCT
ALLFCT
GM3D
DATAIN
AFABTA
FLUX
ASEMBL
SFLOW
THNODE
I.M.O
M
M
0
I
0
I
M
M
I
I
I
I
I
231
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
V a r i a b 1 dJnits TVHP Description
PROP (2, 1 M/Lg Array Bulk Density
THNODE I
PROP (3, 1) L Array Longitudinal Dispersivity
PROP (4, 1) L Array Transverse Dispersivity
PROP (5,1) L2/T Array Molecular Diffusion
Coefficient
PROP (6, 1) -- Array Tortuosity
PROP (7, 1) 1/L Array Decay Constant
Sub- Common
routine Block
GM3D
DATAIN
AFABTA
FLUX
ASEMBL
SFLOW
GM3D
DATAIN
AFABTA
FLUX
ASEMBL
SFLOW
THNODE
GM3D
DATAIN
AFABTA
FLUX
ASEMBL
SFLOW
THNODE
GM3D
DATAIN
AFABTA
FLUX
ASEMBL
SFLOW
THNODE
GM3D
DATAIN
AFABTA
FLUX
ASEMBL
SFLOW
THNODE
GM3D
DATAIN
AFABTA
FLUX
ASEMBL
SFLOW
THNODE
I.M.O
M
M
I
I
I
I
M
M
I
I
I
I
I
M
M
I
I
I
I
I
M
M
I
I
I
I
I
M
M
I
I
I
I
I
M
M
I
I
I
I
I
232
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Unite
PROP(8,I)
QA(8,8)
Q,AA(8,8)
QB(8,8)
QBMP M/LVT
QC(8,8)
QCB(I) M/L2/T
QCBF(I,J) M/T/L2
QDM
QNB(I) M/LVT
QNBF(I,J) WT/L2
QR(8)
QRM
QRX(8)
QRY(8)
QRZ(8)
Type
Array
Array
Array
Array
Scalar
Array
Array
Array
Scalar
Array
Array
Array
Scalar
Array
Array
Array
en
Freundlich N or Langmuir
SMAX
An Element Matrix
An Element Matrix
An Element Matrix
Flux or Concentration of
the Boundary Side
An Element Matrix
Value of Cauchy Flux at
the Present Time of the
I-th Cauchy Flux Profile
Flux of the I-th Data
Point in the J-th Cauchy
Flux vs. Time Profile
Integration of Local
Variable S
Value of Neumann Flux at
the Present Time of the
I-th Neumann Flux Profile
Flux of the I-th Data
Point in the J-th Neumam
Flux vs. Time Profile
An Element Load Vector
Integration of TH*C
Element Lead Vector for
x-Flux
Element Load Vector for
Y-Flux
Element Load Vector for
Sub- Common
routine Block
GM3D
DATAIN
AFABTA
FLUX
ASEMBL
SFLOW
THNODE
Q8
Q8
Q8DV
Q8
Q4CNVB
Q4ADB
Q8
GM3D
BC
ADVBC
GM3D
DATAIN
Q8R
GM3D
BC
GM3D
DATAIN
Q8
Q8R
Q8DV
Q8DV
Q8DV
I.M.O
M
M
I
I
I
I
I
0
0
0
0
I
I
0
M
I
I
M
M
0
M
I
M
M
0
0
0
0
0
z-Flux
233
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
QV(8,8)
R(MAXNP)
RHOB M/L3
RI(N)
RL(N)
RLD(N)
RLDG
MAXNP)
RLDL(N)
RQ(4) M/T
RQI(4) M/L2/T
RQL(4) L3/L2/T
SOS(Ll) L3/L2/T
SOS(I,2) L7L7T
SOSC M/L3
SOSCP M/L3
SOSF L7TIL'
Type
Array
Array
Scalar
Array
Array
Array
Array
Array
Array
Array
Array
Array
Array
Scalar
Scalar
Array
Description
An Element Matrix
Load Vector
Bulk Density of the
Material in Element
Working Array Used in
Subroutines BLKITR and
ADVBC
Working Array Used in
Subroutine ADVBC
An Array to Store the
Right Hand Side of the
N-th Equation of the
Assembled Global Load
Vector
Global Load Vector
Assembled Load Vector for
a Subregion
Integrated Flux at Four
Nodes of the Element
Surface
Material-Flux at Four
Nodes of the Surface
Flow-Flux at Four Nodes
of the Surface
Source Flow Rate of the
I-th Profile at Time t
Source Concentration of
the I-th Profile at Time
t
Source Concentration
Concentration in Element-
Source
Source Flow Rate of the
I-th Data Point in the
Sub- Common
routine Block
Q8
SOLVE
Q8
GM3D
BLKITR
ADVBC
GM3D
ADVBC
GM3D
ASEMBL
BC
BLKITR
GM3D
BLKITR
Q4CNVB
Q4BB
Q4ADB
Q4ADB
GM3D
ASEMBL
SFLOW
GM3D
ASEMBL
SFLOW
Q8
Q8R
GM3D
DATAIN
I.M.O
0
M
I
M
0
M
M
M
M
0
0
I
M
I
0
0
0
0
M
I
I
M
I
I
I
I
M
M
J-th Profile
234
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
SOSF M/L'
(U,2)
SOSM M/T
SOSQ L3/T
SOSQP LVT
SS
SWQ(8) WT
T T
TAU
TCDBF(U) T
TCVBF(IJ) T
TDTCH(I) T
TFLOW ML
TH(I,M)
THG(8)
THN(N)
Type
Array
Scalar
Scalar
Scalar
Scalar
Array
Scalar
Scalar
Array
Array
Array
Scalar
Array
Array
Array
Description
Source Concentration of
the I-th Data Point in
the J-th Profile
Integration of Q*Cin
Element-Source Flow Rate
Element-Source Flow Rate
XSI-Coordinate of the
Gaussian Point
Iterate of the Adsorbed
Concentration at Eight
Gaussian Points of the
Element
Time
Tortuosity
Time of the I-th Data
Point in the J-th
Dirichlet Concentration
vs. Time Profile
Time of the I-th Data
Point in the J-th Vari-
able Concentration vs.
Time Profile
Time of the I-th Time
to Reset Time-Step Size
( = DELTO)
Total Flow
Moisture Content at the
I-th Node of the M-th
Element
Moisture Content at Eight
Gaussian Points of the
Element
Moisture Content at the
N-th Node
Sub- Common
routine Block
GM3D
DATAIN
Q8R
Q8
Q8R
SHAPE
Q8
ALLFCT
Q8
GM3D
DATAIN
GM3D
DATAIN
GM3D
DATAIN
CFLOW
GM3D
FLUX
ASEMBL
SFLOW
THNODE
Q8
Q8R
GM3D
THNODE
NDTAU
I.M.O
M
M
0
I
I
I
I
I
I
M
M
M
M
M
M
M
I
I
I
I
I
I
M
0
I
235
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
THP(I,M)
THQ(8)
TIME T
TITLE
TMAX T
TOLA L
TOLB L
TPRF( T
MXDP,
MXPR)
TQCBF(LJ) T
TQNBF(LJ) T
TSOSF(IJ) T
TT
TWSSF(IJ) T
UU
VX(N) UT
Type
Array
Array
Scalar
Scalar
Scalar
Scalar
Array
bray
Array
Array
Scalar
Array
Scalar
Array
Description
Value of TH(I,M) at the
Previous Time
Moisture Content at Eight
Points of the Element
Time
Title of the Problem
Maximum Value of Time
Steady-State Tolerance
Transient-State Tolerance
Time of the Data Point on
the Profile
Time of the I-th Data
Point in the J-th Cauchy
Flux vs. Time Profile
Time of the I-th Data
Point in the J-th Neumam
Flux vs. Time Profile
Time of the I-th Data
Point in the J-th
Element Source Profile
Eta-Coordinate of the
Gaussian Point
Time of the I-th Data
Point in J-th Well
Source Profile
Zeta-Coordinate of the
Gaussian Point
X-Component Velocity at
the N-th Node
Sub- Common
routine Block
ADVTRN
ADVBC
GM3D
ASEMBL
THNODE
Q8DV
DATAIN
PRINTT
STORE
STORE
CREAL
CREAL
BLKITR CREAL
ALLFCT
GM3D
DATAIN
GM3D
DATAIN
GM3D
DATAIN
SHAPE
GM3D
DATAIN
SHAPE
GM3D
FLUX
I.M.O
I
I
M
I
I
I
M
I
I
I
I
I
M
M
M
M
M
M
I
M
M
I
M
I
236
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
VXP(N) L/T
VXQ(8) L/T
VY(N) L/T
VYP(N) L/T
VYQ(8) L/T
VZ(N) L/T
Type Description
Array Value of VX(N) at the
Previous Time
Array X-Velocity of Eight Nodes
of the Element
Array Y-Component Velocity at
the N-th Node
Array Value of VY(N) at the
Previous Time
Array Y- Velocity of Eight Nodes
of the Element
Array Z-Component Velocity at
the N-th Node
Sub- Common
routine Block
AFABTA
ASEMBL
BC
NDTAU
ADVTRN
ADVBC
GM3D
AFABTA
ASEMBL
BC
NDTAU
ADVTRN
ADVBC
Q8DV
Q8
Q4CNVB
Q4ADB
GM3D
AFABTA
FLUX
ASEMBL
BC
NDTAU
ADVTRN
ADVBC
GM3D
AFABTA
ASEMBL
BC
NDTAU
ADVTRN
ADVBC
Q8DV
Q8
Q4CNVB
Q4ADB
GM3D
AFABTA
FLUX
ASEMBL
BC
NDTAU
ADVTRN
ADVBC
I.M.O
I
I
I
I
I
I
M
I
I
I
I
I
I
I
I
I
I
M
I
I
I
I
I
I
I
M
I
I
I
I
I
I
I
I
I
I
M
I
I
I
I
I
I
I
237
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TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units
VZP(N) L/T
VZQ(8) L/T
W(8)
WETAB(J,
M)
WSS(Ll) LVT
WSS(I,2) M/L3
WSSF L3/T
(JJ.1)
WSSF M/L3
(J.1,2)
WWRK(N)
X(N) L
Tvoe
J r
Array
Array
Array
Array
Array
Array
Array
Array
Array
Array
Description
Value of VZ(N) at the
Previous Time
Z-Velocity of Eight Nodes
of the Element
Weighting Function at
Eight Points of the
Element
Weighting Factor for the
J-th Side of the M-th
Element
Well Source Flow Rate of
the I-th Profile at Time
t
Well Source Concentration
at the I-th Profile
Well Source Flow Rate of
the I-th Data Point in
the J-th Profile
Well Source Concentration
of the I-th Data Point in
the J-th Profile
Working Array Used in
Subroutine THNODE
X-Coordinate of the N-th
Node
Sub- Common
routine Block
GM3D
AFABTA
ASEMBL
BC
NDTAU
ADVTRN
ADVBC
Q8DV
Q8
Q4CNVB
Q4ADB
SHAPE
GM3D
AFABTA
FLUX
ASEMBL
GM3D
ASEMBL
SFLOW
GM3D
ASEMBL
SFLOW
GM3D
DATAIN
GM3D
DATAIN
GM3D
THNODE
GM3D
DATAIN
SURF
AFABTA
FLUX
ASEMBL
BC
SFLOW
STORE
THNODE
LMQ
M
I
I
I
I
I
0
M
0
I
I
M
I
I
M
I
I
M
M
M
M
M
0
M
M
I
I
I
I
I
I
I
I
238
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Units Tvnp Description
XP L Scalar X-Coordinate of the
Fictitious Particle
XQ(8) L Array X-Coordinate at Eight
Points of the Element
Sub- Common
routine Block
NDTAU
ADVTRN
MPLOC
XS13D
ADVBC
MPLOC
FCOS
XSI3D
Q8DV
Q8
SHAPE
Q4CNVB
Q4BB
Q8R
Q4ADB
I.M.O
I
I
I
I
I
I
I
I
I
0
I
I
I
I
I
XSI
Y(N)
Scalar
Array
Local Coordinate of the
Particle
Y-Coordinate of the N-th
Node
YP
YQ(8)
Scalar
Array
Y-Coordinate of the
Fictitious
Y-Coordinate at Eight
Points of the Element
XSI3D
GM3D
DATAIN
SURF
AFABTA
FLUX
ASEMBL
BC
SFLOW
STORE
THNODE
NDTAU
ADVTRN
MPLOC
FCOS
XSI 3D
ADVBC
MPLOC
FCOS
XSI3D
Q8DV
Q8
SHAPE
Q4CNVB
Q4BB
Q8R
Q4ADB
M
M
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
0
I
I
I
I
I
239
-------
TABLE D-2. 3DLEWASTE PROGRAM VARIABLES, UNITS, LOCATION, AND
VARIABLE DESIGNATION (continued)
Variable Unite TVDP Description
Z(N) L Array Z-Coordinate of the N-th
Node
ZP L Scalar Z-Coordinate of the
Fictitious Particle
ZQ(8) L Array Z-Coordinate at Eight
Points of the Element
Sub- Common
routine Block
GM3D
DATAIN
SURF
AFABTA
FLUX
ASSEMBL
BC
SFLOW
STORE
THNODE
NDTAU
ADVTRN
MPLOC
FCOS
XSI3D
ADVBC
MPLOC
FCOS
XSI3D
Q8DV
Q8
SHAPE
Q4CNVB
Q4BB
Q8R
Q4ADB
I.M.O
M
M
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
0
I
I
I
I
I
ZTA
Scalar Local Coordinate of the
Particle
XSI3D
0
U.S. GOVERNMENT PRINTING OFFICE: 1992-750-002/60,103
240
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